Res publica Dominiciana

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      <comment>Imago DM16Ago.png deleta est ex Communibus ab Fitindia. Ille hanc rationem dedit: No permission since 9 August 2022</comment>
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{{Capsa civitatis Vicidata}}
[[Fasciculus:Tabula Rei publicae Dominicianae.png|thumb|Tabula Rei publicae Dominicianae.]]
[[Fasciculus:SantoDomingoedit.JPG|thumb|[[Dominicopolis]], caput Reipublicae Dominicanae]]
'''Res publica Dominiciana'''&lt;ref&gt;[[Paulus VI]], &quot;[https://w2.vatican.va/content/paul-vi/la/letters/1965/documents/hf_p-vi_let_19650202_in-dominiciana-republica.html In Dominiciana Republica]&quot; (incipit epistula anni 1965). Ita &quot;''Dominicana (República)'': Res pública Dominiciana&quot; apud [http://www.juan23.edu.ar/latin/ Diccionario auxiliar: español-latino para el uso moderno del Latín Von José Juan Del Col] Dictionarium Auxiliare Hispano-Latinum ab Ioanne Del Col&lt;/ref&gt;&lt;ref&gt;[[Carolus Egger]], ''Regionum et urbium Americae Mediae et Meridianae nomina Latina''. In: ''[[Latinitas (periodicum)|Latinitas]]'' 1972, 2, [https://books.google.de/books?id=0WdfAAAAMAAJ&amp;focus=searchwithinvolume&amp;q=%22Res+publica+Dominiciana%2C+f.%22 p. 111]&lt;/ref&gt; seu '''Dominicana'''&lt;ref name=&quot;DL43&quot;&gt;{{Egger DL|43}}&lt;/ref&gt;&lt;ref&gt;[http://www.vatican.va/news_services/press/sinodo/documents/bollettino_20_x-ordinaria-2001/xx_plurilingue/b01_xx.html Synodus Episcoporum Bollettino 2001]/[http://www.vatican.va/news_services/press/sinodo/documents/bollettino_25_xiii-ordinaria-2012/xx_plurilingue/b01_xx.html 2012], e situ interretiale [[Vaticanus|Vaticani]]&lt;/ref&gt; ([[Lingua Hispanica|Hispanice]] ''República Dominicana'') est [[civitas sui iuris]], quae occupat orientales quinque octavas partes insulae [[Hispaniola]]e, in [[Antillae Maiores|Antillarum Maiorum]] [[archipelagus|archipelago]] apud regionem [[Mare Caribaeum|Caribaeam]]. Occidentales tres huius [[insula]]e octantes ab [[Haitia]] civitate occupantur,&lt;ref&gt;Alan Dardik, ed. (2016), &quot;[https://books.google.es/books?id=de9NDQAAQBAJ&amp;printsec=frontcover&amp;redir_esc=y#v=onepage&amp;q&amp;f=false Vascular Surgery: A Global Perspective]&quot; (Anglice) (Springer), 341. ISBN 978-3-319-33745-6.&lt;/ref&gt;&lt;ref&gt;Jagran Josh, ed. (2016). &quot;[https://books.google.es/books?id=5wBsDQAAQBAJ&amp;printsec=frontcover&amp;redir_esc=y#v=onepage&amp;q&amp;f=false Current Affairs November 2016 eBook]&quot;. p. 93.&lt;/ref&gt; quod facit Hispaniolam unam inter duas Caribaeas insulas, iuxta cum [[Insula Sancti Martini|Sancto Martino]], quae divisae sunt duabus a nationibus. Respublica Dominiciana est secunda maxima Caribaea natio secundum aream (post [[Cuba]]m), 48 445 chiliometra quadrata seu 18&amp;thinsp;705 milia passuum quadratorum lata, et tertia secundum numerum incolarum cum circiter decem millionibus incolarum, quorum propemodum tres milliones territorium urbis [[Dominicopolis]]&lt;ref name=&quot;DL43&quot;/&gt; (Hispanice ''Santo Domingo''), [[caput (urbs)|capitis]], incolunt.&lt;ref&gt;[http://data.worldbank.org/country/dominican-republic&quot;Dominican Republic|Data]&quot; (Anglice). data.worldbank.org.&lt;/ref&gt;&lt;ref&gt;[https://web.archive.org/web/20110511103811/http://www.conapofa.gov.do/__estimaciones_y_proyecciones/Estimacionesyproyecciones2008.zip &quot;Estimaciones y Proyecciones de la Población Dominicana por Regiones, Provincias, Municipios y Distritos Municipales, 2008]&quot; (Hispanice). Archivatum ex originali die 11 Maii anni 2011. Inspectum die 25 Decembris anni 2008. Contextus: [https://web.archive.org/web/20110808193423/http://www.conapofa.gov.do/estimaciones.asp &quot;Estimaciones; Población en Tiempo Real&quot;] (Hispanice). Archivatum ex originali die 8 Augusti anni 2011. Inspectum die 13 Ianuarii anni 2008.&lt;/ref&gt;

[[Christophorus Columbus]] in occidentali Hispaniolae parte die 6 Decembris anni 1492 appulit, quae nunc Haitia est. Insula prima [[Colonizatio Americae Hispanica|Hispanicae dicionis colonialis]] sedes in [[Mundus novus|Mundo Novo]] facta est. Dominiciani [[libertas|libertatem]] Novembri mense anni [[1821]] declaraverunt, sed a potentiore vicina Haitia Februario anni [[1822]] dynamice annexi sunt. Post anni [[1844]] victoriam in bello libertatis Dominicianae Haitianam contra dicionem, res publica rursus sub Hispanicam dicionem [[colonia]]lem usque ad bellum restitutionis Dominicianae die [[16 Augusti]] anni [[1865]] cecidit.&lt;ref&gt;Franco, Caesar A. (César A.) [https://web.archive.org/web/20150624072932/https://www.dgii.gov.do/et/nivelBasico/Documents/Guerra%20de%20la%20Restauraci%C3%B3n%2C%2016%20de%20agosto%201863.pdf &quot;La guerra de la Restauración Dominicana, el 16 de agosto de 1863&quot;] (PDF) (Hispanice). dgii.gov.do. Archivatum ex originali (PDF) die 24 Iunii anni 2015.&lt;/ref&gt;&lt;ref&gt;Guerrero, Ioannes (Johnny) (die 16 Augusti anni 2011). [http://eldia.com.do/la-restauracion-de-la-republica-como-referente-historico/ &quot;La Restauración de la República como referente histórico&quot;] (Hispanice). ''El Día.''&lt;/ref&gt;&lt;ref&gt;Sagas, Ernestus (Ernesto). [http://faculty.webster.edu/corbetre/haiti/misctopic/dominican/conception.htm &quot;An Apparent Contradiction? Popular Perceptions of Haiti and the Foreign policy of the Dominican Republic&quot;] (Anglice). Lehman College (Praesentatum in Congressione Annua Sexta Consociationis Studiorum Haitianorum, Bostoniae, in Massachusetta).&lt;/ref&gt;

Res publica Dominiciana plerumque [[bellum civile|intestina proelia]] (per Rem publicam Secundam) usque ad annum [[1916]] experta est. Rem publicam [[Civitates Foederatae]] annos octo inter 1916 et [[1924]] occupabat, et consequentem tranquillam et prosperam sexenii periodum, praeside [[Horatius Vásquez|Philippo Horatio Vásquez Lajara]], secuta est dictatura [[Raphael Trujillo|Raphaëlis Leonidae Trujillo Molina]] usque ad [[1961]]. [[Bellum civile]] anno [[1965]], rei publicae recentissimum, finitum est alia militari occupatione, quam secutum est auctoritarium regimen [[Ioachimus Balaguer|Ioachimi Antonii Balaguer Ricardo]] ab anno [[1966]] ad [[1978]]. Ex quo tempore, res publica [[democratia|democratiam repraesentativam]] migravit&lt;ref name=&quot;CIA&quot;&gt;[https://www.cia.gov/library/publications/the-world-factbook/geos/dr.html &quot;CIA – The World Factbook – Dominican Republic&quot;] (Anglice). [[Procuratio Nuntiorum Centralis]] (CIA). Inspectum [[4 Iunii|pridie Nonas Iunias]] anni 2007.&lt;/ref&gt; et ducta est a [[Leonellus Fernández|Leonello Fernández]] per plerumque tempus e [[1996]]. [[Danilo Medina|Danilus Medina]], praesens rei publicae praeses, Leonello Fernández successit anno [[2012]], cum abstulisset 51 centesimas suffragiorum contra [[Hippolytus Mejía|Hippolytum Mejía]], priorem praesidem.&lt;ref&gt;Ben Fox et Ezequiel Abiu Lopez (die 20 Maii anni 2012), &quot;Dominican Republic Elections: Ex-President Hipolito Mejia Challenges Danilo Medina&quot; (Anglice). ''Huffington Post.'' Archivatum ex originali pridie Kalendas Februarias anni 2016.&lt;/ref&gt;

Res publica Dominiciana nonam oeconomiam in [[Latinamerica]] habet, maximam [[oeconomia]]m in [[Mare Caribaeum|Caribaea]] et regione [[America Centralis|Mesoamericana]].&lt;ref&gt;[https://www.cia.gov/library/publications/the-world-factbook/rankorder/2001rank.html &quot;CIA – The World Factbook – Rank Order – GDP (purchasing power parity)&quot;] (Anglice).&lt;/ref&gt;&lt;ref&gt;[http://www.worldbank.org/en/country/dominicanrepublic &quot;Dominican Republic&quot;] (Anglice). www.worldbank.org.&lt;/ref&gt; Quamquam multo tempore clara propter suam [[agricultura]]m et metalleuticen, in oeconomiam nunc domitantur diaconiae.&lt;ref&gt;[https://www.cia.gov/library/publications/the-world-factbook/geos/dr.html &quot;CIA – The World Factbook – Dominican Republic&quot;] (Anglice). [[Procuratio Nuntiorum Centralis]] (CIA).&lt;/ref&gt; Ultima per decennia duo, Res publica Dominiciana eminuit sicuti una apud oeconomias maxime crescentes in [[America|Americis]], cum medio indice proventus domestici grossi realis 5,4% inter 1992 et 2014.&lt;ref name=&quot;Argentaria&quot;&gt;[http://www.worldbank.org/en/country/dominicanrepublic/overview &quot;Dominican Republic Overview&quot;] (Anglice). [[Argentaria mundana]]. Inspectum 2016-04-29.&lt;/ref&gt; [[PDG]] incrementum annis 2014 et 2015 7,3 et 7,0% proprie, maximum in Hemisphaerio Occidentali, attigit.&lt;ref name=&quot;:Argentaria&quot;&gt;[http://www.worldbank.org/en/country/dominicanrepublic/overview &quot;Dominican Republic Overview&quot;] (Anglice). [[Argentaria mundana]]. Inspectum 2016-04-29.&lt;/ref&gt; In primo anni 2016 dimidio oeconomia Dominiciana 7,4% crevit, rapidi oeconomici incrementi propensione pergente.&lt;ref name=&quot;:1&quot;&gt; [https://web.archive.org/web/20160826201017/http://www.dominicantoday.com/dr/economy/2016/8/25/60419/Dominican-economy-grows-74-in-first-half-paced-by-construction &quot;Dominican economy grows 7.4% in first half, paced by construction&quot;] (Anglice). Dominican Today. Archivatum ex originali die 26 Augusti anni 2016. Inspectum die 27 Augusti anni 2016.&lt;/ref&gt;

Recens incrementum constructione, fabricatione periegesique ductum est. Privata consumptio fortis fuit, propter parvam inflationem (sub 1% mediae quantitatis anni 2015), quaestus creationem non minus quam altam remissionum summam. Res publica Dominiciana chrematisterium, Chrematisterium Rei publicae Dominicianae (Hispanice ''Bolsa de Valores de la República Dominicana''; acronymo ''BVRD''),&lt;ref&gt;[http://www.bvrd.com.do/quienes-somos &quot;¿Quiénes Somos?&quot;] (Hispanice). Chrematisterium Rei publicae Dominicianae. Inspectum 2016-03-03.&lt;/ref&gt; et provectum telecommunicationis systema et vecturae infrastructuram&lt;ref name=consulatus&gt;[http://www.dominicanconsulate.org/gralinfo.htm &quot;Consulatus Generalis Rei publicae Dominicianae Bancoci in Thailandia&quot;] (Anglice). Inspectum die 27 Februarii anni 2009.&lt;/ref&gt; habet. Nihilominus, inopia quaestus,&lt;ref name=&quot;CIA&quot;&gt;[https://www.cia.gov/library/publications/the-world-factbook/geos/dr.html &quot;CIA – The World Factbook – Dominican Republic&quot;] (Anglice). [[Procuratio Nuntiorum Centralis]] (CIA). Inspectum [[4 Iunii|pridie Nonas Iunias]] anni 2007.&lt;/ref&gt; gubernii corruptio et inconsistens electrica diakonia praecipua problemata manent. Natio significativam emolumentorum inaequalitatem quoque habet.&lt;ref name=&quot;CIA&quot;&gt;[https://www.cia.gov/library/publications/the-world-factbook/geos/dr.html &quot;CIA – The World Factbook – Dominican Republic&quot;] (Anglice). [[Procuratio Nuntiorum Centralis]] (CIA). Inspectum [[4 Iunii|pridie Nonas Iunias]] anni 2007.&lt;/ref&gt; Internationalis migratio valde Rem publicam Dominicianam afficit, quoniam magnos migrantium fluxus accipit mittitque. Gregaria illegalis [[Haitia]]na immigratio et Haitianae originis Dominicianos integrare praecipua problemata sunt.&lt;ref name=&quot;Diógenes&quot;&gt;Diogenes (Diógenes) Pina (die 21 Martii anni 2007). [https://web.archive.org/web/20080109194929/http://www.ipsnews.net/news.asp?idnews=37018 &quot;Dominican Republic: Deport Thy (Darker-Skinned) Neighbour&quot; (Anglice). Inter Press Service (IPS). Archivatum ex originali die 9 Ianuarii anni 2008. Inspectum die 14 Ianuarii anni 2008.&lt;/ref&gt; Magna Dominiciana diaspora, praesertim in [[Civitates Foederatae Americae|Civitatibus Foederatis]],&lt;ref&gt;&quot;United States – Selected Population Profile in the United States (Dominican (Dominican Republic))&quot;] (Anglice). ''2008 American Community Survey 1-Year Estimates''. U.S. Census Bureau. Archivatum ex originali die 2 Decembris anni 2010. Inspectum die 10 Ianuarii anni 2010.&lt;/ref&gt; est, quae evolutioni contribuit, mittendis [[billio]]nibus [[dollarium (CFA)|dollariorum]] Dominicianis familiis per remissiones.&lt;ref name=&quot;CIA&quot;&gt;[https://www.cia.gov/library/publications/the-world-factbook/geos/dr.html &quot;CIA – The World Factbook – Dominican Republic&quot;] (Anglice). [[Procuratio Nuntiorum Centralis]] (CIA). Inspectum [[4 Iunii|pridie Nonas Iunias]] anni 2007.&lt;/ref&gt;&lt;ref name=&quot;Relations&quot;&gt;[https://www.state.gov/r/pa/ei/bgn/35639.htm &quot;U.S. Relations With the Dominican Republic&quot;] (Anglice). [[Ministerium Rerum Externarum Civitatum Foederatarum]]. Die 22 Octobris anni 2012.&lt;/ref&gt;

Res publica Dominiciana destinatio visitatissima in Caribaeo est. Annui campi [[pillamalleus|pillamallei]] praecipuae illecebrae sunt.&lt;ref name=consulatus&gt;[http://www.dominicanconsulate.org/gralinfo.htm &quot;Consulatus Generalis Rei publicae Dominicianae Bancoci in Thailandia&quot;] (Anglice). Inspectum die 27 Februarii anni 2009.&lt;/ref&gt; Sicuti geographice varia natio, Res publica Dominiciana non solum excelsissimi montani culminis in Caribaeo, [[Duarte (culmen)|Culminis Duartis]] (Hispanice ''Pico Duarte''), sed etiam maximi lacus et loci minime elevati in Caribaeo, [[Lacus Enriquillensis]]&lt;ref name=&quot;Enriquillense&quot;&gt; Nomini lacus Hispanice ''Enriquillo'' nuncupato, confer binomen &quot;Chondropoma enriquillense&quot;.&lt;/ref&gt; seu [[Lacus Henriculus]]&lt;ref&gt;Nomen lacus nominatum est a regulo Henriculo (Hispanice ''Enriquillo'') qui apparet in libro &quot;[https://books.google.es/books?id=vobbcIgAeK0C&amp;pg=PA80&amp;dq=bohechius&amp;hl=es&amp;sa=X&amp;ved=0ahUKEwidtq3pxvjWAhXLDcAKHY1zD5MQ6AEIPjAD#v=onepage&amp;q=bohechius&amp;f=false Annales minorum seu Trium Ordinum a S. Francisco institutorum, Volumen 15]&quot;, ubi dicitur: &quot;(...) Henriculum, Indiarum nobilem, a se suisque Consodalibus optime a juventute apud Conventum Verae Pacis in provincia Xarague (in qua Bohequius, unus ex quinque Regibus Hispaniolae imperabat).&quot;&lt;/ref&gt; (Hispanice ''Lago Enriquillo''), domus est.&lt;ref&gt;Baker, Christophorus P. (Christopher P.); Mingasson, Aegidius (Gilles) (2008). ''[https://books.google.es/books?id=toEFe48MD1IC&amp;pg=PA190&amp;redir_esc=y&amp;hl=es Dominican Republic]'' (Hispanice). National Geographic Books, p. 190. ISBN 978-1-4262-0232-2.&lt;/ref&gt; Insula mediam temperaturam 26&amp;nbsp;°C seu 78,8&amp;nbsp;°F et magnam climaticam biologicamque [[diversitas biologica|diversitatem]] habet.&lt;ref name=consulatus&gt;[http://www.dominicanconsulate.org/gralinfo.htm &quot;Consulatus Generalis Rei publicae Dominicianae Bancoci in Thailandia&quot;] (Anglice). Inspectum die 27 Februarii anni 2009.&lt;/ref&gt; Natio quoque primorum [[cathedralis]], [[castellum|castelli]], [[monasterium|monasterii]], [[fortalitia]]e in [[America|Americis]] aedificatorum locus est, sitorum Urbi Coloniali [[Dominicopolis|Dominicopolitana]] (Hispanice ''Ciudad Colonial de Santo Domingo''), [[Patrimonium totius mundi|Patrimonio Totius Mundi]].&lt;ref&gt;[http://whc.unesco.org/en/list/526 &quot;Urbs Colonialis Dominicopolitana]&quot;] (Variis linguis). Centrum [[Patrimonium totius mundi|Patrimonii Totius Mundi]] ab [[UNESCO]]. Inspectum die 24 Augusti anni 2016.&lt;/ref&gt;&lt;ref&gt;[http://www.unesco.org/nac/geoportal.php?country=DO&amp;language=S UNESCO per totum mundum|Res publica Dominiciana] (Anglice, Hispanice, Francice). Unesco.org (die 14 Novembris anni 1957). Inspectum die 2014-04-02.&lt;/ref&gt; Musica et ars athletica magni ponderis Dominiciana in cultura sunt, cum [[Meringinis (genus musicum)|meringine]] ac [[bachata]] sicuti nationali saltatione ac musica et cum [[basipila]] sicuti praedilecta arte athletica.&lt;ref name=&quot;Ambasciata&quot;&gt;[https://web.archive.org/web/20150626100357/http://www.domrep.org/gen_info.html &quot;Ambasciata Rei publicae Dominicianae in Civitatibus Foederatis&quot;] (Anglice). Archivatum ex originali die 2015-06-26. Inspectum die 27 Februarii anni 2009.&lt;/ref&gt;

== Nomina et etymologia ==
Per plerumque historiae tempus, usque ad independentiam, natio '''Sanctus Dominicus'''&lt;ref&gt;Verbi gratia, apud librum &quot;[https://www.amazon.com/Praedonibus-Insulam-Dominici-Celebrantibus-Saeculo/dp/0259387452 De Praedonibus Insulam Sancti Dominici Celebrantibus Saeculo Septimo Decimo]&quot; ab Henrico Lorin.&lt;/ref&gt; (Hispanice ''Santo Domingo'') nuncupata est&lt;ref&gt;&quot;[http://countrystudies.us/dominican-republic/3.htm Dominican Republic – The first colony]&quot; (Anglice). ''Country Studies''. Bibliotheca Congressus; Federal Research Divisio Investigatio Foederalis. Inspectum die 19 Iunii anni 2008.&lt;/ref&gt; —aliud nomen praesentis capitis, [[Dominicopolis]], et [[Dominicus Oxomensis|patroni sancti]]— et sic communiter nuncupari perrexit velut Anglice usque ad ineuns saeculum XX.&lt;ref&gt;[https://books.google.es/books?id=CLpoeD3cbBkC&amp;pg=PA1&amp;redir_esc=y#v=onepage&amp;q&amp;f=false Hand Book of Santo Domingo:] (Anglice) Commentariolus nº 52. U.S. Government Printing Office, 1892. Digitizatum die 14 Augusti anni 2012. p. 3. &quot;''...the Republic of Santo Domingo or República Dominicana (Dominican Republic) as it is officially designated'''.&quot;&lt;/ref&gt; Incolae Hispanice ''Dominicanos'' (i.e. &quot;Dominiciani&quot; seu &quot;Dominicani&quot;) nucupabantur, quod est adiectivalis forma &quot;''Domingo''&quot; (i.e. &quot;Dominicus&quot;) Hispanice et revolutionarii suam nove indepedentem nationem Hispanice ''La República Dominicana'' (i.e. &quot;Res publice Dominiciana&quot;) nominavere.

In [[hymnus nationalis|hymno nationali]] Rei publicae Dominicianae (Hispanice ''Himno Nacional'') vocabulum &quot;Dominicianus&quot; non apparet. Verborum cantatorum auctor, [[Aemilius Prud’Homme]], consistenter poëticum terminum Hispanice ''Quisqueyanos'' (i.e. &quot;Guisqueiani&quot;&lt;ref name=&quot;Lexicon&quot;&gt;[http://www.uni-mannheim.de/mateo/camenaref/hofmann/hof2/s0525b.html Lexicon Universale Hofmann] ubi dicitur &quot;Hispaniola (...) Interdum etiam Guisqueia (...)&quot;.&lt;/ref&gt;) utitur. Verbum Hispanice &quot;''Quisqueya''&quot; (Latine &quot;Guisqueia&quot;&lt;ref name=&quot;Lexicon&quot;&gt;[http://www.uni-mannheim.de/mateo/camenaref/hofmann/hof2/s0525b.html Lexicon Universale Hofmann] ubi dicitur &quot;Hispaniola (...) Interdum etiam Guisqueia (...)&quot;.&lt;/ref&gt;) ex autochthonica [[Indi Americani|Indorum]] [[Taini|Tainorum]] lingua derivat et &quot;matrem omnium terrarum significat. Quo crebro utuntur carmina sicuti alio nomine nationi. Nationis nomen crebro &quot;R.D.&quot; breviatur.&lt;ref&gt;Kraft, Randy (die 27 August anni 2000). &quot;[http://articles.mcall.com/2000-08-27/entertainment/3318706_1_punta-cana-caribbean-sea-white-beach Paradise On The Beach Resorts Are Beautiful In Caribbean's Punta Cana, But Poverty Is Outside The Gates]&quot; (Anglice). ''The Morning Call''. Archivatum ex originali die 21 September anni 2013.&lt;/ref&gt;

== Historia ==
=== Historia Praeeuropaea ===
[[Fasciculus:Cacicatus Hispaniolae.png|thumb|350px|Hispaniolae cacicatus quinque Christophoro Columbo advento.]]
[[Fasciculus:Cueva El Pomier.jpg|thumb|350px|Specus Pomierenses (Hispanice ''Cuevas de El Pomier'') specuum quinque et quinquaginta series [[Fanum Sancti Christophori (Res publica Dominicana)|Fani Sancti Christophori]] (Hispanice ''San Cristóbal'') septentrione in Re publica Dominiciana sita sunt. Maximam 2000 annos antiquae artis rupestris collectionem in Caribaeo continent.]]
[[Linguae Arawakanae|Arawakanophoni]]&lt;ref&gt;Vbb. adiectt. &quot;Arawakanus&quot;, &quot;Arawakensis&quot;: cf. nomina biologica e.g. ''[[Drosophila arawakana]]'', ''[[Furculanurida arawakensis]]''.&lt;/ref&gt; [[Taini]] [[Hispaniola]]m e boreorientali regione eius quod nunc America Australis nuncupatur migraverunt, loco movendis prioribus incolis,&lt;ref name=&quot;Luna&quot;&gt;Luna Calderón, Ferdinandus (Fernando) (Decembri mense anni 2002). [https://web.archive.org/web/20081001151311/http://www.kacike.org/CalderonEspanol.pdf &quot;ADN Mitocondrial Taíno en la República Dominicana&quot;] (PDF). ''Kacike'' (Hispanice) (Speciale). ISSN 1562-5028. Archivatum ex originali (PDF) Kalendis Octobribus anni 2008.&lt;/ref&gt; circa annum 650. Agriculturae piscatuique&lt;ref name=&quot;Encarta&quot;&gt;[https://web.archive.org/web/20071114170306/http://encarta.msn.com/encyclopedia_761563569_7/Dominican_Republic.html &quot;Dominican Republic&quot;] (Anglice). ''Encarta''. Microsoft Corporation. Archivatum ex originali die 14 Novembris anni 2007. Inspectum die 6 Iunii anni 2007.&lt;/ref&gt; et venationi collectionique&lt;ref name=&quot;Luna&quot;&gt;Luna Calderón, Ferdinandus (Fernando) (Decembri mense anni 2002). [https://web.archive.org/web/20081001151311/http://www.kacike.org/CalderonEspanol.pdf &quot;ADN Mitocondrial Taíno en la República Dominicana&quot;] (PDF). ''Kacike'' (Hispanice) (Speciale). ISSN 1562-5028. Archivatum ex originali (PDF) Kalendis Octobribus anni 2008.&lt;/ref&gt; se dederunt. Feri [[Caribae]]&lt;ref name=&quot;Lexicon Universale&quot;&gt;Vide &quot;Caribae&quot; apud [http://www.uni-mannheim.de/mateo/camenaref/hofmann/hof1/s0726b.html Lexicon Universale Hofmann].&lt;/ref&gt; Tainos versus boreorientale [[Mare Caribaeum|Caribaeum]] per plerumque [[saeculum 15|saeculum XV]] pepulere.&lt;ref&gt;Royal, Robertus (Robert) (vere anni 1992). [https://web.archive.org/web/20090216092556/http://www.millersville.edu/~columbus/data/ant/ROYAL-01.ANT &quot;1492 and Multiculturalism&quot;]. ''The Intercollegiate Review''. 27 (2): 3–10. Archivatum ex originali die 16 Februarii anni 2009.&lt;/ref&gt; Aestimationes de numero incolarum in Hispaniola anno 1492 ample variat e centum milibus&lt;ref name=&quot;Rawley&quot;&gt;Rawley, Iacobus A. (James A.); Behrendt, Stephanus (Stephen) D. (2005). ''[https://books.google.es/books?id=Sn5pK8rbR5MC&amp;pg=PA49&amp;redir_esc=y&amp;hl=es#v=onepage&amp;q&amp;f=false The Transatlantic Slave Trade: A History]'' (Anglice). University of Nebraska Press. p. 49. ISBN 0-8032-3961-0.&lt;/ref&gt;, trecentis milibus&lt;ref name=&quot;Luna&quot;&gt;Luna Calderón, Ferdinandus (Fernando) (Decembri anni 2002). [https://web.archive.org/web/20081001151311/http://www.kacike.org/CalderonEspanol.pdf &quot;ADN Mitocondrial Taíno en la República Dominicana&quot;] (PDF). ''Kacike'' (Hispanice) (Speciale). ISSN 1562-5028. Archivatum ex originali (PDF) Kalendis Octobribus anni 2008.&lt;/ref&gt;, quadrigentis milibus ad duos milliones.&lt;ref&gt; Keegan, Gulielmus (William). [https://web.archive.org/web/20080321191857/http://www.millersville.edu/~columbus/data/ant/KEEGAN08.ANT &quot;Death Toll&quot;] (Anglice). [[Universitas Millersvilla]], ex ephemeride Archaeology (Ianuario/Februario anni 1992, p. 55). Archivatum ex originali die 21 Martii anni 2008. Inspectum die 19 Iunii anni 2008.&lt;/ref&gt; Determinare praecise quomodo multi homines in insula in [[America praecolumbiana|temporibus Praecolumbianis]] viverent fere impossibile est, quia accurata acta non sunt.&lt;ref&gt;Henige, David (1998). ''[https://books.google.es/books?id=1MJ9HPsGsrUC&amp;pg=PA174&amp;redir_esc=y#v=onepage&amp;q&amp;f=false Numbers from nowhere: the American Indian contact population debate]'' (Anglice). University of Oklahoma Press. p. 174. ISBN 0-8061-3044-X.&lt;/ref&gt; Anno 1492 insula in Tainis [[cacicatus|cacicatibus]]&lt;ref&gt;[https://books.google.es/books?id=S0dcAAAAcAAJ&amp;pg=PT90&amp;lpg=PT90&amp;dq=cacicatus&amp;source=bl&amp;ots=oilKvYB7ng&amp;sig=tqtLTWY1v49EpnO6GgXJBjssMVc&amp;hl=es&amp;sa=X&amp;ved=0ahUKEwjdo8Gm-ebVAhXMOxQKHbkPDVgQ6AEIRDAF#v=onepage&amp;q=cacicatus&amp;f=false Tomus alterus De justa Indiarum occidentalium gubernatione] ubi dicitur &quot;(...) et disponunt in successione Cacicatus, (...)&quot;.&lt;/ref&gt; quinque divisa erat.&lt;ref&gt;Robertus (Roberto) Cassá (1992). ''[https://books.google.es/books?id=oJ-wJ49cNwAC&amp;pg=PA126&amp;redir_esc=y#v=onepage&amp;q&amp;f=false Los Indios de Las Antillas]'' (Hispanice). Editorial Abya Yala. pp. 126–. ISBN 978-84-7100-375-1. Inspectum die 15 Augusti anni 2012.&lt;/ref&gt;&lt;ref&gt;Wilson, Samuel M. (1990). ''Hispaniola: Caribbean Chiefdoms in the Age of Columbus'' (Anglice). Univ. of Alabama Press. p. 110. ISBN 0-8173-0462-2.&lt;/ref&gt; [[Taini]]cum nomen toti insulae ''Ayti''&lt;ref&gt;[https://books.google.es/books?id=Lmk8NQJuOYYC&amp;pg=PA541&amp;lpg=PA541&amp;dq=xaragua+quinque+regna&amp;source=bl&amp;ots=sR2UpmmuwN&amp;sig=pgR-Okk_dmk7K16ilQLWebcy6dY&amp;hl=es&amp;sa=X&amp;ved=0ahUKEwjkspX7hefVAhWLQBQKHXyEBWMQ6AEIPTAJ#v=onepage&amp;q&amp;f=false Georgi Horni Arca Noæ, sive Historia imperiorum et regnorum à condito orbe.]&lt;/ref&gt; seu ''[[Quisqueia]]''&lt;ref&gt;[https://books.google.es/books?id=d_OKCTI2Qv0C&amp;pg=PA122&amp;lpg=PA122&amp;dq=regina+Anacaona&amp;source=bl&amp;ots=mJbtv9Ethv&amp;sig=tMIJBtvJDe8mRRK9DmQYr_Ktf9w&amp;hl=es&amp;sa=X&amp;ved=0ahUKEwjx1I-xgI_WAhXHQJoKHY88DEkQ6AEIWTAM#v=onepage&amp;q=regina%20Anacaona&amp;f=false Columbeis] ubi dicitur: &quot;(...) Axis ad oppositi populos, ignota Quisqueiae Littora (...)&quot;.&lt;/ref&gt; erat.&lt;ref&gt;Anglería, Petrus Martyr (Pedro Mártir de) (1949). ''Décadas del Nuevo Mundo, Tercera Década, Libro VII'' (Hispanice). Bonaëropoli: Editorial Bajel.&lt;/ref&gt;

Hispani anno 1492 advenerunt. Benignas post rationes initio, Taini contra occupationem renisi sunt, a femina phylarcha [[Anacaona]]&lt;ref&gt;[https://books.google.es/books?id=d_OKCTI2Qv0C&amp;pg=PA122&amp;lpg=PA122&amp;dq=regina+Anacaona&amp;source=bl&amp;ots=mJbtv9Ethv&amp;sig=tMIJBtvJDe8mRRK9DmQYr_Ktf9w&amp;hl=es&amp;sa=X&amp;ved=0ahUKEwjx1I-xgI_WAhXHQJoKHY88DEkQ6AEIWTAM#v=onepage&amp;q=regina%20Anacaona&amp;f=false Columbeis] ubi dicitur: &quot;(...) Pone subit magna matrum comitante caterua Anacaona soror, qua non prudentior ulla (...)&quot;.&lt;/ref&gt; (Hispanice ''Anacaona'') [[Xaragua]]e&lt;ref&gt;[https://books.google.es/books?id=Lmk8NQJuOYYC&amp;pg=PA541&amp;lpg=PA541&amp;dq=xaragua+quinque+regna&amp;source=bl&amp;ots=sR2UpmmuwN&amp;sig=pgR-Okk_dmk7K16ilQLWebcy6dY&amp;hl=es&amp;sa=X&amp;ved=0ahUKEwjkspX7hefVAhWLQBQKHXyEBWMQ6AEIPTAJ#v=onepage&amp;q&amp;f=false Georgi Horni Arca Noæ, sive Historia imperiorum et regnorum à condito orbe.]&lt;/ref&gt; et eius quondam coniuge Caonabone&lt;ref&gt;[https://books.google.es/books?id=Lmk8NQJuOYYC&amp;pg=PA541&amp;lpg=PA541&amp;dq=xaragua+quinque+regna&amp;source=bl&amp;ots=sR2UpmmuwN&amp;sig=pgR-Okk_dmk7K16ilQLWebcy6dY&amp;hl=es&amp;sa=X&amp;ved=0ahUKEwjkspX7hefVAhWLQBQKHXyEBWMQ6AEIPTAJ#v=onepage&amp;q&amp;f=false ''Georgi Horni Arca Noæ, sive Historia imperiorum et regnorum à condito orbe''] ubi dicitur: &quot;(...) Ultimus Rex ''Caonabo'', qui in Columbi navigatione ''Canoba'' id est ''domus auri'' dicitur. (...)&quot;.&lt;/ref&gt; (Hispanice ''Caonabo'') [[Maguana]]e&lt;ref&gt;[https://books.google.es/books?id=Lmk8NQJuOYYC&amp;pg=PA541&amp;lpg=PA541&amp;dq=xaragua+quinque+regna&amp;source=bl&amp;ots=sR2UpmmuwN&amp;sig=pgR-Okk_dmk7K16ilQLWebcy6dY&amp;hl=es&amp;sa=X&amp;ved=0ahUKEwjkspX7hefVAhWLQBQKHXyEBWMQ6AEIPTAJ#v=onepage&amp;q&amp;f=false ''Georgi Horni Arca Noæ, sive Historia imperiorum et regnorum à condito orbe''] ubi dicitur: &quot;(...) ''Xaragua'' aliis ''Guaccayrima'', Haniguaiaga'' vel ''Maguana'' (...)&quot;.&lt;/ref&gt; non minus quam phylarchis [[Guacanagarix|Guacanagarice]], [[Guama]], [[Hatueius|Hatueio]] et [[Henriculus|Henriculo]]&lt;ref&gt;&quot;[https://books.google.es/books?id=vobbcIgAeK0C&amp;pg=PA80&amp;dq=bohechius&amp;hl=es&amp;sa=X&amp;ved=0ahUKEwidtq3pxvjWAhXLDcAKHY1zD5MQ6AEIPjAD#v=onepage&amp;q=bohechius&amp;f=false Annales minorum seu Trium Ordinum a S. Francisco institutorum, Volumen 15]&quot;, ubi dicitur: &quot;(...) Henriculum, Indiarum nobilem, a se suisque Consodalibus optime a juventute apud Conventum Verae Pacis in provincia Xarague (in qua Bohequius, unus ex quinque Regibus Hispaniolae imperabat) (...)&quot;. &lt;/ref&gt; (Hispanice ''Guacanagarix, Guamá, Hatuey, Enriquillo'') ducti. Cuius ultimi bene gesta suae genti autonomam [[inclavatura]]m in temporis aliquod in insula consecuta sunt. Paucis annis post 1492, Tainorum numerus extreme declinavit, propter [[variola]]m,&lt;ref&gt;[http://www.smithsonianmag.com/people-places/what-became-of-the-taino-73824867/ &quot;What Became of the Taíno?&quot;] (Anglice). ''Smithsonian'' Octobri 2011.&lt;/ref&gt; [[morbillus|morbillum]] et alios [[morbus|morbos]] qui advenerunt cum Europaeis, &lt;ref name=&quot;Smallpox&quot;&gt;[http://www.dshs.texas.gov/preparedness/bt_public_history_smallpox.shtm &quot;History of Smallpox – Smallpox Through the Ages&quot;] (Anglice). ''Texas Department of State Health Services''.&lt;/ref&gt; apud alias res infra explicatas.

Prima relata variolae eruptio in Americis in Hispaniola anno 1507 evenit.&lt;ref name=&quot;Smallpox&quot;&gt;[http://www.dshs.texas.gov/preparedness/bt_public_history_smallpox.shtm &quot;History of Smallpox – Smallpox Through the Ages&quot;] (Anglice). ''Texas Department of State Health Services''.&lt;/ref&gt; Ultima purorum Tainorum acta in natione e 1864 erant. Etiamnunc hereditas biologica longe superfuit propter permixtionem. Census acta e 1514 40% Hispanorum virorum Tainas feminas uxores duxisse in Dominicopoli revelant&lt;ref&gt; Ferbel Azcarate, Petrus (Pedro) J. (Decembri 2002). [https://web.archive.org/web/20040617195321/http://www.kacike.org/FerbelEnglish.pdf &quot;Not Everyone Who Speaks Spanish is from Spain: Taíno Survival in the 21st Century Dominican Republic&quot;] (Anglice) (PDF). ''Kacike'' (extraordinarius numerus). ISSN 1562-5028. Archivatum ex originali (PDF) die 17 Iunii anni 2004. Inspectum die 24 Septembris anni 2009.&lt;/ref&gt; et aliqui hodiernorum Dominicianorum Tainam originem habent.&lt;ref name=&quot;GuitarKac&quot;&gt;Guitar, Lynne (Decembri anni 2012). [https://archive.org/details/KacikeJournal &quot;Documenting the Myth of Taíno Extinction&quot;] (Anglice) (PDF). ''Kacike'' (numerus extraordinarius). ISSN 1562-5028. Inspectum die 24 Augusti anni 2016.&lt;/ref&gt;&lt;ref&gt;Martínez Cruzado, Ioannes Carolus (Juan Carlos) (Decembri anni 2002). [&quot;The Use of Mitochondrial DNA to Discover Pre-Columbian Migrations to the Caribbean: Results for Puerto Rico and Expectations for the Dominican Republic&quot;] (Anglice) (PDF). ''Kacike'' (numerus extraordinarius). ISSN 1562-5028. Inspectus die 24 Augusti anni 2016.&lt;/ref&gt; Apud Tainicas culturas sunt picturae rupestres non minus quam [[figlina]]e designatio quae est in usu adhuc in parvo opificum villagio Higüerito,in [[Moca (Res publica Dominicana)|Moca]].&lt;ref&gt;O'Halloran, Hyacintha (Jacinta) (2007-01-01). [https://books.google.es/books?id=s9fMQbQUp6wC&amp;redir_esc=y Fodor's Budapest]. Fodor's Travel Publications. ISBN 978-1-4000-1740-9.&lt;/ref&gt;

=== Colonizatio Europaea ===
[[5 Decembris|Nonis Decembribus]] anni [[1492]] [[Christophorus Columbus|Christophori Columbi]] [[carabus|carabi]] in insulam advenerunt, quae Hispaniola (Hispanice ''La Española'') nominata est. Columbus ibi nautarum novem et triginta sedem, [[Nativitas (Res publica Dominicana)|Nativitatem]] (Hispanice ''La Navidad'') nominatam, reliquit. Insequenti anno, cum secundo itinere rediret, eam destructam invenit et novam sedem plus ad orientem, in hodiernae Rei publicae Dominicianae territorio, [[Isabella (Res publica Dominicana)|Isabellam]] (Hispanice ''La Isabela'') nuncupatam et primam Europaeam sedem in Americis putatam condere decrevit. Insula colonia Hispanica facta est.
[[Fasciculus:AMH-6754-NA Bird's eye view of the city of Santo Domingo.jpg|thumb|[[Dominicopolis]] circa annum 1665 ab [[Iohannes Vingboons|Iohanne Vingboons]] picta.]]

Primis ex annis Hispanicae dicionis memoratur schema fundorum: quod in [[Lusitania|Lusitanorum]] experientia in occidentali Africae litore fundatur et consistebat ex apparatione laboris solutae ab Hispanis, regimine servitutis autochthonibus gentibus, eandum venditione in Hispania et impositione tributi in pulvere aureo aut [[Gossypium (genus)|gossypio]]. Naturalium divitiarum et virium laboris autochthonum apparatio solum Coronae Hispanicae favere poterat, sed non privatis. Quod multum fastidii Hispanis et mortem, crebro propter tristitiam, Tainorum per iter oceanicum attulit. Modus quo tractabantur indigenae (putati praemium occupationis) effecit collapsionem physicae eorum condicionis et exspectationis vitae. Taini adhuc se coniunctim occiderunt et abortus effecerunt sicuti unicam viam evitandae servitutis; numerus incolarum e circa 400 000 hominum anno 1492 calculatorum ad 60 000 anno 1508 deminuit.

Rara autochthonum operariorum summa et eius condensatio paucis in familiis aristocraticis Hispanos colonos alias versus terras emigrare fecit. Solum introducta intensiva [[Saccharum officinarum|harundinis sacchari]] cultura, numerus incolarum crescere coepit et huius culturae gratia incepit etiam commercium nigricolorum servorum ex Africa. Medio saeculo XVI aestimatur adfuisse in insula 20 000 Afrorum oriundorum dissimilibus e tribubus, dum Taini fere exstincti sunt.

Ineunte anno 1600, ad pugnandum contra subintroductionem et piratarum impetus, Domus Regia Hispanica omnes incolas in occidentali et boreorientali insulae zona viventes in zonas protegibiliores et vicinas capiti, [[Dominicopolis|Dominicopoli]], traiicere decrevit. Quod attulit generalem insularis oeconomiae paupertatem et possibilitatem ut [[lesta]]e et [[bucaner]]es&lt;ref&gt;[[Pirata#De piratarum genera]]&lt;/ref&gt; occidentalem partem ([[Insula Testudinis|Insulam Testudinis]], Hispanice ''Isla Tortuga''; Francice ''Île de la Tortue''; et [[Haitiane]] ''Latòti'') sicuti suam praecipuam discessus sedem ad faciendos impetus contra naves directas et oriundas ex Europa occuparent.

=== Dicio Francica (1795-1809) ===
[[Fasciculus:Battle of Santo Domingo (French and British ships).jpg|thumb|Francicae et Britannicae naves in Proelio Dominicopoleos.]]
Anno [[1801]], [[Francicus Dominicus Toussaint Louverture]] Dominicopolin advenit, servitutis abolitionem pro [[Francia|Rem publicam Francicam]] edicens. Brevi ante, [[Napoleo I (imperator Franciae)|Napoleo]] exercitum misit qui totam insulam domuit et eam paucos per menses rexit. [[Mulatus|Mulati]] et [[nigricolor]]es denuo hos contra Francicos Octobri mense anno [[1802]] consurrexerunt et denique eos vicerunt Novembri mense anni [[1803]]. [[1 Ianuarii|Kalendis Ianuariis]] anni [[1804]] victores [[Sanctus Dominicus (colonia Francica)|Sanctum Dominicum]] (Francice ''Saint-Domingue'') rem publicam independentem [[Haitia]]m esse declaraverunt. Adhuc antequam ab Haitianis victi essent, Francicum praesidiolum Dominicopoli mansit. Servitus rursus statuta est et Hispanorum exsulum multi rediverunt. Anno 1805, postquam se Imperator coronaret, [[Ioannes Iacobus Dessalines]] invasit, attingenda Dominicopoli antequam Francicam ob turmam navalem recederet. Per [[Cibao]]nem&lt;ref name=&quot;Globorotalia&quot;&gt;Confer binomen &quot;Globorotalia cibaoensis&quot;.&lt;/ref&gt; recedentes, Haitiani urbes [[Sanctus Iacobus Equitum|Sanctum Iacobum Equitum]]&lt;ref name=&quot;Hierarchia&quot;&gt;[http://catholic-hierarchy.org/diocese/dsnca.html Hierarchia Catholica].&lt;/ref&gt; (Hispanice ''Santiago de los Caballeros'') et [[Moca (Res publica Dominicana)|Mocam]] (Hispanice ''Moca'') spoliaverunt, fere omnes incolas trucidantes et iuvantes condere bisaecularem inimicitiam duas inter nationes.

Francici in orientali insulae parte reluctati sunt donec ab Hispanis insulae incolis in Proelio Pali Defixi (Hispanice ''Batalla de Palo Hincado'') die [[7 Novembris]] anni 1808 victi sunt. Iuvante [[Regia Classis Britannica|Regia Classi Britannica]], Hispani urbem Dominicopolin obsederunt. Francici obsessa urbe denique deditionem die [[9 Iulii]] anni 1809 fecerunt, incipiente annorum duodecim periodo Hispanicae dicionis, nuncupatae Dominiciana in historia &quot;[[Hispania Stulta|Hispaniae Stultae]]&quot;.

=== Reversio in Hispaniam (1809-1821) ===
In historia Rei publicae Dominicianae, &quot;[[Hispania Stulta|Hispaniae Stultae]]&quot; (Hispanice ''España Boba'') periodus e 1809 ad 1821 duravit, per quam [[Maxima Praefectura Sancti Dominici]] (Hispanice ''Capitanía General de Santo Domingo'') sub dicione Hispanica fuit; nihilominus Hispanicum gubernium minimis potestatibus functum est quoniam eius opes tenues factae sunt propter [[Bellum Independentiae Hispanicae]] et aliquot bella independentiae in America Hispanica.&lt;ref&gt;[https://web.archive.org/web/20160809033901/http://dominicanaonline.org/Portal/espanol/cpo_conquista5.asp Período de “la España Boba”] (Hispanice) - apud Dominicanaonline.org.&lt;/ref&gt; Haec periodus finivit cum Dominicianae auctoritates brevis durationis independentiam [[30 Novembris|pridie Kalendas Decembres]] anni 1821 declaraverunt.

=== Independentia ex Hispania (1821) ===
Post duodecadem annorum fastidii et independentiae propositum casum ab aliquot oppositoribus globis, prior Sancti Dominici Locumtenens-Gubernator (summus administrator) [[Iosephus Núñez de Cáceres]] coloniae independentiam ex [[Imperium Hispanicum|Hispanica corona]] sicuti [[Res publica Haitiae Hispanicae|Haitiam Hispanicam]] [[30 Novembris|pridie Kalendas Decembres]] anni 1821 declaravit. Haec periodus Independentia Ephemera nuncupatur.&lt;ref&gt; H. Hoetink (die 29 Maii 1986). &quot;The Dominican Republic c. 1870–930&quot;. In Leslie Bethell. [https://books.google.es/books?id=X4axAmBS-7wC&amp;pg=PA287&amp;redir_esc=y#v=onepage&amp;q&amp;f=false ''The Cambridge History of Latin America'']. V, Circa 1870 to 1930 (Anglice). Cambridge University Press. p. 287. ISBN 978-0-521-24517-3.&lt;/ref&gt;

=== Unificatio Hispaniolae (1822-1844) ===
[[Fasciculus:President Jean-Pierre Boyer of Haiti (Hispaniola Unification Regime) Portrait.jpg|thumb|180px|[[Ioannes Petrus Boyer]], mulatus rector Haitiae]]
Denuo independens res publica duo menses post finivit causa [[Haitia]]ni gubernii ab [[Ioannes Petrus Boyer|Ioanne Petro Boyer]] ducti.&lt;ref name=&quot;GuitarHist&quot;&gt;Guitar, Lynne. [http://www.hispaniola.com/dominican_republic/info/history.php &quot;History of the Dominican Republic&quot;]. Hola.com. Inspectum die 29 Maii anni 2007.&lt;/ref&gt;

Ut [[Francicus Dominicus Toussaint Louverture]] decennia duo antea fecerat, Haitiani servitutem aboleverunt. Ad colligendam pecuniam magnae indemnitati 150 millionum francorum cui assensit se soluturam esse prioribus Francicis colonis et quae inceps ad 60 milliones francorum deminuit, Haitianum gubernium onerosa vectigalia Dominicianis imposuit. Quia Haitia suo exercitui sufficienter providere nequivit, occupantes vires magna ex parte superfuerunt, cibum et promissa pistolio exquisiverunt et confiscaverunt. Redistribuendorum agrorum conatus conflixit cum communalis agrorum tenurae systemate (Hispanice ''terrenos comuneros''), quod e fundorum pecuariorum oeconomia surrexerat, et aliqui incolae indignati sunt cum culturas mercatorias serere coacti essent, vigente [[Ioannes Petrus Boyer|Ioannis Petri Boyer]] et [[Iosephus Balthasar Inginac|Iosephi Balthasaris Inginac]] Codice Rurali (Francice ''Code Rural'').&lt;ref&gt;Hoetink, ''The Dominican People: Notes for a Historical Sociology'' versum in Anglicam a Stephano Ault Pg. 83. Johns Hopkins Press: Baltimorae, 1982.&lt;/ref&gt; Ruralibus et asperis in areis, Haitiana administratio nimis inefficax esse solebat ad exsequendas suas proprias leges. Fuit urbe Dominicopoli ubi occupationis effectus acutissime sensi sunt et fuit ibi ubi independentiae motus ortus est.

Haitiana constitutio albicolores primores agros tenere vetavit et praecipuae Dominicianae familiae agrorum possessores vi proprietatibus suis privatae sunt. Complures in [[Cuba]]m, [[Portus Dives|Portum Divitem]] (qui ambo [[Imperium Hispanicum|possessiones Hispanicae]] in illo tempore erant) aut in [[Grandis Columbia|Grandem Columbiam]] emigraverunt, plerumque cohortantibus Haitianis officialibus qui agros acquirebant. Haitiani [[Ecclesia Catholica Romana|Ecclesiam Catholicam Romanam]] cum Francicis servorum dominis qui eos sub iugum miserant sociaverunt et omnes Ecclesiae proprietates confiscaverunt, omnes peregrinos clericos deportaverunt et reliquorum clericorum vincula cum [[Sancta Sedes|Vaticano]] secaverunt.

Omnia educativa fastigia collapsa sunt; universitas clausa est, quia et opibus et studentibus carebat, cum Dominiciani iuvenes viri e 16 in 25 annos nati Haitianum in exercitum conscripti essent. [[Ioannes Petrus Boyer|Ioannis Petri Boyer]] copiae occupantes, quae magna ex parte Dominiciani erant, insolutae erant et Dominicianis e civibus pabulari spoliareque debuerunt. Haitia onerosum vectigal Dominiciano populo imposuit.&lt;ref name=&quot;Matibag&quot;&gt;Matibag, Eugenius (Eugenio) (2003). ''Haitian-Dominican Counterpoint: Nation, State, and Race on Hispaniola'' (Anglice). Macmillan. ISBN 0-312-29432-8.&lt;/ref&gt;

Multi albicolores e Dominicopoli in [[Portus Dives|Portum Divitem]] ac [[Cuba]]m (ambos rectos ab Hispania), [[Venetiola]]m et alibi fugerunt. Denique oeconomia defecit et vectigalia onerosa facta sunt. Rebelliones adhuc a libertis gesta sunt, dum Dominiciani Haitianique coniunctim laboraverunt ad eiiciendum [[Ioannes Petrus Boyer|Ioannem Petrum Boyer]] e potestate. Antihaitiani aliquot generum motus -pro independentia -pro [[Hispania]], pro [[Francia|Francicis]], pro [[Britanniarum Regnum|Britannicis]], pro [[Civitates Foederatae Americae|Civitatibus Foederatis]]- corroborati sunt, deiecto Ioanne Petro Boyer anno 1843.&lt;ref name=&quot;Matibag&quot;&gt;Matibag, Eugenius (Eugenio) (2003). ''Haitian-Dominican Counterpoint: Nation, State, and Race on Hispaniola'' (Anglice). Macmillan. ISBN 0-312-29432-8.&lt;/ref&gt;

=== Independentia ex Haitia (1844) ===
[[Fasciculus:Juan pablo duarte diez.jpg|thumb|right|upright|[[Ioannis Paulus Duarte]], pater conditor Rei publicae Dominicianae.]]
Anno 1844 popularis rebellionis motus ab [[Ioannis Paulus Duarte|Ioanne Paulo Duarte]] rectus induxit in Rei publicae Dominicianae independentiam sanctam declaratione ut omnes homines, sine discriminatione, aequi sint. Nascenti in re publica disputabant ei qui absolutam independentiam volebant et ei qui a natione evolutiore protectoratus optionem malebant.

=== Res publica Restitutionis et exeuns saeculum XIX ===
Anno 1860, Dominicianus praeses [[Petrus Santana]] foedus readmissionis in Hispaniam (anno 1861) obsignavit. Tale foedus aliquorum praefectorum rebellionem fecit, et bellum Restitutionis nuncupatum incepit. Anno 1865, res publica independentiam recuperavit, tempus sine gubernatione et continuis cum gubernantium mutationibus transeuns. Hic status duravit donec [[Ulixes Heureaux]] dictaturam per annos duodecim (inter 1887 et 1899) instituit, quae finivit eo occiso.

=== Ineunte saeculo XX (1900-1930) ===
Ineunte [[saeculum 20|saeculo XX]] politica oeconomicaque instabilitas et [[saeculum 19|saeculi XIX]] creditae pecuniae solvendae mora fecerunt ut eveniret quod nuncupatur &quot;Prima Invasio [[Statunitensis]]&quot;, quae duravit e [[1916]] usque ad [[1924]]. Per periodon inter 1924 et [[1930]], Dominiciana oeconomia tempus &quot;Saltationem Millionum&quot; (Hispanice ''Danza de los Millones'') essentialiter propter aucta internationalis [[harundo saccharina|harundinis saccharinae]] pretia nominatum vixit.

=== Aetas Raphaëlis Trujillo (1930-1961) ===
[[Fasciculus:Rafael Trujillo 1952.jpg|thumb|left|[[Raphael Trujillo|Raphaël Trujillo]] anno 1952.]]
E [[1930]] usque ad [[1961]] natio sub [[Raphael Trujillo|Raphaëlis Leonidae Trujillo]] dictatura fuit, quae fuit tempus obscurissimum Dominicianae historiae, cum oppositorum insectatione et caedibus. Sublato Motu die 14 Iunii (Hispanice ''Movimiento 14 de Junio'') anno [[1959]] et interfectis Sororibus Mirabal (Hispanice ''Hermanas Mirabal''), regimen languescere coepit quoad Raphaël Trujillo anno [[1961]] occisus est.

=== Aetas Post Raphaëlem Trujillo (1961-2000) ===
[[Raphael Trujillo|Raphaële Trujillo]] occiso, natio aliquot gubernia transit apud quae inveniuntur id professoris [[Ioannes Bosch|Ioannis Bosch]], menses septem postea prostratum; triumviratus; et [[Statunitensis]] armatus interventus anni [[1965]]. Anno [[1966]] [[Ioachimus Balaguer]] potestatem accepit et ibi mansit per annorum duodecim periodum in gubernio semidictatoriali in quo usus est comitialibus fraudibus et repressionibus contra suos politicos oppositores.

Per anni [[1978]] comitia electus est [[Antonius Guzmán Fernández]], oppositoris Factionis Revolutionariae Dominicianae (Hispanice ''Partido Revolucionario Dominicano''; acronymo ''PRD''). Fuit primum gubernium electum populari suffragio e 1924. Eius muneris tempus depictum est re ut esset unum liberalissimorum quod Res publica Dominiciana habuisset in decenniis. Finivit cum Antonius Guzmán anno [[1982]] se occidit et ei successit praeses vicarius [[Iacobus Majluta]], qui per dies 43 gubernavit.

Anno 1982 vicit in comitiis [[Salvator Georgius Blanco]], tunc gubernantis factionis, PRD. Anno [[1986]] denuo cepit potestatem Ioachimus Balaguer, natus 80 annos.

Anno [[1990]] victor fuit in comitiis circumdatis delationibus fraudum e parte Ioannis Bosch, Factionis Liberationis Dominicianae (Hispanice ''Partido de la Liberación Dominicana''; acronymo ''PLD''). Anno [[1994]], Ioachimus Balaguer denuo in comitiis vicit; sed, allegatis scripto fraude impedimentisque ut militantes oppositores suffragium ferrent, reductam suam praesidialem periodum ad biennium vidit, decernens comitia habere anno [[1996]].

=== Saeculum XXI ===
 
Anno [[2000]] Factionis Revolutionariae Dominicianae (Hispanice ''Partido Revolucionario Dominicano''; acronymo ''PRD'') [[Hippolytus Mejía]] comitiis vicit. Hoc oeconomicorum problematum tempus fuit.&lt;ref name=&quot;Patterson&quot;&gt;Patterson, Claudia (die 4 Octobris anni 2004). [https://web.archive.org/web/20081107015518/http://www.coha.org/2004/10/president-leonel-fernandez-friend-or-foe-of-reform/ &quot;President Leonel Fernández: Friend or Foe of Reform?&quot;] (Anglice). ''Council on Hemispheric Affairs''. Archivatum ex originali die 7 Novembris anni 2008.&lt;/ref&gt; [[Hippolytus Mejía]] in conatu ut eligeretur anno 2004 a Leonello Fernández Factionis Liberationis Dominicianae (Hispanice ''Partido de la Liberación Dominicana''; acronymo ''PLD'') victus est. Anno 2008, Leonellus Fernández denuo electus est tertio munere.&lt;ref name=&quot;Relations&quot;&gt;[https://www.state.gov/r/pa/ei/bgn/35639.htm &quot;U.S. Relations With the Dominican Republic&quot;] (Anglice). [[Ministerium Rerum Externarum Civitatum Foederatarum]]. Die 22 Octobris anni 2012.&lt;/ref&gt; Leonellus Fernández et Factio Liberationis Dominicianae (Hispanice ''Partido de la Liberación Dominicana''; acronymo ''PLD'') creduntur inceptis quae nationem technologice promovit, sicuti constructione ferriviae subterraneae (Hispanice ''El Metro''). Nihilominus eius administrationes de corruptione accusatae sunt.&lt;ref name=&quot;Patterson&quot;&gt;Patterson, Claudia (die 4 Octobris anni 2004). [https://web.archive.org/web/20081107015518/http://www.coha.org/2004/10/president-leonel-fernandez-friend-or-foe-of-reform/ &quot;President Leonel Fernández: Friend or Foe of Reform?&quot;] (Anglice). ''Council on Hemispheric Affairs''. Archivatum ex originali die 7 Novembris anni 2008.&lt;/ref&gt;

[[Danilo Medina]], Factionis Liberationis Dominicianae (Hispanice ''Partido de la Liberación Dominicana''; acronymo ''PLD''), electus est anno 2012 et rursus anno 2016. Ambivit de consilio magis pecuniae collocandae in programmatibus socialibus educationeque et minus in infrastructura.

== Geographia ==
[[Fasciculus:Dominican Republic relief location map.jpg|thumb|350px|Rei publicae Dominicianae topographia.]]
[[Fasciculus:Constanza.jpeg|thumb|300px|Vallis [[Constantia (Res publica Dominicana)|Constantiae]] (Hispanice ''Constanza'').]]
Rei publicae Dominicianae territorium orientalem partem insulae [[Hispaniola]]e, sitae in [[Mare Caribaeum|Mari Caribaeo]] et secundae insulae maximae [[Antillae|Antillarum]] (post [[Cuba]]m), comprehendit. Eius superficies 48 442 km² est. Natio solum unum terrestrem limitem cum Re publica [[Haitia]]na occidente habet et [[Oceanus Atlanticus|Oceano Atlantico]] septentrione et [[Mare Caribaeum|Mari Caribaeo]] meridie alluitur; [[Canalis Monensis]]&lt;ref&gt;[https://web.archive.org/web/20160305191555/http://www.waza.org/es/zoo/elegir-una-especie/reptiles/los-ofidios/epicrates-monensis waza.org] ubi apparet binomen &quot;Epicrates monensis&quot;.&lt;/ref&gt; (Hispanice ''Canal de la Mona'') Rem publicam Dominicianam a [[Portus Dives|Portu Divite]] separat.

=== Morphologia ===
Dominicianum territorium valde montuosum est, cui imminet [[Iugum Centrale (Res publica Dominicana)|Iugum Centrale]] (Hispanice ''Cordillera Central''), in quo exstat [[Duarte (culmen)|Culmen Duarte]] (Hispanice ''Pico Duarte''), maximum culmen in Caribaeo, 3 087 metra altum. Alia iuga montium [[Iugum Septentrionale (Res publica Dominicana)|Iugum Septentrionale]] (Hispanice ''Cordillera Septentrional'') seu [[Iugum Montis Christi]]&lt;ref&gt;[http://www.catholic-hierarchy.org/diocese/dmaom.html Hierarchia Catholica].&lt;/ref&gt; (Hispanice ''Sierra de Monte Cristi''), [[Iugum Orientale (Res publica Dominicana)|Iugum Orientale]] (Hispanice ''Cordillera Oriental''), [[Montes Yamasaenses]] (Hispanice ''Sierra de Yamasá''), [[Montes Samanaenses]]&lt;ref name=&quot;samanaensis&quot;&gt;Confer binomen &quot;Lyria samanaensis&quot;.&lt;/ref&gt; (Hispanice ''Sierra de Samaná''), [[Montes Baorucoense]] (Hispanice ''Sierra de Baoruco''), [[Montes Neyba]] (Hispanice ''Sierra de Neyba'') et [[Montes Martini Garcia]] (Hispanice ''Sierra Martín García'') sunt.

Inter Iugum Centrale et Septentrionale patet [[Vallis Cibaoensis]]&lt;ref&gt;Confer binomen &quot;Globorotalia cibaoensis&quot;.&lt;/ref&gt; (Hispanice ''Valle del Cibao''), ampla uberque planities quae dat nomen toti septentrionali nationis regioni. Austrorientali in zona patet altera ampla litoralis planities.

Maximus lacus Dominiciano in territorio [[Lacus Enriquillensis]]&lt;ref name=&quot;Enriquillense&quot;&gt; Nomini lacus Hispanice ''Enriquillo'' nuncupato, confer binomen &quot;Chondropoma enriquillense&quot;.&lt;/ref&gt; seu [[Lacus Henriculus|Henriculi]]&lt;ref&gt;Nomen lacus nominatum est a regulo Henriculo (Hispanice ''Enriquillo'') qui apparet in libro &quot;[https://books.google.es/books?id=vobbcIgAeK0C&amp;pg=PA80&amp;dq=bohechius&amp;hl=es&amp;sa=X&amp;ved=0ahUKEwidtq3pxvjWAhXLDcAKHY1zD5MQ6AEIPjAD#v=onepage&amp;q=bohechius&amp;f=false Annales minorum seu Trium Ordinum a S. Francisco institutorum, Volumen 15]&quot;, ubi dicitur: &quot;(...) Henriculum, Indiarum nobilem, a se suisque Consodalibus optime a juventute apud Conventum Verae Pacis in provincia Xarague (in qua Bohequius, unus ex quinque Regibus Hispaniolae imperabat) (...)&quot;. &lt;/ref&gt; (Hispanice ''Lago Enriquillo'') est, 265 km2 latus et oceanicae originis et cum superficie 44 metra sita supra aequor maris.

=== Clima ===
[[Fasciculus:Köppen Rei publicae Dominicianae.png|thumb|300px|left|[[Enumeratio climatum Köppen–Geiger|Typi climatum Köppen]] Rei publicae Dominicianae.]]
[[Clima]] [[clima tropicum|tropicum]] Caribicum cum abundantibus pluviis et obviis climaticis laciniis respectu altitudinis. Temperatura media annua (inter minimum et maximum valorem quotidianum) e 19,5&amp;nbsp;°C ad 15&amp;nbsp;°C altitudinum aequarum aut altiorum ad 1000 metra supra aequor maris usque ad 1500 metra supra aequor maris; e 15&amp;nbsp;°C ad 10&amp;nbsp;°C e 1500 metris usque ad 2000 supra aequor maris; e 10&amp;nbsp;°C ad 5&amp;nbsp;°C e 2000 metra usque ad 2500 supra aequor maris; e 5&amp;nbsp;°C ad 0&amp;nbsp;°C e 2500 ad 3000 metra supra aequor maris variant. Ad 200 metra supra aequor maris computantur circa 25,5&amp;nbsp;°C gradus mediae quantitatis annuae, ad transeundum ad 24&amp;nbsp;°C ad 400 metra supra aequor maris, 22,5&amp;nbsp;°C ad 600 metra supra aequor maris et 21&amp;nbsp;°C mediae quantitatis ad 800 metra supra aequor maris. Maximae normaliter 40&amp;nbsp;°C in vallibus calidas per periodos protectis attingere possunt, dum commune est ut attingantur 5&amp;nbsp;°C in montibus [[tempora anni|tempore anni]] minime calido sed haud dubie pluviosissimo, [[Umidum (tempus anni)|tempore pluviarum]] (Octobri, Novembri, Decembri et adhuc Ianuario). Attamen, perceptibiliter, in parte, praecipitationes nivosae exiguissimae sunt propter penuriam montium multum altorum (excepto [[Duarte (culmen)|Culmine Duarti]] solum 3000 metra supra aequor maris eminente).

Secundum tempus (et [[Umidum (tempus anni)|pluviarum]] et [[Siccum (tempus anni)|siccum]]) et respectu altitudinis maximae Solis in caelo, propter mediam nationis [[Latitudo geographica|latitudinem]], dies technice ex horis 11 ad solum minus quam horis 13 et minutis 16 durant.

[[Umidum (tempus anni)|Tempus pluviarum]] Maio incipit et Novembri finitur, quamquam in septentrionali nationis regione pluviae adhuc mense Decembri pergunt. Torrentis indoles nonnullas terrarum labes facit, quamquam maxima damna [[huracanum|huracanis]] et tempestatibus tropicis afferuntur, quorum tempus anni ex Augusto usque ad Octobrem evenire solet.

== Civilitas ==
[[Fasciculus:National Palace Dominican Republic1.jpg|thumb|300px|[[Palatium Nationale (Res publica Dominicana)|Palatium Nationale]] (Hispanice ''Palacio Nacional'') Dominicopoli.]]
Natio generis praesidialis [[res publica]] est. Omnes homines maiorea quam 18 annos nati suffragium ferre possunt, sicuti nupti homines, cuiuslibet aetatis. Nihilominus, [[vigil]]es et [[miles|militares]] suffragium ferre nequeunt.

[[Potestas exsecutiva]] a praeside et praeside vicario quadrienni muneri electis repraesentatur. Comitia praesidialia in singula quadriennia multipla numeri quattuor ([[2008]], [[2004]], [[2000]], etc...) die [[16 Maii]] habentur.

[[Potestas legifera]] a [[Congressus Nationalis Rei publicae Dominicanae|Congressu Nationali Rei publicae Dominicianae]] (Hispanice ''Congreso Nacional de la República Dominicana'') geritur, quem duae camerae componunt: [[Senatus Rei publica Dominicanae|Senatus]] (Hispanice ''Senado'') et [[Camera Deputatorum Rei publicae Dominicanae|Camera Deputatorum]] (Hispanice ''Cámara de Diputados''). Senatus 32 sedes et Camera Deputatorum 178 habet. Comitia legifera in sigulos annos pares haud divisibiles per quattuor habentur per suffragium directum.

=== Rationes internationales ===
Res publica Dominiciana arctas rationes cum [[Civitates Foederatae Americae|Civitatibus Foederatis]] et cum aliis Interamericani systematis civitatibus habet. Res publica Dominiciana fortissima vincula et rationes cum [[Portus Dives|Portu Divite]] habet.

Rei publicae Dominicianae rationes vicina cum [[Haitia]] intentae sunt propter Haitianam gregariam migrationem in Rem publicam Dominicianam, Dominicianis civibus Haitianos ob auctum scelus et alia socialia problemata culpantibus.&lt;ref&gt;Childress, Sara (Sarah) ([[31 Augusti|pridie Kalendas Septembris]] anni 2011). [https://www.pri.org/stories/2011-08-31/dr-haitians-get-lost &quot;DR to Haitians: get lost&quot;] (Anglice). pri.org. Global Post. Inspectum die 24 Augusti anni 2016.&lt;/ref&gt; Res publica Dominiciana ordinaria [[Francophonia|Organizationis Internationalis Francophoniae]] (Francice ''Organisation Internationale de la Francophonie'') socia est.

Res publica Dominiciana foedus liberi commercii cum [[Civitates Foederatae Americae|Civitatibus Foederatis]], [[Costarica]], [[Salvatoria]], [[Guatimalia]], [[Honduria]] et [[Nicaragua]] per [[Foedus Liberi Commercii inter Civitates Foederatas, Americam Centralem et Rem publicam Dominicianam]] (Hispanice ''Tratado de Libre Comercio entre Estados Unidos, Centroamérica y República Dominicana'' seu ''ILC''; Anglice ''Dominican Republic-Central America Free Trade Agreement'' seu ''CAFTA-DR'')&lt;ref&gt;[https://ustr.gov/trade-agreements/free-trade-agreements/cafta-dr-dominican-republic-central-america-fta &quot;CAFTA-DR (Dominican Republic-Central America FTA) | United States Trade Representative&quot;] (Anglice). ''ustr.gov''. Inspectum die 2017-02-08.&lt;/ref&gt; et Foedus Sodalitatis Oeconomicae cum [[Unio Europaea|Unione Europaea]] et [[Communitas Caribica|Communitate Caribica]] (Hispanice ''Comunidad del Caribe''; Anglice ''Caribbean Community''; Francice ''Communauté Caribéenne''; et Batavice ''Caribische Gemeenschap'') per [[Forum Caribicum]] seu CARIFORUM&lt;ref&gt;[http://ec.europa.eu/trade/policy/countries-and-regions/regions/caribbean/ &quot;Caribbean – Trade – European Commission&quot;] (Anglice). ''ec.europa.eu''. Inspectum 2017-02-08.&lt;/ref&gt; habet.

=== Vires armatae ===
[[Fasciculus:Two Commando soldiers of the MEFA provide security.jpg|thumb|300px|Dominiciani milites Dominicopoli se exercentes.]]
[[Congressus Nationalis Rei publicae Dominicanae|Congressus]] (Hispanice ''Congreso'') auctoritatem dat coniunctae vi militari 44 000 activorum laboris legatorum. Vera activa laboris vis circiter 32 000 est. Quorum propemodum 50% sunt in usu haud militaribus activitatibus sicuti praebitoribus securitatis apparatuum haud militarium gubernativorum, portoriis autoviarum, carceribus, silviculturae, publicis et privatis societatibus. Rerum militarium [[archistrategus]] praeses est.

Exercitus maior est quam aliae diaconiae coniunctae cum circiter 20 000 activis laboris legatis, consistens e peditatus cohortibus, cohorte auxiliari proelii una et cohorte auxiliari diaconiae in proelio una. Classis aëria praecipuas stationes duas operatur: alteram australi in regione prope Dominicopolin et alteram septentrione in regione prope [[Portus Argentarius|Portum Argentarium]]&lt;ref name=&quot;Portus Argentarius&quot;&gt;[http://www.catholic-hierarchy.org/diocese/dpupl.html catholic-hierarchy.org].&lt;/ref&gt; (Hispanice ''Puerto Plata''). Classis praecipuas stationes navales duas, alteram Dominicopoli et alteram [[Sinus Caldariarum|Caldariis]] (Hispanice ''Las Calderas'') austroccidentali in litore operatur et operatorias naves duodecim tenet. Res publica Dominiciana secundas maiores res militares Caribaea in regione post Cubam habet.&lt;ref name=&quot;Relations&quot;&gt;[https://www.state.gov/r/pa/ei/bgn/35639.htm &quot;U.S. Relations With the Dominican Republic&quot;] (Anglice). [[Ministerium Rerum Externarum Civitatum Foederatarum]]. Die 22 Octobris anni 2012.&lt;/ref&gt;

Vires armatae Corpus Securitatis Specializatum Aëroportuum (Hispanico acronymo ''CESA'') et Corpus Securitatis Specializatum Portuum (Hispanico acronymo ''CESEP'') instruxerunt ad explendas internationalis securitatis necessitates his in areis. Virium armatarum secretarius etiam destinata ad formandum praesidium limitare (Hispanico acronymo ''CESEF'') nuntiavit. Vires armatae 75% operariorum Directorio Nationali Investigationum (Hispanico acronymo ''DNI'') et Directorio Nationali Contra Drogam (Hispanico acronymo ''DNCD'') praebent.&lt;ref name=&quot;Relations&quot;&gt;[https://www.state.gov/r/pa/ei/bgn/35639.htm &quot;U.S. Relations With the Dominican Republic&quot;] (Anglice). [[Ministerium Rerum Externarum Civitatum Foederatarum]]. Die 22 Octobris anni 2012.&lt;/ref&gt;

Vis Custodiae Nationalis Dominicianae 32 000 custodum continet. Custodia non pertinent ad vires armatas sed aliquae superposita securitatis munera partiuntur. Tres et sexaginta per centum vis in areis munera traditionalem extra custodiam ministrant, simili modo statui eorum militarium parium.&lt;ref name=&quot;Relations&quot;&gt;[https://www.state.gov/r/pa/ei/bgn/35639.htm &quot;U.S. Relations With the Dominican Republic&quot;] (Anglice). [[Ministerium Rerum Externarum Civitatum Foederatarum]]. Die 22 Octobris anni 2012.&lt;/ref&gt;

=== Divisio administrativa ===
{{Vide-etiam|Provinciae Rei publicae Dominicianae}}
{{Vide-etiam|Municipia Rei publicae Dominicianae}}
[[Fasciculus:Administrativae divisiones Rei publicae Dominicianae.png|thumb|center|668x668px|Provinciae Rei publicae Dominicianae.]]
Res publica Dominiciana divisa est in [[Provinciae Rei publicae Dominicianae|provincias]] 31. [[Dominicopolis]], caput, designatum est [[Districtus Nationalis]] (Hispanice ''Distrito Nacional''). Provinciae in [[Municipia Rei publicae Dominicianae|municipia]] divisae sunt, quae sunt secundi gradus politica administrativaque subdivisio hac in natione. Praeses 31 provinciarum gubernatores designat. [[Alcaldis|Alcaldes]] (Hispanice ''alcaldes'' seu ''síndicos'') et coetus municipales districtus municipales et Districtum Nationalem (Dominicopolin) administrant, qui eliguntur eodem tempore quam congressuales repraesentantes.

== Oeconomia ==
[[Fasciculus:View of Santo Domingo Skyline.jpg|thumb|300px|[[Dominicopolis|Dominicopoleos]] centrum pecuniarium Rei publicae Dominicianae est.]]
[[Fasciculus:Index inflationis Rei publicae Dominicianae.jpg|thumb|350px|right|Index inflationis per annum secundum fontem [https://web.archive.org/web/20090608172033/http://www.merkap.com/ Merkap Rei publicae Dominicianae].]]
[[Oeconomia]] praecipue in [[agricultura]] et [[periegesis|periegesi]] nititur. Agricultura 7,1% e [[PDG]] (anno 2010) repraesentat, industriae 28,3% e [[PDG]] (eodem anno) repraesentant, et diakoniae, praesertim periegesis, 64,6 e [[PDG]] (eodem etiam anno) repraesentant. Periegesis crescens pondus ex [[anni 1990|annis 1990]] capit. Quae plus quam [[milliardum]] [[dollarium (CFA)|dollariorum]] repraesentat, et exsulum remissiones, praesertim e [[Civitates Foederatae Americae|Civitatibus Foederatis]], 1,5 milliarda dollariorum (anno 2000) repraesentant. Periegesis et remissiones ex exteris nationibus duos praecipuos pecuniae exterae fontes constituunt.

Anno [[2009]], [[PDG]] realis 46,74 milliarda [[dollarium (CFA)|dollariorum Statunitensium]] erat, dum PIB pro incola 8 300 dollariorum erat et index [[inflatio]]nis 1,4% erat.

Annis [[2003]] et [[2004]], natio, contextu plurium scandalorum inter quae hoc societatis argentariae [[Banco Intercontinental]], gravem crisin pecuniariam cognovit, quae in fortem devaluationem [[pensum Dominicianum|pensi Dominiciani]] (cuius paritas circiter e 16 pensis Dominicianis pro uno dollario Statunitensi ad 50 pensa Dominiciana pro uno dollario Statunitensi transivit), magnam inflationem et debiti contracti crisin induxit. Hanc crisin pecuniariam ampla deminutio fastigii vitalis incolarum comitata est.

Administrationis mutatio post comitia praesidialia anni 2004, exeunte qua praeses exeuns electus non est, denuo dedit fiduciam internationalibus collatoribus, exeunte anno 2004 resiliente [[pensum Dominicianum|penso Dominiciano]] (28,45 pensorum Dominicianorum pro uno dollario Statunitensi). Annis 2005 et 2006, natio forte incrementum redintegravit, macrooeconomico statu stabilito.
[[Fasciculus:Pdg Rei publicae Dominicianae.jpg|thumb|350px|left|[[Proventus domesticus grossus]] nominalis secundum fontem.]]

Die [[22 Novembris]] anni 1916, sacerdos Michael Dominicus Fuertes Loren paroechiae Barahonensis a Ministerio Rei publicae Dominicianae veniam petivit ut exploraretur exercereturque fodinam quam is invenerat et quae continebat quoddam caeruleum saxum, [[larimar]]. Quod de caerulei coloris [[pectolithos|pectolitho]] agitur qui solum in Re publica Dominiciana et Italia invenitur. Quia nemo id de quo is loquebatur cognovit, eius petitio ad irritum cecidit et huius caerulei saxi inventio retardata est.

Larimaris vena rursus inventa anno 1974 sed solum ex anno 1976 execita est. Duo incolae ex villagio Los Chupaderos, circa decem chiliometra ab urbe [[Barahona]]&lt;ref name=&quot;Barahonensis&quot;&gt;[http://www.catholic-hierarchy.org/diocese/dbara.html catholic-hierarchy.org] ubi apparet &quot;Dioecesis Barahonensis&quot;.&lt;/ref&gt;, quemdam colorem caeruleum super orae harena et in fundo fluminis [[Bahoruco (flumen)|Bahoruconis]], in [[Mare Caribaeum]] influentis, animadverterant. Cum fluctum pedibus ascendissent, sic venam super sitam, in cacumine montis tecti luxuriosa tropici generis vegetatione, invenere. Nomen &quot;larimar&quot; a Michaele Méndez datum est, miscendo nomine suae filiae, &quot;Larissa&quot;, cum [[lingua Hispanica|Hispanico]] substantivo &quot;''mar''&quot;, &quot;mare&quot; significante.

Villagii Los Chupaderos fodina sola larimaris fodina in toto orbe terrarum cognita constituit et larimar anno 1979 [[Gemma (lapis)|gemma]] classificatum est. Fodina in partes duas scissa est: hinc fossio a Civitate effecta machinalis reddita est, illinc ea humilium fossorum qui archaicis adiumentis laborant.

Evaluatur qualitas petrae secundum colorationem: petra profundius caerulea, pretiosior est.

Subvirides colorationes quoque cognoscuntur sed non bene aestimantur, nisi viridis intensus sit. Huius pectolithi crystallizatio intra caminos vulcanicos, ubi incandescens materia [[gas]]e pulsa, fit. Quamobrem, larimaris fossio de horum caminorum situ pendet. Etiam sociantur ad alios colores et alia mineralia. Rubrae colorationes in larimari ferri vestigia indicant. Notandum est pectolithos photosensibiles esse, eapropter larimar caeruleam colorationem amittit per annos.

Res publica Dominiciana etiam est locus pracipuarum [[sucinum|sucini]] venarum.

== Infrastructura ==
=== Vectura ===
[[Fasciculus:Obelisco Santo Domingo.jpg|thumb|300px|Via principalis El Malecón [[Dominicopolis|Dominicopoli]].]]
Natio tres truncales vias autocineticas habet, quae omnes praecipuas urbes connectunt. Quae sunt [[DR-1]], [[DR-2]] et [[DR-3]], quae e [[Dominicopolis|Dominicopoli]] versus septentrionales ([[Cibao]]nem&lt;ref name=&quot;cibaoensis&quot;&gt;Confer binomen &quot;Globorotalia cibaoensis&quot;.&lt;/ref&gt;), austroccidentales ([[Meridies (Res publica Dominiciana)|Meridiem]]) et orientales nationis partes ([[Oriens (Res publica Dominiciana)|Orientem]]), respective, discedunt. Haec viae autocineticae consequenter emendatae sunt expansis denuoque constructis multis sectionibus. Aliae duae nationales viae autocineticae divortio ([[DR-5]]) aut aliis tramitibus ([[DR-4]]) inserviunt.

Praeter nationales viae autocineticas, gubernium se in expansiva reconstructione tramitum secundariorum divortii implicavit, qui minores urbes truncalibus cum viis autocineticis connectunt. Ultimis paucis annis gubernium viam cum portorio 106 chiliometra longam Dominicopolin boreorientali nationis paeninsulae connectentem construxit. Viatores nunc in [[Xamana]]m&lt;ref name=&quot;Xamana&quot;&gt;[https://books.google.es/books?id=MTY8AAAAcAAJ&amp;pg=PT36&amp;lpg=PT36&amp;dq=%22primum+Santheremus+Xamanae+latus%22&amp;source=bl&amp;ots=TGz0GECZBL&amp;sig=xqahpRYlRXKa7XjEBLt9G-phaY8&amp;hl=es&amp;sa=X&amp;ved=0ahUKEwjZ6Li8qdXWAhWCIcAKHQqrDpsQ6AEILTAA#v=onepage&amp;q=Hifpaniola&amp;f=false De rebus Oceanis &amp; Orbe novo decades tres a Petro Martyro ab Anghiera], ubi dicitur &quot;(...) In Huhabo provincia sunt regiones Xamana, Canabacoa, Cuhabo (...)&quot;.&lt;/ref&gt; paeninsulam minus quam horas duas venire possunt. Alias apud additiones sunt viarum autocineticarum [[DR-28]] (inter [[Iarabacoa]]m et [[Constantia (Res publica Dominicana)|Constantiam]]) et [[DR-12]] (inter Constantiam et [[Bonao]]nem&lt;ref name=&quot;bonaoensis&quot;&gt;Confer binomen &quot;Meliola bonaoensis&quot;.&lt;/ref&gt;) reconstructio. Quamquam conatus facti sunt, multi tramites secundarii adhuc imbituminati aut conservatione egentes manent. Est in praesenti nationale programma ad bituminandos hos et alios usitatos tramites. Etiam, ferriviae urbicae systema [[Sanctus Iacobus Equitum|Sancti Iacobi Equitum]]&lt;ref name=&quot;Equitum&quot;&gt;[http://www.catholic-hierarchy.org/diocese/dsnca.html catholic-hierarchy.org] ubi apparet &quot;Dioecesis Sancti Iacobi Equitum&quot;.&lt;/ref&gt; rationis phasi sed in praesenti in spe est.

Praecipuae [[Laophorium|laophoriorum]] diaconiae duae in Re publica Dominiciana sunt: altera a gubernio recta, per societates Oficina Técnica de Tránsito Terrestre (OTTT) ac Oficina Metropolitana de Servicios de Autobuses (OMSA) et altera privatis a societatibus recta, apud quas Federación Nacional de Transporte La Nueva Opción (FENATRANO) ac Confederación Nacional de Transporte (CONATRA). Gubernii systema vecturae magnos tramites in territoriis urbis sicuti Dominicopoli et Sancti Iacobi Equitum tegit.

Sunt multae privatae laophoriorum societates, sicuti Metro Servicios Turísticos et Caribe Tours, quae quotidianos tramites operantur.
[[Fasciculus:Stodgo metro.jpg|thumb|300px|9000 serierum paria in ferrivia subterranea Dominicopolitana probantur.]]
Res publica Dominiciana [[Ferrivia subterranea|ferriviae subterraneae]] systema [[Dominicopolis|Dominicopoli]], nationis capite, habet. Est extensissimum ferriviae subterraneae systema insulari in [[Mare Caribaeum|Caribaeo]] et regione [[America Centralis|America Centrali]] per longitudinem et stationum numerum. Ferrivia subterranea Dominicopolitana praecipuae &quot;Rationis Expertae Nationalis&quot; pars est ad emendandam vecturam Dominicopoli non minus quam reliqua in natione. Primus trames designatus est ut levaretur maxima vehiculorum frequentia in viis principalibus Máximo Gómez et Hermanas Mirabal. Secundus trames, qui Aprili anni 2013 apertus est, maximam frequentiam per andronem Duarte-Kennedy-Centenario urbe ex occidente in orientem levare significatur. In praesenti ferrivia subterranea 27,35 chiliometra seu 16,99 milia passuum longa est, apertis duorum tramitum sectionibus Augusto anni 2013. Antequam secundus trames aperiretur, 30 856 515 insessorum in ferrivia subterranea Dominicopolitana anno 2012 vecti sunt.&lt;ref&gt;[http://opret.gob.do/Documentos/Estad%C3%ADsticas%20Institucionales/Estad%C3%ADsticas%20de%20peaje%20y%20tiempo%20de%20recorrido%20al%202013.pdf &quot;Estadísticas de peaje y tiempo de recorrido al 2013&quot;] (PDF). opret.gob.do (Hispanice). Septembri anni 2013. p. 2. Inspectum die 17 Septembris anni 2013.&lt;/ref&gt; Ambobus tramitibus apertis, insessorum quantitas auxit ad 61 270 054 insessorum anno 2014.

=== Instrumenta communicationis socialis ===
Res publica Dominiciana bene evolutam [[telecommunicatio]]num infrastructuram habet, cum diffusis diaconiis telephonorum gestabilium et domesticorum. [[Interrete]] caplare et [[Digitalis subnotatoris linea|DSL]] in plerisque nationis partibus apta sunt et multi praebitores diaconiarum interretialium diaconiam interretialem sine filis [[3G]] offerunt. Res publica Dominiciana secunda natio in [[America Latina]] fuit cui fuit diaconia sine filis 4G LTE. Relatae velocitates sunt e 256 chilio[[bit]]orum pro secundo aut 128 chiliobitorum pro secundo residentialibus diaconiis, usque ad quinque megabitos pro secundo aut unum megabitum pro secundo residentialibus diaconiis.

Commerciali diaconiae sunt velocitates e 256 chiliobitorum pro secundo usque ad 128 chiliobitorum pro secundo (omnes congeries numerorum velocitates pro fluxu et contra fluxum, id est, ad usuarium et ex usuario, denotant). Incepta ad extendendos campos retis [[Wi-Fi]] Dominicopoli facta sunt. Commerciales nationis stationes radiophonicae et televisificae in processu transferendi ad spectrum digitale sunt, via [[HD Radio]] et [[Definitio alta|HDTV]], postquam officialiter adoptatae essent normae [[ATSC]] sicuti digitale instrumentum, exstincta analogica transmissione Septembri anni 2015. Societas telecommunicationes hac in natione regens INDOTEL (Instituto Dominicano de Telecomunicaciones) est.

Maxima telecommunicationum societas [[Claro República Dominicana|Claro]] est, pars societatis [[Carolus Slim|Caroli Slim]] [[América Móvil]], sine filis, cum filis, [[fascia lata|fasciae latae]] et [[IPTV]] diaconias praebens. Iunio anni 2009 erant amplius octo milliones subnotatorum lineae telephonicae (et sine et cum filis) in Re publica Dominiciana, quod 81% incolarum nationalium repraesentat et quinquies incrementum ex anno 2000, cum 1,6 milliones erant. Campus telecommunicationum circa 3.0% [[PDG]] generat.&lt;ref&gt;[http://listindiario.com/la-republica/2009/6/4/103567/Dice-el-806-por-ciento-de-los-dominicanos-tiene-telefonos &quot;Dice el 80,6 por ciento de los dominicanos tiene teléfonos&quot;] (Hispanice). listindiario.com. Die 5 Iunii anni 2009. Archivatum ex originali die 16 Ianuarii anni 2013.&lt;/ref&gt; 2 439 997 usuariorum interretialium Martio anni 2009 erant.&lt;ref&gt;[https://web.archive.org/web/20110226134642/http://www.indotel.gob.do/component/option%2Ccom_docman/task%2Ccat_view/gid%2C110/Itemid%2C757 &quot;Indicadores Telefónicos 2009&quot;] (Hispanice). ''Indotel''. Archivatum ex originali die 26 Februarii anni 2011. Inspectum die 5 Iunii anni 2009.&lt;/ref&gt;

Novembri anni 2009, Res publica Dominiciana prima Latinamericana natio facta est quae &quot;conspectum [[genus (scientiae sociales)|generis]]&quot; includi in omni informationis communicatoriaeque technologiae proposito et programmate evoluto a gubernio pollicita est.&lt;ref&gt;[https://archivo.elnuevodiario.com.do/2009/11/16/indotel-garantiza-igualdad-de-genero-en-proyectos-tecnologicos-realiza-en-todo-el-pais/ &quot;Indotel garantiza igualdad de género en proyectos tecnológicos realiza en todo el país&quot;] (Hispanice). elnuevodiario.com.do. Die 16 Novembris anni 2009.&lt;/ref&gt; Quod est pars regionalis propositi eLAC2010. Instrumentum quod Dominiciani omnes publica proposita designare aestimareque elegerunt methodum aestimationis generis APC est.

=== Electricitas ===
Electricitatis diaconia inconstabilis est e [[Raphael Trujillo|Raphaëlis Trujillo]] aera et tantum quam 75% apparatuum veteres sunt. Antiquum nationis rete electricum transmissionis amissiones affert magnam perscriptae electricitatis partem e generatoriis explicantes. Huius campi privatizatio per praevium [[Leonellus Fernández|Leonelli Fernández]] gubernium coepit.&lt;ref name=&quot;Patterson&quot;&gt;Patterson, Claudia (die 4 Octobris anni 2004). [https://web.archive.org/web/20081107015518/http://www.coha.org/2004/10/president-leonel-fernandez-friend-or-foe-of-reform/ &quot;President Leonel Fernández: Friend or Foe of Reform?&quot;] (Anglice). ''Council on Hemispheric Affairs''. Archivatum ex originali die 7 Novembris anni 2008.&lt;/ref&gt; Recens collocatio pecuniae in &quot;Magnus Electriductus inter [[Dominicopolis|Dominicopolin]] et [[Sanctus Iacobus Equitum|Sanctum Iacobum Equitum]] ad ferendam [[Potentia (physica)|potentiam]] 345 chilio[[vattium|vattiorum]], &lt;ref&gt;[&quot;Dominican Republic north-south power grid almost finished (Correct)&quot;] (Anglice). Dominican Today. Die 29 Aprilis anni 2009. Archivatum ex originali Idibus Octobribus anni 2015. Inspectum Idibus Octobribus anni 2015.&lt;/ref&gt;, reductis transmissionis amissionibus, praecipua collocatio pecuniae nationali in reti e mediis [[anni 1960|annis 1960]] refertur.

Per [[Raphael Leónidas Trujillo Molina|Raphaëlis Trujillo]] regimen, electrica diaconia introducta est multis urbibus. Paene 95% usus in ratione nequamquam relatae sunt. Circa 2,1 millionum domorum dimidium in Re publica Dominiciana indicem mensorium habet et plerique non solvunt aut solvunt fixum menstruale pretium suae electricae diaconiae.&lt;ref&gt;[https://web.archive.org/web/20090603233719/http://www2.dominicantoday.com/dr/economy/2009/6/1/32173/Dominican-Government-hints-at-blackout-to-justify-electricity-hike &quot;Dominican Government hints at blackout to justify electricity hike&quot;] (Anglice). Dominican Today. Kalendis Iuniis anni 2006. Archivatum ex originali die 3 Iunii anni 2009.&lt;/ref&gt;

Domibus et generaliter electrica diaconia praebetur 110 [[voltium|voltiis]] alternans 60 [[hertz]]iis. [[Statunitensis|Statunitenses]] electrina vi actae machinae sine immutatione suo munere funguntur. Plerique Dominiciani ad electricitatem accessum habet. Periegeticae areae firmiorem electricitatis fluxum habere solent, sicuti negotia, itinerum, salutis et vitales infrastructurae.&lt;ref&gt;[https://web.archive.org/web/20120117175316/http://www.cdeee.gov.do/index.php?option=com_content&amp;view=article&amp;id=791%3Aedesur-agrega-3500-familias-a-24-horas-de-luz&amp;catid=6%3Anoticias&amp;Itemid=2 EDESUR agrega 3,500 familias a 24 Horas de Luz] (Hispanice). Cdeee.gov.do. Inspectum die 22 Septembris anni 2011.&lt;/ref&gt; Intensi conatus nuntiati sunt qui efficaciam praebitionis locis ubi collectionis index 70% attingant augeant.&lt;ref&gt;[https://web.archive.org/web/20070927025602/http://listin.com.do/app/article.aspx?id=9006 &quot;Los apagones toman fuerza en circuitos de barrios PRA&quot;] (Hispanice). Die 11 Aprilis anni 2007. Archivatum ex originali die 27 Septembris anni 2007. Inspectum die 24 Maii anni 2007.&lt;/ref&gt; Electrico sectori magna vis politica confertur. Aliquae electrifinae societates satis pecuniae carent et aliquando sufficientem alimenti copiam emere nequeunt.&lt;ref name=&quot;Relations&quot;&gt;[https://www.state.gov/r/pa/ei/bgn/35639.htm &quot;U.S. Relations With the Dominican Republic&quot;] (Anglice). [[Ministerium Rerum Externarum Civitatum Foederatarum]]. Die 22 Octobris anni 2012.&lt;/ref&gt;

=== Suppeditatio aquae et purgatio ===
Res publica Dominiciana speciosa incrementa accessui suppeditationis aquae et purgationis per praeterita duo decennia attigit. Nihilominus, qualitas diaconiarum suppeditandae purgandae aquae egens manet, quamquam oeconomia valde per annos 1990 crevit. Etsi melioratorum fontium aquae et melioratae purgationis opertio 86% et 83% respective relative altior est, sunt substantivae regionales dissimilitudines. Pauperes domi minores accessus gradus exhibent: solum 56% pauperum domorum cum domesticis aquae conexionibus conectuntur, dispar 80% domorum haud pauperum. Solum 20% pauperum domorum accessum ad cloacas habent, dispar 50% haud pauperum.&lt;ref&gt;[&quot;Dominican Republic: Environmental Priorities and Strategic Options Country Environmental Analysis&quot;] (PDF) (Anglice). ''[[Argentaria Mundana]]''. Die 29 Iunii anni 2004. Archivatum ex originali (PDF) die 24 Octobris anni 2015.&lt;/ref&gt;

== Societas ==
=== Demographia ===
[[Fasciculus:Dominican Rep demography.png|thumb|350px|Incolae Rei publicae Dominicianae inter annos 1961 et 2003.]]
Rei publicae Dominicianae incolae 10 648 791 anno [[2016]] erant.&lt;ref&gt;[https://esa.un.org/unpd/wpp/DataQuery/ &quot;World Population Prospects: The 2017 Revision&quot;] (Anglice). ESA.UN.org (ad personam accommodata data acquisita via situs retialis). Ministerium Rerum Oeconomicarum et Socialium Nationum Unitarum, Divisio Incolarum. Inspectum die 10 Septembris anni 2017.&lt;/ref&gt; Anno 2010, 31,2% incolarum minus quam 15 annos nati erant, et 6% incolarum erant magis 65 annos nati.&lt;ref&gt;[https://esa.un.org/unpd/wpp/Publications/Files/WPP2012_Volume-II-Demographic-Profiles.pdf &quot;World Population Prospects: The 2012 Revision&quot;] (Anglice) (PDF). Ministerium Rerum Oeconomicarum et Socialium Nationum Unitarum. 2013. p. 254.Inspectum die 24 Augusti anni 2016.&lt;/ref&gt; Annuus index incrementi incolarum inter annos 2006 et 2007 1,5% erat, proiecto numero incolarum anni 2015 10 121 000.&lt;ref&gt;[https://www.un.org/esa/population/publications/wpp2006/WPP2006_Highlights_rev.pdf &quot;World Population Prospects: The 2006 Revision, Highlights, Working Paper No. ESA/P/WP.202&quot;] (Anglice) (PDF). Ministerium Rerum Oeconomicarum et Socialium Nationum Unitarum, Divisio Incolarum. 2007. Inspectum die 13 Ianuarii anni 2008.&lt;/ref&gt;

[[Spissitudo incolarum]] anno 2007 192 hominum pro chiliometro quadrato (seu 498 hominum pro mille passuum quadratorum) erat et 63% urbanas areas incolabant.&lt;ref&gt;[https://web.archive.org/web/20110808193423/http://www.conapofa.gov.do/estimaciones.asp &quot;Población en Tiempo Real&quot;] (Hispanice). Consejo Nacional de Población y Familia. Archivatum ex originali die 8 Augusti anni 2011. Inspectum die 13 Ianuarii anni 2008.&lt;/ref&gt; Meridionales litorales planities et Vallis Cibaoensis&lt;ref name=&quot;cibaoensis&quot;&gt;Confer binomen &quot;Globorotalia cibaoensis&quot;.&lt;/ref&gt; densissime habitata loca sunt in natione. Caput [[Dominicopolis]] [[numerus incolarum|numerum incolarum]] 2 907 100 habebat anno 2010.&lt;ref name=&quot;Population&quot;&gt;[http://www.nationsencyclopedia.com/Americas/Dominican-Republic-POPULATION.html Dominican Republic – Population] (Anglice). Encyclopedia of the Nations.&lt;/ref&gt;

Apud alias ponderis urbes sunt [[Sanctus Iacobus Equitum]]&lt;ref name=&quot;Hierarchia&quot;&gt;[http://catholic-hierarchy.org/diocese/dsnca.html Hierarchia Catholica].&lt;/ref&gt; (cum 745 293 incolarum), [[Romana (urbs)|Romana]] (cum 214 109 incolarum), [[Sanctus Petrus de Macoris (urbs)|Sanctus Petrus de Macoris]]&lt;ref name=&quot;Macoris&quot;&gt;[http://www.catholic-hierarchy.org/diocese/dsnpm.html catholic-hierarchy.org] ubi apparet &quot;Dioecesis Sancti Petri de Macoris&quot;.&lt;/ref&gt; (cum 185 255 incolarum), [[Hygvei]]&lt;ref name=&quot;Hygvei&quot;&gt;[https://books.google.es/books?id=Lmk8NQJuOYYC&amp;pg=PA541&amp;lpg=PA541&amp;dq=xaragua+quinque+regna&amp;source=bl&amp;ots=sR2UpmmuwN&amp;sig=pgR-Okk_dmk7K16ilQLWebcy6dY&amp;hl=es&amp;sa=X&amp;ved=0ahUKEwjkspX7hefVAhWLQBQKHXyEBWMQ6AEIPTAJ#v=onepage&amp;q&amp;f=false Georgi Horni Arca Noæ, sive Historia imperiorum et regnorum à condito orbe.]&lt;/ref&gt; seu [[Salvaleon]]&lt;ref name=&quot;Salvaleon&quot;&gt;[http://www.e-rara.ch/zut/content/pageview/12351901 Lexicon Universale Hofmann] ubi dicitur: &quot;Hispaniola (...) a Dominicopoli loco memoranda, Salvaleon sacchari proventu nobilis (...)&quot;.&lt;/ref&gt; (cum 153 174 incolarum), [[Sanctus Franciscus de Macoris]]&lt;ref name=&quot;Sancti Francisci Macoris&quot;&gt;[http://www.catholic-hierarchy.org/diocese/dsnfm.html catholic-hierarchy.org] ubi apparet &quot;Dioecesis Sancti Francisci de Macoris&quot;.&lt;/ref&gt; (cum 132 725 incolarum), [[Portus Argentarius]]&lt;ref name=&quot;Portus Argentarius&quot;&gt;[http://www.catholic-hierarchy.org/diocese/dpupl.html catholic-hierarchy.org].&lt;/ref&gt; (cum 118 282 incolarum) et [[Conceptio de Vega]]&lt;ref name=&quot;Conceptio&quot;&gt;[http://www.e-rara.ch/zut/content/pageview/12351901 Lexicon Universale Hofmann] ubi dicitur: &quot;Hispaniola (...) in parte Austr. Conceptio de Vega, sedes ante Episcopi, (...)&quot;.&lt;/ref&gt; (cum 104 536 incolarum). Secundum [[Nationes Unitae|Nationes Unitas]], index incrementi urbanorum incolarum inter annos 2000 et 2005 2,3% erat.&lt;ref name=&quot;Population&quot;&gt;[http://www.nationsencyclopedia.com/Americas/Dominican-Republic-POPULATION.html Dominican Republic – Population] (Anglice). Encyclopedia of the Nations.&lt;/ref&gt;

=== Ethnē ===
[[Fasciculus:Dominican-people-cibao-1.jpg|thumb|300px|Dominiciani [[Moca (Res publica Dominicana)|Mocae]].]]
Incolae Rei publicae Dominicianae 73% ethnice mixtae originis, 16% [[albicolores]] et 11% [[nigricolores]] sunt.&lt;ref name=&quot;CIA&quot;&gt;[https://www.cia.gov/library/publications/the-world-factbook/geos/dr.html &quot;CIA – The World Factbook – Dominican Republic&quot;] (Anglice). [[Procuratio Nuntiorum Centralis]] (CIA). Inspectum [[4 Iunii|pridie Nonas Iunias]] anni 2007.&lt;/ref&gt; Immigrantia apud [[ethnos|ethnē]] hac in natione sunt [[Asia occidentalis|Mesanatolica]]&lt;ref&gt;Confer Graecum vocabulum Μεσανατολικό.&lt;/ref&gt; —praesertim [[Libanus|Libanenses]], [[Syria]]ci et [[Palaestina|Palaestini]]— sunt.&lt;ref name=&quot;Levinson&quot;&gt;Levinson, David (1998). [https://books.google.es/books?id=uwi-rv3VV6cC&amp;pg=PA345&amp;redir_esc=y ''Ethnic groups worldwide: a ready reference handbook''] (Anglice). Greenwood Publishing Group. pp. 345–6. ISBN 1-57356-019-7.&lt;/ref&gt;

Numerosi immigrantes ex aliis Caribicis nationibus venerunt, quia natio oeconomicas opportunitates obtulit. Sunt circa 32 000 [[Iamaica]]norum Rem publicam Dominicianam incolentes.&lt;ref&gt;Joshua Project (2016). [https://joshuaproject.net/people_groups/12316 &quot;The Jamaicans people group is reported in 14 countries&quot;] (Anglice). Joshuaproject.net. Inspectum die 19 October anni 2016.&lt;/ref&gt; Est crescens [[Portoricus|Portoricensium]] immigrantium numerus, praesertim [[Dominicopolis|Dominicopoli]] et in locis circumiectis; Creduntur numerare propemodum 10 000.&lt;ref&gt;[http://www.topix.com/forum/world/dominican-republic/T4ULLRH92RE5AQ2UL &quot;Growing Puerto Rican population in the Dominican Republic1]&quot; (Anglice). Universitas Centralis Orientis. Archivatum ex originali die 17 Martii 2011. Inspectum die 19 Iulii anni 2010.&lt;/ref&gt;&lt;ref&gt;[https://www.diariolibre.com/noticias/ms-de-medio-milln-de-inmigrantes-residen-en-el-pas-EDDL381577 &quot;Más de medio millón de inmigrantes residen en el país&quot;] (Hispanice). diariolibre.com. Kalendis Maiis anni 2013. Inspectum die 19 Octobris anni 2016.&lt;/ref&gt; Sunt circiter 700 000 hominum [[Haitia]]nae originis, inclusa stirpe in Re publica Dominiciana nata.

[[Asia Orientalis|Asiani Orientales]], imprimis ethnice [[Seres]] et [[Iaponia|Nipponenses]], quoque inveniri possunt.&lt;ref name=&quot;Levinson&quot;&gt;Levinson, David (1998). [https://books.google.es/books?id=uwi-rv3VV6cC&amp;pg=PA345&amp;redir_esc=y ''Ethnic groups worldwide: a ready reference handbook''] (Anglice). Greenwood Publishing Group. pp. 345–6. ISBN 1-57356-019-7.&lt;/ref&gt; [[Europa]]ei ab [[Hispania|Hispanis]] maxime repraesentantur, sed etiam sunt minores multitudines [[Aschenates|Iudaeorum Germanicorum]], [[Italia]]norum, [[Lusitania|Lusitanorum]], [[Britanniarum Regnum|Britannicorum]], [[Batavia|Batavorum]], [[Dania|Danorum]] et [[Hungari|Hungarorum]].&lt;ref name=&quot;Levinson&quot;&gt;Levinson, David (1998). [https://books.google.es/books?id=uwi-rv3VV6cC&amp;pg=PA345&amp;redir_esc=y ''Ethnic groups worldwide: a ready reference handbook''] (Anglice). Greenwood Publishing Group. pp. 345–6. ISBN 1-57356-019-7.&lt;/ref&gt;&lt;ref&gt;[http://news.bbc.co.uk/2/shared/spl/hi/in_depth/brits_abroad/html/caribbean.stm &quot;Brits Abroad&quot;] (Anglice). BBC News. Die 6 Decembris anni 2006. Inspectum die 3 Augusti anni 2010.&lt;/ref&gt;&lt;ref name=&quot;CCNY&quot;&gt;&quot;CCNY Jewish Studies Class to Visit Dominican Village that Provided Refuge to European Jews During World War II&quot; (Press release). City College of New York. November 13, 2006. Archived from the original on May 10, 2011. Retrieved August 3, 2010.&lt;/ref&gt; Aliqui conversi [[Sephardim|Sepharditae]] [[Iudaei]] ex Hispania primarum expeditionum pars fuerunt, solum Catholici in [[Mundus novus|Mundum Novum]] venire permissi sunt.&lt;ref&gt;[https://www.jewishvirtuallibrary.org/the-inquisition &quot;Christian-Jewish Relations: The Inquisition&quot;]. ''Encyclopaedia Judaica''. Inspectum Idibus Maiis anni 2013.&lt;/ref&gt; Postea, fuerunt Iudaei migrantes ex Hiberia et Europa [[anni 1700|annis 1700]] venientes.&lt;ref&gt; &quot;Dominican Republic&quot;. ''Encyclopaedia Judaica''. 2008. Inspectum Idibus Maiis anni 2013.&lt;/ref&gt; Aliqui in attingendo Mari Caribaeo successere profugi per et post [[Secundum bellum mundanum|Secundum Bellum Mundanum]]&lt;ref&gt;Levy, Lauren (die 6 Ianuarii anni 1995). [https://www.jewishvirtuallibrary.org/dominican-republic-as-haven-for-jewish-refugees &quot;The Dominican Republic's Haven for Jewish Refugees&quot;] (Anglice). ''Jerusalem Post''.&lt;/ref&gt;&lt;ref&gt;&quot;Jews in Dominican Republic&quot;. ''Encyclopaedia Judaica''. 6. 1971. Archivatum ex originali die 2013-03-10.&lt;/ref&gt;&lt;ref&gt; [https://web.archive.org/web/20131001221327/https://www.biblediscovered.com/jewish-hebrew-people-in-the-world/dominican-republic-jews-2/ &quot;Dominican Republic-Jews&quot;] (Anglice). ''biblediscovered.com''. Archivatum ex originali die Kalendis Octobribus anni 2013. Inspectum Idibus Maiis anni 2013.&lt;/ref&gt; Aliqui Sepharditae Iudaei [[Sosua]]e residunt, dum alii dispersi sunt totam per nationem. Se Iudaei identificantes 3 000 numerantur; alii Dominiciani Iudaicam originem habere possunt propter matrimonia inter conversos Iudaeos Catholicos et alios Dominicianos e colonialibus temporibus. Aliqui Dominiciani in Civitatibus Foederatis Americae nati nunc in Re publica Dominiciana residunt, creantes certam exsilum communitatem.&lt;ref&gt;[https://web.archive.org/web/20110225044115/http://www.aca.ch/amabroad.pdf &quot;American Citizens Living Abroad by Country&quot;] (Anglice) (PDF). Ministerium Civitatis Statunitensis. Archivatum ex originali (PDF) die 25 Februarii anni 2011. Inspectum die 3 Augusti anni 2010.&lt;/ref&gt;

=== Linguae ===
Rei publicae Dominicianae plerique incolae [[Hispanice]] loquuntur. Localis linguae Hispanicae varietas [[lingua Hispanica Dominiciana]] nuncupatur, quae arcte assimulatur aliis vernaculis in Mari Caribaeo et [[lingua Hispanica Canaria|linguae Hispanicae Canariae]]. Praeterea, vocabula ab Indigenicis Caribicis linguis particularibus insulae Hispaniolae mutuata est.&lt;ref&gt; Henríquez Ureña, Petrus (Pedro) (1940). ''El Español en Santo Domingo'' (Hispanice). Bono Aëre: Institutum Philologiae Universitatis Bonaëropolitanae.&lt;/ref&gt;&lt;ref&gt;Deive, Carolus Stephanus (Carlos Esteban) (2002). [https://books.google.es/books?id=tLTyydkMtrIC&amp;pg=PA9&amp;dq=El+%22Espa%C3%B1ol+Dominicano%22&amp;redir_esc=y&amp;hl=es#v=onepage&amp;q=El%20%22Espa%C3%B1ol%20Dominicano%22&amp;f=false ''Diccionario de dominicanismos'']. Dominicopoli: Librería La Trinitaria. pp. 9–16. ISBN 99934-39-07-X.&lt;/ref&gt; Scholae ad instar Hispanicum exemplar educativum fundatur; Lingua Anglica et Francica obligatoriae linguae peregrinae et privatis et publicis scholis sunt,&lt;ref&gt;[https://web.archive.org/web/20110803222706/http://www.see.gob.do/documentosminerd/Desarrollo%20Curricular/guia-didac-inicial.pdf Guía Didáctica. Inicial] (PDF). Ministerium Educationis Rei publicae Dominicianae. I. 2010. ISBN 978-99934-43-26-1. Archivatum ex originali (PDF) die 3 Augusti anni 2011.&lt;/ref&gt; quamquam qualitas docendarum linguarum peregrinarum ieiuna est.&lt;ref&gt; Apolinar, Bethania (die 2 Augusti anni 2015). [https://www.listindiario.com/la-republica/2015/08/02/382666/ensenanza-del-ingles-es-pobre-en-escuelas &quot;Enseñanza del inglés es &quot;pobre&quot; en escuelas&quot;] (Hispanice). Dominicopoli: Listín Diario. Inspectum die 24 Augusti anni 2016.&lt;/ref&gt; Aliquae privata instituta educativa disciplinam aliarum linguarum, insigniter Italianae, Iaponicae et Mandarinae, praebent.&lt;ref&gt;&quot;Especialistas en idiomas&quot; (Hispanice). Hoy digital. Die 28 Iunii anni 2006. Inspectum die 24 Augusti anni 2016.&lt;/ref&gt;&lt;ref&gt;Pujols, Daniela (die 23 Aprilis anni 2015). [https://www.listindiario.com/la-vida/2015/04/23/364463 &quot;Colegio Chino: Cuando el idioma no es limitante&quot;] (Hispanice). Listín Diario. Inspectum die 24 Augusti anni 2016.&lt;/ref&gt;

[[Lingua creola Haitiana]] maxima lingua minoritaria est in Re publica Dominiciana et ea loquuntur [[Haitia]]ni immigrantes et eorum posteri.&lt;ref&gt; Baker, Colin; Prys Jones, Sylvia, eds. (1998). [ttps://books.google.es/books?id=YgtSqB9oqDIC&amp;pg=PA389&amp;dq=second+largest+language+spoken+in+the+dominican+republic&amp;redir_esc=y&amp;hl=es#v=onepage&amp;q&amp;f=false ''Encyclopedia of Bilingualism and Bilingual Education''.] p. 389. ISBN 1-85359-362-1. Inspectum die 20 Novembris anni 2015.&lt;/ref&gt; Est communitas paucorum milium hominum quorum proavi [[Lingua Anglica Samanaensis|Lingua Anglica Samanaensi]] in [[Paeninsula Samanaensis|Paeninsula Samanaensi]]&lt;ref name=&quot;samanaensis&quot;&gt;Confer binomen &quot;Lyria samanaensis&quot;.&lt;/ref&gt; loquebantur. Qui sunt posteri antiquorum addictorum Aframericanorum qui [[saeculum 19|saeculo XIX]] venerant, sed solum pauci senes hac lingua hodie loquuntur.&lt;ref&gt;Davis, Martha Hellena (Martha Ellen) (2011). [https://es.scribd.com/document/248798261/La-Historia-de-Los-Inmigrantes-Afro-Americanos-y-Sus-Iglesias-en-Samana-Segun-El-Reverendo-Nehemiah-Willmore &quot;La Historia de Los Inmigrantes Afro-Americanos Y Sus Iglesias En Samaná Según El Reverendo Nehemiah Willmore&quot;.] ''Boletín Del Archivo General de La Nación''. 36 (129): 237–45.&lt;/ref&gt; Periegesis, cultura [[musica popularis|popularis]] Statunitensis, Dominicianorum Statunitensium affectio et nationales oeconomicae relationes cum Civitatibus Foederatis Americae alios Dominicianos ad discendam linguam Anglicam incitant. Res publica Dominiciana collocata est ex ordine secundo in America Latina et tertio vicesimo in toto orbe terrarum in proficientia linguae Anglicae.&lt;ref&gt; [https://www.weforum.org/agenda/2016/11/which-countries-are-best-at-english-as-a-second-language-4d24c8c8-6cf6-4067-a753-4c82b4bc865b/ Which countries are best at English as a second language?] (Anglice), Forum Oeconomicum Mundiale. Inspectum die 10 Iulii anni 2017&lt;/ref&gt;&lt;ref&gt;[https://www.ef.com/wwpt/epi/regions/latin-america/dominican-republic/ EF English Proficiency Index – Dominican Republic] (Anglice), EF Education First. Inspectum die 10 Iulii anni 2017.&lt;/ref&gt;

{| class=&quot;wikitable sortable&quot;
|+ Dominicianorum [[sermo patrius]] secundum anni 1950 censum&lt;ref&gt;Nicasio Rodríguez, Irma; Jesús de la Rosa (1998). ''Historia, Metodología y Organización de los Censos en República Dominicana: 1920–1993'' (Hispanice). Dominicopoli: Sedes Nationalis Statisticae. pp. 44, 131.&lt;/ref&gt;
|-
! Lingua   !! Totum % !! Urbibus % !! Ruri %
|-|| style=&quot;text-align:center;&quot;|
| [[Lingua Hispanica|Hispanica]] || style=&quot;text-align:center;&quot;|  98,00 || style=&quot;text-align:center;&quot;|  97,82 || style=&quot;text-align:center;&quot;|  98,06
|-
| [[Lingua Francica|Francica]] || style=&quot;text-align:center;&quot;|  1,19 || style=&quot;text-align:center;&quot;|  0,39|| style=&quot;text-align:center;&quot;|  1,44
|-
| [[Lingua Anglica|Anglica]] || style=&quot;text-align:center;&quot;|  0,57 || style=&quot;text-align:center;&quot;|  0,96|| style=&quot;text-align:center;&quot;|  0,45
|-
| [[Lingua Arabica|Arabica]] || style=&quot;text-align:center;&quot;|  0,09 || style=&quot;text-align:center;&quot;|  0,35|| style=&quot;text-align:center;&quot;|  0,01
|-
| [[Lingua Italiana|Italiana]] || style=&quot;text-align:center;&quot;|  0,03 || style=&quot;text-align:center;&quot;|  0,10|| style=&quot;text-align:center;&quot;|  0,006
|-
| Alia lingua  || style=&quot;text-align:center;&quot;|  0,12 || style=&quot;text-align:center;&quot;|  0,35|| style=&quot;text-align:center;&quot;| 0,04
|}

=== Religio ===
[[Fasciculus:Santo_Domingo_-_Catedral_Santa_Maria_La_Menor_02.JPG|thumb|[[Cathedralis Dominicopolitana]], prima cathedralis [[America]]e.]]
{{bar box
|título=Religiones in Re publica Dominiciana
|width=350px
|títulobar=#ddd
|left1=Religio&lt;ref&gt;[ww.thearda.com/internationalData/countries/Country_70_2.asp Dominican Republic (Anglice)]. The Association of Religion Data Archives.&lt;/ref&gt;
|right1=Percentatio
|float=left
|bars=
{{bar percent|[[Religio Christiana|Christiani]]|#9955BB|95.0}}
{{bar percent|[[Irreligio|Sine religione]]|red|2.6}}
{{bar percent|Aliae [[Religio|religiones]]|blue|2.2}}
}}

Res publica Dominiciana [[Libertas religionis|libertatem religionis]], per suam constitutionem, permittit, dum etiam omnibus religionibus tolerationem confirmat. Attamen, [[Ecclesia Catholica Romana]] [[religio publica]] habetur, culta a 68,9% incolarum. Catholicos Romanos sequuntur [[Protestantes]] (18,2%), [[irreligio]]si (10,6%) et aliorum religionum asseclae (2,3%), ubi includuntur [[musulmanus|Musulmani]], [[Iudaei]], religiones Afrocaribicae ([[Palus (religio)|Palus]], [[Vodun]], [[Sanctaria]] et caetera) apud alias. Nuper hac in natione coeperunt coli aliae denominationes religiosae, sicuti [[Spiritismus|Spiritae]] (2,2%), [[Ecclesia Iesu Christi Diebus Ultimis Sanctorum|Mormones]] (1,1%), [[Buddhismus|Buddhistae]] (0,1%), [[Religio Baha'i|Bahaistae]] (0,1%) et religiones traditionales Sinicae (0,1%). Nationi duae [[Sanctus patronus|sanctae patronae]] sunt: [[Domina Nostra de la Altagracia]]&lt;ref name=&quot;Domina&quot;&gt;[http://www.catholic-hierarchy.org/diocese/dnuse.html catholic-hierarchy.org] ubi apparet &quot;Dioecesis A Domina Nostra vulgo de la Altagracia in Higüey, seu Higueyensis&quot;.&lt;/ref&gt; et [[Nostra Domina Mercedis]].&lt;ref&gt;[http://www.thearda.com/internationalData/countries/Country_70_2.asp Dominican Republic] (Anglice). The Association of Religion Data Archives.&lt;/ref&gt;

Ecclesia Catholica exeunte saeculo XIX auram popularem perdere coepit. Quod accidit quia deficiebat collocatio pecuniae sacerdotibus et programmatibus subsidii. Eodem tempore motus Evangelicus Protestans auram popularem adipìsci coepit. Hac in natione religiosae discordiae rarae sunt.&lt;ref&gt;Encyclopedia of World Cultures &quot;Dominicans&quot;, [http://www.encyclopedia.com/philosophy-and-religion/christianity/roman-catholic-orders-and-missions/dominicans] (Anglice).&lt;/ref&gt;

=== Praecipuae urbes ===
Hic inferius est tabula cum urbibus populosissimis in Re publica Dominiciana secundum Sationem Nationalem Statisticae anno 2015.&lt;ref&gt;[&quot;Expansión Urbana de las ciudades capitales de RD: 1988-2010&quot;] (Hispanice). Dominicopoli: Statio Nationalis Statisticae. Kalendis Maiis anni 2015. ISBN 978-9945-8984-3-9. Inspectum die 25 Ianuarii anni 2016.&lt;/ref&gt;:

{| class=&quot;wikitable floatright&quot; style=&quot;text-align:center; width:100%; margin-left:0px; font-size:70%&quot;
|-
! align=center style=&quot;background:#B9D5E4;&quot; | Positio
! align=center style=&quot;background:#B9D5E4;&quot; | [[Urbs]]
! align=center style=&quot;background:#B9D5E4;&quot; | [[Provinciae Rei publicae Dominicianae|Provincia]]
! align=center style=&quot;background:#B9D5E4;&quot; | [[Numerus incolarum|Incolae]]
! align=center style=&quot;background:#B9D5E4;&quot; | Positio
! align=center style=&quot;background:#B9D5E4;&quot; | [[Urbs]]
! align=center style=&quot;background:#B9D5E4;&quot; | [[Provinciae Rei publicae Dominicianae|Provincia]]
! align=center style=&quot;background:#B9D5E4;&quot; | [[Numerus incolarum|Incolae]]
|-
| align=center style=&quot;background:#DCE9F0;&quot; | 1 ||align=left | [[Dominicopolis]] &lt;ref name=&quot;Egger&quot;&gt;{{Egger DL|43}}&lt;/ref&gt; seu [[Sanctus Dominicus (Res publica Dominiciana)|Sanctus Dominicus]]&lt;ref name=&quot;Dominici&quot;&gt;&quot;Civitas S. Dominici&quot;: vide imaginem  nostram anni 1589&lt;/ref&gt; || [[Districtus Nationalis]] || 4 665 802 || align=center style=&quot;background:#f0f0f0;&quot; | 11 || align=left | [[Sanctus Ioannes Maguanensis]]&lt;ref name=&quot;Ioannis&quot;&gt;[http://www.catholic-hierarchy.org/diocese/dsjdr.html catholic-hierarchy.org] ubi apparet &quot;Dioecesis Sancti Ioannis Maguanensis&quot;.&lt;/ref&gt; || [[Sanctus Ioannes (Res publica Dominiciana)|Sanctus Ioannes]]&lt;ref name=&quot;Ioannis&quot;&gt;[http://www.catholic-hierarchy.org/diocese/dsjdr.html catholic-hierarchy.org] ubi apparet &quot;Dioecesis Sancti Ioannis Maguanensis&quot;.&lt;/ref&gt;  || 156 583
|-
| align=center style=&quot;background:#DCE9F0;&quot; | 2 ||align=left | [[Sanctus Iacobus Equitum]]&lt;ref name=&quot;Equitum&quot;&gt;[http://www.catholic-hierarchy.org/diocese/dsnca.html catholic-hierarchy.org] ubi apparet &quot;Dioecesis Sancti Iacobi Equitum&quot;.&lt;/ref&gt;  || [[Sanctus Iacobus (Res publica Dominiciana)|Sanctus Iacobus]]&lt;ref name=&quot;Argenteus&quot;&gt;[http://www.e-rara.ch/zut/content/pageview/12351901 Lexicon Universale Hofmann] ubi dicitur: &quot;Hispaniola (...) nunc Dominicopolitana juncta, Zeibum, ''Zeibo'', Fanum S. Jacobi, ''Sant Jago'', egregio situ inclyta, Portus Argenteus, seu ''Puerto de la Plata'', ob commercii frequentiam, (...)&quot;.&lt;/ref&gt; || 1 301785 || align=center style=&quot;background:#f0f0f0;&quot; | 12 || align=left | [[Cotoy]]&lt;ref name=&quot;Dahabon&quot;&gt;[https://books.google.es/books?id=MTY8AAAAcAAJ&amp;pg=PT36&amp;lpg=PT36&amp;dq=%22primum+Santheremus+Xamanae+latus%22&amp;source=bl&amp;ots=TGz0GECZBL&amp;sig=xqahpRYlRXKa7XjEBLt9G-phaY8&amp;hl=es&amp;sa=X&amp;ved=0ahUKEwjZ6Li8qdXWAhWCIcAKHQqrDpsQ6AEILTAA#v=onepage&amp;q=Hifpaniola&amp;f=false De rebus Oceanis &amp; Orbe novo decades tres a Petro Martyro ab Anghiera], ubi dicitur &quot;(...) Aliae sunt regiones, Dahabon, Cybaho, Manabaho. Cotoy est in insuale medio (...)&quot;.&lt;/ref&gt;&lt;ref&gt;[[:es:Cotuí#cite_ref-2|Vicipaedia Hispanica]], ubi dicitur: &quot;''La ciudad de Cotuí es el municipio cabecera de la provincia de Sánchez Ramírez. Su nombre, escrito antiguamente Cotuy o Cotoy (...)''&quot;.&lt;/ref&gt; || [[Provincia Sanchezramirensis]]&lt;ref name=&quot;sanchezramirense&quot;&gt;E gentilico Hispanico ''sanchezramirense''.&lt;/ref&gt;  || 154 343
|-
| align=center style=&quot;background:#DCE9F0;&quot; | 3 ||align=left | [[Portus Argenteus (urbs)|Portus Argenteus]]&lt;ref name=&quot;Argenteus&quot;&gt;[http://www.e-rara.ch/zut/content/pageview/12351901 Lexicon Universale Hofmann] ubi dicitur: &quot;Hispaniola (...) nunc Dominicopolitana juncta, Zeibum, ''Zeibo'', Fanum S. Jacobi, ''Sant Jago'', egregio situ inclyta, Portus Argenteus, seu ''Puerto de la Plata'', ob commercii frequentiam, (...)&quot;.&lt;/ref&gt; seu [[Portus Argentarius (urbs)|Portus Argentarius]]&lt;ref name=&quot;Argentarii&quot;&gt;[http://www.catholic-hierarchy.org/diocese/dpupl.html catholic-hierarchy.org] ubi apparet &quot;Dioecesis Portus Argentarii&quot;.&lt;/ref&gt; || [[Portus Argenteus]]&lt;ref name=&quot;Argenteus&quot;&gt;[http://www.e-rara.ch/zut/content/pageview/12351901 Lexicon Universale Hofmann] ubi dicitur: &quot;Hispaniola (...) nunc Dominicopolitana juncta, Zeibum, ''Zeibo'', Fanum S. Jacobi, ''Sant Jago'', egregio situ inclyta, Portus Argenteus, seu ''Puerto de la Plata'', ob commercii frequentiam, (...)&quot;.&lt;/ref&gt; seu [[Portus Argentarius]]&lt;ref name=&quot;Argentarii&quot;&gt;[http://www.catholic-hierarchy.org/diocese/dpupl.html catholic-hierarchy.org] ubi apparet &quot;Dioecesis Portus Argentarii&quot;.&lt;/ref&gt;  || 330 783 || align=center style=&quot;background:#f0f0f0;&quot; | 13 || align=left | [[Urbs Baniensis]]&lt;ref name=&quot;Baniensis&quot;&gt;[http://www.catholic-hierarchy.org/diocese/dbani.html catholic-hierarchy.org] ubi apparet &quot;Dioecesis Baniensis&quot;.&lt;/ref&gt; || [[Peravia]] || 145 595
|-
| align=center style=&quot;background:#DCE9F0;&quot; | 4 ||align=left | [[Hygvei]]&lt;ref name=&quot;Hygvei&quot;&gt;[https://books.google.es/books?id=Lmk8NQJuOYYC&amp;pg=PA541&amp;lpg=PA541&amp;dq=xaragua+quinque+regna&amp;source=bl&amp;ots=sR2UpmmuwN&amp;sig=pgR-Okk_dmk7K16ilQLWebcy6dY&amp;hl=es&amp;sa=X&amp;ved=0ahUKEwjkspX7hefVAhWLQBQKHXyEBWMQ6AEIPTAJ#v=onepage&amp;q&amp;f=false Georgi Horni Arca Noæ, sive Historia imperiorum et regnorum à condito orbe.]&lt;/ref&gt; seu [[Salvaleon]]&lt;ref name=&quot;Salvaleon&quot;&gt;[http://www.e-rara.ch/zut/content/pageview/12351901 Lexicon Universale Hofmann] ubi dicitur: &quot;Hispaniola (...) a Dominicopoli loco memoranda, Salvaleon sacchari proventu nobilis (...)&quot;.&lt;/ref&gt; || [[Altagracia]]&lt;ref name=&quot;Domina&quot;&gt;[http://www.catholic-hierarchy.org/diocese/dnuse.html catholic-hierarchy.org] ubi apparet &quot;Dioecesis A Domina Nostra vulgo de la Altagracia in Higüey, seu Higueyensis&quot;.&lt;/ref&gt; || 322 266 || align=center style=&quot;background:#f0f0f0;&quot; | 14 || align=left | [[Bonao]]&lt;ref&gt;Confer binomen &quot;Meliola bonaoensis&quot;.&lt;/ref&gt; || [[Dominus Nouel (provincia)|Dominus Nouel]] || 142 984 
|-
| align=center style=&quot;background:#DCE9F0;&quot; | 5 ||align=left | [[Sanctus Petrus de Macoris (urbs)|Sanctus Petrus de Macoris]]&lt;ref name=&quot;Macoris&quot;&gt;[http://www.catholic-hierarchy.org/diocese/dsnpm.html catholic-hierarchy.org] ubi apparet &quot;Dioecesis Sancti Petri de Macoris&quot;.&lt;/ref&gt; || [[Sanctus Petrus de Macoris (provincia)|Sanctus Petrus de Macoris]]&lt;ref name=&quot;Macoris&quot;&gt;[http://www.catholic-hierarchy.org/diocese/dsnpm.html catholic-hierarchy.org] ubi apparet &quot;Dioecesis Sancti Petri de Macoris&quot;.&lt;/ref&gt;  || 263 077 || align=center style=&quot;background:#f0f0f0;&quot; | 15 || align=left | [[Haina (Res publica Dominicana)|Haina]]&lt;ref&gt;Confer binomen &quot;Teichospora hainensis&quot;.&lt;/ref&gt; || [[Sanctus Christophorus (provincia Dominiciana)|Sanctus Christophorus]] || 142 322
|-
| align=center style=&quot;background:#DCE9F0;&quot; | 6 ||align=left | [[Conceptio de Vega]]&lt;ref name=&quot;Conceptio&quot;&gt;[http://www.e-rara.ch/zut/content/pageview/12351901 Lexicon Universale Hofmann] ubi dicitur: &quot;Hispaniola (...) in parte Austr. Conceptio de Vega, sedes ante Episcopi, (...)&quot;.&lt;/ref&gt; || [[Vega (Res publica Dominiciana)|Vega]]&lt;ref name=&quot;Conceptio&quot;&gt;[http://www.e-rara.ch/zut/content/pageview/12351901 Lexicon Universale Hofmann] ubi dicitur: &quot;Hispaniola (...) in parte Austr. Conceptio de Vega, sedes ante Episcopi, (...)&quot;.&lt;/ref&gt; seu [[Provincia Vegensis]]&lt;ref&gt;[http://www.catholic-hierarchy.org/diocese/dlave.html catholic-hierarchy.org] ubi apparet &quot;Dioecesis Vegensis&quot;.&lt;/ref&gt; || 253 919 || align=center style=&quot;background:#f0f0f0;&quot; | 16 || align=left | [[Barahona]]&lt;ref name=&quot;Barahonensis&quot;&gt;[http://www.catholic-hierarchy.org/diocese/dbara.html catholic-hierarchy.org] ubi apparet &quot;Dioecesis Barahonensis&quot;.&lt;/ref&gt; || [[Provincia Barahonensis]]&lt;ref name=&quot;Barahonensis&quot;&gt;[http://www.catholic-hierarchy.org/diocese/dbara.html catholic-hierarchy.org] ubi apparet &quot;Dioecesis Barahonensis&quot;.&lt;/ref&gt; || 138 470
|-
| align=center style=&quot;background:#DCE9F0;&quot; | 7 ||align=left | [[Sanctus Christophorus (urbs Dominiciana)|Sanctus Christophorus]] || [[Sanctus Christophorus (provincia Dominiciana)|Sanctus Christophorus]] || 240 705 || align=center style=&quot;background:#f0f0f0;&quot; | 17 || align=left | [[Azua Compostellae]]&lt;ref name=&quot;Azua&quot;&gt;[http://www.e-rara.ch/zut/content/pageview/12351901 Lexicon Universale Hofmann] ubi dicitur: &quot;Hispaniola (...) Azua seu Compostella &amp; Mons Christi, salis fodinis fructuosa. (...)&quot;.&lt;/ref&gt; || [[Azua]] &lt;ref name=&quot;Azua&quot;&gt;[http://www.e-rara.ch/zut/content/pageview/12351901 Lexicon Universale Hofmann] ubi dicitur: &quot;Hispaniola (...) Azua seu Compostella &amp; Mons Christi, salis fodinis fructuosa. (...)&quot;.&lt;/ref&gt; || 125 487
|-
| align=center style=&quot;background:#DCE9F0;&quot; | 8 ||align=left | [[Sanctus Franciscus de Macoris]]&lt;ref name=&quot;Sanctus Franciscus de Macoris&quot;&gt;[http://www.catholic-hierarchy.org/diocese/dsnfm.html catholic-hierarchy.org] ubi apparet &quot;Dioecesis Sancti Francisci de Macoris&quot;.&lt;/ref&gt;  || [[Duarte]] &lt;ref name=&quot;duarteensis&quot;&gt;Confer binomen &quot;Ilex duarteensis&quot;.&lt;/ref&gt; || 213 906 || align=center style=&quot;background:#f0f0f0;&quot; | 18 || align=left |[[Mao]]&lt;ref name=&quot;Mao&quot;&gt;[http://www.catholic-hierarchy.org/diocese/dbara.html catholic-hierarchy.org] ubi apparet &quot;Dioecesis Maoënsis-Montis Christi&quot;.&lt;/ref&gt; || [[Valverde (Res publica Dominiciana)|Valverde]] || 117 481
|-
| align=center style=&quot;background:#DCE9F0;&quot; | 9 ||align=left | [[Romana (urbs)|Romana]] || [[Romana (Res publica Dominiciana)|Romana]] || 207 784  || align=center style=&quot;background:#f0f0f0;&quot; | 19 || align=left | [[Nagua]] || [[Maria Trinitas Sánchez (provincia)|Maria Trinitas Sánchez]] || 117 195
|-
| align=center style=&quot;background:#DCE9F0;&quot; | 10 ||align=left | [[Moca (Res publica Dominicana)|Moca]] || [[Provincia Espaillatensis]]&lt;ref name=&quot;espaillatense&quot;&gt;Confer gentilicum Hispanicum &quot;espaillatense&quot;.&lt;/ref&gt; || 186 225 || align=center style=&quot;background:#f0f0f0;&quot; | 20 || align=left | [[Villa Formosa (Res publica Dominiciana)|Villa Formosa]] || [[Romana (Res publica Dominiciana)|Romana]] || 108 563
|}{{clear}}

=== Immigratio et emigratio ===
[[Fasciculus:Colonia Japonesa.jpg|thumb|upright=1.35|[[Iaponia|Iaponiensis]] descendentis familia [[Constantia (Res publica Dominicana)|Constantiae]] vicinitate Coloniae Iaponicae.]]
[[Saeculum 20|Saeculo XX]], multi [[Arabes]] (e [[Libanus|Libano]], [[Syria]] et [[Palaestina]])&lt;ref name=&quot;González&quot;&gt;González Hernández, Iulius Amabilis (Julio Amable) (die 11 Augusti anni 2012). [http://www.idg.org.do/capsulas/agosto2012/agosto201211.htm &quot;Registro de Inmigrantes de El Líbano&quot;]. ''Cápsulas Genealógicas en Areíto'' (Hispanice).&lt;/ref&gt;, [[Iaponia|Iaponienses]] et, minore gradu, [[Corea]]ni in natione agricolae et mercatores consederunt. [[Sinensis|Sinenses]] societates negotia in telecommunicationibus, fodinis et ferrivia invenerunt. Arabica communitas auget crescenti indice et numerantur 80 000.&lt;ref name=&quot;González&quot;&gt;González Hernández, Iulius Amabilis (Julio Amable) (die 11 Augusti anni 2012). [http://www.idg.org.do/capsulas/agosto2012/agosto201211.htm &quot;Registro de Inmigrantes de El Líbano&quot;]. ''Cápsulas Genealógicas en Areíto'' (Hispanice).&lt;/ref&gt;

Praeterea, descendentes immigrantium ex aliis Caribicis insulis venientium sunt, apud quas [[Sanctus Christophorus et Nives]], [[Antiqua (insula)|Antiqua]], [[Insula Sancti Vincentii|Sanctus Vincentius]], [[Mons Serratus (insula)|Mons Serratus]], [[Tortola]], [[Sancta Crux (insula)|Sancta Crux]]&lt;ref&gt;[https://books.google.es/books?id=jrRX72gH89QC&amp;pg=RA4-PA639&amp;lpg=RA4-PA639&amp;dq=insula+%22Montis+serrati%22&amp;source=bl&amp;ots=06X3debBgl&amp;sig=1O6YiPE2OGIncuT8tuQkJ8NVsZQ&amp;hl=es&amp;sa=X&amp;ved=0ahUKEwiH--Gzq-XaAhUDjSwKHdgVCZEQ6AEIQDAJ#v=onepage&amp;q=insula%20%22Montis%20serrati%22&amp;f=false Hübner Ioannes. Kort begryp der oude en nieuwe geographie], ubi dicitur &quot;S. CROIX in't Latijn Insula S. Crucis&quot;.&lt;/ref&gt;, [[Sanctus Thomas (insula Virginea Civitatum Foederatarum)|Sanctus Thomas]]&lt;ref&gt;Cf.
{{CathHierDiocese|stth|Dioecesis Sancti Thomae in Insulis Virgineis}}&lt;/ref&gt; et [[Guadalupia (praefectura)|Guadalupia]]. Qui harundinis saccharinae plantationibus et navalibus [[Sanctus Petrus de Macoris (urbs)|Sancti Petri de Macoris]]&lt;ref name=&quot;Macoris&quot;&gt;[http://www.catholic-hierarchy.org/diocese/dsnpm.html catholic-hierarchy.org] ubi apparet &quot;Dioecesis Sancti Petri de Macoris&quot;.&lt;/ref&gt; et [[Portus Argenteus (urbs)|Portu Argenteo]]&lt;ref name=&quot;Argenteus&quot;&gt;[http://www.e-rara.ch/zut/content/pageview/12351901 Lexicon Universale Hofmann] ubi dicitur: &quot;Hispaniola (...) nunc Dominicopolitana juncta, Zeibum, ''Zeibo'', Fanum S. Jacobi, ''Sant Jago'', egregio situ inclyta, Portus Argenteus, seu ''Puerto de la Plata'', ob commercii frequentiam, (...)&quot;.&lt;/ref&gt; seu [[Portus Argentarius (urbs)|Portu Argentario]]&lt;ref name=&quot;Argentarii&quot;&gt;[http://www.catholic-hierarchy.org/diocese/dpupl.html catholic-hierarchy.org] ubi apparet &quot;Dioecesis Portus Argentarii&quot;.&lt;/ref&gt; laborabant. [[Portoricus|Portoricenses]], et minore gradu, [[Cuba]]ni immigrantes in Rem publicam Dominicianam e mediis [[anni 1800|annis 1800]] usque ad annum [[1940]] propter inopem oeconomiam et socialem perturbationem in suis respectivis nationibus fugerunt. Multi Portoricenses immigrantes [[Hygvei]]s&lt;ref name=&quot;Hygvei&quot;&gt;[https://books.google.es/books?id=Lmk8NQJuOYYC&amp;pg=PA541&amp;lpg=PA541&amp;dq=xaragua+quinque+regna&amp;source=bl&amp;ots=sR2UpmmuwN&amp;sig=pgR-Okk_dmk7K16ilQLWebcy6dY&amp;hl=es&amp;sa=X&amp;ved=0ahUKEwjkspX7hefVAhWLQBQKHXyEBWMQ6AEIPTAJ#v=onepage&amp;q&amp;f=false Georgi Horni Arca Noæ, sive Historia imperiorum et regnorum à condito orbe.]&lt;/ref&gt; seu [[Salvaleon]]e&lt;ref name=&quot;Salvaleon&quot;&gt;[http://www.e-rara.ch/zut/content/pageview/12351901 Lexicon Universale Hofmann] ubi dicitur: &quot;Hispaniola (...) a Dominicopoli loco memoranda, Salvaleon sacchari proventu nobilis (...)&quot;.&lt;/ref&gt;, apud alias urbes, consederunt et rapide assimulati sunt similem ob culturam. Ante [[Secundum bellum mundanum|Secundum Bellum Mundanum]], 800 [[Iudaei|Iudaeorum]] profugorum in Rem publicam Dominicianam venerunt.&lt;ref name=&quot;CCNY&quot;&gt;&quot;CCNY Jewish Studies Class to Visit Dominican Village that Provided Refuge to European Jews During World War II&quot; (Press release). City College of New York. November 13, 2006. Archived from the original on May 10, 2011. Retrieved August 3, 2010.&lt;/ref&gt;

Innumeros immigrantes e [[Mare Caribicum|Caribicis]] nationibus venerunt, quia natio oeconomicas opportunitates ottulit. Sunt 32 000 [[Iamaica]]norum Rem publicam Dominicianam incolentium.&lt;ref&gt;Joshua Project (2016). [https://joshuaproject.net/people_groups/12316 &quot;The Jamaicans people group is reported in 14 countries&quot;] (Anglice). Joshuaproject.net. Inspectum die 19 Octobris anni 2016.&lt;/ref&gt; Est crescens [[Portoricus|Portoricensium]] immigrantium numerus, praesertim [[Dominicopolis|Dominicopoli]] et locis circumiectis; Creduntur numerare circa 10 000.&lt;ref&gt;[https://web.archive.org/web/20110317050440/http://www.topix.com/forum/world/dominican-republic/T4ULLRH92RE5AQ2UL &quot;Growing Puerto Rican population in the Dominican Republic&quot;] (Anglice). Universitas Centralis Orientis. Archivatum ex originali die 17 Martii 2011. Inspectum die 19 Iulii anni 2010..&lt;/ref&gt;&lt;ref&gt;[https://www.diariolibre.com/noticias/ms-de-medio-milln-de-inmigrantes-residen-en-el-pas-EDDL381577 &quot;Más de medio millón de inmigrantes residen en el país&quot;] (Hispanice). diariolibre.com. Kalendis Maiis anni 2013. Inspectum die 19 Octobris anni 2016.&lt;/ref&gt;

==== Haitiana immigratio ====
[[Fasciculus:Haiti deforestation.jpg|thumb|upright=1.35|Satellitalis imago finium inter nudum [[Haitia]]e prospectum (laeva) et Rei publicae Dominicianae (dextera), in promptu posita [[desilvatio]]ne Haitiani lateris.]]
[[Fasciculus:Dominicans and Haitians Braving the Weather.jpg|thumb|upright=1.35|Dominiciani et Haitiani lineati ad frequentandas medicos praebitores e Subsidiis Exercitus [[Statunitensis]].]]
[[Haitia]] Rei publicae Dominicianae vicina natio et valde inopior, minus evoluta et praeterea minime evoluta natio in Hemisphaerio Occidentali est. Anno 2003, 80% Haitianorum pauperes (54% abiecta paupertate viventes) et 47,1% analphabeti erant. Novem millionis hominum natio quoque rapide crescentem numerum incolarum habet, sed circiter duae tertiae partes operariorum formalibus laboribus carent. Haitianus per capita [[PDG]] 1 300 [[dollarium (CFA)|$]] anno 2008 aut minus quam una sexta pars Dominiciani erat.&lt;ref name=&quot;CIA&quot;&gt;[https://www.cia.gov/library/publications/the-world-factbook/geos/dr.html &quot;CIA – The World Factbook – Dominican Republic&quot;] (Anglice). [[Procuratio Nuntiorum Centralis]] (CIA). Inspectum [[4 Iunii|pridie Nonas Iunias]] anni 2007.&lt;/ref&gt;&lt;ref&gt;[https://www.cia.gov/library/publications/the-world-factbook/geos/ha.html &quot;CIA – The World Factbook – Dominican Republic&quot;] (Anglice). [[Procuratio Nuntiorum Centralis]] (CIA). Inspectum [[10 Ianuarii]] anni 2010.&lt;/ref&gt;

Ideo hecatontades milium Haitianorum in Rem publicam Dominicianam migraverunt, aestimatis 800 000 Haitianorum hac in natione.&lt;ref name=&quot;Diógenes&quot;&gt;Diogenes (Diógenes) Pina (die 21 Martii anni 2007). [https://web.archive.org/web/20080109194929/http://www.ipsnews.net/news.asp?idnews=37018 &quot;Dominican Republic: Deport Thy (Darker-Skinned) Neighbour&quot; (Anglice). Inter Press Service (IPS). Archivatum ex originali die 9 Ianuarii anni 2008. Inspectum die 14 Ianuarii anni 2008.&lt;/ref&gt;, dum alii in Haitia natos tam numerosos quam millio putant.&lt;ref&gt;[http://www.hrw.org/reports/2002/domrep/domrep0402-02.htm &quot;Illegal people&quot;] (Anglice). [[Custodia Iurum Humanorum]]. Archivatum ex originali die 2002-04-21. Inspectum die 29 Maii anni 2007.&lt;/ref&gt; In demissi salarii et haud idoneis laboribus in re exstructoria, domestica purgatione et harundinetis saccharis laborare solent.&lt;ref&gt;Iacobus (James) Ferguson. [http://minorityrights.org/publications/migration-in-the-caribbean-haiti-the-dominican-republic-and-beyond/ &quot;Migration in the Caribbean: Haiti, the Dominican Republic and Beyond&quot;] (PDF) (Anglice). Minority Rights Group International. Inspectum die 14 Ianuarii anni 2008.&lt;/ref&gt; Sunt accusationes ut aliqui Haitiani immigrantes condicionibus similibus servitudini laborent et valde sub iugum mittantur.&lt;ref&gt;[ttps://www.huffingtonpost.com/richard-morse/haitian-cane-workers-in-t_b_626610.html?ref=twitter Ricardus (Richard) Morse: Haitian Cane Workers in the Dominican Republic] (Anglice). Huffingtonpost.com. Inspectum die 22 Septembris anni 2011.&lt;/ref&gt;

Propter basicarum opium et medicarum facultatum carentiam in Haitia, magnus Haitianarum feminarum numerus, crebro cum aliquot salutis problematibus advenientes, Dominicianum versus solum limites transeunt. Consulto adveniunt per ultimas gravitatis hebdomades medicam curam partui impetratum, quia Dominiciana publica nosocomia medicas curas non recusant propter nationalitatem aut legalem statum. Statistica Dominicopolitano e nosocomio circa 22% partuum Haitianarum matrum esse referunt.&lt;ref&gt;Pantaleón, Doris (die 20 Ianuarii anni 2008). [https://web.archive.org/web/20101013090741/http://listin.com.do/la-republica/2008/1/20/45034/El-22-de-los-nacimientos-son-de-madres-haitianas &quot;El 22% de los nacimientos son de madres haitianas&quot;] (Hispanice). Listin Diario. Archivatum ex originali die 13 Octobris anni 2010.&lt;/ref&gt;

Haitia quoque severam ambitalem degradationem patitur. Desilvatio in Haitia divulgata est; hodie minus quam quaternae centesimae partes Haitianarum silvarum manent, et pluribus in locis solum usque ad saxum basale valde erodatum est.&lt;ref&gt;&quot;Dirt Poor — Haiti has lost its soil and the means to feed itself&quot; (Anglice).&lt;/ref&gt; Haitiani carbonem ligni urunt ut 60% domesticae energiae producantur. Cum Haitia virentem materiem urendam consumpsisset, Haitiani illegalem carbonis mercatum Dominiciano in latere creaverunt. Consertoriae aestimationes illegalem motum 115 [[tonna]]rum carbonis per hebdomadem e Re publica Dominiciana in Haitiam calculant. Dominiciani officiales saltem decem autoplautra per hebdomadem carbone onerata fines transire aestimant.&lt;ref&gt;[http://latinamericanscience.org/2014/03/the-charcoal-war/ &quot;The charcoal war&quot;] (Anglice).&lt;/ref&gt;

Anno 2005, Dominicianus Praeses Leonellus Fernández collectivas Haitianorum expulsiones impugnant, quia factae sunt &quot;abusivo inhumanoque modo&quot;.&lt;ref&gt;[https://web.archive.org/web/20070422232810/http://web.amnesty.org/library/Index/ENGAMR270012007 &quot;Dominican Republic: A Life in Transit&quot;]. [[Amnestia Internationalis]]. Die 21 Martii anni 2007. Archivatum ex originali die 22 Aprilis anni 2007. Inspectum die 3 Iunii anni 2007.&lt;/ref&gt; Cum Nationum Unitarum delegatio praeliminariam relationem se grave problema rassismi discriminationisque contra Haitianae originis homines invenisse affirmantem edixisset, Dominicianus [[Minister rerum externarum|Minister Rerum Externarum]] [[Carolus Morales Troncoso]] formalem affirmationem id denuntians edixit, asserverans: &quot;nostri fines cum Haitia problemata habent; hoc est nostra realitas et intellegenda est. Praecipuum suverenitatem cum indifferentia et securitatem cum [[xenophobia]] non confundere est&quot;.&lt;ref&gt;Diogenes (Diógenes) Pina (Pridie Kalendas Novembres anni 2007). &quot;[https://web.archive.org/web/20080109074036/http://www.ipsnews.net/news.asp?idnews=39867 Republic: Gov't Turns Deaf Ear to UN Experts on Racism&quot;] (Anglice). Inter Press Service (IPS). Archivatum ex originali die 9 Ianuarii anni 2008. Inspectum die 14 Ianuarii anni 2008.&lt;/ref&gt;

Haitianorum immigrantium pueri crebro discives sunt et diakoniae eis denegantur, quoniam etiam parentibus denegatur Dominiciana nationalitas, et transeuntes residentes propter suum illegalem aut indocumentatum statum putantur; pueris, quamquam crebro eligibiles sunt Haitianae nationalitati&lt;ref&gt;[http://www.oas.org/juridico/MLA/en/hti/en_hti-int-const.html &quot;Constitutio Haitianae anni 1987&quot;] (Anglice). Inspectum die 16 Octobris anni 2010. &quot;ARTICULUS 11: Quilibet homo e Haitiano patre aut Haitiana matre qui nativi Haitiani essent et nunquam suam nationalitatem renuntiaverint natus Haitianam nationalitatem partus tempore possidet&quot;&lt;/ref&gt;, haec negatur ab Haitia, ob carentiam propriorum documentorum aut testium.&lt;ref&gt;Maureen Lynch (Kalendis Novembribus anni 2007). [https://web.archive.org/web/20080708224221/http://www.refugeesinternational.org/content/article/detail/9770 &quot;Dominican Republic, Haiti, and the United States: Protect Rights, Reduce Statelessness&quot;] (Anglice). Refugees International. Archivatum ex originali die 8 Iulii anni 2008.&lt;/ref&gt;&lt;ref&gt;Andreas (Andrew) Grossman (die 11 Octobris anni 2004). [http://www.uniset.ca/naty/maternity/ &quot;Birthright citizenship as nationality of convenience&quot;] (Anglice). ''Acta Tertii Colloquii de Nationalitate''. Consilium Europaeum. Inspectum die 3 Iunii anni 2007.&lt;/ref&gt;&lt;ref&gt; [https://web.archive.org/web/20080708193320/http://www.caribbeannetnews.com/cgi-script/csArticles/articles/000052/005242.htm &quot;Dominican Republic, Haiti, and the United States: Protect rights, reduce statelessness&quot;]. Reuters. Die 19 Ianuarii anni 2007. Archivatum ex originali die 8 Iulii anni 2008. Inspectum die 29 Maii anni 2007.&lt;/ref&gt;&lt;ref&gt;Michaela (Michelle) García (2006). [https://web.archive.org/web/20070807031700/http://www.amnestyusa.org/Fall_2006/No_Papers_No_Rights/page.do?id=1105216&amp;n1=2&amp;n2=19&amp;n3=358 &quot;No Papers, No Rights&quot;] (Anglice). [[Amnestia Internationalis]]. Archivatum ex originali die 2007-08-07. Inspectum die 29 Maii anni 2007.&lt;/ref&gt;

=== Emigratio ===
[[Fasciculus:Dominican people at Dominican parade, New York City.jpg|thumb|upright=1.25|Pompa Diei Dominiciani [[Novum Eboracum (urbs)|Novi Eboraci]] anno 2014.]]
Trium exeuntis [[saeculum 20|saeculi XX]] undarum prima anno 1961 coepit, interfecto dictatore Trujillo,&lt;ref&gt;Iacobus (James) A. Wilderotter (die 3 Ianuarii nni 1975). [https://nsarchive2.gwu.edu//NSAEBB/NSAEBB222/family_jewels_wilderotter.pdf &quot;Memorandum for the File, &quot;CIA Matters&quot;&quot;] (Anglice) (PDF). National Security Archive.&lt;/ref&gt; propter metum talionis a dictatoris Trujillo sociatis et politicam incertitudinem generatim. Anno [[1965]] Civitates Foederatae Americae militarem Rei publicae Dominicianae occupationem inceperunt ut finiretur bellum civile. Praeter haec, Civitates Foederatae itinerum restrictiones leniverunt, quod facilius ut Dominiciani Statunitensia visa impetrarent fecit.&lt;ref&gt;Morrison, Thomas K.; Sinkin, Ricardus (Richard) (hieme anni 1982). &quot;International Migration in the Dominican Republic&quot; (Anglice). ''International Migration Review''. 16 (4, Special Issue: International Migration and Development): 819–836. doi:10.2307/2546161. JSTOR 2546161.&lt;/ref&gt; E 1966 in 1978, exodus perrexit, propulsus alta vacuitate laboris et politica repressione. Communitates prima immigrantium unda in CFA sessae rete quod posteriores adventus adiuvaret creaverunt.&lt;ref name=&quot;Annenberg&quot;&gt;[http://www.learner.org/libraries/socialstudies/9_12/weir/background.html &quot;Migration Trends in Six Latin American Countries&quot;] (Anglice). Annenberg Foundation.&lt;/ref&gt;

Ineuntibus anni 1980, horarium deminutum operis, inflatio et incrementum valoris dollarii, omnia ad tertiam emigrationis undam e Re publica Dominiciana contribuerunt. Hodie, emigratio e Re publica Dominiciana alta manet.&lt;ref name=&quot;Annenberg&quot;&gt;[http://www.learner.org/libraries/socialstudies/9_12/weir/background.html &quot;Migration Trends in Six Latin American Countries&quot;] (Anglice). Annenberg Foundation.&lt;/ref&gt; Anno 2012 erant propemodum 1,7 milliones Dominiciani descendentes in CFA, computatis et nativis et peregre natis.&lt;ref&gt;US Census Bureau 2012 American Community Survey B03001 1-Year Estimates HISPANIC OR LATINO ORIGIN BY SPECIFIC ORIGIN (Anglice). Archivatum die 15 August anni 2014, in Wayback Machine. Inspectum die 20 Septembris anni 2013.&lt;/ref&gt; Erat etiam crescens Dominiciana immigratio in [[Portoricus|Portoricum]], cum prope 70 000 Dominicianorum ibi anno 2010 viventium. Quamquam iste numerus lente decrevit et immigratio motus inversus est ob Portoricensem crisin oeconomicam anni 2016.

=== Valetudo ===
[[Fasciculus:Homs links 179.jpg|thumb|Valetudinarium Metropolitanum [[Sanctus Iacobus Equitum|Sancti Iacobi Equitum]].]]
Anno [[2007]], Res publica Dominiciana natalium indicem 22,91 pro 1000 et mortuorum indicem 5,32 pro 1000 habuit.&lt;ref name=&quot;CIA&quot;&gt;[https://www.cia.gov/library/publications/the-world-factbook/geos/dr.html &quot;CIA – The World Factbook – Dominican Republic&quot;] (Anglice). [[Procuratio Nuntiorum Centralis]] (CIA). Inspectum [[4 Iunii|pridie Nonas Iunias]] anni 2007.&lt;/ref&gt; Dominiciana iuventus in Re publica Dominiciana aetatis grex sanissimus est.

[[Sida]]e et viri [[VIDH]] praevalentia in Re publica Dominiciana anno 2011 et propemodum in 0,7% collocatur, quae est relative demissa Caribicas intra formas, cum circiter 62 milliones Dominicianorum positivorum sidae/viro VIDH.&lt;ref&gt;[https://data.worldbank.org/indicator/SH.DYN.AIDS.ZS Prevalence of HIV, total (% of population ages 15–49) | Data | Table] (Anglice). Data.worldbank.org. Inspectum die 2 Aprilis anni 2014.&lt;/ref&gt; Per contra, vicina Haitia sidae viri VIDH indicem ad Rem publicam Dominicianam correspondentem duplicat. Missio in CFA fundata ad pugnandum contra sidam hac in natione adiuvat.&lt;ref&gt;«The President's Emergency Plan for AIDS Relief» (Anglice) (PDF). Office of the U.S. Global AIDS Coordinator. Aprili anni 2005. Archivatum forma (PDF) die 15 Februarii anni 2009.&lt;/ref&gt; [[Febris dengue]] [[endemismus|endemica]] in Re publica facta est et sunt casus [[malaria]]e et [[Zika virus|Zikae viri]].&lt;ref&gt;[https://wwwnc.cdc.gov/travel/page/risk-of-zika-selected-destination «Zika Virus in the Dominican Republic»] (Anglice). ''CDC''. [[5 Augusti|Nonis Augustis]] anni 2016. Inspectum die 24 Augusti anni 2016.&lt;/ref&gt;&lt;ref&gt;[https://travel.state.gov/content/travel/en/international-travel/International-Travel-Country-Information-Pages/DominicanRepublic.html «Dominican Republic».] ''[[Ministerium Rerum Externarum Civitatum Foederatarum]]''. nspectum die 24 Augusti anni 2016.&lt;/ref&gt;

[[Abortus|Abruptio graviditatis]] illegalis est omnibus casibus in Re publica Dominiciana, prohibitio quae includit conceptiones post [[stuprum]] aut [[incestus|incestum]] et condicionibus quibus matris valetudo in periculo versatur, quamquam sit letalis.&lt;ref&gt;Romo, Raphaël (Rafael) (die 18 Augusti anni 2012). [https://edition.cnn.com/2012/08/18/world/americas/dominican-republic-abortion/index.html Pregnant teen dies after abortion ban]. ''CNN''.&lt;/ref&gt; Haec prohibitio reiterata est a Gubernio Dominiciano in rogatione constitutionalis emendationis Septembri anni 2009.&lt;ref&gt;[http://www.foxnews.com/story/2009/09/18/dominican-republic-reaffirms-commitment-against-legalizing-abortion.html «Dominican Republic Reaffirms Commitment Against Legalizing Abortion».] (Anglice) Fox News. Die 18 Septembris anni 2009. Inspectum anno 10 Septembris anni 2010.&lt;/ref&gt;

=== Educatio ===
[[Fasciculus:UASD Santiago - 031.jpg|thumb|[[Universitas Autonoma Dominicopolitana]] (Hispanice: ''Universidad Autónoma de Santo Domingo'').]]
Ludus litterarius a Ministerio Educationis temperatur, cum educatio omnium civium et iuvenum ius in Re publica Dominiciana sit.&lt;ref&gt;[http://www.oas.org/juridico/spanish/mesicic2_repdom_sc_anexo_7_sp.pdf &quot;LEY 66–97 Ley General de Educación&quot;] (Hispanice) (PDF).&lt;/ref&gt;

Puerorum ludus dissimilibus in cyclis ordinatur et prodest ad circulum aetatis inter 2 et 4 ac ad circulum aetatis inter 4 et 6. Puerorum ludus obligatorius non est, excepto ultimo anno. Basica educatio obligatoria est et prodest pueris inter 6 et 14 annos natis. Educatio Secundaria obligatoria non est, quamquam est munus Civitatis eam offerre gratis. Prodest pueris inter 14 et 18 annos natis et ordinatur communi in nucleo quattuor annorum et trium modorum bimorum studiorum quae offeruntur in tribus dissimilibus optionibus: generali seu academico, polytechnico (industriali, agrario et diaconiis) et artificioso.

Educationis Superioris systema in institutis et universitatibus consistit. Instituta maioris technici gradus cursus offerunt. Universitates curricula technica, graduum et postgraduum offerunt; temperantur a Ministerio Educationis Superioris, Scientiae et Technologiae.&lt;ref&gt;[http://www.seescyt.gov.do/baseconocimiento/Leyes y reglamentos/Ley139-01 Educación Superior.pdf &quot;Ley 139-01 de Educación Superior, Ciencia y Tecnología&quot;] (Hispanice) (PDF). Archivatum ex originali (PDF) Kalendis Maiis anni 2015.&lt;/ref&gt;

=== Crimen ===
Anno 2012 Res publica Dominiciana [[index caedium|indicem caedium]] 22,1 pro 100 000 hominum habebat.&lt;ref name=&quot;UNODC&quot;&gt;[https://www.unodc.org/gsh/en/index.html &quot;UNODC: Global Study on Homicide&quot;] (Anglice). ''[[Officium Nationum Unitarum drogis et sceleribus cohibendis]]''. Anno 2013. Inspectum die 24 Augusti anni 2016.&lt;/ref&gt; Erat summa 2 268 caedium in Re publica Dominiciana anno 2012.&lt;ref name=&quot;UNODC&quot;&gt;[https://www.unodc.org/gsh/en/index.html &quot;UNODC: Global Study on Homicide&quot;] (Anglice). ''[[Officium Nationum Unitarum drogis et sceleribus cohibendis]]''. Anno 2013. Inspectum die 24 Augusti anni 2016.&lt;/ref&gt;

Res publica Dominiciana locus [[Transitus (commercium)|transitus]] [[Columbia]]nis [[droga|drogis]] destinatis ad Europam non minus quam Civitates Foederatas Canadamque facta est.&lt;ref name=&quot;CIA&quot;&gt;[https://www.cia.gov/library/publications/the-world-factbook/geos/dr.html &quot;CIA – The World Factbook – Dominican Republic&quot;] (Anglice). [[Procuratio Nuntiorum Centralis]] (CIA). Inspectum [[4 Iunii|pridie Nonas Iunias]] anni 2007.&lt;/ref&gt;&lt;ref&gt;Michael Winerip (die 9 Iulii anni 2000). [https://www.nytimes.com/2000/07/09/us/why-harlem-drug-cops-don-t-discuss-race.html?scp=2&amp;sq=Why%20Harlem%20Drug%20Cops%20Don%27t%20Discuss%20Race&amp;st=cse &quot;Why Harlem Drug Cops Don't Discuss Race&quot;]. ''[[The New York Times]]''.&lt;/ref&gt; [[Lavatio pecuniae]] a [[Columbia]]nis [[droga]]e [[chartelum|chartellis]] propter illicitarum pecuniariarum transactionum facilitatem secundatur.&lt;ref name=&quot;CIA&quot;&gt;[https://www.cia.gov/library/publications/the-world-factbook/geos/dr.html &quot;CIA – The World Factbook – Dominican Republic&quot;] (Anglice). [[Procuratio Nuntiorum Centralis]] (CIA). Inspectum [[4 Iunii|pridie Nonas Iunias]] anni 2007.&lt;/ref&gt; Anno 2004, aestimabatur 8% cocaini furtim in Civitates Foederatas importati per Rem publicam Dominicianam venisse.&lt;ref&gt;Ribando, Clara (Claire) (die 5 Martii anni 2005). [https://fpc.state.gov/documents/organization/46402.pdf &quot;Dominican Republic: Political and Economic Conditions and Relations with the United States&quot;] (Anglice) (PDF). CRS Report for Congress. Retrieved May 29, 2007.&lt;/ref&gt; Res publica Dominiciana respondit cretis conatibus prehendendorum drogiae [[Transitus (commercium)|transituum]], illorum involutorum comprehendorum tradendorumque et confligendae lavationis pecuniae.

Crebro levis tractatio violentorum scelerorum continuus localis controversiae fons fuit. Aprili anni 2010, quinque adulescentes, 15 et 17 annos nati, telo duos autocineti meritorii rectores transfixerunt et interfecerunt et alios quinque interfecerunt cum coegissent eos acidum purgatorium bibere. Die 24 Septembris anni 2010, adulescentes ad catenase tribus ad quinque annos damnati sunt, quamquam autocineti meritorii rectorum familiae reclamitaverunt.&lt;ref&gt;[http://www.bbc.com/news/world-latin-america-11404313 &quot;Teenagers jailed for taxi drivers' murder&quot;] (Anglice). ''BBC News''. die 24 Septembris anni 2010.&lt;/ref&gt;

== Cultura ==
Propter [[syncretismus|culturalem syncretismum]], Dominicianorum cultura moresque [[Europa]]eam culturalem basin habet, et [[Africa|Africis]] et autochthonicis [[Taini]]cis elementis affectam, quamquam endogena elementa Dominicianam intra culturam orta sunt;&lt;ref name=&quot;Esteva&quot;&gt;Esteva Fabregat, Claudius (Claudio) (1981). &quot;La hispanización del mestizaje cultural en América&quot; (PDF). ''Revista Complutense de Historia de América'' (Hispanice). [[Universitas Complutensis Matritensis]]. 1: 133. ISSN 0211-6111. Inspectum die 26 Augusti anni 2016.&lt;/ref&gt; culturaliter Res publica Dominiciana est apud Europaissimas nationes in [[America Hispanica]], iuxta cum [[Portoricus|Portorico]], [[Cuba]], [[Chilia|Chilia Media]], [[Argentina]] et [[Uraquaria]].&lt;ref name=&quot;Esteva&quot;&gt;Esteva Fabregat, Claudius (Claudio) (1981). &quot;La hispanización del mestizaje cultural en América&quot; (PDF). ''Revista Complutense de Historia de América'' (Hispanice). [[Universitas Complutensis Matritensis]]. 1: 133. ISSN 0211-6111. Inspectum die 26 Augusti anni 2016.&lt;/ref&gt; Hispanica instituta coloniali tempore in Dominicianae culturae creatione praevalere potuerunt, sicuti relativus successus in [[acculturatio]]ne et [[assimilatio culturalis|assimilatione culturali]] Africorum servorum Africum culturalem pondus deminuerunt, comparatione aliarum Caribicarum nationum.
[[Fasciculus:Campesino cibaeño.png|thumb|upright=0.9|&quot;Agricola Cibaoenis&quot; (Hispanice ''Campesino cibaeño''), anno 1941 in Museo Artis Modernae (Hispanice ''Museo de Arte Moderno'') [[Dominicopolis|Dominicopoli]].]]
Musica et artes athleticae praecipuae Dominiciana in cultura sunt, cum essent [[Meringinis (genus musicum)|meringinis]] ac [[bachata]] (Hispanice ''merengue'' ac ''bachata'', respective) nationales saltatio musicaque et [[basipila]] percara ars athletica.&lt;ref name=&quot;Ambasciata&quot;&gt;[https://web.archive.org/web/20150626100357/http://www.domrep.org/gen_info.html &quot;Ambasciata Rei publicae Dominicianae in Civitatibus Foederatis&quot;] (Anglice). Archivatum ex originali die 2015-06-26. Inspectum die 27 Februarii anni 2009.&lt;/ref&gt;

=== Artes oculorum ===
Dominiciana ars forsitan nitidorum ardentiumque colorum imaginibusque quae in omnibus periegeticis munusculorum tabernis totam per nationem venduntur adiuncta est. Nihilominus, natio longam [[bellae artes|bellarum artium]] historiam in medios [[anni 1800|annos 1800]] regressam habet, cum natio independens facta est et nationalis artis scaena oriri coepit.

Historice, huius temporis picturae in imaginibus conexis ad nationalem independentiam, historicas scaenas, efficies sed etiam topia et naturae mortuae imagines versae sunt. Picturae genera inter [[neoclassicismus|neoclassicismum]] et [[romanticismus|romanticismum]] extenduntur. Inter annos 1920 et 1940 artis scaena [[Realismus (artes)|realismi]] et [[impressionismus|impressionismi]] generibus affecta est. Dominiciani artifices in frangenda priora academica genera incubuerunt ut evolverentur independentiora individualiaque genera.

=== Architectura ===
[[Fasciculus:DOMREP-s-dom-panteon-innen.jpg|thumb|left|Pantheum nationale [[Dominicopolis|Dominicopoli]].]]
[[Architectura]] in Re publica Dominiciana complexam dissimilium culturarum permixtionem repraesentat. Profundum [[Europa]]eorum colonorum pondus evidentissimum totam per nationem est. Ornatis designationibus et [[Barocus|barocis]] structuris distinctum, hoc genus urbe capite [[Dominicopolis|Dominicopoli]] melius videri potest, quae est domus primae cathedralis, castelli, monasterii et fortalitiae in tota [[America]], sitorum Urbe Coloniali (Hispanice ''Ciudad Colonial'') huius urbis, area declarata [[Patrimonium totius mundi|patrimonio totius mundi]] ab [[UNESCO]].&lt;ref&gt;[http://whc.unesco.org/en/list/526 &quot;Colonial City of Santo Domingo&quot;] (Anglice et Francice). Centrum [[Patrimonium totius mundi|patrimonii totius mundi]] ab [[UNESCO]].&lt;/ref&gt;&lt;ref&gt;[http://www.unesco.org/nac/geoportal.php?country=DO&amp;language=E &quot;Dominican Republic National Commission for UNESCO&quot;] (Anglice et Francice). ''UNESCO''. Die 14 Novembris anni 1957. Consultum die 24 Augusti anni 2016.&lt;/ref&gt; Designationes extensae sunt per villas et aedificia totam per nationem. Animadverti potest quoque in aedificiis exteriores partes tectorio politas, flexas ianuas fenestrasque et rubrarum tegularum tecta continentibus.

Autochthones Rei publicae Dominicianae gentes etiam significativum pondus in nationis architectura habuit. [[Taini]] magnopere anacartio et guano (siccis arecaceais foliis) nitebantur ad componenda opera manualia, opera artis, supellectiles et domos. Luto, culmatis tectis et anacartio, dederunt aedificiis interioribusque supellectilibus naturalem aspectum, continenter mixtum cum insulae circumiectis.

Nuper, orta periegesi et crescente favore populi sicuti Caribica vacationis destinatione, architecti in Re publica Dominiciana nunc innivatorias designationes emphasim in luxu ponentes incorporare coepit. Multimodis architecturalia loca ludrica, villae et deversoria nova genera implent, dum exsecutiones veteribus de rebus offeruntur. Hoc novum genus simplificatis angularibus angulis et magnis fenestris exteriores interioresque partes commiscentibus depingitur. Habita cultura sicuti toto, praesentes architecti locupletem Rei publicae Dominicianae historiam et aliquot culturas complectuntur ad creandum aliquod novum. Inspectis novis villis, inveniri potest trium praecipuorum generum coniunctio: villa angularem, modernisticam aedificationem, flexas Hispanici Colonialis generis fenestras et traditionalem Tainicum lectulum pensilem in cubiculi maeniano continere potest.

=== Coquina ===
[[Fasciculus:ChicharronMixto.JPG|thumb|Tzitzarro mixtus, commune ferculum hac in natione ortum ex [[Vandalitia]] in meridionali Hispania.]]
Dominiciana coquina praesertim [[Gastronomia Hispanica|Hispanica]], [[Taini]]ca, et [[Africa]] est. Usitata coquina simillima inventae{{dubsig}} in aliis [[America Latina|Latinamericanis]] civitatibus est, sed multa e ferculorum nominibus dissimilia sunt. Ientaculi ferculum ex [[ovum|ovis]] et [[mangu]] (Hispanice ''mangú'') sive aulicoctorum [[plantanus|plantanorum]] pulticula consistit. Vigoratiores{{dubsig}} mangus versiones sociantur frictae carni, praesertim salumini&lt;ref&gt;[[:en:Salami#cite_ref-2|Vicipaedia Anglica]]&lt;/ref&gt; Dominiciano (Hispanice ''salami Dominicano''); caseo; aut ambobus. Prandii, generatim maximus et praecipuus cibus quotidianus, plerumque ex [[oryza]], carne, is et acetariis consistens. [[Vexillum (ferculum)|Vexillum]] (Hispanice ''La Bandera'') prandii ferculum cum maximo populi favore est; consistens e carne et rubris fabis super alba oryza. [[Sancoctum]] (Hispanice ''sancocho'') est minutal crebro carnis varietatibus septem factum.
[[Fasciculus:Patacones.JPG|thumb|upright=0.9|Tostones, frictorum [[plantanus|plantanorum]] ferculum.]]

Cibi carnibus amylisque pro lacticiniis holeribusque favere solent. Multa fercula [[condimentum frictum|condimento fricto]] (Hispanice ''sofrito'') fiunt, quod est mixtura localium herbarum quibus utuntur sicuti umido condimento carnibus et assatura quae augeat omnes sapores ferculorum. Austrocentrale per litus, [[purgurium]] sive integrum [[Triticum (alimentum)|triticum]] praecipuum elementum [[tabbula|tabbūla]] seu purgurii acetaria (Hispanice ''quipes'' seu ''tipili'') est. Apud alios praedilectos cibos sunt [[tzitzarro]]nes (Hispanice ''chicharrones''), [[manihot esculenta]] (Hispanice ''yuca''), [[tapioca]] (Hispanice ''casabe''), [[pastelletti]] (Hispanice ''pastelitos'') seu [[artocreas|artocreata]] (Hispanice ''empanadas''), [[Ipomoea batatas|batatae]], [[pastelli]] (Hispanice ''pasteles en hoja''), [[isicium Dominicianum|isicia Dominiciana]] (Hispanice ''chimichurris'') et [[tosto]]nes (Hispanice ''tostones'').

Aliquae bellaria quibus Dominiciani fruuntur sunt [[oryza cum lacte]] (Hispanice ''arroz con leche''), [[scriblita Dominiciana]] (Hispanice ''bizcocho dominicano''), [[fabae cum dulci]] (Hispanice ''habichuelas con dulce''), [[pulticula tumida]] (Hispanice ''flan''), [[nivata sorbitio]] (Hispanice ''frío frío''), [[dulce lactis]] (Hispanice ''dulce de leche''), [[harundo saccharina]] (Hispanice ''caña''). Potiones quibus Dominiciani fruuntur sunt [[morir soñando]], [[rhomium]], [[cervesia]], [[mamma ioanna]] (Hispanice ''Mamá Juana''),&lt;ref&gt;[http://www.republicadominicana.net/4-bebidas-tipicas-de-republica-dominicana/ &quot;Bebidas típicas de República Dominicana&quot;] Archivatum die 4 Martii anni 2016, in Wayback Machine. RepublicaDominicana.net (Hispanice).&lt;/ref&gt; potiones spumeae (Hispanice '''batidas''), [[sucus|suci naturales]] (Hispanice ''jugos naturales''), [[potio e colubrinā ellipticā]] (Hispanice ''mabí''), [[coffeum]] et [[maizium cum lacte]] (Hispanice ''chaca, maíz caqueao, maíz casqueado, maíz con dulce'' seu ''maíz con leche''), ultima potio inventa solum meridionalibus in provinciis huius nationis sicuti in [[Sanctus Ioannes (Res publica Dominiciana)|Sancto Ioanne]]&lt;ref name=&quot;Ioannis&quot;&gt;[http://www.catholic-hierarchy.org/diocese/dsjdr.html catholic-hierarchy.org] ubi apparet &quot;Dioecesis Sancti Ioannis Maguanensis&quot;.&lt;/ref&gt;

=== Musica et saltatio ===
[[Fasciculus:Merengue dancing.jpg|thumb|upright=0.8|left|[[Meringinis Dominicanus|Meringinis]] musica genus autochthonum Rei publicae est.]]
Musice, Res publica Dominiciana nota est propter popularrimum{{dubsig}} in toto orbe terrarum [[musica|genus musicum]] [[meringinis Dominicanus|meringinis]] (Hispanice ''merengue'') nuncupatum,&lt;ref name=&quot;Harvey&quot;&gt; Harvey, Ioannes (Sean) (2006). ''The Rough Guide to The Dominican Republic'' (Anglice). Rough Guides. ISBN 1-84353-497-5.&lt;/ref&gt; genus vivacis, rapidorum motuum, rhythmi et saltationis musica consistens e [[tempo]]{{dubsig}} circiter 120 in 160 numerorum pro minuta (quamquam mutat) fundata musicis in elementis sicuti [[apparatus tympanorum|apparatu tympanorum]], [[aërophonum|aërophonis]], [[chordophonum|chordophonis]], et [[harmonium|harmonio]], non minus quam aliquibus elementis unicis [[lingua Hispanica|Hispanophono]] in Caribico, sicuti [[tympanum Dominicianum|tympano Dominiciano]] (Hispanice ''tambora'') et [[guira]] (Hispanice ''güira'').
[[Fasciculus:Juan Luis Guerra en Acceso Total (6).jpg|thumb|Dominicianus cantor [[Ioannes Ludovicus Guerra]] icon meringinis musicae generis.]]

Cuius syncopati numeri membranophonis Latinamericanis, [[aërophonum|aërophonis metallicis]], gravi sono et [[clavicymbalum|clavicybalo]] aut pinnaria utuntur. Inter [[1937]] et [[1950]], musica [[meringinis Dominicanus|meringinis]] (Hispanice ''merengue'') internationaliter provecta est Dominicianis a gregibus sicuti Billo's Caracas Boys, Chapuseaux &amp; Damiron &quot;Los Reyes del Merengue,&quot; Joseito Mateo, apud alios. Radiophoni, televisio et media internationalia eam longius pervulgaverunt. Notissimos apud meringinis exsecutores [[Wilfridus Vargas]], [[Ioannes Ventura]], [[suorum carminum actor]]es [[Fratres Rosario]], [[Ioannes Ludovicus Guerra]], [[Ferdinandus Villalona]], [[Eduardus Herrera]], [[Sergius Vargas]], [[Antonius Rosario]], [[Milly Quezada]] et [[Chichí Peralta]] sunt.

Meringinis popularis in Civitatibus Foederatis, praesertim in [[Ora Orientalis Civitatum Foederatarum|Ora Orientali]], per [[anni 1980|annos 1980]] et [[anni 1990|annos 1990]]&lt;ref name=&quot;Harvey&quot;&gt; Harvey, Ioannes (Sean) (2006). ''The Rough Guide to The Dominican Republic'' (Anglice). Rough Guides. ISBN 1-84353-497-5.&lt;/ref&gt; factus est, cum multi Dominiciani cantores in Civitatibus Foederatis, imprimis [[Novum Eboracum (urbs)|Novi Eboraci]], residentibus in Latinamericanae consociationis scaena acta ponere coeperunt et consecuti sunt radiophonicam transmissionem. Apud quos sunt Victor Roque y La Gran Manzana, Henry Hierro, Zacarias Ferreira, Aventura, et Milly Jocelyn Y Los Vecinos. [[Bachata]]e ortus, iuxta cum crescenti numero Dominicanorum apud alios Latinamericanos greges Novum Eboracum, [[Nova Caesarea|Novam Caesaream]] et [[Florida]]m incolentium, generale popularitatis incrementum Dominicianae musicae contribuit.&lt;ref name=&quot;Harvey&quot;&gt; Harvey, Ioannes (Sean) (2006). ''The Rough Guide to The Dominican Republic'' (Anglice). Rough Guides. ISBN 1-84353-497-5.&lt;/ref&gt;

[[Bachata]], musicae saltationisque genus in ruralibus marginalibusque Rei publicae Dominicianae vicinitatibus ortum, valde popularis facta est recentibus annis. Cuius argumentum crebro amatorium est, praesertim praevalent amatorii angoris tristitiaeque narrationes. Revera, originale generis nomen ''amargue'' erat (scilicet &quot;acritudo&quot; seu &quot;acris musica&quot;), donec satis ambiguum (et affectionem neutrum) vocabulum bachata populare factum est. Bachata orta ex et adhuc arcte relata ad Panlatinamericanum amatorium genus [[musica bolerica|bolericum]]&lt;ref&gt;Cf. &quot;Institutum internationale nómine Forum Oeconómicum Mundi&quot; in J. J. del Col, ''[https://web.archive.org/web/20110531174857/http://www.juan23.edu.ar/latin/download/diccionario_latin.pdf Diccionario Auxiliar Español-Latino]'' ([[Diccionario Auxiliar Español-Latino]])&lt;/ref&gt; (Hispanice ''bolero'') nuncupatum. Per tempus, meringine et Latinamericanorum citharae Hispanicae generorum varietate affecta est.

[[Palus (musica)|Palus]] (Hispanice ''palo'') Afrodominiciana sacra musica quae totam per insulam inveniri potest est. Tympanum humanaque vox praecipua instrumenta sunt. Palus religiosis caerimoniis —sanctorum sollemnibus plerumque coincidentes— non minus quam saecularibus festis et peculiaribus occasionibus canitur. Cuius radices in [[Congo (flumen)|Congi]] regione Centroccidentalis Africae sunt, sed Europaeis ponderibus in melodiis mixtus est.&lt;ref&gt;[http://www.iasorecords.com/music/palo-drum-afro-dominican-tradition Palo Drum: Afro-Dominican Tradition] (Anglice). iasorecords.com.&lt;/ref&gt;

Musica [[salsa]] plurimam popularitatem hac in natione habuit. Exeuntibus annis 1960 Dominiciani musici sicuti [[Ioannes Pacheco]], gregis [[Fania All-Stars]] creator, significativo munere in evolvendo pervulgandoque hoc genere functi sunt.

[[Musica rockiana]] Dominiciana etiam popularis est. Multi, si non plerique, eius cantorum Dominicopolin et [[Sanctus Iacobus Equitum|Sanctum Iacobum]] incolunt.

=== Moda ===
[[Fasciculus:Oscar de la Renta by foto di matti.jpg|thumb|upright=0.9|Dominicianus autochthon, modae designator et liquoris odorati artifex [[Anscharius de la Renta]].]]
Natio se iactat quia habet una e decem insignissimis designationis scholis in regione, La Escuela de Diseño de Altos de Chavón, quae hanc nationem praecipuum actorem in [[moda]]e et designationis mundo facit. Notus modae designator [[Anscharius de la Renta]] in Re publica Dominiciana anno 1932 natus et [[Statunitensis]] civis anno 1971 factus est. Ducente Hispano designatore [[Christophorus Balenciaga|Christophoro Balenciaga]] et postea cum domo [[Lanvin]] [[Lutetia]]e laboravit. Anno 1963, proprium pittacium habens designavit. Suo domicilio in Civitatibus Foederatis posito, [[Anscharius de la Renta]] tabernulas exquisitas per nationem aperuit. Cuius opus Francicam Hispanicamque modam Statunitensibus cum generibus commiscet.&lt;ref&gt;Fashion: Oscar de la Renta (Dominican Republic) (Anglice). Archivatum die [[16 Ianuarii]] anni 2013, apud Wayback Machine. WCAX.com – Inspectum [[31 Octobris|pridie Kalendas Novembres]] anni 2012.&lt;/ref&gt;&lt;ref name=&quot;De la Renta&quot;&gt;[https://www.britannica.com/biography/Oscar-de-la-Renta Oscar de la Renta] (Anglice). [[Encyclopædia Britannica]]. Inspectum [[31 Octobris|pridie Kalendas Novembres]] anni 2012.&lt;/ref&gt; Quamquam suum domicilium Novi Eboraci posuit, Anscharius de la Renta etiam suum opus in [[America Latina]] promovit et activus in sua autochthonica Re publica Dominiciana mansit, ubi caritativae activitates et personales adeptiones ei Ordinem Meriti Ioannis Pauli Duarte et Ordinem Christophori Columbi meritae sunt.&lt;ref name=&quot;De la Renta&quot;&gt;[https://www.britannica.com/biography/Oscar-de-la-Renta Oscar de la Renta] (Anglice). [[Encyclopædia Britannica]]. Inspectum [[31 Octobris|pridie Kalendas Novembres]] anni 2012.&lt;/ref&gt; Anscharius de la Renta [[Cancer (morbus)|cancri]] implicationibus die 20 Octobris anni 2014 mortuus est.

=== Nationalia symbola et dies festi ===
[[Fasciculus:Pereskia quisqueyana.JPG|thumb|upright=0.7|''[[Pereskia quisqueyana]].'']]
Aliquae e praecipuis Dominicianis symbolis [[vexillum]]; [[insigne]]; et [[hymnus nationalis]], ''[[Hymnus Nationalis Rei publicae Dominicianae|Himno Nacional]]'' intitulatus, sunt. Vexillum magnam albam crucem id in partes quattuor dividentem habet. Duae partes rubrae et duae cyaneae sunt. Ruber color sanguinem a liberatoribus fusum repraesentat. Cyaneus Dei protectionem nationi exprimit. Alba crux liberatorum pugnam ad relinquendam futuris aetatibus liberam nationem symbolizat. Alternativa interpretatio est ut cyaneus desideratum progressus libertatisque repraesentet, dum albus pacem unitatemque apud Dominicianos symbolizet.&lt;ref&gt;[https://web.archive.org/web/20090113181033/http://www.ejercito.mil.do/index.php?option=com_content&amp;task=view&amp;id=165&amp;Itemid=132 &quot;Ejército Nacional de la República Dominicana – Bandera Nacional&quot;] (Hispanice). Exercitus Nationalis Rei Publicae Dominicianae. Archivatum ex originali die 13 Ianuarii anni 2009. Recuperatum die 20 Octobris anni 2008.&lt;/ref&gt;

Media in cruce est insigne Dominicianum, eisdem coloribus quam in vexillo nationali. Insigne rubrum, album cyaneumque scutum vexillo indutum cum Bibliis, aurea cruce et sagittis pingit; scutum oleae ramo (a sinistra) et arecaceo ramo (a dextera) circumdatur. Biblia traditionaliter veritatem lucemque repraesentant. Aurea crux redemptionem e servitudine repraesentat et sagittae nobiles milites celsasque res militares symbolizant. Cyanea taeniola super scuto dicit &quot;''Dios, Patria, Libertas''&quot; (scilicet &quot;Deum, Patriam, Libertatem&quot;). Rubra taeniola sub scuto dicit, &quot;''República Dominicana''&quot; (scilicet &quot;Rem publicam Dominicianam&quot;). Ex omnibus mundi vexillis, Bibliorum depictio unica vexillo Dominiciano est.

[[Flos nationalis]] ''[[Pereskia quisqueyana]] et [[arbor nationalis]] ''[[Swietenia mahagoni]]'' est.&lt;ref&gt;López, Yaniris (die 17 Iulii anni 2011). [https://www.listindiario.com/la-vida/2011/7/16/196080/La-rosa-de-Bayahibe-nuestra-flor-nacional &quot;La rosa de Bayahíbe, nuestra flor nacional&quot;] (Hispanice). ''Listin Diario''.&lt;/ref&gt; [[Avis nationalis]] ''[[Dulus dominicus]]'' est.&lt;ref&gt;Pérez, Faustinus (Faustino). &quot;El jardín Botánico Nacional&quot;. ''DiarioDigitalRD.com'' (Hispanice). Archivatum ex originali die 23 Octobris anni 2008. Inspectum die 20 Octobris anni 2008.&lt;/ref&gt;

Res publica Dominiciana Sollemnitatem [[Domina Nostra de la Altagracia|Dominae Nostrae de la Altagracia]]&lt;ref name=&quot;Domina&quot;&gt;[http://www.catholic-hierarchy.org/diocese/dnuse.html catholic-hierarchy.org] ubi apparet &quot;Dioecesis A Domina Nostra vulgo de la Altagracia in Higüey, seu Higueyensis&quot;.&lt;/ref&gt; (Hispanice ''Día de la Altagracia'') die [[21 Ianuarii]] ad honorem sanctae patronae; Diem [[Juan Pablo Duarte|Ioannis Pauli Duarte]] die 26 Ianuarii ad honorem unius e patribus conditoribus; Diem Independentiae die 27 Februarii; Diem Restitutionis die 16 Augusti; Sollemnitatem [[Nostra Domina Mercedis|Nostrae Dominae Mercedis]] (Hispanice ''Día de la Altagracia'') die 24 Septembris; et Diem Constitutionis die 6 Novembris celebrat.

=== Artes athleticae ===
[[Fasciculus:DSC00621 Albert Pujols.jpg|thumb|upright=0.7|Dominicianus autochthon et [[Maioris Ligae Basipila]]e (Anglice ''Major League Baseball'' seu ''MLB'') lusor [[Albertus Pujols]].]]
[[Basipila]] longe popularissima ars athletica in Re publica Dominiciana est.&lt;ref name=&quot;Harvey&quot;&gt; Harvey, Ioannes (Sean) (2006). ''The Rough Guide to The Dominican Republic'' (Anglice). Rough Guides. ISBN 1-84353-497-5.&lt;/ref&gt; Natio basipilae ligam cum turmis sex habet. Cuius tempus ex more Octobri mense incipit et Ianuario finit. Post Civitates Foederatas, Res publica Dominiciana secundum maximum numerum lusorum in [[Maioris Ligae Basipila]] (Anglice ''Major League Baseball'' seu ''MLB'') habet. [[Olvadus Virgil Senior]] primus lusor in MLB in Re publica Dominiciana natus die 23 Septembris anni 1956 factus est. [[Ioannes Marichal]] et [[Petrus Martínez]] unici lusores in Re publica Dominiciana nati in [[Aula Famae Basipilae]] (Anglice ''Baseball Hall of Fame'') sunt.&lt;ref&gt;&quot;Marichal, Juan&quot; (Anglice). ''Aula Famae Basipilae''. Inspectum die 19 Iulii anni 2010.&lt;/ref&gt; Apud claros basipilae lusores in Re publica Dominiciana natos [[Hadrianus Beltré]], [[Robinson Canó]],[[Ricardus Carty]], [[Starling Marte]], [[Vladimirus Guerrero]], [[Georgius Antonius Bell]], [[Iulianus Javier]], [[Franciscus Liriano]], [[Manuel Ramírez]], [[Iosephus Bautista]], [[Edwinus Encarnación]],[[Hanleyus Ramírez]], [[David Ortiz]], [[Albertus Pujols]], [[Nelson Cruz]], [[Ubaldus Jiménez]], [[Iosephus Reyes (obstructor citerior)|Iosephus Citerior]], [[Placidus Polanco]] et [[Samuel Sosa]] sunt. [[Philippus Alou]] quoque successu sicuti procurator&lt;ref&gt;Puesan, Antonius (Antonio) (die 2 Martii anni 2009). [https://web.archive.org/web/20130116214549/http://www.sobreeldiamante.com/dominicana-busca-corona-en-clasico-mundial.html &quot;Dominicana busca corona en el clásico mundial&quot;] (Hispanice). Sobre el Diamante. Archivatum ex originali die 16 Ianuarii anni 2013. Inspectum die 22 Octobris anni 2012.&lt;/ref&gt; et [[Omar Minaya]] sicuti procurator generalis fructus est. Anno 2013, turma Dominiciana invicta ivit in via vincendi in [[Classicum certamen mundanum basipilae|Classico certamine mundano basipilae]] anni 2013.

In [[Pugilatio (moderna)|pugilatione]], natio plurimos pugiles in toto orbe terrarum notos et aliquot principes in toto terrarum orbe victores protulit,&lt;ref&gt;Fleischer, Nat; Sam Andre; Don Rafael (2002). ''An Illustrated History of Boxing'' (Anglice). Citadel Press. pp. 324, 362, 428. ISBN 0-8065-2201-1.&lt;/ref&gt; tales [[Carolus Cruz (pugil)|Carolus Cruz]], eius frater [[Leonardus Cruz|Leonardus]], [[Ioannes Antonius Guzmán (pugil)|Ioannes Antonius Guzmán]] et [[Ioannes Guzmán (pugil)|Ioannes Guzmán]]. [[Canistriludium]] quoque relative alto popularitatis gradu fruitur. [[Tito Horford]], eius filius [[Al Horford|Al]], [[Philippus López (canistrilusor)|Philippus López]] et [[Franciscus García (canistrilusor)|Franciscus García]] sunt apud Dominicianos natos lusores in praesenti aut olim in [[Sodalitas Nationalis Canistriludii|Sodalitate Nationali Canistriludii]] (Anglice ''National Basketball Association'' seu ''NBA''). Olympionices cum clipeo aureo et princeps in toto terrarum orbe victor [[Cursus impedimentorum|cursor impedimentorum]] [[Felix Sánchez]] est e Re publica Dominiciana, velut [[Foedus Pediludii Nationale|Foederis Pediludii Nationalis]] (Anglice ''National Football League'' seu ''NFL'') cornu defensivum [[Ludovicus Castillo (harpastum Americanum)|Ludovicus Castillo]].&lt;ref&gt;Shanahan, Thomas (Tom) (die 24 Martii anni 2007). [https://web.archive.org/web/20070505132520/http://www.sdhoc.com/main/articles/sportsatlunch/Sportsatlunch2007/Sanchezcastillo &quot;San Diego Hall of Champions – Sports at Lunch, Luis Castillo and Felix Sanchez&quot;] (Anglice). Aula Campionum Didacopolitana. Archivatum ex originali die 5 Maii anni 2007. Inspectum die 29 Maii anni 2007.&lt;/ref&gt;

Aliae praecipuae artes athleticae [[follis volatilis]], anno 1916 Statunitensibus a classiariis introductus et a [[Foederatio Dominiciana Follis Vollatilis|Foederatione Dominiciana Follis Vollatilis]] (Hispanice ''Federación Dominicana de Voleibol'' seu ''FEDOVOLI'') rectus; [[taequondo]], ubi [[Gabriel Mercedes]] Olympicam medaliam argenteam anno 2008 adeptus est; et [[iudo]] sunt.&lt;ref&gt;[https://web.archive.org/web/20101206101315/http://www.fedojudo.org/fedojudo/index.cfm &quot;Fedujudo comparte con dirigentes provinciales&quot;] (Hispanice). fedojudo.org. Archivatum ex originali die 6 Decembris anni 2010. Inspectum die 15 Septembris anni 2010.&lt;/ref&gt;

== Codices ==
Rei publicae Dominicianae hi codices sunt:
* DOM, secundum normam [[ISO 3166-1]] alpha-3 (catalogi codicum nationum);
* DOM, secundum [[Catalogus codicum nationum Consilii Olympici Internationalis|catalogum codicum nationum Consilii Olympici Internationalis]];
* DOM, secundum [[Catalogus codicum internationalium notaculi autocineti|catalogum codicum internationalium notaculi autocineti]];
* DOM, secundum [[Catalogus codicum nationum quibus OTAN utitur|catalogum codicum nationum quibus OTAN utitur]] codicem alpha-3;
* DR, secundum [[Catalogus codicum nationum quibus OTAN utitur|catalogum codicum nationum quibus OTAN utitur]];
* DO, secundum normam [[ISO 3166-1]] (catalogi codicum rerum publicarum), codicem alpha-2 (catalogi codicum civitatis);
* DO, secundum [[Dominium summum|dominium interretiale]];

== Coniunctio communium ==
* [[Miamia]] ([[CFA]])
* [[Sarasota]] ([[CFA]])
* [[Novum Eboracum (urbs)|Novum Eboracum]] ([[CFA]]) (anno 1983)
* [[Matritum]] ([[Hispania]])

== Notae ==
&lt;references/&gt;

== Nexus externi ==
{{Fontes geographici}}
{{CommuniaCat|Dominican Republic|Rem publicam Dominicianam}}
* [http://www.presidencia.gob.do/ Praesidatus Rei publicae Dominicanae] (Hispanice)
* [http://www.dominicanrepublic.com/ Situs interretialis officialis huius nationis] (Anglice, Hispanice, Francice, Italiane, Theodisce, Russice et Arabice)
* [https://web.archive.org/web/20080705060056/http://ucblibraries.colorado.edu/govpubs/for/dominicanrepublic.htm Res publica Dominiciana] apud ''UCB Libraries GovPubs'' (Anglice)
* [http://news.bbc.co.uk/2/hi/americas/country_profiles/1216926.stm Rei publicae Dominicianae descriptio] e [[BBC News]] (Anglice)
* [http://www.godominicanrepublic.com/ Situs interretialis officialis Ministerii Periegeseos Rei publicae Dominicianae] (Anglice, Hispanice, Lusice, Francice, Italiane, Theodisce, Russice et Sinice)
* [http://www.iddi.org/ Situs interretialis officialis IDDI seu Instituti Dominiciani Progressus Integri]
* [https://socialjusticebooks.org/store/dominican-republic/ Conexus Caribici: Res publica Dominiciana] didascalius libellus mediae scholae et lycei studentibus (Anglice)

{{America}}
{{Capsae collectae|Res publica Dominiciana|politica|{{Praesides rei publicae Dominicanae}}{{Cancellarii rei publicae Dominicanae}}}}

{{Myrias|Geographia}}
{{FA stella}}

[[Categoria:Condita 1844]]
[[Categoria:Civitates sui iuris|Dominiciana res]]
[[Categoria:Res publica Dominiciana|!]]</text>
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  <page>
    <title>Franciscus Drakus</title>
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      <text bytes="14075" xml:space="preserve">[[Fasciculus:1590 or later Marcus Gheeraerts, Sir Francis Drake Buckland Abbey, Devon.jpg|thumb|Effigies Francisci Draki a [[Marcus Gheeraerts|Marco Gheeraerts]] post annum 1590 picta, apud [[Abbatia de Boclanda|Abbatiam de Boclanda]] ostentata]]
[[Fasciculus:Sir Francis Drake And His Coat Of Arms.gif|thumb|AUXILIO DIVINO - SIC PARVIS MAGNA]]
'''Franciscus Drakus''',&lt;ref&gt;Nomen gentilicium varie scribitur:
* '''Drake''': 1582: ''Aliqvot notae in Garciae Aromatum historiam. Eivsdem descriptiones non-nullarum stirpium, &amp; aliarum exoticarum rerum, que à generoso viro Francisco Drake equite anglo, &amp; his obseruatae sunt, qui cum in longa illa nauigatione,qua proximis annis vniuersum orbem circumiuit, comitati sunt: &amp; quorundam peregrinorum fructuum quos Londini ab amicis accepit.''
* '''Dracus''': 1587: [[Ioannes Hercusanus|Hercusani]] [http://eebo.chadwyck.com.proxy.uchicago.edu/search/full_rec?SOURCE=pgimages.cfg&amp;ACTION=ByID&amp;ID=V21198 ''Magnifico ac strenuo viro D. Francisco Draco Anglo equiti aurato.''] Carmen.
* '''Drakus''': (1) 1588: [http://memory.loc.gov/cgi-bin/ampage?collId=rbdk&amp;fileName=d020//rbdkd020.db&amp;recNum=0&amp;itemLink=r?intldl/rbdkbib:@field(NUMBER+@band(rbdk+d020))&amp;linkText=0 ''Expeditio F. Draki in Indias occidentales.''] (2) 1589: [http://eebo.chadwyck.com.proxy.uchicago.edu/search/full_rec?SOURCE=pgimages.cfg&amp;ACTION=ByID&amp;ID=V13649 ''Ephemeris expeditionis Norreysij &amp; Draki in Lusitaniam.'']
* '''Draken''': 1599: [[Theodorus a Bry|Theodoro à Bry]] compositore, [[Gotardus Artus|Gotardi Artus]] ''Americae pars VIII: Continens primo, descriptionem trivm itinervm Francisci Draken, qvi peragrato primvm vniverso terrarvm orbe, postea cum . . . Iohanne Havckens, ad expugnandum ciuitatem Panama, in Indiam nauigauit . . . Secvndo, iter . . . Thomae Candisch . . . Tertio, duo itinera . . . Gvaltheri Ralegh . . . nec non . . . capitanei Lavrentii Keyms. Qvibvs . . . describitvr . . . regnum Gviana . . . Primo Anglicana lingva sparsim consignata: iam verò in vnum corpus redacta, &amp; in Latinum sermonem conuersa, auctore Gotardo Artvs.''
* '''Draco''': ibidem pp. 3. 4.
* '''Draeck''': circa 1595: [[Iodocus Hondius|Hondii]] [http://memory.loc.gov/cgi-bin/ampage?collId=rbdk&amp;fileName=d055/rbdkd055.db&amp;recNum=0&amp;itemLink=r?intldl/rbdkbib:@field(DOCID%2B@lit(rbdk000058)) imago Francisci Draek:] ''FRANCISCVS DRAECK NOBILIMVS EQVES ANGLIÆ. AN° ÆT. SVE 45.&quot;&lt;/ref&gt; [[Lingua Anglica|Anglice]] ''Sir Francis Drake'' (natus circa 1540 [[Tavistochia]]e; mortuus prope [[Portus Bellus|Portum Bellum]] [[Panama]]e die [[28 Ianuarii]] [[1596]]) fuit [[cursarius]] [[nauta|navigatorque]], primus [[Anglia|Anglorum]] qui orbem terrarum [[circumnavigatio|circumnavigaverit]].

Circumnavigationem anno 1577 suscepit, die [[26 Septembris]] [[1580]] perfecit.

== Notae ==
&lt;references/&gt;

== Bibliographia ==
; Fontes fere coaevi
* 1582 : &lt;span id=&quot;Clusius (1582)&quot;&gt;&lt;/span&gt;Carolus Clusius, ''Aliquot notae in Garciae Aromatum historiam; Descriptiones nonnullarum stirpium, et aliarum exoticarum rerum, quae à ... Francisco Drake ... observatae sunt'' (Antverpiae: Plantin) [https://www.e-rara.ch/zuz/nagezh/content/titleinfo/9431363 Textus] apud ''e-Rara'' {{Google Books|sF5WAAAAcAAJ}} [https://archive.org/details/Caroli-Clusii-Atreb-Aliquot-notae-in-Garciae-Aromatum-historiam-Eiusdem-Descript-PHAIDRA_o_383729/mode/2up apud ''Internet Archive'']
* c. 1587 : ''Vera descriptio expeditionis nauticae Francisci Draci Angli, cognitis aurati, circa 1587'' [tabula geographica manu scripta, ''Drake-Mellon map'' nuncupata]. [https://collections.britishart.yale.edu/catalog/orbis:9579023 imago]
* 1588 : &lt;span id=&quot;Bigges (1588)&quot;&gt;&lt;/span&gt;Walterus Bygges, ''Expeditio Francisci Draki equitis Angli in Indias Occidentales anno MDLXXXV''. Leydae: apud Fr. Raphelengium [http://hdl.loc.gov/loc.rbc/rbdk.d020 Textus] [https://archive.org/details/narrationesduadm00bigg Editio 1590] [http://www.philological.bham.ac.uk/bigges/ Editio interretialis] [http://www.ancienttexts.org/library/latinlibrary/biggs.html Editio] apud ''[[The Latin Library]]'' [https://archive.org/details/levoyagedemessir00bigg versio Francogallica 1588] [https://archive.org/details/relationoderbesc00bigg Versio Theodisca 1589] Versio Anglica:
** 1589 : Walter Bygges; Thomas Cates, ed., ''A Summarie and True Discourse of Sir Frances Drakes West Indian Voyage wherein were taken the townes of Saint Iago, Sancto Domingo, Cartagena and Saint Augustine''. Londinii: Roger Ward [http://hdl.loc.gov/loc.rbc/rbdk.d022 Textus] 
** 1589 : Walter Bygges; Thomas Cates, ed., ''A Summarie and True Discourse of Sir Frances Drakes West Indian Voyage where in were taken the townes of Saint Iago, Sancto Domingo, Cartagena &amp; Saint Augustine''. Londinii: Richard Field [https://wellcomecollection.org/works/uvg54jd9 Textus cum tabulis?] [https://quod.lib.umich.edu/e/eebo/a68945.0001.001?rgn=main;view=fulltext editio interretialis]; [https://quod.lib.umich.edu/e/eebo/A68946.0001.001?view=toc alia]; [https://archive.org/details/summarietruedisc00bigg Editio 1596] [https://archive.org/details/summarieandtrued00biggrich editio 1652]
* 1589 : ''Ephemeris expeditionis Norreysii &amp; Draki in Lusitaniam''. Londinii, 1589 [https://archive.org/details/ephemerisexpedit00unse Textus] apud ''Internet Archive'' [https://archive.org/details/narrationesduadm00bigg editio 1590]
* 1589 : &quot;[https://archive.org/details/cihm_35668/page/n631/mode/1up  The first voyage attempted and set foorth by ... M. Francis Drake ... to Nombre de Dios ... 1572]&quot; (p. 594 sqq.), &quot;[https://archive.org/details/cihm_35668/page/n684/mode/1up The famous voyage of Sir Francis Drake into the South Sea, and there hence about the whole globe of the Earth]&quot; (post p. 643 insertam), in [[Ricardus Hakluytus|Richard Hakluyt]], ''The Principall Navigations, Voiages and Discoveries of the English Nation, made by sea or over land, to the most remote and farthest distant quarters of the earth at any time within the compasse of these 1500. yeeres''. 1a ed. Londinii: George Bishop and Ralph Newberie
* c. 1595 : [[Iodocus Hondius]], ''Vera totius expeditionis nauticae descriptio D. Franc. Draci ...'' [tabula geographica]. Amstelodami? [[:Fasciculus:Vera totius expeditionis nauticæ - descriptio D. Franc. Draci ... LOC 92680608.jpg|Imago apud Communia]]
* 1598-1600 : &quot;[https://quod.lib.umich.edu/e/eebo/A02495.0001.001/1:84.6?rgn=div2;view=fulltext A briefe relation of the notable seruice performed by Sir Francis Drake ... in the Road of Cadiz ... 1587]&quot;, &quot;[https://quod.lib.umich.edu/e/eebo/A02495.0001.001/1:123?rgn=div1;view=fulltext The first and second discouery of the gulfe of California]&quot;, &quot;[https://quod.lib.umich.edu/e/eebo/A02495.0001.001/1:126.1?rgn=div2;view=fulltext The first voyage attempted and set foorth by ... M. Francis Drake ... to Nombre de Dios ... 1572]&quot;, &quot;[https://quod.lib.umich.edu/e/eebo/A02495.0001.001/1:126.5?rgn=div2;view=fulltext A summarie and true discourse of sir Francis Drakes West Indian voyage ... 1585]&quot;, &quot;[https://quod.lib.umich.edu/e/eebo/A02495.0001.001/1:140.7?rgn=div2;view=fulltext The voyage truely discoursed, made by sir Francis Drake and sir Iohn Hawkins ... 1595]&quot;, &quot;[https://quod.lib.umich.edu/e/eebo/A02495.0001.001/1:161?rgn=div1;view=fulltext The two famous voyages happily perfourmed round about the world, by Sir Francis Drake, and M. Thomas Candish Esquire ... The voyage of M. Iohn Winter into the South sea by the Streight of Magellan ... by Edward Cliffe (etc.)]&quot; in [[Ricardus Hakluytus|Richard Hakluyt]], ''The Principal Navigations, Voiages, Traffiques and Discoveries of the English Nation, made by sea or over-land, to the remote and farthest distant quarters of the earth, at any time within the compasse of these 1600 yeres''. 2a ed. 3 voll. Londinii: George Bishop, Ralph Newberie and Robert Barker [https://archive.org/details/principalnavigat1and2hakl voll. 1-2] [https://archive.org/details/cihm_94220 vol. 3]
* saec. XVII ineunte? : Nicola van Sype, ''La [heroike] enterprinse faict par le Signeur Draeck d'avoir cirquit toute la terre'' [tabula geographica]. Antverpiae? [http://hdl.loc.gov/loc.gmd/g3201s.rb000011 Imago]
* 1626 : Philip Nichols; {{Creanda|en|Sir Francis Drake, 1st Baronet|Franciscus Drake (baronettus I)|Francis Drake (nepos)}}, ed., ''Sir Francis Drake revived''. Londinii: Bourne, 1626 [https://archive.org/details/sirfrancisdrakea00quin/page/n17/mode/2up? Titulus huius editionis]
* 1628 : {{Creanda|en|Sir Francis Drake, 1st Baronet|Franciscus Drake (baronettus I)|Francis Drake (nepos)}}, ed., ''The World Encompassed by Sir Francis Drake''. Londinii: Bourne, 1628 [https://www.wdl.org/en/item/624/ Textus]; [http://hdl.loc.gov/loc.rbc/rbdk.d042 Textus] apud Bibliotheca Congressus; [http://nationalhumanitiescenter.org/pds/amerbegin/contact/text5/drake.pdf Editio interretialis]
* 1652-1653 : Philip Nichols, Francis Fletcher et al.; R. D., ed., ''Sir Francis Drake revived; The world encompassed by Sir Francis Drake; A summarie and true discourse; A full relation of another voyage into the West Indies''. Londinii: Bourne, 1652-1653 [https://archive.org/details/sirfrancisdraker00bourrich/page/n5/mode/2up fasc. 1 i] [https://archive.org/details/sirfrancisdraker00nichrich/page/n5/mode/2up fasc. 1 ii] [https://archive.org/details/sirfrancisdraker00nichrich/page/n5/mode/2up fasc. 2] [https://archive.org/details/summarieandtrued00biggrich fasc. 3] [https://archive.org/details/fullrelationofan00bourrich fasc. 4]

; Editiones fontium
* Mary Frear Keeler, ed., ''Sir Francis Drake's West Indian Voyage, 1585-86''. Londinii, 1981 {{Google Books|DhWfsDLFC34C|Paginae selectae}}
* W. D. Cooley, ed., ''Sir Francis Drake his Voyage, 1595, by Thomas Maynarde, together with the Spanish account of Drake's attack on Puerto Rico''. Londinii: Hakluyt Society, 1849 [http://www.mdz-nbn-resolving.de/urn/resolver.pl?urn=urn:nbn:de:bvb:12-bsb10465824-6 Textus] apud Monacenses
* Zelia Nuttall, ed., ''New Light on Drake: a collection of documents relating to his voyage of circumnavigation''. Londinii: Hakluyt Society, 1914 [https://archive.org/details/newlightondrakec34nuttuoft Textus] apud ''Internet Archive''
* &lt;span id=&quot;O'Brian et al. (1996)&quot;&gt;&lt;/span&gt;Patrick O'Brian et al., edd., ''The Drake Manuscript''. Londinii: André Deutsch, 1996 (editio fac-simile manuscripti ''[[Histoire naturelle des Indes]]'') 
* W. S. W. Vaux, ed., ''The World Encompassed by Sir Francis Drake ... collated with an unpublished manuscript of Francis Fletcher''. Londinii: Hakluyt Society, 1854 [https://archive.org/details/worldencompassed16drak Textus] apud ''Internet Archive'' 
* R. B. Wernham, ed., ''The expedition of Sir John Norris and Sir Francis Drake to Spain and Portugal, 1589'' (Navy Records Society). Londinii: Temple Smith, 1988

; Eruditio
* Kenneth R. Andrews, ''Drake's voyages: a re-assessment of their place in Elizabethan maritime expansion''. Novi Eboraci: Scribner, 1968 [https://archive.org/details/drakesvoyagesrea00andr exemplar mutuabile]
* Laurence Bergreen, ''In Search of a Kingdom: Francis Drake, Elizabeth I, and the Perilous Birth of the British Empire''. Custom House, 2021. ISBN 978-0062875358 [http://www.washingtonindependentreviewofbooks.com/index.php/bookreview/in-search-of-a-kingdom-francis-drake-elizabeth-i-and-the-perilous-birth-of-the-british-empire recensio huius operis]
* Julian S. Corbett, ''Drake and the Tudor navy, with a history of the rise of England as a maritime power''. 2 voll. Londinii: Longmans Green, 1898 [https://archive.org/details/draketudornavy01corb vol. 1] [https://archive.org/details/draketudornavy02corb vol. 2] apud ''Internet Archive'' 
* James B. Davidson, &quot;[https://archive.org/details/westernantiquar06wriggoog/page/n173/mode/2up On some points in natural history first made known by Sir Francis Drake]&quot; in ''Western Antiquary'' vol. 4 (1884/1885) pp. 134-137
* Harry Kelsey, &quot;Drake, Sir Francis (1540–1596)&quot; in {{ODNB}}
* Harry Kelsey, ''Sir Francis Drake: The Queen's Pirate''. Novo Portu: Yale University Press. 2000. ISBN 978-0-300-08463-4
* Hans P. Kraus, ''Sir Francis Drake: A Pictorial Biography''. Amsterdam: N. Israel, 1970 [https://www.loc.gov/rr/rarebook/catalog/drake/index.html Editio interretialis]
* David Beers Quinn, ''Sir Francis Drake as seen by his contemporaries: an essay''. Providentiae: John Carter Brown Library, 1996 [https://archive.org/details/sirfrancisdrakea00quin Textus] apud ''Internet Archive''
* John Sugden, ''Sir Francis Drake''. Londinii: Barrie &amp; Jenkins, 1990 {{Google Books|2CEgmN-3VcMC|Paginae selectae editionis recentioris}}
* Norman J. W. Thrower, &quot;Drake on the Pacific Coast of North America&quot; in Cecil H. Clough, P. E. H. Hair, edd., ''The European Outthrust and Encounter: The First Phase c.1400-c.1700: essays in tribute to David Beers Quinn'' (Liverpolii: Liverpool University Press, 1994) pp. 167-190 {{Google Books|Dg-8ZOeBqcYC|paginae selectae}}
* Norman J. W. Thrower, ''Sir Francis Drake and the Famous Voyage, 1577-1580: Essays Commemorating the Quadricentennial of Drake's Circumnavigation of the Earth''. Berkeleiae: University of California Press, 1984 {{Google Books|tPJVxZu8btoC|Paginae selectae}}

== Nexus externi ==
* ''[http://www.indrakeswake.co.uk/Society/index.htm The Drake Exploration Society]''
** &quot;[http://www.indrakeswake.co.uk/Society/research.htm The society's research]&quot;
* John Thrower, &quot;[http://www.indrakeswake.co.uk/Society/Research/botanical.htm Some Botanical Discoveries Made on Francis Drake's World Voyage]&quot; 
* Martha Doerr Toppin, &quot;[https://open.conted.ox.ac.uk/sites/open.conted.ox.ac.uk/files/resources/Create%20Document/%28Pages%204-13%29%20MARTHA%20%28Group%201%29.pdf Nova Albion Revealed: Drake’s claim of California celebrated by Hakluyt and Hondius]&quot;

{{bio-stipula}}

{{DEFAULTSORT:Drakus, Franciscus}}
[[Categoria:Exploratores Britannici]]
[[Categoria:Exploratores Californiae]]
[[Categoria:Exploratores Oceani Pacifici]]
[[Categoria:Exploratores Oregoniae]]
[[Categoria:Nati saeculo 16]]
[[Categoria:Mortui 1596]]
[[Categoria:Elisabetha I (regina Angliae)]]

{{Myrias|Homines}}</text>
      <sha1>lkdp8c5y0a72t2yzrnl9f2y6rsz2s15</sha1>
    </revision>
  </page>
  <page>
    <title>Navis actuaria</title>
    <ns>0</ns>
    <id>9388</id>
    <revision>
      <id>3697740</id>
      <parentid>3113652</parentid>
      <timestamp>2022-08-17T10:13:19Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Naves]]; +[[Categoria:Genera navium]]</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="768" xml:space="preserve">'''Navis actuaria''' est [[navis]] quae celeriter movetur.

== Antiquitates ==
Verbum ''Navis actuaria'' ad genera navium remis et velis actarum apud [[Classis|classem]] [[Imperium Romanum|Romanam]] usitatum fuit. Designavit naves [[Navis oneraria|onerarias]] et alias.

== Aevo moderno ==
Inter naves modernas variae formae celeriter moventes inveniunter qualibus hoc verbum idoneum esse videtur (e.g. [[Anglice]] ''hydrofoil'').

{{NexInt}}
*[[Navis]]
*[[Navigatio]]

== Bibliographia ==
* Emil Luebeck: ''Actuariae''. In: Paulys Realencyclopädie der classischen Altertumswissenschaft tomus I,1, (Stutgardiae 1893), 331.
* Ioannes Viereck: ''Die römische Flotte. Classis romana'' (Hamburgum 1996).

[[Categoria:Genera navium|actuaria]]
[[Categoria:Classis Romana]]</text>
      <sha1>i63fhb64pvp8ftkrpi8w6hzu8kvu5gf</sha1>
    </revision>
  </page>
  <page>
    <title>Navis oneraria</title>
    <ns>0</ns>
    <id>9390</id>
    <revision>
      <id>3697747</id>
      <parentid>3181676</parentid>
      <timestamp>2022-08-17T10:14:43Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Naves]]; +[[Categoria:Genera navium]]</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="467" xml:space="preserve">[[Image:Lyttelton New Zealand 1968.jpg|thumb|Navis oneraria]]
'''Navis oneraria''' est [[navis]] onera portans. Ordinarie [[commercium|commercio]] adhibetur ut [[merx|merces]] transferat quae vendantur, alias tamen et praecipue antiquis temporibus [[navis bellica|naves bellicas]] comitabat, [[cibus|cibos]], onera utiliaque pro iis ferens.           

{{NexInt}}
*[[Navis vincta]]
*[[Societas mercium navigandarum]]

{{stipula}}

[[Categoria:Genera navium|oneraria]]</text>
      <sha1>5nwymajxaoz6yscabm6kem409tswwc0</sha1>
    </revision>
  </page>
  <page>
    <title>Guinea Bissaviensis</title>
    <ns>0</ns>
    <id>10457</id>
    <revision>
      <id>3697703</id>
      <parentid>3347911</parentid>
      <timestamp>2022-08-16T20:28:48Z</timestamp>
      <contributor>
        <ip>84.78.253.101</ip>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2062" xml:space="preserve">{{Capsa civitatis Vicidata}}
[[Fasciculus:Guinea bissau sm03.png|thumbnail|200 px|Tabula Guineae Bissaviensis]]

'''Guinea Bissaviensis'''&lt;ref&gt;&quot;Guinea Bissaviensis&quot;: ''[http://www.gcatholic.org/dioceses/country/GW.htm Giga-Catholic]''&lt;/ref&gt; seu '''Guinea-Bissavia'''&lt;ref&gt;[http://www.vatican.va/news_services/press/sinodo/documents/bollettino_25_xiii-ordinaria-2012/xx_plurilingue/b01_xx.html Bollettino Synodus Episcoporum]&lt;/ref&gt; est civitas in [[Africa]], cuius caput est [[urbs]] [[Bissavia]]&lt;ref&gt;&quot;Bissavia, (-ae, ''f'')&quot;: a Petro Lucusaltiano Latinophilo, [http://www.lateinlexikon.com ''Lexicon Latinum Hodiernum vel Vocabularium Latinitatis Huius Aetatis,''] apud www.lateinlexikon.com.&lt;/ref&gt; seu '''Urbs Bissagensis'''&lt;ref&gt;[https://www.catholic-hierarchy.org/country/dgw.html Hierarchia Catholica].&lt;/ref&gt; ([[Lusice]] ''Bissau''). Guineae Bissaviensi finitimae civitates sunt: [[Guinea]] et [[Senegalia]].

Lingua sollemnis Guineae Bissaviensis est [[Lingua Lusitana|sermo Lusitanus]].

== Geographia ==
=== Flumina ===
* [[Geba flumen|Geba]]

== Historia ==

Guinea Bissaviensis fuit a [[saeculum|saeculo]] [[saeculum 15|XV]], accurate ab anno 1446 usque ad annum [[1974]], [[colonia]] [[Lusitania]]e. At die [[24 Septembris]] [[1973]] suam [[independentia]]m declaravit et anno 1974 admissa est.
== Notae ==
&lt;div class=&quot;references-small&quot;&gt;&lt;references /&gt;&lt;/div&gt;

== Nexus externi ==
{{Communia|Guiné-Bissau|Guineam Bissaviensem}}

* [http://www.abenaa.de/guineabissau.htm Narrationes peregrinationum atque imagines]
* [https://www.ensp.unl.pt/luis.graca/guine_guerracolonial10_mapageral.html Tabula geographica Guineae Portugallensis 1:500 000]

{{natio-stipula}}

{{Africa}}
{{Capsae collectae|Guinea Bissaviensis|politica|{{Praesides Guineae Bissaviensis}}{{Primi ministri Guineae Bissaviensis}}{{Ministri rerum externarum Guineae Bissaviensis}}}}
{{Capsae collectae|Guinea Bissaviensis|historica|{{PALOP}}}}

[[Categoria:Guinea Bissaviensis|!]]
[[Categoria:Condita 1975]]
[[Categoria:Respublicae]]
[[Categoria:Civitates sui iuris]]

{{Myrias|Geographia}}</text>
      <sha1>71a8mdamdituq9r5j26rmcbzjoad5vg</sha1>
    </revision>
    <revision>
      <id>3697704</id>
      <parentid>3697703</parentid>
      <timestamp>2022-08-16T20:29:23Z</timestamp>
      <contributor>
        <ip>84.78.253.101</ip>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2060" xml:space="preserve">{{Capsa civitatis Vicidata}}
[[Fasciculus:Guinea bissau sm03.png|thumbnail|200 px|Tabula Guineae Bissaviensis]]

'''Guinea Bissaviensis'''&lt;ref&gt;&quot;Guinea Bissaviensis&quot;: ''[http://www.gcatholic.org/dioceses/country/GW.htm Giga-Catholic]''&lt;/ref&gt; seu '''Guinea-Bissavia'''&lt;ref&gt;[http://www.vatican.va/news_services/press/sinodo/documents/bollettino_25_xiii-ordinaria-2012/xx_plurilingue/b01_xx.html Bollettino Synodus Episcoporum]&lt;/ref&gt; est civitas in [[Africa]], cuius caput est [[urbs]] [[Bissavia]]&lt;ref&gt;&quot;Bissavia, (-ae, ''f'')&quot;: a Petro Lucusaltiano Latinophilo, [http://www.lateinlexikon.com ''Lexicon Latinum Hodiernum vel Vocabularium Latinitatis Huius Aetatis,''] apud www.lateinlexikon.com.&lt;/ref&gt; seu [[Urbs Bissagensis]]&lt;ref&gt;[https://www.catholic-hierarchy.org/country/dgw.html Hierarchia Catholica].&lt;/ref&gt; ([[Lusice]] ''Bissau''). Guineae Bissaviensi finitimae civitates sunt: [[Guinea]] et [[Senegalia]].

Lingua sollemnis Guineae Bissaviensis est [[Lingua Lusitana|sermo Lusitanus]].

== Geographia ==
=== Flumina ===
* [[Geba flumen|Geba]]

== Historia ==

Guinea Bissaviensis fuit a [[saeculum|saeculo]] [[saeculum 15|XV]], accurate ab anno 1446 usque ad annum [[1974]], [[colonia]] [[Lusitania]]e. At die [[24 Septembris]] [[1973]] suam [[independentia]]m declaravit et anno 1974 admissa est.
== Notae ==
&lt;div class=&quot;references-small&quot;&gt;&lt;references /&gt;&lt;/div&gt;

== Nexus externi ==
{{Communia|Guiné-Bissau|Guineam Bissaviensem}}

* [http://www.abenaa.de/guineabissau.htm Narrationes peregrinationum atque imagines]
* [https://www.ensp.unl.pt/luis.graca/guine_guerracolonial10_mapageral.html Tabula geographica Guineae Portugallensis 1:500 000]

{{natio-stipula}}

{{Africa}}
{{Capsae collectae|Guinea Bissaviensis|politica|{{Praesides Guineae Bissaviensis}}{{Primi ministri Guineae Bissaviensis}}{{Ministri rerum externarum Guineae Bissaviensis}}}}
{{Capsae collectae|Guinea Bissaviensis|historica|{{PALOP}}}}

[[Categoria:Guinea Bissaviensis|!]]
[[Categoria:Condita 1975]]
[[Categoria:Respublicae]]
[[Categoria:Civitates sui iuris]]

{{Myrias|Geographia}}</text>
      <sha1>kbm5k3qu91p5058mes1smgbf5ywdey9</sha1>
    </revision>
    <revision>
      <id>3697707</id>
      <parentid>3697704</parentid>
      <timestamp>2022-08-16T21:05:12Z</timestamp>
      <contributor>
        <ip>84.78.253.101</ip>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2943" xml:space="preserve">{{Capsa civitatis Vicidata}}
[[Fasciculus:Guinea bissau sm03.png|thumbnail|200 px|Tabula Guineae Bissaviensis]]

'''Guinea Bissaviensis'''&lt;ref&gt;&quot;Guinea Bissaviensis&quot;: ''[http://www.gcatholic.org/dioceses/country/GW.htm Giga-Catholic]''&lt;/ref&gt; seu '''Guinea-Bissavia'''&lt;ref&gt;[http://www.vatican.va/news_services/press/sinodo/documents/bollettino_25_xiii-ordinaria-2012/xx_plurilingue/b01_xx.html Bollettino Synodus Episcoporum]&lt;/ref&gt;, rite ''Res publica Guinea Bissaviensis'' ([[Lusice]] ''República da Guiné-Bissau''), est civitas in [[Africa Occidentalis|Africā Occidentali], una apud parvissimas hac in continente. Finitima est [[Senegalia]]e Septentrione, [[Guinea]]e Meridiē Orienteque et [[Oceanus Atlanticus|Oceano Atlantico]] Occidente. Ante caput, [[Bissavia]]&lt;ref&gt;&quot;Bissavia, (-ae, ''f'')&quot;: a Petro Lucusaltiano Latinophilo, [http://www.lateinlexikon.com ''Lexicon Latinum Hodiernum vel Vocabularium Latinitatis Huius Aetatis,''] apud www.lateinlexikon.com.&lt;/ref&gt; seu [[Urbs Bissagensis]]&lt;ref&gt;[https://www.catholic-hierarchy.org/country/dgw.html Hierarchia Catholica].&lt;/ref&gt; ([[Lusice]] ''Bissau''), situm est [[archipelagus Bijagós]], compositum centenariis insularum dissimilium magnitudinum, quarum multae incolis orbatae sunt. Olim [[Lusitania]]e seu [[Portugallia]]e [[colonia]] cum nomine &quot;[[Guinea Lusitanica|Guineae Lusitanicae]] ([[Lusice]] ''Guiné Portuguesa''). Guinea Bissavia suam independentiam die [[24 Septembris]] anni [[1973]] proclamavit et, postea, agnita est internationaliter die [[10 Septembris]] anni [[1974]]. Primigenio nomini additum est id capitis, Bissaviae, ad evitandam confusionem cum [[Guinea|Guineā]], priore [[Francia|Francicā]] coloniā.

Cuius caput est [[urbs]]  . Guineae Bissaviensi finitimae civitates sunt: [[Guinea]] et [[Senegalia]]. 

Lingua sollemnis Guineae Bissaviensis est [[Lingua Lusitana|sermo Lusitanus]].

== Geographia ==
=== Flumina ===
* [[Geba flumen|Geba]]

== Historia ==

Guinea Bissaviensis fuit a [[saeculum|saeculo]] [[saeculum 15|XV]], accurate ab anno 1446 usque ad annum [[1974]], [[colonia]] [[Lusitania]]e. At die [[24 Septembris]] [[1973]] suam [[independentia]]m declaravit et anno 1974 admissa est.
== Notae ==
&lt;div class=&quot;references-small&quot;&gt;&lt;references /&gt;&lt;/div&gt;

== Nexus externi ==
{{Communia|Guiné-Bissau|Guineam Bissaviensem}}

* [http://www.abenaa.de/guineabissau.htm Narrationes peregrinationum atque imagines]
* [https://www.ensp.unl.pt/luis.graca/guine_guerracolonial10_mapageral.html Tabula geographica Guineae Portugallensis 1:500 000]

{{natio-stipula}}

{{Africa}}
{{Capsae collectae|Guinea Bissaviensis|politica|{{Praesides Guineae Bissaviensis}}{{Primi ministri Guineae Bissaviensis}}{{Ministri rerum externarum Guineae Bissaviensis}}}}
{{Capsae collectae|Guinea Bissaviensis|historica|{{PALOP}}}}

[[Categoria:Guinea Bissaviensis|!]]
[[Categoria:Condita 1975]]
[[Categoria:Respublicae]]
[[Categoria:Civitates sui iuris]]

{{Myrias|Geographia}}</text>
      <sha1>otww3ka8gi5koy5abb8ningrvv4mnsy</sha1>
    </revision>
    <revision>
      <id>3697708</id>
      <parentid>3697707</parentid>
      <timestamp>2022-08-16T21:05:38Z</timestamp>
      <contributor>
        <ip>84.78.253.101</ip>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2838" xml:space="preserve">{{Capsa civitatis Vicidata}}
[[Fasciculus:Guinea bissau sm03.png|thumbnail|200 px|Tabula Guineae Bissaviensis]]

'''Guinea Bissaviensis'''&lt;ref&gt;&quot;Guinea Bissaviensis&quot;: ''[http://www.gcatholic.org/dioceses/country/GW.htm Giga-Catholic]''&lt;/ref&gt; seu '''Guinea-Bissavia'''&lt;ref&gt;[http://www.vatican.va/news_services/press/sinodo/documents/bollettino_25_xiii-ordinaria-2012/xx_plurilingue/b01_xx.html Bollettino Synodus Episcoporum]&lt;/ref&gt;, rite ''Res publica Guinea Bissaviensis'' ([[Lusice]] ''República da Guiné-Bissau''), est civitas in [[Africa Occidentalis|Africā Occidentali], una apud parvissimas hac in continente. Finitima est [[Senegalia]]e Septentrione, [[Guinea]]e Meridiē Orienteque et [[Oceanus Atlanticus|Oceano Atlantico]] Occidente. Ante caput, [[Bissavia]]&lt;ref&gt;&quot;Bissavia, (-ae, ''f'')&quot;: a Petro Lucusaltiano Latinophilo, [http://www.lateinlexikon.com ''Lexicon Latinum Hodiernum vel Vocabularium Latinitatis Huius Aetatis,''] apud www.lateinlexikon.com.&lt;/ref&gt; seu [[Urbs Bissagensis]]&lt;ref&gt;[https://www.catholic-hierarchy.org/country/dgw.html Hierarchia Catholica].&lt;/ref&gt; ([[Lusice]] ''Bissau''), situm est [[archipelagus Bijagós]], compositum centenariis insularum dissimilium magnitudinum, quarum multae incolis orbatae sunt. Olim [[Lusitania]]e seu [[Portugallia]]e [[colonia]] cum nomine &quot;[[Guinea Lusitanica|Guineae Lusitanicae]] ([[Lusice]] ''Guiné Portuguesa''). Guinea Bissavia suam independentiam die [[24 Septembris]] anni [[1973]] proclamavit et, postea, agnita est internationaliter die [[10 Septembris]] anni [[1974]]. Primigenio nomini additum est id capitis, Bissaviae, ad evitandam confusionem cum [[Guinea|Guineā]], priore [[Francia|Francicā]] coloniā.

Lingua sollemnis Guineae Bissaviensis est [[Lingua Lusitana|sermo Lusitanus]].

== Geographia ==
=== Flumina ===
* [[Geba flumen|Geba]]

== Historia ==

Guinea Bissaviensis fuit a [[saeculum|saeculo]] [[saeculum 15|XV]], accurate ab anno 1446 usque ad annum [[1974]], [[colonia]] [[Lusitania]]e. At die [[24 Septembris]] [[1973]] suam [[independentia]]m declaravit et anno 1974 admissa est.
== Notae ==
&lt;div class=&quot;references-small&quot;&gt;&lt;references /&gt;&lt;/div&gt;

== Nexus externi ==
{{Communia|Guiné-Bissau|Guineam Bissaviensem}}

* [http://www.abenaa.de/guineabissau.htm Narrationes peregrinationum atque imagines]
* [https://www.ensp.unl.pt/luis.graca/guine_guerracolonial10_mapageral.html Tabula geographica Guineae Portugallensis 1:500 000]

{{natio-stipula}}

{{Africa}}
{{Capsae collectae|Guinea Bissaviensis|politica|{{Praesides Guineae Bissaviensis}}{{Primi ministri Guineae Bissaviensis}}{{Ministri rerum externarum Guineae Bissaviensis}}}}
{{Capsae collectae|Guinea Bissaviensis|historica|{{PALOP}}}}

[[Categoria:Guinea Bissaviensis|!]]
[[Categoria:Condita 1975]]
[[Categoria:Respublicae]]
[[Categoria:Civitates sui iuris]]

{{Myrias|Geographia}}</text>
      <sha1>o6vvngb98lwsopfh8j9hkphvtzilltf</sha1>
    </revision>
    <revision>
      <id>3697709</id>
      <parentid>3697708</parentid>
      <timestamp>2022-08-16T21:06:01Z</timestamp>
      <contributor>
        <ip>84.78.253.101</ip>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2837" xml:space="preserve">{{Capsa civitatis Vicidata}}
[[Fasciculus:Guinea bissau sm03.png|thumbnail|200 px|Tabula Guineae Bissaviensis]]

'''Guinea Bissaviensis'''&lt;ref&gt;&quot;Guinea Bissaviensis&quot;: ''[http://www.gcatholic.org/dioceses/country/GW.htm Giga-Catholic]''&lt;/ref&gt; seu '''Guinea-Bissavia'''&lt;ref&gt;[http://www.vatican.va/news_services/press/sinodo/documents/bollettino_25_xiii-ordinaria-2012/xx_plurilingue/b01_xx.html Bollettino Synodus Episcoporum]&lt;/ref&gt;, rite ''Res publica Guinea Bissaviensis'' ([[Lusice]] ''República da Guiné-Bissau''), est civitas in [[Africa Occidentalis|Africā Occidentali], una apud parvissimas hac in continente. Finitima est [[Senegalia]]e Septentrione, [[Guinea]]e Meridiē Orienteque et [[Oceanus Atlanticus|Oceano Atlantico]] Occidente. Ante caput, [[Bissavia]]&lt;ref&gt;&quot;Bissavia, (-ae, ''f'')&quot;: a Petro Lucusaltiano Latinophilo, [http://www.lateinlexikon.com ''Lexicon Latinum Hodiernum vel Vocabularium Latinitatis Huius Aetatis,''] apud www.lateinlexikon.com.&lt;/ref&gt; seu [[Urbs Bissagensis]]&lt;ref&gt;[https://www.catholic-hierarchy.org/country/dgw.html Hierarchia Catholica].&lt;/ref&gt; ([[Lusice]] ''Bissau''), situm est [[archipelagus Bijagós]], compositum centenariis insularum dissimilium magnitudinum, quarum multae incolis orbatae sunt. Olim [[Lusitania]]e seu [[Portugallia]]e [[colonia]] cum nomine &quot;[[Guinea Lusitanica|Guineae Lusitanicae]] ([[Lusice]] ''Guiné Portuguesa''). Guinea Bissavia suam independentiam die [[24 Septembris]] anni [[1973]] proclamavit et, postea, agnita est internationaliter die [[10 Septembris]] anni [[1974]]. Primigenio nomini additum est id capitis, Bissaviae, ad evitandam confusionem cum [[Guinea|Guineā]], priore [[Francia|Francicā]] coloniā. Lingua sollemnis Guineae Bissaviensis est [[Lingua Lusitana|sermo Lusitanus]].

== Geographia ==
=== Flumina ===
* [[Geba flumen|Geba]]

== Historia ==

Guinea Bissaviensis fuit a [[saeculum|saeculo]] [[saeculum 15|XV]], accurate ab anno 1446 usque ad annum [[1974]], [[colonia]] [[Lusitania]]e. At die [[24 Septembris]] [[1973]] suam [[independentia]]m declaravit et anno 1974 admissa est.
== Notae ==
&lt;div class=&quot;references-small&quot;&gt;&lt;references /&gt;&lt;/div&gt;

== Nexus externi ==
{{Communia|Guiné-Bissau|Guineam Bissaviensem}}

* [http://www.abenaa.de/guineabissau.htm Narrationes peregrinationum atque imagines]
* [https://www.ensp.unl.pt/luis.graca/guine_guerracolonial10_mapageral.html Tabula geographica Guineae Portugallensis 1:500 000]

{{natio-stipula}}

{{Africa}}
{{Capsae collectae|Guinea Bissaviensis|politica|{{Praesides Guineae Bissaviensis}}{{Primi ministri Guineae Bissaviensis}}{{Ministri rerum externarum Guineae Bissaviensis}}}}
{{Capsae collectae|Guinea Bissaviensis|historica|{{PALOP}}}}

[[Categoria:Guinea Bissaviensis|!]]
[[Categoria:Condita 1975]]
[[Categoria:Respublicae]]
[[Categoria:Civitates sui iuris]]

{{Myrias|Geographia}}</text>
      <sha1>sre4to61vzi15qckxekj73a32t8zibc</sha1>
    </revision>
  </page>
  <page>
    <title>Linguae Romanicae</title>
    <ns>0</ns>
    <id>15086</id>
    <revision>
      <id>3697716</id>
      <parentid>3552502</parentid>
      <timestamp>2022-08-17T02:17:08Z</timestamp>
      <contributor>
        <ip>2601:140:8C80:AD00:F0CE:897:CA43:E0A</ip>
      </contributor>
      <comment>/* Comparatio */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9188" xml:space="preserve">{{Capsa familiae linguarum
|nomen= Linguae Romancae
|regio= Primum in [[Europa meridiana|Europa meridiana]], hodie etiam in [[America]], [[Africa]], partibusque [[Asia]]e et [[Oceania]]e
|familia=[[Linguae Indoeuropaeae|Indoeuropaea]]&lt;br&gt;
&amp;nbsp;&amp;nbsp;[[linguae Italicae|Italica]]&lt;br&gt;
&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;Linguae Romanicae
|filia1= L. Romanicae occidentales
|filia2= L. Romanicae orientales
|filia3= Lingua Sarda
|iso2=roa
|tabula= [[Fasciculus:Map-Romance Language World.png|center|255px]]
|syllabus= Ubi linguae Romanicae reperiuntur: Italiana (flavo), Hispanica (viridi obscuro et claro), Francogallica (caeruleo), Lusitana (flammeo), Dacoromanica (rubro)}}
'''Linguae Romanicae'''&lt;ref&gt;{{DMLBS|Romanicus}}&lt;/ref&gt; sunt [[lingua]]e nostri temporis quae ex [[Sermo vulgaris Latinus|sermone vulgari Latino]] inter saecula [[saeculum 6|VI]] et [[saeculum 9|IX]] creverunt et inter [[linguae Italicae|linguas Italicas]] familiae [[Linguae Indoeuropaeae|Indoeuropaeae]] enumerantur.

Hodie fere 800 [[millio]]nes hominum habent [[sermo patrius|patrium sermonem]] Romanicum, praecipue in [[Europa]], [[Africa]], [[America]]; etiam multi alii Romanicas linguas discunt et iis in negotiis, peregrinatione, et alibi utuntur.

Quod ad numerum attinet eorum, qui Romanice sermone patrio loquuntur, quinque linguae Romanicae maximae sunt [[lingua Hispanica|Hispanica]] (470 milliones), [[lingua Francogallica|Francogallica]] (300 milliones),  [[lingua Lusitanica|Lusitanica]] (250 milliones), [[lingua Italiana|Italiana]] (70 milliones), [[lingua Dacoromanica|Dacoromanica]] (25 milliones).&lt;ref&gt;[[Nationalencyklopedin]] &quot;Världens 100 största språk 2007&quot; The World's 100 Largest Languages in 2007/2010&lt;/ref&gt;

== Divisio linguarum Romanicarum ==
* [[Lingua Latina]]
* [[Sermo vulgaris]]
** [[lingua Sarda]]
** linguae Romanicae orientales
*** [[lingua Dacoromanica]]
*** [[lingua Moldavica]]
*** [[lingua Valachica]]
*** [[lingua Moeso-Romanica]]
*** [[lingua Istroromanica]]
*** [[lingua Macedoromanica]] sive Aromena
*** [[lingua Meglenoromanica]]
** linguae Romanicae Italo-occidentales
*** [[Linguae Italoromanicae]]
**** [[linguae Italoromanicae mediae]]
***** [[lingua Italiana|lingua Tusca]] (quae et [[lingua Italiana|Italiana sive Italica]])
***** [[lingua Corsa]]
***** [[dialectus Romanensis]]
**** [[lingua Neapolitana]] sive linguae Italoromanicae meridianae
***** [[dialectus Campana]]
***** [[dialectus Aprutina]]
***** [[dialectus Fodiana]]
***** [[dialectus Bariana]]
***** [[dialectus Lucana]]
**** [[lingua Sicula]] sive linguae Italoromanicae meridionalissimae
***** [[dialectus Sallentina]]
***** [[dialectus Bruttia]]
***** [[dialectus Sicula]]
*** Linguae Romanicae occidentales
**** [[linguae Gallaico-Lusitanae]]
***** [[lingua Gallaica]]
***** [[lingua Lusitana]]
**** [[linguae Hiberoromanicae]]
***** [[lingua Hispanica]]
***** [[lingua Iudaeo-Hispanica]] sive Ladina
***** [[lingua Aragonica]]
***** [[lingua Mosarabica]]
**** [[Dominium linguisticum Asturicum|Complexus Asturo-Legionicus]]
***** [[lingua Asturiana]]
***** [[lingua Legionica]]
***** [[lingua Mirandica]]
***** [[lingua Extremaduriana]]
***** [[lingua Cantabrica]] sive Montana
***** [[lingua Fala]]
**** [[Linguae Galloromanicae]]
***** [[lingua Raetoromanica|Linguae Raetoromanicae]]
****** [[Lingua Rhaetica (Romanica)]]
****** [[Lingua Ladina (Dolomiana)]]
****** [[Lingua Foroiuliensis]]
***** [[Gallicoitalicae dialecti|linguae Galloitalicae]]
****** [[Dialectos Langobarda|lingua Langobarda]]
****** [[lingua Ligustica]]
****** [[lingua Monoecica]]
****** [[lingua Pedemontana]]
****** [[lingua Aemiliensis]]
****** [[lingua Venetica]] (cf. [[Venetae dialecti]])
***** linguae Francogallicae
****** [[lingua Francogallica]]
****** [[lingua Pictavino-Santonica]]
****** [[lingua Burgundica]]
****** [[lingua Campanica]]
****** [[lingua Franco-Comitatica]]
****** [[lingua Lotharingica]]
****** [[lingua Gallonica]]
****** [[lingua Normannica]]
******* [[lingua Anglo-Normannica]] †
******* [[lingua Auranica]]
******* [[lingua Guernesica]]
******* [[lingua Caesariaca]]
******* [[lingua Sercquiaca]]
****** [[lingua Picardica]]
****** [[lingua Vallonica]]
***** [[linguae Occitano-Catalanae]]
****** [[lingua Catalana]]
******* [[Dialectus Russinica]]
******* [[Dialectus Balaearensis]]
******* [[Dialectus Valenciana]]
****** [[lingua Occitanica]]
******* [[lingua Gasconica]]
******** [[lingua Aranica]]
******* [[lingua Lemovicensis]]
******* [[lingua Limanica]]
******* [[lingua Languedociana]]
******* [[lingua Provincialis]]
******* [[lingua Vivariensis-Alpensis]]
**** [[lingua Arpitanica]]
** [[lingua Dalmatica]] †
** [[lingua Afroromanica]] †

† [[Lingua exstincta|Exstincta]]

== Comparatio ==
Linguae Romanicae inter se comparatae.

Sententia est '(Illa) fenestram semper claudit antequam cenat'.

:{| cellspacing=&quot;3px&quot;
|-
| [[Lingua Latina]] || ''(Illa) claudit semper fenestram antequam cēnat.''
|-
| [[Lingua Aragonensis]] || ''(Ella) zarra siempre a finestra antes de cenar.''
|-
| [[Lingua Aromanica]] || ''(Ea/Nâsa) încljidi/nkidi totna firida ninti di tsinâ.''
|-
| [[Lingua Arpitanica]] || ''(Le) sarre toltin/tojor la fenétra avan de goutâ/dinar/sopar.''
|-
| [[Lingua Asturiana]] || ''(Ella) pieslla siempre la feniestra/ventana enantes de cenar.''
|-
| [[Eastern Lombard language|Bergamasque]] || ''(Lé) la sèra sèmper sö la finèstra prima de senà.''
|-
| [[Bolognese dialect|Bolognese]] || ''(Lî) la sèra sänper la fnèstra prémma ed dsnèr.''
|-
| [[Lingua Catalana]] || ''(Ella) tanca sempre la finestra abans de sopar.''
|-
| [[Lingua Corsa]] || ''(Ella chjode sempre a finestra prima di cena.''
|-
| [[Emilian language|Emilian]] || ''(Lē) la sèra sèmpar sù la fnèstra prima ad snàr.''
|-
| [[Lingua Dacoromanica]] || ''(Ea) închide totdeauna fereastra înainte de a cina.''
|-
| [[Lingua Extremadurensis]] || ''(Ella) afecha siempri la ventana antis de cenal.''
|-
| [[Lingua Foroiuliensis]] || ''(Jê) e siere simpri il barcon prin di cenâ.''
|-
| [[Lingua Francogallica]] || ''Elle ferme toujours la fenêtre avant de dîner/souper.''
|-
| [[Lingua Gallaica]] || ''(Ela) pecha/fecha sempre a fiestra/xanela antes de cear.''
|-
| [[Lingua Hispanica]] || ''(Ella) siempre cierra la ventana antes de cenar.''
|-
| [[Lingua Iudaeo-Hispanica]] || ''Eya serra syempre la ventana antes de senar.''
|-
| [[Lingua Italiana]] || ''(Ella/Lei) chiude sempre la finestra prima di cenare.''
|-
| [[Mediae Italiae dialecti]] || ''Essa chjude sempre la finestra prima de cena'.''
|-
| [[Lingua Ladina (Dolomiana)|Lingua Ladina]] || ''(Ëra) stlüj dagnora la finestra impröma de cenè.'' (badiot) ''(Ëila) stluj for l viere dan maië da cëina'' (gherdëina)
|-
| [[Lingua Legionensis]] || ''(Eilla) pecha siempres la finiestra cabeiru de cenare.''
|-
| [[Lingua Ligustica]] || ''(Le) saera sempre u balcun primma de cenà.''
|-
| [[Lingua Lusitana]] || ''(Ela) fecha/serra sempre a janela antes de jantar/cear''
|-
| [[Mediolanensis dialectus]] || ''(Le) la sara semper sü la finestra prima de disnà.''
|-
| [[Lingua Mirandica]] || ''(Eilha) cerra siempre la bentana/jinela atrás de jantar.''
|-
| [[Lingua Mosarabica]] || ''Ella cloudet sempre la fainestra abante da cenare.'' (reconstructed{{dubsig}})
|-
| [[Lingua Neapolitana]] || ''Essa nzerra sempe 'a fenesta primma 'e magnà.''
|-
| [[Lingua Normannica]] || ''Lli barre tréjous la crouésie devaunt de daîner.''
|-
| [[Lingua Occitanica]] || ''(Ela) barra sempre/totjorn la fenèstra abans de sopar.''
|-
| [[Lingua Pedemontica]] || ''Chila a sara sèmper la fnestra dnans ëd fé sin-a/dnans ëd siné.''
|-
| [[Lingua Picardica]] || ''Ale frunme tojours l’ creusèe édvint éd souper.''
|-
| [[Lingua Puericisca]] || ''(Ena) cerovâ suempre la velustra atratès dî zzenar.''
|-
| [[Lingua Rhaetica (Romanica)|Lingua Rhaetica]] || ''Ella clauda/serra adina la fanestra avant ch'ella tschainia.''
|-
| [[Lingua Sarda]] || ''Issa serrat semper sa bentana innantis 'e chenare.''
|-
| [[Sassarese language|Sassarese]] || ''Edda sarra sempri lu balchoni primma di zinà.''
|-
| [[Lingua Sicula]] || ''Idda chiui sempri la finestra prima di pistiari/manciari.''
|-
| [[Lingua Vallonica]] || ''Ele sere todi li finiesse divant di soper.''
|-
| [[Lingua Veneta]] || ''Eła ła sara/sera sempre ła fenestra vanti de xenàr/disnar.''
|-
|}

== Notae ==
&lt;references /&gt;

{{NexInt}}
* [[Ars Romanica]]
* [[Linguae Iudaeo-Romanicae]]
* [[Linguae mundi]]
* [[Unio Latina]]

== Bibliographia ==
* Dalby, Andrew. &quot;Romance Languages&quot;. In ''Dictionary of Languages: The Definitive Reference to More Than 400 Languages'', 512–14. Novi Eboraci: Columbia University Press, 1998.

== Nexus externi ==
* [https://www.ethnologue.com/subgroups/romance De linguis Romanicis] apud [[Ethnologue]] {{ling|Anglice}}
* [https://bigenc.ru/linguistics/text/4323600 De linguis Romanicis] in Большая российская энциклопедия {{ling|Russice}}
* [https://www.britannica.com/topic/Romance-languages De linguis Romanicis] in [[Encyclopaedia Britannica]] {{ling|Anglice}}

{{Linguae Romanicae}}

[[Categoria:Indices linguarum]]
[[Categoria:Linguae Romanicae|!]]

{{Myrias|Anthropologia}}</text>
      <sha1>7q0l5b2auj8pawf31342zg862kar0wr</sha1>
    </revision>
  </page>
  <page>
    <title>Olympia aestiva 1984</title>
    <ns>0</ns>
    <id>15850</id>
    <revision>
      <id>3697772</id>
      <parentid>2996278</parentid>
      <timestamp>2022-08-17T11:15:57Z</timestamp>
      <contributor>
        <username>IacobusAmor</username>
        <id>1163</id>
      </contributor>
      <comment>Augenda &amp;c.</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1090" xml:space="preserve">{{Augenda|2022|08|17}}
{{Olympiaebox |
  Nomen = Ludi XXIII Olympici |
  Logo                   = Olympic flag.svg |
  Mag                   = 200 |
  Optional caption       = |
  Urbs                   = [[Angelopolis]], [[Civitates Foederatae Americae]] |
  Numerus nationum       = 140 |
  Numerus athletarum       = 6797 |
  Eventa                 = 221 in 23 [[deportus]] |
  Caerimonia initialis= [[28 Iulii]], [[1984]] |
  Caerimonia finalis     = [[12 Augusti]], [[1984]] |
  Inaugurator      = [[Ronaldus Reagan]] |
  Hymnus athletae        = Edwin Moses |
  Hymnus iudicis         = Sharon Weber |
  Fax Olympiae           = Rafer Johnson |
  Stadium                = [[Angelopolis Memoriae Coliseum]] |
}}
[[Fasciculus:Olympic Torch Tower of the Los Angeles Coliseum.jpg|left|thumb|Caerimonia initialis. Nota &quot;[[Citius, altius, fortius]]&quot; sententia a sinistra.]] 
'''Olympia aestiva 1984''', sive '''ludi XXIII Olympici''', [[Angelopolis|Angelopoli]] in [[Civitates Foederatae|Civitatibus Foederatis]] facti sunt. 

{{Olympia}}

[[Categoria:Olympia Certamina]]
[[Categoria:1984]]</text>
      <sha1>luxlv81c0jp7il1a3lwj55iohvnsney</sha1>
    </revision>
  </page>
  <page>
    <title>Ministerium Saeptorum Nationalium (Civitates Foederatae)</title>
    <ns>0</ns>
    <id>19786</id>
    <revision>
      <id>3697640</id>
      <parentid>2908958</parentid>
      <timestamp>2022-08-16T11:59:14Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>[[Project:HotCat|HotCat]]: +[[Categoria:Oecologia]]; ±[[Categoria:Civitates Foederatae Americae]]→[[Categoria:Ministeria Civitatum Foederatarum Americae]]</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5743" xml:space="preserve">{{latinitas|-2}}
'''Ministerium Saeptorum Nationalium''' ([[Anglice]]: National ''Park Service''; NPS) dicitur foederalis argentura creata anno [[1916]], quae 390  saeptis nationalibus, monumentis, et aedificiis significantibus, et naturalibus et historicis, CFA praesidet et operatur. Inter quae saepta includuntur [[Saeptum Iosemitum]], [[Saeptum Vallis Grandis]],  [[Statua Libertatis]], et varia magna monumenta Vasingtoniae. 

Anno [[1872]] Saeptum Croceapetrum{{dubsig}} (''Yellowstone'') conditum est primum saeptum nationale. Instituens Ministerium Monumentorum Nationalium, ''Acta Antiquitatum'' anno [[1906]] protectionem ad perietinas Amerindicas dedit. 

Ministerium Saeptorum Nationalium demum anno [[1916]] conditum est, ut haec saepta et monumenta melius administrarentur. Subsequenter, multa nova saepta et loca historica crebiter addita sunt. Anno [[1980]], iam 312 loca numerantur, anno [[2006]], 390. 

Magna saepta naturalia hodie vocantur [[Saeptum Nationale|Saepta Nationalia]] (National Park), dum aedificia historica vocantur [[Situs Historicus Nationalis]] (National Historic Site) saepe. Saepta et situs in 49 civitatibus inveniuntur (omnes extra Delavaria), et in Guama, Samoa Americana, Portu Diviti, et Insulis Virgineis.

==Index Saeptorum Nationalium==
Ecce index omnium Saeptorum Nationalium, iuxta civitatem et annum institutum

* '''Acadia''', Maina (1916/1929)
* '''Samoa Americana''' (1988)
* '''Arci''', Uta (1929/1971)
* '''Terrae Malae''', Dakota Meridoinalis (1929/1978)
* '''Flexus Magnus''', Texia (1935)
* '''Biscaina''', Florida (1968/1980)
* '''Vallis Nigra Gunnisonensis''', Coloratum (1933/1999)
* '''Vallis Bryce''', Uta (1923/1928)
* '''Terrae Vallium''', Uta (1964)
* '''Scopulus Capitolinus''', Uta (1937/1971)
* '''Spenulcae Carlsbad''', Novum Mexicum (1923/1930)
* '''Insulae Canalis''', California (1938/1980)
* '''Colles Arenosae Coloratensis''' (2000/2005)
* '''Lacus Calderae''', Oregonia (1902)
* '''Vallis Cuiahogae''', Ohium (1974/2000)
* '''Vallis Mortis''', California-Nevada (1933/1994)
* '''Denali''', Alaska (1917/1980)
* '''Tortugae Secae''', Florida (1935/1992)
* '''Evergladae''', Florida (1934)
* '''Portae Arcticae''', Alaska (1978/1984)
* '''Glaciatum''', Montana (1910)
* '''Sinus Glaciatorum''', Alaska (1925/1986)
* '''Vallis Grandis''', Arizona (1893/1919)
* '''Teton Grandis''', Vaiominga (1929)
* '''Trulleum Magnum''', Nevada (1922/1986)
* '''Montes Fungosae Magnae''', Carolina Septentrionalis-Tennesia (1926/1934)
* '''Montes Guadalupae''', Texia (1966)
* '''Haleakala''', Hawaia (1916/1960)
* '''Vulcani Hawaienses''' (1916/1961)
* '''Fontes Caldae''', Arkansas (1832/1921)
* '''Insula Regalis''', Misigania (1931)
* '''Arbor Ioshui''', California (1936/1994)
* '''Katmai''', Alaska (1918/1980)
* '''Fiorda Kenaienses''', Alaska (1978/1980)
* '''Vallis Regum''', California (1890/1940)
* '''Vallis Kobuk''', Alaska (1978/1980)
* '''Lacus Clark''', Alaska (1978/1980)
* '''Vulcanicus Lassen''', California (1907/1916)
* '''Spelunca Mammathorum''', Kentukia (1926/1941)
* '''Mensa Viridis''', Coloratum (1906)
* '''Mons Rainier''', Vasingtonia (1899)
* '''Cascadae Septentionales''', Vasingtonia (1968)
* '''Olympicus''', Vasingtonia (1909/1938)
* '''Boscus Petrificatus''', Arizona (1906/1962)
* '''Sempervivens''', California (1968)
* '''Montes Petrosae''', Coloratum (1915)
* '''Saguarum''', Arizona (1933/1994)
* '''Sequoia''', California (1890)
* '''Senandoa''', Virginia (1926)
* '''Theodorus Roosevelt''', Dakota Septentrionalis (1947/1978)
* '''Insulae Virgineae''' (1956)
* '''Navigatores''', Minnesota (1971)
* '''Spenulca Ventosa''', Dakota Meridionalis (1903)
* '''Wrangell–Sanctus Elias''', Alaska (1978/1980)
* '''Croceapetrum'''{{dubsig}}, Vyomina/Idahum/Montana (1872)
* '''Iosemita''', California (1890)
* '''Ziona''', Uta (1909/1919)

==Alii loci ab Ministerio Saeptorum Nationalium administrati==
'''Areae Recreationales Nationales''' (NRA)

* '''Pons Aurea''', Sanctus Franciscus
* '''Montes Sanctae Monicae''', Angelopolis
* '''Lacus Mead''', Nevada-Arizona
* '''Vallis Glenica''', Arizona-Uta
* '''Postis''', Novum Eboracum–Nova Caesarea

'''Monumenta Nationalia''' (NM)
* '''Bosci Muir''', California
* '''Turrus Diaboli''', Vaiominga
* '''Pontes Naturales''', Uta
* '''Statua Libertatis''', Novum Eboracum

'''Saepta Nationalia Historica''' (NHP)
* '''Colonialis''', Virginia
* '''Furnus Fabrilis Vallis''', Pennsylvania
* '''Independentia''', Philadelphia
* '''Pugnator in Notitia Immediata''' (Minute Man), Massachusetta

'''Monumenta Nationalia''' (NMEM)
* '''USS Arizona''', Hawaia
* '''Mons Rushmore''', Dakota Meridionalis
* '''Expansio Jeffersonis''', Sanctus Ludovicus
* '''Monumentum Vasingtoni''', Vasingtonia, DC
* '''Memorialis Lincolnis''', Vasingtonia, DC
* '''Memorialis Jeffersonis''', Vasingtonia, DC
* '''Veterani Belli Vietnami''', Vasingtonia, DC
* '''Veterani Belli Coreani''', Vasingtonia, DC
* '''Secundum bellum mundanum''', Vasingtonia, DC

'''Alii'''
* '''Gettysburg''' (Saeptum Nationale Militare), Pennsylvania
* '''Vicksburg''' (Saeptum Nationale Militare), Mississippia
* '''Tractus Natchetii''' (Via Nationalis; National Parkway), Mississippia/Alabama/Tennesia
* '''Iugum Ceruleum''' (Via Nationalis), Virginia/ Carolina Septentrionalis
* '''Appalachianus''' (Ductus Scaenicus Nationalis; National Scenic Trail), Maina ad Georgiam
* '''Caput Cod''' (Litus Marinum Nationale; National Seashore), Massachusetta
* '''Insula Ignis''' (Litus Marinum Nationale), Novum Eboracum
* '''Ursa Dormiens''' (Litus Marinum Nationale), Misigania
* '''Malleum Nationale''', Vasingtonia, DC
* '''Casa Blanca''', Vasingtonia, DC

{{stipula}}

[[Categoria:Ministeria Civitatum Foederatarum Americae]]
[[Categoria:Oecologia]]</text>
      <sha1>k0mk7a2sswkizezm5daehtmymk2jsgy</sha1>
    </revision>
  </page>
  <page>
    <title>Hypercosmos</title>
    <ns>0</ns>
    <id>28752</id>
    <revision>
      <id>3697655</id>
      <parentid>3651795</parentid>
      <timestamp>2022-08-16T13:24:29Z</timestamp>
      <contributor>
        <username>LilyKitty</username>
        <id>18316</id>
      </contributor>
      <comment>de studia futurorum</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2441" xml:space="preserve">'''Hypercosmos''' (-i, ''m'') vel '''superspatium''' fere usque ab [[decennium 194|decennio IV]] [[saeculum 20|saeculi XX]] in [[Litterae utopicae|libris de rebus futuris]] apparet. Quaestio fuit quomodo [[navis|naves]] inter [[stella]]s [[cosmos|cosmi]] citius [[lux|luce]] vehere possint, cum secundum leges naturales ab [[Albertus Einstein|Alberto Einstein]] inventas nihil nostro in mundo citius luce movi potest. 

In libris de [[studia futurorum|rebus futuris]] respondebatur nostrum mundum in tres spatii dimensiones extendi. Praeter has dimensiones autem etiam quartam esse videlicet hypercosmi. In [[imperium Galacticum|imperio Galactico]] exempli gratia librorum quos [[Isaac Asimov]] de rebus futuris scripsit naves inter stellas vehentes ex nostro mundo saliunt, ut ita dicam, trans hypercosmum, quo nostrum tempus non valet, vehunt et eodem temporis puncto in longinqua [[via lactea|Galaxiae]] parte iterum apparent. Ita basis ut ita dicam materialis efficitur ut imperium Galactium omnino esse possit.

== In cetera cultura populari ==
[[Fasciculus:Planetary System in Gliese 581 (artist's impression).jpg|thumb|300px|Imago illustrata.]]
Superspatium multum in cultura populari commemoratum est, primum in [[scientia ficticia]], sicut universo ficto ''[[Stargate]]'' (&quot;Astrorum Porta&quot;), ''[[Star Trek]]'' (&quot;Bellis Stellarum&quot;), et cetera. Superspatium in scientia ficticia saepe alia dimensio in universo est ubi celerrime veheris, celerior quam [[lux]]. In recta vita, scienta nostra id dicat si celerior quam lucem veheris, tempus afficit ad lentiorem ire vehitori, dum nobis qui in eo non vehimur ad normam tempus intersunt. Ita extendere alios [[planeta]]s ab aliis [[Systema solare|systematibus solaribus]], per superspatium esse necesse est. Alioquin multa centena sive milliones annorum capere potest, et etiam in [[velocitas lucis|velocitate lucis]] ingentissimum tempus caperet.

Intrare superspatium saepe modis monstrati sunt ubi creas propylaeum superspatiale et deinde intras, sive aperis fenestram superspatialem et deinde intras. Quando ad locum destinatum termini advenisti, creas sive aperis propylaeum sive fenestram extrandae.

{{NexInt}}
* [[Astrophysica]]
* [[Foramen vermis]]
* [[Astronavis|Navis sideralis]]
* [[Scientia ficticia]]
* [[Spatium (astronomia)]]
* [[Teleportatio]]

[[Categoria:Astrophysica]]
[[Categoria:Litterae rerum futurarum]]
[[Categoria:Loci ficticii]]
[[Categoria:Fictio scientifica]]</text>
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  </page>
  <page>
    <title>Felis silvestris</title>
    <ns>0</ns>
    <id>31485</id>
    <revision>
      <id>3697765</id>
      <parentid>2415974</parentid>
      <timestamp>2022-08-17T11:04:36Z</timestamp>
      <contributor>
        <username>IacobusAmor</username>
        <id>1163</id>
      </contributor>
      <comment>Augenda &amp;c.</comment>
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      <text bytes="999" xml:space="preserve">{{Augenda|2022|08|17}}
{{Titulus italicus}}
[[Fasciculus:Wildkatze_002.jpg|thumb|upright=1|Felis silvestris.]]

'''''Felis silvestris''''' est [[felis]] [[subfamilia]]e [[Felinae|Felinarum]], [[praedator]] quae indigena in [[Europa]] et in [[Africa]] et orientalibus [[Asia]]e partibus [[habitatio|invenitur].

== Subspecies ==
* ''[[Felis silvestris brockmani]]''
* ''[[Felis silvestris cafra]]''
* ''[[Felis silvestris foxi]]''
* ''[[Felis silvestris griselda]]''
* ''[[Felis silvestris lybica]]''
* ''[[Felis silvestris ocreata]]''
* ''[[Felis silvestris pyrrhus]]''
* ''[[Felis silvestris caudata]]''
* ''[[Felis silvestris ornata]]''
* ''[[Felis silvestris shawiana]]''
* ''[[Felis silvestris cretensis]]''
* ''[[Felis silvestris caucasia]]''
* ''[[Felis silvestris grampia]]''
* ''[[Felis silvestris jordansi]]''
* ''[[Felis silvestris reyi]]''
* ''[[Felis silvestris sarda]]''
* ''[[Felis silvestris silvestris]]''
* ''[[Felis silvestris catus]]''

{{biologia-stipula}}

[[Categoria:Felidae]]</text>
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    <revision>
      <id>3697767</id>
      <parentid>3697765</parentid>
      <timestamp>2022-08-17T11:08:36Z</timestamp>
      <contributor>
        <username>IacobusAmor</username>
        <id>1163</id>
      </contributor>
      <minor />
      <comment>~</comment>
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      <text bytes="1030" xml:space="preserve">{{Augenda|2022|08|17}}
{{Titulus italicus}}
[[Fasciculus:Wildkatze_002.jpg|thumb|upright=1|Felis silvestris.]]

'''''Felis silvestris''''' est [[felis]] [[subfamilia]]e [[Felinae|Felinarum]], [[praedator]] quae indigena in [[Europa]] et in [[Africa]] et orientalibus [[Asia]]e partibus [[habitatio|invenitur]].

== Subspecies ==
{{div col|3}}
* ''[[Felis silvestris brockmani]]''
* ''[[Felis silvestris cafra]]''
* ''[[Felis silvestris foxi]]''
* ''[[Felis silvestris griselda]]''
* ''[[Felis silvestris lybica]]''
* ''[[Felis silvestris ocreata]]''
* ''[[Felis silvestris pyrrhus]]''
* ''[[Felis silvestris caudata]]''
* ''[[Felis silvestris ornata]]''
* ''[[Felis silvestris shawiana]]''
* ''[[Felis silvestris cretensis]]''
* ''[[Felis silvestris caucasia]]''
* ''[[Felis silvestris grampia]]''
* ''[[Felis silvestris jordansi]]''
* ''[[Felis silvestris reyi]]''
* ''[[Felis silvestris sarda]]''
* ''[[Felis silvestris silvestris]]''
* ''[[Felis silvestris catus]]''
{{div col end}}

{{biologia-stipula}}

[[Categoria:Felidae]]</text>
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  </page>
  <page>
    <title>Bissavia</title>
    <ns>0</ns>
    <id>38138</id>
    <revision>
      <id>3697701</id>
      <parentid>3189698</parentid>
      <timestamp>2022-08-16T20:23:00Z</timestamp>
      <contributor>
        <ip>84.78.253.101</ip>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="977" xml:space="preserve">{{Capsa urbis Vicidata}}
'''Bissavia'''&lt;ref&gt;&quot;Bissavia, (-ae, ''f'')&quot;: a Petro Lucusaltiano Latinophilo, [http://www.lateinlexikon.com ''Lexicon Latinum Hodiernum vel Vocabularium Latinitatis Huius Aetatis,''] apud www.lateinlexikon.com.&lt;/ref&gt; seu '''Urbs Bissagensis'''&lt;ref&gt;[https://www.catholic-hierarchy.org/country/dgw.html Hierarchia Catholica].&lt;/ref&gt; ([[Lusice]] ''Bissau'') est urbs circa 388.000 (die [[1 Ianuarii]] [[2005]]) maxima ac [[caput]] [[Guinea Bissaviensis|Guineae Bissaviensis]].
 
== Aer ==
[[Fasciculus:Klimadiagramm-deutsch-Bissau-Guinea-Bissau.png|thumb|left|Diagramma aeris Bissau]]

== Historia ==
[[Fasciculus:Bissau4.jpg|thumb|Via in Bissau]]

Urbs anno [[1687]] a [[Lusitania|Lusitanis]] condita est et anno [[1942]] caput eorum [[colonia]]e facta est. 

==Notae==
&lt;references /&gt;

{{urbs-stipula}}

{{Urbes Africae capitales}}

[[Categoria:Capita]]
[[Categoria:Urbes Africae]]
[[Categoria:Condita 1687]]
[[Categoria:Bissau|!]]

{{Myrias|Geographia}}</text>
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    <revision>
      <id>3697705</id>
      <parentid>3697701</parentid>
      <timestamp>2022-08-16T20:51:44Z</timestamp>
      <contributor>
        <username>IacobusAmor</username>
        <id>1163</id>
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      <minor />
      <comment>IacobusAmor movit paginam [[Bissau]] ad [[Bissavia]] praeter redirectionem: Lemma</comment>
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      <text bytes="977" xml:space="preserve">{{Capsa urbis Vicidata}}
'''Bissavia'''&lt;ref&gt;&quot;Bissavia, (-ae, ''f'')&quot;: a Petro Lucusaltiano Latinophilo, [http://www.lateinlexikon.com ''Lexicon Latinum Hodiernum vel Vocabularium Latinitatis Huius Aetatis,''] apud www.lateinlexikon.com.&lt;/ref&gt; seu '''Urbs Bissagensis'''&lt;ref&gt;[https://www.catholic-hierarchy.org/country/dgw.html Hierarchia Catholica].&lt;/ref&gt; ([[Lusice]] ''Bissau'') est urbs circa 388.000 (die [[1 Ianuarii]] [[2005]]) maxima ac [[caput]] [[Guinea Bissaviensis|Guineae Bissaviensis]].
 
== Aer ==
[[Fasciculus:Klimadiagramm-deutsch-Bissau-Guinea-Bissau.png|thumb|left|Diagramma aeris Bissau]]

== Historia ==
[[Fasciculus:Bissau4.jpg|thumb|Via in Bissau]]

Urbs anno [[1687]] a [[Lusitania|Lusitanis]] condita est et anno [[1942]] caput eorum [[colonia]]e facta est. 

==Notae==
&lt;references /&gt;

{{urbs-stipula}}

{{Urbes Africae capitales}}

[[Categoria:Capita]]
[[Categoria:Urbes Africae]]
[[Categoria:Condita 1687]]
[[Categoria:Bissau|!]]

{{Myrias|Geographia}}</text>
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  </page>
  <page>
    <title>Capsicum (fructus)</title>
    <ns>0</ns>
    <id>42428</id>
    <revision>
      <id>3697676</id>
      <parentid>3696938</parentid>
      <timestamp>2022-08-16T16:20:19Z</timestamp>
      <contributor>
        <username>Andrew Dalby</username>
        <id>1084</id>
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      <minor />
      <comment>/* Bibliographia */</comment>
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      <text bytes="64203" xml:space="preserve">[[Fasciculus:Chilli tribute from Tuchpa Codex Mendoza 52r detail.jpg|thumb|&quot;DCCC siccorum capsici fructuum sarcinae&quot;: tributum urbis [[Tuchpa]] in [[Codex Mendoza|Codice Mendoza]] ante annum 1542 enumeratum]]
{{Videhom|Capsicum (genus)}}
'''Capsicum'''&lt;ref&gt;&quot;Capsicum&quot;: [[#Gesnerus (1561)]]; [[#Gregorius de Regio (1611)]]&lt;/ref&gt; seu '''chilli'''&lt;ref&gt;&quot;Chilli&quot; (indecl.): [[#Nierembergius (1635)]]&lt;/ref&gt;&lt;ref&gt;&quot;lada Chili&quot; nomen Malaium: [[#Bontius (1642)]]&lt;/ref&gt; est [[fructus]] [[Angiospermae|plantarum florentium]] [[genus (taxinomia)|generis]] ''[[Capsicum (genus)|Capsici]],'' in [[familia (taxinomia)|familia]] [[Solanaceae|Solanacearum]]. Huic generi sunt quinque fere [[species (taxinomia)|species]], quarum [[Varietates capsicorum|cultivarietatibus]] [[homo|homines]] pro [[cibus|cibo]] et [[medicamentum|medicamentis]] uti solent, videlicet ''[[Capsicum annuum]]'', ''[[Capsicum baccatum|C. baccatum]]'', ''[[Capsicum chinense|C. chinense]]'', ''[[Capsicum frutescens|C. frutescens]]'', ''[[Capsicum pubescens|C. pubescens]]''.

Illae ''[[Capsicum annuum|C. annui]]'' [[cultivarietas|varietates]], quibus [[sapor]] non pungens sed dulcis sit, a productoribus comestoribusque nomine semper distinguuntur. Latine &quot;[[capsicum dulce (fructus)|capsicum dulce]]&quot; appellantur.&lt;ref name=&quot;Dunal (1813)&quot;&gt;Michel Félix Dunal, ''Histoire naturelle, médicale et économique des Solanum'' (Lutetiae: Koenig, 1813) {{Google Books|Dc5LAAAAYAAJ}}&lt;/ref&gt;

[[Aves]] fructibus capsicorum pabulantur, [[animalia]] rarissime quod apud ea [[capsaicinum]] dolorem stimulat.&lt;ref&gt;Dan Gleason, &quot;[https://eugene.wbu.com/birds-and-hot-pepper The Story of Birds and Hot Pepper]&quot;&lt;/ref&gt; [[Homo sapiens|Homines]] tantum se capsicis medent.&lt;ref&gt;[[#Pickersgill (2016)]]&lt;/ref&gt;

== De nominibus ==
[[Fasciculus:Conquistador on chilli.jpg|thumb|upright=2|''... una sorte di pepe da condire che si chiama Chil ...'': vocabulum ''chilli'' primum in linguam Europaeam mutuatum est. [[#Conquisitator anonymus (post 1519)]]]]
[[Fasciculus:Petrus Martyr on chilli.jpg|thumb|upright=2|&quot;... vocant ipsi haxi ultima acuta ...&quot;: vocabulum ''ají'', iam a [[Christophorus Columbus|Columbo]] relatum, [[Petrus Martyr ab Angleria]] primum Latine recitavit. [[#Petrus Martyr (1530)]]]]
''Capsicum'' e verbo Graecitatis Byzantinae κάψικον derivatur, quod invicem e vocabulo Latino ''capsa'' mutuatur. [[Ioannes Actuarius]], medicus Graecus saeculi XIII, κάψικον pluries, semel κάψικον Ἰνδικόν (i.e. &quot;capsicum Indicum&quot;) in praeceptis medicamentorum scripsit, sine explicatione. Mathisius, versionis Latinae Actuarii confector, &quot;capsicum&quot; et &quot;capsicum Indicum&quot; sine pluribus scripsit.&lt;ref&gt;Corn. Henricus Mathisius, ''Actuarii Joannis filii Zachariae operum ... Methodi medendi libri VI'' (Lugduni, 1556) [https://archive.org/details/BIUSante_33457/page/446/mode/2up p. 446]; ''Medicae artis principes post Hippocratem et Galenum'' (Genavae: Henricus Stephanus, 1567) [https://www.e-rara.ch/gep_g/content/zoom/4144666 vol. 2 col. 277]&lt;/ref&gt; [[Ioannes Ruellius]] (''De natura stirpium'') &quot;[[cardamomum]]&quot; hoc vocabulo Graeco intellegendum esse censuit,&lt;ref&gt;[[Ioannes Ruellius]], ''De natura stirpium libri III'' (Basileae, 1537) [https://gallica.bnf.fr/ark:/12148/bpt6k1520724r/f383 p. 287]&lt;/ref&gt; quem [[Carolus du Fresne, Dominus du Cange|Carolus du Cange]] secutus est.&lt;ref&gt;{{DuCangeGr}} col. 628 s.v. κάψικον&lt;/ref&gt; Sed [[Casparus Bauhinus]] in ''Phytopinace'', Actuario per editionem 1567 citato, idem vocabulum &quot;capsicum/κάψικον&quot; sicut synonymum &quot;piperis Indici&quot; intellegit, etymologiae falsae utens (non e vocabulo Latino ''capsa'' sed e Graeco κάπτω &quot;haurio&quot; deduci suasit)&lt;ref&gt;[[#Bauhinus (1596)]]; Casparus Bauhinus, ''Pinax Theatri botanici'' (Basileae: sumptibus Ludovici Regis) [https://archive.org/details/mobot31753000495769/page/101/mode/2up pp. 101-103]&lt;/ref&gt; geographiamque et historiam deviam assumens: hoc enim genus, non in [[India]] sensu normali sed in [[Indiae occidentales|Indis occidentalibus]] [[America]]que continente ortum, Actuarius scriptor saeculi XIII Europaeus cognoscere nequibat. Rubrica Bauhini &quot;Capsicum seu Piper Indicum&quot; nihilominus ab aliis botanistis accepta (praesertim a [[Gregorius de Regio|Gregorio de Regio]] qui de his fructibus mox utilissime disseruerit) omnes fere eruditi recentiores hoc nomen adhibent.

''Ají'', nomen origine [[lingua Arawakensis|Arawakense]], [[Christophorus Columbus]] primus Europaeorum anno [[1493]] audivit:&lt;ref name=&quot;Columbi ephemeris (1493)&quot; /&gt; quod nomen ex [[Hispaniola]] in usum Hispanorum omnium Novi Mundi incolarum apprehensum est. [[Petrus Martyr ab Angleria]] (''[[De orbe novo decades]]'') hoc vocabulum ad Latinitatem suam anno [[1530]] accommodavit: &quot;De pipere insulari continentique nunc parum ... dico piper quum non sit piper, quia piperis habeat vim et aroma ... vocant ipsi ''haxi'' ultima acuta&quot;.&lt;ref&gt;[[#Petrus Martyr (1530)]]: [[:Fasciculus:Petrus Martyr on chilli.jpg|vide imaginem]]&lt;/ref&gt;&lt;ref&gt;&quot;Ají&quot; in [[#Friederici (1947)]] p. 46&lt;/ref&gt;

''Chilli'' fuit nomen apud [[Azteci|Aztecos]] [[lingua Navatlaca|linguá Navatlacá]] usitatum. Ita iam rettulit [[Relatione di alcune cose della Nuova Spagna e della gran città di Temestitan Messico|conquisitator anonymus]] in ''Relatione'' anno circiter 1521 scripta.&lt;ref&gt;&quot;Chile&quot; in [[#Friederici (1947)]] pp. 174-175&lt;/ref&gt;

''Piper'' (ut qui ad aroma ''[[Piper nigrum|Piperis nigri]]'' alluderet) est nomen in linguis Europaeis iam ab anno [[1493]] saepissime adhibitum quia capsicum locum piperis in arte coquinaria Americanorum habuisse censebatur (ita iam Columbus et Petrus Martyr supra citati) atque mox in gastronomia Europaea talem usum usurpare incipiebat. ''Piper Indicum'' plures saeculo XVI scripserunt, ea ratione quod &quot;[[Indiae occidentales]]&quot; et &quot;[[Asia austro-orientalis|orientales]]&quot; ab initio apud Europaeos haud distinguebantur; alii inter quos [[Carolus Clusius]], ambiguitate huius nominis suasi, ''Piper Americanum'' scribere maluerunt.

''Pigmentum'' (i.e. tinctura, conditura) est origo Latinus nominis in variis linguis hodiernis adhibiti sicut synonymum piperis vel speciarii.

''Quiya'' atque recentius ''cayenne'', nomina tam [[Pulvis cayennae|pulveris culinarii]] quam {{Creanda|en|Cayenne pepper|Cayenne (varietas)|cultivarietatis}}, e nuncupationibus in [[lingua Tupi]] capsicorum in [[Brasilia]] saeculo XVII cognitorum derivantur, sit ''quiyaqui, quiya-apua, quiya-cumati, quiya-carapo'' apud  [[Iacobus Bontius|Bontium]];&lt;ref&gt;[[#Bontius ed. Piso (1658)]]&lt;/ref&gt; ''quiya-apua, quiya-cumeri, quiya-uca'';&lt;ref&gt;[[#Raius (1693)]]&lt;/ref&gt; ''kyynha'', varietates ''kyynhavi, kyynhai'';&lt;ref&gt;Christoph Gottlieb von Murr, ''Reisen einiger Missionarien der Gesellschaft Jesu in Amerika''. Norimbergae, 1785 {{Google Books|8748AAAAcAAJ|p. 519}}&lt;/ref&gt; ''quiya, quiynhá''.&lt;ref&gt;Theodoro J. H. Langgaard, ''Novo formulario medico e pharmaceutico'' (1868) {{Google Books|QNk8AAAAcAAJ|p. 537}}&lt;/ref&gt; Unde [[Anglice]] a principio ''kian, chian, chyan, kayan, cayan'' appellabatur, denuo ab anno 1783 ''cayenne pepper''.&lt;ref&gt;&quot;cayan-butter&quot;, &quot;cayan-pepper&quot; in &lt;span id=&quot;Cassidy et Le Page (1967)&quot;&gt;&lt;/span&gt;F. G. Cassidy, R. B. Le Page, ''Dictionary of Jamaican English'' (Cantabrigiae: Cambridge University Press, 1967) p. 97; &quot;cayenne&quot; in {{OED}}&lt;/ref&gt;

''Uchu'' est nomen in [[lingua Quechua]] adhibita.&lt;ref&gt;&quot;Uchu&quot; in [[#Friederici (1947)]] p. 639&lt;/ref&gt; Hodie apud [[Peruvia|Peruanos]] varietates Peruvianas denotat, ''ají'' varietates Mexicanas et Caribicas. 

== Capsicum in America et Europa ==
[[Fasciculus:German pepper described by Hieronymus Bock 1539 fasc 2 f 86v.jpg|thumb|upright=1.25|Capsici in Europa culti notitia antiquissima ([[Hieronymus Bock]], 1539)&lt;ref name=&quot;Bock (1539)&quot;&gt;[[#Bock (1539)]]&lt;/ref&gt;]]
Primus Europaeorum [[Christophorus Columbus]] de capsico fructu in [[ephemeris|ephemeride]] primae [[navigatio]]nis sub die [[15 Ianuarii]] [[1493]] refert: &quot;Abundat etiam ''ají'', videlicet piper huius gentis e quorum magis quam piper valeant, sine quo nullus unquam cenat, quod saluberrimum est; quo quinquaginta caravellae quotannis in hac [[Hispaniola]] onerari possunt.&quot;&lt;ref name=&quot;Columbi ephemeris (1493)&quot;&gt;''Tambien hay mucho ají, ques su pimienta, della que vale mas que pimienta, y toda la gente no come sin ella, que la halla muy sana; puédense cargar 50 carabelas cada año en aquella Española'': [[#Columbi ephemeris (1493)]] [https://archive.org/stream/coleccindelosv01nava#page/286/mode/2up p. 286 editionis 1858]&lt;/ref&gt; Ipse semina huius fructus ad reges Hispanicos statim reddidit, sicut ex epistula anno insequenti in insula Isabella scripta constat: &quot;illius ''axí'' quod nos piper nuncupamus, de quo e prima navigatione ad Exc. vestr. adduximus, hinc recipietis quantum mandabitis; cuius semina in hortis serta crescuntur.&quot;&lt;ref&gt;''del axí, a qui deçimos pimienta, del que truxe el otro viaje a V. Al., aquí ay y abrá cuanto V. Al. mandare, que les siembran y naçen en huertas'': [[#Columbi liber copiarum (1493-1503)]] [http://www.biblioteca.tv/artman2/publish/1494_259/El_segundo_viaje_a_las_Indias_Fragmento_de_la_cart_441.shtml no. 2]&lt;/ref&gt; [[Didacus Alvares Chanca]] etiam, particeps huius navigationis, in epistula anno [[1494]] scripta ait: &quot;Tanquam speciem ad condimentum habent speciem ''agí'' appellatam, e qua tam pisces quam aves quantas capiant comeduntur; quae multis varietatibus prodit&quot;&lt;ref&gt;''Tienen por especia, por lo adobar, una especia que se llama Agí con la cual comen también el pescado, como aves cuando las pueden haber'': [[#Chanca (1494)]] [https://archive.org/stream/coleccindelosv01nava#page/370/mode/2up p. 370 editionis 1858]&lt;/ref&gt; {{Creanda|it|Relazione d'alcune cose della Nuova Spagna e della gran città di Temestitan Messico|Conquisitator anonymus}} (sic nuncupatus) qui cum [[Ferdinandus Cortesius|Cortesio]] Mexicopolim anno [[1519]] venit de &quot;pipere&quot; nundinis huius urbis venditato ait:&lt;ref&gt;[[#Conquisitator anonymus (post 1519)]] [https://archive.org/stream/dellenavigationi00ramu#page/n605/mode/2up f. 258v editionis 1606]&lt;/ref&gt; &quot;Habent varietatem piperis condimentarii, ''chil'' appellati, sine quo nullum cibum comedunt.&quot;&lt;ref&gt;''Hanno una sorte di pepe da condire, che si chiama Chil, che niuna cosa mangiano senza esso'': [[#Conquisitator anonymus (post 1519)]]: [[:Fasciculus:Conquistador on chilli.jpg|vide imaginem]]&lt;/ref&gt;

[[Fasciculus:Calechutischer Pfeffer Leonhart Fuchs 1543.jpg|thumb|&quot;Siliquastrum maius et minus&quot; ab [[Leonhartus Fuchsius|Leonharto Fuchsio]] depicta (exemplum coloratum)&lt;ref&gt;[[#Fuchsius (1542)]] p. 732&lt;/ref&gt;]]
[[Fasciculus:Albrecht Meyer25.jpg|thumb|&quot;Siliquastrum tertium&quot; ab [[Leonhartus Fuchsius|Leonharto Fuchsio]] depicta (exemplum coloratum)&lt;ref&gt;[[#Fuchsius (1542)]] p. 733&lt;/ref&gt;]]
[[Fasciculus:Breyter Indianischer Pfeffer Leonhart Fuchs 1543.jpg|thumb|&quot;Siliquastrum quartum&quot; (an ''[[Capsicum chinense]]''?) ab [[Leonhartus Fuchsius|Leonharto Fuchsio]] depicta (exemplum coloratum)&lt;ref&gt;[[#Fuchsius (1542)]] p. 734&lt;/ref&gt;]]
Seminibus a Columbo in Hispaniam remissis, cultus in hortis usque in [[Germania]]m anno [[1539]] penetraverat, ubi [[Hieronymus Bock]] [[Spira]]e in urbe observans originem speciei nescivit: ''deutscher Pfeffer'' nuncupavit.&lt;ref name=&quot;Bock (1539)&quot; /&gt; E talibus observationibus [[Leonhartus Fuchsius]] anno [[1542]], nomina vernacularia ''indianischer'' et ''Calecutischer Pfeffer'' citans, [[species|speciebus]] a maioribus definitis &quot;siliquastro&quot; et &quot;piperitide&quot; frustra attribuere temptans, optimas imagines varietatum quattuor (inter quas fortasse ''[[Capsicum chinense|Capsici chinensis]]'') divulgavit.&lt;ref&gt;[[#Fuchsius (1542)]]; imagines ab [[Albertus Meyer|Alberto Mayer]] pictas, cf. H. Walter Lack, &quot;[https://www.zobodat.at/pdf/ANNA_104B_0463-0478.pdf Eine unbekannte Wiener Bilderhandschrift: Der Codex Amphibiorum]&quot; in ''Annalen des Naturhistorischen Museums in Wien'' vol. 104 B (2003) pp. 463-478&lt;/ref&gt; Quem sequens [[Guilielmus Turnerus]] anno [[1548]] capsica in hortis [[Anglia]]e aliquibus repperit eisdemque nominibus usus est.&lt;ref&gt;''Piperitis called also siliquastram after the judgemente of Fuchsius . . . called in Englishe Indishe peper. . . . The herbe groweth in certeyne gardines in Englande'': [[Guilielmus Turnerus|William Turner]], ''The Names of Herbes in Greke, Latin, Englishe, Duch and Frenche'' (Londinii: John Day, 1548) s.v. &quot;Piperitis&quot; (James Britten, ed., ''The Names of Herbes, by William Turner'' [Londinii: English Dialect Society, 1881] [https://archive.org/details/namesofherbesad100turnuoft/page/62/mode/2up p. 63])&lt;/ref&gt; [[Conradus Gesnerus]] anno [[1561]] e scientia horticultorum Germanorum &quot;Aestiva est herba&quot; recte ait &quot;et nisi primo vere apud nos seratur, siliquas suas pulcherrime rubentes non perficit&quot;.&lt;ref&gt;[[#Gesnerus (1561)]]&lt;/ref&gt; Ergo [[Iohannes Gerardus]], de Anglia anno [[1597]] loquens, has plantas e terris extraneis in Hispaniam [[Italia]]mque importatas &quot;unde nos&quot; ait &quot;semina in nostros hortos Anglicos accepimus, cuius capsicum non iam ad illum splendidum colorem rubrum maturabat quem suá naturá possidet&quot; causa adversitatis climaticae.&lt;ref&gt;''These plants are brought from forren countries, as Ginnee, India and those parts, into Spaine and Italy; from whence wee have received seede for our English gardens, where they come to fruit bearing, but the cod doth not come to that bright red colour which naturally it is possessed with . . . by reason of these unkindely yeeres that are past'': [[#Gerard (1597)]]&lt;/ref&gt;

Interea [[Carolus Clusius]], botanistarum [[saeculum 16|saeculi XVI]] princeps, cultum capsicorum in [[Hispania]] et [[Portugallia]] rettulit tam ex itinere suo annis [[1564]] et [[1565]] suscepto, quam e descriptione [[Nicolaus Monardes|Nicolai Monardes]] quam ipse e [[lingua Hispanica]] Latine verterit:
:Nec praetermittendum est piper ex Indiis nostris missum, quandoquidem non modo in medicum usum receptum est, sed planta est excellentissima et totae Hispaniae notissima: nam nullus est hortus, in quo non seratur ob fructus [[pulchritudo|pulchritudinem]]. Vidi aliquando in [[Hispalis|hac urbe]] ad arboris altitudinem excrescentem. [[Folium|Folio]] est [[viridis|viridi]], ocymo latifolio simili, [[flos|flore]] [[albus|albo]], ex quo pullulat [[fructus]] diversae formae, oblongus, rotundus, melope[po]nis forma aut ceraseorum, sed immaturius viridis est, maturus vero colore rubro gratissimo. Particulatim concisum et iusculo maceratum, meliorem saporem edulus conciliat, quam piper vulgare, ideoque eius usus est in omnibus, in quibus [[aroma]]ta ex [[Insulae Moluccae|Maluccis insulis]] et Calecutio delata commendantur, in eo solum differens, quod illa multis aureis emuntur, hoc sola satione adquiritur: nam in una planta colliguntur aromata in totius anni usum, minore dispendio et maiore commodo. Flatus discutit, utlie pectori et perfrictionibus, calefacitque et roborat partes internas. Caliditatis et siccitatis in quarto fere gradu particeps:&lt;ref&gt;[[#Clusius (1574)]] [https://archive.org/stream/mobot31753003541627#page/70/mode/2up p. 71]. ''No quiero dexir de dezir de la pimienta que traen delas Indias, que non solo sirve a medicina, pero es especie excelentissima, la qual es conocida en toda España, porque no ay jardin ni huerta ni maceton que no la tenga sembrada por la hermosura del fructo que lleva. Es planta grande, tanto que yo he visto en esta ciudad alguna que ygualava con algunes arboles. Echa las hojas verdes a modo de albahaca de la ancha que llaman charanfoli. Echa unas flores blancas de que sale el fructo, que es en diversas formas: unos pimientos son largos, otros redondos, otros de hechura de melones, otros de ceresas, pero todos son al principio quando no estan maduros muy verdes, et maduros muy colorados, con un color muy gracioso. Usan dellos en todos los guisados y potages, porque haze mejor gusto que la pimienta comun: hecho tajadas y echadas en caldo, es salsa excelentissima, usan dellos en todo aquello que sirven las especies que traen de Maluco y de Calicud. Desieren en que las della India cuestan muchos ducados: estotra no cuesta mas que sembrarla, porque en una planta ay especias para todo el año, con menos daño y mas provecho nostro. Conforta mucho, ressuelve ventosidades, son buenos para el pecho, y para los frios de complexion, calienta y conforta, corroborando los miembros principales. Es caliente y seca casi en quarto grado'': [[Nicolaus Monardes|Nicolás Monardes]], ''[[Simplicium medicamentorum ex novo orbe delatorum historia|Dos libros. El uno trata de todas las cosas que traen de nuestras Indias Occidentales]]'' (Hispali, 1565) [https://archive.org/stream/hin-wel-all-00002448-001#page/n100/mode/2up quaternion f 6]&lt;/ref&gt;
Litteris parvis addidit ipse Clusius: &quot;Capsicum hoc seu piper Indicum (Americum potius) diligentissime colitur tota Castella cum ab hortulanis, tum a mulieribus in fenestris aedium suarum. Etenim eo utuntur per totum annum, cum virente tum sicco, pro condimento et pipere. Spectatur varia forma ... sed et haec omnia genera aliquando vidi colore flavescente, in Lusitania, monasterio quodam circa Olysiponem&quot;.&lt;ref&gt;[[#Clusius (1574)]] [https://archive.org/stream/mobot31753003541627#page/74/mode/2up p. 74]&lt;/ref&gt; 

Clusius postea se &quot;Ulyssipone Conimbricam&quot; proficiscente hanc speciem &quot;nonnullis Lusitaniae locis&quot; vidisse meminit, &quot;caule in cubitales ramos diviso, graciliores tamen quam in vulgari&quot;, fructu exiguo &quot;tam acris et fervidi saporis, ut gullatum fauces incenderet&quot;.&lt;ref&gt;[[#Gregorius de Regio (1611)]] [https://archive.org/stream/bub_gb_18RbAAAAQAAJ#page/n117/mode/1up pp. 104-105]&lt;/ref&gt; Quod ille piper Brasilianum appellabat, nos sub nomine ''[[Capsicum baccatum|Capsici baccati]]'' recognoscere possumus.

Idem Clusius primus omnium de cultu prope [[Hungaria]]m hac annotatione relatus est: &quot;Memini etiam videre anno Christi [[1585|M.D.XXCV]] magná copiá cultum in suburbanis [[Brunna]]e celebris [[marchionatus]] [[Moravia]]e urbis hortis, e quo cultores non contemnendum quaestum faciebant; erat enim apud vulgus frequens eius usus&quot;. Hodie eadem fere regione tritura [[paprica]] nomine e varietatibus capsicorum admodum mitioribus producitur.

[[Garcias Lasus Inca]] primus omnium, in [[Hispania]] exsul saeculo XVII ineunte scribens, species tres [[Peruvia]]nas cultas generis ''Capsici'' distinxit, videlicet ''[[rocoto|rocot uchu]]'' (''C. pubescens''), ''[[chinchi-uchu|chinchi uchu]]'' (''C. chinense''), tertiaque cuius nominis oblitus est, id est, ''[[Ají amarillo|kellu-uchu]]'' (''C. baccatum'' var. ''pendulum''). ''Los de mi tierra'', ait, ''son tan amigos del uchu, que no comeron sin el, aunque non sea sino unas yervas crudas'' (&quot;gens patriae meae capsicum tam amant ne ullus cibus sine illud comedatur etiamsi nihil nisi paucae herbae crudae&quot;).&lt;ref&gt;[[#Garcias Lasus (1609)]]&lt;/ref&gt;

Cultus capsicorum a scriptoribus saeculo XVIII describitur, generis &quot;climatum calidissimorum [[America]]e et [[Africa]]e oriundi&quot; cui tempus sationis in Europa ultima hebdomas mensis Februarii esse debet;&lt;ref&gt;''It is a native of the warmest climates of America and Africa ... The last week of [February] will be the proper time for sowing them'': [[#Hill (1757)]]&lt;/ref&gt; solam plantam esse ([[Vicia faba|fabis]] exceptis) asseveratur &quot;cui rustici Provinciae Francicae Occitaniaeque curas minutiores volenter dabunt: promptiores mense Februario, alii Martio serunt&quot;.&lt;ref&gt;''C’est la seule plante, après les fèves, pour laquelle les paysans de Provence et de Languedoc ne plaignent pas les petits soins ... Les plus pressés sèment en février, les autres en mars'': [[#Rozier (1789)]]&lt;/ref&gt; Saeculo XIX ineunte botanistae separatim species tres, videlicet annuam, frutescentem, baccatam recognoverunt,&lt;ref&gt;[[#Descourtilz (1828)]]&lt;/ref&gt; saepe et quartam fructibus dulcibus.&lt;ref&gt;[[Michael Felix Dunal|Michel Félix Dunal]], ''Histoire naturelle, médicale et économique des Solanum''. Lutetiae: Koenig, 1813 {{Google Books|Dc5LAAAAYAAJ}}&lt;/ref&gt; Speciem ''[[Capsicum chinense|C. chinense]]'' [[Nicolaus Iosephus Jacquin]] anno [[1776]] ex horto botanico imperiali [[Vindobona|Vindobonensi]] descripsit.&lt;ref&gt;[[Nicolaus Iosephus Jacquin]], ''Hortus botanicus Vindobonensis'' (Vindobonae: typis Leopoldi Joannis Kaliwoda aulae imperialis typographi, 1770-1776) vol. 3 [https://www.biodiversitylibrary.org/item/10251#page/42/mode/1up p. 38], [https://www.biodiversitylibrary.org/item/10251#page/189/mode/1up tab. 67]&lt;/ref&gt;

== Capsicum in Asia et Africa ==
[[Hancheum|Hanchei]] in urbe [[Sinae|Sinarum]] iam anno [[1591]] ad hortos ornandos cultum est.&lt;ref&gt;''Foreign pepper (''fanjiao'' 䔐㢺): it has dense growth. The flowers are white. The fruits look just like the worn-out tip of a writing brush. Their flavor is hot/spicy (''la'' 彋). Their color is red. They are very pleasing to look at. They grow from seeds'': [[#Gao Lian (1591)]]: vide [[#Dott (2020)]] p. 132&lt;/ref&gt; 

Capsicum &quot;ubique&quot; in [[Corea]], secundum [[Yi Su-gwang]], anno [[1614]] colebatur: &quot;piper barbarorum meridianorum&quot;, i.e. [[Portugallia|Portugallensium]], nuncupatum est; ab [[Iaponia]] in Coream introductum erat.&lt;ref&gt;[[#Yi Su-gwang (1614)]]: vide [[#Dott (2020)]] p. 24&lt;/ref&gt; 

== Usus ==
Fructus [[capsaicinum]], substantiam facile in pingue dissolventem (''lipophilicam'') continet, quod [[dolor]]em ingentem et acerbum per [[receptorium capsaicini]] (TrpV&lt;sub&gt;1&lt;/sub&gt;) inurit.&lt;ref&gt;{{cite journal |authors=Caterina M.J., Schumacher M. A., et al. |title=The capsaicin receptor: a heat-activated ion channel in the pain pathway |journal=Nature |year=1997 |month=Oct |volume=389 |issue=6653 |pages=816-24 |url=https://pubmed.ncbi.nlm.nih.gov/9349813/}}&lt;/ref&gt;. Antagonista TrpV&lt;sub&gt;1&lt;/sub&gt; dolores mitigare queant.&lt;ref&gt;{{cite journal |authors=Cui M., Honore P., et al. |title=TRPV1 receptors in the CNS play a key role in broad-spectrum analgesia of TRPV1 antagonists |journal=The journal of neuroscience |year=2006 |month=Sep |volume=26 |issue=37 |pages=9385-93 |url=https://pubmed.ncbi.nlm.nih.gov/16971522/}}&lt;/ref&gt;

Tali sensu statim agnito, diaetetici Europaei iam medio saeculo XVI capsicum calidum ad gradum tertium vel et quartum ordinabant. Usque in medium saeculum XVIII, sicut antea, auctores hoc aroma hominibus sub climate tripico habitantibus, neque aliis, utile esse aiunt: ita [[Vincentius La Chapelle]], scriptor de re coquinaria Francicus, habitudines &quot;Indicas&quot;, scilicet Indiarum occidentalium, explicavit: ''C'est la manière Indienne. Au lieu de safran ce sont des racines de safran dont ils se servent, et à la place de poivre, c'est du piment'' (&quot;mos est Indorum: loco [[crocus sativus|croci]] [[curcuma]]m, loco [[piper nigrum|piperis]] capsicum adhibent&quot;).&lt;ref&gt;[[Vincentius La Chapelle|Vincent La Chapelle]], ''Le cuisinier moderne'' (2a ed. 5 voll. Hagae Comitum, 1742) 
[https://archive.org/details/lecuisiniermode03chapgoog/page/n106/mode/2up vol. 5 p. 90]&lt;/ref&gt;

Eodem fere tempore horticultor quidam Anglus primus omnium utilitatem capsicorum aceto conditorum confessus est: ''The flower is inconsiderable, but the fruit is conspicuous in the highest degree ... it affords an excellent pickle'' (&quot;Flos eius haud notabilis, fructus autem conspicuissimus condituram optimam praebet&quot;).&lt;ref&gt;[[#Hill (1757)]]&lt;/ref&gt; Distinctio saeculo XVIII exeunte inter provincias septentrionales mediasque [[Francia]]e relata est, capsicis hic ad comesum cottidianum et loco piperis, illic ad hortos ornandos cultis. Insuper provinciis meridianis capsica ad [[ientaculum]] loco cepae et allii consumebantur. Idem auctor condituram capsicorum sicut [[cucumis anguria|anguriorum]] refert atque (primus omnium) confectionem [[acetum e capsicis|aceti e capsicis]] per iniectionem capsicorum in cupas acetarias &quot;ut vim aceti maturescentis augetur&quot;.&lt;ref&gt;''Dans la majeure partie de nos provinces du nord, on ne cultive cette plante que pour la décoration des potagers ... Il n’en est pas ainsi dans les provinces de l’intérieur, son fruit dans la maturité et quand il est sec tient complètement lieu de poivre dans les cuisines des grandes et petites fermes. Dans nos provinces du midi, leurs habitans préfèrent un poivron à l’oignon et à l’ail pour le repas du matin. Le poivron est ce fruit encore petit et vert, et qui n’a pas encore changé de couleur. Lorsque sa robe a pris la teinte du corail, il ne sert plus que pour la cuisine ... Le fruit tient lieu de poivre à une très-grande partie des habitans de ce royaume. Quelques personnes font confire dans le vinaigre les poivrons, de la même manière que les cornichons. Les marchands de vinaigre ont grand soin d’ajouter une certaine quantité de poivrons murs et secs dans leurs barriques de vinaigre, dont ils augmentent singulièrement la &quot;force&quot;'': [[#Rozier (1789)]]&lt;/ref&gt; Illi autem, qui usus &quot;Indorum&quot; [[Mare Caribaeum|Caribicorum]] Hispanorumque et Lusitanorum descripserunt, alias confectiones varias enumeraverunt: capsica e saccharo condita; sorbitiones patinasque et elixa e capsicis accommodata; parationem [[butyrum cayennae|butyri cayennae]].&lt;ref&gt;''Les Indiens les préfèrent au poivre ordinaire, et les mangent crus. On les confit aussi au sucre et l'on en porte sur mer pour servir dans les voyages de long cours ... On les cueille aussi en vert ... La plupart des autres espèces de piment sont en usage chez les Indiens, qui en mêlent dans leurs ragoûts [et] en font des espèces de bouillons ou de décoctions très-fortes ... Les Portugais établis dans ces contrées appellent ces potions stomachiques ''caldo di pimento'' ... C'est particulièrement avec le piment à petites baies que les Indiens préparent leur beurre de cayan'': [[#Descourtilz (1828)]]&lt;/ref&gt;

Fructus capsicorum inter pabula salubriora gallinis dari solet.&lt;ref&gt;A. A. El-Deek et al., &quot;[https://www.researchgate.net/publication/288974515_Hot_pepper_Capsicum_Annum_as_an_alternative_to_oxytetracycline_in_broiler_diets_and_effects_on_productive_traits_meat_quality_immunological_responses_and_plasma_lipids Hot pepper (Capsicum Annum) as an alternative to oxytetracycline in broiler diets and effects on productive traits, meat quality, immunological responses and plasma lipids]&quot; in ''Archiv fur Geflügelkunde'' vol. 76  (2012) pp. 73-80&lt;/ref&gt; In medicina veterinaria aut [[capsaicinum]] aut &quot;capsici fructus acer&quot; (i.e. fructus siccus [[capsicum cayennense|capsici cayennensis]]) aut ''[[Capsicum frutescens]]'' praescribitur ad {{Creanda|en|Virulent Newcastle disease|morbus Newcastle|morbum Newcastle}} avium, praesertim [[gallus gallinaceus|gallinarum]] curandum.&lt;ref&gt;&quot;[https://www.ema.europa.eu/en/documents/mrl-report/capsici-fructus-acer-summary-report-committee-veterinary-medicinal-products_en.pdf Capsici fructus acer]&quot; (1999) apud ''European Agency for the Evaluation of Medicinal Products''; M. M. A. Mtambo et al., &quot;[https://www.suaire.sua.ac.tz/bitstream/handle/123456789/4104/biblio-vm-17-mtambo.pdf Evaluation of the efficacy of the crude extracts of Capsicum frutescens, Citrus limon and Opuntia vulgaris against Newcastle disease in domestic fowl in Tanzania]&quot; in ''Journal of Ethnopharmacology'' vol. 68 (1999) pp. 55–61; Cheryl Lans et al., &quot;[https://www.academia.edu/28751035/Ethnoveterinary_Medicine_Potential_Solutions_for_Large_Scale_Problems Ethnoveterinary Medicine: Potential Solutions for Large-Scale Problems?]&quot; in ''Veterinary Herbal Medicine'' (2007)&lt;/ref&gt; Fructus capsicorum praescribuntur a populo [[Aguaruna]] [[Peruvia]]e ad [[diarrhoea]]m [[canis|canium]] venatorum, sicut et hominum, medendam.&lt;ref&gt;Kevin A. Jernigan, &quot;[https://d-nb.info/1109301847/34 Barking up the same tree: a comparison of ethnomedicine and canine ethnoveterinary medicine among the Aguaruna]&quot; in ''Journal of Ethnobiology and Ethnomedicine'' vol. 5 (2009) no. 33&lt;/ref&gt;

== Varietates ==
{{Vide-etiam|Varietates capsicorum}}

== Pulveres ==
Hae [[tritura]]e de fructibus capsicorum a coquis saepe adhibentur:
* [[Paprica]]
* [[Pulvis cayennae]]
* [[Pulvis capsici]]
* [[Pulvis caril]] (aromatum variorum tritura)

== Notae ==
&lt;references/&gt;

==Bibliographia==
; Fontes antiquiores
* 1493 : &lt;span id=&quot;Columbi ephemeris (1493)&quot;&gt;&lt;/span&gt;[[Christophorus Columbus]], Ephemeris primae navigationis a [[Bartholomaeus Casaus|Bartholomaeo Casao]] rescripta, in Martin Fernandez de Navarrete, ed., ''Colección de los viages y descubrimientos'' vol. 1 (2a ed. Matriti, 1858) [https://archive.org/stream/coleccindelosv01nava#page/152/mode/2up pp. 153-313]
* 1493-1503 : &lt;span id=&quot;Columbi liber copiarum (1493-1503)&quot;&gt;&lt;/span&gt;[[Christophorus Columbus]], Epistulae ineditae in Antonio Rumeu de Armas, ed., ''Libro copiador de Cristóbal Colón: correspondencia inédita con los Reyes Católicos sobre los viajes a América'' (Matriti, 1989) [[:es:Libro copiador de Colón#Enlaces externos|Editiones interretiales]]
* 1494 : &lt;span id=&quot;Chanca (1494)&quot;&gt;&lt;/span&gt;[[Didacus Álvarez Chanca]], Epistula de secunda navigatione Christophori Columbi, in Martin Fernandez de Navarrete, ed., ''Colección de los viages y descubrimientos'' vol. 1 (2a ed. Matriti, 1858) [https://archive.org/stream/coleccindelosv01nava#page/370/mode/2up pp. 370-371]
* post 1519 : &lt;span id=&quot;Conquisitator anonymus (post 1519)&quot;&gt;&lt;/span&gt; &quot;[[Relatione di alcune cose della Nuova Spagna e della gran città di Temestitan Messico]]&quot; in [[Ioannes Baptista Ramusius|Giovanni Battista Ramusio]], ed., ''[[Navigationi et viaggi (Ramusius)|Navigationi et viaggi]]'' (Venetiis, 1555-1559) vol. 3 [https://archive.org/details/terzovolumedelle32ramu/page/n693/mode/2up vol. 3 f. 306r]; [https://archive.org/details/dellenavigationi00ramu/page/n599/mode/2up vol. 3 f. 255v editionis 1606]
* 1526/1557 : &lt;span id=&quot;Oviedo (ante 1557)&quot;&gt;&lt;/span&gt;[[Gundisalvus Fernández de Oviedo y Valdés|Gonzalo Fernandez de Oviedo y Valdés]], ''[[Historia general y natural de las Indias, islas y tierra-firme del mar Océano]]'' (José Amador de los Rios, ed. 4 voll. Matriti: Real Academia de la Historia, 1851-1855 [https://archive.org/details/gri_33125007267921/page/274/mode/2up vol. 1 p. 275])
* 1530 : &lt;span id=&quot;Petrus Martyr (1530)&quot;&gt;&lt;/span&gt;[[Petrus Martyr ab Angleria]], ''[[De orbe novo decades]]'' lib. 5 cap. 9 [https://archive.org/stream/deorbenouopetrim00angh#page/n173/mode/2up f. 83r]
* c. 1536 : [[Bartholomaeus Casaus]], ''Apologética historia sumaria'' cap. 10 (M. Serrano y Sanz, ed., ''Historiadores de Indias'' vol. 1 [Matriti: Bailly-Baillière, 1909] [https://archive.org/details/historiadoresdei01serr/page/26/mode/2up p. 27])
* 1539 : &lt;span id=&quot;Bock (1539)&quot;&gt;&lt;/span&gt;[[Hieronymus Bock]], ''[[Kreutterbuch (Bock)|New Kreütter Buch]]'' [https://www.digitale-sammlungen.de/de/view/bsb11069345?page=546 pars 2 f. 86v]; eiusdem ''Kreüterbuch'' (1551) [https://bildsuche.digitale-sammlungen.de/index.html?c=viewer&amp;bandnummer=bsb00091270&amp;pimage=741 ff. 350r-351r]; eiusdem ''Kreutterbuch'' (1560) [https://archive.org/stream/mobot31753000820529#page/n725/mode/2up ff. 342-344]
* ante 1542 : ''[[Codex Mendoza]]'' (liber pictus, c. 1534/1542) f. 51v-52r etc.; cf. Frances Berdan, Patricia Anawalt, ''The Essential Codex Mendoza'' (Berkeleiae, 1997) pp. 132-133 etc.
* 1542 : &lt;span id=&quot;Fuchsius (1542)&quot;&gt;&lt;/span&gt;[[Leonhartus Fuchsius]], ''[[De historia stirpium commentarii insignes]]'' [https://archive.org/stream/Dehistoriastirp00Fuch#page/730/mode/2up pp. 731-735]
* 1550 : [[Hieronymus Cardanus]], ''De subtilitate'' [https://archive.org/details/bub_gb_Tmf3wRsurVsC/page/n235 pp. 197-198]
* 1554 : &lt;span id=&quot;Dodonaeus (1554)&quot;&gt;&lt;/span&gt;[[Rembertus Dodonaeus]], ''Posteriorum trium de stirpium historia commentariorum imagines'' {{Google Books|fCo6AAAAcAAJ|pp. 181-183}}; &lt;span id=&quot;Dodonaeus (1557)&quot;&gt;&lt;/span&gt;eiusdem ''Histoire des plantes'' (1557) [https://archive.org/stream/hin-wel-all-00000412-001#page/n470/mode/2up pp. 441-442]
* 1557 : [[Ioannes Stadius|Hans Staden]], ''Warhaftige Historia und Beschreibung eyner Landtschafft der wilden, nacketen, grimmigen menschfresser Leuthen in der Newenwelt America gelegen'' [https://archive.org/details/staden/page/n145/mode/2up lib. 2 cap. 11], [https://archive.org/details/staden/page/n179/mode/2up cap. 38]; [https://archive.org/details/americaetertiapa00stad/page/110/mode/2up p. 110], [https://archive.org/details/americaetertiapa00stad/page/132/mode/2up p. 132 versionis Latinae 1592]
* 1557 : [[Iulius Caesar Scaliger]], ''Exotericarum exercitationum liber quintus decimus, de subtilitate, ad Hieronymum Cardanum'' {{Google Books|LB88AAAAcAAJ|ff. 200v-201r}}
* 1561 : &lt;span id=&quot;Gesnerus (1561)&quot;&gt;&lt;/span&gt;[[Conradus Gesnerus]], &quot;De hortis Germaniae&quot; in [[Valerius Cordus]], ''Annotationes in Dioscoridis ... [etc.]'' [http://dl.ub.uni-freiburg.de/diglit/cordus1561/0566 f. 272b] &quot;Piper Indicum, capsicum ... piper Hispanicum, Calecuticum, Bresilicum&quot;
* 1565 : [[Nicolaus Monardes]], ''[[Simplicium medicamentorum ex novo orbe delatorum historia|Dos libros. El uno trata de todas las cosas que traen de nuestras Indias Occidentales]]'' [https://archive.org/stream/hin-wel-all-00002448-001#page/n100/mode/2up quaternion f 6]
* ante 1566 : [[Bartholomaeus Casaus|Bartolomé de las Casas]], ''Apologética historia sumaria'' [https://archive.org/stream/historiaindias04casarich#page/304/mode/2up pp. 304-305 editionis 1876]
* 1573 : [[Petrus Andreas Matthiolus]], ''[[Commentarii (Matthiolus)|I discorsi ... nelle sei libri di Pedacio Dioscoride Anazarbeo della materia medicinale]]'' (editio 1573) lib. 2 cap. 148 {{Google Books|w8JCAAAAcAAJ|p. 405}}; idem, ''Commentarii in libros sex Pedacii Dioscoridis Anazarbei de Materia Medica'' (editio 1574) lib. 2 cap. 153 [https://archive.org/stream/mobot31753000819257#page/433/mode/2up pp. 434-437]
* 1574 : &lt;span id=&quot;Clusius (1574)&quot;&gt;&lt;/span&gt;[[Nicolaus Monardes]]; [[Carolus Clusius]] (ed.), ''[[Simplicium medicamentorum ex novo orbe delatorum historia|De simplicibus medicamentis ex occidentali India delatis quorum in medicina usus est]]'' [https://archive.org/stream/mobot31753003541627#page/71/mode/2up pp. 71-74]
* ante 1581 : &lt;span id=&quot;Duran (ante 1581)&quot;&gt;&lt;/span&gt; [[Didacus Duran]], ''[[Codex Duran|Historia de las Indias de Nueva-España y islas de Tierra Firme]]'' (Mexicopoli, 1867-1880) vol. 1 [https://archive.org/stream/historiadelasind01dur#page/210/mode/2up p. 211] etc.
* ante 1584 : &lt;span id=&quot;Díaz (a. 1584)&quot;&gt;&lt;/span&gt;[[Bernardus Díaz del Castillo]], ''[[Historia verdadera (Díaz)|La historia verdadera de la conquista de la Nueva España]]'' (manuscriptum, ante 1584) cap. 83 [http://www.rae.es/sites/default/files/Aparato_de_variantes_Historia_verdadera_de_la_conquista_de_la_Nueva_Espana.pdf p. 254 editionis interretialis Serés]
* ante 1585 : &lt;span id=&quot;Sahagún (ante 1585)&quot;&gt;&lt;/span&gt;[[Bernardinus de Sahagun]], ''[[Historia general de las cosas de Nueva España]]'' lib. 10 [https://www.loc.gov/resource/gdcwdl.wdl_10621/?sp=100&amp;st=image f. 48v-49r] (Charles E. Dibble, Arthur J. O. Anderson, edd., ''Florentine Codex: Book 10: The People'' [Santa Fe: School of American Research, 1961] pp. 67-68)
* 1590 : &lt;span id=&quot;I. Acosta (1590)&quot;&gt;&lt;/span&gt;[[Iosephus de Acosta]], ''[[Naturalis et moralis Indiae Occidentalis historia|Historia natural y moral de las Indias]]'' (Hispali: en casa de Juan de Leon) [https://archive.org/details/historianaturaly00acos_2/page/246/mode/2up pp. 246-247]
* 1590 : &lt;span id=&quot;Tabernaemontanus (1590)&quot;&gt;&lt;/span&gt;[[Iacobus Theodorus Tabernaemontanus]], ''[[Eicones plantarum seu stirpium]]'' [https://archive.org/stream/Eiconesplantaru00Theo#page/858/mode/2up p. 859]
* 1591 : &lt;span id=&quot;Gao Lian (1591)&quot;&gt;&lt;/span&gt;[[Gao Lian]], ''[[Commentarii octo de vitae principiis]]'' (遵生八牋: vide [[#Dott (2020)]] p. 132)
* 1591 : [[Ioannes de Cárdenas|Juan de Cárdenas]], ''Primera parte de los problemas y secretos maravillosos de las Indias'' (Mexicopoli) [https://archive.org/stream/primerapartedelo00cr#page/112/mode/2up pp. 113-115 editionis 1913]
* ante 1595 : &lt;span id=&quot;Recchus (ante 1595)&quot;&gt;&lt;/span&gt;[[Franciscus Hernandez]]; [[Nardus Antonius Recchus]], scriba, ''[[Rerum medicarum Novae Hispaniae thesaurus|De materia medica Novae Hispaniae, Philippi Secundi Hispaniarum ac Indiarum regis invictissimi iussu]]'' (manuscriptum, ante 1595) [https://archive.org/stream/demateriamedican00hern#page/n189 ff. 92v-94r]
* 1596 : &lt;span id=&quot;Bauhinus (1596)&quot;&gt;&lt;/span&gt;[[Casparus Bauhinus]], ''[[Phytopinax]]''. Basileae: per Sebastianum Henricpetri [https://archive.org/stream/mobot31753000815909#page/154/mode/2up pp. 155-156]
* 1597 : &lt;span id=&quot;Gerard (1597)&quot;&gt;&lt;/span&gt;[[Iohannes Gerardus|John Gerard]], ''[[Herbal (Gerard)|The Herball, or generall historie of plantes]]'' [https://archive.org/stream/mobot31753000817749#page/292/mode/2up pp. 292-293 &quot;Of Ginny or Indian pepper&quot;]
* 1601 : &lt;span id=&quot;Spachius (1601)&quot;&gt;&lt;/span&gt;[[Israel Spachius]], interpr.; [[Ioannes Fragosus]], ''Aromatum, fructuum et simplicium aliquot medicamentorum ex India utraque .. in Europam delatorum ... historia brevis''. Argentinae {{Google Books|0oo6AAAAcAAJ|f. 33r}} &quot;Piper Indorum Occidentalium, quod vocant Axi&quot; 
* 1609 : &lt;span id=&quot;Garcias Lasus (1609)&quot;&gt;&lt;/span&gt;[[Garcias Lasus Inca]], ''[[Comentarios Reales de los Incas]]'' [https://archive.org/stream/primerapartedelo00vega#page/n445/mode/2up vol. 1 f. 210]; [http://shemer.mslib.huji.ac.il/lib/W/ebooks/001531300.pdf recensio interretialis]
* 1611 : &lt;span id=&quot;Gregorius de Regio (1611)&quot;&gt;&lt;/span&gt;[[Gregorius de Regio]]; Carolus Clusius, interpr., &quot;[[De varietate capsicorum]]&quot; in [[Carolus Clusius]], ''Curae posteriores'' [https://archive.org/stream/bub_gb_18RbAAAAQAAJ#page/n107/mode/2up pp. 95-108]
* 1614 : &lt;span id=&quot;Yi Su-gwang (1614)&quot;&gt;&lt;/span&gt;[[Yi Su-gwang]], ''[[Variae commentationes Jibong]]'' (''Jibong yuseol'': vide [[#Dott (2020)]] p. 24) 
* 1615 : &lt;span id=&quot;Ximenez (1615)&quot;&gt;&lt;/span&gt;[[Franciscus Hernandez]]; Francisco Ximenez, interpr., ''[[Rerum medicarum Novae Hispaniae thesaurus|Quatro libros de la naturaleza y virtudes de las plantas y animales que estan recevidos en el uso de medicina en la Nueva Espana]]'' (1615) lib. 2 cap. 3 [http://bibdigital.rjb.csic.es/spa/Libro.php?Libro=4961 ff. 72r-74r], ''Rerum medicarum Novae Hispaniae thesaurus'' (1628) lib. 5 cap. 3 [https://archive.org/stream/rerummedicarumno00hern#page/134/mode/2up pp. 134-138]
* c. 1622 : [[Ioannes Smith (explorator)|Ioannes Smith]], ''The Historye of the Bermudaes or Summer Islands'' (ed. J. Henry Lefroy. Londinii: Hakluyt Society, 1889 [https://archive.org/details/historyebermuda00unkngoog/page/n309/mode/2up p. 277])
* 1630 : Bartholomaeus Ambrosinus, ''De capsicorum varietate cum suis iconibus brevis historia''. Bononiae
* 1635 : &lt;span id=&quot;Nierembergius (1635)&quot;&gt;&lt;/span&gt;[[Ioannes Eusebius Nierembergius]], ''Historia naturae maxime peregrinae'' (Antverpiae: ex officina Plantiniana, 1635) [https://archive.org/details/IoannisEvsebiiN00Nier/page/362/mode/2up pp. 363-364]
* 1640 : [[Ioannes Parkinsonus]], ''Theatrum botanicum'' [https://babel.hathitrust.org/cgi/pt?id=ucm.5325114272;view=1up;seq=405 pp. 355-359] {{Google Books|5m72g_lC-RcC}}
* 1640 : [[Basilius Besler]], ''Hortus Eystettensis''. 2a ed. Norimbergae [https://archive.org/details/mobot31753003651095/page/6/mode/1up classis autumnalis tabb. 6-13]
* 1642 : &lt;span id=&quot;Bontius (1642)&quot;&gt;&lt;/span&gt;[[Iacobus Bontius]], ''De medicina Indorum libri IV'' (Lugduni Batavorum: apud Franciscum Hackium) {{Google Books|Es-IGwAACAAJ|pp. 90-91}} &quot;An nescis, eos addere fructum ricini Americani, quod lada Chili Malaii vocant, quasi dicas Piper è Chile, Brasiliae contermina regione ...&quot;
* 1648 : [[Gulielmus Piso]], [[Georgius Marcgravius]], ''[[Historia naturalis Brasiliae]]''. Lugduni Batavorum: apud Franciscum Hackium [https://archive.org/details/mobot31753000818648/page/n121/mode/2up pars i (Piso) pp. 107-108], [https://archive.org/details/McGillLibrary-osl_historia-naturalis-brasiliae_folioP678h1648-20882/page/38/mode/2up pars ii (Marcgravius) pp. 39-41]
* 1648 : [[Thomas Gage (peregrinator)|Thomas Gage]], ''The English-American his Travail by Sea and Land, or A new survey of the West India's''. Londinii: John Sweeting [https://archive.org/details/graff_1470/page/n115/mode/2up pp. 98-101, 108-111, 140-143]
* ante 1653 : [[Barnabas Cobo|Bernabé Cobo]], ''Historia del Nuevo Mundo'' (1890-1893) ([https://archive.org/details/historiadelnuev00cobogoog/page/370/mode/2up vol. 1 pp. 371-374]; [https://archive.org/details/A331002/page/n463/mode/2up libri manu scripti ff. 228r-229v]
* 1658 : &lt;span id=&quot;Bontius ed. Piso (1658)&quot;&gt;&lt;/span&gt;[[Iacobus Bontius]]; [[Gulielmus Piso]], ed., ''De Indiae utriusque re naturali et medica libri'' (Amstelaedami: apud Elzevirios) [https://archive.org/details/mobot31753002909064/page/n245/mode/2up pars i (Piso) pp. 225-226] (&quot;Quiya sive piper Brasiliense ... Teste Ximene Mexicani hanc plantam ''Chilli'' vocant, quae fert siliquas illas Hispaniolae Incolae ''Axi'' et Antiqui Siliquastrum, Hispani Piper vocant Americanum, et Auctarius Capsicum&quot;), [https://archive.org/details/mobot31753002909064/page/200/mode/2up pars iii (Bontius) p. 200]
* 1699 : [[Robertus Morison]], ''Plantarum historiae universalis Oxoniensis''. Oxoniae {{Google Books|lqPQH-5-d-oC|pars iii pp. 528-531; sectionis 13 tabula 2}} (&quot;Solanum urens Capsicum dictum&quot;)
* 1693 : &lt;span id=&quot;Raius (1693)&quot;&gt;&lt;/span&gt;[[Ioannes Raius]], ''Historia plantarum generalis'' (1693) {{Google Books|btw-AAAAcAAJ|vol. 1 pp. 676-679}}
* 1699 : [[Lionel Wafer]], ''A New Voyage and Description of the Isthmus of America''. Londinii (George Parker Winship, ed., 1903 [https://archive.org/details/newvoyagedescrip00wafe/page/106/mode/2up p. 107])
* 1700 : [[Iosephus Pitton Tournefort]], ''[[Institutiones rei herbariae (Tournefort)|Institutiones rei herbariae]]'' [https://archive.org/details/mobot31753000521648/page/151/mode/2up pp. 152-153], [https://archive.org/details/mobot31753000521655/page/65/mode/2up tabula 66]
* 1678-1703 : [[Henricus van Rheede|Henricus van Rhede tot Draakestein]], ''[[Hortus Malabaricus|Horti Malabarici]] pars prima [... duodecima]''. Amstelaedami: sumptibus Johannis van Someren, et Joannis van Dyck [https://www.biodiversitylibrary.org/item/14375#page/342/mode/1up pars 2 tab. 56, p. 109]; [https://www.biodiversitylibrary.org/item/14379#page/88/mode/1up pars 9 tab. 35]
* c. 1722 : [[Franciscus Ximénez de Quesada]], ''Historia natural de la provincia de San Vicente de Chiapas y Guatemala'' (fide [[#Andrews (1995)]] p. 32)
* 1723-1735 : Aerae Yongzheng ''Chorographia Shandong'' (山东通志): 秦椒，色红有子与花椒味俱辛 (&quot;Pipera ''qin'' colore rubra, granis plena, tam calida quam [[zanthoxyli fructus]]&quot;) 
* ante 1725 : Henry Barham, ''Hortus Americanus'' (Kingston Iamaicae, 1794) [https://archive.org/details/b29319870/page/30/mode/2up pp. 30-31]
* 1707-1725 : &lt;span id=&quot;Sloane (1707-1725)&quot;&gt;&lt;/span&gt;[[Ioannes Sloane|Hans Sloane]], ''A Voyage to the Islands Madera, Barbadoes, Nieves, S. Christophers and Jamaica'' (2 voll. Londinii) [https://www.biodiversitylibrary.org/item/11242#page/411/mode/1up vol. 1 pp. 240-243] et [https://www.biodiversitylibrary.org/item/11242#page/581/mode/1up tab. 146], [https://www.biodiversitylibrary.org/item/11241#page/401/mode/1up vol. 2 p. 378]
* 1737 : [[Ioannes Burmannus]], ''Thesaurus Zeylanicus''. Amstelaedami: apud Janssonio-Waesbergios &amp; Salomonem Schouten [https://www.biodiversitylibrary.org/item/14648#page/94/mode/1up p. 53]
* 1737 : [[Carolus Linnaeus]], ''[[Hortus Cliffortianus]]''. Amstelodami [https://www.biodiversitylibrary.org/item/13838#page/105/mode/1up pp. 59-60]
* 1747 : {{Rumphius}} [https://archive.org/details/mobot31753000819455/page/247/mode/2up vol. 5 pp. 247-252 et tab. 88]; cf. {{Merrill}} [http://www.biodiversitylibrary.org/page/38882613#page/470/mode/1up p. 462]
*:&quot;Malaice ''Lada Tschili'', ac tantummodo ''Tschili'', unde multi crediderunt, hoc Piperis genus primum ex regno Chili fuisse translatum, quod cum vero non videtur convenire&quot; (lib. 8, cap. 50).
* 1752 : &quot;Brasilien-Pfeffer&quot; in Carl Günther Ludovici, ''Eröffnete Akademie der Kaufleute, oder vollständiges Kaufmanns-Lexicon'' vol. 1 (Lipsiae, 1752) {{Google Books|dbxRAAAAcAAJ|coll. 2091-2095}}
* 1753-1761 : [[Petrus Kalm|Pehr Kalm]], ''En resa til Norra America'' [https://www.biodiversitylibrary.org/bibliography/35535 Textus] (fide [[#Andrews (1995)]] p. 32)
* 1756 : Patrick Browne, ''The Civil and Natural History of Jamaica''. Londinii: White {{Google Books|lX7LZEdcZ7YC|pp. 176-177}}
* 1757 : &lt;span id=&quot;Hill (1757)&quot;&gt;&lt;/span&gt;John Hill, ''Eden, or a compleat body of gardening'' (Londinii: Osborne, 1757) [https://archive.org/details/mobot31753000809647/page/n24/mode/1up pp. 13-14]
* 1768 : [[Philippus Miller|Philip Miller]], ''The Gardener's Dictionary''. 8a ed. Londinii [https://archive.org/details/b30454190/page/n214/mode/1up s.v. &quot;Capsicum&quot;]
* 1789 : &lt;span id=&quot;Rozier (1789)&quot;&gt;&lt;/span&gt;Abbé Rozier, ''Cours d'agriculture'' (12 voll. Lutetiae, 1781-1805) [https://archive.org/details/bub_gb_6vHvDhT7xc8C/page/n219/mode/2up vol. 8 pp. 170-171] (&quot;Poivre d'Inde ou de Guinée, ou poivre long, ou corail des jardins&quot;)
* 1794 : {{Creanda|de|Ignaz Pfefferkorn|Ignatius Pfefferkorn|Ignaz Pfefferkorn}}, ''Beschreibung der Landschaft Sonora'' (fide [[#Andrews (1995)]] p. 33) 
* 1796 : [[Yuan Mei]], ''[[Praecepta ex horto plenitudinis|Suiyuan shidan]]'' (Sean J. S. Chen, ed. et interpr., [https://wayoftheeating.wordpress.com/2019/04/26/side-dishes-10-lahu-sauce/ 喇虎醬 = Lahu Sauce])
* 1807 : [[Grimod de la Reynière]] et al., ''[[Almanach des gourmands]]'' vol. 5 (1807) [https://archive.org/details/b21525250_0005 p. 183] (&quot;piment ou poivre-long&quot;); vol. 7 (1810) pp. 143-147, 185-186 (&quot;piment enragé ou poivre de Guinée&quot;)
* 1820 : [[Ioannes Crawfurd|John Crawfurd]], ''History of the Indian Archipelago''. Edinburgi {{Google Books|8EdSAAAAcAAJ|vol. 1 p. 377}}
* 1826 : [[Whitelaw Ainslie]], ''Materia Indica''. Londinii: Longman [https://archive.org/details/materiaindicaors01ains/page/306/mode/2up vol. 1 pp. 306-308]
* 1828 : &lt;span id=&quot;Descourtilz (1828)&quot;&gt;&lt;/span&gt;Michel Étienne Descourtilz, ''Flore médicale des Antilles'' fasc. 6 (1828) [https://www.biodiversitylibrary.org/item/21850#page/216/mode/1up pp. 172-181, tabb. 422-423] (&quot;vulg. Poivre d'Inde; Piment zozo, piment enragé, poivre d'oiseau, piment caraìbe&quot;)
* 1832 : Edwin Lankester, ''Vegetable Substances Used for the Food of Man'' [https://archive.org/details/vegetablesubstan00lank/page/312/mode/2up pp. 313-314]
* 1852 : [[Michael Felix Dunal|M. F. Dunal]], &quot;Solanaceae&quot; in [[Alphonsus Pyramus de Candolle|A. de Candolle]], ''[[Prodromus systematis naturalis regni vegetabilis]]'' vol. 13 fasc. 1 (Lutetiae: Masson, 1852) [https://www.biodiversitylibrary.org/page/56755458#page/432/mode/1up p. 428]
* 1868 : &quot;Pimenteira da terra&quot; in Theodoro J. H. Langgaard, ''Novo formulario medico e pharmaceutico'' (1868) {{Google Books|QNk8AAAAcAAJ|pp. 537-538}}
* 1883 : [[Alphonsus Pyramus de Candolle|Alphonse de Candolle]], ''Origine des plantes cultivées'' (Lutetiae: Baillière) [https://archive.org/details/originedesplant02candgoog/page/n241/mode/2up pp. 229-230]
* 1885 : William Dymock, ''The Vegetable Materia Medica of Western India''. Bombay {{Google Books|0ygJAAAAIAAJ|pp. 640-643}}
* 1889 : George Watt, ''A Dictionary Of The Economic Products Of India'' vol. 2 (Calcuttae) [https://archive.org/details/DictionaryOfTheEconomicProductsOfIndia2/page/n137/mode/2up pp. 134-140]
* 1919 : [[Eduardus Ludovicus Sturtevant|E. L. Sturtevant]]; U. P. Hedrick, ed., ''Sturtevant's notes on edible plants'' (''Report of the New York Agricultural Experiment Station'', 1919, pars 2. Albaniae, 1919) [https://archive.org/details/sturtevantsnotes00sturuoft/page/134/mode/2up pp. 134-140]

; Lexicographica
* &quot;ají&quot;, &quot;chile&quot;, &quot;cumarí&quot;, &quot;malagueta&quot;, &quot;uchu&quot; in Georg Friederici, ''Amerikanistisches Wörterbuch'' (Hamburgi: Cram, De Gruyter, 1947) pp. 46, 174-175, 225, 369-371, 639 {{Google Books|j7J6DwAAQBAJ|Paginae selectae}} 

; Eruditio recentior
* Heather Arndt Anderson, ''Chilli: a global history''. Londinii: Reaktion Books, 2016. ISBN 978 1 78023 635 3
* &lt;span id=&quot;Andrews (1995)&quot;&gt;&lt;/span&gt;[[Ioanna Andrews|Jean Andrews]], ''Peppers: the domesticated capsicums''. 2a ed. Austin: University of Texas Press, 1995
* Katherine L. Chiou, [[Christina Hastorf|Christine A. Hastorf]], &quot;A Systematic Approach to Species-Level Identification of Chile Pepper (Capsicum spp.) Seeds: Establishing the Groundwork for Tracking the Domestication and Movement of Chile Peppers through the Americas and Beyond&quot; in ''Economic Botany'' vol. 68 (2014) pp. 316-336 [https://www.jstor.org/stable/43305668 JSTOR]
* D. J. Cotter, ''[http://contentdm.nmsu.edu/cdm/compoundobject/collection/AgCircs/id/30347/rec/1 A Review of Studies on Chile]''. NMSU Cooperative Extension Service and Agricultural Experiment Station, 1980
* &lt;span id=&quot;Davidson (1999)&quot;&gt;&lt;/span&gt;&quot;Chilli&quot; in [[Alanus Davidson|Alan Davidson]], ''The Oxford Companion to Food'' (Oxonii: Oxford University Press, 1999. ISBN 0-19-211579-0) pp. 169-171 
* &lt;span id=&quot;De, ed. (2003)&quot;&gt;&lt;/span&gt;Amit Krishna De, ed., ''Capsicum: The genus Capsicum''. Londinii: Taylor &amp; Francis, 2003 {{Google Books|zepxG-bjUYUC|Paginae selectae}}
* David C. Haak et al., &quot;[https://royalsocietypublishing.org/doi/10.1098/rspb.2011.2091 Why are not all chilies hot? A trade-off limits pungency]&quot; in ''Proceedings of the Royal Society B: Biological Sciences'' vol. 279 (2011/2012)
* &lt;span id=&quot;Ramalho do Rêgo et al, edd. (2016)&quot;&gt;&lt;/span&gt;Elizanilda Ramalho do Rêgo, Mailson Monteiro do Rêgo, Fernando Luiz Finger, edd., ''Production and Breeding of Chilli Peppers (Capsicum spp.)''. Springer, 2016
* &lt;span id=&quot;Russo, ed. (2012)&quot;&gt;&lt;/span&gt;Vincent M. Russo, ed., ''Peppers: Botany, Production and Uses''. Wallingford: CAB International, 2012 {{Google Books|dA5mkzfmATEC|Paginae selectae}}
* Joshua Tewksbury et al., &quot;[https://www.researchgate.net/publication/7207709_Where_did_the_Chili_Get_its_Spice_Biogeography_of_Capsaicinoid_Production_in_Ancestral_Wild_Chili_Species Where did the Chili Get its Spice? Biogeography of Capsaicinoid Production in Ancestral Wild Chili Species]&quot; in ''Journal of Chemical Ecology'' vol. 32 (2006) pp. 547-564

; De origine et historia
* Araceli Aguilar-Meléndez et al., &quot;[https://www.researchgate.net/publication/51180103_Genetic_diversity_and_structure_in_semiwild_and_domesticated_chiles_Capsicum_Annuum_Solanaceae_from_Mexico Genetic diversity and structure in semiwild and domesticated chiles (Capsicum Annuum; Solanaceae) from Mexico]&quot; in ''American Journal of Botany'' vol. 96 (2009) pp. 1190-1202
* [[Ioanna Andrews|Jean Andrews]], &quot;The Peripatetic Chilli Pepper: diffusion of the domesticated capsicums since Columbus&quot; in Nelson Foster, Linda S. Cordell, edd., ''Chilies to Chocolate: Food the Americas Gave the World'' (Tucson: University of Arizona Press, 1992) pp. 81-94 {{Google Books|VsQevENGMr0C|Paginae selectae}} [https://archive.org/details/chiliestochocola00fost Exemplar mutuabile]
* Cecil H. Brown et al., &quot;The Paleobiolinguistics of Domesticated Chili Pepper (Capsicum spp.)&quot; in ''Ethnobiology Letters'' vol. 4 (2013) pp. 1-11 [https://www.jstor.org/stable/26423551 JSTOR]
* &lt;span id=&quot;Carvalho et al. (2014)&quot;&gt;&lt;/span&gt;Sabrina Isabel C. de Carvalho et al., &quot;[https://www.researchgate.net/publication/265649073_Morphological_and_genetic_relationships_between_wild_and_domesticated_forms_of_peppers_Capsicum_frutescens_L_and_C_chinense_Jacquin Morphological and genetic relationships between wild and domesticated forms of peppers (Capsicum frutescens L. and C. chinense Jacquin)]&quot; in ''Genetics and molecular research'' vol. 13 (2014) pp. 7447-7464
* &lt;span id=&quot;Chiou et Hastorf (2014)&quot;&gt;&lt;/span&gt;Katherine L. Chiou, [[Christina Hastorf|Christine A. Hastorf]], &quot;A Systematic Approach to Species-Level Identification of Chile Pepper (Capsicum spp.) Seeds: Establishing the Groundwork for Tracking the Domestication and Movement of Chile Peppers through the Americas and Beyond&quot; in ''Economic Botany'' vol. 68 (2014) pp. 316-336 [https://www.jstor.org/stable/43305668 JSTOR]
* [[Carolus Clement|Charles R. Clement]], Michelly de Cristo-Araújo, Geo Coppens d’Eeckenbrugge, Alessandro Alves Pereira, Doriane Picanço-Rodrigues, &quot;[https://pdfs.semanticscholar.org/7844/1b92b93c09a9b1da1aad7f6bafaa68f0907e.pdf Origin and Domestication of Native Amazonian Crops]&quot; in ''Diversity'' vol. 2 (2010) pp. 72-106
* [[Sophia Coe|Sophie D. Coe]], ''America's First Cuisines'' (Austinopoli: University of Texas Press, 1994) pp. 60-65
* &lt;span id=&quot;Cote Roman et al. (2020)&quot;&gt;&lt;/span&gt;André Luís Cote Roman, Lin Chau Ming, Maria das Graças Piras Sablayrolles, ''Pimentas Capsicum L. no Brasil: notas sobre botânica, histórica, concepções indígenas e folclore]''. Porto Alegre, 2020 [https://drive.google.com/file/d/17VmhkE-NmX0DSaT7Ru_PS8Fr30-Fc1nw/view Textus]
* Marie-Christine Daunay, Henri Laterrot, [[Iulius Janick|Jules Janick]], &quot;Iconography and History of Solanaceae: Antiquity to the XVIIth Century', in: Jules Janick, ''Horticultural Reviews'', vol. 34 (2007), pp. 1-112 {{Google Books|FBa-E96pQ0kC|Paginae selectae}}
* Marie-Christine Daunay, Henri Laterrot, Jules Janick, &quot;[http://www.hort.purdue.edu/newcrop/actahort745.pdf  Iconography of the Solanaceae from Antiquity to the XVIIth Century: a Rich Source of Information on Genetic Diversity and Uses]&quot; in ''Acta Hort.'' no. 745 (2007) pp. 59-88
* &lt;span id=&quot;Dott (2020)&quot;&gt;&lt;/span&gt;Brian R. Dott, ''The Chile Pepper in China: a cultural biography''. Novi Eboraci: Columbia University Press, 2020
* [[Gulielmus Hardy Eshbaugh|W. H. Eshbaugh]], &quot;[https://hort.purdue.edu/newcrop/proceedings1993/v2-132.html Peppers: history and exploitation of a serendipitous new crop discovery]&quot; in [[Iulius Janick|Jules Janick]], J. E. Simon, edd., ''New crops'' (Novi Eboraci: Wiley, 1993) pp. 132–139
* Carlos A. García-González, Cristina Silvar, &quot;[https://www.mdpi.com/2223-7747/9/8/986 Phytochemical Assessment of Native Ecuadorian Peppers (Capsicum spp.) and Correlation Analysis to Fruit Phenomics]&quot; in ''Plants'' vol. 9 viii no. 986 (4 Augusti 2020)
* Zoltán Halász, ''Hungarian Paprika Through the Ages''. Budapestini, 1968
* [[Carolus Bixler Heiser|Charles B. Heiser, Jr.]], [[Barbara Pickersgill]], &quot;Names for the Cultivated Capsicum Species (Solanaceae)&quot; in ''Taxon'' vol. 18 (1969) pp. 277-283 [https://www.jstor.org/stable/1218828 JSTOR]
* [[Carolus Bixler Heiser|Charles B. Heiser, Jr.]], Paul G. Smith, &quot;The Cultivated Capsicum Peppers&quot; in ''Economic Botany'' vol. 7 (1953) pp. 214-227 [https://www.jstor.org/stable/4287775 JSTOR]
* Angela Jianu, Violeta Barbu, edd., ''Earthly Delights: Economies and cultures of food in Ottoman and Danubian Europe, c. 1500-1900'' (Leiden: Brill, 2018) pp. 107-108, 162-163
* Seungill Kim et al., &quot;[https://www.nature.com/articles/ng.2877 Genome sequence of the hot pepper provides insights into the evolution of pungency in Capsicum species]&quot; in ''[[Nature]] Genetics'' vol. 46 (2014) pp. 270–278
* Janet Long-Solís, ''Capsicum y cultura: la historia del chilli''. 2a ed. Mexicopoli: Fondo de Cultura Económica, 1998
* Janet Long, &quot;Orígenes, rutas y evolución del Capsicum&quot; in ''Artes de México'' no. 126 (2017) pp. 8-17 [https://www.jstor.org/stable/45198984 JSTOR]
* Linda Perry et al., &quot;[http://www.bio-nica.info/Biblioteca/Perry2007Capsicum.pdf Starch Fossils and the Domestication and Dispersal of Chili Peppers (Capsicum spp. L.) in the Americas]&quot; in ''Science'' vol. 315 (2007) pp. 986-988
* [[Barbara Pickersgill]], &quot;Relationships Between Weedy and Cultivated Forms in Some Species of Chili Peppers (Genus capsicum)&quot; in ''Evolution'' vol. 25 (1971) pp. 683-691 [https://www.jstor.org/stable/2406949 JSTOR]
* [[Barbara Pickersgill]], &quot;Migrations of chili peppers, Capsicum spp., in the Americas&quot; in D. Stone, ed., ''Pre-Columbian plant migration'' (''Papers of the Peabody Museum of Archeology and Ethnology'' vol. 76. Cantabrigiae Massachusettensium: Harvard University Press, 1984) pp. 105-123
* Divya Schäfer, &quot;[https://crossasia-books.ub.uni-heidelberg.de/xasia/reader/download/366/366-43-81613-1-10-20180704.pdf Exotic Tastes, Familiar Flavours. Transcultural Culinary Interactions in Early Modern India]&quot; in Rafael Klöber, Manju Ludwig, edd., ''HerStory: Historical Scholarship between South Asia and Europe. Festschrift in Honour of Gita Dharampal-Frick'' (Heidelbergae: CrossAsia eBooks, 2018. ISBN 978-3-946742-44-9) pp. 43-64
* Frederick J. Simoons, ''Food in China: A Cultural and Historical Inquiry'' (CRC Press, 2014) pp. 385-386 {{Google Books|H0JZDwAAQBAJ|Paginae selectae}}
* Vito Teti, ''Storia del peperoncino: un protagonista delle culture mediterranee''. Romae: Donzelli, 2007
* Pasquale Tripodi et al., &quot;[https://www.pnas.org/doi/full/10.1073/pnas.2104315118 Global range expansion history of pepper (Capsicum spp.) revealed by over 10,000 genebank accessions]&quot; in ''Proceedings of the National Academy of Sciences'' vol. 118 (34) e2104315118 (16 Augusti 2021)
* Maarten van Zonneveld et al., &quot;[https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0134663 Screening Genetic Resources of Capsicum Peppers in Their Primary Center of Diversity in Bolivia and Peru]&quot; in ''PLoS One'' vol. 10 ix (2015) no. e0134663
* Clifford A. Wright, &quot;[http://www.cliffordawright.com/caw/food/entries/display.php/topic_id/18/id/113/ The Medieval Spice Trade and the Diffusion of the Chile]&quot; in ''Gastronomica'' vol. 7 no. 2 (2007) pp. 35-43 [https://www.jstor.org/stable/10.1525/gfc.2007.7.2.35 JSTOR]

; De cultu et cultivarietatibus
* &lt;span id=&quot;Bosland et al. (1998)&quot;&gt;&lt;/span&gt;[[Paulus Bosland|Paul W. Bosland]], Alton L. Bailey, Jaime Iglesias-Olivas, ''[http://contentdm.nmsu.edu/cdm/ref/collection/AgCircs/id/12518 Capsicum pepper varieties and classification]''. NMSU Cooperative Extension Service and Agricultural Experiment Station, 1998
* &lt;span id=&quot;Bosland et Votava (2012)&quot;&gt;&lt;/span&gt;[[Paulus Bosland|Paul W. Bosland]], Eric J. Votava, ''Peppers: Vegetable and Spice Capsicums''. Oxoniae: CABI, 2012. ISBN 978-1-84593-825-3
* &lt;span id=&quot;DeWitt et Bosland (2014)&quot;&gt;&lt;/span&gt;[[David DeWitt]], [[Paulus Bosland|Paul W. Bosland]], ''The Complete Chile Pepper Book: A Gardener’s Guide to Choosing, Growing, Preserving, and Cooking''. Novi Eboraci, 2014
* A. T. Erwin, &quot;The peppers of America&quot; in ''Iowa Agricultural Experiment Station: Bulletin'' no. 293 (1932) pp. 121–152
* &lt;span id=&quot;Libreros et al. (2013)&quot;&gt;&lt;/span&gt;Dimary Libreros et al., ''Catalogo de ajies (Capsicum spp.) peruanos''. Romae: Bioversity International, 2013
* Anelise Macedo, &quot;[https://www.embrapa.br/documents/1355126/2250572/EDIÇÃO+18i.pdf/82e27ace-f1a5-4f95-9215-9207a6c1b7de Pimentas Capsicum: uma história de sucesso na cadeia produtiva de hortaliças]&quot; in ''Hortaliças em revista'' vol. 4 no. 18 (2015)
* Sven W. Meckelmann et al., &quot;Compositional Characterization of Native Peruvian Chili Peppers (Capsicum spp.)&quot; in ''Journal of Agricultural and Food Chemistry'' vol. 61 no. 10 (2013) pp. 2530–2537
* Mario Parisi, Daniela Alioto, Pasquale Tripodi, &quot;[https://www.mdpi.com/1422-0067/21/7/2587 Overview of Biotic Stresses in Pepper (Capsicum spp.): Sources of Genetic Resistance, Molecular Breeding and Genomics]&quot; in ''International Journal of Molecular Sciences'' vol. 21 vii no. 2587 (8 Aprilis 2020)
* Catherine Parry et al., &quot;[https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0243689 Reproductive compatibility in Capsicum is not necessarily reflected in genetic or phenotypic similarity between species complexes]&quot; in ''PLoS One'' (24 Martii 2021)
* &lt;span id=&quot;Ribeiro et al. (2008)&quot;&gt;&lt;/span&gt;Cláudia S. da C. Ribeiro et al., ''Pimentas Capsicum''. Brasiliopoli: Embrapa Hortaliças, 2008 (liber in interrete reperiendum)
* A. Terpó, &quot;Kritische Revision der wildwachsenden Arten und der kultivierten Sorten der Gattung Capsicum L.&quot; in ''Feddes repertorium'' vol. 72 (1966) pp. 155-191

; De re medica et diaetetica
* Gaber El-Saber Batiha et al., &quot;[https://www.mdpi.com/1422-0067/21/15/5179 Biological Properties, Bioactive Constituents, and Pharmacokinetics of Some Capsicum spp. and Capsaicinoids]&quot; in ''International Journal of Molecular Sciences'' vol. 21 xv no. 5179 (22 Iulii 2020)
* Mustafa Chopan, Benjamin Littenberg, &quot;[https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0169876 The Association of Hot Red Chili Pepper Consumption and Mortality: A Large Population-Based Cohort Study]&quot; in ''PLoS One'' vol. 12 i no. e0169876 (9 Ianuarii 2017)
* Morrine A. Omolo et al., &quot;[https://www.omicsonline.org/open-access/antimicrobial-properties-of-chili-peppers-2332-0877.1000145.php?aid=27614 Antimicrobial Properties of Chili Peppers]&quot; in ''Journal of Infectious Diseases &amp; Therapy'' vol. 2 (2014)
* &lt;span id=&quot;Pickersgill (2016)&quot;&gt;&lt;/span&gt;[[Barbara Pickersgill]], &quot;Chile Peppers&quot; in Rafael Lira, Alejandro Casas, José Blancas, edd., ''Ethnobotany of Mexico: Interactions of People and Plants in Mesoamerica'' (Novi Eboraci: Springer, 2016) pp. 417-438 {{Google Books|iUEWDAAAQBAJ|Paginae selectae}}

; De gastronomia
* James D. Campbell, ''Mr Chilehead: adventures in the taste of pain''. Toronti, 2003
* Kurt Michael Friese, Kraig Kraft, Gary Paul Nabhan, ''Chasing Chiles: Hot Spots Along the Pepper Trail''. White River Junction, 2011
* &quot;The Chiles of Oaxaca&quot; in Diana Kennedy, ''Oaxaca al gusto: an infinite gastronomy'' (Austinopoli: University of Texas Press, 2010) pp. xix-xxii 
* Jay M. Lillywhite, Jennifer E. Simonsen, Mark E. Uchanski, &quot;[https://journals.ashs.org/horttech/view/journals/horttech/23/6/article-p868.xml Spicy Pepper Consumption and Preferences in the United States]&quot; in ''HortTechnology'' vol. 23 (2013) pp. 868–876
* Amal Naj, ''Peppers: A Story of Hot Pursuits''. Novi Eboraci, 1992
* Sota Yamamoto, Eiji Nawata, &quot;Use of Capsicum frutescens L. by the Indigenous Peoples of Taiwan and the Batanes Islands&quot; in ''Economic Botany'' vol. 63 (2009) pp. 43-59 [https://www.jstor.org/stable/40390434 JSTOR]

; Praecepta culinaria
* 1898 : [[Incarnatio Pinedo|Encarnación Pinedo]], ''El cocinero español''. Franciscopoli (Dan Strehl, Victor Valle, edd., ''Encarnación’s kitchen : Mexican recipes from nineteenth-century California'' [Berkeleiae: University of California Press, 2003] pp. 120-128)

{{NexInt}}
{{Aromata}}
* [[Capsicum e carne]]
* [[Restula]]
* [[Sambal]]

== Nexus externi ==
{{CommuniaCat|Chili peppers|Capsica}}
* [http://www.plantcultures.org.uk/plants/chilli_pepper_landing.html De chili (historia, botanica etc.)]
* [http://www2.dpi.qld.gov.au/horticulture/18717.html &quot;Capsicum and Chillies: Commercial Cultivation&quot;] apud ''DPI&amp;F Queensland, Australia''
* [http://www.hort.purdue.edu/newcrop/med-aro/factsheets/CAPSICUM_PEPPER.html &quot;Capsicum pepper factsheet&quot;] apud ''Purdue Guide to Medicinal and Aromatic Plants''
* [http://www.hort.purdue.edu/newcrop/proceedings1996/V3-479.html &quot;Capsicums: Innovative Uses of an Ancient Crop: History, Botany, Breeding, and Pungency&quot;]
* [http://www.bioversityinternational.org/fileadmin/bioversity/publications/pdfs/345.pdf?cache=1245945997 &quot;Descriptors for Capsicum (Capsicum spp.)&quot;] apud ''Bioversity International''
* &quot;[http://gernot-katzers-spice-pages.com/engl/Caps_fru.html Chile]&quot; apud ''Gernot Katzer's Spice Pages''
* Dave Bliss, ''[https://www.academia.edu/8773753/Adoption_and_Use_of_the_Tomato_and_Pepper_in_England_through_the_Industrial_Revolution Adoption and Use of the Tomato and Pepper in England through the Industrial Revolution]'' (dissertatio universitatis Leicestriensis)

==Pinacotheca==
&lt;gallery&gt;
Fasciculus:Cachi 02.jpg|Capsica in Argentina
Fasciculus:Pickled friggitelli.jpg|[[Capsicum annuum|Capsica annua]] varietatis ''[[friggitello]]''
Fasciculus:Fefferoni.jpg|Varietas ''[[friggitello]]'' in popina [[Suecia|Suecica]] ''[[gyrus (ferculum)|döner kebap]]'' 
Fasciculus:HotPeppersinMarket.jpg|''[[Capsicum chinense]]'' varietatis ''[[Scotch bonnet]]'' in macello [[Mare Caribicum|Caribico]] venditatum
Fasciculus:African red devil peppers.jpg|''[[Capsicum frutescens]]'' varietatis ''African Devil''
Fasciculus:Naga Jolokia Peppers.jpg|''[[Capsicum chinense]] x C. frutescens'' varietatis ''[[Naga Jolokia]]'' vel ''bhut jolokia''
Fasciculus:Capsicum_Annum_Flower_Closeup.JPG|[[Flos]] ''[[Capsicum annuum|Capsici annui]]
&lt;/gallery&gt;

{{Myrias|Biologia}}

[[Categoria:Aromata]]
[[Categoria:Capsicum|!]]
[[Categoria:Condimenta]]</text>
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  <page>
    <title>Iuncus (navis)</title>
    <ns>0</ns>
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      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Naves]]; +[[Categoria:Genera navium]]</comment>
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      <text bytes="2254" xml:space="preserve">[[Fasciculus:SongJunk.jpg|thumb|Iuncus [[saeculum 13|saeculi 13]], regnante [[domus Song|domu Song]]]]{{videdis|Iunca (discretiva)}}
'''Iuncus,'''&lt;ref&gt;'''1543''' [[Franciscus Xaverius|Francisci Xaverii]] ''Epistulae'' 3.10: 
:&quot;Exclusus isto concilio, applicuit animum ad navem Sinensis formae, quam iuncum vocant.&quot;&lt;/ref&gt; seu '''iunkus,'''&lt;ref&gt;'''Saec. 14''' [[Iohannes de Marignolis|Iohannis de Marignolis]] ''Chronicon Bohemorum'':
[[Fasciculus:-HK CityHall Seaview 51217 5.png|thumb|Iuncus hodiernus [[Honcongum|Honcongi]] fluctuans]]
: &quot;Deinde volentes navigare ad Sanctum Thomam apostolum et inde ad Terram sanctam ascendentes Junkos de inferiori India, que Mimbar vocatur, in vigilia sancti Georgii tot procellis ferebamur, quod sexaginta vicibus vel amplius fuimus quasi demersi sub aqua usque ad profundum maris, et solo divino miraculo evadentes tot mirabilia vidimus, scilicet ardere mare, dracones ignivomos volantes et occidentes aliorum Junkorum personas in suo transitu, nostro divina ope manente illeso virtute corporis Christi, quod portabam, et meritis Virginis gloriose et sancte Clare, et quia omnes christianos induxeram ad lamentum et penitenciam. Ipsaque procella durante dedimus vela ventis, nos divino regimini committentes, de solis animabus curantes.&quot;&lt;/ref&gt; '''zuncus'''&lt;ref&gt;'''1331''' Frater Odoricus:
&quot;Cum autem illic in Polumbo fuimus nos ad portum, aliam navim nomine Zuncum...&quot;&lt;/ref&gt;, '''iunca'''&lt;ref&gt;[[Maximilianus Transylvanus]], ''De Moluccis insulis'' : &quot;Hoc genus nauium iuncas appellant&quot;[https://books.google.be/books?id=4h8qwRQkNMoC&amp;printsec=frontcover&amp;hl=fr#v=onepage&amp;q&amp;f=false  Legendum in interreti].&lt;/ref&gt; est navigium [[Sina|Sinicum]], quod nomen de voce [[Lingua Iavanica|Iavanica]] ''djong'', [[Lingua Malaica|Malaia]] ''adjong'' (nomen enim Sinicum vel 中国帆船 ''Zhōngguó fānchuán'', vel 戎克船 ''róngkè chuán,'' sed ''djong'' ipsa fortasse a [[wikt:船|船]] ''chuán,'' Minnane ''chun&lt;sup&gt;5&lt;/sup&gt;''). Id genus [[navis|navigii]] primum effectum est regnante [[domus Han|domu Han]] ([[220 a.C.n.]]—[[200]] p.C.n.), et per annos et mutationes factum est secundissimum totius historiae genus. 

== Notae ==
&lt;references/&gt;

[[Categoria:Genera navium]]
[[Categoria:Sinae]]


{{Myrias|Technologia}}</text>
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  <page>
    <title>Bloemfontein</title>
    <ns>0</ns>
    <id>44181</id>
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        <ip>84.78.253.101</ip>
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      <text bytes="851" xml:space="preserve">[[File:Bloemfontein_panorama.jpg|thumb|upright=1.5|Despectus in urbem Bloemfontein.]]
'''Bloemfontein'''&lt;ref&gt;[[Nomen adiectivum]] ''Bloemfonteinensis'': {{Graesse}}.&lt;/ref&gt; ([[Linguae Nigro-Congenses|Sotho]]: ''Mangaung'') sive '''Bloemfonteinum'''&lt;ref&gt;&quot;Bloemfonteinum, (-i, n)&quot;: a Petro Lucusaltiano Latinophilo, [http://www.lateinlexikon.com ''Lexicon Latinum Hodiernum vel Vocabularium Latinitatis Huius Aetatis,''] apud www.lateinlexikon.com.&lt;/ref&gt; est iudiciale [[Africa Australis|Africae Australis]] [[caput (urbs)|caput]], quod ad [[provincia]]m [[Civitas Libera|Civitatem Liberam]] pertinet. Anno [[1846]] condita est. Ibi etiam sedes [[Universitas Civitatis Liberae|Universitatis Civitatis Liberae]] est.

== Notae ==
&lt;div class=&quot;references-small&quot;&gt;&lt;references /&gt;&lt;/div&gt;

{{urbs-stipula}}

[[Categoria:Condita 1846]]
[[Categoria:Bloemfontein|!]]</text>
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  <page>
    <title>Linter</title>
    <ns>0</ns>
    <id>46449</id>
    <revision>
      <id>3697739</id>
      <parentid>2743248</parentid>
      <timestamp>2022-08-17T10:13:08Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Naves]]; +[[Categoria:Genera navium]]</comment>
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      <text bytes="2724" xml:space="preserve">[[Fasciculus:Boating in fair weather.jpg|thumb|Linter prope litus]]

'''Linter,''' etiam '''cymba, navicula, navigium, scapha,''' est vehiculum aquaticum, plerumque parva res constructa ut potest aquae innare, et praesertim lacus, flumina, vel protectas litoris aquas navigare. Confer [[Navis]] et [[caudica]], quidam linter parvus.&lt;!--A '''boat''' is a [[watercraft]] designed to float or plane on water, and provide transport over it. Usually this water will be inland (lakes) or in protected coastal areas. However, boats such as the [[whaleboat]] were historically designed to be operated from a [[ship]] in an offshore environment. In naval terms, a boat is something small enough to be carried aboard another vessel (a [[ship]]). Some boats too large for the naval definition include the [[lake freighter|Great Lakes freighter]], [[riverboat]], [[narrowboat]] and [[ferry]]boat. Modern [[submarines]] can also be called boats, despite their underwater capabilities and size.  This may be because the first submarines could be carried by a ship and were not capable of making independent offshore passages. Boats may be used by the military or other government interests, or for research or commercial purposes; but regardless of size, a vessel in private, non-commercial usage is almost certainly a boat.--&gt;

[[Fasciculus:EgyptTombOarboat.jpg|thumb|Linter in tumulo in Aegypo, circa annum 1450 a.C.n. pictus]]
[[Fasciculus:DerelictBoatFollyIs.jpg|thumb|[[Linter vitae]], ferro constructa.&lt;!--rusting away in the wetlands of [[Folly Island]], [[South Carolina]], [[United States]].--&gt;]]

== Historia ==
Antiquissima recuperata linter est &quot;cymba of Pesse.&quot; Secundum explicationem C&lt;sub&gt;14&lt;/sub&gt;, hoc cymba est inter annos 8200 et 7600 a.C.n. constructa. Id in [[Museum Drents|Museo Drents]] in urbe [[Assen]] in [[Nederlandia]] nunc exhibita est.

== Nexus externi ==
* [http://www.sailingtheweb.net Sailboats database]
* [http://www.boatfax.com/international-hin-formats.html Hull Identification Numbers] Explicatio Internationalium HIN formarum
* University of Washington Libraries Digital Collections -- [http://content.lib.washington.edu/cgi-bin/queryresults.exe?CISOOP=adv&amp;CISORESTMP=%2Fsite-templates%2Fsearch_results-sub.html&amp;CISOVIEWTMP=%2Fsite-templates%2Fitem_viewer.html&amp;CISOMODE=thumb&amp;CISOGRID=thumbnail%2CA%2C1%3Btitle%2CA%2C1%3Bsubjec%2CA%2C0%3Bdescri%2C200%2C0%3B0%2CA%2C0%3B10&amp;CISOBIB=title%2CA%2C1%2CN%3Bsubjec%2CA%2C0%2CN%3Bdescri%2CK%2C0%2CN%3B0%2CA%2C0%2CN%3B0%2CA%2C0%2CN%3B10&amp;CISOTHUMB=3%2C5&amp;CISOTITLE=10&amp;CISOPARM=%2Ffishimages%3Asubjec%3Avessels&amp;x=51&amp;y=5 Freshwater and Marine Image Bank -- Vessels] Imagines vehiculorum aquaticorum

{{stipula}}
[[Categoria:Genera navium]]

{{Myrias|Technologia}}</text>
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  <page>
    <title>Heraclidae (Euripides)</title>
    <ns>0</ns>
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        <username>Marcus Terentius Bibliophilus</username>
        <id>2059</id>
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      <comment>/* Personae */</comment>
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      <text bytes="11763" xml:space="preserve">{{Titulus italicus}}
[[Fasciculus:Herakles and Telephos Louvre MR219.jpg|thumb|Heracles et [[Telephus]] [[filius]]. [[Museum Lupariense]].]]
'''''Heraclidae''''' ([[Graece]] {{Polytonic|Ἡρακλεῖδαι}}) est [[tragoedia]] quam [[Euripides]] circa annum [[430 a.C.n.]] docuit. [[Poeta]] fabulam Atheniensium Herculis proli ([[Heracleidae|Heraclidae]]) opem contra [[Eurystheus (mythologia)|Eurystheum]] ferentium scaenice tractabat. Textus traditus nonnullas difficultates praebet, et multi [[critica litterarum|critici]] magnam lacunam suspicantur (brevissima enim ex Euripideis tragoediis est et partes lyricae in primis solito breviores sunt)&lt;ref&gt;v. Wilamowitz-Möllendorff, &quot;[https://www.jstor.org/stable/4471814 Excurse zu Euripides Herakliden]&quot;, ''Hermes'', 1882ː 337-364, 496.&lt;/ref&gt;. Aeschylus quoque tragoediam ''Heraclidae'' inscriptam hodie deperditam docuerat; nescitur tamen an argumentum idem fuerit quia multae fabulae ad Heraclidas pertinent.

== Personae ==
*[[Iolaus]], Herculis propinquus et comes, iam senex
* Praeco Eurysthei
* [[Chorus Graecus|Chorus]] [[Athenae|Atheniensium]]
* [[Demophoon|Demophon]], rex Atheniensium, [[Theseus|Thesei]] filius
* Herculis filia ([[Macaria (Herculis filia)|Macaria]])
* [[Hyllus|Hylli]] famulus
*[[Alcmene|Alcmena]], Herculis mater iam vetula
*Nuntius, Alcmenae servus
*[[Eurystheus (mythologia)|Eurystheus]], Argivorum rex
* [[Acamas (filius Thesei)|Acamas]], Demophontis frater et Herculis parvi filii sunt personae mutae.
Res [[Marathon (Graecia)|Marathone]] in Attica geruntur ubi cum liberis Herculis Iolaus ad Iovis aram confugit.

== Summarium ==
*'''Prologus'''ː Iolaus spectatoribus narrat quomodo Eurystheum tyrannum per plures Graeciae civitates fugiens qui omnem Herculis stirpem exstirpare decrevit cum liberis amici nunc in caelo manentis Marathonem advenerit et ad Iovis aram confugerit. Mox procedit praeco ab Eurystheo missus  se [[Mycenae|Mycenas]] sequi iubens ubi lapidabuntur. Iolaum recusantem vi protrahit atque humi sternit. Qui fidem incolarum invocat.
*'''Parodos'''&lt;ref&gt;Jean Irigoin, &quot;La parodos des Héraclides d'Euripide&quot; in ''[https://openlibrary.org/books/OL2589017M/Me%CC%81morial_Andre%CC%81-Jean_Festugie%CC%80re Mémorial André-Jean Festugière]'', [[Genava|Genavae]], 1984ː 13-21&lt;/ref&gt;ː clamoribus accitus Atheniensium chorus senem iacentem miseratur. Suos homines esse respondet praeco et iure a se abigi. Chorus regi de his rebus esse cognoscendum existimat.
*'''Primum episodium'''ː accedunt rex Demophon cum fratre Acamante qui barbare agere peregrinum praeconem existimat&lt;ref&gt;Versus 130-1ː  ''Καὶ μὴν στολήν γ' Ἕλληνα καὶ ῥυθμὸν πέπλων / ἔχει, τὰ δ' ἔργα βαρβάρου χερὸς τάδε''&lt;/ref&gt;. Sequitur agon inter Iolaum cum alia tum etiam adfinitatem quandam inter Thesei domum et Herculem invocantem necnon beneficia a Theseo olim accepta et Argivum praeconem militarem Eurysthei potentiam minitantem. Quamvis paci pronior Demophon supplices tueri constituit quia deorum iura respicit et timet ne Athenae in posterum libera civitas non iam  dicatur si Eurysthei iussis obtemperet. Minabundus abit praeco et Iolaus gratias agit.
*'''Primum stasimon'''ː chorus Argivorum adrogantiam vituperat et Atheniensium virtutem extollit.
*'''Secundum episodium'''ː iam adest Argivorum exercitus. Demophon omnia sibi ad proelium parata quidem esse Iolao nuntiat sed oracula consulta negant victuros esse Athenienses nisi filiae Demetris virgo nobili sanguine sacrificetur. Ipse filiam suam immolare  aspernatur nec quod sibi recusat alii imponere vult. Omnem spem salutis iam amiserat Iolaus cum Herculis filia maxima natu, quam argumentum Macariam appellat, in hac tragoedia Euripides non nominat, e templo exitː nam lamentationes audivit et postquam rem cognovit sua sponte pro fratribus vitam dare non dubitat Atheniensium exemplo qui pro hospitibus bello perire parati erant. Maeret quidem Iolaus atque sorte unam ex Herculis filiabus ducendam maluisset nisi pertinacia Macariae non sortis necessitate sed sua libera voluntate perire cupientis vicisset. Propinquis valere dicit virgo et ad sacrificium pergit.
*'''Secundum stasimon'''ː chorus Iolaum solatur gloriosam mortem Macariam adire et patre Hercule dignam demonstrans.
*'''Tertium episodium'''ː advenit [[Hyllus|Hylli]] (qui maximus natu e filiis Herculis erat)  servus qui Alcmenen flagitat. Quae e templo ubi parvas Herculis filias tuebatur exit et Hyllum cum magnis sociorum copiis adesse certior fit. Laetatur quidem sed Iolaos quamvis senex proelio interesse vult. Arma consecrata e templo sumit et bracchio comitis innixus quamvis reclamante Alcmena frustra dissuadentem servum sequitur.
*'''Tertium stasimon'''ː chorus Iovis et Athenae auxilium in bello adversus [[Mycenae|Mycenas]] invocat quia Atheniensium causa iusta et pia erat.
*'''Quartum episodium'''ː nuntius e proelio pugnam et Argivorum cladem narratː ipsum Iolaum cuius preces exaudientes dei iuventutem ad hunc diem reddiderunt ducem hostium Eurystheum captivum fecisse nec occidisse ut Alcmene prostratum et humilem videre posset quem antea triumphantem et potentem vidisset.
*'''Quartum stasimon'''ː chorus victoriam celebrat et deis gratiam agit.
*'''Exodos'''ː Eurystheus ante Alcmenen ducitur quae odio exasperata malae morti eum dare praecipit. Cui respondetur occidi qui vivus in proelio captus sit Atheniensibus legibus vetari. Tum Alcmene inimicum manu sua interficere constituit ne quis alius homicidio contaminaretur. Eurystheus mori non recusat  et Apollinis oraculo fretus praenuntiat corpus suum prope Athenae fanum humatum saluti Atheniensibus et Heraclidis exitio quondam fore cum Herculis progenies beneficiorum oblita Atticam invadet (ita Euripides ad [[Bellum Peloponnesiacum|Bellum Peloponnesium]] alludebat quia [[Dores|Dorica]] invasio Heraclidarum in Peloponnesum reditus esse credebaturː ita omnes Dores Heraclidae lato sensu dici possunt). Nam quamquam ab Atheniensibus impeditus ne Herculis liberis potiretur eis gratiam agit quod vitam sibi reliquisse voluerint.

== Quid sibi proposuerit poeta ==
Euripides ''Heraclidas'' ineunte Bello Peloponnesiaco scripsit. Itaque argumentum patriam suam glorificans elegit. Nam Athenienses gloriari solebant quod Heraclidis sive Theseo sive Demophonte regnante saluti fuissent  nec dubitabant quin eventum historicum memorarent&lt;ref&gt;[[Herodotus]], ''[[Historiae (Herodotus)|Historiae]]'' IX.27. [[Lysias]] ''Epitaphium'' 11. [[Isocrates]] ''[[Panegyricus|Panegyricum]]'' 56-60, ''Archidamos'' 42, ''Panathenaica'' 194. [[Demosthenes]], ''[[De corona (Demosthenes)|De Corona]]'' 186 etc. Cf {{Strabo}} VIII.6.19.&lt;/ref&gt;. Ita magnanimitatem maiorum suorum et benignitatem erga profugos asylum rogantes ostentabant, quod idem in hac tragoedia effecit Euripides. Simul per Eurysthei ultima verba&lt;ref&gt;Versus 1035/7ː  ''ὅταν μόλωσι δεῦρο σὺν πολλῇ χερὶ /
χάριν προδόντες τήνδε. Τοιούτων ξένων / προύστητε'' = (inimicus ero horum posteris) cum huc magna manu venient praesens beneficium prodentes. Tales sunt hospites quos tuemini.&lt;/ref&gt; Lacedaemonios et eorum socios, qui Dores erant, ingratitudinis accusat, praesertim [[Reges Spartae|reges Spartae]] qui se Herculis sanguine natos iactabant. Econtra Argivi qui per maximam tragoediae partem Atheniensium hostes sunt velut post mortem amici fiunt. Tempore quo scribebat Euripides Argos neutras partes sequebatur (triginta annorum indutias cum Lacedaemoniis observabant), sed adsiduas inimicitias cum [[Sparta|Spartanis]] iamdiu gerebant; quocirca Athenienses eorum societatem sperabant. 

Duo episodia tamen novasse videtur Euripidesː nam [[Diodorus Siculus]], [[Strabo]], [[Pausanias (scriptor)|Pausanias]]&lt;ref&gt;''[[Graeciae descriptio|Descriprio Graeciae]]'' I.44.10 qui locum eius tumuli indicat.&lt;/ref&gt; [[Apollodori bibliotheca|Apollodorique bibliotheca]] Eurystheum pugnando perisse volunt. Econtra in ''Heraclidis'' Eurystheus captivus spreto belli iure Alcmenae voluntate interficitur. Ita Alcmenae saevitia aliquam invidiam Heraclidis faciebat. Similiter episodium Macariae addidit ut qui pathetico adulescentium mortem voluntariam pro propinquis vel pro patria fortiter adeuntium libenter utebatur&lt;ref&gt;P. Roussel, &quot;[https://www.persee.fr/doc/rbph_0035-0818_1922_num_1_2_6168 Le thème du sacrifice volontaire dans la tragédie d'Euripide]&quot;, ''Revue belge de Philologie et d'Histoire'', 1922ː 225-240&lt;/ref&gt; (e.g. [[Iphigenia]], Menoeceus, [[Polyxena]]...).

''Heraclidae'' inter optimas Euripidis fabulas non numerantur et plerumque neglectae sunt. Exempli gratia miramur cur Macariae sacrificium a versu 630 non iam memoretur nisi fortasse obscuris verbis versus 822&lt;ref&gt;''λαιμῶν βροτείων εὐθὺς οὔριον φόνον.'' Sed sunt qui corrigant et legant βοείων (iugulum bovis).&lt;/ref&gt; qui ad sacrificium humanum alludere videtur. Quae - praeter nonnullos versus a posterioribus auctoribus ''Heraclidis'' attributos qui in hodierno texto non leguntur - praecipua causa fuit cur nonnulli philologi totum episodium excidisse arbitrentur in quo nuntius Macariae mortem, ut in multis aliis Euripideis tragoediis fit,  referebat necnon sequens stasimon quo chorus virginem deplorabat. Alii textum quod hodie legimus fere idem esse atque illud quod Euripides olim docuit putant&lt;ref&gt;Exempli gratia Ludovicus Méridier in notitia ad editionem [[Collection des Universités de France]].&lt;/ref&gt;.

== Notae ==
&lt;references/&gt;

== Editiones et commentarii ==
*William Allan (Cur.), ''Euripides. The Children of Heracles with an introduction, translation and commentary'', Aris and Philips, 2001 [https://www.persee.fr/doc/antiq_0770-2817_2003_num_72_1_2530_t1_0332_0000_1 Recensio critica]
*Maria Grazia Fileni, ''Eraclidi : i canti'', Romae, 2006 [https://bmcr.brynmawr.edu/2008/2008.04.27 Recensio critica]
*John Wilkins, Euripides, ''Heraclidae. With Introduction and Commentary'', Oxonii, Clarendon press, 1993 [https://www.persee.fr/doc/antiq_0770-2817_1995_num_64_1_1237_t1_0274_0000_3 Recensio critica]

== Plura legere si cupis ==
*Harry C. Avery, &quot;[https://www.jstor.org/stable/292663 Euripides' &quot;Heracleidai.&quot;]&quot;, ''The American Journal of Philology'', 1971ː 539-565
*Peter Burian, &quot;[https://www.jstor.org/stable/267645 Euripides' Heraclidae: An Interpretation]&quot;, ''Classical Philology'', 1977ː  1-21
*Anne Burnett, &quot;[https://www.jstor.org/stable/268514 Tribe and City, Custom and Decree in Children of Heracles]&quot;, ''Classical Philology'', 1976ː 4-26
*John H. McLean, &quot;[https://www.jstor.org/stable/289771 The Heraclidae of Euripides]&quot;, ''The American Journal of Philology'', 1934ː 197-224
*Sophie Mills, &quot;[https://onlinelibrary.wiley.com/doi/abs/10.1002/9781118351222.wbegt2600 Euripides: Children of Heracles (Ἡρακλεῖδαι)]&quot; in ''Encyclopedia of Greek tragedy'', Wileyː 2013.
*Florence Yoon, ''Euripides &quot;Children of Heracles&quot;'', Londini, Bloomsbury, 2020 [https://books.google.fr/books?id=ayO9DwAAQBAJ&amp;pg=PT57&amp;hl=fr&amp;source=gbs_selected_pages&amp;cad=2#v=onepage&amp;q&amp;f=false Nonnullae paginae apud Guglum librorum]

== Nexus externi ==
*[https://el.wikisource.org/wiki/%CE%97%CF%81%CE%B1%CE%BA%CE%BB%CE%B5%CE%AF%CE%B4%CE%B1%CE%B9 Textus Graecus apud Vicifontem]
*[https://www.greekmythology.com/Plays/Euripides/Children_of_Heracles/children_of_heracles.html ''Children of Heracles'' apud Greekmythology.com]
*[http://classics.mit.edu/Euripides/heracleidae.html Versio] ab [[Eduardus Coleridge|Eduardo Coleridge]] confecta anno 1891 {{Ling|Anglice}}

[[Categoria:Euripides]]
[[Categoria:Graeciae scripta]]
[[Categoria:Hercules]]
[[Categoria:Litterae Graecae antiquae]]
[[Categoria:Scripta saeculo 5 a.C.n.]]
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      <text bytes="12928" xml:space="preserve">{{Videdis|Hercules (discretiva)}}
[[Fasciculus:Ugo_Pagliai_Anfitrione_Eracle_2007.jpg|thumb|200 px|[[Histrio]] [[Italia]]nus [[Hugo Pagliai]] ut [[Amphitruo]] in Hercule  [[Syracusae|Syracusis]] anno [[2007]] agit]]

'''''Hercules''''' ([[Graece]] {{Polytonic|Ἡρακλῆς}} aut ''Ἡρακλῆς μαινόμενος'') est [[tragoedia]] [[Euripides|Euripidea]] circa annum [[416 a.C.n.]] docta qua poeta fabulam Herculis furore excaecati et propriam uxorem cum liberis interficientis in scaenam proferebat. Quam tragoediam Latine imitatus est Seneca in ''[[Hercules furens|Hercule furente]]''. Ut in [[Aiax (Sophocles)|Sophocleo ''Aiace'']] heros furore excaecatus turpes res commisit sed ad sanitatem reversus alia viam atque Aiax adiit.

== Personae ==
* [[Amphitryon|Amphitruo]], Herculis pater mortalis
*[[Megara (mythologia)|Megara]], Herculis uxor
* [[Chorus Graecus|Chorus]] senum
* Lycos [[Tyrannis|tyrannus]]
*[[Hercules]]
*[[Iris (mythologia)|Iris]], deorum nuntia
*Lyssa, &quot;Rabies&quot;&lt;ref&gt;Jacqueline Duchemin, &quot;[https://www.persee.fr/doc/reg_0035-2039_1967_num_80_379_3928 Le personnage de Lyssa dans Héraclès d'Euripide]&quot;, ''Revue des Études Grecques'', 1967ː 130-139&lt;/ref&gt;, Noctis filia
*Nuntius
*[[Theseus]], Atheniensium rex
* Herculis et Megarae tres filii et dea Athena sunt personae mutae.
Res [[Thebae (Boeotia)|Thebis]] in [[Boeotia]] geruntur. Spectatores aram pro Amphitruonis domo vident ad quam Herculis familia confugit.

== Summarium ==
[[Fasciculus:Madrid Krater Asteas MAN Inv11094.jpg|thumb|right|300px|Hercules furens in [[Crater|cratere]]  ex [[Salernum|Salerno]] in [[Campania]]. IV saeculo a.C.n.]]
*'''Prologusː''' Amphitruo spectatoribus narrat cur ipse et Megara, [[Creon Thebanus|Creontis]] regis defuncti filia, ad aram confugerintː tyrannus ex Euboea Lycos Creontem interfecit et regno potitus est. Nunc omnem stirpem regiam delere cupit, quia sibi persuasum habet Herculem ex Inferis quo [[Cerberus|Cerberi]] in lucem trahendi causa descendit numquam rediturum esse. Megara solationem et auxilium apud socerum quaerit qui spem abiciendam negat&lt;ref&gt;Versus 105-6ː  ''οὗτος δ᾽ ἀνὴρ ἄριστος ὅστις ἐλπίσι / πέποιθεν αἰεί: τὸ δ᾽ ἀπορεῖν ἀνδρὸς κακοῦ'' = vir optimus numquam desperat; ignavi viri est abdicare.&lt;/ref&gt;.
*'''Parodosː''' chorus senum Thebanorum intrat infirmitatem et senium deplorans, Herculis liberis  iam auxilio esse non possunt quorum eos miseret.
*'''Primum episodiumː''' advenit Lycus qui ab eis quaerit an mortem vitaturos esse adhuc sperentː nam Herculem nunc in Inferis iacere adfirmabat et vivum timidum bellatorem fuisse ut qui eminus sagittis potius quam gladio pugnaret. Amphitruo filii honorem vindicat sagittarios victoriae saepe utiliores fuisse quam qui comminus pugnabant demonstransː praeterea hoplitae fortuna e virtute circumiacentium militum pendet qui agmen efficiunt. Lycus non respondet sed comitibus imperat ut materiam congerant ad supplices circa aram concremandos. Tum Megara mortem non iam fugere decernit&lt;ref&gt;Versus 309-10ː ''τὰς τῶν θεῶν γὰρ ὅστις ἐκμοχθεῖ τύχας, / πρόθυμός ἐστιν, ἡ προθυμία δ᾽ ἄφρων'' = qui sortibus deorum impugnat strenuus quidem est sed illa strenuitas amens.&lt;/ref&gt; atque tantummodo petit ut sibi liberos in domo funebri apparatu vestire interim liceat. Quod concessit Lycus et omnes scaenam relinquunt.
*'''Primum stasimonː'''  chorus [[Labores Herculis|Herculis labores]] celebrat et senii imbellicitatem deplorat.
*'''Secundum episodiumː''' Herculis familia e domo exit, ad mortem parata. Megara praeclaram vitam describit quam tribus filiis speraverat et iterum Herculem precatur ut appareat etiam sub umbrae forma. Quod mirabiliter evenitː nam Herculem inopinate ex Inferis redeuntem adspiciunt qui eis narrat quomodo Cerberum super terram traxerit atque simul [[Theseus|Theseum]] ibi captivum in lucem reduxerit. Horrescit cum liberos suos funebribus stolis vestitos adspicit. Lyci scelera postquam audiit ad ultionem paratus cum familia domum intrat.
*'''Secundum stasimonː'''  chorus senectutem suam vituperans iuventutem celebrat. Deos si humana sentirent secundam iuventutem bonis viris daturos fuisse existimat ut boni a malis certo signo secernerentur. Ipsum numquam [[Musae (deae)|Musarum]] ministerium deserturum pollicetur quia  [[Mnemosyne|Mnemosynen]] colere etiam seni licet atque ita hymno Herculem celebrabit&lt;ref&gt;Versus 676-9ː ''μὴ ζῴην μετ' ἀμουσίας, /
αἰεὶ δ' ἐν στεφάνοισιν εἴην. / Ἔτι τοι γέρων ἀοιδὸς / κελαδεῖ Μναμοσύναν·'' = Ne vivam sine Musis, semper sub coronis verser. Senex quoque vates Mnemosynen canit...&lt;/ref&gt;. Multi [[Critica litterarum|critici]] hos versus ipsius Euripidis de se verba esse arbitrantur.
*'''Tertium episodiumː''' Lycus redux Amphitruonem solum invenit quem ubi sint ceteri morituri rogat. Postquam intra domum eos manere accepit cum satellitibus domum ingreditur. Mox lugubres clamores audiuntur et chori senes  tyrannum Herculis ictibus perire intelligunt.
*'''Tertium stasimonː''' chorus Iovi ceterisque deis gratiam agit quod opem iustis tulerint et de impiis poenas sumpserint. Totam Boeotiam cum montibus et fluminibus ad gaudendum provocat. Ultimis versibus iam horrendis visis in caelo manifestis terretur.
*'''Quartum episodiumː''' advenae caelestes sunt Iris, Iunonis nuntia et Lyssa, Noctis filia cuius aspectus [[Furiae]] similis est et munus in furore pectori eorum incutiendo quos dei perdere volunt constat. Iris consilium suum senibus aperitː Herculem  usque ad caedem innocentium filiorum furore insanientem  faciet. Nisi enim Hercules hanc poenam  dabit dei in posterum spernentur quos homines vicisse sese iactabunt. Lyssa primum Herculem, nobilem heroem, adoriri aspernatur. At cum irascitur Iris iussa facere incipit. Mox lamentationes Amphitruonis e domo audiuntur et nuntius accurrit quomodo Hercules liberos et uxorem immaniter interfecerit nuntiaturus quos Eurysthei familiam esse falso credebat. Cum autem patrem sagittis iam peteret dea Athena domum diruens saxum in herois pectus proiecit qui profundo somno statim obrutus est.
*'''Quartum stasimonː''' chorus caedem et Herculis fatum flebiliter deplorat dum machina ''ekkuklêma'' vocata domum interiorem et cadavera et Herculem ligatum dormientemque in scaenam profert et spectatoribus ostendit.
*'''Exodosː''' Hercules columnae fragmento Amphitruonis iussu adligatus expergiscitur ruinis et cadaveribus undique circumdatus. Quid evenerit omnino oblitus est et postquam pater eum de liberorum caede paulatim certiorem fecit ad mortem voluntariam decurrere vult scelerum conscientiam ut effugeret. Tum vero Theseus advenit qui beneficii memor Athenarum hospitalitatem heroi polliceturː ubi post mortem quoque lapideis monumentis coletur. Quin etiam exemplo deorum eum solatur qui nefanda facinora volentes committere solent nec minus dei sunt. Hercules respondet fabulis a poetis de deis excogitatis se non credere&lt;ref&gt;Versus 1340-6.&lt;/ref&gt;  sed miseram vitam tolerare fortius esse existimans&lt;ref&gt;Versus 1347-52ː ''Ἐσκεψάμην δὲ καίπερ ἐν κακοῖσιν ὤν, / μὴ δειλίαν ὄφλω τιν' ἐκλιπὼν φάος· ... ἐγκαρτερήσω θάνατον'' = 'quamquam miser sum timui ne ignaviae accusarer si lucem liquissem... Morti resistam' (sunt qui textum manuscriptorum ita emendentː ''ἐγκαρτερήσω βίοτον'' = vitam tolerabo).&lt;/ref&gt; quam ad mortem confugere gratias agit et Theseum sequitur postquam liberos sepeliendos patri Amphitruoni tradidit.
== De argumento ==
Herculis insaniae mythus iam ante Euripidem innotuerat&lt;ref&gt;Iam in ''Carminibus Cypriis'' narrabatur.&lt;/ref&gt;; at ille multa novavit atque mutavit in primis ut heroem minus rudem et agrestem efficeret quam in fabulis ostendebatur. Ita haec caedes quae vulgo ante [[Labores Herculis|labores]] a prioribus poetis ponebatur post labores ad ultimam vitam transtulit&lt;ref&gt;Ita Euripides [[Deianira|Deianiram]] et pyram in monte [[Oeta]] ignorabat.&lt;/ref&gt; ita ut poenam eiusmodi minime meruisse videretur. In mytho communi liberos tantum necabat atque deinde Megaram propinquo [[Iolaus|Iolao]] in matrimonium tradebat quod spectatoribus quinti saeculi nimis durum videri potuit. Tyrannus Lycus quoque Euripidis inventio fuisse creditur quae invidiam in Herculis inimicos moveret. Deam poliadem Athenam et Atheniensium regem Theseum in mythum introduxit in maiorem patriae gloriam spectatorumque benevolentiae captandae causa. Simul non malum hominem esse Herculem magis  apparebat siquidem amicitiam in infortunio tam fidelem servare meruisset. Praeterea Hercules in Atticam proficiscitur velut ibi moriturus&lt;ref&gt;Versus 1331-7.&lt;/ref&gt; et insuper eodem patrem Amphitruonem arcessiturus ad sepulturam&lt;ref&gt;Versus 1421.&lt;/ref&gt;. Quod nusquam alibi legitur. Quin etiam labores suscepisse dicitur non tam Iunonis ob iram coactus quam ut patris exsulis reditum in patriam Argivam redimeret&lt;ref&gt;Verba  Amphitruonis versibus 16-21.&lt;/ref&gt;, quod a vulgata versione discrepat.

Diu critici in hac tragoedia actionem velut in duas partes distractam reprehenderuntː prima enim parte Lyci saevitia timetur et  periculo illo Herculis reditu vix suppresso iam chorus providentiam deorum iustitiam in terris restituentium celebrabat, quae felix conclusio fuisset,  cum duae saevae deae  in scaenam inrumpunt et qui liberis prius saluti fuit eorum interfector per voluntatem deorum fit. Itaque reiectis deis Hercules Iovis paternitatem aspernatur atque solius Amphitrionis filius esse decernit&lt;ref&gt;Versus 1265ː ''πατέρα γὰρ ἀντὶ Ζηνὸς ἡγοῦμαι σὲ ἐγώ''. &lt;/ref&gt;ː simul ad humanam amicitiam decurrit. Non apud deos enim sed apud homines auxilium et misericordia inveniuntur.

==Notae==
&lt;references/&gt;

== Editiones et commentarii ==
[[Fasciculus:Mosaic panel depicting the madness of Heracles (Hercules furens), from the Villa Torre de Palma near Monforte, 3rd-4th century AD, National Archaeology Museum of Lisbon, Portugal (12973806145).jpg|thumb|[[Opus tessellatum]] in [[Lusitania]] repertumː Hercules furens parvum filium coram Megara taeda petit.]]
*''Euripides, Heracles with introduction and commentary by Godfrey W. Bond'', Oxonii, Clarendon Press, 1982 [https://www.persee.fr/doc/reg_0035-2039_1983_num_96_455_1366_t2_0320_0000_2 Recensio critica]

== Plura legere si cupis ==
*Jacqueline Assaël, &quot;[https://journals.openedition.org/rursus/563 La violence dans l'Héraclès furieux d’Euripide. Lecture girardienne]&quot;, ''Rursus'', 2011.
*Gerhard J. Baudy, &quot;[https://www.jstor.org/stable/4476951 Die Herrschaft des Wolfes: Das Thema der 'verkehrten Welt' in Euripides' 'Herakles']&quot;, ''Hermes'', 1993ː 159-180
*E. M. Griffiths, &quot;[https://www.jstor.org/stable/4433378 Euripides' &quot;Herakles&quot; and the Pursuit of Immortality]&quot;, ''Mnemosyne'', 2002ː 641-656
*W. E. Higgins, &quot;[https://www.jstor.org/stable/20538843 Deciphering Time in the &quot;Herakles&quot; of Euripides]&quot;, ''Quaderni Urbinati di Cultura Classica'', 1984ː 89-109
*F. Jouan, &quot;[https://www.persee.fr/doc/rea_0035-2004_1970_num_72_3_3872 Le « Prométhée» d'Eschyle et l'« Héraclès » d'Euripide]&quot;, ''Revue des Études Anciennes'', 1970ː 317-331
*Brooke Holmes, &quot;[https://www.jstor.org/stable/10.1525/ca.2008.27.2.231 Euripides' Heracles in the Flesh]&quot;, ''Classical Antiquity'', 2008ː 231-281
*Rocco Marseglia, &quot;[https://journals.openedition.org/pallas/16361?lang=fr Mythe et dramaturgie dans l’Héraclès d’Euripide]&quot;, ''Pallas'', 2019ː  17-31
*Thalia Papadopoulou, &quot;[https://www.jstor.org/stable/4433556 Herakles and Hercules: The Hero's Ambivalence in Euripides and Seneca]&quot;, ''Mnemosyne'', 2004ː 257-283 
*Cecilia J. Perczyk, &quot;[https://ri.conicet.gov.ar/bitstream/handle/11336/62117/El_ritual_y_la_locura_en_Heracles_de_Euripides_-__Cecilia_J._Perczyk.pdf?sequence=5&amp;isAllowed=y El ritual y la locura en Heracles de Eurípides]&quot;, ''Phoenix'', 2015ː 41-58
*Kathleen Riley, &quot;[https://onlinelibrary.wiley.com/doi/10.1002/9781118351222.wbegt2670 Euripides: Heracles (Ἡρακλῆς μαινόμενος)]&quot; in ''Encyclopedia of Greek Tragedy'', Wiley, 2013
*Renate Schlesier, &quot;[https://www.jstor.org/stable/24307066 &quot;Héraclès&quot; et la critique des dieux chez Euripide]&quot;, ''Annali della Scuola Normale Superiore di Pisa. Classe di Lettere e Filosofia'', 1985ː 7-40

== Nexus externi ==
*[http://hodoi.fltr.ucl.ac.be/concordances/euripide_hercule_furieux/lecture/1.htm Textus Graecus et versio Francogallica apud Itinera electronica]
*[https://www.greekmythology.com/Plays/Euripides/Heracles/heracles.html Herakles apud Greekmythology.com]

[[Categoria:Tragoediae]]
[[Categoria:Litterae Graecae antiquae]]
[[Categoria:Graeciae scripta]]
[[Categoria:Euripides]]
[[Categoria:Scripta saeculo 5 a.C.n.]]
[[Categoria:Hercules]]</text>
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      <text bytes="13060" xml:space="preserve">{{Videdis|Hercules (discretiva)}}
[[Fasciculus:Ugo_Pagliai_Anfitrione_Eracle_2007.jpg|thumb|200 px|[[Histrio]] [[Italia]]nus [[Hugo Pagliai]] ut [[Amphitruo]] in Hercule  [[Syracusae|Syracusis]] anno [[2007]] agit]]

'''''Hercules''''' ([[Graece]] {{Polytonic|Ἡρακλῆς}} aut ''Ἡρακλῆς μαινόμενος'') est [[tragoedia]] [[Euripides|Euripidea]] circa annum [[416 a.C.n.]] docta qua poeta fabulam Herculis furore excaecati et propriam uxorem cum liberis interficientis in scaenam proferebat. Quam tragoediam Latine imitatus est Seneca in ''[[Hercules furens|Hercule furente]]''. Ut in [[Aiax (Sophocles)|Sophocleo ''Aiace'']] heros furore excaecatus turpes res commisit sed ad sanitatem reversus alia viam atque Aiax adiit.

== Personae ==
* [[Amphitryon|Amphitruo]], Herculis pater mortalis
*[[Megara (mythologia)|Megara]], Herculis uxor
* [[Chorus Graecus|Chorus]] senum
* Lycos [[Tyrannis|tyrannus]]
*[[Hercules]]
*[[Iris (mythologia)|Iris]], deorum nuntia
*Lyssa, &quot;Rabies&quot;&lt;ref&gt;Jacqueline Duchemin, &quot;[https://www.persee.fr/doc/reg_0035-2039_1967_num_80_379_3928 Le personnage de Lyssa dans Héraclès d'Euripide]&quot;, ''Revue des Études Grecques'', 1967ː 130-139&lt;/ref&gt;, Noctis filia
*Nuntius
*[[Theseus]], Atheniensium rex
* Herculis et Megarae tres filii et dea Athena sunt personae mutae.
Res [[Thebae (Boeotia)|Thebis]] in [[Boeotia]] geruntur. Spectatores aram pro Amphitruonis domo vident ad quam Herculis familia confugit.

== Summarium ==
[[Fasciculus:Madrid Krater Asteas MAN Inv11094.jpg|thumb|right|300px|Hercules furens in [[Crater|cratere]]  ex [[Salernum|Salerno]] in [[Campania]]. IV saeculo a.C.n.]]
*'''Prologusː''' Amphitruo spectatoribus narrat cur ipse et Megara, [[Creon Thebanus|Creontis]] regis defuncti filia, ad aram confugerintː tyrannus ex Euboea Lycos Creontem interfecit et regno potitus est. Nunc omnem stirpem regiam delere cupit, quia sibi persuasum habet Herculem ex Inferis quo [[Cerberus|Cerberi]] in lucem trahendi causa descendit numquam rediturum esse. Megara solationem et auxilium apud socerum quaerit qui spem abiciendam negat&lt;ref&gt;Versus 105-6ː  ''οὗτος δ᾽ ἀνὴρ ἄριστος ὅστις ἐλπίσι / πέποιθεν αἰεί: τὸ δ᾽ ἀπορεῖν ἀνδρὸς κακοῦ'' = vir optimus numquam desperat; ignavi viri est abdicare.&lt;/ref&gt;.
*'''Parodosː''' chorus senum Thebanorum intrat infirmitatem et senium deplorans, Herculis liberis  iam auxilio esse non possunt quorum eos miseret.
*'''Primum episodiumː''' advenit Lycus qui ab eis quaerit an mortem vitaturos esse adhuc sperentː nam Herculem nunc in Inferis iacere adfirmabat et vivum timidum bellatorem fuisse ut qui eminus sagittis potius quam gladio pugnaret. Amphitruo filii honorem vindicat sagittarios victoriae saepe utiliores fuisse quam qui comminus pugnabant demonstransː praeterea hoplitae fortuna e virtute circumiacentium militum pendet qui agmen efficiunt. Lycus non respondet sed comitibus imperat ut materiam congerant ad supplices circa aram concremandos. Tum Megara mortem non iam fugere decernit&lt;ref&gt;Versus 309-10ː ''τὰς τῶν θεῶν γὰρ ὅστις ἐκμοχθεῖ τύχας, / πρόθυμός ἐστιν, ἡ προθυμία δ᾽ ἄφρων'' = qui sortibus deorum impugnat strenuus quidem est sed illa strenuitas amens.&lt;/ref&gt; atque tantummodo petit ut sibi liberos in domo funebri apparatu vestire interim liceat. Quod concessit Lycus et omnes scaenam relinquunt.
*'''Primum stasimonː'''  chorus [[Labores Herculis|Herculis labores]] celebrat et senii imbellicitatem deplorat.
*'''Secundum episodiumː''' Herculis familia e domo exit, ad mortem parata. Megara praeclaram vitam describit quam tribus filiis speraverat et iterum Herculem precatur ut appareat etiam sub umbrae forma. Quod mirabiliter evenitː nam Herculem inopinate ex Inferis redeuntem adspiciunt qui eis narrat quomodo Cerberum super terram traxerit atque simul [[Theseus|Theseum]] ibi captivum in lucem reduxerit. Horrescit cum liberos suos funebribus stolis vestitos adspicit. Lyci scelera postquam audiit ad ultionem paratus cum familia domum intrat.
*'''Secundum stasimonː'''  chorus senectutem suam vituperans iuventutem celebrat. Deos si humana sentirent secundam iuventutem bonis viris daturos fuisse existimat ut boni a malis certo signo secernerentur. Ipsum numquam [[Musae (deae)|Musarum]] ministerium deserturum pollicetur quia  [[Mnemosyne|Mnemosynen]] colere etiam seni licet atque ita hymno Herculem celebrabit&lt;ref&gt;Versus 676-9ː ''μὴ ζῴην μετ' ἀμουσίας, /
αἰεὶ δ' ἐν στεφάνοισιν εἴην. / Ἔτι τοι γέρων ἀοιδὸς / κελαδεῖ Μναμοσύναν·'' = Ne vivam sine Musis, semper sub coronis verser. Senex quoque vates Mnemosynen canit...&lt;/ref&gt;. Multi [[Critica litterarum|critici]] hos versus ipsius Euripidis de se verba esse arbitrantur.
*'''Tertium episodiumː''' Lycus redux Amphitruonem solum invenit quem ubi sint ceteri morituri rogat. Postquam intra domum eos manere accepit cum satellitibus domum ingreditur. Mox lugubres clamores audiuntur et chori senes  tyrannum Herculis ictibus perire intelligunt.
*'''Tertium stasimonː''' chorus Iovi ceterisque deis gratiam agit quod opem iustis tulerint et de impiis poenas sumpserint. Totam Boeotiam cum montibus et fluminibus ad gaudendum provocat. Ultimis versibus iam horrendis visis in caelo manifestis terretur.
*'''Quartum episodiumː''' advenae caelestes sunt Iris, Iunonis nuntia et Lyssa, Noctis filia cuius aspectus [[Furiae]] similis est et munus in furore pectori eorum incutiendo quos dei perdere volunt constat. Iris consilium suum senibus aperitː Herculem  usque ad caedem innocentium filiorum furore insanientem  faciet. Nisi enim Hercules hanc poenam  dabit dei in posterum spernentur quos homines vicisse sese iactabunt. Lyssa primum Herculem, nobilem heroem, adoriri aspernatur. At cum irascitur Iris iussa facere incipit. Mox lamentationes Amphitruonis e domo audiuntur et nuntius accurrit quomodo Hercules liberos et uxorem immaniter interfecerit nuntiaturus quos Eurysthei familiam esse falso credebat. Cum autem patrem sagittis iam peteret dea Athena domum diruens saxum in herois pectus proiecit qui profundo somno statim obrutus est.
*'''Quartum stasimonː''' chorus caedem et Herculis fatum flebiliter deplorat dum machina ''ekkuklêma'' vocata domum interiorem et cadavera et Herculem ligatum dormientemque in scaenam profert et spectatoribus ostendit.
*'''Exodosː''' Hercules columnae fragmento Amphitruonis iussu adligatus expergiscitur ruinis et cadaveribus undique circumdatus. Quid evenerit omnino oblitus est et postquam pater eum de liberorum caede paulatim certiorem fecit ad mortem voluntariam decurrere vult scelerum conscientiam ut effugeret. Tum vero Theseus advenit qui beneficii memor Athenarum hospitalitatem heroi polliceturː ubi post mortem quoque lapideis monumentis coletur. Quin etiam exemplo deorum eum solatur qui nefanda facinora volentes committere solent nec minus dei sunt. Hercules respondet fabulis a poetis de deis excogitatis se non credere&lt;ref&gt;Versus 1340-6.&lt;/ref&gt;  sed miseram vitam tolerare fortius esse existimans&lt;ref&gt;Versus 1347-52ː ''Ἐσκεψάμην δὲ καίπερ ἐν κακοῖσιν ὤν, / μὴ δειλίαν ὄφλω τιν' ἐκλιπὼν φάος· ... ἐγκαρτερήσω θάνατον'' = 'quamquam miser sum timui ne ignaviae accusarer si lucem liquissem... Morti resistam' (sunt qui textum manuscriptorum ita emendentː ''ἐγκαρτερήσω βίοτον'' = vitam tolerabo).&lt;/ref&gt; quam ad mortem confugere gratias agit et Theseum sequitur postquam liberos sepeliendos patri Amphitruoni tradidit.
== De argumento ==
Herculis insaniae mythus iam ante Euripidem innotuerat&lt;ref&gt;Iam in ''Carminibus Cypriis'' narrabatur.&lt;/ref&gt;; at ille multa novavit atque mutavit in primis ut heroem minus rudem et agrestem efficeret quam in fabulis ostendebatur. Ita haec caedes quae vulgo ante [[Labores Herculis|labores]] a prioribus poetis ponebatur post labores ad ultimam vitam transtulit&lt;ref&gt;Ita Euripides [[Deianira|Deianiram]] et pyram in monte [[Oeta]] ignorabat.&lt;/ref&gt; ita ut poenam eiusmodi minime meruisse videretur. In mytho communi liberos tantum necabat atque deinde Megaram propinquo [[Iolaus|Iolao]] in matrimonium tradebat quod spectatoribus quinti saeculi nimis durum videri potuit. Tyrannus Lycus quoque Euripidis inventio fuisse creditur quae invidiam in Herculis inimicos moveret. Deam poliadem Athenam et Atheniensium regem Theseum in mythum introduxit in maiorem patriae gloriam spectatorumque benevolentiae captandae causa. Simul non malum hominem esse Herculem magis  apparebat siquidem amicitiam in infortunio tam fidelem servare meruisset. Praeterea Hercules in Atticam proficiscitur velut ibi moriturus&lt;ref&gt;Versus 1331-7.&lt;/ref&gt; et insuper eodem patrem Amphitruonem arcessiturus ad sepulturam&lt;ref&gt;Versus 1421.&lt;/ref&gt;. Quod nusquam alibi legitur. Quin etiam labores suscepisse dicitur non tam Iunonis ob iram coactus quam ut patris exsulis reditum in patriam Argivam redimeret&lt;ref&gt;Verba  Amphitruonis versibus 16-21.&lt;/ref&gt;, quod a vulgata versione discrepat.

Diu critici in hac tragoedia actionem velut in duas partes distractam reprehenderuntː prima enim parte Lyci saevitia timetur et  periculo illo Herculis reditu vix suppresso iam chorus providentiam deorum iustitiam in terris restituentium celebrabat, quae felix conclusio fuisset,  cum duae saevae deae  in scaenam inrumpunt et qui liberis prius saluti fuit eorum interfector per voluntatem deorum fit. Itaque reiectis deis Hercules Iovis paternitatem aspernatur atque solius Amphitrionis filius esse decernit&lt;ref&gt;Versus 1265ː ''πατέρα γὰρ ἀντὶ Ζηνὸς ἡγοῦμαι σὲ ἐγώ''. &lt;/ref&gt;ː simul ad humanam amicitiam decurrit. Non apud deos enim sed apud homines auxilium et misericordia inveniuntur.

==Notae==
&lt;references/&gt;

== Editiones et commentarii ==
[[Fasciculus:Mosaic panel depicting the madness of Heracles (Hercules furens), from the Villa Torre de Palma near Monforte, 3rd-4th century AD, National Archaeology Museum of Lisbon, Portugal (12973806145).jpg|thumb|[[Opus tessellatum]] in [[Lusitania]] repertumː Hercules furens parvum filium coram Megara taeda petit.]]
*''Euripides, Heracles with introduction and commentary by Godfrey W. Bond'', Oxonii, Clarendon Press, 1982 [https://www.persee.fr/doc/reg_0035-2039_1983_num_96_455_1366_t2_0320_0000_2 Recensio critica]

== Plura legere si cupis ==
*Jacqueline Assaël, &quot;[https://journals.openedition.org/rursus/563 La violence dans l'Héraclès furieux d’Euripide. Lecture girardienne]&quot;, ''Rursus'', 2011.
*Gerhard J. Baudy, &quot;[https://www.jstor.org/stable/4476951 Die Herrschaft des Wolfes: Das Thema der 'verkehrten Welt' in Euripides' 'Herakles']&quot;, ''Hermes'', 1993ː 159-180
*E. M. Griffiths, &quot;[https://www.jstor.org/stable/4433378 Euripides' &quot;Herakles&quot; and the Pursuit of Immortality]&quot;, ''Mnemosyne'', 2002ː 641-656
*W. E. Higgins, &quot;[https://www.jstor.org/stable/20538843 Deciphering Time in the &quot;Herakles&quot; of Euripides]&quot;, ''Quaderni Urbinati di Cultura Classica'', 1984ː 89-109
*F. Jouan, &quot;[https://www.persee.fr/doc/rea_0035-2004_1970_num_72_3_3872 Le « Prométhée» d'Eschyle et l'« Héraclès » d'Euripide]&quot;, ''Revue des Études Anciennes'', 1970ː 317-331
*Brooke Holmes, &quot;[https://www.jstor.org/stable/10.1525/ca.2008.27.2.231 Euripides' Heracles in the Flesh]&quot;, ''Classical Antiquity'', 2008ː 231-281
*Rocco Marseglia, &quot;[https://journals.openedition.org/pallas/16361?lang=fr Mythe et dramaturgie dans l’Héraclès d’Euripide]&quot;, ''Pallas'', 2019ː  17-31
*Mark W. Padilla, &quot;[https://www.jstor.org/stable/26309583 Heroic paternity in Euripides' &quot;Heracles&quot;]&quot;, ''Arethusa'', 1994ː 279-302
*Thalia Papadopoulou, &quot;[https://www.jstor.org/stable/4433556 Herakles and Hercules: The Hero's Ambivalence in Euripides and Seneca]&quot;, ''Mnemosyne'', 2004ː 257-283 
*Cecilia J. Perczyk, &quot;[https://ri.conicet.gov.ar/bitstream/handle/11336/62117/El_ritual_y_la_locura_en_Heracles_de_Euripides_-__Cecilia_J._Perczyk.pdf?sequence=5&amp;isAllowed=y El ritual y la locura en Heracles de Eurípides]&quot;, ''Phoenix'', 2015ː 41-58
*Kathleen Riley, &quot;[https://onlinelibrary.wiley.com/doi/10.1002/9781118351222.wbegt2670 Euripides: Heracles (Ἡρακλῆς μαινόμενος)]&quot; in ''Encyclopedia of Greek Tragedy'', Wiley, 2013
*Renate Schlesier, &quot;[https://www.jstor.org/stable/24307066 &quot;Héraclès&quot; et la critique des dieux chez Euripide]&quot;, ''Annali della Scuola Normale Superiore di Pisa. Classe di Lettere e Filosofia'', 1985ː 7-40

== Nexus externi ==
*[http://hodoi.fltr.ucl.ac.be/concordances/euripide_hercule_furieux/lecture/1.htm Textus Graecus et versio Francogallica apud Itinera electronica]
*[https://www.greekmythology.com/Plays/Euripides/Heracles/heracles.html Herakles apud Greekmythology.com]

[[Categoria:Tragoediae]]
[[Categoria:Litterae Graecae antiquae]]
[[Categoria:Graeciae scripta]]
[[Categoria:Euripides]]
[[Categoria:Scripta saeculo 5 a.C.n.]]
[[Categoria:Hercules]]</text>
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[[Fasciculus:Ugo_Pagliai_Anfitrione_Eracle_2007.jpg|thumb|200 px|[[Histrio]] [[Italia]]nus [[Hugo Pagliai]] ut [[Amphitruo]] in Hercule  [[Syracusae|Syracusis]] anno [[2007]] agit]]

'''''Hercules''''' ([[Graece]] {{Polytonic|Ἡρακλῆς}} aut ''Ἡρακλῆς μαινόμενος'') est [[tragoedia]] [[Euripides|Euripidea]] circa annum [[416 a.C.n.]] docta qua poeta fabulam Herculis furore excaecati et propriam uxorem cum liberis interficientis in scaenam proferebat. Quam tragoediam Latine imitatus est Seneca in ''[[Hercules furens|Hercule furente]]''. Ut in [[Aiax (Sophocles)|Sophocleo ''Aiace'']] heros furore excaecatus turpes res commisit sed ad sanitatem reversus alia viam atque Aiax adiit.

== Personae ==
* [[Amphitryon|Amphitruo]], Herculis pater mortalis
*[[Megara (mythologia)|Megara]], Herculis uxor
* [[Chorus Graecus|Chorus]] senum
* Lycos [[Tyrannis|tyrannus]]
*[[Hercules]]
*[[Iris (mythologia)|Iris]], deorum nuntia
*Lyssa, &quot;Rabies&quot;&lt;ref&gt;Jacqueline Duchemin, &quot;[https://www.persee.fr/doc/reg_0035-2039_1967_num_80_379_3928 Le personnage de Lyssa dans Héraclès d'Euripide]&quot;, ''Revue des Études Grecques'', 1967ː 130-139&lt;/ref&gt;, Noctis filia
*Nuntius
*[[Theseus]], Atheniensium rex
* Herculis et Megarae tres filii et dea Athena sunt personae mutae.
Res [[Thebae (Boeotia)|Thebis]] in [[Boeotia]] geruntur. Spectatores aram pro Amphitruonis domo vident ad quam Herculis familia confugit.

== Summarium ==
[[Fasciculus:Madrid Krater Asteas MAN Inv11094.jpg|thumb|right|300px|Hercules furens in [[Crater|cratere]]  ex [[Salernum|Salerno]] in [[Campania]]. IV saeculo a.C.n.]]
*'''Prologusː''' Amphitruo spectatoribus narrat cur ipse et Megara, [[Creon Thebanus|Creontis]] regis defuncti filia, ad aram confugerintː tyrannus ex Euboea Lycos Creontem interfecit et regno potitus est. Nunc omnem stirpem regiam delere cupit, quia sibi persuasum habet Herculem ex Inferis quo [[Cerberus|Cerberi]] in lucem trahendi causa descendit numquam rediturum esse. Megara solationem et auxilium apud socerum quaerit qui spem abiciendam negat&lt;ref&gt;Versus 105-6ː  ''οὗτος δ᾽ ἀνὴρ ἄριστος ὅστις ἐλπίσι / πέποιθεν αἰεί: τὸ δ᾽ ἀπορεῖν ἀνδρὸς κακοῦ'' = vir optimus numquam desperat; ignavi viri est abdicare.&lt;/ref&gt;.
*'''Parodosː''' chorus senum Thebanorum intrat infirmitatem et senium deplorans, Herculis liberis  iam auxilio esse non possunt quorum eos miseret.
*'''Primum episodiumː''' advenit Lycus qui ab eis quaerit an mortem vitaturos esse adhuc sperentː nam Herculem nunc in Inferis iacere adfirmabat et vivum timidum bellatorem fuisse ut qui eminus sagittis potius quam gladio pugnaret. Amphitruo filii honorem vindicat sagittarios victoriae saepe utiliores fuisse quam qui comminus pugnabant demonstransː praeterea hoplitae fortuna e virtute circumiacentium militum pendet qui agmen efficiunt. Lycus non respondet sed comitibus imperat ut materiam congerant ad supplices circa aram concremandos. Tum Megara mortem non iam fugere decernit&lt;ref&gt;Versus 309-10ː ''τὰς τῶν θεῶν γὰρ ὅστις ἐκμοχθεῖ τύχας, / πρόθυμός ἐστιν, ἡ προθυμία δ᾽ ἄφρων'' = qui sortibus deorum impugnat strenuus quidem est sed illa strenuitas amens.&lt;/ref&gt; atque tantummodo petit ut sibi liberos in domo funebri apparatu vestire interim liceat. Quod concessit Lycus et omnes scaenam relinquunt.
*'''Primum stasimonː'''  chorus [[Labores Herculis|Herculis labores]] celebrat et senii imbellicitatem deplorat.
*'''Secundum episodiumː''' Herculis familia e domo exit, ad mortem parata. Megara praeclaram vitam describit quam tribus filiis speraverat et iterum Herculem precatur ut appareat etiam sub umbrae forma. Quod mirabiliter evenitː nam Herculem inopinate ex Inferis redeuntem adspiciunt qui eis narrat quomodo Cerberum super terram traxerit atque simul [[Theseus|Theseum]] ibi captivum in lucem reduxerit. Horrescit cum liberos suos funebribus stolis vestitos adspicit. Lyci scelera postquam audiit ad ultionem paratus cum familia domum intrat.
*'''Secundum stasimonː'''  chorus senectutem suam vituperans iuventutem celebrat. Deos si humana sentirent secundam iuventutem bonis viris daturos fuisse existimat ut boni a malis certo signo secernerentur. Ipsum numquam [[Musae (deae)|Musarum]] ministerium deserturum pollicetur quia  [[Mnemosyne|Mnemosynen]] colere etiam seni licet atque ita hymno Herculem celebrabit&lt;ref&gt;Versus 676-9ː ''μὴ ζῴην μετ' ἀμουσίας, /
αἰεὶ δ' ἐν στεφάνοισιν εἴην. / Ἔτι τοι γέρων ἀοιδὸς / κελαδεῖ Μναμοσύναν·'' = Ne vivam sine Musis, semper sub coronis verser. Senex quoque vates Mnemosynen canit...&lt;/ref&gt;. Multi [[Critica litterarum|critici]] hos versus ipsius Euripidis de se verba esse arbitrantur.
*'''Tertium episodiumː''' Lycus redux Amphitruonem solum invenit quem ubi sint ceteri morituri rogat. Postquam intra domum eos manere accepit cum satellitibus domum ingreditur. Mox lugubres clamores audiuntur et chori senes  tyrannum Herculis ictibus perire intelligunt.
*'''Tertium stasimonː''' chorus Iovi ceterisque deis gratiam agit quod opem iustis tulerint et de impiis poenas sumpserint. Totam Boeotiam cum montibus et fluminibus ad gaudendum provocat. Ultimis versibus iam horrendis visis in caelo manifestis terretur.
*'''Quartum episodiumː''' advenae caelestes sunt Iris, Iunonis nuntia et Lyssa, Noctis filia cuius aspectus [[Furiae]] similis est et munus in furore pectori eorum incutiendo quos dei perdere volunt constat. Iris consilium suum senibus aperitː Herculem  usque ad caedem innocentium filiorum furore insanientem  faciet. Nisi enim Hercules hanc poenam  dabit dei in posterum spernentur quos homines vicisse sese iactabunt. Lyssa primum Herculem, nobilem heroem, adoriri aspernatur. At cum irascitur Iris iussa facere incipit. Mox lamentationes Amphitruonis e domo audiuntur et nuntius accurrit quomodo Hercules liberos et uxorem immaniter interfecerit nuntiaturus quos Eurysthei familiam esse falso credebat. Cum autem patrem sagittis iam peteret dea Athena domum diruens saxum in herois pectus proiecit qui profundo somno statim obrutus est.
*'''Quartum stasimonː''' chorus caedem et Herculis fatum flebiliter deplorat dum machina ''ekkuklêma'' vocata domum interiorem et cadavera et Herculem ligatum dormientemque in scaenam profert et spectatoribus ostendit.
*'''Exodosː''' Hercules columnae fragmento Amphitruonis iussu adligatus expergiscitur ruinis et cadaveribus undique circumdatus. Quid evenerit omnino oblitus est et postquam pater eum de liberorum caede paulatim certiorem fecit ad mortem voluntariam decurrere vult scelerum conscientiam ut effugeret. Tum vero Theseus advenit qui beneficii memor Athenarum hospitalitatem heroi polliceturː ubi post mortem quoque lapideis monumentis coletur. Quin etiam exemplo deorum eum solatur qui nefanda facinora volentes committere solent nec minus dei sunt. Hercules respondet fabulis a poetis de deis excogitatis se non credere&lt;ref&gt;Versus 1340-6.&lt;/ref&gt;  sed miseram vitam tolerare fortius esse existimans&lt;ref&gt;Versus 1347-52ː ''Ἐσκεψάμην δὲ καίπερ ἐν κακοῖσιν ὤν, / μὴ δειλίαν ὄφλω τιν' ἐκλιπὼν φάος· ... ἐγκαρτερήσω θάνατον'' = 'quamquam miser sum timui ne ignaviae accusarer si lucem liquissem... Morti resistam' (sunt qui textum manuscriptorum ita emendentː ''ἐγκαρτερήσω βίοτον'' = vitam tolerabo).&lt;/ref&gt; quam ad mortem confugere gratias agit et Theseum sequitur postquam liberos sepeliendos patri Amphitruoni tradidit.
== De argumento ==
Herculis insaniae mythus iam ante Euripidem innotuerat&lt;ref&gt;Iam in ''Carminibus Cypriis'' narrabatur.&lt;/ref&gt;; at ille multa novavit atque mutavit in primis ut heroem minus rudem et agrestem efficeret quam in fabulis ostendebatur. Ita haec caedes quae vulgo ante [[Labores Herculis|labores]] a prioribus poetis ponebatur post labores ad ultimam vitam transtulit&lt;ref&gt;Ita Euripides [[Deianira|Deianiram]] et pyram in monte [[Oeta]] ignorabat.&lt;/ref&gt; ita ut poenam eiusmodi minime meruisse videretur. In mytho communi liberos tantum necabat atque deinde Megaram propinquo [[Iolaus|Iolao]] in matrimonium tradebat quod spectatoribus quinti saeculi nimis durum videri potuit. Tyrannus Lycus quoque Euripidis inventio fuisse creditur quae invidiam in Herculis inimicos moveret. Deam poliadem Athenam et Atheniensium regem Theseum in mythum introduxit in maiorem patriae gloriam spectatorumque benevolentiae captandae causa. Simul non malum hominem esse Herculem magis  apparebat siquidem amicitiam in infortunio tam fidelem servare meruisset. Praeterea Hercules in Atticam proficiscitur velut ibi moriturus&lt;ref&gt;Versus 1331-7.&lt;/ref&gt; et insuper eodem patrem Amphitruonem arcessiturus ad sepulturam&lt;ref&gt;Versus 1421.&lt;/ref&gt;. Quod nusquam alibi legitur. Quin etiam labores suscepisse dicitur non tam Iunonis ob iram coactus quam ut patris exsulis reditum in patriam Argivam redimeret&lt;ref&gt;Verba  Amphitruonis versibus 16-21.&lt;/ref&gt;, quod a vulgata versione discrepat.

Diu critici in hac tragoedia actionem velut in duas partes distractam reprehenderuntː prima enim parte Lyci saevitia timetur et  periculo illo Herculis reditu vix suppresso iam chorus providentiam deorum iustitiam in terris restituentium celebrabat, quae felix conclusio fuisset,  cum duae saevae deae  in scaenam inrumpunt et qui liberis prius saluti fuit eorum interfector per voluntatem deorum fit. Itaque reiectis deis Hercules Iovis paternitatem aspernatur atque solius Amphitrionis filius esse decernit&lt;ref&gt;Versus 1265ː ''πατέρα γὰρ ἀντὶ Ζηνὸς ἡγοῦμαι σὲ ἐγώ''. &lt;/ref&gt;ː simul ad humanam amicitiam decurrit. Non apud deos enim sed apud homines auxilium et misericordia inveniuntur.

==Notae==
&lt;references/&gt;

== Editiones et commentarii ==
[[Fasciculus:Mosaic panel depicting the madness of Heracles (Hercules furens), from the Villa Torre de Palma near Monforte, 3rd-4th century AD, National Archaeology Museum of Lisbon, Portugal (12973806145).jpg|thumb|[[Opus tessellatum]] in [[Lusitania]] repertumː Hercules furens parvum filium coram Megara taeda petit.]]
*''Euripides, Heracles with introduction and commentary by Godfrey W. Bond'', Oxonii, Clarendon Press, 1982 [https://www.persee.fr/doc/reg_0035-2039_1983_num_96_455_1366_t2_0320_0000_2 Recensio critica]

== Plura legere si cupis ==
*Jacqueline Assaël, &quot;[https://journals.openedition.org/rursus/563 La violence dans l'Héraclès furieux d’Euripide. Lecture girardienne]&quot;, ''Rursus'', 2011.
*Gerhard J. Baudy, &quot;[https://www.jstor.org/stable/4476951 Die Herrschaft des Wolfes: Das Thema der 'verkehrten Welt' in Euripides' 'Herakles']&quot;, ''Hermes'', 1993ː 159-180
*E. M. Griffiths, &quot;[https://www.jstor.org/stable/4433378 Euripides' &quot;Herakles&quot; and the Pursuit of Immortality]&quot;, ''Mnemosyne'', 2002ː 641-656
*W. E. Higgins, &quot;[https://www.jstor.org/stable/20538843 Deciphering Time in the &quot;Herakles&quot; of Euripides]&quot;, ''Quaderni Urbinati di Cultura Classica'', 1984ː 89-109
*F. Jouan, &quot;[https://www.persee.fr/doc/rea_0035-2004_1970_num_72_3_3872 Le « Prométhée» d'Eschyle et l'« Héraclès » d'Euripide]&quot;, ''Revue des Études Anciennes'', 1970ː 317-331
*Brooke Holmes, &quot;[https://www.jstor.org/stable/10.1525/ca.2008.27.2.231 Euripides' Heracles in the Flesh]&quot;, ''Classical Antiquity'', 2008ː 231-281
*Rocco Marseglia, &quot;[https://journals.openedition.org/pallas/16361?lang=fr Mythe et dramaturgie dans l’Héraclès d’Euripide]&quot;, ''Pallas'', 2019ː  17-31
*Mark W. Padilla, &quot;[https://www.jstor.org/stable/26309583 Heroic paternity in Euripides' &quot;Heracles&quot;]&quot;, ''Arethusa'', 1994ː 279-302
*Thalia Papadopoulou, &quot;[https://www.jstor.org/stable/4433556 Herakles and Hercules: The Hero's Ambivalence in Euripides and Seneca]&quot;, ''Mnemosyne'', 2004ː 257-283 
*Cecilia J. Perczyk, &quot;[https://ri.conicet.gov.ar/bitstream/handle/11336/62117/El_ritual_y_la_locura_en_Heracles_de_Euripides_-__Cecilia_J._Perczyk.pdf?sequence=5&amp;isAllowed=y El ritual y la locura en Heracles de Eurípides]&quot;, ''Phoenix'', 2015ː 41-58
*Antonietta Provenza, &quot;[https://www.jstor.org/stable/23470076 Madness and bestialization in Euripides' &quot;Heracles&quot;]&quot;, ''The Classical Quarterly'', 2013ː 68-93 
*Kathleen Riley, &quot;[https://onlinelibrary.wiley.com/doi/10.1002/9781118351222.wbegt2670 Euripides: Heracles (Ἡρακλῆς μαινόμενος)]&quot; in ''Encyclopedia of Greek Tragedy'', Wiley, 2013
*Renate Schlesier, &quot;[https://www.jstor.org/stable/24307066 &quot;Héraclès&quot; et la critique des dieux chez Euripide]&quot;, ''Annali della Scuola Normale Superiore di Pisa. Classe di Lettere e Filosofia'', 1985ː 7-40

== Nexus externi ==
*[http://hodoi.fltr.ucl.ac.be/concordances/euripide_hercule_furieux/lecture/1.htm Textus Graecus et versio Francogallica apud Itinera electronica]
*[https://www.greekmythology.com/Plays/Euripides/Heracles/heracles.html Herakles apud Greekmythology.com]

[[Categoria:Tragoediae]]
[[Categoria:Litterae Graecae antiquae]]
[[Categoria:Graeciae scripta]]
[[Categoria:Euripides]]
[[Categoria:Scripta saeculo 5 a.C.n.]]
[[Categoria:Hercules]]</text>
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      <text bytes="13350" xml:space="preserve">{{Videdis|Hercules (discretiva)}}
[[Fasciculus:Ugo_Pagliai_Anfitrione_Eracle_2007.jpg|thumb|200 px|[[Histrio]] [[Italia]]nus [[Hugo Pagliai]] ut [[Amphitruo]] in Hercule  [[Syracusae|Syracusis]] anno [[2007]] agit]]

'''''Hercules''''' ([[Graece]] {{Polytonic|Ἡρακλῆς}} aut ''Ἡρακλῆς μαινόμενος'') est [[tragoedia]] [[Euripides|Euripidea]] circa annum [[416 a.C.n.]] docta qua poeta fabulam Herculis furore excaecati et propriam uxorem cum liberis interficientis in scaenam proferebat. Quam tragoediam Latine imitatus est Seneca in ''[[Hercules furens|Hercule furente]]''. Ut in [[Aiax (Sophocles)|Sophocleo ''Aiace'']] heros furore excaecatus turpes res commisit sed ad sanitatem reversus alia viam atque Aiax adiit.

== Personae ==
* [[Amphitryon|Amphitruo]], Herculis pater mortalis
*[[Megara (mythologia)|Megara]], Herculis uxor
* [[Chorus Graecus|Chorus]] senum
* Lycos [[Tyrannis|tyrannus]]
*[[Hercules]]
*[[Iris (mythologia)|Iris]], deorum nuntia
*Lyssa, &quot;Rabies&quot;&lt;ref&gt;Jacqueline Duchemin, &quot;[https://www.persee.fr/doc/reg_0035-2039_1967_num_80_379_3928 Le personnage de Lyssa dans Héraclès d'Euripide]&quot;, ''Revue des Études Grecques'', 1967ː 130-139&lt;/ref&gt;, Noctis filia
*Nuntius
*[[Theseus]], Atheniensium rex
* Herculis et Megarae tres filii et dea Athena sunt personae mutae.
Res [[Thebae (Boeotia)|Thebis]] in [[Boeotia]] geruntur. Spectatores aram pro Amphitruonis domo vident ad quam Herculis familia confugit.

== Summarium ==
[[Fasciculus:Madrid Krater Asteas MAN Inv11094.jpg|thumb|right|300px|Hercules furens in [[Crater|cratere]]  ex [[Salernum|Salerno]] in [[Campania]]. IV saeculo a.C.n.]]
*'''Prologusː''' Amphitruo spectatoribus narrat cur ipse et Megara, [[Creon Thebanus|Creontis]] regis defuncti filia, ad aram confugerintː tyrannus ex Euboea Lycos Creontem interfecit et regno potitus est. Nunc omnem stirpem regiam delere cupit, quia sibi persuasum habet Herculem ex Inferis quo [[Cerberus|Cerberi]] in lucem trahendi causa descendit numquam rediturum esse. Megara solationem et auxilium apud socerum quaerit qui spem abiciendam negat&lt;ref&gt;Versus 105-6ː  ''οὗτος δ᾽ ἀνὴρ ἄριστος ὅστις ἐλπίσι / πέποιθεν αἰεί: τὸ δ᾽ ἀπορεῖν ἀνδρὸς κακοῦ'' = vir optimus numquam desperat; ignavi viri est abdicare.&lt;/ref&gt;.
*'''Parodosː''' chorus senum Thebanorum intrat infirmitatem et senium deplorans, Herculis liberis  iam auxilio esse non possunt quorum eos miseret.
*'''Primum episodiumː''' advenit Lycus qui ab eis quaerit an mortem vitaturos esse adhuc sperentː nam Herculem nunc in Inferis iacere adfirmabat et vivum timidum bellatorem fuisse ut qui eminus sagittis potius quam gladio pugnaret. Amphitruo filii honorem vindicat sagittarios victoriae saepe utiliores fuisse quam qui comminus pugnabant demonstransː praeterea hoplitae fortuna e virtute circumiacentium militum pendet qui agmen efficiunt. Lycus non respondet sed comitibus imperat ut materiam congerant ad supplices circa aram concremandos. Tum Megara mortem non iam fugere decernit&lt;ref&gt;Versus 309-10ː ''τὰς τῶν θεῶν γὰρ ὅστις ἐκμοχθεῖ τύχας, / πρόθυμός ἐστιν, ἡ προθυμία δ᾽ ἄφρων'' = qui sortibus deorum impugnat strenuus quidem est sed illa strenuitas amens.&lt;/ref&gt; atque tantummodo petit ut sibi liberos in domo funebri apparatu vestire interim liceat. Quod concessit Lycus et omnes scaenam relinquunt.
*'''Primum stasimonː'''  chorus [[Labores Herculis|Herculis labores]] celebrat et senii imbellicitatem deplorat.
*'''Secundum episodiumː''' Herculis familia e domo exit, ad mortem parata. Megara praeclaram vitam describit quam tribus filiis speraverat et iterum Herculem precatur ut appareat etiam sub umbrae forma. Quod mirabiliter evenitː nam Herculem inopinate ex Inferis redeuntem adspiciunt qui eis narrat quomodo Cerberum super terram traxerit atque simul [[Theseus|Theseum]] ibi captivum in lucem reduxerit. Horrescit cum liberos suos funebribus stolis vestitos adspicit. Lyci scelera postquam audiit ad ultionem paratus cum familia domum intrat.
*'''Secundum stasimonː'''  chorus senectutem suam vituperans iuventutem celebrat. Deos si humana sentirent secundam iuventutem bonis viris daturos fuisse existimat ut boni a malis certo signo secernerentur. Ipsum numquam [[Musae (deae)|Musarum]] ministerium deserturum pollicetur quia  [[Mnemosyne|Mnemosynen]] colere etiam seni licet atque ita hymno Herculem celebrabit&lt;ref&gt;Versus 676-9ː ''μὴ ζῴην μετ' ἀμουσίας, /
αἰεὶ δ' ἐν στεφάνοισιν εἴην. / Ἔτι τοι γέρων ἀοιδὸς / κελαδεῖ Μναμοσύναν·'' = Ne vivam sine Musis, semper sub coronis verser. Senex quoque vates Mnemosynen canit...&lt;/ref&gt;. Multi [[Critica litterarum|critici]] hos versus ipsius Euripidis de se verba esse arbitrantur.
*'''Tertium episodiumː''' Lycus redux Amphitruonem solum invenit quem ubi sint ceteri morituri rogat. Postquam intra domum eos manere accepit cum satellitibus domum ingreditur. Mox lugubres clamores audiuntur et chori senes  tyrannum Herculis ictibus perire intelligunt.
*'''Tertium stasimonː''' chorus Iovi ceterisque deis gratiam agit quod opem iustis tulerint et de impiis poenas sumpserint. Totam Boeotiam cum montibus et fluminibus ad gaudendum provocat. Ultimis versibus iam horrendis visis in caelo manifestis terretur.
*'''Quartum episodiumː''' advenae caelestes sunt Iris, Iunonis nuntia et Lyssa, Noctis filia cuius aspectus [[Furiae]] similis est et munus in furore pectori eorum incutiendo quos dei perdere volunt constat. Iris consilium suum senibus aperitː Herculem  usque ad caedem innocentium filiorum furore insanientem  faciet. Nisi enim Hercules hanc poenam  dabit dei in posterum spernentur quos homines vicisse sese iactabunt. Lyssa primum Herculem, nobilem heroem, adoriri aspernatur. At cum irascitur Iris iussa facere incipit. Mox lamentationes Amphitruonis e domo audiuntur et nuntius accurrit quomodo Hercules liberos et uxorem immaniter interfecerit nuntiaturus quos Eurysthei familiam esse falso credebat. Cum autem patrem sagittis iam peteret dea Athena domum diruens saxum in herois pectus proiecit qui profundo somno statim obrutus est.
*'''Quartum stasimonː''' chorus caedem et Herculis fatum flebiliter deplorat dum machina ''ekkuklêma'' vocata domum interiorem et cadavera et Herculem ligatum dormientemque in scaenam profert et spectatoribus ostendit.
*'''Exodosː''' Hercules columnae fragmento Amphitruonis iussu adligatus expergiscitur ruinis et cadaveribus undique circumdatus. Quid evenerit omnino oblitus est et postquam pater eum de liberorum caede paulatim certiorem fecit ad mortem voluntariam decurrere vult scelerum conscientiam ut effugeret. Tum vero Theseus advenit qui beneficii memor Athenarum hospitalitatem heroi polliceturː ubi post mortem quoque lapideis monumentis coletur. Quin etiam exemplo deorum eum solatur qui nefanda facinora volentes committere solent nec minus dei sunt. Hercules respondet fabulis a poetis de deis excogitatis se non credere&lt;ref&gt;Versus 1340-6.&lt;/ref&gt;  sed miseram vitam tolerare fortius esse existimans&lt;ref&gt;Versus 1347-52ː ''Ἐσκεψάμην δὲ καίπερ ἐν κακοῖσιν ὤν, / μὴ δειλίαν ὄφλω τιν' ἐκλιπὼν φάος· ... ἐγκαρτερήσω θάνατον'' = 'quamquam miser sum timui ne ignaviae accusarer si lucem liquissem... Morti resistam' (sunt qui textum manuscriptorum ita emendentː ''ἐγκαρτερήσω βίοτον'' = vitam tolerabo).&lt;/ref&gt; quam ad mortem confugere gratias agit et Theseum sequitur postquam liberos sepeliendos patri Amphitruoni tradidit.
== De argumento ==
Herculis insaniae mythus iam ante Euripidem innotuerat&lt;ref&gt;Iam in ''Carminibus Cypriis'' narrabatur.&lt;/ref&gt;; at ille multa novavit atque mutavit in primis ut heroem minus rudem et agrestem efficeret quam in fabulis ostendebatur. Ita haec caedes quae vulgo ante [[Labores Herculis|labores]] a prioribus poetis ponebatur post labores ad ultimam vitam transtulit&lt;ref&gt;Ita Euripides [[Deianira|Deianiram]] et pyram in monte [[Oeta]] ignorabat.&lt;/ref&gt; ita ut poenam eiusmodi minime meruisse videretur. In mytho communi liberos tantum necabat atque deinde Megaram propinquo [[Iolaus|Iolao]] in matrimonium tradebat quod spectatoribus quinti saeculi nimis durum videri potuit. Tyrannus Lycus quoque Euripidis inventio fuisse creditur quae invidiam in Herculis inimicos moveret. Deam poliadem Athenam et Atheniensium regem Theseum in mythum introduxit in maiorem patriae gloriam spectatorumque benevolentiae captandae causa. Simul non malum hominem esse Herculem magis  apparebat siquidem amicitiam in infortunio tam fidelem servare meruisset. Praeterea Hercules in Atticam proficiscitur velut ibi moriturus&lt;ref&gt;Versus 1331-7.&lt;/ref&gt; et insuper eodem patrem Amphitruonem arcessiturus ad sepulturam&lt;ref&gt;Versus 1421.&lt;/ref&gt;. Quod nusquam alibi legitur. Quin etiam labores suscepisse dicitur non tam Iunonis ob iram coactus quam ut patris exsulis reditum in patriam Argivam redimeret&lt;ref&gt;Verba  Amphitruonis versibus 16-21.&lt;/ref&gt;, quod a vulgata versione discrepat.

Diu critici in hac tragoedia actionem velut in duas partes distractam reprehenderuntː prima enim parte Lyci saevitia timetur et  periculo illo Herculis reditu vix suppresso iam chorus providentiam deorum iustitiam in terris restituentium celebrabat, quae felix conclusio fuisset,  cum duae saevae deae  in scaenam inrumpunt et qui liberis prius saluti fuit eorum interfector per voluntatem deorum fit. Itaque reiectis deis Hercules Iovis paternitatem aspernatur atque solius Amphitrionis filius esse decernit&lt;ref&gt;Versus 1265ː ''πατέρα γὰρ ἀντὶ Ζηνὸς ἡγοῦμαι σὲ ἐγώ''. &lt;/ref&gt;ː simul ad humanam amicitiam decurrit. Non apud deos enim sed apud homines auxilium et misericordia inveniuntur.

==Notae==
&lt;references/&gt;

== Editiones et commentarii ==
[[Fasciculus:Mosaic panel depicting the madness of Heracles (Hercules furens), from the Villa Torre de Palma near Monforte, 3rd-4th century AD, National Archaeology Museum of Lisbon, Portugal (12973806145).jpg|thumb|[[Opus tessellatum]] in [[Lusitania]] repertumː Hercules furens parvum filium coram Megara taeda petit.]]
*''Euripides, Heracles with introduction and commentary by Godfrey W. Bond'', Oxonii, Clarendon Press, 1982 [https://www.persee.fr/doc/reg_0035-2039_1983_num_96_455_1366_t2_0320_0000_2 Recensio critica]

== Plura legere si cupis ==
*Jacqueline Assaël, &quot;[https://journals.openedition.org/rursus/563 La violence dans l'Héraclès furieux d’Euripide. Lecture girardienne]&quot;, ''Rursus'', 2011.
*Gerhard J. Baudy, &quot;[https://www.jstor.org/stable/4476951 Die Herrschaft des Wolfes: Das Thema der 'verkehrten Welt' in Euripides' 'Herakles']&quot;, ''Hermes'', 1993ː 159-180
*E. M. Griffiths, &quot;[https://www.jstor.org/stable/4433378 Euripides' &quot;Herakles&quot; and the Pursuit of Immortality]&quot;, ''Mnemosyne'', 2002ː 641-656
*W. E. Higgins, &quot;[https://www.jstor.org/stable/20538843 Deciphering Time in the &quot;Herakles&quot; of Euripides]&quot;, ''Quaderni Urbinati di Cultura Classica'', 1984ː 89-109
*F. Jouan, &quot;[https://www.persee.fr/doc/rea_0035-2004_1970_num_72_3_3872 Le « Prométhée» d'Eschyle et l'« Héraclès » d'Euripide]&quot;, ''Revue des Études Anciennes'', 1970ː 317-331
*Brooke Holmes, &quot;[https://www.jstor.org/stable/10.1525/ca.2008.27.2.231 Euripides' Heracles in the Flesh]&quot;, ''Classical Antiquity'', 2008ː 231-281
*J. C. Kamerbeek, &quot;[https://www.jstor.org/stable/4429190 Unity and Meaning of Euripides' &quot;Heracles&quot;]&quot;, ''Mnemosyne'', 1966ː 1-16
*Rocco Marseglia, &quot;[https://journals.openedition.org/pallas/16361?lang=fr Mythe et dramaturgie dans l’Héraclès d’Euripide]&quot;, ''Pallas'', 2019ː  17-31
*Mark W. Padilla, &quot;[https://www.jstor.org/stable/26309583 Heroic paternity in Euripides' &quot;Heracles&quot;]&quot;, ''Arethusa'', 1994ː 279-302
*Thalia Papadopoulou, &quot;[https://www.jstor.org/stable/4433556 Herakles and Hercules: The Hero's Ambivalence in Euripides and Seneca]&quot;, ''Mnemosyne'', 2004ː 257-283 
*Cecilia J. Perczyk, &quot;[https://ri.conicet.gov.ar/bitstream/handle/11336/62117/El_ritual_y_la_locura_en_Heracles_de_Euripides_-__Cecilia_J._Perczyk.pdf?sequence=5&amp;isAllowed=y El ritual y la locura en Heracles de Eurípides]&quot;, ''Phoenix'', 2015ː 41-58
*Antonietta Provenza, &quot;[https://www.jstor.org/stable/23470076 Madness and bestialization in Euripides' &quot;Heracles&quot;]&quot;, ''The Classical Quarterly'', 2013ː 68-93 
*Kathleen Riley, &quot;[https://onlinelibrary.wiley.com/doi/10.1002/9781118351222.wbegt2670 Euripides: Heracles (Ἡρακλῆς μαινόμενος)]&quot; in ''Encyclopedia of Greek Tragedy'', Wiley, 2013
*Renate Schlesier, &quot;[https://www.jstor.org/stable/24307066 &quot;Héraclès&quot; et la critique des dieux chez Euripide]&quot;, ''Annali della Scuola Normale Superiore di Pisa. Classe di Lettere e Filosofia'', 1985ː 7-40

== Nexus externi ==
*[http://hodoi.fltr.ucl.ac.be/concordances/euripide_hercule_furieux/lecture/1.htm Textus Graecus et versio Francogallica apud Itinera electronica]
*[https://www.greekmythology.com/Plays/Euripides/Heracles/heracles.html Herakles apud Greekmythology.com]

[[Categoria:Tragoediae]]
[[Categoria:Litterae Graecae antiquae]]
[[Categoria:Graeciae scripta]]
[[Categoria:Euripides]]
[[Categoria:Scripta saeculo 5 a.C.n.]]
[[Categoria:Hercules]]</text>
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      <text bytes="13501" xml:space="preserve">{{Videdis|Hercules (discretiva)}}
[[Fasciculus:Ugo_Pagliai_Anfitrione_Eracle_2007.jpg|thumb|200 px|[[Histrio]] [[Italia]]nus [[Hugo Pagliai]] ut [[Amphitruo]] in Hercule  [[Syracusae|Syracusis]] anno [[2007]] agit]]

'''''Hercules''''' ([[Graece]] {{Polytonic|Ἡρακλῆς}} aut ''Ἡρακλῆς μαινόμενος'') est [[tragoedia]] [[Euripides|Euripidea]] circa annum [[416 a.C.n.]] docta qua poeta fabulam Herculis furore excaecati et propriam uxorem cum liberis interficientis in scaenam proferebat. Quam tragoediam Latine imitatus est Seneca in ''[[Hercules furens|Hercule furente]]''. Ut in [[Aiax (Sophocles)|Sophocleo ''Aiace'']] heros furore excaecatus turpes res commisit sed ad sanitatem reversus alia viam atque Aiax adiit.

== Personae ==
* [[Amphitryon|Amphitruo]], Herculis pater mortalis
*[[Megara (mythologia)|Megara]], Herculis uxor
* [[Chorus Graecus|Chorus]] senum
* Lycos [[Tyrannis|tyrannus]]
*[[Hercules]]
*[[Iris (mythologia)|Iris]], deorum nuntia
*Lyssa, &quot;Rabies&quot;&lt;ref&gt;Jacqueline Duchemin, &quot;[https://www.persee.fr/doc/reg_0035-2039_1967_num_80_379_3928 Le personnage de Lyssa dans Héraclès d'Euripide]&quot;, ''Revue des Études Grecques'', 1967ː 130-139&lt;/ref&gt;, Noctis filia
*Nuntius
*[[Theseus]], Atheniensium rex
* Herculis et Megarae tres filii et dea Athena sunt personae mutae.
Res [[Thebae (Boeotia)|Thebis]] in [[Boeotia]] geruntur. Spectatores aram pro Amphitruonis domo vident ad quam Herculis familia confugit.

== Summarium ==
[[Fasciculus:Madrid Krater Asteas MAN Inv11094.jpg|thumb|right|300px|Hercules furens in [[Crater|cratere]]  ex [[Salernum|Salerno]] in [[Campania]]. IV saeculo a.C.n.]]
*'''Prologusː''' Amphitruo spectatoribus narrat cur ipse et Megara, [[Creon Thebanus|Creontis]] regis defuncti filia, ad aram confugerintː tyrannus ex Euboea Lycos Creontem interfecit et regno potitus est. Nunc omnem stirpem regiam delere cupit, quia sibi persuasum habet Herculem ex Inferis quo [[Cerberus|Cerberi]] in lucem trahendi causa descendit numquam rediturum esse. Megara solationem et auxilium apud socerum quaerit qui spem abiciendam negat&lt;ref&gt;Versus 105-6ː  ''οὗτος δ᾽ ἀνὴρ ἄριστος ὅστις ἐλπίσι / πέποιθεν αἰεί: τὸ δ᾽ ἀπορεῖν ἀνδρὸς κακοῦ'' = vir optimus numquam desperat; ignavi viri est abdicare.&lt;/ref&gt;.
*'''Parodosː''' chorus senum Thebanorum intrat infirmitatem et senium deplorans, Herculis liberis  iam auxilio esse non possunt quorum eos miseret.
*'''Primum episodiumː''' advenit Lycus qui ab eis quaerit an mortem vitaturos esse adhuc sperentː nam Herculem nunc in Inferis iacere adfirmabat et vivum timidum bellatorem fuisse ut qui eminus sagittis potius quam gladio pugnaret. Amphitruo filii honorem vindicat sagittarios victoriae saepe utiliores fuisse quam qui comminus pugnabant demonstransː praeterea hoplitae fortuna e virtute circumiacentium militum pendet qui agmen efficiunt. Lycus non respondet sed comitibus imperat ut materiam congerant ad supplices circa aram concremandos. Tum Megara mortem non iam fugere decernit&lt;ref&gt;Versus 309-10ː ''τὰς τῶν θεῶν γὰρ ὅστις ἐκμοχθεῖ τύχας, / πρόθυμός ἐστιν, ἡ προθυμία δ᾽ ἄφρων'' = qui sortibus deorum impugnat strenuus quidem est sed illa strenuitas amens.&lt;/ref&gt; atque tantummodo petit ut sibi liberos in domo funebri apparatu vestire interim liceat. Quod concessit Lycus et omnes scaenam relinquunt.
*'''Primum stasimonː'''  chorus [[Labores Herculis|Herculis labores]] celebrat et senii imbellicitatem deplorat.
*'''Secundum episodiumː''' Herculis familia e domo exit, ad mortem parata. Megara praeclaram vitam describit quam tribus filiis speraverat et iterum Herculem precatur ut appareat etiam sub umbrae forma. Quod mirabiliter evenitː nam Herculem inopinate ex Inferis redeuntem adspiciunt qui eis narrat quomodo Cerberum super terram traxerit atque simul [[Theseus|Theseum]] ibi captivum in lucem reduxerit. Horrescit cum liberos suos funebribus stolis vestitos adspicit. Lyci scelera postquam audiit ad ultionem paratus cum familia domum intrat.
*'''Secundum stasimonː'''  chorus senectutem suam vituperans iuventutem celebrat. Deos si humana sentirent secundam iuventutem bonis viris daturos fuisse existimat ut boni a malis certo signo secernerentur. Ipsum numquam [[Musae (deae)|Musarum]] ministerium deserturum pollicetur quia  [[Mnemosyne|Mnemosynen]] colere etiam seni licet atque ita hymno Herculem celebrabit&lt;ref&gt;Versus 676-9ː ''μὴ ζῴην μετ' ἀμουσίας, /
αἰεὶ δ' ἐν στεφάνοισιν εἴην. / Ἔτι τοι γέρων ἀοιδὸς / κελαδεῖ Μναμοσύναν·'' = Ne vivam sine Musis, semper sub coronis verser. Senex quoque vates Mnemosynen canit...&lt;/ref&gt;. Multi [[Critica litterarum|critici]] hos versus ipsius Euripidis de se verba esse arbitrantur.
*'''Tertium episodiumː''' Lycus redux Amphitruonem solum invenit quem ubi sint ceteri morituri rogat. Postquam intra domum eos manere accepit cum satellitibus domum ingreditur. Mox lugubres clamores audiuntur et chori senes  tyrannum Herculis ictibus perire intelligunt.
*'''Tertium stasimonː''' chorus Iovi ceterisque deis gratiam agit quod opem iustis tulerint et de impiis poenas sumpserint. Totam Boeotiam cum montibus et fluminibus ad gaudendum provocat. Ultimis versibus iam horrendis visis in caelo manifestis terretur.
*'''Quartum episodiumː''' advenae caelestes sunt Iris, Iunonis nuntia et Lyssa, Noctis filia cuius aspectus [[Furiae]] similis est et munus in furore pectori eorum incutiendo quos dei perdere volunt constat. Iris consilium suum senibus aperitː Herculem  usque ad caedem innocentium filiorum furore insanientem  faciet. Nisi enim Hercules hanc poenam  dabit dei in posterum spernentur quos homines vicisse sese iactabunt. Lyssa primum Herculem, nobilem heroem, adoriri aspernatur. At cum irascitur Iris iussa facere incipit. Mox lamentationes Amphitruonis e domo audiuntur et nuntius accurrit quomodo Hercules liberos et uxorem immaniter interfecerit nuntiaturus quos Eurysthei familiam esse falso credebat. Cum autem patrem sagittis iam peteret dea Athena domum diruens saxum in herois pectus proiecit qui profundo somno statim obrutus est.
*'''Quartum stasimonː''' chorus caedem et Herculis fatum flebiliter deplorat dum machina ''ekkuklêma'' vocata domum interiorem et cadavera et Herculem ligatum dormientemque in scaenam profert et spectatoribus ostendit.
*'''Exodosː''' Hercules columnae fragmento Amphitruonis iussu adligatus expergiscitur ruinis et cadaveribus undique circumdatus. Quid evenerit omnino oblitus est et postquam pater eum de liberorum caede paulatim certiorem fecit ad mortem voluntariam decurrere vult scelerum conscientiam ut effugeret. Tum vero Theseus advenit qui beneficii memor Athenarum hospitalitatem heroi polliceturː ubi post mortem quoque lapideis monumentis coletur. Quin etiam exemplo deorum eum solatur qui nefanda facinora volentes committere solent nec minus dei sunt. Hercules respondet fabulis a poetis de deis excogitatis se non credere&lt;ref&gt;Versus 1340-6.&lt;/ref&gt;  sed miseram vitam tolerare fortius esse existimans&lt;ref&gt;Versus 1347-52ː ''Ἐσκεψάμην δὲ καίπερ ἐν κακοῖσιν ὤν, / μὴ δειλίαν ὄφλω τιν' ἐκλιπὼν φάος· ... ἐγκαρτερήσω θάνατον'' = 'quamquam miser sum timui ne ignaviae accusarer si lucem liquissem... Morti resistam' (sunt qui textum manuscriptorum ita emendentː ''ἐγκαρτερήσω βίοτον'' = vitam tolerabo).&lt;/ref&gt; quam ad mortem confugere gratias agit et Theseum sequitur postquam liberos sepeliendos patri Amphitruoni tradidit.
== De argumento ==
Herculis insaniae mythus iam ante Euripidem innotuerat&lt;ref&gt;Iam in ''Carminibus Cypriis'' narrabatur.&lt;/ref&gt;; at ille multa novavit atque mutavit in primis ut heroem minus rudem et agrestem efficeret quam in fabulis ostendebatur. Ita haec caedes quae vulgo ante [[Labores Herculis|labores]] a prioribus poetis ponebatur post labores ad ultimam vitam transtulit&lt;ref&gt;Ita Euripides [[Deianira|Deianiram]] et pyram in monte [[Oeta]] ignorabat.&lt;/ref&gt; ita ut poenam eiusmodi minime meruisse videretur. In mytho communi liberos tantum necabat atque deinde Megaram propinquo [[Iolaus|Iolao]] in matrimonium tradebat quod spectatoribus quinti saeculi nimis durum videri potuit. Tyrannus Lycus quoque Euripidis inventio fuisse creditur quae invidiam in Herculis inimicos moveret. Deam poliadem Athenam et Atheniensium regem Theseum in mythum introduxit in maiorem patriae gloriam spectatorumque benevolentiae captandae causa. Simul non malum hominem esse Herculem magis  apparebat siquidem amicitiam in infortunio tam fidelem servare meruisset. Praeterea Hercules in Atticam proficiscitur velut ibi moriturus&lt;ref&gt;Versus 1331-7.&lt;/ref&gt; et insuper eodem patrem Amphitruonem arcessiturus ad sepulturam&lt;ref&gt;Versus 1421.&lt;/ref&gt;. Quod nusquam alibi legitur. Quin etiam labores suscepisse dicitur non tam Iunonis ob iram coactus quam ut patris exsulis reditum in patriam Argivam redimeret&lt;ref&gt;Verba  Amphitruonis versibus 16-21.&lt;/ref&gt;, quod a vulgata versione discrepat.

Diu critici in hac tragoedia actionem velut in duas partes distractam reprehenderuntː prima enim parte Lyci saevitia timetur et  periculo illo Herculis reditu vix suppresso iam chorus providentiam deorum iustitiam in terris restituentium celebrabat, quae felix conclusio fuisset,  cum duae saevae deae  in scaenam inrumpunt et qui liberis prius saluti fuit eorum interfector per voluntatem deorum fit. Itaque reiectis deis Hercules Iovis paternitatem aspernatur atque solius Amphitrionis filius esse decernit&lt;ref&gt;Versus 1265ː ''πατέρα γὰρ ἀντὶ Ζηνὸς ἡγοῦμαι σὲ ἐγώ''. &lt;/ref&gt;ː simul ad humanam amicitiam decurrit. Non apud deos enim sed apud homines auxilium et misericordia inveniuntur.

==Notae==
&lt;references/&gt;

== Editiones et commentarii ==
[[Fasciculus:Mosaic panel depicting the madness of Heracles (Hercules furens), from the Villa Torre de Palma near Monforte, 3rd-4th century AD, National Archaeology Museum of Lisbon, Portugal (12973806145).jpg|thumb|[[Opus tessellatum]] in [[Lusitania]] repertumː Hercules furens parvum filium coram Megara taeda petit.]]
*''Euripides, Heracles with introduction and commentary by Godfrey W. Bond'', Oxonii, Clarendon Press, 1982 [https://www.persee.fr/doc/reg_0035-2039_1983_num_96_455_1366_t2_0320_0000_2 Recensio critica]

== Plura legere si cupis ==
*Jacqueline Assaël, &quot;[https://journals.openedition.org/rursus/563 La violence dans l'Héraclès furieux d’Euripide. Lecture girardienne]&quot;, ''Rursus'', 2011.
*Shirley A. Barlow, &quot;[https://www.jstor.org/stable/642338 Structure and Dramatic Realism in Euripides' 'Heracles']&quot;, ''Greece &amp; Rome'', 1982ː 115-125
*Gerhard J. Baudy, &quot;[https://www.jstor.org/stable/4476951 Die Herrschaft des Wolfes: Das Thema der 'verkehrten Welt' in Euripides' 'Herakles']&quot;, ''Hermes'', 1993ː 159-180
*E. M. Griffiths, &quot;[https://www.jstor.org/stable/4433378 Euripides' &quot;Herakles&quot; and the Pursuit of Immortality]&quot;, ''Mnemosyne'', 2002ː 641-656
*W. E. Higgins, &quot;[https://www.jstor.org/stable/20538843 Deciphering Time in the &quot;Herakles&quot; of Euripides]&quot;, ''Quaderni Urbinati di Cultura Classica'', 1984ː 89-109
*F. Jouan, &quot;[https://www.persee.fr/doc/rea_0035-2004_1970_num_72_3_3872 Le « Prométhée» d'Eschyle et l'« Héraclès » d'Euripide]&quot;, ''Revue des Études Anciennes'', 1970ː 317-331
*Brooke Holmes, &quot;[https://www.jstor.org/stable/10.1525/ca.2008.27.2.231 Euripides' Heracles in the Flesh]&quot;, ''Classical Antiquity'', 2008ː 231-281
*J. C. Kamerbeek, &quot;[https://www.jstor.org/stable/4429190 Unity and Meaning of Euripides' &quot;Heracles&quot;]&quot;, ''Mnemosyne'', 1966ː 1-16
*Rocco Marseglia, &quot;[https://journals.openedition.org/pallas/16361?lang=fr Mythe et dramaturgie dans l’Héraclès d’Euripide]&quot;, ''Pallas'', 2019ː  17-31
*Mark W. Padilla, &quot;[https://www.jstor.org/stable/26309583 Heroic paternity in Euripides' &quot;Heracles&quot;]&quot;, ''Arethusa'', 1994ː 279-302
*Thalia Papadopoulou, &quot;[https://www.jstor.org/stable/4433556 Herakles and Hercules: The Hero's Ambivalence in Euripides and Seneca]&quot;, ''Mnemosyne'', 2004ː 257-283 
*Cecilia J. Perczyk, &quot;[https://ri.conicet.gov.ar/bitstream/handle/11336/62117/El_ritual_y_la_locura_en_Heracles_de_Euripides_-__Cecilia_J._Perczyk.pdf?sequence=5&amp;isAllowed=y El ritual y la locura en Heracles de Eurípides]&quot;, ''Phoenix'', 2015ː 41-58
*Antonietta Provenza, &quot;[https://www.jstor.org/stable/23470076 Madness and bestialization in Euripides' &quot;Heracles&quot;]&quot;, ''The Classical Quarterly'', 2013ː 68-93 
*Kathleen Riley, &quot;[https://onlinelibrary.wiley.com/doi/10.1002/9781118351222.wbegt2670 Euripides: Heracles (Ἡρακλῆς μαινόμενος)]&quot; in ''Encyclopedia of Greek Tragedy'', Wiley, 2013
*Renate Schlesier, &quot;[https://www.jstor.org/stable/24307066 &quot;Héraclès&quot; et la critique des dieux chez Euripide]&quot;, ''Annali della Scuola Normale Superiore di Pisa. Classe di Lettere e Filosofia'', 1985ː 7-40

== Nexus externi ==
*[http://hodoi.fltr.ucl.ac.be/concordances/euripide_hercule_furieux/lecture/1.htm Textus Graecus et versio Francogallica apud Itinera electronica]
*[https://www.greekmythology.com/Plays/Euripides/Heracles/heracles.html Herakles apud Greekmythology.com]

[[Categoria:Tragoediae]]
[[Categoria:Litterae Graecae antiquae]]
[[Categoria:Graeciae scripta]]
[[Categoria:Euripides]]
[[Categoria:Scripta saeculo 5 a.C.n.]]
[[Categoria:Hercules]]</text>
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      <text bytes="3930" xml:space="preserve">{{L}}[[Fasciculus:Seliga.jpg|thumb|upright=1|Caudica.]]
'''Caudica'''&lt;ref&gt;Beatrice Adelaide Lees Stevenson, [http://books.google.com/books?id=ymINAAAAIAAJ&amp;pg=PA263&amp;amp;lpg=PA263&amp;dq=caudica+latin&amp;source=web&amp;ots=RkLttWVOfz&amp;sig=re6S_X49GGMunrBcpzG4tDK5Zg0&amp;hl=en&amp;sa=X&amp;oi=book_result&amp;resnum=1&amp;ct=result &quot;Alfred the Great: The Truth Teller, Maker of England, 848-899,&quot; ''Asser,'' p.322. The mediaeval latin &quot;caudica&quot; meant a boat formed from a single hollow tree-trunk, like an Indian &quot;dugout&quot;']; sive:&lt;br&gt; '''cumbula''' (''Oxford Latin Dictionary'': &quot;Cumbula: a small boat; Plin. Ep.8.20.7&quot;), &lt;br&gt;'''scapha''' ([http://www.josephsusanka.com/silva An Online Dictionary of Modern Latin Terms] ''.boat canoe scapha, cymbula; - vi scapham impello; go -ing scapha gestor (Lev.); boat canoe, dug-out / Einbaum: linter monxyla [Plin. N.H.]; alveus -- Kanu: caudica [Gell.]; lembus caudiceus [Auson.] (Helf.)'', &lt;br&gt;'''canoa''' ({{DAEL|Canoa|178}}.&lt;/ref&gt; est [[linter]] quidam parvus, plerumque [[remus (navigatio)|remis]] acta, etsi nonnumquam [[velum|velo]] vel [[motrum|motro]] citata. Caudicae generatim acutae [[puppis|puppi]] [[prora]]que sunt. Etiam tegi possunt.

Viribus humanis citatur, et remis saepe duobus remiges usi sunt, remorum enim numerus a magnitudine caudicae dependet. Remiges sellis carinae iunctis sedentes vel ad carinam ipsam adnitentes ad cursum aspiciunt. Motus quoque contrarius ad cursum effici potest, si remis in columbariis fixis remigantur. Remi una vel duabus aciebus sunt.

== Historia ==
Veterrima caudica usquam reperta in [[Hollandia]]e oppido [[Assen]] est. Secundum [[computatio radiocarbonica|computationem radiocarbonicam]], haec caudica inter 8.200 et 7.600 a.C.n. confixa est. Haec hoc in loco in [[Musaeum Drents|Musaeo Drents]] nunc exhibetur.

== Figura et comparatio ==
=== Partes caudicae ===
[[Fasciculus:canoe.png|right|Adspectus insuper caudicam.]]
# [[Prora]]
# [[Puppis]]
# [[Carina]]
# [[Sella]]
# [[Transtrum]]
# [[Margo]] - labrum carinae.
# [[Stega]]
Apparatus additicii (non in diagrammate ostensi):
# [[Iugum (pars caudicae)|Iugum]] - Transtrum ad caudicam gerendam factum.
# Saci fluitando - ne caudica undis demergatur.
# [[Tegmen]] - ne caudica aquis impleatur.

== Genera caudicarum ==
Homines ubique varias caudicas conficiunt: et simplices uno ex caudice factas (Anglice ''dugout''), et magnas multiplicesque. Hodie quoque caudicae speciales ludis certando confectae sunt.

=== Figurae adsuetae ===
Caudicae antiquae semper ex idoneis materiis naturalibus confectae sunt. Caudicae variis in regionibus factae haec sunt:

{| class=&quot;wikitable&quot;
! [[Fasciculus:Dugout canoe on land.jpg|180px|Caudica simplex]] &lt;br /&gt;Caudica (simplex)
| Ex caudice uno facta; alibi adminicula quoque iuncta habere potest. In ora occidentali [[America Septentrionalis|Americae Septentrionalis]] [[Indi Americani]] caudicis simplicibus in [[Oceanus Pacificus|Oceano Pacifico]] ad [[balaena]]s venandas usi sunt.
|-
! [[Fasciculus:OjIBWE BIRCH BARK CANOE 1910 mINNESOTA.jpg|180px|Caudica cortice betullae tecta]] &lt;br /&gt;Caudica cortice betullae tecta
| In orientalibus regionibus placidis [[America Septentrionalis|Americae Septentrionalis]], caudicae ligneo replo confectae atque cortice [[betulla]]e tectae sunt; [[Bitumen|bitumine]] ad aquas obsistendas lita est.
|-
! [[Fasciculus:FAHopkins Shooting Rapids.jpg|180px|Caudica navigationis]] &lt;br /&gt;Caudica navigationis
| Haec caudicis e cortice betullae confectis similes fuerunt sed maiores atque ad onera hominesque vehendos (usque ad viginti homines et 1.400 kg) confectae sunt.
|-
! [[Fasciculus:Woodcanoe.jpg|150px|Caudica lignea]] &lt;br /&gt;Caudica lignea
| Caudica e ligno et velo in [[Cenomannica]] saeculo 19 confecta, cum velum facilius quam cortex betullae inveniri posset.
!
|}

{{NexInt}}
* [[Linter]]
* [[Navis]]
* [[Ratis]]
* [[Remigatio]]
* [[Umiak]]

== Notae ==
&lt;references /&gt;

[[Categoria:Genera navium|caudica]]
{{Myrias|Technologia}}</text>
      <sha1>qnh03twybeiqo0k13k8tr1bondoxp4e</sha1>
    </revision>
  </page>
  <page>
    <title>Gabriel Lippmann</title>
    <ns>0</ns>
    <id>66646</id>
    <revision>
      <id>3697674</id>
      <parentid>3168961</parentid>
      <timestamp>2022-08-16T15:57:11Z</timestamp>
      <contributor>
        <username>LilyKitty</username>
        <id>18316</id>
      </contributor>
      <comment>de vita academica</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1690" xml:space="preserve">[[Fasciculus:Gabriel_Lippmann2.jpg|thumb|Gabriel Lippmann]]

'''Gabriel Lippmann''' (natus die [[16 Augusti]] [[1845]] in [[Bonnevoie]],&lt;ref&gt;Cf. p. 82: J.A. Massard (1997): [http://massard.info/pdf/lippmann_massard.pdf ''Gabriel Lippmann et le Luxembourg''.] in: J.P. Pier &amp; J.A. Massard (éds): ''Gabriel Lippmann: Commémoration par la section des sciences naturelles, physiques et mathématiques de l’Institut grand-ducal de Luxembourg du 150e anniversaire du savant né au Luxembourg, lauréat du prix Nobel en 1908.'' Luxembourg, Section des sciences naturelles, physiques et mathématiques de l’Institut grand-ducal de Luxembourg en collaboration avec le Séminaire de mathématique et le Séminaire d’histoire des sciences et de la médecine du centre universitaire de Luxembourg: 81-111.&lt;/ref&gt; [[Luxemburgum]]- mortuus est die [[13 Iulii]] [[1921]] ) fuit [[physicus]] [[Francia|Francicus]]. Anno 1908 [[Praemium Nobelianum Physicae|Nobelianum Physicae Praemium]] abstulit artis [[photographia]]e [[color]]is ab [[unda]] [[lux|lucis]] causa.

Alma mater eius [[Schola normalis superior (Lutetia)|Schola normalis superior]] fuit.

== Notae ==
&lt;div class=&quot;references-small&quot;&gt;&lt;references/&gt;&lt;/div&gt;

== Nexus externus ==
* [http://nobelprize.org/nobel_prizes/physics/laureates/1908/ De Gabriel Lippmann] in pagina Nobeliano Physicae Praemio dicata {{Ling|Anglice}}

{{scien-bio-stipula}}
{{DEFAULTSORT:Lippmann, Gabriel}}

[[Categoria:Physici Franciae]]
[[Categoria:Physici Luxemburgi]]
[[Categoria:Socii Academiae Scientiarum Francicae]]
[[Categoria:Socii Regalis Societatis Londiniensis]]
[[Categoria:Nati 1845]]
[[Categoria:Mortui 1921]]
[[Categoria:Praemium Nobelianum Physicae]]</text>
      <sha1>ht9bohttpkm04grwpuul7987du7zfy4</sha1>
    </revision>
  </page>
  <page>
    <title>Triremis</title>
    <ns>0</ns>
    <id>71331</id>
    <revision>
      <id>3697761</id>
      <parentid>3697616</parentid>
      <timestamp>2022-08-17T10:17:29Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Naves]]; +[[Categoria:Genera navium]]</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1250" xml:space="preserve">{{L}}[[Fasciculus:ACMA Relief Lenormant.jpg|thumb|Sculptura saeculi 5 a.C.n.:  triremis Atheniensis]]
'''Triremis''' (Graece:  τριήρης) est [[navis remivaga]] antiqua adhibita praesertim in [[Graecia]] ab anno [[600 a.C.n.]] usque ad annum [[100 a.C.n.]]. [[Remus (navigatio)|Remorum]] tres ordines habuit. Celeris et mobilis est navis (vide e.g. Xen. Oec. 8.8).

Classis Athenensis saeculo quinto plus quam 300 triremes habuit.  [[Themistocles]] oraculum cognovit, quod &quot;murum ligneum&quot; construi debere dicit, qui murus est classis (Hdt. 7.141-143).

[[Fasciculus:Η τριήρης «Ολυμπιάς» εκτίθεται στον ειδικά διαμορφωμένο στεγασμένο χώρο στο Πάρκο Ναυτικής Παράδοσης, στο Τροκαντ - panoramio.jpg|thumb|Triremis ''Olympias,'' saeculo 20 facta, quae antiquas triremes aemulatur.]]

== Bibliographia ==
* Casson, Lionel.  ''Ships and Seamanship in the Ancient World.'' Princeton:  1971.  ISBN 9780691035369
* Morrison, J. S., J. F. Coates, N. B. Rankov.  ''The Athenian Trireme: the history and reconstruction of an ancient Greek warship.''  Cantabridgiae:  2000.  ISBN 0521564565

{{hist-stipula}}
[[Categoria:Genera navium]]

{{Myrias|Technologia}}</text>
      <sha1>f1hj2nmgvavuh49gyoznkd8x3c57vrz</sha1>
    </revision>
  </page>
  <page>
    <title>Navis velifera</title>
    <ns>0</ns>
    <id>72537</id>
    <revision>
      <id>3697755</id>
      <parentid>3697606</parentid>
      <timestamp>2022-08-17T10:16:46Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Naves]]; +[[Categoria:Genera navium]]</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10188" xml:space="preserve">{{L}}
'''Navis velifera'''&lt;ref&gt;V. ''sailboat'' in: {{Morgan}}&lt;/ref&gt; est [[navis]] quae a [[ventus|vento]] [[velum (navigatio)|velis]] reprehenso admota per [[mare]], [[lacus|lacum]] aut [[flumen]] navigat. Est haec navis saepissime [[lignum|lignea]], hodie tamen nonnullae eius partes [[metallum|metallo]], fibra [[vitrum|vitrea]] aut [[plasticum|materia synthetica]] instrui possunt. Veliferarum navium nonnulla genera sunt inter sese diversa magnitudine, celeritate, forma, usu atque ornamento. Primum ab [[Aegyptus|Aegyptiis]] [[Phoenices|Phoenicibusque]] perfectae, dein autem apud cunctas maritimas nationes omnis aetatis hae floruerunt naves ab antiquo usque ad [[Saeculum 19|saeculum undevicesimum]], ubi totaliter superatae ac substitutae sunt a velocioribus [[ferrum|ferreis]] navibus [[vapor]]e actis. Veliferis usus nonnulli collati sunt, cum omni aevo ad piscandum, ad [[mercatura]]s implendas, itinera maritima aut fluvialia perficienda atque [[bellum|bella]] gerenda naves huius speciei extruxissent.        

== Quomodo naves veliferae instruebantur et quibusnam partibus ==
Quaeque lignea navis velifera multas in partes dividitur, quarum alterae fixae, alterae mobiles sunt. Prora appellatur anterior navis pars, sustinens ligneum castellum quod ''casteria prorae'' nomen habet, contra posterior navis pars puppis appellatur, in qua castellum alterum (''casteria puppis'') consurgit. Eminentes partes fixae quae in navibus distingui possunt sunt:

[[Fasciculus:CantiereOlandese.jpg|thumb|Navale [[Nederlandia|Nederlandicum]] saeculi XVII]]

*'''Carina''': solidissimum infimumque fundamentum ligneum navis.
*'''Casteria prorae''': tectum ligneum saepe quasi celeumatis dormitorium habitum.  
*'''Casteria puppis''': elevatum planumque aedificium ex quo praefecti celeuma ducunt, quod se ab extremo puppis ad mediam navem extendit.     
*'''Malus ligneus''': arbor qui vela antemnasque suffert.
*'''Antemna''': trabs transversa malo fixa, in qua vela affiguntur.
*'''Cornu''': altissima sectio mali.
*'''Alveus''' ac '''sentina''': interiora navis. Alveus, a parietibus ligneis in particulas divisus, est locus sub transtris ubi onera servantur.
*'''Arbor prorae''':  arbor quae in extrema prora ponitur, sub quo saepe statua tutelaris affigitur.

=== Modus instruendi ===
Navem instruere statuto, primum ponenda carina, quae numquam postea extolli potest cum totum aedificium navis sufferat. Adduntur dein complures arcus et fasciarum tegumen ligneum quod pingui [[pix|pice]] aspergendum est. Nisi id fieret, navis inferioribus insidiaretur maris aqua in fascias ligneas penetrans ac eas facile erodens. Scaphio elevato transtris, fasciis ac statuminis, exstruuntur casteriae, latumque constratum ligneum id est pons quod alveum a superiora separat. Locutio ''sub constrato'' nautarum sermone enim alveum atque omnia interiora navis designat. Instruuntur dein mali saepe usque ad tres, eorumque antemnae, atque arbor prorae. Ad velificandum tamen multa cetera mobilia addere necesse est, scilicet gubernaculum, quod directionem navigii administret, vela complura et rudentes ad ea tendenda ac regulanda, [[ancora]] atque remi, qui vela substituant cum ventum deficiat. Denique cum navis ad velificium parata sit, de [[navale|navali]] exiens per varum trabem in mare immittitur.

== Navigatio velifica antiqua ==
[[Fasciculus:NaveGreca.jpg|thumb|Navis velifera Graeca]]
Antiquitus naves veliferae instruebantur apud [[Mare Mediterraneum|mediterraneorum]] litorum nationes, scilicet Phoenices, Aegyptios, [[Persia|Persas]] et [[Carthago|Carthaginienses]]. Naves veliferae huius aetatis quidem frequentissimae usu bellico fuerunt [[Graecia|Graecae]], quae tres in species dividebantur: naves longae, parvae celeresque, [[triremis|triremes]], ordines tres remorum habentes, atque naves quinquaginta remorum ([[Lingua Graeca antiqua|Graece]] ''Pentecontèreis''), maximae omnium trium. Erant enim his navibus praeter remos a remigantibus actos etiam singula in malo affixa quadra vela, quae venta capientia laboriosam remigationem adiuvarent. Traditur primum triremes [[Saeculum 6 a.C.n.|saeculo sexto a.C.n.]] currente instructas esse a quodam [[Amynocles|Amynocle]] [[Corinthus|Corinthio]]&lt;ref&gt;De hac re [[Thucydides]] scripsit ([[De bello Peloponnesiaco]], ''I'', 13).&lt;/ref&gt;, reliquaque duo genera etiam antiquiora esse. Apud [[Roma|Romanos]] a [[Saeculum 4 a.C.n.|saeculo quarto a.C.n.]] usque ad [[Saeculum 5|saeculum quintum p.C.n.]] naves longae illis quas diximus similes in usu fuerunt, utrisque remis ac velis quadris navigantes.

== Navigatio velifica mediaevalis ==
[[Medium Aevum|Medio Aevo]] vero usus velificae navigationis non plus quam antiquitate impetraverat obtinuit, ob parvum mercaturae incrementum ac imperitiam in navibus extruendis. Distingui hoc tempore possunt eminentes navium species duae: [[galea (navis)|galeae]] sive galerae, longae ac subtiles naves praecipue per Mare Mediterraneum velificantes a [[cursarius|cursariis]] [[Europa]]eis aut [[Arabes|Arabicis]] ornatae, et naves [[Scandinavia|scandinavae]], breviores crassioresque, alveos amplos qui merces ingentis magnitudinis continerent habentes. Commixtio horum generum duorum quidem multas variasque naves [[Novum Aevum|Novi Aevi]] genuit. Expletorum itinerum Medii Aevi unum tantum ab aliis cunctis eiusdem temporis ardore atque periculis diversum commemorandum esse videtur. Anno [[1291]] Ugolinus et Vadinus Vivaldi [[Italia|Italici]] fratres iter arduum ad [[Oceanus Atlanticus]] inceperunt duabus galeis paratis, cum Africae litora circumnavigare aut simpliciter quales quaque forma terrae extra [[Columnae Herculis|Columnas Herculis]] essent cognoscere vellent. Quae expeditio nullum bonum habuit exitum quod Vivaldi fratres umquam revertere nequiverunt, nec scimus quo pervenerint, at post eorum velicationem usque ad [[Saeculum 15|saeculum decimum quintum]] nemo certe Oceanum temptare conatus est.

== Aevum aureum navigationis velificae ==
[[Fasciculus:PortoAmsterdam.jpg|thumb|Navis in portu [[Amstelodamum|Amstelodami]], a [[Ludolf Backhuyzen]] picta]]
Principium aevi aurei navigationis velificae circiter idem ac terminus Medii Aevi est. A quo naves extrui formis proprietatibusque inceperunt et galearum et scandinavarum navium, cum primarum longum scaphium, alterarum autem magnum alveum pontemque elevatum haberent. Praeterea  cetera nautica inventa ad velificationem incrementandam multa contributa sunt, scilicet vela triangularia, quae praeter vela quadra navi addita venta optime etiam [[Oceanus|oceanica]] capere quebant, dein [[astrolabium]] ad accurate velificandum sideribus inspectis. Navigatione valide prodeunte nonnullae subtiles apertaeque nauticae depictae sunt chartae, ex quibus omnibus Arabicae hoc tempore diligentissime perfectae esse videntur. Patebant enim non solum in Arabicis regnis sed etiam in Hispania conventa in qua dissertari de re nautica atque chartas consultare licebat, quibus nomen erat Alcasarium ([[Lingua Arabica|Arabice]] ''Alcazàr''). Navigationis Europaeae [[Aevum Novum|Aevi Novi]] fructus numerari complures possunt, in iis America inventa a [[Christophorus Columbus|Christophoro Columbo]] (anno [[1492]]), commercium [[India]]e a [[Lusitania|Lusitanis]] actum, primum anno [[1497]] [[Valascus de Gama|Valasco de Gama]] navium duce, atque adventum fortuitum in [[Brasilia|Brasilem]] Lusitani navigatoris [[Alvaresius Cabral|Alvaresii Cabral]], anno [[1500]]. De Valasci de Gama itinere  praesertim narratiuncula nobis pervenit: cum [[classis]] commercialis Lusitana ab India in patriam reverteret, [[portus|portu]] conspecto, nauarchus in malis extollere iussit vela damascena, pretiosissimis pannis quos in India mercatus erat suta, ad optimum expeditionis exitum compatriotis cunctis etiam ante eum appellentem monstrandum. 

=== Imperium marium: coloniae ac societates mercatoriae ===
Lusitania circiter triginta annos id est usque ad [[1530]] maximam inter civitates potestatem marium commercique retinuit, dein autem [[Regnum Britanniarum]] ac [[Nederlandia]] de imperio inter se decertaverunt, postquam Lusitaniam usu cursariorum, [[colonia|coloniis]] perpetuis atque incremento mercaturae superaverant. Etenim annis [[1600]] ac [[1602]] duae floruerunt potissimae praeclaraeque [[Societates Indiae orientalis|societates mercatoriae Indiae Orientalis]] Aevi Novi, altera condita a Regno Britanniarum, [[Lingua Anglica|Anglice]] ''East India Company'' appellata, altera a [[Nederlandia]], [[Lingua Nederlandica|Nederlandice]] ''Vereenigde Ostindische Compagnie''.

=== Pirateria et naves bellicae ===
Saeculis [[Saeculum 16|sexto decimo]] atque [[Saeculum 17|septimo decimo]] igitur maximae nationes Europaeae inter se in praeeminentiam commercialem concurrerunt, [[cursarius|cursariis]] [[navis bellica|navibusque bellicis]] et generaliter vi militari saepius quam mercatoria arte utentes. Serviebant naves veliferae etiam [[Pirata|pirateriae]], quae formidulosa pestis coloniarum praecipue [[America]]narum hoc tempore fuit, indistincte [[Hispania|Hispanicas]], [[Anglia|Anglicas]], Lusitanas atque Nederlandicas mercaturas graviter affligens. Naves veliferae bellicae quae contra piratas aut ad proelia maritima pugnanda mittebantur insertae sex in ordinibus sunt, quorum quisque secundum rationem numeri [[canno]]num statuebatur. Hic ordo tamen plurimum ad [[Saeculum 18|saeculum duodevicesimum]] ineuntem spectat, quod temporibus antiquioribus numeratio varia diversaque fuisse videtur&lt;ref&gt;Hoc admonet D.Cordingly, in libro [http://books.google.com/books?id=slQSAAAAYAAJ&amp;q=Under+the+Black+Flag&amp;dq=Under+the+Black+Flag&amp;ei=JxY-St3bF5viygSPhcG6BQ&amp;hl=it&amp;pgis=1|''Under the Black Flag''].&lt;/ref&gt;.


[[Fasciculus:PortoOslo.jpg|thumb|left|Naves veliferae in portu [[Asloa]]e]] 
{| border=&quot;border&quot; class=&quot;wikitable sortable&quot;
|-
! Ordo
! Numerus cannonum
|-
| Primus
| 100
|-
| Secundus
| 90
|-
| Tertius
| 80-70
|-
| Quartus
| 64-50
|-
| Quintus
| 40-28
|-
| Sextus
| 24-12
|}

{{NexInt}}
* ''[[Carabella]]''
* [[Galeo]]
*[[Technologia navalis]]
*[[Glossarium nauticum]]

== Notae ==

&lt;references /&gt;

== Nexus externi ==
{{commons|sailing|navigationem velificam}}

[[Categoria:Navigatio]]
[[Categoria:Genera navium|velifera]]</text>
      <sha1>1iq766tvf3b8482pj68ia68ojbhaklj</sha1>
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  </page>
  <page>
    <title>Navis piscatoria</title>
    <ns>0</ns>
    <id>73982</id>
    <revision>
      <id>3697751</id>
      <parentid>3412802</parentid>
      <timestamp>2022-08-17T10:15:47Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Naves]]; +[[Categoria:Genera navium]]</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1166" xml:space="preserve">[[Fasciculus:Fishing Boat.JPG|thumb|Navis piscatoria [[Civitates Foederatae Americae|Civitatum Foederatarum]]]]
'''Navis piscatoria''' est navis quae ad piscandum navigat. Saepissime non ingentis est magnitudinis, et ornata latis retibus per quae [[piscis|pisces]] aut alia aquaria [[animal]]ia reprehenduntur ac ab aqua extolluntur. Operam dant in navibus huius generis nautae et [[piscator]]es. 

Sunt piscatoriarum navium genera diversa materie ac magnitudine, cum aliquae e parvulis [[linter|scaphis]] ligneis remis permotis constent, aliae autem e maioribus ferreis navibus motoriis actis. Primae iuxta litora, alterae etiam longius decessae pisces, [[crustacea]] aut [[mollusca]] excipiunt, qui postea directe consumantur aut in [[mercatus|mercatis]] vendantur. 

{{NexInt}} 
* [[Piscator]]
* [[Piscatus]]

== Nexus externus ==
{{CommuniaCat|Fishing boats|Navem piscatoriam}}

[[Categoria:Genera navium|piscatoria]]

[[en:Fishing vessel]]
[[es:Barco pesquero]]
[[fr:Navire de pêche]]
[[he:כלי שיט לדיג]]
[[ja:漁船]]
[[ko:어선]]
[[lv:Zvejas kuģis]]
[[no:Fiskebåt]]
[[pt:Barco de pesca]]
[[ru:Рыболовное судно]]
[[sv:Fiskefartyg]]</text>
      <sha1>5gja0glio4lw3akdyuab8tnr7tvajxk</sha1>
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  </page>
  <page>
    <title>Categoria:Naves praedatoriae</title>
    <ns>14</ns>
    <id>75813</id>
    <revision>
      <id>3697732</id>
      <parentid>2645126</parentid>
      <timestamp>2022-08-17T10:11:35Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Naves]]; +[[Categoria:Genera navium]]</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="73" xml:space="preserve">[[Categoria:Genera navium|praedatoriae]]
[[Categoria:Vehicula militaria]]</text>
      <sha1>0sbsausd2dqdialxbjfhhve8igelog5</sha1>
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  </page>
  <page>
    <title>Hiranuma Kiichirō</title>
    <ns>0</ns>
    <id>77981</id>
    <revision>
      <id>3697771</id>
      <parentid>2965385</parentid>
      <timestamp>2022-08-17T11:13:51Z</timestamp>
      <contributor>
        <username>IacobusAmor</username>
        <id>1163</id>
      </contributor>
      <comment>Augenda &amp;c.</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="699" xml:space="preserve">{{Augenda|2022|08|17}}
[[Fasciculus:Kiichiro Hiranuma.jpg|thumb|Hiranuma Kiichirō.]]
{{Capsa hominis Vicidata}}
'''Hiranuma Kiichirō''' ([[Iaponice]] 沼 騏一郎 ''Hiranuma Kiichirōnatus'' die [[28 Septembris]] [[1867]] natus; [[Tocio]]ne mortuus die [[22 Augusti]] [[1952]]) fuit [[iuris consultus]], politicorum peritus, et [[Iaponia]]e [[primus minister]] a die [[5 Ianuarii]] [[1939]] usque ad diem [[30 Augusti]] [[1939]].

==Nexus externi==
{{CommuniaCat|Kiichiro Hiranuma|Hiranuma Kiichirō}}

{{bio-stipula}}

{{Primi Ministri Iaponiae}}

{{Lifetime|1867|1952|Kiichiro, Hiranuma}}
[[Categoria:Iuris consulti]]
[[Categoria:Primi ministri Iaponiae]]
[[Categoria:Secundum bellum mundanum]]</text>
      <sha1>kcd0ly6y0ayhwan0lqy0j7ohgt4mliu</sha1>
    </revision>
  </page>
  <page>
    <title>Wendell Meredith Stanley</title>
    <ns>0</ns>
    <id>80167</id>
    <revision>
      <id>3697644</id>
      <parentid>3553279</parentid>
      <timestamp>2022-08-16T12:45:51Z</timestamp>
      <contributor>
        <username>LilyKitty</username>
        <id>18316</id>
      </contributor>
      <comment>de crystallizatione</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1111" xml:space="preserve">[[File:Wendell Meredith Stanley.jpg|right|210px|thumb|Wendell Meredith Stanley  (1946)]]
''' Wendell Meredith Stanley''' (natus [[Ridgeville]], [[Indiana]] die [[16 Augusti]] [[1904]] – mortuus est [[Salmantica (Hispania)|Salmanticae]] die [[15 Iunii]] [[1971]]) fuit [[biochemicus]] et [[virologia|virorum peritus]] [[CFA|Americanus]]. Stanley anno [[1946]] una cum [[Iacobus Batcheller Sumner|Iacobo Batcheller Sumner]] et [[Ioannes Howard Northrop|Ioanne Howard Northrop]] [[Praemium Nobelianum Chemiae|Nobelianum Chemiae Praemium]] de studio [[crystallum|crystallizationis]] [[enzymum|enzymi]] [[proteinum|proteinique]] [[virus biologicum|virorum]] accepit.

== Nexus externi ==

* [http://nobelprize.org/nobel_prizes/chemistry/laureates/1946/ De Wendell Meredith Stanley] in pagina Nobeliano Chemiae Praemio dicata {{Ling|Anglice}}

{{scien-bio-stipula}}

{{DEFAULTSORT:Stanley, Wendell Meredith }}

[[Categoria:Nati 1904]]
[[Categoria:Mortui 1971]]
[[Categoria:Chemici Civitatum Foederatarum]]
[[Categoria:Virologi]]
[[Categoria:Eruditi Civitatum Foederatarum]]
[[Categoria:Praemium Nobelianum Chemiae]]</text>
      <sha1>3ivum34tf9ejhtyyml9agxv5w5yc96q</sha1>
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  </page>
  <page>
    <title>Navis submarina</title>
    <ns>0</ns>
    <id>84683</id>
    <revision>
      <id>3697753</id>
      <parentid>3659623</parentid>
      <timestamp>2022-08-17T10:16:19Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Naves]]; +[[Categoria:Genera navium]]</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5541" xml:space="preserve">[[Fasciculus:Guido da Vigevano submarine 1280 1349.jpg|thumb|Hanc navem submarinam descripsit [[Guido Vigevanensis]] anno [[1280]].]]
[[Fasciculus:Roberto Valturio submarine De Re Militari 1472.jpg|thumb|Hanc navem submarinam descripsit [[Robertus Valturius]] anno [[1472]].]]
'''Navis submarina'''&lt;ref&gt;{{TraupmanLatinEng3}}.&lt;/ref&gt; seu '''navis subaquanea'''&lt;ref&gt;Bacci.&lt;/ref&gt;&lt;ref&gt;Caroli Egger, {{LRL}}.&lt;/ref&gt;&lt;ref&gt;{{Morgan}}.&lt;/ref&gt;&lt;ref&gt;Confer et [http://www.culturaclasica.com/lingualatina/lexicon_latinum_morgan.pdf].&lt;/ref&gt; seu '''navis epibathica'''&lt;ref&gt;{{Vox Latina}}&lt;/ref&gt; est [[navis]] quae sub [[maris aequor]]e moveri potest cum [[periscopium|periscopio]] et [[echometrum|echometro]]. Quod genus navium ab [[Isaacus Peral|Isaaco Peral]] [[Hispania|Hispanico]] anno [[1885]] excogitatum est. 

==Comparatio==
{{Vehicula}}
Plurima tegumina navium subaquanearum ex [[metallum|metallis]] fiunt.

==Usus==
[[Batalaria subaquanea|Batalariae subaquaneae]] in [[primum bellum mundanum|primo]] et [[secundum bellum mundanum|secundo bellis mundanis]] nec non in [[Bellum frigidum|bello frigido]] adhibitae sunt.

Hae naves, cum submersae sint, [[aqua|aquis]] celantur ab aliis navibus quae aequore navigantur; verumtamen naves quae [[Sonicum instrumentum detectorium|SID]] habent naves subaquaneas detegere possunt.

==Historia==
===''Lusitania''===
''[[RMS Lusitania]]'' fuit magna [[navis]] [[Anglia|Anglica]], quae in itinere ad [[Novum Eboracum (urbs)|Novum Eboracum]], oppugnata et submersa anno 1915 a navi submarina Germanica est [[Oceanus Atlanticus|Oceano Atlantico]] prope meridianum [[Hibernia|Hiberniae]] litus. Ob perditionem navis et bellum Germaniae submarinum contra naves mercatorias armatas Civitates Foederatae Americae die 1 Februarii 1917 [[Imperium Germanicum|imperio Germanico]] bellum indixerunt.&lt;ref&gt;Bailey, Thomas A. (October 1935), &quot;The Sinking of the Lusitania&quot;, ''The American Historical Review'' 41(1): 54–73.&lt;/ref&gt;

==Notae==
&lt;references/&gt;

==Bibliographia==
;Historia generalis
* [[Ioannes Maria Mathey|Mathey, Jean-Marie]], et Alexandre Sheldon-Duplaix. [[2002]]. ''Histoire des sous-marins: des origines à nos jours.'' Boulogne-Billancourt: ETAI.

;Cultura
* Redford, Duncan. [[2010]]. ''The Submarine: A Cultural History From the Great War to Nuclear Combat.'' I. B. Tauris.

;Submarinae ante 1914
* {{cite book | last = Gardiner | first = Robert | title = Steam, Steel and Shellfire, The steam warship 1815-1905 | year = [[1992]] | location = Annapoli, Terrae Mariae | publisher = Naval Institute Press | isbn = 9781557507747 | oclc = 30038068 | unused_data = |&lt;!-- authorlink = Robert Gardiner (author) --&gt; }}

;Bellum Russo-Iaponiense, 1904–1905
* {{cite book | last = Jentschura | first = Hansgeorg | coauthors = Dieter Jung, Peter Mickel | title = Warships of the Imperial Japanese Navy 1869-1945 | year = [[1977]] | publisher = United State Naval Institute | location = Annapoli, Terrae Mariae | isbn = 0-87021-893-X }}
*{{cite book | last = Olender | first = Piotr | title = Russo-Japanese Naval War 1904-1905 Vol. 2 Battle of Tsushima | year = [[2010]] | publisher = Stratus s.c. | location = Sandomierz 1, Poloniae | isbn = 978-83-61421-02-3 }}
* {{cite book | last = Showell | first = Jak | title = The U-Boat Century-German Submarine Warfare 1906-2006 | year = [[2006]] | location = Magna Britannia | publisher = Chatham Publishing | isbn = 1-86176241-0 | oclc = | unused data = }}
* {{cite book | last = Simmons | first = Jacques | title = A Grosset All-Color Guide WARSHIPS | year = [[1971]] | location = Civitates Foederatae | publisher = Grosset &amp; Dunlap, Inc. | isbn = 0-448-04165-0 }}
* {{cite book | last = Watts | first = Anthony J. | title = The Imperial Russian Navy | year = [[1990]] | location = Londinii | publisher = Arms and Armour Press | isbn = 0-85368-912-1}}

;Bellum mundanum secundum
* {{cite book | last = Blair | first = Clay | authorlink = Clay Blair | title = Silent Victory: The U.S. Submarine War Against Japan | year = [[1975]] | location = Philadelphiae | publisher = Lippincott | isbn = 9780397007530 | oclc = 821363 }}
* {{cite book | last = Lockwood | first = Charles A. | authorlink = Charles A. Lockwood | title = Sink 'Em All: Submarine Warfare in the Pacific | year = [[1951]] | location = Novi Eboraci | publisher = Dutton | isbn = | oclc = 1371626 }}
* {{cite book | last = O'Kane | first = Richard H. | authorlink = Richard O'Kane | title = Clear the Bridge!: The War Patrols of the USS Tang | year = [[1977]] | location = Sicagi | publisher = Rand McNally | isbn = 9780528810589 | oclc = 2965421 }}
* {{cite book | last = O'Kane | first = Richard H. | authorlink = Richard O'Kane | title = Wahoo: The Patrols of America's Most Famous World War II Submarine | year = [[1987]] | location = Novato, Californiae | publisher = Presidio Press | isbn = 9780891413011 | oclc = 15366413 }}
* {{cite book | last = Werner | first = Herbert A. | authorlink = Herbert A. Werner | title = Iron coffins: a personal account of the German U-Boat battles of World War II | year = [[1999]] | location = Londinii | publisher = Cassell Military | isbn = 9780304353309 | oclc = 41466905 }}

;Bellum Frigidum
*[[Petrus Huchthausen|Huchthausen, Peter]], et [[Alexander Sheldon-Duplaix|Alexandre Sheldon-Suplaix]]. [[2008]]. ''Hide and Seek: The Untold Story of Cold War Espionage at Sea.'' Hoboken, N.C.: J. Wiley &amp; Sons.

[[Categoria:Bella]]
[[Categoria:Inventiones Angliae]]
[[Categoria:Genera navium|submarina]]
[[Categoria:Vehicula]]

{{1000 paginae}}
{{Myrias|Technologia}}</text>
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  <page>
    <title>Batalaria subaquanea</title>
    <ns>0</ns>
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    <revision>
      <id>3697734</id>
      <parentid>3623300</parentid>
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      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Naves]]; +[[Categoria:Genera navium]]</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="707" xml:space="preserve">[[Fasciculus:HMCS Windsor SSK 877.jpg|thumb|Batalaria Subaquanea [[Canada|Canadensis]] ''HMCS Windsor'']]
'''Batalaria&lt;ref&gt;[http://www.latin-dictionary.org/JM-Latin-English-Dictionary/batalaria JM Latin English Dictionary] [http://athirdway.com/glossa Glossa]&lt;/ref&gt; subaquanea''' (-ae. f) est [[navis subaquanea]] quae in bello adhibetur.  Est ergo species [[navis bellica]]e.  Sunt autem naves submarinae quae in scientia vel aliis actionibus pacificibus adhibentur.
[[Fasciculus:SNLE-NG noir.gif|thumb|left|Exemplum animatum batalariae subaqueanae operationem monstrans]] 

== Notae ==
&lt;references /&gt;

{{Nexus desiderati}}

[[Categoria:Vehicula militaria]]
[[Categoria:Genera navium|batalaria subaquanea]]</text>
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  <page>
    <title>Batalaria</title>
    <ns>0</ns>
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        <username>Demetrius Talpa</username>
        <id>81729</id>
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      <model>wikitext</model>
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      <text bytes="419" xml:space="preserve">{{Contribuenda|Navis bellica}}
Batalaria&lt;ref&gt;[http://www.latin-dictionary.org/JM-Latin-English-Dictionary/batalaria JM Latin English Dictionary] [http://athirdway.com/glossa Glossa]&lt;/ref&gt; (-ae f.) est nomen navis quae in bellis adhibetur. 

{{NexInt}}
* [[Batalaria subaquanea]]
* [[Astrobatalaria]]
== Notae ==
&lt;references /&gt;
{{Nexus desiderati}}

[[Categoria:Genera navium|batalaria]]
[[Categoria:Vehicula militaria]]</text>
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  <page>
    <title>Depstio (felis)</title>
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      <text bytes="3255" xml:space="preserve">{{Latinitas|-4}}
[[Fasciculus:Cat claws.ogg|thumb|Felis depsens]]

'''Depstio''' est actio omnibus [[felis|felibus]] domesticis communis, animi aequitatem et tranquillitatem ostendens. Felis depsens [[Pes (animal)|ungues]] priores invicem expellit retrahitque, saepe etiam dextros et sinistros ungues alternans.

==Causa==
Multae theoriae causam depstionis explicant. Depstio ab [[Maiores|maioribus]] feris felium fortasse exoritur, quibus opus erat graminem vel [[folium|folia]] obterere ut [[nidus|nidum]] in tempus in quo quiescere faciant. Ita, depsens saepe est praecursor [[Somnus|dormiendis]]. Etiam actio fortasse est vestigium recentis nati [[mamma]]s matris depsendi ut lac secernendum excitent.&lt;ref&gt;{{cite web|url=http://www.cat-health-behavior.com/cat-behavior-kneading.html|title=Cat behavior kneading|publisher=cat-health-behavior.com|accessdate=2 Mai 2010|language=Anglice}}&lt;/ref&gt;&lt;ref&gt;{{cite web|url=http://www.pethealthandcare.com/askquestion/2978/why-do-cats-knead.html|title=Why Do Cats Knead|accessdate=2 Mai 2010|language=Anglice|year=2010|publisher=Pet Health and Care}}&lt;/ref&gt;&lt;ref&gt;{{cite web|url=http://www.catchannel.com/experts/pam_johnson_bennett/article_PamJB0020.aspx|title=Why Do Cats Knead?|first=Pam|last=Johnson-Bennett|publisher=CatChannel.com|language=Anglice|accessdate=2 Mai 2010}}&lt;/ref&gt;

Multae feles [[Murmurillum (felis)|submurmurant]] depsentes.

==Actio==
[[Image:2003-08-10 feather  04.JPG|thumb|Felis stragulum mollum depset.]]

Felis pede firme depremet, [[Digitus|digitos]] separat ut [[unguis|ungues]] exponet, et tunc ungues claudat dum pedem tollet. Illa actio invicem pedibus, unus vel dui secundi intervallum, acta est. Si felis ageat dum in domino sedet, fortasse sit acerbus si felis est magna vel valida vel ungues acutos habeat. Quamquam feles in superficie dura cum gaudio sedent, feles solum superficiem mollem vel flexibilem depsent. Nonnulla feles in superficie dura &quot;nusquam iter faciunt&quot; pro depsent.

Si felis depsens interrumpitur, interdum inritata vel oppressa visa erit.&lt;ref&gt;{{cite  web|url=http://catfactsblog.info/cat-communications-purring-and-padding|title=Cat  Communcations: Purring and  Padding|year=2007|publisher=CatFacts|language=Anglice|accessdate=3 Mai  2010}}&lt;/ref&gt;

In horta in qua feles adsunt, loci protecti effectus depsentis &quot;feri&quot; saepe aperiunt: nidi rotundi et instar felis in gramine longo obtriti. Et feles domesticae &quot;nidos&quot; ex arcis chartae densatae vel rebus similibus faciunt. Unguibus expositis depsent ut materiam radeat emolliatque. Ita, illa actio est lingua corporalis institutoque absimilis depsentis &quot;felicis&quot;.

Feles saepe stragulum ut [[stragulum salutis]] adsument. Sunt multum depsens, submurmurans, et stragulum sugens. Feles motiones sexuales monstrare cum depsente sugenteque saepe visa sunt, haud absimilis canis crus hominum &quot;futuendum&quot;. Catuli felium qui ex matre ante lacte depellantem capiuntur fortasse consuetudinem hominum, quem adsumptus est ut [[figura materna]], depsendi et aurem, oculum, nasum, digitum, vel tunicam sugendi educent. Et feles ad animalia texti agunt. Feles maxime agunt dum catuli felium sunt, sed actio in adulto saepe extendet.

{{NexInt}}
* [[Murmurillum (felis)]]

== Notae ==
&lt;references/&gt;

[[Categoria:Mores felium]]</text>
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      <text bytes="3237" xml:space="preserve">{{Latinitas|-4}}
[[Fasciculus:Cat claws.ogg|thumb|Felis depsens]]

'''Depstio''' est actio omnibus [[felis|felibus]] domesticis communis, animi aequitatem et tranquillitatem ostendens. Felis depsens [[Pes (animal)|ungues]] priores invicem expellit retrahitque, saepe etiam dextros et sinistros ungues alternans.

==Causa==
Multae theoriae causam depstionis explicant. Depstio ab [[Maiores|maioribus]] feris felium fortasse exoritur, quibus opus erat graminem vel [[folium|folia]] obterere ut [[nidus|nidum]] in tempus in quo quiescere faciant. Ita, depsens saepe est praecursor [[Somnus|dormiendis]]. Etiam actio fortasse est vestigium recentis nati [[mamma]]s matris depsendi ut lac secernendum excitent.&lt;ref&gt;{{cite web|url=http://www.cat-health-behavior.com/cat-behavior-kneading.html|title=Cat behavior kneading|publisher=cat-health-behavior.com|accessdate=2 Mai 2010|language=Anglice}}&lt;/ref&gt;&lt;ref&gt;{{cite web|url=http://www.pethealthandcare.com/askquestion/2978/why-do-cats-knead.html|title=Why Do Cats Knead|accessdate=2 Mai 2010|language=Anglice|year=2010|publisher=Pet Health and Care}}&lt;/ref&gt;&lt;ref&gt;{{cite web|url=http://www.catchannel.com/experts/pam_johnson_bennett/article_PamJB0020.aspx|title=Why Do Cats Knead?|first=Pam|last=Johnson-Bennett|publisher=CatChannel.com|language=Anglice|accessdate=2 Mai 2010}}&lt;/ref&gt;

Multae feles [[Murmurillum (felis)|submurmurant]] depsentes.

==Actio==
[[Image:2003-08-10 feather  04.JPG|thumb|Felis stragulum mollum depset.]]

Felis pede firme depremet, [[Digitus|digitos]] separat ut [[unguis|ungues]] exponet, et tunc ungues claudat dum pedem tollet. Illa actio invicem pedibus, unus vel dui secundi intervallum, acta est. Acerbum est, si felis magna vel valida vel ungues acutos habens, hoc agit dum in domino sedet. Quamquam feles in superficie dura cum gaudio sedent, feles solum superficiem mollem vel flexibilem depsent. Nonnulla feles in superficie dura &quot;nusquam iter faciunt&quot; pro depsent.

Si felis depsens interrumpitur, interdum inritata vel oppressa visa erit.&lt;ref&gt;{{cite  web|url=http://catfactsblog.info/cat-communications-purring-and-padding|title=Cat  Communcations: Purring and  Padding|year=2007|publisher=CatFacts|language=Anglice|accessdate=3 Mai  2010}}&lt;/ref&gt;

In horta in qua feles adsunt, loci protecti effectus depsentis &quot;feri&quot; saepe aperiunt: nidi rotundi et instar felis in gramine longo obtriti. Et feles domesticae &quot;nidos&quot; ex arcis chartae densatae vel rebus similibus faciunt. Unguibus expositis depsent ut materiam radeat emolliatque. Ita, illa actio est lingua corporalis institutoque absimilis depsentis &quot;felicis&quot;.

Feles saepe stragulum ut [[stragulum salutis]] adsument. Sunt multum depsens, submurmurans, et stragulum sugens. Feles motiones sexuales monstrare cum depsente sugenteque saepe visa sunt, haud absimilis canis crus hominum &quot;futuendum&quot;. Catuli felium qui ex matre ante lacte depellantem capiuntur fortasse consuetudinem hominum, quem adsumptus est ut [[figura materna]], depsendi et aurem, oculum, nasum, digitum, vel tunicam sugendi educent. Et feles ad animalia texti agunt. Feles maxime agunt dum catuli felium sunt, sed actio in adulto saepe extendet.

{{NexInt}}
* [[Murmurillum (felis)]]

== Notae ==
&lt;references/&gt;

[[Categoria:Mores felium]]</text>
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    <title>Categoria:Naves longae</title>
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        <username>Demetrius Talpa</username>
        <id>81729</id>
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      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Naves]]; +[[Categoria:Genera navium]]</comment>
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      <text bytes="34" xml:space="preserve">[[Categoria:Genera navium|Longae]]</text>
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  </page>
  <page>
    <title>Carolus Otfried Müller</title>
    <ns>0</ns>
    <id>102675</id>
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      <id>3697696</id>
      <parentid>3586525</parentid>
      <timestamp>2022-08-16T18:45:19Z</timestamp>
      <contributor>
        <username>Giorno2</username>
        <id>30162</id>
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      <text bytes="1108" xml:space="preserve">[[Fasciculus:Karl Otfried Müller - Imagines philologorum.jpg|thumb|Carolus Otfried Müller]]
'''Carolus Otfried Müller''' (natus [[Brzeg]] in [[Silesia]] die [[28 Augusti]] [[1797]]; [[Athenae|Athenis]] die [[1 Augusti]] [[1840]] mortuus), [[Ludovicus Fridericus Heindorf|Ludovici Friderici Heindorf]]  discipulus, fuit sermonum antiquorum [[philologus]] et [[archaeologus]] [[Germania|Germanicus]]. Ab anno 1819 apud [[Universitas Regia Georgia Augusta|Universitatem Goettingensem]] docuit.

== Opera selecta ==
* ''Geschichten Hellenischer Stämme und Städte''. 3 tomi, Vratislaviae 1820/1824.
* ''Die Etrusker''. 2 tomi, Vratislaviae 1828.
* ''Handbuch der Archäologie der Kunst''. Vratislaviae 1830

== Nexus externi ==
{{Communia|Karl Otfried Müller|Carolum Otfried Müller}}
*{{PND|11954153X}}

{{bio-stipula}}
{{Lifetime|1797|1840|Muller, Carolus Otfried}}
[[Categoria:Auctores Theodisci]]
[[Categoria:Philologi Germaniae]]
[[Categoria:Professores Universitatis Goettingensis]]
[[Categoria:Interpretes Graeco-Latini]]
[[Categoria:Interpretes Graeco-Theodisci]]
[[Categoria:Archaeologi Germaniae]]</text>
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    <title>Categoria:Naves bellicae</title>
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        <id>81729</id>
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      <text bytes="100" xml:space="preserve">{{Principalis|Navis bellica}}

[[Categoria:Vehicula militaria]]
[[Categoria:Genera navium|bellicae]]</text>
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  <page>
    <title>Ordo Fratrum Minorum Conventualium</title>
    <ns>0</ns>
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      <comment>pag. auxi</comment>
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      <text bytes="1259" xml:space="preserve">[[Fasciculus:Monaco andante.JPG|thumb|Franciscanus Conventualis [[Asisium|Asisii]]]]
'''Ordo Fratrum Minorum Conventualium''' ([[Index siglorum ordinum ecclesiae catholicae|Sigla]]: '''O.F.M.Conv.''' sive '''O.Min.Conv.'''), ad * [[Ordines Franciscani|Ordines Franciscanos]] pertinet. [[Religiosus|Religiosi]] huius ordinis etiam ''Minoriti'' (in terris linguae Theodiscae), ''Fratres Grisei'' (''Grey Friars'' in terris linguae Anglicae) vel ''Chordati'' (''Cordelier'' lingua Francogallica) appellantur.

Ordines Franciscani ex tribus ramis compositi sunt, scilicet ex ordinibus virorum, feminarum et tertio ordine Franciscano. Anno [[1517]] ordo virorum olim a [[Sanctus Franciscus Assisiensis|Sancto Francisco Assisiensi]] conditus in duos (et postea in tres) ordines [[sui iuris]] divisus est,  in ''Minoritas'' et ''Observantes'',  qui nomen [[Ordo Fratrum Minorum|Fratrum Minorum]] retinuerunt. His diebus Franciscanis Conventualibus sunt fere 4.200 fratres (anno 2015). Communitas minor est quam illa Fratrum Minorum aut [[Ordo Fratrum Minorum Capuccinorum|Cappuccinorum]], qui postea ab ordine Fratrum Minorum separati sunt. 

{{NexInt}}
* [[Ordines Mendicantes]]

{{reli-stipula}}

[[Categoria:Ordines Franciscani]]
[[Categoria:Ordines mendicantes]]</text>
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        <username>Utilo</username>
        <id>18337</id>
      </contributor>
      <comment>pag. auxi</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2585" xml:space="preserve">[[Fasciculus:Monaco andante.JPG|thumb|Franciscanus Conventualis [[Asisium|Asisii]]]]
'''Ordo Fratrum Minorum Conventualium''' ([[Index siglorum ordinum ecclesiae catholicae|Sigla]]: '''O.F.M.Conv.''' sive '''O.Min.Conv.'''), ad * [[Ordines Franciscani|Ordines Franciscanos]] pertinet. [[Religiosus|Religiosi]] huius ordinis etiam ''Minoriti'' (in terris linguae Theodiscae), ''Fratres Grisei'' (''Grey Friars'' in terris linguae Anglicae) vel ''Chordati'' (''Cordelier'' lingua Francogallica) appellantur.

Ordines Franciscani ex tribus ramis compositi sunt, scilicet ex ordinibus virorum, feminarum et tertio ordine Franciscano. Anno [[1517]] ordo virorum olim a [[Sanctus Franciscus Assisiensis|Sancto Francisco Assisiensi]] conditus in duos (et postea in tres) ordines [[sui iuris]] divisus est,  in ''Minoritas'' et ''Observantes'',  qui nomen [[Ordo Fratrum Minorum|Fratrum Minorum]] retinuerunt. 

== Structura ordinis ==
Curia ordinis generalis est [[Roma]]e in conventu prope [[Ecclesia Sanctorum XII Apostolorum|ecclesiam Sanctorum XII Apostolorum]], locus medius spiritalis est ''Sacer Conventus Assisiensis'' ([[monasterium]] Conventualium [[Asisium|Asisii]]). Minister Generalis ex mense Maio [[2019]] est [[Carolus Trovarelli]] [[Argentinia|Argentinus]]. 

His diebus (status 2018) ordini sunt 30 provinciae, 18 custodiae, 460 monasteria et 4048 fratres toto orbe terrarum. Communitas ergo minor est quam illa Fratrum Minorum aut [[Ordo Fratrum Minorum Capuccinorum|Cappuccinorum]], qui postea ab ordine Fratrum Minorum separati sunt. Fratres in [[paroecia|paroeciis]], [[schola|scholis]] et multis aliis locis serviunt. Proprium ordinis est vita communis et apostolatus urbanus.

Conventuales privilegio gaudent sepulcra Sancti Francisci Asisii et [[Antonius Patavinus|Sancti Antonii]] [[Patavium|Patavii]] colendi. 

== Habitus ==
Conventualium habitus ex tunica tenui funiculo (sive ''chorda'', inde ''Chordati'') albo circa ventrem alligata constat, paenula in fronte rotunda pone in acutum desinente addita. Color sive niger sive griseus esse potest.

== Conventuales noti ==
Poeta [[Angelus Silesius]] (1624–1677), postquam se [[Ecclesia Catholica Romana|Ecclesiae Cathlicae Romanae]] adiunxit, in ordinem Fratrum Minorum conventualium intravit. Inter Conventuales notissimos est [[Maximilianus Kolbe]] (1894–1941), qui in [[castra carceralia|castris carceralibus]] [[Auschwitz (castra carceralia)|Auschwitz]] pro patre familiae sua sponte mortem obiit. 


{{NexInt}}
* [[Ordines Mendicantes]]

[[Categoria:Ordines Franciscani]]
[[Categoria:Ordines mendicantes]]</text>
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    </revision>
    <revision>
      <id>3697661</id>
      <parentid>3697657</parentid>
      <timestamp>2022-08-16T13:34:26Z</timestamp>
      <contributor>
        <username>Utilo</username>
        <id>18337</id>
      </contributor>
      <comment>+nexus externi</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2661" xml:space="preserve">[[Fasciculus:Monaco andante.JPG|thumb|Franciscanus Conventualis [[Asisium|Asisii]]]]
'''Ordo Fratrum Minorum Conventualium''' ([[Index siglorum ordinum ecclesiae catholicae|Sigla]]: '''O.F.M.Conv.''' sive '''O.Min.Conv.'''), ad * [[Ordines Franciscani|Ordines Franciscanos]] pertinet. [[Religiosus|Religiosi]] huius ordinis etiam ''Minoriti'' (in terris linguae Theodiscae), ''Fratres Grisei'' (''Grey Friars'' in terris linguae Anglicae) vel ''Chordati'' (''Cordelier'' lingua Francogallica) appellantur.

Ordines Franciscani ex tribus ramis compositi sunt, scilicet ex ordinibus virorum, feminarum et tertio ordine Franciscano. Anno [[1517]] ordo virorum olim a [[Sanctus Franciscus Assisiensis|Sancto Francisco Assisiensi]] conditus in duos (et postea in tres) ordines [[sui iuris]] divisus est,  in ''Minoritas'' et ''Observantes'',  qui nomen [[Ordo Fratrum Minorum|Fratrum Minorum]] retinuerunt. 

== Structura ordinis ==
Curia ordinis generalis est [[Roma]]e in conventu prope [[Ecclesia Sanctorum XII Apostolorum|ecclesiam Sanctorum XII Apostolorum]], locus medius spiritalis est ''Sacer Conventus Assisiensis'' ([[monasterium]] Conventualium [[Asisium|Asisii]]). Minister Generalis ex mense Maio [[2019]] est [[Carolus Trovarelli]] [[Argentinia|Argentinus]]. 

His diebus (status 2018) ordini sunt 30 provinciae, 18 custodiae, 460 monasteria et 4048 fratres toto orbe terrarum. Communitas ergo minor est quam illa Fratrum Minorum aut [[Ordo Fratrum Minorum Capuccinorum|Cappuccinorum]], qui postea ab ordine Fratrum Minorum separati sunt. Fratres in [[paroecia|paroeciis]], [[schola|scholis]] et multis aliis locis serviunt. Proprium ordinis est vita communis et apostolatus urbanus.

Conventuales privilegio gaudent sepulcra Sancti Francisci Asisii et [[Antonius Patavinus|Sancti Antonii]] [[Patavium|Patavii]] colendi. 

== Habitus ==
Conventualium habitus ex tunica tenui funiculo (sive ''chorda'', inde ''Chordati'') albo circa ventrem alligata constat, paenula in fronte rotunda pone in acutum desinente addita. Color sive niger sive griseus esse potest.

== Conventuales noti ==
Poeta [[Angelus Silesius]] (1624–1677), postquam se [[Ecclesia Catholica Romana|Ecclesiae Cathlicae Romanae]] adiunxit, in ordinem Fratrum Minorum conventualium intravit. Inter Conventuales notissimos est [[Maximilianus Kolbe]] (1894–1941), qui in [[castra carceralia|castris carceralibus]] [[Auschwitz (castra carceralia)|Auschwitz]] pro patre familiae sua sponte mortem obiit. 

{{NexInt}}
* [[Ordines Mendicantes]]

== Nexus externi ==
*[https://www.ofmconv.net/en/ pagina ordinis domestica]

[[Categoria:Ordines Franciscani]]
[[Categoria:Ordines mendicantes]]</text>
      <sha1>2zmz2n5bftmlonbhndx10b55an4xj70</sha1>
    </revision>
  </page>
  <page>
    <title>Scapha Polynesia</title>
    <ns>0</ns>
    <id>116326</id>
    <revision>
      <id>3697759</id>
      <parentid>3134377</parentid>
      <timestamp>2022-08-17T10:17:16Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Naves]]; +[[Categoria:Genera navium]]</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1231" xml:space="preserve">[[Imago:Pirogue Nouvelle Calédonie.JPG|thumb|Scapha Polynesia in [[Nova Caledonia]].]][[Fasciculus:Outrigger canoe in Kenya.jpg|thumb|Scapha Polynesia in [[Kenia]].]]

'''Scapha Polynesia'''  praecipua [[Oceanus Pacificus|Pacifici]] et [[Oceanus Indicus|Indici]] [[Navis velifera|velifera]] est, cum paulo onerosae fabricationis et paucis instrumentis. Inter dissimiles ragionales varietates sine dubio velifera difusissima est.

Solitissimum exemplar praecipua [[compago|compagine]] constituunt cuius inferior pars trunco (proprio [[linter|lintrum]]), aliquando cum tabulis levata, transitur. Stabilitas [[pendulum|pendulo]] affirmatur: crebro simplex levis ligneus truncus parvae diametri. Instrumenta navi regendae simplicissima sunt, [[velum quadratum]] in pino.

Vide quoque [[vaa]]m, lintrem cursus sine velo (cum remis bipalibus).

==Historia==
[[Imago:Pōpao.jpg|thumb|right|[[Pōpao]] traditionalis [[Tonga]]nus.]]
Scapha Polynesia inter [[Linguae Austronesiae|linguarum Austronesiarum]] populos orta est. Quoque in [[Insulindia]], in [[Oceanus Pacificus|Pacifico]] et [[Madagascaria]], regione ubi [[Austronesii]] sedem collocavere, invenitur.

{{NexInt}}
*[[Linter]]
*[[Navis]]
*[[Caudica]]

[[Categoria:Genera navium]]</text>
      <sha1>3ftfnkyc5dkqdynrgn4jgdhww2ftw3d</sha1>
    </revision>
  </page>
  <page>
    <title>Navis lusoria</title>
    <ns>0</ns>
    <id>119814</id>
    <revision>
      <id>3697746</id>
      <parentid>3512423</parentid>
      <timestamp>2022-08-17T10:14:34Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Naves]]; +[[Categoria:Genera navium]]</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="732" xml:space="preserve">{{non stipula}}
[[Fasciculus:Rusalka.jpg|thumb|Navis lusoria in lacu]]
'''Navis lusoria''', aut simpliciter '''lusoria''' (-ae, ''f''), est navis plurimum ad delectationem adhibita, non ingens magnitudine, sic ut non solum per [[mare]], immo etiam per [[flumen|flumina]] [[lacus]]que navigare possit. Privati naves huius generis gubernant ut se cum familia aut amicis navigando delectent.

Antiquitus parvae [[navis velifera|naves veliferae]] e [[lignum|ligno]] factae pro ludo in usu erant. Hodie [[velum|velis]] aut [[motor]]is electricis ornantur naves lusoriae, ligno aut [[metallum|metallo]] instructae.        

{{NexInt}}
*[[Paulus Elvstrøm]]
*[[Navis velifera]]
*[[Glossarium nauticum]]

[[Categoria:Genera navium|lusoria]]</text>
      <sha1>isr71he5w7wa3mczlzp9ex84v0lgdt7</sha1>
    </revision>
  </page>
  <page>
    <title>Dorna</title>
    <ns>0</ns>
    <id>120618</id>
    <revision>
      <id>3697737</id>
      <parentid>2454847</parentid>
      <timestamp>2022-08-17T10:12:46Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Naves]]; +[[Categoria:Genera navium]]</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="637" xml:space="preserve">[[Fasciculus:Nasa11dorna05eue.jpg|thumb|right|300px|Dorna sirpiculos portans.]]
'''Dorna''' (-ae f.) vocabulum [[lingua Gallaica|Gallaicum]] est parvam [[cymba]]m ad [[piscatio]]nem destinatam designans, quae generaliter [[malus (navis)|malum]] fert et remo sive velo propelli potest [[prora]]m rotundam, puppem planam et [[carina (navis)|carinam]] prominentem habens. Dorna typica [[Gallaecia (Communitas autonoma)|Gallaicarum]] [[ria]]rum cymba est.&lt;ref&gt;[http://www.edu.xunta.es/diccionarios/BuscaTermo.jsp Definitio dornae in Dictionario Regiae Academiae Linguae Gallaicae]&lt;/ref&gt;

==Notae==
&lt;references /&gt;

[[Categoria:Genera navium]]</text>
      <sha1>bq0go7japyuq1i9df8cdnmx6o6l7qgv</sha1>
    </revision>
  </page>
  <page>
    <title>Minority Report (pellicula)</title>
    <ns>0</ns>
    <id>120647</id>
    <revision>
      <id>3697695</id>
      <parentid>3697636</parentid>
      <timestamp>2022-08-16T18:43:22Z</timestamp>
      <contributor>
        <username>Neander</username>
        <id>1973</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5514" xml:space="preserve">{{Titulus italicus}}
[[File:ExpoSYFY - Minority Report (10825723756).jpg|thumb|Ioannes Anderton, nomen vigilis principalis]]
'''''Minority Report''''' ('Renuntiatio minoris partis') est [[pellicula]] anno [[2002]] a [[Stephanus Spielberg|Stephano Spielberg]], [[moderator cinematographicus|moderatore cinematographico]] [[Civitates Foederatae|Americano]], creata et ducta.
 
== De historia ==
Quae pellicula, summam [[technologia]]m futuram squalori praeteriti temporis intermiscens, est magna futurorum visio [[dystopia|dystopica]], quae aliquatenus cum ''[[Blade Runner]]'' ([[1982]]) et ''[[Children of Men]]'' ([[2006]]) comparari potest. Spielberg in pellicula conficienda studebat, ut [[Vasingtonia (DC)|Vasingtonia]] anni 2054 a Vasingtonia sui temporis satis differret. Ex altera parte tamen nimiam dystopiam vitabat. 

In praeparanda pellicula anno 1999 Spielberg officinam cogitatoriam constituit
ex quindecim artificibus scriptoribusque et [[architectus|architectis]] coactam. Ex quibus cogitationibus [[Alex McDowell]] [[scaenographia|scaenograhus]] libellum 80 paginarum composuit, quo mundus huius pelliculae aperiretur. 

''Minority Report'' pellicula ducta est ex narratiuncula eiusdem nominis a [[Philippus Kindred Dick|Philippo K. Dick]] anno [[1956]] scripto.&lt;ref&gt;Kimbell 2011.&lt;/ref&gt;
 
== De argumento ==
[[Vasingtonia (D.C.)|Vasingtonia]]e, in capite [[Civitates Foederatae Americae|Civitatum Foederatarum Americae]], anno 2054 usu recipitur '''Praecrimen''' (''Precrime'') novum programma, cuius ope scelera, quae in futuris committentur, praedici praevenirique possint. Nam Praecrimen praedictiones haurit ex tribus hariolis (''Precogs'') — Agatha et geminis Arthuro et Dashiell — qui in brevi stagno iacent medicamentis somnificis torpentes externisque stumulis surdi et caeci. Cogitata eorum in [[monitorium|monitorio]] proiciuntur ac in [[datorum repositorium|receptacula datorum]] deponuntur. Cum triga illa homicidium factum iri praevidit, magistratus, ubi fiat, sciscitantur, homicidam ante facinus capturi. 

Ioannes Anderton, praefectus collegio sceleribus prohibendis, ipse aliquo die reus fit homicidii futuri. Programmati se adiunxerat, cum filius eius raptus nec umquam repertus esset. Itaque demisso animo est, vitabundus, adpetens neuroini,&lt;ref&gt;''Neuroinum'', commenticium medicamentum psychotropicum durum.&lt;/ref&gt; ab uxore Lara relictus. Praedicunt igitur hariolani quendam Leonem Crow ab Anderton intra 36 horas occisum iri. Anderton de hac re certior factus in fugam se dat, etsi de isto viro nihil noverit. Danny Witwer, legatus ministerii iustitiae, e vestigio initium persecutionis facit. 

In fuga Anderton, cum Iridem Hineman conditricem Praecriminis adit, certior fit Agatham hariolam eventa futura interdum alio modo videre quam ceteros. Quae discrepantia visionis, '''renuntiatio minoris partis''' dicta, conformationis causa ex tabulis pro auctoritate habitis tollitur. At harioli suam quisque memoriam retinent. Itaque Anderton ad Praecrimen reversus Agatham surripit, quo facto clauditur concordia qua Praecrimen continetur. 

Anderton cum Agatha Leonem Crow in conclavi cuiusdam deversorii indagine includit, ubi multas imagines puerorum nec non sui filii videt. Anderton Crow homicidii sui filii insimulat, Crow autem se ad picturas inserendas conductum esse dicit, poscitque ab Anderton occidi, ut hoc quidem modo familiae suae utilis sit. Recusante Anderton, Crow ipse se occidit. 

Anderton, cum renuntiationem minoris partis in mente Agathae non reperit, tamen memoriam reperit homicidii quinque annis ante facti, cuius victima est Agathae mater, quae neurono devota filiam suam Praecrimini vendiderat. Postremo Anderton comprehenditur, et de homicidio Leonis Crow suspectus in carcerem conicitur. Agatha denuo in Praecrimen restituitur. Quo facto nova creatur renuntiatio: Anderton a Lamar Burgess, Praecrimini praeposito, occisum iri. Conveniunt illi, sed Burgess se occidit oppressus [[dilemma]]te, quod solvere nequit. 

== Partium distributio ==
In pellicula hi agunt [[histrio]]nes:

* [[Thomas Cruise]]: Ioannes (&quot;John&quot;) Anderton
* [[Colin Farrell]]: deprehensor Danny Witwer
* [[Samantha Morton]]: Agatha
* [[Maximus de Sydow]]: Director Lamar Burgess
* [[Stephanus Harris]]: Jad 
* [[Catharina Morris]]: Lara Anderton
* [[Jessica Capshaw]]: Evanna
* [[Neal McDonough]]: centurio Gordon Fletcher
* [[Ricardus Coca]]: militiae vir
* [[Patricus Kilpatrick]]: centurio Jeff Knott
* [[Franciscus Grillo]]: militiae vir
* [[Timotheus Blake Nelson]]: Gideon
* [[Arye Gross]]: Howard Marks
* [[Joel Gretsch]]: Donald Dubin
* [[Jessica Harper]]: Anne, Agathae mater
* [[Petrus Stormare]]: Dr. Solomon Eddie
* [[Victor Raider-Wexler]]: civitatis causidicus Nash
* [[Gulielmus Mapother]]: operarius deversorii 
* [[Meredith Monroe]]: 
* [[Paulus Thomas Anderson]]: viator in
* [[Cameron Crowe]]: viator in
* [[Cameron Diaz]]: Frau in ferrivia subterranea
* [[Paulus Wesley]]: Nathan cum birota

== Notae ==
&lt;references /&gt;

== Bibliographia ==
* Kimbell, Keith (2011) [https://www.metacritic.com/feature/movies-based-on-philip-k-dick-stories Ranked: Movies Based on Philip K. Dick Stories]. ''Metacritic'' 2 Martii.

== Nexus externus ==
* {{Imdb title|0181689}}
*[https://cinephiliabeyond.org/minority-report-steven-spielbergs-proof-dont-need-sacrifice-substance-produce-spectacle/ pellicula quomodo creata sit]
{{Stephani Spielberg pelliculae}}
[[Categoria:Pelliculae 2002]]
[[Categoria:Pelliculae Civitatum Foederatarum]]
[[Categoria:Stephanus Spielberg]]</text>
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  </page>
  <page>
    <title>Hyllus</title>
    <ns>0</ns>
    <id>127903</id>
    <revision>
      <id>3697687</id>
      <parentid>3697299</parentid>
      <timestamp>2022-08-16T16:44:31Z</timestamp>
      <contributor>
        <username>Marcus Terentius Bibliophilus</username>
        <id>2059</id>
      </contributor>
      <comment>/* Fontes */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1302" xml:space="preserve">'''Hyllus''' in [[mythologia Graeca]] est filius [[Hercules|Herculis]] et [[Deianira|Deianirae]]. In [[Thessalia]] a [[Ceyx|Ceyce]] [[Trachis|Trachiniorum]] rege eruditur, dum pater labores suos perficit. Postquam [[Deianira]] patrem quaesitum eum mittit, Hyllus Herculem in monte Oeta morientem invenit, [[Nessus|Nessi]] tunica combustum. Tum rogo construendo adest, atque ultimas voluntates patris accipit: cuius iussu [[Iole|Iolen]] uxorem ducit, e qua Cleodaeum gignit.

Postea [[Eurystheus|Eurystheum]] necat, qui Hercule mortuo, matrem eius [[Alcmene|Alcmenen]] et liberos vexare non cessat. Mox dux [[Heracleidae|Heracleidarum]] fit, qui [[Peloponnesus|Peloponnesum]] armis recuperare conantur, sed a [[Tegea|Tegeaeorum]] rege victus interficitur.

Hyllus persona est in tragoediis de morte Herculis: ''[[Trachiniae|Trachiniis]]'' [[Sophocles|Sophocleis]] et ''[[Hercules Oetaeus|Hercule Oetaeo]]'' [[Seneca|Senecano]].

==Fontes==
* [[Apollodori bibliotheca|Apollodori Bibliotheca]] II,7,7 et 8,1-2.
*[[Euripides]], ''[[Heraclidae (Euripides)|Heraclidae]]'' 639sqq
*[[Herodotus]] IX,26.
* [[Pausanias]], ''[[Graeciae descriptio]]'' I.41.2
*[[Seneca]], ''[[Hercules Oetaeus]].''
*[[Sophocles]], ''[[Trachiniae]].''

[[Categoria:Heraclidae]]
[[Categoria:Hercules]]
[[Categoria:Mythologia Graeca]]</text>
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  </page>
  <page>
    <title>Pheluca</title>
    <ns>0</ns>
    <id>146655</id>
    <revision>
      <id>3697756</id>
      <parentid>2646822</parentid>
      <timestamp>2022-08-17T10:16:54Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Naves]]; +[[Categoria:Genera navium]]</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="663" xml:space="preserve">[[Fasciculus:feluka1.jpg|thumb|Duae phelucae in flumine [[Nilus|Nilo]]]]
[[Fasciculus:Feluka7.JPG|thumb|Phelucae in flumine Nilo prope [[Syene]]n]]
'''Pheluca'''&lt;ref&gt;Latina nominis transcriptio secundum exemplar Graecum φελούκα&lt;/ref&gt; ([[Arabice]]: فلوكة) fuit species navis mercatoriae parvae in regionibus circum [[Mare Mediterraneum]] usitatae uno vel duobus malis instructae, quae remis ac velis ([[Vela Latina]]) incitari potuit. [[Aegyptus|Aegyptii]] phelucis adhuc utuntur.

== Nexus externi ==
{{CommuniaCat|Feluccas|Phelucas}}

== Notae ==
&lt;div class=&quot;references-small&quot;&gt;&lt;references /&gt;&lt;/div&gt;

[[Categoria:Navigatio]]
[[Categoria:Genera navium]]</text>
      <sha1>gkc0d78etu8zam7jj9e3ylrny7ktadp</sha1>
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  </page>
  <page>
    <title>Functio zeta Riemanniana</title>
    <ns>0</ns>
    <id>147763</id>
    <revision>
      <id>3697662</id>
      <parentid>3630589</parentid>
      <timestamp>2022-08-16T13:34:31Z</timestamp>
      <contributor>
        <username>LilyKitty</username>
        <id>18316</id>
      </contributor>
      <comment>de quantitate variabili</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1290" xml:space="preserve">{{L}}
[[Fasciculus:Riemann-Zeta-Func.png|thumb|Function zeta Riemanniana]]
'''Functio zeta Riemanniana''' est 

&lt;math&gt;\zeta(s) = \sum_{n=1}^{\infty} n^{-s} = 1 + \frac{1}{2^s} + \frac{1}{3^s} + \dots&lt;/math&gt;

Si [[quantitas variabilis]] ''s'' est [[numerus realis]], necesse est s &gt; 1 esse, ut [[series (mathematica)|series]] ad valorem convergat.  Sed quia functio est [[functio differentiabilis|differentiabilis]] etiamsi ''s'' est [[numerus complexus]], extensionem [[analysis|analyticam]] ad omnes numeros complexos habet.

Functio magni momenti est in [[theoria numerorum]].  [[Euler]] demonstravit

&lt;math&gt;\zeta(s) = \prod_{p}(1 - p^{-s})^{-1}, p \text{ primus}, s \text{ realis} &gt; 1&lt;/math&gt;

[[Bernardus Riemann]] (1826-1866) etiam huic [[functio]]ni studuit.  [[Hypothesis Riemanniana]] dicit omnes quantitates ''s'' ut &lt;math&gt;\zeta(s) = 0&lt;/math&gt; (praeter valores triviales) partem realem 1/2 habere.

== Bibliographia ==
* Peter Borwein, Stephen Choi, Brendan Rooney, Andrea Weirathmueller. ''The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike.'' Novi Eboraci:  Springer, 2008.  ISBN  978-0-387-72126-2

== Nexus Externi ==
{{CommuniaCat|Riemann zeta function|functionem zeta Riemanniana}}

[[Categoria:Theoria numerorum]]
[[Categoria:Analysis]]
{{math-stipula}}</text>
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  <page>
    <title>Galeones Manilenses</title>
    <ns>0</ns>
    <id>153255</id>
    <revision>
      <id>3697651</id>
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      <contributor>
        <username>IacobusAmor</username>
        <id>1163</id>
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      <minor />
      <comment>/* Bibliographia */ +Bib ex en (10K)</comment>
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      <text bytes="4731" xml:space="preserve">[[Fasciculus:Spanish Galleon.jpg|thumb|Galeo Hispanicus.]]
'''Galeones Manilenses,''' pleniore '''Galeones Manilenses et Acapulcenses''' ([[Hispanice]] ''Galeones de Manila-Acapulco,'' [[Lingua Philippinica|Tagalice]] ''Kalakalang Galyon ng Maynila at Acapulco'') fuerunt [[galeo]]nes [[navis oneraria|onerarii]] [[Hispania|Hispanici]] qui [[unus|semel]] aut [[duo|bis]] per [[annus|annum]] trans [[Oceanus Pacificus|Oceanum Pacificum]] inter [[Manila]]m in [[Indiae Orientales Hispanicae|Indiis Orientalibus Hispanicis]] (hodie [[Philippinae]]) et [[Acapulcum]] in [[Nova Hispania]] (hodie [[Mexicum]]) [[navigatio|navigabant]]. Mutari solebat nomen cuiusque galeonis secundum nomen [[urbs|urbis]] ex qua [[navis]] navigabat.&lt;ref&gt;Glyn Williams, ''The Prize of All the Oceans'' (Novi Eboraci: Viking, 1999), ISBN 0-670-89197-5, p. 4.&lt;/ref&gt; Hoc [[iter commercii]]&lt;!--trade route--&gt; anno [[1568]] coepit, via [[oceanus|oceanica]] ab [[Andreas de Urdaneta|Andrea de Urdaneta]] anno [[1565]] inventa, et usque ad [[1815]] persistebat, cum [[Bellum Libertatis Mexicanum]] hoc [[commercium]] perpetuo interrumperet.

[[Fasciculus:16th century Portuguese Spanish trade routes.png|thumb|400px|[[Iter commercii]] Manila-Acapulco anno [[1568]] coepit, cum [[classis divitiarum Hispanica|classes divitiarum Hispanicae]]&lt;!--Spanish treasure fleets--&gt; itineribus hic [[albus|albis]] [[color]]atis utebantur, dum earum aemulae orientales, [[armata Indica Portugallica|armatae Indicae Portugallicae]],&lt;!--Portuguese India Armadas--&gt; itineribus [[caeruleus|caeruleis]] annis [[1498]]–[[1640]] utebantur.]]
{{NexInt}}
*[[Bernardus de la Torre]]
*[[Classis divitiarum Hispanica]]&lt;!--Spanish treasure fleet--&gt;
*[[Historia litoris occidentalis Americae Septentrionalis]]
*[[Historia Philippinarum (1521–1898)]]
*[[Indiae Orientales Hispanicae]]
*[[Proelia La Naval de Manila]]

==Notae==
&lt;references/&gt;

==Bibliographia==
* Bjork, Katharine. [[1998]]. &quot;The Link that Kept the Philippines Spanish: Mexican Merchant Interests and the Manila Trade, 1571–1815.&quot; ''Journal of World History'' 9 (1): 25–50.
* Carrera Stampa, Manuel. [[1959]]. &quot;La Nao de la China.&quot; ''Historia Mexicana'' 9 (33): 97-118.
* Fish, Shirley. [[2011]]. ''The Manila-Acapulco Galleons: The Treasure Ships of the Pacific, with an Annotated List of the Transpacific Galleons 1565–1815.'' Central Milton Keynes in Anglia: Authorhouse.
*Flynn, Dennis Owen, Arturo Giráldez, et James Sobredo, eds. [[2001]]. ''European Entry into the Pacific: Spain and the Acapulco-Manila Galleons.'' Aldershot Angliae et Burlington Montis Viridis: Ashgate. ISBN 0754601528.
* Gasch-Tomás, José Luis. [[2018]]. ''The Atlantic World and the Manila Galleon: Circulation, Market, and Consumption of Asian Goods in the Spanish Empires, 1565-1650.'' Lugduni Batavorum: Brill.
* Giraldez, Arturo. [[2015]]. ''The Age of Trade: The Manila Galleons and the Dawn of the Global Economy.'' Lanhamiae in Terra Maria: Rowman &amp; Littlefield.
* Luengo, Josemaria Salutan. [[1996]]. ''A History of the Manila-Acapulco Slave Trade, 1565–1815.'' Tubigon, Bohol: Mater Dei Publications.
* McCarthy, William J. [[1993]]. &quot;Between Policy and Prerogative: Malfeasance in the Inspection of the Manila Galleons at Acapulco, 1637.&quot; ''Colonial Latin American Historical Review'' 2 (2): 163–83.
* Oropeza Keresey, Deborah. [[2007]]. &quot;Los 'indios chinos' en la Nueva España: la inmigración de la Nao de China, 1565–1700.&quot; Dissertatio PhD, El Colegio de México, Centro de Estudios Históricos.
* Rogers, R. [[1999]]. ''Shipwreck of Hawai'i: A Maritime History of the Big Island.'' Haleiwa Hawaiorum: Pilialoha Publishing. ISBN 0967346703.
* Schurz, William Lytle. [[1917]]. [http://www.tshaonline.org/publications/journals/shq/online/v021/n2/article_1.html The Manila Galleon and California.] ''Southwestern Historical Quarterly'' 21 (2): 107–26.
* Schurz, William Lytle. [[1939]]. ''The Manila Galleon.'' Novi Eboraci: E. P. Dutton &amp; Co.

==Nexus externi==
{{CommuniaCat|Galleon|Galleon}}
*[http://www.galeondemanila.org &quot;Asociación Cultural Galeón de Manila,&quot;] apud http://www.galeondemanila.org/
*[http://ns.gov.gu/galleon/ &quot;Findings from the wreck of ''Nuestra Senora de la Concepcion'' in the Marianas, 1638,&quot;] apud ns.gov.gu
*[http://www.mms.gov/omm/pacific/kids/manilagalleons.htm &quot;Manila Galleons along the Californian coasts,&quot;] apud http://www.mms.gov/
*[http://www.metmuseum.org/toah/hd/mgtr/hd_mgtr.htm &quot;Metropolitan Museum: Manila Galleon,&quot;] apud http://www.metmuseum.org/

[[Categoria:Naves]]
[[Categoria:Imperium Hispanicum]]
[[Categoria:Historia Oceaniae]]
[[Categoria:Itinera mercatoria]]
[[Categoria:Historia Mexici]]
[[Categoria:Oceanus Pacificus]]
[[Categoria: Hispania apud Philippinas]]</text>
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    <revision>
      <id>3697653</id>
      <parentid>3697651</parentid>
      <timestamp>2022-08-16T13:10:39Z</timestamp>
      <contributor>
        <username>IacobusAmor</username>
        <id>1163</id>
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      <minor />
      <comment>+Osborne in bib</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4937" xml:space="preserve">[[Fasciculus:Spanish Galleon.jpg|thumb|Galeo Hispanicus.]]
'''Galeones Manilenses,''' pleniore '''Galeones Manilenses et Acapulcenses''' ([[Hispanice]] ''Galeones de Manila-Acapulco,'' [[Lingua Philippinica|Tagalice]] ''Kalakalang Galyon ng Maynila at Acapulco'') fuerunt [[galeo]]nes [[navis oneraria|onerarii]] [[Hispania|Hispanici]] qui [[unus|semel]] aut [[duo|bis]] per [[annus|annum]] trans [[Oceanus Pacificus|Oceanum Pacificum]] inter [[Manila]]m in [[Indiae Orientales Hispanicae|Indiis Orientalibus Hispanicis]] (hodie [[Philippinae]]) et [[Acapulcum]] in [[Nova Hispania]] (hodie [[Mexicum]]) [[navigatio|navigabant]]. Mutari solebat nomen cuiusque galeonis secundum nomen [[urbs|urbis]] ex qua [[navis]] navigabat.&lt;ref&gt;Glyn Williams, ''The Prize of All the Oceans'' (Novi Eboraci: Viking, 1999), ISBN 0-670-89197-5, p. 4.&lt;/ref&gt; Hoc [[iter commercii]]&lt;!--trade route--&gt; anno [[1568]] coepit, via [[oceanus|oceanica]] ab [[Andreas de Urdaneta|Andrea de Urdaneta]] anno [[1565]] inventa, et usque ad [[1815]] persistebat, cum [[Bellum Libertatis Mexicanum]] hoc [[commercium]] perpetuo interrumperet.

[[Fasciculus:16th century Portuguese Spanish trade routes.png|thumb|400px|[[Iter commercii]] Manila-Acapulco anno [[1568]] coepit, cum [[classis divitiarum Hispanica|classes divitiarum Hispanicae]]&lt;!--Spanish treasure fleets--&gt; itineribus hic [[albus|albis]] [[color]]atis utebantur, dum earum aemulae orientales, [[armata Indica Portugallica|armatae Indicae Portugallicae]],&lt;!--Portuguese India Armadas--&gt; itineribus [[caeruleus|caeruleis]] annis [[1498]]–[[1640]] utebantur.]]

{{NexInt}}
*[[Bernardus de la Torre]]
*[[Classis divitiarum Hispanica]]&lt;!--Spanish treasure fleet--&gt;
*[[Historia litoris occidentalis Americae Septentrionalis]]
*[[Historia Philippinarum (1521–1898)]]
*[[Indiae Orientales Hispanicae]]
*[[Proelia La Naval de Manila]]

==Notae==
&lt;references/&gt;

==Bibliographia==
* Bjork, Katharine. [[1998]]. &quot;The Link that Kept the Philippines Spanish: Mexican Merchant Interests and the Manila Trade, 1571–1815.&quot; ''Journal of World History'' 9 (1): 25–50.
* Carrera Stampa, Manuel. [[1959]]. &quot;La Nao de la China.&quot; ''Historia Mexicana'' 9 (33): 97-118.
* Fish, Shirley. [[2011]]. ''The Manila-Acapulco Galleons: The Treasure Ships of the Pacific, with an Annotated List of the Transpacific Galleons 1565–1815.'' Central Milton Keynes in Anglia: Authorhouse.
*Flynn, Dennis Owen, Arturo Giráldez, et James Sobredo, eds. [[2001]]. ''European Entry into the Pacific: Spain and the Acapulco-Manila Galleons.'' Aldershot Angliae et Burlington Montis Viridis: Ashgate. ISBN 0754601528.
* Gasch-Tomás, José Luis. [[2018]]. ''The Atlantic World and the Manila Galleon: Circulation, Market, and Consumption of Asian Goods in the Spanish Empires, 1565-1650.'' Lugduni Batavorum: Brill.
* Giraldez, Arturo. [[2015]]. ''The Age of Trade: The Manila Galleons and the Dawn of the Global Economy.'' Lanhamiae in Terra Maria: Rowman &amp; Littlefield.
* Luengo, Josemaria Salutan. [[1996]]. ''A History of the Manila-Acapulco Slave Trade, 1565–1815.'' Tubigon, Bohol: Mater Dei Publications.
* McCarthy, William J. [[1993]]. &quot;Between Policy and Prerogative: Malfeasance in the Inspection of the Manila Galleons at Acapulco, 1637.&quot; ''Colonial Latin American Historical Review'' 2 (2): 163–83.
* Oropeza Keresey, Deborah. [[2007]]. &quot;Los 'indios chinos' en la Nueva España: la inmigración de la Nao de China, 1565–1700.&quot; Dissertatio PhD, El Colegio de México, Centro de Estudios Históricos.
* Osborne, Thomas J. [[2013]]. ''Pacific Eldorado: A History of Greater California.'' Novi Eboraci: John Wiley &amp; Sons. ISBN 978-1-4051-9454-9. [https://books.google.com/books?id=oqFtdhdbt7UC Google Books.]
* Rogers, R. [[1999]]. ''Shipwreck of Hawai'i: A Maritime History of the Big Island.'' Haleiwa Hawaiorum: Pilialoha Publishing. ISBN 0967346703.
* Schurz, William Lytle. [[1917]]. [http://www.tshaonline.org/publications/journals/shq/online/v021/n2/article_1.html The Manila Galleon and California.] ''Southwestern Historical Quarterly'' 21 (2): 107–26.
* Schurz, William Lytle. [[1939]]. ''The Manila Galleon.'' Novi Eboraci: E. P. Dutton &amp; Co.

==Nexus externi==
{{CommuniaCat|Galleon|Galleon}}
*[http://www.galeondemanila.org &quot;Asociación Cultural Galeón de Manila,&quot;] apud http://www.galeondemanila.org/
*[http://ns.gov.gu/galleon/ &quot;Findings from the wreck of ''Nuestra Senora de la Concepcion'' in the Marianas, 1638,&quot;] apud ns.gov.gu
*[http://www.mms.gov/omm/pacific/kids/manilagalleons.htm &quot;Manila Galleons along the Californian coasts,&quot;] apud http://www.mms.gov/
*[http://www.metmuseum.org/toah/hd/mgtr/hd_mgtr.htm &quot;Metropolitan Museum: Manila Galleon,&quot;] apud http://www.metmuseum.org/

[[Categoria:Hispania apud Philippinas]]
[[Categoria:Historia Oceaniae]]
[[Categoria:Imperium Hispanicum]]
[[Categoria:Itinera mercatoria]]
[[Categoria:Historia Mexici]]
[[Categoria:Naves]]
[[Categoria:Oceanus Pacificus]]</text>
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  <page>
    <title>Trainera</title>
    <ns>0</ns>
    <id>167014</id>
    <revision>
      <id>3697762</id>
      <parentid>2496937</parentid>
      <timestamp>2022-08-17T10:17:37Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Naves]]; +[[Categoria:Genera navium]]</comment>
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      <text bytes="991" xml:space="preserve">[[Fasciculus:Trainiere Croquis.jpg|thumb|Trainera [[piscatus|piscatoria]] cum remis&lt;!--ubi suntremi?--&gt;.]]
'''Trainera''' (-ae f.)&lt;ref&gt;{{Fontes desiderati}}.&lt;/ref&gt; est [[traditio]]nalis [[linter]] [[litus|litoris]] [[Mare Cantabricum|Maris Cantabrici]] in meridiano [[Sinus Cantabrorum]] ([[Hispanice]] ''Golfo de Vizcaya''; [[Francice]] ''Golfe de Gascogne'') limite, [[remus (navigatio)|remis]] et olim [[velum (navigatio)|velis]] propulsus. Est tenuum linearum linter, cum elata [[prora]] et convexa [[puppis|puppi]], ut [[unda|undis]] resistat. Trainerae olim ad afferendas [[Engraulidae|engraulidas]] atque [[Sardina pilchardus|sardinas]] diarias captas e [[mare|mari]] in [[portus|portum]] dedicabantur, ex [[consuetudo|consuetudine]] contendentes inter se de vendendis captis [[piscis|piscibus]] antea. Hodie, haec [[mos]] praecipua [[ars athleica]] certaminis nautici factus est.

==Notae==
&lt;div class=&quot;references-small&quot;&gt;&lt;references /&gt;&lt;/div&gt;

{{stipula}}

[[Categoria:Genera navium]]</text>
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  <page>
    <title>Educatio Romana</title>
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      <text bytes="5715" xml:space="preserve">{{Latinitas|-6}}
{{vicificanda}}
{{contribuenda|Schola Romana}}
[[File:Roman school.jpg|thumb|600px|Toreum reppertum in Neumagen circa Trier,cum doctore et tres discipulis (180-185 AD)]]

Discere Romae erat luxuriam.  Non omnes liberi ierunt ad scholas.  Saepe liberi docti sunt domi a suis parentibus.  Modum liberi ex familiis beatissimis ierunt ad scholas.  Pauci liberi conpleverunt totum cursum.  Sed [[schola]] dicitur fuisse gravissimus pars Romarum vitarum.   Pueri docti sunt in schola aut in casa magistrorum.  Parentes credider magistrum bene docturum esse suos liberos.  Discipuli ierunt in casa magistrorum rogatum auxilium.   Schola paravit pueros esse meliores cives.  Puellae doctae sunt domi.  Puellae non eunt ad scholam sine permissum suorum patrim.    

Romani formaverunt sui scholarum systema secundam scholarum systema Graecorum.  Graeca scholarum systema erat divisus in tres partes: ludus et grammaticus et rhetoricus.  Nescimus multum de disciplina liberorum Romanorum.   Sciemus duas res fuisse negui neqotii.   Philologi credunt numerus dierum schola frequentare multigener.  Schola semper incepit die Martii XXIV.  Secundo rhetoric erat amplissimum scholam.

== Primarium educatio ==
Discipuli magistro docebantur. Saepe erat Graecus servus, et dimissus. Erat non amicibilis et discipuli qui non complebant voluntas sui, puniebantur. Hoc educationem secutus est, donec alumnus XII annorum est.  Alumni comitabantur per paedagogum. Ducebat ei ab scolam, et iuvabat re, qui debunt fieri.

Locus ubi docebantur, plerumque non bene erat. Hic locus saepe in aere aperto erat, aut in loco cum paulo populi. Dies scolae duravit 6 horas, cum tempore ubi tu nihil fecis. Ibi fuerunt dies, in quo non docebatur. Ibi fuerunt multae feriae, in his diebus nullus in scola erat. In aestivis quoque nullus in scola erat , quia in diebus illis admodum calidum erat.
Liberi discebant legere et scribere Latine. Etiam discebant linguam Graecam, secunda lingua Romano imperio. Lingua Graeca erat admodum maximus. Etiam discebant numerare. Ut hoc educationem sequebant, poterant melius opus invenire.

In ludo liberi didicerunt de scribendo et [[arithmetica]].  Scripserunt in cera et docti sunt ab litteratoribus.    Tabulae delebantur a liberis.  Cera tabularum liquata est et usurpata est.  

== Secundarium educationem ==

Secundarium educationem non erat pro omnibus. Solum filii locupletium patentum sunt concessa ad (permisit) secundarium educationem. 
Eorum annum quintum decimum, erant ad Grammaticus. Ibi didicerunt graeco. Et legerunt graece et letine textuum. Sed etiam aliis subjectis, quasi: historia et geographia.

In grammatica, pueri reppererunt Latina et Graecam.  Graecae datus est locus primus in grammatica.  Puellae non ierunt ad scholam.   undecem anno quod saepe puellas nuptae sunt  duodecino anno.  Pueri docti sunt aestinare magnos viros Romanos fabularum historiaeque.  Romani heroes erant verisimilor quam Graeci.  Necesse est pueris dictare Graece quod scientia Graecae linguae significavit nobilitatem.  Multi pueri reppererunt ab Graecis servis  domi.   

== In retorum scolis ==
Si puer de copiosis parentes erat, ad scolam rhetoris venire poterat. In hac scola puer discebat loquere in publico. Educatio in scola rhetoris constabat multa pecuniae. Ideo non omnibus licet, ad scolam rhetoris ire.

Facilis dictu rhetorica erat gravissimus pars eruditionis in Romae.  Paratis a grammatica,  discipulis intraverunt rhetoricam.  Iuvenes uixi sunt prosae in rhetorica.  Noverunt quoque geographiam, musicam, philosophiam, mythologiam, geometriamque.  Temptaverunt sibi scientiam scholasticam et inceperunt discere quomodo haberent orationes.  Pueri amant vocandum.  Pueris dicituris magister dedit consilium.  Ad Excoleudam bonam publicam artem rhetoricam magister dedit discipulis materiam pro orationibus.  

Discipuli studuerunt in altera ex duabus areis de rhetorica.  Altera erat deliberativa discipulina de studii.  Haec disciplina docta est  discipulis quaereatibus in locum Romano senatu.  Secunda disciplina erat iudicialis rhetorica.  Hoc exemplum duxit ad curricula inudicis vel patronae sicut recens iurisconsultus.    

== finis iusta educationis ==
Composita disciplina Romani pueri finite est  sexton et decino anno.  Scholis finitis discipuli gesserunt togas habitum adultum.   Adultas vestimenta erant longam album et lana facta.  Pueri visi sunt postquam gesserant adulta vestimenta.  Pueris finituris consilia  futura incepta sunt.  Multi iuvenes impleverunt unum annum tirones; transcenderunt ab inopibus liberis ad rectos cives tirocinio; possunt nunc habere officium aut militiam.   

Oratores erant magnis momenti quod adiuverunt sententiam mutare civitatis.  Romana civitas  saepe erat in res civilibus malis.  Bonus orator sentiam mutavit civitarum.  Rhetorica sivi iuvenes incipere curricula in republica.  Curricula publica saepe praestiterunt divitias et vim.  CursusPublicus etiam praestit magnum civilem ordinem.  Sine oratoribus et discipulis qui rhetiorae  studerunt Romana civitas non potuisset eadem.

== Scripturam materialis ==
Materialis quod solebat, erat [[Papyrus (materies)|papyrus]] et pergamenarii. Pupillas solebant papyrus. Quia pergamenarij erat carus. Sed potuit pergamenarij diutius uti quam papyrus. Pupillas scripserunt cum [[atramentum]].

[[Categoria:Roma antiqua]]
[[Categoria:Studiorum universitas]]

== Bibliographia ==
*&quot;Education in Ancient Rome.&quot; Education in Ancient Rome.  Web. 22 Sept. 2013.
*&quot;School in Ancient Rome.&quot; ThinkQuest. Oracle Foundation. Web. 22 Sept. 2013.
*Bonner, Stanley Frederick. Education in Ancient Rome: From the Elder Cato to the Younger Pliny. Berkeley: University of California, 1977. Print.</text>
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  <page>
    <title>Ieremias Bonomelli</title>
    <ns>0</ns>
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        <id>81729</id>
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      <text bytes="839" xml:space="preserve">{{Capsa hominis Vicidata}}
'''Ieremias Bonomelli''' (natus in ''[[Nigoline]]'' ([[Curtis Franca]]) die [[22 Septembris]] [[1831]]; [[Cremona]]e mortuus die [[3 Augusti]] [[1914]]) fuit [[episcopus]] [[Italia]]nus [[Dioecesis Cremonensis]]. Inter alia Bonomelli notus est opera sua pro migrantibus et quia acri subtilique ingenio et iudicio percensuit [[Sancta Sedes|Sanctae Sedis]] inimicitiam contra [[Regnum Italiae]] post quam die [[20 Septembris]] [[1870]] [[Roma]] ab exercitu [[rex|regis]] [[Victorius Emanuel II|Victorii Emanuelis II]] [[Italia]]e capta, [[Civitas Ecclesiae]] finem habuit.

== Nexus externus ==

* {{CathHierBishop|bonom|De Ieremia Bonomelli}}
{{bio-stipula}}

{{Lifetime|1831| 1905|Bonomelli, Ieremias}}

[[Categoria:Episcopi Ecclesiae Catholicae]]
[[Categoria:Incolae Italiae]]
[[Categoria:Episcopi Cremonenses]]</text>
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    <revision>
      <id>3697694</id>
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      <text bytes="866" xml:space="preserve">{{Capsa hominis Vicidata}}
'''Ieremias Bonomelli''' (natus in ''[[Nigoline]]'' ([[Curtis Franca]]) die [[22 Septembris]] [[1831]]; [[Cremona]]e mortuus die [[3 Augusti]] [[1914]]) fuit [[episcopus]] [[Italia]]nus [[Dioecesis Cremonensis]]. Inter alia Bonomelli notus est opera sua pro migrantibus et quia acri subtilique ingenio et iudicio percensuit [[Sancta Sedes|Sanctae Sedis]] inimicitiam contra [[Regnum Italiae (1861-1946)|Regnum Italiae]] post quam die [[20 Septembris]] [[1870]] [[Roma]] ab exercitu [[rex|regis]] [[Victorius Emanuel II|Victorii Emanuelis II]] [[Italia]]e capta, [[Civitas Ecclesiae]] finem habuit.

== Nexus externus ==

* {{CathHierBishop|bonom|De Ieremia Bonomelli}}
{{bio-stipula}}

{{Lifetime|1831| 1905|Bonomelli, Ieremias}}

[[Categoria:Episcopi Ecclesiae Catholicae]]
[[Categoria:Incolae Italiae]]
[[Categoria:Episcopi Cremonenses]]</text>
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  <page>
    <title>Mayflower</title>
    <ns>0</ns>
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      <format>text/x-wiki</format>
      <text bytes="4433" xml:space="preserve">[[Fasciculus:Mayflower in Plymouth Harbor, by William Halsall.jpg|thumb|350px|''Mayflower in Plymouth Harbor'' a [[Gulielmus Halsall|Gulielmo Halsall]] picta (1882).]]
'''''Mayflower'''''&lt;ref&gt;Nomen saepe [[saeculum 17|saeculo septimo decimo]] ''May-flower'' scriptum.&lt;/ref&gt; fuit [[navis]] [[Peregrinatores (Colonia Plimmuta)|Peregrinatorum]] quae anno [[1620]] [[navigatio]]nem [[historia|historicam]] ab [[Anglia]] ad [[Novus Mundus|Novum Mundum]] fecit. Navis [[centum]] et [[duo]]s vectores in duobus gregibus transportavit: [[religio]]sos [[Separatismus|Separatistas]] [[Nederlandia]], et plerumque non religiosos [[colonia|colonos]]&lt;!--nexus: settlers--&gt; [[Londinium|Londinio]] migrantes. Haec [[navigatio]] facta est [[icon|iconica]]&lt;!--nexus recte: icon culturalis--&gt; in primis [[historia]]e [[Americani|Americanae]] annalibus [[narratio]] relata, [[tragoedia|tragicam]] [[mors|mortis]] [[vita]]eque in difficillimis circumiectis [[hiems|hiemis]] [[Nova Anglia|Novae Angliae]] [[fabula]]m narrans. Fastigium navigationis, [[Pactio Mayflower]] signata, late habetur unum ex gravissimorum momentorum temporis in [[historia]] Americae et [[Civitates Foederatae|Civitatum Foederatarum]], fundamenta [[civitas|civitatis]] hodiernae formae [[civitas sui iuris|gubernationis sui]] [[democratia|democraticae]] et principalium [[libertas|libertatum]] praebens. &lt;!--PLUS IN EN:--&gt; &lt;!--

==Vide etiam==
* [[Billericay]], where the Pilgrim Fathers met prior to the voyage
* [[Leigh-on-Sea]], where the ''Mayflower'' was outfitted
* [[Taylor-Bray Farm]], a farm in southeastern Massachusetts owned by descendants of ''Mayflower'' passengers
* [[Thanksgiving (United States)]]
* ''[[The Ark (ship)|The Ark]]'' and ''[[Maryland Dove|The Dove]]'', ships that settled [[Maryland]] in 1634
* [[Pilgrim (Plymouth Colony)]]
* [[Puritan migration to New England (1620–1640)]]
* ''[[Plymouth Adventure]]'' (directed by Clarence Brown, 1952)
* ''[[Mayflower: The Pilgrims' Adventure]]'' (1979)
* ''[[Mayflower II]]'', a replica of the Mayflower in Plymouth, Massachusetts --&gt;

==Adnotationes==
&lt;references/&gt;

==Bibliographia==
* Amez, Azel. [[1901]]. [http://www.archive.org/details/mayflowerherlogj00ames ''The May-flower and Her Log, July 15, 1620—May 6, 1621.''] Bostoniae et Novi Eboraci: Houghton Mifflin Company. Editio Internet Archive.
* Ames, Azel. [[1901]]. [http://www.gutenberg.net/browse/BIBREC/BR4107.HTM ''The May-flower and Her Log, July 15, 1620—May 6, 1621.''] Bostoniae et Novi Eboraci: Houghton Mifflin Company. Editio Project Gutenberg.
* Bradford, William. [[1908]]. [http://books.google.com/books?id=Sd9BAAAAIAAJ|ref=Bradford ''Bradford's History of Plymouth Plantation, 1606–1646.''] Ed. William T. Davis. Novi Eboraci: Scribners. (Sola navigationis relatio scripta.)
* Marsden, R, G, [[1904]]. The 'Mayflower. ''English Historical Review'' 19:669–680.
* Philbrick, Nathaniel. [[2006]]. ''Mayflower: A Story of Courage, Community, and War.'' Novi Eboraci: Viking. ISBN 0670037605.
* Usher, Roland G. [[1918]], [[1984]]. ''The Pilgrims and Their History.'' Williamstown Massachusettae: Corner House Publishers. ISBN 0879280824.

==Nexus externi==
{{CommuniaCat|Mayflower|''Mayflower''}}&lt;!--
{{Wikisource1911Enc|Mayflower|''Mayflower''}}--&gt;
* [http://bloosee.com/r/im8FVs &quot;Exact arrival site of the Mayflower on Satellite Map and NOAA Chart,&quot;] bloosee.com (BlooSee)
* [http://www.mayflowerhistory.com/History/history.php Historia navis,] www.mayflowerhistory.com
* [http://www.plimoth.org/features/mayflower-2/ Mayflower II,] www.plimoth.org
* [http://www.pilgrimhall.org/ Pilgrim Hall Museum,] www.pilgrimhall.org (Plimmutae Massachusettae)
* [http://www.themayflowersociety.com Societas Generalis Progenierum Mayflower,] www.themayflowersociety.com&lt;!--
* [http://www.mayflowersteps.co.uk/ Contemporary photos of Plymouth's Barbican and the Mayflower Steps]
* [http://www.highton-ridley.co.uk/blog/2009/07/pilgrims-point-rare-one-in-colour.html Pilgrims Point, Plymouth (UK)] A photo of the modern-day Mayflower Steps Arch and Pilgrims Point--&gt;

{{hist-stipula}}

[[Categoria:Mayflower|*]]
[[Categoria:1620]]
&lt;!--recte: Historia T. C.--&gt; &lt;!--
[[Category:English emigration]]
[[Category:Exploration ships of England]]
[[Category:Pre-statehood history of Massachusetts]]
[[Category:Nautical lore]]
[[Category:Individual sailing vessels]]
[[Category:Sailing ships]]
[[Category:Ships of England]]--&gt;
{{titulus italicus}}</text>
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  <page>
    <title>Of Plimmoth Plantation</title>
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      <text bytes="2534" xml:space="preserve">{{Titulus italicus}}
[[Fasciculus:Of Plimoth Plantation First 1900.jpg|thumb|[[Pagina titularis]] [[manuscriptum|manuscripti]] [[Gulielmus Bradford|Gulielmi Bradford]].]]

'''''Of Plimmoth Plantation''''' ('De Plimmuta deducta' vel 'De plantario Plimmutensi') est [[liber]] per multos [[annus|annos]] a [[Gulielmus Bradford|Gulielmo Bradford]] [[gubernator]]is [[colonia|colonici]] de [[historia]] deductionis [[Colonia Plimmuta|Plimmutae Coloniae]] in [[Nova Anglia]] factus, [[auctoritas]] singula et plenissima [[narratio]]nis [[Peregrinatores (Colonia Plimmuta)|Peregrinatorum]] primorumque [[colonia]]e ab eis conditae annorum, atque ad ultimum [[unus|unum]] ex fundamentalibus [[litterae Civitatum Foederatarum|litterarum]] [[historia Civitatum Foederatarum|historiaeque]] [[Civitates Foederatae|Civitatum Foederatarum]] instrumentis. [[Vocabulum|Verba]], inter [[1630]] et [[1651]] composita, [[vita]]s pregrinatorum ab anno [[1608]], cum se in [[Respublica Nederlandica|Republica Nederlandica]] considerent, per [[navigatio]]nem trans [[Oceanus Atlanticus|Oceanum Atlanticum]] ad [[America]]m in ''[[Mayflower]]'' anno [[1620]], usque ad annum [[1647]] describit. [[Manuscriptum]], indice anno [[1651]] perscripto, vectorum in  ''Mayflower'' et eorum rerum gestarum finitur.

{{NexInt}}
* [[Caedes Mystica]]
* [[Foedus Mayfloweranum]]
* [[Saxum Plimmutense]]&lt;!--
 
==Notae==
&lt;references/&gt;--&gt;

==Editiones==
*[[Gulielmus Bradford|Bradford, William]]. [[1856]]. ''History of Plymouth Plantation,'' ed. Charles Deane Bostoniae: Little Brown and Company. Editio princeps.
*[[Gulielmus Bradford|Bradford, William]]. [[1952]]. ''Of Plymouth Plantation, 1620–1647: The Complete Text, with Notes and an Introduction by [[Samuel Eliot Morison]].'' Novi Eboraci: Knopf.
*[[Gulielmus Bradford|Bradford, William]]. [[2003]]. ''William Bradford's Books: Of Plimmoth Plantation and the Printed Word,'' ed. Douglas Anderson. Baltimorae: Johns Hopkins University Press. ISBN 0801870747. Editio recentissima.

==Nexus==
*[http://books.google.com/books?id=tYecOAN1cwwC&amp;printsec=titlepage ''Of Plymouth Plantation,] books.google.com.
*[http://faculty.gordon.edu/hu/bi/ted_hildebrandt/NEReligiousHistory/NEReligiousHistory.html Religiosa Novae Angliae historia,] faculty.gordon.edu (Collegium Gordonianum Massachusettae).

[[Categoria:Civitatum Foederatarum scripta]]
[[Categoria:Colonia Plimmuta]]
[[Categoria:Historia Coloniarum Tredecim]]
[[Categoria:Historia Massachusettae]]
[[Categoria:Litterae Anglicae]]
[[Categoria:Scripta saeculo 17]]</text>
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    <title>Sapor</title>
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      <text bytes="10134" xml:space="preserve">{{Videdis|Sapor (discretiva)}}
[[Fasciculus:Adriaen Brouwer - The Bitter Potion - Google Art Project.jpg|thumb|Sapor acerbus. (Pictura &quot;Het bittere drankje&quot; ab [[Adrianus Brouwer|Adriano (Adriaen) Brouwer]] anno 1637 facta.]]

'''Sapor''' est gustus naturalis et proprius [[pomum|pomorum]], [[holus|holerum]], [[caro|carnium]], [[fungus|fungorum]] aliorumque edulium. [[Caseus]], qui iucunde sapit, saporem iucundum habet, et [[piscis]], qui [[mare]] sapit, saporem marinum habet. Sapor et [[gustatus]] non idem est. Eadem est enim inter gustatum et saporem differentia, quae generaliter inter [[sensus (biologia)|sensus]] et [[sensibilia]] est. Sapor est, qui sua [[vis|vi]] propria in [[alimentum|alimentis]] exstat. Quem sensum obiectivum [[mel]]li ingenitum [[Marcus Tullius Cicero]] ita describit: &quot;Ut enim mel, etsi dulcissimum est, suo tamen proprio genere saporis, non comparatione cum aliis dulce esse sentitur.&quot;&lt;ref&gt;''[[De finibus bonorum et malorum|Fin.]]'' 3.34.&lt;/ref&gt; Eandem sapiendi vim in pomis tempore maturatum [[Ovidius]] describit: &quot;[Tempus,] ne sint tristi poma sapore, cavet.&quot;&lt;ref&gt;''[[Tristia|Trist.]]'' 4.6.12.&lt;/ref&gt; Interdum '''gustus''' eodem sensu adhibetur, sicut apud [[Aulus Cornelius Celsus|Cornelium Celsum]] legimus: &quot;Adtrahaturque spiritu is sucus, donec in ore gustus [= sapor] eius sentiatur.&quot;&lt;ref&gt;''[[De medicina|Med.]]'' 6.8.1b.&lt;/ref&gt; 

An sapores similibus contrariisve discernamus philosophi Graeci antiqui differebant. [[Anaxagoras]] (saeculo V a.C.n.) contrariis: &quot;id enim quod calore frigoreve par sit neque calefacere, neque refrigerare; neque vero dulce acidumve per ea ipsa cognosci, sed frigidum per calidum, potabile per salsum, dulce per acidum secundum cuiusque defectum (ista enim omnia in nobis adesse) ... omnem vero sensum coniunctum esse cum dolore ... omne enim dissimile tangendo incommodum affert&quot;.&lt;ref&gt;[[Theophrastus]], ''De sensu'' 27-29; emendata Friderici Wimmer versio&lt;/ref&gt;

Numerum saporum alii aliter recensuerunt. Inter primos, qui de hac re scripserint, [[Menestor]] (eodem fere saeculo) sapores infinitos exstitisse scripserit si iterum doxographiam Theophrasti accipimus.&lt;ref&gt;Menestor 7 {{DK}} apud [[Theophrastus|Theophrastum]], ''De causis plantarum'' 6.3.5&lt;/ref&gt; Huic fortasse coaevus [[Empedocles]] multa de sensu, pauca de sapore cecinit, sed quattuor sapores nominavit, videlicet dulcem, amarum, acidum, pungentem, qui secundum similitudines in corpora humana suscipiuntur:
:ὣς γλυκὺ μὲν γλυκὺ μάρπτε, πικρὸν δ' ἐπὶ πικρὸν ὄρουσεν,&lt;br /&gt;
:ὀξὺ δ' ἐπ' ὀξὺ ἔβη, δαερὸν δ' ἐποχεῖτο δαηρῶι.&lt;br /&gt;
:Sic dulcia quaerunt dulcia, amara advolant ad amara,&lt;br /&gt;
:acuta petunt acuta, calida cum calidis coeunt.&lt;ref&gt;Empedocles B 90 {{DK}} apud [[Plutarchus|Plutarchum]], ''[[Quaestiones convivales]]'' 663a; versio F. G. A. Mullachii leviter emendata&lt;/ref&gt;
[[Plato]] (voce Timaei) χυμούς seu sapores permultos exstitisse censuit quorum typos e mixturis aquosis derivatos quattuor nominat, οἶνος vinum, ἐλαιηρὸν εἶδος genus oleosum, μέλι mel, ὀπός sucus;&lt;ref&gt;Plato, ''[[Timaeus (Plato)|Timaeus]]'' 59e-60a, cf. Theophrastus, ''De sensu'' 89&lt;/ref&gt; mox genera saporum septem enumerat, στρυφνά acerba, αὐστηρά austera, πικρά amara, ἁλυκά salsa, δριμέα acra, ὀξύ acidum, γλυκύ dulce.&lt;ref&gt;Plato, ''Timaeus'' 65c-66c&lt;/ref&gt; A Theophrasto sapores septem enumerantur: γλυκύς dulcis, λιπαρός pinguis, πικρός amarus, αὐστηρός austerus, δριμύς acer, ὀξύς acidus, στρυφνός acerbus, dubitante octavus ἁλμυρός salsus, repugnante nonus decimusque οἰνώδης vinosus et γαλακτώδης lacteus.&lt;ref&gt;Theophrastus, ''De causis plantarum'' 6.4.1-2, cf. 6.3.3&lt;/ref&gt; Ille enim, sicut Plato, interdum alimentis obrussis utitur ad sapores definiendos, e.g. de fructibus scribens:
:Sucorum vero alii sunt vinosi [οἱνώδεις], ut vitis, mori, myrti; alii oleosi [ἑλαώδεις], ut oleae, lauri, nucis, amygdalae, piceae, pini, abietis; alii mellei [μελιτώδεις], ut fici, palmae, castaneae; alii acres, ut origani, thymbrae, cardami, sinapis; alii amari, ut apsinthii, centaurii. Tum etiam odoribus suavibus [εὑωδίαις] insignes, ut [[Pimpinella anisum|anisi]], cedridis; alii aquei [ὐδαρεῖς] esse videntur, ut cotoneorum; alii acidi, ut punicorum et quorundam malorum, huic generi autem omnes vinosi adnumerandi; alii denique sunt alius generis, de quibus omnibus accuratius disseremus in libello de saporibus&quot; [nobis deperdito].&lt;ref&gt;Theophrastus, ''[[Historia plantarum]]'' 1.12.1; emendata Friderici Wimmer versio&lt;/ref&gt;

Eruditi Sanscrite scribentes sex sapores enumerare solent, videlicet ''madhura'' dulcis, ''lavaṇa'' salsus, ''kaśaya'' acerbus, ''amla'' acidus, ''kaṭu'' acer, ''tikta'' amarus.&lt;ref&gt;Vide e.g. ''[[Suśrutasaṃhitā]]'' 1.42 [https://archive.org/stream/englishtranslati01susruoft#page/382/mode/2up versio Anglica]; ''[[Carakasaṃhitā]]'' 26.40 sqq.[https://archive.org/details/BIUSante_47357/page/n421/mode/2up Kaviraj Avinash Chandra Kaviratna, interpr.] [https://archive.org/details/in.ernet.dli.2015.326551/page/n217/mode/2up Shree Gulabkunverba Ayurvedic Society, interprr.]&lt;/ref&gt;

Apud Sinas quinque sapores enumerantur, qui sunt 酸 ''suān'' amarus, 苦 ''kǔ'' acer, 甜 ''tián'' dulcis, 辣 ''là'' pungens vel 熱 ''rè'' calidus, 咸 ''xián'' salsus. Enumeratio iam abhinc duo milia ducentos annos reperitur in ''[[Veres autumnique domini Lü|Veribus autumnisque domini Lü]]'', a [[Lü Buwei]] editis: ibi philosophica de arte coquinaria dissertatio ad coquum mythicum [[Yi Yin]] attribuitur.&lt;ref&gt;''[[Veres autumnique domini Lü]]'' 14/2.1, 14/3.4 (John Knoblock, Jeffrey Riegel, edd. et interprr., ''The Annals of Lü Buwei'' [Stanfordiae, 2000] pp. 306-311). Cf. K. C. Chang, ''Food in Chinese Culture'' (Novo Portu, 1977) p. 68&lt;/ref&gt; Eodem aevo liber ''[[Laocius]]'' ait per paradoxum asceticum: 五味令人口爽 ''wǔ wèi lìng rén kǒu shuǎng'' &quot;Quinque sapores os hebetant&quot;; in ''[[Traditio Zuo|Traditio Zuo]]'' suadetur indoles hominum per harmoniam aut dissonantiam quinque saporum mutari.&lt;ref&gt;''[[Laocius]]'' [http://www.tao-te-king.org/12.htm 12]; ''[[Traditio Zuo|Zuo zhuan]]'' Zhao 9.5, 20.8a (Stephen Durrant, Wai-yee Li, David Schaberg, interprr., ''Zuo Tradition'' [Seattli: University of Washington Press, 2016] pp. 1448-1451, 1586-1587&lt;/ref&gt; Coqui autem [[Sichuan|provinciae Sichuanensis]] sextum saporem 麻 ''má'' &quot;torporentem&quot; addunt, ab aromate eorum indigena [[Zanthoxyli fructus|zanthoxylo]] effectum, cum sapore pungenti aequilibrem (麻辣 ''má là'').&lt;ref&gt;[[Fuchsia Dunlop]], ''The Food of Sichuan'' (Londinii: Bloomsbury, 2019) pp. 21-24, cf. 471-474; Gernot Katzer, &quot;[http://gernot-katzers-spice-pages.com/engl/Zant_pip.html Sichuan pepper]&quot;&lt;/ref&gt;

Europaeorum recentiorum [[Ioannes Anthelmus Brillat-Savarin|Brillat-Savarin]] sapores infinitos esse agnovit;&lt;ref&gt;[[Ioannes Anthelmus Brillat-Savarin]], ''[[Physiologie du goût]]'' (1826) [https://gallica.bnf.fr/ark:/12148/btv1b8626673x/f90 vol. 1 p. 72]&lt;/ref&gt; plurimi autem aut quattuor accipiunt aut ([[osmazomum|osmazomo]] vel ''umami'' addito) quinque.
 
== Notae ==
&lt;references/&gt;

== Bibliographia ==
[[File:Breastfeeding infant.jpg|thumb|150px|Lac maternum gustum pueruli formans]]
; De sapore
* Linda M. Bartoshuk, &quot;[https://www.researchgate.net/publication/22788130_The_psychophysics_of_taste The psychophysics of taste]&quot; in ''American Journal of Clinical Nutrition'' vol. 31 (1978) pp. 1068-1077
* Nirupa Chaudhari, Stephen D. Roper, &quot;[http://jcb.rupress.org/content/jcb/190/3/285.full.pdf The cell biology of taste]&quot; in ''Journal of Cell Biology'' vol. 190 pp. 285-296 (9 Augusti 2010)
* Virginia B. Collings, &quot;[https://link.springer.com/article/10.3758/BF03203270 Human taste response as a function of locus of stimulation on the tongue and soft palate]&quot; in ''Perception &amp; Psychophysics'' vol. 16 (1974) pp. 169–174
* D. P. Hanig, &quot;[http://vlp.mpiwg-berlin.mpg.de/library/data/lit4562/index_html?pn=1&amp;ws=1.5 Zur Psychophysik des Geschmackssinnes]&quot; in ''Philosophische Studien'' vol. 17 (1901) pp. 576-623
* Kim Hullot-Guiot, &quot;[https://www.liberation.fr/lifestyle/gastronomie/au-royaume-uni-le-vin-aurait-un-gout-de-vacances-20220815_UXFYGVNWCZEQJMWXHH7QOVJ3GA/ Au Royaume-Uni, le vin aurait un goût de vacances]&quot; in ''[[Libération]]'' (15 Augusti 2022)
* Thomas Hummel, J. F. Delwiche, C. Schmidt, K.-B. Hüttenbrink, &quot;[https://www.academia.edu/12756273/Effects_of_the_form_of_glasses_on_the_perception_of_wine_flavors_a_study_in_untrained_subjects Effects of the form of glasses on the perception of wine flavors: a study in untrained subjects]&quot; in ''Appetite'' vol. 41 (2003) pp. 197–202
* Carl Pfaffmann, &quot;Neurophysiological mechanisms of taste&quot; in ''American Journal of Clinical Nutrition'' vol. 31 (1978) pp. 1058–1067 [https://academic.oup.com/ajcn/article/31/6/1058/4650650?searchresult=1 Situs venalis]

; De saporibus enumeratis
* Annick Faurion, &quot;[https://www.persee.fr/doc/jatba_0183-5173_1988_num_35_1_6674 Naissance et obsolescence du concept de quatre qualités en gustation]&quot; in ''Journal d'agriculture traditionnelle et de botanique appliquée'' vol. 35 (1988) pp. 21-40
* Cordelia A. Running, Bruce A. Craig, Richard D. Mattes, &quot;[https://www.researchgate.net/publication/279753553_Oleogustus_The_Unique_Taste_of_Fat Oleogustus: The Unique Taste of Fat]&quot; in ''Chemical Senses'' vol. 40 (2015) pp. 507-516

; De singulis saporibus 
* Jennifer Billing, Paul W. Sherman, &quot;Antimicrobial functions of spices: why some like it hot&quot; in ''Quarterly review of biology'' vol. 73 (1998) pp. 3-49 [https://www.jstor.org/stable/3036683 JSTOR]

== Nexus externi ==
* &quot;[https://www.psychologicalscience.org/observer/classical-blunders Classical Blunders]&quot; apud ''Association for Psychological Science''
* Richard Gawel, &quot;[https://www.aromadictionary.com/articles/tonguemap_article.html Challenging the Tongue Taste Map]&quot;

[[Categoria:Scientia cibaria]]
[[Categoria:Sensus]]</text>
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    <title>Reges Spartae</title>
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      <text bytes="8679" xml:space="preserve">Velut collegium duorum '''regum [[Sparta|Spartae]]''' fuit usque ad [[saeculum 2 a.C.n.|II saec. a.C.n.]], quorum alter e gente Eurypontidum&lt;ref&gt;Res gestas et seriem illorum regum breviter enumerat Pausanias libro tertio ''[[Descriptio Graeciae|Descriptionis Graeciae]]'', cap.7-10.&lt;/ref&gt;, alter e gente [[Agiades|Agiadum]]&lt;ref&gt;Res gestas et seriem regum Agiadum enumerat [[Pausanias]] libro tertio ''[[Descriptio Graeciae|Descriptionis Graeciae]]'', cap.2-6.&lt;/ref&gt; ortus esse debebat. Quae gentes ad duos fratres gemellos [[Heracleidae|Heracleidas]], id est ab [[Hercules|Hercule]] originem trahentes,  Eurysthenem, patrem Agiadum, et Proclem, patrem Eurypontidum, mythice referebantur&lt;ref&gt;{{Strabo}} VIII.5.5. Pausanias III.1.6-9.&lt;/ref&gt;. Genealogiae autem separatae erant ([[conubium]] inter se non habebant) nec fas erat alteram in regnum alteri succedere. 

Iam sexto saeculo a.C.n. potentia regum imminuebatur  et summum imperium penes [[Ephori|ephoros]] et ''gerusian'' (senatum) paulatim transferebatur. Reges  exercitus ducere solebant atque sacra religionis tamquam summi sacerdotes administrare&lt;ref&gt;[[De republica Lacedaemoniorum (Xenophon)|De rep. Lac.]] XIII,11 et XIV(XV),2.&lt;/ref&gt;.
==An regnum duplex quid profecerit==
Reges Spartae velut imago in terris erant geminorum deorum, [[Dioscuri|Dioscurum]] [[Castor et Pollux|Castoris Pollucisque]] qui Spartae nati ibi maximi dei aestimabantur : nonne reges quotiencumque ad bellum proficiscebantur simulacra Dioscurum secum portabant ? Haec ipsa duplicitas ad stabilitatem institutorum Lacedaemoniorum aliquid contulisse traditur :  nam altero ex eis mortuo regnum non vacabat et si quis minor natu patri successerat collega officio eius fungi poterat. Itaque a sexto saeculo ineunte  regibus simul eundem exercitum ducere non licebat ne eidem periculo subicerentur. [[Aristoteles]]&lt;ref&gt;''Pol''. II,9.&lt;/ref&gt; hoc institutum hac de causa quoque laudat quod duo reges aemuli inter se et adversarii  esse solent atque ita potentiam alterius comminuunt (quod idem de [[Consul|consulibus]] [[Roma antiqua|Romanis]] dici potest&lt;ref&gt;[[Dionysius Halicarnassensis|Dion. Hal]]. ''Ant. Rom''. IV,73,4.&lt;/ref&gt;). Certe ante secundam partem tertii saeculi a.C.n. atque reges [[Cleomenes III|Cleomenem III]] et [[Nabis(rex Spartae)|Nabidem]] regnum Spartae numquam in [[Tyrannis|tyrannidem]] degeneravit. Itaque diuturnitatem illius regni duarum gentium multi auctores antiqui mirati sunt.

==De militari imperio==
Antiquitus fortasse reges de bello decernebant ; temporibus autem de quibus quidquam certi accepimus concilium populi et sociorum de pace belloque decernebat, [[Ephori|ephori]] dilectus habebant (quod dicebatur φρουρὰν φαίνειν). Sed in quocumque exercitu versabatur  rex, ibi dux summus erat, etiamsi saepe a duobus ephoris comitabatur qui eum postea accusare poterant. [[Lysander]] et ipse, victoriis praeclarus vir, sese eis submittere debebat. Legatos ad hostes de indutiis, ad socios de auxiliis accersendis mittere poterant.

==De sacerdotali officio==
Reges ipsi sacri sacerdotes sunt ''Iovis Lacedaemonii'' et ''Iovis Quranii''. Domi militiaeque sacra administrant atque pellibus omnium victimarum et in sequentibus epulis optimis partibus honoris causa donantur. &quot;Pythios&quot; creant ac secum habent quos ad Pythiam consulendam [[Delphi (urbs Graeciae)|Delphos]], ut libitum est, mittere possunt. Illi ''Pythii'' corpus responsorum [[Pythia|Pythiae]] custodiunt, quibus consilia ducis firmantur si necesse est.

Funera magnifica eis apparantur qualia Spartae nisi in regibus fieri mos non est. Ita origo divina omnibus ostentatur.

==De vera potentia==
Reges ipso facto in gerusia (senatu) sedebant. Mense unoquoque ineunte [[Ephori|ephori]] et reges sibi invicem sacramenta tralaticia praestabant : reges pollicebantur sese legibus obtemperaturos esse, ephori autem, si quidem reges iurata servassent regnum firmaturos atque perpetuum facturos. Non supra leges igitur erant reges Spartae et a magistratibus coercebantur. Nihilominus nonnullis eorum rempublicam aliquatenus regere contigit, ut [[Cleomenes I|Cleomeni I]] (520-483) et [[Agesilaus II (rex Spartae)|Agesilao II]] (400-360 a.C.n.). Quinto saeculo contra potentia regum imminuta est : in ius ab ephoris vel etiam aliis civibus saepe trahebantur et ob multitudinem bellorum navarchi non solum classibus sed etiam exercitibus praeficiebantur. Ita gloriosissimus dux bellicus quinti saeculi non rex fuit sed Lysander. Potestas regia maior vel minor erat prout ipsius regis auctoritas  et concordia cum ephoris et gerusia valebat. Nihilominus aliquid sacri Heraclidis regibus inerat nec eis manus inferre fas erat ː primum [[Agis IV|Agidem IV]] anno [[241 a.C.n.]] ab ephoris et parte senatus damnatum supplicio adfectum esse memorabat [[Plutarchus]]&lt;ref&gt;[[ Vita Agidis (Plutarchus)|Agis]]'' 21&lt;/ref&gt; nec multo diutius superfuit regia dignitas quoniam iam a tertio saeculuo exeunte [[Tyrannis|tyrannorum]] aetas Spartae floruit.

Quin etiam potentiam e divitiis ducebant : ditissimi homines in [[Laconia]] erant reges Spartae qui plurimos agros possidebant, quibus tributum regium a Lacedaemoniis pendebatur, quibus non tantum maxima pars praedae bello captae sed etiam pelles omnium victimarum atque, ne victimae umquam sacris deessent, unus fetus ex omnibus scrofis parturientibus servabatur. Ita clientelam in civitate vel factionem sibi parare poterant&lt;ref&gt;[[Xenophon (scriptor)|Xenophon]], ''[[Agesilaus (Xenophon)|Ages]]''. I,19.&lt;/ref&gt;. Agesilaus II qui magna reverentia erga magistratus et senatores semper utebatur, donis novos senatores honorabat&lt;ref&gt;[[Plutarchus|Plut]]., ''Ages''. IV,5.&lt;/ref&gt; vel beneficiis etiam adversarios politicos sibi obstringebat&lt;ref&gt;[[Plutarchus|Plut]]., ''Ages''. XX,6.&lt;/ref&gt;.

==Fasti regum Spartae (V-III saec. a.C.n.)==
{| class=&quot;wikitable&quot;
|-
! Tempora !! Agiades !! Tempora !! Eurypontides
|-
| ca. 520–490  || [[Cleomenes I]] || 510-491 || [[Demaratus|Damaratos]]
|-
| 490-480 || [[Leonidas I]] || 491-469 || [[Leotychidas II]]
|-
| 480-458 || Pleistarchos || 469-427 ||  [[Archidamus II|Archidamos II]].
|-
| 458-445|| [[Plistoanax|Pleistoanax]]  || 427-400  || [[Agis II]]
|-
| 445-426 || [[Pausanias (rex Spartae)|Pausanias]] || 399-360 || [[Agesilaus II (rex Spartae)|Agesilaus II]]
|-
| 426-408 ||[[Plistoanax|Pleistoanax]] iterum ||360-336 || [[Archidamus III|Archidamos III]].
|-
| 408-395 || [[Pausanias (rex Spartae)|Pausanias iterum]] || 336-331 || [[Agis III]]
|-
| 395-381 || [[Agesipolis I (rex Spartae)|Agesipolis I]] || 331-305 || Eudamidas I
|-
| 380-371 || [[Cleombrotus I]] || 305-275 || Archidamos IV
|-
| 371-369 || [[Agesipolis II]] || 275-244 || Eudamidas II
|-
| 370-309 || Cleomenes II || 244-241 || [[Agis IV]]
|-
| 309-265 || Areus I ||241-228 || [[Eudamidas III]]
|-
| 265-262 || Akrotatos || 228-227 || [[Archidamus V|Archidamos V]]
|-
| 262-254 || Areus II || 227-222 || [[Eucleidas|Eukleidas]]
|-
| 254-242 || [[Leonidas II]] || 219-211 || Lycurgus
|-
| 242-241 || [[Cleombrotus II]] || 211-207 || Pelops
|-
| 241-235 || [[Leonidas II]] iterum || 207-192 || Nabis
|-
| 235-222 || [[Cleomenes III]] ||||
|-
| 219-215 || Agesipolis III||||
|-
|||||||
|}

==Fontes==
*[[Aristoteles]], ''Politica'' II,9 et III,4.
*[[Cornelius Nepos]], ''Liber de excellentibus ducibus exterarum gentium'' XVII, capitulo primo.
*[[Dionysius Halicarnassensis]], libro secundo ''Antiquitatum Romanarum'' XIII,4 et XIV,2.
**[[Herodotus]] libro sexto [[Historiae (Herodotus)|Historiarum]] 51-58
* [[Pausanias]] libro primo ''[[Graeciae descriptio|Graeciae descriptionis]]'' XIII.4 necnon libro tertio cap. 1-11
*[[Plutarchus]], &quot;Agis et Cleomenes&quot; &quot;Agesilaus&quot; &quot;Lycurgus&quot; in ''[[Vitae parallelae|Vitis Parallelis]]''
*[[Polybius]], libro quarto ''Historiarum'' cap. 34-36 et 81.
* [[Strabo]], libro decimo [[Geographica (Strabo)|Geographicorum]] capite quarto 19
*[[Thucydides]], libro quinto ''[[De bello Peloponnesiaco]]'' 59-63 et 71-72
* [[Xenophon (scriptor)|Xenophon]] ''[[De republica Lacedaemoniorum (Xenophon)|De republica Lacedaemoniorum]]'' cap. XIII et XV (in nonnullis editionibus XIII-XIV): [[Wikisource:el:Λακεδαιμονίων Πολιτεία| Apud Vicifontem Graecum]]
**  ''[[Agesilaus (Xenophon)|Agesilaus]]''

== Plura legere si cupis ==
*Edmundus Lévy, ''Sparte: histoire politique et sociale jusqu'à la conquête romaine''. Parisiis, Seuil, 2003. ISBN 9782020324533.
*Bernard Sergent, &quot;[https://www.persee.fr/doc/rhr_0035-1423_1976_num_189_1_6283 La représentation spartiate de la royauté.]&quot;, ''Revue de l'histoire des religions'',1976ː 3-52

== Notae ==
&lt;div class=&quot;references-small&quot;&gt;&lt;references /&gt;&lt;/div&gt;

[[Categoria:Reges Spartae|!]]
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      <text bytes="38903" xml:space="preserve">::''Transclusio paginae Vicipaedia:Paginae quas omnibus Wikipediis contineri oportet/Expansio/Technologia''

== Technologia (800 tituli) ==

=== Res generales ===

{{Div col|cols=4}}
# [[Technologia]]
# [[Ars ingeniaria]]
# [[Intellegentia artificialis]]
# [[Biotechnologia]]
# [[Clonizatio]]
# [[Organismus genetice mutatus]]
# [[Technologia genetica]]
# [[Nanotechnologia]]
# [[Technologia nuclearis]]
# [[Transhumanismus]]
# [[Ingeniaria moderationis]]
# [[Normalizatio]]
# [[Innovatio disruptiva]]
# [[Ingeniaria transportationis]]
# [[Ingeniaria constructionis]]
# [[Ingeniaria petrolei]]
# [[:en:Naval architecture]] [[:fr:Architecture navale]] [[:it:Architettura navale]] [[:de:Schiffstechnik]]
# [[Innovatio]]
# [[Timor technologiae]]
# [[Microtechnologia]]
# [[Pictura technica]]
# [[Automatio]]
# [[Convergentia technologica]]
# [[:en:Science studies]] [[:fr:Science studies]] [[:de:Wissenschaftsforschung]]
# [[:en:Science and technology studies]] [[:fr:Études des sciences et des techniques]] [[:it:Studi su scienza e tecnologia]] [[:de:Science and Technology Studies]]
# [[:en:High tech]] [[:fr:Techniques de pointe]] [[:it:Alta tecnologia]] [[:de:Spitzentechnologie]]
# [[Administratio ingeniariae]]
# [[Ingeniaria cibaria]]
# [[Mutatio technologica]]
# [[:en:Human reliability]] [[:fr:Facteur humain]] [[:de:Menschlicher Faktor]]
# [[Determinismus technologicus]]
# [[Ingeniaria urbana]]
# [[Ingeniaria terrae motuum]]
# [[Administratio litoralis]]
# [[:en:Ceramic engineering]] [[:fr:Céramique technique]] [[:it:Tecnica ceramica]] [[:de:Technische Keramik]]
# [[Ethica ingeniariae]]
# [[Ingeniaria agriculturalis]]
# [[Ingeniaria aërospatialis]]
# [[Ingeniaria biomedica]]
# [[Ars ingeniaria civilis]]
# [[Ars electrica]]
# [[Ingeniaria industrialis]]
# [[Ingeniaria militaris]]
# [[Ingeniaria nuclearis]]
# [[Ingeniaria corporum programmatum]]

{{Div col end}}

=== Agricultura ===

{{Div col|cols=4}}
# [[Scientia agriculturalis]]
# [[Agronomia]]
# [[Domitus]]
# [[Fundus]]
# [[Agricola (munus)]]
# [[Res novae virides]]
# [[:en:Selective breeding]] [[:fr:Sélection artificielle]] [[:it:Selezione artificiale]] [[:de:Zucht]]
# [[Kolkhoz]]
# [[Horreum]]
# [[Granarium]]
# [[:en:Grain elevator]] [[:fr:Élévateur à grain]] [[:de:Getreideheber]]
# [[:en:Stable]] [[:fr:Écurie]] [[:it:Scuderia (equini)]] [[:de:Pferdestall]]
# [[Caldarium (hortorum cultus)]]
# [[Messor compositus]]
# [[Cultivarietas]]
# [[Laetamen]]
# [[Hortorum cultus]]
# [[Messis]]
# [[Hydroponica]]
# [[Insecticida]]
# [[Inrigatio]]
# [[Pesticida]]
# [[Herbicida]]
# [[:en:Plant breeding]] [[:fr:Culture sélective des plantes]] [[:de:Pflanzenzüchtung]]
# [[Aratio]]
# [[Vannus]]
# [[Educatio animalium]]
# [[Apicultura]]
# [[Pecus]]
# [[Seminium]]
# [[Aves cohortales]]
# [[Hortus]]
# [[Fruges]]
# [[Linum xylinum]]
# [[Faenum]]
# [[:en:Weed]] [[:fr:Adventice]] [[:it:Piante infestanti]] [[:de:Unkraut]]
# [[Officina lactaria]]
# [[:en:Extensive farming]] [[:fr:Agriculture extensive]] [[:it:Agricoltura estensiva]] [[:de:Extensive Landwirtschaft]]
# [[Piscicultura]]
# [[:en:Industrial agriculture]] [[:fr:Agriculture industrielle]] [[:de:Industrielle Landwirtschaft]]
# [[Agricultura intensiva]]
# [[Arbustum]]
# [[Agricultura organica]]
# [[Permacultura]]
# [[Sericultura]]
# [[:en:Subsistence agriculture]] [[:fr:Agriculture vivrière]] [[:it:Agricoltura di sussistenza]] [[:de:Subsistenzwirtschaft]]
# [[:en:Sustainable agriculture]] [[:fr:Agriculture durable]] [[:it:Agricoltura sostenibile]] [[:de:Nachhaltige Landwirtschaft]]
# [[Agricultura urbana]]
# [[:en:Agricultural policy]] [[:it:Politica agraria]] [[:de:Agrarpolitik]]

{{Div col end}}

=== Constructio ===

{{Div col|cols=4}}
# [[:en:Bitumen]] [[:fr:Bitume]] [[:it:Bitume]] [[:de:Bitumen]]
# [[:en:Adobe]] [[:fr:Adobe (brique)]] [[:it:Adobe (mattone)]]
# [[Pons pendulus]]
# [[:en:Reinforced concrete]] [[:fr:Béton armé]] [[:it:Calcestruzzo armato]] [[:de:Stahlbeton]]
# [[Fossa castrensis]]
# [[Lutum]]
# [[Chamulcus automatarius]]
# [[:en:Mortar (masonry)]] [[:fr:Mortier (matériau)]] [[:it:Malta (materiale)]] [[:de:Mörtel]]
# [[:en:Plaster]] [[:fr:Plâtre]] [[:it:Gesso (materiale)]] [[:de:Putz (Bauteil)]]
# [[Quadra fictilis]]
# [[:en:Building material]] [[:fr:Matériau de construction]] [[:it:Materiale da costruzione]] [[:de:Baustoff]]
# [[:en:Foundation (engineering)]] [[:fr:Fondation (construction)]] [[:it:Fondazione (edilizia)]] [[:de:Gründung (Bauwesen)]]
# [[:en:Numerical control]] [[:it:Controllo numerico computerizzato]] [[:de:Computerized Numerical Control]]
# [[:en:Stucco]] [[:fr:Stuc]] [[:it:Stucco]] [[:de:Stuck]]
# [[Moles fluctifraga]]

==== Res generales ====
# [[Aedificium]]
# [[:en:Architectural engineering]] [[:it:Ingegneria edile]] [[:de:Bauwesen]]
# [[:en:Building code]] [[:fr:Document technique unifié]]
# [[Infrastructura]]
# [[Ingeniaria structuralis]]

==== Materies constructionis ====
# [[Clavus]]
# [[Later]]
# [[Ferrumen]]
# [[Rudus concretum]]
# [[:en:Masonry]] [[:fr:Maçonnerie]] [[:it:Muratura]] [[:de:Mauerwerk]]
# [[:en:Scaffolding]] [[:fr:Échafaudage]] [[:it:Ponteggio]] [[:de:Gerüst]]

==== Genera aedificiorum ====
# [[Arcus (architectura)]]
# [[:en:Ceiling]] [[:fr:Plafond (architecture)]] [[:it:Soffitto]] [[:de:Decke (Bauteil)]]
# [[Columna]]
# [[Tholus]]
# [[Ianua]]
# [[:en:Façade]] [[:fr:Façade]] [[:it:Facciata]] [[:de:Fassade]]
# [[:en:Floor]] [[:fr:Plancher]] [[:it:Pavimento]] [[:de:Fußboden]]
# [[Tectum]]
# [[Conclave (architectura)]]
# [[Scalae]]
# [[Murus]]
# [[Fenestra]]
# [[:en:Harbor]] [[:fr:Havre]]
# [[Pharus]]
# [[:en:Pier]] [[:fr:Estacade]] [[:it:Pontile]] [[:de:Seebrücke]]
# [[:en:Office]] [[:fr:Bureau (immobilier)]] [[:it:Ufficio (locale)]] [[:de:Büro]]
# [[:en:Warehouse]] [[:fr:Entrepôt]] [[:it:Magazzino]] [[:de:Lagerhaltung]]
# [[Diaeta (aedificium)]]
# [[:en:Barracks]] [[:fr:Caserne (militaire)]] [[:it:Caserma]] [[:de:Kaserne]]
# [[Domus]]
# [[Pagoda]]
# [[:en:Palace]] [[:fr:Palais]] [[:it:Palazzo]] [[:de:Palast]]
# [[:en:Penthouse apartment]] [[:fr:Penthouse (appartement)]] [[:it:Attico (abitazione)]] [[:de:Penthouse]]
# [[Pyramis (aedificium)]]
# [[Caeliscalpium]]
# [[Turris]]
# [[:en:Tower block]] [[:fr:Immeuble de grande hauteur]] [[:it:Casa a torre]] [[:de:Hochhaus]]
# [[Villa]]
# [[Casa]]
# [[Iglu]]
# [[:en:Basement]] [[:fr:Sous-sol (architecture)]] [[:it:Seminterrato]] [[:de:Keller]]
# [[Balneum]]
# [[Cubiculum]]
# [[:en:Dining room]] [[:fr:Salle à manger]] [[:it:Sala da pranzo]] [[:de:Esszimmer]]
# [[Stabulum autocineticum]]
# [[Culina]]
# [[Exedrium]]
# [[:en:Pantry]] [[:fr:Cellier (architecture)]] [[:de:Speisekammer]]
# [[:en:Tent]] [[:fr:Tente]] [[:it:Tenda (abitazione)]] [[:de:Zelt]]
# [[Yurt]]

==== Scientia urbium ====
# [[Ars urbium disponendarum]]
# [[Designatio urbana]]
# [[:en:Campus]] [[:fr:Campus]] [[:it:Campus]] [[:de:Campus]]
# [[:en:City block]] [[:fr:Îlot urbain]] [[:it:Isolato]] [[:de:Häuserblock]]
# [[:en:Downtown]] [[:de:Downtown]]
# [[Ghectum]]
# [[:en:Industrial park]] [[:fr:Parc industriel]] [[:de:Industriepark]]
# [[Viridarium]]
# [[Via urbis]]
# [[Forum (architectura)]]
# [[:en:Zoning]] [[:fr:Zonage (urbanisme)]] [[:it:Zonizzazione]]

==== Moles ====
# [[Moles (agger)]]
# [[Moles Hooverana]]
# [[Moles Trium Angustiarum]]

==== Aedificia ad homines vehendos ====
# [[Pons]]
# [[:en:Akashi Kaikyo Bridge]] [[:fr:Pont du détroit d'Akashi]] [[:it:Ponte dello stretto di Akashi]] [[:de:Akashi-Kaikyō-Brücke]]
# [[:en:Bang Na Expressway]] [[:fr:Bang Na Expressway]] [[:de:Bang Na Expressway]]
# [[Pons Brooklynensis]]
# [[Viaeductus Danyang–Kunshan]]
# [[Cuniculus Freti Femerae]]
# [[:en:George Washington Bridge]] [[:fr:Pont George-Washington]] [[:it:Ponte George Washington]] [[:de:George-Washington-Brücke]]
# [[Pons Aureae Portae]]
# [[Pons Baltici Freti Maioris]]
# [[:en:Hong Kong–Zhuhai–Macau Bridge]] [[:fr:Pont Hong Kong-Zhuhai-Macao]] [[:it:Ponte Hong Kong-Zhuhai-Macao]] [[:de:Hongkong-Zhuhai-Macau-Brücke]]
# [[:en:Lake Pontchartrain Causeway]] [[:fr:Chaussée du lac Pontchartrain]] [[:it:Lake Pontchartrain Causeway]] [[:de:Lake Pontchartrain Causeway]]
# [[:en:London Bridge]] [[:fr:Pont de Londres]] [[:it:London Bridge]] [[:de:London Bridge]]
# [[Pons Oresundianus]]
# [[Pons Russicus]]
# [[:en:Tower Bridge]] [[:fr:Tower Bridge]] [[:it:Tower Bridge]] [[:de:Tower Bridge]]
# [[Cuniculus (aedificium)]]
# [[Cuniculus Freti Gallici]]
# [[Cuniculus Seikan]]

{{Div col end}}

=== Industria ===

{{Div col|cols=4}}
# [[:en:Industrial robot]] [[:fr:Robotique industrielle]] [[:it:Robot industriale]] [[:de:Industrieroboter]]
# [[:en:Mass production]] [[:fr:Production en série]] [[:it:Produzione di massa]] [[:de:Massenproduktion]]
# [[Machinarum inductio]]
# [[:en:Waste]] [[:fr:Déchet]] [[:it:Rifiuto]] [[:de:Abfall]]
# [[:en:Landfill]] [[:fr:Décharge (déchet)]] [[:it:Discarica]] [[:de:Deponie]]
# [[Anacyclismus]]
# [[:en:Waste management]] [[:fr:Gestion des déchets]] [[:it:Gestione dei rifiuti]]
# [[:en:Woodworking]] [[:fr:Travail du bois]] [[:de:Holzverarbeitung]]

==== Energia ====
# [[Energia geothermica]]
# [[Energia hydroelectrica]]
# [[Energia nuclearis]]
# [[Energia solaris]]
# [[Electricitas solaris]]
# [[Energia renovabilis]]
# [[Energia ventosa]]
# [[Cereus]]
# [[Ventimola]]
# [[:en:Hydropower]] [[:fr:Énergie hydraulique]] [[:it:Energia idroelettrica]] [[:de:Wasserkraft]]
# [[:en:Electricity generation]] [[:fr:Production d'électricité]] [[:it:Produzione di energia elettrica]] [[:de:Stromerzeugung]]
# [[Electrificina]]
# [[:en:Cooling tower]] [[:fr:Tour aéroréfrigérante]] [[:it:Torre di raffreddamento]] [[:de:Kühlturm]]
# [[:en:Solar cell]] [[:fr:Cellule photovoltaïque]] [[:it:Cella solare]] [[:de:Solarzelle]]
# [[:en:Turbine]] [[:fr:Turbine]] [[:it:Turbina]] [[:de:Turbine]]
# [[Mola pneumatica]]
# [[:en:Electrical grid]] [[:fr:Réseau électrique]] [[:it:Rete elettrica]] [[:de:Stromnetz]]
# [[:en:Electric power transmission]] [[:fr:Transport d'énergie électrique]] [[:it:Trasmissione di energia elettrica]]
# [[:en:Electric power distribution]] [[:fr:Réseau de distribution électrique]] [[:it:Distribuzione di energia elettrica]]
# [[Fomes fossilis]]
# [[Carbo]]
# [[Benzinum]]
# [[Ceroleum]]
# [[Gasium naturale]]
# [[Petroleum]]

{{Div col end}}

==== Cibus et salus ====

{{Div col|cols=4}}
# [[Furnulus undarum brevium]]
# [[Pasteurizatio]]
# [[Armarium frigidarium]]
# [[Sapo]]
# [[Furnulus]]
# [[Vasa Scubae]]
# [[Lagoena caldaria]]
# [[:en:Laundry]] [[:fr:Entretien des textiles]] [[:de:Wäsche]]
# [[:en:Sanitation]] [[:fr:Assainissement]] [[:de:Sanitärtechnik]]
# [[Systema cloacarum]]
# [[:en:Sewage treatment]] [[:fr:Épuration des eaux]] [[:de:Abwasserentgiftung]]
# [[:en:Water purification]] [[:fr:Purification de l'eau]] [[:it:Potabilizzazione dell'acqua]] [[:de:Wasseraufbereitung]]
# [[Aquae ductus]]
# [[:en:Plumbing]] [[:fr:Plomberie]] [[:it:Impianto idraulico]]
# [[:en:Pumping station]] [[:fr:Station de pompage]] [[:it:Idrovora]] [[:de:Pumpwerk]]
# [[Turris aquaria]]
# [[Puteus]]

{{Div col end}}

==== Technologia chemica ====

{{Div col|cols=4}}
# [[Materia displosiva]]
# [[Dynamites]]
# [[Metallurgia]]
# [[Plastica (materia)]]
# [[Cummis]]
# [[Gluten]]
# [[:en:Ore]] [[:fr:Minerai]] [[:it:Minerale grezzo]] [[:de:Erz]]
# [[Charta]]
# [[Papyrus (materies)]]
# [[:en:Lumber]] [[:fr:Bois (matériau de construction)]] [[:it:Legname]] [[:de:Bauholz]]
# [[:en:Petrochemical]]

==== Technologia metallorum ====
# [[:en:Metalworking]] [[:it:Lavorazione dei metalli]] [[:de:Metallverarbeitung]]
# [[Faber ferrarius]]
# [[:en:Die casting]] [[:fr:Moulage métallique]] [[:it:Pressofusione]] [[:de:Druckguss]]
# [[:en:Drilling]] [[:fr:Perçage]] [[:it:Perforazione (ingegneria)]] [[:de:Bohren]]
# [[:en:Extrusion]] [[:fr:Extrusion]] [[:it:Estrusione]] [[:de:Extrusion (Verfahrenstechnik)]]
# [[:en:Laser cutting]] [[:fr:Découpe laser]] [[:de:Laserschneiden]]
# [[:en:Machine press]] [[:it:Pressa]] [[:de:Presse (Fertigungsmaschine)]]
# [[:en:Quenching]] [[:fr:Trempe (métallurgie)]] [[:it:Trattamento di tempra]] [[:de:Abschrecken (Metallurgie)]]
# [[:en:Rolling (metalworking)]] [[:fr:Laminage]] [[:it:Laminazione]] [[:de:Walzen]]
# [[:en:Soldering]] [[:it:Brasatura]] [[:de:Löten]]
# [[:en:Welding]] [[:fr:Soudage]] [[:it:Saldatura]] [[:de:Schweißen]]

{{Div col end}}

=== Machinae ===

{{Div col|cols=4}}

==== Res generales ====
# [[Ars mechanica]]
# [[Machina]]
# [[Centrifugium]]
# [[Ingenium (machina)]]
# [[Motrum electricum]]
# [[Motrum combustionis internae]]
# [[Machina pyraulocinetica]]
# [[Machina vaporaria]]
# [[Robotum]]
# [[Machina simplex]]
# [[:en:Inclined plane]] [[:fr:Plan incliné]] [[:it:Piano inclinato]] [[:de:Schiefe Ebene]]
# [[Vectis]]
# [[Trochlea]]
# [[Clavus cochleatus]]
# [[Cuneus]]
# [[Rota]]
# [[:en:Wheel and axle]] [[:de:Wellrad]]
# [[Robotica]]
# [[Systema]]
# [[Tolleno]]

==== Partes machinarum ====
# [[:en:Axle]] [[:fr:Axe (mécanique)]] [[:it:Asse (meccanica)]] [[:de:Achse (Maschinenelement)]]
# [[:en:Bearing (mechanical)]] [[:fr:Palier (mécanique)]] [[:de:Lager (Maschinenelement)]]
# [[:en:Clutch]] [[:fr:Embrayage]] [[:it:Frizione (meccanica)]] [[:de:Kupplung]]
# [[Rota dentata]]
# [[:en:Worm drive]] [[:fr:Vis sans fin]] [[:de:Schneckengetriebe]]
# [[:en:Electrical cable]] [[:fr:Fil électrique]] [[:it:Cavo elettrico]] [[:de:Kabel]]
# [[Catena]]
# [[Funis]]
# [[Rete (instrumentum)]]
# [[:en:Twine]] [[:fr:Ficelle (corde)]] [[:it:Spago]] [[:de:Zwirn]]
# [[:en:Fastener]]
# [[Matrix (mechanica)]]
# [[Nodus]]
# [[:en:Rivet]] [[:fr:Rivet]] [[:it:Rivetto]] [[:de:Niet]]
# [[:en:Washer (hardware)]] [[:fr:Rondelle (mécanique)]] [[:it:Rondella (meccanica)]] [[:de:Unterlegscheibe]]
# [[:en:Kinematic pair]] [[:it:Coppia cinematica]] [[:de:Gelenk (Technik)]]
# [[:en:Lock and key]] [[:fr:Serrure]] [[:it:Serratura]] [[:de:Schloss (Technik)]]
# [[Clavis]]
# [[Tubus]]
# [[:en:Gasket]] [[:fr:Joint (étanchéité)]] [[:it:Guarnizione]] [[:de:Flachdichtung]]
# [[Elater]]

==== Machinae magnae ====
# [[:en:Chainsaw]] [[:fr:Tronçonneuse]] [[:it:Motosega]] [[:de:Kettensäge]]
# [[:en:Cotton gin]] [[:fr:Cotton gin]] [[:it:Sgranatrice di cotone]] [[:de:Egreniermaschine]]
# [[:en:Cultivator]] [[:fr:Cultivateur (outil)]] [[:de:Grubber]]
# [[:en:Irrigation sprinkler]] [[:fr:Arrosage automatique]] [[:it:Irrigatore]] [[:de:Sprinkler (Beregnung)]]
# [[Herbisectrum]]
# [[Machina sartoria]]
# [[Tractorium]]
# [[:en:Air conditioning]] [[:fr:Climatisation]] [[:it:Condizionatore d'aria]]
# [[Machina siccatoria volubilis]]
# [[Machinula coffearia]]
# [[Machina elutoria]]
# [[Furnus]]
# [[Trulleum]]
# [[:en:Electric stove]] [[:fr:Plaque électrique]] [[:it:Forno elettrico da cucina]] [[:de:Elektroherd]]
# [[Latrina]]
# [[Pulveris hauritorium]]
# [[Machina lavatoria]]

==== Instrumenta ====
# [[Instrumentum]]
# [[Cochlea Archimedea]]
# [[Securis]]
# [[Malleus]]
# [[Serra (instrumentum)]]
# [[Scalprum dentatum]]
# [[:en:Clamp (tool)]] [[:fr:Serre-joint]] [[:it:Morsetto]] [[:de:Schraubzwinge]]
# [[Tornus]]
# [[Runcina]]
# [[Charta aspera]]
# [[Forfex]]
# [[Cochleatorstrum]]
# [[Scopae]]
# [[Regula (instrumentum)]]
# [[Scalae portatiles]]
# [[Peniculus dentarius]]
# [[Terebra]]
# [[Incus (instrumentum)]]
# [[Complanatorium]]
# [[Gummi deletile]]
# [[Crates dentatae]]
# [[Rastrum]]
# [[:en:Machete]] [[:fr:Machette]] [[:it:Machete]] [[:de:Machete]]
# [[Furca]]
# [[Aratrum]]
# [[Pecten (instrumentum)]]
# [[:en:Roller (agricultural tool)]] [[:fr:Rouleau agricole]] [[:it:Rullo compattatore]] [[:de:Walze (Landtechnik)]]
# [[Falx fenatoria]]
# [[Falx]]
# [[Pala]]
# [[Nasiterna]]

{{Div col end}}

=== Technologia optica ===

{{Div col|cols=4}}
# [[Machina photographica]]
# [[Perspicillum]]
# [[Instrumentum lasericum]]
# [[:en:Lens]] [[:fr:Lentille optique]] [[:it:Lente]] [[:de:Linse (Optik)]]
# [[Microscopium]]
# [[Prisma]]
# [[Telescopium]]
# [[Holographia]]
# [[Fibra optica]]
# [[:en:Photodetector]] [[:fr:Photodétecteur]] [[:it:Fotorivelatore]] [[:de:Photodetektor]]
# [[:en:Optical illusion]] [[:fr:Illusion d'optique]] [[:it:Illusione ottica]] [[:de:Optische Täuschung]]
# [[Albedo]]
# [[:en:Depth of field]] [[:fr:Profondeur de champ]] [[:it:Profondità di campo]] [[:de:Schärfentiefe]]
# [[:en:Photodiode]] [[:fr:Photodiode]] [[:it:Fotodiodo]] [[:de:Photodiode]]
# [[Camera obscura]]
# [[Periscopium]]
# [[:en:Anti-reflective coating]] [[:fr:Traitement antireflet]] [[:it:Trattamento antiriflesso]] [[:de:Antireflexbeschichtung]]
# [[:en:Diffraction grating]] [[:fr:Réseau de diffraction]] [[:it:Reticolo di diffrazione]] [[:de:Optisches Gitter]]
# [[:en:Focal length]] [[:fr:Focale]] [[:it:Lunghezza focale]] [[:de:Brennweite]]
# [[:en:Focus (optics)]] [[:fr:Foyer (optique)]] [[:it:Fuoco (ottica)]] [[:de:Fokus]]
# [[:en:Fresnel lens]] [[:fr:Lentille de Fresnel]] [[:it:Lente di Fresnel]] [[:de:Fresnel-Linse]]
# [[Speculum]]
# [[:en:Polarizer]] [[:fr:Polariseur]] [[:it:Polarizzatore]] [[:de:Polarisator]]
# [[:en:Rayleigh scattering]] [[:fr:Diffusion Rayleigh]] [[:it:Scattering di Rayleigh]] [[:de:Rayleigh-Streuung]]
# [[:en:Dispersion (optics)]] [[:fr:Dispersion (mécanique ondulatoire)]] [[:it:Dispersione ottica]] [[:de:Dispersion (Physik)]]
# [[:en:Binoculars]] [[:fr:Jumelles]] [[:it:Binocolo]] [[:de:Fernglas]]
# [[:en:Interferometry]] [[:fr:Interférométrie]] [[:it:Interferometria]] [[:de:Interferometrie]]
# [[:en:Catadioptric system]] [[:fr:Objectif catadioptrique]] [[:de:Spiegellinsenobjektiv]]
# [[:en:Reflecting telescope]] [[:fr:Télescope réflecteur]] [[:it:Telescopio riflettore]] [[:de:Spiegelteleskop]]
# [[:en:Refracting telescope]] [[:fr:Lunette astronomique]] [[:it:Telescopio rifrattore]] [[:de:Fernrohr]]

{{Div col end}}

=== Electronica ===

{{Div col|cols=4}}
# [[Electronica]]
# [[:en:Alternating current]] [[:fr:Courant alternatif]] [[:it:Corrente alternata]] [[:de:Wechselstrom]]
# [[:en:Audio power amplifier]] [[:fr:Amplificateur audio]] [[:it:Finale di potenza]] [[:de:Endstufe]]
# [[Ampflificator operationalis]]
# [[:en:Circuit design]]
# [[:en:Direct current]] [[:fr:Courant continu]] [[:it:Corrente continua]] [[:de:Gleichstrom]]
# [[Amplificator electronicus]]
# [[Oscilloscopium]]
# [[Commutamen]]
# [[:en:Signal processing]] [[:fr:Traitement du signal]] [[:it:Teoria dei segnali]] [[:de:Signalverarbeitung]]
# [[:en:Mechatronics]] [[:fr:Mécatronique]] [[:it:Meccatronica]] [[:de:Mechatronik]]
# [[Elementum electronicum]]
# [[Antenna (radiophonia)]]
# [[Nucleus electricus]]
# [[Condensatrum]]
# [[Diodus]]
# [[LED]]
# [[:en:Flat-panel display]] [[:fr:Écran plat]] [[:it:Schermo piatto]] [[:de:Flachbildschirm]]
# [[Globulus electricus]]
# [[Inductorium]]
# [[Circuitus integratus]]
# [[:en:Cavity magnetron]] [[:fr:Magnétron]] [[:it:Magnetron]] [[:de:Magnetron]]
# [[Restitorium]]
# [[Semiconductrum]]
# [[Transformatrum]]
# [[Transistrum]]
# [[:en:Vacuum tube]] [[:fr:Tube électronique]] [[:it:Valvola termoionica]] [[:de:Elektronenröhre]]
# [[:en:Waveguide]] [[:de:Wellenleiter]]
# [[:en:Cathode-ray tube]] [[:fr:Tube cathodique]] [[:it:Tubo a raggi catodici]] [[:de:Kathodenstrahlröhre]]
# [[PCB]]
# [[:en:Electrical connector]] [[:fr:Connectique]] [[:it:Connettore elettrico]] [[:de:Steckverbinder]]
# [[Megaphonum]]
# [[:en:Power supply]] [[:fr:Alimentation électrique]] [[:it:Alimentazione elettrica]] [[:de:Labornetzteil]]
# [[:en:Switch]] [[:fr:Interrupteur]] [[:it:Interruttore]] [[:de:Schalter (Elektrotechnik)]]
# [[:en:Circuit breaker]] [[:fr:Disjoncteur]] [[:de:Leitungsschutzschalter]]
# [[:en:Fuse (electrical)]] [[:fr:Fusible (électricité)]] [[:it:Fusibile elettrico]] [[:de:Schmelzsicherung]]
# [[:en:Wire]] [[:fr:Fil métallique]] [[:it:Filo]] [[:de:Draht]]
# [[:en:Flip-flop (electronics)]] [[:fr:Bascule (circuit logique)]] [[:it:Flip-flop]] [[:de:Flipflop]]
# [[:en:Thyristor]] [[:fr:Thyristor]] [[:it:Tiristore]] [[:de:Thyristor]]

{{Div col end}}

=== Navigatio et mensura temporis ===

{{Div col|cols=4}}
# [[:en:Atomic clock]] [[:fr:Horloge atomique]] [[:it:Orologio atomico]] [[:de:Atomuhr]]
# [[:en:Celestial navigation]] [[:fr:Navigation astronomique]] [[:it:Navigazione astronomica]] [[:de:Astronomische Navigation]]
# [[Horologium]]
# [[Pyxis nautica]]
# [[:en:Dead reckoning]] [[:fr:Navigation à l'estime]] [[:it:Navigazione stimata]] [[:de:Koppelnavigation]]
# [[GPS (Systema localizationis globale)]]
# [[Gyroscopium]]
# [[Pendulum]]
# [[Radarium]]
# [[Echometrum]]
# [[Solarium]]
# [[Ars navigandi]]
# [[:en:Sextant]] [[:fr:Sextant]] [[:it:Sestante]] [[:de:Sextant]]
# [[:en:Watch]] [[:fr:Montre (horlogerie)]]
# [[:en:Azimuth]] [[:fr:Azimut]] [[:it:Azimut]] [[:de:Azimut]]
# [[Dies sideralis]]
# [[:en:Astrolabe]] [[:fr:Astrolabe]] [[:it:Astrolabio]] [[:de:Astrolabium]]
# [[:en:Water clock]] [[:fr:Horloge hydraulique]] [[:it:Orologio idraulico]] [[:de:Wasseruhr]]
# [[:en:Satellite navigation]] [[:fr:Système de positionnement par satellites]] [[:it:Sistema satellitare globale di navigazione]] [[:de:Globales Navigationssatellitensystem]]
# [[Clepsydra (horologium)]]

{{Div col end}}

=== Technologia informationis ===

{{Div col|cols=4}}

==== Res generales ====
# [[Technologia informationis]]
# [[Abacus]]
# [[Calculatrum]]
# [[Computatio]]
# [[Data (computatio)]]
# [[:en:Expert system]] [[:fr:Système expert]] [[:it:Sistema esperto]] [[:de:Expertensystem]]
# [[:en:Internationalization and localization]] [[:fr:Internationalisation et localisation]]

==== Technologia computatrorum ====
# [[Informatica]]
# [[Algorithmus]]
# [[Compilatrum]]
# [[Compressio datorum]]
# [[Structura datorum]]
# [[Deprehensio et correctio mendorum]]
# [[Quadratura (mathematica)]]
# [[:en:Search engine (computing)]] [[:fr:Moteur de recherche]] [[:it:Motore di ricerca]] [[:de:Suchmaschine]]
# [[Cryptographia]]
# [[:en:Authentication]] [[:fr:Authentification]] [[:it:Autenticazione]] [[:de:Authentifizierung]]
# [[:en:Encryption]] [[:fr:Chiffrement]] [[:de:Verschlüsselung]]
# [[Tessera (solutio)]]
# [[Programmatura computatralis]]
# [[Paradigma programmandi]]
# [[Programmatura functionalis]]
# [[Programmatura ad res directa]]
# [[Programmatura ordinata]]

==== Machinae computatrorum ====
# [[Computatrum]]
# [[Initiatio systematis]]
# [[Discus compactus]]
# [[ENIAC]]
# [[Discus fixus]]
# [[Tabula mater]]
# [[Procestrum princeps]]
# [[Memoria volatilis]]
# [[Memoria volatilis dynamica]]
# [[Memoria fixa]]
# [[Periphericum]]
# [[Interfacies utentium]]
# [[Claviatura]]
# [[Monitorium]]
# [[Quadrum crystallorum liquidorum]]
# [[Mus (computatralis)]]
# [[Quadrum tactile]]

==== Programmata computatralia ====
# [[Corpus programmatum]]
# [[Programma computatrale]]
# [[Datorum repositorium]]
# [[:en:Data warehouse]] [[:fr:Entrepôt de données]] [[:it:Data warehouse]] [[:de:Data Warehouse]]
# [[GUI]]
# [[Malware]]
# [[Programma fontium apertorum]]
# [[Tabula calculatoria]]
# [[Programma editorium]]
# [[Navigatrum]]
# [[Realitas virtualis]]
# [[Systema internum]]
# [[Berkeley Software Distribution]]
# [[Linux]]
# [[Systemata interna Macintosh]]
# [[Microsoft Windows]]
# [[MS-DOS]]
# [[:en:Multics]] [[:fr:Multics]] [[:it:Multics]] [[:de:Multics]]
# [[OpenVMS]]
# [[Unix]]
# [[Lingua programmandi]]
# [[Ada (lingua programmandi)]]
# [[Lingua coactionis]]
# [[Basic]]
# [[C (lingua programmandi)]]
# [[C++]]
# [[COBOL]]
# [[FORTRAN]]
# [[Java (lingua programmandi)]]
# [[JavaScript]]
# [[Lisp]]
# [[PHP]]
# [[Python (lingua programmandi)]]
# [[SQL]]

==== Retia computatrorum ====
# [[Reticulum computatrale]]
# [[Ethernet]]
# [[Collineatorium]]
# [[Wi-Fi]]
# [[Interrete]]
# [[Cursus electronicus]]
# [[:en:Transmission Control Protocol]] [[:fr:Transmission Control Protocol]] [[:it:Transmission Control Protocol]] [[:de:Transmission Control Protocol]]
# [[Tela totius terrae]]
# [[Protocollum translationis hypertextuum]]
# [[HTML]]
# [[:en:Internet protocol suite]] [[:fr:Suite des protocoles Internet]] [[:it:Suite di protocolli Internet]] [[:de:Internetprotokollfamilie]]
# [[Machina quaesitoria]]
# [[Situs interretialis]]
# [[Vici]]

{{Div col end}}

=== Communicatio ===

{{Div col|cols=4}}
# [[:en:Radio broadcasting]] [[:it:Radio (mass media)]] [[:de:Hörfunk]]
# [[:en:Communications satellite]] [[:fr:Satellite de télécommunications]] [[:it:Telecomunicazioni satellitari]] [[:de:Kommunikationssatellit]]
# [[Photographia]]
# [[Impressio]]
# [[Typographum]]
# [[Prelum typographicum]]
# [[Editio]]
# [[Radiophonia]]
# [[:en:Amateur radio]] [[:fr:Radioamateurisme]] [[:it:Radiantismo]] [[:de:Amateurfunkdienst]]
# [[:en:Radio station]]
# [[Telecommunicatio]]
# [[Telegraphia]]
# [[Telephonum]]
# [[Televisio]]
# [[:en:Cable television]] [[:fr:Télévision par câble]] [[:it:Televisione via cavo]] [[:de:Kabelfernsehen]]
# [[:en:Satellite television]] [[:fr:Télévision par satellite]] [[:it:Televisione satellitare]] [[:de:Satellitenrundfunk]]
# [[Dactylographium]]
# [[:en:Video]] [[:fr:Vidéo]] [[:it:Video]] [[:de:Videotechnik]]
# [[:en:Video camera]] [[:it:Telecamera]] [[:de:Videokamera]]
# [[:en:Videotape]] [[:fr:Cassette vidéo]] [[:it:Videocassetta]] [[:de:Videokassette]]
# [[Coacervatio datorum]]
# [[Quadrum computatrale]]
# [[Taeniola magnetica]]
# [[Charta memorialis]]
# [[Discus opticus]]
# [[Cursus publicus]]
# [[Pittacium cursuale]]
# [[:en:Courier]] [[:fr:Coursier (métier)]] [[:it:Corriere espresso]] [[:de:Bote]]
# [[:en:Mobile device]] [[:fr:Appareil mobile]] [[:it:Dispositivo mobile]] [[:de:Mobilgerät]]
# [[Telephonum gestabile]]
# [[:en:Photocopier]] [[:fr:Photocopieur]] [[:it:Fotocopiatrice]]
# [[Media socialia]]
# [[Blog]]
# [[:en:Sound recording]] [[:fr:Enregistrement sonore]] [[:it:Registrazione sonora]] [[:de:Tonaufnahme]]
# [[Microphonum]]
# [[:en:Speech synthesis]] [[:fr:Synthèse vocale]] [[:it:Sintesi vocale]] [[:de:Sprachsynthese]]
# [[:en:Coding theory]] [[:fr:Théorie des codes]] [[:it:Teoria dei codici]] [[:de:Kodierungstheorie]]
# [[Transmissio datorum]]
# [[:en:Forward error correction]] [[:it:Forward Error Correction]] [[:de:Vorwärtsfehlerkorrektur]]
# [[:en:Modulation]] [[:fr:Modulation du signal]] [[:it:Modulazione]] [[:de:Modulation (Technik)]]
# [[Cartibulum visificum lusorium]]

{{Div col end}}

=== Technologica spatialis ===

{{Div col|cols=4}}
# [[:en:Atacama Large Millimeter Array]] [[:fr:Grand réseau d'antennes millimétrique/submillimétrique de l'Atacama]] [[:it:Atacama Large Millimeter Array]] [[:de:Atacama Large Millimeter/submillimeter Array]]
# [[Ordo Europaeus spatio cosmico investigando]]
# [[:en:Great Observatories program]] [[:fr:Programme des Grands Observatoires]] [[:it:Grandi Osservatori della NASA]]
# [[:en:Herschel Space Observatory]] [[:fr:Herschel (télescope spatial)]] [[:it:Herschel Space Observatory]] [[:de:Herschel-Weltraumteleskop]]
# [[Telescopium spatiale Hubbleanum]]
# [[Statio Spatialis Internationalis]]
# [[Telescopium spatiale Webbianum]]
# [[:en:Moon landing]] [[:fr:Alunissage]] [[:it:Allunaggio]] [[:de:Mondlandung]]
# [[Administratio nationalis aeronautica et spatialis]]
# [[Programma Apollo]]
# [[Apollo 11]]
# [[Rocheta]]
# [[Satelles artificialis]]
# [[:en:Soyuz (spacecraft)]] [[:fr:Soyouz (véhicule spatial)]] [[:it:Sojuz (veicolo spaziale)]] [[:de:Sojus (Raumschiff)]]
# [[Volatus spatialis]]
# [[Navicula sideralis]]
# [[Statio spatialis]]
# [[:en:Spitzer Space Telescope]] [[:fr:Spitzer (télescope spatial)]] [[:it:Telescopio spaziale Spitzer]] [[:de:Spitzer-Weltraumteleskop]]
# [[:en:Very Large Telescope]] [[:fr:Très Grand Télescope]] [[:it:Very Large Telescope]]
# [[:en:Wilkinson Microwave Anisotropy Probe]] [[:fr:Wilkinson Microwave Anisotropy Probe]] [[:it:WMAP]] [[:de:Wilkinson Microwave Anisotropy Probe]]
# [[Astronauta]]
# [[:en:Launch vehicle]] [[:fr:Lanceur (astronautique)]] [[:it:Vettore (astronautica)]] [[:de:Trägerrakete]]
# [[:en:Rocket engine]] [[:fr:Moteur-fusée]] [[:it:Motore a razzo]] [[:de:Raketentriebwerk]]
# [[:en:Space capsule]] [[:fr:Capsule spatiale]]
# [[:en:Spaceport]] [[:fr:Base de lancement]] [[:it:Spazioporto]] [[:de:Weltraumbahnhof]]
# [[Speculatrum spatiale]]
# [[:en:Space suit]] [[:fr:Combinaison spatiale]] [[:it:Tuta spaziale]] [[:de:Raumanzug]]
# [[Baikonur]]
# [[:en:China National Space Administration]] [[:fr:Administration spatiale nationale chinoise]] [[:it:Agenzia spaziale cinese]] [[:de:Nationale Raumfahrtbehörde Chinas]]
# [[:en:JAXA]] [[:fr:Agence d'exploration aérospatiale japonaise]] [[:it:JAXA]] [[:de:Japan Aerospace Exploration Agency]]
# [[:en:Kennedy Space Center]] [[:fr:Centre spatial Kennedy]] [[:it:John F. Kennedy Space Center]] [[:de:Kennedy Space Center]]
# [[:en:Geosynchronous orbit]] [[:fr:Orbite géosynchrone]] [[:it:Orbita geosincrona]] [[:de:Geosynchrone Umlaufbahn]]
# [[Sputnik 1]]
# [[:en:Voyager program]] [[:fr:Programme Voyager]] [[:it:Programma Voyager]] [[:de:Voyager-Programm]]
# [[:en:Proton (rocket family)]] [[:fr:Proton (fusée)]] [[:it:Proton (lanciatore)]] [[:de:Proton (Rakete)]]
# [[:en:Saturn V]] [[:fr:Saturn V]] [[:it:Saturn V]]
# [[:en:Cerro Tololo Inter-American Observatory]] [[:fr:Observatoire interaméricain du Cerro Tololo]] [[:it:Osservatorio di Cerro Tololo]] [[:de:Cerro Tololo Inter-American Observatory]]
# [[:en:Mauna Kea Observatories]] [[:fr:Observatoires du Mauna Kea]] [[:it:Osservatorio di Mauna Kea]] [[:de:Mauna-Kea-Observatorium]]
# [[Observatorium Montis Wilson]]
# [[Observatorium Palomar]]
# [[Jet Propulsion Laboratory]]
# [[:en:Ion thruster]] [[:fr:Moteur ionique]] [[:it:Propulsore ionico]] [[:de:Ionenantrieb]]
# [[Astronautica]]
# [[Vostok (programma)]]
# [[Vostok 1]]

{{Div col end}}

=== Textilia ===

{{Div col|cols=4}}

==== Fili et panni ====
# [[Corium]]
# [[Sericum]]
# [[Lana]]

{{Div col end}}

=== Vectura ===

{{Div col|cols=4}}

==== Res generalis ====
# [[:en:Cargo]] [[:fr:Cargaison]] [[:it:Carico (trasporti)]] [[:de:Frachtgut]]
# [[:en:Conveyor belt]] [[:fr:Bande transporteuse]] [[:it:Nastro trasportatore]] [[:de:Förderband]]
# [[Anabathrum]]
# [[Scalae versatiles]]
# [[:en:Intermodal container]] [[:fr:Conteneur]] [[:it:Container]] [[:de:ISO-Container]]
# [[:en:Logistics]] [[:fr:Logistique]] [[:it:Logistica]] [[:de:Logistik]]
# [[:en:Passenger]] [[:fr:Passager (transport)]] [[:it:Passeggero]] [[:de:Passagier]]
# [[:en:Pipeline transport]] [[:fr:Pipeline]] [[:it:Condotta (idraulica)]] [[:de:Rohrleitungstransport]]
# [[:en:Public transport]] [[:fr:Transport en commun]] [[:it:Trasporto pubblico]] [[:de:Öffentlicher Verkehr]]
# [[:en:Rush hour]] [[:fr:Heure de pointe]] [[:de:Verkehrszeiten]]
# [[Vectura]]
# [[Iter]]
# [[Vehiculum]]

==== Aerinavigatio ====
# [[Vehiculum aërium]]
# [[:en:Airline]] [[:fr:Compagnie aérienne]] [[:it:Compagnia aerea]] [[:de:Fluggesellschaft]]
# [[Aëroplanum]]
# [[Aëroportus]]
# [[Navis aeria]]
# [[Navigatio aerea]]
# [[Globus aerostaticus]]
# [[Helicopterum]]
# [[Draco volans]]
# [[Umbrella descensoria]]
# [[Teleplanum]]
# [[Airbus]]
# [[:en:Bombardier Aviation]] [[:fr:Bombardier Aéronautique]] [[:it:Bombardier Aerospace]] [[:de:Bombardier Aerospace]]
# [[Boeing]]
# [[:en:Embraer]] [[:fr:Embraer]] [[:it:Embraer]] [[:de:Embraer]]
# [[Tupolev]]
# [[:en:Beijing Capital International Airport]] [[:fr:Aéroport international de Pékin-Capitale]] [[:it:Aeroporto di Pechino-Capitale]] [[:de:Flughafen Peking-Hauptstadt]]
# [[:en:Hartsfield–Jackson Atlanta International Airport]] [[:fr:Aéroport international Hartsfield-Jackson d'Atlanta]] [[:it:Aeroporto Internazionale di Atlanta-Hartsfield-Jackson]] [[:de:Hartsfield–Jackson Atlanta International Airport]]
# [[:en:Los Angeles International Airport]] [[:fr:Aéroport international de Los Angeles]] [[:it:Aeroporto Internazionale di Los Angeles]] [[:de:Los Angeles International Airport]]

==== Vectura navalis ====
# [[:en:Barge]] [[:fr:Barge (bateau)]] [[:it:Chiatta]] [[:de:Schute (Schiffstyp)]]
# [[Linter]]
# [[Caudica]]
# [[Carabella]]
# [[Navis exceptaculifera]]
# [[:en:Cruise ship]] [[:fr:Navire de croisière]] [[:it:Nave da crociera]] [[:de:Kreuzfahrtschiff]]
# [[Navis traiectoria]]
# [[:en:Hovercraft]] [[:fr:Aéroglisseur]] [[:it:Hovercraft]] [[:de:Luftkissenfahrzeug]]
# [[Iuncus (navis)]]
# [[Portus]]
# [[Ratis]]
# [[Navis remivaga]]
# [[Velum (navigatio)]]
# [[:en:Sailing]] [[:fr:Navigation à voile]] [[:de:Segeln]]
# [[Navis]]
# [[Navis cisternata]]
# [[Navigium]]

==== Vectura in terra ====

===== Vectura per ferrivias =====
# [[:en:High-speed rail]] [[:fr:Grande vitesse ferroviaire]] [[:it:Alta velocità ferroviaria]] [[:de:Hochgeschwindigkeitsverkehr]]
# [[Currus tractorius]]
# [[Vaporitraha]]
# [[:en:Maglev]] [[:fr:Train à sustentation magnétique]] [[:it:Treno a levitazione magnetica]] [[:de:Magnetschwebebahn]]
# [[:en:Rail transport]] [[:fr:Transport ferroviaire]] [[:it:Trasporto ferroviario]] [[:de:Bahn (Verkehr)]]
# [[:en:Railway signal]] [[:fr:Signalisation ferroviaire]] [[:it:Segnale ferroviario]] [[:de:Eisenbahnsignal]]
# [[Ferrivia]]
# [[Tramen]]
# [[Statio ferriviaria]]
# [[Currus electricus]]
# [[Ferrivia subterranea]]
# [[Ferrivia subterranea Londiniensis]]
# [[Ferrivia subterranea Moscuensis]]
# [[Ferrivia metropolitana Neo-Eboracensis]]
# [[Ferrivia metropolitana Parisiensis]]
# [[Ferrivia Transsibirica]]

===== Vectura per vias =====
# [[:en:Auto rickshaw]] [[:fr:Tuk-tuk]] [[:it:Tuk-tuk]] [[:de:Autorikscha]]
# [[Birota]]
# [[Laophorium]]
# [[Filoraeda]]
# [[:en:Bus station]] [[:fr:Gare routière]] [[:it:Autostazione]] [[:de:Busbahnhof]]
# [[:en:Carriage]] [[:fr:Voiture (hippomobile)]] [[:it:Carrozza]] [[:de:Wagen]]
# [[Birota automataria]]
# [[Via]]
# [[Autovia]]
# [[:en:Saddle]] [[:fr:Selle (équitation)]] [[:de:Reitsattel]]
# [[Stapes]]
# [[Commeatus]]
# [[:en:Traffic collision]] [[:fr:Accident de la route]] [[:it:Incidente stradale]] [[:de:Straßenverkehrsunfall]]
# [[Signum viale]]
# [[Semita]]
# [[Autocinetum onerarium]]
# [[Autocurrus sarcinarius]]

====== Autocinetum ======
# [[Autocinetum]]
# [[:en:Automotive industry]] [[:fr:Construction automobile]] [[:it:Industria automobilistica]] [[:de:Automobilindustrie]]
# [[:en:Electric car]] [[:fr:Voiture électrique]] [[:it:Auto elettrica]] [[:de:Elektroauto]]
# [[Taxiraeda]]
# [[Ford Motor Company]]
# [[:en:General Motors]] [[:fr:General Motors]] [[:it:General Motors]] [[:de:General Motors]]
# [[Honda]]
# [[:en:Hyundai Motor Company]] [[:fr:Hyundai Motor]] [[:it:Hyundai Motor Company]] [[:de:Hyundai Motor Company]]
# [[:en:Nissan]] [[:fr:Nissan]] [[:it:Nissan Motor]] [[:de:Nissan]]
# [[:en:Renault]] [[:fr:Renault]] [[:it:Renault]] [[:de:Renault]]
# [[:en:Stellantis]] [[:fr:Stellantis]] [[:it:Stellantis]] [[:de:Stellantis]]
# [[Toyota]]
# [[Volkswagen]]
# [[:en:Ford Model T]] [[:fr:Ford T]] [[:it:Ford Model T]] [[:de:Ford Modell T]]
# [[VW Käfer]]
# [[:en:Toyota Corolla]] [[:fr:Toyota Corolla]] [[:it:Toyota Corolla]] [[:de:Toyota Corolla]]
# [[:en:VAZ-2101]] [[:fr:Lada 2101/2102]] [[:de:WAS-2101]]

{{Div col end}}

=== Arma ===

{{Div col|cols=4}}
# [[Arma]]
# [[Raeda]]
# [[:en:Stealth technology]] [[:fr:Furtivité]] [[:it:Tecnologia stealth]] [[:de:Tarnkappentechnik]]
# [[Autocurrus armatus]]
# [[:en:Torpedo]] [[:fr:Torpille]] [[:it:Siluro]] [[:de:Torpedo]]
# [[:en:War elephant]] [[:fr:Éléphant de guerre]] [[:it:Elefante da guerra]] [[:de:Kriegselefant]]
# [[:en:Ammunition]] [[:fr:Munition]] [[:it:Munizione]] [[:de:Munition]]
# [[:en:Cartridge (firearms)]] [[:it:Cartuccia (munizione)]] [[:de:Patrone (Munition)]]
# [[Pulvis pyrius]]
# [[Sagitta]]
# [[Glans (arma)]]
# [[Calibra]]
# [[:en:Body armor]] [[:fr:Armure (équipement)]] [[:it:Armatura]] [[:de:Rüstung]]
# [[:en:Bulletproof vest]] [[:fr:Gilet pare-balles]] [[:it:Giubbotto antiproiettile]] [[:de:Beschusshemmende Weste]]
# [[:en:Combat helmet]] [[:fr:Casque de combat]] [[:it:Elmetto]] [[:de:Militärische Kopfbedeckung]]
# [[Cassis]]
# [[Lorica hamata]]
# [[:en:Plate armour]] [[:fr:Plate (armure)]] [[:it:Armatura a piastre]] [[:de:Plattenpanzer]]
# [[Scutum]]
# [[Bomba]]
# [[:en:Cluster munition]] [[:fr:Arme à sous-munitions]] [[:it:Bomba a grappolo]] [[:de:Streumunition]]
# [[:en:Improvised explosive device]] [[:fr:Engin explosif improvisé]] [[:it:Ordigno esplosivo improvvisato]] [[:de:Unkonventionelle Spreng- und Brandvorrichtung]]
# [[Granata manuaria]]
# [[Mina terrestris]]
# [[Missile]]
# [[Ignis Graecus]]
# [[Cocktail Molotov]]
# [[:en:Napalm]] [[:fr:Napalm]] [[:it:Napalm]] [[:de:Napalm]]
# [[:en:Nunchaku]] [[:fr:Nunchaku]] [[:it:Nunchaku]] [[:de:Nunchaku]]
# [[:en:Bayonet]] [[:fr:Baïonnette (arme)]] [[:it:Baionetta]] [[:de:Bajonett]]
# [[:en:Battle axe]] [[:fr:Hache de guerre]] [[:it:Ascia da battaglia]] [[:de:Streitaxt]]
# [[:en:Dagger]] [[:fr:Poignard]] [[:it:Pugnale]] [[:de:Dolch]]
# [[:en:Épée]] [[:fr:Épée (escrime)]] [[:it:Spada (sport)]]
# [[Katana]]
# [[Culter]]
# [[Rapperia]]
# [[:en:Sabre]] [[:fr:Sabre]] [[:it:Sciabola]] [[:de:Säbel]]
# [[Hasta]]
# [[Gladius]]
# [[:en:Tomahawk]] [[:fr:Tomahawk (hache)]] [[:it:Tomahawk]] [[:de:Tomahawk]]
# [[Fustis]]
# [[Sclopetum]]
# [[:en:Assault rifle]] [[:fr:Fusil d'assaut]] [[:it:Fucile d'assalto]] [[:de:Sturmgewehr]]
# [[AK-47]]
# [[Pistolium]]
# [[Sclopetum polybolicum]]
# [[:en:Uzi]] [[:fr:Uzi]] [[:it:IMI Uzi]] [[:de:Uzi]]
# [[Sclopetum striatum]]
# [[Tormenta]]
# [[:en:Battering ram]] [[:fr:Bélier (machine de guerre)]] [[:it:Ariete (arma)]] [[:de:Rammbock]]
# [[Canno]]
# [[:en:Howitzer]] [[:fr:Obusier]] [[:it:Obice]] [[:de:Haubitze]]
# [[:en:Mortar (weapon)]] [[:fr:Mortier (arme)]] [[:it:Mortaio]] [[:de:Mörser (Geschütz)]]
# [[:en:Military aircraft]] [[:fr:Avion militaire]] [[:it:Aeromobile militare]] [[:de:Militärflugzeug]]
# [[:en:Bomber]] [[:fr:Bombardier (avion)]] [[:it:Bombardiere]] [[:de:Bomber]]
# [[Aeroplanum insectatorium]]
# [[:en:Attack aircraft]] [[:fr:Avion d'attaque au sol]] [[:it:Aereo da attacco al suolo]] [[:de:Erdkampfflugzeug]]
# [[:en:Battleship]] [[:fr:Cuirassé]] [[:it:Nave da battaglia]] [[:de:Schlachtschiff]]
# [[Navis aëroplanigera]]
# [[:en:Dreadnought]] [[:fr:Dreadnought]] [[:it:Dreadnought]] [[:de:Dreadnought]]
# [[:en:Dromon]] [[:fr:Dromon]] [[:it:Dromone]] [[:de:Dromone]]
# [[:en:Galley]] [[:fr:Galère (navire)]] [[:it:Galea]] [[:de:Galeere]]
# [[Navis submarina]]
# [[:en:Capital ship]] [[:fr:Capital ship]] [[:de:Großkampfschiff]]
# [[:en:Ship of the line]] [[:fr:Navire de ligne]] [[:it:Vascello]] [[:de:Linienschiff]]
# [[Triremis]]
# [[Navis bellica]]
# [[Ballista]]
# [[Boomerang]]
# [[Arcus (arma)]]
# [[Catapulta]]
# [[Manuballista (sagittaria)]]
# [[Mosquetum]]
# [[Sclopetum dispergens]]
# [[:en:Sling (weapon)]] [[:fr:Fronde (arme)]] [[:it:Frombola]] [[:de:Schleuder (Waffe)]]
# [[:en:Weapon of mass destruction]] [[:fr:Arme de destruction massive]] [[:it:Arma di distruzione di massa]] [[:de:Massenvernichtungswaffe]]
# [[:en:Biological weapon]] [[:fr:Arme biologique]] [[:it:Arma biologica]] [[:de:Biologische Waffe]]
# [[Arma chemica]]
# [[Arma nuclearia]]
# [[:en:Thermonuclear weapon]] [[:fr:Bombe H]] [[:it:Bomba all'idrogeno]] [[:de:Wasserstoffbombe]]

{{Div col end}}


&lt;noinclude&gt;
[[Categoria:Myrias]]
[[Categoria:Vicipaedia]]
&lt;/noinclude&gt;</text>
      <sha1>409ribxfouw8dju180li7i0w3cdozuz</sha1>
    </revision>
  </page>
  <page>
    <title>Chymatoploion</title>
    <ns>0</ns>
    <id>218686</id>
    <revision>
      <id>3697768</id>
      <parentid>2786121</parentid>
      <timestamp>2022-08-17T11:09:21Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Naves]]; +[[Categoria:Genera navium]]</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="825" xml:space="preserve">[[Imago:Sabrina I cropped.jpg|300px|thumb|Sabrina I modernum [[Handymax]] chymatoploion est.]]
'''Bulk carrier''', sive '''chymatoploion'''&lt;ref&gt;{{Fontes desiderati}}&lt;/ref&gt; (-ou ''n.'', e [[lingua Graeca|Graeco]] χύμα, &quot;fusa&quot;, et πλοῖον, &quot;navis&quot;), est [[navis]] specializata in [[vectura|vehendis]] [[merx|mercibus]] affatim. Generatim magnum [[coëfficiens molis|coëfficientem molis]] (Cb&gt;0,75) habent.

Inter genera excellunt:
* [[Metallochymatopetrelaioploion]], quod siccas merces, [[Hydrocarboneum|hydrocarbonea]] aut [[minerale|mineralia]] affatim alternatim vehit;
* [[Metalloploion]], quod graves merces velut [[minerale|mineralia]] vehit;
* [[Xeroploion]], quod siccas merces affatim vehit.

==Notae==
&lt;div class=&quot;references-small&quot;&gt;&lt;references /&gt;&lt;/div&gt;

[[Categoria:Navigatio]]
[[Categoria:Genera navium]]</text>
      <sha1>7ogv1v7168gr7nod8vpsbs7p3f8ztx7</sha1>
    </revision>
    <revision>
      <id>3697770</id>
      <parentid>3697768</parentid>
      <timestamp>2022-08-17T11:12:32Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="800" xml:space="preserve">[[Imago:Sabrina I cropped.jpg|300px|thumb|Sabrina I modernum [[Handymax]] chymatoploion est.]]
'''''Chymatoploion'''''&lt;ref&gt;{{Fontes desiderati}}&lt;br&gt;[[Anglice]] ''bulk carrier''.&lt;/ref&gt; (-u ''n.'', e [[lingua Graeca|Graeco]] χύμα, &quot;fusa&quot;, et πλοῖον, &quot;navis&quot;), est [[navis]] specializata in [[vectura|vehendis]] [[merx|mercibus]] affatim. Generatim magnum [[coëfficiens molis|coëfficientem molis]] (Cb&gt;0,75) habent.

Inter genera excellunt:
* [[Metallochymatopetrelaioploion]], quod siccas merces, [[Hydrocarboneum|hydrocarbonea]] aut [[minerale|mineralia]] affatim alternatim vehit;
* [[Metalloploion]], quod graves merces velut [[minerale|mineralia]] vehit;
* [[Xeroploion]], quod siccas merces affatim vehit.

==Notae==
&lt;references /&gt;

[[Categoria:Navigatio]]
[[Categoria:Genera navium]]</text>
      <sha1>7ya3tc0rbcrts5btpup6xu8rvzr6o8u</sha1>
    </revision>
    <revision>
      <id>3697773</id>
      <parentid>3697770</parentid>
      <timestamp>2022-08-17T11:19:32Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <minor />
      <comment>Demetrius Talpa movit paginam [[Bulk carrier]] ad [[Chymatoploion]] praeter redirectionem: mmm... </comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="800" xml:space="preserve">[[Imago:Sabrina I cropped.jpg|300px|thumb|Sabrina I modernum [[Handymax]] chymatoploion est.]]
'''''Chymatoploion'''''&lt;ref&gt;{{Fontes desiderati}}&lt;br&gt;[[Anglice]] ''bulk carrier''.&lt;/ref&gt; (-u ''n.'', e [[lingua Graeca|Graeco]] χύμα, &quot;fusa&quot;, et πλοῖον, &quot;navis&quot;), est [[navis]] specializata in [[vectura|vehendis]] [[merx|mercibus]] affatim. Generatim magnum [[coëfficiens molis|coëfficientem molis]] (Cb&gt;0,75) habent.

Inter genera excellunt:
* [[Metallochymatopetrelaioploion]], quod siccas merces, [[Hydrocarboneum|hydrocarbonea]] aut [[minerale|mineralia]] affatim alternatim vehit;
* [[Metalloploion]], quod graves merces velut [[minerale|mineralia]] vehit;
* [[Xeroploion]], quod siccas merces affatim vehit.

==Notae==
&lt;references /&gt;

[[Categoria:Navigatio]]
[[Categoria:Genera navium]]</text>
      <sha1>7ya3tc0rbcrts5btpup6xu8rvzr6o8u</sha1>
    </revision>
  </page>
  <page>
    <title>Navis oneraria chemicorum</title>
    <ns>0</ns>
    <id>218703</id>
    <revision>
      <id>3697748</id>
      <parentid>2939534</parentid>
      <timestamp>2022-08-17T10:15:07Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Naves]]; +[[Categoria:Genera navium]]</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1558" xml:space="preserve">[[Fasciculus:Fiona-swan.jpg|thumb|Navis oneraria chemicorum.]]
'''Navis oneraria chemicorum'''{{convertimus}} est navalis evolutio classicae [[navis petrolearia|navis petroleariae]] ad vehenda producta [[chemia|chemica]] indigeste.

Naves chemicorum multo sophisticatae sunt, in multas cisternas subdivisae quae varia producta vehere permittunt. Omnes cisternae habent [[Antlia (mechanica)|antliam onerationis]] et limites ad oneranda ac exoneranda producta chemica modo omnino segregato si opus sit, quia varia producta chemica incompatibilia esse possunt et, inter se attingentes, periculose reagere aut ceterum se in alias substantias vertere possint, perditis eorum specificis characteribus.

Velut navibus [[navis petrolearia|petroleariis]] et [[aerioploion|aerioploiis]], navibus chemicorum est classificatio secundum cisternas oneris et secundum ipsas naves.

Cisternae esse possunt: independentes, integrales, gravitatis et pressionis, dum naves typi 1, typi 2, typi 3 esse possunt: cuique typo 
respondet, secundum hunc ordinem, gradus protectionis semper minor, cum possibilitate vehendorum productorum cum periculositate semper minore. Cisternae systemata calefactionis oneris ad vitandas crystallizationes aut solidificationes habent.

== Notae ==
&lt;div class=&quot;references-small&quot;&gt;&lt;references /&gt;&lt;/div&gt;

== Bibliographia ==
* International Maritime Organization, ''[https://books.google.com/books?id=9hapFPcD81oC Specialized Training for Chemical Tankers]''. London: IMO, 2006.

[[Categoria:Navigatio]]
[[Categoria:Genera navium|oneraria chemicorum]]</text>
      <sha1>582pa7gdga678c2r32q1leosq78t1gi</sha1>
    </revision>
  </page>
  <page>
    <title>Navis exceptaculifera</title>
    <ns>0</ns>
    <id>218704</id>
    <revision>
      <id>3697744</id>
      <parentid>3673598</parentid>
      <timestamp>2022-08-17T10:14:11Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Naves]]; +[[Categoria:Genera navium]]</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1235" xml:space="preserve">[[Fasciculus:Container ship loading-700px.jpg|thumb|200px|Navis exceptaculifera &quot;Rita&quot;  [[Hafnia]]e exoneratur; manipulus in stega et [[exceptaculum|exceptaculorum]] strues in terra notandae sunt.]]
'''Navis exceptaculifera'''{{Convertimus}} est [[navis]] specifica ad vehenda [[exceptaculum|exceptacula]] cuius stiva crepidines aut cellas ad accipienda et ad contrudenda [[exceptaculum|exceptacula]] consonanter habet, quod operationes onerationis et exonerationis festinat. Generatim propria instrumenta onerandi, [[tolleno]]nes seu [[tolleno|grues]] non habet et diaconiae [[velocitas|velocitatem]] maiorem quam traditionales [[navis oneraria|naves onerariae]] habere solent. Praesens tendentia harum navium gigantismus est et iam designata est unitates capacitatibus maioribus quam decem milia exceptaculorum viginti [[Pes (mensura)|pedum]] seu [[teu]]a ([[lingua Anglica|Anglicum]] [[acronymum]] ''twenty foot equivalent unit''). Navis exceptaculifera etiam est in usu ad vehenda exceptacula in [[portus|portum]].

==Notae==
&lt;references /&gt;

== Nexus externi ==
* [http://www.containershipregister.nl/ Registrum navium exceptaculiferarum]

[[Categoria:Navigatio]]
[[Categoria:Genera navium|exceptaculifera]]

{{Myrias|Technologia}}</text>
      <sha1>gkuzxqt0n0jp2d63r5cxtyz7w83zion</sha1>
    </revision>
  </page>
  <page>
    <title>Navis oneraria gasis</title>
    <ns>0</ns>
    <id>218706</id>
    <revision>
      <id>3697749</id>
      <parentid>2939301</parentid>
      <timestamp>2022-08-17T10:15:20Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Naves]]; +[[Categoria:Genera navium]]</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="510" xml:space="preserve">[[Fasciculus:LNG-carrier.Galea.wmt.jpg|thumb|Navis oneraria gasis ''Galea''.]]
'''Navis oneraria gasis'''{{convertimus}} (in pluribus linguis adhibentur nomina Anglica ''gas carrier'' aut ''gas tanker'') est genus [[navis]] specialis constructionis accomodatae ad [[gas]]ia liquefacta vehenda. Cuius praecipuus character formae rotundae [[gasarium|gasaria]] super [[stega]] sunt. 

==Notae==
&lt;div class=&quot;references-small&quot;&gt;&lt;references /&gt;&lt;/div&gt;

[[Categoria:Navigatio]]
[[Categoria:Genera navium|oneraria gasis]]</text>
      <sha1>22bwpiws1fwsxa3yumeabpaz2esqqu6</sha1>
    </revision>
  </page>
  <page>
    <title>Navis epibatica</title>
    <ns>0</ns>
    <id>218785</id>
    <revision>
      <id>3697743</id>
      <parentid>2762149</parentid>
      <timestamp>2022-08-17T10:13:56Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Naves]]; +[[Categoria:Genera navium]]</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="836" xml:space="preserve">[[Fasciculus:Costserana.JPG|thumb|upright=1.4|[[Navis circumvectans]] [[Costa Serena]], societatis [[Costa Crociere]], est navis epibaticae genus.]]
'''Navis epibatica'''&lt;ref&gt;[https://archive.org/stream/TheLexiconAnglumEtLatinumByDavidMorgan/MorganAndSilvaFurmanUniversityLexicon_djvu.txt Lexicon Latinum Morgan]&lt;/ref&gt; (a [[lingua Graeca|Graeco]] ἐπιβάτης) est [[navis]] ad [[viator]]es vehendos accommodata. Sunt varia navium epibaticarum genera, inter quae communissima sunt [[navis circumvectans|naves circumvectantes]], de more accomodatae viatorum [[periegesis|periegesi]], et [[navis traiectoria|naves traiectoriae]], quae ceterum viatoribus simplicem [[vectura]]e diaconiam praebent.

==Notae==
&lt;div class=&quot;references-small&quot;&gt;&lt;references/&gt;&lt;/div&gt;

{{stipula}}

[[Categoria:Navigatio]]
[[Categoria:Genera navium|epibatica]]</text>
      <sha1>a5jh6slyta9b3004kfupj06bxt0k6x7</sha1>
    </revision>
  </page>
  <page>
    <title>Navis petrolearia</title>
    <ns>0</ns>
    <id>218845</id>
    <revision>
      <id>3697750</id>
      <parentid>3379522</parentid>
      <timestamp>2022-08-17T10:15:33Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Naves]]; +[[Categoria:Genera navium]]</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1418" xml:space="preserve">[[Fasciculus:Brosen tugboats tanker.jpg|thumb|Superpetrolearia vacua in [[portus|portum]] [[Gedanum|Genadensem]] veniens.]]
'''Navis petrolearia''' sive '''olearia'''&lt;ref&gt;[https://archive.org/stream/TheLexiconAnglumEtLatinumByDavidMorgan/MorganAndSilvaFurmanUniversityLexicon_djvu.txt Lexicon Latinum Morgan].&lt;/ref&gt; est [[navis cisternata]] accomodata ad vehenda [[petroleum]] et subfructus ([[benzinum]]). Ad transportanda alia [[liquidum|liquida]], [[navis|naves]] alias nuncupationes habent: [[navis methanaria|naves methanariae]] naves [[gasium naturale]] vehentes, [[navis butanaria|naves butanariae]] naves [[butanum]] vehentes, [[chemicoploion|chemicoploia]] naves [[chemia|chemica]] producta vehentes nuncupantur. Petroleariae etiam [[navis cisternata|naves cisternatae]] aut supercisternatae maximae nuncupantur.

Petroleariae seu oleariae fundamentale elementum in mundano [[petroleum|petrolei]] commercio repraesentant; aliquae giganteas magnitudines attingunt, veram provocationem ad [[ingeniarius|ingeniarios]] repraesentantes. Hae naves etiam notae sunt [[petrelaeocēlis|petrelaeocēlidibus]] quas petroleariae afferunt.

[[1 Septembris|Kalendis Septembribus]] anni [[2013]], [[datorum repositorium]] maritimum 17 476 petroleariarum activarum in toto orbe terrarum computavit.

== Notae ==
&lt;references /&gt;

[[Categoria:Inventiones Scotiae]]
[[Categoria:Genera navium|petrolearia]]
[[Categoria:Navigatio]]</text>
      <sha1>dvx1ovjs73174v369v1clsaa5158vw3</sha1>
    </revision>
  </page>
  <page>
    <title>Navis cisternata</title>
    <ns>0</ns>
    <id>218849</id>
    <revision>
      <id>3697742</id>
      <parentid>2827892</parentid>
      <timestamp>2022-08-17T10:13:46Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Naves]]; +[[Categoria:Genera navium]]</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1329" xml:space="preserve">'''Navis cisternata'''&lt;ref&gt;[http://es.pons.com/traducci%C3%B3n/lat%C3%ADn-alem%C3%A1n/cisternata Dictionarium Latinum Theodiscum Pons]&lt;/ref&gt; est genus [[navis oneraria]]e accomodata ad vehenda [[liquidum|liquida]] affatim in magnis [[cisterna|cisternis]] aut [[cupa|cupis]].

==Genera==
Distinguuntur haec naves secundum genus oneris vecti:
* [[Navis petrolearia|Petroleariae]] [[petroleum]] et eorum subfructus vehunt. Sunt maximae naves in toto orbe terrarum.
* [[Chemicoploion|Chemicoploia]] [[chemia|chemica]] producta vehunt et draconianis normis subiecta sunt.
* [[Navis butanaria|Butanariae]] et [[Navis methanaria|Methanariae]] [[hygraerion]] seu liquefactum [[gasium]] in cupis designatis ad resistendum magnis [[pressio]]nibus vehunt.
* [[Navis vinaria|Vinariae]], fere amissae, [[vinum]] vehebant.
* Aliquae [[navis frigorifica|naves frigorificae]], v.gr. naves [[potio malosinensis|potionem malosinensem]] vehentes, in naves cisternatas conversae sunt et refrigerationis instrumenta habent. 

==Pinacotheca==
&lt;gallery&gt;
Imago:Batillus tanker in Saint-Nazaire.jpg|Superpetrolearia Batillus
Imago:Fiona-swan.jpg|Chemicoploion Fiona Swan
Imago:Methanier aspher LNGRIVERS.jpg|Methanaria LNG Rivers
&lt;/gallery&gt;

==Notae==
&lt;references /&gt;

[[Categoria:Navigatio]]
[[Categoria:Genera navium|cisternata]]

{{Myrias|Technologia}}</text>
      <sha1>pn5mhmztar95asb99h89pa8msjdvpzx</sha1>
    </revision>
  </page>
  <page>
    <title>Studia futurorum</title>
    <ns>0</ns>
    <id>227549</id>
    <revision>
      <id>3697656</id>
      <parentid>3488045</parentid>
      <timestamp>2022-08-16T13:26:05Z</timestamp>
      <contributor>
        <username>LilyKitty</username>
        <id>18316</id>
      </contributor>
      <comment>de hypercosmos</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8092" xml:space="preserve">[[Fasciculus:PPTMooresLawai.jpg|thumb|300px|[[Lex Mooreana]] est exemplum studiorum futurorum; quod est [[statistica]] inclinationum praeteritarum et praesentium congeries, cuius finis est inclinationes futurae accurate [[extrapolatio|extrapolandae]].]]
[[Fasciculus:H G Wells pre 1922.jpg|thumb|upright|[[Herbertus Georgius Wells|H. G. Wells]] studia futura in acroasi anno [[1902]] habita primum suasit.]]
[[Fasciculus:Michio Kaku in 2012.jpg|thumb|Michio Kaku.]]
[[Fasciculus:Raymond Kurzweil Fantastic Voyage.jpg|thumb|200px|[[Raimundus Kurzweil]].]]

'''Studia futurorum,''' etiam '''studia futura,''' '''studiologia,'''{{dubsig}} et '''futurismus''' &lt;ref&gt;Sermone quotidiano tantum futura a multis exercitatoribus appellata.&lt;/ref&gt; appellata, est disciplina quae [[tempus futurum|futura]] possibilia, probabilia, potiora postulat, cum [[cosmotheoria|cosmotheoriis]] et [[mythologia|mythis]] quae ea sustinent. [[Educatio|Eruditi]] inter se disputant num haec disciplina [[ars]] aut [[scientia (ratio)|scientia]] sit. Studia futura pars [[scientia socialis|scientiarum socialium]] generatim putari possunt, [[historia]]e consimilia. Historia quidem praeteritum, futurismus futura investigat. Studia futurorum res intellegere volunt quae [[probabilitas|probabilius]] persistent, ac res quae verisimiliter commutabuntur. Haec disciplina ergo rationalem et in exemplaribus conditam praeteriti et praesentis [[intellegentia]]m petit, cum futura eventuum et inclinationum verisimilitudine.&lt;ref&gt;[http://wordnetweb.princeton.edu/perl/webwn?s=futurology&amp;sub=Search+WordNet&amp;o2=&amp;o0=1&amp;o8=1&amp;o1=1&amp;o7=&amp;o5=&amp;o9=&amp;o6=&amp;o3=&amp;o4=&amp;h= &quot;Futurology,&quot;] Wordnet Search 3.1 (Princeton University).&lt;/ref&gt; Studia futura, scientiarum physicarum (ubi ratio angustior et subtilior investigatur) dissimilia, multo maius et multiplicius systema mundanum tractant. Eorum [[methodologia]] et scientia sunt multo minus certa quam [[ratio]]nes et [[veritas|veritates]] [[scientia naturalis|scientiae naturalis]] vel adeo [[scientia socialis|scientiarum socialium]], sicut [[sociologia]], [[oeconomica]], [[scientia socialis]] et [[civilitas|scientia civilis]].

==Futuristae notabiles==
{{div col|cols=2}}
*[[Daniel Bell]]&lt;!--
*[[Petrus C. Bishop]]
*[[Nicolaus Bostrom]]--&gt;
*[[Rachel Carson]]&lt;!--
*[[Jamais Cascio]]--&gt;
*[[Arthurus C. Clarke]]&lt;ref&gt;[http://query.nytimes.com/gst/fullpage.html?sec=technology&amp;res=9E07E4DC143AF932A35757C0A9649C8B63&amp;n=Top%2fReference%2fTimes%20Topics%2fPeople%2fC%2fClarke%2c%20Arthur%20C%2e &quot;Compressed Data; On a Futurists' Forum, Money Backs Up Predictions,&quot; ''The New York Times,'' 1 Aprilis 2002].&lt;/ref&gt;&lt;!--
*[[Iacobus Dator]]
*[[Nicolaus De Santis]]
*[[Petrus Diamandis]]
*[[Mahdi Elmandjra]]
*[[Iacobus Fresco]]&lt;ref&gt;Jacque Fresco, [http://www.thevenusproject.com/ &quot;The Venus Project,&quot;] ''The Venus Project: Beyond Politics, Poverty, and War.''&lt;/ref&gt; 
*[[Georgius Friedman]]
*[[Hugo de Garis]]
*[[Jennifer M. Gidley]]
*[[Beniaminus Goertzel]]
*[[Arthurus Harkins]]--&gt;
*[[Stephanus Hawking]]&lt;ref&gt;[http://observer.guardian.co.uk/uk_news/story/0,6903,545653,00.html Alter our DNA or robots will take over, warns Hawking,] ''The Guardian,'' 2 Septembris 2001.&lt;/ref&gt;&lt;ref&gt;[http://news.bbc.co.uk/2/hi/uk_news/6158855.stm Our species must move to another planet,] BBC News, 30 Novembris 2006.&lt;/ref&gt;
*[[Aldous Huxley]]&lt;!--
*[[Sohail Inayatullah]]
*[[Mitchell Joachim]]
*[[Gulielmus Joy]]
*[[Robertus Jungk]]--&gt;
*[[Hermannus Kahn]]
*[[Michio Kaku]]
*[[Raimundus Kurzweil]]
*[[Leonardus Vincius]]
*[[Carolus Marx]]&lt;!--
*[[Maximus More]]--&gt;
*[[Georgius Orwell]]&lt;!--
*[[David Passig]]
*[[Kim Stanley Robinson]]
*[[Michael Saloff Coste]]
*[[Andreas Sandberg]]
*[[Petrus Schwartz (futurista)|Petrus Schwartz]]
*[[Ioannes Smart]]
*[[Marcus Stevenson]]--&gt;
*[[Alvin Toffler]]
*[[Iulius Verne]]&lt;!--
*[[Natasha Vita-More]]--&gt;
*[[Herbertus Georgius Wells|H. G. Wells]]&lt;!--
*[[Eliezer Yudkowsky]]--&gt;
{{div col end}}

==Libri notabiles de futuris==
*[[Petrus Diamandis|Diamandis, Peter]]. ''Abundance: The Future Is Better Than You Think.''
*[[Patricius Dixon|Dixon, Patrick]]. ''Futurewise: Six Faces of Global Change.''
*[[Iacobus Fresco|Fresco, James]]. ''Looking Forward.''
*[[Georgius Friedman|Friedman, George]]. ''The Next 100 Years: A Forecast for the 21st Century.''
*[[Albertus Gore|Gore, Albert]]. ''The Future: Six Drivers of Global Change.''
*[[Aldous Huxley|Huxley, Aldous]]. ''[[Brave New World]].''
*[[Michio Kaku|Kaku, Michio]]. ''Visions: How Science Will Revolutionize the 21st Century.''
*[[Michio Kaku|Kaku, Michio]]. ''Physics of the Impossible: A Scientific Exploration into the World of Phasers, Force Fields, Teleportation, and Time Travel.''
*[[Michio Kaku|Kaku, Michio]]. ''Physics of the Future: How Science Will Shape Human Destiny and Our Daily Lives by the Year 2100.''
*[[Michio Kaku|Kaku, Michio]]. ''The Future of the Mind: The Scientific Quest to Understand, Enhance, and Empower the Mind.''
*[[Raimundus Kurzweil|Kurzweil, Ray]]. ''The Age of Intelligent Machines''
*[[Raimundus Kurzweil|Kurzweil, Ray]]. ''The Age of Spiritual Machines|The Age of Spiritual Machines: When Computers Exceed Human Intelligence.''
*[[Raimundus Kurzweil|Kurzweil, Ray]]. ''The Singularity Is Near: When Humans Transcend Biology.''
*[[Bjørn Lomborg|Lomborg, Bjørn]]. ''The Skeptical Environmentalist.''
*[[Iacobus Lovelock|Lovelock, James]]. ''The Revenge of Gaia.''
*[[Carolus Marx|Marx, Karl]], et [[Fridericus Engels]]. ''[[Praeconium Communisticum]].''
*[[Ioannes McKay|McKay, John]]. ''An Anarchist FAQ.''
*[[Donella Meadows|Meadows, Donella]]. ''The Limits to Growth.''
*[[Iacobus Ogilvy|Ogilvy, James]]. ''Creating Better Futures: Scenario Planning As A Tool For A Better Tomorrow.''
*[[Martinus Rees|Rees, Martin]]. ''Our Final Hour.''
*[[Byron Reese|Reese, Byron]]. ''Infinite Progress: How the Internet and Technology Will End Ignorance, Disease, Poverty, Hunger, and War.''
*[[Alvinus Toffler|Toffler, Alvin]]. ''Future Shock.''
*[[Alvinus Toffler|Toffler, Alvin]]. ''The Third Wave.''
*[[Iulius Verne|Verne, Jules]]. ''Paris in the Twentieth Century.''
*[[Ioannes Zerzan|Zerzan, John]]. ''Future Primitive and Other Essays.''

===Periodica et monographa===
*''[[International Journal of Forecasting]]''
*''[[Journal of Futures Studies]]''
*''[[Technological Forecasting and Social Change]]''
*''The Futurist'' ([[World Future Society]])

==Organizationes notabiles==
{{div col|cols=2}}
*[[Futurum (situs interretialis)]]
*[[Institutum Arlingtoniense]]
*[[Institutum Hudsonianum]]
*[[Institutum Telluris]]&lt;!--Anglice: Tellus Institute--&gt;
*[[Propositum Veneris]]&lt;!--Anglice: The Venus Project--&gt;
*[[Sodalitas Romana]]
*[[Universitas Singularitatis]]&lt;!--
*[[Acceleration Studies Foundation]]
*[[Applied Foresight Network]]
*[[Association of Professional Futurists]]
*[[Global Business Network]]
*[[Global Scenario Group]]
*[[Future of Humanity Institute]]
*[[Gold Mercury International]]
*[[Long Now Foundation]]
*The Millennium Project
*[[NASA Institute for Advanced Concepts]]
*[[Project 2049 Institute]]
*[[RAND Corporation]]
*[[Strategic Foresight Group]]
*[[TechCast Project]]
*[[World Future Society]]
*[[World Futures Studies Federation]]--&gt;
{{div col end}}

{{NexInt}}
{{div col|2}}
*[[Ars praenoscendi futura]]
*[[Coniectura]]
*[[Evolutio bioculturalis]]
*[[Hypercosmos]]
*[[Meteorologia]]
*[[Nebula utilis]]
*[[Rota futurorum]]
*[[Simulatio computatralis]]
*[[Technologia opinabilis]]
*[[Temporalitas]]
*[[Transhumanismus]]&lt;!--
*[[List of emerging technologies]]
*[[Human overpopulation]]
*[[Outline of future studies]]
*[[Sustainocene]]
*The [[Human genetic engineering]], [[cyborg]] technology, and other hypothetical forms of [[future human evolution]].--&gt;
{{div col end}}

==Notae==
&lt;div class=&quot;references-small&quot;&gt;&lt;references/&gt;&lt;/div&gt;

==Nexus externi==&lt;!--
{{wikibooks|Futurology}}--&gt;
{{wikiquote}}
*{{dmoz|Society/Future|Future}}

[[Categoria:Futurologia| ]]
[[Categoria:Praedictio]]
[[Categoria:Technologia]]
[[Categoria:Theoriae]]&lt;!--
[[Category:Systems thinking]]
[[Category:Environmental economics]]
[[Category:Technology assessment]]
[[Category:Technology forecasting]]--&gt;</text>
      <sha1>rqgaruo9e3j2tlclrhoaowvwcxontap</sha1>
    </revision>
  </page>
  <page>
    <title>Pussay</title>
    <ns>0</ns>
    <id>233735</id>
    <revision>
      <id>3697769</id>
      <parentid>3442852</parentid>
      <timestamp>2022-08-17T11:10:49Z</timestamp>
      <contributor>
        <username>IacobusAmor</username>
        <id>1163</id>
      </contributor>
      <minor />
      <comment>~</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="827" xml:space="preserve">[[Fasciculus: Mairie-Pussay.jpg |thumb|[[Vici conciliabulum]].]]
[[Fasciculus: Map commune FR insee code 91511.png|thumb|upright=0.8|left|Communis tabula.]]

'''Pussay''' est [[commune]] [[Francia|Francicum]] 2011 incolarum (anno [[2012]]) [[Tabula administrativa Franciae|praefecturae]] [[Exona (praefectura Franciae)|Exonae]] in regione [[Insula Franciae]].

{{NexInt}}
*[[Index communium praefecturae Exonae|Indicem communium praefecturae Exonae]]

==Nexus externi==
{{CommuniaCat|Pussay|Pussay}}

* [http://www.mairiepussay.fr/ Huius communis pagina interretialis]
* {{INSEE commune|91511|de=Pussay}}
* [http://cassini.ehess.fr/cassini/fr/html/fiche.php?select_resultat=28239 De hoc commune apud cassini.ehess.fr.]

{{geo-stipula}}

[[Categoria:Communia praefecturae Exonae]]
[[Categoria:Loci habitati praefecturae Exonae]]</text>
      <sha1>c5cw630dxc0gjkmxsxhyrrtznwt52uh</sha1>
    </revision>
  </page>
  <page>
    <title>Danilo Medina</title>
    <ns>0</ns>
    <id>241306</id>
    <revision>
      <id>3697719</id>
      <parentid>3357953</parentid>
      <timestamp>2022-08-17T05:49:49Z</timestamp>
      <contributor>
        <username>CommonsDelinker</username>
        <id>1422</id>
      </contributor>
      <comment>Imago DM16Ago.png deleta est ex Communibus ab Fitindia. Ille hanc rationem dedit: No permission since 9 August 2022</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1563" xml:space="preserve">{{L}}
{{Capsa hominis Vicidata}}
[[Fasciculus:Presidente Lula recibe a Danilo Medina.jpg|thumb|upright=1.25|[[Lula da Silva]] [[Brasilia]]nus et Danilo Medina anno fere 2010 picti]]
 
'''Danilo Medina Sánchez''' (in Arroyo Cano iuxta [[Bohechío]] natus die [[10 Novembris]] [[1951]]) est [[politicus]] [[res publica Dominicana|rei publicae Dominicanae]] et a die [[16 Augusti]] [[2012]] praeses patriae. Cancellarios (i.e. ministros a rebus externis) selegit [[Andreas Navarro|Andream Navarro]] ab anno [[2014]] usque in [[2016]], deinde [[Michael Vargas Maldonado|Michaelem Vargas Maldonado]].

Alumnus est [[Universitas Autonoma Sancti Dominici|Universitatis Autonomae Sancti Dominici]] et [[Institutum Technologicum Sancti Dominici|Instituti Technologici Sancti Dominici]] ubi [[oeconomia|rebus oeconomicis]] studuit.

== Bibliographia ==
* &quot;[http://www.economist.com/blogs/americasview/2014/08/dominican-republic A popular president]&quot; in ''[[The Economist]]: Americas View'' (14 Augusti 2014) [situs difficilis]

== Nexus externi ==
{{CommuniaCat|Danilo Medina}}
* {{CIDOB|http://www.cidob.org/biografias_lideres_politicos/america_central_y_caribe/republica_dominicana/danilo_medina_sanchez/(language)/esl-ES|Danilo Medina Sánchez}}

{{Praesides rei publicae Dominicanae}}

{{DEFAULTSORT:Medina, Danilo}}

[[Categoria:Nati 1951]]
[[Categoria:Homines vivi]]
[[Categoria:Duces civitatum]]
[[Categoria:Politici rei publicae Dominicianae]]
[[Categoria:Alumni Universitatis Autonomae Sancti Dominici]]
[[Categoria:Alumni Instituti Technologici Sancti Dominici]]</text>
      <sha1>r7eaa4rj709kkzpjqm4vxnp1o4pduqk</sha1>
    </revision>
  </page>
  <page>
    <title>Navis vaporaria</title>
    <ns>0</ns>
    <id>254584</id>
    <revision>
      <id>3697754</id>
      <parentid>3325264</parentid>
      <timestamp>2022-08-17T10:16:31Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Naves]]; +[[Categoria:Genera navium]]</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2772" xml:space="preserve">[[Fasciculus:Columbia (steamship 1877) 03.jpg|thumb|''Columbia'' navis vaporaria.]]
[[Fasciculus:Germania (Schiff 1842).jpg|thumb|''Germania'' navis vaporaria anno [[1842]] constructa super flumen [[Rhenus|Rhenum]] apud urbem [[Colonia Agrippina|Coloniam Agrippinam]].]]
[[Fasciculus:Ds lötschberg.jpg|thumb|[[Lötschberg (navis)|''Lötschberg,'']] navis vaporaria anno [[1914]] constructa, super lacum [[Lacus Briensis|Briensem]] in [[Helvetia]].]]
'''Navis vaporaria''' est [[vehiculum]] [[navigatio]]nis cuius [[motrum]] [[vis|vi]] [[machina]]e [[Machina vaporaria|vaporariae]] agitur.

== Historia ==
Initium [[historia]]e [[navis|navium]] vaporariarum fuit constructio talis navis a [[Claudio Francisco Jouffroy d’Abbans]] anno [[1783]]. [[1788]] [[Isaac Briggs]] et [[Gulielmus Longstreet]] [[Diploma inventionis|diplomam inventionis]] navium vaporariarum receperunt. Prima a [[Robertus Fulton|Roberto Fulton]] [[ingeniaria|ingeniario]] [[Civitates Foederatae|Americano]]  anno [[1807]] constructa navis nominis ''[[North River Steam Boat]]'', postea ''Clermont'' vocata, inter urbes [[Novum Eboracum]] et [[Albania (Novum Eboracum)|Albaniam]] vecturam praebuit.

Navis ''United States'' anno 1952 ab officina navium urbis [[Newport News]] constructa' fuit navis vaporaria maximae [[tellus|orbis terrarum]] [[velocitas|velocitatis]].

== Technologia ==
Naves vaporariae in [[aqua]] aut [[propulsorium|propulsorio]] aut rota tabulata (vel rota palata) vehuntur.

== Naves ==
*''[[Stella Occidentis]]''
*[[Navis vaporaria 'Regina Maria'|''Regina Maria'']] (Anglice 'Queen Mary')
*[[RMS Titanic|''Titanic'']]
*[[Aurora (navis)|''Aurora'']]
*[[Lusitania (navis)|''Lusitania'']]

{{NexInt}}
* [http://de.wikipedia.org/wiki/Dampfschiffahrts-Gesellschaft_für_den_Nieder-_und_Mittelrhein Societas Navigationis vaporariae Rheni medii et inferioris]

== Pinacotheca ==
&lt;gallery&gt;
Clermont illustration - Robert Fulton - Project Gutenberg eText 15161.jpg|''Clermont,'' navis Roberti Fulton.
RMS Queen Mary Long Beach January 2011 view.jpg|''Queen Mary,'' navis vaporaria
Paddleboat Natchez.jpg|''Natchez,'' navis vaporaria, super flumen [[Flumen Mississippiense|Mississippium]], rota tabulata agita, apud pontes [[Crescent City Connection (pontes)|Crescent City Connection]].
CGN-GenDufour-CartePostale2.jpeg|[[Gulielmus Henricus Dufour|''General Dufour,'']] navis vaporaria, super [[Lacus Lemanus|Lacum Lemanum]] apud urbem [[Genava]]m.
DS GALLIA 011.jpg|''Gallia,'' navis vaporaria anno 1913 constructa, in lacu [[Quattuor Regionum Lacus|quattuor Regionum Helvetiae]].
&lt;/gallery&gt;

== Nexus externi ==
{{CommuniaCat|Steamboats|Navem vaporariam}}
* [http://www.beo-news.ch/dampf/inhalt.htm Index navium vaporariarum.]

[[Categoria:Genera navium|vaporaria]]

[[en:Steamship]]</text>
      <sha1>9z9uf7glbf8bwjy3zqyp63sm4qdvxsx</sha1>
    </revision>
  </page>
  <page>
    <title>Ecclesiazusae</title>
    <ns>0</ns>
    <id>256928</id>
    <revision>
      <id>3697645</id>
      <parentid>3695591</parentid>
      <timestamp>2022-08-16T12:48:41Z</timestamp>
      <contributor>
        <username>Marcus Terentius Bibliophilus</username>
        <id>2059</id>
      </contributor>
      <comment>/* Plura legere si cupis */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2581" xml:space="preserve">{{titulus italicus}}{{L1}}
[[Fasciculus:Pnyx-berg2.png|thumb|Collis ubi Ecclesia Athenensis convocatur]]
'''''Ecclesiazusae''''' (litteris Graecis ''{{lang|grc| Ἐκκλησιάζουσαι}}''), sive Latine '''''Contionatrices''''',&lt;ref&gt;Aliter ''Concionatrices'', ''Contionantes'', ''Concionantes''. ''[https://www.google.com/books/edition/Comoediae_undecim/CWZWCelPUU4C Aristophanis facetissimi Comoediae undecim]'' (Basileae: Cratander, 1532); vide paginam tituli et synopsen manu scriptam.&lt;/ref&gt; est [[comoedia]] [[Aristophanes|Aristophanis]] anno fere 392 acta.

Feminae Athenarum, [[vestimentum|vestes]] virorum gerentes, in concilium civium eunt, ubi res novas proponunt et legem perferunt. Omnes omnia communia tenebunt.

In hac fabula reperitur longissimum linguae Graecae vocabulum, [[λοπαδοτεμαχοσελαχογαλεο...]], nomen ferculi.

== Notae ==
&lt;references /&gt;

== Bibliographia ==
=== Editiones et commentarii ===
* Berglerus, Stephanus et Dukerus, Carolus Andreas. ''[https://www.google.com/books/edition/Aristophanis_Comoediae_Undecim_Graece_Et/Z1LPe6w2F1gC Aristophanis Comoediae Undecim], Graece et '''Latine'''''. Lugduni Batavorum: apud Samuelem et Ioannem Luchtmans, 1760.
* Ussher, R. G. ''Ecclesiazusae,'' commentarium. Oxoniae: Oxford University Press, 1973, ISBN 0198141912 [https://www.persee.fr/doc/antiq_0770-2817_1973_num_42_2_1727_t1_0608_0000_2 Recensio critica]
* Wilson, N. G. ''[https://www.oxfordscholarlyeditions.com/view/10.1093/actrade/9780198721819.book.1/actrade-9780198721819-work-4 Aristophanis Fabulae]'', tomus 2. Oxoniae: Oxford University Press, 2007. ISBN 9780198721802
=== Plura legere si cupis ===
*Helene P. Foley, &quot;[https://www.jstor.org/stable/269802 The &quot;Female Intruder&quot; Reconsidered: Women in Aristophanes' Lysistrata and Ecclesiazusae]&quot;, ''Classical Philology'', 1982ː 1-21 
*Edmond Lévy, &quot;[https://www.persee.fr/doc/ktema_0221-5896_1976_num_1_1_1778 Les femmes chez Aristophane]&quot;, ''Ktèma'', 1976ː 99-112
* Rothwell, Kenneth S., Jr. ''[https://www.academia.edu/20655284/Rothwell_Politics_and_Persuasion_in_Aristophanes_Ecclesiazusae_Brill_1990_ Politics and Persuasion in Aristophanes' Ecclesiazusae.]'' Lugduni Batavorum: Brill, 1990. ISBN 9004091858
== Nexus externi ==
*[https://el.wikisource.org/wiki/%CE%95%CE%BA%CE%BA%CE%BB%CE%B7%CF%83%CE%B9%CE%AC%CE%B6%CE%BF%CF%85%CF%83%CE%B1%CE%B9 Textus Graecus apud Vicifontem]
{{Aristophanis fabulae}}

{{lit-stipula}}

[[Categoria:Comoediae Graecae antiquae]]
[[Categoria:Graeciae scripta]]
[[Categoria:Aristophanes]]
[[Categoria:Scripta saeculo 4 a.C.n.]]</text>
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    <revision>
      <id>3697646</id>
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      <contributor>
        <username>Marcus Terentius Bibliophilus</username>
        <id>2059</id>
      </contributor>
      <comment>/* Nexus externi */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2711" xml:space="preserve">{{titulus italicus}}{{L1}}
[[Fasciculus:Pnyx-berg2.png|thumb|Collis ubi Ecclesia Athenensis convocatur]]
'''''Ecclesiazusae''''' (litteris Graecis ''{{lang|grc| Ἐκκλησιάζουσαι}}''), sive Latine '''''Contionatrices''''',&lt;ref&gt;Aliter ''Concionatrices'', ''Contionantes'', ''Concionantes''. ''[https://www.google.com/books/edition/Comoediae_undecim/CWZWCelPUU4C Aristophanis facetissimi Comoediae undecim]'' (Basileae: Cratander, 1532); vide paginam tituli et synopsen manu scriptam.&lt;/ref&gt; est [[comoedia]] [[Aristophanes|Aristophanis]] anno fere 392 acta.

Feminae Athenarum, [[vestimentum|vestes]] virorum gerentes, in concilium civium eunt, ubi res novas proponunt et legem perferunt. Omnes omnia communia tenebunt.

In hac fabula reperitur longissimum linguae Graecae vocabulum, [[λοπαδοτεμαχοσελαχογαλεο...]], nomen ferculi.

== Notae ==
&lt;references /&gt;

== Bibliographia ==
=== Editiones et commentarii ===
* Berglerus, Stephanus et Dukerus, Carolus Andreas. ''[https://www.google.com/books/edition/Aristophanis_Comoediae_Undecim_Graece_Et/Z1LPe6w2F1gC Aristophanis Comoediae Undecim], Graece et '''Latine'''''. Lugduni Batavorum: apud Samuelem et Ioannem Luchtmans, 1760.
* Ussher, R. G. ''Ecclesiazusae,'' commentarium. Oxoniae: Oxford University Press, 1973, ISBN 0198141912 [https://www.persee.fr/doc/antiq_0770-2817_1973_num_42_2_1727_t1_0608_0000_2 Recensio critica]
* Wilson, N. G. ''[https://www.oxfordscholarlyeditions.com/view/10.1093/actrade/9780198721819.book.1/actrade-9780198721819-work-4 Aristophanis Fabulae]'', tomus 2. Oxoniae: Oxford University Press, 2007. ISBN 9780198721802
=== Plura legere si cupis ===
*Helene P. Foley, &quot;[https://www.jstor.org/stable/269802 The &quot;Female Intruder&quot; Reconsidered: Women in Aristophanes' Lysistrata and Ecclesiazusae]&quot;, ''Classical Philology'', 1982ː 1-21 
*Edmond Lévy, &quot;[https://www.persee.fr/doc/ktema_0221-5896_1976_num_1_1_1778 Les femmes chez Aristophane]&quot;, ''Ktèma'', 1976ː 99-112
* Rothwell, Kenneth S., Jr. ''[https://www.academia.edu/20655284/Rothwell_Politics_and_Persuasion_in_Aristophanes_Ecclesiazusae_Brill_1990_ Politics and Persuasion in Aristophanes' Ecclesiazusae.]'' Lugduni Batavorum: Brill, 1990. ISBN 9004091858
== Nexus externi ==
*[https://el.wikisource.org/wiki/%CE%95%CE%BA%CE%BA%CE%BB%CE%B7%CF%83%CE%B9%CE%AC%CE%B6%CE%BF%CF%85%CF%83%CE%B1%CE%B9 Textus Graecus apud Vicifontem]
*[https://www.greekmythology.com/Plays/Aristophanes/Assemblywomen_/assemblywomen_.html ''Assemblywomen'' apud Greekmythology.com]
{{Aristophanis fabulae}}

{{lit-stipula}}

[[Categoria:Comoediae Graecae antiquae]]
[[Categoria:Graeciae scripta]]
[[Categoria:Aristophanes]]
[[Categoria:Scripta saeculo 4 a.C.n.]]</text>
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        <id>2059</id>
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      <comment>/* Plura legere si cupis */</comment>
      <model>wikitext</model>
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      <text bytes="2911" xml:space="preserve">{{titulus italicus}}{{L1}}
[[Fasciculus:Pnyx-berg2.png|thumb|Collis ubi Ecclesia Athenensis convocatur]]
'''''Ecclesiazusae''''' (litteris Graecis ''{{lang|grc| Ἐκκλησιάζουσαι}}''), sive Latine '''''Contionatrices''''',&lt;ref&gt;Aliter ''Concionatrices'', ''Contionantes'', ''Concionantes''. ''[https://www.google.com/books/edition/Comoediae_undecim/CWZWCelPUU4C Aristophanis facetissimi Comoediae undecim]'' (Basileae: Cratander, 1532); vide paginam tituli et synopsen manu scriptam.&lt;/ref&gt; est [[comoedia]] [[Aristophanes|Aristophanis]] anno fere 392 acta.

Feminae Athenarum, [[vestimentum|vestes]] virorum gerentes, in concilium civium eunt, ubi res novas proponunt et legem perferunt. Omnes omnia communia tenebunt.

In hac fabula reperitur longissimum linguae Graecae vocabulum, [[λοπαδοτεμαχοσελαχογαλεο...]], nomen ferculi.

== Notae ==
&lt;references /&gt;

== Bibliographia ==
=== Editiones et commentarii ===
* Berglerus, Stephanus et Dukerus, Carolus Andreas. ''[https://www.google.com/books/edition/Aristophanis_Comoediae_Undecim_Graece_Et/Z1LPe6w2F1gC Aristophanis Comoediae Undecim], Graece et '''Latine'''''. Lugduni Batavorum: apud Samuelem et Ioannem Luchtmans, 1760.
* Ussher, R. G. ''Ecclesiazusae,'' commentarium. Oxoniae: Oxford University Press, 1973, ISBN 0198141912 [https://www.persee.fr/doc/antiq_0770-2817_1973_num_42_2_1727_t1_0608_0000_2 Recensio critica]
* Wilson, N. G. ''[https://www.oxfordscholarlyeditions.com/view/10.1093/actrade/9780198721819.book.1/actrade-9780198721819-work-4 Aristophanis Fabulae]'', tomus 2. Oxoniae: Oxford University Press, 2007. ISBN 9780198721802
=== Plura legere si cupis ===
Simon Byl, &quot;[https://www.persee.fr/doc/rbph_0035-0818_1982_num_60_1_3361 La métis des femmes dans l'Assemblée des femmes d'Aristophane]&quot;, ''Revue belge de Philologie et d'Histoire'',  1982ː 33-40
*Helene P. Foley, &quot;[https://www.jstor.org/stable/269802 The &quot;Female Intruder&quot; Reconsidered: Women in Aristophanes' Lysistrata and Ecclesiazusae]&quot;, ''Classical Philology'', 1982ː 1-21 
*Edmond Lévy, &quot;[https://www.persee.fr/doc/ktema_0221-5896_1976_num_1_1_1778 Les femmes chez Aristophane]&quot;, ''Ktèma'', 1976ː 99-112
* Rothwell, Kenneth S., Jr. ''[https://www.academia.edu/20655284/Rothwell_Politics_and_Persuasion_in_Aristophanes_Ecclesiazusae_Brill_1990_ Politics and Persuasion in Aristophanes' Ecclesiazusae.]'' Lugduni Batavorum: Brill, 1990. ISBN 9004091858

== Nexus externi ==
*[https://el.wikisource.org/wiki/%CE%95%CE%BA%CE%BA%CE%BB%CE%B7%CF%83%CE%B9%CE%AC%CE%B6%CE%BF%CF%85%CF%83%CE%B1%CE%B9 Textus Graecus apud Vicifontem]
*[https://www.greekmythology.com/Plays/Aristophanes/Assemblywomen_/assemblywomen_.html ''Assemblywomen'' apud Greekmythology.com]
{{Aristophanis fabulae}}

{{lit-stipula}}

[[Categoria:Comoediae Graecae antiquae]]
[[Categoria:Graeciae scripta]]
[[Categoria:Aristophanes]]
[[Categoria:Scripta saeculo 4 a.C.n.]]</text>
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      <id>3697648</id>
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        <username>Marcus Terentius Bibliophilus</username>
        <id>2059</id>
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      <text bytes="2912" xml:space="preserve">{{titulus italicus}}{{L1}}
[[Fasciculus:Pnyx-berg2.png|thumb|Collis ubi Ecclesia Athenensis convocatur]]
'''''Ecclesiazusae''''' (litteris Graecis ''{{lang|grc| Ἐκκλησιάζουσαι}}''), sive Latine '''''Contionatrices''''',&lt;ref&gt;Aliter ''Concionatrices'', ''Contionantes'', ''Concionantes''. ''[https://www.google.com/books/edition/Comoediae_undecim/CWZWCelPUU4C Aristophanis facetissimi Comoediae undecim]'' (Basileae: Cratander, 1532); vide paginam tituli et synopsen manu scriptam.&lt;/ref&gt; est [[comoedia]] [[Aristophanes|Aristophanis]] anno fere 392 acta.

Feminae Athenarum, [[vestimentum|vestes]] virorum gerentes, in concilium civium eunt, ubi res novas proponunt et legem perferunt. Omnes omnia communia tenebunt.

In hac fabula reperitur longissimum linguae Graecae vocabulum, [[λοπαδοτεμαχοσελαχογαλεο...]], nomen ferculi.

== Notae ==
&lt;references /&gt;

== Bibliographia ==
=== Editiones et commentarii ===
* Berglerus, Stephanus et Dukerus, Carolus Andreas. ''[https://www.google.com/books/edition/Aristophanis_Comoediae_Undecim_Graece_Et/Z1LPe6w2F1gC Aristophanis Comoediae Undecim], Graece et '''Latine'''''. Lugduni Batavorum: apud Samuelem et Ioannem Luchtmans, 1760.
* Ussher, R. G. ''Ecclesiazusae,'' commentarium. Oxoniae: Oxford University Press, 1973, ISBN 0198141912 [https://www.persee.fr/doc/antiq_0770-2817_1973_num_42_2_1727_t1_0608_0000_2 Recensio critica]
* Wilson, N. G. ''[https://www.oxfordscholarlyeditions.com/view/10.1093/actrade/9780198721819.book.1/actrade-9780198721819-work-4 Aristophanis Fabulae]'', tomus 2. Oxoniae: Oxford University Press, 2007. ISBN 9780198721802
=== Plura legere si cupis ===
*Simon Byl, &quot;[https://www.persee.fr/doc/rbph_0035-0818_1982_num_60_1_3361 La métis des femmes dans l'Assemblée des femmes d'Aristophane]&quot;, ''Revue belge de Philologie et d'Histoire'',  1982ː 33-40
*Helene P. Foley, &quot;[https://www.jstor.org/stable/269802 The &quot;Female Intruder&quot; Reconsidered: Women in Aristophanes' Lysistrata and Ecclesiazusae]&quot;, ''Classical Philology'', 1982ː 1-21 
*Edmond Lévy, &quot;[https://www.persee.fr/doc/ktema_0221-5896_1976_num_1_1_1778 Les femmes chez Aristophane]&quot;, ''Ktèma'', 1976ː 99-112
* Rothwell, Kenneth S., Jr. ''[https://www.academia.edu/20655284/Rothwell_Politics_and_Persuasion_in_Aristophanes_Ecclesiazusae_Brill_1990_ Politics and Persuasion in Aristophanes' Ecclesiazusae.]'' Lugduni Batavorum: Brill, 1990. ISBN 9004091858

== Nexus externi ==
*[https://el.wikisource.org/wiki/%CE%95%CE%BA%CE%BA%CE%BB%CE%B7%CF%83%CE%B9%CE%AC%CE%B6%CE%BF%CF%85%CF%83%CE%B1%CE%B9 Textus Graecus apud Vicifontem]
*[https://www.greekmythology.com/Plays/Aristophanes/Assemblywomen_/assemblywomen_.html ''Assemblywomen'' apud Greekmythology.com]
{{Aristophanis fabulae}}

{{lit-stipula}}

[[Categoria:Comoediae Graecae antiquae]]
[[Categoria:Graeciae scripta]]
[[Categoria:Aristophanes]]
[[Categoria:Scripta saeculo 4 a.C.n.]]</text>
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    <revision>
      <id>3697679</id>
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      <timestamp>2022-08-16T16:24:43Z</timestamp>
      <contributor>
        <username>Bavarese</username>
        <id>43163</id>
      </contributor>
      <comment>typo</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2912" xml:space="preserve">{{titulus italicus}}{{L1}}
[[Fasciculus:Pnyx-berg2.png|thumb|Collis ubi Ecclesia Athenensis convocatur]]
'''''Ecclesiazusae''''' (litteris Graecis ''{{lang|grc| Ἐκκλησιάζουσαι}}''), sive Latine '''''Contionatrices''''',&lt;ref&gt;Aliter ''Concionatrices'', ''Contionantes'', ''Concionantes''. ''[https://www.google.com/books/edition/Comoediae_undecim/CWZWCelPUU4C Aristophanis facetissimi Comoediae undecim]'' (Basileae: Cratander, 1532); vide paginam tituli et synopsin manu scriptam.&lt;/ref&gt; est [[comoedia]] [[Aristophanes|Aristophanis]] anno fere 392 acta.

Feminae Athenarum, [[vestimentum|vestes]] virorum gerentes, in concilium civium eunt, ubi res novas proponunt et legem perferunt. Omnes omnia communia tenebunt.

In hac fabula reperitur longissimum linguae Graecae vocabulum, [[λοπαδοτεμαχοσελαχογαλεο...]], nomen ferculi.

== Notae ==
&lt;references /&gt;

== Bibliographia ==
=== Editiones et commentarii ===
* Berglerus, Stephanus et Dukerus, Carolus Andreas. ''[https://www.google.com/books/edition/Aristophanis_Comoediae_Undecim_Graece_Et/Z1LPe6w2F1gC Aristophanis Comoediae Undecim], Graece et '''Latine'''''. Lugduni Batavorum: apud Samuelem et Ioannem Luchtmans, 1760.
* Ussher, R. G. ''Ecclesiazusae,'' commentarium. Oxoniae: Oxford University Press, 1973, ISBN 0198141912 [https://www.persee.fr/doc/antiq_0770-2817_1973_num_42_2_1727_t1_0608_0000_2 Recensio critica]
* Wilson, N. G. ''[https://www.oxfordscholarlyeditions.com/view/10.1093/actrade/9780198721819.book.1/actrade-9780198721819-work-4 Aristophanis Fabulae]'', tomus 2. Oxoniae: Oxford University Press, 2007. ISBN 9780198721802
=== Plura legere si cupis ===
*Simon Byl, &quot;[https://www.persee.fr/doc/rbph_0035-0818_1982_num_60_1_3361 La métis des femmes dans l'Assemblée des femmes d'Aristophane]&quot;, ''Revue belge de Philologie et d'Histoire'',  1982ː 33-40
*Helene P. Foley, &quot;[https://www.jstor.org/stable/269802 The &quot;Female Intruder&quot; Reconsidered: Women in Aristophanes' Lysistrata and Ecclesiazusae]&quot;, ''Classical Philology'', 1982ː 1-21 
*Edmond Lévy, &quot;[https://www.persee.fr/doc/ktema_0221-5896_1976_num_1_1_1778 Les femmes chez Aristophane]&quot;, ''Ktèma'', 1976ː 99-112
* Rothwell, Kenneth S., Jr. ''[https://www.academia.edu/20655284/Rothwell_Politics_and_Persuasion_in_Aristophanes_Ecclesiazusae_Brill_1990_ Politics and Persuasion in Aristophanes' Ecclesiazusae.]'' Lugduni Batavorum: Brill, 1990. ISBN 9004091858

== Nexus externi ==
*[https://el.wikisource.org/wiki/%CE%95%CE%BA%CE%BA%CE%BB%CE%B7%CF%83%CE%B9%CE%AC%CE%B6%CE%BF%CF%85%CF%83%CE%B1%CE%B9 Textus Graecus apud Vicifontem]
*[https://www.greekmythology.com/Plays/Aristophanes/Assemblywomen_/assemblywomen_.html ''Assemblywomen'' apud Greekmythology.com]
{{Aristophanis fabulae}}

{{lit-stipula}}

[[Categoria:Comoediae Graecae antiquae]]
[[Categoria:Graeciae scripta]]
[[Categoria:Aristophanes]]
[[Categoria:Scripta saeculo 4 a.C.n.]]</text>
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  </page>
  <page>
    <title>Navis catenaria</title>
    <ns>0</ns>
    <id>262688</id>
    <revision>
      <id>3697741</id>
      <parentid>3340333</parentid>
      <timestamp>2022-08-17T10:13:34Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Naves]]; +[[Categoria:Genera navium]]</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4175" xml:space="preserve">[[Fasciculus:Toueur.jpg|thumb|Navis catenaria pontones trahens in [[Sequana]] (ineunte saeculo XX).]]

'''Navis catenaria''',{{Convertimus}} vel '''navis tractoria''',{{Convertimus}} fuit genus [[navis fluviatilis]] in extrema parte [[saeculum 19|saeculi XIX]] ineunteque [[saeculum 20|saeculo XX]] in multis [[flumen|fluminibus]] [[Europa]]eis adhibitae, quae [[catena]] [[chalybs|chalybea]] in longitudinem alvei fluminis inmersa protrahebatur.&lt;ref&gt;MacGregor 1867: 97-99; Zimmermann 1979; Friedrichson 2012: 149.&lt;/ref&gt; Quae catena vi [[machina vaporaria|machinae vaporariae]] in nave positae trahebatur. Qua ratione factum est, ut series [[ponto]]num remulcro trahi posset.&lt;ref&gt;[https://books.google.com/books?id=NyOO7aI4crUC&amp;q=%22chain+boat%22+barges&amp;dq=%22chain+boat%22+barges&amp;hl=en&amp;sa=X&amp;ei=ws0gU5vdBIiChQfb1IGwDg&amp;ved=0CHMQ6AEwCQ ''Hearst's International, Volume 3''], 1902, p. 1782.&lt;/ref&gt;&lt;ref&gt;MacGregor (1867: 97–98): &quot;The power of this chain-boat is so great that it will pull along, and that too against the rapid stream, a whole string of barges, several of them of 300 tons' burthen, while the long fleet advances steadily though slowly, and the irresistible engine works with smokeless funnels, but with groanings within, telling of tight-strained iron, and loud undertoned breaths of confined steam.&quot;&lt;/ref&gt;

Navis huius generis in [[Germania]] varie ''Kettenschleppschiff,'' ''Kettenschlepper,'' ''Kettendampfer,'' et ''Kettenschiff,'' et in [[Francia]] ''toueur'' appellabatur.&lt;ref&gt;Pilkington 1965: 33; 1969.&lt;/ref&gt;

==Notae==
&lt;references /&gt;
 
==Bibliographia==
* Friedrichson, J. (2012) ''[https://books.google.fi/books?id=8pY1pMxg8HYC&amp;dq=kettenschifffahrt&amp;source=gbs_navlinks_s Schifffahrts-Lexikon]''. BoD – Books on Demand.
* Grötschel, Theodor, et Helmut Düntzsch (2003). Betriebsmittelverzeichnis der &lt;small&gt;Kette&lt;/small&gt;: Deutsche Elbschiffahrts-Gesellschaft. In 'Ein Leben für die Schiffahrt&lt;!--sic--&gt;,'' ed. Ewald Bellingrath. Lauenburg: Schriften des Vereins zur Förderung des Lauenburger Elbschiffahrtsmuseums e.V., vol. 4.
* Kettenschleppschiffahrt (1907). In ''Lexikon der gesamten Technik und ihrer Hilfswissenschaften,'' vol. 5, ed 2a retractata, ed. Otto Lueger, 460–462. Stuttgartiae et Lipsiae: Deutsche Verlagsanstalt. [http://www.zeno.org/Lueger-1904/A/Kettenschleppschiffahrt Textus in zeno.org.]
* MacGregor, John (1867) ''The voyage alone in the yawl &quot;Rob Roy&quot;: from London to Paris, and back''. Londinii: Maranda merrill, Son and Marston. 
* McKnight, Hugh (1985) ''Cruising French Canals and Rivers.'' Seven Seas Press.
* Pilkington, Roger (1965) ''Small boat in Southern France.'' Macmillan.
* Pilkington, Roger (1969) ''Small boat to Northern Germany.'' Macmillan.
* Zimmermann, Willi (1979) Über Seil und Kettenschiffahrt, ''Beiträge zur Rheinkunde''. Rheinmuseum Koblenz.
*Schanz, Georg (1893). Die Kettenschleppschiffahrt auf dem Main. In ''Studien über die bay. Wasserstraßen,'' vol. 1. Bamberg: C.C. Buchner Verlag. [http://www.digitalis.uni-koeln.de/Schanz/schanz_index.html Textus, Library of the Seminar for Economic and Social History of the University of Cologne.]
* Suppán, Carl Victor (1902) ''Wasserstrassen und Binnenschiffahrt.'' Berlin-Grunewald: A. Troschel.&lt;!--Section: ''Dampfschiffahrt''&lt;!--sic--&gt;. (''Ketten- und Seiltauer''261/262, ''Tauereibetrieb'' 262–265, ''Auf- und Abnehmen der Kette''. 265, ''Kettenrolle mit Fingerlingen''. 266, ''Elektrische Kettenrolle'' 266, ''Vor- und Nachtheile der Tauerei'' 266–269, ''Versuche mittels endloser Kette''269/270.--&gt;
* Zesewitz, Sigbert, Helmut Düntzsch, et Theodor Grötschel (1987). ''Kettenschiffahrt.'' Berolini: VEB Verlag Technik. ISBN 3341002820.

==Nexus externi==
{{CommuniaCat|Chain ships|navem catenariam}}
{{Wikisource|de:Tauerei}}
*[http://www.youtube.com/watch?v=-KW2Qaqycpo Pellicula de navigatione in navi catenaria electrica per Cuniculum Riqueval.] Youtube.com.
*[http://www.vagus-vagrant.fr/forum/viewforum.php?f=12 De multis photographematibus navium catenariarum Francicarum.] vagus-vagrant.fr

[[Categoria:Industriae]]
[[Categoria:Instrumenta]]
[[Categoria:Navigatio]]
[[Categoria:Genera navium|catenaria]]
[[Categoria:Vectura]]</text>
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  </page>
  <page>
    <title>Usor:Andrew Dalby/Cenae notabiles</title>
    <ns>2</ns>
    <id>266950</id>
    <revision>
      <id>3697660</id>
      <parentid>3694856</parentid>
      <timestamp>2022-08-16T13:33:57Z</timestamp>
      <contributor>
        <username>Andrew Dalby</username>
        <id>1084</id>
      </contributor>
      <minor />
      <comment>/* Cena Divione die 12 Novembris 2021 data */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="71212" xml:space="preserve">= [[Oetsius|Oetsii]] ultima cena =
* Frank Maixner et al., &quot;The 5300-year-old Helicobacter pylori genome of the Iceman&quot; in ''Science'' vol. 351 (2016) pp. 162–165
* Frank Maixner et al., &quot;[https://www.cell.com/current-biology/fulltext/S0960-9822(18)30703-6 The Iceman’s Last Meal Consisted of Fat, Wild Meat, and Cereals]&quot; in ''Current Biology'' (12 Iulii 2018)
* &quot;[http://www.theguardian.com/science/2016/jan/08/otzi-the-icemans-5000-year-old-stomach-bug-sheds-light-on-human-migration Ötzi the Iceman's 5,000-year-old stomach bug sheds light on human migration]&quot; in ''[[The Guardian]]'' (8 Ianuarii 2016)

= [[Cena Ulixi ab Achille oblata]] =
* ''[[Ilias]]'' 9.202-217
* [[Ioannes Anthelmus Brillat-Savarin]], ''[[Physiologie du goût]]'' meditatio 27

= Year-long Galatian feast =
Phylarch., FGrH81F2 in Athen., Deipn. 4.150d–f

= Lovernius’s feast =
Posid., Hist. 23, fr. 170 Theiler in Athen., Deipn. 4.152d–e 

= [[Cena Lentuli Nigri aditialis]] anno 70 a.C.n. =

= Convivium quod Hengistus Vortigerno anno 449 obtulerit =
* [[Nennius]], ''[[Historia Brittonum]]'' 37 (vide [[sicera]])
* [[Galfridus Monemutensis]], ''[[De gestis Britonum]]''
* [[Wace]], ''[[Roman de Brut]]''
* [[Layamon]], ''[[Historia Britonum]]'' 7121-7170

* ''OED'' s.v. wassail
* [[Andreas Dalby|Andrew Dalby]], &quot;[https://www.researchgate.net/publication/360427392_Biogastronomica_II Biogastronomica II. When Vortigern dined with Hengist]&quot; in ''Petits propos culinaires'' no. 122 (2022) pp. 100-105

= Brugis 1374 =
[[Brugae|Brugis]] annis 1374 et 1375 [[Philippus Audax]] bonitatem vinorum comitatus et ducatus [[Burgundia]]e regibus Francorum [[Carolus VI (rex Francorum)|Carolo VI]] et Anglorum [[Eduardus III (rex Angliae)|Eduardo III]] magnifice demonstravit, inter quae Belnensis:
:&quot;hos omnes principes inter se disputantes sua in urbe Brugensi recepit eosque per totam, quam ibi consumpsere, hiemem ad convivia opipare apparata accersivit, in iisque epulis maximam vini Belnensis, [[vinum Arbosiense|Arbosiensis]] et [[vinum Castrocharolini|Castro-charolini]] suppeditavit copiam. Sic nimirum Burgundiae dux esse se suavium vinorum (tam comitatus quam ducatus) facile principem voluit convivis suis perspectum&quot;.&lt;ref&gt;''... parquoy furent tous ces princes à Bruges sans pouvoir tomber en aucun accord. Auquel lieu ils sejournerent tout cest hyver, et y tint le duc de Bourgogne grand estat, où ne furent épargnés les bons vins de Beaune, Arboys, et chasteau Charlon, voulant bien monstrer que le duc de Bourgogne est le prince des bons vins, tant au duché qu'au comté'': [[#Paradinus (1566)]]; versio Latina e [[#Salins (1702)]] p. 9 dempta, admodum emendata&lt;/ref&gt;

; Fontes antiquiores
* 1566 : &lt;span id=&quot;Paradinus (1566)&quot;&gt;&lt;/span&gt;[[Gulielmus Paradinus|Guillaume Paradin]], ''Annales de Bourgongne'' (Lugduni: par Antoine Gryphius, 1566) [https://numelyo.bm-lyon.fr/f_view/BML:BML_00GOO0100137001101351745# p. 369]
* 1702 : &lt;span id=&quot;Salins (1702)&quot;&gt;&lt;/span&gt;Ioannes Baptista de Salins, ''Defensio vini Burgundiani adversus vinum Campanum''. Parisiis {{Google Books|RSZhAAAAcAAJ|pp. 8-9}}

= 1399 =
* Rosamond Joscelyne Mitchell, ''The Medieval Feast: The Story of the Coronation Banquet of King Henry IV in Westminster Hall, 13th October 1399''. Londinii: Longmans, 1958

= Cena qua Henricus V et Catharina Valesia Londinium die 24 Februarii 1421 recepti sunt =
* [[Robertus Fabyan]], ''Chronicon''
* Dalby, ''The Shakespeare Cookbook'' p. 39

= Cenae anni 1427/8 =
* L. Jefferson, &quot;[https://oxoniensia.org/oxo_volume.php?vol=63 Neville, Babthorpe and the Serjeants: Three fifteenth-century feast menus]&quot; in ''Oxoniensia'' vol. 63 (1998) pp. 241-249

= Cenae annis 1452 et 1465 a Georgio Neville datae =
* T. Wright, J. Halliwell, edd., ''Reliquiae antiquae'' (1841-1843) vol. 1 p. 88
* R. Warner, ed., ''Antiquitates culinariae'' (1791) pp. 93-106
* L. Jefferson, &quot;[https://oxoniensia.org/oxo_volume.php?vol=63 Neville, Babthorpe and the Serjeants: Three fifteenth-century feast menus]&quot; in ''Oxoniensia'' vol. 63 (1998) pp. 241-249 (vide p. 243 nn. 18-19)

= Epulum nuptiale Mediolani ab Ioanne Trivulitio de 22 Aprilis 1487 celebratum =
* [[Angelus Politianus]], ''Epistolae'' lib. 12 [https://archive.org/details/bub_gb_ZrvUeY61aSwC/page/n173/mode/2up Textus] (verificandum)
* Paolo Morigia, ''Historia dell' antichità di Milano'' [https://archive.org/details/historiadellanti00mori Textus] (verificandum)
* David S. Waldon, &quot;From Menu to Recipe to Meal: a Renaissance wedding banquet&quot; in Harlan Walker, ed., ''The Meal: Proceedings of the Oxford Symposium on Food and Cookery, 2001'' (Prospect Books, 2002. ISBN 1903018242) {{Google Books|GsNyprRS7EIC}}

= Cenae Shakesperianae =
* Henry VIII (I.iv): [[Epulum a Thoma Wolseio celebratum (1526)|Wolsey's feast]]
* Macbeth (III.iv): [[Cena Banchoni a Macbetho oblata|supper attended by Banquo's ghost]] 
* The Winter's Tale (IV.3-4): [[Feriae tonsurae ovium (Shakesperius)|the sheep-shearing festival]]
* Romeo and Juliet (I.5): [[Epulum apud Cappellettos celebratum|supper at the Capulets']]
* Antony and Cleopatra (II.2): [[Cenae Antonio a Cleopatra datae (41 a.C.n.)|Cleopatra's dinners for Antony]]
* The Merry Wives of Windsor (off stage I.i): dinner at the Pages'
* The Tempest (III.iii): disappearing banquet
* Hamlet: Claudius and Gertrude's wedding feast and Hamlet's funeral feast in different versions of the story (Dalby, ''The Shakespeare Cookbook'' pp. 76-79)

= [[Cena Motezumae]] mense Novembri 1519 a conquisitatoribus Hispanis observata =

= Cena a Ferdinando Cortesio anno 1521 apud Coyoacán celebrata =
* Ray Sokolov, Why we eat what we eat p. 29
* [[Bernardus Díaz del Castillo]], ''[[Historia verdadera (Díaz)|La historia verdadera de la conquista de la Nueva España]]'' cap. 156 ([http://www.rae.es/sites/default/files/Aparato_de_variantes_Historia_verdadera_de_la_conquista_de_la_Nueva_Espana.pdf pp. 625-7, nota 10867 editionis interretialis Serés]

= Cena a Ferdinando Cortesio die 1 Novembris 1522 celebrata =
* Thomas 1993 pp. 580, 635-636
* &quot;Acusación ... contra Hernando Cortés, sobre haber muerto éste a su muxer ... Catalina&quot; in Torres de Mendoza et al, edd., ''Colección de documentos inéditos relativos al descubrimiento'' (Matriti, 1864-) vol. 26 [https://archive.org/details/coleccindedocum24ultrgoog/page/n301 pp. 298-351]

= Cena imperatore Carolo V a cardinale [[Laurentius Campeggius|Campeggio]] Romae mense Aprili 1536 oblatum =
* 1570 : [[Bartholomaeus Scappi|Bartolomeo Scappi]], ''Opera'' (Venetiis: Tramezzino, 1570) lib. 3 cap. 169-179 [https://archive.org/details/operavenetiascap00scap f. 320r]

= Epulae Mexicanae quibus foedus Francorum Hispanorumque anno 1538 celebratum est =
; Fons primarius
* [[Bernardus Díaz del Castillo]], ''[[Historia verdadera (Díaz)|La historia verdadera de la conquista de la Nueva España]]'' cap. 201/198 [http://www.rae.es/sites/default/files/Aparato_de_variantes_Historia_verdadera_de_la_conquista_de_la_Nueva_Espana.pdf pp. 910-918 editionis interretialis Serés]

; Eruditio
* Raymond Sokolov, ''Why We Eat What We Eat. How Columbus Changed the Way the World Eats'' (Novi Eboraci: Simon &amp; Schuster, 1991) pp. 29-31 {{Google Books|EH9LqhXuJ3QC|Paginae selectae}}

= Feriae Romae die 14 Martii 1549 ab Ioanne Bellaio de Ludovico Aurelianensi nuper nato celebratae ==
* [[Copia d'una lettera sopra la festa per il nascimento del duca d'Orliens]] (''La Sciomachie'')
ibi vide

= Pompa imperatoris Ferdinandi Pragae die 8 Novembris 1558 =
* [[Petrus Andreas Matthiolus]], ''Le solenni pompe, i superbi et gloriosi apparati, i trionfi, i fuochi et gli altri splendidi &amp; dilettevoli spettacoli fatti alla venuta dell'invittissimo imperadore Ferdinando primo ... nella real cittá di Praga ... 1558'' {{Google Books|vs1dAAAAcAAJ}}

= Cena nuptialis regis Francorum Caroli IX anno 1570 celebrata =
* Grimod, ''Manuel'' [https://archive.org/details/b28521985/page/55/mode/2up p. 55]
* Contradictory sources on the turkey {{Google Books|sQZLAAAAIAAJ|p. 407}}
* [https://archive.org/details/defaustocarol00mass/mode/2up Dinner described on penultimate page of text: no food details]

= Cena Londinii die cinerum 1583 vel 1584 celebrata =
* [[Iordanus Brunus]], [http://www.filosofico.net/cena.htm La cena delle ceneri] (1584) 
* Frances Yates, ''Giordano Bruno and the Hermetic Tradition''
* Frances Yates, &quot;The Religious Policy of Giordano Bruno&quot;
* Michel Jeanneret, ''Des Mets et des mots'' (Lutetiae: Corti, 1987) Versio Anglica: ''A Feast of Words'' (Cantabrigiae: Polity Press, 1991) pp. 191-198 

= Cena Romae Galilaeo Galilaei a Friderico Caesio die 16 Aprilis 1611 oblata =
* [https://archive.org/stream/ldpd_7219594_000#page/n451 Fons]
* {{Google Books|c3ljJpB2NM0C|pp. 253-4}}
* {{Google Books|6jkhgzgc_OQC|pp. 108-110}}
* [[:en:Giovanni Demisiani]]
* Stillman Drake, ''Galileo At Work'' (Chicago: University of Chicago Press, 1978) pp. 158-168
* [https://www-jstor-org.ezproxy2.londonlibrary.co.uk/stable/227204 JSTOR]
* Edward Rosen, ''The Naming of the Telescope''. 1947
* Cf. [https://archive.org/stream/microrevelations00carprich#page/124 De microscopio]

= Cena scabinorum urbis Sancti Maxentii die 12 Iunii 1661 habita =
* [[Ioannes Drouhet|Jean Drouhet]], ''La Moirie de Sen-Moixont''. Pictavii: par Pierre Amassard, 1661 [https://gallica.bnf.fr/ark:/12148/btv1b86120666 Textus]

= [[Cena apud Vallem Vicecomitis (1661)|Cena apud Vallem Vicecomitis]] die 17 Augusti 1661 a Francisco Vatel parata =
* Primary sources linked [https://www.chateauversailles-recherche.fr/francais/ressources-documentaires/corpus-electroniques/corpus-raisonnes/sources-des-fetes-et-des-306/la-reception-du-roi-par-nicolas.html here]
* Jean Cordey, Vaux-le-Vicomte, Paris, Albert Morancé, 1924
* Gaëlle Lafage, ''Charles Le Brun décorateur de fêtes'' (Presses universitaires de Rennes, 2015)
* &lt;span id=&quot;Wheaton (1983)&quot;&gt;&lt;/span&gt;Barbara Ketcham Wheaton, ''Savouring the Past'' (Londinii: Chatto &amp; Windus, 1983) p. 130
* [https://www.chateauversailles-recherche.fr/francais/ressources-documentaires/corpus-electroniques/corpus-raisonnes/sources-des-fetes-et-des-306/la-reception-du-roi-par-nicolas.html La réception du roi par Nicolas Fouquet à Vaux-le-Vicomte, le 17 août 1661]
* Patrice de Vogüé, &quot;[https://francearchives.fr/commemo/recueil-2011/39048 Fête de Vaux-le-Vicomte]&quot;

= cena Nanchini a {{Creanda|en|Zhou Lianggong}} 周亮工 anno 1669 data =
* [http://asiapacific.anu.edu.au/cap-events/2017-05-04/splendid-party-zhou-lianggong-1612-1672-and-his-friends-literary-nanjing-1669 A Splendid Party: Zhou Lianggong (1612–1672) and His Friends in Literary Nanjing, 1669]&quot;
* [[:zh|黃虞稷|Huang Yuji]] 黃虞稷 (1629–1691) cenam descripsit
 
= [[cena apud Cantiliacum (1671)|cena apud Cantiliacum]] die 24 Aprilis 1671 a Francisco Vatel parata =
* &lt;span id=&quot;Wheaton (1983)&quot;&gt;&lt;/span&gt;Barbara Ketcham Wheaton, ''Savouring the Past'' (Londinii: Chatto &amp; Windus, 1983) pp. 143-147
* [[Maria de Sévigné]], epistulae 24 et 26 Aprilis 1671 (Roger Duchêne, ''Madame de Sévigné: Correspondance'' [2a ed., vol. 1. Lutetiae: Gallimard, 1973] p. 1071)

= cenae 1664 1666 1725 =
* 1804 : [[Grimod de la Reynière]] et al., ''[[Almanach des gourmands]]'' vol. 2 (2a ed. 1804) [http://www.archive.org/details/almanachdesgour00costgoog pp. 65-74]

= Cena Manchu-Han anno 1694 habita =
* Yao Tinglin 姚廷遴, ''Linian ji'' 歷年記 [''Annals''] (Shanghai shi wenwu baoguan weiyuanhui, 1962)
* Isaac Yue, &quot;[https://brill.com/view/journals/mqyj/22/1/article-p93_5.xml?language=en The Comprehensive Manchu–Han Banquet: History, Myth, and Development]&quot; in ''Ming Qing Yanjiu'' vol. 22 (2018) pp. 93–111, vide pp. 97-98

= Epulum Manchu-Han saeculo XVIII habita =
vide {{Creanda|en|Manchu–Han Imperial Feast|Epulum Manchu-Han}} inter [[Usor:Andrew Dalby/Cibi|Cibos]]

= Cena Graeca dominae Vigée-Lebrun =

= Cena-collatio nocturna a Grimod de la Reynière die 1 Februarii 1783 data =
:''Vous êtes prié d'assister au souper-collation de Me Alexandre-Balthazar-Laurent Grimod de La Reynière, écuyer, avocat au Parlement, membre de l'Académie des Arcades de Rome, associé libre du Musée de Paris, et rédacteur de la partie dramatique du ''Journal de Neufchâtel''; qui se fera en son domicile, rue des Champs-Elysées, paroisse de la Madeleine-de-la-Ville-l'Evéque, le premier jour du mois de février 1783. On fera son possible pour vous recevoir selon vos mérites; et, sans se flatter encore que vous soyez pleinement satisfait, on ose vous assurer dès aujourd'hui que du côté de l'huile et du cochon vous n'aurez rien à désirer. On s'assemblera à neuf heures et demie, pour souper à dix. Vous êtes instamment supplié de n'amener ni chien ni valet, le service devant être fait par des servantes ad hoc''.&lt;ref&gt;''Correspondance littéraire, philosophique et critique de Grimm'' vol. 11, 1782/1783 (1830) [https://archive.org/details/correspondancel02chaugoog/page/n380/mode/2up p. 364]; [[#Desnoiresterres (1877)]] pp. 73-74&lt;/ref&gt;
* ''Correspondance littéraire, philosophique et critique de Grimm'' vol. 11, 1782/1783 (1830) [https://archive.org/details/correspondancel02chaugoog/page/n378/mode/2up pp. 363-366]
* ''Mémoires secrets'' vol. 22 (1783) {{Google Books|ol8HAAAAQAAJ|pp. 72-73, 76-79, 81-83}} 
* ''Correspondance secrète, politique et littéraire'' vol. 14 (Londinii: John Adamson, 1783) {{Google Books|uzsVAAAAQAAJ|pp. 137-138}}
* Pierre-Jean-Baptiste Nougaret, ''Tableau mouvant de Paris ou Variétés amusantes'' (Londinii, 1787) vol. 1 [https://gallica.bnf.fr/ark:/12148/bpt6k6438400v/f315 pp. 295-297]
*: ''Cet Olibrius, écrit La Reynière à Rétif, ne s'est-il pas avisé de parler de mon souper? Il a copié les Mémoires secrets, et a encore ajouté mille absurdités à toutes celles qu'il y a trouvées. Vous avez aussi votre coup de patte dans sa préface: c'est le coup de pied de l'âne. Je vous gronderai toute ma vie de m'avoir amené chez moi une pareille espèce'' (fide Desnoiresterres (1877) p. 75)
* Petrus de Boisgelin de Kerdu, [[Alphonsus de Fortia de Piles]], ''Correspondance philosophique de Caillot-Duval'' (1795) no. 8 et 18, pp. [https://archive.org/details/correspondanceph00fort/page/12/mode/2up 13-15] et [https://archive.org/details/correspondanceph00fort/page/30/mode/2up 31-38] (epistula Barth)

* Samuel-Henry Berthoud, ''Les mémoires de ma cuisinière'' (1846) [http://www.blamont.info/textes944.html narratio cenae-collationis mythistorica]
* André Delrieu, &quot;Les Masques parisiens au 18e siècle&quot; in ''Revue de Paris'' (Februario 1835) {{Google Books|L3gPAAAAQAAJ|pp. 107-120 editionis Bruxellensis}}
* &lt;span id=&quot;Lacroix (1856)&quot;&gt;&lt;/span&gt;Paul Lacroix, ''Histoire des mystificateurs et des mystifiés''. Bruxellis: Lebègue, 1856 {{GB|4B0PAAAAQAAJ|fasc. 1 pp. 110-118}}
* &lt;span id=&quot;Desnoiresterres (1877)&quot;&gt;&lt;/span&gt;Gustave Desnoiresterres, ''Grimod de la Reynière et son groupe''. Paris, Didier, 1877 [https://archive.org/details/grimoddelareyni00desn pp. 73-90

= Cena die 18 Novembris 1792 apud White’s Hotel Lutetiae habita =
* Mathieu Ferradou, &quot;[https://journals.openedition.org/ahrf/13560 Histoire d’un « festin patriotique » à l’hôtel white (18 novembre 1792): les irlandais patriotes à paris, 1789-1795]&quot; in ''Annales historiques de la Révolution française'' no. 382 (2015)
* Richard Hayes, Ireland and Irishmen in the French Revolution, London, Ernest Benn Limited, 1932, y consacre un chapitre (p. 99-111) mais son utilisation cavalière des sources pose souvent problème
* Jonathan Israel, « Celebrating Modern Democracy’s Beginning : the “British Club” in Paris (1789-1793) », conférence donnée à l’Institute for Advanced Studies, Princeton, 7 mars 2012 [En ligne]
* Richard J. Hooker, « The American Revolution Seen through a Wine Glass », The William and Mary Quarterly, Third Series, vol. 11, n° 1, January 1954, p. 52-77
* Le Patriote Français (ci-après LPF), 21 novembre 1792 ; Les Nouvelles politiques, nationales et étrangères, 22 novembre 1792 ; Le Moniteur universel, 26 novembre 1792 ; The Morning Chronicle (ci-après TMC), 26 novembre 1792 ; The Manchester Herald, 1er décembre 1792 ; The Northern Star, 6 décembre 1792 ; The Dublin Evening Post, 6 décembre 1792

= Collatio Napoleoni apud ''hôtel de ville'' data =
* 1804 : [[Grimod de la Reynière]] et al., ''[[Almanach des gourmands]]'' vol. 2 (2a ed. 1804) [http://www.archive.org/details/almanachdesgour00costgoog p. 81]

= Epulum matrimoniale Hieronymi Bonaparte et Catharinae Wurtembergensis die 22 Augusti 1807 celebratum =
* 1815 : &lt;span id=&quot;Carême (1815)&quot;&gt;&lt;/span&gt;[[Maria Antonius Carême|M. A. Carême]], ''Le Pâtissier royal parisien'' [https://gallica.bnf.fr/ark:/12148/bpt6k852237p/f12 vol. 2 pp. 4-6]
* 1822 : &lt;span id=&quot;Carême (1822)&quot;&gt;&lt;/span&gt;[[Maria Antonius Carême|M. A. Carême]], ''Le Maître d'hôtel français'' Vol. 1 pp. [https://gallica.bnf.fr/ark:/12148/bpt6k1040003h/f106 78-88] 
* 1828 : &lt;span id=&quot;Carême (1828)&quot;&gt;&lt;/span&gt;[[Maria Antonius Carême|M. A. Carême]], ''Le Cuisinier parisien, ou l'art de la cuisine française au XIXe siècle'' (2a ed. 1828) pp. [https://gallica.bnf.fr/ark:/12148/btv1b86172102/f61 53-55]
** De Riquette vide Carême (1815) vol. 2 p. 5
= ''Grands bals de 1810 et 1811'' =
* 1815 : &lt;span id=&quot;Carême (1815)&quot;&gt;&lt;/span&gt;[[Maria Antonius Carême|M. A. Carême]], [https://gallica.bnf.fr/ark:/12148/bpt6k852237p/f458 vol. 2 pp. 374 ff.]

= Epulum matrimoniale Napoleonis et Mariae Ludovicae Austriacae anno 1810 celebratum =
* &lt;span id=&quot;Carême (1815)&quot;&gt;&lt;/span&gt;[[Maria Antonius Carême|M. A. Carême]], ''Le Pâtissier royal parisien'' (1815)[https://gallica.bnf.fr/ark:/12148/bpt6k852237p/f12 vol. 2 pp. 4-6]
* [[Maria Antonius Carême|M. A. Carême]], ''Le Cuisinier parisien, ou l'art de la cuisine française au XIXe siècle'' (1828) [https://gallica.bnf.fr/ark:/12148/btv1b86172102/f61 pp. 53-55]

= &quot;Obsequia&quot; sua a Grimod de la Reynière die 7 Iulii 1812 celebrata =

= Cenae sociorum victorum Durocatalauni ad [[Vertus]] mense Septembri 1815 habita =
Hic circum diem [[10 Septembris]] [[1815]] conventum campestrem rex Francorum [[Ludovicus XVIII]] imperatorque Russiae [[Alexander I (imperator Russiae)|Alexander I]] tenuerunt.&lt;ref&gt;[[Maria Antonius Carême]], ''Le Maître d'hôtel français'' (1822) [https://gallica.bnf.fr/ark:/12148/bpt6k1040004x/f148 vol. 2 pp. 110-111], [https://gallica.bnf.fr/ark:/12148/bpt6k1040004x/f165 127-133]&lt;/ref&gt;

= Convivium inaugurale Andreae Jackson die 4 Martii 1829 datum =
* [https://www.mentalfloss.com/article/61313/5-presidents-who-fought-their-right-party Brian Abrams], cf. liber eiusdem ''PARTY LIKE A PRESIDENT'' (2015)

= Cena Lutetiae dominae de Morgan a barone de Rothschild die 6 Iulii 1829 data =
* 1830 : [[Sydney, domina Morgan|Lady Morgan]], ''France in 1829-30'' (Londinii: Saunders and Otley) [https://archive.org/details/francein03sydgoog/page/n413/mode/2up vol. 2 pp. 402-420] 

* Ian Kelly, ''Cooking For Kings: The Life of Antoine Carême, the First Celebrity Chef'' (Novi Eboraci: Walker, 2004. ISBN 0-8027-1436-6) chapter 1
:''I picture to myself the perplexity and despair of the greatest cook of this century, Carême ... in the household of Baron Rothschild in Paris. All the traditions of his art assured him that soups and sauces are nothing without ham — that ham is the trumpet obligato in the symphony of a sauce — and ham was denied to him. It was due to the genius of Carême that the Baron's dinner-table became the most refined in Europe; but it did not require the genius of Carême to prove that the absence of ham in the sauce made no difference''.&lt;ref&gt;{{Creanda|en|Eneas Sweetland Dallas|Aeneas Sweetland Dallas|Eneas Sweetland Dallas}}, ''Kettner's Book of the Table'' (Londinii: Dulau, 1877) https://archive.org/details/b2153794x/page/418/mode/2up p. 418]&lt;/ref&gt;

= Prandium Helmuti de Moltke die 16 Iunii 1836 prope Prusam sumptum =
Moltke contubernalisque suus die [[16 Iunii]] [[1836]] prandium modo Turcicissimo sub [[Olympus (mons Mysiae)|Olympo Mysio]] apud ''Kiebabtschi'' (''kabābjī'', &quot;carnis veru assae venditor&quot;) pretio 120 ''para'' avido appetitu sumpserunt, videlicet chebab agninum, oliva sale condita, [[chaloe]], [[sorbet]] cum glacie.&lt;ref&gt;''Unser Mittagsmahl nahmen wir ganz türkisch beim Kiebabtschi ein; nachdem wir die Hände gewaschen, setzten wir uns nicht an, sondern auf den Tisch, wobei mir meine Beine schrecklich im Wege waren. Dann erschien auf einer hölzernen Scheibe der Kiebab oder kleine Stückchen Hammelfleisch am Spieß gebraten und in Brotteig eingewickelt, ein sehr gutes schmackhaftes Gericht; darauf eine Schüssel mit gesalzenen Oliven, die ganz vortrefflich sind, der Helwa, oder die beliebte süße Schüssel, und eine Schaale mit Scherbett (ein Aufguß von Wasser auf Trauben mit einem Stückchen Eis darin), zusammen ein Diner, welches für zwei herzhafte Esser 120 Para oder 5 Silbergroschen kostete'': {{Creanda|de|Helmuth von Moltke (Generalfeldmarschall)|Helmuthus de Moltke|Helmuth, Graf von Moltke}}, ''Briefe über Zustände und Begebenheiten in der Türkei aus den Jahren 1835 bis 1839'' (2a ed. Berolini: Mittler, 1876) {{Google Books|cXkJAAAAQAAJ|p. 66}} [https://www.projekt-gutenberg.org/moltke/halbmond/halbmo14.html recensio interretialis]&lt;/ref&gt;

&lt;references /&gt;

= Cena apud Reform Club, coquo Alexio Soyer, die 9 Maii 1846 parata =
* Soyer, Gastronomic regenerator

= Comissatio praesidis Franklini Pierce de 23 Octobris 1857 celebrata =
* [https://www.mentalfloss.com/article/61313/5-presidents-who-fought-their-right-party Brian Abrams], cf. liber eiusdem ''PARTY LIKE A PRESIDENT'' (2015)

= Cena ad honorem Alexandri Dumas a Dionysio Iosepho Vuillemot anno 1863 parata =
* François Callais, &quot;[https://histoire-compiegne.com/wp-content/uploads/BULLETINS/27-09.pdf Alexandre Dumas et Vuillemot à l'Hôtel de la Cloche]&quot; in ''Bulletin: Société de l'Histoire de Compiègne'' no. 28 (1980) pp. 183-195
* &quot;[https://archive.org/details/b28092818/page/1128/mode/2up Vuillemot (Denis-Joseph)]&quot; in &lt;span id=&quot;Dumas (1873)&quot;&gt;&lt;/span&gt;[[Alexander Dumas (pater)|Alexandre Dumas]], ''Grand Dictionnaire de cuisine'' (Lutetiae: Lemerre, 1873) pp. 1129-1134
* &quot;[https://forge-de-laguiole.com/en/post/news/histoire-coutellerie/ Un Caucasien très gascon, le couteau d’Alexandre Dumas]&quot; apud ''Forge de Laguiole''; cf. [https://www.gazette-drouot.com/lots/3466743 hic]

= Cenae praesidum Civitatum Foederatarum inaugurales 1865-2013 =
* [http://wtop.com/food-restaurant/2017/01/all-the-presidents-meals-the-history-of-inaugural-food/slide/1/ WTOP]

= Cena Alexandro Dumas e Russia redito oblata [1869] =
* 1873 : &quot;Vuillemot (Denis-Joseph)&quot; in &lt;span id=&quot;Dumas (1873)&quot;&gt;&lt;/span&gt;[[Alexander Dumas (pater)|Alexandre Dumas]], ''Grand Dictionnaire de cuisine'' (Lutetiae: Lemerre, 1873) [https://archive.org/details/b28092818/page/634/mode/2up pp. 634-635]

= Cena anarchistarum ab Iosepho Favre parata [1875/1876] =
* [[Iosephus Favre]], [[Michael Bakunin]], [[Elisaeus Reclus]], [[Henricus Malatesta]], [[Arthurus Arnould]]
* [[Pudingum montis Salvatoris]]
* 1905 : [[Iosephus Favre]], ''Dictionnaire universel de cuisine pratique'' (2a ed.) [https://gallica.bnf.fr/ark:/12148/bpt6k57300438/f267 vol. 4 pp. 1646-1647]

= Cenae societatis Gridiron 1885- =
* [[:en:Gridiron Club]]
* [https://www.theguardian.com/us-news/2018/mar/04/donald-trump-gridiron-dinner-jared-kushner-melania 2018 Guardian]

= Cena inaugurationis viae Antonii Carême Lutetiae die 21 Iunii 1894 habita =
* [[Iosephus Favre]], [[Aemilius Bernard]]
Emile Darenne, &quot;Carême: inauguration de la rue Antoine-Carême&quot; in Joseph Favre, ed., ''Dictionnaire universel de cuisine pratique'' (2a ed. 1905) [https://gallica.bnf.fr/ark:/12148/bpt6k57317645 vol. 2 pp. 488-491]

= Cena abscessionis sesquicentennialis Universitatis Princetoniensis peractis die 22 Octobris 1896 =
* [https://archive.org/stream/memorialbookofse00prin#page/n217/mode/2up liber]

= Cena natalicia Franklini Roosevelt anno 1934 data =
* [https://www.mentalfloss.com/article/61313/5-presidents-who-fought-their-right-party Brian Abrams], cf. liber eiusdem ''PARTY LIKE A PRESIDENT'' (2015)

= Cena Coloniae Allobrogum apud villam Deodati a Diana Cooper die 17 Septembris 1938 data =
* Diana Cooper, ''The light of common day''

= Cena iuvenum victorum die 30 Iulii 1945 =
* [https://books.google.fr/books?id=2fHXCQAAQBAJ&amp;pg=PT60&amp;lpg=PT60&amp;dq=Young+Victors+hugh+dalton+dinner&amp;source=bl&amp;ots=MHaEkv0fI0&amp;sig=JvxRzhbi8pnybRcVXoIgCq44TCo&amp;hl=en&amp;sa=X&amp;ved=0ahUKEwjf4JLXxerTAhUJ1RoKHVlWA5UQ6AEIJzAA#v=onepage&amp;q=Young%20Victors%20hugh%20dalton%20dinner&amp;f=false Hugh Purcell]
* Pimlott, Ben &quot;Harold Wilson&quot; Harper Collins (1993) p. 93
* [[:en:Evan Durbin]]

= [[:en:Laffer curve]] apud Two Continents Restaurant anno 1974 monstratum =
* [http://www.heritage.org/taxes/report/the-laffer-curve-past-present-and-future Laffer ipse]
* [https://www.nytimes.com/2017/10/13/us/politics/arthur-laffer-napkin-tax-curve.html New York Times]

= [[:fr:Soupe aux truffes noires VGE|Cena die 25 Februarii 1975 qua Pauli Bocuse receptio in Legionem honoris celebrata est]] =
* Cf. [http://premium.lefigaro.fr/mon-figaro/2013/11/08/10001-20131108ARTFIG00490-bernard-vaussion-le-chef-du-palais.php Le Figaro]

= Cena praemiorum Caesarum =
* [http://madame.lefigaro.fr/celebrites/soiree-des-cesar-diner-au-fouquets-penelope-cruz-marion-cotillard-vanessa-paradis-lanniversaire-de-javier-bardem-on-y-etait-030318-147592 43e 2018 Le figaro]
* [http://premium.lefigaro.fr/lifestyle/2018/03/02/30001-20180302ARTFIG00234-dans-les-coulisses-du-diner-des-cesar-avec-le-chef-pierre-gagnaire.php Figaro]
* [https://journalduluxe.fr/diner-cesar-fouquets-2018/ Journal du luxe]
* [https://france3-regions.francetvinfo.fr/paris-ile-de-france/paris/autre-menu-gustatif-ceremonie-cesar-2018-1433623.html FranceInfo]

= The Edge Annual Dinner, “The Billionaire’s Dinner” =
* [https://www.edge.org/the-billionaires-dinner Edge]

= Convivium certaminis ''Look of the Year'' in nave ''Spirit of New York'' die 1 Septembris 1991 celebratum =
* [https://www.theguardian.com/us-news/2020/mar/14/teen-models-powerful-men-when-donald-trump-hosted-look-of-the-year The Guardian]

= Cena Galfrido Epstein a Donaldo Trump anno 1992 data =
* [https://www.nytimes.com/2019/07/09/us/politics/trump-epstein.html New York Times]
* [https://www.thedailybeast.com/trump-hosted-party-with-jeffrey-epstein-and-28-girls-report-claims Daily Beast]
* [https://www.theguardian.com/us-news/2019/jul/17/trump-jeffrey-epstein-party-footage-surfaces Guardian] et [https://www.youtube.com/watch?v=ad1ysX2iLmA YouTube] (eodem anno, haud eodem die)

= [[:en:Blair–Brown deal]] apud tabernam Granita die 31 Maii 1994 =
* [http://www.independent.co.uk/news/uk/politics/granita-a-byword-for-the-pact-that-has-hung-over-new-labour-for-a-decade-107817.html Independent]
* [http://www.telegraph.co.uk/news/uknews/1420356/New-Labour-pact-over-dinner-for-two-has-left-a-bitter-taste.html Telegraph]
* [http://www.telegraph.co.uk/news/uknews/1432329/The-night-that-power-was-on-the-menu.html Telegraph]
* [https://www.theguardian.com/politics/2003/jun/06/labour.uk Guardian]
* contra [https://www.theguardian.com/politics/2013/sep/19/tony-blair-gordon-brown BBC]
* cf. [https://www.theguardian.com/politics/2013/sep/19/tony-blair-gordon-brown Guardian]
* cf. [http://www.telegraph.co.uk/news/2017/11/05/gordon-brown-tony-blair-suggested-would-stand-pm-five-years/ Telegraph]

= Cena ducibus civitatum ASEAN oblata =
* XXVII die 21 Novembris 2015 Kuala Lumpur [https://www.thestar.com.my/news/nation/2015/11/21/asean-summit-leaders-feted-at-gala-dinner/ The Star]
** [http://www.mondodr.com/27th-asean-summit-gala-dinner/ Mondo DR]
* XXVIII/XXIX die 7 Septembris 2016 Vientiane [http://www.straitstimes.com/asia/se-asia/in-pictures-asean-summit-gala-dinner Straits Times]
** [http://globalnation.inquirer.net/144240/duterte-obama-briefly-talk-before-asean-dinner Inquirer]
* XXX die 29 Aprilis 2017 Manila [http://globalnation.inquirer.net/144240/duterte-obama-briefly-talk-before-asean-dinner Inquirer]
** [https://www.youtube.com/watch?v=ZWVNkJ_5IQA YouTube]
* XXXI die 12 Novembris 2017 Manila [http://www.reuters.com/article/us-asean-summit/gala-glitz-masks-asias-tensions-as-trump-winds-up-tour-idUSKBN1DC07P Reuters]
** [http://news.abs-cbn.com/life/11/12/17/look-whats-on-the-menu-for-the-asean-gala-dinner ABS/CBN]
** [http://globalnation.inquirer.net/161621/chef-jessie-sofitel-prep-filipino-asian-feast-asean-gala-dinner Inquirer]
** [https://www.rappler.com/life-and-style/best-eats/187894-chef-jessie-sincioco-sofitel-cater-asean-2017-gala-dinner Rappler]

= Vigilium Omnium Sanctorum anno 2000 Novi Eboraci ab Heidi Klum celebratum =
* [https://www.dailymail.co.uk/news/article-2900787/Prince-Andrew-Heidi-Klum-Hookers-Pimps-party-New-York-socialite-accused-procuring-underage-girls-billionaire-pedophile-Jeffrey-Epstein.html 2015]
* [https://www.dailymail.co.uk/news/article-7695307/Prince-Andrew-partied-Heidi-Klum-Ghislaine-Maxwell-New-York-Exclusive-photos.html Daily Mail 2019]
* princeps Andreas hic et alibi:
** [https://www.dailymail.co.uk/news/article-2900632/Pictured-Prince-Andrew-surrounded-topless-women-Thai-holiday-paedophile-billionaire-Epstein-friend-says-Duke-tits-bums-man.html Daily Mail]
** [https://www.theguardian.com/uk-news/2019/nov/18/the-party-prince-how-andrew-got-his-bad-reputation The Guardian]
** [https://www.thesun.co.uk/news/10367796/prince-andrew-party-girls-photos-saint-tropez/ The Sun]
** [https://thenewdaily.com.au/entertainment/celebrity/2019/11/19/prince-andrew-never-partied/ New Daily]
** [https://www.thedailybeast.com/inside-jeffrey-epsteins-creepy-parties-with-prince-andrew Beast Inside]
** [https://jezebel.com/who-let-prince-andrew-give-this-interview-about-jeffery-1839912130 Jezebel]
* idem vigilium aliis annis: [https://time.com/5686487/best-halloween-costumes-heidi-klum-party-2019/ Time]
** [https://www.yahoo.com/lifestyle/heidi-klum-halloween-2019-100751939.html?guccounter=1&amp;guce_referrer=aHR0cHM6Ly93d3cuZ29vZ2xlLmZyLw&amp;guce_referrer_sig=AQAAAAcisvPDx0F6H_B7HxtUrVXYiUlHWr4PxcEnw2KbVuTryanO58CSeIQj2CSfJDnsRuB6QSq570ydiCr3k6niXUGdc5V7BjJdLEpYYubHFXPHqN2DrqHrF8gSbZt8hnc-s6aNCthzHnnpj9gUlWBTSNAx2Xuc-duSlidUc8gCqzmq 2019]
** [https://www.hollywoodreporter.com/news/heidi-klums-halloween-party-supermodel-talks-evolution-epic-bash-1156051 2018]

= [[:fr:Soirée du Fouquet's du 6 mai 2007]] =
* [http://www.closermag.fr/article/nicolas-sarkozy-le-fouquet-s-cecila-m-a-fait-attendre-jusqu-a-22-heures-405524 Closermag]
* [http://leplus.nouvelobs.com/contribution/345593-les-revelations-sur-la-soiree-du-fouquet-s-annoncent-la-defaite-de-sarkozy.html Nouvel Observateur]
* [http://www.voici.fr/news-people/actu-people/pour-nicolas-sarkozy-le-diner-au-fouquet-s-a-ete-un-cauchemar-a-cause-de-cecilia-542909 Voici]

= Cenae maecenatum Bibliothecae Nationalis Francicae 2008- =
* 2008 : [http://agenda.germainpire.info/view_entry.php?id=17645&amp;date=20080623 agence]
* 2009 : [http://www.lemonde.fr/culture/article/2009/06/13/deux-cents-personnes-dinent-ensemble-pour-garder-en-france-l-oeuvre-de-debord_1206504_3246.html Le Monde]
* [http://next.liberation.fr/culture/2011/02/24/le-fonds-guy-debord-se-situe-a-la-bnf_717134 Libération]
* 2012 : [http://www.bnf.fr/documents/cp_mecenes_foucault.pdf BnF]
* [http://www.lexpress.fr/culture/livre/un-diner-a-la-bnf-pour-les-archives-de-michel-foucault_1120262.html L'Express]
* 2013 : [http://www.bnf.fr/documents/cp_mecenes_proust.pdf BnF]
* 2016 : [https://www.actualitte.com/article/patrimoine-education/la-bnf-veut-acquerir-le-manuscrit-de-nadja-d-andre-breton/65621 ActuaLitté]
* 2017 : [http://premium.lefigaro.fr/culture/2017/06/07/03004-20170607ARTFIG00289--la-bnf-ministre-et-mecenes-au-chevet-des-livres.php Le Figaro]
* [http://www.ladepeche.fr/article/2017/06/07/2589138-bnf-reunit-mecenes-renover-historique-salle-lecture.html La Dépêche]

= Cena principi Andreae a Galfrido Epstein die 2 Decembri 2010 data =
* [https://www.thedailybeast.com/katie-couric-woody-allen-jeffrey-epsteins-society-friends-close-ranks Daily Beast]
* [https://www.nbcnews.com/think/opinion/jeffrey-epstein-s-downfall-crumbling-america-s-elites-ncna1029966 Think]
* [https://www.hollywoodreporter.com/news/jeffrey-epstein-moved-freely-hollywood-circles-2008-conviction-1223336 Hollywood Reporter]
* [https://www.insider.com/report-jeffrey-epstein-threw-dinner-party-with-prince-andrew-2019-7 Insider]
* [http://www.businessinsider.fr/us/famous-people-jeffery-epstein-money-manager-sexual-trafficking-connected-2019-7 Business Insider]
* [https://www.dailymail.co.uk/news/article-7238769/Jeffrey-Epstein-threw-party-honor-Prince-Andrew-2010.html Daily Mail]
* [https://www.dailymail.co.uk/news/article-7367511/Prince-Andrew-pictured-inside-paedophile-Jeffrey-Epsteins-63million-mansion-depravity.html Daily Mail]
* [https://www.thesun.co.uk/news/9794821/prince-andrew-jeffrey-epstein-six-days/ The Sun]
* de Trump in coquina cenante
** [https://www.buzzfeednews.com/article/alexcampbell/court-papers-trump-ate-at-jeffrey-epsteins-house Buzzfeed]
* de Epstein hospite
** [http://nymag.com/nymag/features/n_8672/index2.html New York Magazine] (2 May 2003)
** [http://nymag.com/intelligencer/2019/07/jeffrey-epstein-high-society-contacts.html New York Magazine]
** [https://www.dailymail.co.uk/news/article-2900632/Pictured-Prince-Andrew-surrounded-topless-women-Thai-holiday-paedophile-billionaire-Epstein-friend-says-Duke-tits-bums-man.html Daily Mail]
** [https://www.thedailybeast.com/i-tried-to-warn-you-about-sleazy-billionaire-jeffrey-epstein-in-2003?ref=scroll Vicky Ward] in ''Daily Beast''
** [https://www.thedailybeast.com/inside-jeffrey-epsteins-creepy-parties-with-prince-andrew Beast Inside]

= Cena narratorum Domus Albae 2011- =
(et antea?)
* [https://www.washingtonpost.com/video/entertainment/trump-jokes-from-the-2011-white-house-correspondents-dinner/2016/04/27/6a4384de-0bec-11e6-bc53-db634ca94a2a_video.html?utm_term=.770e47e1e7fc 2011 et al. Washington Post]
* [https://www.washingtonpost.com/lifestyle/style/i-sat-next-to-donald-trump-at-the-infamous-2011-white-house-correspondents-dinner/2016/04/27/5cf46b74-0bea-11e6-8ab8-9ad050f76d7d_story.html?utm_term=.1ee31b43e023 2011 Washington Post] 
* [https://obamawhitehouse.archives.gov/blog/2011/05/01/president-s-speech-white-house-correspondents-dinner White House]

= Cena Francisco Hollande Berolini die 15 Maii 2012 data =
* [http://premium.lefigaro.fr/mon-figaro/2012/05/14/10001-20120514ARTFIG00655-merkel-hollande-les-secrets-d-un-premier-rendez-vous.php Le Figaro]
* [http://www.planet.fr/politique-francois-hollande-un-incident-diplomatique-a-cause-dasperges.1107557.29334.html Planet]
* Cf. [http://premium.lefigaro.fr/mon-figaro/2013/11/08/10001-20131108ARTFIG00490-bernard-vaussion-le-chef-du-palais.php Le Figaro] de Vaussion
* Cf. [http://www.planet.fr/politique-noel-a-lelysee-tous-les-presidents-voulaient-un-repas-comme-a-la-maison.746363.29334.html Planet] de Vaussion
* Cf. [http://www.huffingtonpost.fr/bernard-vaussion/le-chef-des-cuisines-de-l-elysee-est-briefe-des-le-lendemain-de_a_22081184/ Huffpost] de Vaussion

= Convivium Perugiae in palatio Terranova ab Eugenio Lebedev die 13 Octobris 2012 habitum =
* [http://www.umbria24.it/attualita/il-sindaco-di-londra-boris-johnson-in-umbria-ospite-delloligarca-evgeny-lebedev Umbria 24]
* [http://snipelondon.com/scoop/boris-johnson-flown-to-italy-by-russian-oligarch Snipe London]
* [https://www.opendemocracy.net/en/opendemocracyuk/revealed-boris-russian-oligarch-and-page-3-model/ Open Democracy] (de variis conviviis ibidem habitis)

= Cena narratorum Vestmonasteriensium 1920s-1974, 2014- =
* [https://www.ft.com/content/48e29882-7f6e-11e3-b6a7-00144feabdc0 FT]
* [http://www.bbc.com/news/uk-25771647 2014 BBC]
* [http://www.itv.com/news/2014-01-17/the-prime-ministers-priority-for-2014-hiding-his-bald-spot/ ITV]
* [http://www.independent.co.uk/voices/labour-needs-an-angry-leader-its-time-for-ed-miliband-to-go-to-war-10015910.html 2015 Independent]
* [http://www.independent.co.uk/news/uk/home-news/george-osborne-jokes-funny-parliamentary-dinner-westminster-john-whittingdale-a7006771.html 2016 Independent]
* [http://www.huffingtonpost.co.uk/entry/george-osborne-jokes_uk_572326aae4b0d6f7bed5c263 Huff]
* [https://www.politico.eu/blogs/on-media/2016/04/george-osborne-cosies-up-to-political-journalists-at-westminster-correspondents-dinner/ Politico]
* [https://order-order.com/2018/03/01/mays-best-gags-from-westminster-correspondents-dinner/ 2018 Order Order]
* [https://blogs.spectator.co.uk/2018/03/theresa-mays-westminster-correspondents-dinner-speech-cameron-rudd-and-press-commissar-milne/ Spectator]
* [https://www.theguardian.com/politics/2018/mar/01/maybot-jokes-boris-johnson-david-mad-max-davis-snap-election-westminster-correspondents-dinner Guardian]

= Cena Borisii Johnson et Michaelis Gove  die 23 Februarii 2016 habita =
* [https://www.theguardian.com/politics/2016/feb/24/boris-johnson-dinner-party-not-eu-referendum-coup Guardian]

= Comissatio Alexandri Downer et Georgii Papadopuli mense Maio 2016 =
* [https://www.theguardian.com/us-news/2017/dec/30/donald-trump-russia-inquiry-george-papadopoulos-australian-diplomat The Guardian]

= Convivium Perugiae in palatio Terranova ab Eugenio Lebedev mense Octobri 2016 habitum =
* [https://www.thesun.co.uk/news/9453136/katie-price-flashed-boobs-boris-johnson/ The Sun]
* [https://www.opendemocracy.net/en/opendemocracyuk/revealed-boris-russian-oligarch-and-page-3-model/ Open Democracy]

= Cena Donaldi Trump apud 21 Club die 15 Novembris 2016 =
** [https://www.usatoday.com/story/travel/destinations/2017/01/22/donald-trumps-new-york-favorite-restaurants/96723736/# De consuetudinibus]
** [https://www.theguardian.com/us-news/2017/dec/03/trump-mcdonalds-binges-screaming-fits-book-lewandowski-bossie De consuetudinibus]
* [http://www.dailymail.co.uk/news/article-3940430/Donald-Trump-ditches-press-eat-famous-New-York-steak-house-21-Club.html Daily Mail]
* [https://www.bloomberg.com/news/articles/2017-02-23/eat-like-donald-trump-at-his-favorite-restaurant-new-york-s-21-club Bloomberg]
* [https://ny.eater.com/2016/11/16/13649792/donald-trump-21-club Eater]
* [http://www.nydailynews.com/life-style/eats/eat-president-elect-donald-trump-21-article-1.2876384 Daily News]
* [http://www.politico.com/story/2016/11/donald-trump-press-231458 Politico]

= Cena Nigelli Farage die 24 Novembris 2016 =
* [https://www.nytimes.com/2017/05/10/opinion/nigel-farage-ukip-and-the-revenge-of-the-fruitcakes.html NYT]

= Cena Donaldi Trump cum Mitt Romney apud Jean-Georges die 29 Novembris 2016 =
* [https://qz.com/848808/who-paid-for-donald-trumps-600-dinner-with-mitt-romney-at-jean-georges/ QZ]
* [http://www.cnn.com/2016/11/29/politics/donald-trump-mitt-romney-jean-georges/ CNN]
* [http://www.vanityfair.com/news/2016/11/mitt-romney-donald-trump-dinner-secretary-of-state Vanity Fair]
* [http://ny.eater.com/2016/11/30/13791892/trump-jean-georges-romney Eater NY]
* [https://www.thedailybeast.com/cheats/2016/11/29/romney-praises-trump-after-their-dinner Daily Beast]

= Cena Stephano Bannon et Rogero Ailes a Michaele Wolff die 3 Ianuarii 2017 data =
* [http://nymag.com/daily/intelligencer/2018/01/michael-wolff-fire-and-fury-book-donald-trump.html Wolff]
* [https://splinternews.com/who-were-the-mystery-guests-at-michael-wolffs-exclusive-1821785114 Splinter]
* [http://edition.cnn.com/videos/politics/2018/01/04/anthony-scaramucci-bannon-intv-newday.cnn/video/playlists/donald-trump-and-steve-bannon/ Janice Min apud CNN]
* [https://www.theguardian.com/us-news/2018/jan/14/michael-wolff-interview-fire-and-fury-donald-trump The Guardian]

= Convivium Vasingtoniae die 19 Ianuarii 2017 habitum =
* [https://www.theguardian.com/world/2017/jan/20/too-much-love-nigel-farage-and-friends-have-a-bad-boys-ball-in-dc The Guardian]

= Cena Donaldi Trump et Iacobi Comey die 27 Ianuarii 2017 habita =
* [https://www.nytimes.com/2017/05/11/us/politics/trump-comey-firing.html NYT]
* [https://www.theguardian.com/us-news/2017/jun/07/james-comey-trump-congress-statement The Guardian]
* [https://www.theguardian.com/us-news/2021/jan/19/the-capitol-riot-was-our-chernobyl-james-comey-on-trump-the-pee-tape-and-clintons-emails The Guardian Comey recalls]

= Cena a Donaldo Trump die 25 Februarii 2017 celebrata =
* [http://ijr.com/2017/02/810965-trump-ditched-the-press-to-have-dinner-heres-how-the-president-acts-when-no-one-is-watching/ IJR]
* [https://www.theguardian.com/politics/2017/feb/26/nigel-farage-dinner-with-the-donald-joins-trumps-table-at-washington-hotel Guardian]
* [http://www.express.co.uk/news/uk/773020/nigel-farage-praises-donald-trump-chance-dinner-washington-hotel Daily Express]

= Cena a Donaldo Trump apud Mar-a-Lago die 7 Aprilis 2017 celebrata =
* [https://www.bostonglobe.com/news/nation/2017/04/06/here-what-menu-for-trump-dinner-tonight/MgPjLdgPiyE9xzkfxcEhIK/story.html Boston Globe]
* [http://www.palmbeachpost.com/news/new-the-mar-lago-dining-room-where-trump-will-have-dinner/7s1mGg64EdrtT1DxIZ6dMN/ Palm Beach Post]
* [https://www.theguardian.com/us-news/2017/apr/07/trumps-dinner-of-steak-and-carrots-then-the-cruise-missiles-struck-syria The Guardian]
* [http://www.huffingtonpost.com/entry/maralago-health-violations_us_58efc30ee4b0da2ff85f1a9e Huffington Post]
* [https://www.theguardian.com/us-news/2017/may/02/trumps-attack-on-syria-after-dinner-entertainment-wilbur-ross-commerce-secretary The Guardian]

= [[:fr:Soirée du Fouquet's du 6 mai 2007#Postérité|Cena Emmanuelis Macron apud La Rotonde die 23 Aprilis 2017 celebrata]] =
* [http://premium.lefigaro.fr/elections/presidentielles/2017/04/24/35003-20170424ARTFIG00296-radis-people-et-politiques8230-une-soiree-a-la-rotonde-qui-fait-jaser.php Figaro]
* [http://premium.lefigaro.fr/culture/2017/04/24/03004-20170424ARTFIG00120-presidentielle-le-diner-de-macron-avec-line-renaud-stephane-bern-et-erik-orsenna.php Figaro]

= Cena a Theresa May die 26 Aprilis 2017 data =
* [http://www.faz.net/aktuell/brexit/juncker-bei-may-das-desastroese-brexit-dinner-14993605.html FAZ]
* [https://www.forbes.com/sites/francescoppola/2017/04/30/the-uk-government-is-completely-deluded-about-brexit/#1c6222b54f04 Forbes]
* [https://www.theguardian.com/politics/2017/may/01/jean-claude-juncker-to-theresa-may-on-brexit-im-10-times-more-sceptical-than-i-was-before Guardian]
* [http://www.reuters.com/article/us-britain-eu-meeting-idUSKBN17S2TK Reuters]
* [https://www.theguardian.com/world/2017/may/01/how-junckers-downing-street-dinner-turned-sour Guardian]

= Cena Emmanuelis Macron excubitorumque Pictavii habita die 28 Aprilis 2017 =
* [https://www.capital.fr/economie-politique/benalla-arme-sans-permis-en-2017-selon-mediapart-une-enquete-ouverte-1308502 Capital]
* [https://www.ledauphine.com/france-monde/2018/09/24/quand-benalla-degainait-son-arme-pour-un-selfie-en-plein-restaurant Le Dauphiné]
* [https://blogs.mediapart.fr/pascal-b/blog/250918/benalla-passa-par-poitiers-avec-son-flingue Mediapart]
* [https://reve86.org/benalla-de-paris-a-paris-en-passant-par-poitiers/ Reve86] (imagines)
* [https://www.francebleu.fr/infos/politique/cette-photo-est-vraie-la-serveuse-poitevine-qui-a-pose-avec-benalla-arme-au-poing-confirme-1537887408 France Bleu]
* [https://www.google.fr/maps/@46.5826902,0.3399958,18z Google Maps]
* [https://www.lemonde.fr/police-justice/article/2019/03/20/alexandre-benalla-de-nouveau-mis-en-examen-notamment-dans-l-affaire-du-selfie-arme_5438981_1653578.html Le Monde 20 Martii 2019]

= Prandium ad honorem principum G7 die 26 Maii 2017 Tauromenii celebrata =
* [http://premium.lefigaro.fr/international/2017/06/02/01003-20170602ARTFIG00277-dans-les-coulisses-du-diner-des-chefs-d-etat-au-sommet-du-g7.php Le Figaro]

= Cena G20 mense Iulio 2017 celebrata =
* [https://www.theguardian.com/world/gallery/2017/jul/08/dinner-diplomacy-melania-trump-sits-next-to-vladimir-putin-at-g20-banquet Guardian] (imagines)

= Prandium Ieremia Corbyn a Michaele Barnier Londinii die 13 Iunii 2017 datum =
* Barnier pp. 100-102

= Cena Donaldo Trump ab Emmanuele Macron die 14 Iulii 2017 data =
* [http://premium.lefigaro.fr/international/2017/07/14/01003-20170714ARTFIG00140-diner-entre-amis-au-sommet-du-monde.php Le Figaro]
* [http://premium.lefigaro.fr/vox/politique/2017/07/12/31001-20170712ARTFIG00191-trump-et-macron-au-restaurant-de-la-tour-eiffel-un-diner-presque-parfait.php Le Figaro]
* cf. [http://premium.lefigaro.fr/gastronomie/2017/03/30/30005-20170330ARTFIG00274-alain-ducasse-et-christian-regouby-le-pouvoir-du-citoyen-est-dans-l-assiette.php Le Figaro] de Alano Ducasse
* cf. [https://www.nytimes.com/2017/07/14/fashion/melania-trump-brigitte-macron-paris.html Le Monde] de vestimentis

= Cena a Donaldo Trump die 26 Iulii 2017 data =
* [http://www.newyorker.com/news/ryan-lizza/anthony-scaramucci-called-me-to-unload-about-white-house-leakers-reince-priebus-and-steve-bannon New Yorker]
* [http://thehill.com/homenews/administration/344068-trump-dines-with-sean-hannity-scaramucci-at-white-house The Hill]

= Cena de Brexitu Bruxellis die 16 Octobris 2017 sumpta =
* [https://www.theguardian.com/politics/2017/oct/16/whos-who-on-the-brexit-dinner-party-guest-list The Guardian]
* [http://www.faz.net/aktuell/politik/ausland/brexit-verhandlungen-ohne-qualen-geht-es-nicht-15257859.html FAZ]
* [https://www.theguardian.com/politics/2017/oct/22/mays-ex-policy-chief-claims-juncker-aide-leaked-brexit-dinner-details The Guardian]
* Barnier pp. 125-126 describes a dinner on 12 October: same?

= Prandium a Donaldo Trump et Mitch McConnell die 16 Octobris 2017 sumptum =
* [https://www.nytimes.com/2017/10/16/us/politics/trump-mcconnell-bannon.html New York Times]

= Cena Seuli Donaldo Trump die 7 Novembris 2017 oblata =
* [https://www.theguardian.com/world/2017/nov/07/seoul-food-for-trump-shrimp-beef-and-360-year-old-soy-sauce The Guardian]
* [http://www.dailymail.co.uk/news/article-5058273/South-Korean-president-hosts-Trump-lavish-state-dinner.html Daily Mail]
* [https://www.eater.com/2017/11/7/16617400/donald-trump-south-korea-dinner Eater]
* [http://dailycaller.com/2017/11/07/melania-dazzles-in-lace-sequins-dress-at-state-dinner-in-south-korea-photos/ Daily Caller]
* [https://www.theguardian.com/world/2017/nov/10/japan-anger-south-koreas-shrimp-surprise-menu-donald-trump-sex-slave The Guardian]

* Prandium eodem die sumptum: [https://www.stripes.com/news/trump-joins-us-s-korean-troops-for-taco-tuesday-at-camp-humphreys-1.496581 Stars and Stripes]

= Prandium Bruxellis die 4 Decembris 2017 habitum =
* Barnier pp. 134-138

= Cena nativitatis Christi inter legatos a Brexitu Europaeos die 15 Decembris 2017 celebrata =
* Barnier pp. 143-146

= Cena ''Presidents Club'' die 18 Ianuarii 2018 celebrata =
* [https://www.theguardian.com/society/2018/jan/24/great-ormond-street-return-presidents-club-donations-harassment-claims Guardian]
* [https://www.theguardian.com/uk-news/2018/jan/24/claret-and-short-skirts-just-what-is-the-presidents-club Guardian]
* [https://www.theguardian.com/world/2018/jan/24/guest-list-presidents-club-all-male-charity-gala Guardian] (convocati)
* [https://www.ft.com/content/075d679e-0033-11e8-9650-9c0ad2d7c5b5 FT]
* [http://www.independent.co.uk/news/business/news/slackberry-hornby-considers-guru-to-aid-rehab-1876933.html Independent] (2010)

= Cena in urbe Pyeongchang de Olympiis hiemalibus die 9 Februarii 2018 celebrata =
* [https://www.afp.com/en/news/23/us-vp-pence-skips-olympic-dinner-seoul-doc-z74gx3 AFP]
* [http://nymag.com/daily/intelligencer/2018/02/pence-avoids-kim-jong-uns-sister-at-olympics.html NY Magazine]
* [http://www.bbc.com/news/world-asia-43003564 BBC]
*[http://thehill.com/homenews/administration/373111-pence-briefly-attends-olympic-reception-with-north-korea-leader The Hill]

= Prandium Michaeli Barnier Londinii die 5 Februarii 2018 datum =
* Barnier pp. 155-157

= Cena Olivarii Robbins et Michaelis Barnier in Hibernia die 5 Martii 2018 celebrata =
* Barnier pp. 169-171

= Cena Londinii ad Muḥammad ibn Salmān ab Eugenio Lebedev die 7/9 Martii 2018 data =
* [https://www.theguardian.com/media/2019/sep/03/media-mogul-evgeny-lebedev-dinner-saudi-leader-mohammed-bin-salman The Giardian], cf. [https://www.theguardian.com/uk-news/2018/mar/07/saudi-crown-prince-uk-visit]
** cf. [https://insidearabia.com/saudi-involvement-bezos-blackmail-mbs-preoccupation-western-media/ Inside Arabia]
** cf. [https://www.newstatesman.com/politics/uk/2018/10/first-thoughts-saudi-propaganda-and-independent-slow-news-and-why-ft-doesn-t New Statesman]
** cf. [https://www.craigmurray.org.uk/archives/2018/03/saudi-evil/comment-page-3/  Craig Murray]

= Cena a Donaldo Trump apud Mount Vernon die 23 Aprilis 2018 celebrata =
* [http://premium.lefigaro.fr/international/2018/04/22/01003-20180422ARTFIG00171-diner-a-mount-vernon-discours-devant-le-congres-le-programme-de-macron-aux-etats-unis.php Figaro]
* [https://www.washingtonpost.com/world/national-security/the-latest-french-president-arrives-for-state-visit/2018/04/23/2b387584-4721-11e8-8082-105a446d19b8_story.html?noredirect=on&amp;utm_term=.6bc36aa43677 Washington Post]
* [https://www.washingtonpost.com/world/national-security/first-lady-chooses-lamb-and-opera-for-macron-state-dinner/2018/04/23/84ead2ea-4710-11e8-8082-105a446d19b8_story.html?utm_term=.230823a17586 Washington Post]
* [http://www.straitstimes.com/world/united-states/trumps-host-macrons-for-a-glitzy-mount-vernon-evening Straits Times]
* [https://www.lexpress.fr/actualites/1/monde/le-diner-glamour-des-trump-et-des-macron-a-mount-vernon_2002286.html L'Express]

= Cena Vasingtoniae a Donaldo et Melania Trump die 24 Aprilis 2018 data =
* [https://www.theguardian.com/us-news/2018/apr/23/melania-trump-macron-state-dinner-menu-new-orleans Guardian]
* [http://premium.lefigaro.fr/international/2018/04/24/01003-20180424ARTFIG00073-emmanuel-macron-aux-etats-unis-les-details-du-diner-d-etat-de-ce-mardi.php Figaro] [https://wtop.com/white-house/2018/04/first-lady-chooses-lamb-and-opera-for-macron-state-dinner/slide/1/ idem apud WTOP]
* [http://premium.lefigaro.fr/international/2018/04/25/01003-20180425ARTFIG00006-macron-a-washington-le-glamour-du-diner-d-etat-apres-le-marathon-politique.php Le Figaro]
* [http://www.straitstimes.com/world/united-states/over-a-year-into-presidency-trump-hosts-first-state-dinner Straits Times]
* [https://www.nytimes.com/2018/04/23/us/politics/trump-state-dinner-france.html New York Times]
** [https://www.nytimes.com/2017/02/27/dining/trump-white-house-food-policy.html De coquina] 
** [http://www.washingtonpost.com/wp-dyn/content/article/2005/08/14/AR2005081400625.html De coquo] [[:en:Cristeta Comerford]]
* [https://edition.cnn.com/2019/07/22/politics/trump-state-dinner-australia/index.html De cenis civicis]

= Cena ad Panmunjom a Moon Jae-in die 27 Aprilis 2018 celebrata =
* [https://www.theguardian.com/world/2018/apr/24/swiss-rosti-with-a-twist-north-and-south-korea-summit-menus-diplomacy The Guardian]
* [https://www.theguardian.com/world/2018/apr/28/cold-noodles-are-peace-symbol-summit-to-savour-for-euphoric-koreans De itriis Pyeongyangensibus]
* [https://www.straitstimes.com/asia/east-asia/south-koreans-lunch-today-pyongyang-cold-noodles idem]

= Convivium Perugiae in palatio Terranova ab Eugenio Lebedev die 28 Aprilis 2018 habitum =
* [https://www.theguardian.com/politics/2019/jul/18/boris-johnson-refuses-to-answer-questions-over-party-in-lebedev-mansion The Guardian]
* [https://www.theguardian.com/politics/2019/jul/26/boris-johnson-security-evgeny-lebedev-perugia-party The Guardian]
* [https://www.opendemocracy.net/en/opendemocracyuk/revealed-boris-russian-oligarch-and-page-3-model/ Open Democracy] (de variis conviviis ibidem habitis)
* [https://www.theguardian.com/politics/2020/oct/21/parties-politics-peerages-boris-johnson-evgeny-lebedev-friendship The Guardian]  (de variis conviviis)
* [https://www.theguardian.com/media/2019/nov/17/boris-johnson-met-alexander-lebedev-without-security-after-nato-summit The Guardian]
* [https://www.theguardian.com/politics/2020/sep/21/no-10-denies-reports-boris-johnson-went-on-secret-italy-trip De 2020]
* [https://www.theguardian.com/politics/video/2022/jul/06/boris-johnson-pressed-on-meeting-ex-kgb-agent-while-foreign-secretary-video Yes]
* [https://www.theguardian.com/politics/2022/jul/16/carole-cadwalladr-boris-johnson-lebedevs-prime-ministers-defining-scandal Carole Cadwalladr]
* [https://www.theguardian.com/politics/2022/jul/26/boris-johnson-says-2018-lebedev-visit-was-in-line-with-security-protocols Guardian]

= Cena a Donaldo Trump diurnariis quattuor die 8 Maii 2018 data =
* [https://time.com/donald-trump-after-hours/ Time]

= Cena apud Chequers die 6 Iulii 2018 habita =
* [https://www.theguardian.com/politics/2018/jun/29/chequers-dinner-could-end-in-on-the-brexit-express Guardian 29 Iunii]
* [https://www.theguardian.com/politics/2018/jul/06/brexit-means-beef-fillet-how-mays-day-at-chequers-played-out Guardian]
* [https://www.theguardian.com/politics/2018/jul/07/brexit-chequers-summit-compromise-theresa-may-reprieve Guardian]
* [https://www.theguardian.com/politics/live/2018/jul/09/david-davis-resigns-as-brexit-secretary-live-updates?page=with:block-5b43940be4b0f86cea746286#block-5b43940be4b0f86cea746286 Sunday Times via Guardian]
* mentioned by Barnier pp. 213-214

= Cena Londinii a Stephano Bannon mense Iulio 2018 data (?= prandium die 15 Iulii habitum) =
* [https://www.theguardian.com/world/2018/nov/21/secret-rightwing-gathering-europe-steve-bannon Guardian]
* youtu.be/SX2twSMMdHs
* [https://www.thedailybeast.com/inside-bannons-plan-to-hijack-europe-for-the-far-right Daily Beast]
* [https://www.hopenothate.org.uk/2018/07/14/international-far-right-gather-london-free-tommy-robinson-demonstration/ Hope not hate]&quot; (ante cenam)
* [https://formiche.net/2018/07/bannon-conquista-ue-the-movement/ Formiche] (fonte Il Ghibellino?)
* [http://blog.ilgiornale.it/puglisi/2018/07/23/linternazionale-populista-di-bannon-guarda-a-washington-e-tel-aviv/ Il Ghibellino]
* [http://viralcontentclub.com/inside-bannons-plan-to-hijack-europe-for-the-far-right/ Viral Content Club] (fonte Daily Beast?)
* [https://www.tichyseinblick.de/daili-es-sentials/kann-steve-bannon-gegenspieler-von-george-soros-werden/ Tichys Einblick]
* [https://www.elindependiente.com/politica/2018/08/05/movimiento-steve-bannon-cuenta-atras-populistas-europeos-hacia-2019/ El Independiente] (fonte Daily Beast?)
* [https://www.thetimes.co.uk/article/steve-bannon-bags-a-belgian-buddy-to-launch-far-right-assault-on-brussels-7d8rgrt57 Sunday Times] (illustratur prandium die 15 Iulii habitum)
* [https://www.reuters.com/article/us-eu-parliament-bannon/belgian-lawyer-launches-trump-inspired-anti-eu-movement-idUSKBN1KE2BJ Reuters]
* [http://www.leparisien.fr/politique/populisme-en-europe-bannon-n-est-pas-un-sous-marin-des-etats-unis-18-08-2018-7856739.php Le Parisien]  (prandium die 15 Iulii habitum)

= Cena Donaldo Trump et Theresae May apud legationem Americanam Londiniensem die 12 Iulii 2018 (?) data =
* [https://www.express.co.uk/news/uk/1134486/Donald-Trump-UK-visit-US-Theresa-May-Boris-Johnson-Nigel-Farage Express]

= Cena Pechingi ab Europaeis et Sinis die 16 Iulio 2018 habita =
* [https://www.theguardian.com/world/2018/dec/19/eu-cables-hack-reveals-no-bombshells-but-many-insights The Guardian]
* [https://miningawareness.wordpress.com/2018/12/19/report-describes-thousands-of-hacked-eu-diplomatic-cables/ Mining Awareness]
* [https://int.nyt.com/data/documenthelper/540-read-the-diplomatic-cables/27bc7c9cfe024869481d/optimized/full.pdf NYTimes data] (quaere &quot;16 July&quot;)

= Cena gubernatorum civitatum Europaearum Bruxellis die 17 Octobris 2018 habita =
* [https://www.theguardian.com/politics/2018/oct/18/brexit-unpalatable-truths-for-dinner-at-eu-summit Guardian]
* Barnier pp. 258-260

= Cena a Donaldo Trump apud Mar-a-Lago die 22 Novembris 2018 celebrata =
* [https://www.theguardian.com/us-news/2018/nov/25/donald-trump-democrats-subpoenas-house Guardian]
* [https://thehill.com/homenews/administration/418005-trump-and-family-hold-thanksgiving-dinner-at-mar-a-lago The Hill]
* [https://www.independent.co.uk/news/world/americas/us-politics/donald-trump-thanksgiving-dinner-turkey-day-2018-president-first-family-florida-mar-a-lago-security-a8645276.html Independent]
* [https://www.newsweek.com/trump-thanksgiving-dinner-mar-lago-feast-featured-more-turkey-1228471 Newsweek]
* [https://www.washingtontimes.com/news/2018/nov/22/trump-thanksgiving-dinner-menu-mar-lago-includes-c/ Washington Times]

= Cena a Donaldo Trump et Xi Jinping Bonaëropoli die 1 Decembris 2018 habita =
* [https://www.businessinsider.fr/us/heres-what-trump-and-xi-jinping-ate-at-their-formal-dinner-2018-12 Business Insider]
* [https://www.elintransigente.com/politica/2018/12/1/donald-trump-xi-jinping-acordaron-una-tregua-comercial-526979.html El Intransigente]
* [https://www.theguardian.com/world/2018/dec/02/donald-trump-and-xi-jinping-declare-trade-truce-at-g20 Guardian]
* [https://www.scmp.com/news/china/diplomacy/article/2175995/xi-trump-dinner-some-key-takeaways South China Morning Post]

= Cena a Donaldo Trump athletis Clemson Tigers die 14 Ianuarii 2019 (?) data =
* [https://www.nbcnews.com/politics/white-house/trump-welcomes-clemson-tigers-white-house-american-fast-food-paid-n958661 NBC News]
* [https://bleacherreport.com/articles/2815986-michael-strahan-ayesha-curry-offer-to-feed-clemson-after-donald-trump-dinner Bleacher Report]

= Cena diei S. Valentini 2019 a Francisco de Rugy celebrata =
* Haec cena aliaeque: [https://www.liberation.fr/politiques/2019/07/16/des-homards-aux-frais-de-mandat-les-affaires-de-rugy-en-cinq-actes_1740326/ Libération]
* [https://www.mediapart.fr/journal/france/100719/la-vie-de-chateau-sur-fonds-publics-des-epoux-de-rugy Mediapart]
* [https://www.liberation.fr/france/2019/07/12/intolerance-aux-crustaces-degout-du-champagne-rugy-se-defend-sur-bfm-tv_1739623/ Libération]

= Cena Donaldo Trump et Shinzo Abe Tokii die 26 Maii 2019 celebrata =
* [https://english.kyodonews.net/news/2019/05/1264d3cbc103-abe-trump-relax-over-traditional-charcoal-grill-dinner-in-tokyo.html Kyodo News]
* [https://www.thenational.ae/lifestyle/travel/a-traditional-hibachi-dinner-what-restaurant-did-trump-eat-at-in-japan-1.866395 The National]
* [http://www.businessinsider.fr/us/trump-japan-visit-menu-for-dinner-prime-minister-abe-shinzo-2019-5 Business Insider]

= Cena a regina Elizabetha Donaldo Trump die 3 Iunii 2019 data =
* [https://www.insider.com/prince-harry-dodges-trump-after-he-called-meghan-markles-comments-nasty-2019-6 De prandio eodem die habita: Insider]
* [https://www.today.com/food/what-did-trumps-queen-elizabeth-eat-dinner-buckingham-palace-t155480 Today]
* [https://www.express.co.uk/news/uk/1135833/trump-state-banquet-what-s-on-the-menu-queen-donald-trump-uk-state-visit Daily Express]
* [https://www.express.co.uk/life-style/style/1135794/Melania-Trump-state-dinner-donald-latest-news-kate-middleton Daily Express]
* [https://www.townandcountrymag.com/style/fashion-trends/a27758010/queen-elizabeth-kate-middleton-princess-anne-camilla-melania-trump-white-dresses-state-banquet/ Town &amp; Country]
* [https://www.insider.com/kate-middleton-white-gown-trump-visit-uk-2019-6 Insider]
* [https://www.thecut.com/2019/06/trump-wears-tiny-jacket-to-state-dinner-with-queen.html The Cut]
* [https://www.insider.com/trump-state-visit-what-queen-served-buckingham-palace-banquet-2019-6 De ferculis: Insider]

= Cena a Donaldo Trump principi Carolo die 4 Iunii 2019 data =
* [https://www.townandcountrymag.com/society/tradition/a27791605/prince-charles-camilla-trump-dinner-chris-jackson-photos/ Town &amp; Country]
* [https://www.townandcountrymag.com/society/tradition/a27751165/prince-charles-donald-trump-piers-morgan-climate-change-talk/ Town &amp; Country]
* [https://www.indy100.com/article/trump-uk-visit-prince-charles-royal-family-dinner-menu-steak-and-chips-opinion-8945711 Indy100]
* [https://metro.co.uk/2019/06/04/donald-melania-trump-welcome-camilla-charles-thank-dinner-9811662/ Metro]
* [https://www.insider.com/donald-trump-served-beef-and-vanilla-ice-cream-dinner-royals-2019-6 Insider]
* [https://news.sky.com/story/trumps-dinner-menu-what-has-the-president-laid-on-for-charles-11735002 De ferculis: Sky]
* [https://www.pressdemocrat.com/article/business/iron-horse-vineyards-wine-served-during-trumps-elaborate-dinner-for-prince/ De vinis]

= Cena a Donaldo Trump ad Doonbeg die 6 Iunii 2o19 habita =
* [https://www.irishtimes.com/news/politics/trump-has-relaxed-dinner-in-doonbeg-with-fish-and-chips-on-the-menu-1.3917403 Irish Times]

= Cena Vasingtoniae a Stephano Mnuchin regi Tamim bin Hamad Al Thani Qatarensi die 8 Iulii 2019 data =
* [https://www.tmz.com/2019/07/09/robert-kraft-donald-trump-dinner-dc/ TMZ]
* [https://www.whitehouse.gov/briefings-statements/remarks-president-trump-dinner-hosted-secretary-treasury-honor-amir-state-qatar/ Contiones: White House]
* [https://www.bloomberg.com/news/articles/2019-07-09/patriots-owner-is-trump-dinner-guest-despite-prostitution-charge absente Kim Darroch: Bloomberg]

= Prandium Luxemburgi Borisio Johnson die 16 Septembris 2019 data =
* [https://www.theguardian.com/politics/2019/sep/13/boris-johnson-to-meet-juncker-for-brexit-talks-in-luxembourg Guardian]
* [https://www.theguardian.com/politics/video/2019/sep/16/boris-johnson-cautious-brexit-lunch-jean-claude-juncker-video Guardian video]
* [https://www.theguardian.com/politics/audio/2017/dec/05/what-went-wrong-crucial-brexit-lunch-teresa-may-juncker-arlene-foster-brexit-means-podcast Guardian podcast]
* [https://www.theguardian.com/politics/live/2019/sep/16/brexit-latest-news-boris-johnson-talks-juncker-eu-must-show-flexibility-says-raab-ahead-of-boris-johnsons-key-meeting-with-juncker-live-news-latest-news?page=with:block-5d7fa71f8f0834740f3bc411#block-5d7fa71f8f0834740f3bc411 Guardian live] [https://www.theguardian.com/politics/live/2019/sep/16/brexit-latest-news-boris-johnson-talks-juncker-eu-must-show-flexibility-says-raab-ahead-of-boris-johnsons-key-meeting-with-juncker-live-news-latest-news?page=with:block-5d7f95408f08f9df5bdd5607#liveblog-navigation see 13.46 for photo of menu]
* [https://www.theguardian.com/politics/2019/sep/16/johnson-humiliated-by-luxembourg-pm-at-empty-chair-press-conference Guardian]
* [https://www.theguardian.com/politics/2019/sep/17/luxembourg-pms-treatment-of-johnson-may-harm-brexit-talks Guardian]
* [https://www.reuters.com/article/uk-britain-eu-johnson-juncker/johnson-buffeted-in-luxembourg-says-brexit-deal-emerging-idUSKBN1W10OO Reuters brief menu]
* [http://www.lebouquetgarni.lu/menu-lunch/ This is the restaurant]
* Barnier pp. 353-355

= Cena Vasingtoniae a Donaldo et Melania Trump Scott Morrison die 20 Septembris 2019 danda =
* [https://edition.cnn.com/2019/07/22/politics/trump-state-dinner-australia/index.html De cenis civicis]

=  Cena Bruxellis Borisio Johnson die 17 Octobris 2019 data  =
* [https://www.thisismoney.co.uk/wires/pa/article-7585625/Prime-Minister-dines-scallops-veal-clinching-Brexit-deal.html This is money]
* Barnier, busy with negotiations, doesn't mention the dinner

= Convivium Londinii ab Alexandro Lebedev die 13 Decembris 2019 celebratum =
* [https://www.theguardian.com/politics/2019/dec/22/johnson-visit-to-lebedev-party-after-victory-odd-move-for-peoples-pm Guardian]
* [https://www.eurotrib.com/story/2019/12/22/202549/32 European Tribune]
* [https://www.businessinsider.sg/boris-johnson-partied-with-mick-jagger-after-election-win-2019-12/ Business Insider]
* [https://www.dailymail.co.uk/tvshowbiz/article-7792479/Anna-Friel-43-dazzles-alongside-Penny-Lancaster-48-attend-star-studded-Christmas-Party.html Mail Online] [https://www.dailymail.co.uk/news/article-7792671/Princesses-Beatrice-Eugenie-head-media-tycoon-Evgeny-Lebedevs-Christmas-party.html ibidem]
* [https://www.thesun.co.uk/tvandshowbiz/7934341/barbara-windsor-alzheimers-husband-scott/ Sun]
* [https://www.alamy.com/stock-photo-various-celebrities-attend-evgeny-lebedev-christmas-party-the-owner-172019688.html Alamy]
* [https://www.thetimes.co.uk/article/a-servant-of-the-people-at-a-super-rich-bash-r9jkw58b8 Times]

* [https://www.theguardian.com/media/2019/nov/17/boris-johnson-met-alexander-lebedev-without-security-after-nato-summit The Guardian: De Alexandro Lebedev conviva]
* [https://www.theguardian.com/politics/2020/oct/21/parties-politics-peerages-boris-johnson-evgeny-lebedev-friendship Boris Johnson and Evgeny Lebedev: a decade of politics, parties and peerages] (Guardian)

= Prandium ad nuptias futuras Petri Alberti Aurelianensis celebrandas die 7 Martii 2020 datum =
* [https://www.theguardian.com/world/2020/apr/04/brazils-super-rich-and-the-exclusive-club-at-the-heart-of-a-coronavirus-hotspot Guardian]
* [https://pleno.news/brasil/tres-membros-da-familia-imperial-estao-com-covid-19.html Pleno]

= Convivia apud 10 Downing Street sub CoVID annis 2020 et 2021 habita =
* [https://www.theguardian.com/politics/2022/may/25/named-and-shamed-who-are-the-politicians-and-aides-in-sue-gray-report Homines]
* [https://www.theguardian.com/politics/2022/may/25/sue-gray-report-full-breakdown-findings-no-10-parties Epitome]
* [https://www.theguardian.com/politics/2022/may/25/read-sue-grays-full-report-into-downing-street-parties?CMP=twt_gu&amp;utm_source=Twitter&amp;utm_medium#Echobox=1653475233 Commentarium] (cf. /My downloads/Gray Report.pdf)

= Cena Bruxellis Borisio Johnson ab Ursula von der Leyen die 9 Decembris 2020 data =
* [https://www.theguardian.com/politics/2020/dec/09/brexits-fishy-business-called-for-turbot-charged-talks The Guardian]
* [https://www.theguardian.com/politics/2020/dec/10/the-brexit-brussels-dinner-fish-and-frank-talk-but-no-one-left-satisfied The Guardian]
* Barnier pp. 511-514

= Cena Lutetiae ab Emmanuele Macron die 16 Decembris 2020 data =
* [https://www.lepoint.fr/politique/macron-positif-au-covid-mercredi-soir-le-long-diner-de-la-majorite-a-l-elysee-17-12-2020-2406311_20.php Le Point]
* [https://www.lemonde.fr/politique/article/2020/12/17/emmanuel-macron-a-ete-diagnostique-positif-au-covid-19_6063701_823448.html Le Monde]
* [https://www.sfchronicle.com/news/article/French-President-Macron-tests-positive-for-15810326.php San Francisco Chronicle]
* [https://www.dailymail.co.uk/news/article-9063043/Emmanuel-Macron-tests-positive-Covid-19-self-isolating.html Daily Mail]

= Cena Londinii Catharinae Tai a Lynn Truss mense Iunio 2021 data =
* [https://www.dailymail.co.uk/debate/article-10173453/ANNA-MIKHAILOVA-Filthy-Fifty-MPs-mission-silence-sleaze-busters.html Trencher Truss takes the biscuit]
* [https://www.mirror.co.uk/news/politics/liz-truss-insisted-luxury-lunch-25834430 Mirror]
* [https://www.dailymail.co.uk/news/article-10362187/Liz-Truss-overruled-officials-demand-used-Tory-donors-Mayfair-club-meet-diplomat.html Mail]
* [https://www.theguardian.com/politics/2022/jan/02/liz-truss-hosted-3k-lunch-for-us-envoy-over-civil-service-objections Guardian]
* [https://sputniknews.com/20220103/liz-truss-reportedly-insisted-on-taxpayer-funded-luxury-lunch-at-tory-donors-mayfair-club-1091989659.html Sputnik]

* [https://spearswms.com/how-5-hertford-st-became-the-most-influential-club-in-the-world/ De societate] (How 5 Hertford St became the most influential club in the world)
* [https://www.dailymail.co.uk/news/article-10324529/Mayfair-club-thats-Ground-Zero-anti-Boris-Johnson-plotters.html?ns_mchannel=rss&amp;ns_campaign=1490&amp;ito=1490 De societate]
* [https://www.dailymail.co.uk/femail/article-10364269/Inside-one-Londons-secretive-private-members-club.html De societate]

= Cena die 18 Octobris 2021 Londinii a Borisio Johnson data =
* Rupert Neate, Peter Walker, &quot;[https://www.theguardian.com/business/2021/oct/18/jellied-eel-canapes-and-venison-no-10-hosts-biggest-names-in-business-boris-johnson-bill-gates Jellied eel canapés and venison: No 10 hosts biggest names in business]&quot; in ''[[The Guardian]]'' (18 Octobris 2021)

= Cena apud societatem Garrick die 2 Novembris 2021 habita =
* Andrew Pierce, [https://www.dailymail.co.uk/debate/article-10167609/ANDREW-PIERCE-political-fiasco-humbled-Boris-Johnson.html A political fiasco that started over claret and pheasant at the garrick... and ended in humiliation]&quot;
* [https://www.thearticle.com/boris-johnsons-dinner-at-the-garrick The Article]
* [https://www.mirror.co.uk/news/politics/boris-johnson-races-back-cop-25371485 Mirror]

= Cena Divione die 12 Novembris 2021 data =
* Kim Hullot-Guiot, &quot;Plats pour la planète&quot; via email, saved under &quot;Queries on food&quot; (cf. ''[https://www.liberation.fr/dossier/tu-mitonnes/ Tu mitonnes]'')

= Nuptiae et otia Borisii Johnson aestate 2022 =
* &quot;[https://www.theguardian.com/politics/2022/aug/15/boris-johnsons-summer-of-fun-what-has-the-pm-been-doing Guardian]&quot;
* &quot;[https://www.huffingtonpost.co.uk/entry/boris-johnson-is-planning-a-summer-sausage-offensive-to-shore-up-support_uk_62b8070de4b0cf43c865e56d summer sausage offensive]&quot;
* &quot;[https://www.dailymail.co.uk/news/article-11064941/A-family-affair-Boris-sister-Rachel-arrives-Carries-wedding-celebration-party.html Daily Mail]&quot;
* &quot;[https://order-order.com/2022/08/05/watch-boris-and-carrie-dance-to-sweet-caroline/ Guido Fawkes]&quot;
* &quot;[https://www.theguardian.com/politics/2022/jul/29/boris-carrie-johnson-to-hold-wedding-party-tory-donor-estate Guardian]&quot;
* &quot;[https://www.dailymail.co.uk/news/article-11090815/Boris-Carrie-Johnson-spend-mini-moon-540-night-Slovenian-eco-resort.html Daily Mail]&quot;
* &quot;[https://the-slovenia.com/gastronomy/boris-johnson-in-slovenia/ The Slovenia]&quot;
* &quot;[https://www.dailymail.co.uk/news/article-11112545/Boris-allies-deny-thrown-towel-heading-holiday.html Daily Mail]&quot;
* &quot;[https://greekcitytimes.com/2022/08/14/boris-johnson-supermarket/ Greek City Times]&quot;

= Gastrodiplomatia =
* [http://www.lemonde.fr/m-actu/article/2013/09/06/toques-au-sommet_3471672_4497186.html Le Monde]

= Gastrobiographia =
* [https://www.lemonde.fr/international/article/2019/10/28/les-gouts-simples-du-president-nouille-bolsonaro_6017133_3210.html Le Monde] (Bolsonaro)
* [http://www.leparisien.fr/politique/arretez-d-emmerder-les-francais-pourquoi-macron-cite-du-georges-pompidou-23-02-2018-7575666.php Pompidou: n'emmerdez pas les Français]
* [http://www.rtl.fr/actu/politique/je-bois-du-vin-midi-et-soir-macron-exclut-tout-durcissement-de-la-loi-evin-7792401192 Macron: Moi, je bois du vin le midi et le soir]

= Cena ficta factionis Momenti =
* [https://www.theguardian.com/politics/2017/jul/28/momentum-video-jeremy-corbyn-labour-university-tuition The Guardian]
* [https://www.theguardian.com/commentisfree/2017/jul/28/they-just-dont-get-it-video-upset-corbyn-deniers-labour-momentum The Guardian]

= Instrumentum cenaticum =
* [http://premium.lefigaro.fr/arts-expositions/2017/12/29/03015-20171229ARTFIG00205-table-imperiale-a-champs-sur-marne.php Le Figaro]

= De rebus politicis =
* [https://www.theguardian.com/commentisfree/2018/nov/22/pizza-plot-balti-bailout-politics-food-granita-politicians Guardian]

= Cenae ficticiae =
* [https://archive.org/details/remainsofearlypo0003hazl/page/92/mode/2up The Turnament of Totenham: The feest]</text>
      <sha1>n14r9a5hmn3m1i7ypvno2im3zp71x29</sha1>
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  </page>
  <page>
    <title>Hypothesis Riemanniana</title>
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      <contributor>
        <username>LilyKitty</username>
        <id>18316</id>
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      <comment>de coniectura et functione zeta</comment>
      <model>wikitext</model>
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      <text bytes="3358" xml:space="preserve">{{L}}
[[Fasciculus:RiemannCriticalLine.svg|thumb|Linea rufa partem realem, linea caerulea partem imaginariam valorum functionis ζ(s) monstrat, in linea &quot;criticale&quot; dicta ubi Re(s) = 1/2.  ζ(s) = 0 ubi lineae ambo axem horizontalem transeunt:  ±14.35, ±21.022, etc.]]
'''Hypothesis Riemanniana,''' est [[coniectura]] vel [[hypothesis]] in [[theoria numerorum]], dicit omnes [[numerus complexus|numeros complexos]] ''s'' ut ζ(s) = 0, praeter valores triviales, partem realem 1/2 habere;  ζ(s) = [[functio zeta Riemanniana]].  Si vera est, possumus aestimare quot [[numerus primus|numeri primi]] sint minores quam numero quolibet ''n,'' h.e. π(n).

== Quantitas primorum ==
[[Theorema]] clarissimum de numeris primis, &quot;theorema numerorum primorum&quot; dictum, probaverunt [[Iacobus Hadamard]] et [[Carolus Ioannis De La Vallée Poussin]] anno 1896.  Sit π(''x'') = quot numeri primi minores sunt quam ''x,'' et sit Li(''x'') = &quot;[[logarithmus|'''l'''ogarithmicum]] [[integrale|'''i'''ntegrale]]&quot;:

:&lt;math&gt; {\rm Li} (x) = \int_{2}^{x} \frac{dt}{\log t}&lt;/math&gt;

Tunc 

:&lt;math&gt;\lim_{x\to \infty}\frac{\pi(x)}{{\rm Li}(x)}=1&lt;/math&gt;

Hoc est, Li(''x'') bene approximat π(''x'').  Hypothesis Riemanniana autem dicit hanc approximationem etiam meliorem esse:  hypothesis implicat:

:&lt;math&gt;\forall n \in \mathbb Z &gt; 3, |\pi(n) - {\rm Li}(n)| \le \sqrt{n} \log(n)&lt;/math&gt;

Quia Li(''n'') ≈ n/log(n), quae quantitas maior est quam &lt;math&gt;\sqrt{n} \log(n)&lt;/math&gt;, vidimus errorem approximationis multo minorem esse quam aut π(n) aut Li(n).&lt;ref&gt;Mazur et Stein, p. 41;  Gowers et al., p. 715&lt;/ref&gt;

== Functio zeta ==

[[Functio zeta Riemanniana]] haec est:

:&lt;math&gt;\zeta(s) = \sum_{n=1}^{\infty} n^{-s} = 1 + \frac{1}{2^s} + \frac{1}{3^s} + \dots&lt;/math&gt;

[[Leonhardus Eulerus]] demonstravit:

:&lt;math&gt;\zeta(s) = \prod_{p}(1 - p^{-s})^{-1}, p \text{ primus}, s \text{ realis} &gt; 1&lt;/math&gt;

[[Bernardus Riemann]] functionem in numeros complexos extendit (praeter ''s'' = 1, scilicet).  Nunc, si pars realis ''s'' &gt; 1,

:&lt;math&gt;\log \zeta(s) = \sum_{p primus} -\log(1 - p^{-s})&lt;/math&gt;
:&lt;math&gt; \frac{d}{ds} \frac{s}{\zeta(s)} = - \sum_{n=1}^\infty \frac{\Lambda(n)}{n^s}&lt;/math&gt;

ubi functio Λ haec est:

::Λ(n) = log(''p''), si n = p&lt;sup&gt;k&lt;/sup&gt;, p primus, k &gt; 0
::: = 0 si n nec numerus primus nec potestas numeri primi est

Hoc est, [[Functio zeta Riemanniana|functio zeta]] positionem numerorum primorum repraesentare videtur.&lt;ref&gt;Mazur e Stein, p. 123&lt;/ref&gt;

== Notae ==
&lt;references/&gt;

== Bibliographia ==
* Timothy Gowers, June Barrow-Green, Imre Leader, edd. ''The Princeton Companion to Mathematics.'' Princetoniae: Princeton University Press, 2008. ISBN 978-0-691-11880-2
* Marcus du Sautoy.  ''The Music of the Primes:  Searching to Solve the Greatest Mystery in Mathematics.'' Novi Eboraci:  HarperCollins, 2003.  ISBN 0-06-621070-4
* Barry Mazur et William Stein. ''Prime Numbers and the Riemann Hypothesis.'' Cantabridgiae:  Cambridge University Press, 2016.  ISBN 978-1-107-49943-0

== Nexus externi ==
* [http://www.claymath.org/millennium-problems/riemann-hypothesis De Hypothesi], Clay Mathematics Institute
* [https://primes.utm.edu/notes/rh.html Prime Pages]
* [http://www.pourlascience.fr/ewb_pages/a/article-l-hypothese-de-riemann-19849.php Pour la science]

{{math-stipula}}

{{Myrias|Mathematica}}
[[Categoria:Theoria numerorum]]
[[Categoria:Analysis]]</text>
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  <page>
    <title>Usor:Andrew Dalby/Fontes</title>
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      <text bytes="265834" xml:space="preserve">* Vide [[Usor:Andrew Dalby/Fontes Orta et Clusii]]

== ante Christum natum ==
* saec. VII a.C.n. : [[Hesiodus]], ''[[Opera et dies]]'' [http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3atext%3a1999.01.0131 Textus]
* saec. VII a.C.n. : [[Hesiodus]], ''[[Theogonia]]'' [http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0129%3Acard%3D453 vv. 453-491]
* ante saec. VI a.C.n. : ''[[Liber carminum (Sinae)|Shijing]]'' ([[Iacobus Legge|James Legge]], ed. et interpr., ''The Chinese Classics'' vol. 4 i-ii: ''The first part of the She-king ... The second, third and fourth parts of the She-king''. Hongcongi: Lane, Crawford &amp; Co., 1871 [https://archive.org/details/chineseclassics41legg_0 pars i] [https://archive.org/details/chineseclassics42legg_0 pars ii]
* saec. V a.C.n. : [[Mithaecus]], praecepta (maxima parte deperdita)
* saec. V-III a.C.n. : ''[[Ver et autumnus|Chunqiu]]'' ([[Iacobus Legge|James Legge]], ed. et interpr., ''The Chinese Classics''. Vol. 5. ''The Ch'un Ts'ew, with the Tso Chuen'' (2 partes. Hongcongi: Lane, Crawford, 1872) [https://archive.org/details/chineseclassics51legg i] [https://archive.org/details/chineseclassics52legg ii])
* saec. IV a.C.n.? : ''[[Regimen (Hippocrates)|Regimen]]'' e [[Corpus Hippocraticum|corpore Hippocratico]]
* saec. IV a.C.n.? : ''[[De regimine acutorum]]'' e [[Corpus Hippocraticum|corpore Hippocratico]]
* saec. IV/I a.C.n. : ''[[Carmina regni Chu|Chu ci]]'' (楚辭)
* saec. IV a.C.n./III p.C.n. : ''[[Commentarius Zuo|Zuo zhuan]]'' ([[Iacobus Legge|James Legge]], ed. et interpr., ''The Chinese Classics''. Vol. 5. ''The Ch'un Ts'ew, with the Tso Chuen'' (2 partes. Hongcongi: Lane, Crawford, 1872) [https://archive.org/details/chineseclassics51legg i] [https://archive.org/details/chineseclassics52legg ii])
* c. 350 a.C.n. : [[Archestratus]], ''[[Hedypathia]]'' (S. Douglas Olson, Alexander Sens, edd., ''Archestratos of Gela: Greek culture and cuisine in the fourth century BCE'' (Oxonii: Oxford University Press, 2000)
* c. 310 a.C.n. : &lt;span id=&quot;Theophrastus&quot;&gt;&lt;/span&gt;[[Theophrastus]], ''[[Historia plantarum (Theophrastus)|Historia plantarum]]'' 6.6.11, 6.8.1-2 (ed. F. Wimmer [Lutetiae: Didot, 1866] [https://archive.org/details/bub_gb_6dXhmh761sYC/page/n143/mode/2up pp. 106-108])
* saec. III a.C.n.? : &quot;Di Yuan&quot; (地員)&lt;ref&gt;[https://ctext.org/guanzi/di-yuan?searchu=其種忍蘟，忍葉如雚葉，以長狐茸，黃莖黑莖黑秀，其粟大，無不宜也。&amp;searchmode=showall#result Textus]&lt;/ref&gt; in ''[[Guanzi]]'' 
* saec. III/I a.C.n. : ''[[Ratiocinantium colloquia|Analecta Confucii]]'' 12.22 ([[Iacobus Legge|James Legge]], ed. et interpr., ''The Chinese Classics''. Vol. 1: ''Confucian Analects'' [etc.]. 2a ed. Oxonii: Clarendon Press, 1893 [https://archive.org/details/chineseclassics01legg/page/260/mode/2up p. 261])
* c. 300 a.C.n. : [[Diphilus Siphnius]], ''De iis quae vescenda bene valentibus et aegris dantur'' (fragmentis servatum)
* 299 a.C.n./281 p.C.n. : ''[[Annales bambuseis laciniis scriptae|Zhushu jinian]]'' (&quot;[https://archive.org/details/chineseclassics31legg/page/n129/mode/2up The Annals of the Bamboo Books]&quot; in James Legge, ed. et interpr., ''The Chinese Classics''. Vol. 3 i: ''The first part of the Shoo King'' [Hongcongi, 1865] pp. 105-183)
* post 289 a.C.n. : ''[[Mencius (liber)|Mencius]]'' 2.B.2.8, 5.A.6.4-5.A.7.9, 5.B.1.2, 6.B.1.6 et alibi ([[Iacobus Legge|James Legge]], ed. et interpr., ''The Chinese Classics''. Vol. 2: ''The Works of Mencius''. 2a ed. Oxonii: Clarendon Press, 1895 [https://archive.org/details/chineseclassics02legg/page/214/mode/2up pp. 214], [https://archive.org/details/chineseclassics02legg/page/360/mode/2up 360-364], [https://archive.org/details/chineseclassics02legg/page/370/mode/2up 370], [https://archive.org/details/chineseclassics02legg/page/432/mode/2up 433] et alibi)
* c. 239 a.C.n. : [[Lü Buwei]], ed., ''[[Veres autumnique domini Lü|Lüshi Chunqiu]]'' 14/2.1, 14/3.4 (John Knoblock, Jeffrey Riegel, edd. et interprr., ''The Annals of Lü Buwei'' [Stanfordiae, 2000] pp. 306-311)
* saec. III a.C.n. exeunte : ''[[Laocius]]'' [http://www.tao-te-king.org/12.htm 12]
* saec. II a.C.n.? : [[Paxamus]], ''De re culinaria ordine alphabetico''
* saec. II a.C.n.? : [[Paxamus]], ''Georgica''
* saec. II a.C.n. : [[Nicander Colophonius]], ''[[Georgica (Nicander)|Georgica]]''
* saec. II a.C.n. ineunte? : ''[[Traditio Zuo|Zuo zhuan]]'' Zhao 9.5, 20.8a (Stephen Durrant, Wai-yee Li, David Schaberg, interprr., ''Zuo Tradition'' [Seattli: University of Washington Press, 2016] pp. 1448-1451, 1586-1587)
* saec. II a.C.n. ineunte : [[Quintus Ennius|Ennius]], ''[[Hedyphagetica]]'' (E. Courtney, ''The fragmentary Latin poets [Oxonii, 1993] Ennius fr. 28)
* c. 170 a.C.n./320 p.C.n. : ''[[Liber documentorum|Shujing]]'' (James Legge, ed. et interpr., ''The Chinese Classics''. Vol. 3 i-ii: ''The Shoo King'' [Hongcongi, 1865] [https://archive.org/details/chineseclassics31legg pars i] [https://archive.org/details/chineseclassics32legg ii]
* c. 160 a.C.n. : [[Marcus Porcius Cato maior|Cato]], ''[[De agri cultura]]''
* c. 150 a.C.n. : ''[[Commentarius Gongyang|Gongyang zhuan]]'' (Harry Miller, ''The Gongyang Commentary on the Spring and Autumn Annals''. Novi Eboraci: Palgrave Macmillan, 2015)
* ante 146 a.C.n. : [[Mago Carthaginiensis]], opus de re rustica [[Punice]] scriptum
* ante 139 a.C.n. : ''[[Magistri Huainan|Huainanzi]]'' 11.16, 19.1, 20.25, 20.26 (John S. Major et al., interprr., ''The Huainanzi: a guide to the theory and practice of government in early Han China'' [Novi Eboraci: Columbia University Press, 2010)]
* ante 122 a.C.n. : ''[[Magister Zhuang|Zhuangzi]]'' 23, 28 ([[Iacobus Legge|James Legge]], interpr., ''The Sacred Books of China: The texts of Taoism'' vol. 2 [Oxonii: Clarendon Press, 1891] [https://archive.org/details/sacredbooksofchi028287mbp/page/n99/mode/2up pp. 80], [https://archive.org/details/sacredbooksofchi028287mbp/page/n173/mode/2up 162]
* saec. II a.C.n. exeunte? : ''[[Ritus regni Zhou|Zhou li]]'' (Edouard Biot, interpr., ''Le Tcheou-li, ou Rites des Tcheou''. 3 voll. Lutetiae: Imprimerie Nationale, 1851 [https://archive.org/details/letcheouliourite1t3chou 3 voll. in 1] apud ''Internet Archive'' [https://gallica.bnf.fr/ark:/12148/bpt6k5038370 voll. 1-2] [https://gallica.bnf.fr/ark:/12148/bpt6k503972s vol. 3] apud Gallica
* saec. I a.C.n.? : Fan Shengzhi, ''[[Liber Fan Shengzhi|Fan Shengzhi shu]]'' (fragmentis servatum) 
* saec. I a.C.n. ineunte : [[Cienus|Sima Cian]], ''[[Opus chronographicum]]''
* 59 a.C.n. : [[Wang Bao]], ''[[Syngrapha servi|Tong yue]]'' (&quot;Syngrapha servi&quot;)
* c. 36 a.C.n. : [[Marcus Terentius Varro|Varro]], ''[[De re rustica (Varro)|De re rustica]]''
* c. 35 a.C.n. : &lt;span id=&quot;Horatius&quot;&gt;&lt;/span&gt;[[Quintus Horatius Flaccus|Horatius]], ''[[Sermones (Horatius)|Sermones]]'' [http://www.thelatinlibrary.com/horace/serm2.shtml#2.3 2.3.150-155]
* 29 a.C.n. : [[Publius Vergilius Maro|Vergilius]], ''[[Georgica]]''
* ante 8 a.C.n. : [[Liu Xiang]], ''[[Strategemata civitatum bellatricum|Zhan guo ce]]''
* ante saec. I : ''[[Rituum et officiorum memoriale|Liji]]'' (Séraphin Couvreur, ed. et interpr., ''Li ki, ou Mémoires sur les bienséances et les cérémonies ... avec une double traduction en français et en latin'', 2a ed. [Ho Kien Fu: Mission Catholique, 1913] [https://archive.org/details/likioummoiress01couvuoft vol. 1] [https://archive.org/details/likioummoiress02couvuoft 2]) ([[Iacobus Legge|James Legge]], interpr., ''The Sacred Books of China: The texts of Confucianism'' partes 3-4: ''The Lî Kî'' [Oxonii: Clarendon Press, 1885] [https://archive.org/details/sacredbooksofchi3conf i-x] [https://archive.org/details/sacredbooksofchi04conf xi-xlvi]

== saec. I-X ==
* 5 : ''[[Decretum praeceptorum menstruorum]]'' (Charles Sanft, interpr., &quot;Edict of Monthly Ordinances for the Four Seasons in Fifty Articles&quot; in ''Early China'' vol. 32 [2008/2009] pp. 125-208 [https://www.jstor.org/stable/23351787 JSTOR])
* c. 55/92 : [[Ban Gu]], ''[[Han shu]]''
* c. 60 : [[Lucius Iunius Moderatus Columella|Columella]], ''[[Rei rusticae libri (Columella)|Rei rusticae libri]]'' 11.3.19-23
* ante 79 : &lt;span id=&quot;Plinius maior&quot;&gt;&lt;/span&gt;{{Pnh}} [https://penelope.uchicago.edu/Thayer/L/Roman/Texts/Pliny_the_Elder/21*.html#65 21.65], [https://penelope.uchicago.edu/Thayer/L/Roman/Texts/Pliny_the_Elder/21*.html#167 167]
* 79/saec. III : ''[[Bai Hu Tong]]'' (Tjan Tjoe Som, ''Po hu t'ung: the comprehensive discussions in the White Tiger Hall'' [2 voll. Lugduni Batavorum: Brill, 1949-1952] {{GB|nQgVAAAAIAAJ|vol. 1}} {{Google Books|nzgVAAAAIAAJ|vol. 2: paginae selectae}} [https://www.jstor.org/stable/609660 recensio apud JSTOR]
* c. 80 : &lt;span id=&quot;Dioscorides&quot;&gt;&lt;/span&gt;{{Dioscorides}} 3.120 (ed. [[Max Wellmann]] [Berolini: Weidmann, 1906-1914] [https://gallica.bnf.fr/ark:/12148/bpt6k6228793s/f170.item vol. 2 pp. 130-131])
* c. 100 : [[Marcus Valerius Martialis|Martialis]], ''Epigrammata'' [https://www.hs-augsburg.de/~harsch/Chronologia/Lspost01/Martialis/mar_ep04.html 4.44]
* c. 120 : [[Publius Annius Florus|Florus]], ''[[Epitome rerum Romanarum (Florus)|Epitome rerum Romanarum]]'' [https://penelope.uchicago.edu/Thayer/L/Roman/Texts/Florus/Epitome/1C*.html#XI 1.xi.16.4]
* saec. II : [[Cui Shi]], ''[[Mensualia ordinum quattuor populi praecepta|Simin yueling]]'', calendarium rusticum
* saec. II ineunte? : Pseudo-[[Dioscorides]], ''[[Synonyma plantarum barbara]]''
* saec. II medio : &lt;span id=&quot;Liu Xi&quot;&gt;&lt;/span&gt;[[Liu Xi]], ''[[Expositio nominum|Shiming]]'' cap. 10 (vide [[#Shurtleff et Aoyagi (2012)]] p. 21)
* saec. II exeunte : [[Sextus Quintilius Valerius Maximus]] et [[Sextus Quintilius Condianus]], ''[[Georgica (Quintilii)|Georgica]]'' (partim in ''[[Geoponica|Geoponicis]]'' servata)
* c. 160 : &lt;span id=&quot;Cui Shi&quot;&gt;&lt;/span&gt;[[Cui Shi]], ''[[Praecepta plebis mensualia|Simin Yueling]]'' (vide [[#Shurtleff et Aoyagi (2012)]] p. 21; cf. ''[http://www.chinaknowledge.de/Literature/Science/siminyueling.html China Knowledge]'') 
* saec. II/VI : ''[[Carakasaṃhitā]]'' (Avinash Chandra Kaviratna, interpr. Calcuttae, 1890-1908 [https://archive.org/details/BIUSante_47357 Textus] [https://www.biusante.parisdescartes.fr/histoire/medica/resultats/index.php?do=chapitre&amp;cote=47357 alibi]; Shree Gulabkunverba Ayurvedic Society, interprr. 6 voll. Jamnagar, 1949 [https://archive.org/details/in.ernet.dli.2015.326549 1] [https://archive.org/details/charakasamhitagulakunvebaaurvedicsocietyvol2_167_F 2] [https://archive.org/details/charakasamhitagulakunvebaaurvedicsocietyvol3_422_c 3] [https://archive.org/details/in.ernet.dli.2015.326550 4] [https://archive.org/details/in.ernet.dli.2015.326551 5] [https://archive.org/details/charakasamhitagulakunvebaaurvedicsocietyvol6_130_M 6])
* c. 200 : [[Sextus Iulius Africanus|Iulius Africanus]], ''[[Cesti (Iulius Africanus)|Cesti]]'' et ''Paradoxa''
* c. 200 : [[Claudius Galenus|Galenus]], ''[[De facultatibus simplicium medicamentorum]]'' 9.109 [https://archive.org/details/klaudiougalenou00khgoog/page/n133/mode/2up vol. 12 p. 123 editionis 1826] 
* c. 200 : [[Claudius Galenus|Galenus]], ''[[De alimentorum facultatibus]]''  [https://archive.org/details/hapantaoperaomni06galeuoft/page/734/mode/1up vol. 6 p. 734 Kühn]
* c. 200 : [[Claudius Galenus|Galenus]], ''[[De victu attenuante]]''
* c. 200 : [[Claudius Galenus|Galenus]], ''[[De rebus boni malique suci]]''
* saec. III : [[Gargilius Martialis]], ''[[Medicinae ex holeribus et pomis]]''
* saec. III/IV? : [[Quintus Serenus]], ''[[De medicina praecepta (Serenus)|De medicina praecepta]]''
* c. 220 : &lt;span id=&quot;Deipnosophistae&quot;&gt;&lt;/span&gt;[[Athenaeus Naucratites|Athenaeus]], ''[[Deipnosophistae]]'' [http://digitalathenaeus.org/tools/KaibelText/search.php?what=μύραινα 312b-e (7.90) et alibi]
* c. 220 : [[Yang Fu]], ''[[异物志|Yiwu zhi]]''
* c. 300 : [[Ji Han]], ''[[Descriptio plantarum arborumque regionis meridianae|Nanfang caomu zhuang]]''
* saec. IV? : [[Palladius Rutilius Taurus Aemilianus|Palladius]], ''[[Opus agriculturae]]'' 12.6
* saec. IV? : [[Palladius Rutilius Taurus Aemilianus|Palladius]], ''[[De insitione (Palladius)|De insitione]]''
* saec. IV? : Pseudo-[[Antonius Musa]], ''[[De herba vettonica liber]]''
* saec. IV? : Pseudo-[[Apuleius (scriptor pseudonymus)|Apuleius]], ''[[Herbarius (pseudo-Apuleius)|Herbarius]]''
* saec. IV? : Pseudo-[[Dioscorides]], ''[[Liber medicinae ex herbis femininis]]''
* saec. IV? : &lt;span id=&quot;Apicius&quot;&gt;&lt;/span&gt;''[[Apicius sive De re coquinaria]]'' [http://www.fh-augsburg.de/~harsch/Chronologia/Lspost04/Apicius/api_re00.html textus] (Christopher Grocock, Sally Grainger, edd., ''Apicius. A critical edition with an introduction and an English translation'' [Totenais: Prospect Books, 2006. ISBN 1903018137])
* saec. IV? : [[Vindonius Anatolius]] Berytius, ''Synagoge consuetudinum rei rusticae'' (partim in ''[[Geoponica|Geoponicis]]'' servata)
* 312/315 : [[Eusebius Pamphili]], ''[[Praeparatio evangelica|Evangelicae praeparationis libri XV]]'' (E. H. Gifford, ed. et interpr., ''Eusebii Pamphili Evangelicae praeparationis libri XV'' [4 voll. in 5. Oxonii: e typographeo Clarendoniano, 1903] [https://archive.org/details/evangelicaepraep01euse 1] [https://archive.org/details/evangelicaepraep02euse 2] [https://archive.org/details/p1evangelicaepra03euse 3 i] [https://archive.org/details/p2evangelicaepra03euse 3 ii] [https://archive.org/details/eusebioutoupamp00eusegoog 4] [https://www.tertullian.org/fathers/index.htm#Praeparatio_Evangelica_(The_Preparation_of_the_Gospel) editio interretialis versionis Anglicae]
* 320 p.C.n. nisi antea : ''[[Liber documentorum|Shujing]]'', libri Shang, 3-6 ([[Iacobus Legge|James Legge]], ed. et interpr., ''The Chinese Classics''. Vol. 3 i-ii: ''The Shoo King'' [Hongcongi, 1865] [https://archive.org/details/chineseclassics31legg/page/184/mode/2up pp. 184-219])
* saec. IV exeunte : [[Decimus Magnus Ausonius|Ausonius]], ''Epistulae''
* c. 371 : [[Decimus Magnus Ausonius|Ausonius]], ''[[Mosella (Ausonius)|Mosella]]'' [https://www.hs-augsburg.de/~harsch/Chronologia/Lspost04/Ausonius/aus_mose.html Textus]
* c. 400 : [[Ambrosius Theodosius Macrobius|Macrobius]], ''[[Saturnalia (Macrobius)|Saturnalia]]'' [https://penelope.uchicago.edu/Thayer/L/Roman/Texts/Macrobius/Saturnalia/3*.html#15 3.15]
* saec. V? : [[Vinidarius]], ''[[Apici excerpta]]'' (Christopher Grocock, Sally Grainger, edd., ''Apicius. A critical edition with an introduction and an English translation'' [Totenais: Prospect Books, 2006. ISBN 1903018137] pp. 312-325)
* saec. V/VI : ''[[Suśrutasaṃhitā]]'' (Kunja Lal Bhushagratna, interpr., ''An English translation of the Sushruta samhita''. 3 voll. Calcuttae, 1907 [https://archive.org/details/englishtranslati01susruoft 1] [https://archive.org/details/b24758619_0002 2] [https://archive.org/details/b24758619_0003 3]) 
* post 511 : [[Anthimus (legatus)|Anthimus]], ''[[De observatione ciborum]]'' 61
* c. 544 : [[Jia Sixie]], ''[[Principales populo favendi artes|Qimin yaoshu]]''
* 587/588 : [[Venantius Fortunatus]], ''[[De navigio suo (Venantius)|De navigio suo]]'' [https://www.hs-augsburg.de/~harsch/Chronologia/Lspost06/Venantius/ven_ca10.html#09 vv. 29-42]
* c. 700 : &lt;span id=&quot;Meng Shen&quot;&gt;&lt;/span&gt;[[Meng Shen]], ''[[Materia diaetetica compendiosa|Shiliao bencao]]'' (vide [[#Shurtleff et Aoyagi (2012)]] p. 28)
* saec. VIII/X : [[Liber Exoniensis|Libri Exoniensis]] aenigma 74 vel 77 [https://web.archive.org/web/20060903091110/http://www2.kenyon.edu/AngloSaxonRiddles/Riddles/Riddle74.htm Textus]
* saec. VIII exeunte : [[Lu Yu]], ''[[Canon theae|Cha jing]]'' (&quot;Canon theae&quot;)
* saec. IX : {{Creanda|fr|Masawaiyh|Mesue}}
* saec. IX : {{Creanda|fr|Yuhanna ibn Masawaih|Ioannes Mesue}} (cf. Q62052886)
* ante 863 : [[Duan Chengshi]], ''[[Miscellaneae montis You offae|Youyang zazu]]''
* saec. IX/X : ''[[Medicinale Anglicum]]''
* saec. X : [[Isaac Iudaeus]], ''[[De diaetis universalibus et particularibus]]'' [http://daten.digitale-sammlungen.de/0004/bsb00042770/images/index.html Manuscriptum]
* saec. X : &lt;span id=&quot;Warrāq)&quot;&gt;&lt;/span&gt;Ibn Sayyār al-Warrāq, ''[[Liber coquinarius (Warrāq)|Kitāb al-ṭabīḫ]]'' (Kaj Öhrnberg, Sahban Mroueh, edd., ''Ibn Sayyār al-Warrāq: Kitāb al-ṭabīkh'' [Helsingiae: Finnish Oriental Society, 1987]; Nawal Nasrallah, interpr., ''Annals of the Caliphs' Kitchens: Ibn Sayyār al-Warrāq's Tenth-Century Baghdadi Cookbook'' [Lugduni Batavorum: Brill, 2007] {{Google Books|sQCwCQAAQBAJ|Paginae selectae}})
* saec. X/XI : ''[[Lacnunga]]''
* c. 1000 : {{Creanda|fr|Masawaih (Mésué le Jeune)|Ioannes Mesue iunior}} (cf. Q24794616)

== Saec. XI ==
* saec. XI : &lt;span id=&quot;al-Kāšġarī&quot;&gt;&lt;/span&gt;{{Creanda|en|Mahmud al-Kashgari|Mahometus al-Kāšġarī}}, ''[[Dīwān Luġāt al-Turk]]'' (Robert Dankoff, ed. et interpr., ''Maḥmūd al-Kāšγarī: Compendium of The Turkic Dialects (Dīwān Luγāt at-Turk)''. 3 voll. Cantabrigiae Massachusettensium, 1982)  [https://archive.org/details/CompendiumOfTheTurkicDialectsPart1-MahmudAl-Kashghari 1] [https://archive.org/details/CompendiumOfTheTurkicDialectsPartII-MahmudAl-Kashghari 2] [https://archive.org/details/CompendiumOfTheTurkicDialectsPartIII-MahmudAl-Kashghari 3]
* saec. XI : &lt;span id=&quot;Ḫāṣṣ Ḥājib&quot;&gt;&lt;/span&gt;{{Creanda|en|Yūsuf Balasaguni|Iosephus Ḫāṣṣ Ḥājib}}, ''[[Qutadğu Bilig]]'' (Robert Dankoff, interpr., ''Yusuf Khass Hajib: Wisdom of Royal Glory (Kutadgu Bilig): a Turko-Islamic mirror for princes''. Chicagine: University of Chicago Press, 1983)
* 1025 : [[Avicenna]], ''[[Liber canonis medicinae]]'' [https://archive.org/details/hin-wel-all-00002046-001/page/n198/mode/2up f. 78v editionis 1527]

== Saec. XII ==
* saec. XII? : ''[[Pākadarpaṇam]]'' (Madhulika, interpr.; Jay Ram Nadav, ed. Benaris, 2013 [http://www.saujanyabooks.com/details.aspx?id=41411 De hac editione] [https://www.ancientscienceoflife.org/article.asp?issn=0257-7941;year=2014;volume=33;issue=4;spage=259;epage=262;aulast=Kodlady Recensio])
* saec. XII : [[Someśvara III]], ''{{Creanda|en|Manasollasa|Mānasollāsa}}''
* saec. XII : ''[[Antidotarium Mesue]]'' seu ''Grabadin''
* saec. XII medio : &lt;span id=&quot;Luo Yuan&quot;&gt;&lt;/span&gt;[[Luo Yuan]], ''[[Alae expositoris|Erya yi]]'' (vide [[#Shurtleff et Aoyagi (2012)]] pp. 30-31)
* 1150/1175 : ''[[Salsamenta Pictavensium]]'' (Giles E. M. Gasper, Faith Wallis, &quot;Salsamenta Pictavensium: gastronomy and medicine in twelfth-century England&quot; in ''English Historical Review'' vol. 131 [2016] pp. 1353-1385 [https://dro.dur.ac.uk/19826/1/19826.pdf textus recensionis praeliminaris])
* 1150/1180 : [[Matthaeus Platearius]], ''[[Practica (Platearius)|Practica]]'' (cf. 1497 et 1582)
* 1160/1170 : ''[[Circa instans]]'' (cf. 1497 et 1582)
* ante 1187 : [[Gerardus Cremonensis]], Versio Latina Avicennae ''Canonis Medicinae'', [http://alfama.sim.ucm.es/dioscorides/consulta_libro.asp?ref=X532932098&amp;idioma=0] II 'Littera M', cpt. 451 'Melongena' p.353
* c. 1190 : &lt;span id=&quot;Nequam&quot;&gt;&lt;/span&gt;[[Alexander Nequam]], ''[[De nominibus utensilium]]'' ([[Thomas Wright]], ed., ''A Volume of Vocabularies'' [Liverpolii, 1882] pp. 96-119 ad [https://archive.org/details/b2487341x_0001/page/102/mode/2up p. 102] (&quot;viridis sapor, verde sause&quot;)
* saeculo XII exeunte : [[Radulphus de Diceto]], ''[[Ymagines historiarum]]'' (William Stubbs, ed., ''Radulfi de Diceto decani Lundoniensis opera historica'' [''[[Rerum Britannicarum Medii Aevi scriptores]]''. Londinii, 1876] [https://babel.hathitrust.org/cgi/pt?id=wu.89095820973&amp;view=1up&amp;seq=404&amp;skin=2021 vol. 1 p. 294] [https://archive.org/details/radulfidedicetod02dice vol. 2] [https://babel.hathitrust.org/cgi/pt?id=wu.89090758863&amp;view=1up&amp;seq=9&amp;skin=2021 vol. 2 alibi])
* c. 1250 : Lin Hong, ''{{Creanda|de|Shanjia qinggong|Simplicia montani victualia}}'' (Françoise Sabban, &quot;[https://www.persee.fr/doc/etchi_0755-5857_1997_num_16_1_1254 La diète parfaite d'un lettré retiré sous les Song du Sud]&quot; in ''Études chinoises'' vol. 16 (1997) pp. 7-57)

== Saec. XIII ==
* saec. XIII : ''Wuṣla ilā al-ḥabīb'' (Maxime Rodinson, ''Studies in Arabic Manuscripts'' in Maxime Rodinson, A. J. Arberry, Charles Perry, ''Medieval Arab Cookery'' [Totnes: Prospect Books, 2001] pp. 131-148)
* saec. XIII : Serapionis iunioris ''[[De simplici medicina (Serapio)|De simplici medicina]]'' iv.241 (''assa'')[https://archive.org/details/mobot31753000818200/page/160/mode/2up pp. 160-162 editionis 1531]
* saec. XIII : [[Nicolaus Myrepsus]], ''[[Antidotarium|Dynameron]]'' seu ''Antidotarium'' (vide 1549)
* saec. XIII : &lt;span id=&quot;Tujibi&quot;&gt;&lt;/span&gt;[[Ibn Razīn al-Tujībī]], ''Fiḍālat al-ḫiwān'' (Peter Heine, ''Kulinarische Studien: Untersuchungen zur Kochkunst im arabisch-islamischen Mittelalter'' (1988) p. 128 (&quot;qaṭāʾif&quot;))
* saec. XIII : &lt;span id=&quot;Andalusiensis&quot;&gt;&lt;/span&gt;''Kitāb al-ṭabīḫ fī'l-Maǧrīb wa'l-Andalūs'' (A. Huici Miranda, ed., ''La cocina Hispano-Magrebi en la España almohade'' [Matriti, 1965] f. 68v; Charles Perry, interpr., ''An Anonymous Andalusian Cookbook of the 13th Century'' [http://www.daviddfriedman.com/Medieval/Cookbooks/Andalusian/andalusian9.htm Textus]) (&quot;sanbûsak&quot;)
* saec. XIII : ''Wuṣla ilā al-ḥabīb'' (Maxime Rodinson, ''Studies in Arabic Manuscripts'' in Maxime Rodinson, A. J. Arberry, Charles Perry, ''Medieval Arab Cookery'' [Totnes: Prospect Books, 2001] pp. 131-148) cap. 7 no. 5, p. 141
* saec. XIII : ''[[La Bataille de Caresme et de Charnage]]'' (Grégoire Lozinski, ed., ''La Bataille de Caresme et de Charnage'' [Lutetiae: Honoré Champion, 1933])
* saec. XIII/XIV : &lt;span id=&quot;Kitāb waṣf&quot;&gt;&lt;/span&gt;''[[Descriptio ciborum communiorum|Kitāb waṣf al-ʿaṭima al-muʿtada]]'' (Charles Perry, &quot;The Description of Familiar Foods&quot; in Maxime Rodinson, A. J. Arberry, Charles Perry, ''Medieval Arab Cookery'' [Totnes: Prospect Books, 2001] pp. 273-465) 
* saec. XIII/XV : ''[[Tacuinum sanitatis]]'' (vide hanc paginam) {{Google Books|ggFcAAAAcAAJ|p. 87 editionis 1531}}
* saec. XIII ineunte : ''[[Egils saga Skalla-Grímssonar]]'' cap. 71
* 1223/1224 : [[Henricus de Andeliaco]], ''[[La Bataille des vins]]'' (Alexandre Héron, ''Oeuvres de Henri d'Andeli, trouvère normand du 13e siècle'' [Rotomagi, 1880] pp. [https://archive.org/details/oeuvresdehenrida00henruoft/page/22/mode/2up 23-30], [https://archive.org/details/oeuvresdehenrida00henruoft/page/86/mode/2up 87-129])
* 1226 : &lt;span id=&quot;Baghdadi&quot;&gt;&lt;/span&gt;[[Baghdādī]], ''[[Liber coquinarius Bagdatensis|Kitāb al-ṭabīḫ]]'' (Charles Perry, interpr., ''A Baghdad Cookery Book'' [''Petits propos culinaires'', no. 79. Totnes: Prospect Books, 2005])
* ante 1240 : [[Iacobus Vitriacensis]], ''[[Historia Orientalis et Occidentalis|Libri duo quorum prior Orientalis sive Hierosolymitanae, alter Occidentalis historiae nomine inscribitur]]''. Duaci: ex officina Balthazaris Belleri, 1597 [http://gallica.bnf.fr/ark:/12148/bpt6k54504q Textus apud Gallica] [https://archive.org/details/IacobiDeVitriacoCardinalisLibriDuo1596 textus] apud ''Internet Archive'' [http://mdz-nbn-resolving.de/urn:nbn:de:bvb:12-bsb10189102-1 textus] apud Monacenses
* c. 1240 : [[Gualterus de Bibbesworth]], ''[[Le Tretiz (Bibbesworth)|Le Tretiz]]'' (MS G) 465-466 ([[Andreas Dalby|Andrew Dalby]], interpr., ''The Treatise of Walter of Bibbesworth'' [Totenais: Prospect Books, 2012. ISBN 978-1-903018-86-6] pp. 82-83)
* ante 1250 : ''[[Libellus de arte coquinaria]]'' (M. Kristensen, ed., ''The socalled Harpestreng cookbook (13th century) = Das sog. Harpestreng-Kochbuch [13. Jh.]: Codex K: Harpestræng''. Hauniae, 1908-1920) [https://www.uni-giessen.de/fbz/fb05/germanistik/absprache/sprachverwendung/gloning/tx/harp-kkr.htm Textus]
* c. 1250 : Lin Hong, ''{{Creanda|de|Shanjia qinggong|Simplicia montani victualia}}'' (Françoise Sabban, &quot;[https://www.persee.fr/doc/etchi_0755-5857_1997_num_16_1_1254 La diète parfaite d'un lettré retiré sous les Song du Sud]&quot; in ''Études chinoises'' vol. 16 (1997) pp. 7-57)
* c. 1250 : &lt;span id=&quot;La Villeneuve&quot;&gt;&lt;/span&gt;Guillaume de La Villeneuve, &quot;Crieries de Paris&quot; vv. 16-22 (Étienne Barbazan, Dominique Martin Méon, edd., ''Fabliaux et contes des poètes françois des XIe, XIIe, XIIIe, XIVe et XVe siècles'' [Lutetiae, 1808] [https://gallica.bnf.fr/ark:/12148/bpt6k79574/f293 vol. 2 p. 277]
* 1250/1270 : [[Ioannes XXI|Petrus Hispanus]], ''[[Thesaurus pauperum]]''
* post 1254 : [[Gulielmus de Rubruquis]], ''Itinerarium'' (Peter Jackson, interpr., ''The Mission of Friar William of Rubruck''. Londinii: Hakluyt Society, 1990)
* 1259 : [[Albertus Magnus]], ''De vegetabilibus'' vi.2.2.279 {{Google Books|euAHAAAAIAAJ|pp. 279-280 editionis 1867}}
* saec. XIII exeunte : &lt;span id=&quot;Iamboninus&quot;&gt;&lt;/span&gt;[[Iamboninus Cremonensis]], ''[[Liber de ferculis|Liber de ferculis et condimentis]]'' (Anna Martellotti, ''Il Liber de ferculis di Giambonino da Cremona'' [Fasano: Schena, 2001])
* saec. XIII exeunte : &lt;span id=&quot;Wu Zimu&quot;&gt;&lt;/span&gt;Wu Zimu, ''Mengliang lu'' (vide [[#Shurtleff et Aoyagi (2012)]] pp. 33-34)
* saec. XIII exeunte : ''{{Creanda|fr|Tractatus de herbis}}'' (vide et {{Creanda|fr|Liste des simples traités dans le Tractatus de herbis}})
* c. 1275 : ''[[Viandier]]'': manuscriptum Sedunense (Terence Scully, ed., ''The Viandier of Taillevent: an edition of all extant manuscripts'' [Ottavae: University of Ottawa Press, 1988]) [https://histolf.ulb.be/index.php/textes-gh/251-1275-viandier-de-sion-texte-et-traduction Textus cum versione] ab Annick Englebert recensus
* c. 1282 : &lt;span id=&quot;Memoriali bolognesi&quot;&gt;&lt;/span&gt;''Memoriali bolognesi'' (Gianfranco Contini, ed., ''Poeti del Duecento''. 2 voll. Mediolani: Ricciardo Ricciardi, 1960)
* 1287 : [[Salimbene de Adam]], ''Cronica'' 786, sub die 11 Augusti 1284 (ed. Ferdinando Bernini [Baro: Laterza, 1942] p. 264)
* saeculo XIII exeunte : ''[[La Bataille de Caresme et de Charnage]]'' vv. [https://gallica.bnf.fr/ark:/12148/bpt6k40533/f21 249], [https://gallica.bnf.fr/ark:/12148/bpt6k40533/f28 433] [https://www.arlima.net/ad/bataille_de_caresme_et_de_charnage.html De opere]
* ante 1300 : ''[[Viande e claree|Coment l'en deit fere viande e claree]]'' (Constance B. Hieatt, R. Jones, &quot;Two Anglo-Norman culinary collections&quot; in ''Speculum'' vol. 61 (1986) pp. 859-882 [https://www.jstor.org/stable/2853971 JSTOR])
* c. 1300 : ''[[Enseignements]]'' [https://www.uni-giessen.de/fbz/fb05/germanistik/absprache/sprachverwendung/gloning/tx/1300ens.htm no. 62] (Grégoire Lozinski, ed., ''La Bataille de Caresme et de Charnage'' [Lutetiae: Honoré Champion, 1933] [https://gallica.bnf.fr/ark:/12148/bpt6k40533/f191.item pp. 181-187]) [https://histolf.ulb.be/index.php/textes-gh/254-1300-les-enseingnemenz-texte-et-traduction alibi]

== Saec. XIV ==
* saec. XIV : &lt;span id=&quot;Thesaurus&quot;&gt;&lt;/span&gt;''[[Thesaurus beneficiorum mensaeque varietatum|Kanz al-fawāʾid fī tanwīʿ al-mawāʾid]]'' (Manuela Marin, David Waines, edd. [Stutgardiae: Steiner, 1993]; Nawal Nasrallah, interpr., ''Treasure Trove of Benefits and Variety at the Table'' [Lugduni Batavorum: Brill, 2018])
* saec. XIV : &lt;span id=&quot;Sent soví&quot;&gt;&lt;/span&gt;''[[Llibre de sent soví]]'' (Joan Santanach i Suñol, ed., Robin M. Vogelzang, interpr., ''The book of Sent Soví: medieval recipes from Catalonia'' [2008] pp. 68-69, no. xiii) (&quot;escabetx a peix frit&quot;)
* saec. XIV : &lt;span id=&quot;Libro per cuoco&quot;&gt;&lt;/span&gt;''[[Libro per cuoco]]'' vel ''Anonimo Veneziano'' (Ludovico Frati, ed., ''Libro di cucina del secolo XIV'' (Liburni, 1899) no. 80 [https://www.uni-giessen.de/fbz/fb05/germanistik/absprache/sprachverwendung/gloning/tx/frati.htm Textus a Thoma Gloning divulgatus])
* saec. XIV ineunte : [[BL Royal 12.C.xii]] f. 11r (Constance B. Hieatt, R. Jones, &quot;Two Anglo-Norman culinary collections&quot; in ''Speculum'' vol. 61 (1986) pp. 859-882 [https://www.jstor.org/stable/2853971 JSTOR])
* c. 1300/1320 : [[Matthaeus Silvaticus]], ''[[Liber pandectarum medicinae]]'' (vide infra)
* 1300/1309 : &lt;span id=&quot;Liber de coquina&quot;&gt;&lt;/span&gt;''[[Liber de coquina|Liber de coquina ubi diversitates ciborum docentur]]'' (Marianne Mulon, &quot;Deux traités inédits d'art culinaire médiéval&quot; in ''Bulletin philologique et historique (jusqu'à 1610) du Comité des Travaux Historiques et Scientifiques'' [1968 (1971)] vol. 1 pp. 369-435) [http://www.uni-giessen.de/gloning/tx/mul2-lib.htm Textus]
* 1305/1308 : &lt;span id=&quot;Regimen sanitatis&quot;&gt;&lt;/span&gt;[[Arnaldus de Villa Nova]], ''[[Regimen sanitatis ad regem Aragonum]]'' ([https://daten.digitale-sammlungen.de/0006/bsb00065328/images/index.html?id=00065328&amp;groesser=&amp;fip=xdsydfsdryztsyztsxdsydsdasxdsyd&amp;no=64&amp;seite=54 editio 1474])
* ante 1309 : ''[[Tractatus de modo praeparandi et condiendi omnia cibaria]]'' (Marianne Mulon, &quot;Deux traités inédits d'art culinaire médiéval&quot; in ''Bulletin philologique et historique (jusqu'à 1610) du Comité des Travaux Historiques et Scientifiques'' [1968 (1971)] vol. 1 pp. 369-435) [https://www.uni-giessen.de/fbz/fb05/germanistik/absprache/sprachverwendung/gloning/tx/mul1-tra.htm Textus]
* 1309 : [[Petrus de Crescentiis]], ''[[Ruralia commoda]]'' (vide hanc paginam)
* ante 1325 : ''[[Doctrina faciendi diversa cibaria]]'' ([[Constantia Hieatt|Constance B. Hieatt]], Sharon Butler, edd., ''Curye on Inglysch'' [Londinii: Oxford University Press, 1985] pp. 43-58)
* c. 1325 : ''[[Tractatus de vinis]]'' [[Arnaldus de Villa Nova|Arnaldo de Villa Nova]] falso adscriptum
* 1330 : &lt;span id=&quot;Hu&quot;&gt;&lt;/span&gt;Hu Si-hui, ''[[Propria ad mensam Imperatoris principia]]'' (Paul D. Buell, Eugene N. Anderson, edd. et interprr., ''A Soup for the Qan: Chinese dietary medicine of the Mongol era as seen in Hu Szu-hui's Yin-shan cheng-yao'' [Londinii: Kegan Paul, 2000] no. 10, p. 282)
* c. 1335 : [[Maynus de Mayneriis]], ''[[Opusculum de saporibus]]'' ([[Lynn Thorndike]], &quot;A Medieval Sauce-Book&quot; in ''Speculum'' vol. 9 (1934) pp. 183-190 [https://www.jstor.org/stable/2846594 JSTOR]; cf. [[Terentius Scully|Terence Scully]], &quot;The Opusculum de saporibus of Magninus Mediolanensis&quot; in ''Medium aevum'' vol. 54 (1985) pp. 178-207 [https://www.jstor.org/stable/43628892 JSTOR])
* c. 1340 : [[Ibn Baṭṭūṭa]], ''[[Peregrinatio (Ibn Baṭṭūṭa)|Peregrinatio]]'' (C. Defrémery, B. R. Sanguinetti, edd. et interprr., ''Voyages d'Ibn Batoutah'' [5 voll. Lutetiae: Société Asiatique,  1853-1859] [https://archive.org/details/ldpd_6017227_000 Vol. 1] [https://archive.org/details/voyagestextearab02ibnb vol. 2] [https://archive.org/details/voyagestextearab03ibnb vol. 3] [https://archive.org/details/voyagestextearab04ibnb vol. 4] [https://archive.org/details/ldpd_5998015_000 Index])
* medio saeculo XIV : Jia Ming, ''[[De potu ciboque scienda|Yinshi xuzhi]]'' (T.T. Chang, &quot;Chia Ming's Elements of Dietetics: a summary of the first volume with an introduction&quot; in '' Isis'' vol. 20 (1933) pp. 324-334
* 1353 : [[Iohannes Boccacius]], ''[[Decameron]]''
* c. 1355 : &lt;span id=&quot;Boccacius&quot;&gt;&lt;/span&gt;[[Iohannes Boccacius|Boccaccio]], ''Corbaccio''
* ante 1374 : &lt;span id=&quot;Ni Zan&quot;&gt;&lt;/span&gt;[[Ni Zan]], ''[[Aedis silvarum nebulosarum regimentarium|Yunlintang yinshi zhidu shi]]'' (Teresa Wang, E. N. Anderson, &quot;Ni Tsan and his Cloud Forest Hall Collection of Rules for Drinking and Eating&quot; in ''Petits Propos Culinaires'' no. 60 (1998) pp. 24-41) 
* 1377/1400 : ''[[Forme of cury]]'' no. 22, 202 ([[Constantia Hieatt|Constance B. Hieatt]], Sharon Butler, edd., ''Curye on Inglysch'' [Londinii: Oxford University Press, 1985] pp. 93-145) (&quot;mawmenee, mawmenny&quot;) 
* 1381 : ''[[Diversa servicia]]'' no. 30 ([[Constantia Hieatt|Constance B. Hieatt]], Sharon Butler, edd., ''Curye on Inglysch'' [Londinii: Oxford University Press, 1985] pp. 59-79) (&quot;maumene&quot;)
* ante 1390 : &lt;span id=&quot;Modus&quot;&gt;&lt;/span&gt;''[[Modus viaticorum preparandorum et salsarum]]'' 
* saec. XIV exeunte : &lt;span id=&quot;Llibre d'aparellar&quot;&gt;&lt;/span&gt;''[[Llibre d'aparellar de menjar]]'', [https://mdc.csuc.cat/digital/collection/manuscritBC/id/290716 Bibliotheca de Catalunya MS. 2112] ff. 25r-35r (Joan Santanach i Suñol, ed., Robin M. Vogelzang, interpr., ''The book of Sent Soví: medieval recipes from Catalonia'' [2008] pp. 190-191) (&quot;menjar blanc&quot;)
* saec. XIV exeunte : ''[[Utilis coquinario]]'' no. 25 ([[Constantia Hieatt|Constance B. Hieatt]], Sharon Butler, edd., ''Curye on Inglysch'' [Londinii: Oxford University Press, 1985] pp. 61-91) (&quot;mawmene&quot;)
* saec. XIV exeunte : ''[[Grettis saga Ásmundarsonar]]'' cap. 28
* c. 1390/1404 : ''[[Herbarium Carrarense]]''
* c. 1393 : &lt;span id=&quot;Mesnagier&quot;&gt;&lt;/span&gt;''[[Le Mesnagier de Paris]]'' (Jérôme Pichon, ed., ''Le ménagier de Paris ... par un bourgeois parisien'' (Lutetiae, 1846-1847) [https://archive.org/details/lemnagierdepari01renagoog/page/n111/mode/2up vol. 2 pp. 100], [https://archive.org/details/lemnagierdepari01renagoog/page/n185/mode/2up 175]

== Saec. XV ==
* saec. XV : [[Ibn al-Mubarrad]], ''Kitāb al-Ṭibāḫa'' (Charles Perry, &quot;''Kitāb al-Ṭibāḫa'': a fifteenth-century cookbook&quot; in Maxime Rodinson, A. J. Arberry, Charles Perry, ''Medieval Arab Cookery'' (Totnes: Prospect Books, 2001) p. 473 
* saec. XV : ''{{Creanda|fr|Livre des simples médecines}}''
* saec. XV : ''[[Mensura omnium Oceani litorum|Yingyai shenglan]]'' {{Google Books|DjQ9AAAAIAAJ|p. 118 versionis Anglicae}}
* saec. XV ineunte : ''[[Viandier]]'': manuscriptum Vaticanum (Terence Scully, ed., ''The Viandier of Taillevent: an edition of all extant manuscripts'' [Ottavae: University of Ottawa Press, 1988] no. 95, 199) (&quot;blanc mengier d'un chappon pour ung malade; blanc menger party&quot;)
* c. 1420 : &lt;span id=&quot;Liber cure cocorum&quot;&gt;&lt;/span&gt;''[[Liber cure cocorum]]'' (Richard Morris, ed., ''Liber cure cocorum'' [Londinii: Philological Society, 1862] [https://archive.org/details/libercurecocorum00morr/page/18/mode/2up p. 19]) (&quot;blanc maungere of fysshe&quot;) 
* c. 1420 : ''Potage Dyvers I'' (British Library, MS Harl. 279, c.1420) (Thomas Austin, ed., ''Two Fifteenth-Century Cookery Books'' [EETS OS. 91] 1888, pp. 5-56) [https://archive.org/details/twofifteenthcent00austuoft/page/22/mode/2up p. 23] (&quot;oystrys in bruette&quot;)
* c. 1420 : [[Chiquardus]], ''[[Du fait de cuisine]]'' ([[Terentius Scully]], ed., &quot;[https://doc.rero.ch/record/21865/files/I-N-268_1985_06_00.pdf?version=1 Du fait de cuisine par Maistre Chiquart, 1420]&quot; in ''Vallesia'' [1985] pp. 101-231)
* ante 1431 : [[Iohannes de Bockenheim]], ''[[Registrum coquinae]]'' [https://gallica.bnf.fr/ark:/12148/btv1b10035736w/f66 MS BNF latin 7054] [https://www.persee.fr/doc/mefr_0223-5110_1988_num_100_2_2987 Editio]
* c. 1433 : [[Bertrandon de la Broquière]], ''Voyage d'outremer'' [http://gallica.bnf.fr/ark:/12148/btv1b8449038d Liber manuscriptus] (Ch. Schefer, ed., ''Le voyage d'outremer de Bertrandon de La Broquière'' (Lutetiae, 1892) [https://archive.org/details/levoyagedoutreme00labruoft/page/130/mode/2up p. 130])
* 1433 : [[Ma Huan]], ''[[Mensura omnium Oceani litorum|Yingyai shenglan]]'' (J. V. G. Mills, interpr., ''Ying-yai Sheng-lan: 'The Overall Survey of the Ocean's Shores' [1433]'' [Cantabrigiae: Cambridge University Press, 1970. [[Societas Hakluyt|Hakluyt Society]] ] [https://faculty.washington.edu/qing/huan_ying-yai_sheng-lan%5B1%5D.pdf Textus pp. 137-178] {{Google Books|DjQ9AAAAIAAJ|Paginae selectae}})
* 1430/1440 : [[Bibliotheca Britannica|BL]] MS Harleianus 279 [https://www.gutenberg.org/files/24790/24790-h/keruyng.html Textus]
* c. 1440 : ''Potage Dyvers II'' (British Library, MS Harl. 4016, c.1440) (Thomas Austin, ed., ''Two Fifteenth-Century Cookery Books'' [EETS OS. 91] 1888, pp. 69-107) [https://archive.org/details/twofifteenthcent00austuoft/page/100/mode/2up p. 100] (&quot;oystres in gvey&quot;)
* c. 1440 : ''[[Promptorium parvulorum]]'' (Albertus Way, ed., ''Promptorium parvulorum'' [Londinii: Societas Camdenensis, 1865] [https://archive.org/details/promptoriumparvu00camduoft/page/30/mode/2up p. 31])
* c. 1440 : ''[[Horae Catharinae Cliviensis]]'' [https://www.themorgan.org/collection/Hours-of-Catherine-of-Cleves/thumbs Textus]
* c. 1450 : ''[[Vivendier]]'' ([[Terentius Scully|Terence Scully]], ''The Vivendier ... A critical edition with English translation''. Totnes: Prospect Books, 1997)
* c. 1450 : ''An Ordinance of Pottage'' (MS. Yalensis Beinecke 163: Constance B. Hieatt, ed., ''An Ordinance of Pottage'' [Londinii: Prospect Books, 1988]) no. 32 (&quot;conyngges in grave&quot;)
* c. 1450 : &lt;span id=&quot;Savonarola&quot;&gt;&lt;/span&gt;[[Michaël Savonarola]], ''Libreto de tutte le cosse che se magnano'' (ed. Jane Nystedt [Holmiae: Almqvist &amp; Wiksell, 1988] p. 62) (&quot;Fasse de farina lassagne, lassagnole, menudelli, dicti tri in medesina&quot;)
* post 1467 : ''[[Noble Book of Cookery|A Noble Boke off Cookry]]'' (MS. Holkham 674) (Robina Napier, ed., ''A Noble Boke off Cookry ffor a prynce houssolde or eny other estately houssolde''. Londinii: Elliot Stock, 1882 [https://archive.org/details/b21529565 Textus] apud ''Internet Archive''; [http://www.medievalcookery.com/notes/napier.txt recensio interretialis]; [https://quod.lib.umich.edu/c/cme/CME00005  recensio interretialis]); cf. 1500
* c. 1470 : &lt;span id=&quot;Martino&quot;&gt;&lt;/span&gt;[[Martinus Comensis (magister coquinarius)|Maestro Martino]] (Gillian Riley, interpr., ''Maestro Martino: Libro de Arte Coquinaria'' (CD-ROM. Quercupoli: Octavo, 2005. ISBN 1-891788-83-3) p. 9 versionis Anglicae
* c. 1480 : [[Bibliotheca Britannica|BL]] MS Harleianus 5401 (Constance B. Hieatt, &quot;The Middle English Culinary Recipes in MS Harley 5401: an Edition and Commentary&quot; in ''Medium Aevum'' vol. 65 (1996) pp. 54-71) [https://www.gutenberg.org/files/24790/24790-h/keruyng.html Textus]
* saec. XV exeunte : ''[[Cuoco Napoletano]]'' (Terence Scully, [[Rudolphus Grewe|Rudolf Grewe]], edd., ''The Neapolitan Recipe Collection: Cuoco Napoletano'' (Ann Arbor: University of Michigan Press, 2000) p. 41 {{Google Books|5p_bcD4uUMYC|Paginae selectae}}
* c. 1500 : MS [[Bibliotheca universitatis Gandavensis|Gandavensis]] 476 (Ria Jansen-Sieben, Johanna Maria van Winter, edd., ''De keuken van de late Middeleeuwen. Een kookboek uit de Lage Landen''. Amstelodami: Bert Bakker, 1998) [https://www.dbnl.org/tekst/jans016keuk01_01/jans016keuk01_01.pdf recensio interretialis])

== post Gutenberg ==
* 1473 : ''[[De simplici medicina (Serapio)|Liber Serapionis agregatus in medicinis simplicibus]]''. Mediolani [http://daten.digitale-sammlungen.de/0007/bsb00070094/images/index.html Textus]; Venetiis, 1479 [http://daten.digitale-sammlungen.de/0006/bsb00064401/images/index.html Textus]
* 1474 : [[Matthaeus Silvaticus]], ''[[Liber pandectarum medicinae]]'' Bononiae [http://daten.digitale-sammlungen.de/~db/0006/bsb00069437/images/ Textus]; [http://daten.digitale-sammlungen.de/bsb00073212/image_1 Argentorati 1480]; [http://daten.digitale-sammlungen.de/bsb00060727/image_1 Venetiis 1492]; [http://daten.digitale-sammlungen.de/bsb00060715/image_1 Venetiis 1498]; [http://daten.digitale-sammlungen.de/bsb00060630/image_1 Venetiis 1499]; [http://mdz-nbn-resolving.de/urn:nbn:de:bvb:12-bsb10147814-6 Venetiis 1507]; [http://mdz-nbn-resolving.de/urn:nbn:de:bvb:12-bsb11200086-7 Papiae 1508]; [http://mdz-nbn-resolving.de/urn:nbn:de:bvb:12-bsb10147815-2 Venetiis 1523]; [http://mdz-nbn-resolving.de/urn:nbn:de:bvb:12-bsb10147816-7 Taurinis 1526]; [http://mdz-nbn-resolving.de/urn:nbn:de:bvb:12-bsb10147817-2 Venetiis 1540]
* 1470/1475 : &lt;span id=&quot;Platina (1471?)&quot;&gt;&lt;/span&gt;[[Bartholomaeus Platina]], ''[[De honesta voluptate et valetudine]]'' [https://archive.org/details/Platine-De-honesta-voluptate-et-valitudine-ad-amplissimum-ac-doctissimum-D-B-Rou-PHAIDRA_o_359459 editio sine anno] [https://archive.org/details/ita-bnc-in2-00000024-001 editio 1475]
* c. 1477 : [[Robertus de Nola]], ''Libre del coch'' (vide 1520, 1525)
* 1484 : ''[[Herbarius (Moguntiae, 1484)|Herbarius]]'' (vide hanc paginam)
* 1485 : Ioannes de Cuba, ''[[Hortus sanitatis]]'' seu ''Gart der Gesundheit'' (vide hanc paginam)
* 1487 : [[Isaac Iudaeus]], ''[[De diaetis universalibus et particularibus|De particularibus diaetis]]''. Patavii [http://daten.digitale-sammlungen.de/0007/bsb00070131/images/index.html Textus]
* 1455/1488 : &lt;span id=&quot;Cadamustus&quot;&gt;&lt;/span&gt;[[Aloisius Cadamustus]], ''Libro de la prima nauigatione per locceano a le terre de Nigri'' in [[Francanus Montalbodus|Fracanzano Montalboddo]], ed., ''Paesi novamente retrovati et nouo mondo da Alberico Vesputio Florentino'' [http://gallica.bnf.fr/ark:/12148/bpt6k58988n Textus 1507 apud Gallica] [https://archive.org/details/paesinouamentere00frac Editio 1508 rursus impressa 1916] apud ''Internet Archive''
* 1491 : [[Antonius Gazius]], ''Corona florida medicinae''. Venetiis, 1491 [https://gallica.bnf.fr/ark:/12148/bpt6k604578/ Textus aegre legibilis] [https://archive.org/details/hin-wel-all-00000949-001/ editio 1534] [https://archive.org/details/hin-wel-all-00000948-001 alia]
* 1492 : [[Nicolaus Leonicenus]], ''De Plinii et aliorum in medicina erroribus''. Ferrariae [http://daten.digitale-sammlungen.de/~db/0006/bsb00067914/images/index.html Textus]
* ante 1493 : [[Hermolaus Barbarus]], ''Corollarium ad Dioscoridem in quinque Libros priores de materia Medica'' [http://alfama.sim.ucm.es/dioscorides/consulta_libro.asp?ref=X532556568] Liber IV, cpt.79 'De Mandragora' p.239r
* 1493 : &lt;span id=&quot;Columbi ephemeris (1493)&quot;&gt;&lt;/span&gt;[[Christophorus Columbus]], Ephemeris primae navigationis a [[Bartholomaeus Casaus|Bartholomaeo Casao]] rescripta, in Martin Fernandez de Navarrete, ed., ''Colección de los viages y descubrimientos'' vol. 1 (2a ed. Matriti, 1858) [https://archive.org/stream/coleccindelosv01nava#page/152/mode/2up pp. 153-313]
* 1493-1505 : &lt;span id=&quot;Columbi epistulae (1493-1505)&quot;&gt;&lt;/span&gt;[[Christophorus Columbus]], Epistulae de navigationibus in Martin Fernandez de Navarrete, ed., ''Colección de los viages y descubrimientos que hicieron por mar los españoles desde fines del siglo XV'' vol. 1 (Matriti, 1825) [http://bdh.bne.es/bnesearch/detalle/bdh0000052618 Textus] (2a ed. Matriti, 1858) [https://archive.org/details/coleccindelosv01nava Textus]
* 1493-1503 : &lt;span id=&quot;Columbi liber copiarum (1493-1503)&quot;&gt;&lt;/span&gt;[[Christophorus Columbus]], Epistulae ineditae in Antonio Rumeu de Armas, ed., ''Libro copiador de Cristóbal Colón: correspondencia inédita con los Reyes Católicos sobre los viajes a América'' (Matriti, 1989) [[:es:Libro copiador de Colón#Enlaces externos|Editiones interretiales]]
* 1494 : &lt;span id=&quot;Chanca (1494)&quot;&gt;&lt;/span&gt;[[Didacus Álvarez Chanca]], Epistula de secunda navigatione Christophori Columbi in Martín Fernández de Navarrete, ed., ''Coleccion de los viages y descubrimientos que hicieron por mar los españoles desde fines del siglo XV'' vol. 1 (Matriti: Imprenta Real) pp. 198-224 ([http://bdh.bne.es/bnesearch/detalle/bdh0000052618 imagines 362-388]; (2a ed. Matriti, 1858) [https://archive.org/stream/coleccindelosv01nava#page/346/mode/2up pp. 347-372]; versio Anglica: A. M. Fernandez de Ybarra, &quot;[https://repository.si.edu/handle/10088/26153 The letter of Dr. Diego Alvarez Chanca, dated 1494, relating to the second voyage of Columbus to America (being the first written document on the natural history, ethnography and ethnology of America)]&quot; in ''Smithsonian Miscellaneous Collections'' vol. 48 (1907) pp. 428-457
* 1494-1530 : &lt;span id=&quot;Petrus Martyr (1530)&quot;&gt;&lt;/span&gt;[[Petrus Martyr ab Angleria]], ''[[De orbe novo decades]]'' lib. 5 cap. 9 [https://archive.org/stream/deorbenouopetrim00angh#page/n173/mode/2up f. 83r]
* 1497 : ''Practica [[Ioannes Serapio|Jo. Serapionis]] dicta Breviarium; Liber Serapionis [[De simplici medicina (Serapio)|de simplici medicina]]; Liber de simplici medicina dicta [[Circa instans]]; Practica [[Matthaeus Platearius|Platearii]]''. Venetiis: mandato Octaviani Scoti per Bonetum Locatellum [http://beta.biblissima.fr/ark:/43093/edataaed898ad32478b39e3ed4c0ff68f7a0597ab9fc4 Textus] [https://daten.digitale-sammlungen.de/0006/bsb00061068/images/index.html Textus] [https://gallica.bnf.fr/ark:/12148/bpt6k58567k Textus]
* 1497 : [[Ioannes Michael Savonarola]],''Practica medicinae''. Venetiis: mandato Octaviani Scoti per Bonetum Locatellum [http://daten.digitale-sammlungen.de/0006/bsb00061120/images/index.html]
* 1499 : [[Polydorus Vergilius]], ''De inventoribus rerum libri tres'' {{Google Books|LEg8AAAAcAAJ|editio Veneta 1503}} (cf. 1521) 
* c. 1500 : &lt;span id=&quot;Nimmatnama&quot;&gt;&lt;/span&gt;''{{Creanda|en|Nimmatnama-i-Nasiruddin-Shahi}}'' (Norah M. Titley, ed. et interpr., ''The Ni'matnama Manuscript of the Sultans of Mandu: The Sultan's Book of Delights'' [Londinii: Routledge, 2004] {{Google Books|LGp_AgAAQBAJ|Fragmentum}})
* 1500 : ''[[Noble Book of Cookery|A Noble Boke off Cookry]]''. Londinii: Richard Pynson, 1500; cf. 1467

== 1501-1550 ==
* saec. 16 : Kṣemaśarmā, ''Kṣemakutūhalam'' (R. Shankar, interpr. Bengaluri, 2009)
* saec. 16 : Lolimbarāja, ''Vaidyajivana'' (Nirmal Saxena, interpr. 2000 [https://www.vedicbooks.net/vaidya-jivana-lolimbaraja-p-14132.html De hoc libro])
* saec. 16 : ''The Image of Hypocrisy'' (F. J. Furnivall, ed., ''Ballads from Manuscripts'' vol. 1 [1868] p. 218)
* 1502 : Benjamin F. Stevens, ed., ''Christopher Columbus: his own book of privileges''. Londinii, 1893 [https://archive.org/details/christophercolum00colu Textus] apud ''Internet Archive''
* 1503 : [[Americus Vesputius|Albericus Vespuccius]]; Ioannes Iocundus, interpr., ''Mundus novus'' [https://archive.org/details/mundusnouus00vesp Editio 1504] (Augustae Vindelicae: Iohannes Otmar) apud ''Internet Archive''
* 1506/1508 : [[Eduardus Pacheco Pereira|Duarte Pacheco Pereira]]; Raphael Eduardo de Azevedo Basto, ed., ''Esmeraldo de situ orbis'' (Olisipone, 1892) [http://purl.pt/28964/1/index.html#/112-113/html pp. 57-58]  [https://archive.org/details/esmeraldodesituorbisporduartepachecopereira/page/n95/mode/2up p. 52]
* c. 1508 : [[Maṅgarāsa III]], ''Sūpaśastra'' (Madhukar Konantambigi, interpr., ''Culinary Traditions of Medieval Karnataka: the Soopa Shastra of Mangarasa III'' (Dellii, 2012)
* 1508 : &lt;span id=&quot;Madrignanus (1508)&quot;&gt;&lt;/span&gt;Archangelus Madrignanus, interpr.; Francanus Montalbodus, ed., ''Itinerarium Portugallensium e Lusitania in Indiam et inde in occidentem et demum ad aquilonem'' [https://archive.org/details/bub_gb_OEBMA3dMZcMC Textus] apud ''Internet Archive''
* 1508 : [[Ioannes Bourdichon]], picturae botanicae ''[[Horae Annae Britanniae|horarum Annae Britanniae]]'' (manuscriptum, 1503-1508) [http://gallica.bnf.fr/ark:/12148/btv1b52500984v/f344.image f. 168r]
* 1511 : [[Aegidius Gallus]], ''De viridario Augustini Chigii'' [https://www.jstor.org/stable/23973712 Editio 1989]
* 1511 : [[Nicolaus de la Chesnaye|Nicole de la Chesnaye]], ''La Condamnacion de bancquet''. Lutetiae, 1511 (P. L. Jacob, ed., ''Recueil de farces, soties et moralités du quinzième siècle'' [Lutetiae, 1859] pp. 269-454 ad p. 310) (&quot;Tout premier vous sera donnée Saulce robert et cameline ...&quot;) [https://archive.org/details/RecueilFarcesSotiesJacob1876/ aliud exemplar]
* 1512 : [[Blosius Palladius]], ''Suburbanum Augustini Chisii'' {{Google Books|ipNWAAAAcAAJ}}
* 1515 : [[Isaac Iudaeus]], ''Omnia opera Ysaac''. Lugduni [https://archive.org/details/1856324.med.yale.edu Textus]
* ante 1516 : [[Eduardus Barbosa|Duarte Barbosa]] ''Livro de Duarte Barbosa'' in ''Collecção de noticias para a historia e geografia das nações ultramarinas'' vol. 2 no. 7 (Olisipone, 1813) [https://archive.org/details/collecodenot10lisb/page/n287 Textus] [https://babel.hathitrust.org/cgi/pt?id=nyp.33433082405238;view=1up;seq=287 alius]; Augusto Reis Machado, ed., ''Livro em que dá relação do que viu e ouviu no Oriente''. Olisipone, 1946 [https://web.archive.org/web/20090303053302/http://purl.pt/435 Textus]
* 1516 : [[Ioannes Ruellius]], interpr., ''Pedacii Dioscoridis Anazarbei de medicinali materia libri quinque De virulentis animalibus, et venenis canerabioso, et corum notis, ac remediis libri quattuor. Joanne Ruellio Suessionensi interprete''. Lutetiae apud [[Henricus Stephanus|Henricum Stephanum]] {{Ling|Latine}} [http://www.biusante.parisdescartes.fr/histmed/medica/cote?00815 Textus] [https://archive.org/details/dioscoridislibr00ruelgoog Editio bilinguis 1549] [http://daten.digitale-sammlungen.de/bsb00011921/image_5 Editio cum aliorum adnotationibus 1549] [https://archive.org/details/mobot31753003467922 Editio 1552]
* 1518 : [[Ioannes Nannius Utinensis]], picturae botanicae [[porticus Amoris et Psychae]] in [[viridarium Augustini Chigii|viridarii Augustini Chigii]] [https://hort.purdue.edu/newcrop/udine/ quaere speciem no. 68]
* 1519-1544 : [[Ferdinandus Cortesius]], Epistulae, in Pascual de Gayangos, ed., ''Cartas y relaciones de Hernan Cortés al emperador Carlos V''. Lutetiae, 1866 [https://archive.org/details/gri_cartasyrelac00cort Textus] apud ''Internet Archive'' {{Google Books|e7IGAAAAQAAJ}}
* post 1519 : &lt;span id=&quot;Relazione (post 1519)&quot;&gt;&lt;/span&gt; &quot;{{Creanda|it|Relazione d'alcune cose della Nuova Spagna e della gran città di Temestitan Messico|Relatione d'alcune cose della Nuova Spagna e della gran città di Temestitan Messico}}&quot; in [[Ioannes Baptista Ramusius|Giovanni Battista Ramusio]], ed., ''[[Navigationi et viaggi (Ramusius)|Navigationi et viaggi]]'' (Venetiis, 1550-1559) vol. 3 [https://archive.org/details/terzovolumedelle32ramu/page/n691/mode/2up ff. 304v-310r] [https://archive.org/stream/dellenavigationi00ramu#page/n595/mode/2up ff. 254r-259r editionis 1606] 
* 1480/1520 : [[Eustathius de la Fosse|Eustache de la Fosse]]; R. Fourché-Delbosc, ed., &quot;Voyage à la côte occidentale d'Afrique&quot; in ''Revue hispanique'' vol. 4 (1897) [https://archive.org/details/revuehispaniquer04hispuoft/page/174/mode/2up pp. 174-201]
* 1520 : &lt;span id=&quot;Robertus (1520)&quot;&gt;&lt;/span&gt;[[Robertus de Nola]], ''[[Libre del coch|Libre de doctrina per a ben servir, de tallar, y del art de coch]]''. Barcinone, 1520 {{Google Books|rF-PPdZ_SjIC}} [http://www.cervantesvirtual.com/obra-visor/libre-de-doctrina-per-a-ben-servir-de-tallar-y-del-art-de-coch--1/html/ series paginarum]; [[:File:Llibre del Coch (1520).djvu|alibi]]; [http://www.cervantesvirtual.com/obra-visor/libre-de-doctrina-per-a-ben-servir-de-tallar-y-del-art-de-coch-transcripcio--0/html/ff1e2444-82b1-11df-acc7-002185ce6064_2.html#I_211_ recensio interretialis] {{Google Books|xO85L2pjvDgC|editio 1568}}
* 1521 : [[Polydorus Vergilius]], ''Adagiorum liber ... De inventoribus rerum libri octo''. Basileae: Frobenius {{Google Books|vs9WAAAAcAAJ}}
* 1524 : [[Paulus Iovius]], ''[[De Romanis piscibus libellus]]''. Romae: F.M. Calvo [http://archive.org/details/pauliiouiinouoco00giov Textus] apud ''Internet Archive''  {{Google Books|yW08AAAAcAAJ|ed. 1527 titulo ''De piscibus marinis ...''}} {{Google Books|yzBOAAAAcAAJ|ed. 1531}}; Carlo Zancaruolo, interpr., ''Libro ... de' pesci romani''. Venetiis, 1560 [https://archive.org/details/bub_gb_9O5eSH2-bTgC Textus] apud ''Internet Archive'' [https://archive.org/details/depesciromani00giov Textus]
* 1525 : &lt;span id=&quot;Robertus (1525)&quot;&gt;&lt;/span&gt;[[Robertus de Nola]], ''[[Libre del coch|Libro de cozina]]''. Toleti: Ramon de Petras, 1525 {{Ling|Hispanice}}  [http://bdh.bne.es/bnesearch/detalle/bdh0000061324 Textus]
* 1526 : &lt;span id=&quot;Oviedo (1526)&quot;&gt;&lt;/span&gt;[[Gundisalvus Fernández de Oviedo y Valdés|Gonzalo Fernandez de Oviedo y Valdés]], ''Dela natural hystoria delas Indias''. Toleti [https://www.wdl.org/en/item/7331/view/1/1/ Textus] [[:Fasciculus:About the Natural History of the Indies WDL7331.pdf|textus]]
* 1529 : [[Ioannes Ruellius]], interpr.; [[Otho Brunfelsius]], ed., ''P. Dioscoridae pharmacorum simplicium reique medicae libri VIII'' [https://archive.org/details/mobot31753000817921 Textus]
* 1530 : [[Otho Brunfelsius]], ''Herbarum vivae eicones'' [https://archive.org/details/mobot31753003125165 Textus]
* 1531 : [[Otho Brunfelsius]], ed., ''... Ioan. Serapionis Arabis [[De simplici medicina (Serapio)|De simplicibus medicinis]] opus praeclarum et ingens, Averrois Arabis de eisdem liber eximius, Rasis filii Zachariae de eisdem opusculum perutile, incerti item autoris de centaureo libellus hactenus Galeno inscriptus ...'' Argentorati: Georgius Ulricher [https://archive.org/details/mobot31753000818200 Textus]
* 1533 (interpres) : ''Ex Aeliani Historia per Petrum Gyllium latini facti, itemque ex Porphyrio, Heliodoro, Oppiano, tum eodem Gyllio luculentis accessionibus aucti libri XVI de vi et natura animalium; eiusdem Gyllii liber unus de gallicis et latinis nominibus piscium''. Lugduni: apud Sebastianum Gryphium [https://opacplus.bsb-muenchen.de/title/BV004256028 apud Monacenses] [http://gallica.bnf.fr/ark:/12148/bpt6k9905508 editio 1535 apud gallica]
* 1533 : &quot;Liber summarius de Gallicis et Latinis nominibus piscium Massiliensium&quot; in [[Petrus Gyllius]], interpr., ''Ex Aeliani historia per Petrvm Gyllivm latini facti'' (Lugduni: apud Seb. Gryphium, 1533) [https://www.digitale-sammlungen.de/en/view/bsb11216933?page=578 pp. 546-598]
* 1533 : ''Tacuini sanitatis Elluchasem Elimithar; Albengnefit De virtutibus medicinarum et ciborum; Iac. Alkindus De rerum gradibus''. Argentorati: apud Ioannem Schottum [http://mdz-nbn-resolving.de/urn:nbn:de:bvb:12-bsb10197837-7 Textus] apud Monacenses
* 1534 : [[Otho Brunfelsius]], ''Kreüterbuch contrafayt'' [https://archive.org/details/mobot31753000811650 Textus]
* 1534 : [[Franciscus Rabelaesus]],''[[Gargantua (Rabelaesus)|La Vie très horrificque du grand Gargantua]]'' (Lugduni) [https://gallica.bnf.fr/ark:/12148/btv1b8609586k/f21 cap. 4]
* post 1534 : ''Prima relatione di Iacques Carthier della Terra nuoua detta la Nuoua Francia, trouata nell'anno 1534'' in [[Ioannes Baptista Ramusius|Giovanni Battista Ramusio]], ''[[Navigationi et viaggi (Ramusius)|Navigationi et viaggi]]'' (Venetiis, 1550-1559) vol.  3 [https://archive.org/stream/dellenauigationi03ramu#page/n809/mode/2up Textus editionis 1606 apud ''Internet Archive'']
* 1535 : [[Gundisalvus Fernández de Oviedo y Valdés|Gonzalo Fernandez de Oviedo y Valdés]], ''Historia general de las Indias'' [https://archive.org/details/mobot31753000819539 Textus]
* c. 1536 : [[Bartholomaeus Casaus]], ''''Apologética historia sumaria'' in M. Serrano y Sanz, ed., ''Historiadores de Indias'' (Matriti: Bailly-Baillière, 1909) [https://archive.org/details/historiadoresdei01serr vol. 1]
* 1536 : ''Constantini Africani ... opera''. Basileae [http://mdz-nbn-resolving.de/urn:nbn:de:bvb:12-bsb10140123-5 Textus]
* 1536 : [[Ioannes Ruellius]], ''De natura stirpium libri III''. 1536 [https://gallica.bnf.fr/ark:/12148/bpt6k1520724r Editio Basileensis 1537] apud Gallica
* 1536 : [[Carolus Stephanus]], ''[[De re hortensi libellus]]''
* 1537 : [[Carolus Stephanus]], ''[[Vinetum (Stephanus)|Vinetum]]'' [https://archive.org/details/hin-wel-all-00003077-001 Textus] apud ''Internet Archive''
* 1538 : [[Carolus Stephanus]], ''Sylva, frutetum, collis'' {{Google Books|K5nY6jwUeF4C}}
* 1538 : [[Carolus Stephanus]], ''Arbustum, fonticulus, spinetum'' {{Google Books|OjM6AAAAcAAJ}}
* c. 1539 : ''[[Livre de cuisine très utile et profitable|Livre de cuysine tres utille et prouffitable]]'' (Lutetiae) [https://gallica.bnf.fr/ark:/12148/btv1b105020786/f107 f. 41r] (&quot;saulce barbe robert&quot;)
* 1539 : [[Antonius Gazius]], ''De vino et cerevisia tractatio'' {{Google Books|lGFWAAAAcAAJ|editio 1546 ad finem libri}} {{Google Books|-_9mAAAAcAAJ|editio 1549 ad finem libri}}
* 1539 : &lt;span id=&quot;Bock (1539)&quot;&gt;&lt;/span&gt;[[Hieronymus Bock]], ''[[Kreutterbuch (Bock)|New Kreütter Buch]]'' [http://reader.digitale-sammlungen.de/de/fs1/object/display/bsb11069345_00590.html pars 2 f. 86v]; eiusdem ''Kreüterbuch'' (1546) [https://archive.org/details/mobot31753000815859 Textus]; eiusdem ''Kreüterbuch'' (1551) [https://bildsuche.digitale-sammlungen.de/index.html?c=viewer&amp;bandnummer=bsb00091270&amp;pimage=741 ff. 350r-351r]; eiusdem ''Kreutterbuch'' (1560) [https://archive.org/stream/mobot31753000820529#page/n725/mode/2up ff. 342-344]
* circa 1540 : &lt;span id=&quot;Codex Tudela&quot;&gt;&lt;/span&gt;''[[Codex Tudela]]'' (liber pictus, c. 1530/1554) [http://ceres.mcu.es/pages/Viewer?accion=4&amp;Museo=MAM&amp;AMuseo=MAM&amp;Ninv=70400&amp;txt_id_imagen=4&amp;txt_rotar=0&amp;txt_contraste=0 f. 3r]
* circa 1540 : [[Toribius Motolinia]], ''[[Historia de los Indios de la Nueva España]]'' [https://archive.org/stream/historiadelosind00moto#page/192/mode/2up pp. 193-194 editionis 1914] etc.
* 1540 : [[Carolus Stephanus]], ''Seminarium'' {{Google Books|xjsPAAAAQAAJ}}
* 1541 : [[Conradus Gesnerus]], ''Historia plantarum ...'' [https://archive.org/details/historiaplantar00gess Textus]
* ante 1542 : ''[[Codex Mendoza]]'' (liber pictus, c. 1534/1542) f. 51v-52r etc.; cf. Frances Berdan, Patricia Anawalt, ''The Essential Codex Mendoza'' (Berkeleiae, 1997) pp. 132-133 etc.
* 1542 : [[Bartholomaeus Casaus|Bartolomé de las Casas]], ''[[Brevissima relación (Casaus)|Brevissima relación de la destruyción de las Indias]]'' (Hispali, 1552) [http://uvadoc.uva.es/handle/10324/17058 Textus] [http://bdh-rd.bne.es/viewer.vm?id=0000172864&amp;page=1 Versio Latina 1614]
* 1542 : &lt;span id=&quot;Fuchsius (1542)&quot;&gt;&lt;/span&gt;[[Leonhartus Fuchsius]], ''[[De historia stirpium commentarii insignes]]'' [https://archive.org/stream/Dehistoriastirp00Fuch#page/730/mode/2up pp. 731-735];  editio altera (1549), [http://books.google.com/books?id=3fWFctV3hWIC&amp;pg=PT574&amp;source=gbs_selected_pages&amp;cad=0_0#PPT545,M1 cpt. 203 'De malis insanis']
* 1542 : [[Andreas Boorde|Andrew Boorde]], ''[[A Compendious Regiment|A Compendyous Regyment or a Dyetary of Helth]]'' ([[Fridericus Iacobus Furnivall|F. J. Furnivall]], ed., ''The fyrst boke of the introduction of knowledge, made by Andrew Borde, of physycke doctor; a compendyous regiment ...'' (Londinii, 1870) [https://archive.org/details/fyrstbokeofintro00boorrich/page/222/mode/2up p. 223 ff.]
* post 1542 : [[Casparus de Carvajal]], ''Relación del nuevo descubrimiento del famoso Río Grande'' in José Toribio Medina, ed., ''Descubrimiento del río de las Amazonas según la relación de ... Gaspar de Carvajal'' (Hispali: E. Rasco, 1894) [https://archive.org/details/raha_103310 Textus] apud ''Internet Archive''
* 1543 : [[Carolus Stephanus]], ''Pratum, lacus, arundinetum'' {{Google Books|ul46AAAAcAAJ}} 
* ante 1544 : &lt;span id=&quot;Cordus (ante 1544)&quot;&gt;&lt;/span&gt;[[Valerius Cordus]]; [[Conradus Gesnerus]], ed., ''Annotationes in Dioscoridis; Historiae stirpium libri IV [etc.]''. 1561 [https://www.biodiversitylibrary.org/bibliography/8036#/summary Textus]
* 1544 : &lt;span id=&quot;Cervantes (1544)&quot;&gt;&lt;/span&gt;[[Franciscus Cervantes de Salazar]], ''[[Cronica de la Nueva España]]'' [https://archive.org/stream/cronicadelanueva00cervuoft#page/470/mode/2up lib. 4 cap. 108]
* 1544 : [[Petrus Andreas Matthiolus]], ''[[Commentarii (Matthiolus)|Di Pedacio Dioscoride libri cinque]]'' [https://archive.org/details/ita-bnc-pos-0000072-001 Mantuae 1549], longius in editione 1573 {{Google Books|w8JCAAAAcAAJ|p. 689}}
* 1544 : [[Guilielmus Turnerus]], ''Avium praecipuarum, quarum apud Plinium et Aristotelem mentio est, brevis et succincta historia''. Coloniae {{Google Books|s106AAAAcAAJ}} [https://gallica.bnf.fr/ark:/12148/bpt6k74821j Textus exemplaris peioris] apud Gallica
* 1545 : ''[[A Proper New Book of Cookery|A Propre New Booke of Cokerye]]''. Londinii: Richard Lant &amp; Richarde Bankes [http://www.uni-giessen.de/gloning/tx/bookecok.htm editio interretialis]; cf. 1575
* 1547? : [[Valerius Cordus]], ''Pharmacorum conficiendorum ratio. Vulgo uocant Dispensatorium''. Norimbergae {{Google Books|2NBaAAAAcAAJ}}
* post 1547 : ''[[Livre de cuisine très utile et profitable|Le grant cuysinier de toute cuysine tres utille et proffitable]]'' {{Google Books|M2MgN7-H1c0C|ff. 48v-49r}} (&quot;saulce barbe robert&quot;)
* 1548 : [[Petrus de Medina|Pedro de Medina]], ''Libro de grandezas y Cosas memorables de Espana''. Matriti {{Google Books|Uh9JAAAAcAAJ}} [https://catalog.hathitrust.org/Record/009300157 editio 1595]
* 1548 : &lt;span id=&quot;Turner (1548)&quot;&gt;&lt;/span&gt;[[Guilielmus Turnerus|William Turner]], ''The Names of Herbes in Greke, Latin, Englishe, Duch and Frenche''. Londinii: John Day, 1548 {{Google Books|quQql5HMlmAC}}
* 1549 : [[Leonhartus Fuchsius]], ''Plantarum effigies'' [https://archive.org/details/mobot31753000812732 Textus]
* 1549 : [[Leonhartus Fuchsius]], ed., ''[[Antidotarium|Nicolai Myrepsi Alexandrini medicamentorum opus ... a Leonharto Fuchsio medico ... recens conversum]]''. Basileae: apud Io. Oporinum, 1549 {{Google Books|EN5aAAAAcAAJ}}
* 1549 : [[Christophorus de Messisbugo]], ''[[Banchetti (Messisbugo)|Banchetti]]'' (1549) {{Google Books|J9g2AQAAMAAJ}} (cf. 1564)
* 1550 : [[Hieronymus Cardanus]], ''De subtilitate'' [https://archive.org/details/bub_gb_Tmf3wRsurVsC/page/n235 pp. 197-198]
* 1550 : [[Carolus Stephanus]], ''[[De nutrimentis (Stephanus)|De nutrimentis ad Baillyum libri tres]]''. Lutetiae, 1550 [http://www.archive.org/details/carolistephanide00esti Textus] apud ''Internet Archive''
* 1550 : Eucharius Roeslin, ''Kreuterbuch'' [https://archive.org/details/mobot31753000818796 Textus]
* 1550 : [[Georgius Vasarius|Giorgio Vasari]], ''Le vite de piu eccellenti architetti, pittori, et scvltori italiani''. Florentiae [https://archive.org/details/gri_vitedepivecc01vasa Partes 1-2] [https://archive.org/details/gri_vitedepivecc03vasa Pars 3]
* 1550-1559 : [[Ioannes Baptista Ramusius|Giovanni Battista Ramusio]], ed., ''[[Navigationi et viaggi (Ramusius)|Navigationi et viaggi]]'' (Venetiis)

== 1551-1600 ==
* 1551 : [[Guilielmus Turnerus|William Turner]], ''[[A New Herbal (Turner)|A New Herball]]''. Londinii: Steven Mierdman {{Google Books|_QZfAAAAcAAJ}} [https://archive.org/details/b30342053_0002 Textus voll. 1-2] apud ''Internet Archive'' [https://archive.org/details/b30342053_0001 alius]
* 1551 : [[Petrus Bellonius|Pierre Belon]], ''L'Histoire naturelle des estranges poissons marins''. Lutetiae [https://archive.org/details/lhistoirenaturel00belo Textus][http://gallica.bnf.fr/ark:/12148/btv1b86083054 Textus apud gallica]
* 1551 : [[Helius Eobanus Hessus]], ''[[De tuenda bona valetudine]]'' (1551) {{Google Books|SWZWAAAAcAAJ|f. 40v}}
* 1551 : [[Conradus Gesnerus]], ''Historiae animalium liber I'' [https://archive.org/details/ConradiGesnerimIGess Textus]
* 1551-1555 : [[Adamus Lonicer]], ''Naturalis historiae opus novum'', lemma [http://www.uni-mannheim.de/mateo/camenaref/lonitzer/loni1/jpg/s075a.html 'mala insana']
* 1552 : Theodorus Gaza, interpr., ''Theophrasti ... De historia plantarum'' [https://archive.org/details/mobot31753000811833 Textus]
* 1552 : [[Hieronymus Bock|Hieronymus Tragus]], ''De stirpium maxime earum quae in Germania nostra nascuntur ... commentariorum libri tres'' [https://archive.org/details/bub_gb_nDCPb-dIeqYC Textus]
* 1552 : [[Rembertus Dodonaeus]], ''De frugum historia''. Antverpiae [https://archive.org/details/mobot31753000810710 Textus] [https://www.biodiversitylibrary.org/bibliography/7103#/summary Textus]
* 1552 : [[Martinus de la Cruz]]; [[Ioannes Badianus]], interpr., ''[[Libellus de medicinalibus Indorum herbis]]'' [https://www.academia.edu/2777939/Libellus_de_Medicinalibus_Indorum_Herbis_Digital_facsimile_ f. 13v]
* 1552 : [[Ioannes Ruellius]], interpr., ''Pedanii Dioscoridis Anazarbei De medicinali materia'' [https://archive.org/details/mobot31753003467922 Textus]
* 1552-1563 : [[Ioannes de Barros|João de Barros]], ''[[Décadas da Ásia]]'' (Olisipone) [http://archive.org/details/decadaprimeirate00barr 1] [http://archive.org/details/decadaprimeirate01barr 2] [http://archive.org/details/decadaprimeirate02barr 3] editionis 1628
* circa 1553 : [[Petrus de Cieza de León|Pedro de Cieza de León]]; ''[[Chronica del Peru|Las guerras civiles Peruanas]]'' in Carmelo Saenz de Santa Maria, ed., ''Obras completas'' (Matriti, 1984-1985)
* 1553 : [[Petrus Bellonius|Pierre Belon]], ''[[Observationes (Bellonius)|Les observations de plusieurs singularitez et choses mémorables trouvées en Grèce, Asie, Judée, Égypte, Arabie et autres pays estranges]]''. Lutetiae. [https://archive.org/details/gri_observations00belo Textus apud archive.org]  [http://gallica.bnf.fr/ark:/12148/bpt6k53619s Textus apud gallica] [https://archive.org/details/obseruationsdep00belo apud Smithsonianum]
* 1553 : [[Petrus Bellonius]], ''[[De arboribus coniferis resiniferis|De arboribus coniferis resiniferis, aliis quoque nonnullis sempiterna fronde virentibus]]'' [http://gallica.bnf.fr/ark:/12148/btv1b8608304q Textus apud gallica]
* 1553 : [[Petrus Bellonius]], ''De admirabili operum antiquorum et rerum suspiciendarum praestantia liber primus. De medicato funere seu cadavere condito et lugubri defunctorum ejulatione liber secundus. De medicamentis nonnullis servandi cadaveris vim obtinentibus liber tertius'' [http://gallica.bnf.fr/ark:/12148/btv1b86083039 Textus apud gallica]
* 1553 : [[Petrus Bellonius]], ''De aquatilibus libri duo cum iconibus ad vivam ipsorum effigiem quoad ejus fieri potuit expressis''. Lutetiae: Ch. Estienne [http://www.biodiversitylibrary.org/bibliography/5765#/summary Textus apud Biodiversity Library]
* 1553 : [[Petrus de Cieza de León|Pedro de Cieza de León]], ''[[Chronica del Peru|Parte primera dela chronica del Peru]]'' [https://archive.org/stream/parteprimeradela00ciez#page/n271/mode/2up ]
* 1553 : &lt;span id=&quot;Dodonaeus (1553)&quot;&gt;&lt;/span&gt;[[Rembertus Dodonaeus]], ''Trium priorum De stirpium historia commentariorum imagines'' [https://www.biodiversitylibrary.org/bibliography/7109#/summary Textus] {{Google Books|RCo6AAAAcAAJ}}
* 1554 : [[Carolus Stephanus]], ''[[Praedium rusticum]]''
* 1554 : &lt;span id=&quot;Matthiolus (1554)&quot;&gt;&lt;/span&gt;[[Petrus Andreas Matthiolus]], ''Commentarii'' [https://archive.org/stream/petriandreaematt00matt#page/536/mode/2up ed. 1558]; [https://archive.org/details/petriandreaemat00matt ed. 1560]; [https://archive.org/details/PetriAndreaMatt00Matt ed. 1565]; &lt;span id=&quot;Matthiolus (1574)&quot;&gt;&lt;/span&gt;longius editione 1574 [https://archive.org/stream/mobot31753000819257#page/759/mode/2up lib. 4 cap. 71]
* 1554 : &lt;span id=&quot;Dodonaeus (1554)&quot;&gt;&lt;/span&gt;[[Rembertus Dodonaeus]], ''Posteriorum trium De stirpium historia commentariorum imagines'' {{Google Books|fCo6AAAAcAAJ|pp. 181-183}}
* 1554 : [[Conradus Gesnerus]], ''Appendix historiae quadrupedum'' [https://archive.org/details/AppendixhistoriIIIAGess Textus]
* 1554 : [[Gulielmus Rondeletius]], ''[[De piscibus marinis (Rondeletius)|De piscibus marinis]], libri XVIII, in quibus verae piscium effigies expressae sunt''. Lugduni: apud Matthiam Bonhomme. Insunt: &quot;De testaceis libri II&quot;, &quot;De insectis et zoophytis liber&quot;, &quot;De piscibus stagni marini liber&quot;, &quot;De piscibus lacustribus liber&quot;, &quot;De piscibus fluviatilibus liber&quot;, &quot;De palustribus liber&quot;, &quot;De amphibiis liber&quot; [https://archive.org/details/gvlielmirondelet00rond Textus apud archive.org] [https://archive.org/details/GulielmiRondele00Rond alius]; Versio Francogallica a [[Laurentius Ioubertus|Laurent Joubert]] facta: ''L'Histoire entiere des poissons, composee premierement en Latin par Maistre Guillaume Rondelet ... maintenant traduites en françois. Avec leurs pourtraits au naïf''. Lugduni, 1558 [http://gallica.bnf.fr/ark:/12148/btv1b86261840 Textus apud Gallica]
* 1554 : [[Hippolytus Salvianus]], ''Aquatilium animalium historiae'' [https://archive.org/details/Aquatiliumanima00Salv Textus]
* 1554: [[Andreas Lacuna Segobiensis]], ''Annotationes in Dioscoridem Anazarbeum ... iuxta vetustissimorum codicum elaboratae''. Lugduni, 1554 [http://www.mdz-nbn-resolving.de/urn/resolver.pl?urn=urn:nbn:de:bvb:12-bsb10313120-9 vide p. 827 huius exemplaris] {{Google Books|Lvw8AAAAcAAJ|Vide fasciculum alterum huius exemplaris}}
* [[Simon Grynaeus]], praef., ''[[Novus orbis regionum ac insularum veteribus incognitarum]]''. Basileae: Hervagius, 1555 [https://www.digitale-sammlungen.de/en/view/bsb10140943?page=1 Textus] apud Monacenses
* 1555: [[Andreas Lacuna Segobiensis]], interpr., ''Pedacio Dioscorides Anazarbeo Acerca de la materia medicinal y de los venenos mortiferos''. Antverpiae {{Ling|Hispanice}} {{Google Books|hG0rnSNIDykC}} [http://www.wdl.org/en/item/10632/ Exemplar manu tinctum] apud ''World Digital Library'' [https://archive.org/details/hin-wel-all-00000404-001 Editio Salmanticensis 1566] apud archive.org {{Google Books|xzxnLxvzvHUC|Editio Salmanticensis 1570}}
* 1555 : [[Gulielmus Rondeletius]], ''Universae aquatilium historiae pars altera, cum veris ipsorum imaginibus'' [http://mdz-nbn-resolving.de/urn:nbn:de:bvb:12-bsb10140050-0 Textus] apud Monacenses [https://gallica.bnf.fr/ark:/12148/bpt6k97272x.image apud Gallica] {{Google Books|lKNCAAAAcAAJ}} 
* 1555 : [[Petrus Bellonius|Pierre Belon]], ''La nature et diversité des poissons, avec leurs pourtraictz représentez au plus près du naturel''. Lutetiae: Ch. Estienne [http://gallica.bnf.fr/ark:/12148/btv1b550056516 Textus apud gallica]
* 1555 : [[Petrus Bellonius|Pierre Belon]], ''L'Histoire de la nature des oyseaux, avec leurs descriptions et naïfs portraicts retirez du naturel'' [https://archive.org/details/hin-wel-all-00001828-001 Textus apud archive.org] [http://www2.biusante.parisdescartes.fr/livanc/?do=livre&amp;cote=00907 Textus apud BiuSante] [http://gallica.bnf.fr/ark:/12148/btv1b8608302w Textus apud gallica]
* 1555 : ''[[Livre fort excellent de cuisine|Livre fort excellent de cuysine tres utille et proffitable]]''. Lugduni: Arnoullet, 1555 [https://gallica.bnf.fr/ark:/12148/btv1b8626163t Textus] apud Gallica
* 1556 : [[Conradus Gesnerus]], ''De piscibus et aquatilibus omnibus libelli III novi''. Tiguri [https://www.e-rara.ch/doi/10.3931/e-rara-4928 Textus]; alio titulo ''[[Publius Ovidius Naso|Ovidii]] Halieuticon, hoc est, De piscibus libellus; accedit Aquatilium animantium enumeratio iuxta Plinium'' [https://www.e-rara.ch/zuz/content/titleinfo/215501 Textus] apud ''e-Rara''
* 1526/1557 : &lt;span id=&quot;Oviedo (ante 1557)&quot;&gt;&lt;/span&gt;[[Gundisalvus Fernández de Oviedo y Valdés|Gonzalo Fernandez de Oviedo y Valdés]], ''[[Historia general y natural de las Indias, islas y tierra-firme del mar Océano]]'' (José Amador de los Rios, ed. 4 voll. Matriti: Real Academia de la Historia, 1851-1855 [https://archive.org/details/gri_33125007267921 Pars 1] [https://archive.org/details/gri_33125007267988 Pars 2 vol. 1] [https://archive.org/details/historiageneral02fernguat pars 2 vol. 2] [https://archive.org/details/historiageneral04fernguat Pars 3]
* 1557 : &lt;span id=&quot;Clusius (1557)&quot;&gt;&lt;/span&gt;[[Rembertus Dodonaeus|Rembert Dodoëns]]; [[Carolus Clusius|Charles de l'Escluse]], interpr., ''Histoire des plantes'' [https://archive.org/stream/hin-wel-all-00000412-001#page/n470/mode/2up pp. 441-442]
* 1557 : [[Iulius Caesar Scaliger]], ''Exotericarum exercitationum liber quintus decimus, de subtilitate, ad Hieronymum Cardanum''. 1557 {{Google Books|LB88AAAAcAAJ}}
* 1557 : [[Petrus Bellonius|Pierre Belon]], ''Portraicts d’oyseaux, animaux, serpens, herbes, arbres, hommes et femmes d’Arabie et d'Égypte''. Lutetiae [http://gallica.bnf.fr/ark:/12148/bpt6k85310z Textus apud gallica] [http://www2.biusante.parisdescartes.fr/livanc/?do=livre&amp;cote=07744 Textus apud biuSante]
* 1557 : [[Ioannes Stadius|Hans Staden]], ''Warhaftige Historia und Beschreibung eyner Landtschafft der wilden, nacketen, grimmigen menschfresser Leuthen in der Newenwelt America gelegen''. Marburgiae [https://archive.org/details/staden Textus] [https://www.deutschestextarchiv.de/book/view/staden_landschafft_1557 alibi]; [https://archive.org/details/americaetertiapa00stad/page/108/mode/2up p. 109 versionis Latinae]
* 1558 : [[Petrus Bellonius|Pierre Belon]], ''[[De neglecta stirpium cultura|Les Remonstrances sur le défaut du labour et culture des plantes]]''. Lutetiae {{Google Books|HjBeAAAAcAAJ}}
* 1558 : [[Andreas Thevetus|André Thevet]], ''[[Les Singularités de la France antarctique|Les Singularitez de la France antarctique, autrement nommée Amérique]]''. Lutetiae [https://archive.org/details/lessingularitezd00thev Textus] apud ''Internet Archive''
* 1558 : [[Amatus Lusitanus]], ''In Dioscoridis Anazarbei De medica materia libros quinque'' (1558), [http://books.google.com/books?id=FoWjfrJFmd0C&amp;pg=PT560&amp;vq=melongena&amp;source=gbs_search_r&amp;cad=0_1 p.648]
* 1558 : [[Guilielmus Turnerus]], &quot;Epistola&quot; [de piscibus] in [[Conradus Gesnerus]], ''Historiae animalium liber IIII, qui est de piscium et aquatilium animantium natura'' (Tiguri: apud Christoph. Froschouerum, 1558) [https://archive.org/details/BIUSante_pharma_res000050x04/page/1294/mode/1up pp. 1294-1297]
* 1558 : ''[[En tibi perpetuis ridentem floribus hortum]]'' (herbarium manuscriptum apud [[Museum Historiae Naturalis Lugduno-Batavum|Naturalem]])
* 1559 : [[Conradus Gesnerus]], ed.: [[Ianus Dubravius]], ''De piscinis et piscium qui in eis aluntur naturis libri quinque''; [[Xenocrates]], ''De Alimento ex aquatilibus'', cum versione Latina Ioannis Baptista Rasari [http://doi.org/10.3931/e-rara-8219 Textus]
* 1559 : [[Iacobus Praefectus]], ''[[De diversorum vini generum natura liber]]''. Venetiis [https://archive.org/details/hin-wel-all-00002781-001 Textus] {{Google Books|LT46AAAAcAAJ}}
* 1560 : &lt;span id=&quot;Gesnerus (1560)&quot;&gt;&lt;/span&gt;[[Conradus Gesnerus]], ''Icones animalium quadrupedum'' [https://archive.org/details/iconesanimaliumq00gess Textus]
* 1561? : [[Georgius Pictorius]], ''Sanitatis tuendae methodus carmine elegiaco conscripta'' (1561?) {{Google Books|Q0uZo4CJTQsC|p. 13}}
* 1561 : &lt;span id=&quot;Gesnerus (1561)&quot;&gt;&lt;/span&gt;[[Conradus Gesnerus]], ed., ''In hoc volumine continentur [[Valerius Cordus|Valerii Cordii]] Annotationes in Dioscoridis ... [etc.]'' [http://dl.ub.uni-freiburg.de/diglit/cordus1561/0566 f. 272b] [https://archive.org/details/mobot31753000817848 Textus]
* 1561 : &lt;span id=&quot;Gesnerus (1561)&quot;&gt;&lt;/span&gt;[[Conradus Gesnerus]], &quot;De hortis Germaniae&quot; in [[Valerius Cordus]], ''Annotationes in Dioscoridis ... [etc.]'' [http://dl.ub.uni-freiburg.de/diglit/cordus1561/0566 f. 272b] [https://archive.org/details/mobot31753000817848 Texztus]
* 1561 : ''Polybi De diaeta salubri sive de victu privatorum libellus'' (1561) {{Google Books|-WoGnOScNPEC|f. 58r et index}}
* post 1561 : [[Franciscus Vásquez]], ''Relación de todo lo que sucedío en la jornada de Omagua y Dorado hecho per el gobernador Pedro de Orsúa'' in M. Serrano y Sanz, ed., ''Historiadores de Indias'' (Matriti: Bailly-Baillière, 1909) [https://archive.org/details/historiadoresdei02serr_0/page/422 vol. 2 p. 422 ff.]
* 1562 : [[Guilielmus Turnerus|William Turner]], ''[[A New Herbal (Turner)|The Seconde Parte of William Turners Herball]]''. Coloniae: Arnold Birckman [https://archive.org/details/b30342053_0002/page/n216/mode/1up Textus] [https://archive.org/details/b30342053_0001 alius]
* 1562 : [[Guilielmus Turnerus|William Turner]], ''A Booke of the Natures and Properties as Well of the Bathes in England as of Other Bathes in Germany and Italy''. Coloniae [https://archive.org/details/b30342053_0001/page/n594/mode/1up Textus]; [https://archive.org/details/englishmanstreas00vica editio 1599]
* c. 1562-1592 : [[Georgius Bocskay]], [[Georgius Hoefnagel]], ''[[Mira calligraphiae monumenta]]'' (manuscriptum) [http://www.getty.edu/publications/virtuallibrary/089236212X.html Editio facsimile 1992]
* 1563 : &lt;span id=&quot;Orta (1563)&quot;&gt;&lt;/span&gt;[[Garcias ab Orta|Garcia de Orta]], ''[[Aromatum apud Indos nascentium historia|Colóquios dos simples e drogas da Índia]]'' (Goae) [http://purl.pt/22937/3/#/524 textus] [http://digitarq.arquivos.pt/viewer?id=4614066 alius]
* 1563 : [[Conradus Gesnerus]], ed.; [[Iodocus Willichius]], ''Ars magirica, hoc est, coquinaria''; [[Iacobus Bifrons]], ''De operibus lactariis epistola''. Tiguri: apud Iacobum Gesnerum [http://doi.org/10.3931/e-rara-5266 Textus]
* 1563 : [[Guilielmus Gratarolus]], ''Regimen omnium iter agentium''. Argentorati: per Wendelinum Rihelium, 1563 {{Google Books|HBA8AAAAcAAJ|pp. 29-30}}
* 1564 : [[Carolus Stephanus|Charles Estienne]], ''[[Praedium rusticum|L'Agriculture et maison rustique]]'' [https://archive.org/details/lagricultureetma00esti Textus]
* 1564 : [[Christophorus de Messisbugo]], ''Libro novo nel qual s'insegna a far d'ogni sorte di vivanda'' {{Google Books|MCU6AAAAcAAJ}} (cf. 1549)
* ante 1565 : [[Petrus Antonius Michiel]], ''I cinque libri di piante'' (manuscriptum 1553-1565) in Ettore de Toni, ed., ''Pietro Antonio Michiel, I cinque libri di piante: codice Marciano'' (Venetiis, 1940) [https://www.scribd.com/document/288397333/Pietro-Antonio-Michiel-I-cinque-libri-di-piante Textu]s apud ''Scribd''
* 1565 : [[Guilielmus Gratarolus]], ''De vini natura''. Argentorati: Theodosius Rihelius, 1565 {{Google Books|IEM6AAAAcAAJ}}
* 1565 : &lt;span id=&quot;Benzo (1565)&quot;&gt;&lt;/span&gt;[[Hieronymus Benzo|Girolamo Benzoni]], ''[[Novae Novi orbis historiae|La historia del Mondo nuovo]]'' (Venetiis) [https://archive.org/stream/hin-wel-all-00001834-001#page/n218/mode/2up ff. 102r-103v editionis 1572]
* 1565 : &lt;span id=&quot;Monardes (1565)&quot;&gt;&lt;/span&gt;[[Nicolaus Monardes|Nicolás Monardes]], ''[[Simplicium medicamentorum ex novo orbe delatorum historia|Dos libros. El uno trata de todas las cosas que traen de nuestras Indias Occidentales]]'' [https://archive.org/stream/hin-wel-all-00002448-001#page/n100/mode/2up quaternion f 6]
* ante 1566 : [[Bartholomaeus Casaus|Bartolomé de las Casas]]; Marqués de la Fuensanta del Valle, José Sanchez Rayon, edd., ''Historia de las Indias''. 5 voll. Matriti, 1875-1876 [https://archive.org/details/historiaindias01casarich Vol. 1] [https://archive.org/details/historiaindias02casarich 2] [https://archive.org/details/HistoriaDeLasIndias.TomoIIIFrayBartolomDeLasCasas 3] [https://archive.org/details/historiaindias03casarich 4] [https://archive.org/details/historiaindias04casarich 5]
* ante 1566 : [[Bartholomaeus Casaus|Bartolomé de las Casas]], ''Apologética historia sumaria'' [https://archive.org/stream/historiaindias04casarich#page/304/mode/2up p. 304 editionis 1876]
* 1566 : [[Rembertus Dodonaeus]], ''Frumentorum, leguminum ...'' [https://archive.org/details/mobot31753003488142 Textus]
* 1566 : &lt;span id=&quot;Fragosus (1566)&quot;&gt;&lt;/span&gt;[[Ioannes Fragosus]], ''Catalogus simplicium medicamentorum, quae in usitatis huius temporis compositionibus praesertim Mesuaei &amp; Nicolai aliorum penuria inuicem supponuntur''. Compluti [http://bibliotecavirtual.ranf.com/i18n/consulta/registro.cmd?id=6191 Textus]
* 1566 : Francesco Calzolari, ''Il viaggio di Monte Baldo della magnifica città di Verona, nel quale si descrive con maraviglioso ordine il sito di detto Monte, &amp; d'alcune altre parti ad esso contigue, et etiandio si narra d'alcune segnalate piante &amp; herbe che ivi nascono, &amp; che nell'uso della medicina più di tutte l'altre conferiscono''. Venetiis: Valgrisio [https://archive.org/details/bub_gb_xcKYxLXyoAIC Textus]
* 1567 : &lt;span id=&quot;Clusius (1567)&quot;&gt;&lt;/span&gt;[[Carolus Clusius]], interpr.; [[Garcias ab Orta]], ''[[Aromatum apud Indos nascentium historia]]'' [https://www.biodiversitylibrary.org/bibliography/7096#/summary Textus] [https://archive.org/details/mobot31753003488159 Textus] [https://archive.org/details/mobot31753003634315 4a ed. 1593]
* 1568 : [[Guilielmus Turnerus|William Turner]], ''[[A New Herbal (Turner)|The First and Seconde Partes of the Herbal ... with the Third Parte]]''. Coloniae Agrippinae [https://quod.lib.umich.edu/e/eebo2/A14059.0001.001 Recensio interretialis]
* 1568 : &lt;span id=&quot;Epistula ad Monardem (1568)&quot;&gt;&lt;/span&gt;&quot;Carta: al muy magnifico señor, mi señor Doctor Monardes, medico en Sevilla&quot; (1568) in [[Nicolaus Monardes]], ''[[Simplicium medicamentorum ex novo orbe delatorum historia|Segunda parte del libro des las cosas que se traen de nuestras Indias Occidentales]]'' {{Google Books|yP19XmWVSMwC|ff. 82r-83v}}
* 1568 : &lt;span id=&quot;Dodonaeus (1568)&quot;&gt;&lt;/span&gt;[[Rembertus Dodonaeus]], ''Florum et coronariarum odoratarumque nonnullarum herbarum historia''. Antverpiae: Plantin [https://archive.org/details/bub_gb_ryRhpBQMI5gC/ Textus] apud ''Internet Archive'' [http://docnum.u-strasbg.fr/cdm/ref/collection/coll13/id/89898 Textus] apud universitatem Argentoratensem [https://archive.org/details/bub_gb_L5Y1sZFjYLYC/ Editio 2a 1569] [https://archive.org/details/mobot31753003131585 2a ed.]
* 1568 : [[Georgius Vasarius|Giorgio Vasari]], ''Le vite de' piv eccellenti pittori, scvltori, et architettori''. Florentiae: I Giunti (editio aucta) [https://archive.org/details/levitedepiveccel01vasa Partes 1-2] [https://archive.org/details/levitedepiveccel02vasa pars 3 vol. 1]  [https://archive.org/details/levitedepiveccel03vasa pars 3 vol. 2]
* circa 1569 : [[Carolus Clusius]] et al., ''[[Libri picturati A 16-30|Libri picturati]]'' (manuscriptum) 
* 1570 : [[Bartholomaeus Scappi|Bartolomeo Scappi]], ''Opera'' (Venetiis: Tramezzino, 1570) lib. 3 cap. 169-179 [https://archive.org/details/operavenetiascap00scap/page/n291/mode/2up ff. 139r-141v]
* 1570 : &lt;span id=&quot;Matthiolus (1570)&quot;&gt;&lt;/span&gt;[[Petrus Andreas Matthiolus]], ''Commentarii in libros sex Pedacii Dioscoridis'' (Venetiis: ex officina Valgrisiana) [https://archive.org/details/petriandreaemat01mattgoog/page/n468/mode/1up p. 305] (&quot;Frumentum Indicum&quot;)
* 1571 : [[Petrus Pena]], [[Matthias de Lobel]], ''Stirpium adversaria nova'' [http://bibdigital.rjb.csic.es/ing/Libro.php?Libro=4137 Textus]; anno 1576 titulo ''Nova stirpium adversaria'' [https://archive.org/details/gri_33125009313756 rursus edita]
* 1571 : &lt;span id=&quot;Monardes (1571)&quot;&gt;&lt;/span&gt;[[Nicolaus Monardes]], ''[[Simplicium medicamentorum ex novo orbe delatorum historia|Segunda parte del libro des las cosas que se traen de nuestras Indias Occidentales]]'' {{Google Books|yP19XmWVSMwC|ff. 82r-83v}}
* 1571 : &lt;span id=&quot;Ulloa (1571)&quot;&gt;&lt;/span&gt; Alphonsus Ulloa, interpr.; [[Ferdinandus Columbus]], ''[[Historie (Ferdinandus Columbus)|Historie]]'' [https://archive.org/details/historiedelsdfer00coln Textus] apud ''Internet Archive''
* 1571 : &lt;span id=&quot;Ulloa (1571)&quot;&gt;&lt;/span&gt; Alphonsus Ulloa, interpr.; [[Romanus Pané]], &quot;[https://archive.org/stream/historiedelsdfer00coln#page/n295/mode/2up Scrittura di Fra Roman delle antichità degl'Indiani]&quot; in [[Ferdinandus Columbus]], ''[[Historie (Ferdinandus Columbus)|Historie]]'' ff. 124v-145v
* 1571 : &lt;span id=&quot;Matthiolus (1571)&quot;&gt;&lt;/span&gt;[[Petrus Andreas Matthiolus]], ''Compendium de plantis omnibus, unà cum earum iconibus, de quibus scripsit suis in commentariis in Dioscoridem editis''. Venetiis [https://www.biodiversitylibrary.org/item/209997 Textus] 
* 1571 : Hieremias Martius, interpr.; [[Iacobus Grevinus]], ''De venenis'' [https://archive.org/details/mobot31753000817822 Textus]
* 1571 : [[Nicander Colophonius]], ''Theriaca'', ''Alexipharmaca'' in Hieremias Martius, interpr.; [[Iacobus Grevinus]], ''De venenis'' [https://archive.org/details/mobot31753000817822 Textus]
* ante 1572 : &lt;span id=&quot;Felici (ante 1572)&quot;&gt;&lt;/span&gt;[[Constantius Felici]], &quot;De l'insalata e piante che in qualunque modo vengono per cibo de l'homo&quot; (manu scriptum 1565-1572). Enzo Cecchini, Guido Arbizzoni et al., edd., ''Costanzo Felici Da Piobbico. Lettere sulle Insalate. Lectio Nona De Fungis'' (Urbini, 1977) [http://www.medievalcookery.com/helewyse/files/newworld.pdf vide p. 9]
* 1572 : &lt;span id=&quot;Fragosus (1572)&quot;&gt;&lt;/span&gt;[[Ioannes Fragosus]], ''[[Discursos de las cosas aromáticas|Discursos de las cosas aromáticas, árboles y frutales, y de otras muchas medicinas simples que se traen de la India Oriental, y siruen al uso de medicina]]'' (Matriti: Francisco Sánchez) {{Google Books|ZGpIEb3BRfAC|f. 25r}} &quot;Flor del sol, gigantea, sol de las Indias&quot; 
* 1572 : Henry Hawks in [[Ricardus Hakluytus|Richard Hakluyt]], ''[[Principal Navigations (Hakluytus)|The Principall Navigations, Voiages and Discoveries of the English Nation]]'' (1a ed. Londinii: George Bishop, 1589) [https://archive.org/details/cihm_35668/page/n820/mode/1up p. 547] (''sapotes'')
* 1573 : &lt;span id=&quot;Matthiolus (1573)&quot;&gt;&lt;/span&gt;[[Petrus Andreas Matthiolus]], ''[[Commentarii (Matthiolus)|Commentarii ... di Pedacio Dioscoride Anazarbeo libri cinque della historia &amp; materia medicinale]]'' (editio 1573) lib. 2 cap. 148 {{Google Books|w8JCAAAAcAAJ|p. 405}} sive et ''Commentarii in libros sex Pedacii Dioscoridis Anazarbei de Materia Medica'' (editio 1574) lib. 2 cap. 153 [https://archive.org/stream/mobot31753000819257#page/433/mode/2up pp. 434-437]
* 1573 : Thomas Tusser, ''Five Hundreth Points of Good Husbandry''. Londinii: Tottell, 1573
* 1573 : John Partridge, ''[[The Treasury of Hidden Secrets|The Treasurie of Commodious Conceits and Hidden Secrets]]''. Londinii: Richard Jones [http://www.medievalcookery.com/notes/treasurie.pdf Textus]
* 1574 : &lt;span id=&quot;Monardes (1574)&quot;&gt;&lt;/span&gt;[[Nicolaus Monardes]], ''[[Simplicium medicamentorum ex novo orbe delatorum historia|Primera, segunda y tercera partes de la historia medicinal de las cosas que se traen de nuestras Indias Occidentales, que sirven en medicina]]'' {{Google Books|pEHeQNiTzA0C|f. 109v}}
* 1574 : &lt;span id=&quot;Clusius (1574)&quot;&gt;&lt;/span&gt;[[Nicolaus Monardes]]; [[Carolus Clusius]], interpr., ''[[Simplicium medicamentorum ex novo orbe delatorum historia|De simplicibus medicamentis ex occidentali India delatis quorum in medicina usus est]]'' [https://archive.org/stream/mobot31753003541627#page/71/mode/2up pp. 71-74] [https://archive.org/details/mobot31753003530810 2a ed. 1593]
* 1574 : &lt;span id=&quot;Dodonaeus (1574)&quot;&gt;&lt;/span&gt;[[Rembertus Dodonaeus]], ''Purgantium ...''. Antverpiae: Plantin [https://archive.org/details/mobot31753004146418 Textus]
* 1575 : &lt;span id=&quot;Fragosus (1575)&quot;&gt;&lt;/span&gt;[[Ioannes Fragosus]], ''De succedaneis medicamentis liber denuo auctus''. Mantuae ''[https://archive.org/details/hin-wel-all-00000717-001 Textus] apud ''Internet Archive''
* 1575 : ''[[A Proper New Book of Cookery|A Proper New Booke of Cookery]]''. Londinii: William How [http://www.medievalcookery.com/notes/pnboc1575.txt editio interretialis]; cf. 1545
* 1575 : Leonard Mascall, ''A booke of the arte and maner how to plant and graffe all sortes of trees, how to set stones, and sowe pepins, to make wylde trees to graffe on ...'' Londinii: John Wight [https://archive.org/details/bookeartmanerho00masc Editio 1582]
* ante 1576 : [[Gerardus Cibo]], ''Codices botanici'' [http://www.bl.uk/manuscripts/FullDisplay.aspx?ref=Add_MS_22332 vol. 1] [http://www.bl.uk/manuscripts/FullDisplay.aspx?ref=Add_MS_22333 vol. 2]
* 1576 : [[Matthias de Lobel]], ''Plantarum seu stirpium historia'' [https://archive.org/details/gri_33125009313699 Textus] [https://archive.org/details/mobot31753003488167 Textus]
* 1576 : [[Reginaldus Scot|Reginald Scot]], ''A Perfite Platforme of a Hoppe Garden''. Londinii: Henrie Denham [http://www.loc.gov/item/2004574086/ Textus]
* 1576 : [[Carolus Clusius]], ''Rariorum aliquot stirpium per Hispanias ...'' [https://archive.org/details/mobot33768000399827/page/n2 Textus]
* 1577 : [[Thomas Hyll]], ''The Gardener’s Labyrinth''. Londinii [https://archive.org/details/CAT10889940 Textus]
* 1577 : [[Raphael Holinshed]], ''[[Chronicles of England, Scotland, and Ireland]]''. Londinii [https://archive.org/details/firstevolumeofch01wolf vol. 1] [https://archive.org/details/firstevolumeofch02wolf vol. 2]
* 1577 : John Frampton, interpr.; [[Nicolaus Monardes]], ''Ioyfull newes out of the newe founde worlde''. Londinii {{Google Books|OatkAAAAcAAJ| editio 1580}} [https://archive.org/details/mobot31753000810678 editio 1596]
* 1578 : [[Ioannes Lerius|Jean Léry]], ''[[Historia navigationis in Brasiliam|Histoire d'un voyage fait en la terre du Bresil]]'' (Rupellae: Antoine Chuppin) [https://archive.org/details/histoiredunvoyag01lryj Textus]
* 1578 : Calveto, interpr.; [[Hieronymus Benzo|Girolamo Benzoni]], ''[[Novae Novi orbis historiae|Novae Novi orbis historiae, id est, rerum ab Hispanis in India occidentali hactenus gestarum, et acerbo illorum in eas gentes dominatu, libri tres]]'' (Genavae) [https://www.e-rara.ch/gep_g/content/titleinfo/1752097 Textus]
* 1578 : &lt;span id=&quot;C. Acosta (1578)&quot;&gt;&lt;/span&gt;[[Christophorus Acosta]], ''[[Aromatum et medicamentorum in orientali India nascentium historia|Tractado delas drogas y medicinas de las Indias Orientales]]'' [http://bibdigital.rjb.csic.es/ing/Libro.php?Libro=4587 Textus]
* 1578 : &lt;span id=&quot;Lyte (1578)&quot;&gt;&lt;/span&gt;[[Henricus Lyte|Henry Lyte]], interpr., [[Rembertus Dodonaeus]], ''A Niewe Herball, or historie of plantes'' (Antverpiae) [https://archive.org/details/hin-wel-all-00000717-001 Textus] [https://archive.org/details/mobot31753000811155 Textus] [https://archive.org/details/mobot31753000811148 Editio 1586]
* c. 1580 : [[Prosper Alpinus]]: vide 1735
* 1580 : &lt;span id=&quot;Dodonaeus (1580)&quot;&gt;&lt;/span&gt;[[Rembertus Dodonaeus]], ''[[Historia vitis vinique|Historia vitis vinique et stirpium nonnullarum aliarum]]''. Coloniae {{Google Books|EBBO60Ljh9sC}}
* 1580 : ''Pharmacopoea seu de usitatiorum medicamentorum componendorum ratione liber''. Bergomi {{Google Books|XnZVAAAAcAAJ}}
* post 1580 : &lt;span id=&quot;O'Brian et al. (1996)&quot;&gt;&lt;/span&gt;''[[Histoire naturelle des Indes]]'' (manuscriptum) in Patrick O'Brian et al., ''The Drake Manuscript'' (Londinii: André Deutsch, 1996)
* ante 1581 : &lt;span id=&quot;Duran (ante 1581)&quot;&gt;&lt;/span&gt; [[Didacus Duran]], ''[[Codex Duran|Historia de las Indias de Nueva-España y islas de Tierra Firme]]'' (Mexicopoli, 1867-1880) vol. 1 [https://archive.org/stream/historiadelasind01dur#page/210/mode/2up p. 211] etc.
* 1581 : [[Alexander Traianus Petronius]], ''De victu Romanorum et de sanitate tuenda'' (1581) {{Google Books|0UtBAAAAcAAJ|in indice}}
* 1581 : [[Matthias de Lobel]], ''Kruydtboeck''
* 1581-1592 : [[Augerius Gislenius Busbequius]], ''[[Legationis Turcicae epistolae|Itinera Constantinopolitanum et Amasianum]]'' (= ''Legationis Turcicae epistolae'' I-II) {{Google Books|nv0QS19NhJEC|Editio 1581}} {{Google Books|AMXFsrOw1zcC|Editio 1582}} [https://archive.org/details/bub_gb_nv0QS19NhJEC Textus apud archive.org];  ''[[Legationis Turcicae epistolae|Legationis Turcicae epistolae quatuor]]'' (editio perfecta, 1595) [http://www.uni-mannheim.de/mateo/camenahist/autoren/busbecq_hist.html Textus] apud [[CAMENA]]m
* 1582 : &lt;span id=&quot;Donno, ed. (1976)&quot;&gt;&lt;/span&gt;Richard Madox, Ephemeris (Elizabeth Story Donno, ed., ''An Elizabethan in 1582: the diary of Richard Madox, Fellow of All Souls'' [Londinii: Hakluyt Society, 1976] p. 176)
* 1582 : &lt;span id=&quot;Clusius (1582)&quot;&gt;&lt;/span&gt;[[Carolus Clusius]], ''Aliquot notae in Garciae Aromatum historiam; Descriptiones nonnullarum stirpium, et aliarum exoticarum rerum, quae à ... Francisco Drake ... observatae sunt'' (Antverpiae: Plantin) [https://www.e-rara.ch/zuz/nagezh/content/pageview/9431392 Textus] pp. 28-30 {{Google Books|sF5WAAAAcAAJ}}
* 1582 : &lt;span id=&quot;Clusius (1582)&quot;&gt;&lt;/span&gt;[[Carolus Clusius]], interpr., ''[[Aromatum et medicamentorum in orientali India nascentium historia|Christophori a Costa ... Aromatum et medicamentorum in Orientali India nascentium liber]]'' (Antverpiae: Plantin) [https://archive.org/details/mobot31753000812856 Textus] apud ''Internet Archive''; [https://archive.org/details/mobot31753003634646 2a ed. 1593]
* 1582 : [[Nicolaus Praepositus]], ''Dispensarium Nicolai Praepositi ad aromatarios''; [[Matthaeus Platearius|Platearius]] vulgo ''[[Circa instans]]'' nuncupatus [https://archive.org/details/BIUSante_pharma_res011020 Textus] [https://www.arlima.net/mp/nicole_prevost.html De Nicolao Praeposito]
* 1582-1583 : [[Leonhartus Rauwolfius]], ''Aigentliche Beschreibung der Raiß, so er vor diser Zeit gegen Auffgang inn die Morgenländer ... selbs volbracht''. Laugingen: Rainmichel, Willers [http://reader.digitale-sammlungen.de/de/fs1/object/display/bsb10901685_00005.html Impressum 1582] [http://reader.digitale-sammlungen.de/de/fs1/object/display/bsb10206788_00005.html Impressum 1583]
* 1583 : &lt;span id=&quot;Clusius (1583)&quot;&gt;&lt;/span&gt;[[Carolus Clusius]], ''Rariorum aliquot stirpium per Pannoniam ...''. Antverpiae [https://archive.org/details/mobot31753000811379 Textus]
* 1583 : &lt;span id=&quot;Clusius (1583)&quot;&gt;&lt;/span&gt;[[Carolus Clusius]], interpr.; [[Rembertus Dodonaeus]], ''Stirpium historiae pemptades sex''. Antverpiae [https://archive.org/details/mobot31753000817947 Textus] [https://archive.org/stream/hin-wel-all-00000420-001#page/n289/mode/2up p. 263] &quot;Chrysanthemum Peruvianum&quot; [https://archive.org/details/mobot31753000817939 Editio 1616]
* 1583 : &lt;span id=&quot;Caesalpinus (1583)&quot;&gt;&lt;/span&gt;[[Andreas Caesalpinus]], ''De plantis libri XVI'' (Florentiae) {{Google Books|Iw2Z4FlnR2YC}} [https://archive.org/details/deplantislibrixv00cesa Textus]
* ante 1584 : &lt;span id=&quot;Díaz (ante 1584)&quot;&gt;&lt;/span&gt;[[Bernardus Díaz del Castillo]], ''[[Historia verdadera (Díaz)|La historia verdadera de la conquista de la Nueva España]]'' (manuscriptum, ante 1584) cap. 83 [http://www.rae.es/sites/default/files/Aparato_de_variantes_Historia_verdadera_de_la_conquista_de_la_Nueva_Espana.pdf p. 254 editionis interretialis Serés]
* 1584 : [[Ioannes Baptista Rossetti|Giovan Battista Rossetti]], ''Dello scalco''. Ferrariae [https://www.mori.bz.it/gastronomia/Rossetti%20-%20Dello%20scalco.pdf Textus]
* 1584 : A. W., ''[[A Book of Cookery (A. W.)|A Book of Cookrye]]''; cf. 1591
* 1584-1598 : [[Gulielmus Bouchet|Guillaume Bouchet]], ''Les serées''. 3 voll. Pictaviae, Lutetiae, 1584-1598 [https://archive.org/details/hin-wel-all-00000003-001 vol. 1 ed. 1585] [https://gallica.bnf.fr/ark:/12148/bpt6k87071084 alia] [https://numelyo.bm-lyon.fr/f_view/BML:BML_00GOO0100137001102193823 vol. 1 ed. 1586] [https://archive.org/details/premierliuredess00bouc vol. 1 ed. 1608] [https://archive.org/details/secondliuredesse00bouc vol. 2 ed. 1608] [https://archive.org/details/troisiesmeliured00bouc vol. 3 ed. 1598] [vol. 3 ed. 1608]
* ante 1585 : &lt;span id=&quot;Sahagún (ante 1585)&quot;&gt;&lt;/span&gt;[[Bernardinus de Sahagun]], ''[[Historia general de las cosas de Nueva España]]'' (manuscriptum, ante 1585) lib. 10 f. 49r [https://www.wdl.org/es/item/10621/ quaere paginam 101] (Arthur J. O. Anderson, Charles E. Dibble, interprr., ''Bernardino de Sahagún: The Florentine Codex; General History of the Things of New Spain'' [University of Utah Press, 1950-1975. 12 partes]); cf. &lt;span id=&quot;Codices Matritenses&quot;&gt;&lt;/span&gt;''Codices Matritenses'' apud [[#Durand-Forest (1967)]] [http://www.historicas.unam.mx/publicaciones/revistas/nahuatl/pdf/ecn07/092.pdf pp. 159-162] 
* 1585 : &lt;span id=&quot;Mendoza (1585)&quot;&gt;&lt;/span&gt;{{Creanda|en|Juan González de Mendoza|Ioannes Consalvus Mendoza|Juan González de Mendoza}}, ''Historia de las cosas mas notables, ritos y costumbres, del gran reyno dela China'' (Romae: a costa de Bartholome Grassi) [https://archive.org/details/historiadelascos00gonz/page/6/mode/2up p. 7]. Versio Latina: ''Nova et succincta, vera tamen historia de amplissimo potentissimoque ... regno China'' (Francofurdi ad Moenum, 1589) [https://archive.org/details/nouaetsuccinctav00gonz/page/n39/mode/2up p. 36]
* 1585 : &quot;De gallinacei generis totius in cibis usu&quot; in [[Conradus Gesnerus]], ''Historiae animalium liber III'' (Francofurdi, 1585) [https://archive.org/details/bub_gb_OKGaf7B1PFsC/page/n411/mode/1up pp. 403-408]
* 1585 : &lt;span id=&quot;Durante (1585)&quot;&gt;&lt;/span&gt;{{Creanda|en|Castore Durante|Castor Durante}}, ''Herbario nuovo'' [http://dfg-viewer.de/show/?tx_dlf%5Bid%5D=http%3A%2F%2Fdigital.ub.uni-duesseldorf.de%2Foai%2F%3Fverb%3DGetRecord%26metadataPrefix%3Dmets%26identifier%3D2434775&amp;tx_dlf%5Bpage%5D=5&amp;tx_dlf%5Bdouble%5D=0&amp;cHash=070eae8db1735f8b28140bb175c89ce8 Editio 1602] [https://archive.org/details/bub_gb_v7cQtey_AvwC Editio 1617] apud ''Internet Archive'' [http://bibdigital.rjb.csic.es/ing/Libro.php?Libro=4714 Editio 1617]
* 1585/1587 : [[Thomas Dawson]], ''[[The Good Housewife's Jewel]]''; cf. 1596
* 1586 : [[Renatus de Goulaine de Laudonnière|René de Goulaine de Laudonnière]], ''L'histoire notable de la Floride située ès Indes Occidentales''. Lutetiae: chez Guillaume Auvray [https://catalog.hathitrust.org/Record/100828509 Textus] apud ''Hathi Trust'' [https://gallica.bnf.fr/ark:/12148/btv1b86107660?rk=64378;0 Textus] apud ''Gallica''
* 1586: [[Ioannes Lerius]], ''[[Historia navigationis in Brasiliam]]'' (Genavae: Eustathius Vignon) [https://archive.org/stream/historianauigati00lryj#page/154/mode/2up pp. 155-156] [https://archive.org/stream/historianavigati00lryj#page/154/mode/2up 2a ed.]
* 1586 : {{Creanda|en|Castore Durante|Castor Durante}}, ''Il tesoro della sanità'' [https://archive.org/details/iltesorodellasa01duragoog Editio 1596] apud ''Internet Archive''
* 1586 : &lt;span id=&quot;Dalecampius et al. (1586)&quot;&gt;&lt;/span&gt;[[Iacobus Dalecampius]]; Gulielmus Rouillius, ed., ''Historia generalis plantarum''. Lugduni {{Google Books|eP5u41mhCRoC|pars prima}} {{Google Books|ZGuAqsKxwNkC|pars altera}}
* 1586 : &lt;span id=&quot;Camerarius (1586)&quot;&gt;&lt;/span&gt;[[Ioachimus Camerarius (iunior)|Ioachimus Camerarius]], ed.; [[Petrus Andreas Matthiolus]], ''De plantis epitome utilissima ... diligenter aucta'' [http://daten.digitale-sammlungen.de/bsb00089694/image_1 Textus]
* 1586 : &lt;span id=&quot;Camerarius (1586)&quot;&gt;&lt;/span&gt;[[Ioachimus Camerarius (iunior)|Ioachimus Camerarius]], interpr.; [[Petrus Andreas Matthiolus]], ''[[Commentarii (Matthiolus)|Kreutterbuch]]'' [http://daten.digitale-sammlungen.de/0009/bsb00091089/images/ Textus]
* 1587 : William Harrison, &quot;The Description of England&quot; in [[Raphael Holinshed]], ''[[Chronicles of England, Scotland, and Ireland]]'' (2a ed. Londinii) [https://archive.org/details/firstsecondvolum00holi Textus]
* 1588 : ''The Good Hous-wives Treasurie''. Londinii: Edwarde Allde, 1588
* 1588 : &lt;span id=&quot;Hariot (1588)&quot;&gt;&lt;/span&gt;[[Thomas Hariot]], ''A briefe and true report of the new found land of Virginia, of the commodities there found and to be raysed ...''. Londinii [https://archive.org/details/abriefeandtruer00harigoog/page/n24 Textus editionis fac-simile 1903]; [https://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1020&amp;context=etas Recensio interretialis]; [https://archive.org/details/briefetruereport00harr/page/n9 Textus editionis illustratae 1590 Francofurti ad Moenum impressae]
* 1588 : [[Iulianus Palmarius]], ''[[De vino et pomaceo]]''. Parisiis: apud Guillelmum Auvray [https://archive.org/details/hin-wel-all-00002644-001 Textus] apud ''Internet Archive''
* 1588 : &lt;span id=&quot;Bigges (1588)&quot;&gt;&lt;/span&gt;Walterus Bygges, ''Expeditio Francisci Draki equitis Angli in Indias Occidentales anno MDLXXXV''. Leydae: apud Fr. Raphelengium [http://hdl.loc.gov/loc.rbc/rbdk.d020 Textus] [https://archive.org/details/narrationesduadm00bigg Editio 1590] [http://www.philological.bham.ac.uk/bigges/ Editio interretialis] [http://www.ancienttexts.org/library/latinlibrary/biggs.html Editio] apud ''[[The Latin Library]]'' [https://archive.org/details/levoyagedemessir00bigg versio Francogallica 1588] [https://archive.org/details/relationoderbesc00bigg Versio Theodisca 1589]
* 1589 : [[Carolus Clusius]], interpr.; [[Petrus Bellonius]], ''Plurimarum singularium et memorabilium rerum observationes''. Antverpiae, 1589 [https://archive.org/details/hin-wel-all-00001827-001 Textus apud archive.org] [http://gallica.bnf.fr/ark:/12148/bpt6k536173 Textus apud gallica]
* 1589 : Walter Bygges; Thomas Cates, ed., ''A Summarie and True Discourse of Sir Frances Drakes West Indian Voyage wherein were taken the townes of Saint Iago, Sancto Domingo, Cartagena and Saint Augustine''. Londinii: Roger Ward [http://hdl.loc.gov/loc.rbc/rbdk.d022 Textus] 
* 1589 : [[Carolus Clusius]], interpr.; [[Petrus Bellonius]], ''De neglecta stirpium cultura atque earum cognitione libellus'' {{Google Books|VCM6AAAAcAAJ}} [https://archive.org/details/hin-wel-all-00001827-002 Textus apud archive.org]
* 1589 : [[Ricardus Hakluytus|Richard Hakluyt]], ''[[Principal Navigations (Hakluytus)|The Principall Navigations, Voiages and Discoveries of the English Nation]]'' (1a ed. Londinii: George Bishop, 1589) [https://archive.org/details/cihm_35668 Textus]
* 1589 : ''Ephemeris expeditionis Norreysii &amp; Draki in Lusitaniam''. Londinii, 1589 [https://archive.org/details/ephemerisexpedit00unse Textus] apud ''Internet Archive'' [https://archive.org/details/narrationesduadm00bigg editio 1590]
* ante 1590 : [[Carolus Clusius]], Esaya le Gillon, ''[[Codex Clusii|Icones fungorum in Pannoniis observatorum]]'' (manuscriptum)  [https://www.biodiversitylibrary.org/bibliography/7131#/summary Textus]  [https://www.biodiversitylibrary.org/item/30616#page/297/mode/1up Editio fac-simile 1900]
* 1590 : &lt;span id=&quot;I. Acosta (1590)&quot;&gt;&lt;/span&gt;[[Iosephus de Acosta]], ''[[Naturalis et moralis Indiae Occidentalis historia|Historia natural y moral de las Indias]]''. Hispali: en casa de Juan de Leon [https://archive.org/details/historianaturaly00acos_2 Textus] [https://archive.org/details/historianatural01acosrich Editio 1894]
* 1590 : [[Thomas Hariot]]; [[Carolus Clusius]], interpr.; [[Theodorus de Bry]], ed., ''[[America (series librorum)|Admiranda narratio fida tamen, de commodis et incolarum ritibus Virginiae]]''. Francofurti ad Moenum [https://archive.org/details/admirandanarrati00harr Textus] [https://archive.org/details/Wunderbarliched00BryT Editio Theodisca 1593]
* 1590 : &lt;span id=&quot;Camerarius (1590)&quot;&gt;&lt;/span&gt;[[Ioachimus Camerarius (iunior)|Ioachim Camerarius]], ''[[Symbolorum et emblematum centuriae|Symbolorum et emblematum ex re herbaria desumtorum centuria una]]'' [https://archive.org/details/symbolorvmetembl01came f. 59] (no. 49) &quot;Chrysanthemum Perunianum, planta maxima, flos solis&quot;
* 1590 : &lt;span id=&quot;Tabernaemontanus (1590)&quot;&gt;&lt;/span&gt;[[Iacobus Theodorus Tabernaemontanus]], ''[[Eicones plantarum seu stirpium]]'' [https://archive.org/stream/Eiconesplantaru00Theo#page/858/mode/2up p. 859]
* c. 1590 : [[Abū al-Faḍl al-Mubārak]], ''[[Āīn-i-Akbarī]]'' (H. Blochmann, H. S. Jarrett, interprr., ''The Aín i Akbari by Abul Fazl Allámi'' [Calcuttae: Asiatic Society, 1873-1894] [https://archive.org/details/ainiakbari00jarrgoog vol. 1] [https://archive.org/details/ainiakbarivolum00mubgoog 2] [https://archive.org/details/in.ernet.dli.2015.46757 3])
* 1591 : [[Matthias de Lobel]], ed., ''Icones stirpium seu plantarum tam exoticarum quam indigenarum''. Antverpiae: Plantin [https://www.biodiversitylibrary.org/bibliography/9308 Textus] [https://archive.org/details/mobot31753000812435 alibi]
* 1591 : &lt;span id=&quot;Cardenas (1591)&quot;&gt;&lt;/span&gt;[[Ioannes de Cardenas|Juan de Cárdenas]], ''Primera parte de los problemas y secretos maravillosos de las Indias'' (Mexicopoli) [https://archive.org/stream/primerapartedelo00cr#page/112/mode/2up pp. 113-115 editionis 1913]
* 1591 : [[Philippus Pigafetta]], ''Relatione del reame di Congo et delle circonvicine contrade, tratta dalli scritti &amp; ragionamenti di Odoardo Lopez''. Romae: Bartolomeo Grassi {{Google Books|-aurfsHGclwC}} [https://archive.org/details/regnvmcongohoces00piga_2 versio Latina 1598]
* 1591 : [[Iacobus Le Moyne de Morgues]]; [[Theodorus de Bry]], ed., ''[[America (series librorum)|Brevis narratio eorum quae in Florida Americae provicia Gallis acciderunt, secunda in illam navigatione duce Renato de Laudoniere... anno MDXIV, quae est secunda pars Americae]]''. Francofurti ad Moenum [https://archive.org/details/brevisnarratio00debry Textus apud ''Internet Archive''] [https://gallica.bnf.fr/ark:/12148/bpt6k109491c?rk=42918;4 Textus] apud ''Gallica''
* 1591 : &lt;span id=&quot;Hortop (1591)&quot;&gt;&lt;/span&gt;Job Hortop, ''The Travailes of an English Man''. Londinii, 1591 [https://quod.lib.umich.edu/e/eebo/A03702.0001.001 editio interretialis]
* 1591 : A. W., ''[[A Book of Cookery (A. W.)|A Book of Cookrye]]''. 2a ed. Londinii: Edward Allde, 1591 (prima editio, 1584) [http://jducoeur.org/Cookbook/Cookrye.html Textus] (&quot;To make Farts of Portingale&quot;)
* 1591 : [[Gao Lian]], ''[[Commentarii octo de vitae principiis]]'' (遵生八牋) (Sumei Yi, Eugene Anderson, interprr., &quot;[https://www.happygoatproductions.com/blog/2015/8/26/guest-blog-a-translation-by-sumei-yi-edited-by-eugene-anderson-of-gao-lians-writings-on-food-and-drink Gao Lian (Ming dynasty): Essays on Drinks and Delicacies for Medicinal Eating]&quot;. 2008/2015)
* ante 1592 : [[Augerius Gislenius Busbequius]], ''[[Epistolarum legationis Gallicae libri II|Epistolae ad Rudolphum II Imperatorem e Gallia scriptae]]'' (= ''Epistolarum legationis Gallicae'' liber I) [http://gallica.bnf.fr/ark:/12148/bpt6k6434677v Editio 1630] {{Google Books|n_9GAAAAcAAJ|Editio 1631}}; ''[[Epistolarum legationis Gallicae libri II]]'' (editio perfecta, 1632) {{Google Books|OqDGk6ENA4YC}}
* 1593 : Marco Bussato, ''Giardino di agricoltura'' [https://archive.org/details/giardinodiagric00buss Textus]
* 1594 : Laurentius Scholzius, ''Catalogus arborum, fruticum ac plantarum, tam indigenarum quam exoticarum, horti medici D. Laurentii Scholzii'' (Vratislaviae, 1594) [https://gallica.bnf.fr/ark:/12148/bpt6k851094q/f10.item s.v.]
* 1594 : ''[[The Good Housewife's Handmaid for the Kitchen|The Good Huswifes Handmaide for the Kitchin]]'' (Londinii: Richard Jones, 1594) [https://www.uni-giessen.de/fbz/fb05/germanistik/absprache/sprachverwendung/gloning/ghhk/ Textus]
* ante 1595 : &lt;span id=&quot;Recchus (ante 1595)&quot;&gt;&lt;/span&gt;[[Franciscus Hernandez]]; [[Nardus Antonius Recchus]], scriba, ''[[Rerum medicarum Novae Hispaniae thesaurus|De materia medica Novae Hispaniae, Philippi Secundi Hispaniarum ac Indiarum regis invictissimi iussu]]'' (manuscriptum, ante 1595) [https://archive.org/stream/demateriamedican00hern#page/n101 Textus]
* 1595 : [[Iohannes Hugonis Linscotanus|Jan Huygen van Linschoten]], ''Reys-gheschrift vande navigatien der Portugaloysers in Orienten''. Amstelodami: Cornelis Claesz. {{Google Books|bbVOAAAAcAAJ}}
* 1595-1600 : {{Creanda|en|Thomas Platter the Younger|Thomas Platerus (medicus)}}, ''Itinera''. [https://www.e-manuscripta.ch/bau/doi/10.7891/e-manuscripta-18678 Textus libri manuscripti] Editiones: Hans Hecht, ed., ''Thomas Platters des Jüngeren Englandfahrt im Jahre 1599''. Halle, 1929. Rut Keiser, ed., &quot;[https://www.e-periodica.ch/digbib/view?rid=bzg-002:1963:63::94#94 Aus dem Tagebuch von Thomas Platter dem Jüngeren]&quot; in ''Basler Zeitschrift für Geschichte und Altertumskunde'' vol. 63. (1963) pp. 75–111 Rut Keiser, ed., ''Thomas Platter d. J.: Beschreibung der Reisen durch Frankreich, Spanien, England und die Niederlande (1595-1600)''. 2 voll. Basileae: Schwabe, 1968. Versiones: ''Félix et Thomas Platter à Montpellier 1552-1559--1595-1599: notes de voyage de deux étudiants Bâlois''. Monspessulani: Société de Bibliophiles de Montpellier, 1892 [https://archive.org/details/flixetthomaspla00platgoog Textus] apud ''Internet Archive'' Clare Williams, interpr., ''Thomas Platter's Travels in England''. Londinii: Cape, 1937. [https://archive.org/details/in.ernet.dli.2015.506933 Textus] apud ''Internet Archive'' Seán Jennett, interpr., ''Journal of a Younger Brother: the life of Thomas Platter as a medical student in Montpellier''. Londinii. Muller, 1963. Ludovic Legré; M. Kieffer, interpr., ''La Botanique en Provence au XVIe siècle: Félix et Thomas Platter, avec extraits relatifs à la Provence des mémoires de Félix et de Thomas Platter''. Marseille: Aubertin &amp; Rolle, 1900 [https://archive.org/details/labotaniqueenpr00legr Textus] apud ''Internet Archive''
* 1596 : [[Iohannes Hugonis Linscotanus|Jan Huygen van Linschoten]], ''Itinerario, Voyage ofte schipvaert''. Amstelodami: Cornelis Claesz. {{Google Books|UbVOAAAAcAAJ}} [https://archive.org/details/itinerariovoyage00lins Textus] [https://archive.org/details/andertheilderori00lins_0 versio pars 2 1598] [https://archive.org/details/drittertheilindi00lins_0 versio pars 3 1599] [https://archive.org/details/vierdertheildero00lins_0 versio pars 4 1600] [https://archive.org/details/quintaparsindiae00neck_0 versio pars 5]
* 1596 : &lt;span id=&quot;Baccius (1596)&quot;&gt;&lt;/span&gt;[[Andreas Baccius]], ''[[De naturali vinorum historia]]''. Romae: ex officina Nicholai Mutii [http://www.mdz-nbn-resolving.de/urn/resolver.pl?urn=urn:nbn:de:bvb:12-bsb10148134-6 Textus apud MDZ] {{Google Books|H31DAAAAcAAJ}} [https://archive.org/details/bub_gb_RPdmcHgp1QEC/page/n321/mode/2up p. 299]
* 1596 : &lt;span id=&quot;Bauhinus (1596)&quot;&gt;&lt;/span&gt;[[Casparus Bauhinus]], ''[[Phytopinax]]''. Basileae: per Sebastianum Henricpetri [https://archive.org/stream/mobot31753000815909#page/154/mode/2up pp. 155-156]
* 1596 : &lt;span id=&quot;Li (1596)&quot;&gt;&lt;/span&gt;{{Creanda|en|Li Shizhen|Li Shizhen}}, ''{{Creanda|en|Compendium of Materia Medica|Compendium materiae medicae|Bencao gangmu}}'' (vide [[#Shurtleff et Aoyagi (2012)]] pp. 40-45)
* 1596 : [[Thomas Dawson]], ''[[The Good Housewife's Jewel|The Good Huswifes Jewell]]'' [http://www.medievalcookery.com/notes/ghj1596.txt editio interretialis]; cf. 1585
* 1597 : ''The Second Part of the Good Hus-wives Jewell''. Londinii: Edward White, 1597 [https://wellcomecollection.org/works/cg64by9v Catalogus]
* 1597 : &lt;span id=&quot;Gerard (1597)&quot;&gt;&lt;/span&gt;[[Iohannes Gerardus|John Gerard]], ''[[Herbal (Gerard)|The Herball, or generall historie of plantes]]''. Londinii: John Norton [https://www.biodiversitylibrary.org/bibliography/51606#/summary Textus] [https://archive.org/details/mobot31753000817749 Textus]
* 1597/1626 : &lt;span id=&quot;McClure, ed. (1939)&quot;&gt;&lt;/span&gt;[[Ioannes Chamberlain]], epistolae (N. E. McClure, ed., ''The Letters of John Chamberlain''. 2 voll. Philadelphiae: American Philosophical Society, 1939 [https://babel.hathitrust.org/cgi/pt?id=uc1.32106005854481 1] [https://babel.hathitrust.org/cgi/pt?id=uc1.32106005854473 2])  
* ante 1598 : [[Ferdinandus Alvarado Tezozómoc|Hernando Alvarado Tezozómoc]], ''Crónica Mexicana'' (c. 1598) (Mexicopoli: Editorial Leyenda, 1944) [http://www.cervantesvirtual.com/obra/cronica-mexicana-escrita-hacia-el-ano-de-1598-929707/ Textus]
* 1598 : &lt;span id=&quot;Camerarius (1598)&quot;&gt;&lt;/span&gt;[[Ioachimus Camerarius (iunior)|Ioachimus Camerarius]], ''Hortus medicus et philosophicus, in quo plurimarum stirpium breves descriptiones, novae icones non paucae, indicationes locorum natalium, observationes de cultura earum peculiares, atque insuper nonnulla remedia euporista, nec non philologica quaedam continentur'' [http://daten.digitale-sammlungen.de/bsb00087678/image_1 Textus]
* 1598 : &lt;span id=&quot;Bauhinus (1598)&quot;&gt;&lt;/span&gt;[[Casparus Bauhinus]], ed.; [[Petrus Andreas Matthiolus]], ''Petri Andreae Matthioli Opera quae extant omnia: hoc est, commentarii in VI libros Pedacii Dioscoridis Anazarbei De medica materia, adiectis in margine variis Graeci textus lectionibus ... De ratione distillandi ... Apologia in Amatum Lusitanum ... Epistolarum medicinalium libri quinque ... Dialogus de morbo Gallico''. Francofurti {{Ling|Latine}} [https://opacplus.bsb-muenchen.de/search?oclcno=634151153&amp;db=100 Textus]
* 1598-1600 : [[Ricardus Hakluytus|Richard Hakluyt]], ''[[Principal Navigations (Hakluytus)|The Principal Navigations, Voyages, Traffiques and Discoveries of the English Nation]]'' (2a ed. Londinii: George Bishop, 1598-1600) [https://archive.org/details/principalnavigat1and2hakl voll. 1-2] [https://archive.org/details/cihm_94220 vol. 3] [https://quod.lib.umich.edu/cgi/t/text/text-idx?c=eebo;idno=A02495.0001.001 editio interretialis]
* ante 1599 : ''[[Naturstudien]]'' (manuscriptum) [http://data.onb.ac.at/rec/AC14451685 Series imaginum]
* 1599 : [[Iohannes Hugonis Linscotanus]], ''Navigatio ac itinerarium Iohannis Hugonis Linscotani in orientalem sive Lusitanorum Indiam''. Hagae Comitum, 1599 [https://archive.org/details/Nauigatio-ac-itinerarium-Iohannis-Hugonis-Linscotani-in-orientalem-siue-Lusitanoru-PHAIDRA_o_386621/page/n6/mode/1up Textus]
* 1599 : [[Iohannes Gerard]], ''Catalogus arborum, fruticum, ac plantarum tam indigenarum, quam exoticarum, in horto Johannis Gerardi civis et chirurgi Londinensis nascentium'' {{Google Books|RcIYAAAAIAAJ|Textus editionis fac-simile 1962}}
* c. 1600 : [[Anselmus De Boodt]], Album quadripedum [https://www.rijksmuseum.nl/en/collection/RP-T-BR-2017-1-2 Textus] [https://www.rijksmuseum.nl/en/press/press-releases/tefafs-top-masterpiece-to-the-rijksmuseum De collectu]
* ante 1600 : [[Ioannes Victorius Soderini]], ''Trattato di agricoltura'' [https://archive.org/details/bub_gb_WFYxxvLvl6cC Editio 1850]
* ante 1600 : [[Ioannes Victorius Soderini]], ''Della cultura degli orti e giardini'' in Giuseppe Sarchiani, ed., ''Della cultura degli orti e giardini: trattato di Giovanvettorio Soderini'' (Florentiae, 1814) [https://archive.org/details/dellaculturadegl00sode Textus]
* ante 1600 : [[Ioannes Victorius Soderini]], ''Trattato degli animali domestici'' in Alberto Bacchi della Lega, ed., ''Il trattato degli animali domestici di Giovanvettorio Soderini'' (Bononiae, 1907) [https://archive.org/details/iltrattatodegli00sodegoog Textus]
* 1600 : [[Ioannes Victorius Soderini]], ''Trattato della coltivazione delle viti''. Florentiae: Giunti [https://archive.org/stream/trattatodellacol00sode#page/n195 Editio 1806]
* 1600 : [[Olivarius de Serres|Olivier de Serres]], ''[[Le Théâtre d'agriculture|Le Theatre d'agriculture et mesnage de champs]]'' (Lutetiae) [https://gallica.bnf.fr/ark:/12148/bpt6k1510895w/f167 Textus]

== 1601-1650 ==
* 1601 : [[Iohannes Hugonis Linscotanus|Jan Huygen van Linschoten]], ''Voyagie, ofte schip-vaert ... van by Noorden ...'' [https://archive.org/details/itinerariovoyage00lins Textus]
* 1601 : &lt;span id=&quot;Clusius (1601)&quot;&gt;&lt;/span&gt;[[Carolus Clusius]], ''[[Rariorum plantarum historia]]'' (Antverpiae, 1601) [https://archive.org/stream/mobot31753000810538#page/n577/mode/2up appendix p. ccii]
* 1601 : &lt;span id=&quot;Spachius (1601)&quot;&gt;&lt;/span&gt;[[Israel Spachius]], interpr.; [[Ioannes Fragosus]], ''Aromatum, fructuum et simplicium aliquot medicamentorum ex India utraque .. in Europam delatorum ... historia brevis''. Argentinae {{Google Books|0oo6AAAAcAAJ}}
* circa 1602 : &lt;span id=&quot;Champlain (c. 1602)&quot;&gt;&lt;/span&gt;[[Samuel Champlain]], ''Brief Discours des choses plus remarquables que Sammuel Champlain de Brouage a reconneues aux Indes Occidentalles'' in C.-H. Laverdière, ed., ''Oeuvres de Champlain'' vol. 1 (Quebeci, 1870) [https://archive.org/stream/uvresdechamplai01cham#page/n91/mode/2up Textus] [https://archive.org/details/narrativeofvoyag00chamrich Versio Anglica 1859]
* 1603 : &lt;span id=&quot;Champlain (1603)&quot;&gt;&lt;/span&gt;[[Samuel Champlain]], ''Des Sauvages, ou Voyage de Samuel Champlain de Brouage'' (Lutetiae) [http://gallica.bnf.fr/ark:/12148/btv1b8626417m/f1.image Textus]
* 1603 : &lt;span id=&quot;Vocabulario (1603)&quot;&gt;&lt;/span&gt;''Vocabulario da lingoa de Japam, com adeclaração em portugues, feito por alguns padres e irmaõs da Companhia de Jesu''. Em Nengasaqui [Nagasaki], 1603 [https://archive.org/details/nippojishovocabv06doit Exemplare mutuabile] 1604 : 
* ante 1604 : [[Thomas Moffetus]]; ''[[Health's Improvement|Healths Improvement]]'' (Christopher Bennet, ed. Londinii: Samuel Thomson, 1655 [https://www.loc.gov/item/07018644/ Textus] [https://quod.lib.umich.edu/cgi/t/text/text-idx?c=eebo;idno=A89219.0001.001 editio interretialis]) 
* 1604 : &lt;span id=&quot;Rios (1604)&quot;&gt;&lt;/span&gt;[[Gregorius de los Ríos|Gregorio de los Rios]], ''Agricultura de iardines'' s.v. &quot;pomates&quot; {{Google Books|M6br_Cu7EZ4C|Fasciculus ultimus}}
* 1604 : [[Iosephus de Acosta|José de Acosta]], Eduardus Grimston, interpr., ''[[Naturalis et moralis Indiae Occidentalis historia|The naturall and moral historie of the East and West Indies]]'' (Londinii, 1604) [https://archive.org/stream/naturallmoralhis00acos#page/n297/mode/2up pp. 270-273]
* 1605 : &lt;span id=&quot;Clusius (1605)&quot;&gt;&lt;/span&gt;[[Carolus Clusius]], ''[[Exoticorum libri decem]]'' (Antverpiae) [https://archive.org/stream/mobot31753000811387#page/n69/mode/1up pp. 55-56]
* 1607 : Gianfranco Angelita, ''I pomi d'oro''
* 1609 : [[Marcus Lescarbot|Marc Lescarbot]], ''Histoire de la Nouvelle France, contenant les navigations, découvertes, et habitations faites par les François és Indes Occidentales et Nouvelle-France''. Lutetiae: Jean Milot  [https://archive.org/details/lesmusesdelanouv00lesc Textus]; 3a ed. Lutetiae: Adrian Perier, 1617-1618 [https://archive.org/details/histoiredelanouu00lesc/page/930 p. 931]
* 1609 : Juan Barrios, ''Libro en el cual se trata del chocolate''. Mexicopoli (de hoc libro vide [[#Pinelo (1636)]] f. 120r; [http://www.scielo.org.mx/pdf/ecn/v46/v46a3.pdf p. 70])
* 1609-1617 : [[Garcias Lasus Inca]], ''[[Comentarios Reales de los Incas]]''. Olisipone: en la officina de Pedro Crasbeeck, 1609 [https://archive.org/details/primerapartedelo00vega pars 1]; [https://archive.org/details/bub_gb_cRWQ_k1TXmUC pars 2]; [http://shemer.mslib.huji.ac.il/lib/W/ebooks/001531300.pdf recensio interretialis] &lt;!-- standard title from part 2, f. 1 --&gt;
* 1611 : &lt;span id=&quot;Gregorius de Regio (1611)&quot;&gt;&lt;/span&gt;[[Gregorius de Regio]], &quot;[[De varietate capsicorum]]&quot; in [[Carolus Clusius]], ''Curae posteriores'' [https://archive.org/stream/bub_gb_18RbAAAAQAAJ#page/n107/mode/2up pp. 95-108]
* 1611 : [[Theodorus de Bry]], ed., ''Florilegium novum'' [https://archive.org/details/florilegiumnovu00bryj Textus]
* 1612 : [[Paulus Hentznerus]], ''Itinerarium Germaniae, Galliae, Angliae, Italiae''. Norimbergae [https://archive.org/details/itinerariumgerma00hent Textus]
* 1612 : &lt;span id=&quot;Champlain (1612)&quot;&gt;&lt;/span&gt;[[Samuel Champlain]], &quot;Carte géographique de la Nouvelle Franse faicte par le Sieur de Champlain Saint Tongois&quot; [http://gallica.bnf.fr/ark:/12148/btv1b53098793g/f1 imago]
* 1613 : &lt;span id=&quot;Champlain (1613)&quot;&gt;&lt;/span&gt;[[Samuel Champlain]], ''Les Voyages du Sieur de Champlain Xaintongeois'' (Lutetiae) [https://archive.org/details/lesvoyagesdusieu00cham Textus] [http://gallica.bnf.fr/ark:/12148/bpt6k1094934/f1.image Textus]
* 1613 : [[Basilius Besler]], ''[[Hortus Eystettensis|Hortus Eystettensis, sive Diligens et accurata omnium plantarum, florum, stirpium... quae in celeberrimis viridariis arcem episcopalem ibidem cingentibus, hoc tempore conspiciuntur, delineatio et ad vivum repraesentatio]]'' [http://bibdigital.rjb.csic.es/ing/Libro.php?Libro=1617 vol. 1] [http://bibdigital.rjb.csic.es/ing/Libro.php?Libro=1623 vol. 2] [https://www.biodiversitylibrary.org/creator/94396#/titles voll. 1-2 alio itinere] [https://archive.org/details/HortusEystettensisSiveDiligensEtAccurataOmniumPlantarumFlorumStirpium1613BasiliusBeslerIII vol. 1 alibi]
* 1613 : [[Gervasius Markham|Gervasius Markham]], ''The English Husbandman''. Londinii [http://www.gutenberg.org/files/22973/22973-h/22973-h.htm Recensio interretialis]
* 1613 : [[Ulysses Aldrovandus]]; Ioannes Cornelius Uterverius, Hieronymus Tamburinus, edd., ''De piscibus libri V, et De cetis lib. unus'' (Bononiae) [http://amshistorica.cib.unibo.it/diglib.php?inv=17 pp. 692-696]
* 1614 : &lt;span id=&quot;Yi Su-gwang (1614)&quot;&gt;&lt;/span&gt;[[Yi Su-gwang]], ''[[Variae commentationes Jibong]]'' (''Jibong yuseol'': vide [[#Dott (2020)]] p. 24) 
* 1614 : Diego Granado, ''Libro del arte de cozina''. Lerida: Manescal, 1614 [https://archive.org/details/hin-wel-all-00001044-001/ Textus]
* 1615 : &lt;span id=&quot;Dalecampius et al. (1615)&quot;&gt;&lt;/span&gt;[[Iacobus Dalecampius|Jacques Daléchamps]], Jean Des Moulins, ''Histoire générale des plantes''. Lugduni {{Google Books|QZgOp_vxrEkC|pars prima}} {{Google Books|tJRDAAAAcAAJ|pars altera}} &lt;!-- authorship to verify --&gt;
* 1615 : &lt;span id=&quot;Ximenez (1615)&quot;&gt;&lt;/span&gt;[[Franciscus Hernandez]]; Francisco Ximenez, interpr., ''[[Rerum medicarum Novae Hispaniae thesaurus|Quatro libros de la naturaleza y virtudes de las plantas y animales que estan recevidos en el uso de medicina en la Nueva Espana]]'' (1615) lib. 2 cap. 3 [http://bibdigital.rjb.csic.es/spa/Libro.php?Libro=4961 ff. 72r-74r] {{Google Books|qQpFp_bImPMC}}
* 1615 : [[Gervasius Markham|Gervasius Markham]], ''Countrey Contentments ... [The English huswife]'' [https://archive.org/details/b30337628 Textus]
* 1615 : [[Ioannes Murrell|I. M.]], ''A New Booke of Cookerie ... London Cookerie''. Londinii: John Browne, 1615 fol. B 8 [https://www.uni-giessen.de/fbz/fb05/germanistik/absprache/sprachverwendung/gloning/tx/1615murr.htm Editio interretialis] (&quot;To make an oyster pye&quot;)
* 1615 : [[Ioannes de Torquemada|Juan de Torquemada]], ''Monarquía Indiana''. Hispali, 1615 {{Google Books|0O5cTpgOIpQC&amp;|pp. 550, 620 editionis 1723}} &quot;chocolate&quot;)
* 1616 : [[Philippus Guamán Poma de Ayala]], ''[[El primer nueva corónica y buen gobierno]]'' (manuscriptum) [http://www.kb.dk/permalink/2006/poma/info/en/frontpage.htm Recensio interretialis] 
* 1616 : [[Fabius Columna]], ''Minus cognitarum rariorumque nostro coelo orientium stirpium ecphrasis'' [https://archive.org/details/mobot31753000810439 Textus]
* 1616 : [[Martinus de Murua]], ''[[Historia general del Piru]]'' (manuscriptum) [http://www.getty.edu/publications/virtuallibrary/9780892368952.html Editio facsimile 2008]
* 1617 : [[Ioannes Murrell|John Murrell]], ''A Daily Exercise for Ladies and Gentlewomen, whereby they may learne and practice the whole art of making pastes, preserves, marmalades, conserves, tartstuffes, gellies, breads, sucket-candies, cordiall waters, conceits in sugar-workes of severall kindes: as also to dry lemonds, orenges, or other fruits''. Londinii: The widow Helm [https://www.loc.gov/item/65059509/ Textus]; [https://quod.lib.umich.edu/cgi/t/text/text-idx?c=eebo2;idno=A07931.0001.001 editio interretialis]
* 1618 : Bartolomé Marradón, ''Diálogos del uso del tabaco y los daños que causa ... y del chocolate y otras bebidas''. Sevilla: Gabriel Ramos [https://babel.hathitrust.org/cgi/pt?id=uc1.31822031019177;view=1up;seq=65 Versio Francogallica imperfecta, 1643]
* 1618 : William Lawson, ''A New Orchard and Garden''
* 1619 : &lt;span id=&quot;Champlain (1619)&quot;&gt;&lt;/span&gt;[[Samuel Champlain]], ''Voyages et descouvertures faites en la Nouvelle France'' (Lutetiae) [http://gallica.bnf.fr/ark:/12148/bpt6k8705145s Textus]
* 1620 : &lt;span id=&quot;Bauhinus (1620)&quot;&gt;&lt;/span&gt;[[Casparus Bauhinus]], ''Prodromus Theatri botanici'' [https://archive.org/details/mobot31753000495744 Textus] [http://www.botanicus.org/title/b11921341 pp. 90-91]
* 1620 : Tobias Venner, ''Via recta ad vitam longam''. Londonii, 1620 (vide et 1638)
* c. 1622 : [[Ioannes Smith (explorator)|Ioannes Smith]], ''The Historye of the Bermudaes or Summer Islands'' (ed. J. Henry Lefroy. Londinii: Hakluyt Society, 1889 [https://archive.org/details/historyebermuda00unkngoog/ Textus]
* 1623 : &lt;span id=&quot;Bauhinus (1623)&quot;&gt;&lt;/span&gt;[[Casparus Bauhinus]], ''Pinax Theatri botanici''. Basileae: sumptibus Ludovici Regis [https://archive.org/details/mobot31753000495769/page/101/mode/2up pp. 101-103]
* 1623 : [[Gervasius Markham|Gervasius Markham]], ''Countrey Contentments, or The English huswife'' (editio separata et aucta: olim pars secunda operis ''Countrey Contentments'', 1615) [https://digital.library.lse.ac.uk/objects/lse:heh898zor Textus]
* 1623 : Pierre Vallet, ''Le Jardin du Roy tres chrestien, Loys XIII, Roy de France et de Navare'' [https://archive.org/details/mobot31753003039788 Textus]
* 1624 : [[Henricus Wotton|Henry Wotton]], ''[[The Elements of Architecture (Wotton)|The Elements of Architecture]]''. Londinii [https://archive.org/details/architectureelem00wott Textus]
* 1624 : Santiago Valverde Turices, ''Un Discurso del chocolate''. Seville (citatio verificanda)
* 1625 : [[Franciscus Bacon|Francis Bacon]], &quot;Of Gardens&quot; in ''[[Essays or Counsels|The Essayes Or Counsels, Civill and Morall]&quot;. Londinii {{Google Books|JQwzAQAAMAAJ}}
* 1626 : Philip Nichols; {{Creanda|en|Sir Francis Drake, 1st Baronet|Franciscus Drake (baronettus I)|Francis Drake (nepos)}}, ed., ''Sir Francis Drake revived''. Londinii: Bourne, 1626 [https://archive.org/details/sirfrancisdrakea00quin/page/n17/mode/2up? Titulus huius editionis]
* 1627 : Daniel Rabel, ''Theatrum Florae''
* 1628 : {{Creanda|en|Sir Francis Drake, 1st Baronet|Franciscus Drake (baronettus I)|Francis Drake (nepos)}}, ed., ''The World Encompassed by Sir Francis Drake''. Londinii: Bourne, 1628 [https://www.wdl.org/en/item/624/ Textus]; [http://hdl.loc.gov/loc.rbc/rbdk.d042 Textus] apud Bibliotheca Congressus; [http://nationalhumanitiescenter.org/pds/amerbegin/contact/text5/drake.pdf Editio interretialis]
* 1628 : &lt;span id=&quot;Sala (1628)&quot;&gt;&lt;/span&gt;Ioannes Dominicus Sala, ''De alimentis et eorum recta administratione'' (Patavii: Giovanni Batista Martino) [http://www.mdz-nbn-resolving.de/urn/resolver.pl?urn=urn:nbn:de:bvb:12-bsb11105714-6 Textus]
* 1629 : [[Ioannes Parkinsonus|John Parkinson]], ''[[Paradisi in sole Paradisus terrestris]]'' (Londinii: Humphrey Lownes) [https://archive.org/details/mobot31753000703782/page/586 pp. 596-598]
* 1630 : [[Ioannes Murrell|John Murrell]], ''Murrels Two Books of Cookerie and Carving''. Londinii {{Google Books|s69jMjBUQ4kC|Pars prima editionis 1630}} {{GB|2EB9-2tcdq4C|Pars altera editionis 1650}} {{GB|wAaYygGsMQsC|Pars tertia editionis 1641 cum libri omnis indice}} [https://quod.lib.umich.edu/e/eebo2/A51636.0001.001?view=toc recensio interretialis editionis 1641] 
* 1630 : Bartholomaeus Ambrosinus, ''De capsicorum varietate cum suis iconibus brevis historia''. Bononiae
* ante 1631 : &lt;span id=&quot;Bontius (1642)&quot;&gt;&lt;/span&gt;[[Iacobus Bontius]], ''De medicina Indorum libri quattuor'' (Lugduni Batavorum, 1642) [http://data.onb.ac.at/rep/1088AC21 Textus]
* ante 1631 : [[Iacobus Bontius]], &quot;Historiae naturalis et medicae Indiae orientalis libri sex&quot; in [[Gulielmus Piso]], ''De Indiae utriusque re naturali et medica'' (1658) [https://archive.org/stream/mobot31753002909064#page/n395/mode/2up Textus]
* ante 1631 : L. S. A. M. von Römer, ed., ''Epistolae Jacobi Bontii'' (Bataviae, 1921)
* 1631 : [[Antonius Colmenero de Ledesma]], ''Curioso tratado de la naturaleza y calidad del chocolate''. Matriti {{Google Books|1vhmAAAAcAAJ}}; Diego de Vades-forte [i.e. James Wadsworth], interpr., ''A curious treatise of the nature and quality of chocolate''. Londinii: J. Okes, 1640 [https://quod.lib.umich.edu/e/eebo/A19160.0001.001?view=toc Textus] {{Google Books|UWRpAAAAcAAJ}}; René Moreau, interpr., ''Du chocolate: discours curieux''. Parisiis: Sebastien Cramoisy, 1643 [https://catalog.hathitrust.org/Record/010746929 textus]; Marcus Aurelius Severinus Tarsensis, interpr., ''Chocolata inda: opusculum de qualitate et natura chocolatae''. Sevilla, 1644 [https://catalog.hathitrust.org/Record/009281759 textus] [https://archive.org/details/chocolataindaopu00colm/page/n6 alibi]
* 1632 : &lt;span id=&quot;Champlain (1632)&quot;&gt;&lt;/span&gt;[[Samuel Champlain]] et al., ''Les Voyages de la Nouvelle-France occidentale, dicte Canada, faits par le Sr de Champlain'' (Lutetiae) [http://gallica.bnf.fr/ark:/12148/bpt6k5685518m/f4 Textus]
* 1633 : [[Ioannes Baptista Ferrarius]], ''De florum cultura''. Romae [https://www.biodiversitylibrary.org/bibliography/102250#/summary Textus]; ''Flora overo Cultura di fiori'' (1638) [https://www.biodiversitylibrary.org/bibliography/104943#/summary Textus]
* 1634 : [[Theodorus Mayernius]], ed., ''[[Insectorum theatrum|Insectorum sive minimorum animalium theatrum]], olim ab Edoardo Wottono, Conrado Gesnero, Thomaque Pennio inchoatum, tandem {{Creanda|en|Thomas Muffet|Thomas Moffetus|Tho. Moufeti}} opera ... concinnatum''. Londini: Thomas Cates, 1634 [https://archive.org/details/insectorumsivemi00moff Textus]
* 1635 : &lt;span id=&quot;Nierembergius (1635)&quot;&gt;&lt;/span&gt;[[Ioannes Eusebius Nierembergius]], ''Historia naturae maxime peregrinae'' (Antverpiae: ex officina Plantiniana, 1635) [https://archive.org/details/IoannisEvsebiiN00Nier/page/362/mode/2up pp. 363-364]
* 1636 : &lt;span id=&quot;Pinelo (1636)&quot;&gt;&lt;/span&gt;Antonio de León Pinelo, ''Question moral si el chocolate quebranta el ayuno eclesiástico''. Matriti [https://catalog.hathitrust.org/Record/009285452 Textus]
* 1638 : &lt;span id=&quot;Calancha (1628)&quot;&gt;&lt;/span&gt;Antonio de la Calancha, ''Corónica moralizada del órden de San Agustín en el Peru'' (Barcinone)
* 1638 : [[Thomas Herbert]], ''Some Yeares Travels into Divers Parts of Asia and Afrique''. Londinii {{Google Books|mlJBAAAAcAAJ}}
* 1638 : Tobias Venner, ''Via recta ad vitam longam''. Londonii, 1638 (vide et 1620)
* 1639 : [[Theodorus Mayernius|Theodore de Mayerne]], Thomas Cademan, ''The Distiller of London, compiled and set forth by the speciall licence and command of the Kings most excellent Majesty, for the sole use of the Company of Distillers of London''. Londinii: Richard Bishop {{Google Books|bF1jAAAAcAAJ}} [https://quod.lib.umich.edu/e/eebo2/A06304.0001.001?view=toc Editio interretialis]
* 1640 : &lt;span id=&quot;Parkinsonus (1640)&quot;&gt;&lt;/span&gt;[[Ioannes Parkinsonus]], ''Theatrum botanicum'' [https://babel.hathitrust.org/cgi/pt?id=ucm.5325114272;view=1up;seq=405 pp. 355-359] {{Google Books|5m72g_lC-RcC}}
* 1640 : [[Basilius Besler]], ''Hortus Eystettensis''. 2a ed. Norimbergae [https://archive.org/details/mobot31753003651095/page/n339/mode/2up classis autumnalis tabb. 1-2]
* 1640 : Walter Stonehouse, &quot;A Modell of my Garden at Darfield, 1640&quot; in R. T. Gunther, ''Early British Botanists and their Gardens'' (Oxoniae, 1922) pp. 348-351 [https://archive.org/details/earlybritishbota1922gunt Textus]
* 1642 : [[Iacobus Bontius]], ''De medicina Indorum libri IV'' (Lugduni Batavorum: apud Franciscum Hackium) {{Google Books|Es-IGwAACAAJ}}
* 1646 : [[Alphonsus de Ovalle]], ''Historica relacion del Reyno de Chile'' [http://www.memoriachilena.cl/602/w3-article-8380.html pp. 4-8, 51-58 editionis 1646]
* 1646 : [[Ioannes Baptista Ferrarius]], ''[[Hesperides (Ferrarius)|Hesperides]]''. Romae [https://www.biodiversitylibrary.org/bibliography/365#/summary Textus]
* 1648 : [[Gulielmus Piso]], [[Georgius Marcgravius]], ''[[Historia naturalis Brasiliae]]''. Lugduni Batavorum: apud Franciscum Hackium [https://archive.org/details/McGillLibrary-osl_historia-naturalis-brasiliae_folioP678h1648-20882/page/n3/mode/2up Textus] [https://archive.org/details/mobot31753000818648 Exemplar manu tinctum]
* 1648 : [[Thomas Gage (peregrinator)|Thomas Gage]], ''The English-American his Travail by Sea and Land, or A new survey of the West India's''. Londinii: John Sweeting [https://archive.org/details/graff_1470/page/n7/mode/2up Textus] (ed. Thompson (1958) p. 145)
* 1650 : Melchior Sebizius, ''De alimentorum facultatibus libri quinque'' (1650 {{Google Books|5GfgnMMEemYC|in indice}}
* 1650 : [[Samuel Hartlib]], ''A Discours of Husbandrie used in Brabant and Flanders''. 2a ed., 1652: ''''Samuel Hartlib his Legacie''. 3a ed., 1655: ''Samuel Hartlib his Legacy of Husbandry''. [https://archive.org/details/b30333453 2a ed.] {{Google Books|RRpN3QEZueQC|3a ed}}

== 1651-1700 ==
* 1651 : &lt;span id=&quot;Hernandez (1651)&quot;&gt;&lt;/span&gt;[[Franciscus Hernandez]]; [[Fridericus Caesius]] et al., edd., ''[[Rerum medicarum Novae Hispaniae thesaurus]]'' (Romae) lib. 8 cap. 50 [https://archive.org/stream/rerummedicarumno00hern#page/294/mode/2up pp. 295-296]
* 1651 : Pierre Morin, ''Catalogues de quelques plantes à fleurs qui sont de present au jardin de Pierre Morin le jeune''. Lutetiae
* 1651 : &lt;span id=&quot;La Varenne (1651)&quot;&gt;&lt;/span&gt;[[Franciscus Petrus de La Varenne|La Varenne]], ''Le Cuisinier françois'' (Lutetiae: David, 1651) [https://gallica.bnf.fr/ark:/12148/bpt6k114423k/ Textus]; (2a ed. Lutetiae: David, 1652) [https://archive.org/details/lecuisinierfranc00lava/page/210/mode/2up pp. 210-211] (&quot;aumare ... langouste&quot;)
* 1652 : [[Nicholaus Culpeper|Nicholas Culpeper]], ''[[The English Physitian]]'' (Londinii, 1652) [https://archive.org/details/b30335310/page/n9/mode/1up Textus]
* 1652 : James Wadsworth, ''Chocolate: or, An Indian Drinke''. Londinii: John Dakins, 1652
* ante 1653 : [[Barnabas Cobo|Bernabé Cobo]], ''[[Historia del Nuevo Mundo]]'' (1890-1893) ([https://archive.org/details/historiadelnuev00cobogoog vol. 1][http://bibliotecadigital.aecid.es/bibliodig/es/consulta/registro.cmd?id=579 vol. 2] [https://archive.org/details/historiadelnuevo0002unse vol. 2 alibi] [https://archive.org/details/historiadelnuev00espagoog vol. 3] [https://archive.org/details/historiadelnuev01cobogoog vol. 4]; [http://fondosdigitales.us.es/fondos/libros/2423/1012/historia-del-nuevo-mundo-por-el-padre-bernabe-cobo-de-la-compania-de-jesus/ pp. 1012-1164 libri manu scripti])
* 1652-1653 : Philip Nichols, Francis Fletcher et al.; R. D., ed., ''Sir Francis Drake revived; The world encompassed by Sir Francis Drake; A summarie and true discourse; A full relation of another voyage into the West Indies''. Londinii: Bourne, 1652-1653 [https://archive.org/details/sirfrancisdraker00bourrich/page/n5/mode/2up fasc. 1 i] [https://archive.org/details/sirfrancisdraker00nichrich/page/n5/mode/2up fasc. 1 ii] [https://archive.org/details/sirfrancisdraker00nichrich/page/n5/mode/2up fasc. 2] [https://archive.org/details/summarieandtrued00biggrich fasc. 3] [https://archive.org/details/fullrelationofan00bourrich fasc. 4]
* 1653 : Ralph Austen, ''A Treatise of Fruit Trees''
* 1653 : W. J., ''A True Gentlewomans Delight''. Londinii: R. Norton, 1653
* 1654 : [[Iosephus Cooper]], ''The Art of Cookery Refin'd and Augmented, containing an Abstract of Some Rare and Rich Unpublished Receipts of Cookery collected from the Practise of that Incomparable Master of These Arts, Mr. Jos. Cooper, Chiefe Cook to the Late King''. Londinii: R. Lowndes {{Google Books|CY9mAAAAcAAJ}}
* 1654 : ''The Ladies Companion''. Londinii: W. Bentley
* 1655 : [[Samuel Hartlib]], ''The Reformed Virginian Silk-worm'' {{Google Books|HlHjw-JHMI0C}}
* 1655 : [[Samuel Hartlib]], ''The Reformed Common-wealth of Bees'' [https://archive.org/details/reformedcommonwe00hart Textus]
* 1655 : [[Olaus Wormius]], ''Museum Wormianum''. Lugduni Batavorum: ex officina Elseviriorum [https://www.biodiversitylibrary.org/page/51143811#page/216/mode/1up p. 192]
* 1656 : [[Petrus de Lune|Pierre de Lune]], ''Le Cuisinier''. Lutetiae: David, 1656 [https://gallica.bnf.fr/ark:/12148/bpt6k9819294k Textus] apud Gallica (cf. 1659)
* 1656 : [[Michaël Boym]], ''[[Flora Sinensis]]''. Viennae Austriae: typis Matthaei Rictii, 1656 [https://neptun.unamur.be/s/neptun/item/8941 Imagines]
* 1656 : ''[[The Queen's Closet Opened]]''. Londinii {{Google Books|Lw1o1y349vQC|editio 1662}} [https://archive.org/details/101493846.nlm.nih.gov De hoc opere] [https://www.erudit.org/en/journals/cuizine/1900-v1-n1-cuizine0888/1019319ar/ De hoc opere]
* 1658 : [[Iacobus Bontius]], [[Gulielmus Piso]], ''[[De Indiae utriusque re naturali et medica]]'' (Amstelaedami: apud Elzevirios) [https://archive.org/details/mobot31753002909064 textus] {{Google Books|k_L7yEyzJ-wC|pars iii p. 200}} (&quot;tlilxochitl ... vaynillae&quot;)
* 1658 : [[Theodorus Mayernius]], ''Archimagirus Anglo-Gallicus, or Excellent and approved receipts and experiments in cookery''. Londinii: G. Bodell {{Google Books|EI9mAAAAcAAJ}} [https://ota.bodleian.ox.ac.uk/repository/xmlui/handle/20.500.12024/A50384 Editio interretialis]
* 1658 : [[Thomas Moffetus|Thomas Moffet]]; [[Theodorus Mayernius|Theodore de Mayerne]], ed., &quot;[[Insectorum theatrum|The Theater of Insects, or Lesser living creatures]] ... by Tho. Mouffet&quot; in Edward Topsel, ed., ''The History of Four-Footed Beasts and Serpents'' (Londinii: G. Sawbridge) [https://www.biodiversitylibrary.org/item/151513#page/841/mode/1up Textus]
* 1659 : Thomas Hanmer, ''The Garden Book''
* 1659 : [[Petrus de Lune|Pierre de Lune]], ''Le nouveau cuisinier''. Lutetiae: Pierre David, 1659 [https://archive.org/details/lenouueaucuisini00lune Textus] apud ''Internet Archive'' {{Google Books|oo1mAAAAcAAJ|2a ed. 1660}}
* 1659 : [[Petrus de Lune|Pierre de Lune]]? ''Le Maistre d'hostel ... ensemble Le Sommelier ... [Le Confiturier de la cour]''. Lutetiae: Pierre David, 1659 [https://archive.org/details/lemaistredhostel00unse Textus] apud ''Internet Archive''
* 1660 : &lt;span id=&quot;Almeida (1660)&quot;&gt;&lt;/span&gt;[[Manuel de Almeida]], ''Historia geral de Ethiopia a alta'' (Conimbrigae, 1660) {{Google Books|uSacgFzlt7gC|pp. 42-43}}, cf. C. F. Beckingham, G. W. B. Huntingford, ''Some Records of Ethiopia'' (Londinii: Hakluyt Society, 1954) 
* 1660 : [[Ioannes Raius]], ''Catalogus plantarum circa Cantabrigiam nascentium''. Cantabrigiae: impensis Gulielmi Nealand {{Google Books|qlc-AAAAcAAJ}}
* 1661 : Simon Paulli (titulus verificandus); Robert James, interpr., ''A Treatise on Tobacco, Coffee, Tea, and Chocolate''. Londinii: T. Osborne, 1746
* 1661 : [[Ioannes Drouhet|Jean Drouhet]], ''La Moirie de Sen-Moixont''. Pictavii: par Pierre Amassard, 1661 [https://gallica.bnf.fr/ark:/12148/btv1b86120666 Textus]
* 1662 : [[Henricus Stubbe|Henry Stubbe]], ''[[The Indian Nectar]], or a Discourse Concerning Chocolata''. Londinii [https://quod.lib.umich.edu/e/eebo/A61881.0001.001?view=toc Textus]
* 1662 : [[Petrus de Lune|Pierre de Lune]], ''Le nouveau et parfait maistre d'hostel royal''. Lutetiae: Loyson, 1662 [https://gallica.bnf.fr/ark:/12148/bpt6k1339465 Textus] apud Gallica
* 1662 : ''[[Le Cuisinier méthodique]]'' (Lutetiae: Promé, 1662) {{Google Books|W41mAAAAcAAJ}}
* 1662 : ''[[The Queen's Closet Opened|A Queens Delight, or The art of preserving, conserving, and candying, as also a right knowledge of making perfumes, and distilling the most excellent waters]]''. Londinii [https://archive.org/details/queensdelightora00unse Textus]
* ante 1663 : &lt;span id=&quot;Della Valle (1663)&quot;&gt;&lt;/span&gt;[[Petrus della Valle]]; Biagio Deversino, ed., ''Viaggio di Pietro della Valle il pellegrino''. 2 voll. Romae: Mascardi, 1657-1663 ([https://archive.org/details/bub_gb_Rf_QVsjgjocC vol. 1] [https://archive.org/details/viaggidipietrode02dell vol. 2] editionis 1843; [https://archive.org/details/travelsofpietrod00dell vol. 1] [https://archive.org/details/in.gov.ignca.13145 vol. 2] versionis Anglicae)
* 1663 : &lt;span id=&quot;Vasconcellos (1663)&quot;&gt;&lt;/span&gt;[[Simon de Vasconcellos|Simão de Vasconcellos]], ''Chronica da Companhia de Jesu do estado do Brasil'' (Olisipone) lib. 1 cap. 141 {{Google Books|bHT08G-vNSIC|pp. 86-87}} [https://archive.org/details/chronicadacompan00vasc vol. 1] [https://archive.org/details/chronicadacompan01vasc vol. 2 editionis 1865] 
* 1664 : [[Athanasius Kircherus]], ''Mundus subterraneus''. Amstelodami [https://archive.org/details/mundussubterrane00unse Textus] [https://archive.org/details/mundussubterrane02kirc 3a ed. 1678]
* 1664 : [[Ioannes Evelyn|John Evelyn]], ''[[Sylva (Evelyn)|Sylva, or a discourse of forest-trees and the propagation of timber in His Majesty's dominions]]'' (Londinii: [[Regalis Societas Londiniensis|Royal Society]], 1664) [http://www.archive.org/details/sylvaordiscourse00eveluoft Textus apud archive.org]
* 1664 : [[Ioannes Evelyn|John Evelyn]] et alii, &quot;[[Pomona (Evelyn)|Pomona]], or an appendix concerning fruit-trees in relation to cider&quot; in John Evelyn, ''[[Sylva (Evelyn)|Sylva, or a discourse of forest-trees and the propagation of timber in His Majesty's dominions]]'' (Londinii: [[Regalis Societas Londiniensis|Royal Society]], 1664) [http://www.archive.org/details/sylvaordiscourse00eveluoft Textus apud archive.org]
* 1665 : [[Ioannes Rea|John Rea]], ''Flora, seu De florum cultura, or A complete florilege, furnished with all the requisites belonging to a florist''. Londinii: Thomas Clarke. [https://archive.org/details/mobot31753000815081 Textus]
* 1665 : &lt;span id=&quot;May (1665)&quot;&gt;&lt;/span&gt;Robert May, ''The Accomplisht Cook'' (Londinii: Wood) {{Google Books|7dhopy-AJ98C}}
* 1667 : [[Athanasius Kircherus]], ''China ... illustrata''. Amstelodami [https://archive.org/details/bub_gb_CSjQ6j3dWcsC Textus] [https://htext.stanford.edu/content/kircher/china/kircher.pdf Versio Anglica]
* 1668 : [[Ioannes Henricus Meibomius]], ''[[De cervisiis potibusque et ebriaminibus extra vinum aliis]]'' (Helmstadii, 1668) [https://archive.org/details/bub_gb_e3LsZz6vApMC Textus] apud ''Internet Archive''
* 1668 : &lt;span id=&quot;Ecole (1668)&quot;&gt;&lt;/span&gt;''L'ecole des ragoust, ou Le chef-d'euvre du cuisinier, du patissier et du confiturier''. Lugduni, 1668 [https://archive.org/details/lecoledesragoust00lava/page/84/mode/2up pp. 84-85], [https://archive.org/details/lecoledesragoust00lava/page/330/mode/2up 331-332] (&quot;Bisque de poisson, bisque aux oeufs&quot;)
* 1668 : [[Petrus de Lune|Pierre de Lune]], ''Le Nouveau et Parfait Cuisinier''. Nova ed. Lutetiae [http://terroirs.denfrance.free.fr/p/frameset/07.html Fragmentum interretiale]
* 1669 : [[Kenelmus Digbius|Kenelm Digby]]; George Hartman, ed.? ''[[The Closet of Sir Kenelme Digbie Opened|The Closet of the Eminently Learned Sir Kenelme Digbie Kt. Opened]]''. Londinii: Henry Brome, 1669 {{Google Books|yc1EAQAAMAAJ|Textus e fac-simile demptus}} [https://www.gutenberg.org/ebooks/16441 editio interretialis] apud ''Project Gutenberg'' [https://archive.org/details/closetofeminentl00digb editio 1677]
* 1670 : [[Ioannes Raius]], ''Catalogus plantarum Angliae et insularum adiacentium, tum indigenas, tum in agris passim cultas complectens''. Londinii: impensis J. Martyn [https://gallica.bnf.fr/ark:/12148/bpt6k98508w Textus]; [https://www.biodiversitylibrary.org/bibliography/82234 2a ed. 1677]
* 1671 : Philippe Sylvestre Dufour, ''De l'Usage du caphé, du thé, et du chocolate''. Lugduni: J. Girin, B. Riviere {{Google Books|Ed0EPN_iakUC}}; John Chamberlayne, interpr., ''The Manner of Making Coffee, Tea, and Chocolate, as it is Used in Most Parts of Europe, Asia, Africa, and America''. Londinii: Christopher Wilkinson, 1685
* 1671 : W. M., ''The Compleat Cook'' (one of the three parts of The Queens Closet Opened). Londinii: E. Tyler &amp; R. Holt, 1671 (facs. repr. London, Prospect, 1984. The text remained the same through all editions from 1655)
* 1671 : [[Li Yu (scriptor)|Li Yu]], ''[[Fortuitae affectuum otiosorum adumbrationes|Xianqing ouji]]''
* ante 1672 : [[Anna Fanshawe]], ''Memoirs''. Editio: ''The Memoirs of Ann Lady Fanshawe, wife of the Right Honble. Sir Richard Fanshawe, Bart., 1600-72''. (Herbert Charles Fanshawe, ed.) Londinii: John Lane, 1907. [https://archive.org/details/memoirsofannlady00fansuoft Textus]
* 1672 : William Hughes, ''The American Physitian, or, A Treatise of the Roots, Plants, Trees, Shrubs, Fruit, Herbs, &amp;c. Growing in the English Plantations in America''. Londinii: William Crook
* 1673 : William Rabisha, ''The Whole Body of Cookery Dissected''. Londinii: E. Calvert, 1673 [https://www.loc.gov/item/44028918/ Textus] (1st ed. 1661?)
* 1673 : [[Ioannes Raius|John Ray]], ''Observations topographical, moral, and physiological, made in a journey through part of the Low-Countries, Germany, Italy, and France, with a catalogue of plants ...'' Londinii: John Martyn [https://archive.org/details/observationstopo00rayj Textus]
* 1674 : [[Ioannes Raius|John Ray]], ''A Collection of English Words not generally used ... with catalogues of English birds and fishes''. Londinii {{Google Books|njdWAAAAYAAJ}}
* 1675 : [[Ioannes Raius|John Ray]], ''Dictionariolum Trilingue Secundum Locos Communes, Nominibus Usitatioribus Anglicis, Latinis, Graecis, Ordine Parallelo Dispositis''. Londini: Burrel
* 1675 : [[Hadrianus Valesius]], ''Notitia Galliarum ordine litterarum digesta''. Lutetiae, 1675 {{Google Books|mSVENQUSrioC|p. 538}}
* 1675 : [[Franciscus van Sterbeeck]], ''Theatrum fungorum''
* 1676 : Denis Dodart, ''Mémoires pour servir à l’histoire des plantes''
* 1676-1689 : [[Ioannes Worlidge|John Worlidge]], ''[[Vinetum Britannicum]], or, A treatise of cider and other wines and drinks extracted from fruits growing in this kingdom with the method of propagating all sorts of vinous fruit-trees, and a description of the new-invented ingenio or mill for the more expeditious making of cider''. Londinii: T. Dring, 1676; 2a ed., 1678; 3a ed., 1691; ''The second parts of Systema agriculturae, or, The mystery of husbandry; and, Vinetum Britannicum, or, A treatise of cider; wherein are contained many selected and curious observations ... with the best and most natural rules and methods for the making of cider, and other English liquors''. Londinii, 1689
* 1675 : Hannah Wolley, ''The Queen-Like Closet''. Londinii: Richard Lownes [https://digital.library.lse.ac.uk/objects/lse:mir865luj Textus]
* 1676-1679 : [[Ioannes Baptista Tavernier|Jean-Baptiste Tavernier]], ''Les Six Voyages'' (Lutetiae, 1676-1679) {{GB|_cgM18MuVgsC|vol. 1}} {{GB|mV5CAAAAcAAJ|vol. 2}} {{Google Books|edIPyR4KgrsC|''Recueil'' [vol. 3]}} [https://archive.org/details/travelsinindia01tave Version Anglica vol. 1] [https://archive.org/details/travelsinindia02tave 2]
* 1677 : ''{{Creanda|en|Bak tongsa|Pak interpres}}'' (''Bak tongsa'' vel ''Piao tongshi'') (Svetlana R. Dyer, interpr., ''Pak the Interpreter: An Annotated Translation and Literary-cultural Evaluation of the Piao Tongshi of 1677''. Pandanus Books, 2006)
* 1678 : [[Ioannes Raius|John Ray]], ''A Collection of English Proverbs''. Cantabrigiae {{Google Books|rnlQoxh95VMC}}
* 1678 : [[Ioannes Raius|John Ray]], ed., ''The Ornithology of Francis Willughby of Middleton in the county of Warwick, esq.'' Londinii [https://archive.org/details/ornithologyFran00Will Textus]
* 1678-1703 : [[Henricus van Rheede|Henricus van Rhede tot Draakestein]], ''[[Hortus Malabaricus|Horti Malabarici]] pars prima [... duodecima]''. Amstelaedami: sumptibus Johannis van Someren et Joannis van Dyck [https://www.biodiversitylibrary.org/bibliography/707 Textus]
* 1680-1699 : [[Robertus Morison]], ''Plantarum historiae universalis Oxoniensis'' ... [pars i-iii]. Oxoniae {{GB|kc4wCzH05d4C|i}} {{GB|j0Ks4ncT6QEC|ii}} {{Google Books|lqPQH-5-d-oC|iii}}
* 1681 : Robert Knox, ''An Historical Relation of the Island Ceylon in the East-Indies'' (Londinii: Chiswell, 1681) {{Google Books|yFpTAAAAcAAJ|p. 12}}
* 1682 : [William Rabisha], ''The Whole Body of Cookery Dissected''. Londinii: George Calvert &amp; Ralph Simpson, 1682 (facs. repr. Totnes, Prospect, 2003; the text remained essentially unchanged from the first edition of 1661) cf. 1673
* 1682 : John Chamberlayne, ''The Natural History of Coffee, Thee', Chocolate, Tobacco''. Londinii: Christopher Wilkinson [https://archive.org/details/naturalhistoryof00chamuoft Textus]
* 1682 : ''Methodus plantarum nova : brevitatis &amp; perspicuitatis causa synoptice in tabulis exhibita''. Londinii: Faithorne [https://www.biodiversitylibrary.org/bibliography/37647 Textus]
* 1683 : ''The Young Cook's Monitor, or Directions for cookery and distilling, by M. H.'' Londinii: William Downing {{Google Books|ivlmAAAAcAAJ}}
* 1685 : Robert May, ''The Accomplisht Cook'' (5a ed. Londinii) [http://www.gutenberg.org/ebooks/22790 Textus] 
* 1685 : Philippe Sylvestre Dufour, ''Traitez nouveaux et curieux du café, du thé et du chocolate''. Lugduni: J. Girin, B. Riviere [https://archive.org/details/bub_gb_XDU97KHMFu4C Textus] [https://catalog.hathitrust.org/Record/009349246 Editio 1693]
* 1686-1704 : [[Ioannes Raius]], ''[[Historia plantarum generalis (Raius)|Historia plantarum generalis]]'' {{GB|btw-AAAAcAAJ|vol. 1 2a ed.}} {{GB|PXoxquU_jgUC|vol. 2 2a ed.}} {{Google Books|vKzKmrmHNfsC|vol. 3}}
* 1687 : Nicolas de Blegny, ''Le Bon Usage du thé, du caffé, et du chocolat''. Lugduni: T. Amaulry [https://catalog.hathitrust.org/Record/009281765 Textus]
* 1687 : [[Ioannes Worlidge|John Worlidge]], ''The most easie method for making the best cyder''. Londinii: George Graston
* 1688 : [[Ioannes Raius]], ''Fasciculus Stirpium Britannicarum, post editum plantarum Angliae catalogum observatarum''. Londinii: Faithorne {{Google Books|pxyYUGCMZs4C}}
* ante 1689 : [[Ioannes Reresby|John Reresby]], ''Memoirs''. Editio: James J. Cartwright, ed., ''The Memoirs of Sir John Reresby, of Thrybergh, bart., M. P. for York, &amp;c., 1634-1689''. Londinii: Longmans Green, 1875 [https://gallica.bnf.fr/ark:/12148/bpt6k105383c Textus]
* 1690 : [[Ioannes Raius]], ''Synopsis methodica stirpium Britannicarum''. Londini [https://www.biodiversitylibrary.org/bibliography/63346 Textus]
* 1691 : &lt;span id=&quot;Massialot (1691)&quot;&gt;&lt;/span&gt;[[Franciscus Massialot|François Massialot]], ''Le Cuisinier roial et bourgeois''. Lutetiae {{Google Books|HXwOAQAAIAAJ}}
* 1691 : Richard Ames, ''The Search after Claret''. Londinii: E. Hawkins, 1691 [https://quod.lib.umich.edu/e/eebo/A25274.0001.001/1:3?rgn=div1;view=fulltext Editio interretialis]
* 1691 : Richard Ames, ''A Farther Search after Claret''. Londinii: E. Hawkins, 1691 {{Google Books|hKbb2W5vyj4C}}
* 1693 : [[Franciscus Massialot|François Massialot]], ''Le Cuisinier royal et bourgeois''. 2a ed. Lutetiae [https://archive.org/details/lecuisinierroyal00mass Textus]
* 1693 : [[Ioannes Raius]], ''Synopsis methodica animalium quadrupedum et serpentini generis''. Londinii [https://archive.org/details/bub_gb_OnACLEEY8WQC Textus]
* 1693 : [[Ioannes Raius|John Ray]], ed., ''A Collection of Curious Travels and Voyages, in two tomes''. Londinii {{GB|8oMrAQAAMAAJ|vol. 1}} {{Google Books|DytDAAAAcAAJ|vol. 2}}
* 1693 : [[Ioannes Raius]], ''Historia plantarum generalis'' (1693) {{Google Books|btw-AAAAcAAJ|vol. 1 pp. 676-679}}
* 1694 : [[Antonius Latini|Antonio Latini]], ''Lo scalco alla moderna''. Neapoli, 1694 {{GB|edZMAAAAcAAJ|vol. 1}} {{Google Books|VHxPSH5JFxAC|vol. 2}} (Tommaso Astarita, interpr., ''Antonio Latini's The Modern Steward'' [Ledesiae: ARC Humanities Press, 2019] [https://assets.ctfassets.net/4wrp2um278k7/3Ok5PSNdcLjxMWsT3lJ1yP/533ac771a1f48e08efb4c2232e7d1813/Antonio_Latini___s_The_Modern_Steward__or_The_Art_of_Preparing_Banquets_Well_ToC___Intro.pdf Textus praefationis])
* 1694 : [[Ioannes Raius]], ''Stirpium Europaearum extra Britannias nascentium sylloge''. Londini: Smith &amp; Walford {{Google Books|8Fc-AAAAcAAJ}}
* ante 1696 : [[Ioannes Aubreius|John Aubrey]], ''[[Brief Lives]]'' (Andrew Clark, ed., ''Brief Lives, chiefly of contemporaries, set down by John Aubrey between the years 1669 and 1696'' [2 voll. Oxonii: Clarendon Press, 1898] [https://archive.org/details/briefliveschiefl01aubr 1] [https://archive.org/details/briefliveschiefl02aubr 2]) [https://archive.org/details/briefliveschiefl01aubr/page/224/mode/2up vol. 1 pp. 224-233 et alibi]
* 1696 : J. Ovington, ''A Voyage to Suratt in the Year 1689'' (Londinii: Tonson, 1696) [https://archive.org/details/AVoyageToSurattInTheYear1689/page/n419/mode/2up p. 397]; [https://archive.org/details/in.ernet.dli.2015.79805/page/n261/mode/2up p. 231 editionis 1929]
* 1698 : [[Ioannes Fryer|John Fryer]], ''A New Account of East-India and Persia in Eight Letters'' (Londinii: Chiswell, 1698) [https://archive.org/details/dli.granth.85618/page/404/mode/2up p. 404]
* 1699 : [[Martinus Lister|Martin Lister]], ''A journey to Paris in the year 1698'' (Londinii) [https://archive.org/details/McGillLibrary-osl_journey-paris_BO5070_L77325j1698-1699-20094/page/n75/mode/2up pp. 60-63]
* 1699 : [[Lionel Wafer]], ''A New Voyage and Description of the Isthmus of America''. Londinii (George Parker Winship, ed., 1903 [https://archive.org/details/newvoyagedescrip00wafe/page/106/mode/2up p. 107])
* 1700 : [[Iosephus Pitton de Tournefort]], ''[[Institutiones rei herbariae (Tournefort)|Institutiones rei herbariae]]'' [https://archive.org/details/mobot31753000521648 vol. 1] [https://archive.org/details/mobot31753000521655 vol. 2] [https://archive.org/details/mobot31753000521663 vol. 3]

== 1701-1750 ==
* 1702 : [[Franciscus Massialot]]; J. K., interpr., ''The Court and Country Cook, giving new and plain directions how to order all manner of entertainments''. Londinii: A. &amp; J. Churchill {{Google Books|paFhAAAAcAAJ}}
* 1703 : [[Ioannes Raius]], ''Nomenclator Classicus''. 4a ed. 1703 {{Google Books|gVFgAAAAcAAJ|p. 38}}
* 1703 : Iosephus Browne, ed., ''[[Theodorus Mayernius|Theo. Turquet Mayernii]] ... Opera medica''. Londinii, 1703 {{Google Books|EvRbAAAAcAAJ|pp. 65, 177}}
* 1705 : &lt;span id=&quot;Dale (1705)&quot;&gt;&lt;/span&gt;{{Creanda|en|Samuel Dale (physician)|Samuel Dale}}, ''Pharmacologiae, seu manuductionis ad materiam medicam, supplementum'' (1705) p. 184 {{Google Books|TapdAAAAcAAJ|cf. editio 1718 p. 193}}
* 1705 : {{Creanda|fr|Charles de Saint-Évremond|Sancteuremondus}}, ''Oeuvres meslées'' (Londinii: Tonson, 1705) {{GB|6wv7uLCSHiwC|vol. 1}} {{GB|e3rGj2_xGPEC|vol. 2 pars 1 p. 36}} {{Google Books|yj9tGOFo8TAC|vol. 2 pars 2}}
* 1706 : &quot;[[Acetaria (Evelyn)|Acetaria: a discourse of sallets]]&quot; in [[Ioannes Evelyn|John Evelyn]], ''[[Sylva (Evelyn)|Silva, or a discourse of forest-trees and the propagation of timber in His Majesty's dominions]]'' (4a ed. Londinii, 1706) [https://archive.org/details/b30414155/page/130/mode/2up pars 2 pp. 131-213]
* 1707 : [[Ioannes Christophorus Becmanus]], ''Historia orbis terrarum geographica et civilis''. 6a ed. 1707 {{Google Books|F4dBAAAAcAAJ|p. 420}}
* 1707-1725 : [[Ioannes Sloane|Hans Sloane]], ''A Voyage to the Islands Madera, Barbadoes, Nieves, S. Christophers and Jamaica'' (2 voll. Londinii, 1707–1725) [https://www.biodiversitylibrary.org/item/11242#page/411/mode/1up vol. 1 p. 240], [https://www.biodiversitylibrary.org/item/11241#page/401/mode/1up vol. 2 p. 378]
* 1711 : Charles Lockyer, ''An account of the trade in India''. Londinii, 1711 [https://archive.org/details/pli.kerala.rare.11854/ Textus]
* 1712 : &lt;span id=&quot;Kaempfer (1712)&quot;&gt;&lt;/span&gt;[[Engelbertus Kaempfer]], ''Amoenitatum exoticarum politico-physico-medicarum fasciculi V'' [https://archive.org/details/b30500606_0001/page/836/mode/2up pp. 837-840]
* 1713 : [[Ioannes Raius]], ''Synopsis methodica avium et piscium: opus posthumum''. Londini. Londinii [https://archive.org/details/bub_gb_d5NnSulin5oC Textus]
* 1714 : Mary Kettilby, ''A collection of above three hundred receipts in cookery, physick and surgery'' (Londinii, 1714) [https://archive.org/details/b30514976_0001/page/42/mode/2up p. 43]
* 1719 : Giovanni Francesco Upezzinghi, ''Il cuoco in villa''. Urbini, 1719 [https://archive.org/details/bub_gb_7WS5bUYnlckC Textus]
* c. 1722 : [[Franciscus Ximénez de Quesada]], ''Historia natural de la provincia de San Vicente de Chiapas y Guatemala''
* 1723 : John Nott, ''The Cooks and Confectioners Dictionary'' (1723) [https://archive.org/details/cooksandconfect00nottgoog/page/n399/mode/2up 152: To dress pikes a la Sainte Robert] (&quot;make your Sauce Robert in the following manner&quot;)
* 1723-1735 : Aerae Yongzheng ''Chorographia Shandong'' (山东通志): 秦椒，色红有子与花椒味俱辛 (&quot;Pipera ''qin'' colore rubra, granis plena, tam calida quam [[zanthoxyli fructus]]&quot;) 
* ante 1725 : Henry Barham, ''Hortus Americanus'' (Kingston Iamaicae, 1794) [https://archive.org/details/b29319870/page/30/mode/2up pp. 30-31]
* 1725 : Noël Chomel; R. Bradley, ed., ''Dictionaire oeconomique, or The family dictionary''. Londinii: D. Midwinter, 1725 {{Google Books|iXIiAQAAMAAJ|s.v. Civet}}
* 1726 : &lt;span id=&quot;Sanfelicius (1726)&quot;&gt;&lt;/span&gt;Antonius Sanfelicius, ''Campania notis illustrata''. Neapoli: Johannes-Franciscus Paci {{Google Books|Y4UqAO-bSwIC|pp. 121-123}}
* 1727 : &lt;span id=&quot;Smith (1729)&quot;&gt;&lt;/span&gt;[[Eliza Smith]], ''The Compleat Housewife''. Londinii {{Google Books|h_ZAAQAAMAAJ|3a ed. 1729}} [https://www.loc.gov/item/48039324/ 4a ed. 1730] [https://archive.org/details/b30498090_0001 6a ed. 1734] {{Google Books|7OJQAQAAIAA|9a ed. 1739}} [https://quod.lib.umich.edu/cgi/t/text/text-idx?c=evans;idno=N04107.0001.001 Editio interretialis editonis Virginianae 1742]
* 1731-1754 : &lt;span id=&quot;Zedler&quot;&gt;&lt;/span&gt;Johann Heinrich Zedler, ed., ''Grosses vollständiges Universal-Lexicon aller Wissenschaften und Künste'' [https://www.zedler-lexikon.de/index.html?c=zedlerinfo&amp;l=de editio interretialis]
* 1733 : Sarah Harrison, ''The house-keeper's pocket-book, and compleat family cook''. Londinii {{Google Books|8z-xmsHoqPQC|2a ed. 1739}} {{GB|yE3ncQhWcEsC|&quot;3a ed.&quot; Dublini, 1738}} {{GB|77FH9N99Yk4C|5a ed. 1751}}
* 1733 : &lt;span id=&quot;La Chapelle (1733)&quot;&gt;&lt;/span&gt;[[Vincentius La Chapelle|Vincent La Chapelle]], ''The Modern Cook'' (3 voll. Londinii: Nicolas Prevost, 1733) [https://archive.org/details/moderncook01lach vol. 1] [https://archive.org/details/moderncook02lach 2] [https://archive.org/details/moderncook03lach 3]
* 1735 : &lt;span id=&quot;La Chapelle (1735)&quot;&gt;&lt;/span&gt;[[Vincentius La Chapelle|Vincent La Chapelle]], ''Le cuisinier moderne'' (4 voll. Hagae Comitum, 1735) [https://gallica.bnf.fr/ark:/12148/bpt6k1042599r vol. 1] [https://gallica.bnf.fr/ark:/12148/bpt6k1042601c 2] [https://gallica.bnf.fr/ark:/12148/bpt6k10426036 3] [https://gallica.bnf.fr/ark:/12148/bpt6k10426051 4]
* 1735 : &lt;span id=&quot;Alpinus, ed. Veslingius (1735)&quot;&gt;&lt;/span&gt;[[Prosper Alpinus]]; Ioannes Veslingius, ed., ''Prosperi Alpini ... Historiae Aegypti naturalis pars prima [... secunda]'' (1735) [https://archive.org/details/prosperialpinima11735alpi/page/228/mode/2up p. 229]
* 1737 : [[Ioannes Raius|John Ray]], ''A Compleat Collection of English Proverbs; also the most celebrated proverbs of the Scotch, Italian, French, Spanish, and other languages ... A Collection of English Words''. Londinii: J. Hughs {{Google Books|VuhDAAAAcAAJ}}
* 1737 : [[Carolus Linnaeus]], ''Critica Botanica'' (Lugduni Batavorum, 1737) [https://bibdigital.rjb.csic.es/viewer/11541/?offset=#page=107&amp;viewer=picture&amp;o=bookmark&amp;n=0&amp;q= p. 91] [http://linnean-online.org/120000/ alibi]
* 1737 : [[Carolus Linnaeus]], ''[[Hortus Cliffortianus]]''. Amstelodami [https://www.biodiversitylibrary.org/bibliography/690 Textus]
* 1737 : [[Ioannes Burmannus]], ''Thesaurus Zeylanicus''. Amstelaedami: apud Janssonio-Waesbergios &amp; Salomonem Schouten [https://www.biodiversitylibrary.org/bibliography/734 Textus]
* 1738 : [[Ioannes Raius|John Ray]], ed.,  ''Travels through the Low Countries ... vol. 1'', ''A Collection of Curious Travels and Voyages ... vol. 2''. 2a ed. Londinii: J. Walthoe [etc.] [https://archive.org/details/travelsthroughlo01rayj vol. 1] [https://archive.org/details/travelsthroughlo02rayj vol. 2]
* 1738 : Joseph Pitts, ''A faithful account of the religion and manners of the Mahometans'' (4a ed. Londinii: Longman, 1738) {{Google Books|swthAAAAcAAJ|pp. 23-24}}
* 1738 : Thomas Shaw, ''Travels, or observations relating to several parts of Barbary and the Levant'' (Oxoniae, 1738) [https://archive.org/details/travelsorobserva00shaw/page/n15/mode/2up p. v]
* 1738 : [[Carolus Linnaeus]], ''Classes plantarum''
* 1739 : [Menon], ''Nouveau traité de la cuisine''. Lutetiae [https://gallica.bnf.fr/ark:/12148/bpt6k1511758k vol. 1] [https://gallica.bnf.fr/ark:/12148/bpt6k1511760n 2] apud Gallica
* 1739 : [[Franciscus Marin|François Marin]], ''Les Dons de Comus''. Lutetiae: Prault, 1739 {{Google Books|DhH_o1xyRaMC}}
* 1740 : ''Le Cuisinier gascon''. Lutetiae [https://gallica.bnf.fr/ark:/12148/bpt6k1511832p Textus]
* 1742 : [[Franciscus Marin|François Marin]], ''Suite des Dons de Comus''. 3 voll. Lutetiae: veuve Pissot, 1742 [https://gallica.bnf.fr/ark:/12148/bpt6k1512116g vol. 1] [https://gallica.bnf.fr/ark:/12148/bpt6k15121189 vol. 2] apud Gallica {{Google Books|7yA_clSTZc0C|vol. 3}}
* 1742 : [Menon], ''La Nouvelle Cuisine ... continuation au Nouveau traité de la cuisine''. Lutetiae {{Google Books|9YU7AAAAcAAJ}}
* 1742 : [[Vincentius La Chapelle|Vincent La Chapelle]], ''Le cuisinier moderne'' (2a ed. 5 voll. Hagae Comitum, 1742) [https://archive.org/details/lecuisiniermode04chapgoog vol. 1] [https://archive.org/details/lecuisiniermode01chapgoog 2] [https://archive.org/details/lecuisiniermode00chapgoog 3] [https://archive.org/details/lecuisiniermode02chapgoog 4] [https://archive.org/details/lecuisiniermode03chapgoog 5]
* 1742 : &lt;span id=&quot;Labat (1742)&quot;&gt;&lt;/span&gt;[[Ioannes Baptista Labat|Jean Baptiste Labat]], ''Nouveau voyage aux isles de l'Amerique'' (Lutetiae: Théodore Le Gras, 1742) [https://archive.org/details/nouveauvoyagea02laba/page/228/mode/2up pp. 228-229]
* 1743 : ''Voyages de Monsr. Shaw, M. D. dans plusieurs provinces de la Barbarie et du Levant'' (2 voll. Hagae Comitum: Neaulme, 1743) [https://archive.org/details/bub_gb_JnvRiNp8-AEC vol. 1] [https://archive.org/details/bub_gb_GTXTRek7HHQC 2]
* 1745 : [[Ioannes Altamiras|Juan Altamiras]], ''Nuevo arte de la cocina española''
* 1745 : {{Rumphius}} vol. 5 p. 247 et tab. 88; cf. {{Merrill}} [http://www.biodiversitylibrary.org/page/38882613#page/470/mode/1up p. 462]
* 1746 : Menon, ''La Cuisinière bourgeoise''. Lutetiae [https://gallica.bnf.fr/ark:/12148/bpt6k1512126v Textus] aud Gallica; [https://gallica.bnf.fr/ark:/12148/bpt6k9793731w ed. 1748] {{Google Books|DpNhwxHGEcEC|ed. 1750}} {{Google Books|4CGDiYkKP1kC|ed. 1769}}
* 1747 : &lt;span id=&quot;Glasse (1747)&quot;&gt;&lt;/span&gt;[[Anna Glasse|Hannah Glasse]], ''[[The Art of Cookery Made Plain and Easy]]'' (Londinii, 1747) [https://archive.org/details/artofcookerymade00glas/page/130/mode/2up pp. 130-131] (&quot;To pickle mackrel, call'd caveach&quot;)
* 1748 : &lt;span id=&quot;Behr (1748)&quot;&gt;&lt;/span&gt;Georg Heinrich Behr, ''Zwey Bücher Von der Materia medica''. Argentorati {{Google Books|P_JaAAAAcAAJ|p. 506}} (&quot;''Vinum Andegaviense'' oder ''Vin d'Anjou'')
* 1748-1799 : [[Georgius Washingtonius]], ''[[Georgii Washingtonii Ephemerides|Diaries]]'' (Donald Jackson, Dorothy Twohig, edd., ''The Diaries of George Washington'' vol. 4, 1784-1786 [Charlottesville, 1978] [https://tile.loc.gov/storage-services/service/mss/mgw/mgwd/wd04/wd04.pdf diebus 13 et 28 Iunii 1785])
* 1749 : ''[[L'Ecole du jardin potager]]''. Lutetiae [https://www.e-rara.ch/zut/doi/10.3931/e-rara-21828 Textus] apud ''e-Rara''
* 1749 : [Menon], ''La Science du maître d'hôtel cuisinier''. Lutetiae [https://gallica.bnf.fr/ark:/12148/bpt6k54549881 Textus]; [https://gallica.bnf.fr/ark:/12148/bpt6k15119766 ed. 1768] apud Gallica

== 1751-1800 ==
* 1752 : &quot;Brasilien-Pfeffer&quot; in Carl Günther Ludovici, ''Eröffnete Akademie der Kaufleute, oder vollständiges Kaufmanns-Lexicon'' vol. 1 (Lipsiae, 1752) {{Google Books|dbxRAAAAcAAJ|coll. 2091-2095}}
* 1753 : &lt;span id=&quot;Linnaeus (1753)&quot;&gt;&lt;/span&gt;{{Linnaeus SP|185}}
* 1753-1761 : [[Petrus Kalm|Pehr Kalm]], ''En resa til Norra America'' [https://www.biodiversitylibrary.org/bibliography/35535 Textus]
* 1755 : [[Ioannes Fridericus Gronovius|Iohannes Fredericus Gronovius]], ''Flora orientalis, sive recensio plantarum, quas ... [[Leonhartus Rauwolfius|Leonhardus Rauwolffus]] ... annis 1573-1574 et 1575 in Syria, Arabia, Mesopotamia, Babylonia, Assyria, Armenia &amp; Iudaea crescentes observavit''. Lugduni Batavorum: de Groot [http://www.mdz-nbn-resolving.de/urn/resolver.pl?urn=urn:nbn:de:bvb:12-bsb10302832-1 Textus]
* 1755 : Menon, ''[[Les Soupers de la cour]]''. 4 voll. Lutetiae: Guillyn, 1755 [https://gallica.bnf.fr/ark:/12148/bpt6k6468593p vol. 1] [https://gallica.bnf.fr/ark:/12148/bpt6k64685943 2] [https://gallica.bnf.fr/ark:/12148/bpt6k6463215w 3] [https://gallica.bnf.fr/ark:/12148/bpt6k64524528 4] apud Gallica{{GB|yzE7AAAAcAAJ|vol. 1}} {{GB|QTI7AAAAcAAJ|2}} {{GB|VDI7AAAAcAAJ|3}} {{Google Books|dTI7AAAAcAAJ|4}}; [https://gallica.bnf.fr/ark:/12148/bpt6k1511948w ed. 1778 vol. 1] [https://gallica.bnf.fr/ark:/12148/bpt6k1511950z 2] [https://gallica.bnf.fr/ark:/12148/bpt6k15119781 3]
* 1755 : &lt;span id=&quot;Glasse (1755)&quot;&gt;&lt;/span&gt;[[Anna Glasse|Hannah Glasse]], ''[[The Art of Cookery Made Plain and Easy]]'' (5a ed. Londinii, 1755) [https://archive.org/details/b30501842/page/334/mode/2up p. 334] (&quot;To make India pickle&quot;)
* 1756 : [[Carolus Linnaeus]], ''Centuria II. Plantarum ...'' (Upsaliae, 1756) [https://www.biodiversitylibrary.org/page/36017975#page/14/mode/1up pp. 10-11]
* 1757 : ''Hygiene dogmatico-practica rationem conservandae sanitatis corporis humani ... exponens''. Francofurti: Gaum {{Google Books|YN6CCrfVIlMC|p. 199}}
* 1757 : &lt;span id=&quot;Hill (1757)&quot;&gt;&lt;/span&gt;John Hill, ''Eden, or a compleat body of gardening'' (Londinii: Osborne, 1757) [https://archive.org/details/mobot31753000809647/page/n64/mode/1up p. 47]
* 1757 : [[Ricardus Bentley|Richard Bentley]], interpr.; [[Horatius Walpole|Horace Walpole]], ed. [[Paulus Hentznerus]], ''A Journey into England by Paul Hentzner in the year MDXCVIII''. Strawberry Hill. [https://archive.org/details/gri_33125011039175 Textus]
* 1758 : Juan Altamiras, ''Nuevo arte de cocina''. Barcinone: Bezàres, 1758 [http://bdh.bne.es/bnesearch/detalle/bdh0000098942 Textus]
* 1758 : &lt;span id=&quot;Glasse (1758)&quot;&gt;&lt;/span&gt;[[Anna Glasse|Hannah Glasse]], ''[[The Art of Cookery Made Plain and Easy]]''. 6a ed. Londinii {{Google Books|8I9cAAAAcAAJ|pp. 353, 374}} (&quot;How to make mead; To make white mead&quot;)
* 1758 : [Menon], ''Cuisine et office de santé''. Lutetiae [1757] [https://gallica.bnf.fr/ark:/12148/bpt6k15117649 Textus] apud Gallica {{Google Books|vNlQAQAAIAAJ}}
* 1758 : &lt;span id=&quot;Dombi (1758)&quot;&gt;&lt;/span&gt;Samuel Dombi, ''Dissertatio inauguralis physico-chemico-medica de vino Tokaiensi''. Traiecti ad Rhenum: Broedelet {{Google Books|afpAAQAAMAAJ}}
* 1759 : [[Carolus Linnaeus]], ''Amoenitates academicae'' vol. 4 (Holmiae, 1759) [https://www.biodiversitylibrary.org/item/15496#page/310/mode/1up p. 307]
* 1762-1778 : Christopher Sauer, &quot;Kurtzgefasstes Kräuterbuch&quot; (William Woys Weaver, interpr., ''Sauer's Herbal Cures: America's First Book of Botanic Healing, 1762-1778'' [Novi Eboraci: Routledge, 2001]) {{Google Books|OAMA51H9tfUC|Paginae selectae}}
* 1763 : Francisco Martínez Montiño, ''Arte de cocina, pastelerìa, vizcocherìa, y conserverìa''. Matriti, 1763 [http://bdh.bne.es/bnesearch/detalle/bdh0000202557 Textus];  [http://bdh.bne.es/bnesearch/detalle/bdh0000091117 Textus editionis 1822]
* 1764-1767 : J. de Blainville, ''Reisebeschreibung durch Holland, Oberdeutschland und die Schweiz, besonders aber durch Italien''. 4 voll. in 8 fascc. Lemgo: Meyer [https://www.digitale-sammlungen.de/en/view/bsb10366570 1.i] [https://www.digitale-sammlungen.de/en/view/bsb10366571 1.ii] [https://www.digitale-sammlungen.de/en/view/bsb10366572 2.i] [https://www.digitale-sammlungen.de/en/view/bsb10366573 2.ii] [https://www.digitale-sammlungen.de/en/view/bsb10366574 3.i] [https://www.digitale-sammlungen.de/en/view/bsb11450065 3.ii] [https://www.digitale-sammlungen.de/en/view/bsb10366576 4.i] [https://www.digitale-sammlungen.de/en/view/bsb10366577 4.ii] [https://www.digitale-sammlungen.de/en/view/bsb10366578 5]; alia exemplaria  [https://www.digitale-sammlungen.de/en/view/bsb11222371 1] [https://www.digitale-sammlungen.de/en/view/bsb11222372 2] [https://www.digitale-sammlungen.de/en/view/bsb11222373 3] [https://www.digitale-sammlungen.de/en/view/bsb11222374 4] [https://www.digitale-sammlungen.de/en/view/bsb11222375 5]
* 1765 : Louis de Chambray, ''L'Art de cultiver les pommiers, les poiriers et de faire des cidres selon l'usage de la Normandie''. Lutetiae: chez Ganeau [http://www.bmlisieux.com/normandie/chambray.htm Textus]
* 1766-1824 : [[Thomas Jeffersonius|Thomas Ieffersonius]], ''[[Thomas Ieffersonii Liber hortensis|Garden Book]]'' (Edwin Morris Betts, ed., ''Thomas Jefferson's Garden Book 1766-1824'' [Philadelphiae: American Philosophical Society, 1944] [https://archive.org/details/in.ernet.dli.2015.503395/ Textus] apud ''Internet Archive'')
* 1767 : Ioannes Michael Schosulan, ''Dissertatio inauguralis medica De vinis''. Viennae [Vindobonae]: Trattnern, 1767 {{Google Books|emg5AAAAcAAJ}}
* 1767 : ''Dictionnaire portatif de cuisine, d'office et de distillation''. Lutetiae: Vincent, 1767 {{Google Books|-HA9AAAAcAAJ}}
* 1768 : [[Philippus Miller|Philip Miller]], ''The Gardener's Dictionary''. 8a ed. Londinii [https://archive.org/details/b30454190/page/n214/mode/1up s.v. &quot;Capsicum&quot; ad finem] (&quot;Cayan butter, or what the inhabitants of America call Pepper-pots&quot;)
* 1768 : &lt;span id=&quot;Burmannus (1768)&quot;&gt;&lt;/span&gt;[[Nicolaus Laurentius Burmannus]], ''Flora Indica'' (Lugduni Batavorum: Haek, 1768) [https://www.biodiversitylibrary.org/page/39900544#page/147/mode/1up p. 57]
* 1769 : [[Ioannes Fridericus Zückert]], ''Materia alimentaria.'' Berolini: Augustus Mylius {{Google Books|zbVEAAAAcAAJ|pp. 341-353}}
* 1769 : [[Elizabeth Raffald]],&lt;ref&gt;Roy Shipperbottom, &quot;Elizabeth Raffald (1733-1781)&quot; in Harlan Walker, ed., ''Cooks and other people: proceedings of the Oxford Symposium on Food and Cookery'' (Totenais: Prospect Books, 1996) {{Google Books|lpOqTUucwhUC|pp. 233-236}}
&lt;/ref&gt; ''The experienced English housekeeper''. Mancunii [https://archive.org/details/b30522134/ Textus] [https://archive.org/details/b2150541x/ editio 1798]
* 1770 : Harriott Horry, ''Receipt Book'' (Richard J. Hooker, ed., ''A Colonial Plantation Cookbook: the Receipt Book of Harriott Pinckney Horry''. University of South Carolina Press, 1984) p.58 (&quot;to caveach mackrel&quot;) {{Google Books|feG7xCABZ44C|Paginae selectae}}
* 1770 : Lieutaud, interpr., ''Précis de la matiere médicale''. Nova ed. Lutetiae: Didot, 1770 [https://archive.org/details/b30535967_0001 vol. 1] [https://archive.org/details/b30535967_0002 vol. 2]
* 1770-1776 : [[Nicolaus Iosephus Jacquin]], ''Hortus botanicus Vindobonensis'' (Vindobonae: typis Leopoldi Joannis Kaliwoda aulae imperialis typographi, 1770-1776) [https://www.biodiversitylibrary.org/item/10249 vol. 1] [https://www.biodiversitylibrary.org/item/10250 vol. 2] [https://www.biodiversitylibrary.org/item/10251 vol. 3]
* 1771 : &lt;span id=&quot;Linnaeus (1771)&quot;&gt;&lt;/span&gt;[[Carolus Linnaeus]], ''Mantissa plantarum altera'' (Holmiae: impensis Laurentii Salvii, 1771) [https://www.biodiversitylibrary.org/page/42945260#page/73/mode/1up p. 205]
* 1773 : Vincenzo Corrado, ''Il cuoco galante''. Neapoli, 1773 [https://archive.org/details/bub_gb__Hov_shu2IgC/page/n3/mode/2up Textus] [https://archive.org/details/b21532497 3a ed. 1786] [https://archive.org/details/b22008561/ 6a ed. 1820]
* 1775 : Cristoforo Pilati, &quot;Aggiunta sopra il formentone&quot; in [[Augustinus Gallus|Agostino Gallo]], ''[[De recto agrorum cultu et voluptate rustica|Le venti giornate dell' agricoltura e de'piaceri della villa]]'' (nova editio. Brixiae) {{Google Books|je4jxXNpeLYC|pp. 533-558}}
* 1775 : [[Petrus Forskål]], ''Flora Aegyptiaco-Arabica'' (Hauniae: ex officina Mölleri, 1775) [https://www.biodiversitylibrary.org/item/122#page/205/mode/1up centuria ii p. 47]
* ante 1779 : Dom Denise; [[Marcus Lastri|Marco Lastri]], interpr., ''[[Delle viti e dei vini di Borgogna]]: memoria di un monaco cisterciense tradotta in italiano sur un manoscritto franzese''. Florentiae: Gaetano Cambiagi, 1779 {{Google Books|mTHnIWPFh8IC}}
* 1779-1784 : [[Dominicus Sestini]], ''Lettere del signor abate Domenico Sestini scritte dalla Sicilia e dalla Turchia a diversi suoi amici in Toscana''. 7 voll. Florentiae etc., 1779-1784
* 1781-1805 : &lt;span id=&quot;Rozier (1781-1805)&quot;&gt;&lt;/span&gt;Abbé Rozier, ''Cours d'agriculture'' (12 voll. Lutetiae, 1781-1805) [https://archive.org/details/bub_gb_6vHvDhT7xc8C/page/n225/mode/2up vol. 8 pp. 176-178] [[:s:fr:Cours d’agriculture (Rozier)|textus apud Vicifontem]]
* 1782 : [[Petrus Sonnerat|Pierre Sonnerat]], ''Voyage aux Indes Orientales'' (Lutetiae) [https://gallica.bnf.fr/ark:/12148/bpt6k1518236f vol. 1] [https://gallica.bnf.fr/ark:/12148/bpt6k15182388/f351 vol. 2 pp. 230-231, tab. 129] (&quot;Le litchi, ''Litchi chinensis''&quot;) 
* 1782 : [[Petrus Le Grand d'Aussy|Le Grand d'Aussy]], ''Histoire de la vie privée des français depuis l'origine de la nation jusqu'à nos jours''. 3 voll. Lutetiae: Pierres, 1782 [https://archive.org/details/b2152550x_0001 vol. 1] [https://archive.org/details/b2152550x_0002 2] [https://archive.org/details/b2152550x_0003 3]
* 1782 : [[Georgius Colman|George Colman]], &quot;Prologue to the new comedy of the ''East-Indian''&quot; in ''The Lady's Magazine'' vol. 13, 1782 {{Google Books|RJtPAQAAMAAJ|p. 383}}; George Colman, ''Prose upon Several Occasions'' vol. 3 (1787) [https://archive.org/details/proseonseveraloc03colmuoft/page/234/mode/2up p. 235]
* 1783 : John Farley, ''The London Art of Cookery''. Londinii [https://archive.org/details/b2152984x/page/144/mode/2up 2a ed. (1784) pp. 144-145] [https://archive.org/details/b29351145 12th ed. (1811)]
* 1784 : &lt;span id=&quot;Plenck (1784)&quot;&gt;&lt;/span&gt;[[Iosephus Iacobus Plenck]], ''Bromatologia seu doctrina de esculentis et potulentis.'' Viennae [i.e. Vindobonae]: Graeffer {{Google Books|RDpfAAAAcAAJ|p. 401}} &quot;vinum Graecum di Somma&quot;
* 1785 : Christoph Gottlieb von Murr, ''Reisen einiger Missionarien der Gesellschaft Jesu in Amerika''. Norimbergae, 1785 {{Google Books|8748AAAAcAAJ|p. 519}}
* 1787 : [[Thomas Jeffersonius|Thomas Jefferson]], ''[[Memorandums Taken on a Journey from Paris into the Southern Parts of France and Northern of Italy]]'' (Julian P. Boyd, ed., ''[[The Papers of Thomas Jefferson]] vol. 11, 1 January–6 August 1787'' [Princetoniae: Princeton University Press, 1955] pp. 415–464) [https://founders.archives.gov/documents/Jefferson/01-11-02-0389#TSJN-01-11-0389-ks-9901 textus]; [https://www.loc.gov/item/mtjbib002614/ manuscriptum]
* 1788 : {{Creanda|sv|Olof Swartz|Olavus Swartz|Olof Swartz}}, ''Nova genera et species plantarum, seu Prodromus descriptionum vegetabilium'' (Holmiae: in bibliopoliis M. Swederi, 1788) [https://www.biodiversitylibrary.org/page/376766#page/55/mode/1up p. 47]
* 1788 : Richard Briggs, ''The English Art of Cookery''. Londinii: G. G. J. and J. Robinson, 1788 {{Google Books|zZIEAAAAYAAJ}}
* 1789 : [[Dominicus Sestini]]; Pingeron, interpr., ''Lettres de Monsieur l'Abbé Dominique Sestini écrites à ses amis en Toscane, pendant le cours de ses voyages en Italie, en Sicile et en Turquie, sur l'histoire naturelle, l'industrie &amp; le commerce de ces différentes contrées'' (Lutetiae: veuve Duchesne &amp; fils, 1789) {{Google Books|q_sOAAAAQAAJ|vol. 1}} {{GB|tba3qNX6vcgC|vol. 2}} {{GB|gPBaAAAAQAAJ|vol. 3}}; cf. 1779-1784
* 1790 : [[Casimirus Gomezius Ortega]], ed., ''Francisci Hernandi, medici atque historici Philippi II, hispan. et indiar. Regis, et totius novi orbis archiatri opera cum edita tum inedita, ad autographi fidem''. 3 voll. (Matriti: ex typographia Ibarrae heredum) {{Google Books|ujT4W9CZuY8C|vol. 1}} {{Google Books|0foeAAAAYAAJ|vol. 2}} {{Google Books|tB4fAAAAYAAJ|vol. 3}}
* 1790 : [[Ioannes de Loureiro]], ''Flora cochinchinensis, sistens plantas in regno Cochinchina nascentes''. Ulyssipone: typis academicis, 1790 [https://www.biodiversitylibrary.org/bibliography/560#/summary vol. 1, vol. 2]
* 1792 : Francis Collingwood, John Woollams, ''The Universal Cook'' (Londinii, 1792) [https://archive.org/details/b2152970x/page/178/mode/2up p. 179]
* 1793 : ''L'arte di far cucina di buon gusto''. Taurinis [https://archive.org/details/b28765552 Textus]
* 1793 : [[Bryan Edwards]], ''The history, civil and commercial, of the British colonies in the West Indies'' (Londinii: Stockdale, 1793) [https://archive.org/details/historyciviland00brougoog vol. 1] {{Google Books|_70NAAAAQAAJ|vol. 2}}; 3a ed. 1801 [https://archive.org/details/historycivilcomm01edwa_0 vol. 1] [https://archive.org/details/civilcommercialo02edwa 2] [https://archive.org/details/historycivilcomm03edwa_0 3]
* 1794 : Henry Barham, ''Hortus Americanus''. Kingston Iamaicae, 1794 [https://archive.org/details/b29319870 Textus]
* 1794 : Francis Collingwood, John Woollams; Fridericus Gotthelf Baumgärtner, interpr., ''Neues Londner Kochbuch''. Lipsiae, 1794 {{Google Books|kqlgAAAAcAAJ|p. 220}}
* 1794 : {{Creanda|de|Ignaz Pfefferkorn|Ignatius Pfefferkorn|Ignaz Pfefferkorn}}, ''Beschreibung der Landschaft Sonora''. 1794 {{Google Books|2BvEmHRIme4C|vol. 2 p. 134}} [haven't found vol. 1 yet] 
* 1795 : &lt;span id=&quot;Schrader et Wendland (1795)&quot;&gt;&lt;/span&gt;[[Henricus Adolphus Schrader]], [[Ioannes Christophorus Wendland]], ''Sertum Hannoveranum'' (Goettingae, 1795-1798) [http://digitale-sammlungen.gwlb.de/resolve?PPN=742569349 Tab ii textus, tab. ii imago]
* 1795-1819 : [[Gulielmus Roxburgh|William Roxburgh]]; Joseph Banks, ed., ''Plants of the coast of Coromandel'' (Londinii: Nicol) [https://www.biodiversitylibrary.org/bibliography/467 Textus]
* 1796 : [[Amelia Simmons]], ''American cookery''. Printed by Hudson &amp; Goodwin for the Author, 1796 [https://www.loc.gov/item/96126967/ Textus] [https://d.lib.msu.edu/fa/1 editio 1798]
* 1796 : [[Yuan Mei]], ''[[Praecepta ex horto plenitudinis|Suiyuan shidan]]'' (Wolfram Eberhard, interpr., &quot;Die chinesische Küche. Die Kochkunst des Herrn von Sui-Yüan&quot; in ''Sinica'' vol. 15 (1940) pp. 190–228; Sean J. S. Chen, ed. et interpr., ''Recipes from the Garden of Contentment'' [Great Barrington Massachusettensium: Berkshire Publishing, 2018] {{Google Books|iGi9DwAAQBAJ|p. 11}} [https://wayoftheeating.wordpress.com/order-of-translation/ textus praeliminaris])
* 1797 : &lt;span id=&quot;Plenck (1797)&quot;&gt;&lt;/span&gt;[[Iosephus Iacobus Plenck]], ''Bromatologia seu doctrina de esculentis et potulentis.'' Lovanii: van Overbeke {{Google Books|6YREAAAAcAAJ|pp. 318-332}} prefer 1784
* 1797 : [[Ludovicus Sebastianus Mercier|Louis-Sébastien Mercier]], ''Le nouveau Paris'' (Lutetiae: Fuchs, 1797) [https://gallica.bnf.fr/ark:/12148/bpt6k64572183 vol. 1] [https://gallica.bnf.fr/ark:/12148/bpt6k64383950 vol. 2] [https://gallica.bnf.fr/ark:/12148/bpt6k6438369h/f116 vol. 3 pp. 108-110]
* 1797 : [[Thomas Andreas Knight|T. A. Knight]], ''A Treatise on the Culture of the Apple and Pear and on the manufacture of cider and perry''. Ludlow: H. Procter {{Google Books|Km8FAAAAQAAJ|Textus}} [http://www.archive.org/details/atreatiseoncult01kniggoog Editio 1801 apud archive.org]
* &lt;span id=&quot;Ortega (1798)&quot;&gt;&lt;/span&gt;1797-1800 : {{Creanda|es|Casimiro Gómez Ortega|Casimirus Gomezius Ortega}}, ''Novarum aut rariorum plantarum Horti Reg. Botan. Matrit. descriptionum decades''. Matriti: ex typographia Mariniana, 1797-1800 {{Google Books|W-o9AAAAYAAJ|p. 56}} 
* 1798 : &lt;span id=&quot;Mayr (1798)&quot;&gt;&lt;/span&gt;Christophorus Mayr, ''Dispensatorium universale''. 1798 {{Google Books|8kpfAAAAcAAJ|p. 213}}
* 1800 : [[Ioannes Antonius Chaptal|Jean-Antoine-Claude Chaptal]], ''Essai sur le vin''. Lutetiae: Delalain {{Google Books|GfMTAAAAQAAJ}}; editio 1801 sub titulo ''L'Art de faire, gouverner et perfectionner les vins'' {{Google Books|NPMTAAAAQAAJ}}

== 1801-1850 ==
* 1801 : {{Creanda|fr|Joseph de Berchoux|Iosephus de Berchoux|Joseph de Berchoux}}, ''[[La Gastronomie (Berchoux)|La Gastronomie]]''. 1801&lt;ref&gt;Julia Abramson, &quot;Legitimacy and Nationalism in the Almanach des Gourmands (1803-1812)&quot; in ''Journal for Early Modern Cultural Studies'' vol. 3 (2003) pp. 101-135, vide pp. 117-118 [https://www.jstor.org/stable/27793769 JSTOR]&lt;/ref&gt; 
* 1801 : [[Ioannes Antonius Chaptal|Jean-Antoine-Claude Chaptal]] et al., ''Traité théorique et practique sur la culture de la vigne''. 2a ed. 2 voll. Lutetiae: {{GB|ttw6AAAAcAAJ|vol. 1}} {{Google Books|99w6AAAAcAAJ|vol. 2}} vide etiam 1808
* 1802 : ''A Practical Guide during a Journey from London to Paris''. Londinii: Phillips, 1802 {{Google Books|ds6SNok78loC|2a ed., 1803}}
* 1803 : [[Susanna Carter|Sussannah Carter]], ''The frugal housewife, or Complete woman cook''. Novi Eboraci: G. &amp; R. Waite, 1803 [https://d.lib.msu.edu/fa/32 Txtus]
* 1803-1812 : [[Grimod de la Reynière]] et al., ''[[Almanach des gourmands]]'' (Lutetiae, 1803-1812) [https://archive.org/details/gri_33125008532588 Textus anni 1803, 2a ed.] {{Google Books|Wr06AAAAcAAJ|Textus anni 1803, 3a ed.}} [http://www.archive.org/details/almanachdesgour00costgoog Textus anni 1804, 2a ed.] {{Google Books|hBsJAAAAQAAJ|Textus anni 1805}} [https://archive.org/details/b21525250_0003 Textus anni 1805, 2a ed.] {{Google Books|kRsJAAAAQAAJ|Textus anni 1806}} [https://archive.org/details/b21525250_0004 Textus anni 1806] [https://archive.org/details/b21525250_0005 Textus anni 1807] [https://archive.org/details/almanachdesgour02costgoog alius] {{Google Books|1xsJAAAAQAAJ|Textus anni 1808}} [https://archive.org/details/b21525250_0006 Textus anni 1808] {{Google Books|-RsJAAAAQAAJ|Textus anni 1810}} [https://archive.org/details/b21525250_0007 Textus anni 1810] [https://archive.org/details/b21525250_0008 Textus anni 1812] [http://www.archive.org/details/almanachdesgour01costgoog alius]
* 1804 : Louis-Marie Prudhomme, ''Miroir de l'ancien et du nouveau Paris, avec treize voyages en vélocifères dans ses environs''. Lutetiae) [https://gallica.bnf.fr/ark:/12148/bpt6k64720830 vol. 1] [https://gallica.bnf.fr/ark:/12148/bpt6k6472262h vol. 2] apud Gallica 
* 1804 : &quot;Soy&quot; in A. F. M. Willich; James Mease, ed., ''The Domestic Encyclopedia'' (Philadelphiae: Birch &amp; Small) vol. 5 pp. 12-13 (fide [[#Shurtleff et Aoyagi (2012)]] pp. 154-155) 
* 1804 : Alexander Hunter, ''Culina famulatrix medicinae, or Receipts in cookery ... by Ignotus'' (Eboraci: Mawman, 1804) [https://archive.org/details/b21527581/page/128/mode/2up p. 128] (&quot;Tomata sauce&quot;); 2a ed., titulo ''Culina famulatrix medicinae, or Receipts in modern cookery ... by Ignotus; and revised by A. Hunter'' (Eboraci, 1805) [https://archive.org/details/b2152743x/page/150/mode/2up p. 150]
* 1804 : [[Nicolaus Iosephus Jacquin]], ''Plantarum rariorum Horti Caesarei Schoenbrunnensis descriptiones et icones'' vol. 4 (1804) [https://www.biodiversitylibrary.org/bibliography/332 series librorum]
* 1805-1817 : [[Amatus Bonpland]], [[Alexander de Humboldt]], ''Plantae aequinoctiales'' [https://www.biodiversitylibrary.org/bibliography/75757 vol. 1] [https://www.biodiversitylibrary.org/bibliography/75758 vol. 2]
* 1806 : [[Ioannes Shecut|John L. E. W. Shecut]], ''Flora Carolinæensis''. Vol. 1. Charleston, 1806 {{Google Books|cElHAAAAYAAJ}}
* 1806 : Maria Rundell, ''A new system of domestic cookery''. Londinii: John Murray [https://archive.org/details/b21526321/page/n7/mode/2up textus] [https://archive.org/details/newsystemofdomes01rund/ editio Bostoniensis 1807] [https://archive.org/details/anewsystemdomes01rundgoog/page/n114/mode/2up &quot;New ed., corrected&quot; 1808]
* 1806-1807 : ''[[Journal des gourmands et des belles]]''. Lutetiae: Capelle et Renand, 1806-1807 [http://www.archive.org/details/journaldesgourm01unkngoog Ian.-Mar. 1807] [http://www.archive.org/details/journaldesgourm00unkngoog Apr.-Iun. 1807] [http://www.archive.org/details/journaldesgourm02unkngoog Iul.-Sep. 1807] apud archive.org {{Google Books|KowBAAAAYAAJ|Oct.-Dec. 1807}}
* 1807-1808 : Francesco Leonardi, ''Apicio moderno''. Romae [https://archive.org/details/b21525225_0001 1] [https://archive.org/details/b21525225_0002 2] [https://archive.org/details/b21525225_0003 3] [https://archive.org/details/b21525225_0004 4] [https://archive.org/details/b21525225_0005 5] [https://archive.org/details/b21525225_0006/ 6] [https://archive.org/details/b22026228_0001 pars II 1] [https://archive.org/details/b22026228_0002 pars II 2]
* 1808 : Lucy Emerson, ''The New-England cookery''. Montpelier: Josiah Parks, 1808 [https://d.lib.msu.edu/fa/25 Textus]
* 1808 : [[Grimod de la Reynière]], ''[[Manuel des amphitryons]]: contenant un traité de la dissection des viandes à table, la nomenclature des menus les plus nouveaux pour chaque saison, et des élémens de politesse gourmande'' (Lutetiae: Capelle et Renand, 1808) [https://archive.org/details/b28521985/page/n9/mode/2up Textus] apud ''Internet Archive''
* 1808 : &lt;span id=&quot;Chaptal et al. (1808)&quot;&gt;&lt;/span&gt;[[Ioannes Antonius Chaptal|Jean-Antoine-Claude Chaptal]] et al.; Iosephus Voltiggi, interpr., ''Tractatus de vitis cultura arteque parandi vinum''. Viennae [i.e. Vindobonae] 1808 {{GB|SetMAAAAcAAJ|vol. 1}} {{Google Books|CGBt8wHcW3sC|vol. 2}}
* 1808 : Charles-Louis Cadet de Gassicourt, ''Cours gastronomique''. 1808&lt;ref&gt;Julia Abramson, &quot;Legitimacy and Nationalism in the Almanach des Gourmands (1803-1812)&quot; in ''Journal for Early Modern Cultural Studies'' vol. 3 (2003) pp. 101-135, vide pp. 118-119 [https://www.jstor.org/stable/27793769 JSTOR]&lt;/ref&gt;
* 1810 : [[Maria Antonius Carême|M. A. Carême]], ''Le Pâtissier royal parisien'' [https://gallica.bnf.fr/ark:/12148/bpt6k852393j Vol. 1] [https://gallica.bnf.fr/ark:/12148/bpt6k852237p vol. 2] apud Gallica
* 1811 : ''Manuel de la cuisine, ou, L'art d'irriter la gueule'' (Lutetiae) [https://archive.org/details/manueldelacuisin00unse/page/2/mode/2up p. 3] (&quot;Achia ou Achiar&quot;)
* 1811 : [[Alexander de Humboldt]], ''Essai politique sur le royaume de la Nouvelle-Espagne''&lt;ref&gt;Fide [[#Andrews (1995)]] p. 33: fortasse et in aliis operibus?&lt;/ref&gt;
* 1813 : &lt;span id=&quot;Forbes (1813)&quot;&gt;&lt;/span&gt;James Forbes, ''Oriental Memoirs'' (4 voll. Londinii, 1813) [https://archive.org/details/dli.venugopal.506 1] [https://archive.org/details/dli.venugopal.507 2] [https://archive.org/details/dli.venugopal.508 3] [https://archive.org/details/dli.venugopal.509 4]
* 1813 : [[Ludovicus Eustathius Ude|Louis Eustache Ude]], ''The French Cook''. Londinii [https://archive.org/details/b21471721 Textus] apud ''Internet Archive''
* 1813 : [[Michael Felix Dunal|Michel Félix Dunal]], ''Histoire naturelle, médicale et économique des Solanum'' (Lutetiae: Koenig, 1813)  [https://www.biodiversitylibrary.org/page/59103808#page/232/mode/1up p. 222] {{Google Books|Dc5LAAAAYAAJ}}
* 1814 : [[Gulielmus Roxburgh|William Roxburgh]], ''Hortus Bengalensis : or, A catalogue of the plants growing in the ... Botanic Garden at Calcutta'' (Serampore) [https://www.biodiversitylibrary.org/bibliography/95337 Textus]
* 1814 : [[Antonius Beauvilliers]], ''L'Art du cuisinier'' [https://gallica.bnf.fr/ark:/12148/bpt6k109846s Vol. 1] [https://gallica.bnf.fr/ark:/12148/bpt6k1098475 2] apud Gallica; [https://archive.org/details/lartducuisinier01beau vol. 1] [https://archive.org/details/lartducuisinier02beau 2] apud ''Internet Archive''
* 1814 : Vincenzo Agnoletti, ''La nuovissima cucina economica''. Romae [https://archive.org/details/b29321451 Textus] apud ''Internet Archive''
* 1815 : [[Maria Antonius Carême|M. A. Carême]], ''Le Pâtissier pittoresque'' [https://gallica.bnf.fr/ark:/12148/btv1b8541000q Textus] apud Gallica; [https://gallica.bnf.fr/ark:/12148/bpt6k1025012g 4a ed. (1842)]
* 1816 : &lt;span id=&quot;Jullien (1816)&quot;&gt;&lt;/span&gt;[[Andreas Jullien|André Jullien]], ''Topographie de tous les vignobles connus'' (Lutetiae, 1816) [https://archive.org/details/b21504660 Textus] apud ''Internet Archive''
* 1817 : [[Gulielmus Kitchiner|William Kitchiner]], ''Apicius Redivivus, or The Cook's Oracle''. Londinii: Bagster, 1817 [https://archive.org/details/b21533908 Textus] apud ''Internet Archive'' [https://archive.org/details/cooksoracleconta00kitc 4a ed. 1822] [https://archive.org/details/b22016430 Nova ed. 1827]
* 1819 : [[Andreas Duncan]], ''The New Edinburgh Dispensatory'' (Edinburgi, 1819) [http://nbn-resolving.de/urn:nbn:de:hbz:061:2-10611 Textus] apud Dusseldorpenses
* 1820 : [[Gulielmus Roxburgh|William Roxburgh]], Nathaniel Wallich; William Carey, ed., ''Flora indica, or, Descriptions of Indian plants'' vol. 1 (Serampore) [https://archive.org/details/mobot21753000002930 Textus]
* 1820 : [[Ioannes Crawfurd|John Crawfurd]], ''History of the Indian Archipelago''. Edinburgi {{Google Books|8EdSAAAAcAAJ|vol. 1}} {{GB|FOlAAAAAcAAJ|vol. 2}} {{GB|iY0AtXJsdbUC|vol. 3}}
* 1821 : George Pearson, ''Arranged catalogues of the articles of food, seasonings and drinks'' (Londinii, 1821) [https://archive.org/details/b29316108/page/12/mode/2up p. 13]
* 1821 : [[Gulielmus Martinus Leake|W. M. Leake]], ''[[Topography of Athens (Leake)|The Topography of Athens, with some remarks on its antiquities]]'' (1821) [http://www.archive.org/details/topographyofathe00leakuoft Textus] apud ''Internet Archive''
* 1821-1828 : &lt;span id=&quot;Descourtilz (1821-1828)&quot;&gt;&lt;/span&gt;Michel Étienne Descourtilz, ''Flore médicale des Antilles'' fasc. 1 (1821), fasc. 6 (1828) [https://www.biodiversitylibrary.org/item/21850#page/120/mode/1up tab. 405], [https://www.biodiversitylibrary.org/item/21850#page/124/mode/1up pp. 95-97] (&quot;vulg. Tomate à côtes&quot;)
* 1822 : [[Maria Antonius Carême|M. A. Carême]], ''Le Maître d'hôtel français'' [https://gallica.bnf.fr/ark:/12148/bpt6k1040003h Vol. 1] [https://gallica.bnf.fr/ark:/12148/bpt6k1040004x vol. 2] apud Gallica
* 1822 : Charles Yves Cousin, ''Le Parfait Cuisinier [... Le Cuisinier étranger; Le Patissier impérial]''. Lutetiae, 1822 [https://archive.org/details/b21532564 Textus] apud ''Internet Archive''
* 1822 : &lt;span id=&quot;Jullien (1822)&quot;&gt;&lt;/span&gt;[[Andreas Jullien|André Jullien]], ''Topographie de tous les vignobles connus'' (2a ed. Lutetiae, 1822) [https://numelyo.bm-lyon.fr/f_view/BML:BML_00GOO0100137001102535122 Textus]
* 1822 : {{Creanda|de|Carl Friedrich von Rumohr|Carolus Fridericus de Rumohr|Carl Friedrich von Rumohr}}, ''Geist der Kochkunst''
* 1824 : [[Gulielmus Martinus Leake|W. M. Leake]], ''[[Journal of a Tour in Asia Minor (Leake)|Journal of a Tour in Asia Minor, with comparative remarks on the ancient and modern geography of that country]]'' (1824) [http://www.archive.org/details/journalatourina01leakgoog Textus] apud ''Internet Archive''
* 1824 : [[Antonius Beauvilliers|A. B. Beauvilliers]], ''The Art of French Cookery'' by A. B. Beauvilliers. Londinii: Longman [https://archive.org/details/b21504751 Textus] apud ''Internet Archive''
* 1824 : [[Maria Randolph|Mary Randolph]], ''The Virginia House-wife''. Vasingtoniae: Printed by Davis and Force, 1824 [https://www.loc.gov/resource/rbc0001.2015pennell17897/?sp=119 pp. 103-104 (&quot;To caveach fish&quot;) (cf. 1836)
* 1824 : Alexander Moon, ''Catalogue of Ceylon Plants''
* 1825 : [[Ioannes Anthelmus Brillat-Savarin]], ''[[Physiologie du goût]]'' (Lutetiae, 1826) [https://gallica.bnf.fr/ark:/12148/btv1b8626673x vol. 1] [https://gallica.bnf.fr/ark:/12148/btv1b8626674b vol. 2] apud Gallica; [https://archive.org/details/b28748177_0001 vol. 1] [https://archive.org/details/b28748177_0002 vol. 2] apud ''Internet Archive''
* 1825 : Julien Archambault, ''Le Cuisinier économe ou Élémens nouveaux de cuisine, de pâtisserie et d'office'' (Lutetiae: Librairie du Commerce, 1825) p. 346
* 1825 : &lt;span id=&quot;Blume (1825)&quot;&gt;&lt;/span&gt;[[Carolus Ludovicus Blume|C. L. Blume]], ''Bijdragen tot de flora van Nederlandsch Indië'' fasc. 13 (Bataviae: ter Lands Drukkerij, 1825) [https://www.biodiversitylibrary.org/page/429393#page/65/mode/1up Textus]
* 1826 : Margaret Dods [i.e. Christian Isobel Johnstone], ''The Cook and Housewife's Manual'' (Edinburgi) [https://archive.org/details/b21505366/page/82/mode/2up pp. 83-85] (&quot;Mock turtle soup&quot;) [https://www.surugadai.ac.jp/sogo/media/bulletin/Ronso35/Ronso35matsui.pdf De hoc opere]
* 1826 : [[Whitelaw Ainslie]], ''Materia Indica'' [https://archive.org/details/materiaindicaors01ains vol. 1] [https://archive.org/details/materiaindicaors02ains vol. 2]
* 1828 : &lt;span id=&quot;Carême (1828)&quot;&gt;&lt;/span&gt;[[Maria Antonius Carême|M. A. Carême]], ''Le Cuisinier parisien, ou l'art de la cuisine française au XIXe siècle'' (2a ed. 1828) [https://gallica.bnf.fr/ark:/12148/btv1b86172102 Textus] apud Gallica [https://archive.org/details/b29298581/ 2a ed. 1828] [https://archive.org/details/b21525407/ 2a ed. 1842 (?)] [https://archive.org/details/b29300125/ 3a ed. 1842]
* 1828 : &lt;span id=&quot;Descourtilz (1828)&quot;&gt;&lt;/span&gt;Michel Étienne Descourtilz, ''Flore médicale des Antilles'' fasc. 6 (1828) [https://www.biodiversitylibrary.org/item/21850#page/216/mode/1up pp. 172-181, tabb. 422-423] (&quot;vulg. Poivre d'Inde; Piment zozo, piment enragé, poivre d'oiseau, piment caraìbe&quot;)
* 1828 : &quot;Repas normand&quot; in Amand Masson de Saint-Amand, ''Lettres d'un voyageur à l'embouchure de la Seine'' (1828) {{Google Books|Gj5YAAAAcAAJ|pp. 226-236}}
* 1830 : [[Sydney, domina Morgan|Lady Morgan]], ''France in 1829-30'' (Londinii: Saunders and Otley) [https://archive.org/details/francein01sydgoog vol. 1] [https://archive.org/details/francein03sydgoog/page/n413/mode/2up vol. 2 pp. 402-420] 
* 1830-1846 : [[Gulielmus Martinus Leake|W. M. Leake]], ''[[Travels in the Morea (Leake)|Travels in the Morea]]'' (1830); cum supplemento ''Peloponnesiaca'' (1846) [http://www.archive.org/details/travelsinmoreawi01leak Vol. 1] [http://www.archive.org/details/travelsinmoreawi02leak 2] [http://www.archive.org/details/travelsinmoreawi03leak 3] [http://www.archive.org/details/peloponnesiacaa00leakgoog Supplementum]
* 1831 : Sandford Arnot, interpr., &quot;Indian Cookery, as practised and described by the natives of the East&quot; in ''Miscellaneous Translations from Oriental Languages'' vol. 1 (Londinii) [https://archive.org/details/32882019070997-miscellaneoustr/page/n275/mode/2up fasc. 5] {{Google Books|pSctAAAAYAAJ}}
* 1832 : ''Il Cuoco Piemontese ridotto all'ultimo gusto''. 6a ed. Mediolani, 1832 [http://www.mori.bz.it/gastronomia/Cuoco%20piemontese.pdf Textus] [https://archive.org/details/b22020767 alius]
* 1832 : N. K. M. Lee, ''The cook's own book, being a complete culinary encyclopedia''. Bostoniae: Munroe and Francis, 1832 [https://d.lib.msu.edu/fa/18 Textus]
* 1832 : [[Gulielmus Roxburgh|William Roxburgh]]; William Carey, ed., ''Flora indica, or, Descriptions of Indian plants'' (2a ed. 3 voll. Serampore) [https://www.biodiversitylibrary.org/bibliography/590 Textus]
* 1832 : &lt;span id=&quot;Jullien (1832)&quot;&gt;&lt;/span&gt;[[Andreas Jullien|André Jullien]], ''Topographie de tous les vignobles connus'' (3a ed. Lutetiae, 1832) [https://gallica.bnf.fr/ark:/12148/bpt6k3056299n Textus] apud Gallica
* 1832 : Antonius Fingerhuth, ''Monographia generis Capsici''. Düsseldorpii {{Google Books|U2PRSAAACAAJ}} [https://archive.org/details/monographiagene00finggoog/ apud] ''Internet Archive''
* 1833 : &lt;span id=&quot;Carême (1833)&quot;&gt;&lt;/span&gt;[[Maria Antonius Carême|M. A. Carême]], ''L'Art de la cuisine française au dix-neuviême siècle'' [https://gallica.bnf.fr/ark:/12148/bpt6k1510011f vol. 1] [https://gallica.bnf.fr/ark:/12148/bpt6k853460z vol. 2] apud Gallica (vide et 1847)
* 1834 : &lt;span id=&quot;Nees (1834)&quot;&gt;&lt;/span&gt;[[Christianus Godofredus Nees ab Esenbeck|Christian Godfrey Nees von Esenbeck]], &quot;Monograph of the East Indian Solaneae&quot; in ''Transactions of the Linnean Society of London'' vol. 17 (1834/1837) p.37 sqq., vide [https://www.biodiversitylibrary.org/page/13786867#page/85/mode/1up pp. 59-60] 
* 1835 : Abraham Hayward, &quot;Gastronomy and Gastronomers&quot; in ''Quarterly Review'' vol. 44, 1835 {{Google Books|Dc6jvFGZ4qwC|pp. 117-155}}
* 1835 : ''[[Travels in Northern Greece (Leake)|Travels in Northern Greece]]'' (1835) [http://www.archive.org/details/travelsinnorthe06leakgoog Vol. 1] [http://www.archive.org/details/travelsinnorthe07leakgoog 2] [http://www.archive.org/details/travelsinnorthe05leakgoog 3] [http://www.archive.org/details/travelsinnorthe04leakgoog 4]
* 1836 : [[Maria Randolph|Mary Randolph]], ''The Virginia Housewife, or Methodical cook''. Philadelphiae: Plaskitt, 1836 [https://archive.org/details/virginiahousewif00randrich/page/64/mode/2up p. 64] (&quot;to caveach fish&quot;) {{GB|R4YEAAAAYAAJ|editio 1838}} [https://d.lib.msu.edu/fa/71#page/1/mode/2up editio 1838]; [https://www.gutenberg.org/files/12519/12519-h/12519-h.htm editionis 1860 editio interretialis]; [https://archive.org/details/virginiahousewif00rand_1 editio 1871] (1a ed. 1824)
* 1837 : [[Eliza Leslie]], ''Directions for cookery, being a system of the art in its various branches]]. Philadelphiae: Carey &amp; Hart, 1837 [https://archive.org/details/directionsforcoo00lesl Textus]
* 1837 : [[Eduardus Gulielmus Lane|Edward William Lane]], ''An Account of the Manners and Customs of the Modern Egyptians''. 2 voll. Londinii: Knight, 1837 [https://archive.org/details/anaccountmanner09lanegoog vol. 1] [https://archive.org/details/anaccountmanner08lanegoog 2] aegre legibiles; E. S. Poole, ed. [https://archive.org/details/accountofmanners00lane 5a ed. (1860)]
* 1837 : Joseph Roques, ''Nouveau traité des plantes usuelles, spécialement appliqué à la médecine domestique, et au régime alimentaire de l'homme sain ou malade''. Lutetiae: Dufart {{GB|Yb5Jdw5gwssC|vol. 1}} {{GB|8Bc9VsIiXPIC|vol. 2 pp. 12-25}} {{GB|t3iwjQGCe8wC|vol. 3}} 
* 1837 : Ludovicus Prodan, ''Vina. Dissertatio inauguralis medica''. Pestini: Beimel, 1837 {{Google Books|omBUAAAAcAAJ|p. 11}}
* 1839 : Mountstuart Elphinstone, ''Account of the Kingdom of Caubul'' (Nova ed. Londinii: Bentley, 1839) [https://archive.org/details/b29331791_0001/page/394/mode/2up vol. 1 p. 395] [https://archive.org/details/b29331791_0002 vol. 2]
* 1839 : ''La cuynera catalana''. 2a ed. Barcinone [http://bdh.bne.es/bnesearch/detalle/bdh0000125370 Textus]
* 1839 : &lt;span id=&quot;Courchamps (1839)&quot;&gt;&lt;/span&gt;[[Mauritius de Courchamps]], ''Néo-physiologie du gout par ordre alphabétique, ou, Dictionnaire génerál de la cuisine française ancienne et moderne''. Lutetiae, 1839 {{Google Books|ts9QAQAAIAAJ}}
* 1839 : Ippolito Cavalcanti, ''Cucina teorica-pratica''. Neapoli, 1839 [http://www.mori.bz.it/gastronomia/Cavalcanti%20-%20Cucina%20casereccia%20napoletana.pdf Textus]
* 1839 : ''Le Cuisinier méridional d'après la méthode provençale et languedocienne''
* 1839 : [[Ioannes Paget|John Paget]], ''Hungary and Transylvania'' (Londinii) [https://archive.org/details/agw0321.0002.001.umich.edu/page/226/mode/2up vol. 2 p. 227]
* 1840 : &quot;Nourriture&quot; in Ami Boué, ''La Turquie d'Europe'' (4 voll. Lutetiae: Bertrand, 1840) [https://babel.hathitrust.org/cgi/pt?id=chi.65535022&amp;view=1up&amp;seq=242&amp;skin=2021 vol. 2 pp. 234-258] [https://babel.hathitrust.org/cgi/pt?id=chi.65535159 vol. 1] [https://babel.hathitrust.org/cgi/pt?id=chi.65535084 vol. 3] {{Google Books|g15cAAAAcAAJ|vol. 4}}
* 1841 : [[Gulielmus Makepeace Thackeray]], &quot;Memorials of Gormandising ... by M. A. Titmarsh&quot; in ''Fraser's Magazine'' (Iunio 1841) [https://archive.org/details/sim_frasers-magazine_1841-06_23_138/page/710/mode/2up pp. 710-725] {{Google Books|_1tAAAAAYAAJ|pp. 375-404 in editione 1885}}
* 1842/1844 : &lt;span id=&quot;Bollaert (c. 1842)&quot;&gt;&lt;/span&gt;William Bollaert, commentaria (W. Eugene Hollon, Ruth Lapham Butler, edd., ''William Bollaert's Texas'' [Norman: University of Oklahoma, 1956] p. 218)
* 1843 : &lt;span id=&quot;Wight (1843)&quot;&gt;&lt;/span&gt;Robert Wight, ''Icones plantarum Indiae Orientalis'' vol. 2 (Maderaspatani: J. B. Pharoah, 1843) tab. 345
* 1843 : &quot;Un souper de M. le marquis de Cussy à Rouen&quot; in C. F. A. Fayot, ed., ''Les Classiques de la table'' (Lutetiae, 1843) [https://archive.org/details/b21525869/page/514/mode/2up pp. 515-518] 
* 1843-1844 : Armand Plumerey, ''Le principal de la cuisine de Paris''. 2 voll. Lutetiae [https://archive.org/details/leprincipaldelac00plum vol. 1] (vide et 1847)
* 1845 : Philippus Laurentius Geiger, Carolus Fridericus Mohr, ''Pharmacopoea universalis''. Heidelbergae: Winter, 1845 {{Google Books|8EKWfK507b0C}}
* 1845 : Giovanni Brizzi, ''La cuciniera moderna'' Senae, 1845 [http://www.mori.bz.it/gastronomia/Brizzi%20-%20La_cuciniera_moderna_1845.pdf Textus]
* 1845 : &lt;span id=&quot;Davidis (1845)&quot;&gt;&lt;/span&gt;Henriette Davidis, ''Praktisches Kochbuch für die bürgerliche und feine Küche'' (1845) [http://digital.ub.uni-duesseldorf.de/urn/urn:nbn:de:hbz:061:1-40760 3a ed. 1847] [https://archive.org/details/b28108309 25a ed. 1882][https://archive.org/details/b21525961/page/n3/mode/2up 37a ed. 1898]
* 1845 : &lt;span id=&quot;Acton (1845)&quot;&gt;&lt;/span&gt;&quot;Curries ...&quot; in Eliza Acton, ''Modern cookery in all its branches'' (Londinii: Longmans) [https://archive.org/details/b21531857/page/342/mode/2up pp. 343-353] [https://archive.org/details/moderncookeryin00actogoog 2a ed.] [https://archive.org/details/b21531869/page/n5/mode/2up 3a ed.] [https://archive.org/details/b21531870/page/n5/mode/2up 4a ed.] [https://archive.org/details/b21531882/page/n5/mode/2up 5a ed., 1846] [https://archive.org/details/b21533283/page/n7/mode/2up 6a ed., 1847]
* 1845 : &quot;Au feu marquis de Cussy&quot; in C. F. A. Fayot, {{Creanda|fr|Elzéar Blaze|Elzear Blaze|Elzéar Blaze}}, ''Causeries de gourmets et de chasseurs'' (Lutetiae: Martinon, 1845) [https://gallica.bnf.fr/ark:/12148/bpt6k9612601n/f17 pp. 5-15]
* 1846 : &quot;Les Restaurants de Paris&quot; in Eugène Briffault, ''Paris à table'' (Lutetiae: Hetzel, 1846) [https://gallica.bnf.fr/ark:/12148/bpt6k108869b/f150 pp. 144-178]
* 1846 : &quot;Quelques explications préliminaires&quot; in C. F. A. Fayot, ed., ''Les Classiques de la table: édition usuelle'' (Lutetiae, 1846) [https://archive.org/details/lesclassiquesdel00unse/page/n25/mode/2up pp. ix-xii]
* 1846 : &lt;span id=&quot;Francatelli (1846)&quot;&gt;&lt;/span&gt;[[Carolus Elmé Francatelli|Charles Elmé Francatelli]], ''The Modern Cook'' {{Google Books|3p9kAAAAcAAJ|p. 401}}
* 1846 : [[Alexis Soyer]], ''The Gastronomic Regenerator''. Londinii: Simpkin, Marshall, 1846 [https://archive.org/details/gastronomicregen00soye_0 Textus]
* 1847 : &lt;span id=&quot;Carême (1847)&quot;&gt;&lt;/span&gt;[[Antoninus Carême|Antonin Carême]], Armand Plumerey; C. F. A. Fayot, ed., ''L'Art de la cuisine française au dix-neuviême siècle'' (Lutetiae, 1847) [https://archive.org/details/b21526047_0001/ vol. 1] [https://archive.org/details/b21526047_0002/ vol. 2] [https://archive.org/details/b21525687_0003/ vol. 3] [https://archive.org/details/b21526047_0004/ vol. 4] [https://archive.org/details/b21526047_0005/page/252/mode/2up vol. 5 pp. 252-254]
* 1847 : Susan Shelby Magoffin, ephemerides (Stella M. Drumm, ed., ''Down the Santa Fe Trail and into Mexico: The Diary of Susan Shelby Magoffin, 1846–1847'' [Novo Portu: Yale University Press, 1962] [http://cdigital.dgb.uanl.mx/la/1020000885/1020000885.PDF pp. 65, 94]
* 1848 : &quot;Da dou&quot; in {{Creanda|de|Wu Qijun}}, ''Tractatus nominum naturaeque plantarum'' [植物名實圖攷] (Taiyuan, 1848) (fide [[#Shurtleff et Aoyagi (2012)]] pp. 220-221)
* 1848 : ''La Cuisine ordinaire'' (congeries operum [[Antonius Beauvilliers|Beauvilliers]], [[Maria Antonius Carême|Carême]], [[Ioannes Anthelmus Brillat-Savarin|Brillat-Savarin]], [[Iosephus Gastaldy|Gastaldy]], aliorum) [https://gallica.bnf.fr/ark:/12148/bpt6k3041516k vol. 1] [https://gallica.bnf.fr/ark:/12148/bpt6k30415170 vol. 2] apud Gallica
* 1848 : &lt;span id=&quot;Jullien (1848)&quot;&gt;&lt;/span&gt;[[Andreas Jullien|André Jullien]], ''Topographie de tous les vignobles connus''. 4a ed. Lutetiae, 1848 {{Google Books|nLNgAAAAcAAJ}}
* 1849 : &quot;Soya&quot; in Isidore Hedde et al., ''Étude pratique du commerce d'exportation de la Chine'' (Lutetiae: Librairie du Commerce, 1849) [https://numelyo.bm-lyon.fr/f_view/BML:BML_00GOO0100137001102439556 pp. 188-190]
* 1850 : Gideon Nye, ''Tea and the tea trade''. Novi Eboraci: G. W. Wood, 1850 [https://archive.org/details/teateatradeparts00nyegricht Textus] apud ''Internet Archive''

== 1851-1900 ==
* 1851 : Giovanni Rajberti, ''L'arte di convitare''. 1851 [http://www.mori.bz.it/gastronomia/Rajberti%20-%20L'Arte%20di%20Convitare.pdf Textus]
* 1852 : &lt;span id=&quot;Riddell (1852)&quot;&gt;&lt;/span&gt;&quot;Curries&quot; in Robert F. Riddell, ''Indian domestic economy and receipt book'' (3a ed. Bombayae: Bombay Gazette Press) [https://archive.org/details/b21529942/page/380/mode/2up pp. 380-406]
* 1852 : [[Sara Iosepha Hale|Sarah Josepha Hale]], ''The ladies' new book of cookery''. 1852 [https://n2t.net/ark:/85335/m5467w Textus]
* 1852 : Abraham Hayward, ''The Art of Dining, or, Gastronomy and gastronomers''. Londinii: John Murray, 1852 [https://archive.org/details/b21526746 Textus] apud ''Internet Archive''
* 1832 : Edwin Lankester, ''Vegetable Substances Used for the Food of Man'' [https://archive.org/details/vegetablesubstan00lank/page/312/mode/2up pp. 313-314]
* 1852 : [[Michael Felix Dunal|M. F. Dunal]], &quot;Solanaceae&quot; in [[Alphonsus Pyramus de Candolle|A. de Candolle]], ''[[Prodromus systematis naturalis regni vegetabilis]]'' vol. 13 fasc. 1 (Lutetiae: Masson, 1852) [https://www.biodiversitylibrary.org/page/56755458#page/432/mode/1up p. 428]
* 1853 : [[Alexis Soyer]], ''The Pantropheon, or History of food and its preparation from the earliest ages of the world''. Londinii: Simpkin, Marshall [https://archive.org/details/b2152953x Textus] apud ''Internet Archive''
* 1854 : Giovanni Vialardi, ''Tratto di cucina, pasticceria moderna, credenza''. Taurinis, 1854 [http://www.mori.bz.it/gastronomia/Vialardi%20-Trattato%20Di%20Cucina%20-%201854.pdf Textus]
* 1855 : &lt;span id=&quot;Améro (1855)&quot;&gt;&lt;/span&gt;&quot;Notice sur Brillat-Savarin, Grimod de la Reynière, Cussy, Roques, Berchoux et Colnet&quot; in Justin Améro, ed., ''Les Classiques de la table: Nouvelle édition refondue et complétée'' (Lutetiae: Didot, 1855) [https://archive.org/details/b28125551_0001/page/n11/mode/2up vol. 1 pp. v-xv]
* 1855/1861 : {{Creanda|fr|Jacques Marie Cyprien Victor Coste|Victor Coste}}, ''Voyage d'exploration sur le littoral de la France et de l'Italie''. [https://archive.org/details/bub_gb_d5LomsVLskoC Textus]; [https://archive.org/details/voyagedexplorati00cost 2a ed., 1861] [https://wwz.ifremer.fr/archives/Portraits/V.-Coste De auctore]
* 1856 : [[Urbanus Dubois|Urbain Dubois]], [[Aemilius Bernard|Emile Bernard]], ''La cuisine classique: études pratiques, raisonnées et démonstratives de l'école française appliquée au service à la russe''. Lutetiae, 1856 [https://archive.org/details/bub_gb_pBo-AAAAcAAJ Textus] apud ''Internet Archive''
* 1856 : [[Fredericus Antonius Guilielmus Miquelius|Fred. Ant. Guilielmi Miquel.]] ''Flora Indiae Batavae = Flora van Nederlandsch Indië'' vol. 2 pars 2 (Lipsiae: apud Frid. Fleischer, 1856) {{Google Books|dzY-AAAAcAAJ|p. 656}}
* 1858 : ''Nuevo cocinero mejicano en forma de diccionario''. Lutetiae: Rosa y Bouret, 1858 [http://bdh.bne.es/bnesearch/detalle/bdh0000040797 Textus]
* 1858 : &quot;[https://archive.org/details/irishquarterlyr00unkngoog/page/n523/mode/2up Brillat-Savarin]&quot; in ''Irish Quarterly Review'' vol. 8 no. 30 (1858) pp. 461-492
* 1858 : ''Il cuciniere italiano moderno''. Liburni [https://archive.org/details/b28136469/ Textus] [https://archive.org/details/b21538372/ Editio recentior, anno incerto]
* 1859 : ''Il cuoco milanese e la cuciniera piemontese'' (Mediolani) [http://www.mori.bz.it/gastronomia/Cuoco%20milanese.pdf pp. 302-303] (&quot;Risotto alla milanese&quot;)
* 1859 : ''The habits of good society: a handbook of etiquette for ladies and gentlemen''. Londinii: Hogg [https://archive.org/details/habitsofgoodsoci00unse Textus] apud ''Internet Archive''
* 1859 : [[Carolus Monselet|Charles Monselet]] et al., ''La Cuisinière poétique''. Lutetiae: Lévy [https://archive.org/details/b28062346/ Textus]; Lipsiae: Dürr [https://gallica.bnf.fr/ark:/12148/bpt6k9682092b Textus] 
* 1860 : [[Iacobus Emerson Tennant|James Emerson Tennant]], ''Ceylon: an account of the island, physical, historical, and topographical''. 4a ed. 2 voll. Londinii: Longman, 1860 [https://archive.org/details/ceylonaccountofi01tenn vol. 1] [https://archive.org/details/ceylonaccountofi02tenn vol. 2]
* 1860 : Emile de La Bédollière, ''Le nouveau Paris: histoire de ses 20 arrondissements''. Lutetiae: Barba {{Google Books|iboNAAAAIAAJ|p. 273}} 
* 1861 : [[Isabella Beeton]], ''[[Beeton's Book of Household Management]]'' (Londinii, 1861) [https://archive.org/details/b20392758/page/136/mode/2up pp. 135-136] &quot;Kegeree&quot;
* 1861 : Herbert Byng Hall, ''The Oyster: where, how, and when to find, breed, cook, and eat it''. Londinii: Trübner, 1861 [https://archive.org/details/b21526862 Textus]
* 1862 : [[Alfredus Delvau|Alfred Delvau]], ''Histoire anecdotique des cafés et cabarets de Paris''. Lutetiae, 1862 [https://gallica.bnf.fr/ark:/12148/bpt6k1025028b Textus] apud Gallica
* 1863 : Thomas George Shaw, ''Wine, the Vine and the Cellar''. Londinii: Longman, Green, 1863 [https://archive.org/details/b28128230 Textus]
* 1864 : ''Cookery for English households by a French lady'' (1864) [https://archive.org/details/cookeryforengli00housgoog/page/n108/mode/2up pp. 93-94] sub titulo ''Poulet au vin blanc''
* 1865 : Emanuele Rossi, ''La vera cuciniera genovese'' (Genuae) [http://www.mori.bz.it/gastronomia/E.%20Rossi%20-%20La%20vera%20cuciniera%20genovese.pdf Textus]
* 1866 : &lt;span id=&quot;Jullien (1866)&quot;&gt;&lt;/span&gt;[[Andreas Jullien|André Jullien]]; C. E. Jullien, ed., ''Topographie de tous les vignobles connus'' (5a ed. Lutetiae, 1866) [https://archive.org/details/topographiedeto00jullgoog Textus] apud ''Internet Archive''
* 1867 : [[Leo Brisse]], ''Le calendrier gastronomique pour l’année 1867: les 365 menus du baron Brisse: un menu par jour'' [https://www.loc.gov/item/87103777/ Textus] [https://catalog.hathitrust.org/Record/011530851 alibi] [https://gallica.bnf.fr/ark:/12148/bpt6k6568678k editio 1868]
* 1867 : &quot;Restaurants&quot; in [[Alfredus Delvau|Alfred Delvau]], ''Les plaisirs de Paris'' (Lutetiae, 1867) [https://gallica.bnf.fr/ark:/12148/bpt6k1522335s/f109 pp. 109-144]
* 1867 : [[Iulius Gouffé|Jules Gouffé]], ''Le Livre de cuisine'' (Lutetiae: Hachette) [https://gallica.bnf.fr/ark:/12148/bpt6k1080414 Textus] [https://gallica.bnf.fr/ark:/12148/bpt6k3411211m 2a ed. 1870] [https://archive.org/details/b21525778/page/633/mode/2up 4a ed. 1877, pp. 633-634]
* 1868 : [[Iulius Gouffé|Jules Gouffé]]; Alphonse Gouffé, interpr., ''The royal cookery book (Le livre de cuisine)''. Londinii: Sampson Low, 1868 [https://archive.org/details/b21505093/ Textus] apud ''Internet Archive''
* 1868 : Theodoro J. H. Langgaard, ''Novo formulario medico e pharmaceutico'' (1868) {{Google Books|QNk8AAAAcAAJ|p. 537}}
* 1868? : [[Leo Brisse]], ''La Cuisine à l'usage des ménages bourgeois et des petits ménages'' [https://gallica.bnf.fr/ark:/12148/bpt6k110689f editio 1884] 
* 1869 : &lt;span id=&quot;Indian Cookery Book (1869)&quot;&gt;&lt;/span&gt;&quot;Curries&quot; in ''The Indian Cookery Book: a practical handbook to the kitchen in India'' (Calcuttae: Wyman) [https://archive.org/details/b2152824x/page/14/mode/2up pp. 14-34]
* 1869 : William Terrington, ''Cooling cups and dainty drinks''. Londinii: Routledge [https://archive.org/details/coolingcupsandd00terrgoog Textus] apud ''Internet Archive''
* 1869 : [[Iulius Gouffé|Jules Gouffé]], ''Le livre des conserves''. Lutetiae: Hachette, 1869 [https://archive.org/details/b2152578x Textus] apud ''Internet Archive''
* 1870 : [[Urbanus Dubois|Urbain-Dubois]], ''Cosmopolitan cookery: popular studies'' (Londinii: Longmans, 1870) [https://archive.org/details/b28131241/page/11/mode/1up p. 11] (&quot;'Cuscus' of the Arabs&quot;)
* 1872 : [[Urbanus Dubois|Urbain Dubois]], ''Cuisine de tous les pays: études cosmopolites''. 3a ed. (Lutetiae: Dentu, 1872) [https://gallica.bnf.fr/ark:/12148/bpt6k65442310/f91 pp. 69-70] (&quot;Soupe maigre aux huîtres; Soupe aux huîtres à l'américaine&quot;)
* 1873 : &lt;span id=&quot;Dumas (1873)&quot;&gt;&lt;/span&gt;[[Alexander Dumas (pater)|Alexandre Dumas]], ''Grand Dictionnaire de cuisine'' (Lutetiae: Lemerre, 1873) [https://archive.org/details/b28092818/page/634/mode/2up pp. 634-635]
* 1873 : [[Aemilius Zola|Emile Zola]], ''[[Le Ventre de Paris]]'' (Lutetiae: Charpentier) [https://gallica.bnf.fr/ark:/12148/bpt6k6213165k/f162 pp. 154-155]
* 1873? : [[Leo Brisse]], ''Cuisine en carême du baron Brisse'' [https://gallica.bnf.fr/ark:/12148/bpt6k9205980 2a ed. 1874]
* 1875 : [[Henricus Vizetelly|Henry Vizetelly]], ''The wines of the world characterized and classed, with some particulars respecting the beers of Europe'' (Londinii: Ward, Lock, 1875) [https://archive.org/details/b21528287/page/50/mode/2up pp. 51-57]
* 1876 : [[Henricus Vizetelly|Henry Vizetelly]], ''Facts about sherry''. Londinii: Ward, Lock, 1876 [https://archive.org/details/factsaboutsherr00vizegoog Textus]
* 1876 : [[Ludovicus Henricus Morgan|Lewis Henry Morgan]], &quot;[montezumasdinne00morggoog/page/n45/mode/2up Montezuma's Dinner]&quot; [commentatio sine titulo divulgata] in ''North American Review'' (Aprilis 1876) pp. 265-308; postea forma libri reimpressa Novi Eboraci: New York Labor News Company, 1950
* 1877 : [[Iulius Gouffé|Jules Gouffé]], ''Le Livre de pâtisserie''. Lutetiae: Hachette [https://archive.org/details/GoufPa/ Textus] apud ''Internet Archive''
* 1877 : &quot;Sauce&quot; in {{Creanda|en|Eneas Sweetland Dallas|Aeneas Sweetland Dallas|Eneas Sweetland Dallas}}, ''Kettner's Book of the Table'' (Londinii: Dulau, 1877) [https://archive.org/details/b2153794x/page/408/mode/2up pp. 409-419]
* 1877 : &lt;span id=&quot;Blanco (1877)&quot;&gt;&lt;/span&gt;Manuel Blanco, ''Flora de Filipinas'' (4 voll. Manilae: Plana, 1877-1883) [https://www.biodiversitylibrary.org/item/277741#page/217/mode/1up vol. 1 p. 177] et tabula 49
* 1878 : A. R. Kenney-Herbert, ''Culinary Jottings for Madras ... by &quot;Wyvern&quot;''. Maderaspatani: Higginbotham [https://archive.org/details/b28119617/ Textus]; titulo ''Culinary Jottings'' [https://archive.org/details/b28089583 5a ed. 1885]
* 1879 : &lt;span id=&quot;Davidis (1879)&quot;&gt;&lt;/span&gt;Henriette Davidis, ''Praktisches Kochbuch für die Deutschen in Amerika'' (Milwaukee: Brumber, 1879) [https://archive.org/details/praktischeskochb00davi/ Textus] [https://archive.org/details/praktischeskochb01davi/ 2a ed. 1897]
* 1879 : [[Henricus Vizetelly|Henry Vizetelly]], ''Facts about champagne and other sparkling wines''. Londinii: Ward, Lock, 1879 [https://archive.org/details/b21528317 Textus]
* 1880 : [[Henricus Vizetelly|Henry Vizetelly]], ''Facts about Port and Madeira''. Londinii: Ward, Lock, 1880 {{Giigle Books|dlA7AQAAMAAJ}}
* 1882 : [[Henricus Vizetelly|Henry Vizetelly]], ''A history of champagne''. Londinii: Vizetelly, 1882 [https://archive.org/details/historyofchampag00vize Textus]
* 1882 : [[Leo Brisse]]; Edith Matthew Clark, ed. et interpr., ''366 menus and 1200 recipes of the Baron Brisse in French and English''. Londinii: Low, Marston, Searle, &amp; Rivington [https://catalog.hathitrust.org/Record/008611336 Textus] 
* 1883 : Samuel Wells Williams, ''The Middle Kingdom''. Ed. recensa. Londinii: Allen, 1883 [https://archive.org/details/middlekingdomsur01will_2 vol. 1] [https://archive.org/details/middlekingdomsur02will_2 vol. 2]
* 1883 : [[Alphonsus Pyramus de Candolle|Alphonse de Candolle]], ''Origine des plantes cultivées''. Lutetiae: Baillière [https://archive.org/details/originedesplant02candgoog Textus]
* 1883 : &lt;span id=&quot;Dickens (1883)&quot;&gt;&lt;/span&gt;&quot;Tripes à la mode de Caen&quot; in [[Carolus Dickens|Charles Dickens]], ''Dickens's Dictionary of Paris, 1883''. Macmillan {{Google Books|UW0DAAAAQAAJ|p. 268}}
* 1885 : [[Lafcadio Hearn]], ''La Cuisine creole: a collection of culinary recipes from leading chefs and noted Creole housewives who have made New Orleans famous for its cuisine''. 2a ed. Novae Aureliae, 1885 [https://archive.org/details/lacuisinecreolec00hearrich Textus]
* 1885 : [[Agnes Marshall|A. B. Marshall]], ''The Book of Ices''. Londinii [https://archive.org/details/b21528068/ Textus priscus]; [https://archive.org/details/b21539613/ editio aucta]
* 1885 : William Dymock, ''The Vegetable Materia Medica of Western India''. Bombay, 1885 {{Google Books|0ygJAAAAIAAJ|pp. 640-643}}
* 1886 : [[William Carew Hazlitt|Gulielmus Carew Hazlitt]], ''Old Cookery Books and Ancient Cuisine'' (Londinii: Elliot Stock, 1886) [https://archive.org/details/b21525079 Textus] apud ''Internet Archive''; [https://www.gutenberg.org/files/12293/12293-h/12293-h.htm editio 1902] apud ''Project Gutenberg''; [[:s:en:Old Cookery Books and Ancient Cuisine|apud Vicifontem]] 
* 1886 : J. E. T. Aitchison, &quot;Some Plants of Afghanistan and Their Medicinal Products&quot; in ''Pharmaceutical Journal and Transactions'' (Decembri 1886); reimpressum in eiusdem ''Notes on the Products of Western Afghanistan and North-Eastern Persia'' (Edinburgi, 1890) [https://archive.org/details/dli.pahar.1203/page/69/mode/2up pp. 69-73]
* 1887 : Maria Parloa, ''Miss Parloa's Kitchen Companion'' (Bostoniae: Estes and Lauriat) [https://archive.org/details/missparloaskitch00parlrich/page/224/mode/2up p. 225]
* 1887-1890 : [[Eduardus Ludovicus Sturtevant|E. L. Sturtevant]], &quot;The History of Garden Vegetables&quot; in ''American Naturalist'' voll. 21-24 (1887-1890): vol. 21 pp. 49-59, 125-133, 321-333, 433-444, 701-712, 826-833, 903-912, 975-985; vol. 22 pp. 420-433, 802-808, 979-987; vol. 24 pp. 30-48, 143-157, 629-656, 719-744 (cf. 1919)
* 1888 : Doctor [[Marianus Pardo de Figueroa|Thebussem]], ''La mesa moderna''. Matriti [https://archive.org/details/lamesamodernaca00conggoog Textus] [https://archive.org/details/lamesamodernacar00theb 2a ed. eiusdem anni]
* 1888 : ''Dainty Dishes for Indian Tables'' (2a ed. Calcuttae: Newman, 1888) [https://archive.org/details/daintydishesfor00unkngoog/page/n174/mode/2up pp. 158-161]
* 1888 : W. H. Dawe, ''The Wife's Help to Indian Cookery'' (Londinii: Elliot Stock, 1888) [https://archive.org/details/b21528378 Textus]
* 1889 : Gustave Garlin, ''Le cuisinier moderne''. Lutetiae: Garnier, 1889 [https://archive.org/details/lecuisiniermoder01garl vol. 1] [https://archive.org/details/lecuisiniermoder02garl 2]
* 1889-1893 : George Watt, ''A Dictionary Of The Economic Products Of India'' 6 voll. (Calcuttae) [https://archive.org/details/DictionaryOfTheEconomicProductsOfIndia1 vol. 1] [https://archive.org/details/DictionaryOfTheEconomicProductsOfIndia2 2] [https://archive.org/details/DictionaryOfTheEconomicProductsOfIndia3 3] [https://archive.org/details/DictionaryOfTheEconomicProductsOfIndia4 4] [https://archive.org/details/DictionaryOfTheEconomicProductsOfIndia5 5] [https://archive.org/details/DictionaryOfTheEconomicProductsOfIndia61 6 i] [https://archive.org/details/DictionaryOfTheEconomicProductsOfIndia62 6 ii] [https://archive.org/details/DictionaryOfTheEconomicProductsOfIndia63 6 iii] [https://archive.org/details/DictionaryOfTheEconomicProductsOfIndia64 6 iv]
* 1889-1891 : [[Iosephus Favre|Joseph Favre]], ''Dictionnaire universel de cuisine et d'hygiène alimentaire'' [https://gallica.bnf.fr/ark:/12148/bpt6k65654445 vol. 1] [https://gallica.bnf.fr/ark:/12148/bpt6k6565413m vol. 2]
* 1890-1891 : John R. Philpots, ''Oysters, and all about them''. 2 voll. Londinii: John Richardson, 1890-1891 [https://archive.org/details/oystersallaboutt01phil Textus]
* 1891 : Grace Johnson, ''Anglo-Indian And Oriental Cookery''. Londinii: Allen [https://archive.org/details/in.ernet.dli.2015.501516/ Textus]
* 1891 : Gabrielle Le Brasseur; Mary Hooper, interpr., ''Hints on cookery and management of the table'' (Londinii: Spencer Blackett, 1891) [https://archive.org/details/b28125770/page/n41/mode/2up pp. 27-28] (titulus priscus: ''Ma cuisine'')
* 1891 : [[Peregrinus Artusi|Pellegrino Artusi]], ''[[La scienza in cucina e l'arte di mangiar bene]]''. Florentiae: Landi, 1891 [https://archive.org/details/artusi-1891 Textus] apud ''Internet Archive''
* 1892 : [[Carolus Herman Senn|Charles Herman Senn]], ''Practical gastronomy and culinary dictionary'' (Londinii: Spottiswoode) [https://archive.org/details/b28107822/page/n7/mode/2up Textus] apud ''Internet Archive''
* 1893 : Caroline Sullivan, ''The Jamaica Cookery Book'' (Kingston Iamaicae: Gardner, 1893) p. 115 (&quot;scaveeched king-fish&quot;)
* 1893 : Flora Annie Steel, Grace Gardiner, ''The complete Indian housekeeper and cook'' (3a ed. Edinburgi: Edinburgh Press, 1893) [https://archive.org/details/b28081663 Textus] [https://archive.org/details/b21528640 Nova ed. 1909]
* 1893 : Jessup Whitehead, ''Cooking for profit: A new American cook book''. Chicagine {{Google Books|UDhEAAAAYAAJ|p. 193}} (&quot;Oyster brochettes à la creole&quot;)
* 1894 : Charles Ranhofer, ''The Epicurean'' (Novi Eboraci, 1894) [https://d.lib.msu.edu/fa/26#page/422/mode/2up pp. 408-413]
* 1894 : ''Spons' Household Manual: a treasury of domestic receipts and guide for home management'' (Londinii: Spon) [https://archive.org/details/b21539303/page/502/mode/2up p. 500] (&quot;Malay chicken (Doopiazeh curry)&quot;)
* 1894 : [[Agnes Marshall|A. B. Marshall]], ''Fancy Ices''. Londinii [https://archive.org/details/b29314501/ Textus]
* 1894 : Chatillon-Plessis, ''La Vie à table à la fin du 19e siècle''. Lutetiae: Diderot {{Google Books|JFQ4-9XhE-gC|pp. 273-275}}
* 1895 : Thomas J. Murrey, ''The Murrey Collection of Cookery Books'' (Novi Eboraci) [https://archive.org/details/murreycollectio00murrgoog/page/n424/mode/2up &quot;Oysters and Fish&quot; pp. 27-28] (&quot;Oysters en brochette&quot;)
* 1895 : George J. Kappeler, ''Modern American Drinks: How to Mix and Serve All Kinds of Cups and Drinks''. Novi Eboraci: Merriam, 1895 [https://archive.org/details/modernamericandr00kapp Textus]
* 1895 : &lt;span id=&quot;Sala (1895)&quot;&gt;&lt;/span&gt;[[Georgius Augustus Sala|George Augustus Sala]], ''The Thorough Good Cook'' [https://archive.org/details/b21530014/page/343/mode/2up p. 343 editionis Neo-Eboracensis 1896] 
* 1895 : Jean De Gouy, ''La Cuisine et la patisserie bourgeoises en Belgique et à l'étranger''. Bruxellis {{Google Books|n8xHAQAAMAAJ|p. 157}} (petits pâtés au godiveau&quot;)
* 1896 : Louis Fouquet, ''Bariana: Recueil de toutes boissons americains et anglaises'' (Lutetiae) [http://www.euvs.org/pdf/BARIANA-WEB.pdf p. 25 no. 18] (&quot;martini cocktail&quot;)
* 1897 : [[Abrahamus Stoker|Bram Stoker]], ''[[Dracula (Stoker)|Dracula]]'' (Londinii) p. 1
* 1898 : [[Carolus Herman Senn|Charles Herman Senn]], ''Senn's culinary encyclopaedia'' (Londinii: Spottiswoode) [https://archive.org/details/sennsculinaryenc00senn Textus] apud ''Internet Archive''
* 1898 : [[Georgius Watt|George Watt]], ''The pests and blights of the tea plant being a report of investigations conducted in Assam and to some extent also in Kangra''. Calcuttae: Superintendent, Government Printing [https://archive.org/details/cu31924024006623/page/n6 Textus] apud ''Internet Archive''
* 1898 : {{Creanda|es|Henry Clay Irish|Henricus Clay Irish|H. C. Irish}}, &quot;[https://archive.org/details/revisionofgenusc00irisrich/ A revision of the genus Capsicum, with especial reference to garden varieties]&quot; in ''Missouri Botanical Garden Annual Report'' (1898) pp. 53-110 [https://www.jstor.org/stable/2992137 JSTOR]
* 1898 : {{Creanda|d|Q100233613|Incarnatio Pinedo|Encarnación Pinedo}}, ''El cocinero español''. Franciscopoli (Dan Strehl, Victor Valle, edd., ''Encarnación’s kitchen : Mexican recipes from nineteenth-century California'' [Berkeleiae: University of California Press, 2003])
* 1899 : [[Nathaniel Newnham-Davis]], ''Dinners and Diners: where and how to dine in London''. Londinii: Grant Richards, 1899 [https://archive.org/details/b21528974 Textus] apud ''Internet Archive''
* 1900 : ''The Nabob's Cookery Book: a manual of East and West Indian recipes ... by P. O. P.'' (Londinii: Warne) [https://archive.org/details/b28060799/page/n15/mode/2up no. 1-24]
* 1900 : ''The Picayune's Creole Cook Book'' Novae Aureliae [https://www.nola.com/entertainment_life/eat-drink/article_a067a27c-91f6-538a-af50-0be0eca945d1.html de editione prima]; [https://archive.org/details/picayunescreolec0000unse_n7m7/mode/2up ed. 2ae (1901) reimpressae exemplar mutuabile]; [https://archive.org/details/cu31924073878708/page/n5/mode/2up ed. 4a (1910)]
* 1900 : Rowland Strong, ''Where and How to Dine in Paris''. Londinii: Grant Richards, 1900 [https://archive.org/details/whereandhowtodi00strogoog Textus] apud ''Internet Archive''
* 1900 : Pierre Lacam, ''Le mémorial historique et géographique de la pâtisserie'' (Lutetiae) [https://archive.org/details/b28054234/page/170/mode/2up p. 171]

== saeculum XX ==
* 1901 : &lt;span id=&quot;Holle (1901)&quot;&gt;&lt;/span&gt;Henriette Davidis; Luise Holle, ed., ''Henriette Davidis-Holle: Praktisches Kochbuch für die bürgerliche und feine Küche'' (Bielefeld: Velhagen &amp; Klasing, 1901) [https://archive.org/details/b21536004/ Textus] apud ''Internet Archive''
* 1903 : [[Nathaniel Newnham-Davis]], Algernon Bastard, ''The Gourmet's Guide to Europe''. Londinii: Grant Richards, 1903 [https://archive.org/details/b21504829 Textus] apud ''Internet Archive'' 
* 1903 : Adolphe Meyer, ''The post-graduate cookery book'' (Novi Eboraci: Caterer, 1903) [https://archive.org/details/postgraduatecook00meye/page/30/mode/2up pp. 30-31]
* 1903 : Ketab, ''Indian dishes for English tables''. Londinii: Chapman &amp; Hall, 1903 [https://archive.org/details/b2812277x Textus] apud ''Internet Archive''
* 1904 : A. R. Kenney-Herbert, ''Wyvern's Indian Cookery Book''. Maderaspatani: Higginbotham, 1904 [https://archive.org/details/in.ernet.dli.2015.221260 Textus] apud ''Internet Archive''
* 1904 : Edmond Richardin, ed., ''L'Art du bien manger'' (Lutetiae, 1904) [https://gallica.bnf.fr/ark:/12148/bpt6k15261857 Textus]
* 1905 : [[Iosephus Favre|Joseph Favre]], ''Dictionnaire universel de cuisine pratique''. 2a ed. [https://gallica.bnf.fr/ark:/12148/bpt6k57300060 vol. 1] [https://gallica.bnf.fr/ark:/12148/bpt6k57317645 vol. 2] [https://gallica.bnf.fr/ark:/12148/bpt6k57280474 vol. 3] [https://gallica.bnf.fr/ark:/12148/bpt6k57300438 vol. 4]
* 1906 : &lt;span id=&quot;Richardin, ed. (1906)&quot;&gt;&lt;/span&gt;Edmond Richardin, ed., ''L'art du bien manger'' (Lutetiae, 1906) [https://gallica.bnf.fr/ark:/12148/bpt6k203897f.image.f252 p. 252]
* 1906 : &lt;span id=&quot;Senn (1906)&quot;&gt;&lt;/span&gt;&quot;Indian Cookery&quot; in [[Isabella Beeton]]; [[Carolus Herman Senn|Charles Herman Senn]], ed., ''[[Beeton's Book of Household Management|Mrs. Beeton's Book of Household Management]]'' (Londinii: Ward, Lock, 1906) [https://archive.org/details/b21530105/page/1598/mode/2up pp. 1599-1613]
* 1906 : Frank Schloesser, ''The Greedy Book: a gastronomical anthology'' (Londinii: Gay and Bird, 1906) [https://archive.org/details/b21505007/page/166/mode/2up p. 167]
* 1907 : &lt;span id=&quot;Escoffier (1907)&quot;&gt;&lt;/span&gt;[[Augustus Escoffier|Auguste Escoffier]], ''Le Guide culinaire'' (2a ed. Lutetiae, 1907) [https://archive.org/details/b21525912/page/524/mode/2up pp. 525-526]
* 1907 : &lt;span id=&quot;Escoffier (1907)&quot;&gt;&lt;/span&gt;[[Augustus Escoffier|Auguste Escoffier]], ''A Guide to Modern Cookery''. Londinii: Heinemann [https://archive.org/details/cu31924000610117 Textus]; 2a ed. Londinii, 1909; [https://archive.org/details/b21530142 Textus impressionis 1926]
* 1907? : &lt;span id=&quot;Escoffier (1907)&quot;&gt;&lt;/span&gt;[[Augustus Escoffier|Auguste Escoffier]], ''A Few Recipes'' (Londinii: Escoffier, 1907?) [https://archive.org/details/fewrecipesbymons00esco Textus] apud ''Internet Archive''
* 1907 : &lt;span id=&quot;Wiemann (1907)&quot;&gt;&lt;/span&gt;Henriette Davidis; Gertrude Wiemann, ed., ''Praktisches Kochbuch für die bürgerliche und feine Küche. Neue illustrierte Ausgabe'' (Berolini: W. Herlet, 1907) [https://www.projekt-gutenberg.org/davidis/kochbuch/kochbuch.html Recensio interretialis]
* 1908 : [[Nathaniel Newnham-Davis]], ''The Gourmet's Guide to Europe''. 2a ed. Londinii: Grant Richards, 1908 [https://archive.org/details/b2152709x Textus] apud ''Internet Archive'' 
* 1909 : Eleanor Jenkinson, ''The Ocklye Cookery Book: a book of recipes by a lady and her cook''. Crowborough: H. Wilkins, 1909 [https://archive.org/details/b28065311/page/n79/mode/2up Textus] apud ''Internet Archive'' 
* 1911 : Robert H. Christie, ''Banquets of the Nations: eighty-six dinners characteristic and typical each of its own country''. Edinburgh: Gray, 1911 [https://archive.org/details/b21528676 Textus] apud ''Internet Archive''
* 1911 : [[Nathaniel Newnham-Davis]], ''The Gourmet's Guide to Europe''. 3a ed. Londinii: Grant Richards, 1911 [https://archive.org/details/b2804938x Textus] apud ''Internet Archive'' 
* 1912 : &lt;span id=&quot;Escoffier (1912)&quot;&gt;&lt;/span&gt;[[Augustus Escoffier|Auguste Escoffier]], ''Le Livre des menus''. Lutetiae, 1912 [https://gallica.bnf.fr/ark:/12148/bpt6k9629814d Textus]
* 1912 : [[Franciscus Jammes|Francis Jammes]], ''Les Géorgiques chrétiennes'' (1912) [https://archive.org/details/lesgorgiquesch00jammuoft/page/144/mode/2up p. 145 editionis 1914]
* 1913 : Emilia Pardo Bazán, ''La cocina española antigua''. Matriti: Renacimiento, 1913 [http://bdh.bne.es/bnesearch/detalle/bdh0000244326 Textus]
* 1914 : May Byron, ''Pot-luck, or The British home cookery book''. Londinii: Hodder &amp; Stoughton, 1914 [https://archive.org/details/cu31924001893241 2a ed. 1915]
* 1914 : [[Nathaniel Newnham-Davis]], ''The Gourmet's Guide to London''. Londinii: Grant Richards [https://archive.org/details/b28107548 Textus] apud ''Internet Archive''; Novi Eboraci: Brentano's [https://www.gutenberg.org/files/53304/53304-h/53304-h.htm Textus] apud ''Project Gutenberg''
* 1919 : [[Eduardus Nignon|Édouard Nignon]] et al., ''L'Heptameron des gourmets''. Lutetiae, 1909 [https://gallica.bnf.fr/ark:/12148/bpt6k315710q Textus] apud Gallica
* 1919 : [[Eduardus Ludovicus Sturtevant|E. L. Sturtevant]]; U. P. Hedrick, ed., ''Sturtevant's notes on edible plants''. (''Report of the New York Agricultural Experiment Station'', 1919, pars 2.) Albaniae, 1919 [https://archive.org/details/sturtevantsnotes00sturuoft/ Textus]
* 1920 : Carmen de Burgos, ''La cocina práctica'' (Valentiae: Sempere, 1920) [http://bdh.bne.es/bnesearch/detalle/bdh0000161962 Textus]
* 1920 : [[Marcellus Rouff|Marcel Rouff]], ''La Vie et la passion de Dodin-Bouffant gourmet''. Lutetiae: Société littéraire de France (editio lacunosa); editio perfecta, 1924 [http://dodinbouffant.com/site_images/telechargement/La%20vie%20et%20passion%20de%20dodin-Bouffant.pdf Recensio interretialis editionis 1924]
* 1922 : &lt;span id=&quot;Zäh (1922)&quot;&gt;&lt;/span&gt;Henriette Davidis; Rudolf Zäh, ed., ''Praktisches Kochbuch für die bürgerliche und feine Küche. Volksausgabe'' (1922) [https://d-nb.info/575461721/04 Index capitulorum]
* 1923 : {{Creanda|fr|Victor Thomas|Victor Thomas|Thomas Gringoire}}, [[Ludovicus Saulnier|Louis Saulnier]], ''Le Répertoire de la cuisine''. 3a ed. Londinii: Allard, 1923 [https://gallica.bnf.fr/ark:/12148/bpt6k9638563t Textus] apud Gallica (1a ed. 1914)
* 1927 : &lt;span id=&quot;Escoffier (1927)&quot;&gt;&lt;/span&gt;[[Augustus Escoffier|Auguste Escoffier]], ''Le Riz''. Lutetiae: Flammarion, 1927 [https://gallica.bnf.fr/ark:/12148/bpt6k858014w Textus]
* 1928 : &lt;span id=&quot;Escoffier (1928)&quot;&gt;&lt;/span&gt;[[Augustus Escoffier|Auguste Escoffier]], ''L'Aide-mémoire culinaire''. Lutetiae: Flammarion, 1928 [https://gallica.bnf.fr/ark:/12148/bpt6k1265661m Textus]
* 1928 : [[Augustinus de Croze|Austin de Croze]], ''Les Plats régionaux de France. 1400 succulentes recettes traditionnelles de toutes les provinces françaises''. Lutetiae: Editions Montaigne [https://gallica.bnf.fr/ark:/12148/bpt6k323158z Textus]
* 1929 : Dionisio Pérez, ''Guía del buen comer español''. Matriti, 1929 [http://bdh.bne.es/bnesearch/detalle/bdh0000204799 Textus]
* 1930 : Gerardo Corrales, ''Club de cantiñeros de la República de Cuba: Manual oficial'' (Havanae, 1930) [https://euvs-vintage-cocktail-books.cld.bz/1930-Club-de-Cantineros-de-la-Republica-de-Cuba-Manual-Oficial/50/ p. 51] (&quot;Martini cocktail ... fórmula original creación de la case Martini &amp; Rossi&quot;)
* 1931 : &lt;span id=&quot;Grieve (1931)&quot;&gt;&lt;/span&gt;[[Magdalena Grieve|Maud Grieve]]; [[Hilda Leyel]], ed., ''A modern herbal'' (Londinii: Jonathan Cape, 1931) p. 974 (&quot;Meadowsweet&quot;) [http://botanical.com/botanical/mgmh/mgmh.html Recensio interretialis]
* 1931 : ''Poesia nascosta: seicento ricette di cucina ebraica in Italia'' (Patavii, 1931) [http://www.archivio-torah.it/ebooks/poesianascosta/ Textus]
* 1932 : &lt;span id=&quot;Sloppy Joe (1932)&quot;&gt;&lt;/span&gt;''Sloppy Joe's Cocktails Manual'' (Havanae, 1932) [https://euvs-vintage-cocktail-books.cld.bz/1932-Sloppy-Joe-s/8 pp. 9, 13] (&quot;Bacardi cocktails: mojito; Gordon Dry Gin cocktails: mojito&quot;)
* 1932 : Filippo Tommaso Marinetti, ''La cucina futurista''. Mediolani: Sonzogno, 1932 [https://archive.org/details/marin_cucina_1932_images/ Textus]
* 1932 : Florence White, ''Good Things in England'' (Londinii: Jonathan Cape, 1932) (&quot;caramel cream&quot;) 
* 1933 : &lt;span id=&quot;Sloppy Joe (1933)&quot;&gt;&lt;/span&gt;''Sloppy Joe's Cocktails Manual'' (Havanae, 1933) [https://euvs-vintage-cocktail-books.cld.bz/1933-Sloppy-Joe-s-Cocktails-Manual/II p. 18] (&quot;dry martini, sweet martini&quot;)
* 1933 : [[Gualterus Starkie|Walter Starkie]], ''Raggle-Taggle: adventures with a fiddle in Hungary and Roumania'' (Londinii) [https://archive.org/details/in.ernet.dli.2015.208611/page/n303/mode/2up p. 289]
* 1934 : &lt;span id=&quot;Escoffier (1934)&quot;&gt;&lt;/span&gt;[[Augustus Escoffier|Auguste Escoffier]], ''Ma Cuisine''. Lutetiae: Flammarion, 1934 [https://archive.org/details/AugusteEscoffier1934FlamarionMaCuisineINDEXP677 Textus]
* 1935 : {{Creanda|es|Marquesa de Parabere|María Mestayer de Echagüe}}, ''Platos escogidos de la cocina vasca''. Bilbao: Grijelmo, 1935 [http://www.kmliburutegia.net/Record/230001 Textus]
* 1938 : &lt;span id=&quot;Montagné (1938)&quot;&gt;&lt;/span&gt;[[Prosper Montagné]], ''Larousse gastronomique'' (Lutetiae: Larousse, 1938) p. 574
* 1939 : ''Floridita cock-tails'' (Havanae, 1939) [https://euvs-vintage-cocktail-books.cld.bz/1939-Floridita-Cock-tails/46 pp. 47-48] (&quot;mojito criollo no. 1, no. 2, no.3&quot;)
* 1939 : ''Sloppy Joe's Cocktail Manual'' (Havanae, 1939) [https://euvs-vintage-cocktail-books.cld.bz/1939-Sloppy-Joe-s-Season-1939/10 pp. 10, 14] (&quot;Sloppy Joe's ron drinks: mojito; jin mojito&quot;)
* 1939 : Charles H. Baker Jr, ''The Gentleman’s Companion, Volume II: being an exotic drinking book''. Novi Eboraci: Derrydale Press, 1939 [https://euvs-vintage-cocktail-books.cld.bz/1939-The-Gentleman-s-Companion-volume-II-Beeing-an-Exotic-Drinking-Book/X Textus] 
* 1939 : Stanley Clisby Arthur, ''Famous New Orleans Drinks and How to Mix 'Em''. Novae Aureliae: Harmanson, 1939 [https://euvs-vintage-cocktail-books.cld.bz/1938-Famous-New-Orleans-Drinks-and-how-to-mix-em-3rd-printing-by-Stanley-Clisby-Arthur/2 Textus]
* 1940 : Claud Bald, ''Indian Tea: 'A Textbook on the Culture and Manufacture of Tea.'' Ed. 5a. Calcuttae: Thacker, Spink &amp; Co. [https://archive.org/details/dli.bengal.10689.3392 4a ed., 1922] apud ''Internet Archive''
* 1945 : Buwei Yang Chao, [[Yuen Ren Chao]], ''{{Creanda|en|How to Cook and Eat in Chinese}}''. Novi Eboraci: John Day&lt;ref&gt;Charles W. Hayford, &quot;Open Recipes Openly Arrived At&quot; in ''Journal of Oriental Studies'' vol. 45 (2012) pp. 67-87 [https://www.jstor.org/stable/43498205 JSTOR]&lt;/ref&gt; [https://archive.org/details/howtocookeatinch00cha0 exemplar mutuabilis editionis 1949]; [https://archive.org/details/howtocookeatinch00bych editionis 1963]; [https://archive.org/details/howtocookeatinch00chao editionis Britannicae]
* 1948 : David A. Embury, ''The Fine Art of Mixing Drinks''. Novi Eboraci: Doubleday, 1948 
* 1948 : Hilario Alonso Sánchez, ''El Arte del Cantinero'' (Havanae, 1948) [https://euvs-vintage-cocktail-books.cld.bz/1948-El-Arte-del-Cantinero-Mixellany/468/ p. 436] (&quot;mojito&quot;)
* 1948 : ''Ron Daiquiri coctelera cocktail book'' (Havanae, 1948) [https://euvs-vintage-cocktail-books.cld.bz/1948-Ron-Daiquiri-Coctelera-Cocktail-Book-Habana-Cuba/18/ p. 20] (&quot;Cuban drinks: mojito&quot;)
* 1949 : &lt;span id=&quot;Fisher (1949)&quot;&gt;&lt;/span&gt;[[M. F. K. Fisher]], interpr., ''Jean Anthelme Brillat-Savarin: The Physiology of Taste; or, Meditations on transcendental gastronomy''. Novi Eboraci: Limited Editions Club, 1949
* medio saeculo XX : [[Phia Sing]] ([[Alanus Davidson|Alan Davidson]], Jennifer Davidson, edd., ''Phia Sing: Traditional Recipes of Laos'' [Londinii: Prospect Books, 1981] pp. 48, 183, 197 et passim} 
* 1954 : [[Elizabeth David]], ''Italian Food''. Londinii: Macdonald
* c. 1960 : Felice Cùnsolo, ''Cucina milanese''. Mediolani: Novedit, s.d. [https://archive.org/details/digitami_LO11012739/ Textus]
* 1960 : &lt;span id=&quot;David (1960)&quot;&gt;&lt;/span&gt;[[Elizabeth David]], ''[[French Provincial Cooking]]''. Harmondsworth: Penguin, 1960
* 1963 : [[Elizabeth David]], ''Italian Food''. Nova ed. (Harmondsworth: Penguin, 1963) pp. 123-125
* 1975 : [[Alanus Davidson|Alan Davidson]], ''Fish and Fish Dishes of Laos'' (Rutland: Tuttle, 1975) p. 104
* 1975 : &lt;span id=&quot;Lambert Ortiz (1975)&quot;&gt;&lt;/span&gt;Elisabeth Lambert Ortiz, ''Caribbean Cooking'' (Londinii: Deutsch, 1975) pp. 76-77 et passim
* 1979 : [[Alanus Davidson|Alan Davidson]], ''North Atlantic Seafood'' (Londinii: Macmillan, 1979) pp. 186-187
* 1980 : Meera Taneja, ''Indian Regional Cookery'' (Londinii: Mills and Boon) pp. 18-19 (&quot;aaloo samosa, ''pastries filled with spicy potato''&quot;)
* 1981 : [[Alanus Davidson|Alan Davidson]], ''Mediterranean Seafood'' (2a ed. Londinii: Penguin, 1981) p. 178
* 1983 : &lt;span id=&quot;David (1983)&quot;&gt;&lt;/span&gt;[[Elizabeth David]], ''[[French Provincial Cooking]]''. Nova ed. Londinii: Penguin, 1983
* 1993 : Rena Salaman, ''Greek Food''. Nova ed. (Londinii: Harper Collins, 1993) pp. 26, 66-67 (&quot;mayiritsa&quot;)
* 1995 : Maria Kaneva Johnson, ''The Melting Pot: Balkan food and cookery'' (Totenais: Prospect Books, 1995) pp. 78-79, cf. 63 
* 1999 : ''Pat Chapman's Curry Bible''. Londinii: Hodder &amp; Stoughton. ISBN 0-340-68037-7

== Saeculum XXI ==
* 2001 : Diane Kochilas, ''The Glorious Foods of Greece'' (Novi Eboraci: William Morrow, 2001) p. 125 (&quot;Greek Easter innards sausage: kokkoretsi&quot;)
* 2002 : Eliana Thibaut i Comalada, ''La cuina tradicional de la Catalunya Nord'' (Cossetània Edicions, 2002. ISBN 84-95684-67-5) {{Google Books|34mqMjgDK7EC|p. 143}} (&quot;llagostada o civet de llagosta de Cotlliure&quot;)
* 2002 : William Rubel, ''The Magic of Fire'' (Berkeleiae: Ten Speed Press) pp. 112-113 (&quot;lamb kebab&quot;)
* 2003 : [[Alanus Davidson|Alan Davidson]], ''Seafood of South-East Asia'' (2a ed. Totenais: Prospect Books, 2003) p. 139
* 2003 : Clifford Wright, ''The Little Foods of the Mediterranean'' (Cantabrigiae Mass.: Harvard Common Press, 2003) {{Google Books|x3t2IJeFIh8C|pp. 451-452}} (&quot;mulinciana a schibecci&quot; et &quot;cucuzzeddi fritti cu schibecci&quot;, paropsides duae de melongena fricta e sicbeis)
* 2006 : Christopher Grocock, Sally Grainger, edd., ''Apicius. A critical edition with an introduction and an English translation''. Totenais: Prospect Books. ISBN 1903018137 {{Ling|Latine|Anglice}}
* 2006 : [[Fuchsia Dunlop]], ''Revolutionary Chinese Cookbook: recipes from Hunan province'' (Londinii: Ebury Press, 2006) pp. 76, 78-80
* 2010 : &quot;Crab paste jeow: jeow nam bpoo&quot; in Dorothy Culloty, ''Food from Northern Laos: the Boat Landing cookbook'' (Te Awamutu Novae Zelandiae: Galangal Press, 2010) [https://ffnlblog.files.wordpress.com/2019/03/food-from-n-laos-pdf-2019.pdf pp. 82-91, cf. p. 39]
* 2019 : [[Fuchsia Dunlop]], ''The Food of Sichuan'' (Londinii: Bloomsbury, 2019) pp. 342-367

== Eruditio ==
* K. T. Achaya, ''Indian Food: a historical companion'' (Dellii: Oxford University Press, 1994) 
* &lt;span id=&quot;Lambert (2002)&quot;&gt;&lt;/span&gt;Carole Lambert, &quot;Medieval France: B. The South&quot; in Melitta Weiss Adamson, ed., ''Regional Cuisines of Medieval Europe: A Book of Essays'' (Londinii: Routledge, 2002) pp. 67-84
* &lt;span id=&quot;Varey (2002)&quot;&gt;&lt;/span&gt;Simon Varey, &quot;Medieval and Renaissance Italy: A. The Peninsula&quot; in Melitta Weiss Adamson, ed., ''Regional Cuisines of Medieval Europe: A Book of Essays'' (Londinii: Routledge, 2002) pp. 85-112
* E. N. Anderson, ''Food and Environment in Early and Medieval China'' (Philadelphiae: University of Pennsylvania Press, 2014) pp. 85-87
* &lt;span id=&quot;Andrews (1980)&quot;&gt;&lt;/span&gt;A. C. Andrews, &quot;Index of plants&quot; in W. H. S. Jones, ''Pliny: Natural History, books XXIV-XXVII'' (Cantabrigiae Mass., editio recensa, 1980) pp. 485-557 s.v. &quot;oenanthe&quot;
* Luc Bihl-Willette, ''Des tavernes aux bistrots. Une histoire des cafés'' (Lutetiae: L'Âge d'Homme, 1997. ISBN 9782825107737)
* Eric Block, ''Garlic and Other Alliums: The Lore and the Science''. Royal Society of Chemistry, 2010 {{Google Books|6AB89RHV9ucC|Paginae selectae}}
* Hector Bolitho, ''The Glorious Oyster: His History in Rome and in Britain, His Anatomy and Reproduction, how to Cook Him, and what Various Writers and Poets Have Written in His Praise''. Novi Eboraci: Knopf, 1929
* &lt;span id=&quot;Bosland et al. (1998)&quot;&gt;&lt;/span&gt;[[Paulus Bosland|Paul W. Bosland]], Alton L. Bailey, Jaime Iglesias-Olivas, ''[http://contentdm.nmsu.edu/cdm/ref/collection/AgCircs/id/12518 Capsicum pepper varieties and classification]''. NMSU Cooperative Extension Service and Agricultural Experiment Station, 1998
* &lt;span id=&quot;Burnett et Saberi (2008)&quot;&gt;&lt;/span&gt;David Burnett, Helen Saberi, ''The Road to Vindaloo: curry cooks and curry books''. Totnes: Prospect Books, 2008. ISBN 978-1-903018-57-6
* David Burton, ''The Raj at Table: A Culinary History of the British in India''. Londinii: Faber, 1994
* Kwang-chi Chang, &quot;Food and Food Vessels in Ancient China&quot; in ''Transactions of the New York Academy of Medicine'' ser. 2a vol. 35 (1973)
* Kwang-chi Chang, ed., ''Food in Chinese Culture''. Novo Portu: Yale University Press, 1977
* [[Eduardus Hetzel Schafer|Edward H. Schafer]], &quot;T'ang&quot; in Kwang-chi Chang, ed., ''Food in Chinese Culture'' (Novo Portu: Yale University Press, 1977) pp. 122-124
* Claire Clifton, Colin Spencer, edd., ''The Faber Book of Food''. Londinii: Faber, 1993
* [[Sophia Coe|Sophie D. Coe]], ''America's First Cuisines''. Austinopoli: University of Texas Press, 1994 [https://archive.org/details/americasfirstcui00coes Exemplar mutuabile]
* J. Michelle Coghlan, ed., ''The Cambridge Companion to Literature and Food''. Cantabrigiae: Cambridge University Press, 2020 {{Google Books|acfVDwAAQBAJ|Paginae selectae}}
* William J. Darby, Paul Ghalioungui, Louis Grivetti, ''Food: gift of Osiris'' (Londinii: Academic Press, 1977) pp. 656-660
* &quot;The True Emulsion&quot; in [[Elizabeth David]], ''An Omelette and a Glass of Wine'' (Londinii, 1984) {{Google Books|2aBCBAAAQBAJ|Fragmenta editionis recentioris}}
* [[Alanus Davidson|Alan Davidson]], ''Mediterranean Seafood'' (2a ed. Harmondsworth: Penguin, 1981) pp. 50-54
* [[Alanus Davidson|Alan Davidson]], ed., ''National and Regional Styles of Cookery: Proceedings: Oxford Symposium 1981'' (Londinii: Prospect Books, 1981) {{Google Books|zcNdB_sl2JkC|textus}}
* Keiko Ohnuma, &quot;Curry Rice: Gaijin Gold: how the British version of an Indian dish turned Japanese&quot; in ''Petits propos culinaires'' no. 52; reimpressum in  [[Alanus Davidson|Alan Davidson]], Helen Saberi, edd., ''The Wilder Shores of Gastronomy'' (Berkeleiae: Ten Speed Press, 2002) pp. 160-167
* Antoine De Baecque, ''La France Gastronome: comment le restaurant est entré dans notre histoire''. Lutetiae: Payot, 2019. ISBN 978-2228922647
* &quot;Interior Regulations&quot;: A chapter in the ''Classic of Rites'' instructs us how to prepare food&quot; in Daniel Ding, ''The Historical Roots of Technical Communication in the Chinese Tradition''. Cantabrigiae: Cambridge Scholars Publishing, 2020 {{Google Books|kSn_DwAAQBAJ|pp. 131-146}}
* Lou DiPalo, Rachel Wharton, &quot;Mozzarella (Burrata)&quot; in ''DiPalo's Guide to the Essential Foods of Italy'' (Novi Eboraci: Ballantine, 2014)
* &lt;span id=&quot;Dott (2020)&quot;&gt;&lt;/span&gt;Brian R. Dott, ''The Chile Pepper in China: a cultural biography''. Novi Eboraci: Columbia University Press, 2020
* &lt;span id=&quot;Downie (2017)&quot;&gt;&lt;/span&gt;David Downie, ''A Taste of Paris'' (Novi Eboraci: St Martin's Press, 2017) pp. 184-201
* R. L. Dressler, &quot;The pre-Columbian cultivated plants of Mexico&quot; in ''Botanical Museum Leaflets, Harvard University'' vol. 16 (1953) pp. 115–172
* &lt;span id=&quot;Egmond (2010)&quot;&gt;&lt;/span&gt;Florike Egmond, ''The World of Carolus Clusius: natural history in the making, 1550-1610'' (Londinii: Pickering and Chatto, 2010)
* &quot;Soy Sauce&quot; in Edward R. Farnworth, ed., ''Handbook of Fermented Functional Foods'' (2a ed. CRC Press, 2008) pp. 17-18, 442-448 {{Google Books|gpcXqE-j6gEC|Paginae selectae}}
* Priscilla Parkhurst Ferguson, ''Accounting for Taste: The Triumph of French Cuisine'' (Chicagi: University of Chicago Press, 2004) pp. 30-33 et alibi
* Joan Fitzpatrick, ''Food in Shakespeare''. Londinii: Ashgate, 2007. ISBN 9780754684145
* [[Ioanna Andrews|Jean Andrews]], &quot;The Peripatetic Chilli Pepper: diffusion of the domesticated capsicums since Columbus&quot; in Nelson Foster, Linda S. Cordell, edd., ''Chilies to Chocolate: Food the Americas Gave the World'' (Tucson: University of Arizona Press, 1992) pp. 81-94 {{Google Books|VsQevENGMr0C|Paginae selectae}} [https://archive.org/details/chiliestochocola00fost Exemplar mutuabile]
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* David Gentilcore, ''Food and Health in Early Modern Europe: diet, medicine and society, 1450–1800'' (Londinii: Bloomsbury, 2016) pp. 139-144
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* Eric Glatré, ''Histoire(s) de vin: 33 dates qui façonnèrent les vignobles français''. Paris: Le Félin, 2020 {{Google Books|Vg0HEAAAQBAJ|Paginae selectae}}
* Carrington Goodrich, &quot;Chinese Food over the Millennia&quot; in ''Journal of the American Oriental Society'' vol. 99 (1979) 
* Emily Gowers, ''The Loaded Table'' (Oxonii: Oxford University Press, 1993) pp. 180-210
* Peter Heine, ''Kulinarische Studien: Untersuchungen zur Kochkunst im arabisch-islamischen Mittelalter'' (1988) pp. 101-102
* Karen Hess, ed., ''The Virginia House-wife, by Mary Randolph'' (University of South Carolina Press, 1984) pp. 263-264 {{Google Books|oszKiYe2RyAC|Paginae selectae}} 
* Thomas O. Höllmann, ''Schlafender Lotus, trunkenes Huhn. Kulturgeschichte der chinesischen Küche''. Monaci: C. H. Beck, 2010. ISBN 978-3-406-60539-0
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* Pierre Hurtubise, &quot;[https://www.persee.fr/doc/hes_0752-5702_1994_num_13_2_1695 «De Honesta Voluptate» ou l'art de bien manger à Rome pendant la Renaissance]&quot; in ''Annales: Histoire, économie &amp; société'' vol. 13 no. 2 (1994) pp. 235-247
* Pierre Hurtubise, &quot;[https://www.persee.fr/doc/mefr_0223-5110_1980_num_92_1_2546 La &quot;Table&quot; d'un cardinal de la Renaissance]&quot; in ''Mélanges de l'Ecole fraiçaise de Rome: Moyen-Age'' vol. 92 (1980) pp. 249-282
* Pierre Hurtubise, &quot;Une vie de palais: la cour du cardinal Alexandre Farnèse vers 1563&quot; in ''Renaissance and Reformation'' vol. 16 (1992) pp. 37-54 [https://www.jstor.org/stable/43445591 JSTOR]
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* &lt;span id=&quot;Hyman et Hyman (2001)&quot;&gt;&lt;/span&gt;Philip Hyman, Mary Hyman, &quot;Les Livres de cuisine imprimés en France: du règne de Charles VIII à la fin de l'Ancien régime&quot; in ''Livres en bouche: cinq siècles d'art culinaire français'' (Lutetiae: Bibliothèque nationale de France, 2001) pp. 55-75 
* &quot;Sake Brewing&quot; in Naomichi Ishige, ''The History and Culture of Japanese Food'' (Londinii: Kegan Paul, 2001) pp. 32-35
* Alexandra Hicks, &quot;The Mystique of Garlic: history, uses, superstitions, and scientific revelations&quot;  in Tom Jaine, ed., ''Oxford Symposium on Food &amp; Cookery 1984 &amp; 1985: Cookery: Science, Lore &amp; Books'' (Londinii: Prospect Books, 1986) pp. 140-161
* &lt;span id=&quot;Johnson (1971)&quot;&gt;&lt;/span&gt;[[Hugo Johnson|Hugh Johnson]], ''The World Atlas of Wine'' (Londinii: Mitchell Beazley, 1971) pp. 128-135. 7a ed., 2013
* &lt;span id=&quot;Johnson (2013)&quot;&gt;&lt;/span&gt;[[Hugo Johnson|Hugh Johnson]], Jancis Robinson, ''The World Atlas of Wine'' (7a ed. Londinii, 2013) pp. 258-259
* &lt;span id=&quot;Jurafsky (2014)&quot;&gt;&lt;/span&gt;Dan Jurafsky, ''The Language of Food''. Novi Eboraci: Norton, 2014 {{Google Books|7BF0AwAAQBAJ|quaere &quot;sik&quot;}} 
* David B. Goldstein, &quot;[https://journals.openedition.org/shakespeare/1702 Failures of Eating in The Merchant of Venice]&quot; in Pierre Kapitaniak, Christophe Hauserman, Dominique Goy-Blanquet, edd., ''Shakespeare et les arts de la table: Actes du congrès organisé par la Société Française Shakespeare les 17, 18 et 19 mars 2011'' (OpenEdition Journals, 2012 [https://journals.openedition.org/shakespeare/1684 ~])
* &quot;The Chiles of Oaxaca&quot; in Diana Kennedy, ''Oaxaca al gusto: an infinite gastronomy'' (Austinopoli: University of Texas Press, 2010) pp. xix-xxii 
* Kian Lam Kho, &quot;Dragon on a Platter: the art of naming Chinese dishes&quot; in Mark McWilliams, ed., ''Food and Communication: proceedings of the Oxford Symposium on Food and Cookery 2015'' (Londinii: Prospect Books, 2016) 
* David R. Knechtges, &quot;Food in Early Chinese Literature&quot; in ''Journal of the American Oriental Society'' vol. 106 (1986) pp. 49-63 [http://www.jstor.com/stable/602363 JSTOR]
* David R. Knechtges, &quot;Gradually Entering the Realm of Delight: Food and Drink in Early Medieval China&quot; in ''Journal of the American Oriental Society'' vol. 117 (1997) pp. 229-239 [https://www.jstor.org/stable/605487 JSTOR]
* T. C. Lai, ''Chinese Food for Thought''. Hongcongi: Hong Kong Book Centre, 1978 
* {{Laufer SI}} pp. 376-377
* Bruno Laurioux, &quot;[https://books.openedition.org/psorbonne/21145 Le latin de la cuisine]&quot; in Monique Goullet, Michel Parisse, edd., ''Les Historiens et le latin médiéval'' (Lutetiae: Editions de la Sorbonne, 2001. ISBN 9782859444204
* Bruno Laurioux, ''Le Règne de Taillevent: livres et pratiques culinaires à la fin du moyen age'' (Lutetiae: Publications de la Sorbonne, 1997) pp. 155, 170-172 [https://books.openedition.org/psorbonne/34357 Textus]
* R. A. Lawrie, D.  A. Ledward, ''Lawrie's meat science''. 7a ed. Cantabrigiae: Woodhead, 2006. ISBN 978-1-84569-159-2
* Cecilia Leong-Salobir, ''Food Culture in Colonial Asia: a taste of empire'' (Londinii: Routledge, 2011) p. 18
* &quot;Foodiegate: Ritz and Escoffier at the Savoy&quot; in [[Paulus Levy|Paul Levy]], ''Out to Lunch'' (Londinii: Chatto &amp; Windus, 1986) pp. 225-230
* Gilly Lehmann, &quot;[https://journals.openedition.org/shakespeare/1693 At The Dramatists’ Table: The Climax and Decline of a Mannerist Cuisine in England, 1580-1630]&quot; in Pierre Kapitaniak et al., edd., ''Shakespeare et les arts de la table: Actes du congrès organisé par la Société Française Shakespeare les 17, 18 et 19 mars 2011'' (OpenEdition Journals, 2012 [https://journals.openedition.org/shakespeare/1684 ~])
* Manuel Martínez Llopis, Simone Ortega, ''La Cocina típica de Madrid'' (Matriti: Alianza Editorial, 1987) pp. 95-101
* Janet Long-Solís, ''Capsicum y cultura: la historia del chilli'' (2a ed. Mexicopoli: Fondo de Cultura Económica, 1998) pp. 91-93, 163
* Alessandro Lovatelli, ''[https://www.fao.org/3/AB716E/AB716E01.htm Status of oyster culture in selected Asian countries]''. Bancoci: National Inland Fisheries Institute, 1988
* A. A. Macdonell, A. B. Keith, ''Vedic index of names and subjects'' (Londinii: Murray, 1912) [https://archive.org/details/vedicindexofname01macduoft/page/338/mode/2up vol. 1 p. 338], [https://archive.org/details/vedicindexofname02macduoft/page/476/mode/2up vol. 2 pp. 477-478]
* &quot;Meat&quot; in Harold McGee, ''On Food and Cooking'' (Novi Eboraci: Scribner, 1984; Londinii: Allen &amp; Unwin, 1986) pp. 82-122
* &lt;span id=&quot;McKnight (2004)&quot;&gt;&lt;/span&gt;Justine Woodard McKnight, “A Study of Native American Plant Use, based on a review of early historic comments on subsistence and technology” in Charles LeeDecker et al., Archaeology of the Puncheon Run Site (Dover: Delaware Department of Transportation, 2004/2005) [http://deldot.gov/environmental/archaeology/puncheon_run/index.shtml vol. 2 pp. E1-E123]
* Clyde L. MacKenzie et al., edd., ''The history, present condition, and future of the molluscan fisheries of North and Central America and Europe''. 3 voll. (''NOAA technical report NMFS'' no. 127-129. US Department of Commerce, 2017) [https://repository.library.noaa.gov/view/noaa/3011 vol. 1] [https://repository.library.noaa.gov/view/noaa/3028 vol. 2] [https://repository.library.noaa.gov/view/noaa/2997 vol. 3]
* &lt;span id=&quot;Young (2013)&quot;&gt;&lt;/span&gt;Carolin C. Young, &quot;The Soup that Went Into the Tureen&quot; in Mark McWilliams, ed., ''Food &amp; Material Culture: Proceedings of the Oxford Symposium on Food and Cookery 2013''. Totenais: Prospect Books, 2014 {{Google Books|yj8QDgAAQBAJ|pp. 33-47}} (vide &quot;Vincent La Chapelle and the Chesterfield Tureen&quot;, pp. 33-43)
* [[Raimundus Sokolov|Ray Sokolov]], &quot;Secrets of the Great Chefs: decrypting untrustworthy communications from the kitchens of Carême, Escoffier and Guérard&quot; in Mark McWilliams, ed., ''Food &amp; Communication: proceedings of the Oxford Symposium in Food and Cookery 2015'' (Londinii: Prospect Books, 2015) pp. 20-28
* &quot;Cheddar Cheese&quot; in Laura Mason, Catherine Brown, ''Traditional Foods of Britain: an inventory'' (Totnes: Prospect Books, 1999) pp. 118-120
* &lt;span id=&quot;Mencken (1945)&quot;&gt;&lt;/span&gt;[[H. L. Mencken]], ''The American Language: supplement 1'' (Novi Eboraci: Knopf, 1945) [https://archive.org/details/in.ernet.dli.2015.157922/page/n280/mode/1up pp. 256-260]
* &lt;span id=&quot;Mennell (1985)&quot;&gt;&lt;/span&gt; Stephen Mennell, ''All Manners of Food'' (Oxoniae: Blackwell, 1985) pp. 134-165
* Marie-Josèphe Moncorgé, ''Lyon 1555: capitale de la culture gourmande au XVIe siècle'' (Lugduni: Editions Lyonnaises d'Art et d'Histoire, 2008) pp. 115-126, 227-228
* &quot;Kiwifruit&quot; in Julia F. Morton, ''Fruits of warm climates'' (Miami, 1987) pp. 287-300 [https://hort.purdue.edu/newcrop/morton/kiwifruit_ars.html Textus]
* S. Douglas Olson, Alexander Sens, ''Archestratos of Gela: Greek Culture and Cuisine in the Fourth Century BCE''. Oxonii: Oxford University Press, 2000. {{Ling|Graece|Anglice}}
* Om Prakash, ''Economy and food in ancient India. Part 2: Food''. Dellii: Bharatiya Vidya, 1987; 1a ed., titulo ''Food and drinks in ancient India'' (1961) [https://archive.org/details/in.gov.ignca.10960/page/n47/mode/2up pp. 22-24 et passim]
* José Pardo-Tomás, María Luz López Terrada, ''[https://www.academia.edu/508720/Las_primeras_noticias_sobre_plantas_americanas_en_las_relaciones_de_viajes_y_crónicas_de_Indias_1493-1553 Las primeras noticias sobre plantas americanas en las relaciones de viajes y crónicas de Indias (1493-1553)]''. Valencia : Instituto de Estudios Documentales e Históricos sobre la Ciencia, 1993 [http://digital.csic.es/handle/10261/91333 alibi]
* Jeffrey M. Pilcher,''Planet Taco: a global history of Mexican food'' (Novi Eboraci: Oxford University Press, 2012) pp. 69-71 
* Liliane Plouvier, &quot;La gastronomie dans les Pays-Bas méridionaux sous les ducs de Bourgogne: le témoignage des livres de cuisine&quot; in ''Publications du Centre Européen d'Etudes Bourguignonnes'' vol. 47 (2007)
* Liliane Plouvier, &quot;Spécialités bourguignonnes dans les Pays-Bas méridionaux (XVe siècle)&quot; in Paul Janssens, Siger Zeischka, edd., ''La Noblesse à table: des ducs de Bourgogne aux rois des Belges'' (2008) pp. 285-306 {{Google Books|UsW-9qOt5IgC|Paginae selectae}}
* Julius Preuss, ''Biblisch-talmudische Medizin''. Berolini: Karger, 1911 [https://nbn-resolving.de/urn:nbn:de:hbz:061:2-22092 Textus] [https://archive.org/details/biblischtalmudis00preu/page/n3/mode/2up 1928 reimpressum]
* Mudaliyar C. Rasanayagam, ''Ancient Jaffna'' (Maderaspatani: Everyman's Publishers, 1926) [https://archive.org/details/in.ernet.dli.2015.281961/page/n185/mode/2up p. 152]
* [[Gulielmus Woodville Rockhill|W. W. Rockhill]], &quot;Notes on the Relations and Trade of China with the Eastern Archipelago and the Coasts of the Indian Ocean during the Fourteenth Century&quot; in ''T'oung Pao'' 2a ser. vol. 14 (1913) pp. 473-476; vol. 15 (1914) pp. 419-447; vol. 16 (1915) pp. 61-159, 236-271, 374-392, 435-467, 604-626
* Françoise Sabban, &quot;Insights into the problem of preservation by fermentation in 6th century China&quot; in A. Riddervold, A. Ropeid, edd., ''Food conservation'' (Londinii: Prospect Books, 1988) pp. 45-55
* Maxime Rodinson, A. J. Arberry, Charles Perry, ''Medieval Arab Cookery''. Totnes: Prospect Books, 2001
* Roberto Rubino et al., ''Italian Cheese: a guide to its discovery and appreciation''. Bra: Slow Food, 2005
* &lt;span id=&quot;Sanchez (2008)&quot;&gt;&lt;/span&gt;Priscilla C. Sanchez, ''Philippine Fermented Foods: Principles and Technology'' (Manilae: UP Press, 2008) {{Google Books|smfr-KYgtWkC|Paginae selectae}}
* [[Eduardus Hetzel Schafer|Edward Schafer]], ''[[The Golden Peaches of Samarkand]]'' (Berkeleiae: University of California Press, 1963) p. 20
* [[Eduardus Hetzel Schafer|Edward Schafer]], B. E. Wallacker, &quot;Local tribute products of the T'ang dynasty&quot; in ''Journal of Oriental Studies'' vol. 4 (1957/1958) pp. 213-248
* &lt;span id=&quot;Sen (2015)&quot;&gt;&lt;/span&gt;Colleen Taylor Sen, ''Feasts and Fasts: a history of food in India'' (Londinii: Reaktion Books, 2015) pp. 153, 193-195, 204-206
* &lt;span id=&quot;Shurtleff et Aoyagi (2012)&quot;&gt;&lt;/span&gt;William Shurtleff, Akiko Aoyagi, ''History of soy sauce (160 CE to 2012) ... bibliography and sourcebook''. Soyinfo Center, 2012 [https://archive.org/details/bub_gb_-M5t2nXIXi4C/mode/2up Textus] apud ''Internet Archive'' 
* Frederick J. Simoons, ''Food in China: A Cultural and Historical Inquiry'' (CRC Press, 2014) pp. 378-381 {{Google Books|H0JZDwAAQBAJ|Paginae selectae}}
* &lt;span id=&quot;Smith (1918)&quot;&gt;&lt;/span&gt;W. F. Smith, ''Rabelais in his Writings''. Cantabrigiae: Cambridge University Press, 1918 [https://archive.org/details/b2993011x/ Textus] apud ''Internet Archive''
* [[Raimundus Sokolov|Raymond Sokolov]], ''Why We Eat What We Eat. How Columbus Changed the Way the World Eats'' (Novi Eboraci: Simon &amp; Schuster, 1991) pp. 29-31 {{Google Books|EH9LqhXuJ3QC|Paginae selectae}}
* Yan-kit So, ''Classic Food of China'' (Londinii: Macmillan, 1992) pp. 27-28
* &lt;span id=&quot;Spang (2020)&quot;&gt;&lt;/span&gt;Rebecca Spang, ''The Invention of the Restaurant: Paris and modern gastronomic culture''. 2a ed. Cantabrigiae Massachusettensium: Harvard University Press, 2020
* &lt;span id=&quot;Srinivasa Iyengar (1930)&quot;&gt;&lt;/span&gt;&quot;Food&quot; in P. T. Srinivasa Iyengar, ''Pre-Aryan Tamil Culture'' (Maderaspatani: University of Madras, 1930) [https://archive.org/details/prearyantamilculturesrinivasaiyengarp.t._206_u/page/n57/mode/2up pp. 57-65]
* Keith Steinkraus, ''Handbook of Indigenous Fermented Foods'' (2a ed. Novi Eboraci: Marcel Dekker, 1996) {{GB|VKlHKIvrogUC|Paginae selectae}} {{Google Books|AUxaDwAAQBAJ|Paginae selectae}}
* Roel Sterckx, ''Food, Sacrifice, and Sagehood in Early China'' (Cantabrigiae: Cambridge University Press, 2011) pp. 65-76
* Roel Sterckx, ed., ''Of Tripod and Palate: Food, Politics, and Religion in Traditional China''. Londinii: Palgrave Macmillan, 2005
* Timothy J. Tomasik, &quot;Cuisine by the Cut of One's Trousers. Cookbook Marketing in Early Modern France&quot; in ''Food and History'' vol. 14 (2016) pp. 223-247
* &lt;span id=&quot;Tomasik et Albala (2014)&quot;&gt;&lt;/span&gt;Timothy J. Tomasik, interpr.; Ken Albala, ed., ''The Most Excellent Book of Cookery: an edition and translation of the 16th century &quot;Livre fort excellent de cuysine&quot;''. Totnes: Prospect Books, 2014. ISBN 978-1-903018-96-5
* Gillian Riley, &quot;Platina, Martino, and Their Circle&quot; in Harlan Walker, ed., ''Cooks and other people: proceedings of the Oxford Symposium on Food and Cookery'' (Totenais: Prospect Books, 1996) {{Google Books|lpOqTUucwhUC}}
* Andrew Dalby, &quot;Silphium and asafoetida: evidence from Greek and Roman writers&quot; in Harlan Walker, ed., ''Spicing up the palate: proceedings of the Oxford Symposium on Food and Cookery, 1992'' (Londinii: Prospect Books, 1993) pp. 67-72
* Priscilla Bain, Maggie Black, &quot;A Glance at the Life of Sir Kenelm Digby, Knight, at his cookbook called The Closet Opened ...&quot; in Harlan Walker, ed., ''Cooks and Other People: proceedings of the Oxford Symposium on Food and Cookery 1995'' (Londinii: Prospect Books, 1996) {{Google Books|lpOqTUucwhUC|pp. 27-34}}
* William Woys Weaver, interpr., ''Sauer's Herbal Cures: America's First Book of Botanic Healing, 1762-1778''. Novi Eboraci: Routledge, 2001 {{Google Books|OAMA51H9tfUC|Paginae selectae}}
* Paul Wheatley, &quot;Geographical notes on some commodities involved in Sung maritime trade&quot; in ''Journal of the Malayan Branch of the Royal Asiatic Society'' vol. 32 no. 2 (1959) pp. 1-140 [https://www.jstor.org/stable/41503160 JSTOR]
* &lt;span id=&quot;Wheaton (1983)&quot;&gt;&lt;/span&gt;Barbara Ketcham Wheaton, ''Savouring the past'' (London: Chatto &amp; Windus, 1983) p. 128
* &lt;span id=&quot;Zancani (2017)&quot;&gt;&lt;/span&gt;Diego Zancani, &quot;[https://www.tandfonline.com/doi/pdf/10.1080/02614340.2010.11917482 Notes on the vocabulary of gastronomy in literary works from Boccaccio to Giulio Cesare Croce]&quot; in ''The Italianist'' vol. 30 suppl. 2 (2010) pp. 132-148, praecipue pp. 136-137
* Heinrich Zimmer, ''Altindisches Leben'' (Berolini: Weidmann, 1879) [https://archive.org/details/altindischeslebe00zimm/page/272/mode/2up pp. 272-280]
* {{Zohary}} pp. 195-197

== Lexica ==
* &lt;span id=&quot;Allsopp (1996)&quot;&gt;&lt;/span&gt;Richard Allsopp, ''Dictionary of Caribbean English Usage'' (Oxonii: Oxford University Press, 1996) s.v. cabeche
* Mary Ella Milham, &quot;Platina&quot; in {{Arndt}} pp. 288-291
* Albert Barrère, Charles Godfrey Leland, ''A Dictionary of Slang, Jargon and Cant'' (Ballantyne Press, 1889-1890) [https://archive.org/details/adictionaryofsla01barriala 1] [https://archive.org/details/dictionaryofslan02barriala 2]
* Paul Martin Blüher, ''Blühers Rechtschreibung der Speisen und Getränke : alphabetisches Fachlexikon''. 1899 [https://archive.org/details/b21504696 Textus]
* Marianne Bouchon, &quot;[http://documents.irevues.inist.fr/bitstream/handle/2042/2928/04%20TEXTE.pdf Latin de cuisine]&quot; in ''Archivum Latinitatis Medii Aevi'' vol. 22 (1952) pp. 63-76
* &lt;span id=&quot;Cassidy et Le Page (1967)&quot;&gt;&lt;/span&gt;F. G. Cassidy, R. B. Le Page, ''Dictionary of Jamaican English'' (Cantabrigiae: Cambridge University Press, 1967) s.vv. cabeach, scaveeched fish
* &lt;span id=&quot;Davidson (1999)&quot;&gt;&lt;/span&gt;Alan Davidson, ''The Oxford Companion to Food'' (Oxonii: Oxford University Press, 1999. ISBN 0-19-211579-0) pp. 623-624; Tom Jaine, ed., 2a ed. 2006; 3a ed. 2014 
* &lt;span id=&quot;Dinaux (1867)&quot;&gt;&lt;/span&gt;&quot;Jury dégustateur&quot; in Arthur Martin Dinaux; Gustave Brunet, ed., ''Les sociétés badines, bachiques, chantantes et littéraires'' (2 voll. Lutetiae: Bachelin-Deflorenne, 1867) vol. 1 {{Google Books|ew8PAAAAQAAJ|vol. 1 pp. 428-434}} {{GB|jQ8PAAAAQAAJ|vol. 2}}
* Bronwen Percival, &quot;Cheddar&quot;; Val Bines, &quot;Cheddaring&quot; in Catherine Donnelly, ed., ''The Oxford Companion to cheese'' (Novi Eboraci: Oxford University Press, 2016) pp. 131-136
* &quot;Chile&quot; in Georg Friederici, ''Amerikanistisches Wörterbuch'' (Hamburgi: Cram, De Gruyter, 1947) pp. 174-175 {{Google Books|j7J6DwAAQBAJ|Paginae selectae}}
* Charles Perry, &quot;Qaṭāʾif&quot; in Darra Goldstein, ed., ''The Oxford Companion to Sugar and Sweets'' (Oxonii: Oxford University Press, 2015) p. 569
* Samuel Halkett, John Laing; James Kennedy et al., edd., ''[[Dictionary of Anonymous and Pseudonymous English Literature]]'' (editio aucta. 9 voll. Edinburgi: Oliver and Boyd, 1926-1963 [https://archive.org/details/b31359681_0001 1] [https://archive.org/details/b31359681_0002 2] [https://archive.org/details/b31359681_0003 3] [https://archive.org/details/b31359681_0004 4] [https://archive.org/details/b31359681_0005 5] [https://archive.org/details/b31359681_0006 6] [https://archive.org/details/b31359681_0007 7] [https://archive.org/details/b31359681_0008 8] [https://archive.org/details/b31359681_0009 9])
* Richard Hosking, ''A Dictionary of Japanese Food: Ingredients and Culture''. Tuttle, 1996
* Man-kuei Li, &quot;[[:s:en:Page:Eminent Chinese of the Ch'ing Period/Yüan Mei|Yüan Mei]]&quot; in Arthur W. Hummel et al., ''Eminent Chinese of the Ch'ing Period'' (Vasingtoniae, 1943) vol. 2 pp. 955-957
* [[Andreas Dalby|Andrew Dalby]], &quot;Stimulants&quot; in Solomon H. Katz, ed., ''Encyclopedia of Food and Culture'' (Novi Eboraci: Scribner, 2003) vol. 3 pp. 346-349
* &quot;Glossaire technique&quot; in Grégoire Lozinski, ed., ''[[La Bataille de Caresme et de Charnage]]'' (Lutetiae: Honoré Champion, 1933) [https://gallica.bnf.fr/ark:/12148/bpt6k40533/f131 pp. 121-190]
* &quot;[[:s:fr:Page:Michaud - Biographie universelle ancienne et moderne - 1843 - Tome 2.djvu/565|Bacci (André)]]&quot; in Louis-Gabriel Michaud, ed., ''Biographie universelle ancienne et moderne'' 2a ed. vol. 2 (Lutetiae, 1843) p. 560 ([[:s:fr:Biographie universelle ancienne et moderne/2e éd., 1843|index]])
* &lt;span id=&quot;Rey (1998)&quot;&gt;&lt;/span&gt;Alain Rey, ed., ''Dictionnaire historique de la langue française'' (2a ed. Lutetiae: Le Robert, 1998) vol. 1 p. 407
* &quot;Risotto&quot; in Gillian Riley, ''The Oxford Companion to Italian Food'' (Novi Eboraci: Oxford University Press, 2007) pp. 444-445
* Jancis Robinson, ''The Oxford Companion to Wine''. Oxonii: Oxford University Press, 1994. 3a ed., 2006. [https://web.archive.org/web/20080721091842/http://www.winepros.com.au/jsp/cda/reference/oxford_index.jsp?alphabet=b pirated]
* Andrew F. Smith, ed., ''The Oxford Companion to American Food and Drink''. Oxford University Press, 2007 {{Google Books|AoWlCmNDA3QC|Paginae selectae}}
* ''The Wealth of India: Raw Materials''. Dillii, 1948- [https://archive.org/details/in.ernet.dli.2015.46097 A-B] [https://archive.org/details/in.ernet.dli.2015.73201 A-B] [https://archive.org/details/in.ernet.dli.2015.46098 C] [https://archive.org/details/in.ernet.dli.2015.46099 F-G] [https://archive.org/details/in.ernet.dli.2015.23547 H-K] [https://archive.org/details/in.ernet.dli.2015.46100 L-M] [https://archive.org/details/in.ernet.dli.2015.46101 N-Pe] [https://archive.org/details/in.ernet.dli.2015.39929 Ph-Re]; nova editio [https://archive.org/details/wealthofindiadic0000unse Ca-Ci] [https://kipdf.com/the-wealth-of-india-raw-materials-series_5ac5ff871723dddc5d22ae66.html De serie]
* D'Arcy Wentworth Thompson, ''Glossary of Greek Fishes'' (Londinii: Oxford University Press, 1947) p. 132 s.v. &quot;κραγγών&quot;

== Mote ==
* p. 216 ''[[大明會典|Da-Ming huidian]]'' [http://www.chinaknowledge.de/Literature/Historiography/minghuidian.html CK] ''[[:en:Collected Statutes of the Ming Dynasty]]
* p. 216, 219 ''[[明實錄|Ming shilu]]'' [http://www.chinaknowledge.de/Literature/Historiography/mingshilu.html CK] ''[[:en:Ming Shilu]]''
** ''Ming Taizu shilu''
* p. 221, 248 ''[[金瓶梅|Jin ping mei]]'' [http://www.chinaknowledge.de/Literature/Novels/jinpingmei.html CK] ''[[:en:Jin Ping Mei]]'' David Tod Roy translation at LL
* p. 227 [[Jia Ming]], ''[[饮食须知|Yinshi xuzhi]]'' ''[[:de:Yinshi xuzhi (Jia Ming)]]''
* p. 236, 245 [[Gu Qiyuan]] (Ku ch'i-yuan) 1565-1628, ''[[客座赘语|Kezuo zhuiyu]]'' ''K'e-tso chui-yü'' 1618. [[:zh:顾起元]], ''[[:zh:客座赘语]]'' (cf. [http://www.chinaknowledge.de/Literature/Science/shuolve.html CK])

== Bibliographiae ==
* Georges Vicaire, ''Bibliographie gastronomique''. Lutetiae: Rouquette, 1890 {{Google Books|FZY9AAAAYAAJ}} [https://archive.org/details/b21779442 apud ''Internet Archive'']
* [http://terroirs.denfrance.free.fr/p/frameset/07.html Bibliographia Francogallica]

== Achaya ==
* [[Ludovicus Vartomannus]] 1510 ([https://archive.org/details/bub_gb_vwzaAGAfAKMC Italiane] [https://archive.org/details/itinerario00vartuoft reimpressum] [http://reader.digitale-sammlungen.de/de/fs1/object/display/bsb11196587_00001.html versio Latina] [https://archive.org/details/travelsofludovic00vartrich versio Jones &amp; Badger pp. 15, ]) [[:en:Ludovico di Varthema]]
* [[Dominicus Paes]] c. 1522 ([https://archive.org/stream/chronicadosreisd00nune#page/80/mode/2up Lusitane] [https://archive.org/details/aforgottenempir00paesgoog Sewell pp. 237, 249, 255-6, 258-9]). [[:en:Domingo Paes]]. See [https://www.academia.edu/838106/Travel_and_ethnology_in_the_Renaissance_South_India_through_European_eyes_1250-1625 Rubiés] p. 20 and passim
* [[Ferdinandus Nuniz]] c. 1537 ([https://archive.org/details/itinerario00vartuoft Lusitane] [https://archive.org/details/aforgottenempir00paesgoog Sewell] p. 375). [[:en:Fernão Nunes]]. See [https://www.academia.edu/838106/Travel_and_ethnology_in_the_Renaissance_South_India_through_European_eyes_1250-1625 Rubiés] p. 20 and passim
* [[Antonio Monserrate]] c. 1580 ([https://archive.org/stream/memoirsofasiatic03asia#page/n597/mode/2up text] [https://archive.org/details/commentaryoffath00monsuoft transl. pp. 25, 149, 199, 214] [[:fr:Antonio Monserrate]]
* [[Radulphus Fitch]] c. 1591 ([https://archive.org/details/earlytravelsinin00fostuoft Foster] pp. 38 [Chiangmai], 45-47) [[:en:Ralph Fitch]]
* [[Gulielmus Finch]] c. 1611 ([https://archive.org/details/earlytravelsinin00fostuoft Foster] pp. 150-151, 166) [[:en:William Finch (merchant)]]
* [[Eduardus Terry]] c. 1619 ([https://archive.org/details/earlytravelsinin00fostuoft Foster] p. 297) and in ''Purchas his Pilgrimes'' [[:en:Edward Terry (author)]]

== Vide ==
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      <text bytes="2194" xml:space="preserve">[[Fasciculus:USNS Vanguard.jpg|thumb|[[USNS Mission San Fernando (T-AO-122)|USNS ''Vanguard'']] speculatoria quae etiam caelum indigat.&lt;!--While [[USNS Mission San Fernando (T-AO-122)|USNS ''Vanguard'']] was not strictly a spy ship, being used for space tracking, there is some overlap between her capabilities and those of a spy ship.-- &gt;{{In  progressu}}A spy ship or reconnaissance vessel is a dedicated ship intended to gather intelligence, usually by means of sophisticated electronic eavesdropping. In a wider sense, any ship intended to gather information could be considered a spy ship.
Spy ships are usually controlled by a nation's government, due to the high costs and advanced equipment required. They tend to be parts of the nation's navy, though they may also be operated by secret services.

A exploratorem Navis vel vasis ad exploranda circa dedicated est in animo navem colligentes intelligentia, plerumque per sophisticated electronic auscultatio. In latiori sensu, quid in animo navem exploratorem Navis colligentes notitia considerari potest.
Exploratorem sunt naves plerumque regit imperium de gente, ex provectus armorum altum sumptibus et requiritur. Tendunt partes gentis classis quoque licet occultus operatur elit. --&gt; ]] 
'''Speculatoria'''&lt;ref&gt;{{TraupmanLatinEng3|397}}&lt;/ref&gt; est navis dedicata &lt;!--cui consilium est --&gt; indicia colligere speculatoris, saepe per auscultationem electronicam adparatam. In lato sensu, ulla [[navis]] consilio indicia colligere haberi potest speculatoriam esse. Speculatoriae saepe gubernari [[rectio|dicione]] civitatis, enim sumptuosae sunt et exigunt provectum artificiosum. Plerique sunt partes [[Classis (nautica)|classicarii]] nationis, quamquam possunt etiam gubernari ab ministerio curiosi.
== Dictum ==
&quot;Quod cum animadvertisset Caesar, scaphas longarum navium, item speculatoria navigia militibus compileri iussit et, quos laborantes conspexerat, his subsidia submittebat.&quot;
== Bibliographia ==
&lt;references /&gt;
*[https://www.encyclopedia.com/politics/encyclopedias-almanacs-transcripts-and-maps/ships-designed-intelligence-collection De speulatoria ex situ encyclopedia.com]
[[Categoria:Genera navium]]
[[Categoria:Speculatio]]</text>
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  <page>
    <title>Vicipaedia:Pagina prima/Pagina cottidiana</title>
    <ns>4</ns>
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      <id>3697670</id>
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      <contributor>
        <username>Giorno2</username>
        <id>30162</id>
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      <text bytes="200" xml:space="preserve">[[File:San Francesco d'Assisi in preghiera.jpg|right|140px]]
'''[[Ordines Franciscani]]''', &lt;br /&gt;Instituts religiosa&lt;div style=&quot;clear:both;&quot;&gt;&lt;/div&gt;&lt;noinclude&gt;
[[Categoria:Pagina prima]]

&lt;/noinclude&gt;</text>
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      <id>3697673</id>
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        <username>Giorno2</username>
        <id>30162</id>
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      <text bytes="200" xml:space="preserve">[[File:San Francesco d'Assisi in preghiera.jpg|right|140px]]
'''[[Ordines Franciscani]]''', &lt;br /&gt;Instituta religiosa&lt;div style=&quot;clear:both;&quot;&gt;&lt;/div&gt;&lt;noinclude&gt;
[[Categoria:Pagina prima]]

&lt;/noinclude&gt;</text>
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      <text bytes="200" xml:space="preserve">[[File:San Francesco d'Assisi in preghiera.jpg|right|140px]]
'''[[Ordines Franciscani]]''', &lt;br /&gt;instituta religiosa&lt;div style=&quot;clear:both;&quot;&gt;&lt;/div&gt;&lt;noinclude&gt;
[[Categoria:Pagina prima]]

&lt;/noinclude&gt;</text>
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  <page>
    <title>Vicipaedia:Index paginarum cottidianarum</title>
    <ns>4</ns>
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    <revision>
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      <contributor>
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      <text bytes="1651" xml:space="preserve">&lt;div style=&quot;-moz-box-shadow:10px 10px 10px #CCC; -webkit-box-shadow:6px 6px 6px #CCC; box-shadow:8px 8px 8px #CCC; float:right; border:1px solid #BEBEBE; padding:1em&quot;&gt; 
{{Vicipaedia:Pagina prima/Capsa prima| titulus = [[Vicipaedia:Pagina prima/Pagina cottidiana|Pagina cottidiana]] | color tituli = white | color vexilli = #D61D00 | margo cacuminis = 0 | corpus = {{Vicipaedia:Pagina prima/Pagina cottidiana}}}}
&lt;/div&gt;

Ut paginam cottidianam hodiernam promoveas, [[Vicipaedia:Pagina prima/Pagina cottidiana|formulam paginae cottidianae]] in fenestram adiunctam aperi, duobusque locis emenda.
# Loco tituli anterioris titulum novum insere
# Loco imaginis anterioris imaginem novam a te selectam insere. Cura ut faciem imaginis ad paginam primam restringas: non |thumb|, non |upright|, sed (exempli gratia) |right|100px|
Quo facto, titulum paginae a te promotae e rubrica inferiori &quot;Promovendae&quot; in subpaginam &quot;[[Vicipaedia:Index paginarum cottidianarum/Anno 2022 promotae]]&quot; move. Rem perfecisti!
== Promovendae == 
* [[Capsicum annuum var. glabriusculum]]
* [[Carmen solutum]]
* [[Desertum Victoria]]
* [[Dionysia]]
* [[Fasti regum Lydiae]] 
* [[Fiunt oratores: nascuntur poëtae]]
* [[Heraclidae (Euripides)]]
* [[Iugum Atlanticum medium]]
* [[Supplices (Euripides)]] 
* [[Tiresias]]
* [[Valerius Cordus]]

== Iam promotae ==
* [[Vicipaedia:Index paginarum cottidianarum/Anno 2019 promotae]]
* [[Vicipaedia:Index paginarum cottidianarum/Anno 2020 promotae]]
* [[Vicipaedia:Index paginarum cottidianarum/Anno 2021 promotae]]
* [[Vicipaedia:Index paginarum cottidianarum/Anno 2022 promotae]]

[[Categoria:Pagina prima]]
[[Categoria:Paginae honoratae]]</text>
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    <revision>
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      <contributor>
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      <comment>/* Promovendae */</comment>
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{{Vicipaedia:Pagina prima/Capsa prima| titulus = [[Vicipaedia:Pagina prima/Pagina cottidiana|Pagina cottidiana]] | color tituli = white | color vexilli = #D61D00 | margo cacuminis = 0 | corpus = {{Vicipaedia:Pagina prima/Pagina cottidiana}}}}
&lt;/div&gt;

Ut paginam cottidianam hodiernam promoveas, [[Vicipaedia:Pagina prima/Pagina cottidiana|formulam paginae cottidianae]] in fenestram adiunctam aperi, duobusque locis emenda.
# Loco tituli anterioris titulum novum insere
# Loco imaginis anterioris imaginem novam a te selectam insere. Cura ut faciem imaginis ad paginam primam restringas: non |thumb|, non |upright|, sed (exempli gratia) |right|100px|
Quo facto, titulum paginae a te promotae e rubrica inferiori &quot;Promovendae&quot; in subpaginam &quot;[[Vicipaedia:Index paginarum cottidianarum/Anno 2022 promotae]]&quot; move. Rem perfecisti!
== Promovendae == 
* [[Bessastadae]]
* [[Capsicum annuum var. glabriusculum]]
* [[Carmen solutum]]
* [[Desertum Victoria]]
* [[Dionysia]]
* [[Fasti regum Lydiae]] 
* [[Fiunt oratores: nascuntur poëtae]]
* [[Heraclidae (Euripides)]]
* [[Iolaus]]
* [[Iugum Atlanticum medium]]
* [[Supplices (Euripides)]] 
* [[Tiresias]]
* [[Valerius Cordus]]

== Iam promotae ==
* [[Vicipaedia:Index paginarum cottidianarum/Anno 2019 promotae]]
* [[Vicipaedia:Index paginarum cottidianarum/Anno 2020 promotae]]
* [[Vicipaedia:Index paginarum cottidianarum/Anno 2021 promotae]]
* [[Vicipaedia:Index paginarum cottidianarum/Anno 2022 promotae]]

[[Categoria:Pagina prima]]
[[Categoria:Paginae honoratae]]</text>
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  </page>
  <page>
    <title>Candida Siu</title>
    <ns>0</ns>
    <id>289571</id>
    <revision>
      <id>3697712</id>
      <parentid>3541180</parentid>
      <timestamp>2022-08-17T01:42:13Z</timestamp>
      <contributor>
        <username>Uriel1022</username>
        <id>61003</id>
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      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1655" xml:space="preserve">[[Fasciculus:Candida Xu.jpg|thumb|Candida Siu]]
'''Candida Siu'''&lt;ref&gt;Rarissima est Candida natu Sinica quae [[consuetudo nominationum Sinarum|Sinicum nomen]] non habebat, sed Latinum tantum; idcirco vocata est a Sinicis {{lang|zh|徐甘地大}} ''{{lang|zh-Latn|Xú Gāndìdà}}'', quod eorum sermone enuntiatum simile videtur ac Latinum nomen.&lt;/ref&gt; ([[Sinice]] {{lang|zh|徐甘第大}}; nata {{die|4|9|1607}}, mortua {{die|24|7|1680}}), [[Sinae (gens)|Sinica]] patrona [[ecclesia Catholica|Catholicae ecclesiae]], [[Paulus Siu|Pauli Siu]] neptis, [[Philippus Couplet|Philippi Couplet]] discipula ac fautrix.

Nata die natali [[Sancta Candida senior|Sanctae Candidae senioris]], cuius gratia est nominata, a teneris unguiculis vitam pro salute ecclesiae ea in regione vovens privata cura impensaque et ecclesias aedificavit ac sodalibus [[societas Iesu|societatis Iesu]] quicumque [[Sciamhaevum|Sciamhaevi]] erant auxilium praebuit, quorum libros Sinice edidit; propter res gestas ab Europaeis Catholicis ut apostola Sinica laudabatur, cum Philippus Couplet confessor eius librum titulo ''Historie van eene groote christene Mevrouwe van China met naeme Mevrouw Candida Hiu'' (id est ''Historia probae Christianae feminae Sinarum, nomine Candida Siu'') publicasset.&lt;ref name=&quot;Anderson1999&quot;&gt;{{cite book|author=Gerald H. Anderson|title=Biographical Dictionary of Christian Missions|url=https://books.google.com/books?id=oQ8BFk9K0ToC&amp;pg=PA596|year=1999|publisher=Wm. B. Eerdmans Publishing|isbn=978-0-8028-4680-8|page=752}}.&lt;/ref&gt;

==Notae==
&lt;references /&gt;

{{NexInt}}
* [[Matthaeus Riccius]]
* [[Michael Shen]]

{{Lifetime|1607|1680}}
[[Categoria:Feminae]]</text>
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  <page>
    <title>Ponto</title>
    <ns>0</ns>
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      <id>3697757</id>
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      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Naves]]; +[[Categoria:Genera navium]]</comment>
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      <text bytes="1471" xml:space="preserve">{{L|-1}}
[[Fasciculus:Cambridge uni math bridge.JPG|thumb|Pontones [[Cantabrigia]]e.]]
'''Ponto'''&lt;ref&gt;{{TLL}} {{DuCange}} s.v. &quot;pontones&quot;. {{DMLBS|ponto}}&lt;/ref&gt; est unum ex diversis generibus [[navigium|navigiorum]].

Forma pontonis Romanis antiquis noti non manifesta est. [[Iulius Caesar]] in commentariis ''[[De bello civili]]'' scripsit pontonem esse &quot;genus navium [[Galli]]carum&quot;. Adhibebantur ad flumina transeunda, ut [[navis traiectoria]]. Fortasse plurima navigia coniungebatur ad [[pons|pontem]] temporarium creandum. Vocabulum a ''ponte'' derivatum est ''pontonium'', quod [[Isidorus]] scripsit esse &quot;navigium fluminale tardum et grave, quod non nisi [[remigium|remigio]] progredi potest&quot;.&lt;ref&gt;{{TLL}}&lt;/ref&gt;

Apud [[Angli|Anglos]] hodiernos ponto (''punt'') est navigium alveo plano et prurâ quadratâ, [[contus|conto]] propulsus. Eos comune utuntur Cantabrigiae Oxoniaeque vehere turistas aut pupilos cuisque Universitatis. Profecto dum pupili Universitatis finem examinum suorum celebrant a nomine [[Hebdomada Mai]] Anglice &quot;May Week&quot; saepe ii utituntur pontes ad navidandum per flumen Grantam aut ad Castras Grantae Anlgice &quot;Granchester&quot; aut centrum urbi veheri. Discrimen autem est inter modos utiendi inter quidamque universitatem nam Cantabrigiani puppi navem gubernant et Oxonienses prora naveque inversa.

== Notae ==
&lt;references /&gt;

{{NexInt}}
* [[Scapha]]
* [[Cymba]]

== Nexus externi ==
{{CommuniaCat|Punts|pontones}}

[[Categoria:Genera navium]]</text>
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  <page>
    <title>Shrek (2001 pellicula)</title>
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        <username>Turpilius Morologus</username>
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      <text bytes="1175" xml:space="preserve">{{Vicificanda}}
{{Pagina non annexa}}
{{Latinitas|-2}}
''Shrek'' (Anglice; Latine, ''Shrek'') est [[pellicula]], directa a moderatore Andreas Adamson et soluta in [[Annus|anno]] 2001. Haec pellicula [[Fabula|fabulam]] de belua, quae heroicum iter facit ut [[Principissa|principisam]] liberet narrat. ''Shrek'' tantam prosperitatem apud spectatores habuit ut DreamWorks tres alias pelliculas fecerit.

== Principales dramatis personae ==
* Mike Myers: Shrek, [[viridis]] belua qui egreditur ut Principissa liberet et paludem eius servet
* Eddie Murphy: Asinus, fortissimus et heroicus [[Equus asinus|asinus]]
* Cameron Diaz: Principissa Fiona, regina quae [[Nox|nocte]] belua fit
* John Lithgow: Dominus Farquaad, homunculus qui Fionae nubere vult
Thelonius: Tortor et carnifex verisimilis, Gingerbread cruciatus, et magicae speculum frangere minatus est, si reginae non ostendisset.Tandem cum Gingerbread amicitia factus est.

== Nexus externi ==
* [[imdbtitle:0126029|Res de hac pellicula]]
* [https://www.rottentomatoes.com/m/shrek Sententiae de hac pellicula]

[[Categoria:Pelliculae Civitatum Foederatarum]]
[[Categoria:Pelliculae 2001]]
[[Categoria:Pelliculae animatae]]</text>
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    <title>Usor:Tchougreeff/QUOMODO sive HOW TO/ELEMENTORUM PHYSICAE MATHEMATICAE VOLUMEN PRIMUM AUCTORE ANDREA CARAFFA E SOCIETATE JESU IN COLLEGIO ROMANO PHYSICAE MATHEMATICAE PROFESSORE</title>
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      <text bytes="729247" xml:space="preserve">== PRAEFATIO ==
Rerum naturalium ordinem considerare, Deumque in iis mirifice operantem intueri, proprium est verae sapientiae, quam Philosophia profitetur. Haec scientia, quae dicitur Physica , inter scientias homine dignissimas. atque inter praecipua Dei dona jure commendatur: ecquid enim potest esse praestantius aut utilius quam divinae sapientiae opera, Deumque ipsum suas in natura perfectiones ostentantem contemplari? An quod Deus omnipotentia sua non judicavit indignum in iis quae creavit , quod in iis quae regit et gubernat attentione sua dignatur Providentia Dei, hoc nos meditari supervacaneum atque otiosum iudicabimus? Otiosam illam dicerem Physicam, quae ita immoraretur in Operis consideratione, ut opificis non perpetue suspicere! industriam: caecus est, qui Deum non videt in natura ejusque providentiam ac sapientiam non admiratur. Similem illum dixerim homini, qui librum ob Oculos apertum tenens characterum elegantiam contemplatur, numerat verba; sensum non penetrat. ⋅Neque vero minus utilis Naturae cognitio ministris Ecclesiae quam caeteris hominibus existimanda est: imo et hanc ipsis maxime necessariam duxerim hoc praesertim tempore cum homines vano inflati doctrinae apparatu scientias pro viribus adversus Religionem convertant , et Phyicam praecipue revelationi satagant opponere , vereque Opponi non desinant clamare eoram ignaris. Cum igitur se linguae impiae in injuriam Religionis armant, pudeat hominem Religionis amantem, et eo charactere insignitum qui ipsum Religionis statuat defensorem, aut turpiter obmutescere, aut Religionem. male defensam hominibus impiis vanum jactantibus triumphum, et ministrorum ignorantiam in Religionis opprobrium vertentium, deridendam proponere. Quod si nihil a viro ecclesiastico quaereretur aliud in Physica quam honesta mentis recreatio, justaque serii laboris intermissio, quid potest esse homini laboribus sacris defuncto aut jucundius aut dignius quam otium inutile, ac saepae periculosum, otio erudito et physico commutare? Quam multa offeret naturae spectaculum , ipsiusque arcanorum inquisitio, quae laudabilem nutriant curiositatem ,utiles praebeant considerationes, suaves atque innoxias pariant delicias! Etenim nullam esse in Philosophia partem quae majori voluptate quam naturae contemplatio animum compleat, Tullii auctoritate facile possem confirmare. Haec nihilominus voluptas non sine studio ac labore comparari potest: sapientissimus enim rerum omnium creator et rector Deus, quamvis sensuum nostrorum perceptioni corpora haec sensibilia subjecerit, illorum tamen naturam et vim mira quadam sepsit caligine, ut quicumque ad eam penitus scrutandam accedunt, in media luce densis veluti tenebris repente se obrui sentiant; quas tenebras etiamsi non ita discutere possimus ut claram ac certam rerum omnium scientiam assequamur, attamen si nos studii, diligentiae ac laboris non piguerit, ita tenebras attenuari experiemur ut multarum rerum certam cognitionem , plurimarum admodum probabilem obtineamus . Ad occulta Naturae arcana inquirenda duae sunt viae, quas eximii ingenii vir Franciscus Baconus de Verulanio notavit in novo scientiarum organo lib . 1. aphor, 19. Prima, qua a sensu et particularibus incipientes advolamus ad axiomata maxime generalia; atque ex iis principiis, eorumque immota veritate judicamus et invenimus axiomata media . Altera a sensu et particularibus excitat axiomata ascendendo continenter et gradatim , ut ultimo loco perveniatur ad ma
xime generalia. Primam viam plures arripuerunt, qui conjecturas non admodum graves sequuti , atque experientia
non satis accurata innixi generalia axiomata nimia festina
tione constituerunt , iisque naturalium causarum et effe
ctyum vim omnem contineri voluerunt; atque in iis tuen
∼∣⋁in Physica quam honesta mentis recreatio, iustaque serii
laboris intermissio, quid potest esse homini laboribus sacris defuncto aut iucundius aut dignius quam Otium inutile,
ac saepae periculosum, Otioterudito et physico commutare?
Quam multa offeret naturae speCtaculum, ipsiusque arca-
norum inquisitio, quae laudabilem nutriant curiositatem,
utiles praebeant considerationes, suaves atque innoxias pariant delicias! Etenim nullam esse in Philosophia partem
quae majoriivoluptate quam naturae contemplatio animum
compleat, Tullii auctoritate facile possem confirmare.
Haec nihilominus voluptas non' sine studio. ac labore
Comparari potest: sapientissimus enim rerum omnium creator
et rector Deus, quamvis sensuum nostrorum perceptioni
corpora haec sensibilia subiecerit, illorum tamen naturam
et vim miraaquadam sepsit caligine, ut quicumque ad eam
penitus scrutantium accedunt, in media luce densis veluti
tenebris repente se obrui sentiant; quas tenebras etiamsi
non ita discutere possimus ut claram ac certam rerum o-
mnium scientiam assequamur, attamen si nos studii. dili-
gentiae ac laboris non piguerit, ita tenebras attenuari ex-
periemur ut multarum rerum certam-cognitionem , pluri-
marum admodum probabileur Obtineamus. Ad Occulta Na-
turae arcana inquirenda duae sunt viae, quas eximii inge-
nii vir Franciscus Baconus de,.Verulamio notavit&quot; in novo
scientiarum organo lib. ∎∙ aphor. 19. Prima, qua a sensu
et particularibus incipientes advolamus.ad axiomata-; mas-
xime generalia; atque ex iis principiis, eorumque-[immota
veritate iudicamus et invenimus axiomata 'media. :Altera'a
sensu et particularibus excitat axiomata ascendendo contio
nenter et gradatim, ut ultimo loco perveniatur adfusa-i-
xime generalis. Primam viam plures arripueruut, qui' con-
iecturas non admodum graves,,s'equuti , atque experientia
non satis accurata innixi generalia axiomata nimia festina-
tione constituerunt,, iisque naturalium causarum et eil'e-
ctuum vim omnem contineri voluerunt; atque in iis tuen-dis totam ingenii aciem intendentes inciderunt in perver
sam philosophandi rationem , adeo ut rerum universitatem
commenti sint omnino aliam ac éa est. Altera aliis placuit
via, qui rerum naturam in rebus ipsis longa observatione
atque accurata experientia quaerendam esse statuerunt; isti
effectuum et causarum naturalium indolem singillatim in
quirere coeperunt, corporum texturám intimam , configu
rationem, motum scrutati sunt; atque ex his; aliisque in
numeris diligentissime observatis Naturae leges deduxerunt,
verasque rerum causas deprehenderunt. ' Hoc pacto plura
nostris temporibus certissima sunt , quae olim ignoraban
tur : alia probabili conjectura assecuti sumus : adhuc ta
men non pauca restant ambigua et involuta ; sed non de
erunt fortasse, qui aliquando novas in lucem eruant veritates.
Notandum vero Naturam ubique mathematicam esse ,
eamque velle absque Mathesi expiscari perinde fore, ait Gul
lielminus , ac sine cruribus ambulare. Porro tota Naturae
compago soliditate constal geometrica, resque physica rei geo
metricae unitur mystico quodam nexu, quem soli mathe
maticae Analysi datum est reserare: Analyseos ductu ex ob
servationibus et experimentis ab iis quae in promptu sunt
ad reconditiora deferimur, et ab exteriori venustate ad in
ternos naturae sinus. Observationes quidem virium exi
stentiam demonstrant, sed proprium est Analyseos pate
facere virium leges, curvas a corporibus in gyrum actis
descriptas, circumvolutionum ac motuum inaequalitates et
aberrationes; quae omnia aspectabilem hanc Mundi ma
chinam maxime illustrant . Quid ab Analyseos indole ma
gis abhorrere videbatur quam electricorum phoenomenorum
congeries, cum et innumera prope sint, et innumeris ob
noxia conditionibus? Ad electricas tamen vires expenden
das accessit Analysis , earumque non paucos effectus leges
que aequationibus definivit.
Ut Tyronum , qui physicis praelectionibus in Romano
Soc. Jesu Collegio dant operam, commodo utilitatique ser
'
dis totam ingenii aciem intendentes inciderunt 'inrïperwe'r.»
sam philosophandi rationem, adeo ut rerum.:nniversitatem
commenti sint omnino aliam-ac ea est. .Altera aliis placuit
via, qui rerum naturamin: rebus ipsis longa-ObservatiOne
atque- accurata - experientia quaerendam, 'esse' statuerunt: :.i'sd
effectuum. ïet; causarum. naturalium 'indolem tsin'gillat'im in—
quirere coeperunt, corporumf-textuttam--imimdmf, configu-
rationem, motum scrutati sunt; atque ex his, aliisque-.in-
numeris diligentissime observatis Naturae leges deduxerunt,
verasque rerum causas deprehenderunt. 'Hoc pacto plura
nostris tempOribus certissima sunt, quae Olim ignoraban-
tur: alia probabili coniectura assecuti sumus : adhuc ta-
men non pauca restant— ambigua et.-involuta; sed non de-
erunt fortasse, qui aliquando novas in lucem eruant veritates.
Notandum vero Naturam ubique mathematicam esse,
eamque velle absque Mathesi expiscari perinde fore, ait Gul-
lielminus, ac sine cruribus ambulare. Porro tota Naturae
compago soliditate constat geometrica, resque physica rei geo-
metricae unitur mystico quodam nexu, quem soli mathe-
maticae Analysi datum est reserare: Analyseos ductu ex Ob-
servationibus et experimentis ab iis quae in promptu sunt
ad reconditiora deferimur, et ab exteriori venustate ad in-
ternqs naturae sinus. Observationes quidem virium exi-
stentiam demonstrant, sed prOprium est Analyseos pate-
facere virium leges, curvas a corporibus in gyrum actis
descriptas, circumvolutionum se motuum inaequalitates et
aberrationes; quae omnia aspectabilem hanc Mundi mn-
chinam maxime illustrant. Quid ab Analyseos indole ma-
gis abhorrere videbatur quam electricorum phoenomenorum
congeries, cum et innumera prope sint, et innumeris Ob-
noxia conditionibus? Ad electricas tamen vires eXpenden-
das accessit Analysis, earumque non paucos eil'ectus' leges-
que aequationibus definivit.
Ut Tyronum, qui physicis praelectionibus in Romano
Soc. Iesu Collegio dant operam, commodo utilitatique ser-VI
viam , in lucem edo illas Physicae partes , quae intimiori
vinculo nectuntur mathesi, quarumque explicandarum cu
ra mihi est demandata. A Mechanica exordior ; siquidem
reliquarum est veluti basis et fundamentum : caeterum sic
Physicae mathematicae tractationem concinnavi, ut quae aste
risco inveniuntur signata, possint ab iis Tyronibus omitti,
qui primo duntaxat Philosophiae anno mathematicis insti
tutionibus studuerunt.
VI
viam
,
in lucem edo illas Physicae partes
, quae intimiori
vinculo nectuntur mathesi, quarumque explicandarum cu-
ra mihi est demandata. A Mechanica exordiar.; siquidem
reliquarum est veluti basiset fundamentum: caeterum sic
Physicae mathematicae tractationem concinnavi, ut quae aste-
risco inveniuntur signata, pOssint ab iis Tyron'ibus omitti,
qui primo duntaxat Philosophiae anno mathematicis insti-
tutionibus studuerunt.

== MECHANICAE PRINCIPIA ==

=== Notiones Praeambulae ===
[[1|1]]. Moto puncto materiali, si ratio inter numericos spatii percursi ac respondentis temporis valores &lt;math&gt;s&lt;/math&gt; ac &lt;math&gt;t&lt;/math&gt; permanet eadem, motus dicitur uniformis; quod si ratio illa jugiter mutetur, motus dicitur varius, acceleratus nempe vel
retardatus, prout crescente
e crescit vel decrescit ipsa &lt;math&gt;\frac s t &lt;/math&gt;: porro motus rectilineus atque uniformis est simplicissimus
omnium motuum, quorum exsistit capax punctum materiale. In &lt;u&gt;motu uniformi&lt;/u&gt; ratio &lt;math&gt;\frac s t &lt;/math&gt; dicitur &lt;u&gt;velocitas&lt;/u&gt;; qua designata per &lt;math&gt;v&lt;/math&gt;, erit

&lt;math&gt;v = \frac s t .&lt;/math&gt;

Quoad punctum materiale, cujus massa seu quantitas materiae (&lt;math&gt;=m&lt;/math&gt;), et velocitas &lt;math&gt;=v&lt;/math&gt;, factum &lt;math&gt;mv&lt;/math&gt; appellatur quantitas motus.

[[2|2]]. Corpus de se est indifferens ad motam et ad quietem. Haec indifferentia sic probari potest ex natura loci: nequit corpus de se postulare at localiter moveatur nisi exigat natura sua non esse in loco ubi est, et locum in quo non est occupare; contra nequit corpus de se quietem
exigere nisi exigat natura sua esse potius in loco ubi est
quam in loco quem occuparet si moveretur. Neutrum vero ex natura sua exigit corpus; cum enim omnia loca sint
ejusdem rationis, jam nulla datur ratio cur corporea substantia exigat esse potius uno in loco quam in alio: propterea etc.

[[3|3]]. Quae causae motum gignunt, accelerant, retardant, detorquent, eae vocantur potentiae seu vires. Plures potentiae corpori aut corporum systemati applicitae sese ita possunt impedire, ut nullus inde oriatur motus; tunc vero potentiae dicuntur constitutae in aequilibrio. Fac ut duae vires punctum materiale sollicitent in partes contrarias ; si
eae sunt in aequilibrio, dicentur aequales: pone duas, tres
etc. . . : . ex ejusmodi viribus aequalibus applicari puncto materiali ita , ut in unam eamdemque rectam conspirent; inde habebis vim duplam , triplam etc. . . . Poterunt nempe vires omnes exprimi per numeros ; et consequenter repraesentari per lineas rectas istis numeris proportionales, quarum directiones cum ipsarum virium directionibus congruant. Mechanica tota est in aequilibrii ac
motus doctrina consideranda.

[[4|4]]. Finge tibi globum &lt;math&gt;A&lt;/math&gt; quiescentem e filo pendulum,
in quem impingat globus &lt;math&gt;B&lt;/math&gt; cum certo quodam velocitatis
gradu. Si nullam motui resistentiam afferret globus &lt;math&gt;A&lt;/math&gt;, eadem velocitate pergeret moveri &lt;math&gt;B&lt;/math&gt;, qua movebatur antea , secum pertrahendo globum &lt;math&gt;A&lt;/math&gt;: cur enim minueretur motus in &lt;math&gt;B&lt;/math&gt;, cum globus &lt;math&gt;A&lt;/math&gt; nihil obstaret illius motui , et ipse loco suo facile cederet? Iamvero si experientiam consulimus, multo aliter rem evenire comperiemus: cedit quidem loco suo globus &lt;math&gt;A&lt;/math&gt;, sed non sine detrimento motus
in &lt;math&gt;B&lt;/math&gt;, eoque majori quo majorem globus &lt;math&gt;A&lt;/math&gt; opponit massam impellenti se globo &lt;math&gt;B&lt;/math&gt;. Resistere igitur motui , status
que mutationi obniti concipitur &lt;math&gt;A&lt;/math&gt;, acquisitumque motum
resistentia sua destruere in &lt;math&gt;B&lt;/math&gt;. Motus habetur tamquam vis
activae effectus; quod autem vis activae effectum destruit,
potest et ipsum verae vis nomen accipere. In ipsis etiam
corporibus motis sese prodit ejusmodi vis: corpus certo quodam velocitatis gradu donatum, eumdem servabit nisi quem
inveniat obicem , nec ullum sui motus augmentum patietur nisi cum vis alienae in ipsum agentis detrimento; haud
aliter ac restitit primo motui dum quiesceret; ipso in motu resistit majori motui.

Non ergo praefata indifferentia sita est in non renitentia ad motum ex quiete, vel in non renitentia ad quietem ex mota, sed in eo quod corpus de se non magis ad
motum quam ad quietem tendat, nec magis resistat quieti
si fuerit in motu quam molui renitatur si fuerit in quiete.
Quoniam igitur ab ipsa materia nequit oriri ulla de
terminatio ( huc pertinet materiae inertia ) ad novum statum sive quietis, sive motas; profecto deficiente causa quae
materiale punctum determinet ad hunc potius quam ad
illum novum statum, punctum ipsum si in quiete sit quiescet semper, si ad motum semel fuit excitatum perget moveri cum eadem perpetuo velocitate et directione: porro
motus directio est recta linea, quam mobile aut describit,
aut describere nititur; primum obtinet in motibus rectilineis, secundum in curvilineis. 

[[5|5]]. Duo puncta materialia &lt;math&gt;H&lt;/math&gt; et &lt;math&gt;K&lt;/math&gt; ( fig 1. ) eamdem
massam habentia, eamdemque lineam communi vi &lt;math&gt;P&lt;/math&gt; incedentia, haud dubie conjunctim procedent: verum ubi puncto &lt;math&gt;K&lt;/math&gt; praeter &lt;math&gt;P&lt;/math&gt; applicetur et vis &lt;math&gt;Q&lt;/math&gt;, disjungetur illico &lt;math&gt;K&lt;/math&gt;
ab &lt;math&gt;H&lt;/math&gt;, et observator constitutus in &lt;math&gt;H&lt;/math&gt; deprehendet: motum puncti &lt;math&gt;K&lt;/math&gt; perinde ac deprehenderet si &lt;math&gt;H&lt;/math&gt; quiesceret et &lt;math&gt;K&lt;/math&gt;
moveretur sola &lt;math&gt;Q&lt;/math&gt;: sive nimirum ponatur &lt;math&gt;H&lt;/math&gt; moveri vi &lt;math&gt;P&lt;/math&gt;
et &lt;math&gt;K&lt;/math&gt; viribus &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt;, sive &lt;math&gt;H&lt;/math&gt; quiescere et &lt;math&gt;K&lt;/math&gt; moveri unica
&lt;math&gt;Q&lt;/math&gt;, idem in utroque casu, experientia teste , prodibit motus puncti &lt;math&gt;K&lt;/math&gt; quoad &lt;math&gt;H&lt;/math&gt;: huc spectat principium motus
relativi . Jamvero in secundo casu motus relativus soli &lt;math&gt;Q&lt;/math&gt; est
manifeste, adscribendus; idipsum ergo dicendum et in primo. Effectus videlicet a nova vi &lt;math&gt;Q&lt;/math&gt; genitus in puncto materiali &lt;math&gt;K&lt;/math&gt; idem est utcumque caeteroqui se habeat praecedens status ipsius &lt;math&gt;K&lt;/math&gt;: quod consequi videtur ex materiei
inertia. Etenim si variato statu praecedente variaret effectus ille, non aeque se haberet materia ad status omnes ,
punctumque materiale sibi commissum rediret tandem in
statum illum , ad quem magis tendit; sicque ab ipsa materia oriretur determinatio ad novum statum.

[[6|6]]. Exhibeant
&lt;math&gt;v&lt;/math&gt; et
&lt;math&gt;v^\prime&lt;/math&gt; velocitates, quas gignunt vires
&lt;math&gt;P&lt;/math&gt; et
&lt;math&gt;Q&lt;/math&gt;, sitque
&lt;math&gt;u&lt;/math&gt; velocitas
, quam generat vis assumpta pro
communi mensura (3) ipsarum
&lt;math&gt;P&lt;/math&gt; et
&lt;math&gt;Q&lt;/math&gt;; erunt (5)
&lt;math&gt;v = Pu,
v^\prime = Qu&lt;/math&gt;, unde:
&lt;math&gt;v:v^\prime=Pu: Qu=P: Q&lt;/math&gt;.
Permanente videlicet massa, vires erunt ut simplices velocitates: et quoniam permanente velocitate et variata massa, vis est ut massa ipsa; inferimus vires esse ut motus quantitates.

[[7|7]]. Dixi ([[4]]) tantam motus quantitatem excitari in globo
&lt;math&gt;A&lt;/math&gt; quantam ipse
&lt;math&gt;A&lt;/math&gt; resistendo destruit in globo
&lt;math&gt;B&lt;/math&gt;: atque huc spectat illud de actione et reactione principium,
quod sic enunciari solet &quot;actioni contraria semper et aequalis est reactio, sive duorum corporum actiones in se mutuo semper sunt aequales, et in contrarias partes diriguntur&quot;. Huic autem principio locus est in rerum natura sive corpora in contactu agant in se mutuo, sive dissitis e locis sese invicem ad status mutationem quocumque modo determinent.

Notetur illud: cum corpus omne obnitatur semper sui statos mutationi, inferimus ipsam status mutationem haud repente gigni a viribus extrinsecis, sed per gradus indefinitae attenuationis capaces: secus enim dicendum foret inesse materiei vim quamdam infinitam. Siquidem in hypothesi finitae mutationis instantaneae materies valeret opponere resistentiam finitam, labente tempusculo infinite quod nequit admitti. Verum quia vires quaedam tam cito gignunt mutationem status, ut eam in istanti videantur absolvere; inde fit ut vires dividi soleant in instantaneas, et continuas.

=== De virium compositione et resolutione, deque earum momentis et aequilibrio: aliquid quoque notatur de vecte, axe in peritrochio , trochlea etc. . . . ===
[[8|8]]. Fac ut per communem vim &lt;math&gt;P&lt;/math&gt; puncta &lt;math&gt;H&lt;/math&gt; et &lt;math&gt;K&lt;/math&gt; (fig. 2.)
determinentur ambo ad percurrendam motu uniformi rectam lineam &lt;math&gt;AB&lt;/math&gt; intra tempus &lt;math&gt;t&lt;/math&gt; , per &lt;math&gt;Q&lt;/math&gt; vero determinetur
&lt;math&gt;K&lt;/math&gt; ad percurrendam motu pariter uniformi rectam lineam 
&lt;math&gt;AD&lt;/math&gt; intra idem tempus &lt;math&gt;t&lt;/math&gt; ; et comple parallelogrammum &lt;math&gt;BD&lt;/math&gt;.
Ex principio motus relativi punctum &lt;math&gt;K&lt;/math&gt; in fine temporis &lt;math&gt;t&lt;/math&gt;
reperietur in &lt;math&gt;C&lt;/math&gt; ; ac proinde intra tempus &lt;math&gt;t&lt;/math&gt; percurret motu
uniformi diagonalem &lt;math&gt;AC&lt;/math&gt; : idem nimirum existet motus sive mobile feratur per diagonalem &lt;math&gt;AC&lt;/math&gt; velocitate
&lt;math&gt;\frac{AC}{t}&lt;/math&gt; ex vi 
unica impressa &lt;math&gt;R&lt;/math&gt;, sive conjunctis viribus &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; impellatur per latera &lt;math&gt;AB&lt;/math&gt; et &lt;math&gt;AD&lt;/math&gt; velocitatibus &lt;math&gt;\frac{AB}{t}&lt;/math&gt; et &lt;math&gt;\frac{AD}{t}&lt;/math&gt;; eritque (6)

&lt;math&gt;
R : P : Q : =AC: AB: AD.
&lt;/math&gt;

Hinc pro duabus viribus &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; poterit, substitui vis
&lt;math&gt;R&lt;/math&gt;; quae substitutio dicitur virium compositio : et viceversa
pro &lt;math&gt;R&lt;/math&gt; poterunt substitui duae &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt;; quae substitutio dicitur virium resolutio : &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; vocantur componentes, &lt;math&gt;R&lt;/math&gt;
resultans, vel etiam composita.

[[9|9]]. Haec notentur. 
1º. ex tribus &lt;math&gt;R&lt;/math&gt; , &lt;math&gt;P&lt;/math&gt; , &lt;math&gt;Q&lt;/math&gt; unaquae
vis potest repraesentari per sinum anguli, qui sub aliarum
directionibus continetur ; nam
&lt;math&gt;
R : P : Q = AC : DC: AD = \sin BAD : \sin CAD : \sin BAC .
&lt;/math&gt;

2°. Hinc &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; sunt reciproce ut perpendicula ,
quae a puncto quolibet resultantis &lt;math&gt;R&lt;/math&gt; ducuntur ad ipsarum
&lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; directiones .

3º. Denotante &lt;math&gt;i&lt;/math&gt; angulum interceptum directionibus
virium &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt;, triangulum &lt;math&gt;A C D&lt;/math&gt; praebebit

&lt;math&gt;
RP = P^2 + Q^2 - 2PQ \cos(180^{\circ} - i) = P^2 + Q^2 + 2PQ \cos i.
&lt;/math&gt;

4°. Si punctum &lt;math&gt;K&lt;/math&gt; ( fig. 3. ) urgetur viribus &lt;math&gt;KA, KB, KC, KD&lt;/math&gt; etc. . . , ducantur autem &lt;math&gt;Aa&lt;/math&gt; parallela et
&lt;math&gt;= KB&lt;/math&gt; , &lt;math&gt;Aa'&lt;/math&gt; &lt;math&gt;Aa''&lt;/math&gt; parallela et &lt;math&gt;= KC&lt;/math&gt; , &lt;math&gt;a'' a''' &lt;/math&gt; parallela et&lt;math&gt; =
KD&lt;/math&gt; , etc. vis cunctis aequivalens exhibebitur manifeste per lineam rectam &lt;math&gt;Ka'''&lt;/math&gt;, quae jungit punctum &lt;math&gt;K&lt;/math&gt;
et extremitatem &lt;math&gt;a'''&lt;/math&gt; ultimae &lt;math&gt;a''a'''&lt;/math&gt; . Porro linearum rectarum aequalium et parallelarum projectiones sive in recta
quavis &lt;math&gt;EE'&lt;/math&gt;, sive in plano quovis , sunt aequales et parallelae: hinc virium &lt;math&gt;KA, KB, KC, KD&lt;/math&gt;, etc. . . projectiones in recta &lt;math&gt;EE'&lt;/math&gt; simul sumptae aequabuntur projectionibus rectarum &lt;math&gt;KA, Aa', a'a'', a'' a'''&lt;/math&gt; etc. , in eadem &lt;math&gt;EE'&lt;/math&gt; simul pariter sumptis. Harum vero projectionum summa nihil est aliud nisi projectio resultantis &lt;math&gt;Ka'''&lt;/math&gt; : igitur projectio resultantis aequabitur projectionibus componentium &lt;math&gt;KA, KB, KC, KD&lt;/math&gt;, etc. , in summam contractis , si modo habeatur ratio signorum, ut censeantur negativae, quae vergunt v. gr. ad &lt;math&gt;E&lt;/math&gt;, habitis pro positivis, quae versus &lt;math&gt;E'&lt;/math&gt; se dirigunt.

5°. In hypothesi trium duntaxat virium &lt;math&gt;KA, KB , KC&lt;/math&gt;, quisque videt aequipollentem vim repraesentatum iri
per diagonalem parallelepipedi sub lateribis &lt;math&gt;KA, KB, KC&lt;/math&gt;.

6°. Si punctum &lt;math&gt;K&lt;/math&gt; urgetur vi &lt;math&gt;Ka'''&lt;/math&gt;, constructo ad
libitum polygono &lt;math&gt;a''' a'' ... K&lt;/math&gt;, ductaque &lt;math&gt;KD&lt;/math&gt; parallela et &lt;math&gt;=a''' a''&lt;/math&gt; , &lt;math&gt;KC&lt;/math&gt; parallela et &lt;math&gt;= a'' a'&lt;/math&gt;, &lt;math&gt;KB&lt;/math&gt; parallela et &lt;math&gt;= a' A&lt;/math&gt; etc.
resolvetur &lt;math&gt;Ka'''&lt;/math&gt; in &lt;math&gt;KD, KC, KB&lt;/math&gt;, etc ....

7°. Ad resolvendam &lt;math&gt;Ka'''&lt;/math&gt; in ternas sese dirigentes
juxta datas rectas &lt;math&gt;KB, KC, KD&lt;/math&gt;, satis erit per &lt;math&gt;a'''&lt;/math&gt; ducere
tria plana parallela planis &lt;math&gt;BKC, CKD, BKD&lt;/math&gt;; hoc pacto
exsurget parallelepipedum , cujus latera apud &lt;math&gt;K&lt;/math&gt; exhibebunt
( 5°) quaesitas vires componentes.

8°.  Puncta &lt;math&gt;B, C, D, K&lt;/math&gt;, ponantur inter se rigidis
lineis connexa: manentibus virium directionibus, si ternae
componentes intelliguntur applicitae punctis &lt;math&gt;B, C, D&lt;/math&gt;, adhuc
iis manifeste aequipollebit &lt;math&gt;Ka'''&lt;/math&gt; . Inferimus vim quamvis
&lt;math&gt;Ka'''&lt;/math&gt; resolvi posse in ternas, quae et sint applicitae tribus punctis ad libitum sumptis ( si sumuntur ita , ut in
eorum plano inveniatur etiam punctum &lt;math&gt;K&lt;/math&gt;, non debebit
&lt;math&gt;Ka'''&lt;/math&gt; esse extra id planum ) et sese dirigant juxta rectas ab istiusmodi punctis ductas ad punctum &lt;math&gt;K&lt;/math&gt; , cui applicatur ipsa &lt;math&gt;Ka'''&lt;/math&gt;.

9º. Dato systemate punctorum materialium rigidis
lineis inter se firmiter connexorum ( huc spectat corpus solidum ) respondentibusque viribus sollicitatorum; quia possunt (8º. ) singulae vires resolvi in cernas applicitas tribus
punctis &lt;math&gt;A , B, C&lt;/math&gt; ad libitum sumptis, poterunt ( 4°) omnes
traduci ad aequipollens trium virium systema.

10° . Per unam ex hisce tribus viribus duc planum , quod secet reliquas duas : vis , per quam ducitur planum , poterit resolvi ( 4° ) in binas , applicitas intersectionum punctis. Inde fit, ut vires omnes solidum corpus sollicitantes
traduci etiam possint ad aequipollens duarum virium systema.

[[10|10]]. Facile est determinare quandonam plures potentiae eidem puncto applicitae in aequilibrio permaneant.
Binas potentias pro lubito sumptas compone, et pro illis
aequipollentem substitue , atque id iterato donec ad duas
devenias. Si hae directe contrariae et aequales inveniuntur, constabit omnes potentias in aequilibrio consistere .

Facile etiam intelliges quanam ratione inveniri possit potentia duabus &lt;math&gt;AH, BF&lt;/math&gt; ( fig. 4. ) in eodem plano jacentibus,
rectaeque rigidae &lt;math&gt;AB&lt;/math&gt; applicatis aequivalens, et aequilibrium
obtineri; productis (?) enim directionibus &lt;math&gt;AH, BF&lt;/math&gt; donec concurrant
in &lt;math&gt;C&lt;/math&gt;, transferantur potentiae in punctum &lt;math&gt;C&lt;/math&gt;. Sumptis in earum directionibus &lt;math&gt;CH' = AH&lt;/math&gt;, et &lt;math&gt;CF' = BF&lt;/math&gt;, istae componantur. Facto parallelogrammo &lt;math&gt;CF'KH'&lt;/math&gt;, cujus diameter &lt;math&gt;CK&lt;/math&gt; equivalentem vim repraesentabit, haec producatur donec concurrat in &lt;math&gt;D&lt;/math&gt; cum &lt;math&gt;AB&lt;/math&gt;; perspicuum est potentiam &lt;math&gt;KC&lt;/math&gt; translatam in &lt;math&gt;DL&lt;/math&gt; et rectae &lt;math&gt;AB&lt;/math&gt; applicitam in D aequipollere duabus &lt;math&gt;AH , BF&lt;/math&gt;. Quare si &lt;math&gt;AB&lt;/math&gt; in puncto &lt;math&gt;D&lt;/math&gt; sustentetur, potentiae &lt;math&gt;AH, BF&lt;/math&gt; in aequilibrio quiescent; et constabit quam potentiam exerceat punctum &lt;math&gt;D&lt;/math&gt;, nimirum aequalem et oppositam potentiae aequivalenti &lt;math&gt;DL&lt;/math&gt;. Ad positionem puncti &lt;math&gt;D&lt;/math&gt; quod pertinet, concipiamus ex eo duci duo perpendicula &lt;math&gt;p&lt;/math&gt; et &lt;math&gt;q&lt;/math&gt; , alterum in &lt;math&gt;AH&lt;/math&gt; , alterum
in &lt;math&gt;BF&lt;/math&gt; ; sintque &lt;math&gt;AH = P , BF = Q&lt;/math&gt;, longitudo &lt;math&gt;AB = h ,
AD = x&lt;/math&gt;, angulus &lt;math&gt;BAC =a&lt;/math&gt;, angulus &lt;math&gt;ABC = b&lt;/math&gt; : erunt &lt;math&gt;p =
x \sin a, q = ( h- x ) \sin b &lt;/math&gt;, ideoque &lt;math&gt;\frac p q = \frac{x \sin a}{(h - x ) \sin b}  &lt;/math&gt;
Sed( 9.2º )
&lt;math&gt;\frac p q = \frac Q P &lt;/math&gt;;
igitur
&lt;math&gt; \frac Q P = 
\frac{x \sin a}{
(h- x ) \sin b} &lt;/math&gt;, unde &lt;math&gt; \frac{x }{
(h- x ) } = \frac{Q \sin b}{P \sin a }.   &lt;/math&gt;

Quod spectat ad angulum interceptum resultante &lt;math&gt;CK&lt;/math&gt; et
data recta &lt;math&gt;AC&lt;/math&gt; , is dicatur &lt;math&gt;\alpha&lt;/math&gt; : erit ( 9. 1º )
&lt;math&gt;P : Q = \sin BCD : \sin ACD= \sin ( 180^{\circ}- a - b- \alpha) : \sin \alpha&lt;/math&gt;,
unde

&lt;math&gt;\tan \alpha =\frac{
Q \sin ( a + b )}{
P - Q \cos ( a + b )}.&lt;/math&gt;

Quod vero spectat ad resultantem
&lt;math&gt;CK ( = R )&lt;/math&gt; , habemus ( 9. 3º )
&lt;math&gt;R^2 = P^2 + Q^2 - 2P Q\cos ( a + b )&lt;/math&gt;.

Penultima formula traduci potest ad

&lt;math&gt;\cos \alpha = \frac{P - Q \cos ( a + b )}{
R}.   &lt;/math&gt;

Haec subjungimus. 1º. Recta &lt;math&gt;AB&lt;/math&gt; rotetur circa &lt;math&gt;D&lt;/math&gt;, ut ejus extrema puncta &lt;math&gt;A&lt;/math&gt; et &lt;math&gt;B&lt;/math&gt; eodem tempusculo infinitesimo
describant circulares arcus infinitesimos &lt;math&gt;Aa', Bb'&lt;/math&gt;; ex &lt;math&gt;a'&lt;/math&gt; et &lt;math&gt;b'&lt;/math&gt;
duc perpendicula &lt;math&gt;a'a'', b'b''&lt;/math&gt; in directiones virium &lt;math&gt;AH ,
BF&lt;/math&gt; ; sintque &lt;math&gt;Aa'' = p' , Bb'' = q'&lt;/math&gt;: erunt &lt;math&gt;p' = Aa' \cos a'Aa''
= Aa' \cos ( DAa'' - 90^{\circ} ) = Aa' \sin DAa'' = Aa' \sin a ,
q'= Bb'\cos b'Bb'' = Bb'\cos (90^{\circ}-b) = Bb'\sin b&lt;/math&gt;; et consequenter
&lt;math&gt;\frac{p'}{q'}= \frac{Aa' \sin a}{
Bb' \sin b}=
\frac{AD \sin a}{BD \sin b} = \frac{x \sin a}{(h - x ) \sin b} =
\frac{Q}{P} .&lt;/math&gt;

Nihil sunt aliud &lt;math&gt;Aa'&lt;/math&gt; et &lt;math&gt;Bb'&lt;/math&gt; nisi spatiola tempusculo infinitesimo circa immobile punctum &lt;math&gt;D&lt;/math&gt; simul describenda ab &lt;math&gt;A&lt;/math&gt;
et &lt;math&gt;B&lt;/math&gt; in hypothesi turbati aequilibrii; quibus punctis &lt;math&gt;A&lt;/math&gt; et &lt;math&gt;B&lt;/math&gt;
applicantur vires &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt;: exhibent &lt;math&gt;p', q'&lt;/math&gt; illorum spatiolorum projectiones super ipsarum virium directionibus. Vires igitur &lt;math&gt;P, Q&lt;/math&gt; sese mutuo librantes circa &lt;math&gt;D&lt;/math&gt; erunt reciproce ut eae projectiones.

2º. Etiam sic : triangula &lt;math&gt;Aa'a'', DAh&lt;/math&gt; , itemque
&lt;math&gt;Bb'b'', DBh'&lt;/math&gt; habent latera sibi respective perpendicularia ; igitur &lt;math&gt;\frac {DA} {Aa'} = \frac{p}{p'} , \frac{DB} {Bb'} = \frac{q}{q'}&lt;/math&gt;.

Denotet &lt;math&gt;i&lt;/math&gt; valorem rationum aequalium &lt;math&gt; \frac{DA}{Aa'} , \frac {DB}{Bb'}   &lt;/math&gt;, projectio insuper &lt;math&gt;p'&lt;/math&gt; computata in ipsa directione respondentis potentiae &lt;math&gt;P&lt;/math&gt; censeatur positiva; projectio vero &lt;math&gt;q'&lt;/math&gt; computata in directione contraria illi , quam obtinet respondens  potentia &lt;math&gt;Q&lt;/math&gt; , censeatur negativa: erunt
&lt;math&gt;p = ip' , q = -
iq'&lt;/math&gt; ;
propterea
&lt;math&gt; \frac QP = \frac pq = -\frac{ip'}{iq'}= -\frac{p'}{q'}
, Pp' + Qq' = 0 .&lt;/math&gt; 

Huc spectat principium velocitatum &lt;u&gt;virtualium&lt;/u&gt;.

3º. Ex quovis puncto (&lt;math&gt;  M&lt;/math&gt; ) sive intra , sive extra
angulum &lt;math&gt; ACB&lt;/math&gt; , duc perpendicula &lt;math&gt; p'', q'' , r''&lt;/math&gt;  ad &lt;math&gt; P, Q, R&lt;/math&gt;  ; duc
quoque ab (&lt;math&gt;  M&lt;/math&gt; ) ad &lt;math&gt; C &lt;/math&gt; rectam ( &lt;math&gt; MC = c &lt;/math&gt; ), cui normaliter
insistat alia recta (&lt;math&gt;  E E' &lt;/math&gt; ) transiens per &lt;math&gt; C &lt;/math&gt;: singulis &lt;math&gt; P , Q,
R &lt;/math&gt; resolutis in duas , alteram juxta (&lt;math&gt;  CM &lt;/math&gt; ) , alteram juxta
( &lt;math&gt; EE'&lt;/math&gt;  ), expriment
&lt;math&gt; P\frac{p''}{c},Q\frac{q''}{c},R\frac{r''}{c} &lt;/math&gt; 
componentes juxta (&lt;math&gt;  EE'&lt;/math&gt;  ) . Quoad (&lt;math&gt;M&lt;/math&gt;) situm extra angulum &lt;math&gt;ACB&lt;/math&gt;, primae duae erant conspirantes; quoad (&lt;math&gt;M &lt;/math&gt;)
situm intra &lt;math&gt;ACB&lt;/math&gt; erunt contrariae : cum igitur &lt;math&gt;R&lt;/math&gt; resultet ex &lt;math&gt;P&lt;/math&gt;
et &lt;math&gt;Q&lt;/math&gt;, prodibit ( 9. 4° ) in primo casu
&lt;math&gt; P\frac{p''}{c}+Q\frac{q''}{c}=R\frac{r''}{c} &lt;/math&gt; et consequenter &lt;math&gt;Pp'' + Qq'' = Rr''&lt;/math&gt;,
in secundo.

&lt;math&gt; \pm(P\frac{p''}{c}-Q\frac{q''}{c}) = R\frac{r''}{c} &lt;/math&gt;, ideoque &lt;math&gt; \pm(Pp''-Qq'') = Rr'', &lt;/math&gt; sumptis signis vel superioribus , vel inferioribus , prout 
&lt;math&gt; P\frac{p''}{c} &gt;  &lt;/math&gt; vel &lt;math&gt; &lt;Q\frac{q''}{c} &lt;/math&gt;: rectangula &lt;math&gt; Pp'',Qq'', Rr'' &lt;/math&gt;  dicuntur momenta virium &lt;math&gt;P, Q, R&lt;/math&gt; quoad punctum (&lt;math&gt;M&lt;/math&gt;). Hinc stabilitur
illud: momentum resultantis &lt;math&gt;R&lt;/math&gt; aequatur summae ex momentis componentium &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; si &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; in eamdem plagam circa (&lt;math&gt;M&lt;/math&gt;) nituntur movere puncta , quibus applicitae
sunt; aequatur differentiae si nituntur movere in plagas
contrarias.

4° . Idipsum facile extenditur ad quemvis numerum
virium &lt;math&gt;P, Q, S, V, U&lt;/math&gt;, ... in dato plano jacentium : fac
v. gr. ut ternae &lt;math&gt;P, Q, S&lt;/math&gt;, in unam eamdemque plagam circa ( &lt;math&gt;M&lt;/math&gt; ) nitantur movere puncta, ad quae sunt applicitae; caeterae vero &lt;math&gt;V, U&lt;/math&gt;, ... in plagam contrariam ; sitque &lt;math&gt;L&lt;/math&gt; resultans ex &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt;; &lt;math&gt;N&lt;/math&gt; resultans ex &lt;math&gt;L&lt;/math&gt; et &lt;math&gt;S&lt;/math&gt;, ac proinde ex
&lt;math&gt;P, Q, S&lt;/math&gt;; &lt;math&gt;O&lt;/math&gt; resultans ex reliquis &lt;math&gt;V, U&lt;/math&gt;. . . Erurt
&lt;math&gt;Ll''= Pp'' + Qq'', Nn'' = Ll'' +Ss''&lt;/math&gt; ;
et consequenter
&lt;math&gt;Nn'' = Pp'' + Qq'' + Ss''&lt;/math&gt; :
simili modo obtinetur
&lt;math&gt;Oo'' = Vv'' + Uu''+&lt;/math&gt; .
Iam si &lt;math&gt;R&lt;/math&gt; exhibet resultantem ex &lt;math&gt;N&lt;/math&gt; et &lt;math&gt;O&lt;/math&gt; , ideoque ex &lt;math&gt;P,
Q, S, V , U &lt;/math&gt;, ... ; cum sit 
fist the
&lt;math&gt;Rr'' = \pm ( Nn'' - Oo'' ) &lt;/math&gt;,
erit quoque
&lt;math&gt;Rr'' = ( Pp'' + Qq'' + Ss'' - Vv'' - Uu'' - ... ) .&lt;/math&gt;

5º. Fac ut &lt;math&gt;R&lt;/math&gt; transeat per (&lt;math&gt;M&lt;/math&gt;) ; erit &lt;math&gt;r'' = 0&lt;/math&gt;: propterea
&lt;math&gt;Pp'' + Qq'' + Ss'' - Vv'' - Uu'' - ... = 0&lt;/math&gt; ;
viriumque systema consistet in aequilibrio circa immobile
punctum (&lt;math&gt;M&lt;/math&gt;) . Vocatur (&lt;math&gt;M&lt;/math&gt;) centrum momentorum.

6º. Habemus ( 2 )
&lt;math&gt;p'' = ip' , q'' = iq' , s'' = is' , v'' = -iv', u'' = - iu', ...&lt;/math&gt;
Traducetur igitur aequatio ( 5°) ad
&lt;math&gt;Pp' + Qq' + Ss' + Vv' + Uu' + ... = 0&lt;/math&gt;

7° Vires &lt;math&gt;AH, BF&lt;/math&gt; haud jacentes in eodem plano nequeunt ad unicam vim aequipollentem traduci. Si enim tradu
ad aequipollentem &lt;math&gt;DL&lt;/math&gt;, poterit etiam ex quodam
istius puncto ad quoddam punctum componentis v . gr.
&lt;math&gt;BF&lt;/math&gt; duci recta linea haud occurrens alteri componenti &lt;math&gt;AH&lt;/math&gt;:
fac ut haec recta linea evadat immobilis ; elisa &lt;math&gt;DL&lt;/math&gt;, emerget aequilibrium; sed elisa quoque &lt;math&gt;BF&lt;/math&gt;, et salva &lt;math&gt;AH&lt;/math&gt;, ex hac ultima emerget motus. In ea ergo qua sumus hypothesi de
traductione virium &lt;math&gt;AH, BF&lt;/math&gt; ad unicam &lt;math&gt;DL&lt;/math&gt; obtinebit simul
aequilibrium et motus in eodem systemate: quod nequit esse; ideoque etc. ... 

8°. Patet solidum liberumque corpus haud consistere
in aequilibrio, nisi binae aequipollentes ( 8. 10° ) vires ,
ad quas traducuntur vires omnes corpus ipsum sollicitantes , sint aequales, contrariae, jaceantque in directum .

9°. Patet quoque solidum corpus, mobile dumtaxat
circa punctum fixum , consistere in aequilibrio, si eae binae
vires aequipollentes et jaceant in eodem plano ( 7º) , et suppeditent resultantem , quae transeat per punctum illud .

10°. Solidum corpus ponatur mobile dumtaxat circa
rectam fixam &lt;math&gt;AZ&lt;/math&gt; ( fig.5 ), sintque &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; binae aequipollentes
vires, ad quas traducuntur ( 9. 10° ) vires omnes corpus ipsum sollicitantes. Duc planum &lt;math&gt;XOY&lt;/math&gt; et normaliter insistens
rectae &lt;math&gt;AZ&lt;/math&gt;, et secans in punctis v. gr. &lt;math&gt;B, C&lt;/math&gt; directiones virium &lt;math&gt;P ( = BB' ), Q ( = CC' )&lt;/math&gt;: poterit &lt;math&gt;P&lt;/math&gt; resolvi in duas , alteram &lt;math&gt;BB'''&lt;/math&gt; perpendicularem plano &lt;math&gt;XO&lt;/math&gt;Y , alteram &lt;math&gt;BB''&lt;/math&gt; jacentem in ipso &lt;math&gt;XOY&lt;/math&gt;; similiter &lt;math&gt;Q&lt;/math&gt; poterit resolvi in duas ,
alteram &lt;math&gt;CC'''&lt;/math&gt; perpendicularem eidem &lt;math&gt;XOY&lt;/math&gt;, alteram &lt;math&gt;CC''&lt;/math&gt; in
eo jacentem . Binae &lt;math&gt;BB''', CC'''&lt;/math&gt;, utpote parallelae ad rectam fixam &lt;math&gt;AZ&lt;/math&gt;, peribunt elisae : in ea igitur qua sumus hypothesi
haud consistet solidum corpus in aequilibrio, nisi resultans
ex &lt;math&gt;BB'' , CC''&lt;/math&gt; transeat per aliquod punctum &lt;math&gt;O&lt;/math&gt; rectae fixae
&lt;math&gt;AZ&lt;/math&gt; ; et consequenter ( 9. 2° ) , ductis ex &lt;math&gt;O&lt;/math&gt; in istas vires
perpendicalis &lt;math&gt;b, c&lt;/math&gt;, nisi valeat aequatio

&lt;math&gt;\frac{b}{c} = \frac{CC''}{BB''} &lt;/math&gt;:

producta ex &lt;math&gt;b&lt;/math&gt; in &lt;math&gt;BB''&lt;/math&gt; et ex &lt;math&gt;c&lt;/math&gt; in &lt;math&gt;CC''&lt;/math&gt; dicuntur momenta virium &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; quoad &lt;math&gt;AZ&lt;/math&gt; . Si &lt;math&gt;P&lt;/math&gt; v. gr. , applicita ad punctum
&lt;math&gt;B'&lt;/math&gt;, est parallela plano &lt;math&gt;XOY&lt;/math&gt;, applicabuntur ad &lt;math&gt;B&lt;/math&gt; duae quaelibet vires &lt;math&gt;H &lt;/math&gt; et &lt;math&gt;- H&lt;/math&gt; aequales, contrariae et parallelae axi &lt;math&gt;AZ&lt;/math&gt;; tum una ex iis v. gr. &lt;math&gt;H&lt;/math&gt; componetur cum &lt;math&gt;P&lt;/math&gt; : vis inde
resultans poterit transferri in punctum v. gr. &lt;math&gt;B&lt;/math&gt; plani &lt;math&gt;XOY&lt;/math&gt;,
ibique resolvi in binas, alteram &lt;math&gt;BB''' ( = H )&lt;/math&gt; parallelam rectae &lt;math&gt;AZ&lt;/math&gt;, alteram &lt;math&gt;BB'' ( = P )&lt;/math&gt; jacentem in &lt;math&gt;XOY&lt;/math&gt;; eritque
&lt;math&gt;b. BB ' ( = b. P )&lt;/math&gt; momentum vis &lt;math&gt;P&lt;/math&gt; quoad &lt;math&gt;AZ&lt;/math&gt;. Quisque autem videt , si per &lt;math&gt;B '&lt;/math&gt; ducitur planum parallelum plano &lt;math&gt;XOY&lt;/math&gt;,
et ex pancto, ubi istud novum planum secat rectam &lt;math&gt;AZ&lt;/math&gt;, demittitur perpendiculum in vim &lt;math&gt;P&lt;/math&gt; applicitam ad &lt;math&gt;B '&lt;/math&gt;, ejusmodi
perpendiculum nihil fore aliud nisi &lt;math&gt;b&lt;/math&gt;; ita ut, sive momen
tum sumatur apud planum &lt;math&gt;XOY&lt;/math&gt;, sive apud illud alterum
planum parallelum ipsi &lt;math&gt;XOY&lt;/math&gt;, perinde sit.

[[11|11]]. Fac ut vis ( 10) &lt;math&gt;BF&lt;/math&gt; (fig. 4) revolvatur circa punctum &lt;math&gt;B&lt;/math&gt;,
donec evadat parallela vi &lt;math&gt;AH&lt;/math&gt;; erit &lt;math&gt;a + b = 180^{\circ}&lt;/math&gt;, ideo
que &lt;math&gt;\sin b = \sin (180^{\circ} - a ) = \sin a&lt;/math&gt; si vires ad eamdem
plagam obvertantur ; &lt;math&gt;a + b = 360^{\circ}&lt;/math&gt;, ideoque &lt;math&gt;\sin b =
\sin ( 360^{\circ} - a ) = - \sin a &lt;/math&gt; si ad contrarias plagas. In primo
igitur casu exsistent.

&lt;math&gt;\frac{x}{h-x} = \frac{Q}{P}, x=  \frac{hQ}{P+Q}, R = P + Q , \cos \alpha =\frac{P+Q}{R}=1.&lt;/math&gt;

In secundo

&lt;math&gt;\frac{x}{h-x} = -\frac{Q}{P}, x=  \frac{hQ}{Q-P}, R = \pm(P - Q) , \cos \alpha =\frac{P-Q}{R}=\pm 1.&lt;/math&gt;

valet signum superius ubi &lt;math&gt; P &gt; Q&lt;/math&gt;, inſerius ubi &lt;math&gt;P &lt; Q&lt;/math&gt;;
siquidem &lt;math&gt;P, Q, R&lt;/math&gt; denotant hic virium dumtaxat intensitates.
Inferimus illud; resultans ex duabus parallelis viribus adaequat istarum vel summam, vel differentiam , prout
vel ambae conspirant in eamdem plagam, vel altera in unam
et altera in contrariam plagam; ipsis insuper componentibus
viribus est parallela , et ad eam plagam semper obversa ,
quam respicit major ex componentibus illis ; transit denique
per ejusmodi punctum in directione &lt;math&gt;AB&lt;/math&gt;, quod distet a
punctis applicationis componentium in reciproca earum ratione : istud punctum appellari solet centrum virium parallelarum ; estque invariabile, modo et respectiva virium
positio et ipsarum ratio non mutentur.
Si &lt;math&gt;P = Q&lt;/math&gt;, in secundo casu nulla exsistet resultans.
Non est enim ratio in ea qua sumus hypothesi cur ad
plagam unius potius componentis quam ad alterius componentis plagam sese dirigat resultans. Formulae praebent &lt;math&gt;x= \infty, R =0.&lt;/math&gt;

Etsi vires &lt;math&gt;AH&lt;/math&gt; et &lt;math&gt;BF&lt;/math&gt; (fig.6) parallelae, aequales et contrariae nequeunt librari unica vi , utpote omni resultante destitutae; librabuntur nihilominus duabus aliis viribus &lt;math&gt;AH'&lt;/math&gt; et &lt;math&gt;BF'&lt;/math&gt; parallelis, aequalibus, contrariis, et in plano &lt;math&gt;HABF&lt;/math&gt; iacentibus, dummodo ductis ex &lt;math&gt;A&lt;/math&gt; in &lt;math&gt;BF BF'&lt;/math&gt;
perpendiculis &lt;math&gt;AO&lt;/math&gt; et &lt;math&gt;AO'&lt;/math&gt;, exsistat
&lt;math&gt;BF. AO=BF'. AO'&lt;/math&gt;:
tunc enim , ductis ex &lt;math&gt;B&lt;/math&gt; in &lt;math&gt;AH&lt;/math&gt; et &lt;math&gt;AH'&lt;/math&gt; perpendiculis &lt;math&gt;BC&lt;/math&gt; et
&lt;math&gt;BC'&lt;/math&gt;, ob &lt;math&gt;BF =  AH , BF' = AH' , AO = BC , AO' = BC'&lt;/math&gt;
erit quoque
&lt;math&gt;AH. BC=AH'. BC'&lt;/math&gt;;
et consequenter ( 9. 2°) resultans ex &lt;math&gt;AH&lt;/math&gt; et &lt;math&gt;AH'&lt;/math&gt; sese diriget a puncto &lt;math&gt;A&lt;/math&gt; ad punctum &lt;math&gt;B&lt;/math&gt;, simulque resultans ex &lt;math&gt;BF&lt;/math&gt; et
&lt;math&gt;BF'&lt;/math&gt; sese diriget a puncto &lt;math&gt;B&lt;/math&gt; ad punctum &lt;math&gt;A&lt;/math&gt; ; istiusmodi praeterea resultantes sunt manifeste aequales: iccirco etc. ... Systema itaque virium &lt;math&gt;AH', AF'&lt;/math&gt; aequipollebit systemati virium
&lt;math&gt;AH , AF&lt;/math&gt; ; poteritque alterum ( mutatis ejus directionibus
in contrarias partes ) alteri substitui. Consequitur posse binas vires parallelas, aequales et contrarias transferri ab
una positione ad alteram in proprio ipsarum plano, variata simul virium et magnitudine , et directione ; modo tamen productum ex communi earum valore in mutuam distantiam maneat constans.

[[12|12]]. Sint nunc plures vires parallelae &lt;math&gt;P, P ', P ''&lt;/math&gt;, ... variis
solidi corporis punctis applicitae , quarum aliae conspirent
in unam plagam , aliae in plagam contrariam . Componendo &lt;math&gt;P&lt;/math&gt; v . gr. et &lt;math&gt;P'&lt;/math&gt; in unicam &lt;math&gt;R '&lt;/math&gt;, &lt;math&gt;R'&lt;/math&gt; et &lt;math&gt;P'&lt;/math&gt; in unicam &lt;math&gt;R''&lt;/math&gt; , &lt;math&gt;R''&lt;/math&gt; et &lt;math&gt;P'''&lt;/math&gt;
in unicam &lt;math&gt;R''' &lt;/math&gt;, etc. , ... facile devenies ( 11 ) ad illud : resultans
&lt;math&gt;R&lt;/math&gt; ex pluribus viribus parallelis adaequat differentiam inter
summam conspirantium in unam plagam et summam conspirantium in plagam contrariam ; ipsis insuper componentibus viribus est parallela , et ad eam plagam obvertitur , quam
respicit major ex illis summis .

Hinc si vires tendentes in unam plagam censentur positivae , tendentes vero in plagam contrariam negativae , obtinebit aequatio
&lt;math&gt;R = P + P' + P'' + ... (a )&lt;/math&gt;.
Ad haec : denotantibus (fig .7) &lt;math&gt;A, B, D &lt;/math&gt;, ... puncta , quibus applicantur parallelae vires &lt;math&gt;P , P ', P''&lt;/math&gt;, ... , et &lt;math&gt;AB , BD &lt;/math&gt;. ..
rigidas rectas jungentes puncta illa , cum transeant &lt;math&gt;R ', R '' &lt;/math&gt;, ...
per ejusmodi puncta &lt;math&gt;K , H &lt;/math&gt;, ... , quorum positiones sive in
rectis &lt;math&gt;AB , KD &lt;/math&gt;, ... sive in earum prolongationibus unice pendent a conditionibus
&lt;math&gt;P ' :R'= AK :AB , P'' : R'' = HK : KD,&lt;/math&gt; etc. ... ,

seu

&lt;math&gt;P: P'+P= AK : AB , P'' : P + P' + P'' = HK : KD&lt;/math&gt;, etc. ... ,

devenietur etiam ad illud : in systemate parallelarum viriam
habetur constans et immutabile centrum , per quod semper transit resultans &lt;math&gt;R&lt;/math&gt; , quacumque ceteroqui ratione componentes vires volvantur circa puncta quibus applicitae sunt ,
modo et maneant parallelae , et applicitae iisdem punctis in
iisdem respective distantiis.

[[13|13]]. Ducto quolibet plano &lt;math&gt;MQ&lt;/math&gt;, demittantur in illud ex
punctis &lt;math&gt;A , B , D,&lt;/math&gt; ... perpendicula &lt;math&gt;AM ( =z) , BN ( = z; ) , DQ
( = z''), ...&lt;/math&gt; ; sive ( 12) &lt;math&gt;K , H &lt;/math&gt;, ... sint in rectis &lt;math&gt;AB , KD &lt;/math&gt;, ... .
sive in earum prolongationibus , demittantur quoque in
idem &lt;math&gt;MQ&lt;/math&gt; ex istis punctis perpendicula &lt;math&gt;KL , HO &lt;/math&gt;, ... ; per
ipsa &lt;math&gt;K , H &lt;/math&gt;, ... agantur rectae &lt;math&gt;RS , TU &lt;/math&gt;, ... , prima rectae
MN parallela et perpendiculis &lt;math&gt;AM , BN&lt;/math&gt; occurrens in &lt;math&gt;R , S &lt;/math&gt;,
secunda rectae &lt;math&gt;LQ&lt;/math&gt; parallela et perpendiculis &lt;math&gt;KL , DQ&lt;/math&gt; occurrens in &lt;math&gt;T , U &lt;/math&gt;, etc ... Erunt
&lt;math&gt;AR = MR - AM = KL - z, BS = BN - NS =z' - KL,
DU=UQ-DQ=HO-z'', KT = KL - LT = KL - HO &lt;/math&gt;; etc ....
Jamvero ( 11 )
&lt;math&gt;BS:AR = BK :AK = P : P' ,DU :KT = DH :HK = P + P':P''&lt;/math&gt;,etc ...,
ideoque
&lt;math&gt;AR.P = BS.P', DU.P'' = KT (P + P'), &lt;/math&gt;etc....
Igitur
&lt;math&gt;(KL- z) P = (z' -KL )P',(HO- z'') P'' = (KL-HO)(P + P'),&lt;/math&gt;etc....
unde
&lt;math&gt;KL (P + P') = zP + z'P', HO (P + P + P'' ) = KL (P + P') +
z'' P '' = zP + z' P' +z'' P'',&lt;/math&gt; etc.
seu
&lt;math&gt;KL. R ' = zP + z' P', HO. R'' = zP + z'P' + z''P'' , &lt;/math&gt;etc....

Generatim exhibente &lt;math&gt;z_{\mathrm I}&lt;/math&gt;, perpendiculum ex centro omnium datarum virium parallelarum ductum in &lt;math&gt;MQ&lt;/math&gt; , habebimus

&lt;math&gt;z_{\mathrm I} R = zP + z' P' + z'' P'' + z''' P ''' + ... :&lt;/math&gt;
rectangula &lt;math&gt;z_{\mathrm I} R , zP&lt;/math&gt;, dicuntur momenta virium &lt;math&gt;R , P&lt;/math&gt;, ...
quoad plapum &lt;math&gt;MQ&lt;/math&gt;.

Haec notentur: 1° Etsi non omnia puncta , quibus applicantur parallelae vires &lt;math&gt;P , P', P'' &lt;/math&gt;... sita sunt supra planum &lt;math&gt;MQ&lt;/math&gt;
adhuc tamen algebraica summa rectangulorum sub &lt;math&gt;P , P'&lt;/math&gt; ... et
respondentibus perpendiculis ductis in &lt;math&gt;MQ&lt;/math&gt; ex punctis illis '''adaequabit''' rectangulum sub resultante &lt;math&gt;R&lt;/math&gt; et perpendiculo ducto ex centro ipsarum &lt;math&gt;P, P' , &lt;/math&gt;... in idem &lt;math&gt;MQ&lt;/math&gt;; moto enim &lt;math&gt;MQ&lt;/math&gt; versus ea puncta ita , ut maneat sibi parallelum , atque a primitiva positione recedat intervallo &lt;math&gt;h&lt;/math&gt; , si nova perpendicula exhibentur per &lt;math&gt;k, k', k '', ... k_{\mathrm I}&lt;/math&gt; erunt &lt;math display=''inline''&gt;k = z - h , k' = z'- h , k'' = z'' - h, ...  k_{\mathrm I} = z_{\mathrm I} - h &lt;/math&gt;; hinc &lt;math&gt;(k_{\mathrm I} +h) R = (k + h) P + ( k' + h) P' + (k'' + h ) P'' + &lt;/math&gt;... 

est autem ( 12.''a'') &lt;math&gt;hR =h (P + P' + P'' + ...) = hP + hP' + hP'' + ...&lt;/math&gt;;

igitur

&lt;math&gt;k_{\mathrm I} R = kP +k'P' + k'' P'' + ... &lt;/math&gt;

ubi &lt;math&gt;k, k ', k'', ... k_{\mathrm I}&lt;/math&gt; possunt esse vel positiva , vel negativa.

2° Praeter &lt;math&gt;MQ&lt;/math&gt; seu &lt;math&gt;XOY&lt;/math&gt; ( Fig.8 ) concipiantur duo alia plana &lt;math&gt;XOZ , YOZ&lt;/math&gt;; quod autem in ordine ad &lt;math&gt;XOY&lt;/math&gt; est, sit &lt;math&gt;z, z',... z_{\mathrm I} &lt;/math&gt;, sit &lt;math&gt;x, x',... x_{\mathrm I} &lt;/math&gt; in ordine ad &lt;math&gt;YOZ &lt;/math&gt;, et &lt;math&gt;y, y',... y_{\mathrm I} &lt;/math&gt; in ordine ad &lt;math&gt;XOZ&lt;/math&gt;; qua ratione assequuti sumus
&lt;math&gt;z_{\mathrm I}R=zP+z'P'+z''P'' + ...,&lt;/math&gt;
eadem assequemur (a')
&lt;math&gt;x_{\mathrm I}R=xP+x'P'+x''P'' + ...
y_{\mathrm I}R=yP+y'P'+y''P'' + ...&lt;/math&gt;

3° Si compendii causa per  &lt;math&gt;\Sigma P &lt;/math&gt; exprimitur summa
potentiarum &lt;math&gt;P, P', P'', &lt;/math&gt;
et per &lt;math&gt;\Sigma_x P, \Sigma_y  P, \Sigma_z P &lt;/math&gt; designantur summae rectangulorum sub potentiis et respectivis 
perpendiculis , formulae ( a' ) scribi poterunt in hunc modum ( 12. ''a'')
&lt;math&gt;x_{\mathrm I}\Sigma P = \Sigma_x P, y_{\mathrm I}\Sigma P = \Sigma_y P ,z_{\mathrm I}\Sigma P = \Sigma_z P, &lt;/math&gt;
unde
&lt;math&gt;x_{\mathrm I} = \frac{\Sigma_x P}{ \Sigma P} , y_{\mathrm I}= \frac{\Sigma_y P}{ \Sigma P},z_{\mathrm I}= \frac{\Sigma_z P}{ \Sigma P} &lt;/math&gt;
In hypothesi planorum &lt;math&gt;XOY , XOZ , YOZ&lt;/math&gt; orthogonalium ,
&lt;math&gt;x_{\mathrm I}, y_{\mathrm I}&lt;/math&gt; et &lt;math&gt;z_{\mathrm I}&lt;/math&gt;, erunt orthogonales coordinatae , quibus determinatur positio centri parallelarum virium .

4.° Aequatio
P + P + P + ... ... = o ( a &lt;nowiki&gt;''&lt;/nowiki&gt; )&lt;nowiki&gt;''&lt;/nowiki&gt;
manifeste denotat unam quamvis ex datis potentiis v. gr. P
esse aequalem et contrariam resultanti R, ex reliquis omni
bus P' , P &lt;nowiki&gt;''&lt;/nowiki&gt; , ... Ponamus XOY perpendiculare , et XOZ , YOZ&lt;nowiki&gt;''&lt;/nowiki&gt;
parallela directioni potentiarum ; in hac hypothesi erunt
P et R, directe contrariae si perpendicula x et y spectantia
ad punctum , cui applicalur P , spectent ambo ad centrum
quoque virium p ', P &lt;nowiki&gt;''&lt;/nowiki&gt;, ... , si nempe habeantur&lt;nowiki&gt;''&lt;/nowiki&gt;
x R , = x'P' + x &lt;nowiki&gt;''&lt;/nowiki&gt; P t ... ,&lt;nowiki&gt;''&lt;/nowiki&gt;
y R, =ÝP' +y&lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; + .
seu , ob R,
x P + x' P ' + x &lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; +
yP + ' P ' + y &lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; +
-P
= 0,
0; }(cm

5. ° Quibus positis , stabilitur illud : parallelarum
virium systema consistit in aequilibrio sub duabus conditio
nibus simul explendis ; altera est , ut evanescat earum sum
ma : altera ut evanescat summa ex earum momentis in ordi
ne ad duo plana ipsis viribus parallela. Si parallelae vires
inveniuntur omnes in eodem plano, secunda conditio jam de
18
tur summae rectangulorum sub potentiis et reSpectivis perpen-
diculis, formulae (a') scribi poterunt in hunc modum (12. a)
a:, EP :ZxP,y,ZxP: ZJP, z.l 2P:ZzP,
unde u
∙∙∙ zxp ∙∙ \sum∫ M) (0 )
∙−− ⋅ −\sum−⇂⋅−↗∫≖ \sum⇂≀ .z,--—— ZP
ln hypothesi planorum XOT, XOZ , TOZ orthogonalium ,
x, ,y, , et 2! erunt orthogonales coordinatae, quibus deter-
minatur positio centri parallelarum Vtrium.
43 Aequatio
P gr ≖⋡⋅−⊦∙∙⋅−−∙∶∘ (a&lt;nowiki&gt;'''&lt;/nowiki&gt;)&lt;nowiki&gt;'''&lt;/nowiki&gt;
manifeste denotat uuam quamvis-ex datis potentiis v. gr. P
esse aequalem et contrariam resultanti R, ex reliquis omni-
bus P', P&lt;nowiki&gt;''&lt;/nowiki&gt;, Ponamus XOV perpendiculare , et XOZ ,&lt;nowiki&gt;''&lt;/nowiki&gt;
ïOZ
parallela directioni potentiarum; in hac hypothesi erunt
P et B[ directe contrariae si perpendicula x et y spectantia
ad punctum , cui applicatur P ,
spectent ambo ad centrum
quoque virium P',P&lt;nowiki&gt;''&lt;/nowiki&gt;, , si nempe habeantur&lt;nowiki&gt;''&lt;/nowiki&gt;

&lt;nowiki&gt;::&lt;/nowiki&gt; Bl :x'F—I-x&lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; ⊣−∙∙∙∙ Ja, ∶−−∫∣⊉≀−⊢∜∣∣≖≻∥−⊢∙∙∙∙

seu,ob B' :—P,
xP—- x'P'-- x&lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; −∙∙ :o,
yp ——y' PI ...—7&lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; −∙∙ ∙∙∙ :0' )
(a&lt;nowiki&gt;''&lt;/nowiki&gt;)
5.&lt;nowiki&gt;''&lt;/nowiki&gt; Quibus positis
, stabilitur illud : parallelarum
virium systema consistit in aequilibrio sub duabus conditio-
nibus simul explendis; altera est
, ut evanescat earum sum-
ma :altera ut evanescat summa ex earum momentis in ordi-
ne ad duo plana ipsis viribus parallela. Si parallelae vires
inveniuntur omnes in eodem plano, secunda conditio iam de19
9
se explebitur quoad istud planum , satisque erit ut explea
tur quoad aliud tantummodo planum .
6. Etsi vires P, P' , P &lt;nowiki&gt;''&lt;/nowiki&gt;, ... non sunt parallelae , pos
sunt tamen reduci ad terna ejusmodi systemata , quorum pri.
mum coalescat ex viribus parallelis axi OZ , secundum ex
viribus jacentibus in plano XOY simulque parallelis axi OY ,
tertium ex viribus agentibus juxta axem OX. 

Ut demonstretur assertio , resolve unam quamvis ex datis viribus v. gr.

P applicitam puncto A in tres X, Y, Z, respective parallelas
axibus Ox, OY, OZ; ad punctum A applica duas vires H et
- H aequales , contrarias , et parallelas axi OZ ; compone
X ( = AC ) et H sese dirigentem juxta AE , sitque AB dire
ctio resultantis ; produc BA donec in N occurrat plano XOY ;
transfer in N resultantem illam , et sic translatam resolve in
binas , alteram parallelam axi OX , alteram axi OZ ; prodi
bunt componentes X ( = NC = AC ) , H ( = ND ) , qua
rum primam transfer in C ut sit C'C' ( = NC' ) = X; ad C
applica binas vires K et — K aequales , 'contrarias et pa
rallelas axi OY ; compone X ( = .CC ') et K sese dirigen
tem juxia C'F , sitque C'L directio resultantis ; produc LC
donec in V occurrat axi OX ; transfer in V novam istam re
sultantem , et sic translatam resolve in binas , alteram juxta
ox , alteram parallelam axi OY ; emergent componentes
X ( = VV' = CC&lt;nowiki&gt;''&lt;/nowiki&gt; ) , K ( = VF '): compone nunc Y et - H ;
produc directionem resultantis donec rectae C' F occurrat
v . gr. in N ' ; hanc resultantem transfer in N ' , et sic traus
latam resolve in duas , alteram parallelam axi OY , alteram
axi OZ ; exurgent componentes Y et -H applicitae puucto N:
hoc pacto vi P poterunt substitui sex vires
Z, H, — H applicitae punctis A, N, N' et parallelae axi Oz,
K, Y - K applicitae punctis V, C' et parallelae axi OY ,
X applicita puncto V et agens juxta OX .
Consimiles operationes cum possint instaurari quoad P', P ” ...
non pluribus opus est , at pateat veritas assertionis .
19
se explebitur quoad istud planum , satisque erit ut explea-
tur quoad aliud tantummodo planum .
6.o Etsi vires P, P', P&lt;nowiki&gt;''&lt;/nowiki&gt;, non sunt parallelae ,pos-
sunt tamen reduci ad terna eiusmodi systemata
, quorum pri-
mum coalescat ex viribus parallelis axi OZ , secundum ex
viribus jacentibus in plano XOT simulque parallelis axi Oï
,
tertium ex viribus agentibus juxta axem OX. Ut demon-
stretur assertio , resolve unam quamvis ex datis viribus v. gr.
P applicitam puncto A in tres X, T, Z, respective parallelas
axibus OX, OT, OZ; ad punctum A applica duas vires H et
∙∙∙ H aequales , contrarias ,
et parallelas axi OZ ; compone
X (: AC) et H sese dirigentem iuxta AE .sitque AB dire-
ctio resultantis; produc BA donec in N occurrat planc XOT;
transfer in N resultantem illam
,
et sic translatam resolve in
binas , alteram parallelam axi OX , alteram axi OZ; prodi-
bunt componentes X (: NC':AC )
, H (: ND) , qua-
rum primam transfer in C' ut sit C'C&lt;nowiki&gt;''&lt;/nowiki&gt; (: NC' :) X; ad C'
applica binas vires K et —K aequales ,
'contrarias et parallelas axi OV; compone X (:.C' C&lt;nowiki&gt;''&lt;/nowiki&gt;) et K sese dirigen-&lt;nowiki&gt;''&lt;/nowiki&gt;
tem juxt'a C'F , sitque C'L directio resultantis ; produc LC'
donec in V occurrat axi OX ; transfer in V novam istam re-
sultantem ,
et sic translatam resolve in binas , alteram juxta
OX, alteram parallelam axi OV ; emergent componentes
X (:VV':C' C&lt;nowiki&gt;''&lt;/nowiki&gt;) ,K (:VF'): compone nunc V et —H;
produc directionem resultantis donec rectae C' F occurrat
v. gr. in N'; hanc resultantem transfer inN' , et sic traus-
latam resolve' tn duas
, alteram parallelam axi OV, alteram
axi OZ ; exurgeut componentes ?et —H applicitae puncto N':
hoc pacto vi P poterunt substitui sex vires
Z,,H — H applicitae punctis A, N, Net parallelae axi OZ,
K, ï— K applicitae punctis V, C' et parallelae axi OV,
X applicita puncto V et agens juxta OX.
Consimiles operationes cum possint instaurari quoadP' ,P&lt;nowiki&gt;''&lt;/nowiki&gt;,... non pluribus opus est
,
ut pateat veritas assertionis.20
7. Axes OX , OY, OZ sumantur orthogonales ; erit
H : X = ND : NC' NC zX Z :
H ,
et consequenter perpendicula ducta ex N in plana YOZ , XOZ
exprimentur per
2X
H
g ;
erit quoque
H : Y = AC ' : C'N ' = 2 : C'N' =
2Y
H
ac proinde perpendicula ducta ex N' in eadem plana YOZ ,
XOZ exprimentur per
x18+1;
insuper
Vi : Ci = VV' : VF' , seu x - OV : y = X , K ,
ex qua eruitur perpendiculum ductum ex Vin planum YOZ,
nempe
OV =
y X
K
8 . '* Quod in ordine ad Pest X, Y, Z, H, K, sit X ', Y ,
Z ', H , K ' in ordine ad P ', sit X ”, Y &lt;nowiki&gt;''&lt;/nowiki&gt;, Z&lt;nowiki&gt;''&lt;/nowiki&gt;, H ” , K &lt;nowiki&gt;''&lt;/nowiki&gt; in or&lt;nowiki&gt;''&lt;/nowiki&gt;
dine ad P, etc. ... Systema ( 6&lt;nowiki&gt;''&lt;/nowiki&gt;) virium parallelarum axi OZ
consistet in aequilibrio sub tribus istis conditionibns ( 59)
2 + Z ' + Z &lt;nowiki&gt;''&lt;/nowiki&gt; +... + H + HP + H &lt;nowiki&gt;''&lt;/nowiki&gt; + .- H - H²- H &lt;nowiki&gt;''&lt;/nowiki&gt; -... = 0 ,
x2+x+2 + .. + ( x -7 ) +la
ZX
H
- ) H +
'
x H - X'H '-...
20
7
∙∘∙ Axes OX, 07, OZ sumantur orthogonales ;erit
H:X:ND:NC': -Nc': f—X
...-7
et consequenter perpendicula ducta ex N in plana ïOZ, XOZ
exprimentur per
zX x——s.7-i
eritquoque
H. r:.tcx ea:: aut: 2?
—, H
ac proinde perpendicula ducta ex N' in eadem plana TOZ,
XOZ exprimentur per
T
xsf'l'ïïi—i insuper
Vi:C'i:VV':VF',senx—OV:J:X,K. ↴
ex qua eruitur perpendiculum ductum ex Vin planum TOZ, )-
nempe )
∘∇∶∙≖−⋅\sum⋮∙ K
8. 01: Quod in ordine adPestX, T, Z, H, K, sit X'.ï',
Z', H', K' in ordine ad P', sit X&lt;nowiki&gt;''&lt;/nowiki&gt;, T', Z&lt;nowiki&gt;''&lt;/nowiki&gt;, H&lt;nowiki&gt;''&lt;/nowiki&gt;, K&lt;nowiki&gt;'''&lt;/nowiki&gt;m or-
dine ad P&lt;nowiki&gt;''&lt;/nowiki&gt; ,
etc.. «Systema (60) virium parallelamm axi OZ
consistet in aequilibrio sub tribus istis conditionibus (50)
z −⊦ ⊠∣⊣−≀∥⊹∙∙∙−⊦∐⊣−∐∣⊣−∐∥−⊦ ∙∙⋅− ⊟∙↧∓∣∙⊟∥∙∙∙∙ :
xZ-l—x'ZH—&lt;nowiki&gt;''&lt;/nowiki&gt;xl-(x- fl—iï' H—1-(x'
-
)&lt;nowiki&gt;''&lt;/nowiki&gt;IX, H'—)- ..

&lt;nowiki&gt;:&lt;/nowiki&gt; r H—x'H'-.. .
&lt;nowiki&gt;:&lt;/nowiki&gt; o.21

y2 +y2 + ... + 38+y'! '+ ..- ( o + #) : -
(-+ -+* ) r -...--.
seu
2 + 2 + Z&lt;nowiki&gt;''&lt;/nowiki&gt; + ... = 0 ,
x2–2x + x2–5x' + x Z&lt;nowiki&gt;''&lt;/nowiki&gt; _z&lt;nowiki&gt;''&lt;/nowiki&gt; X &lt;nowiki&gt;''&lt;/nowiki&gt; + ... = o,
y2 - zY + y'Z' — zY + y&lt;nowiki&gt;''&lt;/nowiki&gt; Z&lt;nowiki&gt;''&lt;/nowiki&gt; —z&lt;nowiki&gt;''&lt;/nowiki&gt; Y&lt;nowiki&gt;''&lt;/nowiki&gt; + ... :. =0.
360&lt;nowiki&gt;''&lt;/nowiki&gt;

Systema (69) coalescens ex viribus jacentibus in plano XOY
simulque parallelis axi OY consistet in aequilibrio 

sub duabus istis conditionibus ( 5° ) . 

Y - K + Y - K + Y&lt;nowiki&gt;''&lt;/nowiki&gt; _K&lt;nowiki&gt;''&lt;/nowiki&gt; + . + K + K + K + ... = a,
2{Y -K)+7 (9 –K)+- + (3 - X) +(37 )K +
seu
Y + Y + Y &lt;nowiki&gt;''&lt;/nowiki&gt; + ... = 0,
xY4yX + x'Y' — y'X ' + x &lt;nowiki&gt;''&lt;/nowiki&gt; Y&lt;nowiki&gt;''&lt;/nowiki&gt; -- y&lt;nowiki&gt;''&lt;/nowiki&gt; X &lt;nowiki&gt;''&lt;/nowiki&gt; +... =0.
0.}10&lt;nowiki&gt;''&lt;/nowiki&gt;)
Systema ( 6°) virium agentium juxta OX 

consistet in aequilibrio sub ista tantum conditione

X + X+X&lt;nowiki&gt;''&lt;/nowiki&gt;+... = o ( a &lt;nowiki&gt;''&lt;/nowiki&gt; &lt;nowiki&gt;''&lt;/nowiki&gt; ).
Inferimus solidum liberumque corpus viribus P , P' ,
P&lt;nowiki&gt;''&lt;/nowiki&gt; , ... sollicitatum haud mansurum in aequilibrio, nisi ex
pletis conditionibus ( a' ) , ( a &lt;nowiki&gt;''&lt;/nowiki&gt; ) , ( a &lt;nowiki&gt;''&lt;/nowiki&gt; &lt;nowiki&gt;''&lt;/nowiki&gt; ); quas ita scri
bimus ( 30 )
21
⊺∄⊹↗⋅∄∣−⊢∙∙∙⊹∫∐⊹∫∣∐∣⊹∙⋅∙− (
∫⊹.äï.) H —
(y'-]- ⋮⋅≨⋚∣⇀∙≻ H'—. ..

&lt;nowiki&gt;:&lt;/nowiki&gt; o,

seu
∅⊣−∅∣⊣−⊈∥⊹∙∙∙∶∘∙ ;
(a') xZ—zX-I-x'Z'— z'X'-l-x&lt;nowiki&gt;''&lt;/nowiki&gt;Z&lt;nowiki&gt;''&lt;/nowiki&gt;— z'X&lt;nowiki&gt;''&lt;/nowiki&gt;—-]-. ..

&lt;nowiki&gt;:&lt;/nowiki&gt; o,

yz -— zV-l—J'Z'—z'ï'—I- y&lt;nowiki&gt;''&lt;/nowiki&gt;Z&lt;nowiki&gt;''&lt;/nowiki&gt;—z&lt;nowiki&gt;''&lt;/nowiki&gt;ï&lt;nowiki&gt;''&lt;/nowiki&gt;-l— ..

&lt;nowiki&gt;:&lt;/nowiki&gt;

.
0.
Systema (60) coalescens ex viribus jacentibus in plano XOV
simulque parallelis axi OV consistet in aequilibrio sub dua-
bus istis conditionibus ( 5o )-
r—x-t-x'—x'—l-1z&lt;nowiki&gt;''&lt;/nowiki&gt;—xq-.. —[-K-]-K'-)-K&lt;nowiki&gt;''&lt;/nowiki&gt;—]—. ..

&lt;nowiki&gt;:&lt;/nowiki&gt;

0,
I
'
X
&lt;nowiki&gt;!
IX ∣
..
ï—KH—x (r—x ⊢⊢⋅∙∙−⊢ xli?) x-l-(x ïk.-')K ∙⊦∙∙≔∶⋅∘⋅
seu .
y—I—T-I- ï''&lt;/nowiki&gt;—l— ..

&lt;nowiki&gt;:&lt;/nowiki&gt;

.
0, '
h
0&lt;nowiki&gt;''&lt;/nowiki&gt;)
xï—yX-l—x'ïL-y'X' x&lt;nowiki&gt;''&lt;/nowiki&gt;ï&lt;nowiki&gt;''&lt;/nowiki&gt;-y&lt;nowiki&gt;''&lt;/nowiki&gt; &lt;nowiki&gt;''&lt;/nowiki&gt;
∙⊦∙∙∙∶−−∙ ∙
Systema (60) virium agentium iuxta OX consistet in ae-
quilibrio sub ista tantum conditione '
,x-t—X'-l-X&lt;nowiki&gt;''&lt;/nowiki&gt;-1-...:o (a&lt;nowiki&gt;'''&lt;/nowiki&gt;).
Inferimus solidum liberumque corpus viribus P, P',
P&lt;nowiki&gt;''&lt;/nowiki&gt;, .. . sollicitatum haud mansurum in aequilibrio, nisi ex-
pletis conditionibus (a' )
,
( a&lt;nowiki&gt;''&lt;/nowiki&gt; )
,
(av&lt;nowiki&gt;''&lt;/nowiki&gt; ); quas ita scri-
bimus ( 3&lt;nowiki&gt;''&lt;/nowiki&gt;)22
EX
=
0
, EY
=
0
, E2
=
0
,
}
(
a
&lt;nowiki&gt;''&lt;/nowiki&gt;
)
2
( zYX)
= 0,2
( x2–2X
)
= 0,2
(x2
– zY
)
=
0..
9
'

&lt;nowiki&gt;#&lt;/nowiki&gt; Denotet

R
' resultantem ex viribus primi syste
matis
(
6
°
)
,
R
&lt;nowiki&gt;''&lt;/nowiki&gt; ex viribus secundi
,
R
&lt;nowiki&gt;''&lt;/nowiki&gt; ex viribus tertii&lt;nowiki&gt;''&lt;/nowiki&gt;

&lt;nowiki&gt;;&lt;/nowiki&gt;

erunt
( 12

&lt;nowiki&gt;:&lt;/nowiki&gt;

a
)
R
= EZ
,
R
= EY
,
R
&lt;nowiki&gt;''&lt;/nowiki&gt;
= EX
.
Recta
, in qua agit
R
&lt;nowiki&gt;''&lt;/nowiki&gt;
, occurrit axi OX sub angulo recto

&lt;nowiki&gt;;&lt;/nowiki&gt;

et disignante
r
&lt;nowiki&gt;''&lt;/nowiki&gt; distantiam inter
O et punctum occursus
erit
( 2º
.
7
°
)
.
r
“
R
”
=
x
(Y
—
K
)
+
x'(
Y
’
–
K
”
+ ...
XK
(
s
–
&lt;nowiki&gt;''&lt;/nowiki&gt;
)
k
'
+ ...,ideoque
?&lt;nowiki&gt;''&lt;/nowiki&gt;=
EfxY
-yX
)
.
R
&lt;nowiki&gt;''&lt;/nowiki&gt;
tra
potest
R
'
&lt;nowiki&gt;''&lt;/nowiki&gt; transferri in illud punctum occursus sicque
componi cum
R
&lt;nowiki&gt;''&lt;/nowiki&gt; ut inde obtineatur resultans VR
&lt;nowiki&gt;''&lt;/nowiki&gt;
2
+
R
&lt;nowiki&gt;''&lt;/nowiki&gt;
3.
Iterum
( 9. 9º
. 10
°
.
) patet ergo vires
P
P
'
,
P
'
,
, ...
duci vel ad ternas
, vel ad binas aequipollentes
.
10.
°

&lt;nowiki&gt;#&lt;/nowiki&gt; Recta

, in qua agit
R
'
, occurrit normaliter plano
XOY

&lt;nowiki&gt;;&lt;/nowiki&gt; et designantibus

a
'
,
b
' coordinatas istius occursus
,
erunt
( 2º
. 7º
.
)
a
'
$
(xZ
- zX
)
R
'
6
Egy
Z
-
Y
) 1 R
Occurrent sibi mutuo R’et VR
”
?
+
R
&lt;nowiki&gt;''&lt;/nowiki&gt;
2, ac proinde
jacebunt in eodem plano
, quotiescumque
a
' et
b
' recident
in duas quasvis ex coordinatis illius rectae in qua agit
VR'
2
+
R
'&lt;nowiki&gt;'''&lt;/nowiki&gt;
2

&lt;nowiki&gt;;&lt;/nowiki&gt; propterea

22
ZX:0,Zï:o,ZZ:o,

&lt;nowiki&gt;;&lt;/nowiki&gt; (aVIII)

\sum (xï—ïyX):o.Z(xZ—zX):0,2(yZ—zï): 0.
9. 01: Denotet B' resultantem ex viribus primi syste-
matis '(60 ), B&lt;nowiki&gt;''&lt;/nowiki&gt; ex viribus secundi , B&lt;nowiki&gt;'''&lt;/nowiki&gt; ex viribus tertii;
erunt
( 12. a)
R,:Z Z, B&lt;nowiki&gt;''&lt;/nowiki&gt;:Zï, R&lt;nowiki&gt;'''&lt;/nowiki&gt;:ZX.
Recta, in qua agit R&lt;nowiki&gt;''&lt;/nowiki&gt;, occurrit axi OX sub angulo recto

&lt;nowiki&gt;;&lt;/nowiki&gt;

et disignante r&lt;nowiki&gt;''&lt;/nowiki&gt; distantiam inter
0 et punctum occursus,
ertt∙
(
2
∘
∘
. 7. ).
r&lt;nowiki&gt;''&lt;/nowiki&gt;R&lt;nowiki&gt;''&lt;/nowiki&gt;:x(ï—K)—)—x'(ï'—K')—-)—...-)- (x... JKX.)
K −⊦
(x'-— 2274.) K' −∙⊢∙
∙ .,ideoque r&lt;nowiki&gt;''&lt;/nowiki&gt;-— xwy-FK)

&lt;nowiki&gt;:&lt;/nowiki&gt;

potest B&lt;nowiki&gt;'''&lt;/nowiki&gt; transferri in illud punctum occursus
, sicque
componi cum B&lt;nowiki&gt;''&lt;/nowiki&gt; ut inde obtineatur resultans l/B&lt;nowiki&gt;''&lt;/nowiki&gt;3-l—B'&lt;nowiki&gt;'''&lt;/nowiki&gt;. Iterum (9. 90.100.) patet ergo vires P P', P', , .. .
tra-
duci vel ad ternas, vel ad binas aequipollentes.
↿∘∙∘⋕ Recta, in qua agit B', occurrit normaliter plano
XOT; et designantibus a', b' coordinatas istius occursus,
erunt (20. 70.)
↙⋮∣∙− X(xZ—zX) b' ∙∙∙
\sum (yZ—zï)
B'
' R'
⋅
Occurrent sibi mutuo B' et l/B&lt;nowiki&gt;''&lt;/nowiki&gt;2-)-B&lt;nowiki&gt;'''&lt;/nowiki&gt;2, ac proinde
iacebunt in eodem plano, quotiescumque a' et b' recident
in duas quasvis ex coordinatis illius rectae in qua agit
⇂∕ B&lt;nowiki&gt;''&lt;/nowiki&gt;2-I-B&lt;nowiki&gt;'''&lt;/nowiki&gt;2; propterea23
a ' - p &lt;nowiki&gt;''&lt;/nowiki&gt; : 6 = R : R &lt;nowiki&gt;''&lt;/nowiki&gt;
et consequenter
b' R' + ( r &lt;nowiki&gt;''&lt;/nowiki&gt; – a ' ) R &lt;nowiki&gt;''&lt;/nowiki&gt; = 0 ;
quae , adhibitis substitutionibus, traducitur ad
EXE(yZ — ZY) + EYXzX— « Z ) + EZE (xY yX ) = 0.
Sub hac ilaque conditione occurrent sibi mutuo vires
R' , V R &lt;nowiki&gt;''&lt;/nowiki&gt;2+ R &lt;nowiki&gt;''&lt;/nowiki&gt; ), dabuntque resultantem
VR2+ R &lt;nowiki&gt;''&lt;/nowiki&gt;2 + R &lt;nowiki&gt;'''&lt;/nowiki&gt; a = V (EX)2 + (PY )2+ ( EZ )2.
11 . '* Si nequeunt vires alium gignere motum ni
si circa immobilem axem Oz , quisque videt aequilibrii
conditiones redactum iri ad unicam r ' = 0 , seu ad quar
tam ( a &lt;nowiki&gt;''&lt;/nowiki&gt; ), Ad haec si nequeunt vires alium gignere mo
tum nisi circa immobile punctum 0 , redigentur aequili
brii conditiones ad r&lt;nowiki&gt;''&lt;/nowiki&gt; = 0 , a' = 0,6 = 0 , seu ad quar
tam , quintam et sextam ( a )
12. '* Fac ut duo solida corpora A et B ( Fig. 9) ,
alterum viribus P , P , P &lt;nowiki&gt;''&lt;/nowiki&gt;... sollicitatum , alterum viri
bus Q , , Q &lt;nowiki&gt;''&lt;/nowiki&gt; , ... , sese invicem aeque premendo apud da
lum mutui contactus punctum C maneant in aequilibrio ;
quaeritur istiusmodi pressionis magnitudo w. Duc per C pla
num tangens DD' , cui normaliter insistat recta ECE': de
notent fig, h coordinatas puncti C ; a , á , a &lt;nowiki&gt;''&lt;/nowiki&gt; angulos
interceptos recta CE axibusque orthogonalibus OX , OY ,
OZ ; et quod in ordine ad P' , P' , P &lt;nowiki&gt;''&lt;/nowiki&gt; , ... est X, Y , Z,
á , : , X , Y , Z , X ', . . . sit a , b , c , a , ... A , B ,
C , A ', ... in ordine ad C , Q , ... Pressio agens
versus E resolvetur in ternas
23
a'—- r&lt;nowiki&gt;''&lt;/nowiki&gt;: 6':a&lt;nowiki&gt;'''&lt;/nowiki&gt;: a&lt;nowiki&gt;''&lt;/nowiki&gt;
et consequenter
↘∙∙
b' B&lt;nowiki&gt;'''&lt;/nowiki&gt;—i— ( r&lt;nowiki&gt;''&lt;/nowiki&gt;-—a') B&lt;nowiki&gt;''&lt;/nowiki&gt;: 0

&lt;nowiki&gt;;&lt;/nowiki&gt;

quae , adhibitis substitutionibus, traducitur ad
ZXZUZ—zTH-ZïXzX—xZH-ZZZ (a.-T —JX):o.
Sub hac itaque conditione occurrent sibi mutuo vires
B', l/ B&lt;nowiki&gt;''&lt;/nowiki&gt;2-)- B&lt;nowiki&gt;'''&lt;/nowiki&gt;, dabuntque resultantem
⇂∕↓↖⋅≖−⊦↓⊰⋅⋅≖−⊢∐⋯≖∶ ⇂∕ (mun-)- (zx) ≕⊣−≺ \sum∣∠≻≖∙
11.&lt;nowiki&gt;''&lt;/nowiki&gt;; Si nequeunt vires alium gignere motnm ni-
si circa immobilem axem OZ, quisque videt aequilibrii
conditiones redactum iri ad unicam r&lt;nowiki&gt;''&lt;/nowiki&gt; :o
, seu ad quar-
tam (a'&lt;nowiki&gt;'''&lt;/nowiki&gt; ). Ad haec si nequeunt vires alium gignere mo-
tum nisi circa immobile punctum 0 , redigentur aequili-
brii conditiones ad r&lt;nowiki&gt;''&lt;/nowiki&gt;:0, a':o, b':o, seu ad quar-
tam, quintam et sextam ( am&lt;nowiki&gt;''&lt;/nowiki&gt;)
12.&lt;nowiki&gt;''&lt;/nowiki&gt;: Fac ut duo solida corpora A et B (Fig. 9),
alterum viribus P , P', P&lt;nowiki&gt;''&lt;/nowiki&gt;. .. sollicitatum , alterum viri-
bus Q, Q' , Q&lt;nowiki&gt;''&lt;/nowiki&gt;, .. ., sese invicem aeque premendo apud da-
tum mutui contactus punctum C maneant in aequilibrio;
quaeritur istiusmodi pressionis magnitudo 'a'. Duc per C pla-
num tangens DD', cui normaliter insistet recta ECE': de-
notent f, g ,
]: coordinatas puncti C; at, a', a&lt;nowiki&gt;''&lt;/nowiki&gt; angulos
interceptus recta CE axibusque orthogonalibns OX, Of
, OZ; et quod in ordine ad P'
, P', P&lt;nowiki&gt;''&lt;/nowiki&gt;,
...
est a:, 7, z,
x', . . X,ï, Z, X',. . . sita,b, c, a,... A,. B,
C, A', . . .
in ordine ad Q', Q, . . . Pressio :: agens
versus E resolvetur in ternas24
cosa , cose , a cos &lt;nowiki&gt;''&lt;/nowiki&gt; ,
agens vero versus E resolvetur in ternas
w cos ( 180 ° - « ) = - COS Q, a cos ( 180 ° - = - a coseć,
cos ( 180º – Ø&lt;nowiki&gt;''&lt;/nowiki&gt; ) W cos a :
in primo casu w librat ex hypothesi vires P, P,
in secundo vires Q, C, ... Igitur
EX +w cosa = 0, Erto cosá = 0 , xZ + w cosa &lt;nowiki&gt;''&lt;/nowiki&gt; = 0 ,
Σ Α W cosa = 0 ,
EB - cosa = 0,8C — a cos &lt;nowiki&gt;''&lt;/nowiki&gt; = 0 , E ( «Y -y X ) +
W ( f cos ' - g cosc) =0 ,
ElxZ - 2X ) + o ( f cosc &lt;nowiki&gt;''&lt;/nowiki&gt; — h cosc ) =0 , Ely2 -zY) +
wig cosa &lt;nowiki&gt;''&lt;/nowiki&gt; -hcosé ) =0 ,
(aB - 6A ) - ( fcos - g cos ) = 0 ,E (aC - A )
a ( f cosa &lt;nowiki&gt;''&lt;/nowiki&gt; -hcosa) = 0 , E (6C - cB ) - ( g cosa &lt;nowiki&gt;''&lt;/nowiki&gt; - h cosa') = 0 .
Eliminata , prodibunt undecim aequationes , inde
pendentes ab ipsa a , inter quantitates datas ; quibus ae
quationibus expletis, habebitur aequilibrium , poteritque ab
una quavis ex duodecim praecedentibus erui valor u .
13.0# Solidum corpus sollicitatum viribus , P P ',
P &lt;nowiki&gt;''&lt;/nowiki&gt; , ... delineatur duobus punctis fixis , sumptis in axe
v. gr. OZ ; sic facile determinabuntur pressiones M, N , L
et M ', N ', L' exercitae in puncta illa juxta coordinatos a.
xes Ox , OY, OZ. Exprimant m, n , l coordinatas unius
ex duobus panciis , et m ', ní, ľ coordinatas alterius. Quo
uiam spectari debent
24
a: cosa, a cosa', wcosac' ,
agens vero versus E' resolvetur in ternas
m cos(1800—a): — arcus a, acos (1800—at'):—w cosa',
a cos ( 180o -— ac&lt;nowiki&gt;''&lt;/nowiki&gt;) ∶≖ −meos ac&lt;nowiki&gt;''&lt;/nowiki&gt;:
in primo casu ut librat ex bypOthesi vires P, P', .
∙ ∙
,
in secundo vires Q, Q', . . .
Igitur
2X —l—w cosa::o, Zy-l—a cosa':o,ZZ—l-ar cosa:&lt;nowiki&gt;''&lt;/nowiki&gt;:o,
EA — z: cosa: :0,
2B —a cosa':o,ZC—z.ïcos at&lt;nowiki&gt;''&lt;/nowiki&gt;:o, Z (xï —7 X) −−∣−
15 (fcosa&lt;nowiki&gt;''&lt;/nowiki&gt;—-g cos ac) :0,
2( a:Z—zX)-I—w(fcosa&lt;nowiki&gt;''&lt;/nowiki&gt;—hcosa):o, XOZ—z?) −⊦
w(g cosa&lt;nowiki&gt;''&lt;/nowiki&gt;--hcosat'):o,
E( aB—bA) —w(fcos a'—gcosa):o,2 (aC—cA)—-
a(fcosa&lt;nowiki&gt;''&lt;/nowiki&gt;-h cos a):o,2 (bC-cB) -zz(gcos a&lt;nowiki&gt;''&lt;/nowiki&gt;- hcosac'):o.
Eliminata a, prodibunt undecim aequationes, inde-
pendentes ab ipsa a'
,
inter quantitates datas.; quibus ae-
quationibus expletis, habebitur aequilibrium, poteritque ab
una quavis ex duodecim praecedentibus erui valor a.
1394: Solidum corpus sollicitatum viribus, P P',
P&lt;nowiki&gt;''&lt;/nowiki&gt;, . . . detineatur duobus punctis fixis, sumptis, in axe
v. gr. OZ; sic facile determinabantur pressiones M, N, L
et M', N', L' exercitae in puncta illa juxta coordinatos a-
xes OX, DV, 02. Exprimant m, n, !coordinatas unius
ex duobus punctis, et m', n'. [ coordinatas alterius. Quo-
niam spectari debent25
M, N , -L, — M ', - N - L'
tanquam vires , quibus librantur caeterae P , P , P' ... ,
ac insuper m = 0 , n = o , m' =0 , n = 0 , necnon ( 110. )
(xY - yX ) = 0 : iccirco ( 8º. a &lt;nowiki&gt;''&lt;/nowiki&gt; )
EX - M - M ' = 0 , EY -N - N = 0,8Z -L - L ' = 0 ,
{ ( xZ - 2X ) + 2M + l'M' = 0 , E ( yZ – zY) +IN +
Ľ N' = 0 ;
quarum tertia nos edocet axem OZ premi vi XZ in dire
ctione z , reliquae vero suppeditant M , M' , N , N' . Si P ,
P ' , P &lt;nowiki&gt;''&lt;/nowiki&gt; , ... evadunt parallelae axi OZ, et se dirigunt ad
plagam negativam , erunt ( 12: 13, 2° ) ,
EX = 0 , EY = 0 , EZ = P - P - P ' -... = - R,
ExZ- X ) = - xP - X'P' - &lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; --... X, R,
(y2 — zY) = - ype ' P' y &lt;nowiki&gt;''&lt;/nowiki&gt; P &lt;nowiki&gt;''&lt;/nowiki&gt; —... = - y . R;
hic denotant P, P ', P &lt;nowiki&gt;''&lt;/nowiki&gt; , ... virium duntaxat intensitates. Quare
M + M ' = 0 , N + N = 0 , L + L + R = 0,2M +
I'M – x, R = 0 , 2N + IN - Y , R = 0 ;
unde
M = -M'
1, R
1 - T '
N = -N y R ,
, L
L +
+ LEL' = - R.
3
25
—M,-FNg-Lg—M'g—N' '..L,
tanquam vires
, quibus librantur caeterae P, P', P&lt;nowiki&gt;''&lt;/nowiki&gt;. ..,
ac insuper m::o, &lt;nowiki&gt;''&lt;/nowiki&gt;:D, in'-:(), &lt;nowiki&gt;'''&lt;/nowiki&gt; ∶−−⋅ o, necngn ( 110.)
Xxï—yX :) o:
iccirco( 80. a'&lt;nowiki&gt;'''&lt;/nowiki&gt; )
EX—M—M':o,2ï-—N-—N' :o,ZZ—L-—L' :0,
Si xZ—zX)—I-lM-I— l'M':o,Z(yZ—-zï)—l-IN-l-
&lt;nowiki&gt;!' N':o;
quarum tertia nos edocet axem OZ premi vi ZZ in dire-
ctione&lt;/nowiki&gt; :, reliquae vero suppeditant M, M' , N ,N'. Si P,
P'
,P&lt;nowiki&gt;''&lt;/nowiki&gt; ,... evadunt parallelae axi OZ, et se dirigunt ad
plagam negativam, erunt (12: 13. 20 ),
ZX:o, Zïzo,zz :P—P'-—P&lt;nowiki&gt;''&lt;/nowiki&gt;-—. ..

&lt;nowiki&gt;:&lt;/nowiki&gt; — R,

XxZ—QX):—xP -x'P'
-— x&lt;nowiki&gt;''&lt;/nowiki&gt;P&lt;nowiki&gt;''&lt;/nowiki&gt; —-. .

&lt;nowiki&gt;:&lt;/nowiki&gt;

. —a:. B,.
2(yz ∙∙∙ zï):—yP—— r' P' —.7&lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; ∙∙∙ ∙ ∙ ∙ ∶−∙ ∙−−∫∎ R;
bic denotant P, P', P&lt;nowiki&gt;''&lt;/nowiki&gt;, ... virium duntaxat intensitates. Quare
∐−⊦∐∣∶∘∙∾⊣−∐∙−−∶∘∙ ↧⋅−↽↧⋅∙−↽≖↸≓∘∙≀∐⊣⊸
I'M' — x,R:o, lN-i-l'N'—-y,R:o;
nnde
M: x,B -—M':—
---—- r—z '
Nz—N' :-—l',y'—-—R—2,L-I—L':—R- 326
14. Ex dictis de virium aequilibrio deduci possunt
aequilibrii leges in Machinis. Sic v. gr. in vecte poten
tia librabit resistentiam seu pondus, quotiescumque ( sumptis
( 10. 10 ) momentis quoad axem immobilem, circa quem po
test vectis moveri ) momentum potentiae aequatur momento
resistentiae.Idipsum obtinet quoad Axem in peritrochio ; idi
psum quoad trochleam fixam . Potentia et resisteutia istis
machinis applicantur in directione parallela planis perpen
dicularibus axi immobili; perinde igitur ( 10. 10 ° ) erit si
ve in eorum uno sive in altero accipiantur momenta ;
poteritque vectis repraesentari per lineam mobilem circa
punctum fixum , quod dicitur fulcrum , hypomoclion : axis
in peritrochio per circulares projectiones rotae ac cylin
dri in uno quovis ex dictis planis , mobiles circa com
mune immobile centrum : trochlea fixa per circulum ro
tatilem circa suum centrum , cujus circuli radius sit ipse
trochleae radius; verum quia in trochlea fixa nihil sunt
aliud perpendicula momentorum propria nisi trochleae
radii, ducti ad puncta ubi funis desinit ipsam trochleam
tangere, ideo erunt aequalia , et consequenter aequilibrium
in trochlea fixa importat potentiae ac resistentiae ae
qualitatem.
Ad trochleam mobilem quod spectat , sit P (Fig. 10)
pondus sustinendum a potentia Q : quoniam in casu aequi
librii , P et Q praebeant oportet resultantem R transeuntem
per punctum fixum 0, iccirco ( 9. 10. ) Q : P = sin \beta : sin a
= sin i : sin 2i = cos x : sin 2x = cos x : 2sin x cos
1 : 2 sin ; ac proinde
P
Q
2 sin s
Posuimus angulum OaQ dividi aequaliter directione
ponderis P : id vero facile intelligemus animadvertendo , si
filum OaQ fixum in 0 et Q , tenditur vi applicita puncto
26

14. Ex dictis de virium aequilibrio deduci possunt
aequilibrii leges in Machinis. Sic v. gr. in vecte poten-
tia librabit resistentiam seu pondus, quotiescumque ( sumptis
(10. 100) momentis quoad axem immobilem, circa quem po-
test vectis moveri ) momentum potentiae aequatur momento
resistentix-Idipsum obtinet quoad Axem in peritrochio ; idi-
psum quoad trocbleam lixam. Potentia et resistentia istis
machinis applicantur in directione parallela planis perpen-
dicularibus axi immobili; perinde igitur( 10. 100) erit si-
ve in eorum- uno sive in altero accipiantur momenta;
poteritque vectis repraesentari per lineam mobilem circa
punctum fixum, quod dicitur fulcrum, hypomoclion: axis
in peritrochio per circulares proiectiones rotae ac cylin-
dri in uno quovis ex dictis planis, mobiles circa com-
mune immobile centrum: trochlea lixa per circulum ro-
tatilem circa suum centrum,cuius circuli radius sit ipse
trochleae radius; verum quia in trochlea fixa nihil sunt
aliud perpendicula momentorum propria nisi trochleae
radii, ducti ad puncta ubi funis desinit ipsam trocbleam
tangere, ideo erunt aequalia ,
et consequenter aequilibrium
in trochlea fixa importat potentiae ac resistentiae ae-
qualitatem.
Ad trocbleam mobilem quod spectat , sit P (Fig. 10)
pondus sustinendum a potentia Q: quoniam in casu aequi-
librii , P et Q praebeant oportet resultantem R transeuntem
per punctum fixum 0, iccirco (9. 10.) Q: P:sin 13: sin «

&lt;nowiki&gt;:&lt;/nowiki&gt; sin i :sin 2i:cos x : sin Zx:cos x: Zsinxcosx
&lt;nowiki&gt;:&lt;/nowiki&gt; 1: 2 sin a:; ac proinde

P Q&lt;nowiki&gt;''&lt;/nowiki&gt;üü'
Posuimus angulum OaQ dividi aequaliter directione
ponderis P: id vero facile intelligemus animadvertendo, si
iilum OaQ fixum in 0et Q ,
tenditur vi applicita puncto
∙∙∙ '. 'una-
,.. ↙∙∙∎⋅−27
a libere excurrenti juxta ipsum Oal , punctum a necessa
rio permansurum in perimetro ellipseos , cujas foci O et Q;
ideoque in casu aequilibrii vim illam fore perimetro elli
pseos normalem ; quod certe importat praedictam anguli
OaQ divisionem . Vide n. 55. 13 .
Etiam sic : cum in casu aequilibrii funis ubique ma
neat aeque distentus , pondus P librabit resultantem R'
ex duabus viribus aequalibus Q et concurrentibus apud
punctum a ; et quoniam R' aequaliter dividit angulum Oal ,
idipsum dicendum erit de ponderis directione. Jamvero
R ' ( = P2) = Q + + Q2 + 2QQ cos 2i =2Q ( 1 +cos 2i) =
4 Q* cos 2i = 4 Q* sinºx :
rursus igitur
P
-
2sin x
angulo x = 90° respondebit minimal ; erit Q = P
2
si x = 30° ; vergente x ab 30° ad 09 , verget Q ab P ad co .
15. Vectis primi generis nuncupatur , si fulcrum sit
inter potentiam et pondus ; dicitur secundi generis si pon
dus sit inter fulcrum et potentiam ; denique si potentin me.
dium locum teneat inter fulcrum et pondus , vectis tertii ge
neris vocatur. Hinc vectes primi et secundi generis poten
tiam juvant , quatenus eo minor requiritur potentia ad da
tum pondus sustinendum , quo major est potentiae distantia
a fulcro relate ad ponderis distantiam ab eodem fulcro ;
adeo ut quodvis pondus utcumque ingens possit vectis ope
a quacumque potentia utcumque exigua sustineri : quod cum
bene nosset Archimedes , illud dixisse fertur Hieroni Regi ..
dic ubi consistam , coelum , terramque movebo ,, : vectis au
tem tertii generis potentiam non juvat , quia in hoc vecte
potentia minus distat a fulcro quam resistentia seu pondus.
27

&lt;nowiki&gt;::&lt;/nowiki&gt; libere excurrenti juxta ipsum OaQ , punctum :: necessa-

rio permansurum in perimetro ellipseos, cuius foci O et Q;
ideoque in' casu aequilibrii vim illam fore perimetro elli-
pseos normalem; quod certe importat praedictam anguli
OaQ divisionem . Vide n. 55. 13.0
Etiam sic :cum in casu aequilibrii funis ubique ma-
neat aeque distentus , pondus P librabit resultantem R'
ex duabus viribus aequalibus Q et Q concurrentibus apud
punctum a; et quoniam R' aequaliter dividit angulum OaQ,
idipsum dicendum erit de ponderis directione. Iamvero
a'
∙≺⇌−− re:? -1-Q*-l—2QQcos2i:2Q'(1-l-cva 20:
4Q' cos 3i:4Q3 sin'x:
rursus igitur
P
Q— 2sinx,
angulo x:900 respondebit minima Q

&lt;nowiki&gt;:&lt;/nowiki&gt; ä; erit Q:P

si a: 300 ,- vergente :: ab 300 ad 00
,
verget Qab P ad 00
.

15. Vectis primi generis nuncupatur, si fulcrum sit
inter potentiam et pondus; dicitur secundi generis si pon-
dus sit inter fulcrum et potentiam ;denique si potentia me-
dium locum teneat inter fulcrum et pondus , vectis tertii ge-
neris vocatur. Hinc vectes primi et secundi generis poten-
tiam iuvant, quatenus eo, minor requiritur potentia ad d'a-
tum pondus sustinendum , quo maior est potentiae distantia
a fulcro relate ad ponderis distantiam ab eodem fulcro;
adeo ut quodvis pondus utcumque ingens possit vectis ope
a quacumque potentia utcumque exigua sustineri :quod cum
bene nosset Archimedes , illnd dixisse fertur Hieroni Regi ,,
dic ubi consistam ,coelum ,terramque movebo ,, :vectis an-
tem tertii generis potentiam non juvat , quia in hoc vecte
potentia minus distat a fulcro quam resistentia seu pondus.28
Ex indicata vectis theoria redditur ratio innumerabi
liam effectuum quos quotidie cernimus fieri ; ac primo qui
dem quicumque ex lapidicinis extrahunt lapides utuntur ut
plurimum vecte ferreo : quoties autein multum resistit la
pis sive propter magnitudinem sive quod nimis firmiter aliis
adhaereat , tunc hypomoclion quam proxime ponderi admo
vent , ut facilius moveant , quod vulgo dicitur ,, dar la
leva ,, . Pro hypomoclio antem utuntur quovis sustentaculo
v . gr. lapide ; saepe etiam cum duo lapides ab invicem
sejungendi sunt , unus respectu alterius habet rationem hy
pomoclii . Hic notandum est maxillas quoque esse vectes
secundi generis quum cibi dentibus molaribus comminuen
di traduntur . Secundo : si avellendus est clavus ope mal
lei , quanto clavus , qui ponderis vicem obtinet , propior
fuerit hypomoclio , eo facilius educetur ; unde cum jam tan
tisper eductus est , ita ut extremitas mallei nequeat am
plius insistere subjectae tabulae aut parieti e quo est dedu
cendus , solemus aliud corpus interserere ut quam minima
sit distantia. Tertio : in forcipibus quoque duplex est vectis
primi generis , quorum unum est commune hypomoclion ,
clavus nempe circa quem uterque ramus volvitur , eoque va
lidius stringetur corpus quo rami , qua parte secant , brevio
res , qua parte vero applicatur potentia seu manus , longiores
erunt . Quarto : cum portas aperimus aut claudimus , eo
facilius id praestamas , quo longius a cardinibus eas impel
Iimus , nempe janua est vectis secundi generis , cujas hy
pomoclion sunt cardines. Quinto : remorum motu cymba
promovetur , quia remi sunt vectes secundi generis , quo
rum bypomoclion est aqua , cymba est pondus seu resi
stentia , manus hominis sunt potentia applicata : hinc quo
magis ab aqua remotae sunt manus quam punctum cym
bae , cui remi insistunt , eo majus est potentiae momen
ium. Sexto : ex his etiam intelligitur cur difficillima sit
bacali oblongi elevatio si per extremitatem accipiatur , el
cur quo longior fuerit ipse baculus , eo facilius curvetur
aut frangatur.
28
Ex indicata vectis theoria redditur ratio innumerabi-
lium efi'ectuum quos quotidie cernimus iieri ; ac primo qui-
dem quicumque ex lapidicinis extrahunt lapides utuntur ut
plurimum vecte ferreo :quoties autem multum resistit la-
pis sive prOpter magnitudinem sive quod nimis firmiter aliis
adhaereat ,
tuuc hypomoclion quam proxime ponderi admo-
vent
,
ut facilius moveant, quod vulgo dicitur
,, der in
leva ,, . Pro hypomocliol autem utuutur quovis sustentaculo
v. gr. lapide; saepe etiam cum duo lapides ab invicem
sejungendi sunt
, unus respectu alterius habet rationem hy-
pomoclii. Hic notandum est maxillas quoque esse vectes
secundi generis quum cibi dentibus molaribus comminuen-
di traduntur. Secundo: si avellendus est clavus ope mal-
lei, quanto clavus, qui ponderis vicem obtinet, propior
fuerit hypomoclio , eo facilius educetur ;unde cum iam tan-
tisper eductus est, ita ut extremitas mallei nequeat am-
plius insistere subjectae tabulae aut parieti e quo est dedu-
cendus , solemus aliud corpus interserere ut quam minima
sit distantia. Tertio :in forcipibus quoque duplex est vectis
primi generis, quorum unum est commune hypomoclion,
clavus nempe circa quem uterque ramus volvitur, eoque va-
lidius stringetur corpus quo rami
, qua parte secant
, brevio-
res, qua parte vero applicatur potentia seu manus ,
longiores
erunt. Quarto: cum portas aperimus aut claudimus , eo
facilius id praestamus
, quo longius a cardinibus eas impel-
limus , nempe janua est vectis secundi generis , cujus hy-
pomoclion sunt cardines. Quinto : remorum motu cymba
promovetur
, quia remi sunt vectes secundi generis , quo-
rum bypomoclion est aqua, cymba est pondus seu resi-
stentia , manus hominis sunt potentia applicata: hinc quo
magis ab aqua remotae sunt manus quam punctum cym-
hae, cui remi insistunt , eo majus est potentiae momen-
tum. Sexto : ex his etiam intelligitur cur difficillima sit
baculi oblongi elevatio si per extremitatem accipiatur ,
et
cur quo longior fuerit ipse baculus, eo facilius curvetur
aut frangatur.29
16. Supponantur nunc plures quotcumque vectes ita
dispositi , ut potentia Q in 0 ( Fig. 11 ) magis , puta decu
plo distet a fulcro A quam resistentia in L , quae simili
ter magis distet , puta noncuplo a fulcro C quam resisten
tia in K , quae rursus magis distet a fulcro D puta quin
tuplo quam resistentia in E , et haec similiter magis puta
quadruplo distet a fulcro G quam resistentia in F , haec
denique duplo magis distet a fulcro H quam pondus P in B.
Si res ita se habet , atque insuper potentia et pondus di
rectiones habeant perpendiculares ad respectivos vectes
factis AO = a , CL = a ', DK = a &lt;nowiki&gt;''&lt;/nowiki&gt; , GE = a &lt;nowiki&gt;''&lt;/nowiki&gt; , HF - a &lt;nowiki&gt;''&lt;/nowiki&gt; ,
AL = 6, CK = b' , DE = 6&lt;nowiki&gt;''&lt;/nowiki&gt; ,GF = 6 &lt;nowiki&gt;''&lt;/nowiki&gt; , HB = 6 &lt;nowiki&gt;''&lt;/nowiki&gt;
b &lt;nowiki&gt;''&lt;/nowiki&gt; , erunt
in casu aequilibrii,
L. 6 E. 6 &lt;nowiki&gt;''&lt;/nowiki&gt;
Q
F.6&lt;nowiki&gt;''&lt;/nowiki&gt; &lt;nowiki&gt;''&lt;/nowiki&gt;
il
K =
Kiba,K
E F
P. &lt;nowiki&gt;''&lt;/nowiki&gt;

&lt;nowiki&gt;;&lt;/nowiki&gt;

a
a&lt;nowiki&gt;''&lt;/nowiki&gt;
a '
IV
ex quarum multiplicatione prodibit
b 6'6&lt;nowiki&gt;''&lt;/nowiki&gt; 6 &lt;nowiki&gt;''&lt;/nowiki&gt; 8 &lt;nowiki&gt;''&lt;/nowiki&gt; P
Q
α α' α
P
a &lt;nowiki&gt;''&lt;/nowiki&gt; a &lt;nowiki&gt;''&lt;/nowiki&gt; 3600&lt;nowiki&gt;''&lt;/nowiki&gt;
Quisque videt haec applicari systemati cuicumque rotarum
dentatarum.
Supponantur quoque plures trochleae mobiles v.gr. tres
(Fig. 12) ; erunt ( 14)
Q
L
2 sin r &lt;nowiki&gt;''&lt;/nowiki&gt;
K р
LE 2 sin ac '
&gt; K = ;
2 sin x
et consequenter
Q =
P
23 sin x sin a ' sipx&lt;nowiki&gt;''&lt;/nowiki&gt;
Quod si detur systema coalescens ex trochleis fixis
v. gr , C , C' , C &lt;nowiki&gt;''&lt;/nowiki&gt;, C &lt;nowiki&gt;'''&lt;/nowiki&gt; ( Fig . 13 ) et ex mobilibus F, E, K
29
16. Supponantur nunc plures quotcumque vectes ita
dispositi , ut potentia Q in O (Fig. 11 ) magis , puta decu.
plo distet a fulcro A quam resistentia in L , quae simili-
ter magis distet
, puta noncuplo a fulcro C quam resisten-
tia in K, quae rursus magis distet a fulcro D piita quin-
tuplo quam resistentia in E, et haec similiter magis puta
quadruplo distet a fulcro G quam resistentia in F, haec
denique duplo magis distet a fulcro H quam pondus P in B.
Si res ita se habet , atque insuper potentia et pondus di-
rectiones habeant perpendiculares ad respectivos vectes
,
factis AO:a,CL :a'
, DK:
a&lt;nowiki&gt;''&lt;/nowiki&gt;, GE :a&lt;nowiki&gt;'''&lt;/nowiki&gt;,HF :a&lt;nowiki&gt;''&lt;/nowiki&gt;,
AL:&amp;, CK:6', DE :6&lt;nowiki&gt;''&lt;/nowiki&gt;, GF:b&lt;nowiki&gt;'''&lt;/nowiki&gt;, HB:ö&lt;nowiki&gt;''&lt;/nowiki&gt;
, erunt
in casu aequilibrii,
'
'
'. ∙
'&lt;nowiki&gt;'''&lt;/nowiki&gt; Q—qy'b,L—K'£.,K:E'f ∙ !' ,E—Eb ,F—Pf ;
a a a a a
ex quarum multiplicatione prodibit
Q 6 b' 1)&lt;nowiki&gt;''&lt;/nowiki&gt; b&lt;nowiki&gt;'''&lt;/nowiki&gt; 6&lt;nowiki&gt;''&lt;/nowiki&gt;P P ⇠
a
.: ∙
as an aut alv 3600
Quisque Videt baec applicari systemati cuicumque rotarum
dentatarum.
. Su pponantur quoque plures trochleae mobiles v. gr. tres
(Fig. 12) ; erunt (14). ⋅
et consequenter
Q.... 23 sinu: sinx' sin x&lt;nowiki&gt;''&lt;/nowiki&gt;
Quod si detur systema coalescens ex trochleis fixis
&lt;nowiki&gt;''&lt;/nowiki&gt;
gr, C ∙∁⋅∣ C&lt;nowiki&gt;''&lt;/nowiki&gt;. 0&lt;nowiki&gt;''&lt;/nowiki&gt; (Fig. 13) et ex mobilibus F, E, K30
uno eodemque fane conjunctis ; quoniam , librato systemate ,
funis ubique manet aeque tensus , ideo Q : Q = Q &lt;nowiki&gt;''&lt;/nowiki&gt;
Q &lt;nowiki&gt;''&lt;/nowiki&gt; = Q &lt;nowiki&gt;''&lt;/nowiki&gt; = Q = Q &lt;nowiki&gt;''&lt;/nowiki&gt; = Q &lt;nowiki&gt;''&lt;/nowiki&gt; &lt;nowiki&gt;''&lt;/nowiki&gt; . Jamvero
F E K
Q Q '&lt;nowiki&gt;'''&lt;/nowiki&gt; QP 2sin x' ' 2 sin x
2 sin 2
et consequenter
F = 2Q ' sin &lt;nowiki&gt;''&lt;/nowiki&gt; 2Q sin x &lt;nowiki&gt;''&lt;/nowiki&gt; , E = 2Q sin ü&lt;nowiki&gt;''&lt;/nowiki&gt; , K =
2Q sin x ;
cum igitur sint
L = Q&lt;nowiki&gt;''&lt;/nowiki&gt; &lt;nowiki&gt;''&lt;/nowiki&gt; , F +E + K +L = P ,
iccirco
2 Q sin x &lt;nowiki&gt;''&lt;/nowiki&gt; + 2 Q sin x' + 2 Q sin x +Q = P :
unde
P
Q =
1 +2 (sin x +sin x ' + sin x &lt;nowiki&gt;''&lt;/nowiki&gt; )
Fac demum ut puncta materialia K , K ', K &lt;nowiki&gt;''&lt;/nowiki&gt; , K '&lt;nowiki&gt;'''&lt;/nowiki&gt;,
( fig. 14 ) jungantur Glis K K' , K'K &lt;nowiki&gt;''&lt;/nowiki&gt; determinatae
quidem longitudinis, sed mobilibus circa K , K &lt;nowiki&gt;''&lt;/nowiki&gt; . Si pun
cta illa sollicitantur viribus Q , Q , Q &lt;nowiki&gt;''&lt;/nowiki&gt; , Q &lt;nowiki&gt;'''&lt;/nowiki&gt; , ad aequi
librium haec manifeste requirentur : potentia Q in di
rectione K'K tendens ab K' versus K ; resultans R' ex
Q et Q' in directione K &lt;nowiki&gt;''&lt;/nowiki&gt; K ' tendens ab K &lt;nowiki&gt;''&lt;/nowiki&gt; versus K' ; re
sultans R &lt;nowiki&gt;''&lt;/nowiki&gt; ex R' et Q &lt;nowiki&gt;''&lt;/nowiki&gt; in directione K &lt;nowiki&gt;''&lt;/nowiki&gt; K &lt;nowiki&gt;''&lt;/nowiki&gt; tendens ab K&lt;nowiki&gt;''&lt;/nowiki&gt; &lt;nowiki&gt;''&lt;/nowiki&gt; '
versus K &lt;nowiki&gt;''&lt;/nowiki&gt; ; potentia Q &lt;nowiki&gt;'''&lt;/nowiki&gt; in directione K &lt;nowiki&gt;''&lt;/nowiki&gt; K' ' ' tendens ab K &lt;nowiki&gt;''&lt;/nowiki&gt;
versus K' ' ' : demum ipsa Q's aequalis resultanti R &lt;nowiki&gt;''&lt;/nowiki&gt; .

&lt;nowiki&gt;*&lt;/nowiki&gt; Denotantibus X , Y , Z componentes coordi

natis orthogonalibusque axibus parallelas , in quas resolvi
tur Q, erunt
30
uno eodemque fune coniunctis; quoniam .
librato systemate,
funis ubique manet— aeque tensus
, ideo, Q:Q' ∶⋅−−−−∙ Q&lt;nowiki&gt;''&lt;/nowiki&gt; ∙∙∙−∙∶
Qu::
le:
Qv :va :Qvu ∙
Iamvero
F ∙∙∙ E v K
Q −⇀⋅⋅ 2 SQ..— sin m' 2 sinx
Q'—
−⋅ Zsin x&lt;nowiki&gt;''&lt;/nowiki&gt; ∙
et consequenter
F:
2Q'Isin a:&lt;nowiki&gt;''&lt;/nowiki&gt; ZQ sin x&lt;nowiki&gt;''&lt;/nowiki&gt;, E:2Q sin x', K:
2Q sinx; ⋅
cum igitur sint
LSva'sF4-E—FK—FL2P,
iccirco—
2Qsinx&lt;nowiki&gt;''&lt;/nowiki&gt;—I-2Qsinx'—]-2Qsinx—l-AQ:P:
nnde
P
1—l-2 (sinx-l—sin x' ∙−⊢ sin x&lt;nowiki&gt;''&lt;/nowiki&gt;)
.
Fac demum nt puncta materialia K,K' ,K&lt;nowiki&gt;''&lt;/nowiki&gt;, K&lt;nowiki&gt;'''&lt;/nowiki&gt;, ..:
(Gg. 14 ) iungantur filis K K', K' K&lt;nowiki&gt;''&lt;/nowiki&gt; , ... determinatae
quidem longitudinis, sed mobilibus circa K', K&lt;nowiki&gt;''&lt;/nowiki&gt;. Si pun-
cta illa sollicitantur viribus Q, Q' , Q&lt;nowiki&gt;''&lt;/nowiki&gt; , Q&lt;nowiki&gt;''&lt;/nowiki&gt; , ad aequi-
librium haec manifeste requirentur: potentia Q in di-
rectione K'K tendens ab K' versus K; resultans R' ex
Q et Q' in directione K&lt;nowiki&gt;''&lt;/nowiki&gt;K' tendens ab K&lt;nowiki&gt;''&lt;/nowiki&gt; versus K'; re-
sultans R&lt;nowiki&gt;''&lt;/nowiki&gt; ex B' et Q&lt;nowiki&gt;''&lt;/nowiki&gt; in directione K&lt;nowiki&gt;'''&lt;/nowiki&gt;K&lt;nowiki&gt;''&lt;/nowiki&gt; tendens .ab K&lt;nowiki&gt;''&lt;/nowiki&gt;
versus K&lt;nowiki&gt;''&lt;/nowiki&gt;; potentia Q&lt;nowiki&gt;'''&lt;/nowiki&gt; in directione K&lt;nowiki&gt;''&lt;/nowiki&gt; K&lt;nowiki&gt;'''&lt;/nowiki&gt; tendens ab K&lt;nowiki&gt;''&lt;/nowiki&gt;
versus K&lt;nowiki&gt;''&lt;/nowiki&gt;: demum ipsa Q&lt;nowiki&gt;'''&lt;/nowiki&gt; aequalis resultanti R&lt;nowiki&gt;''&lt;/nowiki&gt;.
&amp; Denotantibus X , T, Z componentes coordi-
natis orthogonalibusque axibus parallelas, in quas resolvi-
⋅ tur Q, erunt
Q;:31
X Y
ē
z
Q
cosinus angulorum , quos cum iis axibus intercipit l; de
notantibus insuper 2 , y , z coordinatas puncti K , et x' ,
j ', z coordinatas puncti K' , erunt
2x yay 22
KKKK KK
cosinus angulorum, quos cum ipsis axibus efficit K'K ; ob
tinebit itaque primum ex requisitis ad aequilibrium, quoties
cumque fuerint
X XX Y
DKKKK .
yg Z
KÖK &gt;
K’K &lt;nowiki&gt;''&lt;/nowiki&gt;
seu
X Y Z
(h ) .
Quod in ordine ad Q est X , Y , Z , sit X', Y ', Z ' in or
dine ad Q ' : si resolvitur l' in ternas coordinalis axibus
parallelas, eae erunt ( 9. 40. )
x + X ' , Y + Y ' , 2 + Z ';
hinc designantibus a&lt;nowiki&gt;''&lt;/nowiki&gt;, y ', z &lt;nowiki&gt;''&lt;/nowiki&gt; coordinatas puncti K&lt;nowiki&gt;''&lt;/nowiki&gt; , ob
tinebit secundum ex requisitis ad aequilibrium , ubi fuerint
X + X __ * ' - &lt;nowiki&gt;''&lt;/nowiki&gt; Y + Y_y_y&lt;nowiki&gt;''&lt;/nowiki&gt; 2 + 2_z'- &lt;nowiki&gt;''&lt;/nowiki&gt;
R ? KK R' K ” K R K&lt;nowiki&gt;''&lt;/nowiki&gt;K&lt;nowiki&gt;'''&lt;/nowiki&gt;
.
seu
X + * _ * + Y_2_Z
x - x yay 22
( h ').
31
X ? Z
Q Q Q
cosinus angulorum, quos cum iis axibus intercipit Q; de-
notantibus insuper a: , y , :: coordinatas puncti K,, et x',
y', s' coordinatas puncti K' ,
erunt
⋅⇂⋅−−⋅⊴⇂∙∣ .7-7&lt;nowiki&gt;''&lt;/nowiki&gt; z—z'
K'K , K'K . K'K
cosinus angulorum, quos cum ipsis axibus efficit K'K: ob-
tinebit itaque primum ex requisitis ad aequilibrium, quoties-
cumque fuerint '
≟−−−⋅−∝−−≄∣ it.s,-ï Z
Q K'K&lt;nowiki&gt;''&lt;/nowiki&gt; 'Q K'K &lt;nowiki&gt;''&lt;/nowiki&gt;G'ka' ↽−≖∙⊍↼∙≕∣
seu gx z
r—x' y—y' x—z'
Quod in ordine ad Q est X , T, Z , sit X', ï', Z' in or-
dine ad Q':si resolvitur Q' in ternas coordinatis axibus
parallelas, eae erunt (9. 40.)
X—FX' , T—Fï' , Z—l-Z' ; ↽
hinc designantibus z', y&lt;nowiki&gt;''&lt;/nowiki&gt;, :&lt;nowiki&gt;''&lt;/nowiki&gt; coordinatas puncti K&lt;nowiki&gt;''&lt;/nowiki&gt;, ob-
tinebit secundum ex requisitis ad aequilibrium , ubi fuerint ⋅
X—l—X' x'-x&lt;nowiki&gt;''&lt;/nowiki&gt; T—l-Tl—TI—j&lt;nowiki&gt;'''&lt;/nowiki&gt; ∅⊣−⊈∣↼↼≂∣∙ z&lt;nowiki&gt;''&lt;/nowiki&gt;
B' ⋅⋅⇀∣⋦∣∣↓⊊∣ ∙ nf- KI/KT '-T—KHK' '—
....t ∙⇁−⋅∣ ↖↽∙∣ ∣
X X T
T—Z-Z.(h).
/ II I I/32
non pluribus opus est ut intelligamus quod, expleta
X + X + X &lt;nowiki&gt;''&lt;/nowiki&gt; _Y + Y + Y &lt;nowiki&gt;''&lt;/nowiki&gt; _Z + Z + Z &lt;nowiki&gt;''&lt;/nowiki&gt;
x ' - 0 &lt;nowiki&gt;''&lt;/nowiki&gt; g'my &lt;nowiki&gt;''&lt;/nowiki&gt; z &lt;nowiki&gt;'''&lt;/nowiki&gt; - &lt;nowiki&gt;''&lt;/nowiki&gt;
( h&lt;nowiki&gt;''&lt;/nowiki&gt; ) ,&lt;nowiki&gt;''&lt;/nowiki&gt;
obtinebit tertium ex requisitis illis ; componentes X” , Y&lt;nowiki&gt;''&lt;/nowiki&gt; ,
Z&lt;nowiki&gt;''&lt;/nowiki&gt; spectant ad vim Q &lt;nowiki&gt;''&lt;/nowiki&gt;, coordinatae z ' ', y, pun.
clum K &lt;nowiki&gt;'''&lt;/nowiki&gt; . Designantibus demum X '&lt;nowiki&gt;'''&lt;/nowiki&gt; , Y
Y ' &lt;nowiki&gt;''&lt;/nowiki&gt;,
&lt;nowiki&gt;''&lt;/nowiki&gt; , Z &lt;nowiki&gt;''&lt;/nowiki&gt;
componen
tes in ordine ad Q&lt;nowiki&gt;''&lt;/nowiki&gt; , expletisque
X + X + X &lt;nowiki&gt;''&lt;/nowiki&gt; + X &lt;nowiki&gt;''&lt;/nowiki&gt; = 0 ,
Y + r' + &lt;nowiki&gt;''&lt;/nowiki&gt; + I&lt;nowiki&gt;''&lt;/nowiki&gt; = 0 ,
2 + 2 +2&lt;nowiki&gt;''&lt;/nowiki&gt; + Z &lt;nowiki&gt;''&lt;/nowiki&gt; = 0 ,
( h &lt;nowiki&gt;''&lt;/nowiki&gt; )
manifeste obtinebit quartum simulque quintum ex requisi
tis ad aequilibrium. Sub novem igitur distinctis condi
tionibus propositum quatuor virium systema consistet in
aequilibrio: si quinque darentur vires , undecim prodirent
conditiones; generatim 2 n + 1 conditiones quoad n vires.
Collatis primis ac secundis membris formularum ( h) , (h') ,
( h&lt;nowiki&gt;''&lt;/nowiki&gt;) , emergent
Y ( 2 - x ) – X (y - ) = 0 ,
( Y + Y') (a' - &lt;nowiki&gt;''&lt;/nowiki&gt; ) – ( X + X ') ( 7'- , ' ) = 0 ,
( X + Y' + Y &lt;nowiki&gt;''&lt;/nowiki&gt;) ( ' &lt; &lt;nowiki&gt;''&lt;/nowiki&gt; ) — ( X + x ' + X &lt;nowiki&gt;''&lt;/nowiki&gt;) (y &lt;nowiki&gt;''&lt;/nowiki&gt; , ' ') = 0;&lt;nowiki&gt;''&lt;/nowiki&gt;
quarum summa praebet
xY_yXfwY — y'X ' + x &lt;nowiki&gt;''&lt;/nowiki&gt; Y &lt;nowiki&gt;''&lt;/nowiki&gt; —y &lt;nowiki&gt;''&lt;/nowiki&gt; X &lt;nowiki&gt;''&lt;/nowiki&gt; + &lt;nowiki&gt;''&lt;/nowiki&gt; ( X + X' + x &lt;nowiki&gt;''&lt;/nowiki&gt;) —
x &lt;nowiki&gt;''&lt;/nowiki&gt; ( Y + Y ' + Y &lt;nowiki&gt;''&lt;/nowiki&gt; ) = 0 ,
∃⊈∙
non pluribus Opus est ut intelligamus quod, expleta
x-1-xq-x'Q—v-1-rq-rff—z-i-zq-z&lt;nowiki&gt;''&lt;/nowiki&gt; W,),
xli—xlli J/l ∙∙⇁ 7&lt;nowiki&gt;''&lt;/nowiki&gt;, z&lt;nowiki&gt;''&lt;/nowiki&gt;—z&lt;nowiki&gt;'''&lt;/nowiki&gt;
obtinebit tertium ex requisitis illis; componentes X&lt;nowiki&gt;''&lt;/nowiki&gt;, ?&lt;nowiki&gt;''&lt;/nowiki&gt;
,
⋅
Z&lt;nowiki&gt;''&lt;/nowiki&gt; spectant ad vim Q&lt;nowiki&gt;''&lt;/nowiki&gt;, coordinatae x&lt;nowiki&gt;'''&lt;/nowiki&gt;. &lt;nowiki&gt;''&lt;/nowiki&gt;, z&lt;nowiki&gt;'''&lt;/nowiki&gt; ad pun-
ctum K&lt;nowiki&gt;''&lt;/nowiki&gt; . Designentibus demnm X&lt;nowiki&gt;'''&lt;/nowiki&gt;, ï&lt;nowiki&gt;'''&lt;/nowiki&gt;
, Z&lt;nowiki&gt;'''&lt;/nowiki&gt; componen-
tes in ordine ad Q&lt;nowiki&gt;'''&lt;/nowiki&gt; , expletisque
\sum∙⊦\sum∣∙⊢\sum∦⊹\sum∣∥∶∶∘∙
T .l-T-l-TII—l- III,: 0 , (hi/I)
Z-i-ZIä-le-l—ZIflzo'
manifeste obtinebit quartum simulque quintum ex requisi-
tis ad aequilibrium: Sub novem igitur distinctis condi-*
tionibus propositum quatuor virium systema consistet in
aequilibrio: si quinque darentur vires, undecim prodirent
conditiones; generatim 2 n ⊣− ↿conditiones quoad :: vires.
Collatis primis ac secundis membris formularum (I:), (b'),
U;&lt;nowiki&gt;''&lt;/nowiki&gt; )
, emergent
?(x—x') —X (?'—?') −∙−−−∘ ∙
(
ï—l— !' )(x' ∙∙∙ x&lt;nowiki&gt;''&lt;/nowiki&gt;)—( X-l-X') (r'—y&lt;nowiki&gt;''&lt;/nowiki&gt;) :o
,
(HF-IJ&lt;nowiki&gt;''&lt;/nowiki&gt;) (x&lt;nowiki&gt;''&lt;/nowiki&gt;-— ∣∣∣≻⊣≖≖−⊦\sum∣−⊦\sum∥≻ (y'—y&lt;nowiki&gt;'''&lt;/nowiki&gt;)
−−− .;
quarum summa praebet
xy-Jx-Jlïl—y/X/ :&lt;nowiki&gt;''&lt;/nowiki&gt; ï&lt;nowiki&gt;''&lt;/nowiki&gt;—y&lt;nowiki&gt;''&lt;/nowiki&gt;X&lt;nowiki&gt;''&lt;/nowiki&gt;—l—y&lt;nowiki&gt;'''&lt;/nowiki&gt;(X XLI-X&lt;nowiki&gt;''&lt;/nowiki&gt;) ∙−⋅
↕∣∣∣≼↕⊹⊺∣⊹↕∥≻ :0 ,33
seu , ob primam et secundam ( hm) ,
-Y yXTY'y'x + x'Y&lt;nowiki&gt;''&lt;/nowiki&gt; _7 / X &lt;nowiki&gt;''&lt;/nowiki&gt; + x &lt;nowiki&gt;''&lt;/nowiki&gt; I &lt;nowiki&gt;''&lt;/nowiki&gt; —7&lt;nowiki&gt;'''&lt;/nowiki&gt;X &lt;nowiki&gt;'''''&lt;/nowiki&gt;
Simili modo collatis primis ac tertiis membris ipsarum ( h) ,
( h') , ( h&lt;nowiki&gt;''&lt;/nowiki&gt; ), attentisque prima ac tertia ( h '&lt;nowiki&gt;'''&lt;/nowiki&gt;) ; itemque col
latis secundis ac tertiis membris earumdem ( h ) , ( h ) , ( h&lt;nowiki&gt;''&lt;/nowiki&gt; ) ,&lt;nowiki&gt;''&lt;/nowiki&gt;
attentisque secunda ac tertia ( h &lt;nowiki&gt;'''&lt;/nowiki&gt;) , assequemur&lt;nowiki&gt;'''&lt;/nowiki&gt;
xZ - 2X + « Z_z'X' + x'Z&lt;nowiki&gt;''&lt;/nowiki&gt; _z&lt;nowiki&gt;''&lt;/nowiki&gt;X &lt;nowiki&gt;''&lt;/nowiki&gt; Tx &lt;nowiki&gt;''&lt;/nowiki&gt; Z '&lt;nowiki&gt;'''&lt;/nowiki&gt; —Z'y&lt;nowiki&gt;''&lt;/nowiki&gt; = 0 ,&lt;nowiki&gt;''&lt;/nowiki&gt;
32—3Y + y^2?–49 + &lt;nowiki&gt;''&lt;/nowiki&gt;Z&lt;nowiki&gt;''&lt;/nowiki&gt; &gt;&lt;nowiki&gt;''&lt;/nowiki&gt;Y &lt;nowiki&gt;''&lt;/nowiki&gt; +y &lt;nowiki&gt;''&lt;/nowiki&gt; Z&lt;nowiki&gt;''&lt;/nowiki&gt; — ;&lt;nowiki&gt;''&lt;/nowiki&gt; Y&lt;nowiki&gt;''&lt;/nowiki&gt; = 0 .
Conditiones videlicet aequilibrii ( 13. 8º. ) quoad systema
punctorum lineis rigidis inter se firmiter connexorum in
cluduntur in conditionibus aequilibrii quoad propositum
systema habens formam variabilem .

=== De centro gravitatis. ===
[[17|17]]. Constat experimentis corpora jugiter sic tendere, seu gravitare in tellurem, ut sibi commissa descendant verticaliter in eius superficiem, gravitas ergo, seu vis unde provenit iste verticalis descensus, eatenus haberi poterit pro sibi ad sensum parallela, quatenus licebit superficiem illam habere pro physice plana: constat insuper experimentis omnia quaevis corpora eodem tempore idem spatium verticaliter in vacuo percurrere, idest aequali velocitate ex aequali altitudine perpendiculariter ad horizontem descendere. Inde sequitur vires gravitatis in diversis corporibus esse illorum massis proportionales, et corpus quodlibet spectari posse tanquam aggregatum materialium graviumque particularum, quae gaudeant parallelarum virium proprietatibus: centrum virium parallelarum (12) in casu dicitur centrum gravitatis. Resultans ex omnibus gravitatis viribus, quae vigent in corporis particulis, vocatur corporis pondus; transit constanter per gravitatis centrum, et directionem obtinet horizonti perpendicularem.

Porro si massula indefinite parva &lt;math&gt;\nu&lt;/math&gt; apud datum corporis punctum dividitur per respondens volumen &lt;math&gt;\beta&lt;/math&gt;, ratio &lt;math&gt;\frac{\nu}{\beta} (= \mu ) &lt;/math&gt; vocatur corporis densitas apud illud punctum; diciturque corpus vel homogeneum, vel heterogeneum prout &lt;math&gt;\mu&lt;/math&gt; apud singula corporis puncta est vel eadem, vel diversa; in corporibus homogeneis ratio &lt;math&gt;= \mu&lt;/math&gt;  est eadem ac ratio inter totalem corporis massam et ejus totale volumen; pondusculum massulae &lt;math&gt;= \nu&lt;/math&gt;, utpote proportionale ipsi &lt;math&gt;= \mu&lt;/math&gt;, exprimitur per &lt;math&gt;= \mu&lt;/math&gt; ductam in quandam constantem &lt;math&gt;c&lt;/math&gt;; ratio &lt;math&gt;\frac{c \nu}{\beta} (= c \mu ) &lt;/math&gt;  appellatur specifica corporis gravitas apud praefatum punctum; estque densitati proportionalis.

[[18|18]]. Notetur illud: etsi corpus gravitate sua jugiter sollicitatur deorsum; hoc tamen non officit quominus adhuc (2) dicatur corpus de se et natura sua indifferens ad quietem vel motum. Gravitas enim est dumtaxat vel aliquid extrinsecum corpori, vel illi intrinsecus additum, non autem aliquid eidem essentiale. Patet, quia vel nomine gravitatis intelligitur vis quaedam, qua corpora versus terram urgentur, vel vis qua tendunt ad determinatam quamdam spatii immobilis partem. Non hoc secundum, quia eo ipso casus purus admitteretur contra principium rationis sufficientis, cum nulla appareat ratio cur mobile ad hanc potius partem ferri debeat quam ad illam, cum spatium ubique sit homogeneum; ergo primum erit dicendum: sed si ita est, certe gravitas non est corporibus essentialis; nulli enim corpori essentiale est ut sibi caetera coexistant, ac proinde unum potest existere quin existant caetera, et consequenter etiam quin existat terra.

[[19|19]]. Dato centro gravitatis corporis, facile definitur utrum corpus in dato situ extra lapsus periculum constitui possit. Nam ex eo centro demissa ad planum horizontale recta perpendiculari, quae vocatur linea directionis, si haec intra basim cadat, corpus extra lapsus periculum erit positum, secus ruet in eam partem in quam perpendicularis recta dirigitur. Hinc patet ratio cur turres aliquae &lt;u&gt;inclinatae&lt;/u&gt; non cadant, ut sunt Bononiensis, Pisana etc: linea scilicet directionis extra ipsarum basim non excurrit. Hinc etiam valde pingues, et qui magnum aliquod onus brachiis complectuntur, retrorsum; gibbosi autem et bajuli antrorsum; qui dextra pondus aliquod sustinent, sinistrorsum; qui vero sinistra, dextrorsum &lt;u&gt;inflectuntur&lt;/u&gt;. Per hanc scilicet declinationem efficiunt ut linea directionis transeat per spatium, quod inter pedes continetur; quod spatium est basis corporis humani. Eamdem ob caussam si quis velit ex. gr. dextero pede stare, crus &lt;u&gt;inclinat&lt;/u&gt; paullulum dexteram partem versus, nec diu haerere potest in eo statu , quia cum basis totius corporis sit unus dumtaxat pes, linea directiouis facile potest basis tam anguslae limites praetergredi. His autem corporis nostri flexibus ac librationibus ita ab infantia assuevimus usu continuo ut nec advertentes recto illas ordine peragamus. Patet hinc denique cur aves uni pedi insistentes dormire solent capite sub ala recondito; id nempe faciunt ut linea directionis intra pedis cui insistunt latitudinem servetur.

[[20|20]]. Centrum gravitatis inveniri potest vel ratione mechanica, vel ratione, algebraica. Ad primam quod attinet, si corpus aliquod filo suspendas, volvetur converteturque donec in aequilibrio tandem consistat, et filum ad terrae superficiem perpendiculariter dirigatur. In hac perpendiculari, quae est linea directionis per quam centrum gravitatis corporis tendit, erit centrum ipsum. Iam notetur linea a filo perpendiculari in corpore designata, rursusque ex alio puncto suspendatur corpus, et facto aequilibrio linea perpendicularis pariter notetur. In communi duarum linearum intersectione reperietur quaesitum centrum. Ratio algebraica desumitur ex dictis ( 13.2.º''a''&quot; ): sumantur nempe vires proportionales massis &lt;math&gt;m, m' , m''&lt;/math&gt;, ..... punctorum, quibus applicitae sunt; hoc pacto, ad positionem centri gravitatis determinandam exsistent
&lt;math display=&quot;block&quot;&gt;x_{\mathrm I}=\frac{\sum m x}{\sum m}, y_{\mathrm I}=\frac{\sum m y}{\sum m}, z_{\mathrm I}=\frac{\sum m z}{\sum m}  (b) &lt;/math&gt;Si corpus intelligitur divisum in varias portiones dimensionis finitae , et earum massae denotantur per  &lt;math&gt;m, m' , m''&lt;/math&gt;, adhuc valebunt formulae (b); nihilque aliud erunt &lt;math&gt;x , y , z ,x' y ',z',x''&lt;/math&gt;, ... nisi coordinatae centrorum gravitatis illarum portionum. Si corpus ponitur insuper homogeneum quoad omnes partes, erunt massae ut respondentia volumina; poteruntque haec illis substitui in formulis (''b'') : quisque videt coordinatas &lt;math&gt;x_{\mathrm I}, y_{\mathrm I}, z_{\mathrm I}&lt;/math&gt;, ex (''b'') haud pendere ab intensitate gravitatis. Caeterum plures sunt casus, in quibus centrum gravitatis absque formularum subsidio immediate cognoscitur. Sic in linea recta centrum gravitatis est medium ipsius rectae punctum: in parallelogrammo punctum, ubi binae diagonales se mutuo secant: in circulo centrum figurae: in cylindro habente bases parallelas punctum medium axeos:
in parallelepipedo punctum, ubi quatuor diagonales se mutuo secant: in sphaera ipsum magnitudinis centrum.

In triangulo centrum gravitatis est punctum illud, ubi sese invicem secant rectae lineae, quae a duobus trianguli verticibus ducuntur ad puncta media laterum oppositorum: cum enim &lt;math&gt;AD&lt;/math&gt; (Fig. 15) dividat aequaliter rectas omnes lateri &lt;math&gt;BC&lt;/math&gt; parallelas, et &lt;math&gt;BE&lt;/math&gt; rectas omnes lateri &lt;math&gt;AC&lt;/math&gt; parallelas, reperietur centrum gravitatis areae triangularis tam in &lt;math&gt;AD&lt;/math&gt; quam in &lt;math&gt;BE&lt;/math&gt;; ideoque erit in &lt;math&gt;H&lt;/math&gt;. Jamvero ducta &lt;math&gt;DE&lt;/math&gt;, ea exsistet parallela lateri &lt;math&gt;AB&lt;/math&gt;; et consequenter triangula &lt;math&gt;ABH , DEH&lt;/math&gt; erunt similia; hinc&lt;math display=&quot;block&quot;&gt;\frac{DE}{AB}=\frac{DH}{AH}&lt;/math&gt;sed, ob &lt;math&gt;CE = \frac12  AC&lt;/math&gt; et &lt;math&gt;CE = \frac12 CD = BC&lt;/math&gt;, est DE = &lt;math&gt;CE = \frac12 AB&lt;/math&gt;; igitur &lt;math&gt;DH = \frac12  AH&lt;/math&gt;; ac proinde &lt;math&gt;DH = \frac12 AD&lt;/math&gt;; et &lt;math&gt;AH = \frac23 AD&lt;/math&gt;.

In pyramide triangulari &lt;math&gt;ABCO&lt;/math&gt; (Fig. 16) erit &lt;math&gt;G&lt;/math&gt; centrum gravitatis; ubi nempe se mutuo secant binae rectae &lt;math&gt;OH , CK&lt;/math&gt;, quae ex &lt;math&gt;O&lt;/math&gt; et &lt;math&gt;C&lt;/math&gt; ducuntur ad centra gravitatis
&lt;math&gt;H&lt;/math&gt; et &lt;math&gt;K&lt;/math&gt; triangulorum &lt;math&gt;ABC , ABO&lt;/math&gt;. Secetur enim pyramis,

1.º planis parallelis triangulo &lt;math&gt;ABC&lt;/math&gt;, 

2.º planis parallelis triangulo &lt;math&gt;ABO&lt;/math&gt;; transibit &lt;math&gt;OH&lt;/math&gt; per centra gravitatis omnium illarum sectionum triangularium; transibit &lt;math&gt;CK&lt;/math&gt; per centra gravitatis omnium harum. Ergo pyramis habebit suum gravitatis centrum tam in &lt;math&gt;OH&lt;/math&gt; quam in &lt;math&gt;CK&lt;/math&gt;, et consequenter in &lt;math&gt;G&lt;/math&gt;. Ducatur nunc &lt;math&gt;HK&lt;/math&gt;; erit &lt;math&gt;HK&lt;/math&gt; parallela rectae &lt;math&gt;CO&lt;/math&gt;, et triangula similia &lt;math&gt;HKG , CGO&lt;/math&gt; praebebunt
&lt;math&gt;\frac{HK}{CO}=\frac{HG}{OG}.&lt;/math&gt;
Sed, ob &lt;math&gt;MH =\frac13 CM&lt;/math&gt; et &lt;math&gt;MK = \frac13 OM&lt;/math&gt;, est &lt;math&gt;HK = \frac13 OC&lt;/math&gt;;
ideoque &lt;math&gt;HG =\frac13 OG&lt;/math&gt;; igitur &lt;math&gt;HG = \frac14 OH&lt;/math&gt;, et &lt;math&gt;OG = \frac34 OH&lt;/math&gt;.

=== De corporum collisione ===
[[21|21]]. Quaestio de corporum collisione eo redit, ut datis velocitatibus ante collisionem, determinentur velocitates post collisionem. Corpora sese collidentia assumimus sphaerica, et in singulis stratis concentricis homogenea; in quibus proinde corporibus centrum gravitatis erit ipsum magnitudinis centrum. Corporum sese collidentium centra vel moventur in eadem recta, vel in diversis rectis; in primo casu collisio dicitur normalis, in secundo obliqua.
[[Fasciculus:Inelastischer stoß.gif|thumb]]
[[22]]. Invenire velocitatem &lt;math&gt;v''&lt;/math&gt;, quam habent duo data corpora non elastica post normalem collisionem, datis eorum
velocitatibus &lt;math&gt;v'&lt;/math&gt; et &lt;math&gt;v&lt;/math&gt; ante collisionem. Dicantur &lt;math&gt;m', m&lt;/math&gt; corporum massae; erunt &lt;math&gt;mv , m'v'&lt;/math&gt; quantitates motus ante collisionem: eatenus corpus subsequens agit in antecedens quatenus hoc lentius illo movetur, adeo ut perseveret actio donec ad aequalitatem velocitatis deveniatur; unde velocitas &lt;math&gt;v''&lt;/math&gt; post collisionem erit communis, et aequalis in utroque: summa praeterea quantitatum motus est eadem ante et post collisionem; velocitas autem obtinetur dividendo quantitatem motus per massam. Ergo demum&lt;math display=&quot;block&quot;&gt; v'' =\frac{mv + m'v'}{m + m'}&lt;/math&gt;

Haec observentur: 

1.° &lt;math&gt; v'' - v &lt;/math&gt; exprimit quantum velocitatis acquisierit corpus antecedens, quod ponimus esse
&lt;math&gt;m&lt;/math&gt;; et &lt;math&gt; v' - v'' &lt;/math&gt; quantum amiserit impellens &lt;math&gt;m'&lt;/math&gt;. 

2.° consideranda erit pro lubito alterutra velocitas tamquam negativa, si corpora ex oppositis plagis adveniunt; hinc in formulis ubicumque ea inveniatur, signo contrario erit adhibenda - Sic v. gr. si massae &lt;math&gt;m'&lt;/math&gt; directio habeatur pro positiva, sumenda erit &lt;math&gt;v&lt;/math&gt; negative, ac proinde &lt;math&gt; v'' =\frac{m'v'- mv}{m + m'}&lt;/math&gt;.

3.° ponetur &lt;math&gt;v = 0&lt;/math&gt;, si corpus impellendum &lt;math&gt;m&lt;/math&gt; quiescit; erit &lt;math&gt; v'' =\frac{m'v'}{m + m'}&lt;/math&gt;: hinc &lt;math&gt;v''&lt;/math&gt; ferme evanescet si massa &lt;math&gt;m&lt;/math&gt; sit physice infinita respectu &lt;math&gt;m'&lt;/math&gt;. 

4.º numquam habebitur perfecta quies post collisionem si &lt;math&gt;m&lt;/math&gt; et &lt;math&gt;m'&lt;/math&gt; in easdem partes oppositas, et velocitates sint reciproce ut massae, tunc &lt;math&gt; v'' = 0&lt;/math&gt;, et consequenter habebitur perfecta quies.

23. Invenire velocitates corporum perfecte elasticorum post normalem collisionem, datis velocitatibus &lt;math&gt;v', v&lt;/math&gt; ante collisionem. Perspicuum est hujusmodi corpora sequi leges non elasticorum toto compressionis tempore, tum restitutionis tempore effectum hunc instaurari. Ergo facta partium restitutione inveniri debet in corpore impulso dupla velocitatis acquisitio; dupla vero celeritatis amissio in impellente. Itaque si dicantur r' ' et &quot; velocitates corporis im
pellentis et corporis impulsi post factam restitutionem ,
erunt ( 22)
u &quot; = V - 2 ( 0--0&quot; ) = v - 2
my + ms
mtm
2 mv tv (m ' — m)
( 9 ),
m + m
(
1
vi &quot; = 0 + 2 (0 &quot; ~ v ) = 2 + 2 ( -v)
mv + m's
m + m
2 m ' ú tu (m - m ')
(9) .
mtm

24. Haec ex formulis (9) et (q' ) deducuntur .

1.• Si massae sunt aequales , elastica corpora post colli
sionem movebuntur .facta velocitatum permutatione, Nam
moveantur primo in eamdem plagam ; propter m = m' , for
mula (9) abit in
2 m v'
et ( 9 ') in 3,10 v' ; ergo etc.
Rursus praeter m = m ' habeatur etiam v = 0 , hoc est cor
2 mo
pus percussum quiescat; erit v = 0 , et v ' . = V ' ;
corpus nempe percutiens post collisionem quiescet , et per
2 mv
2 m 2 m
2 m
1
39
moveantur, vel* alterutra solum quiescat :quod si collisio
liat ad partes oppositas ,
et velocitates sint reciproce ut mas-
sae, tunc v&quot;:o ,
et consequenter habebitur perfecta quies.

23. Invenire velocitates corporum perfecte elasticorum
post normalcm collisionem, datis velocitatibus v', 0 ante col-
lisionem. Perspicuum est huiusmodi corpora sequi leges non
elasticorum tOto compressionis tempore, tum restitutionis
tempore effectum hunc instaurari. Ergo facta partium re-
stitutione inveniri debet in corpore impulso dupla velo-
citatis acquisitio; dupla vero celeritatis amissio in impel-
lente. ltaque si dicantur v'&quot; et v&quot; velocitates corporis im-
pellentis et corporis impulsi post factam restitutionem ,
erunt (22)
'
'
um:-D' --2 ('n'—v&quot;) :'--2 (,; ∙−−−−−−−−⋯⇂↓−⊢⋯∣∣↗ m −−⊢ m )
..2 mv −⊢∣v (m' −∙∙ m)
'm ∙−∙∙ m' ∓∎∎∎∎∎∎ (9):
W:w—l-2(v&quot;—v):v-l—2 Maii:-31; -v)
-&quot;2mv—l-v(m-—m) (qr).
m-l-m' J—
24. Haec ex formulis (q) et (q') deducuntur.
↿∙∘ Si massae sunt aequales, elastica corpora post colli-
sionem movebuntur.fdcta velocitatum permutatione, Nam
moveantur primo in eamdem plagam; propter m:m', for-
mula (q) abit in
2 mv ' ∙
2
'&quot;
'v'&quot;::«v p. . ,
et (q ) 111 10&quot;: :v ; ergo etc.
a m
'
-
Rursus praeter m:m' habeatur etiam :::o , hoc est cor-
- -
∣∣∣ tv 2 m &quot;( pus percussum qutescat; er1t a::o, et a:::
'v ; m
corpus nempe percutiens post collisionem quiescet , et per-40
cussum movebitur velocitate , quam percutiens habebat ante
collisionem . Demum sibi mutuo occurrant : ubicumque ergo
invenitur v , sumenda erit negative ; qua mutatione facta ,
habebuntur
2 mv 2 mv'
2 m
v, et viv v' .
2 m
Jam vides mutationes velocitatum exhiberi per ipsas litte
ras , et ubi debeat etiam mutari directio , regressus expri
mitur per mutationem signorum.
2.• Si statuatur series corporum perfecte elasticorum , ae
qualium , se mutuo tangentium , et quorum centra unam
rectam constituant , in primum autem quacumque velocitate
incidat alterum corpus aequale , movebitur tantummodo cor
pus ultimum , quiescentibus omnibus aliis . Quod si statua
tur series corporum habentium massas in progressione geo
m3,
metrica m' , m, ... ; et caeteris quiescentibus, pri
mum m' incidat in secundum velocitate v' , expriment
m2
m ? m
2 m
v' .
m +m (m *:)*,~(m2 I )
m
velocitates excitatas a primo in secundo , a secundo in ter
tio , a tertio in quarto etc. Denotante igitur n numerum cor
porum , movebitur ultimum velocitate
2 m'
N- 1
I
Cena ntmi
).
3. Quotiescumque corpus impellens minus erit corpore
impacto quiescente , toties illud regredietur , uti patet ex
formula ( 9 ) , posita v = o et m &gt; m' . Quod si m =
et v =0 , prodibit v'' = -1 , nimirum si globus minor''
∢⋅∘cussum movebitur velocitate ,quam percutiens habebat ante
collisionem. Demum sibi mutuo occurrant :ubicumque ergo
invenitur :: , sumenda erit negative; qua mntatione facta.,
habebuntur
2mv &quot; 2mv'

: —-v,et'v :

2m ∙∙−−−∙⋅∙≀≀∙ 2m
Iam vides mutationes velocitatum exhiberi per ipsas litte-
ras , et'ubi debeat etiam mutari directio , regressus expri-
mttur per mutat1onem signorum.
2.&quot; Si statuatur series corporum perfecte elasticorum, ae-
qualium, se mutuo tangentium, et quorum centra unam
rectam constituant
,
in primum autem quacumque velocitate
incidat alterum corpus aequale , movebitur tantummodo cor-
pus ultimum
, quiescentibus omnibus aliis. Quod si statua-
tur series corporum habentium massas in progressione geo-
∙ m! ma. metrtca m', m, ∙ ∙ ∙ ∙ &quot;7, , m .; et caeterts qmescenttbus , prt-
mum m' incidat in secundum velocitate v', expriment
v,2m' ⋅∙ 2m' :
,( 2m' 3
m—l—m&quot; 'v (m—l—m')&quot;, m-l-m'
velocitates excitatas a primo in secundo , a secundo in ter-
tio
, a tertio in quarto etc. Deuotante igitur n numerum cor-
porum , movebitur ultimum
⋅⋅⋮ velocitate ea&quot;
3.&quot; Quotiescumque corpus impellens minus erit corpore
impacto quiescente ,
toties illud regredietur , uti patet ex
formula (q) , posita v :o et m m'
. Quod si ut:ea
et 9

: o

, prodibit v'&quot;:
-— v' , nimirnm si globus minor41
incurrat in globum immensae massae quiescentem , resiliet
cum velocitate eadem , cum qua advenerat .
4.• Si duo corpora elastica occurrant sibi velocita
tibus v , v ', quae massis m, m ' reciproce sint proportionales ,
eadem qua venerant velocitate ambo resilient. Etenim cum
ex oppositis partibus corpora congrediantur , ac praeterea
m : m ' :: v ' : v , in formulis ( 9) , ( 9' ) sumenda erit » negative ,
et ponendum mv = m' '; quibus peractis , obtinebitur
v ' &quot; = &gt; &quot; (m + m )
et viv=v
Im + m
no-tm
Imtin
-- 5. ° Ex ipsis ( 9) et ( 9' ) eruitur m'y's mula
m 'ustomus: factum ex massa in quadratum respondentis
velocitatis dicitur vis viva ; hinc in corporibus perfecte
elasticis summa virium vivarum permanet eadem ante et post
collisiopem .

25. Formulae ( 9) , (9 ') aptari possunt etiam corporibus,
imperfecte elasticis , modo quantitatibus
2(v— mm Imus)
my tms
mtm
et 2
( mahu
my + mv
m + m
--)
substituantur
(n+ m ( = m **)e (1+- ( Inv—-).
denotante r rationem inter vim , qua partes sese resti
tuunt , et vim comprimentem. Quantitas r experimentis de
terminanda est in singulis corporum speciebus : fac ut m
quiescat , sitque co ; erit post collisionem '&quot; = -ru':
unde , cognita velocitate v' ., qua m ' offendit in m , et velo
citate negativa v'' , qua post impactum resilit , habebitur''
-
4
⋅↣
' 41
incurrat in globum immensae massae quiescentem ,
resiliat
cum velocitate eadem, cum qua advenerat.
49 Si duo corpora elastica occurrant sibi velocita-
tibus v, v', quae massis m, m' reciproce sint proportionales ,
eadem qua venerant velocitate ambo resilient. Etenim cum
ex oppositis partibus corpora congrediantur , ac .praeterea
ut :m'::v':
9, in formulis (q) ,(q' ) sumenda erit .9 negative ,
et ponendum m 9:m' v'; quibus peractis , obtinebitur
v'&quot;:
— v' (Z.—lm,) :-v'. et v&quot;:v (m )

:

v.
∙
∙⊢⋯⋅ ⋯∙−⊦⋯
∂∙∘ Ex ipsis (q)et.(q') eruitur m' v'&quot;3-l- mv&quot;::
m'∎∣∣≖ -l-m vi: factum ex massa in quadratum respondentis
velocitatis dicitur vis viva; hinc in corporibus perfecte
elasticis summa virium vivarum permanet eadem ante et post
collisionem. . 

25. Formulae (q), (q')'aptari possunt etiam corporibus ,
imperfecte elasticis , modo quantitatibus .
2(v,—nw—-mlv) et/2 (mv-l-mlv m-l-m —v) mm
substituantur (1 44) (v'. m.,-(.m'v') et≰↿⊹↗⋝⋅≼⋯⇂≩−−⋯⋅∣⇂≀∣ ∙∙∙∙∙ v) ∙
∦⇂⊣−⋯≳ m—l—m
denotante r rationem inter vim
, qua partes sese resti-
tuunt , et vim comprimentem. Quantitas r experimentis de-
terminanda est in singulis corpürum speciebus :fac ut 11:
quiescat , sitque :co ; erit post collisionem v'&quot;:
-
r v':
unde , cognita velocitate v' ., qua m' olfendit in m, et velo-
citate negativa v'&quot;
, ua post impactum resilit, habebitur '42

26. Ad collisionem obliquam quod pertinet , si corpora sibi mutuo occurrunt directionibus convergentibus bm , b'm
( Fig.17 ) et velocitatibus expressis per easdem rectas bm ,b'm ', resolvantur bm , b'm ', altera in duas by, ba, altera in duas b'y ', b'a', ita ut by, b'y' existant normales , ba vero et bá parallelae sint rectae m m corporum centra jungenti. Quoniam componentes b y , b'y' parallelae sunt tangenti TT ductae per punctum contactus, ab ipsis nullo pacto pendebit collisio, nullamque in collisione subibunt mutationem . Cor
pora igitur sese collident velocitatibus ba = ym, b'a' =
y'm '. Inventis itaque ( 23 ) v &quot; , et v '' , sumptisque ex. gr.'' mf = y &quot; , mi = &quot; in recta y r', et ductis mv = by , m'ú
= bóý , si complentur parallelogramma fv, iv', exprimentur per diagonales mf, m'i' tum velocitates , tum directiones corporum post collisionem. Haec autem ex modo dictis facile colliguntur; 

1.º Si globus minime elasticus iacidit oblique in planum immobile, progredietur secundum directionem plani cum velocitate m'v ' ( = a'm '), quae ad velocitatem priorem b'm ' erit ut sinus anguli incidentiae b'm'y' ad radium. 

2.º si globus fuerit perfecte elasticus, resiliet per m'z efficiendo angulum reflexionis z míy aequalem angulo incidentiae b'm'ý . 

3.° quod si globus incidens sit imperfecte elasticus, resiliet ad angu
lom i'm'y ', cujus cotangens ad cotangentem anguli inciden
liae b'm'y ' ut r : 1 .

=== De motu rectilineo utcumque vario.=== 
27. Nonnulla hic praemittimus ex analysi infinitesimali.
1.o Quantitas iniinitesima a: (minor videlicet qua-
cumque data utcumque parva) censeatur esse primi ordinis ;
«2 erit inlinitesima secundi ordinis; «3 iniiuitesima tertii; etc.
2.&quot; Inlinitesima a) dicetur esse primi ordinis si ra-
∙ G) ∙ ∙ ∙
a tno ∙
.. valorem habet (imtum
, secund1 s1 ∙−− valorem obtinet
ac «:43
similiter finitum , atque ita porro . Denotante generatim k
valorem illum finitum , poterit infinitesima quantitas ordinis
msimi exhiberi per
w kam
3. Sumptis aliis valoribus finitis k,; ka, ... km ,
habentur pro aequalibus
kmetkam tk , an- tkzam- ² + ... + kmiat kma
km_ ,a et kam + kamer + ... + km_, &amp; ,
kmed k * et kam + kamer t . tkm -rQ ?.
etc .... ;
admittuntur nimirum aequationes
kam tka&quot;-t ... tkm , at km
km
kam +k ,am -s +... +kimeza? + kmail
km ,
51
etc.
quatenus differentia inter utrumque membrum est minor
quacumque data quantitate alcunique parva. Huc spectat
illud : quantitates infinitesimae , quaecumque eae sint, et quo
rumcumque ordinum , absque ullo errore negliguntur prae
quantitate finita : itemque infinitesimae quantitates altiorum
ordinum absque ullo pariter errore negliguntur prae infini
tesima quantitate inferioris ordinis.
4.0 Quantitates infinitae ( majores videlicet qua
cumque data utcamque magna) cum possint exprimi person
43
similiter tinitum
, atque ita porro . Deuotante generatim k
valorem illnm finitum
, poterit intinitesima quantitas ordinis
msimi exhiberi per
m::ka&quot;
3.&quot; Sumptis aliis valoribus finitis k,
,
It,,
... k,&quot;,
habentur pro aequalibus

: , et kat'-I-k, ∝⋅⋅−≖⊣−≀∣≖∝∙−⋅≖−⊦ −⊦↗⊏⋅∙∙⋅∝−⊦∣⊏⋅⋅∙

∄⊄⋅⋅∙≖∘≖ et ka&quot; ⊣− 1, ∘⋍⋅∙∙⋅⋅⊳⋅ ⊣− ⊣− r.,, a: -
It,... «* et kat&quot; −⊦ kp?&quot; -]— ∙∙∙−∣⋅⋅ km., æ.
etc-eoo ;
.
admittuntur nimirum aequationes ⋅ '
↗⊄⊧∘↙⋅⊣−∣∁⋅⊶⋅∙−⋅⊣−∙∙∙∣−⊦↗≂⋅∙∙⋅⊄−⊦↳
↿ ⋅
km .
l
kan-l-Ic,ac&quot;&quot;-l--. &quot;'l-kaum&quot;,
hngua −⊦⋠⋅∙∙∙∸⇂⇉⊄∙−⋡↿ ⋅ «
⋅
etc. ∙ ∙ , . .
.. ⋮∙ ;
,-
∣

;

quatenus differentia, inter utrumque membrum'est'minor
quacumque data quantitate utcumque parva. Huc. &quot;spectat
illud :quantitates inünitesimae, quaecumque eae sint. et quo- ∙
rumcumque ordinum
, absque ullo errore negliguntur prae
quantitate finita :itemque infinitesimae quantitates altiorum
ordinum absque ullo pariter errore negliguntur prae ⋮≖≖∅≖∙≖∃∙∙ tesima quantitate inferioris ordinis.
∙∡∙∘ Quantitates infinitae (majores videlicet qua-
cumque data utcumque magna) cum possint exprimi per-;,44
tribuentur et ipsae in varios ordines ; illudque facile stabi
lietur : quotcumque finitae quantitates tuto : negliguntur
prae quantitate infinita ; quantitatesque infinitae ordinum in
feriorum tuto etiam negliguntur prae quantitate infinita altio
ris ordinis. Facto enim \beta . , et designantibus a, b,c, ... , 9
valores finitos , habebitur
1
. 0
a \betam + bBm - tom-&gt; + ... +9\betato
1
EL
-la + bw twat..tqomat ww .
5.- Si variabiles x, y sunt inter se per certam quam
dam relationem ita connexae ut data v. g. X , inde possit
valor y determinari , y vocatur functio quantitatis x ; ipsa
vero x dicitur independens. Si relatio inter x et y expri
mitur aequatione minime resoluta quoad functionem y habi
tam pro incognita , y appellatur functio implicita ; quod si
valor y detur expressus immediate per independentem x,
vel talis obtineatur per aequationis resolutionem , y dicitur
functio explicita. In aequatione v, g. yo -2xy + m2 =0
y functio implicita quantitatis variabilis x ; at facta re
solutione , evadet y functio explicita ipsius x , duplicemque
habebit valorem , scilicet y = x + Vx? m2 , Functio
nes explicitae quantitatis x designari solent in hunc modum
est
-
F ( x) , f ( x) , ..
6.0 Differenziale dx quantitatis x est incrementum
infinitesimum , quod ipsi x adscribitur : differentiale vero dy
functionis y = f (x ) est respondens incrementum f ( x + dx)
- f (x ) .quod ob variatam x recipit in se functio illa : pro
ponantur v. gr. invenienda differentialia functionum
44
tribuentur et ipsae in varios ordines ; illudque-facile stabi-
lietur :quotcumque finitae quantitates tuto.: negliguntur
prae quantitate infinita; quantitatesque infinitae Ordinum in-
feriorum tuto etiam negliguntur prae quantitate infinita altio-
ris ordinis. Facto enim þ:
S;, et designantibus a,b,c, ..., q
valores linitos , habebitur
∘∣⊰⋅∙⊣−∂↙⊰⋅⋅∙⋅≖−⊦∘≀⊰⊶−≖ −⊦⋅⋅⋅ ⊣−⊄∣⊰−⊦ .
∸−−
te». &quot;(a ∙⊸⊦bæ—l— ccc&quot; −⊦∙∙ .-l—qm&quot;&quot;' ∎∙−∣− r m'&quot;).
5.&quot; Si variabiles æ,y sunt inter se per certam quam-
dam relatidnem ita connexae, ut data v. g. a:
,
inde possit
valor ]determinari ,;
vocatur functio quantitatis se: ipsa
vero «: dicitur independens. Si relatio intern- et y expri-
mitur aequatione minime resoluta quoad functionem ]habi-
tam pro incognita ,
]appellatur functio implicita ; quod si
valor y detur expressus immediate per independentem :,
vel talis obtineatur per aequationis resolutionem ,
]dicitur
functio explicita. ln aeqnatione v,- g. ;&quot; -—,2ay −⊢ ⇑∙∅:o
est 7 functio implicita quantitatis variabilis z'; at facta re-
solutione
, evadet ]
functio explicita ipsius a:, duplicemque
habebit valorem, scilicet 7—:a:∶⊨ ⇂⋅∕⋅↕∙≖ −− m'. anctio-
nes explicitae quantitatis a: designari solent 1n hunc modum
,F (x), f(x),... ⋅∙
6.&quot;Dill'erentiale dx quantitatis x est incrementum
infinitesimum , quod ipsi :: adseribitnr :differentiale 'vero dy
functionis y:--f (x) est respondens incrementum f
(x dx)
—f
(a:)
, quod ob variatum se recipit in se functio illa: pro-
ponantur v. gr. inVenienda difi'erentialia functionum45
at +6,9 +0,24+ Cisin x + C , cos x+c ,
tang x + C, log x + C , a ' tc ,
ubi a et C sunt quantitates constantes.
Erit I. dy = [ alx + dx ) + ] - [ax + C ] = adx.
a
II.dy = [ f'da+ c ]- [* + c]atda X
adot
adx
x2 + xdx
.
III.dy = [ ( x + dx)* + C]-[x4 +C]=ax“-'dx + 29,
a'a- 1 )
24-2d.22
t . ax-' dx .
IV.dy = [ sin ( x + dx ) + c ]- [sinx + C ) = sin ( x + dx ) — sinx
2 cos 2xdx)sinh dz = 2cosx sind = cos xdx .
V.dy = [cos ( c + dº + C ]- [cos.FC ] = cos ( c - day -cosx =
2 sin - (2x +dx)sin __ (x -x -dx) = sin xdx.
VI.dy = [tang ( xtdx )+c] - [langat.C ]=
sin ( x + 2x)
cos(x + dx )
sinx cosxsin (xtdx)-sinxcos( x + dx ) sin ( x + dx - x )
cos2 cosx cos ( cdc )
cos2x
45
a'−⊦∁∙−⋮∙−∙⊹ C, æa-l- C,'sin a: ∙⋅⊢ C,cos æ-l-C,
tang æ-l—C, logæ-l— C, ar-l-C,
ubi a et C suntquantitates constantes.
Erit l. dy: [a( æ-l—dx) ——C ]— [ux—I»- C ]:adæ.
∥⋅↙∣↗↗⊣⋤⋮⋅−−⊦ (i]—[?&quot; C]— jd,— :—
∥∣∙↙∄∫⇋∶∐↕⊹≴≀↕≻⋅⊹∁⊐∙⋢∞⋅⊹∁∃≕∞≕∙∣↙≀↝⋍⊹↽∘↙↙⊑⋅∣≱↶∶⊄−≖∠≀↓⋅≖
&quot;I- ∙ ∙∼ ∙ :aæ&quot;-' dx.
IV. a];:[sin(æ-l-dæHCI-[sinæ-l-C] :sin(æ-l—dæ)— aina:

: 2cos—;..(Zæ-l-dæ)sin-—;. dæ: a cosa: sin-;— dx:cos ædæ .

V.dJ:[cos(æ-l—dæ)-l-C]- [cosa: −⊢∁↥ ∙−−− cos (æ—l—dæþcosæ:
2sin :(ZPFdrþin-i—(æ-x-dx):— sin ædæ.
Vl.dy::[tang (
.z—l-dæ ≻−∣−∁⋮∣ ∙
⇂⊏∄∐⊰⊅⇥∙∁⊐ —8lll(æ..ll-rlæ) cos(æ—-dæ)
aina: cosæsin(æ-l-dæ)-sinæcos(H—dx)— sin(.i—l—dx-æ)
cosa: cosæ cos( æ
⊹∠≀∙↧⋅ ) coszx ⇁⇁−∙↼46
dx
cos2 x
VII.dy= [log(x-tdx)+ c]-[logo +C ]= log ( + )
dit 15 ( 1 +4x)dx _d2log [2 + } (1- dot) +
23 (1-4 )(1-2dt)+... ]
det 103 ( 2 + + 43 + 234 + ...)
dxlog [ 2 , 718281828
dx
] ;
sumptisque logarithmis quoad basim 2 , 718281828
dx dy X
istiusmodi basis solet exprimi per e.
VIII.dy = [a ++dx + C ] - [ a * + C ] = a *+ x_qt = da? =
a* d log (a *) = a * d [ x log (a )] = a * log (a ) dx.
70. Quantitas constans C, quaecumque ea sit, non in
venitur in differentialibus: idemque proveniet differentiale
sive differentietur v. g. sin x + C, sive sin x.
dy 8º. In primo exemplo habemus a,
dz
cundo axe- ', in dy quarto dx dx
- in se
dy
a
in tertio dy dx
x2
46
da:
∙∙∙∎∙∙↼⇁∙−⇁
∞∘⇄∙⋍∙∙
∇∥∙ ↙≀↨↶−−−∏∘⊰≺⊿↾∶∙∔⋞≀∙↕≻−⊦∁∃⋅⊏∣∘∷∞⊣−∁↿⇌ log (
↿
.? −⊢↙∙⇣⋮⋮⋟⋮
dx —log (HE ↙↿−⋤⋅∶∙↙≀−≟−∅∣∘⊰∣∶≆⊣⊸≑−≺↿− ff):
⋮⇡↽≐≺↿−⋛≣≤ (fi-:): --]—
da: ↿ ↿
⊺⊅−∣∘⊰≺⊈⋅∙⊦−≆−⊣−≐−∙≡ 2..3
∢⊯∎⊦∙∙∙ '):
≦−↕∣∘∥⋣∙ 718281828 ...
];
sumptisque logaritbmis quoad basim 2, 718281828 ..., .
(II:—;:
istiusmodi basis solet exprimi per e.
Vlll'dy :[a&quot;dx—I—C ]—[ax ∙−⋅⋅⊢∁⋮∣∶∅≖↤≖− ar: daJr:

: a'd log (a'):axd [æ log (a)]:axlog (a) dt.

⋅70. Quantitas constans C,quaecumque ea sit, non in-
venitur iu differentialibus: idemqne proveniet dili'erentiale
sive dilferentietur v. g. sinæ ∙−∣⋅− C, sive siuæ.
80. In primo exemplo habemus ?: a, in se-
x
d d cundo a- . J
⊋−⋮⋮⋮ :— &amp;, 1n tertio 23:01: ', 111 quarto 71:47
in COST
in octavo
dy dy cost , in quinto sin x, in sexto dx dx
1
septimo di die= a * loga . Quisque videt
dy fore generatim novam functionem variabilis z : si ea
denotatur per f(x) , erit
2
dx
de = f( ), et dy = f ( z )dx .
Functio f '(x ) appellari solet derivata ex primitiva f( x) :
caeterum dx, seu differentiale variabilis independentis x,
spectatur quidem ut quantitas infinite parva ; sed simul con
stans atque arbitraria,
9º. Ex ivº, vº, et viº exemplo habemus
d sinx d sinx da dx :
dcosx
COS X V
sinx 1 - sinar
dcosx
77- cosa a '
dx = cosa x d tang x =
d tanga
sec2 x
dtang x
1 + langa x
Aequationes istae in hunc modum scribi possunt
dz dz
darc (sin = z ) = darc(cos = 2 ) =
V1 - Z
V 1-22
dz darc ( tang = 2 )
1 + z2
47
cosa: ,
in quinto ?; −∙−∙−∙ -—sinx, in sexto ⋛⋚∶≎∙⊂≐⊭−∙ in
septimo g

:

-.::— ,
in octavo :::-ï :axloga. Quisque videt
217— fore generatim novam functionem variabilis :: si ea

denotatur per f(æ) , erit
ngþ), et 47 :f(æ)dæ .
l
Functio f(æ) appellari solet derivata ex primitiva f(æ):
caeterum dx, seu differentiale variabilis independentis x,
spectatur quidem ut quantitas infinite parva; sed simul con-
stans atque arbitraria.
go, Ex &quot;o, vo, et vi&quot; exemplo habemus
dsinæ dsinæ dccsæ dar.... ∶−−−−−⇀⋅−⇁−⋅ :
.

: ⋅ cos 3&quot; l/ 1

-
sm'æ '&quot;&quot;:
&amp;
, dx:cos3xd tangæ:
deosæ
Vi-oos'x
dtangx ∙∙∙ dtangx
secaæ 1—i—tang3x
Aequationes istae in hunc modum scribi possunt
'
d
dare (sin— !.):sz ,darc(cos——z)-—- V
132 ,
-zz ∙ 2
darc(tang:z): T'↶−≀≘≖−−?'48
10 ° . Sicuti ex y = f( x) obtinuimus ( 8&quot; ) dy
f ( x )dx, sic ex hac obtinebimus ddy = f' ( x) dxdx
f '(x )dx?, ex qua rursus dddy = f &quot; (x )dx dx ' = f &quot; (x ) dr },
atque ita porro ; denotant fif &quot;, ... novas functiones
variabilis independentis x. Itaque si compendii causa e
xhibentur ddy, dddy, ... per dy, dy,.. , profluent
d
d’y = f '( x ) dx?,dy = f &quot; ( x ) dx3,
dy= f (x) ,
da²
d3y
= f'&quot; (r ) , ... : d.x3
assumpta v.gr.y = x ^, erunt f( x) = x ^ , f ( x ) = axa if '( ')
=a ( a - 1 ) 219-2, f ( x ) = a (a - 1 ) ( a - 2 ) x4-3, .
Differentialia dy , dºy , dy ,. . , itemque functiones deri
vatae f (x ), f ' (x) , f &quot; (x ), ... dicuntur primi, secundi, ter
tii , ... ordinis respectu functionis primitivae y = f (x ).
11 ° . Quemadmodum data functione possunt quaeri
ejus differentialia , ita vicissim dato differentiali quaeri po
test functio unde illud promanal. Sint F (x ), f (x ) ejusmo
di functiones variabilis x , ut exsistat F' ( x) =f( x) : quan
titas F ( x) + C vocatur integrale indefinitum differentia
lis f ( ) dx, designaturque praefigendo litteram ſipsi differen
tiali , ut scribatur
ſf(x) dx F ( x) + C ;
exprimit C quantitatem ( 7 &quot; ) constantem atque arbitrariam.
12° . Formula f ( x )dx ita sese aliquando exhibet,
ut statim appareat eam esse differentiale cujusdam da
tae functionis ; tunc vero in promptu est integrale: atque
hoc pacto habemus ( 6º . 9° )
f (a + 1)x*dx ******+. C,unde fredr =
xati
atito
48
100. Sicuti ex 7:f(x) obtinuimus (80) d] ∶∙∙−−⋅
f
(æ)d.r, sic ex bac obtinebimus ddj :
f
'(æ) dædæ :
f'(x)dæ', ex qua rursus dild]:f&quot;(æ)dx dæ' :f'&quot;(æ) dx3,
atque ita porro; denotant f,f &quot;,
... novas functiones
variabilis independentis æ. Itaque si compendii causa e-
xhibentur ddy, dddy, ... per dy, d37 ,. ., profluent
dïy
&amp;? d')&quot; :f'(x) da.&quot;,dfly :f&quot; (æ) das-3, ...,

: f'(æ),

113!
dæ3 :f'&quot;(.r),...:
∙∙⋅⋅∙⋅
assumpta v. gr.y:æ&quot;, erunt f
(æ):æ',f' (x):ax&quot;',f'(a-)

:: a(a — 1) x&quot;&quot;,f&quot; (x):a(a—1)(a—-2)x&quot;3, ....

Difаerentialia dy, diy, d3y,. . , itemque functiones derivatae f(x), f'(x) , f&quot;(æ) , ... dicuntur primi, secundi, tertii, ...ordinis respectu functionis primitivae y:f(x).
110. Quemadmodum data functione possunt quaeri
eius differentialia, ita vicissim dato differentiali quaeri po-
test functio unde illud promanat. SintF(x),f(æ) ejusmo-
di functiones variabilis x, ut exsistat F'(æ):f(x): quan-
titas F(x) −−∣− C vocatur integrale indefinitum differentia-
lis f
(.r) dar, designaturque praefigendo litteram ]ipsi differen-
tiali, ut scribatur
ff(æ)dæ:-—F(æ)—1-C;
exprimit C quantitatem (70) constantem atque arbitrariam.
120. Formula f(x)dx ita sese aliquando exhibet,
ut statim appareat eam esse dili'erentiale cujusdam da-
tae functionis; tunc vero in promptu est integi—ale: atque
hoc pacto habemus (60. 90)
a
&amp;
a-l-l C :: xtt-H f(a—1-1)æ dx:x ∙⋅∣− ,undefæ dx: ∉⊋∙∙⊦∙∙∙∓ ⊹∁⊒ï49
QCx
ſalog/a)d(c== q** + C, unde ſe*dx =clogiastc

;

dx
S
= arc ( sin = x ) +C ;
V 1 - 22
Sa
dx
1 + x2
= arc ( tang = x ) + C.
130. Interdum formula f (x )dz, de cujus integra
tione non constat , per quasdam substitationes transfor
matur in aliam , cujus integrale illico cognoscitur. Sic.
v . gr. positis ax = 2 , - = z ,assequimur a
dx dz 1
Si Salita
=
14a²x² arc ( tang == z) + C =
1
arc ( tang = ax ) + C ,
Sa
dix
22 ta
1
Sat
dz
a (1 + z2)
arc ( tang = 2 ) + c = a
-a arc tang * + c,
-Svador - Svado --Svet
( cos = ) + c
arc ( cos = z ) + c
= arc
fa&quot;log(a)d(cæ):a&quot; —]—C,undefa&quot; dx: -ac
dx
- ⇂∕↿∙−⋅⋥∎⊑ :arc (sin :x) −⊢∁≂
f 1112 :arc( tang:x )-l-C.
130. Interdum formula f(æ)dx, de cujus integra-
tione non constat, per quasdam substitutiones transfor-
matur in aliam, cuius integrale illico ougnoscitur. Sic.
.. æ . '. gl'. POSIUS nær-z.;— Zoasaequlmur
dæ ⇀∙∙− dz 1 — ∙−−
fl'l'a'æ' a(1-I-za) a
&quot;c (tang—z)-[-C—..
dx
xï-l-a
dz 1
faU-l—z') − a arc(tangzz)-I-C:
↿
—a.arc (tang :ax)-[- C,] ——
.—
—1-arc( tang :
−⋅⋮− ⋟⊹ C,
et
f
dx ]
adz ]
dz ∙∙∙
[fas-xa ∣∕ ∅≖∙∅≖≖≖ −⇀ ∣∕↿ -zz
arc(cos:z)-1-C :
arc(cos :?) ⊹∁∙50
140. In integrali indefinito ( 11 °) adhibeantur suc
cessive pro x peculiares valores xo, x n , ac dein ab F ( zn )
+ C subtrahatur F ( x ) +C ut , eliminata C , prodeat
F (xn) - F ( xo) : ejusmodi differentia vocatur integrale de
finitum differentialis f (x ) dx , sumptum videlicet ab x =
а &quot;
x
Xo
xh
ſ p(x)dx = F(wow )— F( xo ) .
Xo
Hinc v. gr.
a
dx
jederati
7T
a
o
Variato altero ex binis limitibus v. gr. x ny variabit
ipsum quoque integrale ; et adhibita x pro xmo erit
X
ſ f(x)dx= F ( x) — F ( xo ) :
Xo
habebitur videlicet integrale illud , quod incipit ab xo ,
quodque evanescit facto x = x,: et quoniam
aff(x)dx = d [F(x)-F(xs)] =dF( x) =f ( x) dx ;
X.
iccirco
X
S SP(x)dx = Sp«x ) dx + c .
X.
15 °. Sit arcus infinitesimus ABEH ( Fig. 18 ) , et
in eo chordae infinitesimae AB , BE , EH , quarum prima
50
140. In integrali indefinito (110) adhibeantur suc-
cessive pro x peculiares valores xo, x,, , ac dein ab F (x,)
−⋅∣− C subtrahatur F(xo)-I-C ut, eliminata C, prodeat
F(x,,)— F (x,): eiusmodi differentia vocatur integrale de-
finitum differentialis f(x)dx, sumptum videlicet ab x:
x&quot;
x, ad x:æ, ,designaturque per [f(x) dx, ut scribatur
æo
xn
]
f(æ)dx :P(æ.) — P(æ. )-
æo Hinc v. gr.
[ a
fa,-' dx: ..-.-..-—1 J'EL :
-E- a-l—1 ' xï-l-aa a.
0 0
Variato altero ex binis limitibus v. gr. x,, variabit
ipsum quoque integrale; et adhibita x pro x,, erit
.?
faa-w.r: ∌⇁≺∙↿∶≻−−∙≖⊸⇁≺∞∘≻≃⋍∙
æo
habebitur videlicet integrale illud, quod incipit ab x.,
,
quodque evanescit facto x: x,: et quoniam
&amp;?
df/(x)dæ:d[F(x)-F(æ.) ]:dF(æ) :f(æ) dx;
xo
iccirco
fff(-1')dx:ff( x
)
dxH—FC.
↿⋅⇂⋝∘⋅ x.,
Sit arcus iniinitesimus ABEH( Fig. 18 ), et
in eo chordae infinitesimae AB, BE, EH, quarum prima51
SUC:
ac tertia producantur donec concurrant in D. Quoniam an
guli DBE, DEB sunt infinitesimi, angulus quoque ODE
erit infinitesimus: designetur iste angulus per i , et fiant odest
e de
BD =
es
BE = c , DE b ;
habebimus
lur
62 =a +62 – 2ab cos ( 180° -1i ) = a + b2 +
2 ab cos i = a : + 62 + 2 ab- 2ab + 2 abcosi = (a + b )2
2 ab ( 1 — cosi ) =( a + b )2 – 4ab sin ’ şi ,
unde

: 1

4ab
sin _ i = 1
(a + b ) ( a + b )2
ariabi
et consequenter
[1 - ( +5)*] sinº in
= - = [ - (-3 ) ]su'_

: [&quot; - )*]*sist i -....

2 + b
ban
Differentia nimirum inter unitatem et rationem c ad
a + b consistet in terminis duntaxat infinitesimis , quorum
ordines excedunt omnes ordinem primum .
16º. Idipsum a fortiori dicendum de differentia
inter unitatem et rationem ipsius c ad subtensum arcum
BmE ; siquidem BmE &lt;a + b et &gt; c. Inde fit ut et ar
cus infinite parvus censealur aequalis respondenti chor
dae , et curva quaevis spectetur tamquam polygonum coa
lescens ex laterculis infinitesimis numero infinitis, et isto.
rum laterculorum prolongationes habeantur pro totidem
tangentibus apud varia curvae puncta.
rini
⋅ 500
(I.]
0an
ede-
lur
anali
bf&quot;
Lr;
(im!
51
ac tertia producantur donec concurrant in D. Quoniam an-
guli DBE, DEB sunt infinitesimi, angulus quoque ODE
erit infinitesimus: designetur iste angulus per i, et fiant
BD—fd, BE:c, DE:6;
habebimus
ea :a: ⊹∂≖ —2abcos(180'-'—-i) :03 ∙−⊦ 63 −∣−
Zabensiz—maa-i-ba -l-Zab—Zab-l-2abcosi :(cs-Fb):—
Zab,(1—- cosi):( a --[-b):
-
4absin* −≧−≀⋅∙
nnde
c*— 406 (a- -b)' ∙∙∎∙∙∙∙∙∶↿∙∙∙ . (a -l-b)3 sin ∙⋮−∎
a—ö a .

: ,

.
[1 (—r—b)]sm;-h et consequenter
c
⋍↿∙−−∶∙−∣∶↿ −≺∅≆≴≻≖∃ aina-Li—
a b a
∸⋇⋅∣∶↿ 3
−≺⋮−⋮−−⇣∙≑≻≏∃≏∘⋮∎≖∣⇩ ..;-i ∙−− ∙ ∙ ∙ ∙
DiEerentia nimirum inter unitatem et rationem c ad
a −⊦ &amp; consistet in terminis duntaxat inünitesimis, quorum
ordines excedunt omnes ordinem primum.
160. Idipsum a fortiori dicendum de differentia
inter unitatem et rationem ipsius (: ad subtensam arcum BmE ; siquidem BmE a −↿− 6 et 0. Inde Et ut et ar-
cus infinite parvus censeatur aequalis respondenti chor-
dae, et curva quaevis spectetur tamquam polygonum coa-
lescens ex laterculis infinitesimis numero infinitis, et isto-
rum laterculorum prolongationes habeantur pro totidem'
tangentibus apud varia curvae puncta.52
17º. Fac ut aequatio y f( x) pertineat ad cor
vam ABD ( Fig. 19 ) et sumptis coordinatis orthogonali
bus, sit abscissa OG = x, ordinata CB = y , infinitesimum
abscissae incrementum CC = dx : ducta per C' alia ordinata
C'B' , et per B lineola recta Bm parallela axi abscissarum
OX, erunt B'm = dy , Bm = CC = dx. Pone tangentem
BE occurrere abscissarum axi in E , normalem vero BH in H;
triangula rectangula et similia BEC , B'Bm , BCH dabunt
ydy

:

tang E
-
tang B'Bm dy, ce = ydx CH dx dy dx
CE dicitur subtangens, CH subnormalis.
18º. Ob auctam x area curvilinea BCa'a recipit
incrementum infinitesimum BB'C'C; est autem
BB'C'C =
dx
(rty + dy ) =
dxdy ydx +
2

&lt;math&gt;= ydx + f (x)dx =&lt;/math&gt;
ydr:
2
facta igitur Oa' = xo , erit
BCa'a- j^ydx = ${( )dx Xo Xo
Area BCa'a manifeste traduci polest ad rectangularem a
ream sub ejusmodi lateribus , quorum alterum sit differen
alterum vero ordinata quaedam ym media in
ter ordinatam aa' respondentem abscissae xo et ordina
tam BC respondentcm abscissae x : propterea
tia c Xo ,
X
ſ ydx = ( x - X . \ 'm , seu S f (x )dx = ( x - x . ) f ( xm ) .
X. Xo
Eadem area BCa'a spectari potest veluti summa ex infini
tis numero infinitesimis areolis rectangularibus
52
170. Fac ut aequatioy :
f
(et) pertineat ad cor--
vam ABD( Fig. 19) et sumptis coordinatis orthogonali—
bus, sit abscissa OG:x. ordinata CB: , infinitesimum

abscissae incrementum CC':dx :ducta per 0alia ordinata
C'B', et per B lineola rectaBm parallela axi abscissarum
OX, erunt B'm:dy
, Bm:CC':dx. Pone tangentem
BF. occurrere abscissarum axi in E, normalem vero BH in H;
triangula rectangula et similia BEC , B'Bm, BCH dabunt
J—— , tang E: tang B'Bm :
.. £,CF—Jjæ (31:731: '
]
.L'
CE dicitur subtangens, CH subnormalis.
180. Ob auctam x area curvilinea BCa'a recipit
incrementum infinitas-imum BB'C'C; est autem
⊞∍⋅∁∙∁−−−−↙⋚∁≺⊺ −⊢∫ ⊣−↙≀∫ )
∙−−∶
ydx −⊢ ∂⋅⋅↕−⋮↨−↗− :ydx-l— [figi-£ :ydx:
facta igitur Oa':x.,
, erit
x x
BCa'a:fydx :ff(x)dx.
xo xo
Area BCa'a manifeste traduci potest ad rectangularem a-
ream sub eiusmodi lateribus , quorum alterum sit differen-
tia x —-xo , alterum vero ordinata quaedamym media in-
ter ordinatam aa' respondentem abscissae an. et ordina-
tam BC respondentem abscissae x: propterea
x ⋅ x
fydx: (x -e-x., ly,&quot;, seuff(x)dx:(x—- x.,)f(x,,, ).
x., . xo
Eadem area BCa'a spectari potest veluti summa ex infini-
tis numero inlinitesimis areolis rectangularibus53
f ( x ) dx , f ( x +dx ) dx , COP
f ( xo +2dx ) dx f ( x — dx ) dx ;
nali
imum
Binala
.
sarum ubi nibil sunt aliud f (xo) , f( x + dx ), f (xo + 2dx), ...
nisi ordinatae respondentes abscissis xo , xo + dx , to +
2 dx ,
Quare
entem
in Hi;
bunt C
ſ f(x).lx = f(x )dx + f( xo + dx)dx +
Y :
Xo
fl xo + 2 dx )dc + .
+ f ( x -dx )dx.
recipé 19º. Ponatur arcus aB = s , ejusque incrementum
infinitesimum BB' = ds; quoniam BB'2 = Bm2 + B'ma, erit
2
ds = dx= + dy ,ideoque s= V dx=+dya =
X.
jäevitro
Xo
Tema
iffere
dia is
ordin
200. Circulus habens communia cum curva CC
( Fig. 18 ) duo proxima latcrcula v. gr. AB et BE, dici
tur osculator: sit O centrum istius circuli, BO ( r) ra
dius, OʻK et O'K' perpendicula ex O ducta in AB et BE ,
i angulus OBE , ds' et ds infinitesimi arcus laterculis AB
et BE subtensi, alter spectans ad circulum osculatorem , al
ler ad curyam CC' . Quadrilaterum KOʻKB praebet angu
lum KO'K ' = 180° — KBK' ; sed KBK' = 180°-OBE =
180° -1 ; igitur KOK' = , et consequenter ds' =
r( KOK' ) = ri' . Est autem ( 16 ° ) ds' = ds : propterea
infini
mali-
imum
linat:
arua
entem
Liuii; ↽
bum
⇟⇁∙∎↘⊰
..
recipi
rem ?
illerä
dia i?
orzlïm'
inüw'
53
f(xo)dx,f(xo-I-dx)dx,
f(xo—l—2dx)dx,. . ..f(x—dx)dx;
ubi nihil sunt. aliud f
(..-.,)
,
f(xo—l-dx), f(æQ—l-2dæ). .. . nisi ordinatae respondentes abscissis xo , xo -l-.dx , xo −∣−
de, . .. . Quare
æ
J. f(x)dx :f(xo)dx −⊢∙∣≼ xo-l-dx )dx ∙−∣−
∙↾≀⋅⋅∘
f(xo-l-2dx)dx −⊦ ∙ ∙
..
-I-f(æ-dx)dx.
190. Ponatur arcus aB: :. ejusque incrementum
iniinitesimum BB':ds; quoniam BB'a :Bma—l-B'ma ∙ erit
x
d:&quot;:dxï-l-dyïddeoque s:f V de-l-dy ∶−∙⋅−∙
æo
x
⋅∣∙↙≢∙↿∶⇂∕↿∙∙⊢∣⇃≖≼⋅≖⋅⋟∙
. xn ⋮⋅⋅
200. Circulus habens communia cum curva CG'
( Fig. 18 ) duo proxima latercula v. gr. AB et BE, dici-
tur osculator: sit 0' centrum istius circuli, BO' (:) ra-
dius, O'K et O'K' perpendicula ex 0' ducta in AB et BE,

: &quot; angulus OBE, 'ds' etïds-iniinitesimi arcus later-culis AB

et BE subtensi, alter spectans ad circulum osculatorem, al-
ter ad curvam CC'. Quadrilaterum KO'K'B praebet angu-
lum KO'K':1800 −− KBK'; sed KBK':1800—OBE:
180o — i' ; igitur KOK':i'
,
et consequenter ds&quot;:
r( KOK') :ri'. Est autem ( 16o ) ds':ds: propterea54
ds
21.• Curva CC' sit plana ; exhibeaturque per y =
f (x ), sumptis abscissis x in RX ( Fig. 20) . Erit i = Q
a = - (-a) = - dx , ideoque ( 170)
ds
ds
d x darc ( tang dy -
dx
)
Jamvero (90 )
dy
darc ( tang )
a dy
dr
dy²
dx²
df ( 30)
1 + f ? (x )
f (x ) dx ;
1 + f ? ( x )
dx
igitur
[1 + F2(x) ] }
f &quot; ( 3)
22.• Si ordinata y in curva y =f ( x) fit alicubi
maxima vel minima, exhibeaturque respondens abscissa per
Xn , quisque videt angulum interceptum tangente geometrica
et positivo abscissarum axe fore acutum vel obtusum pro
ut punctum contactus habuerit abscissam x &lt; vel &gt; xn in
casu maximi , &gt; vel &lt; x , in casu minimi , fore autem in
utroque casu = o ubi punctum contactus habuerit abscissam
x = x , Inferimus illud ( 8º. 170) : functio f (xn) est maxi
ma quotiescumque
f ( x) &lt; o quoad x = x + w ( denotat a quantitatem
infinite parvam &gt;0 ) , et f ( 2) &gt; o quoad x = xn - W ;
est minima quotiescumque
54
21 ∙∘ Curva CC' sit plana ;exhibeaturque per :7
f
(x), sumptis abscissis x
in RX (Fig. 20). Erit :&quot; a— a':
—(a'- a):
— dx , ideoque (170)
ds- ds
↗−− dx— dy
darc(tang:ä-;
Iamvero (90)
- si! darc(tang:i-'r .— dx ∙− ↙≀∣↬≺∙↿∶∟∙∙− f (adde;
dx −−↿ ,dJ' 1-t-f'(æ) l*f'ix)
dx'
igitur
3
[1 ⊣∙↾↔≖ (æ) ]
∶⊸∙ f&quot; r— (æ)
22.0 Si ordinata ;- in curva ;-:[(x) (it alicubi
maxima vel minima, exhibeaturque respondens abscissa per
x,, , quisque videt angulum interceptum tangente geometrica
et positivo abscissarum axe fore acutum vel obtusum pro-
ut punctum contactus habuerit abscissam x(vel )x,, in
casu maximi ,
)vel (x,, in casu minimi , fore autem in
utroque casu:
0 ubi punctum contactus habuerit abscissum
x:x,. Inferimus illud ( 80. 170) :functio f
(x,) est maxi-
ma quotiescumque
f
(x) (o quoad x :. x,, ↼⊢ co (denotat a quantitatem
infinite parvam )o )
.
et f' (x) )o quoad x :x. — a) ;
est minima quotiescumque55
f (x ) &lt; o quoad x = x, — W, et f ( x ) &gt; o quoad x =
X'n tw ;
valores X c.quibus respondet maxima vel minima f( xr ), quae
rendi sunt inter radices aequationis
p' ( x)
In
Si f ( x) maneret aut constanter negativa , aut constan
ter positiva, dum x versatur in viciniis x m , certe f ( x ) neque
maxima esset , neque minima .
Ad haec : quoad casum maximi, crescente x in viciniis
decrescit f' ( oc) , decrescente x decrescit f ( x) ; ideoque
df ( x)
&lt; 0 , seu f&quot; ( 30 ) &lt;0 . Quoad casum vero minimi ,
dx
crescente x crescit f (x) , decrescente x decrescit f ' (x ), et
af' ( x)
consequenter &gt; o seu f ( x) &gt;o .
23. Functiones plurium variabilium independen
tium x , 2 , u , ... designantur in hunc modum
dx
F ( x, 2, Ú, ... ) _f ( x, 2, U, ... ) , ...
Ponatur j = f (x ,2 , 9-9.) : si quaevis una ex quan
titatibus x, 2, u, spectetur uti variabilis et habeantur cae
terae pro constantibus , poterunt differentialia functionis u
eodem manifeste modo determinari ac differentialia functio
num quae ab unica pendent variabili. Ejusmodi differentialia
dicuntur partialia , ipsaque sic exhiberi queunt ,
ut
det , draf . d. , dal , ...
denotent differentialia functionis fe , primi , secundi ... ordi
nis quoad x , quoad 2 , ... Ad partiales functiones derivatas
quod pertinet , eae poterunt sic exprimi , ut per
55
f(x)(oquoad x:x,,—-o),etf (x))o quoad x:
a'.—FG);
valores x,,,quibus respondet maxima vel minima f
(x,,)
, quae-
rendi sunt inter radices aequationis
,'(æ)::00
Si f
(x) maneret aut constanter negativa , aut constan-
ter positiva,-dum x versatur in viciniis xn, certe f
(x,) neque
maxima esset
, neque minima.
Ad haec :quoad casum maximi, crescente x in viciniis
x,, decrescit ]&quot; (x) ,decrescente x decrescit f' (x); ideoque
df (x)
dx 0
.- seu f&quot; (x) (o. Quoad casum vero minimi ,
crescente x crescit f
(x) , decrescente x decrescit f' (x) ,
et
consequenter (IS .(ræ) )o seu f&quot; (x) )o.
23.&quot; Functiones plurium variabilium independen-
tium x ,z , u, designantur in hunc modum
F (x, :, ti, ...) ,f( x, :, u, ...
)
,
Ponatur p.:
f
(x, :, a.,.,.) :si quaevis una ex quan-
titatibus x, z,u. spectetur uti variabilis et habeantur cae-
terae pro constantibus
, poterunt differentialia functionis p. ↴
eodem manifeste modo determinari ac differentialia functio-
num quae ab unicapendent variabili. Eiusmodi dill'ereutialia
dicuntur partialia , ipsaque sic exhiberi queunt , ut
dxld-1 dxaPQ'&quot; ∂∷⊬∙∠↨≖≖⊬∙∙∙∙ 
denotent differentialia functionis 9. ,primi , secundi ordi-
nis quoad x , quoad :, Ad partiales functiones derivatas
quod pertinet , eae poterunt sic exprimi ,
ut per56
doll darf
dx dx2
dou , dazle
dedz dza
vel
per
fx(X , Z, Up... ) , f&quot; , (3 , 2, U, ... ) , . f : (3 , 2, U, ... ) ,
fo( %, 2 , Wo...) , ...
designentur functiones , primi , secundi ... ordinis derivatae
ex M = f ( x , % , U. ... ) quoad x , quoad 2 , ... Plerumque
tamen in his derivatis functionibus exprimendis detrahuntor ,
compendii causa , litterae d signa x , % , U 7 .** , et pro
dal d , ² l
dx dx2 d,I
d², M
dz dz
adhibentur
du del i
dx dx2
du dele
dz dz ?
9
24º . Totale functionis pe differentiale due ( quum
nempe x
spectantur omnes ut simul variabiles )
eruitur ex partialibus dx f , d , f , dul , ... ; sunt enim
% , U ,
f ( x + dx, 2, 1, ... ) - f ( x , ,U, ...) = fx ( x ,2 ,4, ... ) dx,
f ( x + dx, atdz, u, ... ) -f( x + dx, 2, U, ... ) =
f: ( x + dx, 2, u, ...) dz = f : ( x, z, u, ... ) dz,
f ( x + dx, atdz, utdu, ... ) — ( x + dx, atdz, u , ...) —
f'u ( x + dx, z + dz, il ... ) du = f ( x, 2, u, ... ) du, etc... ,
)
1
.—
56
≀≀≖≀∸ −−↙≀⇄↕≴∸ .⋅≤≀−⊦∸
−∙∙ −−−⊓≀⇄≖≴∸ dx dx&quot; , dz dza '
vel per
fx(x, :, uh&quot;) ,
f&quot;: (x, :, u, a&quot;) , ∙∙∙ f, (x' z' u, a.) ∙
f',(x, :, u, ...)
,
designentur functiones , primi , secundi .. ordinis derivatae
ex &quot;. f
(æ.:, u.
...
) quoad x, quoad :∙ ∙∙∙ Plerumque
tamen in his derivatis functionibus exprimendis detrahantur,
compendii causa ,
litterae d signa :, a, «,... ,
et pro
de- dx'P- (!sz (I,,[L da:
∙ m ⋅⋅⋅⊤ ⋅−∂⋅⊒⋅−
adhibentur
≴≀−⋅≖∸− ↙≀≖⋅⊀↓ de dw
dæ'dx' ,' de, dza
240. Totale functionis p. dilferentiale dp. (qumn
nempe x , z , n , spectantur omnes ut simul variabiles )
eruitur ex partialibus d, (1.
,d, pt
, d,, p. , ; sunt enim
fl xhi—dx, 39 ut ...) ↼f( æ, :, u, ...):
f, (æ, :, ., ...) dx,
f(æ-l—dæ, t—l—rlz, u,...) — f(x-[r-dx, :, n, ...):
f: (xä-dx, :, &quot;, mida: f, (x, s, u, ...) dz,
f( æ-I- dx, z-l—dz, u-l-du, ...) ∙− ≼∙↧∙⋅∙∣−↙∣∙↧⋅∙ z—i-dz, u, ...) ::
f,&quot; (x-i—dæi z-l-dzo nus) du:f,, (æ, Z, &quot;, ,,.) du, etc-0-
'57
quarum summa praebet
p ( x + dx, atdz, utdu , ...) — f ( x, z , l , ... ) =
fr ( x, 2, U, ...)dx + f : (x , 2, u, ...)dz + f'u ( x,
Z, U, ... ) dut ... , seu dų = d .; + d ,l + d.le + ...
25.• Potest etiam functio pe differentiari successi
ve quoad binas, lernas , ... variabiles v . gr. quoad x, z, quoad
X , 2, u ; etc. ... Id genus partialia secundi , tertii , ... ordinis
differentialia designari queunt per
d, dx M , d , d , dal , ...
sive autem functio u prius differentietur v . gr. quoad x
deinde quoad z , sive prius quoad z , deinde quoad x ,
paallulum attendenti patebit idem in utroque casu pro
venturum differentiale .
26. Detur nunc differentialis aequatio primi ordinis
dy - cydx f ( x ) dx ;
facta y = zu, et adhibita substitutione, emerget zdu + ud:
czudx = f ( x) dx . Pone udz – czudr = 0 ; habebis
dz
= cdx , log ( x ) = cx = cx log ( e) = log ( eⓇx ) ; unde 7
z &gt; eºx :
in ea qua sumus hypothesi
zdu = f(x) dx ;
igilur du = f ( x ) dx
f (x) dx , u Sf (x)dx + G ; et 7 es
ex
1
5
quarum summa praebet
f(x-f—dx, z-l—dz, (kl—du,...) —f(x,'z, u, ...):
f: (æs 31 ut ⋅∙ ')dæ—I—fg (æ, :, II,. ..)dz—l—f'u (æ,
.z, u, .,.) du-l—n., seu dy.— −∙∙ d,.p. :i- dyp. ∙−⊦ d,); ∙∣−∙∙∙
25. ∘ Potest etiam functio p. diil'erentiari successi-
vequoad binas, ternas... .variabiles v. gr. quoad x,z, quoad
æ, :, u; etc.
... Id genus partialia secundi ,
tertii,... ordinis
diii'erentialia designari queunt per
ds dxp'adudadxp-vmi
sive autem functio p. prius differentietur 11. gr. quoad ac
deinde quoad :, sive prius quoad z, deinde quoad x ,
paullulum attendenti patebit idem in utroque casu p1o-
venturum differentiale.
26.&quot; Detur nunc differentialis aequatio primi ordinis
,dy— cydx :f(x) dx;
facta ]: zu, et adhibita substitutione, emerget zdu −⋅⊢ ud:
— czudx :
f
(x) dx
. Pone udz —. czud-r :o ; habebis
dz
Z
∙−−− ∖∙∘⊄≀⋅⋍∙⋅ ,
log (z):cx:cx log (e):
log (e&quot;); unde
∙−−− cx
, z....e :
«
in ea qua sumus hypothesi
zdu :f(x) dx; ∙ ∙
'
igitur du:
,
(x) dx *fbl'c) dx :
&quot;:M—i—C; et 2 0 .: et.:
5 d58
consequenter
y = eriſ f x)dx = C ] :
integratio videlicet dalae aequationis differentialis traducitur
ad integrationem functionis f (x ) dx Porro absoluta aequa er
tionum differentialium integratio eo redit , ut quae relatio
inter quantitatem et quantitatem per eas exprimitur , aequi
valenter exprimatur per aequationes differentialibus liberalas.
27 .. Si dalur differentialis aequatio secundi or
dinis
day dy ta dx +
0 ,
dxt
by:
designantibus k et k' radices aequationis 32 taz +b =0 ,
traducelur illius integratio ad integrationem binarum pri
mi ordinis
dy ' dy - ky ' = 0 ,
dx dx
siquidem , eliminata y' , prodibit
- ky = y ' ;
a
dy
-ky)
dx dy
dx – k G - hy) == 0 ;
quae , ob k tok = -a et kk' = b , recidit in datam. Jam
vero ( 260 )
dx y ' = Cetry = e ** C
elix

:

ergo
y = ek's
es [ foe-tyde +c ] - [ * +c ]=
Ceks + C'ek's .
58
consequenter
yzccxiffiæidæzcl: Bex
integratio videlicet datae aequationis differen'tialis traducitur
f
(x) dx ad integrationem functionis
ac:
. Porro absoluta aequa-
tionum differentialium integratio: eo redit ,
ut quae relatio
inter quantitatem et quantitatem per eas exprimitur , aequi-
valenter exprimatur per aequationes differentialibus liberatas.
27.0 Si datur differentialis aequatio secundi or-
diuis
?;: 437 dr −⊦ &quot;ï; −⊢ b,! −−∶ ∘⋅
designantibus I: et k' radices aequationis z' −⊸⊢ az —]-b:0.
traducetur illius integratio ad integrationem binarum pri-
mi ordinis ⋅
alt&quot; ∙⋅− df
↙↙−↜↕∶∎∎−∎↗⋮∫−−∘∣∠≀↜↿∶−↻ ⋅⋅⋅⋅∙∙⋅−−−−−↗ '
siqnidem , eliminata y'
, prodibit
d d ∠−− '
(dx kf) A(g—F):0.
d.; ⋅ dx ] '
quae ∙Ob k −⋅⊢ ∣⊏∎:
— a et kk':b, recidit in datam. Jam-
vero (260)
.)&quot;: Ce&quot;'.y:e*&quot;[ ∫−⋅≤∎⇂−⊺∶⋮⇆∙⊹∁∙ ]

:

ek'x
ergo
∙∙∙ ': ∙∙ ∙ r ∙ rr Ceu—H).:
..7— e*. [Ca,/460 &amp;) dx—l—C] ∙−−∶ e* k—k. *C]:
- Ce&quot; ∙⋅∣−∁∎ e*&quot; .59
28.- Si daretur d²y dxata
dr. + by = f(x),tra
duceretur integratio ad integrationem binarum dy'
-ky'
da
P(z) Tipo - Ky = y'; sicque prodirent ( 260)
[Sl + c]
e** [ S ,* + c ]
y' = etxe
k et k 'sunt , ut supra, radices aequationis z2 taz + b = 0.
29.• Resumentes functionem f ( x ) , ponamus
f(x) = a, tax taqx? +R3 2 :3 + 04x4 +

:

exsurgent
f ( x) = a + 2a , x + 3az x2 + 404 x3 + ... , f&quot; ( x)
2a + 3.2a3 x + 4.394 va t ... if&quot; ( 0) = 3.2a3 +
4.3.204 x +
Facto x = 0 , emergent ao f (o) , a, =: f ( 0) , a,
i f' (o) , az =-3f&quot; (o), etc.... Hinc
etc...
f(x) = f(0)+xf ( 0) + 1&quot; (0) + &quot;(o) + ...
Sint v. gr. f (x) = e*. f (x) sinx , f (x) = cosx : quoad
primam f (o ) = 1, f (0 ) = 1 , f ' (o) = 1,8 &quot; (0 ) = 1 , etc...;
quoad secundam , f (o) = 0 , f ( 0) = 1 , p (o ) = 0 ,
fr (0 ) • , 1 , f (0) = 0, f ( 0) = 1 , etc...; quoad ter
59
. dfy dy 28.0 St daretur . (: d −⊦∙ 6]:
f
(x) ,.tra-
dxa x
duceretur integratio ad integrationem binarum Si.-..;. ':
dx '7
f
(a:), £
—)(]:
!' &lt;nowiki&gt;; sicque prodirent (260) www-rc]&lt;/nowiki&gt;
730, reli]dx
11 et k' sunt
,
ut supra ,
radices aequationis z' -l—az—-I—-b :o.
29.&quot; Resumentes functionem f
(x) , ponamus
f(x):ao—I&quot;alx −⊦∁≖∙↕≖∙−∣⋅−↷∍ ∷∙⋅∍⊣−∦∣∙∙≂↙∣−⊦∙∙∙⇋
exsurgent
f(x):a, -l-Za,x-l-3a3x3 404 ∞∍−⊢∙∙⋅ ,f&quot; (x):
2a3-1l-3. Zaax-i— 4. 3a(,x2 -[-...,f&quot;(x) :3.2a3—[—
4. 3. 244x—l-..,
,etc...
Facto x:a, emergent a,:
f(o) , a,:
f
(a) , a,:

;

f. (0) , 03 3— f&quot; (0) ' :.
etc-00. Hinc 3 '
f(x): f(0)-l-xf(0)-i-—-f'(0)'l&quot;——f (o)-b&quot;-
Sint v. gr. f(x):ex.f(x) :sinx ,f
(x):
coax :quoad
primam f(o):1, f
(0):1, f' 'o):1,f&quot; (o):1, etc...;
quoad secundam , f(o):0, f(o) :1 , f&quot; (a):0 ,
f&quot; (Ol—&quot;'
-— ∙−− 1, f' (0):0, f' (0):1, etc...; quoad ter-60
23
tiam , f (o ) = 1 , f ( 0) = 0.8&quot; ( ) 1,8 &quot; ( 0 ) = 0 ,
f (o ) = 1 , ' (o) = 0 , p (0 ) 1 , etc... ; ideoque
x2 24 3 e* : 1 tox +*+ +
sin u = r
2.3.4 2.3
x2 8: 4
cos = 1
2.3.4.5 2
+
2.3.4 2.3.4.5.6
1+
ar5
x6
1.5.0 + ...
30.• Adhibita xV - 1 pro x in istarum prima,
emerget
= 1
x2
e **vi .x4
+
2.3.4
Xc6
2 2.3.4.5.6 +
r3 xc5
+
2.3 2.3.4.5 -.)v = 7.
ܪ
unde , ob secundam ac tertiam ,
e #xVST = cos x + V1 sio x .

28. Fac nunc ut punctum materiale vi qualibet continua
sollicitetur ad motum rectilineum: sit »» velocitas puncti
in fine temporis t,,.s spatium percursum , et ds spatiolum
percurrendam subsequente tempusculo dt. Perinde spectari
poterit ds ac si motu uniformi conficeretur , sola nimirum
velocitate praeconcepta 1); siquidem nova velocitas dv, quae
labente d:accedit materiali puncto, utpote infinitesima. ne-
gligenda.Hi11c (1 )
s
[
∥
Motus rectilineus puncti materialis iugiter sollicitati vi
constanter eadem, dicitur uniformiter varius. Per ep desi-61
/
gnetur velocitas, quam vis constanter eadem gignit intra
tempus 1 , erit qe velocitas ( 6 ) genita intra tempus t :
propterea denotante vo velocitatem initialem , qua nimirum
donatur materiale punctum quum t = 0, existet
v =v, +9
ds Hinc dt votoe : fac ut tempori 1 = o respondeat
So ; habebis
s -8 = v. i + 902 ?

;

2
1
et eliminato t ,
v2— v.2 = 29 / s - s . ) :
positis v ,30,
0, erunt
V = pt , s =
- Det
, v?= 205 ,
o dicitur vis acceleratrix : el designante m
massam puno
cti materialis, m q appellatur vis motrix : insuper spatium s in
aequatione ultima vocatur allitudo debila velocitati v.
Ad motum rectilineum utcumque varium quod spe
ctat , nomine vis , acceleratricis apud terminum spatii per
carsi s nihil aliud intelligitur nisi velocitas q , quam gi.
gneret vis conversa in constantem, constantique energia
qua inibi pollet , agens loto tempore 1. lamvero exhibet
do numerum tempusculorum , ex quibus coalescit tempus
1 ; ergo velocitas illa exprimetar per dv; nimirum
1
61
gnetur velocitas, quam vis constanter eadem gignit intra
tempus 1, erit got velocitas (6) genita intra tempus ::
propterea denotante 'Uo velocitatem initialem, qua nimirum
donatur materiale punctum quum :

: o, existet

v:v, ⊣∙− cp :.
Hincd ∙−∙−:v.,-l—got: fac ut tempori :

: o respondeat

,,- s,;habebis
(2
⋅⇟−⋅⋅≖∘∶∶
'&quot;o t&quot;l&quot; 92&quot;;
et eliminato t,
vï—vo*:2?( s—s, ):
positis v,: o ,r,: 0, erunt
v:g0t, :: gt: ,
v': 291,
q; dicitur vis acceleratrix: et designante m massam pun-
cti materialis, m ? appellatur vis motrix: insuper spatium .: in
aeqnatione ultima vocatur altitudo debita velocitati 9.
Ad motum rectilineum utcumque varium quod spe-
ctat
, nomine visacceleratricis apud terminum spatii per-
cursi :nihil aliud intelligitur nisi velocitas ga, quam gi-
gneret vis conversa in constantem, constantique Aenergia ,
qua inibi pollet, agens toto tempore 1. Iamvero exhibet
−↿−∙ numerum tempuscu'lorum, ex'quibus coalescit tempus
dt
1 .: ergo velocitas illa exprimatur per Tit-dv; nimirum62
dv
dt .
et quia dy
d's d
dt ; idcirco erit quoque
dès
d12
habetur pro variabili atque independente quantitate.
29. Fac v. gr. ut materiale punctum sollicitetur versus
datum centrum vi acceleratrice, quae distantiae z' ab eo cen
tro exsistat proportionalis , ut , denotante C ' quantitatem
constantem , habeamus q =C'z' ; sit z, initialis distantia ,
ibique vo =0 , t =0 ; sit insuper v ' velocitas in distantia
z' : erit ( 28 )
v = d (20-3')
dc
dz
dt
du'
ideoque C'z' =
dt
v'dz'
dzi
Hinc 19. Cʻz'dz' = -v'dv'; ex cujus integratione prodit
C- W'2
C'z'2 =C -2'2 , z = ve C'
facto z' =0, erit v' velocitas punci materialisió centro virium ;
exprimit igitur C hujusce velocitatis quadratum : quod si fiat
z' =2 . , erit ex hypothesi v = v = o, ideoque
VT= 2.VC ;
velocitas nempe puncti materialis in centro virium est ut ipsa
initialis distancia zo.
62
0:32- ' dt -'
et quia xlv::! g:- ; idcirco erit quoque
d3s
(:):
d::
'
habetur :pro variabili atque independente quantitate.

29. Fac v. gr. ut materiale punctum sollicitetur versus
datum centrum vi acceleratrice, quae distantiae z' ab eo cen-
tro exsistat proportionalis , ut, denotante C&quot; quantitatem
constantem , habeamus q) :C'z'; sit zoïinitialis distantia ,
ibique v.:o, t:o; sit insuper v' velocitas in distantia
z' : erit ( 28 )
d(zo-z')-——dz' .d ∙
∙−− dp'— v'dz' d: d:
&quot;
eoque c.. d. dz' '
I,.....
Hinc ↿∘∙ C'z'dz':
—- v'dv'; ex cuius integratione prodit
C— 'v'ï cause—w.r: ⇂∕−∁∼⊤−⋮
facto z':o,erit v' velocitas puncti materialisin centro virium;
exprimit igitur(] hujusce velocitatis quadratum: quod si fiat
z':z,, erit ex hypothesi v':v,;—..-o, ideoque
⇂∕⋜⋮∶−−⊸−≖∘⇂∕−∁−⋮≂
velocitas nempe puncti materialis in centro virium est ut ipsa
initialis distantia z..63
2.º du
1 Tc di=
C'zi v CV C -via VC
Vic С
suinptisque integralibus ,
i = C &quot; +
ve are (sin = vo ):
v = o quando i = 0 , proinde
Vc are ( sin = o), exquav = VC sinero.
3º. Cum in centro virium sit v = VC, erit ibi
1 = sint y C , et consequenter t = Inferimus pun
n
2V0
a
1
ctum materiale eodem semper tempore quacumque 2VC
distantia z . perventurum ad centrum illud .
4º. Si materiale punctum movetur vi accelera
trice, quae distantiae a dato centro sit proportionalis , sic
absque formularum subsidio polest ostendi eodem semper
tempore punctum ipsum peryenturum ad centrum illud :
concipiantur duo puncta, quorum primum triplo magis i
nitio molus distet a centro quam secundum : quoniam ex
hypothesi vis est proportionalis distantiae a centro, erit vis
primi triplo major quam secundi , ideoque triplam velo
citatem primo tempuscalo illud acquiret, et triplum spa
lium percurret; quare etiam tripla ibidem residua erit di
stantia. Igitur et secundo tempusculo triplam velocitatem
novam acquiret, et triplum spatiolum tum praecedente, imm
⊖⊰∆
2.-
−≀≀∙↗⋅ ∙∙∙ dv' '
dv' 1 C'z' yel/CT?&quot;— ⇂∕⋅∁⋅ ⇂∕↿−−−∙∙−−∙∽⋅∙∑−⋅ :
sumptisque integralibus ,
1
(
. v' )
are sm −−− ;
C' y'C
v':o quando :: o , proinde

: z ∁∙∙−⊢ ⇂∕

!
(z.—1.-.— arc (
sin: v
), ex qua ≸↗⋅∶∶⇂∕ ⇂∕∁ ⇂∕∁ sint;/CZ
30. Cum in centro virium sit v': l/C,erit ibi
'io
. n ↿∶∶ . sunl/C, et consequenter :: ï— . Infenmus pun-
ctum materiale eodem semper tempore a quacumque 21/ C
distantia za perventurum ad centrum illud.
40. Si materiale punctum movetur vi accelera-
trice, quae distantiae a dato centro sit propmtionalis, sic
absque formula1um subsidio potest ostendi eodem semper
tempore punctum ipsum perventurum ad centrum illud:
concipiantur duo puncta, quorum primum triplo magis i-
nitio motus distet a centro quam secundum: quoniam ex
hypothesi vis est proportionalis distantiae a centro, erit vis
primi triplo maior quam secundi, ideoque triplam velo-
citatem primo tempusculo illud nequiret, et triplum spa-
tium percurret; quare etiam tripla ibidem residua erit di-
stantia. lgitur et secundo tempusculo triplam velocitatem
novam acquiret, et triplum spatiolum tum praecedente, tam64
nova vi et velocitate percurret: unde consequitur ut tripla
pariter sit lota velocitas jam acquisita , triplum totum spa
tium percursum, tripla distantia residua. Propterea et no
vo tempusculo tripla erit nova velocitas acquisita , tri
plum spatium novum percursum , tripla nova distantia; atque
ita porro. Patet igitur post tempus quodvis distantiam
primi fore triplam distantiae secundi, ac proinde imminu
ta in infinitum ac demum evanescente hujus secundi di
stantia, illius quoque primi distantiam in infinitum immi
nui ac simul evanescere: haud poterit ergo secundum pun.
clum ad centrum pervenire, quin simul cum secundo ipso
primum punctum perveniat. Hoc tantummodo discrimen e
rit, quod primum eo deveniet velocitate tripla secundi ; ex
quo manifeste consequitur , quod si primum illud punctum
ex centro cum illa tripla velocitate projicitur , debebit ad
triplam distantiam pervenire; nam vis in recessu velocita
tem codem ordine extinguit , quo generat in accesso. Por
ro quod diximus de ratione tripla , patet generatim conve
nire rationi cuicumque ; nimirum in quacumque propor
tione fuerit distantia prini punci major quam secundi ,
eodem tempore semper ambo ad centrum devenient cum
velocitalibus , quae distantiis initio habitis sint proportio
nales; et si inde discedant cum velocitatibus quibuscumque,
pervenient eodem pariter tempore ad distantias ipsis ve
locitatibus proportionales.
5º. Dicatur tempus quo materiale punctum it ac
redit uude primo discessit; erit ( 3º. )
471 276 271 0
276 VC
Quare ( 1º, 2º. )
2750 C ,
6
0 2751
0
G
220
210
VO TT
z = VC
COS 277
64
nova vi et velocitate percurret: tinde consequitur ut tripla
pariter sit tota velocitas iam acquisita, triplum totum spa-
tium percursum, tripla distantia, residua. Propterea et no-
vo tempusculo tripla erit nova velocitas acquisita, tri-
plum spatium novum percursum, tripla nova distantia; atque
ita porro. Patet igitur post tempus quodvis distantiam
primi fore triplam distantiae secundi, ac proinde imminu-
ta in infinitum ac demum evanescente huius secundi di-
stantia, illius quoque primi distantiam in infinitum immi-
nui ac simul evanescere: haud poterit ergo secundum pun-
ctum ad centrum pervenire, quin simul cum secundo ipso
primum punctum perveniat. Hoc tantummodo discrimen e-
rit, quod primum eo deveniet velocitate tripla secundi; ex
quo manifeste consequitur, quod si primum illud punctum
ex centro cum illa tripla velocitate projicitur, debebit ad
triplam distantiam pervenire; nam vis in recessu velocita-
tem eodem ordine extinguit , quo generat in accessu. Por-
ro quod diximus de ratione tripla, 'patet generatim conve-
nire rationi cuicumque; nimirum in quacumque propor-
tione fuerit distantia prinii puncti maior quam secundi,
eodem tempore semper ambo ad centrum devenient cum
velocitatibus, quae distantiis initio habitis sint proportio-
nales; et si inde discedant cum velocitatibus quibuscumque,
pervenient eodem pariter tempore ad distantias ipsis ve-
locitatibus proportionales.
50. Dicatur 9 tempus quo materiale punctum it ac
redit uude primo discessit; erit (30.)

:

∡⊺≖∙∙∙∙∙⊸ 211 ,—21t
9
⊋⇂↗⇠∁⋮↼−⇀∣−∕−⋐⇀∶⋅−∙⋅⇂∕∁−⇀⊺⋅∙ Quare (10. 20.)
∙∙∙⊇⇂∕∁ 9
220 ∙∙∙⊓∙ ⇂∕∐⋅ ∶∶↼⋤⋮−⇂∕∁∙≀≀↗⋅∶⇂∕∁⊱⋮∥ −−−−⊖−⋅ ⊋⋯⋅ ⋅− 9 271! z': l/C -—-co −−−−∙
271 s 965

=== De verticali gravium descensu atque ascensu. ===
[[30|30]]. Si gravitas aequaliter semper ad sensum corpora decidentia sollicitare intelligitur, motus erit uniformiter varius (28): positis igitur &lt;math&gt;v_0=0,s_0=0&lt;/math&gt;, et denotante &lt;math&gt;g&lt;/math&gt; vim acceleratricem ex gravitate, in ea qua sumas hypothesi determinabitur motus per formulas

&lt;math&gt;v =gt,s=gt^2/2, v^2 =2gs (b) , &lt;/math&gt;

legibusque sequentibus subjicietar.

1&lt;sup&gt;a&lt;/sup&gt;. Spatium &lt;math&gt;s&lt;/math&gt; percursum intra tempus &lt;math&gt;t&lt;/math&gt; est dimidia pars illus spatii &lt;math&gt;s'&lt;/math&gt;, quod percurreretur si grave aequali tempore pergeret moveri uniformiter cum velocitate &lt;math&gt;v&lt;/math&gt; in
fine temporis &lt;math&gt;t&lt;/math&gt; acquisita; nam (1)

&lt;math&gt;s' = tv = tgt = gt^2 = 2s.&lt;/math&gt;

2&lt;sup&gt;a&lt;/sup&gt;. Spatia totalia a gravibus libere decidentibus percursa,
sunt ut quadrata temporum quibus eadem spatia conficiuntur: item ut quadrata velocitatum tempore descensus acquisitarum  Nam

&lt;math&gt;s=gt^2/2=\frac{v^2}{2g}.&lt;/math&gt;

3&lt;sup&gt;a&lt;/sup&gt;. Spatia a gravibus libere decidentibus percorsa
aequalibus et successivis temporibus sequuntur progressio
numerorum imparium 1,3,5,7, ... ; assumpto enim
&lt;math&gt;t = 1,2,3,4 &lt;/math&gt;, ... spatia illa exprimentur per &lt;math&gt;\frac{g}{2},
\frac{4g-g}{2},\frac{9g - 4g}{2},\frac{16g-9g}{2}, \mathrm{seu}\,
\frac{3g}{2}, \frac{5g}{2}, \frac{7g}{2}.
&lt;/math&gt; 

Hae leges experientiae cum sin &lt;math&gt;t&lt;/math&gt; consentaneae, hypothesis gravitatis aequaliter semper ad sensum agentis prope
telluris superficiem existimanda est naturae conveniens: et
quoniam experimentis saepe iteratis apud nostras regiones compertum est, grave sibi relictum percurrere pedes
15, 0915 ... intervallo unius minuti secundi, erit

&lt;math&gt;g = \frac{2s}{t^2} = 2\times 15,0915 ... = 30,183 ... &lt;/math&gt;&lt;ref&gt;9,78:30,183=0,324 m/pes&lt;/ref&gt;

Eam nimirum velocitatem gravitas valet mobili communicare intervallo unius secundi, qua si mobile pergeret uniformiter moveri, absolveret singulis secundis pedes 30,2 circiter.

Deprehenderunt quidem Physici gravitatem esse diversam tum ad diversas supra terrestrem superficiem altitudines, tum ad diversas ab aequatore terrestri distantias: verum ejusmodi variationes in corporum gravitate haud fiunt sensibiles nisi sub differentiis admodum grandibus sive inter altitudines illas, sive inter illas distantias; propterea absque sensibili errore contemni poterunt in ordine ad singula corpora terrestria, quae ut plurimum veniunt consideranda.
Si retenta &lt;math&gt;s_0=0&lt;/math&gt;, ponitur &lt;math&gt;v = a&lt;/math&gt;, exsurgent (28) &lt;math display=&quot;block&quot;&gt;v=a+gt, s = at + gt^2/2, v^2-a^2 = 2gs (b').&lt;/math&gt;

[[31|31]]. Assumpta &lt;math&gt;g&lt;0&lt;/math&gt; in (b'), prodibunt&lt;math display=&quot;block&quot;&gt;v = a-gt, s = at - gt^2/2, a^2-v^2 = 2gs (b'');&lt;/math&gt;

quae formulae manifeste determinant verticalem gravium ascensum. Facta &lt;math&gt;v=0&lt;/math&gt; in tertia ac prima (b&quot;), emergent
&lt;math&gt;
s=\frac{a^2}{2g}, 
t= \frac{a}{g} (b'''),
&lt;/math&gt;
maxima nempe altitudo ad quam ascendit grave, tempusque respondens.

Obiter hic notamus illud: Si datur ejusmodi potentia &lt;math&gt;R&lt;/math&gt;, quae agendo ad modum vis instantaneae valeat massae &lt;math&gt;M'&lt;/math&gt; communicare velocitatem &lt;math&gt;a&lt;/math&gt;, ut sit (6) &lt;math&gt;R= M'a&lt;/math&gt;, ipsa &lt;math&gt;R&lt;/math&gt; agendo ad modum vis continuae per gradus infinitesimos poterit ponderosam massam &lt;math&gt;M&lt;/math&gt; sustentare libratam per totum tempus &lt;math&gt;t = \frac{M'a}{Mg}&lt;/math&gt;

Cum enim singulis tempusculis infinitesimis &lt;math&gt;dt&lt;/math&gt; gignat
gravitas in massa &lt;math&gt;M&lt;/math&gt; quantitatem motus &lt;math&gt;Mgdt&lt;/math&gt;, certe singulis &lt;math&gt;dt&lt;/math&gt; debebit &lt;math&gt;R&lt;/math&gt; ad librandam &lt;math&gt;M&lt;/math&gt; exerere actionem infinite parvam &lt;math&gt;=Mgdt&lt;/math&gt;; proinde totalis actio respondens toti &lt;math&gt;t&lt;/math&gt; erit &lt;math&gt;\int Mgdt = Mgt&lt;/math&gt;: igitur &lt;math&gt;Mgt=M'a&lt;/math&gt;; ideoque etc.

Quisque nunc videt posse vim &lt;math&gt;R&lt;/math&gt; exhiberi non solum per
&lt;math&gt;M'a&lt;/math&gt;, sed etiam per &lt;math&gt;Mgt&lt;/math&gt;.
[[Fasciculus:Atwoods machine.png|thumb]]
[[32]]. Ad motum gravium determinandum in machina Atwoodi, sint &lt;math&gt;m&lt;/math&gt; et &lt;math&gt;m +m'&lt;/math&gt; massae filo appensae: quisque videt motricem systematis vim exhiberi per &lt;math&gt;
g ( m +m' ) - gm =gm'&lt;/math&gt;; unde profluit vis acceleratrix &lt;math&gt;g\frac{m'}{2m + m'}&lt;/math&gt; substituenda loco &lt;math&gt;g&lt;/math&gt; in formulis (b). Quoniam vis ista potest pro lubito attenuari, sequitur in Machina Alwoodi posse motus velocitatem imminui quantum libuerit; quod maxime conducit et ad accuratius definienda spatia percursa, et ad aeris resistentiam tuto negligendam. Sicuti enim corpus, quod movetur in medio aliquo materiali, agit in ipsum medium, ejus particulas expellendo, exerceturque corporis actio juxta motus directionem, ita medii particulae juxta contrariam directionem reagunt (7) in corpus atque resistunt; inde oritur quidem imminutio virium in corpore, sed major vel minor, prout major vel minor velocitas communicatur medio expellendo; et consequenter prout major vel minor est velocitas corporis expellentis.

[[33|33]]. Haec notamus circa gravium motum in medio resistente.

1º. Constat gravia decidentia in pleno homogeneo motum suum vi gravitatis sic accelerare ut paullatim evadat proxime et sensibiliter uniformis. Dum nempe corpus initio movetur, primumque velocitatis gradum acquirit, aliquam hujus gradus jacturam pati debet ex opposita medii resistentia. Sed quia velocitas corporis in progressu semper augetur, multo magis augeri etiam debet medii resistentia; siquidem major corporis velocitas non solum importat ut major quoque velocitas communicetur singulis particulis removendis, sed praeterea ut major quoque resistentis materiae quantitas dato tempore dimoveatur. Ergo velocitatis gradus semper magis imminuetur: unde fit quod velocitas corporis ad valorem constantem propius semper accedat, ejusque motus paullatim evadat proxime et sensibiliter uniformis.
[[Fasciculus:Atwood.svg|thumb]]
2º. Medii resistentia cum tota exerceatur contra corporis superficiem, vis motrix inde resultans haud pendebit ab ipsius corporis massa, eritque eadem utcumque sub eadem et forma, et amplitudine superficiei, crescat vel decrescat massa: non sic dicendum de respondente vi acceleratrice, quae cum obtineatur dividendo vim motricem per corporis massam, permanente et forma, et amplitudine superficiei, erit reciproce ut ipsa massa. Hinc patet cur, caeteris paribus, quo major est massa corporis in medio resistente decidentis, eo etiam rapidior sit motus finalis.

3º. Si concipimus planum variis resistentis medii stratis normaliter occurrens velocitate &lt;math&gt;v&lt;/math&gt;, ponimusque et plani actionem in medii particulas intra singula tempuscula infinitesima sese protendere ad respondentia duntaxat strata dimovenda, et haec eadem strata illico sic dimoveri ut statim atque dimota sunt nullam praeterea actionem sive immediatam, sive medialam exerceant in dimovens planum; expressa per &lt;math&gt;ds&lt;/math&gt; crassitudine strati dimovendi intra tempusculum &lt;math&gt;dt&lt;/math&gt;, per &lt;math&gt;\mu&lt;/math&gt; densitate medii, et per &lt;math&gt;A&lt;/math&gt; area dimoventis plani, orietur inde (28) resistentia &lt;math&gt;A\mu v ds \frac{1}{dt}&lt;/math&gt; seu &lt;math&gt;A \mu v^2&lt;/math&gt;. Duplicatur resistentia in casu medii elastici (23).

4°* Si vis acceleratrix ex medii resistentia assumitur proportionalis quadrato velocitatis, ut denotante &lt;math&gt;\mathrm{k}&lt;/math&gt; quantitatem constantem (experimentis determinandam), exhiberi possit vis illa per  &lt;math&gt;g\frac{v^2}{\mathrm{k}^2}&lt;/math&gt; gravia descendentia sollicitabuntur vi acceleratrice &lt;math&gt;g-g\frac{v^2}{\mathrm{k}^2}&lt;/math&gt; ascendentia vi acceleratrice  &lt;math&gt;-\left(g+g\frac{v^2}{\mathrm{k}^2}\right)&lt;/math&gt;: proinde (28) quoad gravium descensum
 &lt;math&gt;\frac{dv}{dt}=g-g\frac{v^2}{\mathrm{k}^2}&lt;/math&gt; 
quod ascensum
 &lt;math&gt;\frac{dv}{dt}=-g-g\frac{v^2}{\mathrm{k}^2}&lt;/math&gt; 

5°* In primo casu 
 &lt;math&gt;gdt=\frac{{\mathrm{k}^2}dv}{{\mathrm{k}^2}-v^2}=\frac{\mathrm{k}}{2}\left(\frac{dv}{\mathrm{k}+v} + \frac{dv}{\mathrm{k}-v} \right)&lt;/math&gt; 
sumptisque integralibus:(27.6 °) in hypothesi velocitatis &lt;math&gt;v_0=0&lt;/math&gt;,
 &lt;math&gt;gt=\frac{\mathrm{k}}{2}\ln\left(\frac{\mathrm{k}+v}{\mathrm{k}-v} \right)&lt;/math&gt; 
unde 
 &lt;math&gt;e^{\frac{ngt}{\mathrm{k}}}=\frac{\mathrm{k}+v}{\mathrm{k}-v}&lt;/math&gt;
Primum membrum est necessario &lt;math&gt;&gt;0&lt;/math&gt;; ergo et secundum: crescente igitur &lt;math&gt;t&lt;/math&gt; crescet quidem &lt;math&gt;v&lt;/math&gt;; ita tamen ut nunquam fiat &lt;math&gt;v &gt; k&lt;/math&gt;: quod consentit cum dictis (10). Ad haec : quoniam (28)
&lt;math&gt;
dt=\frac{ds}{v}&lt;/math&gt; erit &lt;math&gt;gds=\frac{{\mathrm{k}^2}vdv}{{\mathrm{k}^2}-v^2}&lt;/math&gt; quam integrantes assequemur &lt;math&gt;gs = C - \mathrm{k}^2\ln(\mathrm{k}^2 -v^2)&lt;/math&gt;: in initio motus ex hypothesi &lt;math&gt;v =0 , s =0&lt;/math&gt;, ac proinde &lt;math&gt;C = \frac{\mathrm{k}^2}{2}\ln \mathrm{k}^2&lt;/math&gt;; igitur
 &lt;math&gt;gs= \frac{\mathrm{k}^2}{2}\ln\frac{{\mathrm{k}^2}}{{\mathrm{k}^2}-v^2}&lt;/math&gt;
6°* In secundo casu
 &lt;math&gt;gdt=\frac{{\mathrm{k}^2}dv}{{\mathrm{k}^2}+v^2}&lt;/math&gt;
ideoque (27.13°)
 &lt;math&gt;gt = C - \mathrm{k}\arctan(\frac{v}{\mathrm{k}})&lt;/math&gt;
tempori &lt;math&gt;t = 0&lt;/math&gt; respondet &lt;math&gt;v =v_0&lt;/math&gt;, et consequenter &lt;math&gt;C
= \mathrm{k}\arctan(\frac{v_0}{\mathrm{k}})&lt;/math&gt;; igitur (tang- ):
5l = k [ arc(tang = :) - arc (ranga ) ] .
ds Ad haec : ob de habemus


s
V71
ndum :
gds =
kavdv ; propterea gs = C—
kat va -- 105 (1º + vw).
log ( Kº +w.), ce
ka In initio motus s = 0 , v = Vo;hinc CF
2
gs =
log k2+0.2
katua 2
Facta v = 0 , prodibunt
k2 proind k
log
ktve t
2g 8 are (tang = ).

maxima videlicet altitudo ad quam in medio resistente ascendit grave, tempasque respondens.

7º. Fac ut , exhibente YM ( Fig. 17) directionem
normalem stratis TT ''medii resistentis , planum A oblique''
occurrat stratis ipsis sub angulo BMY ( =\beta ) . Recta bc
parallela rectae YM repraesentet velocitatem v , qua move
tur A : resoluta bc in Kc perpendicularem et BK parallelam
plano A , exprimet Aje . KC2 resistentiam medii ; et quo
niam KC bc . sin Kbc = vsin \beta , iccirco resistentia ista
Ajwa , sin a\beta .
J =

=== De gravium descensu atque ascensu per plana inclinata; de attritu; deque cochlea, et cuneo.===
[[Fasciculus:Free body.svg|thumb|Planum inclinatum]]
[[34]]. Super plano ad horizontem &lt;u&gt;inclinato&lt;/u&gt; collocetur corpus quod habeat centrum gravitatis in &lt;math&gt;G&lt;/math&gt; (Fig. 21) et massam &lt;math&gt;M&lt;/math&gt;; ex &lt;math&gt;G&lt;/math&gt; horizontem demittatur perpendiculum &lt;math&gt;GH&lt;/math&gt;; et ex &lt;math&gt;H&lt;/math&gt; ducatur alterum perpendiculum &lt;math&gt;HB&lt;/math&gt; in communem plani horizontalis et plani inclinati intersectionem; vis motrix ex corporis pondere jacebit in plano perpendiculornm &lt;math&gt;GH , HB&lt;/math&gt;; demisso enim ex &lt;math&gt;G&lt;/math&gt; perpendiculo &lt;math&gt;Gi&lt;/math&gt; in planum inclinatum, vis illa invenietur in plano &lt;math&gt;iGH&lt;/math&gt; normaliter insistente intersectioni plani inclinati et plani horizontalis; quod planum &lt;math&gt;iGH&lt;/math&gt; manifeste recidit in planum perpendiculorum &lt;math&gt;GH , HB&lt;/math&gt;. Sit &lt;math&gt;AB&lt;/math&gt; communis intersectio istius plani et plani inclinati; &lt;math&gt;AC&lt;/math&gt; perpendiculum ex &lt;math&gt;A&lt;/math&gt; demissum in &lt;math&gt;BH ... ; c&lt;/math&gt; angulus &lt;math&gt;ABC&lt;/math&gt;: recta &lt;math&gt;AB&lt;/math&gt; vocatur longitudo plani inclinati, &lt;math&gt;AC&lt;/math&gt; altitudo, &lt;math&gt;c&lt;/math&gt; &lt;u&gt;angulus inclinationis&lt;/u&gt;. Vim motricem per &lt;math&gt;GK&lt;/math&gt; repraesentatam resolve in duas &lt;math&gt;Gi , Gh&lt;/math&gt;, quarum altera sit perpendicularis, altera parallela rectae &lt;math&gt;AB&lt;/math&gt;; erunt

&lt;math&gt;Gi = gM \cos c , Gh = gM \sin c&lt;/math&gt;.

Cadat &lt;math&gt;Gi&lt;/math&gt; intra corporis basim; elisa &lt;math&gt;Gi&lt;/math&gt; a resistentia plani inclinati, gignetur motus a sola &lt;math&gt;Gh&lt;/math&gt;; quae cum maneat constanter eadem, non alium pariet motum nisi uniformiter varium. His positis, ad determinandum gravium motum per plana inclinata satis erit in (6,6' . 30) et in ( 6 &quot; . 31 ) substituere &lt;math&gt;g \sin c&lt;/math&gt; pro &lt;math&gt;g&lt;/math&gt;: denotantibus itaque &lt;math&gt;\theta&lt;/math&gt; tempus, &lt;math&gt;u&lt;/math&gt; velocitatem, et &lt;math&gt;z&lt;/math&gt; spatium, erunt quoad gravium descensum per plana inclinata u = g 9 sin c, z = * gga sin c, u = 2gz sin c ( 6 &quot; )
si tempori 0 = o respondent u = 0,2 = 0 ; et
u = u + go sinc,z = altiglasin c,u ? —a? = 2g zsin c (6 )
si tempori 0 =o respondent u = a, z = 0 : quoad ascen
sum vero
u =a -g6 sinc, z =a9— 1 g 2sinc, a ?—u? = 2gz sin c (65 &quot; )
&lt;u&gt;Componens&lt;/u&gt; &lt;math&gt;Gi&lt;/math&gt; exhibet pressionem, quam exercet grave contra planum inclinatum .

et :spatium , eruntxquoad gravium descen-
sum per plana inclinata
u:g93inc, :

: äggï sinc, 113:2gz sinc (ö&quot;)

si tempori 6:o respondent u:o,:

: o; et

u:u-1-g 9ainc.z:a G—i-äggasin c,u'—a*:Zgzsin c(b')
si tempori 9:o respondent u:a, s:o :quoad ascen-
∙ sum vero
u :::—gg sinc, :349— äggaslncaaa—uzzzgz Sine (b'-l)&quot; r Componeus Gi exhibet pressionem, quam exercet grave
contra planum inclinatum .73

35. Comparantes ( 6 ' ' ) cum (6 ) haec facile stabiliemus.
1. ' Si licals t . erunt i GH
plaui 1 : sin c , s : 2 = 1 : sin c ;
pla
inter
cula
Br
noguls
si duo nempe gravia eodem tempore delabuntur, alterum
verticaliter , alterum per planum inclinatum AB , tam ve
locitates v , u ab ipsis gravibus in descensu perpendiculari
et obliquo acquisitae , quam spatia s , z descripta , erunt
ut longitudo plani ad ejus altitudinem .
2. ° Hinc ubi ex puncto C concursus rectae verti
calis com horizontali ducatur perpendiculam CE ad plani
inclinati lougitudinem AB , grave percurret lapsu obliquo
spatium = AE eo tempore , quo percurreret lapsu verticali
totam altitudinem AC ; nam AC : AE - AB : AC.
3. ° Inde sequitur chordas omnes circuli ad supre
mam , vel infimam diametri verticalis extremitalem pertin
gentes describi eodem tempore ; eo nimirum , quo descri
beretur ipsa circali diameter.
4. ° Velocitates u , v gravium in plano ioclinato
et in recta verticali aequales sunt si gravia ex punctis aeque
altis ad eamdem rectam horizontalem pervenerint : in ea
sumus hypothesi est s = zsinc , ac proinde
a pla
ifors
16 :3
cempo
enim qua

: u2 V =U .

5.° Tempus descensus per longitudinem plani in
clinati ad tempus descensus per altitudinem est ut ipsa
longitudo ad altitudinem : nam in casu ( 4º ) u = v ; ideoque
SIDEN
g9 sin gt , et 0 : t 1 : sin c .

36. Sint nunc plura plana sibi contigua ( fig. 22. * )
diversimode ad horizontem inclinata . Si grave ab AB
transit ad planum BD , in eo transitu non retinebit in
initio plani BD totam velocitatem , quam habebat in
fine plani AB. Si enim concipitur recta AC perpendi
6 et ?
73
35. Comparantes (b &quot;') cum (6) haec facile stabiliemus.
1.&quot; Si 9:t . erunt
v:u:1:sinc,s:z:1:sinc;'
si duo nempe gravia eodem tempore delebuntur, alterum
verticaliter , alterum per planum inclinatum AB, tam ve-
locitates v ,
1: ab ipsis gravibus in descensu perpendiculari
et obliquo acquisitae , quam spatia s
, z descripta , erunt
ut longitudo plani ad ejus altitudinem .
2? Hinc ubi ex puncto G concursus rectae verti-
calis cum horizontali ducatur perpendiculum CE ad plani
inclinati longitudinem AB, grave percurret lapsu obliquo
spatium :AE eo tempore , quo percurreret lapsu verticali
totam altitudinem AC; uam AC :AE :- AB :AC.
3.&quot; Inde sequitur chordas omnes circuli ad supre-
mam , vel infimam diametri verticalis extremitatem pertin-
gentes describi eodem tempore; eo nimirum , quo descri-
beretur ipsa circnli diameter.
. 4.&quot; Velocitates 11 .'v gravium in plano inclinato
et in recta verticali aequales sunt si gravia ex punctis aeque
altis ad eamdem rectam horizontalem pervenerint: in ea
enim qua sumus hypothesi est s:zsinc
, ac proinde
v&quot;:
uz , v :u .
'
5.&quot; Tempus descensus per longitudinem plani in-
clinati ad tempus descensus per altitudinem est ut ipsa
longitudo ad altitudinem: nam in casu (40) u:v ; ideoque
g95inc:gt,et-9:t:1:sinc.
36. Sint nunc plura plana sibi contigua (fig. 22.')
diversimode ad horizontem inclinata. Si grave ab AB
transit ad planum BD, in eo transitu non retinebit in
initio plani BD totam velocitatem, quam habebat -in
fine plani AB. Si enim concipitur recta AC perpendi-
6
- .... ↹∙∙∙↽∙⊾ −↿−⇀⋅⋅⋅⋅↽∙⋅↽ f.:-.. ∙−←−−− ↘−∼∙⋅ ,. ∙∙⋅∙∙∙⇁ . ∙∙
'1
cularis plano BD producto
, et velocitas in fine plani ha
bens directionem AB concipitur resoluta in duas AC
,
CB

; illa prior AC

a novo plano BD elidetur
, utpote quae
tota insumitar in eo normaliter percutiendo
, ac seclusa
0
mois elasticitatis consideratione
, sola altera CB urgebit cor
pus per novum planum BD
, eritque velocitas prior ad no
vam
, qua nempe ingreditur novum planum ut AB

: CB sive ut

radius ad cosinum anguli ABC
, et prior velocitas ad amissam
erit ut radius ad sinum versum ejusdem anguli ABC

; cum

nempe
, si centro
B et radio BA describatur semicirculus
EAE
'
, sit velocitas prior ad amissam ul AB

: CE

.
Erraverunt igitur qui banc velocitatis jacturam minime
considerantes falsum hoc theorema confecerunt,, Ex aliitu
dine quacumque descendens grave per quotlibet ac quaeli
bet plana AB
, BD sibi contigua utcumque inclinata eamdem
in puncto infimo
D velocitatem acquiret ac cadendo perpen
diculariter ex eorum omnium altitudine,, Erit tamen veris
simum theorema si non ad plana contigua quaecumque scd
ad curvas, quae ex infinitis numero rectis lineis et infinite
parvis
( 27. 16
°
) coalescere intelliguntur
, applicetur et poterit verissime sic enunciari ,, Quodlibet grave ex quacum
que altitudine cadens supra superficiem curvam quamcum
que
, eamdem in puncto infimo velocitatem acquiret ac ca
dendo perpendiculariter ex ipsius curvae altitudine ,, Ratio
est quia velocitatum jactura in transitu de uno in aliud
planum
, decrescente angulo quem continet planum alterum
AB cum altero DB producto
, decrescit siquidem decrescente angulo ABC decrescet sinus versus CE repraesentans
velocitatem amissam
. Quare faclo infinite parvo angulo ABC
,
uti contingit in curvis
, velocitas quoque amissa fiet infinite
parva
, ac proinde grave ingredietur planum BD cum ve
locitate acquisita in descensu per planum AB
. Porro sinus
versus CE
' ita decrescit ut, facto infinite parvo primi or
dinis angulo ABC
, ipse CE
' evadat infinitesimus secundi or
dinis

; nam EC
: AC

= AC

: CE

'.
74
cularis plano BD producto
,
et velocitas in fine plani ha-
bens directionem AB concipitur resoluta in duas AC
,
CB; illa prior AC :: novo plano BD elidetur, utpote quae
tota insumitur in eo normaliter percutiendo, ac seclusao-
mnis elasticitatis consideratione, sola altera CB urgebit cor-
pus per novum planum BD, eritque veloeitas prior ad no-
vam, qua nempe ingreditur novum planum ut AB:CB sive ut
radius ad cosinum anguli ABC, et prior velocitas ad amissam
erit ut radius ad sinum versum ejusdem anguli ABC; cum
nempe, si centro
B et radio BA describatur semicirculus
EAE', sit velocitas prior ad amissam ut AB: CE'.
Erraverunt igitur qui hanc velocitatis jacturam minime
considerantes falsum hoc theorema coufecerunt,, Ex altitu-
dine qnacumque descendens grave per quotlibet ac quaeli-
bet plana AB, BD sibi contigua utcumque inclinata eamdem
in puncto infimo
D velocitatem acquirat ac cadendo perpen-
diculariter ex eorum omnium altitudine,, Erit tamen veris-
simum theorema si non ad plana contigua quaecumque sed
ad curvas, quae ex infinitis numero rectis lineis-et infinite
parvis (27. 16&quot;) coalescere intelliguntur, applicetnr; et po-
terit verissime sic enunciari ,, Quodlibet grave ex quacum-
que altitudine cadens supra superficiem curvam quamcum—
que, eamdem in puncto infimo velocitatem acquiret ac ca-
dendo pan-pendiculariter ex ipsius curvae altitudine ,, Ratio
est quia velocitatum jactura in transitu de uno in aliud
planum, decrescente angulo quem continet planum alterum
AB cum altero DB producto, decrescit; siquidem decrescen-
te angulo ABC decrescet sinus versus CE' repraesentans
velocitatem amissam. Quare facto infinite parvo angulo ABC,
nti contingit in curvis, velocitas quoque amissa fiet infinite
parva, ac proinde grave ingredietur planum BD cum ve-
locitate acquisita in descensu, per planum AB. Porro sinus
versus CE' ita decrescit ut, facto infinite parvo primi or-
dinis angulo ABC, ipse CE' evadat infiuitesimus secundi or-
diuis; nam EC: AC:AC: CE'.75
1

37. Hactenus nullam habuimus rationem attritus , seu
resistentiae ex asperitate superficierum : prominentes nem
pe unius superficiei denticuli foveas alterius ingrediun
tur ; sicque haud poterit una superficies alteri superposita
promoveri, nisi ipsi denticuli vel frangantur, vel inflectan
tur, vel , saperiori superficie identidem elevata parumper ,
e foveolis egrediantur: possunt quidem denticuli ita poli
tione imminui , ut sensum inermem effugiant, sed penitus
tolli nequeunt.Statue corpus super plano horizontali ; tum pla
num istud eousque sensim inclina , donec sub quodam angulo
c=c'corpus tantum non incipiat descendere, incipiat vero cre
scente utcumque parum c ultra c' . Attritus respondens angulo
c = c dicatur f: quoniam f accurate librat vim gM sinc'
erit f =g Msinc' ; hinc si per r exprimitur ratio attritus f ad
pressionem gM cosc' ut sit fer. GM cosc ', habebitur
.
r.gM cosc = gM sinc' ,
ideoque
r = tang c' .
0
5
Permanente qualitate massae M, itemque politionis gra
du , constat experimentis quod permanet quoque angulus c' ,
et consequenter ratio r, licet quantitas ipsius M augeatur,
vel minuatur. Inde sequitur attritum f, caeteris paribus, fo
re proportionalem pressioni r.gM cosc' .
Si ponimus attritum adhuc pressioni proportionalem
quum angulus c superat angulum c'; ad habendam ratio
nem attritus in motu gravium per plana inclinata , pro
gsinc substituetur g sin c - rg cosc in ( b ), et gsinc +
rg cosc in ( 6 &quot; ); caeterum in casu motus videtur f non a so
la pressione , sed a corporis quoque velocitate haud pa
rum pendere.
Haec subjungimus.
&quot;
75

37. Hactenus nullam habuimus rationem attritus, seu
resistentiae ex asperitate superficierum :prominentes nem-
pe unius superficiei denticuli foveas' alterius ingrediun-
tur ; sicque haud poterit una superficies alteri superposita-
promoveri, nisi ipsi denticuli vel frangantur, vel mflectan-
tur, vel, superiori superficie identidem elevata parumper ,
e foveolis egrediantur: possunt quidem denticuli ita poli-
tione imminui, ut sensum inermem effugiam, sed penitus
tolli nequeunt.Statue corpus super plano horizontali; tum pla-
num istud eousque sensim inclina
, donec sub quodam angulo
c:c' corpus tantum non incipiat descendere, incipiat vero cre-
scente utcumque parum c ultra c'. Attritus respondens angulo
c:c' dicatur f: quoniam f
accurate librat vim nginc'
erit f

: g Msinc'; hinc si perr exprimitur ratio attritus f

ad
pressionem gM cosc' ut sit:r. gM cosc', habebitur
r. gM cosc': gM sinc' ,
ideoque
r:tang c'
.
Permanente qualitate massae M, itemque politionis gra-
du, constat experimentis quod permanet quoque angulus c',
et consequenter ratio r, licet quantitas ipsius M augeatur,

: vel minuatur. Inde sequitur-attritum f,'caeteris paribus, fo-

1e proportionalem pressioni ngM cosc'.
Si ponimus attritum adhuc pressioni proportionalem
↴⋅ quum angulus c superat angulum ∁∙∍ ad habendam ratio-
lnem attritus in motu gravium per plana inclinata
, pro
igsinc substituetur gsinc—rgcosc in (b' ), et gsinc −∣−
' rgcosc tn ( b &quot;); caeterum in casu motus videtur fnon a so-
lla pressione, sed a corporis quoque velocitate haud pa-
rum pendere.
Haec subjungimus.76
1º . Si corpus in plano inclinato constitutum li
brandum sit potential applicita ( Fig. 21 ) puncto G, quae
potentia et sollicitat ad ascensum, et efficit angulum &amp; cum
AB, gignitque propterea pressionem Qsind, satis erit ut re
sultans ex viribus Q et M ( g sinc F rg cosc ) Fr (sin
exsistat ipsi plano perpendicularis , sese videlicet diri
gat juxta Gi: continet autem Q cum Gi angulum 900
An et vis Mg ( sinc F r cosc ) FrQsinc angulum
cum eadem Gi. Igitur ( 9.10 )
= 90
Q: Mg( sincar cosc ) FrQsing = sin 90 ° ; sin ( 90 °
a ) = 1 : cosa ;
ideoque
sinc Frcosc OSCMS
Q
cos a Es since
secun Sumpio superiori signo, nequit Q esse minor
do membro quin corpus descendat; sumplo inferiori si
gno, nequit Q esse major secundo membro quin corpus
ascendat; perstabit aequilibrium intra limites
sinc - rcosc sinc torcose Mg, el &lt;
cosa + rsing
Mg.
cosu - osinc
2º. In hypothesi nullius attritus erit r = 0 ; et
consequenter
sin c
Q
Mg COSU.
3º. Si Q est insuper parallela horizontali BC, e
rit a = c ; ideoque
76
1&quot;. Si corpus in plano inclinato constitutum li-
brandum sit potentia Q applicita ( Fig'. 21) puncto G, quae
potentia et sollicitat ad ascensum, et eilicit anguluma cum
AB, gignitque propterea pressionem Qsinac, satis erit ut re-
sultans ex viribus Q et M (gsinc :rgcosc ):F r Qsin a
exsistat ipsi plano perpendicularis , sese videlicet diri-
gat juxta Gi: continet autem Q cum Gi angulum :90&quot;—
a, et vis Mg ( sine: rcosc) :rQsina angulum :90&quot;
cum eadem Gi. Igitur ( 9. 1&quot; )
Q: Mgüincqzr 0050 ):t:rQsinat:sin 90&quot; :sin ( 90&quot;—
at:) 1:cosa:;
ideoque
sinc ∓r cosc
−∙∙ Mo
cos a:
r siua
Sumpto superiori signo, nequit Q esse minor secun-
do membro quin corpus descendat; sumpto inferiori si-
gno, nequit Q esse maior secundo membro quia corpus
ascendat; perstabit aequilibrium intra limites
sinc—rcosc sinc rrnsr Q)...— Mg, et Q(
−⊢ Mg.
cosa rsiuat cosa—rsiua
2&quot;. In hypothesi nullius attritus erit r: o ; et
consequenter
sin 0 M
g. szz cosa:
3&quot;. Si Q est insuper parallela horizontali BC, e-
rit at:c ; ideoque77
sipc :
Mg COSC
potentia videlicet ad pondus ut plani altitudo AC ad hori
zontaleon BC. Hinc facile deducuntur aequilibrii leges in
cochlea et cuneo.
4º. Cum cochlea non sit nisi planum inclina
tum ABC, quod circum cylindruni ducitur; dum vero co
chlea agit , potentia sit rectae lineae BC parallela, erit
potentia ad pondus seu resistentiam superandam , ut al
titudo plani seu helicam distantia ( =h ) ad basim plani
seu cylindri peripheriam ( = k ). Hinc
Q
hP ;
k
quae formula supponit distantiam inter cylindri axem et pun .
ctum cui applicatur potentia , esse ipsius cylindri radium
( = m ) : quod si distantia illa fiat alia ab r', et exhibea
tur per R' ; denotante e potentiam respondentem novae
distantiac, exsistet
mi?
R'
ac proinde Q
-
hP
R '
LP
25R .
k
In ordine ad cochleam infinitam , dicatur A radius ma
joris rotae , a radius minoris , et P' pondus seu poten
tia apud dentes ipsius rotae majoris; erunt
ар P =
Q
api A
hP
27.R '
ideoque
Q =
h a P
21AR'
77
Q sine. NT: ⋅⇀ SE.—.'
potentia videlicet ad pondus ut plani altitudo AC ad hori-
zoutalem BC. Hinc facile deducuntur aequilibrii leges in
cochlea et cuneo.
4&quot;. Cum cochlea non sit nisi planum inclina-
tum ABC, quod circum cylindrum ducitur; dum vero eo-
chlea agit, potentia sit rectae lineae BC parallela, erit
potentia ad pondus seu resistentiam superandam, ut al-
titudo plani seu helicum distantia ( :h)ad basim plani
seu cylindri peripheriam :( k). Hinc
Qz—k-i
quae formula supponit distantiam inter cylindri axem et pun-
ctum cui applicatur potentia, esse ipsius cylindri radium
(: r' ): quod si distantia illa fiat alia ab r', et exhibea-
tur per B'; denotante Q' potentiam respondentem novae
distantiae, exsistet
Q'—r' ∙∙ ∙∙∙∣≖∣⊃⋅↿⋅⋅∙−∣≀∌ ∙≺⋮−−−−∙↓⊤∙ ac ptomde QI—B— . ∣∙⋮−−−−∙ ⊋∙⋮⋮⋅⋮↸↽∙
In ordine ad cochleam infinitam, dicatur A radius ma-
ioris rotae
, a radius minoris , et P' pondus seu poten-
tia apud dentes ipsius rotae maioris; erunt
aP , hP'
P::ï'Q—an' '
ide ue
Oq haP78
1
5 ° Cuneus spectari potest tamquam coalescens ex
duobus planis inclinatis ut ABC, tum ob figuram suam ,
tum quia idem est sive pondus per planum inclinatum
trahatur sursum , sive planum sub pondere promoveatur.
Agit autem potentia in cuneo juxta CB; quoad igitur u
1
nam cunei partem ABC respondens potentia Qerit ad
m
1
respondentem resistentiam P ut AC ( = D ) , sen di
midia cunei crassities ad BC ( = H ) , idest ad altitudinem
1
Q
1
ad respondentem resistentiam P P erit pariter ut
į D ad H. Igitur
m
LQ.H - 1P.HD, Q (m - 1 ) .
A
m2 m m
P (m - 1 )
mi '
· D ;
quibus aequationibus in summam collectis ,
Q. H = P. , D ,
et consequenter
D
H totalis videlicet potentia Q ad totalem resistentiam P ut
dimidia cunei crassities D ad ejus altitudinem H ; mo
do tamen exerceatur resistentia normaliter ad H.
6º . Si in cochlea v . gr. considerandus esset at
tritus , haberetur ( 10.40.) ,
1
!
1
5&quot; Cuneus spectari potest tamquam coalescens ex
duobus planis inclinatis ut ABC, tum ob figuram suam,
tum quia idem est sive pondus per planum inclinatum
trahatur sursum, sive planum sub pondere promoveatur.
Agit autem potentia in cuneo iuxta CB; quoad igitur u-
. . 1 nam cune1 partem ABC . respondens potentta —Qer1t ad
.
'
m
respondentem resistentiam −↿−∙∶ P ut AC (: äD ), seu di-
↾
m
midia cunei crassities ad BC (: H ), idest ad altitudinem
. . 1 cunei. Quoad alteram partem respondens poteutta Q—- −− Q
m
. . 1 ad . . respondentem rc51stent1am P
-— −−∙ P er1t partter ut
171
&amp; D ad H. Igitur
D, Q—(m-1).H: −↿−↽≺≀∙∥∶∶∙−↿∙−↕⊃∙ ;.
m m m
P
∙∙
-
(m'1) -äD;
,- ut
quibus aequationibus in summam collectis,
QaHzpaL'D, et consequenter
≟≺−≀∙∙− :D
.
P −⋅⋅ H ⋅
totalis videlicet potentia Q ad totalem resistentiam P ut
dimidia cunei crassities &amp; D ad ejus altitudinem H; mo-
do tamen exerceatur resistentia normaliter ad H.
6&quot;. Si in cochlea v. gr. considerandus esset at-
tritus , haberetur (10. 4&quot;.),
≁−−−−∎⋅−−
−−⋅⋅...-—79
sinc FrcoscP =
cosc trsinc
h = 2 te r'r P ;
h Erk
P
k trh 2 trh
ideoque
Q
Qr Pr' h = 27r's
R ? -R 2 r'trh
0
70. Veniat quoque considerandus attritus in ae- ,
quilibrio corporis AB ( Fig. 23: 24 ) , quod ad rolatilem
motum circa fixum cylindrum sollicitatur vi Rjacente in
plano, cui normaliter insistit axis cylindri.
Ac primo cylindrus impleat accurate circularem cor
poris aperturam DE ( Fig. 23 ) , in quam inseritur: per cy
lindri centrum O duc rectam OEE' parallelam vi R , et
pancto E corporis AB applica duas . vires Q ', Q' aequa
les eidem R, et contrarias, alteram nempe tendentem, ab
E versus E' , alteram ab E versus O; vi R licebit substi
tuere systema virium R , Q ', Q &quot; : et cum possint absque sy
stematis turbatione sic transferri ( 11 ) R et l ' ut aequi
distent ab O, eae nitentur dumtaxat gignere motum ro
tatilem circa cylindrum quin ullam pariant pressionem a
pud ipsius cylindri superficiem ; pressio igitur in hanc su
perficiem redigetur ad solam
୧ = R , ideoque f = Rr.
Attritus fest vis tangentialis respectu superficiei cylin
dricae; hinc denotante a radium OE cylindri , et p per
pendiculum Oi ex O ductum in directionem potentiae R,
ad aequilibrium satis erit, ut exsistat ( 9. 2° )
R 1
2
р
Rr
. 79
Q-—sinc:r:rc.oscP 11:er P—h:t:2nr'rp
cosczbrsmc R::brh 2nr':t:rh ,
ideoque
—Qr' Pr' II::ZRr'r
a' &quot;B' 'an'äzrh Q! '
70. Veniat quoque considerandus attritus in ae- ↗
qnilibrio corporis AB ( Fig. 23: 24 ), quod ad rotatilem
motnm circa fixum cylindrum sollicitatur vi R iacente in
plano, cui normaliter insistit axis cylindri.
Ac primo cylindrus impleat accurate circularem ocr-
poris aperturam DE (Fig. 23), in quam inseritur: per cy- ⋅
lindri centrum O duc rectam OEE' parallelam vi B, et
pnncto E corporis AB applica duas, vires Q', Q&quot; aequa-
les eidem R, et contrarias, alteram nempe tendentem, ab
E versns E', alteram ab E versus O; vi R licebit substi-
tuere system virium R, Q', Q&quot;: et cum possint absque sy—
stematis turbatione sic transferri (11) B et Q'0ut aequi-
distent ab 0, eae uitentur dumtaxat gignere motum ro-
tatilem circa cylindrum quin ullam pariant pressionem a-
pud ipsius cylindri superficiem; pressio igitur in hanc su-
perficiem redigetur ad solam
Q&quot; −∙∙−− R, ideoque f: R r.
Attritus fest vis tangentialis respectu superficiei cylin-
dricae; hinc denotante a radium OE cylindri, et p per-
pendiculum Oi ex Oductum in directionem potentiae Pt,
ad aequilibrium satis erit, ut exsistat ( 9. 20)80
et consequenter
P
facto p &gt; ar , disrumpetur aequilibrium ; facto p &lt; ar ,
subsistet .
Ponatur secundo circularis apertura corporis baud
impleri accurate cylindro ( Fig.24) : vis R manifeste trans
ibit per contactum E cylindri et corporis AB . Resolve R
in duas EF, et ED' , quarum altera transeat per centrum 0 ,
altera tangat cylindrum : per EF exprimetur pressio ; ac
proinde
f = r.EF .
Obtinebit igitur aequilibrium quotiescumque
ED ' &lt; r. EF ,
vel ED' = r.EF :
cum autem ( 9. 1. ° ) .
ED' : R = sin FER ; sin D'EF = sin FER : 1 ,
EF : R = sin D'ER ; sin D'EF = cos FER : 1 ,
cumque ducto perpendiculo Oi ex O in ER ,
Oi Ei voa ?
OE sin FER Р cos FER 22 -
p2
a OE
iccirco praefatac aequilibrii conditiones vertentur in
Rp Rr Vap2 Rp a
Rr Va - p ?
a a
quae traducuntur ad
80
et consequenter
&quot;' p :: ar :
facto p ar, disrumpetur aequilibrium; facto p ar
,
subsistet .
l
Ponatur secundo circularis apertura corporis baud
impleri accurate cylindro (Fig.24): vis B manifeste trans-
ibit per contactum E cylindri et corporis AB
. Resolve B
in duas EF, et ED' , quarum altera transeat per centrum O,
altera tangat cylindrum: per EF exprimetur pressio; ac
proinde
f

: r. EF.

Obtinebit igitur aequilibrium quotiescumque
ED' (r. EF , vel ED' −−∶ r. EF :.
cum autem (9. 1.0).
.'
ED': R ::sin FER : sin D'EF :sin FER : 1 ,
EF fii ∙−−∶ sin D'ER; sin D'EF: cos FER : 1,
cumque ducto perpendiculo Oi ex 0 in EB
.
Oi p Et. ⇂∣ (13 ∙−− :;
' :∙−−− :... ∙ FER ↽− −∙ p
sin FER 08 a 005 OF. a ,
iccirco praefatae aequilibrii conditiones vertentur in
n,,(RrI/aa—pz ↧≹∣↗∙∙∙↧≹≀⋅ Wiz—pa
7.&quot; −−−−↴∶∎−−∙−↙≀∎ ⇀⇀ a '
quae traducuntur ad
⇁−∙↱⇁≓≓81 1
ar 2
p &lt;
р
1 + 12 vit ?
8.• Si ponitur R nihil esse aliud nisi resultans
ex datis viribus P' , Pi ad puncta data v . gr. A , B appli
citis , innotescet R ex dictis ( 10 ) , itemque p. ex ( 10.3° ) .
Sic habetur ratio attritus in vecte : caeterum in machinis
praeter resistentiam ex attritu spectanda etiam est resi
stentia ex funibus . Hi enim inflexioni suae resistunt quum
cylindris vel trochleis circumvolvuntur; et quidem eo ma
gis , quo majori pondere tenduntur , quo insuper crassio
res sunt , et quo minor fuerit trochleae, aut cylindri radius.

=== De motu gravium oblique projectorum.===
[[Fasciculus:Ferde hajitas2.svg|thumb]]
[[38]]. Grave &lt;math&gt;M&lt;/math&gt; (Fig. 25) juxta directionem MG velocitate &lt;math&gt;v_0&lt;/math&gt; projectum urgebitur duplici motu, altero aequabili per &lt;math&gt;MG&lt;/math&gt; ex impetu recepto, altero (nihil est aliud nisi motus relativus mobilis &lt;math&gt;M&lt;/math&gt; quoad ipsum &lt;math&gt;M&lt;/math&gt; iens per &lt;math&gt;MG&lt;/math&gt; sola &lt;math&gt;v_0&lt;/math&gt;) uniformiter accelerato gravitatis proprio per rectam verticalem &lt;math&gt;MR&lt;/math&gt;, vel ipsi &lt;math&gt;MR&lt;/math&gt; parallelam. Sit &lt;math&gt;S&lt;/math&gt; spatium quod cumque &lt;math&gt;MC&lt;/math&gt; primo illo aequabili motu seorsim sumpto percursum, &lt;math&gt;t&lt;/math&gt; tempus impensum ad ejusmodi spatium percurrendum, sitque &lt;math&gt;s&lt;/math&gt; spatium &lt;math&gt;MF&lt;/math&gt; pari tempore percursum secundo motu item seorsim sumpto. Completo parallelogrammo &lt;math&gt;MFQC&lt;/math&gt;, in fine temporis &lt;math&gt;t&lt;/math&gt; grave erit (5) in &lt;math&gt;Q&lt;/math&gt;;
et quia (1:30)
&lt;math&gt;S = v_0 t , s = \frac{gt^2}{2},&lt;/math&gt;
eliminato &lt;math&gt;t&lt;/math&gt;, existet
&lt;math&gt;S^2 = \frac{2 v_0^2}{g}s &lt;/math&gt;
aequatio ad curvam (dicitur parabola) in qua defertur grave. Si altitudo debita (28) velocitati &lt;math&gt;v_0&lt;/math&gt;, dicatur &lt;math&gt;\mathrm{A}&lt;/math&gt;, erit
&lt;math&gt;v_0^2 = 2g\mathrm{A}&lt;/math&gt;, et aequatio transformabitur in
&lt;math&gt;S^2 = 4 As ( c)&lt;/math&gt;.

[[39|39]]. Denotet x horizontalem rectam MK , y vertica
lem KQ , et h angulum CMK ; erunt x = S cosh , y =
CK - CQ = S sin h -5 ; unde
X
X
S =
cosh . sinh :
cosh
quibus valoribus substitutis in (c) , prodibit
x2 rcsinh
4 A
CO -Y) ,
et consequenter cos2 h
cos h
y =xtang h 1 + tang k
4 A
x2 ( c' ) .

[[40|40]]. Haec facile nunc stabiliuntur. 1º facta y = 0 ,
proveniet amplitudo jactus
4 Atangah
1 + tang h
4Asinhcosh = 2 Asin2h.
2.º Inde sequitur maximam jaclus amplitudinem
haberi sub angulo h = 45°.

3. ° Si quaeritur angulus h , sub quo proiicien
dum est grave ut offendat in datum scopum , cujus nempe
dantur coordinatae x et y , erit
2A + V 4A2-4 Ay - x2
tangh
82
aequatio ad curvam (dicitur parabola) in qua defertur
grave. Si altitudo debita (28) velocitati v, dicatur A, erit
vio:2gA, et aequatio trausformabitur in
S':4As (c) Esistente igitur 4 A 2—4 Ay-x &gt;0 , poterit sub duplici
angulo projici grave ut datum -scopum attingat : attinget
autem in fine temporis ( 38 : 39 )
S
ts Vo .
Vo cos h

4.0 in ( c ) pone
2 Atangh ta ;
1 +tangah
babebis
A tangah
1+ tang2h ya
1 + tang 2h W?
4A
( c ' ' ) .
Iam vero maxima y ( dicitur altitudo jactus ) manifeste re
spondet valori w = 0 ; altitudo igitur jactus exhibebitur
per
A tangah seu A sinh.
1 +tangah

5º . Ex eadem ( c &quot; ) quisque colligit parabolam , in qua
defertur grave, dividi a maxima y in duas aequales simi
lesque partes : extremitas maximae y vocatur vertex pa
rabolae; ipsa vero maxima y indefinite producta juxla gra
vitatis directionem appellatur axis parabolae.
[[Fasciculus:Ferde hajitas7.svg|thumb]]
6º Si angulus h fit &lt; o, ut initialis directio cadat iтfra horizontalem rectam ML, jactus amplitudo x (1°) ex &gt; fiet &lt; 0; jactus vero altitudo y ( 40 ) permanebit &gt;o. Quod si fuerit h = o, ut initialis directio recidat in rectam horizontalem ML, nulla erit amplitudo jacеus, nullaque ejus altitudo.

7º. Demittatur perpendiculum QP ex puncto Q
parabolae in axem NI ... , sintque NP = x', Q P =y'; erunt ( 1º . 4º . ) x MI — QP = 2 A sinh con -y'
y=NI — NP = A sin’h— x' : quibus valoribus substitutis
in ( c' : 39 ) , proveniet
y2= 4 A x' cosah
aequatio ad parabolam M N L inter x' ety' computatas a
vertice ; quantitas 4 A cos’h dicitur parameter parabolae ;
quod si in axe sumatur punctum H ita , ut ejus distantia
a vertice sit quarta parametri pars seu A cos ?h , habebitur
punctum illud , quod appellatur parabolae focus.

[[41]]. Cum ad curvam parabolicam describendam, corporis motus, qui fit secundum lineam projectionis, debeat esse aequabilis, qui vero fit secundum lineam verticalem, debeat esse uniformiter acceleratus, cumque hujusmodi certe neuter esse possit si medium utrique motui resistat, iccirco nonnisi in vacuo motus corporis oblique projecti fieri potest per curvam, quae sit perfecte parabolica. In medio resistente curva minus late patet, minusque assurgit quam in vacuo; duobus insuper cruribus dissimilibus &lt;math&gt;AN, NL&lt;/math&gt; (Fig. 26) componitur, quorum descendens &lt;math&gt;NL&lt;/math&gt; ad rectam quamdam &lt;math&gt;FE&lt;/math&gt; ut asymptotum accedit in infinitum, quin unquam congruant. Etenim resoluta projectionis velocitate in duas, alteram verticalem, alteram horizontalem, verticalis tum ab aeris resistentia, tum a gravitate usque ad punctum &lt;math&gt;N&lt;/math&gt; minuetur: propterea punctum &lt;math&gt;N&lt;/math&gt; minus assurget quam in vacuo: postquam grave ad &lt;math&gt;N&lt;/math&gt; pervenerit, descendet ob gravitatis vim damna ex medii resistentia reparantem, et hujusmodi descensus fiet motu verticali ad motum aequabilem (33) semper accedente. At horizontalis velocitas minuitur perpetuo, nulla interim vi iacturam reparante, atque inde fit ut recessus horizontalis a recta verticali &lt;math&gt;NP&lt;/math&gt; certum limitem non praetergrediatur, quem curva habet pro asymptoto. Haec contingunt potissimum corporibus ingenti velocitate in aere projectis.
=== De generalibus quibusdam proprietatibus motus curvilinei, orti a viribus, quarum una determinat materiale punctum ad motum aequabilem, altera ipsi materiali puncto est continue applicata.=== 
[[42|42]]. Concipiamus secundam vim agere solum in initiis quorundam tempusculorum, ac tantam velocitatem unico impulsu valido producere, quantam vis perpetuo agens producit toto illo tempusculo, ut deinde inminuta magnitudine tempusculorum in infinitum, habeatur linea curva orta ex continua vis actione.  

Projecto puncto materiali cum velocitate CB (Fig. 27) simulque illi impressa velocitate CA, abiret punctum per diagonalem CO parallelogrammi AOBC et esset in fine primi tempusculi in O cum determinatione describendi altero aequali tempusculo rectam OL = OC, eique in directum jacenlem. Si hic iterum illi imprimeretur alia velocitas OF, completo parallelogrammo FILO , incederet per diagonalem OI, essetque in fine secundi tempusculi in I cum determinatione describendi tertio tempusculo aequali rectam IM = 10, eique in directum jacentem. Sed ob impressam hic quoque aliam velocitatem IV abiret per novam parallelogrammi diagonalem IH, atque ita porro. Fieret ergo in ejusmodi hypothesi vis agentis per intervalla tempusculorum ut materiale punctum describeret polygonum COIHN etc, cujus latera certam magnitudinem et positionem haberent, definita nempe a directione virium et a ratione velocitatum, quas initio cujusvis tempusculi mobile obtineret. Hinc pro diversis virium ila agentium ordinibus numero infinitis infinita considerari possunt ejusmodi polygona, quorum alia in se ipsa redirent, desinente ultimo latere in puncto C ubi primum inceperat; alia abirent in infinitum. Concipiamus jam numerum tempusculorum augeri, et simul eorum magnitudinem imminui in infinitum, vitum magnitudine tum directione vel constantes manere, vel variare certa quadam lege ad continuam quamdam variationis rationem accedente in infinitum. Augebitur in infinitum numerus laterum polygoni determinato tempore descripti, imminutis interea in infinitum angulis, quos efficit quodlibet latus praecedens cum consequente: cum enim LI debeatur impulsui, qui initio tempusculi 0 eam
velocitatem producere concipitur, quam produceret vis to
to tempusculo agens, cumque per tempusculum infinitesimum vis ista habenda sit pro constante, existet ( 28: 30. 14. ) LI =
092; ideoque ob o finitam, et quadratum 62 infinitesimum secundi ordinis, erit etiam LI infinitesima ordinis secundi, sed OL est infinitesima ordinis primi, utpote quae tempusculo O describitur cum velocitate finita; ergo angulus LOI erit ivfinitesimus: atque eodem pacto demonstrantur infinitesimi anguli MIH , K'HN , etc. Hinc polygonum ad curvam continuam semper magis accedet; et ubi demum continua habealur actio vis, et continuae cuidam legi subjiciantur directio ipsius et magnitudo, obtinebitur curva continua cavam sui partem versus eam plagam obvertens, in quam tendunt vires.

43. Abeunte polygono in curvam , rectae CL , OM' , IH ', HK , etc abeunt in tangentes apud puncta C, O, I, H , etc. Ubi ergo in aliquo curvae puncto vis desinat agere,,
excurret mobile per tangentem apud illud punctum.

44. Sit IM (fig, 28 ) spatiolum quod tempusculo 9
mobile percurreret sola velocitate praeconcepta, et IV
spatiolum respondens vi agenti unico impnlsui valido ; ita
ut existat (42) IV ::99&quot;. Completo parallelogrammo, positis-
que lM:P ,
lH:B, et angulo MIV :i, erit (9. 3.&quot; )
∶∶ Vra-Hæ os −⊢⋅∠⇂⊃⊊↶⊖⋍ cos :.87
Evolvatur quantitas radicalis in seriem : proveniet
R = P + q9 cos i , unde R - P = º02cosi ,
neglectis infinitesimis altioris ordinis. Sit v' velocitas , qua
mobile percurrit laterculum R; erit R = v'0 : sit etiam v
velocitas , qua mobile percurreret laterculum P. si non
adesset impulsus validas in I ; erit P =v @ : hinc R --
P = vv( ) 0 .; et consequenter
v ' - v = q Ocosi.
Ex hac aequatione patet v— esse quantitatem in
finitesimam primi ordinis , positivam vel negativam prout
i &lt;vel &gt; 90° , esse autem =0 si i 90° . Inferimus il
lud : ubi tempore finito angulus , quem efformat vis ac
celeratrix cum directione tangentis , fuerit semper aculus,
acquiret mobile incrementum velocitatis finitum ; si sem
per obtusus , patietur decrementum finitum ; si semper re
lus , velocitas manebit constans.

45. In motu quovis curvilineo quadratum velocitatis aequat vim acceleratricem ductam in dimidium chordae, quae ex ejus directione abscinditur a circulo osculatore. Denotet enim a lineolam infinitesimam IM (Fig. 29) ut sito et consequenter IV = 902
cipiatur circulus , qui transiens per tria puncta 0 , I ,
H ( fig . 27. 29. ) habeat centrum in G , quique erit
circulus osculator apud curvae punctum O ; producantur
IV , MH donec occurrant peripheriae in G &quot; , G '' ; et ex''
G ducatur perpendiculum GGʻad chordam IG &quot; : erunt
IG &quot;
MG &quot; = IG &quot; = ICE Est autem MH . MG ' &quot; : MI. MO;
2
ergo MH . 21Gʻ = MI.MO = MI . 2MI , seu 21G'
2x2. Hinc v2 = . IGʻ ; ideoquc etc. Porro angulus IGG' =
2 Oxa

; con

.
px ?
22
87
Evolvatur quantitas radicalis in seriem : proveniet
B:P −⊢ o9zcos i
, unde B—P:cp92cosi ,
neglectis infiuitesimis altioris ordinis. Sit 'v' velocitas , qua mobile percurrit laterculum R; erit R: 0'9: sit etiam »
velocitas , qua mobile percurreret laterculum P. si non
adesset impulsus validus in I, erit P:-v 9: 'hinc R
-— P:(v'--v)9; et consequenter
v'—v:cp9cosi .
Ex hac aeqnatione patet 'o'—v esse quantitatem in-
fiuitesimam primi ordinis
, positivam vel negativam prout
i(vel 90&quot; , esse autem :0 si 1&quot;: 90&quot;. Inferimus il-
lnd : ubi tempore finito angulus, quem efformat vis ac-
celeratrir cum directione tangentis ,
fuerit semper acutus,
acquiret mobile incrementum velocitatis finitum; si sem-
per obtusus, patietur decrementum finitum; si semper re-
ctus, velocitas manebit constans.

45. In motu quovis curvilineo quadratum velocitatis
aequat vim acceleratricem ductam in dimidium chordae,
quae ex eius directione abscinditur a circulo. osculatore.
Denotet enim a lineolam infiuitesimam IM (fig. 29. )
gox- ; con-
92
cipiatur circulus, qui transiens per tria puncta 0, I,
II (fig. 27. 29..) habeat centrum in G, quique erit
circulus osculator apud curvae punctum 0; producantur
IV, MH donec occurrant peripheriae in G&quot;, G'&quot;; et ex
G ducatur perpendiculum GG' ad chordam IG&quot;: erunt
MG&quot;':IG&quot;, −−−∙−↧∁⇀−− ⇀∸−↧−⊊≩−⋅∎−∙ Est autemMH. MG'&quot;:MI. MO;
ut sit :9 i,
et consequenter IV:
99&quot;:
'»
ergoMH.21G':MI.-:MO MI. 2Ml.seu—-— &quot;&quot;&quot;&quot; ,210':
'v&quot;
.Hiuc v&quot;: 39. lG' ; ideoque etc. Porro angulus IGG'— −∙∙
−∙↼⇀−−
. −↼∙⋅⋅∙∙⋅↼−∎∣ −↼
∙∙∙88
90 ° -GIGʻ = 900 (MIV - MIG ) = 90 ' - ( i - 90 °) = 180 °-i ;
proinde , denotante r radium GI , erit IG ' = rsin IGG' =
rsini , et consequenter
va = grsini ( b ) .

:

46. Haec vera sunt de omni virium genere. Sint nunc
vires acceleratrices directae ' ad centrum datum : in casu,
curva ColH .... ( fig . 27. ) jacebit in plano transeunte
per rectam projectionis et per centrum virium ; quod fa
cile intelligitur , cum in eodem plano agant vires omnes.
Viribus ad punctum aliquod S tendentibus , radius
vector ( est recta , quae ab S ducitur ad mobile ) descri .
bet areas circa idem punctum temporibus proportionales ,
et viceversa.
Quod spectat ad primam assertionis partem , assum
ptis tempusculis aequalibus , et ducta recta SL conside .
rentur triangula SCO , SOL , SOI : est SCO = SOL ,
cum sivt super bases CO , OL aequales ob aequali
tatem tempusculorum , eamdemque habeant altitudinem
est etiam SOL = SOI , quia insistunt ambo eidem basi
SO, et sunt inter easdem parallelas SO , LI : ergo SCO
SOI. Eodem modo ostenditur triangula SOI , SIH aequa
lia esse eidem SIM , et proinde aequalia esse inter se ,
et similiter omnes areas sequentium triangulorum aequa
les esse inter se et cum areis praecedentibus. Quare cum
temporibus finitis quibuscumque contineantur numeri tem
pusculorum aequalium ipsis temporibus proportionales, areae
terminatae polygoni perimetro et rectis ad centrum virium
pertingentibus , hoc est compositae a lot areolis triangu
lorum aequalium quot tempuscula respondent illis tem
poribus , quibus perimetri partes describuntur , erunt
ipsis temporibus proportionales . Cum autem id locum ha
beat quomodocumque augeatur numerus tempusculorum,
eorumque magnitudo imminuatur , consequens est ut, ubi
⇤
88
⊖∘∘∙∁≖↧∁↾⋅ :soc—(MIV—MIG) :90&quot;—(i—gO&quot;):180&quot;—i ;
proinde , denotante r radium GI, erit IG':rsin IGG':
rsini , et consequenter
-v&quot;:g9rsini (6).

46. Haec vera sunt de omni virium genere. Sint nunc
vires acceleratrices directae'ad centrum datum: in casu,
curva COIH
.. ..
(Gg. 27. ) jacebit in plano transeunte
per rectam projectionis et per centrum virium; quod fa-
cile intelligitur , cum in eodem plano agant vires omnes.
Viribus ad punctum aliquod S tendentibus, radius
vector (est recta
, quae ab 5 ducitur ad mobile ) descri-
bet areas circa idem punctum temporibus proportionales,
et viceversa.
Quod spectat ad primam assertionis partem, assum-
ptis tempusculis aequalibus, et ducta recta SL conside-
rentur triangula SCO, SOL , SOI: est SCO:SOL,
cum sint super bases CO, OL aequales ob aequali-
tatem tempusculorum, eamdemque habeant altitudinem:
est etiam SOL :SOI . 'quia insistunt ambo eidem basi
50, et sunt inter easdem parallelas SO, LI : ergo 500:-
SOI. Eodem modo ostenditur triangula SOI , SIH aequa-
lia esse eidem SIM ,
et proinde aequalia esse inter se ,
et similiter omnes areas sequentium triangulorum aequa-
les esse inter se et cum areis praecedentibus. Quare cum
temporibus Gnitis quibuscumque contineantur numeri tem-
pusculorum aequalium ipsis temporibus proportionales, areae
terminatae polygoni perimetro et rectis ad centrum virium
pertingentibus , hoc est compositae a tot areolis triangu-
lorum aequalium quot tempuscula respondent illis tem-
poribus
, quibus perimetri partes describuntur ,
erunt
ipsis temporibus proportionales. Cum autem id locum ha-
beat quomodocumque augeatur numerus tempusculorum,
eorumque magnitudo imminuatur , consequens est ut, ubi89
demum polygonum abit iu curvam continuam , areae ter
minatae arcu curvilineo et rectis ad centrum virium ten
dentibus sint itidem temporibus proportionales.
Ad secundam assertionis partem quod spectat , sint
areae SCO, SOI, aequalibus temporibus confectae , omnino
aequales. Quoniam producta CO in L ita , ut existat OL =
CO, est triangulum SOL = SCO, idcirco SOL =SOI; sed
baec duo triangula habent basim communem SO ; erunt
igitur inter easdem parallelas, ideoque IL erit parallela re
ctae So. Ducatur IF parallela ad OL; motus per Ol com
ponetur ex duobus per OL et OF , quorum prior cum
oriatur a determinatione motum praecedentem continuandi
per C O , certe posterior a vi acceleratrice generabitur ,
quae propterea dirigitur ad centrum S.

47. Velocitas qua pollet mobile in eadem curva , est
reciproce proportionalis perpendiculo e centro virium du
cto in tangentem . Velocitas enim mobilis in quovis latere
polygoni est ut ipsum latus ob aequalia tempuscula , quibus
unumquodque latus percurri supponimus : est autem unum :
quodque ejusmodi latus reciproce ut perpendiculum quod
ex centro virium ducitur in latus ipsum ; siquidem id
perpendiculum habent pro altitudine triangula illa exigua
polygoni , si hujus latera pro eorumdem trianguloruin basi
bus assumantur ; ea insuper triangula sunt aequalia , et in
triangulis aequalibus debent bases esse in ratione recipro
ca altitudinum : est igitur ea velocitas reciproce ut per
pendiculum ductum ex centro virium in latera polygoni.
Sed abeunte polygono in curvam continuam , directiones la
teruın abeunt in tangentes ; ergo velocitas mobilis in quo
vis curvae puncto erit reciproce ut perpendiculum ex cen
tro virium in langentem demissum.

48. Denotet a areolam NSZ , et g perpendiculum SE
ductum ex centro S in laterculum NZ ; describetur NZ ve
NZ 2a ; siquidem NZ.SE=2NSZ: hinc ( 45 ) o
locitate v=
90
7
89
demum polygonum abit iu curvam continuam
, areae ter-
minatae arcu curvilineo et rectis ad centrum virium ten-
dentibus sint itidem temporibus proportionales.
Ad secundam assertionis partem quod spectat, sint
areae SCO, SOI, aequalibus temporibus confectae, omnino
aequales. Quoniam producta CO in L ita, ut existat OL:
CO, est triangulum SOL:SCO, idcirco SOL:SOI; sed .
haec duo triangula habent basim communem SO.; erunt
igitur inter easdem parallelas, ideoque IL erit parallela re-
ctae SO. Ducatur lF parallela ad OL; motus per OI com-
ponetur ex duobus per OL et OF, quorum prior cum
oriatur a determinatione motum praecedentem coutinuaudi
per C 0, certe posterior a vi acceleratrice generabitur ,
quae propterea dirigitur ad centrum S.

49. Quoniam radius vector , juxta quem agit vis con
tinua , potest assumi ut sibi parallelus per tempusculum
quodvis infinitesimum 0 , ipsaque vis ut constans per to
tum illud tempusculum ; ideo si mobile K incedens cur
vam CX ( fig. 30 ) viribus ad centrum S tendentibus de
scribit arcum infinitesimum HN labente , ductis SH ,
SN , et producto SN donec occurrat in H' tangenti HH &quot; ,
lineola recta H'N repraesentabit motum relativum mobi
lis K quoad ipsum Kieps per HH' sola vi praeconcepta in
H. Igitur cum motus iste relativus sit unice repelendus
( 5 ) a vi continuata per tempusculum e , exsistet
H'N son (6&quot;).

50. Haec subiungimus . 

1.&quot; Sive vires tendant ad
centrum datum
, sive non; denotantibus any, :coordinatas
puncti materialis in fine temporis t
, profecto x ,r,:peu-
debunt ab ipso :; erunt videlicet æ, y, :functiones tem-
peris :, ut scribi possit
.
——-—————.——-—-——-——.—.——...———..—91
= f ( ) , y = fi ( ) , z = 12

2. • Si vocatur s arcus a materiali puncto percursus
tempore t, w velocitas ejusdem puncti in fine ipsius t , pe
rinde spectari poterit ds ac si motu uniformi conficeretur ,
sola nimirum velocitate praeconcepta v ; siquidem nova
velocitas, dv , quae labente dt accedit materiali puncto ,
est infinitesima . Propterea hic quoque ( 28 )
ds
dt

3.º Resoluta vi o in duas , quarum altera sese
dirigat juxta tangentem , altera juxta respondentem nor
malem , erit ( 44) prima
des o cos i duษ
dc &gt;
dta
secunda (45 )
2² ♡ sini
ds²
r rdta

4.°# Incedente puncto materiali K per arcum s , mo
vebuntur motu rectilineo projectiones K' , K &quot;, K '' ipsius K in''
coordinatis orthogonalibusque axibus OX , OY, OZ ( Fig.5 ) ,
eruntque ( 28 )
dx dy dz
dt dt dt &gt;
earum velocitates in fine temporis : , quum nempe K ha
ds bet ( 2 ) velocitatem Vi acceleratrice dc
K , resoluta in ternas P ', P &quot; , D' ' ' iisdem axibus parallelas,
. , qua sollicitatur
∙ 91-
x:f(t)-J:fx(t)o 2:130)-
2.&quot; Si vocatur .: arcus a materiali puncto percnrsus
tempore :, v velocitas eiusdem puncti in fine ipsius t, pe-
rinde spectari poterit ds ac si motu uniformi couGeeretnr ,
sola nimirum velocitate praeconcepta v, ∙ siquidem nova
velocitas dv , quae labente dt accedit materiali puncto ,
est infinitesima
. Propterea hic quoque (28)
ds
Pr.—...... dt
3.&quot; Besoluta vi 9 in duas , quarum altera sese
dirigat juxta tangentem , altera juxta respondentem nor-
malem
, erit (44) prima
'n'—'v—d'v-Sd&quot;: cpcost—p ,9 de dt&quot;
secunda (45) ⋅
'
∙⋅ &quot;
'
∙ . ,,,a d;:
cpsmr— r — rdt&quot;'
4.&quot;e Incedente puncto materialiK per arcum :, mo-
vebuntur motu rectilineo projectiones K', K&quot;, K'&quot; ipsius Km
coordinatis orthogonalibusque axibus OX, Oï. OZ (Frg- 5) :
eruntque (28)
'de: (I)-' dz
dt
'
dt '
dt
earum velocitates in Gne temporis :, quum nempe K ha-
bet (2&quot;) velocitatem? .Vi acceleratrice , qua sollicitatur

: - -

K
,
resoluta in ternas P', P&quot;, P'&quot; iisdem axibus-parallelas,92
motus projectionis K' nihil erit aliud nisi motus rela
tivus puncti K quoad ipsum K sollicitatum viribus dum
dx taxat P &quot; , P ''' ; proinde velocitas debelur soli P' ex dt'''
dr ternis P' , P &quot; , P &quot; ; simili ratione ostenditur. deberi soli dt
dz P &quot; ex ternis P' ,P &quot; , P , et soli P&quot; ' ' ex iis 'componenti
dt
bus . Hinc ( 28 )
adx ddy adz
de de dt
P' ,
P &quot; , = P &quot; , dt de dt
seu
dex day daz
dt2 P'
dia
P &quot; ,
di?
= P &quot; .

5. °* Si punctum materiale incedit curvam plagam,
sumptis axibus v. gr . OX , OY in plano curvae , habebuntur
tantummodo
der day
de² P ' ,
dia
= P &quot; .
Fac v. gr. ut vis acceleratrix o sit parallela axi OY , ita
lamen ut sese dirigat ad plagam ordinatae y negativae : erunt
P = 0 , P :
ideoque
d2x
dla 0,
dy
di ?
Istarum prima suppeditat
I
92
motns projectionis K' nihil erit aliud nisi motus rela-
tivus puucti K quoad ipsnm K sollicitatam viribus dum-
taxat P&quot;, P&quot;'; proinde velocitas .j—f. debetur soli P' ex
ternis P', P&quot;, P'&quot; ; simili ratione ostenditur-(g.; deberi soli
&quot;
∙ ∙∙∙ dz ∙ n ∙∙ P ex ternis P', P&quot; ∙ ∙
, P , et −− soh P' ex 11s'componeut1-
dc
bus . Hinc (28)
' ddf ddZ ddi dt dt dt
∙−−− −−∶ '. ∙−−−:
P. −∙∙:
dt dt '&quot;
'
de P '
seu
'
' ⊒ ∙
' '
d3æ (137 d&quot;:
dt&quot; −−−∶ P
'
∙−− ∙−−− ∙−:
P
'di&quot; P '
dt&quot;

5.&quot;; Si punctum materiale incedit curvam planam,
Sumptis axibus v. gr. OX, O? in plano curvae , habebuntur
tantummodo .
⋅ ⋅
dzæ - d
dc&quot; :&quot;)&quot;Zïz'
Fac v. gr. ut vis acceleratrix q; sit parallela axi Oï
,
ita
tamen ut sese dirigat ad plagam ordinatae] negativae :erunt
ideoque
dh: ∙∙ d'] ∙∙∙ ∙∙
dt&quot; —0' −↲⋅≀⋅⇀≖− ?
Istarum prima suppeditat93
dx
dt C , x =Ct +C' ;
secunda, in hypothesi o constantis , praebet
dy ota
dt ot + C &quot;, y = 2 +0&quot; 4 + C &quot; :
eliminato t ,
y
y = c&quot; + * (** ) (* = ) .
Habes itaque, in ea qua sumus hypothesi , coordina
tas x ety expressas ( 10) per t; habes insuper aequatio
nem ad curvam, quam describit materiale punctum : re
stat ut constantes arbitrarias C, C' , C &quot;, C '&quot; determinemus.
Ponatur materiale punctum ex ipsa coordinatarum origi
ne O projici cum velocitate Yo juxta rectam inclinatam
ad OX sub angulo h: resoluta v. in' binas, alteram paral
lelam axi Ox, alteram parallelam axi OY, erit illa = v , cosh,
haec Vo sinh: initio motus obtinent simul
t = 0 , x = y = 0 ,
dx
dt = v , cosh,
dy
dt = V , sinh ;
igitur
C = Vocosh , C = 0.C &quot; = V . sinh , C = 0 ;
et consequenter
012
x = vol cosh ,y = v , sinh -
csinh
cosh
gx2
2v.cosh
93
dr
.
E—:C, æ:Ct-l-C,
secunda, in hypothesi ? constantis , praebet
' d ∙
'

: &quot; 73: :,n—j-c'.7:— ∙≌⇉−−−⊦∁∥≀−⊦ ∁⋯≖ eliminato t ,

∜−⋅−−−≺⋮⋅⋅⋅⊹∁∣⋅ (&quot;€&quot;)— −≣−≺∙≄ ;
&quot;):
.
Habes itaque, in ea qua sumus hypothesi, cbordina-
tas æ ety expressas (1&quot;) per :; habes insuper aequatio-
nem ad curvam, quam describit materiale 'punctum: re-
stat ut constantes arbitrarias C, 0, C&quot;, C'&quot; determinemus.
Ponatur materiale punctum ex ipsa coordinatarum origi-
ne O projici cum velocitate vo juxta rectam inclinatam
ad OX sub angulo h: resoluta v., in' binas., alteram paral-
lelam axi OX, alteram parallelam axi Oï, erit illa:vo cos 11,
haec:vo sin/1: initio motus obtinent simul
da: dy .
t.:o,x:o,y:o, ï

: vo cosh, ?::v., smh;

igitur
C: vo cosh , C': o .C&quot;: vo sin]: ,C&quot; ':o;
et consequenter '
cpt&quot; æsiuh (pa-&quot;
2 &quot;7— cos/1 -21Jo&quot;cos&quot;lt :

: votcOsIt,y:vosiult—94

x tangh -
9 1+ tangah
2 v2.
22.
Recole quae diximus ( 39).

6°# Fac nunc ut, permanentibus caeteris ( 5º. ) , pun
clum materiale moveatur in medio resistente: poterit vis ac
celeratrix ex resistentia medii exprimi ( 32. 33 ) generatim
per f (v ) ; per functionem videlicet velocitatis v
tem , decrescentem , evanescentem simul cum v Sit \beta an
gulus interceptus directione motus et ordinatarum axe OY ;
erunt ( 32 ) P' f (w) sin \beta , P &quot; = -- flv) cos \beta ;
ideoque
crescen
dar d²y : - flv )sin\beta , = -9 - flu) cos\beta ( c ) :
dt2 dla
insuper ( 40)
dx
dt
dy v sin\beta ,
dt = v cos\beta (c' )
quae differentiatae suppeditant
d22 dy d\beta dy do d\beta
dt sin\beta tvcos\beta dt dt2 dt cos\beta — v sin\beta ordt dt2 .
Ergo
dv
sin \beta + y cos \beta
d\beta
dt dt : -f (v )sin\beta,
do
de
d3 cos \beta-usin \beta 0 - f v ) cos\beta: dt
94
x tangh .::
t—ïngïhæt
Recole quae diximus (39).

604: Fac nunc ut, permanentibus caeteris (50.),ptm-
ctum materiale moveatur in medio resistente: poterit vis ac-
⋅ celeratrix ex resistentia medii exprimi (32. 33) generatim
per f(v); per functionem videlicet velocitatis v crescen-
tem
, decrescentem
,
evanescentem simul cum 0Sit B an-
gulus interceptus directione motus et ordinatarum axe Oï;
erunt (32) P':
-
f(v) siuþ ∙P&quot;: −− ?
−f(P) 008 p;
ideoque
d'æ
dt: :—ftv)sinþ,d —:— —f(v) cosþ (c):
insuper '(40)
da: . d .
'at—:&quot;.lnþO £: &quot;waþ (0)
quae diB'erentiatae snppeditant
dzæ
−↙⊼≖−−∶−⋇⋮∐⇪ ⊣−∙≀∘∞⇪⊼ d'B. dz :d—ïcosþ— —vsinþ dþ
dt
Ergo
——sin,8 −⋅⊢ vcos 5—d—-5 −∙−−−∙ —-f(v)sin,8,
dv
Ft— cosþ—vsin B (35—:— ep —f(v) cosþ:95
istarum primam multiplica per sin\beta , secundam per cos\beta,
tum collige in summam; eamdem primam multiplica per
cos\beta , et secundam per sin\beta , cum subtrahe; habebis
dy d\beta
+ fv) =– pcos\beta,
=
Psins
(c' ) .
dc dt
Quibus positis, haec stabilientur: cum nequeat \beta fie
ri &gt; 180° ( siquidem in transitu . per 180° vires omnes e
vaderent verticales, motusque permaneret verticalis ) , cum
que p etv existant perseveranter &gt; 0, ob secundam ( c &quot; ) erit
d\beta constanter 0 ; proinde crescente e crescet semper an dt
gulus \beta accedendo ad quemdam limitem B.
In hypothesi anguli initialis \beta. (=90° - h)&lt;90°, per
get o cos \beta per aliquod tempus esse &gt; o : sed flv ) &gt; 0 ; i
gitur , ob primam ( c''), per totum illud tempus erit de''
et consequenter crescente t decrescet v.
Prima ( c&quot; ) differentiata praebet
du
&lt; o .
d2v dv d\beta gsin\beta ;
dt -
dea + au f '(o )
seu , attenta secunda ( d ''),''
dev dy

dia
+ áf ( ) = q *sin- B
dv
facta igitur dt , emerget
dev oʻsina&gt; o.
dt
95
istarum primam multiplica per sin 13, secundam per cosþ, ⋅
tum collige in summam; eamdem primam multiplica per
cosþ ,
et secundam per. siuþ ,t'um subtrahe; habebis
∙ d d ∙
⊋⋮∙∙⊣−∣↻⇝⇌− ws?- ∙⊺∙↙↙⋛−∶∶∲−−−∘∎⋮∙∂ (a&quot;)-
Quibus positis, haec stabilientur: cum nequeat. þ Ge-
ri )180o ( siquidem in transitu.per 1800 vires omnes e-
vaderent verticales, motusque permaneret verticalis ), cum-
que (p et v existant perseveranter o, ob secundam (e&quot;) erit ↭ ∣
d ∙ ⋅
constanter £
)a; promde crescente :
crescet semper an-
gulus þ accedendo ad quemdam limitem B.
In hypothesi anguli initialis B., (:::90() -H( 90&quot;,per-
get ? .cosp per aliquod tempus esse ∘:sed iv))o'; i-
gitur
, ob primam (e&quot; ), per totum illud tempus erit ⋚∶≺∘∙ et consequenter crescente :decrescet 0.
Prima (e&quot;) differentiam praebet
⋣≖−⊦↙↨−⋛∣≼⋅⇝⇌≡≴∊∹∾⋅≖⋅∣⋮⇋ dav d d . ∖∖ seu, attenta secunda (c' '),
d'v dv ∙∙∙ ∳≖∘⋮∐≏∆⊙ ∙
⊄⋮⋮⋝⊹⊋−∑ f(V)-— v '
. .
dv
facta igitur 22 :o
, emerget
dav cp'sinïþ
üt: ⇀−− v
)0-96
Inferimus ( 27. 22°. ) velocitatem v haud exsistere capacem
maximi: poterit quidem esse capax minimi ; ita tamen , ut
mutato decremento in incrementum, hoc neque vertatur ite
rum in decrementum, neque certum quemdam limitem E
praetergrediatur. Primum patet ex eo quod , posita conver
sione incrementi in decrementum, jam obtineret maximum:
secundum ex eo quod, aucta indefinite v, augeretur indefi
dv nite flv ), simulque foret &gt;0 ; id vero adversatur pri dt
mae ( 6' ) .
Ex ( c &quot;) eruuntur binae
20 21
V2-01 ſię cos$ + fvde,B2- B;= Sosiu\beta dt ;
t t
exprimunt N,, V, velocitates , item B , B, angulos limitibus t,
2t respondentes. Fac
o cos\beta + v ) = f (t) ,
psins = fa (t) :
habebis ( 27. 18º. )
V; - v.--tfittat) • B. - = falttal) ;
exprimunt a et a numeros &gt; o et &lt; 1. Sed crescente t in
definite , vergit fi (t) ad q cosB + f (E ),et fu( t) ad qsinB
E
ac proinde
2 - -V2
limes quantitatis cos B + F( E ) ,
3. - 22 O
limesque quantitatis
sinB
E
96
∙
Inferimus (27. 220.) velocitatem v haud exsistere capacem
maximi: poterit quidem esse capax minimi; ita tamen, ut
mutato decremento in incrementum,hoc neque vertatur ite-
rum in decrementum,- neque certum quemdam limitem E
praetergrediatur. Primum patet ex eo quod, posita conver-
sione incrementi in decrementum, iam obtineret maximum:
secundum ex eo quod, aucta indefinite v, augeretur indefi-
nite f(v). simulque foret-(£)o, ∙ id vero adversatur pri-
mae (c&quot; ). ∙
Ex (e&quot;) eruuntur binae
2t
∙↗−⇂↗≖ −−−∙−− ∙∣ ( ? cosþ-l- fwndz, ↾⊖≖−,B— fra-018 de;
exprimunt v, , v, velocitates,' 1tem (i,, ,H, angulos limitibus !,
2t respondentes. Fac
9) cosB ^v):fd!) ∙∲≊∣∶∁ :
fam
habebis (27. 180.)
'Ur—vzzf— tf1(t&quot;l&quot;at) ∙⇪≖−−⇪≃∶∶⊀≖↸≖⊣−⊄⋅∁⋟⋮
exprimunt a: et «' numeros )b et ↿∙ Sed crescente :in-
≺↿⊜∊⊓⋮⇂∊∙ &quot;&quot;sit fxw ad 90053 —I-f(E).et rm ad ?&quot;&quot;B-
ac proinde
2! -—v

limes quantitatis :

Bos B4-f(E) ,x—Bz ↽− wir-B : .
limes ue uantitatis . q q E97
quoniam igitur
VI - V2 lim.
B - \beta ,
0 lim t t
erunt
Ø cos B + f(E)= 0 ; sin B
E
et consequenter
B = 180° , f(E )= .
Ex istarum prima inferimus motum materialis puncti ver
gere ad rectilineum verticalemque motum; e secunda ( viri
bus p et medii resistentis sese in limite elidentibus, utpo
te aequalibus et contrariis ) ad motum uniformem , proce
dentem videlicet a sola vi praeconcepta.
Divide primam ( c&quot; ) per secundam (c&quot;) : proveniet
dx
d\beta
sie
X-X
B-Brvm?;
iccirco ( 27. 18º. )
i\beta
Spa\beta
Q Q
Bm
exprimit um valorem medium velocitatis v. Haud praeter
greditur ' ' m certum quemdam valorem finitum ; insuper ver
git \beta ad B= 180° : ergo neque x praetergredietur finitum
valorem; ideo que materiale punctum incedet curvam prae
ditam asymptoto verticali. Recole, quae diximus nº. 41 .
Posita ( 33. 4º. ) flv )
formulae ( c) evadent k?
qua 1
quoniam igitur
&quot;r'—Va lim. :o, lim Bi—Ba :.0,
erunt
?
∘∞↿∃⊣−⊀≺≖∙∶⊢− 0
∙
∲≕⋮⋮∶⊔∄−∙−− −−∘⊰
et consequenter
3:180&quot; ,f(E):9.
Ex istarum prima inferimus motum materialis puncti ver-
97
gere ad rectilineum verticalemque motum; esecunda(viri-
bus 91 et medii resistentis sese in limite elidentibus, utpo-
te aequalibus et contrariis ) ad motum uniformem, proce-
dentem videlicet a sola vi praeconcepta.
Divide primam (c') per secundam (e&quot;) :proveniet
iccirco ( 27. 180.)
exprimit v,, valorem medium velocitatis
,,,
Haud praeter-
greditur v,, certum quemdam valorem finitum; insuper ver-
git B ad B: 1800: ergo neque æ praetergrediatur finitum
valorem; ideoque materiale punctum incedet curvam prae-
ditam asymptoto verticali. Recole, quae diximus n&quot;. 

41.
Posita ( 33. 40.) f(v):SE,-2 , formulae (c) evadent
.k?98
dar
di ?
-sing,
day
dla 9
qua
cos\beta : ka
sed haec hactenus.
7º. Intelligantur per coordinatarum orthogonalium
originem O ( Fig. 5 ) duci binae rectae 8,0&quot; intercipien
tes angulum a : earum extremitatibus junctis recta d '&quot;, erit
cosa =
02 +02.02
28 &quot;
Extremitas rectae , habeat coordinatas a ', y, z ', rectae au
tem o coordinatas x &quot;, 1 &quot; , 2 &quot; : paullulum attendenti pate
bit fore
õ = x's + y + 2,0% = &lt; &quot; + ya + z'2 ,
d's = (x - x &quot; )2 + 6 - y &quot; )2+ (z'- z&quot; )?;
adhibitis substitutionibus ,
cosa =
x' x &quot; ta'y &quot; tz'z&quot;
8o&quot;
Sint a' , b' , c' , anguli, quos Ở facit cum axibus OX, OY ,
OZ ; et a &quot; , 1 &quot; , c&quot; anguli quos d &quot; facit cum iisdem axi
bus: erunt 1
x' = cosa' , y ' = ' cos b ', z ' = ' cosc'
x &quot; = &quot; cosa &quot; , y &quot; = 0 &quot; cosb &quot;, z&quot; = 0&quot; cosc&quot; ;
rursusque adhibitis substitutionibus,
98
−∙−≂− −≌≝≖⋅ ∙ 9
9008?-
sed haec hactenus.
70. Intelligentnr per eoordinatarnm orthogonalium
originem O ( Fig. 5 ) duci binae rectae d', d&quot; intercipien-
tes angulum a: earum extremitatibusjunctis recta ö&quot;, erit
ö&quot;:
eo&quot; −⊦∂∣∣∶∎−∂≀∥≖ ⋅−∎ 26' a&quot; '
Extremitas rectae ö' habeat coordinatas 0:231, z', rectae an-
tem d&quot;coordinatas x&quot; , y&quot;, z&quot;: paullulum attendenti pate-
bit fore ⋅
∂∣≏−−∶∞↾≖−⋅⊦∙↗∣≖−⊢≖↾⋩∙ ∂∣⋅≖∙∸⋅∞↾∎≖⊹∕∣≖−∣−≖∥≖ ,
3' ⋅≖−−−−≺∙⊅∣∙∞⋅∣⋟≖⊣⊣∙↗∣⋅∫∎⋅ )'—l-(z'-z&quot; ),:
adhibitis substitutionibns ,
∙−− æ; æ&quot;——)&quot;)'&quot;—l-Z' zn
cosa ∶⋅↳ a, 6&quot;
Sint a', 6', c', anguli, quos 6' facit cum axibus OX, Oï.
OZ; et a&quot;, b&quot;, e&quot; anguli quos 6&quot; facit cum iisdem axi-
bus: erunt
x':d' cosa' ,y*zzd&quot; cosb', z':ö' cosc'
æ&quot;: ö&quot;cosa&quot;, y'': ö&quot; cosb&quot;, 2&quot;: d&quot; cosc&quot;;''
rursusque adhibitis substitutionibus,
−∙∙⋅∙−⋅−−⋅99
cosa = cosa' cosa&quot; -- cosb' cosb&quot; + cose'cosc &quot; .

* His positis, fac ut vis acceleratrix o sese constanter dirigat ad centrum datum : constituta in eo coordinatarum ori gine O, erunt sle D 5.5 cosinus angulorum , quos cum axibus coordinatis efficit ra dius vector D; et P P &quot; P '&quot; P cosinus angulorum , quos cum iisdem axibus efficit . Pro pterea P X op + . $ . Þ==1 , sumpto vel superiore, vel inferiore signo , prout o nititur vel adducere materiale punctum ad centrum illud, vel ab ipso distrahere: in primo enim casu q et D faciunt angu lum a = 180° , in secundo angulum a = 0. Inde profluit ( 49) d2x Ide² dy v + D dia D daz dt2 8.• * Sumptis axibus OX, OY in plano ( 46) cur vae , quam incedit materiale punctum , erit der Q =F Ndt² on the + 5) . 99 cosa:eosa'cosa&quot;-]-cosb' cos6&quot;-1-cose' cosa&quot;. ∙His positis, fac ut vis acceleratrix (p sese constanter di- rigat ad centrum datum: constituta in eo coordinatarum ori- gine 0, erunt æLz D'D'D cosinus angulorum, quos cum axibus coordinatis edicit ra- dius vector D; et P' P&quot; P'&quot; r ' a ' ? cosinus angulorum, quos cum iisdem axibus ellicit ep. Pro- ? se P&quot; 7 p--- ∙∙∙ ∙−−− ' D—&quot;'ï&quot;10 ? sumpto vel superiore, vel inferiore signo, prout ep nititur vel adducere materiale punctum ad centrum illud, vel ab ipso distrahere: in primo enim casu ? et D faciunt angu- lum a:1800, in secundo angulnm a −−−− 0. Inde profluit (40) ≕∙∙∙∙∙∙ dia: :: d'y )- ? D) *(dz : &quot;D'l'dcz ∐↼⊦↲⋮−−≟ ⋅⋅− ⋅ 8.0 «: Sumptis axibus OX, 0? in plano (46) cur- vae , quam incedit materiale punctum , erit100 Ad exprimendamo per coordinatas polares , exhi beat 180°-W angulum interceplum radio vectore D et axe OX ; erunt De = x ? tys , x= - Dcosw , j = Dsina . Prima semel iterumque differentiata dat dDP + Dd D = xd x + ydży + dx2 + dy? ; secunda et tertia praebent dx = Dsiow cosw - coswdD . dy = Dcos wdw tsinwdD , ideoque dsa = dx2 + dyr= D -dw2+ dD2 , Hinc 2 der dia dy a D + dla D d - D dea D 2) ܪ . ac proinde la pa (d- D dla 0 ( ) ). Ad haec : P P &quot; = P P &quot; unde D àla D y et consequenter 1 1 1 100 Ad exprimendam (p per coordinatas polares, exhi- ' beat 1800—0 angulum interceptum radio vectore D et axe OX ; erunt Dï':a:3--l-)'2 , x: — Dcosw .szsinm. Prima semel iterumque differentiam dat dDL-l-DdzDzædïæ-l-yd'y-þdæï-l-dyz .; secunda et tertia praebent dæ:Dsinm cos co —cos ad D. dy:Dcos ædwf-sinædD, ideoque d.,- ∙−−− dx: −⊦ dyaznadæ-l—doa . Hinc dïæ a: dfy ] (PL) ? Dei?-),. ∎⊃⊣−≺∄↙⇄ ⋅∎⊃−−⇤↲⋍≖ dt &quot; ac proinde dzD (deo)!) ∙−−∶ −− D — ? ∓ ∙ (aua dt Ad haec : P' a: P&quot; ⋅∙∙∙∙ )» P ∙∙∙ P .;. :ï,?—q:.ü.,unde-; 7- et con sequenter ∙∙∙∎∙∎⋅∎−⋅101 • dx yd dt rady FO : de quam integrantes assequemur dr V dc dy dt C , seu ydx - xdy = Cdt. Est autem ydxxdy = Dsinud(-Dcosw ) + Dcosad( Dsinw ) Dºdw , propterea с dwla CdtD - da da de ( ) = C2 D D4 insuper AD Code : d d - D dla de dt dD da dt dt . ( dD C do D2 dt 1 . ( D ( ( d da da) da C2 D d d dw ,!... Hit C = as dt aan zoals da ? Coil 100 dwudt da . Da aby boxe parutis 1 C2 D D2 dw² Quare J 101 quam integrantes assequemnr da: dy,&quot; ⋅ ∙∙∙∙ ∙≯≀∙⊋∙↕−− ∙∙∷⊋∙⋮−−− C, seu ydæ—Jt'dj—Cdt- Est autem ydx—ædy:Dsinæd(—DcosmH—Dcosæd(Dsinæ) : D'daii , propterea ∙−− dai—C ∙ de) 3—01 ∙ ∁↙≀⇞−−∐⇟⊄∄∾⋅∙⋅⊋⊼−−−∐−≖⋅∙ (a)—&quot;1373&quot; insuper di? d(dD. 49) d(iD ∙⊆− ≀∄⋅∣⊃∙∙ d ∙∙∙⋅∃⊂∙−⊃⋅ 71? ∙− do) ne). dt'- d; ⋅ d:; dï- ⋪∙−⋅−⊳ ↿ ⋅ ⋯↿ 41 d(ï) d D d(B) 1101 ...-2 d, ⋅ da) ∙−↽∁⋅ ↪↼⋅−↽−⇁∁ ↜⊒⋅∶≥⇀⋍−−⋅−≤⋮∶ a d: dmwdt;,, nad-'O) . ' f&quot; &quot; c: ∐≖⋅↙∄∘−∎⊃−dasz102 D + ) ( 61 ). D2 dw² Fac v. gr. ut, viribus ad datum centrum tendentibus, materiale punctum incedat curvam (dicitur spiralis loga rithmica ) repraesentatam per Draw Habebis ( 27. 6.° ) . el = 2 11름 loga dw ,de 1 D% log ? a dway log.'a ; iccirco go CP D2 a log-a + b) ( logo a+ 1 ) . vis nempe acceleratrix erit in ratione reciproca triplicata distantiarum a dato centro.

9. # . Ad constantem C quod spectat, ex coordina
larum origine 0.(Fig . 19 ) intelligantur duci bini radii ve
clores, alter ad punctum datum a habens coordinatas xo,
Yo, alter ad punctum quodvis B habens coordinatas x, y;
sitque A area sectoris terminati arcu aB et radiis illis, A
area aBca' : erit
A = A +
Xoyo
2
xy
2
&gt;
⇀↿∘⊋∙
du-
−−∶⊨ C, ...—l.).. ..,— (6) ..... i)? da: D !'
Fac 9. gr. nt, viribus ad datum centrum tendentibus,
materiale punctum incedet curvam (dicitur spiralis loga-
rithmica ) repraesentatam per
D:
ac .
Habebis ( 27. 6.&quot; )
1 -ao 1 1
T)- :a ∙↙≀−∣⋝−∙−−−−−− logadæ,d'ï-—
.. ∠≀≖−∣↿⋝∙ ..
a logia dei:-, :logæa ,

dm&quot;
iccirco
1»ng (
−∾∙∣∘⊰⋅∅⊣−∎↿⋥≻−−−∌⋮ ——(l0gi ∅−⊦
1):
vis nempe acceleratrix erit in ratione reciproca triplicata
distantiarum a dato centro.
9011. Ad constantem C quod spectat, ex coordina-
tarum origine O-(Fig. 19) intelligantur duci bini radiive-
ctores, 'alter ad punctum datum et habens coordinatas xo,
yo, alter ad punctum quodvis B habens coordinatas x, y;
sitque A area sectoris terminati arcu aB et radiis illis, A'
area cha': erit '
A—A' : æozïo ?;103
ideoque ( 27. 18° )
dA = dA - d
xy ydxxdy
2
denotante praeterea i angulum interceptum tangente in B
et respondente radio vectore D, est ( 48)
D sinids dA 2
Igitar
ydx = xdy = Dsini ds,
et consequenter ( 70 )
C dt = D sini ds ;
unde ( 20 )
ds
C = D sini - Du sini .
dt
Caetero quantitatem Dvsini esse eamdem ubicumqe suae cur
vae sit materiale punctum, liquet ex dictis ( 47 ) .

10 ° # Habemus ( 2º. 8° )
ds
de²
DP dw2 + dD CP
dc2 (Da dwa + dD&quot;)p4
dwa
dD 2
D2
[ 11
+ 9 seu
de
103
ideoque ( 27. 18o .)
dA ::dA'- J
222 −−−∫↙≀∞−⋍≀↨≧↩ −∫∂∞−≨⋅⇣⇃⋮↙≀∫ ;
denotante praeterea t' angulum interceptum tangente in B
et respondente radio vectore D, est (48)
D sini ds
2 ∙
(IA:
lgitur
ydx':xdy :Dsint' ds,
et consequenter ( 70 )
Gde:D sini ds ;
unde ( 20)
CZDsini £:Dvsiü. dt
Caetera quantitatem Dvsini esse eamdem ubicumqe suae cur-
vae sit materiale punctum, liquet ex dictis (47).
100a Habemus ( 20. 8&quot;)
2
(Isa 02 dGP—xl-JD':(02 da,-1- dDz) c,
.
— —∙∙∙
−⋅ dt&quot; ⋅⋅−⋅ dt: D4 dc.-13:
∁≖
dDa
äirl—(5)], seu
de?104
v2 = C2 -- [ + (3 ]
( m) .
11. # Quemadmoduni , data linea quam incedit
materiale punctum , innotescit q ; sic vicissim , data op , po
terit sciri linea per quam movetur materiale punctum
Denolante B quantitatem constantem et n numerum inte
B
grum , sit v . gr. g = ; erit ( 7° 6. )
D &quot;
B CP D
+ )

;

dwa
1 B
quae ,
facto D = 1
D' et
et og h , vertetur in C2
d2 D'
h D ' r-2 = +
diwa +D) .
Chaton
Haec multiplicata per 2dD ' suppeditat
E12dD' dD d
dw da +
2D'dDdD' ) -2-2 h D'n -2 d D' = 0 ;
sumptisque integralibus ,
= [CD)* + D ] - 2,0-4C = 0;
unde
dw
(6,2
dD'
2h Dina quoad o adducentem
D'2 į ad centrum ,
'? – C ).
∠⊢⋅⋅ ↿
' 2 2
1 D∶∁ Exi-(a)] (&quot;**-

↿↿∙∘∙ Quemadmodum, data linea quam incedit
materiale punctum
,
innotescit ?; sic vicissim , data ep , po-
terit sciri linea per quam movetur materiale punctum
.
Deuotante B quantitatem constantem
, et 11 numerum inte-
grum, sit v. gr. ep:DT; erit ( 70 6.)
l
'l
3— B 02 (...d D.. 57.— 25 .'.)* da: D
⋅ ↿ B ∙
quae, fama-:D et——⋜⋮−:h,vertetur tn

* D' ≀≖≖≖⋅⋅−≖⇌⇀−⊻≐≺∡∽≖ −⊦∘∙≻⋅

Haec multiplicata per 2dD' suppeditat
∶⊨≺∶≳↙⊋≞⇗∠∄−−⊣− 2D' dD')—2h D&quot;'2dD':——o ;
sumptisque integralibus ,
465)&quot; HB ]-—'— ∣⊃⋅⋅−⋅∙−⊦∁⋅−−−∘≅ n—l
unde
da: dD'
2]; quoad ?adducentem
TDV&quot; —-D'3 —C')5 ad centrum,105
dD'
dw 2h quoad o distrahentem
(0 –
a centro :
n =; D**?— D» ) *
quarum integratio praebebit relationem inter w et D' , ideo
que inter coordinatas polares w et D lineae quaesitae .
12.°* In istarum aequationum prima sume v . gr.
n = 2 ; ea sic poterit scribi
D'
doma
V
ha- C
da
h D'
h2_C Hinc
w = C &quot; + arc cos =
h - D'
VhC
cos (6-C' ' ) ;
et restitutis valoribus h , D' ,
D =
C2
B - 1 B2 – C4 C cos (W – C&quot;) ·
Pone
C2 C
= B (1 + €),
=B' ( 1 —E) ,
B-HVB2_C4 C B - V B2 - C4C
quae in summam collectae praebent
B
CPC B '
,
invicem multiplicatae suppeditant --
8
105
&amp;
dm: dD
quoad p distrahentem
(C' - 36- D'&quot;&quot;' −−∙ ∎⊃∎∌≻≩⋅ a centro :
⇀
n—1
quarum integratio praebebit relationem inter 61 et D' ,
ideo-
que inter coordinatas polares &amp;) et D lineae quaesitae .
12.&quot;; In istarum aequationum prima sume v. gr.
n:2 ; ea sic poterit scribi

: .-n

'
↶⋮≼⇂∕−−⊮−∁∙⋟
dæ:- ⇂∕↿
Hinc −≺⊓⋅≻≖∙∣≖≖−∁⋅
G):C&quot;-l— :COS(GO—C&quot;)i arC(cos ∙−∙−−−−
h—D' h—D'
⋅⇂∕∣−≖−−−⋯≖−⇀∁∙ ⇂∕∣≖≖∙−∁∙ '&quot;
et restitutis valbribus I: , D',
B—l/Bz—Clt C' cos (co—C&quot;)
D
Pone
C2 02 −−−−− −−−−−−↧≉⋅↿
). −−⋅⊨
−−−−−−∶ —B'(1—e). B—l/Ba—cac- ≺⊹⋮ ∌−⊢⇂∕∌≖−∁↙∣∁∣
quae in summam collectae praebent
c:c' B -—-−⋅⋅
. B',
invicem multiplicatae suppeditant106
&lt;= B' ? ( 1 —-z );
habebis
1 C2 C' = B
B'2 ( 1
B' ( 1 — 52)
Propterea
D =
B' ( 1 - 2)
E cosWC( '')''
(62) .
1
13.0* Potest C' esse vel &gt; 0 , vel &lt; o , vel == 0;
in primo casu erit B ' &gt; o et € &lt; 1 ; in secundo B' &lt;o et
&gt; 1 ; in tertio B ' = et z = 1. Primum ac secundum
casum alibi considerabimus .
14. * Ad tertium quod pertinet , exhibeat NI...
(Fig . 25) axem parabolae ( 40. 5.º 7.º ) ; sintque NO ( 3x)
et 00' ( =y) orthogonales coordinatae : designante 2p pa
ramelrum , exsistet
ya = 2px .
Substituto x' + ip pro x , transferetur coordinatarum
origo in focum H , eritque quoad novam originem H
ya = 2px' +p .
Duc radium HO =D) ; habebis
NHO x'
--- D cos w , y = D sin w ;
et consequenter
D2 sin ’ w = p - 2pDcosw .
Spectatur autem D ut quantitas constanter positiva ; proinde
106
↿ 'a a .
&quot;ö'.:B (1—£)1
habebis
∙∙∙ ↿ ⋅ ∙∙∙ Ca
B'3(1 -
a&quot;) ' B' (1—5') ⋅
Propterea
B' (1 — a')
D −∙− (b,) . 1— :cos (co— C&quot;)
1391» Potest C' esse vel≻∘ ∙ vel (o , vel:o;
in primo casu erit B' o et e ↿;in secundo B' (0 et s
↿; in tertio B':eo et e:1 . Primum ac secundum
⋅ casum alibi considerabimus .
145): Ad tertium quod pertinet , exhibeat Nl..
.
(Fig.25) axem parabolae (40. 5.&quot; 79); sintque NO (:.r)
et 00' (: y) orthogonales coordinatae :designante 2p pa-
rametrum
, exsistet
y':2pæ .
Substituto x' −⊦ ∙⇡∙↼ ;) pro æ .
transferetur coordinatarum
origo in focum H , eritque quoad novam originem H
7&quot; ⇌ 2pæ' ⊣− r'-
D'uc- radium HO' (:D) ; habebis
NHO':61, uf:—D cos æ,y:D sin(-);
et consequenter
D2 sin2 01:p' −∙∙ 2pDcos co .
Spectatur autem D ut quantitas constanter positiva; proinde107
DE P cosa
+ V
V pa pacos w_P(1 ~ cos )
sin? W sin? W sin4 w sin' w
Sed sin? w = 1 - Cos w = (1 — cosa) (1 + cosw ) : igitur
P
D =
1 +cosa (63) .
Designata nimirum quantitate B '(1 - 6 ) per P , et assumpta
C &quot; = 180° , recidet (62) in (63) ; unde consequitur illud :
iribus ad centrum datum tendentibus in ratione reciproca
duplicala distantiarum ab ipso centro , poterit materiale
punctum describere parabolam habentem suum focnm in
centro illo .
15.0# Quoad parabolam ( 14º. ),
(* )
sinaw 1 1 +cosw cosa a
COS
pa da P р
2 1 2 CM 1 1
D
O
Hinc ( 90.m) va - .
р
D
P
D
Sit E altitudo debita velocitati v ; erit ( 12º. 14º. )
2C E 2C?
v2 = 20E =
2BE
D2
E
B
D2 ' ( 1 -62 ) D2 p
et consequenter
2C2 E 2C 1 E
D

; unde D2 D

.
р P
Inferimus illud : si in distantia D a centro virium proji .
citur materiale punctum , haud describetur parabola nisi
107
D:∙∙∙ ;) cosa) ∙∙∙⊦ Vpa .l.-paene: c.)—p(l—cosï
≖⋮∐⇄
∙
a) s1na a) sint! ea sin' 6)
Sed sinit.):1−cosa a:(1— eos a)) ('l-l- cos a) :igitur
∼ P
D:1—i-cosm ∅⋮⋝⋅ ⋅
Designata nimirnm quantitate B'(1-- 6&quot;) per p , et assumpta
&quot;:180o , recidet (b,) in (63) ; unde consequitur illud :
viribus ad centrum datum tendentibus in ratione reciproca
duplicata distantiarum ab ipso centro
, poterit materiale
punctum describere parabolam habentem suum focum in
centro illo .
1530 Quoad parabolam (Mc,),
∙
∙−−− — ∙−−−
d'
⋅
( D) sin'm 1—cosaæ—1—l-cosæ 1—cosa1
df&quot; P&quot; ?' p P
2 ↿ ↿ ∙ 2
c- 1

;

∙ D ∙∙∙ DQ. Hlnc (90.m) 02: ∙∙∙∎∎∙ ∙
ö
∙
Sit Ealtitudo debita velocitati «a; eri; (1241. 14o.)
2311: 20» E
∙∙∙∶≿∁∶
E
ng—ZQE— Da —B'('l—-£3) ∙ [P p . 02 ,
et consequenter
zcn E—zc: '-dE-1
p.Da—p.D,uneD—. .
Inferimus illud :. si in distantia D a centro virium proii-
citnr materiale punctum , baud describetur parabola nisi108
eidem distantiae D fuerit aequalis altitudo illa , per quam
mobile vi acceleratrice vigente in puncto projectionis ca
dendo motu uniformiter accelerato acquireret velocitatem
ipsius projectionis.

51. Hactenus de motu curvilineo libero, quum nempe nihil obstat quominus mobile obtemperet viribus; fac nunc ut materiale punctudi, cujus massa = m, moveatur motu impedito, sollicitatum videlicet vi acceleratrice q adstringatur moveri vel in data superficie vel in data linea curva. Quoniam ejusmodi superficies et linea nihil praestant aliud nisi exercere in puncto materiali resistentiam m ç sibi perpendicularem, ideo motus perinde fiet ac si punctum materiale esset liberum viribusque acceleratricibus et d', seu quod eodem redit viq &quot; inde resultanti libere obtemperaret.

Pone quod motus impeditus in data linea debeatur
unice vi praeconceplae et vi gp' ut sit 9
habebis
q &quot; = 0 ; i = 90 °; et consequenter ( 45. b)
0 :
2,2
( 6' ' ' ) ;
my?
Precisa nimirum q , exprimet ( 28 ) pressio
nem exercitam a puncto materiali in lineam illam , atque
huc spectat vis centrifuga ; pressio videlicet a puncto ma
teriali exercita in eam lineam , orta e sola inertia ad prae
seulem velocitatis siatum contracta.
Ad haec : in eadem hypothesi vis acceleratricis ♡
facile colligitur ex dictis ( 36) motum impeditum fore u
niformem .
!
108
eidem distantiae D fuerit aequalis altitudo illa , per quam
mobile vi acceleratrice vigente in puncto projectionis ca-
dendo motn uniformiter accelerato acquireret velocitatem
ipsius proiectionis.

===De vi acceleratrice in motu circulari, existente centro virium in centro circuli.===

52. Ex demonstratis (47) patet istiusmodi motum esse uniformem. Sit R radius circuli, per cujus peripheriam incedit mobile: in ( b: 45 ) erant r = R, i = 90° ; in ( b' : 48) vero D =9 = r = R; et denotante A lotam circuli aream, T tempus periodicum, quo nempe mobile conficit integram circuli peripheriam, in eadem ( 8' ) erunt quoque A = n R?, = T. Hinc ex ( 6)
1
RO
et ex ( 6 ) ( c )
4 762 R
T2

53. Haec facile punc stabiliuntur. 

1º. mobile velocitate quadam projectum in distantia R a centro virium von describet circularem curvam nisi velocitas illa tanta sit quantam mobile ipsum acquireret cadendo per { R motu uniformiter accelerato et vi acceleratrice, quae viget in projectionis puncto; siquidem prima (c) suppeditat v = 2 0.4 R.

2º. In circularibus peripheriis eodem tempore
descriptis vires acceleratrices sunt ut respondentes radii: patet ex secunda (c).

3º. Ex eadem secunda (c) inferimus vires acceleratrices fore in ratione reciproca duplicata radiorum quotiescumque quadrata temporum periodicorum fuerint ut radiorum cubi.

54. Obiter haec notamus. 

1º. Ex circulari telluris rotatione circa suum axem oritur vis centrifuga (51) in materialibus punctis tam apud aequatorem quam apud circulos aequatori parallelos, generatim expressa per &lt;math&gt;m\varphi'=\frac{mv^2}{R};&lt;/math&gt; et quia rotatio illa fit motu uniformi, ideo&lt;math display=&quot;block&quot;&gt;v=\frac{2\pi}{T}\,\mathrm{ 
et}\, \varphi'=\frac{4\pi^2 R}{T^2}
&lt;/math&gt;Tempus periodicum &lt;math&gt;T&lt;/math&gt; est ubique idem; &lt;math&gt;R&lt;/math&gt; vero decrescit ab aequatore ad polos; in eadem ergo ratione ab aequatore ad polos descrescet vis centrifuga. 

2º. Exhibeat R , radium aequatoris terrestris (Fig. 31) et a geographicam latitudinem, cui respondet circulus aequatori parallelus habens radium R, erit R =R cosa , et consequenter R , cosa
T2
Resoluta q' in duas, quarum altera sit verticalis, altera horizontalis, existet illa 402R , cosa
D'cosa=
T2
et quoniam q' cosa est vis contraria gravitati, inferimus gravitatem imminui magis semper a polis ad aequatorem, ejusque decrementum fore ut quadratum ex cosinu latitudinis, spectata videlicet tellure instar sphaerae.

3º. Exprimat s altitudinem debitam velocitati rotationis; erit ( 30) 2gs = v ?, ideoque ( 10 ) 2gs = q R, et
consequenter
8
solia
R
.
2s
110
mg': mv&quot;R; )
et quia rotatio illa Et motu uniformi, ideo
27rR et ∙∙∙∙∙ ∢∏≃∣≹
T ' ?&quot; Ta
.-

ecosa: Tat quoniam cp' cosa: est vis contraria gravitati, inferimus gravi-
tatemimminui magis semper a polis ad aequatorem, ejusque
decrementum fore ut quadratum ex cosinu latitudinis ,
spectata videlicet tellure instar sphaerae.

111
Hinc innotescit ratio inter gravitatem et vim centri
fugam : sic apud aequatorem invenitur
8
R,
= 288 circiter;
2s1
inde sequitur quod gravitas sub aequatore in hypothesi tel
luris immotae esset == 1880' + q = 289 .
=== De vi acceleratrice in motu elliptico, existente centro virium in foco ellipsis. ===
55. Haec praemittimus: 

1 °. si ex puncto quovis M (Fig. 32) ducuntur duae rectae MN, MS tangentes sphaeram SN .. , erit MN = MS: ductis enim ex centro C radiis CN, CS ad contactus puncta N et S; itemque CM ad punctum M, triangula CMN, CMS rectangula in N et S habebunt latus CM commune, latera vero CN , CS aequalia; ideoque etc.

2°. Si per tangentes MN , MS ducuntur plana tangentia NMT , SMT ad sphaeram SN .... sese muluose.
cantia juxta rectam MT, angulus NMT aequalis erit an
gulo SMT: nam ex C , N , S ad punctum v . gr. T rectae MT, ductis CT , NT , ST, quoniam NT et ST jacent in planis tangentibus NMT , SMT , iccirco in triangulis CTN , CT'S anguli CNT, CST erunt recti; latera in.
super CN CS sunt aequalia , et CT commune: proinde
NT = ST. Triangala igitur MNT, MST exsistent ( 1 ° ) invicem aequilatera; ideoque etc.

3º. Si denotat p projectionem lineae rectae l in
plano quovis , et a angulum , quem efficit I cum eo plano , erit&lt;math display=&quot;block&quot;&gt;p = l\cos\alpha&lt;/math&gt;: patet ex Trigonometria.
4º. Si denotat P projectionem mn (Fig. 33) areae planae cd ( = A ) in plano quovis gr , et i angulum ,
quem efficit A cum gr , erit,
P = A cosi .
Ducatur enim planum mg parallelum areae A, in quod
demittatur ex d perpendiculum dK ( = x ) ; ducantor quo
que plana gh , de parallela plano qr; ponaturque dg = y .
Quisque videt prisma vel cylindrum md aequari prismati
vel cylindro ge: propterea Ax Py ; unde P A ; est
autem -
sindgK = cosi ; igitur etc. yу

5º. Secetür cylindrus rectus aB ( Fig. 34 ) plano
ad circularem ipsius cylindri basim inclinato; sectio AMBL
dicitur ellipsis ; punctum C, ubi planum secans occurrit axi
cylindri, vocatur ellipseos centrum; punctumque S, ubi se
crio illa tangit sphaeram sambl cylindro inscriptam , appel
latur ellipseos focus; pro cylindri base sumimus circuluin trans
euntem per centrum c sphaerae inscriptae; inde fit, ut ba
seos peripheria sit linea contactuum superficiei sphaericae
et superficiei cylindricae.

6º. Si per C ducitur linea quaevis recta LM ter
minata ad ellipseos perimetrum , ejus projectio in cylindri
base erit ipsius baseos diameter lm , ita at lc sit projectio
portionis LC, et mc projectio portionis MC. Sed lc mc ;
ergo ( 30 ) LC = MG: lineae videlicet rectae transeuntes per
ellipseos centrum , et ad ellipseos perimetrum terminatae ,
dividuntur omnes bifariam in eodem centro.

7º. Per extrema puncta 1 et m diametri lm du
ctis ad circularem cylindri basim tangentibus lh et mt ,
hae utpote perpendiculares ipsi lm erunt parallelae; rectae
quoque IL , mM utpote cylindri basi perpendiculares, erunt
parallelae; ergo plana hll , ImM cylindricam superficiem
112
40. Si denotat P proiectionem mn (Fig. 33 ) a-
reae planae cd:( A ) in plano quovis qr ,
et t' angulum ,
quem eliicit A cum qr', erit,
P:A cost'.
Ducatur enim planum mg parallelum areae A, in quod
demittatur ex d. perpendiculnde ( −−∶ æ ); ducantur quo-
que plana gh, de parallela plano qr; ponaturque liga:-7.
Quisque videt prisma vel cylindrum md aequari prismati
vel cylindro ge: propterea Aa: −−∶ Py: unde P:
.i.-A; est
autem −⋅↕⇣∙ ∶−− siudgK :cosi; igitur etc.
.7
20. Secet'ur cylindrus rectus aB (Fig. 34 )plano
ad circularem ipsius cylindri basim inclinato; sectio AMBL
dicitur ellipsis; punctum C, ubi planum secans occurrit axi
cylindri, vocatur ellipseos centrum; punctumque S, ubi se-
ctio illa tangit sphaeram sambl cylindro inscriptam, appel-
latur ellipseos focus; pro cylindri base sumimus circulum trans-
euntem per centrum c sphaerae inscriptae; inde fit, ut ba-
seos peripheria sit linea contactuum superficiei sphaericae
et superficiei cylindricae.
60. Si per C ducitur linea quaevis recta LM ter—
minata ad ellipseos perimetrum, ejus proiectio in cylindri
base erit ipsins baseos diameter lm, ita ut lc sit projectio
portionis LC, et me projectio portionis MC. Sed lc :: mc;
ergo (30) LC: MC: lineae videlicet rectae transeuntes per
ellipseos centrum
.
et ad ellipseos perimetrum terminatae.
dividuntur omnes bifariam in eodem centro.

113
tangentia existent parallela inter se; et couscquenter inter
sectiones quoque LH, MT istorum planorum cum elipseos
plano exsistent parallelae. Liquet autein intersectiones il
las esse tangentes elipseos in L et M; ellipseos igitur lan
gentes ductae per extrema puocta cujusvis rectae, quae trans
eat per centrum , quaeque terminetur ad curvae perime
trum, erunt inter se parallelae.
Recta LM secat bifariam ( 3º ) chordas omnes paral
lelas tangentibus LH , MT; ejusmodi enim chordarum pro
jectiones nibil sunt aliud nisi circularis baseos chordae pa
rallelae tangentibus lh, mi, atque ideo perpendiculares dia
metro lm , a qua proinde secantur bifariam : inde fit , ut
LM dicatur ellipseos diameter.

8º. Ex M ad focum S ducatur MS; rectae MS ,Mm tangent ( 50 ) sphaeram, altera in S , aliera in punctum lineae contactuum superficiei cylindricae et superficiei sphaericae: ergo ( 19. ) MS = Mm. Simili modo, ex L ad S du
cta LS, erit LS = LI.

9º. Plana TMS, MMT et transeunt per rectas MS,
Mm tangentes sphaeram , et sphaeram tangunt, et sese mutuo
secanı juxta MT; ergo ( 2º )anguli TMm, TMS erunt aequales :
simili ratione ostenditur angulos IILS esse aequales.

10º. Denotet a rectam Cc jungentem centra Cet c: trapezium LMml suppeditat Ll +Mm 2a ; igirur i 80 )
SL + SM 2a . Variala utcumquc positione diametri LM ,
non ideo variabit recta Cc , sed mavebit cousians in ea
dem ellipsi ; ergo summa rectarum SL et SM, quae in ea
dem ellipsi ducuntur a foco ad extrema puncta cujuscum
que diametri LM, erit quantitas constans. Ad haec: rectae
SL, SM efficiunt cum tangentibus LH , MT avgulos aequa
les SLH, SMT; cum enim LH et MTsint parallelae ( 7 °) ,
itemque Ll et Mm parallelae , angulus HLL aequalis erit
angulo TMm; proinde ( 99) etc.

11º. Revolvatur diameter LM donec transeat per focum S, sicque evadal AB: rccidet SL in SA, et SM in
113
tangentia existent parallela inter se; et consequenter inter-
sectiones quoque LH, MT istorum planorum cum elipseos
plano exsistent parallelae. Liquet autem intersectiones il-
las esse tangentes elipseos in L et M; ellipseos igitur tan-
gentes ductae per extrema puncta cuiusvis rectae, quae trans-
eat per centrum, quaeque terminetur ad curvae perime-
trum, erunt inter se parallelae.
Recta LM secat bifariam (30) chordas omnes parallelas tangentibus LH, MT; ejusmodi enim chordarum pro-
jectiones nihil sunt aliud nisi circularis baseos chordae pa-
rallelae tangentibus lh, mt, atque ideo perpendiculares dia-
metro lm, a qua proinde secantur bifariam: inde fit, ut
LM dicaturo ellipseos diameter.
.Ex M ad focum S ducatur MS; rectae MS
. Mm tangeiit (50) sphaeram, altera in S, altera in puncto
m lineae contactuum superficiei cylindricae et superficici sphae-
ricae: ergo (10. ) MS :Mm. Simili modo, ex L ad S du-
cta LS, erit LS:LI.
90. Plana TMS, mMT et transeuntper rectas MS,
Mm tangentes sphaeram, et sphaeram tangunt, et sese mutuo
secant iuxta MT; ergo (2&quot;)anguli TMm, 'I'MS erunt aequales:
simili ratione ostenditur angnlos HLS esse aequales.

12.• Aa est minimum , Bb est maximum omnium
perpendiculorum Ll , Mm , ... quae ex perimetro ellipseos
demittuntur in cylindri basim ; ergo ( 89) SA erit minima ,
SB erit maxima omnium rectarum , quae ex foco S du
cuntur ad ipsam ellipseos perimetrum .

13.• Punctum S' ita determinatum in axe trans
verso AB , ut sit CS' = CS , dicitur alter ellipseos focus.
Jam si ex S' ad M et L ducuntur rectae S'M et S'L , quo
niam SC = S'C et ( 69) LC = MC , iccirco SL et SM
erunt aequales et parallelae ; igitur ( 109)
SL + SM SM + SM = SL + SL = 2a .
Praeterea angulus SLH aequatur angulo SMR ; ergo
( 10 °.) angulus SMT aequabitur angulo SMR.

14°. Producatur MS donec tangenti LH occurrat
in H , erit ( 30. ) angulus LHS aequalis angulo SMT. Sed
( 109. ) SMT = SLH ; ergoò LHS == SLH , ideoque SL=SH:
hinc ( 13. ) HM = 2a .
56. His praemissis venio cum D &quot; o Arpere ad quaestio
nem propositam de invenienda vi acceleratrice o in motu
elliptico , exsistente centro virium in ellipseos foco S. Conci
piantur duo radii vectores SM , SN intercipientes angulum
inGnitesimum MSN , et producatur SN donec occurrat
tangenti TM ... in R ; erit ( 49 , 6 &quot; )
Q
2 NR
62
Binae NR , MH babendae sunt pro parallelis , eruntque
114
SB; ideoque (100) AB:Za. Quoad alias positiones diame-
tri LM habetur semper LM (SL ∙−⊢ SM, et consequen-
ter (100) LM 2a; igitur AB est omnium diametrorum
maxima: AB dicitur axis transversus ellipseos; diameter per-
pendicularis axi transverso dicitur axis conjugatus.

140. Producatur MS donec tangenti LH occurrat
in H , erit (70.) angulus LHS aequalis angulo SMT. Sed
(loo-) SMT:SLH ; ergö LHS:SLH ,
ideoque SL:SH:
hinc (139) HM:20.
56. His praemissis venio cum D'&quot; Atnpere ad quaestio-
nem propositam de invenienda vi acceleratrice ep in motu
elliptico , exsistente centro virium in ellipseos foco S. Conci-
piantur duo radii vectores SM , SN intercipientes angulum
infiuitesimam MSN , et producatur SN donec occurrat
tangenti TM
...
in R; erit (49. b&quot;)
2NR
∙∙∙⇀−−∙ −−⇀∙∙62
Binae NR , MH habendae sunt pro parallelis , eruntque115
proinde ( 55. 3. ) ut respondentes projectiones nr , mh in
cylindri base : hinc ( 55. 14º.)
nr . MH NR =
nr 2a
mh mh
Sit T tempus periodicum , quo nempe materiale pun
ctum totam percurrit ellipticam orbitam ; erit ( 46) ellipseos
area ad aream MSN ut Tad 0 : istae areae sunt ut re
spondentes projectiones ( 55. 4º. ) in cylindri basi , nimirum
ut ipsa cylindri basis ambll =
mila et area msn : ad
haec ; demisso perpendiculo st ex s io tangentem mt ,
erit
msn = j st , mr = 1 st (nr . mg) : quare
( mza)
712 14
ml
16 T2
2
SC
nir , mg
et consequenter
mi ml 62
T2 .
nr T2 2
st mg
Triangula mlh , mlg sunt rectangula , alterum in l ,
alterum in g ; habent insuper communem angulum in m :
iccirco
ml&quot;
= mh .
mg
Anguli mhl et hmt sunt ( 55. 7. &quot; ) aequales ; propterea
triangula mlh , stm rectangula in l ac o dabunt (55.30. 14º.)
115
proinde (55. 39) ut respondentes proiectiones nr, mi: in
cylindri base : hinc (55. 140.)
nr . MH nr

2
−∙− ∙ NR mh (: mh
Sit T tempus periodicum, quo nempe materiale pun-
ctum totam percurrit ellipticam orbitam; erit (46) ellipseos
area ad aream MSN ut T ad 9: istae areae sunt ut re?
spondentes projectiones (55. 40.) in cylindri basi , nimirum
∙ ∙ ∙ ∙ ↿≖ −∎⋅
↴ ut ipsa cylindri basis ambl(:-Z - ml&quot;) et aram nim: ad
haec ; demisso perpendiculo st ex .: in tangentem mt
, erit
mm:&amp;st,mr:äst(nr.mg)iï:quare
l ml ml: 62
nr ∙−−− ∙∙ T! :,- '
—-- '
&quot;rf&quot;; '
st2 mg
Triangula mih , mlg sunt rectangula , alterum in I,
alterum in g; habent insuper communem angulum in m :
iccirco '
tl,
— z'mll.
Anguli mi:! et hmt sunt (55. 73) aequales; propterea
triangula mllt ,
stm rectangula in 1ac :dabunt (55 . 30. 140.)116
Im mh MH 2a
SC si SM SM
Non pluribus opus est , ut assequamur
47' a3 1
( h) ;
T2 SM
vim nempe acceleratricem in ratione reciproca duplicata
radii vectoris .
Quoad aliam ellipsim
4 R² a ,
1
T ;
i
S, MI 2
hinc si
1 1
a3
T2
a ,
T

: erit op :
:

2 SM 2
S, M ,
Si nempe in diversis ellipsibus quadrala temporum pe
riodicorum sunt ut cubi semiaxium transversorum , vires
acceleratrices tendentes ad respectivos focos erunt in sola
ratione reciproca duplicata respondentium radiorum ve
ctorum .

57. Haec subjungimus .

1.º Fiat CS
CA
CS seu a
€ ; numerus &amp; K1 ) dici
lur excentricitas : ex L in axem transversum ducatur per
pendiculam Li , et ponantur Ci = x , Li = r ; erunt
SL = y2 + ( x — $ a) 2, S'L ' =y2 + ( x + ε a) 2 ,
et consequenter ( 55 , 13º. )
↿16
lm mh MH 212
∙−−∙∙−−− −∙∙sm SM SM .

Non pluribus opus est, ut assequamur
47:303 1
−− ∙∙∙ lt ; ? Ta sit-r, (
vimnempe acceleratrieem in ratione reciproca duplicata
radii veetoris.
⋅ Quoad ≘∣⋮∘⊡↾ ellipsim
∙− 4 123 a,3 1
ut ?! Tla 5! M :
hinc si .?- gz.
. −↿− ↿
⊽↓⊽∶⊺∣≖∙∁≖∣⇂∲∙∲∎⇌⇋⊤⊡∶
Si nempe in diversis ellipsibus quadrata temporum pe-
riodicorum sunt ut cubi semiaxium transversorum, vires
acceleratrices tendentes ad respectivos focos erunt in sola
ratione reciproca duplicata respondentium radiorum ve-
ctorum
.

57. Haec subjungimus.
CS 

↿∙∘ Fiat äseuï :8: numerus :
((1) dici-
tur excentricitas : ex L in axem transversum ducatur per-
pendiculum Li
,
et ponantur Ci:æ , Li:7; erunt
ST.&quot; :y2 −⊢ ≼⋅⊅−−∙⋮∠≖≽⇄⋮ ST]? :]! −⊢ ≼⋅≈⋅−⊢⋮∘≻≖ ,
et consequenter (55.130.)117
Vym + (x - ea) + V y2 + (x + ea )
2a ; !
unde
ye + ( x – sa )2 + 2V 99 + (2 - a) Vya + (xta)
ty: + (x + a ) = 4aº ;
ac propterea
V12 + (x - a)2 V y2 + (x +-a)? = 2a? —yox? - ?o ?
ex qua obtinetur
ya = (1-2) (a? – x2) ( o) ;
aequatio ad ellipsim inter x et y computatas a centro C.

2. ° Facta x = o in ( o ) , valor y inde proveniens
nihil erit aliud nisi valor semiaxis conjugati ( 110.) : hinc ,
denotante 6 istiusmodi semiaxem , exsistet
2
62 CS seu ( 10.) 1 -
62 ideoque CS' =a2-6.

al'
a a2
Inferimus distantiam inter focum et punctum illud ,
in quo semiaxis conjugatus occurrit ellipseos perimetro ,
acqnari semiaxi transverso .
39. Loco x substituatur a - ain (o) : emerget
y2 = ((1 — 82 ) ((2ax - x2 ) ( 0' ) ;
aequatio ad ellipsim inter x et y computatas a vertice A.
Jam vergente e ad 1 , simulque crescente a indefinite ver
117
Vr-l—(æ—eaP-l- l/Ja-l—(æ—l-eaPr-h? ⇥
'
nnde
y' −⊦ (æ −∙∙ id? −⊢ 21/7' ∓−⋅⋜∞∶∽≻∙ Vy' −⊢≺∙↿⊏⊹∽⋟≖
−⊦↗≖ −⊦ (..-'.]. ..). ∶−− ta:
.
EC propterea
Vm VW:2(:* —y2—æ2—s*a'
ex qua obtinetur
]&quot; −−−−−− ≺↿∙−∊≖≻ (a' --.r*) (a):
aequatio ad ellipsim inter se et] computatas a centro C.
2.(, Facta a: o in (a) , valor ]inde proveniens
nihil erit aliud nisi valor semiaxis coniugati (HO.) :hinc ,
denotante b istiusmodi semiaxem , exsistet
—2 b' CS &amp;&quot; ∙
..... 1−−∊≖−∙−∶ 23, seu (1 0,)1 ...—a—z- ;; ;1deoque CSa −−∶∅⇄∙− ∂≖⋅
Inferimus distantiam inter focum et punctum illud,
in quo semiaxis conjugatus occurrit ellipseos perimetro,
aequari semiaxi transverso
.

30. Loco a: substituatur a— a: in (0) :emerget

(1—82) (2aæ—æ2) (0') :
aequatio ad ellipsim inter se et y computatas a vertice A .
Jam vergente P. ad 1
, simulque crescente a indefinite ver-118
gat 2 (1 — ?) a ad limitem quemdam finitum B : aequatio
( 0 &quot; ) verget ad
yö = B x (o &quot; ) ,
et consequenter , precedente foco S' indefinite a vertice A ,
ellipsis repraesentata per (o' ) ad parabolam repraesentatam
( 40.70. ) per (o &quot; ) .
Inferimus illud : si a quovis parabolae puncto du
cuntur binae rectae altera ad focum , altera axi paral
lela , eae cum tangente per idem punctum ducta aequa
les ( 55. 130. ) hinc inde continebunt angulos.

4.• Pone conjugatum ellipseos axem fieri imagi
narium ; adhibe nempe 26V - 1 pro 26 : fiet
22 1-62 = , ideoque e &gt; 1 .
Q2
Aequatio nimirum ( 0) novam curvam repraesentabit,
quae dicitur hyperbola , quaeque secat abscissarum axem
in distantiis CA ( = a ) et CB ( =-a) ab G ; inde in infi .
nitum excurrit cum quatuor ramis ab axe illo magis sem
per recedentibus , quorum bini respiciunt partem posi
tivam , bini negativam , habet insuper centrum in C ,
focos in 0 et O' , exsistente CO = CO ' = ɛa .

5. ° * In aequatione ( o) substitue x' + sa pro x;
habebis
ya=( 1—62) ( a2 -x'tea) )
ad ellipsim vel hyperbolam prout &lt;&lt; vel &gt; 1 , exsisten
te coordinatarum origine in respectivo foco S vel 0. As
sumptis nunc ( 7.9 )
x = Dcosw , y = Dsina ,
118
gat 2(t-—£*)a ad limitem quemdam finitum B :aequatio
(a') verget ad
J'2Bæ ⋅ (a&quot;).
et consequenter
, recedente foco S' indefinite a vertice A ,
ellipsis repraesentata per (a') ad parabolam repraesentatam
(40. 70.) per (a&quot;)
.
Inferimus illud: si a quovis parabolae pnncto du-
cuntur binae rectae altera ad focum, altera axi paral-
lela
, eae cum tangente per idem punctum ducta aequa-
les (55.130.) hinc inde continebunt angulos.

4. 0 Pone coniugatum ellipseos axem fieri imagi-
narium; adhibe nempe ⊋∂⇂∕∙−−−−↿ pro 26 :iie't
↿∟∊≖−−∶−⋮⋮ ideoque : ↿∙
∙
Aequatio nimirum (o) novam curvam repraesentabit,
quae dicitur hyperbola , quaeque secat abscissarum axem
in distantiis CA (:a) et CB (:—a) ab C; inde in inli-
nitum excurrit cum quatuor ramis ab axe illa magis sem-
per recedentibus , quorum bini respiciunt partem posi-
tivam, bini negativam, habet insuper centrum in C,
focos in O et O' , exsistente CO:CO':sa.

5.0 11 In aeqnatione (o) substitue x' −∣− Sa pro a:;
habebis ↼
⋅
J*-——(1—8*)(a—(x'-l—w)2)
ad ellipsim vel hyperbolam prout :( vel)1
, exsisten-
te coordinatarum origine in respectivo foco S vel 0. As-
sumptis nunc (7?)
x': -- DCOSGJ
,yzDsinm ,119
erit
Dasin 6) = (1-2)( a ) - (ea - Dcosa)) &quot;) ;
quae traducitur ad
Da 2 ea ( 1-2) cosa a ' (1-2)
D =
1-6 cos26 1 - &amp; cosa
unde c
D :
a (1-2) ( ECOSW +1 ) .
18? cos26 1
Habetur D pro positiva quantitate ; sumpto itaque su
periore signo quoad &lt;&lt; 1 , emerget in ordine ad elli
psim
D
al 1-52) ( 1 t-scosa )
( 1 +acosw) ( 1 -ecosw)
a ( 1-2)
1 -ECOSW ( h) ;
sumpto inferiore signo quoad &gt;1 , prodibit in ordine ad
hyperbolam
a (1-2) ( ECOSW - 1 ) a (621)
D =
( 1 + scos ) (1 - Cosw ) 1 tecosw
(h' )
Non pluribus opus est ut intelligamus in primo ex ca
sibus alibi ( 50. 13.° 14. ) consideratis descriptum iri
ellipsim , in secundo hyperbolam , exsistente focorum al
tero in centro virium : quoad ellipsim , B= a; quoad hy
perbolam, B' = - a.
6. # Ex ( h)
119
erit
Didone-:( 1—s*)(a'—-(ea—chsæ)3) ;
quae traducitur ad
25a(1—s*) cos 6) D∙∙− a'( 1
Da −∙∙ −∊≖≱≖ .
1—szcos2ca 1—e*cos*c.1
unde
∙∙∙ ⇩≺↿∙−⋮⇄⋟ (scusa) :bt)
1----ea cosa:»
D
1
Habetur D pro positiva quantitate; sumpto itaque su-
periore signo quoad e(1
, emerget in ordine ad elli-
psim '
3( l—sï) (1—l—scosa1) —a(1 —e')
D—(l—l-Ecosw) (1—äcosm) 1—scosc1
(71) :
sumpto inferiore signo quoad s)1
, prodibit in ordine ad
hyperbolam
∙∙ -a(1—e*)(scosca——1) —a(sï—1)
(1—I—scosa1) (1—scosm). 1—I—ecosct
Non pluribus Opus est ut intelligamus in primo ex ea-
sibus alibi (50. 13.014.0) consideratis descriptum iri
ellipsim ,
in secundo byperbolam , exsistente iocorum al-
tero in centro virium :quoad ellipsim, B:; quoad hy-
perbolam, B': — a.
69
.
Ex (h)
∙−− .n..- ∙∙ -&quot; ∙∙∙∙∙∙∙−⋅↖∙∙∙− '.120
1 2
a ( 1-2)
sasin '
ECOSA= 1
€2-82cos ?
Ꭰ . dw al( 1 - E22 a-(1-6 )
a (1— $ 2) 2
( 1 - &quot;)
a (1452)
2 1 1 1
a (14 € 2 ) D
bi a - 1—62)
D2
proinde ( 50. 9.º )
02 2C2
a (1-2)
G- ) ( h &quot; ).
Ex ( h' )
€2sin ?
ECOS W =
a( 82-1 )
D
( a2( 1–82) 2
–1).
€ 2 .
a ( 821)
&amp; 2 - cos26 D
42( 1-2)
2 1 .
a (21) D
a’( 1-62) 2
1 1
a2 ( 2-1) Da,
ideoque ( 50. 10.)
V2
2C2
a 2-1 ( + za) ( 17&quot;).
120
∣a(1-52) d 0 eisinïæ sï-sïcosza)
o ' −⋅
' ⇀−
∙∙∙∙−∙ ∈∁∘⊱∞∶ ↿∙− −∙∙
czu-e*)a czu-ez):
proinde ( 50. 99 )
vï— 202 (
1 '1 )
h&quot;
∙ (tU—83) D 20 ( ).
Ex (h')

8003 6) a(83—1) ;
(ï) ∙∙∙ £2sin26)
−− '
dcc D aï(1—82)2
. &amp;:
e* (cuï—1) 1)2 −− ∊≖∁∘⊱≃∾∙∙∙ D
?
2 1
uzu—w?)a t
czu—ez? a(e*—-1) D
↿ ↿ ∙
↙≖≖≺∊≖−↿⋟ ⋅−⋅ ⋅↧⋅⊃−≖∙ '
ideoque (50. 100.)
202 1 1
)
,,, ↗⇩≕−− an:—1 )(D 'l'ïiz (h)121
Sit E altitudo debita velocitati v; erit ( 50. 12º. )
2BE
v=2qE=
Da
2C E
B (1-82) D2
Igitur in ellipsi
1 E
1
B ' D (ó -za),
2
seu ( 50)
olt
E
D
D
2a ( h &quot; );
in hyperbola
1
B'
E
Da - ( + za)
seu ( 5 )
E = 1 + (tha&quot;)
Ex (h &quot; ) et ( h ) consequitur, si in distantia D a cen
tro virium projicitur materiale punctum, haud descriptom
iri ellipsim vel hyperbolam nisi respectu ejusdem distan
tiae D fuerit minor vel major altitudo illa , per quam mo
bile vi acceleratrice vigente in puncto projectionis cadendo
molu uniformiter accelerato acquireret velocitatem ipsius
projectionis.
7 ° * Quoad ellipsim ( 50 , h. 6° )
9
∙
121
Sit E altitudo debita velocitati v.; erit (a 50. 12'.)
2BE— zcn E
D: B'(1-e*) ⋅
ï; ⊍≖∶∃∲⊡∶−∙− ⋅
Igitur in ellipsi
1'Efn'1 1&quot; 1) B&quot;Dï—-a(o za'
seu (50)
in hyperbola
seu (50)
E -D
,
⋮−⇂∃⇌−−⋅⊳⊣−⋅⇄−∅⋅⋅≺≀⋅⋟⋅ ∙ l
Ex (II&quot;) et (h') consequitur, si in distantia D a cen-
tro virium proiicitur materiale punctum, baud descriptum
iri ellipsim vel hyperbolam 'nisi respectu eiusdem distan-
tiae D fuerit minor vel major altitudo illa, per quam mo-
bile vi acceleratrice vigente in þuncto projectionis cadendo
motu uniformiter accelerato acquireret velocitatem ipsius
proiectionis. -
70t Quoad ellipsim (50. I:. 60)122
7
a 옘 E COSQ ) 1
dw² a ( 1-2) Q ( 1-22) - 5
hinc ( 50. 8º. b .)
go
Ca
a ( 1-62 )
1
Da
areo ds D sinids
Est ( 50. 9º . ) C =D sini.

; exhibet

dt 2
lam a radio vectore D descriptam tempusculo de : deno
tante igitur A totam ellipseos aream, T tempus periodi
cum, habebitur
ds C = D sini dt
2A
T
Est ( 27. 18º. )
a
A = 2V 1-* [Vaº-x:dx ;
exprimit 2 | Va?-xă de circularem aream , cujus radius
= a , et consequenter 1
A = Tla ? VT-
Propterea
1
C2 4 A2
T2
4772 24 (1-2)
T2
42 a3
et p =
Ta
0
9 D2
122
≖↿ ⋅ *

:

cosa) 1 1
dm&quot; −⇩≼↿∙∊≖⋗⊽ ⋅⋅∙↽∙↰↿∙∊≖≽∙ D '
liinc ( 50. 80. b,.)
∙−− ∁∙ ↿
,? ↼⇀ (tU-e&quot;)- ⋅
⋅∎⋝≖⋅∙
Est (50. 90.) C :D ciuili-f.; exhibet Egit-If. areo-
£
iam a radio vectore D descriptam tempusculo dt: deno-
tante igitur A totam ellipseos aream, T tempus periodi-
cum, habebitur &quot;
⋅
∙ ∙ ds 2A
C—DSID! 'a'ï—T ∙
Est (27. me.) fZl/l-Ez l/a'-æ' dx;

∽ ∘
exprimit Zf Vaz- ac2 dx circularem aream , cujus radius
o
∙−∙−−−∙− a , et consequenter
A −−∶↿∽≖ ⇂∕↿ ∙a&quot; ∙
Propterea
4A3 47taa4(1-s*) 41:303 1 ∙
Ta&quot; TTL ∅∘⊔⊢−
'1'» C*.—. 'ne'123
prorsus ut supra ( 56).

8º. Obiter notamus illud: si materiale punctum
motu elliptico movetur viribus ad ellipseos centrum len
dentibus, eae erunt in ratione directa distantiarum ab ipso
centro . Assertionis demonstratio eruitur ex dictis ( 56) : sint
enim duo radii vectores CM ', CN' sub angulo infinitesimo
M'ON' , et producatur CN' donec occurrat tangenti M'T
in R' ; erit ( 49. 6' ' )
2N'R'
♡ 02
binae N'R' , M'C censendae sunt parallelae; proinde ( 55.3º. )
m'c : n'r' = M'C : N'R'
M'C . n'r
m'c
area insuper ellipseos ad areolam M'ON' ut tempus pe
riodicum T ad tempusculum 6 ; quae areae cum sint ( 55.4º. )
ut respondentes projectiones in cylindri basi , nimirum
ut ipsa cylindri basis ambl ( = 76. cm ' ) et areola
cm'.r'm' cm '
m'cn'
V r'n'. 2 cm ) , iccirco 2 2 m'
cm'
r'n ' . 2cm
4 02 unde r'n'
762.
cm
272. cm' ; 1 4 T2 T2

et consequenter
M'C . 27. cm'
T2
N'R'
cm'
Ta 272. M'C..
--
123
prorsus ut supra (56).
80. Obiter notamus illud: si materiale punctum
motu elliptico movetur viribus ad ellipseos centrum ,ten-
dentibus, eae erunt in ratione directa distantiarum ab ipso
centro. Assertionis demonstratio eruitur ex dictis (56): sint
enim duo radii vectores CM', CN' sub angulo infinitesimo
M'CN' ,
et producatur CN'donec occurrat tangenti M'T'
in B'; erit (49. b&quot;)
2N'R'
? ∶−⋅− ⊖≖ binae NZR', M'C ceusendae sunt parallelae;proinde (55.30.)
m'c:n'r':M'C:
⋅ 'C. &quot; N'R'-— M nr :
m'c
area insuper ellipseos ad areolam M'CN' ut tempus pe-
riodicum T ad tempusculum 9; quae areae cum sint (55.4&quot;.)
ut respondentes proiectiones in cylindri basi , nimirum
ut ipsa cylindri basis amb! (: Tt. 27:23 et areola
, .
cm'.r'm' cm ∙
∙ ∙
2
...—3
CI&quot;. I o
,
—— - n .Zcm - 4 9: . .
∙ 9: a , .
32. cmlb :T2;undern :::-'F. 212. em,
et consequenter
9:
MC. ∙⊤↓⋅↴∙⋮−⋅∙ ⇄∏≖ cm

NR −− ∙ ∙−−− -;'21t'.M'C.
cm124
Propterea
. M'C :
vis nempe acceleratrix Q directe ut distantia M'C ab el
lipseos centro

* Etiam sic : in ( o. 1º. ) fac

X
Dcosw y = Dsinw ;
prodibil aequatio inter coordinatas polares ab ellipseos cen
tro computatas, nimirum
av182
Dsin ? w = (1-2) (a² - D2cos w ), unde D= V 1-8? cos26
Hinc
at 2
d:2 av1 (via1-2003 (1-8? cosaw ) V 1-2coscosti
D3 1
a* ( 1-2) D
.
ac proinde ( 50. 8º. 3 , )
CP
a4 ( 1482 )
D :
quae ad superiorem expressionem traducitur; nam ( 70. )
4724 (1-2)
C2 =
4A2
T2 T2
124
Propterea
4 ita
?

:

0132 ∙ M'C;
vis nempe acceleratrix go directe ut distantia MC ab el-
lipseos eentro.
&amp; Etiam sic: in (0. 10.) fac'
∶−∙−− -Doosa) ,y
−−−−− Dsinæ;
prodibit aequatio inter coordinatas polares ab ellipseos cen-
tro computatas, nimirum
al/1— ei
Dsin2 a): ( 1—53) (aa.-ul)2 cosm), unde D—
Hinc
([21 «
⋅
') ? cos-36) sium
113&quot; (zl/1—ea ⇂∕ ↿−⋅⋅∊≖∞≘≖∾ ≼↿∙∊≖∘∘≘≖∾⋟⇂∕↿−⋅⋮∅∾∙≖∞≻ ∙∙∙ D3 1
−− a4(1—£2)—ï '
' ⋅ ac proinde ( 50. 823, )
Ca
?
∙−− a4(1—-s2 )
quae ad superiorem expressionem traducitur; nam (72)
Ca— 4A' ∙∙∙ 4n304(1—£2l
T2 T:
⇂∕⋅↿ ∙⊽∊≖∞⊱≖∾ .125

=== De motu relativo punctorum materialium, tendentium in se mutuo viribus acceleratricibus quae sint directe ut massae in quas tenditur, et reciproce ut &lt;u&gt;quadrata&lt;/u&gt; respondentium distantiarum.===
58.* Sint m, m ', m , ... punctorum massae; a, b, c coordinatae orthogonales puncti m in ordine ad axes OX, OY, OZ (Fig. 8); x ', y', z' , x &quot; , y &quot;, z &quot; , x '&quot; , ... Coordinatae reliquorum punctorum in ordine ad novos axes et parallelos axibus Ox, OY, OZ, et habentes originem in m. Factis compendii causa ( 50. 7.0)
x ' ty's tz's =k ?, x &quot; ty's t-z&quot; = k &quot; , etc ...
erunt ( 50. 4.0) quoad motum puncti m
de a m ' x' m' ' Qc &quot; d²b m'
.
g' , m &quot; g
+
k'
&quot; &quot; ) dc2 k2 k' k &quot; 2 hit
d12 ka kita
d2c
m' z'
k'
m &quot;
k' ' ?
.
dc2 ti to...,
seu
d'a d26
dc2
m'x m'z '
Σ
k'3
niy'
Σ
dc
dca &gt; ( o ) .
dt2 k'3
Nunc quod spectat ad aliud punctum v . gr. mi' , pone ( 50.70. )
(.x &quot; —X')2 +6 &quot; -Y')2 + (z&quot; -z&quot;) = 002 ,
( z&quot; &quot; ' —x' ) 2 + 6 — ')2+ ( z' — z &quot;)2 = ' ' , etc... ;
exhibebunt


126
t ... The **** +
en +++
m &quot; yy'
+
d'a + ... ,
+ ..
vires acceleratrices ab m &quot; , m ' exercitas in m' , no
visque axibus parallelas : denotant
ac m
j'
k'a k'
. C
k'a ' ki k'2 k'
vires acceleratrices ab m exercitas in m' , iisdemque novis
axibus parallelas ; sunt insuper ata , bty' , cta' coor
dinatae puncti m' in ordine ad axes OX,OY,02; facto igitur
m &quot; m '&quot;
+ .. =
assequemur quoad motum puncti m'
20 dQ d'a+x' )
dta
mx' d2(6 + y ')
k3 dla
my'
k'3 dx '
dy '
dQ mz' dºlc + z )
dia dzi k'3
d²a d2b
Substitutis valoribus
dac
ex ( 0 ) , prodibunt dca dla dt2
daxi dl mx' m'r'
dxc '
.
day' d my'
dc2 dy
m'y'
Σ
dea k'3
k3
k3 k'3
126
m&quot; .v&quot;--.r' a&quot;; 7 '—:7' F
∙⋅⊱∷−∎∙−⊦∂⋅≖ a ⊣−∙⋅∙∙
vires acceleratrices ab m&quot;, m'&quot;
, ,..
exercitas in m'
, no-
visque axibus parallelas: denotant
ut se' m y' m :
&quot;F' la&quot;—k&quot; k&quot;&quot;'1?'-&quot;£'
vires acceleratrices ab m exercitae in m'
,
iisdemque novis
axibus parallelas ; sunt insuper a-l-z' , (Hl-y' , e—l—e' coor-
dinatae puncti m' in ordine ad axes 0X,OT,OZ; facto igitur
&quot;
m
m m 37 −∂∙−⋅⋅ −⊦ −−∶
9-
assequhmur quoad motum puncti m'
d'(a-[-æ') ∙∙∙ dQ mx' d3(b-l-y') ∙∙∙ dQ my . d,. dx&quot; k-a de dy' k'3
(P(e-l-z') .... di) me'
dt' dz' k'3
∙ ∙ ∙ dia d'b die ∙
∙⊱∎≖∣⋯⋅⋯∎⋯ valonbus dt&quot; ∙ dt' ∙ dt? ex (0) , prodibunt
g'æ/ dQ -mæ' zm'x' d'y' dQ my' zmiy'
dt' dæ' 163 It'3
'
dt2 dj'- k'3- It'3127
daa' d2 mz'
K'3
m'z'
Σ
dta dzi k3
formulae determinantes motum relatiyum puncti m' quoad
punctum m . Quoniam
00 mx' m'x
k'3
mtm x +
k'3 dx ' k'3
zel
2 X
m &quot;
come
-ac '
813 -) +mi&quot;
xc
k3
V3 k'3) +... ,
dQ
,
- - monte +
-&quot; * 7- ) + m.A-A ) +...
en e -maile +
) &quot; V +
d2 mz'
-Σ dzi k3
m &quot;
tom &quot;
t ... ;
03 k3
hinc facto
R = m &quot;
.6. – +) +
(5--**&quot; +jx +e*e&quot;)+ - ( &quot; ),
m &quot;
formulae ( 0' ) vertentur in
127
∙⇌⋅⋮⋅⋮⋅≕↙⇣≴⋅≖−−−∶↗−⋅⋮−−− ∑∶≀−≖⇣ (.,-,,
dt: dzï lt'3 k'3
formulae determinantes. motum relativum puncti m' quoad
punctum m . Quoniam
dQ ⋯⋅∙∙∙∑∽∙∙↼∙⋅⋅≈∙↾−∙ m—I-m' .
⊋⊑⋅∙⋅−⋅⊼∙∶⊤∣ k'3 −−−⋅∎−∎ ↗⊏∙⋮∣ æ
III I'll
.. ∞⋅⋅−−⋅↕∙⇗ æ&quot; ,,, æ —x' x ≺−−⊽⋮−−−⋅−↗⋮⇁⋮⋮−≻ ⊹≖⊷ ⋯⋯≻⋅⊢ df
———— —— k'3 ∙−− 72— k&quot;3
&quot; yn ∙ yl! &quot; yon—70 ..- 70&quot;.
&quot;' (
a&quot;? ≀⊏⊤∍≻−⊦∽ ↾≺↴↼⋮⋅−∣⋮∎∎ ≀∎⊄−⋅∣∎⋅⋮≻∎⊦⋅⋅⋅∙
(19 Mi z m'x' m—l-m' zo ∙⊦
dQ my' Z mfy' m-l-m'y. ∙∙⊦
⊋∎≖∎⋅∎∎∎∎ k'3 15&quot;— ⋅∎∎∎ ↗⊏⋅⋮⇂ a'.—Z&quot; z&quot; &quot;' zIIO—zt all-l .
&quot;'&quot; ea &quot;&quot;17'5) &quot;'&quot;&quot;
«W &quot;:?75) ⊹⋅⋅⋅⋅ hinc facto
1 æoæn ⊣∙∙ o n

: , z'&quot; B: m&quot; (y'—W) ∙∙∣∎∙

1 me xlv ' '&quot; zl zh, &quot;
mm (öt—or— J—æO—ïä—L) ⊣∎∙∙∙∙ (O 2, .
formulae (o') vertentur in128
dax
de2
m -tm '
+ x's K'3
dR day '
dx ' ' de
mtm
+
k'3
g
mtm
dR daz' dR
dy' '
dit de k'3
dz
Porro , cum habeamus
ka + k &quot;? – 02
x ' x &quot; ty'y &quot; + =' z&quot; =
2
k'2 + k ' ' ?
d''2''
x' x'&quot; ty'g '&quot; +z'z' &quot; etc... ;
2
poterit (o&quot; ' ) scribi etiam in hunc modum
(
R = m k'o + k&quot; — 0° ) +
22
in '&quot; . k'2 + k '''2'''
2k &quot; 2 3* 2) + ... ( o &quot; ) .

59 * Fac at systema reducatur ad duo tantum pun
eta m et m' ; habebis R = 0 , et consequenter
der mm x
+
k'2 k'
day' mtm +
dia K2 K
&gt;
dta

* 3&quot;. d2 z' mtm

dt2 +
k'2 k
Relativus videlicet motus puncti mi quoad m proveniet
m +m: (50. 4. 20. ) a vi acceleratrice tendente ad m : pro.
k' ?
⋯⊣−⋯⋮↨↾ '
(0 ∙∎∣).
klö ,d—l;
dR dïz' m—I—m'z, dR ∙
«(y' dc2 k'3 dz'
Porro, cum habeamus
&quot; k': k&quot;: ∙∙∙ ∝↭⊹⊔↤⇥⋠−−⊦⊇ ∂∣∣≖ ⋅
-k'jl −−⊢ k'&quot;a — ö&quot;&quot; x'M* x'&quot; &quot;' z'e &quot;&quot;— 2 ∙ etc... :
poterit (o&quot;) scribi etiam in hunc modum
!, k.: kn; −∙− ux.,, ∂∜≖
.).
21./&quot;a
↿ ⋅ ra −⊦∣⊏⋯≖ −⋅∂∣∙⋅≖⋅≻ ..
∙−∂⋅−∣⋅∣∣ . ka2&quot; .l.-&quot;' (0 )- mllt
59; Fac 'ut systema 'reducaturad duo 'tantum pun-
cta m et m' ;habebis R ∙−−∶ ∘,
et consequenter
d'x' m di' '
'
−⊦⋯P',—mi −⊢⋯⋅−⊣⋤↾⋮⋡−∙≛ −−−−−∘∙ (it—T k'3 k dca k' k'
da z' ∙ ⋯⊣−⋯∣ .'
d:: [ kl; '
kl :::-&quot;'o.
Relativus videlicet motus puncti m' quoad m proveniet
(50 . 40 . 70.) a vi acceleratrice mt;&quot;. tendente ad m :pro.129
&gt; 7
pterea ( 50.13º . 140.57.50. ) describet m' motu relativo
vel parabolain , vel ellipsim , vel hyperbolam , existente foco in m .
dR dR dR 60# Secunda membra formularum
dx' ' dy' ' dz
( o &quot; ) exhibent ( 50 , 4.:) vires turbantes relativum motum
puncti m' determinatum per formulas (o &quot;) . Hinc si membra
illa manent constanter tenuissima , ita ut (o ' ') et ( o&quot;) dif
ferant in terminis exiguissimis , turbationes quoque inde
provenientes manebunt tenuissimae ; lineaque ab m descri
pla circa m poterit adhuc spectari tanquam vel parabolica ,
vel elliptica , vel hyperbolica ; ita tamen , ut gaudeat ele mentis continue mulatis .

61 * Datis tribus punctis m , m ' , m &quot; ( Fig. 35 ) ,
demissoque ex m' in mm &quot; perpendiculo m'A , sint
x' = mA , y' = m'A , X &quot; = mm &quot; , z' = 0, y = 0, z &quot; = 0.
Erit ( 58)
a' x
1
R m'
(-- = m &quot;
k3
ha( x&quot; —x'to) 2ty'a )
unde prodeunt vires distrahentes m' ab m juxta directiones
x' et y' , nimirum
dR x &quot; — x 1
dx = m &quot;
[(x&quot; — x'ja traj
. DR
dy '
m &quot;
[(x“ — x'ja + y'a ] }
Denotet h angulum m'mm &quot; , et D distantiam mm' ; erunt
x ' = D cos h , y = Dsinh , et consequenter
129
pterea (50 . 130. 14&quot; . 57 . ö&quot;.) describet m' motn relativo
vel parabolam , vel ellipsim, vel hyperbolam , existente
foco in m .
dR dR dR
dæ' , d)&quot; , dz'
(o&quot;') exhibent (50 . 40:) vires turbantes relativum motum
puncti m' determinatum per formulas (a')
.Hinc si membra
illa manent constanter tenuissima ,
ita ut (o&quot;') et (a') dif-
ferant in terminis exiguissimis , turbationes quoque inde
provenientes manebunt tenuissimae ; lineaqne ab m' descri-
pta circa n; poterit adhuc spectari tanquam vel parabolica ,
vel elliptica, vel hyperbolica; ita tamen ,
ut gaudeat ele-
mentis continue mutatis .

130
[ ( x &quot; —x ) 2 + y'r] - = ( x &quot; 2—2D.x&quot; cosh + Daj - -
mi [1+ (13–2cori)]- *

* &quot; [ - P2-2.5k)+3 . ) 6–2 cos )&quot;

- 2.7 CM) 6-2005 )'+ ]
Propterea , si D est ita parva prae aut possint omitui
termini includentes factorem exsistet
(2 )&quot;,
[cº= 2') + s] =
1
+3 D cos h
73 &quot;4
ac proinde
dR
dx = m '
3 D cos h 3 D2 cos2 h
x''3''
=
2 2:14
X

m
* +357 Dosh )
( -Dout
mi ( Doco 4-3C )*.)
China + 202 )=
2 m D cos h
dR 3 DP sin h cos h
dy.
m &quot;
24
130
3
∐∙↧⋅∥−−∙↧⋅⋅⋟≖⊹∫⋅≖⋮∣ −'i;:[£&amp;—2th cosh-l— D'] −⋅⋮∎∎ :
∙∙ 3
æ&quot; :
[130 2, äl—Zcosh)]— 7:
.'! 30 D 3.5 D): D
) a': ∣∶↿⋅−⋮⊸⋅⊋∙−⊤≺⋤−∣ 2—-cosh)-i—m (;&quot; (;,—2005&quot;
−∶≣⋅−≣−−∶≟≺⋚⋛∥−≻ ≺−⋅⋅⋅⋛⋮⊽∙−⋮∞≖↗⋮≻⋮⊣−⋅∙∙∃∙
PrOpterea, si D est ita parva prae æ&quot; ut possint omitti
. . . termini D . . includentes factorem (F) ,
exsistet
[(. ' BD-io-sh. «J)- −⊦∂∣≖∃−⋮⋅↼−↿ −−∽ ↽⊦−−−−−−
ac proinde
dR ,, a.,—a.&quot; æ&quot;—æ' 1
ævo, a'./3&quot; −⊦ [[[. ∎∎∎∎∎ II
a: .r.-3 ,,(1 Dcosh BDcosh— BD'cosïh 1)
., 2 D cOs '! (D : cos: !: 2m&quot; Dcos]:
m -—-3 −−
∙ ,, −− ( æ... .) .: .r., a
−∉⋮≹∙∙∙ ∣≺∐∘⋮∐∣⇂⊹∍∘∙∙⋮∐∣≖∞≖∣≖ −∙∙
dy —--—m ∙↿∙∙∦⋮ æ&quot;], )—131
sin h cos m&quot;
-
D sin h
3 +3 m&quot; D sin h
m 

62. Bonum erit alia ratione nonnulla hic stabilire
circa vires in praefato motu relativo . 

↿∙∘ Sint duo puncta
T , P (Fig. 36.)
, quorum massae m, m', distantia vero TP
(: k');et &quot;veniat determinanda vis acceleratrix in motu
relativo puncti P quoad T . Ex hypothesi P tendit in T vi
I
acceleratrice . m . . m

; — ;et T in P v
: acceleratrice ∙−−∣−−≖ sive au.

] 3
I
tem T sollicitetur .. vn. m . m .
−− ∣−⋮∣−≖− et Pv: 17; , sive T quiescat et P
I
sollicitetur vili—ïm]— &amp;, idem in utroque casu (5) habetur
motus relativus puncti P quoad T; vis ergo acceleratrix
in istiusmodi motu erit

2.&quot; Praeter P , T detur et tertium punctum S ,
cuius massa m&quot; , ut determinentur vires iude provenien-
tes, quibus turbatur motus relativus puncti PquoadT
ortus ex vi (0) . Ducta ST
, completoque parallelogrammo
.. STPP' , exhibeat diagonalis SP (: 8&quot;) vim g.: , qua sol-
licitatur P versus S:resolvatur vis ista in duas, quarum al-
tera (: ?') sese dirigat iuxta PT , altera (:f) iuxta PP';
exhibebitur illa (8) per parallelogrammi latus PT (: k')
.
haec per latus PP':ST (: k&quot;); eritque '
m&quot; .,

; n ⇀
: m&quot; IC, '

m&quot; k, ≒≀−∣−⋮∙⊊≱⋅∙∣⇆∶∂ ∶∣⊄∙ ]; ,unde 93:77sz &quot;3 ⋅132
m' '
m &quot;
Sollicitatur T versus S vi ; et attentis f et i motus k''2''
relativus puncti P quoad T eodem prorsus modo fiet ( 5 )
sive T quiescat et P sollicitetur vi f
m '
sive T sollicite k'2
m &quot;
tur vi et P vi f. Propterea vires provenientes ex S ,
et perturbantes motum relativum puncti P quoad T , al
tera juxta PT altera juxta PP' parallelam rectae ST , ex
primentur per
k &quot; 2
ø=73
m &quot; k
&quot;
g = f mi&quot;
&quot; Cess )
( c' ) . k's
3.° Ex puncto S demittatur perpendiculum SS'
( =i) in planum curvae , quam describit P motu relati
vo quoad T; ab S ad T ducatur recta ST ( =n ) , sitque
angulus STP = a : vis q&quot; agens juxta directionem paralle.
lam rectae ST resolvetur in duas, quarum altera q&quot;cosSTS
seu q &quot; . ! existet parallela rectae ST in plano cur
vae , altera q &quot; sinSTS' seu o&quot; . perpendicularis eidem pla k &quot;
resolvetur in duas quarum altera o &quot;
no: rursus onk
cos a
aget in curvae plano juxta TP , altera om. sina in eo k &quot;
dem plano normaliter ad TP. His positis , quisque in
telligit vires perturbantes motum relativum puncti P exhiberi
posse per
132
SollicitaturT versus S vi &quot;'

; et attentis f et −∥↼↕−∙ -, motus 1.&quot;» 1.&quot;

relativus puncti Pquoad T eodem prorsus modo fiet (5)
sive T quiescat et P sollicitetur vi f— 'I—N. ,
∣∣≖ sive T sollicite/-
et P vi f. Propterea vires provenientes ex S ,
∙ m tur '! k&quot;-
et perturbant'es motum relativum puncti P quoad T , al-
tera juxta PT altera juxta PP' parallelam rectae ST, ex-
primentur per
' mllko &quot; &quot;zl! &quot; kl! 1 '
Pf&quot;??- ⊕−−⇌↾−−⊺⋇⊽≏∶⋯ (Fa-&quot;' ia&quot;) &quot;'

3.&quot; Ex puncto S demittatur perpendiculum SS'
(::t') in planum curvae , quam describit P motu relati-
vo quoad T; ab S' ad T ducatur recta S'T (::n)
, sitque
angulus S'TPr-at: vis 9&quot; agens juxta directionem paralle-
lam rectae ST resolvetur in duas, quarum altera 9&quot;cosST5'
seu 9&quot;? existet parallela rectae S'T in plano cur-
vae, altera 9&quot;sinSTS' seu q;&quot;.grperpendicularis eidem pla-
no: rursus ?&quot;]?- resolvetur in duas quarum altera ⊄∙⊅⋅⋅∙⋮∙− eos ::
aget in curvae plano iuxta TP , altera ?&quot;)—;.sinat in eo-
dem plano normaliter ad TP. His positis, quisque. in? telligit vires perturbautes motum relativum puncti P exhiberi
posse per133
COS Q = cosa ,
9 =porn o--&quot; (* - ) .com
Pa = e&quot; sin æ =
= m
m &quot; (- ) snæ ,
93 = ml - )
( c )
i

;

9 , et Q2 agentes in curvae seu orbitae plano ipsam orbi
tam turbant ; 93 perpendicularis plano orbitae turbat
ipsius plani positionem .
4. ° Pone S , T, P esse constanter in uno eo
demque plario ; erunt i = 0 , n=k&quot;, a=S'TP=STP(=h) :
proinde
PI
m &quot;
8'3 -m&quot;( )cosh ,
&quot; sink ,
}
( cm)
Q2 , 93 = 0 .
Pone insuper ST, SP ita magnas prae TP ut , ex P du
clo perpendiculo PQ in ST, assumi possit absque sensi
bili errore SP=SQ , nimirum d&quot; = k&quot; -kcosh ; erit
1
js =(k“&quot; —k'cosh)-3 = 13 +
3k'cosh
+
Hinc proxime
m &quot;
m'k ( 1-3cos'h= ( 1 +3cos2h) ,
k &quot; 3 2K3
( c&quot; )
3m''K'sinhcosh 3m'k'sin2h''
92 k&quot;3
2k'3
133

: n&quot; muli, &quot; k&quot; ! n 913? —Q ?

eos a:
ïïï —m 673 k,,a k,, 0082,
.&quot;
k ⋅
,
93——9 &quot;,;— Blna :m &quot;(ä-3- It.—741) ,——,- sin a ,- (c)
]. LII-. [
i
93:907?sz −−⋮ ⇁≖⊼↗ ;
4). et (p, agentes in curvae seu orbitae plano ipsam orhi-
tam turbant; (pg perpendicularis plano orbitae turbat
ipsius plani positionem.
4.&quot; Pone S, T, P esse constanter in uno eo-
demque platfo; erunt i:o, n.:k&quot;, a:S'TP:STP(:h):
proinde
mrlk' &quot; kn
'
(Pr −−∶ 7873- −−⋅ m
⊱∣−⊵∙−− F,.)COSII,
(e&quot;)
?::m&quot;≣∶⋅⋅⋮∙−⋅ -—k,,,)smh , 93:30.
Pone insuper ST, SP ita magnas prae TP ut. ex P du;
cto perpendiculo PQ in ST, assumi possit absque sensi-
hili errore SP:SQ , nimirum d&quot;:k&quot;—k'cosh ; erit
1
I,
, −∙∙ BkCOBh ∙≦↜−∽⋮⋅−−−−−≺↗⊏ —kcosh) ∍≔−−↼−−⊺⋮−−∣−−−−
k&quot;, −⊦
,
∙∙
Hinc proxime
∙ ?: 2773— (1—3cosïh):— (1-1—3c092h) ,
z—kHS .
(e&quot;)
—3m' &quot;ksinhcosh —3m &quot;k sinZh134
5,9 Fac ut orbita puncti P sit circularis , ipsum .
que P moveatur ad partes N : sive spectentur formulae
( 6 ') , sive (6 &quot;) , sive ( c &quot; ), aget 92 juxta orbitae tangen
tem contra motus directionem : ejus proinde valori erit
praefigendum signum negativum.
=== De pendulis; deque gravium descensu per arcus cycloidales. ===
[[Fasciculus:Simple pendulum generalized coordinates.svg|thumb|Pendulum]]
[[Fasciculus:Pendulum simplicium.svg|thumb]]
63. Pendulum constat filo tenui secundum alteram sui extremitatem fixo, quod tamquam linea recta et gravitatis expers concipitur, ex quo suspensum punctum ponderosum a directione verticali dimotum potest huc et illuc circum punctum illud alterum extremum fixum in motum circinationis per arcum excurrere. Excursio penduli ab uno arcus, quem describit, extremo &lt;math&gt;C&lt;/math&gt; (Fig. 37) ad aliud extremum &lt;math&gt;D&lt;/math&gt; dicitur &lt;u&gt;vibratio&lt;/u&gt; seu &lt;u&gt;oscillatio&lt;/u&gt;: accessus ad verticalem directionem ex &lt;math&gt;C&lt;/math&gt; in punctum infimum &lt;math&gt;B&lt;/math&gt;, vel recessus ex &lt;math&gt;B&lt;/math&gt; in &lt;math&gt;D,&lt;/math&gt; dicitur semivibratio. Si unicum ponderosum punctum pendeat e filo, pendulum dicitur simplex, si plura in diversa a suspensionis puncto distantia pendeant, dicitur compositum.  
[[Fasciculus:Pendulo simples.jpg|thumb]]
Illud facile quisque intelligit, pendulum &lt;math&gt;AB&lt;/math&gt; circa punctum fixum &lt;math&gt;A&lt;/math&gt; eodem motu arcum circuli &lt;math&gt;CBD&lt;/math&gt; descripturum ac si, sublato filo, in superficie sphaerica perfecte dura et levigata punctum ponderosum moveretur motu impedito. Sicut enim adducto puncto illo ad praedictae superficiei punctum &lt;math&gt;C&lt;/math&gt;, et exinde demisso, gravitas &lt;math&gt;CT&lt;/math&gt; horizonti perpendicularis &lt;u&gt;resolveretur&lt;/u&gt; in duas vires, quarum altera &lt;math&gt;CE&lt;/math&gt; ad tangentem &lt;math&gt;CG&lt;/math&gt; normalis insumeretur in premenda superficie, altera expressa ab ipsa &lt;math&gt;CG&lt;/math&gt; sollicitaret punctum ponderosum ad motum per tangentem infinite parvam, ac deinde per aliam atque aliam subsequentem, et sic deinceps per reliquas omnes numero infinitas et infinite parvas tangentes, quibus constare arcus descriplus concipitur; ita a filo resolvetar gravitas eodem prorsus modo , nempe partim in trahendo filo insumpta, partiin ad singulas arcus circularis tangentes infinite parvas subinde determinata, qua deducetur pendulum per arcum circularem motu omnino simili, subeunte filo &lt;math&gt;AG&lt;/math&gt; vices curvilineae superficiei: hinc sicuti punctum illud ponderosum propter suam gravitatem, postquam descendisset ex &lt;math&gt;C&lt;/math&gt; in &lt;math&gt;B&lt;/math&gt;, cogeretur ascendere ex &lt;math&gt;B&lt;/math&gt; versus &lt;math&gt;D&lt;/math&gt;, ita ob rationem similem pendulum post descensum ex &lt;math&gt;C&lt;/math&gt; in &lt;math&gt;B&lt;/math&gt; ascendet ex &lt;math&gt;B&lt;/math&gt; versus &lt;math&gt;D&lt;/math&gt;. Rursus quemadmodum ponderosum punctum in praedicta superficie ascendere inciperet per arcum &lt;math&gt;BD&lt;/math&gt; cum eadem velocitate, quam acquisivisset in puncto infimo &lt;math&gt;B&lt;/math&gt;, et ideo ad eamdem altitudinem, ex qua descendisset, perveniret, nempe usque in &lt;math&gt;D&lt;/math&gt;, ubi extincta omni velocitate, iterum gravitate sua inciperet descendere, et in puncto &lt;math&gt;B&lt;/math&gt; priori velocitate rursus acquisita, cum ea ascenderet iterum in &lt;math&gt;C&lt;/math&gt;, atque ita porro ascendendo et descendendo perpetuas et aequalęs in peripheria &lt;math&gt;CBD&lt;/math&gt; excursiones perficeret, ita ob eamdem rationem penduli oscillationes aequales essent natura sua et perpetuo duraturae, nisi ab aeris &lt;u&gt;resistentia&lt;/u&gt; et &lt;u&gt;frictione&lt;/u&gt; aliqua circa sustentationis punctum &lt;math&gt;A&lt;/math&gt; inaequales primo redderentur, ac denique extinguereatur; adimentibus scilicet ejusmodi causis in singulis oscillationibus aliquid de illa velocitate, quae producitur a gravitate.

64. Velocitates &lt;math&gt;v&lt;/math&gt; et &lt;math&gt;v'&lt;/math&gt; in puncto infimo B acquisitae a gravibus per arcus &lt;math&gt;CB, C'B&lt;/math&gt; descendentibus sunt ut ipsorum arcuum chordae. 

Per &lt;math&gt;B&lt;/math&gt; concipiamus duci tangentem et in eam ex &lt;math&gt;C&lt;/math&gt; et &lt;math&gt;C'&lt;/math&gt; demitti perpendicula &lt;math&gt;z&lt;/math&gt; et &lt;math&gt;z'&lt;/math&gt;: denotante &lt;math&gt;r&lt;/math&gt; radium &lt;math&gt;AB&lt;/math&gt; et denotantibus &lt;math&gt;k, k'&lt;/math&gt; arcus quoad radium 1 similes ''arcubus'' &lt;math&gt;CB, CB'&lt;/math&gt;, erunt 

&lt;math&gt;z = r ( 1 - \cos k ) , z' = r (1 - \cos k' ) ; &lt;/math&gt; 

et quoniam (30: 36)

&lt;math&gt;v^2 = 2gz, v'^2= 2gz';&lt;/math&gt;

propterea

&lt;math&gt;v: v' = \sqrt{2gr (1 - \cos k )} : \sqrt{2gr (1 - \cos k')} = \sin \frac{k}{2} : \sin \frac{k'}{2};&lt;/math&gt;

ideoque etc.

65. Pendulum, quod incipit descendere ex &lt;math&gt;C&lt;/math&gt;, percurrat arcum &lt;math&gt;CM&lt;/math&gt; tempore &lt;math&gt;t&lt;/math&gt;; sitque &lt;math&gt;\alpha&lt;/math&gt; arcus quoad radium 1 similis arcui &lt;math&gt;BM&lt;/math&gt;: erunt &lt;math&gt;CB =rk, BM = r\alpha&lt;/math&gt;; et designante &lt;math&gt;u&lt;/math&gt;velocitatem in puncto &lt;math&gt;M&lt;/math&gt;, exsistet

&lt;math&gt;u = - 2gr (\cos\alpha - \cos k ) = 4gr \sin\frac{k+\alpha}{2} \sin \frac{k-\alpha}{2}.&lt;/math&gt;

Si arcus &lt;math&gt;k&lt;/math&gt; est ita exiguus, ut possit absque sensibili errore substitui respondenti sinui, habebimus
&lt;math&gt;u^2 = gr(k^2 -\alpha^2),&lt;/math&gt; et consequenter (28)

&lt;math&gt;\frac{ds^2}{dt^2} = gr(k^2 -\alpha^2),&lt;/math&gt; 

unde

&lt;math&gt;dt = \frac{ds}{\sqrt{gr(k^2 -\alpha^2)}} = \frac{r\beta}{\sqrt{gr(k^2 -\alpha^2)}}= \frac{\beta}{\sqrt{\frac gr (k^2 -\alpha^2)}};&lt;/math&gt; 

&lt;math&gt;\beta&lt;/math&gt; est arcus quoad radium 1 similis arcui infinitesimo Mm
( = ds ). Nunc centro H ( Fig. 38) et radio HD ( = k) describe circulum DED' ; sume HN : Ν »
B; duc perpen
dicula Ne, ne super HD: et Ey parallelam radio HD. Trian
gula similia HEN, Eey rectangula in N, y praebent

∙∙∙⇀ ,4þf - ⇀∙⋅∙∎∙ .. ⊸∙⋅⋅⋅∙∎∎∣∙ 4.-
∙−
..137
Ey: EN = Ee: HE, seu B: V R2-42 = Ee: k :
hinc
B
Ee 8
dt ;
V R2-42 k
et consequenter
Ee dt
kV
tempusculum videlicet dt impensum ad percurrendum
seu Nn, obtinetur dividendo respondentem arcum Ee per
kV § . Inferimus tempus t impensum ad percurren
dum ka seu ND, obtineri dividendo respondentem ar
cum ED per kV ; nimirum
ED
자
름
k
Quare
VED –are(com);
ideoque
Vare(cový = ) &lt;( a ) .
10
137
Ey:EN :Ee: HE, seu ,8: V kï-aï: Ee: h:
hinc
∙
!
Ee .... B ∙∸−⋅⋅ :: Vii. dt ;
VIR-æ ]: '
r
et consequenter
⋅↙≀↥∶∎∙−−⋮∶∶⇣∶⋮ *Ve-
tempusculum videlicet dt impensum ad percurrendam þ
seu Nn, obtinetur dividendo respondentem arcum Ee per
]; Vi .
Inferimus, tempus :
impensum ad percurren-
'.
dum ]:-
ut seu ND, obtineri dividendo respondentem arcum ED per kI/ £.; nimirum
r
t ∙∙∙ ED
l.V-f,- .
Quare ∙
.yz... −∙−⊡∍ ......
«
.
rf- k —- (eos:.k),
ideoque
≀⇌⇂∕∑− arc (eo: &amp;) (a) ∙
5
10138
Iam vero in puncto infimo B (Fig. 37) exsistit a = 0 ;
erit igitur tempus semioscillationis
TT
ti
V
2
8
tempus integrae oscillationis ( a ' ).
t2 =
V
quas formulas cum non ingrediatur initialis angulus k, patet oscillationes ejusdem penduli, vel plurium pendulorum aequalis longitudinis r per arcus satis exiguos utcumque ceteroquin inaequales, fore ad sensum isochronas seu aequi
diuturnas.
Idipsum facile demonstratur hac alia ratione: angulus GCT
= 90° BAC; hinc vis acceleratrix CG , ex qua sola repetendus est penduli descensus, exhibebitur per gsing: in hypothesi nimirum arcuum satis exiguorum spectari poterit CG tamquam proportionalis distantiae a puncto infimo B, computatae in arcu BC. Ergo ( 29. 4°) etc.... Etiam sic: est ds = d rík - a ) rda ;
et consequenter
rda da
dt
V rg (k -u?)
-Vivok²-u?
factaque integratione ( 27. 13º. 14° ) prius ab a kad
a =0 , dein ab a = k ad a = -k, emergent binae (a' ) .

66. Haec notentur: 

1º: secunda ( a' ) dat
77 r
8 ( a '');''
ta
atque inde innotescit gravitas g.
138
Iam vero in puncto intimo B (Fig. 37) exsistit «:o;
erit igitur tempns semioscillationis. &quot;
∙−∣
⋍⋅∶−−−−⇄⋅−∣∕−≦−∙
tempus integrae oscillationis (,,-)
−∣−∙−
ta:T! −∙− ∣∕ :
,
quas formulas cum non ingrediatur initialis angulus k, patet oscillationes ejusdem penduli, vel plurium pendulorum
aequalis longitudinis :- per arcus satis exiguos utcumque
ceteroquin inaequales, fore ad sensum isochronas seu aequi-
diuturnas-
Idipsum facile demonstratur hac alia ratione: angulus GCT

: 900— BAC; hinc vis acceleratrix CG, ex qua sola re-

petendus est penduli descensus, exhibebitur per gsinat: in
hypothesi nimirum arcuum satis exiguorum spectari poterit
XCG tamquam proportionalis distantiae a puncto infimo B.
computatae in arcu BC. Ergo (29. 40) etc.
...
Etiam sic: est ds:dr(k— a):
-— rda;
et consequenter
rdat dat
dt −∙− ↵
−− '&quot;
VrgUe-æ .? sz-az :
factaque integratione (27. 130. 140 ) prius ab et: I: ad
«:o, dein 'ab a: ∙−−− I, ad ac ∶−−⋅ —-k, emergent binae (a')-

2º. Etsi ponderosa diversae materiei puncta permissa sunt oscillare, attamen idem semper prodiit valor g in eodem terrae loco: rursus ( 17 ) igitur devenitur ad proportionalitatem inter corporum massas et respondentes gravitatis vires.

3º. Constat observationibus longitudinem penduli
simplicis oscillationem absolventis intra mioutum secundum
eo esse minorem, quo magis ad aequatorem acceditur: quoniam ergo, haud variato tz, gravitas est ut longitudo illa, minuetur gravitas a polo ad aequatorem usque ( 30) .

4º. Apud nostras regiones praefata penduli longitudo cum sit = 3ped opol glin, 38 = 440lin, 38, factis
in ( a &quot; ) tz = 1 &quot;, r = 440lin , 38, prodibit respondens gravitatis valor g = 30ped , 183 alibi (30) indicatus.
[[Fasciculus:Double-Pendulum.svg|thumb]]
67. Quod spectat ad pendulum compositum concipiamus (Fig. 39) puncta ponderosa B, B. , B2 , . . filo appensa: invicem disjuncta conficerent haec puncta temporibus inaequalibus oscillationes suas; punctum nempe B, citius  (66) quam B, punctum B, citius quam B, etc: invicem ergo conjuncta agent ita in se mutuo, ut quae, minus distant a puncto suspensionis A retardentur ab iis quae magis distant, et quae magis distant a suspensionis puncto accelerentur ab iis quae minus distant: fiet propterea oscillatio penduli compositi tempore quodam medio inter minimum ac maximum praedictorum temporum inaequalium. Hinc sequitur fore in AB punctum quoddam B.,m suas conficiens oscillationes perinde ac esset solitarium, nulloque nexu caeteris punctis uui
retor: Bm dicitur centrum oscillationis, cujus centri distantia a puncto suspensionis est longitudo penduli simplicis suas perficientis oscillationes eodem tempore ac pendulum compositum. Inferimus oscillationes pendali compositi, et ipsas fore isochronas; modo tamen exsistant satis exiguae.

68*. Facile intelligimus ( 50. 3º. 6° : 66 ) motum penduli simplicis in medio resistente determinari generatim per aequationem

140
das
di?
= gsing -f(v ) .
derka( )
di Ob dc2
dra
dt2 et ( 27. 29º . ) sing
23
2.3 +
aequatio illa eyadit
creat + s ( « - +...)-fo) =
Pone fv) = cv ; et angulum a ita perseveranter exiguum,
ut ejus tertia potentia absque sensibili errore negligi pos
sit : habebis
da с
+
dla
ola
0 .
ds Est autem v =
dt
drak -a)
dt
da
dt ; igitur
d2C da
dt2 to dt + baro:
quam integrantes in hypothesi c constantis assequemur( 27.270. )
... [ :V546.-V21
( 6),
In experimentis, quae pendulorum ope solent institui ,
r est multo minor quam g; item densitas penduli, et con
sequenter ( 33 ) c fractio admodum parva. Fac ergo
140
tiis ∙
de'-* :gsmat —f(v)
d': d2r(k—a) (lioc
↽⇁−−− −∙−
' Ob daz dtz 27. 290. .:
rdtï
, et ( ) stna
'
a3 . '11 d'
rdzatdta-l-g(at—-— ...):fþv) :0.
Pone f(v): cv; et angulum ac ita perseveranter exiguum,
ut ejus tertia potentia absque sensibili errore negligi pos-
sit: habebis
dza g c
dia—FTa—Tv—o
.Et :
−−∠≀≖∙∙↙≀↗≺∣≖⋅⋅∝≻∙− ∎∠≀∘⊏∙⋮ ⋯⋅⋅ s auem'v— dt— dt rd , g1
quam integrantes in hypothesi c constantis assequemur(2 7.2 70.)
&quot; til &quot;'i-i.]c) —l ! —-..—.
2:32 [CG 4
r—l—C'f—B
ln expetimentis, quae pendulorum Ope solent institui,
r est multo minor quam g; item densitas penduli, et con-
sequenter (33) c fractio admodum parva. Fac ergo141
VS
ut sit VA = iVT ;
4 .
vertetur ( 6) in
ti V = 1 -til
+C'e 2 -
&quot; ]
U = e
seu ( 27. 30° )
3 [
e
] ;
C &quot; sin it +C ' ' cosit
unde
cosit
data o - [( c'i - )
(c&quot;: + * ) ainit ]
da
In joitio motus t=0 , a= k , 0 ;
di
propterea C &quot; &quot; k, C '
ck
21 ; et .
= ke - - [
sin it + cosit ] . )
2i
( 6 )
dan
dt = -ke- Ź [ ita sinic.
141
'
. . ∙
c' a . .
ä-ï': - utsit V—--€- :::tl/ −−↿ ∙
r 4 4 :-
vertetur (b) in
−−−∘⋮−≀ — tiV—1 -til/—1
at:6 2
)
C'0 v
380 ( 270 300 )
↴
c ↼
a:
∘−−≖− '[C&quot; sinit ⊣−∁⋯ cosit] ;
unde
∘⋮ C'&quot;
↙⋮⋮∶∶ .
∘∙⋅−
; t [( C'i— 20) cosa
: -— dt ∙

Cnc . .
(
0&quot; i −⊢ —2 )
sunt ].
dat ∙∙∙ ∙
In initio motus t—:o
, at:— ]:
, &quot;dt −∙− 0-
∙∙∙ k
': propterea C Ck et −−−∙−− ∙
∙−− 21' ∙142
с
Ex.Vihabemus zi 11
ll
Hlacin
c2
V C
it =-V
rc?
4g &gt;
2V EV
rc2 1
48
1
i +
VE ;factoigitur V
cr2 =c,
43
&quot; Vi ro2
4g
binae ( 6 ) sic poterunt exprimi
a = ke

* IVE Vētowi.V ]

1.)
(6 )
dm-- .- iv E sinórV.
In fine cujusque oscillationis est da
dt = 0; proinde, ob
= 0: inferimus in fine primae secundam (3&quot;),since V
12 풍VV
2.V
oscillationis fore t = &gt; in fine secundae
8
271 376 in fine tertiae t =T
8
8
gulae itaque oscillationes absolventur aequali tempore
E ,
in
V
, elc.. .. ; sin . с
g. ∘ Ex ::i habemus 21. ∶∶ − c —- .
∙
∙−−−−−−−
I/ ⇂∕−−−⋅↿rc
c ⋅ :! −⋮↓−≔⋤⋅ a
cr&quot; ,
. CZ .g 1;factoigilur V1—-- :c, ↓⊣− ' ⋅ −−∶ ⋅
binae ( b') sic poterunt exprimi
£(sz 2 inc't cosc' g
c[hc V—s Vg ∙−⊢ ::ll/:]
(ö&quot;)
(E;—..., &quot;&quot;T-V—sinc't 5-
dt :-
/
In fine cuiusque oscillationis est ≤−∝⋮∶∶∘⋮ proinde, ob
secundum (b&quot;), sinc': Vi:o: inferimus in fine primae
!'
1:
osc1llat10n1s ∙ ∙ ∙ r ∙ fore :

:

−⋮⇆⊤ V—3,
in fine secundae
271 .
⋅⊤ −∙
,
in fine tertiae :
−−−−−∣−
−− ,etc.. -sm-
gulaec itaque goscillationes absolventur aeqnali tempore143
و
=
ا
( 6 '' ) .''
8
In primo substitutis valoribus 0, 20 , 30 , ... no
pro t , emerge
Qu - ke 2 A2 = ke -
&gt;
929
as= -ke- 30
Q. = 1–1 y&quot; ke – no
hinc successivarum oscillationum amplitudines
0
2
음
k + ke ke - 9 +ke - 2
-ke 39 -
2/2
20
the
ke
seu
1(1 +-2), ( +-3, -2,

* (170-99 . -Ź- 42329......

Decrescunt igitur amplitudines istae in progressione
geometrica : quod cum confirmatum sil experimentis pen
dulorum in aere oscillantium per arcus satis exiguos, haud
majores v . g. tertia parte unius gradus, licebit quoad e
jusmodi oscillationes assumere aeris resistentiam tanquam
proportionalein simplici velocitati.
143
n −−−∽−∣∕∙⊂− (B&quot;)- 0

6
In prim: (. ⋮⋅ substitutis valoribus 9, 29 , 39 , ... 719
pro :, emerge f
⋅∙⇁− -
∙
1- 0
29 ⋅ C
a;:— ke 2 ,agzke 2 ,a3z—ke 2 39
C
119 a.::(—1)&quot; ke −⋮⋅ :
hinc successivarum oscillationum amplitudines
.
c ⋅∙⋮∙∙ ...—£
k-l—Re— i.e.]:e— 294-ke ∙
229
&quot;
ke −⋅⋮⋅⋮∂−⊦∣∥⋝ 2
39,. ;
seu
.- e ∘
9
−−∘ &amp;
Decrescunt igitur amplitudines istae in progressione
geometrica : quod cum confirmatum sit experimentis pen-
dulorum in aere oscillantium per arcus satis exiguos, haud
maiores v. g. tertia parte unius gradus, licebit quoad e-
jusmodi oscillationes assumere aeris resistentiam tanqnam
proportionalem simplici velocitati.144
Experientia insuper docet decrementum illud gradu
admodum lento procedere : sic D. Borda expertas est non
nisi post 1800 oscillationes valorem en converti in k .
Hoc posito, existet
2
1
1800c7V
.18000 e 2
2 2c seu ob (6 ' ') e
3اندبه
et consequenler
1800CTE
= c'log. 13 == c (o , 40546) . 2
( 1800)? c ?72. &quot;
= 1 - c '? , ideoque 8
4g
ro2 Sed
4
= (1800 ) 2772 /1— 2) ; igitur ( 1800)277 ?(1 — c'2)
= c (0 , 40546 ) ; unde
c' ?
( 1800 )2772 1
(1800 )222 I.(0,40546 )
&gt; et e
V
+18007
1+
0,4054612
180076
1
quam proxime.
gua
Si ( 33 , 4.&quot; )
poneretur f (v )
terminandum exsisteret
ad penduli motum de
144 *
Experientia insuper docet decrementum illud gradu
admodum lento procedam : sic D. Borda expertus est non-
. . , 1118! ∙ ⋅ ∙ ∙ 2 post 1800 osmllattones valorem ac,, com-cru mes-k.
Ilnc posito, existet
'
u
1eöocn VL
..
'
5.
20'
≀∙ ⋅
.
...—18009 2
e 2 −−∙−−− −∙− , seu 01) (b&quot;') e
⊏⋅⋮− ,
. 5 ⇁
3
(!l. 000 sequenter

: .—c'(o ∙⇁ '40546) b. &quot; .

1800072l/L .
2 g :c'lo
'.
(1800≻∖⋅:0:712.— g
PC: '
∙
'
Sed −−−−−↿ ---c2
, 1deoque 45
⇌≺↿⋅∂∘⊙≻≃⊺≖≖≺↿−⋅≺∶∣≏≻⇋ igitur ≺↿∂⊙∘≻⇄∏≖≺↿−∘∣≖≻

: c'5(o, 4054673; unde

(1800)??? - -, 1
, ↼−−
≺↿ et -c':
≨∃⊙∘⋟≖⇃∙≖⋍⊣−⋅⊏∘∙∠↥∘⋦∢⊖≻ ↼↼
'3
Vl—l&quot;(7'g55; o.4(l546)z
≖≖−−−↿ quam proxime.
↓
2 '
51 (33. 41.&quot;) poneretur f(v) ∶−−⊸∙⋮⊥⋮− , ad penduli motum de-
02
termiuandum exsisteret145
des
dt?
gsing gu2 d ?
seu de2 + sine —
8r /dala
2
= 0 .
c²lde
Haec prias multiplicata per 2du , ac dein integrata suppe
ditat
' dala Idala
ca ldt
2g
COS O
seu facto Slaa) dx = y , ideoque Coupe ( )
dy
da
dy
28
2gr COS Q
da y = 0 ;
cojas integratio traducitur ( 27. 26 °.) ad integrationem fun
ctionis
2g cosada
2gra
c2 re
Jamvero , facto compendii causa 2gr
= m , habemus c2
dem sina ) coso, da Se
ma
-mu m e sing da ,
d ( e-ma cosa )
-ma sina da - me COSQ da :
igitur
ſe-ma cosa da e -ma sina tm se-ma siac da ,
ſe-ma sina du = me-ma cosa m ſe-me. cosa da ;
ex quibus
145
d's— ∙ g.&quot;: dia g ⋅⋅ . gr
äzäfgsma— Z;- , sendt: da)!— -[-r sma 02 22 —--0.
Haec prius multiplicata per Zda , ac dein integram suppe-
ditat -
(&amp;)2— dt ∙−⊋∊∁∘≘∝−⋅⋮⋚⊆∫≺≦− r
f:) ↙≀⊄∶∘∙∙∙
dat ∙∙∙ ∙ da: a... 47&quot;
seu facto f(ä—t) fia —J—, ,
1deoque (22) −∙− ä; ,
≝⊻−≟≝∁∘⊱⊄≉≣≝⋅∫∶∘⇋ . dat
cuius integratio traducitur (27- 262) ad integrationem fun-
ctionis
Zg cosadat ∙
⋅
2grat .
∘≖ . re :
∙
2 r
'
Iamvero ,facto compendii causa −⋚−⋅:m ,habemus
02
,! (.;-&quot;&quot;&quot; since ):e'ma cosa: d-a −∙∙ m e'm' sinat da
,
d(e'macosat):-e'm ∋⋮∘⊄↙∄∝−⋅⊪∘⋅⋯ &quot;cosada:
igitur
fe'm. cosa da::
efm sinat—lfm fe'm sinat dat
,
fe'm sinat daz ⋅⋅−−− −−∶ e'm cosa: — m fe'm- cosa daz ,-
ex quibus146
ſen-Ma cosa da e -ma sina — те cosa -m2ſe-mecosadu 7,
et consequenter
2g
Sce-ma cosa da
2g ( e -ma since me- mu cosa)
r ( 1 + m2)
Erit itaque (27. 26 °.)
y = Cema +
2g ( sing - m cosa )
r ( 1 +ma )
ex qua differentiata quoad &amp; cum emergat
dy
da
Cmema + 2g (cosx + m sina )
r ( 1 +m2)
restituto valore dy
da habebimus
ca
dal 2g (cosa + m sina)
= - Cmema +
r ( 1 +m2 )
da
In initio motus a = k , = 0 ; hinc dt
Cm 2g ( cosk + m sink) e-mk
r ( 1 + m2)
propterea
-m (k - a )
Cate)
dal 2
ldt
29
r ( 1 +m2)
cosa + msina - cosk + msink)e ( h).
Facto a = o in ( h) , prodibit inde velocitas penduli in pun
cto infimo B ( Fig. 37.) : ascendet pendulum cum velocitate
146
fe'm&quot; cosa da: e'm&quot; sinat —me*mcosoc −∙∙ ⋯≖∫∘⋅⋅⊪∞∽∡∠≀∝ ∙
et consequenter
2g &quot; 2g (e-&quot;W- sinat −∙∙ rne-ma cosa)
∙−∙∣ (.'-'&quot;&quot; cosa dat:
⋅
'
r(1 −⊢ ⋯⋅≀≽ Erit itaque (27 . 260.)
2g (sinat —m cosa)
y.:Cama'i' r(1-l—m') :
ex qua differentiam quoad a: cum emergat
dy ∙− M Zg (com-[- m 5213.)-
da :Cme r (1 —f-m3) '
∙
d
' ∙ resututo valore 1, babebmus ⋅
de:
((!—S :Cma'&quot; &quot;I&quot; 25 (cosa: ∙⊢ 11: sind:? ∙
.. dt r(1-l-m3)
∙ ∙ ∙
'
da ∙
In 1n1t1o motus a:k −∙− −−−−∙ o ; bmc 'dt
2g (cosk −↿− m sinit) er:-&quot;&quot;* ∙
Cm: r(1—l-m2) ..
propterea
(de!)2 28 &quot;[COSa-Hnsina—(cosk-i'msïnk) e-m(k-a)] 32 :r(1—i—m2) (73)-
Facto a:o in (I:), prodibit inde velocitas penduli in pun-
cto infimo B (Fig. 37.) :ascendet pendulum cum velocitate147
ista versus D , conficietque arcum , cui respondebit — Q,; et
quoniam in extremo puncto illius arcus extinguitur tota ve
locitas , iccirco
COS - m sina, · ( cosk + m sink) e -m (4+ 1) = 0 ,
seu
(cosa,
m sing , emai (cosk +m sink) e-m * = 0
( h ' ).
...
mk .
maa ,
Sunt ( 27. 29.° ) emas = 1 + m « . + +
2
mak ?
=1 -mk+ -... ; est insuper m fractio admodum
parva ( 33) : neglectis igitur terminis , ubi invenitur mº ,
traducetur ( h ) ad
2
&gt;
cos@g - m (sina, cosax) = coskt m (sinkkcosk ) (h &quot; ).
Denolante o differentiam inter valores a, et k ut sit Q= k -0 ,
certe ð erit fractio tenuissima : hinc substituto k- loco
Qy in ( &quot; ) , sumpto 1 pro cosd et à pro sind , missisque
öz et mo , assequemur
2m
Osink = 2m ( sink - kcosk) , d = 0
sink (sipk- kcosk ) ;
unde
Uy= h
2m
(sink-kcosk) .
sin k
Si popimus k ita iguum , ut ejus quarta potentia prae
termitti possit , obtinebimus (27. 29.° )
147
ista versus D, &quot;conficietque arcum , cui respondebit — at,; et
quoniam in extremo puncto illius arcus extinguitur tota ve-
locitas
,
iccirco
111 cosa:, −∙∙ m sinatl — (cosk −∣− m sink) e&quot; (b'-3 1): o ,
seu
(cosa, ∙− 11: sind,) a'&quot;! — (cosk —-msink)e&quot;&quot;'* :: o (b').
Sunt (27.29.0) erat: mna? ↿−⊦⋯⊄≖−⊦
2 −⊦ ∙ ∙ ∙
∙ ,∙⋯⋆
⋯≖⇂∙∙≖ ⋅ ∙ ∙ ⋅

: i —mk—l—-—2—-— ...; est 1nsuper m fract1o admodum ∙

parva (33): neglectis igitur terminis ,.
ubi invenitur m',
traducetur (h') ad
tuom,—m (si na,—aleam,:cosk—l-müi nk—kcosk) (h&quot;).
Deuotante ö differentiam inter valores a, et k ut sit ac,: k-ö.
certe d erit fractio tenuissima : hinc substituto k—ö loco
a, in (II&quot;), sumpto 1 pro cosd et 6 pro sind
. missisque
d' et md , asscquemur
ösinkz2m (siuk—kcosk) , ö: ET- (sink—kcosk);
. sink
-
nnde
2m
sin lt at:-k— (siïnk—kcoslc) .
Si ponimus !: ita exiguum , ut eius quarta potentia prae-
termitti possit , obtinebimus (27. 293)148
2m 2m
- (sink - koska gink k
21 k2 1
2.3
2m 2m ka ( 1
k2
2.3 h2 ,
3
ac proinde
Q = k
2m
3
k2 :
quemadmodum valor a, deducitur ex k , sic ,yalor d, ex
valor az ex la , atque ila porro ; erunt nempe
2m Aa =
2.1 az ?; 3=0,- 313, etc...
| Patet illud ; si vis acceleratrix ex medij resistentia sumitur proportionalis quadrato velocitatis, haud subsistet superior lex, experimentis confirmata, de oscillationum amplitudinibus in progressione geometrica decrescentibus.
[[Fasciculus:Cycloid f.gif|thumb]]
[[Fasciculus:Cycloid03d.svg|thumb]]
[[Fasciculus:Cycloide InfinimentPetits.svg|thumb]]
69. ° * Aliquid subjungimus de gravium descensu per arcus cycloidales. Circulus A'D ( Fig. 40 ) tangens rectam A”E in A&quot; revolvatur super ipsa A”E ita, ut eam pergat semper tangere. Punctum A&quot; circuli regredietur ab A&quot; in E, lineamque curvam describet, quae appellatur cyclois: circulus ille mobilis vocatur cycloidis genitor, recta A ” E basis, diameter AB perpendicularis mediae basi dicitur axis, punctum A vertex; patet autem quemvis circuli genitoris arcum B’A' aequari rectae A'B, quae intercipitur duobus punctis A &quot; et B', in quibus extre ma puncta ipsius arcus tanguntur ab A'E; et totam basim AE aequari peripheriae circuli genitoris. Ducantur
148
2
2,&quot; (siuk—kcosk)-— &quot;' .↗⋮∍ mnk ]: (
kz 3 2.) ⋅
ac proinde
2m
'k 3k
quemadmodum valor a; deducitur ex 1: , sic.valor ag, ex
'a, , valor 013 ex ac, , atque ita porro; erunt nempe
a—a—ïaa' a—a—zma' etc - 2—1 3
1
, 3—3 T;, ∙Ducantur149
jam ex cycloidis puncto v. g . A' perpendicula A'rl= y )
et A'C , alterum in basim AE, alterum in axem AB ;
sit A'r = x ; diameter circuli genitoris dicatur 2a; exhibea
turque per &amp; arcus quoad radium -- = 1 similis arcui A'B' .
Erunt
x = A'B - B'r - A'B ' - AM = a5 - asins ,
y = B'M = asin.v.zza( 1 - сoss) .
ex istarum prima assequimur
dx = ads - acoss ds = a ( 1 - cos )de ;
et dividendo per secundam.
dc
de .
y
Est autem arc sin=
are(sin = AMM))—are (sin V Zay —ya )
a
IV2ay - y2
et consequenter de 2
2ay - y
aa
dyZay - y ?
dy
V2ay - y2 ; ergo
2a
dy = dx V?
(at ) ;
y
149
iam ex cycloidis puncto v. g. A' perpendicula A'r(:y) ⋅
et A'C, alterum in basim A&quot;E. alterum in axem .AB;
sit A&quot;r:æ; diameter circuli genitoris dicatur 20; exhibea-
turque per &amp; arcus quoad radium −∶∙−↿ similis arcui A'B'.
Erunt
æ:A&quot;B'——B'r—-A&quot;B'—-A'M:ae-—asins ,
y:B'M:asin.v.:a(1—coss) ∙
ex istarum prima assequimur
dæ:ada—acoss de:-au -cose)d£ ;
et dividendo per secundam.
d
-—æ-- :de ∙ 7,
Est autem :arc(sin: M):arc (sin ∙−−∶ ∣∕⊋∅∫−↗↾≖
a a ),
et consequenter de:
.a— ∙ a
): ∣∕ ↿−− 5351-
02
dl/Zay—yz df
a—y VZaJ—yi 'ergo150
aequatio differentialis ad cycloidem , computatis coordina
tis a baseos initio A &quot;, Quod si computentur a vertice
A , ut novae coordinatae sint AC ( = x ') , et A'C ( =y' ) ,
cum habeamus x = an — y , y = 2a — x', prodibit
-dx'adyV
x'
2a- , seu dy = dxV
2a - x
xช่ (a &quot;) .
Nunc ad gravium descensum quod pertinet per ar.
cum quemvis cycloidalem , cujus vertex in puncto inſimo
B ( Fig. 37), sit C initialis positio puncli ponderosi , quum
nempe t =0 et v = v = 0 , M positio in fine temporis 1;
quibus positionibus respondeant altitudines c et ac' supra
horizontalem rectam transeuntem per B , ut in Mha
beatur v = V 2g(c-x') : denotantibus h , se s' cycloidales
arcus CB , CM , BMBM ,, erit erit dsds== dhd (h -- ss'')) = - ds'; unde''
ds
di
ds'
dt = V2g(c -x '), ex qua obtinelur
ds'
dt
V 28(c — x ')
Formula ( a&quot; ) praebet ( 27. 19.0)
2a -x
do = Vdx =+ody&quot;a= dx V17 = dx '

;

x
hinc
da - c
a
dt dx'
V. 8 V cx' — x'2 -Va
GVFECITATE
150
aequatio diti'erentialis ad cycloidem . computatis coordina-
tis a baseos initio A&quot; Quod si computentur a vertice
A , ut novae coOrdinatae siut AC (:æ') , et A'C (:y'),
cum habeamus x:an'—-7, 1:20—æ', prodibit
I
Za—æ ∙−−−
, seu df:dx I/ æ, (a') .
Nunc ad gravium descensum quod pertinet per ar-
cum quemvis cycloidalem . cujus vertex in puncto infimo
B (Fig. 37), sit C initialis positio puncti ponderosi, quum
nempe t:o et ⇂↾−−∙∶⇂↗∘∶∶∘ , M positio in fine temporis :;
quibus positionibus respondeant altitudines c et æ'supra
horizontalem rectam transeuntem per B , ut in M ba-
beatur :»:V Zg(c-x ':) denotantibus ,: .
s, s' cycloidales
arcus CB, CM, BM , erit ds:d(h—s'):—ds'; uude
ds di' .
v −− dt— dt —l/2g(c-æ'), ex qua obtmetur
ds'
dc ∶−∙−− − ⇂∕−−−−−
⇄∊≺∁−−⋅↿⊏⋅ )
Formula (a') praebet (27. 19!)
d.;— −−∙ ⇂∕∎∎−−∎∎−∎∎∎ ↙↙∙≖⇌ ≖⊣⊸↙↿∫−
hinc
(1&quot;.—— a flx' ;.
(ll:
∙−∙ V
∙−− ∙−− V
a .; 20 .
l
—— g J/Fæ—æ'z ∙ g
∣∕↿ ∙↕⋅∎⋅≩∁≻≖⊽151
sumptisque integralibus ( 27. 9,9 ) ,
= c +Vare (co== **)

;

in positione initiali est t=0, simulque x' =c; igitur C = o, et
Vore (rosa ).
Facta x=0, prodibit tempus descensus usque ad punctum
infimum B, nimirum
11 =T
VO
ubi cum non inveniatur c , patet , ex quocumque cycloi
dis puncto demittatur grave, eodem semper tempore per
venturum ad B. Hanc cycloidis proprietatem posteaquam
detexit Hugenius , cycloidem ad pendolum adhibere cae
pit : quod qua ratione fieri possit , ostendit in parte 3. “
Horologii oscillatorii.

=== De attractione corporum in hypothesi attractionis agentis in ratione directa massarum, et in reciproca duplicata distantiarum.===
70. Pyramis AH (Fig. 41) habens basim GH infinitesimam secetur superficie sphaerica, cujus centrum in A , et radius AZ ( = r ); sit Ky = B ) projectio intersectionis VZ ( = ) in plano AB; supra basim Ky erigatur prisma KyE altitudinis CH ( =x): exprimet KE AZ sumptisque integralibus (27. 93) ,
↥⋅∶∁⊹ l
V— a1c (eo: 200) ;
in positione initiali est t:o. simulque æ':c; igitur (l:-o, et

* x −−∶−∁

Facta x':o. prodibit tempus descensus usque ad punctum
infimum B, nimirum
a
II:&quot; V— :
g .
ubi cum non inveniatur c , patet, ex quocumque cycloidis puncto demittatur grave, eodem semper tempore perventurum ad B. Hanc cycloidis prOprietatem posteaquam detexit Hugenius, cycloidem ad pendulum adhibete caepit: quod qua ratione fieri possit, ostendit in parte 3.'
Horologii oscillatorii. ⋅

lit-F.
AZZ152
vim attractivam ( p) segmenti CG in punctum A juxta directionem perpendicularem plano AB. Intelligatur enim segmentum CG secari sphaericis
superficiebus numero infinitis , quarum centrum in A , et
r2 , 7* 3 sintque eoz , A2 , A3 intersectionum areae. Erit
radii rs .
din 23

; 2

2 r2 3 p2
vis nempe attractiva cujuscumque areolae Qy , da , .
aequabit vim attractivam areolae A. Ex punctis Z, C,
ducantur in AB ... perpendicula Zy ( =n) , CB ( = n ) ...:
singulis viribus resolutis in duas, quarum altera sit paral
lela , altera perpendicularis plano AB , componentes per
pendiculares repraesentabuntur per
ni na n
2
ri ra
et quia
ni n2 n
72
iccirco
li ni 0.2 п,
=
a n
.
t'i p22 ra
his positis , quisque videt fore
n
f
152 .
vim attractivam (:f) segmenti CG in punctum A juxta
directionem perpendicularem plano AB.
Intelligatur enim segmentum CG secari sphaericis
superficiebus numero infinitis. quarum centrum in A, et
radii r; , r,. rg
.... sintque at,
, at,, ata ...
inter-—
sectionum areae. Erit
«! a; «3 ∙
. a
r,: fa, rna ra.
∙
'
vis nempe attractiva cujuscumque areolae at,, at,, ∙∙∙
aequabit vim attractivam areolae at. Ex punctis Z, C,
...
ducantur in AB
... perpendicula Zf (:n)
, CB (:m) ...:
singulis viribus resolutis in duas, quarum altera sit paralf
lela, altera perpendicularis plano AB , componentes per-
pendiculares repraesentabantur per
a! &quot;[ a:; &quot;a a n
. , ∙ , ∙ ∙ . ∙−−− .—,. r,2 r, rf r, :-a r
et quia
—:
&quot;! &quot;2 n
∙−− ∙ ∙ ∙ :∙
—;
r, r, r
iccirco
«! &quot;r a: n, a n
∙
∙−−∎ ∙ ∙ ∙ ∙:∙∙−∎ ∙ ∙∙∙∙ :
rl: rl ,.22 &quot;, &quot;, r153
Jamvero (55.4. )
\beta
= cosyZA 3
igitur &gt;
n
ela
B
sli
oli
a
et consequenter
Bx
r?
KyE f
AZ

71. Singula corporis cuiuscumque KGDH (Fig. 42) puncta trahant punctum C positione datum. Centro C et
radio quolibet CM describatur sphaera MBN; in eius
superficiem incurrat in A recta quaelibet CG permeans
corpus KGDH iuxta DG ; demittatur ex A perpendi-
culum AQ supra planum MCN; capiatur in- AQ pars
TQ aequalis segmento DG intra corpus KGDH demerso;
quod si plura fuerint huiusmodi segmenta, pars in per-
pendiculo accepta sequetur omnium summae, Si per GM?
dividitur solidum ïTXV, quod continetur plano MCN
et superficie ab omnibus punctis T determinata , expri-
met quotus vim, qua totum corpus KGDH trahit punctum
C perpendiculariter ad planum MCN.
Prodeant enim ex C infinitae numero pyramides, qua-
rum segmenta DG impleant totum corpus KGDH; pote-
runt totidem respondentia (69) prismata TQ concipi , quae
totum solidum ïTXV impleant; ergo etc.
Quoniam vires omnes sollicitantes punctum C
possunt traduci ad ternas
, quarum directiones congruant
cum tribus rectis se mutuo ad angulum rectum secantibus in ipso C; ternae vero istiusmodi vires in unam com-
↿↿154
positae dant resultantem ex illis omnibus , inde fit quod
ubi determinentur (70) ternae vires corporis KGDH re
spective perpendiculares tribus planis orthogonalibus per
punctum C traseuntibus , eae in unam contractae suppedi
tabunt et directionem , et intensitatem illius vis , quae re
sultat ex omnibus viribus punctorum constituentium cor
pus ipsum KGDH.
Si punctum C intra - corpus trahens collocaretur
accipienda esset TQ aequalis differentiae inter distantias
ipsius C ab extremis D et G rectae DG transeuntis per
C. Hinc si C fuerit situm intra ejusmodi crustae cavita
tem , ut per C ducta quavis recta , aequales hinc inde par
les illius rectae intra crustae crassitiem intercipiantur, eva
nescentibus omnibus TQ , evanescet etiam omnis vis pla
no cuicumque perpendicularis , et punctum C in aequi.
librio consister.

72. • * Coordinatarum originem O constitue in
quovis corporis puncto ; sin que x, y, z coordinatae pun
cli altrahentis ; a , b , c coordinatae puncti allracti ; '
distantia inter punctum attrahens et punctum altractum :
expriment
b - r
CZ
ba 몇
7
A'
A
cosinus angulorum , quos a continet cum axibus coor
dinatis OX , OY, OZ. Quare denotantibus Hc , H,, H,
componentes iisdem axibus parallelas , in quas rosolvitur
attrahens totius corporis vis H , et dm elementum massae,
eront
H,
- Som dm , 1 , = Sabah dm
(o)
H , Set dm :
A3
154
positae dant resultantem ex illis omnibus; inde Et quod
ubi determinentur (70) ternae vires corporis KGDH re-
spective perpendiculares tribus planis orthOgonalibus per
punctum C traseuntibus, eae in unam contractae suppedi-
tabunt et directionem, et intensitatem illius vis, quae re-
sultat ex omnibus viribus punctorum constituentium cor-
pus ipsum KGDH. ↴
Si punctum C intra -corpus trahens collocaretur ,
accipienda esset TQ aequalis differentiae inter distantias
ipsius C ab extremis D et G rectae DG transeuntis per
C. Hinc si C fuerit situm intra eiusmodi crustae cavita-
tem
, ut per C ducta quavis recta, aequales hinc inde par-
tes illius rectae intra crustae crassitiem intercipiantur, eva-
nescentibus omnibus TQ, evanescet etiam omnis vis pla-
no cuicumque perpendicularis. et punctum C in aequi-
librio consistet. ⋅
⊽∑∙∘∙ Coordinatarum originem O constitue in
quovis corporis puncto; sintque x, ],:coordinatae pun-
cti attrahentis; a, b ,&quot;c coordinatae puncti attracti; A'
distantia inter punctum. attrahens et punctum attractum:
expriment
⊄≖∙∙−−∙∙−−∙⋮≖ b—gr c—z
∆∣∙
'
∆∣
'
∆∣
cosinus angulorum, quos ∆⋅ continet cum. axibus coor-
dinatis OX , Oï, .OZ. Quare denotantibus H, , H, , H,
componentes iisdem axibus parallelas, in quas rosolvitur
attrahens totius corporis vis H ∙ et dm elementum massae,
erunt ⋅
a—æ &quot;bf—7 ⋅ ∏≖∶∶∫ ∆∣⊰ dm,
⊟⋮⇌−∽∙∣∙−⊒↙∙⇁⋮⇀≀≀≀↿∙ .-
(0)
,:szjä-äfdm:155
integralia se se protendunt ad totam corporis massam M.
Pone
Q = Sam
( o ')
habes quidem
A2 = (a — x )&quot; + ( my) + cz( )&quot; ;
sed quia integrationis limites non pendent ab a , b , c ,
ideo ex prima ( o' ) erues
dQ
da ſ
dm ,
dQ
db
den
ES ,
do dm
dQ
dc -dm ;
da
secunda vero (o' ) praebet /
a 영 1 dA a -X a
A' ? da
b
da 4'3
db A'3엷
slot
dc 4'3
traducentar itaque ( o ) ad
H=
dQ
da H ,
do
db H,
dQ
dc ( o &quot; ) ,
componentesque H , H ,, H, pendebunt ab unico integrali
l. Fiat
a : + 62 + c = A2 ,
155
integi-alia se se protendunt ad totam. corporis massam M.,,
Pone
≺≀⇌⇀⋅ dm −∙−≃
habes quidem (O,)
∆≏⇋−∙≺∅∙−∞≻⋍⊣−≼≀↗−∫≻≔⊣⊣∘∙−≖≻⋅ ;
sed quia integrationis limites non pendent ab a, b , c ,
ideo ex prima (a') erues
⋛≣∎∶∆∼∣↙≟ dm' ∎−⋅∫↲∂↙≀⋯⋅−−−−∫ &quot;'&quot;-('m-
secunda vero (a') praebet
↿ ↿
⊄∄−−−↽
∆∙ ↿ siA—, a—æ (LA-7 b—r
de'&quot; A'da A'3 ↞∙ ∠≀∣⊃−−⋅−−−−∆⋅≀∙ '
↙≀∙∙↿⊽ ac.
.... de −⋅⋅ ∆∣∍
traducentur itaque (a) ad
dQ dQ dQ−⊋⋤−∙ Hic—2? ∙ Ha ≔−⋅⊋⊂∙∙− (O&quot;) 1

componentesque H, , H,, H, pendebunt ab unico integrali
Q. Fiat ⋅
∁≖⊹∂≖−⊦∁∶≖−−∆≖ ∙156
ut secunda ( o ' ) scribi possit in hunc modum
A ' = 12—2(axtby tcz) + wa + ya + z2 ;
erit
1
-
[ 12—2( ax + by + cz) + xa + ya + za = +
+
2(ax + by + cz) xtya taza)
+
243
12(ax + by + cz)2- [ 12 (ax + by + cz )-3(x2 + ya + za) ][ x ? tye + z&quot;)
845
+
.
unde, ob prinam ( o' ) ,
m
Q

* ++ ſ(ax + by+ ca)dın

25 /(x +y +z&quot;)du + z flar+ by +czydom.co&quot;.
Sit coordinatarum origo in centro gravitatis massae at
trahentis; erit ( 20. b )
1
43
Slax ( ax + by + cz) dni = ta fædm +
bſydm + ſzam ] = 0 ;
ideoque vertetur ( o '&quot;') in
156
ut secnnda (o') scribi possit in hunc modum
A&quot;::A3—2(aæ—-l-bj-l—cz)*æï-þyl-l-z' ;
erit
−↙∃≃∙⋅⋅ −−−−−
[∆⋅−≆≺∾↼⊦∂∫⊣⊸≉≻⊣−↕⋅⇀⊦∫⋅−⊦≖≖ ⊐⋅− *
↽−⇌−↿≴↸ ⋍≺∅↕⊹≀↗∫⊣⊸∡≻−≺↕∙⊣⇀∫⋅⊹≖≖≻
2A3
12(aæ-l—b.7—l-Cz)'-[1 ⊋≼↙⇂∙↿∙∙⊹⊘↾⊣⊸∅⊢∃≼∞≖⊹∫≖⊹≂≖≻∃ ∣⋮∙∙∁≴⊣↰↾⊣∎≖∶∣∙
8A5 '
'
-I-..;
unde, ob primam (a')-
. 1
1
Q ∙∙∙ Z
∙∣∙− A3]
(aæ-l-bJ-l-czkim—
1
- 3 &quot;
555] (x'-l'f' ∙⊦≖≖≱ dnl-l— üïf(aæ'l'lïï'*l'cz)'dm-n-(0 ').
Sit coordinatarnm origo in centro gravitatis massae
&quot;'
trahentis; erit ( 20. b)
. ↿ ∆↿−⋮∫≺∘∞⊣−≀≀↗−⊢∞≻↙≀⋯∶ 33— ta xdm-i-
bjïydm ⊣− rfzdm ]: 0

:

ideoque vertetur (o&quot;') in157
M 1
Q
Δ 243f\ x3 + y* +32) dmt
3
245 (ax + by + cz)-dm-, .. ( 0 &quot; &quot;).
ca
73. Corpus KGDH Sit sphaericum , ejusque centrum
in puncto extremo B radii CB (Fig. 43) inveniatur ; ipsi corpori occurrat QA in T. et Q '; ducto perpendiculo BE
supra CA , triangula rectangula AQC, BEC propter latus
AC=CB , et angulum QAC=BCA , erunt aequalia , adeo
que QC=BE ; chordae nimirum SD,CT aequidistabunt a
centro B; erunt itaque inter se aequales , ac proinde
OʻT ( Fig. 43 ) , aequabit QT (Fig. 42): quod cum ubique contingat, erit area KGDH (Fig. 43) sic .aequalis
areae XYC (Fig. 42) , ut solida genita ab his areis cir
suos axes revolutis aequalia sint inter se. Vim proinde , qua punctum C tendit in sphaeram KGTH ( Fig. 43 ) exprimet ipsa sphaera divisa per CM (=CB) seu per
quadratum distantiae puncti C ab ' ipsius sphaerae centro; siquidem aliae duae componentes (71) evanescunt:
Sed si sphaera ita condensaretur , ut coiret in centrum ,
eodem prorsus modo exprimeretur ejus attractiva vis; ergo punctum extra sphaeram situm eadem omnino ratio
ne in ipsam tendit , ac si omnia sphaerae puncta in cen
tro compenetrarentur.
Haec vera sunt , licet corpus non sit omnino ho
mogeneum , modo tamen sint ubique bomogeneae ejus par
tes a centro aequidistantes ; quod notandum etiam in se
quenti assertione.


73. Corpus KGDH Sit Sphaericum
. eiusque centrum
in puncto extremo B radii CB (F ig, 43) inveniatur; ipsi
corpori occur1at QA in T
et Q'; ducto perpendiculo BE
snpra CA , triangula rectangula AQC, BEC propter latus
AC;-:: CB, et angulum QAC:BCA ,
erunt aequalia, adeo-
que QC—BE- , chordae nimirum SD Q' T aequidistabunt a
«centro B; erunt 'itaque inter se aequales , ac proinde
Q'T (Fig. 43) aequabit QT (Fig. 42): quod cum ubi-
qne contingat, erit area KGDH (Fig. 43 ) sic .aequalis
areae XTC (Fig. 42) , ut solida genita ab his areis cir-
ca suos axes revolutis aequaha sint inter se. Vim proin-
de ,qua punctum Ctendit in sphaeram KGTH (F 1g 43)
exprimet ipsa sphaera divisa per ∁∾∙≖ (:CB') seu per
quadratum distantiae puncti. C ab ipsius sphaerae een-
tro ; siquidem aliae duaeïcomponentes (71) evanescunt,:
Sed si sphaera ita, condensantur,, ut coiret in centrum,
eodem prorsus modo exprimeretur eius attractiva vis; er-
go punctum extra sphaeram situm eadem omnino ratio-
ne in ipsam tendit
, ac si omnia sphaerae puncta in cen-
tro compenetrarentur. ⋅ ⋅
Haec vera sunt
,
lieet corpus non sit omnino ho-
mogeneam, modo tamen sint ubique homogeneae eius par-
tes a centro aequidistantes; quod notandum etiam in se-
quenti assertione.

74. Si punctum materiae locetur intra crustam sphaericam, sive intra orbem sphaericum intus cavum terminatum binis superficiebus sphaericis concentricis, id punctum, destructis viribus consistet in aequilibrio. Sint ( Fig. 44) NEQ, MFP superficies illae concentricae , punctum vero materiae sit O. Ducta per 0 quavis chorda MNEF, et ex centro K demisso perpendiculo KC supra ipsam chordam, erunt CM=CF, CN=CE;
igitur MN = EF , ac proinde ( 72 ) etc:

75. Ex dictis ( 73. 74 ) sequitur: 

1º . punctum in superficie duarum sphaerarum positum gravitare in ipsas sphaeras in ratione radiorum directa: nam sphaerae sunt ut radiorum cubi, quibus per eorumdem quadrata divisis, prodeunt radii simplices: 

2° . gravitatem puncti intra globum homogeneum pergentis a superficie ad centrum decrescere in ratione directa distantiae a centro ipso. 

76. Haec notentur. 1º. materiale punctum valde distans a corpore attrahente, utcumque se habeat forma corporis, ea proxime ratione tendit in ipsum corpus, qua tenderet si corporis partes in centro gravitatis compenetrarentur; patet tum ex dictis (12: 20) , tum ex eo quod in casu vires attrahentes punctorum constituentium corpus considerari possint tamquani proxime parallelae et proportionales ipsorum punctorum massis.

* Patet etiam ex ( 0 &quot; . 01 .: 72 ) ; nam si A est ila na

gna , ut, retento primo termino in ( o &quot; ), possint caeteri
praetermitti absque sensibili errore , sicque habeatur
M
Q
exsistent
M C
H
M
A2 ., HH ,
M 3
42 : A
H
ac proinde
M
H
ViH + H ,* + H , +
158
Sint (Fig. 44) NEQ, MF P supetticies illae concen.
tricae, punctum vero materiae sit 0. Ducta per 0 qua-
vis chorda MNEF , et ex centro K demisso perpendicu-
lo KC supra ipsam chordam, erunt CMr—CF, CNzCE;
igitur MN— EF
, ac proinde (72) etc:

75. Ex dictis (73. 74) sequitur: 10. punctum in su-
perficie duarum sphaerarum positum gravitare in ipsas sphae-
'ras in ratione radiorum directa: nam sphaerae sunt ut ra-
diorum cubi, quibus per eorumdem quadrata divisis, pro-
deunt radii simplices: 20. gravitatem puncti intra globum
homogeneum pergentis a supe1ticie ad centrum decrescere in ratione directa distantiae a centro ipso.

76. Haec notentur. 10. materiale punctum valde di-
stans a corpore attrahente, utcumque se habeat forma cor-
'poris, ea proxime ratione tendit in ipsum corpus , qua
tenderet si corporis partes in centro gravitatis comPe-
netrarentur', patet tum ex dictis (12: 20), tum exeo quod
ih casu vires ,attrahentes punctorum constituentium cor-
pus considerari possint tamquam proxime parallelae et pro-
portionales ipsorum punctoruin massis.
' ea Patet etiam ex (a&quot; . o&quot; 72); nam si∆∙ est tta ma-
gna , ut, retento primo terminogin (o'f ), possint. caeteri praetermitti 'absque &quot;sensibili 'et-rore , sicque habeatur
.
'
. - «. ∙
⋅
r ↾
1' I
.
. M. 11&quot;
≺≀⇌⋅⊼−↿⋅ ⋅
exsistent '1-
- --M 0
'M 6
&quot;' M
∣⋅
∏⋍−⋅−− −−∶∙−−⋅∙−− ...—...; .' AaA'H' ArA'H': ∣⋅∙↘∆∙
ac proinde
−−−∙∙−−−−−−∙∙∙∙−−∙ M
H:: l/Hil'i'nya'i&quot; He's-A&quot;?159
2º. Non pluribus opus est , ut stabiliatur illud: u
bi dimensiones corporum quorumcumque se matuo attra
hentium in ratione directa massarum, et reciproca duplicata
distantiarum sint admodum exiguae prae distantiis, quibus
ipsa corpora disjunguntur, eorum alterum tendet in alterum
perinde ac si essent 'ambo in suis gravitatis centris compe
netrata . Dicantur enim M , M' massae duorum ejusmodi
corporum , m, massa cujuslibet puncti spectantis ad M , et
A distantia inter m, ac centrum gravitatis massae M ; ex
Mm ,
primet vim attractionis motricem ( 28) , qua m, len.
dit in M, simulque ( 7 ) vim attractionis motricem, qua M tendit
in ma; ideoque merit vis attractionis acceleratrix, qua
M tendit in mo . Atqui hoc pacto M tenderet in mo, si to
la massa M compenetraretur in suo gravitatis centro ; er
go M revera tendit in mi, id est in singula puncta mas
sae M' , perinde ac si tota M foret in suo gravitatis cen
tro compenetrata: cumque ob paritatem rationis idem con
tingat massae M' quoad M , jam patet veritas assertionis.

3º. Quoad sphaerica corpora, quorum partes aequidistantes a suis centris sans homogeneae, obtinet assertio, utcumque caeteroqui se habeat intercedens distantia.

=== De gravitatione universali ===
[[77|77]]. Quae de coelestium corporum motibus, ex astronomicis observationibus hic subjicimus, ad ipsorum gravitatis centra respiciunt.

1º. Areae, quas circa solem describit radius vector uniuscujusque planetae sunt respondentibus temporibus proportionales: idipsum obtinet quoad areas descriptas a radio vectore uniuscujusque satellitis seu planetae secundarii circa suum planetam primarium.

2º. Convertuntur planetae circa solem in orbitis ellipticis ita, ut singularam ellipsium alterum focum occupet sol: convertuntur planetae secundarii circa suos planetas primarios in orbitis ellipticis ita, ut istarum focum occupet respectivus planeta primarius.

3º. Quadrala temporum periodicorum sunt in diversis planetis ut cubi semiaxium transversorum: idipsum obtinet quoad diversos satellites circa respondentem planetam primarium.

[[78|78]]. Hinc 

1º. planetae urgentur vi acceleratrice &lt;u&gt;tendente in solem&lt;/u&gt;; itidem satellites urgentur vi acceleratrice &lt;u&gt;ad respectivos planetas primarios tendente&lt;/u&gt;: plauetae, nimirum gravitant in solem, satellites vero in planetas, quibus adhaerent.

2º. Unusquisque planetarum (56) urgetur in solem vi gravitatis, quae sequitur rationem reciprocam duplicatam distantiarum ab ipso sole: idem dicendum de unoquoque satellite in ordine ad suum planetam primarium .

3º. Collatis inter se viribus acceleratricibus, quibus diversi planetae urgentur in solem, eae erunt (56) in sola ratione reciproca duplicata distantiarum a sole ipso; praecisa igitur projectionis vi, si diversi planetae in aequalibus a sole distantiis constituerentur, aequali tempore in eum descenderent. Idem obtinet in satellitibus quoad respectivos planetas primarios. 

[[79|79]]. Planetae secundarii una cum primariis, quibus adhaerent, in solem urgentur eadem gravitatis lege. Nam corpus omne, quod circa corpas alterum utcumque motum describit areas temporibus proportionales, urgelur duplici vi, altera tendente ad corpus illud utcumque motum, altera utriusque communi (5:46): cum igitur planetae primarii gravitent in solem, cumque planetae secundarii circa suos primarios describant areas temporibus proportionales; propterea etc.

[[80|80]]. Gravitant in se mutuo corpora omnia, ex quibus coalescit planeticum systema. Planetae siquidem omnes cum primarii tum secundarii vi gravitatis urgentur in solem; ergo sol in planetas omnes vi ejusdem gravitatis (7) urgetur: atque hoc argumento ostendes terram gravitare in lunam (id confirmant phoenomena marini aestus) caeterosque planetas primarios in suos satellites. Quod autem planeta quilibet in alium quemvis gravitet, satis e sola comprobaretur analogia, etiamsi nulli essent effectus, ex quibus haec gravitatio immediate detegi posset. Sed ejusmodi effectus non desunt: perturbationes videlicet, quae in recensitis motibus (77) observantur, quaeque per mutuam coelestium corporum gravitatem optime determinantur (62*60). Sic cum lunae motum ad regularis calculi normam ex observationibus exigere se posse Astronomi desperarent, tandem postquam ejusdem perturbationes ex mutua corporum coelestium gravitatione investigare coeperunt, tabulas lunariam motuum potuerunt conficere, quarum tantus est cum coelo consensus, quantum sperare ex observationibus nemo potest.

[[81|81]]. Praecisis perturbationum causis , urgebitar luna in tellurem vi acceleratrice (56):&lt;math display=&quot;block&quot;&gt;
\varphi=\frac{4\pi^2 a'^3}{T^2}\frac{1}{\Delta^2};&lt;/math&gt;


denotat &lt;math&gt;T&lt;/math&gt; tempus periodicum = dieb. 27 , 322 = minut. secund. 60&lt;sup&gt;2&lt;/sup&gt;. 24. 27 , 322; &lt;math&gt;a'&lt;/math&gt; semiaxem transversum orbitae lunaris, &lt;math&gt;\Delta&lt;/math&gt; radium vectorem ipsius orbitae. Iamvero mediocris radius terrestris = 16931100&lt;ref&gt;Error in originale&lt;/ref&gt; ped., mediocris parallaxis lunaris 57' + 11&quot;, unde &lt;math&gt;a' =\frac{16931100}{\sin(57' + 11'')}&lt;/math&gt;facto  igitur &lt;math&gt;\Delta = 19631100&lt;/math&gt;, gravitatis vis qua luna urgetur in terram evadet in ipsius terrae superficie
&lt;math display=&quot;block&quot;&gt;blah blah blah&lt;/math&gt;qui valor cum sit proxime 30,2 ped., inferimus gravitatem qua luna urgetur in terram nihil esse aliud nisi gravitatem ipsam terrestrem imminutam in ratione reciproca duplicata lunaris distantiae a terrae centro.

[[82|82]]. Vis gravitatis, qua lapis v . gr. urgetur in terram, est (80) ejusdem speciei cum illa gravitatis vi, qua corpora mundani systematis in se mutuo tendunt; ergo idem in utraque erit agendi modus. Atqui vis qua totus lapis urgetur in terram resultat ex viribus, quibus singulae lapidis particulae in eamdem nituntur; igitur et vires, quibus corpora mundani systematis gravitant in se mutuo, resultant ex viribus, per quas singulae ipsorum, particulae se mutuo petunt. His positis, stabilietur illud: gravitas ita materiam afficit, ut singulae ejus particulae in alias omnes et singulas gravitent in ratione directa massarum, ad quas tenditur, et reciproca duplicata distantiarum alterius ab altera. Recole quae diximus (76.2º.3º); etenim coelestia corpora et habent dimensiones admodum exiguas prae mutuis distantiis, et induunt formam prope sphaericam.

[[83|83]]. Bonum erit nonulla hic annotare. 

1º. designantibus &lt;math&gt;M&lt;/math&gt; et &lt;math&gt;m&lt;/math&gt; solarem et planeticam massam, ex dictis (56.k, 62.c) eruitur&lt;math display=&quot;block&quot;&gt;
M + m =\frac{4 \pi^2 a^3}{T^2}
&lt;/math&gt;ratio igitur inter cubum semiaxis transversi et quadratum temporis periodici, utpote pendens a massa planetica, nequit esse accurate constans quoad diversas planetarum massas. Atqui tamen ex astronomicis observationibus infertur rationem illam, sin minus accurate, certe esse quamproxime constantem: concludendum itaque planetarum massas admodum exiguas esse, ubi comparentur cum massa solis.
2.º Eodem modo ostenditur, si lunam excipis, satellitum massas fore et ipsas valde parvas prae massis planetarum, quibus adhaerent.

3.° Quae quantitates sunt designatae per &lt;math&gt;m, 
a , T&lt;/math&gt; quoad planetam , eae designentur per &lt;math&gt;m' , a' , T'&lt;/math&gt; quoad satellitem; erit&lt;math display=&quot;block&quot;&gt;
m + m'=\frac{4 \pi^2 a'^3}{T'^2}&lt;/math&gt;Hinc (1°)&lt;math display=&quot;block&quot;&gt;
\frac{m + m'}{M + m}=\frac{T^2}{T'^2}\frac{a'^3}{a^3}
&lt;/math&gt;praetermissa ( 19. 20. ) &lt;math&gt;m'&lt;/math&gt; in numeratore primi membri, itemque m in denominatore , et facta M = 1 , prodibit.
T2
TO
a's
i
quae formula suppeditat rationem inter solarem massam
habitam pro unitate , et massas planetarum ( tellurem ex
cipe) , qui satellitibus stipantür.


4.° Quod spectat ad tellurem consideratam in
star sphaerae habentis radium R , et massam m , sit &amp; gravi .
tas prope ejus superficiem , erit (73) 8 =R. , ideoque (10. )
M +m
4 712 a3
&amp; R2T
et praetermissa ( 19. ) m in numeratore primi membri , factaque solari massa M = 1 , emerget
163
2! Eodem modo ostenditur ,
si lunam excipis,
satellitum massas fore et ipsas valde parvas prae massis planetarum, quibus adhaerent .
-l ⋅ 

:

41:2003 -
∙∙ ↶↿ m-4—m' £S.-&quot;IW.,
Hï'n'c (10)
⇀
..
.m -- in'. T., a'-3-- - M,-—-.nsl ≔−⋅⋅∙−∙− ∙ ∙∙∙ : T'a ::
&quot;
praetermissa (.10 20. ) tu' in numeratore primi membri ,
itemque ut in denominatore ,
et facta M— −∙− ↿, prodibit.
⋅ 1 ↴− ⋅↧⇁≖ -a ∙
−
⋯∙↽↽⊽ .
..;3, .'
∙⊾⊺⋅⋅
quae? .fottmule- suppeditat rationem -
intcr. solarem^ massam
habitam pro unitate ,
et massas planetarum (tellurem ex-
cipe) , qui satellitibus .stipantur.
'i-o Quæ-Spectat ad tellurem' consideratam in-
star sphaerae habentis radium R , et massam 111, sit 3gravi-
tas prope eius superficiem ,
erit (73) gr ≖⋅⇁⋅∙−⋮↾−⋮↾− , ideoque (10.)
IUI—tm,— 4 11:303.
,
gRQTi'
et praetermissa (10.) »: in numeratore primi membri, f.-
ctaque solari massa M ∙−⇁−−−∙ ↿, emerget ⋅164
&amp; R2T2
4 Ti ? a3
PE

5° Media telluris densitas ( = M) determinari potest ex penduli aberratione. Sit CB (Fig. 45) pendulum; a longitudo rectae CB, quae nec distendi possit nec inflecti; S centrum massae sphaericae ( = m' ) ad se trahentis punctum ponderosum B , r radius , M densitas; b recta CS; CD posilio penduli digressi a recta verticali CB;
&amp; angulus BCD; h angulus BCS; k recta SD: centro insuper C et radio CB intelligatur describi circularis arcus
BD , et per D duci tangens nn' . In D sollicitatur materiale
m'
punctum viribus acceleratricibus g et altera juxta ver ka
ticalem DD' , altera juxta rectam DS ; anguli
SU
1
D'Dn = CnD = 90 ° — E , SDn ' = # (CDS - 90° ) ,
to
et consequenter
b sin (h — 5)
cosD'Dn sins , cosSDn ' = sinCDS =
k
Vires igitur motrices respondentes praefatis viribus ac
celeratricibus sese librant in D quotiescumque fuerit
que
bas
bit
ha m'
23 gsine b sin ( h - €) .
sed
Pone longitudinem penduli ita exiguam prae distantia CS ,
ut absque sensibili errore sumi possit k = b ; traduce
tur aequilibrii conditio ad
gb2 sin é = m ' sin (h - ) ;
m
et substitutis ( 4º . ) valoribus 8 T RH, m
R? 3
4
90
paris 75 1'3 je , prodibit
164
g R*T3
m— 4 723 (13
50 Media telluris densitas (:: p.) determinari po-
test ex penduli aberratione . Sit CB (Fig. 45) pendulum longitudo rectae CB, quae nec disteudi possit nec inflecti; S centrum massae sphaericae (: m') ad se trahentis
punctum ponderosum B ,
r' radius
, pf densitas ; 6 recta
CS; CD positio penduli digressi a recta verticali CB;
a angulus BCD :
]: angulus BCS: k recta SD: centro in-
super C et radio CB intelligatur describi circularis arcus
BD , et per D duci tangens nn' . In D sollicitatur materiale
punctum viribus acceleratricibus 3 et ?; , altera juxta ver-
ticalem DD', altera juxta rectam DS : anguli
ix)-D'.. ∙∁∥↧⊃ :. 90o −−∙ e
, sna':∶↿≐ (CDS — 900)
.
et ⋅ consequenter
bsinUt—s) ——k .
Vires igitur motrices respondentes praefatis viribus ac-
celeratricibus sese librant in D quotiescumque fuerit
! r ' ,
!'
cosD'Dn :: sins, cosSDn' ∶∙ sinCDS −−−−⋅−
gsins:
£;- b sin( h — £) .
liane. longitudinem penduli ita exiguam prae distantia CS ,
ut absque sensibili errore sumi possit 1::6; traduce-
tur aequilibt'ii conditio ad
gbï sin ; −−∶ m' sin-(h — a) ;
et substitutis (40.) valoribus g: €; ∶∶ £- 11 R p., m' −∙−∙−−
4
,, ,
'
∙⋅ ; &quot; . .
⋅
∙⋅⋮↿∏⋅ p. , prod1h1t
!
'
I
151
sit
lla165
1
b- Rue sin
{
= 13
M
' sio
(
h
-
E)

unde
i
p3
y sin
h rº
( sinh
– coshtang :)
1
lang
E
=
Ruba tospicosh ji Rba lang

Permanentibus
r
' et
'
, valores
b
=
r
' et
h
= 90
° manife
ste suppeditant maximam penduli aberrationem
&amp;
, ut quoad
istiusmodi aberrationem sint
Se
re tang R pe
3/3
Rtang
s
Densitas fl
, prout colligitur ex aberratione penduli
, cen
setur quater vel quinquies major quam densitas aquae. 6. ** Eadem u determinatur etiam experimentis in
stitutis in libra torsionis
. Sit
( Fig
. 9.
) HH
(
=2a
'
) posi
tio vectis horizontaliter librati

;

E punctum medium, in quo vectis appenditur filo metallico verticali HA circulus horizontalis centro
E et radio EH
=
a
')

; SS

( 26
)
recta similiter horizontalis
, transiens per
E
, ibique secta
bifariam

; sint

S et
S centra sphaerarum inter se aequa
liam et quoad volumen
, et quoad massam
(
=
m
)
, ad se
trahentium massulas sphaericas
m
' et
m
&quot; inter se pariter
aequales
, quarum centra in
H et
H
'
. Movebitur vectis
circa punctum
E immotum
, motusque iste repetendus ab
attrahente sphaerarum vi juxta circuli tangentem
, cui vi
jugiter adversatur vis lorsionis ex filo metallico verticali

et quoniam corpuscula
m
' et
m
&quot; eodem prorsus donantur
motu circa
E
, satis erit alterum dumtaxat
v
. gr
.
m
' con
siderare
. Dicatur itaque
h datus angulus HES

;

&amp; angulus
,
quem in fine temporis
i continet vectis cum initiali po
sitione EH

;

k distantia inter
S et
m
' in five ipsius
t

: sol

licitabitur
m
' juxla circuli tangentem vi attractiva
!
)- a
165
63 Bpain :::/3 (fimul—15);
unde
tan E' ∙∙∙⋅ r'3 pf sin 11 p. '
r'3 (sinit −⋅ coshtang &amp;)
∙
g −⇀ nubi-l- r'3 picas/1
.pf ⇀−− lib2 teng
&amp; .
Permanentibus r' et p!
, valores
b

: r' et 11
: 900 manife-

ste suppeditant maximam penduli aberrationem

:

, ut quoad
istiusmodi aberratiouem sint
tangi—£&quot;.! P',—.
r −⇁∙
p p. Btang
&amp; ,
Densitas p., proutcolligitur ex aberratione penduli.
, cen-
setur quater vel quinquies maior quam densitas aquae.

6?- Eadem p. determinatur etiam experimentis in-
stitutis in libra torsionis
. Sit (Fig.'9*.) HH' (:Za') posi-
tio vectis horizontaliter librati;
E punctum medium
,
in
quo vectis appenditur (ilo, metallico verticali; HA circu-
lus horizontalis centro
E et radio EH (:a'); SS' (:26)
recta similiter horizontalis
,
transiens per
E
,
ibique secta
bifariam

; sint

S et S' centra sphaerarum inter se aequa-
lium et quoad volumen
,
et quoad massam (:m)
, ad se
trahentium massulas sphaericas m' et 111&quot; inter se pariter
aequales
, quarum centra in
H et H' ., Movebitur vectis
circa punctum
E immotum
, motusque iste repetendus ab
attraheute sphaerarum vi juxta circuli tangentem
, cui vi
jugiter adversatur vis torsionis ex filo metallico verticali

:

et quoniam corpuscula tu' et m&quot; eodem prorsus douantur
motu circa
E ,,satis erit alterum dumtaxat v. gr. m' con-
siderare
. Dicatur itaque ]: datus angulus HES

;

&amp; angulus
,
quem in fine temporis

: continet vectis cum initiali po-

sitione EH;
k distantia inter
5 et m' in line ipsius
t

: sol-

licitabitur m' iutta circuli tangentem vi attractiva-166
b
hak sin (h — e), eritque kº = a's - 2a'b cos (h --- e) + 6+;
experimenta insuper praebent vim torsionis proportionalem
angulo e , et consequenter expressam per ce : quoniam igi
tur labente e describit m' arcum ás , iccirco ( 50.3º. )
áre mb
sin (h -€)
dta k3
3
1
&lt;u&gt;aequatio ad motum&lt;/u&gt; corpusculi m' . Ob angulum &amp; valde exi
gaum , sin (h-5~s)) = sin h - e cos h , k - = [ a's -2a'b ( cosh
+sinh) + ] := k +. 2a'besio h ]
R3
3a'b esinh
+
ubi denotat k , valorem k respondentem initio k.
molus , quum nempe E = 0 ; proinde
sin (h-E ) sinh
23
E COS h 3abe sin’h
+
k.
k.
sinh
k.
k .
£ k cosh 3a'be sinh sinh
+
kb.
K.
her )
[ (a's tabo) cos h
k . k5.
sinh 2a'b cosah — 3a' b sin : h]
[laat69)cosh
- 2db -a'b sinä h] :
et factis compendii causa
mo
[ la'a + b ) cosh — 2a'b – a'b sinºh] +c = g' ,
166
m 1)
.
'
.
F.
∙∣⋮∙ nuUi—s) ,entque ka: a'a— 2a'b cos (71—5)-1—63;
experimenta insuper praebent vim torsionis proportionalem
angulo :,
et consequenter expressam per et: quoniam igi-
tur labente :describit m' arcum a's, iccirco (50.?)0.)
,d'38 mb ∙
aa;-a.- Fama—Q—ct
aequatio ad motum corpusculi m'
.Ob angulum :valde exi-
guum ,
sin (It—s) :sin h—s cos 11, k'3: [a&quot; −∙∙ Za'b (cos):
3 . 3
-i-ssinh) ⊣−∂∙∎∙∣− 'a': [le, −∙∙ Za'besin H's—:i?
⊣⋅− a.:-s;&quot;: h, ubi denotat k, valorem k respondentem initio
motus
, quum nempe s.: o ; proinde
sin (I;—s) sin 11 :
cbs Il 30'68 sin'h sin ls
∙−−−⋅∙≖∙−−∙− ∙∙∙⋅ ks' &quot;&quot;&quot;&quot; k3, −∎⋮∣⋮∍∙ fl&quot;, P, P,
sit' eos]: l Bez-'besin'h sinh : .

*

5,
w
cos-1. −−⋅ adb sin-t.] :
822&quot; −⋅⊼⋮−⊏≺∘⋅⋅−⊦∂⋅↗ coslt
∙∙∙⋅ ⊋∘∣∂∙∙− a'b sin' &amp;] :.
et factis compendii causa
iii- [(.-a ⊣⇁ 61) cos h—w—
a'6 .i..- h]
∙⊦≖⋅∸ −−∶
z'-167
mô sinh
wg'
23.
aequatio ad motum corpusculi m ' vertetur in
do e
a' ó (0) -- s ) ;
de²
ex cujus integratione ( 27. 28º. )
(9)*va
' @
'ri
{ = w + Ce + Ce
Sunt autem ( 27.300. )
.
va
cos (9 )* + v = on e(2)
, - ) vi
cose ( ) -va sine ( 2)
propterea
szaf1CTC&quot;) cos( ) +(c —c )V= sine ( :
sumptisque C +C' =C.cos C,, C — C = CV -ī sincs,
= - + 6, co [4 ( 4 ) + c ]
Minima vectis declinatio , í = u - C , ab aequilibrii positio
167
mb sinh
−−∣−∣⋮−≣∶−−≂−↩∾⊰
aequatio admotum c0rpusculi m' vertetur-in
,d'l
(: -d—t;:::g (6)—S):
ex cuius integratione (27 . 280.)
⋅⋅ .

*

.
.'.t ⋅↴∶
≖−↽−−−⇀↠∾−⊦∁∊ :(?) [l:—[— 08 &quot;(ä-') V
..
Sunt autem (27 .300.)
.;.
propterea &quot;
∙⋐⋅∶∶⊙−⊦≼∁−⊢∁⋅≱ ∾≘↙≺−⋚⊑⋅≻⋚⋅⋅−⊢ ((i—C' )
(V:; sint (?);;
sumptisque ∁−⊢ ∁∙∶∁≖∞∙ c, , ∁−∁∽−↽⇌∁≖⇂∕∙⊺∘⋮∥∁≖∙
⋅⋅ ∸ g' ?
≘−∙−−∙∾−⊦∁∙∞∘∣↣⋮≀ ≼⋮⇉⋟ −⊦∁≖∃∙
Minima-vectis declinatio ,
(:o)—G. ab aequilibrii pocitio-168
ue H'H respondet valori ( ) + Cs = ( 2n – 13 ;ma
= + c = 212nt : determinatis
itaque per observationem i'et s&quot; , eruetur inde
te&quot;
et ducta 00 ita , ut sit angulus HEO = w , perget vectis
moveri instar pendali horizontalis circum EO , impendet
que tempus tz = &quot; - t ad integram conficiendam oscilla
tionem , nimirum
ty=T
VAg'
Sit nunc a longitudo penduli simplicis ( 66) , quod intra
idem tempus t absolvit oscillationes suas : cum habeamus
ty = TO Van
a
erit 8 et denotante si densitatem sphaerae m ,

a'
r radium , substitutisque valoribus ( 5.0 )
4
пRр .8 3
mbsinh
wk3
471p3 M'bsinh
3wk. 3
proveuica
Ruwk3 р
p3M'bsinha
unde ar3bsinh
a'Rwk.3 1
Densitas pe sic determinata censetur esse ad
tatem ut 5,48 : 1 .
aquae densi
168
ne H'H respondet valori t'(g——,)ä -I-C,:(2n-1)1r; ma-
xima ⋮∣↾∶∶∾⊹∁≖ valori :
.(gwik) ⊹∁≖:2mr: determinatis
itaque per observationem eet :&quot; , eruetur inde
−−≘⋮−⊢⋮⋅∣∙ −− ⇄ ∙
et ducta O'O ita
,
ut sit angulus HEOzzæ, perget. vectis
moveri instar penduli horizontalis circum EO, impendet-
que tempus :::-:i &quot;—t' ad integram conGciendam oscilla-
tionem , nimirum
a'&quot; Vf- ∙

5
Sit nunc a longitudo penduli simplicis (66), quod intra
idem tempus :, absolvit oscillationes suas: cum habeamus
(3:11 V? .
. g a '
∙
erit?-;? ; et denotante p. denutatem sphaerae m ,

: - radium

. substitutisque valoribus (59)
∙∙∙⋅ ∙≤∙ nR ,— mbsinh— 4nr39'bsinh ∙ ∙
o ∙−−− 3 p. ,g 01:03 30 1:03 ,
provenit...
Rpali-03 a p. arabsinh
r3p'bsinh—c7 '
unde ∙∥∙∽ a'Rmkoï'l '
Densitas p. sic determinata censetur esse ad aquae densi-
tatem ut 5,48:1.169

7. ** Ex mariui aestus phoenomeno deduci pol
est ratio inter massam lunarem m &quot; et terrestrem m. Sit
m' ( Fig. 35 ) quodvis terrae punctum ; lunares vires distrah
entes punctum mé juxta mm&quot; et Am exprimuntur ( 62) per
2m &quot; Dcosh m&quot;Dsinh
(0) ,
( 0' ) :. x'3
X :' 3
quod in ordine ad lunam est h , x&quot; , in ordine ad so
lem sit H , X &quot; ; prodibunt consimiles vires solares
2MDcosH MDsind
X &quot; 3
( a ) , ( a ').
X'3
In casu angulorum h et H aequalium habemus
( 0) m&quot;X &quot;3
(a)
-- ( 0 )
M.2'3 ( a' )
caeteris vero paribus , ratio inter lunares et solares vi
res est eadem ac ratio inter respondentes aquarum ela
tiones ; denotante igitur p hanc secundam rationem , erit
X3
M
р
m '
M
m'
unde m
= P
3
x &quot;' 3
X &quot; 3
Observationes praebent p = 2 , 35333 : vide mechan, coel.
vol , 5. pag . 206.
Aliquid notatur de motu punctorum materialium
utcumque inter se connexorum .

84.* Vires motrices P, P , P&quot; , ... sollicitantes
istiusmodi punctorum massas m ,
mi ,
m' 0 resol
12
169
7.&quot; Ex marini aestus phaenomeno deduci pot-
est 'ratio inter massam lunarem m&quot; et terrestrem m. Sit
m' (Fig. 35) quodvis terrae punctum .; lunares vires distrah-
entes punctum m juxta mm&quot; et Am' exprimuntur (62) per
2m&quot;Dcosh m&quot;Dsinh (0) ∙ ∙−−∙↕∙−∙⋅∃−−⋅ (O,)
xara ⋅
quod in ordine ad lunam est h , a:&quot; ,
in ordine ad so-
lem sit H , X&quot;; prodibunt consimiles vires solares
2MDcosH' ⋅ MDsinH
W (a), ∙∎∎ Xl/3 (a').
In casu angulorum I; et H aequalium habemus
(o) -m&quot;X&quot;3l-(o') ∙
(a) Mx&quot;3 X(a') ⋅
caeteris vero paribus
,
ratio inter lunares et solares vi-
res est eadem ac ratio inter respondentes aquarum ela-
tiones ; denotante igitur p hanc secundam rationem
, erit
X&quot;3 d m&quot; M a:&quot;3
∙ &quot;&quot;;- un B −−∶ ∙ ,. x' 3
' m ? m X'3 31 ≊⋅∙ p:
Observationes praebent p——:2 , 35333 : avide machen. coel.
vol. 5. pag. 206.
Aliquid notatur de motu punctorum materialium
utcumque inter se conus-xarum.

843: Vires motrices P, P&quot;, P&quot;, ... sollicitantes
istiusmodi punctorum massas m , m'
, m&quot; , ...
resol-
12170
vantur singulae in ternas coordinatis axibus OX , OY ,
OZ ( Fig. 8 ) parallelas ; designentur per X , Y , Z, X',
Y , Z ' , X &quot; , . . componentes inde ortae ; sintque x, y,
3 , x ', y ', z' , x &quot; • punctorum coordinatae responden
tes temporit , ut ( 50, 1.º ) per
x == f (1 ), y = f(t), 2 = F (t), ' = fi (t), y = fz(t ), == F ,(t)
x ' = fale) ,y&quot; =f(e), z&quot; = F.(6),7: &quot; = f5e) , ... ) co)
exhibeantur aequationes ad actuales molus ; ad eos nem
pe motus , quos reapse concipiunt massae m , m' , m &quot; ,
ob actiones virium P , P' , P &quot;, ... Quoniam materialia
puncta , etsi mutuis nexibus liberala , viribusque ( 50. 4.0 )
dºx dz m
dạy
de² m
d²x
dla m . dia dla dea
d2z '
dca dc ?
sollicitala , adhuc tamen conciperent motus ( 0 ) ; ideo ,
attentis nexibus , consistent in aequilibrio vires
dez X - m
d2x
de2 Ymdạy
di?
2m X'
der'
di2 &gt;
7
dt2
Y - m
d²ý
dt2
daz'
Z ' - m '?
dt²
X &quot; -m.dºx
&quot;:
.
dt2
Conditiones ( a &quot; 13. 8. ) includuntur in conditionibus
aequilibrii quoad liberum punctorum utcumque connexo
um systema : liquet enim varians systema, semel libra
tum , adhuc permansurum in aequilibrio , etsi ejus pun
cla rigidis lineis immutabiliter connectuntur. Propterea
170
vantur singulae in ternas coordinatis axibus OX, Oï
,
OZ ( Fig. 8 ) parallelas; designentur per X, ? ,- Z, X',
1&quot;, Z' , X&quot; , . . . componentes inde ortae; sintque x,],

: , x', y', z', æ&quot; , ... punctorum coordinatae responden-

tes tempori t
, ut (50. 19) per
x:f(t), 7:112) ,z:F(t),.r ':f,(t),y ':f (t), z':F,(t),
(0)
x&quot;:-f.(t)
.
y&quot;:f.(t)
.
(&quot;zl-&quot;.m. m&quot;':--f3(t)
.
∙ ∙ ∙
exbibeantur aequationes ad actuales motus; ad eos nem-
pe motus
, quos reapse concipiunt massae m , m', m&quot;
,
ob actiones virium P, P', P&quot;, . . . Quoniam materialia
puncta , etsi mutuis nexibus liberata, viribusque (50. 49)
da.. mdzy de.
⋯∽∣↙≀≄∙↿∶ ∙↙∄≖∫⋅
&quot;B'—'— m-——- m— d,.
' md:2 '. d?
'md? ' dta '
,dzz, ad:-I?&quot; m d£2 '.. m dt:- '
∙ ∙ ∙
sollicitata, adhuc tamen conciperent motus (a); ideo ,
attentis nexibus, consistent in aequilibrio vires
(P:: (P] daz ,dzæ'
X
⇁∎−−∙ —p ⋅⋅⇁ ∙∙∙∙ ∙−−− '&quot;sz '
?
&quot;'d'T'z' Z &quot;&quot;da: X &quot;'de '
≀∠⋮⊺ &quot;rad :
&quot;
rnnndaæ ï'-——-—m '
...
dtï' ,..—z ⋯∠↙⊤ 'X— dtz
'
Conditiones (a&quot;f' 13. 8.0) includuntur in conditionibus
aequilibrii quoad liberum punctorum utcumque connexa-
rum systema :liquet enim varians systema, semel libra-
tnm, adhuc permansurum in aequilibrio, etsi eius pun-
cta rigidis lineis immutabiliter connectuntur. Propterea171
(xam )=o, 3(1— )= 0, $ (2- -o,
=[+ (rad ) – (xrm ) ] = o,

* [&gt; (2 mm ) -- ( - ) ]

&lt;math&gt; [ - (-) - = ( x ) ] &lt;/math&gt;
0
seu
daz ΣΖ - Ση
de² EX = sme , Y =sme &gt; ( 0 )
$( wYyX) =Em ( -e
)
3(y2=-1)= sm (voeding - :)
Eml
1
Z
day
de2 (0')
(2X == Z ) = 2m Z
dax
dta -
daz
dea :
formulae (o ') spectant ad translativum punctorum motum,
prima juxta OX , secunda juxta OY , tertia juxta OZ ;
formulae ( o“) ad rotatilem punctorum motum , prima cir
ca OZ , secunda circa OX , tertia circa OY ; eaedem ve
ro (o &quot; ) simul , ad punctorum motum circa fixam coor
dinatarum originem .
Haec facile nunc stabiliuntur. 1 . ** Habemus (20. 6.)
seu
æ ,dþ- ∙−−
∠∄≖⋍ ,
day d3æ ?
−∙∙ − ..(æy—yX) Em (
«: Tt&quot; ]—dt3 )
.
- ∑ dan: daz (zX—æZ): Zm( :217; — æ
—) :
formulae (c')/spectant ad translativum punctOrum motum,
prima juxta OX , secnnda juxta Oï, tertia juxta OZ;
formulae (a&quot;) ad rotatilem punctorum motum, prima' cir-
ca OZ , secunda eirca OX , tertia circa Oï ; eaedem ve-
ro (o&quot;) simul , ad pnnctorum motum circa fixam coor-
dinatarum originem.
Haec facile nunc stabiliuntur. ↿∙∘⋇ Habemus (20. b.)172
dar
Em dea dex,
dla
day Σm.
dt2
Em
daz,
dc2
da ,
Em dla
Em &gt;
Em: dt?
Hinc &gt;, ob (o' ) ,
der ΣΧ daz,
de
ΣΥ
Em ' dt2
ΣΖ
Σm (o' ' ' ) : Am dla
molo videlicet systemate punctorum m , m' , m &quot; ,
perinde ( 50. 4. ) movebitur gravitatis centrum ac si , co
euntibus punctis in ipsum centrum , applicarentur centro
eaedem vires P, P , P &quot; , ... cum iisdem directioni
bus , quibus puncta illa sollicitantur .
2 . '* Fac ut vires nihil sint aliud nisi punclo
rum actiones mutuae : denotante A actionem puncti v.
gr. m in aliud quodvis v . gr. m' , et A' actionem puncti m'
in m , erit ( 7 ) A=A' ; et expressa per D distantia inter
utrumque punctum , resolvetur A' in ternas coordinatis
axibus parallelas
x' #A D
TA
EA ;
ilem A in ternas iisdem axibus parallelas ( o'r )
ŁA to ,
EA D po', -A D
sumpto superiori signo si A , A' sunt vires attrahentes,
inferiori si repellentes . Quare EX =0, EY=0 , &amp;Z=0, et
consequenter
dér,
=0,
adi ?
day1
di?
dz,
-0,
=O ;
di?
in ea scilicet qua sumus hypothesi nullis viribus acce
↙≀⊴⋅↕⋮ d'), inuia—z- '
d'æ, dta (if/y. md:2 dïz, dta
dtz— Em 'dt: Em '
du ïm
Hinc , 06 (o') ,
d'æ, EX (Ph- Zï diru— ZZ
dF—Zm' dt&quot; &quot;Zm dt2 −∑⋯ (0 ):
moto videlicet systemate punctorum m , m' , m , . . ,
perinde (50. .f.&quot;) movebitur gravitatis centrum ac si, eo-
euntibus pnnctis in ipsum centrum
, applicarentur centro
eaedem vires P, P', P&quot; , . . . cum iisdem directioni-
bus ∙ quibus puncta illa sollicitantur.
294: Fac ut vires nihil sint aliud nisi puncto-
rum actiones mutuae :denotante A- actionem puncti v.
gr.m in aliud quodvis v. gr. m', et A' actionem puncti m'
in m, erit (7) A:A'; et expressa per D distantia inter
utrumque punctum, resolvetur A' in ternas coerdinatis
axibus parallelas
3.&quot;—æ ∙−⇠ 7—7 ...-,: z—z .
drA U , A D A D
itcm A in ternas iisdem axibus parallelas (o&quot;)
æ—x' J—y' z—z' ∙
sumpto superiori signo si A, A' sunt vires attrahentes,
inferiori si repellentes. Quare XX :0, ∑∟ -o, ZZ:o, et
consequenter
in ea scilicet qua sumus hypothesi nullis viribus acce-173
leratricibus agetur gravitatis centrum , nulloque ob mutuas
panctorum actiones afficietur motu. Huc spectat princi
pium de conservatione centri gravitatis.
3.°# Super planis XOY, YOZ, XOZ fiant proje
ctiones a, b, c, a' , b' , c' , a &quot; , 6 &quot;, c &quot; , a ' , , . . arearum
descriptarum a radiis vectoribus punctorum m, m' , m&quot;,
computatis radiis ab origine coordinatarum : erunt ( 50. 8. )
xdy — ydx Σmda = Σm ydz - zdy και Σmdb = Σm 2 2
zdxxdz Σmdc = Ση 2 unde
daa 2Em -Σm α2
dta dt
22m d2b
dta
daz
у

;

= Em (:
dla e )

: )

dec dex dez 2Σm -Σm dt2 sm
(
20
de?
et consequenter ( o &quot; )
d'a 22m dt² 8(xY4yX), 28mmdla = Eby2 — zY),
( 0 )
dac 2Em =E( zX-xZ) .
dta
4.0 # Si vires consistunt in mutuis punctorum actio
nibus , erunt ( 2.º o &quot; )
173
leratricibns agetur gravitatis centrum
, nulloque ob mutuas
punctorum actiones afficietur motn. Huc spectat princi-
pium de conservatione centri gravitatis.
3.&quot;; Super planis XOï, ïOZ, XOZ fiant proie-
ctiones a, b, c, a', b'. c', a&quot;, b&quot;, c' ', a'&quot;, ∙ ∙ ∙ arearum
descriptarum a radiis vectoribus punctorum m, m', m&quot; , .. .
computatis radiis ab origine coordinatarum : erunt (50. 8.&quot;)
∑⋅↾⋅⊿↙↓⋅−−≔−∑⋯⊔−−−−∫−≌∙∑⋯↲≀⊨∑⋯⇅−−−≖−≗↶∙ d d d—d
2 ∙ 2
Emma.—zn. fix—?? ,
unde
22m ⋛∙∶−≧∶−−⋅∑⋯≺⊰≵ :::-£v:— 7
id?-:?) ∙ ⋮⋯∶⊜≀≀∶≖∂−↽−−≖⋅⋯⋅↗≺ :: −− ::z ⇋
d'c— d:.r ædaz
et consequenter (o&quot;)
ZZm −⋛⊴⋮↥⋮−⇌∑≺∞⊺−∜∑⋟ ,
ZZmäï—b- :Zþ'Z—zï),
d (a')
220: ⋅⊋≖−∶∶ :2(zX—-æZ).
494: Si vires consistunt in mutuis punctorum actio-
nibus, erunt (2.o o&quot;)174
8 (xY - 7X ) = 0 , (yz - zY) = 0 ; $ (zX -- XZ) = 0;
ideoque
dra dc Emdl2
d2b Σm dc2 Emadt²
}
et computatis areis ab initio temporis t ,
Ema = Ct , Emb = Ct, Emc = C &quot; ! (0 ) :
huc special principium de conservatione arearum. Formu
lae ( o &quot; ) adhuc obstinent , etsi in systemate invenitur pun
ctum fixum , modo tamen in pancto illo collocetur origo
coordinatarum : siquidem vigent in casua equationes ( o&quot; ' ' ) ,
unde profluunt ( o &quot; ).
5.0 * Si arcus s refertur ad tres axes orthogona
les , ejus incrementum infinitesimum ds poterit spectari
tanquam diagonalis parallelepipedi rectanguli sub later
culis dx , dy , dz : hinc.
dsa = dx2 + dyz-tdz?, et consequenter ( 50, 2.0 )
v2= dx2+ dyatdz
dla
Erit itaque
Emvdv = Em
der
de²
d'I
ayt
ar
- .
ac proinde (0 )
Emvdv = E (Xdx + ydy + Zdz) ( o&quot; &quot; ' ) .
Fac ut E (Xdx + Ydy + Zdz) exsistat differentiale exactum ,
!
174 .
£(xï—77X):o , ZUZ—zï): :X(zX—xZ):o;
ideoque
d'a 'dzb (130
dt2
?
et computatis areis ab initio temporis :,
2ma:Ct , 2mb:C't, ch:C&quot;t (o&quot;):
huc spectat principium de conservatione arearum. Formu-
lae (ov') adbuc obstinent
, etsi in systemate invenitur pun-
ctum fixum, modo tamen in puncto illo collocetur origo
coordinatarum: siquidem vigent in casua equationes (o&quot;'),
unde profluunt (o&quot;).
594: Si arcus :refertur ad tres axes orthogona-
les, eius incrementum infiuitesimam ds poterit spectari
tanquam diagonalis parallelepipedi rectanguli sub later-
culis dx
, dy , ds :hinc.
ds':dæï-l-df3-l-dz3 ,
et consequenter (50, 2?)
daga—.dyzudzz
dt2
⋅
02.—
Erit itaque
da a :
Zmpdu:2m(d£fdx : (iyd): ↿ d
zdz) .
ac proinde (o') ∙
vadv :!(de ⊣− ⊺⊄↴⋅⊺∫ ⊣−∅∠∄≂≻⋅⋅
(o&quot;').
Fac. ut XXdæ—fïdJ—l—Zdz) exsistat differentiale exactum.175
prodeat nimirum ex differentiatione cujusdam functionis
F (x , y , z, x ', y , z, x &quot; , ... ) ; habebis
Em (u2 — V.2) = 2F (x ,y ,z,x',...) —2 F (xo,9o , zo , x '. , ... ) ;
quantitates v. , xo, Yo, Zo, x'o, ... respondent initio mo
tus. Consequitur, quod, redeuntibus iisdem coordinatis, ea
dem quoque redibit summa virium vivarum : huc spectat
principium de virium vivarum conservatione.

6. °* Denotent &lt;math&gt;h, i, k , h , i , k ' , h '' ...&lt;/math&gt; coordinatas punctorum &lt;math&gt;m, m', m''&lt;/math&gt; in ordine ad novos axes, qui et paralleli sint axibus &lt;math&gt;OX , OY , OZ ,&lt;/math&gt; et originem habeant in communi gravitatis centro; erunt
x = xrth , y = yiti , z = zetk , x' =xith' ,
y = yiti, z= z+k ', w &quot; = xrth &quot; , ... ;
quibus valoribus substitatis in ( o &quot; ) , attentisque aequatio
nibus ( 20)
dah deh , dai dai Σm Σm=0, Σ . Σm=0
dcz dta des dt2 =
dek Σm-
dt
dakı Em=0
dc2
1
nec non aequationibus ( o &quot;&quot; ), prodibunt
dai dah
2 ( XiY) = Em ( h TI dea dt2
E ( iz - kY) = Em
(ala dih),
(
- ) com
(akone )
( o &quot; )
dah
ElkX_hZ) = Em ( k dt2
175
prodeat nimirum ex differentiatione cujusdam functionis
F(x,y, :, x', y', z', æ&quot;
, ..
.) ; habebis
M(æ—voz):2F(æ,7,z,æ',..-) -2 P(æo,yo , zo , x', , ..
.);
quantitates v,
, æ., y,,zo, æ'o,
... respondent initio mo-
tns. Consequitur, quod, redeuntibus iisdem coordinatis, ea-
dem quoque redibit summa virium vivarum: liuc spectat
principium de virium vivarum conservatione.-
604: Denotent h, i, k, h', i', A', I:&quot; .. . coordi-
natas punctorum m, m', m&quot; , . ..
in ordine ad novos
axes , qui et paralleli sint axibus OX, Oï
, OZ
,
et ori-
ginem habeant in communi gravitatis centro; erunt
r—æl-i-h ,.szl-ï-i !≖∶∅∎⊹∣⊂ 'x':xx-Fll' ,
f:.yg-I-i', z':z,-l—k', x&quot;:æ,-l-h&quot;, ...;
quibus valoribus substitutis in (a&quot;)
, attentisque aequatio-
nibus (20)
d'h dïb, dii dïi,
EMzzï—an—O, zmcïS—dtï Zm--o ∙
d']: dïk,
ZmäF—äz; Zm—o ∙
nec non aequationibus (o&quot;'), prodibunt
E(hX—-iï):2m(hää—ci £b) ∙ dr2
⋅ ∙ dq: ti*i z
(iz—mzn. (.
&amp;? −:. $) ∙
(o....)
d.,, dal. ∑≺⋌⊔∅≻⇌∑⋯≺∣≂−∂−↙⋮−−∣⋅⋮⋮↙⇆⋟∙ .176
Formulae (o &quot; ) se habent ad commune gravitatis cen
trum prorsus ut formulae ( o &quot; ) ad fixam coordinatarum x ,
J, 2, x' , ... originem 0 , respiciuntque relativum syste
matis motum quoad ipsum gravitatis centrum:

7. • * Relativus rigidi liberique systematis motus
quoad gravitatis centrum determinatur per ( o &quot; &quot; &quot; ) ; motus
vero ipsius centri per ( o' ' ' ) Ad haec : si resultans ex omni
bus viribus systemati rigido applicitis transit per gravi
tatis centrum , nullus inde orietur relativus systematis mo
tas quoad ipsum centrum : etenim quoad istiusmodi mo
tum similiter procedet res ac si resultans illa exerceretur
contra punclum fixum ( 6.&quot; ). Eadem de causa , accedentibus
novis viribus , relativus systematis motus quoad gravitatis
centrum nullo pacto turbabitur , ubi eae resultantem sup
peditent transeuntem per centrum illud.
85.&amp; Pauca subjungentes de motu rigidi systematis cir
ca axem fixum praemittimus illud: praeter orthogonales axes   &lt;math&gt;OX , OY , OZ ,&lt;/math&gt;   ( Fig. 9 ) sint alii tres axes similiter orthogonales On, Op, Oq, quibuscum ii angulos efficiant designatos per ( xn) , (xp ) , ( aq) , (yn) , ( yp ) , (99) , (zn) ,
(zp ), ( z9 ) . Si panctum E, quod referebatur ad axes OX,
OY, OZ , referendum sit ad axes On, Op , Og , quaeri
tur relatio inter veteres coordinatas x , y
nip , q. Ponatur OE = a, et per (ax ), (ay ) , (az), ( an ) ,
(ap) , ( aq) exhibeantur anguli , quos OE facit com axibus
OX , OY , OZ , On , Op , Oq: erunt ( 50. 6º . )
ma
z et novas
cos (ax) =cos (an) cos ( xn) +cos ( ap) cos (xp) +
cos (aq) cos (xq) , cos (ay ) =cos ( an ) cos (yn ) +
cos (ap) cos (yp) + cos (aq) cos (79) , cos (az)
eos ( an) cos ( zn) * cos(ap) cos (zp) + cos ( aq ) cos ( 29) .
176
Formulae (o&quot;&quot;) se habent ad commune gravitatis cen-
trum prorsus ut formulae (a&quot;) ad fixam coordinatarum æ,
y, 2, æ',
.. . originem O, respiciantque relativum syste-
matis motum quoad ipsum gravitatis centrum:
73»: Relativus rigidi liberique systematis motus
quoad gravitatis centrum determinatur per (o&quot;&quot;); motus
vero ipsius centri per (o&quot;') Ad haec: si resultans ex omni-
bns viribus systemati rigido applicitis transit per gravi-
tatis centrum
. nullus inde orietur relativus systematis mo-
tus quoad ipsum centrum : etenim quoad istiusmodi mo-
tum similiter procedet res ac si resultans illa exerceretur
contra punctum fixnm (S.&quot;). Eadem de causa , accedentibus
novis viribus
, relativus systematis motns quoad gravitatis
centrum nullo pacto turbabitur , ubi eae resultantem suppeditent transeuntem per centrum illud. '
853 Pauca subiungentes de motu rigidi systematis cir-
ca axem fixum praemittimus illud: praeter orthogonales a-
xes OX, Of, OZ (Fig. 9) sint alii tres axes similiter or-
thogonales On, Op, Oq, quibuscum ii angnlos efficiant de-
signatos Per (æ&quot;)s (æpl- (xq) :(f&quot;) , 07)» (f?) :(znls
(zp), (zq). Si punctum E, quod referebatur ad axes OX,
OV, OZ, referendum sit ad axes On, Op, Oq , quaeri-
tur relatio inter veteres coordinatas æ , y , :. et novas
n , p , q. Ponatur OE :a, et per (aæ)
,
(ay) ,(az), (an),
(ap), (aq) exhibeantur anguli, quos OE facit cum axibus
OX , Oï
, OZ , On. , Op , Oq: erunt ( 50. 60.)
cos (aæ):cos (an) cos (xn) —-[-cos (ap) cos (æp) −⊢ ⋅
cos (aq)cos (xq) , cos (ay) :cos (an) cos (yn) −∙⊢
cos (ap) cos (yp) ∙−⊢ eos (aq) cos (rq) ∙
005 (az) −−∶
eos (an) cos (zn) —,l-cos(ap)cos(zp)-—- eos (aq) cos (zq).
3751177
Sed cos (ax ) =
a , cos(ay) = cos(az ) = a
cos (an) =
, cos ( ap ) = .. cos (aq ) =
= 9
a
adhibitis igitur substitutionibus , provenient
x = ncos( an) + pcos (xp) + qcos(xq) ,
y = ncos(yn) + pcos (yp ) + acos(yq) ,
x = ncos( zn ) + pcos(zp) + qcos(zq) ;
formulae praebentes quaesitam relationem .
Nunc 1 . ** Sit OX rotationis axis, datumque systema
tis punctum reperiatur constanter in plano YOZ: si per OX
et per punctum illud ducitur planum occurrens plano YOZ,
satis erit determinare situm intersectionis istorum plano
rum ut innotescat systematis positio. Concipiamus itaque
novos axes orthogonales On , Op , Oq sic constitutos , ut
firmiter adhaereant systemati , primusque incidat in OX ,
tertius in intersectionem illam ; erunt
y= pcos(zq ) + qsiu(z9) , z =qcos (29) — psin(zg) :
adhibita substitutione in secundo membro secundae ( o &quot; .84)
animadvertendo quod variato e non ideo variant novae co
ordinatae , factoque
2 m ( p2 +9 ) = B ,
proveniet
d ' (29 )
di2 - $ (72—28 )
(o'r) :
∙∙∙⋅ 177
&amp; Sed ∾⋇≺⊄∣∙↿∶≻∶−−⊶⋚ ,cos (ay): a , c08(az):ä— .
']
...
(an):
⋮⋮−∙ costam: g.... (aq) ⇌⋅−− −↙⋅↓−
.
adhibitis igitur substitutionibus , provenient
x:ncos(æn) -l-pcoa (æp) ⊣− 9005(-qu :
)»:ncosU'n) pcos (ïp) −∣⋅− ⊄∾≘∩⊄⋟ '

: &quot;cos(zn) −⊢ pcos(zp) -I-— qcos(zq) :

famulae praebentes (quaesitam relationem.
Nune ↿∙∘∙ Sit OX rotationis axis, datumque systema-
tis punctum reperiatur constanter in plano ïOZ: si per OX
et per punctum illud ducitur planum occurrens plano ïOZ,
satis erit determinare situm intersectionis istorum plano-
rum ut innotescat systematis positio. Concipiamus itaque
novos axes orthogonales'.0n, Op , Oq sic constitutos, ut
firmiter adhaereant systemati, primusque incidat in OX ,
tertius in intersectionem illam; erunt
7: pcos(zq ) ⊣−⊄⊗∃∥≺∅⊄⋟ :
3 −−∶ quos(zq) --
psiu(zq) :
adhibita substitutione in secundo membro secundae (o&quot;.84)
animadvertendo quod variato :non ideo variant novae co-
ordinatae, factoque
. Zm(p'-l-q3)-——-B.
proveniet
(P(zq)
- dt2
z.. 1
-B— ZUZ—zï) (o&quot;):178
d (29 )
velocitas ( 50. 2º BE . ) respondet radio 1 , diciturque dla
velocitas angularis: binomia patq , p'? + 92.. nihil sunt aliud
nisi quadrata perpendiculorum ex m , m' , ... in axem On de
missorum ; summa productorum ex massis m , m' ... in
quadrata respondentium perpendiculorum , seu m (pa+92) +
m ' ( p2t 92) + . . . vocatur momentum inertiae systema
tis m , m' , .... quod axem On .
2. °# Ponamus vires acceleratrices consistere in so
la gravitate g, axesque Ox , OY jacere in horizontali pla
no: erunt Y = 0, Z
8 , et consequenter
7
de( 29 )
1 1
dla B
&amp; Emy =
1
g Em [ p cos ( zq) +qsiu ( zq ) ] E
&amp; [cos(zq) . Emp + sin ( zq) . Emq).
Fac ut illud systematis punctum , quod posuimus ( 10.)
reperiri constanter in axe Oq, sit gravitatis centrum; ex
sistent ( 20)
Σmp = p,Σm = 0 , Σmg = qΣm :
proinde
d? ( )
- 1/3
sin ( zqı ) . Em ;
dt?
B
891
quae prius multiplicata per 2d( 29, ) , ac dein integrata
praebebit
[da ] = - 69.cos/ 291). Em + c
x
178
. ∘
'
d(zq) velocitas ( 50. 2 . ..
.
) 7:2- respondet radio1,d1c1turque
velocitas angularis: binomia phi-qi, p'H—q'æ. nihil sunt aliud
nisi quadrata perpendiculorum ex m, tu',... in axem On de-
missorum ; snmma productorum ex massis m , m'
...
in
quadrata respondentium perpendiculorum, seu m (pi-I—q'H-
m' (p'ï-l- q'3)-i- ....
vocatur momentum inertiae systema-
.tis m, m',
.... quod axem On.
2.0a Ponamus vires acceleratrices consistere in so-
la gravitate g, axesque OX , Oï jacere in horizontali pla-
no: erunt T:o, Z:
-— g ,et consequenter
d3(z ) 1 ↿ ∙
de? ∶−∙−−↕≣− g Em]: —B—g2m[pcos(zq)—l-qstn(zq)]
1 - .
−−−−− -B- g[cos(zq). Zmp −↘∟ stn (zq). qu].
Fac ut illud systematis punctum ,quod posuimus (10.)
reperiri constanter in axe Oq, sit gravitatis centrum; ex-
BlStent (20) ∣
Zmp :plzm:o , qu :q12m :
proinde*
dï ↿ ∙)— B gq, sm (sq,). Em;

quae prius multiplicata per 2d( sq, )
, ac dein integrata
praebebit
. d
Z [ 2
2 .
[ld-g-l ∸∶−∙∙∙⋅ ∙∙∙ ï
gqx 008(zq1). Zm ⊹∁ ∙
iis179
Exsistentibus in initio motus d (291)
= uo et ( 291) = a , erit du
2
C = uo% + B 69 , cosa. Em :
propterea
d (290) 72
=u' . +
dt 2/3
891 [ cosa - cos ( 291) ] Em (o' ) .
Huc spectat theoria penduli compositi.
3.•* Intelligantur m , m' , m &quot; , .... coire in u
nicum punctum annexum axi horizontali Ox ope rectae r;
exsurget pendulum simplex : in casu
p = p = p = ... = 0 , q = 9 = 9 &quot; = ... = 9 = r ,
B = 2m(p + g ”) = 2n ;
et consequenter quoad pendulum simplex
d ( 292) 7
2
[Company *== +
s [ cos a - cos ( 291) ] (o &quot; ).
2
4.0# Facto 8
2
B
89. EmEm , proveniet
B
9 , £ m col) ;
longitudo videlicet penduli simplicis , quod suas perficit
oscillationes eodem tempore ac pendulum compositum . Re
cole quae diximus (67 ) .
5.° * Pone nullas esse vires acceleratrices i
erit ( 1. ° 0 ' )
179
d(zq !)
dc
Exsistentibus in initio motus :u. et (zq.):a, erit
.
2 C:u.,2 −⊦⋅ ïgq, cosa. Em:
prapterea
d(dZQtli:u⋅−⊢ ∙−⋛−∊⊄∙ [cosa — cos (zq.)]2m (a').
Huc spectat theoria penduli compositi.
39»: Intelligantur m , m', m&quot;, ... . . coire in u-
nicum punctum annexum axi horizontali OX Ope rectae r;
exsurget pendulum simplex: in casu
p::p'::p&quot;: ∙ ∙ ∙ −−∙−−∶∘ , qzq'2q&quot;: ∙ ∙ ∙
∶⊄∎∶↿∙ ,
&quot;B::ZmQF-l-qa) claim ;
et consequenter quoad pendulum simplex
↙≀≺≦≦∣≖⋝⊺−−⋅↙∘≖ ; f ,.
(....
...
(..., ].
2 2
4.0a Facto ∙∓− g :ïgq, Em
, proveniet
B .
r'.&quot;—∙−∙∙ q,2m
Om) ;
longitudo videlicet penduli simplicis, quod suas perficit
oscillationes eodem tempore ac pendulum compositum. Re-
cole quae diximus (67).
5.
.,
Pone nullas esse vires acceleratrices: ,
erit (1. ∘ o)180
dº(aq )
dia
d( 24)
unde velocitas angolaris u =
dc = const. = u , .
1
1 Motus igitur exsistet uniformis , eritque velocitas angu
laris ad velocitatem puncti v . gr. m ut 1 ad radium cir
culi descripti ab ipso m , seu
1
u : v =1 : V patqz , ac proinde v = u ? (patoga )
quoad illud itaque punctum obtinebit vis contrifuga expres
sa ( 51 ) per
= u’m V pat92 .
V pr + q2
1
vam 2
Resolvatur haec vis in ternas coordinatis axibus On, Op,
Og parallelas ; prodibunt 1
+
9
р
0 , u²mV p2tga .
V p²ta?
wimb p'tgo.
Foto &gt;
seu
0
0 , ump , u'mg :
1
quoad totum ergo systema habebuntur 2
0 , użEmp , u’Emq ;
ideoque orietur pressio in axem OX. Prima membra formu
larum ( a : 13. 8.° ) in casu fiunt
0 , użEmp, użEmq , użEmnp , użEmng , u’Em (pa - pa ) :
!
hinc ubi fuerint 1
1
Emp= 0 , Emg = 0 , Emnp = 0, Emng = 0 ( o'r) ,
180
(l*(z'q)
d?
d(zq)
dc

: o

, unde velocitas angularis ::: :const.-zuo,
Motus igitur exsistet uniformis , eritque velocitas angu-
laris ad velocitatem puncti v. gr. m ut 1 ad radium cir-
culi descripti ab ipso m, seu
∣ ...—....
a: p −−−−−↿ :Vpl-I—qa , ac proinde V::u' (pH-q2 )
quoad illud itaque punctum obtinebit vis centrifuga expres-
sa (51) per
vam
l/P'&quot;l'qa −−∶
&quot;&quot;&quot; Vlf-*?&quot;
-
Resolvatur haec vis in ternas&quot; coordinatis axibus On. Op,
Oq parallelas; prodibunt
seu
0, uïrnp
,
'u'mq :
quoad totum ergo systema habebuntur
o , u'Zmp , u'qu;
ideoque orietur pressio in axem OX. Prima membra formu-
larum (a'm : 13. 8.&quot;) in casu fiunt
o , ti*Zmp, uazmq , u'Zmnp , u'Zmnq , u'Zmþq—pq) :
hinc ubi fuerint
Zmp:o , M.,—:a, Zmnpzo, zmnqzo (atur)
'181
1
vires centrifugae se muluo librabunt independenter ab axe
Ox
, nullamque iste axis patietur pressionem
. Prima et se
cunda
(oh
) important
( 20. 6.
) transitum axeos On seu
OX per gravitatis centrum tertia vero et quarta important peculiarem quandam axiuin On
, Op, Oq positionem
relate ad punctorum
m
,
m
'
,
m
&quot; systema
. Porro si On
,
Op
, Oq ita sunt positi, ut suppeditent Emnp
=
0
, Emng
=
0
,
Empq
=
o
, appellari solent principales systematis axes in
ordine ad originem
itidem quae momenta ad eos referuntur
, et ipsa dicuntur principalia inertiae momenta
. Ex
pletis tertia et quarta
(
o
&quot;
&quot;&quot;
)
, non autem prima et secun
da
, ex omnibus viribus centrifugis resultabit
( 13. 9.0 10.9
)
vis premens rolationis axem in O. 6. '* Superiores formulae manifeste applicantur
continuo materialium punctorum systemati mutando
&amp; in
ſ et
m in dm
, integrationemque protendendo ad totam
systematis massam
.
7.9 Saepe videmus corpora impulsu aliquo loca
liter mota affici simul rotationis motu

: etiam praecisis

,
quae diximus
( 84
)
, sic ostendi potest motum istum oriri ex
eo quod impulsus non habeat directionem transeuntem per
centra gravitatis corporum
. Sic
G gravitatis centrum cor
poris MM
'
( Fig
. 46
)
, et AZ vis corpori cominunicata
..
Ducatur per
G ad AZL perpendiculum GL dividalur

bifariam AZ in
C
, et resolvatur CA in AD per
G tran
seantem
, et in AB normalem rectae AZ producatur AG

donec GF aequet GA intelligatur AD applicita ad punclum F , sitque FK = AD resolvatur FK in FH parallelam et FI perpendicularem rectae LGN

: quibus posi

tis
, substituti poteront vi AZ quatuor vires CZ
, AB
, FI
,
FH
. Jamvero CZ
, FI utpote aequales
, parallelae et ad
eamdem plagam tendentes contrahuntur in unam reprae
sentatam per GE
( 11
)
=GZ
+Fl
=AZ
, transeuntem per
G
, eidemque AZ parallelam proinde movebitur centrum

G non secus ac vis AZ ipsi esset applicata
. At duae aliae
↿∂⋅↿
vires centrifugae se mutuo librabunt independenter ab axe
OX, nullamque iste axis patietur pressionem. Prima et se-
cunda (o&quot;&quot;) important (20. b.) transitum axeos On seu
OX per gravitatis centrum: tertia vero et quarta impor-
tant peculiarem quandam axium On
, Op, Oq positionem
relate. ad punctorum m, m', m&quot; ,... systema. Porro si On,
Op, Oq ita sunt positi, ut suppeditent Zmnp:o, Zmnq:o,
Zmpq:o, appellari solent principales systematis axes in
ordine ad originem
O

itidem quae momenta ad eos refe-
runtur, et ipsa dicuntur principalia inertiae momenta. Expletis tertia et quarta (on&quot;), non autem prima et secunda, ex omnibus viribus centrifugis resultabit (13. 9310!)
vis premens rotationis axem in O.

63. Superiores formulae manifeste applicantur
continuo materialium punctorum systemati mutando
2 in
]et as in dm
,
integrationemque protendendo ad totam
systematis massam.

7.&quot; Saepe videmus corpora impulsu aliquo loca-
liter mota aflici simul rotationis motu: etiam praecisis,
quae diximus (84), sic ostendi potest motum istum oriri ex
eo quod impulsus non habeat directionem transeuntem per
centra gravitatis corporum. Sit
G gravitatis centrum cor-
poris MM' (Fig. 46)
, et AZ vis corpori communicata.
Ducatur per
G ad AZL perpendiculum GL; dividatur
bifariam AZ in C, et resolvatur CA in AD per
G tran-
seuntem, et in AB normalem rectae AZ; producatur AG
donec GF aequet GA; intelligatur AD applicita ad pun-
ctum
F
, sitque FK: A

resolvatur FK in FH paral-
lelam et FI perpendicularem rectae LGN

: quibus posi-

tis, substituti poterunt vi AZ quatuor vires CZ,AB, FI
,
FH. Iamvero CZ, FI utpote aequales
, parallelae et ad
eamdem plagam tendentes contrahuntur in unam reprae-
sentatam per GE (11):GZ—-FI:AZ, transeuntem per
G, eidemque AZ parallelam proinde movebitur centrum

0 non secus ac vis AZ ipsi esset applicata. At duae aliae182
.AB, FH utpote aequales , parallelae , et ad contrarias par-
tes tendentes , nequeunt gravitatis centrum e suo loco di-
movere : spectatis itaqne istiusmodi viribus, immobile eqn-
sisteret gravitatis centrum; sed eae sese mutuo non de-
struunt, cum e diametro non opponantur. Aliud ergo praestare non poterunt nisi corporis rotationem circa gravitatis centrum. Rotationis motus incipit circa reetam aliquam seu axem, et quoniam in omnes corporis particulas ex rotatione inducitur vis centrifuga; hinc si vires centrifugae inde ortae aequilibrantur circa rectam illam, invariabilis exsistet rotationis axis, defereturque per spatium sibimet semper parallelas; secus, mutabitur indesinenter rotationis axis donec ad aequilibrium deveniatur.

=== De fluidorum corporum aequilibrio. ===
86. Fluida corpora spectamus veluti materialiam punctorum congeries; quae puncta, utpote invicem independentia, vel minimo cedunt impulsui. In massa fluida undique librata sume punctum quodvis [exhibemus per &lt;math&gt;( x , y , z )&lt;/math&gt;, denotantibus &lt;math&gt;x, y, z&lt;/math&gt; ejus coordinatas] sollicitatum vi acceleratrice &lt;math&gt;\varphi&lt;/math&gt; praebente componentes &lt;math&gt;X, Y, Z&lt;/math&gt; coordinatis axibus &lt;math&gt;OX, OY, OZ&lt;/math&gt;,  parallelas et per punctum illud fac ut transeat superficies &lt;math&gt;k&lt;/math&gt; plana, rigida atque infinitesima: consistet &lt;math&gt;k&lt;/math&gt; in aequilibrio; et consequenter pressiones hinc et illinc exercitae in &lt;math&gt;k&lt;/math&gt; ab circumpositis massae fluidae stratis, erunt vires aequales et directe contrariae, simulque normales ipsi &lt;math&gt;k&lt;/math&gt;. Ejusmodi pressionum alteram repraesenta per &lt;math&gt;\varpi k&lt;/math&gt;; ratio &lt;math&gt;\frac{\varpi k}{k}(= \varpi)&lt;/math&gt; dicitur pressio hydrostatica exercita &lt;math&gt;k&lt;/math&gt; apud punctum &lt;math&gt;( x, y , z )&lt;/math&gt; contra aream ( = 1 ) sumptam in plano superficiei &lt;math&gt;k&lt;/math&gt;. In eadem massa fluida fac ut per punctum alterum &lt;math&gt;( x_0, y , z )&lt;/math&gt; transeat talis superficies &lt;math&gt;k_0&lt;/math&gt; plana, rigida et infinitesima, quae communem habeat projectionem cum superficie &lt;math&gt;k&lt;/math&gt; in plano &lt;math&gt;YOZ&lt;/math&gt;; voca &lt;math&gt;h&lt;/math&gt; projectionem illam, et &lt;math&gt;\varpi_0&lt;/math&gt;, hydrostaticam pressionem apud punctum &lt;math&gt;(x_0, y , z)&lt;/math&gt; contra aream ( =1 ) sumptam in plano areae &lt;math&gt;k_0&lt;/math&gt;. Massa fluida adhuc perget esse librata, etsi in qualibet ejus portione intelliguntur puncta rigidis lineolis firmiter connecti, seu, quod eodem redit, etsi quaelibet ejus portio fit solida: ponatur id contingere portioni cylindricae habenti rectam parallelam axi &lt;math&gt;OX&lt;/math&gt; pro generatrice, et &lt;math&gt;k , k_0&lt;/math&gt; pro basibus; denotet &lt;math&gt;\mu&lt;/math&gt; densitatem massae fluidae apud punctum &lt;math&gt;(x , y , z)&lt;/math&gt;; sitque &lt;math&gt;x &gt; x_0&lt;/math&gt; Exprimetur per&lt;math display=&quot;block&quot;&gt;h\int_{x_0}^x \mu X dx
&lt;/math&gt;summa ex viribus motricibus, quibus juxta &lt;math&gt;OX&lt;/math&gt; sollicitantur puncta illius portionis; exprimenlur praeterea per  &lt;math display=&quot;block&quot;&gt;\frac{h}{k_0}\varpi k_0, -\frac{h}{k}\varpi k &lt;/math&gt;pressiones exercitae juxta eumdem OX , altera in basim ko,altera in basim k quod spectat ad pressiones contra lateralis superficiei puncta, eae utpote normales generatrici rectae nullas dabunt componentes axi OX parallelas. Quia igitur solidata portio perseverat in aequilibrio, iccirco

&lt;math display=&quot;block&quot;&gt;h\int_{x_0}^x \mu X dx + h\varpi_0 - h\varpi = 0, \, \mathrm{unde}\,   \varpi = \varpi_0 + \int_{x_0}^x \mu X dx .
&lt;/math&gt;

Haud mutata positione superficiei &lt;math&gt;k_0&lt;/math&gt;, revolvatur utcumque superficies &lt;math&gt;k&lt;/math&gt; circa punctum &lt;math&gt;(x , y ,z)&lt;/math&gt;: permanebit secundum membrum ultimae aequationis; ergo et primum. Quare perseverabit in eodem valore hydrostatica pressio quoad omnia plana per punctum illud utcumque ducta: huc spectat principium de aequalitate pressionis. Consequitur, si recta generatrix sumitur parallela, prius axi &lt;math&gt;OY&lt;/math&gt;, deinde axi &lt;math&gt;OZ&lt;/math&gt;, denotantibus &lt;math&gt;\varpi_0',\varpi_0''&lt;/math&gt; hydrostaticas pressiones apud puncta &lt;math&gt;( x , y_0, z) , (x , y , z_0 )&lt;/math&gt;, fore etiam&lt;math display=&quot;block&quot;&gt;
\varpi = \varpi_0' + \int_{y_0}^y \mu Y dy , \varpi  = \varpi_0''+ \int_{z_0}^z \mu Z dz &lt;/math&gt;Terni valores &lt;math&gt;\varphi&lt;/math&gt; differentiati, primus quoad &lt;math&gt;x&lt;/math&gt;,  secundus quoad &lt;math&gt;y&lt;/math&gt;, tertius quoad &lt;math&gt;z&lt;/math&gt;, praebent
&lt;math display=&quot;block&quot;&gt;
\frac{d\varpi}{dx} = \mu X,  \frac{d\varpi}{dy} = \mu Y, \frac{d\varpi}{dz} = \mu Z. (o) &lt;/math&gt;et consequenter (27.24º)
&lt;math display=&quot;block&quot;&gt;
d\varpi = \mu ( Xdx + Ydy + Zdz). ( o' ) &lt;/math&gt; Itaque conditiones requisitae ad massae fluidae aequilibrium eo redeunt ut exsistat ejusmodi functio &lt;math&gt;\varpi&lt;/math&gt; variabilium &lt;math&gt;x, y, z&lt;/math&gt;, quae expleat sive ternas (o), sive unicam (o').

87. Haec notentur. 

1º. Si fluidum continetur vase undique clauso satisque firmo, utcumque se habeat valor &lt;math&gt;
\varpi &lt;/math&gt; ex (o') quoad superficiem fluidi, is constanter aequivalebit reactioni ex vasis lateribus: at si fluidi superficies sit libera, externisque subjecta pressionibus, ad aequilibrium explenda insuper erit (o') per talem valorem &lt;math&gt;
\varpi &lt;/math&gt;, qui in singulis liberae superficiei punctis aequivaleat respondenti pressioni externae.

2º. Hinc si pressio externa vel ponitur &lt;math&gt;=0&lt;/math&gt; vel ubique eadem, erit &lt;math&gt;d\varpi = 0&lt;/math&gt; quoad superficiem fluidi librati, ideoque
&lt;math display=&quot;block&quot;&gt;Xdx + Ydy + Zdz  = O (o''). &lt;/math&gt;

3º. Traduci potest (o&quot;) ad&lt;math display=&quot;block&quot;&gt;\frac{X}{\varphi} \frac{dx}{ds}+\frac{Y }{\varphi} \frac{dy}{ds} + \frac{Z}{\varphi}\frac{dz}{ds} = 0&lt;/math&gt;exprimunt &lt;math&gt;X/\varphi, Y/\varphi, Z/\varphi&lt;/math&gt; cosinus angulorum, quos efficit vis acceleratrix &lt;math&gt;\varphi&lt;/math&gt; cum axibus coordinatis &lt;math&gt;OX, OY, OZ&lt;/math&gt;; denotant &lt;math&gt;\frac{dx}{ds}, \frac{dy}{ds},\frac{dz}{ds}&lt;/math&gt; cosinus angulorum, quos recta tangens arcum &lt;math&gt;s&lt;/math&gt; apud ejus extremum facit cum iisdem axibus: inferimus (50. 6.) vim &lt;math&gt;\varphi&lt;/math&gt; intercipere angulum = 90° cum rectis omnibus tangentibus ubivis superficiem vel nullo pacto, vel aeque pressam; ac proinde &lt;math&gt;\varphi&lt;/math&gt; sese dirigere normaliter ad istiusmodi superficiem.

4.º Integrata (o&quot;), si constanti arbitrariaeque quantitati tribuuntur alii atque alii valores, emergent aliae atque aliae aequationes, quibus totidem respondebunt distinctae superficies aeque pressae.

5.°* In hypothesi &lt;math&gt;\varphi&lt;/math&gt; tendentis ad punctum fixum, constitue ibi coordinatarum originem: denotante &lt;math&gt;D&lt;/math&gt; distantiam inter punctum illud et &lt;math&gt;( x , y , z)&lt;/math&gt;, erunt (50. 6º)&lt;math display=&quot;block&quot;&gt;X = -\varphi \frac x D, Y= -\varphi \frac y D, Z= -\varphi \frac z D&lt;/math&gt;hinc&lt;math display=&quot;block&quot;&gt;X dx + Ydy + Zdr = - \frac \varphi D (xdx + ydy + zdz).&lt;/math&gt;Est insuper &lt;math&gt;x^2 + y^2 + z^2 = D^2 &lt;/math&gt;, unde &lt;math&gt;xdx + ydy + zdz = DdD;&lt;/math&gt; et consequenter&lt;math display=&quot;block&quot;&gt;Xdx +Ydy + Zdz = -\varphi dD.&lt;/math&gt;In ordine igitur ad superficiem aeque pressam exsistet &lt;math&gt;dD = 0&lt;/math&gt;: propterea &lt;math&gt;D = C&lt;/math&gt;; ex qua &lt;math&gt;x^2 + y^2 + z^2 = C^2 &lt;/math&gt;: massa videlicet fluida atque librata induet sphaericam formam.


6. Quoad fluidum elasticitate pollens, constat experimentis densitatem &lt;math&gt;\mu&lt;/math&gt;, permanente temperie, esse proportionalem respondenti pressioni &lt;math&gt;
\varpi &lt;/math&gt;, nimirum&lt;math display=&quot;block&quot;&gt;\mu = \theta \varpi: (o''')&lt;/math&gt;Eliminata  &lt;math&gt;\mu&lt;/math&gt; ab (o') et (o''&quot;''), proveniet&lt;math display=&quot;block&quot;&gt;\frac{d\varpi}{\varpi}=\theta(Xdx + Ydy + Zdz);&lt;/math&gt;et facto &lt;math&gt;Xdx + Ydy + Zdz = df (x,y,z)&lt;/math&gt;, erit:&lt;math display=&quot;block&quot;&gt;\ln \varpi = \int \theta df + \ln C = \ln (e^{\int \theta df}) + \ln C= \ln (C e^{\int \theta df})&lt;/math&gt;hinc&lt;math display=&quot;block&quot;&gt;\varpi = C e^{\int \theta df}, \mu = C \theta e^{\int \theta df}&lt;/math&gt;coefficiens &lt;math&gt;\theta&lt;/math&gt; pendet a temperie vigente apud &lt;math&gt;(x , y , z)&lt;/math&gt;. Inferimus aequilibrii statum in fluido elastico importare temperiem vel ubique eamdem, vel talem ut sit functio quantitatis &lt;math&gt;f&lt;/math&gt;. Haec insuper quantitas est (2º, 4º) constans in unaquaque superficie aeque pressa; idipsum ergo dicendum de temperie. 


7.º Constat etiam experimentis fluidum elasticitate pollens ita contrahi vel expandi, imminuta vel aucta temperie ac permanente pressione &lt;math&gt;
\varpi' &lt;/math&gt; ut ejus volumen &lt;math&gt;
V &lt;/math&gt;minuatur vel augeatur partibus 0,00375 pro singulis gradibus thermometri centigradi; inde fit, ut posito 0,00375 = &lt;math&gt;
a &lt;/math&gt;, et aucta temperie gradibus &lt;math&gt;n&lt;/math&gt; ultra &lt;math&gt;0^\circ \mathrm{C}&lt;/math&gt; , volumen  &lt;math&gt;V&lt;/math&gt; evadet &lt;math&gt;V ( 1 + an )&lt;/math&gt;; propterea, designantibus  &lt;math&gt;\mu_0&lt;/math&gt;  et   &lt;math&gt;\mu_1&lt;/math&gt;  respondentes densitates, erit &lt;math&gt;\frac{\mu_1}{\mu_0}=\frac{1}{1+an}.&lt;/math&gt; Nunc, permanente temperie &lt;math&gt;n&lt;/math&gt;, crescat pressio ab &lt;math&gt;
\varpi' &lt;/math&gt; ad  &lt;math&gt;
\varpi &lt;/math&gt;;  denotante &lt;math&gt;\mu&lt;/math&gt;  respondentem densitatem, erit  (1º) &lt;math&gt;\frac{\varpi}{\varpi'}=\frac{\mu}{\mu_1},&lt;/math&gt; quocirca&lt;math display=&quot;block&quot;&gt;\varpi = \frac{\varpi'\mu}{\mu_1} =  \frac{\varpi'}{\mu_0} \mu ( 1 + an )&lt;/math&gt;; et facto &lt;math&gt; \frac{\varpi'}{\mu_0} =i&lt;/math&gt;, &lt;math&gt;\varpi = i \mu ( 1 + an ) (o^{(iv)})&lt;/math&gt;.

=== De gravium homogeneorumque liquidorum aequilibrio. ===
88. Planum &lt;math&gt;XOY&lt;/math&gt; sit horizontale, axisque &lt;math&gt;OZ&lt;/math&gt; (Fig. 47) vergat deorsum juxta directionem gravitatis &lt;math&gt;g&lt;/math&gt;; erunt &lt;math&gt;X=0, Y=0, Z = g&lt;/math&gt;: proinde (86. 6), &lt;math display=&quot;block&quot;&gt;d\varpi = g \mu dz ( 0^{v} )&lt;/math&gt;Si pressio externa ponitur vel = 0, vel ubique eadem, erit &lt;math&gt;d\varpi = 0&lt;/math&gt; quoad librati fluidi superficiem, ideoque &lt;math&gt;dz = 0&lt;/math&gt;, et &lt;math&gt;z = Const&lt;/math&gt;: superficies nempe illa existet plana atque horizontalis. 

Pone &lt;math&gt;\mu&lt;/math&gt;  constantem; ex (0&lt;sup&gt;v&lt;/sup&gt;) habebis

&lt;math&gt;\varpi = g \mu z + C_1&lt;/math&gt;,

In fluidi superficie aeque pressa constitue planum horizontale &lt;math&gt;XOY&lt;/math&gt;: quoad eam erit &lt;math&gt;z = 0&lt;/math&gt;; nihilque aliud denotabit &lt;math&gt;C_1&lt;/math&gt; nisi externam pressionem in aream ( = 1 ) quaquaversus per fluidum aequaliter diffusam.

Haec facile nunc stabiliuntur circa pressiones gravium homogeneorumque liquidorum intra vasa in aequilibrio consistentium. 
[[Fasciculus:Hydrostatic-pressure.svg|thumb]]
1º. Si per &lt;math&gt;\Pi&lt;/math&gt; designatur pressio in horizontalem aream &lt;math&gt;A&lt;/math&gt; demersam ad profunditatem &lt;math&gt;z&lt;/math&gt;, exsistet &lt;math&gt;\Pi = A \varpi = A (g\mu z + C_1 ) .&lt;/math&gt;

2º. Si &lt;math&gt;C_1 = 0&lt;/math&gt;, aequivalebit &lt;math&gt;\Pi&lt;/math&gt; ponderi prismatis, cujus basis est &lt;math&gt;A&lt;/math&gt;, altitudo &lt;math&gt;z&lt;/math&gt;, densitas vero eadem ac densitas liquidi.

3º. Exhibente &lt;math&gt;A&lt;/math&gt; horizontalem vasis fundum, ideoque &lt;math&gt;z&lt;/math&gt; altitudinem vasis; quoniam &lt;math&gt;\Pi&lt;/math&gt; nullatenus pendet a vasis figura, iccirco permanentibus &lt;math&gt;A&lt;/math&gt; et eadem perstabit liquidi pressio in horizontalem fundum, utcumque de caetero varient figura et capacitas vasis.

4º. Area &lt;math&gt;A&lt;/math&gt; sit oblique intra liquidum utcumque demersa: divide &lt;math&gt;A&lt;/math&gt; in areolas infinitesimas &lt;math&gt;a , a ', a'' &lt;/math&gt; quarum distantiae ab extima liquidi superficie designentur per &lt;math&gt;z' , z''...;&lt;/math&gt; denotante &lt;math&gt;\Pi'&lt;/math&gt; totalem pressionem, et &lt;math&gt;z_1&lt;/math&gt; perpendiculum ductum ex centro gravitatis areae &lt;math&gt;A&lt;/math&gt; in planum &lt;math&gt;XOY&lt;/math&gt;; erit (20) 

&lt;math&gt;\Pi' = a(g\mu z + C_1) + a'(g\mu z'+ C_1) +... = g\mu(az + a'z' + ...) + C_1( a + a' + ... ) = g\mu z_1 A + C_1 A = A (g\mu z_1 + C_1 )&lt;/math&gt;. 

Hinc si centrum gravitatis manet ad eamdem profunditatem demersum, haud variabit  &lt;math&gt;\Pi'&lt;/math&gt;, utcumque circa illud revolvatur area demersa: potest A repraesentare quamlibet rectilineam portionem internae superficiei vasis.

Ad haec: coordinatae ( 13. 3º. )

&lt;math&gt;b=\frac{\sum ax (g\mu z + C_1)}{ \sum a(g\mu z + C_1)}; b' = \frac{\sum ay (g\mu z + C_1)}{ \sum a(g\mu z + C_1)}, b'' = \frac{\sum az (g\mu z + C_1)}{ \sum a(g\mu z + C_1)} &lt;/math&gt; seu (20) 

&lt;math&gt;b =   \frac{g\mu \sum ax z + C_1 A x_1 }{ A(g\mu z_1 + C_1) }; 
b' =  \frac{g\mu \sum ay z + C_1 A y_1 }{  A(g\mu z_1 + C_1)}, 
b'' = \frac{g\mu \sum a  z^2 + C_1 A z_1 }{ A(g\mu z_1 + C_1)} &lt;/math&gt;

respondent illi puncto areae &lt;math&gt;A&lt;/math&gt;, per quod transit resultans ex parallelis viribus &lt;math&gt;a(g\mu z + C_1), a'(g\mu z'+ C_1), a''(g\mu z''+ C_1)...&lt;/math&gt;; istiusmodi punctum dicitur centrum pressionis.
[[Fasciculus:PolydirectionalPressure.svg|thumb]]
5º . Veniat considerandum solidum liquido immersum: sume apud punctum &lt;math&gt;( x , y , z )&lt;/math&gt; in solidi superficie areolam infinitesimam &lt;math&gt;k&lt;/math&gt; , et apud puncta &lt;math&gt;( x_0, y, z ) , (x , y_0, z ) , ( x , y , z_0 )&lt;/math&gt;in eadem solidi superficie areolae &lt;math&gt;k_0, k'_0, k''_0&lt;/math&gt;, sitque &lt;math&gt;h&lt;/math&gt; projectio areolae &lt;math&gt;k_0&lt;/math&gt; in plano &lt;math&gt;YOZ&lt;/math&gt;, &lt;math&gt;h'&lt;/math&gt; projectio areolae &lt;math&gt;k'_0&lt;/math&gt; in plano &lt;math&gt;XOZ, h''&lt;/math&gt;projectio areolae &lt;math&gt;k''_0&lt;/math&gt; in plano &lt;math&gt;XOY&lt;/math&gt;; congruant vero &lt;math&gt;h, h' , h''&lt;/math&gt; cum projectionibus areolae &lt;math&gt;k&lt;/math&gt; in iisdem planis: per &lt;math&gt;k(g\mu z + C_1), k_0(g\mu z + C_1),k'_0(g\mu z + C_1), k''_0(g\mu z + C_1),&lt;/math&gt;exprimentur pressiones normaliter exercitae in areolas  &lt;math&gt;k, k_0, k'_0, k''_0&lt;/math&gt;; ejusmodi pressionum prima resolvitur in &lt;math&gt;\frac{h}{k}\cdot k(g\mu z + C_1), \frac{h'}{k}\cdot k(g\mu z + C_1),\frac{h''}{k}\cdot k(g\mu z + C_1),&lt;/math&gt;&lt;ref&gt;Figura deest ergo clare non est si aequatio est recte stripta &lt;/ref&gt; parallelas rectis &lt;math&gt;OX , OY , OZ&lt;/math&gt;; secunda praebet componentem

&lt;math&gt;-\frac{h}{k_0}\cdot k_0(g\mu z + C_1)&lt;/math&gt;

parallelam rectae OX, tertia dat componentem

&lt;math&gt;-\frac{h'}{k'_0}\cdot k'_0(g\mu z + C_1)&lt;/math&gt;

parallelam rectae OY; quarta suppeditat componentem

&lt;math&gt;-\frac{h''}{k''_0}\cdot k''_0(g\mu z_0 + C_1),&lt;/math&gt;

parallelam rectae &lt;math&gt;OZ&lt;/math&gt;. His positis, quisque videt areolam &lt;math&gt;k&lt;/math&gt;, elisis componentibus horizontalibus, urgeri sursum verticali pressione&lt;math display=&quot;block&quot;&gt;h'' g \mu ( z - z_0 )&lt;/math&gt;totum igitur demersum solidum ad verticalem ascensum sollicitatur parallelis viribus praebentibus resultantem, quae aequivalet ponderi liquidi expulsi, et transit per punctum illud, ubi erat gravitatis centrum ipsius liquidi expulsi. Itaque si &lt;math&gt;V'&lt;/math&gt; et &lt;math&gt;\mu'&lt;/math&gt; exhibent volumen et densitatem solidi liquido immersi, &lt;math&gt;V&lt;/math&gt; volumen liquidi espulsi; pondus, quod superest solido, exprimelur per &lt;math&gt;g( V'\mu' - V\mu )&lt;/math&gt;: in solidis heterogeneis designat &lt;math&gt;\mu'&lt;/math&gt; densitatem mediam.

89. Sit 1º &lt;math&gt;\mu' &gt; \mu &lt;/math&gt; cum nequeat esse &lt;math&gt;V &gt; V '&lt;/math&gt;, erit semper &lt;math&gt;V'\mu' - V\mu &gt;0&lt;/math&gt;; tamdiu igitur descendet solidum, ubicumque in liquido collocetur, donec aliquod offendat obstaculum, cui adstringatur adhaerere. Si collocatur in liquidi superficie; statim atque totum fuerit demersum, exsistet &lt;math&gt;V = V';&lt;/math&gt; et consequenter perget solidum moveri vi acceleratrice &lt;math&gt;\frac{gV ' ( \mu' - \mu )}{V'\mu'}&lt;/math&gt; seu &lt;math&gt;g\left( 1 - \frac{\mu }{\mu'}\right)&lt;/math&gt;

Ab exploratis solidi ponderibus P et P' in vacuo et in li
quido elici potest ratio inter u et l ; siquidem
P = gV ' ', P = 8 ! V' ' — Vp ) , et V = V :
propterea
P í M
Hop
unde р
P
P - P
[[Fasciculus:EB1911 Hydromechanics - Fig. 3.jpg|thumb]]
Sit 2º. M '= H: tamdiu V'u ' - Vl &gt; o quamdiu &lt;math&gt;V' &gt; V&lt;/math&gt;; solidum nempe collocatum in superficie liquidi eo usque descendet, donec totum demergatur; quod ubi contigerit, evanescente V' M' — Vp , consisteret in aequilibrio nisi urgeretur adhuc vi acquisita descendendo ante et aequivalet ponderi liquidi expulsi, et transit per punctum illud, ubi erat gravitatis centrum ipsius liquidi expulsi. ltaque si V'et p! exhibent volumen et' densitatem
solidi liquido immersi, V volumen liquidi expulsi; pondus,
quod superest solido, exprimetur per
g( V'p.'—Vp.) :
in solidis heterogeneis designat p! densitatem mediam. ⋅
89. Sit. 1041!) p.: cum nequeat esseV) V', erit sem-
per V' pf ∙−− Vp.) o; tamdiu igitur descendet solidum, ubi-'
cumque in liquido collocetur, donec aliquod offendat ob-
staculum
, cui adstringatur adhaerere. Si collocatur in li-
quidi superficie; statim atque totum fuerit demersum, ex-
sistet V:V'; et consequenter perget solidum moveri vi
acceleratrice '
sv. ∣≺⊮∸⋮⋅⋅−⋅∟∸≻ ∘ −.r. v'F-I , .seu :,(1 l*') .
Ab exploratis solidi ponderibus P et'P' in vacuo et in li-
quido elici potest ratio inter p! et p.; siquidem
≖∙⊃−∙−⇀−∊⋁∙⊬↼∙∙ P',—.: g( vir—v,. ), .xv.-: V':
prOpterea
.
P p! p!— P
P' −−−⊬∙∙⊬∙ uude F- P-P'
.
Sit 20. pl: p.: tandiu V'pf -— VP) o quamdiu
V&quot; V ; solidum nempe collocatum in superficie liquidi
eo .usque descendet,, donec totum demergatur; quod ubi
contigerit, evanescentev p! —Vp. , consisteret in aequi-,-
librio nisi urgeretur adbuc vi acquisita descendendo ante192
1
V'de VM
1
1
totalem immersionem ; ad aequilibrium praeterea deberent
gravitatis centra solidi et liquidi expulsi esse in eadem re
cia verticali,

Sit 3º. p &lt; l tandiu . Vil – Ve &lt; o quandiu
V &gt; ; et facto V , erit Vų – VH = 0.
Solidum igitur collocatum intra liquidum ascendet ad li
quidi superficiem ; situm in ipsa superficie supernatabit ;
eritque portio demersa V ad volumen integrum V' ut j ': fl.
Innatantis solidi aequilibrium requirii insuper ut in eadem
recta verticali inveniantur gravitatis centra ipsius solidi et
liquidi quod expellitur.
Itaque positio aequilibrii quoad solidum homogeneum
liquido insideas determinabitur si plano ita secetur soli
dum, ut et alterius segmenti volumen sit ad solidi volu
men ia data ratione pe': fhy et haec volumina habeant sua
gravitatis centra in eadem recta , quae normaliter insistat
plano secanti: rem declaramus exemplo. Determinanda sit
positio aequilibrii in prismate recto ac triangulari , quod
ita demergitur ut et ejus bases maneant verticales, et u
na ex tribus faciebns v. g. BC ( Fig 48 ) exsistat cota ex
tra liquidum.
Quisque videt directionem plani secantis non pende
re a mutua basium distantia, satisque esse ut determine
tur intersectio De illius plani et baseos v . g. ABC. Exhi.
beant a ', a“ latera AB, AC dati trianguli ABC , et a', w &quot;
latera incognita AD, AE crianguli ADE : triangulares areae
ABC, ADE exprimentur per
3
i a'a ' sin A , Law&quot; sin A.
Sed area ABC est ad aream ADE ut integrum prisma ad
prismaticam portionem demersam ; igitur
le
IWW 'sin A: į a' a &quot; sin A = fe':J.,Was&quot;
P.
-a'a' ( k) .
192
totalem immersionem; ad aequilibrium praeterea deberent
gravitatis centra solidi et liquidi expulsi esse in eadem re-
cta verticali. ⋅
'
Sit 3&quot;. p! p. : tandiu. V'pl ∙− Vp.( o quandiu
V P- ;et factoV:V V) P- ,eritV'pf—Vp.:o.
Solidum igitur collocatum intra liquidum ascendet ad li-
quidi superficiem; situm in ipsa superficie superuatabit;
eritque portio demersa V ad volumen integrum V' ut pf: p..
Iunatantis solidi aequilibrium requirit insuper utin eadem
recta verticali inveniantur gravitatis centra ipsius solidi et
liquidi quod expellitur.
ltaque positio aequilibrii quoad solidum homogeneum
liquido insidens determinabitur si plano ita secetur soli-
dum, ut et alterius segmeuti volumen sit ad solidi volu-
men iu data ratione an., et haec volumina habeant sua
gravitatis centra in eadem recta, quae normaliter insistat
plano secanti: rem declaramus exemplo. Determinauda sit
positio aequilibrii in prismate recto ac triangulari, quod
ita demergitur ut et eius bases maneant verticales, et u- ⋅
na ex tribus faciebus v. g. BC ( Fig 48) exsistat tota ex—
tra liquidum.
Quisque videt directionem plani secantis nou pende-
re a mutua basium distantia, satisque esse ut determine-
tur intersectio DE illius plani et baseos v. g. ABC. Exhi-
beant a', a&quot; latera AB. AC dati trianguli ABC, et m', a)&quot;
latera incognita AD, AE trianguli ADE :triangulares areae
ABC, ADE exprimeutur per
∙∙⋅∙ äaa smA I &quot;-
, ämæstu A.
Sed area ABC est ad aream ADE ut integrum prisma ad
prismaticam portionem demersam: igitur
P:.
in' d'siu A: ;a' a&quot; sin A∶∶∶ [1]: .n., 'n' a&quot;:—-—a'a&quot; (k) .193

pla AM
Ž
AH '
Nunc secto bifariam in H latere BC, ducatur AH; sum
2
3
AH , centrum gravitatis trianguli ABC e
rit in M: simili modo, secto bifariam in H ' latere DE, sum
2
ptaque AN = AH', erit N centrum gravitatis trianguli 3
AM AN
ADE. Quia igitur ideo MN et HH' erunt АН
inter se parallelae: sed in casu aequilibrii recta MN, jun
gens gravitatis centra M et N , est perpendicularis rectae
DE ; ergo et HH' erit perpendicularis ipsi DE . Hinc DH=
HE: vicissim si DH =HE, erit HH' ac proinde MN per
pendicularis rectae DE; conditio nimirum necessaria ac
sufficiens ut recta jungens gravitatis centra M et N sit per
pendicularis rectae DE redigetur ad mutuam aequalitatem
rectarum DH, HE. Quibus positis , denotent B, Borangulos
DAH, BAH, et b rectam AH; triangula ADH, AHE dabunt
DA’ = w2762—2wbcos B ,HE' = w &quot; 2 + 62—20 &quot; bcoss *:
propterea
w2 -2bw' cos B = &quot;? - 26w &quot; cos \beta &quot; (k' ) .
Ex duabus ( k) et ( k ' ) eruentur a eta' , uude innotescit
positio intersectionis DE.
Quod si determinanda esset positio aequilibrii in eo
dem prismate quum ita demergitur ut puncta B et C ma
neant infra liquidi superficiem DE, foret
area BCDE : aream ABC = h' : u ': fb,
ideoque
ABC - BCDE ( ADE ): ABC Hope': fl , seu
−−∙≔∎⊾↼−−⇀
193
' Nune secto bifariam in H latere BC, ducatur AH; sum-
pta AM:
∙⋛−⋅ AH, centrum gravitatis trianguli ABC e-
rit in M: simili modo, secto bifariam in H' latere DE, sum-
ptaque AN:
∙−−≣−− AH', erit N centrum gravitatis trianguli
ADE. Quia igitur illi: −∙∙ 23,
inter se parallelae: sed in casu aequilibrii recta MN,iuu- ⋅
gens gravitatis centra M et N , est perpendicularis rectae
DE; ergo et HH' erit perpendicularis ipsi DE. Hinc DEI:-.'
HE: vicissim si DH :HE, erit HH' ac proinde MN per-
pendicularis rectae DE; conditio nimirum necessaria ac
sullicieus ut recta iungens gravitatis centra M et N sit per-
pendicularis rectae DE redigatur ad mutuam aequalitatem
rectarum DH, HE. Quibus positis, denotent B', B&quot;angulos
DAH, BAH, et brectam AH; triangula ADH,AHE dabunt
,
ideo MN et HH' erunt
BB': 'i—l-b' —29'6 cos B', B—Ea ⇌∾∣⋅≖−⊢ &amp;&quot; —20&quot;bcosB&quot;:
propterea
a)&quot; ---260' cos B': si&quot;! -
266)&quot; 'cos B&quot; (k' ).
Ex duabus (I:) et (k') erucutur a' et et&quot;, unde innotescit
positio intersectionis DE.
Quod si determinanda esset positio aequilibrii in eo-
dem prismate quum ita demergitur ut puncta B et C ma-
neant infra liquidi superficiem DE, foret
area BCDE: aream ABC −∙∶−− pl: p.': p.,
ideoque
ABC −∙− BCDE (: ADE ): ABC :p.- p.': p., seu194
Ww&quot; sin A : 1 a'a&quot; sin A = M - pe : plo
et consequenter
s'avº = ( 1- )« a”(k&quot;).
Ad haec : centrum gravitatis trianguli ABC invenilor
in recta jungente centra gravitatis portionum ADE ac BCDE.
Sed ABC et BCDE habent sua gravitatis centra in recta
perpendiculariter insistente rectae DE : adhuc igitur MN
erit perpendicularis ipsi DE ; rursusque prodibit (k' ) : eru
entur videlicet in casu w' et w &quot; ex binis ( k' ) et (k &quot; ) .

90. Determinata aequilibrii positione, restat videndum utrum aequilibrium sit stabile nec ne. Pone v. gr. innatans solidum esse tale, ut secari possit plano verticali AB ( Fig. 49. ) in duas partes omnino symmetricas tum quoad formam, tum quoad densitatem, et in casu aequilibrii sit HK intersectio plani AB et horizontalis plani repraesentantis superficiem liquidi: gravitatis centra M et N innatantis solidi et ejecti liquidi invenientur ambo in plano AB super eadem verticali CD; si solidum est homogeneum exsistet N subter M; si heterogeneum, poterit M esse vel subter N vel supra. Fac ut aliquantulo revolvatur solidum circa axem perpendicularem plano AB, sicque removeatur ab aequilibrii positione; ita tamen ut, exhibente H'K ' (Fig. 50) novam intersectionem plani AB et horizontalis plani repraesentantis superficiem liquidi, segmentum solidi respondens angulo K i K' aequetur constanter segmento quod respondet angulo H i H' ; hoc pacto haud variato ejecti liquidi volumine, permanebit ( 89.30. )
gV'p ' = gVd : proinde solidum absque initiali velocitate sibi
commissum movebitur ( 84 ) circa centrum M immotum.
Jam si ex puncto N' , ubi , amoto solido ab aequilibrii
positione , situm est gravitatis centram liquidi expulsi , du
194
&amp; o'o'f aiu A:) a'a&quot; sin A:p—p:p., /
et consequenter
Ad haec :centrum gravitatis trianguli ABC invenitur
in recta iungente centra gravitatis porticuum ADE ac BCDE.
Sed ABC et BCDE habent sua gravitatis centra in recta
perpendiculariter insistente rectae DE: adhuc&quot; igitur MN
erit perpendicularis ipsi DE; rursusque prodibit (k') :eru-
entur videlicet in casu a' et a)&quot; ex binis (k') et (Is&quot;) . ducatur verticalis recta N'R occurrens rectae CD in R , oc
cursus iste vel fiet supra M , vel infra , vel in ipso M :
in primo casu vis g Vlagens sursum juxta N'R manife
ste nitetur ut CD resumat verticalem positionem, et conse
quenter aequilibrium erit stabile ; in secundo ipsa gVp.
nitetur ut CD magis recedat a verticali positione , ideoque
aequilibrium instabile ; in tertio aequilibrium adhuc ob
tinebit quoad novam positionem .

=== De gravium liquidorum aequilibrio in vasis communicantibus. ===
[[Fasciculus:Communicating vessels.svg|thumb]]
91. Vasa communicantia dicuntur illa, quae ita sunt inter se conjuncta ut ex altero in alterum pateat aditus fluido. In altero contineatur fluidum homogeneum, cujus densitas &lt;math&gt;\mu&lt;/math&gt;; in altero fluidum pariler homogeneum cujus densitas &lt;math&gt;\mu'&lt;/math&gt;; siatque &lt;math&gt;z&lt;/math&gt; et &lt;math&gt;z+ z'&lt;/math&gt; distantiae inter punctum quodvis superficiei communis utrique fluido ac extimas fluidorum superficies. Fluidis se mutuo librantibus, exsistet (88)

&lt;math&gt;g\mu z + C_1 = g\mu ( z + z' ) +C_2.&lt;/math&gt;
[[Fasciculus:11 hidrostatica de 61 a 70.jpg|thumb]]
92. Haec facile nunc stabiliuntur. 

1.º Si vasis communicantibus idem continetur liquidum, ut sit &lt;math&gt;\mu = \mu '&lt;/math&gt;, erit &lt;math&gt;g \mu z = g \mu' z&lt;/math&gt; ideoque &lt;math&gt;z' = \frac{C_1 - C_2}{g \mu'};&lt;/math&gt; emerget ergo &lt;math&gt;z' = 0&lt;/math&gt; vel &lt;math&gt;z' &gt; 0&lt;/math&gt;, prout &lt;math&gt;C_1 = C_2 &lt;/math&gt;vel &lt;math&gt;C_1 &gt; C_2&lt;/math&gt;: in ea videlicet qua sumus hypothesi liquidum sub externis aequalibusque pressionibus manebit in utroque vase aeque altum, sub externis vero inaequalibusque pressionibus altias apud eam partem assurget ubi minor exercetur pressio. Inde profluit explicatio variorum effectuum; cujusmodi sunt hydrargyrum in barometro suspensum, aqua elevata in siphone, in antliis etc.... Sic v. gr. quoad antlias adspirantes, dum attollitur embolus ex &lt;math&gt;H'H''&lt;/math&gt; in &lt;math&gt;HI&lt;/math&gt; (Fig. 51), aer in tubo &lt;math&gt;HB'&lt;/math&gt; confestim fit rarior, et consequenter externus aer densior aquam in receptaculo vel puteo contentam cogit in tubum ascendere usque ad altitudinem v. gr. &lt;math&gt;A' B'&lt;/math&gt;: quam ob causam descendet aqua in receptaculo ab &lt;math&gt;AE&lt;/math&gt; in &lt;math&gt;ii'&lt;/math&gt;. Jam datis &lt;math&gt;H'Q ( = a ')., EQ ( = a '' ) , HH' ( = b) ,&lt;/math&gt;itemque horizontalibus receptaculi, ac tuborum &lt;math&gt;BQFD', FQA'B'&lt;/math&gt; sectionibus &lt;math&gt;\omega, \omega' , \omega ''&lt;/math&gt;, si debeat inveniri altitudo &lt;math&gt;AA'&lt;/math&gt;, pone &lt;math&gt;AA' = \beta&lt;/math&gt; et &lt;math&gt;Ai = \beta'&lt;/math&gt;: densitates aeris interni ante et post emboli elationem sunt in ratione reciproca voluminum &lt;math display=&quot;block&quot;&gt;a'\omega' + a'' \omega'' , a'\omega' + a'' \omega'' + b\omega' - \beta \omega'';&lt;/math&gt;ideoque (87. 6º) in eadem ratione erunt pressiones a et ;
hinc
( a'w ' ta'w ') a
(a' +6) + (a&quot; – 3 ) cs&quot; |
designante m aquae densitatem , aqua elevata supra ii ' exer
cebit ( 88) pressionem
a = gm (B + B ) .
Cum igitor a' to = 5W , cumque Bw &quot;' = f'w , iccirco
( a'w' + aa'') as''
(a + b ) w + la &quot; -B, w 

sia ponitur parvitatis contemnendae prae w , erit
( a'w' ta'a ') as tgms = a .
( a ' + 6) + ( a &quot; -B) w &quot;
196
sunt hydrargyrum iu barometro suspensum , aqua elevata
in siphone ,
in antliis etc.... Sic v. gr. quoad antlias ad- ∙
spirantes , dum attollitur embolus ex H'H&quot; iu Hl (Fig.51.),
aer in tubo HB' confestim Et rarior , et consequenter ex-
ternus aer densior aquam in receptaculo velputeo conten-
tam cogit iu tubum ascendere usque ad altitudinem v. gr.
A' B' : quam ob causam descendet aqua in receptaculo ab
AE in ii'. Jam datis H'Q (: a')... EQ (: a&quot;)
, HH'(:—...- 6),
itemque horizontalibus receptaculi , ac tuberum BQFD' ,
FQA'B' sectionibus a), m', ei&quot;, si debeat inveniri altitudo AA',
pone AA':B et Ai:B' :densitates aeris interni ante et
post emboli elationem sunt in ratione reciproca voluminum
a'm' −↿− a&quot;a)&quot;
, a'æ' a&quot;o)&quot; −∣⋅− ∂∾⋅−Ba)&quot; ;
ideoque (87. 60.) in eadem ratione erunt pressiones a et se' ;
hinc
(a'ai' −⋅∣− d'un&quot;) ur −⇀⋅ (a'-1-b)m'-1-(a&quot;—B)m&quot;
designante m aquae densitatem ,,aqua elevata supra ii' exer-
cebit (88) pressionem
0&quot;: sm (49 ⊣− B')-
Cum igitur a' -l-—a&quot; :0, cumque Ba)&quot;:B'm ,
iccirco
[
U'
(a'æ' ⊣∙⋅ d'ai&quot;) ar −⊢⊣−⊣−⊰⋯≺↿⊣−∾−↜∶≻∣∃⇌≔⇌
si a)&quot; ponitur parvitatis contemnendae prae a)
, erit
(a'æ' −⋅⊢ J'ai&quot;) :: l ∙−−
(a'-l—b) ∾∣∙∙⊢ (avl—þ) 0)&quot; l gmB—a-197
la eadem hypothesi , post iteratos descensus atque ascen
sus , restituto embolo ab altitudine minima H'H ' ad maxi
mam HI , pertingat aqua ad inferiorem superficiem mem
branae G ; descendente rursus embolo et denotante k alti
tudinem aquae de se librantis atmosphaericam pressionem ,
elevabitur membrana D , ac proinde rarescente adhuc aere
in consequenti emboli ascensu , aqua pergei assurgere quo
tiescumque fuerit
EQ . HQ &lt; k (HH') .
Ut enim elevetur membrana D , debei elasticitas k' aeris
HF sic augeri quum aer HF traducitur ad volumen H'F , ut
elasticitas k' ' densati aeris H'F superet elasticitatem k aeris
atmosphaerici embolo superincumbentis . Est aulem ( 87.1º . ) .
k : k &quot; = HQ : HQ ;
vis insuper elastica k' unita ponderi aquae suspens ae EQ
librat pressionem aeris atmosphaerici , nimirum
h' + EQ = k ;
et consequenter
k &quot;
k' (HQ)
H'Q
(k- EQ) (HQ)
H'Q
Igitur
( k — EQ) ( HQ)
&gt; k ; ac proinde etc. ... H'Q

2. Tubus cylindricus longitudinis h , et in una
sui extremitate clausus , impleatur hydrargyro usque ad
197
in eadem hypothesi , post iteratus descensus atque&quot; ascen-
sus , restituto embolo ab altitudine minima H'H&quot; ad maxi-
mam Hl , pertingat aqua ad inferiorem superficiem mem-
branae G; descendente rursus embolo et denotante ]: alti-
tudinem aquae de se librantis atmosphaericam pressionem ,
elevabitur membrana D, ac proinde rarescente adhuc aere
in consequenti emboli ascensu , aqua perget assurgere quo-
tiescumque fuerit
EQ
. HQ h(HH').
Ut enim elevetur membrana D , dabei elasticitas k' aeris
HF sic augeri quum aer HF traducitur ad volumen H'F , ut
elasticitas k&quot;densati aeris H'F superet elasticitatem k aeris
atmosphaerici embolo superiucumbentis .Est autem (87.10.).
k': k&quot;:H'Q :HQ ;
vis insuper elastica k' unita punderi aquae suspensae EQ
librat pressionem aeris atmosphaerici, nimirum
k&quot; -]— EQ:k ;
et consequenter
- k&quot; k' (HQ) −∙∙≺∣⊂− EQ) (HQ). ↼−− l'l'Q HQ
igitur
de −−⋅⋅ EQ) (&quot;Q)
H'Q k; ac proinde etc.
..

2.&quot; Tubus cylindricus longitudinis A, et in una
sui extremitate clausus ,
'impleatur hydrargyro usque ad198
altitudinem hoh , cum digito ad orificium in altera parte
exsistens apposito invertatur tubus ; et remolo digito , col
locetur hoc ipsum orificium in superficie hydrargyri sta
gnantis intra aliquod vas. Ascendet aer l' ad supremam in
versi tubi partem ; augescet h , et fiet = h &quot; . Jam vero ad
inveniendam h &quot; denotante k' altitadinem hydrargyri libran
tis atmosphaericam pressionem , satis erit animadvertere
h'ki
quod exhibet altitudinem hydrargyri librantis rarefa h &quot;
clum aerem h ' ; unde
hk
h - h '
+
To
k ;
ac propterea
h &quot; = h - k' = V Th — kj» + 4hºk
2
signum inferius non pertinet ad praesens problema .
lu formula ( 10)
C, C,
Sle'
Spkk
sunt C, = gu'k ' , C,
z'
'

3º. Pone ple , pe inaequales , et C, = C2 ; habebis
p.z = p ( = + z ) ,
unde
2 : 3+ = M ' : pe ;
diversorum nempe liquidorum altitudines z et ztz' in vasis
communicantibus erunt reciproce ut ipsorum liquidorum
densitates.
198
altitudinem h—h' ,
tum digito ad orificium in altera parte
exsistens apposito invertatur tubus ; et remoto digito , col-
locetur hoc ipsum orificium in superficie hydrargyri sta-
gnantis intra aliquod vas. Ascendet aer b' ad supremam iu-
versi tubi partem ; augescet h' ,
et fiet:h&quot;. Jam vero ad
inveniendam h&quot; denotante k' altitudinim hydrargyri libran-
tis atmos'phaericam pressionem , satis erit animadvertere
quod exhibet (i£—, altitudinem hydrargyri librantis rarefa-
ctum sereni I:&quot; ; unde
h—h&quot; ∙−⊢ h—Ij—L: k' ;
ac propterea
1.&quot; −∣∙ ∣⋅−∣⊏⋅∶⊨∣∕⇀≺∣≖−∣≂⊤≻⋅⊣−⊓≖⋅∣⊏⋮∙ , z
signum inferius non pertinet ad praesens problema.
lu formula (10)
sunt C, :gpjk' , Ca.-:. ∙−−−−

diversorum nempe liquidorum altitudines :
et &lt;math&gt;z+z'&lt;/math&gt; in vasis
communicantibus erunt reciproce ut ipsorum liquidorum densitates.

=== De gravium elasticorumque fluidorum aequilibrio; necnon de altitudinibus dimetiendis ope barometri, et de pondere ac densitate vaporum. ===
93. Binae (o&lt;sup&gt;iii&lt;/sup&gt; 87.), (o&lt;sup&gt;v&lt;/sup&gt; 88.) dant &lt;math display=&quot;block&quot;&gt;\frac{d\varpi}{ \varpi} = g\theta dz;&lt;/math&gt;binae (o&lt;sup&gt;iii&lt;/sup&gt;), (o&lt;sup&gt;iv&lt;/sup&gt;  87) praebent&lt;math display=&quot;block&quot;&gt;\theta=\frac{\mu}{\varpi}=\frac{1}{ i(1 + an )}&lt;/math&gt;propterea&lt;math display=&quot;block&quot;&gt;\frac{d\varpi}{ \varpi} = \frac{gdz}{ i(1 +an )} ( b ).&lt;/math&gt; Assumptis autem logarithmis quoad basim 10,&lt;math display=&quot;block&quot;&gt;d\log_{10}(\varpi)= \frac{d\varpi}{\varpi} \log_{10}[2,718281828 ] = 0 , 4342945 \frac{d\varpi}{\varpi}&lt;/math&gt;ideoque
dos dL(W)
0,4342945
Designante igitur pressionem apud punctum ( x , 0)
in hypothesi temperiei constantis formula (6) suppeditabit.
L - Llw ')
0,4342945 ( 6' ) .
i ( 1 + an )

94. Quoad punctum ( x , y ; '-— z) supra horizontale pla
num XOY ( Fig. 47 ) , aequatio ( 6' ) suppeditat
LULLG )
0.4342945
82
i (1tan)

et inde infertur valor z dimetiendae altitudinis supra XOY
sic expressus
i ( 1 + an )
L
0,4342945g ( 6 '') .''
Haec observentur: 1. ° sub temperie = 0 , et barometrica
hydrargyri elatione =2,33958 ped. apud geographicam lati
tudinem = 48° 50' 14 &quot;, ubi gravitas 30,1959 ped. ,
Biot et Arrago invenerunt densitatem hydrargyri esse ad
aeris densitatem po ut 10467 : 1 ; inde habemus respon
dentem pressionem ( 88 )
Ww=( 30,1959) ( 10467 floo ) ( 2,33958) ,
ideoque
wo -
( 30,1959 ) ( 10467) ( 2,33958)
ро
( 30,1959 ) ( 24488, 38386) =739448, 790198174.

2.o Eo minorem experimur temperiem , quo ma-
gis supra terrestrem superficiem assurgimus ,
at, igno&quot;-
mus qua lege liat ejusmodi imminutio; designantibus ?'
et ': temperies in intimo ac supremo puncto dimetiendae
altitudinis z, solet assumi
.- 'r'-l—r
'
&quot;: 2• 201
poniturque ista temperies media constanter vigere per to
tam 2.
1

3. Singulis gradibus imminutae temperiei respon
det hydrargyri condensatio = ; igitur si M et M
5550
exhibent densitates bydrargyri sub temperiebus t' ; ac to
DY in infimo ac supremo puncto altitudinis , erit
t'
M

: 1 = M' : M, unde M

5550 I' ,-1,
1
5550
rica
ati.
ed.,
e ad
00
Temperies hydrargyri tubo barometrico inclusi nonnisi
post aliquod tempus ad aequalitatem reducitur cum aeris
circumstantis temperie , hinc t'i et t, solent definiri sub
sidio thermometri , quod ad barometrum ipsum adnecti
tar ; aliae vero t ' et determinantur ope thermometri ,
quod cum barometro non communicat.

4.0 Si l' et h exprimunt barometricas altitudi
nes apud infimum et supremum punctum altitudinis
erunt ( 88 )
h' =gM'h , =
&amp;M'h
t',
1
5550
ideoque ma'
/
Jora
us ?
endae

: - * (

I' , - 1 ]
5550

5. ° experimentis pendulorum subsidio institutis
14
'201
∣∙ poniturque ista temperies media constanter vigere per to-
tam :. '
&amp;&quot; Singulis gradibus imminutae temperiei respon-
det hydrargyri condensatio: 5150; igitur si M' et M

; )

exhibent densitates bydrargyri sub temperiebus 'r', ac 't',
ll
⋅
in infimo ac supremo puncto altitudinis , erit
———- f.:—T! M' :1: ': ∙∙∙⋅ J
5550 M M, uudeM T',—Tx
'
5550&quot;
ric-a Temperies hydrargyri tubo -barometrico inclusi nonnisi
all' post aliquod tempus ad amnalitatem reducitur cum aeria
ed.. circumstantis temperie , hinc 'r', et 1.&quot;, isolent definiri sub-
sad
ron-
sidio tbermometri , quod ad barometrnm ipsum adnecti-
tur; aliae vero 1&quot; et ': determinantur ope tbermometri ,
quod cum barometro non communicat.

4.&quot; Si b' et lt exprimunt barometricas altitudi-
nes apud infimum et supremum punctum altitudinis z ,
erunt (88)
.: M'h ∙∣≖∙ ∙∙ g ∙
a *gM .m. fr.—ff: ,
∎∎− 5550
mr ideoque
nora-
0st a. ∙∙∙ h' 1 T.]- T; .»pdæ ⊺≖−−−∣≖ (&quot;5550)'
5.0 experimentis pendulorum subsidio institutis
14
- x'!
.202
probatum est , si gi est gravitas apud geographicam la
titudinem = 45 , apud aliam latitudinem å fore
g = g . (1-0,002589cos22 ) ;
erit igitur ( 1 )
30,1959 = g1 [1–0,002588 cos2 (48° 50'14'') ]''
ac proinde
30,1959 ( 1–0,002588 cos 22 )
1 -0,002588 cos2 (48 ° 50'14 &quot; )
.
6.° Quibus positis , formula ( 6 &quot; , 94) traducetur ad
24488,38( 1–0,002588cos2 [48° 50'14*]/ (1 +0,00395+7 )
X
1-0,002588cos22
L
CO
I '
1
5550
0,4342945
-) ]
ped.
e ,
formu 95. # Sumptis logarithmis quoad basim
la ( 6 &quot; , 94 ) evadet
i (1 + an ) .
,( );
upde
et consequenter ( 87. 7. ) H =
82
e iſitan )
202
probatum est
.
si g. est gravitas apud geographicum la-
titudinem;—4

5. ∘ apud aliam latitudinem ). fore
gzg, (1—0,002588cos2)t) :
erit igitur (10)
30,, 95gzg,[1—o,002588 cosz (48-50'1 4&quot;)1
ac proinde
∙− 30,1959 (1—0,002588 cos zx)
5 1-0,002588 cos2 (4so50'14&quot;) '
6.&quot; Quibus positis, formula (E&quot;, 94) traducatur ad
24488,38(1—0.002588cosz[4so50'14&quot;])(1'-1-o,003757 BH) s—
⇁⋅⊤
' - X '1—0,002588cos2').
h. r.!—T! )]
L I: &quot;( ↿−∎∎ 5550
o,4342945 pcd.
95-0 Sumptis logarithmis quoad basim e ,
formu-
la (6&quot;. 94 ) evadet

: i(1tan)L (jul-) ;

unde
a,
ex --
z
.
et consequenter (87. 7. ) p,:
e i(t-l—an)203
1
g?
i ( 1 + an) e il1 +an)
Denotent V'et i volumen et densitatem corporis aere
demersi , ipsoque aere specifice levioris : urgebitur corpus
ad verticalem ascensum vi acceleratrice
8 (V'4 — V'x ')
Vph Gelee Me gz
if1tan)
i ( 1 + an) e
Facile intelligimus , si denotat densitatem mediam glo
bi aereostatici , verticalem ascensum ipsius globi determi
natum iri per
daz
de2
8
)
(6 '' )''
.
i (1 + an ) e
i(1 + an )
Multiplica (6 &quot; ' ) per 2dz, et sume integralia; habebis ( 27. 12.9)
gz
dz2 i(1 + an)
dla с
2g role
re' f 8 + ks)
In hypothesi velocitatis initialis = o erunt simul z=0
20

o , ideoque C

Hinc do
dz et
20 82 dza
de ² (
1 e
i (1 + an ) — 2g2 (6 &quot;) .
hey 252 &quot;
203
I
3
gz
,
i (1—l—an) e i(1—l-an)
Denotent V' et pf volumen et densitatem corporis aere
demersi, ipsoque aere specifice levioris :urgebitur corpus
ad verticalem ascensum vi acceleratrice
V' —-V' ')
,
'a' , gt P
&quot;
—g,(p p.) g( —H)- VP ⊬ M sz
i (1-l—an) e
t(t-i-an)
Facile intelligimus, si denotat p! densitatem mediam glo-
bi aereostatici, verticalem ascensum ipsius globi determi-
natum iri per
I
daz g es'
dt: p.(
Multiplica (6&quot;') per 2dz, et sume integralis; habebis (27. 12.&quot;)
..
52
dzz— c zg a t(1—l—an) '] '
&amp;f— —F g
'&quot;')'
ln hypothesi velocitatis initialis :
0
erunt simul 220
(12. 20! et ⊼⋅−−−∶∘ ,
ideoque C..:
F.]iinc
d:: 25, ∙−− ...—gj— ⋅ '['
(7:3—?( 1 — 8 :
(l*'-an)) .'.2gz (6 ).204
cto
ex cujus integratione innotescet relatio inter z ac t . Fa
dez
=o, formula ( 6 ' ' ' ) suppeditabit altitudinem 2, apud dia
dz
quam exsistet f = M ; et facto = 0 , formula ( 6 &quot; ) praebe dt
bit maximam globi elationem z.

96. Quod pertinet ad pondus ac densitatem vaporum, haec subjungimus: 

1.0 Si vase undique clauso continetur satis liquidi, ut inde sese possit evolvere tantum vaporis, quantum postulat capacitas vasis, quantitas vaporis sese evolventis pertinget ad quoddam maximum unice pendens a vigente temperie: qua videlicet permanente, istud maxinium perstabit idem aut vas exsistat vacuum ab aere, aut aerem contineat, vel quodvis aliud gas ulcum
que densatum vel rarefactum: sive autem obtinuerit illud maximum, sive non, tantundem augebitur vis elastica sicci aeris, vel gas inclusi, quanta est elasticitas evoluti vaporis.

2.° Si vapor aqueus seorsum spectatus posset sub data temperie, quin ad liquidam formam redigeretur, eam librare pressionem ā, quam sub eadem temperie librat siccus aer, ex Gay-Lussac foret densitas té aquei vaporis ad sicci aeris densitatem / ut 10 : 16 , ideoque
M=
104
16

3.• Permanente temperie , fac ut aqueus vapor seor
sum consideratus libret reipsa pressionem Wri si vaporis
densitas vocatur Hiss erit ( 87 : 1. )
10

: @ = ht ' i theo unde pos = 16 Mo ;

et denotantibus P ac P, pondera aeris ac vaporis sub ae
quali volumine ,
204
ex cuius integratione innotescet relatio inter z ac :. Fa-
dzz . . . cto 27; ::o, formula (F&quot;) suppedttabtt altitudinem :, apud
quam exsistet p.:pl; et facto 5; :o, formula (ö&quot;) praehe-
bit maximam globi elationem :.

96. Quod pertinet ad pondus ac densitatem vaporum,
haec subjungimus: 1.&quot; Si vase undique clauso continetur
.satis liquidi, ut inde sese possit evolvere tantum vapo-
ris
, quantum postulat capacitas vasis , quantitas vaporis
sese evolventis pertingat ad quoddam maximum unice pen-
dens a vigente temperie :qua videlicet permanente
,
istud
maximum perstabit idem aut vas exsistat vacuum ab ae-
re, aut aerem contineat
, vel quodvis aliud gas utcum-
que densatum vel rarefactum : sive autem obtinuerit illud
maximum, sive non, tantundem augebitur vis elastica sicci
aeris, vel gas inclusi, quanta est elasticitas evoluti vaporis.
2.&quot; Si vapor aqueus seorsum spectatus posset sub da-
ta temperie , quin ad liquidam formam redigeretur , eam
librare pressionem 0, quam sub eadem temperie librat
siccus aer, ex Gay- Lussac foret densitas p! aquei va-
poris ad sicci aeris densitatemlp. ut 10 : 16, ideoque

4. • Nunc ex aqueo vapore librante pressionem ,
et ex aere sicco emergat volumen V aeris vaporosi librantis
pressionem , et habentis densitatem &amp; ; istiusmodi aeris
massa erit Vs; aer siccus in aere vaporoso contentus utpote
librans pressionem ( 1.9 ) a— , pollebit ( 87 : 1. ° ) den
( - )
sitate Quoniam igitur ( 39) vapor aequeus in
,
to
10 WI
16 W
aere vaporoso pariter contentus pollet densitate pi
propterea ad
Ve = y (0 ) tv
10 i
16 W
por
ICCI
ris
da unde
bra €
( ---+ -s)= (:-)
1 &quot;
sic v. gr. in ordine ad aerem maxime vaporosum sub
temperie =0 , et barometrica hydrargyri altitudine
2,33958 ped. , quoniam maxima pressio librata ab aqueo
vapore sub temperie = 0 respondet barometricae altitu
dini =0,015638 ped ., erunt ( 95. 1.° )
g = W = ( 10467No) ( 2,33958 )g , w = (10467 /lo) ( 0,015638 ) g;
ac proinde designante eo respondentem valorem €,
seor
pors

; 3

2,33958 0,015638
lo 8
Eo = Too
bi
Wo :)-- (*
2,33958
=0,997495po.
205
'
—Pp.,— 10 ut,
⊬∙⊬≖≔⊉∙⊅∎∙⊉∎ F 16.;—P.
,
∙ 2,33958 — 3- .0,015638 ⇌−⇀
−⋮⊥−∘⇠ −−∃↾− ).. 8
a'., ∘ a m ↼⊣∸∘ 2,33958

: o,997495p.o.

l—xu .206
Hinc
E. 0 , 997495 ;
Ho
ratio videlicet inter densitatem aeris maxime vaporosi et
densitatem sicci aeris in ea qua sumus temperiei ac pres
sionis hypothesi.

5. • Valor i jam inventus ( 94. 1. ° ) spectat ad ae
rem siccum ; quoad aerem v. gr. maxime vaporosum erit
T. .
0,997495 flo
( 30,1959 ) ( 10467 ) ( 2,33958 )
Eo 0,997495

6. Obiter notamus illud : aquam sub satis alta
praesertim temperie in vapores versam conari sese qua
quaversus incredibili vi expandere indubia evincunt expe
rimenta. Hinc usus aquei vaporis in movendis machinis :
certo quodam tuborum valvularumque artificio vapor ex
caldario introducitur in antliam , ita , ut antliae cavitates ,
alteram infra embolum , alteram supra embolum , vicissim
obtineat, vicissimque frigidae suffusione ad pristinam redeat
Jiquiditatis conditionem ; vapor inferiorem cavitatem obtinens,
attollit embolum ; superiorem, deprimit ; embolus adnexus
est alteri ex duabus cujuspiam vectis extremitatibus ; qui
vectis altera sui extremitate vel immediate vel instrumen.
torum apte conjunctorum subsidio motum communicat rotis ,
malleis , elc.... ; prout nempe importat machinae
movendae natura.

Hinc
∙⇣∘−−−∶ o,997495;
p..
ratio videlicet inter densitatem aeris maxime vaporosi et
densitatem sicci aeris in ea qua sumus temperiei ac pres-
sionis hypothesi. ,
'
5.&quot; Valor i iam iuventus (94. 1.&quot;) spectat ad ae-
rem siccum; quoad aerem v. gr. maxime vaporosum erit
↿≖∘∙∙ ar, —(3o,1959) (10467) (2.33958)
s, o,997495 ⊬∘ o,997495 '
i..—

=== De aqua egrediente per angustum foramen e vasis verticalibus sive cylindricis, sive prismaticis.===

97. Haec praemittimus ex pluries iteratis experimentis 
[[Fasciculus:Aqua egrediens.png|thumb]]
1.° Minuta corpuscula disseminata per descendentem aquam verticaliter descendunt commuui ad sensum velocitate usque ad horizontalem &lt;math&gt;HH'&lt;/math&gt; (Fig. 52), cujus distantia ab orificio &lt;math&gt;hh'&lt;/math&gt; aequat triplum radiurn ipsius &lt;math&gt;hh'&lt;/math&gt;; tum cursum flectentia, perque lineas curvas incedentia conspirant versus orificium. Aqueae igitur particulae verticaliter descendunt usque ad &lt;math&gt;HH'&lt;/math&gt;; formaturque ab &lt;math&gt;HH'&lt;/math&gt; ad &lt;math&gt;hh'&lt;/math&gt;conoides aquea &lt;math&gt;Hhh'H'&lt;/math&gt;, quiescentibus portiunculis lateralibus &lt;math&gt;B.B'&lt;/math&gt;.

2.° Adhuc obtinent et verticalis particularum descensus, et earum conspiratio ad formandam conoidem, etsi orificium aperitur in latere vasis.

3.° Aqua ex aperto orificio verticaliter saliens assurgit ad supremam fere prementis aquae superficiem.

98. Denotet &lt;math&gt;\omega&lt;/math&gt; velocitalem aquae egredientis ex orificio &lt;math&gt;hh'&lt;/math&gt;, et &lt;math&gt;z&lt;/math&gt; altitudinem prementis aquae supra orificium, erit proxime (30:31)&lt;math display=&quot;block&quot;&gt;
\omega=\sqrt{2gz}(k) . &lt;/math&gt;Ad haec; si &lt;math&gt;\alpha&lt;/math&gt; denotat horizontalem vasis basim, &lt;math&gt;a&lt;/math&gt; orificium &lt;math&gt;hh'&lt;/math&gt;, &lt;math&gt;v&lt;/math&gt; velocitatem particularum ex quibus coalescit suprema aquae superficies, erit &lt;math&gt;
\alpha v dt= a.\omega dt,&lt;/math&gt; unde &lt;math&gt;w= vi&lt;/math&gt; et facta asna,
a
imen
roting
w = nv ( k' ) .
Hinc (27 )
asis
dz ndy
do 2gz et dt
V28%
ideoque designante 2, initialem valorem » , quum nempe t = 0 ,
inted
ft?
ae-
erit
talil
men&quot;
aul?&quot;
207

98. Denotet &amp;) velocitatem aquae egredientis ex ori-
ficio hls', et : altitudinem prementis aquae supra orifi-
cium , erit proxime (30 :31 )
a) −−∶ Vig.; (k).
Ad haec :
si a denotat horizontalem vasis basim , a ori-
licium hh' , :: velocitatem particularum , ex quibus coa-
lescit suprema aquae superficies, erit
ac «.vdtzamdt, unde a): −⇀ v :
et facta «scita,
a
mzn-v (k').
Hinc (27)
ds ⇂∕ nds
— :: ∙−−−∶ 2 :∙∙∙ ∙ dt ga ,
et dt V—zgs .
ideoque designante s., initialem valorem :, quum nem-
Pe ∁−−−−∘⇟208
i - V7(
..- , ) ( &quot; ) .

99. Sit \beta volumen aquae tempore t egredientis ex
orificio a ; erit ( 98. k . k &quot; )
233= a.orde=a(28)* . * de= a/ 2018 ( 3 - V . Jde
Propterea
B =a/25)*(*.* -VERSI-)
( k' ' ) .

100. Assumpta z = o in ( k &quot;. 98 ) , prodibit tempus
O , quo vas lotum evacuatur ; nimirum
11/ 를
2n 0—
V 29
( k &quot; ) .
In ( k &quot;) et ( K '') substitae valorem molè ex ( k ' ) ; habebis''
2n 을
21
5 B
ag
V28
2n (25–2-ce). ( &quot;) .

101. Ex (k &quot; ) sequitur illud : si duo vasa habuerint
et altitudines zo , zo, el orificia a , a' aequalia , tempo
ra 0 , 0 quibus deplentur , erunt in ratione basium a,a' ,
siquidem
2n 2n '
á
0 : 0 =
V 29

: z ' .

V 28
--- N : n ' = . Q : a' :
a '
208
≖−−−−⇁ 21( soi—1 ii) (li&quot;)
-
l/Zg
99. Sit þ volumen aquae tempore :egredientis ex
orificio a ; erit (98. I:. k&quot;)
.l. s
'
s i— dþzaüdtza (25? s
'dt:a(2g)ir&quot; (
zog— &amp;
t
)dt. ⇂

Propterea
,

100. Assumpta ≖∙∶−−− ∘
in (Is&quot;. 98 )
. prodibit tempus
9 , quo vas totum evacuatur; nimirum
9: 2: ∣∙∘⋚ (z.-&quot;) ⋅
l/Zg
In (It-&quot;) et (I.-&quot;') substitue valorem s.,ïli ex (Is&quot;); habebis

:: 6— 2n 3,- ag( .

⋅−
2-
. — ⋅ 20 t): wg I.&quot;. (3 ,&quot; ( )
101. Ex (It-&quot;) sequitur illud: si duo vasa habuerint
et altitudines s.
, a',, et orificia a
.
a' aequalia. tempo-
ra 9. 9' quibus deplcntur ,
erunt in ratione basium a.d.
siquidem
∙ 2n 2n'

: 9:∶−− zo : ...z'o zn:n'— :—,-—a:a':

l/Zg a a· 209

102. Quantitates aquarum successivis et aequalibus tem
poribus effluentium decrescunt secundum numeros impa-
res ordine retrogrado sumptos. Assertio facile ostenditur
e secunda ( k ) , facto successive t=1,2,3,4, • ; nam
quantitates illae prodibunt expressae per
ag
2n (29-1 ) , L ( 49-4 ) – ( 29-11 , P.(69-9)– ( 49-4) ,
Se
ag ( 80-16)
2n
ag ( 60-9) ... , seu 2n
ag
2n (20-1 ) , 29–3 ) , (29-5 ) , (29–7), - ;
ideoque etc... Idipsum eruitur ex (k &quot; ) et ex prima (k&quot; ) ;
denotantibus enim 21 , 22, 23 , ... valores z respondentes tem
poribus 1 , 2, 3, ... eae praebebunt
&amp; 02 , 2, 3
(0-1) 2,225 2n? 2n2 , =-2,(0-2) », 23 =
S
(0-3 ) , ... 29-1 2n2
8
2n2 ;
unde
6
( 29-1 ) , 21-22 2n?
8
( 29-3 ) , Zz- 23 =
2n2
8
2n2 (26-5) , ... 29-3 8
2n?
et consequenter etc...
Hinc si dividendum sit vas in partes successivis dati tem
'209
102. Quantitates aquarum successivis et aequalibus tem--
poribus ellluentium .decrescunt secundum numeros impa-
res ordine retrogrado sumptos. Assertio facile ostenditur
e secunda (k')
.
facto successive t::1,2,3.4, .; nam
qnantitates illae prodibunt expressae per
Zn (29 'l), 2&quot; (49 4) 2&quot; (29 1) , 2n(69 9) g(49 4) ,
a—g - .. &quot;£ -
∙∙ 2&quot;(89 16) 2&quot; (69 9) . , seu
ag
- es -
es -
∙−−
7:091). 2&quot; (29 3), 2809 5)sa g(29- &quot;711&quot;;
ideoque etc... Idipsum eruitur ex (In&quot;) et exprime (k');
denotantibus enim sus,, &amp;
.... valores :respondentes tem-
poribus 1,2, 3, ... eae praebebunt
'
z.,:
⋚−⊯≖ ⊖⋅∙ z. ↼−− ⊋⋅⋚⇆≺⊖⋅↿ ):,
≖≖−−−∶⇄−⋚⊑≺⊖∙⊋≻≖∣ za⇌∎
- 2 -' ∙−−− i.— ↿∠∏−−≖⋅ i(ä 3) zo Zn' '
↴
uude ⋅
2, ∙z. ∙−−−⋮⋚≔ (29-1),z,-z, −−∶ Zif-;, (za-3), 22-33 −∙−−∙∸−
s- - ...g. .
ZI€3(29 5), ∙∙∙ Zo-x —-2na ,
et consequenter etc..
Hinc si dividendum sit vas in partes successivis dati tem-210
1 poris a unitatibus vacuandas , determinata altima 20-1
ceterae usque ad primanı erunt
320-7.526-4,72 6-7** (29-3 )z 0-10 ( 26-1 ) 0-1 .
1
1
Liquet autem fore
2:6-1 + 326-1 + 520.4 + 720-1 + . + 20-3)26-17
0
(29-1930_1 = {1 + 29-1)
o
2
0-1 = 622
6-41 1
d
.

103. Tria subjungimus, quae certissimis constant experimentis. 

1º. Vena aquae exilientis a foramine aperto in pertenui lamina magis semper contrahitur usque ad ejusmodi distantiam ab orificio, quae vix aequat ipsius orificii radium; estque venae maxime contractae area cc' ad orificii aream ut 5 : 8 circiter. Istius contractionis ratio ex eo desumenda videtur quod aqueae particulae etiam paullo extra vas retinent obliquos convergentesque motus, quibus orificium subierunt.

2.• Tanta effluit aqua intra datum tempus ex fo
ramine aperto in pertenui lamina , quantam suppeditat for
5
mula ( k &quot; ) , modo tamen pro a substituamus 8

3.º Aptatis orificio exterius tubis cylindricis, co
nicis etc., pro varietate tuborum variae habebuntur quan
titates aquae dato tempore exilientis.

104. Haec notentur 

1º. Acceleratio , per quam velocitas aquae admodum exigua usque ad HH' mutatur in finalein satisque grandem effluxus velocitatem, tota manifeste perficitur ab HH ad cc' intra spatium interceptum conoide ac vena contracta, ubi nempe descendentium stratorum amplitudines citissime decrescunt. Vas ergo ABB'A '
a.
210
poris 9 unitatibus vacuaudas
.
determinata ultima &quot;9-1 ,
ceterae usque ad primam erunt&quot;
3z9-1, 529-1,7z ⊖∙↿∙∙∙ (29-3): ∂∙↿ '
(29-1)z 9-1 ∙
Liquet autem fore
ze, ↿
∎∎⊢∍∅∂∙↿⊣−⋮≖∂ ∙↿ ⊣−⋅∄≖∂∙↿∙⊢∙∙∙↤⊋∂∙⊰≻∅∂∙↿−⊢
. 9 ∙∙∙ :
(29-1)z9-1:(1-1—29-1)ïz ⊖∙↿ —9 294 -
7

1 '.
er
perto
ejus
citci
ori.
spectandum erit tamquam terminatum tubo Hcc'll' ad se
ctionem HH ' aptato.

2.0 Motus aquae defluentis in regularibus alveis
traduci potest ad motum aquae prosilientis ex angustis
vasorum orificiis. Concipe regularem alveun secari plano
verticali ; in plano isto insculpi plura foramina , ex qui
bus effluat aqua ; iisque nova addi foramina ut indefinite
crescat foraminum numerus , totaquc sectio veluti unicum
efficiat foramen infinitis numero foraminibus' coalescens :
cum e singulis foraminibus aqua debeat effluere veloci
tate illa , qua et erumperet e vase ad eamdem altitudi
nem pleno , et , sublato plano , Queret in eodem sectionis
loco , idem ferme erit casus aquae defluentis per alveum
et aquae prosilieatis e vase ad eamdem altitudinem pleno.

3. • Si in regulari atque horizontali alveo mo
vetur inferior aqua ob superioris aquae pressionem , nec
directionum obliquitate , et fundi laterumque resistentia
turbatur conceptus motus , apud particulam quamvis de
notante i altitudinem superincumbentis aquae , exprimet
V 2gi particulae velocitatem.

4.° Quod si regularis alveus ad horizontem ex
sistat inclinatus , sitque m altitudo debita velocitati apud
supremam aquae superficiem , cum haec velocitas ( levio
ribus corporibus aquae injectis determinari potest ) utpote
orta ab inclinatione alvei debeat aquae omni esse
munis , exbibebit V 28 (i + m ) particulae velocitatem .

5.° Hinc poterit in utroque casu definiri quan
titas V aquarum intra datum tempus t defluentium apud
quamlibet regularis alvei sectionem ; sic v. gr. in hypo
thesi rectangularis sectionis habentis latitudinem r , erit
in primo casu
i
2tri.
V = tr įdi
3
no es
دالاق
uibus
fo
o for
com
S, CO
paano
relo
in 6
Dani
?ptom
Vžg
I stra
B!!
in secundo
lCP
perta
ejus-
filicii
, ori-
iot!
.aullo
uibui
⊊∣∝⊦
luan-
«de
new&quot;
ipua
! slfl'
BN
↗−
⋅ 211
spectandum erit tamquam terminatum tubo Hcc'll' ad se-
ctionem HH' aptato.
2.&quot; Motus aquae defluentis in regularibus alveis
traduci potest ad motum aquae prosilientis ex angustis
vasorum orificiis. Concipe regularem alveum secari plano
verticali .; in plano isto insculpi plura foramina , ex qui-
bus effluat aqua; iisque nova addi foramina ut indefinite
crescat foraminum numerus , totaque sectio veluti unicum
efficiat foramen inlinitis numero foraminibus' coalescens :
cum e singulis foraminibus aqua debeat eflluere veloci-
tate illa, qua et erumperet e vase ad eamdem altitudi-
nem pleno , et
, sublato plano ,
(lueret in eodem sectionis
loco, idem ferme erit casus aquae defluentis per alveum
et aquae prosilientis e vase ad eamdem altitudinem pleno.

105. Auctores non pauci tractantes de motu li
quidorum ex apertis luminibus effluentium , illud usorpare
solent tanquam principium , quod nempe unumquodque li
quidi in vase quolibet descendentis tenuissimum et hori
zontale stratum coalescat iisdem constanter particulis com
muni , eaque tantum verticali , velocitate donatis . Deno
tante v verticalem velocitalem , qua pollet in fine tempo
ris i quodvis massae liquidae punctum ( x, y, z ) sollicita
tum gravitate g , vis acceleratris de se valens producere
dy
actualem motum exprimetur ( 28) per : et qaoniam ,
dt
praecisis etiam mutuis punctorum pressionibns , adhuc ta du men vis de gigneret actualem motum ; ideo , attentis pres
sionibus , consistet in aequilibrio punctum (x , y , z) solli
du citatum vi g Propterea ( 88 )
dt
do
dz
dvi
dt (kº ) .
Attenta insuper liquidi continuitate ( liquidum ponitur in
capax compressionis ) ; sequitur , si A designat amplitudi
nem cujusvis strati horizontalis , fore ( 98)
viw = a : A , unde v =
Ä ( * &quot; &quot; ) ;
w est functio temporis t ; A distantiae ; ab XOY : sequi
212
∙∙∙ i. &amp; 3
VZU'l/ng (i-l—m) di: Z',..l/Zg g'[(10 m)⇣⇥≖∶∣⋅ a
denotat i., sectionis altitudinem .

1054: Auctores non pauci tractantes de motu li-
quidarum ex apertis luminibus ellluentium. illud usurpare
solent tanquam prineipium . quod nempe unumquodque li-
quidi in vase quolibet descendentis tenuissimum et hori-
zontale stratum coalescat iisdem constanter particulis com-
muni, eaque tantum verticali .
velocitate dona-tis . Deuo-
tante v verticalem velocitatem
, qua pollet in fine tempo-
ris : quodvis massae liquidae punctum (.r.-7, :) sollicita-
tum gravitate g. vis acceleratrix de se valens-producere
. dv ⋅
⋅ ∙
actualem motum exprimetur (28) per .d—t : et quoniam
.
praecisis etiam mutuis punctorum pressionibus , adhuc ta-
dv
men vis —d—£ gigneret actualem motu-m; ideo , attentis pres-
sionibus
.
consistet in aequilibrio 'punctum (.r.-y. :) solli-
citatnm vi g—⋛⋮ ∙ PrOpterea (88)
der . ⋅ dv
'
z,; ∙−−∶ P- (
−−⋅ (17) ('i ')
-
Atteuta insuper liquidi continuitate (liquidum ponitur in-
capax compressionis ) ; sequitur . si A designat amplitudi-
nem cuiusvis strati horizontalis .
fore (98)
psa-ca: A, lel). ,
undevzr-a— A (
cc est functio temporis :; A distantiae :ab XOï: sequi-213
1
i
tur quoque supremam descendentis liquidi superficiem ma
nere horizontalem . Ex kl( ) habemus
dv a da
a do
dt
aw dA dx
A2 dz dc
-
A dt A do
.
aw dA
a da a’w2 dA
A2 dz A di A3 da
iccirco formula (K ™ ) traducelur ad
do dz dz =+ (sds - au de):
1 sumptisque integralibus quoad % ,
==C+u(sma )
Zo
ic
exprimit 200 distantiam inter XOY et supremam liquidi
superficiem A , Denotante w, pressionem v . gr. atmosphae.
ricam in superficiem illam , assequimur
Two
-= C+4 (** 241,3.
)
unde C=0. – (
( 50-100)
.
li propterea
-=o +15(2-)-avenit SA- G --&gt;) ( 47 ).
Zu
je Apud orificium
213
tur quoque supremam descendentis liquidi superficiem ma-
nere horizontalem. Ex (F&quot;) habemus
dv a de.) am dA d: a dm
.dt—A dt A*dz dc-TA dc
as) ubi a da) ama dA ∙
Aza.&quot; Ad: A3 d.'
iccirco formula (Is&quot;) traducetur ad
' dar − das d:. am: dA
- )
⋅⊋−⋮∁≀∅−−∙↱∙≺⊰∠≀∅−−∅∙∣⊺⋮⊺−⊢ A3 d: dz ,
,. sumptisque integralibus quoad s,
⋍≖⇌∁−⊦⊬≺≊≴−∘≤≀≜∫≖≤≀⋮− −
.-)
,.
dt A 2112
zo
.i-
eXprimit s., distantiam inter XOï et supremam liquidi
superficiem A.,. Deuotante wo pressionem v. gr. atmosphae-
ricam in superficiem illam
, assequimur
2 2 2 2
≔∘−−−⋅∁−⊦⇤∸≼∊≴∘−≦⋏∘∶≕≻ ⋅
....
∁−−⇌≖≖∘
..
(g.... &quot;.? ):
[&quot; propterea
d ad:. 2
1 1
w:eod—Pgu-uþauä A a&quot; P:) (A*—. :) (k&quot;&quot;).
zo

; . Apud orificium214

1
Wo
A2 a
designantibus insuper b et i distantias ipsius orificii ab XOY et ab A. ,
m=b , 2 = b - i , 1 - % = i :
facto igitur
b
dz
A
biI !
erit ibi
mode,
gi - sa- (1-4 ) = (A &quot; .
Quoad (k &quot; &quot;&quot; ) et ( k &quot; ) notamus haec tria. 1.0# Si a est
parvitatis contemnendae , ex (k &quot; ) profluet
a = w.tugis mo ) ,
ut in casu liquidi aequilibrari (88) ; ex (k' ) vero emerget
V2gi ,
quae formula recidit in formulam ( k) .
2.0* Si , affluente novo liquido, eadem servatur
in vase altitudo liquoris , quantitates i, A., B exsistent con
stanles ac datae ; et facto
a2 1 h
A.2
↿ 1
≖⋝−−≖≖⋅∙ ∙ :::—:::;
designantibus insuper &amp; et t' distantias ipsius oriücii ab
XOT et ab A.,.
526 , sozb—t' .
s—sozi:
facto igitur
erit ibi
ad!» c.)&quot;( a2
g' Bdc 2.↿∎∎∎⊼∘⊑≻∶∘ (kl
Quoad (k&quot;&quot;) et (Is&quot;) n0tamus haec tria. ↿∙∘∙ Si a est
parvitatis contemnendae
, ex (k&quot;&quot;) profluet
Uzwoillg (z'—*o) )
ut in casu liquidi aequilibrati (88) ; ex ('tu) vero emerget
∾⋅⇌ vra-u
quae formula recidit in formulam' (lt).
2.0a Si
. affluente novo liquido. eadem servatur
in vase altitudo liquoris. quantitates i. A.,. B exsistent con-
stantes ac datae ; et facto
emuli
a ad
Qiiia215
formula ( k &quot; ) praebebit
h
d
2a Bdt V
2gi
2ada
2gi - hwa hV
2gi
h2 .62
2gi
d h
h
d
a V
2gi
V
2gi
hv
2gil
1+
v
2gi
+

: ) h h
;

-
Vzgi
unde , sumptis bogarithmis quoad basim
a
Bta
log
hy
2gi
V 2gi + hw
V
2gi
ha
non additar constans et arbitraria quantitas utpote =0
siquidem tempori t =o respondet w =o. Ex ista aequatio
ne emergit
Bhty2gi
V2gi(1
a
h
1 + e
Bhiv
2gi
a
inferimus , elapso brevi quodam tempore t, fieri ad sensum
1
V 2gii
itemque
21 5
formula (li&quot;) praebebit
d—L
.,
Bdt— iuda −− 21. V? —
⋣∊∙−⋅∣⇂≖∾≖ hl/Zgi IP
1——c.)2
Zgi
d
a ita di;—00 —( ⇂∕2gi ∣
l/2gi ) h
'⋅⊾ ⋅
l/Z-g—l ↿∙∙∙ .b— 6)
,
⇂∕⇄∃−⋮⋅∾ Vzgi
unde
, sumptis bgarithmis quoad basim e ,
'Bt: ]: a— log iii—E? :
l/th' Vzgi −− hæ
non additur constans et arbitraria quantitas utpote ∙−−−∘ ,
siquidem tempori tzo respondet 6) 30. Ex ista aequatio-
ne emergit
⋅≖∃∣≖≀⇂∕⋝⋮⋜
&amp;
l/2-g-i1—e— ::
∎∎⇀ h B'm/223-
1—l—e— a
inferimus
. elapso'brevi quodam tempore :. fieri ad sensum
1 − −−−−−−↗↓−∎∕∑∊≀⋅⊰
itemque216
= w.tuzia - 2 .) – paga?i( 1 hot G1 - ),-- VE
3.•* Si vas consistit in verticali cylindro , vel pri
smate , A erit constans , et A.=A ; insuper
dz 7-zo 1
A A B
ic
zo

=== Aliquid subjungitur circa generalem theoriam motus corporum fluidorum. ===
106.* Velocitas v, qua pollet if fine temporis !
quodvis massae fluidae punctum ( x, y, z) sollicitatum (86)
vi acceleratrice Q , resolvatur in ternas v' , w &quot; , 1 &quot;&quot; coor
dinalis axibus Ox, OY, OZ parallelas ; erunt ( 29 ) dý ,
1
dy' ' vires iisdem axibus parallelae , in dt dt
quas resolvitur' vis acceleratrix q' valens de se produ
cere actualem motum. Quoniam , etsi praecisis pun
cloruni mutuis pressionibus , adhuc tamen gignit actualem
motum ; ideo , attentis pressionibus , consistet in aequi
librio punctum ( 2, y, z ) sollicitatum viribus X
, ) – , dv',
1
Y dy&quot; , 2 dy'&quot;'; ac proinde ( 86. o )
di
1
dt
de = - ( x – do ). -- ( v- à dv&quot; ) , )
de u ( 2-2 &quot; )..
( 6)
216 .
↴
a':' 1 ↿ a Vii—g'
∙
szo-l-pg(2'.—20) uia (Aa Ag) ,
VS—K ∙
T
∙
∶⊰∙∘∙ Si vas consistit in verticali cylindro. vel pri-
smate
, A erit constans, et A.,:zA; insuper

:

Zo
Aliquid subjungitur circa generalem theoriam motus
corporum [[ uidorum.

106: Velocitas 0, qua pollet i! fine temporis :
quodvis massae fluidae punctum ( æ, y, z) sollicitatum (86)
vi acceleratrice ?, resolvatur in ternas v'. 9&quot;
.
v'&quot; coor-
dinatis axibus OK, OT. OZ parallelas ; erunt (29) ;llg-dvl ,
,, 1 ∙∙∙ ∙ ∙∙ ∙ ∙
(Z— dv
, &amp;? dv vires iisdem axibus parallelae ,
tn
quas resolvitur vis acceleratrix ?' valens de se produ-
cere actualem motum. Quoniam (a' .
etsi praecisis pun-
ctorum mutuis pressionibus, adhuc tamen gignit actualem
motum; ideo , attentis pressionibus , consistet in aequi-
librio punctum (z, y, :
) sollicitatam viribus X — −↿− dv'.
dc
y— —dv&quot;,Z—- dc dv' ; ac promde (86. o)217
1
dt ,
dy
du
dx
axt
du' !
(du ?
Habitis v ', &quot;.0&quot;, pro functionibus variabilium x, y, z, t,
exsistent ( 27. 24.0)
dy' du dv du du' dx + dy +
dzt.
dix dz dc
dur dul
dy + dz +
dt,
dy dz de
du du dy't dy + dz + dt ,
dx dy dz de
du du=
dr
seu ,
ob dx v dt , dy udt , dz udt ( 27 ) ,
dv dú dv ' dun du'
dt ,
dx dy dz de
dur dvd dv du'a + dt ,
dy dz de
( 6 ).
dy'll dy dy &quot;
\dx dy dz
dat
di axt du.
du
dyt
dzt dz dt dy
dl ;
at
will +
leo
lesin
' du'
\dx vt
alt
- de
dy ut
21&quot; + .!&quot;
to
at
dt ,
ide
dx du du.
vt
de ede :
w
itot
dy
dz
formulaeque (6) vertentur in
15
1
1
217
Habitis v'. v&quot; . v'&quot;, p. pro functionibus variabilium x,]. z,t,
∙ exsistent (27. 243) '
/
dVr-äï-I-dæ-f-g dy—l—d 7; v/dz—l-dïvt-dth
.
I, I/
dp'p&quot;: &quot;
≤⋮∙−≤⋅⇗↙↙∠∞∙⋅∣− dv ∙−∙↙∣∫⊣−↙≀↥≟ dz-l— ii)—dt,
III '&quot;
dv'&quot;: dv Tdæ—i—djr dv/Il d-v'&quot; dy −⊸⊢−−⋅ zdz—l- -^——--dt,
d d ⊬−− ⊋⊥∸↙≢↕ ↙∄↕≤−⊦ Hari- ⋮⋮↙∄≖⊣−−↙−∣≛⋮∠≀≀⊰
seu , Ob dx:
'v' dt .ei)-':«:;&quot;dt, dz 37)/&quot;dt (2&quot;..
dv! ∣∙∙− d'&quot;, v&quot; ∣∣ (if—,) dv—(ïr-v-l-ï P-l-d—z-IVI-ï-dï— dt,
&quot; &quot;
vl] II
V&quot;'—:(d—; V, &quot;J—d gr.&quot;- .v/j-l—g-z- will-i-ïi'l;-—-)d[,
(V).
dv ∣−− dvlll dui/I
−−−−⊋⋤−≼ .
[v;/1.&quot;, dv'&quot;
,
∣∣
), 1
∎∎⊢∎∎∎−∎−∎ dz 'l&quot; dt —)dt ⋅∣⊹ '
∙−− dP'. ,v/ !
dlu' ut dp' ∣∣∣ dp') ∙
dy. (dæ'v [
dy&quot; ⋅−↱⋅⋅∓⇂≀ −↽⊋−∁⋅− dt.
formulaeque (6) vertentur in
15218
dos deild dy&quot;
dx
dv'
dx dy
dz che si (
(v
do ( 6&quot;) v'
do'
dy
v
du
dz w
dur
dy
dx de - ) ,
dy ')
do dv'

: -(2

v'
dy't
dy
v
du Win
dz dx dz de
107 #. Quae portiuncula infinitesima massae fluidae a
pud punctum ( x , 3 , 2 ) sub volumine V in fine tem
poris i exprimitur per V , eadem sub volumine V+dV
in fine temporis + dt ad punctam aliud translata expri
metur per ( V+dV ) ( pe + du ); ideoque
V = V + dV) (v + dpl)= Vu + udV + Vdp. + dp.dV ,
et consequenter, misso dudv,
Vdp. + pdV= ( 6 &quot; ).
Sumatur V = dxdydz, aequale nimirum parallelepipedo
rectangulo AF ( Fig. 47. ) sub laterculis AD( =dx) , AB( =
dy ) , AH = dz); punctaque A , B, C , D , H , M , F , E po
nantur transferri tempusculo de ad A ' , B , C , D , H' , M',
F' , E , ut sit V + DV = A'F'. Transferetur A in A '
velocitatibus d' , 0 , 2, juxta coordinatos axes ,
runtque
e
x + v'dt, y tv&quot; dt , z tudt
coordinatac puncti A': designatis v ', u ' &quot; per
d7 dx d] dz : '
,
dm' dv&quot;' −∙∙: Z- ∣- dv'&quot; du''' dv'&quot; ∣∣'''
∙∙∙ ∙−−⋁∣∣∣∙− —) ∙
dz F
(
da: v dy 'v dz dt
107-. Quae portiuncula infinitesima massae fluidae a-
pud punctum (æ
,
I,: ) sub volumine V in fine tem-
poris : exprimitur per VP-o eadem sub volumine V-l-dV
in fine temporis t −⊢ dt ad punctum aliud transl'ata expri-
metur per ( V-l—dV) ( p. dy. ); ideoque
Virsz-l-dV) (p.-l—dp.) ∙−−∶ ⋁∣↓∙⊹ ⊦∙∠≀∇−⊢ ∇⊂∣≴⊥∙⋅⊢ dde .
et consequenter, misso dpdV,
. Vdp. −⊦ ⊬↙∣∇−−∶⋄
('b'&quot;).
Sumatur Vzdædydz, aequale nimirum parallelepipedo
rectangulo AF (Fig. 47.) sub laterculis AD(-:-:dæ) , AB(-—--
dy ), AH(-:dz); punctaque A, B, C, D, H, M , F , E po-
nantur transferri tempusculo dt ad A' , B' , C' , D' , H' , M',
F', E' , ut sit V −↿− dV −−∶ A'F'. Transferetur A in A'
velocitatibus v'. a:&quot; ,
∙⇂∙∥⋅ juxta coordinatas axes , e-
runtque ∕∕∕ '
..: ⊣−⋁∣↙∣⊀ ∙
]−⊦ wa: , z −⊢ war:
coordinatae puncti A': designatis v', v&quot;, «a'/' per219
fi( x , y , %, t ) , fa(x , y , z , 1), 13 (x , y , z , t) ,
expriment
fi (x , y , z + d2, e) ,fz(x , y , z + dz, t ), f3( x , y, z + dz, t)
velocitates coordinatis axibas parallelas puncti H euntis in
H '; et cum babeamus ( 27. 24.)
filx9,2 + dz,t) = f (x , y ,z, e)7df1(x,y,z,e)dz = uti
du
dz dz ,
dz
e ao
em dy ”
fa (x , y, 2 + dz, t ) = 0&quot; +
dz ,
dz
spri.
f3( x , y , z + dz, t ) = 0 !!! allt dv !
dz,
dz
IV ,
coordinatae puncti H'erunt
X
+ (v + da )dt,y + (&quot;* + de )de,

: +de+ (** + adaptada dt:

pipedo
AB =
E po inferimus, missis infinitesimis tertii ordinis, fore ( 50. 6º. )
1 , M.
in 4
5 ,
ee A'H' =
[ledesdeu + )de de +
) ]=d =+
dy &quot;
-dz dt dz dt .
dz
Motus puncti Cin C'juxta coordinatos axes fiet velocitatibus
!
15.
C ∎∙
em-
[pl'l'
IV.
. 219
fuci-'s], 3! 1), fa(æs)'s 3! t) ∙ ⊀∍≺⋅↕∎∙∫↿≖∙ :),
expriment
ftlæsfaz-l'dzs 1) ,falæoys z 'l'dz; t)sf3(æs ïs ≖∙∙∣− d:, 1)
velocitates coordinatis axibus parallelas puncti Hieuntis in
H'; et cum habeamus (27. 240.)
dfx(æJ,z,t)d fax-a',: 4—dz,t):f.(æ,y,z, :
dz dz—v ↾−⊦↙↙∙⋚∙ —dz,
&quot;
falæsïsz (I:-',! :):vlf'i'd'ä- dz:
d'UIII fave,], z-l-dz,t ): ⇝∣∣∣−⊢ —zdz.
coordinatae puncti H' erunt ' &quot;'
ahi-(» ∣⊣− −↲≖≻≳≀∙∫−⊦≼↙∣⊣−≝∂≖⋟↙∦⋅

: ↽⊦ dz −⊦ (W.;- ↙⋛↙−≖

d: )dc.
inferimus, missis infinitesimis tertii ordinis, fore (50. 60-)
A'H': RSTV) dz-dz −⊦≺⋅⋮↷⋛↗−−−≖−∥≖⋟↙≀≖≏
d:: −⊦
d.,/II dv!&quot; -
(d:-[- d: dzdi )]ä :dz-F—dzdt-
Motus puncti Cm∁∣ juxta coordinatas axes fiet velocitatibus220
falar + dx , y + dy, zil ) = fi(x , y , 2,1 ) +
afı( 8•
7,5,6)
det dfi (x , y , 2,1) du
dy
dy
dic = tIdxt dx
falx + dx , y tdy, 2, 1 ) = &quot;&quot; + -dat dx
du &quot;
dy ,
dy
dumi
dy !!!
ON +
f3(x + dx ,y + dy , z , 1 ) = &quot;&quot; dat dx dy ;
dy
inde prodeunt coordinatae puncti C
du
d )
dy x + dx + (v + ad det )dt
y + dy + ( ** + na tempat day ) di,

: + ( v&quot; + data darym dy de :

motus puncti F in F ' juxta coordinatos axes fiet velocitatibus
du'
filxtdxy + dy,z + d2,2)= x + xdx + dydy +
du
dz,
dz
dy&quot; du&quot;
falxtdx,y + dy,atdz,t)= &quot; + de + dy dy
du &quot;
da dz,
du &quot;
du
f3( x + dx, y + dy; z + dz,t) = 1 &quot;'
-dxt
dy&quot;
dy dy +
dz;
dx dz
inde exsurgunt coordinatae puncti F
220
fdæ'i'dng—i-dy- ≖∙⋅↕⋟⇌∣≖≺∝⇟∫∙ ze t)&quot;,-
df,(æ.j,z,t) dfl(æsyszvt)
dw'd d &quot; ≀∂≖≺∙↿⊏−≱⊢↙↕↡∫↽⊢∂∫⋅ ≖↿−⊸⋅⋅⇂∙∥⊣−↽⊋− &quot;L &quot;ad; df '
f3(æ-l—dæ,y—]-dy,z, :
)-— v'∣∣⊹−⇁∙
inde prodeunt coordinatae puncti C'
∙↴⊲−⊢∠∄∸≀∶⊣− (⊣−⋅≦⋮∠↴↧⋅↕⊣⇀−− ↙∄⋤↙↿∫⋟≴↙≀⋅
∫∔∂⋮∫⊹ (
∣∣⊣↼ d,,⇡⋮≀−−⋅∶≴←⊦≤−⋚−∥⇩≀ wa.)
vll/ du ≖↽⊦≺⊛ ∣∣∣
w&quot;'-l--d—-æ dr—l— df )dz:
motus puncti Fm F'luxta coordinatas axes Eet velocitatibus
∙ . . . '
' d ,.
fia—W;? ∂∫∙≖−−∠⇣∙≀⋟∶⋁⊹⊼∶↙⊩⊦≣↗ −⋤−∶⊔⊹ −:dz, dv&quot; dv&quot; dv&quot;
ta(æ-i-er-l-dr, a—l—dzn): ⊎∣∣⊣⋅∙⋣∂≛−⊦ df dy ! dz ds,
d&quot; III d..&quot; f3(æ—]—dx, ⊹↙∄∙↗∙≖−⊢∣≂∙ ():—Ju&quot; i-i-d; dælL dr cir—]— —-dz;
inde exsurguut coordinatae puncti F'221
dyn
= da + ( + van de tener tous de Jdeo
s + d3 + ( v +adar an nas tudi nadia )dt,
s + de + ( * + dpt dathetn dy + advan die Jde:
1
inferimus, missis infinitesimis tertii ordinis, fore
CF = [ 'de de + oem )deº de +
( de + de &quot;de de ))]]* = da +
dy&quot;
de dt.
dz
Ad motum puncti B in B ', computatum in coordinatis axi
bus, spectant velocitates
f( x, y +dy, z, t ) , falx , yt dy, z, t), f3(x ,y + dy, z, t ) ;
ad con similem vero motum puncti M in M' velocitates tatibus
fi(x ,y + dy, z + dz, t) , 82(x , y + dy ,ztdz, t ) ,
th
dan
f3(x , y + dy , z + dz , t ) :
dy&quot;.
propterea coordinatae puncti B ’ desi
dz
+ (x + dy dy )de , y + dy + (.* + dar dy ) dt,
du&quot;,
dy
7
+ (*&quot;'+ dydy hdi:
7;
(221'
I
æ—l—dæ-l-(tb -]-d −∙⋮dx-j—d −−∣vlddy-l— ——dz )dt,
,,
'
dv&quot; . dv&quot; y-l-dJ-l-( −⊢−− da.−−∥↙≀↓⊣− ⊒∫−∠≀∫−⊢−− ↙≀≖≻∠≀∁∙
&quot; dv&quot;: z-l—ds-l—(&quot; ⊣−≦−≦⊥∅≀∝−⊦↙∄ dyd ∣
↙↙≖ d: )dt:
unferimus, missis infinitesimis tertii ordinis, fore
∙∙∙⋅ dv) ∙ '
(du&quot;ïd) ∙ es'—[(? d: a: &amp;
∠≀∥≀⋍⋅−⊦
≺∁≀≖−⊢⋛≖ ——dzdt )]; :dz—l-Q—ds dt.
d:.
Ad motnm puncti B in B', computatnm in coordinatis axi-
bus, spectant velocitates
I.i-ïs;)&quot; ⊣∙∙ dy. 39 i)
'
f2(æay—l— d]: 2. t)af3(-'rsy—l—dft Z, !) 3
ad consimilem vero motum &quot;puncti M in M' velocitates
↿∎≺∞∙∙↗↾⊣−∠∄∫∙≖⊣− dzs t) sf2(æ sy'l—fi'r, ≖−⊦∠∄≖ ∙ :) ∙
fam ,y-l-dy.a-1- a.:):
propterea coordinatae puncti B'
æ-i- (&quot;'-l- ⋛⋚∠∄∫≻↙∄∁ ,J—l-dy—l-('
≻≖≀⋅≂⋮ &quot;j,-l— 72:41)!&quot;-
z −∣⋅− (vm-I— dv
dy222
coordinatae puncti M
++ (1 - en deJdt,
y + dy + (** + disa dy + &quot;deJdi.
z + dz + (** +
(** + + en in diehele
hinc
B'M dz +
du
dzdt .
dz
Ad motum puncti D in D ', computatum in coordinatis a
xibus, pertinent velocitates
fi ( x + dx, y , z , 1 ) , fa( x + dx,y ,z, 1), f3(x + dx, y , z ,t ) ;
ad consimilem autem motum puncti E in E' velocitates
filx + dr, y, z + dz, t ) , f (x + dx, y , z + dz , t ) ,
f (x + dx ,y , z + dz , t ):
proinde coordinatae puncti D'
de&quot;

* + dx + (ut ea adx)de,y + ( * + dxdx )dt,

++ (- + de -dx)dici
222
coordinatae puncti M'
x-l—(tf— ⋛≶↙≀∫⊣− ——dz)dt,
maH-( ⇂≀⋅⋛−−⊦ pri—4449
≖−⊦↶≀≖↼⊦≼ ⋮⋅∠⋛⋮−⊣− MH-;'dzdu)
hinc
,B'M'-— ∙−− tis-l- -—-dzdt.
Ad motnm puncti D in D', computatum in coordinatis a-
xibus, pertinent velocitates
ru(æ ∙−⊦ ciæ,], zit)1fa(æ &quot;l'dæofszo t) sf3(x&quot;'i&quot;dæs)'; 2; 1);
ad consimilem autem motum puncti E in E' velocitates
fdx—i-dæqæz-l-dz, t ) 'fa(x-l-dx,y , z—l—dz .
t ).
B(æ-j-dæ ,y, s--]— dz .: ):.
proinde coordinatae puncti D'
∝⋅⊦⊄↿↕∸−⊦≼↩∙−⊦ ——dæ)dc ,y—l—(v' −⊦≤−−⋮⋅⊑⋅↙≀⋅⊐∁⋟↲↥∙

: ∙⋅⊢ ≺∙∽⋯∙⊢ ⋛⋮≽∙⋮⇣↿∙↕≻≺≀∷223

puncti autem E
drt
s + dx +((uv + des de +die dz)dt ,y+ (** + de la
de &quot;a )de, a (* &quot; + dz) dt;
2 + dz to dv&quot;&quot; dy&quot; ,
det da dz
et consequenter
D'E' = dz to
du&quot;.
dz dzdt .
Itaque
A'H ' = C'F' = B'M ' = D'E' = dz +
du 
dzdt :
đz
simili modo eruuntur
AD = B'C ' FM H'E di = dx +
dx
dxdt ,
1t ) ;
A'B' C'D FE H'M ' =
dú'
dy + dydt.
dy es
thi
Ex laterculorum aequalitate manifeste consequitur eorum
parallelismus ; eritque A'F' parallelepipedum obliquangu
lum ; ita tamen , ut ejus anguli infinities parum diffe
rant ab angulis rectis parallelepipedi rectanguli AF ; quan
doquidem AF nonnisi tempusculo infinitesimo transfer
tur in A'F ' . Nunc ex H ' v . gr. due perpendiculum Ha in
areolam A'B'C'D ; erit
A'F ' = H'a . A'B'C'D' = H'a . A'B ' . A'D' sio B'A'D '
A'H ' . A'B' . A'D' sin B'A'D' sip H'A'a :
d:.
223
puncti autem E'
dv' '
dv' )
.. da: −⊢ dz z .7' 41- dæ—t- ∙⋅∎∙⋅−⊢ du:-1- (v' ∙∙⊢ da:
dv&quot; dv'&quot; dv&quot;' )
−∙ dz —-d.r —-d d ;
dz)dt ,
z-t-dz-tï 'v. −⋅⊢ da: ∙−⊢ ds :
f
et consequenter
⋅ dv&quot;:
D'E'c: dz −⊢ 71&quot;—2. dzdt .
Itaque
llo
'
'v
A'H':C'F' −−∶ B'M' ∙−−∶ D'E' :: d: ⊣⋅− ∙−≀⋮⋅≖−↙≀⋍⊄∄↥ :
simili modo eruuntur
A'n' :: B'C' −−∶ F'M' −−∶ H'E':dæ −⊢↙≟⋛ dædt
.
A'B'r: C'D' ::F'E': H'M' ∙⋅−−−∸ dy ⊣⋅−∙≣⊥⋅ ↙≀∙↨↾∠∄≀⋅∙ ]
Ex latel-culorum aequalitate manifeste consequitur eorum
parallelismus; eritque A'F' parallelepipcdum obliquangu—
lum; ita tamen , ut eius anguli infinities parum diffe-
rant ab angulis rectis parallelepipedi rectanguli AF ; quan-
doquidem AF nonnisi tempusculo inünitesimo transfer-
tur in A'F' . Nunc ex H' v. gr. due perpendiculum H'a in
areolam A'B'C'D' ; erit ∙ ∼
A'F' ∙−−− H'a . A'B'C'D' ::
'
'a . A'B' . A'D' sin B'A'D' ∶−−⋅≖
A'H' . A'B' . A'D' sin B'A'D' sin H'A'a :224
denotantibus w et w'angulos infinitesimos , poterunt anguli
B'A'D ' , H'A'a repraesentari per 90º + w , 90 ° +6 ; iccirco
sin B'A'D' sin H'A'a = sin (90 ° + w ) sin ( 90° + W' ) =
62 614 w'4 coswcos6= ( 1 -... ) ( 1 ...) . 2 2.3.4 2 2.3.4
Quare , missis infinitesimis quinti ordinis ,
dv &quot; A'F ' = (dx +
dv' du &quot;
dxdt) (dy +
dx dzdt) x
dy dydı) (dz + az
wa
du
( - -
(1-7 • det dy&quot;
• det du&quot;
dt) dxdydz ;
2 dy
dz
ideoque
dV = AF - V =
-
(dv
dx- det
-det
dy
di) dxdydz.
His positis , vertelur ( 6 '' ) in''
dxdydzdje ele
dv
\dx
dtot dt +
dv&quot;
dy
dz di )dxdyds= 0,
seu ( 106.6' )
dje du ut u'+
dr dz
djelo
du +
dy
dt
dvi dv&quot; du &quot; .
dy + demon dz ) =
(619) .
dx

108. * Si massa fluida est incapax compressionis,
unaquaeque particula immutabilem habebit densitatem
eritque du = o : proinde ( 106 , 6')
224
denotantibus eo et tu'-angulos inünitesimos , poterunt anguli
B'A'D' , H'A'a repraesentari per 900-l-cu , 900-l-co'; iccirco
sin B'A'D' sin H'A'a −−∶⋅ sin (90&quot;-FG) sin (900 ⊣−∙ w') ∶∸⋅
. ↿ a): 034 ↿ tu' te'-': )
cosmcosw—( ∙−⋅⋍−−⊢≳∙∙⋝⋅∕⇂∙−−⋯⋟≺ −−∙⋮∙ m—m .
Quare , missis infinitesimis quinti ordinis
,
I d'u' dr&quot; d'v &quot;
'
'
∙−−−
− −
'
AF ∙−− ∙−− (dx—t- dædxdt,(dy-t—- d] afydt) (dz-t- az dzdr) )(
a 'a .'
&quot; '&quot; (1 &quot;2' −−∘≩≻∙−− ≺↿⊹⋛⊰↲↥⊹≘⋚∂∁⊹↙≩⊤∶∂∊≻∂∅∂∫∂∥
ideoque
(lv-.:A'F' v
(
vd: : dv dc.-92:11) dædyde .
dæ dy dz
His positis , vertetur (b&quot;') in
dt&quot; I),, vl'l dæd.) dzdp-t— &quot;(c'ïx dc ⊣−∙ 217 dc ⊣−∙ 72- dt)dædydz ::o,
seu (106 . b')
&quot;&quot;
↿≀∙⊣⋅− d&quot; ∣∣⊣− a'&quot; &quot;-4-'-'—'-' −⊢

: 17- 2?&quot; f??&quot; dt

d'v- ⋅⋅⊢ dv&quot; dv'&quot; ⋅
⋅⋅∙∙− o
F- b,; ( ⊣⋅−⋮⋤ )
—- ( ).
108.s Si massa fluida est incapax compressionis,
unaquaeque particula immutabilem liabebit densitatem ,
eritque dy.:o : proinde (106
. b')225
de
vt
die du du v &quot; + 2 &quot; +
=
0

;

dy dx dz de
et consequenter
( 107.b
)
(
6
)
dv dy
d.x
+
+
dy dz
Formulae
(
6
&quot;
)
,
(69) suppeditant incognitas
a
,
l
,
v
,
v
&quot;,
v
&quot;
.
expressas per
x
,
y
,
z
,
t

; obtentis autem

v
'
,
v
&quot;
,
?,
&quot; per xy
, zat
,
eruentur
x
,
y
,
z per
t ex formulis
dy dx dz
dc
-
”
dc
ru!!!
dt
Si massa fluida incapas compressionis est insuper ho
mogenea
, prima
(
6
&quot;
) fiet idemtica
, satisque erunt
(
6
&quot;
) et
secunda
(
6
&quot;
)
.ad incognitas
,
u
'
,
v
&quot;
v
'&quot; determinandas
. De
mum si massa fluida pollet elasticitate
, formalis
(
6
&quot;
) et
(61
)
jungenda erit formula
(
o
&quot;
. 87.6
'
)
.

=== De tubis capillaribus. ===
[[Fasciculus:Capillarity.svg|thumb|Capillares]]
109. Etsi liquidum homogeneam in vasis communicantibus (92.1º.) manet aeque altum, iu tubis tamen vitreis admodum angustis (dicuntur capillares) utrinque apertis, et altera extremitate demersis aquae vel hydrargyro, cernimus aquam suprema superficie concava terminatam ascendere supra horizoutalem circumambientis liquidi superficiem, hydrargyrum vero suprema superficie convexa terminatum descendere infra horizontalem circumdantis liquidi superficiem: ad istius modi ascensum descensumque explicandum, haec animadvertimus.

1º. ln phaenomenis gravium liquidorum expendendis gravitatem considerantes haud habuimus rationem sive virium quibus liquidi particulae se mutuo petunt, sive virium quibus vasorum materies ad se trahit particulas illas. Porro materiales particulae duplici pollent vi attractiva; altera se prodit utcumque crescant distantiae, sequiturque (82) rationem reciprocam duplicatam distantiarum; altera se prodit dumtaxat in contactu vel quamproxime contactum, sequiturque rationem quamdam distantiarum nondum compertam. Ubi sermo est de liquorum aequilibrio, possumus ab attractione primi generis absque sensibili errore praescindere: ad attractionem secundi generis quod pertinet; cum in contactu exsistat validissima, inde fit ut suprema liquidi superficies prope vasorum latera induat figuram curvam, modo concavarn, modo convexam, et nonnisi ad aliquam ab ipsis lateribus distantiam dici queat physice horizontalis. Exhibeat TT' (Fig. 53) verticalem tubum v. gr. vitreum, utrinque apertum, et infra horizontalem liquidi superficiem partim demersum; O centrum circularis areae tubo interceptae apud eam superficiem; A particulam liquidi in area ista sub actionem vilreae particulae R; OX rectam transeuntem per A; OY horizontalem rectam perpendiculariter insistentem rectae OX; OZ verticalem rectam. Si denotat vim qua A tendit in R, designatis per h, k, i cosinibus angulorum quos AR facit cum ox, oy,
OZ, resolvetur in ternas
ph , pk , ọ
iisdem OX , OY , OZ parallelas: ex R in planum XOY
ducatur perpendiculum Rp , producaturque in R' donec fiat
R'p = Rp ; teadet A in R' vi aequipollente ternis
ch , pk , - oi : demissis perpendiculis ex R , R' in planum Xoz , iisque productis donec productiones aequentur ipsis perpendicu
226
rium quibus liquidi particulae se mutuo petunt, sive virium quibus vasorum materies ad se trahitparticulas illas. 

manifeste determinabuutur in tubo duo puncta , quorum vires dabunt componentes
gh , - ok , pi ,
sh , - ok , - qi :
in ferimus particulam A , elisis componentibus parallelis
rectae OY , itemque componentibus parallelis rectae OZ ,
sollicitatum iri juxta AX vi
4Σ φh
proveniente ex tubi materia. In OX sume Ab = Aa ; duc
verticalem bb' ; et quod in ordine ad tubi materiam est q,
in ordine ad liquidi materiam sit q' : quisque intelligit par
ticulam An elisis componentibus horizontalibus, trahi ver
ticaliter deorsum vi
4 Epi
promanante ex liquido intercepto superficie cylindrica , quam
general recta bb' dum sibi constanter parallela movetur in
peripheria circuli habentis radium Ab. Liquidum ultra
superficiem cylindricam praebet vires
.
- 2 Eph , 2 &quot;pi,
alteram horizontaliter agentem juxta XO , alteram verti
caliter deorsum. Omnes itaque vires sollicitantes particulam
A traducentur ad horizontalem
4Eph -2Ep'h = 2 [2Eoh —Eph] ,
et ad verticalem
45' pit 23&quot; ' i.
227
lis, manifeste determinabuutur in tubo duo puncta. quo-
rnm vires dabunt componentes
9ht—9k09i'
-ph.-—9k,—qn':
inferimus particulam A .
elisis componentibus parallelis
rectae 0?
,
itemque componentibus parallelis rectae OZ
,
sollicitatumeiri juxta AX vi
4297:
proveniente ex tubi materia. In OX sume Ab: Aa; duc
verticalem 65; et quod in ordine ad tubi materiam est p,
in ordine, ad liquidi materiam sit go' :quisque intelligit par-
ticulam A. elisis componentibus horizontalibus, trahi ver-
ticaliter deorsum vi ↽
42'9'i.
promanante ex liquido intercepto superficie cylindrica, quam
generat recta bb' dum sibi constanter parallela movetur in
peripheria circuli habentis radium Ab. Liquidum ultra
superficiem cylindricam praebet vires
-— 2
297: , 2E'p'i ,
alteram horizontaliter agentem juxta KO, alteram verti-
caliter deorsum. Omnes itaque vires sollicitantes particulam
A traducentur ad horizontalem
4ng −− ⇄∑∲∣∣∣:2929]: −∙− ∑⊈⊅⋅∣⋅⊐ .
et ad verticalem (f)
42: p'i-l- 22&quot;qa' i.228
Potest 2Eph -Eph esse aut &gt; o , velo, vel = 0: in primo
casu vis aequipollens et gravitati , et binis (f ) , deviabit a di
rectione verticali faciendo angulum acutum cum AX; et
quia ( 83.3º. ) vis illa debet normaliter sese dirigere ad libra
tam liquidi superficiem , ideo suprema liquidi superficies in
duet curvam concavamque figuram : in secundo casu vis
aequipollens et gravitati, et binis (f), deviabit quidem a ver
ticali directione, sed faciendo angulum oblusum cum AX ;
propterea ( 87. 3 • ) suprema liquidi superficies induet curyam
convexamque figuram : in tertio denique casu ex duabus (8)
remanebit sola verticalis, et consequenter suprema liquidi
superficies erit plana atque horizontalis.

2º . Massae liquidae OS , OS' (Fig. 54 ) ejusdem
naturae, planisque superficiebus OP , OʻP ' terminatae, ae
qualiter trahunt exilissimas columnellas liquidas AR , A'R'
perpendiculariter superficiebus ipsis insistentes; nisi tamen
columella AR intra massam OS trahitur deorsum , columel
la vero A'R' extra massam O'S' trahitur sursum. Intelligan
tur enim centris A et A ', radiisque aequalibus AB et A'B ',
ultra quos sensibilis attractio liquidi non protenditur, des
cribi hemisphaeria BCD , B'C'D' : si vires , quibus hemi
sphaeria agunt in particulas A, A ', resolvuntur in binas, alte
ram horizontalem , alteram verticalem; elisis horizontalibus,
remanebunt verticales, quarum summa est eadem utrinque
cum hoc tantum discrimine quod A trahitur deorsum et A
sursum. ln columellis sume nunc duo alia puncta E, Eʻae
quidistantia ab A , A' , radiisque aequalibus EL, E'L ' ( = AB)
describe segmenta sphaerica FML, F'M'L' : accepla EV=EA,
ductoque plano horizontali HK, vires ex QHKN , QFMN
in punctum E utpote aequales et contrariae se mutuo de
struent , ipsumque E solo segmento HLK deorsum trahe
tur : vis ex HLK deorsum sollicitans particulam E ae
quatur vi ex F'L'M ' sursum trahenti particulam E'; siqui
dem haec segmenta sunt aequalia et similiter posita ad
contrarias partes quoad E et E '. Cum igitur idem redeat
228
Potest 229h—29'h esse aut) a. vel( a, vel:o: in primo
casu vis aequipollens et gravitati, et binis ([ ), deviabit a di-
rectione verticali faciendo angulum acntum- cum AK; et
quia ( 87. 30.) vis illa debet normaliter sese dirigere ad libra-
tam liquidi superficiem, ideo suprema liquidi superficies in-
duet curvam concavamque figuram: in secundo casu vis '
aequipollens et gravitati, et binis (f), deviabit quidema ver-
ticali directione, sed faciendo angulum obtusum cum ax,
propterea (87. 3&quot;-) suprema liquidi superficies induet curvam
convexamque figuram :in tertio denique casu ex duabus (f)
remanebit sola verticalis, et consequenter suprema liquidi
supedicies erit plana atque horizontalis.
2&quot;. Massae liquidae OS , US' (Fig. 54) eiusdem
naturae, planisque superficiebus OP
, O'P' terminatae, ae-
qualiter trahunt exilissimas columellas liquidas AR , A'R'
perpendiculariter superficiebus ipsis insistentes; nisi tamen
columella AR intra massam OS trahitur deorsum. columel-
la vero A'R' extra massam O'S'trabitur sursum. Intelligan-
tur enim centris A et A', radiisque aequalibus AB et A'B',
ultra quos sensibilis attractio liquidi non protenditur,des-
cribi hemisphaeria BCD , B'C'D' : si vires , quibus hemi-
sphaeria aguntin particulas A, A', resolvuntur in binas, alte-
ram horizontalem , alteram verticalem;elisis horisontalibus,
remanebunt verticales, quarum summa est eadem utrinque
cum hoc tantum discrimine quod A trahitur deorsum et A'
sursum. ln columellis sume nunc duo alia puncta E, E'ae-
quidistantia ab A , A', radiisque aequalibus EL, E'L' (::AB)
describe segmenta sphaerica FML, F'M'L': accepta EVzEA,
doctoque plano horizontali HK, vires ex QHKN , QFMN
in punctum E utpote aequales et contrariae se mutuo de-
struent
,
ipsumque E solo segmenta HLK deorsum trahe-
tnr : vis ex HLK deorsum sollicitans particulam E ae-
quatur vi ex F'L'M' sursum trahenti particulam E'; siqui-
dem haec segmenta sunt aequalia et similiter posita ad
contrarias partes quoad E et E'. Cum igitur idem redeat229
cimo
di.

; et

bra
sin
vis
Ver
LAX;
ryam
argumentum de caeteris particulis inter A et C , necnon
inter A ' et C ' ( ponimus A'C &quot; — A'C' ) , cumque particulae
infra C viribus contrariis et aequalibus urgeantur, infra C
sensibili non subjiciantur actioni, jam patet etc
In eodem liquido vis, qua deorsum vel sursum colamella
trahitur, constans est; eam in sequentibus exhibebimus per K.

3º. Fac ut massa liquida BAB'QQ (Fig. 55) , quae intercipitur superficie sphaerica BAB' et plano tangente
QQ, trahat externam columellam liquidanı AR perpendicula
riter insistentem plano tangenti apud contactum A : quo
niam BAB O'Q gignitur rotatione areae ABQ circa ra
dium AO, perinde erit sive spectetur attractio massae BAB'Q'Q,
sive spectentur attractiones infinitarum numero superficierum
cylindricarum , quae in ea rotatione gignuntur a perpendicu
lis DC , D'C' , ... demissis ex punctis D , D ' . circu
laris arcus BA in rectam QA. Exprimant p , pi ... per
pendicula DC , D'C', . . ; 9.9 , .. perpendiculorum di
stantias AC , AC' . .. ab A computatas in AQ; sitque r
sphaericae superficiei radius OA: ob magnam lineolarum
p , p ', . . . tenuitatem prae q, , ..
quidi
Eden
A'R'
ameo
amel
gaD
AB,
des
erunt
lemi
aleo
р
92
2r ,pa2r
libus
Bogu
et A
et consequenter praefatae superficies cylindricae exhibe buntur per
-AB
EL
MY
2πη 비유
Toq3
,2πα g's Tig'3
seu 9
dem 2r 2r
cabe
ae
aqui.
Atqui ob eamdem illam tenuitatem puncta uniuscujusque su :
perficiei cylindricae haberi possunt pro aequidistantibus ab
unoquoque columellae puncto: designantibus igitur o, o, ..
quantitates pendentes et a certa quadam distantiarum lege,
deal
sit
ac-
qui'
de:!
229
argumentum de caeteris particulis inter A et C- , necnon
inter A' et C' (ponimus A'C&quot; :: A&quot;C), cumque particulae
infra C viribus contrariis et aequalibus urgeantur, infraC&quot;
sensibili non subjiciantur actioni, iam patet etc
..... ∙
.In eodem liquido vis, qua deorsum vel sursum colnmella
trahitur, constans est; eam in sequentibus exhibebimus per K.
30. Fac ut massa liquida BAB'Q'Q (Fig. 55), quae
intercipitur superficie sphaerica BAB' et plano tangente
QQ',trahat externam columellam liquidam AR perpendicula-
riter insistentem plano tangenti apud contactum A*: quo-
niam BAB'Q'Q gignitur rotatione areae ABQ circa ra-
dium AO, perinde erit sive spectetur attractio massae BAB'Q'Q,
sive spectentur attractiones infinitarum numero superficierum
cylindricarum, quae in ea rotatione gignuntura perpendicu-
lis DC , D'C', . . . demissis ex punctis D, D' . . . circu-
laris arcus BA in rectam QA. Exprimant p , p', ... per-
pendicula DC, D'C',. :
.; q, q' , .. perpendiculorum di-
stantias AC, A.C' .. ab A computatas in AQ, sitque :-
sphaericae superficiei radius OA: ob magnam lineolarum
p, p, . ..
tenuitatem prae q, q', ...,
erunt
9' ∣
vf:
?&quot; 2r'p— 2r'...'l
et consequenter praefatae superücies cylindricae exhibe-
buntur per
' q,! &quot;03 &quot;q '3
, ∙ ∙ ∙
'
seu , 21) r r
q?
2:- 2nq ,Zitq ,...
Atqui ob eamdem illam tenuitatem puncta uniuscuiusque su.-
perficiei cylindricae haberi possunt pro aequidistantibus ab
unoquoque columellae puncto: designantibus igitur &amp;, ö'. ..
qnantitates pendentes et a certa quadam distantiarum lege,.
Kn.-&quot;M— . ⇀ ⋅−−∙⇀∙⋅↼ ⋅⋅−↪∎⋅⊾ −−↼↼∎↼ ↽− ↼−⋅−⋅−⇀−⇀−⋅∙∎∙∙↼ −−↼
↰⋅−↽⋅
- −⋍⇂∙⋅−230
et a liquidi densitate, et a cosinibus angulorum quos cum
AO faciunt rectae ab attrahentibus superficierum punctis
ductae ad attracta columellae puncta, eae superficies colu
mellam sursum verticaliter trahent viribus
Teq38 Tog'38
totaque massa BAB'D'Q columellam AR sursum verticali
ter trahet vi
1938 +7.9'38' +
.
Si concipitur altera massa liquida PAP'OʻQ intercepta pla
no QQ et nova superficie sphaerica PAP, cujus radius
O'A = p , simili ratione ostendetur vim ex PAP'Q'Q fore
παδ+πα35 '+ .
.
Vires itaque istae erunt ut
- Eq: 8 : &quot;5q?: = &gt;
erunt nempe reciproce at sphaericarum superficierum ra
dii. Hinc designante H opportunam quantitatem constan
tem , exprimet
H
vim, qua sursum verticaliter massa liquida BAB'Q'Q trahit
.columellam AR : caeterum quisque videt fore H = 12q30.
230
et a liquidi densitate, et a cusinibus angulum quos cum ⋅
AO faciunt rectae ab attrahentibns superficierum punctis
ductae ad attracta columellae pnncta, eae superficies colu-
mellam sursum verticaliter trahent viribus
nq3d th'3ö&quot;
,.... r :-
totaque massa BAB'Q'Q columellam AR sursum verticali-
tcr trahet vi
∏⊄∍∂−⊢∏⊄⋅∃∂⋅⊹ ∙ ∙ ∙ ∙
.
r .
Si concipitur altera massa liquida PAP'Q'Qintcrcepta pla-
no QQ' et nova superficie sphaerica PAP', cuius radius) O'A-z r' , simili ratione ostendetur vim ex PAP'Q'Q fore
12:738 −−⊢ ∏⊄≖∃∂∣⋅−⊢ ∙ ∙ ∙ ∙
r
Vires itaque istae erunt ut
7! :: ↿ ↿
r Zq d . —r,2q ∙−−≀∙ −∙⋮∙∙ ,
erunt nempe reciproce ut sphaericarum superficierum ra-
dii. Hinc designante H Opportunam quantitatem constan-
tem
, exprimet
H
∙∙−∙−
'
vim, qua sursum verticaliter massa liquida BAB'Q'Q trahit
.columellam'AR : caeterum quisque videt fore H::an36.231

: cum e.

tis r
r
H

4. Quantitas K major est quam
nim K exprimat vim ( 2.0) , qua sursum trahitur columella
H
AR a massa liquida LFAF'L' , exprimet K vim
qua sursum trahitur AR a segmento sphaerico MBAB '.
H
Id vero importat K &gt; o ; ergo etc.

5.0 Massa liquida BAB'E'E terminetur superficie
concavo -spherica BAB' : ducto per A plano tangente QQ ,
sollicitabitur columella AR deorsum ( 2.9 ) vi K ex EFAF'E' ,
H
sarsum (3.9) vi ex BAB'F'F ; tota igitur BAB'E'E trahet
deorsum columellam AR vi (4.0)
.
bio
3
13
н
. K
IR
in i, i' , ... ,

6.• Superficies sphaerica NAN' habens radium
O'A = 0A tangatur plano QQ in A ; columella AR ae
que trahetur sursum a massa liquida NAN'Q'Q ac trabi
lur a massa BAB'Q'Q : patet ( 3. ) çum ex eo quod, pro
ductis DC , D'C' , ... donec occurrant arcui circulari AN
exsistunt DC=Ci; D'C' =C'i, ... ; tum ex
eo quod Ci , Ci', ... , sunt tenuissimae prae AC, AC,
si qua pars columellae non trahitur sursum
sit tenuissima prae reliqua parte sursum altracta .

7.º Columella igitur AR magis trahetur deor
sum ab EE'N'AN quam ab EE'F'AF ; excessusque unius
H
attractionis supra alteram erit . Propterea massa liqui
da desinens in superficiem convexo- sphaericam NAN' traliet
deorsum columellam AR vi
ita ut ea
1
K +
1
i.
231
H 4.&quot; Quantitas K major est quam —-: cum e-
- r
nim K exprimat-vim (29)
. qua sursum trahitur columella
AR a massa liqui/da LFAF'L', exprimet K —E r
vim ,
qua sursum trahitur AR a segmento sphaerico MBAB'.
Id veroinrportat'K—g- ≻∘ ∙∙∙ ergo etc.
. . .
59 Massa liquida BAB'E'E terminetur superficie
concavo-spherica BAB' : ducto per A plano tangente QQ',
sollicitabitur columella AR deorsum (2.0) vi K ex EFAF'E',
sursum (3.&quot;) vi!-.;l ex BAB'F'F; tota igitur BAB'E'E trabet
deorsum columellam AR vi (4.0)
∙
K—ll'a r
6.0 Superficies sphaerica NAN' habens radium
()&quot;AzOA tangatur plano QQ' in A; columella AB ae-
que trahetur sursum a massa liquida NAN'Q'Q ac trabi-
tnr a massa BAB'Q'Q: patet (39) tum ex eo quod, pro—
ductis DC, D'C' , ... donec occurrant arcui circulari AN
in i, s&quot;.
..., exsistunt DCxCi; D'C'..-::C't&quot;, ...: tum ex
eo quod Ci, C'i',
. . . ,
sunt tenuissimae prae AC. AC',
ita ut si qua pars columellae non trahitur sursum , ea
sit tenuissima prae reliqua parte sursum attracta.
7.o Columella igitur AR magis trahetur deor-
sum ab EE'N'AN quam ab EE'F'AF; eicessusque unius
H
attractionis snpra alteram erit -— . Propterea massa liqui-
⋅ r
da desinens in superficiem convexo-sphaericam NAN' trahet
deorsum columellam AR vi
n
∣≺⊣−−−⊑−⋅∙232

8.º Pone superficiem BAB' neque esse sphaericam
,
neque gigni rotatione ullius curvae circa AO

; secla BAB

planis transeuntibus per A0
, curvilineae sectiones apud
contaclum
A gaudebunt inaequalibus osculi radiis

; quos

inter
( demonstrationem suo tempore videre erit in parte 3.4 nostrorum elementorum matheseos 0. 118
) bi
ni reperiunlur
, alter minimus
(
=
r
)
, alter maxi
mus
(
=
r
'
), pertinentes ad binas sectiones sub angulo re
cto invicem constitutas
. Iam
, in ea qua sumus hypothe
si
, hoc pacto determinabitur visex BAB'Q'Q sursum
verticaliter trahens columellam AR
. Intelligatur coalesce
re BAB'Q'Q ex infinitis numero superficiebus cylindri
cis normaliter insistentibus plano tangenti QQ
'

:

' unaquae
que superficies cylindrica non eamdem habebit ubique
altitudinem

; sed apud bina puncta

e diametro opposita
,
quibus nempe maximus respondet circựlus osculator
, al
titudo erit minima

; apud bina puncta

e diametro pari
ter opposita
, perque gradus 90 ab illis primis sejuncta
,
quibus videlicet minimus respondet circulus osculator
,
altitudo erit maxima

: apud intermedia puncta altitudines

interjacebunt minimam maximamque
. Quapropter evoluta
superficie cylindrica super aliquo plano
, ea poterit reprae
sentari per aream QNN
&quot;
Q
&quot;
( Fig
. 56
)

; NN

&quot; aequatur
basi superficiei cylindricae

; QN et

Q
&quot;
N
&quot; simul cum Q'N
'
exhibent altitndines minimas

; Fu et F'u

' altitudines ma
ximas hinc Nu

= uN
'
= N'u
'
= u'N
&quot;
. ob perexiguum ba
seos cylindricae radium poterunt QF
, Q'F
, Q'F
'
,
(
&quot;
F
'
haberi pro lineis rectis

; eritque

1
QN
+Fu NN &quot; QʻF'Q'FQ = 1NuFQ = 4 Nu 2 NN ” QN + F4 2
232
8.&quot; Pone superficiem BAB' neque esse sphaericam,
neque gigni rotatione ullius curvae circa AO; secta BAB'
planis transeuntibus per AO, curvilineae sectiones apud
contactum
A gaudebunt inaequalibus osculi radiis quos

inter (demonstrationem suo tempore videre erit in par-
te 3.*' nostrorum elementorum matheseos n. 118) bi-s
ni reperiuntur, alter minimus
( ::r)
, alter maxi-
mus (:r'), pertinentes ad binas sectiones sub angulo re-
cto invicem constitutas. Iam
,
in ea qua sumus hypothe-
si, hoc pacto determinabitur vis et BAB'Q'Q sursum
verticaliter trahens columellam AR. Intelligatur coalesce-
re BABHQQ ex infinitis numero superficiebus cylindri-
cis normaliter insisteutibus plano tangenti QQ' :'unaquae-
que superficies cylindrica- non eamdem habebit ubique
altitudinem; sed apud bina puncta
e diametro opposita,
quibus nempe maximus respondet circulus osculator
, al-
titudo erit minima; apud bina puncta
c diametro pari-
ter opposita, perque gradus 90 ab illis &quot;primis seiuncta,
quibus videlicet minimus respondet circulus osculator
,
altitudo erit maxima: apud intermedia puncta altitudines
interjacebunt minimam maximamque. Quapropter evoluta
superficie cylindrica spper aliquo plano
, ea poterit reprae-
sentari per aream QNN&quot;Q&quot; (Fig. 56); NN&quot; aequatur
basi snperficiei cylindricae QN et Q&quot;N' simul cum Q'N'

exhibent altitudines minimas -; Fa et F'u' altitudines ma-
ximas; biuc NucuN'zN'u'2u'N&quot;
.
ob perexiguum ba-
seos cylindricae radium poterunt QF
, Q'F, Q'F ', Q&quot;F'
habcri pro lineis rectis; eritque
NN&quot;Q'F'Q'FQ::4NuFQ

:

4 Nu 'QN'zl-F&quot;

:

∙ NN&quot; QNj'F&quot;
.233
ericam,
a BAB
es apud
i quoi
Retentis igitur denominationibus ( 3.9 ) , superficies cylin
dricae , ex quibus intelligitur coalescere massa BAB'Q'Q
( Fig. 55 ) , erunt
92 Lo pár.
+
92 q /2
+
2r
2r' 2r'
2πα ,
2r
2tq'
&gt;
seu
mari 2 2
alore
ypothe
Sursa mga (: + ?). mg ( + ),...
Dalesce
linde
naquat
ubige
pposila,
et consequenter massa BAB'Q'Q desinens in superficiem
BAB' utcumque concavam trahet verticaliter sursum co
lumellam AR vi
or , al
πα
2
o pari ?( + 1) + &quot;, ( +3)x + ...
Atqui ( 3.9 ) 7.938 + Teq'38 ' + .... = H :
Ejuncta,
ulator ,
Studios
evolu
reprat
equatur
exprimetur ergo vis illa per
16+)
les m2
um bio
1 07
9. Sume Q '&quot;'N '&quot; et F &quot; u &quot; ( Fig. 56 ) aequidi
stantes ab QN et Fu : erunt Q &quot; N &quot;&quot; , F &quot; u &quot; duae ex al
titudinibus intermediis ( 8. ) respondentes duabus sectio
nibus curvilineis ad angulum rectum invicem constitutis.
Radii circulorum osculantium sectiones istas apud A ( Fig.
55 ) designentar per po&quot; et p' : erit ( Fig. 56 )
9'2
Q&quot; N &quot; +F &quot; u &quot; =
92
2r &quot;.
qo +
27 
+
g'?
2r . ' 2r' '
1
16
233
Retentis' igitur denominationibus (3.&quot;) , superficies cylin-
dricae, ex quibus intelligitur coalescere massa BAB'Q'Q
(Fig. 55 )
,
erunt
/
£ fl&quot; q&quot; a&quot;
2r .l-Zr ∙ 2r
21tq 2 , 27tq .l-Zr' 2 , ∙ , seu
↔⇍≖↙≀⋮↿≺ ,
') &quot;rv ')
⋅⋅
2 rii-&quot; 2
≀∙⊣−∣⋅⋅∅⋅⋅⋅⋅
et consequenter massa BAB'Q'Q desinens in superficiem
BAB' utcumque concavam trahet verticaliter sursum co-
lumellam AR vi
&quot;f(ï-Fl?) ∂⊹∙⋮≖−≣−⋅⋮−≺−∶−−⊦∙≙≻∂↝⊹∙ ∙∙
r
Atqui (3.0) nq3ö -l-7tq'3d' ∙−⊦ ∙ ∙ ∙:∙ H:
exprimetur ergo vis illa per
⋅≣⋅≺⊥⊣−−↿⊺≻ ⋅

: - r

9.0 Sume Q'&quot;&quot;'N et F&quot;u&quot; (Fig. 56) aequidi-
stantes ab QN et Fu : erunt Q'&quot;N&quot;', F&quot;u&quot; duae ex al-
titudinibus intermediis (89) respondentes duabus sectio- .
nibus curvilineis ad angulum ≖⋅∁∁⋯⊞∙ invicem constitutis.
Radii circulorum osculantium sectiones istas apud A ( Fig.
55 ) designentur per r&quot; et r'&quot; : erit (Fig. 56)
0 (
ns '
lr Jr ' ∎∙ lr.
'
a a 'a' ': ≺≀⋅⋅∙↓↜⇃∙⋅∙−⊦∣∂⇁⇈≀↓∙∙∶∶−∙↙−⇃−⋅− ⊄
⋞∣ −⊦⋞∣∙∙234
est autem
mane cc
Q &quot; N '&quot; + F &quot; u&quot; = QN + Fu =
92
2r
9'2
2r +
2r etc.:
1
igitur
+++++
110
et consequenter
16 + *) = &quot; 6 + - ): under
10.º Si superficies concava BAB' ( Fig. 55 ) gi
gnitur rotatione alicujus curyae circa OA , fiet
Cor
ace
r = r = r ' = r '&quot;
I supe
ac proinde vis ex BAB'Q'Q sursum verticaliter trahens
columellam AR erit ban,
zebic
H
reciproce nimirum ut radius osculi apud A.
11.• Facile nunc intelligimus attractionem mas
sae liquidae BAB'E'E, terminatae superficie utcumque con
caya BAB' , in columellam AR fore
K - &quot; ( + -) .velK6+ );
234
est autem
↾∣∣ ⇌∎
!
'2 :
NI&quot; F&quot; &quot;: −∙ Q. ∙−⊦ u QN—l—Fu q2r ↿ q 0
L
I 2r' , 2r ∙−⊦ 2r' ,etc.t
igitur
1 ↿− ↿ ↿
i- ∣
rr ∣⋅⊤⋅ .'&quot;? '
et consequenter
H 1 ↿ H
∎↿ ↿ ⋅
−⋮−≺−≀∶⇀⊦ r' )— 2
(r&quot; hl-r'&quot;) ⋅∙
10.0 Si superficies con-cava BAB' (Fig. 55 ) gi.
gnitur rotatione alicujus curvae circa OA ,
fiet
I ';
∣←−∶≀∙ :r :r '&quot; ;
ac proinde vis ex BAB'Q'Q sursum verticaliter trahens
columellam AR erit
'H-
&quot;,
reciproce nimirum ut radius osculi apud A.

11.(, Facile nunc intelligimus attractionem mas-
sae liquidae BAB'E'E, terminatae superficie utcumque con-
cava BAB' ,
in columellam AR fore
H
1,1) H(1,1 ∙
K
2(r '7 'velK 2r&quot; 'r'&quot;)'
ïfbiu235
massae vero liquidae NAN'E'E, terminatae superficie utcum
que convexa NAN' , in ipsam AR fore
K + 6 + -) .
ved K + &quot;6+ ) :
fiet =r =r&quot; = r&quot; in casu superficiei genitae rotatione li
neae curvae circa OA.

110. His declaratis , venio ad ascensum descensum que liquorum in tubis capillaribus.

1.° Aqua in tubis vitreis diversae capacitatis ad diversas ascendit altitudines, quae sunt reciproce ut tuborum diametri: idipsum contingit oleo, spiritui vini, etc..; ascendentesque liquores terminantur superne concava superficie. Concavae superficiei causam adsignavimus (109 1.): ad ascensum quod pertinet, sit QQ ( Fig. 57 ) suprema superficies aquae circumambientis tubum LE , et I A'B' concava superficies aquae intra tubum; sint insuper A'R, VR' binae columellae verticales, altera intra, altera extra tubum, quas columellas jungat horizontalis columella RR'. Urgebitur A'R deorsum gravitate simulque vi (109. 11. ° )
K - 16 + );
urgebitur VR' deorsum gravitate simulque vi ( 109. 2.° )
K :.
cum igitur
Н
K &gt; K
(
2 + ),
235
massae vero liquidae NAN'E'E, terminatae superficie utcum-
que convexa NAN' ,
in'ipsam 'AR fore
H 1 1 H 1 1 K—(-2(,, ∣∣⋅ .).ve1K( ,(,.,'. ...-)
r r
fiet r:r':r&quot;—-::r&quot;' in casu snperficiei genitae rotatione li-
neae curvae circa OA&quot;.

110. His declaratis, venio ad ascensum descensumque liquorum in tubis capillaribus.

1.&quot; Aqua in tubis vitreis diversae capacitatis ad diversas ascendit altitudines, quae sunt reciproce ut tuborum diametri: idipsum contingit oleo, spiritui vini, etc..; ascendentesque liquores terminantur superne concava superficie. 

Concavae superficiei causam adsignavimus (10913): ↙
ad ascensum quod pertinet , sit QQ' (Fig. 57) supre-
ma superficies aquae circumambientis tubum LE , et I'A'B'
concava superficies aquae intra tubum; sint insuper A'R,
VR' binae columellae verticales, altera intra, altera extra
tabum, quas columellas iungat horizontalis columella RR'.
Urgebitur A'R deorsum gravitate simulque vi ( 109. 11.&quot;)
H1711,
K
2(rlr1)a
urgebitur VR' deorsum gravitate simulque vi ( 109. Z.&quot;)
∕
cum igitur236
haud poterunt A'R , VR' consistere in aequilibrio nisi A'R
ascendat supra QQ . Denotet z altitudinem AA , ad quam ascendit columella A'R supra QQ'; sitque c gravitas specifica liquoris: fiet eousque columellae ascensio donec habeatur
H
K = K
-16 + )+c=;unde == 2c ( + ).

Vires ex materia tubi eas tantum liquidi particulas afficiant, quae ad internam ipsius tubi superficiem maxime accedunt; iccirco liquidum perinde trahetur, a tubo ac si interna superficies esset plana: permanente igitur tubi ac liquoris qualitate, etsi variat tubi diameter, eodem tamen pacto trahentur liquoris particulae versus tubum; quae attractio quia cum constanti attractione liquoris erga seipsum consociatur, extrema latercula curvae BAB' aeque inclinabuntur ubilibet ad internam tuborum superficiem. Arcus itaque omnes BAB' erunt similes in diversis tubis, ipsorumque arcuum rotatione circa tubi axem gignetur superficies concava superne terminans liquorem : bini videlicet
r et r' exsistent aequales, simulque tuborum diametris pro
portionales; ideoque altitudo z reciproce ut eae diametri.

2. • Hydrargyrum in tubis vitreis descendit in
fra circumambientis hydrargyri superficiem QQ ad ejus
modi altitudines , quae sunt tuborum diametris recipro
ce proportionales ; descendensque liquidum terminatur su
perne convexa superficie NOM.
Convexitatis causam adsignavimus ( 109. 1.0 ) : ad de
scensnm quod spectat , columellarum OR et VR' altera
sollicitatur deorsum gravitate simulque vi ( 109. 11.° )
0
{
K + C + ) ,
236
haud poterunt A'R, VR' consistere in aequilibrio nisi A'R
ascendat supra QQ'. Denotet :altitudinem AA' , ad quam
ascendit columella A'R supra QQ'; sitque c gravitas spe-
cifica liquoris: fiet eousque columellae ascensio donec ha-
beatur
H ↿ ↿ H 1 ↿
∣≮≓⋅∶∶∣⊊∙− −∙≨∙⋖−−∙−⊢⊤≻−⊢∶≖∙ under—' 2c(r ≓≀∙∙≻ .
r .
!
Vires ex materia tubi eas tantum liquidi particulas af-
ficiunt, quae ad internam ipsius tubi superficiem maxi-
me accedunt; iccirco liquidum perinde trahetur, a tuba ac si
interna superficies esset plana : permanente igitur tubi ite
liquoris qualitate, etsi variat tubi diameter, eadem tamen
pacto trahentur liquoris particulae versus tubum; quae
attractio quia cum constanti attractione liquoris erga se-
ipsum consociatnr, extrema latercula curvae BAB' aeque
incl'uabuntur ubilibet ad internam tubarum superficiem.
Arcus itaque omnes BAB' erunt similes in diversis tubis,
ipsorumque arcuum rotatione circa tubi axem gignetursuper-
iicies concava superne terminans liquorem : bini videlicet
r et r' exsistent aequales, simulque tubarum diametris pro-
portionales; ideoque altitudo :reciproce ut eae diametri.

2.0 Hydrargyrum in tubis vitreis descendit in-
fra circumambientis hydrargyri superficiem QQ' ad eius-
modi altitudines , quae sunt tubarum diametris recipro-
ce proportionales; descendensque liquidum terminatur su-
perne convexa superficie NOM.
Convexitatis causam adsignavimus (109. 1.(, ) : ad de-
scensnm quod spectat, columellarum OR et VR' altera
sollicitatur deorsum gravitate simulque vi (109. 11.&quot;)
Hi 1
K&quot;.'a(1&quot;'r&quot;)'
I:
'P.?237
altera sollicitatur deorsum gravitate simulque vi ( 109. 2.0)
K :
cum igitur
K &lt; K + -( + )
haud 'poterunt OR et VR' sese librarè nisi OR descen
dat infra QQ. Designet é altitudinem AO , ad quam
deprimitur columella OR infra QQ' ; sitque c' gravitas
specifica hydrargyri : eousque fiet columellae depressio do
nec habeatur,
H
K=K + ( + )- c'z' , unde z' = 2c'
(
+ &gt;&gt;).
Ut supra ( 1. ) ostenditur binos r , r' fore aequales, simul
que proportionales tuborum diametris ; iccirco etc.

111. Nonnulla subjungimas, quorum ratio desumitur ex animadversionibus (109).

1.º Duae laminae vitreae et parallelae PP ', SS ' demergantur verticaliter in aquam, earumque mutua distantia aequetur diametro tubi capillaris LE: suprema aquae superficies B &quot; A &quot; B &quot; inter laminas evadet concava instar canalis horizontalis; altitudo vero A'al= x ), ad quam attollitur aqua, erit duplo minor altitudine ad quam attollitur in tubo LE.: Infima superficiei B &quot; A &quot; B '&quot;' puncta jacent omnia inea
dem recta A'A'&quot;: secetur B &quot; A &quot; B &quot; &quot; plano perpendiculari ad A &quot; A &quot; '; sectio erit ubilibet arcus arcui BA'B' similis et
aequalis : istorun arcuum radius osculi apud puncta infima
dicatur r ; in tubo LE erit p = r , in laminis r= -0 . Colamellarum igitur A'R , VR aequilibrium praebebit ..
237
altera sollicitatur deorsum gravitate simulque vi (109. 23)
K :
cum igitur
X(K '
H( ↿
r ral,).r
baud' poterunt OR et VR' sese librare nisi OR descen-
dat infra -'QQ. Designet z' altitudinem AO,- ad quam
deprimitur columella OR infra QQ'; sitque c' gravitas
specifica hydrargyri: eousque fiet columellae depressio do-
nec habeatur.
1
∙
↿
'
∣∟−∣≖⊹−−⊸ 2(-—--]——-)-—c&quot;,z undez'—.2[:7(1 [
r').
Ut supra (1 .&quot;) ostenditur binos r, 'r' fore aequales, simul-
que praportionales tubarum diametris; iccirco etc.
,.,11.1 Nonnulla subjungimtts ,- quorum ratio desu-mitur ex animadversionibus (109).↿∙∘ ∐∎≖≔∘∙ laminae vitreae et parallelae PP', SS'demergantur verticaliter in aquam, earumque mutua distantiasequetur diametro tubi capillaris LE : suprema aquae super-ficies B&quot;A&quot;B&quot;' inter laminas evadet concava instar canalishorizontalis; altitudo vero A&quot;a(:.r) , ad quam attollituraqua , erit duplo minor altitudine , ad quam attolliturintnhoIfE.- ⋅∶∶⊸∙Infima superficiei B&quot;A&quot;B&quot;' puncta iacent omnia infen-dem recta A&quot;A'&quot;: secetur B&quot;A&quot;B&quot;' plano perpendiculariad A&quot;A&quot;'; sectio erit ubilibet arcus arcui BA'B' similis etaequalis: istorum arcuum radius osculi apud puncta infimadicatur r; in tuba LE erit r'zzr, in laminis r'zoo; Cos-lumellarnm igitur A'R , VR' aequilibrium praebebit't-dïff2382c ( + ) &quot; ;Tetscolumellarum autem A ” R &quot; , VR aequilibrium suppeditabitH101 IK -K - I ( + ) +re , f =.2c1Hinc xai 2ż z ; ideoque etc.....2.0 Laminae PP ' , SS , sibi commissae ad sematuo accedunt.Sit P&quot; punctum quodvis laminae PP: infra QQ adprofunditatem Alla &quot; : columella verticalis Alla&quot; transmittetpuncto P vim ( 1.0 ) .K- ( + ) +ostan&quot;) = KH+2r Tersus1 .C2c 5+c. a a'&quot; = K +0. aa!&quot;dicenndnetfenotaversus Qt : attenta columella horizontali . a &quot; ' P &quot; , urgebi amelltur P vi seu pressione externa + traiKversus Q't': colamella verticalis V'a transmittet puncto P &quot;vimK+c.aa &quot; &quot;versus Qi' : attenta columella horizontali a'P ' , solicitabitur P &quot; vi seu pressione interna 16TSU238H(t 1) H 1z— ⇂ ∙−−− ∙⋅ ;20 r !' C !'columellarum autem A&quot;R&quot;', VR' aequilibrium suppeditabit!H 1 1 H 1− ↿ ∣ ⊫−∙− −⋅⋅ ∙!cx . ∙∖Hlnc ∙ æ;—2—z;l ∙ ⋅ 1deoqueetc......⋅⋅ n .'2.o Laminae 'PP. SS', sibi commissae ad sea') 'At mutuo aeeedunt. )Sit P&quot; punctum quodvis laminae PP' infra QQ' adprofunditatem A&quot;a &quot;: columella verticalis A&quot;a transmittetpuncto P&quot; vim (1.). &amp; ↽(⋅. H 1 1⋅ &quot; HK(cæ—-—aa&quot;' :K −−−∙−−∙ 2 ≀⋅∙⋮∙∙∞ .'. (M' . 2r. . '&quot;1ciii .-—--)-c.aa :::K'I—ILmaa'&quot;.⋅.. .: '&quot;.versuth : attenta columella horizantali- a.'&quot;P&quot; , urgebi-tur P&quot; vi seu pressione externa,.. ⋅∙ ⋅('. a∙ t '. '(1
..

infima
dicatur r; in tuba LE erit r'zzr, in laminis r'zoo; Cos-
lumellarnm igitur A'R , VR' aequilibrium praebebit't-dïff238
2c ( + ) &quot; ;
Tets
columellarum autem A ” R &quot; , VR aequilibrium suppeditabit
H
101 I
K -K - I ( + ) +re , f =
.
2c
1
Hinc xai 2ż z ; ideoque etc.....
2.0 Laminae PP ' , SS , sibi commissae ad se
matuo accedunt.
Sit P&quot; punctum quodvis laminae PP: infra QQ ad
profunditatem Alla &quot; : columella verticalis Alla&quot; transmittet
puncto P vim ( 1.0 ) .
K
- ( + ) +ostan&quot;) = K
H
+
2r Tersus
1 .
C
2c 5+c. a a'&quot; = K +0. aa!&quot;
dicen
ndnet
fenota
versus Qt : attenta columella horizontali . a &quot; ' P &quot; , urgebi amell
tur P vi seu pressione externa + trai
K
versus Q't': colamella verticalis V'a transmittet puncto P &quot;
vim
K+c.aa &quot; &quot;
versus Qi' : attenta columella horizontali a'P ' , solicita
bitur P &quot; vi seu pressione interna 16TSU
238
H(t 1) H 1
z— ⇂ ∙−−− ∙⋅ ;
20 r !' C !'
columellarum autem A&quot;R&quot;', VR' aequilibrium suppeditabit
!
H 1 1 H 1
− ↿ ∣ ⊫−∙− −⋅⋅ ∙
!
cx . ∙∖
Hlnc ∙ æ;—2—z;l ∙ ⋅ 1deoqueetc......
⋅⋅ n .
'2.o Laminae 'PP. SS', sibi commissae ad se
a
') 'At mutuo aeeedunt. )
Sit P&quot; punctum quodvis laminae PP' infra QQ' ad
profunditatem A&quot;a &quot;: columella verticalis A&quot;a transmittet
puncto P&quot; vim (1.). &amp; ↽
(
⋅
. H 1 1
⋅ &quot; H
K
(
cæ—-—aa&quot;' :K −−−∙−−∙ 2 ≀⋅∙⋮∙∙∞ .'. (M' . 2r
. . '
&quot;
1
ciii .-—--)-c.aa :::K'I—ILmaa'&quot;
.
⋅
.
. .: '
&quot;
.versuth : attenta columella horizantali- a.'&quot;P&quot; , urgebi-
tur P&quot; vi seu pressione externa
,
.. ⋅∙ ⋅
('. a
∙ t '. '
(1
..

: - ∎∙ 3 ' ∙

.
versus Q't': columella verticalis V'a' transmittat puncto P&quot;
visu ⋅⋅ ' ⇀
'
'
'
⋅
⋅⋅
K—l—caa ?
.
versus Q't': attenta columella horizontali a'P&quot;
, sollicita-
bitur P&quot; vi seu :pressione interna &quot;
'
'
-
⇂⇣ r 'a'
ms
1111:
&quot;But
&quot;fin239
K
versas Qt : erit igitur p &quot; aeque sollicitatum hinc et illinc,
nullusque motus gignetur ab istis viribus.
Sit P' &quot; ' punctum ipsius PP' inter QQ et A &quot; A '' : ver''
ricalis columella A &quot; a &quot; transmittet puncto P &quot; ' vim
к
-16 + ) + ( 6 – aa &quot;) = K -
H
+
2r
H
S c.aa&quot; = K- c.aa' '
2r
versus Qt : ob columellam horizontalem P '&quot; a &quot; urgebitur
P &quot; &quot; vi seu pressione externa
K
versus Qt : cum igitur K &gt; K - c.aa &quot; , nitelur &quot; mo
veri ad plagam (t' .
Ascendet aliquantulum aqua externa prope laminam ÞP
induetque ( 109. 1. ) figuram concavam ee'e ' ; propterea,
denotante &amp; radium osculi apud punctum v . gr. e' , co
lumella e'b normaliter insistens superficiei curvae ee'e&quot; in
e' transmittet puncto v. gr. b vim
H
K
29 ( +4) = K 2 €
versus Qt’ : attenta columella horizontali e'b ', urgebi
tur b ' vi seu pressione interna
K
versus Qt : ex aqua éb'é' proveniet in bi vis
∙ 239
K
Versus Qt: erit igitur P&quot; aeque sollicitatum hinc et illinc,
nullusque motus gignetur ab istis .viribus. Sit P'&quot; punctum ipsius PP' inter QQ' et A&quot;A&quot;': ver-
ticalis columella A&quot; a&quot; transmittet puncto P&quot;'v vim
K
--—-(—1--—l--—) —[—c(x—-aa&quot; ∙−−∶ ∣⊂−∙≗ ∙⋅⊢
H
—— c.aa&quot;
−−−−− K— c.aa&quot; 2r ∕
versus Qt'
- ob columellam horizontalem P&quot;'a &quot;
P'&quot; vi sen pressione externa
&quot;—
urgebitur
K
versus Q't'. ∙ cum igitur K)K—c.aa&quot; , uitetur P&quot;' mo-
veri ad plagam Q't' .
Ascendet aliquantulum aqua externa prope laminam PP' induetque (109.1.?) figuram concavam ee'e&quot; ; propterea,
denotante &amp; radium osculi apud punctum v. gr. e'
, co-
lumella e'b normaliter insistens superficiei curvae ee'e&quot; in
e' transmittet puncto v. gr. b' vim
H 1 1 H K—
⊸≳↽≼−⋮⇀⊣−∘−∘−⊢ ≖⊊−−⊸⇄−∊
versus Q't'- attenta columella horizontali e',b' urgebi-
tur b' vi seu pressione interna '-
. K
8
versus Qt: ex aqua e'b'e&quot; proveniet in b' vis240
c.e&quot; 6
versus Qit' : columella verticalis A &quot; 6 &quot; transmitiet puncto
b' yim
H
K
25
+ c . A &quot; 6 &quot;
versus Qt : attenta columella horizontali b'b' impelletur 6
vi seu pressione esterna
K
versus Qt . Est
H
2r = cx = C . A'a ;
librato insuper liquido , pressiones apud V' et é' sunt ae
quales , et consequenter
K = K
H
28 to.e&quot; 6 ,
H
= c.eb' ;
2 €
detractisque proinde viribus versus Qt ex viribus versus
Q'ť , emerget
H н
K
2e-K + c.e^ 6—K+
-c.A &quot;b&quot; + K = c.eb - c.A &quot;6&quot; +
H H
2r 2 €
c (e&quot; b' — A &quot; 6&quot; + 1&quot; a - c'b') = c.b &quot;a &gt; o :
sollicitabitur ergo b' vi c.6 &quot; a versus Q't' .
Veniat denique spectandum in lamina PP punctum p '
inter A &quot; A &quot; et B &quot; B &quot; : sit P'a&quot; columella horizontalis ; a'i
columella perpendicularis superficiei curvae B’A &quot; B &quot; &quot; apud
a &quot; ; dicaturque é radius osculi in a ' ' . Transmittet a'i
puncto Ph vim
240
c . e&quot;b'
versus Q't' :columella verticalis A&quot;b&quot; transmittet puncto
b' vim
. H &quot;&quot;
K—Z—i—c'Ab
versus Qt :attenta columella horizontali b'b&quot; impelletur b'
vi seu pressione externa
↴
K
versus Q't'. Est
H
∙∠−≀∙−∶∶∘∙↿∽⋅∶∘∙∆∎∣∅
librato insuper liquido , pressiones apud V' et a' sunt ac-
quales , et consequenter
-——-K——'l&quot;c. e&quot;6'. Eli-:o- e&quot;b':
detractisque proinde viribus versus Qt ex viribus versus
Q&quot;: , emerget ⋅
K—is'&quot;. -K—]—
.- .∘∣∙∣↗∣−−⋅↧≮−⊦ ≛↿−⊑∙⊸∙∆∦∣⊃⋅∣−⊢↧⊊∶⊸⋅⊜∣⋅∣⊃⋅−∘⋅∆⋅⋅≀≀⋅∣−⊦
H H &quot;
.!
.,
h 1
∙−−
&quot;
.
ii.—22&quot;— ..—c(&quot;eb'- Ab -I-Aa-c.b)—-c.b a)o.
sollicit'abitur ergo b' vi c.b&quot;a vcrsus Q't' .
Veniat denique Spectandum in lamina PP' punctum P&quot;
inter A&quot;A&quot;' et B&quot;B&quot; : sit P&quot;'a&quot; columella horizontalis; a&quot;i
columella perpendicularis snperficiei curvae B&quot;A&quot;B&quot;' apud
a&quot; ; diceturque e' radius osculi in a&quot;. Transmittet a&quot;:
puncto P&quot; vim.241
K
H
2 €
.
versus Qt : ex liquido superincumbente proveniet in p ' ' vis
B ' P &quot;
versus Qc : attenta P &quot; a &quot; urgebitur p &quot; vi seu pressione
externa
K
versus Qit' : librato liquido , pressiones apud a &quot; et A ” sunt
aequales ; proinde ducta horizontali Allu ,
H H
K tc.P&quot; u = K = K- C. A'a ,
2 € 2r
H
= c ( P''u + A &quot; u ) = c.P'' ' :
26
detractis igitur primis duabus viribus ex tertia , assequemur
H
K - K +
c . B ' piv H
-C.B&quot; piv
2€ 22'
c ( Piu' B ' p ' ) &gt; 0.
Lamina itaque PP' movebitur versus Q'C' : eadem de causa
movebitur lamina SS' versus Qt ; ideoque etc...

3. Si aquae substituitur hydrargyrum , supre
ma liquidi superficies inter laminas PP' , SS ' fiet con
vexa instar horizontalis inversique canalis ; deprimetur li
quidam ad altitudinem duplo minorem quam in tubo LE ;
ipsae insuper laminae adhuc ad se mutuo accedent . Haec
explicantur simili ratione ac ( 10, 20. )..
241
versus Qt :ex liquido superincumbente proveniet in P&quot; vis
&amp; ∙ B1v Ptv
versus Qt : attenta P&quot; a&quot;
externa
urgebitur P&quot; vi seu pressione
K
versus Q't' :librato liquido . pressiones apud a&quot; et A&quot; sunt
aequales .; proinde ducta horizontali A&quot;u,
H &quot;
∙∙∙ H ∙∙
↿⊂−∙−∙⋮≳−⋮∙−⊢∘∙↧∙ ∥⋅−−↿⊊∙−⋮≳≀∙−−∙−−↧⊊−∁∙ A a
.
H ,,
⊓∣
27-—-c(P&quot;u-l—Au)——-—c.P u:
detractis igitur primis duabus viribus ex tertia , assequemur
⊏−≖⊂−⊢⋮−⋮∶∶−∘∙∌∏ ↕⊃≖⊽∶∶∶ IST—c .B&quot; Piv:
c (P&quot;'u' — Blv P&quot;) ∘∙
Lumina itaque PP' movebitur versus Q't' : eadem de causa
movebitur lamina SS' versus Qt ; ideoque etc...
3.&quot; Si aquae substituitur hydrargyrum , supre-
ma liquidi superficies inter laminas PP' , SS' fiet con-
vexa instar horizontalis inversique canalis; deprimetur li-
quidam ad altitudinem duplo minorem quam in tubo LE;
ipsae insuper laminae adhuc ad se mutuo accedent. Haec
explicantur simili ratione ac (10. 20.)
.242

4. • Super vitream laminam horizontalem AA'B'B
( Fig . 58 ) affunde gattam olei terebinthini mm ' ; tum al
teram laminam vitream A &quot; A'B'B &quot; priori AA'B'B impone
sub angulo sic exiguo , ut imposita lamina gutlam le
viter attingat ; conspicies mm' , instar trochleae , termi
natam quodam canaliculo ; qui canaliculus plano hori
zontali sectus dabit curvilineam convexamque sectionem
plano verticali sectus curvilineam concavamque sectionem .
Radius convexitatis ( € ) manet proxime idem in
punctis m et m' e diametro oppositis ; radius vero con
cavitatis ( = r' ) in puncto m' magis accedente ad A'B'
minor erit quam radius concavitatis ( = r) in puncto m
minus accedente ad ipsam A'B ' . Spectantes columellam
mam ' perpendicularem rectae A'B' , quoniam r et é ' apud m
obverluntur ad plagas contrarias , itemque r et ê &quot; apud m' ,
facile intelligemus ( 109. 110. ) sollicitatum iri mam ver
sus A'B' vi
H
K
2
simulque versus AB vi
K
16-&gt;).
Cum igitur &gt; m , prima vis erit major quam secunda ;
columellaque mam , et una cum mam' tola gutta mo
vebitur versus A'B' motu accelerato : idipsum contingit
guttaeaqueae . At si ejusmodi guttis substituatur gutta
hydrargyri , haec movebitur versus AB ; ratio est quia
gutta hydrargyri tam in sectione horizontali quam in
verticali praebet curvam convexam , radiusque novae con
vexitatis in m superat radium novae convexitatis in m .

5.° Capillaris tubus in aquam QQtt ( Fig. 57. ) demergatur; tum, apposito digito ad orificium inferius extrahatur : remoto digito , aqua jam elevata eflaet ali
quantulum ex orificio illo , ibique demum haerebit sus
pensa in guttam rotundam conformata ; residuae vero
aquae altitudo in tubo extracto invenitur major quam
altitudo ( 110, 1º.)
H
Z =
16 + 3) = * ( + ) cr
supra QQ in tubo demerso.
Exprimant et w , altera radium convexitatis apud infi
mam aquae superficiem in tubo extracto ,
altera ipsius
aquae altitudinem : ex aqueae columnae aequilibrio pro
fluit&quot; ( 110. 1 ° 2°. )
H H
K
+++
www =
= ktö : +
H
co ;
cr
ideoque w &gt; z . Si aquae substituitur hydrargyrum , tam
suprema quam infima superficies liquidi exsistet conve
x2 ; ex aequilibrio igitur hydrargyri in tubo extracto
emerget ( 110. 20. )
H H
H
K + tow == Kt H
co CI
et consequenter a = 0 si r = 0 .

112. Quae diximus de liquorum ascensu tubulis vitreis, applicari possunt ascensui liquorum in tenuibus cujuscumque materiei tabulis: hinc patet cur liquida ascendendo imbuant spongias, saccharum, ellychnia etc: cur succus inserviens plantarum vegetationi sursum ex terra eluctetur; etc... Istiusmodi corpora vel constant exilissimis fibris, in quibus tanquam in totidem capillaribus tubis ascendit liquidum, vel innumeros habent angustos meatus vicem tubulorum varie flexorum supplentes. 

Caeterum methodo inhaerentes, qua D. Pessuti LaPlacianam theoriam ad faciliorem formam traduxit, capillarium luborum phoenomena explicavimus in hypothesi liquidorum eamdem usque ad extimas omuino superficies obtinentium densitatem: non enim nobis in animo est vel leviter attingere novam theoriam, quam de actione capillari anno 1831 edidit D. Poisson.

== ACUSTICAE PRINCIPIA ==

=== Notiones preambulae ===
[[113|113]]. Acustica agit de sono: non defuerunt, qui sonum consistere putabant in efluviorum a soporo corpore vibratorum motu quae efluvia ex affrictu, vel contusione sonori corporis ejaculantur atque huic affinis est alia quaedam sententia, quod contusione illa vel affrictu particulae aeris purioris in eo corpore absconditi, vel ipsum circumdantis, expellantur et ad aures usque excurrant. Verum experimento machinae pneumaticae compertum est, quod incluso tintinnabulo vel horologio horas personante in recipiente, ubi aer incipit exhauriri, incipit sonus minui; ubi autem totus exhaustus est aer, nihil jam soni auditur, utcumque pergat tintinnabulum concuti, aut horologium pulsibus affici. Hoc probat sonum non consistere in effluviis a sonoro corpore vibrati cur enim non emittuntur amplius, aut ad aures non permeant, cum imo liberius ob minora obstacula deberent? &lt;u&gt;Ad majorem rei evidentiam&lt;/u&gt; ita hoc experimentum instituitur horologium in vitro aere pleno ac probe clauso reponitur, ne aer scilicet inde possit exhauriri tum in recipiente pneumatico collocatur, atque ex hoc educentes aerem animadvertimus sonum nullum audiri. Machina horaria aere circumsepta est ergo nullimode suspicari licet aliquid deesse circa ipsum corpus sonorum quominus sonus exaudiatur. 

Dicendum potius non audiri sonum propter defectum aeris intermedii inter utrumque corpus. Porro corpus cum resonat, motu tremulo atque &lt;u&gt;oscillatorio&lt;/u&gt; minimarum partium afficitur singulis autem oscillationibus aer corpus tremulam circumdans concutitur, similesque recipit vibrationes, quas in ulteriores particulas aereas pariter defert nisi quod impulsus in circumfusum aerem delapsi atque auditus organum afficientes eo minores ac debiliores fiunt quo magis a fonte recedunt. Enimvero corpora, quae sonora dicantur, tunc sonum excitant quando ita agitantur, ut illorum partes tremulo ac vibratorio satisque rapido concutiantur motu: sic campanae malleo percussae, ratione elasticitatis concipiunt tremoris motum, qui major fit postquam vehementius ac diutius agitatae fuerint: instrumenta musica, dum illorum fides agitantur, simili pariter tremore concutiuntur; hinc est quod chartae frustula resonanti corpori imposita subsultare cernuntur. His positis certum est tremorem similem communicari debere aeri immediate ambienti, et deinde tremorem late &lt;u&gt;diffundi&lt;/u&gt; per &lt;u&gt;aereas particulas&lt;/u&gt;; nam particulae aeris sonoro corpori proximae illius impulsu comprimuntur; et cum sint elasticae , post &lt;u&gt;compressionem&lt;/u&gt; &lt;u&gt;dilatantur&lt;/u&gt;, aliasque sibi proximas urgent; atque hoc pacto et illae vibrant, et longe lateque in particulas aereas similis vibratio omni ex parte discurrit. Hinc pulvisculi aeri innatantes, qui radio solis in obscurum conclave intromisso conspicui sunt, agitari videntur si sonus proxime intendatur: pulsato prope stagnantem aquam tympano, validius et crispari, et subsultare aqua pariter cernitur.

Haec notentur. 1º . Vis acceleratrix &lt;math&gt;\varphi&lt;/math&gt; in vibrante particula resonantis corporis ita pendet a spatiolo &lt;math&gt;z'&lt;/math&gt; quod particulae superest excurrendum usque ad nativam aequilibrii positionem, ut crescente, decrescente , vel evanescente &lt;math&gt;z'&lt;/math&gt; crescat simul, decrescat, vel evanescat; propterea&lt;math display=&quot;block&quot;&gt;\varphi = C'z' + C'' z'^2 + C''' z'^3 + ...;&lt;/math&gt;et quia particularum excursiones exsistunt exiguissimae, erit,&lt;math display=&quot;block&quot;&gt;\varphi = C'z':&lt;/math&gt;vis nempe acceleratrix assumi potest proportionalis spatiolo &lt;math&gt;z'&lt;/math&gt;. 

2º. Non pluribus opus est ut intelligamus (29.4º) vibrationes omnes, sive majores, sive minores ejusdem particulae fore aequidiuturnas.

[[114|114]]. Progignitur quoque sonus ab aere vehementer compresso seseque statim restituente: etenim propter impetum in restitutione conceptum ad majorem, quam in statu naturali occupabat, extensionem perveniet; ac proinde cogetur se rursus contrahere, minusque naturali spatio tenere. His autem successivis contractionibus et expansionibus in reliquo aere pulsus &lt;u&gt;excitantur&lt;/u&gt;: sic producitur sonus v. gr. virgae aerem celerrime perstringentis: simili modo qui in tibiam insufflat, sonum gignit; dum nempe per tubi orificium aer insufflatione intromittitur, ille, qui continebatur in tubo, necessario secundum longitudinem comprimitur; unde fit ut is iterum expandatur, tum denuo coarctetur; atque hoc pacto, quamdiu perseverat inflatio, perficiantur oscillationes, hisque sonus progignatur. Certe si aerea columna tubo &lt;u&gt;inclusa&lt;/u&gt; non afficiatur nisi motu totius, sonus minime obtinebitur; utcumque vero &lt;u&gt;excitentur vibrationes&lt;/u&gt;; ut &lt;u&gt;perceptibilem&lt;/u&gt; sonum edant, earum numerus intra minutum secundum non debet praetergredi quosdam certos limites, videlicet 6 circiter et amplius 24000; uti compertum est experimentis D&lt;sup&gt;''ni''&lt;/sup&gt; Savart.

[[115|115]]. Saepe contingit nos voce elatiori quibusdam in locis loquentes, aliquo tempore postquam siluimus repente audire rursus verba a nobis antea prolata; atque haec est illa echo, de qua plura fabulantur poetae. Philosophi in hoc conveniunt, quod echo sit motus reflexus aeris, qui &lt;u&gt;motu ondulatorio&lt;/u&gt; affectus obici incurrens resilit consimili motu, et rursum aures nostras afficiens nos determinat ad eumdem sonum audiendum, quem antea audivimus: ut autem effectus iste contingat, necesse est aliquanto longius a loquente obicem existere. Ratio est quia si &lt;u&gt;obex&lt;/u&gt; proximior fuerit, sonus reflexus efficiet in auribus impressionem suam antequam impressio soni directi defecerit; tunc vero non poterit secunda impressio a prima discerni. Aliquando semel tantum, aliquando saepius eadem vox per reflexionem auditur: primam contingit quando ab unico loco vox collecta rejicitur, vel a pluribus, sed ad eamdem distantiam: secundum quando vox in pluribus locis ad diversas distantias collecta revertitur ad aures sensibili successione. Hinc intelligitur quare in vallibus, quas undique colles cingunt, echo saepius iteretur.

[[116|116]]. Non solus aer est &lt;u&gt;medium&lt;/u&gt; idoneum transmissioni sonorum: nam per alia quoque elastica fluida propagatur sonus. Vapores ipsi, in quos aqua, spiritus vini etc. attenuantur, sonum transmittunt; etenim si recipiens pneumaticum aere atmosphaerico evacuetur, tum aliquo ex dictis fluidis repleatur, sonus campanae vel horologii adhuc bene audietur: quin et liquores, aqua v. gr. sonum non intercipiunt, sed ipsum debilitatum licet propagant; qui enim intra aquam sunt, audiunt sonos extra aquam editos; et qui extra aquam sunt, audiunt sonos editos intra aquam. Tandem etiam corpora solida deferunt sonos ad ingentes distantias: celebre est apud milites ita terram excavare donec strato alicui bene solido aurem applicare possint, ut ex reboatu agnoscant adventum hostilis legionis, praesertim equitatus; huic strato non raro tympanum imponunt, atque levia corpora tympano imposita ex sonoris tremoribus subsultant.

=== De intensitate soni; deque ejus gravitate, et acutie. ===

[[117|117]]. &lt;u&gt;Intensitas&lt;/u&gt; major vel minor soni importat majorem vel minorem ejusdem soni vim ad sensationem excitandam, quae proinde in intensiore sono vehementior est, ita ut aures prae violentia laedat aliquando; in remissiore ita debilis, ut vix aliquando audiatur. Iamvero evidens est quod quo plures sunt partes sive in corpore sonoro, sive in aere simul oscillantes, eo plus motus atque activitatis, caeteris paribus, habent; ac proinde vehementius organum auditus pulsare poterunt: quo singularum partium itus et reditus major est, seu quo fortius singulae particulae comprimuntur et restituantur in unaquaque oscillatione sive in corpore sonoro, sive deinde in aere, fortiori item impressione aptae erunt organum auditus afficere. Contra, quo pauciores partes sonori corporis oscillant, eo minus communicabitur motus particulis aeris, et consequenter ab his minus afficietor auditus organum: quo singulae sonori corporis partes unamquamque oscillationem minorem habent, eo minorem item oscillationem in aeris particulis producent, ac proinde impressione minus valida auditus organum concutient. Quod ratione perspectum est, &lt;u&gt;experientia quoque confirmatur&lt;/u&gt;; et quod ad sonum, quem vocant primitivum, attinet, corpora densiora, caeteris paribus, magis sopora sunt quam quae ex opposito; atqui hoc nonnisi quia plures particulae in his oscillant simul; ergo ex numero particularum oscillantium sonus major vel minor pendet. Rursum inter corpora aeque densa, atqae elastica, quod validius percutitur validiorem profecto sonum excitat et &lt;u&gt;magnitudo&lt;/u&gt; soni &lt;u&gt;magnitudini&lt;/u&gt; percussionis est proportionalis: undenam hoc repetendum est nisi ex eo quod validior percussio fortias comprimit atque oscillare vehementius cogit particulas elasticas? Quoad derivatum sonum res constat experimento machinae pneumaticae (113): cum enim exhauriri aer incipit, sonus incipit imminui; atqui hoc est quia  aeris quantitas in excipulo imminuitur; et cum rarior evadat aer, minus valide comprimi et restitui ejus particulae debent; neque enim ulla alia &lt;u&gt;probabilis&lt;/u&gt; causa est. Condensando insuper aerem in eodem excipulo ultra &lt;u&gt;statum ordinarium&lt;/u&gt;, quem tenet in almosphaera, compertum est quod condensatus aer sopam reddit intensiorem; atque hoc quidem ita, ut intensitatis augmentum proportionem servet cum augmento condensationis. Franciscus M. Zannotti diligentius rem exploravit: aerem inclusum vase calefecit; quo pacto aeris &lt;u&gt;elasticitatem&lt;/u&gt; auxit, &lt;u&gt;densitate&lt;/u&gt; eadem servata, cum nullus permitteretur aeri exitas; et tunc sonus intendebatur, At rima aliqua in vase relicta, per quam aer posset erumpere, tum igne admoto, sonus multo minor visus est quam antea fuerat. Cum igitur, permanente aeris elasticitate, non idem permanserit sonus, rursus patet quod soni intensitas non solum ab elasticitate, et consequenter a magnitudine vibrationum, sed a densitate, id est a numero particularum vibrantium dependet. Nec arte solum ex rarefacto vel condensato aere intensitas soni mutata deprehenditur , sed naturali etiam aeris rarefacti vel condensati constituțione idem evenit: hinc in altissimis montibus, ubi aer rarior est, ac proinde minus elasticus, sonus multo est remissior quam in planitie , ubi condensatione atque elasticitate pollet majori.

118. Ex his explicantur sequentia circa soni intensitatem. 

1º. In aperto aere sonus calore minuitur, in clauso vero calore augetur: apertus enim aer, ubi calore afficitur, sese continuo dilatat, adeoque ejus intensitas minuitur, quin &lt;u&gt;elasticitas&lt;/u&gt; augeri debeat; quia nempe habet quo se rarefactus recipiat; ergo minor numerus particularum oscillat, adeoque remissior sonus. Contra, si aer undique clausus est, cum densitas eadem manere debeat, elasticitas autem ex calore crescat, idem erit particularum numerus, sed singularum oscillatio propter auctam elasticitatem augebitur; ergo intensior sonus.

2°. Sunt qui dicunt, aestate sonum intensiorem esse, caeteris paribus, quam hyeme; alii contra opponunt, quod hyeme intensior sit sonus quam aestate. Si in re incerta quoad factum et ex circumstantiarum varietate adeo varia ut fortasse determinari non possit , si inquam ratio reddenda esset, ajendum sonum aestate imminui debere, quia aer terram ambiens calore rarefactus minori densitate pollet, ac proinde minor erit numerus particularum oscillantium. Cum autem ex calore elasticitas crescat , hoc capite augeri debet sonus , cum nempe singularum particularum oscillationes validiores debeant. Videndum igitur quid praevaleat; et juxta vel densitatem hyeine praevalentem imminutioni elasticitatis, vel elasticitatem praevalentem aestate imminutioni densitatis, qui effectus sequi debeat.

3º. Hinc etiam explicant nonnulli cur nocte, caeteris paribus, soni majores sint quam interdiu; quia nempe densior est per noctem aer ob calorem minorem; at hujus rei explicatio verior est, quod per noctem, cessante ea aeris commotione quae per diem habetur ex multiplici strepitu, magis aptus sit aer ad soni vibrationes concipiendas et deferendas, organumque auditus nulla alia sensatione percussum aptius sit ad peculiarem aliquem sonum exaudiendum.

119. Discrimen inter gravem et acutum in sono importare profecto debet diversitatem aliquam in motu aeris, quo afficitur organum auditus, atque adeo in motu sonori corporis ex quo in aere motus hujusmodi derivatur; nam cum sensatio sonii ex impressione organi auditorii oriatur, at omnis alia sensatio ex impressione organi proportionati, et impressio ista per motum aeris ad organum appellentis fiat, profecto diversa impressio, quae a sono gravi atque acuto fit, diversum motum exigit tum in aere ex quo immediate producitur, tum in corpore sonoro a quo mediate progignitur; atqui ista diversitas non ex validiori vibratione seu oscillatione majori provenit; ex hac enim quantitas sive intensitas soni (117), non autem qualitas seu tonus procedit; ergo diversitas ista in celeriori seu crebriori vibratione partium aeris, et consequenter sonori corporis, derivanda videtur. Ratio consequentiae est, quia non alia diversitas saltem probabilior in oscillatione partium concipi potest quam, ut haec sit vel major ut scilicet quisque itus et reditus spatium majus percurrat, vel quod sit celerior ut scilicet eodem tempore plures situs ac reditus habeantur. Ergo cum ex primo capite discrimen acuti et gravis repeti nequeat, nihil afferri probabilius potest: quam celeritas oscillationum, quae certe in satione diversitatem afferre debet. 

Quoniam vero in rebus physicis natura explorari maxime debet experimentis atque observationibus, ita prosequor. Constat in chordis musicis, eas quae vel breviores sunt, vel magis tensae , vel minoris diametri (nam ex hoc triplici capite diversitas tonorum habetur in fidibus) acutius resonare; contra graviorem sonum reddere eas, quae longiores sunt, vel minus tensae vel majoris diametri: atqui chordae breviores vel magis tensae etc. percussae, plures numero vibrationes pari temporis intervallo producunt, pauciores aliae; hoc patet ex ipso sensuum testimonio: ergo sonus acutus habetur in chordis, quae frequentius dato tempore oscillant; gravis autem etc. In ea insuper proportione, in qua frequentiores aut rariores sunt vibrationes chordae musicae, est etiam magis vel minus acutus sonus: ergo frequentior aut rarior vibratio omnino connexionem habet cum tono per chordam musicam producto; pendetque tonus ex illa &lt;u&gt;frequentia&lt;/u&gt; aut raritate vibrationum, tamquam effectus a sua causa. Quod dictum est de chordis musicis, valet etiam in campanis et pocalis vitreis, aliisque id genus sonoris corporibus; haec enim percussa figuram rotundam in ovalem mutant, eorumque proinde fibrae eundo et redeundo oscillare debent, atque ex hac oscillatione sonus oriri colligitur; ut autem gravior vel acutior est sonus corporis, ita in figura immutatio et restitutio seu fibrarum ítus ac reditus rariores sunt aut crebriores. Porro si id in sonoro corpore contingit, ut gravior sonus obtineatur quando minor vibrationum numerus habetur in corpore, jam tunc in aere quoque minor vibrationum numerus haberi dicendus est: siquidem tot numero vibrationes dato temporis intervallo producuntur iu aereis particulis a tremulo motu corporis resonantis, quot ab ipsius corporis sonori fibris seu particulis eodem tempore peraguntur; et vice versa quot in aere gigni ac propagari vibrationes constat, totidem in ipso corpore resonante produci dicendum est.

120. Dum plurium corporum sonus ita temperatur ut gratus sit auribus, dicitur consonantia seu concentus, si ingratum  sonum produxerint, appellamus dissonantiam: in sonis ita temperandis ut sint jucundi, ars musica versatur. Tonus musicus seu consonantia pendet ex eo quod certo tempore certus vibrationum numerus a pluribus sonoris corporibus peragatur, et particulis aereis communicetur. Si duo vel plura corpora sonora intra idem tempus vibrationem absolverint , consonantia est omnium perfectissima , et sonus dicitur unisonus ; si eodem tempore unum corpus unam , aliud duas vibrationes expleat, consonantia haec dicitur octava: ita appellatur ex eo quod per quandam tonorum seriem ascendendo hic tonus a musicis octavo loco constituitur. Si eo tempore quo unum duas vibrationes, aliud tres absolvat, adeoque secunda unius cum tertia alterius concurrat, dicitur quinta: si eo tempore quo unum tres, aliud quatuor vibrationes conficiat, quarta nuncupatur; atque istae sunt consonantiae illae, quas Pythagoras advertisse traditur, dum quinque fabri malleis ferreis massam ferream contunderent. Consonantiae istae in vibrationibus chordarum inventae sunt ; imo etiam alii successu temporis consonantiae gradus additi , quos diligenter musicae scriptores explicant. Si videlicet numeri vibrationum , quas dato tempore chordae  musicae efficiunt , sunt ut &lt;math&gt;1 , \frac9 8, \frac5 4, \frac4 3, \frac 3 2 ,\frac5 3,  \frac{15}8, 2&lt;/math&gt; chordae illae edent notissimos tonos ''do, re , mi , fa , sol , la , si , do'': constat experimentis saepissime iteratis; etenim chordae homogeneae , aeque crassae , eodemque pondere tensae , quarum longitudines sint uti &lt;math&gt;1 , \frac89, \frac 4 5, \frac 3 4, \frac 2 3, \frac 3 5,  \frac8{15}, \frac12&lt;/math&gt;praefatos tonos edunt. 

Haec subjungimus circa exiguissimas chordarum vibrationes. 

1°. Chorda homogenea &lt;math&gt;AB&lt;/math&gt; (Fig. 59) uniformiter crassa ubique tensa aequaliter, punctisque &lt;math&gt;A&lt;/math&gt; et &lt;math&gt;B&lt;/math&gt; fixa, traducatur ad datam formam curvilineam &lt;math&gt;AC''B&lt;/math&gt;; tum sibi relinquatur: pro quovis temporis momento determinanda proponitur curva &lt;math&gt;AC'''B&lt;/math&gt;, in quam abit chorda. Sint &lt;math&gt;AO ( =x)&lt;/math&gt; et &lt;math&gt;S'O ( = y )&lt;/math&gt; coordinatae orthogonales; &lt;math&gt;h&lt;/math&gt; longitudo chordae &lt;math&gt;AB&lt;/math&gt;; &lt;math&gt;M&lt;/math&gt; massa; &lt;math&gt;\theta&lt;/math&gt; tensio: in ea qua sumus exiguissimarum vibrationum hypothesi, maxima chordae elongatio ab aequilibrii positione cum sit ferme insensibilis, haec obtinebunt quamproxime. Primo: apud quodvis chordae vibrantis punctum Seadem vigebit constanter tensio &lt;math&gt;\theta&lt;/math&gt;. Secundo: movebitur &lt;math&gt;S&lt;/math&gt; juxta directionem &lt;math&gt;SO&lt;/math&gt; respondentis ordinatae. Tertio: denotante a angulum tenuissimum &lt;math&gt;S'EA&lt;/math&gt; interceptum tangente &lt;math&gt;S'E&lt;/math&gt; et abscissarum axe &lt;math&gt;AB&lt;/math&gt;, erunt &lt;math&gt;\alpha = \sin \alpha = \tan\alpha ;\, \cos \alpha =1&lt;/math&gt;. Quoniam exercetur &lt;math&gt;\theta&lt;/math&gt; juxta vibrantis chordae longitudinem; sumptis arcubus infinitesimis &lt;math&gt;S'i , Si&lt;/math&gt;, denotabunt &lt;math&gt;S'i\, \mathrm{et}\, S'i'&lt;/math&gt; directiones tensionum apud &lt;math&gt;S'&lt;/math&gt;: resolvatur tensio juxta &lt;math&gt;Si&lt;/math&gt; in duas, quarum altera existat parallela rectae &lt;math&gt;AB&lt;/math&gt;, altera perpendicularis eidem &lt;math&gt;AB&lt;/math&gt;; et idipsum fiat quoad tensionem juxta &lt;math&gt;S'i'&lt;/math&gt;. Componentes parallelae axi &lt;math&gt;AB&lt;/math&gt; se mutuo destruent; componentes vero perpendiculares ipsi &lt;math&gt;AB&lt;/math&gt; exprimentur per &lt;math&gt;\theta\sin\alpha&lt;/math&gt; versus &lt;math&gt;O&lt;/math&gt;, et Osini atda ) versus S , seu per Ox et Oatd « ). Superest igitur vis
- Oda
gignens motum juxta SO : differentiale da sumendum quoad
x tantum, utpote denotans variationem anguli a in eadem curva AC &quot; B. Quisque videt --Oda esse vim motricem, cujusmodi est tensio  &lt;math&gt;\theta&lt;/math&gt;: propterea, designante dm elementum massae , exprimetur per
Oda
dm
∶≀≤↓⇟≓
miter crassa, ubique tensa aequaliter, punctisque A et B fixa, traducatur ad datam formam curvilineam AC&quot;B;
tum sibi relinquatur: pro quovis temporis momento de-
terminanda proponitur curva AC&quot;'B, in quam abit chorda. Siut AO (:æ) et S'O (::y) coordinatae orthogonales; '
h longitudo chordae AB; M massa;9 tensio: in ea qua
sumus exiguissimarnm vibratiouum hypothesi, maxima chordae elongatio ab aequilibrii positione cum- sit ferme in-
sensibilis , haee obtinebunt quamproxime. Primo: apud
quodvis chordae «vibrantis punctum S' eadem vigebit constanter tensio 9. Secundo: movebitur S' inxta directio-
nem SO respondentis ordinatae. Tertio :denotante et an-
gulnm tenuissimum S'EA interceptum tangente S*E et
abscissarum axe AB, erunt ut :sina −∙−−−− tangat ; cos at −−−∶↿∙
Quoniam exercetur 9 iuxta vibrantis chordae longitudinem :
[snmptis arcubus infinitesimi: S'i
, S'i', denotabunt S'i et S'i'
directiones tensionum apud S': resolvatur tensio iuxta Si
in duas
, quarum altera existat parallela rectae AB, alte-
ra perpendicularis eidem AB; et idipsum fiat quoad ten-
sionem iuxta S'i'. Componentes parallelae axi AB se mu-
tuo destruent ; componentes vero perpendiculares ipsi AB
exprimentur per Osina versus O, et 9sin( at-l-dat) ver-
sus S , seu per 90: et B(a-I—dat). Superest igitur vis
—9dat
gignens motum juxta SOI: dili'erentiale dat sumendum quoad, utpote denotans variationem anguli &amp; in ea-
dem cnrva AC'&quot;B. Quisque videt —-9dat esse vim motri-
cem , cuiusmodi est tensio 9: prapterea
, designante dm
elementum massae , exprimatur per
Gala
&quot;2711-255
respondens vis acceleratrix. Ob uniformem chordae cras
sitiem ,
dx
h
dm
M
Mar
ideoque dm =
h

;

insuper
a = tang a =
dy
dx ;
sumptisque differentialibus quoad x ,
da dany
dx dx²
Facto itaque compendii causa
on
M
superior expressio vis acceleratricis traducetur ad
d²y C2 dx²
unde ( 28 )
day
da(SS)
dia
d ” (SO - SO )
dc2
d²y
dx² de²
seu
1
255
respondens vis acceleratrix. Ob uniformem chordae cras-
sitiem ,
, ideoque dni ∙∙∶−− —-—de ;
9.&quot;M :.
da: ∙∙∙⋅ h −∙−
insuper
at:—— tangat-agi ;
sumptisque diiferentialibus quoad a:
,
data:-dv dx dx: ⋅
Facto itaque compendii causa
911
∙∙−∙∙↽−∙∶∘∙ ∙ M
superior expressio vis acceleratricis traducetur ad
da
⋅−∘⋅⊒≀−⋛−⋮⋅ ∙
nnde (28 )
,↶≀≖↗∙∙ irss; -dz(so-s'b) ∙∙∙ a.,
c dx&quot; dt: d? d;:
— '
sen256
day c2 day
dia (a) .
dx2
Formula ( a) suppeditat quaesitam problematis solutionem.
2. • Fac ut vis acceleratrix sit ut 1 , nimirum
day C'y ; erit dta
der
· c?
day
dx2 C'y
1 seu tör
so . dra
Inde habemus ( 27 , 27.0 )
VVT
y=CC + C, e
с - CV-4
evanescente X , evanescit et y ; hinc C = -C, , et con
sequenter ( 27. 30. )

* V0V1

y = C , [e
- *70V1
1 =
2011'sin rc=2C1V= 1sin
2 VMCMC
h9
facta x = h , evanescet y ; proinde
sin k V MC
ik V
MC'
ho TT C' =
OTE2
LM .
ho
9
ordinata CC &quot; respondens abscissae AC ( **)
256 ∙
da,, ⋅
−−↙⋮∎⋅≒↿∣
dx? :d—t; (a) . .
jl
C:
Formula (a) suppeditat quaesitam problematis solutionem. *
2.&quot; Fac ut vis acceleratrix sit uty
, nimirum '
lude habemus (27. 27.?)
f.. t/CV −−↿
-..-z ∁∣⇂∕∶⋅↿⋅
yiL-3010 c
⊣⋅∙ O; 6 - c :
evanescente a:, evanescit et )&quot;
, hinc C,: ---Cl , et con-
sequenter (27. 30. ∘ )
⋅−∙⊽⋮⋅∣∕∁⋅⇂∕∶↿ ... ⋅⋮∸−⇂∕∁⋅⇂∕∶↿
c c
y-—-—C,[e -—e ]∶∶
2C1V —-1 'sin −⋅⋮−− ∣∕∎∁⋅∶⊋∁∎≖∣∕∶−↴ sinx 9:709. :
facta x:h , evanescet y; proinde
. ⋅∙∥⋅∪⋅∙∙ VH?&quot; 97:-
&quot;[III 119 ∙−−−∘∙≀∙ WC;&quot;, (:::-IIM:
,
ordinata CC'&quot; respondens abscissae AC (;.-ä 11) ex-
'257
hibeatur per y ' , erit

;; = - 20, vt in . V MC =20,V = tsin.V MORE

TT
2C,V -1 sin î
2C, V 31 .
Propterea
2 = sin -
77 ()a' ) ;
aequatio ad curvam AC''B.''
3.° Per ty denotetur tempus unius semivibra
tionis ; erit ( 29. 3.° )
TT
1,5
2V C
VhM

;

0
et consequenter tempus unius vibrationis
hM ta VRM
Ad haec : designante n numerum vibrationum , quae ip
tra temporis unitatem absolvuntur , exsistet
1
V
TANTE
12
In hypothesi chordae cylindricae habentis radium
r el densitatem , erit M = fErPhò ; ideoque
257
hibeatur per J&quot;, erit
. -—- .
': MC' −−
∙ h M9112
r;.—20. ⇂∕⋅−↿ sm ∙− ∙−−−⇌⊋∁≖⇂∕⋅−∙↿ 810 2— IPGM :: 2 116
∙−− 7! −∙∙ ≢∁⋅↾∕∙−↿ sin ∙−⇇∋∙− −−−∙⇌ my.—1 .
Propterea
yzy' sin :; 71 (a') ;
aequatio ad curvam AC'&quot;B,
3.0 Per t. denotetur tempus unius semivibra—
tionis; erit ( 29. 33 )
et consequenter tempus unius vibrationis
.:Vg.
Ad haec: designante n uumerum vibratiouum, quae in-
tra temporis unitatem absolvuntur , exsistet
In hypothesi chordae cylindricae habentis radium
r et densitatem 8 , erit Mr.-:Ttrïhö ; ideoque558
13=rkVis, n - EVO
4. • Facta Osy. , velocitas puncti S in fine
temporis ( erit ( 29. 1.° 2.° )
v = y.Vī sine VC -Yosin
hinc ( 29. 1. ° )
=V
9. C - 02
C
yo V1 - sin’LVA
yo coseV C sy= . COS cos r.
Simili modo , facta CC“ =jo , velocitas pancti C in
fine temporis ( erit
7 t
yo sin Ti; simulquey'= y's cososeme- T ;
ta
et aequatio (a' ) ad chordam vibrantem poterit scribi in
hunc modum
y = yo com-A sin
C
-TT h
( á ').
5.° Si abscissae x in ( a'') substiluitur vel anh''
vel ( 2n + 1) h, prodibit y = o quotiescumque n aut erit =0,
aut erit quivis numerus integer : binae videlicet
558
9
≄≖−⊣⋅↗≖⇂∕∂ ∙
∙≖⋮−−⇀⊑⋅−−≀≖ '?Eä'
4.0 Facta OS;-:]. , velocitas puncti S in line
temporis : erit (29. 1.&quot; 2.')
v': J/ö' sint t/&quot;ä ∶∶−∶−≖−∫∘ sin-;- tt
!- 8
hinc (29. 1.&quot;-)
∙⊺∶−−−∙∙⇂∕⋅↗⇗ (S'—v ∙−−− J. l/1—s1n'q/ Q'::
j'. costV C' :y, cos ∙⋮− 11.
2
Simili modo
,
facta CC
∙−−∶ y'.. , velocitas puncti C&quot; in
fine temporis :erit
' n
s
∙ t ∙
' ' :
si:—y., s1n——1r;stmnlquey-:yocos—1t ;
t

: , :,

et aequatio (a') ad chordam vibrantem poterit scribi in
hunc modum
∙−− '
cst nsinæn ⋅ (a&quot;) J—yo O.t2 h .
5.0 Si abscissae a: in (a&quot;) substituitur vel an]:
vel (a'n-l-nh, prodibit yzo quotiescumque 11 aut erit 20,
aut erit quivis numerus integer: binae videlicet
Je!259
x = 2nh ,
( 2n+1 ) h
spectabunt ad quiescentia chordae vibrantis puncta. In
ferimus illud : chorda AB produci potest ultra limites A
et B quin puncta A et B per iteratas chordae vibra
tiones a statu quietis dimoveantur , etsi puncta illa poo
nuntur de se mobilia ; modo tamen AB in eamdem ac
antea conformelur initialem curvam , eidemque subjiciatur
tensioni : imo sumpta BH = HH' = =h , ita vibra
tiones suas conficiet chorda ABHH ' .
ut puncta A,
B , H , H ', ... in quiete persistant. Ad istiusmodi vi
brantis chordae figuram quod pertinet , sit v . gr. HD
HD = AO = x ; erunt AD = AH -HD = 2h - x , AD = 2h + x :
in la &quot; ) substitue prius 2h-x , deinde 2hta loco x ;
provenient ordinatae yı ety respondentes punctis D et D ',
nimirum
visy.cos Ti sin ( 2
a sin 16 는 (2-m ) R = my'o cos t2
sa= com sin ( 2+ )n = foco na sio ža .
Igitur y = -y, ya= y : ordinatae scilicet y , y , sunt
aequales , et ad eamdem plagam obversae ; ordinatae ve
ro y , y sunt quidem aequales , sed obversae ad con
trarias plagas. Chorda itaque dividitur in partes alterna
tim vibrantes supra et infra rectam AH'.
6.** Quoad (a) generatim spectatam ; denotanti
bus f et F binas functiones arbitrarias , satis hic erit
animadvertere eam expletum iri per
y =-flatct) + F ( x - ct) (a ' ' ) ;
siquidem
259
a.:2n71 ∙ :r (2n—l-1)I;
spectabunt ad quiescentia chordae vibrantis puncta. ln-
ferimus illnd: chorda AB produci potest ultra limites A
et B quin puncta, A et B per iteratas chordae vibra-
tiones a statu quietis dimoveantnr, etsi pnncta illa po-
nantur de se mobilia; modo tamen AB in eamdem ac
antea conformetur initialem curvam, eidemque subjiciatnr
tensioni: imo sumpta BH −−−−− HH' ::
...
zh
,
ita vibra-
tiones suas conficiet chorda ABHH' . .. ,
ut puncta A,
B , H , H', ...
in quiete persistant. Ad istiusmodi vi-
brantis chordae figuram quod pertinet ,
sit v. gr. Hl):
HD'zAOsæ; erunt AD;:AH-HDzah-æ
, AD':.2h-l—æ :
in (a&quot;) substitue prins alz—a:, deinde Zh—l-æ loco :;
provenient ordinatae y. etj, respondentes-punctis D et D',
nimirnm
'
tnsin(2 −⋅⋮≻↿∎∎∶ 'cos tu' æ
fac:-Tou.&quot; (: 'l &quot;70 :: Olli-i:&quot;.
, s ∙ æ ,
t
. æ
Jar—jre cos-1t am (2 −⋅⊢ --)1t :yocos —-1t sm --1t
.
- :, h : 11
Igitur y. ∙∶−−−∙ −∫∙ ∙↗≀≏−−−−−⋮↗↟⇌ ordinatae scilicet y , ;, sunt
aeciuales ,
et ad eamdem plagam obversae; ordinatae ve-
ro y, y, sunt quidem aequales, sed obversae ad cou-
trarias plagas. Chorda itaque dividitur in partes alterna-
tim vibrantes supra et infra rectam AH'.
6
∙∘∙ Quoad (a) generatim spectatam; denotanti-
bus f
et E binas functiones arbitrarias
,
satis hic erit
animadvertere eam expletum iri per
Fnæ-l-ct) −∣⋅− P(æ—ct) (if&quot;) ;
siquidem'260
dº[fix + c ) + F(x – ct) ]_da[fixtet) + F (x – ct )] .
do[ ) dt2
7 . ** Velocilas puncli S in fine temporis i prodit
expressa ( 28) per
dOS - OS')
dt
dy
dt [flatct)-F&quot;(x – ct)]:
initio motus , quum nempe t = 0 , est v=0 ; iccirco
c [ f '( x )-F'(x )] = 0, $' ( x)=F&quot; ( x) , et f (x ) = F (x ) ;
aequationes igitur determinantes et curvam ASB , et ve
locitatem traducentur ad
y = f(x + ct) + f(xớctct), v '= -c[ f '( x + chf'( x - ct) ] .
Facto t = 0 , istarum prima praebebit
y = 2f \x ) ,
aequationem videlicet ad curvam datam ACSB : ex hac
itaque curva pendet natura functionis f. Caeterum , ge
neralem de integratione differentialium partialiumque ae
quationum doctrinam suo tempore videre erit in parte 3.4
nostrorum elementorum Matheseos n. 200 , 201 ,
121. Si chorda instrumenti musici percutiatur , et pro
pe adsit instrumentum aliud , in quo chorda sit ad aniso
num cum priore tensa , baec alterius instrumenti chorda
sensim tremere incipiet , et undulationes sensim majores
concipiendo ad sonum ipsa quoque excitabitur eumdem to
num reddendo quem prior illa chorda percussa reddit .
Jam vero si ad hujus rei rationem attendas, plana erit juxta
theoriam traditam : sicut enim pendulum quiescens etiam
minimo impulsu , uti est qui ex insufflatione procedit , ad
260
⊄≀⇟⊏∣≺↕⊣⊸↥≻ −⊢ Für—ct) ],. «l*[m—l-aH-Fw—ctü.
dt:
(
7. ∙∙∙ Velocitas puncti S'111 fine temporis :prodit
expressa (28) per
os.-os d
v. ∙∙∙⋅ &amp; dt ):... .... :]? ∶−∙−− —c[f(æ-l-ct)—-F'(:r—ct)]:
initio motus , qunm nempe t::o
,est ⇂↓−∙−−−∘⋅ , iccirco
c[f (æ)—-F' (x)] ∙−−− o, f(xrr—F' (x)
, etfix):F(æ)' ,
aequationes igitur determinantes et curvam AS'B ,
et ve—
locitatem v' traducentur ad
y-fþ—I—ct) *Aæ—ct) , ∙≀⋅−−− ∙−−− ∙−− c[f '(æ-i-ctF-f'w—ct) ].
Facto t--—-o
, istarum prima praebebit
F2nx) :
aequationem videlicet: ad curvam datam AG&quot;B : ex haei
itaqua curva pendet natura functionis f. Caeternm, ge-
neralem de integratione differentialium partialinmqne ae-
qnationnm doctrinam suo tempore videre erit in parte 3.&quot;
nostrorum elementorum Matheseos n. 200, 201, .-. .
.-

121. Si chorda instrumenti musici percutiatur, et prope adsit instrumentum aliud, in quo chorda sit ad unisonum cum, priore tensa, haec alterius instrumenti chorda sensim tremere incipiet, et undulationes sensim maiores concipiendo ad sonum ipsa quoque excitabitur eumdem tonum reddendo quem prior illa chorda percussa reddit. 

Jam vero si ad huius rei rationem attendas, plana erit iuxta theoriam traditam: sicut enim pendulum quiescens etiam minimo impulsu , uti est qui ex iusufilatione procedit , ad motum oscillatorium minimum primo concitabitur , et si in
suflationem saepius repetas , poteris sensim oscillationes
majores , ac majores perficere (tunc tamen id fiel quando novi isti impulsus certa periodo, parique intervallo habeantur; si enim pendulum contra insufflantem venit, insufflantes rursum potius motum impediemus quam adjuvabimus, atque idet' finita una oscillatione debet opportune rur
sus alius impulsus addi , sicque ubi oscillationes penduli
et novi impulsus certa periodo sibi respondeant , effectus
habebitur) ita in chorda praefata evenit ut trémitns con
cipiatur et augeatur ; donec excitetur sonus ; quia nempe
Oscillationes unius chordae consentiunt cum oscillationibus
ad quas altera determinabilis est , iccirco ex repetitis chor
dae percussae uşdulationibus , quae sunt isochronae undulationibus alterius , obtinebitur ut hae augeantar donec so
nus excitetur in chorda etiam plectro minime percussa. Ex hac doctrina infero: ergo in utraque chorda oscillationes sunt pares numero; ergo cum tonus ab utraque redditus idem sit, tonus igitur a numero vibrationum hujusmodi pendet. Ad magis declarandam traditam doctrinam de acutie et gravitate sonorum, utile erit non nulla hic explicanda proponere circa chordas vibrantes. 

Ac 1º. quamvis chordae non sint unisonae, attamen una percussa, alia sonum edit, si modo tensae sint ad octavam, aut alias quasdam habeant armonicas proportiones. 

2º. Si duae chordae tensae sint ad octavam, et pulsetur chorda longior; quae dimidia ejus est, reddet tonum sui proprium, scilicet octavam acutam; at si pulsetur chorda brevior, excitabitur in longiore tonus non sui proprius, scilicet ad octavam gravem, sed tonus chordae brevioris. 

3º, Refert Sauverius hoc phoenomenon: chorda longa 5 ped. percutiatar, et notetur tonus;
tum ad distantiam unius pedis ponatur supra chordam le
ve aliquod obstaculum velati plumae frustulum , quod ta
men non impediat molus communicationem : si quinta haec
1
1
1
261
motum oscillatorinm minimum primo concitabitur ,
et si in-
snæationem saepius repetas, poteris sensimf oscillationes
maiores , ac maiores perficere (tunc tamen id fiet quando
novi isti impulsus certa periodo», parique intervallo babe-
autur; si enim pendulum contra'insumantem venit ,
insuf-
Hantes rursum Potius motum impediemus quam' adiuvabi-
mns , atque idet-' finita nna'osci'llatione dehet opportune rur-
sns alius' impulsnsgaddi, sicque ubi oscillationes penduli
et novi impulsus certa periodo sibi respondeant , effectus
habebitur) ita in chorda praefata evenit ut tremitus con-
cipiatur et augeatur, donec excitetur sonus; quia nempe
oscillationes unins chordae consentiunt cum oscillationibus
ad quas altera determinabilia est
,
iccirco ex repetitis chor-
dae percussae undulationibus , quae sunt isochronae undu-
lationibus ulterius-, obtinebitur ut hac augeantur donec ac-
nos excitetur in chorda: etiam plectro minime percussa. Ex
hac&quot; doctrina infero: ergo in utraque chorda 'oscillationes
sunt pares numero; &quot;ergo cum tonus ab utraque redditus
idem sit, to'nus igitur a numero vibratiouum hujusmodi
pendet. , - - -
'
'
'
Ad magis declarandam- traditam doctrinam de acutie
et gravitate sonorum, utile erit non nulla hic &quot;explicanda
prcponere circa chordas vibrantes.

q'uod ta-
men non impediat motus communicationem: si quinta haec
.*262
P
-chordae pars pulselur, tongm efficiet proprium chordae d
- nias pedis; hunc autem ipsum tonum reddit etiam reliqua
chordae pars, etsi quadrupla. Rursum si obstaculum pona
tur post duos pedes , eveniet ut pars brevior citius oscil
let, et longioris motum perturbet; subinde utraque chor
dae pars ita , sese componet; ut vibrationes eodem tempo
re compleat: tunc vero tonus reddetur neq w proprius chor
dae duorum pedum , neque trium, sed proprius chordae u
nius pedis.
Ad primum quod attinet , quoties duae chordae len
sae sunt ad octavam, jam vibrationi unius chordae ,respon
dent duae vibrationes alterius; ergo quamvis singulae , O
scillationes non conveniant, adeoque tremitus aeris non re
novet impulsum in alia chorda post singulas ejusdem oscil
lationes, renovari tamen potest impulsus hic post binas ;
eo ipso poterit chorda ad octavam tensa , etsi difficilius ,
ad oscillandum determinari ex alterins oscillationibus. Idem
valet de aliis chordis quae eam habent proportionem ut
oscillationes recurrere possint post aliquem ipsarum nu
merum: ac proinde illae, quae vel ejusmodi recursum non
admittuut , vel quarum recursus majorem postulat' quam
par est vibrationum numerum, non ita invicem ad reso
nandum poterunt determinari.
Ad secundum : quod chorda brevior resonans ad pulsa
tionem longioris reddat tonum sui proprium , cohaeret cum
doctrina jam tradita : quod autem chorda longior reddat
Lonum proprium chordae brevioris non officit; etenim si
chorda sit dupla , quasi in duas dividetur, neque tota oscil
Jabit ( 120..5º. ) per modum unius, sed habens in medio
punclum quiescens, seu nodúm, oscillabit seorsim in sin
gulis dimidiis partibus, ac si, scamould adjecto , bifariam
arte divisa esset ; , et si chorda ' triplo sit longior , ia - tres
partes aequales dividetur: quo posito , nil mirum quod chor
da dupla non sui proprium tonum , sed tonum subduplae
reddat, et tripla sonum subtriplae.
ic
0
at
LE
262
chordae pars pulsetur, tonum efficiet proprium chordae n-
onius&quot; pedis; hunc autem ipsum tonum reddit etiam reliqua
chordae pars,; etsi quadrupla. Rursum siobstacnlum pona-
tur post duos pedes'. eveniet ut pars brevior. citius oscil-
.let,. et, longioris motnm perturbet; subinde utraque-. chor-
dae pars ita sese componet&quot;, ut vibrationes eodem tempo-
re compleat: tunc vero tonus reddetur neq ,.: proprius chor-
dae duorum pedum, neque trium, sed proprius [chordae u-
,niuspedis. ↴
⋅⋅∶⇟⋡∢⋅ ∙

Ad primum quod attinet, quoties duae chordae tensae sunt ad octavam, iam vibratioui unins chordae respondent duae vibrationes alterius; ergo quamvis singulae oscillatioueslnon conveniant adeoque tremitus aeris non renovet impulsum in alia chordae post singulas ejusdem oscillationes, renovari tamen potest impulsus hic post binas; eo ipso poterit chorda ad octavam tensa, etsi difficilius, ad oscillandum determinari. ex&quot; alterius. oscillationibus. Idem
.valet de aliis chordis-anae-eam habent prOportionem ut
oscillationes recurrere possint post- aliquem - ipsarnm nn-
merum: ac proinde illae, quae vel ejusmodi recursum non
admittunt
,vel quarum recursus majoreni- pastulat' quam
par est vibratiouum numerum, non ita invicem a'd reso-
.nandum poterunt determinari.

Ad secundum: quod chorda brevior resonans ad pulsationem longioris reddat tonum sui pmprium cohaeret cum doctrina iam tradita: quod autem chorda longior reddattonum - proprium  chordae brevioris non officit; (etenim-si
chorda sit dupla, quasi in &quot;duas dividetur,- neque- tota oscil-
Jabit (120. 50.) per modum unins, bed habens in medio
punctum quiescens, s'eu nodnm, oscillabit seorsim in sin-
gulis dimidiis partibus, ac si, scamnnlb adiecto , bifariam
arte .'divisa esset;. et: si chorda' triplo sit longior, in- tres
partes aequales dividetur: quo posito, nil mirum-quod chor-
da dupla non aui proprium tionnm, sed tonum'subduplae
reddat, et tripla sonum subtriplae.
n-rts lar-Q .-263

Ad tertium: idem Sauverius hanc in Academia Pari
siensi explicationem attulit . Dum chorda nullo obstaculo
apposito pulsatur, vibrationes efficit toti suae longitudini
proportionales: at dum leve illud obstaculum apponitur post
pedem unum , undulatio totalis chordae dividitur ; prima
enim pars chordae , utpote quinta chordae totius ,
quinquies citius oscillare debet quam oscillaret integra
chorda : sic citius oscillando abripiet partem sibi proxi
mam in vibrationes aequales ; secunda pars tertiam, atque
ita singulae quinque partes seorsum oscillationes pera
geat. Alterum vero, quod magis est admirabile, ila ab eo
dem auctore explicatur ; pars brevior chordae, scilicet duo
rum pedum, citius oscillans quam reliqua , secum abripit
per sui motus communicationem partem sibi similem, nem
pe duorum pedum; in quinto autem pede oscillationes e
tiam communicantur, quae cum esse debeant longitudini
proportionales, duplo crebrius oscillabit extrema haec chor
dae pars quam reliquae; proinde ista sibi proximam unius
pedis partem trahet ad analogas oscillationes , et secunda
tertiam atque ita de reliquis , donec in hoc etiam casu quin
que chordae partes oscillent juxta longitudinem propriam ,
et consequenter sonum reddant respondentem longitudini
upius pedis.

122. Quaeri potest quomodo sonus trans obicem queat communicari ita, ut tonus proprius sonori corporis permaneat; nam fibrae, seu partes elasticae obicis puta parietis aut cancelli vitrei, ad motum concitatae vel sui proprium tonum reddere debent, vel si dissimiles sint, plurium tonorum mixturam, quod non accidit. Respondeo nullam esse difficultatem, si immediate per aerem soni propagatio habeatur, etiam intermedio exsistente obice. Quod si per obicem sonus diffunditur, in ipso admitti possunt partes aptae diversos sonos reddere aerique transposito communicare; atque ita, ut ille sensibilis sit trans obicem tonus, qui a partibus analogam oscillationem habentibus cum sonoro corpore communicatur. Forte etiam dici potest, quod si fibrae non habentur aptae eum tonum reddere, dividantur, ut in chorda non unisona contingit, adeo ut idem tonus transmitti possit.

123. Quoniam de tonis, ex quibus qualitas soni denominatur, egimus; quaerendum esset unde asperitas aut lenitas, quae pariter ad qualitatem quamdam soni pertinet, proficiscatur. Animadverte sonum quemcumque non esse simplicem, sed compositum e sono plurimarum sonori corporis partium: sic chorda musica percussa non simplicem edit sonum, sed quemdam veluti concentum edicit , qui a
peritioribus musicis probe dignoscitur; in quo tamen cum
fortior tonus praevaleat, alios minores obruit : coexsistunt
videlicet in chorda sonora, et generatim in quovis particu-
larum s'ystemate, variae exiguarum oscillationum species.
Imo vero non tantum sonorum ipsum corpus attendendum
est plerumque v. gr.-chorda musica, sed instrumentum i-
psum cui chorda adhaeret: variae insuper reflexiones ani-
madverti debent, quibus aer ad aurem deveniens diversas
subit modificationes. Itaque si vibrationes partium sonori
corporis sint bomologae, sonus lenis erit; si contra, asper:
atque hinc aspere sonant chordae inaequales in materia ,
crassitie etc; item ex reflexione aequabili atque uniformi
sive instrumenti, cui chorda adhaeret, sive circumstantium
corporum, lenitas soni orietur, asperitas ex opposito. 'Bo-
num erit observare quod chorda musica vehementius quam
par est distraCta stridet; quia videlicet valde percussa non
eam' servat legem quam in moderatis percussionibus obti-
net ut sub eodem tempore oscillationes suas sive majores,
sive minores dbsolvat; sed continget ut tempora oscillationum
inordinate mutentur, stridorque pro tono solito erumpat.

124. Haec notentur 

1º. chordarum vibrationes hactenus consideratae, dicuntur transversae: quae nimirum- obtinen-
tur chordam percutiendo in directione ad ejus axem perpendiculari: quod si atteratur chorda in directione ad e
jus axem parallela, adhuc sodos edet, sed , caeteris pari
bus, multo acutiores quam qui ex vibrationibus transver
sis progignuntur ; idque ex eo repetendum esse videtur
quod elasticitas propria chordae in vibrationibus longitudi
nalibus validior sit quam in transversis.

2.° Ubi in longitudinalibus vibrationibus chorda
rum obtineant &lt;u&gt;nodi&lt;/u&gt; , molus ita fiet ut partes hinc illinc cira
ca podum quemlibet positae simul ad ipsum nodum accedant, simulque alternatim recedant.

3º. Corpus omne, dum resonat, dividitur in plu
res partes vibrantes invicem ' separatas lineis , quae vocan
tur &lt;u&gt;nodales&lt;/u&gt;, quaeque oculis subjiciuntur spargendo per su
perficiem corporis minutissima arenae grana: haec enim su :
per lineis illis acervari observantur. Nodales propterea li
neae modo' sunt rectae, modo curyae, modo ex rectis si
mul et curvis coalescunt.

4.º  Malála nodalium linearum figura, plerumque mutatur et sonus; semper autem acutior vel gravior evadet sonus, prout corporis superficies in majores vel minores numero parles vibrantes dividetur ab ipsis noda libus lineis.

5.° Laminae rigidae ex ferro, vitro etc. in transversis vibrationibus absolvendis sequuntur leges alias ab illis, quas sequuntur chordae.

=== De directa soni propagatione per aerem. ===
125. Experientia nos edocet quod in iisdem circumstantiis sonus aequabili velocitate in toto decursu devehiеur; atque omnes soni , sive intensi , sive remissi , sive graves, sive acuti eadem velocitate diffunduntur. Nam 1.º Academici Florentini ad percurrendam distantiam unius milliaris sonum tormenti bellici impendisse quinque secundorum tempus experti sunt, ejusdem vero tormenti sonum ad conficiendum dimidium milliare impendisse dimidium tempus testantur aequabili nimirum velocitate perrexit sonus. Derhamus saepius repetitis experimentis idipsum invenit, adeo ut ab uno ad duodecim milliaria sumens intervalla invenerit aequale spatium aequali tempore in quavis a sonoro corpore distantia confici. 

2.º Prope sonorum corpus intensior est sonus, remissior in majore a sonoro corpore distantia atqui tam prope quam procul a sonoro corpore aequali velocitate pergit sonus ergo tam intensus, quam remissus etc. Hoc ipsum institutis ad id experimentis etiam constat Gassendus sclopeti et tormenti bellici fragorem eodem tempore pervenisse affirmat, cum eodem tempore exploderentur. Florentini et Derhamus in diversi generis tormentis idipsum evenisse notant itemque tormenti bellici minoris et mallei fragorem idem unius milliaris intervallum confecisse eodem tempore. Certum est ergo tam intensum, quam remissum etc. Huc spectat quod Derhamus quoque notat post Florentinos, scilicet eodem tempore sonum ad aures pervenire sive tormentum ad observatorem convertatur, sive ad contrariam plagam videtur enim intensior in eam partem, in quam tormentum dirigitur, esse debere sonus.

3.° In concentu sive ex instrumentorum pulsatione, si malleorum ictibus etiam ad satis notabilem distantiam dignoscitur tonorum successio eo praecise ordine, quo ictus varios tonos producentes habentur successive, et quidem sine sensibili temporis mora atqui si toni diversi non eadem propagarentur velocitate, jam qui toni successive habentur, non successive atque ordine illo ad aures venirent ergo etc. Erit fortasse qui quaerat qua ratione fieri possit ut sonus in quavis distantia, sive intensus, sive remissus, uniformiter &lt;u&gt;propagetur&lt;/u&gt;. Respondeo: eadem materiae quantitas eodem tempore, tum ex vi majore, tum ex minore, undulare potest ergo eadem aeris portio, seu &lt;u&gt;unda ejusdem latitudinis&lt;/u&gt;, eodem tempore potest undulationem perficere, sive ex majori, sive ex minori vi impellente. Antecedens est evidens; pendulum enim idem , adeoque eadem massa , eodem tempore oscillationes peragit sive magis , sive minus impellatur ad oscillandum: ergo a pari eadem aeris quantitas oscillare potest sub eodem tempore sive ex majori , sive ex minori impulsu. Sed si eadem aeris quantitas aequali tempore potest comprimi et restitui , jam eodem tempore potest sonus, ad datam distantiam pervenire , sive intensior , sive remissior: haec minor est evidens; si enim eadem est &lt;u&gt;latitudo undae&lt;/u&gt; , idemque tempus, jam eodem intervallo temporis spatium datum a sono conficietur; ergo sive intensus sit , sive remissus , seu vi majori aut minori aereae undae propellantor, eadem esse potest soni velocitas.

Quid ergo provenit ex hoc quod in sono intensiore vis major aerem impellat? Nempe quod ejusdem latitudinis unda, licet eodem tempore conficiatur , compressionem tamen ac restitutionem patiatur validiorem , vel languidiorem; sicut in pendulo accidit , quod eodem tempore oscillans ex impulsione maiori oscillationem concipit magis validam , et minus ex vi minori. Atqui hoc idem praestat minorem intensitatem , non autem minorem soni velocitatem . Ostendo: intensitas soni pendet a vi , qua in organum appellunt aeris particulae ; ergo si vi majore condensantur , et restituuntur , intensiorem efficient soni sensationem; at velocitas ex dictis pendet a latitudine undae, et tempore quo perficitur: neque latitudo immutatur , neque tempus; ergo non mutatur velocitas. Quod autem neque latitudo , neque tempus mutetur , ita probari potest. Latitudo enim undae , seu aeris quantitas ad oscillandum per modum unius determinata , ea esse debet quae potest obtemperare vibrationibus sonori corporis , a quo unda producitur , quaeque potest oscillationes suas eodem tempore complere quo sonorum corpus oscillationes suas perficit: ergo latitudo undae proportionari debet tempori quo sonorum corpus perficit vibrationes suas. Atqui sive intensior , sive remissior sit sonus, tempus quo sonorum corpus vibrationes suas complet , est ( 113. 2.°) semper idem; ergo item latitudo undae aereae eadem esse semper debet. Idem probat simul, quod sicut eadem latitudo, ita idem esse debet tempus quo unda perficitur. Et sane si tempus mutaretur , deberet quoque mutari tonus: atqui idem manet tonus in quacumque distantia a sonoro corpore , et quidem sive corpus resonet intensius , sive remissius; ergo etc.   

Hinc dum de sono agitur duplex in motu undae aereae velocitas distinguenda est: altera importat tempus quo unda conficitur , seu quo segmentum aeris datae latitudinis oscillat ; altera importat motum particularum aerearum itum et reditum perficientium in ejusdem undae efformatione. 

Quaeri hic potest in quanam ratione intensitas soni minuatur in progressu . Reponunt communiter quod intensitas soni est in ratione duplicata distantiarum inversa a centro soni : rationem afferunt , quia sonus quantum est de se aequabiliter undequaque diffunditur in modum sphaerae. Atqui ex hac aequabili in modum sphaerae diffusione sequitur decrementum in ratione praedicta ; nam si ita diffunditur , debet in ea proportione intensive decrescere , qua extensive augetur , sea qua latius materia , cui communicatur motus , sese expandit ; sed hujusmodi extensionis augmentum est in ratione duplicata distantiarum ; hanc enim rationem sequuntur sphaericae superficies : ergo etc... . 

126. Sit c velocitas , qua propagatur sonus ; &lt;math&gt;\Delta&lt;/math&gt; distantia inter vibrantem sonori corporis particulam et particulam aeream : exprimet tempus a sono impensum ad percurrendam distantiam &lt;math&gt;\Delta&lt;/math&gt; ; motusque particulae vibrantis nonnisi post tempus I = pertinget ad aeream particulam: propterea substituto 2— —c- 'a duabus ulti-
mis formulis(29. 5.&quot;), si :
∙−−≜−⋅ incipit ab 0
, ultraque
progreditur, determinabitur aereae particulae motus per»
∙ 271: A , 9 211 .A.
∦⋅⇋↙∁∘∎∐∙−⊖−⋅≺∁−−∘−−≻ ,
szC-Z-n' 008
! j(t—z).
F30i20,1t,2,3,4,-...,act—-e—:i9,tln- c
de habes A:c(t—-i9): erit
⇂↓∣∶⇂∕∁ sin 21'12:o.
Sumptis ergo distantiis Azct, c(t—G), c(t—29), c(t—BG), ...,
uulla velocitas v' ibi invenietur : aer proinde in locisi il-
lis omnino quiescet quando desinit tempus :; eritque n—
sque ad Ar.-ct in plureswundas distinctum similes et aequa-
les ; quarum communis latitudo ::09 ; numerus vero

:

∆−∘−⊖− ⋅
∆
Quantitas l/C sin −⋛≖−≺ t — A;) manet positiva ab
t — 30- :::-id ad :
2 — (i ] &amp;) 6 ; manet-negativa
A . A .
⋅ ∙ ∙ ∙
ab t—-—c- :::(12-l-ä)9 ad t— -c—-.-:(1-l-1)9. Ertt 1g1-
tur v' positiva inter A———-0(t—i9) et A −−∶ c [t—(i—i—ä— )9];
erit negativa inter Ach-t—(i—i-ä-W] et A:c[t—(i-l—1)9].
in tribus hisce distantiis est praeterea v':o. Ergo quae-
libet ex dictis undis constat duabus partibus aequalibus ;
recedit aereum fluidum ab oscillante sonori corporis par-
ticula in anteriora parte, accedit in posteriore; quiescit stra-270
tum medium ; maxima viget aerearum particularum velo
citas in medio semiundae anterioris ; maxima item in me
die semiundae posterioris.

127. Soni velocitas augetur a vento secundo, minui
tur ab adverso. Derhamus videns ab aliis affirmari nullam
mutationem afferri a ventis circa soni velocitatem , hanc
rem statuit explorare ita exacte et diu , ut ambigendi lo
cus omnis tolleretur. Ad hoc autem summa ipse fruens
opportunitate experimenta habebat omnino in promptu .
Nam cum ex arce Blancheath , ubi tyrones rei tormenta
riae exercebantur , saepe exploderentur tormenta bellica ,
ipse e sua Ecclesia in agro Upminsther ad 13 milliaria
distante flammam advertere poterat ; animadvertit autem
optimo usus chronometro non semel aut iterum , sed
triennio integro. Porro ex tabula , quam observationum
suarum confecit, quaeque habetur in Transactionibus An
glicanis, et a Masschembroekio descripta fuit in suis com
mentariis ad lentamina Florentinorum , constat quod so
ni velocitas inter tempus quo ventus favens spirabat ,
et contra venius sono adversus erat, cum scilicet in utro
que casu yentus validus admodum esset , discrepat un
decim semisecundis circiter in praedicto intervallo. Ergo
experimentis hisce insistendo dicendum augeri secundo ven
to soni velocitatem , imminui autem etc.
Derhami observationibus consentiunt observationes Aca
demicorum Parisiensium , qui anno 1738 exploraturi ve
locitatem soni jussu Regiae Academiae pariter testantur
non eandem esse adverso ac secundo vento velocitatem
qua propagatur. Rationis momentum experientiae suffra
gatur : nam ventus transfert loco aerem ; ergo undas so
noras ad oscillationem a sonoro corpore impulsas trans
fert ; ergo tantum accelerari debet propagatio soni , quan
tum aeris sonori translatio ratione venti importat.
Opporluna est comparatio circulorum in aqua exci
latorum ope lapilli decidentis : si enim aqua non sit sta
270
tum medium; maxima viget aerearnm particularum velo-
citas in medio semiundae anterioris; maxima item in me-
die semiundae posterioris.

271
SUS asserue
gnans sed fluens aequabili motu ; jam dum post lapidis
descensum circuli successive efformantur , lota ipsa aqua,
in qua efformantur circuli , localiter transfertur ; ergo
circuli appellent ad datum locum citius in eam partem
versus quam defluit aqua quam ad alteram in eadem
distantia ex opposita parte : ita paritate rationis in sono.
Iis , quae . modo diximus , objici possunt experimenta
Cl. virorum qui de velocitate soni nihil immutari pror
sive secundus sit , sive adversus ventus
runt. Gassendus enim , et Mersennus id sibi accidisse te
stantar ; et Academici Florentini , collocatis observatori
bus inter se duo milliaria distantibus , dum ventus spi
raret , asserunt tormenti bellici , quod medio illo inter
vallo situm erat , fragorem pervenisse eodem tempore ad
utrosque , etsi aliis faveret ventus , aliis esset contrarius.
At cum experimenta recte instituta aliis item recte
institutis nequeant esse contraria , yidendum quaenam
praevaleant. ' Florentini , quorum tentamen prae caeteris
in medium afferri solet , unica nocte , et in distantia pau
corum milliarium experimentum instituerunt . Derhamus
triennio experimenta iteravit , et in 13 milliarium distan
tia ; haec autem distantia in experimentis Derhami eadem
erat semper , a sua scilicet Ecclesia ad arcem ; in ten
tamine Florentinorum tormentum constitutum est medio
inter duos observatores intervallo ; quod intervallum utrin
que aequale asseritur , sed fortasse non ita esse potuit.
Ergo apparet quam recte Derhami observationes antepo
nantur observationibus Florentinorum , atque eodem jure
Gassendi et Mersenni . Reipsa etsi experimento quodam
Musschembroekius quoque deprehendere sibi visus fuerit
aequalem velocitatem tam secundo quam adverso spirante
vento , tamen Derhamo assentitur , et Florentinis quo
rum sagacitatem saepe alibi commendat , minime in hoc
adstipulatur.
Obiter hic notamus quod juxta auctores ferme omnes
etiam intensitatem sąni auget ventus secundas , et minuit
.
1
271-
gnans 'sed fluens aequabili motn; jam dum post lapidis
descensum circuli successive eil'ormantur ,
tota ipsa aqua,
in qua eB'ormantur circuli ,
localiter transfertur; ergo
circuli appellent ad datum locnm citius in eam partem
versus quam defluit aqua quam ad alteram in eadem
distantia ex opposita parte: ita paritate rationis in sono.
Iis
, quae-modo diximus, objici possunt experimenta
Cl. virorum qui de velocitate soni nihil immutari pror-
sus , sive secundus sit , sive adversus ventus
, asserue-
runt. Gassendus enim-, et Mersennus id sibi accidisse te-
stantur; et Academici Florentini, collocatis observatori-
bus inter se duo milliaria distantibus , dum ventus spi-
raret
, asserunt tormenti bellici , quod medio illo inter-
vallo situm erat
,
fragorem pervenisse eodem tempore ad
utrosque, etsi aliis faveret ventus
, aliis esset contrarius.
At cum experimenta recte instituta aliis item recte
institutis nequeant esse contraria , videndum quaenam
praevaleant.x Florentini , quorum tentamen prae caeteris
in medium afferri solet , unica nocte
.,
et in distantia pau-
corum milliarium experimentum instituerunt. Derhamus
triennio experimenta iteravit, et in 13 milliarium distan-
tia; haec autem distantia in experimentis Derhami eadem
erat semper, a sua scilicet Ecclesia ad arcem; in ten-
tamine Florentinorum tormentum constitutum est medio
inter duos observatores intervallo; quod intervallum utrin-
que aequale asseritur , sed fortasse non ita esse potuit.
Ergo apparet quam recte Derhami observationes antepo-
nantur observationibus Florentinorum , atque eodem iure
Gassendi et Mersenni . Reipsa etsi experimento quodam
Musschembroekius quoque deprehendere sibixvisus fuerit
aequalem velocitatem tam secundo quam adverso spirante
vento, tamen Derbamo assentitur , et Florentinis , quo-
rum sagacitatem saepe alibi commendat, minime in hoc
adstipulatur.
Obiter hic notamus quod iuxta auctores ferme omnes
etiam intensitatem soni auget ventus secundus
,
et minuit
.272
1
P
TE
8
ta
11
11
adversus. Hoc , ajunt , experientia vulgari notum est : si
quidem campanae sonus , aut tormenti explosi fragor multo
melius auditur si conspiret in eam partem ventus quan
si contrarius sit ; et saepe ad aliquam distantiam auditar
ope venti secundi, ad quam , cum ventus est adversas ,
minime audiri potest : auget ergo ventus soni intensita
tem. Ratio quoque idipsum suadet : nam vencus secundus
undas sonoras transfert ; ergo ad aurem appellent undae
sonorae , quae a centro minus remotae erunt , adeoque
intensiorem deyehunt sonum .

128. Ad soni velocitatem determinandam multa in
stituta sunt experimenta , quae tamen non satis conve
niunt : experimenta instituta ab Academicis Parisiensibus
anno 1738 praebuerunt soni velocitatem , seu spatium minuto
secundo a sono percursum = 172 , 56 hexap. = 336 , 32
metr. Apud Madras in India orientali D. Goldingham ex
perimentis per annum integrum multoties repetitis ( Annal.
de Plays . et de Chim . tom. 23. pag. 12 ) exploravit soni ve
locitatem : prodiit mediocris velocitas 1134 , 33 ped.
Britan . = 345 , 74 metr. Varias hujusmodi mensuras vi
dere est in tabella , quam protulere DD Moll , Van-Beek
etc. ( Bibliotheque universelle tom. 30) : qui Auctores opus
definiendae velocitatis soni susceperunt anno 1823 , perfe
ceruntque in Hollandia , assumpto ad observationes eo spa
lio , quod Zevenboompies et Koolijesberg interjacet. Ten
tamiva sumpta die 28 Junii praebuerunt soni velocitatem
339 , 34 metr.
Hujus diversitatis plures esse possunt rationes : ac 19. In
strumenti aut attentionis exquisitae ad instrumentum deſe
ctus ; cum enim flamma attendi debeat simulque penduli
oscillatio , jam facile est ut vibratio aliqua initio non nu
meretur. 2. Spatium exiguum ab aliquibus assumptum ;
minimus enim error facilius est contemaibilis , si ingens
intermediet spatium. 3.° Venti qui aut retardant , aut ac
celerant souum . llaec variationis causa attenuari potest , ac
PL
M
ti
.
0
272 '
adversus. Hoc , aiunt , experientia vulgari notum est: si-
quidem campanae sonus ,
aut tormenti explosi fragor multe
melius - auditur si couspiret in eam partem ventus quam
'si comrarius sit :
et saepe ad aliquam distantiam auditur
Ope venti secundi, ad quam, cum ventus est adversus,
minime audiri potest: auget ergo ventus soni intensita-
tem. Batio quoque idipsum suadet: nam ventus secundas
undas sonoras transfert; ergo ad aurem appellent undae
sonorae , quae a centro minus remotae erunt, adeoque
intensiorem devehunt sonum.
128. Ad soni velocitatem determinandam multa in-
stituta sunt experimenta , quae tamen non 'satis conve-
niunt : experimenta ⋅ instituta ab Academicis Parisiensibus
anno 1738 praebuerunt soni velocitatem
, seu spatium minuto
secundo a sono percursum ∙−−∶ 172 , 56 hexap. −−∶ 336, 32
metr. Apud Madras in India Orientali D. Goldingham ex-
perimentis per annum integrum multoties repetitis (Annal.
de Phys. et de Chim. tom. 23. pag. 12) exploravit soni ve-
locitatem :prodiit mediocris velocitas :: 1134
, 33 ped.
Britan. −−∙− 345 , 74 metr. Varias hujusmodi mensuras vi-
dere est* in tabella , quam protulere DD Moll
, Van-Beelt
etc. ( Bibliotheque unive'rselle tom. 30) :qui Auctores opus
definiendae velocitatis soni susceperunt anno 1823
, perfe-
ceruntque in Hollandia , assumpto ad observationes eo spa-
tio , quod Zevenboompics et Kooltjesberg interiacet. Ten-
tamiua sumpta die 28 Junii praebuerunt soni velocitatem
−∸−⇁∙ 339 , 34 metr.
Huius diversitatis plures esse possunt rationes: ac ↿∘∙ In-
strumenti aut attentionis exquisitae ad instrumentum defe-
ctus; cum enim flamma attendi debeat simulque penduli
oscillatio , iam facile est ut vibratio aliqua initio non nu-
meretur. 2.*' Spatium exiguum ab aliquibus assumptum;
minimus enim error facilius est contemnibilis , si ingens
intermediet spatium. 39 Venti qui aut retardant ,
aut ac-
celerent sonum. llaec variationis causa attenuari potest
, ac
J
.
maälzz—äwæ-EL'T-aa
&amp;.273
ferme destrui , si collocatis tormentis bellicis in utraque
extremitate A et B illius distantiae , quam debet sonus
percurrere , tormenta ipsa eodem temporis momento ex
plodantur ; tunc enim si determinetur velocitas , qua per
venit sonus ex A in B , itemque velocitas qua pervenit ex
B in A , harum velocitatum semisumma erit velocitas illa ,
qua propagaretur sonus in aere tranquillo. 4.º Animadvertit
Musschembroekius quod cum sonus non in instanti audia
tur , sed initio minus , subinde organum aliquanto vehe
mentius percellat, hinc quidam ad initium , alii ad progres
sum sensationem soni potuerunt animadvertere , atque inde
inter se discrepare. 5.9 Varia atmosphaerae temperies.
Determinatio velocitatis , qua sonus propagatur , utilis
est nautis ut agnoscant quantum littus , aut alia navis
distet ; militibus ut quantam oppugnata urbs distet ; geo
graphis item ut quantum inter duo loca , praecipue cum
intervallum hexapeda metiri non licet , intersit . Etenim nu
merando minuta secunda ab erumpente flamma ad usque
audiendum sonum tormenti bellici , distantia loci colligi
potest . Quod si ea sit locorum constitutio ut flamma videri
non possit,o res ita supplenda est , ut cum ad aurem per
venit souüs , exploso statim alio tormento bellico , alter hic
sonus ad primum observatorem perveniat : si hic nume
ravit minuta secunda ab eo puncto , quo explosit suum
tormentum usque ad punctum quo audivit sonum alterius
tormenti , et haec minuta bifariam dividantur , habebitur
tempus propagationis soni inter duo illa loca : ita etiam nu
bis distantiam aliqui metiri docent , numerando scilicet mi
nuta secunda , quae inter fulgur emicans et auditionem to
nitrus intersunt .

129.# Nonnulla subjicimus ex theoria fluidorum
( 106. 107 ) ad soni propagationem applicata ; ita tamen ,
ut ad aeris gravitatem minime attendamus , libratu mque
aerem spectemus tanquam elasticum fluidum eadem ubique
pollens et densitate p' , et pressione a' , et temperie n.
273
ferme destrui
.
si collocatis tormentis bellicis in utraque
extremitate A et B illius distantiae . quam debet sonus
percurrere ,
tormenta ipsa eodem temporis momento explodantur; tunc enim si determinetur velocitas , qua pervenit sonus ex A in B , itemque velocitas qua pervenit;:
B in A , harum velocitatum semisumma erit velocitas illa,
qua propagaretur sonus in aere tranquillo. 49 Animadvertit
Musschembroeltius quod cum sonus non in instanti audia-
tur, sed initio minus , subinde organum aliquanto vebe-
mentius percellat, hinc quidam ad initium , alii ad progres-
sum sensationem soni potuerunt animadvertere , atque inde
inter se discrepare. 5.o Varia atmosphaerae temperies.
Determinatio velocitatis
, qua sonus propagatur
, utilis
est nautis ut agnoscant quantum littus ', aut alia navis
distet: militibus ut quantum oppugnata urbs distet; geographis item ut quantum inter duo loca , praecipue cum
intervallum bexapeda metiri non licet ,
intersit. Etenim nu-
merando minuta-secunda ab erumpente flamma ad usque
audiendum sonum tormenti bellici
, distantia loci colligi
potest . Quod si ea sit locorum constitutio ut flamma videri
non possit A, res ita supplenda est
,
ut cum ad aurem per-
venit somä , exploso statim alio tormento bellico
, alter bie
sonus ad primum observatorem perveniat : si bic numeravit minuta secunda ab eo puncto
, quo explosiot suum
tormentum usque ad punctum quo audivit sonum alterius
tormenti , et haec minuta bifariam dividantur , habebitur
tempus prcpagationis soni inter duo illa loca :ita etiam nu-
bis distantiam aliqui metiri docent , numerando-scilicet minuta secunda , quae inter fulgur emicans et auditionem to-
nitrus intersunt.
1294 Nonnulla snbiicimus ex theoria fluidorum
(106 . 107) ad soni propagationem applicata ; ita tamen ,
ut ad aeris gravitatem minime attendamus
,
libratumque
aerem spectemus tanquam elasticum fluidum eadem ubique
pollens et densitate ≀⊥⋅ ,
et pressione a: ,-et tmperie n.
..—274
10 Fac ut concutiantur librati aeris particulae
comprehensae sphaerico spatiolo habente radianı = (y , et
centrum in coordinatarum origine 0 ; talem vero patiantur
in densitate variationem , et velocitatem recipiant juxta re
spondentes radios vectores a , ut utraque exsistat admodum
exigua , et altera queat repraesentari per f ( ) , altera per
f ( Q) , evanescentibus fg , f quoad a = o et « &gt; « ,: sit r
distantia puncti ( x , y , z) ab 0 , ut obtineant
i
x2 + y2 + z = p2 xdx + ydy + zdz = rdr ,
Propagato motu per reliquum fluidum ; quoniam v' , v &quot; , 20 &quot;
sunt constanter parvissimae , et ad fluidi gravitatem minime
attendimus , iccirco formulae ( 6 &quot; , 106 ) , missis terminis
exiguissimis secundi ordinis , factisque X = 0 , Y = o, Z=0,
dabunt quoad punctum ( aco y, z)
1 do dui 1 do dv &quot; 1 das
de
dv'&quot;
dt &gt;
M dx dt I dy to da
et consequenter
lo
I can do to edip dy+
dz dz
du dvi'
dy + dt dt (©) .
Jam vero
dic dir -dx do
dy
do dr
dr dx &amp; ar ፊ dydy 9
dy
dosdz dz
da dr
dr de dz 2
.
274
↿∘∙≖∎⊀ Fac' ut concutiuntur Iibrati aeris particulae
comprehensae sphaerico spatiolo habente radium −−−−≖ a, ,
'et
centrum in coordinatarum origine 0; talem vero patiantur -
iu densitate variationem , et velocitatem recipiant iuxta re-
spondentes radios vectores &amp;!
, ut utraque exsistat admodum
exigua, et altera queat repraesentari per f! (a) , altera per
f (a) , evanescentibus !; ,
f quoad ac −−∶ ∘
et a) 0:' :. sit :-
distantia puncti (æ
,y, :) ab 0, ut obtineant
x' ∙−⊢∫∙⊣−≖≏∶−−≀∙≖ , ædx-i—ydy-t-zdzzzrdr,
PrOpag'ato .motu per reliquum Huidum ; quoniam v', 11&quot;, v'&quot;
sunt constanter parvissimae ,
et ad fluidi gravitatem minime
attendimus ,
iccirco formulae (ö&quot; . 106 ), missis terminis
exiguissimis secundi ordinis
,factisque X::o, ïze, Zzo,
dabunt quoad punctum (æ, y, :)
1 da der ↿⊄∄↑≖⋅∙∙∙∙ dv&quot; 1
ftdæ— dt'pdj&quot; dt'p.
et consequenter
1 der da der − − ... −−⋍≀ )
↽−
⊬
(dxdx—i- dy d),—i- dz :.
eiu' dv&quot; dv'&quot; )
∙
—- de—k-äuy—F—ät— d: (i)-
Jam vero
(la-: (lux dr dar das dr
,
(Erit: ∙−−−−∙⊋−∣∙∙ (?;-lld? ïydj :
z; gd)»- ,
dadz—dw drdz ,275
ac proinde
du do dos
dx + dy +
dz =
dx dz 1
do Idr dr dr
• dxt
dr dy
do-dr
) = dr
v , &quot; =
insuper v ' un
v , ideoque
dv' dv &quot; dv '' d (v'da tudytou '&quot; dz) dx +''
-dyti dz =
dt dc dc dt
dfædx + ydy + zdz
d (vdr)
dc dt
traducetur igitur ( i) ad
1 do dr d ( vdr)
dt -
( i ) .
f . dr
Ponentes
dQ u'dx + u'dy + v &quot;da = dQ ,ut sint v'=
dx 10&quot; :
dQ
dy
2011 !
dQ
dz
assequimur
dQ d
To
d
Come)
vdr d (vr)
dr,
dc dr -
' de
dr dr :
dr dt
vertelur itaque ( i ) in
275
ac proinde
Heia-4- — ;d; -]-d 2; ad;:
≤↾−⋮⋅↾ ïta.-.- −∙⋅⊄∄↗∙⋅∹−∙≦− ∙⋅−⋤−↙∄⇝⇌∶−∙⋡−−↙≀≀∙ ;
'
∙−−−⋅⋮∙⋅ ∙−− £ &quot;zl. lnsuPero—rv,-v' '.—r-v,-v&quot; rc:,ideoque
dv' *d-v&quot; ...-'de: &quot;' d(v'dx -1-v&quot;dy—1-v'&quot;da)
ïdïdæ'l' dc ↙↡↗⋅⋅⊢ ⋅⊋−⋮⋅∂≖ dt ∙−−−⋅
d
(ædæ A-ydy ∙−⊢ zdz 0)

: - d(wdr) ∙↗

d: ∙
&quot;' dt
traducetur igitur (i) ad
1 du! ∙∙∙ d(vdr) .,dr ∙−−− −− dt (( ).

Ponentes
u m ∙
l
∙
'o v'dx-t-m dy—t—v ds:dQ,ut sint d
∙↗∶⋛−≣−∙⇝ :::-g. *v
⋅⋅∙−−−∶∙−↿⋚≳−∙ '
assequimur
⋅
dQ d(vr) d —&quot;'Q) d (....) — (dr ∙∙∙⋅ ⋅ dt
.
⇀
mi'-&quot;ïd&quot; d: −⇀ dt 4&quot; ∙− dr 4&quot;
vertetur itaque (t&quot;) in276
1 do Cena
( i &quot; ) .
hdr dr
Pertingente motu ad punctum (x , y , z) , crescit ibi librati
aeris densitas M , et evadit
l = h' ( 1 + $) ;
augetur aliquantulum etiam temperies n in ipso condensa
tionis actu , fitque ntv : pressio , quae ob auctam den
sitatem evaderet a' ( 1 + 8) , augescit adhuc propter incre
mentum v ; et cum v pendeat ab € , novum pressionis in
crementum pendebit rursus ab z , eritque ob incremento
rum tenuitatem ipsi &amp; ad sensum proportionale ; iccirco ,
praetermisso é , emerget pressio ex duplici capite aucta
m = (1 + 5) (1 - +-AE) w [1+ (1 + A ) £] .
Poterit ergo ( i&quot; ) sic scribi
1
ale
de de .
( 1 + A
M
17 € dr dr &gt;
seu
dt
is 13
( 1 +A)
dL ( 1 + -E)
dr dr
Hinc
Bis
( 1 - +- A ) L ( 1 + E)
dQ
dt
276
Pertingente motu ad punctum (a:, y, :)
, crescit ibi librati
aeris densitas p! ∙ et evadit
it:-&quot;a' ≺↿−⊦⋮≻⋮
augetur aliquantulum etiam temperies 1: in ipso condensa-
tionis .actu
,
(itque n—l— »: pressio , quae ob auctam den-
sitatem evaderet m' (1 ∙−⊦ e) , augescit adhuc propter incre-
mentum »; et cum 9 pendeat ab a , novum pressionis in-
crementum pendebit rursus ab a , eritque ob incremento-
rum tenuitatem ipsi a ad sensum proportionale: iccirco
,
praetermisso ?
, emerget pressio ex duplici capite aucta
a:d(1-—1—s)(1-1-Aa) −−∶ a'[1-t-(1-t-A)s] .
Poterit ergo (i&quot;) sic scribi ∙
' d(ig) a' 1 de dt −− ↿⊣⇁∆∼ ∙−− — pii '
↿⊣−∙∊ dr dr
'
seu dc.-112) . :
p.
liinc
' d
⋮⋝−∽ ≺↿⊣⇁⋀≻↧∙≺↿⊣−⊽∊≻↽−∙−−−∙− 3- - p. dt277
est autem ( 27.29º. )
?
L ( 1 + E) = E
+ - +
Propterea , facto ( 1 + A ) = C ,
A dQ
&quot; .
ca do
Ad haec :
dv ' dy&quot;
dx
dur
dz
d’Q
dx² +
d’Q
dy ?
+
d2Q
dz² ;
dy
formula igitur ( 619. 107) , substituto p. ( 1 + €) loco fe , mis
sis terminis exiguissimis secundi ordinis, atque attenta ( i'''),'''
praebebit
d2Q daQ
dea = ca
e
d Q
dy ?
det
d2Q
da ?
( it ) ;
\ dx²
et quoniam
dQ dQdr dQ y
dr dx
dQ dQ x
dQz
dr
of
dQ_dQ
dxi
dz dr dy dr
unde
d’Q
dx²
daQ xa
dra 2
dQy? +z2 d2Q
dr p3 dy?
d’Qys , dQ x2 + z3
dr ra dr p3
d'Q
d22
d2Q 22
dr.2 p2
dQ x2 +y2
dr 产 产
277
est autem (27 .290.)
e* 53 si
1 :−∙− ∙−−− Ou: ∙∙ ↥⋅≺−⊢∊≻ s ⇄⊣⋅∙∃ 4(.,, :
Proptereü , factO :
(1 :A) −∙− c,,
1 dQ
ca dt
Ad haec :
d'v' dv&quot; ⊣⇀ ↙∣⊛∣∦ ↙≀≏⊄⊋ sz dïQ .
da: d] dz −⇀⋅ dx: d),: d:&quot; a
formula igitur (ö&quot; . 107) , substituto p: (ii-145) loco p. , mis-
sis terminis exiguissimis secundi ordinis, atque attenta (zw'),
' ∙ ?' praebebit
sz sz daQ sz .&quot;
.
(.i—t;. ⇀−− c&quot; da,-3 ∙−⊦ ∠∄∫≖ .* dzg) (: la.
et quoniam ⋅
'
∙ '
dQ dQ dr-—dQæ dQ—JQJ, iq—æi
dæ' drdæ dr-r'äy dr r'dz—drr'
unde ( ⋅ ⇁ ⋅ ,
dj—æ i,*deail'zz dag—dïQlyiA-iQxa—an dat.:—dr: rr: dr &quot;3 ,dyl d'.) rg ∙ &amp; r3
dQdeina *igæ'ä-J' .
d:&quot; dr: ra dr 'a &quot;.278
ideo traducetur ( i&quot; ) ad
d’Q
dia coloro
d-Q
( dra
2 dQ
r Thedrbest seu da (rQ )
dla ca d ( )
dra
Ex (i) habemus ( 120. 6º. )
Q = -- [80+ c ) + F(r — ct)] ;
et consequenter
dQ 1
dr [ f'ir tt) + F' ( r - ct )]
) — ]
( i&quot; )
Ar + c ) + F (r - ce}]
- [f(r + c )—– F&quot;( – ce)].
1 dQ
c2 dt
Ad f et F determinandas , sume t=0 ; habebis
f(a ) f (a ) : . E =
proinde
a&gt; f( x ) = af ( a ) + aF'( a ) f « ) - F( a ) ,
- caf( a ) = f ( ) – F' ( « ) .
Pone
fa) +F(a) = w , fra ) — F( X) = w ;
erunt
.
278
ideo traducetur (i&quot; ) ad
432—
. «PQ-,. 2
sit'—c &amp;? 747↲≺≀≻∙⊷≖∂↿≺↾≬⋗−≖∙↲≖≺↗≺≀≻ de'—' dn
Ex (.&quot;) habemus 120. 60.)
- 1
Q ∙−−− ;- [f(r.:i- ct) −⊦ F(r— et)] :
et consequenter−∙∙ :? ∙−−⋮∙ [f'(r-l-ct)-]-—F'(r-—ct)]

—--—1r;-[f(r-]-ct)-]—F(Qr—ct)] ,
1 dQ— ↿
s —
⊑
ca d: ;S.-[f(r-t-cn— F'(r— cs )].
Ad f
et F determinandus, sume :::-o; habebis
w:f(a)
, s:f,(a):
proinde ⋅
æf(a):af(a)—1—al-&quot;(a) -—f( cc)—P(a),
—eaf,(a)——:f(a)—F'(a).
Pone
fe) -t-F(a) :::.)
,f(a)- F(ac) ∶−∙−∾⋅
erunt
0&quot;)279
d @ = f( ) + F&quot;(x)= f(a) —F(«) da = f( x )da ;
dw ' = [f ( ) — F ' ( ) ] da = -ca f ( ) da ;
unde
a fixdx , w ==
cfafica) da :
hae suppeditant
f(Q )
w -two
2
1
2 frazda - of facada,
F(x)= afscada + ; fafceda;
ideoque ( iº )
f(x)= ff( )fat a pascafica),
Standa+ af )+ caf,ca).
F ( a )
2
20# Secunda membra (2011) evanescunt quoad a &gt;
Az ; ut igitur functiones
flrtct) , f'(x + ct) , Fr — ct ) , F &quot; (r — ct)
sint aliquae , non debet r ct esse &gt; &amp; : atqui in ordi .
ne ad fluidi particulas ultra Qi , cum e sit quantitas posi
tiva, est semper s + ct &gt; As ; ad has ergo particulas quod
attinet, erunt constanter
279
d(a-i)— af(a)—t-aF'(ac) —f(a) --F(a) dae— f(a)da;
« - æ
dar.-:. [f(az) —- F' (et)] da :: — cat &amp; (et) da;
uude '
∙∾−−−∶∶∝ «a)daz , Q':-irc af,(a)daz:
bae— suppeditant
aH—a' 1 - 1
f(a)-— 2 ∙−− ⋣∙ ⊄∫⇟↸∝⋟↙≀∝−−−⋮−−∘∫∝ f,(a)da,
c.)—of 1 1
Hall- 2
&quot;*.2 «li(alda—r—ïcfafdaW-ï &amp;
ideoque (i'&quot;)
1 1 1
f(a): -2- f(a') fat—1- ä-a ((a)—ïm f,(a) ,
↿ 1 1
F'(a) ∶−∙−−∙ ∙⋮⋅≳−∫∫≼∝⋝↙≀∘⊢⊢ -2-af(a)-1- -2—- caf,(a).
2011 Secunda membra (im) evanescunt quoad ac)
«,.; ut igitur functiones ⇀
f(r-t-ct) , f(f-Jf- ct)
, F(r -
ct )
,F'(r
-
et)
sint aliquae , non debet r

: b et esse )a, :atqui in ordine ad fluidi particulas ultra et, , cum t sit quantitas posi- tiva, est semper :- −⊢ ct a, ; ad has ergo particulas quod attinet, erunt constanter280 fir + ct ) = 0 , f ( r + c ) = 0 ; et consequenter -F(r —c)F( r -ce ) , 6 = 1.- F &quot; (r — ce) (**** ) . 30 Aereae particulae respondentes radio vectorir non incipiunt moveri nisi quum tempus sic increvit , ut habeatur rct = ly , seu r = ctt cy : inferimus sonum propagatum iri uniformiter velocitate V ( 11 + A ) Quod spectat ad numerum A, habemus (87. 70. ) a = im [1 + a (n + v)] = im '(1 + E)[ 1-+ an + ») ] , itemque ( 10.) 5 '[1+ (1 + A )ɛ] =; if' ( 1+ an) [1 + ( 1 + A ) ]: hinc i '(1 + E)[ 1-+-ant-v) ] = iu'1 + an ) [1+ ( 1 + A )ɛ ]; ex qua eruitur av A av( 17) El 1 + an ) $ ( 1 + an ) Ponamus vase aliquo accurate obserato aerem conti neri ejusdem densitatis pé ac temperiei n cum aere exter• no; sitque h altitudo barometrica utrique communis : con . ⋀≀∙⊣∙∙∘⊔≔≖∘∙⊓≀⋅⇀⊢∝⋟∶∘⋮ et consequenter 1 1 ⇀ ↿ ∙ −⋅−−−− —F'(r-ct)-—;F( r—ct). : ⋅−−−− -—F'(r—ct)(t&quot;&quot;). r r cr 3&quot;:- Aereae particulae respondentes radio vectori r non 1nc1piunt moveri nisi quum tempus sic increvit, ut babeatur r— ct:ac, , seu ::- ct ∙⊦∙ at, :inferimus sonum prcpagatum iri uniformiter velocitate 'c: Vä- (1—1-A) (i&quot;) - Quod spectat ad numerum A, habemns (87. 70.) ∙ saiw-1-a(n-1-v)]:zp'u-u-e)[1-.-a(nM)]. itemque (10.) 6 −∙−−−⊤ w'[1—t-(1 a—A)e] :; ip'U—t— an)[1 ∙−⊦⋅ (1 ∙⊢ A)e]: bino - ⋅ ↴ 's ip'(1-1-s)[1-1-a(n-1-v)] ∶−− ≀⋅⊬⋅≺↿−⊢⊄⋯≻⊏↿⊣−≺↿−⊢∆≻∊⊐⋮ ex qua eruitur ↼ . cru-H:) av ∙−−− −∙∙ e(1-1-an) s(1-1-an) Ponamus vase aliquo accurate obserat'o aerem conti-- neri eiusdem densitatis pf ac temperiei iz cum aere exter- no; sitque !: altitudo barometrica utrique communis: con-281 1 11 cipiatur extrahi e vase aliquantulum inclusi aeris, vel qui erat inclusus aliquantulo magis comprimi , et denotet d'1 Fé) densitatem , h' altitudinem barometricam, postquam aer in tra vas ad pristinam redierit temperiem n. Tum constituta parumper communicatione cum externo aere, donec nimirum redigaturad h, mutationem quandam suscipiet lam p' ( 13) quam n; et illa quidem transformabitur in u'1 *8' ) (18&quot; ), haec autem in ny. Sed cum v' brevi evanescat, et so la n supersit quin variet MIFÉ' ) (1 # &quot; ) , mutabitur iterum h et evadet h &quot; . Alteram instituendi experimenti rationem sequuti sunt Desormes et Clement , alteram Gay - Lussac et Welter: inspiciatur sequens tabella . torie pun Desormes et Clement. n = 12 , 5 , heo” , 7665 , h - hs o ” , 01381 , 11 h - h&quot; = 0 , 003611 ; 2 &quot; hi sese restituit ad h intra tempus &lt; &lt; 5 Gay - Lussac et Welter. n = 13° , h = omom,, 757 757 ,, hh -- hh : = 0 &quot; , 0163644 , h &quot; - h = 0 , 0044409 ; q &quot; h sese restituit ad h intra tempus 6 Iam vero, depolante D densitatem hydrargyri , sunt conti erter Dgh = ip (176) (1 + an ) , 000 19 villi] 1 !' torir L, 111 )num conti- erit?' con- 281 cipiatur extrabi ei vase aliquantulum inclusi aeris,'*vel qui eratinclusus aliquantulo magis comprimi, et denotet p.'(1::1:s') densitatem, h' altitudinem barometricam, postquam aer in- tra vas ad pristinam redierit temperiem 11. Tum constituta parumper communicatione cum externo aere, donec nimirum h' redigatur ad h, mutationem quamdam suscipiet tam (if( quam ∎∶∙∶∔⋅∶∊⋅⋟ .: et illa quidem transformabitur 111 p.'(1.-.;:s')(1:£ e&quot;), haec autem in :::». Sed cum v' brevi evanescat, et so- la n supersit quin variat p.'(1:1: a' ) (1 :1: e&quot; ) , mutabitur iterum I: et evadet h&quot;. Alteram instituendi experimenti rationem sequuti sunt Desormes et Clement , alteram Gay- Lussac et Welter: inspiciatur sequens tabella. Desormes et Clement. 11:12&quot;, 5 , h:o&quot;,i7665 , 11 -h': 0&quot;, 01381 , h — h&quot;:o&quot;, 003611 ; h' sese restituit ad h intra tempus ≺∙−≣− ∙ Gay - Lussac et Welter. ' n:130, h:o'&quot; , 757 ,h'-— h:d&quot;,0163644 , h&quot; — h:o'&quot;, 0044409 ; 1&quot; h' sese restituit ad h intra tempus (—6—- . hm vero, denotante D densitatem bydrargyri , sunt Dgh':i;t'(1q:e')(1—t-an), ∎⊨∎ 'i i ! 19282 Dgh = id'l 176) ( 1 #t&quot; ) ( 1 + a nv( ) ) , $ Dgh &quot; = id'l 176 ) ( 1 + &quot;) ( 1 + an ) : hinc h &quot; = 1 &amp; € &quot; , h h &quot; 1+ anty') 1+ an = 1 + R 1 + an h &quot; 'h αν&quot;' h hh&quot; h&quot;

1 tan
ideoque }
ań
= ( 1 - an )
h hh&quot;
h &quot; 7 &quot; -h*
Substitatis valoribus ex Gay - Lussac et Welter ,
αν
€&quot; ( 1 + an)
=0, 3785020934 :
1
R
et quoniam iste numerus neque ex temperie neque ex pres
sione pendere videtur, iccirco poterit generatim assumi 1
A= 0, 3785020934 ;
sicque soni velocitas prodibit expressa per ( 94. 1 ° )
V
1 , 3785020934
to
fe
-V 1,3785020934i(1+ an) = 1009 , 614V1+ an (i&quot; ).
282
1131. −−−−⋅⋅⋅⊬∣≺↿∓⋮⋮≻ ≺↿ :::&quot; ) ( ↿
....
(a:-:») )
.
Dgh&quot;:ip.'(1q:s-' ) (1:t:€&quot;) (1 qum):
tibine
−≸⋮−⋤⋅−⋅
≕↿
∙∙⋅⊧∙≘⋅⋅ ,
∣∣⋮∙⋅ −−⋅↿−⊦⋅↿∘∙≦↾∙≔⋮∙⋓⇗≱ −−⋅↿∙−⋅⊦−∙↿−−∙⋮⋮≔−∙
zh :&quot; ∙∙∙ l:&quot; --l:' ,.4, .av' −∣∎ −∦∣⇂∙∙ ;
↙
h' 1 −⊢⋅⋯∎ &amp;
ideoque
av' h' b—h&quot;
e&quot;(1-t-an)— h&quot; h&quot;—-h'
⋅
Substitutis valoribus ex Gay-Lussac et Welter
,
av'
et quoniam iste. numerus neque ex temperieneque ex pres-
sione pendere videtur, iccirco poterit generatim assumi
sicque soni velocitas prodibit expressa per (94. 10)
I
c ∸−−−⇀ ↿∙ 3785020934 1;—
wjt—lV1, 3785020934 ⋅⋅≺↿∙⊢ an) ∙−−∶ 1009,- 614 ∣∕↿⊣⇀∘≀∎ tc&quot;)- H283

Si attendenda est quoque bygrometrica aeris constitutio, de
notante 6, pressionem libratam ab aqueo vapore , pro ui'
substituendum erit ( 96. 4º. )
1
seu
i( 1+ an)
exsistet nempe
V
11
w' il 1 to an )
1 , 3785020934 3
--8
W1
009 , 614
V
8 ã' (1+ an )
80-30 ,
(i &quot; ) .
In soni velocitatem diligentissime inquisiverunt an
no 1822 DD. Arago, Prony , Mathieu , Bouvard, Humboldt
et Gay - Lussac: distantia, ad quam observationes de cor
ruscatione flammae et fragore instituebantur in explosionibus
Lormenti bellici, ea fuit quae Monthlery et Villejuif inter
jacet ; velocitas inde deducta, seu spatium iolra 1&quot; a so
no percursum, 89 Erat autemn =15°, 9; unde
Vitan = 1 , 029 : dabit igitur formula ( it )
340metr.
103gped
.
893 metr .
337 , 432 . &gt;
Hygrometricam quoque aeris constitutionem notarunt
Auctores Cl . Sub mediocri videlicet altitudine barometrica
metr .
0 76 index hygrometri, quod vocant a capello, o
slendebat grad . 72 : in hac vero hygrometrica aeris consti
lutione, et sub temperie 15° , 9 ,pressioni , respóndet ba
metr.
rometrica aliiludo 0 00679; hinc
283
Si attendenda est quoque bygrometrica aeris constitutio, de-
notante u', pressionem libratam ab aqueo vapore , pro pf
substituendum erit (96. 40.)
exsistet nempe
' ∙
1
T
∘⋅−−− ∣∕ 1, 3785020934 &quot;'
'(
a, ↼⋅⊢ s
'&quot;&quot;- ..
8
1009 614⇂∕ afuit—13:111) (i&quot;)-
In soni velocitatem diligentissime inquisiverunt au-
no 1822 00. Arago, Prony, Mathieu, Bouvard, Humboldt
et Gay-Lussac: distantia, ad quam observationes de cor-
ruscatione dammae et fragore instituebantur-in explosionibus
tormenti bellici, ea fuit quae Montblery et Villejuif inter-
iacet :
velocitas iude deducta, seu spatium intra 1&quot; a so-
no percursum, :340'm&quot; ,89 Erat autemn:150,9; unde
l/1-t-an :1, 029: dabit igitur formula (ix)
0:1038ped' ,
893 :..- 337'&quot;&quot;' ,432.
Hygrometricam quoque aeris constitutionem natarunt
Auctores Cl. Sub mediocri videlicet altitudine barometrica
Gum. , 76 index bygrometri, quod vocant :: capella, o-
stendebat grad. 72:' m hac vero hygrometrica aeris consti-
tutione, et sub temperie 150, 9 ,pressioni a', respöndet ba-
rometrica altitudo Omm ,00679; binc284
v
80
8w' 30,
=1,002 ;
et consequenter ex (3 &quot; ) eruetur
1040ped ., 97 = 338metr .
11 .
Consensus itaque experientiam inter et expositam theo .
riam tantus invenitur , ut major profecto desiderari non
debeat in praesenti argumento : difficile admodum est in id
genus observationibus ventorum vim prorsus eludere, alias
que causas declinare quae huic consensui multipliciter no
cere possunt : mirum deinde quantum ardua res sit va
lorem A experimentis accurate determinare.

4. °* Evanescunt secunda membra (iº !! ) etiam quoad
a = o : in distantia igitur r evanescent &amp; , v statim atque,
labente tempore , eo devenitur ut sit rect = o . Quia er
go in distantia illa incipiunt , v esse aliquae quum
rct = lg, sequitur motum in distantia illa minime du
raturum ultra tempus Eaedem itaque &amp; , v evanescent
in distantia r- , statim atque incipiunt esse aliquae in di
stantia r : propterea non cientur una nisi particulae con
stituentes stratum crassiliei

5.° Velocitas v duabus ( 2.º į&quot; ) constat parti
bus , quarum altera sequitur rationem reciprocam distan
tiae a centro unde promanat sonus , altera rationem re
ciprocam duplicatam ejusdem distantiae : functiones prae
terea F, Fmanent constanter parvolae. Quia igitur im
pulsio in datum obicem facta pendet a velocitate v , pa
let , quo longius propagatur sonus , eo magis ipsum de bilitatum audiri. Quum sonus ad modicam pervenerit
distantiam, licebit secundam illam partem negligere; eritque
C
284
V
85' —1 002 ∙
∂≖≖⋅ ∙− 30, ' '
et consequenter ex (2&quot;) eruetur
ed- . (2:10le ,
97:agam ∙ ↿↿∙
Consensus itaque experientiam inter et expositam tbeo-
riam tantus invenitur ,
ut maior profecto desiderari non
debeat in praesenti argumento: difficile admodum est in id
genus observationibus ventorum vim prorsus eludere, alias-
que causas declinare quae huic consensui multipliciter uo-
cere possunt : mirum deinde- quantum ardua res sit va-
lorem A experimentis accurate determinare.

:

4.0a Evanescunt secunda membra (t'&quot;') etiam quoadin distantia igitur :- evanescent :, v statim atque.

labente tempore , eo devenitur ut sit r—ct—o. Quia er-
go in distantia illa incipiunt :, 9
esse aliquae quam
r—ct:a,, sequitur motum in distantia illa minime du-
raturum ultra a: . tempus −⋮ . Eaedem ltaque :, v evanescent
in distantia r—a, statim atque incipiunt esse aliquae in di-
stantia :- : pr0pterea non cientur una nisi particulae con-
st1tuentes stratum crassitiei a,.
5-0 Velocitas v duabus (2..) i&quot;&quot;) constat parti-
bus , quarum altera sequitur rationem reciprocam distan-
tiae a centro nnde promanat sonus , altera rationem re-
ciprocam duplicatam eiusdem distantiae : functiones prae-
terea F, F' manent constanter parvulae. Quia igitur im-
pulsio in datum obicem facta pendet a velocitate P. P'-
tet, quo longius .prOpagatur sonus , eo magis ipsum de-
bilitatum audiri. Quum sonus ad modicam pervenerit
distantiam, licebit secundam illam partem negligere; eritque285
V =CES F ( r - ct) .
Inferimus illud : si impulsio in obicem facta quadrato ve.
locitatis v sumitur proportioualis , rationem duplicatam di
stantiarum sequetur soni debilitatio ( 125 ) .

6.°* Fac ut librati aeris particulae concutiantur
una circum plura puncta O , 0 &quot; , ... ; quorum distan
tiae ab ( x , y, z ) exhibeantur per r' , o&quot; .... ; ipsis.
que O' , 0 ' , ... , tanquam originibu's respondeant sua
axium systemata parallela systemati habeati originem O.
Quoniam novae coordinatae s ', x &quot;, ...5,0 &quot; , ... é , z &quot; ..
constantibus quantitatibus differunt ab x, y, z ; ideo
dr ' dr dr &quot; ar 
dx doc ' F ' da d.x &quot;
!
y' g &quot;
dr'
dy
&gt;
dy '
p&quot;
dr
dy &quot;
dr
dz&quot;
dr dr
dy
dr'i
dz
2
dz dz'
el consequenter
dQ _dQ dr dQdr&quot;
tar
dx
+ .. dx dr' dx
dQ x'
dQ y
dr +
dQxt&quot;
dr'' gli t....''
dQ_ dQ y
+
dy dr p ' dr to.
dQ
dz
dQ á
dr '
+
dQ
di &quot;
.
. ilemque
to
d²Q x 2 dQ 7/ 2+22
dx² dr'a g'a + +
285
1 '
r
inferimus illud : si impulsio in obicem facta quadrato ve-
locitatis v sumitur proportionalis, ratiunem duplicatam di-
stantiarum sequetur soni debilitatio (125).
691» Fac ut librati aeris particulae concutientur
una circum 'plura puncta O', 0&quot;, ... ; quorum distan-
tiae .ab (æ, y, :) exbibeantur per r'
, r&quot; . ...; ipsis-
que O'
, O&quot;, ,
tanquam originibus respondeant sua
axium systemata parallela systemati habenti originem 0.
Quoniam novae coordinatae se', a:&quot;,
...y' ,y&quot;, ... z', :&quot;
..
constantibus quantitatibus differunt ab a:, y, :;
ideo
dr' dr' æ' dr&quot; ∙∙∙ dr&quot; ∙∙∙⋅ æ&quot; ⇀
dx daf—r dx' dr&quot; ≀⋅∎⋅↬⋅⋅⋅
et consequenter
dQ 'der- −⊦↙≀≺≀∂∙↾∙∙⋅ du:- dr'dæ dr&quot; da:
dQ æ' dQ æ&quot; &quot;
dQ ∙−−↙≀≺⊇∙⊺ d.QJ
&quot;?;/7 21.-717 −⊢∙∙ 4&quot;?!— −−⊣∎∎∙∙
': &quot;dy— dr'r'
.dQ —dQ f:: dQ ∙⋮↾∙⋅ ∙−⊦ '
dz —dr' l&quot;-1 dr&quot; r&quot; ⋅ ..
itemque
'PQ −− ↨≖≬x&quot; dQ ∟∣≖⊣−≖∙∙∣∷ .
⋅
'
dæ' dr'3 :&quot;2 −⊦⋅−−(Ti—' −−⋅∣⋅∙−⋅↾⊰ ..,-286
daQ x2 dQ &quot; 272&quot;
+
dr''2 p &quot; 2 dri p/13 + ... ,''
da d’Qy'a
dya drar'a tari
dQ x2+22 +
p3
d'Q.7 &quot;?, dQ x&quot; : + z'2 + ti. dr&quot; ' a p &quot; 2
lo:
dri
d2Q
ddza
daQ z'2
dr2 p'2
dQ x's + y'2 dQ 242
dr' 3
+
dril2 pll2 +
dQ x2+ y'a
ti ..
Adhibitis substitutionibus in ( i ' ', 1.0 ) ,
d'Q
de2
( d - Q
= c2 Adr'a +
2 dQ d2Q 2 dQ
z dr + dra +pdr&quot; + ... )

;

ex cujus forma intelligimus fore
Q = [filr'tou + F (r — ct)]+
[far&quot; +41++ F.(r&quot; ct) ] + . ( * &quot; ).
Nunc facile stabilitur illud : in hypothesi plurium concus
sionum simultanearum , ubi eae ad punctum ( x , y , z )
eodem temporis momento una perlingant , numerus e ni
hil erit aliud nisi summa consimilium numerorum re
spondentium iisdem concussionibus seorsum spectatis ; si
quidem ( 1.0 ) .
286
(PQ ∙⋅⇂⋅∥∙ ↿ dQ
dr&quot;: ∙↗≀∦≖∙⊦≖∥≖⊹ r'&quot; ∣
dr&quot; r&quot;3 ⋅⋅
'
'
- «PQ— d'Q 7&quot; ∙⊦↙∄≺≀∙−−−−∙−−−−−⊦⋅ æ'2-l-z'2
d]:—
d'Q )&quot; dQ æ'ä-l—z&quot;; .
dr&quot; ≀⋅∥⋮⊹↲≀⋅∙∣∣ r' '34- ⊣− ⋅⋅. '
d-Q Adeo ∷⋅≖ ∣dQ ⊴↾∶∣≖−⊦∜∣⋅ æno z.&quot;:
dza 'di'/3 r'&quot;
'
dr' r'3 dr&quot;3 r&quot;'
dQ ∙⋅≖∥≖⊣−∜∥≕ ↿
dr&quot; rl'3. 'l .
.. ∙ ∙
Adbibitis substitutionibus in (i&quot;. 1.&quot;).
duo (PQ 2 dQ 2 dQ ∙
∙−−− ∘≺↙↙↾∣≏−⊣−≀⋅∣ ∡≔∣∙⊦≤∶−−⊽− ⋖⋮≀≕−⊽∣−⊋−↾⊽⊣−⋯≻
ex cuius forma intelligimus fore
Q ∶−∎⋅
⋅↗⋮⊤⋅∐⋩≖≺⋅∦⊣⊸⊩⊢−∶⋮∙− ∇≖≪↗⋅⊣⊸≀⊢⊢ F,(r'—-ct)]-l—
F.(r⋅⋅⋯ ]-l—-
Nunc facile stabilitur illud :, in hypothesi plurium concus-
sionnm simultanearum , ubi eae ad punctum. (a: ,
J
,

: )

eodem temporis momento una pertingant, numerus :ni-
hil erit aliud nisi summa consimilium numerorum re-
spondentium iisdem concussionibus seorsum spectatis; si-
quidem (1.&quot;).287
DP zo al
- F&quot;
['r( + ce)-F'(x'ct)
[facr &quot;+ c8)— F'xr&quot; —cr) ] - ...
Insuper
DP
dQx' dQx&quot;
+ t . dx dr ' dr&quot; r &quot; + ...
G [r« tch+F&quot; ret) ]– 16 +6 +
F.( c )]) + ( - &quot;+e +F',(==ci)] –
[for&quot;tor)+F60—60)) + ....
vº dQ
dy
dQ r' dl go &quot;
+ + .. dr ' r ' + dr &quot; r &quot;
G - triktet)tF'(x - ce )]= i [ fim'tot +
Fa(r = -1)]) + ( -186 *408)+ F&quot;(&quot; –ce)] -
wraca&quot; terhFall -ct)]) + .
287
⋅⇌⊐ ∙−−≕↿−∙⋅ ?,?&quot;-- --',..'[f .
(r -!-c:)—F'.(r -—-c:)1—-
⋅≺∽⋅−∎↿⋅⊤∶∁↿⋮⊅⋍∣⋅∥⊹⋯∙− ↧⋅⋅∣∙≺≀∙∙⋅∙⊳∙−∙∘≀≻ ]— ....
lnsuper
&quot;- «me ⋅≄⋅ .... −−∶ .. ↼−−⋍⋜⊑⋅∙−−⋅∡⋰−∙⋅⊤−⊢⊿−≀⋅−∙⇉↗⊷ −⊦ ∙
⋅⋅
⋅
&quot;
, 1! ' '
⋅∎ ≺∎≙∶∎↾⋅⊀∎≺∣⋅⋅−∣∎⊸∘⊣−∏⋅∎ (' -—0t )]-— ∙≀−∙⋅∙−∙−∣⋅∫∎≼∣∙∎∙⊦⊸↥⊢∣−
,
' ⋅ æ' .
⋅ ↿ ⋅
⋅ ⋅
'
∙
Fl(r &quot;'&quot;'ct)
&quot;
])&quot;T'f'l'(—: [f,(r'Lï-CO—l-F', ∎⋅ ( r—ct )] ∙∙∙⋅
r
∙
∙≖−⋅≟⊑ ⊏∣≖≺↗⋅⋅−∣⊸≀≻⊣−↿⋮⋅≖≺⋅∙⋅⋅∙−−∝≻∃ )£f.-.- −⊦ ∙
⋅∙∙
. ∂≺≀∙∙−⋅∠≀≖≀∜⋅
lu.—dy dr' r' ä-l-dr&quot;— r&quot;∶∣⋅−⊦∎⋅ .. 1
(£.- ⊏⊀⋅≺≀⊤∙∙⇀⊸⋅≀⊢⊢≖⋅⋅∙⋅≺ r'-ct )1: ;; [f.(f-l-cu-l-
F.(r'— et)] ≻∙∜−⋮−⊦r ≺↿⊤↕∣∣≖≺↗⊓⊣⊸⋍⋝⊣−⇂⋅⇁⋅⋅≺∣∙⊷∙−∘≀∏ :- - ⋅
⋝∙≟≟∁∣≖≼↾∣⋅⊹≕⊢∣⋅⇁≖≺∙⋅≀∙∙⊸∁∏ ⋟∑≖∙⊤⋅⋮− −⊦∢ ∙ ∙ ∙288
dQ
dz
dQz
dr
dQz&quot;
t .
dr '' r &quot;''
( -fr.( tre)+F ;(r = cr)] - Pfalriteest
Fu
F.(x*—- )]) + ( far&quot;+c)+ F &quot;–ce)] -
pen
na[ far tcent Ffrº-cr)]) + ...;
UI
De
go
inferimus velocitatem v debitam simultaneis concussioni
bus circum 0,0 eodem temporis momento ad
punctum ( x , y , z ) una pertingentibus nihil fore aliud
nisi resultantem ex velocitatibus , quae debentur iisdem
concussionibus seorsim spectatis : atque hinc facile intel
ligimus cur, pluribus corporibus simul resonantibus , inter
oscillationes in aere excitalas non habeatur confusio ,
omnesque diversi soni inde orti ad aures distincte per
veniant. Huc spectat principium de superpositione exiguo
rum motuum.
7.04 Redeuntes ad unicam concussionem in 0 ,
ponamus aerem contineri tubo cylindrico , cujus axis ox,
motumque particularum esse ipsi OX parallelum : erunt
v&quot; = 0, v&quot; = 0; propterea formula ( i&quot; ) evadet
d2Q daQ
de unde Q = f ( x + .ct) + F ( x - ct ) ; ረder2 1
et consequenter
288
... dQ- JQ :
&quot;'l-(,Q' z'—dz −−↲≀⋅⋅ r' dr&quot; r'

ll ≼⋅≟≑⊏∣∣∙≺↗⋅⊣⇥≻⊣−≖⋅⇁⋅≖≺↗⋅⋅−⋅∘≀∏ ∙−⋅⋅⋮−⋅⋮⋅⇆⋅⋅⊔≖⋅∊⋅↾⋅⊣↽⊸≀⊢⊦ ∙
,
⋅ mo*—cn] )f— ⊣−≺⊽⊏ ↑∼≖≼↗⋅∙−⊦∘↥≻−⊦ ≖∸⋅∙≖≺↗∙∙−∘≀∏ −
⋮∙−⊦∘≀≻⊹↧⊸⇁≖≺≀∙↝−−∘≀∏⋟−⋮⊽ −↿− ∙ ∙
⋅
inferimus velocitatem v debitam simultaneis concussioni-
bus circum O'. 0&quot;
, ... eodem temporis momento ad
punctum ( x . y ,

: ) una pertingentibus nihil fore aliud nisi resultantem ex velocitatibus , quae debentur iisdem concussionibus seorsim spectatis: atque hinc facile intel- ligimus cur, pluribus corporibus simul resonantibus . inter oscillationes in aere excitatas non babeatur confusio. omnesque diversi soni inde orti ad aures distincte per- veniant. Huc spectat principium de superpositione exiguo- rum motuum. 7.&quot; Redeuntes ad unicam concussionem in 0, ponamus aerem contineri tubo cylindrico, cuius axis OX. motumque particularum esse ipsi OX parallelum: erunt 0' '::--0. 0&quot; ':o; propterea formula (i&quot;) evadet ⋅ 32? —-c £?. unde Q—−−⋅∣↗≼∶∁−∔⊸⇂⋟ -l-F(x—-ct); et consequenter289 dQ 1 dQ dx = pilatot) + F '( x - 1), E = - ca do [fotot) – F (x – ct )] . Functiones f et F absque ulla difficultate determinantur: sunt enim ( 1.9). f( x) = f(@ + F'(Q ), - cf:(Q ) = f (a) F '( ) ; ideoque f'( X) = f (Q )-cfi(Q ) 2 f(@ t-of ( ) F (a ) = 2 Ultimae ac penultimae aequationis secunda membra eva nescisnt statim ac a fil &gt;Oto : erit itaque f ( t ) = 0 quoad -aereas particulas ultra azi proinde quoad ejusmodi particulas F ' ( x-ct ) . Hinc sequitur souum adhuc ( 3. ) propagatum iri unifor miter velocitate се V 11 + 4 ). • De reflexa soni propagatione per aerem .
: 130. Cam in directa propagatione sonoras aer offen dit obicem aptum, reflectitur; hinc echo ( 115 ) progignitur; assertio sic probatur . Constat quod corpus in motu positum , si in obstacu lum incidit , quod elasticum sit , vel durum , et corpus ipsum ⋅ 289 v −∸−≖ B:] ')(æ-l-ct -l-F'(æ—ct), a:.— — ∙−∙−−−⋅⋅∶ ⋅↿ ∙

: - [f(ar-l—ct)

-
F'(x—-ct)] .
Functiones f
et P absque ulla didicu-l'tate determinantur:
sunt enim (1.&quot;). ⋅
⊞≀∝≻−−−↿≺⊄⊢⊦⋮⇁≀≺∝≻∙ ∙− cf.(a)-—:f(a1—F'(a) :
ideoque f(ao ⇌≖
aa)-zcnm) ∙ Ha): Karl-faa) ∙
Ultimae ac penultimae aequationis secunda membra eva-
nescunt statim ac « Et )a.. : erit itaque fur-H:):o
quoad aereas particulas ultra «.'
,proinde quoad eiusmodi
particulas ⋅-cs :: F' (a:—ct ).

Hinc sequitur sonum adbuc (3.&quot;) prcpagatum iri unifor-
miter velocitate
C::V-z-I—(i-I—A). ' De reflexa soni prcpagatione per aerem ∙⋅

130. Cum in directa propagatione sonoras aer oü'en-
dit obicem aptum, reflectitur: binc echo (115) progiguitur: assertio sic probatur.
Constat quod corpus in motu positum, si in obstacu-
lum incidit ,quod elasticum sit
, vel durum, et corpus ipsum290
impingens elasticilate gaudet , debet molus directionem mu
tare ac reflecti : ergo aer , elasticus cum sit , ubi in ob
staculum offendit , quod vel elasticum sit , vel certe non
molle , reflecti debet ; undae videlicet aereae, quae ex so
noro corpore progignantur ac propagantur directe , debent
obicem offendendo regredi , sonumque reflexum progignere .
Exemplo circulorum in aqua ex injecto lapide excitatorum
res oculis subjicitur : circuli enim isti ubi ad ripam appel
lunt , reflectuntur inde eo ordine , quo appulerunt . Aliter
sic: ejusdem naturae est 'echo cum sono ipso directo ; 0
btinet enim utrinque sonus eodem generatim tono, iisdem
que affectionibus praeditus ; ergo echo gigni debet eodem
modo quo sonus directas : atqui hic per undas aereas suc
cessive a sonori corporis motu genitas procreatur; ergo per
similes undas etc. Hinc in aperta planitie, ubi nullas est
obex, sono directo minime Echo respondet. Gohaeret do
ctrina com Echo phoenomenis. Nam 

1° redit reflexa vox
duplo temporis intervallo: ab experientia doctus sum , in
quit Derhamus, Echo redire duplo intervallo, quo vox pri
maria ad objectum phonocanticum pertingebat; scilicet tem
pus requiritur ut ad obicem vox primaria deveniat, et rur
sum tantumdem temporis exigitur ut reflexa ab obice redeat ad loquentem . 

2º. Remissior plerumque est Echo quam
vox directa audiri soleat; aliquando tamen intensius reso
nat Echo quam sonus directus audiatur. Ralio primi est :
cum soni intensitas decrescat pro aucta distuntia a sonoro
corpore, jam decrescit sonus ad obicem pergens; inde autem
regrediens , et novas undas progignens, iterum decrescere
debet intensitas: ratio secundi, quia si obstaculum concavum
sit, plures colligere poterit radios phonicos , quos unitos si
mul in uno loco regerat. 

3º. Aliquando ( 115 ) seinel vox refle
ctitur, aliquando saepius: prima dicitur Echo monophona,
altera polyphona. Si enim obstaculum unicum sit , jam nonnisi
semel potest vocem remittere; contra saepius remittitur dupli
ci ex causa. Prima est cum iu variis distantiis plura habentar
290
impingens elasticitate gaudet , debet motus directionem mu-
tare ac reflecti :ergo aer , elasticus cum sit ∙ ubi in ob-
staculum offendit
, quod vel elasticum sit, vel certe non
molle
,
reflecti debet; undae videlicet aereae, quae ex so-
noro corpore prOgignuntnr ac propagantur directe , debent
obicem oll'endendo regredi , sonumque reflexum progignere .
Exemplo circulorum in aqua ex iniecto lapide excitatorum
res oculis subiicitur: circuli enim isti ubi ad ripam appel—
lunt , reflectuntur inde eo ordine , quo appulerunt . Aliter
sic: eiusdem naturae est 'echo cum sono ipso directo: o-
btinet enim utrinque sonus eodem generatim tono, iisdem-
que affectionibus praeditus; ergo echo gigni debet eodem
modo quo sonus directus: atqui hic per undas aereas suc-
cessive a sonori corporis motu genitas procreatur; ergo per
similes undas etc. Hinc in aperta planitie, ubi nullus est
obex, sono directo minime Echo respondet. Cobaeret do-
ctrina cum Echo phoeuomenis. 

Nam ↿∘ redit reflexa vox
duplo temporis intervallo: ab experientia doctus sum ,
in-
quit Derhamus, Echo redire duplo intervallo, quo vox pri-
maria ad obiectum phonocanticum pertingebat; scilicet tcm-
pus requiritur ut.ad obicem vox primaria deveniat, et rur-
sum tantumdem temporis exigitur ut reflexa ab obice re-
deat ad loquentem.- 20. Remissior plerumque est Echo quam
vox directa audiri soleat; aliquando tamen intensius reso-
nat Ecbo quam sonus directus audiatur. Ratio primi est :
cum soni intensitas decrescat pro aucta distantia a sonoro
corpore, iam decrescit sonus ad- obicem pergens: inde autem
regrediens, et novas undas prOgignens, iterum decrescere
debet intensitas: ratio secundi, quia si obstaculum concavum
sit, plures colligere poterit radios phonicos, quos unitos si-
mul in uno loco regerat. 3&quot;. Aliquando (1 15) semel vox refle-
ctitur,aliquando saepius: prima dicitur Echo monopbona,
altera polyphona. Si enim obstaculum unicum sit. iam nonnisi
semel potest vocem remittere; contra saepius remittitur dupli-
ci ex causa. Prima ut cum iu variis distantiis plura habentur291
obstacula: altera causa est, cum duo sunt obices e regione col
locati; vox enim ex uno reflexa in alterum incidit, atque ex
hoc iterum reflexa iucidit in primum, et ita porro. Apud
veteres memoratur Olympiae porticus ; quam eplaphonam
dicebant, quod septies eamdem vocem redderet , ut tradit
Plinius. Prope Mediolanum celebris est Echo in palatio
Simonetta, in quo ope duorum parietum parallelorum fra
gor minoris fistulae bellicae vicies, et aliquando tricies re
petitur teste Schoto. 40. Echo saepius unam tantum syl
labam, aliquando plures refert: echo monosyllaba prima, et
polysyllaba altera dicitur: habentur loca , ex quibus integer
versus hexameter repetitur. Ea nempe est obicis ( 115) di
stantia, ut sonus reflexus primarum syllabarum tunc demum
ad aures regrediendo perveniat quando vocis directae im
pressio jam desinit; ac tunc sonus primae syllabae, qui op:
portune regreditur jam expleto versu , poterit esse sepsi
bilis, itemque aliarum successive. 5º. Echo redditur ali
quando a silyis ; imo etiam a campis sulco exasperatis et a
planitie cespitibus ac virgultis inspersa reverberalur vox ;
reverberari autem a sulcis ac cespitibus animadvertit Kir
cherus , quia quando sulci eversi, ac virgulta praecisa fue
runt Echo nulla reddebatur: talis nempe esse potest irre
gularis partium reflectentium dispositio, ut etiamsi plures ra
dii phonici dispergantur, non pauci tamen in eumdem lo
cum collineent.

131. Reflexio soni fil ad angulos incidentiae et refle
xionis aequales : quod sic explicamus . Sit AB ( Fig. 60. ) fir
ma , planaque superficies ; KCK' recta perpendicularis su
perficiei AB ; K centrum sonorum , ex quo propagantur
sphaericae undae CDD', C'EE', etc... appellentes ad AB in C,
C'ete ... Fiet soni reflexio in C, C , ..; ethabitis C , C ... pro noris
secundariarum undarum centris , ipsae secundariae undae
remittentur cum eadem principalis undae velocitate. Pro
grediente unda principali ab CDD' usque ad BB ' , unda
manans ex C progredietur ab C usque ad Q ; repraesenta
291 obstacula: altera-causa est, cum duo sunt obices e regione col-
locati; vox enim ex uno reflexa in alterum incidit, atque ex
hoc iterum reflexa incidit in primum, et ita porro. Apud
veteres memoratur Olympiae porticus; quam eptaphonam
dicebant, quod septies eamdem vocem redderet, ut tradit
Plinius. PrOpe Mediolanum celebris est Echo in palatio
Simonetta, in quo ope duorum parietum parallelorum fra-
gor minoris fistulae bellicae vicies, et aliquando tricies re-
petitur teste Scboto. 40. Echo saepius unam tantum syl-
labam, aliquando plures refert: echo monosyllaba prima, et
polysyllaba altera dicitur: habentur loca, ex quibus integer
versus hexameter repetitur. Ea nempe est obicis (115) di-
stantia, nt sonus reflexus primarum syllabarum tunc demum
ad aures regredieodo perveniat quando vocis directae im-
pressio- iam desinit; ac tunc sonus primae syllabae, qui op-
portune regreditur iam expleto versu, poterit esse sensi-
bilis, itemque aliarum successive. 50. Echo redditur ali-
quando &amp;
silvis; iuno etiam a campis sulco exasperatis et a
planitie cespitibus ac virgultis inspersa reverberatur vox ;
reverberari autem a sulcis ac cespitibus animadvertit Kir-
cberus. quia quando sulci eversi, ac virgulta praecisa fue-
runt Echo nulla reddebatur: talis nempe esse potest irre-
gularis partium reflectendum dispositio, ut etiamsi plures ra-
dii phonici dispergentur, non pauci tamen in eumdem lo-
cum collineent.

131. Beflexio soni fit ad angulos incidentiae et refle-
xionis aequales: quod sic explicamus .Sit AB (Fig. 60.) Gr-
ma, plenaque superficies; KCK' recta perpendicularis su-
perficiei AB ; K centrum sonorum , ex quo propagantur
sphaericae undae CDD', C'EE', etc... appellentes ad AB in C,
B' etc... Fiet soni reflexio in C, C',..; et habitis C,C'... pro novis
secundariarum undarum centris, ipsae secundariae undae
remittentur cum eadem principalis undae velocitate. Pro-x grcdieute unda principali ab CDD' usque ad BB' , unda
manans ex C prOgredietur ab C usqæ ad Q; repraesenta-292
biturque hemisphaerio , cujas semidiameter CQ = D'B ' :
item progrediente unda principali ab C'EE usque ad BB” ,
unda manans ex C' progredietur ab C usque ad C &quot; : re
praesentabiturque hemisphaerio , cujus semidiameter CC&quot;
E'B' ; alque ita porro. Inferimus , si concipitur superficies
curva AQC &quot; B tangens omnia haec hemisphaeria in Q, C &quot; ....,
in ea fore puncta illa , quae a secundariis andis reflexis
attinguntur eodem instanti , quaeque tunc incipient concuti
quum principalis unda pervenerit ad BB' ; exhibebit nimi
rum AQC&quot;B superficiem undae reflexae; et quoniam productis
infra AB superficiebus BB' , Qa, C'a ' , ..., recta KA' exsistit per:
pendicularis ad primam et secundam, recta KH ad primam et
tertiam , etc ... ; ac proinde sphaerica superficies B'BA'A tan
git sphaericas superficies QaA' , C'a'H , . ; sequitur super
ficiem AQB undae reflexae fore sphaericam , ejusque cen
trum in K , et semidiametrum K'Q = KA' . Jamvero quem
admodum auris collocata v. gr. in C deprehendit sonum
directum venire juxta KC' perpendicularem undae incidenti ,
sic auris in C' deprehendet sonum reflexum venire juxta
K'C &quot; perpendicularem updae reflexae ; et cum , ob mutuum
sphaericarum superficierum AQB , C'a'H contactum , recta
K'C &quot; transeat per C' ; cumque , ob latus KC = K'C , et
latus CC commune , triangula rectangula KCC , KCC' dent
angulum KCC aequalem angulo K'C'C , erit angulus KCC
angulo C &quot; CB ; ideoque angulus incidentiae aequalis an
gulo reflexionis .
Sit nunc firma curvilineaque superficies AB ( Fig 61. ) ,
in quam incidant undae CE , HE&quot; , ... BB' propagatae ex
centro sonoro K ; si centris C , H , ... describuntur sphae
rae , quarum semidiametri ( KB-KC' ) , ( KB - KH ) , ... ,
aereae particulae sitae in superficie BD tangente sphaeras
istas incipient simul affici motu reflexo stalia ab adventu
undac ex K in B. Erit igitur BD superficies undae refle
xae : quam superficiem pon esse sphaericam nemo est qui
non videat. Fac ut puncta C , H sint inter se infinite vi
292
biturqne hemisphaerio , cuins semidiameter CQ ∶⋅−⋅ D'B' :
itcm progrediente unda principali ab C'EE' usque ad BB',
unda manans ex 0 progredietur ab C' usque ad C&quot;; re-
praesentabitnrque hemisphaerio
.cnius semidiameter C'C' ::
E'B' ; atque ita porro. Inferimus ,si concipitur superficies
curva AQC&quot;B tangens omnia haec hemisphaeria in Q, C&quot;,...,
in ea fore puncta illa , quae a secundariis undis reflexis
attinguntur eodem instanti , quaeque tunc incipient concuti -
quum principalis unda pervenerit ad BB' ;exhibebit nimi-
rum AQC&quot;B superficiem undae reflexae; et quoniam productis
infra AB superficiebus BB',Qa, C'a',..., recta KA' exsistit per-
pendicularis ad primam et secundam, recta KH ad primam et
tertiam
,
etc... ; ac proinde sphaerica superficies B'BA'A tan-
git sphaericas superficies QaA', C&quot;a'H .∙∙∙ ;sequitur super-
ficiem AQB undae reflexae fore sphaericam, eiusque cen-
trum in K', et semidiametrum K'Q ∶⋅−∙⋅ KA'. Iamvero qnem-
admodum auris collocata v. gr. in C' deprehendit sonum
directum venire iuxta KC' perpendicularem nudae incidenti ,
sic auris in C&quot; deprehendet sonum reflexum venire iuxta
K'C&quot; perpendicularem undae reflexae ; et cum , ob mutuum
sphaericarum superficierum AQB , C&quot;a'H contactum
,
recta
K'C&quot; transeat per C'; cumque , ob latus KC −∙−∸− K'C , et
latus CC' commune ,
triangula rectangula KCC', K'CC' dent
angulum KC'C aequalem angulO'K'C'C , erit angulus KC'C

: angulo C&quot;CB; ideoque angulus incidentiae aequalis angulo reflexionis . Sit nunc firma curvilineaqne superficies AB (Fig GI.), in quam incidant undae C'E' , HE&quot; ,... BB' propagatae ex centro sonoro K; si centris C' , H , ... describantur sphae- rae , quarum semidiametri ( KB—KC') , (KB—KH) , , aereae particulae sitae in superficie BD tangente sphaeras istas incipient simul affici motu reflexo statim ab adventu undae ex K in B. Erit igitur BD superficies undae refle- xae : quam superficiem non esse Sphaericam nemo est qui non videat. Fac ut puncta C' , H sint inter se infinite vi-293 cina , sintque C'C &quot; , HQ normales ad BD : ex H ductis per pendiculis Ha , Ha' in KC , C'C &quot; , erit Ca ' = CC &quot;—HQ = KB - KC ) - (KB - KH ) = KH - KC = Ca. Quoniam igitur triangula rectangula Cal , Ca'H habent latera aequalia C'a , C'a ', latusque C'H commuue , habebunt ae quales angulos ac'h , a'C'H : hinc sequitur , etsi unda re flexa non est sphaerica , adhuc tamen reflexionis angulum fore aequalem angulo incidentiae.

132. * Haec deducimus ex ( 129) in ordine ad aereum
fluidum concussum in K ( Fig. 60 ) , planoque fixo AB ter
minatum. 1 °* Sumpta x in KC normaliter ad AB, peribit
apud AB tota componens v' ; erit nempe ( 129. 10. )
dQ
dxdo O ( a )
quoad x = KC ( = h ). ProducaturKC donec KC = KC;
radius vector r' computetur ab K' ; et x ab eodem K' in
K'C ; explebitur (a) per
Q = --[Pr + c ) + F(ra) ] +
[fri + ce ) + F(x – ċe)]
( a ) ;
siquidem quoad puncta sita in AB
dQ dQ r=r' ,
dr x = h , it's - h ,
dr
dris
dx dx
Determinatis praeterea f et F ex ( i&quot; &quot; . 129. 10. ) , re
praesentabit ( a' ) initialem fluidi statum: quoniain igitur ( a' )
a— 293
cina , sintque C',C&quot; HQ normales ad BD: ex H ductis per-
pendiculis Ha ,
[in' in KC', 0C.&quot; , erit
∁≮≖⇌∁⋅∙∁ ∙∶−⇀−∐≺≀ (KB'—-KC';-(KB-—Kll)—-KH-—KC':C a.
Quoniam igitur triangula rectangulaC aH, C:: 'H habent latera
aequalia C'a
, C'a', latusque C'H commune , habebant ae-
quales angulos aC',H a'C' H hinc sequitur , etsi unda re-
flexa non est sphaerica , adhuc tamen reflexionis angulum
fore aequalem angulo incidentiae.
1324: Haec deducimus ex (129) in ordine ad aereum
fluidum concussam in K (Fig. 60), planoque fixo AB ter-
minutum. ↿∘∙ Sumpta æ in KC normaliter ad AB, peribit
apud AB tota componens v'; erit nempe (129. 10.)
19.
da: :0 (a)
quoad .c— KC (: It ). Producatur KC donec K' C:: KC:
radius vector r' computetur ab K'; et .r' ab eodem K' in
K'C; explebitur (a) per
≬⇌−⋮−∥↸≀∙⊣−∘≖⋮⋟⊣−⊏⋅⇁≺↗⋅−⋅−∘∩⊐−⊦
↿
−≀−∙−∙⋅−∣⋮⋀≀∙⋅−∣⋅−∘≀⊅⊹⊞↱⋅−−⊄⋮↕∙⋟⋅∙∣ (a');
siquidem quoad puncta sita in AB
∙∙ dQ dQ ∙∙∙
↙≀↾∙∙∙⊲ dr'
⋅⋅−—&quot;'-27—27 ***-&quot; ∙⋅↕−⇀−∣⋅∙⋅↴∙⋮⋮⊒−− 2;-
Determinatis praeterea f
et F ex (i'&quot;. 129. 10.), re-
praesentabit (a') initialem fluidi statum: quoniam igitur (a')291 !
1
1
1
1
1
et satisfacit conditioni ( a ) , et exprimit initialem fluidi statum,
poterunt per ( a' ) definiri, quae spectant ad motus propaga
tionem, attento obstaculo AB.
2º . * Punctum C &quot; , ad quod pertinent radii vecto
res r et r seu KC&quot; et K'C &quot; , perinde motum concipiet ac
si ( 129. 6. &quot; ) , sublato plano AB, fluidum eadem omnino
ratione concuteretur simul circa duo puncta K et K' . Per
tinget itaque ( 229. 4º. ) concussio ad C &quot;, primum in fine
temporis deinde in fine temporis : hinc bi
ni successive motus in C &quot; , alter directus, alter reflexus ; et
quia secunda concussio non pervenit ad C &quot; nisi quum tempus
sic invrevit, ut habeatur r = ct + a,, iccirco eadem velo
citate c regredietur motus, qua incedebat antequam in obi
cem impiogeret. Ad haec : cum anguli KC'C, K'C'C sint
aequales, rursus ( 131 ) patet sonum illisum obici AB re
gressurum efficiendo angulum reflexionis aequalem angulo
incidentiae.
C. с
De instrumentis pneumaticis.

133. In instrumentis pneumaticis soni genesis repe
tenda non est saltem praecipue ex oscillatione partium so
lidarum ipsius instrumenti. Etenim si in hisce instrumentis
dicatur soous creari eodem modo ac in instrumentis per
cussione resonantibus, jam sonus ipse connexionem haberet
maximam cum materia qua instrumentum compactum est ,
nec non cum ejusdem crassitie; quod tum ratione verissi
mum apparet , lum etiam constat ex vi paritatis. Ratione
quidem: nam in instrumentis ex diversa materia compactis,
quorum proinde particulae non aeque elasticae sunt, et ad mo.
cum oscillatorium non aeque aptae, non eodem modo tremu.
lus ille motus per insufflationem excitari debet ; quo vero
crassius instrumentum est, eo in plures continuas partes 0
294
et satisfacit conditioni (a), et exprimit initialem fluidi statum,
poterunt per (a') definiri, quae spectant ad motus prcpaga-
tionem, attento obstaculo AB.
20.a Punctum C&quot;, ad quod pertinent radii vecto-
res r et r' seu KC&quot; et K'C&quot;, perinde motum concipiet ac
si (129. 6.0), sublato plano AB, fluidum eadem omnino
ratione concuteretur simul circa duo puncta K et K'. Per-
tinget itaque (229. 40.) concussio ad C&quot;, primum in fine
tempons c
, deinde in fine temporis c : hinc bi-
ni successive motus in C&quot;, alter directus, alter reflexus; et
quia secunda concussio non pervenit ad C&quot; nisi quam tempus
sic iuvrevit, ut habeatur r': ct ⊣−∙ a. ,
iccirco eadem velo-
citate c regredietur motus, qua incedebat antequam in obi-
cem impingeret. Ad haec : cum anguli KC'C, K'C'C sint
aequales, rursus (131) patet sonum illisum obici AB re-
gressurum efficiendo angulum reflexionis aequalem angulo
incidentiae.
∙ r—al r'e—a
De instrumentis pneumatict's.

133. In instrumentis pneumaticis soni genesis repe-
tenda non est saltem praecipue ex oscillatione partium so-
lidaram ipsius instrumenti. Etenim si in hisce instrumentis
dicatur sonus creari eodem modo ac in instrumentis per-
cussione resonantibus, jam sonus ipse connexionem haberet
mammam cum materia qua instrumentum compactum est,
nec non cum eiusdem crassitie; quod tum ratione verissi-
mum apparet ,
tum etiam constat ex vi paritatis. Ratione
quidem: nam in instrumentis ex diversa materia compactis,
quorum proinde particulae non aeque elasticae sunt, et ad mo-
tum oscillatorium non aeque aptae, non eodem modo tremu-
lus ille motus per insufilationem excitari debet ; quo vero
crassius instrumentum est, eo in plures continuas partes o-295
scillatorius molus dispesci debet. Vi paritatis autem : nam
reipsa instrumenta, quae percussione sopant, pro materiae di
versitate etiam in pari crassitudine diversimode contremiscunt;
et si ejusdem sint materiae, pro diversa crassitie diversum
item sonum edunt. Ergo sonus in instrumentis pneumaticis
maximam connexionem etc. si in his soni genesis etc. Atqui
hoc est falsum : in tibiis enim cylindricis ejusdem longitu
dinis idem habetur sonus aut fere idem , nullo respectu
habito ad materiam aut crassitiem ipsius instrumenti , ut
constat experimentis; totumque artificium pro varietate to
norum pendet ex instrumenti variata longitudine: propte
rea etc.
Quonam igitur pacto in hujusmodi instrumentis erit
soni genesis explicanda ? Eo videlicet , quem indicavimus
(114.). In interna instrumenti capacitate aeris columna includitur, quae externi aeris pressione urgetur: dum igitur
per exiguum orificium fistulae alius aer insufflatione intro
mittitur, aer ille inclusus condensari debet , atque urgeri
contra aerem externum; externus autem vi suae pressionis
resistere; et quum aeris interni aucta pressio vincit, hic se
dilatando condensabit externum aerem , repelletque; externus
aer ita densatus ut ejus elasticitas praevaleat, se restituendo
rursum comprimet internum aerem : in columna videlicet illa
fiei compressio et restitutio, sicque in aeris particulis oscillato
rius motus excitabitur; qui motus communicabitur externo
aeri contiguo, et ad aures perveniet. Aer itaque secundum lon
gitudinem fistulae se habet instar chordae peragentis longita
dinales vibrationes: quo tibia et consequenter columna aerea
longior est, eo etiam longiores undae efformantur; longius
que erit tempus compressionis et restitutionis , ac proinde
Lonus gravior. Hinc in instrumentis, quae secundum longi
Ludinem sunt foraminibus instructa, modo hoc et modo il
lud foramen aperiendo, sublato digito, varii obtinentur to
ni; siquidem externum aerem sic admittendo , modo ma
jorem et modo minorem columnae aereae longitudinem ha
295
scillatdrius motus dispesci debet. Vi paritatis autem: nam
reipsa instrumenta, quae percussione sonant, pro materiae di-
versitate etiam in pari crassitudine diversimode contremiscunt;
et si ejusdem sint materiae, pro 'diversa crassitie diversum
item sonum edunt. Ergo sonus in iustrumentispneumaticis
maximam connexionem etc. si in his soni genesis etc. Atqui
'
hoc est falsum: in tibiis enim cylindricis ejusdem longitu-
dinis idem habetar sonus aut fere idem , nullo respectn
habito ad materiam aut crassitiem ipsius instrumenti ,
ut
constat experimentis; totumque artificium pro varietate to-
norum pendet ex instrumenti variata longitudine: propte-
rea etc.
Quonam igitur pacto in hujusmodi instrumentis erit
soni genesis explicanda ? Eo videlicet
, quem iudicavimus
(114.).ln interna instrumenti-capacitate aeris columna in-
cluditur, quae externi aeris pressione urgetur: dum igitur
per exiguum orificium fistulae alius aer iusufflatione intro-
mittitur, aer ille inclusus condensari debet, atque urgeri
contra aerem externum; externus autem vi suae pressionis
resistere; et quam aeris interni aucta pressio vincit, hic se
dilatando condensabit externum aerem, repelletque; externus
aer ita densatus ut ejus elasticitas praevaleat, se restituendo
rursum comprimet internum aerem: in columna videlicet illa
fiet compressio et restitutio, sicque in aeris particulis oscillato-
rius motus excitabitur; qui motus communicabitur externo
aeri contiguo, et ad aures perveniet. Aer itaque secundum lon-
gitudinem fistulae se habet instar chordae peragentia longitu-
dinales vibrationes: quo tibia et consequenter columna aerea
longior est, eo etiam longiores undae efi'ormantur; longius-
que erit tempus' compressionis et restitutionis , ac proinde
tonus gravior. Hinc in instrumentis, quae secundum lougi-
tudinem sunt foraminibus instructa, modo hoc et modo il-
lud foramen aperiendo, sublato digito, varii obtineatur to-
ni; siquidem externum aerem sic admittendo , modo ma-
iorem et modo minorem columnae aereae longitudinem ha-296
benius. Ita in chordis, pro majori chordae longitudine gra
vior est tonus, acutior pro minori; et digitis comprimendo
camdem chordam, ut evadat plus aut minus longa , varios
assequimur tonos .
Dixi soni genesim repetendam non esse saltem prae
cipue ex oscillatione solidarum partium etc. Etsi enim ex
materia instrumenti non habetur varietas quoad soni qua
litatem , aut valde notabilem intensitatem ; varietas tamen
habelur quoad meliorem aliquam resonantiam; idque ex eo
desumendum videtur quod aer inclusus pro diversitate cor
poris includentis melius aut minus bene oscillare potest ;
magis nimirum aut minus impeditus adhaesione ad ipsum
corpus et scabritie aliqua. Ad haec; si instrumentum pneu
maticum sit compactum ex materia non resistente, quale v.g.
esset instrumentum membranaceum , tunc per vibrationes
columnae aereae excitari poterit sensibilis motus oscillato
rius in partibus instrumenti; quae partes vicissim in aerem
vibrantem reagendo valebunt sonum ipsum modificari etiam
quoad tonum; et quidem plurimum si instrumentum sit val
de breve, quemadmodum expertus est D. Savarı; adeo ut
brevi tubo membranaceo obtineri possil magna varietas
lonorum , qui eo graviores erunt quo minus tenditur mem
brana.

134. Haec proponimus explicanda circa instrumenta
pneumatica. 

1º. Aperto aliquo foramine ex. gr. tertio, cae
lerisque clausis, ac deinde aperto alio puta quinto , variat
lonus: at si ' per aperitionem tertii inducitur communicatio
interni aeris cum externo, nonne ex dictis ( 133 ) audiri de
beret idem tonus sive apertum sive clausum sit quintam
foramen ? 

2º. Sola inflationis intensione mutantur toni , e
tiam servata eadem internae columnae longitudine 

3º. In
canna organi ejusdem diametri superius clausa, si subdupla
sit longitudo , idem redditur tonus qui obtinetur ex can
na superius aperta, et longitudinis duplae.
Ad 1. Cum varia in instrumento pneumatico fora
mina aperiuntur, variae interni aeris columnae communi
296
hemas. ita in chordis, pro maiori chordae longitudine g'ra-
vior est tonus, acutior pro minori; et digitis comprimendo
eamdem chordam, ut evadat plus aut minus longa , varios
assequimur tonos.
Dixi soni geneaim repetendam non esse saltem prae-
cipue ex oscillatione solidarum partium ctc. Etsi enim ex
materia instrumenti non habetur varietas quoad soni qua-
litatem , aut valde notabilem intensitatem ; varietas tamen
habetur quoad meliorem aliquam resonantiam; idque exeo
desumendum videtur quod aer inclusus pro diversitate cor-
poris incladentis melius aut minus bene oscillare potest;
magis nimirum aut minus impeditus adbaesione ad ipsum
corpus et scabritia aliqua. Ad haec; si instrumentum pneu-
maticum sit compactum ex materia non resistente, quale v.g.
esset instrumentum membranaceum ,
tunc per vibrationes
columnae aerea'e excitari poterit sensibilis motus oscillato-
rius in partibus instrumenti; quae partes vicissim in aerem
vibrantem reagendo valebunt sonum ipsum modificari etiam
quoad tonum; et quidem plurimum si instrumentum sit val-
de breve, quemadmodum expertus est D. Savart; adeo ut
brevi tubo membranacea obtineri possit magna varietas
tonorum, qui eo graviores erunt quo minus tenditur mem-
liraua.

134. Haec proponimus explicanda cirea instrumenta
pneumatica. 10. Aperto aliquo foramine ex. gr. tertio, cae-
terisque clausis, ac deinde aperto alio puta quinto. variat
tonus: at si' per aperitionem tertii inducitur communicatio
interni aeris cum externo, nonne ex dictis (133) audiri de-
beret idem tonus sive apertum sive clausum sit quintum
foramen? 20. Sola inflationis intensione, mutantur toni. e-
tiam servata eadem internae columnae longitudine 3&quot;. In
canna organi eiusdem diametri superius clausa, si subdupla
sit longitudo, idem redditur tonus qui obtinetur ex can-
na superius aperta, et longitudinis duplae.
Ad 1.&quot; Cum varia in instrumento pneumatico forf- mina aperiuntur, variae interni aeris columnae communi-297
cantes çum aere externo excitantur; non ita tamen commu
nicantes, ut simul non etiam inter se communicent; ergo
looi variare per plurium foraminum aperitionem debent ,
etsi exquisitam ejus rei rationem aegerrime quis reddere possit.
Ad 2.&quot; Ut per vehementiorem percussionem in chor
da instrumenti fidicularis contingit ut ea resonet ad oclavam,
ita in columna aerea per variam inflationis intensionem con
tingit ut tonus mutetur; et sicut certum est in chorda mu
sica quod ea tunc dividitur in duas partes separatim oscil
lantes, ita eadem asserenda est fieri divisio et oscillatio in
columna aerea sub tempore, quod sił proportionale tono
quem reddit. Hinc deducitur explicatio saltus ut ajunt tu
bae v. gr. ad octayam: cam paulo vehementius inspiralur tu
ba, cogitur aer ad celeriorem motum , quem tamen colu
mnae aereae jam vibrantes , utpote nimis longae, praesta
re non possunt. Dividitur igitur columna per medium ita ,
ut duo nova segmenta aeris aequalia suas vibrationes separatim
peragant. Ubi vero saltus non sit ad octavam , alia divi
sio fieri dicenda est .
Ad 3.&quot; Ia medio cannae duplae efformátur nodus ,
habetur aereum stratum quiescens , quemadmodum
habetur in orificio clauso cannae subduplae ; adeoque ea
dem undae aereae longitudo in utraque canna , idemque
proinde tonus .

135.* Sit tubus cylindricus determinatae longitudinis 1,
firmiter obseratus apud alterum orificium , aperius apnd al
terum : aequilibrium aereae columnae inclosae ita turbari
pono, ut qui aer orificio aperto ( ubi initium distantiae x
consliluo ) respondet, nullam densitatis variationem subeat,
et qui orificio clauso, nullatenus moveatur.Functiones ( 129.7 °) .
f, fx , ac proinde f , F ' tanquam datas assumo ab x = 0
ad x = l. E statu aeris apud extremitates tubi habemus
= o si x = 0, v = 0 si x = l; hinc
seu
fl + 1) + F'll — cl) = 0 ( 0 ) ,
20
297
can'tcs cum aere externo excitantur; non ita tamen cdmmu-
nicantes, ut simul non etiam inter se communicent; ergo
toni variare per plurium foraminum aperitionem debent
,
etsi exquisitam eius rei rationem aegerrime quis reddere possit.
Ad 2.&quot; Ut per vehementiorem percussionem in chor-
da instrumenti fidicularis cdntingit ut ea resonet ad octavam,
ita in columna aerea per variam inflationis intensionem cou-
ting'it nt tonus mutetur; et sicut certum est in chorda mu-
sica quod ea tunc dividitur in duas partes separatim oscil-
lantes, ita eadem assereuda est fieri divisio-et oscillatio in
columna aerea sub tempore, quod sit proportionale tono
quem reddit. Hinc deducitur explicatio saltus ut aiunt tu-
bae v. gr. ad octavam: cum paulo vehementius inspiratur tu-
ba, cOgitur aer ad celeriorem motum, quem tamen colu-
mnae aereae iam vibrantes, utpote nimis longae, praesta-
re non possunt. Dividitur igitur columna per medium ita,
ut duo nova segmenta aeris aequalia suas vibrationes separatim
peragant. Ubi vero saltus non sit ad octavam, alia divi-
sio fieri dicenda est, ⋅ '
Ad 3.&quot; In medio cannae duplae eEorm'atur nodus ,
seu habetur aereum stratum quiescens, quemadmodum
habetur in orificio clauso cannae subduplae: adeoque ea-
dem uudae aereae longitudo in utraque canna
,
idemque
proinde tonus.

135 Sit tubus cylindricns determinatae longitudinis !,
firmiter obseratus apud alterum orificium, apertas apud al-
tequm : aequilibrium aereae columnae inclusae ita turbari
pono, ut qui aer orificio aperto ( ubi initium distantiae .a-
constituo ) respondet, nullam densitatis variationem subeat,
et quiorificio clauso, nullatenus moveatur.F unctiones (129.7&quot;).
f, f. , ac proinde f, F' tanquam datas assume ab a: 30
ad .r.-zl. E statu aeris apud extremitates tubi habemus

:: osi æzo,v:——osiæ:-:l; hinc

≀≖∣⋅∶≀−∙⊢≀∶∠⊢⊢ F'(l—c1):o ( o )
.
20298
Fll — ct) - f ( c ) = 0 ( o' ) .
In (0 ) substituatur ct +1- x in locum ct : prodibit
f (21 + ci - x) = - F '(x - 1) ( 0&quot; ) ;
unde
= f'( x + cl) – f'( 21+ ct - x ) ,
c = -f(x + ct) -f(21 +ct - x) :
( o ' ')
in ( o &quot; ) fiat x = 0 ; erit ob ( o')
f (c + 2) = -F ( - ct) = -f (c ) (0&quot; &quot; );
subrogato ct +21 in locum ct, habebitur
f '( c +4 ) = -f( ct + 2 ) = f(t ) (o ');
denique si in ( 0 &quot;) ponitur ci=0, emerget
f( 21 — x ) = - F ( x ) ( 0 &quot; ) .
.
Aequationes ( o' : 0' ! ) satis sunt, ut functionem f con
siderare possimus veluti datam quoad omnes positivos ya.
lores quantitatis variabilis , ad quam respicit ipsa functio.
Etenim e statu initiali atque arbitrario aereae columnae ad
motum incitatae , data est F' ab x = o ad x = l ; ergo
ob (o &quot; ) data erit fab x = l ad x = 21 : ex eodem sta
tu jam dala erat fab x = o ad x = l ; ergo dabiturf
ab x = o ad x = 21. Aequatio autem ( o&quot;) rem evidentissime
absolvit quoad caeteros quantitatis variabilis positivos va
lores. Ergo etc.
298
F'(—ct)——f(ct):-to (c').
In (o) substituetur et −⋅⊢≀ —- a: in locum et: prodibit
f(2l −∣⋅− ct — a:) ∶−∙−− F'(x—ct) (o&quot; );
uude
v:f(æ-i-ct)—f(2l—l—ct—æ).
etc:—f(x-i-cz)—f(2l -i-ct—æ):
111 (o&quot;) fiat m::- o; erit Ob (0')
(o&quot;')
f(cs −↿− 21) ∶−∙− −F'( —ct) −−∶ —f(ct) (o&quot;):
subrogato et —-[-21 in locum et, habebitur
f(cz ↽⊢ 4!) ∙−−− —-f(ct −⊦ 21) ↽↼−−⋅≖ f(ct) (a';
denique si in (a'—') ponitur ctzo, emerget
f(ZI—æ) z—F'Lr) (o&quot;).
Aequationes (a': a'!) satis sunt, ut functionem f coa-
siderare possimus veluti datam quoad omnes positivos ,va-
lores quantitatis variabilis, ad quamrespicit ipsa functio.
Etenim e statu initiali atque arbitrario aereae columnae ad
motum incitatae
, data est F' ab a: a ad w:! : ergo
ob (o&quot;) data erit f
ab æ-—-:l ad se:21 :ex eodem sta-
tu iam data erat f' ab a: :: o ada::1; ergo dabitur ['
ab a: :: 0 ad se:21. Aequatio autem (o&quot;) rem evidentissime
absolvit quoad caeteros quantitatis variabilis positivos va-
lores. Ergo etc.299
Quoniam ab x = o ad x = 21 dependet f' ab ini
tiali atque arbitrario statu aereae columnae ; poterit igitur
sic assumi inter illos limites , ut facto i = 1,3,5,7 ,..., sit
21
f'(c + % -f(cc + = p&quot;(ce) ( 0 &quot; &quot; );
numeri pares = 2, 4, 6 ... debent excludi ob aequatio
dem (o &quot; ). Instaurantur ergo iidem functionis f valores
42 quotiescumque tempus t evadit it ; sed ( o &quot; ) a functio ic
ne f unice dependent v, E. Columna igitur aerea in eum
dem restituitur statum per aequalia intervalla, suasque com
41
plet oscillationes intra tempus ; quarum propterea nu
merus intra q ' erit
ic
ic
41
136.. Evanescet (135.0 '&quot;') velocitas v ubi fuerit f ( x + cos
= f (21 + ct - x ) ; evanescet e si f'( x + ct) - f (21+
ci - x ). Primum contingit ( 135 : 0 &quot; ) quando (22 +ic - x )
41'2 - ( x + cl) seu 21 2x ; secundum quando 21—
21&quot; i 1
2x= • Hinc 1º. facto i = 0 , 1 , 2 , 3 .. i
scet aer in distantiis
2 , quie
lli - 21 )
X
2º . Facto i&quot; =1 , 3 , 5 .... 1 ; movebitur aer in
locis
299
Quoniam ab a: a ad a: 2! dependet ;&quot; ab ini-
tiali atque arbitrario statu aereae columnae ; poterit igitur
sic assumi inter'illos limites, ut facto i:1,3,5,7,...., sit
41 21 ∣
'&quot;
∣⇃≺∁⊢⊢ ∙
−−⋮∙⊣∶∶≕−∣↙≼⋄⊢⊢ 7): f
(ct) (0 ):
numeri pares r':2, 4, 6 ...
debent excludi ob aequatio-
nem (o&quot;). Instaurantur ergo iidem functionis f' valores ,
quotiescumque tempus :
evadit t −∣∙⋅ &amp; sed (o&quot;')a functio-ne )&quot; unice dependent v, :. Columne igitur aerea in eum--
dem restituitur statum per aequalia intervalla, suasque com-
. . . plet 4! osc1llat1ones intra tempus ∙≀⋅−≔∙ ; quarum propterea nu-
merus intra 1&quot; erit
t'

136. a Evanescet (135.o&quot;')velocitas :: ubi fuerit f' (æ—l—ct

f(ZH— ct —- æ ); evanescet :si f(x −∣⋅− et):
— f(ZI-i-
ct ∙−− a: ). Primum contingit (135: a'&quot;) quando (21—1—4 cs --' a:)
41&quot;! −∙− ( æ ⋅−⊢ ct) seu 2! -- 21: −∙−−−−− T;secundum quando 21—
2
∙∣∣ ∙−↿ .
2x: −⋮∙−↨ ∙Hinc ↿∘∙ facto 1&quot;: a, 1, 2, 3 ....
'T, qu1e-
scet. aer in distantiis' -
∙∙∙ [( t' — 21&quot;)
æ
!
20. Facto 1'&quot; :1, 3, 5 ....
i; movebitur aer in
locis300
llimi)
quin tamen ullam patiatur densitatis variationem, Aper
tis itaque foraminibus in hisce postremis locis , nullo pa
cto sonus mutari debet ; quod experientiae consonum re
peritur: imo non mutabitur sonus, licet lubo abscindatur
pars 1- x , quae ultra locum x ad fundum usque pro
tenditur. Atqui pars reliqua nihil aliud est nisi tubus
in utraque patens extremitate: ergo si de hujusnuodi cubis
sermo sit, posita e = o apud unum orificium erit quo
que apnd alterum { =0.
137. # In tubis itaque cylindricis, quorum ambo ori
ficia libera omnino sunt, habetur ( 129.7 .)

#

Fl—ct) — 9 (2+ cl ) = 0, F1 — ct) - f'(C ) = 0.
Hinc facile deducuntur ( 135 ) sequentes aequationes
f (21 + cix) = F ' x - ct), v = P ( x + ct) + P (21 + c1 - ),
c : = f ( 21 + (1 - x ) — f (x + c ! ), f (ce + 21 = F (-1)= f (c ),
f'(21 — * ) = F ( x ).
Quia vero ab x = o ad x = 21 rursus dependet p
ab initiali atque arbitrario statu 'aereae columnae , ic
circo poterit etiam asseri sequens aequatio.
f (ce + *-) = f ( c )
in praesenti est i = 1. 2, 3, 4 ......
æs. lu.—'r')
1
,
quia tamen ullam patiatur densitatis variationem. Aper-
tis itaque foraminibus in hisce postremis locis, nullo pa-
cto sonus mutari debet; quod experientiae consonumre-
peritur: imo non mutabitur sonus, licet tubo abscindatur
pars l—æ, quae ultra ↙
locum a: ad fundum usque pro-
tenditur. Atqui pars reliqua nihil aliud est nisi tubas
in utraque patens extremitate: ergo si de huiusmodi tubis
sermo sit, posita :

: o apud/uuum oriücium erit quo-

que apud alterum::o.

137.a In tubis itaque cylindricis, quorum ambo ori-
licia libera omnino sunt, habetur (129. 70.)
F'U—ct) -
f(l −↿− ct) ∙−−−−∙∙ o,F'( -
ct) --f(ct): 0.
Hinc facile deducuntur (135) sequentes aequationes
f(21 -l-ct -æ):F'(æ—-ct), v ::f'(æ ⊣− et)-t— f(ZI-i-ct—x).
es:/(21 -t-ct —x)-— f(æ-t—ct), f(ct-t-Zl):F'(—cc):f(c1),
f(2l — a: ): F' (æ)-
Quia vero ab a: 0 ad ..r:21 rursus dependet f
ab initiali atque arbitraria statu 'aereae columnae
,
ic-
circo poterit etiam asseri sequens aequatio.
f(ct—i- -—2'-£-) :f'(ct )

:

in praesenti est i: 1. 2, 3, 4 ......
≡⊲∙⋅⇀≣∎ lJ-r

: 301

22
Iterat ergo aerea columna per aequalia intervalla ic
oscillationes suas , quarum proinde numerus intra 1 &quot; erit
ic n =
21
Haud immoror inquisitioni distantiarum , ubi a
er vel quiescit, vel nativam retinet densitatem : hujusmo.
di namque investigatio similiter perficitur ac in Lubis,
quorum unum orificium apertum est . Satius forsan e
rit adnotare quod, facto i = 1 , exhibet ( 137 ) aequatio n '
relationem inter principalem tonum n ', redditum ab
elastico fluido intra tubum oscillante, et velocitatem c qua
sonus incederet si per ipsum fluidum propagaretur. Hinc
patet quomodo experimentis indagari possit velocitas c in
aliis elasticis flaidis ab aere atmosphaerico diversis : ex
tentaminibus Van - Rees, Frammeyer, et Moll prodiit so
ni velocitas sub temperie 10.° C
21
io gas oxigenio 3,7m, 9 :
bydrogenio 1233 , 3 ,
nitrogenio .
.
339 .
oxido nitrico 317 , 4 ,
acido salphuroso 229 , 2 ,
acido carbonico 370 , 7 ,
. .
suboxido carbonico . . 341,1
etc. etc.
301
. . 2! [terat ergo aerea columna per sequsl1a1ntervalla ∙∙∙∙⋮∙⋅−
oscillationes suas ∙ quarum ⋅proinde numerus intra 1&quot; crit
'
IC
Haud immoror inquisitioni distantiarum , ubi a-
er vel quiescit, vel nativam retinet densitatem: huiusmo-
di namque investigatio similiter perficitur ac in tubis,
quorum unum orificium apertum est. Satius forsan e-
rit adnotare quod, facto 1':1,exhibet (157) aequatio n'
0
∙−−∶ -2-l- relationem inter principalem tonum n', redditum ab
clastico flaido intra tubam oscillante, et velocitatem e qua
sonus incederet si per ipsum fluidum prcpagaretar. Hinc
patet quomodo experimentis indagari possit velocitasc in
aliis elasticis fluidis ab aere atmosphaerico diversis: ex
tentaminibas Van— Bees, Frammeyer, et Moll prodiit so-
ni velocitas sub temperie 10.0C
in gas oxigenio . . . . . . 317',g,
hydrogenio . . . . . 1233,3,
nitrogenio. . . . . . 339 . ∙
oxido uitrico . . . . 317 ,4
acido sulphuroso . . -
229 , 2
,
acido carbonico . . . . 370 , 7 ,
suboxido carbonico . . . 341 ,
1
,
etc. etc.302

138. Si tubus proponilur utrinque obseratus , quis
que videt fore v = o apud ambas extremitates; unde (129.7°)
f (c ) + F ( -ct)= 0,flfct) + F (l ct) = 0,
quarum ope determinatur motus inclusi aeris,
De propagatione soni per liquida , et per solida corpora.
139.* Quod spectat ad liquida corpora , in comperlo
est aquam v. g. contrahi perparum posse atque restitui
in suis partibus : itaque qua ratione turbatum posuimus
( 129..1 . ° ) aequilibrium , eadem in praesenti imaginemur
turbari . Propagato motu , densitas ré aquae libratae ver
tetur in je = pili + :) apud (2. , y , z ) ; et pressio o'
in w= '+Ae ; exprimit A numerum experimentis deter
minandum. Sumptis hic quoque X=0, Y=0, Z=o, et ra
tiocinando ut in citato n. ° assequemur
d
dQ
1 do dt (
dQ
dt
dr
A dL {1+ :)
dr seu
р. dr pi
A
tum facto c ” , perveniemus ad formulas (i' '.;&quot; . .
ji
į &quot; : 129, 1.0 ) . Non pluribus opus est ut intelligamus
( 129. 2.° 3,0 ) sonum per aquam diffundi aequabiliter ve
locitate.
VA
Numerus A potest determinari ex parvula contractione ,
quam juxta longitudinem à ( haud variata diametro ) pa
302
1381: Si tnbus praponitnr utrinqne obseratus , quis-
que videt fore v:o apud ambas extremitates; unde (129.?)
f(ctH-FX --ct):o,f7(l—i-ct)-i-F'(l—ct):a,
quarum Ope determinatur motus inclusi aeris.
De propagatione soni per liquida, et per solida corpora.
139:- Quod spectat ad liquida corpora, in comperto
est aquam v. g. contrahi perparum posse atque restitui
in suis partibus :itaqne qua ratione turbatum posuimus
( 129. ⋅↿∙∘ ) aequilibrium, eadem in praesenti imaginemur
turbari. Pr0pagato motn, densitas pf aquae libratae ver-
tetur in þ.:yJU—I—s) apud (.x.-,,] ,
z) :
et pressio a'
in ≔≖⇌≖⋝∣↰∟⋀⋮⋅⋮ eXprimit A numerum experimentis deter-
minandum. Sumptis hic quoque X:0, ↧↗−−−−⋅∘∙ Z:o, et ru-
tiocinando ut in citato 11.0 assequemur
d
dQ) ↼ 1 de ∙− dQ) (Et? A; JLu-Jr-s) ∙− &quot;( dt ∙
a d?
.
dr ' se.. a' d.- 4.-
A -
∙∣∣ -1 tum facto ?

: cz, perveniamus ad formulas (:

'.
t
'.
i'
i&quot;: 129, ↿∙∘ )
. Non pluribus opus est ut intelligamus
( 129. 2.0 3.&quot;) sonum per aquam diffundi aequabiliter ve-
locitate. ⋅
.: ⇂∕⋅−⋮∶⇡∣−⋅
Numerus A potest determinari'ex parvula contractione f:,
quam iuxta longitudinem l (haud variata diametro) pa-303
tilur columna aquea ob incrementum 5. superadditum
pressioni o '. Nam 1 : 1 - B = + ): , ideoque &lt; =
B \beta

: ' sed o=u'two=a' +As, igitur スー B 2

6.
A :
σολ
E \beta
In hypothesi pressionis . = 0 &quot; , 76) g, ac temperiei n=
10.• C, experimenta Dni Canton suppeditant B = 0,000046 ),
inter quem valorem et quos invenerunt DD . Parkins et
Oersted , nimirum
B=0,0000452 , B=0,0000482 ,
parvula est differentia. Ponatur hydrargyri densitas 1 ;
erit proxime u'= . : assumpta igitur g=9m, 8088,
13,5819
1
emerget
c=1483&quot; , 59.
Sonus videlicet propagatur per aquam plus quadruplo ce
lerius quam per aerem. D. Beudant dicit in hac se fuis
se sententia , ut e suis experimentis in mari institutis ta
lem deduceret soni velocitatem , quae 1500m saltem aequaret.
140.* Quisque videt soni velocitatem eadem ratione
posse in caeteris corporibus , sive liquidis , sive solidis ,
determinari , modo eorum partes contrahi perparum queant
atque restitui . Sic , manente 5 . = (0,76 ) 8 , obtinuit idem
ipse Canton hydrargyri contractionem B = 0,0000032 : as
sumpla igitur u = 1 , erit
c = 1576m , 35
1
303
titur columna aquea' ob incrementum m superadditum
pressioni w'- Nam 71: l—þ:p.'(1-l-s): p!
,
ideoque ::
P −⇀ 13 ⇤
⋅
m−⊸T;'sed ∏∙−−∶∏∎∙⊦∏∘∶−−∸⋅∄≖⋅−⊦∆⋮∙ igitur
'
A ∙− a'., ∙∙∙ wo).
8
5
In hypothesi pressionis uro :( o'&quot;, 76) g, ac temperiei :::,
10.(, C. experimenta Dni Cauton suppeditant þ:0,0000461,
inter quem valorem et quos invenerunt DD. Parltins et
Oersted , nimirum
,ezo,oooo45) ∙ þ:0,000048).
.
parvula est differentia. Ponatur hydrargyri densitas :1;
↿
15.5819
erit proxime pf: : assumpta igitur g:9&quot;?,,8088,
emerget ⋅
c:1483&quot;'
. 59.
Sonus videlicet prcpagatur per aquam plus quadruplo cc-
lerius quam per aerem. D. Bendant dicit in hac se fuis-
se seateutia , ut e suis exPerimentis in mari institutis ta-
lem deduceret soni velocitatem, quae 1500&quot; saltem aequaret.
1403 Quisque videt soni velocitatem eadem ratione
posse in caeteris corporibus
.
sive liquidis , sive soli-dis ,
determinari , modo eorum partes contrahi perparum queant-
atque restitui. Sic, manente wo:(0,76) g , obtinuit idem
ipse Canton hydrargyri contractionem [5:0,0000037t : as-
sumpta igitur pf:1 , erit
0:1576,&quot; ∙ 35304
velocitas , qua per hydrargyrom diffunditur sonus. Ante
quam usum contractionis \beta animadverteret Laplace ad de finiendam soni velocitatem per liquida et solida corpora ,
exhibuerat Chladni in sua Acustica aliam methodum
sane ingeniosam , ejusdem velocitatis investigandae in cor
poribus solidis.
141. # Innititur ista methodus analogiae, quam norunt
Physici inter oscillationes aeris in tubo cylindrico apud
ambas extremitales aperto et longitudinales oscillationes
virgae rigidae , cujus ambo extrema omnino libera sint.
Exprimat enimvero l oscillantis virgae longitudinem ; n'
principalem tonum , quem edit resonans virga ; c' quae
sitam velocitatem . Erit ( 137 )
n &quot; ; unde n ' : n &quot; = 0 : c' , ' =
21
Iam si velocitas soni per aerem repraesenterar per &quot; , ex
perimenta D.ni Chladoi praebent soni velocitatem c
per stannum . 717 를
per argentum
per cuprum .
12
per ferrum et vitrum ... 17
per varia lignorum genera 11 ad 17 , . .
Ad explorandam soni velocitatem per ferruin fusionis , in
promptu habebat D. Biot 376 tubos ex hoc metallo com .
pactos ; quibus singulis mediocris erat longitudo duorum
304
velocitas , qua per hydrargyrnm diffunditur sonus. Ante-
quam usum contractionis þ animadverteret Laplace ad de-
finiendum soni velocitatem per liquida et solida corpora ,
cxbibuerat Chladni in sua Acustica aliam methodum,
sane ingeniosam .
ejusdem velocitatis investigandae in cor-
poribus solidis.
1414: lnnititur ista methodus analogiae, quam norunt
Physici inter oscillationes aeris in tabo cylindrico apud
ambas extremitates aperto et lougitudiuales' oscillationes
virgae rigidae, cuius ambo extrema omnino libera sint.
Exprimat enimvero! oscillantis virgae longitudinem ; n&quot;
principalem tonum
, quem edit resonans virga ; c' quae-
sitam velocitatem. Erit (137) '
c, '
nn 11&quot;:-2-i-;nuden:n&quot;:c:c', c':c-—J.
»
Iam si velocitas soni per aerem repraesentetur per 1, ex-
perimenta D.!d Chladni praebent soni velocitatem c'
per stannum . ∙ ∙ ∙ ∙
, 7
vet--
per argentnm . . . . . . 9 ,
per cuprum . . . . .
.. . 12 ,
per ferrum et vitrum . . .
17
.
per varia lignorum genera . .
11 ad 17,
Ad explorandam soni velocitatem per ferrum' fusionis ,
in
promptu habebat. D. Biot 376 tubos ex hoc metallo com'
pactos ; quibus singulis mediocris erat longitudo duorum

* a305

metr. cum partibus millesimis 515. Sumptis experimentis,
prodiit soni velocitas 104 ; nisi quod jungebantur ii tu
bi ope plumbi, quod aliquanto sonum retardare videtur.

=== De vocis humanae origine. ===
142. Vocis humanae organum etsi considerari maxi
me debet tamquam instrumentum pneumaticum ftexili
et elastica materia ex parte compactum , non tamen ita
est ut cum instrumentis etiam fidicularibus aliquam non
babeat analogiam. Quod ut melius intelligatur , nonnulla
ex anatomicis sunt hic afferenda.
Palmo est viscus respirationi inserviens: in duas par
tes distinguitur , dexteram et sinistram , et duo magni lo
bi dicuntur , etsi quivis ex his duobus dividitur mino
ribus aliis. Substantia constat molli , spongiosa , rara et
vessiculosa ita ut ad aerem excipiendum aptissimus sit :
motu ergo dilatationis aere impletur , et constrictionis
motu eundem expellit ; atque aer ita expulsus primo per
multiplices canaliculos lobis interspereos , qui bronchia
dicuntur ; tum per duos ex utroque lobo emergentes ; de.
mum per ampliorem canalem emergit , qui ex praefa
tis duobus in unum conjunctis coalescit. Hic canalis seu tubus
ad oris usque radices ascendens trachea seu aspera arteria
nuncupatur ; in summitate asperae arteriae brevis canaliculus
habetur , qui larynx dicitur , cujus summitatem facit rima
ovalis a duabus membranis horizontaliter jacentibus relicta ;
quae rima glottis dicitur : atque huic superposita est epi
glottis ; tenuis scilicet et mobilis cartilago glottidem te
gens , quae ad hoc praecipue statuta esse videtur, ut dum
aliquid deglutimus , quidquam cibi aut potus in asperam
arteriam minime irruat, sed per contiguum canalem, qui
exophagus dicitur, et cujus orificium pharynx vocatur , de
more in stomachum demittatur. Itaque lobi pulmonis instar
}
305
metr. cum partibus millesimis 515. Sumptis experimentis.
prodiit soni velocitas 10;- ; nisi quod iungebantur ii ftu-
bi »ope plumbi, quod aliquanto sonum retardare videtur.
De vocis humanae origine.


142.1Vocis humanae organum etsi considerari maxi-
me debet tamquam instrumentum pneumaticum fiexili
et elastica materia ex parte compactam, non tamen ita
est ut cum instrumentis etiam fidicularibus aliquam non
habeat analogiam. Quod ut melius intelligatur, nonnulla
ex anatomicis sunt hic aderenda.
⋅ Palmo est viscus respirationi inserviens: in duas par-
tcc distinguitur , dexteram et sinistram ,
et duo magni lo-
bi dicuntur , etsi quivis ex his duobus dividitur mino-
ribus aliis. Substantia constat molli , spongiosa , rara et
vesaiculosa ita ut ad aerem excipiendum aptissimus sit:
motu ergo dilatationis aere impletur , et constrictioais
motu eumdem expellit; atque aer ita expulsus primo per
multiplices canaliculos' lobis interspereos , qui bronchia
dicuntur; tum per duos ex utroque lobo emergentes :dc-
mum per ampliorem canalem emergit , qui ex praefa-
tis duobus in unum coniunctis coalescit. Hic canalis seu tubas
ad oris usqne radices ascendens tracbea seu aspera arteria
nuncupatur; in summitate asperae arteriae brevis canaliculus
habetur , qui laryux dicitur, cuius summitatem facit rima
ovalis a duabus membranis horizontaliter jacentibus relicta;
quae rima glottis dicitur : atque huic superposita est epi-
glottis; tenuis scilicet et mobilis cartilago glottidem te-
gens, quae ad hoc praecipue statuta esse videtur, ut dum
aliquid deglutimus, quidquam cibi aut potus in asperam
arteriam minime irruat, sed per contiguum canalem, qui
cxophagus dicitur, et cuius orificium pharynx vocatur
, de
more in stomachum demittatur. Itaque lobi pulmonis instar306
re cur com
follium aerem excipiunt, cum compressi illum emittunt per
asperam arteriam : aer ita expulsus per asperam arteriam
in arctiorem laryngis canalem irroit , atque ita ex am
pliori in angustius spatium redactus compressionem pati
debet , oscillatoriumque motum concipere. Sed quia la
rynx flexili et elastica materia compingitur, iccirco ( 133)
ad motum tremulum ab aere vibrante excitabitur. Deinde
vero in eumdem aerem diversimode reagendo, prout magis
vel minus erit tensa , ejus Oscillationes diversimode quoque
modificabitur.
Obiter notamus antiquos et cum iis Galenum male
organum vocis humanae in trachea constituisse ; quam
arbitrabantur vices gerere tubi, per quem aer ad sonum
jam excitatus excurrit. Refelles hanc opinionem consi
derans aerem qui tracheam ascendit , libere ascende
et liberius habere spatium ; unde non est
primi debeat et oscillatorium motum habere : cum autem
glottis sit multo arctior quam trachea , per glottidem
transiens habet quidem unde comprimi possit. Haec de
voce humana : ad vocem enim quod spectat quorumdam
animalium , uti sunt multae aves ; hae cum etiam exse
cto collo , sola ventris compressione sonum edant , in his
utique trachea concurrit ad sonum ipsum modificandum .
Sed nil hinc eruitur contra jam dicta: in istis namque avi
bus trachea habetur supra glottidem , seu gloutis esse obser
vatur non ad summitatem , sed infra tracheam ; contra
ac est in homine , et plerisque aliis animalibus.
In monumentis Academiae Parisiensis ad an. 1741
observat Ferreinius intra laryngem duas haberi fibras ad
labrum glottidis ; quae fibrae ex impetu aeris per angu
stiorem laryngis canaliculum irrumpentis ad tremitum con
citantur , atque hoc tremitu resonant , quemadmodum in
fidibus contingit ; unde dictum est vocis humanae organum
analogiam habere aliquam ad instrumenta fidicularia . Sum
psit ille plures laryages cum sua glottide ; dunque insuf
306
folliam aerem excipiunt. tam compressi illam emittunt per
asperam arteriam: aer ita expulsus per asperam arteriam
in arctiorem laryngis canalem irruit ,. atque ita 'ex am-
pliori in angustias spatium redactus compressionem pati
debet, oscillatoriamque motum concipere. Sed quia la-
rynx flexili et elastica materia compingitur. iccirco (133)
ad motum tremulum ab aere vibrante excitabitur. Deinde
vero in eamdem aerem diversimode reagendo, prout magis
vel minus erit tensa, eius oscillationes diversimode quoque
modificabitur. ∙
Obiter notamus antiquos ,et cum iis Galenum male
organum vocis humanae in trachea constituisse; quam
arbitrabantur vices gerere tubi, per quem aer ad sonum
iam excitatus excurrit. Refelles hanc opinionem consi-
derans aerem. qui tracheam ascendit
,
libere ascende-
re,'et liberius habere spatium ; unde non est cur ,com-
primi debeat et oscillatorium motum habere: cum autem
glottis sit multo arctior quam trachea , per glottidem
transiens habet quidem unde comprimi possit. Haec de
voce humana : ad vocem enim quod spectat quorumdam
animalium , uti, sunt multae aves; hac cum etiam exse-
cto collo, sola ventris compressione sonum edant, in his
utique-trachea concurrit ad sonum ipsum modificandam.
Sed nil hinc eruitur contra iam dicta: in istis namque avi-
bus tracbea habetur supra glottidem , sen glottis esse obser-
vatur non ad summitatem, (sed infra tracheam ; contra
ac est in homine , et plerisque aliis animalibus. .
In. monumentis Academiae Parisiensis ad an. 1741
observat .Ferreinius intra laryngem duas haberi, fibras ad
labrum glottidis ; quae fibrae ex impetu aeris per angu-
stiorem laryugis canaliculata irrumpentis ad tremitum.con-
citantur, atque hoc tremitu resonant
, quemadmodum in
fidibus contingit; unde dictum est .vocis humanae organum
analogiam habere aliquam ad instrumenta fidicularia. Sum-
psit illa plures' larynges cum sua glottidc; dumque insuf-307
Aando sonus vocis animalis excitabatur, microscopio Gibras
praedictas inspiciendo tremor et vibratio in iisdem cerne
batur , prout in chordis musicis habetur dum resonant.
Atqui eo ipso ex earum tremitu ab irruente aere tamquam
a vi percutiente excitato sonus gigni debet, vel jam geni
tus modificationem quamdam recipere.
Rursus , sicut in chordis musicis contingit ' ut chorda
brevior det sonum acutiorem , graviorem longior : ita ani
madvertendum hic fuit an fibrarum illarum major minor
ve longitudo toni mutationem induceret. Compertum au
tem est quod , impedita illarum fibrarum parte ne tre
meret , tonus prodibat acutior.
Sumpsit etiam larynges bovis , canis, aliorumque ani
malium , deinde insufflando excitabatur mugitus bovis , et
conformis aliis animalibus sonus. Movendo autem musculos
ita , ut traherentur et distenderentur fibrae, excitabantur
mutationes soni , quae haberi solent in varia horum ani
malium voce.
Notetur illud : cum tensio vel remissio fibrarum glot
tidis et cartilagineae substantiue , qua larynx constat , ab
eodem musculo dependeat , ut notat Savart , consequitur
una cum fibris illis etiam laryngem tendi vel remitti.
Laxatis fibris, orificium glottidis ampliatur , et sonus pro
dit gravior ; tensis vero , orificium restringitur, et sonus
evadit acutior , ut in canibus observavit D. Magendie.

143. Si vox in larynge et glottide tanquam in proprio
organo fit ; quid ergo, inquies, os atque ejus partes con
ferunt ad formationem vocis ?
Respondeo oris cavitatem , linguam, dentes, labia con
currere ad modificationem perfectionemque vocis ; quae
larynge et glottide incipit quidem , sed non omnimode
ibi perficitur : nam quod in illis partibus sufficiens habea.
tur organum quin prorsus necessaria sint oris et linguae or
gana ad exhibendum aliquo modo sonum animalis pro
prium , apparet ex eo quod grues abscisso
in
et anseres 1
307
flando sonus vocis animalis excitabatur, microscupio fibras
praedictas inspiciendo tremor et vibratio in iisdem cerne-
batur, prout in chordis musicis habetur dum resonant.
Atqui eo ipso ex earum trcmitu ab irruente aere tamquam
a vi percutiente excitato sonus gigni debet. vel iam geni-
tus modificationem quamdam recipere.
Rursus , sicut in chordis musicis contingit 'ut chorda
brevior det sonum acutiorem, graviorem longior :
ita ani-
madvertendum hic fuit an librarum illarum maior minor-
ve longitudo toni mutationem induceret. Compertum au-
tem est quod ,
impedita illarum fibrarum parte ac tre-
meret. tonus prodibat acutior.
Sumpsit etiam larynges bovis , canis. aliorumque sni-
malium, deinde insufflaudo excitabatur mugitus bovis ,
et
conformis aliis animalibus sonus. Movendo autem musculos
ita, ut traherentur et distenderentur fibrae, excitabantur
'mutationes soni, quae haberi solent in varia horum ani-
malium voce.
Notetur illud: cum tensio vel remissio librarum glot—
tidis et cartilagineae substantiae, qua larynx constat , ab
eodem musculo dependeat , ut notat Savart, consequitur
una cum fibris illis etiam laryngem tendi vel remitti.
Laxatis' fibris, ,orificiu-m glottidis ampliatur, et sonus pro-
dit gravior; tensis vero , orificium restringitur. et sonus
evadit acutior, ut in canibus observavit D. Magendie.

143. Si vox in larynge et glottide tanquam in proprio
organo fit; quid ergo, inquies, os atque eius partes cou-
ferunt ad formationem vocis?
Respondeo oris cavitatem. linguam, dentes. labia con-
currere ad modificationem perfectionemque'vocis; (quae in
⇁
larynge et glottide incipit ⋅ quidem , sed non omnimode
⋅⋅
ibi perficitur: nam quod in illis partibus sufficiens habea-
tur organum qain prorsus necessaria sint oris et linguae or-
,gana ad exhibendum aliquo modo sonum animalis pro-
prium ,,apparet ex eo quod grues et anseres , abscisso308
capite , ex ventris compressione sonos edere possint iis si
miles, quos viventes edebant. Ad modificationem igitur per
fectionemque vocis in larynge et glottide inchoatae caete
ra concurrunt : neque haec modificatio in mera reflexione
consistit, sed in resonantia proportionata tono soni a
glottide emissi .
Ad articulatarum vocum formationem quod attinet ,
ea praecipue a mota linguae et labiorum repeti solet :
inter caeteros P. Fabri diligenter expendit quo pacto lin
gua et labia componantur ad cujusque syllabae efforma
tionem .

144. Dices: potest sonus excitari aerem expellendo
per angustius spatium ; atque ita sibilus per labiorum com
pressionem excitatur. Ergo dicendum videtur quod ex 90
la emissione aeris per angustius glottidis spatium vox effor
inari possit quin confugiamus ad tremitum laryngis et
fibrarum glottidis ; qui tremitus effectus erit soni quin in
sonum ipsum influat.
Respondeo : etsi sonus aliquis obtineri praecise pos
sit per hoc quod ex ampliore in angustius spatium aer
cogatar transire ; attamen quae hactenus diximus suadent
tremitum laryngis et fibrarum ad vocis formationem con .
currere; attenta praecipue varietate maxima , quae in vo
cis modificatione habetur. Novimus enim et singulos ho
mines modificari quam maxime vocem , et in diversis ho
minibus quam maxime diversum esse vocis sonum . Iam ve
ro cum habeatur sibilus per solam labiorum compressio
nem , inde expulso violenter aere , exigua est hujusmodi
soni diversitas; et omnes fere homines eumdem sonum ef
ficiunt , licet in diversa intensione : ergo cum contra in
voce diversitas sit maxima et proprius cuique sit homini so
nus , ad diversam fibrarum et laryngis materiam ac ten
sionem recurrendum potius videtur. Scio equidem ab in
strumentis pneumaticis etiam materia resistente compactis
et lingula instructis magnam edi posse varietatem sono
308
capite , ex ventris compressione sonos edere possint iis si-
miles, quos viventes edebant. Ad modificationem igitur per-
fectionemque vocis in laryuge et glottide inchoatae caete-
ra concurrunt: neque haec modificatio in mera reflexione
consistit, sed in resonantia prOportionata tono soni a
glottide emissi.
Ad articulatarum vocum formationem quod attinet ,
ea praecipue a motu linguae et labiorum repeti solet:
inter caeteros P. Fabri diligenter expendit quo pacto lin-
gua et labia componantur ad cuiusque syllabae efforma-
tionem.

144. Dices: potest sonus excitari aerem eXpellendo
per angustius spatium : atque ita sibilus per labiorum com-
pressionem excitatur. Ergo dicendum videtur quod ex so-
la emissione aeris per angustius glottidis spatium vox effor-
mari possit' quin confugiamus ad tremitum laryngis et
fibrarnm glottidis; qui tremitus effectus erit soni quin in'
sonum ipsum influat.
Respondeo: etsi sonus aliquis obtineri praecise pos-
sit per hoc quod ex ampliore in angustius spatium aer
cogatur transire; attamen quae hactenus diximus suadent
tremitum laryngis et librarum ad vocis formationem cou-
1:11rrere; attenta praecipue varietate maxima, quae in vo-
cis modificatione habetur. Novimus enim et singulos bo-
mines modificari quam maxime vocem, et in diversis ho-
minibus quam maxime diversum esse vocis sonum. Iam ve-
ro cum habeatur sibilus per solam labiorum compressio-
nem ,
inde expulso violenter aere , exigua est huiusmodi
soni diversitas; et omnes fere homines eumdem sonum ef-
ficiunt
,
licet in diversa intensione : ergo cum contra in
voce diversitas sit maxima et proprius cuique sit homini so-
nus , ad diversam librarum et laryngis materiam ac ten-
sionem recurrendum potius videtur. Scio equidem ab in-
strumentis pneumaticis etiam materia resistente compactis
et lingula instructis magnam edi posse varietatem sono-309
recessum
rum , atque ad instrumenta ista referri organum vocis ab
auctoribus non paucis. Verum non video quomodo glotti
dis fibrae se habeant ad vocis organum perinde ac lin
gula : si non ita haec movetur , ut epistomium alterne
aperiatur claudaturque ; licet ea citissime oscillet , nullus
inde prodibit sensibilis sonus . Iam vero glottidis fibrae
non sic oscillant , ut per mutuum accessum et
alterne claudatur aperiaturque ipsius glottidis foramen .
In glottidis fibris aeris irrumpentis impetu ad tremitum
concitalis auctores aliqui cum Ferreinio organum vocis ma
xime constituunt , illudque ad instrumenta fidicularia po
tissime revocant , minime considerantes quod hujusmodi fi
brae careant ea longitudine et crassitie , quae necessaria
esset ad graves atque ingentes humanae vocis tonos effi
ciendos,

145. Quaeres 

1.º qui sit defectus, ob quem mutis loquela deest. Cum plerique muti sint, quia sunt a nativitate surdi, quique proinde cum non possint alios loquentes audire ne loqui quidem discunt unquam; attamen sunt qui auditu pollent, at loquendi facultate destituuntur. Vitium in his multiplex esse potest; aut ex humorum nimietate et crassitie; aut ex fibrarum inelasticitate, qua etiam fit ut, timore insolito obrigescentibus fibris, vox impediatur in iis qui caeterum muti non sunt; vel ex nimia linguae turgescentia; vel alio vitio: adeoque non desunt exempla mutorum arte medica, aut etiam solius naturae auxilio loquelam adipiscentium. 

2.º Cur aves aliquae humanam vocem aemulentur, pleraeque non item. In psitlacis diligenter rem inspexit Kircherus, atque animadvertit pro corporis quantitate os satis magnum et excavatum, maxillas turgentes, linguam maxime flexilem, et rostrum superius contra indolem aliarum avium mobile; unde bruta pro majore vel minore aptitudine ad oris dilatationem, flexilitatem linguae, labiorum, vel rostri modificationem apta erunt ad sovum humanae vocis imitandum. Picae io
309
rum , atque ad instrumenta ista referri organum, vocis. ab
auctoribus non paucis. Verum non video quomodo glottidis fibrae se habeant ad vocis organum perinde ac lingula: si non ita haec movetur, ut epistomium alterne aperiatur claudaturque; licet ea citissime oscillet, nullus inde prodibit sensibilis sonus. Iam vero glottidis fibrae non sic oscillant, ut per mutuum accessum et recessum alterne claudatur aperiaturque ipsius glottidis foramen. In glottidis fibris aeris irrumpentis impetu ad tremitum concitatis auctores aliqui cum Ferreinio organum vocis maxime constituunt, illudque ad instrumenta fidicularia potissime revocant, minime considerantes quod huiusmodi fibrae careant ea longitudine et crassitie, quae necessaria esset ad graves atque ingentes humanae vocis tonos efiiciendos. 

145. Quaeres ↿∙∘ qui sit defectus , ob quem mutis
loquela deest. Cum plerique muti sint, quia sunt a nati-
vitate surdi
, quique proinde cum non possint alios loquen-
tes audire ne loqui quidem discunt unquam; attamen sunt
qui auditu pollent, at loquendi facultate destituuntur. Vitium
in his multiplex esse potest: aut ex humOrum nimietate
et crassitie; aut ex fibrarum inelasticitate , qua etiam fit
ut
,
timore insolito obrigescentibus fibris
, vox impedia-
'tur in iis qui caeterum muti non sunt; vel ex nimia lin-
guae turgescentia; vel alio vitio: adeoque non desunt exem-
pla mutorum arte medica
,
aut etiam solius naturae auxi-
lio loquelam adipiscentium. 2.(' Cur aves aliquae humanam
vocem aemulentur , pleraeque non item. In psittacis dili-
genter rem inspexit Kircherus , atque animadvertit pro
corporis quantitate os satis magnum et excavatum, maxil-
las turgentes, linguam maxime flexilem , et rostrum su-
perius contra indolem aliarum avium mobile; unde bru-
te pro majore vel minore aptitudine ad oris dilatationem
,
⋅ Hexilitaïem linguae ,
labiorum , vel rostri modificationem
apta'erunt ad sonum humanae vocis imitandum. Picae- iu-310
a
ter caeteras aves , et corvi antiquitus etiam ad voces hu
manas formandas instituebantur. 

3. ° An verum sit quod vox ita procreari possit ut infra laryngem genita videatur ,
ideoque sonus audiatur non tanquam ex ore , sed tanquam
ex alvo veniens. Ita de facto Pythonissae , quas Ethnicorum
historiae , et sacra scriptura etiam memorat, loquebantur:
aliquos tamen sine ulla suspicione daemoniaci operis ven
triloquos esse exemplis pluribus compertum est. Sicut enim
loquitur aerem expellendo , qui in larynge et glottide vo
cem excitat ; ita fieri potest ut aerem ore ac naribus at
lrahendo in gloutide item parem molum excitemus , sicque
non ex ore sed infra laryngem vox orta videatur,
प
be
AL

=== De auditus organo. ===
146. Externa auris pars palula est; et ex cartilagine intus concava atque elastica constat; quae in concham
sea cavitatem referentem conchae figuram desinit. Inser
vit ad colligendas uudas soni : hinc quasi natura duce qui
minus acuto pollet auditu , aut ad vocein nimis e lon
ginquo attendit , manum ad aures applicat , ut eo pacto
plures colligat aeris undas. Externa haec auris pars , quae
auricula simpliciter dicitur , musculis adornatur , quorum
ope sunt aliqui homines qui auriculam ad libitum mo
vent ; oves autem , equi et bruta alia multo facilius :
adnotant nonnulli Analomici ila necessariam esse exter
banc partem ut sonorus lenius allabatur
in internas cavitates, ut nonnisi confusa et quasi cum
inurmure fluentis aquae audiant ii, quibus auriculae abscis
sau sint. Animadvertendum tamen reptilia et aves hoc ex
lerno adminiculo carere.
Ad fundum conchae incipit meatus auditorius , qui est
canaliculus aliquanto tortuosus ; et ex majori latitudine in
minorem paullatim coarctator. Ita factum notat Val
9
nam aer 1
1
310.
ter caeteras aves , 'et corvi antiquitus etiam ad voces hn-
manas formandasinstituebantur. 3.0 An verum sit quod
'vox ita proci-cari possit ut infra laryngem genita videatur,
ideoque sonus audiatur non tanquam ex ore , sed tanquam
ex alvo veniens. Ita de facto Pythonissae , quas Ethnicorum
historiae ,
et sacra scriptura etiam memorat, loquebantur:
aliquos tamen sine ulla suspicione daemoniaci operis ven-
triloquos esse exemplis pluribus compertum est. Sicut enim
loquitur aerem expellendo , qui in larynge et glottide vo-
cem excitat; ita fieri potest ut aerem ore ac naribus at-
trahendo in glottide item parem motum excitemus, sicque
non ex ore sed infra laryngem vox orta videatur,
De auditu: organo.
146. Externa auris pars patula est, et ex cartilagi-
ne iutus concava atque elastica constat; quae in concham
sen cavitatem referentem conchae figuram desinit. Inser-
vit,ad colligendas undas soni: hinc qnasi natura duce qui
minus acuto pollet auditu
,
aut ad vocem nimis e lon-
giuquo attendit, manum ad aures applicat , ut eo pacto
plures colligat aeris undas. Externa haec auris pars, quae
auricula simpliciter dicitur
, musculis adornatur , quorum
0pe sunt aliqui homines qui auriculam ad Hibitum mo-
vent; oves autem
, equi et bruta alia multo facilius :
adnotaut nonnulli Anatomici itaqnecessariam esSe exter-
nam lianc partem ut aer sonorus lenius allahatur
in internas cavitates, ut nonnisi confusa et quasi- cum
murmure fluentis aquae audiant ii, quibus auriculae abscis-
sae sint. Animadverteudum tamen reptilia et aves hoc ex-
terno adminiculo carere.
Ad fundum conchae incipit meatus auditorins , qui est
canaliculus aliquanto tortuosus; et ex maiori latitudine in
minorem paullatim coarctatur. Ita factum notat Val-311
sa
salva at sonus intendatur magis , sicuti in recurvis lubis
a surdastris adhiberi solitis intenditur ; alii potius ad im
minuendum aeris impetum , ne in auris interiora fortius
impellat , has tortuositates in organo auditus a natura in
stilutas putant. In auditorio meatu humor quidam amarus
ac viscosus habetur , seu cerumen ; exsudat e glandulis
quas sebaceas vocant , et institutum est ut minima ani
malcula ab ingressu ad interiora auris arceantur .
Ad finem hujus canalis habetur membrana tympani:
haec membrana tenuissima , sicca , pellucida et valde ela
stica , obtensa est annullo , qui tamen totum circuitum
non complet ; et fere ad similitudinem pellis tympani mi
litaris cavitatem interiorem superambit : non est recte exten
sed curva nonnihil ; coacava scilicet respectu auris
externae , convexa ad partes internas . Fuit acerrima quae
stio , an membrana tympani omnem communicationem in
ter externam internamve aurem excludat , an contra per
via sit aeri externo. Argumentum pro communicatione va
lidum est , quod aliqui fumum ore exceptum per aurem
emittunt ; neque id semper imposturae vertendam est ,
ut compertum fuisse Nolletus ait a viro , cni Academia
regia jussum fecerat facti veritatem explorare. Argumen
tum contra communicationem est , quod Valsava , immis
so in aurem internam hydrargyro , quantumvis excute .
retur , nihil unquam per externam aurem defluxit ; quam
quam reponere soleat qui communicationem tuetur , quod
in cadaveris organo non est necesse partium structuram sal
vari.
Post pellem tympani habetur cavitas aere plena , quae
capsula dicitur , quaeque cum membrana praedicta tym
panum constituit. In hac sunt quatuor ossicula quae ap
pellantur malleus , incus , os orbiculare , stapia. Alii tria
tantum numerant , omisso osse orbiculari , vel quia, cum
ompium humani corporis ossiam minimum sit , adeo ut
non superet dimidium grani millii , animadversionem fu
31↿⋮ salva ut sonus intendatur magis , sicuti in recurvis tubis
'a snrdastris adhiberi solitis intenditur; alii potius ad im-
miuuendum aeris impetum , ne in auris interiora fortius
impellat, has tortuositates in organo auditus a natura in-
stitutas putant. In auditorio meatu humor quidam amarus
ac viscosus habetur , seu cerumen; exsudat e glandulis,
quas sebaceas vocant , et institutum est ut minima ani-
malcula ab ingressu ad interiora auris arceantur.
Ad finem hujus canalis habetur membrana tympani:
haec membrana tenuissima , sicca , pellucida et valde ela-
stica
, obtensa est annnllo , qui tamen totnm circuitum
non complet; et fere ad similitudinem pellis tympani «mi—
litaris cavitatem interiorem superambit: non est recte exten-
sa , sed 'curva nonnihil : concava scilicet respectu auris.
externae
, convexa ad partes internas. Fuit acerrima qnae-
stio , an membrana tympani omnem communicationem in-
ter externam internamve aurem excludat , an contra -per-
via sit aeri externo. Argumentum pro-communicatione va-
lidum est
, quod aliqui fumum ore exceptum per aurem
emittunt; neque id semper imposturae vertendam est,
ut compertum fuisse 'Nolletus ait a- viro, cni Academia
regia iussum fecerat facti veritatem explorare. Argumen-
tum eontra communicationem est
, quod Valsava ,
immis-
so in aurem internam hydrargyro , quantumvis excute-
retur, nihil unquam per externam aurem defluxit; quam-
quam reponere soleat qui communicationem tuetur
, quod
in cadaveris organo non est necesse partinm structuram sal-
var]. '
Post pellem tympani habetur- cavitas aere plena , quae
capsula dicitur
, quaeque eum membrana praedicta tym-
panum constituit. In hae sunt quatuor ossicula quae ap-
pellantur mallens
.
incus , os orbiculare
, stapia. Alii tria
tantum numerant
, omisso osse orbiculari, vel quia, cum
omnium humani uerporis ossium minimum sit, adeo ut
non superet dimidium grani millii
, animadversionem fu-312
1
1
gerit : vel quia ita adhaeret slapiae et incudi , at cum al
tero ex his confundi potuerit, Circa haec ossicula nolan
dum , quod ejusdem magnitudinis sint tam in infante quam
in adulto viro contra aliarum omnium corporis partium in- .
dolem , quae aetatis progressu augentur. Hoc ideo factum
esse docet Valsalva , ne augmento partium auditui inservien
tium alia sit sonorum ratio adulla aetate ac fuit ab ini
tio ; et ideas gravis atque acuti quas pueri imbibimus, ma
tare aetate proficiente cogamur.
In tympani cavitate habetur canalis quidam seu lu
ba Eustachiana dicta ab ipsius inventore : per hanc tu
bam ab interna auris cavitate obtinet communicatio cum
cavitate oris; desinit enim ista tuba ad radices uvae; quem
prope locum etiam interiora nasi communicant : hujus tu
bae ope fit , ut sonus ex oris cavitate auri communicetur,
ideoque qui dentibus stringit corpus resonans sobum au.
dit etiam auribus impeditis ; et surdastri hiante ore so
nos excipere solent , ut tali pacto juvelur melius auditio.
Praeter foramen ex quo tuba Eustachiana procedit ,
duo alia babentur in tympani cavitate foramina , quorum
unum dicitur fenestra ovalis , allerum fenestra rotunda.
Feuestra ovalis basi slapiae occluditur, rotunda solo mem
branulae tegumento obtegitur. .
Ex tympani cavitate per duo praedicta foramina , ovale
scilicet ac rotundum , itur in labyriothum , qui - est inte
rior alia cavitas in osse petroso ulterius excavata , et quo
dam liquido plena : in hac tres partes distingui solent ;
prima est vestibulum labyrinthi ; secunda constat tribus
ossiculis semicircularibus , quibus nomen labyrinthi pecu
liarins aliqui tribuunt ; tertia est cochlea seu limax, quae
ex osse constat in cochleae modum conlorto duos gyros
cum dimidio faciente. Elsi cochlea unus canalis videri
possit , est lameu revera duplex : dividitur enim secun
dum longitudinem medio segmento , parim osseo , partim
membranaceo , quod dicitur lamina spiralis. Cochlea in
1
1
1
i
1
1
1
•
2
0
1
1
1
.
312
gerit: vel quia ita adhaeret stapiae et incudi , at cum al-
tero ex his confundi potuerit. Circa haec ossicula notan-
dum , quod eiusdem magnitudinis sint tam in infante quam
in adulto viro contra aliarum omnium corporis partium in--
dolem, quae aetatis progressu augentur. Hoc ideo factum
esse docet Valsalva, ne augmento partium auditui inservien-
tium alia sit sonorum ratio adulta aetate ac fuit ab iui-
tio; et ideas gravis atque acuti quas pueri imbibitüus, mu-
tare aetate proficiente cogamur.
ln tympani cavitate habetur canalis quidam seu tu-
ba Eustachiaua dicta ab ipsius inventore: per hanc tu-
bam ab interna auris cavitate obtinet communicatio cum
cavitate oris; desinit enim ista tuba ad radices uvae; quem
prope locum etiam interiora nasi communicant :-huius tu-
bae upe fit
, ut sonus ex oris cavitate auri .communicetur,
ideoque qui dentibus stringit corpus resonans sonum su-
dit etiam auribus impeditis ; et surdastri hiante ore so-
nos excipere solent , ut tali pacto iuvetur melius auditio-
Praeter foramen ex quo tuba Eustachiana procedit,
duo alia habentur in tympani cavitate foramina , quorum
unum dicitur fenestra ovalis, alterum fenestra rotunda.
Feuestra ovalis basi stapiae occluditur, rotunda solo mem-
branulae tegumento obtegitur. .
Ex tympani cavitate per duo praedicta foramina
, ovsle
scilicet ac rotundum ,
itur in labyrinthum
, qui-est inte-
- rior alia cavitas in esse petroso ulterius excavata
,
et quo-
dam liquido plena: in hac,tres partes distingui solent;
prima est vestibulum labyrinthi ; secunda constat tribus
ossiculis semicircularibus , quibus nomen labyrinthi pecu-
liarius aliqui tribuunt; tertia est cochlea seu limax, quae
ex osse constat in cochleae modum contorto duos gyros
cum dimidio. faciente. Etsi cochlea unus canalis videri
possit ,
est tamen revera duplex: dividitur enim secun-
dum longitudinem medio segmento, partim osseo , partim
membranacea , quod dicitur lamina spiralis. Cochlea in313
avibus deest , si vera refert Boyle ; at ipsemet notat de
fectum hunc suppleri per cavitatem oblongam instar sacci.
Plures rami nervei per foramina perexigua ex eo ,
qui dicitur uervus auditorius , propagati per totam fere
aurem distribuuntur : in labyrinthum per quinque fora
mina ingrediuntur Gibrae nerveae , et ejus cavitatem inves
tiunt ; sed de nervis totum auris organum permeantibus
non vacat disserere. Id unum adnoto ex Mairano , spira
lem laminam fibrillis ita instructam esse ut quemadmo
dum ipsa ascendens ad cochleae apicem . semper angustior
fit , ita fibrae ipsae breviores semper evadunt.
147. Quaenam vero ex hactenus descriptis auris par
tibus pro praecipuo atque immediato auditionis organo sta
tueuda est ?. Aliqui membranam tympani assignarunt : at
experientia constat quod hac membrana lacerata vel erepta
adhuc manet auditio aliqua. Alii inepte in ossiculis intra
capsulam contentis organom auditus statuerunt , et sonum
ab anima immediate perceptibilem ex horum collisione
nasci affirmarunt. Praeter quam quod solus malleus in a
nimantibus quibusdam habeatur, falsum omnino est collidi
invicem haec ossicula atque inde sonum creari : adnexum
enim est caput mallei firmiter corpori incudis , et hujus
processus alter stapiae; adeoque cum aer exterous tympa
ni membranam impellit, omnia per modum unius intromit
tuntur et conjuncta simul sese restituunt ad locum pristi
num. Magis autem absona est illorum sententia , qui in
aere per capsam et labyrinthum existente auditus organum
collocabant: aerem hunc, quem innatum, insitum, vernacu
lum, implantatum dicebant, animatum statuere non vere
bantur. Communiter nunc auditus organum collocatur in
fibrillis nerveis per aurem internam, ac praesertim intra co
chleam disseminatis. Tremores itaque a corpore
excitati communicantur membranae tympani; tum per aea
rem in tympano existentem , nec non per ossiculorum se
riem, ad parietes asque labyrinthi et praecipue ad dupli
Sonoro
21
313
avibus deest , si vera refert Boyle ;
'at ipsemet notat de-
fectum hunc suppleri per cavitatem oblongam instar sacci.
Plures rami nervei per foramina perexigua ex eo ,- qui dicitur nervus auditorius, prcpagati per totam fere
aurem distribuuntur: in labyrinthum per quinque fora-
mina ingrediuntur fibrae nerveae, et eius cavitatem inves-
tiunt; sed de nervis totum auris organum permeantibus
non vacat disserere. Id unum adnoto ex Mairano , spira-
lem laminam fibrillis ita instructam esse ut quemadmo-
dum ipse ascendens ad cochleae apicem- semper angustior
fit
,
ita fibrae ipsae breviores semper evadunt.

147. Quaenam vero ex hactenus descriptis auris par-
tibus pro praecipuo atque immediatoauditionis organo sta-
tuenda est ?. Aliqui membranam tympani assignarent : at
experientia constat quod hac membrana lacerata vel erepta
adhuc manet auditio aliqua. Alii inepte in ossiculis intra
capsulam contentis organum auditus statuerunt
,
et sonum
ab anima- immediate perceptibilem ex horum collisione
nasci affirmarunt. Praeter quam quod solus malleus in a-
nimantibus quibusdam habeatur, falsum omnino est collidi
invicem haec ossicula atque inde sonum creari: adnexum
enim est caput mallei firmiter corpori incudis ,
et huius
processus alter stapiae; adeoque cum aer externus tympa-
ni membranam impellit, omnia per modum uniua intromit-
tuntnr et coniuncta simul sese restituunt ad, locum pristi-
num. Magis autem absona est illorum sententia , qui in..
aere per capsam et labyrinthum existente auditus organum
collocabant: aerem hunc, quem innatum, insitum, vernacu-
lum, implantatum dicebant, animatum statuere non vere-
bantur. Communiter nunc auditus organum collocatur in
fibrillis nerveis per aurem internam, ac praesertim intra co-
chleam disseminatis. Tremores itaque a sonoro corpore
excitati commnnicantur membranae tympani; tum per aes-.
rem in tympano existentem, nec non per ossiculorum se-
riem, ad parietes usque labyrinthi et praecipue ad dupli-
21
is314
cem fenestram , ovalem ac rolundam , transmissi deducuntur
ad liquidum cavitate labyrinthi contenlum ; inde vero ad fi
brillas nerveas praedictas, atque ad nervum ipsum audito
rium: unde fit, ut ex lege commercii anima ad sensationem
soni determinetur. Animadvertit Mairanus, quod sicut in
instrumentis quibusdam musicis binae et binae chordae pro
tonis singulis disponuntur, ita in lamina spirali nerveae fi
brillae dispositae sint : ex quo infert huc potius spectare
organum quam ad alias auris internas partes.
148. Quaeres 1º. Cur quibusdam grata , aliis pene ni
hil, aut etiam molesta sit harmonia. Alibi ( 121 ) dictumn
est chordam upisonam facile ad tremitum concitari: aliam
item , sed difficilius prout majorem minoremve cum chor
da percussa harmonicam proportionem habet. Alert Kir
cherus aliud experimentum , quod ad rem aptum est etiam
magis : experimentum ita se habet. Quinque sumantur
scyphi vitrei.ejusdem magnitudinis et capacitatis , et unus
quidem liquore impleatur, qui acquavite dicitur; alter vi
no ; tertius aqua puriori; quartus aqua communi ; quintus
crassiori aqua: tum ora cujusque poculi confricando sonus
quam fieri potest acntissimus excitetur. In primo quidem
• scypho spiritus ille maxime subsultabit; vinum moderatam su
bibit concitationem ; adhuc moderatior erit molus purio
ris aquae, et ita porro . Ex his intelligitur quomodo variae
in variis hominibus partium corporis commotiones orian
tur e sono. Cum autem animi molus, in quibus voluptas
consistit vel molestia , pendeant ex partium corporis affe
ctionibus; iis gratissima accidere poterit harmonia, quibus
ea solidorum ac fluidorum constitutio est , ut in iisdem com
motio consequatur impressionem factam in organo auditus
satis . vivida et animi moribus cum voluptate conjunctis ex
citandis apta: ii erunt ad harmoniam indifferentes, in qui
bus impressionem factam in organo auditus vix ulla con
sequitur alteratio solidarum fuidarumve corporis partium
quae pariat animi motus vel consonos, vel incongruos: iis
314
cem fenestram, ovalem ac rotundam, transmissi deducuntur
ad liquidum cavitate labyrinthi contentum; inde vero ad fi-
brillas nerveas praedictas, atque ad nervum ipsum audito-
rium: nnde fit, ut ex lege commerciianima ad sensationem
aoni determinetur. Animadvertit Mairanus, quod sicut in
instrumentis quibusdam musicis binae et binae chordae pro
tonis singulis disponuntur, ita ip lamina spirali nerveae fi-
brillae dispositae sint: ex quo infert huc potius spectare
organum quam ad alias auris internas partes.
148. Quaeres 1&quot;. Cur quibusdam grata, aliis pene ni-
hil, aut etiam molesta sit harmonia. Alibi (121 ) dictum
est chordam unisonam facile ad tremitum concitari: aliam
item, sed difficilius prout majorem minoremve cum chor-
da percussa harmonicum proportionem habet. Affert Kir-
cherus aliud experimentum, quod .ad rem aptum est etiam
magis : experimentum ita se habet. Quinque sumantur
scyphi vitrei-ejusdem magnitudinis et capacitatis, et unus
quidem liqum'e impleatur, qui acquavite dicitur; alter vi-
no; tertius aqua PUI'lOl'i; quartus aqua communi; quintus
crassiori aqua: tum ora cujusque poculi confricando sonus
quam fieri potest acutissimus excitetur. In primo quidem
scypho spiritus ille maxime subsultahit; vinum moderatam su-
bibit concitationem; adhuc moderatior erit motus purio-
ris aquae, et ita porro. Ex his intelligitur quomodo variae
in variis hominibus partium corporis commotiones orian-
tur e sono. Cum autem animi motus, in quibus voluptas
consistit vel molestia, pendeant ex partium corporis affe-
ctionibus; iis gratissima accidere poterit harmonia, quibus
easolidorum ac fluidorum constitutio est, ut in iisdem com-
motio consequatur impressionem factam in organo auditus
satis.vivida et animi motibus cum voluptate conjunctis ex-
citandis apta: ii erunt ad harmoniam indifferentes. tu qui-
bus impressionem factam in Organo auditus vix ulla con-
sequitur alteratio solidarum fluidarumve corporis partium ∙
quae pariat animi motus vel consouos, vel incongruos: iis1
1
315
denique molestia etiam accidet, quibus ex impressione ner
vorum acusticorum contingat incongrua motuum alteratio
in partibus corporis ad pracfatos animi molus inservienti
bus: quo fit etiam mechanice ut alii aliis sonorum gene
ribus vel delectentur magis, vel contra. Hanc tamen me
chanicam causam non arbitror esse sufficientem atque adae
quatam: admittenda est causa ex rationali natura hominis
pendens. Consistit harmonia in proportione illa , quam so
ni habent inter se ; unde fit ut in organo auditus vibra
tiones diversi generis, aliae frequentiores, aliaė tardio
res efficiantur: dum vibrationes istae organum anditus af
ficiunt, mens easdein comparat inter se, earumque propor
tionem animadvertit : si haec proportio ejusmodi sit ut fa
cile possit a mente percipi, et vibrationes facile comparari
queant, ex hoc gaudet mens; si confusa et inordinata vi
brationum sit comparatio , neque has mens facile con
ferre inter se potest, obruelur taedio: et quia imperi
tas in musica facilius comparat simpliciores consonantias quam
magis compositas, ideo musica planissima vulgo arridet ; qui
vero periti sunt et copiosioribus compositiouibus assueti, vix
patiuntur musicam nimis simplicem: item cum ex consue
tadine pendeat ut aliquas harmonicas proportiones faci
lius mens assequatur quam alias ; inde oritur at volu
ptas ex eo musices genere major sit, cui quis sit assue
tus; adeo ut ex hoc capite diversis nationibus diversae
placeant musicae species. Porro ab exercitatione etiam
profluit ut gratior accidat musica , etiam qua ex parte
mechanice voluptatem parit; ex assuetudine enim in fi
brillas nerveas docilitas inducitur ad recipiendas facilius
impressiones harmonicas.
2º. Cur duabus auribus unus idemque sonus au
diatur. Communis responsio est hujusmodi : cum in utra.
que aure creetur simillima impressio; non duplicem , sed
voam sensationem ab anima haber¡ necesse est. Qua in
re scite animadvertit Valsalva , summa industria provisum
315
denique molestia etiam accidet, quibus ex impressione ner-
vorum acnsticorum contingat incongrua motuum alteratio
in partibus corporis ad praefatos animi motus inservienti-
bus: quo fit etiam mechanica ut alii aliis sonorum gene-
ribus vel delectentur magis, 'vel contra. Hanc tamen me-
chanicam causam non arbitror esse sufficientem atque adae-i
quatam: admittenda est causa ex rationali natura hominis
pendens. Consistit harmonia in praportione illa, quam so-
ni habent inter se; unde fit ut in organo auditusvibraP-
tiones diversi generis, aliae frequentiores, aliae tardio-
res efficiantur: dum vibrationes istae organum auditus af-
Hciunt, mens easdem comparat inter se, earumque propor-
tionem animadvertit: si haec proportio ejusmodi sit ut fa-
cile possit a mente percipi, et vibrationes facile comparari
queant, ex hoc gaudet mens; si confusa et inordinata vi-
bratiouum sit comparatio , neque has mens facile con-
ferre inter se potest, obruetur taedio: et quia imperi-
tus in musica facilius comparat simpliciores consonantias quam
magis compositas, ideo musica planissima vulgo arridet ; qui
vero periti sunt et c0piosioribus compositionibus assueti, vix
patiuntur musicam nimis simplicem: item cum ex consue-,
tudine pendeat ut aliquas harmonicas preportiones faci-
lius mens assequetur quam alias; inde oritur ut volu-
ptas ex eo mus1ces genere major sit, cui quis sit assue-
tus; adeo ut ex hoc capite diversis nationibus diversae
placeant musicae species. Porro ab exercitatione etiam
profluit ut gratior accidat musica, etiam qua ex parte
mechanica voluptatem parit; ex assuetudine enim in fi-
brillas nerveas docilitas inducitur ad recipiendas facilius
impressiones harmonicas. ⋅
20. Cur duabus auribus unus idemque sonus au-
diatur. Communis responsio est huiusmodi: cum in utra-
que aure creetur simillima impressio; non duplicem, sed,
unam sensationem ab anima haberi necesse est. Qua in
re scite animadvertit Valsalva, summa industria provisum316
fuisse a natura ut in utraque aure quam simillima es
sent organa omnia ; adeo ut, dum in diversis hominibus
structura partium nonnihil variat, in eodem tamen bomi
mine nulla prorsus sit utriusque auris vel minima variatio .
Notetur illud : quemadmodum eadem chorda varios
tonos varia tensione edere potest, ita membrana tympani
lenditur diversimode ut variis tonis aple accomodetur ;
eapropter manubrium mallei eidem adnexum est, et ba
sis stapiae eodem modo membranae fenestrae ovalis: ten
sio autem et relaxatio membranae, nobis insciis , potest na
turaliter fieri. Itaque, ut primae vibrationes membranam
feriunt, admonita natura organum opportuna tensione aptat
ut respondens tonus in successivis vibrationibus fiat ma
gis vel minus sensibilis.
316
fuisse a natura ut in utraque aure quam simillima es-
sent organa omnia; adeo ut, dum in diversis hominibus
structura partium nonnihil variat, in eodem tamen homi-
miue nulla prorsus sit utriusque auris vel minima variatio.
Notetur illud: quemadmodum eadem chorda varios
tonos varia tensione edere potest, ita membrana tympani
tenditur diversimode ut variis tonis apte accomodetur;
eapropter manubrium mallei eidem adnexum est, et ba-
sis stapiae eodem modo membranae fenestrae ovalis: ten-
sio autem et relaxatio membranae, nobis insciis,potest na-
turaliter fieri. Itaque, ut primae vibrationes membranam
feriunt, admonita natura organum opportuna tensione aptat
ut respondens tonus in successivis vibrationibus fiat ma-
gis vel minus sensibilis.INDEX
RERUM QUAE IN PRIMO VOLUMINE CONTINENTUR.
MECHANICA E PRINCIPIA
Notiones praeambulae. pag. 1 .
Molus uniformis et varius : velocitas et quantitas mo
tas in motu uniformi. num . 1 .
Corporum indifferentia ad motum et ad quietem: quid
vires : quid earum aequilibrium ; et quomodo repraesen
tentur sive per lineas rectas, sive per numeros . n. 2, 3, 4.
Principiom motus • relativi : vires sunt ut quantitates
motus , n. 5, 6 .
Principium actionis et reactionis : mutatio status in
corpore haud repente gignitur a viribus extrinsecis , sed
per gradus indefinitae attenuationis capaces: quid vires in
stanlaneae et continuae. n. 7 .
De virium compositione et resolutione , deque earum
momentis et aequilibrio : aliquid quoque notatur de
vecte, axe in peritrochio, trochlca etc. pag. 6.
Compositio virium materiali puncto applicatarum: ae
quilibrium: varia circa virium resolutionem .. n. 8. 9. 10.
⋅ N D EX
RERUM QUAE IN ramo VOLUMINE CONTINENTUR.
' MECHANICAE PRINCIPIA
∙
W
⋅∙
Nott'ones praeambulae. pag. 1.
Motus uniformis et varius: velocitas et quantitas mo-
tus in motu uniformi. . . . . . . . . . num-1.
Corporum indifferentia ad motum et ad quietem: quid
vires: quid earum aequilibrium; et quomodo repraesen-
tentur sive per lineas rectas, sive per numeros. n.2, 3, 4.
. Principium motus 'relativi: vires sunt ut quantitates.
motus. ∙ ∙ ∙ ∙↴ ∙ ∙
.' ∙ ∙ ∙ ∙ ∙ ∙ n. 5, 6.
Principium actionis et reactionis: mutatio status in
corpore haud repente gignitur a viribus extrinsecis ,
sed
per gradus indefinitae attenuationis capaces: quid vires in-
stantaneae et-continuae.
. . . . . . . . . n. 7.»
De virium compositione et resolutione , deque earum
momentis et aequilibrio : aliquid quoque notatur de
vecte, axe in peritrochio, trochlea etc. pag. 6.
Compositio virium materiali puncto applicatarum: ae-
quilibrium: varia circa virium resolutionem.. n. 8. 9. 10.318
Compositio duarum virium extremis rectae rigidae punctis
applicatarum, et in eodem plano jacentium: aequilibrium circa
immobile punctum: principiam velocitatum virtualium in ordi
ne ad istiusmodi vires: momenta virium quoad punctum ( M) :
momentum resultantis aequatur summae ex momentis com
ponentium si hae in eamdem plagam circa ( M ) nituntur
movere puncta , quibus applicitae sunt; aequatur differentiae
si nituntur movere in plagas contrarias. n. 10. 10. 20.30,
Haec ipsa extenduntur ad quemvis numerum virium in
dato plano jacentium.
n. 10. 4º , 5º , 6º.
Binae vires haud jacentes in eodem plano nequeunt
ad unicam vim aequipollentem traduci : vires in aequili
brio constitutae; sollicitantesque vel solidum liberumque cor
pus, vel solidam corpus mobile duntaxat circa punctum fi
xum, vel solidum corpus mobile tantummodo circa asem
fixum : momenta quoad axem . n. 10: 70. ... 10 °.
Vires parallelae: vis inde resaltans: earum centrum :
momenta quoad planum: respondens theorema n . 11 , 12 ,
13. 10. 2º. 3º.
Parallelarum virium systema consistit in aequilibrio sub
duabus conditionibus simul explendis; altera est ut evane
scat earum summa; altera ut evanescat summa ex earum
momentis in ordine ad duo plana ipsis viribus paral
lela . n. 13. 4º. 5º.
.
Etsi vires non sunt parallelae, possunt tamen rednci
ad terna ejusmodi systemata, quorum primum coalescat ex
viribus parallelis axi OZ, secundum ex viribus jacentibus
in plano XOY simulque parallelis axi Qy, tertium ex vi
ribus agentibus juxta axem OX: aequilibrii conditiones quoad
systemata rigida viribus minime parallelis sollicitata; 1º , in
318
Compositio duarum virium extremis rectae rigidae punctis
applicatarum,etin eodem plano jacentium: aequilibrium circa
immobile punctum: princi pium velocitatum virtualium in ordi-
ne ad istiusmodi vires: momenta virium quoad punctum (M):
momentum resultantis aequatur summae ex momentis com-
ponentium si hae in eamdem plagam circa (M ) nituhtur
movere puncta, quibus applicitae sunt; aequatur differentiae
si nituntur movere in plagas contrarias. n. 10. 10. 2230.
∙ Haec ipsa extenduntur ad quemvis numerum virium in
dato plano jacentium.
. . . . . . n. 10. 40. 50. 6&quot;.
Binae vires haud jacentes in eodem plano nequeunt
ad unicam vim aequipollentem traduci : vires in aequili-
brio constitutae; sollicitantesque vel solidum liberumque cor-
pus, vel solidum corpus mobile duntaxat circa punctum fi-
xum, vel solidum corpus mobile tantummodo circa axem
fixum: momenta quoad axem. .'
. . n. 10: 70. 10&quot;-
Vires parallelae: vis inde resultans: earum centrum:
momenta quoad planum: respondens theorema &quot;. 11, 121
13. 10. 20. 30.
Parallelarum virium systema consistit in aequilibrio sub
duabus conditionibus simul explendis; altera est ut evane-
scat earum summa; altera ut evanescat summa ex earum
momentis in ordine ad duo plana ipsis viribus paral-
lela.
. . . . . . . . . . . . . n.13.4&quot;.5'-
Etsi vires non sunt parallelae, possunt tamen -reduci
ad terna eiusmodi systemata. quorum primum coalescat ex
viribus parallelis axi OZ, secundum ex viribus jacentibus
in plano XOV simulque parallelis axi QT, tertium ex vi-
ribus agentibus juxta axem OX: aequilibrii conditiones quoad
systemata rigida viribus minime parallelis sollicitata;1&quot;.in319
hypothesi systematis liberi; 2 °. in hypothesi systematis de
tenti puncto fixo; 3º . in hypothesi systematis detenti axe
fixo: conditio explenda ut plures vires minime parallelae
traduci possint ad unicam aequipollentem . n. 13. 6 ... 11º.
Duo solida corpora , datis viribus sollicitata , sese in
vicem aeque premendo apud datum mutui contaclus pan
ctum manent in aequilibrio : determinatur istiusmodi pres
sionis magnitudo. n. 13. 12 .
Solidum corpus , datis viribus sollicitatum, detinetur
duobus punctis fixis, sumptis in axe v. gr. OZ: determi
nantur pressiones exercitae in puncta illa juxta coordi
nalos axes OX, OY, OZ. n. 13. 13 .
Exempla aequilibrii in quibusdam machinis, praeci
so attritu : aequilibrium punctoruni materialium juncto
rum flis determinatae quidem longitudinis sed mobili
bas circa data puncta. n. 14. 15. 16 .
De centro gravitatis. pag. 33.
Varia de vi gravitatis, de corporum densitate , deque
specifica eorum gravitate: linea directionis. n . 17 , 18 , 19.
Generales formulae determinantes centrum gravita
tis: inveniri potest ratione mechanica: peculiari metho
do determinalur in triangulo et pyramide triangulari, n. 20.
De corporum collisione. pag . 37 .
Normalis collisio : 1º. corporum non elasticorum : 2 ° .
corporum perfecte elasticorum : 3º . corporum imperfe
cte elasticorum . n. 21 , 22, ... 25 .
319
hypothesi systematis liberi: &quot;. in hypothesi systematis de-
⋅ teuti puncto fixo: 30. in hypothesi systematis detenti axe
fixo: conditio explenda ut plures vires minime parallelae
traduci possint ad unicam aequipollentem. n. 13. 60... 1'l0.
Duo solida corpora, datis viribus sollicitata, sese in-
vicem aeque premendo apud datum mutui contactus pun-
ctum manent in aequilibrio: determinatur istiusmodi pres-
sionis magnitudo. . . . . . . . . . n. 13. 120.
Solidum corpus . datis viribus sollicitatum. detinetur
duobus punctis fixis, sumptis in axe v- gr. OZ: determi-
nantur pressiones exercitae in puncta illa iuxta coordi-'
natos axes OX, 0ï,OZ. . . . ∎∙ ∙ ∙ n.13.130.
Exempla aequilibrii. in quibusdam machinis, praeci-
so attritu : aequilibrium punctorum materialium iuncto-
rum filis determinatae quidem longitudinis sed mobili-
bus circa data puncta.
. . . . . . n.14.15.'16.
De centro gravitatis. pag. 33.
Varia de vi gravitatis, de corporum densitate.deque
specifica eorum gravitate: linea directionis. n. 17, 18, 19.
Geuerales formulae determinantes centrum gravita-
tis: inveniri potest ratione mechanica: peculiari metho-
do determinatur in triangulo et pyramide triangulari. n. 20-
Dä corporum collisione. pag. 37-
Normalis collisio: 10. corporum non elasticorum: 2&quot;.
corporum perfecte elasticorum : 3&quot;. corporum imperfe-
cte elasticorum. . .
-
. . . . . n.21,22,...25.320
Obliqua eorumdem corporum collisio. n . 26.
De motu rectilineo utcumque vario. pag. 42
Praemittantur nonnulla ex analysi infinitesimali, e
jusque ad res geometricas applicatione. n. 27. 10.2 ... 300.
Formulae spectantes ad motum rectilineum utcumque
varium : formulae quoad motum rectilineum uniformiter
varium: vis acceleratrix : vis motrix. n. 28.
Formulae pertinentes ad motum rectilineum utcum
que varium applicantur ad materiale punctum sollicita
tum vi acceleratrice, quae sit distantiae a dato centro pro
portionalis. n. 29.

De verticali gravium descensu atque ascensu . pag.
65.
Formulae, legesque huc spectantes: motus gravium in
machina Atwoodi. n . 30, 31 , 32 .
Quid si gravium descensus vel ascensus fiat in me
dio resistente sub ea conditione, ut resistentia medii sit pro .
portionalis quadrato velocitatis. n. 33.
De gravium descensu per plana inclinala ; de attritu ;
deque cochlea, et cuneo. pag. 71.
Formulae determinantes descensum gravium per plana
inclinata: gravium descensus per plana inclinata compara
tur cum verticali eorum descensu. n . 34, 35.
320
⋅∡ Obliqua eorumdem corporum collisio. . ∙⋅ n. 26.
De motu rectilineo utcumque uario. pag. 42
Praemittuntur nonnulla ex analysi infiuitesimali, e-
iusque ad res geometricas applicatione. n. 27. 10. 2&quot;....300.
Formulae spectantes ad motum rectilineum utcumque
varium: formulae quoad mo'tum rectilineum uniformiter
varium: vis acceleratrix: vis motrix. . . . . . n. 28.
Formulae pertinentes ad motum rectilineum utcum-
que varium applicantur ad materiale punctum sollicita-
tnm vi acceleratrice. quae sit distantiae a dato centro pro-
portionalis. ............n.ag.
!
De verticali gravium descensu atque ascensu. pag. 65.
Formulae, legesque huc spectantes: motus gravium in
machina Atwoodi. . . . . . . . . n. 30,31,32-
Quid si gravium descensus vel ascensus liat in me-
dio resistente sub ea conditione, ut resistentia medii sit pro-
portionalis quadrato velocitatis. . . . . . . n. 33-
De gravium descensu per plana inclinata; de attritu; ⇥
deque cochlea, et cuneo. ∙ pag. 71.
Formulae determinantes descensum gravium per plana
inclinata: gravium descensus per plana inclinata compara-
tur cum verticali eorum descensu. . . . n. 34, 35-321
Gravium descensus per plura plana inclinata sibi con
rigua . n. 36.
non.
Unde orialur attritus , caeteris paribus , est proportio
nalis pressioni : quomodo habeatur ratio attritus in motu
gravium per plana inclinata : grave in plano inclinato li
brandum potentia aliqua, sive habeatur ratio attritus , sive
n. 37. 10. 20 30
Aequilibrii leges in cochlea, et cuneo. n. 37. 4º. 5º.
Spectatur attritus in aequilibrio cochleae: itemque in
aequilibrio corporis ad rotatilem motum circa fixum cy
lindrum sollicitati : in machinis praeter resistentiam ex at
tritu spectanda etiam est resistentia ex funibus n. 37.6º.70.8° .
De motu gravium oblique projectorum . pag . 81 ,
Aequatio ad curvam, quam describunt gravia oblique
projecta; istiusmodi curva dicitur parabola. n. 38, 39.
Amplitudo jactus: maxima jactus amplitudo habetur sub
angulo projectionis semirecto: sub quo angulo projiciendum sit
grave ut offendat in datum scopum : altitudo jactus : ali
quid subjungitur de proprietatibus praefatae curvae. n. 40. 1º.
2 ° .... 70
Quid si gravia oblique projiciantur in medio resi
n. 41 .
stente.
De generalibus quibusdam proprietatibus motus curvili
nei, orti a viribus quarum una determinat materiale
punctum ad motum uniformem , altera ipsi materia
li puncto est continue applicata .
.
pag. 85.
Ubi in aliquo curvae puncto vis acceleratrix desinat
agere, excurret mobile per tangentem în punclo illo : ubi
321
Gnavium descensus-per plura plana inclinata sibi con-
ligua...............n.36.
Unde oriatur attritus. caeteris paribus, est proportio-
nalis pressioni: quomodo habeatur ratio attritus in motu
gravium per plana inclinata: grave in plano inclinato li-
brandum potentia aliqua, sive habeatur ratio attritus, sive
non. . , . . . . . . . . . . n. 37.10.2030.
Aequilibrii leges in cochlea, et cuneo. n. 37. 40. 50.
Spectatur attritus in aequilibrio cochleae: itemque in
aequilibrio corporis ad rotatilem motum circa fixum cy-
lindrum sollicitati: in machinis praeter resistentiam ex et-
tritu spectanda etiam est resistentia ex funibus n. 37.60.70.80.
De motu gravium oblique projectorum: pag. 81,
∙ Aequatio ad curvam, quam describunt gravia oblique
proiecta; istiusmodi curva dicitur parabola. . n. 38. 39.
Amplitudo iactus: maxima jactus amplitudo habetur sub
angulo projectiouis semirecto: sub quo angulo proiiciendum sit
grave ut offendat in datum scopum : altitudo jactus: ali-
quid subiungitur de proprietatibus praefatae curvae. n. 40. 10.
20 .... 70.
Quid si gravia oblique projiciantnr in medio resi-
stente. ↖∙∙∙∙∙∙∙∙∙∙⋅∙∙∙∥∙∡∎∙
De generalibus quibusdam praprietatibus motus curvili-
'nei, orti a viribus quarum una determinat materiale
punctum ad motum uniformem. altera ipsi materia-
li puncto est continue applicata.
. . . . pag. 85-
Ubi in aliquo curvae puncto vis acceleratrix desinat
agere, excurret mobile per tangentem in puncto illo: ubi322
tempore finito angulus, quem efformat vis acceleratrix cum
directione tangentis , fuerit semper acutus, acquiret mo
bile incrementum velocitatis finitum ; si semper obtusus,
patietur decrementum finitum ; si semper rectus , veloci
tas manebit constans: quadratum velocitatis adaequat vim
acceleratricem ductam in dimidium chordae, quae ex ejus
directione abscinditur ab osculatore circulo. n . 42.... 45.
Haec vera sunt de omnium virium genere; ponantur
vires acceleratrices ad centrum datum directae : jacebit cur
va in plano transeunte per rectam projectionis et per cen
trum virium: radius vector describet areas circa virium cen
trum temporibus proportionales: viceversa si radius ve
ctor describit areas circa punctum aliquod temporibus pro
portionales, vis acceleratrix erit constanter directa ad pun
ctum illud: velocitas, qua pollet mobile in eadem curva ,
exsistit reciproce proportionalis perpendiculo ducto a centro
virium in tangentem: vis acceleratrix in quovis curvae pun:
cto est directe ul radius vector, et reciproce ut factum
ex osculi radio in cnbum praefati perpendiculi : si ultra
punctum contactus sumitur arcus infinitesimus, a materiali
puncto describendus subsequente tempusculo, radiusque ve
ctor pertingens ad hujus arcus extremitatem producitur
donec occurrat tangenti, vis acceleralrix in contactus pun
cto erit directe ut pars radii vectoris producti intercepta
ac tangente , et reciproce ut quadratum tempuscu
li .
arcu
n. 46,
49.
Sive vires tendant ad centrum datum, sive non ; coor
dinatae puncti materialis in fine temporis e spectandae sunt
tanquam functiones ipsius t : formulae respicientes et veloci
tatem in quolibet curyae puncto, et binas componentes, al
teram juxta tangentem , alteram juxta normalem , in quas
resolvitur yis acceleratrix. n, 50. 10. 2º . 3º.
.
322
tempore linito angulus, quem etl'ormat- vis acceleratrix cum
directione tangentis ,
fuerit semper acutus, acquirat mo-
bile incrementum velocitatis Gnitum; si semper obtusus,
patietur decrementum (initum: si semper rectus, veloci-
tas mauebit constans: quadratum velocitatis adaequat vim.
acceleratricem ductam in dimidium chordae, quae ex eius
directione abscinditur ab osculatore circulo. n. 42.... 45.
Haec vera sunt de omnium virium genere; ponantur
vires acceleratrices ad centrum datum directae: iacebit cur-
va in plano transeunte per rectam projectiouis et per cen-
trum virium: radius vector describet areas circa virium cen-
trum temporibus proportionales: viceversa si radius ve-
ctor describit areas circa punctum aliquod temporibus pro-
portionales, vis acceleratrix erit constanter directa ad pun-
ctum illud: velocitas, qua pollet mobile in eadem curva
.
exsistit reciproce proportionalis perpendiculo ducto a centro
virium in tangentem: vis acceleratrix in quovis curvae pung-
cto est directe ut radius vector, et reciproce ut factum
ex osculi radio iu cubum praefati perpendiculi: si ultra
punctum contactus sumitur arcusiufiuitesimus, a materiali
puncto describendus subsequente tempusculo, radiosque ve-
ctor pertingens-ad huius arcus extremitatem producitur
donec occurrat tangenti, vis acceleratrix in contactus pun-
cto erit directe ut pars radii vectoris producti intercepta
arcu ac tangente ,
et reciproce ut quadratum tempuscu-
li. ∙∙∙∙∙⋅∙∙∙ ∙ ∙∙ ..n.46,...49-
Sive vires tendant ad centrum datum, sive non; coor-
dinatae puncti materialis in fine temporis t spectandae sunt
tanquam functiones ipsius :: formulae respicientes et veloci-
tatem in quolibet curvae puncto, et binas componentes, al-
teram juxta tangentem, alteram juxta normalcm. in. qu&quot;.
resolvitur vis acceleratrix. . . . . . n. 50.1'-2&quot;- 30,323
Resolata vi acceleratrice in ternas componentes axi
bus coordinatis parallelas, stabiliuntur formulae huc per
tinentes: applicantur formulae ad duas quaestiones, quarum al
tera respicit gravia oblique projecta in vacuo, altera respicit
gravia oblique projecta in medio resistente. n. 50. 4º. 5º . 6º.
Quomodo vis acceleratrix directa ad centrum expri
matur generatim per coordinatas polares : quomodo, data vi
acceleratrice directa ad centrum , inveniri possit aequatio
inter coordinatas polares ad lineam per quam movetur ma
teriale punclum: exemplum desumptum a vi acceleratrice ,
quae sit reciproce ut quadratum radii vectoris: sub hac le
ge poterit materiale punctum describere parabolam haben
tem suum focum in centro virium:
quaenam velocitas pro
jectionis ad id sit necessaria. n. 50. 7º. 8º... 15 °
Motus curvilineus impeditus : vis centrifuga. n. 51 .
De vi acceleratrice in motu circulari, existente
centro virium in centro circuli. pag . 109,
Istiusmodi motus ' est uniformis: vis acceleratrix obti
netur dividendo quadratum velocitatis per curvae circularis
radium: varia inde inferuntur et quoad projectionis velo citatem necessariam ad describendam cicularem curvam , et
quoad vires acceleratrices in diversis peripheriis circula
ribus. n. 52 , 53.
Vis centrifuga orta ex circulari telluris rotatione cir
ca suum axem : qua ratione decrescat ab aequatore ad po
los: qua ratione vis centrifuga imminuat gravitatem a po
lis ad aequatoren . n. 54.
323
Resoluta vi acceleratrice in ternas componentes axi-
bus coordiuatis parallelas, stabiliuntur formulae huc per-
tinentes: applicantur formulae ad duas quaestiones, quarum al-
tera respicitgravia oblique projecta in vacuo, altera respicit
gravia oblique proiecta in medio resistente. n. 50. 40. 50. 60.
Quomodo vis acceleratrix directa ad centrum lexpri- matur generatim per coordinatas polares: quomodo, data vi
acceleratrice directa ad centrum
.
inveniri possit aequatio
inter coordinatas polares ad lineam per quam movetur ma-
teriale punctum: exemplum desumptum a vi acceleratrice,
quae sit reciproce'ut quadratum radii vectoris: sub hac le-
ge poterit materiale punctum describere parabolam haben-
tem suum focum in centro virium: quaenam velocitas pro-
fectionis ad id sit necessaria. '
. . n. 50. 7&quot;. 80...150.
Motus curvilineus impeditus: vis centrifuga. n. St.
De vi acceleratrice in motu circulari, existente
centro m'rium' in centro circuli . pag. 109.
Istiusmodi motus 'est uniformis: vis acceleratrix obti-
netur dividendo quadratum velocitatis per curvae circularis
radium: varia iude inferuntur et quoad proiectionis velo-
citatem necessariam ad&quot; describendam cicularem curvam, et
quoad vires acceleratrices in diversis peripheriis circula-
ribus-.............n.52,53.
.
Vis centrifuga orta ex circulari telluris rotatione cir-
ca suum axem: qua ratione decrescat ab aequatore ad po-
los: qua ratione vis centrifuga imminuat gravitatem a po-
lis ad aequatorem.
. . .. . . . . ∙∙ ∙ n. 54-324
De vi acceleratrice in motu elliptico, existente
centro virium in foco ellipsis pag. 111,
Varia praemittuntur et circa rectas ita ductas ex puncto
quovis, ut tangant datam sphaeram ; et circa plaua tangentia
ducta per ejusmodi rectas ; et circa rectarum , arearumque
planarum projectiones in plano quolibet ; sed praecipue
circa ellipsim. n. 55. 1º, 2º ...14 °. . .
Quibus praemissis, demonstratur illud : existente cen
tro virium in foco ellipseos , vis acceleratrix in motu el
liptico est reciproce ut quadratum radii vectoris : quid in
duabus ellipsibus si quadrata temporum periodicorum sint ut cubi semiaxiun transversorum . n. 56.
Paucis subjunctis de ellipsi , parabola , et hyperbo
la, demonstratur quod, agentibus viribus in ratione reci
proca duplicata distantiarum a dato centro, praeter para
bolam poterit quoque mobile describere vel ellipsim
vel hyperbolam, existente focorum altero in centro virium:
quaenam projectionis velocitas requiratur ad ellipsim de
scribendam , quaenam ad hyperbolam. n, 67. 1.2.7 .
Obiter de lege virium in motu elliptico, ubi eae ten
dant ad ellipseos centrum . n. 57 , 8 .
De motu relativo punctorum materialium , tendentium
in se mutuo viribus acceleratricibus quae sint di
recte ut massae in quas tenditur, et reciproce ut qua
drata respondentium distantiarum . pag. 125.
Generales ad istiusmodi motum aequationes differen
tiales. n, 58,
324
- De ui acceleratrice in motu elliptica. existente
centro virium in foco ellipsis pag. 111.
Varia praemittuntur et circa rectas ita ductas ex puncto
quovis, ut tangant datam sphaeram; et circa plana tangentia
ducta per ejusmodi rectas; et circa rectarum, arearumque
planarum proiectiones in plano quolibet ; sed praecipue
circa ellipsim. . . . . . . . . n.55.10. 20 ...140.
Quibus praemissis, demonstratur illud: existente cen-
tro virium in foco ellipseos , vis acceleratrix in motu el-
liptico est reciproce ut quadratum radii vectoris: quid in
duabus ellipsibus si quadrata temporum periodicorum sint
ut cubi semiaxium transversorum . . . . . . n- 56.
Paucis subjunctis de ellipsi, parabola ,
et hyperbo-
la, demonstratur quod, agentibus viribus in ratione reci-
proca duplicata distantiarum a dato centro, praeter para-
bolam poterit quoque mobile describere vel ellipsim ,
vel hyperbolam, existente focorum altero in centro virium:
quaenam proiectiouis velocitas requiratur ad ellipsim de-
scribendam, quaenam ad hyperbolam. . n. 57. ↿∘∙ ⋍∘∙∙∙ 70.
Obiter de lege virium in motu elliptica, ubi eae ten- dant ad ellipseos centrum.
. . . . . . n. 57. 8&quot;.
De motu relativo punctorum &quot;materialium
,
tendentium
in se mutuo viribus acceleratricibus quae sint di-
recte ut massae in quas tenditur, et reciproce ut quab
drata respondentium distantiarum. pag. 125.
Gener-ales ad istiusmodi motum aequationes dideren-
tiüles- ∙∎∎ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ &quot;a 58.325
Spectantur duo tantum materialia puncta: vires per
turbantes ex reliquis punctis. n. 59, ... 62.
De pendulis ; deque gravium descensu per
arcus cycloidales. pag . 134.
Quid pendulum simplex ; quid compositum : vires
gignentes motum penduli simplicis n . 63.
Velocitates in puncto infimo acquistae a gravibus per
inaequales ejusdem circuli arcus descendentibus sunt ut
ipsorum arcuum chordae .
. n. 64.
Oscillationes penduli simplicis per arcus satis exi
guos , ulcumque ceteroquin inaequales , sunt ad sensum
isochronae seu aequidiuturnae : quid ex doctrina penduli
simplicis circa terrestrem gravitatem n. 65 , 66.
Centrum oscillationis in pendulo composito : etiam
oscillationes penduli compositi suņt isochronae, modo ta
men existant satis exiguae .
n. 67.
Oscillationes penduli simplicis in medio resistente :
primo in hypothesi resistentiae proportionalis simplici ve
locitati; deinde in hypothesi resistentiae proportionalis qua
drato velocitatis .
n. 68.
n . 69 .
Paucis praemissis de cycloide, demonstratur illud : ex
quocumque cycloidis puncto demittatur grave , eodem sem
per tempore perveniet ad punctum infimum
De attractione corporum in hypothesi attractionis agentis
in ratione directa massarum , et in reciproca
duplicata distantiarum .
Attractio corporum quorumcumque in materiale pun
clum situm sive extra corpus attrahens, sive intra. n . 70,71,72.
pag . 151 .
325
Spectautur duo tantum materialia puncta: vires per-
turbantes ex reliquis punctis. . . . . . n. 59....62.
De pendulis; deque gravium descensu per
arcus cycloidales. pag. 134.
Quid pendulum simplex; quid compositum : vires
gignentes motum penduli simplicis
. . . . . n. 63.
Velocitates in puncto intimo acquistae a gravibus per
inaequales ejusdem circuli arcus descendentibus sunt ut
ipsorum arcuum chordae
. . . . . .- . . n. 64.
Oscillationes penduli simplicis per arcus satis exi-
guos, utcumque ceteroquin iuaequales ,
sunt ad sensum
isochrouae seu aequidiuturuae: quid ex doctrina penduli
simplicis circa terrestrem gravitatem . . . n. 65
, 66.
Centrum oscillationis in pendulo composito: etiam
oscillationes penduli compositi sunt isochrouae, modo ta-
men existant satis exiguae . . . . . . . . n. 67.
Oscillationes penduli simplicis in medio resistente:
primo in hypothesi resistentiae proportionalis simplici ve-
locitati; deinde in hypothesi resistentiae proportionalis qua-
drato velocitatis
. . . .. . . . . . . n. 68.
Paucis praemissis de cycloide, demonstratur illud : ex
quocumque cycloidis puncto demittatur grave , eodem sem-
per tempore perveniet ad punctum infimum
. . n.. 69.
De attractione corporum in hypothesi attractionis agentis
in ratione directa massarum, et in reciproca
duplicata distantiarum. pag. 151 .
Attractio corporum quorumcumque in materiale pun-
ctum situm sive extra corpus attrahens, sireintrafn. 70,71,72.326
Expediuntur quae pertinent ad attractionem corpo
rum sphaericorum in punctum materiale n. 73,74,75.
Materiale punctum valde distans a corpore attrahente,
utcumque se habeat forma corporis, ea proxime ratione tendit
in ipsum corpus, qua tenderet si corporis partes in centro
gravitatis compenetrarentur: ubi dimensiones corporum quo
rumcuinque se mutuo attrahentium sint admodum exiguae
prae distantiis , quibus ipsa corpora disjunguntur , eorum
alterum tendet in alterum perinde ac si essent ambo in
suis gravitatis centris compenetrata ; haec assertio quoad
sphaerica corpora valet utcumque se habeat intercedens
distantia n. 76.
De gravitatione universali. pag. 159.
Ex mutua coelestium corporum gravitatione collata
cum terrestrii gravitate inferimus illud : gravitas ita ma
teriam afficit , ut singulae ejus particulae in alias omnes
et singulas gravitent in ratione directa massarum ad quas
tenditur , et reciproca duplicata distantiarum alterius ab
altera n . 77 , ...82.
.
Aliquid circa solarem et planeticas massas... n.83.10... 4.
Media telluris densitas determinata ex penduli aber
ratione ; itemque experimentis institutis in libra
siouis n. 83. 5. 6.°
tor
Quomodo ex marini aestus phoenomeno deduci pos
sit ratio inter lunarem ac terrestrem massam . n. 83.7 . °
326
Expediuntur quae pertinent ad attractionem corpo-
rum sphaericorum iu punctum materiale
. n. 73,74,75.
Materiale punctum- valde distans a corpore attraheute,
utcumque se habeat forma corporis, ea proxime ratione tendit
in ipsum corpus, qua tenderet si corporis partes in centro
gravitatis compenetrarentur: ubi dimensiones corporum quo-
rumcumque se mutuo attrahentium sint admodum exiguae
prae distantiis, quibus ipsa corpora disiunguntur , eorum
alterum tendet in alterum perinde ac si essent ambo in
suis gravitatis centris compenetrata ; haec assertio quoad
sphaerica corpora valet utcumque se habeat intercedens
distantia ...............n76
De gravitatione universali. pag. 159.
Ex mutua coelestium corporum gravitatione collata
cum terrestrii gravitate.iuferimus illud: gravitas ita'ma-
teriam allicit, ut singulae eius particulae in alias omnes
et singulas graviteut in ratione directa massarum ad quas
tenditur, et reciproca duplicata distantiarum alterius ab
altera . . . . . . . . . ∎∙ ∙ ∙ n. 77,...82.
Aliquid circa solarem et plaueticas massas...n.83.10...4.'
Media telluris densitas determinata ex penduli aber-
ratione : itemque experimentis institutis in libra tor-
Sioni. ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙∙ ∙ ∙ n. 830 5-0 S..
Quomodo ex marini aestus phoenomeuo deduci pos-
sit ratio inter lunarem ac terrestrem massam. n. 83. 7!327
Aliquid notatur de motu punctorum materialium
utcumque inter se connexorum pag. 169.
Formulae spectantes et ad translativum punctorum
motum juxta coordinatos axes, et ad rotatilem eorum mo
tum circum axes ipsos n. 84.
Moto punctorum systemate, perinde movebitur com
mune gravitatis centrum ac si , coeuntibus punctis in
ipsum centrum , applicarentur centro eaedem vires cum
iisdem directionibus , quibus pancta sollicitantur. n. 84.1.6
Principium de conservatione centri gravitatis : item
de conservatione arearum : necnon de conservatione vi
rium vivarum n. 84. 2.0 ... 5 ..
Relativus rigidi liberique systematis motus quoad
gravitatis centrum n. 84. 6. ° 7.0
Motus rigidi systematis circa axem fixum ; quibus
cuinque caeteroquin viribus acceleratricibus sollicitetur sy
stema : quid si vires acceleratrices consistant in sola gra
vitate ; huc spectat theoria penduli compositi : longitudo
penduli simplicis , quod suas perficit oscillationes eodem
tempore ac pendulum compositum : quid si nullae sint
vires acceleratrices : inertiae momenta quoad axem
principales systematis axes : principalia inertiae momen
n. 85. 1.° 2.° ... 7.0
.
ta .
De fluidorum corporum aequilibrio pag. 182.
Ex perfecta mobilitate , qua ponuntur gaudere flui
dorum corporum particulae , ostenditur principium de
aequalitate pressionis , atque inde eruuntur conditiones
requisitae ad aequilibrium cujusvis massae fluidae. n. 86.
327
Aliquid notatur de motu punctorum materialium
utcumque inter se connexorum pag. 169.
Formulae spectantes et ad translativum punctorum
motum iuxta coordinatos axes, et ad rotatilem eorum mo-
tum circum axes ipsos . . . . . . . . . n. 84.
Moto punctorum, systemate, perinde movebitur com-
mune gravitatis centrum ac si
, coeuntibus punctis in
ipsum centrum, applicarentur centro eaedem vires cum
iisdem directionibus , quibus puncta sollicitantur. n. 84.1.'
Principium de conservatione centri gravitatis: item
de conservatione arearum : necnon de conservatione vi-
rium vivarum . . . .
'. ⋅∙ ∙ ∙ n. 84. Z.&quot;...Sæ
Belativus rigidi liberique systematis motus quoad
gravitatis centrum
. . . . . . . . n.84. 6.&quot; 79
Motus rigidi systematis circa axem fixum .: quibus-
cumque caeteroquin viribus acceleratricibus sollicitetur sy-
stema :quid si vires acceleratrices consistant in sola gra-
vitate; huc spectat theoria penduli compositi: longitudo
penduli simplicis
, quod suas perficit oscillationes eodem
tempore ac pendulum compositum .: quid si nullae sint
vires acceleratrices : inertiae momenta quoad axem :
principales systematis axes : principalia inertiae momen-
ta.
. . . . . . . . . . n.85.1.0 Z.&quot;... 7.&quot;
De fluidorum corporum aequilibrio pag. 182.
Ex perfecta 'mobilitate
. qua ponuntur gaudere Hui-
dorum corporum particulae ,, ostenditur principium de
aequalitate pressionis , atque inde eruuntur conditiones
requisitae ad aequilibrium cujusvis massae Huidae. n. 86.328
Quid notandum circa superficiem massae fluidae li
bratae n. 87, 1. ° 2.° ... 5 .
Quid circa fluidum elasticitate pollens, ni 87. 6.0 7 .
De gravium homogeneorumque liquidorum
aequilibrio. pag. 187.
Liquida constituta in aequilibrio intra vasa : pres.
siones in areas sive horizontaliter , sive oblique demer
sas : centrum pressionis .
n. 98. 1. ° . , . 4.0
Solida liquidis immersa : aequilibrii positiones quoad
solidum liquido insidens n. 88.5 . , 89. 1.° 2. ° 3.°
Utrum aequilibrium sit stabile , nec ne. n. 90.
.
De gravium liquidorum aequilibrio in vasis
communicantibus.
pag. 195.
Quid si vasis communicantibus idem contineatur li
quidum : explicatio variorum effectuum ; antliae adspi
ranles , etc n. 91 , 92. 1.° 2.°
Quid si diversa contineantur liquida.
. n. 92. 3.0
De gravium elasticorumque fluidorum aequilibrio nec
non de altitudinibus dimetiendis ope barometri, et de
pondere ac densitate vaporum . pag. 199.
Conditio aequilibrii expressa per aequationem dif
ferentialem : perficitur integratio in hypothesi temperiei
constantis n. 93.
Inde eruitur formula inserviens ad altitudines di
328
Quid notandum circa superüciemi massae liuidae li-
bratae
. . . . . . . . . n. 87.1.02.&quot;...5.0
Quid circa fluidum elasticitate pollens. n: 87. 6.&quot; 7.&quot;
-De gravium homogeneorumque liquidarum
aequilibrio. pag. 187.
Liquida constituta in aequilibrio intra vasa: pres- ⋅
tiones in areas sive horizontaliter , sive oblique demer-
sas: centrum pressionis . . . . . n. 88. 1.&quot;
..,.
4.&quot;
∙
!'
Solida liquidis immersa : aequilibrii positiones quoad
solidum liquido insidens . . n. 88.5.&quot;, 89. 1.&quot; 2.0 3.&quot;
Utrum aequilibrium sit stabile, nec ne. . . n. 90.
De gravium liquidarum aequilibrio in 'vasis
communicantibus. pag. 195.
Quid si vasis communicantibus idem continaptur li-
quidum: explicatio variorum effectuum : antliae adspi-
TODIBBQ etc ∙ ∙ ∙ ∙ ∙
↼
∙ ∙ a ∙ &quot;.
5 91. 92. 1.02.0
Quid si diversa cbntiueantur liquida. . . n. 92. 3.&quot;
De gravium elasticorumque fluidorum aequilibrio nec
non de altitudinibus dimetiendis ope barometri, et de
pondere ac densitate. naporum. pag. 199.
Conditio aequilibrii expressa per aequationem dif-
ferentialem : perficitur integratio in hypothesi temperiei
constantis . . . .
.'&quot;.
. . ⋅ n. 93.
Inde eruitur formula inservieus ad altitudines di-329
metiendas ope barometri : varia observantur pro commo
diori formulae usu n. 94. 1. ° 2.° ... 6.•
Verticalis ascensus globi aereostatici : maxima glo
bi elatio . n. 95.
Maxima quantitas vaporis sese evolventis in vase un
dique clauso : vis elastica sicci aeris aucta ob evolu
tum vaporem : ratio inter densitatem aquei vaporis ac
densitatem sicci aeris sub eadem temperie eademque pres
sione : ratio inter eorum densitates ac pondera sub ea
dem temperie et diversis pressionibus : densitas aeris va
porosi librantis datam pressionem sub temperie data. n. 96.1 . °
2 .
Usus aquei vaporis in movendis machinis. n . 99.6. •
De aqua egrediente per angustum foramen
e vasis verticalibus sive cylindricis,
sive prismaticis. pag. 206 .
Nonnulla praemittuntur ex pluries iteratis experimen
tis .
n . 97.
Quaenam velocitas aquae egredientis: tempus impen
sum in descensu usque ad quamlibet altitudinem datam . n.98.
Quantitas aquae dato tempore egredientis : tempus
quo vas totum evacualur n. 99, 100,
Ratio inter tempora , quibus deplentur duo vasa ha
bentia et altitudines et orificia aequalia : quantitales aqua
rum successivis ' et aequalibus temporibus ex vasis ori
ficio efluentium : divisio vasorum in partes successivis dati
temporis unitatibus vacuandas n. 101 , 102.
22
' 329
metiendus ope barometri : varia observantur pro commo-
diori formulae usu . . . . . n. 94. 1..) .&quot;
... 6.&quot;
Verticalis ascensus globi aereostatici : maxima glo-
bi elatio. ∙ ∙ ∙ ∙ ∙ ' ∙ ∙ ∙ ∙ ∙ ∙
∙∎∎ ∙ n- 950
-Maxima quantitas vaporis sese evolventis in vase uu-
dique clauso : vis elastica sicci aeris aucta&quot; ob evolu-
tum vaporem : ratio inter densitatem aquei vaporis ac
densitatem sicci aeris sub eadem temperie eademque pres-
sione: ratio inter eorum densitates ac pondera sub ea-
dem temperie' et diva-sis- pressionibus: densitas aeris va-
porosi librantis datam pressionem sub temperie data. n. 961.&quot;
2.0
...
5.&quot;
Usus aquei vaporis in movendis machinis. n. 99. 6.&quot;
De aqua egrediente per angustum foramen
e vasis «verticalibus sive cylindricis,
sive prismaticis. pag. 206.
Nonnulla praemittuntur ex pluries iteratis experimen-
tis ∙ ∙ ∙ ∙ ∙ ⋅∙⋅ ∙ ∙ . ∙ ∙ ∙ ∙∎∎ ∙ ∙ ∙ &quot;o 970
Quaenam velocitas aquae egredientis: tempus impen-
sum in descensu usque ad quamlibet altitudinem datam. n.98.
,. Quantitas aquae dato tempore egredientis: tempus
quo vas tatum evacuatur . . . . . . n. 99,100.
Ratio inter tempora, quibus deplentur duo vasa ha-
bentia et altitudines 'et oriiicia aequalia : quantitates aqua-
rum successivis' et aequalibus temporibus ex vasis ori-
iicio efluentium: divisio vasorum in partes successivis dati
temporis unitatibus vacuandas . . . ∙∙ n. 101, 102.
22330
Contractio venae aqueae n. 103.
Ubinam perficiatur acceleratio , per quam velocitas
aquae descendentis admodum exigua mutatur in finalem
satisque grandem effluxus velocitatem . n . 104. 1.0
Quomodo motus aquae defluentis in regularibus al
veis traduci possit ad motum aquae prosilientis ex an
gustis vasorum orificiis n. 104. 2.• , ..5.
Illud cum Auctoribus non paucis assumitur tanquam
principium , quod nempe unumquodque liquidi in vase
quolibet descendentis tenuissimum et horizontale stra
tum coalescat iisdem constanter particulis communi , ea
que tantum verticali , velocitale donatis ; inde vero eruun
tur , quae pertinent ad ipsius liquidi motum n . 105.
Aliquid subjungiur circa generalem theoriam
motus corporum fluidorum . pag. 216.
Aequationes ad istiusmodi motum et quum massa
fluida homogenea vel heterogenea est incapax compressio
nis, et quum massa fluida pollet elasticitate. n. 106,107,108.
De tubis capillaribus. pag. 225.
sol Vires ex materia tubi , et ex materia liquidi ,
licitantes datam ipsius liquidi particulam : attentis viri
bus istis , suprema liquidi superficies vel induet curvam
concavamque figuram , vel curvam convexamque , vel ma
nebit, plana atque horizontalis, n. 109,1.9
330
Contractio venae aqueae . . . . -. . n. 103.
Ubinam perficiatur acceleratio, per quam velocitas
aquae descendentis admodum exigua mutatur in finalem
satisque grandem effluxus velocitatem. .
-
.n. 104. 1.&quot;
Quomodo motus aquae defluentis in regularibus al-
veis traduci possit ad motum aquae prosilientis ex au-
gustis vasorum orificiis
. . . . . n. 104. Z.&quot;. ..5.&quot;
Illud cum Auctoribus non paucis assumitur tanquam
principium, quod nempe unumquodque liquidi in vase
quolibet descendentis tenuissimum et horizontale stra-
tum coalescat iisdem constanter particulis communi , ea-
que tantum verticali, velocitate donatis :
inde vero eruun-
tur, qnae pertinent ad ipsius liquidi motum
. ∙⋅ n. 105-
Aliquid subjungiur circa generalem theoriam
motus corporum fluidorum. pag. 215-
Aequationes ad istiusmodi motum et quum massa
fluida homogenea vel heterogenea est incapax compressio-
nis, et quum massa fluida pollet elasticitate. n. 106,107,108.
De tubis capillaribus. pag. 225.
Vires ex materia tubi
,
et ex materia liquidi .
sol-
licitantes datam ipsius liquidi particulam: attentis viri-
bus istis , suprema liquidi superficies vel induet curvam
concavamque figuram , vel curvam couvexamque , vel ma-
nebit, plana atque horizontalis, ,. . . .. . n. 109.1.&quot;331
Quam attractionem exerceat massa liquida , cujus su
prema superficies est plana , in columellam liquidam per
pendiculariter illi superficiei planae insistentem . n. 109.2 .
Quam attractionem exerceat massa liquida , cujus su.
prema superficies est vel concavo -sphaerica vel convexo
sphaerica , in columellam liquidam perpendiculariter in
sistentem plano tangenti , dactó vel per punctum infimum
superficiei concavo-sphaericae vel per supremum superfi
ciei convexo -sphaericae. n. 109. 3.° 4.0 ... 70
Quid si massa liquida terminetur superficie concaya
vel convexa , quae non sit sphaerica. n. 109. 8.° ... 11.º
His declaratis , explicamus ascensum descensumque
liquorum in lubis capillaribus n. 110,
.
Nonnalla subjunguntur , quorum ratio desumitur ab
actione capillari . n. 111. 1.° 2.° ... 5 ° , 112
)
ACUSTICAE PRINCIPIA
Notiones praeambulae.
1
pag . 245.
Corpora, quae sonora dicuntur tunc sonum exci
tant quando ita agitantur , ut illorum partes tremulo ac
vibratorio satisque rapido concutiantur motu ; qui motus
communicatus aeri ambienti , et late diffusus afficit orga
nym auditus: vis acceleratrix in vibrante particula resonan
tis corporis.
. n . 113. 10. 20.
331
Quam attractiduem exerceat massa liquida , cuius su-
prema superficies est plana ,
in columellam liquidam per-
pendiculariter illi superficiei planae insistentem. n. 1092.&quot;
Quam attractionem exerceat massa liquida , cuius su-
prema- superficies est vel concavo-sphaerica vel convexo-
sphaerica, in columellam liquidam peu-pendiculariter in-
sistentem plano tangenti , dnctö vel per punctum infimum
superficiei concavo-sphaericae vel per supremum superfi-
ciei convexo-sphaericae. . . . n. 109. 3.&quot; 4.&quot;
.. . 7.&quot;
Quid si massa liquida terminetur superficie concava
vel convexa, quae non sit sphaerica. n. 109. 8.&quot;. ..
11.&quot;
His declaratis , explicamus ascensum descensumque
liquorum iu .tubis capillaribus . . . . . . n. 110.
Nonnulla subjunguntur ∙ quorum -ratio desumitur ab
actione capillari. . . . . n. 111. 1.&quot; 2.&quot;
. . . 5&quot;,112
AOUSTIGAE
W
PRINCIPIA
Notiones praeambulae. ∣
pag. 245.
Corpora, quae sonora dicuntur
,
tunc sonum exci-
tant quando ita agitantur, ut illorum partes tremulo ac
vibratorio satisque rapido concutiuntur motu; qui motus
communicatus aeri ambienti, et late diffusus afficit orga-
num auditus: vis acceleratrix in vibrante particula resonan-
tis corporis. . . . . . . '. . . . n. 113.1&quot;. 2&quot;.332
Progignitur quoque sonus ab aere vehementer compres
so , seseque statim restituente , n. 114.
.
Soni reflexio; inde echo. n . 115 .
Non solus aer est medium ideoneum transmissioni
sonorum. n. 116.
De intensitate soni; deque ejus gravitate,
et acutie .
pag. 248.
Sonus intensior ex eo gignitur quod in sonoro cor
pore plures ejusdem partes simul oscillant, et majus spa
tium singulis oscillationibus dato tempusculo percurrunt;
atque ita in aere ex numero item et majori oscillatione
partium aeris intensitas soni dependet ; remissior autem
sonus ex opposito.
n. 117.
Nonnulla explicantur circa soni intensitatem. n . 118.
ex Soni gravis et acuti discrimen repetendum est
numero vibrationum in partibus sonori corporis, ita ut
sonus gravior oriatur ex minus frequentibus vibrationibus so
nori corporis, ex crebrioribus contra sonus acutus ; idem
que de oscillationibus aeris in sono derivato. n. 119.
Quid consonantia , et quid dissonantia: varii conso
nantiae gradus: theoria chordaram vibrantium in hypothe
si vibrationum admodum exiguarum. n. 120. 1 ” 2 ”... 7 .
Varia proponuntur explicanda circa chordas vibran
tes . n. 121 .
332
Progignitur quoque sonus ab aere vdhemeuter compres-
so, seseque statim restituente. . . . . . . n. 114.
Soni reflexio; inde echo. . . . . . . n. 115.
Non solus aer est medium ideoneum transmissioni
SODOmm. ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ &quot;. 116.
De intensitate soni; deque eius gravitate,
et acutie. pag. 248.
Sonu's intensior ex eo gignitur quod in sonoro cor-
pore plures eiusdem partes simul oscillaut, et maius spa-
tium singulis oscillationibus dato tempusculo percurrunt:
atque ita in aere ex numero item et maiori oscillatione
partium aeris intensitas soni dependet; remissior autem
sonus ex opposito. . . . . . . . . . ,n. 117.
Nonnulla explicantur circa soni intensitatem.
. .n. 118.
Soni' gravis et acuti discrimen repetendum est ex
numero vibratiouum in partibus sonori corporis, ita ut
sonus gravior oriatur ex minus frequentibus vibrationibus so-
nori corporis, ex crebrioribus contra sonus acutus,- idem-
que de oscillationibus aeris in sono derivato. . n. 119.
Quid consonantia, et quid dissonantia: varii conso-
nantiae gradus: theoria chordarum vibrantium in hypothe-
si vibrationum admodum exiguarum. n- 120. 1&quot; 2&quot;... 7&quot;.
Varia proponuntur explicanda circa chordas vibran-
tes. ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙
Q &quot;. 1210333
Quomodo sonus trans obicem possit communicari ita,
ut tonus proprius sonori corporis permaneat. n. 122.
Unde asperitas aut lenitas soni proficiscatur. n. 123.
Transversae et longitudinales chordarum vibratio
nes: nodi in chordis vibrantibus: lineae nodales in super
ficiebus corporum resonantium : vibrationes laminarum ri
gidarum .
n. 124 .
De directa soni propagatione per aerem . pag. 265.
In iisdem circumstantiis sonus aequabili velocitate
in toto decursu devehitur; omnesque soni sive intensi ,
sive remissi, sive graves, sive acuti eadem velocitate dif
funduntur: qua ratione intensitas soni minuatur in pro
gressu . n. 125,
Undae sonorae constitutio. n. 126,
Soni et velocitas, et intensitas augetur a vento se
cundo, minuitur ab adverso . n . 127.
Experimenta instituta ad soni velocitatem determi
nandam; quae tamen experimenta non satis conveniunt :
hujus diversitatis rationes : quaenam utilitas ex determi
natione velocitatis qua propagatur sonus. .
n. 128
Generalis de fluidorum motu theoria applicatur ad
soni propagationem : soni velocitas eruta ex applicatione
theoriae comparatur cum velocitate quam praebent ex
perimenta .
n. 129. 10. 2º. 3º.
Crassities aerei strati, in quo particulae cientur una :
si impulsio in obicem facta quadrato velocitatis sumitur
-
22&quot;
333
Quqmodo sonus trans obicem possit communicari ita,
ut tonus proprius sonori corporis permaneat.
. n. 122.
Unde asperitas aut leuitas soni proficiscatur. n. 123.
Trausversae et longitudinales chordarum vibratio,-
nes: nodi in chordis vibrantibns: lineae nodales in super-'-
iiciebus corporum resonantium: vibrationes laminarum ri-
gidarum...........;..n.124.
De directa soni propagatione per aerem. pag. 265.
In iisdem circumstantiis sonus aequabili velocitate
in toto decursu devehitur; omnesque soni sive intensi ,
sive remissi, sive graves, sive acuti eadem velocitate dif-
fuuduntur: qua ratione intensitas soni minuatur in pro-
gressu..............-n.125.
Undae sonorae constitutio. . . . . . . n.126,
Soni et velocitas, et intensitas augetur a vento se-
eundi), mall!!! EI) adverw. ∙ ∙ ∙ ∙ ∙ ∙ n- 127.
Experimenta instituta ad soni velocitatem determi-
nandam; quae tamen experimenta non satis conveniunt:
hujus diversitatis rationes: quaenam utilitas ex determi-
natione velocitatis qua prOpagatur sonus. . . n. 128.
Generalis de fluidorum motu theoria applicatur ad
soni propagationem: soni velocitas eruta ex applicatione
theoriae comparatur cum velocitate quam praebent ex-
perimenta . . . . . . . . . . n.129.1o.2&quot;.3&quot;.
Crassities aerei strati, in quo particulae cientur uua:
si impulsio iu obicem facta quadrato velocitatis sumitur
22'334
proportionalis, rationem duplicatam distantiarum .sequetur
soni debilitatio.
n. 129. 4. 5 .
Cur pluribus corporibus simul resonantibus , inter
oscillationes in aere excitatas non habeatur confusio , omnes
que diversi soni inde orti ad aures distincte perveniant:
huc spectat principium de superpositione exiguorum mo
tuum. n. 129. 6.
Propagatio soni in cubis cylindricis indefinitae lon
gitudinis. n. 129.7 .
J
De reflexa soni propagatione per aerem pag. 289.
Cum in directa propagatione sonorus aer . offendit o
bicem aptum , reflectitur : varia ad echo spectantia ex
plicantur. n. 130.
Reflexio soni fit ad angulos incidentiae et reflexionis
aequales; regrediturque sonus eadem velocitate qua incedebat
antequam in obicem impingeret. n . 131 , 132, 1º. 2º
De instrumentis pneumaticis. pag. 294.
In instrumentis pneumaticis soni genesis repetenda non
est saltem praecipue ex oscillatione partium solidarum i
psius instrumenti : quo pacto sit explicanda : aer secun
dum fistulae longitudinem se habet instar chordae peragen
tis longitudinales vibrationes: etsi ex materia instrumenti
non habetur varietas quoad soni qualitatem, aut valde uota
bilem intensitatem ; varietas tamen habetur quoad meliorem
334
proportionalis, rationem duplicatam distantiarum .sequetur
soni dehilitatio. . . . . . . . ∙ ∙ n.129.40.50.
Cur pluribus corporibus simul resonantibns ,
inter
oscillationes in aere excitatas non habeatur confusio,omues-
que diversi soni inde orti ad aures distincte perveniant:
huc spectat principium de superpositione exiguorum mo-
tuum. ∙⋅∙ '.
. . . . . . . . . . n.129.6&quot;.
Propagatioi soni in. tubis cylindricis indefinitae lou-
gitudinisa, ∙ ∙ ∙ . . . . . .. . . n.129.7&quot;.
]
De refleæa soni propagatione per aerem pag. 289.
!. ∙ . .
Cum indirecta prOpagatione sonorus aer .oii'eudit o-
bicem aptum, reflectitur : varia ad echo spectantia ex-
Plimnturo. ∙∙ ∙∙ ∙∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ n. 1300
. Reflexio soni iit ad angulos incidentiae et reflexionis
aequales: regrediturque sonus eadem velocitate qua incedebat
antequam in obicem impingeret. . . .&quot;' 131, 132.1&quot;.2&quot;.
'pul '
De instrumentis pneumaticis. pag. 294.
In instrumentis pneumaticis soni genesis repetenda non
est saltem praecipue ex oscillatione partium solidarum i-
psius instrumenti: quo pacto sit explicanda: aer secun-
dum fistulae longitudinem se habet instar chordae peragen-
tis lougitudinales vibrationes: etsi ex materia instrumenti
non habetur varietas quoad soni qualitatem, aut valde uota-
bilem intensitatem; varietas tamen habetur quoad meliorem335
aliquam resonantiam : quid si intrumentum pneumaticum sit
compactum ex materia non resistente , quale v. g. esset in
strumentum membranaceum .. . n. 133.
Nonnulla proponuntur explicanda circa instrumenta
pneumatica. n . 134.
Tremulus aeris motus in tubis cylindricis determinatae
longitudinis :
1º. Quum tubus est firmiter obseratus apud alterum
orificium simulque apertus apud alterum n. 135, 136.
2°. Quum tubus est patens in utraque extremitate: in
de eruitur ratio investigandi velocitatem , qua propagatur
sonus in aliis fluidis elasticis diversis ab aere. n. 137.
3 °. Quum tubus est utrinque obseratus. n. 138.
De propagatione soni per liquida, et per solida
corpora . pag. 302.
Formulae huc spectantes: parvula contractio aquae et
hydrargiri ob auctam pressionem: usus istius contractionis in
determinanda velocitate soni per haec duo liquida. n . 139,140 .
Analogia inter oscillationes aeris in tubo cylindrico a
pud ambas extremitates aperto et longitudinales oscillationes
virgae rigidae suppeditat peculiarem methodum investigandi
velocitatem propagationis per solida corpora. n. 141 .
De vocis humanae origine. pag. 305.
Nonnulla ex anatomicis praemittuntur; quibus praemis
sis , stabilitur illud : vocis humanae organum etsi conside
rari maxime debeat tanquam instrumentum pneumaticum
335
aliquam resonantiam: quid si intrumentnm pneumaticum sit
compactum ex materia nou'reaistente, quale v.. g. esset in-
strumentum membranaceum. .. .. .. .. .. .. . n. 133.
Nonnulla proponuntur explicanda circa instrumenta
pneumatica. .. . .. .. .. .. .. .. .. .. .. ..
.n. 134.
Tremulus aeris motus'iu tubis cylindricis determinatae
longitudinis :
⇝
↿∘∙ ⊄⊇⇂⋯⋯∙⋯∣⋯∘⋅⊖⊱⇂ firmiter ohseratus apud alternm
orificium simulque apertus apud alterum . n. 135,136.
20. Quum tuhus est patens in utraque extremitate: in-
de eruitur ratio investigandi velocitatem
, qua propagatur
sonus in aliis fluidis elasticis diversis ab aere. n. 137.
(
3&quot;. Quum tubus est utrinque ohseratus. . n. 138.
i '
⋅
⋅ ↼
De prapagau'one soni per liquida, ettper &quot;solida ⊳∣ corpora. pag. 302.
.Fornrnlae huc spectantes: parvula contractio ailuae et
hydrargiri ob auctam pressionem: ususistius contractionis in
determinanda velocitate soni per haec duo liquida.ia.139, 140.
Analogia inter oscillationes aeris in tuho cylindrico a-
pud ambas extremitates aperto et lougitudinales oscillationes
virgae rigidae suppeditat peculiarem methodum investigandi
velocitatem propagatiouis per solida corpora. . n. 141.
De 'vocis humanae origine. pag. 305.
Nonuulla ex anatomicis praemittuntur; quibus praemis-
sis, stahilitur illud: vocis humanae organum etsi conside-
rari maxime debeat tanquam instrumentum pneumaticum
∩336
flexili et elastica materia ex parte compactum , non tamen ita
est ut cum instrumentis fidicularibus aliquam non habeat
analogiam . n. 142.
Quid, os atque ejus partes conferant ad formationem
vocis.
n. 143.
Variae refellantur sententiae de humanae vocis ori
gine; variaeque circa vocem humanam proponuntur quae
stiones. n . 144 , 145 .

De auditus organo . pag. 310.
Auris descriptio.
n. 146.
Quaenam ex auris partibus pro praecipuo atque im
mediato auditionis organo statuenda sit. n. 147.
Cur quibusdam grata, aliis pene nihil, aut etiam mo lesta sit harmonia ,
n. 148. 19.
Cur daabus auribus unus idemque sonus audiatur n.148.2 °.
1
336 '
Bexiliot'elgstica materia ex parte compactum, non tamen ita
eat ut cum, instrumentis iidicularibus aliquam non habeat
malogihmoo-o-0 ∙⋅∙∙∙⋅∙ ∙ ∙ ∙ ∙ ∙ ∙
&quot;0142.
Quid, os atque eius partes conferant ad formationem
'owa ∙∙∙ ∙ ∙ ∙ ∙ ∙ ∙∙∙∙∙ ∙ ∙
&quot;.1430
Variae refelluntur sententiae de humanae vocis ori-
gine, variaeque circa vocem humanam proponuntur quae-
'none'- ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ &quot;- 14401450
De auditus organo. pag. 310.
Auriadeacriptio. . . . . . . . . . n.146.
Quaenam ex auris partibus pro praecipuo atque im-
mediato auditionia organo statuenda- sit.
. . . n. 147.
Cur quibusdam grata, aliis pene nihil, aut etiam mo-
lestasit harmonia. ∙ ∙ ∙
. . . . . . n. 148. ∎∘∙
Cur duabus auribus unus idemque sonus audiatur n.148.2'.ERRATA CORRIGE
pag. lin .
1. 4. saepae
1. 5. decresit
4. 28. istanti
5. 13. rive
7. 29. poductis
8. 14. sin a
14 29. AH'.BC
15. 3. AF
saepe
decrescit
instanti .
siye .
prodactis
sin a .
AH'.BC
BF'
BF .
Y.
4. AF
24, 7. Sy
50. 6 et 7.
S * S

*

17. ſsfla)dx Sfaxdx.
52. 14.
f (x )dx f '( x )d.x2
2
2
56. 18.- ( tdx ,z + dz,u,...) -f(xtdx , z + de, u, ...).
dull .
Sfx )dx 22. ( x) dx
eck
58. 1.-C
+0 .
57. 4. del
1
Sfaydar
62. 3. W
v'dz'
11. dzi
63. 8. sint va
69. 12. quod
... 17.
v'du'
da
sintVC .
quoad
ngt 2gt
70. 7.- 7 . kalog(k2—12)
.
.log(k ?-- ).
1
-
1
!
ERBATA CORRIGE
pag. lin.
&quot;. 4. saepae saepe .
1. 5. decresit decrescit
4. 28. istanti instanti
.
5. 13. rive sive
.
7. 29. poductis productis .
8. 14. aina sin at
.
14 29. AH'.BC AH'.BC'
.
15. 3. AF' BF' .
⋅
∙ ∙ ∙ 4. AF BF
. 24, 7. &lt;nowiki&gt;:] z?&lt;/nowiki&gt;
.
50. 6et 7. f:&quot; f:&quot;
... 17. JfftæMx [f(xkiæ .
52. 14. figit&quot; f'(æ2)dæ* .
56. 18.—(æ-]-dæ,z-l—dz,u,...) --f(a—-[-dx , z—l-dz, u, -
.
)- - 57. 4. d,,p. dup.
.
.. .
22. 111-2635 f(ældæ
.
803 803
58. 1.-:.-C —]—G .
62. 3. 9) p
↿↿∙∙∙ v'dz' til—tf
.
d:, dz'
.
63. 8. siun/C sint;/C .
69. 12. quod ' quoad .
⋅ n : 2 c
...
17. ∘−⋚∟ ∉−≓∙∙
k: 70. 7. −∙∙ 2 2
∣⊂≄∣∘⊰≼∣∁≖−⋁≖⋟ −−⋅⊋−⋅ lOg(k —P )
--
. ∙∙∽∙∙⋅ −∙− ↼∙ -
∙−⊣ERRATA CORRIGE
pag. lin .
kdv Ka dy
70. 12 .
katus kype &quot;
71. 13 et 14. KC Kc
72. 23. u = ułgosinc u = a + g9 sinc .
75. 23. pressioni r.gMcosc' pressioni gMcosc' .
87. 2. Denotet enim a Denotet enim x .
IG &quot; IG &quot;
27. = IC &quot; : IC =
2 2
110. 9. R = Rcosa R = R , cosa
111. 5. 1880 to 288q'to .
da dala 146. 8 .
idt
148. 12. 69.º* 69. *
149. 6. x = A'B' - B'r - A'B ' - A'M x=A''B'-B'r=A'B'-A'M .''
x' ? c
x
151. 2 .
ic
(de)
Centre Ide
i
152. 78 et 20. r2
153. 22. (69)
154. 17.
72.°*
157. 8. SD
161. 26. 16931100
193. 23. u : M ' : fle ..
205. 7. aequeus
208. 14. aia
r2
( 70 ) .
72.*
GD .
19631100 .
Me : No :
aqueus ,
Q:.
i
3
ERRATA CORRIGE
∙∙∙ ∙∙∙∄≾≖∠≀⇂↗ *
kæ-I—uz ⋅
Kc
uza-l—gg sinc .
23. pressioni ngMcosc' pressioni gMcosc' .
pag. lin
70. 12:
liti—v- kZ.-v2
71. 13 et 14. KC
72. 23. uzu-l—gasinc '
75.
87. 2. Denotet enim a
∙⋅≆↴ IG&quot; ∙ IG&quot;
. 27.:IC :::—2— . 10: -—2-— .
110. 9- R::Rcosa: BzB, cos a .
111. 5. 1889'—]—-p' ∙ 28897'—-q)' .
' doc 146. 8. ; daz :
(2? (22?
148. 12. 6994! 6931:
149. 6. a::A&quot;B'-B'r-A&quot;B'-A'M sz&quot;B'-B'r:-A&quot;B'-A'M .
.... .. ⋅↕⋅≟≣∁ ⋅
...-7...
50 ac
152. 78 et20. fi ∙∘−⋮⋅⋅∙ ra .rz 153. 22. (69) (70)
.
154. 17. 7291: 724
157. 8. SD .GD .
161. 26. 16931100 - 19631100 .
193.23. p.':p.':p.., php.
205. 7. aequeus aqueus ,
208.1.£. a:a' «:a' .
Denotet enim æ .</text>
      <sha1>9mmv64dwr4bevdi02v5idx9hyiy1zde</sha1>
    </revision>
    <revision>
      <id>3697721</id>
      <parentid>3697681</parentid>
      <timestamp>2022-08-17T07:23:36Z</timestamp>
      <contributor>
        <ip>2A00:1FA0:463D:49B:1C13:8621:AED2:2</ip>
      </contributor>
      <comment>/* De directa soni propagatione per aerem. */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="727054" xml:space="preserve">== PRAEFATIO ==
Rerum naturalium ordinem considerare, Deumque in iis mirifice operantem intueri, proprium est verae sapientiae, quam Philosophia profitetur. Haec scientia, quae dicitur Physica , inter scientias homine dignissimas. atque inter praecipua Dei dona jure commendatur: ecquid enim potest esse praestantius aut utilius quam divinae sapientiae opera, Deumque ipsum suas in natura perfectiones ostentantem contemplari? An quod Deus omnipotentia sua non judicavit indignum in iis quae creavit , quod in iis quae regit et gubernat attentione sua dignatur Providentia Dei, hoc nos meditari supervacaneum atque otiosum iudicabimus? Otiosam illam dicerem Physicam, quae ita immoraretur in Operis consideratione, ut opificis non perpetue suspicere! industriam: caecus est, qui Deum non videt in natura ejusque providentiam ac sapientiam non admiratur. Similem illum dixerim homini, qui librum ob Oculos apertum tenens characterum elegantiam contemplatur, numerat verba; sensum non penetrat. ⋅Neque vero minus utilis Naturae cognitio ministris Ecclesiae quam caeteris hominibus existimanda est: imo et hanc ipsis maxime necessariam duxerim hoc praesertim tempore cum homines vano inflati doctrinae apparatu scientias pro viribus adversus Religionem convertant , et Phyicam praecipue revelationi satagant opponere , vereque Opponi non desinant clamare eoram ignaris. Cum igitur se linguae impiae in injuriam Religionis armant, pudeat hominem Religionis amantem, et eo charactere insignitum qui ipsum Religionis statuat defensorem, aut turpiter obmutescere, aut Religionem. male defensam hominibus impiis vanum jactantibus triumphum, et ministrorum ignorantiam in Religionis opprobrium vertentium, deridendam proponere. Quod si nihil a viro ecclesiastico quaereretur aliud in Physica quam honesta mentis recreatio, justaque serii laboris intermissio, quid potest esse homini laboribus sacris defuncto aut jucundius aut dignius quam otium inutile, ac saepae periculosum, otio erudito et physico commutare? Quam multa offeret naturae spectaculum , ipsiusque arcanorum inquisitio, quae laudabilem nutriant curiositatem ,utiles praebeant considerationes, suaves atque innoxias pariant delicias! Etenim nullam esse in Philosophia partem quae majori voluptate quam naturae contemplatio animum compleat, Tullii auctoritate facile possem confirmare. Haec nihilominus voluptas non sine studio ac labore comparari potest: sapientissimus enim rerum omnium creator et rector Deus, quamvis sensuum nostrorum perceptioni corpora haec sensibilia subjecerit, illorum tamen naturam et vim mira quadam sepsit caligine, ut quicumque ad eam penitus scrutandam accedunt, in media luce densis veluti tenebris repente se obrui sentiant; quas tenebras etiamsi non ita discutere possimus ut claram ac certam rerum omnium scientiam assequamur, attamen si nos studii, diligentiae ac laboris non piguerit, ita tenebras attenuari experiemur ut multarum rerum certam cognitionem , plurimarum admodum probabilem obtineamus . Ad occulta Naturae arcana inquirenda duae sunt viae, quas eximii ingenii vir Franciscus Baconus de Verulanio notavit in novo scientiarum organo lib . 1. aphor, 19. Prima, qua a sensu et particularibus incipientes advolamus ad axiomata maxime generalia; atque ex iis principiis, eorumque immota veritate judicamus et invenimus axiomata media . Altera a sensu et particularibus excitat axiomata ascendendo continenter et gradatim , ut ultimo loco perveniatur ad ma
xime generalia. Primam viam plures arripuerunt, qui conjecturas non admodum graves sequuti , atque experientia
non satis accurata innixi generalia axiomata nimia festina
tione constituerunt , iisque naturalium causarum et effe
ctyum vim omnem contineri voluerunt; atque in iis tuen
∼∣⋁in Physica quam honesta mentis recreatio, iustaque serii
laboris intermissio, quid potest esse homini laboribus sacris defuncto aut iucundius aut dignius quam Otium inutile,
ac saepae periculosum, Otioterudito et physico commutare?
Quam multa offeret naturae speCtaculum, ipsiusque arca-
norum inquisitio, quae laudabilem nutriant curiositatem,
utiles praebeant considerationes, suaves atque innoxias pariant delicias! Etenim nullam esse in Philosophia partem
quae majoriivoluptate quam naturae contemplatio animum
compleat, Tullii auctoritate facile possem confirmare.
Haec nihilominus voluptas non' sine studio. ac labore
Comparari potest: sapientissimus enim rerum omnium creator
et rector Deus, quamvis sensuum nostrorum perceptioni
corpora haec sensibilia subiecerit, illorum tamen naturam
et vim miraaquadam sepsit caligine, ut quicumque ad eam
penitus scrutantium accedunt, in media luce densis veluti
tenebris repente se obrui sentiant; quas tenebras etiamsi
non ita discutere possimus ut claram ac certam rerum o-
mnium scientiam assequamur, attamen si nos studii. dili-gentiae ac laboris non piguerit, ita tenebras attenuari ex-
periemur ut multarum rerum certam-cognitionem , pluri-
marum admodum probabileur Obtineamus. Ad Occulta Naturae arcana inquirenda duae sunt viae, quas eximii inge-
nii vir Franciscus Baconus de,.Verulamio notavit&quot; in novo
scientiarum organo lib. ∎∙ aphor. 19. Prima, qua a sensu et particularibus incipientes advolamus.ad axiomata-; mas-
xime generalia; atque ex iis principiis, eorumque-[immota
veritate iudicamus et invenimus axiomata 'media. :Altera'a
sensu et particularibus excitat axiomata ascendendo contio
nenter et gradatim, ut ultimo loco perveniatur adfusa-i-
xime generalis. Primam viam plures arripueruut, qui' con-
iecturas non admodum graves,,s'equuti , atque experientia
non satis accurata innixi generalia axiomata nimia festina-
tione constituerunt,, iisque naturalium causarum et eil'e-
ctuum vim omnem contineri voluerunt; atque in iis tuen-dis totam ingenii aciem intendentes inciderunt in perver
sam philosophandi rationem , adeo ut rerum universitatem
commenti sint omnino aliam ac éa est. Altera aliis placuit
via, qui rerum naturam in rebus ipsis longa observatione
atque accurata experientia quaerendam esse statuerunt; isti
effectuum et causarum naturalium indolem singillatim in
quirere coeperunt, corporum texturám intimam , configu
rationem, motum scrutati sunt; atque ex his; aliisque in
numeris diligentissime observatis Naturae leges deduxerunt,
verasque rerum causas deprehenderunt. ' Hoc pacto plura
nostris temporibus certissima sunt , quae olim ignoraban
tur : alia probabili conjectura assecuti sumus : adhuc ta
men non pauca restant ambigua et involuta ; sed non de
erunt fortasse, qui aliquando novas in lucem eruant veritates.
Notandum vero Naturam ubique mathematicam esse ,
eamque velle absque Mathesi expiscari perinde fore, ait Gul
lielminus , ac sine cruribus ambulare. Porro tota Naturae
compago soliditate constal geometrica, resque physica rei geo
metricae unitur mystico quodam nexu, quem soli mathe
maticae Analysi datum est reserare: Analyseos ductu ex ob
servationibus et experimentis ab iis quae in promptu sunt
ad reconditiora deferimur, et ab exteriori venustate ad in
ternos naturae sinus. Observationes quidem virium existentiam demonstrant, sed proprium est Analyseos pate
facere virium leges, curvas a corporibus in gyrum actis
descriptas, circumvolutionum ac motuum inaequalitates et
aberrationes; quae omnia aspectabilem hanc Mundi ma
chinam maxime illustrant . Quid ab Analyseos indole magis abhorrere videbatur quam electricorum phoenomenorum congeries, cum et innumera prope sint, et innumeris obnoxia conditionibus? Ad electricas tamen vires expendendas accessit Analysis , earumque non paucos effectus leges
que aequationibus definivit.
Ut Tyronum , qui physicis praelectionibus in Romano
Soc. Jesu Collegio dant operam, commodo utilitatique ser
'
dis totam ingenii aciem intendentes inciderunt 'inrïperwe'r.»
sam philosophandi rationem, adeo ut rerum.:nniversitatem
commenti sint omnino aliam-ac ea est. .Altera aliis placuit
via, qui rerum naturamin: rebus ipsis longa-ObservatiOne
atque- accurata - experientia quaerendam, 'esse' statuerunt: :.i'sd
effectuum. ïet; causarum. naturalium 'indolem tsin'gillat'im in—
quirere coeperunt, corporumf-textuttam--imimdmf, configu-
rationem, motum scrutati sunt; atque ex his, aliisque-.in-
numeris diligentissime observatis Naturae leges deduxerunt,
verasque rerum causas deprehenderunt. 'Hoc pacto plura
nostris tempOribus certissima sunt, quae Olim ignoraban-
tur: alia probabili coniectura assecuti sumus : adhuc ta-
men non pauca restant— ambigua et.-involuta; sed non de-
erunt fortasse, qui aliquando novas in lucem eruant veritates.
Notandum vero Naturam ubique mathematicam esse,
eamque velle absque Mathesi expiscari perinde fore, ait Gul-
lielminus, ac sine cruribus ambulare. Porro tota Naturae
compago soliditate constat geometrica, resque physica rei geo-
metricae unitur mystico quodam nexu, quem soli mathe-
maticae Analysi datum est reserare: Analyseos ductu ex Ob-
servationibus et experimentis ab iis quae in promptu sunt
ad reconditiora deferimur, et ab exteriori venustate ad in-
ternqs naturae sinus. Observationes quidem virium exi-
stentiam demonstrant, sed prOprium est Analyseos pate-
facere virium leges, curvas a corporibus in gyrum actis
descriptas, circumvolutionum se motuum inaequalitates et
aberrationes; quae omnia aspectabilem hanc Mundi mn-
chinam maxime illustrant. Quid ab Analyseos indole ma-
gis abhorrere videbatur quam electricorum phoenomenorum
congeries, cum et innumera prope sint, et innumeris Ob-
noxia conditionibus? Ad electricas tamen vires eXpenden-
das accessit Analysis, earumque non paucos eil'ectus' leges-
que aequationibus definivit.
Ut Tyronum, qui physicis praelectionibus in Romano
Soc. Iesu Collegio dant operam, commodo utilitatique ser-VI
viam , in lucem edo illas Physicae partes , quae intimiori
vinculo nectuntur mathesi, quarumque explicandarum cu
ra mihi est demandata. A Mechanica exordior ; siquidem
reliquarum est veluti basis et fundamentum : caeterum sic
Physicae mathematicae tractationem concinnavi, ut quae aste
risco inveniuntur signata, possint ab iis Tyronibus omitti,
qui primo duntaxat Philosophiae anno mathematicis insti
tutionibus studuerunt.
VI
viam
,
in lucem edo illas Physicae partes
, quae intimiori
vinculo nectuntur mathesi, quarumque explicandarum cu-
ra mihi est demandata. A Mechanica exordiar.; siquidem
reliquarum est veluti basiset fundamentum: caeterum sic
Physicae mathematicae tractationem concinnavi, ut quae aste-
risco inveniuntur signata, pOssint ab iis Tyron'ibus omitti,
qui primo duntaxat Philosophiae anno mathematicis insti-
tutionibus studuerunt.

== MECHANICAE PRINCIPIA ==

=== Notiones Praeambulae ===
[[1|1]]. Moto puncto materiali, si ratio inter numericos spatii percursi ac respondentis temporis valores &lt;math&gt;s&lt;/math&gt; ac &lt;math&gt;t&lt;/math&gt; permanet eadem, motus dicitur uniformis; quod si ratio illa jugiter mutetur, motus dicitur varius, acceleratus nempe vel
retardatus, prout crescente
e crescit vel decrescit ipsa &lt;math&gt;\frac s t &lt;/math&gt;: porro motus rectilineus atque uniformis est simplicissimus
omnium motuum, quorum exsistit capax punctum materiale. In &lt;u&gt;motu uniformi&lt;/u&gt; ratio &lt;math&gt;\frac s t &lt;/math&gt; dicitur &lt;u&gt;velocitas&lt;/u&gt;; qua designata per &lt;math&gt;v&lt;/math&gt;, erit

&lt;math&gt;v = \frac s t .&lt;/math&gt;

Quoad punctum materiale, cujus massa seu quantitas materiae (&lt;math&gt;=m&lt;/math&gt;), et velocitas &lt;math&gt;=v&lt;/math&gt;, factum &lt;math&gt;mv&lt;/math&gt; appellatur quantitas motus.

[[2|2]]. Corpus de se est indifferens ad motam et ad quietem. Haec indifferentia sic probari potest ex natura loci: nequit corpus de se postulare at localiter moveatur nisi exigat natura sua non esse in loco ubi est, et locum in quo non est occupare; contra nequit corpus de se quietem
exigere nisi exigat natura sua esse potius in loco ubi est quam in loco quem occuparet si moveretur. Neutrum vero ex natura sua exigit corpus; cum enim omnia loca sint
ejusdem rationis, jam nulla datur ratio cur corporea substantia exigat esse potius uno in loco quam in alio: propterea etc.

[[3|3]]. Quae causae motum gignunt, accelerant, retardant, detorquent, eae vocantur potentiae seu vires. Plures potentiae corpori aut corporum systemati applicitae sese ita possunt impedire, ut nullus inde oriatur motus; tunc vero potentiae dicuntur constitutae in aequilibrio. Fac ut duae vires punctum materiale sollicitent in partes contrarias; si eae sunt in aequilibrio, dicentur aequales: pone duas, tres etc. . . : . ex ejusmodi viribus aequalibus applicari puncto materiali ita , ut in unam eamdemque rectam conspirent; inde habebis vim duplam , triplam etc. . . . Poterunt nempe vires omnes exprimi per numeros ; et consequenter repraesentari per lineas rectas istis numeris proportionales, quarum directiones cum ipsarum virium directionibus congruant. Mechanica tota est in aequilibrii ac
motus doctrina consideranda.

[[4|4]]. Finge tibi globum &lt;math&gt;A&lt;/math&gt; quiescentem e filo pendulum,
in quem impingat globus &lt;math&gt;B&lt;/math&gt; cum certo quodam velocitatis
gradu. Si nullam motui resistentiam afferret globus &lt;math&gt;A&lt;/math&gt;, eadem velocitate pergeret moveri &lt;math&gt;B&lt;/math&gt;, qua movebatur antea , secum pertrahendo globum &lt;math&gt;A&lt;/math&gt;: cur enim minueretur motus in &lt;math&gt;B&lt;/math&gt;, cum globus &lt;math&gt;A&lt;/math&gt; nihil obstaret illius motui , et ipse loco suo facile cederet? Iamvero si experientiam consulimus, multo aliter rem evenire comperiemus: cedit quidem loco suo globus &lt;math&gt;A&lt;/math&gt;, sed non sine detrimento motus
in &lt;math&gt;B&lt;/math&gt;, eoque majori quo majorem globus &lt;math&gt;A&lt;/math&gt; opponit massam impellenti se globo &lt;math&gt;B&lt;/math&gt;. Resistere igitur motui , status
que mutationi obniti concipitur &lt;math&gt;A&lt;/math&gt;, acquisitumque motum
resistentia sua destruere in &lt;math&gt;B&lt;/math&gt;. Motus habetur tamquam vis
activae effectus; quod autem vis activae effectum destruit,
potest et ipsum verae vis nomen accipere. In ipsis etiam
corporibus motis sese prodit ejusmodi vis: corpus certo quodam velocitatis gradu donatum, eumdem servabit nisi quem
inveniat obicem , nec ullum sui motus augmentum patietur nisi cum vis alienae in ipsum agentis detrimento; haud
aliter ac restitit primo motui dum quiesceret; ipso in motu resistit majori motui.

Non ergo praefata indifferentia sita est in non renitentia ad motum ex quiete, vel in non renitentia ad quietem ex mota, sed in eo quod corpus de se non magis ad
motum quam ad quietem tendat, nec magis resistat quieti
si fuerit in motu quam molui renitatur si fuerit in quiete.
Quoniam igitur ab ipsa materia nequit oriri ulla de
terminatio ( huc pertinet materiae inertia ) ad novum statum sive quietis, sive motas; profecto deficiente causa quae
materiale punctum determinet ad hunc potius quam ad
illum novum statum, punctum ipsum si in quiete sit quiescet semper, si ad motum semel fuit excitatum perget moveri cum eadem perpetuo velocitate et directione: porro
motus directio est recta linea, quam mobile aut describit,
aut describere nititur; primum obtinet in motibus rectilineis, secundum in curvilineis. 

[[5|5]]. Duo puncta materialia &lt;math&gt;H&lt;/math&gt; et &lt;math&gt;K&lt;/math&gt; ( fig 1. ) eamdem
massam habentia, eamdemque lineam communi vi &lt;math&gt;P&lt;/math&gt; incedentia, haud dubie conjunctim procedent: verum ubi puncto &lt;math&gt;K&lt;/math&gt; praeter &lt;math&gt;P&lt;/math&gt; applicetur et vis &lt;math&gt;Q&lt;/math&gt;, disjungetur illico &lt;math&gt;K&lt;/math&gt;
ab &lt;math&gt;H&lt;/math&gt;, et observator constitutus in &lt;math&gt;H&lt;/math&gt; deprehendet: motum puncti &lt;math&gt;K&lt;/math&gt; perinde ac deprehenderet si &lt;math&gt;H&lt;/math&gt; quiesceret et &lt;math&gt;K&lt;/math&gt;
moveretur sola &lt;math&gt;Q&lt;/math&gt;: sive nimirum ponatur &lt;math&gt;H&lt;/math&gt; moveri vi &lt;math&gt;P&lt;/math&gt;
et &lt;math&gt;K&lt;/math&gt; viribus &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt;, sive &lt;math&gt;H&lt;/math&gt; quiescere et &lt;math&gt;K&lt;/math&gt; moveri unica
&lt;math&gt;Q&lt;/math&gt;, idem in utroque casu, experientia teste , prodibit motus puncti &lt;math&gt;K&lt;/math&gt; quoad &lt;math&gt;H&lt;/math&gt;: huc spectat principium motus
relativi . Jamvero in secundo casu motus relativus soli &lt;math&gt;Q&lt;/math&gt; est
manifeste, adscribendus; idipsum ergo dicendum et in primo. Effectus videlicet a nova vi &lt;math&gt;Q&lt;/math&gt; genitus in puncto materiali &lt;math&gt;K&lt;/math&gt; idem est utcumque caeteroqui se habeat praecedens status ipsius &lt;math&gt;K&lt;/math&gt;: quod consequi videtur ex materiei
inertia. Etenim si variato statu praecedente variaret effectus ille, non aeque se haberet materia ad status omnes ,
punctumque materiale sibi commissum rediret tandem in
statum illum , ad quem magis tendit; sicque ab ipsa materia oriretur determinatio ad novum statum.

[[6|6]]. Exhibeant
&lt;math&gt;v&lt;/math&gt; et
&lt;math&gt;v^\prime&lt;/math&gt; velocitates, quas gignunt vires
&lt;math&gt;P&lt;/math&gt; et
&lt;math&gt;Q&lt;/math&gt;, sitque
&lt;math&gt;u&lt;/math&gt; velocitas
, quam generat vis assumpta pro
communi mensura (3) ipsarum
&lt;math&gt;P&lt;/math&gt; et
&lt;math&gt;Q&lt;/math&gt;; erunt (5)
&lt;math&gt;v = Pu,
v^\prime = Qu&lt;/math&gt;, unde:
&lt;math&gt;v:v^\prime=Pu: Qu=P: Q&lt;/math&gt;.
Permanente videlicet massa, vires erunt ut simplices velocitates: et quoniam permanente velocitate et variata massa, vis est ut massa ipsa; inferimus vires esse ut motus quantitates.

[[7|7]]. Dixi ([[4]]) tantam motus quantitatem excitari in globo
&lt;math&gt;A&lt;/math&gt; quantam ipse
&lt;math&gt;A&lt;/math&gt; resistendo destruit in globo
&lt;math&gt;B&lt;/math&gt;: atque huc spectat illud de actione et reactione principium,
quod sic enunciari solet &quot;actioni contraria semper et aequalis est reactio, sive duorum corporum actiones in se mutuo semper sunt aequales, et in contrarias partes diriguntur&quot;. Huic autem principio locus est in rerum natura sive corpora in contactu agant in se mutuo, sive dissitis e locis sese invicem ad status mutationem quocumque modo determinent.

Notetur illud: cum corpus omne obnitatur semper sui statos mutationi, inferimus ipsam status mutationem haud repente gigni a viribus extrinsecis, sed per gradus indefinitae attenuationis capaces: secus enim dicendum foret inesse materiei vim quamdam infinitam. Siquidem in hypothesi finitae mutationis instantaneae materies valeret opponere resistentiam finitam, labente tempusculo infinite quod nequit admitti. Verum quia vires quaedam tam cito gignunt mutationem status, ut eam in istanti videantur absolvere; inde fit ut vires dividi soleant in instantaneas, et continuas.

=== De virium compositione et resolutione, deque earum momentis et aequilibrio: aliquid quoque notatur de vecte, axe in peritrochio , trochlea etc. . . . ===
[[8|8]]. Fac ut per communem vim &lt;math&gt;P&lt;/math&gt; puncta &lt;math&gt;H&lt;/math&gt; et &lt;math&gt;K&lt;/math&gt; (fig. 2.)
determinentur ambo ad percurrendam motu uniformi rectam lineam &lt;math&gt;AB&lt;/math&gt; intra tempus &lt;math&gt;t&lt;/math&gt; , per &lt;math&gt;Q&lt;/math&gt; vero determinetur
&lt;math&gt;K&lt;/math&gt; ad percurrendam motu pariter uniformi rectam lineam 
&lt;math&gt;AD&lt;/math&gt; intra idem tempus &lt;math&gt;t&lt;/math&gt; ; et comple parallelogrammum &lt;math&gt;BD&lt;/math&gt;.
Ex principio motus relativi punctum &lt;math&gt;K&lt;/math&gt; in fine temporis &lt;math&gt;t&lt;/math&gt;
reperietur in &lt;math&gt;C&lt;/math&gt; ; ac proinde intra tempus &lt;math&gt;t&lt;/math&gt; percurret motu
uniformi diagonalem &lt;math&gt;AC&lt;/math&gt; : idem nimirum existet motus sive mobile feratur per diagonalem &lt;math&gt;AC&lt;/math&gt; velocitate
&lt;math&gt;\frac{AC}{t}&lt;/math&gt; ex vi 
unica impressa &lt;math&gt;R&lt;/math&gt;, sive conjunctis viribus &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; impellatur per latera &lt;math&gt;AB&lt;/math&gt; et &lt;math&gt;AD&lt;/math&gt; velocitatibus &lt;math&gt;\frac{AB}{t}&lt;/math&gt; et &lt;math&gt;\frac{AD}{t}&lt;/math&gt;; eritque (6)

&lt;math&gt;
R : P : Q : =AC: AB: AD.
&lt;/math&gt;

Hinc pro duabus viribus &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; poterit, substitui vis
&lt;math&gt;R&lt;/math&gt;; quae substitutio dicitur virium compositio : et viceversa
pro &lt;math&gt;R&lt;/math&gt; poterunt substitui duae &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt;; quae substitutio dicitur virium resolutio : &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; vocantur componentes, &lt;math&gt;R&lt;/math&gt;
resultans, vel etiam composita.

[[9|9]]. Haec notentur. 
1º. ex tribus &lt;math&gt;R&lt;/math&gt; , &lt;math&gt;P&lt;/math&gt; , &lt;math&gt;Q&lt;/math&gt; unaquae
vis potest repraesentari per sinum anguli, qui sub aliarum
directionibus continetur ; nam
&lt;math&gt;
R : P : Q = AC : DC: AD = \sin BAD : \sin CAD : \sin BAC .
&lt;/math&gt;

2°. Hinc &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; sunt reciproce ut perpendicula ,
quae a puncto quolibet resultantis &lt;math&gt;R&lt;/math&gt; ducuntur ad ipsarum
&lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; directiones .

3º. Denotante &lt;math&gt;i&lt;/math&gt; angulum interceptum directionibus
virium &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt;, triangulum &lt;math&gt;A C D&lt;/math&gt; praebebit

&lt;math&gt;
RP = P^2 + Q^2 - 2PQ \cos(180^{\circ} - i) = P^2 + Q^2 + 2PQ \cos i.
&lt;/math&gt;

4°. Si punctum &lt;math&gt;K&lt;/math&gt; ( fig. 3. ) urgetur viribus &lt;math&gt;KA, KB, KC, KD&lt;/math&gt; etc. . . , ducantur autem &lt;math&gt;Aa&lt;/math&gt; parallela et
&lt;math&gt;= KB&lt;/math&gt; , &lt;math&gt;Aa'&lt;/math&gt; &lt;math&gt;Aa''&lt;/math&gt; parallela et &lt;math&gt;= KC&lt;/math&gt; , &lt;math&gt;a'' a''' &lt;/math&gt; parallela et&lt;math&gt; =
KD&lt;/math&gt; , etc. vis cunctis aequivalens exhibebitur manifeste per lineam rectam &lt;math&gt;Ka'''&lt;/math&gt;, quae jungit punctum &lt;math&gt;K&lt;/math&gt;
et extremitatem &lt;math&gt;a'''&lt;/math&gt; ultimae &lt;math&gt;a''a'''&lt;/math&gt; . Porro linearum rectarum aequalium et parallelarum projectiones sive in recta
quavis &lt;math&gt;EE'&lt;/math&gt;, sive in plano quovis , sunt aequales et parallelae: hinc virium &lt;math&gt;KA, KB, KC, KD&lt;/math&gt;, etc. . . projectiones in recta &lt;math&gt;EE'&lt;/math&gt; simul sumptae aequabuntur projectionibus rectarum &lt;math&gt;KA, Aa', a'a'', a'' a'''&lt;/math&gt; etc. , in eadem &lt;math&gt;EE'&lt;/math&gt; simul pariter sumptis. Harum vero projectionum summa nihil est aliud nisi projectio resultantis &lt;math&gt;Ka'''&lt;/math&gt; : igitur projectio resultantis aequabitur projectionibus componentium &lt;math&gt;KA, KB, KC, KD&lt;/math&gt;, etc. , in summam contractis , si modo habeatur ratio signorum, ut censeantur negativae, quae vergunt v. gr. ad &lt;math&gt;E&lt;/math&gt;, habitis pro positivis, quae versus &lt;math&gt;E'&lt;/math&gt; se dirigunt.

5°. In hypothesi trium duntaxat virium &lt;math&gt;KA, KB , KC&lt;/math&gt;, quisque videt aequipollentem vim repraesentatum iri
per diagonalem parallelepipedi sub lateribis &lt;math&gt;KA, KB, KC&lt;/math&gt;.

6°. Si punctum &lt;math&gt;K&lt;/math&gt; urgetur vi &lt;math&gt;Ka'''&lt;/math&gt;, constructo ad
libitum polygono &lt;math&gt;a''' a'' ... K&lt;/math&gt;, ductaque &lt;math&gt;KD&lt;/math&gt; parallela et &lt;math&gt;=a''' a''&lt;/math&gt; , &lt;math&gt;KC&lt;/math&gt; parallela et &lt;math&gt;= a'' a'&lt;/math&gt;, &lt;math&gt;KB&lt;/math&gt; parallela et &lt;math&gt;= a' A&lt;/math&gt; etc.
resolvetur &lt;math&gt;Ka'''&lt;/math&gt; in &lt;math&gt;KD, KC, KB&lt;/math&gt;, etc ....

7°. Ad resolvendam &lt;math&gt;Ka'''&lt;/math&gt; in ternas sese dirigentes
juxta datas rectas &lt;math&gt;KB, KC, KD&lt;/math&gt;, satis erit per &lt;math&gt;a'''&lt;/math&gt; ducere
tria plana parallela planis &lt;math&gt;BKC, CKD, BKD&lt;/math&gt;; hoc pacto
exsurget parallelepipedum , cujus latera apud &lt;math&gt;K&lt;/math&gt; exhibebunt
( 5°) quaesitas vires componentes.

8°.  Puncta &lt;math&gt;B, C, D, K&lt;/math&gt;, ponantur inter se rigidis
lineis connexa: manentibus virium directionibus, si ternae
componentes intelliguntur applicitae punctis &lt;math&gt;B, C, D&lt;/math&gt;, adhuc
iis manifeste aequipollebit &lt;math&gt;Ka'''&lt;/math&gt; . Inferimus vim quamvis
&lt;math&gt;Ka'''&lt;/math&gt; resolvi posse in ternas, quae et sint applicitae tribus punctis ad libitum sumptis ( si sumuntur ita , ut in
eorum plano inveniatur etiam punctum &lt;math&gt;K&lt;/math&gt;, non debebit
&lt;math&gt;Ka'''&lt;/math&gt; esse extra id planum ) et sese dirigant juxta rectas ab istiusmodi punctis ductas ad punctum &lt;math&gt;K&lt;/math&gt; , cui applicatur ipsa &lt;math&gt;Ka'''&lt;/math&gt;.

9º. Dato systemate punctorum materialium rigidis
lineis inter se firmiter connexorum ( huc spectat corpus solidum ) respondentibusque viribus sollicitatorum; quia possunt (8º. ) singulae vires resolvi in cernas applicitas tribus
punctis &lt;math&gt;A , B, C&lt;/math&gt; ad libitum sumptis, poterunt ( 4°) omnes
traduci ad aequipollens trium virium systema.

10° . Per unam ex hisce tribus viribus duc planum , quod secet reliquas duas : vis , per quam ducitur planum , poterit resolvi ( 4° ) in binas , applicitas intersectionum punctis. Inde fit, ut vires omnes solidum corpus sollicitantes
traduci etiam possint ad aequipollens duarum virium systema.

[[10|10]]. Facile est determinare quandonam plures potentiae eidem puncto applicitae in aequilibrio permaneant.
Binas potentias pro lubito sumptas compone, et pro illis
aequipollentem substitue , atque id iterato donec ad duas
devenias. Si hae directe contrariae et aequales inveniuntur, constabit omnes potentias in aequilibrio consistere .

Facile etiam intelliges quanam ratione inveniri possit potentia duabus &lt;math&gt;AH, BF&lt;/math&gt; ( fig. 4. ) in eodem plano jacentibus,
rectaeque rigidae &lt;math&gt;AB&lt;/math&gt; applicatis aequivalens, et aequilibrium
obtineri; productis (?) enim directionibus &lt;math&gt;AH, BF&lt;/math&gt; donec concurrant
in &lt;math&gt;C&lt;/math&gt;, transferantur potentiae in punctum &lt;math&gt;C&lt;/math&gt;. Sumptis in earum directionibus &lt;math&gt;CH' = AH&lt;/math&gt;, et &lt;math&gt;CF' = BF&lt;/math&gt;, istae componantur. Facto parallelogrammo &lt;math&gt;CF'KH'&lt;/math&gt;, cujus diameter &lt;math&gt;CK&lt;/math&gt; equivalentem vim repraesentabit, haec producatur donec concurrat in &lt;math&gt;D&lt;/math&gt; cum &lt;math&gt;AB&lt;/math&gt;; perspicuum est potentiam &lt;math&gt;KC&lt;/math&gt; translatam in &lt;math&gt;DL&lt;/math&gt; et rectae &lt;math&gt;AB&lt;/math&gt; applicitam in D aequipollere duabus &lt;math&gt;AH , BF&lt;/math&gt;. Quare si &lt;math&gt;AB&lt;/math&gt; in puncto &lt;math&gt;D&lt;/math&gt; sustentetur, potentiae &lt;math&gt;AH, BF&lt;/math&gt; in aequilibrio quiescent; et constabit quam potentiam exerceat punctum &lt;math&gt;D&lt;/math&gt;, nimirum aequalem et oppositam potentiae aequivalenti &lt;math&gt;DL&lt;/math&gt;. Ad positionem puncti &lt;math&gt;D&lt;/math&gt; quod pertinet, concipiamus ex eo duci duo perpendicula &lt;math&gt;p&lt;/math&gt; et &lt;math&gt;q&lt;/math&gt; , alterum in &lt;math&gt;AH&lt;/math&gt; , alterum
in &lt;math&gt;BF&lt;/math&gt; ; sintque &lt;math&gt;AH = P , BF = Q&lt;/math&gt;, longitudo &lt;math&gt;AB = h ,
AD = x&lt;/math&gt;, angulus &lt;math&gt;BAC =a&lt;/math&gt;, angulus &lt;math&gt;ABC = b&lt;/math&gt; : erunt &lt;math&gt;p =
x \sin a, q = ( h- x ) \sin b &lt;/math&gt;, ideoque &lt;math&gt;\frac p q = \frac{x \sin a}{(h - x ) \sin b}  &lt;/math&gt;
Sed( 9.2º )
&lt;math&gt;\frac p q = \frac Q P &lt;/math&gt;;
igitur
&lt;math&gt; \frac Q P = 
\frac{x \sin a}{
(h- x ) \sin b} &lt;/math&gt;, unde &lt;math&gt; \frac{x }{
(h- x ) } = \frac{Q \sin b}{P \sin a }.   &lt;/math&gt;

Quod spectat ad angulum interceptum resultante &lt;math&gt;CK&lt;/math&gt; et
data recta &lt;math&gt;AC&lt;/math&gt; , is dicatur &lt;math&gt;\alpha&lt;/math&gt; : erit ( 9. 1º )
&lt;math&gt;P : Q = \sin BCD : \sin ACD= \sin ( 180^{\circ}- a - b- \alpha) : \sin \alpha&lt;/math&gt;,
unde

&lt;math&gt;\tan \alpha =\frac{
Q \sin ( a + b )}{
P - Q \cos ( a + b )}.&lt;/math&gt;

Quod vero spectat ad resultantem
&lt;math&gt;CK ( = R )&lt;/math&gt; , habemus ( 9. 3º )
&lt;math&gt;R^2 = P^2 + Q^2 - 2P Q\cos ( a + b )&lt;/math&gt;.

Penultima formula traduci potest ad

&lt;math&gt;\cos \alpha = \frac{P - Q \cos ( a + b )}{
R}.   &lt;/math&gt;

Haec subjungimus. 1º. Recta &lt;math&gt;AB&lt;/math&gt; rotetur circa &lt;math&gt;D&lt;/math&gt;, ut ejus extrema puncta &lt;math&gt;A&lt;/math&gt; et &lt;math&gt;B&lt;/math&gt; eodem tempusculo infinitesimo
describant circulares arcus infinitesimos &lt;math&gt;Aa', Bb'&lt;/math&gt;; ex &lt;math&gt;a'&lt;/math&gt; et &lt;math&gt;b'&lt;/math&gt;
duc perpendicula &lt;math&gt;a'a'', b'b''&lt;/math&gt; in directiones virium &lt;math&gt;AH ,
BF&lt;/math&gt; ; sintque &lt;math&gt;Aa'' = p' , Bb'' = q'&lt;/math&gt;: erunt &lt;math&gt;p' = Aa' \cos a'Aa''
= Aa' \cos ( DAa'' - 90^{\circ} ) = Aa' \sin DAa'' = Aa' \sin a ,
q'= Bb'\cos b'Bb'' = Bb'\cos (90^{\circ}-b) = Bb'\sin b&lt;/math&gt;; et consequenter
&lt;math&gt;\frac{p'}{q'}= \frac{Aa' \sin a}{
Bb' \sin b}=
\frac{AD \sin a}{BD \sin b} = \frac{x \sin a}{(h - x ) \sin b} =
\frac{Q}{P} .&lt;/math&gt;

Nihil sunt aliud &lt;math&gt;Aa'&lt;/math&gt; et &lt;math&gt;Bb'&lt;/math&gt; nisi spatiola tempusculo infinitesimo circa immobile punctum &lt;math&gt;D&lt;/math&gt; simul describenda ab &lt;math&gt;A&lt;/math&gt;
et &lt;math&gt;B&lt;/math&gt; in hypothesi turbati aequilibrii; quibus punctis &lt;math&gt;A&lt;/math&gt; et &lt;math&gt;B&lt;/math&gt;
applicantur vires &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt;: exhibent &lt;math&gt;p', q'&lt;/math&gt; illorum spatiolorum projectiones super ipsarum virium directionibus. Vires igitur &lt;math&gt;P, Q&lt;/math&gt; sese mutuo librantes circa &lt;math&gt;D&lt;/math&gt; erunt reciproce ut eae projectiones.

2º. Etiam sic : triangula &lt;math&gt;Aa'a'', DAh&lt;/math&gt; , itemque
&lt;math&gt;Bb'b'', DBh'&lt;/math&gt; habent latera sibi respective perpendicularia ; igitur &lt;math&gt;\frac {DA} {Aa'} = \frac{p}{p'} , \frac{DB} {Bb'} = \frac{q}{q'}&lt;/math&gt;.

Denotet &lt;math&gt;i&lt;/math&gt; valorem rationum aequalium &lt;math&gt; \frac{DA}{Aa'} , \frac {DB}{Bb'}   &lt;/math&gt;, projectio insuper &lt;math&gt;p'&lt;/math&gt; computata in ipsa directione respondentis potentiae &lt;math&gt;P&lt;/math&gt; censeatur positiva; projectio vero &lt;math&gt;q'&lt;/math&gt; computata in directione contraria illi , quam obtinet respondens  potentia &lt;math&gt;Q&lt;/math&gt; , censeatur negativa: erunt
&lt;math&gt;p = ip' , q = -
iq'&lt;/math&gt; ;
propterea
&lt;math&gt; \frac QP = \frac pq = -\frac{ip'}{iq'}= -\frac{p'}{q'}
, Pp' + Qq' = 0 .&lt;/math&gt; 

Huc spectat principium velocitatum &lt;u&gt;virtualium&lt;/u&gt;.

3º. Ex quovis puncto (&lt;math&gt;  M&lt;/math&gt; ) sive intra , sive extra
angulum &lt;math&gt; ACB&lt;/math&gt; , duc perpendicula &lt;math&gt; p'', q'' , r''&lt;/math&gt;  ad &lt;math&gt; P, Q, R&lt;/math&gt;  ; duc
quoque ab (&lt;math&gt;  M&lt;/math&gt; ) ad &lt;math&gt; C &lt;/math&gt; rectam ( &lt;math&gt; MC = c &lt;/math&gt; ), cui normaliter
insistat alia recta (&lt;math&gt;  E E' &lt;/math&gt; ) transiens per &lt;math&gt; C &lt;/math&gt;: singulis &lt;math&gt; P , Q,
R &lt;/math&gt; resolutis in duas , alteram juxta (&lt;math&gt;  CM &lt;/math&gt; ) , alteram juxta
( &lt;math&gt; EE'&lt;/math&gt;  ), expriment
&lt;math&gt; P\frac{p''}{c},Q\frac{q''}{c},R\frac{r''}{c} &lt;/math&gt; 
componentes juxta (&lt;math&gt;  EE'&lt;/math&gt;  ) . Quoad (&lt;math&gt;M&lt;/math&gt;) situm extra angulum &lt;math&gt;ACB&lt;/math&gt;, primae duae erant conspirantes; quoad (&lt;math&gt;M &lt;/math&gt;)
situm intra &lt;math&gt;ACB&lt;/math&gt; erunt contrariae : cum igitur &lt;math&gt;R&lt;/math&gt; resultet ex &lt;math&gt;P&lt;/math&gt;
et &lt;math&gt;Q&lt;/math&gt;, prodibit ( 9. 4° ) in primo casu
&lt;math&gt; P\frac{p''}{c}+Q\frac{q''}{c}=R\frac{r''}{c} &lt;/math&gt; et consequenter &lt;math&gt;Pp'' + Qq'' = Rr''&lt;/math&gt;,
in secundo.

&lt;math&gt; \pm(P\frac{p''}{c}-Q\frac{q''}{c}) = R\frac{r''}{c} &lt;/math&gt;, ideoque &lt;math&gt; \pm(Pp''-Qq'') = Rr'', &lt;/math&gt; sumptis signis vel superioribus , vel inferioribus , prout 
&lt;math&gt; P\frac{p''}{c} &gt;  &lt;/math&gt; vel &lt;math&gt; &lt;Q\frac{q''}{c} &lt;/math&gt;: rectangula &lt;math&gt; Pp'',Qq'', Rr'' &lt;/math&gt;  dicuntur momenta virium &lt;math&gt;P, Q, R&lt;/math&gt; quoad punctum (&lt;math&gt;M&lt;/math&gt;). Hinc stabilitur
illud: momentum resultantis &lt;math&gt;R&lt;/math&gt; aequatur summae ex momentis componentium &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; si &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; in eamdem plagam circa (&lt;math&gt;M&lt;/math&gt;) nituntur movere puncta , quibus applicitae
sunt; aequatur differentiae si nituntur movere in plagas
contrarias.

4° . Idipsum facile extenditur ad quemvis numerum
virium &lt;math&gt;P, Q, S, V, U&lt;/math&gt;, ... in dato plano jacentium : fac
v. gr. ut ternae &lt;math&gt;P, Q, S&lt;/math&gt;, in unam eamdemque plagam circa ( &lt;math&gt;M&lt;/math&gt; ) nitantur movere puncta, ad quae sunt applicitae; caeterae vero &lt;math&gt;V, U&lt;/math&gt;, ... in plagam contrariam ; sitque &lt;math&gt;L&lt;/math&gt; resultans ex &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt;; &lt;math&gt;N&lt;/math&gt; resultans ex &lt;math&gt;L&lt;/math&gt; et &lt;math&gt;S&lt;/math&gt;, ac proinde ex
&lt;math&gt;P, Q, S&lt;/math&gt;; &lt;math&gt;O&lt;/math&gt; resultans ex reliquis &lt;math&gt;V, U&lt;/math&gt;. . . Erurt
&lt;math&gt;Ll''= Pp'' + Qq'', Nn'' = Ll'' +Ss''&lt;/math&gt; ;
et consequenter
&lt;math&gt;Nn'' = Pp'' + Qq'' + Ss''&lt;/math&gt; :
simili modo obtinetur
&lt;math&gt;Oo'' = Vv'' + Uu''+&lt;/math&gt; .
Iam si &lt;math&gt;R&lt;/math&gt; exhibet resultantem ex &lt;math&gt;N&lt;/math&gt; et &lt;math&gt;O&lt;/math&gt; , ideoque ex &lt;math&gt;P,
Q, S, V , U &lt;/math&gt;, ... ; cum sit 
fist the
&lt;math&gt;Rr'' = \pm ( Nn'' - Oo'' ) &lt;/math&gt;,
erit quoque
&lt;math&gt;Rr'' = ( Pp'' + Qq'' + Ss'' - Vv'' - Uu'' - ... ) .&lt;/math&gt;

5º. Fac ut &lt;math&gt;R&lt;/math&gt; transeat per (&lt;math&gt;M&lt;/math&gt;) ; erit &lt;math&gt;r'' = 0&lt;/math&gt;: propterea
&lt;math&gt;Pp'' + Qq'' + Ss'' - Vv'' - Uu'' - ... = 0&lt;/math&gt; ;
viriumque systema consistet in aequilibrio circa immobile
punctum (&lt;math&gt;M&lt;/math&gt;) . Vocatur (&lt;math&gt;M&lt;/math&gt;) centrum momentorum.

6º. Habemus ( 2 )
&lt;math&gt;p'' = ip' , q'' = iq' , s'' = is' , v'' = -iv', u'' = - iu', ...&lt;/math&gt;
Traducetur igitur aequatio ( 5°) ad
&lt;math&gt;Pp' + Qq' + Ss' + Vv' + Uu' + ... = 0&lt;/math&gt;

7° Vires &lt;math&gt;AH, BF&lt;/math&gt; haud jacentes in eodem plano nequeunt ad unicam vim aequipollentem traduci. Si enim tradu
ad aequipollentem &lt;math&gt;DL&lt;/math&gt;, poterit etiam ex quodam
istius puncto ad quoddam punctum componentis v . gr.
&lt;math&gt;BF&lt;/math&gt; duci recta linea haud occurrens alteri componenti &lt;math&gt;AH&lt;/math&gt;:
fac ut haec recta linea evadat immobilis ; elisa &lt;math&gt;DL&lt;/math&gt;, emerget aequilibrium; sed elisa quoque &lt;math&gt;BF&lt;/math&gt;, et salva &lt;math&gt;AH&lt;/math&gt;, ex hac ultima emerget motus. In ea ergo qua sumus hypothesi de
traductione virium &lt;math&gt;AH, BF&lt;/math&gt; ad unicam &lt;math&gt;DL&lt;/math&gt; obtinebit simul
aequilibrium et motus in eodem systemate: quod nequit esse; ideoque etc. ... 

8°. Patet solidum liberumque corpus haud consistere
in aequilibrio, nisi binae aequipollentes ( 8. 10° ) vires ,
ad quas traducuntur vires omnes corpus ipsum sollicitantes , sint aequales, contrariae, jaceantque in directum .

9°. Patet quoque solidum corpus, mobile dumtaxat
circa punctum fixum , consistere in aequilibrio, si eae binae
vires aequipollentes et jaceant in eodem plano ( 7º) , et suppeditent resultantem , quae transeat per punctum illud .

10°. Solidum corpus ponatur mobile dumtaxat circa
rectam fixam &lt;math&gt;AZ&lt;/math&gt; ( fig.5 ), sintque &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; binae aequipollentes
vires, ad quas traducuntur ( 9. 10° ) vires omnes corpus ipsum sollicitantes. Duc planum &lt;math&gt;XOY&lt;/math&gt; et normaliter insistens
rectae &lt;math&gt;AZ&lt;/math&gt;, et secans in punctis v. gr. &lt;math&gt;B, C&lt;/math&gt; directiones virium &lt;math&gt;P ( = BB' ), Q ( = CC' )&lt;/math&gt;: poterit &lt;math&gt;P&lt;/math&gt; resolvi in duas , alteram &lt;math&gt;BB'''&lt;/math&gt; perpendicularem plano &lt;math&gt;XO&lt;/math&gt;Y , alteram &lt;math&gt;BB''&lt;/math&gt; jacentem in ipso &lt;math&gt;XOY&lt;/math&gt;; similiter &lt;math&gt;Q&lt;/math&gt; poterit resolvi in duas ,
alteram &lt;math&gt;CC'''&lt;/math&gt; perpendicularem eidem &lt;math&gt;XOY&lt;/math&gt;, alteram &lt;math&gt;CC''&lt;/math&gt; in
eo jacentem . Binae &lt;math&gt;BB''', CC'''&lt;/math&gt;, utpote parallelae ad rectam fixam &lt;math&gt;AZ&lt;/math&gt;, peribunt elisae : in ea igitur qua sumus hypothesi
haud consistet solidum corpus in aequilibrio, nisi resultans
ex &lt;math&gt;BB'' , CC''&lt;/math&gt; transeat per aliquod punctum &lt;math&gt;O&lt;/math&gt; rectae fixae
&lt;math&gt;AZ&lt;/math&gt; ; et consequenter ( 9. 2° ) , ductis ex &lt;math&gt;O&lt;/math&gt; in istas vires
perpendicalis &lt;math&gt;b, c&lt;/math&gt;, nisi valeat aequatio

&lt;math&gt;\frac{b}{c} = \frac{CC''}{BB''} &lt;/math&gt;:

producta ex &lt;math&gt;b&lt;/math&gt; in &lt;math&gt;BB''&lt;/math&gt; et ex &lt;math&gt;c&lt;/math&gt; in &lt;math&gt;CC''&lt;/math&gt; dicuntur momenta virium &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; quoad &lt;math&gt;AZ&lt;/math&gt; . Si &lt;math&gt;P&lt;/math&gt; v. gr. , applicita ad punctum
&lt;math&gt;B'&lt;/math&gt;, est parallela plano &lt;math&gt;XOY&lt;/math&gt;, applicabuntur ad &lt;math&gt;B&lt;/math&gt; duae quaelibet vires &lt;math&gt;H &lt;/math&gt; et &lt;math&gt;- H&lt;/math&gt; aequales, contrariae et parallelae axi &lt;math&gt;AZ&lt;/math&gt;; tum una ex iis v. gr. &lt;math&gt;H&lt;/math&gt; componetur cum &lt;math&gt;P&lt;/math&gt; : vis inde
resultans poterit transferri in punctum v. gr. &lt;math&gt;B&lt;/math&gt; plani &lt;math&gt;XOY&lt;/math&gt;,
ibique resolvi in binas, alteram &lt;math&gt;BB''' ( = H )&lt;/math&gt; parallelam rectae &lt;math&gt;AZ&lt;/math&gt;, alteram &lt;math&gt;BB'' ( = P )&lt;/math&gt; jacentem in &lt;math&gt;XOY&lt;/math&gt;; eritque
&lt;math&gt;b. BB ' ( = b. P )&lt;/math&gt; momentum vis &lt;math&gt;P&lt;/math&gt; quoad &lt;math&gt;AZ&lt;/math&gt;. Quisque autem videt , si per &lt;math&gt;B '&lt;/math&gt; ducitur planum parallelum plano &lt;math&gt;XOY&lt;/math&gt;,
et ex pancto, ubi istud novum planum secat rectam &lt;math&gt;AZ&lt;/math&gt;, demittitur perpendiculum in vim &lt;math&gt;P&lt;/math&gt; applicitam ad &lt;math&gt;B '&lt;/math&gt;, ejusmodi
perpendiculum nihil fore aliud nisi &lt;math&gt;b&lt;/math&gt;; ita ut, sive momen
tum sumatur apud planum &lt;math&gt;XOY&lt;/math&gt;, sive apud illud alterum
planum parallelum ipsi &lt;math&gt;XOY&lt;/math&gt;, perinde sit.

[[11|11]]. Fac ut vis ( 10) &lt;math&gt;BF&lt;/math&gt; (fig. 4) revolvatur circa punctum &lt;math&gt;B&lt;/math&gt;,
donec evadat parallela vi &lt;math&gt;AH&lt;/math&gt;; erit &lt;math&gt;a + b = 180^{\circ}&lt;/math&gt;, ideo
que &lt;math&gt;\sin b = \sin (180^{\circ} - a ) = \sin a&lt;/math&gt; si vires ad eamdem
plagam obvertantur ; &lt;math&gt;a + b = 360^{\circ}&lt;/math&gt;, ideoque &lt;math&gt;\sin b =
\sin ( 360^{\circ} - a ) = - \sin a &lt;/math&gt; si ad contrarias plagas. In primo
igitur casu exsistent.

&lt;math&gt;\frac{x}{h-x} = \frac{Q}{P}, x=  \frac{hQ}{P+Q}, R = P + Q , \cos \alpha =\frac{P+Q}{R}=1.&lt;/math&gt;

In secundo

&lt;math&gt;\frac{x}{h-x} = -\frac{Q}{P}, x=  \frac{hQ}{Q-P}, R = \pm(P - Q) , \cos \alpha =\frac{P-Q}{R}=\pm 1.&lt;/math&gt;

valet signum superius ubi &lt;math&gt; P &gt; Q&lt;/math&gt;, inſerius ubi &lt;math&gt;P &lt; Q&lt;/math&gt;;
siquidem &lt;math&gt;P, Q, R&lt;/math&gt; denotant hic virium dumtaxat intensitates.
Inferimus illud; resultans ex duabus parallelis viribus adaequat istarum vel summam, vel differentiam , prout
vel ambae conspirant in eamdem plagam, vel altera in unam
et altera in contrariam plagam; ipsis insuper componentibus
viribus est parallela , et ad eam plagam semper obversa ,
quam respicit major ex componentibus illis ; transit denique
per ejusmodi punctum in directione &lt;math&gt;AB&lt;/math&gt;, quod distet a
punctis applicationis componentium in reciproca earum ratione : istud punctum appellari solet centrum virium parallelarum ; estque invariabile, modo et respectiva virium
positio et ipsarum ratio non mutentur.
Si &lt;math&gt;P = Q&lt;/math&gt;, in secundo casu nulla exsistet resultans.
Non est enim ratio in ea qua sumus hypothesi cur ad
plagam unius potius componentis quam ad alterius componentis plagam sese dirigat resultans. Formulae praebent &lt;math&gt;x= \infty, R =0.&lt;/math&gt;

Etsi vires &lt;math&gt;AH&lt;/math&gt; et &lt;math&gt;BF&lt;/math&gt; (fig.6) parallelae, aequales et contrariae nequeunt librari unica vi , utpote omni resultante destitutae; librabuntur nihilominus duabus aliis viribus &lt;math&gt;AH'&lt;/math&gt; et &lt;math&gt;BF'&lt;/math&gt; parallelis, aequalibus, contrariis, et in plano &lt;math&gt;HABF&lt;/math&gt; iacentibus, dummodo ductis ex &lt;math&gt;A&lt;/math&gt; in &lt;math&gt;BF BF'&lt;/math&gt;
perpendiculis &lt;math&gt;AO&lt;/math&gt; et &lt;math&gt;AO'&lt;/math&gt;, exsistat
&lt;math&gt;BF. AO=BF'. AO'&lt;/math&gt;:
tunc enim , ductis ex &lt;math&gt;B&lt;/math&gt; in &lt;math&gt;AH&lt;/math&gt; et &lt;math&gt;AH'&lt;/math&gt; perpendiculis &lt;math&gt;BC&lt;/math&gt; et
&lt;math&gt;BC'&lt;/math&gt;, ob &lt;math&gt;BF =  AH , BF' = AH' , AO = BC , AO' = BC'&lt;/math&gt;
erit quoque
&lt;math&gt;AH. BC=AH'. BC'&lt;/math&gt;;
et consequenter ( 9. 2°) resultans ex &lt;math&gt;AH&lt;/math&gt; et &lt;math&gt;AH'&lt;/math&gt; sese diriget a puncto &lt;math&gt;A&lt;/math&gt; ad punctum &lt;math&gt;B&lt;/math&gt;, simulque resultans ex &lt;math&gt;BF&lt;/math&gt; et
&lt;math&gt;BF'&lt;/math&gt; sese diriget a puncto &lt;math&gt;B&lt;/math&gt; ad punctum &lt;math&gt;A&lt;/math&gt; ; istiusmodi praeterea resultantes sunt manifeste aequales: iccirco etc. ... Systema itaque virium &lt;math&gt;AH', AF'&lt;/math&gt; aequipollebit systemati virium
&lt;math&gt;AH , AF&lt;/math&gt; ; poteritque alterum ( mutatis ejus directionibus
in contrarias partes ) alteri substitui. Consequitur posse binas vires parallelas, aequales et contrarias transferri ab
una positione ad alteram in proprio ipsarum plano, variata simul virium et magnitudine , et directione ; modo tamen productum ex communi earum valore in mutuam distantiam maneat constans.

[[12|12]]. Sint nunc plures vires parallelae &lt;math&gt;P, P ', P ''&lt;/math&gt;, ... variis
solidi corporis punctis applicitae , quarum aliae conspirent
in unam plagam , aliae in plagam contrariam . Componendo &lt;math&gt;P&lt;/math&gt; v . gr. et &lt;math&gt;P'&lt;/math&gt; in unicam &lt;math&gt;R '&lt;/math&gt;, &lt;math&gt;R'&lt;/math&gt; et &lt;math&gt;P'&lt;/math&gt; in unicam &lt;math&gt;R''&lt;/math&gt; , &lt;math&gt;R''&lt;/math&gt; et &lt;math&gt;P'''&lt;/math&gt;
in unicam &lt;math&gt;R''' &lt;/math&gt;, etc. , ... facile devenies ( 11 ) ad illud : resultans
&lt;math&gt;R&lt;/math&gt; ex pluribus viribus parallelis adaequat differentiam inter
summam conspirantium in unam plagam et summam conspirantium in plagam contrariam ; ipsis insuper componentibus viribus est parallela , et ad eam plagam obvertitur , quam
respicit major ex illis summis .

Hinc si vires tendentes in unam plagam censentur positivae , tendentes vero in plagam contrariam negativae , obtinebit aequatio
&lt;math&gt;R = P + P' + P'' + ... (a )&lt;/math&gt;.
Ad haec : denotantibus (fig .7) &lt;math&gt;A, B, D &lt;/math&gt;, ... puncta , quibus applicantur parallelae vires &lt;math&gt;P , P ', P''&lt;/math&gt;, ... , et &lt;math&gt;AB , BD &lt;/math&gt;. ..
rigidas rectas jungentes puncta illa , cum transeant &lt;math&gt;R ', R '' &lt;/math&gt;, ...
per ejusmodi puncta &lt;math&gt;K , H &lt;/math&gt;, ... , quorum positiones sive in
rectis &lt;math&gt;AB , KD &lt;/math&gt;, ... sive in earum prolongationibus unice pendent a conditionibus
&lt;math&gt;P ' :R'= AK :AB , P'' : R'' = HK : KD,&lt;/math&gt; etc. ... ,

seu

&lt;math&gt;P: P'+P= AK : AB , P'' : P + P' + P'' = HK : KD&lt;/math&gt;, etc. ... ,

devenietur etiam ad illud : in systemate parallelarum viriam
habetur constans et immutabile centrum , per quod semper transit resultans &lt;math&gt;R&lt;/math&gt; , quacumque ceteroqui ratione componentes vires volvantur circa puncta quibus applicitae sunt ,
modo et maneant parallelae , et applicitae iisdem punctis in
iisdem respective distantiis.

[[13|13]]. Ducto quolibet plano &lt;math&gt;MQ&lt;/math&gt;, demittantur in illud ex
punctis &lt;math&gt;A , B , D,&lt;/math&gt; ... perpendicula &lt;math&gt;AM ( =z) , BN ( = z; ) , DQ
( = z''), ...&lt;/math&gt; ; sive ( 12) &lt;math&gt;K , H &lt;/math&gt;, ... sint in rectis &lt;math&gt;AB , KD &lt;/math&gt;, ... .
sive in earum prolongationibus , demittantur quoque in
idem &lt;math&gt;MQ&lt;/math&gt; ex istis punctis perpendicula &lt;math&gt;KL , HO &lt;/math&gt;, ... ; per
ipsa &lt;math&gt;K , H &lt;/math&gt;, ... agantur rectae &lt;math&gt;RS , TU &lt;/math&gt;, ... , prima rectae
MN parallela et perpendiculis &lt;math&gt;AM , BN&lt;/math&gt; occurrens in &lt;math&gt;R , S &lt;/math&gt;,
secunda rectae &lt;math&gt;LQ&lt;/math&gt; parallela et perpendiculis &lt;math&gt;KL , DQ&lt;/math&gt; occurrens in &lt;math&gt;T , U &lt;/math&gt;, etc ... Erunt
&lt;math&gt;AR = MR - AM = KL - z, BS = BN - NS =z' - KL,
DU=UQ-DQ=HO-z'', KT = KL - LT = KL - HO &lt;/math&gt;; etc ....
Jamvero ( 11 )
&lt;math&gt;BS:AR = BK :AK = P : P' ,DU :KT = DH :HK = P + P':P''&lt;/math&gt;,etc ...,
ideoque
&lt;math&gt;AR.P = BS.P', DU.P'' = KT (P + P'), &lt;/math&gt;etc....
Igitur
&lt;math&gt;(KL- z) P = (z' -KL )P',(HO- z'') P'' = (KL-HO)(P + P'),&lt;/math&gt;etc....
unde
&lt;math&gt;KL (P + P') = zP + z'P', HO (P + P + P'' ) = KL (P + P') +
z'' P '' = zP + z' P' +z'' P'',&lt;/math&gt; etc.
seu
&lt;math&gt;KL. R ' = zP + z' P', HO. R'' = zP + z'P' + z''P'' , &lt;/math&gt;etc....

Generatim exhibente &lt;math&gt;z_{\mathrm I}&lt;/math&gt;, perpendiculum ex centro omnium datarum virium parallelarum ductum in &lt;math&gt;MQ&lt;/math&gt; , habebimus

&lt;math&gt;z_{\mathrm I} R = zP + z' P' + z'' P'' + z''' P ''' + ... :&lt;/math&gt;
rectangula &lt;math&gt;z_{\mathrm I} R , zP&lt;/math&gt;, dicuntur momenta virium &lt;math&gt;R , P&lt;/math&gt;, ...
quoad plapum &lt;math&gt;MQ&lt;/math&gt;.

Haec notentur: 1° Etsi non omnia puncta , quibus applicantur parallelae vires &lt;math&gt;P , P', P'' &lt;/math&gt;... sita sunt supra planum &lt;math&gt;MQ&lt;/math&gt;
adhuc tamen algebraica summa rectangulorum sub &lt;math&gt;P , P'&lt;/math&gt; ... et
respondentibus perpendiculis ductis in &lt;math&gt;MQ&lt;/math&gt; ex punctis illis '''adaequabit''' rectangulum sub resultante &lt;math&gt;R&lt;/math&gt; et perpendiculo ducto ex centro ipsarum &lt;math&gt;P, P' , &lt;/math&gt;... in idem &lt;math&gt;MQ&lt;/math&gt;; moto enim &lt;math&gt;MQ&lt;/math&gt; versus ea puncta ita , ut maneat sibi parallelum , atque a primitiva positione recedat intervallo &lt;math&gt;h&lt;/math&gt; , si nova perpendicula exhibentur per &lt;math&gt;k, k', k '', ... k_{\mathrm I}&lt;/math&gt; erunt &lt;math display=''inline''&gt;k = z - h , k' = z'- h , k'' = z'' - h, ...  k_{\mathrm I} = z_{\mathrm I} - h &lt;/math&gt;; hinc &lt;math&gt;(k_{\mathrm I} +h) R = (k + h) P + ( k' + h) P' + (k'' + h ) P'' + &lt;/math&gt;... 

est autem ( 12.''a'') &lt;math&gt;hR =h (P + P' + P'' + ...) = hP + hP' + hP'' + ...&lt;/math&gt;;

igitur

&lt;math&gt;k_{\mathrm I} R = kP +k'P' + k'' P'' + ... &lt;/math&gt;

ubi &lt;math&gt;k, k ', k'', ... k_{\mathrm I}&lt;/math&gt; possunt esse vel positiva , vel negativa.

2° Praeter &lt;math&gt;MQ&lt;/math&gt; seu &lt;math&gt;XOY&lt;/math&gt; ( Fig.8 ) concipiantur duo alia plana &lt;math&gt;XOZ , YOZ&lt;/math&gt;; quod autem in ordine ad &lt;math&gt;XOY&lt;/math&gt; est, sit &lt;math&gt;z, z',... z_{\mathrm I} &lt;/math&gt;, sit &lt;math&gt;x, x',... x_{\mathrm I} &lt;/math&gt; in ordine ad &lt;math&gt;YOZ &lt;/math&gt;, et &lt;math&gt;y, y',... y_{\mathrm I} &lt;/math&gt; in ordine ad &lt;math&gt;XOZ&lt;/math&gt;; qua ratione assequuti sumus
&lt;math&gt;z_{\mathrm I}R=zP+z'P'+z''P'' + ...,&lt;/math&gt;
eadem assequemur (a')
&lt;math&gt;x_{\mathrm I}R=xP+x'P'+x''P'' + ...
y_{\mathrm I}R=yP+y'P'+y''P'' + ...&lt;/math&gt;

3° Si compendii causa per  &lt;math&gt;\Sigma P &lt;/math&gt; exprimitur summa
potentiarum &lt;math&gt;P, P', P'', &lt;/math&gt;
et per &lt;math&gt;\Sigma_x P, \Sigma_y  P, \Sigma_z P &lt;/math&gt; designantur summae rectangulorum sub potentiis et respectivis 
perpendiculis , formulae ( a' ) scribi poterunt in hunc modum ( 12. ''a'')
&lt;math&gt;x_{\mathrm I}\Sigma P = \Sigma_x P, y_{\mathrm I}\Sigma P = \Sigma_y P ,z_{\mathrm I}\Sigma P = \Sigma_z P, &lt;/math&gt;
unde
&lt;math&gt;x_{\mathrm I} = \frac{\Sigma_x P}{ \Sigma P} , y_{\mathrm I}= \frac{\Sigma_y P}{ \Sigma P},z_{\mathrm I}= \frac{\Sigma_z P}{ \Sigma P} &lt;/math&gt;
In hypothesi planorum &lt;math&gt;XOY , XOZ , YOZ&lt;/math&gt; orthogonalium ,
&lt;math&gt;x_{\mathrm I}, y_{\mathrm I}&lt;/math&gt; et &lt;math&gt;z_{\mathrm I}&lt;/math&gt;, erunt orthogonales coordinatae , quibus determinatur positio centri parallelarum virium .

4.° Aequatio
P + P + P + ... ... = o ( a &lt;nowiki&gt;''&lt;/nowiki&gt; )&lt;nowiki&gt;''&lt;/nowiki&gt;
manifeste denotat unam quamvis ex datis potentiis v. gr. P
esse aequalem et contrariam resultanti R, ex reliquis omni
bus P' , P &lt;nowiki&gt;''&lt;/nowiki&gt; , ... Ponamus XOY perpendiculare , et XOZ , YOZ&lt;nowiki&gt;''&lt;/nowiki&gt;
parallela directioni potentiarum ; in hac hypothesi erunt
P et R, directe contrariae si perpendicula x et y spectantia
ad punctum , cui applicalur P , spectent ambo ad centrum
quoque virium p ', P &lt;nowiki&gt;''&lt;/nowiki&gt;, ... , si nempe habeantur&lt;nowiki&gt;''&lt;/nowiki&gt;
x R , = x'P' + x &lt;nowiki&gt;''&lt;/nowiki&gt; P t ... ,&lt;nowiki&gt;''&lt;/nowiki&gt;
y R, =ÝP' +y&lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; + .
seu , ob R,
x P + x' P ' + x &lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; +
yP + ' P ' + y &lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; +
-P
= 0,
0; }(cm

5. ° Quibus positis , stabilitur illud : parallelarum
virium systema consistit in aequilibrio sub duabus conditio
nibus simul explendis ; altera est , ut evanescat earum sum
ma : altera ut evanescat summa ex earum momentis in ordi
ne ad duo plana ipsis viribus parallela. Si parallelae vires
inveniuntur omnes in eodem plano, secunda conditio jam de
18
tur summae rectangulorum sub potentiis et reSpectivis perpen-
diculis, formulae (a') scribi poterunt in hunc modum (12. a)
a:, EP :ZxP,y,ZxP: ZJP, z.l 2P:ZzP,
unde u
∙∙∙ zxp ∙∙ \sum∫ M) (0 )
∙−− ⋅ −\sum−⇂⋅−↗∫≖ \sum⇂≀ .z,--—— ZP
ln hypothesi planorum XOT, XOZ , TOZ orthogonalium ,
x, ,y, , et 2! erunt orthogonales coordinatae, quibus deter-
minatur positio centri parallelarum Vtrium.
43 Aequatio
P gr ≖⋡⋅−⊦∙∙⋅−−∙∶∘ (a&lt;nowiki&gt;'''&lt;/nowiki&gt;)&lt;nowiki&gt;'''&lt;/nowiki&gt;
manifeste denotat uuam quamvis-ex datis potentiis v. gr. P
esse aequalem et contrariam resultanti R, ex reliquis omni-
bus P', P&lt;nowiki&gt;''&lt;/nowiki&gt;, Ponamus XOV perpendiculare , et XOZ ,&lt;nowiki&gt;''&lt;/nowiki&gt;
ïOZ
parallela directioni potentiarum; in hac hypothesi erunt
P et B[ directe contrariae si perpendicula x et y spectantia
ad punctum , cui applicatur P ,
spectent ambo ad centrum
quoque virium P',P&lt;nowiki&gt;''&lt;/nowiki&gt;, , si nempe habeantur&lt;nowiki&gt;''&lt;/nowiki&gt;

&lt;nowiki&gt;::&lt;/nowiki&gt; Bl :x'F—I-x&lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; ⊣−∙∙∙∙ Ja, ∶−−∫∣⊉≀−⊢∜∣∣≖≻∥−⊢∙∙∙∙

seu,ob B' :—P,
xP—- x'P'-- x&lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; −∙∙ :o,
yp ——y' PI ...—7&lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; −∙∙ ∙∙∙ :0' )
(a&lt;nowiki&gt;''&lt;/nowiki&gt;)
5.&lt;nowiki&gt;''&lt;/nowiki&gt; Quibus positis
, stabilitur illud : parallelarum
virium systema consistit in aequilibrio sub duabus conditio-
nibus simul explendis; altera est
, ut evanescat earum sum-
ma :altera ut evanescat summa ex earum momentis in ordi-
ne ad duo plana ipsis viribus parallela. Si parallelae vires
inveniuntur omnes in eodem plano, secunda conditio iam de19
9
se explebitur quoad istud planum , satisque erit ut explea
tur quoad aliud tantummodo planum .
6. Etsi vires P, P' , P &lt;nowiki&gt;''&lt;/nowiki&gt;, ... non sunt parallelae , pos
sunt tamen reduci ad terna ejusmodi systemata , quorum pri.
mum coalescat ex viribus parallelis axi OZ , secundum ex
viribus jacentibus in plano XOY simulque parallelis axi OY ,
tertium ex viribus agentibus juxta axem OX. 

Ut demonstretur assertio , resolve unam quamvis ex datis viribus v. gr.

P applicitam puncto A in tres X, Y, Z, respective parallelas
axibus Ox, OY, OZ; ad punctum A applica duas vires H et
- H aequales , contrarias , et parallelas axi OZ ; compone
X ( = AC ) et H sese dirigentem juxta AE , sitque AB dire
ctio resultantis ; produc BA donec in N occurrat plano XOY ;
transfer in N resultantem illam , et sic translatam resolve in
binas , alteram parallelam axi OX , alteram axi OZ ; prodi
bunt componentes X ( = NC = AC ) , H ( = ND ) , qua
rum primam transfer in C ut sit C'C' ( = NC' ) = X; ad C
applica binas vires K et — K aequales , 'contrarias et pa
rallelas axi OY ; compone X ( = .CC ') et K sese dirigen
tem juxia C'F , sitque C'L directio resultantis ; produc LC
donec in V occurrat axi OX ; transfer in V novam istam re
sultantem , et sic translatam resolve in binas , alteram juxta
ox , alteram parallelam axi OY ; emergent componentes
X ( = VV' = CC&lt;nowiki&gt;''&lt;/nowiki&gt; ) , K ( = VF '): compone nunc Y et - H ;
produc directionem resultantis donec rectae C' F occurrat
v . gr. in N ' ; hanc resultantem transfer in N ' , et sic traus
latam resolve in duas , alteram parallelam axi OY , alteram
axi OZ ; exurgent componentes Y et -H applicitae puucto N:
hoc pacto vi P poterunt substitui sex vires
Z, H, — H applicitae punctis A, N, N' et parallelae axi Oz,
K, Y - K applicitae punctis V, C' et parallelae axi OY ,
X applicita puncto V et agens juxta OX .
Consimiles operationes cum possint instaurari quoad P', P ” ...
non pluribus opus est , at pateat veritas assertionis .
19
se explebitur quoad istud planum , satisque erit ut explea-
tur quoad aliud tantummodo planum .
6.o Etsi vires P, P', P&lt;nowiki&gt;''&lt;/nowiki&gt;, non sunt parallelae ,pos-
sunt tamen reduci ad terna eiusmodi systemata
, quorum pri-
mum coalescat ex viribus parallelis axi OZ , secundum ex
viribus jacentibus in plano XOT simulque parallelis axi Oï
,
tertium ex viribus agentibus juxta axem OX. Ut demon-
stretur assertio , resolve unam quamvis ex datis viribus v. gr.
P applicitam puncto A in tres X, T, Z, respective parallelas
axibus OX, OT, OZ; ad punctum A applica duas vires H et
∙∙∙ H aequales , contrarias ,
et parallelas axi OZ ; compone
X (: AC) et H sese dirigentem iuxta AE .sitque AB dire-
ctio resultantis; produc BA donec in N occurrat planc XOT;
transfer in N resultantem illam
,
et sic translatam resolve in
binas , alteram parallelam axi OX , alteram axi OZ; prodi-
bunt componentes X (: NC':AC )
, H (: ND) , qua-
rum primam transfer in C' ut sit C'C&lt;nowiki&gt;''&lt;/nowiki&gt; (: NC' :) X; ad C'
applica binas vires K et —K aequales ,
'contrarias et parallelas axi OV; compone X (:.C' C&lt;nowiki&gt;''&lt;/nowiki&gt;) et K sese dirigen-&lt;nowiki&gt;''&lt;/nowiki&gt;
tem juxt'a C'F , sitque C'L directio resultantis ; produc LC'
donec in V occurrat axi OX ; transfer in V novam istam re-
sultantem ,
et sic translatam resolve in binas , alteram juxta
OX, alteram parallelam axi OV ; emergent componentes
X (:VV':C' C&lt;nowiki&gt;''&lt;/nowiki&gt;) ,K (:VF'): compone nunc V et —H;
produc directionem resultantis donec rectae C' F occurrat
v. gr. in N'; hanc resultantem transfer inN' , et sic traus-
latam resolve' tn duas
, alteram parallelam axi OV, alteram
axi OZ ; exurgeut componentes ?et —H applicitae puncto N':
hoc pacto vi P poterunt substitui sex vires
Z,,H — H applicitae punctis A, N, Net parallelae axi OZ,
K, ï— K applicitae punctis V, C' et parallelae axi OV,
X applicita puncto V et agens juxta OX.
Consimiles operationes cum possint instaurari quoadP' ,P&lt;nowiki&gt;''&lt;/nowiki&gt;,... non pluribus opus est
,
ut pateat veritas assertionis.20
7. Axes OX , OY, OZ sumantur orthogonales ; erit
H : X = ND : NC' NC zX Z :
H ,
et consequenter perpendicula ducta ex N in plana YOZ , XOZ
exprimentur per
2X
H
g ;
erit quoque
H : Y = AC ' : C'N ' = 2 : C'N' =
2Y
H
ac proinde perpendicula ducta ex N' in eadem plana YOZ ,
XOZ exprimentur per
x18+1;
insuper
Vi : Ci = VV' : VF' , seu x - OV : y = X , K ,
ex qua eruitur perpendiculum ductum ex Vin planum YOZ,
nempe
OV =
y X
K
8 . '* Quod in ordine ad Pest X, Y, Z, H, K, sit X ', Y ,
Z ', H , K ' in ordine ad P ', sit X ”, Y &lt;nowiki&gt;''&lt;/nowiki&gt;, Z&lt;nowiki&gt;''&lt;/nowiki&gt;, H ” , K &lt;nowiki&gt;''&lt;/nowiki&gt; in or&lt;nowiki&gt;''&lt;/nowiki&gt;
dine ad P, etc. ... Systema ( 6&lt;nowiki&gt;''&lt;/nowiki&gt;) virium parallelarum axi OZ
consistet in aequilibrio sub tribus istis conditionibns ( 59)
2 + Z ' + Z &lt;nowiki&gt;''&lt;/nowiki&gt; +... + H + HP + H &lt;nowiki&gt;''&lt;/nowiki&gt; + .- H - H²- H &lt;nowiki&gt;''&lt;/nowiki&gt; -... = 0 ,
x2+x+2 + .. + ( x -7 ) +la
ZX
H
- ) H +
'
x H - X'H '-...
20
7
∙∘∙ Axes OX, 07, OZ sumantur orthogonales ;erit
H:X:ND:NC': -Nc': f—X
...-7
et consequenter perpendicula ducta ex N in plana ïOZ, XOZ
exprimentur per
zX x——s.7-i
eritquoque
H. r:.tcx ea:: aut: 2?
—, H
ac proinde perpendicula ducta ex N' in eadem plana TOZ,
XOZ exprimentur per
T
xsf'l'ïïi—i insuper
Vi:C'i:VV':VF',senx—OV:J:X,K. ↴
ex qua eruitur perpendiculum ductum ex Vin planum TOZ, )-
nempe )
∘∇∶∙≖−⋅\sum⋮∙ K
8. 01: Quod in ordine adPestX, T, Z, H, K, sit X'.ï',
Z', H', K' in ordine ad P', sit X&lt;nowiki&gt;''&lt;/nowiki&gt;, T', Z&lt;nowiki&gt;''&lt;/nowiki&gt;, H&lt;nowiki&gt;''&lt;/nowiki&gt;, K&lt;nowiki&gt;'''&lt;/nowiki&gt;m or-
dine ad P&lt;nowiki&gt;''&lt;/nowiki&gt; ,
etc.. «Systema (60) virium parallelamm axi OZ
consistet in aequilibrio sub tribus istis conditionibus (50)
z −⊦ ⊠∣⊣−≀∥⊹∙∙∙−⊦∐⊣−∐∣⊣−∐∥−⊦ ∙∙⋅− ⊟∙↧∓∣∙⊟∥∙∙∙∙ :
xZ-l—x'ZH—&lt;nowiki&gt;''&lt;/nowiki&gt;xl-(x- fl—iï' H—1-(x'
-
)&lt;nowiki&gt;''&lt;/nowiki&gt;IX, H'—)- ..

&lt;nowiki&gt;:&lt;/nowiki&gt; r H—x'H'-.. .
&lt;nowiki&gt;:&lt;/nowiki&gt; o.21

y2 +y2 + ... + 38+y'! '+ ..- ( o + #) : -
(-+ -+* ) r -...--.
seu
2 + 2 + Z&lt;nowiki&gt;''&lt;/nowiki&gt; + ... = 0 ,
x2–2x + x2–5x' + x Z&lt;nowiki&gt;''&lt;/nowiki&gt; _z&lt;nowiki&gt;''&lt;/nowiki&gt; X &lt;nowiki&gt;''&lt;/nowiki&gt; + ... = o,
y2 - zY + y'Z' — zY + y&lt;nowiki&gt;''&lt;/nowiki&gt; Z&lt;nowiki&gt;''&lt;/nowiki&gt; —z&lt;nowiki&gt;''&lt;/nowiki&gt; Y&lt;nowiki&gt;''&lt;/nowiki&gt; + ... :. =0.
360&lt;nowiki&gt;''&lt;/nowiki&gt;

Systema (69) coalescens ex viribus jacentibus in plano XOY
simulque parallelis axi OY consistet in aequilibrio 

sub duabus istis conditionibus ( 5° ) . 

Y - K + Y - K + Y&lt;nowiki&gt;''&lt;/nowiki&gt; _K&lt;nowiki&gt;''&lt;/nowiki&gt; + . + K + K + K + ... = a,
2{Y -K)+7 (9 –K)+- + (3 - X) +(37 )K +
seu
Y + Y + Y &lt;nowiki&gt;''&lt;/nowiki&gt; + ... = 0,
xY4yX + x'Y' — y'X ' + x &lt;nowiki&gt;''&lt;/nowiki&gt; Y&lt;nowiki&gt;''&lt;/nowiki&gt; -- y&lt;nowiki&gt;''&lt;/nowiki&gt; X &lt;nowiki&gt;''&lt;/nowiki&gt; +... =0.
0.}10&lt;nowiki&gt;''&lt;/nowiki&gt;)
Systema ( 6°) virium agentium juxta OX 

consistet in aequilibrio sub ista tantum conditione

X + X+X&lt;nowiki&gt;''&lt;/nowiki&gt;+... = o ( a &lt;nowiki&gt;''&lt;/nowiki&gt; &lt;nowiki&gt;''&lt;/nowiki&gt; ).
Inferimus solidum liberumque corpus viribus P , P' ,
P&lt;nowiki&gt;''&lt;/nowiki&gt; , ... sollicitatum haud mansurum in aequilibrio, nisi ex
pletis conditionibus ( a' ) , ( a &lt;nowiki&gt;''&lt;/nowiki&gt; ) , ( a &lt;nowiki&gt;''&lt;/nowiki&gt; &lt;nowiki&gt;''&lt;/nowiki&gt; ); quas ita scri
bimus ( 30 )
21
⊺∄⊹↗⋅∄∣−⊢∙∙∙⊹∫∐⊹∫∣∐∣⊹∙⋅∙− (
∫⊹.äï.) H —
(y'-]- ⋮⋅≨⋚∣⇀∙≻ H'—. ..

&lt;nowiki&gt;:&lt;/nowiki&gt; o,

seu
∅⊣−∅∣⊣−⊈∥⊹∙∙∙∶∘∙ ;
(a') xZ—zX-I-x'Z'— z'X'-l-x&lt;nowiki&gt;''&lt;/nowiki&gt;Z&lt;nowiki&gt;''&lt;/nowiki&gt;— z'X&lt;nowiki&gt;''&lt;/nowiki&gt;—-]-. ..

&lt;nowiki&gt;:&lt;/nowiki&gt; o,

yz -— zV-l—J'Z'—z'ï'—I- y&lt;nowiki&gt;''&lt;/nowiki&gt;Z&lt;nowiki&gt;''&lt;/nowiki&gt;—z&lt;nowiki&gt;''&lt;/nowiki&gt;ï&lt;nowiki&gt;''&lt;/nowiki&gt;-l— ..

&lt;nowiki&gt;:&lt;/nowiki&gt;

.
0.
Systema (60) coalescens ex viribus jacentibus in plano XOV
simulque parallelis axi OV consistet in aequilibrio sub dua-
bus istis conditionibus ( 5o )-
r—x-t-x'—x'—l-1z&lt;nowiki&gt;''&lt;/nowiki&gt;—xq-.. —[-K-]-K'-)-K&lt;nowiki&gt;''&lt;/nowiki&gt;—]—. ..

&lt;nowiki&gt;:&lt;/nowiki&gt;

0,
I
'
X
&lt;nowiki&gt;!
IX ∣
..
ï—KH—x (r—x ⊢⊢⋅∙∙−⊢ xli?) x-l-(x ïk.-')K ∙⊦∙∙≔∶⋅∘⋅
seu .
y—I—T-I- ï''&lt;/nowiki&gt;—l— ..

&lt;nowiki&gt;:&lt;/nowiki&gt;

.
0, '
h
0&lt;nowiki&gt;''&lt;/nowiki&gt;)
xï—yX-l—x'ïL-y'X' x&lt;nowiki&gt;''&lt;/nowiki&gt;ï&lt;nowiki&gt;''&lt;/nowiki&gt;-y&lt;nowiki&gt;''&lt;/nowiki&gt; &lt;nowiki&gt;''&lt;/nowiki&gt;
∙⊦∙∙∙∶−−∙ ∙
Systema (60) virium agentium iuxta OX consistet in ae-
quilibrio sub ista tantum conditione '
,x-t—X'-l-X&lt;nowiki&gt;''&lt;/nowiki&gt;-1-...:o (a&lt;nowiki&gt;'''&lt;/nowiki&gt;).
Inferimus solidum liberumque corpus viribus P, P',
P&lt;nowiki&gt;''&lt;/nowiki&gt;, .. . sollicitatum haud mansurum in aequilibrio, nisi ex-
pletis conditionibus (a' )
,
( a&lt;nowiki&gt;''&lt;/nowiki&gt; )
,
(av&lt;nowiki&gt;''&lt;/nowiki&gt; ); quas ita scri-
bimus ( 3&lt;nowiki&gt;''&lt;/nowiki&gt;)22
EX
=
0
, EY
=
0
, E2
=
0
,
}
(
a
&lt;nowiki&gt;''&lt;/nowiki&gt;
)
2
( zYX)
= 0,2
( x2–2X
)
= 0,2
(x2
– zY
)
=
0..
9
'

&lt;nowiki&gt;#&lt;/nowiki&gt; Denotet

R
' resultantem ex viribus primi syste
matis
(
6
°
)
,
R
&lt;nowiki&gt;''&lt;/nowiki&gt; ex viribus secundi
,
R
&lt;nowiki&gt;''&lt;/nowiki&gt; ex viribus tertii&lt;nowiki&gt;''&lt;/nowiki&gt;

&lt;nowiki&gt;;&lt;/nowiki&gt;

erunt
( 12

&lt;nowiki&gt;:&lt;/nowiki&gt;

a
)
R
= EZ
,
R
= EY
,
R
&lt;nowiki&gt;''&lt;/nowiki&gt;
= EX
.
Recta
, in qua agit
R
&lt;nowiki&gt;''&lt;/nowiki&gt;
, occurrit axi OX sub angulo recto

&lt;nowiki&gt;;&lt;/nowiki&gt;

et disignante
r
&lt;nowiki&gt;''&lt;/nowiki&gt; distantiam inter
O et punctum occursus
erit
( 2º
.
7
°
)
.
r
“
R
”
=
x
(Y
—
K
)
+
x'(
Y
’
–
K
”
+ ...
XK
(
s
–
&lt;nowiki&gt;''&lt;/nowiki&gt;
)
k
'
+ ...,ideoque
?&lt;nowiki&gt;''&lt;/nowiki&gt;=
EfxY
-yX
)
.
R
&lt;nowiki&gt;''&lt;/nowiki&gt;
tra
potest
R
'
&lt;nowiki&gt;''&lt;/nowiki&gt; transferri in illud punctum occursus sicque
componi cum
R
&lt;nowiki&gt;''&lt;/nowiki&gt; ut inde obtineatur resultans VR
&lt;nowiki&gt;''&lt;/nowiki&gt;
2
+
R
&lt;nowiki&gt;''&lt;/nowiki&gt;
3.
Iterum
( 9. 9º
. 10
°
.
) patet ergo vires
P
P
'
,
P
'
,
, ...
duci vel ad ternas
, vel ad binas aequipollentes
.
10.
°

&lt;nowiki&gt;#&lt;/nowiki&gt; Recta

, in qua agit
R
'
, occurrit normaliter plano
XOY

&lt;nowiki&gt;;&lt;/nowiki&gt; et designantibus

a
'
,
b
' coordinatas istius occursus
,
erunt
( 2º
. 7º
.
)
a
'
$
(xZ
- zX
)
R
'
6
Egy
Z
-
Y
) 1 R
Occurrent sibi mutuo R’et VR
”
?
+
R
&lt;nowiki&gt;''&lt;/nowiki&gt;
2, ac proinde
jacebunt in eodem plano
, quotiescumque
a
' et
b
' recident
in duas quasvis ex coordinatis illius rectae in qua agit
VR'
2
+
R
'&lt;nowiki&gt;'''&lt;/nowiki&gt;
2

&lt;nowiki&gt;;&lt;/nowiki&gt; propterea

22
ZX:0,Zï:o,ZZ:o,

&lt;nowiki&gt;;&lt;/nowiki&gt; (aVIII)

\sum (xï—ïyX):o.Z(xZ—zX):0,2(yZ—zï): 0.
9. 01: Denotet B' resultantem ex viribus primi syste-
matis '(60 ), B&lt;nowiki&gt;''&lt;/nowiki&gt; ex viribus secundi , B&lt;nowiki&gt;'''&lt;/nowiki&gt; ex viribus tertii;
erunt
( 12. a)
R,:Z Z, B&lt;nowiki&gt;''&lt;/nowiki&gt;:Zï, R&lt;nowiki&gt;'''&lt;/nowiki&gt;:ZX.
Recta, in qua agit R&lt;nowiki&gt;''&lt;/nowiki&gt;, occurrit axi OX sub angulo recto

&lt;nowiki&gt;;&lt;/nowiki&gt;

et disignante r&lt;nowiki&gt;''&lt;/nowiki&gt; distantiam inter
0 et punctum occursus,
ertt∙
(
2
∘
∘
. 7. ).
r&lt;nowiki&gt;''&lt;/nowiki&gt;R&lt;nowiki&gt;''&lt;/nowiki&gt;:x(ï—K)—)—x'(ï'—K')—-)—...-)- (x... JKX.)
K −⊦
(x'-— 2274.) K' −∙⊢∙
∙ .,ideoque r&lt;nowiki&gt;''&lt;/nowiki&gt;-— xwy-FK)

&lt;nowiki&gt;:&lt;/nowiki&gt;

potest B&lt;nowiki&gt;'''&lt;/nowiki&gt; transferri in illud punctum occursus
, sicque
componi cum B&lt;nowiki&gt;''&lt;/nowiki&gt; ut inde obtineatur resultans l/B&lt;nowiki&gt;''&lt;/nowiki&gt;3-l—B'&lt;nowiki&gt;'''&lt;/nowiki&gt;. Iterum (9. 90.100.) patet ergo vires P P', P', , .. .
tra-
duci vel ad ternas, vel ad binas aequipollentes.
↿∘∙∘⋕ Recta, in qua agit B', occurrit normaliter plano
XOT; et designantibus a', b' coordinatas istius occursus,
erunt (20. 70.)
↙⋮∣∙− X(xZ—zX) b' ∙∙∙
\sum (yZ—zï)
B'
' R'
⋅
Occurrent sibi mutuo B' et l/B&lt;nowiki&gt;''&lt;/nowiki&gt;2-)-B&lt;nowiki&gt;'''&lt;/nowiki&gt;2, ac proinde
iacebunt in eodem plano, quotiescumque a' et b' recident
in duas quasvis ex coordinatis illius rectae in qua agit
⇂∕ B&lt;nowiki&gt;''&lt;/nowiki&gt;2-I-B&lt;nowiki&gt;'''&lt;/nowiki&gt;2; propterea23
a ' - p &lt;nowiki&gt;''&lt;/nowiki&gt; : 6 = R : R &lt;nowiki&gt;''&lt;/nowiki&gt;
et consequenter
b' R' + ( r &lt;nowiki&gt;''&lt;/nowiki&gt; – a ' ) R &lt;nowiki&gt;''&lt;/nowiki&gt; = 0 ;
quae , adhibitis substitutionibus, traducitur ad
EXE(yZ — ZY) + EYXzX— « Z ) + EZE (xY yX ) = 0.
Sub hac ilaque conditione occurrent sibi mutuo vires
R' , V R &lt;nowiki&gt;''&lt;/nowiki&gt;2+ R &lt;nowiki&gt;''&lt;/nowiki&gt; ), dabuntque resultantem
VR2+ R &lt;nowiki&gt;''&lt;/nowiki&gt;2 + R &lt;nowiki&gt;'''&lt;/nowiki&gt; a = V (EX)2 + (PY )2+ ( EZ )2.
11 . '* Si nequeunt vires alium gignere motum ni
si circa immobilem axem Oz , quisque videt aequilibrii
conditiones redactum iri ad unicam r ' = 0 , seu ad quar
tam ( a &lt;nowiki&gt;''&lt;/nowiki&gt; ), Ad haec si nequeunt vires alium gignere mo
tum nisi circa immobile punctum 0 , redigentur aequili
brii conditiones ad r&lt;nowiki&gt;''&lt;/nowiki&gt; = 0 , a' = 0,6 = 0 , seu ad quar
tam , quintam et sextam ( a )
12. '* Fac ut duo solida corpora A et B ( Fig. 9) ,
alterum viribus P , P , P &lt;nowiki&gt;''&lt;/nowiki&gt;... sollicitatum , alterum viri
bus Q , , Q &lt;nowiki&gt;''&lt;/nowiki&gt; , ... , sese invicem aeque premendo apud da
lum mutui contactus punctum C maneant in aequilibrio ;
quaeritur istiusmodi pressionis magnitudo w. Duc per C pla
num tangens DD' , cui normaliter insistat recta ECE': de
notent fig, h coordinatas puncti C ; a , á , a &lt;nowiki&gt;''&lt;/nowiki&gt; angulos
interceptos recta CE axibusque orthogonalibus OX , OY ,
OZ ; et quod in ordine ad P' , P' , P &lt;nowiki&gt;''&lt;/nowiki&gt; , ... est X, Y , Z,
á , : , X , Y , Z , X ', . . . sit a , b , c , a , ... A , B ,
C , A ', ... in ordine ad C , Q , ... Pressio agens
versus E resolvetur in ternas
23
a'—- r&lt;nowiki&gt;''&lt;/nowiki&gt;: 6':a&lt;nowiki&gt;'''&lt;/nowiki&gt;: a&lt;nowiki&gt;''&lt;/nowiki&gt;
et consequenter
↘∙∙
b' B&lt;nowiki&gt;'''&lt;/nowiki&gt;—i— ( r&lt;nowiki&gt;''&lt;/nowiki&gt;-—a') B&lt;nowiki&gt;''&lt;/nowiki&gt;: 0

&lt;nowiki&gt;;&lt;/nowiki&gt;

quae , adhibitis substitutionibus, traducitur ad
ZXZUZ—zTH-ZïXzX—xZH-ZZZ (a.-T —JX):o.
Sub hac itaque conditione occurrent sibi mutuo vires
B', l/ B&lt;nowiki&gt;''&lt;/nowiki&gt;2-)- B&lt;nowiki&gt;'''&lt;/nowiki&gt;, dabuntque resultantem
⇂∕↓↖⋅≖−⊦↓⊰⋅⋅≖−⊢∐⋯≖∶ ⇂∕ (mun-)- (zx) ≕⊣−≺ \sum∣∠≻≖∙
11.&lt;nowiki&gt;''&lt;/nowiki&gt;; Si nequeunt vires alium gignere motnm ni-
si circa immobilem axem OZ, quisque videt aequilibrii
conditiones redactum iri ad unicam r&lt;nowiki&gt;''&lt;/nowiki&gt; :o
, seu ad quar-
tam (a'&lt;nowiki&gt;'''&lt;/nowiki&gt; ). Ad haec si nequeunt vires alium gignere mo-
tum nisi circa immobile punctum 0 , redigentur aequili-
brii conditiones ad r&lt;nowiki&gt;''&lt;/nowiki&gt;:0, a':o, b':o, seu ad quar-
tam, quintam et sextam ( am&lt;nowiki&gt;''&lt;/nowiki&gt;)
12.&lt;nowiki&gt;''&lt;/nowiki&gt;: Fac ut duo solida corpora A et B (Fig. 9),
alterum viribus P , P', P&lt;nowiki&gt;''&lt;/nowiki&gt;. .. sollicitatum , alterum viri-
bus Q, Q' , Q&lt;nowiki&gt;''&lt;/nowiki&gt;, .. ., sese invicem aeque premendo apud da-
tum mutui contactus punctum C maneant in aequilibrio;
quaeritur istiusmodi pressionis magnitudo 'a'. Duc per C pla-
num tangens DD', cui normaliter insistet recta ECE': de-
notent f, g ,
]: coordinatas puncti C; at, a', a&lt;nowiki&gt;''&lt;/nowiki&gt; angulos
interceptus recta CE axibusque orthogonalibns OX, Of
, OZ; et quod in ordine ad P'
, P', P&lt;nowiki&gt;''&lt;/nowiki&gt;,
...
est a:, 7, z,
x', . . X,ï, Z, X',. . . sita,b, c, a,... A,. B,
C, A', . . .
in ordine ad Q', Q, . . . Pressio :: agens
versus E resolvetur in ternas24
cosa , cose , a cos &lt;nowiki&gt;''&lt;/nowiki&gt; ,
agens vero versus E resolvetur in ternas
w cos ( 180 ° - « ) = - COS Q, a cos ( 180 ° - = - a coseć,
cos ( 180º – Ø&lt;nowiki&gt;''&lt;/nowiki&gt; ) W cos a :
in primo casu w librat ex hypothesi vires P, P,
in secundo vires Q, C, ... Igitur
EX +w cosa = 0, Erto cosá = 0 , xZ + w cosa &lt;nowiki&gt;''&lt;/nowiki&gt; = 0 ,
Σ Α W cosa = 0 ,
EB - cosa = 0,8C — a cos &lt;nowiki&gt;''&lt;/nowiki&gt; = 0 , E ( «Y -y X ) +
W ( f cos ' - g cosc) =0 ,
ElxZ - 2X ) + o ( f cosc &lt;nowiki&gt;''&lt;/nowiki&gt; — h cosc ) =0 , Ely2 -zY) +
wig cosa &lt;nowiki&gt;''&lt;/nowiki&gt; -hcosé ) =0 ,
(aB - 6A ) - ( fcos - g cos ) = 0 ,E (aC - A )
a ( f cosa &lt;nowiki&gt;''&lt;/nowiki&gt; -hcosa) = 0 , E (6C - cB ) - ( g cosa &lt;nowiki&gt;''&lt;/nowiki&gt; - h cosa') = 0 .
Eliminata , prodibunt undecim aequationes , inde
pendentes ab ipsa a , inter quantitates datas ; quibus ae
quationibus expletis, habebitur aequilibrium , poteritque ab
una quavis ex duodecim praecedentibus erui valor u .
13.0# Solidum corpus sollicitatum viribus , P P ',
P &lt;nowiki&gt;''&lt;/nowiki&gt; , ... delineatur duobus punctis fixis , sumptis in axe
v. gr. OZ ; sic facile determinabuntur pressiones M, N , L
et M ', N ', L' exercitae in puncta illa juxta coordinatos a.
xes Ox , OY, OZ. Exprimant m, n , l coordinatas unius
ex duobus panciis , et m ', ní, ľ coordinatas alterius. Quo
uiam spectari debent
24
a: cosa, a cosa', wcosac' ,
agens vero versus E' resolvetur in ternas
m cos(1800—a): — arcus a, acos (1800—at'):—w cosa',
a cos ( 180o -— ac&lt;nowiki&gt;''&lt;/nowiki&gt;) ∶≖ −meos ac&lt;nowiki&gt;''&lt;/nowiki&gt;:
in primo casu ut librat ex bypOthesi vires P, P', .
∙ ∙
,
in secundo vires Q, Q', . . .
Igitur
2X —l—w cosa::o, Zy-l—a cosa':o,ZZ—l-ar cosa:&lt;nowiki&gt;''&lt;/nowiki&gt;:o,
EA — z: cosa: :0,
2B —a cosa':o,ZC—z.ïcos at&lt;nowiki&gt;''&lt;/nowiki&gt;:o, Z (xï —7 X) −−∣−
15 (fcosa&lt;nowiki&gt;''&lt;/nowiki&gt;—-g cos ac) :0,
2( a:Z—zX)-I—w(fcosa&lt;nowiki&gt;''&lt;/nowiki&gt;—hcosa):o, XOZ—z?) −⊦
w(g cosa&lt;nowiki&gt;''&lt;/nowiki&gt;--hcosat'):o,
E( aB—bA) —w(fcos a'—gcosa):o,2 (aC—cA)—-
a(fcosa&lt;nowiki&gt;''&lt;/nowiki&gt;-h cos a):o,2 (bC-cB) -zz(gcos a&lt;nowiki&gt;''&lt;/nowiki&gt;- hcosac'):o.
Eliminata a, prodibunt undecim aequationes, inde-
pendentes ab ipsa a'
,
inter quantitates datas.; quibus ae-
quationibus expletis, habebitur aequilibrium, poteritque ab
una quavis ex duodecim praecedentibus erui valor a.
1394: Solidum corpus sollicitatum viribus, P P',
P&lt;nowiki&gt;''&lt;/nowiki&gt;, . . . detineatur duobus punctis fixis, sumptis, in axe
v. gr. OZ; sic facile determinabantur pressiones M, N, L
et M', N', L' exercitae in puncta illa juxta coordinatos a-
xes OX, DV, 02. Exprimant m, n, !coordinatas unius
ex duobus punctis, et m', n'. [ coordinatas alterius. Quo-
niam spectari debent25
M, N , -L, — M ', - N - L'
tanquam vires , quibus librantur caeterae P , P , P' ... ,
ac insuper m = 0 , n = o , m' =0 , n = 0 , necnon ( 110. )
(xY - yX ) = 0 : iccirco ( 8º. a &lt;nowiki&gt;''&lt;/nowiki&gt; )
EX - M - M ' = 0 , EY -N - N = 0,8Z -L - L ' = 0 ,
{ ( xZ - 2X ) + 2M + l'M' = 0 , E ( yZ – zY) +IN +
Ľ N' = 0 ;
quarum tertia nos edocet axem OZ premi vi XZ in dire
ctione z , reliquae vero suppeditant M , M' , N , N' . Si P ,
P ' , P &lt;nowiki&gt;''&lt;/nowiki&gt; , ... evadunt parallelae axi OZ, et se dirigunt ad
plagam negativam , erunt ( 12: 13, 2° ) ,
EX = 0 , EY = 0 , EZ = P - P - P ' -... = - R,
ExZ- X ) = - xP - X'P' - &lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; --... X, R,
(y2 — zY) = - ype ' P' y &lt;nowiki&gt;''&lt;/nowiki&gt; P &lt;nowiki&gt;''&lt;/nowiki&gt; —... = - y . R;
hic denotant P, P ', P &lt;nowiki&gt;''&lt;/nowiki&gt; , ... virium duntaxat intensitates. Quare
M + M ' = 0 , N + N = 0 , L + L + R = 0,2M +
I'M – x, R = 0 , 2N + IN - Y , R = 0 ;
unde
M = -M'
1, R
1 - T '
N = -N y R ,
, L
L +
+ LEL' = - R.
3
25
—M,-FNg-Lg—M'g—N' '..L,
tanquam vires
, quibus librantur caeterae P, P', P&lt;nowiki&gt;''&lt;/nowiki&gt;. ..,
ac insuper m::o, &lt;nowiki&gt;''&lt;/nowiki&gt;:D, in'-:(), &lt;nowiki&gt;'''&lt;/nowiki&gt; ∶−−⋅ o, necngn ( 110.)
Xxï—yX :) o:
iccirco( 80. a'&lt;nowiki&gt;'''&lt;/nowiki&gt; )
EX—M—M':o,2ï-—N-—N' :o,ZZ—L-—L' :0,
Si xZ—zX)—I-lM-I— l'M':o,Z(yZ—-zï)—l-IN-l-
&lt;nowiki&gt;!' N':o;
quarum tertia nos edocet axem OZ premi vi ZZ in dire-
ctione&lt;/nowiki&gt; :, reliquae vero suppeditant M, M' , N ,N'. Si P,
P'
,P&lt;nowiki&gt;''&lt;/nowiki&gt; ,... evadunt parallelae axi OZ, et se dirigunt ad
plagam negativam, erunt (12: 13. 20 ),
ZX:o, Zïzo,zz :P—P'-—P&lt;nowiki&gt;''&lt;/nowiki&gt;-—. ..

&lt;nowiki&gt;:&lt;/nowiki&gt; — R,

XxZ—QX):—xP -x'P'
-— x&lt;nowiki&gt;''&lt;/nowiki&gt;P&lt;nowiki&gt;''&lt;/nowiki&gt; —-. .

&lt;nowiki&gt;:&lt;/nowiki&gt;

. —a:. B,.
2(yz ∙∙∙ zï):—yP—— r' P' —.7&lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; ∙∙∙ ∙ ∙ ∙ ∶−∙ ∙−−∫∎ R;
bic denotant P, P', P&lt;nowiki&gt;''&lt;/nowiki&gt;, ... virium duntaxat intensitates. Quare
∐−⊦∐∣∶∘∙∾⊣−∐∙−−∶∘∙ ↧⋅−↽↧⋅∙−↽≖↸≓∘∙≀∐⊣⊸
I'M' — x,R:o, lN-i-l'N'—-y,R:o;
nnde
M: x,B -—M':—
---—- r—z '
Nz—N' :-—l',y'—-—R—2,L-I—L':—R- 326
14. Ex dictis de virium aequilibrio deduci possunt
aequilibrii leges in Machinis. Sic v. gr. in vecte poten
tia librabit resistentiam seu pondus, quotiescumque ( sumptis
( 10. 10 ) momentis quoad axem immobilem, circa quem po
test vectis moveri ) momentum potentiae aequatur momento
resistentiae.Idipsum obtinet quoad Axem in peritrochio ; idi
psum quoad trochleam fixam . Potentia et resisteutia istis
machinis applicantur in directione parallela planis perpen
dicularibus axi immobili; perinde igitur ( 10. 10 ° ) erit si
ve in eorum uno sive in altero accipiantur momenta ;
poteritque vectis repraesentari per lineam mobilem circa
punctum fixum , quod dicitur fulcrum , hypomoclion : axis
in peritrochio per circulares projectiones rotae ac cylin
dri in uno quovis ex dictis planis , mobiles circa com
mune immobile centrum : trochlea fixa per circulum ro
tatilem circa suum centrum , cujus circuli radius sit ipse
trochleae radius; verum quia in trochlea fixa nihil sunt
aliud perpendicula momentorum propria nisi trochleae
radii, ducti ad puncta ubi funis desinit ipsam trochleam
tangere, ideo erunt aequalia , et consequenter aequilibrium
in trochlea fixa importat potentiae ac resistentiae ae
qualitatem.
Ad trochleam mobilem quod spectat , sit P (Fig. 10)
pondus sustinendum a potentia Q : quoniam in casu aequi
librii , P et Q praebeant oportet resultantem R transeuntem
per punctum fixum 0, iccirco ( 9. 10. ) Q : P = sin \beta : sin a
= sin i : sin 2i = cos x : sin 2x = cos x : 2sin x cos
1 : 2 sin ; ac proinde
P
Q
2 sin s
Posuimus angulum OaQ dividi aequaliter directione
ponderis P : id vero facile intelligemus animadvertendo , si
filum OaQ fixum in 0 et Q , tenditur vi applicita puncto
26

14. Ex dictis de virium aequilibrio deduci possunt
aequilibrii leges in Machinis. Sic v. gr. in vecte poten-
tia librabit resistentiam seu pondus, quotiescumque ( sumptis
(10. 100) momentis quoad axem immobilem, circa quem po-
test vectis moveri ) momentum potentiae aequatur momento
resistentix-Idipsum obtinet quoad Axem in peritrochio ; idi-
psum quoad trocbleam lixam. Potentia et resistentia istis
machinis applicantur in directione parallela planis perpen-
dicularibus axi immobili; perinde igitur( 10. 100) erit si-
ve in eorum- uno sive in altero accipiantur momenta;
poteritque vectis repraesentari per lineam mobilem circa
punctum fixum, quod dicitur fulcrum, hypomoclion: axis
in peritrochio per circulares proiectiones rotae ac cylin-
dri in uno quovis ex dictis planis, mobiles circa com-
mune immobile centrum: trochlea lixa per circulum ro-
tatilem circa suum centrum,cuius circuli radius sit ipse
trochleae radius; verum quia in trochlea fixa nihil sunt
aliud perpendicula momentorum propria nisi trochleae
radii, ducti ad puncta ubi funis desinit ipsam trocbleam
tangere, ideo erunt aequalia ,
et consequenter aequilibrium
in trochlea fixa importat potentiae ac resistentiae ae-
qualitatem.
Ad trocbleam mobilem quod spectat , sit P (Fig. 10)
pondus sustinendum a potentia Q: quoniam in casu aequi-
librii , P et Q praebeant oportet resultantem R transeuntem
per punctum fixum 0, iccirco (9. 10.) Q: P:sin 13: sin «

&lt;nowiki&gt;:&lt;/nowiki&gt; sin i :sin 2i:cos x : sin Zx:cos x: Zsinxcosx
&lt;nowiki&gt;:&lt;/nowiki&gt; 1: 2 sin a:; ac proinde

P Q&lt;nowiki&gt;''&lt;/nowiki&gt;üü'
Posuimus angulum OaQ dividi aequaliter directione
ponderis P: id vero facile intelligemus animadvertendo, si
iilum OaQ fixum in 0et Q ,
tenditur vi applicita puncto
∙∙∙ '. 'una-
,.. ↙∙∙∎⋅−27
a libere excurrenti juxta ipsum Oal , punctum a necessa
rio permansurum in perimetro ellipseos , cujas foci O et Q;
ideoque in casu aequilibrii vim illam fore perimetro elli
pseos normalem ; quod certe importat praedictam anguli
OaQ divisionem . Vide n. 55. 13 .
Etiam sic : cum in casu aequilibrii funis ubique ma
neat aeque distentus , pondus P librabit resultantem R'
ex duabus viribus aequalibus Q et concurrentibus apud
punctum a ; et quoniam R' aequaliter dividit angulum Oal ,
idipsum dicendum erit de ponderis directione. Jamvero
R ' ( = P2) = Q + + Q2 + 2QQ cos 2i =2Q ( 1 +cos 2i) =
4 Q* cos 2i = 4 Q* sinºx :
rursus igitur
P
-
2sin x
angulo x = 90° respondebit minimal ; erit Q = P
2
si x = 30° ; vergente x ab 30° ad 09 , verget Q ab P ad co .
15. Vectis primi generis nuncupatur , si fulcrum sit
inter potentiam et pondus ; dicitur secundi generis si pon
dus sit inter fulcrum et potentiam ; denique si potentin me.
dium locum teneat inter fulcrum et pondus , vectis tertii ge
neris vocatur. Hinc vectes primi et secundi generis poten
tiam juvant , quatenus eo minor requiritur potentia ad da
tum pondus sustinendum , quo major est potentiae distantia
a fulcro relate ad ponderis distantiam ab eodem fulcro ;
adeo ut quodvis pondus utcumque ingens possit vectis ope
a quacumque potentia utcumque exigua sustineri : quod cum
bene nosset Archimedes , illud dixisse fertur Hieroni Regi ..
dic ubi consistam , coelum , terramque movebo ,, : vectis au
tem tertii generis potentiam non juvat , quia in hoc vecte
potentia minus distat a fulcro quam resistentia seu pondus.
27

&lt;nowiki&gt;::&lt;/nowiki&gt; libere excurrenti juxta ipsum OaQ , punctum :: necessa-

rio permansurum in perimetro ellipseos, cuius foci O et Q;
ideoque in' casu aequilibrii vim illam fore perimetro elli-
pseos normalem; quod certe importat praedictam anguli
OaQ divisionem . Vide n. 55. 13.0
Etiam sic :cum in casu aequilibrii funis ubique ma-
neat aeque distentus , pondus P librabit resultantem R'
ex duabus viribus aequalibus Q et Q concurrentibus apud
punctum a; et quoniam R' aequaliter dividit angulum OaQ,
idipsum dicendum erit de ponderis directione. Iamvero
a'
∙≺⇌−− re:? -1-Q*-l—2QQcos2i:2Q'(1-l-cva 20:
4Q' cos 3i:4Q3 sin'x:
rursus igitur
P
Q— 2sinx,
angulo x:900 respondebit minima Q

&lt;nowiki&gt;:&lt;/nowiki&gt; ä; erit Q:P

si a: 300 ,- vergente :: ab 300 ad 00
,
verget Qab P ad 00
.

15. Vectis primi generis nuncupatur, si fulcrum sit
inter potentiam et pondus; dicitur secundi generis si pon-
dus sit inter fulcrum et potentiam ;denique si potentia me-
dium locum teneat inter fulcrum et pondus , vectis tertii ge-
neris vocatur. Hinc vectes primi et secundi generis poten-
tiam iuvant, quatenus eo, minor requiritur potentia ad d'a-
tum pondus sustinendum , quo maior est potentiae distantia
a fulcro relate ad ponderis distantiam ab eodem fulcro;
adeo ut quodvis pondus utcumque ingens possit vectis ope
a quacumque potentia utcumque exigua sustineri :quod cum
bene nosset Archimedes , illnd dixisse fertur Hieroni Regi ,,
dic ubi consistam ,coelum ,terramque movebo ,, :vectis an-
tem tertii generis potentiam non juvat , quia in hoc vecte
potentia minus distat a fulcro quam resistentia seu pondus.28
Ex indicata vectis theoria redditur ratio innumerabi
liam effectuum quos quotidie cernimus fieri ; ac primo qui
dem quicumque ex lapidicinis extrahunt lapides utuntur ut
plurimum vecte ferreo : quoties autein multum resistit la
pis sive propter magnitudinem sive quod nimis firmiter aliis
adhaereat , tunc hypomoclion quam proxime ponderi admo
vent , ut facilius moveant , quod vulgo dicitur ,, dar la
leva ,, . Pro hypomoclio antem utuntur quovis sustentaculo
v . gr. lapide ; saepe etiam cum duo lapides ab invicem
sejungendi sunt , unus respectu alterius habet rationem hy
pomoclii . Hic notandum est maxillas quoque esse vectes
secundi generis quum cibi dentibus molaribus comminuen
di traduntur . Secundo : si avellendus est clavus ope mal
lei , quanto clavus , qui ponderis vicem obtinet , propior
fuerit hypomoclio , eo facilius educetur ; unde cum jam tan
tisper eductus est , ita ut extremitas mallei nequeat am
plius insistere subjectae tabulae aut parieti e quo est dedu
cendus , solemus aliud corpus interserere ut quam minima
sit distantia. Tertio : in forcipibus quoque duplex est vectis
primi generis , quorum unum est commune hypomoclion ,
clavus nempe circa quem uterque ramus volvitur , eoque va
lidius stringetur corpus quo rami , qua parte secant , brevio
res , qua parte vero applicatur potentia seu manus , longiores
erunt . Quarto : cum portas aperimus aut claudimus , eo
facilius id praestamas , quo longius a cardinibus eas impel
Iimus , nempe janua est vectis secundi generis , cujas hy
pomoclion sunt cardines. Quinto : remorum motu cymba
promovetur , quia remi sunt vectes secundi generis , quo
rum bypomoclion est aqua , cymba est pondus seu resi
stentia , manus hominis sunt potentia applicata : hinc quo
magis ab aqua remotae sunt manus quam punctum cym
bae , cui remi insistunt , eo majus est potentiae momen
ium. Sexto : ex his etiam intelligitur cur difficillima sit
bacali oblongi elevatio si per extremitatem accipiatur , el
cur quo longior fuerit ipse baculus , eo facilius curvetur
aut frangatur.
28
Ex indicata vectis theoria redditur ratio innumerabi-
lium efi'ectuum quos quotidie cernimus iieri ; ac primo qui-
dem quicumque ex lapidicinis extrahunt lapides utuntur ut
plurimum vecte ferreo :quoties autem multum resistit la-
pis sive prOpter magnitudinem sive quod nimis firmiter aliis
adhaereat ,
tuuc hypomoclion quam proxime ponderi admo-
vent
,
ut facilius moveant, quod vulgo dicitur
,, der in
leva ,, . Pro hypomocliol autem utuutur quovis sustentaculo
v. gr. lapide; saepe etiam cum duo lapides ab invicem
sejungendi sunt
, unus respectu alterius habet rationem hy-
pomoclii. Hic notandum est maxillas quoque esse vectes
secundi generis quum cibi dentibus molaribus comminuen-
di traduntur. Secundo: si avellendus est clavus ope mal-
lei, quanto clavus, qui ponderis vicem obtinet, propior
fuerit hypomoclio , eo facilius educetur ;unde cum iam tan-
tisper eductus est, ita ut extremitas mallei nequeat am-
plius insistere subjectae tabulae aut parieti e quo est dedu-
cendus , solemus aliud corpus interserere ut quam minima
sit distantia. Tertio :in forcipibus quoque duplex est vectis
primi generis, quorum unum est commune hypomoclion,
clavus nempe circa quem uterque ramus volvitur, eoque va-
lidius stringetur corpus quo rami
, qua parte secant
, brevio-
res, qua parte vero applicatur potentia seu manus ,
longiores
erunt. Quarto: cum portas aperimus aut claudimus , eo
facilius id praestamus
, quo longius a cardinibus eas impel-
limus , nempe janua est vectis secundi generis , cujus hy-
pomoclion sunt cardines. Quinto : remorum motu cymba
promovetur
, quia remi sunt vectes secundi generis , quo-
rum bypomoclion est aqua, cymba est pondus seu resi-
stentia , manus hominis sunt potentia applicata: hinc quo
magis ab aqua remotae sunt manus quam punctum cym-
hae, cui remi insistunt , eo majus est potentiae momen-
tum. Sexto : ex his etiam intelligitur cur difficillima sit
baculi oblongi elevatio si per extremitatem accipiatur ,
et
cur quo longior fuerit ipse baculus, eo facilius curvetur
aut frangatur.29
16. Supponantur nunc plures quotcumque vectes ita
dispositi , ut potentia Q in 0 ( Fig. 11 ) magis , puta decu
plo distet a fulcro A quam resistentia in L , quae simili
ter magis distet , puta noncuplo a fulcro C quam resisten
tia in K , quae rursus magis distet a fulcro D puta quin
tuplo quam resistentia in E , et haec similiter magis puta
quadruplo distet a fulcro G quam resistentia in F , haec
denique duplo magis distet a fulcro H quam pondus P in B.
Si res ita se habet , atque insuper potentia et pondus di
rectiones habeant perpendiculares ad respectivos vectes
factis AO = a , CL = a ', DK = a &lt;nowiki&gt;''&lt;/nowiki&gt; , GE = a &lt;nowiki&gt;''&lt;/nowiki&gt; , HF - a &lt;nowiki&gt;''&lt;/nowiki&gt; ,
AL = 6, CK = b' , DE = 6&lt;nowiki&gt;''&lt;/nowiki&gt; ,GF = 6 &lt;nowiki&gt;''&lt;/nowiki&gt; , HB = 6 &lt;nowiki&gt;''&lt;/nowiki&gt;
b &lt;nowiki&gt;''&lt;/nowiki&gt; , erunt
in casu aequilibrii,
L. 6 E. 6 &lt;nowiki&gt;''&lt;/nowiki&gt;
Q
F.6&lt;nowiki&gt;''&lt;/nowiki&gt; &lt;nowiki&gt;''&lt;/nowiki&gt;
il
K =
Kiba,K
E F
P. &lt;nowiki&gt;''&lt;/nowiki&gt;

&lt;nowiki&gt;;&lt;/nowiki&gt;

a
a&lt;nowiki&gt;''&lt;/nowiki&gt;
a '
IV
ex quarum multiplicatione prodibit
b 6'6&lt;nowiki&gt;''&lt;/nowiki&gt; 6 &lt;nowiki&gt;''&lt;/nowiki&gt; 8 &lt;nowiki&gt;''&lt;/nowiki&gt; P
Q
α α' α
P
a &lt;nowiki&gt;''&lt;/nowiki&gt; a &lt;nowiki&gt;''&lt;/nowiki&gt; 3600&lt;nowiki&gt;''&lt;/nowiki&gt;
Quisque videt haec applicari systemati cuicumque rotarum
dentatarum.
Supponantur quoque plures trochleae mobiles v.gr. tres
(Fig. 12) ; erunt ( 14)
Q
L
2 sin r &lt;nowiki&gt;''&lt;/nowiki&gt;
K р
LE 2 sin ac '
&gt; K = ;
2 sin x
et consequenter
Q =
P
23 sin x sin a ' sipx&lt;nowiki&gt;''&lt;/nowiki&gt;
Quod si detur systema coalescens ex trochleis fixis
v. gr , C , C' , C &lt;nowiki&gt;''&lt;/nowiki&gt;, C &lt;nowiki&gt;'''&lt;/nowiki&gt; ( Fig . 13 ) et ex mobilibus F, E, K
29
16. Supponantur nunc plures quotcumque vectes ita
dispositi , ut potentia Q in O (Fig. 11 ) magis , puta decu.
plo distet a fulcro A quam resistentia in L , quae simili-
ter magis distet
, puta noncuplo a fulcro C quam resisten-
tia in K, quae rursus magis distet a fulcro D piita quin-
tuplo quam resistentia in E, et haec similiter magis puta
quadruplo distet a fulcro G quam resistentia in F, haec
denique duplo magis distet a fulcro H quam pondus P in B.
Si res ita se habet , atque insuper potentia et pondus di-
rectiones habeant perpendiculares ad respectivos vectes
,
factis AO:a,CL :a'
, DK:
a&lt;nowiki&gt;''&lt;/nowiki&gt;, GE :a&lt;nowiki&gt;'''&lt;/nowiki&gt;,HF :a&lt;nowiki&gt;''&lt;/nowiki&gt;,
AL:&amp;, CK:6', DE :6&lt;nowiki&gt;''&lt;/nowiki&gt;, GF:b&lt;nowiki&gt;'''&lt;/nowiki&gt;, HB:ö&lt;nowiki&gt;''&lt;/nowiki&gt;
, erunt
in casu aequilibrii,
'
'
'. ∙
'&lt;nowiki&gt;'''&lt;/nowiki&gt; Q—qy'b,L—K'£.,K:E'f ∙ !' ,E—Eb ,F—Pf ;
a a a a a
ex quarum multiplicatione prodibit
Q 6 b' 1)&lt;nowiki&gt;''&lt;/nowiki&gt; b&lt;nowiki&gt;'''&lt;/nowiki&gt; 6&lt;nowiki&gt;''&lt;/nowiki&gt;P P ⇠
a
.: ∙
as an aut alv 3600
Quisque Videt baec applicari systemati cuicumque rotarum
dentatarum.
. Su pponantur quoque plures trochleae mobiles v. gr. tres
(Fig. 12) ; erunt (14). ⋅
et consequenter
Q.... 23 sinu: sinx' sin x&lt;nowiki&gt;''&lt;/nowiki&gt;
Quod si detur systema coalescens ex trochleis fixis
&lt;nowiki&gt;''&lt;/nowiki&gt;
gr, C ∙∁⋅∣ C&lt;nowiki&gt;''&lt;/nowiki&gt;. 0&lt;nowiki&gt;''&lt;/nowiki&gt; (Fig. 13) et ex mobilibus F, E, K30
uno eodemque fane conjunctis ; quoniam , librato systemate ,
funis ubique manet aeque tensus , ideo Q : Q = Q &lt;nowiki&gt;''&lt;/nowiki&gt;
Q &lt;nowiki&gt;''&lt;/nowiki&gt; = Q &lt;nowiki&gt;''&lt;/nowiki&gt; = Q = Q &lt;nowiki&gt;''&lt;/nowiki&gt; = Q &lt;nowiki&gt;''&lt;/nowiki&gt; &lt;nowiki&gt;''&lt;/nowiki&gt; . Jamvero
F E K
Q Q '&lt;nowiki&gt;'''&lt;/nowiki&gt; QP 2sin x' ' 2 sin x
2 sin 2
et consequenter
F = 2Q ' sin &lt;nowiki&gt;''&lt;/nowiki&gt; 2Q sin x &lt;nowiki&gt;''&lt;/nowiki&gt; , E = 2Q sin ü&lt;nowiki&gt;''&lt;/nowiki&gt; , K =
2Q sin x ;
cum igitur sint
L = Q&lt;nowiki&gt;''&lt;/nowiki&gt; &lt;nowiki&gt;''&lt;/nowiki&gt; , F +E + K +L = P ,
iccirco
2 Q sin x &lt;nowiki&gt;''&lt;/nowiki&gt; + 2 Q sin x' + 2 Q sin x +Q = P :
unde
P
Q =
1 +2 (sin x +sin x ' + sin x &lt;nowiki&gt;''&lt;/nowiki&gt; )
Fac demum ut puncta materialia K , K ', K &lt;nowiki&gt;''&lt;/nowiki&gt; , K '&lt;nowiki&gt;'''&lt;/nowiki&gt;,
( fig. 14 ) jungantur Glis K K' , K'K &lt;nowiki&gt;''&lt;/nowiki&gt; determinatae
quidem longitudinis, sed mobilibus circa K , K &lt;nowiki&gt;''&lt;/nowiki&gt; . Si pun
cta illa sollicitantur viribus Q , Q , Q &lt;nowiki&gt;''&lt;/nowiki&gt; , Q &lt;nowiki&gt;'''&lt;/nowiki&gt; , ad aequi
librium haec manifeste requirentur : potentia Q in di
rectione K'K tendens ab K' versus K ; resultans R' ex
Q et Q' in directione K &lt;nowiki&gt;''&lt;/nowiki&gt; K ' tendens ab K &lt;nowiki&gt;''&lt;/nowiki&gt; versus K' ; re
sultans R &lt;nowiki&gt;''&lt;/nowiki&gt; ex R' et Q &lt;nowiki&gt;''&lt;/nowiki&gt; in directione K &lt;nowiki&gt;''&lt;/nowiki&gt; K &lt;nowiki&gt;''&lt;/nowiki&gt; tendens ab K&lt;nowiki&gt;''&lt;/nowiki&gt; &lt;nowiki&gt;''&lt;/nowiki&gt; '
versus K &lt;nowiki&gt;''&lt;/nowiki&gt; ; potentia Q &lt;nowiki&gt;'''&lt;/nowiki&gt; in directione K &lt;nowiki&gt;''&lt;/nowiki&gt; K' ' ' tendens ab K &lt;nowiki&gt;''&lt;/nowiki&gt;
versus K' ' ' : demum ipsa Q's aequalis resultanti R &lt;nowiki&gt;''&lt;/nowiki&gt; .

&lt;nowiki&gt;*&lt;/nowiki&gt; Denotantibus X , Y , Z componentes coordi

natis orthogonalibusque axibus parallelas , in quas resolvi
tur Q, erunt
30
uno eodemque fune coniunctis; quoniam .
librato systemate,
funis ubique manet— aeque tensus
, ideo, Q:Q' ∶⋅−−−−∙ Q&lt;nowiki&gt;''&lt;/nowiki&gt; ∙∙∙−∙∶
Qu::
le:
Qv :va :Qvu ∙
Iamvero
F ∙∙∙ E v K
Q −⇀⋅⋅ 2 SQ..— sin m' 2 sinx
Q'—
−⋅ Zsin x&lt;nowiki&gt;''&lt;/nowiki&gt; ∙
et consequenter
F:
2Q'Isin a:&lt;nowiki&gt;''&lt;/nowiki&gt; ZQ sin x&lt;nowiki&gt;''&lt;/nowiki&gt;, E:2Q sin x', K:
2Q sinx; ⋅
cum igitur sint
LSva'sF4-E—FK—FL2P,
iccirco—
2Qsinx&lt;nowiki&gt;''&lt;/nowiki&gt;—I-2Qsinx'—]-2Qsinx—l-AQ:P:
nnde
P
1—l-2 (sinx-l—sin x' ∙−⊢ sin x&lt;nowiki&gt;''&lt;/nowiki&gt;)
.
Fac demum nt puncta materialia K,K' ,K&lt;nowiki&gt;''&lt;/nowiki&gt;, K&lt;nowiki&gt;'''&lt;/nowiki&gt;, ..:
(Gg. 14 ) iungantur filis K K', K' K&lt;nowiki&gt;''&lt;/nowiki&gt; , ... determinatae
quidem longitudinis, sed mobilibus circa K', K&lt;nowiki&gt;''&lt;/nowiki&gt;. Si pun-
cta illa sollicitantur viribus Q, Q' , Q&lt;nowiki&gt;''&lt;/nowiki&gt; , Q&lt;nowiki&gt;''&lt;/nowiki&gt; , ad aequi-
librium haec manifeste requirentur: potentia Q in di-
rectione K'K tendens ab K' versus K; resultans R' ex
Q et Q' in directione K&lt;nowiki&gt;''&lt;/nowiki&gt;K' tendens ab K&lt;nowiki&gt;''&lt;/nowiki&gt; versus K'; re-
sultans R&lt;nowiki&gt;''&lt;/nowiki&gt; ex B' et Q&lt;nowiki&gt;''&lt;/nowiki&gt; in directione K&lt;nowiki&gt;'''&lt;/nowiki&gt;K&lt;nowiki&gt;''&lt;/nowiki&gt; tendens .ab K&lt;nowiki&gt;''&lt;/nowiki&gt;
versus K&lt;nowiki&gt;''&lt;/nowiki&gt;; potentia Q&lt;nowiki&gt;'''&lt;/nowiki&gt; in directione K&lt;nowiki&gt;''&lt;/nowiki&gt; K&lt;nowiki&gt;'''&lt;/nowiki&gt; tendens ab K&lt;nowiki&gt;''&lt;/nowiki&gt;
versus K&lt;nowiki&gt;''&lt;/nowiki&gt;: demum ipsa Q&lt;nowiki&gt;'''&lt;/nowiki&gt; aequalis resultanti R&lt;nowiki&gt;''&lt;/nowiki&gt;.
&amp; Denotantibus X , T, Z componentes coordi-
natis orthogonalibusque axibus parallelas, in quas resolvi-
⋅ tur Q, erunt
Q;:31
X Y
ē
z
Q
cosinus angulorum , quos cum iis axibus intercipit l; de
notantibus insuper 2 , y , z coordinatas puncti K , et x' ,
j ', z coordinatas puncti K' , erunt
2x yay 22
KKKK KK
cosinus angulorum, quos cum ipsis axibus efficit K'K ; ob
tinebit itaque primum ex requisitis ad aequilibrium, quoties
cumque fuerint
X XX Y
DKKKK .
yg Z
KÖK &gt;
K’K &lt;nowiki&gt;''&lt;/nowiki&gt;
seu
X Y Z
(h ) .
Quod in ordine ad Q est X , Y , Z , sit X', Y ', Z ' in or
dine ad Q ' : si resolvitur l' in ternas coordinalis axibus
parallelas, eae erunt ( 9. 40. )
x + X ' , Y + Y ' , 2 + Z ';
hinc designantibus a&lt;nowiki&gt;''&lt;/nowiki&gt;, y ', z &lt;nowiki&gt;''&lt;/nowiki&gt; coordinatas puncti K&lt;nowiki&gt;''&lt;/nowiki&gt; , ob
tinebit secundum ex requisitis ad aequilibrium , ubi fuerint
X + X __ * ' - &lt;nowiki&gt;''&lt;/nowiki&gt; Y + Y_y_y&lt;nowiki&gt;''&lt;/nowiki&gt; 2 + 2_z'- &lt;nowiki&gt;''&lt;/nowiki&gt;
R ? KK R' K ” K R K&lt;nowiki&gt;''&lt;/nowiki&gt;K&lt;nowiki&gt;'''&lt;/nowiki&gt;
.
seu
X + * _ * + Y_2_Z
x - x yay 22
( h ').
31
X ? Z
Q Q Q
cosinus angulorum, quos cum iis axibus intercipit Q; de-
notantibus insuper a: , y , :: coordinatas puncti K,, et x',
y', s' coordinatas puncti K' ,
erunt
⋅⇂⋅−−⋅⊴⇂∙∣ .7-7&lt;nowiki&gt;''&lt;/nowiki&gt; z—z'
K'K , K'K . K'K
cosinus angulorum, quos cum ipsis axibus efficit K'K: ob-
tinebit itaque primum ex requisitis ad aequilibrium, quoties-
cumque fuerint '
≟−−−⋅−∝−−≄∣ it.s,-ï Z
Q K'K&lt;nowiki&gt;''&lt;/nowiki&gt; 'Q K'K &lt;nowiki&gt;''&lt;/nowiki&gt;G'ka' ↽−≖∙⊍↼∙≕∣
seu gx z
r—x' y—y' x—z'
Quod in ordine ad Q est X , T, Z , sit X', ï', Z' in or-
dine ad Q':si resolvitur Q' in ternas coordinatis axibus
parallelas, eae erunt (9. 40.)
X—FX' , T—Fï' , Z—l-Z' ; ↽
hinc designantibus z', y&lt;nowiki&gt;''&lt;/nowiki&gt;, :&lt;nowiki&gt;''&lt;/nowiki&gt; coordinatas puncti K&lt;nowiki&gt;''&lt;/nowiki&gt;, ob-
tinebit secundum ex requisitis ad aequilibrium , ubi fuerint ⋅
X—l—X' x'-x&lt;nowiki&gt;''&lt;/nowiki&gt; T—l-Tl—TI—j&lt;nowiki&gt;'''&lt;/nowiki&gt; ∅⊣−⊈∣↼↼≂∣∙ z&lt;nowiki&gt;''&lt;/nowiki&gt;
B' ⋅⋅⇀∣⋦∣∣↓⊊∣ ∙ nf- KI/KT '-T—KHK' '—
....t ∙⇁−⋅∣ ↖↽∙∣ ∣
X X T
T—Z-Z.(h).
/ II I I/32
non pluribus opus est ut intelligamus quod, expleta
X + X + X &lt;nowiki&gt;''&lt;/nowiki&gt; _Y + Y + Y &lt;nowiki&gt;''&lt;/nowiki&gt; _Z + Z + Z &lt;nowiki&gt;''&lt;/nowiki&gt;
x ' - 0 &lt;nowiki&gt;''&lt;/nowiki&gt; g'my &lt;nowiki&gt;''&lt;/nowiki&gt; z &lt;nowiki&gt;'''&lt;/nowiki&gt; - &lt;nowiki&gt;''&lt;/nowiki&gt;
( h&lt;nowiki&gt;''&lt;/nowiki&gt; ) ,&lt;nowiki&gt;''&lt;/nowiki&gt;
obtinebit tertium ex requisitis illis ; componentes X” , Y&lt;nowiki&gt;''&lt;/nowiki&gt; ,
Z&lt;nowiki&gt;''&lt;/nowiki&gt; spectant ad vim Q &lt;nowiki&gt;''&lt;/nowiki&gt;, coordinatae z ' ', y, pun.
clum K &lt;nowiki&gt;'''&lt;/nowiki&gt; . Designantibus demum X '&lt;nowiki&gt;'''&lt;/nowiki&gt; , Y
Y ' &lt;nowiki&gt;''&lt;/nowiki&gt;,
&lt;nowiki&gt;''&lt;/nowiki&gt; , Z &lt;nowiki&gt;''&lt;/nowiki&gt;
componen
tes in ordine ad Q&lt;nowiki&gt;''&lt;/nowiki&gt; , expletisque
X + X + X &lt;nowiki&gt;''&lt;/nowiki&gt; + X &lt;nowiki&gt;''&lt;/nowiki&gt; = 0 ,
Y + r' + &lt;nowiki&gt;''&lt;/nowiki&gt; + I&lt;nowiki&gt;''&lt;/nowiki&gt; = 0 ,
2 + 2 +2&lt;nowiki&gt;''&lt;/nowiki&gt; + Z &lt;nowiki&gt;''&lt;/nowiki&gt; = 0 ,
( h &lt;nowiki&gt;''&lt;/nowiki&gt; )
manifeste obtinebit quartum simulque quintum ex requisi
tis ad aequilibrium. Sub novem igitur distinctis condi
tionibus propositum quatuor virium systema consistet in
aequilibrio: si quinque darentur vires , undecim prodirent
conditiones; generatim 2 n + 1 conditiones quoad n vires.
Collatis primis ac secundis membris formularum ( h) , (h') ,
( h&lt;nowiki&gt;''&lt;/nowiki&gt;) , emergent
Y ( 2 - x ) – X (y - ) = 0 ,
( Y + Y') (a' - &lt;nowiki&gt;''&lt;/nowiki&gt; ) – ( X + X ') ( 7'- , ' ) = 0 ,
( X + Y' + Y &lt;nowiki&gt;''&lt;/nowiki&gt;) ( ' &lt; &lt;nowiki&gt;''&lt;/nowiki&gt; ) — ( X + x ' + X &lt;nowiki&gt;''&lt;/nowiki&gt;) (y &lt;nowiki&gt;''&lt;/nowiki&gt; , ' ') = 0;&lt;nowiki&gt;''&lt;/nowiki&gt;
quarum summa praebet
xY_yXfwY — y'X ' + x &lt;nowiki&gt;''&lt;/nowiki&gt; Y &lt;nowiki&gt;''&lt;/nowiki&gt; —y &lt;nowiki&gt;''&lt;/nowiki&gt; X &lt;nowiki&gt;''&lt;/nowiki&gt; + &lt;nowiki&gt;''&lt;/nowiki&gt; ( X + X' + x &lt;nowiki&gt;''&lt;/nowiki&gt;) —
x &lt;nowiki&gt;''&lt;/nowiki&gt; ( Y + Y ' + Y &lt;nowiki&gt;''&lt;/nowiki&gt; ) = 0 ,
∃⊈∙
non pluribus Opus est ut intelligamus quod, expleta
x-1-xq-x'Q—v-1-rq-rff—z-i-zq-z&lt;nowiki&gt;''&lt;/nowiki&gt; W,),
xli—xlli J/l ∙∙⇁ 7&lt;nowiki&gt;''&lt;/nowiki&gt;, z&lt;nowiki&gt;''&lt;/nowiki&gt;—z&lt;nowiki&gt;'''&lt;/nowiki&gt;
obtinebit tertium ex requisitis illis; componentes X&lt;nowiki&gt;''&lt;/nowiki&gt;, ?&lt;nowiki&gt;''&lt;/nowiki&gt;
,
⋅
Z&lt;nowiki&gt;''&lt;/nowiki&gt; spectant ad vim Q&lt;nowiki&gt;''&lt;/nowiki&gt;, coordinatae x&lt;nowiki&gt;'''&lt;/nowiki&gt;. &lt;nowiki&gt;''&lt;/nowiki&gt;, z&lt;nowiki&gt;'''&lt;/nowiki&gt; ad pun-
ctum K&lt;nowiki&gt;''&lt;/nowiki&gt; . Designentibus demnm X&lt;nowiki&gt;'''&lt;/nowiki&gt;, ï&lt;nowiki&gt;'''&lt;/nowiki&gt;
, Z&lt;nowiki&gt;'''&lt;/nowiki&gt; componen-
tes in ordine ad Q&lt;nowiki&gt;'''&lt;/nowiki&gt; , expletisque
\sum∙⊦\sum∣∙⊢\sum∦⊹\sum∣∥∶∶∘∙
T .l-T-l-TII—l- III,: 0 , (hi/I)
Z-i-ZIä-le-l—ZIflzo'
manifeste obtinebit quartum simulque quintum ex requisi-
tis ad aequilibrium: Sub novem igitur distinctis condi-*
tionibus propositum quatuor virium systema consistet in
aequilibrio: si quinque darentur vires, undecim prodirent
conditiones; generatim 2 n ⊣− ↿conditiones quoad :: vires.
Collatis primis ac secundis membris formularum (I:), (b'),
U;&lt;nowiki&gt;''&lt;/nowiki&gt; )
, emergent
?(x—x') —X (?'—?') −∙−−−∘ ∙
(
ï—l— !' )(x' ∙∙∙ x&lt;nowiki&gt;''&lt;/nowiki&gt;)—( X-l-X') (r'—y&lt;nowiki&gt;''&lt;/nowiki&gt;) :o
,
(HF-IJ&lt;nowiki&gt;''&lt;/nowiki&gt;) (x&lt;nowiki&gt;''&lt;/nowiki&gt;-— ∣∣∣≻⊣≖≖−⊦\sum∣−⊦\sum∥≻ (y'—y&lt;nowiki&gt;'''&lt;/nowiki&gt;)
−−− .;
quarum summa praebet
xy-Jx-Jlïl—y/X/ :&lt;nowiki&gt;''&lt;/nowiki&gt; ï&lt;nowiki&gt;''&lt;/nowiki&gt;—y&lt;nowiki&gt;''&lt;/nowiki&gt;X&lt;nowiki&gt;''&lt;/nowiki&gt;—l—y&lt;nowiki&gt;'''&lt;/nowiki&gt;(X XLI-X&lt;nowiki&gt;''&lt;/nowiki&gt;) ∙−⋅
↕∣∣∣≼↕⊹⊺∣⊹↕∥≻ :0 ,33
seu , ob primam et secundam ( hm) ,
-Y yXTY'y'x + x'Y&lt;nowiki&gt;''&lt;/nowiki&gt; _7 / X &lt;nowiki&gt;''&lt;/nowiki&gt; + x &lt;nowiki&gt;''&lt;/nowiki&gt; I &lt;nowiki&gt;''&lt;/nowiki&gt; —7&lt;nowiki&gt;'''&lt;/nowiki&gt;X &lt;nowiki&gt;'''''&lt;/nowiki&gt;
Simili modo collatis primis ac tertiis membris ipsarum ( h) ,
( h') , ( h&lt;nowiki&gt;''&lt;/nowiki&gt; ), attentisque prima ac tertia ( h '&lt;nowiki&gt;'''&lt;/nowiki&gt;) ; itemque col
latis secundis ac tertiis membris earumdem ( h ) , ( h ) , ( h&lt;nowiki&gt;''&lt;/nowiki&gt; ) ,&lt;nowiki&gt;''&lt;/nowiki&gt;
attentisque secunda ac tertia ( h &lt;nowiki&gt;'''&lt;/nowiki&gt;) , assequemur&lt;nowiki&gt;'''&lt;/nowiki&gt;
xZ - 2X + « Z_z'X' + x'Z&lt;nowiki&gt;''&lt;/nowiki&gt; _z&lt;nowiki&gt;''&lt;/nowiki&gt;X &lt;nowiki&gt;''&lt;/nowiki&gt; Tx &lt;nowiki&gt;''&lt;/nowiki&gt; Z '&lt;nowiki&gt;'''&lt;/nowiki&gt; —Z'y&lt;nowiki&gt;''&lt;/nowiki&gt; = 0 ,&lt;nowiki&gt;''&lt;/nowiki&gt;
32—3Y + y^2?–49 + &lt;nowiki&gt;''&lt;/nowiki&gt;Z&lt;nowiki&gt;''&lt;/nowiki&gt; &gt;&lt;nowiki&gt;''&lt;/nowiki&gt;Y &lt;nowiki&gt;''&lt;/nowiki&gt; +y &lt;nowiki&gt;''&lt;/nowiki&gt; Z&lt;nowiki&gt;''&lt;/nowiki&gt; — ;&lt;nowiki&gt;''&lt;/nowiki&gt; Y&lt;nowiki&gt;''&lt;/nowiki&gt; = 0 .
Conditiones videlicet aequilibrii ( 13. 8º. ) quoad systema
punctorum lineis rigidis inter se firmiter connexorum in
cluduntur in conditionibus aequilibrii quoad propositum
systema habens formam variabilem .

=== De centro gravitatis. ===
[[17|17]]. Constat experimentis corpora jugiter sic tendere, seu gravitare in tellurem, ut sibi commissa descendant verticaliter in eius superficiem, gravitas ergo, seu vis unde provenit iste verticalis descensus, eatenus haberi poterit pro sibi ad sensum parallela, quatenus licebit superficiem illam habere pro physice plana: constat insuper experimentis omnia quaevis corpora eodem tempore idem spatium verticaliter in vacuo percurrere, idest aequali velocitate ex aequali altitudine perpendiculariter ad horizontem descendere. Inde sequitur vires gravitatis in diversis corporibus esse illorum massis proportionales, et corpus quodlibet spectari posse tanquam aggregatum materialium graviumque particularum, quae gaudeant parallelarum virium proprietatibus: centrum virium parallelarum (12) in casu dicitur centrum gravitatis. Resultans ex omnibus gravitatis viribus, quae vigent in corporis particulis, vocatur corporis pondus; transit constanter per gravitatis centrum, et directionem obtinet horizonti perpendicularem.

Porro si massula indefinite parva &lt;math&gt;\nu&lt;/math&gt; apud datum corporis punctum dividitur per respondens volumen &lt;math&gt;\beta&lt;/math&gt;, ratio &lt;math&gt;\frac{\nu}{\beta} (= \mu ) &lt;/math&gt; vocatur corporis densitas apud illud punctum; diciturque corpus vel homogeneum, vel heterogeneum prout &lt;math&gt;\mu&lt;/math&gt; apud singula corporis puncta est vel eadem, vel diversa; in corporibus homogeneis ratio &lt;math&gt;= \mu&lt;/math&gt;  est eadem ac ratio inter totalem corporis massam et ejus totale volumen; pondusculum massulae &lt;math&gt;= \nu&lt;/math&gt;, utpote proportionale ipsi &lt;math&gt;= \mu&lt;/math&gt;, exprimitur per &lt;math&gt;= \mu&lt;/math&gt; ductam in quandam constantem &lt;math&gt;c&lt;/math&gt;; ratio &lt;math&gt;\frac{c \nu}{\beta} (= c \mu ) &lt;/math&gt;  appellatur specifica corporis gravitas apud praefatum punctum; estque densitati proportionalis.

[[18|18]]. Notetur illud: etsi corpus gravitate sua jugiter sollicitatur deorsum; hoc tamen non officit quominus adhuc (2) dicatur corpus de se et natura sua indifferens ad quietem vel motum. Gravitas enim est dumtaxat vel aliquid extrinsecum corpori, vel illi intrinsecus additum, non autem aliquid eidem essentiale. Patet, quia vel nomine gravitatis intelligitur vis quaedam, qua corpora versus terram urgentur, vel vis qua tendunt ad determinatam quamdam spatii immobilis partem. Non hoc secundum, quia eo ipso casus purus admitteretur contra principium rationis sufficientis, cum nulla appareat ratio cur mobile ad hanc potius partem ferri debeat quam ad illam, cum spatium ubique sit homogeneum; ergo primum erit dicendum: sed si ita est, certe gravitas non est corporibus essentialis; nulli enim corpori essentiale est ut sibi caetera coexistant, ac proinde unum potest existere quin existant caetera, et consequenter etiam quin existat terra.

[[19|19]]. Dato centro gravitatis corporis, facile definitur utrum corpus in dato situ extra lapsus periculum constitui possit. Nam ex eo centro demissa ad planum horizontale recta perpendiculari, quae vocatur linea directionis, si haec intra basim cadat, corpus extra lapsus periculum erit positum, secus ruet in eam partem in quam perpendicularis recta dirigitur. Hinc patet ratio cur turres aliquae &lt;u&gt;inclinatae&lt;/u&gt; non cadant, ut sunt Bononiensis, Pisana etc: linea scilicet directionis extra ipsarum basim non excurrit. Hinc etiam valde pingues, et qui magnum aliquod onus brachiis complectuntur, retrorsum; gibbosi autem et bajuli antrorsum; qui dextra pondus aliquod sustinent, sinistrorsum; qui vero sinistra, dextrorsum &lt;u&gt;inflectuntur&lt;/u&gt;. Per hanc scilicet declinationem efficiunt ut linea directionis transeat per spatium, quod inter pedes continetur; quod spatium est basis corporis humani. Eamdem ob caussam si quis velit ex. gr. dextero pede stare, crus &lt;u&gt;inclinat&lt;/u&gt; paullulum dexteram partem versus, nec diu haerere potest in eo statu , quia cum basis totius corporis sit unus dumtaxat pes, linea directiouis facile potest basis tam anguslae limites praetergredi. His autem corporis nostri flexibus ac librationibus ita ab infantia assuevimus usu continuo ut nec advertentes recto illas ordine peragamus. Patet hinc denique cur aves uni pedi insistentes dormire solent capite sub ala recondito; id nempe faciunt ut linea directionis intra pedis cui insistunt latitudinem servetur.

[[20|20]]. Centrum gravitatis inveniri potest vel ratione mechanica, vel ratione, algebraica. Ad primam quod attinet, si corpus aliquod filo suspendas, volvetur converteturque donec in aequilibrio tandem consistat, et filum ad terrae superficiem perpendiculariter dirigatur. In hac perpendiculari, quae est linea directionis per quam centrum gravitatis corporis tendit, erit centrum ipsum. Iam notetur linea a filo perpendiculari in corpore designata, rursusque ex alio puncto suspendatur corpus, et facto aequilibrio linea perpendicularis pariter notetur. In communi duarum linearum intersectione reperietur quaesitum centrum. Ratio algebraica desumitur ex dictis ( 13.2.º''a''&quot; ): sumantur nempe vires proportionales massis &lt;math&gt;m, m' , m''&lt;/math&gt;, ..... punctorum, quibus applicitae sunt; hoc pacto, ad positionem centri gravitatis determinandam exsistent
&lt;math display=&quot;block&quot;&gt;x_{\mathrm I}=\frac{\sum m x}{\sum m}, y_{\mathrm I}=\frac{\sum m y}{\sum m}, z_{\mathrm I}=\frac{\sum m z}{\sum m}  (b) &lt;/math&gt;Si corpus intelligitur divisum in varias portiones dimensionis finitae , et earum massae denotantur per  &lt;math&gt;m, m' , m''&lt;/math&gt;, adhuc valebunt formulae (b); nihilque aliud erunt &lt;math&gt;x , y , z ,x' y ',z',x''&lt;/math&gt;, ... nisi coordinatae centrorum gravitatis illarum portionum. Si corpus ponitur insuper homogeneum quoad omnes partes, erunt massae ut respondentia volumina; poteruntque haec illis substitui in formulis (''b'') : quisque videt coordinatas &lt;math&gt;x_{\mathrm I}, y_{\mathrm I}, z_{\mathrm I}&lt;/math&gt;, ex (''b'') haud pendere ab intensitate gravitatis. Caeterum plures sunt casus, in quibus centrum gravitatis absque formularum subsidio immediate cognoscitur. Sic in linea recta centrum gravitatis est medium ipsius rectae punctum: in parallelogrammo punctum, ubi binae diagonales se mutuo secant: in circulo centrum figurae: in cylindro habente bases parallelas punctum medium axeos:
in parallelepipedo punctum, ubi quatuor diagonales se mutuo secant: in sphaera ipsum magnitudinis centrum.

In triangulo centrum gravitatis est punctum illud, ubi sese invicem secant rectae lineae, quae a duobus trianguli verticibus ducuntur ad puncta media laterum oppositorum: cum enim &lt;math&gt;AD&lt;/math&gt; (Fig. 15) dividat aequaliter rectas omnes lateri &lt;math&gt;BC&lt;/math&gt; parallelas, et &lt;math&gt;BE&lt;/math&gt; rectas omnes lateri &lt;math&gt;AC&lt;/math&gt; parallelas, reperietur centrum gravitatis areae triangularis tam in &lt;math&gt;AD&lt;/math&gt; quam in &lt;math&gt;BE&lt;/math&gt;; ideoque erit in &lt;math&gt;H&lt;/math&gt;. Jamvero ducta &lt;math&gt;DE&lt;/math&gt;, ea exsistet parallela lateri &lt;math&gt;AB&lt;/math&gt;; et consequenter triangula &lt;math&gt;ABH , DEH&lt;/math&gt; erunt similia; hinc&lt;math display=&quot;block&quot;&gt;\frac{DE}{AB}=\frac{DH}{AH}&lt;/math&gt;sed, ob &lt;math&gt;CE = \frac12  AC&lt;/math&gt; et &lt;math&gt;CE = \frac12 CD = BC&lt;/math&gt;, est DE = &lt;math&gt;CE = \frac12 AB&lt;/math&gt;; igitur &lt;math&gt;DH = \frac12  AH&lt;/math&gt;; ac proinde &lt;math&gt;DH = \frac12 AD&lt;/math&gt;; et &lt;math&gt;AH = \frac23 AD&lt;/math&gt;.

In pyramide triangulari &lt;math&gt;ABCO&lt;/math&gt; (Fig. 16) erit &lt;math&gt;G&lt;/math&gt; centrum gravitatis; ubi nempe se mutuo secant binae rectae &lt;math&gt;OH , CK&lt;/math&gt;, quae ex &lt;math&gt;O&lt;/math&gt; et &lt;math&gt;C&lt;/math&gt; ducuntur ad centra gravitatis
&lt;math&gt;H&lt;/math&gt; et &lt;math&gt;K&lt;/math&gt; triangulorum &lt;math&gt;ABC , ABO&lt;/math&gt;. Secetur enim pyramis,

1.º planis parallelis triangulo &lt;math&gt;ABC&lt;/math&gt;, 

2.º planis parallelis triangulo &lt;math&gt;ABO&lt;/math&gt;; transibit &lt;math&gt;OH&lt;/math&gt; per centra gravitatis omnium illarum sectionum triangularium; transibit &lt;math&gt;CK&lt;/math&gt; per centra gravitatis omnium harum. Ergo pyramis habebit suum gravitatis centrum tam in &lt;math&gt;OH&lt;/math&gt; quam in &lt;math&gt;CK&lt;/math&gt;, et consequenter in &lt;math&gt;G&lt;/math&gt;. Ducatur nunc &lt;math&gt;HK&lt;/math&gt;; erit &lt;math&gt;HK&lt;/math&gt; parallela rectae &lt;math&gt;CO&lt;/math&gt;, et triangula similia &lt;math&gt;HKG , CGO&lt;/math&gt; praebebunt
&lt;math&gt;\frac{HK}{CO}=\frac{HG}{OG}.&lt;/math&gt;
Sed, ob &lt;math&gt;MH =\frac13 CM&lt;/math&gt; et &lt;math&gt;MK = \frac13 OM&lt;/math&gt;, est &lt;math&gt;HK = \frac13 OC&lt;/math&gt;;
ideoque &lt;math&gt;HG =\frac13 OG&lt;/math&gt;; igitur &lt;math&gt;HG = \frac14 OH&lt;/math&gt;, et &lt;math&gt;OG = \frac34 OH&lt;/math&gt;.

=== De corporum collisione ===
[[21|21]]. Quaestio de corporum collisione eo redit, ut datis velocitatibus ante collisionem, determinentur velocitates post collisionem. Corpora sese collidentia assumimus sphaerica, et in singulis stratis concentricis homogenea; in quibus proinde corporibus centrum gravitatis erit ipsum magnitudinis centrum. Corporum sese collidentium centra vel moventur in eadem recta, vel in diversis rectis; in primo casu collisio dicitur normalis, in secundo obliqua.
[[Fasciculus:Inelastischer stoß.gif|thumb]]
[[22]]. Invenire velocitatem &lt;math&gt;v''&lt;/math&gt;, quam habent duo data corpora non elastica post normalem collisionem, datis eorum
velocitatibus &lt;math&gt;v'&lt;/math&gt; et &lt;math&gt;v&lt;/math&gt; ante collisionem. Dicantur &lt;math&gt;m', m&lt;/math&gt; corporum massae; erunt &lt;math&gt;mv , m'v'&lt;/math&gt; quantitates motus ante collisionem: eatenus corpus subsequens agit in antecedens quatenus hoc lentius illo movetur, adeo ut perseveret actio donec ad aequalitatem velocitatis deveniatur; unde velocitas &lt;math&gt;v''&lt;/math&gt; post collisionem erit communis, et aequalis in utroque: summa praeterea quantitatum motus est eadem ante et post collisionem; velocitas autem obtinetur dividendo quantitatem motus per massam. Ergo demum&lt;math display=&quot;block&quot;&gt; v'' =\frac{mv + m'v'}{m + m'}&lt;/math&gt;

Haec observentur: 

1.° &lt;math&gt; v'' - v &lt;/math&gt; exprimit quantum velocitatis acquisierit corpus antecedens, quod ponimus esse
&lt;math&gt;m&lt;/math&gt;; et &lt;math&gt; v' - v'' &lt;/math&gt; quantum amiserit impellens &lt;math&gt;m'&lt;/math&gt;. 

2.° consideranda erit pro lubito alterutra velocitas tamquam negativa, si corpora ex oppositis plagis adveniunt; hinc in formulis ubicumque ea inveniatur, signo contrario erit adhibenda - Sic v. gr. si massae &lt;math&gt;m'&lt;/math&gt; directio habeatur pro positiva, sumenda erit &lt;math&gt;v&lt;/math&gt; negative, ac proinde &lt;math&gt; v'' =\frac{m'v'- mv}{m + m'}&lt;/math&gt;.

3.° ponetur &lt;math&gt;v = 0&lt;/math&gt;, si corpus impellendum &lt;math&gt;m&lt;/math&gt; quiescit; erit &lt;math&gt; v'' =\frac{m'v'}{m + m'}&lt;/math&gt;: hinc &lt;math&gt;v''&lt;/math&gt; ferme evanescet si massa &lt;math&gt;m&lt;/math&gt; sit physice infinita respectu &lt;math&gt;m'&lt;/math&gt;. 

4.º numquam habebitur perfecta quies post collisionem si &lt;math&gt;m&lt;/math&gt; et &lt;math&gt;m'&lt;/math&gt; in easdem partes oppositas, et velocitates sint reciproce ut massae, tunc &lt;math&gt; v'' = 0&lt;/math&gt;, et consequenter habebitur perfecta quies.

23. Invenire velocitates corporum perfecte elasticorum post normalem collisionem, datis velocitatibus &lt;math&gt;v', v&lt;/math&gt; ante collisionem. Perspicuum est hujusmodi corpora sequi leges non elasticorum toto compressionis tempore, tum restitutionis tempore effectum hunc instaurari. Ergo facta partium restitutione inveniri debet in corpore impulso dupla velocitatis acquisitio; dupla vero celeritatis amissio in impellente. Itaque si dicantur r' ' et &quot; velocitates corporis im
pellentis et corporis impulsi post factam restitutionem ,
erunt ( 22)
u &quot; = V - 2 ( 0--0&quot; ) = v - 2
my + ms
mtm
2 mv tv (m ' — m)
( 9 ),
m + m
(
1
vi &quot; = 0 + 2 (0 &quot; ~ v ) = 2 + 2 ( -v)
mv + m's
m + m
2 m ' ú tu (m - m ')
(9) .
mtm

24. Haec ex formulis (9) et (q' ) deducuntur .

1.• Si massae sunt aequales , elastica corpora post colli
sionem movebuntur .facta velocitatum permutatione, Nam
moveantur primo in eamdem plagam ; propter m = m' , for
mula (9) abit in
2 m v'
et ( 9 ') in 3,10 v' ; ergo etc.
Rursus praeter m = m ' habeatur etiam v = 0 , hoc est cor
2 mo
pus percussum quiescat; erit v = 0 , et v ' . = V ' ;
corpus nempe percutiens post collisionem quiescet , et per
2 mv
2 m 2 m
2 m
1
39
moveantur, vel* alterutra solum quiescat :quod si collisio
liat ad partes oppositas ,
et velocitates sint reciproce ut mas-
sae, tunc v&quot;:o ,
et consequenter habebitur perfecta quies.

23. Invenire velocitates corporum perfecte elasticorum
post normalcm collisionem, datis velocitatibus v', 0 ante col-
lisionem. Perspicuum est huiusmodi corpora sequi leges non
elasticorum tOto compressionis tempore, tum restitutionis
tempore effectum hunc instaurari. Ergo facta partium re-
stitutione inveniri debet in corpore impulso dupla velo-
citatis acquisitio; dupla vero celeritatis amissio in impel-
lente. ltaque si dicantur v'&quot; et v&quot; velocitates corporis im-
pellentis et corporis impulsi post factam restitutionem ,
erunt (22)
'
'
um:-D' --2 ('n'—v&quot;) :'--2 (,; ∙−−−−−−−−⋯⇂↓−⊢⋯∣∣↗ m −−⊢ m )
..2 mv −⊢∣v (m' −∙∙ m)
'm ∙−∙∙ m' ∓∎∎∎∎∎∎ (9):
W:w—l-2(v&quot;—v):v-l—2 Maii:-31; -v)
-&quot;2mv—l-v(m-—m) (qr).
m-l-m' J—
24. Haec ex formulis (q) et (q') deducuntur.
↿∙∘ Si massae sunt aequales, elastica corpora post colli-
sionem movebuntur.fdcta velocitatum permutatione, Nam
moveantur primo in eamdem plagam; propter m:m', for-
mula (q) abit in
2 mv ' ∙
2
'&quot;
'v'&quot;::«v p. . ,
et (q ) 111 10&quot;: :v ; ergo etc.
a m
'
-
Rursus praeter m:m' habeatur etiam :::o , hoc est cor-
- -
∣∣∣ tv 2 m &quot;( pus percussum qutescat; er1t a::o, et a:::
'v ; m
corpus nempe percutiens post collisionem quiescet , et per-40
cussum movebitur velocitate , quam percutiens habebat ante
collisionem . Demum sibi mutuo occurrant : ubicumque ergo
invenitur v , sumenda erit negative ; qua mutatione facta ,
habebuntur
2 mv 2 mv'
2 m
v, et viv v' .
2 m
Jam vides mutationes velocitatum exhiberi per ipsas litte
ras , et ubi debeat etiam mutari directio , regressus expri
mitur per mutationem signorum.
2.• Si statuatur series corporum perfecte elasticorum , ae
qualium , se mutuo tangentium , et quorum centra unam
rectam constituant , in primum autem quacumque velocitate
incidat alterum corpus aequale , movebitur tantummodo cor
pus ultimum , quiescentibus omnibus aliis . Quod si statua
tur series corporum habentium massas in progressione geo
m3,
metrica m' , m, ... ; et caeteris quiescentibus, pri
mum m' incidat in secundum velocitate v' , expriment
m2
m ? m
2 m
v' .
m +m (m *:)*,~(m2 I )
m
velocitates excitatas a primo in secundo , a secundo in ter
tio , a tertio in quarto etc. Denotante igitur n numerum cor
porum , movebitur ultimum velocitate
2 m'
N- 1
I
Cena ntmi
).
3. Quotiescumque corpus impellens minus erit corpore
impacto quiescente , toties illud regredietur , uti patet ex
formula ( 9 ) , posita v = o et m &gt; m' . Quod si m =
et v =0 , prodibit v'' = -1 , nimirum si globus minor''
∢⋅∘cussum movebitur velocitate ,quam percutiens habebat ante
collisionem. Demum sibi mutuo occurrant :ubicumque ergo
invenitur :: , sumenda erit negative; qua mntatione facta.,
habebuntur
2mv &quot; 2mv'

: —-v,et'v :

2m ∙∙−−−∙⋅∙≀≀∙ 2m
Iam vides mutationes velocitatum exhiberi per ipsas litte-
ras , et'ubi debeat etiam mutari directio , regressus expri-
mttur per mutat1onem signorum.
2.&quot; Si statuatur series corporum perfecte elasticorum, ae-
qualium, se mutuo tangentium, et quorum centra unam
rectam constituant
,
in primum autem quacumque velocitate
incidat alterum corpus aequale , movebitur tantummodo cor-
pus ultimum
, quiescentibus omnibus aliis. Quod si statua-
tur series corporum habentium massas in progressione geo-
∙ m! ma. metrtca m', m, ∙ ∙ ∙ ∙ &quot;7, , m .; et caeterts qmescenttbus , prt-
mum m' incidat in secundum velocitate v', expriment
v,2m' ⋅∙ 2m' :
,( 2m' 3
m—l—m&quot; 'v (m—l—m')&quot;, m-l-m'
velocitates excitatas a primo in secundo , a secundo in ter-
tio
, a tertio in quarto etc. Deuotante igitur n numerum cor-
porum , movebitur ultimum
⋅⋅⋮ velocitate ea&quot;
3.&quot; Quotiescumque corpus impellens minus erit corpore
impacto quiescente ,
toties illud regredietur , uti patet ex
formula (q) , posita v :o et m m'
. Quod si ut:ea
et 9

: o

, prodibit v'&quot;:
-— v' , nimirnm si globus minor41
incurrat in globum immensae massae quiescentem , resiliet
cum velocitate eadem , cum qua advenerat .
4.• Si duo corpora elastica occurrant sibi velocita
tibus v , v ', quae massis m, m ' reciproce sint proportionales ,
eadem qua venerant velocitate ambo resilient. Etenim cum
ex oppositis partibus corpora congrediantur , ac praeterea
m : m ' :: v ' : v , in formulis ( 9) , ( 9' ) sumenda erit » negative ,
et ponendum mv = m' '; quibus peractis , obtinebitur
v ' &quot; = &gt; &quot; (m + m )
et viv=v
Im + m
no-tm
Imtin
-- 5. ° Ex ipsis ( 9) et ( 9' ) eruitur m'y's mula
m 'ustomus: factum ex massa in quadratum respondentis
velocitatis dicitur vis viva ; hinc in corporibus perfecte
elasticis summa virium vivarum permanet eadem ante et post
collisiopem .

25. Formulae ( 9) , (9 ') aptari possunt etiam corporibus,
imperfecte elasticis , modo quantitatibus
2(v— mm Imus)
my tms
mtm
et 2
( mahu
my + mv
m + m
--)
substituantur
(n+ m ( = m **)e (1+- ( Inv—-).
denotante r rationem inter vim , qua partes sese resti
tuunt , et vim comprimentem. Quantitas r experimentis de
terminanda est in singulis corporum speciebus : fac ut m
quiescat , sitque co ; erit post collisionem '&quot; = -ru':
unde , cognita velocitate v' ., qua m ' offendit in m , et velo
citate negativa v'' , qua post impactum resilit , habebitur''
-
4
⋅↣
' 41
incurrat in globum immensae massae quiescentem ,
resiliat
cum velocitate eadem, cum qua advenerat.
49 Si duo corpora elastica occurrant sibi velocita-
tibus v, v', quae massis m, m' reciproce sint proportionales ,
eadem qua venerant velocitate ambo resilient. Etenim cum
ex oppositis partibus corpora congrediantur , ac .praeterea
ut :m'::v':
9, in formulis (q) ,(q' ) sumenda erit .9 negative ,
et ponendum m 9:m' v'; quibus peractis , obtinebitur
v'&quot;:
— v' (Z.—lm,) :-v'. et v&quot;:v (m )

:

v.
∙
∙⊢⋯⋅ ⋯∙−⊦⋯
∂∙∘ Ex ipsis (q)et.(q') eruitur m' v'&quot;3-l- mv&quot;::
m'∎∣∣≖ -l-m vi: factum ex massa in quadratum respondentis
velocitatis dicitur vis viva; hinc in corporibus perfecte
elasticis summa virium vivarum permanet eadem ante et post
collisionem. . 

25. Formulae (q), (q')'aptari possunt etiam corporibus ,
imperfecte elasticis , modo quantitatibus .
2(v,—nw—-mlv) et/2 (mv-l-mlv m-l-m —v) mm
substituantur (1 44) (v'. m.,-(.m'v') et≰↿⊹↗⋝⋅≼⋯⇂≩−−⋯⋅∣⇂≀∣ ∙∙∙∙∙ v) ∙
∦⇂⊣−⋯≳ m—l—m
denotante r rationem inter vim
, qua partes sese resti-
tuunt , et vim comprimentem. Quantitas r experimentis de-
terminanda est in singulis corpürum speciebus :fac ut 11:
quiescat , sitque :co ; erit post collisionem v'&quot;:
-
r v':
unde , cognita velocitate v' ., qua m' olfendit in m, et velo-
citate negativa v'&quot;
, ua post impactum resilit, habebitur '42

26. Ad collisionem obliquam quod pertinet , si corpora sibi mutuo occurrunt directionibus convergentibus bm , b'm
( Fig.17 ) et velocitatibus expressis per easdem rectas bm ,b'm ', resolvantur bm , b'm ', altera in duas by, ba, altera in duas b'y ', b'a', ita ut by, b'y' existant normales , ba vero et bá parallelae sint rectae m m corporum centra jungenti. Quoniam componentes b y , b'y' parallelae sunt tangenti TT ductae per punctum contactus, ab ipsis nullo pacto pendebit collisio, nullamque in collisione subibunt mutationem . Cor
pora igitur sese collident velocitatibus ba = ym, b'a' =
y'm '. Inventis itaque ( 23 ) v &quot; , et v '' , sumptisque ex. gr.'' mf = y &quot; , mi = &quot; in recta y r', et ductis mv = by , m'ú
= bóý , si complentur parallelogramma fv, iv', exprimentur per diagonales mf, m'i' tum velocitates , tum directiones corporum post collisionem. Haec autem ex modo dictis facile colliguntur; 

1.º Si globus minime elasticus iacidit oblique in planum immobile, progredietur secundum directionem plani cum velocitate m'v ' ( = a'm '), quae ad velocitatem priorem b'm ' erit ut sinus anguli incidentiae b'm'y' ad radium. 

2.º si globus fuerit perfecte elasticus, resiliet per m'z efficiendo angulum reflexionis z míy aequalem angulo incidentiae b'm'ý . 

3.° quod si globus incidens sit imperfecte elasticus, resiliet ad angu
lom i'm'y ', cujus cotangens ad cotangentem anguli inciden
liae b'm'y ' ut r : 1 .

=== De motu rectilineo utcumque vario.=== 
27. Nonnulla hic praemittimus ex analysi infinitesimali.
1.o Quantitas iniinitesima a: (minor videlicet qua-
cumque data utcumque parva) censeatur esse primi ordinis ;
«2 erit inlinitesima secundi ordinis; «3 iniiuitesima tertii; etc.
2.&quot; Inlinitesima a) dicetur esse primi ordinis si ra-
∙ G) ∙ ∙ ∙
a tno ∙
.. valorem habet (imtum
, secund1 s1 ∙−− valorem obtinet
ac «:43
similiter finitum , atque ita porro . Denotante generatim k
valorem illum finitum , poterit infinitesima quantitas ordinis
msimi exhiberi per
w kam
3. Sumptis aliis valoribus finitis k,; ka, ... km ,
habentur pro aequalibus
kmetkam tk , an- tkzam- ² + ... + kmiat kma
km_ ,a et kam + kamer + ... + km_, &amp; ,
kmed k * et kam + kamer t . tkm -rQ ?.
etc .... ;
admittuntur nimirum aequationes
kam tka&quot;-t ... tkm , at km
km
kam +k ,am -s +... +kimeza? + kmail
km ,
51
etc.
quatenus differentia inter utrumque membrum est minor
quacumque data quantitate alcunique parva. Huc spectat
illud : quantitates infinitesimae , quaecumque eae sint, et quo
rumcumque ordinum , absque ullo errore negliguntur prae
quantitate finita : itemque infinitesimae quantitates altiorum
ordinum absque ullo pariter errore negliguntur prae infini
tesima quantitate inferioris ordinis.
4.0 Quantitates infinitae ( majores videlicet qua
cumque data utcamque magna) cum possint exprimi person
43
similiter tinitum
, atque ita porro . Deuotante generatim k
valorem illnm finitum
, poterit intinitesima quantitas ordinis
msimi exhiberi per
m::ka&quot;
3.&quot; Sumptis aliis valoribus finitis k,
,
It,,
... k,&quot;,
habentur pro aequalibus

: , et kat'-I-k, ∝⋅⋅−≖⊣−≀∣≖∝∙−⋅≖−⊦ −⊦↗⊏⋅∙∙⋅∝−⊦∣⊏⋅⋅∙

∄⊄⋅⋅∙≖∘≖ et ka&quot; ⊣− 1, ∘⋍⋅∙∙⋅⋅⊳⋅ ⊣− ⊣− r.,, a: -
It,... «* et kat&quot; −⊦ kp?&quot; -]— ∙∙∙−∣⋅⋅ km., æ.
etc-eoo ;
.
admittuntur nimirum aequationes ⋅ '
↗⊄⊧∘↙⋅⊣−∣∁⋅⊶⋅∙−⋅⊣−∙∙∙∣−⊦↗≂⋅∙∙⋅⊄−⊦↳
↿ ⋅
km .
l
kan-l-Ic,ac&quot;&quot;-l--. &quot;'l-kaum&quot;,
hngua −⊦⋠⋅∙∙∙∸⇂⇉⊄∙−⋡↿ ⋅ «
⋅
etc. ∙ ∙ , . .
.. ⋮∙ ;
,-
∣

;

quatenus differentia, inter utrumque membrum'est'minor
quacumque data quantitate utcumque parva. Huc. &quot;spectat
illud :quantitates inünitesimae, quaecumque eae sint. et quo- ∙
rumcumque ordinum
, absque ullo errore negliguntur prae
quantitate finita :itemque infinitesimae quantitates altiorum
ordinum absque ullo pariter errore negliguntur prae ⋮≖≖∅≖∙≖∃∙∙ tesima quantitate inferioris ordinis.
∙∡∙∘ Quantitates infinitae (majores videlicet qua-
cumque data utcumque magna) cum possint exprimi per-;,44
tribuentur et ipsae in varios ordines ; illudque facile stabi
lietur : quotcumque finitae quantitates tuto : negliguntur
prae quantitate infinita ; quantitatesque infinitae ordinum in
feriorum tuto etiam negliguntur prae quantitate infinita altio
ris ordinis. Facto enim \beta . , et designantibus a, b,c, ... , 9
valores finitos , habebitur
1
. 0
a \betam + bBm - tom-&gt; + ... +9\betato
1
EL
-la + bw twat..tqomat ww .
5.- Si variabiles x, y sunt inter se per certam quam
dam relationem ita connexae ut data v. g. X , inde possit
valor y determinari , y vocatur functio quantitatis x ; ipsa
vero x dicitur independens. Si relatio inter x et y expri
mitur aequatione minime resoluta quoad functionem y habi
tam pro incognita , y appellatur functio implicita ; quod si
valor y detur expressus immediate per independentem x,
vel talis obtineatur per aequationis resolutionem , y dicitur
functio explicita. In aequatione v, g. yo -2xy + m2 =0
y functio implicita quantitatis variabilis x ; at facta re
solutione , evadet y functio explicita ipsius x , duplicemque
habebit valorem , scilicet y = x + Vx? m2 , Functio
nes explicitae quantitatis x designari solent in hunc modum
est
-
F ( x) , f ( x) , ..
6.0 Differenziale dx quantitatis x est incrementum
infinitesimum , quod ipsi x adscribitur : differentiale vero dy
functionis y = f (x ) est respondens incrementum f ( x + dx)
- f (x ) .quod ob variatam x recipit in se functio illa : pro
ponantur v. gr. invenienda differentialia functionum
44
tribuentur et ipsae in varios ordines ; illudque-facile stabi-
lietur :quotcumque finitae quantitates tuto.: negliguntur
prae quantitate infinita; quantitatesque infinitae Ordinum in-
feriorum tuto etiam negliguntur prae quantitate infinita altio-
ris ordinis. Facto enim þ:
S;, et designantibus a,b,c, ..., q
valores linitos , habebitur
∘∣⊰⋅∙⊣−∂↙⊰⋅⋅∙⋅≖−⊦∘≀⊰⊶−≖ −⊦⋅⋅⋅ ⊣−⊄∣⊰−⊦ .
∸−−
te». &quot;(a ∙⊸⊦bæ—l— ccc&quot; −⊦∙∙ .-l—qm&quot;&quot;' ∎∙−∣− r m'&quot;).
5.&quot; Si variabiles æ,y sunt inter se per certam quam-
dam relatidnem ita connexae, ut data v. g. a:
,
inde possit
valor ]determinari ,;
vocatur functio quantitatis se: ipsa
vero «: dicitur independens. Si relatio intern- et y expri-
mitur aequatione minime resoluta quoad functionem ]habi-
tam pro incognita ,
]appellatur functio implicita ; quod si
valor y detur expressus immediate per independentem :,
vel talis obtineatur per aequationis resolutionem ,
]dicitur
functio explicita. ln aeqnatione v,- g. ;&quot; -—,2ay −⊢ ⇑∙∅:o
est 7 functio implicita quantitatis variabilis z'; at facta re-
solutione
, evadet ]
functio explicita ipsius a:, duplicemque
habebit valorem, scilicet 7—:a:∶⊨ ⇂⋅∕⋅↕∙≖ −− m'. anctio-
nes explicitae quantitatis a: designari solent 1n hunc modum
,F (x), f(x),... ⋅∙
6.&quot;Dill'erentiale dx quantitatis x est incrementum
infinitesimum , quod ipsi :: adseribitnr :differentiale 'vero dy
functionis y:--f (x) est respondens incrementum f
(x dx)
—f
(a:)
, quod ob variatum se recipit in se functio illa: pro-
ponantur v. gr. inVenienda difi'erentialia functionum45
at +6,9 +0,24+ Cisin x + C , cos x+c ,
tang x + C, log x + C , a ' tc ,
ubi a et C sunt quantitates constantes.
Erit I. dy = [ alx + dx ) + ] - [ax + C ] = adx.
a
II.dy = [ f'da+ c ]- [* + c]atda X
adot
adx
x2 + xdx
.
III.dy = [ ( x + dx)* + C]-[x4 +C]=ax“-'dx + 29,
a'a- 1 )
24-2d.22
t . ax-' dx .
IV.dy = [ sin ( x + dx ) + c ]- [sinx + C ) = sin ( x + dx ) — sinx
2 cos 2xdx)sinh dz = 2cosx sind = cos xdx .
V.dy = [cos ( c + dº + C ]- [cos.FC ] = cos ( c - day -cosx =
2 sin - (2x +dx)sin __ (x -x -dx) = sin xdx.
VI.dy = [tang ( xtdx )+c] - [langat.C ]=
sin ( x + 2x)
cos(x + dx )
sinx cosxsin (xtdx)-sinxcos( x + dx ) sin ( x + dx - x )
cos2 cosx cos ( cdc )
cos2x
45
a'−⊦∁∙−⋮∙−∙⊹ C, æa-l- C,'sin a: ∙⋅⊢ C,cos æ-l-C,
tang æ-l—C, logæ-l— C, ar-l-C,
ubi a et C suntquantitates constantes.
Erit l. dy: [a( æ-l—dx) ——C ]— [ux—I»- C ]:adæ.
∥⋅↙∣↗↗⊣⋤⋮⋅−−⊦ (i]—[?&quot; C]— jd,— :—
∥∣∙↙∄∫⇋∶∐↕⊹≴≀↕≻⋅⊹∁⊐∙⋢∞⋅⊹∁∃≕∞≕∙∣↙≀↝⋍⊹↽∘↙↙⊑⋅∣≱↶∶⊄−≖∠≀↓⋅≖
&quot;I- ∙ ∙∼ ∙ :aæ&quot;-' dx.
IV. a];:[sin(æ-l-dæHCI-[sinæ-l-C] :sin(æ-l—dæ)— aina:

: 2cos—;..(Zæ-l-dæ)sin-—;. dæ: a cosa: sin-;— dx:cos ædæ .

V.dJ:[cos(æ-l—dæ)-l-C]- [cosa: −⊢∁↥ ∙−−− cos (æ—l—dæþcosæ:
2sin :(ZPFdrþin-i—(æ-x-dx):— sin ædæ.
Vl.dy::[tang (
.z—l-dæ ≻−∣−∁⋮∣ ∙
⇂⊏∄∐⊰⊅⇥∙∁⊐ —8lll(æ..ll-rlæ) cos(æ—-dæ)
aina: cosæsin(æ-l-dæ)-sinæcos(H—dx)— sin(.i—l—dx-æ)
cosa: cosæ cos( æ
⊹∠≀∙↧⋅ ) coszx ⇁⇁−∙↼46
dx
cos2 x
VII.dy= [log(x-tdx)+ c]-[logo +C ]= log ( + )
dit 15 ( 1 +4x)dx _d2log [2 + } (1- dot) +
23 (1-4 )(1-2dt)+... ]
det 103 ( 2 + + 43 + 234 + ...)
dxlog [ 2 , 718281828
dx
] ;
sumptisque logarithmis quoad basim 2 , 718281828
dx dy X
istiusmodi basis solet exprimi per e.
VIII.dy = [a ++dx + C ] - [ a * + C ] = a *+ x_qt = da? =
a* d log (a *) = a * d [ x log (a )] = a * log (a ) dx.
70. Quantitas constans C, quaecumque ea sit, non in
venitur in differentialibus: idemque proveniet differentiale
sive differentietur v. g. sin x + C, sive sin x.
dy 8º. In primo exemplo habemus a,
dz
cundo axe- ', in dy quarto dx dx
- in se
dy
a
in tertio dy dx
x2
46
da:
∙∙∙∎∙∙↼⇁∙−⇁
∞∘⇄∙⋍∙∙
∇∥∙ ↙≀↨↶−−−∏∘⊰≺⊿↾∶∙∔⋞≀∙↕≻−⊦∁∃⋅⊏∣∘∷∞⊣−∁↿⇌ log (
↿
.? −⊢↙∙⇣⋮⋮⋟⋮
dx —log (HE ↙↿−⋤⋅∶∙↙≀−≟−∅∣∘⊰∣∶≆⊣⊸≑−≺↿− ff):
⋮⇡↽≐≺↿−⋛≣≤ (fi-:): --]—
da: ↿ ↿
⊺⊅−∣∘⊰≺⊈⋅∙⊦−≆−⊣−≐−∙≡ 2..3
∢⊯∎⊦∙∙∙ '):
≦−↕∣∘∥⋣∙ 718281828 ...
];
sumptisque logaritbmis quoad basim 2, 718281828 ..., .
(II:—;:
istiusmodi basis solet exprimi per e.
Vlll'dy :[a&quot;dx—I—C ]—[ax ∙−⋅⋅⊢∁⋮∣∶∅≖↤≖− ar: daJr:

: a'd log (a'):axd [æ log (a)]:axlog (a) dt.

⋅70. Quantitas constans C,quaecumque ea sit, non in-
venitur iu differentialibus: idemqne proveniet dili'erentiale
sive dilferentietur v. g. sinæ ∙−∣⋅− C, sive siuæ.
80. In primo exemplo habemus ?: a, in se-
x
d d cundo a- . J
⊋−⋮⋮⋮ :— &amp;, 1n tertio 23:01: ', 111 quarto 71:47
in COST
in octavo
dy dy cost , in quinto sin x, in sexto dx dx
1
septimo di die= a * loga . Quisque videt
dy fore generatim novam functionem variabilis z : si ea
denotatur per f(x) , erit
2
dx
de = f( ), et dy = f ( z )dx .
Functio f '(x ) appellari solet derivata ex primitiva f( x) :
caeterum dx, seu differentiale variabilis independentis x,
spectatur quidem ut quantitas infinite parva ; sed simul con
stans atque arbitraria,
9º. Ex ivº, vº, et viº exemplo habemus
d sinx d sinx da dx :
dcosx
COS X V
sinx 1 - sinar
dcosx
77- cosa a '
dx = cosa x d tang x =
d tanga
sec2 x
dtang x
1 + langa x
Aequationes istae in hunc modum scribi possunt
dz dz
darc (sin = z ) = darc(cos = 2 ) =
V1 - Z
V 1-22
dz darc ( tang = 2 )
1 + z2
47
cosa: ,
in quinto ?; −∙−∙−∙ -—sinx, in sexto ⋛⋚∶≎∙⊂≐⊭−∙ in
septimo g

:

-.::— ,
in octavo :::-ï :axloga. Quisque videt
217— fore generatim novam functionem variabilis :: si ea

denotatur per f(æ) , erit
ngþ), et 47 :f(æ)dæ .
l
Functio f(æ) appellari solet derivata ex primitiva f(æ):
caeterum dx, seu differentiale variabilis independentis x,
spectatur quidem ut quantitas infinite parva; sed simul con-
stans atque arbitraria.
go, Ex &quot;o, vo, et vi&quot; exemplo habemus
dsinæ dsinæ dccsæ dar.... ∶−−−−−⇀⋅−⇁−⋅ :
.

: ⋅ cos 3&quot; l/ 1

-
sm'æ '&quot;&quot;:
&amp;
, dx:cos3xd tangæ:
deosæ
Vi-oos'x
dtangx ∙∙∙ dtangx
secaæ 1—i—tang3x
Aequationes istae in hunc modum scribi possunt
'
d
dare (sin— !.):sz ,darc(cos——z)-—- V
132 ,
-zz ∙ 2
darc(tang:z): T'↶−≀≘≖−−?'48
10 ° . Sicuti ex y = f( x) obtinuimus ( 8&quot; ) dy
f ( x )dx, sic ex hac obtinebimus ddy = f' ( x) dxdx
f '(x )dx?, ex qua rursus dddy = f &quot; (x )dx dx ' = f &quot; (x ) dr },
atque ita porro ; denotant fif &quot;, ... novas functiones
variabilis independentis x. Itaque si compendii causa e
xhibentur ddy, dddy, ... per dy, dy,.. , profluent
d
d’y = f '( x ) dx?,dy = f &quot; ( x ) dx3,
dy= f (x) ,
da²
d3y
= f'&quot; (r ) , ... : d.x3
assumpta v.gr.y = x ^, erunt f( x) = x ^ , f ( x ) = axa if '( ')
=a ( a - 1 ) 219-2, f ( x ) = a (a - 1 ) ( a - 2 ) x4-3, .
Differentialia dy , dºy , dy ,. . , itemque functiones deri
vatae f (x ), f ' (x) , f &quot; (x ), ... dicuntur primi, secundi, ter
tii , ... ordinis respectu functionis primitivae y = f (x ).
11 ° . Quemadmodum data functione possunt quaeri
ejus differentialia , ita vicissim dato differentiali quaeri po
test functio unde illud promanal. Sint F (x ), f (x ) ejusmo
di functiones variabilis x , ut exsistat F' ( x) =f( x) : quan
titas F ( x) + C vocatur integrale indefinitum differentia
lis f ( ) dx, designaturque praefigendo litteram ſipsi differen
tiali , ut scribatur
ſf(x) dx F ( x) + C ;
exprimit C quantitatem ( 7 &quot; ) constantem atque arbitrariam.
12° . Formula f ( x )dx ita sese aliquando exhibet,
ut statim appareat eam esse differentiale cujusdam da
tae functionis ; tunc vero in promptu est integrale: atque
hoc pacto habemus ( 6º . 9° )
f (a + 1)x*dx ******+. C,unde fredr =
xati
atito
48
100. Sicuti ex 7:f(x) obtinuimus (80) d] ∶∙∙−−⋅
f
(æ)d.r, sic ex bac obtinebimus ddj :
f
'(æ) dædæ :
f'(x)dæ', ex qua rursus dild]:f&quot;(æ)dx dæ' :f'&quot;(æ) dx3,
atque ita porro; denotant f,f &quot;,
... novas functiones
variabilis independentis æ. Itaque si compendii causa e-
xhibentur ddy, dddy, ... per dy, d37 ,. ., profluent
dïy
&amp;? d')&quot; :f'(x) da.&quot;,dfly :f&quot; (æ) das-3, ...,

: f'(æ),

113!
dæ3 :f'&quot;(.r),...:
∙∙⋅⋅∙⋅
assumpta v. gr.y:æ&quot;, erunt f
(æ):æ',f' (x):ax&quot;',f'(a-)

:: a(a — 1) x&quot;&quot;,f&quot; (x):a(a—1)(a—-2)x&quot;3, ....

Difаerentialia dy, diy, d3y,. . , itemque functiones derivatae f(x), f'(x) , f&quot;(æ) , ... dicuntur primi, secundi, tertii, ...ordinis respectu functionis primitivae y:f(x).
110. Quemadmodum data functione possunt quaeri
eius differentialia, ita vicissim dato differentiali quaeri po-
test functio unde illud promanat. SintF(x),f(æ) ejusmo-
di functiones variabilis x, ut exsistat F'(æ):f(x): quan-
titas F(x) −−∣− C vocatur integrale indefinitum differentia-
lis f
(.r) dar, designaturque praefigendo litteram ]ipsi differen-
tiali, ut scribatur
ff(æ)dæ:-—F(æ)—1-C;
exprimit C quantitatem (70) constantem atque arbitrariam.
120. Formula f(x)dx ita sese aliquando exhibet,
ut statim appareat eam esse dili'erentiale cujusdam da-
tae functionis; tunc vero in promptu est integi—ale: atque
hoc pacto habemus (60. 90)
a
&amp;
a-l-l C :: xtt-H f(a—1-1)æ dx:x ∙⋅∣− ,undefæ dx: ∉⊋∙∙⊦∙∙∙∓ ⊹∁⊒ï49
QCx
ſalog/a)d(c== q** + C, unde ſe*dx =clogiastc

;

dx
S
= arc ( sin = x ) +C ;
V 1 - 22
Sa
dx
1 + x2
= arc ( tang = x ) + C.
130. Interdum formula f (x )dz, de cujus integra
tione non constat , per quasdam substitationes transfor
matur in aliam , cujus integrale illico cognoscitur. Sic.
v . gr. positis ax = 2 , - = z ,assequimur a
dx dz 1
Si Salita
=
14a²x² arc ( tang == z) + C =
1
arc ( tang = ax ) + C ,
Sa
dix
22 ta
1
Sat
dz
a (1 + z2)
arc ( tang = 2 ) + c = a
-a arc tang * + c,
-Svador - Svado --Svet
( cos = ) + c
arc ( cos = z ) + c
= arc
fa&quot;log(a)d(cæ):a&quot; —]—C,undefa&quot; dx: -ac
dx
- ⇂∕↿∙−⋅⋥∎⊑ :arc (sin :x) −⊢∁≂
f 1112 :arc( tang:x )-l-C.
130. Interdum formula f(æ)dx, de cujus integra-
tione non constat, per quasdam substitutiones transfor-
matur in aliam, cuius integrale illico ougnoscitur. Sic.
.. æ . '. gl'. POSIUS nær-z.;— Zoasaequlmur
dæ ⇀∙∙− dz 1 — ∙−−
fl'l'a'æ' a(1-I-za) a
&quot;c (tang—z)-[-C—..
dx
xï-l-a
dz 1
faU-l—z') − a arc(tangzz)-I-C:
↿
—a.arc (tang :ax)-[- C,] ——
.—
—1-arc( tang :
−⋅⋮− ⋟⊹ C,
et
f
dx ]
adz ]
dz ∙∙∙
[fas-xa ∣∕ ∅≖∙∅≖≖≖ −⇀ ∣∕↿ -zz
arc(cos:z)-1-C :
arc(cos :?) ⊹∁∙50
140. In integrali indefinito ( 11 °) adhibeantur suc
cessive pro x peculiares valores xo, x n , ac dein ab F ( zn )
+ C subtrahatur F ( x ) +C ut , eliminata C , prodeat
F (xn) - F ( xo) : ejusmodi differentia vocatur integrale de
finitum differentialis f (x ) dx , sumptum videlicet ab x =
а &quot;
x
Xo
xh
ſ p(x)dx = F(wow )— F( xo ) .
Xo
Hinc v. gr.
a
dx
jederati
7T
a
o
Variato altero ex binis limitibus v. gr. x ny variabit
ipsum quoque integrale ; et adhibita x pro xmo erit
X
ſ f(x)dx= F ( x) — F ( xo ) :
Xo
habebitur videlicet integrale illud , quod incipit ab xo ,
quodque evanescit facto x = x,: et quoniam
aff(x)dx = d [F(x)-F(xs)] =dF( x) =f ( x) dx ;
X.
iccirco
X
S SP(x)dx = Sp«x ) dx + c .
X.
15 °. Sit arcus infinitesimus ABEH ( Fig. 18 ) , et
in eo chordae infinitesimae AB , BE , EH , quarum prima
50
140. In integrali indefinito (110) adhibeantur suc-
cessive pro x peculiares valores xo, x,, , ac dein ab F (x,)
−⋅∣− C subtrahatur F(xo)-I-C ut, eliminata C, prodeat
F(x,,)— F (x,): eiusmodi differentia vocatur integrale de-
finitum differentialis f(x)dx, sumptum videlicet ab x:
x&quot;
x, ad x:æ, ,designaturque per [f(x) dx, ut scribatur
æo
xn
]
f(æ)dx :P(æ.) — P(æ. )-
æo Hinc v. gr.
[ a
fa,-' dx: ..-.-..-—1 J'EL :
-E- a-l—1 ' xï-l-aa a.
0 0
Variato altero ex binis limitibus v. gr. x,, variabit
ipsum quoque integrale; et adhibita x pro x,, erit
.?
faa-w.r: ∌⇁≺∙↿∶≻−−∙≖⊸⇁≺∞∘≻≃⋍∙
æo
habebitur videlicet integrale illud, quod incipit ab x.,
,
quodque evanescit facto x: x,: et quoniam
&amp;?
df/(x)dæ:d[F(x)-F(æ.) ]:dF(æ) :f(æ) dx;
xo
iccirco
fff(-1')dx:ff( x
)
dxH—FC.
↿⋅⇂⋝∘⋅ x.,
Sit arcus iniinitesimus ABEH( Fig. 18 ), et
in eo chordae infinitesimae AB, BE, EH, quarum prima51
SUC:
ac tertia producantur donec concurrant in D. Quoniam an
guli DBE, DEB sunt infinitesimi, angulus quoque ODE
erit infinitesimus: designetur iste angulus per i , et fiant odest
e de
BD =
es
BE = c , DE b ;
habebimus
lur
62 =a +62 – 2ab cos ( 180° -1i ) = a + b2 +
2 ab cos i = a : + 62 + 2 ab- 2ab + 2 abcosi = (a + b )2
2 ab ( 1 — cosi ) =( a + b )2 – 4ab sin ’ şi ,
unde

: 1

4ab
sin _ i = 1
(a + b ) ( a + b )2
ariabi
et consequenter
[1 - ( +5)*] sinº in
= - = [ - (-3 ) ]su'_

: [&quot; - )*]*sist i -....

2 + b
ban
Differentia nimirum inter unitatem et rationem c ad
a + b consistet in terminis duntaxat infinitesimis , quorum
ordines excedunt omnes ordinem primum .
16º. Idipsum a fortiori dicendum de differentia
inter unitatem et rationem ipsius c ad subtensum arcum
BmE ; siquidem BmE &lt;a + b et &gt; c. Inde fit ut et ar
cus infinite parvus censealur aequalis respondenti chor
dae , et curva quaevis spectetur tamquam polygonum coa
lescens ex laterculis infinitesimis numero infinitis, et isto.
rum laterculorum prolongationes habeantur pro totidem
tangentibus apud varia curvae puncta.
rini
⋅ 500
(I.]
0an
ede-
lur
anali
bf&quot;
Lr;
(im!
51
ac tertia producantur donec concurrant in D. Quoniam an-
guli DBE, DEB sunt infinitesimi, angulus quoque ODE
erit infinitesimus: designetur iste angulus per i, et fiant
BD—fd, BE:c, DE:6;
habebimus
ea :a: ⊹∂≖ —2abcos(180'-'—-i) :03 ∙−⊦ 63 −∣−
Zabensiz—maa-i-ba -l-Zab—Zab-l-2abcosi :(cs-Fb):—
Zab,(1—- cosi):( a --[-b):
-
4absin* −≧−≀⋅∙
nnde
c*— 406 (a- -b)' ∙∙∎∙∙∙∙∙∶↿∙∙∙ . (a -l-b)3 sin ∙⋮−∎
a—ö a .

: ,

.
[1 (—r—b)]sm;-h et consequenter
c
⋍↿∙−−∶∙−∣∶↿ −≺∅≆≴≻≖∃ aina-Li—
a b a
∸⋇⋅∣∶↿ 3
−≺⋮−⋮−−⇣∙≑≻≏∃≏∘⋮∎≖∣⇩ ..;-i ∙−− ∙ ∙ ∙ ∙
DiEerentia nimirum inter unitatem et rationem c ad
a −⊦ &amp; consistet in terminis duntaxat inünitesimis, quorum
ordines excedunt omnes ordinem primum.
160. Idipsum a fortiori dicendum de differentia
inter unitatem et rationem ipsius (: ad subtensam arcum BmE ; siquidem BmE a −↿− 6 et 0. Inde Et ut et ar-
cus infinite parvus censeatur aequalis respondenti chor-
dae, et curva quaevis spectetur tamquam polygonum coa-
lescens ex laterculis infinitesimis numero infinitis, et isto-
rum laterculorum prolongationes habeantur pro totidem'
tangentibus apud varia curvae puncta.52
17º. Fac ut aequatio y f( x) pertineat ad cor
vam ABD ( Fig. 19 ) et sumptis coordinatis orthogonali
bus, sit abscissa OG = x, ordinata CB = y , infinitesimum
abscissae incrementum CC = dx : ducta per C' alia ordinata
C'B' , et per B lineola recta Bm parallela axi abscissarum
OX, erunt B'm = dy , Bm = CC = dx. Pone tangentem
BE occurrere abscissarum axi in E , normalem vero BH in H;
triangula rectangula et similia BEC , B'Bm , BCH dabunt
ydy

:

tang E
-
tang B'Bm dy, ce = ydx CH dx dy dx
CE dicitur subtangens, CH subnormalis.
18º. Ob auctam x area curvilinea BCa'a recipit
incrementum infinitesimum BB'C'C; est autem
BB'C'C =
dx
(rty + dy ) =
dxdy ydx +
2

&lt;math&gt;= ydx + f (x)dx =&lt;/math&gt;
ydr:
2
facta igitur Oa' = xo , erit
BCa'a- j^ydx = ${( )dx Xo Xo
Area BCa'a manifeste traduci polest ad rectangularem a
ream sub ejusmodi lateribus , quorum alterum sit differen
alterum vero ordinata quaedam ym media in
ter ordinatam aa' respondentem abscissae xo et ordina
tam BC respondentcm abscissae x : propterea
tia c Xo ,
X
ſ ydx = ( x - X . \ 'm , seu S f (x )dx = ( x - x . ) f ( xm ) .
X. Xo
Eadem area BCa'a spectari potest veluti summa ex infini
tis numero infinitesimis areolis rectangularibus
52
170. Fac ut aequatioy :
f
(et) pertineat ad cor--
vam ABD( Fig. 19) et sumptis coordinatis orthogonali—
bus, sit abscissa OG:x. ordinata CB: , infinitesimum

abscissae incrementum CC':dx :ducta per 0alia ordinata
C'B', et per B lineola rectaBm parallela axi abscissarum
OX, erunt B'm:dy
, Bm:CC':dx. Pone tangentem
BF. occurrere abscissarum axi in E, normalem vero BH in H;
triangula rectangula et similia BEC , B'Bm, BCH dabunt
J—— , tang E: tang B'Bm :
.. £,CF—Jjæ (31:731: '
]
.L'
CE dicitur subtangens, CH subnormalis.
180. Ob auctam x area curvilinea BCa'a recipit
incrementum infinitas-imum BB'C'C; est autem
⊞∍⋅∁∙∁−−−−↙⋚∁≺⊺ −⊢∫ ⊣−↙≀∫ )
∙−−∶
ydx −⊢ ∂⋅⋅↕−⋮↨−↗− :ydx-l— [figi-£ :ydx:
facta igitur Oa':x.,
, erit
x x
BCa'a:fydx :ff(x)dx.
xo xo
Area BCa'a manifeste traduci potest ad rectangularem a-
ream sub eiusmodi lateribus , quorum alterum sit differen-
tia x —-xo , alterum vero ordinata quaedamym media in-
ter ordinatam aa' respondentem abscissae an. et ordina-
tam BC respondentem abscissae x: propterea
x ⋅ x
fydx: (x -e-x., ly,&quot;, seuff(x)dx:(x—- x.,)f(x,,, ).
x., . xo
Eadem area BCa'a spectari potest veluti summa ex infini-
tis numero inlinitesimis areolis rectangularibus53
f ( x ) dx , f ( x +dx ) dx , COP
f ( xo +2dx ) dx f ( x — dx ) dx ;
nali
imum
Binala
.
sarum ubi nibil sunt aliud f (xo) , f( x + dx ), f (xo + 2dx), ...
nisi ordinatae respondentes abscissis xo , xo + dx , to +
2 dx ,
Quare
entem
in Hi;
bunt C
ſ f(x).lx = f(x )dx + f( xo + dx)dx +
Y :
Xo
fl xo + 2 dx )dc + .
+ f ( x -dx )dx.
recipé 19º. Ponatur arcus aB = s , ejusque incrementum
infinitesimum BB' = ds; quoniam BB'2 = Bm2 + B'ma, erit
2
ds = dx= + dy ,ideoque s= V dx=+dya =
X.
jäevitro
Xo
Tema
iffere
dia is
ordin
200. Circulus habens communia cum curva CC
( Fig. 18 ) duo proxima latcrcula v. gr. AB et BE, dici
tur osculator: sit O centrum istius circuli, BO ( r) ra
dius, OʻK et O'K' perpendicula ex O ducta in AB et BE ,
i angulus OBE , ds' et ds infinitesimi arcus laterculis AB
et BE subtensi, alter spectans ad circulum osculatorem , al
ler ad curyam CC' . Quadrilaterum KOʻKB praebet angu
lum KO'K ' = 180° — KBK' ; sed KBK' = 180°-OBE =
180° -1 ; igitur KOK' = , et consequenter ds' =
r( KOK' ) = ri' . Est autem ( 16 ° ) ds' = ds : propterea
infini
mali-
imum
linat:
arua
entem
Liuii; ↽
bum
⇟⇁∙∎↘⊰
..
recipi
rem ?
illerä
dia i?
orzlïm'
inüw'
53
f(xo)dx,f(xo-I-dx)dx,
f(xo—l—2dx)dx,. . ..f(x—dx)dx;
ubi nihil sunt. aliud f
(..-.,)
,
f(xo—l-dx), f(æQ—l-2dæ). .. . nisi ordinatae respondentes abscissis xo , xo -l-.dx , xo −∣−
de, . .. . Quare
æ
J. f(x)dx :f(xo)dx −⊢∙∣≼ xo-l-dx )dx ∙−∣−
∙↾≀⋅⋅∘
f(xo-l-2dx)dx −⊦ ∙ ∙
..
-I-f(æ-dx)dx.
190. Ponatur arcus aB: :. ejusque incrementum
iniinitesimum BB':ds; quoniam BB'a :Bma—l-B'ma ∙ erit
x
d:&quot;:dxï-l-dyïddeoque s:f V de-l-dy ∶−∙⋅−∙
æo
x
⋅∣∙↙≢∙↿∶⇂∕↿∙∙⊢∣⇃≖≼⋅≖⋅⋟∙
. xn ⋮⋅⋅
200. Circulus habens communia cum curva CG'
( Fig. 18 ) duo proxima latercula v. gr. AB et BE, dici-
tur osculator: sit 0' centrum istius circuli, BO' (:) ra-
dius, O'K et O'K' perpendicula ex 0' ducta in AB et BE,

: &quot; angulus OBE, 'ds' etïds-iniinitesimi arcus later-culis AB

et BE subtensi, alter spectans ad circulum osculatorem, al-
ter ad curvam CC'. Quadrilaterum KO'K'B praebet angu-
lum KO'K':1800 −− KBK'; sed KBK':1800—OBE:
180o — i' ; igitur KOK':i'
,
et consequenter ds&quot;:
r( KOK') :ri'. Est autem ( 16o ) ds':ds: propterea54
ds
21.• Curva CC' sit plana ; exhibeaturque per y =
f (x ), sumptis abscissis x in RX ( Fig. 20) . Erit i = Q
a = - (-a) = - dx , ideoque ( 170)
ds
ds
d x darc ( tang dy -
dx
)
Jamvero (90 )
dy
darc ( tang )
a dy
dr
dy²
dx²
df ( 30)
1 + f ? (x )
f (x ) dx ;
1 + f ? ( x )
dx
igitur
[1 + F2(x) ] }
f &quot; ( 3)
22.• Si ordinata y in curva y =f ( x) fit alicubi
maxima vel minima, exhibeaturque respondens abscissa per
Xn , quisque videt angulum interceptum tangente geometrica
et positivo abscissarum axe fore acutum vel obtusum pro
ut punctum contactus habuerit abscissam x &lt; vel &gt; xn in
casu maximi , &gt; vel &lt; x , in casu minimi , fore autem in
utroque casu = o ubi punctum contactus habuerit abscissam
x = x , Inferimus illud ( 8º. 170) : functio f (xn) est maxi
ma quotiescumque
f ( x) &lt; o quoad x = x + w ( denotat a quantitatem
infinite parvam &gt;0 ) , et f ( 2) &gt; o quoad x = xn - W ;
est minima quotiescumque
54
21 ∙∘ Curva CC' sit plana ;exhibeaturque per :7
f
(x), sumptis abscissis x
in RX (Fig. 20). Erit :&quot; a— a':
—(a'- a):
— dx , ideoque (170)
ds- ds
↗−− dx— dy
darc(tang:ä-;
Iamvero (90)
- si! darc(tang:i-'r .— dx ∙− ↙≀∣↬≺∙↿∶∟∙∙− f (adde;
dx −−↿ ,dJ' 1-t-f'(æ) l*f'ix)
dx'
igitur
3
[1 ⊣∙↾↔≖ (æ) ]
∶⊸∙ f&quot; r— (æ)
22.0 Si ordinata ;- in curva ;-:[(x) (it alicubi
maxima vel minima, exhibeaturque respondens abscissa per
x,, , quisque videt angulum interceptum tangente geometrica
et positivo abscissarum axe fore acutum vel obtusum pro-
ut punctum contactus habuerit abscissam x(vel )x,, in
casu maximi ,
)vel (x,, in casu minimi , fore autem in
utroque casu:
0 ubi punctum contactus habuerit abscissum
x:x,. Inferimus illud ( 80. 170) :functio f
(x,) est maxi-
ma quotiescumque
f
(x) (o quoad x :. x,, ↼⊢ co (denotat a quantitatem
infinite parvam )o )
.
et f' (x) )o quoad x :x. — a) ;
est minima quotiescumque55
f (x ) &lt; o quoad x = x, — W, et f ( x ) &gt; o quoad x =
X'n tw ;
valores X c.quibus respondet maxima vel minima f( xr ), quae
rendi sunt inter radices aequationis
p' ( x)
In
Si f ( x) maneret aut constanter negativa , aut constan
ter positiva, dum x versatur in viciniis x m , certe f ( x ) neque
maxima esset , neque minima .
Ad haec : quoad casum maximi, crescente x in viciniis
decrescit f' ( oc) , decrescente x decrescit f ( x) ; ideoque
df ( x)
&lt; 0 , seu f&quot; ( 30 ) &lt;0 . Quoad casum vero minimi ,
dx
crescente x crescit f (x) , decrescente x decrescit f ' (x ), et
af' ( x)
consequenter &gt; o seu f ( x) &gt;o .
23. Functiones plurium variabilium independen
tium x , 2 , u , ... designantur in hunc modum
dx
F ( x, 2, Ú, ... ) _f ( x, 2, U, ... ) , ...
Ponatur j = f (x ,2 , 9-9.) : si quaevis una ex quan
titatibus x, 2, u, spectetur uti variabilis et habeantur cae
terae pro constantibus , poterunt differentialia functionis u
eodem manifeste modo determinari ac differentialia functio
num quae ab unica pendent variabili. Ejusmodi differentialia
dicuntur partialia , ipsaque sic exhiberi queunt ,
ut
det , draf . d. , dal , ...
denotent differentialia functionis fe , primi , secundi ... ordi
nis quoad x , quoad 2 , ... Ad partiales functiones derivatas
quod pertinet , eae poterunt sic exprimi , ut per
55
f(x)(oquoad x:x,,—-o),etf (x))o quoad x:
a'.—FG);
valores x,,,quibus respondet maxima vel minima f
(x,,)
, quae-
rendi sunt inter radices aequationis
,'(æ)::00
Si f
(x) maneret aut constanter negativa , aut constan-
ter positiva,-dum x versatur in viciniis xn, certe f
(x,) neque
maxima esset
, neque minima.
Ad haec :quoad casum maximi, crescente x in viciniis
x,, decrescit ]&quot; (x) ,decrescente x decrescit f' (x); ideoque
df (x)
dx 0
.- seu f&quot; (x) (o. Quoad casum vero minimi ,
crescente x crescit f
(x) , decrescente x decrescit f' (x) ,
et
consequenter (IS .(ræ) )o seu f&quot; (x) )o.
23.&quot; Functiones plurium variabilium independen-
tium x ,z , u, designantur in hunc modum
F (x, :, ti, ...) ,f( x, :, u, ...
)
,
Ponatur p.:
f
(x, :, a.,.,.) :si quaevis una ex quan-
titatibus x, z,u. spectetur uti variabilis et habeantur cae-
terae pro constantibus
, poterunt differentialia functionis p. ↴
eodem manifeste modo determinari ac differentialia functio-
num quae ab unicapendent variabili. Eiusmodi dill'ereutialia
dicuntur partialia , ipsaque sic exhiberi queunt , ut
dxld-1 dxaPQ'&quot; ∂∷⊬∙∠↨≖≖⊬∙∙∙∙ 
denotent differentialia functionis 9. ,primi , secundi ordi-
nis quoad x , quoad :, Ad partiales functiones derivatas
quod pertinet , eae poterunt sic exprimi ,
ut per56
doll darf
dx dx2
dou , dazle
dedz dza
vel
per
fx(X , Z, Up... ) , f&quot; , (3 , 2, U, ... ) , . f : (3 , 2, U, ... ) ,
fo( %, 2 , Wo...) , ...
designentur functiones , primi , secundi ... ordinis derivatae
ex M = f ( x , % , U. ... ) quoad x , quoad 2 , ... Plerumque
tamen in his derivatis functionibus exprimendis detrahuntor ,
compendii causa , litterae d signa x , % , U 7 .** , et pro
dal d , ² l
dx dx2 d,I
d², M
dz dz
adhibentur
du del i
dx dx2
du dele
dz dz ?
9
24º . Totale functionis pe differentiale due ( quum
nempe x
spectantur omnes ut simul variabiles )
eruitur ex partialibus dx f , d , f , dul , ... ; sunt enim
% , U ,
f ( x + dx, 2, 1, ... ) - f ( x , ,U, ...) = fx ( x ,2 ,4, ... ) dx,
f ( x + dx, atdz, u, ... ) -f( x + dx, 2, U, ... ) =
f: ( x + dx, 2, u, ...) dz = f : ( x, z, u, ... ) dz,
f ( x + dx, atdz, utdu, ... ) — ( x + dx, atdz, u , ...) —
f'u ( x + dx, z + dz, il ... ) du = f ( x, 2, u, ... ) du, etc... ,
)
1
.—
56
≀≀≖≀∸ −−↙≀⇄↕≴∸ .⋅≤≀−⊦∸
−∙∙ −−−⊓≀⇄≖≴∸ dx dx&quot; , dz dza '
vel per
fx(x, :, uh&quot;) ,
f&quot;: (x, :, u, a&quot;) , ∙∙∙ f, (x' z' u, a.) ∙
f',(x, :, u, ...)
,
designentur functiones , primi , secundi .. ordinis derivatae
ex &quot;. f
(æ.:, u.
...
) quoad x, quoad :∙ ∙∙∙ Plerumque
tamen in his derivatis functionibus exprimendis detrahantur,
compendii causa ,
litterae d signa :, a, «,... ,
et pro
de- dx'P- (!sz (I,,[L da:
∙ m ⋅⋅⋅⊤ ⋅−∂⋅⊒⋅−
adhibentur
≴≀−⋅≖∸− ↙≀≖⋅⊀↓ de dw
dæ'dx' ,' de, dza
240. Totale functionis p. dilferentiale dp. (qumn
nempe x , z , n , spectantur omnes ut simul variabiles )
eruitur ex partialibus d, (1.
,d, pt
, d,, p. , ; sunt enim
fl xhi—dx, 39 ut ...) ↼f( æ, :, u, ...):
f, (æ, :, ., ...) dx,
f(æ-l—dæ, t—l—rlz, u,...) — f(x-[r-dx, :, n, ...):
f: (xä-dx, :, &quot;, mida: f, (x, s, u, ...) dz,
f( æ-I- dx, z-l—dz, u-l-du, ...) ∙− ≼∙↧∙⋅∙∣−↙∣∙↧⋅∙ z—i-dz, u, ...) ::
f,&quot; (x-i—dæi z-l-dzo nus) du:f,, (æ, Z, &quot;, ,,.) du, etc-0-
'57
quarum summa praebet
p ( x + dx, atdz, utdu , ...) — f ( x, z , l , ... ) =
fr ( x, 2, U, ...)dx + f : (x , 2, u, ...)dz + f'u ( x,
Z, U, ... ) dut ... , seu dų = d .; + d ,l + d.le + ...
25.• Potest etiam functio pe differentiari successi
ve quoad binas, lernas , ... variabiles v . gr. quoad x, z, quoad
X , 2, u ; etc. ... Id genus partialia secundi , tertii , ... ordinis
differentialia designari queunt per
d, dx M , d , d , dal , ...
sive autem functio u prius differentietur v . gr. quoad x
deinde quoad z , sive prius quoad z , deinde quoad x ,
paallulum attendenti patebit idem in utroque casu pro
venturum differentiale .
26. Detur nunc differentialis aequatio primi ordinis
dy - cydx f ( x ) dx ;
facta y = zu, et adhibita substitutione, emerget zdu + ud:
czudx = f ( x) dx . Pone udz – czudr = 0 ; habebis
dz
= cdx , log ( x ) = cx = cx log ( e) = log ( eⓇx ) ; unde 7
z &gt; eºx :
in ea qua sumus hypothesi
zdu = f(x) dx ;
igilur du = f ( x ) dx
f (x) dx , u Sf (x)dx + G ; et 7 es
ex
1
5
quarum summa praebet
f(x-f—dx, z-l—dz, (kl—du,...) —f(x,'z, u, ...):
f: (æs 31 ut ⋅∙ ')dæ—I—fg (æ, :, II,. ..)dz—l—f'u (æ,
.z, u, .,.) du-l—n., seu dy.— −∙∙ d,.p. :i- dyp. ∙−⊦ d,); ∙∣−∙∙∙
25. ∘ Potest etiam functio p. diil'erentiari successi-
vequoad binas, ternas... .variabiles v. gr. quoad x,z, quoad
æ, :, u; etc.
... Id genus partialia secundi ,
tertii,... ordinis
diii'erentialia designari queunt per
ds dxp'adudadxp-vmi
sive autem functio p. prius differentietur 11. gr. quoad ac
deinde quoad :, sive prius quoad z, deinde quoad x ,
paullulum attendenti patebit idem in utroque casu p1o-
venturum differentiale.
26.&quot; Detur nunc differentialis aequatio primi ordinis
,dy— cydx :f(x) dx;
facta ]: zu, et adhibita substitutione, emerget zdu −⋅⊢ ud:
— czudx :
f
(x) dx
. Pone udz —. czud-r :o ; habebis
dz
Z
∙−−− ∖∙∘⊄≀⋅⋍∙⋅ ,
log (z):cx:cx log (e):
log (e&quot;); unde
∙−−− cx
, z....e :
«
in ea qua sumus hypothesi
zdu :f(x) dx; ∙ ∙
'
igitur du:
,
(x) dx *fbl'c) dx :
&quot;:M—i—C; et 2 0 .: et.:
5 d58
consequenter
y = eriſ f x)dx = C ] :
integratio videlicet dalae aequationis differentialis traducitur
ad integrationem functionis f (x ) dx Porro absoluta aequa er
tionum differentialium integratio eo redit , ut quae relatio
inter quantitatem et quantitatem per eas exprimitur , aequi
valenter exprimatur per aequationes differentialibus liberalas.
27 .. Si dalur differentialis aequatio secundi or
dinis
day dy ta dx +
0 ,
dxt
by:
designantibus k et k' radices aequationis 32 taz +b =0 ,
traducelur illius integratio ad integrationem binarum pri
mi ordinis
dy ' dy - ky ' = 0 ,
dx dx
siquidem , eliminata y' , prodibit
- ky = y ' ;
a
dy
-ky)
dx dy
dx – k G - hy) == 0 ;
quae , ob k tok = -a et kk' = b , recidit in datam. Jam
vero ( 260 )
dx y ' = Cetry = e ** C
elix

:

ergo
y = ek's
es [ foe-tyde +c ] - [ * +c ]=
Ceks + C'ek's .
58
consequenter
yzccxiffiæidæzcl: Bex
integratio videlicet datae aequationis differen'tialis traducitur
f
(x) dx ad integrationem functionis
ac:
. Porro absoluta aequa-
tionum differentialium integratio: eo redit ,
ut quae relatio
inter quantitatem et quantitatem per eas exprimitur , aequi-
valenter exprimatur per aequationes differentialibus liberatas.
27.0 Si datur differentialis aequatio secundi or-
diuis
?;: 437 dr −⊦ &quot;ï; −⊢ b,! −−∶ ∘⋅
designantibus I: et k' radices aequationis z' −⊸⊢ az —]-b:0.
traducetur illius integratio ad integrationem binarum pri-
mi ordinis ⋅
alt&quot; ∙⋅− df
↙↙−↜↕∶∎∎−∎↗⋮∫−−∘∣∠≀↜↿∶−↻ ⋅⋅⋅⋅∙∙⋅−−−−−↗ '
siqnidem , eliminata y'
, prodibit
d d ∠−− '
(dx kf) A(g—F):0.
d.; ⋅ dx ] '
quae ∙Ob k −⋅⊢ ∣⊏∎:
— a et kk':b, recidit in datam. Jam-
vero (260)
.)&quot;: Ce&quot;'.y:e*&quot;[ ∫−⋅≤∎⇂−⊺∶⋮⇆∙⊹∁∙ ]

:

ek'x
ergo
∙∙∙ ': ∙∙ ∙ r ∙ rr Ceu—H).:
..7— e*. [Ca,/460 &amp;) dx—l—C] ∙−−∶ e* k—k. *C]:
- Ce&quot; ∙⋅∣−∁∎ e*&quot; .59
28.- Si daretur d²y dxata
dr. + by = f(x),tra
duceretur integratio ad integrationem binarum dy'
-ky'
da
P(z) Tipo - Ky = y'; sicque prodirent ( 260)
[Sl + c]
e** [ S ,* + c ]
y' = etxe
k et k 'sunt , ut supra, radices aequationis z2 taz + b = 0.
29.• Resumentes functionem f ( x ) , ponamus
f(x) = a, tax taqx? +R3 2 :3 + 04x4 +

:

exsurgent
f ( x) = a + 2a , x + 3az x2 + 404 x3 + ... , f&quot; ( x)
2a + 3.2a3 x + 4.394 va t ... if&quot; ( 0) = 3.2a3 +
4.3.204 x +
Facto x = 0 , emergent ao f (o) , a, =: f ( 0) , a,
i f' (o) , az =-3f&quot; (o), etc.... Hinc
etc...
f(x) = f(0)+xf ( 0) + 1&quot; (0) + &quot;(o) + ...
Sint v. gr. f (x) = e*. f (x) sinx , f (x) = cosx : quoad
primam f (o ) = 1, f (0 ) = 1 , f ' (o) = 1,8 &quot; (0 ) = 1 , etc...;
quoad secundam , f (o) = 0 , f ( 0) = 1 , p (o ) = 0 ,
fr (0 ) • , 1 , f (0) = 0, f ( 0) = 1 , etc...; quoad ter
59
. dfy dy 28.0 St daretur . (: d −⊦∙ 6]:
f
(x) ,.tra-
dxa x
duceretur integratio ad integrationem binarum Si.-..;. ':
dx '7
f
(a:), £
—)(]:
!' &lt;nowiki&gt;; sicque prodirent (260) www-rc]&lt;/nowiki&gt;
730, reli]dx
11 et k' sunt
,
ut supra ,
radices aequationis z' -l—az—-I—-b :o.
29.&quot; Resumentes functionem f
(x) , ponamus
f(x):ao—I&quot;alx −⊦∁≖∙↕≖∙−∣⋅−↷∍ ∷∙⋅∍⊣−∦∣∙∙≂↙∣−⊦∙∙∙⇋
exsurgent
f(x):a, -l-Za,x-l-3a3x3 404 ∞∍−⊢∙∙⋅ ,f&quot; (x):
2a3-1l-3. Zaax-i— 4. 3a(,x2 -[-...,f&quot;(x) :3.2a3—[—
4. 3. 244x—l-..,
,etc...
Facto x:a, emergent a,:
f(o) , a,:
f
(a) , a,:

;

f. (0) , 03 3— f&quot; (0) ' :.
etc-00. Hinc 3 '
f(x): f(0)-l-xf(0)-i-—-f'(0)'l&quot;——f (o)-b&quot;-
Sint v. gr. f(x):ex.f(x) :sinx ,f
(x):
coax :quoad
primam f(o):1, f
(0):1, f' 'o):1,f&quot; (o):1, etc...;
quoad secundam , f(o):0, f(o) :1 , f&quot; (a):0 ,
f&quot; (Ol—&quot;'
-— ∙−− 1, f' (0):0, f' (0):1, etc...; quoad ter-60
23
tiam , f (o ) = 1 , f ( 0) = 0.8&quot; ( ) 1,8 &quot; ( 0 ) = 0 ,
f (o ) = 1 , ' (o) = 0 , p (0 ) 1 , etc... ; ideoque
x2 24 3 e* : 1 tox +*+ +
sin u = r
2.3.4 2.3
x2 8: 4
cos = 1
2.3.4.5 2
+
2.3.4 2.3.4.5.6
1+
ar5
x6
1.5.0 + ...
30.• Adhibita xV - 1 pro x in istarum prima,
emerget
= 1
x2
e **vi .x4
+
2.3.4
Xc6
2 2.3.4.5.6 +
r3 xc5
+
2.3 2.3.4.5 -.)v = 7.
ܪ
unde , ob secundam ac tertiam ,
e #xVST = cos x + V1 sio x .

28. Fac nunc ut punctum materiale vi qualibet continua
sollicitetur ad motum rectilineum: sit »» velocitas puncti
in fine temporis t,,.s spatium percursum , et ds spatiolum
percurrendam subsequente tempusculo dt. Perinde spectari
poterit ds ac si motu uniformi conficeretur , sola nimirum
velocitate praeconcepta 1); siquidem nova velocitas dv, quae
labente d:accedit materiali puncto, utpote infinitesima. ne-
gligenda.Hi11c (1 )
s
[
∥
Motus rectilineus puncti materialis iugiter sollicitati vi
constanter eadem, dicitur uniformiter varius. Per ep desi-61
/
gnetur velocitas, quam vis constanter eadem gignit intra
tempus 1 , erit qe velocitas ( 6 ) genita intra tempus t :
propterea denotante vo velocitatem initialem , qua nimirum
donatur materiale punctum quum t = 0, existet
v =v, +9
ds Hinc dt votoe : fac ut tempori 1 = o respondeat
So ; habebis
s -8 = v. i + 902 ?

;

2
1
et eliminato t ,
v2— v.2 = 29 / s - s . ) :
positis v ,30,
0, erunt
V = pt , s =
- Det
, v?= 205 ,
o dicitur vis acceleratrix : el designante m
massam puno
cti materialis, m q appellatur vis motrix : insuper spatium s in
aequatione ultima vocatur allitudo debila velocitati v.
Ad motum rectilineum utcumque varium quod spe
ctat , nomine vis , acceleratricis apud terminum spatii per
carsi s nihil aliud intelligitur nisi velocitas q , quam gi.
gneret vis conversa in constantem, constantique energia
qua inibi pollet , agens loto tempore 1. lamvero exhibet
do numerum tempusculorum , ex quibus coalescit tempus
1 ; ergo velocitas illa exprimetar per dv; nimirum
1
61
gnetur velocitas, quam vis constanter eadem gignit intra
tempus 1, erit got velocitas (6) genita intra tempus ::
propterea denotante 'Uo velocitatem initialem, qua nimirum
donatur materiale punctum quum :

: o, existet

v:v, ⊣∙− cp :.
Hincd ∙−∙−:v.,-l—got: fac ut tempori :

: o respondeat

,,- s,;habebis
(2
⋅⇟−⋅⋅≖∘∶∶
'&quot;o t&quot;l&quot; 92&quot;;
et eliminato t,
vï—vo*:2?( s—s, ):
positis v,: o ,r,: 0, erunt
v:g0t, :: gt: ,
v': 291,
q; dicitur vis acceleratrix: et designante m massam pun-
cti materialis, m ? appellatur vis motrix: insuper spatium .: in
aeqnatione ultima vocatur altitudo debita velocitati 9.
Ad motum rectilineum utcumque varium quod spe-
ctat
, nomine visacceleratricis apud terminum spatii per-
cursi :nihil aliud intelligitur nisi velocitas ga, quam gi-
gneret vis conversa in constantem, constantique Aenergia ,
qua inibi pollet, agens toto tempore 1. Iamvero exhibet
−↿−∙ numerum tempuscu'lorum, ex'quibus coalescit tempus
dt
1 .: ergo velocitas illa exprimatur per Tit-dv; nimirum62
dv
dt .
et quia dy
d's d
dt ; idcirco erit quoque
dès
d12
habetur pro variabili atque independente quantitate.
29. Fac v. gr. ut materiale punctum sollicitetur versus
datum centrum vi acceleratrice, quae distantiae z' ab eo cen
tro exsistat proportionalis , ut , denotante C ' quantitatem
constantem , habeamus q =C'z' ; sit z, initialis distantia ,
ibique vo =0 , t =0 ; sit insuper v ' velocitas in distantia
z' : erit ( 28 )
v = d (20-3')
dc
dz
dt
du'
ideoque C'z' =
dt
v'dz'
dzi
Hinc 19. Cʻz'dz' = -v'dv'; ex cujus integratione prodit
C- W'2
C'z'2 =C -2'2 , z = ve C'
facto z' =0, erit v' velocitas punci materialisió centro virium ;
exprimit igitur C hujusce velocitatis quadratum : quod si fiat
z' =2 . , erit ex hypothesi v = v = o, ideoque
VT= 2.VC ;
velocitas nempe puncti materialis in centro virium est ut ipsa
initialis distancia zo.
62
0:32- ' dt -'
et quia xlv::! g:- ; idcirco erit quoque
d3s
(:):
d::
'
habetur :pro variabili atque independente quantitate.

29. Fac v. gr. ut materiale punctum sollicitetur versus
datum centrum vi acceleratrice, quae distantiae z' ab eo cen-
tro exsistat proportionalis , ut, denotante C&quot; quantitatem
constantem , habeamus q) :C'z'; sit zoïinitialis distantia ,
ibique v.:o, t:o; sit insuper v' velocitas in distantia
z' : erit ( 28 )
d(zo-z')-——dz' .d ∙
∙−− dp'— v'dz' d: d:
&quot;
eoque c.. d. dz' '
I,.....
Hinc ↿∘∙ C'z'dz':
—- v'dv'; ex cuius integratione prodit
C— 'v'ï cause—w.r: ⇂∕−∁∼⊤−⋮
facto z':o,erit v' velocitas puncti materialisin centro virium;
exprimit igitur(] hujusce velocitatis quadratum: quod si fiat
z':z,, erit ex hypothesi v':v,;—..-o, ideoque
⇂∕⋜⋮∶−−⊸−≖∘⇂∕−∁−⋮≂
velocitas nempe puncti materialis in centro virium est ut ipsa
initialis distantia z..63
2.º du
1 Tc di=
C'zi v CV C -via VC
Vic С
suinptisque integralibus ,
i = C &quot; +
ve are (sin = vo ):
v = o quando i = 0 , proinde
Vc are ( sin = o), exquav = VC sinero.
3º. Cum in centro virium sit v = VC, erit ibi
1 = sint y C , et consequenter t = Inferimus pun
n
2V0
a
1
ctum materiale eodem semper tempore quacumque 2VC
distantia z . perventurum ad centrum illud .
4º. Si materiale punctum movetur vi accelera
trice, quae distantiae a dato centro sit proportionalis , sic
absque formularum subsidio polest ostendi eodem semper
tempore punctum ipsum peryenturum ad centrum illud :
concipiantur duo puncta, quorum primum triplo magis i
nitio molus distet a centro quam secundum : quoniam ex
hypothesi vis est proportionalis distantiae a centro, erit vis
primi triplo major quam secundi , ideoque triplam velo
citatem primo tempuscalo illud acquiret, et triplum spa
lium percurret; quare etiam tripla ibidem residua erit di
stantia. Igitur et secundo tempusculo triplam velocitatem
novam acquiret, et triplum spatiolum tum praecedente, imm
⊖⊰∆
2.-
−≀≀∙↗⋅ ∙∙∙ dv' '
dv' 1 C'z' yel/CT?&quot;— ⇂∕⋅∁⋅ ⇂∕↿−−−∙∙−−∙∽⋅∙∑−⋅ :
sumptisque integralibus ,
1
(
. v' )
are sm −−− ;
C' y'C
v':o quando :: o , proinde

: z ∁∙∙−⊢ ⇂∕

!
(z.—1.-.— arc (
sin: v
), ex qua ≸↗⋅∶∶⇂∕ ⇂∕∁ ⇂∕∁ sint;/CZ
30. Cum in centro virium sit v': l/C,erit ibi
'io
. n ↿∶∶ . sunl/C, et consequenter :: ï— . Infenmus pun-
ctum materiale eodem semper tempore a quacumque 21/ C
distantia za perventurum ad centrum illud.
40. Si materiale punctum movetur vi accelera-
trice, quae distantiae a dato centro sit propmtionalis, sic
absque formula1um subsidio potest ostendi eodem semper
tempore punctum ipsum perventurum ad centrum illud:
concipiantur duo puncta, quorum primum triplo magis i-
nitio motus distet a centro quam secundum: quoniam ex
hypothesi vis est proportionalis distantiae a centro, erit vis
primi triplo maior quam secundi, ideoque triplam velo-
citatem primo tempusculo illud nequiret, et triplum spa-
tium percurret; quare etiam tripla ibidem residua erit di-
stantia. lgitur et secundo tempusculo triplam velocitatem
novam acquiret, et triplum spatiolum tum praecedente, tam64
nova vi et velocitate percurret: unde consequitur ut tripla
pariter sit lota velocitas jam acquisita , triplum totum spa
tium percursum, tripla distantia residua. Propterea et no
vo tempusculo tripla erit nova velocitas acquisita , tri
plum spatium novum percursum , tripla nova distantia; atque
ita porro. Patet igitur post tempus quodvis distantiam
primi fore triplam distantiae secundi, ac proinde imminu
ta in infinitum ac demum evanescente hujus secundi di
stantia, illius quoque primi distantiam in infinitum immi
nui ac simul evanescere: haud poterit ergo secundum pun.
clum ad centrum pervenire, quin simul cum secundo ipso
primum punctum perveniat. Hoc tantummodo discrimen e
rit, quod primum eo deveniet velocitate tripla secundi ; ex
quo manifeste consequitur , quod si primum illud punctum
ex centro cum illa tripla velocitate projicitur , debebit ad
triplam distantiam pervenire; nam vis in recessu velocita
tem codem ordine extinguit , quo generat in accesso. Por
ro quod diximus de ratione tripla , patet generatim conve
nire rationi cuicumque ; nimirum in quacumque propor
tione fuerit distantia prini punci major quam secundi ,
eodem tempore semper ambo ad centrum devenient cum
velocitalibus , quae distantiis initio habitis sint proportio
nales; et si inde discedant cum velocitatibus quibuscumque,
pervenient eodem pariter tempore ad distantias ipsis ve
locitatibus proportionales.
5º. Dicatur tempus quo materiale punctum it ac
redit uude primo discessit; erit ( 3º. )
471 276 271 0
276 VC
Quare ( 1º, 2º. )
2750 C ,
6
0 2751
0
G
220
210
VO TT
z = VC
COS 277
64
nova vi et velocitate percurret: tinde consequitur ut tripla
pariter sit tota velocitas iam acquisita, triplum totum spa-
tium percursum, tripla distantia, residua. Propterea et no-
vo tempusculo tripla erit nova velocitas acquisita, tri-
plum spatium novum percursum, tripla nova distantia; atque
ita porro. Patet igitur post tempus quodvis distantiam
primi fore triplam distantiae secundi, ac proinde imminu-
ta in infinitum ac demum evanescente huius secundi di-
stantia, illius quoque primi distantiam in infinitum immi-
nui ac simul evanescere: haud poterit ergo secundum pun-
ctum ad centrum pervenire, quin simul cum secundo ipso
primum punctum perveniat. Hoc tantummodo discrimen e-
rit, quod primum eo deveniet velocitate tripla secundi; ex
quo manifeste consequitur, quod si primum illud punctum
ex centro cum illa tripla velocitate projicitur, debebit ad
triplam distantiam pervenire; nam vis in recessu velocita-
tem eodem ordine extinguit , quo generat in accessu. Por-
ro quod diximus de ratione tripla, 'patet generatim conve-
nire rationi cuicumque; nimirum in quacumque propor-
tione fuerit distantia prinii puncti maior quam secundi,
eodem tempore semper ambo ad centrum devenient cum
velocitatibus, quae distantiis initio habitis sint proportio-
nales; et si inde discedant cum velocitatibus quibuscumque,
pervenient eodem pariter tempore ad distantias ipsis ve-
locitatibus proportionales.
50. Dicatur 9 tempus quo materiale punctum it ac
redit uude primo discessit; erit (30.)

:

∡⊺≖∙∙∙∙∙⊸ 211 ,—21t
9
⊋⇂↗⇠∁⋮↼−⇀∣−∕−⋐⇀∶⋅−∙⋅⇂∕∁−⇀⊺⋅∙ Quare (10. 20.)
∙∙∙⊇⇂∕∁ 9
220 ∙∙∙⊓∙ ⇂∕∐⋅ ∶∶↼⋤⋮−⇂∕∁∙≀≀↗⋅∶⇂∕∁⊱⋮∥ −−−−⊖−⋅ ⊋⋯⋅ ⋅− 9 271! z': l/C -—-co −−−−∙
271 s 965

=== De verticali gravium descensu atque ascensu. ===
[[30|30]]. Si gravitas aequaliter semper ad sensum corpora decidentia sollicitare intelligitur, motus erit uniformiter varius (28): positis igitur &lt;math&gt;v_0=0,s_0=0&lt;/math&gt;, et denotante &lt;math&gt;g&lt;/math&gt; vim acceleratricem ex gravitate, in ea qua sumas hypothesi determinabitur motus per formulas

&lt;math&gt;v =gt,s=gt^2/2, v^2 =2gs (b) , &lt;/math&gt;

legibusque sequentibus subjicietar.

1&lt;sup&gt;a&lt;/sup&gt;. Spatium &lt;math&gt;s&lt;/math&gt; percursum intra tempus &lt;math&gt;t&lt;/math&gt; est dimidia pars illus spatii &lt;math&gt;s'&lt;/math&gt;, quod percurreretur si grave aequali tempore pergeret moveri uniformiter cum velocitate &lt;math&gt;v&lt;/math&gt; in
fine temporis &lt;math&gt;t&lt;/math&gt; acquisita; nam (1)

&lt;math&gt;s' = tv = tgt = gt^2 = 2s.&lt;/math&gt;

2&lt;sup&gt;a&lt;/sup&gt;. Spatia totalia a gravibus libere decidentibus percursa,
sunt ut quadrata temporum quibus eadem spatia conficiuntur: item ut quadrata velocitatum tempore descensus acquisitarum  Nam

&lt;math&gt;s=gt^2/2=\frac{v^2}{2g}.&lt;/math&gt;

3&lt;sup&gt;a&lt;/sup&gt;. Spatia a gravibus libere decidentibus percorsa
aequalibus et successivis temporibus sequuntur progressio
numerorum imparium 1,3,5,7, ... ; assumpto enim
&lt;math&gt;t = 1,2,3,4 &lt;/math&gt;, ... spatia illa exprimentur per &lt;math&gt;\frac{g}{2},
\frac{4g-g}{2},\frac{9g - 4g}{2},\frac{16g-9g}{2}, \mathrm{seu}\,
\frac{3g}{2}, \frac{5g}{2}, \frac{7g}{2}.
&lt;/math&gt; 

Hae leges experientiae cum sin &lt;math&gt;t&lt;/math&gt; consentaneae, hypothesis gravitatis aequaliter semper ad sensum agentis prope
telluris superficiem existimanda est naturae conveniens: et
quoniam experimentis saepe iteratis apud nostras regiones compertum est, grave sibi relictum percurrere pedes
15, 0915 ... intervallo unius minuti secundi, erit

&lt;math&gt;g = \frac{2s}{t^2} = 2\times 15,0915 ... = 30,183 ... &lt;/math&gt;&lt;ref&gt;9,78:30,183=0,324 m/pes&lt;/ref&gt;

Eam nimirum velocitatem gravitas valet mobili communicare intervallo unius secundi, qua si mobile pergeret uniformiter moveri, absolveret singulis secundis pedes 30,2 circiter.

Deprehenderunt quidem Physici gravitatem esse diversam tum ad diversas supra terrestrem superficiem altitudines, tum ad diversas ab aequatore terrestri distantias: verum ejusmodi variationes in corporum gravitate haud fiunt sensibiles nisi sub differentiis admodum grandibus sive inter altitudines illas, sive inter illas distantias; propterea absque sensibili errore contemni poterunt in ordine ad singula corpora terrestria, quae ut plurimum veniunt consideranda.
Si retenta &lt;math&gt;s_0=0&lt;/math&gt;, ponitur &lt;math&gt;v = a&lt;/math&gt;, exsurgent (28) &lt;math display=&quot;block&quot;&gt;v=a+gt, s = at + gt^2/2, v^2-a^2 = 2gs (b').&lt;/math&gt;

[[31|31]]. Assumpta &lt;math&gt;g&lt;0&lt;/math&gt; in (b'), prodibunt&lt;math display=&quot;block&quot;&gt;v = a-gt, s = at - gt^2/2, a^2-v^2 = 2gs (b'');&lt;/math&gt;

quae formulae manifeste determinant verticalem gravium ascensum. Facta &lt;math&gt;v=0&lt;/math&gt; in tertia ac prima (b&quot;), emergent
&lt;math&gt;
s=\frac{a^2}{2g}, 
t= \frac{a}{g} (b'''),
&lt;/math&gt;
maxima nempe altitudo ad quam ascendit grave, tempusque respondens.

Obiter hic notamus illud: Si datur ejusmodi potentia &lt;math&gt;R&lt;/math&gt;, quae agendo ad modum vis instantaneae valeat massae &lt;math&gt;M'&lt;/math&gt; communicare velocitatem &lt;math&gt;a&lt;/math&gt;, ut sit (6) &lt;math&gt;R= M'a&lt;/math&gt;, ipsa &lt;math&gt;R&lt;/math&gt; agendo ad modum vis continuae per gradus infinitesimos poterit ponderosam massam &lt;math&gt;M&lt;/math&gt; sustentare libratam per totum tempus &lt;math&gt;t = \frac{M'a}{Mg}&lt;/math&gt;

Cum enim singulis tempusculis infinitesimis &lt;math&gt;dt&lt;/math&gt; gignat
gravitas in massa &lt;math&gt;M&lt;/math&gt; quantitatem motus &lt;math&gt;Mgdt&lt;/math&gt;, certe singulis &lt;math&gt;dt&lt;/math&gt; debebit &lt;math&gt;R&lt;/math&gt; ad librandam &lt;math&gt;M&lt;/math&gt; exerere actionem infinite parvam &lt;math&gt;=Mgdt&lt;/math&gt;; proinde totalis actio respondens toti &lt;math&gt;t&lt;/math&gt; erit &lt;math&gt;\int Mgdt = Mgt&lt;/math&gt;: igitur &lt;math&gt;Mgt=M'a&lt;/math&gt;; ideoque etc.

Quisque nunc videt posse vim &lt;math&gt;R&lt;/math&gt; exhiberi non solum per
&lt;math&gt;M'a&lt;/math&gt;, sed etiam per &lt;math&gt;Mgt&lt;/math&gt;.
[[Fasciculus:Atwoods machine.png|thumb]]
[[32]]. Ad motum gravium determinandum in machina Atwoodi, sint &lt;math&gt;m&lt;/math&gt; et &lt;math&gt;m +m'&lt;/math&gt; massae filo appensae: quisque videt motricem systematis vim exhiberi per &lt;math&gt;
g ( m +m' ) - gm =gm'&lt;/math&gt;; unde profluit vis acceleratrix &lt;math&gt;g\frac{m'}{2m + m'}&lt;/math&gt; substituenda loco &lt;math&gt;g&lt;/math&gt; in formulis (b). Quoniam vis ista potest pro lubito attenuari, sequitur in Machina Alwoodi posse motus velocitatem imminui quantum libuerit; quod maxime conducit et ad accuratius definienda spatia percursa, et ad aeris resistentiam tuto negligendam. Sicuti enim corpus, quod movetur in medio aliquo materiali, agit in ipsum medium, ejus particulas expellendo, exerceturque corporis actio juxta motus directionem, ita medii particulae juxta contrariam directionem reagunt (7) in corpus atque resistunt; inde oritur quidem imminutio virium in corpore, sed major vel minor, prout major vel minor velocitas communicatur medio expellendo; et consequenter prout major vel minor est velocitas corporis expellentis.

[[33|33]]. Haec notamus circa gravium motum in medio resistente.

1º. Constat gravia decidentia in pleno homogeneo motum suum vi gravitatis sic accelerare ut paullatim evadat proxime et sensibiliter uniformis. Dum nempe corpus initio movetur, primumque velocitatis gradum acquirit, aliquam hujus gradus jacturam pati debet ex opposita medii resistentia. Sed quia velocitas corporis in progressu semper augetur, multo magis augeri etiam debet medii resistentia; siquidem major corporis velocitas non solum importat ut major quoque velocitas communicetur singulis particulis removendis, sed praeterea ut major quoque resistentis materiae quantitas dato tempore dimoveatur. Ergo velocitatis gradus semper magis imminuetur: unde fit quod velocitas corporis ad valorem constantem propius semper accedat, ejusque motus paullatim evadat proxime et sensibiliter uniformis.
[[Fasciculus:Atwood.svg|thumb]]
2º. Medii resistentia cum tota exerceatur contra corporis superficiem, vis motrix inde resultans haud pendebit ab ipsius corporis massa, eritque eadem utcumque sub eadem et forma, et amplitudine superficiei, crescat vel decrescat massa: non sic dicendum de respondente vi acceleratrice, quae cum obtineatur dividendo vim motricem per corporis massam, permanente et forma, et amplitudine superficiei, erit reciproce ut ipsa massa. Hinc patet cur, caeteris paribus, quo major est massa corporis in medio resistente decidentis, eo etiam rapidior sit motus finalis.

3º. Si concipimus planum variis resistentis medii stratis normaliter occurrens velocitate &lt;math&gt;v&lt;/math&gt;, ponimusque et plani actionem in medii particulas intra singula tempuscula infinitesima sese protendere ad respondentia duntaxat strata dimovenda, et haec eadem strata illico sic dimoveri ut statim atque dimota sunt nullam praeterea actionem sive immediatam, sive medialam exerceant in dimovens planum; expressa per &lt;math&gt;ds&lt;/math&gt; crassitudine strati dimovendi intra tempusculum &lt;math&gt;dt&lt;/math&gt;, per &lt;math&gt;\mu&lt;/math&gt; densitate medii, et per &lt;math&gt;A&lt;/math&gt; area dimoventis plani, orietur inde (28) resistentia &lt;math&gt;A\mu v ds \frac{1}{dt}&lt;/math&gt; seu &lt;math&gt;A \mu v^2&lt;/math&gt;. Duplicatur resistentia in casu medii elastici (23).

4°* Si vis acceleratrix ex medii resistentia assumitur proportionalis quadrato velocitatis, ut denotante &lt;math&gt;\mathrm{k}&lt;/math&gt; quantitatem constantem (experimentis determinandam), exhiberi possit vis illa per  &lt;math&gt;g\frac{v^2}{\mathrm{k}^2}&lt;/math&gt; gravia descendentia sollicitabuntur vi acceleratrice &lt;math&gt;g-g\frac{v^2}{\mathrm{k}^2}&lt;/math&gt; ascendentia vi acceleratrice  &lt;math&gt;-\left(g+g\frac{v^2}{\mathrm{k}^2}\right)&lt;/math&gt;: proinde (28) quoad gravium descensum
 &lt;math&gt;\frac{dv}{dt}=g-g\frac{v^2}{\mathrm{k}^2}&lt;/math&gt; 
quod ascensum
 &lt;math&gt;\frac{dv}{dt}=-g-g\frac{v^2}{\mathrm{k}^2}&lt;/math&gt; 

5°* In primo casu 
 &lt;math&gt;gdt=\frac{{\mathrm{k}^2}dv}{{\mathrm{k}^2}-v^2}=\frac{\mathrm{k}}{2}\left(\frac{dv}{\mathrm{k}+v} + \frac{dv}{\mathrm{k}-v} \right)&lt;/math&gt; 
sumptisque integralibus:(27.6 °) in hypothesi velocitatis &lt;math&gt;v_0=0&lt;/math&gt;,
 &lt;math&gt;gt=\frac{\mathrm{k}}{2}\ln\left(\frac{\mathrm{k}+v}{\mathrm{k}-v} \right)&lt;/math&gt; 
unde 
 &lt;math&gt;e^{\frac{ngt}{\mathrm{k}}}=\frac{\mathrm{k}+v}{\mathrm{k}-v}&lt;/math&gt;
Primum membrum est necessario &lt;math&gt;&gt;0&lt;/math&gt;; ergo et secundum: crescente igitur &lt;math&gt;t&lt;/math&gt; crescet quidem &lt;math&gt;v&lt;/math&gt;; ita tamen ut nunquam fiat &lt;math&gt;v &gt; k&lt;/math&gt;: quod consentit cum dictis (10). Ad haec : quoniam (28)
&lt;math&gt;
dt=\frac{ds}{v}&lt;/math&gt; erit &lt;math&gt;gds=\frac{{\mathrm{k}^2}vdv}{{\mathrm{k}^2}-v^2}&lt;/math&gt; quam integrantes assequemur &lt;math&gt;gs = C - \mathrm{k}^2\ln(\mathrm{k}^2 -v^2)&lt;/math&gt;: in initio motus ex hypothesi &lt;math&gt;v =0 , s =0&lt;/math&gt;, ac proinde &lt;math&gt;C = \frac{\mathrm{k}^2}{2}\ln \mathrm{k}^2&lt;/math&gt;; igitur
 &lt;math&gt;gs= \frac{\mathrm{k}^2}{2}\ln\frac{{\mathrm{k}^2}}{{\mathrm{k}^2}-v^2}&lt;/math&gt;
6°* In secundo casu
 &lt;math&gt;gdt=\frac{{\mathrm{k}^2}dv}{{\mathrm{k}^2}+v^2}&lt;/math&gt;
ideoque (27.13°)
 &lt;math&gt;gt = C - \mathrm{k}\arctan(\frac{v}{\mathrm{k}})&lt;/math&gt;
tempori &lt;math&gt;t = 0&lt;/math&gt; respondet &lt;math&gt;v =v_0&lt;/math&gt;, et consequenter &lt;math&gt;C
= \mathrm{k}\arctan(\frac{v_0}{\mathrm{k}})&lt;/math&gt;; igitur (tang- ):
5l = k [ arc(tang = :) - arc (ranga ) ] .
ds Ad haec : ob de habemus


s
V71
ndum :
gds =
kavdv ; propterea gs = C—
kat va -- 105 (1º + vw).
log ( Kº +w.), ce
ka In initio motus s = 0 , v = Vo;hinc CF
2
gs =
log k2+0.2
katua 2
Facta v = 0 , prodibunt
k2 proind k
log
ktve t
2g 8 are (tang = ).

maxima videlicet altitudo ad quam in medio resistente ascendit grave, tempasque respondens.

7º. Fac ut , exhibente YM ( Fig. 17) directionem
normalem stratis TT ''medii resistentis , planum A oblique''
occurrat stratis ipsis sub angulo BMY ( =\beta ) . Recta bc
parallela rectae YM repraesentet velocitatem v , qua move
tur A : resoluta bc in Kc perpendicularem et BK parallelam
plano A , exprimet Aje . KC2 resistentiam medii ; et quo
niam KC bc . sin Kbc = vsin \beta , iccirco resistentia ista
Ajwa , sin a\beta .
J =

=== De gravium descensu atque ascensu per plana inclinata; de attritu; deque cochlea, et cuneo.===
[[Fasciculus:Free body.svg|thumb|Planum inclinatum]]
[[34]]. Super plano ad horizontem &lt;u&gt;inclinato&lt;/u&gt; collocetur corpus quod habeat centrum gravitatis in &lt;math&gt;G&lt;/math&gt; (Fig. 21) et massam &lt;math&gt;M&lt;/math&gt;; ex &lt;math&gt;G&lt;/math&gt; horizontem demittatur perpendiculum &lt;math&gt;GH&lt;/math&gt;; et ex &lt;math&gt;H&lt;/math&gt; ducatur alterum perpendiculum &lt;math&gt;HB&lt;/math&gt; in communem plani horizontalis et plani inclinati intersectionem; vis motrix ex corporis pondere jacebit in plano perpendiculornm &lt;math&gt;GH , HB&lt;/math&gt;; demisso enim ex &lt;math&gt;G&lt;/math&gt; perpendiculo &lt;math&gt;Gi&lt;/math&gt; in planum inclinatum, vis illa invenietur in plano &lt;math&gt;iGH&lt;/math&gt; normaliter insistente intersectioni plani inclinati et plani horizontalis; quod planum &lt;math&gt;iGH&lt;/math&gt; manifeste recidit in planum perpendiculorum &lt;math&gt;GH , HB&lt;/math&gt;. Sit &lt;math&gt;AB&lt;/math&gt; communis intersectio istius plani et plani inclinati; &lt;math&gt;AC&lt;/math&gt; perpendiculum ex &lt;math&gt;A&lt;/math&gt; demissum in &lt;math&gt;BH ... ; c&lt;/math&gt; angulus &lt;math&gt;ABC&lt;/math&gt;: recta &lt;math&gt;AB&lt;/math&gt; vocatur longitudo plani inclinati, &lt;math&gt;AC&lt;/math&gt; altitudo, &lt;math&gt;c&lt;/math&gt; &lt;u&gt;angulus inclinationis&lt;/u&gt;. Vim motricem per &lt;math&gt;GK&lt;/math&gt; repraesentatam resolve in duas &lt;math&gt;Gi , Gh&lt;/math&gt;, quarum altera sit perpendicularis, altera parallela rectae &lt;math&gt;AB&lt;/math&gt;; erunt

&lt;math&gt;Gi = gM \cos c , Gh = gM \sin c&lt;/math&gt;.

Cadat &lt;math&gt;Gi&lt;/math&gt; intra corporis basim; elisa &lt;math&gt;Gi&lt;/math&gt; a resistentia plani inclinati, gignetur motus a sola &lt;math&gt;Gh&lt;/math&gt;; quae cum maneat constanter eadem, non alium pariet motum nisi uniformiter varium. His positis, ad determinandum gravium motum per plana inclinata satis erit in (6,6' . 30) et in ( 6 &quot; . 31 ) substituere &lt;math&gt;g \sin c&lt;/math&gt; pro &lt;math&gt;g&lt;/math&gt;: denotantibus itaque &lt;math&gt;\theta&lt;/math&gt; tempus, &lt;math&gt;u&lt;/math&gt; velocitatem, et &lt;math&gt;z&lt;/math&gt; spatium, erunt quoad gravium descensum per plana inclinata u = g 9 sin c, z = * gga sin c, u = 2gz sin c ( 6 &quot; )
si tempori 0 = o respondent u = 0,2 = 0 ; et
u = u + go sinc,z = altiglasin c,u ? —a? = 2g zsin c (6 )
si tempori 0 =o respondent u = a, z = 0 : quoad ascen
sum vero
u =a -g6 sinc, z =a9— 1 g 2sinc, a ?—u? = 2gz sin c (65 &quot; )
&lt;u&gt;Componens&lt;/u&gt; &lt;math&gt;Gi&lt;/math&gt; exhibet pressionem, quam exercet grave contra planum inclinatum .

et :spatium , eruntxquoad gravium descen-
sum per plana inclinata
u:g93inc, :

: äggï sinc, 113:2gz sinc (ö&quot;)

si tempori 6:o respondent u:o,:

: o; et

u:u-1-g 9ainc.z:a G—i-äggasin c,u'—a*:Zgzsin c(b')
si tempori 9:o respondent u:a, s:o :quoad ascen-
∙ sum vero
u :::—gg sinc, :349— äggaslncaaa—uzzzgz Sine (b'-l)&quot; r Componeus Gi exhibet pressionem, quam exercet grave
contra planum inclinatum .73

35. Comparantes ( 6 ' ' ) cum (6 ) haec facile stabiliemus.
1. ' Si licals t . erunt i GH
plaui 1 : sin c , s : 2 = 1 : sin c ;
pla
inter
cula
Br
noguls
si duo nempe gravia eodem tempore delabuntur, alterum
verticaliter , alterum per planum inclinatum AB , tam ve
locitates v , u ab ipsis gravibus in descensu perpendiculari
et obliquo acquisitae , quam spatia s , z descripta , erunt
ut longitudo plani ad ejus altitudinem .
2. ° Hinc ubi ex puncto C concursus rectae verti
calis com horizontali ducatur perpendiculam CE ad plani
inclinati lougitudinem AB , grave percurret lapsu obliquo
spatium = AE eo tempore , quo percurreret lapsu verticali
totam altitudinem AC ; nam AC : AE - AB : AC.
3. ° Inde sequitur chordas omnes circuli ad supre
mam , vel infimam diametri verticalis extremitalem pertin
gentes describi eodem tempore ; eo nimirum , quo descri
beretur ipsa circali diameter.
4. ° Velocitates u , v gravium in plano ioclinato
et in recta verticali aequales sunt si gravia ex punctis aeque
altis ad eamdem rectam horizontalem pervenerint : in ea
sumus hypothesi est s = zsinc , ac proinde
a pla
ifors
16 :3
cempo
enim qua

: u2 V =U .

5.° Tempus descensus per longitudinem plani in
clinati ad tempus descensus per altitudinem est ut ipsa
longitudo ad altitudinem : nam in casu ( 4º ) u = v ; ideoque
SIDEN
g9 sin gt , et 0 : t 1 : sin c .

36. Sint nunc plura plana sibi contigua ( fig. 22. * )
diversimode ad horizontem inclinata . Si grave ab AB
transit ad planum BD , in eo transitu non retinebit in
initio plani BD totam velocitatem , quam habebat in
fine plani AB. Si enim concipitur recta AC perpendi
6 et ?
73
35. Comparantes (b &quot;') cum (6) haec facile stabiliemus.
1.&quot; Si 9:t . erunt
v:u:1:sinc,s:z:1:sinc;'
si duo nempe gravia eodem tempore delebuntur, alterum
verticaliter , alterum per planum inclinatum AB, tam ve-
locitates v ,
1: ab ipsis gravibus in descensu perpendiculari
et obliquo acquisitae , quam spatia s
, z descripta , erunt
ut longitudo plani ad ejus altitudinem .
2? Hinc ubi ex puncto G concursus rectae verti-
calis cum horizontali ducatur perpendiculum CE ad plani
inclinati longitudinem AB, grave percurret lapsu obliquo
spatium :AE eo tempore , quo percurreret lapsu verticali
totam altitudinem AC; uam AC :AE :- AB :AC.
3.&quot; Inde sequitur chordas omnes circuli ad supre-
mam , vel infimam diametri verticalis extremitatem pertin-
gentes describi eodem tempore; eo nimirum , quo descri-
beretur ipsa circnli diameter.
. 4.&quot; Velocitates 11 .'v gravium in plano inclinato
et in recta verticali aequales sunt si gravia ex punctis aeque
altis ad eamdem rectam horizontalem pervenerint: in ea
enim qua sumus hypothesi est s:zsinc
, ac proinde
v&quot;:
uz , v :u .
'
5.&quot; Tempus descensus per longitudinem plani in-
clinati ad tempus descensus per altitudinem est ut ipsa
longitudo ad altitudinem: nam in casu (40) u:v ; ideoque
g95inc:gt,et-9:t:1:sinc.
36. Sint nunc plura plana sibi contigua (fig. 22.')
diversimode ad horizontem inclinata. Si grave ab AB
transit ad planum BD, in eo transitu non retinebit in
initio plani BD totam velocitatem, quam habebat -in
fine plani AB. Si enim concipitur recta AC perpendi-
6
- .... ↹∙∙∙↽∙⊾ −↿−⇀⋅⋅⋅⋅↽∙⋅↽ f.:-.. ∙−←−−− ↘−∼∙⋅ ,. ∙∙⋅∙∙∙⇁ . ∙∙
'1
cularis plano BD producto
, et velocitas in fine plani ha
bens directionem AB concipitur resoluta in duas AC
,
CB

; illa prior AC

a novo plano BD elidetur
, utpote quae
tota insumitar in eo normaliter percutiendo
, ac seclusa
0
mois elasticitatis consideratione
, sola altera CB urgebit cor
pus per novum planum BD
, eritque velocitas prior ad no
vam
, qua nempe ingreditur novum planum ut AB

: CB sive ut

radius ad cosinum anguli ABC
, et prior velocitas ad amissam
erit ut radius ad sinum versum ejusdem anguli ABC

; cum

nempe
, si centro
B et radio BA describatur semicirculus
EAE
'
, sit velocitas prior ad amissam ul AB

: CE

.
Erraverunt igitur qui banc velocitatis jacturam minime
considerantes falsum hoc theorema confecerunt,, Ex aliitu
dine quacumque descendens grave per quotlibet ac quaeli
bet plana AB
, BD sibi contigua utcumque inclinata eamdem
in puncto infimo
D velocitatem acquiret ac cadendo perpen
diculariter ex eorum omnium altitudine,, Erit tamen veris
simum theorema si non ad plana contigua quaecumque scd
ad curvas, quae ex infinitis numero rectis lineis et infinite
parvis
( 27. 16
°
) coalescere intelliguntur
, applicetur et poterit verissime sic enunciari ,, Quodlibet grave ex quacum
que altitudine cadens supra superficiem curvam quamcum
que
, eamdem in puncto infimo velocitatem acquiret ac ca
dendo perpendiculariter ex ipsius curvae altitudine ,, Ratio
est quia velocitatum jactura in transitu de uno in aliud
planum
, decrescente angulo quem continet planum alterum
AB cum altero DB producto
, decrescit siquidem decrescente angulo ABC decrescet sinus versus CE repraesentans
velocitatem amissam
. Quare faclo infinite parvo angulo ABC
,
uti contingit in curvis
, velocitas quoque amissa fiet infinite
parva
, ac proinde grave ingredietur planum BD cum ve
locitate acquisita in descensu per planum AB
. Porro sinus
versus CE
' ita decrescit ut, facto infinite parvo primi or
dinis angulo ABC
, ipse CE
' evadat infinitesimus secundi or
dinis

; nam EC
: AC

= AC

: CE

'.
74
cularis plano BD producto
,
et velocitas in fine plani ha-
bens directionem AB concipitur resoluta in duas AC
,
CB; illa prior AC :: novo plano BD elidetur, utpote quae
tota insumitur in eo normaliter percutiendo, ac seclusao-
mnis elasticitatis consideratione, sola altera CB urgebit cor-
pus per novum planum BD, eritque veloeitas prior ad no-
vam, qua nempe ingreditur novum planum ut AB:CB sive ut
radius ad cosinum anguli ABC, et prior velocitas ad amissam
erit ut radius ad sinum versum ejusdem anguli ABC; cum
nempe, si centro
B et radio BA describatur semicirculus
EAE', sit velocitas prior ad amissam ut AB: CE'.
Erraverunt igitur qui hanc velocitatis jacturam minime
considerantes falsum hoc theorema coufecerunt,, Ex altitu-
dine qnacumque descendens grave per quotlibet ac quaeli-
bet plana AB, BD sibi contigua utcumque inclinata eamdem
in puncto infimo
D velocitatem acquirat ac cadendo perpen-
diculariter ex eorum omnium altitudine,, Erit tamen veris-
simum theorema si non ad plana contigua quaecumque sed
ad curvas, quae ex infinitis numero rectis lineis-et infinite
parvis (27. 16&quot;) coalescere intelliguntur, applicetnr; et po-
terit verissime sic enunciari ,, Quodlibet grave ex quacum-
que altitudine cadens supra superficiem curvam quamcum—
que, eamdem in puncto infimo velocitatem acquiret ac ca-
dendo pan-pendiculariter ex ipsius curvae altitudine ,, Ratio
est quia velocitatum jactura in transitu de uno in aliud
planum, decrescente angulo quem continet planum alterum
AB cum altero DB producto, decrescit; siquidem decrescen-
te angulo ABC decrescet sinus versus CE' repraesentans
velocitatem amissam. Quare facto infinite parvo angulo ABC,
nti contingit in curvis, velocitas quoque amissa fiet infinite
parva, ac proinde grave ingredietur planum BD cum ve-
locitate acquisita in descensu, per planum AB. Porro sinus
versus CE' ita decrescit ut, facto infinite parvo primi or-
dinis angulo ABC, ipse CE' evadat infiuitesimus secundi or-
diuis; nam EC: AC:AC: CE'.75
1

37. Hactenus nullam habuimus rationem attritus , seu
resistentiae ex asperitate superficierum : prominentes nem
pe unius superficiei denticuli foveas alterius ingrediun
tur ; sicque haud poterit una superficies alteri superposita
promoveri, nisi ipsi denticuli vel frangantur, vel inflectan
tur, vel , saperiori superficie identidem elevata parumper ,
e foveolis egrediantur: possunt quidem denticuli ita poli
tione imminui , ut sensum inermem effugiant, sed penitus
tolli nequeunt.Statue corpus super plano horizontali ; tum pla
num istud eousque sensim inclina , donec sub quodam angulo
c=c'corpus tantum non incipiat descendere, incipiat vero cre
scente utcumque parum c ultra c' . Attritus respondens angulo
c = c dicatur f: quoniam f accurate librat vim gM sinc'
erit f =g Msinc' ; hinc si per r exprimitur ratio attritus f ad
pressionem gM cosc' ut sit fer. GM cosc ', habebitur
.
r.gM cosc = gM sinc' ,
ideoque
r = tang c' .
0
5
Permanente qualitate massae M, itemque politionis gra
du , constat experimentis quod permanet quoque angulus c' ,
et consequenter ratio r, licet quantitas ipsius M augeatur,
vel minuatur. Inde sequitur attritum f, caeteris paribus, fo
re proportionalem pressioni r.gM cosc' .
Si ponimus attritum adhuc pressioni proportionalem
quum angulus c superat angulum c'; ad habendam ratio
nem attritus in motu gravium per plana inclinata , pro
gsinc substituetur g sin c - rg cosc in ( b ), et gsinc +
rg cosc in ( 6 &quot; ); caeterum in casu motus videtur f non a so
la pressione , sed a corporis quoque velocitate haud pa
rum pendere.
Haec subjungimus.
&quot;
75

37. Hactenus nullam habuimus rationem attritus, seu
resistentiae ex asperitate superficierum :prominentes nem-
pe unius superficiei denticuli foveas' alterius ingrediun-
tur ; sicque haud poterit una superficies alteri superposita-
promoveri, nisi ipsi denticuli vel frangantur, vel mflectan-
tur, vel, superiori superficie identidem elevata parumper ,
e foveolis egrediantur: possunt quidem denticuli ita poli-
tione imminui, ut sensum inermem effugiam, sed penitus
tolli nequeunt.Statue corpus super plano horizontali; tum pla-
num istud eousque sensim inclina
, donec sub quodam angulo
c:c' corpus tantum non incipiat descendere, incipiat vero cre-
scente utcumque parum c ultra c'. Attritus respondens angulo
c:c' dicatur f: quoniam f
accurate librat vim nginc'
erit f

: g Msinc'; hinc si perr exprimitur ratio attritus f

ad
pressionem gM cosc' ut sit:r. gM cosc', habebitur
r. gM cosc': gM sinc' ,
ideoque
r:tang c'
.
Permanente qualitate massae M, itemque politionis gra-
du, constat experimentis quod permanet quoque angulus c',
et consequenter ratio r, licet quantitas ipsius M augeatur,

: vel minuatur. Inde sequitur-attritum f,'caeteris paribus, fo-

1e proportionalem pressioni ngM cosc'.
Si ponimus attritum adhuc pressioni proportionalem
↴⋅ quum angulus c superat angulum ∁∙∍ ad habendam ratio-
lnem attritus in motu gravium per plana inclinata
, pro
igsinc substituetur gsinc—rgcosc in (b' ), et gsinc −∣−
' rgcosc tn ( b &quot;); caeterum in casu motus videtur fnon a so-
lla pressione, sed a corporis quoque velocitate haud pa-
rum pendere.
Haec subjungimus.76
1º . Si corpus in plano inclinato constitutum li
brandum sit potential applicita ( Fig. 21 ) puncto G, quae
potentia et sollicitat ad ascensum, et efficit angulum &amp; cum
AB, gignitque propterea pressionem Qsind, satis erit ut re
sultans ex viribus Q et M ( g sinc F rg cosc ) Fr (sin
exsistat ipsi plano perpendicularis , sese videlicet diri
gat juxta Gi: continet autem Q cum Gi angulum 900
An et vis Mg ( sinc F r cosc ) FrQsinc angulum
cum eadem Gi. Igitur ( 9.10 )
= 90
Q: Mg( sincar cosc ) FrQsing = sin 90 ° ; sin ( 90 °
a ) = 1 : cosa ;
ideoque
sinc Frcosc OSCMS
Q
cos a Es since
secun Sumpio superiori signo, nequit Q esse minor
do membro quin corpus descendat; sumplo inferiori si
gno, nequit Q esse major secundo membro quin corpus
ascendat; perstabit aequilibrium intra limites
sinc - rcosc sinc torcose Mg, el &lt;
cosa + rsing
Mg.
cosu - osinc
2º. In hypothesi nullius attritus erit r = 0 ; et
consequenter
sin c
Q
Mg COSU.
3º. Si Q est insuper parallela horizontali BC, e
rit a = c ; ideoque
76
1&quot;. Si corpus in plano inclinato constitutum li-
brandum sit potentia Q applicita ( Fig'. 21) puncto G, quae
potentia et sollicitat ad ascensum, et eilicit anguluma cum
AB, gignitque propterea pressionem Qsinac, satis erit ut re-
sultans ex viribus Q et M (gsinc :rgcosc ):F r Qsin a
exsistat ipsi plano perpendicularis , sese videlicet diri-
gat juxta Gi: continet autem Q cum Gi angulum :90&quot;—
a, et vis Mg ( sine: rcosc) :rQsina angulum :90&quot;
cum eadem Gi. Igitur ( 9. 1&quot; )
Q: Mgüincqzr 0050 ):t:rQsinat:sin 90&quot; :sin ( 90&quot;—
at:) 1:cosa:;
ideoque
sinc ∓r cosc
−∙∙ Mo
cos a:
r siua
Sumpto superiori signo, nequit Q esse minor secun-
do membro quin corpus descendat; sumpto inferiori si-
gno, nequit Q esse maior secundo membro quia corpus
ascendat; perstabit aequilibrium intra limites
sinc—rcosc sinc rrnsr Q)...— Mg, et Q(
−⊢ Mg.
cosa rsiuat cosa—rsiua
2&quot;. In hypothesi nullius attritus erit r: o ; et
consequenter
sin 0 M
g. szz cosa:
3&quot;. Si Q est insuper parallela horizontali BC, e-
rit at:c ; ideoque77
sipc :
Mg COSC
potentia videlicet ad pondus ut plani altitudo AC ad hori
zontaleon BC. Hinc facile deducuntur aequilibrii leges in
cochlea et cuneo.
4º. Cum cochlea non sit nisi planum inclina
tum ABC, quod circum cylindruni ducitur; dum vero co
chlea agit , potentia sit rectae lineae BC parallela, erit
potentia ad pondus seu resistentiam superandam , ut al
titudo plani seu helicam distantia ( =h ) ad basim plani
seu cylindri peripheriam ( = k ). Hinc
Q
hP ;
k
quae formula supponit distantiam inter cylindri axem et pun .
ctum cui applicatur potentia , esse ipsius cylindri radium
( = m ) : quod si distantia illa fiat alia ab r', et exhibea
tur per R' ; denotante e potentiam respondentem novae
distantiac, exsistet
mi?
R'
ac proinde Q
-
hP
R '
LP
25R .
k
In ordine ad cochleam infinitam , dicatur A radius ma
joris rotae , a radius minoris , et P' pondus seu poten
tia apud dentes ipsius rotae majoris; erunt
ар P =
Q
api A
hP
27.R '
ideoque
Q =
h a P
21AR'
77
Q sine. NT: ⋅⇀ SE.—.'
potentia videlicet ad pondus ut plani altitudo AC ad hori-
zoutalem BC. Hinc facile deducuntur aequilibrii leges in
cochlea et cuneo.
4&quot;. Cum cochlea non sit nisi planum inclina-
tum ABC, quod circum cylindrum ducitur; dum vero eo-
chlea agit, potentia sit rectae lineae BC parallela, erit
potentia ad pondus seu resistentiam superandam, ut al-
titudo plani seu helicum distantia ( :h)ad basim plani
seu cylindri peripheriam :( k). Hinc
Qz—k-i
quae formula supponit distantiam inter cylindri axem et pun-
ctum cui applicatur potentia, esse ipsius cylindri radium
(: r' ): quod si distantia illa fiat alia ab r', et exhibea-
tur per B'; denotante Q' potentiam respondentem novae
distantiae, exsistet
Q'—r' ∙∙ ∙∙∙∣≖∣⊃⋅↿⋅⋅∙−∣≀∌ ∙≺⋮−−−−∙↓⊤∙ ac ptomde QI—B— . ∣∙⋮−−−−∙ ⊋∙⋮⋮⋅⋮↸↽∙
In ordine ad cochleam infinitam, dicatur A radius ma-
ioris rotae
, a radius minoris , et P' pondus seu poten-
tia apud dentes ipsius rotae maioris; erunt
aP , hP'
P::ï'Q—an' '
ide ue
Oq haP78
1
5 ° Cuneus spectari potest tamquam coalescens ex
duobus planis inclinatis ut ABC, tum ob figuram suam ,
tum quia idem est sive pondus per planum inclinatum
trahatur sursum , sive planum sub pondere promoveatur.
Agit autem potentia in cuneo juxta CB; quoad igitur u
1
nam cunei partem ABC respondens potentia Qerit ad
m
1
respondentem resistentiam P ut AC ( = D ) , sen di
midia cunei crassities ad BC ( = H ) , idest ad altitudinem
1
Q
1
ad respondentem resistentiam P P erit pariter ut
į D ad H. Igitur
m
LQ.H - 1P.HD, Q (m - 1 ) .
A
m2 m m
P (m - 1 )
mi '
· D ;
quibus aequationibus in summam collectis ,
Q. H = P. , D ,
et consequenter
D
H totalis videlicet potentia Q ad totalem resistentiam P ut
dimidia cunei crassities D ad ejus altitudinem H ; mo
do tamen exerceatur resistentia normaliter ad H.
6º . Si in cochlea v . gr. considerandus esset at
tritus , haberetur ( 10.40.) ,
1
!
1
5&quot; Cuneus spectari potest tamquam coalescens ex
duobus planis inclinatis ut ABC, tum ob figuram suam,
tum quia idem est sive pondus per planum inclinatum
trahatur sursum, sive planum sub pondere promoveatur.
Agit autem potentia in cuneo iuxta CB; quoad igitur u-
. . 1 nam cune1 partem ABC . respondens potentta —Qer1t ad
.
'
m
respondentem resistentiam −↿−∙∶ P ut AC (: äD ), seu di-
↾
m
midia cunei crassities ad BC (: H ), idest ad altitudinem
. . 1 cunei. Quoad alteram partem respondens poteutta Q—- −− Q
m
. . 1 ad . . respondentem rc51stent1am P
-— −−∙ P er1t partter ut
171
&amp; D ad H. Igitur
D, Q—(m-1).H: −↿−↽≺≀∙∥∶∶∙−↿∙−↕⊃∙ ;.
m m m
P
∙∙
-
(m'1) -äD;
,- ut
quibus aequationibus in summam collectis,
QaHzpaL'D, et consequenter
≟≺−≀∙∙− :D
.
P −⋅⋅ H ⋅
totalis videlicet potentia Q ad totalem resistentiam P ut
dimidia cunei crassities &amp; D ad ejus altitudinem H; mo-
do tamen exerceatur resistentia normaliter ad H.
6&quot;. Si in cochlea v. gr. considerandus esset at-
tritus , haberetur (10. 4&quot;.),
≁−−−−∎⋅−−
−−⋅⋅...-—79
sinc FrcoscP =
cosc trsinc
h = 2 te r'r P ;
h Erk
P
k trh 2 trh
ideoque
Q
Qr Pr' h = 27r's
R ? -R 2 r'trh
0
70. Veniat quoque considerandus attritus in ae- ,
quilibrio corporis AB ( Fig. 23: 24 ) , quod ad rolatilem
motum circa fixum cylindrum sollicitatur vi Rjacente in
plano, cui normaliter insistit axis cylindri.
Ac primo cylindrus impleat accurate circularem cor
poris aperturam DE ( Fig. 23 ) , in quam inseritur: per cy
lindri centrum O duc rectam OEE' parallelam vi R , et
pancto E corporis AB applica duas . vires Q ', Q' aequa
les eidem R, et contrarias, alteram nempe tendentem, ab
E versus E' , alteram ab E versus O; vi R licebit substi
tuere systema virium R , Q ', Q &quot; : et cum possint absque sy
stematis turbatione sic transferri ( 11 ) R et l ' ut aequi
distent ab O, eae nitentur dumtaxat gignere motum ro
tatilem circa cylindrum quin ullam pariant pressionem a
pud ipsius cylindri superficiem ; pressio igitur in hanc su
perficiem redigetur ad solam
୧ = R , ideoque f = Rr.
Attritus fest vis tangentialis respectu superficiei cylin
dricae; hinc denotante a radium OE cylindri , et p per
pendiculum Oi ex O ductum in directionem potentiae R,
ad aequilibrium satis erit, ut exsistat ( 9. 2° )
R 1
2
р
Rr
. 79
Q-—sinc:r:rc.oscP 11:er P—h:t:2nr'rp
cosczbrsmc R::brh 2nr':t:rh ,
ideoque
—Qr' Pr' II::ZRr'r
a' &quot;B' 'an'äzrh Q! '
70. Veniat quoque considerandus attritus in ae- ↗
qnilibrio corporis AB ( Fig. 23: 24 ), quod ad rotatilem
motnm circa fixum cylindrum sollicitatur vi R iacente in
plano, cui normaliter insistit axis cylindri.
Ac primo cylindrus impleat accurate circularem ocr-
poris aperturam DE (Fig. 23), in quam inseritur: per cy- ⋅
lindri centrum O duc rectam OEE' parallelam vi B, et
pnncto E corporis AB applica duas, vires Q', Q&quot; aequa-
les eidem R, et contrarias, alteram nempe tendentem, ab
E versns E', alteram ab E versus O; vi R licebit substi-
tuere system virium R, Q', Q&quot;: et cum possint absque sy—
stematis turbatione sic transferri (11) B et Q'0ut aequi-
distent ab 0, eae uitentur dumtaxat gignere motum ro-
tatilem circa cylindrum quin ullam pariant pressionem a-
pud ipsius cylindri superficiem; pressio igitur in hanc su-
perficiem redigetur ad solam
Q&quot; −∙∙−− R, ideoque f: R r.
Attritus fest vis tangentialis respectu superficiei cylin-
dricae; hinc denotante a radium OE cylindri, et p per-
pendiculum Oi ex Oductum in directionem potentiae Pt,
ad aequilibrium satis erit, ut exsistat ( 9. 20)80
et consequenter
P
facto p &gt; ar , disrumpetur aequilibrium ; facto p &lt; ar ,
subsistet .
Ponatur secundo circularis apertura corporis baud
impleri accurate cylindro ( Fig.24) : vis R manifeste trans
ibit per contactum E cylindri et corporis AB . Resolve R
in duas EF, et ED' , quarum altera transeat per centrum 0 ,
altera tangat cylindrum : per EF exprimetur pressio ; ac
proinde
f = r.EF .
Obtinebit igitur aequilibrium quotiescumque
ED ' &lt; r. EF ,
vel ED' = r.EF :
cum autem ( 9. 1. ° ) .
ED' : R = sin FER ; sin D'EF = sin FER : 1 ,
EF : R = sin D'ER ; sin D'EF = cos FER : 1 ,
cumque ducto perpendiculo Oi ex O in ER ,
Oi Ei voa ?
OE sin FER Р cos FER 22 -
p2
a OE
iccirco praefatac aequilibrii conditiones vertentur in
Rp Rr Vap2 Rp a
Rr Va - p ?
a a
quae traducuntur ad
80
et consequenter
&quot;' p :: ar :
facto p ar, disrumpetur aequilibrium; facto p ar
,
subsistet .
l
Ponatur secundo circularis apertura corporis baud
impleri accurate cylindro (Fig.24): vis B manifeste trans-
ibit per contactum E cylindri et corporis AB
. Resolve B
in duas EF, et ED' , quarum altera transeat per centrum O,
altera tangat cylindrum: per EF exprimetur pressio; ac
proinde
f

: r. EF.

Obtinebit igitur aequilibrium quotiescumque
ED' (r. EF , vel ED' −−∶ r. EF :.
cum autem (9. 1.0).
.'
ED': R ::sin FER : sin D'EF :sin FER : 1 ,
EF fii ∙−−∶ sin D'ER; sin D'EF: cos FER : 1,
cumque ducto perpendiculo Oi ex 0 in EB
.
Oi p Et. ⇂∣ (13 ∙−− :;
' :∙−−− :... ∙ FER ↽− −∙ p
sin FER 08 a 005 OF. a ,
iccirco praefatae aequilibrii conditiones vertentur in
n,,(RrI/aa—pz ↧≹∣↗∙∙∙↧≹≀⋅ Wiz—pa
7.&quot; −−−−↴∶∎−−∙−↙≀∎ ⇀⇀ a '
quae traducuntur ad
⇁−∙↱⇁≓≓81 1
ar 2
p &lt;
р
1 + 12 vit ?
8.• Si ponitur R nihil esse aliud nisi resultans
ex datis viribus P' , Pi ad puncta data v . gr. A , B appli
citis , innotescet R ex dictis ( 10 ) , itemque p. ex ( 10.3° ) .
Sic habetur ratio attritus in vecte : caeterum in machinis
praeter resistentiam ex attritu spectanda etiam est resi
stentia ex funibus . Hi enim inflexioni suae resistunt quum
cylindris vel trochleis circumvolvuntur; et quidem eo ma
gis , quo majori pondere tenduntur , quo insuper crassio
res sunt , et quo minor fuerit trochleae, aut cylindri radius.

=== De motu gravium oblique projectorum.===
[[Fasciculus:Ferde hajitas2.svg|thumb]]
[[38]]. Grave &lt;math&gt;M&lt;/math&gt; (Fig. 25) juxta directionem MG velocitate &lt;math&gt;v_0&lt;/math&gt; projectum urgebitur duplici motu, altero aequabili per &lt;math&gt;MG&lt;/math&gt; ex impetu recepto, altero (nihil est aliud nisi motus relativus mobilis &lt;math&gt;M&lt;/math&gt; quoad ipsum &lt;math&gt;M&lt;/math&gt; iens per &lt;math&gt;MG&lt;/math&gt; sola &lt;math&gt;v_0&lt;/math&gt;) uniformiter accelerato gravitatis proprio per rectam verticalem &lt;math&gt;MR&lt;/math&gt;, vel ipsi &lt;math&gt;MR&lt;/math&gt; parallelam. Sit &lt;math&gt;S&lt;/math&gt; spatium quod cumque &lt;math&gt;MC&lt;/math&gt; primo illo aequabili motu seorsim sumpto percursum, &lt;math&gt;t&lt;/math&gt; tempus impensum ad ejusmodi spatium percurrendum, sitque &lt;math&gt;s&lt;/math&gt; spatium &lt;math&gt;MF&lt;/math&gt; pari tempore percursum secundo motu item seorsim sumpto. Completo parallelogrammo &lt;math&gt;MFQC&lt;/math&gt;, in fine temporis &lt;math&gt;t&lt;/math&gt; grave erit (5) in &lt;math&gt;Q&lt;/math&gt;;
et quia (1:30)
&lt;math&gt;S = v_0 t , s = \frac{gt^2}{2},&lt;/math&gt;
eliminato &lt;math&gt;t&lt;/math&gt;, existet
&lt;math&gt;S^2 = \frac{2 v_0^2}{g}s &lt;/math&gt;
aequatio ad curvam (dicitur parabola) in qua defertur grave. Si altitudo debita (28) velocitati &lt;math&gt;v_0&lt;/math&gt;, dicatur &lt;math&gt;\mathrm{A}&lt;/math&gt;, erit
&lt;math&gt;v_0^2 = 2g\mathrm{A}&lt;/math&gt;, et aequatio transformabitur in
&lt;math&gt;S^2 = 4 As ( c)&lt;/math&gt;.

[[39|39]]. Denotet x horizontalem rectam MK , y vertica
lem KQ , et h angulum CMK ; erunt x = S cosh , y =
CK - CQ = S sin h -5 ; unde
X
X
S =
cosh . sinh :
cosh
quibus valoribus substitutis in (c) , prodibit
x2 rcsinh
4 A
CO -Y) ,
et consequenter cos2 h
cos h
y =xtang h 1 + tang k
4 A
x2 ( c' ) .

[[40|40]]. Haec facile nunc stabiliuntur. 1º facta y = 0 ,
proveniet amplitudo jactus
4 Atangah
1 + tang h
4Asinhcosh = 2 Asin2h.
2.º Inde sequitur maximam jaclus amplitudinem
haberi sub angulo h = 45°.

3. ° Si quaeritur angulus h , sub quo proiicien
dum est grave ut offendat in datum scopum , cujus nempe
dantur coordinatae x et y , erit
2A + V 4A2-4 Ay - x2
tangh
82
aequatio ad curvam (dicitur parabola) in qua defertur
grave. Si altitudo debita (28) velocitati v, dicatur A, erit
vio:2gA, et aequatio trausformabitur in
S':4As (c) Esistente igitur 4 A 2—4 Ay-x &gt;0 , poterit sub duplici
angulo projici grave ut datum -scopum attingat : attinget
autem in fine temporis ( 38 : 39 )
S
ts Vo .
Vo cos h

4.0 in ( c ) pone
2 Atangh ta ;
1 +tangah
babebis
A tangah
1+ tang2h ya
1 + tang 2h W?
4A
( c ' ' ) .
Iam vero maxima y ( dicitur altitudo jactus ) manifeste re
spondet valori w = 0 ; altitudo igitur jactus exhibebitur
per
A tangah seu A sinh.
1 +tangah

5º . Ex eadem ( c &quot; ) quisque colligit parabolam , in qua
defertur grave, dividi a maxima y in duas aequales simi
lesque partes : extremitas maximae y vocatur vertex pa
rabolae; ipsa vero maxima y indefinite producta juxla gra
vitatis directionem appellatur axis parabolae.
[[Fasciculus:Ferde hajitas7.svg|thumb]]
6º Si angulus h fit &lt; o, ut initialis directio cadat iтfra horizontalem rectam ML, jactus amplitudo x (1°) ex &gt; fiet &lt; 0; jactus vero altitudo y ( 40 ) permanebit &gt;o. Quod si fuerit h = o, ut initialis directio recidat in rectam horizontalem ML, nulla erit amplitudo jacеus, nullaque ejus altitudo.

7º. Demittatur perpendiculum QP ex puncto Q
parabolae in axem NI ... , sintque NP = x', Q P =y'; erunt ( 1º . 4º . ) x MI — QP = 2 A sinh con -y'
y=NI — NP = A sin’h— x' : quibus valoribus substitutis
in ( c' : 39 ) , proveniet
y2= 4 A x' cosah
aequatio ad parabolam M N L inter x' ety' computatas a
vertice ; quantitas 4 A cos’h dicitur parameter parabolae ;
quod si in axe sumatur punctum H ita , ut ejus distantia
a vertice sit quarta parametri pars seu A cos ?h , habebitur
punctum illud , quod appellatur parabolae focus.

[[41]]. Cum ad curvam parabolicam describendam, corporis motus, qui fit secundum lineam projectionis, debeat esse aequabilis, qui vero fit secundum lineam verticalem, debeat esse uniformiter acceleratus, cumque hujusmodi certe neuter esse possit si medium utrique motui resistat, iccirco nonnisi in vacuo motus corporis oblique projecti fieri potest per curvam, quae sit perfecte parabolica. In medio resistente curva minus late patet, minusque assurgit quam in vacuo; duobus insuper cruribus dissimilibus &lt;math&gt;AN, NL&lt;/math&gt; (Fig. 26) componitur, quorum descendens &lt;math&gt;NL&lt;/math&gt; ad rectam quamdam &lt;math&gt;FE&lt;/math&gt; ut asymptotum accedit in infinitum, quin unquam congruant. Etenim resoluta projectionis velocitate in duas, alteram verticalem, alteram horizontalem, verticalis tum ab aeris resistentia, tum a gravitate usque ad punctum &lt;math&gt;N&lt;/math&gt; minuetur: propterea punctum &lt;math&gt;N&lt;/math&gt; minus assurget quam in vacuo: postquam grave ad &lt;math&gt;N&lt;/math&gt; pervenerit, descendet ob gravitatis vim damna ex medii resistentia reparantem, et hujusmodi descensus fiet motu verticali ad motum aequabilem (33) semper accedente. At horizontalis velocitas minuitur perpetuo, nulla interim vi iacturam reparante, atque inde fit ut recessus horizontalis a recta verticali &lt;math&gt;NP&lt;/math&gt; certum limitem non praetergrediatur, quem curva habet pro asymptoto. Haec contingunt potissimum corporibus ingenti velocitate in aere projectis.
=== De generalibus quibusdam proprietatibus motus curvilinei, orti a viribus, quarum una determinat materiale punctum ad motum aequabilem, altera ipsi materiali puncto est continue applicata.=== 
[[42|42]]. Concipiamus secundam vim agere solum in initiis quorundam tempusculorum, ac tantam velocitatem unico impulsu valido producere, quantam vis perpetuo agens producit toto illo tempusculo, ut deinde inminuta magnitudine tempusculorum in infinitum, habeatur linea curva orta ex continua vis actione.  

Projecto puncto materiali cum velocitate CB (Fig. 27) simulque illi impressa velocitate CA, abiret punctum per diagonalem CO parallelogrammi AOBC et esset in fine primi tempusculi in O cum determinatione describendi altero aequali tempusculo rectam OL = OC, eique in directum jacenlem. Si hic iterum illi imprimeretur alia velocitas OF, completo parallelogrammo FILO , incederet per diagonalem OI, essetque in fine secundi tempusculi in I cum determinatione describendi tertio tempusculo aequali rectam IM = 10, eique in directum jacentem. Sed ob impressam hic quoque aliam velocitatem IV abiret per novam parallelogrammi diagonalem IH, atque ita porro. Fieret ergo in ejusmodi hypothesi vis agentis per intervalla tempusculorum ut materiale punctum describeret polygonum COIHN etc, cujus latera certam magnitudinem et positionem haberent, definita nempe a directione virium et a ratione velocitatum, quas initio cujusvis tempusculi mobile obtineret. Hinc pro diversis virium ila agentium ordinibus numero infinitis infinita considerari possunt ejusmodi polygona, quorum alia in se ipsa redirent, desinente ultimo latere in puncto C ubi primum inceperat; alia abirent in infinitum. Concipiamus jam numerum tempusculorum augeri, et simul eorum magnitudinem imminui in infinitum, vitum magnitudine tum directione vel constantes manere, vel variare certa quadam lege ad continuam quamdam variationis rationem accedente in infinitum. Augebitur in infinitum numerus laterum polygoni determinato tempore descripti, imminutis interea in infinitum angulis, quos efficit quodlibet latus praecedens cum consequente: cum enim LI debeatur impulsui, qui initio tempusculi 0 eam
velocitatem producere concipitur, quam produceret vis to
to tempusculo agens, cumque per tempusculum infinitesimum vis ista habenda sit pro constante, existet ( 28: 30. 14. ) LI =
092; ideoque ob o finitam, et quadratum 62 infinitesimum secundi ordinis, erit etiam LI infinitesima ordinis secundi, sed OL est infinitesima ordinis primi, utpote quae tempusculo O describitur cum velocitate finita; ergo angulus LOI erit ivfinitesimus: atque eodem pacto demonstrantur infinitesimi anguli MIH , K'HN , etc. Hinc polygonum ad curvam continuam semper magis accedet; et ubi demum continua habealur actio vis, et continuae cuidam legi subjiciantur directio ipsius et magnitudo, obtinebitur curva continua cavam sui partem versus eam plagam obvertens, in quam tendunt vires.

43. Abeunte polygono in curvam , rectae CL , OM' , IH ', HK , etc abeunt in tangentes apud puncta C, O, I, H , etc. Ubi ergo in aliquo curvae puncto vis desinat agere,,
excurret mobile per tangentem apud illud punctum.

44. Sit IM (fig, 28 ) spatiolum quod tempusculo 9
mobile percurreret sola velocitate praeconcepta, et IV
spatiolum respondens vi agenti unico impnlsui valido ; ita
ut existat (42) IV ::99&quot;. Completo parallelogrammo, positis-
que lM:P ,
lH:B, et angulo MIV :i, erit (9. 3.&quot; )
∶∶ Vra-Hæ os −⊢⋅∠⇂⊃⊊↶⊖⋍ cos :.87
Evolvatur quantitas radicalis in seriem : proveniet
R = P + q9 cos i , unde R - P = º02cosi ,
neglectis infinitesimis altioris ordinis. Sit v' velocitas , qua
mobile percurrit laterculum R; erit R = v'0 : sit etiam v
velocitas , qua mobile percurreret laterculum P. si non
adesset impulsus validas in I ; erit P =v @ : hinc R --
P = vv( ) 0 .; et consequenter
v ' - v = q Ocosi.
Ex hac aequatione patet v— esse quantitatem in
finitesimam primi ordinis , positivam vel negativam prout
i &lt;vel &gt; 90° , esse autem =0 si i 90° . Inferimus il
lud : ubi tempore finito angulus , quem efformat vis ac
celeratrix cum directione tangentis , fuerit semper aculus,
acquiret mobile incrementum velocitatis finitum ; si sem
per obtusus , patietur decrementum finitum ; si semper re
lus , velocitas manebit constans.

45. In motu quovis curvilineo quadratum velocitatis aequat vim acceleratricem ductam in dimidium chordae, quae ex ejus directione abscinditur a circulo osculatore. Denotet enim a lineolam infinitesimam IM (Fig. 29) ut sito et consequenter IV = 902
cipiatur circulus , qui transiens per tria puncta 0 , I ,
H ( fig . 27. 29. ) habeat centrum in G , quique erit
circulus osculator apud curvae punctum O ; producantur
IV , MH donec occurrant peripheriae in G &quot; , G '' ; et ex''
G ducatur perpendiculum GGʻad chordam IG &quot; : erunt
IG &quot;
MG &quot; = IG &quot; = ICE Est autem MH . MG ' &quot; : MI. MO;
2
ergo MH . 21Gʻ = MI.MO = MI . 2MI , seu 21G'
2x2. Hinc v2 = . IGʻ ; ideoquc etc. Porro angulus IGG' =
2 Oxa

; con

.
px ?
22
87
Evolvatur quantitas radicalis in seriem : proveniet
B:P −⊢ o9zcos i
, unde B—P:cp92cosi ,
neglectis infiuitesimis altioris ordinis. Sit 'v' velocitas , qua mobile percurrit laterculum R; erit R: 0'9: sit etiam »
velocitas , qua mobile percurreret laterculum P. si non
adesset impulsus validus in I, erit P:-v 9: 'hinc R
-— P:(v'--v)9; et consequenter
v'—v:cp9cosi .
Ex hac aeqnatione patet 'o'—v esse quantitatem in-
fiuitesimam primi ordinis
, positivam vel negativam prout
i(vel 90&quot; , esse autem :0 si 1&quot;: 90&quot;. Inferimus il-
lnd : ubi tempore finito angulus, quem efformat vis ac-
celeratrir cum directione tangentis ,
fuerit semper acutus,
acquiret mobile incrementum velocitatis finitum; si sem-
per obtusus, patietur decrementum finitum; si semper re-
ctus, velocitas manebit constans.

45. In motu quovis curvilineo quadratum velocitatis
aequat vim acceleratricem ductam in dimidium chordae,
quae ex eius directione abscinditur a circulo. osculatore.
Denotet enim a lineolam infiuitesimam IM (fig. 29. )
gox- ; con-
92
cipiatur circulus, qui transiens per tria puncta 0, I,
II (fig. 27. 29..) habeat centrum in G, quique erit
circulus osculator apud curvae punctum 0; producantur
IV, MH donec occurrant peripheriae in G&quot;, G'&quot;; et ex
G ducatur perpendiculum GG' ad chordam IG&quot;: erunt
MG&quot;':IG&quot;, −−−∙−↧∁⇀−− ⇀∸−↧−⊊≩−⋅∎−∙ Est autemMH. MG'&quot;:MI. MO;
ut sit :9 i,
et consequenter IV:
99&quot;:
'»
ergoMH.21G':MI.-:MO MI. 2Ml.seu—-— &quot;&quot;&quot;&quot; ,210':
'v&quot;
.Hiuc v&quot;: 39. lG' ; ideoque etc. Porro angulus IGG'— −∙∙
−∙↼⇀−−
. −↼∙⋅⋅∙∙⋅↼−∎∣ −↼
∙∙∙88
90 ° -GIGʻ = 900 (MIV - MIG ) = 90 ' - ( i - 90 °) = 180 °-i ;
proinde , denotante r radium GI , erit IG ' = rsin IGG' =
rsini , et consequenter
va = grsini ( b ) .

:

46. Haec vera sunt de omni virium genere. Sint nunc
vires acceleratrices directae ' ad centrum datum : in casu,
curva ColH .... ( fig . 27. ) jacebit in plano transeunte
per rectam projectionis et per centrum virium ; quod fa
cile intelligitur , cum in eodem plano agant vires omnes.
Viribus ad punctum aliquod S tendentibus , radius
vector ( est recta , quae ab S ducitur ad mobile ) descri .
bet areas circa idem punctum temporibus proportionales ,
et viceversa.
Quod spectat ad primam assertionis partem , assum
ptis tempusculis aequalibus , et ducta recta SL conside .
rentur triangula SCO , SOL , SOI : est SCO = SOL ,
cum sivt super bases CO , OL aequales ob aequali
tatem tempusculorum , eamdemque habeant altitudinem
est etiam SOL = SOI , quia insistunt ambo eidem basi
SO, et sunt inter easdem parallelas SO , LI : ergo SCO
SOI. Eodem modo ostenditur triangula SOI , SIH aequa
lia esse eidem SIM , et proinde aequalia esse inter se ,
et similiter omnes areas sequentium triangulorum aequa
les esse inter se et cum areis praecedentibus. Quare cum
temporibus finitis quibuscumque contineantur numeri tem
pusculorum aequalium ipsis temporibus proportionales, areae
terminatae polygoni perimetro et rectis ad centrum virium
pertingentibus , hoc est compositae a lot areolis triangu
lorum aequalium quot tempuscula respondent illis tem
poribus , quibus perimetri partes describuntur , erunt
ipsis temporibus proportionales . Cum autem id locum ha
beat quomodocumque augeatur numerus tempusculorum,
eorumque magnitudo imminuatur , consequens est ut, ubi
⇤
88
⊖∘∘∙∁≖↧∁↾⋅ :soc—(MIV—MIG) :90&quot;—(i—gO&quot;):180&quot;—i ;
proinde , denotante r radium GI, erit IG':rsin IGG':
rsini , et consequenter
-v&quot;:g9rsini (6).

46. Haec vera sunt de omni virium genere. Sint nunc
vires acceleratrices directae'ad centrum datum: in casu,
curva COIH
.. ..
(Gg. 27. ) jacebit in plano transeunte
per rectam projectionis et per centrum virium; quod fa-
cile intelligitur , cum in eodem plano agant vires omnes.
Viribus ad punctum aliquod S tendentibus, radius
vector (est recta
, quae ab 5 ducitur ad mobile ) descri-
bet areas circa idem punctum temporibus proportionales,
et viceversa.
Quod spectat ad primam assertionis partem, assum-
ptis tempusculis aequalibus, et ducta recta SL conside-
rentur triangula SCO, SOL , SOI: est SCO:SOL,
cum sint super bases CO, OL aequales ob aequali-
tatem tempusculorum, eamdemque habeant altitudinem:
est etiam SOL :SOI . 'quia insistunt ambo eidem basi
50, et sunt inter easdem parallelas SO, LI : ergo 500:-
SOI. Eodem modo ostenditur triangula SOI , SIH aequa-
lia esse eidem SIM ,
et proinde aequalia esse inter se ,
et similiter omnes areas sequentium triangulorum aequa-
les esse inter se et cum areis praecedentibus. Quare cum
temporibus Gnitis quibuscumque contineantur numeri tem-
pusculorum aequalium ipsis temporibus proportionales, areae
terminatae polygoni perimetro et rectis ad centrum virium
pertingentibus , hoc est compositae a tot areolis triangu-
lorum aequalium quot tempuscula respondent illis tem-
poribus
, quibus perimetri partes describuntur ,
erunt
ipsis temporibus proportionales. Cum autem id locum ha-
beat quomodocumque augeatur numerus tempusculorum,
eorumque magnitudo imminuatur , consequens est ut, ubi89
demum polygonum abit iu curvam continuam , areae ter
minatae arcu curvilineo et rectis ad centrum virium ten
dentibus sint itidem temporibus proportionales.
Ad secundam assertionis partem quod spectat , sint
areae SCO, SOI, aequalibus temporibus confectae , omnino
aequales. Quoniam producta CO in L ita , ut existat OL =
CO, est triangulum SOL = SCO, idcirco SOL =SOI; sed
baec duo triangula habent basim communem SO ; erunt
igitur inter easdem parallelas, ideoque IL erit parallela re
ctae So. Ducatur IF parallela ad OL; motus per Ol com
ponetur ex duobus per OL et OF , quorum prior cum
oriatur a determinatione motum praecedentem continuandi
per C O , certe posterior a vi acceleratrice generabitur ,
quae propterea dirigitur ad centrum S.

47. Velocitas qua pollet mobile in eadem curva , est
reciproce proportionalis perpendiculo e centro virium du
cto in tangentem . Velocitas enim mobilis in quovis latere
polygoni est ut ipsum latus ob aequalia tempuscula , quibus
unumquodque latus percurri supponimus : est autem unum :
quodque ejusmodi latus reciproce ut perpendiculum quod
ex centro virium ducitur in latus ipsum ; siquidem id
perpendiculum habent pro altitudine triangula illa exigua
polygoni , si hujus latera pro eorumdem trianguloruin basi
bus assumantur ; ea insuper triangula sunt aequalia , et in
triangulis aequalibus debent bases esse in ratione recipro
ca altitudinum : est igitur ea velocitas reciproce ut per
pendiculum ductum ex centro virium in latera polygoni.
Sed abeunte polygono in curvam continuam , directiones la
teruın abeunt in tangentes ; ergo velocitas mobilis in quo
vis curvae puncto erit reciproce ut perpendiculum ex cen
tro virium in langentem demissum.

48. Denotet a areolam NSZ , et g perpendiculum SE
ductum ex centro S in laterculum NZ ; describetur NZ ve
NZ 2a ; siquidem NZ.SE=2NSZ: hinc ( 45 ) o
locitate v=
90
7
89
demum polygonum abit iu curvam continuam
, areae ter-
minatae arcu curvilineo et rectis ad centrum virium ten-
dentibus sint itidem temporibus proportionales.
Ad secundam assertionis partem quod spectat, sint
areae SCO, SOI, aequalibus temporibus confectae, omnino
aequales. Quoniam producta CO in L ita, ut existat OL:
CO, est triangulum SOL:SCO, idcirco SOL:SOI; sed .
haec duo triangula habent basim communem SO.; erunt
igitur inter easdem parallelas, ideoque IL erit parallela re-
ctae SO. Ducatur lF parallela ad OL; motus per OI com-
ponetur ex duobus per OL et OF, quorum prior cum
oriatur a determinatione motum praecedentem coutinuaudi
per C 0, certe posterior a vi acceleratrice generabitur ,
quae propterea dirigitur ad centrum S.

49. Quoniam radius vector , juxta quem agit vis con
tinua , potest assumi ut sibi parallelus per tempusculum
quodvis infinitesimum 0 , ipsaque vis ut constans per to
tum illud tempusculum ; ideo si mobile K incedens cur
vam CX ( fig. 30 ) viribus ad centrum S tendentibus de
scribit arcum infinitesimum HN labente , ductis SH ,
SN , et producto SN donec occurrat in H' tangenti HH &quot; ,
lineola recta H'N repraesentabit motum relativum mobi
lis K quoad ipsum Kieps per HH' sola vi praeconcepta in
H. Igitur cum motus iste relativus sit unice repelendus
( 5 ) a vi continuata per tempusculum e , exsistet
H'N son (6&quot;).

50. Haec subiungimus . 

1.&quot; Sive vires tendant ad
centrum datum
, sive non; denotantibus any, :coordinatas
puncti materialis in fine temporis t
, profecto x ,r,:peu-
debunt ab ipso :; erunt videlicet æ, y, :functiones tem-
peris :, ut scribi possit
.
——-—————.——-—-——-——.—.——...———..—91
= f ( ) , y = fi ( ) , z = 12

2. • Si vocatur s arcus a materiali puncto percursus
tempore t, w velocitas ejusdem puncti in fine ipsius t , pe
rinde spectari poterit ds ac si motu uniformi conficeretur ,
sola nimirum velocitate praeconcepta v ; siquidem nova
velocitas, dv , quae labente dt accedit materiali puncto ,
est infinitesima . Propterea hic quoque ( 28 )
ds
dt

3.º Resoluta vi o in duas , quarum altera sese
dirigat juxta tangentem , altera juxta respondentem nor
malem , erit ( 44) prima
des o cos i duษ
dc &gt;
dta
secunda (45 )
2² ♡ sini
ds²
r rdta

4.°# Incedente puncto materiali K per arcum s , mo
vebuntur motu rectilineo projectiones K' , K &quot;, K '' ipsius K in''
coordinatis orthogonalibusque axibus OX , OY, OZ ( Fig.5 ) ,
eruntque ( 28 )
dx dy dz
dt dt dt &gt;
earum velocitates in fine temporis : , quum nempe K ha
ds bet ( 2 ) velocitatem Vi acceleratrice dc
K , resoluta in ternas P ', P &quot; , D' ' ' iisdem axibus parallelas,
. , qua sollicitatur
∙ 91-
x:f(t)-J:fx(t)o 2:130)-
2.&quot; Si vocatur .: arcus a materiali puncto percnrsus
tempore :, v velocitas eiusdem puncti in fine ipsius t, pe-
rinde spectari poterit ds ac si motu uniformi couGeeretnr ,
sola nimirum velocitate praeconcepta v, ∙ siquidem nova
velocitas dv , quae labente dt accedit materiali puncto ,
est infinitesima
. Propterea hic quoque (28)
ds
Pr.—...... dt
3.&quot; Besoluta vi 9 in duas , quarum altera sese
dirigat juxta tangentem , altera juxta respondentem nor-
malem
, erit (44) prima
'n'—'v—d'v-Sd&quot;: cpcost—p ,9 de dt&quot;
secunda (45) ⋅
'
∙⋅ &quot;
'
∙ . ,,,a d;:
cpsmr— r — rdt&quot;'
4.&quot;e Incedente puncto materialiK per arcum :, mo-
vebuntur motu rectilineo projectiones K', K&quot;, K'&quot; ipsius Km
coordinatis orthogonalibusque axibus OX, Oï. OZ (Frg- 5) :
eruntque (28)
'de: (I)-' dz
dt
'
dt '
dt
earum velocitates in Gne temporis :, quum nempe K ha-
bet (2&quot;) velocitatem? .Vi acceleratrice , qua sollicitatur

: - -

K
,
resoluta in ternas P', P&quot;, P'&quot; iisdem axibus-parallelas,92
motus projectionis K' nihil erit aliud nisi motus rela
tivus puncti K quoad ipsum K sollicitatum viribus dum
dx taxat P &quot; , P ''' ; proinde velocitas debelur soli P' ex dt'''
dr ternis P' , P &quot; , P &quot; ; simili ratione ostenditur. deberi soli dt
dz P &quot; ex ternis P' ,P &quot; , P , et soli P&quot; ' ' ex iis 'componenti
dt
bus . Hinc ( 28 )
adx ddy adz
de de dt
P' ,
P &quot; , = P &quot; , dt de dt
seu
dex day daz
dt2 P'
dia
P &quot; ,
di?
= P &quot; .

5. °* Si punctum materiale incedit curvam plagam,
sumptis axibus v. gr . OX , OY in plano curvae , habebuntur
tantummodo
der day
de² P ' ,
dia
= P &quot; .
Fac v. gr. ut vis acceleratrix o sit parallela axi OY , ita
lamen ut sese dirigat ad plagam ordinatae y negativae : erunt
P = 0 , P :
ideoque
d2x
dla 0,
dy
di ?
Istarum prima suppeditat
I
92
motns projectionis K' nihil erit aliud nisi motus rela-
tivus puucti K quoad ipsnm K sollicitatam viribus dum-
taxat P&quot;, P&quot;'; proinde velocitas .j—f. debetur soli P' ex
ternis P', P&quot;, P'&quot; ; simili ratione ostenditur-(g.; deberi soli
&quot;
∙ ∙∙∙ dz ∙ n ∙∙ P ex ternis P', P&quot; ∙ ∙
, P , et −− soh P' ex 11s'componeut1-
dc
bus . Hinc (28)
' ddf ddZ ddi dt dt dt
∙−−− −−∶ '. ∙−−−:
P. −∙∙:
dt dt '&quot;
'
de P '
seu
'
' ⊒ ∙
' '
d3æ (137 d&quot;:
dt&quot; −−−∶ P
'
∙−− ∙−−− ∙−:
P
'di&quot; P '
dt&quot;

5.&quot;; Si punctum materiale incedit curvam planam,
Sumptis axibus v. gr. OX, O? in plano curvae , habebuntur
tantummodo .
⋅ ⋅
dzæ - d
dc&quot; :&quot;)&quot;Zïz'
Fac v. gr. ut vis acceleratrix q; sit parallela axi Oï
,
ita
tamen ut sese dirigat ad plagam ordinatae] negativae :erunt
ideoque
dh: ∙∙ d'] ∙∙∙ ∙∙
dt&quot; —0' −↲⋅≀⋅⇀≖− ?
Istarum prima suppeditat93
dx
dt C , x =Ct +C' ;
secunda, in hypothesi o constantis , praebet
dy ota
dt ot + C &quot;, y = 2 +0&quot; 4 + C &quot; :
eliminato t ,
y
y = c&quot; + * (** ) (* = ) .
Habes itaque, in ea qua sumus hypothesi , coordina
tas x ety expressas ( 10) per t; habes insuper aequatio
nem ad curvam, quam describit materiale punctum : re
stat ut constantes arbitrarias C, C' , C &quot;, C '&quot; determinemus.
Ponatur materiale punctum ex ipsa coordinatarum origi
ne O projici cum velocitate Yo juxta rectam inclinatam
ad OX sub angulo h: resoluta v. in' binas, alteram paral
lelam axi Ox, alteram parallelam axi OY, erit illa = v , cosh,
haec Vo sinh: initio motus obtinent simul
t = 0 , x = y = 0 ,
dx
dt = v , cosh,
dy
dt = V , sinh ;
igitur
C = Vocosh , C = 0.C &quot; = V . sinh , C = 0 ;
et consequenter
012
x = vol cosh ,y = v , sinh -
csinh
cosh
gx2
2v.cosh
93
dr
.
E—:C, æ:Ct-l-C,
secunda, in hypothesi ? constantis , praebet
' d ∙
'

: &quot; 73: :,n—j-c'.7:— ∙≌⇉−−−⊦∁∥≀−⊦ ∁⋯≖ eliminato t ,

∜−⋅−−−≺⋮⋅⋅⋅⊹∁∣⋅ (&quot;€&quot;)— −≣−≺∙≄ ;
&quot;):
.
Habes itaque, in ea qua sumus hypothesi, cbordina-
tas æ ety expressas (1&quot;) per :; habes insuper aequatio-
nem ad curvam, quam describit materiale 'punctum: re-
stat ut constantes arbitrarias C, 0, C&quot;, C'&quot; determinemus.
Ponatur materiale punctum ex ipsa coordinatarum origi-
ne O projici cum velocitate vo juxta rectam inclinatam
ad OX sub angulo h: resoluta v., in' binas., alteram paral-
lelam axi OX, alteram parallelam axi Oï, erit illa:vo cos 11,
haec:vo sin/1: initio motus obtinent simul
da: dy .
t.:o,x:o,y:o, ï

: vo cosh, ?::v., smh;

igitur
C: vo cosh , C': o .C&quot;: vo sin]: ,C&quot; ':o;
et consequenter '
cpt&quot; æsiuh (pa-&quot;
2 &quot;7— cos/1 -21Jo&quot;cos&quot;lt :

: votcOsIt,y:vosiult—94

x tangh -
9 1+ tangah
2 v2.
22.
Recole quae diximus ( 39).

6°# Fac nunc ut, permanentibus caeteris ( 5º. ) , pun
clum materiale moveatur in medio resistente: poterit vis ac
celeratrix ex resistentia medii exprimi ( 32. 33 ) generatim
per f (v ) ; per functionem videlicet velocitatis v
tem , decrescentem , evanescentem simul cum v Sit \beta an
gulus interceptus directione motus et ordinatarum axe OY ;
erunt ( 32 ) P' f (w) sin \beta , P &quot; = -- flv) cos \beta ;
ideoque
crescen
dar d²y : - flv )sin\beta , = -9 - flu) cos\beta ( c ) :
dt2 dla
insuper ( 40)
dx
dt
dy v sin\beta ,
dt = v cos\beta (c' )
quae differentiatae suppeditant
d22 dy d\beta dy do d\beta
dt sin\beta tvcos\beta dt dt2 dt cos\beta — v sin\beta ordt dt2 .
Ergo
dv
sin \beta + y cos \beta
d\beta
dt dt : -f (v )sin\beta,
do
de
d3 cos \beta-usin \beta 0 - f v ) cos\beta: dt
94
x tangh .::
t—ïngïhæt
Recole quae diximus (39).

604: Fac nunc ut, permanentibus caeteris (50.),ptm-
ctum materiale moveatur in medio resistente: poterit vis ac-
⋅ celeratrix ex resistentia medii exprimi (32. 33) generatim
per f(v); per functionem videlicet velocitatis v crescen-
tem
, decrescentem
,
evanescentem simul cum 0Sit B an-
gulus interceptus directione motus et ordinatarum axe Oï;
erunt (32) P':
-
f(v) siuþ ∙P&quot;: −− ?
−f(P) 008 p;
ideoque
d'æ
dt: :—ftv)sinþ,d —:— —f(v) cosþ (c):
insuper '(40)
da: . d .
'at—:&quot;.lnþO £: &quot;waþ (0)
quae diB'erentiatae snppeditant
dzæ
−↙⊼≖−−∶−⋇⋮∐⇪ ⊣−∙≀∘∞⇪⊼ d'B. dz :d—ïcosþ— —vsinþ dþ
dt
Ergo
——sin,8 −⋅⊢ vcos 5—d—-5 −∙−−−∙ —-f(v)sin,8,
dv
Ft— cosþ—vsin B (35—:— ep —f(v) cosþ:95
istarum primam multiplica per sin\beta , secundam per cos\beta,
tum collige in summam; eamdem primam multiplica per
cos\beta , et secundam per sin\beta , cum subtrahe; habebis
dy d\beta
+ fv) =– pcos\beta,
=
Psins
(c' ) .
dc dt
Quibus positis, haec stabilientur: cum nequeat \beta fie
ri &gt; 180° ( siquidem in transitu . per 180° vires omnes e
vaderent verticales, motusque permaneret verticalis ) , cum
que p etv existant perseveranter &gt; 0, ob secundam ( c &quot; ) erit
d\beta constanter 0 ; proinde crescente e crescet semper an dt
gulus \beta accedendo ad quemdam limitem B.
In hypothesi anguli initialis \beta. (=90° - h)&lt;90°, per
get o cos \beta per aliquod tempus esse &gt; o : sed flv ) &gt; 0 ; i
gitur , ob primam ( c''), per totum illud tempus erit de''
et consequenter crescente t decrescet v.
Prima ( c&quot; ) differentiata praebet
du
&lt; o .
d2v dv d\beta gsin\beta ;
dt -
dea + au f '(o )
seu , attenta secunda ( d ''),''
dev dy

dia
+ áf ( ) = q *sin- B
dv
facta igitur dt , emerget
dev oʻsina&gt; o.
dt
95
istarum primam multiplica per sin 13, secundam per cosþ, ⋅
tum collige in summam; eamdem primam multiplica per
cosþ ,
et secundam per. siuþ ,t'um subtrahe; habebis
∙ d d ∙
⊋⋮∙∙⊣−∣↻⇝⇌− ws?- ∙⊺∙↙↙⋛−∶∶∲−−−∘∎⋮∙∂ (a&quot;)-
Quibus positis, haec stabilientur: cum nequeat. þ Ge-
ri )180o ( siquidem in transitu.per 1800 vires omnes e-
vaderent verticales, motusque permaneret verticalis ), cum-
que (p et v existant perseveranter o, ob secundam (e&quot;) erit ↭ ∣
d ∙ ⋅
constanter £
)a; promde crescente :
crescet semper an-
gulus þ accedendo ad quemdam limitem B.
In hypothesi anguli initialis B., (:::90() -H( 90&quot;,per-
get ? .cosp per aliquod tempus esse ∘:sed iv))o'; i-
gitur
, ob primam (e&quot; ), per totum illud tempus erit ⋚∶≺∘∙ et consequenter crescente :decrescet 0.
Prima (e&quot;) differentiam praebet
⋣≖−⊦↙↨−⋛∣≼⋅⇝⇌≡≴∊∹∾⋅≖⋅∣⋮⇋ dav d d . ∖∖ seu, attenta secunda (c' '),
d'v dv ∙∙∙ ∳≖∘⋮∐≏∆⊙ ∙
⊄⋮⋮⋝⊹⊋−∑ f(V)-— v '
. .
dv
facta igitur 22 :o
, emerget
dav cp'sinïþ
üt: ⇀−− v
)0-96
Inferimus ( 27. 22°. ) velocitatem v haud exsistere capacem
maximi: poterit quidem esse capax minimi ; ita tamen , ut
mutato decremento in incrementum, hoc neque vertatur ite
rum in decrementum, neque certum quemdam limitem E
praetergrediatur. Primum patet ex eo quod , posita conver
sione incrementi in decrementum, jam obtineret maximum:
secundum ex eo quod, aucta indefinite v, augeretur indefi
dv nite flv ), simulque foret &gt;0 ; id vero adversatur pri dt
mae ( 6' ) .
Ex ( c &quot;) eruuntur binae
20 21
V2-01 ſię cos$ + fvde,B2- B;= Sosiu\beta dt ;
t t
exprimunt N,, V, velocitates , item B , B, angulos limitibus t,
2t respondentes. Fac
o cos\beta + v ) = f (t) ,
psins = fa (t) :
habebis ( 27. 18º. )
V; - v.--tfittat) • B. - = falttal) ;
exprimunt a et a numeros &gt; o et &lt; 1. Sed crescente t in
definite , vergit fi (t) ad q cosB + f (E ),et fu( t) ad qsinB
E
ac proinde
2 - -V2
limes quantitatis cos B + F( E ) ,
3. - 22 O
limesque quantitatis
sinB
E
96
∙
Inferimus (27. 220.) velocitatem v haud exsistere capacem
maximi: poterit quidem esse capax minimi; ita tamen, ut
mutato decremento in incrementum,hoc neque vertatur ite-
rum in decrementum,- neque certum quemdam limitem E
praetergrediatur. Primum patet ex eo quod, posita conver-
sione incrementi in decrementum, iam obtineret maximum:
secundum ex eo quod, aucta indefinite v, augeretur indefi-
nite f(v). simulque foret-(£)o, ∙ id vero adversatur pri-
mae (c&quot; ). ∙
Ex (e&quot;) eruuntur binae
2t
∙↗−⇂↗≖ −−−∙−− ∙∣ ( ? cosþ-l- fwndz, ↾⊖≖−,B— fra-018 de;
exprimunt v, , v, velocitates,' 1tem (i,, ,H, angulos limitibus !,
2t respondentes. Fac
9) cosB ^v):fd!) ∙∲≊∣∶∁ :
fam
habebis (27. 180.)
'Ur—vzzf— tf1(t&quot;l&quot;at) ∙⇪≖−−⇪≃∶∶⊀≖↸≖⊣−⊄⋅∁⋟⋮
exprimunt a: et «' numeros )b et ↿∙ Sed crescente :in-
≺↿⊜∊⊓⋮⇂∊∙ &quot;&quot;sit fxw ad 90053 —I-f(E).et rm ad ?&quot;&quot;B-
ac proinde
2! -—v

limes quantitatis :

Bos B4-f(E) ,x—Bz ↽− wir-B : .
limes ue uantitatis . q q E97
quoniam igitur
VI - V2 lim.
B - \beta ,
0 lim t t
erunt
Ø cos B + f(E)= 0 ; sin B
E
et consequenter
B = 180° , f(E )= .
Ex istarum prima inferimus motum materialis puncti ver
gere ad rectilineum verticalemque motum; e secunda ( viri
bus p et medii resistentis sese in limite elidentibus, utpo
te aequalibus et contrariis ) ad motum uniformem , proce
dentem videlicet a sola vi praeconcepta.
Divide primam ( c&quot; ) per secundam (c&quot;) : proveniet
dx
d\beta
sie
X-X
B-Brvm?;
iccirco ( 27. 18º. )
i\beta
Spa\beta
Q Q
Bm
exprimit um valorem medium velocitatis v. Haud praeter
greditur ' ' m certum quemdam valorem finitum ; insuper ver
git \beta ad B= 180° : ergo neque x praetergredietur finitum
valorem; ideo que materiale punctum incedet curvam prae
ditam asymptoto verticali. Recole, quae diximus nº. 41 .
Posita ( 33. 4º. ) flv )
formulae ( c) evadent k?
qua 1
quoniam igitur
&quot;r'—Va lim. :o, lim Bi—Ba :.0,
erunt
?
∘∞↿∃⊣−⊀≺≖∙∶⊢− 0
∙
∲≕⋮⋮∶⊔∄−∙−− −−∘⊰
et consequenter
3:180&quot; ,f(E):9.
Ex istarum prima inferimus motum materialis puncti ver-
97
gere ad rectilineum verticalemque motum; esecunda(viri-
bus 91 et medii resistentis sese in limite elidentibus, utpo-
te aequalibus et contrariis ) ad motum uniformem, proce-
dentem videlicet a sola vi praeconcepta.
Divide primam (c') per secundam (e&quot;) :proveniet
iccirco ( 27. 180.)
exprimit v,, valorem medium velocitatis
,,,
Haud praeter-
greditur v,, certum quemdam valorem finitum; insuper ver-
git B ad B: 1800: ergo neque æ praetergrediatur finitum
valorem; ideoque materiale punctum incedet curvam prae-
ditam asymptoto verticali. Recole, quae diximus n&quot;. 

41.
Posita ( 33. 40.) f(v):SE,-2 , formulae (c) evadent
.k?98
dar
di ?
-sing,
day
dla 9
qua
cos\beta : ka
sed haec hactenus.
7º. Intelligantur per coordinatarum orthogonalium
originem O ( Fig. 5 ) duci binae rectae 8,0&quot; intercipien
tes angulum a : earum extremitatibus junctis recta d '&quot;, erit
cosa =
02 +02.02
28 &quot;
Extremitas rectae , habeat coordinatas a ', y, z ', rectae au
tem o coordinatas x &quot;, 1 &quot; , 2 &quot; : paullulum attendenti pate
bit fore
õ = x's + y + 2,0% = &lt; &quot; + ya + z'2 ,
d's = (x - x &quot; )2 + 6 - y &quot; )2+ (z'- z&quot; )?;
adhibitis substitutionibus ,
cosa =
x' x &quot; ta'y &quot; tz'z&quot;
8o&quot;
Sint a' , b' , c' , anguli, quos Ở facit cum axibus OX, OY ,
OZ ; et a &quot; , 1 &quot; , c&quot; anguli quos d &quot; facit cum iisdem axi
bus: erunt 1
x' = cosa' , y ' = ' cos b ', z ' = ' cosc'
x &quot; = &quot; cosa &quot; , y &quot; = 0 &quot; cosb &quot;, z&quot; = 0&quot; cosc&quot; ;
rursusque adhibitis substitutionibus,
98
−∙−≂− −≌≝≖⋅ ∙ 9
9008?-
sed haec hactenus.
70. Intelligentnr per eoordinatarnm orthogonalium
originem O ( Fig. 5 ) duci binae rectae d', d&quot; intercipien-
tes angulum a: earum extremitatibusjunctis recta ö&quot;, erit
ö&quot;:
eo&quot; −⊦∂∣∣∶∎−∂≀∥≖ ⋅−∎ 26' a&quot; '
Extremitas rectae ö' habeat coordinatas 0:231, z', rectae an-
tem d&quot;coordinatas x&quot; , y&quot;, z&quot;: paullulum attendenti pate-
bit fore ⋅
∂∣≏−−∶∞↾≖−⋅⊦∙↗∣≖−⊢≖↾⋩∙ ∂∣⋅≖∙∸⋅∞↾∎≖⊹∕∣≖−∣−≖∥≖ ,
3' ⋅≖−−−−≺∙⊅∣∙∞⋅∣⋟≖⊣⊣∙↗∣⋅∫∎⋅ )'—l-(z'-z&quot; ),:
adhibitis substitutionibns ,
∙−− æ; æ&quot;——)&quot;)'&quot;—l-Z' zn
cosa ∶⋅↳ a, 6&quot;
Sint a', 6', c', anguli, quos 6' facit cum axibus OX, Oï.
OZ; et a&quot;, b&quot;, e&quot; anguli quos 6&quot; facit cum iisdem axi-
bus: erunt
x':d' cosa' ,y*zzd&quot; cosb', z':ö' cosc'
æ&quot;: ö&quot;cosa&quot;, y'': ö&quot; cosb&quot;, 2&quot;: d&quot; cosc&quot;;''
rursusque adhibitis substitutionibus,
−∙∙⋅∙−⋅−−⋅99
cosa = cosa' cosa&quot; -- cosb' cosb&quot; + cose'cosc &quot; .

* His positis, fac ut vis acceleratrix o sese constanter dirigat ad centrum datum : constituta in eo coordinatarum ori gine O, erunt sle D 5.5 cosinus angulorum , quos cum axibus coordinatis efficit ra dius vector D; et P P &quot; P '&quot; P cosinus angulorum , quos cum iisdem axibus efficit . Pro pterea P X op + . $ . Þ==1 , sumpto vel superiore, vel inferiore signo , prout o nititur vel adducere materiale punctum ad centrum illud, vel ab ipso distrahere: in primo enim casu q et D faciunt angu lum a = 180° , in secundo angulum a = 0. Inde profluit ( 49) d2x Ide² dy v + D dia D daz dt2 8.• * Sumptis axibus OX, OY in plano ( 46) cur vae , quam incedit materiale punctum , erit der Q =F Ndt² on the + 5) . 99 cosa:eosa'cosa&quot;-]-cosb' cos6&quot;-1-cose' cosa&quot;. ∙His positis, fac ut vis acceleratrix (p sese constanter di- rigat ad centrum datum: constituta in eo coordinatarum ori- gine 0, erunt æLz D'D'D cosinus angulorum, quos cum axibus coordinatis edicit ra- dius vector D; et P' P&quot; P'&quot; r ' a ' ? cosinus angulorum, quos cum iisdem axibus ellicit ep. Pro- ? se P&quot; 7 p--- ∙∙∙ ∙−−− ' D—&quot;'ï&quot;10 ? sumpto vel superiore, vel inferiore signo, prout ep nititur vel adducere materiale punctum ad centrum illud, vel ab ipso distrahere: in primo enim casu ? et D faciunt angu- lum a:1800, in secundo angulnm a −−−− 0. Inde profluit (40) ≕∙∙∙∙∙∙ dia: :: d'y )- ? D) *(dz : &quot;D'l'dcz ∐↼⊦↲⋮−−≟ ⋅⋅− ⋅ 8.0 «: Sumptis axibus OX, 0? in plano (46) cur- vae , quam incedit materiale punctum , erit100 Ad exprimendamo per coordinatas polares , exhi beat 180°-W angulum interceplum radio vectore D et axe OX ; erunt De = x ? tys , x= - Dcosw , j = Dsina . Prima semel iterumque differentiata dat dDP + Dd D = xd x + ydży + dx2 + dy? ; secunda et tertia praebent dx = Dsiow cosw - coswdD . dy = Dcos wdw tsinwdD , ideoque dsa = dx2 + dyr= D -dw2+ dD2 , Hinc 2 der dia dy a D + dla D d - D dea D 2) ܪ . ac proinde la pa (d- D dla 0 ( ) ). Ad haec : P P &quot; = P P &quot; unde D àla D y et consequenter 1 1 1 100 Ad exprimendam (p per coordinatas polares, exhi- ' beat 1800—0 angulum interceptum radio vectore D et axe OX ; erunt Dï':a:3--l-)'2 , x: — Dcosw .szsinm. Prima semel iterumque differentiam dat dDL-l-DdzDzædïæ-l-yd'y-þdæï-l-dyz .; secunda et tertia praebent dæ:Dsinm cos co —cos ad D. dy:Dcos ædwf-sinædD, ideoque d.,- ∙−−− dx: −⊦ dyaznadæ-l—doa . Hinc dïæ a: dfy ] (PL) ? Dei?-),. ∎⊃⊣−≺∄↙⇄ ⋅∎⊃−−⇤↲⋍≖ dt &quot; ac proinde dzD (deo)!) ∙−−∶ −− D — ? ∓ ∙ (aua dt Ad haec : P' a: P&quot; ⋅∙∙∙∙ )» P ∙∙∙ P .;. :ï,?—q:.ü.,unde-; 7- et con sequenter ∙∙∙∎∙∎⋅∎−⋅101 • dx yd dt rady FO : de quam integrantes assequemur dr V dc dy dt C , seu ydx - xdy = Cdt. Est autem ydxxdy = Dsinud(-Dcosw ) + Dcosad( Dsinw ) Dºdw , propterea с dwla CdtD - da da de ( ) = C2 D D4 insuper AD Code : d d - D dla de dt dD da dt dt . ( dD C do D2 dt 1 . ( D ( ( d da da) da C2 D d d dw ,!... Hit C = as dt aan zoals da ? Coil 100 dwudt da . Da aby boxe parutis 1 C2 D D2 dw² Quare J 101 quam integrantes assequemnr da: dy,&quot; ⋅ ∙∙∙∙ ∙≯≀∙⊋∙↕−− ∙∙∷⊋∙⋮−−− C, seu ydæ—Jt'dj—Cdt- Est autem ydx—ædy:Dsinæd(—DcosmH—Dcosæd(Dsinæ) : D'daii , propterea ∙−− dai—C ∙ de) 3—01 ∙ ∁↙≀⇞−−∐⇟⊄∄∾⋅∙⋅⊋⊼−−−∐−≖⋅∙ (a)—&quot;1373&quot; insuper di? d(dD. 49) d(iD ∙⊆− ≀∄⋅∣⊃∙∙ d ∙∙∙⋅∃⊂∙−⊃⋅ 71? ∙− do) ne). dt'- d; ⋅ d:; dï- ⋪∙−⋅−⊳ ↿ ⋅ ⋯↿ 41 d(ï) d D d(B) 1101 ...-2 d, ⋅ da) ∙−↽∁⋅ ↪↼⋅−↽−⇁∁ ↜⊒⋅∶≥⇀⋍−−⋅−≤⋮∶ a d: dmwdt;,, nad-'O) . ' f&quot; &quot; c: ∐≖⋅↙∄∘−∎⊃−dasz102 D + ) ( 61 ). D2 dw² Fac v. gr. ut, viribus ad datum centrum tendentibus, materiale punctum incedat curvam (dicitur spiralis loga rithmica ) repraesentatam per Draw Habebis ( 27. 6.° ) . el = 2 11름 loga dw ,de 1 D% log ? a dway log.'a ; iccirco go CP D2 a log-a + b) ( logo a+ 1 ) . vis nempe acceleratrix erit in ratione reciproca triplicata distantiarum a dato centro.

9. # . Ad constantem C quod spectat, ex coordina
larum origine 0.(Fig . 19 ) intelligantur duci bini radii ve
clores, alter ad punctum datum a habens coordinatas xo,
Yo, alter ad punctum quodvis B habens coordinatas x, y;
sitque A area sectoris terminati arcu aB et radiis illis, A
area aBca' : erit
A = A +
Xoyo
2
xy
2
&gt;
⇀↿∘⊋∙
du-
−−∶⊨ C, ...—l.).. ..,— (6) ..... i)? da: D !'
Fac 9. gr. nt, viribus ad datum centrum tendentibus,
materiale punctum incedet curvam (dicitur spiralis loga-
rithmica ) repraesentatam per
D:
ac .
Habebis ( 27. 6.&quot; )
1 -ao 1 1
T)- :a ∙↙≀−∣⋝−∙−−−−−− logadæ,d'ï-—
.. ∠≀≖−∣↿⋝∙ ..
a logia dei:-, :logæa ,

dm&quot;
iccirco
1»ng (
−∾∙∣∘⊰⋅∅⊣−∎↿⋥≻−−−∌⋮ ——(l0gi ∅−⊦
1):
vis nempe acceleratrix erit in ratione reciproca triplicata
distantiarum a dato centro.
9011. Ad constantem C quod spectat, ex coordina-
tarum origine O-(Fig. 19) intelligantur duci bini radiive-
ctores, 'alter ad punctum datum et habens coordinatas xo,
yo, alter ad punctum quodvis B habens coordinatas x, y;
sitque A area sectoris terminati arcu aB et radiis illis, A'
area cha': erit '
A—A' : æozïo ?;103
ideoque ( 27. 18° )
dA = dA - d
xy ydxxdy
2
denotante praeterea i angulum interceptum tangente in B
et respondente radio vectore D, est ( 48)
D sinids dA 2
Igitar
ydx = xdy = Dsini ds,
et consequenter ( 70 )
C dt = D sini ds ;
unde ( 20 )
ds
C = D sini - Du sini .
dt
Caetero quantitatem Dvsini esse eamdem ubicumqe suae cur
vae sit materiale punctum, liquet ex dictis ( 47 ) .

10 ° # Habemus ( 2º. 8° )
ds
de²
DP dw2 + dD CP
dc2 (Da dwa + dD&quot;)p4
dwa
dD 2
D2
[ 11
+ 9 seu
de
103
ideoque ( 27. 18o .)
dA ::dA'- J
222 −−−∫↙≀∞−⋍≀↨≧↩ −∫∂∞−≨⋅⇣⇃⋮↙≀∫ ;
denotante praeterea t' angulum interceptum tangente in B
et respondente radio vectore D, est (48)
D sini ds
2 ∙
(IA:
lgitur
ydx':xdy :Dsint' ds,
et consequenter ( 70 )
Gde:D sini ds ;
unde ( 20)
CZDsini £:Dvsiü. dt
Caetera quantitatem Dvsini esse eamdem ubicumqe suae cur-
vae sit materiale punctum, liquet ex dictis (47).
100a Habemus ( 20. 8&quot;)
2
(Isa 02 dGP—xl-JD':(02 da,-1- dDz) c,
.
— —∙∙∙
−⋅ dt&quot; ⋅⋅−⋅ dt: D4 dc.-13:
∁≖
dDa
äirl—(5)], seu
de?104
v2 = C2 -- [ + (3 ]
( m) .
11. # Quemadmoduni , data linea quam incedit
materiale punctum , innotescit q ; sic vicissim , data op , po
terit sciri linea per quam movetur materiale punctum
Denolante B quantitatem constantem et n numerum inte
B
grum , sit v . gr. g = ; erit ( 7° 6. )
D &quot;
B CP D
+ )

;

dwa
1 B
quae ,
facto D = 1
D' et
et og h , vertetur in C2
d2 D'
h D ' r-2 = +
diwa +D) .
Chaton
Haec multiplicata per 2dD ' suppeditat
E12dD' dD d
dw da +
2D'dDdD' ) -2-2 h D'n -2 d D' = 0 ;
sumptisque integralibus ,
= [CD)* + D ] - 2,0-4C = 0;
unde
dw
(6,2
dD'
2h Dina quoad o adducentem
D'2 į ad centrum ,
'? – C ).
∠⊢⋅⋅ ↿
' 2 2
1 D∶∁ Exi-(a)] (&quot;**-

↿↿∙∘∙ Quemadmodum, data linea quam incedit
materiale punctum
,
innotescit ?; sic vicissim , data ep , po-
terit sciri linea per quam movetur materiale punctum
.
Deuotante B quantitatem constantem
, et 11 numerum inte-
grum, sit v. gr. ep:DT; erit ( 70 6.)
l
'l
3— B 02 (...d D.. 57.— 25 .'.)* da: D
⋅ ↿ B ∙
quae, fama-:D et——⋜⋮−:h,vertetur tn

* D' ≀≖≖≖⋅⋅−≖⇌⇀−⊻≐≺∡∽≖ −⊦∘∙≻⋅

Haec multiplicata per 2dD' suppeditat
∶⊨≺∶≳↙⊋≞⇗∠∄−−⊣− 2D' dD')—2h D&quot;'2dD':——o ;
sumptisque integralibus ,
465)&quot; HB ]-—'— ∣⊃⋅⋅−⋅∙−⊦∁⋅−−−∘≅ n—l
unde
da: dD'
2]; quoad ?adducentem
TDV&quot; —-D'3 —C')5 ad centrum,105
dD'
dw 2h quoad o distrahentem
(0 –
a centro :
n =; D**?— D» ) *
quarum integratio praebebit relationem inter w et D' , ideo
que inter coordinatas polares w et D lineae quaesitae .
12.°* In istarum aequationum prima sume v . gr.
n = 2 ; ea sic poterit scribi
D'
doma
V
ha- C
da
h D'
h2_C Hinc
w = C &quot; + arc cos =
h - D'
VhC
cos (6-C' ' ) ;
et restitutis valoribus h , D' ,
D =
C2
B - 1 B2 – C4 C cos (W – C&quot;) ·
Pone
C2 C
= B (1 + €),
=B' ( 1 —E) ,
B-HVB2_C4 C B - V B2 - C4C
quae in summam collectae praebent
B
CPC B '
,
invicem multiplicatae suppeditant --
8
105
&amp;
dm: dD
quoad p distrahentem
(C' - 36- D'&quot;&quot;' −−∙ ∎⊃∎∌≻≩⋅ a centro :
⇀
n—1
quarum integratio praebebit relationem inter 61 et D' ,
ideo-
que inter coordinatas polares &amp;) et D lineae quaesitae .
12.&quot;; In istarum aequationum prima sume v. gr.
n:2 ; ea sic poterit scribi

: .-n

'
↶⋮≼⇂∕−−⊮−∁∙⋟
dæ:- ⇂∕↿
Hinc −≺⊓⋅≻≖∙∣≖≖−∁⋅
G):C&quot;-l— :COS(GO—C&quot;)i arC(cos ∙−∙−−−−
h—D' h—D'
⋅⇂∕∣−≖−−−⋯≖−⇀∁∙ ⇂∕∣≖≖∙−∁∙ '&quot;
et restitutis valbribus I: , D',
B—l/Bz—Clt C' cos (co—C&quot;)
D
Pone
C2 02 −−−−− −−−−−−↧≉⋅↿
). −−⋅⊨
−−−−−−∶ —B'(1—e). B—l/Ba—cac- ≺⊹⋮ ∌−⊢⇂∕∌≖−∁↙∣∁∣
quae in summam collectae praebent
c:c' B -—-−⋅⋅
. B',
invicem multiplicatae suppeditant106
&lt;= B' ? ( 1 —-z );
habebis
1 C2 C' = B
B'2 ( 1
B' ( 1 — 52)
Propterea
D =
B' ( 1 - 2)
E cosWC( '')''
(62) .
1
13.0* Potest C' esse vel &gt; 0 , vel &lt; o , vel == 0;
in primo casu erit B ' &gt; o et € &lt; 1 ; in secundo B' &lt;o et
&gt; 1 ; in tertio B ' = et z = 1. Primum ac secundum
casum alibi considerabimus .
14. * Ad tertium quod pertinet , exhibeat NI...
(Fig . 25) axem parabolae ( 40. 5.º 7.º ) ; sintque NO ( 3x)
et 00' ( =y) orthogonales coordinatae : designante 2p pa
ramelrum , exsistet
ya = 2px .
Substituto x' + ip pro x , transferetur coordinatarum
origo in focum H , eritque quoad novam originem H
ya = 2px' +p .
Duc radium HO =D) ; habebis
NHO x'
--- D cos w , y = D sin w ;
et consequenter
D2 sin ’ w = p - 2pDcosw .
Spectatur autem D ut quantitas constanter positiva ; proinde
106
↿ 'a a .
&quot;ö'.:B (1—£)1
habebis
∙∙∙ ↿ ⋅ ∙∙∙ Ca
B'3(1 -
a&quot;) ' B' (1—5') ⋅
Propterea
B' (1 — a')
D −∙− (b,) . 1— :cos (co— C&quot;)
1391» Potest C' esse vel≻∘ ∙ vel (o , vel:o;
in primo casu erit B' o et e ↿;in secundo B' (0 et s
↿; in tertio B':eo et e:1 . Primum ac secundum
⋅ casum alibi considerabimus .
145): Ad tertium quod pertinet , exhibeat Nl..
.
(Fig.25) axem parabolae (40. 5.&quot; 79); sintque NO (:.r)
et 00' (: y) orthogonales coordinatae :designante 2p pa-
rametrum
, exsistet
y':2pæ .
Substituto x' −⊦ ∙⇡∙↼ ;) pro æ .
transferetur coordinatarum
origo in focum H , eritque quoad novam originem H
7&quot; ⇌ 2pæ' ⊣− r'-
D'uc- radium HO' (:D) ; habebis
NHO':61, uf:—D cos æ,y:D sin(-);
et consequenter
D2 sin2 01:p' −∙∙ 2pDcos co .
Spectatur autem D ut quantitas constanter positiva; proinde107
DE P cosa
+ V
V pa pacos w_P(1 ~ cos )
sin? W sin? W sin4 w sin' w
Sed sin? w = 1 - Cos w = (1 — cosa) (1 + cosw ) : igitur
P
D =
1 +cosa (63) .
Designata nimirum quantitate B '(1 - 6 ) per P , et assumpta
C &quot; = 180° , recidet (62) in (63) ; unde consequitur illud :
iribus ad centrum datum tendentibus in ratione reciproca
duplicala distantiarum ab ipso centro , poterit materiale
punctum describere parabolam habentem suum focnm in
centro illo .
15.0# Quoad parabolam ( 14º. ),
(* )
sinaw 1 1 +cosw cosa a
COS
pa da P р
2 1 2 CM 1 1
D
O
Hinc ( 90.m) va - .
р
D
P
D
Sit E altitudo debita velocitati v ; erit ( 12º. 14º. )
2C E 2C?
v2 = 20E =
2BE
D2
E
B
D2 ' ( 1 -62 ) D2 p
et consequenter
2C2 E 2C 1 E
D

; unde D2 D

.
р P
Inferimus illud : si in distantia D a centro virium proji .
citur materiale punctum , haud describetur parabola nisi
107
D:∙∙∙ ;) cosa) ∙∙∙⊦ Vpa .l.-paene: c.)—p(l—cosï
≖⋮∐⇄
∙
a) s1na a) sint! ea sin' 6)
Sed sinit.):1−cosa a:(1— eos a)) ('l-l- cos a) :igitur
∼ P
D:1—i-cosm ∅⋮⋝⋅ ⋅
Designata nimirnm quantitate B'(1-- 6&quot;) per p , et assumpta
&quot;:180o , recidet (b,) in (63) ; unde consequitur illud :
viribus ad centrum datum tendentibus in ratione reciproca
duplicata distantiarum ab ipso centro
, poterit materiale
punctum describere parabolam habentem suum focum in
centro illo .
1530 Quoad parabolam (Mc,),
∙
∙−−− — ∙−−−
d'
⋅
( D) sin'm 1—cosaæ—1—l-cosæ 1—cosa1
df&quot; P&quot; ?' p P
2 ↿ ↿ ∙ 2
c- 1

;

∙ D ∙∙∙ DQ. Hlnc (90.m) 02: ∙∙∙∎∎∙ ∙
ö
∙
Sit Ealtitudo debita velocitati «a; eri; (1241. 14o.)
2311: 20» E
∙∙∙∶≿∁∶
E
ng—ZQE— Da —B'('l—-£3) ∙ [P p . 02 ,
et consequenter
zcn E—zc: '-dE-1
p.Da—p.D,uneD—. .
Inferimus illud :. si in distantia D a centro virium proii-
citnr materiale punctum , baud describetur parabola nisi108
eidem distantiae D fuerit aequalis altitudo illa , per quam
mobile vi acceleratrice vigente in puncto projectionis ca
dendo motu uniformiter accelerato acquireret velocitatem
ipsius projectionis.

51. Hactenus de motu curvilineo libero, quum nempe nihil obstat quominus mobile obtemperet viribus; fac nunc ut materiale punctudi, cujus massa = m, moveatur motu impedito, sollicitatum videlicet vi acceleratrice q adstringatur moveri vel in data superficie vel in data linea curva. Quoniam ejusmodi superficies et linea nihil praestant aliud nisi exercere in puncto materiali resistentiam m ç sibi perpendicularem, ideo motus perinde fiet ac si punctum materiale esset liberum viribusque acceleratricibus et d', seu quod eodem redit viq &quot; inde resultanti libere obtemperaret.

Pone quod motus impeditus in data linea debeatur
unice vi praeconceplae et vi gp' ut sit 9
habebis
q &quot; = 0 ; i = 90 °; et consequenter ( 45. b)
0 :
2,2
( 6' ' ' ) ;
my?
Precisa nimirum q , exprimet ( 28 ) pressio
nem exercitam a puncto materiali in lineam illam , atque
huc spectat vis centrifuga ; pressio videlicet a puncto ma
teriali exercita in eam lineam , orta e sola inertia ad prae
seulem velocitatis siatum contracta.
Ad haec : in eadem hypothesi vis acceleratricis ♡
facile colligitur ex dictis ( 36) motum impeditum fore u
niformem .
!
108
eidem distantiae D fuerit aequalis altitudo illa , per quam
mobile vi acceleratrice vigente in puncto projectionis ca-
dendo motn uniformiter accelerato acquireret velocitatem
ipsius proiectionis.

===De vi acceleratrice in motu circulari, existente centro virium in centro circuli.===

52. Ex demonstratis (47) patet istiusmodi motum esse uniformem. Sit R radius circuli, per cujus peripheriam incedit mobile: in ( b: 45 ) erant r = R, i = 90° ; in ( b' : 48) vero D =9 = r = R; et denotante A lotam circuli aream, T tempus periodicum, quo nempe mobile conficit integram circuli peripheriam, in eadem ( 8' ) erunt quoque A = n R?, = T. Hinc ex ( 6)
1
RO
et ex ( 6 ) ( c )
4 762 R
T2

53. Haec facile punc stabiliuntur. 

1º. mobile velocitate quadam projectum in distantia R a centro virium von describet circularem curvam nisi velocitas illa tanta sit quantam mobile ipsum acquireret cadendo per { R motu uniformiter accelerato et vi acceleratrice, quae viget in projectionis puncto; siquidem prima (c) suppeditat v = 2 0.4 R.

2º. In circularibus peripheriis eodem tempore
descriptis vires acceleratrices sunt ut respondentes radii: patet ex secunda (c).

3º. Ex eadem secunda (c) inferimus vires acceleratrices fore in ratione reciproca duplicata radiorum quotiescumque quadrata temporum periodicorum fuerint ut radiorum cubi.

54. Obiter haec notamus. 

1º. Ex circulari telluris rotatione circa suum axem oritur vis centrifuga (51) in materialibus punctis tam apud aequatorem quam apud circulos aequatori parallelos, generatim expressa per &lt;math&gt;m\varphi'=\frac{mv^2}{R};&lt;/math&gt; et quia rotatio illa fit motu uniformi, ideo&lt;math display=&quot;block&quot;&gt;v=\frac{2\pi}{T}\,\mathrm{ 
et}\, \varphi'=\frac{4\pi^2 R}{T^2}
&lt;/math&gt;Tempus periodicum &lt;math&gt;T&lt;/math&gt; est ubique idem; &lt;math&gt;R&lt;/math&gt; vero decrescit ab aequatore ad polos; in eadem ergo ratione ab aequatore ad polos descrescet vis centrifuga. 

2º. Exhibeat R , radium aequatoris terrestris (Fig. 31) et a geographicam latitudinem, cui respondet circulus aequatori parallelus habens radium R, erit R =R cosa , et consequenter R , cosa
T2
Resoluta q' in duas, quarum altera sit verticalis, altera horizontalis, existet illa 402R , cosa
D'cosa=
T2
et quoniam q' cosa est vis contraria gravitati, inferimus gravitatem imminui magis semper a polis ad aequatorem, ejusque decrementum fore ut quadratum ex cosinu latitudinis, spectata videlicet tellure instar sphaerae.

3º. Exprimat s altitudinem debitam velocitati rotationis; erit ( 30) 2gs = v ?, ideoque ( 10 ) 2gs = q R, et
consequenter
8
solia
R
.
2s
110
mg': mv&quot;R; )
et quia rotatio illa Et motu uniformi, ideo
27rR et ∙∙∙∙∙ ∢∏≃∣≹
T ' ?&quot; Ta
.-

ecosa: Tat quoniam cp' cosa: est vis contraria gravitati, inferimus gravi-
tatemimminui magis semper a polis ad aequatorem, ejusque
decrementum fore ut quadratum ex cosinu latitudinis ,
spectata videlicet tellure instar sphaerae.

111
Hinc innotescit ratio inter gravitatem et vim centri
fugam : sic apud aequatorem invenitur
8
R,
= 288 circiter;
2s1
inde sequitur quod gravitas sub aequatore in hypothesi tel
luris immotae esset == 1880' + q = 289 .
=== De vi acceleratrice in motu elliptico, existente centro virium in foco ellipsis. ===
55. Haec praemittimus: 

1 °. si ex puncto quovis M (Fig. 32) ducuntur duae rectae MN, MS tangentes sphaeram SN .. , erit MN = MS: ductis enim ex centro C radiis CN, CS ad contactus puncta N et S; itemque CM ad punctum M, triangula CMN, CMS rectangula in N et S habebunt latus CM commune, latera vero CN , CS aequalia; ideoque etc.

2°. Si per tangentes MN , MS ducuntur plana tangentia NMT , SMT ad sphaeram SN .... sese muluose.
cantia juxta rectam MT, angulus NMT aequalis erit an
gulo SMT: nam ex C , N , S ad punctum v . gr. T rectae MT, ductis CT , NT , ST, quoniam NT et ST jacent in planis tangentibus NMT , SMT , iccirco in triangulis CTN , CT'S anguli CNT, CST erunt recti; latera in.
super CN CS sunt aequalia , et CT commune: proinde
NT = ST. Triangala igitur MNT, MST exsistent ( 1 ° ) invicem aequilatera; ideoque etc.

3º. Si denotat p projectionem lineae rectae l in
plano quovis , et a angulum , quem efficit I cum eo plano , erit&lt;math display=&quot;block&quot;&gt;p = l\cos\alpha&lt;/math&gt;: patet ex Trigonometria.
4º. Si denotat P projectionem mn (Fig. 33) areae planae cd ( = A ) in plano quovis gr , et i angulum ,
quem efficit A cum gr , erit,
P = A cosi .
Ducatur enim planum mg parallelum areae A, in quod
demittatur ex d perpendiculum dK ( = x ) ; ducantor quo
que plana gh , de parallela plano qr; ponaturque dg = y .
Quisque videt prisma vel cylindrum md aequari prismati
vel cylindro ge: propterea Ax Py ; unde P A ; est
autem -
sindgK = cosi ; igitur etc. yу

5º. Secetür cylindrus rectus aB ( Fig. 34 ) plano
ad circularem ipsius cylindri basim inclinato; sectio AMBL
dicitur ellipsis ; punctum C, ubi planum secans occurrit axi
cylindri, vocatur ellipseos centrum; punctumque S, ubi se
crio illa tangit sphaeram sambl cylindro inscriptam , appel
latur ellipseos focus; pro cylindri base sumimus circuluin trans
euntem per centrum c sphaerae inscriptae; inde fit, ut ba
seos peripheria sit linea contactuum superficiei sphaericae
et superficiei cylindricae.

6º. Si per C ducitur linea quaevis recta LM ter
minata ad ellipseos perimetrum , ejus projectio in cylindri
base erit ipsius baseos diameter lm , ita at lc sit projectio
portionis LC, et mc projectio portionis MC. Sed lc mc ;
ergo ( 30 ) LC = MG: lineae videlicet rectae transeuntes per
ellipseos centrum , et ad ellipseos perimetrum terminatae ,
dividuntur omnes bifariam in eodem centro.

7º. Per extrema puncta 1 et m diametri lm du
ctis ad circularem cylindri basim tangentibus lh et mt ,
hae utpote perpendiculares ipsi lm erunt parallelae; rectae
quoque IL , mM utpote cylindri basi perpendiculares, erunt
parallelae; ergo plana hll , ImM cylindricam superficiem
112
40. Si denotat P proiectionem mn (Fig. 33 ) a-
reae planae cd:( A ) in plano quovis qr ,
et t' angulum ,
quem eliicit A cum qr', erit,
P:A cost'.
Ducatur enim planum mg parallelum areae A, in quod
demittatur ex d. perpendiculnde ( −−∶ æ ); ducantur quo-
que plana gh, de parallela plano qr; ponaturque liga:-7.
Quisque videt prisma vel cylindrum md aequari prismati
vel cylindro ge: propterea Aa: −−∶ Py: unde P:
.i.-A; est
autem −⋅↕⇣∙ ∶−− siudgK :cosi; igitur etc.
.7
20. Secet'ur cylindrus rectus aB (Fig. 34 )plano
ad circularem ipsius cylindri basim inclinato; sectio AMBL
dicitur ellipsis; punctum C, ubi planum secans occurrit axi
cylindri, vocatur ellipseos centrum; punctumque S, ubi se-
ctio illa tangit sphaeram sambl cylindro inscriptam, appel-
latur ellipseos focus; pro cylindri base sumimus circulum trans-
euntem per centrum c sphaerae inscriptae; inde fit, ut ba-
seos peripheria sit linea contactuum superficiei sphaericae
et superficiei cylindricae.
60. Si per C ducitur linea quaevis recta LM ter—
minata ad ellipseos perimetrum, ejus proiectio in cylindri
base erit ipsins baseos diameter lm, ita ut lc sit projectio
portionis LC, et me projectio portionis MC. Sed lc :: mc;
ergo (30) LC: MC: lineae videlicet rectae transeuntes per
ellipseos centrum
.
et ad ellipseos perimetrum terminatae.
dividuntur omnes bifariam in eodem centro.

113
tangentia existent parallela inter se; et couscquenter inter
sectiones quoque LH, MT istorum planorum cum elipseos
plano exsistent parallelae. Liquet autein intersectiones il
las esse tangentes elipseos in L et M; ellipseos igitur lan
gentes ductae per extrema puocta cujusvis rectae, quae trans
eat per centrum , quaeque terminetur ad curvae perime
trum, erunt inter se parallelae.
Recta LM secat bifariam ( 3º ) chordas omnes paral
lelas tangentibus LH , MT; ejusmodi enim chordarum pro
jectiones nibil sunt aliud nisi circularis baseos chordae pa
rallelae tangentibus lh, mi, atque ideo perpendiculares dia
metro lm , a qua proinde secantur bifariam : inde fit , ut
LM dicatur ellipseos diameter.

8º. Ex M ad focum S ducatur MS; rectae MS ,Mm tangent ( 50 ) sphaeram, altera in S , aliera in punctum lineae contactuum superficiei cylindricae et superficiei sphaericae: ergo ( 19. ) MS = Mm. Simili modo, ex L ad S du
cta LS, erit LS = LI.

9º. Plana TMS, MMT et transeunt per rectas MS,
Mm tangentes sphaeram , et sphaeram tangunt, et sese mutuo
secanı juxta MT; ergo ( 2º )anguli TMm, TMS erunt aequales :
simili ratione ostenditur angulos IILS esse aequales.

10º. Denotet a rectam Cc jungentem centra Cet c: trapezium LMml suppeditat Ll +Mm 2a ; igirur i 80 )
SL + SM 2a . Variala utcumquc positione diametri LM ,
non ideo variabit recta Cc , sed mavebit cousians in ea
dem ellipsi ; ergo summa rectarum SL et SM, quae in ea
dem ellipsi ducuntur a foco ad extrema puncta cujuscum
que diametri LM, erit quantitas constans. Ad haec: rectae
SL, SM efficiunt cum tangentibus LH , MT avgulos aequa
les SLH, SMT; cum enim LH et MTsint parallelae ( 7 °) ,
itemque Ll et Mm parallelae , angulus HLL aequalis erit
angulo TMm; proinde ( 99) etc.

11º. Revolvatur diameter LM donec transeat per focum S, sicque evadal AB: rccidet SL in SA, et SM in
113
tangentia existent parallela inter se; et consequenter inter-
sectiones quoque LH, MT istorum planorum cum elipseos
plano exsistent parallelae. Liquet autem intersectiones il-
las esse tangentes elipseos in L et M; ellipseos igitur tan-
gentes ductae per extrema puncta cuiusvis rectae, quae trans-
eat per centrum, quaeque terminetur ad curvae perime-
trum, erunt inter se parallelae.
Recta LM secat bifariam (30) chordas omnes parallelas tangentibus LH, MT; ejusmodi enim chordarum pro-
jectiones nihil sunt aliud nisi circularis baseos chordae pa-
rallelae tangentibus lh, mt, atque ideo perpendiculares dia-
metro lm, a qua proinde secantur bifariam: inde fit, ut
LM dicaturo ellipseos diameter.
.Ex M ad focum S ducatur MS; rectae MS
. Mm tangeiit (50) sphaeram, altera in S, altera in puncto
m lineae contactuum superficiei cylindricae et superficici sphae-
ricae: ergo (10. ) MS :Mm. Simili modo, ex L ad S du-
cta LS, erit LS:LI.
90. Plana TMS, mMT et transeuntper rectas MS,
Mm tangentes sphaeram, et sphaeram tangunt, et sese mutuo
secant iuxta MT; ergo (2&quot;)anguli TMm, 'I'MS erunt aequales:
simili ratione ostenditur angnlos HLS esse aequales.

12.• Aa est minimum , Bb est maximum omnium
perpendiculorum Ll , Mm , ... quae ex perimetro ellipseos
demittuntur in cylindri basim ; ergo ( 89) SA erit minima ,
SB erit maxima omnium rectarum , quae ex foco S du
cuntur ad ipsam ellipseos perimetrum .

13.• Punctum S' ita determinatum in axe trans
verso AB , ut sit CS' = CS , dicitur alter ellipseos focus.
Jam si ex S' ad M et L ducuntur rectae S'M et S'L , quo
niam SC = S'C et ( 69) LC = MC , iccirco SL et SM
erunt aequales et parallelae ; igitur ( 109)
SL + SM SM + SM = SL + SL = 2a .
Praeterea angulus SLH aequatur angulo SMR ; ergo
( 10 °.) angulus SMT aequabitur angulo SMR.

14°. Producatur MS donec tangenti LH occurrat
in H , erit ( 30. ) angulus LHS aequalis angulo SMT. Sed
( 109. ) SMT = SLH ; ergoò LHS == SLH , ideoque SL=SH:
hinc ( 13. ) HM = 2a .
56. His praemissis venio cum D &quot; o Arpere ad quaestio
nem propositam de invenienda vi acceleratrice o in motu
elliptico , exsistente centro virium in ellipseos foco S. Conci
piantur duo radii vectores SM , SN intercipientes angulum
inGnitesimum MSN , et producatur SN donec occurrat
tangenti TM ... in R ; erit ( 49 , 6 &quot; )
Q
2 NR
62
Binae NR , MH babendae sunt pro parallelis , eruntque
114
SB; ideoque (100) AB:Za. Quoad alias positiones diame-
tri LM habetur semper LM (SL ∙−⊢ SM, et consequen-
ter (100) LM 2a; igitur AB est omnium diametrorum
maxima: AB dicitur axis transversus ellipseos; diameter per-
pendicularis axi transverso dicitur axis conjugatus.

140. Producatur MS donec tangenti LH occurrat
in H , erit (70.) angulus LHS aequalis angulo SMT. Sed
(loo-) SMT:SLH ; ergö LHS:SLH ,
ideoque SL:SH:
hinc (139) HM:20.
56. His praemissis venio cum D'&quot; Atnpere ad quaestio-
nem propositam de invenienda vi acceleratrice ep in motu
elliptico , exsistente centro virium in ellipseos foco S. Conci-
piantur duo radii vectores SM , SN intercipientes angulum
infiuitesimam MSN , et producatur SN donec occurrat
tangenti TM
...
in R; erit (49. b&quot;)
2NR
∙∙∙⇀−−∙ −−⇀∙∙62
Binae NR , MH habendae sunt pro parallelis , eruntque115
proinde ( 55. 3. ) ut respondentes projectiones nr , mh in
cylindri base : hinc ( 55. 14º.)
nr . MH NR =
nr 2a
mh mh
Sit T tempus periodicum , quo nempe materiale pun
ctum totam percurrit ellipticam orbitam ; erit ( 46) ellipseos
area ad aream MSN ut Tad 0 : istae areae sunt ut re
spondentes projectiones ( 55. 4º. ) in cylindri basi , nimirum
ut ipsa cylindri basis ambll =
mila et area msn : ad
haec ; demisso perpendiculo st ex s io tangentem mt ,
erit
msn = j st , mr = 1 st (nr . mg) : quare
( mza)
712 14
ml
16 T2
2
SC
nir , mg
et consequenter
mi ml 62
T2 .
nr T2 2
st mg
Triangula mlh , mlg sunt rectangula , alterum in l ,
alterum in g ; habent insuper communem angulum in m :
iccirco
ml&quot;
= mh .
mg
Anguli mhl et hmt sunt ( 55. 7. &quot; ) aequales ; propterea
triangula mlh , stm rectangula in l ac o dabunt (55.30. 14º.)
115
proinde (55. 39) ut respondentes proiectiones nr, mi: in
cylindri base : hinc (55. 140.)
nr . MH nr

2
−∙− ∙ NR mh (: mh
Sit T tempus periodicum, quo nempe materiale pun-
ctum totam percurrit ellipticam orbitam; erit (46) ellipseos
area ad aream MSN ut T ad 9: istae areae sunt ut re?
spondentes projectiones (55. 40.) in cylindri basi , nimirum
∙ ∙ ∙ ∙ ↿≖ −∎⋅
↴ ut ipsa cylindri basis ambl(:-Z - ml&quot;) et aram nim: ad
haec ; demisso perpendiculo st ex .: in tangentem mt
, erit
mm:&amp;st,mr:äst(nr.mg)iï:quare
l ml ml: 62
nr ∙−−− ∙∙ T! :,- '
—-- '
&quot;rf&quot;; '
st2 mg
Triangula mih , mlg sunt rectangula , alterum in I,
alterum in g; habent insuper communem angulum in m :
iccirco '
tl,
— z'mll.
Anguli mi:! et hmt sunt (55. 73) aequales; propterea
triangula mllt ,
stm rectangula in 1ac :dabunt (55 . 30. 140.)116
Im mh MH 2a
SC si SM SM
Non pluribus opus est , ut assequamur
47' a3 1
( h) ;
T2 SM
vim nempe acceleratricem in ratione reciproca duplicata
radii vectoris .
Quoad aliam ellipsim
4 R² a ,
1
T ;
i
S, MI 2
hinc si
1 1
a3
T2
a ,
T

: erit op :
:

2 SM 2
S, M ,
Si nempe in diversis ellipsibus quadrala temporum pe
riodicorum sunt ut cubi semiaxium transversorum , vires
acceleratrices tendentes ad respectivos focos erunt in sola
ratione reciproca duplicata respondentium radiorum ve
ctorum .

57. Haec subjungimus .

1.º Fiat CS
CA
CS seu a
€ ; numerus &amp; K1 ) dici
lur excentricitas : ex L in axem transversum ducatur per
pendiculam Li , et ponantur Ci = x , Li = r ; erunt
SL = y2 + ( x — $ a) 2, S'L ' =y2 + ( x + ε a) 2 ,
et consequenter ( 55 , 13º. )
↿16
lm mh MH 212
∙−−∙∙−−− −∙∙sm SM SM .

Non pluribus opus est, ut assequamur
47:303 1
−− ∙∙∙ lt ; ? Ta sit-r, (
vimnempe acceleratrieem in ratione reciproca duplicata
radii veetoris.
⋅ Quoad ≘∣⋮∘⊡↾ ellipsim
∙− 4 123 a,3 1
ut ?! Tla 5! M :
hinc si .?- gz.
. −↿− ↿
⊽↓⊽∶⊺∣≖∙∁≖∣⇂∲∙∲∎⇌⇋⊤⊡∶
Si nempe in diversis ellipsibus quadrata temporum pe-
riodicorum sunt ut cubi semiaxium transversorum, vires
acceleratrices tendentes ad respectivos focos erunt in sola
ratione reciproca duplicata respondentium radiorum ve-
ctorum
.

57. Haec subjungimus.
CS 

↿∙∘ Fiat äseuï :8: numerus :
((1) dici-
tur excentricitas : ex L in axem transversum ducatur per-
pendiculum Li
,
et ponantur Ci:æ , Li:7; erunt
ST.&quot; :y2 −⊢ ≼⋅⊅−−∙⋮∠≖≽⇄⋮ ST]? :]! −⊢ ≼⋅≈⋅−⊢⋮∘≻≖ ,
et consequenter (55.130.)117
Vym + (x - ea) + V y2 + (x + ea )
2a ; !
unde
ye + ( x – sa )2 + 2V 99 + (2 - a) Vya + (xta)
ty: + (x + a ) = 4aº ;
ac propterea
V12 + (x - a)2 V y2 + (x +-a)? = 2a? —yox? - ?o ?
ex qua obtinetur
ya = (1-2) (a? – x2) ( o) ;
aequatio ad ellipsim inter x et y computatas a centro C.

2. ° Facta x = o in ( o ) , valor y inde proveniens
nihil erit aliud nisi valor semiaxis conjugati ( 110.) : hinc ,
denotante 6 istiusmodi semiaxem , exsistet
2
62 CS seu ( 10.) 1 -
62 ideoque CS' =a2-6.

al'
a a2
Inferimus distantiam inter focum et punctum illud ,
in quo semiaxis conjugatus occurrit ellipseos perimetro ,
acqnari semiaxi transverso .
39. Loco x substituatur a - ain (o) : emerget
y2 = ((1 — 82 ) ((2ax - x2 ) ( 0' ) ;
aequatio ad ellipsim inter x et y computatas a vertice A.
Jam vergente e ad 1 , simulque crescente a indefinite ver
117
Vr-l—(æ—eaP-l- l/Ja-l—(æ—l-eaPr-h? ⇥
'
nnde
y' −⊦ (æ −∙∙ id? −⊢ 21/7' ∓−⋅⋜∞∶∽≻∙ Vy' −⊢≺∙↿⊏⊹∽⋟≖
−⊦↗≖ −⊦ (..-'.]. ..). ∶−− ta:
.
EC propterea
Vm VW:2(:* —y2—æ2—s*a'
ex qua obtinetur
]&quot; −−−−−− ≺↿∙−∊≖≻ (a' --.r*) (a):
aequatio ad ellipsim inter se et] computatas a centro C.
2.(, Facta a: o in (a) , valor ]inde proveniens
nihil erit aliud nisi valor semiaxis coniugati (HO.) :hinc ,
denotante b istiusmodi semiaxem , exsistet
—2 b' CS &amp;&quot; ∙
..... 1−−∊≖−∙−∶ 23, seu (1 0,)1 ...—a—z- ;; ;1deoque CSa −−∶∅⇄∙− ∂≖⋅
Inferimus distantiam inter focum et punctum illud,
in quo semiaxis conjugatus occurrit ellipseos perimetro,
aequari semiaxi transverso
.

30. Loco a: substituatur a— a: in (0) :emerget

(1—82) (2aæ—æ2) (0') :
aequatio ad ellipsim inter se et y computatas a vertice A .
Jam vergente P. ad 1
, simulque crescente a indefinite ver-118
gat 2 (1 — ?) a ad limitem quemdam finitum B : aequatio
( 0 &quot; ) verget ad
yö = B x (o &quot; ) ,
et consequenter , precedente foco S' indefinite a vertice A ,
ellipsis repraesentata per (o' ) ad parabolam repraesentatam
( 40.70. ) per (o &quot; ) .
Inferimus illud : si a quovis parabolae puncto du
cuntur binae rectae altera ad focum , altera axi paral
lela , eae cum tangente per idem punctum ducta aequa
les ( 55. 130. ) hinc inde continebunt angulos.

4.• Pone conjugatum ellipseos axem fieri imagi
narium ; adhibe nempe 26V - 1 pro 26 : fiet
22 1-62 = , ideoque e &gt; 1 .
Q2
Aequatio nimirum ( 0) novam curvam repraesentabit,
quae dicitur hyperbola , quaeque secat abscissarum axem
in distantiis CA ( = a ) et CB ( =-a) ab G ; inde in infi .
nitum excurrit cum quatuor ramis ab axe illo magis sem
per recedentibus , quorum bini respiciunt partem posi
tivam , bini negativam , habet insuper centrum in C ,
focos in 0 et O' , exsistente CO = CO ' = ɛa .

5. ° * In aequatione ( o) substitue x' + sa pro x;
habebis
ya=( 1—62) ( a2 -x'tea) )
ad ellipsim vel hyperbolam prout &lt;&lt; vel &gt; 1 , exsisten
te coordinatarum origine in respectivo foco S vel 0. As
sumptis nunc ( 7.9 )
x = Dcosw , y = Dsina ,
118
gat 2(t-—£*)a ad limitem quemdam finitum B :aequatio
(a') verget ad
J'2Bæ ⋅ (a&quot;).
et consequenter
, recedente foco S' indefinite a vertice A ,
ellipsis repraesentata per (a') ad parabolam repraesentatam
(40. 70.) per (a&quot;)
.
Inferimus illud: si a quovis parabolae pnncto du-
cuntur binae rectae altera ad focum, altera axi paral-
lela
, eae cum tangente per idem punctum ducta aequa-
les (55.130.) hinc inde continebunt angulos.

4. 0 Pone coniugatum ellipseos axem fieri imagi-
narium; adhibe nempe ⊋∂⇂∕∙−−−−↿ pro 26 :iie't
↿∟∊≖−−∶−⋮⋮ ideoque : ↿∙
∙
Aequatio nimirum (o) novam curvam repraesentabit,
quae dicitur hyperbola , quaeque secat abscissarum axem
in distantiis CA (:a) et CB (:—a) ab C; inde in inli-
nitum excurrit cum quatuor ramis ab axe illa magis sem-
per recedentibus , quorum bini respiciunt partem posi-
tivam, bini negativam, habet insuper centrum in C,
focos in O et O' , exsistente CO:CO':sa.

5.0 11 In aeqnatione (o) substitue x' −∣− Sa pro a:;
habebis ↼
⋅
J*-——(1—8*)(a—(x'-l—w)2)
ad ellipsim vel hyperbolam prout :( vel)1
, exsisten-
te coordinatarum origine in respectivo foco S vel 0. As-
sumptis nunc (7?)
x': -- DCOSGJ
,yzDsinm ,119
erit
Dasin 6) = (1-2)( a ) - (ea - Dcosa)) &quot;) ;
quae traducitur ad
Da 2 ea ( 1-2) cosa a ' (1-2)
D =
1-6 cos26 1 - &amp; cosa
unde c
D :
a (1-2) ( ECOSW +1 ) .
18? cos26 1
Habetur D pro positiva quantitate ; sumpto itaque su
periore signo quoad &lt;&lt; 1 , emerget in ordine ad elli
psim
D
al 1-52) ( 1 t-scosa )
( 1 +acosw) ( 1 -ecosw)
a ( 1-2)
1 -ECOSW ( h) ;
sumpto inferiore signo quoad &gt;1 , prodibit in ordine ad
hyperbolam
a (1-2) ( ECOSW - 1 ) a (621)
D =
( 1 + scos ) (1 - Cosw ) 1 tecosw
(h' )
Non pluribus opus est ut intelligamus in primo ex ca
sibus alibi ( 50. 13.° 14. ) consideratis descriptum iri
ellipsim , in secundo hyperbolam , exsistente focorum al
tero in centro virium : quoad ellipsim , B= a; quoad hy
perbolam, B' = - a.
6. # Ex ( h)
119
erit
Didone-:( 1—s*)(a'—-(ea—chsæ)3) ;
quae traducitur ad
25a(1—s*) cos 6) D∙∙− a'( 1
Da −∙∙ −∊≖≱≖ .
1—szcos2ca 1—e*cos*c.1
unde
∙∙∙ ⇩≺↿∙−⋮⇄⋟ (scusa) :bt)
1----ea cosa:»
D
1
Habetur D pro positiva quantitate; sumpto itaque su-
periore signo quoad e(1
, emerget in ordine ad elli-
psim '
3( l—sï) (1—l—scosa1) —a(1 —e')
D—(l—l-Ecosw) (1—äcosm) 1—scosc1
(71) :
sumpto inferiore signo quoad s)1
, prodibit in ordine ad
hyperbolam
∙∙ -a(1—e*)(scosca——1) —a(sï—1)
(1—I—scosa1) (1—scosm). 1—I—ecosct
Non pluribus Opus est ut intelligamus in primo ex ea-
sibus alibi (50. 13.014.0) consideratis descriptum iri
ellipsim ,
in secundo byperbolam , exsistente iocorum al-
tero in centro virium :quoad ellipsim, B:; quoad hy-
perbolam, B': — a.
69
.
Ex (h)
∙−− .n..- ∙∙ -&quot; ∙∙∙∙∙∙∙−⋅↖∙∙∙− '.120
1 2
a ( 1-2)
sasin '
ECOSA= 1
€2-82cos ?
Ꭰ . dw al( 1 - E22 a-(1-6 )
a (1— $ 2) 2
( 1 - &quot;)
a (1452)
2 1 1 1
a (14 € 2 ) D
bi a - 1—62)
D2
proinde ( 50. 9.º )
02 2C2
a (1-2)
G- ) ( h &quot; ).
Ex ( h' )
€2sin ?
ECOS W =
a( 82-1 )
D
( a2( 1–82) 2
–1).
€ 2 .
a ( 821)
&amp; 2 - cos26 D
42( 1-2)
2 1 .
a (21) D
a’( 1-62) 2
1 1
a2 ( 2-1) Da,
ideoque ( 50. 10.)
V2
2C2
a 2-1 ( + za) ( 17&quot;).
120
∣a(1-52) d 0 eisinïæ sï-sïcosza)
o ' −⋅
' ⇀−
∙∙∙∙−∙ ∈∁∘⊱∞∶ ↿∙− −∙∙
czu-e*)a czu-ez):
proinde ( 50. 99 )
vï— 202 (
1 '1 )
h&quot;
∙ (tU—83) D 20 ( ).
Ex (h')

8003 6) a(83—1) ;
(ï) ∙∙∙ £2sin26)
−− '
dcc D aï(1—82)2
. &amp;:
e* (cuï—1) 1)2 −− ∊≖∁∘⊱≃∾∙∙∙ D
?
2 1
uzu—w?)a t
czu—ez? a(e*—-1) D
↿ ↿ ∙
↙≖≖≺∊≖−↿⋟ ⋅−⋅ ⋅↧⋅⊃−≖∙ '
ideoque (50. 100.)
202 1 1
)
,,, ↗⇩≕−− an:—1 )(D 'l'ïiz (h)121
Sit E altitudo debita velocitati v; erit ( 50. 12º. )
2BE
v=2qE=
Da
2C E
B (1-82) D2
Igitur in ellipsi
1 E
1
B ' D (ó -za),
2
seu ( 50)
olt
E
D
D
2a ( h &quot; );
in hyperbola
1
B'
E
Da - ( + za)
seu ( 5 )
E = 1 + (tha&quot;)
Ex (h &quot; ) et ( h ) consequitur, si in distantia D a cen
tro virium projicitur materiale punctum, haud descriptom
iri ellipsim vel hyperbolam nisi respectu ejusdem distan
tiae D fuerit minor vel major altitudo illa , per quam mo
bile vi acceleratrice vigente in puncto projectionis cadendo
molu uniformiter accelerato acquireret velocitatem ipsius
projectionis.
7 ° * Quoad ellipsim ( 50 , h. 6° )
9
∙
121
Sit E altitudo debita velocitati v.; erit (a 50. 12'.)
2BE— zcn E
D: B'(1-e*) ⋅
ï; ⊍≖∶∃∲⊡∶−∙− ⋅
Igitur in ellipsi
1'Efn'1 1&quot; 1) B&quot;Dï—-a(o za'
seu (50)
in hyperbola
seu (50)
E -D
,
⋮−⇂∃⇌−−⋅⊳⊣−⋅⇄−∅⋅⋅≺≀⋅⋟⋅ ∙ l
Ex (II&quot;) et (h') consequitur, si in distantia D a cen-
tro virium proiicitur materiale punctum, baud descriptum
iri ellipsim vel hyperbolam 'nisi respectu eiusdem distan-
tiae D fuerit minor vel major altitudo illa, per quam mo-
bile vi acceleratrice vigente in þuncto projectionis cadendo
motu uniformiter accelerato acquireret velocitatem ipsius
proiectionis. -
70t Quoad ellipsim (50. I:. 60)122
7
a 옘 E COSQ ) 1
dw² a ( 1-2) Q ( 1-22) - 5
hinc ( 50. 8º. b .)
go
Ca
a ( 1-62 )
1
Da
areo ds D sinids
Est ( 50. 9º . ) C =D sini.

; exhibet

dt 2
lam a radio vectore D descriptam tempusculo de : deno
tante igitur A totam ellipseos aream, T tempus periodi
cum, habebitur
ds C = D sini dt
2A
T
Est ( 27. 18º. )
a
A = 2V 1-* [Vaº-x:dx ;
exprimit 2 | Va?-xă de circularem aream , cujus radius
= a , et consequenter 1
A = Tla ? VT-
Propterea
1
C2 4 A2
T2
4772 24 (1-2)
T2
42 a3
et p =
Ta
0
9 D2
122
≖↿ ⋅ *

:

cosa) 1 1
dm&quot; −⇩≼↿∙∊≖⋗⊽ ⋅⋅∙↽∙↰↿∙∊≖≽∙ D '
liinc ( 50. 80. b,.)
∙−− ∁∙ ↿
,? ↼⇀ (tU-e&quot;)- ⋅
⋅∎⋝≖⋅∙
Est (50. 90.) C :D ciuili-f.; exhibet Egit-If. areo-
£
iam a radio vectore D descriptam tempusculo dt: deno-
tante igitur A totam ellipseos aream, T tempus periodi-
cum, habebitur &quot;
⋅
∙ ∙ ds 2A
C—DSID! 'a'ï—T ∙
Est (27. me.) fZl/l-Ez l/a'-æ' dx;

∽ ∘
exprimit Zf Vaz- ac2 dx circularem aream , cujus radius
o
∙−∙−−−∙− a , et consequenter
A −−∶↿∽≖ ⇂∕↿ ∙a&quot; ∙
Propterea
4A3 47taa4(1-s*) 41:303 1 ∙
Ta&quot; TTL ∅∘⊔⊢−
'1'» C*.—. 'ne'123
prorsus ut supra ( 56).

8º. Obiter notamus illud: si materiale punctum
motu elliptico movetur viribus ad ellipseos centrum len
dentibus, eae erunt in ratione directa distantiarum ab ipso
centro . Assertionis demonstratio eruitur ex dictis ( 56) : sint
enim duo radii vectores CM ', CN' sub angulo infinitesimo
M'ON' , et producatur CN' donec occurrat tangenti M'T
in R' ; erit ( 49. 6' ' )
2N'R'
♡ 02
binae N'R' , M'C censendae sunt parallelae; proinde ( 55.3º. )
m'c : n'r' = M'C : N'R'
M'C . n'r
m'c
area insuper ellipseos ad areolam M'ON' ut tempus pe
riodicum T ad tempusculum 6 ; quae areae cum sint ( 55.4º. )
ut respondentes projectiones in cylindri basi , nimirum
ut ipsa cylindri basis ambl ( = 76. cm ' ) et areola
cm'.r'm' cm '
m'cn'
V r'n'. 2 cm ) , iccirco 2 2 m'
cm'
r'n ' . 2cm
4 02 unde r'n'
762.
cm
272. cm' ; 1 4 T2 T2

et consequenter
M'C . 27. cm'
T2
N'R'
cm'
Ta 272. M'C..
--
123
prorsus ut supra (56).
80. Obiter notamus illud: si materiale punctum
motu elliptico movetur viribus ad ellipseos centrum ,ten-
dentibus, eae erunt in ratione directa distantiarum ab ipso
centro. Assertionis demonstratio eruitur ex dictis (56): sint
enim duo radii vectores CM', CN' sub angulo infinitesimo
M'CN' ,
et producatur CN'donec occurrat tangenti M'T'
in B'; erit (49. b&quot;)
2N'R'
? ∶−⋅− ⊖≖ binae NZR', M'C ceusendae sunt parallelae;proinde (55.30.)
m'c:n'r':M'C:
⋅ 'C. &quot; N'R'-— M nr :
m'c
area insuper ellipseos ad areolam M'CN' ut tempus pe-
riodicum T ad tempusculum 9; quae areae cum sint (55.4&quot;.)
ut respondentes proiectiones in cylindri basi , nimirum
ut ipsa cylindri basis amb! (: Tt. 27:23 et areola
, .
cm'.r'm' cm ∙
∙ ∙
2
...—3
CI&quot;. I o
,
—— - n .Zcm - 4 9: . .
∙ 9: a , .
32. cmlb :T2;undern :::-'F. 212. em,
et consequenter
9:
MC. ∙⊤↓⋅↴∙⋮−⋅∙ ⇄∏≖ cm

NR −− ∙ ∙−−− -;'21t'.M'C.
cm124
Propterea
. M'C :
vis nempe acceleratrix Q directe ut distantia M'C ab el
lipseos centro

* Etiam sic : in ( o. 1º. ) fac

X
Dcosw y = Dsinw ;
prodibil aequatio inter coordinatas polares ab ellipseos cen
tro computatas, nimirum
av182
Dsin ? w = (1-2) (a² - D2cos w ), unde D= V 1-8? cos26
Hinc
at 2
d:2 av1 (via1-2003 (1-8? cosaw ) V 1-2coscosti
D3 1
a* ( 1-2) D
.
ac proinde ( 50. 8º. 3 , )
CP
a4 ( 1482 )
D :
quae ad superiorem expressionem traducitur; nam ( 70. )
4724 (1-2)
C2 =
4A2
T2 T2
124
Propterea
4 ita
?

:

0132 ∙ M'C;
vis nempe acceleratrix go directe ut distantia MC ab el-
lipseos eentro.
&amp; Etiam sic: in (0. 10.) fac'
∶−∙−− -Doosa) ,y
−−−−− Dsinæ;
prodibit aequatio inter coordinatas polares ab ellipseos cen-
tro computatas, nimirum
al/1— ei
Dsin2 a): ( 1—53) (aa.-ul)2 cosm), unde D—
Hinc
([21 «
⋅
') ? cos-36) sium
113&quot; (zl/1—ea ⇂∕ ↿−⋅⋅∊≖∞≘≖∾ ≼↿∙∊≖∘∘≘≖∾⋟⇂∕↿−⋅⋮∅∾∙≖∞≻ ∙∙∙ D3 1
−− a4(1—£2)—ï '
' ⋅ ac proinde ( 50. 823, )
Ca
?
∙−− a4(1—-s2 )
quae ad superiorem expressionem traducitur; nam (72)
Ca— 4A' ∙∙∙ 4n304(1—£2l
T2 T:
⇂∕⋅↿ ∙⊽∊≖∞⊱≖∾ .125

=== De motu relativo punctorum materialium, tendentium in se mutuo viribus acceleratricibus quae sint directe ut massae in quas tenditur, et reciproce ut &lt;u&gt;quadrata&lt;/u&gt; respondentium distantiarum.===
58.* Sint m, m ', m , ... punctorum massae; a, b, c coordinatae orthogonales puncti m in ordine ad axes OX, OY, OZ (Fig. 8); x ', y', z' , x &quot; , y &quot;, z &quot; , x '&quot; , ... Coordinatae reliquorum punctorum in ordine ad novos axes et parallelos axibus Ox, OY, OZ, et habentes originem in m. Factis compendii causa ( 50. 7.0)
x ' ty's tz's =k ?, x &quot; ty's t-z&quot; = k &quot; , etc ...
erunt ( 50. 4.0) quoad motum puncti m
de a m ' x' m' ' Qc &quot; d²b m'
.
g' , m &quot; g
+
k'
&quot; &quot; ) dc2 k2 k' k &quot; 2 hit
d12 ka kita
d2c
m' z'
k'
m &quot;
k' ' ?
.
dc2 ti to...,
seu
d'a d26
dc2
m'x m'z '
Σ
k'3
niy'
Σ
dc
dca &gt; ( o ) .
dt2 k'3
Nunc quod spectat ad aliud punctum v . gr. mi' , pone ( 50.70. )
(.x &quot; —X')2 +6 &quot; -Y')2 + (z&quot; -z&quot;) = 002 ,
( z&quot; &quot; ' —x' ) 2 + 6 — ')2+ ( z' — z &quot;)2 = ' ' , etc... ;
exhibebunt


126
t ... The **** +
en +++
m &quot; yy'
+
d'a + ... ,
+ ..
vires acceleratrices ab m &quot; , m ' exercitas in m' , no
visque axibus parallelas : denotant
ac m
j'
k'a k'
. C
k'a ' ki k'2 k'
vires acceleratrices ab m exercitas in m' , iisdemque novis
axibus parallelas ; sunt insuper ata , bty' , cta' coor
dinatae puncti m' in ordine ad axes OX,OY,02; facto igitur
m &quot; m '&quot;
+ .. =
assequemur quoad motum puncti m'
20 dQ d'a+x' )
dta
mx' d2(6 + y ')
k3 dla
my'
k'3 dx '
dy '
dQ mz' dºlc + z )
dia dzi k'3
d²a d2b
Substitutis valoribus
dac
ex ( 0 ) , prodibunt dca dla dt2
daxi dl mx' m'r'
dxc '
.
day' d my'
dc2 dy
m'y'
Σ
dea k'3
k3
k3 k'3
126
m&quot; .v&quot;--.r' a&quot;; 7 '—:7' F
∙⋅⊱∷−∎∙−⊦∂⋅≖ a ⊣−∙⋅∙∙
vires acceleratrices ab m&quot;, m'&quot;
, ,..
exercitas in m'
, no-
visque axibus parallelas: denotant
ut se' m y' m :
&quot;F' la&quot;—k&quot; k&quot;&quot;'1?'-&quot;£'
vires acceleratrices ab m exercitae in m'
,
iisdemque novis
axibus parallelas ; sunt insuper a-l-z' , (Hl-y' , e—l—e' coor-
dinatae puncti m' in ordine ad axes 0X,OT,OZ; facto igitur
&quot;
m
m m 37 −∂∙−⋅⋅ −⊦ −−∶
9-
assequhmur quoad motum puncti m'
d'(a-[-æ') ∙∙∙ dQ mx' d3(b-l-y') ∙∙∙ dQ my . d,. dx&quot; k-a de dy' k'3
(P(e-l-z') .... di) me'
dt' dz' k'3
∙ ∙ ∙ dia d'b die ∙
∙⊱∎≖∣⋯⋅⋯∎⋯ valonbus dt&quot; ∙ dt' ∙ dt? ex (0) , prodibunt
g'æ/ dQ -mæ' zm'x' d'y' dQ my' zmiy'
dt' dæ' 163 It'3
'
dt2 dj'- k'3- It'3127
daa' d2 mz'
K'3
m'z'
Σ
dta dzi k3
formulae determinantes motum relatiyum puncti m' quoad
punctum m . Quoniam
00 mx' m'x
k'3
mtm x +
k'3 dx ' k'3
zel
2 X
m &quot;
come
-ac '
813 -) +mi&quot;
xc
k3
V3 k'3) +... ,
dQ
,
- - monte +
-&quot; * 7- ) + m.A-A ) +...
en e -maile +
) &quot; V +
d2 mz'
-Σ dzi k3
m &quot;
tom &quot;
t ... ;
03 k3
hinc facto
R = m &quot;
.6. – +) +
(5--**&quot; +jx +e*e&quot;)+ - ( &quot; ),
m &quot;
formulae ( 0' ) vertentur in
127
∙⇌⋅⋮⋅⋮⋅≕↙⇣≴⋅≖−−−∶↗−⋅⋮−−− ∑∶≀−≖⇣ (.,-,,
dt: dzï lt'3 k'3
formulae determinantes. motum relativum puncti m' quoad
punctum m . Quoniam
dQ ⋯⋅∙∙∙∑∽∙∙↼∙⋅⋅≈∙↾−∙ m—I-m' .
⊋⊑⋅∙⋅−⋅⊼∙∶⊤∣ k'3 −−−⋅∎−∎ ↗⊏∙⋮∣ æ
III I'll
.. ∞⋅⋅−−⋅↕∙⇗ æ&quot; ,,, æ —x' x ≺−−⊽⋮−−−⋅−↗⋮⇁⋮⋮−≻ ⊹≖⊷ ⋯⋯≻⋅⊢ df
———— —— k'3 ∙−− 72— k&quot;3
&quot; yn ∙ yl! &quot; yon—70 ..- 70&quot;.
&quot;' (
a&quot;? ≀⊏⊤∍≻−⊦∽ ↾≺↴↼⋮⋅−∣⋮∎∎ ≀∎⊄−⋅∣∎⋅⋮≻∎⊦⋅⋅⋅∙
(19 Mi z m'x' m—l-m' zo ∙⊦
dQ my' Z mfy' m-l-m'y. ∙∙⊦
⊋∎≖∎⋅∎∎∎∎ k'3 15&quot;— ⋅∎∎∎ ↗⊏⋅⋮⇂ a'.—Z&quot; z&quot; &quot;' zIIO—zt all-l .
&quot;'&quot; ea &quot;&quot;17'5) &quot;'&quot;&quot;
«W &quot;:?75) ⊹⋅⋅⋅⋅ hinc facto
1 æoæn ⊣∙∙ o n

: , z'&quot; B: m&quot; (y'—W) ∙∙∣∎∙

1 me xlv ' '&quot; zl zh, &quot;
mm (öt—or— J—æO—ïä—L) ⊣∎∙∙∙∙ (O 2, .
formulae (o') vertentur in128
dax
de2
m -tm '
+ x's K'3
dR day '
dx ' ' de
mtm
+
k'3
g
mtm
dR daz' dR
dy' '
dit de k'3
dz
Porro , cum habeamus
ka + k &quot;? – 02
x ' x &quot; ty'y &quot; + =' z&quot; =
2
k'2 + k ' ' ?
d''2''
x' x'&quot; ty'g '&quot; +z'z' &quot; etc... ;
2
poterit (o&quot; ' ) scribi etiam in hunc modum
(
R = m k'o + k&quot; — 0° ) +
22
in '&quot; . k'2 + k '''2'''
2k &quot; 2 3* 2) + ... ( o &quot; ) .

59 * Fac at systema reducatur ad duo tantum pun
eta m et m' ; habebis R = 0 , et consequenter
der mm x
+
k'2 k'
day' mtm +
dia K2 K
&gt;
dta

* 3&quot;. d2 z' mtm

dt2 +
k'2 k
Relativus videlicet motus puncti mi quoad m proveniet
m +m: (50. 4. 20. ) a vi acceleratrice tendente ad m : pro.
k' ?
⋯⊣−⋯⋮↨↾ '
(0 ∙∎∣).
klö ,d—l;
dR dïz' m—I—m'z, dR ∙
«(y' dc2 k'3 dz'
Porro, cum habeamus
&quot; k': k&quot;: ∙∙∙ ∝↭⊹⊔↤⇥⋠−−⊦⊇ ∂∣∣≖ ⋅
-k'jl −−⊢ k'&quot;a — ö&quot;&quot; x'M* x'&quot; &quot;' z'e &quot;&quot;— 2 ∙ etc... :
poterit (o&quot;) scribi etiam in hunc modum
!, k.: kn; −∙− ux.,, ∂∜≖
.).
21./&quot;a
↿ ⋅ ra −⊦∣⊏⋯≖ −⋅∂∣∙⋅≖⋅≻ ..
∙−∂⋅−∣⋅∣∣ . ka2&quot; .l.-&quot;' (0 )- mllt
59; Fac 'ut systema 'reducaturad duo 'tantum pun-
cta m et m' ;habebis R ∙−−∶ ∘,
et consequenter
d'x' m di' '
'
−⊦⋯P',—mi −⊢⋯⋅−⊣⋤↾⋮⋡−∙≛ −−−−−∘∙ (it—T k'3 k dca k' k'
da z' ∙ ⋯⊣−⋯∣ .'
d:: [ kl; '
kl :::-&quot;'o.
Relativus videlicet motus puncti m' quoad m proveniet
(50 . 40 . 70.) a vi acceleratrice mt;&quot;. tendente ad m :pro.129
&gt; 7
pterea ( 50.13º . 140.57.50. ) describet m' motu relativo
vel parabolain , vel ellipsim , vel hyperbolam , existente foco in m .
dR dR dR 60# Secunda membra formularum
dx' ' dy' ' dz
( o &quot; ) exhibent ( 50 , 4.:) vires turbantes relativum motum
puncti m' determinatum per formulas (o &quot;) . Hinc si membra
illa manent constanter tenuissima , ita ut (o ' ') et ( o&quot;) dif
ferant in terminis exiguissimis , turbationes quoque inde
provenientes manebunt tenuissimae ; lineaque ab m descri
pla circa m poterit adhuc spectari tanquam vel parabolica ,
vel elliptica , vel hyperbolica ; ita tamen , ut gaudeat ele mentis continue mulatis .

61 * Datis tribus punctis m , m ' , m &quot; ( Fig. 35 ) ,
demissoque ex m' in mm &quot; perpendiculo m'A , sint
x' = mA , y' = m'A , X &quot; = mm &quot; , z' = 0, y = 0, z &quot; = 0.
Erit ( 58)
a' x
1
R m'
(-- = m &quot;
k3
ha( x&quot; —x'to) 2ty'a )
unde prodeunt vires distrahentes m' ab m juxta directiones
x' et y' , nimirum
dR x &quot; — x 1
dx = m &quot;
[(x&quot; — x'ja traj
. DR
dy '
m &quot;
[(x“ — x'ja + y'a ] }
Denotet h angulum m'mm &quot; , et D distantiam mm' ; erunt
x ' = D cos h , y = Dsinh , et consequenter
129
pterea (50 . 130. 14&quot; . 57 . ö&quot;.) describet m' motn relativo
vel parabolam , vel ellipsim, vel hyperbolam , existente
foco in m .
dR dR dR
dæ' , d)&quot; , dz'
(o&quot;') exhibent (50 . 40:) vires turbantes relativum motum
puncti m' determinatum per formulas (a')
.Hinc si membra
illa manent constanter tenuissima ,
ita ut (o&quot;') et (a') dif-
ferant in terminis exiguissimis , turbationes quoque inde
provenientes manebunt tenuissimae ; lineaqne ab m' descri-
pta circa n; poterit adhuc spectari tanquam vel parabolica ,
vel elliptica, vel hyperbolica; ita tamen ,
ut gaudeat ele-
mentis continue mutatis .

130
[ ( x &quot; —x ) 2 + y'r] - = ( x &quot; 2—2D.x&quot; cosh + Daj - -
mi [1+ (13–2cori)]- *

* &quot; [ - P2-2.5k)+3 . ) 6–2 cos )&quot;

- 2.7 CM) 6-2005 )'+ ]
Propterea , si D est ita parva prae aut possint omitui
termini includentes factorem exsistet
(2 )&quot;,
[cº= 2') + s] =
1
+3 D cos h
73 &quot;4
ac proinde
dR
dx = m '
3 D cos h 3 D2 cos2 h
x''3''
=
2 2:14
X

m
* +357 Dosh )
( -Dout
mi ( Doco 4-3C )*.)
China + 202 )=
2 m D cos h
dR 3 DP sin h cos h
dy.
m &quot;
24
130
3
∐∙↧⋅∥−−∙↧⋅⋅⋟≖⊹∫⋅≖⋮∣ −'i;:[£&amp;—2th cosh-l— D'] −⋅⋮∎∎ :
∙∙ 3
æ&quot; :
[130 2, äl—Zcosh)]— 7:
.'! 30 D 3.5 D): D
) a': ∣∶↿⋅−⋮⊸⋅⊋∙−⊤≺⋤−∣ 2—-cosh)-i—m (;&quot; (;,—2005&quot;
−∶≣⋅−≣−−∶≟≺⋚⋛∥−≻ ≺−⋅⋅⋅⋛⋮⊽∙−⋮∞≖↗⋮≻⋮⊣−⋅∙∙∃∙
PrOpterea, si D est ita parva prae æ&quot; ut possint omitti
. . . termini D . . includentes factorem (F) ,
exsistet
[(. ' BD-io-sh. «J)- −⊦∂∣≖∃−⋮⋅↼−↿ −−∽ ↽⊦−−−−−−
ac proinde
dR ,, a.,—a.&quot; æ&quot;—æ' 1
ævo, a'./3&quot; −⊦ [[[. ∎∎∎∎∎ II
a: .r.-3 ,,(1 Dcosh BDcosh— BD'cosïh 1)
., 2 D cOs '! (D : cos: !: 2m&quot; Dcos]:
m -—-3 −−
∙ ,, −− ( æ... .) .: .r., a
−∉⋮≹∙∙∙ ∣≺∐∘⋮∐∣⇂⊹∍∘∙∙⋮∐∣≖∞≖∣≖ −∙∙
dy —--—m ∙↿∙∙∦⋮ æ&quot;], )—131
sin h cos m&quot;
-
D sin h
3 +3 m&quot; D sin h
m 

62. Bonum erit alia ratione nonnulla hic stabilire
circa vires in praefato motu relativo . 

↿∙∘ Sint duo puncta
T , P (Fig. 36.)
, quorum massae m, m', distantia vero TP
(: k');et &quot;veniat determinanda vis acceleratrix in motu
relativo puncti P quoad T . Ex hypothesi P tendit in T vi
I
acceleratrice . m . . m

; — ;et T in P v
: acceleratrice ∙−−∣−−≖ sive au.

] 3
I
tem T sollicitetur .. vn. m . m .
−− ∣−⋮∣−≖− et Pv: 17; , sive T quiescat et P
I
sollicitetur vili—ïm]— &amp;, idem in utroque casu (5) habetur
motus relativus puncti P quoad T; vis ergo acceleratrix
in istiusmodi motu erit

2.&quot; Praeter P , T detur et tertium punctum S ,
cuius massa m&quot; , ut determinentur vires iude provenien-
tes, quibus turbatur motus relativus puncti PquoadT
ortus ex vi (0) . Ducta ST
, completoque parallelogrammo
.. STPP' , exhibeat diagonalis SP (: 8&quot;) vim g.: , qua sol-
licitatur P versus S:resolvatur vis ista in duas, quarum al-
tera (: ?') sese dirigat iuxta PT , altera (:f) iuxta PP';
exhibebitur illa (8) per parallelogrammi latus PT (: k')
.
haec per latus PP':ST (: k&quot;); eritque '
m&quot; .,

; n ⇀
: m&quot; IC, '

m&quot; k, ≒≀−∣−⋮∙⊊≱⋅∙∣⇆∶∂ ∶∣⊄∙ ]; ,unde 93:77sz &quot;3 ⋅132
m' '
m &quot;
Sollicitatur T versus S vi ; et attentis f et i motus k''2''
relativus puncti P quoad T eodem prorsus modo fiet ( 5 )
sive T quiescat et P sollicitetur vi f
m '
sive T sollicite k'2
m &quot;
tur vi et P vi f. Propterea vires provenientes ex S ,
et perturbantes motum relativum puncti P quoad T , al
tera juxta PT altera juxta PP' parallelam rectae ST , ex
primentur per
k &quot; 2
ø=73
m &quot; k
&quot;
g = f mi&quot;
&quot; Cess )
( c' ) . k's
3.° Ex puncto S demittatur perpendiculum SS'
( =i) in planum curvae , quam describit P motu relati
vo quoad T; ab S ad T ducatur recta ST ( =n ) , sitque
angulus STP = a : vis q&quot; agens juxta directionem paralle.
lam rectae ST resolvetur in duas, quarum altera q&quot;cosSTS
seu q &quot; . ! existet parallela rectae ST in plano cur
vae , altera q &quot; sinSTS' seu o&quot; . perpendicularis eidem pla k &quot;
resolvetur in duas quarum altera o &quot;
no: rursus onk
cos a
aget in curvae plano juxta TP , altera om. sina in eo k &quot;
dem plano normaliter ad TP. His positis , quisque in
telligit vires perturbantes motum relativum puncti P exhiberi
posse per
132
SollicitaturT versus S vi &quot;'

; et attentis f et −∥↼↕−∙ -, motus 1.&quot;» 1.&quot;

relativus puncti Pquoad T eodem prorsus modo fiet (5)
sive T quiescat et P sollicitetur vi f— 'I—N. ,
∣∣≖ sive T sollicite/-
et P vi f. Propterea vires provenientes ex S ,
∙ m tur '! k&quot;-
et perturbant'es motum relativum puncti P quoad T , al-
tera juxta PT altera juxta PP' parallelam rectae ST, ex-
primentur per
' mllko &quot; &quot;zl! &quot; kl! 1 '
Pf&quot;??- ⊕−−⇌↾−−⊺⋇⊽≏∶⋯ (Fa-&quot;' ia&quot;) &quot;'

3.&quot; Ex puncto S demittatur perpendiculum SS'
(::t') in planum curvae , quam describit P motu relati-
vo quoad T; ab S' ad T ducatur recta S'T (::n)
, sitque
angulus S'TPr-at: vis 9&quot; agens juxta directionem paralle-
lam rectae ST resolvetur in duas, quarum altera 9&quot;cosST5'
seu 9&quot;? existet parallela rectae S'T in plano cur-
vae, altera 9&quot;sinSTS' seu q;&quot;.grperpendicularis eidem pla-
no: rursus ?&quot;]?- resolvetur in duas quarum altera ⊄∙⊅⋅⋅∙⋮∙− eos ::
aget in curvae plano iuxta TP , altera ?&quot;)—;.sinat in eo-
dem plano normaliter ad TP. His positis, quisque. in? telligit vires perturbautes motum relativum puncti P exhiberi
posse per133
COS Q = cosa ,
9 =porn o--&quot; (* - ) .com
Pa = e&quot; sin æ =
= m
m &quot; (- ) snæ ,
93 = ml - )
( c )
i

;

9 , et Q2 agentes in curvae seu orbitae plano ipsam orbi
tam turbant ; 93 perpendicularis plano orbitae turbat
ipsius plani positionem .
4. ° Pone S , T, P esse constanter in uno eo
demque plario ; erunt i = 0 , n=k&quot;, a=S'TP=STP(=h) :
proinde
PI
m &quot;
8'3 -m&quot;( )cosh ,
&quot; sink ,
}
( cm)
Q2 , 93 = 0 .
Pone insuper ST, SP ita magnas prae TP ut , ex P du
clo perpendiculo PQ in ST, assumi possit absque sensi
bili errore SP=SQ , nimirum d&quot; = k&quot; -kcosh ; erit
1
js =(k“&quot; —k'cosh)-3 = 13 +
3k'cosh
+
Hinc proxime
m &quot;
m'k ( 1-3cos'h= ( 1 +3cos2h) ,
k &quot; 3 2K3
( c&quot; )
3m''K'sinhcosh 3m'k'sin2h''
92 k&quot;3
2k'3
133

: n&quot; muli, &quot; k&quot; ! n 913? —Q ?

eos a:
ïïï —m 673 k,,a k,, 0082,
.&quot;
k ⋅
,
93——9 &quot;,;— Blna :m &quot;(ä-3- It.—741) ,——,- sin a ,- (c)
]. LII-. [
i
93:907?sz −−⋮ ⇁≖⊼↗ ;
4). et (p, agentes in curvae seu orbitae plano ipsam orhi-
tam turbant; (pg perpendicularis plano orbitae turbat
ipsius plani positionem.
4.&quot; Pone S, T, P esse constanter in uno eo-
demque platfo; erunt i:o, n.:k&quot;, a:S'TP:STP(:h):
proinde
mrlk' &quot; kn
'
(Pr −−∶ 7873- −−⋅ m
⊱∣−⊵∙−− F,.)COSII,
(e&quot;)
?::m&quot;≣∶⋅⋅⋮∙−⋅ -—k,,,)smh , 93:30.
Pone insuper ST, SP ita magnas prae TP ut. ex P du;
cto perpendiculo PQ in ST, assumi possit absque sensi-
hili errore SP:SQ , nimirum d&quot;:k&quot;—k'cosh ; erit
1
I,
, −∙∙ BkCOBh ∙≦↜−∽⋮⋅−−−−−≺↗⊏ —kcosh) ∍≔−−↼−−⊺⋮−−∣−−−−
k&quot;, −⊦
,
∙∙
Hinc proxime
∙ ?: 2773— (1—3cosïh):— (1-1—3c092h) ,
z—kHS .
(e&quot;)
—3m' &quot;ksinhcosh —3m &quot;k sinZh134
5,9 Fac ut orbita puncti P sit circularis , ipsum .
que P moveatur ad partes N : sive spectentur formulae
( 6 ') , sive (6 &quot;) , sive ( c &quot; ), aget 92 juxta orbitae tangen
tem contra motus directionem : ejus proinde valori erit
praefigendum signum negativum.
=== De pendulis; deque gravium descensu per arcus cycloidales. ===
[[Fasciculus:Simple pendulum generalized coordinates.svg|thumb|Pendulum]]
[[Fasciculus:Pendulum simplicium.svg|thumb]]
63. Pendulum constat filo tenui secundum alteram sui extremitatem fixo, quod tamquam linea recta et gravitatis expers concipitur, ex quo suspensum punctum ponderosum a directione verticali dimotum potest huc et illuc circum punctum illud alterum extremum fixum in motum circinationis per arcum excurrere. Excursio penduli ab uno arcus, quem describit, extremo &lt;math&gt;C&lt;/math&gt; (Fig. 37) ad aliud extremum &lt;math&gt;D&lt;/math&gt; dicitur &lt;u&gt;vibratio&lt;/u&gt; seu &lt;u&gt;oscillatio&lt;/u&gt;: accessus ad verticalem directionem ex &lt;math&gt;C&lt;/math&gt; in punctum infimum &lt;math&gt;B&lt;/math&gt;, vel recessus ex &lt;math&gt;B&lt;/math&gt; in &lt;math&gt;D,&lt;/math&gt; dicitur semivibratio. Si unicum ponderosum punctum pendeat e filo, pendulum dicitur simplex, si plura in diversa a suspensionis puncto distantia pendeant, dicitur compositum.  
[[Fasciculus:Pendulo simples.jpg|thumb]]
Illud facile quisque intelligit, pendulum &lt;math&gt;AB&lt;/math&gt; circa punctum fixum &lt;math&gt;A&lt;/math&gt; eodem motu arcum circuli &lt;math&gt;CBD&lt;/math&gt; descripturum ac si, sublato filo, in superficie sphaerica perfecte dura et levigata punctum ponderosum moveretur motu impedito. Sicut enim adducto puncto illo ad praedictae superficiei punctum &lt;math&gt;C&lt;/math&gt;, et exinde demisso, gravitas &lt;math&gt;CT&lt;/math&gt; horizonti perpendicularis &lt;u&gt;resolveretur&lt;/u&gt; in duas vires, quarum altera &lt;math&gt;CE&lt;/math&gt; ad tangentem &lt;math&gt;CG&lt;/math&gt; normalis insumeretur in premenda superficie, altera expressa ab ipsa &lt;math&gt;CG&lt;/math&gt; sollicitaret punctum ponderosum ad motum per tangentem infinite parvam, ac deinde per aliam atque aliam subsequentem, et sic deinceps per reliquas omnes numero infinitas et infinite parvas tangentes, quibus constare arcus descriplus concipitur; ita a filo resolvetar gravitas eodem prorsus modo , nempe partim in trahendo filo insumpta, partiin ad singulas arcus circularis tangentes infinite parvas subinde determinata, qua deducetur pendulum per arcum circularem motu omnino simili, subeunte filo &lt;math&gt;AG&lt;/math&gt; vices curvilineae superficiei: hinc sicuti punctum illud ponderosum propter suam gravitatem, postquam descendisset ex &lt;math&gt;C&lt;/math&gt; in &lt;math&gt;B&lt;/math&gt;, cogeretur ascendere ex &lt;math&gt;B&lt;/math&gt; versus &lt;math&gt;D&lt;/math&gt;, ita ob rationem similem pendulum post descensum ex &lt;math&gt;C&lt;/math&gt; in &lt;math&gt;B&lt;/math&gt; ascendet ex &lt;math&gt;B&lt;/math&gt; versus &lt;math&gt;D&lt;/math&gt;. Rursus quemadmodum ponderosum punctum in praedicta superficie ascendere inciperet per arcum &lt;math&gt;BD&lt;/math&gt; cum eadem velocitate, quam acquisivisset in puncto infimo &lt;math&gt;B&lt;/math&gt;, et ideo ad eamdem altitudinem, ex qua descendisset, perveniret, nempe usque in &lt;math&gt;D&lt;/math&gt;, ubi extincta omni velocitate, iterum gravitate sua inciperet descendere, et in puncto &lt;math&gt;B&lt;/math&gt; priori velocitate rursus acquisita, cum ea ascenderet iterum in &lt;math&gt;C&lt;/math&gt;, atque ita porro ascendendo et descendendo perpetuas et aequalęs in peripheria &lt;math&gt;CBD&lt;/math&gt; excursiones perficeret, ita ob eamdem rationem penduli oscillationes aequales essent natura sua et perpetuo duraturae, nisi ab aeris &lt;u&gt;resistentia&lt;/u&gt; et &lt;u&gt;frictione&lt;/u&gt; aliqua circa sustentationis punctum &lt;math&gt;A&lt;/math&gt; inaequales primo redderentur, ac denique extinguereatur; adimentibus scilicet ejusmodi causis in singulis oscillationibus aliquid de illa velocitate, quae producitur a gravitate.

64. Velocitates &lt;math&gt;v&lt;/math&gt; et &lt;math&gt;v'&lt;/math&gt; in puncto infimo B acquisitae a gravibus per arcus &lt;math&gt;CB, C'B&lt;/math&gt; descendentibus sunt ut ipsorum arcuum chordae. 

Per &lt;math&gt;B&lt;/math&gt; concipiamus duci tangentem et in eam ex &lt;math&gt;C&lt;/math&gt; et &lt;math&gt;C'&lt;/math&gt; demitti perpendicula &lt;math&gt;z&lt;/math&gt; et &lt;math&gt;z'&lt;/math&gt;: denotante &lt;math&gt;r&lt;/math&gt; radium &lt;math&gt;AB&lt;/math&gt; et denotantibus &lt;math&gt;k, k'&lt;/math&gt; arcus quoad radium 1 similes ''arcubus'' &lt;math&gt;CB, CB'&lt;/math&gt;, erunt 

&lt;math&gt;z = r ( 1 - \cos k ) , z' = r (1 - \cos k' ) ; &lt;/math&gt; 

et quoniam (30: 36)

&lt;math&gt;v^2 = 2gz, v'^2= 2gz';&lt;/math&gt;

propterea

&lt;math&gt;v: v' = \sqrt{2gr (1 - \cos k )} : \sqrt{2gr (1 - \cos k')} = \sin \frac{k}{2} : \sin \frac{k'}{2};&lt;/math&gt;

ideoque etc.

65. Pendulum, quod incipit descendere ex &lt;math&gt;C&lt;/math&gt;, percurrat arcum &lt;math&gt;CM&lt;/math&gt; tempore &lt;math&gt;t&lt;/math&gt;; sitque &lt;math&gt;\alpha&lt;/math&gt; arcus quoad radium 1 similis arcui &lt;math&gt;BM&lt;/math&gt;: erunt &lt;math&gt;CB =rk, BM = r\alpha&lt;/math&gt;; et designante &lt;math&gt;u&lt;/math&gt;velocitatem in puncto &lt;math&gt;M&lt;/math&gt;, exsistet

&lt;math&gt;u = - 2gr (\cos\alpha - \cos k ) = 4gr \sin\frac{k+\alpha}{2} \sin \frac{k-\alpha}{2}.&lt;/math&gt;

Si arcus &lt;math&gt;k&lt;/math&gt; est ita exiguus, ut possit absque sensibili errore substitui respondenti sinui, habebimus
&lt;math&gt;u^2 = gr(k^2 -\alpha^2),&lt;/math&gt; et consequenter (28)

&lt;math&gt;\frac{ds^2}{dt^2} = gr(k^2 -\alpha^2),&lt;/math&gt; 

unde

&lt;math&gt;dt = \frac{ds}{\sqrt{gr(k^2 -\alpha^2)}} = \frac{r\beta}{\sqrt{gr(k^2 -\alpha^2)}}= \frac{\beta}{\sqrt{\frac gr (k^2 -\alpha^2)}};&lt;/math&gt; 

&lt;math&gt;\beta&lt;/math&gt; est arcus quoad radium 1 similis arcui infinitesimo Mm
( = ds ). Nunc centro H ( Fig. 38) et radio HD ( = k) describe circulum DED' ; sume HN : Ν »
B; duc perpen
dicula Ne, ne super HD: et Ey parallelam radio HD. Trian
gula similia HEN, Eey rectangula in N, y praebent

∙∙∙⇀ ,4þf - ⇀∙⋅∙∎∙ .. ⊸∙⋅⋅⋅∙∎∎∣∙ 4.-
∙−
..137
Ey: EN = Ee: HE, seu B: V R2-42 = Ee: k :
hinc
B
Ee 8
dt ;
V R2-42 k
et consequenter
Ee dt
kV
tempusculum videlicet dt impensum ad percurrendum
seu Nn, obtinetur dividendo respondentem arcum Ee per
kV § . Inferimus tempus t impensum ad percurren
dum ka seu ND, obtineri dividendo respondentem ar
cum ED per kV ; nimirum
ED
자
름
k
Quare
VED –are(com);
ideoque
Vare(cový = ) &lt;( a ) .
10
137
Ey:EN :Ee: HE, seu ,8: V kï-aï: Ee: h:
hinc
∙
!
Ee .... B ∙∸−⋅⋅ :: Vii. dt ;
VIR-æ ]: '
r
et consequenter
⋅↙≀↥∶∎∙−−⋮∶∶⇣∶⋮ *Ve-
tempusculum videlicet dt impensum ad percurrendam þ
seu Nn, obtinetur dividendo respondentem arcum Ee per
]; Vi .
Inferimus, tempus :
impensum ad percurren-
'.
dum ]:-
ut seu ND, obtineri dividendo respondentem arcum ED per kI/ £.; nimirum
r
t ∙∙∙ ED
l.V-f,- .
Quare ∙
.yz... −∙−⊡∍ ......
«
.
rf- k —- (eos:.k),
ideoque
≀⇌⇂∕∑− arc (eo: &amp;) (a) ∙
5
10138
Iam vero in puncto infimo B (Fig. 37) exsistit a = 0 ;
erit igitur tempus semioscillationis
TT
ti
V
2
8
tempus integrae oscillationis ( a ' ).
t2 =
V
quas formulas cum non ingrediatur initialis angulus k, patet oscillationes ejusdem penduli, vel plurium pendulorum aequalis longitudinis r per arcus satis exiguos utcumque ceteroquin inaequales, fore ad sensum isochronas seu aequi
diuturnas.
Idipsum facile demonstratur hac alia ratione: angulus GCT
= 90° BAC; hinc vis acceleratrix CG , ex qua sola repetendus est penduli descensus, exhibebitur per gsing: in hypothesi nimirum arcuum satis exiguorum spectari poterit CG tamquam proportionalis distantiae a puncto infimo B, computatae in arcu BC. Ergo ( 29. 4°) etc.... Etiam sic: est ds = d rík - a ) rda ;
et consequenter
rda da
dt
V rg (k -u?)
-Vivok²-u?
factaque integratione ( 27. 13º. 14° ) prius ab a kad
a =0 , dein ab a = k ad a = -k, emergent binae (a' ) .

66. Haec notentur: 

1º: secunda ( a' ) dat
77 r
8 ( a '');''
ta
atque inde innotescit gravitas g.
138
Iam vero in puncto intimo B (Fig. 37) exsistit «:o;
erit igitur tempns semioscillationis. &quot;
∙−∣
⋍⋅∶−−−−⇄⋅−∣∕−≦−∙
tempus integrae oscillationis (,,-)
−∣−∙−
ta:T! −∙− ∣∕ :
,
quas formulas cum non ingrediatur initialis angulus k, patet oscillationes ejusdem penduli, vel plurium pendulorum
aequalis longitudinis :- per arcus satis exiguos utcumque
ceteroquin inaequales, fore ad sensum isochronas seu aequi-
diuturnas-
Idipsum facile demonstratur hac alia ratione: angulus GCT

: 900— BAC; hinc vis acceleratrix CG, ex qua sola re-

petendus est penduli descensus, exhibebitur per gsinat: in
hypothesi nimirum arcuum satis exiguorum spectari poterit
XCG tamquam proportionalis distantiae a puncto infimo B.
computatae in arcu BC. Ergo (29. 40) etc.
...
Etiam sic: est ds:dr(k— a):
-— rda;
et consequenter
rdat dat
dt −∙− ↵
−− '&quot;
VrgUe-æ .? sz-az :
factaque integratione (27. 130. 140 ) prius ab et: I: ad
«:o, dein 'ab a: ∙−−− I, ad ac ∶−−⋅ —-k, emergent binae (a')-

2º. Etsi ponderosa diversae materiei puncta permissa sunt oscillare, attamen idem semper prodiit valor g in eodem terrae loco: rursus ( 17 ) igitur devenitur ad proportionalitatem inter corporum massas et respondentes gravitatis vires.

3º. Constat observationibus longitudinem penduli
simplicis oscillationem absolventis intra mioutum secundum
eo esse minorem, quo magis ad aequatorem acceditur: quoniam ergo, haud variato tz, gravitas est ut longitudo illa, minuetur gravitas a polo ad aequatorem usque ( 30) .

4º. Apud nostras regiones praefata penduli longitudo cum sit = 3ped opol glin, 38 = 440lin, 38, factis
in ( a &quot; ) tz = 1 &quot;, r = 440lin , 38, prodibit respondens gravitatis valor g = 30ped , 183 alibi (30) indicatus.
[[Fasciculus:Double-Pendulum.svg|thumb]]
67. Quod spectat ad pendulum compositum concipiamus (Fig. 39) puncta ponderosa B, B. , B2 , . . filo appensa: invicem disjuncta conficerent haec puncta temporibus inaequalibus oscillationes suas; punctum nempe B, citius  (66) quam B, punctum B, citius quam B, etc: invicem ergo conjuncta agent ita in se mutuo, ut quae, minus distant a puncto suspensionis A retardentur ab iis quae magis distant, et quae magis distant a suspensionis puncto accelerentur ab iis quae minus distant: fiet propterea oscillatio penduli compositi tempore quodam medio inter minimum ac maximum praedictorum temporum inaequalium. Hinc sequitur fore in AB punctum quoddam B.,m suas conficiens oscillationes perinde ac esset solitarium, nulloque nexu caeteris punctis uui
retor: Bm dicitur centrum oscillationis, cujus centri distantia a puncto suspensionis est longitudo penduli simplicis suas perficientis oscillationes eodem tempore ac pendulum compositum. Inferimus oscillationes pendali compositi, et ipsas fore isochronas; modo tamen exsistant satis exiguae.

68*. Facile intelligimus ( 50. 3º. 6° : 66 ) motum penduli simplicis in medio resistente determinari generatim per aequationem

140
das
di?
= gsing -f(v ) .
derka( )
di Ob dc2
dra
dt2 et ( 27. 29º . ) sing
23
2.3 +
aequatio illa eyadit
creat + s ( « - +...)-fo) =
Pone fv) = cv ; et angulum a ita perseveranter exiguum,
ut ejus tertia potentia absque sensibili errore negligi pos
sit : habebis
da с
+
dla
ola
0 .
ds Est autem v =
dt
drak -a)
dt
da
dt ; igitur
d2C da
dt2 to dt + baro:
quam integrantes in hypothesi c constantis assequemur( 27.270. )
... [ :V546.-V21
( 6),
In experimentis, quae pendulorum ope solent institui ,
r est multo minor quam g; item densitas penduli, et con
sequenter ( 33 ) c fractio admodum parva. Fac ergo
140
tiis ∙
de'-* :gsmat —f(v)
d': d2r(k—a) (lioc
↽⇁−−− −∙−
' Ob daz dtz 27. 290. .:
rdtï
, et ( ) stna
'
a3 . '11 d'
rdzatdta-l-g(at—-— ...):fþv) :0.
Pone f(v): cv; et angulum ac ita perseveranter exiguum,
ut ejus tertia potentia absque sensibili errore negligi pos-
sit: habebis
dza g c
dia—FTa—Tv—o
.Et :
−−∠≀≖∙∙↙≀↗≺∣≖⋅⋅∝≻∙− ∎∠≀∘⊏∙⋮ ⋯⋅⋅ s auem'v— dt— dt rd , g1
quam integrantes in hypothesi c constantis assequemur(2 7.2 70.)
&quot; til &quot;'i-i.]c) —l ! —-..—.
2:32 [CG 4
r—l—C'f—B
ln expetimentis, quae pendulorum Ope solent institui,
r est multo minor quam g; item densitas penduli, et con-
sequenter (33) c fractio admodum parva. Fac ergo141
VS
ut sit VA = iVT ;
4 .
vertetur ( 6) in
ti V = 1 -til
+C'e 2 -
&quot; ]
U = e
seu ( 27. 30° )
3 [
e
] ;
C &quot; sin it +C ' ' cosit
unde
cosit
data o - [( c'i - )
(c&quot;: + * ) ainit ]
da
In joitio motus t=0 , a= k , 0 ;
di
propterea C &quot; &quot; k, C '
ck
21 ; et .
= ke - - [
sin it + cosit ] . )
2i
( 6 )
dan
dt = -ke- Ź [ ita sinic.
141
'
. . ∙
c' a . .
ä-ï': - utsit V—--€- :::tl/ −−↿ ∙
r 4 4 :-
vertetur (b) in
−−−∘⋮−≀ — tiV—1 -til/—1
at:6 2
)
C'0 v
380 ( 270 300 )
↴
c ↼
a:
∘−−≖− '[C&quot; sinit ⊣−∁⋯ cosit] ;
unde
∘⋮ C'&quot;
↙⋮⋮∶∶ .
∘∙⋅−
; t [( C'i— 20) cosa
: -— dt ∙

Cnc . .
(
0&quot; i −⊢ —2 )
sunt ].
dat ∙∙∙ ∙
In initio motus t—:o
, at:— ]:
, &quot;dt −∙− 0-
∙∙∙ k
': propterea C Ck et −−−∙−− ∙
∙−− 21' ∙142
с
Ex.Vihabemus zi 11
ll
Hlacin
c2
V C
it =-V
rc?
4g &gt;
2V EV
rc2 1
48
1
i +
VE ;factoigitur V
cr2 =c,
43
&quot; Vi ro2
4g
binae ( 6 ) sic poterunt exprimi
a = ke

* IVE Vētowi.V ]

1.)
(6 )
dm-- .- iv E sinórV.
In fine cujusque oscillationis est da
dt = 0; proinde, ob
= 0: inferimus in fine primae secundam (3&quot;),since V
12 풍VV
2.V
oscillationis fore t = &gt; in fine secundae
8
271 376 in fine tertiae t =T
8
8
gulae itaque oscillationes absolventur aequali tempore
E ,
in
V
, elc.. .. ; sin . с
g. ∘ Ex ::i habemus 21. ∶∶ − c —- .
∙
∙−−−−−−−
I/ ⇂∕−−−⋅↿rc
c ⋅ :! −⋮↓−≔⋤⋅ a
cr&quot; ,
. CZ .g 1;factoigilur V1—-- :c, ↓⊣− ' ⋅ −−∶ ⋅
binae ( b') sic poterunt exprimi
£(sz 2 inc't cosc' g
c[hc V—s Vg ∙−⊢ ::ll/:]
(ö&quot;)
(E;—..., &quot;&quot;T-V—sinc't 5-
dt :-
/
In fine cuiusque oscillationis est ≤−∝⋮∶∶∘⋮ proinde, ob
secundum (b&quot;), sinc': Vi:o: inferimus in fine primae
!'
1:
osc1llat10n1s ∙ ∙ ∙ r ∙ fore :

:

−⋮⇆⊤ V—3,
in fine secundae
271 .
⋅⊤ −∙
,
in fine tertiae :
−−−−−∣−
−− ,etc.. -sm-
gulaec itaque goscillationes absolventur aeqnali tempore143
و
=
ا
( 6 '' ) .''
8
In primo substitutis valoribus 0, 20 , 30 , ... no
pro t , emerge
Qu - ke 2 A2 = ke -
&gt;
929
as= -ke- 30
Q. = 1–1 y&quot; ke – no
hinc successivarum oscillationum amplitudines
0
2
음
k + ke ke - 9 +ke - 2
-ke 39 -
2/2
20
the
ke
seu
1(1 +-2), ( +-3, -2,

* (170-99 . -Ź- 42329......

Decrescunt igitur amplitudines istae in progressione
geometrica : quod cum confirmatum sil experimentis pen
dulorum in aere oscillantium per arcus satis exiguos, haud
majores v . g. tertia parte unius gradus, licebit quoad e
jusmodi oscillationes assumere aeris resistentiam tanquam
proportionalein simplici velocitati.
143
n −−−∽−∣∕∙⊂− (B&quot;)- 0

6
In prim: (. ⋮⋅ substitutis valoribus 9, 29 , 39 , ... 719
pro :, emerge f
⋅∙⇁− -
∙
1- 0
29 ⋅ C
a;:— ke 2 ,agzke 2 ,a3z—ke 2 39
C
119 a.::(—1)&quot; ke −⋮⋅ :
hinc successivarum oscillationum amplitudines
.
c ⋅∙⋮∙∙ ...—£
k-l—Re— i.e.]:e— 294-ke ∙
229
&quot;
ke −⋅⋮⋅⋮∂−⊦∣∥⋝ 2
39,. ;
seu
.- e ∘
9
−−∘ &amp;
Decrescunt igitur amplitudines istae in progressione
geometrica : quod cum confirmatum sit experimentis pen-
dulorum in aere oscillantium per arcus satis exiguos, haud
maiores v. g. tertia parte unius gradus, licebit quoad e-
jusmodi oscillationes assumere aeris resistentiam tanqnam
proportionalem simplici velocitati.144
Experientia insuper docet decrementum illud gradu
admodum lento procedere : sic D. Borda expertas est non
nisi post 1800 oscillationes valorem en converti in k .
Hoc posito, existet
2
1
1800c7V
.18000 e 2
2 2c seu ob (6 ' ') e
3اندبه
et consequenler
1800CTE
= c'log. 13 == c (o , 40546) . 2
( 1800)? c ?72. &quot;
= 1 - c '? , ideoque 8
4g
ro2 Sed
4
= (1800 ) 2772 /1— 2) ; igitur ( 1800)277 ?(1 — c'2)
= c (0 , 40546 ) ; unde
c' ?
( 1800 )2772 1
(1800 )222 I.(0,40546 )
&gt; et e
V
+18007
1+
0,4054612
180076
1
quam proxime.
gua
Si ( 33 , 4.&quot; )
poneretur f (v )
terminandum exsisteret
ad penduli motum de
144 *
Experientia insuper docet decrementum illud gradu
admodum lento procedam : sic D. Borda expertus est non-
. . , 1118! ∙ ⋅ ∙ ∙ 2 post 1800 osmllattones valorem ac,, com-cru mes-k.
Ilnc posito, existet
'
u
1eöocn VL
..
'
5.
20'
≀∙ ⋅
.
...—18009 2
e 2 −−∙−−− −∙− , seu 01) (b&quot;') e
⊏⋅⋮− ,
. 5 ⇁
3
(!l. 000 sequenter

: .—c'(o ∙⇁ '40546) b. &quot; .

1800072l/L .
2 g :c'lo
'.
(1800≻∖⋅:0:712.— g
PC: '
∙
'
Sed −−−−−↿ ---c2
, 1deoque 45
⇌≺↿⋅∂∘⊙≻≃⊺≖≖≺↿−⋅≺∶∣≏≻⇋ igitur ≺↿∂⊙∘≻⇄∏≖≺↿−∘∣≖≻

: c'5(o, 4054673; unde

(1800)??? - -, 1
, ↼−−
≺↿ et -c':
≨∃⊙∘⋟≖⇃∙≖⋍⊣−⋅⊏∘∙∠↥∘⋦∢⊖≻ ↼↼
'3
Vl—l&quot;(7'g55; o.4(l546)z
≖≖−−−↿ quam proxime.
↓
2 '
51 (33. 41.&quot;) poneretur f(v) ∶−−⊸∙⋮⊥⋮− , ad penduli motum de-
02
termiuandum exsisteret145
des
dt?
gsing gu2 d ?
seu de2 + sine —
8r /dala
2
= 0 .
c²lde
Haec prias multiplicata per 2du , ac dein integrata suppe
ditat
' dala Idala
ca ldt
2g
COS O
seu facto Slaa) dx = y , ideoque Coupe ( )
dy
da
dy
28
2gr COS Q
da y = 0 ;
cojas integratio traducitur ( 27. 26 °.) ad integrationem fun
ctionis
2g cosada
2gra
c2 re
Jamvero , facto compendii causa 2gr
= m , habemus c2
dem sina ) coso, da Se
ma
-mu m e sing da ,
d ( e-ma cosa )
-ma sina da - me COSQ da :
igitur
ſe-ma cosa da e -ma sina tm se-ma siac da ,
ſe-ma sina du = me-ma cosa m ſe-me. cosa da ;
ex quibus
145
d's— ∙ g.&quot;: dia g ⋅⋅ . gr
äzäfgsma— Z;- , sendt: da)!— -[-r sma 02 22 —--0.
Haec prius multiplicata per Zda , ac dein integram suppe-
ditat -
(&amp;)2— dt ∙−⊋∊∁∘≘∝−⋅⋮⋚⊆∫≺≦− r
f:) ↙≀⊄∶∘∙∙∙
dat ∙∙∙ ∙ da: a... 47&quot;
seu facto f(ä—t) fia —J—, ,
1deoque (22) −∙− ä; ,
≝⊻−≟≝∁∘⊱⊄≉≣≝⋅∫∶∘⇋ . dat
cuius integratio traducitur (27- 262) ad integrationem fun-
ctionis
Zg cosadat ∙
⋅
2grat .
∘≖ . re :
∙
2 r
'
Iamvero ,facto compendii causa −⋚−⋅:m ,habemus
02
,! (.;-&quot;&quot;&quot; since ):e'ma cosa: d-a −∙∙ m e'm' sinat da
,
d(e'macosat):-e'm ∋⋮∘⊄↙∄∝−⋅⊪∘⋅⋯ &quot;cosada:
igitur
fe'm. cosa da::
efm sinat—lfm fe'm sinat dat
,
fe'm sinat daz ⋅⋅−−− −−∶ e'm cosa: — m fe'm- cosa daz ,-
ex quibus146
ſen-Ma cosa da e -ma sina — те cosa -m2ſe-mecosadu 7,
et consequenter
2g
Sce-ma cosa da
2g ( e -ma since me- mu cosa)
r ( 1 + m2)
Erit itaque (27. 26 °.)
y = Cema +
2g ( sing - m cosa )
r ( 1 +ma )
ex qua differentiata quoad &amp; cum emergat
dy
da
Cmema + 2g (cosx + m sina )
r ( 1 +m2)
restituto valore dy
da habebimus
ca
dal 2g (cosa + m sina)
= - Cmema +
r ( 1 +m2 )
da
In initio motus a = k , = 0 ; hinc dt
Cm 2g ( cosk + m sink) e-mk
r ( 1 + m2)
propterea
-m (k - a )
Cate)
dal 2
ldt
29
r ( 1 +m2)
cosa + msina - cosk + msink)e ( h).
Facto a = o in ( h) , prodibit inde velocitas penduli in pun
cto infimo B ( Fig. 37.) : ascendet pendulum cum velocitate
146
fe'm&quot; cosa da: e'm&quot; sinat —me*mcosoc −∙∙ ⋯≖∫∘⋅⋅⊪∞∽∡∠≀∝ ∙
et consequenter
2g &quot; 2g (e-&quot;W- sinat −∙∙ rne-ma cosa)
∙−∙∣ (.'-'&quot;&quot; cosa dat:
⋅
'
r(1 −⊢ ⋯⋅≀≽ Erit itaque (27 . 260.)
2g (sinat —m cosa)
y.:Cama'i' r(1-l—m') :
ex qua differentiam quoad a: cum emergat
dy ∙− M Zg (com-[- m 5213.)-
da :Cme r (1 —f-m3) '
∙
d
' ∙ resututo valore 1, babebmus ⋅
de:
((!—S :Cma'&quot; &quot;I&quot; 25 (cosa: ∙⊢ 11: sind:? ∙
.. dt r(1-l-m3)
∙ ∙ ∙
'
da ∙
In 1n1t1o motus a:k −∙− −−−−∙ o ; bmc 'dt
2g (cosk −↿− m sinit) er:-&quot;&quot;* ∙
Cm: r(1—l-m2) ..
propterea
(de!)2 28 &quot;[COSa-Hnsina—(cosk-i'msïnk) e-m(k-a)] 32 :r(1—i—m2) (73)-
Facto a:o in (I:), prodibit inde velocitas penduli in pun-
cto infimo B (Fig. 37.) :ascendet pendulum cum velocitate147
ista versus D , conficietque arcum , cui respondebit — Q,; et
quoniam in extremo puncto illius arcus extinguitur tota ve
locitas , iccirco
COS - m sina, · ( cosk + m sink) e -m (4+ 1) = 0 ,
seu
(cosa,
m sing , emai (cosk +m sink) e-m * = 0
( h ' ).
...
mk .
maa ,
Sunt ( 27. 29.° ) emas = 1 + m « . + +
2
mak ?
=1 -mk+ -... ; est insuper m fractio admodum
parva ( 33) : neglectis igitur terminis , ubi invenitur mº ,
traducetur ( h ) ad
2
&gt;
cos@g - m (sina, cosax) = coskt m (sinkkcosk ) (h &quot; ).
Denolante o differentiam inter valores a, et k ut sit Q= k -0 ,
certe ð erit fractio tenuissima : hinc substituto k- loco
Qy in ( &quot; ) , sumpto 1 pro cosd et à pro sind , missisque
öz et mo , assequemur
2m
Osink = 2m ( sink - kcosk) , d = 0
sink (sipk- kcosk ) ;
unde
Uy= h
2m
(sink-kcosk) .
sin k
Si popimus k ita iguum , ut ejus quarta potentia prae
termitti possit , obtinebimus (27. 29.° )
147
ista versus D, &quot;conficietque arcum , cui respondebit — at,; et
quoniam in extremo puncto illius arcus extinguitur tota ve-
locitas
,
iccirco
111 cosa:, −∙∙ m sinatl — (cosk −∣− m sink) e&quot; (b'-3 1): o ,
seu
(cosa, ∙− 11: sind,) a'&quot;! — (cosk —-msink)e&quot;&quot;'* :: o (b').
Sunt (27.29.0) erat: mna? ↿−⊦⋯⊄≖−⊦
2 −⊦ ∙ ∙ ∙
∙ ,∙⋯⋆
⋯≖⇂∙∙≖ ⋅ ∙ ∙ ⋅

: i —mk—l—-—2—-— ...; est 1nsuper m fract1o admodum ∙

parva (33): neglectis igitur terminis ,.
ubi invenitur m',
traducetur (h') ad
tuom,—m (si na,—aleam,:cosk—l-müi nk—kcosk) (h&quot;).
Deuotante ö differentiam inter valores a, et k ut sit ac,: k-ö.
certe d erit fractio tenuissima : hinc substituto k—ö loco
a, in (II&quot;), sumpto 1 pro cosd et 6 pro sind
. missisque
d' et md , asscquemur
ösinkz2m (siuk—kcosk) , ö: ET- (sink—kcosk);
. sink
-
nnde
2m
sin lt at:-k— (siïnk—kcoslc) .
Si ponimus !: ita exiguum , ut eius quarta potentia prae-
termitti possit , obtinebimus (27. 293)148
2m 2m
- (sink - koska gink k
21 k2 1
2.3
2m 2m ka ( 1
k2
2.3 h2 ,
3
ac proinde
Q = k
2m
3
k2 :
quemadmodum valor a, deducitur ex k , sic ,yalor d, ex
valor az ex la , atque ila porro ; erunt nempe
2m Aa =
2.1 az ?; 3=0,- 313, etc...
| Patet illud ; si vis acceleratrix ex medij resistentia sumitur proportionalis quadrato velocitatis, haud subsistet superior lex, experimentis confirmata, de oscillationum amplitudinibus in progressione geometrica decrescentibus.
[[Fasciculus:Cycloid f.gif|thumb]]
[[Fasciculus:Cycloid03d.svg|thumb]]
[[Fasciculus:Cycloide InfinimentPetits.svg|thumb]]
69. ° * Aliquid subjungimus de gravium descensu per arcus cycloidales. Circulus A'D ( Fig. 40 ) tangens rectam A”E in A&quot; revolvatur super ipsa A”E ita, ut eam pergat semper tangere. Punctum A&quot; circuli regredietur ab A&quot; in E, lineamque curvam describet, quae appellatur cyclois: circulus ille mobilis vocatur cycloidis genitor, recta A ” E basis, diameter AB perpendicularis mediae basi dicitur axis, punctum A vertex; patet autem quemvis circuli genitoris arcum B’A' aequari rectae A'B, quae intercipitur duobus punctis A &quot; et B', in quibus extre ma puncta ipsius arcus tanguntur ab A'E; et totam basim AE aequari peripheriae circuli genitoris. Ducantur
148
2
2,&quot; (siuk—kcosk)-— &quot;' .↗⋮∍ mnk ]: (
kz 3 2.) ⋅
ac proinde
2m
'k 3k
quemadmodum valor a; deducitur ex 1: , sic.valor ag, ex
'a, , valor 013 ex ac, , atque ita porro; erunt nempe
a—a—ïaa' a—a—zma' etc - 2—1 3
1
, 3—3 T;, ∙Ducantur149
jam ex cycloidis puncto v. g . A' perpendicula A'rl= y )
et A'C , alterum in basim AE, alterum in axem AB ;
sit A'r = x ; diameter circuli genitoris dicatur 2a; exhibea
turque per &amp; arcus quoad radium -- = 1 similis arcui A'B' .
Erunt
x = A'B - B'r - A'B ' - AM = a5 - asins ,
y = B'M = asin.v.zza( 1 - сoss) .
ex istarum prima assequimur
dx = ads - acoss ds = a ( 1 - cos )de ;
et dividendo per secundam.
dc
de .
y
Est autem arc sin=
are(sin = AMM))—are (sin V Zay —ya )
a
IV2ay - y2
et consequenter de 2
2ay - y
aa
dyZay - y ?
dy
V2ay - y2 ; ergo
2a
dy = dx V?
(at ) ;
y
149
iam ex cycloidis puncto v. g. A' perpendicula A'r(:y) ⋅
et A'C, alterum in basim A&quot;E. alterum in axem .AB;
sit A&quot;r:æ; diameter circuli genitoris dicatur 20; exhibea-
turque per &amp; arcus quoad radium −∶∙−↿ similis arcui A'B'.
Erunt
æ:A&quot;B'——B'r—-A&quot;B'—-A'M:ae-—asins ,
y:B'M:asin.v.:a(1—coss) ∙
ex istarum prima assequimur
dæ:ada—acoss de:-au -cose)d£ ;
et dividendo per secundam.
d
-—æ-- :de ∙ 7,
Est autem :arc(sin: M):arc (sin ∙−−∶ ∣∕⊋∅∫−↗↾≖
a a ),
et consequenter de:
.a— ∙ a
): ∣∕ ↿−− 5351-
02
dl/Zay—yz df
a—y VZaJ—yi 'ergo150
aequatio differentialis ad cycloidem , computatis coordina
tis a baseos initio A &quot;, Quod si computentur a vertice
A , ut novae coordinatae sint AC ( = x ') , et A'C ( =y' ) ,
cum habeamus x = an — y , y = 2a — x', prodibit
-dx'adyV
x'
2a- , seu dy = dxV
2a - x
xช่ (a &quot;) .
Nunc ad gravium descensum quod pertinet per ar.
cum quemvis cycloidalem , cujus vertex in puncto inſimo
B ( Fig. 37), sit C initialis positio puncli ponderosi , quum
nempe t =0 et v = v = 0 , M positio in fine temporis 1;
quibus positionibus respondeant altitudines c et ac' supra
horizontalem rectam transeuntem per B , ut in Mha
beatur v = V 2g(c-x') : denotantibus h , se s' cycloidales
arcus CB , CM , BMBM ,, erit erit dsds== dhd (h -- ss'')) = - ds'; unde''
ds
di
ds'
dt = V2g(c -x '), ex qua obtinelur
ds'
dt
V 28(c — x ')
Formula ( a&quot; ) praebet ( 27. 19.0)
2a -x
do = Vdx =+ody&quot;a= dx V17 = dx '

;

x
hinc
da - c
a
dt dx'
V. 8 V cx' — x'2 -Va
GVFECITATE
150
aequatio diti'erentialis ad cycloidem . computatis coordina-
tis a baseos initio A&quot; Quod si computentur a vertice
A , ut novae coOrdinatae siut AC (:æ') , et A'C (:y'),
cum habeamus x:an'—-7, 1:20—æ', prodibit
I
Za—æ ∙−−−
, seu df:dx I/ æ, (a') .
Nunc ad gravium descensum quod pertinet per ar-
cum quemvis cycloidalem . cujus vertex in puncto infimo
B (Fig. 37), sit C initialis positio puncti ponderosi, quum
nempe t:o et ⇂↾−−∙∶⇂↗∘∶∶∘ , M positio in fine temporis :;
quibus positionibus respondeant altitudines c et æ'supra
horizontalem rectam transeuntem per B , ut in M ba-
beatur :»:V Zg(c-x ':) denotantibus ,: .
s, s' cycloidales
arcus CB, CM, BM , erit ds:d(h—s'):—ds'; uude
ds di' .
v −− dt— dt —l/2g(c-æ'), ex qua obtmetur
ds'
dc ∶−∙−− − ⇂∕−−−−−
⇄∊≺∁−−⋅↿⊏⋅ )
Formula (a') praebet (27. 19!)
d.;— −−∙ ⇂∕∎∎−−∎∎−∎∎∎ ↙↙∙≖⇌ ≖⊣⊸↙↿∫−
hinc
(1&quot;.—— a flx' ;.
(ll:
∙−∙ V
∙−− ∙−− V
a .; 20 .
l
—— g J/Fæ—æ'z ∙ g
∣∕↿ ∙↕⋅∎⋅≩∁≻≖⊽151
sumptisque integralibus ( 27. 9,9 ) ,
= c +Vare (co== **)

;

in positione initiali est t=0, simulque x' =c; igitur C = o, et
Vore (rosa ).
Facta x=0, prodibit tempus descensus usque ad punctum
infimum B, nimirum
11 =T
VO
ubi cum non inveniatur c , patet , ex quocumque cycloi
dis puncto demittatur grave, eodem semper tempore per
venturum ad B. Hanc cycloidis proprietatem posteaquam
detexit Hugenius , cycloidem ad pendolum adhibere cae
pit : quod qua ratione fieri possit , ostendit in parte 3. “
Horologii oscillatorii.

=== De attractione corporum in hypothesi attractionis agentis in ratione directa massarum, et in reciproca duplicata distantiarum.===
70. Pyramis AH (Fig. 41) habens basim GH infinitesimam secetur superficie sphaerica, cujus centrum in A , et radius AZ ( = r ); sit Ky = B ) projectio intersectionis VZ ( = ) in plano AB; supra basim Ky erigatur prisma KyE altitudinis CH ( =x): exprimet KE AZ sumptisque integralibus (27. 93) ,
↥⋅∶∁⊹ l
V— a1c (eo: 200) ;
in positione initiali est t:o. simulque æ':c; igitur (l:-o, et

* x −−∶−∁

Facta x':o. prodibit tempus descensus usque ad punctum
infimum B, nimirum
a
II:&quot; V— :
g .
ubi cum non inveniatur c , patet, ex quocumque cycloidis puncto demittatur grave, eodem semper tempore perventurum ad B. Hanc cycloidis prOprietatem posteaquam detexit Hugenius, cycloidem ad pendulum adhibete caepit: quod qua ratione fieri possit, ostendit in parte 3.'
Horologii oscillatorii. ⋅

lit-F.
AZZ152
vim attractivam ( p) segmenti CG in punctum A juxta directionem perpendicularem plano AB. Intelligatur enim segmentum CG secari sphaericis
superficiebus numero infinitis , quarum centrum in A , et
r2 , 7* 3 sintque eoz , A2 , A3 intersectionum areae. Erit
radii rs .
din 23

; 2

2 r2 3 p2
vis nempe attractiva cujuscumque areolae Qy , da , .
aequabit vim attractivam areolae A. Ex punctis Z, C,
ducantur in AB ... perpendicula Zy ( =n) , CB ( = n ) ...:
singulis viribus resolutis in duas, quarum altera sit paral
lela , altera perpendicularis plano AB , componentes per
pendiculares repraesentabuntur per
ni na n
2
ri ra
et quia
ni n2 n
72
iccirco
li ni 0.2 п,
=
a n
.
t'i p22 ra
his positis , quisque videt fore
n
f
152 .
vim attractivam (:f) segmenti CG in punctum A juxta
directionem perpendicularem plano AB.
Intelligatur enim segmentum CG secari sphaericis
superficiebus numero infinitis. quarum centrum in A, et
radii r; , r,. rg
.... sintque at,
, at,, ata ...
inter-—
sectionum areae. Erit
«! a; «3 ∙
. a
r,: fa, rna ra.
∙
'
vis nempe attractiva cujuscumque areolae at,, at,, ∙∙∙
aequabit vim attractivam areolae at. Ex punctis Z, C,
...
ducantur in AB
... perpendicula Zf (:n)
, CB (:m) ...:
singulis viribus resolutis in duas, quarum altera sit paralf
lela, altera perpendicularis plano AB , componentes per-
pendiculares repraesentabantur per
a! &quot;[ a:; &quot;a a n
. , ∙ , ∙ ∙ . ∙−−− .—,. r,2 r, rf r, :-a r
et quia
—:
&quot;! &quot;2 n
∙−− ∙ ∙ ∙ :∙
—;
r, r, r
iccirco
«! &quot;r a: n, a n
∙
∙−−∎ ∙ ∙ ∙ ∙:∙∙−∎ ∙ ∙∙∙∙ :
rl: rl ,.22 &quot;, &quot;, r153
Jamvero (55.4. )
\beta
= cosyZA 3
igitur &gt;
n
ela
B
sli
oli
a
et consequenter
Bx
r?
KyE f
AZ

71. Singula corporis cuiuscumque KGDH (Fig. 42) puncta trahant punctum C positione datum. Centro C et
radio quolibet CM describatur sphaera MBN; in eius
superficiem incurrat in A recta quaelibet CG permeans
corpus KGDH iuxta DG ; demittatur ex A perpendi-
culum AQ supra planum MCN; capiatur in- AQ pars
TQ aequalis segmento DG intra corpus KGDH demerso;
quod si plura fuerint huiusmodi segmenta, pars in per-
pendiculo accepta sequetur omnium summae, Si per GM?
dividitur solidum ïTXV, quod continetur plano MCN
et superficie ab omnibus punctis T determinata , expri-
met quotus vim, qua totum corpus KGDH trahit punctum
C perpendiculariter ad planum MCN.
Prodeant enim ex C infinitae numero pyramides, qua-
rum segmenta DG impleant totum corpus KGDH; pote-
runt totidem respondentia (69) prismata TQ concipi , quae
totum solidum ïTXV impleant; ergo etc.
Quoniam vires omnes sollicitantes punctum C
possunt traduci ad ternas
, quarum directiones congruant
cum tribus rectis se mutuo ad angulum rectum secantibus in ipso C; ternae vero istiusmodi vires in unam com-
↿↿154
positae dant resultantem ex illis omnibus , inde fit quod
ubi determinentur (70) ternae vires corporis KGDH re
spective perpendiculares tribus planis orthogonalibus per
punctum C traseuntibus , eae in unam contractae suppedi
tabunt et directionem , et intensitatem illius vis , quae re
sultat ex omnibus viribus punctorum constituentium cor
pus ipsum KGDH.
Si punctum C intra - corpus trahens collocaretur
accipienda esset TQ aequalis differentiae inter distantias
ipsius C ab extremis D et G rectae DG transeuntis per
C. Hinc si C fuerit situm intra ejusmodi crustae cavita
tem , ut per C ducta quavis recta , aequales hinc inde par
les illius rectae intra crustae crassitiem intercipiantur, eva
nescentibus omnibus TQ , evanescet etiam omnis vis pla
no cuicumque perpendicularis , et punctum C in aequi.
librio consister.

72. • * Coordinatarum originem O constitue in
quovis corporis puncto ; sin que x, y, z coordinatae pun
cli altrahentis ; a , b , c coordinatae puncti allracti ; '
distantia inter punctum attrahens et punctum altractum :
expriment
b - r
CZ
ba 몇
7
A'
A
cosinus angulorum , quos a continet cum axibus coor
dinatis OX , OY, OZ. Quare denotantibus Hc , H,, H,
componentes iisdem axibus parallelas , in quas rosolvitur
attrahens totius corporis vis H , et dm elementum massae,
eront
H,
- Som dm , 1 , = Sabah dm
(o)
H , Set dm :
A3
154
positae dant resultantem ex illis omnibus; inde Et quod
ubi determinentur (70) ternae vires corporis KGDH re-
spective perpendiculares tribus planis orthOgonalibus per
punctum C traseuntibus, eae in unam contractae suppedi-
tabunt et directionem, et intensitatem illius vis, quae re-
sultat ex omnibus viribus punctorum constituentium cor-
pus ipsum KGDH. ↴
Si punctum C intra -corpus trahens collocaretur ,
accipienda esset TQ aequalis differentiae inter distantias
ipsius C ab extremis D et G rectae DG transeuntis per
C. Hinc si C fuerit situm intra eiusmodi crustae cavita-
tem
, ut per C ducta quavis recta, aequales hinc inde par-
tes illius rectae intra crustae crassitiem intercipiantur, eva-
nescentibus omnibus TQ, evanescet etiam omnis vis pla-
no cuicumque perpendicularis. et punctum C in aequi-
librio consistet. ⋅
⊽∑∙∘∙ Coordinatarum originem O constitue in
quovis corporis puncto; sintque x, ],:coordinatae pun-
cti attrahentis; a, b ,&quot;c coordinatae puncti attracti; A'
distantia inter punctum. attrahens et punctum attractum:
expriment
⊄≖∙∙−−∙∙−−∙⋮≖ b—gr c—z
∆∣∙
'
∆∣
'
∆∣
cosinus angulorum, quos ∆⋅ continet cum. axibus coor-
dinatis OX , Oï, .OZ. Quare denotantibus H, , H, , H,
componentes iisdem axibus parallelas, in quas rosolvitur
attrahens totius corporis vis H ∙ et dm elementum massae,
erunt ⋅
a—æ &quot;bf—7 ⋅ ∏≖∶∶∫ ∆∣⊰ dm,
⊟⋮⇌−∽∙∣∙−⊒↙∙⇁⋮⇀≀≀≀↿∙ .-
(0)
,:szjä-äfdm:155
integralia se se protendunt ad totam corporis massam M.
Pone
Q = Sam
( o ')
habes quidem
A2 = (a — x )&quot; + ( my) + cz( )&quot; ;
sed quia integrationis limites non pendent ab a , b , c ,
ideo ex prima ( o' ) erues
dQ
da ſ
dm ,
dQ
db
den
ES ,
do dm
dQ
dc -dm ;
da
secunda vero (o' ) praebet /
a 영 1 dA a -X a
A' ? da
b
da 4'3
db A'3엷
slot
dc 4'3
traducentar itaque ( o ) ad
H=
dQ
da H ,
do
db H,
dQ
dc ( o &quot; ) ,
componentesque H , H ,, H, pendebunt ab unico integrali
l. Fiat
a : + 62 + c = A2 ,
155
integi-alia se se protendunt ad totam. corporis massam M.,,
Pone
≺≀⇌⇀⋅ dm −∙−≃
habes quidem (O,)
∆≏⇋−∙≺∅∙−∞≻⋍⊣−≼≀↗−∫≻≔⊣⊣∘∙−≖≻⋅ ;
sed quia integrationis limites non pendent ab a, b , c ,
ideo ex prima (a') erues
⋛≣∎∶∆∼∣↙≟ dm' ∎−⋅∫↲∂↙≀⋯⋅−−−−∫ &quot;'&quot;-('m-
secunda vero (a') praebet
↿ ↿
⊄∄−−−↽
∆∙ ↿ siA—, a—æ (LA-7 b—r
de'&quot; A'da A'3 ↞∙ ∠≀∣⊃−−⋅−−−−∆⋅≀∙ '
↙≀∙∙↿⊽ ac.
.... de −⋅⋅ ∆∣∍
traducentur itaque (a) ad
dQ dQ dQ−⊋⋤−∙ Hic—2? ∙ Ha ≔−⋅⊋⊂∙∙− (O&quot;) 1

componentesque H, , H,, H, pendebunt ab unico integrali
Q. Fiat ⋅
∁≖⊹∂≖−⊦∁∶≖−−∆≖ ∙156
ut secunda ( o ' ) scribi possit in hunc modum
A ' = 12—2(axtby tcz) + wa + ya + z2 ;
erit
1
-
[ 12—2( ax + by + cz) + xa + ya + za = +
+
2(ax + by + cz) xtya taza)
+
243
12(ax + by + cz)2- [ 12 (ax + by + cz )-3(x2 + ya + za) ][ x ? tye + z&quot;)
845
+
.
unde, ob prinam ( o' ) ,
m
Q

* ++ ſ(ax + by+ ca)dın

25 /(x +y +z&quot;)du + z flar+ by +czydom.co&quot;.
Sit coordinatarum origo in centro gravitatis massae at
trahentis; erit ( 20. b )
1
43
Slax ( ax + by + cz) dni = ta fædm +
bſydm + ſzam ] = 0 ;
ideoque vertetur ( o '&quot;') in
156
ut secnnda (o') scribi possit in hunc modum
A&quot;::A3—2(aæ—-l-bj-l—cz)*æï-þyl-l-z' ;
erit
−↙∃≃∙⋅⋅ −−−−−
[∆⋅−≆≺∾↼⊦∂∫⊣⊸≉≻⊣−↕⋅⇀⊦∫⋅−⊦≖≖ ⊐⋅− *
↽−⇌−↿≴↸ ⋍≺∅↕⊹≀↗∫⊣⊸∡≻−≺↕∙⊣⇀∫⋅⊹≖≖≻
2A3
12(aæ-l—b.7—l-Cz)'-[1 ⊋≼↙⇂∙↿∙∙⊹⊘↾⊣⊸∅⊢∃≼∞≖⊹∫≖⊹≂≖≻∃ ∣⋮∙∙∁≴⊣↰↾⊣∎≖∶∣∙
8A5 '
'
-I-..;
unde, ob primam (a')-
. 1
1
Q ∙∙∙ Z
∙∣∙− A3]
(aæ-l-bJ-l-czkim—
1
- 3 &quot;
555] (x'-l'f' ∙⊦≖≖≱ dnl-l— üïf(aæ'l'lïï'*l'cz)'dm-n-(0 ').
Sit coordinatarnm origo in centro gravitatis massae
&quot;'
trahentis; erit ( 20. b)
. ↿ ∆↿−⋮∫≺∘∞⊣−≀≀↗−⊢∞≻↙≀⋯∶ 33— ta xdm-i-
bjïydm ⊣− rfzdm ]: 0

:

ideoque vertetur (o&quot;') in157
M 1
Q
Δ 243f\ x3 + y* +32) dmt
3
245 (ax + by + cz)-dm-, .. ( 0 &quot; &quot;).
ca
73. Corpus KGDH Sit sphaericum , ejusque centrum
in puncto extremo B radii CB (Fig. 43) inveniatur ; ipsi corpori occurrat QA in T. et Q '; ducto perpendiculo BE
supra CA , triangula rectangula AQC, BEC propter latus
AC=CB , et angulum QAC=BCA , erunt aequalia , adeo
que QC=BE ; chordae nimirum SD,CT aequidistabunt a
centro B; erunt itaque inter se aequales , ac proinde
OʻT ( Fig. 43 ) , aequabit QT (Fig. 42): quod cum ubique contingat, erit area KGDH (Fig. 43) sic .aequalis
areae XYC (Fig. 42) , ut solida genita ab his areis cir
suos axes revolutis aequalia sint inter se. Vim proinde , qua punctum C tendit in sphaeram KGTH ( Fig. 43 ) exprimet ipsa sphaera divisa per CM (=CB) seu per
quadratum distantiae puncti C ab ' ipsius sphaerae centro; siquidem aliae duae componentes (71) evanescunt:
Sed si sphaera ita condensaretur , ut coiret in centrum ,
eodem prorsus modo exprimeretur ejus attractiva vis; ergo punctum extra sphaeram situm eadem omnino ratio
ne in ipsam tendit , ac si omnia sphaerae puncta in cen
tro compenetrarentur.
Haec vera sunt , licet corpus non sit omnino ho
mogeneum , modo tamen sint ubique bomogeneae ejus par
tes a centro aequidistantes ; quod notandum etiam in se
quenti assertione.


73. Corpus KGDH Sit Sphaericum
. eiusque centrum
in puncto extremo B radii CB (F ig, 43) inveniatur; ipsi
corpori occur1at QA in T
et Q'; ducto perpendiculo BE
snpra CA , triangula rectangula AQC, BEC propter latus
AC;-:: CB, et angulum QAC:BCA ,
erunt aequalia, adeo-
que QC—BE- , chordae nimirum SD Q' T aequidistabunt a
«centro B; erunt 'itaque inter se aequales , ac proinde
Q'T (Fig. 43) aequabit QT (Fig. 42): quod cum ubi-
qne contingat, erit area KGDH (Fig. 43 ) sic .aequalis
areae XTC (Fig. 42) , ut solida genita ab his areis cir-
ca suos axes revolutis aequaha sint inter se. Vim proin-
de ,qua punctum Ctendit in sphaeram KGTH (F 1g 43)
exprimet ipsa sphaera divisa per ∁∾∙≖ (:CB') seu per
quadratum distantiae puncti. C ab ipsius sphaerae een-
tro ; siquidem aliae duaeïcomponentes (71) evanescunt,:
Sed si sphaera ita, condensantur,, ut coiret in centrum,
eodem prorsus modo exprimeretur eius attractiva vis; er-
go punctum extra sphaeram situm eadem omnino ratio-
ne in ipsam tendit
, ac si omnia sphaerae puncta in cen-
tro compenetrarentur. ⋅ ⋅
Haec vera sunt
,
lieet corpus non sit omnino ho-
mogeneam, modo tamen sint ubique homogeneae eius par-
tes a centro aequidistantes; quod notandum etiam in se-
quenti assertione.

74. Si punctum materiae locetur intra crustam sphaericam, sive intra orbem sphaericum intus cavum terminatum binis superficiebus sphaericis concentricis, id punctum, destructis viribus consistet in aequilibrio. Sint ( Fig. 44) NEQ, MFP superficies illae concentricae , punctum vero materiae sit O. Ducta per 0 quavis chorda MNEF, et ex centro K demisso perpendiculo KC supra ipsam chordam, erunt CM=CF, CN=CE;
igitur MN = EF , ac proinde ( 72 ) etc:

75. Ex dictis ( 73. 74 ) sequitur: 

1º . punctum in superficie duarum sphaerarum positum gravitare in ipsas sphaeras in ratione radiorum directa: nam sphaerae sunt ut radiorum cubi, quibus per eorumdem quadrata divisis, prodeunt radii simplices: 

2° . gravitatem puncti intra globum homogeneum pergentis a superficie ad centrum decrescere in ratione directa distantiae a centro ipso. 

76. Haec notentur. 1º. materiale punctum valde distans a corpore attrahente, utcumque se habeat forma corporis, ea proxime ratione tendit in ipsum corpus, qua tenderet si corporis partes in centro gravitatis compenetrarentur; patet tum ex dictis (12: 20) , tum ex eo quod in casu vires attrahentes punctorum constituentium corpus considerari possint tamquani proxime parallelae et proportionales ipsorum punctorum massis.

* Patet etiam ex ( 0 &quot; . 01 .: 72 ) ; nam si A est ila na

gna , ut, retento primo termino in ( o &quot; ), possint caeteri
praetermitti absque sensibili errore , sicque habeatur
M
Q
exsistent
M C
H
M
A2 ., HH ,
M 3
42 : A
H
ac proinde
M
H
ViH + H ,* + H , +
158
Sint (Fig. 44) NEQ, MF P supetticies illae concen.
tricae, punctum vero materiae sit 0. Ducta per 0 qua-
vis chorda MNEF , et ex centro K demisso perpendicu-
lo KC supra ipsam chordam, erunt CMr—CF, CNzCE;
igitur MN— EF
, ac proinde (72) etc:

75. Ex dictis (73. 74) sequitur: 10. punctum in su-
perficie duarum sphaerarum positum gravitare in ipsas sphae-
'ras in ratione radiorum directa: nam sphaerae sunt ut ra-
diorum cubi, quibus per eorumdem quadrata divisis, pro-
deunt radii simplices: 20. gravitatem puncti intra globum
homogeneum pergentis a supe1ticie ad centrum decrescere in ratione directa distantiae a centro ipso.

76. Haec notentur. 10. materiale punctum valde di-
stans a corpore attrahente, utcumque se habeat forma cor-
'poris, ea proxime ratione tendit in ipsum corpus , qua
tenderet si corporis partes in centro gravitatis comPe-
netrarentur', patet tum ex dictis (12: 20), tum exeo quod
ih casu vires ,attrahentes punctorum constituentium cor-
pus considerari possint tamquam proxime parallelae et pro-
portionales ipsorum punctoruin massis.
' ea Patet etiam ex (a&quot; . o&quot; 72); nam si∆∙ est tta ma-
gna , ut, retento primo terminogin (o'f ), possint. caeteri praetermitti 'absque &quot;sensibili 'et-rore , sicque habeatur
.
'
. - «. ∙
⋅
r ↾
1' I
.
. M. 11&quot;
≺≀⇌⋅⊼−↿⋅ ⋅
exsistent '1-
- --M 0
'M 6
&quot;' M
∣⋅
∏⋍−⋅−− −−∶∙−−⋅∙−− ...—...; .' AaA'H' ArA'H': ∣⋅∙↘∆∙
ac proinde
−−−∙∙−−−−−−∙∙∙∙−−∙ M
H:: l/Hil'i'nya'i&quot; He's-A&quot;?159
2º. Non pluribus opus est , ut stabiliatur illud: u
bi dimensiones corporum quorumcumque se matuo attra
hentium in ratione directa massarum, et reciproca duplicata
distantiarum sint admodum exiguae prae distantiis, quibus
ipsa corpora disjunguntur, eorum alterum tendet in alterum
perinde ac si essent 'ambo in suis gravitatis centris compe
netrata . Dicantur enim M , M' massae duorum ejusmodi
corporum , m, massa cujuslibet puncti spectantis ad M , et
A distantia inter m, ac centrum gravitatis massae M ; ex
Mm ,
primet vim attractionis motricem ( 28) , qua m, len.
dit in M, simulque ( 7 ) vim attractionis motricem, qua M tendit
in ma; ideoque merit vis attractionis acceleratrix, qua
M tendit in mo . Atqui hoc pacto M tenderet in mo, si to
la massa M compenetraretur in suo gravitatis centro ; er
go M revera tendit in mi, id est in singula puncta mas
sae M' , perinde ac si tota M foret in suo gravitatis cen
tro compenetrata: cumque ob paritatem rationis idem con
tingat massae M' quoad M , jam patet veritas assertionis.

3º. Quoad sphaerica corpora, quorum partes aequidistantes a suis centris sans homogeneae, obtinet assertio, utcumque caeteroqui se habeat intercedens distantia.

=== De gravitatione universali ===
[[77|77]]. Quae de coelestium corporum motibus, ex astronomicis observationibus hic subjicimus, ad ipsorum gravitatis centra respiciunt.

1º. Areae, quas circa solem describit radius vector uniuscujusque planetae sunt respondentibus temporibus proportionales: idipsum obtinet quoad areas descriptas a radio vectore uniuscujusque satellitis seu planetae secundarii circa suum planetam primarium.

2º. Convertuntur planetae circa solem in orbitis ellipticis ita, ut singularam ellipsium alterum focum occupet sol: convertuntur planetae secundarii circa suos planetas primarios in orbitis ellipticis ita, ut istarum focum occupet respectivus planeta primarius.

3º. Quadrala temporum periodicorum sunt in diversis planetis ut cubi semiaxium transversorum: idipsum obtinet quoad diversos satellites circa respondentem planetam primarium.

[[78|78]]. Hinc 

1º. planetae urgentur vi acceleratrice &lt;u&gt;tendente in solem&lt;/u&gt;; itidem satellites urgentur vi acceleratrice &lt;u&gt;ad respectivos planetas primarios tendente&lt;/u&gt;: plauetae, nimirum gravitant in solem, satellites vero in planetas, quibus adhaerent.

2º. Unusquisque planetarum (56) urgetur in solem vi gravitatis, quae sequitur rationem reciprocam duplicatam distantiarum ab ipso sole: idem dicendum de unoquoque satellite in ordine ad suum planetam primarium .

3º. Collatis inter se viribus acceleratricibus, quibus diversi planetae urgentur in solem, eae erunt (56) in sola ratione reciproca duplicata distantiarum a sole ipso; praecisa igitur projectionis vi, si diversi planetae in aequalibus a sole distantiis constituerentur, aequali tempore in eum descenderent. Idem obtinet in satellitibus quoad respectivos planetas primarios. 

[[79|79]]. Planetae secundarii una cum primariis, quibus adhaerent, in solem urgentur eadem gravitatis lege. Nam corpus omne, quod circa corpas alterum utcumque motum describit areas temporibus proportionales, urgelur duplici vi, altera tendente ad corpus illud utcumque motum, altera utriusque communi (5:46): cum igitur planetae primarii gravitent in solem, cumque planetae secundarii circa suos primarios describant areas temporibus proportionales; propterea etc.

[[80|80]]. Gravitant in se mutuo corpora omnia, ex quibus coalescit planeticum systema. Planetae siquidem omnes cum primarii tum secundarii vi gravitatis urgentur in solem; ergo sol in planetas omnes vi ejusdem gravitatis (7) urgetur: atque hoc argumento ostendes terram gravitare in lunam (id confirmant phoenomena marini aestus) caeterosque planetas primarios in suos satellites. Quod autem planeta quilibet in alium quemvis gravitet, satis e sola comprobaretur analogia, etiamsi nulli essent effectus, ex quibus haec gravitatio immediate detegi posset. Sed ejusmodi effectus non desunt: perturbationes videlicet, quae in recensitis motibus (77) observantur, quaeque per mutuam coelestium corporum gravitatem optime determinantur (62*60). Sic cum lunae motum ad regularis calculi normam ex observationibus exigere se posse Astronomi desperarent, tandem postquam ejusdem perturbationes ex mutua corporum coelestium gravitatione investigare coeperunt, tabulas lunariam motuum potuerunt conficere, quarum tantus est cum coelo consensus, quantum sperare ex observationibus nemo potest.

[[81|81]]. Praecisis perturbationum causis , urgebitar luna in tellurem vi acceleratrice (56):&lt;math display=&quot;block&quot;&gt;
\varphi=\frac{4\pi^2 a'^3}{T^2}\frac{1}{\Delta^2};&lt;/math&gt;


denotat &lt;math&gt;T&lt;/math&gt; tempus periodicum = dieb. 27 , 322 = minut. secund. 60&lt;sup&gt;2&lt;/sup&gt;. 24. 27 , 322; &lt;math&gt;a'&lt;/math&gt; semiaxem transversum orbitae lunaris, &lt;math&gt;\Delta&lt;/math&gt; radium vectorem ipsius orbitae. Iamvero mediocris radius terrestris = 16931100&lt;ref&gt;Error in originale&lt;/ref&gt; ped., mediocris parallaxis lunaris 57' + 11&quot;, unde &lt;math&gt;a' =\frac{16931100}{\sin(57' + 11'')}&lt;/math&gt;facto  igitur &lt;math&gt;\Delta = 19631100&lt;/math&gt;, gravitatis vis qua luna urgetur in terram evadet in ipsius terrae superficie
&lt;math display=&quot;block&quot;&gt;blah blah blah&lt;/math&gt;qui valor cum sit proxime 30,2 ped., inferimus gravitatem qua luna urgetur in terram nihil esse aliud nisi gravitatem ipsam terrestrem imminutam in ratione reciproca duplicata lunaris distantiae a terrae centro.

[[82|82]]. Vis gravitatis, qua lapis v . gr. urgetur in terram, est (80) ejusdem speciei cum illa gravitatis vi, qua corpora mundani systematis in se mutuo tendunt; ergo idem in utraque erit agendi modus. Atqui vis qua totus lapis urgetur in terram resultat ex viribus, quibus singulae lapidis particulae in eamdem nituntur; igitur et vires, quibus corpora mundani systematis gravitant in se mutuo, resultant ex viribus, per quas singulae ipsorum, particulae se mutuo petunt. His positis, stabilietur illud: gravitas ita materiam afficit, ut singulae ejus particulae in alias omnes et singulas gravitent in ratione directa massarum, ad quas tenditur, et reciproca duplicata distantiarum alterius ab altera. Recole quae diximus (76.2º.3º); etenim coelestia corpora et habent dimensiones admodum exiguas prae mutuis distantiis, et induunt formam prope sphaericam.

[[83|83]]. Bonum erit nonulla hic annotare. 

1º. designantibus &lt;math&gt;M&lt;/math&gt; et &lt;math&gt;m&lt;/math&gt; solarem et planeticam massam, ex dictis (56.k, 62.c) eruitur&lt;math display=&quot;block&quot;&gt;
M + m =\frac{4 \pi^2 a^3}{T^2}
&lt;/math&gt;ratio igitur inter cubum semiaxis transversi et quadratum temporis periodici, utpote pendens a massa planetica, nequit esse accurate constans quoad diversas planetarum massas. Atqui tamen ex astronomicis observationibus infertur rationem illam, sin minus accurate, certe esse quamproxime constantem: concludendum itaque planetarum massas admodum exiguas esse, ubi comparentur cum massa solis.
2.º Eodem modo ostenditur, si lunam excipis, satellitum massas fore et ipsas valde parvas prae massis planetarum, quibus adhaerent.

3.° Quae quantitates sunt designatae per &lt;math&gt;m, 
a , T&lt;/math&gt; quoad planetam , eae designentur per &lt;math&gt;m' , a' , T'&lt;/math&gt; quoad satellitem; erit&lt;math display=&quot;block&quot;&gt;
m + m'=\frac{4 \pi^2 a'^3}{T'^2}&lt;/math&gt;Hinc (1°)&lt;math display=&quot;block&quot;&gt;
\frac{m + m'}{M + m}=\frac{T^2}{T'^2}\frac{a'^3}{a^3}
&lt;/math&gt;praetermissa ( 19. 20. ) &lt;math&gt;m'&lt;/math&gt; in numeratore primi membri, itemque m in denominatore , et facta M = 1 , prodibit.
T2
TO
a's
i
quae formula suppeditat rationem inter solarem massam
habitam pro unitate , et massas planetarum ( tellurem ex
cipe) , qui satellitibus stipantür.


4.° Quod spectat ad tellurem consideratam in
star sphaerae habentis radium R , et massam m , sit &amp; gravi .
tas prope ejus superficiem , erit (73) 8 =R. , ideoque (10. )
M +m
4 712 a3
&amp; R2T
et praetermissa ( 19. ) m in numeratore primi membri , factaque solari massa M = 1 , emerget
163
2! Eodem modo ostenditur ,
si lunam excipis,
satellitum massas fore et ipsas valde parvas prae massis planetarum, quibus adhaerent .
-l ⋅ 

:

41:2003 -
∙∙ ↶↿ m-4—m' £S.-&quot;IW.,
Hï'n'c (10)
⇀
..
.m -- in'. T., a'-3-- - M,-—-.nsl ≔−⋅⋅∙−∙− ∙ ∙∙∙ : T'a ::
&quot;
praetermissa (.10 20. ) tu' in numeratore primi membri ,
itemque ut in denominatore ,
et facta M— −∙− ↿, prodibit.
⋅ 1 ↴− ⋅↧⇁≖ -a ∙
−
⋯∙↽↽⊽ .
..;3, .'
∙⊾⊺⋅⋅
quae? .fottmule- suppeditat rationem -
intcr. solarem^ massam
habitam pro unitate ,
et massas planetarum (tellurem ex-
cipe) , qui satellitibus .stipantur.
'i-o Quæ-Spectat ad tellurem' consideratam in-
star sphaerae habentis radium R , et massam 111, sit 3gravi-
tas prope eius superficiem ,
erit (73) gr ≖⋅⇁⋅∙−⋮↾−⋮↾− , ideoque (10.)
IUI—tm,— 4 11:303.
,
gRQTi'
et praetermissa (10.) »: in numeratore primi membri, f.-
ctaque solari massa M ∙−⇁−−−∙ ↿, emerget ⋅164
&amp; R2T2
4 Ti ? a3
PE

5° Media telluris densitas ( = M) determinari potest ex penduli aberratione. Sit CB (Fig. 45) pendulum; a longitudo rectae CB, quae nec distendi possit nec inflecti; S centrum massae sphaericae ( = m' ) ad se trahentis punctum ponderosum B , r radius , M densitas; b recta CS; CD posilio penduli digressi a recta verticali CB;
&amp; angulus BCD; h angulus BCS; k recta SD: centro insuper C et radio CB intelligatur describi circularis arcus
BD , et per D duci tangens nn' . In D sollicitatur materiale
m'
punctum viribus acceleratricibus g et altera juxta ver ka
ticalem DD' , altera juxta rectam DS ; anguli
SU
1
D'Dn = CnD = 90 ° — E , SDn ' = # (CDS - 90° ) ,
to
et consequenter
b sin (h — 5)
cosD'Dn sins , cosSDn ' = sinCDS =
k
Vires igitur motrices respondentes praefatis viribus ac
celeratricibus sese librant in D quotiescumque fuerit
que
bas
bit
ha m'
23 gsine b sin ( h - €) .
sed
Pone longitudinem penduli ita exiguam prae distantia CS ,
ut absque sensibili errore sumi possit k = b ; traduce
tur aequilibrii conditio ad
gb2 sin é = m ' sin (h - ) ;
m
et substitutis ( 4º . ) valoribus 8 T RH, m
R? 3
4
90
paris 75 1'3 je , prodibit
164
g R*T3
m— 4 723 (13
50 Media telluris densitas (:: p.) determinari po-
test ex penduli aberratione . Sit CB (Fig. 45) pendulum longitudo rectae CB, quae nec disteudi possit nec inflecti; S centrum massae sphaericae (: m') ad se trahentis
punctum ponderosum B ,
r' radius
, pf densitas ; 6 recta
CS; CD positio penduli digressi a recta verticali CB;
a angulus BCD :
]: angulus BCS: k recta SD: centro in-
super C et radio CB intelligatur describi circularis arcus
BD , et per D duci tangens nn' . In D sollicitatur materiale
punctum viribus acceleratricibus 3 et ?; , altera juxta ver-
ticalem DD', altera juxta rectam DS : anguli
ix)-D'.. ∙∁∥↧⊃ :. 90o −−∙ e
, sna':∶↿≐ (CDS — 900)
.
et ⋅ consequenter
bsinUt—s) ——k .
Vires igitur motrices respondentes praefatis viribus ac-
celeratricibus sese librant in D quotiescumque fuerit
! r ' ,
!'
cosD'Dn :: sins, cosSDn' ∶∙ sinCDS −−−−⋅−
gsins:
£;- b sin( h — £) .
liane. longitudinem penduli ita exiguam prae distantia CS ,
ut absque sensibili errore sumi possit 1::6; traduce-
tur aequilibt'ii conditio ad
gbï sin ; −−∶ m' sin-(h — a) ;
et substitutis (40.) valoribus g: €; ∶∶ £- 11 R p., m' −∙−∙−−
4
,, ,
'
∙⋅ ; &quot; . .
⋅
∙⋅⋮↿∏⋅ p. , prod1h1t
!
'
I
151
sit
lla165
1
b- Rue sin
{
= 13
M
' sio
(
h
-
E)

unde
i
p3
y sin
h rº
( sinh
– coshtang :)
1
lang
E
=
Ruba tospicosh ji Rba lang

Permanentibus
r
' et
'
, valores
b
=
r
' et
h
= 90
° manife
ste suppeditant maximam penduli aberrationem
&amp;
, ut quoad
istiusmodi aberrationem sint
Se
re tang R pe
3/3
Rtang
s
Densitas fl
, prout colligitur ex aberratione penduli
, cen
setur quater vel quinquies major quam densitas aquae. 6. ** Eadem u determinatur etiam experimentis in
stitutis in libra torsionis
. Sit
( Fig
. 9.
) HH
(
=2a
'
) posi
tio vectis horizontaliter librati

;

E punctum medium, in quo vectis appenditur filo metallico verticali HA circulus horizontalis centro
E et radio EH
=
a
')

; SS

( 26
)
recta similiter horizontalis
, transiens per
E
, ibique secta
bifariam

; sint

S et
S centra sphaerarum inter se aequa
liam et quoad volumen
, et quoad massam
(
=
m
)
, ad se
trahentium massulas sphaericas
m
' et
m
&quot; inter se pariter
aequales
, quarum centra in
H et
H
'
. Movebitur vectis
circa punctum
E immotum
, motusque iste repetendus ab
attrahente sphaerarum vi juxta circuli tangentem
, cui vi
jugiter adversatur vis lorsionis ex filo metallico verticali

et quoniam corpuscula
m
' et
m
&quot; eodem prorsus donantur
motu circa
E
, satis erit alterum dumtaxat
v
. gr
.
m
' con
siderare
. Dicatur itaque
h datus angulus HES

;

&amp; angulus
,
quem in fine temporis
i continet vectis cum initiali po
sitione EH

;

k distantia inter
S et
m
' in five ipsius
t

: sol

licitabitur
m
' juxla circuli tangentem vi attractiva
!
)- a
165
63 Bpain :::/3 (fimul—15);
unde
tan E' ∙∙∙⋅ r'3 pf sin 11 p. '
r'3 (sinit −⋅ coshtang &amp;)
∙
g −⇀ nubi-l- r'3 picas/1
.pf ⇀−− lib2 teng
&amp; .
Permanentibus r' et p!
, valores
b

: r' et 11
: 900 manife-

ste suppeditant maximam penduli aberrationem

:

, ut quoad
istiusmodi aberratiouem sint
tangi—£&quot;.! P',—.
r −⇁∙
p p. Btang
&amp; ,
Densitas p., proutcolligitur ex aberratione penduli.
, cen-
setur quater vel quinquies maior quam densitas aquae.

6?- Eadem p. determinatur etiam experimentis in-
stitutis in libra torsionis
. Sit (Fig.'9*.) HH' (:Za') posi-
tio vectis horizontaliter librati;
E punctum medium
,
in
quo vectis appenditur (ilo, metallico verticali; HA circu-
lus horizontalis centro
E et radio EH (:a'); SS' (:26)
recta similiter horizontalis
,
transiens per
E
,
ibique secta
bifariam

; sint

S et S' centra sphaerarum inter se aequa-
lium et quoad volumen
,
et quoad massam (:m)
, ad se
trahentium massulas sphaericas m' et 111&quot; inter se pariter
aequales
, quarum centra in
H et H' ., Movebitur vectis
circa punctum
E immotum
, motusque iste repetendus ab
attraheute sphaerarum vi juxta circuli tangentem
, cui vi
jugiter adversatur vis torsionis ex filo metallico verticali

:

et quoniam corpuscula tu' et m&quot; eodem prorsus douantur
motu circa
E ,,satis erit alterum dumtaxat v. gr. m' con-
siderare
. Dicatur itaque ]: datus angulus HES

;

&amp; angulus
,
quem in fine temporis

: continet vectis cum initiali po-

sitione EH;
k distantia inter
5 et m' in line ipsius
t

: sol-

licitabitur m' iutta circuli tangentem vi attractiva-166
b
hak sin (h — e), eritque kº = a's - 2a'b cos (h --- e) + 6+;
experimenta insuper praebent vim torsionis proportionalem
angulo e , et consequenter expressam per ce : quoniam igi
tur labente e describit m' arcum ás , iccirco ( 50.3º. )
áre mb
sin (h -€)
dta k3
3
1
&lt;u&gt;aequatio ad motum&lt;/u&gt; corpusculi m' . Ob angulum &amp; valde exi
gaum , sin (h-5~s)) = sin h - e cos h , k - = [ a's -2a'b ( cosh
+sinh) + ] := k +. 2a'besio h ]
R3
3a'b esinh
+
ubi denotat k , valorem k respondentem initio k.
molus , quum nempe E = 0 ; proinde
sin (h-E ) sinh
23
E COS h 3abe sin’h
+
k.
k.
sinh
k.
k .
£ k cosh 3a'be sinh sinh
+
kb.
K.
her )
[ (a's tabo) cos h
k . k5.
sinh 2a'b cosah — 3a' b sin : h]
[laat69)cosh
- 2db -a'b sinä h] :
et factis compendii causa
mo
[ la'a + b ) cosh — 2a'b – a'b sinºh] +c = g' ,
166
m 1)
.
'
.
F.
∙∣⋮∙ nuUi—s) ,entque ka: a'a— 2a'b cos (71—5)-1—63;
experimenta insuper praebent vim torsionis proportionalem
angulo :,
et consequenter expressam per et: quoniam igi-
tur labente :describit m' arcum a's, iccirco (50.?)0.)
,d'38 mb ∙
aa;-a.- Fama—Q—ct
aequatio ad motum corpusculi m'
.Ob angulum :valde exi-
guum ,
sin (It—s) :sin h—s cos 11, k'3: [a&quot; −∙∙ Za'b (cos):
3 . 3
-i-ssinh) ⊣−∂∙∎∙∣− 'a': [le, −∙∙ Za'besin H's—:i?
⊣⋅− a.:-s;&quot;: h, ubi denotat k, valorem k respondentem initio
motus
, quum nempe s.: o ; proinde
sin (I;—s) sin 11 :
cbs Il 30'68 sin'h sin ls
∙−−−⋅∙≖∙−−∙− ∙∙∙⋅ ks' &quot;&quot;&quot;&quot; k3, −∎⋮∣⋮∍∙ fl&quot;, P, P,
sit' eos]: l Bez-'besin'h sinh : .

*

5,
w
cos-1. −−⋅ adb sin-t.] :
822&quot; −⋅⊼⋮−⊏≺∘⋅⋅−⊦∂⋅↗ coslt
∙∙∙⋅ ⊋∘∣∂∙∙− a'b sin' &amp;] :.
et factis compendii causa
iii- [(.-a ⊣⇁ 61) cos h—w—
a'6 .i..- h]
∙⊦≖⋅∸ −−∶
z'-167
mô sinh
wg'
23.
aequatio ad motum corpusculi m ' vertetur in
do e
a' ó (0) -- s ) ;
de²
ex cujus integratione ( 27. 28º. )
(9)*va
' @
'ri
{ = w + Ce + Ce
Sunt autem ( 27.300. )
.
va
cos (9 )* + v = on e(2)
, - ) vi
cose ( ) -va sine ( 2)
propterea
szaf1CTC&quot;) cos( ) +(c —c )V= sine ( :
sumptisque C +C' =C.cos C,, C — C = CV -ī sincs,
= - + 6, co [4 ( 4 ) + c ]
Minima vectis declinatio , í = u - C , ab aequilibrii positio
167
mb sinh
−−∣−∣⋮−≣∶−−≂−↩∾⊰
aequatio admotum c0rpusculi m' vertetur-in
,d'l
(: -d—t;:::g (6)—S):
ex cuius integratione (27 . 280.)
⋅⋅ .

*

.
.'.t ⋅↴∶
≖−↽−−−⇀↠∾−⊦∁∊ :(?) [l:—[— 08 &quot;(ä-') V
..
Sunt autem (27 .300.)
.;.
propterea &quot;
∙⋐⋅∶∶⊙−⊦≼∁−⊢∁⋅≱ ∾≘↙≺−⋚⊑⋅≻⋚⋅⋅−⊢ ((i—C' )
(V:; sint (?);;
sumptisque ∁−⊢ ∁∙∶∁≖∞∙ c, , ∁−∁∽−↽⇌∁≖⇂∕∙⊺∘⋮∥∁≖∙
⋅⋅ ∸ g' ?
≘−∙−−∙∾−⊦∁∙∞∘∣↣⋮≀ ≼⋮⇉⋟ −⊦∁≖∃∙
Minima-vectis declinatio ,
(:o)—G. ab aequilibrii pocitio-168
ue H'H respondet valori ( ) + Cs = ( 2n – 13 ;ma
= + c = 212nt : determinatis
itaque per observationem i'et s&quot; , eruetur inde
te&quot;
et ducta 00 ita , ut sit angulus HEO = w , perget vectis
moveri instar pendali horizontalis circum EO , impendet
que tempus tz = &quot; - t ad integram conficiendam oscilla
tionem , nimirum
ty=T
VAg'
Sit nunc a longitudo penduli simplicis ( 66) , quod intra
idem tempus t absolvit oscillationes suas : cum habeamus
ty = TO Van
a
erit 8 et denotante si densitatem sphaerae m ,

a'
r radium , substitutisque valoribus ( 5.0 )
4
пRр .8 3
mbsinh
wk3
471p3 M'bsinh
3wk. 3
proveuica
Ruwk3 р
p3M'bsinha
unde ar3bsinh
a'Rwk.3 1
Densitas pe sic determinata censetur esse ad
tatem ut 5,48 : 1 .
aquae densi
168
ne H'H respondet valori t'(g——,)ä -I-C,:(2n-1)1r; ma-
xima ⋮∣↾∶∶∾⊹∁≖ valori :
.(gwik) ⊹∁≖:2mr: determinatis
itaque per observationem eet :&quot; , eruetur inde
−−≘⋮−⊢⋮⋅∣∙ −− ⇄ ∙
et ducta O'O ita
,
ut sit angulus HEOzzæ, perget. vectis
moveri instar penduli horizontalis circum EO, impendet-
que tempus :::-:i &quot;—t' ad integram conGciendam oscilla-
tionem , nimirum
a'&quot; Vf- ∙

5
Sit nunc a longitudo penduli simplicis (66), quod intra
idem tempus :, absolvit oscillationes suas: cum habeamus
(3:11 V? .
. g a '
∙
erit?-;? ; et denotante p. denutatem sphaerae m ,

: - radium

. substitutisque valoribus (59)
∙∙∙⋅ ∙≤∙ nR ,— mbsinh— 4nr39'bsinh ∙ ∙
o ∙−−− 3 p. ,g 01:03 30 1:03 ,
provenit...
Rpali-03 a p. arabsinh
r3p'bsinh—c7 '
unde ∙∥∙∽ a'Rmkoï'l '
Densitas p. sic determinata censetur esse ad aquae densi-
tatem ut 5,48:1.169

7. ** Ex mariui aestus phoenomeno deduci pol
est ratio inter massam lunarem m &quot; et terrestrem m. Sit
m' ( Fig. 35 ) quodvis terrae punctum ; lunares vires distrah
entes punctum mé juxta mm&quot; et Am exprimuntur ( 62) per
2m &quot; Dcosh m&quot;Dsinh
(0) ,
( 0' ) :. x'3
X :' 3
quod in ordine ad lunam est h , x&quot; , in ordine ad so
lem sit H , X &quot; ; prodibunt consimiles vires solares
2MDcosH MDsind
X &quot; 3
( a ) , ( a ').
X'3
In casu angulorum h et H aequalium habemus
( 0) m&quot;X &quot;3
(a)
-- ( 0 )
M.2'3 ( a' )
caeteris vero paribus , ratio inter lunares et solares vi
res est eadem ac ratio inter respondentes aquarum ela
tiones ; denotante igitur p hanc secundam rationem , erit
X3
M
р
m '
M
m'
unde m
= P
3
x &quot;' 3
X &quot; 3
Observationes praebent p = 2 , 35333 : vide mechan, coel.
vol , 5. pag . 206.
Aliquid notatur de motu punctorum materialium
utcumque inter se connexorum .

84.* Vires motrices P, P , P&quot; , ... sollicitantes
istiusmodi punctorum massas m ,
mi ,
m' 0 resol
12
169
7.&quot; Ex marini aestus phaenomeno deduci pot-
est 'ratio inter massam lunarem m&quot; et terrestrem m. Sit
m' (Fig. 35) quodvis terrae punctum .; lunares vires distrah-
entes punctum m juxta mm&quot; et Am' exprimuntur (62) per
2m&quot;Dcosh m&quot;Dsinh (0) ∙ ∙−−∙↕∙−∙⋅∃−−⋅ (O,)
xara ⋅
quod in ordine ad lunam est h , a:&quot; ,
in ordine ad so-
lem sit H , X&quot;; prodibunt consimiles vires solares
2MDcosH' ⋅ MDsinH
W (a), ∙∎∎ Xl/3 (a').
In casu angulorum I; et H aequalium habemus
(o) -m&quot;X&quot;3l-(o') ∙
(a) Mx&quot;3 X(a') ⋅
caeteris vero paribus
,
ratio inter lunares et solares vi-
res est eadem ac ratio inter respondentes aquarum ela-
tiones ; denotante igitur p hanc secundam rationem
, erit
X&quot;3 d m&quot; M a:&quot;3
∙ &quot;&quot;;- un B −−∶ ∙ ,. x' 3
' m ? m X'3 31 ≊⋅∙ p:
Observationes praebent p——:2 , 35333 : avide machen. coel.
vol. 5. pag. 206.
Aliquid notatur de motu punctorum materialium
utcumque inter se conus-xarum.

843: Vires motrices P, P&quot;, P&quot;, ... sollicitantes
istiusmodi punctorum massas m , m'
, m&quot; , ...
resol-
12170
vantur singulae in ternas coordinatis axibus OX , OY ,
OZ ( Fig. 8 ) parallelas ; designentur per X , Y , Z, X',
Y , Z ' , X &quot; , . . componentes inde ortae ; sintque x, y,
3 , x ', y ', z' , x &quot; • punctorum coordinatae responden
tes temporit , ut ( 50, 1.º ) per
x == f (1 ), y = f(t), 2 = F (t), ' = fi (t), y = fz(t ), == F ,(t)
x ' = fale) ,y&quot; =f(e), z&quot; = F.(6),7: &quot; = f5e) , ... ) co)
exhibeantur aequationes ad actuales molus ; ad eos nem
pe motus , quos reapse concipiunt massae m , m' , m &quot; ,
ob actiones virium P , P' , P &quot;, ... Quoniam materialia
puncta , etsi mutuis nexibus liberala , viribusque ( 50. 4.0 )
dºx dz m
dạy
de² m
d²x
dla m . dia dla dea
d2z '
dca dc ?
sollicitala , adhuc tamen conciperent motus ( 0 ) ; ideo ,
attentis nexibus , consistent in aequilibrio vires
dez X - m
d2x
de2 Ymdạy
di?
2m X'
der'
di2 &gt;
7
dt2
Y - m
d²ý
dt2
daz'
Z ' - m '?
dt²
X &quot; -m.dºx
&quot;:
.
dt2
Conditiones ( a &quot; 13. 8. ) includuntur in conditionibus
aequilibrii quoad liberum punctorum utcumque connexo
um systema : liquet enim varians systema, semel libra
tum , adhuc permansurum in aequilibrio , etsi ejus pun
cla rigidis lineis immutabiliter connectuntur. Propterea
170
vantur singulae in ternas coordinatis axibus OX, Oï
,
OZ ( Fig. 8 ) parallelas; designentur per X, ? ,- Z, X',
1&quot;, Z' , X&quot; , . . . componentes inde ortae; sintque x,],

: , x', y', z', æ&quot; , ... punctorum coordinatae responden-

tes tempori t
, ut (50. 19) per
x:f(t), 7:112) ,z:F(t),.r ':f,(t),y ':f (t), z':F,(t),
(0)
x&quot;:-f.(t)
.
y&quot;:f.(t)
.
(&quot;zl-&quot;.m. m&quot;':--f3(t)
.
∙ ∙ ∙
exbibeantur aequationes ad actuales motus; ad eos nem-
pe motus
, quos reapse concipiunt massae m , m', m&quot;
,
ob actiones virium P, P', P&quot;, . . . Quoniam materialia
puncta , etsi mutuis nexibus liberata, viribusque (50. 49)
da.. mdzy de.
⋯∽∣↙≀≄∙↿∶ ∙↙∄≖∫⋅
&quot;B'—'— m-——- m— d,.
' md:2 '. d?
'md? ' dta '
,dzz, ad:-I?&quot; m d£2 '.. m dt:- '
∙ ∙ ∙
sollicitata, adhuc tamen conciperent motus (a); ideo ,
attentis nexibus, consistent in aequilibrio vires
(P:: (P] daz ,dzæ'
X
⇁∎−−∙ —p ⋅⋅⇁ ∙∙∙∙ ∙−−− '&quot;sz '
?
&quot;'d'T'z' Z &quot;&quot;da: X &quot;'de '
≀∠⋮⊺ &quot;rad :
&quot;
rnnndaæ ï'-——-—m '
...
dtï' ,..—z ⋯∠↙⊤ 'X— dtz
'
Conditiones (a&quot;f' 13. 8.0) includuntur in conditionibus
aequilibrii quoad liberum punctorum utcumque connexa-
rum systema :liquet enim varians systema, semel libra-
tnm, adhuc permansurum in aequilibrio, etsi eius pun-
cta rigidis lineis immutabiliter connectuntur. Propterea171
(xam )=o, 3(1— )= 0, $ (2- -o,
=[+ (rad ) – (xrm ) ] = o,

* [&gt; (2 mm ) -- ( - ) ]

&lt;math&gt; [ - (-) - = ( x ) ] &lt;/math&gt;
0
seu
daz ΣΖ - Ση
de² EX = sme , Y =sme &gt; ( 0 )
$( wYyX) =Em ( -e
)
3(y2=-1)= sm (voeding - :)
Eml
1
Z
day
de2 (0')
(2X == Z ) = 2m Z
dax
dta -
daz
dea :
formulae (o ') spectant ad translativum punctorum motum,
prima juxta OX , secunda juxta OY , tertia juxta OZ ;
formulae ( o“) ad rotatilem punctorum motum , prima cir
ca OZ , secunda circa OX , tertia circa OY ; eaedem ve
ro (o &quot; ) simul , ad punctorum motum circa fixam coor
dinatarum originem .
Haec facile nunc stabiliuntur. 1 . ** Habemus (20. 6.)
seu
æ ,dþ- ∙−−
∠∄≖⋍ ,
day d3æ ?
−∙∙ − ..(æy—yX) Em (
«: Tt&quot; ]—dt3 )
.
- ∑ dan: daz (zX—æZ): Zm( :217; — æ
—) :
formulae (c')/spectant ad translativum punctOrum motum,
prima juxta OX , secnnda juxta Oï, tertia juxta OZ;
formulae (a&quot;) ad rotatilem punctorum motum, prima' cir-
ca OZ , secunda eirca OX , tertia circa Oï ; eaedem ve-
ro (o&quot;) simul , ad pnnctorum motum circa fixam coor-
dinatarum originem.
Haec facile nunc stabiliuntur. ↿∙∘⋇ Habemus (20. b.)172
dar
Em dea dex,
dla
day Σm.
dt2
Em
daz,
dc2
da ,
Em dla
Em &gt;
Em: dt?
Hinc &gt;, ob (o' ) ,
der ΣΧ daz,
de
ΣΥ
Em ' dt2
ΣΖ
Σm (o' ' ' ) : Am dla
molo videlicet systemate punctorum m , m' , m &quot; ,
perinde ( 50. 4. ) movebitur gravitatis centrum ac si , co
euntibus punctis in ipsum centrum , applicarentur centro
eaedem vires P, P , P &quot; , ... cum iisdem directioni
bus , quibus puncta illa sollicitantur .
2 . '* Fac ut vires nihil sint aliud nisi punclo
rum actiones mutuae : denotante A actionem puncti v.
gr. m in aliud quodvis v . gr. m' , et A' actionem puncti m'
in m , erit ( 7 ) A=A' ; et expressa per D distantia inter
utrumque punctum , resolvetur A' in ternas coordinatis
axibus parallelas
x' #A D
TA
EA ;
ilem A in ternas iisdem axibus parallelas ( o'r )
ŁA to ,
EA D po', -A D
sumpto superiori signo si A , A' sunt vires attrahentes,
inferiori si repellentes . Quare EX =0, EY=0 , &amp;Z=0, et
consequenter
dér,
=0,
adi ?
day1
di?
dz,
-0,
=O ;
di?
in ea scilicet qua sumus hypothesi nullis viribus acce
↙≀⊴⋅↕⋮ d'), inuia—z- '
d'æ, dta (if/y. md:2 dïz, dta
dtz— Em 'dt: Em '
du ïm
Hinc , 06 (o') ,
d'æ, EX (Ph- Zï diru— ZZ
dF—Zm' dt&quot; &quot;Zm dt2 −∑⋯ (0 ):
moto videlicet systemate punctorum m , m' , m , . . ,
perinde (50. .f.&quot;) movebitur gravitatis centrum ac si, eo-
euntibus pnnctis in ipsum centrum
, applicarentur centro
eaedem vires P, P', P&quot; , . . . cum iisdem directioni-
bus ∙ quibus puncta illa sollicitantur.
294: Fac ut vires nihil sint aliud nisi puncto-
rum actiones mutuae :denotante A- actionem puncti v.
gr.m in aliud quodvis v. gr. m', et A' actionem puncti m'
in m, erit (7) A:A'; et expressa per D distantia inter
utrumque punctum, resolvetur A' in ternas coerdinatis
axibus parallelas
3.&quot;—æ ∙−⇠ 7—7 ...-,: z—z .
drA U , A D A D
itcm A in ternas iisdem axibus parallelas (o&quot;)
æ—x' J—y' z—z' ∙
sumpto superiori signo si A, A' sunt vires attrahentes,
inferiori si repellentes. Quare XX :0, ∑∟ -o, ZZ:o, et
consequenter
in ea scilicet qua sumus hypothesi nullis viribus acce-173
leratricibus agetur gravitatis centrum , nulloque ob mutuas
panctorum actiones afficietur motu. Huc spectat princi
pium de conservatione centri gravitatis.
3.°# Super planis XOY, YOZ, XOZ fiant proje
ctiones a, b, c, a' , b' , c' , a &quot; , 6 &quot;, c &quot; , a ' , , . . arearum
descriptarum a radiis vectoribus punctorum m, m' , m&quot;,
computatis radiis ab origine coordinatarum : erunt ( 50. 8. )
xdy — ydx Σmda = Σm ydz - zdy και Σmdb = Σm 2 2
zdxxdz Σmdc = Ση 2 unde
daa 2Em -Σm α2
dta dt
22m d2b
dta
daz
у

;

= Em (:
dla e )

: )

dec dex dez 2Σm -Σm dt2 sm
(
20
de?
et consequenter ( o &quot; )
d'a 22m dt² 8(xY4yX), 28mmdla = Eby2 — zY),
( 0 )
dac 2Em =E( zX-xZ) .
dta
4.0 # Si vires consistunt in mutuis punctorum actio
nibus , erunt ( 2.º o &quot; )
173
leratricibns agetur gravitatis centrum
, nulloque ob mutuas
punctorum actiones afficietur motn. Huc spectat princi-
pium de conservatione centri gravitatis.
3.&quot;; Super planis XOï, ïOZ, XOZ fiant proie-
ctiones a, b, c, a', b'. c', a&quot;, b&quot;, c' ', a'&quot;, ∙ ∙ ∙ arearum
descriptarum a radiis vectoribus punctorum m, m', m&quot; , .. .
computatis radiis ab origine coordinatarum : erunt (50. 8.&quot;)
∑⋅↾⋅⊿↙↓⋅−−≔−∑⋯⊔−−−−∫−≌∙∑⋯↲≀⊨∑⋯⇅−−−≖−≗↶∙ d d d—d
2 ∙ 2
Emma.—zn. fix—?? ,
unde
22m ⋛∙∶−≧∶−−⋅∑⋯≺⊰≵ :::-£v:— 7
id?-:?) ∙ ⋮⋯∶⊜≀≀∶≖∂−↽−−≖⋅⋯⋅↗≺ :: −− ::z ⇋
d'c— d:.r ædaz
et consequenter (o&quot;)
ZZm −⋛⊴⋮↥⋮−⇌∑≺∞⊺−∜∑⋟ ,
ZZmäï—b- :Zþ'Z—zï),
d (a')
220: ⋅⊋≖−∶∶ :2(zX—-æZ).
494: Si vires consistunt in mutuis punctorum actio-
nibus, erunt (2.o o&quot;)174
8 (xY - 7X ) = 0 , (yz - zY) = 0 ; $ (zX -- XZ) = 0;
ideoque
dra dc Emdl2
d2b Σm dc2 Emadt²
}
et computatis areis ab initio temporis t ,
Ema = Ct , Emb = Ct, Emc = C &quot; ! (0 ) :
huc special principium de conservatione arearum. Formu
lae ( o &quot; ) adhuc obstinent , etsi in systemate invenitur pun
ctum fixum , modo tamen in pancto illo collocetur origo
coordinatarum : siquidem vigent in casua equationes ( o&quot; ' ' ) ,
unde profluunt ( o &quot; ).
5.0 * Si arcus s refertur ad tres axes orthogona
les , ejus incrementum infinitesimum ds poterit spectari
tanquam diagonalis parallelepipedi rectanguli sub later
culis dx , dy , dz : hinc.
dsa = dx2 + dyz-tdz?, et consequenter ( 50, 2.0 )
v2= dx2+ dyatdz
dla
Erit itaque
Emvdv = Em
der
de²
d'I
ayt
ar
- .
ac proinde (0 )
Emvdv = E (Xdx + ydy + Zdz) ( o&quot; &quot; ' ) .
Fac ut E (Xdx + Ydy + Zdz) exsistat differentiale exactum ,
!
174 .
£(xï—77X):o , ZUZ—zï): :X(zX—xZ):o;
ideoque
d'a 'dzb (130
dt2
?
et computatis areis ab initio temporis :,
2ma:Ct , 2mb:C't, ch:C&quot;t (o&quot;):
huc spectat principium de conservatione arearum. Formu-
lae (ov') adbuc obstinent
, etsi in systemate invenitur pun-
ctum fixum, modo tamen in puncto illo collocetur origo
coordinatarum: siquidem vigent in casua equationes (o&quot;'),
unde profluunt (o&quot;).
594: Si arcus :refertur ad tres axes orthogona-
les, eius incrementum infiuitesimam ds poterit spectari
tanquam diagonalis parallelepipedi rectanguli sub later-
culis dx
, dy , ds :hinc.
ds':dæï-l-df3-l-dz3 ,
et consequenter (50, 2?)
daga—.dyzudzz
dt2
⋅
02.—
Erit itaque
da a :
Zmpdu:2m(d£fdx : (iyd): ↿ d
zdz) .
ac proinde (o') ∙
vadv :!(de ⊣− ⊺⊄↴⋅⊺∫ ⊣−∅∠∄≂≻⋅⋅
(o&quot;').
Fac. ut XXdæ—fïdJ—l—Zdz) exsistat differentiale exactum.175
prodeat nimirum ex differentiatione cujusdam functionis
F (x , y , z, x ', y , z, x &quot; , ... ) ; habebis
Em (u2 — V.2) = 2F (x ,y ,z,x',...) —2 F (xo,9o , zo , x '. , ... ) ;
quantitates v. , xo, Yo, Zo, x'o, ... respondent initio mo
tus. Consequitur, quod, redeuntibus iisdem coordinatis, ea
dem quoque redibit summa virium vivarum : huc spectat
principium de virium vivarum conservatione.

6. °* Denotent &lt;math&gt;h, i, k , h , i , k ' , h '' ...&lt;/math&gt; coordinatas punctorum &lt;math&gt;m, m', m''&lt;/math&gt; in ordine ad novos axes, qui et paralleli sint axibus &lt;math&gt;OX , OY , OZ ,&lt;/math&gt; et originem habeant in communi gravitatis centro; erunt
x = xrth , y = yiti , z = zetk , x' =xith' ,
y = yiti, z= z+k ', w &quot; = xrth &quot; , ... ;
quibus valoribus substitatis in ( o &quot; ) , attentisque aequatio
nibus ( 20)
dah deh , dai dai Σm Σm=0, Σ . Σm=0
dcz dta des dt2 =
dek Σm-
dt
dakı Em=0
dc2
1
nec non aequationibus ( o &quot;&quot; ), prodibunt
dai dah
2 ( XiY) = Em ( h TI dea dt2
E ( iz - kY) = Em
(ala dih),
(
- ) com
(akone )
( o &quot; )
dah
ElkX_hZ) = Em ( k dt2
175
prodeat nimirum ex differentiatione cujusdam functionis
F(x,y, :, x', y', z', æ&quot;
, ..
.) ; habebis
M(æ—voz):2F(æ,7,z,æ',..-) -2 P(æo,yo , zo , x', , ..
.);
quantitates v,
, æ., y,,zo, æ'o,
... respondent initio mo-
tns. Consequitur, quod, redeuntibus iisdem coordinatis, ea-
dem quoque redibit summa virium vivarum: liuc spectat
principium de virium vivarum conservatione.-
604: Denotent h, i, k, h', i', A', I:&quot; .. . coordi-
natas punctorum m, m', m&quot; , . ..
in ordine ad novos
axes , qui et paralleli sint axibus OX, Oï
, OZ
,
et ori-
ginem habeant in communi gravitatis centro; erunt
r—æl-i-h ,.szl-ï-i !≖∶∅∎⊹∣⊂ 'x':xx-Fll' ,
f:.yg-I-i', z':z,-l—k', x&quot;:æ,-l-h&quot;, ...;
quibus valoribus substitutis in (a&quot;)
, attentisque aequatio-
nibus (20)
d'h dïb, dii dïi,
EMzzï—an—O, zmcïS—dtï Zm--o ∙
d']: dïk,
ZmäF—äz; Zm—o ∙
nec non aequationibus (o&quot;'), prodibunt
E(hX—-iï):2m(hää—ci £b) ∙ dr2
⋅ ∙ dq: ti*i z
(iz—mzn. (.
&amp;? −:. $) ∙
(o....)
d.,, dal. ∑≺⋌⊔∅≻⇌∑⋯≺∣≂−∂−↙⋮−−∣⋅⋮⋮↙⇆⋟∙ .176
Formulae (o &quot; ) se habent ad commune gravitatis cen
trum prorsus ut formulae ( o &quot; ) ad fixam coordinatarum x ,
J, 2, x' , ... originem 0 , respiciuntque relativum syste
matis motum quoad ipsum gravitatis centrum:

7. • * Relativus rigidi liberique systematis motus
quoad gravitatis centrum determinatur per ( o &quot; &quot; &quot; ) ; motus
vero ipsius centri per ( o' ' ' ) Ad haec : si resultans ex omni
bus viribus systemati rigido applicitis transit per gravi
tatis centrum , nullus inde orietur relativus systematis mo
tas quoad ipsum centrum : etenim quoad istiusmodi mo
tum similiter procedet res ac si resultans illa exerceretur
contra punclum fixum ( 6.&quot; ). Eadem de causa , accedentibus
novis viribus , relativus systematis motus quoad gravitatis
centrum nullo pacto turbabitur , ubi eae resultantem sup
peditent transeuntem per centrum illud.
85.&amp; Pauca subjungentes de motu rigidi systematis cir
ca axem fixum praemittimus illud: praeter orthogonales axes   &lt;math&gt;OX , OY , OZ ,&lt;/math&gt;   ( Fig. 9 ) sint alii tres axes similiter orthogonales On, Op, Oq, quibuscum ii angulos efficiant designatos per ( xn) , (xp ) , ( aq) , (yn) , ( yp ) , (99) , (zn) ,
(zp ), ( z9 ) . Si panctum E, quod referebatur ad axes OX,
OY, OZ , referendum sit ad axes On, Op , Og , quaeri
tur relatio inter veteres coordinatas x , y
nip , q. Ponatur OE = a, et per (ax ), (ay ) , (az), ( an ) ,
(ap) , ( aq) exhibeantur anguli , quos OE facit com axibus
OX , OY , OZ , On , Op , Oq: erunt ( 50. 6º . )
ma
z et novas
cos (ax) =cos (an) cos ( xn) +cos ( ap) cos (xp) +
cos (aq) cos (xq) , cos (ay ) =cos ( an ) cos (yn ) +
cos (ap) cos (yp) + cos (aq) cos (79) , cos (az)
eos ( an) cos ( zn) * cos(ap) cos (zp) + cos ( aq ) cos ( 29) .
176
Formulae (o&quot;&quot;) se habent ad commune gravitatis cen-
trum prorsus ut formulae (a&quot;) ad fixam coordinatarum æ,
y, 2, æ',
.. . originem O, respiciantque relativum syste-
matis motum quoad ipsum gravitatis centrum:
73»: Relativus rigidi liberique systematis motus
quoad gravitatis centrum determinatur per (o&quot;&quot;); motus
vero ipsius centri per (o&quot;') Ad haec: si resultans ex omni-
bns viribus systemati rigido applicitis transit per gravi-
tatis centrum
. nullus inde orietur relativus systematis mo-
tus quoad ipsum centrum : etenim quoad istiusmodi mo-
tum similiter procedet res ac si resultans illa exerceretur
contra punctum fixnm (S.&quot;). Eadem de causa , accedentibus
novis viribus
, relativus systematis motns quoad gravitatis
centrum nullo pacto turbabitur , ubi eae resultantem suppeditent transeuntem per centrum illud. '
853 Pauca subiungentes de motu rigidi systematis cir-
ca axem fixum praemittimus illud: praeter orthogonales a-
xes OX, Of, OZ (Fig. 9) sint alii tres axes similiter or-
thogonales On, Op, Oq, quibuscum ii angnlos efficiant de-
signatos Per (æ&quot;)s (æpl- (xq) :(f&quot;) , 07)» (f?) :(znls
(zp), (zq). Si punctum E, quod referebatur ad axes OX,
OV, OZ, referendum sit ad axes On, Op, Oq , quaeri-
tur relatio inter veteres coordinatas æ , y , :. et novas
n , p , q. Ponatur OE :a, et per (aæ)
,
(ay) ,(az), (an),
(ap), (aq) exhibeantur anguli, quos OE facit cum axibus
OX , Oï
, OZ , On. , Op , Oq: erunt ( 50. 60.)
cos (aæ):cos (an) cos (xn) —-[-cos (ap) cos (æp) −⊢ ⋅
cos (aq)cos (xq) , cos (ay) :cos (an) cos (yn) −∙⊢
cos (ap) cos (yp) ∙−⊢ eos (aq) cos (rq) ∙
005 (az) −−∶
eos (an) cos (zn) —,l-cos(ap)cos(zp)-—- eos (aq) cos (zq).
3751177
Sed cos (ax ) =
a , cos(ay) = cos(az ) = a
cos (an) =
, cos ( ap ) = .. cos (aq ) =
= 9
a
adhibitis igitur substitutionibus , provenient
x = ncos( an) + pcos (xp) + qcos(xq) ,
y = ncos(yn) + pcos (yp ) + acos(yq) ,
x = ncos( zn ) + pcos(zp) + qcos(zq) ;
formulae praebentes quaesitam relationem .
Nunc 1 . ** Sit OX rotationis axis, datumque systema
tis punctum reperiatur constanter in plano YOZ: si per OX
et per punctum illud ducitur planum occurrens plano YOZ,
satis erit determinare situm intersectionis istorum plano
rum ut innotescat systematis positio. Concipiamus itaque
novos axes orthogonales On , Op , Oq sic constitutos , ut
firmiter adhaereant systemati , primusque incidat in OX ,
tertius in intersectionem illam ; erunt
y= pcos(zq ) + qsiu(z9) , z =qcos (29) — psin(zg) :
adhibita substitutione in secundo membro secundae ( o &quot; .84)
animadvertendo quod variato e non ideo variant novae co
ordinatae , factoque
2 m ( p2 +9 ) = B ,
proveniet
d ' (29 )
di2 - $ (72—28 )
(o'r) :
∙∙∙⋅ 177
&amp; Sed ∾⋇≺⊄∣∙↿∶≻∶−−⊶⋚ ,cos (ay): a , c08(az):ä— .
']
...
(an):
⋮⋮−∙ costam: g.... (aq) ⇌⋅−− −↙⋅↓−
.
adhibitis igitur substitutionibus , provenient
x:ncos(æn) -l-pcoa (æp) ⊣− 9005(-qu :
)»:ncosU'n) pcos (ïp) −∣⋅− ⊄∾≘∩⊄⋟ '

: &quot;cos(zn) −⊢ pcos(zp) -I-— qcos(zq) :

famulae praebentes (quaesitam relationem.
Nune ↿∙∘∙ Sit OX rotationis axis, datumque systema-
tis punctum reperiatur constanter in plano ïOZ: si per OX
et per punctum illud ducitur planum occurrens plano ïOZ,
satis erit determinare situm intersectionis istorum plano-
rum ut innotescat systematis positio. Concipiamus itaque
novos axes orthogonales'.0n, Op , Oq sic constitutos, ut
firmiter adhaereant systemati, primusque incidat in OX ,
tertius in intersectionem illam; erunt
7: pcos(zq ) ⊣−⊄⊗∃∥≺∅⊄⋟ :
3 −−∶ quos(zq) --
psiu(zq) :
adhibita substitutione in secundo membro secundae (o&quot;.84)
animadvertendo quod variato :non ideo variant novae co-
ordinatae, factoque
. Zm(p'-l-q3)-——-B.
proveniet
(P(zq)
- dt2
z.. 1
-B— ZUZ—zï) (o&quot;):178
d (29 )
velocitas ( 50. 2º BE . ) respondet radio 1 , diciturque dla
velocitas angularis: binomia patq , p'? + 92.. nihil sunt aliud
nisi quadrata perpendiculorum ex m , m' , ... in axem On de
missorum ; summa productorum ex massis m , m' ... in
quadrata respondentium perpendiculorum , seu m (pa+92) +
m ' ( p2t 92) + . . . vocatur momentum inertiae systema
tis m , m' , .... quod axem On .
2. °# Ponamus vires acceleratrices consistere in so
la gravitate g, axesque Ox , OY jacere in horizontali pla
no: erunt Y = 0, Z
8 , et consequenter
7
de( 29 )
1 1
dla B
&amp; Emy =
1
g Em [ p cos ( zq) +qsiu ( zq ) ] E
&amp; [cos(zq) . Emp + sin ( zq) . Emq).
Fac ut illud systematis punctum , quod posuimus ( 10.)
reperiri constanter in axe Oq, sit gravitatis centrum; ex
sistent ( 20)
Σmp = p,Σm = 0 , Σmg = qΣm :
proinde
d? ( )
- 1/3
sin ( zqı ) . Em ;
dt?
B
891
quae prius multiplicata per 2d( 29, ) , ac dein integrata
praebebit
[da ] = - 69.cos/ 291). Em + c
x
178
. ∘
'
d(zq) velocitas ( 50. 2 . ..
.
) 7:2- respondet radio1,d1c1turque
velocitas angularis: binomia phi-qi, p'H—q'æ. nihil sunt aliud
nisi quadrata perpendiculorum ex m, tu',... in axem On de-
missorum ; snmma productorum ex massis m , m'
...
in
quadrata respondentium perpendiculorum, seu m (pi-I—q'H-
m' (p'ï-l- q'3)-i- ....
vocatur momentum inertiae systema-
.tis m, m',
.... quod axem On.
2.0a Ponamus vires acceleratrices consistere in so-
la gravitate g, axesque OX , Oï jacere in horizontali pla-
no: erunt T:o, Z:
-— g ,et consequenter
d3(z ) 1 ↿ ∙
de? ∶−∙−−↕≣− g Em]: —B—g2m[pcos(zq)—l-qstn(zq)]
1 - .
−−−−− -B- g[cos(zq). Zmp −↘∟ stn (zq). qu].
Fac ut illud systematis punctum ,quod posuimus (10.)
reperiri constanter in axe Oq, sit gravitatis centrum; ex-
BlStent (20) ∣
Zmp :plzm:o , qu :q12m :
proinde*
dï ↿ ∙)— B gq, sm (sq,). Em;

quae prius multiplicata per 2d( sq, )
, ac dein integrata
praebebit
. d
Z [ 2
2 .
[ld-g-l ∸∶−∙∙∙⋅ ∙∙∙ ï
gqx 008(zq1). Zm ⊹∁ ∙
iis179
Exsistentibus in initio motus d (291)
= uo et ( 291) = a , erit du
2
C = uo% + B 69 , cosa. Em :
propterea
d (290) 72
=u' . +
dt 2/3
891 [ cosa - cos ( 291) ] Em (o' ) .
Huc spectat theoria penduli compositi.
3.•* Intelligantur m , m' , m &quot; , .... coire in u
nicum punctum annexum axi horizontali Ox ope rectae r;
exsurget pendulum simplex : in casu
p = p = p = ... = 0 , q = 9 = 9 &quot; = ... = 9 = r ,
B = 2m(p + g ”) = 2n ;
et consequenter quoad pendulum simplex
d ( 292) 7
2
[Company *== +
s [ cos a - cos ( 291) ] (o &quot; ).
2
4.0# Facto 8
2
B
89. EmEm , proveniet
B
9 , £ m col) ;
longitudo videlicet penduli simplicis , quod suas perficit
oscillationes eodem tempore ac pendulum compositum . Re
cole quae diximus (67 ) .
5.° * Pone nullas esse vires acceleratrices i
erit ( 1. ° 0 ' )
179
d(zq !)
dc
Exsistentibus in initio motus :u. et (zq.):a, erit
.
2 C:u.,2 −⊦⋅ ïgq, cosa. Em:
prapterea
d(dZQtli:u⋅−⊢ ∙−⋛−∊⊄∙ [cosa — cos (zq.)]2m (a').
Huc spectat theoria penduli compositi.
39»: Intelligantur m , m', m&quot;, ... . . coire in u-
nicum punctum annexum axi horizontali OX Ope rectae r;
exsurget pendulum simplex: in casu
p::p'::p&quot;: ∙ ∙ ∙ −−∙−−∶∘ , qzq'2q&quot;: ∙ ∙ ∙
∶⊄∎∶↿∙ ,
&quot;B::ZmQF-l-qa) claim ;
et consequenter quoad pendulum simplex
↙≀≺≦≦∣≖⋝⊺−−⋅↙∘≖ ; f ,.
(....
...
(..., ].
2 2
4.0a Facto ∙∓− g :ïgq, Em
, proveniet
B .
r'.&quot;—∙−∙∙ q,2m
Om) ;
longitudo videlicet penduli simplicis, quod suas perficit
oscillationes eodem tempore ac pendulum compositum. Re-
cole quae diximus (67).
5.
.,
Pone nullas esse vires acceleratrices: ,
erit (1. ∘ o)180
dº(aq )
dia
d( 24)
unde velocitas angolaris u =
dc = const. = u , .
1
1 Motus igitur exsistet uniformis , eritque velocitas angu
laris ad velocitatem puncti v . gr. m ut 1 ad radium cir
culi descripti ab ipso m , seu
1
u : v =1 : V patqz , ac proinde v = u ? (patoga )
quoad illud itaque punctum obtinebit vis contrifuga expres
sa ( 51 ) per
= u’m V pat92 .
V pr + q2
1
vam 2
Resolvatur haec vis in ternas coordinatis axibus On, Op,
Og parallelas ; prodibunt 1
+
9
р
0 , u²mV p2tga .
V p²ta?
wimb p'tgo.
Foto &gt;
seu
0
0 , ump , u'mg :
1
quoad totum ergo systema habebuntur 2
0 , użEmp , u’Emq ;
ideoque orietur pressio in axem OX. Prima membra formu
larum ( a : 13. 8.° ) in casu fiunt
0 , użEmp, użEmq , użEmnp , użEmng , u’Em (pa - pa ) :
!
hinc ubi fuerint 1
1
Emp= 0 , Emg = 0 , Emnp = 0, Emng = 0 ( o'r) ,
180
(l*(z'q)
d?
d(zq)
dc

: o

, unde velocitas angularis ::: :const.-zuo,
Motus igitur exsistet uniformis , eritque velocitas angu-
laris ad velocitatem puncti v. gr. m ut 1 ad radium cir-
culi descripti ab ipso m, seu
∣ ...—....
a: p −−−−−↿ :Vpl-I—qa , ac proinde V::u' (pH-q2 )
quoad illud itaque punctum obtinebit vis centrifuga expres-
sa (51) per
vam
l/P'&quot;l'qa −−∶
&quot;&quot;&quot; Vlf-*?&quot;
-
Resolvatur haec vis in ternas&quot; coordinatis axibus On. Op,
Oq parallelas; prodibunt
seu
0, uïrnp
,
'u'mq :
quoad totum ergo systema habebuntur
o , u'Zmp , u'qu;
ideoque orietur pressio in axem OX. Prima membra formu-
larum (a'm : 13. 8.&quot;) in casu fiunt
o , ti*Zmp, uazmq , u'Zmnp , u'Zmnq , u'Zmþq—pq) :
hinc ubi fuerint
Zmp:o , M.,—:a, Zmnpzo, zmnqzo (atur)
'181
1
vires centrifugae se muluo librabunt independenter ab axe
Ox
, nullamque iste axis patietur pressionem
. Prima et se
cunda
(oh
) important
( 20. 6.
) transitum axeos On seu
OX per gravitatis centrum tertia vero et quarta important peculiarem quandam axiuin On
, Op, Oq positionem
relate ad punctorum
m
,
m
'
,
m
&quot; systema
. Porro si On
,
Op
, Oq ita sunt positi, ut suppeditent Emnp
=
0
, Emng
=
0
,
Empq
=
o
, appellari solent principales systematis axes in
ordine ad originem
itidem quae momenta ad eos referuntur
, et ipsa dicuntur principalia inertiae momenta
. Ex
pletis tertia et quarta
(
o
&quot;
&quot;&quot;
)
, non autem prima et secun
da
, ex omnibus viribus centrifugis resultabit
( 13. 9.0 10.9
)
vis premens rolationis axem in O. 6. '* Superiores formulae manifeste applicantur
continuo materialium punctorum systemati mutando
&amp; in
ſ et
m in dm
, integrationemque protendendo ad totam
systematis massam
.
7.9 Saepe videmus corpora impulsu aliquo loca
liter mota affici simul rotationis motu

: etiam praecisis

,
quae diximus
( 84
)
, sic ostendi potest motum istum oriri ex
eo quod impulsus non habeat directionem transeuntem per
centra gravitatis corporum
. Sic
G gravitatis centrum cor
poris MM
'
( Fig
. 46
)
, et AZ vis corpori cominunicata
..
Ducatur per
G ad AZL perpendiculum GL dividalur

bifariam AZ in
C
, et resolvatur CA in AD per
G tran
seantem
, et in AB normalem rectae AZ producatur AG

donec GF aequet GA intelligatur AD applicita ad punclum F , sitque FK = AD resolvatur FK in FH parallelam et FI perpendicularem rectae LGN

: quibus posi

tis
, substituti poteront vi AZ quatuor vires CZ
, AB
, FI
,
FH
. Jamvero CZ
, FI utpote aequales
, parallelae et ad
eamdem plagam tendentes contrahuntur in unam reprae
sentatam per GE
( 11
)
=GZ
+Fl
=AZ
, transeuntem per
G
, eidemque AZ parallelam proinde movebitur centrum

G non secus ac vis AZ ipsi esset applicata
. At duae aliae
↿∂⋅↿
vires centrifugae se mutuo librabunt independenter ab axe
OX, nullamque iste axis patietur pressionem. Prima et se-
cunda (o&quot;&quot;) important (20. b.) transitum axeos On seu
OX per gravitatis centrum: tertia vero et quarta impor-
tant peculiarem quandam axium On
, Op, Oq positionem
relate. ad punctorum m, m', m&quot; ,... systema. Porro si On,
Op, Oq ita sunt positi, ut suppeditent Zmnp:o, Zmnq:o,
Zmpq:o, appellari solent principales systematis axes in
ordine ad originem
O

itidem quae momenta ad eos refe-
runtur, et ipsa dicuntur principalia inertiae momenta. Expletis tertia et quarta (on&quot;), non autem prima et secunda, ex omnibus viribus centrifugis resultabit (13. 9310!)
vis premens rotationis axem in O.

63. Superiores formulae manifeste applicantur
continuo materialium punctorum systemati mutando
2 in
]et as in dm
,
integrationemque protendendo ad totam
systematis massam.

7.&quot; Saepe videmus corpora impulsu aliquo loca-
liter mota aflici simul rotationis motu: etiam praecisis,
quae diximus (84), sic ostendi potest motum istum oriri ex
eo quod impulsus non habeat directionem transeuntem per
centra gravitatis corporum. Sit
G gravitatis centrum cor-
poris MM' (Fig. 46)
, et AZ vis corpori communicata.
Ducatur per
G ad AZL perpendiculum GL; dividatur
bifariam AZ in C, et resolvatur CA in AD per
G tran-
seuntem, et in AB normalem rectae AZ; producatur AG
donec GF aequet GA; intelligatur AD applicita ad pun-
ctum
F
, sitque FK: A

resolvatur FK in FH paral-
lelam et FI perpendicularem rectae LGN

: quibus posi-

tis, substituti poterunt vi AZ quatuor vires CZ,AB, FI
,
FH. Iamvero CZ, FI utpote aequales
, parallelae et ad
eamdem plagam tendentes contrahuntur in unam reprae-
sentatam per GE (11):GZ—-FI:AZ, transeuntem per
G, eidemque AZ parallelam proinde movebitur centrum

0 non secus ac vis AZ ipsi esset applicata. At duae aliae182
.AB, FH utpote aequales , parallelae , et ad contrarias par-
tes tendentes , nequeunt gravitatis centrum e suo loco di-
movere : spectatis itaqne istiusmodi viribus, immobile eqn-
sisteret gravitatis centrum; sed eae sese mutuo non de-
struunt, cum e diametro non opponantur. Aliud ergo praestare non poterunt nisi corporis rotationem circa gravitatis centrum. Rotationis motus incipit circa reetam aliquam seu axem, et quoniam in omnes corporis particulas ex rotatione inducitur vis centrifuga; hinc si vires centrifugae inde ortae aequilibrantur circa rectam illam, invariabilis exsistet rotationis axis, defereturque per spatium sibimet semper parallelas; secus, mutabitur indesinenter rotationis axis donec ad aequilibrium deveniatur.

=== De fluidorum corporum aequilibrio. ===
86. Fluida corpora spectamus veluti materialiam punctorum congeries; quae puncta, utpote invicem independentia, vel minimo cedunt impulsui. In massa fluida undique librata sume punctum quodvis [exhibemus per &lt;math&gt;( x , y , z )&lt;/math&gt;, denotantibus &lt;math&gt;x, y, z&lt;/math&gt; ejus coordinatas] sollicitatum vi acceleratrice &lt;math&gt;\varphi&lt;/math&gt; praebente componentes &lt;math&gt;X, Y, Z&lt;/math&gt; coordinatis axibus &lt;math&gt;OX, OY, OZ&lt;/math&gt;,  parallelas et per punctum illud fac ut transeat superficies &lt;math&gt;k&lt;/math&gt; plana, rigida atque infinitesima: consistet &lt;math&gt;k&lt;/math&gt; in aequilibrio; et consequenter pressiones hinc et illinc exercitae in &lt;math&gt;k&lt;/math&gt; ab circumpositis massae fluidae stratis, erunt vires aequales et directe contrariae, simulque normales ipsi &lt;math&gt;k&lt;/math&gt;. Ejusmodi pressionum alteram repraesenta per &lt;math&gt;\varpi k&lt;/math&gt;; ratio &lt;math&gt;\frac{\varpi k}{k}(= \varpi)&lt;/math&gt; dicitur pressio hydrostatica exercita &lt;math&gt;k&lt;/math&gt; apud punctum &lt;math&gt;( x, y , z )&lt;/math&gt; contra aream ( = 1 ) sumptam in plano superficiei &lt;math&gt;k&lt;/math&gt;. In eadem massa fluida fac ut per punctum alterum &lt;math&gt;( x_0, y , z )&lt;/math&gt; transeat talis superficies &lt;math&gt;k_0&lt;/math&gt; plana, rigida et infinitesima, quae communem habeat projectionem cum superficie &lt;math&gt;k&lt;/math&gt; in plano &lt;math&gt;YOZ&lt;/math&gt;; voca &lt;math&gt;h&lt;/math&gt; projectionem illam, et &lt;math&gt;\varpi_0&lt;/math&gt;, hydrostaticam pressionem apud punctum &lt;math&gt;(x_0, y , z)&lt;/math&gt; contra aream ( =1 ) sumptam in plano areae &lt;math&gt;k_0&lt;/math&gt;. Massa fluida adhuc perget esse librata, etsi in qualibet ejus portione intelliguntur puncta rigidis lineolis firmiter connecti, seu, quod eodem redit, etsi quaelibet ejus portio fit solida: ponatur id contingere portioni cylindricae habenti rectam parallelam axi &lt;math&gt;OX&lt;/math&gt; pro generatrice, et &lt;math&gt;k , k_0&lt;/math&gt; pro basibus; denotet &lt;math&gt;\mu&lt;/math&gt; densitatem massae fluidae apud punctum &lt;math&gt;(x , y , z)&lt;/math&gt;; sitque &lt;math&gt;x &gt; x_0&lt;/math&gt; Exprimetur per&lt;math display=&quot;block&quot;&gt;h\int_{x_0}^x \mu X dx
&lt;/math&gt;summa ex viribus motricibus, quibus juxta &lt;math&gt;OX&lt;/math&gt; sollicitantur puncta illius portionis; exprimenlur praeterea per  &lt;math display=&quot;block&quot;&gt;\frac{h}{k_0}\varpi k_0, -\frac{h}{k}\varpi k &lt;/math&gt;pressiones exercitae juxta eumdem OX , altera in basim ko,altera in basim k quod spectat ad pressiones contra lateralis superficiei puncta, eae utpote normales generatrici rectae nullas dabunt componentes axi OX parallelas. Quia igitur solidata portio perseverat in aequilibrio, iccirco

&lt;math display=&quot;block&quot;&gt;h\int_{x_0}^x \mu X dx + h\varpi_0 - h\varpi = 0, \, \mathrm{unde}\,   \varpi = \varpi_0 + \int_{x_0}^x \mu X dx .
&lt;/math&gt;

Haud mutata positione superficiei &lt;math&gt;k_0&lt;/math&gt;, revolvatur utcumque superficies &lt;math&gt;k&lt;/math&gt; circa punctum &lt;math&gt;(x , y ,z)&lt;/math&gt;: permanebit secundum membrum ultimae aequationis; ergo et primum. Quare perseverabit in eodem valore hydrostatica pressio quoad omnia plana per punctum illud utcumque ducta: huc spectat principium de aequalitate pressionis. Consequitur, si recta generatrix sumitur parallela, prius axi &lt;math&gt;OY&lt;/math&gt;, deinde axi &lt;math&gt;OZ&lt;/math&gt;, denotantibus &lt;math&gt;\varpi_0',\varpi_0''&lt;/math&gt; hydrostaticas pressiones apud puncta &lt;math&gt;( x , y_0, z) , (x , y , z_0 )&lt;/math&gt;, fore etiam&lt;math display=&quot;block&quot;&gt;
\varpi = \varpi_0' + \int_{y_0}^y \mu Y dy , \varpi  = \varpi_0''+ \int_{z_0}^z \mu Z dz &lt;/math&gt;Terni valores &lt;math&gt;\varphi&lt;/math&gt; differentiati, primus quoad &lt;math&gt;x&lt;/math&gt;,  secundus quoad &lt;math&gt;y&lt;/math&gt;, tertius quoad &lt;math&gt;z&lt;/math&gt;, praebent
&lt;math display=&quot;block&quot;&gt;
\frac{d\varpi}{dx} = \mu X,  \frac{d\varpi}{dy} = \mu Y, \frac{d\varpi}{dz} = \mu Z. (o) &lt;/math&gt;et consequenter (27.24º)
&lt;math display=&quot;block&quot;&gt;
d\varpi = \mu ( Xdx + Ydy + Zdz). ( o' ) &lt;/math&gt; Itaque conditiones requisitae ad massae fluidae aequilibrium eo redeunt ut exsistat ejusmodi functio &lt;math&gt;\varpi&lt;/math&gt; variabilium &lt;math&gt;x, y, z&lt;/math&gt;, quae expleat sive ternas (o), sive unicam (o').

87. Haec notentur. 

1º. Si fluidum continetur vase undique clauso satisque firmo, utcumque se habeat valor &lt;math&gt;
\varpi &lt;/math&gt; ex (o') quoad superficiem fluidi, is constanter aequivalebit reactioni ex vasis lateribus: at si fluidi superficies sit libera, externisque subjecta pressionibus, ad aequilibrium explenda insuper erit (o') per talem valorem &lt;math&gt;
\varpi &lt;/math&gt;, qui in singulis liberae superficiei punctis aequivaleat respondenti pressioni externae.

2º. Hinc si pressio externa vel ponitur &lt;math&gt;=0&lt;/math&gt; vel ubique eadem, erit &lt;math&gt;d\varpi = 0&lt;/math&gt; quoad superficiem fluidi librati, ideoque
&lt;math display=&quot;block&quot;&gt;Xdx + Ydy + Zdz  = O (o''). &lt;/math&gt;

3º. Traduci potest (o&quot;) ad&lt;math display=&quot;block&quot;&gt;\frac{X}{\varphi} \frac{dx}{ds}+\frac{Y }{\varphi} \frac{dy}{ds} + \frac{Z}{\varphi}\frac{dz}{ds} = 0&lt;/math&gt;exprimunt &lt;math&gt;X/\varphi, Y/\varphi, Z/\varphi&lt;/math&gt; cosinus angulorum, quos efficit vis acceleratrix &lt;math&gt;\varphi&lt;/math&gt; cum axibus coordinatis &lt;math&gt;OX, OY, OZ&lt;/math&gt;; denotant &lt;math&gt;\frac{dx}{ds}, \frac{dy}{ds},\frac{dz}{ds}&lt;/math&gt; cosinus angulorum, quos recta tangens arcum &lt;math&gt;s&lt;/math&gt; apud ejus extremum facit cum iisdem axibus: inferimus (50. 6.) vim &lt;math&gt;\varphi&lt;/math&gt; intercipere angulum = 90° cum rectis omnibus tangentibus ubivis superficiem vel nullo pacto, vel aeque pressam; ac proinde &lt;math&gt;\varphi&lt;/math&gt; sese dirigere normaliter ad istiusmodi superficiem.

4.º Integrata (o&quot;), si constanti arbitrariaeque quantitati tribuuntur alii atque alii valores, emergent aliae atque aliae aequationes, quibus totidem respondebunt distinctae superficies aeque pressae.

5.°* In hypothesi &lt;math&gt;\varphi&lt;/math&gt; tendentis ad punctum fixum, constitue ibi coordinatarum originem: denotante &lt;math&gt;D&lt;/math&gt; distantiam inter punctum illud et &lt;math&gt;( x , y , z)&lt;/math&gt;, erunt (50. 6º)&lt;math display=&quot;block&quot;&gt;X = -\varphi \frac x D, Y= -\varphi \frac y D, Z= -\varphi \frac z D&lt;/math&gt;hinc&lt;math display=&quot;block&quot;&gt;X dx + Ydy + Zdr = - \frac \varphi D (xdx + ydy + zdz).&lt;/math&gt;Est insuper &lt;math&gt;x^2 + y^2 + z^2 = D^2 &lt;/math&gt;, unde &lt;math&gt;xdx + ydy + zdz = DdD;&lt;/math&gt; et consequenter&lt;math display=&quot;block&quot;&gt;Xdx +Ydy + Zdz = -\varphi dD.&lt;/math&gt;In ordine igitur ad superficiem aeque pressam exsistet &lt;math&gt;dD = 0&lt;/math&gt;: propterea &lt;math&gt;D = C&lt;/math&gt;; ex qua &lt;math&gt;x^2 + y^2 + z^2 = C^2 &lt;/math&gt;: massa videlicet fluida atque librata induet sphaericam formam.


6. Quoad fluidum elasticitate pollens, constat experimentis densitatem &lt;math&gt;\mu&lt;/math&gt;, permanente temperie, esse proportionalem respondenti pressioni &lt;math&gt;
\varpi &lt;/math&gt;, nimirum&lt;math display=&quot;block&quot;&gt;\mu = \theta \varpi: (o''')&lt;/math&gt;Eliminata  &lt;math&gt;\mu&lt;/math&gt; ab (o') et (o''&quot;''), proveniet&lt;math display=&quot;block&quot;&gt;\frac{d\varpi}{\varpi}=\theta(Xdx + Ydy + Zdz);&lt;/math&gt;et facto &lt;math&gt;Xdx + Ydy + Zdz = df (x,y,z)&lt;/math&gt;, erit:&lt;math display=&quot;block&quot;&gt;\ln \varpi = \int \theta df + \ln C = \ln (e^{\int \theta df}) + \ln C= \ln (C e^{\int \theta df})&lt;/math&gt;hinc&lt;math display=&quot;block&quot;&gt;\varpi = C e^{\int \theta df}, \mu = C \theta e^{\int \theta df}&lt;/math&gt;coefficiens &lt;math&gt;\theta&lt;/math&gt; pendet a temperie vigente apud &lt;math&gt;(x , y , z)&lt;/math&gt;. Inferimus aequilibrii statum in fluido elastico importare temperiem vel ubique eamdem, vel talem ut sit functio quantitatis &lt;math&gt;f&lt;/math&gt;. Haec insuper quantitas est (2º, 4º) constans in unaquaque superficie aeque pressa; idipsum ergo dicendum de temperie. 


7.º Constat etiam experimentis fluidum elasticitate pollens ita contrahi vel expandi, imminuta vel aucta temperie ac permanente pressione &lt;math&gt;
\varpi' &lt;/math&gt; ut ejus volumen &lt;math&gt;
V &lt;/math&gt;minuatur vel augeatur partibus 0,00375 pro singulis gradibus thermometri centigradi; inde fit, ut posito 0,00375 = &lt;math&gt;
a &lt;/math&gt;, et aucta temperie gradibus &lt;math&gt;n&lt;/math&gt; ultra &lt;math&gt;0^\circ \mathrm{C}&lt;/math&gt; , volumen  &lt;math&gt;V&lt;/math&gt; evadet &lt;math&gt;V ( 1 + an )&lt;/math&gt;; propterea, designantibus  &lt;math&gt;\mu_0&lt;/math&gt;  et   &lt;math&gt;\mu_1&lt;/math&gt;  respondentes densitates, erit &lt;math&gt;\frac{\mu_1}{\mu_0}=\frac{1}{1+an}.&lt;/math&gt; Nunc, permanente temperie &lt;math&gt;n&lt;/math&gt;, crescat pressio ab &lt;math&gt;
\varpi' &lt;/math&gt; ad  &lt;math&gt;
\varpi &lt;/math&gt;;  denotante &lt;math&gt;\mu&lt;/math&gt;  respondentem densitatem, erit  (1º) &lt;math&gt;\frac{\varpi}{\varpi'}=\frac{\mu}{\mu_1},&lt;/math&gt; quocirca&lt;math display=&quot;block&quot;&gt;\varpi = \frac{\varpi'\mu}{\mu_1} =  \frac{\varpi'}{\mu_0} \mu ( 1 + an )&lt;/math&gt;; et facto &lt;math&gt; \frac{\varpi'}{\mu_0} =i&lt;/math&gt;, &lt;math&gt;\varpi = i \mu ( 1 + an ) (o^{(iv)})&lt;/math&gt;.

=== De gravium homogeneorumque liquidorum aequilibrio. ===
88. Planum &lt;math&gt;XOY&lt;/math&gt; sit horizontale, axisque &lt;math&gt;OZ&lt;/math&gt; (Fig. 47) vergat deorsum juxta directionem gravitatis &lt;math&gt;g&lt;/math&gt;; erunt &lt;math&gt;X=0, Y=0, Z = g&lt;/math&gt;: proinde (86. 6), &lt;math display=&quot;block&quot;&gt;d\varpi = g \mu dz ( 0^{v} )&lt;/math&gt;Si pressio externa ponitur vel = 0, vel ubique eadem, erit &lt;math&gt;d\varpi = 0&lt;/math&gt; quoad librati fluidi superficiem, ideoque &lt;math&gt;dz = 0&lt;/math&gt;, et &lt;math&gt;z = Const&lt;/math&gt;: superficies nempe illa existet plana atque horizontalis. 

Pone &lt;math&gt;\mu&lt;/math&gt;  constantem; ex (0&lt;sup&gt;v&lt;/sup&gt;) habebis

&lt;math&gt;\varpi = g \mu z + C_1&lt;/math&gt;,

In fluidi superficie aeque pressa constitue planum horizontale &lt;math&gt;XOY&lt;/math&gt;: quoad eam erit &lt;math&gt;z = 0&lt;/math&gt;; nihilque aliud denotabit &lt;math&gt;C_1&lt;/math&gt; nisi externam pressionem in aream ( = 1 ) quaquaversus per fluidum aequaliter diffusam.

Haec facile nunc stabiliuntur circa pressiones gravium homogeneorumque liquidorum intra vasa in aequilibrio consistentium. 
[[Fasciculus:Hydrostatic-pressure.svg|thumb]]
1º. Si per &lt;math&gt;\Pi&lt;/math&gt; designatur pressio in horizontalem aream &lt;math&gt;A&lt;/math&gt; demersam ad profunditatem &lt;math&gt;z&lt;/math&gt;, exsistet &lt;math&gt;\Pi = A \varpi = A (g\mu z + C_1 ) .&lt;/math&gt;

2º. Si &lt;math&gt;C_1 = 0&lt;/math&gt;, aequivalebit &lt;math&gt;\Pi&lt;/math&gt; ponderi prismatis, cujus basis est &lt;math&gt;A&lt;/math&gt;, altitudo &lt;math&gt;z&lt;/math&gt;, densitas vero eadem ac densitas liquidi.

3º. Exhibente &lt;math&gt;A&lt;/math&gt; horizontalem vasis fundum, ideoque &lt;math&gt;z&lt;/math&gt; altitudinem vasis; quoniam &lt;math&gt;\Pi&lt;/math&gt; nullatenus pendet a vasis figura, iccirco permanentibus &lt;math&gt;A&lt;/math&gt; et eadem perstabit liquidi pressio in horizontalem fundum, utcumque de caetero varient figura et capacitas vasis.

4º. Area &lt;math&gt;A&lt;/math&gt; sit oblique intra liquidum utcumque demersa: divide &lt;math&gt;A&lt;/math&gt; in areolas infinitesimas &lt;math&gt;a , a ', a'' &lt;/math&gt; quarum distantiae ab extima liquidi superficie designentur per &lt;math&gt;z' , z''...;&lt;/math&gt; denotante &lt;math&gt;\Pi'&lt;/math&gt; totalem pressionem, et &lt;math&gt;z_1&lt;/math&gt; perpendiculum ductum ex centro gravitatis areae &lt;math&gt;A&lt;/math&gt; in planum &lt;math&gt;XOY&lt;/math&gt;; erit (20) 

&lt;math&gt;\Pi' = a(g\mu z + C_1) + a'(g\mu z'+ C_1) +... = g\mu(az + a'z' + ...) + C_1( a + a' + ... ) = g\mu z_1 A + C_1 A = A (g\mu z_1 + C_1 )&lt;/math&gt;. 

Hinc si centrum gravitatis manet ad eamdem profunditatem demersum, haud variabit  &lt;math&gt;\Pi'&lt;/math&gt;, utcumque circa illud revolvatur area demersa: potest A repraesentare quamlibet rectilineam portionem internae superficiei vasis.

Ad haec: coordinatae ( 13. 3º. )

&lt;math&gt;b=\frac{\sum ax (g\mu z + C_1)}{ \sum a(g\mu z + C_1)}; b' = \frac{\sum ay (g\mu z + C_1)}{ \sum a(g\mu z + C_1)}, b'' = \frac{\sum az (g\mu z + C_1)}{ \sum a(g\mu z + C_1)} &lt;/math&gt; seu (20) 

&lt;math&gt;b =   \frac{g\mu \sum ax z + C_1 A x_1 }{ A(g\mu z_1 + C_1) }; 
b' =  \frac{g\mu \sum ay z + C_1 A y_1 }{  A(g\mu z_1 + C_1)}, 
b'' = \frac{g\mu \sum a  z^2 + C_1 A z_1 }{ A(g\mu z_1 + C_1)} &lt;/math&gt;

respondent illi puncto areae &lt;math&gt;A&lt;/math&gt;, per quod transit resultans ex parallelis viribus &lt;math&gt;a(g\mu z + C_1), a'(g\mu z'+ C_1), a''(g\mu z''+ C_1)...&lt;/math&gt;; istiusmodi punctum dicitur centrum pressionis.
[[Fasciculus:PolydirectionalPressure.svg|thumb]]
5º . Veniat considerandum solidum liquido immersum: sume apud punctum &lt;math&gt;( x , y , z )&lt;/math&gt; in solidi superficie areolam infinitesimam &lt;math&gt;k&lt;/math&gt; , et apud puncta &lt;math&gt;( x_0, y, z ) , (x , y_0, z ) , ( x , y , z_0 )&lt;/math&gt;in eadem solidi superficie areolae &lt;math&gt;k_0, k'_0, k''_0&lt;/math&gt;, sitque &lt;math&gt;h&lt;/math&gt; projectio areolae &lt;math&gt;k_0&lt;/math&gt; in plano &lt;math&gt;YOZ&lt;/math&gt;, &lt;math&gt;h'&lt;/math&gt; projectio areolae &lt;math&gt;k'_0&lt;/math&gt; in plano &lt;math&gt;XOZ, h''&lt;/math&gt;projectio areolae &lt;math&gt;k''_0&lt;/math&gt; in plano &lt;math&gt;XOY&lt;/math&gt;; congruant vero &lt;math&gt;h, h' , h''&lt;/math&gt; cum projectionibus areolae &lt;math&gt;k&lt;/math&gt; in iisdem planis: per &lt;math&gt;k(g\mu z + C_1), k_0(g\mu z + C_1),k'_0(g\mu z + C_1), k''_0(g\mu z + C_1),&lt;/math&gt;exprimentur pressiones normaliter exercitae in areolas  &lt;math&gt;k, k_0, k'_0, k''_0&lt;/math&gt;; ejusmodi pressionum prima resolvitur in &lt;math&gt;\frac{h}{k}\cdot k(g\mu z + C_1), \frac{h'}{k}\cdot k(g\mu z + C_1),\frac{h''}{k}\cdot k(g\mu z + C_1),&lt;/math&gt;&lt;ref&gt;Figura deest ergo clare non est si aequatio est recte stripta &lt;/ref&gt; parallelas rectis &lt;math&gt;OX , OY , OZ&lt;/math&gt;; secunda praebet componentem

&lt;math&gt;-\frac{h}{k_0}\cdot k_0(g\mu z + C_1)&lt;/math&gt;

parallelam rectae OX, tertia dat componentem

&lt;math&gt;-\frac{h'}{k'_0}\cdot k'_0(g\mu z + C_1)&lt;/math&gt;

parallelam rectae OY; quarta suppeditat componentem

&lt;math&gt;-\frac{h''}{k''_0}\cdot k''_0(g\mu z_0 + C_1),&lt;/math&gt;

parallelam rectae &lt;math&gt;OZ&lt;/math&gt;. His positis, quisque videt areolam &lt;math&gt;k&lt;/math&gt;, elisis componentibus horizontalibus, urgeri sursum verticali pressione&lt;math display=&quot;block&quot;&gt;h'' g \mu ( z - z_0 )&lt;/math&gt;totum igitur demersum solidum ad verticalem ascensum sollicitatur parallelis viribus praebentibus resultantem, quae aequivalet ponderi liquidi expulsi, et transit per punctum illud, ubi erat gravitatis centrum ipsius liquidi expulsi. Itaque si &lt;math&gt;V'&lt;/math&gt; et &lt;math&gt;\mu'&lt;/math&gt; exhibent volumen et densitatem solidi liquido immersi, &lt;math&gt;V&lt;/math&gt; volumen liquidi espulsi; pondus, quod superest solido, exprimelur per &lt;math&gt;g( V'\mu' - V\mu )&lt;/math&gt;: in solidis heterogeneis designat &lt;math&gt;\mu'&lt;/math&gt; densitatem mediam.

89. Sit 1º &lt;math&gt;\mu' &gt; \mu &lt;/math&gt; cum nequeat esse &lt;math&gt;V &gt; V '&lt;/math&gt;, erit semper &lt;math&gt;V'\mu' - V\mu &gt;0&lt;/math&gt;; tamdiu igitur descendet solidum, ubicumque in liquido collocetur, donec aliquod offendat obstaculum, cui adstringatur adhaerere. Si collocatur in liquidi superficie; statim atque totum fuerit demersum, exsistet &lt;math&gt;V = V';&lt;/math&gt; et consequenter perget solidum moveri vi acceleratrice &lt;math&gt;\frac{gV ' ( \mu' - \mu )}{V'\mu'}&lt;/math&gt; seu &lt;math&gt;g\left( 1 - \frac{\mu }{\mu'}\right)&lt;/math&gt;

Ab exploratis solidi ponderibus P et P' in vacuo et in li
quido elici potest ratio inter u et l ; siquidem
P = gV ' ', P = 8 ! V' ' — Vp ) , et V = V :
propterea
P í M
Hop
unde р
P
P - P
[[Fasciculus:EB1911 Hydromechanics - Fig. 3.jpg|thumb]]
Sit 2º. M '= H: tamdiu V'u ' - Vl &gt; o quamdiu &lt;math&gt;V' &gt; V&lt;/math&gt;; solidum nempe collocatum in superficie liquidi eo usque descendet, donec totum demergatur; quod ubi contigerit, evanescente V' M' — Vp , consisteret in aequilibrio nisi urgeretur adhuc vi acquisita descendendo ante et aequivalet ponderi liquidi expulsi, et transit per punctum illud, ubi erat gravitatis centrum ipsius liquidi expulsi. ltaque si V'et p! exhibent volumen et' densitatem
solidi liquido immersi, V volumen liquidi expulsi; pondus,
quod superest solido, exprimetur per
g( V'p.'—Vp.) :
in solidis heterogeneis designat p! densitatem mediam. ⋅
89. Sit. 1041!) p.: cum nequeat esseV) V', erit sem-
per V' pf ∙−− Vp.) o; tamdiu igitur descendet solidum, ubi-'
cumque in liquido collocetur, donec aliquod offendat ob-
staculum
, cui adstringatur adhaerere. Si collocatur in li-
quidi superficie; statim atque totum fuerit demersum, ex-
sistet V:V'; et consequenter perget solidum moveri vi
acceleratrice '
sv. ∣≺⊮∸⋮⋅⋅−⋅∟∸≻ ∘ −.r. v'F-I , .seu :,(1 l*') .
Ab exploratis solidi ponderibus P et'P' in vacuo et in li-
quido elici potest ratio inter p! et p.; siquidem
≖∙⊃−∙−⇀−∊⋁∙⊬↼∙∙ P',—.: g( vir—v,. ), .xv.-: V':
prOpterea
.
P p! p!— P
P' −−−⊬∙∙⊬∙ uude F- P-P'
.
Sit 20. pl: p.: tandiu V'pf -— VP) o quamdiu
V&quot; V ; solidum nempe collocatum in superficie liquidi
eo .usque descendet,, donec totum demergatur; quod ubi
contigerit, evanescentev p! —Vp. , consisteret in aequi-,-
librio nisi urgeretur adbuc vi acquisita descendendo ante192
1
V'de VM
1
1
totalem immersionem ; ad aequilibrium praeterea deberent
gravitatis centra solidi et liquidi expulsi esse in eadem re
cia verticali,

Sit 3º. p &lt; l tandiu . Vil – Ve &lt; o quandiu
V &gt; ; et facto V , erit Vų – VH = 0.
Solidum igitur collocatum intra liquidum ascendet ad li
quidi superficiem ; situm in ipsa superficie supernatabit ;
eritque portio demersa V ad volumen integrum V' ut j ': fl.
Innatantis solidi aequilibrium requirii insuper ut in eadem
recta verticali inveniantur gravitatis centra ipsius solidi et
liquidi quod expellitur.
Itaque positio aequilibrii quoad solidum homogeneum
liquido insideas determinabitur si plano ita secetur soli
dum, ut et alterius segmenti volumen sit ad solidi volu
men ia data ratione pe': fhy et haec volumina habeant sua
gravitatis centra in eadem recta , quae normaliter insistat
plano secanti: rem declaramus exemplo. Determinanda sit
positio aequilibrii in prismate recto ac triangulari , quod
ita demergitur ut et ejus bases maneant verticales, et u
na ex tribus faciebns v. g. BC ( Fig 48 ) exsistat cota ex
tra liquidum.
Quisque videt directionem plani secantis non pende
re a mutua basium distantia, satisque esse ut determine
tur intersectio De illius plani et baseos v . g. ABC. Exhi.
beant a ', a“ latera AB, AC dati trianguli ABC , et a', w &quot;
latera incognita AD, AE crianguli ADE : triangulares areae
ABC, ADE exprimentur per
3
i a'a ' sin A , Law&quot; sin A.
Sed area ABC est ad aream ADE ut integrum prisma ad
prismaticam portionem demersam ; igitur
le
IWW 'sin A: į a' a &quot; sin A = fe':J.,Was&quot;
P.
-a'a' ( k) .
192
totalem immersionem; ad aequilibrium praeterea deberent
gravitatis centra solidi et liquidi expulsi esse in eadem re-
cta verticali. ⋅
'
Sit 3&quot;. p! p. : tandiu. V'pl ∙− Vp.( o quandiu
V P- ;et factoV:V V) P- ,eritV'pf—Vp.:o.
Solidum igitur collocatum intra liquidum ascendet ad li-
quidi superficiem; situm in ipsa superficie superuatabit;
eritque portio demersa V ad volumen integrum V' ut pf: p..
Iunatantis solidi aequilibrium requirit insuper utin eadem
recta verticali inveniantur gravitatis centra ipsius solidi et
liquidi quod expellitur.
ltaque positio aequilibrii quoad solidum homogeneum
liquido insidens determinabitur si plano ita secetur soli-
dum, ut et alterius segmeuti volumen sit ad solidi volu-
men iu data ratione an., et haec volumina habeant sua
gravitatis centra in eadem recta, quae normaliter insistat
plano secanti: rem declaramus exemplo. Determinauda sit
positio aequilibrii in prismate recto ac triangulari, quod
ita demergitur ut et eius bases maneant verticales, et u- ⋅
na ex tribus faciebus v. g. BC ( Fig 48) exsistat tota ex—
tra liquidum.
Quisque videt directionem plani secantis nou pende-
re a mutua basium distantia, satisque esse ut determine-
tur intersectio DE illius plani et baseos v. g. ABC. Exhi-
beant a', a&quot; latera AB. AC dati trianguli ABC, et m', a)&quot;
latera incognita AD, AE trianguli ADE :triangulares areae
ABC, ADE exprimeutur per
∙∙⋅∙ äaa smA I &quot;-
, ämæstu A.
Sed area ABC est ad aream ADE ut integrum prisma ad
prismaticam portionem demersam: igitur
P:.
in' d'siu A: ;a' a&quot; sin A∶∶∶ [1]: .n., 'n' a&quot;:—-—a'a&quot; (k) .193

pla AM
Ž
AH '
Nunc secto bifariam in H latere BC, ducatur AH; sum
2
3
AH , centrum gravitatis trianguli ABC e
rit in M: simili modo, secto bifariam in H ' latere DE, sum
2
ptaque AN = AH', erit N centrum gravitatis trianguli 3
AM AN
ADE. Quia igitur ideo MN et HH' erunt АН
inter se parallelae: sed in casu aequilibrii recta MN, jun
gens gravitatis centra M et N , est perpendicularis rectae
DE ; ergo et HH' erit perpendicularis ipsi DE . Hinc DH=
HE: vicissim si DH =HE, erit HH' ac proinde MN per
pendicularis rectae DE; conditio nimirum necessaria ac
sufficiens ut recta jungens gravitatis centra M et N sit per
pendicularis rectae DE redigetur ad mutuam aequalitatem
rectarum DH, HE. Quibus positis , denotent B, Borangulos
DAH, BAH, et b rectam AH; triangula ADH, AHE dabunt
DA’ = w2762—2wbcos B ,HE' = w &quot; 2 + 62—20 &quot; bcoss *:
propterea
w2 -2bw' cos B = &quot;? - 26w &quot; cos \beta &quot; (k' ) .
Ex duabus ( k) et ( k ' ) eruentur a eta' , uude innotescit
positio intersectionis DE.
Quod si determinanda esset positio aequilibrii in eo
dem prismate quum ita demergitur ut puncta B et C ma
neant infra liquidi superficiem DE, foret
area BCDE : aream ABC = h' : u ': fb,
ideoque
ABC - BCDE ( ADE ): ABC Hope': fl , seu
−−∙≔∎⊾↼−−⇀
193
' Nune secto bifariam in H latere BC, ducatur AH; sum-
pta AM:
∙⋛−⋅ AH, centrum gravitatis trianguli ABC e-
rit in M: simili modo, secto bifariam in H' latere DE, sum-
ptaque AN:
∙−−≣−− AH', erit N centrum gravitatis trianguli
ADE. Quia igitur illi: −∙∙ 23,
inter se parallelae: sed in casu aequilibrii recta MN,iuu- ⋅
gens gravitatis centra M et N , est perpendicularis rectae
DE; ergo et HH' erit perpendicularis ipsi DE. Hinc DEI:-.'
HE: vicissim si DH :HE, erit HH' ac proinde MN per-
pendicularis rectae DE; conditio nimirum necessaria ac
sullicieus ut recta iungens gravitatis centra M et N sit per-
pendicularis rectae DE redigatur ad mutuam aequalitatem
rectarum DH, HE. Quibus positis, denotent B', B&quot;angulos
DAH, BAH, et brectam AH; triangula ADH,AHE dabunt
,
ideo MN et HH' erunt
BB': 'i—l-b' —29'6 cos B', B—Ea ⇌∾∣⋅≖−⊢ &amp;&quot; —20&quot;bcosB&quot;:
propterea
a)&quot; ---260' cos B': si&quot;! -
266)&quot; 'cos B&quot; (k' ).
Ex duabus (I:) et (k') erucutur a' et et&quot;, unde innotescit
positio intersectionis DE.
Quod si determinanda esset positio aequilibrii in eo-
dem prismate quum ita demergitur ut puncta B et C ma-
neant infra liquidi superficiem DE, foret
area BCDE: aream ABC −∙∶−− pl: p.': p.,
ideoque
ABC −∙− BCDE (: ADE ): ABC :p.- p.': p., seu194
Ww&quot; sin A : 1 a'a&quot; sin A = M - pe : plo
et consequenter
s'avº = ( 1- )« a”(k&quot;).
Ad haec : centrum gravitatis trianguli ABC invenilor
in recta jungente centra gravitatis portionum ADE ac BCDE.
Sed ABC et BCDE habent sua gravitatis centra in recta
perpendiculariter insistente rectae DE : adhuc igitur MN
erit perpendicularis ipsi DE ; rursusque prodibit (k' ) : eru
entur videlicet in casu w' et w &quot; ex binis ( k' ) et (k &quot; ) .

90. Determinata aequilibrii positione, restat videndum utrum aequilibrium sit stabile nec ne. Pone v. gr. innatans solidum esse tale, ut secari possit plano verticali AB ( Fig. 49. ) in duas partes omnino symmetricas tum quoad formam, tum quoad densitatem, et in casu aequilibrii sit HK intersectio plani AB et horizontalis plani repraesentantis superficiem liquidi: gravitatis centra M et N innatantis solidi et ejecti liquidi invenientur ambo in plano AB super eadem verticali CD; si solidum est homogeneum exsistet N subter M; si heterogeneum, poterit M esse vel subter N vel supra. Fac ut aliquantulo revolvatur solidum circa axem perpendicularem plano AB, sicque removeatur ab aequilibrii positione; ita tamen ut, exhibente H'K ' (Fig. 50) novam intersectionem plani AB et horizontalis plani repraesentantis superficiem liquidi, segmentum solidi respondens angulo K i K' aequetur constanter segmento quod respondet angulo H i H' ; hoc pacto haud variato ejecti liquidi volumine, permanebit ( 89.30. )
gV'p ' = gVd : proinde solidum absque initiali velocitate sibi
commissum movebitur ( 84 ) circa centrum M immotum.
Jam si ex puncto N' , ubi , amoto solido ab aequilibrii
positione , situm est gravitatis centram liquidi expulsi , du
194
&amp; o'o'f aiu A:) a'a&quot; sin A:p—p:p., /
et consequenter
Ad haec :centrum gravitatis trianguli ABC invenitur
in recta iungente centra gravitatis porticuum ADE ac BCDE.
Sed ABC et BCDE habent sua gravitatis centra in recta
perpendiculariter insistente rectae DE: adhuc&quot; igitur MN
erit perpendicularis ipsi DE; rursusque prodibit (k') :eru-
entur videlicet in casu a' et a)&quot; ex binis (k') et (Is&quot;) . ducatur verticalis recta N'R occurrens rectae CD in R , oc
cursus iste vel fiet supra M , vel infra , vel in ipso M :
in primo casu vis g Vlagens sursum juxta N'R manife
ste nitetur ut CD resumat verticalem positionem, et conse
quenter aequilibrium erit stabile ; in secundo ipsa gVp.
nitetur ut CD magis recedat a verticali positione , ideoque
aequilibrium instabile ; in tertio aequilibrium adhuc ob
tinebit quoad novam positionem .

=== De gravium liquidorum aequilibrio in vasis communicantibus. ===
[[Fasciculus:Communicating vessels.svg|thumb]]
91. Vasa communicantia dicuntur illa, quae ita sunt inter se conjuncta ut ex altero in alterum pateat aditus fluido. In altero contineatur fluidum homogeneum, cujus densitas &lt;math&gt;\mu&lt;/math&gt;; in altero fluidum pariler homogeneum cujus densitas &lt;math&gt;\mu'&lt;/math&gt;; siatque &lt;math&gt;z&lt;/math&gt; et &lt;math&gt;z+ z'&lt;/math&gt; distantiae inter punctum quodvis superficiei communis utrique fluido ac extimas fluidorum superficies. Fluidis se mutuo librantibus, exsistet (88)

&lt;math&gt;g\mu z + C_1 = g\mu ( z + z' ) +C_2.&lt;/math&gt;
[[Fasciculus:11 hidrostatica de 61 a 70.jpg|thumb]]
92. Haec facile nunc stabiliuntur. 

1.º Si vasis communicantibus idem continetur liquidum, ut sit &lt;math&gt;\mu = \mu '&lt;/math&gt;, erit &lt;math&gt;g \mu z = g \mu' z&lt;/math&gt; ideoque &lt;math&gt;z' = \frac{C_1 - C_2}{g \mu'};&lt;/math&gt; emerget ergo &lt;math&gt;z' = 0&lt;/math&gt; vel &lt;math&gt;z' &gt; 0&lt;/math&gt;, prout &lt;math&gt;C_1 = C_2 &lt;/math&gt;vel &lt;math&gt;C_1 &gt; C_2&lt;/math&gt;: in ea videlicet qua sumus hypothesi liquidum sub externis aequalibusque pressionibus manebit in utroque vase aeque altum, sub externis vero inaequalibusque pressionibus altias apud eam partem assurget ubi minor exercetur pressio. Inde profluit explicatio variorum effectuum; cujusmodi sunt hydrargyrum in barometro suspensum, aqua elevata in siphone, in antliis etc.... Sic v. gr. quoad antlias adspirantes, dum attollitur embolus ex &lt;math&gt;H'H''&lt;/math&gt; in &lt;math&gt;HI&lt;/math&gt; (Fig. 51), aer in tubo &lt;math&gt;HB'&lt;/math&gt; confestim fit rarior, et consequenter externus aer densior aquam in receptaculo vel puteo contentam cogit in tubum ascendere usque ad altitudinem v. gr. &lt;math&gt;A' B'&lt;/math&gt;: quam ob causam descendet aqua in receptaculo ab &lt;math&gt;AE&lt;/math&gt; in &lt;math&gt;ii'&lt;/math&gt;. Jam datis &lt;math&gt;H'Q ( = a ')., EQ ( = a '' ) , HH' ( = b) ,&lt;/math&gt;itemque horizontalibus receptaculi, ac tuborum &lt;math&gt;BQFD', FQA'B'&lt;/math&gt; sectionibus &lt;math&gt;\omega, \omega' , \omega ''&lt;/math&gt;, si debeat inveniri altitudo &lt;math&gt;AA'&lt;/math&gt;, pone &lt;math&gt;AA' = \beta&lt;/math&gt; et &lt;math&gt;Ai = \beta'&lt;/math&gt;: densitates aeris interni ante et post emboli elationem sunt in ratione reciproca voluminum &lt;math display=&quot;block&quot;&gt;a'\omega' + a'' \omega'' , a'\omega' + a'' \omega'' + b\omega' - \beta \omega'';&lt;/math&gt;ideoque (87. 6º) in eadem ratione erunt pressiones a et ;
hinc
( a'w ' ta'w ') a
(a' +6) + (a&quot; – 3 ) cs&quot; |
designante m aquae densitatem , aqua elevata supra ii ' exer
cebit ( 88) pressionem
a = gm (B + B ) .
Cum igitor a' to = 5W , cumque Bw &quot;' = f'w , iccirco
( a'w' + aa'') as''
(a + b ) w + la &quot; -B, w 

sia ponitur parvitatis contemnendae prae w , erit
( a'w' ta'a ') as tgms = a .
( a ' + 6) + ( a &quot; -B) w &quot;
196
sunt hydrargyrum iu barometro suspensum , aqua elevata
in siphone ,
in antliis etc.... Sic v. gr. quoad antlias ad- ∙
spirantes , dum attollitur embolus ex H'H&quot; iu Hl (Fig.51.),
aer in tubo HB' confestim Et rarior , et consequenter ex-
ternus aer densior aquam in receptaculo velputeo conten-
tam cogit iu tubum ascendere usque ad altitudinem v. gr.
A' B' : quam ob causam descendet aqua in receptaculo ab
AE in ii'. Jam datis H'Q (: a')... EQ (: a&quot;)
, HH'(:—...- 6),
itemque horizontalibus receptaculi , ac tuberum BQFD' ,
FQA'B' sectionibus a), m', ei&quot;, si debeat inveniri altitudo AA',
pone AA':B et Ai:B' :densitates aeris interni ante et
post emboli elationem sunt in ratione reciproca voluminum
a'm' −↿− a&quot;a)&quot;
, a'æ' a&quot;o)&quot; −∣⋅− ∂∾⋅−Ba)&quot; ;
ideoque (87. 60.) in eadem ratione erunt pressiones a et se' ;
hinc
(a'ai' −⋅∣− d'un&quot;) ur −⇀⋅ (a'-1-b)m'-1-(a&quot;—B)m&quot;
designante m aquae densitatem ,,aqua elevata supra ii' exer-
cebit (88) pressionem
0&quot;: sm (49 ⊣− B')-
Cum igitur a' -l-—a&quot; :0, cumque Ba)&quot;:B'm ,
iccirco
[
U'
(a'æ' ⊣∙⋅ d'ai&quot;) ar −⊢⊣−⊣−⊰⋯≺↿⊣−∾−↜∶≻∣∃⇌≔⇌
si a)&quot; ponitur parvitatis contemnendae prae a)
, erit
(a'æ' −⋅⊢ J'ai&quot;) :: l ∙−−
(a'-l—b) ∾∣∙∙⊢ (avl—þ) 0)&quot; l gmB—a-197
la eadem hypothesi , post iteratos descensus atque ascen
sus , restituto embolo ab altitudine minima H'H ' ad maxi
mam HI , pertingat aqua ad inferiorem superficiem mem
branae G ; descendente rursus embolo et denotante k alti
tudinem aquae de se librantis atmosphaericam pressionem ,
elevabitur membrana D , ac proinde rarescente adhuc aere
in consequenti emboli ascensu , aqua pergei assurgere quo
tiescumque fuerit
EQ . HQ &lt; k (HH') .
Ut enim elevetur membrana D , debei elasticitas k' aeris
HF sic augeri quum aer HF traducitur ad volumen H'F , ut
elasticitas k' ' densati aeris H'F superet elasticitatem k aeris
atmosphaerici embolo superincumbentis . Est aulem ( 87.1º . ) .
k : k &quot; = HQ : HQ ;
vis insuper elastica k' unita ponderi aquae suspens ae EQ
librat pressionem aeris atmosphaerici , nimirum
h' + EQ = k ;
et consequenter
k &quot;
k' (HQ)
H'Q
(k- EQ) (HQ)
H'Q
Igitur
( k — EQ) ( HQ)
&gt; k ; ac proinde etc. ... H'Q

2. Tubus cylindricus longitudinis h , et in una
sui extremitate clausus , impleatur hydrargyro usque ad
197
in eadem hypothesi , post iteratus descensus atque&quot; ascen-
sus , restituto embolo ab altitudine minima H'H&quot; ad maxi-
mam Hl , pertingat aqua ad inferiorem superficiem mem-
branae G; descendente rursus embolo et denotante ]: alti-
tudinem aquae de se librantis atmosphaericam pressionem ,
elevabitur membrana D, ac proinde rarescente adhuc aere
in consequenti emboli ascensu , aqua perget assurgere quo-
tiescumque fuerit
EQ
. HQ h(HH').
Ut enim elevetur membrana D , dabei elasticitas k' aeris
HF sic augeri quum aer HF traducitur ad volumen H'F , ut
elasticitas k&quot;densati aeris H'F superet elasticitatem k aeris
atmosphaerici embolo superiucumbentis .Est autem (87.10.).
k': k&quot;:H'Q :HQ ;
vis insuper elastica k' unita punderi aquae suspensae EQ
librat pressionem aeris atmosphaerici, nimirum
k&quot; -]— EQ:k ;
et consequenter
- k&quot; k' (HQ) −∙∙≺∣⊂− EQ) (HQ). ↼−− l'l'Q HQ
igitur
de −−⋅⋅ EQ) (&quot;Q)
H'Q k; ac proinde etc.
..

2.&quot; Tubus cylindricus longitudinis A, et in una
sui extremitate clausus ,
'impleatur hydrargyro usque ad198
altitudinem hoh , cum digito ad orificium in altera parte
exsistens apposito invertatur tubus ; et remolo digito , col
locetur hoc ipsum orificium in superficie hydrargyri sta
gnantis intra aliquod vas. Ascendet aer l' ad supremam in
versi tubi partem ; augescet h , et fiet = h &quot; . Jam vero ad
inveniendam h &quot; denotante k' altitadinem hydrargyri libran
tis atmosphaericam pressionem , satis erit animadvertere
h'ki
quod exhibet altitudinem hydrargyri librantis rarefa h &quot;
clum aerem h ' ; unde
hk
h - h '
+
To
k ;
ac propterea
h &quot; = h - k' = V Th — kj» + 4hºk
2
signum inferius non pertinet ad praesens problema .
lu formula ( 10)
C, C,
Sle'
Spkk
sunt C, = gu'k ' , C,
z'
'

3º. Pone ple , pe inaequales , et C, = C2 ; habebis
p.z = p ( = + z ) ,
unde
2 : 3+ = M ' : pe ;
diversorum nempe liquidorum altitudines z et ztz' in vasis
communicantibus erunt reciproce ut ipsorum liquidorum
densitates.
198
altitudinem h—h' ,
tum digito ad orificium in altera parte
exsistens apposito invertatur tubus ; et remoto digito , col-
locetur hoc ipsum orificium in superficie hydrargyri sta-
gnantis intra aliquod vas. Ascendet aer b' ad supremam iu-
versi tubi partem ; augescet h' ,
et fiet:h&quot;. Jam vero ad
inveniendam h&quot; denotante k' altitudinim hydrargyri libran-
tis atmos'phaericam pressionem , satis erit animadvertere
quod exhibet (i£—, altitudinem hydrargyri librantis rarefa-
ctum sereni I:&quot; ; unde
h—h&quot; ∙−⊢ h—Ij—L: k' ;
ac propterea
1.&quot; −∣∙ ∣⋅−∣⊏⋅∶⊨∣∕⇀≺∣≖−∣≂⊤≻⋅⊣−⊓≖⋅∣⊏⋮∙ , z
signum inferius non pertinet ad praesens problema.
lu formula (10)
sunt C, :gpjk' , Ca.-:. ∙−−−−

diversorum nempe liquidorum altitudines :
et &lt;math&gt;z+z'&lt;/math&gt; in vasis
communicantibus erunt reciproce ut ipsorum liquidorum densitates.

=== De gravium elasticorumque fluidorum aequilibrio; necnon de altitudinibus dimetiendis ope barometri, et de pondere ac densitate vaporum. ===
93. Binae (o&lt;sup&gt;iii&lt;/sup&gt; 87.), (o&lt;sup&gt;v&lt;/sup&gt; 88.) dant &lt;math display=&quot;block&quot;&gt;\frac{d\varpi}{ \varpi} = g\theta dz;&lt;/math&gt;binae (o&lt;sup&gt;iii&lt;/sup&gt;), (o&lt;sup&gt;iv&lt;/sup&gt;  87) praebent&lt;math display=&quot;block&quot;&gt;\theta=\frac{\mu}{\varpi}=\frac{1}{ i(1 + an )}&lt;/math&gt;propterea&lt;math display=&quot;block&quot;&gt;\frac{d\varpi}{ \varpi} = \frac{gdz}{ i(1 +an )} ( b ).&lt;/math&gt; Assumptis autem logarithmis quoad basim 10,&lt;math display=&quot;block&quot;&gt;d\log_{10}(\varpi)= \frac{d\varpi}{\varpi} \log_{10}[2,718281828 ] = 0 , 4342945 \frac{d\varpi}{\varpi}&lt;/math&gt;ideoque
dos dL(W)
0,4342945
Designante igitur pressionem apud punctum ( x , 0)
in hypothesi temperiei constantis formula (6) suppeditabit.
L - Llw ')
0,4342945 ( 6' ) .
i ( 1 + an )

94. Quoad punctum ( x , y ; '-— z) supra horizontale pla
num XOY ( Fig. 47 ) , aequatio ( 6' ) suppeditat
LULLG )
0.4342945
82
i (1tan)

et inde infertur valor z dimetiendae altitudinis supra XOY
sic expressus
i ( 1 + an )
L
0,4342945g ( 6 '') .''
Haec observentur: 1. ° sub temperie = 0 , et barometrica
hydrargyri elatione =2,33958 ped. apud geographicam lati
tudinem = 48° 50' 14 &quot;, ubi gravitas 30,1959 ped. ,
Biot et Arrago invenerunt densitatem hydrargyri esse ad
aeris densitatem po ut 10467 : 1 ; inde habemus respon
dentem pressionem ( 88 )
Ww=( 30,1959) ( 10467 floo ) ( 2,33958) ,
ideoque
wo -
( 30,1959 ) ( 10467) ( 2,33958)
ро
( 30,1959 ) ( 24488, 38386) =739448, 790198174.

2.o Eo minorem experimur temperiem , quo ma-
gis supra terrestrem superficiem assurgimus ,
at, igno&quot;-
mus qua lege liat ejusmodi imminutio; designantibus ?'
et ': temperies in intimo ac supremo puncto dimetiendae
altitudinis z, solet assumi
.- 'r'-l—r
'
&quot;: 2• 201
poniturque ista temperies media constanter vigere per to
tam 2.
1

3. Singulis gradibus imminutae temperiei respon
det hydrargyri condensatio = ; igitur si M et M
5550
exhibent densitates bydrargyri sub temperiebus t' ; ac to
DY in infimo ac supremo puncto altitudinis , erit
t'
M

: 1 = M' : M, unde M

5550 I' ,-1,
1
5550
rica
ati.
ed.,
e ad
00
Temperies hydrargyri tubo barometrico inclusi nonnisi
post aliquod tempus ad aequalitatem reducitur cum aeris
circumstantis temperie , hinc t'i et t, solent definiri sub
sidio thermometri , quod ad barometrum ipsum adnecti
tar ; aliae vero t ' et determinantur ope thermometri ,
quod cum barometro non communicat.

4.0 Si l' et h exprimunt barometricas altitudi
nes apud infimum et supremum punctum altitudinis
erunt ( 88 )
h' =gM'h , =
&amp;M'h
t',
1
5550
ideoque ma'
/
Jora
us ?
endae

: - * (

I' , - 1 ]
5550

5. ° experimentis pendulorum subsidio institutis
14
'201
∣∙ poniturque ista temperies media constanter vigere per to-
tam :. '
&amp;&quot; Singulis gradibus imminutae temperiei respon-
det hydrargyri condensatio: 5150; igitur si M' et M

; )

exhibent densitates bydrargyri sub temperiebus 'r', ac 't',
ll
⋅
in infimo ac supremo puncto altitudinis , erit
———- f.:—T! M' :1: ': ∙∙∙⋅ J
5550 M M, uudeM T',—Tx
'
5550&quot;
ric-a Temperies hydrargyri tubo -barometrico inclusi nonnisi
all' post aliquod tempus ad amnalitatem reducitur cum aeria
ed.. circumstantis temperie , hinc 'r', et 1.&quot;, isolent definiri sub-
sad
ron-
sidio tbermometri , quod ad barometrnm ipsum adnecti-
tur; aliae vero 1&quot; et ': determinantur ope tbermometri ,
quod cum barometro non communicat.

4.&quot; Si b' et lt exprimunt barometricas altitudi-
nes apud infimum et supremum punctum altitudinis z ,
erunt (88)
.: M'h ∙∣≖∙ ∙∙ g ∙
a *gM .m. fr.—ff: ,
∎∎− 5550
mr ideoque
nora-
0st a. ∙∙∙ h' 1 T.]- T; .»pdæ ⊺≖−−−∣≖ (&quot;5550)'
5.0 experimentis pendulorum subsidio institutis
14
- x'!
.202
probatum est , si gi est gravitas apud geographicam la
titudinem = 45 , apud aliam latitudinem å fore
g = g . (1-0,002589cos22 ) ;
erit igitur ( 1 )
30,1959 = g1 [1–0,002588 cos2 (48° 50'14'') ]''
ac proinde
30,1959 ( 1–0,002588 cos 22 )
1 -0,002588 cos2 (48 ° 50'14 &quot; )
.
6.° Quibus positis , formula ( 6 &quot; , 94) traducetur ad
24488,38( 1–0,002588cos2 [48° 50'14*]/ (1 +0,00395+7 )
X
1-0,002588cos22
L
CO
I '
1
5550
0,4342945
-) ]
ped.
e ,
formu 95. # Sumptis logarithmis quoad basim
la ( 6 &quot; , 94 ) evadet
i (1 + an ) .
,( );
upde
et consequenter ( 87. 7. ) H =
82
e iſitan )
202
probatum est
.
si g. est gravitas apud geographicum la-
titudinem;—4

5. ∘ apud aliam latitudinem ). fore
gzg, (1—0,002588cos2)t) :
erit igitur (10)
30,, 95gzg,[1—o,002588 cosz (48-50'1 4&quot;)1
ac proinde
∙− 30,1959 (1—0,002588 cos zx)
5 1-0,002588 cos2 (4so50'14&quot;) '
6.&quot; Quibus positis, formula (E&quot;, 94) traducatur ad
24488,38(1—0.002588cosz[4so50'14&quot;])(1'-1-o,003757 BH) s—
⇁⋅⊤
' - X '1—0,002588cos2').
h. r.!—T! )]
L I: &quot;( ↿−∎∎ 5550
o,4342945 pcd.
95-0 Sumptis logarithmis quoad basim e ,
formu-
la (6&quot;. 94 ) evadet

: i(1tan)L (jul-) ;

unde
a,
ex --
z
.
et consequenter (87. 7. ) p,:
e i(t-l—an)203
1
g?
i ( 1 + an) e il1 +an)
Denotent V'et i volumen et densitatem corporis aere
demersi , ipsoque aere specifice levioris : urgebitur corpus
ad verticalem ascensum vi acceleratrice
8 (V'4 — V'x ')
Vph Gelee Me gz
if1tan)
i ( 1 + an) e
Facile intelligimus , si denotat densitatem mediam glo
bi aereostatici , verticalem ascensum ipsius globi determi
natum iri per
daz
de2
8
)
(6 '' )''
.
i (1 + an ) e
i(1 + an )
Multiplica (6 &quot; ' ) per 2dz, et sume integralia; habebis ( 27. 12.9)
gz
dz2 i(1 + an)
dla с
2g role
re' f 8 + ks)
In hypothesi velocitatis initialis = o erunt simul z=0
20

o , ideoque C

Hinc do
dz et
20 82 dza
de ² (
1 e
i (1 + an ) — 2g2 (6 &quot;) .
hey 252 &quot;
203
I
3
gz
,
i (1—l—an) e i(1—l-an)
Denotent V' et pf volumen et densitatem corporis aere
demersi, ipsoque aere specifice levioris :urgebitur corpus
ad verticalem ascensum vi acceleratrice
V' —-V' ')
,
'a' , gt P
&quot;
—g,(p p.) g( —H)- VP ⊬ M sz
i (1-l—an) e
t(t-i-an)
Facile intelligimus, si denotat p! densitatem mediam glo-
bi aereostatici, verticalem ascensum ipsius globi determi-
natum iri per
I
daz g es'
dt: p.(
Multiplica (6&quot;') per 2dz, et sume integralis; habebis (27. 12.&quot;)
..
52
dzz— c zg a t(1—l—an) '] '
&amp;f— —F g
'&quot;')'
ln hypothesi velocitatis initialis :
0
erunt simul 220
(12. 20! et ⊼⋅−−−∶∘ ,
ideoque C..:
F.]iinc
d:: 25, ∙−− ...—gj— ⋅ '['
(7:3—?( 1 — 8 :
(l*'-an)) .'.2gz (6 ).204
cto
ex cujus integratione innotescet relatio inter z ac t . Fa
dez
=o, formula ( 6 ' ' ' ) suppeditabit altitudinem 2, apud dia
dz
quam exsistet f = M ; et facto = 0 , formula ( 6 &quot; ) praebe dt
bit maximam globi elationem z.

96. Quod pertinet ad pondus ac densitatem vaporum, haec subjungimus: 

1.0 Si vase undique clauso continetur satis liquidi, ut inde sese possit evolvere tantum vaporis, quantum postulat capacitas vasis, quantitas vaporis sese evolventis pertinget ad quoddam maximum unice pendens a vigente temperie: qua videlicet permanente, istud maxinium perstabit idem aut vas exsistat vacuum ab aere, aut aerem contineat, vel quodvis aliud gas ulcum
que densatum vel rarefactum: sive autem obtinuerit illud maximum, sive non, tantundem augebitur vis elastica sicci aeris, vel gas inclusi, quanta est elasticitas evoluti vaporis.

2.° Si vapor aqueus seorsum spectatus posset sub data temperie, quin ad liquidam formam redigeretur, eam librare pressionem ā, quam sub eadem temperie librat siccus aer, ex Gay-Lussac foret densitas té aquei vaporis ad sicci aeris densitatem / ut 10 : 16 , ideoque
M=
104
16

3.• Permanente temperie , fac ut aqueus vapor seor
sum consideratus libret reipsa pressionem Wri si vaporis
densitas vocatur Hiss erit ( 87 : 1. )
10

: @ = ht ' i theo unde pos = 16 Mo ;

et denotantibus P ac P, pondera aeris ac vaporis sub ae
quali volumine ,
204
ex cuius integratione innotescet relatio inter z ac :. Fa-
dzz . . . cto 27; ::o, formula (F&quot;) suppedttabtt altitudinem :, apud
quam exsistet p.:pl; et facto 5; :o, formula (ö&quot;) praehe-
bit maximam globi elationem :.

96. Quod pertinet ad pondus ac densitatem vaporum,
haec subjungimus: 1.&quot; Si vase undique clauso continetur
.satis liquidi, ut inde sese possit evolvere tantum vapo-
ris
, quantum postulat capacitas vasis , quantitas vaporis
sese evolventis pertingat ad quoddam maximum unice pen-
dens a vigente temperie :qua videlicet permanente
,
istud
maximum perstabit idem aut vas exsistat vacuum ab ae-
re, aut aerem contineat
, vel quodvis aliud gas utcum-
que densatum vel rarefactum : sive autem obtinuerit illud
maximum, sive non, tantundem augebitur vis elastica sicci
aeris, vel gas inclusi, quanta est elasticitas evoluti vaporis.
2.&quot; Si vapor aqueus seorsum spectatus posset sub da-
ta temperie , quin ad liquidam formam redigeretur , eam
librare pressionem 0, quam sub eadem temperie librat
siccus aer, ex Gay- Lussac foret densitas p! aquei va-
poris ad sicci aeris densitatemlp. ut 10 : 16, ideoque

4. • Nunc ex aqueo vapore librante pressionem ,
et ex aere sicco emergat volumen V aeris vaporosi librantis
pressionem , et habentis densitatem &amp; ; istiusmodi aeris
massa erit Vs; aer siccus in aere vaporoso contentus utpote
librans pressionem ( 1.9 ) a— , pollebit ( 87 : 1. ° ) den
( - )
sitate Quoniam igitur ( 39) vapor aequeus in
,
to
10 WI
16 W
aere vaporoso pariter contentus pollet densitate pi
propterea ad
Ve = y (0 ) tv
10 i
16 W
por
ICCI
ris
da unde
bra €
( ---+ -s)= (:-)
1 &quot;
sic v. gr. in ordine ad aerem maxime vaporosum sub
temperie =0 , et barometrica hydrargyri altitudine
2,33958 ped. , quoniam maxima pressio librata ab aqueo
vapore sub temperie = 0 respondet barometricae altitu
dini =0,015638 ped ., erunt ( 95. 1.° )
g = W = ( 10467No) ( 2,33958 )g , w = (10467 /lo) ( 0,015638 ) g;
ac proinde designante eo respondentem valorem €,
seor
pors

; 3

2,33958 0,015638
lo 8
Eo = Too
bi
Wo :)-- (*
2,33958
=0,997495po.
205
'
—Pp.,— 10 ut,
⊬∙⊬≖≔⊉∙⊅∎∙⊉∎ F 16.;—P.
,
∙ 2,33958 — 3- .0,015638 ⇌−⇀
−⋮⊥−∘⇠ −−∃↾− ).. 8
a'., ∘ a m ↼⊣∸∘ 2,33958

: o,997495p.o.

l—xu .206
Hinc
E. 0 , 997495 ;
Ho
ratio videlicet inter densitatem aeris maxime vaporosi et
densitatem sicci aeris in ea qua sumus temperiei ac pres
sionis hypothesi.

5. • Valor i jam inventus ( 94. 1. ° ) spectat ad ae
rem siccum ; quoad aerem v. gr. maxime vaporosum erit
T. .
0,997495 flo
( 30,1959 ) ( 10467 ) ( 2,33958 )
Eo 0,997495

6. Obiter notamus illud : aquam sub satis alta
praesertim temperie in vapores versam conari sese qua
quaversus incredibili vi expandere indubia evincunt expe
rimenta. Hinc usus aquei vaporis in movendis machinis :
certo quodam tuborum valvularumque artificio vapor ex
caldario introducitur in antliam , ita , ut antliae cavitates ,
alteram infra embolum , alteram supra embolum , vicissim
obtineat, vicissimque frigidae suffusione ad pristinam redeat
Jiquiditatis conditionem ; vapor inferiorem cavitatem obtinens,
attollit embolum ; superiorem, deprimit ; embolus adnexus
est alteri ex duabus cujuspiam vectis extremitatibus ; qui
vectis altera sui extremitate vel immediate vel instrumen.
torum apte conjunctorum subsidio motum communicat rotis ,
malleis , elc.... ; prout nempe importat machinae
movendae natura.

Hinc
∙⇣∘−−−∶ o,997495;
p..
ratio videlicet inter densitatem aeris maxime vaporosi et
densitatem sicci aeris in ea qua sumus temperiei ac pres-
sionis hypothesi. ,
'
5.&quot; Valor i iam iuventus (94. 1.&quot;) spectat ad ae-
rem siccum; quoad aerem v. gr. maxime vaporosum erit
↿≖∘∙∙ ar, —(3o,1959) (10467) (2.33958)
s, o,997495 ⊬∘ o,997495 '
i..—

=== De aqua egrediente per angustum foramen e vasis verticalibus sive cylindricis, sive prismaticis.===

97. Haec praemittimus ex pluries iteratis experimentis 
[[Fasciculus:Aqua egrediens.png|thumb]]
1.° Minuta corpuscula disseminata per descendentem aquam verticaliter descendunt commuui ad sensum velocitate usque ad horizontalem &lt;math&gt;HH'&lt;/math&gt; (Fig. 52), cujus distantia ab orificio &lt;math&gt;hh'&lt;/math&gt; aequat triplum radiurn ipsius &lt;math&gt;hh'&lt;/math&gt;; tum cursum flectentia, perque lineas curvas incedentia conspirant versus orificium. Aqueae igitur particulae verticaliter descendunt usque ad &lt;math&gt;HH'&lt;/math&gt;; formaturque ab &lt;math&gt;HH'&lt;/math&gt; ad &lt;math&gt;hh'&lt;/math&gt;conoides aquea &lt;math&gt;Hhh'H'&lt;/math&gt;, quiescentibus portiunculis lateralibus &lt;math&gt;B.B'&lt;/math&gt;.

2.° Adhuc obtinent et verticalis particularum descensus, et earum conspiratio ad formandam conoidem, etsi orificium aperitur in latere vasis.

3.° Aqua ex aperto orificio verticaliter saliens assurgit ad supremam fere prementis aquae superficiem.

98. Denotet &lt;math&gt;\omega&lt;/math&gt; velocitalem aquae egredientis ex orificio &lt;math&gt;hh'&lt;/math&gt;, et &lt;math&gt;z&lt;/math&gt; altitudinem prementis aquae supra orificium, erit proxime (30:31)&lt;math display=&quot;block&quot;&gt;
\omega=\sqrt{2gz}(k) . &lt;/math&gt;Ad haec; si &lt;math&gt;\alpha&lt;/math&gt; denotat horizontalem vasis basim, &lt;math&gt;a&lt;/math&gt; orificium &lt;math&gt;hh'&lt;/math&gt;, &lt;math&gt;v&lt;/math&gt; velocitatem particularum ex quibus coalescit suprema aquae superficies, erit &lt;math&gt;
\alpha v dt= a.\omega dt,&lt;/math&gt; unde &lt;math&gt;w= vi&lt;/math&gt; et facta asna,
a
imen
roting
w = nv ( k' ) .
Hinc (27 )
asis
dz ndy
do 2gz et dt
V28%
ideoque designante 2, initialem valorem » , quum nempe t = 0 ,
inted
ft?
ae-
erit
talil
men&quot;
aul?&quot;
207

98. Denotet &amp;) velocitatem aquae egredientis ex ori-
ficio hls', et : altitudinem prementis aquae supra orifi-
cium , erit proxime (30 :31 )
a) −−∶ Vig.; (k).
Ad haec :
si a denotat horizontalem vasis basim , a ori-
licium hh' , :: velocitatem particularum , ex quibus coa-
lescit suprema aquae superficies, erit
ac «.vdtzamdt, unde a): −⇀ v :
et facta «scita,
a
mzn-v (k').
Hinc (27)
ds ⇂∕ nds
— :: ∙−−−∶ 2 :∙∙∙ ∙ dt ga ,
et dt V—zgs .
ideoque designante s., initialem valorem :, quum nem-
Pe ∁−−−−∘⇟208
i - V7(
..- , ) ( &quot; ) .

99. Sit \beta volumen aquae tempore t egredientis ex
orificio a ; erit ( 98. k . k &quot; )
233= a.orde=a(28)* . * de= a/ 2018 ( 3 - V . Jde
Propterea
B =a/25)*(*.* -VERSI-)
( k' ' ) .

100. Assumpta z = o in ( k &quot;. 98 ) , prodibit tempus
O , quo vas lotum evacuatur ; nimirum
11/ 를
2n 0—
V 29
( k &quot; ) .
In ( k &quot;) et ( K '') substitae valorem molè ex ( k ' ) ; habebis''
2n 을
21
5 B
ag
V28
2n (25–2-ce). ( &quot;) .

101. Ex (k &quot; ) sequitur illud : si duo vasa habuerint
et altitudines zo , zo, el orificia a , a' aequalia , tempo
ra 0 , 0 quibus deplentur , erunt in ratione basium a,a' ,
siquidem
2n 2n '
á
0 : 0 =
V 29

: z ' .

V 28
--- N : n ' = . Q : a' :
a '
208
≖−−−−⇁ 21( soi—1 ii) (li&quot;)
-
l/Zg
99. Sit þ volumen aquae tempore :egredientis ex
orificio a ; erit (98. I:. k&quot;)
.l. s
'
s i— dþzaüdtza (25? s
'dt:a(2g)ir&quot; (
zog— &amp;
t
)dt. ⇂

Propterea
,

100. Assumpta ≖∙∶−−− ∘
in (Is&quot;. 98 )
. prodibit tempus
9 , quo vas totum evacuatur; nimirum
9: 2: ∣∙∘⋚ (z.-&quot;) ⋅
l/Zg
In (It-&quot;) et (I.-&quot;') substitue valorem s.,ïli ex (Is&quot;); habebis

:: 6— 2n 3,- ag( .

⋅−
2-
. — ⋅ 20 t): wg I.&quot;. (3 ,&quot; ( )
101. Ex (It-&quot;) sequitur illud: si duo vasa habuerint
et altitudines s.
, a',, et orificia a
.
a' aequalia. tempo-
ra 9. 9' quibus deplcntur ,
erunt in ratione basium a.d.
siquidem
∙ 2n 2n'

: 9:∶−− zo : ...z'o zn:n'— :—,-—a:a':

l/Zg a a· 209

102. Quantitates aquarum successivis et aequalibus tem
poribus effluentium decrescunt secundum numeros impa-
res ordine retrogrado sumptos. Assertio facile ostenditur
e secunda ( k ) , facto successive t=1,2,3,4, • ; nam
quantitates illae prodibunt expressae per
ag
2n (29-1 ) , L ( 49-4 ) – ( 29-11 , P.(69-9)– ( 49-4) ,
Se
ag ( 80-16)
2n
ag ( 60-9) ... , seu 2n
ag
2n (20-1 ) , 29–3 ) , (29-5 ) , (29–7), - ;
ideoque etc... Idipsum eruitur ex (k &quot; ) et ex prima (k&quot; ) ;
denotantibus enim 21 , 22, 23 , ... valores z respondentes tem
poribus 1 , 2, 3, ... eae praebebunt
&amp; 02 , 2, 3
(0-1) 2,225 2n? 2n2 , =-2,(0-2) », 23 =
S
(0-3 ) , ... 29-1 2n2
8
2n2 ;
unde
6
( 29-1 ) , 21-22 2n?
8
( 29-3 ) , Zz- 23 =
2n2
8
2n2 (26-5) , ... 29-3 8
2n?
et consequenter etc...
Hinc si dividendum sit vas in partes successivis dati tem
'209
102. Quantitates aquarum successivis et aequalibus tem--
poribus ellluentium .decrescunt secundum numeros impa-
res ordine retrogrado sumptos. Assertio facile ostenditur
e secunda (k')
.
facto successive t::1,2,3.4, .; nam
qnantitates illae prodibunt expressae per
Zn (29 'l), 2&quot; (49 4) 2&quot; (29 1) , 2n(69 9) g(49 4) ,
a—g - .. &quot;£ -
∙∙ 2&quot;(89 16) 2&quot; (69 9) . , seu
ag
- es -
es -
∙−−
7:091). 2&quot; (29 3), 2809 5)sa g(29- &quot;711&quot;;
ideoque etc... Idipsum eruitur ex (In&quot;) et exprime (k');
denotantibus enim sus,, &amp;
.... valores :respondentes tem-
poribus 1,2, 3, ... eae praebebunt
'
z.,:
⋚−⊯≖ ⊖⋅∙ z. ↼−− ⊋⋅⋚⇆≺⊖⋅↿ ):,
≖≖−−−∶⇄−⋚⊑≺⊖∙⊋≻≖∣ za⇌∎
- 2 -' ∙−−− i.— ↿∠∏−−≖⋅ i(ä 3) zo Zn' '
↴
uude ⋅
2, ∙z. ∙−−−⋮⋚≔ (29-1),z,-z, −−∶ Zif-;, (za-3), 22-33 −∙−−∙∸−
s- - ...g. .
ZI€3(29 5), ∙∙∙ Zo-x —-2na ,
et consequenter etc..
Hinc si dividendum sit vas in partes successivis dati tem-210
1 poris a unitatibus vacuandas , determinata altima 20-1
ceterae usque ad primanı erunt
320-7.526-4,72 6-7** (29-3 )z 0-10 ( 26-1 ) 0-1 .
1
1
Liquet autem fore
2:6-1 + 326-1 + 520.4 + 720-1 + . + 20-3)26-17
0
(29-1930_1 = {1 + 29-1)
o
2
0-1 = 622
6-41 1
d
.

103. Tria subjungimus, quae certissimis constant experimentis. 

1º. Vena aquae exilientis a foramine aperto in pertenui lamina magis semper contrahitur usque ad ejusmodi distantiam ab orificio, quae vix aequat ipsius orificii radium; estque venae maxime contractae area cc' ad orificii aream ut 5 : 8 circiter. Istius contractionis ratio ex eo desumenda videtur quod aqueae particulae etiam paullo extra vas retinent obliquos convergentesque motus, quibus orificium subierunt.

2.• Tanta effluit aqua intra datum tempus ex fo
ramine aperto in pertenui lamina , quantam suppeditat for
5
mula ( k &quot; ) , modo tamen pro a substituamus 8

3.º Aptatis orificio exterius tubis cylindricis, co
nicis etc., pro varietate tuborum variae habebuntur quan
titates aquae dato tempore exilientis.

104. Haec notentur 

1º. Acceleratio , per quam velocitas aquae admodum exigua usque ad HH' mutatur in finalein satisque grandem effluxus velocitatem, tota manifeste perficitur ab HH ad cc' intra spatium interceptum conoide ac vena contracta, ubi nempe descendentium stratorum amplitudines citissime decrescunt. Vas ergo ABB'A '
a.
210
poris 9 unitatibus vacuaudas
.
determinata ultima &quot;9-1 ,
ceterae usque ad primam erunt&quot;
3z9-1, 529-1,7z ⊖∙↿∙∙∙ (29-3): ∂∙↿ '
(29-1)z 9-1 ∙
Liquet autem fore
ze, ↿
∎∎⊢∍∅∂∙↿⊣−⋮≖∂ ∙↿ ⊣−⋅∄≖∂∙↿∙⊢∙∙∙↤⊋∂∙⊰≻∅∂∙↿−⊢
. 9 ∙∙∙ :
(29-1)z9-1:(1-1—29-1)ïz ⊖∙↿ —9 294 -
7

1 '.
er
perto
ejus
citci
ori.
spectandum erit tamquam terminatum tubo Hcc'll' ad se
ctionem HH ' aptato.

2.0 Motus aquae defluentis in regularibus alveis
traduci potest ad motum aquae prosilientis ex angustis
vasorum orificiis. Concipe regularem alveun secari plano
verticali ; in plano isto insculpi plura foramina , ex qui
bus effluat aqua ; iisque nova addi foramina ut indefinite
crescat foraminum numerus , totaquc sectio veluti unicum
efficiat foramen infinitis numero foraminibus' coalescens :
cum e singulis foraminibus aqua debeat effluere veloci
tate illa , qua et erumperet e vase ad eamdem altitudi
nem pleno , et , sublato plano , Queret in eodem sectionis
loco , idem ferme erit casus aquae defluentis per alveum
et aquae prosilieatis e vase ad eamdem altitudinem pleno.

3. • Si in regulari atque horizontali alveo mo
vetur inferior aqua ob superioris aquae pressionem , nec
directionum obliquitate , et fundi laterumque resistentia
turbatur conceptus motus , apud particulam quamvis de
notante i altitudinem superincumbentis aquae , exprimet
V 2gi particulae velocitatem.

4.° Quod si regularis alveus ad horizontem ex
sistat inclinatus , sitque m altitudo debita velocitati apud
supremam aquae superficiem , cum haec velocitas ( levio
ribus corporibus aquae injectis determinari potest ) utpote
orta ab inclinatione alvei debeat aquae omni esse
munis , exbibebit V 28 (i + m ) particulae velocitatem .

5.° Hinc poterit in utroque casu definiri quan
titas V aquarum intra datum tempus t defluentium apud
quamlibet regularis alvei sectionem ; sic v. gr. in hypo
thesi rectangularis sectionis habentis latitudinem r , erit
in primo casu
i
2tri.
V = tr įdi
3
no es
دالاق
uibus
fo
o for
com
S, CO
paano
relo
in 6
Dani
?ptom
Vžg
I stra
B!!
in secundo
lCP
perta
ejus-
filicii
, ori-
iot!
.aullo
uibui
⊊∣∝⊦
luan-
«de
new&quot;
ipua
! slfl'
BN
↗−
⋅ 211
spectandum erit tamquam terminatum tubo Hcc'll' ad se-
ctionem HH' aptato.
2.&quot; Motus aquae defluentis in regularibus alveis
traduci potest ad motum aquae prosilientis ex angustis
vasorum orificiis. Concipe regularem alveum secari plano
verticali .; in plano isto insculpi plura foramina , ex qui-
bus effluat aqua; iisque nova addi foramina ut indefinite
crescat foraminum numerus , totaque sectio veluti unicum
efficiat foramen inlinitis numero foraminibus' coalescens :
cum e singulis foraminibus aqua debeat eflluere veloci-
tate illa, qua et erumperet e vase ad eamdem altitudi-
nem pleno , et
, sublato plano ,
(lueret in eodem sectionis
loco, idem ferme erit casus aquae defluentis per alveum
et aquae prosilientis e vase ad eamdem altitudinem pleno.

105. Auctores non pauci tractantes de motu li
quidorum ex apertis luminibus effluentium , illud usorpare
solent tanquam principium , quod nempe unumquodque li
quidi in vase quolibet descendentis tenuissimum et hori
zontale stratum coalescat iisdem constanter particulis com
muni , eaque tantum verticali , velocitate donatis . Deno
tante v verticalem velocitalem , qua pollet in fine tempo
ris i quodvis massae liquidae punctum ( x, y, z ) sollicita
tum gravitate g , vis acceleratris de se valens producere
dy
actualem motum exprimetur ( 28) per : et qaoniam ,
dt
praecisis etiam mutuis punctorum pressionibns , adhuc ta du men vis de gigneret actualem motum ; ideo , attentis pres
sionibus , consistet in aequilibrio punctum (x , y , z) solli
du citatum vi g Propterea ( 88 )
dt
do
dz
dvi
dt (kº ) .
Attenta insuper liquidi continuitate ( liquidum ponitur in
capax compressionis ) ; sequitur , si A designat amplitudi
nem cujusvis strati horizontalis , fore ( 98)
viw = a : A , unde v =
Ä ( * &quot; &quot; ) ;
w est functio temporis t ; A distantiae ; ab XOY : sequi
212
∙∙∙ i. &amp; 3
VZU'l/ng (i-l—m) di: Z',..l/Zg g'[(10 m)⇣⇥≖∶∣⋅ a
denotat i., sectionis altitudinem .

1054: Auctores non pauci tractantes de motu li-
quidarum ex apertis luminibus ellluentium. illud usurpare
solent tanquam prineipium . quod nempe unumquodque li-
quidi in vase quolibet descendentis tenuissimum et hori-
zontale stratum coalescat iisdem constanter particulis com-
muni, eaque tantum verticali .
velocitate dona-tis . Deuo-
tante v verticalem velocitatem
, qua pollet in fine tempo-
ris : quodvis massae liquidae punctum (.r.-7, :) sollicita-
tum gravitate g. vis acceleratrix de se valens-producere
. dv ⋅
⋅ ∙
actualem motum exprimetur (28) per .d—t : et quoniam
.
praecisis etiam mutuis punctorum pressionibus , adhuc ta-
dv
men vis —d—£ gigneret actualem motu-m; ideo , attentis pres-
sionibus
.
consistet in aequilibrio 'punctum (.r.-y. :) solli-
citatnm vi g—⋛⋮ ∙ PrOpterea (88)
der . ⋅ dv
'
z,; ∙−−∶ P- (
−−⋅ (17) ('i ')
-
Atteuta insuper liquidi continuitate (liquidum ponitur in-
capax compressionis ) ; sequitur . si A designat amplitudi-
nem cuiusvis strati horizontalis .
fore (98)
psa-ca: A, lel). ,
undevzr-a— A (
cc est functio temporis :; A distantiae :ab XOï: sequi-213
1
i
tur quoque supremam descendentis liquidi superficiem ma
nere horizontalem . Ex kl( ) habemus
dv a da
a do
dt
aw dA dx
A2 dz dc
-
A dt A do
.
aw dA
a da a’w2 dA
A2 dz A di A3 da
iccirco formula (K ™ ) traducelur ad
do dz dz =+ (sds - au de):
1 sumptisque integralibus quoad % ,
==C+u(sma )
Zo
ic
exprimit 200 distantiam inter XOY et supremam liquidi
superficiem A , Denotante w, pressionem v . gr. atmosphae.
ricam in superficiem illam , assequimur
Two
-= C+4 (** 241,3.
)
unde C=0. – (
( 50-100)
.
li propterea
-=o +15(2-)-avenit SA- G --&gt;) ( 47 ).
Zu
je Apud orificium
213
tur quoque supremam descendentis liquidi superficiem ma-
nere horizontalem. Ex (F&quot;) habemus
dv a de.) am dA d: a dm
.dt—A dt A*dz dc-TA dc
as) ubi a da) ama dA ∙
Aza.&quot; Ad: A3 d.'
iccirco formula (Is&quot;) traducetur ad
' dar − das d:. am: dA
- )
⋅⊋−⋮∁≀∅−−∙↱∙≺⊰∠≀∅−−∅∙∣⊺⋮⊺−⊢ A3 d: dz ,
,. sumptisque integralibus quoad s,
⋍≖⇌∁−⊦⊬≺≊≴−∘≤≀≜∫≖≤≀⋮− −
.-)
,.
dt A 2112
zo
.i-
eXprimit s., distantiam inter XOï et supremam liquidi
superficiem A.,. Deuotante wo pressionem v. gr. atmosphae-
ricam in superficiem illam
, assequimur
2 2 2 2
≔∘−−−⋅∁−⊦⇤∸≼∊≴∘−≦⋏∘∶≕≻ ⋅
....
∁−−⇌≖≖∘
..
(g.... &quot;.? ):
[&quot; propterea
d ad:. 2
1 1
w:eod—Pgu-uþauä A a&quot; P:) (A*—. :) (k&quot;&quot;).
zo

; . Apud orificium214

1
Wo
A2 a
designantibus insuper b et i distantias ipsius orificii ab XOY et ab A. ,
m=b , 2 = b - i , 1 - % = i :
facto igitur
b
dz
A
biI !
erit ibi
mode,
gi - sa- (1-4 ) = (A &quot; .
Quoad (k &quot; &quot;&quot; ) et ( k &quot; ) notamus haec tria. 1.0# Si a est
parvitatis contemnendae , ex (k &quot; ) profluet
a = w.tugis mo ) ,
ut in casu liquidi aequilibrari (88) ; ex (k' ) vero emerget
V2gi ,
quae formula recidit in formulam ( k) .
2.0* Si , affluente novo liquido, eadem servatur
in vase altitudo liquoris , quantitates i, A., B exsistent con
stanles ac datae ; et facto
a2 1 h
A.2
↿ 1
≖⋝−−≖≖⋅∙ ∙ :::—:::;
designantibus insuper &amp; et t' distantias ipsius oriücii ab
XOT et ab A.,.
526 , sozb—t' .
s—sozi:
facto igitur
erit ibi
ad!» c.)&quot;( a2
g' Bdc 2.↿∎∎∎⊼∘⊑≻∶∘ (kl
Quoad (k&quot;&quot;) et (Is&quot;) n0tamus haec tria. ↿∙∘∙ Si a est
parvitatis contemnendae
, ex (k&quot;&quot;) profluet
Uzwoillg (z'—*o) )
ut in casu liquidi aequilibrati (88) ; ex ('tu) vero emerget
∾⋅⇌ vra-u
quae formula recidit in formulam' (lt).
2.0a Si
. affluente novo liquido. eadem servatur
in vase altitudo liquoris. quantitates i. A.,. B exsistent con-
stantes ac datae ; et facto
emuli
a ad
Qiiia215
formula ( k &quot; ) praebebit
h
d
2a Bdt V
2gi
2ada
2gi - hwa hV
2gi
h2 .62
2gi
d h
h
d
a V
2gi
V
2gi
hv
2gil
1+
v
2gi
+

: ) h h
;

-
Vzgi
unde , sumptis bogarithmis quoad basim
a
Bta
log
hy
2gi
V 2gi + hw
V
2gi
ha
non additar constans et arbitraria quantitas utpote =0
siquidem tempori t =o respondet w =o. Ex ista aequatio
ne emergit
Bhty2gi
V2gi(1
a
h
1 + e
Bhiv
2gi
a
inferimus , elapso brevi quodam tempore t, fieri ad sensum
1
V 2gii
itemque
21 5
formula (li&quot;) praebebit
d—L
.,
Bdt— iuda −− 21. V? —
⋣∊∙−⋅∣⇂≖∾≖ hl/Zgi IP
1——c.)2
Zgi
d
a ita di;—00 —( ⇂∕2gi ∣
l/2gi ) h
'⋅⊾ ⋅
l/Z-g—l ↿∙∙∙ .b— 6)
,
⇂∕⇄∃−⋮⋅∾ Vzgi
unde
, sumptis bgarithmis quoad basim e ,
'Bt: ]: a— log iii—E? :
l/th' Vzgi −− hæ
non additur constans et arbitraria quantitas utpote ∙−−−∘ ,
siquidem tempori tzo respondet 6) 30. Ex ista aequatio-
ne emergit
⋅≖∃∣≖≀⇂∕⋝⋮⋜
&amp;
l/2-g-i1—e— ::
∎∎⇀ h B'm/223-
1—l—e— a
inferimus
. elapso'brevi quodam tempore :. fieri ad sensum
1 − −−−−−−↗↓−∎∕∑∊≀⋅⊰
itemque216
= w.tuzia - 2 .) – paga?i( 1 hot G1 - ),-- VE
3.•* Si vas consistit in verticali cylindro , vel pri
smate , A erit constans , et A.=A ; insuper
dz 7-zo 1
A A B
ic
zo

=== Aliquid subjungitur circa generalem theoriam motus corporum fluidorum. ===
106.* Velocitas v, qua pollet if fine temporis !
quodvis massae fluidae punctum ( x, y, z) sollicitatum (86)
vi acceleratrice Q , resolvatur in ternas v' , w &quot; , 1 &quot;&quot; coor
dinalis axibus Ox, OY, OZ parallelas ; erunt ( 29 ) dý ,
1
dy' ' vires iisdem axibus parallelae , in dt dt
quas resolvitur' vis acceleratrix q' valens de se produ
cere actualem motum. Quoniam , etsi praecisis pun
cloruni mutuis pressionibus , adhuc tamen gignit actualem
motum ; ideo , attentis pressionibus , consistet in aequi
librio punctum ( 2, y, z ) sollicitatum viribus X
, ) – , dv',
1
Y dy&quot; , 2 dy'&quot;'; ac proinde ( 86. o )
di
1
dt
de = - ( x – do ). -- ( v- à dv&quot; ) , )
de u ( 2-2 &quot; )..
( 6)
216 .
↴
a':' 1 ↿ a Vii—g'
∙
szo-l-pg(2'.—20) uia (Aa Ag) ,
VS—K ∙
T
∙
∶⊰∙∘∙ Si vas consistit in verticali cylindro. vel pri-
smate
, A erit constans, et A.,:zA; insuper

:

Zo
Aliquid subjungitur circa generalem theoriam motus
corporum [[ uidorum.

106: Velocitas 0, qua pollet i! fine temporis :
quodvis massae fluidae punctum ( æ, y, z) sollicitatum (86)
vi acceleratrice ?, resolvatur in ternas v'. 9&quot;
.
v'&quot; coor-
dinatis axibus OK, OT. OZ parallelas ; erunt (29) ;llg-dvl ,
,, 1 ∙∙∙ ∙ ∙∙ ∙ ∙
(Z— dv
, &amp;? dv vires iisdem axibus parallelae ,
tn
quas resolvitur vis acceleratrix ?' valens de se produ-
cere actualem motum. Quoniam (a' .
etsi praecisis pun-
ctorum mutuis pressionibus, adhuc tamen gignit actualem
motum; ideo , attentis pressionibus , consistet in aequi-
librio punctum (z, y, :
) sollicitatam viribus X — −↿− dv'.
dc
y— —dv&quot;,Z—- dc dv' ; ac promde (86. o)217
1
dt ,
dy
du
dx
axt
du' !
(du ?
Habitis v ', &quot;.0&quot;, pro functionibus variabilium x, y, z, t,
exsistent ( 27. 24.0)
dy' du dv du du' dx + dy +
dzt.
dix dz dc
dur dul
dy + dz +
dt,
dy dz de
du du dy't dy + dz + dt ,
dx dy dz de
du du=
dr
seu ,
ob dx v dt , dy udt , dz udt ( 27 ) ,
dv dú dv ' dun du'
dt ,
dx dy dz de
dur dvd dv du'a + dt ,
dy dz de
( 6 ).
dy'll dy dy &quot;
\dx dy dz
dat
di axt du.
du
dyt
dzt dz dt dy
dl ;
at
will +
leo
lesin
' du'
\dx vt
alt
- de
dy ut
21&quot; + .!&quot;
to
at
dt ,
ide
dx du du.
vt
de ede :
w
itot
dy
dz
formulaeque (6) vertentur in
15
1
1
217
Habitis v'. v&quot; . v'&quot;, p. pro functionibus variabilium x,]. z,t,
∙ exsistent (27. 243) '
/
dVr-äï-I-dæ-f-g dy—l—d 7; v/dz—l-dïvt-dth
.
I, I/
dp'p&quot;: &quot;
≤⋮∙−≤⋅⇗↙↙∠∞∙⋅∣− dv ∙−∙↙∣∫⊣−↙≀↥≟ dz-l— ii)—dt,
III '&quot;
dv'&quot;: dv Tdæ—i—djr dv/Il d-v'&quot; dy −⊸⊢−−⋅ zdz—l- -^——--dt,
d d ⊬−− ⊋⊥∸↙≢↕ ↙∄↕≤−⊦ Hari- ⋮⋮↙∄≖⊣−−↙−∣≛⋮∠≀≀⊰
seu , Ob dx:
'v' dt .ei)-':«:;&quot;dt, dz 37)/&quot;dt (2&quot;..
dv! ∣∙∙− d'&quot;, v&quot; ∣∣ (if—,) dv—(ïr-v-l-ï P-l-d—z-IVI-ï-dï— dt,
&quot; &quot;
vl] II
V&quot;'—:(d—; V, &quot;J—d gr.&quot;- .v/j-l—g-z- will-i-ïi'l;-—-)d[,
(V).
dv ∣−− dvlll dui/I
−−−−⊋⋤−≼ .
[v;/1.&quot;, dv'&quot;
,
∣∣
), 1
∎∎⊢∎∎∎−∎−∎ dz 'l&quot; dt —)dt ⋅∣⊹ '
∙−− dP'. ,v/ !
dlu' ut dp' ∣∣∣ dp') ∙
dy. (dæ'v [
dy&quot; ⋅−↱⋅⋅∓⇂≀ −↽⊋−∁⋅− dt.
formulaeque (6) vertentur in
15218
dos deild dy&quot;
dx
dv'
dx dy
dz che si (
(v
do ( 6&quot;) v'
do'
dy
v
du
dz w
dur
dy
dx de - ) ,
dy ')
do dv'

: -(2

v'
dy't
dy
v
du Win
dz dx dz de
107 #. Quae portiuncula infinitesima massae fluidae a
pud punctum ( x , 3 , 2 ) sub volumine V in fine tem
poris i exprimitur per V , eadem sub volumine V+dV
in fine temporis + dt ad punctam aliud translata expri
metur per ( V+dV ) ( pe + du ); ideoque
V = V + dV) (v + dpl)= Vu + udV + Vdp. + dp.dV ,
et consequenter, misso dudv,
Vdp. + pdV= ( 6 &quot; ).
Sumatur V = dxdydz, aequale nimirum parallelepipedo
rectangulo AF ( Fig. 47. ) sub laterculis AD( =dx) , AB( =
dy ) , AH = dz); punctaque A , B, C , D , H , M , F , E po
nantur transferri tempusculo de ad A ' , B , C , D , H' , M',
F' , E , ut sit V + DV = A'F'. Transferetur A in A '
velocitatibus d' , 0 , 2, juxta coordinatos axes ,
runtque
e
x + v'dt, y tv&quot; dt , z tudt
coordinatac puncti A': designatis v ', u ' &quot; per
d7 dx d] dz : '
,
dm' dv&quot;' −∙∙: Z- ∣- dv'&quot; du''' dv'&quot; ∣∣'''
∙∙∙ ∙−−⋁∣∣∣∙− —) ∙
dz F
(
da: v dy 'v dz dt
107-. Quae portiuncula infinitesima massae fluidae a-
pud punctum (æ
,
I,: ) sub volumine V in fine tem-
poris : exprimitur per VP-o eadem sub volumine V-l-dV
in fine temporis t −⊢ dt ad punctum aliud transl'ata expri-
metur per ( V-l—dV) ( p. dy. ); ideoque
Virsz-l-dV) (p.-l—dp.) ∙−−∶ ⋁∣↓∙⊹ ⊦∙∠≀∇−⊢ ∇⊂∣≴⊥∙⋅⊢ dde .
et consequenter, misso dpdV,
. Vdp. −⊦ ⊬↙∣∇−−∶⋄
('b'&quot;).
Sumatur Vzdædydz, aequale nimirum parallelepipedo
rectangulo AF (Fig. 47.) sub laterculis AD(-:-:dæ) , AB(-—--
dy ), AH(-:dz); punctaque A, B, C, D, H, M , F , E po-
nantur transferri tempusculo dt ad A' , B' , C' , D' , H' , M',
F', E' , ut sit V −↿− dV −−∶ A'F'. Transferetur A in A'
velocitatibus v'. a:&quot; ,
∙⇂∙∥⋅ juxta coordinatas axes , e-
runtque ∕∕∕ '
..: ⊣−⋁∣↙∣⊀ ∙
]−⊦ wa: , z −⊢ war:
coordinatae puncti A': designatis v', v&quot;, «a'/' per219
fi( x , y , %, t ) , fa(x , y , z , 1), 13 (x , y , z , t) ,
expriment
fi (x , y , z + d2, e) ,fz(x , y , z + dz, t ), f3( x , y, z + dz, t)
velocitates coordinatis axibas parallelas puncti H euntis in
H '; et cum babeamus ( 27. 24.)
filx9,2 + dz,t) = f (x , y ,z, e)7df1(x,y,z,e)dz = uti
du
dz dz ,
dz
e ao
em dy ”
fa (x , y, 2 + dz, t ) = 0&quot; +
dz ,
dz
spri.
f3( x , y , z + dz, t ) = 0 !!! allt dv !
dz,
dz
IV ,
coordinatae puncti H'erunt
X
+ (v + da )dt,y + (&quot;* + de )de,

: +de+ (** + adaptada dt:

pipedo
AB =
E po inferimus, missis infinitesimis tertii ordinis, fore ( 50. 6º. )
1 , M.
in 4
5 ,
ee A'H' =
[ledesdeu + )de de +
) ]=d =+
dy &quot;
-dz dt dz dt .
dz
Motus puncti Cin C'juxta coordinatos axes fiet velocitatibus
!
15.
C ∎∙
em-
[pl'l'
IV.
. 219
fuci-'s], 3! 1), fa(æs)'s 3! t) ∙ ⊀∍≺⋅↕∎∙∫↿≖∙ :),
expriment
ftlæsfaz-l'dzs 1) ,falæoys z 'l'dz; t)sf3(æs ïs ≖∙∙∣− d:, 1)
velocitates coordinatis axibus parallelas puncti Hieuntis in
H'; et cum habeamus (27. 240.)
dfx(æJ,z,t)d fax-a',: 4—dz,t):f.(æ,y,z, :
dz dz—v ↾−⊦↙↙∙⋚∙ —dz,
&quot;
falæsïsz (I:-',! :):vlf'i'd'ä- dz:
d'UIII fave,], z-l-dz,t ): ⇝∣∣∣−⊢ —zdz.
coordinatae puncti H' erunt ' &quot;'
ahi-(» ∣⊣− −↲≖≻≳≀∙∫−⊦≼↙∣⊣−≝∂≖⋟↙∦⋅

: ↽⊦ dz −⊦ (W.;- ↙⋛↙−≖

d: )dc.
inferimus, missis infinitesimis tertii ordinis, fore (50. 60-)
A'H': RSTV) dz-dz −⊦≺⋅⋮↷⋛↗−−−≖−∥≖⋟↙≀≖≏
d:: −⊦
d.,/II dv!&quot; -
(d:-[- d: dzdi )]ä :dz-F—dzdt-
Motus puncti Cm∁∣ juxta coordinatas axes fiet velocitatibus220
falar + dx , y + dy, zil ) = fi(x , y , 2,1 ) +
afı( 8•
7,5,6)
det dfi (x , y , 2,1) du
dy
dy
dic = tIdxt dx
falx + dx , y tdy, 2, 1 ) = &quot;&quot; + -dat dx
du &quot;
dy ,
dy
dumi
dy !!!
ON +
f3(x + dx ,y + dy , z , 1 ) = &quot;&quot; dat dx dy ;
dy
inde prodeunt coordinatae puncti C
du
d )
dy x + dx + (v + ad det )dt
y + dy + ( ** + na tempat day ) di,

: + ( v&quot; + data darym dy de :

motus puncti F in F ' juxta coordinatos axes fiet velocitatibus
du'
filxtdxy + dy,z + d2,2)= x + xdx + dydy +
du
dz,
dz
dy&quot; du&quot;
falxtdx,y + dy,atdz,t)= &quot; + de + dy dy
du &quot;
da dz,
du &quot;
du
f3( x + dx, y + dy; z + dz,t) = 1 &quot;'
-dxt
dy&quot;
dy dy +
dz;
dx dz
inde exsurgunt coordinatae puncti F
220
fdæ'i'dng—i-dy- ≖∙⋅↕⋟⇌∣≖≺∝⇟∫∙ ze t)&quot;,-
df,(æ.j,z,t) dfl(æsyszvt)
dw'd d &quot; ≀∂≖≺∙↿⊏−≱⊢↙↕↡∫↽⊢∂∫⋅ ≖↿−⊸⋅⋅⇂∙∥⊣−↽⊋− &quot;L &quot;ad; df '
f3(æ-l—dæ,y—]-dy,z, :
)-— v'∣∣⊹−⇁∙
inde prodeunt coordinatae puncti C'
∙↴⊲−⊢∠∄∸≀∶⊣− (⊣−⋅≦⋮∠↴↧⋅↕⊣⇀−− ↙∄⋤↙↿∫⋟≴↙≀⋅
∫∔∂⋮∫⊹ (
∣∣⊣↼ d,,⇡⋮≀−−⋅∶≴←⊦≤−⋚−∥⇩≀ wa.)
vll/ du ≖↽⊦≺⊛ ∣∣∣
w&quot;'-l--d—-æ dr—l— df )dz:
motus puncti Fm F'luxta coordinatas axes Eet velocitatibus
∙ . . . '
' d ,.
fia—W;? ∂∫∙≖−−∠⇣∙≀⋟∶⋁⊹⊼∶↙⊩⊦≣↗ −⋤−∶⊔⊹ −:dz, dv&quot; dv&quot; dv&quot;
ta(æ-i-er-l-dr, a—l—dzn): ⊎∣∣⊣⋅∙⋣∂≛−⊦ df dy ! dz ds,
d&quot; III d..&quot; f3(æ—]—dx, ⊹↙∄∙↗∙≖−⊢∣≂∙ ():—Ju&quot; i-i-d; dælL dr cir—]— —-dz;
inde exsurguut coordinatae puncti F'221
dyn
= da + ( + van de tener tous de Jdeo
s + d3 + ( v +adar an nas tudi nadia )dt,
s + de + ( * + dpt dathetn dy + advan die Jde:
1
inferimus, missis infinitesimis tertii ordinis, fore
CF = [ 'de de + oem )deº de +
( de + de &quot;de de ))]]* = da +
dy&quot;
de dt.
dz
Ad motum puncti B in B ', computatum in coordinatis axi
bus, spectant velocitates
f( x, y +dy, z, t ) , falx , yt dy, z, t), f3(x ,y + dy, z, t ) ;
ad con similem vero motum puncti M in M' velocitates tatibus
fi(x ,y + dy, z + dz, t) , 82(x , y + dy ,ztdz, t ) ,
th
dan
f3(x , y + dy , z + dz , t ) :
dy&quot;.
propterea coordinatae puncti B ’ desi
dz
+ (x + dy dy )de , y + dy + (.* + dar dy ) dt,
du&quot;,
dy
7
+ (*&quot;'+ dydy hdi:
7;
(221'
I
æ—l—dæ-l-(tb -]-d −∙⋮dx-j—d −−∣vlddy-l— ——dz )dt,
,,
'
dv&quot; . dv&quot; y-l-dJ-l-( −⊢−− da.−−∥↙≀↓⊣− ⊒∫−∠≀∫−⊢−− ↙≀≖≻∠≀∁∙
&quot; dv&quot;: z-l—ds-l—(&quot; ⊣−≦−≦⊥∅≀∝−⊦↙∄ dyd ∣
↙↙≖ d: )dt:
unferimus, missis infinitesimis tertii ordinis, fore
∙∙∙⋅ dv) ∙ '
(du&quot;ïd) ∙ es'—[(? d: a: &amp;
∠≀∥≀⋍⋅−⊦
≺∁≀≖−⊢⋛≖ ——dzdt )]; :dz—l-Q—ds dt.
d:.
Ad motnm puncti B in B', computatnm in coordinatis axi-
bus, spectant velocitates
I.i-ïs;)&quot; ⊣∙∙ dy. 39 i)
'
f2(æay—l— d]: 2. t)af3(-'rsy—l—dft Z, !) 3
ad consimilem vero motum &quot;puncti M in M' velocitates
↿∎≺∞∙∙↗↾⊣−∠∄∫∙≖⊣− dzs t) sf2(æ sy'l—fi'r, ≖−⊦∠∄≖ ∙ :) ∙
fam ,y-l-dy.a-1- a.:):
propterea coordinatae puncti B'
æ-i- (&quot;'-l- ⋛⋚∠∄∫≻↙∄∁ ,J—l-dy—l-('
≻≖≀⋅≂⋮ &quot;j,-l— 72:41)!&quot;-
z −∣⋅− (vm-I— dv
dy222
coordinatae puncti M
++ (1 - en deJdt,
y + dy + (** + disa dy + &quot;deJdi.
z + dz + (** +
(** + + en in diehele
hinc
B'M dz +
du
dzdt .
dz
Ad motum puncti D in D ', computatum in coordinatis a
xibus, pertinent velocitates
fi ( x + dx, y , z , 1 ) , fa( x + dx,y ,z, 1), f3(x + dx, y , z ,t ) ;
ad consimilem autem motum puncti E in E' velocitates
filx + dr, y, z + dz, t ) , f (x + dx, y , z + dz , t ) ,
f (x + dx ,y , z + dz , t ):
proinde coordinatae puncti D'
de&quot;

* + dx + (ut ea adx)de,y + ( * + dxdx )dt,

++ (- + de -dx)dici
222
coordinatae puncti M'
x-l—(tf— ⋛≶↙≀∫⊣− ——dz)dt,
maH-( ⇂≀⋅⋛−−⊦ pri—4449
≖−⊦↶≀≖↼⊦≼ ⋮⋅∠⋛⋮−⊣− MH-;'dzdu)
hinc
,B'M'-— ∙−− tis-l- -—-dzdt.
Ad motnm puncti D in D', computatum in coordinatis a-
xibus, pertinent velocitates
ru(æ ∙−⊦ ciæ,], zit)1fa(æ &quot;l'dæofszo t) sf3(x&quot;'i&quot;dæs)'; 2; 1);
ad consimilem autem motum puncti E in E' velocitates
fdx—i-dæqæz-l-dz, t ) 'fa(x-l-dx,y , z—l—dz .
t ).
B(æ-j-dæ ,y, s--]— dz .: ):.
proinde coordinatae puncti D'
∝⋅⊦⊄↿↕∸−⊦≼↩∙−⊦ ——dæ)dc ,y—l—(v' −⊦≤−−⋮⋅⊑⋅↙≀⋅⊐∁⋟↲↥∙

: ∙⋅⊢ ≺∙∽⋯∙⊢ ⋛⋮≽∙⋮⇣↿∙↕≻≺≀∷223

puncti autem E
drt
s + dx +((uv + des de +die dz)dt ,y+ (** + de la
de &quot;a )de, a (* &quot; + dz) dt;
2 + dz to dv&quot;&quot; dy&quot; ,
det da dz
et consequenter
D'E' = dz to
du&quot;.
dz dzdt .
Itaque
A'H ' = C'F' = B'M ' = D'E' = dz +
du 
dzdt :
đz
simili modo eruuntur
AD = B'C ' FM H'E di = dx +
dx
dxdt ,
1t ) ;
A'B' C'D FE H'M ' =
dú'
dy + dydt.
dy es
thi
Ex laterculorum aequalitate manifeste consequitur eorum
parallelismus ; eritque A'F' parallelepipedum obliquangu
lum ; ita tamen , ut ejus anguli infinities parum diffe
rant ab angulis rectis parallelepipedi rectanguli AF ; quan
doquidem AF nonnisi tempusculo infinitesimo transfer
tur in A'F ' . Nunc ex H ' v . gr. due perpendiculum Ha in
areolam A'B'C'D ; erit
A'F ' = H'a . A'B'C'D' = H'a . A'B ' . A'D' sio B'A'D '
A'H ' . A'B' . A'D' sin B'A'D' sip H'A'a :
d:.
223
puncti autem E'
dv' '
dv' )
.. da: −⊢ dz z .7' 41- dæ—t- ∙⋅∎∙⋅−⊢ du:-1- (v' ∙∙⊢ da:
dv&quot; dv'&quot; dv&quot;' )
−∙ dz —-d.r —-d d ;
dz)dt ,
z-t-dz-tï 'v. −⋅⊢ da: ∙−⊢ ds :
f
et consequenter
⋅ dv&quot;:
D'E'c: dz −⊢ 71&quot;—2. dzdt .
Itaque
llo
'
'v
A'H':C'F' −−∶ B'M' ∙−−∶ D'E' :: d: ⊣⋅− ∙−≀⋮⋅≖−↙≀⋍⊄∄↥ :
simili modo eruuntur
A'n' :: B'C' −−∶ F'M' −−∶ H'E':dæ −⊢↙≟⋛ dædt
.
A'B'r: C'D' ::F'E': H'M' ∙⋅−−−∸ dy ⊣⋅−∙≣⊥⋅ ↙≀∙↨↾∠∄≀⋅∙ ]
Ex latel-culorum aequalitate manifeste consequitur eorum
parallelismus; eritque A'F' parallelepipcdum obliquangu—
lum; ita tamen , ut eius anguli infinities parum diffe-
rant ab angulis rectis parallelepipedi rectanguli AF ; quan-
doquidem AF nonnisi tempusculo inünitesimo transfer-
tur in A'F' . Nunc ex H' v. gr. due perpendiculum H'a in
areolam A'B'C'D' ; erit ∙ ∼
A'F' ∙−−− H'a . A'B'C'D' ::
'
'a . A'B' . A'D' sin B'A'D' ∶−−⋅≖
A'H' . A'B' . A'D' sin B'A'D' sin H'A'a :224
denotantibus w et w'angulos infinitesimos , poterunt anguli
B'A'D ' , H'A'a repraesentari per 90º + w , 90 ° +6 ; iccirco
sin B'A'D' sin H'A'a = sin (90 ° + w ) sin ( 90° + W' ) =
62 614 w'4 coswcos6= ( 1 -... ) ( 1 ...) . 2 2.3.4 2 2.3.4
Quare , missis infinitesimis quinti ordinis ,
dv &quot; A'F ' = (dx +
dv' du &quot;
dxdt) (dy +
dx dzdt) x
dy dydı) (dz + az
wa
du
( - -
(1-7 • det dy&quot;
• det du&quot;
dt) dxdydz ;
2 dy
dz
ideoque
dV = AF - V =
-
(dv
dx- det
-det
dy
di) dxdydz.
His positis , vertelur ( 6 '' ) in''
dxdydzdje ele
dv
\dx
dtot dt +
dv&quot;
dy
dz di )dxdyds= 0,
seu ( 106.6' )
dje du ut u'+
dr dz
djelo
du +
dy
dt
dvi dv&quot; du &quot; .
dy + demon dz ) =
(619) .
dx

108. * Si massa fluida est incapax compressionis,
unaquaeque particula immutabilem habebit densitatem
eritque du = o : proinde ( 106 , 6')
224
denotantibus eo et tu'-angulos inünitesimos , poterunt anguli
B'A'D' , H'A'a repraesentari per 900-l-cu , 900-l-co'; iccirco
sin B'A'D' sin H'A'a −−∶⋅ sin (90&quot;-FG) sin (900 ⊣−∙ w') ∶∸⋅
. ↿ a): 034 ↿ tu' te'-': )
cosmcosw—( ∙−⋅⋍−−⊢≳∙∙⋝⋅∕⇂∙−−⋯⋟≺ −−∙⋮∙ m—m .
Quare , missis infinitesimis quinti ordinis
,
I d'u' dr&quot; d'v &quot;
'
'
∙−−−
− −
'
AF ∙−− ∙−− (dx—t- dædxdt,(dy-t—- d] afydt) (dz-t- az dzdr) )(
a 'a .'
&quot; '&quot; (1 &quot;2' −−∘≩≻∙−− ≺↿⊹⋛⊰↲↥⊹≘⋚∂∁⊹↙≩⊤∶∂∊≻∂∅∂∫∂∥
ideoque
(lv-.:A'F' v
(
vd: : dv dc.-92:11) dædyde .
dæ dy dz
His positis , vertetur (b&quot;') in
dt&quot; I),, vl'l dæd.) dzdp-t— &quot;(c'ïx dc ⊣−∙ 217 dc ⊣−∙ 72- dt)dædydz ::o,
seu (106 . b')
&quot;&quot;
↿≀∙⊣⋅− d&quot; ∣∣⊣− a'&quot; &quot;-4-'-'—'-' −⊢

: 17- 2?&quot; f??&quot; dt

d'v- ⋅⋅⊢ dv&quot; dv'&quot; ⋅
⋅⋅∙∙− o
F- b,; ( ⊣⋅−⋮⋤ )
—- ( ).
108.s Si massa fluida est incapax compressionis,
unaquaeque particula immutabilem liabebit densitatem ,
eritque dy.:o : proinde (106
. b')225
de
vt
die du du v &quot; + 2 &quot; +
=
0

;

dy dx dz de
et consequenter
( 107.b
)
(
6
)
dv dy
d.x
+
+
dy dz
Formulae
(
6
&quot;
)
,
(69) suppeditant incognitas
a
,
l
,
v
,
v
&quot;,
v
&quot;
.
expressas per
x
,
y
,
z
,
t

; obtentis autem

v
'
,
v
&quot;
,
?,
&quot; per xy
, zat
,
eruentur
x
,
y
,
z per
t ex formulis
dy dx dz
dc
-
”
dc
ru!!!
dt
Si massa fluida incapas compressionis est insuper ho
mogenea
, prima
(
6
&quot;
) fiet idemtica
, satisque erunt
(
6
&quot;
) et
secunda
(
6
&quot;
)
.ad incognitas
,
u
'
,
v
&quot;
v
'&quot; determinandas
. De
mum si massa fluida pollet elasticitate
, formalis
(
6
&quot;
) et
(61
)
jungenda erit formula
(
o
&quot;
. 87.6
'
)
.

=== De tubis capillaribus. ===
[[Fasciculus:Capillarity.svg|thumb|Capillares]]
109. Etsi liquidum homogeneam in vasis communicantibus (92.1º.) manet aeque altum, iu tubis tamen vitreis admodum angustis (dicuntur capillares) utrinque apertis, et altera extremitate demersis aquae vel hydrargyro, cernimus aquam suprema superficie concava terminatam ascendere supra horizoutalem circumambientis liquidi superficiem, hydrargyrum vero suprema superficie convexa terminatum descendere infra horizontalem circumdantis liquidi superficiem: ad istius modi ascensum descensumque explicandum, haec animadvertimus.

1º. ln phaenomenis gravium liquidorum expendendis gravitatem considerantes haud habuimus rationem sive virium quibus liquidi particulae se mutuo petunt, sive virium quibus vasorum materies ad se trahit particulas illas. Porro materiales particulae duplici pollent vi attractiva; altera se prodit utcumque crescant distantiae, sequiturque (82) rationem reciprocam duplicatam distantiarum; altera se prodit dumtaxat in contactu vel quamproxime contactum, sequiturque rationem quamdam distantiarum nondum compertam. Ubi sermo est de liquorum aequilibrio, possumus ab attractione primi generis absque sensibili errore praescindere: ad attractionem secundi generis quod pertinet; cum in contactu exsistat validissima, inde fit ut suprema liquidi superficies prope vasorum latera induat figuram curvam, modo concavarn, modo convexam, et nonnisi ad aliquam ab ipsis lateribus distantiam dici queat physice horizontalis. Exhibeat TT' (Fig. 53) verticalem tubum v. gr. vitreum, utrinque apertum, et infra horizontalem liquidi superficiem partim demersum; O centrum circularis areae tubo interceptae apud eam superficiem; A particulam liquidi in area ista sub actionem vilreae particulae R; OX rectam transeuntem per A; OY horizontalem rectam perpendiculariter insistentem rectae OX; OZ verticalem rectam. Si denotat vim qua A tendit in R, designatis per h, k, i cosinibus angulorum quos AR facit cum ox, oy,
OZ, resolvetur in ternas
ph , pk , ọ
iisdem OX , OY , OZ parallelas: ex R in planum XOY
ducatur perpendiculum Rp , producaturque in R' donec fiat
R'p = Rp ; teadet A in R' vi aequipollente ternis
ch , pk , - oi : demissis perpendiculis ex R , R' in planum Xoz , iisque productis donec productiones aequentur ipsis perpendicu
226
rium quibus liquidi particulae se mutuo petunt, sive virium quibus vasorum materies ad se trahitparticulas illas. 

manifeste determinabuutur in tubo duo puncta , quorum vires dabunt componentes
gh , - ok , pi ,
sh , - ok , - qi :
in ferimus particulam A , elisis componentibus parallelis
rectae OY , itemque componentibus parallelis rectae OZ ,
sollicitatum iri juxta AX vi
4Σ φh
proveniente ex tubi materia. In OX sume Ab = Aa ; duc
verticalem bb' ; et quod in ordine ad tubi materiam est q,
in ordine ad liquidi materiam sit q' : quisque intelligit par
ticulam An elisis componentibus horizontalibus, trahi ver
ticaliter deorsum vi
4 Epi
promanante ex liquido intercepto superficie cylindrica , quam
general recta bb' dum sibi constanter parallela movetur in
peripheria circuli habentis radium Ab. Liquidum ultra
superficiem cylindricam praebet vires
.
- 2 Eph , 2 &quot;pi,
alteram horizontaliter agentem juxta XO , alteram verti
caliter deorsum. Omnes itaque vires sollicitantes particulam
A traducentur ad horizontalem
4Eph -2Ep'h = 2 [2Eoh —Eph] ,
et ad verticalem
45' pit 23&quot; ' i.
227
lis, manifeste determinabuutur in tubo duo puncta. quo-
rnm vires dabunt componentes
9ht—9k09i'
-ph.-—9k,—qn':
inferimus particulam A .
elisis componentibus parallelis
rectae 0?
,
itemque componentibus parallelis rectae OZ
,
sollicitatumeiri juxta AX vi
4297:
proveniente ex tubi materia. In OX sume Ab: Aa; duc
verticalem 65; et quod in ordine ad tubi materiam est p,
in ordine, ad liquidi materiam sit go' :quisque intelligit par-
ticulam A. elisis componentibus horizontalibus, trahi ver-
ticaliter deorsum vi ↽
42'9'i.
promanante ex liquido intercepto superficie cylindrica, quam
generat recta bb' dum sibi constanter parallela movetur in
peripheria circuli habentis radium Ab. Liquidum ultra
superficiem cylindricam praebet vires
-— 2
297: , 2E'p'i ,
alteram horizontaliter agentem juxta KO, alteram verti-
caliter deorsum. Omnes itaque vires sollicitantes particulam
A traducentur ad horizontalem
4ng −− ⇄∑∲∣∣∣:2929]: −∙− ∑⊈⊅⋅∣⋅⊐ .
et ad verticalem (f)
42: p'i-l- 22&quot;qa' i.228
Potest 2Eph -Eph esse aut &gt; o , velo, vel = 0: in primo
casu vis aequipollens et gravitati , et binis (f ) , deviabit a di
rectione verticali faciendo angulum acutum cum AX; et
quia ( 83.3º. ) vis illa debet normaliter sese dirigere ad libra
tam liquidi superficiem , ideo suprema liquidi superficies in
duet curvam concavamque figuram : in secundo casu vis
aequipollens et gravitati, et binis (f), deviabit quidem a ver
ticali directione, sed faciendo angulum oblusum cum AX ;
propterea ( 87. 3 • ) suprema liquidi superficies induet curyam
convexamque figuram : in tertio denique casu ex duabus (8)
remanebit sola verticalis, et consequenter suprema liquidi
superficies erit plana atque horizontalis.

2º . Massae liquidae OS , OS' (Fig. 54 ) ejusdem
naturae, planisque superficiebus OP , OʻP ' terminatae, ae
qualiter trahunt exilissimas columnellas liquidas AR , A'R'
perpendiculariter superficiebus ipsis insistentes; nisi tamen
columella AR intra massam OS trahitur deorsum , columel
la vero A'R' extra massam O'S' trahitur sursum. Intelligan
tur enim centris A et A ', radiisque aequalibus AB et A'B ',
ultra quos sensibilis attractio liquidi non protenditur, des
cribi hemisphaeria BCD , B'C'D' : si vires , quibus hemi
sphaeria agunt in particulas A, A ', resolvuntur in binas, alte
ram horizontalem , alteram verticalem; elisis horizontalibus,
remanebunt verticales, quarum summa est eadem utrinque
cum hoc tantum discrimine quod A trahitur deorsum et A
sursum. ln columellis sume nunc duo alia puncta E, Eʻae
quidistantia ab A , A' , radiisque aequalibus EL, E'L ' ( = AB)
describe segmenta sphaerica FML, F'M'L' : accepla EV=EA,
ductoque plano horizontali HK, vires ex QHKN , QFMN
in punctum E utpote aequales et contrariae se mutuo de
struent , ipsumque E solo segmento HLK deorsum trahe
tur : vis ex HLK deorsum sollicitans particulam E ae
quatur vi ex F'L'M ' sursum trahenti particulam E'; siqui
dem haec segmenta sunt aequalia et similiter posita ad
contrarias partes quoad E et E '. Cum igitur idem redeat
228
Potest 229h—29'h esse aut) a. vel( a, vel:o: in primo
casu vis aequipollens et gravitati, et binis ([ ), deviabit a di-
rectione verticali faciendo angulum acntum- cum AK; et
quia ( 87. 30.) vis illa debet normaliter sese dirigere ad libra-
tam liquidi superficiem, ideo suprema liquidi superficies in-
duet curvam concavamque figuram: in secundo casu vis '
aequipollens et gravitati, et binis (f), deviabit quidema ver-
ticali directione, sed faciendo angulum obtusum cum ax,
propterea (87. 3&quot;-) suprema liquidi superficies induet curvam
convexamque figuram :in tertio denique casu ex duabus (f)
remanebit sola verticalis, et consequenter suprema liquidi
supedicies erit plana atque horizontalis.
2&quot;. Massae liquidae OS , US' (Fig. 54) eiusdem
naturae, planisque superficiebus OP
, O'P' terminatae, ae-
qualiter trahunt exilissimas columellas liquidas AR , A'R'
perpendiculariter superficiebus ipsis insistentes; nisi tamen
columella AR intra massam OS trahitur deorsum. columel-
la vero A'R' extra massam O'S'trabitur sursum. Intelligan-
tur enim centris A et A', radiisque aequalibus AB et A'B',
ultra quos sensibilis attractio liquidi non protenditur,des-
cribi hemisphaeria BCD , B'C'D' : si vires , quibus hemi-
sphaeria aguntin particulas A, A', resolvuntur in binas, alte-
ram horizontalem , alteram verticalem;elisis horisontalibus,
remanebunt verticales, quarum summa est eadem utrinque
cum hoc tantum discrimine quod A trahitur deorsum et A'
sursum. ln columellis sume nunc duo alia puncta E, E'ae-
quidistantia ab A , A', radiisque aequalibus EL, E'L' (::AB)
describe segmenta sphaerica FML, F'M'L': accepta EVzEA,
doctoque plano horizontali HK, vires ex QHKN , QFMN
in punctum E utpote aequales et contrariae se mutuo de-
struent
,
ipsumque E solo segmenta HLK deorsum trahe-
tnr : vis ex HLK deorsum sollicitans particulam E ae-
quatur vi ex F'L'M' sursum trahenti particulam E'; siqui-
dem haec segmenta sunt aequalia et similiter posita ad
contrarias partes quoad E et E'. Cum igitur idem redeat229
cimo
di.

; et

bra
sin
vis
Ver
LAX;
ryam
argumentum de caeteris particulis inter A et C , necnon
inter A ' et C ' ( ponimus A'C &quot; — A'C' ) , cumque particulae
infra C viribus contrariis et aequalibus urgeantur, infra C
sensibili non subjiciantur actioni, jam patet etc
In eodem liquido vis, qua deorsum vel sursum colamella
trahitur, constans est; eam in sequentibus exhibebimus per K.

3º. Fac ut massa liquida BAB'QQ (Fig. 55) , quae intercipitur superficie sphaerica BAB' et plano tangente
QQ, trahat externam columellam liquidanı AR perpendicula
riter insistentem plano tangenti apud contactum A : quo
niam BAB O'Q gignitur rotatione areae ABQ circa ra
dium AO, perinde erit sive spectetur attractio massae BAB'Q'Q,
sive spectentur attractiones infinitarum numero superficierum
cylindricarum , quae in ea rotatione gignuntur a perpendicu
lis DC , D'C' , ... demissis ex punctis D , D ' . circu
laris arcus BA in rectam QA. Exprimant p , pi ... per
pendicula DC , D'C', . . ; 9.9 , .. perpendiculorum di
stantias AC , AC' . .. ab A computatas in AQ; sitque r
sphaericae superficiei radius OA: ob magnam lineolarum
p , p ', . . . tenuitatem prae q, , ..
quidi
Eden
A'R'
ameo
amel
gaD
AB,
des
erunt
lemi
aleo
р
92
2r ,pa2r
libus
Bogu
et A
et consequenter praefatae superficies cylindricae exhibe buntur per
-AB
EL
MY
2πη 비유
Toq3
,2πα g's Tig'3
seu 9
dem 2r 2r
cabe
ae
aqui.
Atqui ob eamdem illam tenuitatem puncta uniuscujusque su :
perficiei cylindricae haberi possunt pro aequidistantibus ab
unoquoque columellae puncto: designantibus igitur o, o, ..
quantitates pendentes et a certa quadam distantiarum lege,
deal
sit
ac-
qui'
de:!
229
argumentum de caeteris particulis inter A et C- , necnon
inter A' et C' (ponimus A'C&quot; :: A&quot;C), cumque particulae
infra C viribus contrariis et aequalibus urgeantur, infraC&quot;
sensibili non subjiciantur actioni, iam patet etc
..... ∙
.In eodem liquido vis, qua deorsum vel sursum colnmella
trahitur, constans est; eam in sequentibus exhibebimus per K.
30. Fac ut massa liquida BAB'Q'Q (Fig. 55), quae
intercipitur superficie sphaerica BAB' et plano tangente
QQ',trahat externam columellam liquidam AR perpendicula-
riter insistentem plano tangenti apud contactum A*: quo-
niam BAB'Q'Q gignitur rotatione areae ABQ circa ra-
dium AO, perinde erit sive spectetur attractio massae BAB'Q'Q,
sive spectentur attractiones infinitarum numero superficierum
cylindricarum, quae in ea rotatione gignuntura perpendicu-
lis DC , D'C', . . . demissis ex punctis D, D' . . . circu-
laris arcus BA in rectam QA. Exprimant p , p', ... per-
pendicula DC, D'C',. :
.; q, q' , .. perpendiculorum di-
stantias AC, A.C' .. ab A computatas in AQ, sitque :-
sphaericae superficiei radius OA: ob magnam lineolarum
p, p, . ..
tenuitatem prae q, q', ...,
erunt
9' ∣
vf:
?&quot; 2r'p— 2r'...'l
et consequenter praefatae superücies cylindricae exhibe-
buntur per
' q,! &quot;03 &quot;q '3
, ∙ ∙ ∙
'
seu , 21) r r
q?
2:- 2nq ,Zitq ,...
Atqui ob eamdem illam tenuitatem puncta uniuscuiusque su.-
perficiei cylindricae haberi possunt pro aequidistantibus ab
unoquoque columellae puncto: designantibus igitur &amp;, ö'. ..
qnantitates pendentes et a certa quadam distantiarum lege,.
Kn.-&quot;M— . ⇀ ⋅−−∙⇀∙⋅↼ ⋅⋅−↪∎⋅⊾ −−↼↼∎↼ ↽− ↼−⋅−⋅−⇀−⇀−⋅∙∎∙∙↼ −−↼
↰⋅−↽⋅
- −⋍⇂∙⋅−230
et a liquidi densitate, et a cosinibus angulorum quos cum
AO faciunt rectae ab attrahentibus superficierum punctis
ductae ad attracta columellae puncta, eae superficies colu
mellam sursum verticaliter trahent viribus
Teq38 Tog'38
totaque massa BAB'D'Q columellam AR sursum verticali
ter trahet vi
1938 +7.9'38' +
.
Si concipitur altera massa liquida PAP'OʻQ intercepta pla
no QQ et nova superficie sphaerica PAP, cujus radius
O'A = p , simili ratione ostendetur vim ex PAP'Q'Q fore
παδ+πα35 '+ .
.
Vires itaque istae erunt ut
- Eq: 8 : &quot;5q?: = &gt;
erunt nempe reciproce at sphaericarum superficierum ra
dii. Hinc designante H opportunam quantitatem constan
tem , exprimet
H
vim, qua sursum verticaliter massa liquida BAB'Q'Q trahit
.columellam AR : caeterum quisque videt fore H = 12q30.
230
et a liquidi densitate, et a cusinibus angulum quos cum ⋅
AO faciunt rectae ab attrahentibns superficierum punctis
ductae ad attracta columellae pnncta, eae superficies colu-
mellam sursum verticaliter trahent viribus
nq3d th'3ö&quot;
,.... r :-
totaque massa BAB'Q'Q columellam AR sursum verticali-
tcr trahet vi
∏⊄∍∂−⊢∏⊄⋅∃∂⋅⊹ ∙ ∙ ∙ ∙
.
r .
Si concipitur altera massa liquida PAP'Q'Qintcrcepta pla-
no QQ' et nova superficie sphaerica PAP', cuius radius) O'A-z r' , simili ratione ostendetur vim ex PAP'Q'Q fore
12:738 −−⊢ ∏⊄≖∃∂∣⋅−⊢ ∙ ∙ ∙ ∙
r
Vires itaque istae erunt ut
7! :: ↿ ↿
r Zq d . —r,2q ∙−−≀∙ −∙⋮∙∙ ,
erunt nempe reciproce ut sphaericarum superficierum ra-
dii. Hinc designante H Opportunam quantitatem constan-
tem
, exprimet
H
∙∙−∙−
'
vim, qua sursum verticaliter massa liquida BAB'Q'Q trahit
.columellam'AR : caeterum quisque videt fore H::an36.231

: cum e.

tis r
r
H

4. Quantitas K major est quam
nim K exprimat vim ( 2.0) , qua sursum trahitur columella
H
AR a massa liquida LFAF'L' , exprimet K vim
qua sursum trahitur AR a segmento sphaerico MBAB '.
H
Id vero importat K &gt; o ; ergo etc.

5.0 Massa liquida BAB'E'E terminetur superficie
concavo -spherica BAB' : ducto per A plano tangente QQ ,
sollicitabitur columella AR deorsum ( 2.9 ) vi K ex EFAF'E' ,
H
sarsum (3.9) vi ex BAB'F'F ; tota igitur BAB'E'E trahet
deorsum columellam AR vi (4.0)
.
bio
3
13
н
. K
IR
in i, i' , ... ,

6.• Superficies sphaerica NAN' habens radium
O'A = 0A tangatur plano QQ in A ; columella AR ae
que trahetur sursum a massa liquida NAN'Q'Q ac trabi
lur a massa BAB'Q'Q : patet ( 3. ) çum ex eo quod, pro
ductis DC , D'C' , ... donec occurrant arcui circulari AN
exsistunt DC=Ci; D'C' =C'i, ... ; tum ex
eo quod Ci , Ci', ... , sunt tenuissimae prae AC, AC,
si qua pars columellae non trahitur sursum
sit tenuissima prae reliqua parte sursum altracta .

7.º Columella igitur AR magis trahetur deor
sum ab EE'N'AN quam ab EE'F'AF ; excessusque unius
H
attractionis supra alteram erit . Propterea massa liqui
da desinens in superficiem convexo- sphaericam NAN' traliet
deorsum columellam AR vi
ita ut ea
1
K +
1
i.
231
H 4.&quot; Quantitas K major est quam —-: cum e-
- r
nim K exprimat-vim (29)
. qua sursum trahitur columella
AR a massa liqui/da LFAF'L', exprimet K —E r
vim ,
qua sursum trahitur AR a segmento sphaerico MBAB'.
Id veroinrportat'K—g- ≻∘ ∙∙∙ ergo etc.
. . .
59 Massa liquida BAB'E'E terminetur superficie
concavo-spherica BAB' : ducto per A plano tangente QQ',
sollicitabitur columella AR deorsum (2.0) vi K ex EFAF'E',
sursum (3.&quot;) vi!-.;l ex BAB'F'F; tota igitur BAB'E'E trabet
deorsum columellam AR vi (4.0)
∙
K—ll'a r
6.0 Superficies sphaerica NAN' habens radium
()&quot;AzOA tangatur plano QQ' in A; columella AB ae-
que trahetur sursum a massa liquida NAN'Q'Q ac trabi-
tnr a massa BAB'Q'Q: patet (39) tum ex eo quod, pro—
ductis DC, D'C' , ... donec occurrant arcui circulari AN
in i, s&quot;.
..., exsistunt DCxCi; D'C'..-::C't&quot;, ...: tum ex
eo quod Ci, C'i',
. . . ,
sunt tenuissimae prae AC. AC',
ita ut si qua pars columellae non trahitur sursum , ea
sit tenuissima prae reliqua parte sursum attracta.
7.o Columella igitur AR magis trahetur deor-
sum ab EE'N'AN quam ab EE'F'AF; eicessusque unius
H
attractionis snpra alteram erit -— . Propterea massa liqui-
⋅ r
da desinens in superficiem convexo-sphaericam NAN' trahet
deorsum columellam AR vi
n
∣≺⊣−−−⊑−⋅∙232

8.º Pone superficiem BAB' neque esse sphaericam
,
neque gigni rotatione ullius curvae circa AO

; secla BAB

planis transeuntibus per A0
, curvilineae sectiones apud
contaclum
A gaudebunt inaequalibus osculi radiis

; quos

inter
( demonstrationem suo tempore videre erit in parte 3.4 nostrorum elementorum matheseos 0. 118
) bi
ni reperiunlur
, alter minimus
(
=
r
)
, alter maxi
mus
(
=
r
'
), pertinentes ad binas sectiones sub angulo re
cto invicem constitutas
. Iam
, in ea qua sumus hypothe
si
, hoc pacto determinabitur visex BAB'Q'Q sursum
verticaliter trahens columellam AR
. Intelligatur coalesce
re BAB'Q'Q ex infinitis numero superficiebus cylindri
cis normaliter insistentibus plano tangenti QQ
'

:

' unaquae
que superficies cylindrica non eamdem habebit ubique
altitudinem

; sed apud bina puncta

e diametro opposita
,
quibus nempe maximus respondet circựlus osculator
, al
titudo erit minima

; apud bina puncta

e diametro pari
ter opposita
, perque gradus 90 ab illis primis sejuncta
,
quibus videlicet minimus respondet circulus osculator
,
altitudo erit maxima

: apud intermedia puncta altitudines

interjacebunt minimam maximamque
. Quapropter evoluta
superficie cylindrica super aliquo plano
, ea poterit reprae
sentari per aream QNN
&quot;
Q
&quot;
( Fig
. 56
)

; NN

&quot; aequatur
basi superficiei cylindricae

; QN et

Q
&quot;
N
&quot; simul cum Q'N
'
exhibent altitndines minimas

; Fu et F'u

' altitudines ma
ximas hinc Nu

= uN
'
= N'u
'
= u'N
&quot;
. ob perexiguum ba
seos cylindricae radium poterunt QF
, Q'F
, Q'F
'
,
(
&quot;
F
'
haberi pro lineis rectis

; eritque

1
QN
+Fu NN &quot; QʻF'Q'FQ = 1NuFQ = 4 Nu 2 NN ” QN + F4 2
232
8.&quot; Pone superficiem BAB' neque esse sphaericam,
neque gigni rotatione ullius curvae circa AO; secta BAB'
planis transeuntibus per AO, curvilineae sectiones apud
contactum
A gaudebunt inaequalibus osculi radiis quos

inter (demonstrationem suo tempore videre erit in par-
te 3.*' nostrorum elementorum matheseos n. 118) bi-s
ni reperiuntur, alter minimus
( ::r)
, alter maxi-
mus (:r'), pertinentes ad binas sectiones sub angulo re-
cto invicem constitutas. Iam
,
in ea qua sumus hypothe-
si, hoc pacto determinabitur vis et BAB'Q'Q sursum
verticaliter trahens columellam AR. Intelligatur coalesce-
re BABHQQ ex infinitis numero superficiebus cylindri-
cis normaliter insisteutibus plano tangenti QQ' :'unaquae-
que superficies cylindrica- non eamdem habebit ubique
altitudinem; sed apud bina puncta
e diametro opposita,
quibus nempe maximus respondet circulus osculator
, al-
titudo erit minima; apud bina puncta
c diametro pari-
ter opposita, perque gradus 90 ab illis &quot;primis seiuncta,
quibus videlicet minimus respondet circulus osculator
,
altitudo erit maxima: apud intermedia puncta altitudines
interjacebunt minimam maximamque. Quapropter evoluta
superficie cylindrica spper aliquo plano
, ea poterit reprae-
sentari per aream QNN&quot;Q&quot; (Fig. 56); NN&quot; aequatur
basi snperficiei cylindricae QN et Q&quot;N' simul cum Q'N'

exhibent altitudines minimas -; Fa et F'u' altitudines ma-
ximas; biuc NucuN'zN'u'2u'N&quot;
.
ob perexiguum ba-
seos cylindricae radium poterunt QF
, Q'F, Q'F ', Q&quot;F'
habcri pro lineis rectis; eritque
NN&quot;Q'F'Q'FQ::4NuFQ

:

4 Nu 'QN'zl-F&quot;

:

∙ NN&quot; QNj'F&quot;
.233
ericam,
a BAB
es apud
i quoi
Retentis igitur denominationibus ( 3.9 ) , superficies cylin
dricae , ex quibus intelligitur coalescere massa BAB'Q'Q
( Fig. 55 ) , erunt
92 Lo pár.
+
92 q /2
+
2r
2r' 2r'
2πα ,
2r
2tq'
&gt;
seu
mari 2 2
alore
ypothe
Sursa mga (: + ?). mg ( + ),...
Dalesce
linde
naquat
ubige
pposila,
et consequenter massa BAB'Q'Q desinens in superficiem
BAB' utcumque concavam trahet verticaliter sursum co
lumellam AR vi
or , al
πα
2
o pari ?( + 1) + &quot;, ( +3)x + ...
Atqui ( 3.9 ) 7.938 + Teq'38 ' + .... = H :
Ejuncta,
ulator ,
Studios
evolu
reprat
equatur
exprimetur ergo vis illa per
16+)
les m2
um bio
1 07
9. Sume Q '&quot;'N '&quot; et F &quot; u &quot; ( Fig. 56 ) aequidi
stantes ab QN et Fu : erunt Q &quot; N &quot;&quot; , F &quot; u &quot; duae ex al
titudinibus intermediis ( 8. ) respondentes duabus sectio
nibus curvilineis ad angulum rectum invicem constitutis.
Radii circulorum osculantium sectiones istas apud A ( Fig.
55 ) designentar per po&quot; et p' : erit ( Fig. 56 )
9'2
Q&quot; N &quot; +F &quot; u &quot; =
92
2r &quot;.
qo +
27 
+
g'?
2r . ' 2r' '
1
16
233
Retentis' igitur denominationibus (3.&quot;) , superficies cylin-
dricae, ex quibus intelligitur coalescere massa BAB'Q'Q
(Fig. 55 )
,
erunt
/
£ fl&quot; q&quot; a&quot;
2r .l-Zr ∙ 2r
21tq 2 , 27tq .l-Zr' 2 , ∙ , seu
↔⇍≖↙≀⋮↿≺ ,
') &quot;rv ')
⋅⋅
2 rii-&quot; 2
≀∙⊣−∣⋅⋅∅⋅⋅⋅⋅
et consequenter massa BAB'Q'Q desinens in superficiem
BAB' utcumque concavam trahet verticaliter sursum co-
lumellam AR vi
&quot;f(ï-Fl?) ∂⊹∙⋮≖−≣−⋅⋮−≺−∶−−⊦∙≙≻∂↝⊹∙ ∙∙
r
Atqui (3.0) nq3ö -l-7tq'3d' ∙−⊦ ∙ ∙ ∙:∙ H:
exprimetur ergo vis illa per
⋅≣⋅≺⊥⊣−−↿⊺≻ ⋅

: - r

9.0 Sume Q'&quot;&quot;'N et F&quot;u&quot; (Fig. 56) aequidi-
stantes ab QN et Fu : erunt Q'&quot;N&quot;', F&quot;u&quot; duae ex al-
titudinibus intermediis (89) respondentes duabus sectio- .
nibus curvilineis ad angulum ≖⋅∁∁⋯⊞∙ invicem constitutis.
Radii circulorum osculantium sectiones istas apud A ( Fig.
55 ) designentur per r&quot; et r'&quot; : erit (Fig. 56)
0 (
ns '
lr Jr ' ∎∙ lr.
'
a a 'a' ': ≺≀⋅⋅∙↓↜⇃∙⋅∙−⊦∣∂⇁⇈≀↓∙∙∶∶−∙↙−⇃−⋅− ⊄
⋞∣ −⊦⋞∣∙∙234
est autem
mane cc
Q &quot; N '&quot; + F &quot; u&quot; = QN + Fu =
92
2r
9'2
2r +
2r etc.:
1
igitur
+++++
110
et consequenter
16 + *) = &quot; 6 + - ): under
10.º Si superficies concava BAB' ( Fig. 55 ) gi
gnitur rotatione alicujus curyae circa OA , fiet
Cor
ace
r = r = r ' = r '&quot;
I supe
ac proinde vis ex BAB'Q'Q sursum verticaliter trahens
columellam AR erit ban,
zebic
H
reciproce nimirum ut radius osculi apud A.
11.• Facile nunc intelligimus attractionem mas
sae liquidae BAB'E'E, terminatae superficie utcumque con
caya BAB' , in columellam AR fore
K - &quot; ( + -) .velK6+ );
234
est autem
↾∣∣ ⇌∎
!
'2 :
NI&quot; F&quot; &quot;: −∙ Q. ∙−⊦ u QN—l—Fu q2r ↿ q 0
L
I 2r' , 2r ∙−⊦ 2r' ,etc.t
igitur
1 ↿− ↿ ↿
i- ∣
rr ∣⋅⊤⋅ .'&quot;? '
et consequenter
H 1 ↿ H
∎↿ ↿ ⋅
−⋮−≺−≀∶⇀⊦ r' )— 2
(r&quot; hl-r'&quot;) ⋅∙
10.0 Si superficies con-cava BAB' (Fig. 55 ) gi.
gnitur rotatione alicujus curvae circa OA ,
fiet
I ';
∣←−∶≀∙ :r :r '&quot; ;
ac proinde vis ex BAB'Q'Q sursum verticaliter trahens
columellam AR erit
'H-
&quot;,
reciproce nimirum ut radius osculi apud A.

11.(, Facile nunc intelligimus attractionem mas-
sae liquidae BAB'E'E, terminatae superficie utcumque con-
cava BAB' ,
in columellam AR fore
H
1,1) H(1,1 ∙
K
2(r '7 'velK 2r&quot; 'r'&quot;)'
ïfbiu235
massae vero liquidae NAN'E'E, terminatae superficie utcum
que convexa NAN' , in ipsam AR fore
K + 6 + -) .
ved K + &quot;6+ ) :
fiet =r =r&quot; = r&quot; in casu superficiei genitae rotatione li
neae curvae circa OA.

110. His declaratis , venio ad ascensum descensum que liquorum in tubis capillaribus.

1.° Aqua in tubis vitreis diversae capacitatis ad diversas ascendit altitudines, quae sunt reciproce ut tuborum diametri: idipsum contingit oleo, spiritui vini, etc..; ascendentesque liquores terminantur superne concava superficie. Concavae superficiei causam adsignavimus (109 1.): ad ascensum quod pertinet, sit QQ ( Fig. 57 ) suprema superficies aquae circumambientis tubum LE , et I A'B' concava superficies aquae intra tubum; sint insuper A'R, VR' binae columellae verticales, altera intra, altera extra tubum, quas columellas jungat horizontalis columella RR'. Urgebitur A'R deorsum gravitate simulque vi (109. 11. ° )
K - 16 + );
urgebitur VR' deorsum gravitate simulque vi ( 109. 2.° )
K :.
cum igitur
Н
K &gt; K
(
2 + ),
235
massae vero liquidae NAN'E'E, terminatae superficie utcum-
que convexa NAN' ,
in'ipsam 'AR fore
H 1 1 H 1 1 K—(-2(,, ∣∣⋅ .).ve1K( ,(,.,'. ...-)
r r
fiet r:r':r&quot;—-::r&quot;' in casu snperficiei genitae rotatione li-
neae curvae circa OA&quot;.

110. His declaratis, venio ad ascensum descensumque liquorum in tubis capillaribus.

1.&quot; Aqua in tubis vitreis diversae capacitatis ad diversas ascendit altitudines, quae sunt reciproce ut tuborum diametri: idipsum contingit oleo, spiritui vini, etc..; ascendentesque liquores terminantur superne concava superficie. 

Concavae superficiei causam adsignavimus (10913): ↙
ad ascensum quod pertinet , sit QQ' (Fig. 57) supre-
ma superficies aquae circumambientis tubum LE , et I'A'B'
concava superficies aquae intra tubum; sint insuper A'R,
VR' binae columellae verticales, altera intra, altera extra
tabum, quas columellas iungat horizontalis columella RR'.
Urgebitur A'R deorsum gravitate simulque vi ( 109. 11.&quot;)
H1711,
K
2(rlr1)a
urgebitur VR' deorsum gravitate simulque vi ( 109. Z.&quot;)
∕
cum igitur236
haud poterunt A'R , VR' consistere in aequilibrio nisi A'R
ascendat supra QQ . Denotet z altitudinem AA , ad quam ascendit columella A'R supra QQ'; sitque c gravitas specifica liquoris: fiet eousque columellae ascensio donec habeatur
H
K = K
-16 + )+c=;unde == 2c ( + ).

Vires ex materia tubi eas tantum liquidi particulas afficiant, quae ad internam ipsius tubi superficiem maxime accedunt; iccirco liquidum perinde trahetur, a tubo ac si interna superficies esset plana: permanente igitur tubi ac liquoris qualitate, etsi variat tubi diameter, eodem tamen pacto trahentur liquoris particulae versus tubum; quae attractio quia cum constanti attractione liquoris erga seipsum consociatur, extrema latercula curvae BAB' aeque inclinabuntur ubilibet ad internam tuborum superficiem. Arcus itaque omnes BAB' erunt similes in diversis tubis, ipsorumque arcuum rotatione circa tubi axem gignetur superficies concava superne terminans liquorem : bini videlicet
r et r' exsistent aequales, simulque tuborum diametris pro
portionales; ideoque altitudo z reciproce ut eae diametri.

2. • Hydrargyrum in tubis vitreis descendit in
fra circumambientis hydrargyri superficiem QQ ad ejus
modi altitudines , quae sunt tuborum diametris recipro
ce proportionales ; descendensque liquidum terminatur su
perne convexa superficie NOM.
Convexitatis causam adsignavimus ( 109. 1.0 ) : ad de
scensnm quod spectat , columellarum OR et VR' altera
sollicitatur deorsum gravitate simulque vi ( 109. 11.° )
0
{
K + C + ) ,
236
haud poterunt A'R, VR' consistere in aequilibrio nisi A'R
ascendat supra QQ'. Denotet :altitudinem AA' , ad quam
ascendit columella A'R supra QQ'; sitque c gravitas spe-
cifica liquoris: fiet eousque columellae ascensio donec ha-
beatur
H ↿ ↿ H 1 ↿
∣≮≓⋅∶∶∣⊊∙− −∙≨∙⋖−−∙−⊢⊤≻−⊢∶≖∙ under—' 2c(r ≓≀∙∙≻ .
r .
!
Vires ex materia tubi eas tantum liquidi particulas af-
ficiunt, quae ad internam ipsius tubi superficiem maxi-
me accedunt; iccirco liquidum perinde trahetur, a tuba ac si
interna superficies esset plana : permanente igitur tubi ite
liquoris qualitate, etsi variat tubi diameter, eadem tamen
pacto trahentur liquoris particulae versus tubum; quae
attractio quia cum constanti attractione liquoris erga se-
ipsum consociatnr, extrema latercula curvae BAB' aeque
incl'uabuntur ubilibet ad internam tubarum superficiem.
Arcus itaque omnes BAB' erunt similes in diversis tubis,
ipsorumque arcuum rotatione circa tubi axem gignetursuper-
iicies concava superne terminans liquorem : bini videlicet
r et r' exsistent aequales, simulque tubarum diametris pro-
portionales; ideoque altitudo :reciproce ut eae diametri.

2.0 Hydrargyrum in tubis vitreis descendit in-
fra circumambientis hydrargyri superficiem QQ' ad eius-
modi altitudines , quae sunt tubarum diametris recipro-
ce proportionales; descendensque liquidum terminatur su-
perne convexa superficie NOM.
Convexitatis causam adsignavimus (109. 1.(, ) : ad de-
scensnm quod spectat, columellarum OR et VR' altera
sollicitatur deorsum gravitate simulque vi (109. 11.&quot;)
Hi 1
K&quot;.'a(1&quot;'r&quot;)'
I:
'P.?237
altera sollicitatur deorsum gravitate simulque vi ( 109. 2.0)
K :
cum igitur
K &lt; K + -( + )
haud 'poterunt OR et VR' sese librarè nisi OR descen
dat infra QQ. Designet é altitudinem AO , ad quam
deprimitur columella OR infra QQ' ; sitque c' gravitas
specifica hydrargyri : eousque fiet columellae depressio do
nec habeatur,
H
K=K + ( + )- c'z' , unde z' = 2c'
(
+ &gt;&gt;).
Ut supra ( 1. ) ostenditur binos r , r' fore aequales, simul
que proportionales tuborum diametris ; iccirco etc.

111. Nonnulla subjungimas, quorum ratio desumitur ex animadversionibus (109).

1.º Duae laminae vitreae et parallelae PP ', SS ' demergantur verticaliter in aquam, earumque mutua distantia aequetur diametro tubi capillaris LE: suprema aquae superficies B &quot; A &quot; B &quot; inter laminas evadet concava instar canalis horizontalis; altitudo vero A'al= x ), ad quam attollitur aqua, erit duplo minor altitudine ad quam attollitur in tubo LE.: Infima superficiei B &quot; A &quot; B '&quot;' puncta jacent omnia inea
dem recta A'A'&quot;: secetur B &quot; A &quot; B &quot; &quot; plano perpendiculari ad A &quot; A &quot; '; sectio erit ubilibet arcus arcui BA'B' similis et
aequalis : istorun arcuum radius osculi apud puncta infima
dicatur r ; in tubo LE erit p = r , in laminis r= -0 . Colamellarum igitur A'R , VR aequilibrium praebebit ..
237
altera sollicitatur deorsum gravitate simulque vi (109. 23)
K :
cum igitur
X(K '
H( ↿
r ral,).r
baud' poterunt OR et VR' sese librare nisi OR descen-
dat infra -'QQ. Designet z' altitudinem AO,- ad quam
deprimitur columella OR infra QQ'; sitque c' gravitas
specifica hydrargyri: eousque fiet columellae depressio do-
nec habeatur.
1
∙
↿
'
∣∟−∣≖⊹−−⊸ 2(-—--]——-)-—c&quot;,z undez'—.2[:7(1 [
r').
Ut supra (1 .&quot;) ostenditur binos r, 'r' fore aequales, simul-
que praportionales tubarum diametris; iccirco etc.
,.,11.1 Nonnulla subjungimtts ,- quorum ratio desu-mitur ex animadversionibus (109).↿∙∘ ∐∎≖≔∘∙ laminae vitreae et parallelae PP', SS'demergantur verticaliter in aquam, earumque mutua distantiasequetur diametro tubi capillaris LE : suprema aquae super-ficies B&quot;A&quot;B&quot;' inter laminas evadet concava instar canalishorizontalis; altitudo vero A&quot;a(:.r) , ad quam attollituraqua , erit duplo minor altitudine , ad quam attolliturintnhoIfE.- ⋅∶∶⊸∙Infima superficiei B&quot;A&quot;B&quot;' puncta iacent omnia infen-dem recta A&quot;A'&quot;: secetur B&quot;A&quot;B&quot;' plano perpendiculariad A&quot;A&quot;'; sectio erit ubilibet arcus arcui BA'B' similis etaequalis: istorum arcuum radius osculi apud puncta infimadicatur r; in tuba LE erit r'zzr, in laminis r'zoo; Cos-lumellarnm igitur A'R , VR' aequilibrium praebebit't-dïff2382c ( + ) &quot; ;Tetscolumellarum autem A ” R &quot; , VR aequilibrium suppeditabitH101 IK -K - I ( + ) +re , f =.2c1Hinc xai 2ż z ; ideoque etc.....2.0 Laminae PP ' , SS , sibi commissae ad sematuo accedunt.Sit P&quot; punctum quodvis laminae PP: infra QQ adprofunditatem Alla &quot; : columella verticalis Alla&quot; transmittetpuncto P vim ( 1.0 ) .K- ( + ) +ostan&quot;) = KH+2r Tersus1 .C2c 5+c. a a'&quot; = K +0. aa!&quot;dicenndnetfenotaversus Qt : attenta columella horizontali . a &quot; ' P &quot; , urgebi amelltur P vi seu pressione externa + traiKversus Q't': colamella verticalis V'a transmittet puncto P &quot;vimK+c.aa &quot; &quot;versus Qi' : attenta columella horizontali a'P ' , solicitabitur P &quot; vi seu pressione interna 16TSU238H(t 1) H 1z— ⇂ ∙−−− ∙⋅ ;20 r !' C !'columellarum autem A&quot;R&quot;', VR' aequilibrium suppeditabit!H 1 1 H 1− ↿ ∣ ⊫−∙− −⋅⋅ ∙!cx . ∙∖Hlnc ∙ æ;—2—z;l ∙ ⋅ 1deoqueetc......⋅⋅ n .'2.o Laminae 'PP. SS', sibi commissae ad sea') 'At mutuo aeeedunt. )Sit P&quot; punctum quodvis laminae PP' infra QQ' adprofunditatem A&quot;a &quot;: columella verticalis A&quot;a transmittetpuncto P&quot; vim (1.). &amp; ↽(⋅. H 1 1⋅ &quot; HK(cæ—-—aa&quot;' :K −−−∙−−∙ 2 ≀⋅∙⋮∙∙∞ .'. (M' . 2r. . '&quot;1ciii .-—--)-c.aa :::K'I—ILmaa'&quot;.⋅.. .: '&quot;.versuth : attenta columella horizantali- a.'&quot;P&quot; , urgebi-tur P&quot; vi seu pressione externa,.. ⋅∙ ⋅('. a∙ t '. '(1
..

infima
dicatur r; in tuba LE erit r'zzr, in laminis r'zoo; Cos-
lumellarnm igitur A'R , VR' aequilibrium praebebit't-dïff238
2c ( + ) &quot; ;
Tets
columellarum autem A ” R &quot; , VR aequilibrium suppeditabit
H
101 I
K -K - I ( + ) +re , f =
.
2c
1
Hinc xai 2ż z ; ideoque etc.....
2.0 Laminae PP ' , SS , sibi commissae ad se
matuo accedunt.
Sit P&quot; punctum quodvis laminae PP: infra QQ ad
profunditatem Alla &quot; : columella verticalis Alla&quot; transmittet
puncto P vim ( 1.0 ) .
K
- ( + ) +ostan&quot;) = K
H
+
2r Tersus
1 .
C
2c 5+c. a a'&quot; = K +0. aa!&quot;
dicen
ndnet
fenota
versus Qt : attenta columella horizontali . a &quot; ' P &quot; , urgebi amell
tur P vi seu pressione externa + trai
K
versus Q't': colamella verticalis V'a transmittet puncto P &quot;
vim
K+c.aa &quot; &quot;
versus Qi' : attenta columella horizontali a'P ' , solicita
bitur P &quot; vi seu pressione interna 16TSU
238
H(t 1) H 1
z— ⇂ ∙−−− ∙⋅ ;
20 r !' C !'
columellarum autem A&quot;R&quot;', VR' aequilibrium suppeditabit
!
H 1 1 H 1
− ↿ ∣ ⊫−∙− −⋅⋅ ∙
!
cx . ∙∖
Hlnc ∙ æ;—2—z;l ∙ ⋅ 1deoqueetc......
⋅⋅ n .
'2.o Laminae 'PP. SS', sibi commissae ad se
a
') 'At mutuo aeeedunt. )
Sit P&quot; punctum quodvis laminae PP' infra QQ' ad
profunditatem A&quot;a &quot;: columella verticalis A&quot;a transmittet
puncto P&quot; vim (1.). &amp; ↽
(
⋅
. H 1 1
⋅ &quot; H
K
(
cæ—-—aa&quot;' :K −−−∙−−∙ 2 ≀⋅∙⋮∙∙∞ .'. (M' . 2r
. . '
&quot;
1
ciii .-—--)-c.aa :::K'I—ILmaa'&quot;
.
⋅
.
. .: '
&quot;
.versuth : attenta columella horizantali- a.'&quot;P&quot; , urgebi-
tur P&quot; vi seu pressione externa
,
.. ⋅∙ ⋅
('. a
∙ t '. '
(1
..

: - ∎∙ 3 ' ∙

.
versus Q't': columella verticalis V'a' transmittat puncto P&quot;
visu ⋅⋅ ' ⇀
'
'
'
⋅
⋅⋅
K—l—caa ?
.
versus Q't': attenta columella horizontali a'P&quot;
, sollicita-
bitur P&quot; vi seu :pressione interna &quot;
'
'
-
⇂⇣ r 'a'
ms
1111:
&quot;But
&quot;fin239
K
versas Qt : erit igitur p &quot; aeque sollicitatum hinc et illinc,
nullusque motus gignetur ab istis viribus.
Sit P' &quot; ' punctum ipsius PP' inter QQ et A &quot; A '' : ver''
ricalis columella A &quot; a &quot; transmittet puncto P &quot; ' vim
к
-16 + ) + ( 6 – aa &quot;) = K -
H
+
2r
H
S c.aa&quot; = K- c.aa' '
2r
versus Qt : ob columellam horizontalem P '&quot; a &quot; urgebitur
P &quot; &quot; vi seu pressione externa
K
versus Qt : cum igitur K &gt; K - c.aa &quot; , nitelur &quot; mo
veri ad plagam (t' .
Ascendet aliquantulum aqua externa prope laminam ÞP
induetque ( 109. 1. ) figuram concavam ee'e ' ; propterea,
denotante &amp; radium osculi apud punctum v . gr. e' , co
lumella e'b normaliter insistens superficiei curvae ee'e&quot; in
e' transmittet puncto v. gr. b vim
H
K
29 ( +4) = K 2 €
versus Qt’ : attenta columella horizontali e'b ', urgebi
tur b ' vi seu pressione interna
K
versus Qt : ex aqua éb'é' proveniet in bi vis
∙ 239
K
Versus Qt: erit igitur P&quot; aeque sollicitatum hinc et illinc,
nullusque motus gignetur ab istis .viribus. Sit P'&quot; punctum ipsius PP' inter QQ' et A&quot;A&quot;': ver-
ticalis columella A&quot; a&quot; transmittet puncto P&quot;'v vim
K
--—-(—1--—l--—) —[—c(x—-aa&quot; ∙−−∶ ∣⊂−∙≗ ∙⋅⊢
H
—— c.aa&quot;
−−−−− K— c.aa&quot; 2r ∕
versus Qt'
- ob columellam horizontalem P&quot;'a &quot;
P'&quot; vi sen pressione externa
&quot;—
urgebitur
K
versus Q't'. ∙ cum igitur K)K—c.aa&quot; , uitetur P&quot;' mo-
veri ad plagam Q't' .
Ascendet aliquantulum aqua externa prope laminam PP' induetque (109.1.?) figuram concavam ee'e&quot; ; propterea,
denotante &amp; radium osculi apud punctum v. gr. e'
, co-
lumella e'b normaliter insistens superficiei curvae ee'e&quot; in
e' transmittet puncto v. gr. b' vim
H 1 1 H K—
⊸≳↽≼−⋮⇀⊣−∘−∘−⊢ ≖⊊−−⊸⇄−∊
versus Q't'- attenta columella horizontali e',b' urgebi-
tur b' vi seu pressione interna '-
. K
8
versus Qt: ex aqua e'b'e&quot; proveniet in b' vis240
c.e&quot; 6
versus Qit' : columella verticalis A &quot; 6 &quot; transmitiet puncto
b' yim
H
K
25
+ c . A &quot; 6 &quot;
versus Qt : attenta columella horizontali b'b' impelletur 6
vi seu pressione esterna
K
versus Qt . Est
H
2r = cx = C . A'a ;
librato insuper liquido , pressiones apud V' et é' sunt ae
quales , et consequenter
K = K
H
28 to.e&quot; 6 ,
H
= c.eb' ;
2 €
detractisque proinde viribus versus Qt ex viribus versus
Q'ť , emerget
H н
K
2e-K + c.e^ 6—K+
-c.A &quot;b&quot; + K = c.eb - c.A &quot;6&quot; +
H H
2r 2 €
c (e&quot; b' — A &quot; 6&quot; + 1&quot; a - c'b') = c.b &quot;a &gt; o :
sollicitabitur ergo b' vi c.6 &quot; a versus Q't' .
Veniat denique spectandum in lamina PP punctum p '
inter A &quot; A &quot; et B &quot; B &quot; : sit P'a&quot; columella horizontalis ; a'i
columella perpendicularis superficiei curvae B’A &quot; B &quot; &quot; apud
a &quot; ; dicaturque é radius osculi in a ' ' . Transmittet a'i
puncto Ph vim
240
c . e&quot;b'
versus Q't' :columella verticalis A&quot;b&quot; transmittet puncto
b' vim
. H &quot;&quot;
K—Z—i—c'Ab
versus Qt :attenta columella horizontali b'b&quot; impelletur b'
vi seu pressione externa
↴
K
versus Q't'. Est
H
∙∠−≀∙−∶∶∘∙↿∽⋅∶∘∙∆∎∣∅
librato insuper liquido , pressiones apud V' et a' sunt ac-
quales , et consequenter
-——-K——'l&quot;c. e&quot;6'. Eli-:o- e&quot;b':
detractisque proinde viribus versus Qt ex viribus versus
Q&quot;: , emerget ⋅
K—is'&quot;. -K—]—
.- .∘∣∙∣↗∣−−⋅↧≮−⊦ ≛↿−⊑∙⊸∙∆∦∣⊃⋅∣−⊢↧⊊∶⊸⋅⊜∣⋅∣⊃⋅−∘⋅∆⋅⋅≀≀⋅∣−⊦
H H &quot;
.!
.,
h 1
∙−−
&quot;
.
ii.—22&quot;— ..—c(&quot;eb'- Ab -I-Aa-c.b)—-c.b a)o.
sollicit'abitur ergo b' vi c.b&quot;a vcrsus Q't' .
Veniat denique Spectandum in lamina PP' punctum P&quot;
inter A&quot;A&quot;' et B&quot;B&quot; : sit P&quot;'a&quot; columella horizontalis; a&quot;i
columella perpendicularis snperficiei curvae B&quot;A&quot;B&quot;' apud
a&quot; ; diceturque e' radius osculi in a&quot;. Transmittet a&quot;:
puncto P&quot; vim.241
K
H
2 €
.
versus Qt : ex liquido superincumbente proveniet in p ' ' vis
B ' P &quot;
versus Qc : attenta P &quot; a &quot; urgebitur p &quot; vi seu pressione
externa
K
versus Qit' : librato liquido , pressiones apud a &quot; et A ” sunt
aequales ; proinde ducta horizontali Allu ,
H H
K tc.P&quot; u = K = K- C. A'a ,
2 € 2r
H
= c ( P''u + A &quot; u ) = c.P'' ' :
26
detractis igitur primis duabus viribus ex tertia , assequemur
H
K - K +
c . B ' piv H
-C.B&quot; piv
2€ 22'
c ( Piu' B ' p ' ) &gt; 0.
Lamina itaque PP' movebitur versus Q'C' : eadem de causa
movebitur lamina SS' versus Qt ; ideoque etc...

3. Si aquae substituitur hydrargyrum , supre
ma liquidi superficies inter laminas PP' , SS ' fiet con
vexa instar horizontalis inversique canalis ; deprimetur li
quidam ad altitudinem duplo minorem quam in tubo LE ;
ipsae insuper laminae adhuc ad se mutuo accedent . Haec
explicantur simili ratione ac ( 10, 20. )..
241
versus Qt :ex liquido superincumbente proveniet in P&quot; vis
&amp; ∙ B1v Ptv
versus Qt : attenta P&quot; a&quot;
externa
urgebitur P&quot; vi seu pressione
K
versus Q't' :librato liquido . pressiones apud a&quot; et A&quot; sunt
aequales .; proinde ducta horizontali A&quot;u,
H &quot;
∙∙∙ H ∙∙
↿⊂−∙−∙⋮≳−⋮∙−⊢∘∙↧∙ ∥⋅−−↿⊊∙−⋮≳≀∙−−∙−−↧⊊−∁∙ A a
.
H ,,
⊓∣
27-—-c(P&quot;u-l—Au)——-—c.P u:
detractis igitur primis duabus viribus ex tertia , assequemur
⊏−≖⊂−⊢⋮−⋮∶∶−∘∙∌∏ ↕⊃≖⊽∶∶∶ IST—c .B&quot; Piv:
c (P&quot;'u' — Blv P&quot;) ∘∙
Lumina itaque PP' movebitur versus Q't' : eadem de causa
movebitur lamina SS' versus Qt ; ideoque etc...
3.&quot; Si aquae substituitur hydrargyrum , supre-
ma liquidi superficies inter laminas PP' , SS' fiet con-
vexa instar horizontalis inversique canalis; deprimetur li-
quidam ad altitudinem duplo minorem quam in tubo LE;
ipsae insuper laminae adhuc ad se mutuo accedent. Haec
explicantur simili ratione ac (10. 20.)
.242

4. • Super vitream laminam horizontalem AA'B'B
( Fig . 58 ) affunde gattam olei terebinthini mm ' ; tum al
teram laminam vitream A &quot; A'B'B &quot; priori AA'B'B impone
sub angulo sic exiguo , ut imposita lamina gutlam le
viter attingat ; conspicies mm' , instar trochleae , termi
natam quodam canaliculo ; qui canaliculus plano hori
zontali sectus dabit curvilineam convexamque sectionem
plano verticali sectus curvilineam concavamque sectionem .
Radius convexitatis ( € ) manet proxime idem in
punctis m et m' e diametro oppositis ; radius vero con
cavitatis ( = r' ) in puncto m' magis accedente ad A'B'
minor erit quam radius concavitatis ( = r) in puncto m
minus accedente ad ipsam A'B ' . Spectantes columellam
mam ' perpendicularem rectae A'B' , quoniam r et é ' apud m
obverluntur ad plagas contrarias , itemque r et ê &quot; apud m' ,
facile intelligemus ( 109. 110. ) sollicitatum iri mam ver
sus A'B' vi
H
K
2
simulque versus AB vi
K
16-&gt;).
Cum igitur &gt; m , prima vis erit major quam secunda ;
columellaque mam , et una cum mam' tola gutta mo
vebitur versus A'B' motu accelerato : idipsum contingit
guttaeaqueae . At si ejusmodi guttis substituatur gutta
hydrargyri , haec movebitur versus AB ; ratio est quia
gutta hydrargyri tam in sectione horizontali quam in
verticali praebet curvam convexam , radiusque novae con
vexitatis in m superat radium novae convexitatis in m .

5.° Capillaris tubus in aquam QQtt ( Fig. 57. ) demergatur; tum, apposito digito ad orificium inferius extrahatur : remoto digito , aqua jam elevata eflaet ali
quantulum ex orificio illo , ibique demum haerebit sus
pensa in guttam rotundam conformata ; residuae vero
aquae altitudo in tubo extracto invenitur major quam
altitudo ( 110, 1º.)
H
Z =
16 + 3) = * ( + ) cr
supra QQ in tubo demerso.
Exprimant et w , altera radium convexitatis apud infi
mam aquae superficiem in tubo extracto ,
altera ipsius
aquae altitudinem : ex aqueae columnae aequilibrio pro
fluit&quot; ( 110. 1 ° 2°. )
H H
K
+++
www =
= ktö : +
H
co ;
cr
ideoque w &gt; z . Si aquae substituitur hydrargyrum , tam
suprema quam infima superficies liquidi exsistet conve
x2 ; ex aequilibrio igitur hydrargyri in tubo extracto
emerget ( 110. 20. )
H H
H
K + tow == Kt H
co CI
et consequenter a = 0 si r = 0 .

112. Quae diximus de liquorum ascensu tubulis vitreis, applicari possunt ascensui liquorum in tenuibus cujuscumque materiei tabulis: hinc patet cur liquida ascendendo imbuant spongias, saccharum, ellychnia etc: cur succus inserviens plantarum vegetationi sursum ex terra eluctetur; etc... Istiusmodi corpora vel constant exilissimis fibris, in quibus tanquam in totidem capillaribus tubis ascendit liquidum, vel innumeros habent angustos meatus vicem tubulorum varie flexorum supplentes. 

Caeterum methodo inhaerentes, qua D. Pessuti LaPlacianam theoriam ad faciliorem formam traduxit, capillarium luborum phoenomena explicavimus in hypothesi liquidorum eamdem usque ad extimas omuino superficies obtinentium densitatem: non enim nobis in animo est vel leviter attingere novam theoriam, quam de actione capillari anno 1831 edidit D. Poisson.

== ACUSTICAE PRINCIPIA ==

=== Notiones preambulae ===
[[113|113]]. Acustica agit de sono: non defuerunt, qui sonum consistere putabant in efluviorum a soporo corpore vibratorum motu quae efluvia ex affrictu, vel contusione sonori corporis ejaculantur atque huic affinis est alia quaedam sententia, quod contusione illa vel affrictu particulae aeris purioris in eo corpore absconditi, vel ipsum circumdantis, expellantur et ad aures usque excurrant. Verum experimento machinae pneumaticae compertum est, quod incluso tintinnabulo vel horologio horas personante in recipiente, ubi aer incipit exhauriri, incipit sonus minui; ubi autem totus exhaustus est aer, nihil jam soni auditur, utcumque pergat tintinnabulum concuti, aut horologium pulsibus affici. Hoc probat sonum non consistere in effluviis a sonoro corpore vibrati cur enim non emittuntur amplius, aut ad aures non permeant, cum imo liberius ob minora obstacula deberent? &lt;u&gt;Ad majorem rei evidentiam&lt;/u&gt; ita hoc experimentum instituitur horologium in vitro aere pleno ac probe clauso reponitur, ne aer scilicet inde possit exhauriri tum in recipiente pneumatico collocatur, atque ex hoc educentes aerem animadvertimus sonum nullum audiri. Machina horaria aere circumsepta est ergo nullimode suspicari licet aliquid deesse circa ipsum corpus sonorum quominus sonus exaudiatur. 

Dicendum potius non audiri sonum propter defectum aeris intermedii inter utrumque corpus. Porro corpus cum resonat, motu tremulo atque &lt;u&gt;oscillatorio&lt;/u&gt; minimarum partium afficitur singulis autem oscillationibus aer corpus tremulam circumdans concutitur, similesque recipit vibrationes, quas in ulteriores particulas aereas pariter defert nisi quod impulsus in circumfusum aerem delapsi atque auditus organum afficientes eo minores ac debiliores fiunt quo magis a fonte recedunt. Enimvero corpora, quae sonora dicantur, tunc sonum excitant quando ita agitantur, ut illorum partes tremulo ac vibratorio satisque rapido concutiantur motu: sic campanae malleo percussae, ratione elasticitatis concipiunt tremoris motum, qui major fit postquam vehementius ac diutius agitatae fuerint: instrumenta musica, dum illorum fides agitantur, simili pariter tremore concutiuntur; hinc est quod chartae frustula resonanti corpori imposita subsultare cernuntur. His positis certum est tremorem similem communicari debere aeri immediate ambienti, et deinde tremorem late &lt;u&gt;diffundi&lt;/u&gt; per &lt;u&gt;aereas particulas&lt;/u&gt;; nam particulae aeris sonoro corpori proximae illius impulsu comprimuntur; et cum sint elasticae , post &lt;u&gt;compressionem&lt;/u&gt; &lt;u&gt;dilatantur&lt;/u&gt;, aliasque sibi proximas urgent; atque hoc pacto et illae vibrant, et longe lateque in particulas aereas similis vibratio omni ex parte discurrit. Hinc pulvisculi aeri innatantes, qui radio solis in obscurum conclave intromisso conspicui sunt, agitari videntur si sonus proxime intendatur: pulsato prope stagnantem aquam tympano, validius et crispari, et subsultare aqua pariter cernitur.

Haec notentur. 1º . Vis acceleratrix &lt;math&gt;\varphi&lt;/math&gt; in vibrante particula resonantis corporis ita pendet a spatiolo &lt;math&gt;z'&lt;/math&gt; quod particulae superest excurrendum usque ad nativam aequilibrii positionem, ut crescente, decrescente , vel evanescente &lt;math&gt;z'&lt;/math&gt; crescat simul, decrescat, vel evanescat; propterea&lt;math display=&quot;block&quot;&gt;\varphi = C'z' + C'' z'^2 + C''' z'^3 + ...;&lt;/math&gt;et quia particularum excursiones exsistunt exiguissimae, erit,&lt;math display=&quot;block&quot;&gt;\varphi = C'z':&lt;/math&gt;vis nempe acceleratrix assumi potest proportionalis spatiolo &lt;math&gt;z'&lt;/math&gt;. 

2º. Non pluribus opus est ut intelligamus (29.4º) vibrationes omnes, sive majores, sive minores ejusdem particulae fore aequidiuturnas.

[[114|114]]. Progignitur quoque sonus ab aere vehementer compresso seseque statim restituente: etenim propter impetum in restitutione conceptum ad majorem, quam in statu naturali occupabat, extensionem perveniet; ac proinde cogetur se rursus contrahere, minusque naturali spatio tenere. His autem successivis contractionibus et expansionibus in reliquo aere pulsus &lt;u&gt;excitantur&lt;/u&gt;: sic producitur sonus v. gr. virgae aerem celerrime perstringentis: simili modo qui in tibiam insufflat, sonum gignit; dum nempe per tubi orificium aer insufflatione intromittitur, ille, qui continebatur in tubo, necessario secundum longitudinem comprimitur; unde fit ut is iterum expandatur, tum denuo coarctetur; atque hoc pacto, quamdiu perseverat inflatio, perficiantur oscillationes, hisque sonus progignatur. Certe si aerea columna tubo &lt;u&gt;inclusa&lt;/u&gt; non afficiatur nisi motu totius, sonus minime obtinebitur; utcumque vero &lt;u&gt;excitentur vibrationes&lt;/u&gt;; ut &lt;u&gt;perceptibilem&lt;/u&gt; sonum edant, earum numerus intra minutum secundum non debet praetergredi quosdam certos limites, videlicet 6 circiter et amplius 24000; uti compertum est experimentis D&lt;sup&gt;''ni''&lt;/sup&gt; Savart.

[[115|115]]. Saepe contingit nos voce elatiori quibusdam in locis loquentes, aliquo tempore postquam siluimus repente audire rursus verba a nobis antea prolata; atque haec est illa echo, de qua plura fabulantur poetae. Philosophi in hoc conveniunt, quod echo sit motus reflexus aeris, qui &lt;u&gt;motu ondulatorio&lt;/u&gt; affectus obici incurrens resilit consimili motu, et rursum aures nostras afficiens nos determinat ad eumdem sonum audiendum, quem antea audivimus: ut autem effectus iste contingat, necesse est aliquanto longius a loquente obicem existere. Ratio est quia si &lt;u&gt;obex&lt;/u&gt; proximior fuerit, sonus reflexus efficiet in auribus impressionem suam antequam impressio soni directi defecerit; tunc vero non poterit secunda impressio a prima discerni. Aliquando semel tantum, aliquando saepius eadem vox per reflexionem auditur: primam contingit quando ab unico loco vox collecta rejicitur, vel a pluribus, sed ad eamdem distantiam: secundum quando vox in pluribus locis ad diversas distantias collecta revertitur ad aures sensibili successione. Hinc intelligitur quare in vallibus, quas undique colles cingunt, echo saepius iteretur.

[[116|116]]. Non solus aer est &lt;u&gt;medium&lt;/u&gt; idoneum transmissioni sonorum: nam per alia quoque elastica fluida propagatur sonus. Vapores ipsi, in quos aqua, spiritus vini etc. attenuantur, sonum transmittunt; etenim si recipiens pneumaticum aere atmosphaerico evacuetur, tum aliquo ex dictis fluidis repleatur, sonus campanae vel horologii adhuc bene audietur: quin et liquores, aqua v. gr. sonum non intercipiunt, sed ipsum debilitatum licet propagant; qui enim intra aquam sunt, audiunt sonos extra aquam editos; et qui extra aquam sunt, audiunt sonos editos intra aquam. Tandem etiam corpora solida deferunt sonos ad ingentes distantias: celebre est apud milites ita terram excavare donec strato alicui bene solido aurem applicare possint, ut ex reboatu agnoscant adventum hostilis legionis, praesertim equitatus; huic strato non raro tympanum imponunt, atque levia corpora tympano imposita ex sonoris tremoribus subsultant.

=== De intensitate soni; deque ejus gravitate, et acutie. ===

[[117|117]]. &lt;u&gt;Intensitas&lt;/u&gt; major vel minor soni importat majorem vel minorem ejusdem soni vim ad sensationem excitandam, quae proinde in intensiore sono vehementior est, ita ut aures prae violentia laedat aliquando; in remissiore ita debilis, ut vix aliquando audiatur. Iamvero evidens est quod quo plures sunt partes sive in corpore sonoro, sive in aere simul oscillantes, eo plus motus atque activitatis, caeteris paribus, habent; ac proinde vehementius organum auditus pulsare poterunt: quo singularum partium itus et reditus major est, seu quo fortius singulae particulae comprimuntur et restituantur in unaquaque oscillatione sive in corpore sonoro, sive deinde in aere, fortiori item impressione aptae erunt organum auditus afficere. Contra, quo pauciores partes sonori corporis oscillant, eo minus communicabitur motus particulis aeris, et consequenter ab his minus afficietor auditus organum: quo singulae sonori corporis partes unamquamque oscillationem minorem habent, eo minorem item oscillationem in aeris particulis producent, ac proinde impressione minus valida auditus organum concutient. Quod ratione perspectum est, &lt;u&gt;experientia quoque confirmatur&lt;/u&gt;; et quod ad sonum, quem vocant primitivum, attinet, corpora densiora, caeteris paribus, magis sopora sunt quam quae ex opposito; atqui hoc nonnisi quia plures particulae in his oscillant simul; ergo ex numero particularum oscillantium sonus major vel minor pendet. Rursum inter corpora aeque densa, atqae elastica, quod validius percutitur validiorem profecto sonum excitat et &lt;u&gt;magnitudo&lt;/u&gt; soni &lt;u&gt;magnitudini&lt;/u&gt; percussionis est proportionalis: undenam hoc repetendum est nisi ex eo quod validior percussio fortias comprimit atque oscillare vehementius cogit particulas elasticas? Quoad derivatum sonum res constat experimento machinae pneumaticae (113): cum enim exhauriri aer incipit, sonus incipit imminui; atqui hoc est quia  aeris quantitas in excipulo imminuitur; et cum rarior evadat aer, minus valide comprimi et restitui ejus particulae debent; neque enim ulla alia &lt;u&gt;probabilis&lt;/u&gt; causa est. Condensando insuper aerem in eodem excipulo ultra &lt;u&gt;statum ordinarium&lt;/u&gt;, quem tenet in almosphaera, compertum est quod condensatus aer sopam reddit intensiorem; atque hoc quidem ita, ut intensitatis augmentum proportionem servet cum augmento condensationis. Franciscus M. Zannotti diligentius rem exploravit: aerem inclusum vase calefecit; quo pacto aeris &lt;u&gt;elasticitatem&lt;/u&gt; auxit, &lt;u&gt;densitate&lt;/u&gt; eadem servata, cum nullus permitteretur aeri exitas; et tunc sonus intendebatur, At rima aliqua in vase relicta, per quam aer posset erumpere, tum igne admoto, sonus multo minor visus est quam antea fuerat. Cum igitur, permanente aeris elasticitate, non idem permanserit sonus, rursus patet quod soni intensitas non solum ab elasticitate, et consequenter a magnitudine vibrationum, sed a densitate, id est a numero particularum vibrantium dependet. Nec arte solum ex rarefacto vel condensato aere intensitas soni mutata deprehenditur , sed naturali etiam aeris rarefacti vel condensati constituțione idem evenit: hinc in altissimis montibus, ubi aer rarior est, ac proinde minus elasticus, sonus multo est remissior quam in planitie , ubi condensatione atque elasticitate pollet majori.

118. Ex his explicantur sequentia circa soni intensitatem. 

1º. In aperto aere sonus calore minuitur, in clauso vero calore augetur: apertus enim aer, ubi calore afficitur, sese continuo dilatat, adeoque ejus intensitas minuitur, quin &lt;u&gt;elasticitas&lt;/u&gt; augeri debeat; quia nempe habet quo se rarefactus recipiat; ergo minor numerus particularum oscillat, adeoque remissior sonus. Contra, si aer undique clausus est, cum densitas eadem manere debeat, elasticitas autem ex calore crescat, idem erit particularum numerus, sed singularum oscillatio propter auctam elasticitatem augebitur; ergo intensior sonus.

2°. Sunt qui dicunt, aestate sonum intensiorem esse, caeteris paribus, quam hyeme; alii contra opponunt, quod hyeme intensior sit sonus quam aestate. Si in re incerta quoad factum et ex circumstantiarum varietate adeo varia ut fortasse determinari non possit , si inquam ratio reddenda esset, ajendum sonum aestate imminui debere, quia aer terram ambiens calore rarefactus minori densitate pollet, ac proinde minor erit numerus particularum oscillantium. Cum autem ex calore elasticitas crescat , hoc capite augeri debet sonus , cum nempe singularum particularum oscillationes validiores debeant. Videndum igitur quid praevaleat; et juxta vel densitatem hyeine praevalentem imminutioni elasticitatis, vel elasticitatem praevalentem aestate imminutioni densitatis, qui effectus sequi debeat.

3º. Hinc etiam explicant nonnulli cur nocte, caeteris paribus, soni majores sint quam interdiu; quia nempe densior est per noctem aer ob calorem minorem; at hujus rei explicatio verior est, quod per noctem, cessante ea aeris commotione quae per diem habetur ex multiplici strepitu, magis aptus sit aer ad soni vibrationes concipiendas et deferendas, organumque auditus nulla alia sensatione percussum aptius sit ad peculiarem aliquem sonum exaudiendum.

119. Discrimen inter gravem et acutum in sono importare profecto debet diversitatem aliquam in motu aeris, quo afficitur organum auditus, atque adeo in motu sonori corporis ex quo in aere motus hujusmodi derivatur; nam cum sensatio sonii ex impressione organi auditorii oriatur, at omnis alia sensatio ex impressione organi proportionati, et impressio ista per motum aeris ad organum appellentis fiat, profecto diversa impressio, quae a sono gravi atque acuto fit, diversum motum exigit tum in aere ex quo immediate producitur, tum in corpore sonoro a quo mediate progignitur; atqui ista diversitas non ex validiori vibratione seu oscillatione majori provenit; ex hac enim quantitas sive intensitas soni (117), non autem qualitas seu tonus procedit; ergo diversitas ista in celeriori seu crebriori vibratione partium aeris, et consequenter sonori corporis, derivanda videtur. Ratio consequentiae est, quia non alia diversitas saltem probabilior in oscillatione partium concipi potest quam, ut haec sit vel major ut scilicet quisque itus et reditus spatium majus percurrat, vel quod sit celerior ut scilicet eodem tempore plures situs ac reditus habeantur. Ergo cum ex primo capite discrimen acuti et gravis repeti nequeat, nihil afferri probabilius potest: quam celeritas oscillationum, quae certe in satione diversitatem afferre debet. 

Quoniam vero in rebus physicis natura explorari maxime debet experimentis atque observationibus, ita prosequor. Constat in chordis musicis, eas quae vel breviores sunt, vel magis tensae , vel minoris diametri (nam ex hoc triplici capite diversitas tonorum habetur in fidibus) acutius resonare; contra graviorem sonum reddere eas, quae longiores sunt, vel minus tensae vel majoris diametri: atqui chordae breviores vel magis tensae etc. percussae, plures numero vibrationes pari temporis intervallo producunt, pauciores aliae; hoc patet ex ipso sensuum testimonio: ergo sonus acutus habetur in chordis, quae frequentius dato tempore oscillant; gravis autem etc. In ea insuper proportione, in qua frequentiores aut rariores sunt vibrationes chordae musicae, est etiam magis vel minus acutus sonus: ergo frequentior aut rarior vibratio omnino connexionem habet cum tono per chordam musicam producto; pendetque tonus ex illa &lt;u&gt;frequentia&lt;/u&gt; aut raritate vibrationum, tamquam effectus a sua causa. Quod dictum est de chordis musicis, valet etiam in campanis et pocalis vitreis, aliisque id genus sonoris corporibus; haec enim percussa figuram rotundam in ovalem mutant, eorumque proinde fibrae eundo et redeundo oscillare debent, atque ex hac oscillatione sonus oriri colligitur; ut autem gravior vel acutior est sonus corporis, ita in figura immutatio et restitutio seu fibrarum ítus ac reditus rariores sunt aut crebriores. Porro si id in sonoro corpore contingit, ut gravior sonus obtineatur quando minor vibrationum numerus habetur in corpore, jam tunc in aere quoque minor vibrationum numerus haberi dicendus est: siquidem tot numero vibrationes dato temporis intervallo producuntur iu aereis particulis a tremulo motu corporis resonantis, quot ab ipsius corporis sonori fibris seu particulis eodem tempore peraguntur; et vice versa quot in aere gigni ac propagari vibrationes constat, totidem in ipso corpore resonante produci dicendum est.

120. Dum plurium corporum sonus ita temperatur ut gratus sit auribus, dicitur consonantia seu concentus, si ingratum  sonum produxerint, appellamus dissonantiam: in sonis ita temperandis ut sint jucundi, ars musica versatur. Tonus musicus seu consonantia pendet ex eo quod certo tempore certus vibrationum numerus a pluribus sonoris corporibus peragatur, et particulis aereis communicetur. Si duo vel plura corpora sonora intra idem tempus vibrationem absolverint , consonantia est omnium perfectissima , et sonus dicitur unisonus ; si eodem tempore unum corpus unam , aliud duas vibrationes expleat, consonantia haec dicitur octava: ita appellatur ex eo quod per quandam tonorum seriem ascendendo hic tonus a musicis octavo loco constituitur. Si eo tempore quo unum duas vibrationes, aliud tres absolvat, adeoque secunda unius cum tertia alterius concurrat, dicitur quinta: si eo tempore quo unum tres, aliud quatuor vibrationes conficiat, quarta nuncupatur; atque istae sunt consonantiae illae, quas Pythagoras advertisse traditur, dum quinque fabri malleis ferreis massam ferream contunderent. Consonantiae istae in vibrationibus chordarum inventae sunt ; imo etiam alii successu temporis consonantiae gradus additi , quos diligenter musicae scriptores explicant. Si videlicet numeri vibrationum , quas dato tempore chordae  musicae efficiunt , sunt ut &lt;math&gt;1 , \frac9 8, \frac5 4, \frac4 3, \frac 3 2 ,\frac5 3,  \frac{15}8, 2&lt;/math&gt; chordae illae edent notissimos tonos ''do, re , mi , fa , sol , la , si , do'': constat experimentis saepissime iteratis; etenim chordae homogeneae , aeque crassae , eodemque pondere tensae , quarum longitudines sint uti &lt;math&gt;1 , \frac89, \frac 4 5, \frac 3 4, \frac 2 3, \frac 3 5,  \frac8{15}, \frac12&lt;/math&gt;praefatos tonos edunt. 

Haec subjungimus circa exiguissimas chordarum vibrationes. 

1°. Chorda homogenea &lt;math&gt;AB&lt;/math&gt; (Fig. 59) uniformiter crassa ubique tensa aequaliter, punctisque &lt;math&gt;A&lt;/math&gt; et &lt;math&gt;B&lt;/math&gt; fixa, traducatur ad datam formam curvilineam &lt;math&gt;AC''B&lt;/math&gt;; tum sibi relinquatur: pro quovis temporis momento determinanda proponitur curva &lt;math&gt;AC'''B&lt;/math&gt;, in quam abit chorda. Sint &lt;math&gt;AO ( =x)&lt;/math&gt; et &lt;math&gt;S'O ( = y )&lt;/math&gt; coordinatae orthogonales; &lt;math&gt;h&lt;/math&gt; longitudo chordae &lt;math&gt;AB&lt;/math&gt;; &lt;math&gt;M&lt;/math&gt; massa; &lt;math&gt;\theta&lt;/math&gt; tensio: in ea qua sumus exiguissimarum vibrationum hypothesi, maxima chordae elongatio ab aequilibrii positione cum sit ferme insensibilis, haec obtinebunt quamproxime. Primo: apud quodvis chordae vibrantis punctum Seadem vigebit constanter tensio &lt;math&gt;\theta&lt;/math&gt;. Secundo: movebitur &lt;math&gt;S&lt;/math&gt; juxta directionem &lt;math&gt;SO&lt;/math&gt; respondentis ordinatae. Tertio: denotante a angulum tenuissimum &lt;math&gt;S'EA&lt;/math&gt; interceptum tangente &lt;math&gt;S'E&lt;/math&gt; et abscissarum axe &lt;math&gt;AB&lt;/math&gt;, erunt &lt;math&gt;\alpha = \sin \alpha = \tan\alpha ;\, \cos \alpha =1&lt;/math&gt;. Quoniam exercetur &lt;math&gt;\theta&lt;/math&gt; juxta vibrantis chordae longitudinem; sumptis arcubus infinitesimis &lt;math&gt;S'i , Si&lt;/math&gt;, denotabunt &lt;math&gt;S'i\, \mathrm{et}\, S'i'&lt;/math&gt; directiones tensionum apud &lt;math&gt;S'&lt;/math&gt;: resolvatur tensio juxta &lt;math&gt;Si&lt;/math&gt; in duas, quarum altera existat parallela rectae &lt;math&gt;AB&lt;/math&gt;, altera perpendicularis eidem &lt;math&gt;AB&lt;/math&gt;; et idipsum fiat quoad tensionem juxta &lt;math&gt;S'i'&lt;/math&gt;. Componentes parallelae axi &lt;math&gt;AB&lt;/math&gt; se mutuo destruent; componentes vero perpendiculares ipsi &lt;math&gt;AB&lt;/math&gt; exprimentur per &lt;math&gt;\theta\sin\alpha&lt;/math&gt; versus &lt;math&gt;O&lt;/math&gt;, et Osini atda ) versus S , seu per Ox et Oatd « ). Superest igitur vis
- Oda
gignens motum juxta SO : differentiale da sumendum quoad
x tantum, utpote denotans variationem anguli a in eadem curva AC &quot; B. Quisque videt --Oda esse vim motricem, cujusmodi est tensio  &lt;math&gt;\theta&lt;/math&gt;: propterea, designante dm elementum massae , exprimetur per
Oda
dm
∶≀≤↓⇟≓
miter crassa, ubique tensa aequaliter, punctisque A et B fixa, traducatur ad datam formam curvilineam AC&quot;B;
tum sibi relinquatur: pro quovis temporis momento de-
terminanda proponitur curva AC&quot;'B, in quam abit chorda. Siut AO (:æ) et S'O (::y) coordinatae orthogonales; '
h longitudo chordae AB; M massa;9 tensio: in ea qua
sumus exiguissimarnm vibratiouum hypothesi, maxima chordae elongatio ab aequilibrii positione cum- sit ferme in-
sensibilis , haee obtinebunt quamproxime. Primo: apud
quodvis chordae «vibrantis punctum S' eadem vigebit constanter tensio 9. Secundo: movebitur S' inxta directio-
nem SO respondentis ordinatae. Tertio :denotante et an-
gulnm tenuissimum S'EA interceptum tangente S*E et
abscissarum axe AB, erunt ut :sina −∙−−−− tangat ; cos at −−−∶↿∙
Quoniam exercetur 9 iuxta vibrantis chordae longitudinem :
[snmptis arcubus infinitesimi: S'i
, S'i', denotabunt S'i et S'i'
directiones tensionum apud S': resolvatur tensio iuxta Si
in duas
, quarum altera existat parallela rectae AB, alte-
ra perpendicularis eidem AB; et idipsum fiat quoad ten-
sionem iuxta S'i'. Componentes parallelae axi AB se mu-
tuo destruent ; componentes vero perpendiculares ipsi AB
exprimentur per Osina versus O, et 9sin( at-l-dat) ver-
sus S , seu per 90: et B(a-I—dat). Superest igitur vis
—9dat
gignens motum juxta SOI: dili'erentiale dat sumendum quoad, utpote denotans variationem anguli &amp; in ea-
dem cnrva AC'&quot;B. Quisque videt —-9dat esse vim motri-
cem , cuiusmodi est tensio 9: prapterea
, designante dm
elementum massae , exprimatur per
Gala
&quot;2711-255
respondens vis acceleratrix. Ob uniformem chordae cras
sitiem ,
dx
h
dm
M
Mar
ideoque dm =
h

;

insuper
a = tang a =
dy
dx ;
sumptisque differentialibus quoad x ,
da dany
dx dx²
Facto itaque compendii causa
on
M
superior expressio vis acceleratricis traducetur ad
d²y C2 dx²
unde ( 28 )
day
da(SS)
dia
d ” (SO - SO )
dc2
d²y
dx² de²
seu
1
255
respondens vis acceleratrix. Ob uniformem chordae cras-
sitiem ,
, ideoque dni ∙∙∶−− —-—de ;
9.&quot;M :.
da: ∙∙∙⋅ h −∙−
insuper
at:—— tangat-agi ;
sumptisque diiferentialibus quoad a:
,
data:-dv dx dx: ⋅
Facto itaque compendii causa
911
∙∙−∙∙↽−∙∶∘∙ ∙ M
superior expressio vis acceleratricis traducetur ad
da
⋅−∘⋅⊒≀−⋛−⋮⋅ ∙
nnde (28 )
,↶≀≖↗∙∙ irss; -dz(so-s'b) ∙∙∙ a.,
c dx&quot; dt: d? d;:
— '
sen256
day c2 day
dia (a) .
dx2
Formula ( a) suppeditat quaesitam problematis solutionem.
2. • Fac ut vis acceleratrix sit ut 1 , nimirum
day C'y ; erit dta
der
· c?
day
dx2 C'y
1 seu tör
so . dra
Inde habemus ( 27 , 27.0 )
VVT
y=CC + C, e
с - CV-4
evanescente X , evanescit et y ; hinc C = -C, , et con
sequenter ( 27. 30. )

* V0V1

y = C , [e
- *70V1
1 =
2011'sin rc=2C1V= 1sin
2 VMCMC
h9
facta x = h , evanescet y ; proinde
sin k V MC
ik V
MC'
ho TT C' =
OTE2
LM .
ho
9
ordinata CC &quot; respondens abscissae AC ( **)
256 ∙
da,, ⋅
−−↙⋮∎⋅≒↿∣
dx? :d—t; (a) . .
jl
C:
Formula (a) suppeditat quaesitam problematis solutionem. *
2.&quot; Fac ut vis acceleratrix sit uty
, nimirum '
lude habemus (27. 27.?)
f.. t/CV −−↿
-..-z ∁∣⇂∕∶⋅↿⋅
yiL-3010 c
⊣⋅∙ O; 6 - c :
evanescente a:, evanescit et )&quot;
, hinc C,: ---Cl , et con-
sequenter (27. 30. ∘ )
⋅−∙⊽⋮⋅∣∕∁⋅⇂∕∶↿ ... ⋅⋮∸−⇂∕∁⋅⇂∕∶↿
c c
y-—-—C,[e -—e ]∶∶
2C1V —-1 'sin −⋅⋮−− ∣∕∎∁⋅∶⊋∁∎≖∣∕∶−↴ sinx 9:709. :
facta x:h , evanescet y; proinde
. ⋅∙∥⋅∪⋅∙∙ VH?&quot; 97:-
&quot;[III 119 ∙−−−∘∙≀∙ WC;&quot;, (:::-IIM:
,
ordinata CC'&quot; respondens abscissae AC (;.-ä 11) ex-
'257
hibeatur per y ' , erit

;; = - 20, vt in . V MC =20,V = tsin.V MORE

TT
2C,V -1 sin î
2C, V 31 .
Propterea
2 = sin -
77 ()a' ) ;
aequatio ad curvam AC''B.''
3.° Per ty denotetur tempus unius semivibra
tionis ; erit ( 29. 3.° )
TT
1,5
2V C
VhM

;

0
et consequenter tempus unius vibrationis
hM ta VRM
Ad haec : designante n numerum vibrationum , quae ip
tra temporis unitatem absolvuntur , exsistet
1
V
TANTE
12
In hypothesi chordae cylindricae habentis radium
r el densitatem , erit M = fErPhò ; ideoque
257
hibeatur per J&quot;, erit
. -—- .
': MC' −−
∙ h M9112
r;.—20. ⇂∕⋅−↿ sm ∙− ∙−−−⇌⊋∁≖⇂∕⋅−∙↿ 810 2— IPGM :: 2 116
∙−− 7! −∙∙ ≢∁⋅↾∕∙−↿ sin ∙−⇇∋∙− −−−∙⇌ my.—1 .
Propterea
yzy' sin :; 71 (a') ;
aequatio ad curvam AC'&quot;B,
3.0 Per t. denotetur tempus unius semivibra—
tionis; erit ( 29. 33 )
et consequenter tempus unius vibrationis
.:Vg.
Ad haec: designante n uumerum vibratiouum, quae in-
tra temporis unitatem absolvuntur , exsistet
In hypothesi chordae cylindricae habentis radium
r et densitatem 8 , erit Mr.-:Ttrïhö ; ideoque558
13=rkVis, n - EVO
4. • Facta Osy. , velocitas puncti S in fine
temporis ( erit ( 29. 1.° 2.° )
v = y.Vī sine VC -Yosin
hinc ( 29. 1. ° )
=V
9. C - 02
C
yo V1 - sin’LVA
yo coseV C sy= . COS cos r.
Simili modo , facta CC“ =jo , velocitas pancti C in
fine temporis ( erit
7 t
yo sin Ti; simulquey'= y's cososeme- T ;
ta
et aequatio (a' ) ad chordam vibrantem poterit scribi in
hunc modum
y = yo com-A sin
C
-TT h
( á ').
5.° Si abscissae x in ( a'') substiluitur vel anh''
vel ( 2n + 1) h, prodibit y = o quotiescumque n aut erit =0,
aut erit quivis numerus integer : binae videlicet
558
9
≄≖−⊣⋅↗≖⇂∕∂ ∙
∙≖⋮−−⇀⊑⋅−−≀≖ '?Eä'
4.0 Facta OS;-:]. , velocitas puncti S in line
temporis : erit (29. 1.&quot; 2.')
v': J/ö' sint t/&quot;ä ∶∶−∶−≖−∫∘ sin-;- tt
!- 8
hinc (29. 1.&quot;-)
∙⊺∶−−−∙∙⇂∕⋅↗⇗ (S'—v ∙−−− J. l/1—s1n'q/ Q'::
j'. costV C' :y, cos ∙⋮− 11.
2
Simili modo
,
facta CC
∙−−∶ y'.. , velocitas puncti C&quot; in
fine temporis :erit
' n
s
∙ t ∙
' ' :
si:—y., s1n——1r;stmnlquey-:yocos—1t ;
t

: , :,

et aequatio (a') ad chordam vibrantem poterit scribi in
hunc modum
∙−− '
cst nsinæn ⋅ (a&quot;) J—yo O.t2 h .
5.0 Si abscissae a: in (a&quot;) substituitur vel an]:
vel (a'n-l-nh, prodibit yzo quotiescumque 11 aut erit 20,
aut erit quivis numerus integer: binae videlicet
Je!259
x = 2nh ,
( 2n+1 ) h
spectabunt ad quiescentia chordae vibrantis puncta. In
ferimus illud : chorda AB produci potest ultra limites A
et B quin puncta A et B per iteratas chordae vibra
tiones a statu quietis dimoveantur , etsi puncta illa poo
nuntur de se mobilia ; modo tamen AB in eamdem ac
antea conformelur initialem curvam , eidemque subjiciatur
tensioni : imo sumpta BH = HH' = =h , ita vibra
tiones suas conficiet chorda ABHH ' .
ut puncta A,
B , H , H ', ... in quiete persistant. Ad istiusmodi vi
brantis chordae figuram quod pertinet , sit v . gr. HD
HD = AO = x ; erunt AD = AH -HD = 2h - x , AD = 2h + x :
in la &quot; ) substitue prius 2h-x , deinde 2hta loco x ;
provenient ordinatae yı ety respondentes punctis D et D ',
nimirum
visy.cos Ti sin ( 2
a sin 16 는 (2-m ) R = my'o cos t2
sa= com sin ( 2+ )n = foco na sio ža .
Igitur y = -y, ya= y : ordinatae scilicet y , y , sunt
aequales , et ad eamdem plagam obversae ; ordinatae ve
ro y , y sunt quidem aequales , sed obversae ad con
trarias plagas. Chorda itaque dividitur in partes alterna
tim vibrantes supra et infra rectam AH'.
6.** Quoad (a) generatim spectatam ; denotanti
bus f et F binas functiones arbitrarias , satis hic erit
animadvertere eam expletum iri per
y =-flatct) + F ( x - ct) (a ' ' ) ;
siquidem
259
a.:2n71 ∙ :r (2n—l-1)I;
spectabunt ad quiescentia chordae vibrantis puncta. ln-
ferimus illnd: chorda AB produci potest ultra limites A
et B quin puncta, A et B per iteratas chordae vibra-
tiones a statu quietis dimoveantnr, etsi pnncta illa po-
nantur de se mobilia; modo tamen AB in eamdem ac
antea conformetur initialem curvam, eidemque subjiciatnr
tensioni: imo sumpta BH −−−−− HH' ::
...
zh
,
ita vibra-
tiones suas conficiet chorda ABHH' . .. ,
ut puncta A,
B , H , H', ...
in quiete persistant. Ad istiusmodi vi-
brantis chordae figuram quod pertinet ,
sit v. gr. Hl):
HD'zAOsæ; erunt AD;:AH-HDzah-æ
, AD':.2h-l—æ :
in (a&quot;) substitue prins alz—a:, deinde Zh—l-æ loco :;
provenient ordinatae y. etj, respondentes-punctis D et D',
nimirnm
'
tnsin(2 −⋅⋮≻↿∎∎∶ 'cos tu' æ
fac:-Tou.&quot; (: 'l &quot;70 :: Olli-i:&quot;.
, s ∙ æ ,
t
. æ
Jar—jre cos-1t am (2 −⋅⊢ --)1t :yocos —-1t sm --1t
.
- :, h : 11
Igitur y. ∙∶−−−∙ −∫∙ ∙↗≀≏−−−−−⋮↗↟⇌ ordinatae scilicet y , ;, sunt
aeciuales ,
et ad eamdem plagam obversae; ordinatae ve-
ro y, y, sunt quidem aequales, sed obversae ad cou-
trarias plagas. Chorda itaque dividitur in partes alterna-
tim vibrantes supra et infra rectam AH'.
6
∙∘∙ Quoad (a) generatim spectatam; denotanti-
bus f
et E binas functiones arbitrarias
,
satis hic erit
animadvertere eam expletum iri per
Fnæ-l-ct) −∣⋅− P(æ—ct) (if&quot;) ;
siquidem'260
dº[fix + c ) + F(x – ct) ]_da[fixtet) + F (x – ct )] .
do[ ) dt2
7 . ** Velocilas puncli S in fine temporis i prodit
expressa ( 28) per
dOS - OS')
dt
dy
dt [flatct)-F&quot;(x – ct)]:
initio motus , quum nempe t = 0 , est v=0 ; iccirco
c [ f '( x )-F'(x )] = 0, $' ( x)=F&quot; ( x) , et f (x ) = F (x ) ;
aequationes igitur determinantes et curvam ASB , et ve
locitatem traducentur ad
y = f(x + ct) + f(xớctct), v '= -c[ f '( x + chf'( x - ct) ] .
Facto t = 0 , istarum prima praebebit
y = 2f \x ) ,
aequationem videlicet ad curvam datam ACSB : ex hac
itaque curva pendet natura functionis f. Caeterum , ge
neralem de integratione differentialium partialiumque ae
quationum doctrinam suo tempore videre erit in parte 3.4
nostrorum elementorum Matheseos n. 200 , 201 ,
121. Si chorda instrumenti musici percutiatur , et pro
pe adsit instrumentum aliud , in quo chorda sit ad aniso
num cum priore tensa , baec alterius instrumenti chorda
sensim tremere incipiet , et undulationes sensim majores
concipiendo ad sonum ipsa quoque excitabitur eumdem to
num reddendo quem prior illa chorda percussa reddit .
Jam vero si ad hujus rei rationem attendas, plana erit juxta
theoriam traditam : sicut enim pendulum quiescens etiam
minimo impulsu , uti est qui ex insufflatione procedit , ad
260
⊄≀⇟⊏∣≺↕⊣⊸↥≻ −⊢ Für—ct) ],. «l*[m—l-aH-Fw—ctü.
dt:
(
7. ∙∙∙ Velocitas puncti S'111 fine temporis :prodit
expressa (28) per
os.-os d
v. ∙∙∙⋅ &amp; dt ):... .... :]? ∶−∙−− —c[f(æ-l-ct)—-F'(:r—ct)]:
initio motus , qunm nempe t::o
,est ⇂↓−∙−−−∘⋅ , iccirco
c[f (æ)—-F' (x)] ∙−−− o, f(xrr—F' (x)
, etfix):F(æ)' ,
aequationes igitur determinantes et curvam AS'B ,
et ve—
locitatem v' traducentur ad
y-fþ—I—ct) *Aæ—ct) , ∙≀⋅−−− ∙−−− ∙−− c[f '(æ-i-ctF-f'w—ct) ].
Facto t--—-o
, istarum prima praebebit
F2nx) :
aequationem videlicet: ad curvam datam AG&quot;B : ex haei
itaqua curva pendet natura functionis f. Caeternm, ge-
neralem de integratione differentialium partialinmqne ae-
qnationnm doctrinam suo tempore videre erit in parte 3.&quot;
nostrorum elementorum Matheseos n. 200, 201, .-. .
.-

121. Si chorda instrumenti musici percutiatur, et prope adsit instrumentum aliud, in quo chorda sit ad unisonum cum, priore tensa, haec alterius instrumenti chorda sensim tremere incipiet, et undulationes sensim maiores concipiendo ad sonum ipsa quoque excitabitur eumdem tonum reddendo quem prior illa chorda percussa reddit. 

Jam vero si ad huius rei rationem attendas, plana erit iuxta theoriam traditam: sicut enim pendulum quiescens etiam minimo impulsu , uti est qui ex iusufilatione procedit , ad motum oscillatorium minimum primo concitabitur , et si in
suflationem saepius repetas , poteris sensim oscillationes
majores , ac majores perficere (tunc tamen id fiel quando novi isti impulsus certa periodo, parique intervallo habeantur; si enim pendulum contra insufflantem venit, insufflantes rursum potius motum impediemus quam adjuvabimus, atque idet' finita una oscillatione debet opportune rur
sus alius impulsus addi , sicque ubi oscillationes penduli
et novi impulsus certa periodo sibi respondeant , effectus
habebitur) ita in chorda praefata evenit ut trémitns con
cipiatur et augeatur ; donec excitetur sonus ; quia nempe
Oscillationes unius chordae consentiunt cum oscillationibus
ad quas altera determinabilis est , iccirco ex repetitis chor
dae percussae uşdulationibus , quae sunt isochronae undulationibus alterius , obtinebitur ut hae augeantar donec so
nus excitetur in chorda etiam plectro minime percussa. Ex hac doctrina infero: ergo in utraque chorda oscillationes sunt pares numero; ergo cum tonus ab utraque redditus idem sit, tonus igitur a numero vibrationum hujusmodi pendet. Ad magis declarandam traditam doctrinam de acutie et gravitate sonorum, utile erit non nulla hic explicanda proponere circa chordas vibrantes. 

Ac 1º. quamvis chordae non sint unisonae, attamen una percussa, alia sonum edit, si modo tensae sint ad octavam, aut alias quasdam habeant armonicas proportiones. 

2º. Si duae chordae tensae sint ad octavam, et pulsetur chorda longior; quae dimidia ejus est, reddet tonum sui proprium, scilicet octavam acutam; at si pulsetur chorda brevior, excitabitur in longiore tonus non sui proprius, scilicet ad octavam gravem, sed tonus chordae brevioris. 

3º, Refert Sauverius hoc phoenomenon: chorda longa 5 ped. percutiatar, et notetur tonus;
tum ad distantiam unius pedis ponatur supra chordam le
ve aliquod obstaculum velati plumae frustulum , quod ta
men non impediat molus communicationem : si quinta haec
1
1
1
261
motum oscillatorinm minimum primo concitabitur ,
et si in-
snæationem saepius repetas, poteris sensimf oscillationes
maiores , ac maiores perficere (tunc tamen id fiet quando
novi isti impulsus certa periodo», parique intervallo babe-
autur; si enim pendulum contra'insumantem venit ,
insuf-
Hantes rursum Potius motum impediemus quam' adiuvabi-
mns , atque idet-' finita nna'osci'llatione dehet opportune rur-
sns alius' impulsnsgaddi, sicque ubi oscillationes penduli
et novi impulsus certa periodo sibi respondeant , effectus
habebitur) ita in chorda praefata evenit ut tremitus con-
cipiatur et augeatur, donec excitetur sonus; quia nempe
oscillationes unins chordae consentiunt cum oscillationibus
ad quas altera determinabilia est
,
iccirco ex repetitis chor-
dae percussae undulationibus , quae sunt isochronae undu-
lationibus ulterius-, obtinebitur ut hac augeantur donec ac-
nos excitetur in chorda: etiam plectro minime percussa. Ex
hac&quot; doctrina infero: ergo in utraque chorda 'oscillationes
sunt pares numero; &quot;ergo cum tonus ab utraque redditus
idem sit, to'nus igitur a numero vibratiouum hujusmodi
pendet. , - - -
'
'
'
Ad magis declarandam- traditam doctrinam de acutie
et gravitate sonorum, utile erit non nulla hic &quot;explicanda
prcponere circa chordas vibrantes.

q'uod ta-
men non impediat motus communicationem: si quinta haec
.*262
P
-chordae pars pulselur, tongm efficiet proprium chordae d
- nias pedis; hunc autem ipsum tonum reddit etiam reliqua
chordae pars, etsi quadrupla. Rursum si obstaculum pona
tur post duos pedes , eveniet ut pars brevior citius oscil
let, et longioris motum perturbet; subinde utraque chor
dae pars ita , sese componet; ut vibrationes eodem tempo
re compleat: tunc vero tonus reddetur neq w proprius chor
dae duorum pedum , neque trium, sed proprius chordae u
nius pedis.
Ad primum quod attinet , quoties duae chordae len
sae sunt ad octavam, jam vibrationi unius chordae ,respon
dent duae vibrationes alterius; ergo quamvis singulae , O
scillationes non conveniant, adeoque tremitus aeris non re
novet impulsum in alia chorda post singulas ejusdem oscil
lationes, renovari tamen potest impulsus hic post binas ;
eo ipso poterit chorda ad octavam tensa , etsi difficilius ,
ad oscillandum determinari ex alterins oscillationibus. Idem
valet de aliis chordis quae eam habent proportionem ut
oscillationes recurrere possint post aliquem ipsarum nu
merum: ac proinde illae, quae vel ejusmodi recursum non
admittuut , vel quarum recursus majorem postulat' quam
par est vibrationum numerum, non ita invicem ad reso
nandum poterunt determinari.
Ad secundum : quod chorda brevior resonans ad pulsa
tionem longioris reddat tonum sui proprium , cohaeret cum
doctrina jam tradita : quod autem chorda longior reddat
Lonum proprium chordae brevioris non officit; etenim si
chorda sit dupla , quasi in duas dividetur, neque tota oscil
Jabit ( 120..5º. ) per modum unius, sed habens in medio
punclum quiescens, seu nodúm, oscillabit seorsim in sin
gulis dimidiis partibus, ac si, scamould adjecto , bifariam
arte divisa esset ; , et si chorda ' triplo sit longior , ia - tres
partes aequales dividetur: quo posito , nil mirum quod chor
da dupla non sui proprium tonum , sed tonum subduplae
reddat, et tripla sonum subtriplae.
ic
0
at
LE
262
chordae pars pulsetur, tonum efficiet proprium chordae n-
onius&quot; pedis; hunc autem ipsum tonum reddit etiam reliqua
chordae pars,; etsi quadrupla. Rursum siobstacnlum pona-
tur post duos pedes'. eveniet ut pars brevior. citius oscil-
.let,. et, longioris motnm perturbet; subinde utraque-. chor-
dae pars ita sese componet&quot;, ut vibrationes eodem tempo-
re compleat: tunc vero tonus reddetur neq ,.: proprius chor-
dae duorum pedum, neque trium, sed proprius [chordae u-
,niuspedis. ↴
⋅⋅∶⇟⋡∢⋅ ∙

Ad primum quod attinet, quoties duae chordae tensae sunt ad octavam, iam vibratioui unins chordae respondent duae vibrationes alterius; ergo quamvis singulae oscillatioueslnon conveniant adeoque tremitus aeris non renovet impulsum in alia chordae post singulas ejusdem oscillationes, renovari tamen potest impulsus hic post binas; eo ipso poterit chorda ad octavam tensa, etsi difficilius, ad oscillandum determinari. ex&quot; alterius. oscillationibus. Idem
.valet de aliis chordis-anae-eam habent prOportionem ut
oscillationes recurrere possint post- aliquem - ipsarnm nn-
merum: ac proinde illae, quae vel ejusmodi recursum non
admittunt
,vel quarum recursus majoreni- pastulat' quam
par est vibratiouum numerum, non ita invicem a'd reso-
.nandum poterunt determinari.

Ad secundum: quod chorda brevior resonans ad pulsationem longioris reddat tonum sui pmprium cohaeret cum doctrina iam tradita: quod autem chorda longior reddattonum - proprium  chordae brevioris non officit; (etenim-si
chorda sit dupla, quasi in &quot;duas dividetur,- neque- tota oscil-
Jabit (120. 50.) per modum unins, bed habens in medio
punctum quiescens, s'eu nodnm, oscillabit seorsim in sin-
gulis dimidiis partibus, ac si, scamnnlb adiecto , bifariam
arte .'divisa esset;. et: si chorda' triplo sit longior, in- tres
partes aequales dividetur: quo posito, nil mirum-quod chor-
da dupla non aui proprium tionnm, sed tonum'subduplae
reddat, et tripla sonum subtriplae.
n-rts lar-Q .-263

Ad tertium: idem Sauverius hanc in Academia Pari
siensi explicationem attulit . Dum chorda nullo obstaculo
apposito pulsatur, vibrationes efficit toti suae longitudini
proportionales: at dum leve illud obstaculum apponitur post
pedem unum , undulatio totalis chordae dividitur ; prima
enim pars chordae , utpote quinta chordae totius ,
quinquies citius oscillare debet quam oscillaret integra
chorda : sic citius oscillando abripiet partem sibi proxi
mam in vibrationes aequales ; secunda pars tertiam, atque
ita singulae quinque partes seorsum oscillationes pera
geat. Alterum vero, quod magis est admirabile, ila ab eo
dem auctore explicatur ; pars brevior chordae, scilicet duo
rum pedum, citius oscillans quam reliqua , secum abripit
per sui motus communicationem partem sibi similem, nem
pe duorum pedum; in quinto autem pede oscillationes e
tiam communicantur, quae cum esse debeant longitudini
proportionales, duplo crebrius oscillabit extrema haec chor
dae pars quam reliquae; proinde ista sibi proximam unius
pedis partem trahet ad analogas oscillationes , et secunda
tertiam atque ita de reliquis , donec in hoc etiam casu quin
que chordae partes oscillent juxta longitudinem propriam ,
et consequenter sonum reddant respondentem longitudini
upius pedis.

122. Quaeri potest quomodo sonus trans obicem queat communicari ita, ut tonus proprius sonori corporis permaneat; nam fibrae, seu partes elasticae obicis puta parietis aut cancelli vitrei, ad motum concitatae vel sui proprium tonum reddere debent, vel si dissimiles sint, plurium tonorum mixturam, quod non accidit. Respondeo nullam esse difficultatem, si immediate per aerem soni propagatio habeatur, etiam intermedio exsistente obice. Quod si per obicem sonus diffunditur, in ipso admitti possunt partes aptae diversos sonos reddere aerique transposito communicare; atque ita, ut ille sensibilis sit trans obicem tonus, qui a partibus analogam oscillationem habentibus cum sonoro corpore communicatur. Forte etiam dici potest, quod si fibrae non habentur aptae eum tonum reddere, dividantur, ut in chorda non unisona contingit, adeo ut idem tonus transmitti possit.

123. Quoniam de tonis, ex quibus qualitas soni denominatur, egimus; quaerendum esset unde asperitas aut lenitas, quae pariter ad qualitatem quamdam soni pertinet, proficiscatur. Animadverte sonum quemcumque non esse simplicem, sed compositum e sono plurimarum sonori corporis partium: sic chorda musica percussa non simplicem edit sonum, sed quemdam veluti concentum edicit , qui a
peritioribus musicis probe dignoscitur; in quo tamen cum
fortior tonus praevaleat, alios minores obruit : coexsistunt
videlicet in chorda sonora, et generatim in quovis particu-
larum s'ystemate, variae exiguarum oscillationum species.
Imo vero non tantum sonorum ipsum corpus attendendum
est plerumque v. gr.-chorda musica, sed instrumentum i-
psum cui chorda adhaeret: variae insuper reflexiones ani-
madverti debent, quibus aer ad aurem deveniens diversas
subit modificationes. Itaque si vibrationes partium sonori
corporis sint bomologae, sonus lenis erit; si contra, asper:
atque hinc aspere sonant chordae inaequales in materia ,
crassitie etc; item ex reflexione aequabili atque uniformi
sive instrumenti, cui chorda adhaeret, sive circumstantium
corporum, lenitas soni orietur, asperitas ex opposito. 'Bo-
num erit observare quod chorda musica vehementius quam
par est distraCta stridet; quia videlicet valde percussa non
eam' servat legem quam in moderatis percussionibus obti-
net ut sub eodem tempore oscillationes suas sive majores,
sive minores dbsolvat; sed continget ut tempora oscillationum
inordinate mutentur, stridorque pro tono solito erumpat.

124. Haec notentur 

1º. chordarum vibrationes hactenus consideratae, dicuntur transversae: quae nimirum- obtinen-
tur chordam percutiendo in directione ad ejus axem perpendiculari: quod si atteratur chorda in directione ad e
jus axem parallela, adhuc sodos edet, sed , caeteris pari
bus, multo acutiores quam qui ex vibrationibus transver
sis progignuntur ; idque ex eo repetendum esse videtur
quod elasticitas propria chordae in vibrationibus longitudi
nalibus validior sit quam in transversis.

2.° Ubi in longitudinalibus vibrationibus chorda
rum obtineant &lt;u&gt;nodi&lt;/u&gt; , molus ita fiet ut partes hinc illinc cira
ca podum quemlibet positae simul ad ipsum nodum accedant, simulque alternatim recedant.

3º. Corpus omne, dum resonat, dividitur in plu
res partes vibrantes invicem ' separatas lineis , quae vocan
tur &lt;u&gt;nodales&lt;/u&gt;, quaeque oculis subjiciuntur spargendo per su
perficiem corporis minutissima arenae grana: haec enim su :
per lineis illis acervari observantur. Nodales propterea li
neae modo' sunt rectae, modo curyae, modo ex rectis si
mul et curvis coalescunt.

4.º  Malála nodalium linearum figura, plerumque mutatur et sonus; semper autem acutior vel gravior evadet sonus, prout corporis superficies in majores vel minores numero parles vibrantes dividetur ab ipsis noda libus lineis.

5.° Laminae rigidae ex ferro, vitro etc. in transversis vibrationibus absolvendis sequuntur leges alias ab illis, quas sequuntur chordae.

=== De directa soni propagatione per aerem. ===
125. Experientia nos edocet quod in iisdem circumstantiis sonus aequabili velocitate in toto decursu devehiеur; atque omnes soni , sive intensi , sive remissi , sive graves, sive acuti eadem velocitate diffunduntur.  

Nam 1.º Academici Florentini ad percurrendam distantiam unius milliaris sonum tormenti bellici impendisse quinque secundorum tempus experti sunt, ejusdem vero tormenti sonum ad conficiendum dimidium milliare impendisse dimidium tempus testantur aequabili nimirum velocitate perrexit sonus. Derhamus saepius repetitis experimentis idipsum invenit, adeo ut ab uno ad duodecim milliaria sumens intervalla invenerit aequale spatium aequali tempore in quavis a sonoro corpore distantia confici. 

2.º Prope sonorum corpus intensior est sonus, remissior in majore a sonoro corpore distantia atqui tam prope quam procul a sonoro corpore aequali velocitate pergit sonus ergo tam intensus, quam remissus etc. Hoc ipsum institutis ad id experimentis etiam constat Gassendus sclopeti et tormenti bellici fragorem eodem tempore pervenisse affirmat, cum eodem tempore exploderentur. Florentini et Derhamus in diversi generis tormentis idipsum evenisse notant itemque tormenti bellici minoris et mallei fragorem idem unius milliaris intervallum confecisse eodem tempore. Certum est ergo tam intensum, quam remissum etc. Huc spectat quod Derhamus quoque notat post Florentinos, scilicet eodem tempore sonum ad aures pervenire sive tormentum ad observatorem convertatur, sive ad contrariam plagam videtur enim intensior in eam partem, in quam tormentum dirigitur, esse debere sonus.

3.° In concentu sive ex instrumentorum pulsatione, si malleorum ictibus etiam ad satis notabilem distantiam dignoscitur tonorum successio eo praecise ordine, quo ictus varios tonos producentes habentur successive, et quidem sine sensibili temporis mora atqui si toni diversi non eadem propagarentur velocitate, jam qui toni successive habentur, non successive atque ordine illo ad aures venirent ergo etc. Erit fortasse qui quaerat qua ratione fieri possit ut sonus in quavis distantia, sive intensus, sive remissus, uniformiter &lt;u&gt;propagetur&lt;/u&gt;. Respondeo: eadem materiae quantitas eodem tempore, tum ex vi majore, tum ex minore, undulare potest ergo eadem aeris portio, seu &lt;u&gt;unda ejusdem latitudinis&lt;/u&gt;, eodem tempore potest undulationem perficere, sive ex majori, sive ex minori vi impellente. Antecedens est evidens; pendulum enim idem , adeoque eadem massa , eodem tempore oscillationes peragit sive magis , sive minus impellatur ad oscillandum: ergo a pari eadem aeris quantitas oscillare potest sub eodem tempore sive ex majori , sive ex minori impulsu. Sed si eadem aeris quantitas aequali tempore potest comprimi et restitui , jam eodem tempore potest sonus, ad datam distantiam pervenire , sive intensior , sive remissior: haec minor est evidens; si enim eadem est &lt;u&gt;latitudo undae&lt;/u&gt; , idemque tempus, jam eodem intervallo temporis spatium datum a sono conficietur; ergo sive intensus sit , sive remissus , seu vi majori aut minori aereae undae propellantor, eadem esse potest soni velocitas.

Quid ergo provenit ex hoc quod in sono intensiore vis major aerem impellat? Nempe quod ejusdem latitudinis unda, licet eodem tempore conficiatur , compressionem tamen ac restitutionem patiatur validiorem , vel languidiorem; sicut in pendulo accidit , quod eodem tempore oscillans ex impulsione maiori oscillationem concipit magis validam , et minus ex vi minori. Atqui hoc idem praestat minorem intensitatem , non autem minorem soni velocitatem . Ostendo: intensitas soni pendet a vi , qua in organum appellunt aeris particulae ; ergo si vi majore condensantur , et restituuntur , intensiorem efficient soni sensationem; at velocitas ex dictis pendet a latitudine undae, et tempore quo perficitur: neque latitudo immutatur , neque tempus; ergo non mutatur velocitas. Quod autem neque latitudo , neque tempus mutetur , ita probari potest. Latitudo enim undae , seu aeris quantitas ad oscillandum per modum unius determinata , ea esse debet quae potest obtemperare vibrationibus sonori corporis , a quo unda producitur , quaeque potest oscillationes suas eodem tempore complere quo sonorum corpus oscillationes suas perficit: ergo latitudo undae proportionari debet tempori quo sonorum corpus perficit vibrationes suas. Atqui sive intensior , sive remissior sit sonus, tempus quo sonorum corpus vibrationes suas complet , est ( 113. 2.°) semper idem; ergo item latitudo undae aereae eadem esse semper debet. Idem probat simul, quod sicut eadem latitudo, ita idem esse debet tempus quo unda perficitur. Et sane si tempus mutaretur , deberet quoque mutari tonus: atqui idem manet tonus in quacumque distantia a sonoro corpore , et quidem sive corpus resonet intensius , sive remissius; ergo etc.   

Hinc dum de sono agitur duplex in motu undae aereae velocitas distinguenda est: altera importat tempus quo unda conficitur , seu quo segmentum aeris datae latitudinis oscillat ; altera importat motum particularum aerearum itum et reditum perficientium in ejusdem undae efformatione. 

Quaeri hic potest in quanam ratione intensitas soni minuatur in progressu . Reponunt communiter quod intensitas soni est in ratione duplicata distantiarum inversa a centro soni : rationem afferunt , quia sonus quantum est de se aequabiliter undequaque diffunditur in modum sphaerae. Atqui ex hac aequabili in modum sphaerae diffusione sequitur decrementum in ratione praedicta ; nam si ita diffunditur , debet in ea proportione intensive decrescere , qua extensive augetur , sea qua latius materia , cui communicatur motus , sese expandit ; sed hujusmodi extensionis augmentum est in ratione duplicata distantiarum ; hanc enim rationem sequuntur sphaericae superficies : ergo etc... . 

126. Sit c velocitas , qua propagatur sonus ; &lt;math&gt;\Delta&lt;/math&gt; distantia inter vibrantem sonori corporis particulam et particulam aeream : exprimet tempus a sono impensum ad percurrendam distantiam &lt;math&gt;\Delta&lt;/math&gt; ; motusque particulae vibrantis nonnisi post tempus I = pertinget ad aeream particulam: propterea substituto 2— —c- 'a duabus ulti-
mis formulis(29. 5.&quot;), si :
∙−−≜−⋅ incipit ab 0
, ultraque
progreditur, determinabitur aereae particulae motus per»
∙ 271: A , 9 211 .A.
∦⋅⇋↙∁∘∎∐∙−⊖−⋅≺∁−−∘−−≻ ,
szC-Z-n' 008
! j(t—z).
F30i20,1t,2,3,4,-...,act—-e—:i9,tln- c
de habes A:c(t—-i9): erit
⇂↓∣∶⇂∕∁ sin 21'12:o.
Sumptis ergo distantiis Azct, c(t—G), c(t—29), c(t—BG), ...,
uulla velocitas v' ibi invenietur : aer proinde in locisi il-
lis omnino quiescet quando desinit tempus :; eritque n—
sque ad Ar.-ct in plureswundas distinctum similes et aequa-
les ; quarum communis latitudo ::09 ; numerus vero

:

∆−∘−⊖− ⋅
∆
Quantitas l/C sin −⋛≖−≺ t — A;) manet positiva ab
t — 30- :::-id ad :
2 — (i ] &amp;) 6 ; manet-negativa
A . A .
⋅ ∙ ∙ ∙
ab t—-—c- :::(12-l-ä)9 ad t— -c—-.-:(1-l-1)9. Ertt 1g1-
tur v' positiva inter A———-0(t—i9) et A −−∶ c [t—(i—i—ä— )9];
erit negativa inter Ach-t—(i—i-ä-W] et A:c[t—(i-l—1)9].
in tribus hisce distantiis est praeterea v':o. Ergo quae-
libet ex dictis undis constat duabus partibus aequalibus ;
recedit aereum fluidum ab oscillante sonori corporis par-
ticula in anteriora parte, accedit in posteriore; quiescit stra-270
tum medium ; maxima viget aerearum particularum velo
citas in medio semiundae anterioris ; maxima item in me
die semiundae posterioris.

127. Soni velocitas augetur a vento secundo, minui
tur ab adverso. Derhamus videns ab aliis affirmari nullam
mutationem afferri a ventis circa soni velocitatem , hanc
rem statuit explorare ita exacte et diu , ut ambigendi lo
cus omnis tolleretur. Ad hoc autem summa ipse fruens
opportunitate experimenta habebat omnino in promptu .
Nam cum ex arce Blancheath , ubi tyrones rei tormenta
riae exercebantur , saepe exploderentur tormenta bellica ,
ipse e sua Ecclesia in agro Upminsther ad 13 milliaria
distante flammam advertere poterat ; animadvertit autem
optimo usus chronometro non semel aut iterum , sed
triennio integro. Porro ex tabula , quam observationum
suarum confecit, quaeque habetur in Transactionibus An
glicanis, et a Masschembroekio descripta fuit in suis com
mentariis ad lentamina Florentinorum , constat quod so
ni velocitas inter tempus quo ventus favens spirabat ,
et contra venius sono adversus erat, cum scilicet in utro
que casu yentus validus admodum esset , discrepat un
decim semisecundis circiter in praedicto intervallo. Ergo
experimentis hisce insistendo dicendum augeri secundo ven
to soni velocitatem , imminui autem etc.
Derhami observationibus consentiunt observationes Aca
demicorum Parisiensium , qui anno 1738 exploraturi ve
locitatem soni jussu Regiae Academiae pariter testantur
non eandem esse adverso ac secundo vento velocitatem
qua propagatur. Rationis momentum experientiae suffra
gatur : nam ventus transfert loco aerem ; ergo undas so
noras ad oscillationem a sonoro corpore impulsas trans
fert ; ergo tantum accelerari debet propagatio soni , quan
tum aeris sonori translatio ratione venti importat.
Opporluna est comparatio circulorum in aqua exci
latorum ope lapilli decidentis : si enim aqua non sit sta
270
tum medium; maxima viget aerearnm particularum velo-
citas in medio semiundae anterioris; maxima item in me-
die semiundae posterioris.

271
SUS asserue
gnans sed fluens aequabili motu ; jam dum post lapidis
descensum circuli successive efformantur , lota ipsa aqua,
in qua efformantur circuli , localiter transfertur ; ergo
circuli appellent ad datum locum citius in eam partem
versus quam defluit aqua quam ad alteram in eadem
distantia ex opposita parte : ita paritate rationis in sono.
Iis , quae . modo diximus , objici possunt experimenta
Cl. virorum qui de velocitate soni nihil immutari pror
sive secundus sit , sive adversus ventus
runt. Gassendus enim , et Mersennus id sibi accidisse te
stantar ; et Academici Florentini , collocatis observatori
bus inter se duo milliaria distantibus , dum ventus spi
raret , asserunt tormenti bellici , quod medio illo inter
vallo situm erat , fragorem pervenisse eodem tempore ad
utrosque , etsi aliis faveret ventus , aliis esset contrarius.
At cum experimenta recte instituta aliis item recte
institutis nequeant esse contraria , yidendum quaenam
praevaleant. ' Florentini , quorum tentamen prae caeteris
in medium afferri solet , unica nocte , et in distantia pau
corum milliarium experimentum instituerunt . Derhamus
triennio experimenta iteravit , et in 13 milliarium distan
tia ; haec autem distantia in experimentis Derhami eadem
erat semper , a sua scilicet Ecclesia ad arcem ; in ten
tamine Florentinorum tormentum constitutum est medio
inter duos observatores intervallo ; quod intervallum utrin
que aequale asseritur , sed fortasse non ita esse potuit.
Ergo apparet quam recte Derhami observationes antepo
nantur observationibus Florentinorum , atque eodem jure
Gassendi et Mersenni . Reipsa etsi experimento quodam
Musschembroekius quoque deprehendere sibi visus fuerit
aequalem velocitatem tam secundo quam adverso spirante
vento , tamen Derhamo assentitur , et Florentinis quo
rum sagacitatem saepe alibi commendat , minime in hoc
adstipulatur.
Obiter hic notamus quod juxta auctores ferme omnes
etiam intensitatem sąni auget ventus secundas , et minuit
.
1
271-
gnans 'sed fluens aequabili motn; jam dum post lapidis
descensum circuli successive eil'ormantur ,
tota ipsa aqua,
in qua eB'ormantur circuli ,
localiter transfertur; ergo
circuli appellent ad datum locnm citius in eam partem
versus quam defluit aqua quam ad alteram in eadem
distantia ex opposita parte: ita paritate rationis in sono.
Iis
, quae-modo diximus, objici possunt experimenta
Cl. virorum qui de velocitate soni nihil immutari pror-
sus , sive secundus sit , sive adversus ventus
, asserue-
runt. Gassendus enim-, et Mersennus id sibi accidisse te-
stantur; et Academici Florentini, collocatis observatori-
bus inter se duo milliaria distantibus , dum ventus spi-
raret
, asserunt tormenti bellici , quod medio illo inter-
vallo situm erat
,
fragorem pervenisse eodem tempore ad
utrosque, etsi aliis faveret ventus
, aliis esset contrarius.
At cum experimenta recte instituta aliis item recte
institutis nequeant esse contraria , videndum quaenam
praevaleant.x Florentini , quorum tentamen prae caeteris
in medium afferri solet , unica nocte
.,
et in distantia pau-
corum milliarium experimentum instituerunt. Derhamus
triennio experimenta iteravit, et in 13 milliarium distan-
tia; haec autem distantia in experimentis Derhami eadem
erat semper, a sua scilicet Ecclesia ad arcem; in ten-
tamine Florentinorum tormentum constitutum est medio
inter duos observatores intervallo; quod intervallum utrin-
que aequale asseritur , sed fortasse non ita esse potuit.
Ergo apparet quam recte Derhami observationes antepo-
nantur observationibus Florentinorum , atque eodem iure
Gassendi et Mersenni . Reipsa etsi experimento quodam
Musschembroekius quoque deprehendere sibixvisus fuerit
aequalem velocitatem tam secundo quam adverso spirante
vento, tamen Derbamo assentitur , et Florentinis , quo-
rum sagacitatem saepe alibi commendat, minime in hoc
adstipulatur.
Obiter hic notamus quod iuxta auctores ferme omnes
etiam intensitatem soni auget ventus secundus
,
et minuit
.272
1
P
TE
8
ta
11
11
adversus. Hoc , ajunt , experientia vulgari notum est : si
quidem campanae sonus , aut tormenti explosi fragor multo
melius auditur si conspiret in eam partem ventus quan
si contrarius sit ; et saepe ad aliquam distantiam auditar
ope venti secundi, ad quam , cum ventus est adversas ,
minime audiri potest : auget ergo ventus soni intensita
tem. Ratio quoque idipsum suadet : nam vencus secundus
undas sonoras transfert ; ergo ad aurem appellent undae
sonorae , quae a centro minus remotae erunt , adeoque
intensiorem deyehunt sonum .

128. Ad soni velocitatem determinandam multa in
stituta sunt experimenta , quae tamen non satis conve
niunt : experimenta instituta ab Academicis Parisiensibus
anno 1738 praebuerunt soni velocitatem , seu spatium minuto
secundo a sono percursum = 172 , 56 hexap. = 336 , 32
metr. Apud Madras in India orientali D. Goldingham ex
perimentis per annum integrum multoties repetitis ( Annal.
de Plays . et de Chim . tom. 23. pag. 12 ) exploravit soni ve
locitatem : prodiit mediocris velocitas 1134 , 33 ped.
Britan . = 345 , 74 metr. Varias hujusmodi mensuras vi
dere est in tabella , quam protulere DD Moll , Van-Beek
etc. ( Bibliotheque universelle tom. 30) : qui Auctores opus
definiendae velocitatis soni susceperunt anno 1823 , perfe
ceruntque in Hollandia , assumpto ad observationes eo spa
lio , quod Zevenboompies et Koolijesberg interjacet. Ten
tamiva sumpta die 28 Junii praebuerunt soni velocitatem
339 , 34 metr.
Hujus diversitatis plures esse possunt rationes : ac 19. In
strumenti aut attentionis exquisitae ad instrumentum deſe
ctus ; cum enim flamma attendi debeat simulque penduli
oscillatio , jam facile est ut vibratio aliqua initio non nu
meretur. 2. Spatium exiguum ab aliquibus assumptum ;
minimus enim error facilius est contemaibilis , si ingens
intermediet spatium. 3.° Venti qui aut retardant , aut ac
celerant souum . llaec variationis causa attenuari potest , ac
PL
M
ti
.
0
272 '
adversus. Hoc , aiunt , experientia vulgari notum est: si-
quidem campanae sonus ,
aut tormenti explosi fragor multe
melius - auditur si couspiret in eam partem ventus quam
'si comrarius sit :
et saepe ad aliquam distantiam auditur
Ope venti secundi, ad quam, cum ventus est adversus,
minime audiri potest: auget ergo ventus soni intensita-
tem. Batio quoque idipsum suadet: nam ventus secundas
undas sonoras transfert; ergo ad aurem appellent undae
sonorae , quae a centro minus remotae erunt, adeoque
intensiorem devehunt sonum.
128. Ad soni velocitatem determinandam multa in-
stituta sunt experimenta , quae tamen non 'satis conve-
niunt : experimenta ⋅ instituta ab Academicis Parisiensibus
anno 1738 praebuerunt soni velocitatem
, seu spatium minuto
secundo a sono percursum ∙−−∶ 172 , 56 hexap. −−∶ 336, 32
metr. Apud Madras in India Orientali D. Goldingham ex-
perimentis per annum integrum multoties repetitis (Annal.
de Phys. et de Chim. tom. 23. pag. 12) exploravit soni ve-
locitatem :prodiit mediocris velocitas :: 1134
, 33 ped.
Britan. −−∙− 345 , 74 metr. Varias hujusmodi mensuras vi-
dere est* in tabella , quam protulere DD Moll
, Van-Beelt
etc. ( Bibliotheque unive'rselle tom. 30) :qui Auctores opus
definiendae velocitatis soni susceperunt anno 1823
, perfe-
ceruntque in Hollandia , assumpto ad observationes eo spa-
tio , quod Zevenboompics et Kooltjesberg interiacet. Ten-
tamiua sumpta die 28 Junii praebuerunt soni velocitatem
−∸−⇁∙ 339 , 34 metr.
Huius diversitatis plures esse possunt rationes: ac ↿∘∙ In-
strumenti aut attentionis exquisitae ad instrumentum defe-
ctus; cum enim flamma attendi debeat simulque penduli
oscillatio , iam facile est ut vibratio aliqua initio non nu-
meretur. 2.*' Spatium exiguum ab aliquibus assumptum;
minimus enim error facilius est contemnibilis , si ingens
intermediet spatium. 39 Venti qui aut retardant ,
aut ac-
celerent sonum. llaec variationis causa attenuari potest
, ac
J
.
maälzz—äwæ-EL'T-aa
&amp;.273
ferme destrui , si collocatis tormentis bellicis in utraque
extremitate A et B illius distantiae , quam debet sonus
percurrere , tormenta ipsa eodem temporis momento ex
plodantur ; tunc enim si determinetur velocitas , qua per
venit sonus ex A in B , itemque velocitas qua pervenit ex
B in A , harum velocitatum semisumma erit velocitas illa ,
qua propagaretur sonus in aere tranquillo. 4.º Animadvertit
Musschembroekius quod cum sonus non in instanti audia
tur , sed initio minus , subinde organum aliquanto vehe
mentius percellat, hinc quidam ad initium , alii ad progres
sum sensationem soni potuerunt animadvertere , atque inde
inter se discrepare. 5.9 Varia atmosphaerae temperies.
Determinatio velocitatis , qua sonus propagatur , utilis
est nautis ut agnoscant quantum littus , aut alia navis
distet ; militibus ut quantam oppugnata urbs distet ; geo
graphis item ut quantum inter duo loca , praecipue cum
intervallum hexapeda metiri non licet , intersit . Etenim nu
merando minuta secunda ab erumpente flamma ad usque
audiendum sonum tormenti bellici , distantia loci colligi
potest . Quod si ea sit locorum constitutio ut flamma videri
non possit,o res ita supplenda est , ut cum ad aurem per
venit souüs , exploso statim alio tormento bellico , alter hic
sonus ad primum observatorem perveniat : si hic nume
ravit minuta secunda ab eo puncto , quo explosit suum
tormentum usque ad punctum quo audivit sonum alterius
tormenti , et haec minuta bifariam dividantur , habebitur
tempus propagationis soni inter duo illa loca : ita etiam nu
bis distantiam aliqui metiri docent , numerando scilicet mi
nuta secunda , quae inter fulgur emicans et auditionem to
nitrus intersunt .

129.# Nonnulla subjicimus ex theoria fluidorum
( 106. 107 ) ad soni propagationem applicata ; ita tamen ,
ut ad aeris gravitatem minime attendamus , libratu mque
aerem spectemus tanquam elasticum fluidum eadem ubique
pollens et densitate p' , et pressione a' , et temperie n.
273
ferme destrui
.
si collocatis tormentis bellicis in utraque
extremitate A et B illius distantiae . quam debet sonus
percurrere ,
tormenta ipsa eodem temporis momento explodantur; tunc enim si determinetur velocitas , qua pervenit sonus ex A in B , itemque velocitas qua pervenit;:
B in A , harum velocitatum semisumma erit velocitas illa,
qua propagaretur sonus in aere tranquillo. 49 Animadvertit
Musschembroeltius quod cum sonus non in instanti audia-
tur, sed initio minus , subinde organum aliquanto vebe-
mentius percellat, hinc quidam ad initium , alii ad progres-
sum sensationem soni potuerunt animadvertere , atque inde
inter se discrepare. 5.o Varia atmosphaerae temperies.
Determinatio velocitatis
, qua sonus propagatur
, utilis
est nautis ut agnoscant quantum littus ', aut alia navis
distet: militibus ut quantum oppugnata urbs distet; geographis item ut quantum inter duo loca , praecipue cum
intervallum bexapeda metiri non licet ,
intersit. Etenim nu-
merando minuta-secunda ab erumpente flamma ad usque
audiendum sonum tormenti bellici
, distantia loci colligi
potest . Quod si ea sit locorum constitutio ut flamma videri
non possit A, res ita supplenda est
,
ut cum ad aurem per-
venit somä , exploso statim alio tormento bellico
, alter bie
sonus ad primum observatorem perveniat : si bic numeravit minuta secunda ab eo puncto
, quo explosiot suum
tormentum usque ad punctum quo audivit sonum alterius
tormenti , et haec minuta bifariam dividantur , habebitur
tempus prcpagationis soni inter duo illa loca :ita etiam nu-
bis distantiam aliqui metiri docent , numerando-scilicet minuta secunda , quae inter fulgur emicans et auditionem to-
nitrus intersunt.
1294 Nonnulla snbiicimus ex theoria fluidorum
(106 . 107) ad soni propagationem applicata ; ita tamen ,
ut ad aeris gravitatem minime attendamus
,
libratumque
aerem spectemus tanquam elasticum fluidum eadem ubique
pollens et densitate ≀⊥⋅ ,
et pressione a: ,-et tmperie n.
..—274
10 Fac ut concutiantur librati aeris particulae
comprehensae sphaerico spatiolo habente radianı = (y , et
centrum in coordinatarum origine 0 ; talem vero patiantur
in densitate variationem , et velocitatem recipiant juxta re
spondentes radios vectores a , ut utraque exsistat admodum
exigua , et altera queat repraesentari per f ( ) , altera per
f ( Q) , evanescentibus fg , f quoad a = o et « &gt; « ,: sit r
distantia puncti ( x , y , z) ab 0 , ut obtineant
i
x2 + y2 + z = p2 xdx + ydy + zdz = rdr ,
Propagato motu per reliquum fluidum ; quoniam v' , v &quot; , 20 &quot;
sunt constanter parvissimae , et ad fluidi gravitatem minime
attendimus , iccirco formulae ( 6 &quot; , 106 ) , missis terminis
exiguissimis secundi ordinis , factisque X = 0 , Y = o, Z=0,
dabunt quoad punctum ( aco y, z)
1 do dui 1 do dv &quot; 1 das
de
dv'&quot;
dt &gt;
M dx dt I dy to da
et consequenter
lo
I can do to edip dy+
dz dz
du dvi'
dy + dt dt (©) .
Jam vero
dic dir -dx do
dy
do dr
dr dx &amp; ar ፊ dydy 9
dy
dosdz dz
da dr
dr de dz 2
.
274
↿∘∙≖∎⊀ Fac' ut concutiuntur Iibrati aeris particulae
comprehensae sphaerico spatiolo habente radium −−−−≖ a, ,
'et
centrum in coordinatarum origine 0; talem vero patiantur -
iu densitate variationem , et velocitatem recipiant iuxta re-
spondentes radios vectores &amp;!
, ut utraque exsistat admodum
exigua, et altera queat repraesentari per f! (a) , altera per
f (a) , evanescentibus !; ,
f quoad ac −−∶ ∘
et a) 0:' :. sit :-
distantia puncti (æ
,y, :) ab 0, ut obtineant
x' ∙−⊢∫∙⊣−≖≏∶−−≀∙≖ , ædx-i—ydy-t-zdzzzrdr,
PrOpag'ato .motu per reliquum Huidum ; quoniam v', 11&quot;, v'&quot;
sunt constanter parvissimae ,
et ad fluidi gravitatem minime
attendimus ,
iccirco formulae (ö&quot; . 106 ), missis terminis
exiguissimis secundi ordinis
,factisque X::o, ïze, Zzo,
dabunt quoad punctum (æ, y, :)
1 da der ↿⊄∄↑≖⋅∙∙∙∙ dv&quot; 1
ftdæ— dt'pdj&quot; dt'p.
et consequenter
1 der da der − − ... −−⋍≀ )
↽−
⊬
(dxdx—i- dy d),—i- dz :.
eiu' dv&quot; dv'&quot; )
∙
—- de—k-äuy—F—ät— d: (i)-
Jam vero
(la-: (lux dr dar das dr
,
(Erit: ∙−−−−∙⊋−∣∙∙ (?;-lld? ïydj :
z; gd)»- ,
dadz—dw drdz ,275
ac proinde
du do dos
dx + dy +
dz =
dx dz 1
do Idr dr dr
• dxt
dr dy
do-dr
) = dr
v , &quot; =
insuper v ' un
v , ideoque
dv' dv &quot; dv '' d (v'da tudytou '&quot; dz) dx +''
-dyti dz =
dt dc dc dt
dfædx + ydy + zdz
d (vdr)
dc dt
traducetur igitur ( i) ad
1 do dr d ( vdr)
dt -
( i ) .
f . dr
Ponentes
dQ u'dx + u'dy + v &quot;da = dQ ,ut sint v'=
dx 10&quot; :
dQ
dy
2011 !
dQ
dz
assequimur
dQ d
To
d
Come)
vdr d (vr)
dr,
dc dr -
' de
dr dr :
dr dt
vertelur itaque ( i ) in
275
ac proinde
Heia-4- — ;d; -]-d 2; ad;:
≤↾−⋮⋅↾ ïta.-.- −∙⋅⊄∄↗∙⋅∹−∙≦− ∙⋅−⋤−↙∄⇝⇌∶−∙⋡−−↙≀≀∙ ;
'
∙−−−⋅⋮∙⋅ ∙−− £ &quot;zl. lnsuPero—rv,-v' '.—r-v,-v&quot; rc:,ideoque
dv' *d-v&quot; ...-'de: &quot;' d(v'dx -1-v&quot;dy—1-v'&quot;da)
ïdïdæ'l' dc ↙↡↗⋅⋅⊢ ⋅⊋−⋮⋅∂≖ dt ∙−−−⋅
d
(ædæ A-ydy ∙−⊢ zdz 0)

: - d(wdr) ∙↗

d: ∙
&quot;' dt
traducetur igitur (i) ad
1 du! ∙∙∙ d(vdr) .,dr ∙−−− −− dt (( ).

Ponentes
u m ∙
l
∙
'o v'dx-t-m dy—t—v ds:dQ,ut sint d
∙↗∶⋛−≣−∙⇝ :::-g. *v
⋅⋅∙−−−∶∙−↿⋚≳−∙ '
assequimur
⋅
dQ d(vr) d —&quot;'Q) d (....) — (dr ∙∙∙⋅ ⋅ dt
.
⇀
mi'-&quot;ïd&quot; d: −⇀ dt 4&quot; ∙− dr 4&quot;
vertetur itaque (t&quot;) in276
1 do Cena
( i &quot; ) .
hdr dr
Pertingente motu ad punctum (x , y , z) , crescit ibi librati
aeris densitas M , et evadit
l = h' ( 1 + $) ;
augetur aliquantulum etiam temperies n in ipso condensa
tionis actu , fitque ntv : pressio , quae ob auctam den
sitatem evaderet a' ( 1 + 8) , augescit adhuc propter incre
mentum v ; et cum v pendeat ab € , novum pressionis in
crementum pendebit rursus ab z , eritque ob incremento
rum tenuitatem ipsi &amp; ad sensum proportionale ; iccirco ,
praetermisso é , emerget pressio ex duplici capite aucta
m = (1 + 5) (1 - +-AE) w [1+ (1 + A ) £] .
Poterit ergo ( i&quot; ) sic scribi
1
ale
de de .
( 1 + A
M
17 € dr dr &gt;
seu
dt
is 13
( 1 +A)
dL ( 1 + -E)
dr dr
Hinc
Bis
( 1 - +- A ) L ( 1 + E)
dQ
dt
276
Pertingente motu ad punctum (a:, y, :)
, crescit ibi librati
aeris densitas p! ∙ et evadit
it:-&quot;a' ≺↿−⊦⋮≻⋮
augetur aliquantulum etiam temperies 1: in ipso condensa-
tionis .actu
,
(itque n—l— »: pressio , quae ob auctam den-
sitatem evaderet m' (1 ∙−⊦ e) , augescit adhuc propter incre-
mentum »; et cum 9 pendeat ab a , novum pressionis in-
crementum pendebit rursus ab a , eritque ob incremento-
rum tenuitatem ipsi a ad sensum proportionale: iccirco
,
praetermisso ?
, emerget pressio ex duplici capite aucta
a:d(1-—1—s)(1-1-Aa) −−∶ a'[1-t-(1-t-A)s] .
Poterit ergo (i&quot;) sic scribi ∙
' d(ig) a' 1 de dt −− ↿⊣⇁∆∼ ∙−− — pii '
↿⊣−∙∊ dr dr
'
seu dc.-112) . :
p.
liinc
' d
⋮⋝−∽ ≺↿⊣⇁⋀≻↧∙≺↿⊣−⊽∊≻↽−∙−−−∙− 3- - p. dt277
est autem ( 27.29º. )
?
L ( 1 + E) = E
+ - +
Propterea , facto ( 1 + A ) = C ,
A dQ
&quot; .
ca do
Ad haec :
dv ' dy&quot;
dx
dur
dz
d’Q
dx² +
d’Q
dy ?
+
d2Q
dz² ;
dy
formula igitur ( 619. 107) , substituto p. ( 1 + €) loco fe , mis
sis terminis exiguissimis secundi ordinis, atque attenta ( i'''),'''
praebebit
d2Q daQ
dea = ca
e
d Q
dy ?
det
d2Q
da ?
( it ) ;
\ dx²
et quoniam
dQ dQdr dQ y
dr dx
dQ dQ x
dQz
dr
of
dQ_dQ
dxi
dz dr dy dr
unde
d’Q
dx²
daQ xa
dra 2
dQy? +z2 d2Q
dr p3 dy?
d’Qys , dQ x2 + z3
dr ra dr p3
d'Q
d22
d2Q 22
dr.2 p2
dQ x2 +y2
dr 产 产
277
est autem (27 .290.)
e* 53 si
1 :−∙− ∙−−− Ou: ∙∙ ↥⋅≺−⊢∊≻ s ⇄⊣⋅∙∃ 4(.,, :
Proptereü , factO :
(1 :A) −∙− c,,
1 dQ
ca dt
Ad haec :
d'v' dv&quot; ⊣⇀ ↙∣⊛∣∦ ↙≀≏⊄⊋ sz dïQ .
da: d] dz −⇀⋅ dx: d),: d:&quot; a
formula igitur (ö&quot; . 107) , substituto p: (ii-145) loco p. , mis-
sis terminis exiguissimis secundi ordinis, atque attenta (zw'),
' ∙ ?' praebebit
sz sz daQ sz .&quot;
.
(.i—t;. ⇀−− c&quot; da,-3 ∙−⊦ ∠∄∫≖ .* dzg) (: la.
et quoniam ⋅
'
∙ '
dQ dQ dr-—dQæ dQ—JQJ, iq—æi
dæ' drdæ dr-r'äy dr r'dz—drr'
unde ( ⋅ ⇁ ⋅ ,
dj—æ i,*deail'zz dag—dïQlyiA-iQxa—an dat.:—dr: rr: dr &quot;3 ,dyl d'.) rg ∙ &amp; r3
dQdeina *igæ'ä-J' .
d:&quot; dr: ra dr 'a &quot;.278
ideo traducetur ( i&quot; ) ad
d’Q
dia coloro
d-Q
( dra
2 dQ
r Thedrbest seu da (rQ )
dla ca d ( )
dra
Ex (i) habemus ( 120. 6º. )
Q = -- [80+ c ) + F(r — ct)] ;
et consequenter
dQ 1
dr [ f'ir tt) + F' ( r - ct )]
) — ]
( i&quot; )
Ar + c ) + F (r - ce}]
- [f(r + c )—– F&quot;( – ce)].
1 dQ
c2 dt
Ad f et F determinandas , sume t=0 ; habebis
f(a ) f (a ) : . E =
proinde
a&gt; f( x ) = af ( a ) + aF'( a ) f « ) - F( a ) ,
- caf( a ) = f ( ) – F' ( « ) .
Pone
fa) +F(a) = w , fra ) — F( X) = w ;
erunt
.
278
ideo traducetur (i&quot; ) ad
432—
. «PQ-,. 2
sit'—c &amp;? 747↲≺≀≻∙⊷≖∂↿≺↾≬⋗−≖∙↲≖≺↗≺≀≻ de'—' dn
Ex (.&quot;) habemus 120. 60.)
- 1
Q ∙−−− ;- [f(r.:i- ct) −⊦ F(r— et)] :
et consequenter−∙∙ :? ∙−−⋮∙ [f'(r-l-ct)-]-—F'(r-—ct)]

—--—1r;-[f(r-]-ct)-]—F(Qr—ct)] ,
1 dQ— ↿
s —
⊑
ca d: ;S.-[f(r-t-cn— F'(r— cs )].
Ad f
et F determinandus, sume :::-o; habebis
w:f(a)
, s:f,(a):
proinde ⋅
æf(a):af(a)—1—al-&quot;(a) -—f( cc)—P(a),
—eaf,(a)——:f(a)—F'(a).
Pone
fe) -t-F(a) :::.)
,f(a)- F(ac) ∶−∙−∾⋅
erunt
0&quot;)279
d @ = f( ) + F&quot;(x)= f(a) —F(«) da = f( x )da ;
dw ' = [f ( ) — F ' ( ) ] da = -ca f ( ) da ;
unde
a fixdx , w ==
cfafica) da :
hae suppeditant
f(Q )
w -two
2
1
2 frazda - of facada,
F(x)= afscada + ; fafceda;
ideoque ( iº )
f(x)= ff( )fat a pascafica),
Standa+ af )+ caf,ca).
F ( a )
2
20# Secunda membra (2011) evanescunt quoad a &gt;
Az ; ut igitur functiones
flrtct) , f'(x + ct) , Fr — ct ) , F &quot; (r — ct)
sint aliquae , non debet r ct esse &gt; &amp; : atqui in ordi .
ne ad fluidi particulas ultra Qi , cum e sit quantitas posi
tiva, est semper s + ct &gt; As ; ad has ergo particulas quod
attinet, erunt constanter
279
d(a-i)— af(a)—t-aF'(ac) —f(a) --F(a) dae— f(a)da;
« - æ
dar.-:. [f(az) —- F' (et)] da :: — cat &amp; (et) da;
uude '
∙∾−−−∶∶∝ «a)daz , Q':-irc af,(a)daz:
bae— suppeditant
aH—a' 1 - 1
f(a)-— 2 ∙−− ⋣∙ ⊄∫⇟↸∝⋟↙≀∝−−−⋮−−∘∫∝ f,(a)da,
c.)—of 1 1
Hall- 2
&quot;*.2 «li(alda—r—ïcfafdaW-ï &amp;
ideoque (i'&quot;)
1 1 1
f(a): -2- f(a') fat—1- ä-a ((a)—ïm f,(a) ,
↿ 1 1
F'(a) ∶−∙−−∙ ∙⋮⋅≳−∫∫≼∝⋝↙≀∘⊢⊢ -2-af(a)-1- -2—- caf,(a).
2011 Secunda membra (im) evanescunt quoad ac)
«,.; ut igitur functiones ⇀
f(r-t-ct) , f(f-Jf- ct)
, F(r -
ct )
,F'(r
-
et)
sint aliquae , non debet r

: b et esse )a, :atqui in ordine ad fluidi particulas ultra et, , cum t sit quantitas posi- tiva, est semper :- −⊢ ct a, ; ad has ergo particulas quod attinet, erunt constanter280 fir + ct ) = 0 , f ( r + c ) = 0 ; et consequenter -F(r —c)F( r -ce ) , 6 = 1.- F &quot; (r — ce) (**** ) . 30 Aereae particulae respondentes radio vectorir non incipiunt moveri nisi quum tempus sic increvit , ut habeatur rct = ly , seu r = ctt cy : inferimus sonum propagatum iri uniformiter velocitate V ( 11 + A ) Quod spectat ad numerum A, habemus (87. 70. ) a = im [1 + a (n + v)] = im '(1 + E)[ 1-+ an + ») ] , itemque ( 10.) 5 '[1+ (1 + A )ɛ] =; if' ( 1+ an) [1 + ( 1 + A ) ]: hinc i '(1 + E)[ 1-+-ant-v) ] = iu'1 + an ) [1+ ( 1 + A )ɛ ]; ex qua eruitur av A av( 17) El 1 + an ) $ ( 1 + an ) Ponamus vase aliquo accurate obserato aerem conti neri ejusdem densitatis pé ac temperiei n cum aere exter• no; sitque h altitudo barometrica utrique communis : con . ⋀≀∙⊣∙∙∘⊔≔≖∘∙⊓≀⋅⇀⊢∝⋟∶∘⋮ et consequenter 1 1 ⇀ ↿ ∙ −⋅−−−− —F'(r-ct)-—;F( r—ct). : ⋅−−−− -—F'(r—ct)(t&quot;&quot;). r r cr 3&quot;:- Aereae particulae respondentes radio vectori r non 1nc1piunt moveri nisi quum tempus sic increvit, ut babeatur r— ct:ac, , seu ::- ct ∙⊦∙ at, :inferimus sonum prcpagatum iri uniformiter velocitate 'c: Vä- (1—1-A) (i&quot;) - Quod spectat ad numerum A, habemns (87. 70.) ∙ saiw-1-a(n-1-v)]:zp'u-u-e)[1-.-a(nM)]. itemque (10.) 6 −∙−−−⊤ w'[1—t-(1 a—A)e] :; ip'U—t— an)[1 ∙−⊦⋅ (1 ∙⊢ A)e]: bino - ⋅ ↴ 's ip'(1-1-s)[1-1-a(n-1-v)] ∶−− ≀⋅⊬⋅≺↿−⊢⊄⋯≻⊏↿⊣−≺↿−⊢∆≻∊⊐⋮ ex qua eruitur ↼ . cru-H:) av ∙−−− −∙∙ e(1-1-an) s(1-1-an) Ponamus vase aliquo accurate obserat'o aerem conti-- neri eiusdem densitatis pf ac temperiei iz cum aere exter- no; sitque !: altitudo barometrica utrique communis: con-281 1 11 cipiatur extrahi e vase aliquantulum inclusi aeris, vel qui erat inclusus aliquantulo magis comprimi , et denotet d'1 Fé) densitatem , h' altitudinem barometricam, postquam aer in tra vas ad pristinam redierit temperiem n. Tum constituta parumper communicatione cum externo aere, donec nimirum redigaturad h, mutationem quandam suscipiet lam p' ( 13) quam n; et illa quidem transformabitur in u'1 *8' ) (18&quot; ), haec autem in ny. Sed cum v' brevi evanescat, et so la n supersit quin variet MIFÉ' ) (1 # &quot; ) , mutabitur iterum h et evadet h &quot; . Alteram instituendi experimenti rationem sequuti sunt Desormes et Clement , alteram Gay - Lussac et Welter: inspiciatur sequens tabella . torie pun Desormes et Clement. n = 12 , 5 , heo” , 7665 , h - hs o ” , 01381 , 11 h - h&quot; = 0 , 003611 ; 2 &quot; hi sese restituit ad h intra tempus &lt; &lt; 5 Gay - Lussac et Welter. n = 13° , h = omom,, 757 757 ,, hh -- hh : = 0 &quot; , 0163644 , h &quot; - h = 0 , 0044409 ; q &quot; h sese restituit ad h intra tempus 6 Iam vero, depolante D densitatem hydrargyri , sunt conti erter Dgh = ip (176) (1 + an ) , 000 19 villi] 1 !' torir L, 111 )num conti- erit?' con- 281 cipiatur extrabi ei vase aliquantulum inclusi aeris,'*vel qui eratinclusus aliquantulo magis comprimi, et denotet p.'(1::1:s') densitatem, h' altitudinem barometricam, postquam aer in- tra vas ad pristinam redierit temperiem 11. Tum constituta parumper communicatione cum externo aere, donec nimirum h' redigatur ad h, mutationem quamdam suscipiet tam (if( quam ∎∶∙∶∔⋅∶∊⋅⋟ .: et illa quidem transformabitur 111 p.'(1.-.;:s')(1:£ e&quot;), haec autem in :::». Sed cum v' brevi evanescat, et so- la n supersit quin variat p.'(1:1: a' ) (1 :1: e&quot; ) , mutabitur iterum I: et evadet h&quot;. Alteram instituendi experimenti rationem sequuti sunt Desormes et Clement , alteram Gay- Lussac et Welter: inspiciatur sequens tabella. Desormes et Clement. 11:12&quot;, 5 , h:o&quot;,i7665 , 11 -h': 0&quot;, 01381 , h — h&quot;:o&quot;, 003611 ; h' sese restituit ad h intra tempus ≺∙−≣− ∙ Gay - Lussac et Welter. ' n:130, h:o'&quot; , 757 ,h'-— h:d&quot;,0163644 , h&quot; — h:o'&quot;, 0044409 ; 1&quot; h' sese restituit ad h intra tempus (—6—- . hm vero, denotante D densitatem bydrargyri , sunt Dgh':i;t'(1q:e')(1—t-an), ∎⊨∎ 'i i ! 19282 Dgh = id'l 176) ( 1 #t&quot; ) ( 1 + a nv( ) ) , $ Dgh &quot; = id'l 176 ) ( 1 + &quot;) ( 1 + an ) : hinc h &quot; = 1 &amp; € &quot; , h h &quot; 1+ anty') 1+ an = 1 + R 1 + an h &quot; 'h αν&quot;' h hh&quot; h&quot;

1 tan
ideoque }
ań
= ( 1 - an )
h hh&quot;
h &quot; 7 &quot; -h*
Substitatis valoribus ex Gay - Lussac et Welter ,
αν
€&quot; ( 1 + an)
=0, 3785020934 :
1
R
et quoniam iste numerus neque ex temperie neque ex pres
sione pendere videtur, iccirco poterit generatim assumi 1
A= 0, 3785020934 ;
sicque soni velocitas prodibit expressa per ( 94. 1 ° )
V
1 , 3785020934
to
fe
-V 1,3785020934i(1+ an) = 1009 , 614V1+ an (i&quot; ).
282
1131. −−−−⋅⋅⋅⊬∣≺↿∓⋮⋮≻ ≺↿ :::&quot; ) ( ↿
....
(a:-:») )
.
Dgh&quot;:ip.'(1q:s-' ) (1:t:€&quot;) (1 qum):
tibine
−≸⋮−⋤⋅−⋅
≕↿
∙∙⋅⊧∙≘⋅⋅ ,
∣∣⋮∙⋅ −−⋅↿−⊦⋅↿∘∙≦↾∙≔⋮∙⋓⇗≱ −−⋅↿∙−⋅⊦−∙↿−−∙⋮⋮≔−∙
zh :&quot; ∙∙∙ l:&quot; --l:' ,.4, .av' −∣∎ −∦∣⇂∙∙ ;
↙
h' 1 −⊢⋅⋯∎ &amp;
ideoque
av' h' b—h&quot;
e&quot;(1-t-an)— h&quot; h&quot;—-h'
⋅
Substitutis valoribus ex Gay-Lussac et Welter
,
av'
et quoniam iste. numerus neque ex temperieneque ex pres-
sione pendere videtur, iccirco poterit generatim assumi
sicque soni velocitas prodibit expressa per (94. 10)
I
c ∸−−−⇀ ↿∙ 3785020934 1;—
wjt—lV1, 3785020934 ⋅⋅≺↿∙⊢ an) ∙−−∶ 1009,- 614 ∣∕↿⊣⇀∘≀∎ tc&quot;)- H283

Si attendenda est quoque bygrometrica aeris constitutio, de
notante 6, pressionem libratam ab aqueo vapore , pro ui'
substituendum erit ( 96. 4º. )
1
seu
i( 1+ an)
exsistet nempe
V
11
w' il 1 to an )
1 , 3785020934 3
--8
W1
009 , 614
V
8 ã' (1+ an )
80-30 ,
(i &quot; ) .
In soni velocitatem diligentissime inquisiverunt an
no 1822 DD. Arago, Prony , Mathieu , Bouvard, Humboldt
et Gay - Lussac: distantia, ad quam observationes de cor
ruscatione flammae et fragore instituebantur in explosionibus
Lormenti bellici, ea fuit quae Monthlery et Villejuif inter
jacet ; velocitas inde deducta, seu spatium iolra 1&quot; a so
no percursum, 89 Erat autemn =15°, 9; unde
Vitan = 1 , 029 : dabit igitur formula ( it )
340metr.
103gped
.
893 metr .
337 , 432 . &gt;
Hygrometricam quoque aeris constitutionem notarunt
Auctores Cl . Sub mediocri videlicet altitudine barometrica
metr .
0 76 index hygrometri, quod vocant a capello, o
slendebat grad . 72 : in hac vero hygrometrica aeris consti
lutione, et sub temperie 15° , 9 ,pressioni , respóndet ba
metr.
rometrica aliiludo 0 00679; hinc
283
Si attendenda est quoque bygrometrica aeris constitutio, de-
notante u', pressionem libratam ab aqueo vapore , pro pf
substituendum erit (96. 40.)
exsistet nempe
' ∙
1
T
∘⋅−−− ∣∕ 1, 3785020934 &quot;'
'(
a, ↼⋅⊢ s
'&quot;&quot;- ..
8
1009 614⇂∕ afuit—13:111) (i&quot;)-
In soni velocitatem diligentissime inquisiverunt au-
no 1822 00. Arago, Prony, Mathieu, Bouvard, Humboldt
et Gay-Lussac: distantia, ad quam observationes de cor-
ruscatione dammae et fragore instituebantur-in explosionibus
tormenti bellici, ea fuit quae Montblery et Villejuif inter-
iacet :
velocitas iude deducta, seu spatium intra 1&quot; a so-
no percursum, :340'm&quot; ,89 Erat autemn:150,9; unde
l/1-t-an :1, 029: dabit igitur formula (ix)
0:1038ped' ,
893 :..- 337'&quot;&quot;' ,432.
Hygrometricam quoque aeris constitutionem natarunt
Auctores Cl. Sub mediocri videlicet altitudine barometrica
Gum. , 76 index bygrometri, quod vocant :: capella, o-
stendebat grad. 72:' m hac vero hygrometrica aeris consti-
tutione, et sub temperie 150, 9 ,pressioni a', respöndet ba-
rometrica altitudo Omm ,00679; binc284
v
80
8w' 30,
=1,002 ;
et consequenter ex (3 &quot; ) eruetur
1040ped ., 97 = 338metr .
11 .
Consensus itaque experientiam inter et expositam theo .
riam tantus invenitur , ut major profecto desiderari non
debeat in praesenti argumento : difficile admodum est in id
genus observationibus ventorum vim prorsus eludere, alias
que causas declinare quae huic consensui multipliciter no
cere possunt : mirum deinde quantum ardua res sit va
lorem A experimentis accurate determinare.

4. °* Evanescunt secunda membra (iº !! ) etiam quoad
a = o : in distantia igitur r evanescent &amp; , v statim atque,
labente tempore , eo devenitur ut sit rect = o . Quia er
go in distantia illa incipiunt , v esse aliquae quum
rct = lg, sequitur motum in distantia illa minime du
raturum ultra tempus Eaedem itaque &amp; , v evanescent
in distantia r- , statim atque incipiunt esse aliquae in di
stantia r : propterea non cientur una nisi particulae con
stituentes stratum crassiliei

5.° Velocitas v duabus ( 2.º į&quot; ) constat partibus , quarum altera sequitur rationem reciprocam distan
tiae a centro unde promanat sonus , altera rationem reciprocam duplicatam ejusdem distantiae: functiones praeterea F, Fmanent constanter parvolae. Quia igitur im
pulsio in datum obicem facta pendet a velocitate v , patet , quo longius propagatur sonus , eo magis ipsum debilitatum audiri. Quum sonus ad modicam pervenerit distantiam, licebit secundam illam partem negligere; eritque 

Inferimus illud: si impulsio in obicem facta quadrato ve.
locitatis v sumitur proportioualis , rationem duplicatam di
stantiarum sequetur soni debilitatio ( 125 ) .


6.°* Fac ut librati aeris particulae concutiantur
una circum plura puncta O , 0 &quot; , ... ; quorum distan
tiae ab ( x , y, z ) exhibeantur per r' , o&quot; .... ; ipsis.
que O' , 0 ' , ... , tanquam originibu's respondeant sua
axium systemata parallela systemati habeati originem O.
Quoniam novae coordinatae s ', x &quot;, ...5,0 &quot; , ... é , z &quot; ..
constantibus quantitatibus differunt ab x, y, z ; ideo
dr ' dr dr &quot; ar 
dx doc ' F ' da d.x &quot;
!
y' g &quot;
dr'
dy
&gt;
dy '
p&quot;
dr
dy &quot;
dr
dz&quot;
dr dr
dy
dr'i
dz
2
dz dz'
el consequenter
dQ _dQ dr dQdr&quot;
tar
dx
+ .. dx dr' dx
dQ x'
dQ y
dr +
dQxt&quot;
dr'' gli t....''
dQ_ dQ y
+
dy dr p ' dr to.
dQ
dz
dQ á
dr '
+
dQ
di &quot;
.
. ilemque
to
d²Q x 2 dQ 7/ 2+22
dx² dr'a g'a + +
285
1 '
r
inferimus illud : si impulsio in obicem facta quadrato ve-
locitatis v sumitur proportionalis, ratiunem duplicatam di-
stantiarum sequetur soni debilitatio (125).
691» Fac ut librati aeris particulae concutientur
una circum 'plura puncta O', 0&quot;, ... ; quorum distan-
tiae .ab (æ, y, :) exbibeantur per r'
, r&quot; . ...; ipsis-
que O'
, O&quot;, ,
tanquam originibus respondeant sua
axium systemata parallela systemati habenti originem 0.
Quoniam novae coordinatae se', a:&quot;,
...y' ,y&quot;, ... z', :&quot;
..
constantibus quantitatibus differunt ab a:, y, :;
ideo
dr' dr' æ' dr&quot; ∙∙∙ dr&quot; ∙∙∙⋅ æ&quot; ⇀
dx daf—r dx' dr&quot; ≀⋅∎⋅↬⋅⋅⋅
et consequenter
dQ 'der- −⊦↙≀≺≀∂∙↾∙∙⋅ du:- dr'dæ dr&quot; da:
dQ æ' dQ æ&quot; &quot;
dQ ∙−−↙≀≺⊇∙⊺ d.QJ
&quot;?;/7 21.-717 −⊢∙∙ 4&quot;?!— −−⊣∎∎∙∙
': &quot;dy— dr'r'
.dQ —dQ f:: dQ ∙⋮↾∙⋅ ∙−⊦ '
dz —dr' l&quot;-1 dr&quot; r&quot; ⋅ ..
itemque
'PQ −− ↨≖≬x&quot; dQ ∟∣≖⊣−≖∙∙∣∷ .
⋅
'
dæ' dr'3 :&quot;2 −⊦⋅−−(Ti—' −−⋅∣⋅∙−⋅↾⊰ ..,-286
daQ x2 dQ &quot; 272&quot;
+
dr''2 p &quot; 2 dri p/13 + ... ,''
da d’Qy'a
dya drar'a tari
dQ x2+22 +
p3
d'Q.7 &quot;?, dQ x&quot; : + z'2 + ti. dr&quot; ' a p &quot; 2
lo:
dri
d2Q
ddza
daQ z'2
dr2 p'2
dQ x's + y'2 dQ 242
dr' 3
+
dril2 pll2 +
dQ x2+ y'a
ti ..
Adhibitis substitutionibus in ( i ' ', 1.0 ) ,
d'Q
de2
( d - Q
= c2 Adr'a +
2 dQ d2Q 2 dQ
z dr + dra +pdr&quot; + ... )

;

ex cujus forma intelligimus fore
Q = [filr'tou + F (r — ct)]+
[far&quot; +41++ F.(r&quot; ct) ] + . ( * &quot; ).

Nunc facile stabilitur illud : in hypothesi plurium concus
sionum simultanearum , ubi eae ad punctum ( x , y , z )
eodem temporis momento una perlingant , numerus e ni
hil erit aliud nisi summa consimilium numerorum re
spondentium iisdem concussionibus seorsum spectatis ; si
quidem ( 1.0 ) .
286
(PQ ∙⋅⇂⋅∥∙ ↿ dQ
dr&quot;: ∙↗≀∦≖∙⊦≖∥≖⊹ r'&quot; ∣
dr&quot; r&quot;3 ⋅⋅
'
'
- «PQ— d'Q 7&quot; ∙⊦↙∄≺≀∙−−−−∙−−−−−⊦⋅ æ'2-l-z'2
d]:—
d'Q )&quot; dQ æ'ä-l—z&quot;; .
dr&quot; ≀⋅∥⋮⊹↲≀⋅∙∣∣ r' '34- ⊣− ⋅⋅. '
d-Q Adeo ∷⋅≖ ∣dQ ⊴↾∶∣≖−⊦∜∣⋅ æno z.&quot;:
dza 'di'/3 r'&quot;
'
dr' r'3 dr&quot;3 r&quot;'
dQ ∙⋅≖∥≖⊣−∜∥≕ ↿
dr&quot; rl'3. 'l .
.. ∙ ∙
Adbibitis substitutionibus in (i&quot;. 1.&quot;).
duo (PQ 2 dQ 2 dQ ∙
∙−−− ∘≺↙↙↾∣≏−⊣−≀⋅∣ ∡≔∣∙⊦≤∶−−⊽− ⋖⋮≀≕−⊽∣−⊋−↾⊽⊣−⋯≻
ex cuius forma intelligimus fore
Q ∶−∎⋅
⋅↗⋮⊤⋅∐⋩≖≺⋅∦⊣⊸⊩⊢−∶⋮∙− ∇≖≪↗⋅⊣⊸≀⊢⊢ F,(r'—-ct)]-l—
F.(r⋅⋅⋯ ]-l—-
Nunc facile stabilitur illud :, in hypothesi plurium concus-
sionnm simultanearum , ubi eae ad punctum. (a: ,
J
,

: )

eodem temporis momento una pertingant, numerus :ni-
hil erit aliud nisi summa consimilium numerorum re-
spondentium iisdem concussionibus seorsum spectatis; si-
quidem (1.&quot;).287
DP zo al
- F&quot;
['r( + ce)-F'(x'ct)
[facr &quot;+ c8)— F'xr&quot; —cr) ] - ...
Insuper
DP
dQx' dQx&quot;
+ t . dx dr ' dr&quot; r &quot; + ...
G [r« tch+F&quot; ret) ]– 16 +6 +
F.( c )]) + ( - &quot;+e +F',(==ci)] –
[for&quot;tor)+F60—60)) + ....
vº dQ
dy
dQ r' dl go &quot;
+ + .. dr ' r ' + dr &quot; r &quot;
G - triktet)tF'(x - ce )]= i [ fim'tot +
Fa(r = -1)]) + ( -186 *408)+ F&quot;(&quot; –ce)] -
wraca&quot; terhFall -ct)]) + .
287
⋅⇌⊐ ∙−−≕↿−∙⋅ ?,?&quot;-- --',..'[f .
(r -!-c:)—F'.(r -—-c:)1—-
⋅≺∽⋅−∎↿⋅⊤∶∁↿⋮⊅⋍∣⋅∥⊹⋯∙− ↧⋅⋅∣∙≺≀∙∙⋅∙⊳∙−∙∘≀≻ ]— ....
lnsuper
&quot;- «me ⋅≄⋅ .... −−∶ .. ↼−−⋍⋜⊑⋅∙−−⋅∡⋰−∙⋅⊤−⊢⊿−≀⋅−∙⇉↗⊷ −⊦ ∙
⋅⋅
⋅
&quot;
, 1! ' '
⋅∎ ≺∎≙∶∎↾⋅⊀∎≺∣⋅⋅−∣∎⊸∘⊣−∏⋅∎ (' -—0t )]-— ∙≀−∙⋅∙−∙−∣⋅∫∎≼∣∙∎∙⊦⊸↥⊢∣−
,
' ⋅ æ' .
⋅ ↿ ⋅
⋅ ⋅
'
∙
Fl(r &quot;'&quot;'ct)
&quot;
])&quot;T'f'l'(—: [f,(r'Lï-CO—l-F', ∎⋅ ( r—ct )] ∙∙∙⋅
r
∙
∙≖−⋅≟⊑ ⊏∣≖≺↗⋅⋅−∣⊸≀≻⊣−↿⋮⋅≖≺⋅∙⋅⋅∙−−∝≻∃ )£f.-.- −⊦ ∙
⋅∙∙
. ∂≺≀∙∙−⋅∠≀≖≀∜⋅
lu.—dy dr' r' ä-l-dr&quot;— r&quot;∶∣⋅−⊦∎⋅ .. 1
(£.- ⊏⊀⋅≺≀⊤∙∙⇀⊸⋅≀⊢⊢≖⋅⋅∙⋅≺ r'-ct )1: ;; [f.(f-l-cu-l-
F.(r'— et)] ≻∙∜−⋮−⊦r ≺↿⊤↕∣∣≖≺↗⊓⊣⊸⋍⋝⊣−⇂⋅⇁⋅⋅≺∣∙⊷∙−∘≀∏ :- - ⋅
⋝∙≟≟∁∣≖≼↾∣⋅⊹≕⊢∣⋅⇁≖≺∙⋅≀∙∙⊸∁∏ ⋟∑≖∙⊤⋅⋮− −⊦∢ ∙ ∙ ∙288
dQ
dz
dQz
dr
dQz&quot;
t .
dr '' r &quot;''
( -fr.( tre)+F ;(r = cr)] - Pfalriteest
Fu
F.(x*—- )]) + ( far&quot;+c)+ F &quot;–ce)] -
pen
na[ far tcent Ffrº-cr)]) + ...;
UI
De
go
inferimus velocitatem v debitam simultaneis concussioni
bus circum 0,0 eodem temporis momento ad
punctum ( x , y , z ) una pertingentibus nihil fore aliud
nisi resultantem ex velocitatibus , quae debentur iisdem
concussionibus seorsim spectatis : atque hinc facile intel
ligimus cur, pluribus corporibus simul resonantibus , inter
oscillationes in aere excitalas non habeatur confusio ,
omnesque diversi soni inde orti ad aures distincte per
veniant. Huc spectat principium de superpositione exiguo
rum motuum.
7.04 Redeuntes ad unicam concussionem in 0 ,
ponamus aerem contineri tubo cylindrico , cujus axis ox,
motumque particularum esse ipsi OX parallelum : erunt
v&quot; = 0, v&quot; = 0; propterea formula ( i&quot; ) evadet
d2Q daQ
de unde Q = f ( x + .ct) + F ( x - ct ) ; ረder2 1
et consequenter
288
... dQ- JQ :
&quot;'l-(,Q' z'—dz −−↲≀⋅⋅ r' dr&quot; r'

ll ≼⋅≟≑⊏∣∣∙≺↗⋅⊣⇥≻⊣−≖⋅⇁⋅≖≺↗⋅⋅−⋅∘≀∏ ∙−⋅⋅⋮−⋅⋮⋅⇆⋅⋅⊔≖⋅∊⋅↾⋅⊣↽⊸≀⊢⊦ ∙
,
⋅ mo*—cn] )f— ⊣−≺⊽⊏ ↑∼≖≼↗⋅∙−⊦∘↥≻−⊦ ≖∸⋅∙≖≺↗∙∙−∘≀∏ −
⋮∙−⊦∘≀≻⊹↧⊸⇁≖≺≀∙↝−−∘≀∏⋟−⋮⊽ −↿− ∙ ∙
⋅
inferimus velocitatem v debitam simultaneis concussioni-
bus circum O'. 0&quot;
, ... eodem temporis momento ad
punctum ( x . y ,

: ) una pertingentibus nihil fore aliud nisi resultantem ex velocitatibus , quae debentur iisdem concussionibus seorsim spectatis: atque hinc facile intel- ligimus cur, pluribus corporibus simul resonantibus . inter oscillationes in aere excitatas non babeatur confusio. omnesque diversi soni inde orti ad aures distincte per- veniant. Huc spectat principium de superpositione exiguo- rum motuum. 7.&quot; Redeuntes ad unicam concussionem in 0, ponamus aerem contineri tubo cylindrico, cuius axis OX. motumque particularum esse ipsi OX parallelum: erunt 0' '::--0. 0&quot; ':o; propterea formula (i&quot;) evadet ⋅ 32? —-c £?. unde Q—−−⋅∣↗≼∶∁−∔⊸⇂⋟ -l-F(x—-ct); et consequenter289 dQ 1 dQ dx = pilatot) + F '( x - 1), E = - ca do [fotot) – F (x – ct )] . Functiones f et F absque ulla difficultate determinantur: sunt enim ( 1.9). f( x) = f(@ + F'(Q ), - cf:(Q ) = f (a) F '( ) ; ideoque f'( X) = f (Q )-cfi(Q ) 2 f(@ t-of ( ) F (a ) = 2 Ultimae ac penultimae aequationis secunda membra eva nescisnt statim ac a fil &gt;Oto : erit itaque f ( t ) = 0 quoad -aereas particulas ultra azi proinde quoad ejusmodi particulas F ' ( x-ct ) . Hinc sequitur souum adhuc ( 3. ) propagatum iri uniformiter velocitate се V 11 + 4 ). • De reflexa soni propagatione per aerem .
: 130. Cam in directa propagatione sonoras aer offen dit obicem aptum, reflectitur; hinc echo ( 115 ) progignitur; assertio sic probatur . Constat quod corpus in motu positum , si in obstacu lum incidit , quod elasticum sit , vel durum , et corpus ipsum ⋅ 289 v −∸−≖ B:] ')(æ-l-ct -l-F'(æ—ct), a:.— — ∙−∙−−−⋅⋅∶ ⋅↿ ∙

: - [f(ar-l—ct)

-
F'(x—-ct)] .
Functiones f
et P absque ulla didicu-l'tate determinantur:
sunt enim (1.&quot;). ⋅
⊞≀∝≻−−−↿≺⊄⊢⊦⋮⇁≀≺∝≻∙ ∙− cf.(a)-—:f(a1—F'(a) :
ideoque f(ao ⇌≖
aa)-zcnm) ∙ Ha): Karl-faa) ∙
Ultimae ac penultimae aequationis secunda membra eva-
nescunt statim ac « Et )a.. : erit itaque fur-H:):o
quoad aereas particulas ultra «.'
,proinde quoad eiusmodi
particulas ⋅-cs :: F' (a:—ct ).

Hinc sequitur sonum adbuc (3.&quot;) prcpagatum iri unifor-
miter velocitate
C::V-z-I—(i-I—A). ' De reflexa soni prcpagatione per aerem ∙⋅

130. Cum in directa propagatione sonoras aer oü'en-
dit obicem aptum, reflectitur: binc echo (115) progiguitur: assertio sic probatur.  

Constat quod corpus in motu positum, si in obstaculum incidit, quod elasticum sit, vel durum, et corpus ipsum impingens elasticilate gaudet, debet molus directionem mutare ac reflecti: ergo aer, elasticus cum sit, ubi in obstaculum offendit, quod vel elasticum sit, vel certe non molle, reflecti debet; undae videlicet aereae, quae ex sonoro corpore progignantur ac propagantur directe, debent obicem offendendo regredi, sonumque reflexum progignere. Exemplo circulorum in aqua ex injecto lapide excitatorum res oculis subjicitur: circuli enim isti ubi ad ripam appellunt, reflectuntur inde eo ordine, quo appulerunt . Aliter sic: ejusdem naturae est echo cum sono ipso directo; obtinet enim utrinque sonus eodem generatim tono, iisdemque affectionibus praeditus; ergo echo gigni debet eodem modo quo sonus directas: atqui hic per undas aereas successive a sonori corporis motu genitas procreatur; ergo per similes undas etc. Hinc in aperta planitie, ubi nullas est obex, sono directo minime Echo respondet. Cohaeret doctrina com Echo phoenomenis. Nam 

1° redit reflexa vox duplo temporis intervallo: ab experientia doctus sum, inquit Derhamus, Echo redire duplo intervallo, quo vox primaria ad objectum phonocanticum pertingebat; scilicet tempus requiritur ut ad obicem vox primaria deveniat, et rursum tantumdem temporis exigitur ut reflexa ab obice redeat ad loquentem. 

2º. Remissior plerumque est Echo quam vox directa audiri soleat; aliquando tamen intensius resonat Echo quam sonus directus audiatur. Ralio primi est: cum soni intensitas decrescat pro aucta distuntia a sonoro corpore, jam decrescit sonus ad obicem pergens; inde autem regrediens, et novas undas progignens, iterum decrescere
debet intensitas: ratio secundi, quia si obstaculum concavum
sit, plures colligere poterit radios phonicos , quos unitos simul in uno loco regerat. 

3º. Aliquando ( 115 ) seinel vox reflectitur, aliquando saepius: prima dicitur Echo monophona,
altera polyphona. Si enim obstaculum unicum sit , jam nonnisi
semel potest vocem remittere; contra saepius remittitur duplici ex causa. Prima est cum iu variis distantiis plura habentar
290
impingens elasticitate gaudet , debet motus directionem mu-
tare ac reflecti :ergo aer , elasticus cum sit ∙ ubi in ob-
staculum offendit
, quod vel elasticum sit, vel certe non
molle
,
reflecti debet; undae videlicet aereae, quae ex so-
noro corpore prOgignuntnr ac propagantur directe , debent
obicem oll'endendo regredi , sonumque reflexum progignere . 

Nam ↿∘ redit reflexa vox
duplo temporis intervallo: ab experientia doctus sum ,
in-
quit Derhamus, Echo redire duplo intervallo, quo vox pri-
maria ad obiectum phonocanticum pertingebat; scilicet tcm-
pus requiritur ut.ad obicem vox primaria deveniat, et rur-
sum tantumdem temporis exigitur ut reflexa ab obice re-
deat ad loquentem.- 20. Remissior plerumque est Echo quam
vox directa audiri soleat; aliquando tamen intensius reso-
nat Ecbo quam sonus directus audiatur. Ratio primi est :
cum soni intensitas decrescat pro aucta distantia a sonoro
corpore, iam decrescit sonus ad- obicem pergens: inde autem
regrediens, et novas undas prOgignens, iterum decrescere
debet intensitas: ratio secundi, quia si obstaculum concavum
sit, plures colligere poterit radios phonicos, quos unitos si-
mul in uno loco regerat. 3&quot;. Aliquando (1 15) semel vox refle-
ctitur,aliquando saepius: prima dicitur Echo monopbona,
altera polyphona. Si enim obstaculum unicum sit. iam nonnisi
semel potest vocem remittere; contra saepius remittitur dupli-
ci ex causa. Prima ut cum iu variis distantiis plura habentur291
obstacula: altera causa est, cum duo sunt obices e regione col
locati; vox enim ex uno reflexa in alterum incidit, atque ex
hoc iterum reflexa iucidit in primum, et ita porro. Apud
veteres memoratur Olympiae porticus ; quam eplaphonam
dicebant, quod septies eamdem vocem redderet , ut tradit
Plinius. Prope Mediolanum celebris est Echo in palatio
Simonetta, in quo ope duorum parietum parallelorum fra
gor minoris fistulae bellicae vicies, et aliquando tricies re
petitur teste Schoto. 40. Echo saepius unam tantum syl
labam, aliquando plures refert: echo monosyllaba prima, et
polysyllaba altera dicitur: habentur loca , ex quibus integer
versus hexameter repetitur. Ea nempe est obicis ( 115) di
stantia, ut sonus reflexus primarum syllabarum tunc demum
ad aures regrediendo perveniat quando vocis directae im
pressio jam desinit; ac tunc sonus primae syllabae, qui op:
portune regreditur jam expleto versu , poterit esse sepsi
bilis, itemque aliarum successive. 5º. Echo redditur ali
quando a silyis ; imo etiam a campis sulco exasperatis et a
planitie cespitibus ac virgultis inspersa reverberalur vox ;
reverberari autem a sulcis ac cespitibus animadvertit Kir
cherus , quia quando sulci eversi, ac virgulta praecisa fue
runt Echo nulla reddebatur: talis nempe esse potest irre
gularis partium reflectentium dispositio, ut etiamsi plures ra
dii phonici dispergantur, non pauci tamen in eumdem lo
cum collineent.

131. Reflexio soni fil ad angulos incidentiae et refle
xionis aequales : quod sic explicamus . Sit AB ( Fig. 60. ) fir
ma , planaque superficies ; KCK' recta perpendicularis su
perficiei AB ; K centrum sonorum , ex quo propagantur
sphaericae undae CDD', C'EE', etc... appellentes ad AB in C,
C'ete ... Fiet soni reflexio in C, C , ..; ethabitis C , C ... pro noris
secundariarum undarum centris , ipsae secundariae undae
remittentur cum eadem principalis undae velocitate. Pro
grediente unda principali ab CDD' usque ad BB ' , unda
manans ex C progredietur ab C usque ad Q ; repraesenta
291 obstacula: altera-causa est, cum duo sunt obices e regione col-
locati; vox enim ex uno reflexa in alterum incidit, atque ex
hoc iterum reflexa incidit in primum, et ita porro. Apud
veteres memoratur Olympiae porticus; quam eptaphonam
dicebant, quod septies eamdem vocem redderet, ut tradit
Plinius. PrOpe Mediolanum celebris est Echo in palatio
Simonetta, in quo ope duorum parietum parallelorum fra-
gor minoris fistulae bellicae vicies, et aliquando tricies re-
petitur teste Scboto. 40. Echo saepius unam tantum syl-
labam, aliquando plures refert: echo monosyllaba prima, et
polysyllaba altera dicitur: habentur loca, ex quibus integer
versus hexameter repetitur. Ea nempe est obicis (115) di-
stantia, nt sonus reflexus primarum syllabarum tunc demum
ad aures regredieodo perveniat quando vocis directae im-
pressio- iam desinit; ac tunc sonus primae syllabae, qui op-
portune regreditur iam expleto versu, poterit esse sensi-
bilis, itemque aliarum successive. 50. Echo redditur ali-
quando &amp;
silvis; iuno etiam a campis sulco exasperatis et a
planitie cespitibus ac virgultis inspersa reverberatur vox ;
reverberari autem a sulcis ac cespitibus animadvertit Kir-
cberus. quia quando sulci eversi, ac virgulta praecisa fue-
runt Echo nulla reddebatur: talis nempe esse potest irre-
gularis partium reflectendum dispositio, ut etiamsi plures ra-
dii phonici dispergentur, non pauci tamen in eumdem lo-
cum collineent.

131. Beflexio soni fit ad angulos incidentiae et refle-
xionis aequales: quod sic explicamus .Sit AB (Fig. 60.) Gr-
ma, plenaque superficies; KCK' recta perpendicularis su-
perficiei AB ; K centrum sonorum , ex quo propagantur
sphaericae undae CDD', C'EE', etc... appellentes ad AB in C,
B' etc... Fiet soni reflexio in C, C',..; et habitis C,C'... pro novis
secundariarum undarum centris, ipsae secundariae undae
remittentur cum eadem principalis undae velocitate. Pro-x grcdieute unda principali ab CDD' usque ad BB' , unda
manans ex C prOgredietur ab C usqæ ad Q; repraesenta-292
biturque hemisphaerio , cujas semidiameter CQ = D'B ' :
item progrediente unda principali ab C'EE usque ad BB” ,
unda manans ex C' progredietur ab C usque ad C &quot; : re
praesentabiturque hemisphaerio , cujus semidiameter CC&quot;
E'B' ; alque ita porro. Inferimus , si concipitur superficies
curva AQC &quot; B tangens omnia haec hemisphaeria in Q, C &quot; ....,
in ea fore puncta illa , quae a secundariis andis reflexis
attinguntur eodem instanti , quaeque tunc incipient concuti
quum principalis unda pervenerit ad BB' ; exhibebit nimi
rum AQC&quot;B superficiem undae reflexae; et quoniam productis
infra AB superficiebus BB' , Qa, C'a ' , ..., recta KA' exsistit per:
pendicularis ad primam et secundam, recta KH ad primam et
tertiam , etc ... ; ac proinde sphaerica superficies B'BA'A tan
git sphaericas superficies QaA' , C'a'H , . ; sequitur super
ficiem AQB undae reflexae fore sphaericam , ejusque cen
trum in K , et semidiametrum K'Q = KA' . Jamvero quem
admodum auris collocata v. gr. in C deprehendit sonum
directum venire juxta KC' perpendicularem undae incidenti ,
sic auris in C' deprehendet sonum reflexum venire juxta
K'C &quot; perpendicularem updae reflexae ; et cum , ob mutuum
sphaericarum superficierum AQB , C'a'H contactum , recta
K'C &quot; transeat per C' ; cumque , ob latus KC = K'C , et
latus CC commune , triangula rectangula KCC , KCC' dent
angulum KCC aequalem angulo K'C'C , erit angulus KCC
angulo C &quot; CB ; ideoque angulus incidentiae aequalis an
gulo reflexionis .
Sit nunc firma curvilineaque superficies AB ( Fig 61. ) ,
in quam incidant undae CE , HE&quot; , ... BB' propagatae ex
centro sonoro K ; si centris C , H , ... describuntur sphae
rae , quarum semidiametri ( KB-KC' ) , ( KB - KH ) , ... ,
aereae particulae sitae in superficie BD tangente sphaeras
istas incipient simul affici motu reflexo stalia ab adventu
undac ex K in B. Erit igitur BD superficies undae refle
xae : quam superficiem pon esse sphaericam nemo est qui
non videat. Fac ut puncta C , H sint inter se infinite vi
292
biturqne hemisphaerio , cuins semidiameter CQ ∶⋅−⋅ D'B' :
itcm progrediente unda principali ab C'EE' usque ad BB',
unda manans ex 0 progredietur ab C' usque ad C&quot;; re-
praesentabitnrque hemisphaerio
.cnius semidiameter C'C' ::
E'B' ; atque ita porro. Inferimus ,si concipitur superficies
curva AQC&quot;B tangens omnia haec hemisphaeria in Q, C&quot;,...,
in ea fore puncta illa , quae a secundariis undis reflexis
attinguntur eodem instanti , quaeque tunc incipient concuti -
quum principalis unda pervenerit ad BB' ;exhibebit nimi-
rum AQC&quot;B superficiem undae reflexae; et quoniam productis
infra AB superficiebus BB',Qa, C'a',..., recta KA' exsistit per-
pendicularis ad primam et secundam, recta KH ad primam et
tertiam
,
etc... ; ac proinde sphaerica superficies B'BA'A tan-
git sphaericas superficies QaA', C&quot;a'H .∙∙∙ ;sequitur super-
ficiem AQB undae reflexae fore sphaericam, eiusque cen-
trum in K', et semidiametrum K'Q ∶⋅−∙⋅ KA'. Iamvero qnem-
admodum auris collocata v. gr. in C' deprehendit sonum
directum venire iuxta KC' perpendicularem nudae incidenti ,
sic auris in C&quot; deprehendet sonum reflexum venire iuxta
K'C&quot; perpendicularem undae reflexae ; et cum , ob mutuum
sphaericarum superficierum AQB , C&quot;a'H contactum
,
recta
K'C&quot; transeat per C'; cumque , ob latus KC −∙−∸− K'C , et
latus CC' commune ,
triangula rectangula KCC', K'CC' dent
angulum KC'C aequalem angulO'K'C'C , erit angulus KC'C

: angulo C&quot;CB; ideoque angulus incidentiae aequalis angulo reflexionis . Sit nunc firma curvilineaqne superficies AB (Fig GI.), in quam incidant undae C'E' , HE&quot; ,... BB' propagatae ex centro sonoro K; si centris C' , H , ... describantur sphae- rae , quarum semidiametri ( KB—KC') , (KB—KH) , , aereae particulae sitae in superficie BD tangente sphaeras istas incipient simul affici motu reflexo statim ab adventu undae ex K in B. Erit igitur BD superficies undae refle- xae : quam superficiem non esse Sphaericam nemo est qui non videat. Fac ut puncta C' , H sint inter se infinite vi-293 cina , sintque C'C &quot; , HQ normales ad BD : ex H ductis per pendiculis Ha , Ha' in KC , C'C &quot; , erit Ca ' = CC &quot;—HQ = KB - KC ) - (KB - KH ) = KH - KC = Ca. Quoniam igitur triangula rectangula Cal , Ca'H habent latera aequalia C'a , C'a ', latusque C'H commuue , habebunt ae quales angulos ac'h , a'C'H : hinc sequitur , etsi unda re flexa non est sphaerica , adhuc tamen reflexionis angulum fore aequalem angulo incidentiae.

132. * Haec deducimus ex ( 129) in ordine ad aereum
fluidum concussum in K ( Fig. 60 ) , planoque fixo AB ter
minatum. 1 °* Sumpta x in KC normaliter ad AB, peribit
apud AB tota componens v' ; erit nempe ( 129. 10. )
dQ
dxdo O ( a )
quoad x = KC ( = h ). ProducaturKC donec KC = KC;
radius vector r' computetur ab K' ; et x ab eodem K' in
K'C ; explebitur (a) per
Q = --[Pr + c ) + F(ra) ] +
[fri + ce ) + F(x – ċe)]
( a ) ;
siquidem quoad puncta sita in AB
dQ dQ r=r' ,
dr x = h , it's - h ,
dr
dris
dx dx
Determinatis praeterea f et F ex ( i&quot; &quot; . 129. 10. ) , re
praesentabit ( a' ) initialem fluidi statum: quoniain igitur ( a' )
a— 293
cina , sintque C',C&quot; HQ normales ad BD: ex H ductis per-
pendiculis Ha ,
[in' in KC', 0C.&quot; , erit
∁≮≖⇌∁⋅∙∁ ∙∶−⇀−∐≺≀ (KB'—-KC';-(KB-—Kll)—-KH-—KC':C a.
Quoniam igitur triangula rectangulaC aH, C:: 'H habent latera
aequalia C'a
, C'a', latusque C'H commune , habebant ae-
quales angulos aC',H a'C' H hinc sequitur , etsi unda re-
flexa non est sphaerica , adhuc tamen reflexionis angulum
fore aequalem angulo incidentiae.
1324: Haec deducimus ex (129) in ordine ad aereum
fluidum concussam in K (Fig. 60), planoque fixo AB ter-
minutum. ↿∘∙ Sumpta æ in KC normaliter ad AB, peribit
apud AB tota componens v'; erit nempe (129. 10.)
19.
da: :0 (a)
quoad .c— KC (: It ). Producatur KC donec K' C:: KC:
radius vector r' computetur ab K'; et .r' ab eodem K' in
K'C; explebitur (a) per
≬⇌−⋮−∥↸≀∙⊣−∘≖⋮⋟⊣−⊏⋅⇁≺↗⋅−⋅−∘∩⊐−⊦
↿
−≀−∙−∙⋅−∣⋮⋀≀∙⋅−∣⋅−∘≀⊅⊹⊞↱⋅−−⊄⋮↕∙⋟⋅∙∣ (a');
siquidem quoad puncta sita in AB
∙∙ dQ dQ ∙∙∙
↙≀↾∙∙∙⊲ dr'
⋅⋅−—&quot;'-27—27 ***-&quot; ∙⋅↕−⇀−∣⋅∙⋅↴∙⋮⋮⊒−− 2;-
Determinatis praeterea f
et F ex (i'&quot;. 129. 10.), re-
praesentabit (a') initialem fluidi statum: quoniam igitur (a')291 !
1
1
1
1
1
et satisfacit conditioni ( a ) , et exprimit initialem fluidi statum,
poterunt per ( a' ) definiri, quae spectant ad motus propaga
tionem, attento obstaculo AB.
2º . * Punctum C &quot; , ad quod pertinent radii vecto
res r et r seu KC&quot; et K'C &quot; , perinde motum concipiet ac
si ( 129. 6. &quot; ) , sublato plano AB, fluidum eadem omnino
ratione concuteretur simul circa duo puncta K et K' . Per
tinget itaque ( 229. 4º. ) concussio ad C &quot;, primum in fine
temporis deinde in fine temporis : hinc bi
ni successive motus in C &quot; , alter directus, alter reflexus ; et
quia secunda concussio non pervenit ad C &quot; nisi quum tempus
sic invrevit, ut habeatur r = ct + a,, iccirco eadem velo
citate c regredietur motus, qua incedebat antequam in obi
cem impiogeret. Ad haec : cum anguli KC'C, K'C'C sint
aequales, rursus ( 131 ) patet sonum illisum obici AB re
gressurum efficiendo angulum reflexionis aequalem angulo
incidentiae.
C. с
De instrumentis pneumaticis.

133. In instrumentis pneumaticis soni genesis repe
tenda non est saltem praecipue ex oscillatione partium so
lidarum ipsius instrumenti. Etenim si in hisce instrumentis
dicatur soous creari eodem modo ac in instrumentis per
cussione resonantibus, jam sonus ipse connexionem haberet
maximam cum materia qua instrumentum compactum est ,
nec non cum ejusdem crassitie; quod tum ratione verissi
mum apparet , lum etiam constat ex vi paritatis. Ratione
quidem: nam in instrumentis ex diversa materia compactis,
quorum proinde particulae non aeque elasticae sunt, et ad mo.
cum oscillatorium non aeque aptae, non eodem modo tremu.
lus ille motus per insufflationem excitari debet ; quo vero
crassius instrumentum est, eo in plures continuas partes 0
294
et satisfacit conditioni (a), et exprimit initialem fluidi statum,
poterunt per (a') definiri, quae spectant ad motus prcpaga-
tionem, attento obstaculo AB.
20.a Punctum C&quot;, ad quod pertinent radii vecto-
res r et r' seu KC&quot; et K'C&quot;, perinde motum concipiet ac
si (129. 6.0), sublato plano AB, fluidum eadem omnino
ratione concuteretur simul circa duo puncta K et K'. Per-
tinget itaque (229. 40.) concussio ad C&quot;, primum in fine
tempons c
, deinde in fine temporis c : hinc bi-
ni successive motus in C&quot;, alter directus, alter reflexus; et
quia secunda concussio non pervenit ad C&quot; nisi quam tempus
sic iuvrevit, ut habeatur r': ct ⊣−∙ a. ,
iccirco eadem velo-
citate c regredietur motus, qua incedebat antequam in obi-
cem impingeret. Ad haec : cum anguli KC'C, K'C'C sint
aequales, rursus (131) patet sonum illisum obici AB re-
gressurum efficiendo angulum reflexionis aequalem angulo
incidentiae.
∙ r—al r'e—a
De instrumentis pneumatict's.

133. In instrumentis pneumaticis soni genesis repe-
tenda non est saltem praecipue ex oscillatione partium so-
lidaram ipsius instrumenti. Etenim si in hisce instrumentis
dicatur sonus creari eodem modo ac in instrumentis per-
cussione resonantibus, jam sonus ipse connexionem haberet
mammam cum materia qua instrumentum compactum est,
nec non cum eiusdem crassitie; quod tum ratione verissi-
mum apparet ,
tum etiam constat ex vi paritatis. Ratione
quidem: nam in instrumentis ex diversa materia compactis,
quorum proinde particulae non aeque elasticae sunt, et ad mo-
tum oscillatorium non aeque aptae, non eodem modo tremu-
lus ille motus per insufilationem excitari debet ; quo vero
crassius instrumentum est, eo in plures continuas partes o-295
scillatorius molus dispesci debet. Vi paritatis autem : nam
reipsa instrumenta, quae percussione sopant, pro materiae di
versitate etiam in pari crassitudine diversimode contremiscunt;
et si ejusdem sint materiae, pro diversa crassitie diversum
item sonum edunt. Ergo sonus in instrumentis pneumaticis
maximam connexionem etc. si in his soni genesis etc. Atqui
hoc est falsum : in tibiis enim cylindricis ejusdem longitu
dinis idem habetur sonus aut fere idem , nullo respectu
habito ad materiam aut crassitiem ipsius instrumenti , ut
constat experimentis; totumque artificium pro varietate to
norum pendet ex instrumenti variata longitudine: propte
rea etc.
Quonam igitur pacto in hujusmodi instrumentis erit
soni genesis explicanda ? Eo videlicet , quem indicavimus
(114.). In interna instrumenti capacitate aeris columna includitur, quae externi aeris pressione urgetur: dum igitur
per exiguum orificium fistulae alius aer insufflatione intro
mittitur, aer ille inclusus condensari debet , atque urgeri
contra aerem externum; externus autem vi suae pressionis
resistere; et quum aeris interni aucta pressio vincit, hic se
dilatando condensabit externum aerem , repelletque; externus
aer ita densatus ut ejus elasticitas praevaleat, se restituendo
rursum comprimet internum aerem : in columna videlicet illa
fiei compressio et restitutio, sicque in aeris particulis oscillato
rius motus excitabitur; qui motus communicabitur externo
aeri contiguo, et ad aures perveniet. Aer itaque secundum lon
gitudinem fistulae se habet instar chordae peragentis longita
dinales vibrationes: quo tibia et consequenter columna aerea
longior est, eo etiam longiores undae efformantur; longius
que erit tempus compressionis et restitutionis , ac proinde
Lonus gravior. Hinc in instrumentis, quae secundum longi
Ludinem sunt foraminibus instructa, modo hoc et modo il
lud foramen aperiendo, sublato digito, varii obtinentur to
ni; siquidem externum aerem sic admittendo , modo ma
jorem et modo minorem columnae aereae longitudinem ha
295
scillatdrius motus dispesci debet. Vi paritatis autem: nam
reipsa instrumenta, quae percussione sonant, pro materiae di-
versitate etiam in pari crassitudine diversimode contremiscunt;
et si ejusdem sint materiae, pro 'diversa crassitie diversum
item sonum edunt. Ergo sonus in iustrumentispneumaticis
maximam connexionem etc. si in his soni genesis etc. Atqui
'
hoc est falsum: in tibiis enim cylindricis ejusdem longitu-
dinis idem habetar sonus aut fere idem , nullo respectn
habito ad materiam aut crassitiem ipsius instrumenti ,
ut
constat experimentis; totumque artificium pro varietate to-
norum pendet ex instrumenti variata longitudine: propte-
rea etc.
Quonam igitur pacto in hujusmodi instrumentis erit
soni genesis explicanda ? Eo videlicet
, quem iudicavimus
(114.).ln interna instrumenti-capacitate aeris columna in-
cluditur, quae externi aeris pressione urgetur: dum igitur
per exiguum orificium fistulae alius aer iusufflatione intro-
mittitur, aer ille inclusus condensari debet, atque urgeri
contra aerem externum; externus autem vi suae pressionis
resistere; et quam aeris interni aucta pressio vincit, hic se
dilatando condensabit externum aerem, repelletque; externus
aer ita densatus ut ejus elasticitas praevaleat, se restituendo
rursum comprimet internum aerem: in columna videlicet illa
fiet compressio et restitutio, sicque in aeris particulis oscillato-
rius motus excitabitur; qui motus communicabitur externo
aeri contiguo, et ad aures perveniet. Aer itaque secundum lon-
gitudinem fistulae se habet instar chordae peragentia longitu-
dinales vibrationes: quo tibia et consequenter columna aerea
longior est, eo etiam longiores undae efi'ormantur; longius-
que erit tempus' compressionis et restitutionis , ac proinde
tonus gravior. Hinc in instrumentis, quae secundum lougi-
tudinem sunt foraminibus instructa, modo hoc et modo il-
lud foramen aperiendo, sublato digito, varii obtineatur to-
ni; siquidem externum aerem sic admittendo , modo ma-
iorem et modo minorem columnae aereae longitudinem ha-296
benius. Ita in chordis, pro majori chordae longitudine gra
vior est tonus, acutior pro minori; et digitis comprimendo
camdem chordam, ut evadat plus aut minus longa , varios
assequimur tonos .
Dixi soni genesim repetendam non esse saltem prae
cipue ex oscillatione solidarum partium etc. Etsi enim ex
materia instrumenti non habetur varietas quoad soni qua
litatem , aut valde notabilem intensitatem ; varietas tamen
habelur quoad meliorem aliquam resonantiam; idque ex eo
desumendum videtur quod aer inclusus pro diversitate cor
poris includentis melius aut minus bene oscillare potest ;
magis nimirum aut minus impeditus adhaesione ad ipsum
corpus et scabritie aliqua. Ad haec; si instrumentum pneu
maticum sit compactum ex materia non resistente, quale v.g.
esset instrumentum membranaceum , tunc per vibrationes
columnae aereae excitari poterit sensibilis motus oscillato
rius in partibus instrumenti; quae partes vicissim in aerem
vibrantem reagendo valebunt sonum ipsum modificari etiam
quoad tonum; et quidem plurimum si instrumentum sit val
de breve, quemadmodum expertus est D. Savarı; adeo ut
brevi tubo membranaceo obtineri possil magna varietas
lonorum , qui eo graviores erunt quo minus tenditur mem
brana.

134. Haec proponimus explicanda circa instrumenta
pneumatica. 

1º. Aperto aliquo foramine ex. gr. tertio, cae
lerisque clausis, ac deinde aperto alio puta quinto , variat
lonus: at si ' per aperitionem tertii inducitur communicatio
interni aeris cum externo, nonne ex dictis ( 133 ) audiri de
beret idem tonus sive apertum sive clausum sit quintam
foramen ? 

2º. Sola inflationis intensione mutantur toni , e
tiam servata eadem internae columnae longitudine 

3º. In
canna organi ejusdem diametri superius clausa, si subdupla
sit longitudo , idem redditur tonus qui obtinetur ex can
na superius aperta, et longitudinis duplae.
Ad 1. Cum varia in instrumento pneumatico fora
mina aperiuntur, variae interni aeris columnae communi
296
hemas. ita in chordis, pro maiori chordae longitudine g'ra-
vior est tonus, acutior pro minori; et digitis comprimendo
eamdem chordam, ut evadat plus aut minus longa , varios
assequimur tonos.
Dixi soni geneaim repetendam non esse saltem prae-
cipue ex oscillatione solidarum partium ctc. Etsi enim ex
materia instrumenti non habetur varietas quoad soni qua-
litatem , aut valde notabilem intensitatem ; varietas tamen
habetur quoad meliorem aliquam resonantiam; idque exeo
desumendum videtur quod aer inclusus pro diversitate cor-
poris incladentis melius aut minus bene oscillare potest;
magis nimirum aut minus impeditus adbaesione ad ipsum
corpus et scabritia aliqua. Ad haec; si instrumentum pneu-
maticum sit compactum ex materia non resistente, quale v.g.
esset instrumentum membranaceum ,
tunc per vibrationes
columnae aerea'e excitari poterit sensibilis motus oscillato-
rius in partibus instrumenti; quae partes vicissim in aerem
vibrantem reagendo valebunt sonum ipsum modificari etiam
quoad tonum; et quidem plurimum si instrumentum sit val-
de breve, quemadmodum expertus est D. Savart; adeo ut
brevi tubo membranacea obtineri possit magna varietas
tonorum, qui eo graviores erunt quo minus tenditur mem-
liraua.

134. Haec proponimus explicanda cirea instrumenta
pneumatica. 10. Aperto aliquo foramine ex. gr. tertio, cae-
terisque clausis, ac deinde aperto alio puta quinto. variat
tonus: at si' per aperitionem tertii inducitur communicatio
interni aeris cum externo, nonne ex dictis (133) audiri de-
beret idem tonus sive apertum sive clausum sit quintum
foramen? 20. Sola inflationis intensione, mutantur toni. e-
tiam servata eadem internae columnae longitudine 3&quot;. In
canna organi eiusdem diametri superius clausa, si subdupla
sit longitudo, idem redditur tonus qui obtinetur ex can-
na superius aperta, et longitudinis duplae.
Ad 1.&quot; Cum varia in instrumento pneumatico forf- mina aperiuntur, variae interni aeris columnae communi-297
cantes çum aere externo excitantur; non ita tamen commu
nicantes, ut simul non etiam inter se communicent; ergo
looi variare per plurium foraminum aperitionem debent ,
etsi exquisitam ejus rei rationem aegerrime quis reddere possit.
Ad 2.&quot; Ut per vehementiorem percussionem in chor
da instrumenti fidicularis contingit ut ea resonet ad oclavam,
ita in columna aerea per variam inflationis intensionem con
tingit ut tonus mutetur; et sicut certum est in chorda mu
sica quod ea tunc dividitur in duas partes separatim oscil
lantes, ita eadem asserenda est fieri divisio et oscillatio in
columna aerea sub tempore, quod sił proportionale tono
quem reddit. Hinc deducitur explicatio saltus ut ajunt tu
bae v. gr. ad octayam: cam paulo vehementius inspiralur tu
ba, cogitur aer ad celeriorem motum , quem tamen colu
mnae aereae jam vibrantes , utpote nimis longae, praesta
re non possunt. Dividitur igitur columna per medium ita ,
ut duo nova segmenta aeris aequalia suas vibrationes separatim
peragant. Ubi vero saltus non sit ad octavam , alia divi
sio fieri dicenda est .
Ad 3.&quot; Ia medio cannae duplae efformátur nodus ,
habetur aereum stratum quiescens , quemadmodum
habetur in orificio clauso cannae subduplae ; adeoque ea
dem undae aereae longitudo in utraque canna , idemque
proinde tonus .

135.* Sit tubus cylindricus determinatae longitudinis 1,
firmiter obseratus apud alterum orificium , aperius apnd al
terum : aequilibrium aereae columnae inclosae ita turbari
pono, ut qui aer orificio aperto ( ubi initium distantiae x
consliluo ) respondet, nullam densitatis variationem subeat,
et qui orificio clauso, nullatenus moveatur.Functiones ( 129.7 °) .
f, fx , ac proinde f , F ' tanquam datas assumo ab x = 0
ad x = l. E statu aeris apud extremitates tubi habemus
= o si x = 0, v = 0 si x = l; hinc
seu
fl + 1) + F'll — cl) = 0 ( 0 ) ,
20
297
can'tcs cum aere externo excitantur; non ita tamen cdmmu-
nicantes, ut simul non etiam inter se communicent; ergo
toni variare per plurium foraminum aperitionem debent
,
etsi exquisitam eius rei rationem aegerrime quis reddere possit.
Ad 2.&quot; Ut per vehementiorem percussionem in chor-
da instrumenti fidicularis cdntingit ut ea resonet ad octavam,
ita in columna aerea per variam inflationis intensionem cou-
ting'it nt tonus mutetur; et sicut certum est in chorda mu-
sica quod ea tunc dividitur in duas partes separatim oscil-
lantes, ita eadem assereuda est fieri divisio-et oscillatio in
columna aerea sub tempore, quod sit proportionale tono
quem reddit. Hinc deducitur explicatio saltus ut aiunt tu-
bae v. gr. ad octavam: cum paulo vehementius inspiratur tu-
ba, cOgitur aer ad celeriorem motum, quem tamen colu-
mnae aereae iam vibrantes, utpote nimis longae, praesta-
re non possunt. Dividitur igitur columna per medium ita,
ut duo nova segmenta aeris aequalia suas vibrationes separatim
peragant. Ubi vero saltus non sit ad octavam, alia divi-
sio fieri dicenda est, ⋅ '
Ad 3.&quot; In medio cannae duplae eEorm'atur nodus ,
seu habetur aereum stratum quiescens, quemadmodum
habetur in orificio clauso cannae subduplae: adeoque ea-
dem uudae aereae longitudo in utraque canna
,
idemque
proinde tonus.

135 Sit tubus cylindricns determinatae longitudinis !,
firmiter obseratus apud alterum orificium, apertas apud al-
tequm : aequilibrium aereae columnae inclusae ita turbari
pono, ut qui aer orificio aperto ( ubi initium distantiae .a-
constituo ) respondet, nullam densitatis variationem subeat,
et quiorificio clauso, nullatenus moveatur.F unctiones (129.7&quot;).
f, f. , ac proinde f, F' tanquam datas assume ab a: 30
ad .r.-zl. E statu aeris apud extremitates tubi habemus

:: osi æzo,v:——osiæ:-:l; hinc

≀≖∣⋅∶≀−∙⊢≀∶∠⊢⊢ F'(l—c1):o ( o )
.
20298
Fll — ct) - f ( c ) = 0 ( o' ) .
In (0 ) substituatur ct +1- x in locum ct : prodibit
f (21 + ci - x) = - F '(x - 1) ( 0&quot; ) ;
unde
= f'( x + cl) – f'( 21+ ct - x ) ,
c = -f(x + ct) -f(21 +ct - x) :
( o ' ')
in ( o &quot; ) fiat x = 0 ; erit ob ( o')
f (c + 2) = -F ( - ct) = -f (c ) (0&quot; &quot; );
subrogato ct +21 in locum ct, habebitur
f '( c +4 ) = -f( ct + 2 ) = f(t ) (o ');
denique si in ( 0 &quot;) ponitur ci=0, emerget
f( 21 — x ) = - F ( x ) ( 0 &quot; ) .
.
Aequationes ( o' : 0' ! ) satis sunt, ut functionem f con
siderare possimus veluti datam quoad omnes positivos ya.
lores quantitatis variabilis , ad quam respicit ipsa functio.
Etenim e statu initiali atque arbitrario aereae columnae ad
motum incitatae , data est F' ab x = o ad x = l ; ergo
ob (o &quot; ) data erit fab x = l ad x = 21 : ex eodem sta
tu jam dala erat fab x = o ad x = l ; ergo dabiturf
ab x = o ad x = 21. Aequatio autem ( o&quot;) rem evidentissime
absolvit quoad caeteros quantitatis variabilis positivos va
lores. Ergo etc.
298
F'(—ct)——f(ct):-to (c').
In (o) substituetur et −⋅⊢≀ —- a: in locum et: prodibit
f(2l −∣⋅− ct — a:) ∶−∙−− F'(x—ct) (o&quot; );
uude
v:f(æ-i-ct)—f(2l—l—ct—æ).
etc:—f(x-i-cz)—f(2l -i-ct—æ):
111 (o&quot;) fiat m::- o; erit Ob (0')
(o&quot;')
f(cs −↿− 21) ∶−∙− −F'( —ct) −−∶ —f(ct) (o&quot;):
subrogato et —-[-21 in locum et, habebitur
f(cz ↽⊢ 4!) ∙−−− —-f(ct −⊦ 21) ↽↼−−⋅≖ f(ct) (a';
denique si in (a'—') ponitur ctzo, emerget
f(ZI—æ) z—F'Lr) (o&quot;).
Aequationes (a': a'!) satis sunt, ut functionem f coa-
siderare possimus veluti datam quoad omnes positivos ,va-
lores quantitatis variabilis, ad quamrespicit ipsa functio.
Etenim e statu initiali atque arbitrario aereae columnae ad
motum incitatae
, data est F' ab a: a ad w:! : ergo
ob (o&quot;) data erit f
ab æ-—-:l ad se:21 :ex eodem sta-
tu iam data erat f' ab a: :: o ada::1; ergo dabitur ['
ab a: :: 0 ad se:21. Aequatio autem (o&quot;) rem evidentissime
absolvit quoad caeteros quantitatis variabilis positivos va-
lores. Ergo etc.299
Quoniam ab x = o ad x = 21 dependet f' ab ini
tiali atque arbitrario statu aereae columnae ; poterit igitur
sic assumi inter illos limites , ut facto i = 1,3,5,7 ,..., sit
21
f'(c + % -f(cc + = p&quot;(ce) ( 0 &quot; &quot; );
numeri pares = 2, 4, 6 ... debent excludi ob aequatio
dem (o &quot; ). Instaurantur ergo iidem functionis f valores
42 quotiescumque tempus t evadit it ; sed ( o &quot; ) a functio ic
ne f unice dependent v, E. Columna igitur aerea in eum
dem restituitur statum per aequalia intervalla, suasque com
41
plet oscillationes intra tempus ; quarum propterea nu
merus intra q ' erit
ic
ic
41
136.. Evanescet (135.0 '&quot;') velocitas v ubi fuerit f ( x + cos
= f (21 + ct - x ) ; evanescet e si f'( x + ct) - f (21+
ci - x ). Primum contingit ( 135 : 0 &quot; ) quando (22 +ic - x )
41'2 - ( x + cl) seu 21 2x ; secundum quando 21—
21&quot; i 1
2x= • Hinc 1º. facto i = 0 , 1 , 2 , 3 .. i
scet aer in distantiis
2 , quie
lli - 21 )
X
2º . Facto i&quot; =1 , 3 , 5 .... 1 ; movebitur aer in
locis
299
Quoniam ab a: a ad a: 2! dependet ;&quot; ab ini-
tiali atque arbitrario statu aereae columnae ; poterit igitur
sic assumi inter'illos limites, ut facto i:1,3,5,7,...., sit
41 21 ∣
'&quot;
∣⇃≺∁⊢⊢ ∙
−−⋮∙⊣∶∶≕−∣↙≼⋄⊢⊢ 7): f
(ct) (0 ):
numeri pares r':2, 4, 6 ...
debent excludi ob aequatio-
nem (o&quot;). Instaurantur ergo iidem functionis f' valores ,
quotiescumque tempus :
evadit t −∣∙⋅ &amp; sed (o&quot;')a functio-ne )&quot; unice dependent v, :. Columne igitur aerea in eum--
dem restituitur statum per aequalia intervalla, suasque com-
. . . plet 4! osc1llat1ones intra tempus ∙≀⋅−≔∙ ; quarum propterea nu-
merus intra 1&quot; erit
t'

136. a Evanescet (135.o&quot;')velocitas :: ubi fuerit f' (æ—l—ct

f(ZH— ct —- æ ); evanescet :si f(x −∣⋅− et):
— f(ZI-i-
ct ∙−− a: ). Primum contingit (135: a'&quot;) quando (21—1—4 cs --' a:)
41&quot;! −∙− ( æ ⋅−⊢ ct) seu 2! -- 21: −∙−−−−− T;secundum quando 21—
2
∙∣∣ ∙−↿ .
2x: −⋮∙−↨ ∙Hinc ↿∘∙ facto 1&quot;: a, 1, 2, 3 ....
'T, qu1e-
scet. aer in distantiis' -
∙∙∙ [( t' — 21&quot;)
æ
!
20. Facto 1'&quot; :1, 3, 5 ....
i; movebitur aer in
locis300
llimi)
quin tamen ullam patiatur densitatis variationem, Aper
tis itaque foraminibus in hisce postremis locis , nullo pa
cto sonus mutari debet ; quod experientiae consonum re
peritur: imo non mutabitur sonus, licet lubo abscindatur
pars 1- x , quae ultra locum x ad fundum usque pro
tenditur. Atqui pars reliqua nihil aliud est nisi tubus
in utraque patens extremitate: ergo si de hujusnuodi cubis
sermo sit, posita e = o apud unum orificium erit quo
que apnd alterum { =0.
137. # In tubis itaque cylindricis, quorum ambo ori
ficia libera omnino sunt, habetur ( 129.7 .)

#

Fl—ct) — 9 (2+ cl ) = 0, F1 — ct) - f'(C ) = 0.
Hinc facile deducuntur ( 135 ) sequentes aequationes
f (21 + cix) = F ' x - ct), v = P ( x + ct) + P (21 + c1 - ),
c : = f ( 21 + (1 - x ) — f (x + c ! ), f (ce + 21 = F (-1)= f (c ),
f'(21 — * ) = F ( x ).
Quia vero ab x = o ad x = 21 rursus dependet p
ab initiali atque arbitrario statu 'aereae columnae , ic
circo poterit etiam asseri sequens aequatio.
f (ce + *-) = f ( c )
in praesenti est i = 1. 2, 3, 4 ......
æs. lu.—'r')
1
,
quia tamen ullam patiatur densitatis variationem. Aper-
tis itaque foraminibus in hisce postremis locis, nullo pa-
cto sonus mutari debet; quod experientiae consonumre-
peritur: imo non mutabitur sonus, licet tubo abscindatur
pars l—æ, quae ultra ↙
locum a: ad fundum usque pro-
tenditur. Atqui pars reliqua nihil aliud est nisi tubas
in utraque patens extremitate: ergo si de huiusmodi tubis
sermo sit, posita :

: o apud/uuum oriücium erit quo-

que apud alterum::o.

137.a In tubis itaque cylindricis, quorum ambo ori-
licia libera omnino sunt, habetur (129. 70.)
F'U—ct) -
f(l −↿− ct) ∙−−−−∙∙ o,F'( -
ct) --f(ct): 0.
Hinc facile deducuntur (135) sequentes aequationes
f(21 -l-ct -æ):F'(æ—-ct), v ::f'(æ ⊣− et)-t— f(ZI-i-ct—x).
es:/(21 -t-ct —x)-— f(æ-t—ct), f(ct-t-Zl):F'(—cc):f(c1),
f(2l — a: ): F' (æ)-
Quia vero ab a: 0 ad ..r:21 rursus dependet f
ab initiali atque arbitraria statu 'aereae columnae
,
ic-
circo poterit etiam asseri sequens aequatio.
f(ct—i- -—2'-£-) :f'(ct )

:

in praesenti est i: 1. 2, 3, 4 ......
≡⊲∙⋅⇀≣∎ lJ-r

: 301

22
Iterat ergo aerea columna per aequalia intervalla ic
oscillationes suas , quarum proinde numerus intra 1 &quot; erit
ic n =
21
Haud immoror inquisitioni distantiarum , ubi a
er vel quiescit, vel nativam retinet densitatem : hujusmo.
di namque investigatio similiter perficitur ac in Lubis,
quorum unum orificium apertum est . Satius forsan e
rit adnotare quod, facto i = 1 , exhibet ( 137 ) aequatio n '
relationem inter principalem tonum n ', redditum ab
elastico fluido intra tubum oscillante, et velocitatem c qua
sonus incederet si per ipsum fluidum propagaretur. Hinc
patet quomodo experimentis indagari possit velocitas c in
aliis elasticis flaidis ab aere atmosphaerico diversis : ex
tentaminibus Van - Rees, Frammeyer, et Moll prodiit so
ni velocitas sub temperie 10.° C
21
io gas oxigenio 3,7m, 9 :
bydrogenio 1233 , 3 ,
nitrogenio .
.
339 .
oxido nitrico 317 , 4 ,
acido salphuroso 229 , 2 ,
acido carbonico 370 , 7 ,
. .
suboxido carbonico . . 341,1
etc. etc.
301
. . 2! [terat ergo aerea columna per sequsl1a1ntervalla ∙∙∙∙⋮∙⋅−
oscillationes suas ∙ quarum ⋅proinde numerus intra 1&quot; crit
'
IC
Haud immoror inquisitioni distantiarum , ubi a-
er vel quiescit, vel nativam retinet densitatem: huiusmo-
di namque investigatio similiter perficitur ac in tubis,
quorum unum orificium apertum est. Satius forsan e-
rit adnotare quod, facto 1':1,exhibet (157) aequatio n'
0
∙−−∶ -2-l- relationem inter principalem tonum n', redditum ab
clastico flaido intra tubam oscillante, et velocitatem e qua
sonus incederet si per ipsum fluidum prcpagaretar. Hinc
patet quomodo experimentis indagari possit velocitasc in
aliis elasticis fluidis ab aere atmosphaerico diversis: ex
tentaminibas Van— Bees, Frammeyer, et Moll prodiit so-
ni velocitas sub temperie 10.0C
in gas oxigenio . . . . . . 317',g,
hydrogenio . . . . . 1233,3,
nitrogenio. . . . . . 339 . ∙
oxido uitrico . . . . 317 ,4
acido sulphuroso . . -
229 , 2
,
acido carbonico . . . . 370 , 7 ,
suboxido carbonico . . . 341 ,
1
,
etc. etc.302

138. Si tubus proponilur utrinque obseratus , quis
que videt fore v = o apud ambas extremitates; unde (129.7°)
f (c ) + F ( -ct)= 0,flfct) + F (l ct) = 0,
quarum ope determinatur motus inclusi aeris,
De propagatione soni per liquida , et per solida corpora.
139.* Quod spectat ad liquida corpora , in comperlo
est aquam v. g. contrahi perparum posse atque restitui
in suis partibus : itaque qua ratione turbatum posuimus
( 129..1 . ° ) aequilibrium , eadem in praesenti imaginemur
turbari . Propagato motu , densitas ré aquae libratae ver
tetur in je = pili + :) apud (2. , y , z ) ; et pressio o'
in w= '+Ae ; exprimit A numerum experimentis deter
minandum. Sumptis hic quoque X=0, Y=0, Z=o, et ra
tiocinando ut in citato n. ° assequemur
d
dQ
1 do dt (
dQ
dt
dr
A dL {1+ :)
dr seu
р. dr pi
A
tum facto c ” , perveniemus ad formulas (i' '.;&quot; . .
ji
į &quot; : 129, 1.0 ) . Non pluribus opus est ut intelligamus
( 129. 2.° 3,0 ) sonum per aquam diffundi aequabiliter ve
locitate.
VA
Numerus A potest determinari ex parvula contractione ,
quam juxta longitudinem à ( haud variata diametro ) pa
302
1381: Si tnbus praponitnr utrinqne obseratus , quis-
que videt fore v:o apud ambas extremitates; unde (129.?)
f(ctH-FX --ct):o,f7(l—i-ct)-i-F'(l—ct):a,
quarum Ope determinatur motus inclusi aeris.
De propagatione soni per liquida, et per solida corpora.
139:- Quod spectat ad liquida corpora, in comperto
est aquam v. g. contrahi perparum posse atque restitui
in suis partibus :itaqne qua ratione turbatum posuimus
( 129. ⋅↿∙∘ ) aequilibrium, eadem in praesenti imaginemur
turbari. Pr0pagato motn, densitas pf aquae libratae ver-
tetur in þ.:yJU—I—s) apud (.x.-,,] ,
z) :
et pressio a'
in ≔≖⇌≖⋝∣↰∟⋀⋮⋅⋮ eXprimit A numerum experimentis deter-
minandum. Sumptis hic quoque X:0, ↧↗−−−−⋅∘∙ Z:o, et ru-
tiocinando ut in citato 11.0 assequemur
d
dQ) ↼ 1 de ∙− dQ) (Et? A; JLu-Jr-s) ∙− &quot;( dt ∙
a d?
.
dr ' se.. a' d.- 4.-
A -
∙∣∣ -1 tum facto ?

: cz, perveniamus ad formulas (:

'.
t
'.
i'
i&quot;: 129, ↿∙∘ )
. Non pluribus opus est ut intelligamus
( 129. 2.0 3.&quot;) sonum per aquam diffundi aequabiliter ve-
locitate. ⋅
.: ⇂∕⋅−⋮∶⇡∣−⋅
Numerus A potest determinari'ex parvula contractione f:,
quam iuxta longitudinem l (haud variata diametro) pa-303
tilur columna aquea ob incrementum 5. superadditum
pressioni o '. Nam 1 : 1 - B = + ): , ideoque &lt; =
B \beta

: ' sed o=u'two=a' +As, igitur スー B 2

6.
A :
σολ
E \beta
In hypothesi pressionis . = 0 &quot; , 76) g, ac temperiei n=
10.• C, experimenta Dni Canton suppeditant B = 0,000046 ),
inter quem valorem et quos invenerunt DD . Parkins et
Oersted , nimirum
B=0,0000452 , B=0,0000482 ,
parvula est differentia. Ponatur hydrargyri densitas 1 ;
erit proxime u'= . : assumpta igitur g=9m, 8088,
13,5819
1
emerget
c=1483&quot; , 59.
Sonus videlicet propagatur per aquam plus quadruplo ce
lerius quam per aerem. D. Beudant dicit in hac se fuis
se sententia , ut e suis experimentis in mari institutis ta
lem deduceret soni velocitatem , quae 1500m saltem aequaret.
140.* Quisque videt soni velocitatem eadem ratione
posse in caeteris corporibus , sive liquidis , sive solidis ,
determinari , modo eorum partes contrahi perparum queant
atque restitui . Sic , manente 5 . = (0,76 ) 8 , obtinuit idem
ipse Canton hydrargyri contractionem B = 0,0000032 : as
sumpla igitur u = 1 , erit
c = 1576m , 35
1
303
titur columna aquea' ob incrementum m superadditum
pressioni w'- Nam 71: l—þ:p.'(1-l-s): p!
,
ideoque ::
P −⇀ 13 ⇤
⋅
m−⊸T;'sed ∏∙−−∶∏∎∙⊦∏∘∶−−∸⋅∄≖⋅−⊦∆⋮∙ igitur
'
A ∙− a'., ∙∙∙ wo).
8
5
In hypothesi pressionis uro :( o'&quot;, 76) g, ac temperiei :::,
10.(, C. experimenta Dni Cauton suppeditant þ:0,0000461,
inter quem valorem et quos invenerunt DD. Parltins et
Oersted , nimirum
,ezo,oooo45) ∙ þ:0,000048).
.
parvula est differentia. Ponatur hydrargyri densitas :1;
↿
15.5819
erit proxime pf: : assumpta igitur g:9&quot;?,,8088,
emerget ⋅
c:1483&quot;'
. 59.
Sonus videlicet prcpagatur per aquam plus quadruplo cc-
lerius quam per aerem. D. Bendant dicit in hac se fuis-
se seateutia , ut e suis exPerimentis in mari institutis ta-
lem deduceret soni velocitatem, quae 1500&quot; saltem aequaret.
1403 Quisque videt soni velocitatem eadem ratione
posse in caeteris corporibus
.
sive liquidis , sive soli-dis ,
determinari , modo eorum partes contrahi perparum queant-
atque restitui. Sic, manente wo:(0,76) g , obtinuit idem
ipse Canton hydrargyri contractionem [5:0,0000037t : as-
sumpta igitur pf:1 , erit
0:1576,&quot; ∙ 35304
velocitas , qua per hydrargyrom diffunditur sonus. Ante
quam usum contractionis \beta animadverteret Laplace ad de finiendam soni velocitatem per liquida et solida corpora ,
exhibuerat Chladni in sua Acustica aliam methodum
sane ingeniosam , ejusdem velocitatis investigandae in cor
poribus solidis.
141. # Innititur ista methodus analogiae, quam norunt
Physici inter oscillationes aeris in tubo cylindrico apud
ambas extremitales aperto et longitudinales oscillationes
virgae rigidae , cujus ambo extrema omnino libera sint.
Exprimat enimvero l oscillantis virgae longitudinem ; n'
principalem tonum , quem edit resonans virga ; c' quae
sitam velocitatem . Erit ( 137 )
n &quot; ; unde n ' : n &quot; = 0 : c' , ' =
21
Iam si velocitas soni per aerem repraesenterar per &quot; , ex
perimenta D.ni Chladoi praebent soni velocitatem c
per stannum . 717 를
per argentum
per cuprum .
12
per ferrum et vitrum ... 17
per varia lignorum genera 11 ad 17 , . .
Ad explorandam soni velocitatem per ferruin fusionis , in
promptu habebat D. Biot 376 tubos ex hoc metallo com .
pactos ; quibus singulis mediocris erat longitudo duorum
304
velocitas , qua per hydrargyrnm diffunditur sonus. Ante-
quam usum contractionis þ animadverteret Laplace ad de-
finiendum soni velocitatem per liquida et solida corpora ,
cxbibuerat Chladni in sua Acustica aliam methodum,
sane ingeniosam .
ejusdem velocitatis investigandae in cor-
poribus solidis.
1414: lnnititur ista methodus analogiae, quam norunt
Physici inter oscillationes aeris in tabo cylindrico apud
ambas extremitates aperto et lougitudiuales' oscillationes
virgae rigidae, cuius ambo extrema omnino libera sint.
Exprimat enimvero! oscillantis virgae longitudinem ; n&quot;
principalem tonum
, quem edit resonans virga ; c' quae-
sitam velocitatem. Erit (137) '
c, '
nn 11&quot;:-2-i-;nuden:n&quot;:c:c', c':c-—J.
»
Iam si velocitas soni per aerem repraesentetur per 1, ex-
perimenta D.!d Chladni praebent soni velocitatem c'
per stannum . ∙ ∙ ∙ ∙
, 7
vet--
per argentnm . . . . . . 9 ,
per cuprum . . . . .
.. . 12 ,
per ferrum et vitrum . . .
17
.
per varia lignorum genera . .
11 ad 17,
Ad explorandam soni velocitatem per ferrum' fusionis ,
in
promptu habebat. D. Biot 376 tubos ex hoc metallo com'
pactos ; quibus singulis mediocris erat longitudo duorum

* a305

metr. cum partibus millesimis 515. Sumptis experimentis,
prodiit soni velocitas 104 ; nisi quod jungebantur ii tu
bi ope plumbi, quod aliquanto sonum retardare videtur.

=== De vocis humanae origine. ===
142. Vocis humanae organum etsi considerari maxi
me debet tamquam instrumentum pneumaticum ftexili
et elastica materia ex parte compactum , non tamen ita
est ut cum instrumentis etiam fidicularibus aliquam non
babeat analogiam. Quod ut melius intelligatur , nonnulla
ex anatomicis sunt hic afferenda.
Palmo est viscus respirationi inserviens: in duas par
tes distinguitur , dexteram et sinistram , et duo magni lo
bi dicuntur , etsi quivis ex his duobus dividitur mino
ribus aliis. Substantia constat molli , spongiosa , rara et
vessiculosa ita ut ad aerem excipiendum aptissimus sit :
motu ergo dilatationis aere impletur , et constrictionis
motu eundem expellit ; atque aer ita expulsus primo per
multiplices canaliculos lobis interspereos , qui bronchia
dicuntur ; tum per duos ex utroque lobo emergentes ; de.
mum per ampliorem canalem emergit , qui ex praefa
tis duobus in unum conjunctis coalescit. Hic canalis seu tubus
ad oris usque radices ascendens trachea seu aspera arteria
nuncupatur ; in summitate asperae arteriae brevis canaliculus
habetur , qui larynx dicitur , cujus summitatem facit rima
ovalis a duabus membranis horizontaliter jacentibus relicta ;
quae rima glottis dicitur : atque huic superposita est epi
glottis ; tenuis scilicet et mobilis cartilago glottidem te
gens , quae ad hoc praecipue statuta esse videtur, ut dum
aliquid deglutimus , quidquam cibi aut potus in asperam
arteriam minime irruat, sed per contiguum canalem, qui
exophagus dicitur, et cujus orificium pharynx vocatur , de
more in stomachum demittatur. Itaque lobi pulmonis instar
}
305
metr. cum partibus millesimis 515. Sumptis experimentis.
prodiit soni velocitas 10;- ; nisi quod iungebantur ii ftu-
bi »ope plumbi, quod aliquanto sonum retardare videtur.
De vocis humanae origine.


142.1Vocis humanae organum etsi considerari maxi-
me debet tamquam instrumentum pneumaticum fiexili
et elastica materia ex parte compactam, non tamen ita
est ut cum instrumentis etiam fidicularibus aliquam non
habeat analogiam. Quod ut melius intelligatur, nonnulla
ex anatomicis sunt hic aderenda.
⋅ Palmo est viscus respirationi inserviens: in duas par-
tcc distinguitur , dexteram et sinistram ,
et duo magni lo-
bi dicuntur , etsi quivis ex his duobus dividitur mino-
ribus aliis. Substantia constat molli , spongiosa , rara et
vesaiculosa ita ut ad aerem excipiendum aptissimus sit:
motu ergo dilatationis aere impletur , et constrictioais
motu eumdem expellit; atque aer ita expulsus primo per
multiplices canaliculos' lobis interspereos , qui bronchia
dicuntur; tum per duos ex utroque lobo emergentes :dc-
mum per ampliorem canalem emergit , qui ex praefa-
tis duobus in unum coniunctis coalescit. Hic canalis seu tubas
ad oris usqne radices ascendens tracbea seu aspera arteria
nuncupatur; in summitate asperae arteriae brevis canaliculus
habetur , qui laryux dicitur, cuius summitatem facit rima
ovalis a duabus membranis horizontaliter jacentibus relicta;
quae rima glottis dicitur : atque huic superposita est epi-
glottis; tenuis scilicet et mobilis cartilago glottidem te-
gens, quae ad hoc praecipue statuta esse videtur, ut dum
aliquid deglutimus, quidquam cibi aut potus in asperam
arteriam minime irruat, sed per contiguum canalem, qui
cxophagus dicitur, et cuius orificium pharynx vocatur
, de
more in stomachum demittatur. Itaque lobi pulmonis instar306
re cur com
follium aerem excipiunt, cum compressi illum emittunt per
asperam arteriam : aer ita expulsus per asperam arteriam
in arctiorem laryngis canalem irroit , atque ita ex am
pliori in angustius spatium redactus compressionem pati
debet , oscillatoriumque motum concipere. Sed quia la
rynx flexili et elastica materia compingitur, iccirco ( 133)
ad motum tremulum ab aere vibrante excitabitur. Deinde
vero in eumdem aerem diversimode reagendo, prout magis
vel minus erit tensa , ejus Oscillationes diversimode quoque
modificabitur.
Obiter notamus antiquos et cum iis Galenum male
organum vocis humanae in trachea constituisse ; quam
arbitrabantur vices gerere tubi, per quem aer ad sonum
jam excitatus excurrit. Refelles hanc opinionem consi
derans aerem qui tracheam ascendit , libere ascende
et liberius habere spatium ; unde non est
primi debeat et oscillatorium motum habere : cum autem
glottis sit multo arctior quam trachea , per glottidem
transiens habet quidem unde comprimi possit. Haec de
voce humana : ad vocem enim quod spectat quorumdam
animalium , uti sunt multae aves ; hae cum etiam exse
cto collo , sola ventris compressione sonum edant , in his
utique trachea concurrit ad sonum ipsum modificandum .
Sed nil hinc eruitur contra jam dicta: in istis namque avi
bus trachea habetur supra glottidem , seu gloutis esse obser
vatur non ad summitatem , sed infra tracheam ; contra
ac est in homine , et plerisque aliis animalibus.
In monumentis Academiae Parisiensis ad an. 1741
observat Ferreinius intra laryngem duas haberi fibras ad
labrum glottidis ; quae fibrae ex impetu aeris per angu
stiorem laryngis canaliculum irrumpentis ad tremitum con
citantur , atque hoc tremitu resonant , quemadmodum in
fidibus contingit ; unde dictum est vocis humanae organum
analogiam habere aliquam ad instrumenta fidicularia . Sum
psit ille plures laryages cum sua glottide ; dunque insuf
306
folliam aerem excipiunt. tam compressi illam emittunt per
asperam arteriam: aer ita expulsus per asperam arteriam
in arctiorem laryngis canalem irruit ,. atque ita 'ex am-
pliori in angustias spatium redactus compressionem pati
debet, oscillatoriamque motum concipere. Sed quia la-
rynx flexili et elastica materia compingitur. iccirco (133)
ad motum tremulum ab aere vibrante excitabitur. Deinde
vero in eamdem aerem diversimode reagendo, prout magis
vel minus erit tensa, eius oscillationes diversimode quoque
modificabitur. ∙
Obiter notamus antiquos ,et cum iis Galenum male
organum vocis humanae in trachea constituisse; quam
arbitrabantur vices gerere tubi, per quem aer ad sonum
iam excitatus excurrit. Refelles hanc opinionem consi-
derans aerem. qui tracheam ascendit
,
libere ascende-
re,'et liberius habere spatium ; unde non est cur ,com-
primi debeat et oscillatorium motum habere: cum autem
glottis sit multo arctior quam trachea , per glottidem
transiens habet quidem unde comprimi possit. Haec de
voce humana : ad vocem enim quod spectat quorumdam
animalium , uti, sunt multae aves; hac cum etiam exse-
cto collo, sola ventris compressione sonum edant, in his
utique-trachea concurrit ad sonum ipsum modificandam.
Sed nil hinc eruitur contra iam dicta: in istis namque avi-
bus tracbea habetur supra glottidem , sen glottis esse obser-
vatur non ad summitatem, (sed infra tracheam ; contra
ac est in homine , et plerisque aliis animalibus. .
In. monumentis Academiae Parisiensis ad an. 1741
observat .Ferreinius intra laryngem duas haberi, fibras ad
labrum glottidis ; quae fibrae ex impetu aeris per angu-
stiorem laryugis canaliculata irrumpentis ad tremitum.con-
citantur, atque hoc tremitu resonant
, quemadmodum in
fidibus contingit; unde dictum est .vocis humanae organum
analogiam habere aliquam ad instrumenta fidicularia. Sum-
psit illa plures' larynges cum sua glottidc; dumque insuf-307
Aando sonus vocis animalis excitabatur, microscopio Gibras
praedictas inspiciendo tremor et vibratio in iisdem cerne
batur , prout in chordis musicis habetur dum resonant.
Atqui eo ipso ex earum tremitu ab irruente aere tamquam
a vi percutiente excitato sonus gigni debet, vel jam geni
tus modificationem quamdam recipere.
Rursus , sicut in chordis musicis contingit ' ut chorda
brevior det sonum acutiorem , graviorem longior : ita ani
madvertendum hic fuit an fibrarum illarum major minor
ve longitudo toni mutationem induceret. Compertum au
tem est quod , impedita illarum fibrarum parte ne tre
meret , tonus prodibat acutior.
Sumpsit etiam larynges bovis , canis, aliorumque ani
malium , deinde insufflando excitabatur mugitus bovis , et
conformis aliis animalibus sonus. Movendo autem musculos
ita , ut traherentur et distenderentur fibrae, excitabantur
mutationes soni , quae haberi solent in varia horum ani
malium voce.
Notetur illud : cum tensio vel remissio fibrarum glot
tidis et cartilagineae substantiue , qua larynx constat , ab
eodem musculo dependeat , ut notat Savart , consequitur
una cum fibris illis etiam laryngem tendi vel remitti.
Laxatis fibris, orificium glottidis ampliatur , et sonus pro
dit gravior ; tensis vero , orificium restringitur, et sonus
evadit acutior , ut in canibus observavit D. Magendie.

143. Si vox in larynge et glottide tanquam in proprio
organo fit ; quid ergo, inquies, os atque ejus partes con
ferunt ad formationem vocis ?
Respondeo oris cavitatem , linguam, dentes, labia con
currere ad modificationem perfectionemque vocis ; quae
larynge et glottide incipit quidem , sed non omnimode
ibi perficitur : nam quod in illis partibus sufficiens habea.
tur organum quin prorsus necessaria sint oris et linguae or
gana ad exhibendum aliquo modo sonum animalis pro
prium , apparet ex eo quod grues abscisso
in
et anseres 1
307
flando sonus vocis animalis excitabatur, microscupio fibras
praedictas inspiciendo tremor et vibratio in iisdem cerne-
batur, prout in chordis musicis habetur dum resonant.
Atqui eo ipso ex earum trcmitu ab irruente aere tamquam
a vi percutiente excitato sonus gigni debet. vel iam geni-
tus modificationem quamdam recipere.
Rursus , sicut in chordis musicis contingit 'ut chorda
brevior det sonum acutiorem, graviorem longior :
ita ani-
madvertendum hic fuit an librarum illarum maior minor-
ve longitudo toni mutationem induceret. Compertum au-
tem est quod ,
impedita illarum fibrarum parte ac tre-
meret. tonus prodibat acutior.
Sumpsit etiam larynges bovis , canis. aliorumque sni-
malium, deinde insufflaudo excitabatur mugitus bovis ,
et
conformis aliis animalibus sonus. Movendo autem musculos
ita, ut traherentur et distenderentur fibrae, excitabantur
'mutationes soni, quae haberi solent in varia horum ani-
malium voce.
Notetur illud: cum tensio vel remissio librarum glot—
tidis et cartilagineae substantiae, qua larynx constat , ab
eodem musculo dependeat , ut notat Savart, consequitur
una cum fibris illis etiam laryngem tendi vel remitti.
Laxatis' fibris, ,orificiu-m glottidis ampliatur, et sonus pro-
dit gravior; tensis vero , orificium restringitur. et sonus
evadit acutior, ut in canibus observavit D. Magendie.

143. Si vox in larynge et glottide tanquam in proprio
organo fit; quid ergo, inquies, os atque eius partes cou-
ferunt ad formationem vocis?
Respondeo oris cavitatem. linguam, dentes. labia con-
currere ad modificationem perfectionemque'vocis; (quae in
⇁
larynge et glottide incipit ⋅ quidem , sed non omnimode
⋅⋅
ibi perficitur: nam quod in illis partibus sufficiens habea-
tur organum qain prorsus necessaria sint oris et linguae or-
,gana ad exhibendum aliquo modo sonum animalis pro-
prium ,,apparet ex eo quod grues et anseres , abscisso308
capite , ex ventris compressione sonos edere possint iis si
miles, quos viventes edebant. Ad modificationem igitur per
fectionemque vocis in larynge et glottide inchoatae caete
ra concurrunt : neque haec modificatio in mera reflexione
consistit, sed in resonantia proportionata tono soni a
glottide emissi .
Ad articulatarum vocum formationem quod attinet ,
ea praecipue a mota linguae et labiorum repeti solet :
inter caeteros P. Fabri diligenter expendit quo pacto lin
gua et labia componantur ad cujusque syllabae efforma
tionem .

144. Dices: potest sonus excitari aerem expellendo
per angustius spatium ; atque ita sibilus per labiorum com
pressionem excitatur. Ergo dicendum videtur quod ex 90
la emissione aeris per angustius glottidis spatium vox effor
inari possit quin confugiamus ad tremitum laryngis et
fibrarum glottidis ; qui tremitus effectus erit soni quin in
sonum ipsum influat.
Respondeo : etsi sonus aliquis obtineri praecise pos
sit per hoc quod ex ampliore in angustius spatium aer
cogatar transire ; attamen quae hactenus diximus suadent
tremitum laryngis et fibrarum ad vocis formationem con .
currere; attenta praecipue varietate maxima , quae in vo
cis modificatione habetur. Novimus enim et singulos ho
mines modificari quam maxime vocem , et in diversis ho
minibus quam maxime diversum esse vocis sonum . Iam ve
ro cum habeatur sibilus per solam labiorum compressio
nem , inde expulso violenter aere , exigua est hujusmodi
soni diversitas; et omnes fere homines eumdem sonum ef
ficiunt , licet in diversa intensione : ergo cum contra in
voce diversitas sit maxima et proprius cuique sit homini so
nus , ad diversam fibrarum et laryngis materiam ac ten
sionem recurrendum potius videtur. Scio equidem ab in
strumentis pneumaticis etiam materia resistente compactis
et lingula instructis magnam edi posse varietatem sono
308
capite , ex ventris compressione sonos edere possint iis si-
miles, quos viventes edebant. Ad modificationem igitur per-
fectionemque vocis in laryuge et glottide inchoatae caete-
ra concurrunt: neque haec modificatio in mera reflexione
consistit, sed in resonantia prOportionata tono soni a
glottide emissi.
Ad articulatarum vocum formationem quod attinet ,
ea praecipue a motu linguae et labiorum repeti solet:
inter caeteros P. Fabri diligenter expendit quo pacto lin-
gua et labia componantur ad cuiusque syllabae efforma-
tionem.

144. Dices: potest sonus excitari aerem eXpellendo
per angustius spatium : atque ita sibilus per labiorum com-
pressionem excitatur. Ergo dicendum videtur quod ex so-
la emissione aeris per angustius glottidis spatium vox effor-
mari possit' quin confugiamus ad tremitum laryngis et
fibrarnm glottidis; qui tremitus effectus erit soni quin in'
sonum ipsum influat.
Respondeo: etsi sonus aliquis obtineri praecise pos-
sit per hoc quod ex ampliore in angustius spatium aer
cogatur transire; attamen quae hactenus diximus suadent
tremitum laryngis et librarum ad vocis formationem cou-
1:11rrere; attenta praecipue varietate maxima, quae in vo-
cis modificatione habetur. Novimus enim et singulos bo-
mines modificari quam maxime vocem, et in diversis ho-
minibus quam maxime diversum esse vocis sonum. Iam ve-
ro cum habeatur sibilus per solam labiorum compressio-
nem ,
inde expulso violenter aere , exigua est huiusmodi
soni diversitas; et omnes fere homines eumdem sonum ef-
ficiunt
,
licet in diversa intensione : ergo cum contra in
voce diversitas sit maxima et proprius cuique sit homini so-
nus , ad diversam librarum et laryngis materiam ac ten-
sionem recurrendum potius videtur. Scio equidem ab in-
strumentis pneumaticis etiam materia resistente compactis
et lingula instructis magnam edi posse varietatem sono-309
recessum
rum , atque ad instrumenta ista referri organum vocis ab
auctoribus non paucis. Verum non video quomodo glotti
dis fibrae se habeant ad vocis organum perinde ac lin
gula : si non ita haec movetur , ut epistomium alterne
aperiatur claudaturque ; licet ea citissime oscillet , nullus
inde prodibit sensibilis sonus . Iam vero glottidis fibrae
non sic oscillant , ut per mutuum accessum et
alterne claudatur aperiaturque ipsius glottidis foramen .
In glottidis fibris aeris irrumpentis impetu ad tremitum
concitalis auctores aliqui cum Ferreinio organum vocis ma
xime constituunt , illudque ad instrumenta fidicularia po
tissime revocant , minime considerantes quod hujusmodi fi
brae careant ea longitudine et crassitie , quae necessaria
esset ad graves atque ingentes humanae vocis tonos effi
ciendos,

145. Quaeres 

1.º qui sit defectus, ob quem mutis loquela deest. Cum plerique muti sint, quia sunt a nativitate surdi, quique proinde cum non possint alios loquentes audire ne loqui quidem discunt unquam; attamen sunt qui auditu pollent, at loquendi facultate destituuntur. Vitium in his multiplex esse potest; aut ex humorum nimietate et crassitie; aut ex fibrarum inelasticitate, qua etiam fit ut, timore insolito obrigescentibus fibris, vox impediatur in iis qui caeterum muti non sunt; vel ex nimia linguae turgescentia; vel alio vitio: adeoque non desunt exempla mutorum arte medica, aut etiam solius naturae auxilio loquelam adipiscentium. 

2.º Cur aves aliquae humanam vocem aemulentur, pleraeque non item. In psitlacis diligenter rem inspexit Kircherus, atque animadvertit pro corporis quantitate os satis magnum et excavatum, maxillas turgentes, linguam maxime flexilem, et rostrum superius contra indolem aliarum avium mobile; unde bruta pro majore vel minore aptitudine ad oris dilatationem, flexilitatem linguae, labiorum, vel rostri modificationem apta erunt ad sovum humanae vocis imitandum. Picae io
309
rum , atque ad instrumenta ista referri organum, vocis. ab
auctoribus non paucis. Verum non video quomodo glottidis fibrae se habeant ad vocis organum perinde ac lingula: si non ita haec movetur, ut epistomium alterne aperiatur claudaturque; licet ea citissime oscillet, nullus inde prodibit sensibilis sonus. Iam vero glottidis fibrae non sic oscillant, ut per mutuum accessum et recessum alterne claudatur aperiaturque ipsius glottidis foramen. In glottidis fibris aeris irrumpentis impetu ad tremitum concitatis auctores aliqui cum Ferreinio organum vocis maxime constituunt, illudque ad instrumenta fidicularia potissime revocant, minime considerantes quod huiusmodi fibrae careant ea longitudine et crassitie, quae necessaria esset ad graves atque ingentes humanae vocis tonos efiiciendos. 

145. Quaeres ↿∙∘ qui sit defectus , ob quem mutis
loquela deest. Cum plerique muti sint, quia sunt a nati-
vitate surdi
, quique proinde cum non possint alios loquen-
tes audire ne loqui quidem discunt unquam; attamen sunt
qui auditu pollent, at loquendi facultate destituuntur. Vitium
in his multiplex esse potest: aut ex humOrum nimietate
et crassitie; aut ex fibrarum inelasticitate , qua etiam fit
ut
,
timore insolito obrigescentibus fibris
, vox impedia-
'tur in iis qui caeterum muti non sunt; vel ex nimia lin-
guae turgescentia; vel alio vitio: adeoque non desunt exem-
pla mutorum arte medica
,
aut etiam solius naturae auxi-
lio loquelam adipiscentium. 2.(' Cur aves aliquae humanam
vocem aemulentur , pleraeque non item. In psittacis dili-
genter rem inspexit Kircherus , atque animadvertit pro
corporis quantitate os satis magnum et excavatum, maxil-
las turgentes, linguam maxime flexilem , et rostrum su-
perius contra indolem aliarum avium mobile; unde bru-
te pro majore vel minore aptitudine ad oris dilatationem
,
⋅ Hexilitaïem linguae ,
labiorum , vel rostri modificationem
apta'erunt ad sonum humanae vocis imitandum. Picae- iu-310
a
ter caeteras aves , et corvi antiquitus etiam ad voces hu
manas formandas instituebantur. 

3. ° An verum sit quod vox ita procreari possit ut infra laryngem genita videatur ,
ideoque sonus audiatur non tanquam ex ore , sed tanquam
ex alvo veniens. Ita de facto Pythonissae , quas Ethnicorum
historiae , et sacra scriptura etiam memorat, loquebantur:
aliquos tamen sine ulla suspicione daemoniaci operis ven
triloquos esse exemplis pluribus compertum est. Sicut enim
loquitur aerem expellendo , qui in larynge et glottide vo
cem excitat ; ita fieri potest ut aerem ore ac naribus at
lrahendo in gloutide item parem molum excitemus , sicque
non ex ore sed infra laryngem vox orta videatur,
प
be
AL

=== De auditus organo. ===
146. Externa auris pars palula est; et ex cartilagine intus concava atque elastica constat; quae in concham
sea cavitatem referentem conchae figuram desinit. Inser
vit ad colligendas uudas soni : hinc quasi natura duce qui
minus acuto pollet auditu , aut ad vocein nimis e lon
ginquo attendit , manum ad aures applicat , ut eo pacto
plures colligat aeris undas. Externa haec auris pars , quae
auricula simpliciter dicitur , musculis adornatur , quorum
ope sunt aliqui homines qui auriculam ad libitum mo
vent ; oves autem , equi et bruta alia multo facilius :
adnotant nonnulli Analomici ila necessariam esse exter
banc partem ut sonorus lenius allabatur
in internas cavitates, ut nonnisi confusa et quasi cum
inurmure fluentis aquae audiant ii, quibus auriculae abscis
sau sint. Animadvertendum tamen reptilia et aves hoc ex
lerno adminiculo carere.
Ad fundum conchae incipit meatus auditorius , qui est
canaliculus aliquanto tortuosus ; et ex majori latitudine in
minorem paullatim coarctator. Ita factum notat Val
9
nam aer 1
1
310.
ter caeteras aves , 'et corvi antiquitus etiam ad voces hn-
manas formandasinstituebantur. 3.0 An verum sit quod
'vox ita proci-cari possit ut infra laryngem genita videatur,
ideoque sonus audiatur non tanquam ex ore , sed tanquam
ex alvo veniens. Ita de facto Pythonissae , quas Ethnicorum
historiae ,
et sacra scriptura etiam memorat, loquebantur:
aliquos tamen sine ulla suspicione daemoniaci operis ven-
triloquos esse exemplis pluribus compertum est. Sicut enim
loquitur aerem expellendo , qui in larynge et glottide vo-
cem excitat; ita fieri potest ut aerem ore ac naribus at-
trahendo in glottide item parem motum excitemus, sicque
non ex ore sed infra laryngem vox orta videatur,
De auditu: organo.
146. Externa auris pars patula est, et ex cartilagi-
ne iutus concava atque elastica constat; quae in concham
sen cavitatem referentem conchae figuram desinit. Inser-
vit,ad colligendas undas soni: hinc qnasi natura duce qui
minus acuto pollet auditu
,
aut ad vocem nimis e lon-
giuquo attendit, manum ad aures applicat , ut eo pacto
plures colligat aeris undas. Externa haec auris pars, quae
auricula simpliciter dicitur
, musculis adornatur , quorum
0pe sunt aliqui homines qui auriculam ad Hibitum mo-
vent; oves autem
, equi et bruta alia multo facilius :
adnotaut nonnulli Anatomici itaqnecessariam esSe exter-
nam lianc partem ut aer sonorus lenius allahatur
in internas cavitates, ut nonnisi confusa et quasi- cum
murmure fluentis aquae audiant ii, quibus auriculae abscis-
sae sint. Animadverteudum tamen reptilia et aves hoc ex-
terno adminiculo carere.
Ad fundum conchae incipit meatus auditorins , qui est
canaliculus aliquanto tortuosus; et ex maiori latitudine in
minorem paullatim coarctatur. Ita factum notat Val-311
sa
salva at sonus intendatur magis , sicuti in recurvis lubis
a surdastris adhiberi solitis intenditur ; alii potius ad im
minuendum aeris impetum , ne in auris interiora fortius
impellat , has tortuositates in organo auditus a natura in
stilutas putant. In auditorio meatu humor quidam amarus
ac viscosus habetur , seu cerumen ; exsudat e glandulis
quas sebaceas vocant , et institutum est ut minima ani
malcula ab ingressu ad interiora auris arceantur .
Ad finem hujus canalis habetur membrana tympani:
haec membrana tenuissima , sicca , pellucida et valde ela
stica , obtensa est annullo , qui tamen totum circuitum
non complet ; et fere ad similitudinem pellis tympani mi
litaris cavitatem interiorem superambit : non est recte exten
sed curva nonnihil ; coacava scilicet respectu auris
externae , convexa ad partes internas . Fuit acerrima quae
stio , an membrana tympani omnem communicationem in
ter externam internamve aurem excludat , an contra per
via sit aeri externo. Argumentum pro communicatione va
lidum est , quod aliqui fumum ore exceptum per aurem
emittunt ; neque id semper imposturae vertendam est ,
ut compertum fuisse Nolletus ait a viro , cni Academia
regia jussum fecerat facti veritatem explorare. Argumen
tum contra communicationem est , quod Valsava , immis
so in aurem internam hydrargyro , quantumvis excute .
retur , nihil unquam per externam aurem defluxit ; quam
quam reponere soleat qui communicationem tuetur , quod
in cadaveris organo non est necesse partium structuram sal
vari.
Post pellem tympani habetur cavitas aere plena , quae
capsula dicitur , quaeque cum membrana praedicta tym
panum constituit. In hac sunt quatuor ossicula quae ap
pellantur malleus , incus , os orbiculare , stapia. Alii tria
tantum numerant , omisso osse orbiculari , vel quia, cum
ompium humani corporis ossiam minimum sit , adeo ut
non superet dimidium grani millii , animadversionem fu
31↿⋮ salva ut sonus intendatur magis , sicuti in recurvis tubis
'a snrdastris adhiberi solitis intenditur; alii potius ad im-
miuuendum aeris impetum , ne in auris interiora fortius
impellat, has tortuositates in organo auditus a natura in-
stitutas putant. In auditorio meatu humor quidam amarus
ac viscosus habetur , seu cerumen; exsudat e glandulis,
quas sebaceas vocant , et institutum est ut minima ani-
malcula ab ingressu ad interiora auris arceantur.
Ad finem hujus canalis habetur membrana tympani:
haec membrana tenuissima , sicca , pellucida et valde ela-
stica
, obtensa est annnllo , qui tamen totnm circuitum
non complet; et fere ad similitudinem pellis tympani «mi—
litaris cavitatem interiorem superambit: non est recte exten-
sa , sed 'curva nonnihil : concava scilicet respectu auris.
externae
, convexa ad partes internas. Fuit acerrima qnae-
stio , an membrana tympani omnem communicationem in-
ter externam internamve aurem excludat , an contra -per-
via sit aeri externo. Argumentum pro-communicatione va-
lidum est
, quod aliqui fumum ore exceptum per aurem
emittunt; neque id semper imposturae vertendam est,
ut compertum fuisse 'Nolletus ait a- viro, cni Academia
regia iussum fecerat facti veritatem explorare. Argumen-
tum eontra communicationem est
, quod Valsava ,
immis-
so in aurem internam hydrargyro , quantumvis excute-
retur, nihil unquam per externam aurem defluxit; quam-
quam reponere soleat qui communicationem tuetur
, quod
in cadaveris organo non est necesse partinm structuram sal-
var]. '
Post pellem tympani habetur- cavitas aere plena , quae
capsula dicitur
, quaeque eum membrana praedicta tym-
panum constituit. In hae sunt quatuor ossicula quae ap-
pellantur mallens
.
incus , os orbiculare
, stapia. Alii tria
tantum numerant
, omisso osse orbiculari, vel quia, cum
omnium humani uerporis ossium minimum sit, adeo ut
non superet dimidium grani millii
, animadversionem fu-312
1
1
gerit : vel quia ita adhaeret slapiae et incudi , at cum al
tero ex his confundi potuerit, Circa haec ossicula nolan
dum , quod ejusdem magnitudinis sint tam in infante quam
in adulto viro contra aliarum omnium corporis partium in- .
dolem , quae aetatis progressu augentur. Hoc ideo factum
esse docet Valsalva , ne augmento partium auditui inservien
tium alia sit sonorum ratio adulla aetate ac fuit ab ini
tio ; et ideas gravis atque acuti quas pueri imbibimus, ma
tare aetate proficiente cogamur.
In tympani cavitate habetur canalis quidam seu lu
ba Eustachiana dicta ab ipsius inventore : per hanc tu
bam ab interna auris cavitate obtinet communicatio cum
cavitate oris; desinit enim ista tuba ad radices uvae; quem
prope locum etiam interiora nasi communicant : hujus tu
bae ope fit , ut sonus ex oris cavitate auri communicetur,
ideoque qui dentibus stringit corpus resonans sobum au.
dit etiam auribus impeditis ; et surdastri hiante ore so
nos excipere solent , ut tali pacto juvelur melius auditio.
Praeter foramen ex quo tuba Eustachiana procedit ,
duo alia babentur in tympani cavitate foramina , quorum
unum dicitur fenestra ovalis , allerum fenestra rotunda.
Feuestra ovalis basi slapiae occluditur, rotunda solo mem
branulae tegumento obtegitur. .
Ex tympani cavitate per duo praedicta foramina , ovale
scilicet ac rotundum , itur in labyriothum , qui - est inte
rior alia cavitas in osse petroso ulterius excavata , et quo
dam liquido plena : in hac tres partes distingui solent ;
prima est vestibulum labyrinthi ; secunda constat tribus
ossiculis semicircularibus , quibus nomen labyrinthi pecu
liarins aliqui tribuunt ; tertia est cochlea seu limax, quae
ex osse constat in cochleae modum conlorto duos gyros
cum dimidio faciente. Elsi cochlea unus canalis videri
possit , est lameu revera duplex : dividitur enim secun
dum longitudinem medio segmento , parim osseo , partim
membranaceo , quod dicitur lamina spiralis. Cochlea in
1
1
1
i
1
1
1
•
2
0
1
1
1
.
312
gerit: vel quia ita adhaeret stapiae et incudi , at cum al-
tero ex his confundi potuerit. Circa haec ossicula notan-
dum , quod eiusdem magnitudinis sint tam in infante quam
in adulto viro contra aliarum omnium corporis partium in--
dolem, quae aetatis progressu augentur. Hoc ideo factum
esse docet Valsalva, ne augmento partium auditui inservien-
tium alia sit sonorum ratio adulta aetate ac fuit ab iui-
tio; et ideas gravis atque acuti quas pueri imbibitüus, mu-
tare aetate proficiente cogamur.
ln tympani cavitate habetur canalis quidam seu tu-
ba Eustachiaua dicta ab ipsius inventore: per hanc tu-
bam ab interna auris cavitate obtinet communicatio cum
cavitate oris; desinit enim ista tuba ad radices uvae; quem
prope locum etiam interiora nasi communicant :-huius tu-
bae upe fit
, ut sonus ex oris cavitate auri .communicetur,
ideoque qui dentibus stringit corpus resonans sonum su-
dit etiam auribus impeditis ; et surdastri hiante ore so-
nos excipere solent , ut tali pacto iuvetur melius auditio-
Praeter foramen ex quo tuba Eustachiana procedit,
duo alia habentur in tympani cavitate foramina , quorum
unum dicitur fenestra ovalis, alterum fenestra rotunda.
Feuestra ovalis basi stapiae occluditur, rotunda solo mem-
branulae tegumento obtegitur. .
Ex tympani cavitate per duo praedicta foramina
, ovsle
scilicet ac rotundum ,
itur in labyrinthum
, qui-est inte-
- rior alia cavitas in esse petroso ulterius excavata
,
et quo-
dam liquido plena: in hac,tres partes distingui solent;
prima est vestibulum labyrinthi ; secunda constat tribus
ossiculis semicircularibus , quibus nomen labyrinthi pecu-
liarius aliqui tribuunt; tertia est cochlea seu limax, quae
ex osse constat in cochleae modum contorto duos gyros
cum dimidio. faciente. Etsi cochlea unus canalis videri
possit ,
est tamen revera duplex: dividitur enim secun-
dum longitudinem medio segmento, partim osseo , partim
membranacea , quod dicitur lamina spiralis. Cochlea in313
avibus deest , si vera refert Boyle ; at ipsemet notat de
fectum hunc suppleri per cavitatem oblongam instar sacci.
Plures rami nervei per foramina perexigua ex eo ,
qui dicitur uervus auditorius , propagati per totam fere
aurem distribuuntur : in labyrinthum per quinque fora
mina ingrediuntur Gibrae nerveae , et ejus cavitatem inves
tiunt ; sed de nervis totum auris organum permeantibus
non vacat disserere. Id unum adnoto ex Mairano , spira
lem laminam fibrillis ita instructam esse ut quemadmo
dum ipsa ascendens ad cochleae apicem . semper angustior
fit , ita fibrae ipsae breviores semper evadunt.
147. Quaenam vero ex hactenus descriptis auris par
tibus pro praecipuo atque immediato auditionis organo sta
tueuda est ?. Aliqui membranam tympani assignarunt : at
experientia constat quod hac membrana lacerata vel erepta
adhuc manet auditio aliqua. Alii inepte in ossiculis intra
capsulam contentis organom auditus statuerunt , et sonum
ab anima immediate perceptibilem ex horum collisione
nasci affirmarunt. Praeter quam quod solus malleus in a
nimantibus quibusdam habeatur, falsum omnino est collidi
invicem haec ossicula atque inde sonum creari : adnexum
enim est caput mallei firmiter corpori incudis , et hujus
processus alter stapiae; adeoque cum aer exterous tympa
ni membranam impellit, omnia per modum unius intromit
tuntur et conjuncta simul sese restituunt ad locum pristi
num. Magis autem absona est illorum sententia , qui in
aere per capsam et labyrinthum existente auditus organum
collocabant: aerem hunc, quem innatum, insitum, vernacu
lum, implantatum dicebant, animatum statuere non vere
bantur. Communiter nunc auditus organum collocatur in
fibrillis nerveis per aurem internam, ac praesertim intra co
chleam disseminatis. Tremores itaque a corpore
excitati communicantur membranae tympani; tum per aea
rem in tympano existentem , nec non per ossiculorum se
riem, ad parietes asque labyrinthi et praecipue ad dupli
Sonoro
21
313
avibus deest , si vera refert Boyle ;
'at ipsemet notat de-
fectum hunc suppleri per cavitatem oblongam instar sacci.
Plures rami nervei per foramina perexigua ex eo ,- qui dicitur nervus auditorius, prcpagati per totam fere
aurem distribuuntur: in labyrinthum per quinque fora-
mina ingrediuntur fibrae nerveae, et eius cavitatem inves-
tiunt; sed de nervis totum auris organum permeantibus
non vacat disserere. Id unum adnoto ex Mairano , spira-
lem laminam fibrillis ita instructam esse ut quemadmo-
dum ipse ascendens ad cochleae apicem- semper angustior
fit
,
ita fibrae ipsae breviores semper evadunt.

147. Quaenam vero ex hactenus descriptis auris par-
tibus pro praecipuo atque immediatoauditionis organo sta-
tuenda est ?. Aliqui membranam tympani assignarent : at
experientia constat quod hac membrana lacerata vel erepta
adhuc manet auditio aliqua. Alii inepte in ossiculis intra
capsulam contentis organum auditus statuerunt
,
et sonum
ab anima- immediate perceptibilem ex horum collisione
nasci affirmarunt. Praeter quam quod solus malleus in a-
nimantibus quibusdam habeatur, falsum omnino est collidi
invicem haec ossicula atque inde sonum creari: adnexum
enim est caput mallei firmiter corpori incudis ,
et huius
processus alter stapiae; adeoque cum aer externus tympa-
ni membranam impellit, omnia per modum uniua intromit-
tuntnr et coniuncta simul sese restituunt ad, locum pristi-
num. Magis autem absona est illorum sententia , qui in..
aere per capsam et labyrinthum existente auditus organum
collocabant: aerem hunc, quem innatum, insitum, vernacu-
lum, implantatum dicebant, animatum statuere non vere-
bantur. Communiter nunc auditus organum collocatur in
fibrillis nerveis per aurem internam, ac praesertim intra co-
chleam disseminatis. Tremores itaque a sonoro corpore
excitati commnnicantur membranae tympani; tum per aes-.
rem in tympano existentem, nec non per ossiculorum se-
riem, ad parietes usque labyrinthi et praecipue ad dupli-
21
is314
cem fenestram , ovalem ac rolundam , transmissi deducuntur
ad liquidum cavitate labyrinthi contenlum ; inde vero ad fi
brillas nerveas praedictas, atque ad nervum ipsum audito
rium: unde fit, ut ex lege commercii anima ad sensationem
soni determinetur. Animadvertit Mairanus, quod sicut in
instrumentis quibusdam musicis binae et binae chordae pro
tonis singulis disponuntur, ita in lamina spirali nerveae fi
brillae dispositae sint : ex quo infert huc potius spectare
organum quam ad alias auris internas partes.
148. Quaeres 1º. Cur quibusdam grata , aliis pene ni
hil, aut etiam molesta sit harmonia. Alibi ( 121 ) dictumn
est chordam upisonam facile ad tremitum concitari: aliam
item , sed difficilius prout majorem minoremve cum chor
da percussa harmonicam proportionem habet. Alert Kir
cherus aliud experimentum , quod ad rem aptum est etiam
magis : experimentum ita se habet. Quinque sumantur
scyphi vitrei.ejusdem magnitudinis et capacitatis , et unus
quidem liquore impleatur, qui acquavite dicitur; alter vi
no ; tertius aqua puriori; quartus aqua communi ; quintus
crassiori aqua: tum ora cujusque poculi confricando sonus
quam fieri potest acntissimus excitetur. In primo quidem
• scypho spiritus ille maxime subsultabit; vinum moderatam su
bibit concitationem ; adhuc moderatior erit molus purio
ris aquae, et ita porro . Ex his intelligitur quomodo variae
in variis hominibus partium corporis commotiones orian
tur e sono. Cum autem animi molus, in quibus voluptas
consistit vel molestia , pendeant ex partium corporis affe
ctionibus; iis gratissima accidere poterit harmonia, quibus
ea solidorum ac fluidorum constitutio est , ut in iisdem com
motio consequatur impressionem factam in organo auditus
satis . vivida et animi moribus cum voluptate conjunctis ex
citandis apta: ii erunt ad harmoniam indifferentes, in qui
bus impressionem factam in organo auditus vix ulla con
sequitur alteratio solidarum fuidarumve corporis partium
quae pariat animi motus vel consonos, vel incongruos: iis
314
cem fenestram, ovalem ac rotundam, transmissi deducuntur
ad liquidum cavitate labyrinthi contentum; inde vero ad fi-
brillas nerveas praedictas, atque ad nervum ipsum audito-
rium: nnde fit, ut ex lege commerciianima ad sensationem
aoni determinetur. Animadvertit Mairanus, quod sicut in
instrumentis quibusdam musicis binae et binae chordae pro
tonis singulis disponuntur, ita ip lamina spirali nerveae fi-
brillae dispositae sint: ex quo infert huc potius spectare
organum quam ad alias auris internas partes.
148. Quaeres 1&quot;. Cur quibusdam grata, aliis pene ni-
hil, aut etiam molesta sit harmonia. Alibi (121 ) dictum
est chordam unisonam facile ad tremitum concitari: aliam
item, sed difficilius prout majorem minoremve cum chor-
da percussa harmonicum proportionem habet. Affert Kir-
cherus aliud experimentum, quod .ad rem aptum est etiam
magis : experimentum ita se habet. Quinque sumantur
scyphi vitrei-ejusdem magnitudinis et capacitatis, et unus
quidem liqum'e impleatur, qui acquavite dicitur; alter vi-
no; tertius aqua PUI'lOl'i; quartus aqua communi; quintus
crassiori aqua: tum ora cujusque poculi confricando sonus
quam fieri potest acutissimus excitetur. In primo quidem
scypho spiritus ille maxime subsultahit; vinum moderatam su-
bibit concitationem; adhuc moderatior erit motus purio-
ris aquae, et ita porro. Ex his intelligitur quomodo variae
in variis hominibus partium corporis commotiones orian-
tur e sono. Cum autem animi motus, in quibus voluptas
consistit vel molestia, pendeant ex partium corporis affe-
ctionibus; iis gratissima accidere poterit harmonia, quibus
easolidorum ac fluidorum constitutio est, ut in iisdem com-
motio consequatur impressionem factam in organo auditus
satis.vivida et animi motibus cum voluptate conjunctis ex-
citandis apta: ii erunt ad harmoniam indifferentes. tu qui-
bus impressionem factam in Organo auditus vix ulla con-
sequitur alteratio solidarum fluidarumve corporis partium ∙
quae pariat animi motus vel consouos, vel incongruos: iis1
1
315
denique molestia etiam accidet, quibus ex impressione ner
vorum acusticorum contingat incongrua motuum alteratio
in partibus corporis ad pracfatos animi molus inservienti
bus: quo fit etiam mechanice ut alii aliis sonorum gene
ribus vel delectentur magis, vel contra. Hanc tamen me
chanicam causam non arbitror esse sufficientem atque adae
quatam: admittenda est causa ex rationali natura hominis
pendens. Consistit harmonia in proportione illa , quam so
ni habent inter se ; unde fit ut in organo auditus vibra
tiones diversi generis, aliae frequentiores, aliaė tardio
res efficiantur: dum vibrationes istae organum anditus af
ficiunt, mens easdein comparat inter se, earumque propor
tionem animadvertit : si haec proportio ejusmodi sit ut fa
cile possit a mente percipi, et vibrationes facile comparari
queant, ex hoc gaudet mens; si confusa et inordinata vi
brationum sit comparatio , neque has mens facile con
ferre inter se potest, obruelur taedio: et quia imperi
tas in musica facilius comparat simpliciores consonantias quam
magis compositas, ideo musica planissima vulgo arridet ; qui
vero periti sunt et copiosioribus compositiouibus assueti, vix
patiuntur musicam nimis simplicem: item cum ex consue
tadine pendeat ut aliquas harmonicas proportiones faci
lius mens assequatur quam alias ; inde oritur at volu
ptas ex eo musices genere major sit, cui quis sit assue
tus; adeo ut ex hoc capite diversis nationibus diversae
placeant musicae species. Porro ab exercitatione etiam
profluit ut gratior accidat musica , etiam qua ex parte
mechanice voluptatem parit; ex assuetudine enim in fi
brillas nerveas docilitas inducitur ad recipiendas facilius
impressiones harmonicas.
2º. Cur duabus auribus unus idemque sonus au
diatur. Communis responsio est hujusmodi : cum in utra.
que aure creetur simillima impressio; non duplicem , sed
voam sensationem ab anima haber¡ necesse est. Qua in
re scite animadvertit Valsalva , summa industria provisum
315
denique molestia etiam accidet, quibus ex impressione ner-
vorum acnsticorum contingat incongrua motuum alteratio
in partibus corporis ad praefatos animi motus inservienti-
bus: quo fit etiam mechanica ut alii aliis sonorum gene-
ribus vel delectentur magis, 'vel contra. Hanc tamen me-
chanicam causam non arbitror esse sufficientem atque adae-i
quatam: admittenda est causa ex rationali natura hominis
pendens. Consistit harmonia in praportione illa, quam so-
ni habent inter se; unde fit ut in organo auditusvibraP-
tiones diversi generis, aliae frequentiores, aliae tardio-
res efficiantur: dum vibrationes istae organum auditus af-
Hciunt, mens easdem comparat inter se, earumque propor-
tionem animadvertit: si haec proportio ejusmodi sit ut fa-
cile possit a mente percipi, et vibrationes facile comparari
queant, ex hoc gaudet mens; si confusa et inordinata vi-
bratiouum sit comparatio , neque has mens facile con-
ferre inter se potest, obruetur taedio: et quia imperi-
tus in musica facilius comparat simpliciores consonantias quam
magis compositas, ideo musica planissima vulgo arridet ; qui
vero periti sunt et c0piosioribus compositionibus assueti, vix
patiuntur musicam nimis simplicem: item cum ex consue-,
tudine pendeat ut aliquas harmonicas preportiones faci-
lius mens assequetur quam alias; inde oritur ut volu-
ptas ex eo mus1ces genere major sit, cui quis sit assue-
tus; adeo ut ex hoc capite diversis nationibus diversae
placeant musicae species. Porro ab exercitatione etiam
profluit ut gratior accidat musica, etiam qua ex parte
mechanica voluptatem parit; ex assuetudine enim in fi-
brillas nerveas docilitas inducitur ad recipiendas facilius
impressiones harmonicas. ⋅
20. Cur duabus auribus unus idemque sonus au-
diatur. Communis responsio est huiusmodi: cum in utra-
que aure creetur simillima impressio; non duplicem, sed,
unam sensationem ab anima haberi necesse est. Qua in
re scite animadvertit Valsalva, summa industria provisum316
fuisse a natura ut in utraque aure quam simillima es
sent organa omnia ; adeo ut, dum in diversis hominibus
structura partium nonnihil variat, in eodem tamen bomi
mine nulla prorsus sit utriusque auris vel minima variatio .
Notetur illud : quemadmodum eadem chorda varios
tonos varia tensione edere potest, ita membrana tympani
lenditur diversimode ut variis tonis aple accomodetur ;
eapropter manubrium mallei eidem adnexum est, et ba
sis stapiae eodem modo membranae fenestrae ovalis: ten
sio autem et relaxatio membranae, nobis insciis , potest na
turaliter fieri. Itaque, ut primae vibrationes membranam
feriunt, admonita natura organum opportuna tensione aptat
ut respondens tonus in successivis vibrationibus fiat ma
gis vel minus sensibilis.
316
fuisse a natura ut in utraque aure quam simillima es-
sent organa omnia; adeo ut, dum in diversis hominibus
structura partium nonnihil variat, in eodem tamen homi-
miue nulla prorsus sit utriusque auris vel minima variatio.
Notetur illud: quemadmodum eadem chorda varios
tonos varia tensione edere potest, ita membrana tympani
tenditur diversimode ut variis tonis apte accomodetur;
eapropter manubrium mallei eidem adnexum est, et ba-
sis stapiae eodem modo membranae fenestrae ovalis: ten-
sio autem et relaxatio membranae, nobis insciis,potest na-
turaliter fieri. Itaque, ut primae vibrationes membranam
feriunt, admonita natura organum opportuna tensione aptat
ut respondens tonus in successivis vibrationibus fiat ma-
gis vel minus sensibilis.INDEX
RERUM QUAE IN PRIMO VOLUMINE CONTINENTUR.
MECHANICA E PRINCIPIA
Notiones praeambulae. pag. 1 .
Molus uniformis et varius : velocitas et quantitas mo
tas in motu uniformi. num . 1 .
Corporum indifferentia ad motum et ad quietem: quid
vires : quid earum aequilibrium ; et quomodo repraesen
tentur sive per lineas rectas, sive per numeros . n. 2, 3, 4.
Principiom motus • relativi : vires sunt ut quantitates
motus , n. 5, 6 .
Principium actionis et reactionis : mutatio status in
corpore haud repente gignitur a viribus extrinsecis , sed
per gradus indefinitae attenuationis capaces: quid vires in
stanlaneae et continuae. n. 7 .
De virium compositione et resolutione , deque earum
momentis et aequilibrio : aliquid quoque notatur de
vecte, axe in peritrochio, trochlca etc. pag. 6.
Compositio virium materiali puncto applicatarum: ae
quilibrium: varia circa virium resolutionem .. n. 8. 9. 10.
⋅ N D EX
RERUM QUAE IN ramo VOLUMINE CONTINENTUR.
' MECHANICAE PRINCIPIA
∙
W
⋅∙
Nott'ones praeambulae. pag. 1.
Motus uniformis et varius: velocitas et quantitas mo-
tus in motu uniformi. . . . . . . . . . num-1.
Corporum indifferentia ad motum et ad quietem: quid
vires: quid earum aequilibrium; et quomodo repraesen-
tentur sive per lineas rectas, sive per numeros. n.2, 3, 4.
. Principium motus 'relativi: vires sunt ut quantitates.
motus. ∙ ∙ ∙ ∙↴ ∙ ∙
.' ∙ ∙ ∙ ∙ ∙ ∙ n. 5, 6.
Principium actionis et reactionis: mutatio status in
corpore haud repente gignitur a viribus extrinsecis ,
sed
per gradus indefinitae attenuationis capaces: quid vires in-
stantaneae et-continuae.
. . . . . . . . . n. 7.»
De virium compositione et resolutione , deque earum
momentis et aequilibrio : aliquid quoque notatur de
vecte, axe in peritrochio, trochlea etc. pag. 6.
Compositio virium materiali puncto applicatarum: ae-
quilibrium: varia circa virium resolutionem.. n. 8. 9. 10.318
Compositio duarum virium extremis rectae rigidae punctis
applicatarum, et in eodem plano jacentium: aequilibrium circa
immobile punctum: principiam velocitatum virtualium in ordi
ne ad istiusmodi vires: momenta virium quoad punctum ( M) :
momentum resultantis aequatur summae ex momentis com
ponentium si hae in eamdem plagam circa ( M ) nituntur
movere puncta , quibus applicitae sunt; aequatur differentiae
si nituntur movere in plagas contrarias. n. 10. 10. 20.30,
Haec ipsa extenduntur ad quemvis numerum virium in
dato plano jacentium.
n. 10. 4º , 5º , 6º.
Binae vires haud jacentes in eodem plano nequeunt
ad unicam vim aequipollentem traduci : vires in aequili
brio constitutae; sollicitantesque vel solidum liberumque cor
pus, vel solidam corpus mobile duntaxat circa punctum fi
xum, vel solidum corpus mobile tantummodo circa asem
fixum : momenta quoad axem . n. 10: 70. ... 10 °.
Vires parallelae: vis inde resaltans: earum centrum :
momenta quoad planum: respondens theorema n . 11 , 12 ,
13. 10. 2º. 3º.
Parallelarum virium systema consistit in aequilibrio sub
duabus conditionibus simul explendis; altera est ut evane
scat earum summa; altera ut evanescat summa ex earum
momentis in ordine ad duo plana ipsis viribus paral
lela . n. 13. 4º. 5º.
.
Etsi vires non sunt parallelae, possunt tamen rednci
ad terna ejusmodi systemata, quorum primum coalescat ex
viribus parallelis axi OZ, secundum ex viribus jacentibus
in plano XOY simulque parallelis axi Qy, tertium ex vi
ribus agentibus juxta axem OX: aequilibrii conditiones quoad
systemata rigida viribus minime parallelis sollicitata; 1º , in
318
Compositio duarum virium extremis rectae rigidae punctis
applicatarum,etin eodem plano jacentium: aequilibrium circa
immobile punctum: princi pium velocitatum virtualium in ordi-
ne ad istiusmodi vires: momenta virium quoad punctum (M):
momentum resultantis aequatur summae ex momentis com-
ponentium si hae in eamdem plagam circa (M ) nituhtur
movere puncta, quibus applicitae sunt; aequatur differentiae
si nituntur movere in plagas contrarias. n. 10. 10. 2230.
∙ Haec ipsa extenduntur ad quemvis numerum virium in
dato plano jacentium.
. . . . . . n. 10. 40. 50. 6&quot;.
Binae vires haud jacentes in eodem plano nequeunt
ad unicam vim aequipollentem traduci : vires in aequili-
brio constitutae; sollicitantesque vel solidum liberumque cor-
pus, vel solidum corpus mobile duntaxat circa punctum fi-
xum, vel solidum corpus mobile tantummodo circa axem
fixum: momenta quoad axem. .'
. . n. 10: 70. 10&quot;-
Vires parallelae: vis inde resultans: earum centrum:
momenta quoad planum: respondens theorema &quot;. 11, 121
13. 10. 20. 30.
Parallelarum virium systema consistit in aequilibrio sub
duabus conditionibus simul explendis; altera est ut evane-
scat earum summa; altera ut evanescat summa ex earum
momentis in ordine ad duo plana ipsis viribus paral-
lela.
. . . . . . . . . . . . . n.13.4&quot;.5'-
Etsi vires non sunt parallelae, possunt tamen -reduci
ad terna eiusmodi systemata. quorum primum coalescat ex
viribus parallelis axi OZ, secundum ex viribus jacentibus
in plano XOV simulque parallelis axi QT, tertium ex vi-
ribus agentibus juxta axem OX: aequilibrii conditiones quoad
systemata rigida viribus minime parallelis sollicitata;1&quot;.in319
hypothesi systematis liberi; 2 °. in hypothesi systematis de
tenti puncto fixo; 3º . in hypothesi systematis detenti axe
fixo: conditio explenda ut plures vires minime parallelae
traduci possint ad unicam aequipollentem . n. 13. 6 ... 11º.
Duo solida corpora , datis viribus sollicitata , sese in
vicem aeque premendo apud datum mutui contaclus pan
ctum manent in aequilibrio : determinatur istiusmodi pres
sionis magnitudo. n. 13. 12 .
Solidum corpus , datis viribus sollicitatum, detinetur
duobus punctis fixis, sumptis in axe v. gr. OZ: determi
nantur pressiones exercitae in puncta illa juxta coordi
nalos axes OX, OY, OZ. n. 13. 13 .
Exempla aequilibrii in quibusdam machinis, praeci
so attritu : aequilibrium punctoruni materialium juncto
rum flis determinatae quidem longitudinis sed mobili
bas circa data puncta. n. 14. 15. 16 .
De centro gravitatis. pag. 33.
Varia de vi gravitatis, de corporum densitate , deque
specifica eorum gravitate: linea directionis. n . 17 , 18 , 19.
Generales formulae determinantes centrum gravita
tis: inveniri potest ratione mechanica: peculiari metho
do determinalur in triangulo et pyramide triangulari, n. 20.
De corporum collisione. pag . 37 .
Normalis collisio : 1º. corporum non elasticorum : 2 ° .
corporum perfecte elasticorum : 3º . corporum imperfe
cte elasticorum . n. 21 , 22, ... 25 .
319
hypothesi systematis liberi: &quot;. in hypothesi systematis de-
⋅ teuti puncto fixo: 30. in hypothesi systematis detenti axe
fixo: conditio explenda ut plures vires minime parallelae
traduci possint ad unicam aequipollentem. n. 13. 60... 1'l0.
Duo solida corpora, datis viribus sollicitata, sese in-
vicem aeque premendo apud datum mutui contactus pun-
ctum manent in aequilibrio: determinatur istiusmodi pres-
sionis magnitudo. . . . . . . . . . n. 13. 120.
Solidum corpus . datis viribus sollicitatum. detinetur
duobus punctis fixis, sumptis in axe v- gr. OZ: determi-
nantur pressiones exercitae in puncta illa iuxta coordi-'
natos axes OX, 0ï,OZ. . . . ∎∙ ∙ ∙ n.13.130.
Exempla aequilibrii. in quibusdam machinis, praeci-
so attritu : aequilibrium punctorum materialium iuncto-
rum filis determinatae quidem longitudinis sed mobili-
bus circa data puncta.
. . . . . . n.14.15.'16.
De centro gravitatis. pag. 33.
Varia de vi gravitatis, de corporum densitate.deque
specifica eorum gravitate: linea directionis. n. 17, 18, 19.
Geuerales formulae determinantes centrum gravita-
tis: inveniri potest ratione mechanica: peculiari metho-
do determinatur in triangulo et pyramide triangulari. n. 20-
Dä corporum collisione. pag. 37-
Normalis collisio: 10. corporum non elasticorum: 2&quot;.
corporum perfecte elasticorum : 3&quot;. corporum imperfe-
cte elasticorum. . .
-
. . . . . n.21,22,...25.320
Obliqua eorumdem corporum collisio. n . 26.
De motu rectilineo utcumque vario. pag. 42
Praemittantur nonnulla ex analysi infinitesimali, e
jusque ad res geometricas applicatione. n. 27. 10.2 ... 300.
Formulae spectantes ad motum rectilineum utcumque
varium : formulae quoad motum rectilineum uniformiter
varium: vis acceleratrix : vis motrix. n. 28.
Formulae pertinentes ad motum rectilineum utcum
que varium applicantur ad materiale punctum sollicita
tum vi acceleratrice, quae sit distantiae a dato centro pro
portionalis. n. 29.

De verticali gravium descensu atque ascensu . pag.
65.
Formulae, legesque huc spectantes: motus gravium in
machina Atwoodi. n . 30, 31 , 32 .
Quid si gravium descensus vel ascensus fiat in me
dio resistente sub ea conditione, ut resistentia medii sit pro .
portionalis quadrato velocitatis. n. 33.
De gravium descensu per plana inclinala ; de attritu ;
deque cochlea, et cuneo. pag. 71.
Formulae determinantes descensum gravium per plana
inclinata: gravium descensus per plana inclinata compara
tur cum verticali eorum descensu. n . 34, 35.
320
⋅∡ Obliqua eorumdem corporum collisio. . ∙⋅ n. 26.
De motu rectilineo utcumque uario. pag. 42
Praemittuntur nonnulla ex analysi infiuitesimali, e-
iusque ad res geometricas applicatione. n. 27. 10. 2&quot;....300.
Formulae spectantes ad motum rectilineum utcumque
varium: formulae quoad mo'tum rectilineum uniformiter
varium: vis acceleratrix: vis motrix. . . . . . n. 28.
Formulae pertinentes ad motum rectilineum utcum-
que varium applicantur ad materiale punctum sollicita-
tnm vi acceleratrice. quae sit distantiae a dato centro pro-
portionalis. ............n.ag.
!
De verticali gravium descensu atque ascensu. pag. 65.
Formulae, legesque huc spectantes: motus gravium in
machina Atwoodi. . . . . . . . . n. 30,31,32-
Quid si gravium descensus vel ascensus liat in me-
dio resistente sub ea conditione, ut resistentia medii sit pro-
portionalis quadrato velocitatis. . . . . . . n. 33-
De gravium descensu per plana inclinata; de attritu; ⇥
deque cochlea, et cuneo. ∙ pag. 71.
Formulae determinantes descensum gravium per plana
inclinata: gravium descensus per plana inclinata compara-
tur cum verticali eorum descensu. . . . n. 34, 35-321
Gravium descensus per plura plana inclinata sibi con
rigua . n. 36.
non.
Unde orialur attritus , caeteris paribus , est proportio
nalis pressioni : quomodo habeatur ratio attritus in motu
gravium per plana inclinata : grave in plano inclinato li
brandum potentia aliqua, sive habeatur ratio attritus , sive
n. 37. 10. 20 30
Aequilibrii leges in cochlea, et cuneo. n. 37. 4º. 5º.
Spectatur attritus in aequilibrio cochleae: itemque in
aequilibrio corporis ad rotatilem motum circa fixum cy
lindrum sollicitati : in machinis praeter resistentiam ex at
tritu spectanda etiam est resistentia ex funibus n. 37.6º.70.8° .
De motu gravium oblique projectorum . pag . 81 ,
Aequatio ad curvam, quam describunt gravia oblique
projecta; istiusmodi curva dicitur parabola. n. 38, 39.
Amplitudo jactus: maxima jactus amplitudo habetur sub
angulo projectionis semirecto: sub quo angulo projiciendum sit
grave ut offendat in datum scopum : altitudo jactus : ali
quid subjungitur de proprietatibus praefatae curvae. n. 40. 1º.
2 ° .... 70
Quid si gravia oblique projiciantur in medio resi
n. 41 .
stente.
De generalibus quibusdam proprietatibus motus curvili
nei, orti a viribus quarum una determinat materiale
punctum ad motum uniformem , altera ipsi materia
li puncto est continue applicata .
.
pag. 85.
Ubi in aliquo curvae puncto vis acceleratrix desinat
agere, excurret mobile per tangentem în punclo illo : ubi
321
Gnavium descensus-per plura plana inclinata sibi con-
ligua...............n.36.
Unde oriatur attritus. caeteris paribus, est proportio-
nalis pressioni: quomodo habeatur ratio attritus in motu
gravium per plana inclinata: grave in plano inclinato li-
brandum potentia aliqua, sive habeatur ratio attritus, sive
non. . , . . . . . . . . . . n. 37.10.2030.
Aequilibrii leges in cochlea, et cuneo. n. 37. 40. 50.
Spectatur attritus in aequilibrio cochleae: itemque in
aequilibrio corporis ad rotatilem motum circa fixum cy-
lindrum sollicitati: in machinis praeter resistentiam ex et-
tritu spectanda etiam est resistentia ex funibus n. 37.60.70.80.
De motu gravium oblique projectorum: pag. 81,
∙ Aequatio ad curvam, quam describunt gravia oblique
proiecta; istiusmodi curva dicitur parabola. . n. 38. 39.
Amplitudo iactus: maxima jactus amplitudo habetur sub
angulo projectiouis semirecto: sub quo angulo proiiciendum sit
grave ut offendat in datum scopum : altitudo jactus: ali-
quid subiungitur de proprietatibus praefatae curvae. n. 40. 10.
20 .... 70.
Quid si gravia oblique projiciantnr in medio resi-
stente. ↖∙∙∙∙∙∙∙∙∙∙⋅∙∙∙∥∙∡∎∙
De generalibus quibusdam praprietatibus motus curvili-
'nei, orti a viribus quarum una determinat materiale
punctum ad motum uniformem. altera ipsi materia-
li puncto est continue applicata.
. . . . pag. 85-
Ubi in aliquo curvae puncto vis acceleratrix desinat
agere, excurret mobile per tangentem in puncto illo: ubi322
tempore finito angulus, quem efformat vis acceleratrix cum
directione tangentis , fuerit semper acutus, acquiret mo
bile incrementum velocitatis finitum ; si semper obtusus,
patietur decrementum finitum ; si semper rectus , veloci
tas manebit constans: quadratum velocitatis adaequat vim
acceleratricem ductam in dimidium chordae, quae ex ejus
directione abscinditur ab osculatore circulo. n . 42.... 45.
Haec vera sunt de omnium virium genere; ponantur
vires acceleratrices ad centrum datum directae : jacebit cur
va in plano transeunte per rectam projectionis et per cen
trum virium: radius vector describet areas circa virium cen
trum temporibus proportionales: viceversa si radius ve
ctor describit areas circa punctum aliquod temporibus pro
portionales, vis acceleratrix erit constanter directa ad pun
ctum illud: velocitas, qua pollet mobile in eadem curva ,
exsistit reciproce proportionalis perpendiculo ducto a centro
virium in tangentem: vis acceleratrix in quovis curvae pun:
cto est directe ul radius vector, et reciproce ut factum
ex osculi radio in cnbum praefati perpendiculi : si ultra
punctum contactus sumitur arcus infinitesimus, a materiali
puncto describendus subsequente tempusculo, radiusque ve
ctor pertingens ad hujus arcus extremitatem producitur
donec occurrat tangenti, vis acceleralrix in contactus pun
cto erit directe ut pars radii vectoris producti intercepta
ac tangente , et reciproce ut quadratum tempuscu
li .
arcu
n. 46,
49.
Sive vires tendant ad centrum datum, sive non ; coor
dinatae puncti materialis in fine temporis e spectandae sunt
tanquam functiones ipsius t : formulae respicientes et veloci
tatem in quolibet curyae puncto, et binas componentes, al
teram juxta tangentem , alteram juxta normalem , in quas
resolvitur yis acceleratrix. n, 50. 10. 2º . 3º.
.
322
tempore linito angulus, quem etl'ormat- vis acceleratrix cum
directione tangentis ,
fuerit semper acutus, acquirat mo-
bile incrementum velocitatis Gnitum; si semper obtusus,
patietur decrementum (initum: si semper rectus, veloci-
tas mauebit constans: quadratum velocitatis adaequat vim.
acceleratricem ductam in dimidium chordae, quae ex eius
directione abscinditur ab osculatore circulo. n. 42.... 45.
Haec vera sunt de omnium virium genere; ponantur
vires acceleratrices ad centrum datum directae: iacebit cur-
va in plano transeunte per rectam projectiouis et per cen-
trum virium: radius vector describet areas circa virium cen-
trum temporibus proportionales: viceversa si radius ve-
ctor describit areas circa punctum aliquod temporibus pro-
portionales, vis acceleratrix erit constanter directa ad pun-
ctum illud: velocitas, qua pollet mobile in eadem curva
.
exsistit reciproce proportionalis perpendiculo ducto a centro
virium in tangentem: vis acceleratrix in quovis curvae pung-
cto est directe ut radius vector, et reciproce ut factum
ex osculi radio iu cubum praefati perpendiculi: si ultra
punctum contactus sumitur arcusiufiuitesimus, a materiali
puncto describendus subsequente tempusculo, radiosque ve-
ctor pertingens-ad huius arcus extremitatem producitur
donec occurrat tangenti, vis acceleratrix in contactus pun-
cto erit directe ut pars radii vectoris producti intercepta
arcu ac tangente ,
et reciproce ut quadratum tempuscu-
li. ∙∙∙∙∙⋅∙∙∙ ∙ ∙∙ ..n.46,...49-
Sive vires tendant ad centrum datum, sive non; coor-
dinatae puncti materialis in fine temporis t spectandae sunt
tanquam functiones ipsius :: formulae respicientes et veloci-
tatem in quolibet curvae puncto, et binas componentes, al-
teram juxta tangentem, alteram juxta normalcm. in. qu&quot;.
resolvitur vis acceleratrix. . . . . . n. 50.1'-2&quot;- 30,323
Resolata vi acceleratrice in ternas componentes axi
bus coordinatis parallelas, stabiliuntur formulae huc per
tinentes: applicantur formulae ad duas quaestiones, quarum al
tera respicit gravia oblique projecta in vacuo, altera respicit
gravia oblique projecta in medio resistente. n. 50. 4º. 5º . 6º.
Quomodo vis acceleratrix directa ad centrum expri
matur generatim per coordinatas polares : quomodo, data vi
acceleratrice directa ad centrum , inveniri possit aequatio
inter coordinatas polares ad lineam per quam movetur ma
teriale punclum: exemplum desumptum a vi acceleratrice ,
quae sit reciproce ut quadratum radii vectoris: sub hac le
ge poterit materiale punctum describere parabolam haben
tem suum focum in centro virium:
quaenam velocitas pro
jectionis ad id sit necessaria. n. 50. 7º. 8º... 15 °
Motus curvilineus impeditus : vis centrifuga. n. 51 .
De vi acceleratrice in motu circulari, existente
centro virium in centro circuli. pag . 109,
Istiusmodi motus ' est uniformis: vis acceleratrix obti
netur dividendo quadratum velocitatis per curvae circularis
radium: varia inde inferuntur et quoad projectionis velo citatem necessariam ad describendam cicularem curvam , et
quoad vires acceleratrices in diversis peripheriis circula
ribus. n. 52 , 53.
Vis centrifuga orta ex circulari telluris rotatione cir
ca suum axem : qua ratione decrescat ab aequatore ad po
los: qua ratione vis centrifuga imminuat gravitatem a po
lis ad aequatoren . n. 54.
323
Resoluta vi acceleratrice in ternas componentes axi-
bus coordiuatis parallelas, stabiliuntur formulae huc per-
tinentes: applicantur formulae ad duas quaestiones, quarum al-
tera respicitgravia oblique projecta in vacuo, altera respicit
gravia oblique proiecta in medio resistente. n. 50. 40. 50. 60.
Quomodo vis acceleratrix directa ad centrum lexpri- matur generatim per coordinatas polares: quomodo, data vi
acceleratrice directa ad centrum
.
inveniri possit aequatio
inter coordinatas polares ad lineam per quam movetur ma-
teriale punctum: exemplum desumptum a vi acceleratrice,
quae sit reciproce'ut quadratum radii vectoris: sub hac le-
ge poterit materiale punctum describere parabolam haben-
tem suum focum in centro virium: quaenam velocitas pro-
fectionis ad id sit necessaria. '
. . n. 50. 7&quot;. 80...150.
Motus curvilineus impeditus: vis centrifuga. n. St.
De vi acceleratrice in motu circulari, existente
centro m'rium' in centro circuli . pag. 109.
Istiusmodi motus 'est uniformis: vis acceleratrix obti-
netur dividendo quadratum velocitatis per curvae circularis
radium: varia iude inferuntur et quoad proiectionis velo-
citatem necessariam ad&quot; describendam cicularem curvam, et
quoad vires acceleratrices in diversis peripheriis circula-
ribus-.............n.52,53.
.
Vis centrifuga orta ex circulari telluris rotatione cir-
ca suum axem: qua ratione decrescat ab aequatore ad po-
los: qua ratione vis centrifuga imminuat gravitatem a po-
lis ad aequatorem.
. . .. . . . . ∙∙ ∙ n. 54-324
De vi acceleratrice in motu elliptico, existente
centro virium in foco ellipsis pag. 111,
Varia praemittuntur et circa rectas ita ductas ex puncto
quovis, ut tangant datam sphaeram ; et circa plaua tangentia
ducta per ejusmodi rectas ; et circa rectarum , arearumque
planarum projectiones in plano quolibet ; sed praecipue
circa ellipsim. n. 55. 1º, 2º ...14 °. . .
Quibus praemissis, demonstratur illud : existente cen
tro virium in foco ellipseos , vis acceleratrix in motu el
liptico est reciproce ut quadratum radii vectoris : quid in
duabus ellipsibus si quadrata temporum periodicorum sint ut cubi semiaxiun transversorum . n. 56.
Paucis subjunctis de ellipsi , parabola , et hyperbo
la, demonstratur quod, agentibus viribus in ratione reci
proca duplicata distantiarum a dato centro, praeter para
bolam poterit quoque mobile describere vel ellipsim
vel hyperbolam, existente focorum altero in centro virium:
quaenam projectionis velocitas requiratur ad ellipsim de
scribendam , quaenam ad hyperbolam. n, 67. 1.2.7 .
Obiter de lege virium in motu elliptico, ubi eae ten
dant ad ellipseos centrum . n. 57 , 8 .
De motu relativo punctorum materialium , tendentium
in se mutuo viribus acceleratricibus quae sint di
recte ut massae in quas tenditur, et reciproce ut qua
drata respondentium distantiarum . pag. 125.
Generales ad istiusmodi motum aequationes differen
tiales. n, 58,
324
- De ui acceleratrice in motu elliptica. existente
centro virium in foco ellipsis pag. 111.
Varia praemittuntur et circa rectas ita ductas ex puncto
quovis, ut tangant datam sphaeram; et circa plana tangentia
ducta per ejusmodi rectas; et circa rectarum, arearumque
planarum proiectiones in plano quolibet ; sed praecipue
circa ellipsim. . . . . . . . . n.55.10. 20 ...140.
Quibus praemissis, demonstratur illud: existente cen-
tro virium in foco ellipseos , vis acceleratrix in motu el-
liptico est reciproce ut quadratum radii vectoris: quid in
duabus ellipsibus si quadrata temporum periodicorum sint
ut cubi semiaxium transversorum . . . . . . n- 56.
Paucis subjunctis de ellipsi, parabola ,
et hyperbo-
la, demonstratur quod, agentibus viribus in ratione reci-
proca duplicata distantiarum a dato centro, praeter para-
bolam poterit quoque mobile describere vel ellipsim ,
vel hyperbolam, existente focorum altero in centro virium:
quaenam proiectiouis velocitas requiratur ad ellipsim de-
scribendam, quaenam ad hyperbolam. . n. 57. ↿∘∙ ⋍∘∙∙∙ 70.
Obiter de lege virium in motu elliptica, ubi eae ten- dant ad ellipseos centrum.
. . . . . . n. 57. 8&quot;.
De motu relativo punctorum &quot;materialium
,
tendentium
in se mutuo viribus acceleratricibus quae sint di-
recte ut massae in quas tenditur, et reciproce ut quab
drata respondentium distantiarum. pag. 125.
Gener-ales ad istiusmodi motum aequationes dideren-
tiüles- ∙∎∎ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ &quot;a 58.325
Spectantur duo tantum materialia puncta: vires per
turbantes ex reliquis punctis. n. 59, ... 62.
De pendulis ; deque gravium descensu per
arcus cycloidales. pag . 134.
Quid pendulum simplex ; quid compositum : vires
gignentes motum penduli simplicis n . 63.
Velocitates in puncto infimo acquistae a gravibus per
inaequales ejusdem circuli arcus descendentibus sunt ut
ipsorum arcuum chordae .
. n. 64.
Oscillationes penduli simplicis per arcus satis exi
guos , ulcumque ceteroquin inaequales , sunt ad sensum
isochronae seu aequidiuturnae : quid ex doctrina penduli
simplicis circa terrestrem gravitatem n. 65 , 66.
Centrum oscillationis in pendulo composito : etiam
oscillationes penduli compositi suņt isochronae, modo ta
men existant satis exiguae .
n. 67.
Oscillationes penduli simplicis in medio resistente :
primo in hypothesi resistentiae proportionalis simplici ve
locitati; deinde in hypothesi resistentiae proportionalis qua
drato velocitatis .
n. 68.
n . 69 .
Paucis praemissis de cycloide, demonstratur illud : ex
quocumque cycloidis puncto demittatur grave , eodem sem
per tempore perveniet ad punctum infimum
De attractione corporum in hypothesi attractionis agentis
in ratione directa massarum , et in reciproca
duplicata distantiarum .
Attractio corporum quorumcumque in materiale pun
clum situm sive extra corpus attrahens, sive intra. n . 70,71,72.
pag . 151 .
325
Spectautur duo tantum materialia puncta: vires per-
turbantes ex reliquis punctis. . . . . . n. 59....62.
De pendulis; deque gravium descensu per
arcus cycloidales. pag. 134.
Quid pendulum simplex; quid compositum : vires
gignentes motum penduli simplicis
. . . . . n. 63.
Velocitates in puncto intimo acquistae a gravibus per
inaequales ejusdem circuli arcus descendentibus sunt ut
ipsorum arcuum chordae
. . . . . .- . . n. 64.
Oscillationes penduli simplicis per arcus satis exi-
guos, utcumque ceteroquin iuaequales ,
sunt ad sensum
isochrouae seu aequidiuturuae: quid ex doctrina penduli
simplicis circa terrestrem gravitatem . . . n. 65
, 66.
Centrum oscillationis in pendulo composito: etiam
oscillationes penduli compositi sunt isochrouae, modo ta-
men existant satis exiguae . . . . . . . . n. 67.
Oscillationes penduli simplicis in medio resistente:
primo in hypothesi resistentiae proportionalis simplici ve-
locitati; deinde in hypothesi resistentiae proportionalis qua-
drato velocitatis
. . . .. . . . . . . n. 68.
Paucis praemissis de cycloide, demonstratur illud : ex
quocumque cycloidis puncto demittatur grave , eodem sem-
per tempore perveniet ad punctum infimum
. . n.. 69.
De attractione corporum in hypothesi attractionis agentis
in ratione directa massarum, et in reciproca
duplicata distantiarum. pag. 151 .
Attractio corporum quorumcumque in materiale pun-
ctum situm sive extra corpus attrahens, sireintrafn. 70,71,72.326
Expediuntur quae pertinent ad attractionem corpo
rum sphaericorum in punctum materiale n. 73,74,75.
Materiale punctum valde distans a corpore attrahente,
utcumque se habeat forma corporis, ea proxime ratione tendit
in ipsum corpus, qua tenderet si corporis partes in centro
gravitatis compenetrarentur: ubi dimensiones corporum quo
rumcuinque se mutuo attrahentium sint admodum exiguae
prae distantiis , quibus ipsa corpora disjunguntur , eorum
alterum tendet in alterum perinde ac si essent ambo in
suis gravitatis centris compenetrata ; haec assertio quoad
sphaerica corpora valet utcumque se habeat intercedens
distantia n. 76.
De gravitatione universali. pag. 159.
Ex mutua coelestium corporum gravitatione collata
cum terrestrii gravitate inferimus illud : gravitas ita ma
teriam afficit , ut singulae ejus particulae in alias omnes
et singulas gravitent in ratione directa massarum ad quas
tenditur , et reciproca duplicata distantiarum alterius ab
altera n . 77 , ...82.
.
Aliquid circa solarem et planeticas massas... n.83.10... 4.
Media telluris densitas determinata ex penduli aber
ratione ; itemque experimentis institutis in libra
siouis n. 83. 5. 6.°
tor
Quomodo ex marini aestus phoenomeno deduci pos
sit ratio inter lunarem ac terrestrem massam . n. 83.7 . °
326
Expediuntur quae pertinent ad attractionem corpo-
rum sphaericorum iu punctum materiale
. n. 73,74,75.
Materiale punctum- valde distans a corpore attraheute,
utcumque se habeat forma corporis, ea proxime ratione tendit
in ipsum corpus, qua tenderet si corporis partes in centro
gravitatis compenetrarentur: ubi dimensiones corporum quo-
rumcumque se mutuo attrahentium sint admodum exiguae
prae distantiis, quibus ipsa corpora disiunguntur , eorum
alterum tendet in alterum perinde ac si essent ambo in
suis gravitatis centris compenetrata ; haec assertio quoad
sphaerica corpora valet utcumque se habeat intercedens
distantia ...............n76
De gravitatione universali. pag. 159.
Ex mutua coelestium corporum gravitatione collata
cum terrestrii gravitate.iuferimus illud: gravitas ita'ma-
teriam allicit, ut singulae eius particulae in alias omnes
et singulas graviteut in ratione directa massarum ad quas
tenditur, et reciproca duplicata distantiarum alterius ab
altera . . . . . . . . . ∎∙ ∙ ∙ n. 77,...82.
Aliquid circa solarem et plaueticas massas...n.83.10...4.'
Media telluris densitas determinata ex penduli aber-
ratione : itemque experimentis institutis in libra tor-
Sioni. ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙∙ ∙ ∙ n. 830 5-0 S..
Quomodo ex marini aestus phoenomeuo deduci pos-
sit ratio inter lunarem ac terrestrem massam. n. 83. 7!327
Aliquid notatur de motu punctorum materialium
utcumque inter se connexorum pag. 169.
Formulae spectantes et ad translativum punctorum
motum juxta coordinatos axes, et ad rotatilem eorum mo
tum circum axes ipsos n. 84.
Moto punctorum systemate, perinde movebitur com
mune gravitatis centrum ac si , coeuntibus punctis in
ipsum centrum , applicarentur centro eaedem vires cum
iisdem directionibus , quibus pancta sollicitantur. n. 84.1.6
Principium de conservatione centri gravitatis : item
de conservatione arearum : necnon de conservatione vi
rium vivarum n. 84. 2.0 ... 5 ..
Relativus rigidi liberique systematis motus quoad
gravitatis centrum n. 84. 6. ° 7.0
Motus rigidi systematis circa axem fixum ; quibus
cuinque caeteroquin viribus acceleratricibus sollicitetur sy
stema : quid si vires acceleratrices consistant in sola gra
vitate ; huc spectat theoria penduli compositi : longitudo
penduli simplicis , quod suas perficit oscillationes eodem
tempore ac pendulum compositum : quid si nullae sint
vires acceleratrices : inertiae momenta quoad axem
principales systematis axes : principalia inertiae momen
n. 85. 1.° 2.° ... 7.0
.
ta .
De fluidorum corporum aequilibrio pag. 182.
Ex perfecta mobilitate , qua ponuntur gaudere flui
dorum corporum particulae , ostenditur principium de
aequalitate pressionis , atque inde eruuntur conditiones
requisitae ad aequilibrium cujusvis massae fluidae. n. 86.
327
Aliquid notatur de motu punctorum materialium
utcumque inter se connexorum pag. 169.
Formulae spectantes et ad translativum punctorum
motum iuxta coordinatos axes, et ad rotatilem eorum mo-
tum circum axes ipsos . . . . . . . . . n. 84.
Moto punctorum, systemate, perinde movebitur com-
mune gravitatis centrum ac si
, coeuntibus punctis in
ipsum centrum, applicarentur centro eaedem vires cum
iisdem directionibus , quibus puncta sollicitantur. n. 84.1.'
Principium de conservatione centri gravitatis: item
de conservatione arearum : necnon de conservatione vi-
rium vivarum . . . .
'. ⋅∙ ∙ ∙ n. 84. Z.&quot;...Sæ
Belativus rigidi liberique systematis motus quoad
gravitatis centrum
. . . . . . . . n.84. 6.&quot; 79
Motus rigidi systematis circa axem fixum .: quibus-
cumque caeteroquin viribus acceleratricibus sollicitetur sy-
stema :quid si vires acceleratrices consistant in sola gra-
vitate; huc spectat theoria penduli compositi: longitudo
penduli simplicis
, quod suas perficit oscillationes eodem
tempore ac pendulum compositum .: quid si nullae sint
vires acceleratrices : inertiae momenta quoad axem :
principales systematis axes : principalia inertiae momen-
ta.
. . . . . . . . . . n.85.1.0 Z.&quot;... 7.&quot;
De fluidorum corporum aequilibrio pag. 182.
Ex perfecta 'mobilitate
. qua ponuntur gaudere Hui-
dorum corporum particulae ,, ostenditur principium de
aequalitate pressionis , atque inde eruuntur conditiones
requisitae ad aequilibrium cujusvis massae Huidae. n. 86.328
Quid notandum circa superficiem massae fluidae li
bratae n. 87, 1. ° 2.° ... 5 .
Quid circa fluidum elasticitate pollens, ni 87. 6.0 7 .
De gravium homogeneorumque liquidorum
aequilibrio. pag. 187.
Liquida constituta in aequilibrio intra vasa : pres.
siones in areas sive horizontaliter , sive oblique demer
sas : centrum pressionis .
n. 98. 1. ° . , . 4.0
Solida liquidis immersa : aequilibrii positiones quoad
solidum liquido insidens n. 88.5 . , 89. 1.° 2. ° 3.°
Utrum aequilibrium sit stabile , nec ne. n. 90.
.
De gravium liquidorum aequilibrio in vasis
communicantibus.
pag. 195.
Quid si vasis communicantibus idem contineatur li
quidum : explicatio variorum effectuum ; antliae adspi
ranles , etc n. 91 , 92. 1.° 2.°
Quid si diversa contineantur liquida.
. n. 92. 3.0
De gravium elasticorumque fluidorum aequilibrio nec
non de altitudinibus dimetiendis ope barometri, et de
pondere ac densitate vaporum . pag. 199.
Conditio aequilibrii expressa per aequationem dif
ferentialem : perficitur integratio in hypothesi temperiei
constantis n. 93.
Inde eruitur formula inserviens ad altitudines di
328
Quid notandum circa superüciemi massae liuidae li-
bratae
. . . . . . . . . n. 87.1.02.&quot;...5.0
Quid circa fluidum elasticitate pollens. n: 87. 6.&quot; 7.&quot;
-De gravium homogeneorumque liquidarum
aequilibrio. pag. 187.
Liquida constituta in aequilibrio intra vasa: pres- ⋅
tiones in areas sive horizontaliter , sive oblique demer-
sas: centrum pressionis . . . . . n. 88. 1.&quot;
..,.
4.&quot;
∙
!'
Solida liquidis immersa : aequilibrii positiones quoad
solidum liquido insidens . . n. 88.5.&quot;, 89. 1.&quot; 2.0 3.&quot;
Utrum aequilibrium sit stabile, nec ne. . . n. 90.
De gravium liquidarum aequilibrio in 'vasis
communicantibus. pag. 195.
Quid si vasis communicantibus idem continaptur li-
quidum: explicatio variorum effectuum : antliae adspi-
TODIBBQ etc ∙ ∙ ∙ ∙ ∙
↼
∙ ∙ a ∙ &quot;.
5 91. 92. 1.02.0
Quid si diversa cbntiueantur liquida. . . n. 92. 3.&quot;
De gravium elasticorumque fluidorum aequilibrio nec
non de altitudinibus dimetiendis ope barometri, et de
pondere ac densitate. naporum. pag. 199.
Conditio aequilibrii expressa per aequationem dif-
ferentialem : perficitur integratio in hypothesi temperiei
constantis . . . .
.'&quot;.
. . ⋅ n. 93.
Inde eruitur formula inservieus ad altitudines di-329
metiendas ope barometri : varia observantur pro commo
diori formulae usu n. 94. 1. ° 2.° ... 6.•
Verticalis ascensus globi aereostatici : maxima glo
bi elatio . n. 95.
Maxima quantitas vaporis sese evolventis in vase un
dique clauso : vis elastica sicci aeris aucta ob evolu
tum vaporem : ratio inter densitatem aquei vaporis ac
densitatem sicci aeris sub eadem temperie eademque pres
sione : ratio inter eorum densitates ac pondera sub ea
dem temperie et diversis pressionibus : densitas aeris va
porosi librantis datam pressionem sub temperie data. n. 96.1 . °
2 .
Usus aquei vaporis in movendis machinis. n . 99.6. •
De aqua egrediente per angustum foramen
e vasis verticalibus sive cylindricis,
sive prismaticis. pag. 206 .
Nonnulla praemittuntur ex pluries iteratis experimen
tis .
n . 97.
Quaenam velocitas aquae egredientis: tempus impen
sum in descensu usque ad quamlibet altitudinem datam . n.98.
Quantitas aquae dato tempore egredientis : tempus
quo vas totum evacualur n. 99, 100,
Ratio inter tempora , quibus deplentur duo vasa ha
bentia et altitudines et orificia aequalia : quantitales aqua
rum successivis ' et aequalibus temporibus ex vasis ori
ficio efluentium : divisio vasorum in partes successivis dati
temporis unitatibus vacuandas n. 101 , 102.
22
' 329
metiendus ope barometri : varia observantur pro commo-
diori formulae usu . . . . . n. 94. 1..) .&quot;
... 6.&quot;
Verticalis ascensus globi aereostatici : maxima glo-
bi elatio. ∙ ∙ ∙ ∙ ∙ ' ∙ ∙ ∙ ∙ ∙ ∙
∙∎∎ ∙ n- 950
-Maxima quantitas vaporis sese evolventis in vase uu-
dique clauso : vis elastica sicci aeris aucta&quot; ob evolu-
tum vaporem : ratio inter densitatem aquei vaporis ac
densitatem sicci aeris sub eadem temperie eademque pres-
sione: ratio inter eorum densitates ac pondera sub ea-
dem temperie' et diva-sis- pressionibus: densitas aeris va-
porosi librantis datam pressionem sub temperie data. n. 961.&quot;
2.0
...
5.&quot;
Usus aquei vaporis in movendis machinis. n. 99. 6.&quot;
De aqua egrediente per angustum foramen
e vasis «verticalibus sive cylindricis,
sive prismaticis. pag. 206.
Nonnulla praemittuntur ex pluries iteratis experimen-
tis ∙ ∙ ∙ ∙ ∙ ⋅∙⋅ ∙ ∙ . ∙ ∙ ∙ ∙∎∎ ∙ ∙ ∙ &quot;o 970
Quaenam velocitas aquae egredientis: tempus impen-
sum in descensu usque ad quamlibet altitudinem datam. n.98.
,. Quantitas aquae dato tempore egredientis: tempus
quo vas tatum evacuatur . . . . . . n. 99,100.
Ratio inter tempora, quibus deplentur duo vasa ha-
bentia et altitudines 'et oriiicia aequalia : quantitates aqua-
rum successivis' et aequalibus temporibus ex vasis ori-
iicio efluentium: divisio vasorum in partes successivis dati
temporis unitatibus vacuandas . . . ∙∙ n. 101, 102.
22330
Contractio venae aqueae n. 103.
Ubinam perficiatur acceleratio , per quam velocitas
aquae descendentis admodum exigua mutatur in finalem
satisque grandem effluxus velocitatem . n . 104. 1.0
Quomodo motus aquae defluentis in regularibus al
veis traduci possit ad motum aquae prosilientis ex an
gustis vasorum orificiis n. 104. 2.• , ..5.
Illud cum Auctoribus non paucis assumitur tanquam
principium , quod nempe unumquodque liquidi in vase
quolibet descendentis tenuissimum et horizontale stra
tum coalescat iisdem constanter particulis communi , ea
que tantum verticali , velocitale donatis ; inde vero eruun
tur , quae pertinent ad ipsius liquidi motum n . 105.
Aliquid subjungiur circa generalem theoriam
motus corporum fluidorum . pag. 216.
Aequationes ad istiusmodi motum et quum massa
fluida homogenea vel heterogenea est incapax compressio
nis, et quum massa fluida pollet elasticitate. n. 106,107,108.
De tubis capillaribus. pag. 225.
sol Vires ex materia tubi , et ex materia liquidi ,
licitantes datam ipsius liquidi particulam : attentis viri
bus istis , suprema liquidi superficies vel induet curvam
concavamque figuram , vel curvam convexamque , vel ma
nebit, plana atque horizontalis, n. 109,1.9
330
Contractio venae aqueae . . . . -. . n. 103.
Ubinam perficiatur acceleratio, per quam velocitas
aquae descendentis admodum exigua mutatur in finalem
satisque grandem effluxus velocitatem. .
-
.n. 104. 1.&quot;
Quomodo motus aquae defluentis in regularibus al-
veis traduci possit ad motum aquae prosilientis ex au-
gustis vasorum orificiis
. . . . . n. 104. Z.&quot;. ..5.&quot;
Illud cum Auctoribus non paucis assumitur tanquam
principium, quod nempe unumquodque liquidi in vase
quolibet descendentis tenuissimum et horizontale stra-
tum coalescat iisdem constanter particulis communi , ea-
que tantum verticali, velocitate donatis :
inde vero eruun-
tur, qnae pertinent ad ipsius liquidi motum
. ∙⋅ n. 105-
Aliquid subjungiur circa generalem theoriam
motus corporum fluidorum. pag. 215-
Aequationes ad istiusmodi motum et quum massa
fluida homogenea vel heterogenea est incapax compressio-
nis, et quum massa fluida pollet elasticitate. n. 106,107,108.
De tubis capillaribus. pag. 225.
Vires ex materia tubi
,
et ex materia liquidi .
sol-
licitantes datam ipsius liquidi particulam: attentis viri-
bus istis , suprema liquidi superficies vel induet curvam
concavamque figuram , vel curvam couvexamque , vel ma-
nebit, plana atque horizontalis, ,. . . .. . n. 109.1.&quot;331
Quam attractionem exerceat massa liquida , cujus su
prema superficies est plana , in columellam liquidam per
pendiculariter illi superficiei planae insistentem . n. 109.2 .
Quam attractionem exerceat massa liquida , cujus su.
prema superficies est vel concavo -sphaerica vel convexo
sphaerica , in columellam liquidam perpendiculariter in
sistentem plano tangenti , dactó vel per punctum infimum
superficiei concavo-sphaericae vel per supremum superfi
ciei convexo -sphaericae. n. 109. 3.° 4.0 ... 70
Quid si massa liquida terminetur superficie concaya
vel convexa , quae non sit sphaerica. n. 109. 8.° ... 11.º
His declaratis , explicamus ascensum descensumque
liquorum in lubis capillaribus n. 110,
.
Nonnalla subjunguntur , quorum ratio desumitur ab
actione capillari . n. 111. 1.° 2.° ... 5 ° , 112
)
ACUSTICAE PRINCIPIA
Notiones praeambulae.
1
pag . 245.
Corpora, quae sonora dicuntur tunc sonum exci
tant quando ita agitantur , ut illorum partes tremulo ac
vibratorio satisque rapido concutiantur motu ; qui motus
communicatus aeri ambienti , et late diffusus afficit orga
nym auditus: vis acceleratrix in vibrante particula resonan
tis corporis.
. n . 113. 10. 20.
331
Quam attractiduem exerceat massa liquida , cuius su-
prema superficies est plana ,
in columellam liquidam per-
pendiculariter illi superficiei planae insistentem. n. 1092.&quot;
Quam attractionem exerceat massa liquida , cuius su-
prema- superficies est vel concavo-sphaerica vel convexo-
sphaerica, in columellam liquidam peu-pendiculariter in-
sistentem plano tangenti , dnctö vel per punctum infimum
superficiei concavo-sphaericae vel per supremum superfi-
ciei convexo-sphaericae. . . . n. 109. 3.&quot; 4.&quot;
.. . 7.&quot;
Quid si massa liquida terminetur superficie concava
vel convexa, quae non sit sphaerica. n. 109. 8.&quot;. ..
11.&quot;
His declaratis , explicamus ascensum descensumque
liquorum iu .tubis capillaribus . . . . . . n. 110.
Nonnulla subjunguntur ∙ quorum -ratio desumitur ab
actione capillari. . . . . n. 111. 1.&quot; 2.&quot;
. . . 5&quot;,112
AOUSTIGAE
W
PRINCIPIA
Notiones praeambulae. ∣
pag. 245.
Corpora, quae sonora dicuntur
,
tunc sonum exci-
tant quando ita agitantur, ut illorum partes tremulo ac
vibratorio satisque rapido concutiuntur motu; qui motus
communicatus aeri ambienti, et late diffusus afficit orga-
num auditus: vis acceleratrix in vibrante particula resonan-
tis corporis. . . . . . . '. . . . n. 113.1&quot;. 2&quot;.332
Progignitur quoque sonus ab aere vehementer compres
so , seseque statim restituente , n. 114.
.
Soni reflexio; inde echo. n . 115 .
Non solus aer est medium ideoneum transmissioni
sonorum. n. 116.
De intensitate soni; deque ejus gravitate,
et acutie .
pag. 248.
Sonus intensior ex eo gignitur quod in sonoro cor
pore plures ejusdem partes simul oscillant, et majus spa
tium singulis oscillationibus dato tempusculo percurrunt;
atque ita in aere ex numero item et majori oscillatione
partium aeris intensitas soni dependet ; remissior autem
sonus ex opposito.
n. 117.
Nonnulla explicantur circa soni intensitatem. n . 118.
ex Soni gravis et acuti discrimen repetendum est
numero vibrationum in partibus sonori corporis, ita ut
sonus gravior oriatur ex minus frequentibus vibrationibus so
nori corporis, ex crebrioribus contra sonus acutus ; idem
que de oscillationibus aeris in sono derivato. n. 119.
Quid consonantia , et quid dissonantia: varii conso
nantiae gradus: theoria chordaram vibrantium in hypothe
si vibrationum admodum exiguarum. n. 120. 1 ” 2 ”... 7 .
Varia proponuntur explicanda circa chordas vibran
tes . n. 121 .
332
Progignitur quoque sonus ab aere vdhemeuter compres-
so, seseque statim restituente. . . . . . . n. 114.
Soni reflexio; inde echo. . . . . . . n. 115.
Non solus aer est medium ideoneum transmissioni
SODOmm. ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ &quot;. 116.
De intensitate soni; deque eius gravitate,
et acutie. pag. 248.
Sonu's intensior ex eo gignitur quod in sonoro cor-
pore plures eiusdem partes simul oscillaut, et maius spa-
tium singulis oscillationibus dato tempusculo percurrunt:
atque ita in aere ex numero item et maiori oscillatione
partium aeris intensitas soni dependet; remissior autem
sonus ex opposito. . . . . . . . . . ,n. 117.
Nonnulla explicantur circa soni intensitatem.
. .n. 118.
Soni' gravis et acuti discrimen repetendum est ex
numero vibratiouum in partibus sonori corporis, ita ut
sonus gravior oriatur ex minus frequentibus vibrationibus so-
nori corporis, ex crebrioribus contra sonus acutus,- idem-
que de oscillationibus aeris in sono derivato. . n. 119.
Quid consonantia, et quid dissonantia: varii conso-
nantiae gradus: theoria chordarum vibrantium in hypothe-
si vibrationum admodum exiguarum. n- 120. 1&quot; 2&quot;... 7&quot;.
Varia proponuntur explicanda circa chordas vibran-
tes. ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙
Q &quot;. 1210333
Quomodo sonus trans obicem possit communicari ita,
ut tonus proprius sonori corporis permaneat. n. 122.
Unde asperitas aut lenitas soni proficiscatur. n. 123.
Transversae et longitudinales chordarum vibratio
nes: nodi in chordis vibrantibus: lineae nodales in super
ficiebus corporum resonantium : vibrationes laminarum ri
gidarum .
n. 124 .
De directa soni propagatione per aerem . pag. 265.
In iisdem circumstantiis sonus aequabili velocitate
in toto decursu devehitur; omnesque soni sive intensi ,
sive remissi, sive graves, sive acuti eadem velocitate dif
funduntur: qua ratione intensitas soni minuatur in pro
gressu . n. 125,
Undae sonorae constitutio. n. 126,
Soni et velocitas, et intensitas augetur a vento se
cundo, minuitur ab adverso . n . 127.
Experimenta instituta ad soni velocitatem determi
nandam; quae tamen experimenta non satis conveniunt :
hujus diversitatis rationes : quaenam utilitas ex determi
natione velocitatis qua propagatur sonus. .
n. 128
Generalis de fluidorum motu theoria applicatur ad
soni propagationem : soni velocitas eruta ex applicatione
theoriae comparatur cum velocitate quam praebent ex
perimenta .
n. 129. 10. 2º. 3º.
Crassities aerei strati, in quo particulae cientur una :
si impulsio in obicem facta quadrato velocitatis sumitur
-
22&quot;
333
Quqmodo sonus trans obicem possit communicari ita,
ut tonus proprius sonori corporis permaneat.
. n. 122.
Unde asperitas aut leuitas soni proficiscatur. n. 123.
Trausversae et longitudinales chordarum vibratio,-
nes: nodi in chordis vibrantibns: lineae nodales in super-'-
iiciebus corporum resonantium: vibrationes laminarum ri-
gidarum...........;..n.124.
De directa soni propagatione per aerem. pag. 265.
In iisdem circumstantiis sonus aequabili velocitate
in toto decursu devehitur; omnesque soni sive intensi ,
sive remissi, sive graves, sive acuti eadem velocitate dif-
fuuduntur: qua ratione intensitas soni minuatur in pro-
gressu..............-n.125.
Undae sonorae constitutio. . . . . . . n.126,
Soni et velocitas, et intensitas augetur a vento se-
eundi), mall!!! EI) adverw. ∙ ∙ ∙ ∙ ∙ ∙ n- 127.
Experimenta instituta ad soni velocitatem determi-
nandam; quae tamen experimenta non satis conveniunt:
hujus diversitatis rationes: quaenam utilitas ex determi-
natione velocitatis qua prOpagatur sonus. . . n. 128.
Generalis de fluidorum motu theoria applicatur ad
soni propagationem: soni velocitas eruta ex applicatione
theoriae comparatur cum velocitate quam praebent ex-
perimenta . . . . . . . . . . n.129.1o.2&quot;.3&quot;.
Crassities aerei strati, in quo particulae cientur uua:
si impulsio iu obicem facta quadrato velocitatis sumitur
22'334
proportionalis, rationem duplicatam distantiarum .sequetur
soni debilitatio.
n. 129. 4. 5 .
Cur pluribus corporibus simul resonantibus , inter
oscillationes in aere excitatas non habeatur confusio , omnes
que diversi soni inde orti ad aures distincte perveniant:
huc spectat principium de superpositione exiguorum mo
tuum. n. 129. 6.
Propagatio soni in cubis cylindricis indefinitae lon
gitudinis. n. 129.7 .
J
De reflexa soni propagatione per aerem pag. 289.
Cum in directa propagatione sonorus aer . offendit o
bicem aptum , reflectitur : varia ad echo spectantia ex
plicantur. n. 130.
Reflexio soni fit ad angulos incidentiae et reflexionis
aequales; regrediturque sonus eadem velocitate qua incedebat
antequam in obicem impingeret. n . 131 , 132, 1º. 2º
De instrumentis pneumaticis. pag. 294.
In instrumentis pneumaticis soni genesis repetenda non
est saltem praecipue ex oscillatione partium solidarum i
psius instrumenti : quo pacto sit explicanda : aer secun
dum fistulae longitudinem se habet instar chordae peragen
tis longitudinales vibrationes: etsi ex materia instrumenti
non habetur varietas quoad soni qualitatem, aut valde uota
bilem intensitatem ; varietas tamen habetur quoad meliorem
334
proportionalis, rationem duplicatam distantiarum .sequetur
soni dehilitatio. . . . . . . . ∙ ∙ n.129.40.50.
Cur pluribus corporibus simul resonantibns ,
inter
oscillationes in aere excitatas non habeatur confusio,omues-
que diversi soni inde orti ad aures distincte perveniant:
huc spectat principium de superpositione exiguorum mo-
tuum. ∙⋅∙ '.
. . . . . . . . . . n.129.6&quot;.
Propagatioi soni in. tubis cylindricis indefinitae lou-
gitudinisa, ∙ ∙ ∙ . . . . . .. . . n.129.7&quot;.
]
De refleæa soni propagatione per aerem pag. 289.
!. ∙ . .
Cum indirecta prOpagatione sonorus aer .oii'eudit o-
bicem aptum, reflectitur : varia ad echo spectantia ex-
Plimnturo. ∙∙ ∙∙ ∙∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ n. 1300
. Reflexio soni iit ad angulos incidentiae et reflexionis
aequales: regrediturque sonus eadem velocitate qua incedebat
antequam in obicem impingeret. . . .&quot;' 131, 132.1&quot;.2&quot;.
'pul '
De instrumentis pneumaticis. pag. 294.
In instrumentis pneumaticis soni genesis repetenda non
est saltem praecipue ex oscillatione partium solidarum i-
psius instrumenti: quo pacto sit explicanda: aer secun-
dum fistulae longitudinem se habet instar chordae peragen-
tis lougitudinales vibrationes: etsi ex materia instrumenti
non habetur varietas quoad soni qualitatem, aut valde uota-
bilem intensitatem; varietas tamen habetur quoad meliorem335
aliquam resonantiam : quid si intrumentum pneumaticum sit
compactum ex materia non resistente , quale v. g. esset in
strumentum membranaceum .. . n. 133.
Nonnulla proponuntur explicanda circa instrumenta
pneumatica. n . 134.
Tremulus aeris motus in tubis cylindricis determinatae
longitudinis :
1º. Quum tubus est firmiter obseratus apud alterum
orificium simulque apertus apud alterum n. 135, 136.
2°. Quum tubus est patens in utraque extremitate: in
de eruitur ratio investigandi velocitatem , qua propagatur
sonus in aliis fluidis elasticis diversis ab aere. n. 137.
3 °. Quum tubus est utrinque obseratus. n. 138.
De propagatione soni per liquida, et per solida
corpora . pag. 302.
Formulae huc spectantes: parvula contractio aquae et
hydrargiri ob auctam pressionem: usus istius contractionis in
determinanda velocitate soni per haec duo liquida. n . 139,140 .
Analogia inter oscillationes aeris in tubo cylindrico a
pud ambas extremitates aperto et longitudinales oscillationes
virgae rigidae suppeditat peculiarem methodum investigandi
velocitatem propagationis per solida corpora. n. 141 .
De vocis humanae origine. pag. 305.
Nonnulla ex anatomicis praemittuntur; quibus praemis
sis , stabilitur illud : vocis humanae organum etsi conside
rari maxime debeat tanquam instrumentum pneumaticum
335
aliquam resonantiam: quid si intrumentnm pneumaticum sit
compactum ex materia nou'reaistente, quale v.. g. esset in-
strumentum membranaceum. .. .. .. .. .. .. . n. 133.
Nonnulla proponuntur explicanda circa instrumenta
pneumatica. .. . .. .. .. .. .. .. .. .. .. ..
.n. 134.
Tremulus aeris motus'iu tubis cylindricis determinatae
longitudinis :
⇝
↿∘∙ ⊄⊇⇂⋯⋯∙⋯∣⋯∘⋅⊖⊱⇂ firmiter ohseratus apud alternm
orificium simulque apertus apud alterum . n. 135,136.
20. Quum tuhus est patens in utraque extremitate: in-
de eruitur ratio investigandi velocitatem
, qua propagatur
sonus in aliis fluidis elasticis diversis ab aere. n. 137.
(
3&quot;. Quum tubus est utrinque ohseratus. . n. 138.
i '
⋅
⋅ ↼
De prapagau'one soni per liquida, ettper &quot;solida ⊳∣ corpora. pag. 302.
.Fornrnlae huc spectantes: parvula contractio ailuae et
hydrargiri ob auctam pressionem: ususistius contractionis in
determinanda velocitate soni per haec duo liquida.ia.139, 140.
Analogia inter oscillationes aeris in tuho cylindrico a-
pud ambas extremitates aperto et lougitudinales oscillationes
virgae rigidae suppeditat peculiarem methodum investigandi
velocitatem propagatiouis per solida corpora. . n. 141.
De 'vocis humanae origine. pag. 305.
Nonuulla ex anatomicis praemittuntur; quibus praemis-
sis, stahilitur illud: vocis humanae organum etsi conside-
rari maxime debeat tanquam instrumentum pneumaticum
∩336
flexili et elastica materia ex parte compactum , non tamen ita
est ut cum instrumentis fidicularibus aliquam non habeat
analogiam . n. 142.
Quid, os atque ejus partes conferant ad formationem
vocis.
n. 143.
Variae refellantur sententiae de humanae vocis ori
gine; variaeque circa vocem humanam proponuntur quae
stiones. n . 144 , 145 .

De auditus organo . pag. 310.
Auris descriptio.
n. 146.
Quaenam ex auris partibus pro praecipuo atque im
mediato auditionis organo statuenda sit. n. 147.
Cur quibusdam grata, aliis pene nihil, aut etiam mo lesta sit harmonia ,
n. 148. 19.
Cur daabus auribus unus idemque sonus audiatur n.148.2 °.
1
336 '
Bexiliot'elgstica materia ex parte compactum, non tamen ita
eat ut cum, instrumentis iidicularibus aliquam non habeat
malogihmoo-o-0 ∙⋅∙∙∙⋅∙ ∙ ∙ ∙ ∙ ∙ ∙
&quot;0142.
Quid, os atque eius partes conferant ad formationem
'owa ∙∙∙ ∙ ∙ ∙ ∙ ∙ ∙∙∙∙∙ ∙ ∙
&quot;.1430
Variae refelluntur sententiae de humanae vocis ori-
gine, variaeque circa vocem humanam proponuntur quae-
'none'- ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ &quot;- 14401450
De auditus organo. pag. 310.
Auriadeacriptio. . . . . . . . . . n.146.
Quaenam ex auris partibus pro praecipuo atque im-
mediato auditionia organo statuenda- sit.
. . . n. 147.
Cur quibusdam grata, aliis pene nihil, aut etiam mo-
lestasit harmonia. ∙ ∙ ∙
. . . . . . n. 148. ∎∘∙
Cur duabus auribus unus idemque sonus audiatur n.148.2'.ERRATA CORRIGE
pag. lin .
1. 4. saepae
1. 5. decresit
4. 28. istanti
5. 13. rive
7. 29. poductis
8. 14. sin a
14 29. AH'.BC
15. 3. AF
saepe
decrescit
instanti .
siye .
prodactis
sin a .
AH'.BC
BF'
BF .
Y.
4. AF
24, 7. Sy
50. 6 et 7.
S * S

*

17. ſsfla)dx Sfaxdx.
52. 14.
f (x )dx f '( x )d.x2
2
2
56. 18.- ( tdx ,z + dz,u,...) -f(xtdx , z + de, u, ...).
dull .
Sfx )dx 22. ( x) dx
eck
58. 1.-C
+0 .
57. 4. del
1
Sfaydar
62. 3. W
v'dz'
11. dzi
63. 8. sint va
69. 12. quod
... 17.
v'du'
da
sintVC .
quoad
ngt 2gt
70. 7.- 7 . kalog(k2—12)
.
.log(k ?-- ).
1
-
1
!
ERBATA CORRIGE
pag. lin.
&quot;. 4. saepae saepe .
1. 5. decresit decrescit
4. 28. istanti instanti
.
5. 13. rive sive
.
7. 29. poductis productis .
8. 14. aina sin at
.
14 29. AH'.BC AH'.BC'
.
15. 3. AF' BF' .
⋅
∙ ∙ ∙ 4. AF BF
. 24, 7. &lt;nowiki&gt;:] z?&lt;/nowiki&gt;
.
50. 6et 7. f:&quot; f:&quot;
... 17. JfftæMx [f(xkiæ .
52. 14. figit&quot; f'(æ2)dæ* .
56. 18.—(æ-]-dæ,z-l—dz,u,...) --f(a—-[-dx , z—l-dz, u, -
.
)- - 57. 4. d,,p. dup.
.
.. .
22. 111-2635 f(ældæ
.
803 803
58. 1.-:.-C —]—G .
62. 3. 9) p
↿↿∙∙∙ v'dz' til—tf
.
d:, dz'
.
63. 8. siun/C sint;/C .
69. 12. quod ' quoad .
⋅ n : 2 c
...
17. ∘−⋚∟ ∉−≓∙∙
k: 70. 7. −∙∙ 2 2
∣⊂≄∣∘⊰≼∣∁≖−⋁≖⋟ −−⋅⊋−⋅ lOg(k —P )
--
. ∙∙∽∙∙⋅ −∙− ↼∙ -
∙−⊣ERRATA CORRIGE
pag. lin .
kdv Ka dy
70. 12 .
katus kype &quot;
71. 13 et 14. KC Kc
72. 23. u = ułgosinc u = a + g9 sinc .
75. 23. pressioni r.gMcosc' pressioni gMcosc' .
87. 2. Denotet enim a Denotet enim x .
IG &quot; IG &quot;
27. = IC &quot; : IC =
2 2
110. 9. R = Rcosa R = R , cosa
111. 5. 1880 to 288q'to .
da dala 146. 8 .
idt
148. 12. 69.º* 69. *
149. 6. x = A'B' - B'r - A'B ' - A'M x=A''B'-B'r=A'B'-A'M .''
x' ? c
x
151. 2 .
ic
(de)
Centre Ide
i
152. 78 et 20. r2
153. 22. (69)
154. 17.
72.°*
157. 8. SD
161. 26. 16931100
193. 23. u : M ' : fle ..
205. 7. aequeus
208. 14. aia
r2
( 70 ) .
72.*
GD .
19631100 .
Me : No :
aqueus ,
Q:.
i
3
ERRATA CORRIGE
∙∙∙ ∙∙∙∄≾≖∠≀⇂↗ *
kæ-I—uz ⋅
Kc
uza-l—gg sinc .
23. pressioni ngMcosc' pressioni gMcosc' .
pag. lin
70. 12:
liti—v- kZ.-v2
71. 13 et 14. KC
72. 23. uzu-l—gasinc '
75.
87. 2. Denotet enim a
∙⋅≆↴ IG&quot; ∙ IG&quot;
. 27.:IC :::—2— . 10: -—2-— .
110. 9- R::Rcosa: BzB, cos a .
111. 5. 1889'—]—-p' ∙ 28897'—-q)' .
' doc 146. 8. ; daz :
(2? (22?
148. 12. 6994! 6931:
149. 6. a::A&quot;B'-B'r-A&quot;B'-A'M sz&quot;B'-B'r:-A&quot;B'-A'M .
.... .. ⋅↕⋅≟≣∁ ⋅
...-7...
50 ac
152. 78 et20. fi ∙∘−⋮⋅⋅∙ ra .rz 153. 22. (69) (70)
.
154. 17. 7291: 724
157. 8. SD .GD .
161. 26. 16931100 - 19631100 .
193.23. p.':p.':p.., php.
205. 7. aequeus aqueus ,
208.1.£. a:a' «:a' .
Denotet enim æ .</text>
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      <id>3697722</id>
      <parentid>3697721</parentid>
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      <text bytes="726224" xml:space="preserve">== PRAEFATIO ==
Rerum naturalium ordinem considerare, Deumque in iis mirifice operantem intueri, proprium est verae sapientiae, quam Philosophia profitetur. Haec scientia, quae dicitur Physica , inter scientias homine dignissimas. atque inter praecipua Dei dona jure commendatur: ecquid enim potest esse praestantius aut utilius quam divinae sapientiae opera, Deumque ipsum suas in natura perfectiones ostentantem contemplari? An quod Deus omnipotentia sua non judicavit indignum in iis quae creavit , quod in iis quae regit et gubernat attentione sua dignatur Providentia Dei, hoc nos meditari supervacaneum atque otiosum iudicabimus? Otiosam illam dicerem Physicam, quae ita immoraretur in Operis consideratione, ut opificis non perpetue suspicere! industriam: caecus est, qui Deum non videt in natura ejusque providentiam ac sapientiam non admiratur. Similem illum dixerim homini, qui librum ob Oculos apertum tenens characterum elegantiam contemplatur, numerat verba; sensum non penetrat. ⋅Neque vero minus utilis Naturae cognitio ministris Ecclesiae quam caeteris hominibus existimanda est: imo et hanc ipsis maxime necessariam duxerim hoc praesertim tempore cum homines vano inflati doctrinae apparatu scientias pro viribus adversus Religionem convertant , et Phyicam praecipue revelationi satagant opponere , vereque Opponi non desinant clamare eoram ignaris. Cum igitur se linguae impiae in injuriam Religionis armant, pudeat hominem Religionis amantem, et eo charactere insignitum qui ipsum Religionis statuat defensorem, aut turpiter obmutescere, aut Religionem. male defensam hominibus impiis vanum jactantibus triumphum, et ministrorum ignorantiam in Religionis opprobrium vertentium, deridendam proponere. Quod si nihil a viro ecclesiastico quaereretur aliud in Physica quam honesta mentis recreatio, justaque serii laboris intermissio, quid potest esse homini laboribus sacris defuncto aut jucundius aut dignius quam otium inutile, ac saepae periculosum, otio erudito et physico commutare? Quam multa offeret naturae spectaculum , ipsiusque arcanorum inquisitio, quae laudabilem nutriant curiositatem ,utiles praebeant considerationes, suaves atque innoxias pariant delicias! Etenim nullam esse in Philosophia partem quae majori voluptate quam naturae contemplatio animum compleat, Tullii auctoritate facile possem confirmare. Haec nihilominus voluptas non sine studio ac labore comparari potest: sapientissimus enim rerum omnium creator et rector Deus, quamvis sensuum nostrorum perceptioni corpora haec sensibilia subjecerit, illorum tamen naturam et vim mira quadam sepsit caligine, ut quicumque ad eam penitus scrutandam accedunt, in media luce densis veluti tenebris repente se obrui sentiant; quas tenebras etiamsi non ita discutere possimus ut claram ac certam rerum omnium scientiam assequamur, attamen si nos studii, diligentiae ac laboris non piguerit, ita tenebras attenuari experiemur ut multarum rerum certam cognitionem , plurimarum admodum probabilem obtineamus . Ad occulta Naturae arcana inquirenda duae sunt viae, quas eximii ingenii vir Franciscus Baconus de Verulanio notavit in novo scientiarum organo lib . 1. aphor, 19. Prima, qua a sensu et particularibus incipientes advolamus ad axiomata maxime generalia; atque ex iis principiis, eorumque immota veritate judicamus et invenimus axiomata media . Altera a sensu et particularibus excitat axiomata ascendendo continenter et gradatim , ut ultimo loco perveniatur ad ma
xime generalia. Primam viam plures arripuerunt, qui conjecturas non admodum graves sequuti , atque experientia
non satis accurata innixi generalia axiomata nimia festina
tione constituerunt , iisque naturalium causarum et effe
ctyum vim omnem contineri voluerunt; atque in iis tuen
∼∣⋁in Physica quam honesta mentis recreatio, iustaque serii
laboris intermissio, quid potest esse homini laboribus sacris defuncto aut iucundius aut dignius quam Otium inutile,
ac saepae periculosum, Otioterudito et physico commutare?
Quam multa offeret naturae speCtaculum, ipsiusque arca-
norum inquisitio, quae laudabilem nutriant curiositatem,
utiles praebeant considerationes, suaves atque innoxias pariant delicias! Etenim nullam esse in Philosophia partem
quae majoriivoluptate quam naturae contemplatio animum
compleat, Tullii auctoritate facile possem confirmare.
Haec nihilominus voluptas non' sine studio. ac labore
Comparari potest: sapientissimus enim rerum omnium creator
et rector Deus, quamvis sensuum nostrorum perceptioni
corpora haec sensibilia subiecerit, illorum tamen naturam
et vim miraaquadam sepsit caligine, ut quicumque ad eam
penitus scrutantium accedunt, in media luce densis veluti
tenebris repente se obrui sentiant; quas tenebras etiamsi
non ita discutere possimus ut claram ac certam rerum o-
mnium scientiam assequamur, attamen si nos studii. dili-gentiae ac laboris non piguerit, ita tenebras attenuari ex-
periemur ut multarum rerum certam-cognitionem , pluri-
marum admodum probabileur Obtineamus. Ad Occulta Naturae arcana inquirenda duae sunt viae, quas eximii inge-
nii vir Franciscus Baconus de,.Verulamio notavit&quot; in novo
scientiarum organo lib. ∎∙ aphor. 19. Prima, qua a sensu et particularibus incipientes advolamus.ad axiomata-; mas-
xime generalia; atque ex iis principiis, eorumque-[immota
veritate iudicamus et invenimus axiomata 'media. :Altera'a
sensu et particularibus excitat axiomata ascendendo contio
nenter et gradatim, ut ultimo loco perveniatur adfusa-i-
xime generalis. Primam viam plures arripueruut, qui' con-
iecturas non admodum graves,,s'equuti , atque experientia
non satis accurata innixi generalia axiomata nimia festina-
tione constituerunt,, iisque naturalium causarum et eil'e-
ctuum vim omnem contineri voluerunt; atque in iis tuen-dis totam ingenii aciem intendentes inciderunt in perver
sam philosophandi rationem , adeo ut rerum universitatem
commenti sint omnino aliam ac éa est. Altera aliis placuit
via, qui rerum naturam in rebus ipsis longa observatione
atque accurata experientia quaerendam esse statuerunt; isti
effectuum et causarum naturalium indolem singillatim in
quirere coeperunt, corporum texturám intimam , configu
rationem, motum scrutati sunt; atque ex his; aliisque in
numeris diligentissime observatis Naturae leges deduxerunt,
verasque rerum causas deprehenderunt. ' Hoc pacto plura
nostris temporibus certissima sunt , quae olim ignoraban
tur : alia probabili conjectura assecuti sumus : adhuc ta
men non pauca restant ambigua et involuta ; sed non de
erunt fortasse, qui aliquando novas in lucem eruant veritates.
Notandum vero Naturam ubique mathematicam esse ,
eamque velle absque Mathesi expiscari perinde fore, ait Gul
lielminus , ac sine cruribus ambulare. Porro tota Naturae
compago soliditate constal geometrica, resque physica rei geo
metricae unitur mystico quodam nexu, quem soli mathe
maticae Analysi datum est reserare: Analyseos ductu ex ob
servationibus et experimentis ab iis quae in promptu sunt
ad reconditiora deferimur, et ab exteriori venustate ad in
ternos naturae sinus. Observationes quidem virium existentiam demonstrant, sed proprium est Analyseos pate
facere virium leges, curvas a corporibus in gyrum actis
descriptas, circumvolutionum ac motuum inaequalitates et
aberrationes; quae omnia aspectabilem hanc Mundi ma
chinam maxime illustrant . Quid ab Analyseos indole magis abhorrere videbatur quam electricorum phoenomenorum congeries, cum et innumera prope sint, et innumeris obnoxia conditionibus? Ad electricas tamen vires expendendas accessit Analysis , earumque non paucos effectus leges
que aequationibus definivit.
Ut Tyronum , qui physicis praelectionibus in Romano
Soc. Jesu Collegio dant operam, commodo utilitatique ser
'
dis totam ingenii aciem intendentes inciderunt 'inrïperwe'r.»
sam philosophandi rationem, adeo ut rerum.:nniversitatem
commenti sint omnino aliam-ac ea est. .Altera aliis placuit
via, qui rerum naturamin: rebus ipsis longa-ObservatiOne
atque- accurata - experientia quaerendam, 'esse' statuerunt: :.i'sd
effectuum. ïet; causarum. naturalium 'indolem tsin'gillat'im in—
quirere coeperunt, corporumf-textuttam--imimdmf, configu-
rationem, motum scrutati sunt; atque ex his, aliisque-.in-
numeris diligentissime observatis Naturae leges deduxerunt,
verasque rerum causas deprehenderunt. 'Hoc pacto plura
nostris tempOribus certissima sunt, quae Olim ignoraban-
tur: alia probabili coniectura assecuti sumus : adhuc ta-
men non pauca restant— ambigua et.-involuta; sed non de-
erunt fortasse, qui aliquando novas in lucem eruant veritates.
Notandum vero Naturam ubique mathematicam esse,
eamque velle absque Mathesi expiscari perinde fore, ait Gul-
lielminus, ac sine cruribus ambulare. Porro tota Naturae
compago soliditate constat geometrica, resque physica rei geo-
metricae unitur mystico quodam nexu, quem soli mathe-
maticae Analysi datum est reserare: Analyseos ductu ex Ob-
servationibus et experimentis ab iis quae in promptu sunt
ad reconditiora deferimur, et ab exteriori venustate ad in-
ternqs naturae sinus. Observationes quidem virium exi-
stentiam demonstrant, sed prOprium est Analyseos pate-
facere virium leges, curvas a corporibus in gyrum actis
descriptas, circumvolutionum se motuum inaequalitates et
aberrationes; quae omnia aspectabilem hanc Mundi mn-
chinam maxime illustrant. Quid ab Analyseos indole ma-
gis abhorrere videbatur quam electricorum phoenomenorum
congeries, cum et innumera prope sint, et innumeris Ob-
noxia conditionibus? Ad electricas tamen vires eXpenden-
das accessit Analysis, earumque non paucos eil'ectus' leges-
que aequationibus definivit.
Ut Tyronum, qui physicis praelectionibus in Romano
Soc. Iesu Collegio dant operam, commodo utilitatique ser-VI
viam , in lucem edo illas Physicae partes , quae intimiori
vinculo nectuntur mathesi, quarumque explicandarum cu
ra mihi est demandata. A Mechanica exordior ; siquidem
reliquarum est veluti basis et fundamentum : caeterum sic
Physicae mathematicae tractationem concinnavi, ut quae aste
risco inveniuntur signata, possint ab iis Tyronibus omitti,
qui primo duntaxat Philosophiae anno mathematicis insti
tutionibus studuerunt.
VI
viam
,
in lucem edo illas Physicae partes
, quae intimiori
vinculo nectuntur mathesi, quarumque explicandarum cu-
ra mihi est demandata. A Mechanica exordiar.; siquidem
reliquarum est veluti basiset fundamentum: caeterum sic
Physicae mathematicae tractationem concinnavi, ut quae aste-
risco inveniuntur signata, pOssint ab iis Tyron'ibus omitti,
qui primo duntaxat Philosophiae anno mathematicis insti-
tutionibus studuerunt.

== MECHANICAE PRINCIPIA ==

=== Notiones Praeambulae ===
[[1|1]]. Moto puncto materiali, si ratio inter numericos spatii percursi ac respondentis temporis valores &lt;math&gt;s&lt;/math&gt; ac &lt;math&gt;t&lt;/math&gt; permanet eadem, motus dicitur uniformis; quod si ratio illa jugiter mutetur, motus dicitur varius, acceleratus nempe vel
retardatus, prout crescente
e crescit vel decrescit ipsa &lt;math&gt;\frac s t &lt;/math&gt;: porro motus rectilineus atque uniformis est simplicissimus
omnium motuum, quorum exsistit capax punctum materiale. In &lt;u&gt;motu uniformi&lt;/u&gt; ratio &lt;math&gt;\frac s t &lt;/math&gt; dicitur &lt;u&gt;velocitas&lt;/u&gt;; qua designata per &lt;math&gt;v&lt;/math&gt;, erit

&lt;math&gt;v = \frac s t .&lt;/math&gt;

Quoad punctum materiale, cujus massa seu quantitas materiae (&lt;math&gt;=m&lt;/math&gt;), et velocitas &lt;math&gt;=v&lt;/math&gt;, factum &lt;math&gt;mv&lt;/math&gt; appellatur quantitas motus.

[[2|2]]. Corpus de se est indifferens ad motam et ad quietem. Haec indifferentia sic probari potest ex natura loci: nequit corpus de se postulare at localiter moveatur nisi exigat natura sua non esse in loco ubi est, et locum in quo non est occupare; contra nequit corpus de se quietem
exigere nisi exigat natura sua esse potius in loco ubi est quam in loco quem occuparet si moveretur. Neutrum vero ex natura sua exigit corpus; cum enim omnia loca sint
ejusdem rationis, jam nulla datur ratio cur corporea substantia exigat esse potius uno in loco quam in alio: propterea etc.

[[3|3]]. Quae causae motum gignunt, accelerant, retardant, detorquent, eae vocantur potentiae seu vires. Plures potentiae corpori aut corporum systemati applicitae sese ita possunt impedire, ut nullus inde oriatur motus; tunc vero potentiae dicuntur constitutae in aequilibrio. Fac ut duae vires punctum materiale sollicitent in partes contrarias; si eae sunt in aequilibrio, dicentur aequales: pone duas, tres etc. . . : . ex ejusmodi viribus aequalibus applicari puncto materiali ita , ut in unam eamdemque rectam conspirent; inde habebis vim duplam , triplam etc. . . . Poterunt nempe vires omnes exprimi per numeros ; et consequenter repraesentari per lineas rectas istis numeris proportionales, quarum directiones cum ipsarum virium directionibus congruant. Mechanica tota est in aequilibrii ac
motus doctrina consideranda.

[[4|4]]. Finge tibi globum &lt;math&gt;A&lt;/math&gt; quiescentem e filo pendulum,
in quem impingat globus &lt;math&gt;B&lt;/math&gt; cum certo quodam velocitatis
gradu. Si nullam motui resistentiam afferret globus &lt;math&gt;A&lt;/math&gt;, eadem velocitate pergeret moveri &lt;math&gt;B&lt;/math&gt;, qua movebatur antea , secum pertrahendo globum &lt;math&gt;A&lt;/math&gt;: cur enim minueretur motus in &lt;math&gt;B&lt;/math&gt;, cum globus &lt;math&gt;A&lt;/math&gt; nihil obstaret illius motui , et ipse loco suo facile cederet? Iamvero si experientiam consulimus, multo aliter rem evenire comperiemus: cedit quidem loco suo globus &lt;math&gt;A&lt;/math&gt;, sed non sine detrimento motus
in &lt;math&gt;B&lt;/math&gt;, eoque majori quo majorem globus &lt;math&gt;A&lt;/math&gt; opponit massam impellenti se globo &lt;math&gt;B&lt;/math&gt;. Resistere igitur motui , status
que mutationi obniti concipitur &lt;math&gt;A&lt;/math&gt;, acquisitumque motum
resistentia sua destruere in &lt;math&gt;B&lt;/math&gt;. Motus habetur tamquam vis
activae effectus; quod autem vis activae effectum destruit,
potest et ipsum verae vis nomen accipere. In ipsis etiam
corporibus motis sese prodit ejusmodi vis: corpus certo quodam velocitatis gradu donatum, eumdem servabit nisi quem
inveniat obicem , nec ullum sui motus augmentum patietur nisi cum vis alienae in ipsum agentis detrimento; haud
aliter ac restitit primo motui dum quiesceret; ipso in motu resistit majori motui.

Non ergo praefata indifferentia sita est in non renitentia ad motum ex quiete, vel in non renitentia ad quietem ex mota, sed in eo quod corpus de se non magis ad
motum quam ad quietem tendat, nec magis resistat quieti
si fuerit in motu quam molui renitatur si fuerit in quiete.
Quoniam igitur ab ipsa materia nequit oriri ulla de
terminatio ( huc pertinet materiae inertia ) ad novum statum sive quietis, sive motas; profecto deficiente causa quae
materiale punctum determinet ad hunc potius quam ad
illum novum statum, punctum ipsum si in quiete sit quiescet semper, si ad motum semel fuit excitatum perget moveri cum eadem perpetuo velocitate et directione: porro
motus directio est recta linea, quam mobile aut describit,
aut describere nititur; primum obtinet in motibus rectilineis, secundum in curvilineis. 

[[5|5]]. Duo puncta materialia &lt;math&gt;H&lt;/math&gt; et &lt;math&gt;K&lt;/math&gt; ( fig 1. ) eamdem
massam habentia, eamdemque lineam communi vi &lt;math&gt;P&lt;/math&gt; incedentia, haud dubie conjunctim procedent: verum ubi puncto &lt;math&gt;K&lt;/math&gt; praeter &lt;math&gt;P&lt;/math&gt; applicetur et vis &lt;math&gt;Q&lt;/math&gt;, disjungetur illico &lt;math&gt;K&lt;/math&gt;
ab &lt;math&gt;H&lt;/math&gt;, et observator constitutus in &lt;math&gt;H&lt;/math&gt; deprehendet: motum puncti &lt;math&gt;K&lt;/math&gt; perinde ac deprehenderet si &lt;math&gt;H&lt;/math&gt; quiesceret et &lt;math&gt;K&lt;/math&gt;
moveretur sola &lt;math&gt;Q&lt;/math&gt;: sive nimirum ponatur &lt;math&gt;H&lt;/math&gt; moveri vi &lt;math&gt;P&lt;/math&gt;
et &lt;math&gt;K&lt;/math&gt; viribus &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt;, sive &lt;math&gt;H&lt;/math&gt; quiescere et &lt;math&gt;K&lt;/math&gt; moveri unica
&lt;math&gt;Q&lt;/math&gt;, idem in utroque casu, experientia teste , prodibit motus puncti &lt;math&gt;K&lt;/math&gt; quoad &lt;math&gt;H&lt;/math&gt;: huc spectat principium motus
relativi . Jamvero in secundo casu motus relativus soli &lt;math&gt;Q&lt;/math&gt; est
manifeste, adscribendus; idipsum ergo dicendum et in primo. Effectus videlicet a nova vi &lt;math&gt;Q&lt;/math&gt; genitus in puncto materiali &lt;math&gt;K&lt;/math&gt; idem est utcumque caeteroqui se habeat praecedens status ipsius &lt;math&gt;K&lt;/math&gt;: quod consequi videtur ex materiei
inertia. Etenim si variato statu praecedente variaret effectus ille, non aeque se haberet materia ad status omnes ,
punctumque materiale sibi commissum rediret tandem in
statum illum , ad quem magis tendit; sicque ab ipsa materia oriretur determinatio ad novum statum.

[[6|6]]. Exhibeant
&lt;math&gt;v&lt;/math&gt; et
&lt;math&gt;v^\prime&lt;/math&gt; velocitates, quas gignunt vires
&lt;math&gt;P&lt;/math&gt; et
&lt;math&gt;Q&lt;/math&gt;, sitque
&lt;math&gt;u&lt;/math&gt; velocitas
, quam generat vis assumpta pro
communi mensura (3) ipsarum
&lt;math&gt;P&lt;/math&gt; et
&lt;math&gt;Q&lt;/math&gt;; erunt (5)
&lt;math&gt;v = Pu,
v^\prime = Qu&lt;/math&gt;, unde:
&lt;math&gt;v:v^\prime=Pu: Qu=P: Q&lt;/math&gt;.
Permanente videlicet massa, vires erunt ut simplices velocitates: et quoniam permanente velocitate et variata massa, vis est ut massa ipsa; inferimus vires esse ut motus quantitates.

[[7|7]]. Dixi ([[4]]) tantam motus quantitatem excitari in globo
&lt;math&gt;A&lt;/math&gt; quantam ipse
&lt;math&gt;A&lt;/math&gt; resistendo destruit in globo
&lt;math&gt;B&lt;/math&gt;: atque huc spectat illud de actione et reactione principium,
quod sic enunciari solet &quot;actioni contraria semper et aequalis est reactio, sive duorum corporum actiones in se mutuo semper sunt aequales, et in contrarias partes diriguntur&quot;. Huic autem principio locus est in rerum natura sive corpora in contactu agant in se mutuo, sive dissitis e locis sese invicem ad status mutationem quocumque modo determinent.

Notetur illud: cum corpus omne obnitatur semper sui statos mutationi, inferimus ipsam status mutationem haud repente gigni a viribus extrinsecis, sed per gradus indefinitae attenuationis capaces: secus enim dicendum foret inesse materiei vim quamdam infinitam. Siquidem in hypothesi finitae mutationis instantaneae materies valeret opponere resistentiam finitam, labente tempusculo infinite quod nequit admitti. Verum quia vires quaedam tam cito gignunt mutationem status, ut eam in istanti videantur absolvere; inde fit ut vires dividi soleant in instantaneas, et continuas.

=== De virium compositione et resolutione, deque earum momentis et aequilibrio: aliquid quoque notatur de vecte, axe in peritrochio , trochlea etc. . . . ===
[[8|8]]. Fac ut per communem vim &lt;math&gt;P&lt;/math&gt; puncta &lt;math&gt;H&lt;/math&gt; et &lt;math&gt;K&lt;/math&gt; (fig. 2.)
determinentur ambo ad percurrendam motu uniformi rectam lineam &lt;math&gt;AB&lt;/math&gt; intra tempus &lt;math&gt;t&lt;/math&gt; , per &lt;math&gt;Q&lt;/math&gt; vero determinetur
&lt;math&gt;K&lt;/math&gt; ad percurrendam motu pariter uniformi rectam lineam 
&lt;math&gt;AD&lt;/math&gt; intra idem tempus &lt;math&gt;t&lt;/math&gt; ; et comple parallelogrammum &lt;math&gt;BD&lt;/math&gt;.
Ex principio motus relativi punctum &lt;math&gt;K&lt;/math&gt; in fine temporis &lt;math&gt;t&lt;/math&gt;
reperietur in &lt;math&gt;C&lt;/math&gt; ; ac proinde intra tempus &lt;math&gt;t&lt;/math&gt; percurret motu
uniformi diagonalem &lt;math&gt;AC&lt;/math&gt; : idem nimirum existet motus sive mobile feratur per diagonalem &lt;math&gt;AC&lt;/math&gt; velocitate
&lt;math&gt;\frac{AC}{t}&lt;/math&gt; ex vi 
unica impressa &lt;math&gt;R&lt;/math&gt;, sive conjunctis viribus &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; impellatur per latera &lt;math&gt;AB&lt;/math&gt; et &lt;math&gt;AD&lt;/math&gt; velocitatibus &lt;math&gt;\frac{AB}{t}&lt;/math&gt; et &lt;math&gt;\frac{AD}{t}&lt;/math&gt;; eritque (6)

&lt;math&gt;
R : P : Q : =AC: AB: AD.
&lt;/math&gt;

Hinc pro duabus viribus &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; poterit, substitui vis
&lt;math&gt;R&lt;/math&gt;; quae substitutio dicitur virium compositio : et viceversa
pro &lt;math&gt;R&lt;/math&gt; poterunt substitui duae &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt;; quae substitutio dicitur virium resolutio : &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; vocantur componentes, &lt;math&gt;R&lt;/math&gt;
resultans, vel etiam composita.

[[9|9]]. Haec notentur. 
1º. ex tribus &lt;math&gt;R&lt;/math&gt; , &lt;math&gt;P&lt;/math&gt; , &lt;math&gt;Q&lt;/math&gt; unaquae
vis potest repraesentari per sinum anguli, qui sub aliarum
directionibus continetur ; nam
&lt;math&gt;
R : P : Q = AC : DC: AD = \sin BAD : \sin CAD : \sin BAC .
&lt;/math&gt;

2°. Hinc &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; sunt reciproce ut perpendicula ,
quae a puncto quolibet resultantis &lt;math&gt;R&lt;/math&gt; ducuntur ad ipsarum
&lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; directiones .

3º. Denotante &lt;math&gt;i&lt;/math&gt; angulum interceptum directionibus
virium &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt;, triangulum &lt;math&gt;A C D&lt;/math&gt; praebebit

&lt;math&gt;
RP = P^2 + Q^2 - 2PQ \cos(180^{\circ} - i) = P^2 + Q^2 + 2PQ \cos i.
&lt;/math&gt;

4°. Si punctum &lt;math&gt;K&lt;/math&gt; ( fig. 3. ) urgetur viribus &lt;math&gt;KA, KB, KC, KD&lt;/math&gt; etc. . . , ducantur autem &lt;math&gt;Aa&lt;/math&gt; parallela et
&lt;math&gt;= KB&lt;/math&gt; , &lt;math&gt;Aa'&lt;/math&gt; &lt;math&gt;Aa''&lt;/math&gt; parallela et &lt;math&gt;= KC&lt;/math&gt; , &lt;math&gt;a'' a''' &lt;/math&gt; parallela et&lt;math&gt; =
KD&lt;/math&gt; , etc. vis cunctis aequivalens exhibebitur manifeste per lineam rectam &lt;math&gt;Ka'''&lt;/math&gt;, quae jungit punctum &lt;math&gt;K&lt;/math&gt;
et extremitatem &lt;math&gt;a'''&lt;/math&gt; ultimae &lt;math&gt;a''a'''&lt;/math&gt; . Porro linearum rectarum aequalium et parallelarum projectiones sive in recta
quavis &lt;math&gt;EE'&lt;/math&gt;, sive in plano quovis , sunt aequales et parallelae: hinc virium &lt;math&gt;KA, KB, KC, KD&lt;/math&gt;, etc. . . projectiones in recta &lt;math&gt;EE'&lt;/math&gt; simul sumptae aequabuntur projectionibus rectarum &lt;math&gt;KA, Aa', a'a'', a'' a'''&lt;/math&gt; etc. , in eadem &lt;math&gt;EE'&lt;/math&gt; simul pariter sumptis. Harum vero projectionum summa nihil est aliud nisi projectio resultantis &lt;math&gt;Ka'''&lt;/math&gt; : igitur projectio resultantis aequabitur projectionibus componentium &lt;math&gt;KA, KB, KC, KD&lt;/math&gt;, etc. , in summam contractis , si modo habeatur ratio signorum, ut censeantur negativae, quae vergunt v. gr. ad &lt;math&gt;E&lt;/math&gt;, habitis pro positivis, quae versus &lt;math&gt;E'&lt;/math&gt; se dirigunt.

5°. In hypothesi trium duntaxat virium &lt;math&gt;KA, KB , KC&lt;/math&gt;, quisque videt aequipollentem vim repraesentatum iri
per diagonalem parallelepipedi sub lateribis &lt;math&gt;KA, KB, KC&lt;/math&gt;.

6°. Si punctum &lt;math&gt;K&lt;/math&gt; urgetur vi &lt;math&gt;Ka'''&lt;/math&gt;, constructo ad
libitum polygono &lt;math&gt;a''' a'' ... K&lt;/math&gt;, ductaque &lt;math&gt;KD&lt;/math&gt; parallela et &lt;math&gt;=a''' a''&lt;/math&gt; , &lt;math&gt;KC&lt;/math&gt; parallela et &lt;math&gt;= a'' a'&lt;/math&gt;, &lt;math&gt;KB&lt;/math&gt; parallela et &lt;math&gt;= a' A&lt;/math&gt; etc.
resolvetur &lt;math&gt;Ka'''&lt;/math&gt; in &lt;math&gt;KD, KC, KB&lt;/math&gt;, etc ....

7°. Ad resolvendam &lt;math&gt;Ka'''&lt;/math&gt; in ternas sese dirigentes
juxta datas rectas &lt;math&gt;KB, KC, KD&lt;/math&gt;, satis erit per &lt;math&gt;a'''&lt;/math&gt; ducere
tria plana parallela planis &lt;math&gt;BKC, CKD, BKD&lt;/math&gt;; hoc pacto
exsurget parallelepipedum , cujus latera apud &lt;math&gt;K&lt;/math&gt; exhibebunt
( 5°) quaesitas vires componentes.

8°.  Puncta &lt;math&gt;B, C, D, K&lt;/math&gt;, ponantur inter se rigidis
lineis connexa: manentibus virium directionibus, si ternae
componentes intelliguntur applicitae punctis &lt;math&gt;B, C, D&lt;/math&gt;, adhuc
iis manifeste aequipollebit &lt;math&gt;Ka'''&lt;/math&gt; . Inferimus vim quamvis
&lt;math&gt;Ka'''&lt;/math&gt; resolvi posse in ternas, quae et sint applicitae tribus punctis ad libitum sumptis ( si sumuntur ita , ut in
eorum plano inveniatur etiam punctum &lt;math&gt;K&lt;/math&gt;, non debebit
&lt;math&gt;Ka'''&lt;/math&gt; esse extra id planum ) et sese dirigant juxta rectas ab istiusmodi punctis ductas ad punctum &lt;math&gt;K&lt;/math&gt; , cui applicatur ipsa &lt;math&gt;Ka'''&lt;/math&gt;.

9º. Dato systemate punctorum materialium rigidis
lineis inter se firmiter connexorum ( huc spectat corpus solidum ) respondentibusque viribus sollicitatorum; quia possunt (8º. ) singulae vires resolvi in cernas applicitas tribus
punctis &lt;math&gt;A , B, C&lt;/math&gt; ad libitum sumptis, poterunt ( 4°) omnes
traduci ad aequipollens trium virium systema.

10° . Per unam ex hisce tribus viribus duc planum , quod secet reliquas duas : vis , per quam ducitur planum , poterit resolvi ( 4° ) in binas , applicitas intersectionum punctis. Inde fit, ut vires omnes solidum corpus sollicitantes
traduci etiam possint ad aequipollens duarum virium systema.

[[10|10]]. Facile est determinare quandonam plures potentiae eidem puncto applicitae in aequilibrio permaneant.
Binas potentias pro lubito sumptas compone, et pro illis
aequipollentem substitue , atque id iterato donec ad duas
devenias. Si hae directe contrariae et aequales inveniuntur, constabit omnes potentias in aequilibrio consistere .

Facile etiam intelliges quanam ratione inveniri possit potentia duabus &lt;math&gt;AH, BF&lt;/math&gt; ( fig. 4. ) in eodem plano jacentibus,
rectaeque rigidae &lt;math&gt;AB&lt;/math&gt; applicatis aequivalens, et aequilibrium
obtineri; productis (?) enim directionibus &lt;math&gt;AH, BF&lt;/math&gt; donec concurrant
in &lt;math&gt;C&lt;/math&gt;, transferantur potentiae in punctum &lt;math&gt;C&lt;/math&gt;. Sumptis in earum directionibus &lt;math&gt;CH' = AH&lt;/math&gt;, et &lt;math&gt;CF' = BF&lt;/math&gt;, istae componantur. Facto parallelogrammo &lt;math&gt;CF'KH'&lt;/math&gt;, cujus diameter &lt;math&gt;CK&lt;/math&gt; equivalentem vim repraesentabit, haec producatur donec concurrat in &lt;math&gt;D&lt;/math&gt; cum &lt;math&gt;AB&lt;/math&gt;; perspicuum est potentiam &lt;math&gt;KC&lt;/math&gt; translatam in &lt;math&gt;DL&lt;/math&gt; et rectae &lt;math&gt;AB&lt;/math&gt; applicitam in D aequipollere duabus &lt;math&gt;AH , BF&lt;/math&gt;. Quare si &lt;math&gt;AB&lt;/math&gt; in puncto &lt;math&gt;D&lt;/math&gt; sustentetur, potentiae &lt;math&gt;AH, BF&lt;/math&gt; in aequilibrio quiescent; et constabit quam potentiam exerceat punctum &lt;math&gt;D&lt;/math&gt;, nimirum aequalem et oppositam potentiae aequivalenti &lt;math&gt;DL&lt;/math&gt;. Ad positionem puncti &lt;math&gt;D&lt;/math&gt; quod pertinet, concipiamus ex eo duci duo perpendicula &lt;math&gt;p&lt;/math&gt; et &lt;math&gt;q&lt;/math&gt; , alterum in &lt;math&gt;AH&lt;/math&gt; , alterum
in &lt;math&gt;BF&lt;/math&gt; ; sintque &lt;math&gt;AH = P , BF = Q&lt;/math&gt;, longitudo &lt;math&gt;AB = h ,
AD = x&lt;/math&gt;, angulus &lt;math&gt;BAC =a&lt;/math&gt;, angulus &lt;math&gt;ABC = b&lt;/math&gt; : erunt &lt;math&gt;p =
x \sin a, q = ( h- x ) \sin b &lt;/math&gt;, ideoque &lt;math&gt;\frac p q = \frac{x \sin a}{(h - x ) \sin b}  &lt;/math&gt;
Sed( 9.2º )
&lt;math&gt;\frac p q = \frac Q P &lt;/math&gt;;
igitur
&lt;math&gt; \frac Q P = 
\frac{x \sin a}{
(h- x ) \sin b} &lt;/math&gt;, unde &lt;math&gt; \frac{x }{
(h- x ) } = \frac{Q \sin b}{P \sin a }.   &lt;/math&gt;

Quod spectat ad angulum interceptum resultante &lt;math&gt;CK&lt;/math&gt; et
data recta &lt;math&gt;AC&lt;/math&gt; , is dicatur &lt;math&gt;\alpha&lt;/math&gt; : erit ( 9. 1º )
&lt;math&gt;P : Q = \sin BCD : \sin ACD= \sin ( 180^{\circ}- a - b- \alpha) : \sin \alpha&lt;/math&gt;,
unde

&lt;math&gt;\tan \alpha =\frac{
Q \sin ( a + b )}{
P - Q \cos ( a + b )}.&lt;/math&gt;

Quod vero spectat ad resultantem
&lt;math&gt;CK ( = R )&lt;/math&gt; , habemus ( 9. 3º )
&lt;math&gt;R^2 = P^2 + Q^2 - 2P Q\cos ( a + b )&lt;/math&gt;.

Penultima formula traduci potest ad

&lt;math&gt;\cos \alpha = \frac{P - Q \cos ( a + b )}{
R}.   &lt;/math&gt;

Haec subjungimus. 1º. Recta &lt;math&gt;AB&lt;/math&gt; rotetur circa &lt;math&gt;D&lt;/math&gt;, ut ejus extrema puncta &lt;math&gt;A&lt;/math&gt; et &lt;math&gt;B&lt;/math&gt; eodem tempusculo infinitesimo
describant circulares arcus infinitesimos &lt;math&gt;Aa', Bb'&lt;/math&gt;; ex &lt;math&gt;a'&lt;/math&gt; et &lt;math&gt;b'&lt;/math&gt;
duc perpendicula &lt;math&gt;a'a'', b'b''&lt;/math&gt; in directiones virium &lt;math&gt;AH ,
BF&lt;/math&gt; ; sintque &lt;math&gt;Aa'' = p' , Bb'' = q'&lt;/math&gt;: erunt &lt;math&gt;p' = Aa' \cos a'Aa''
= Aa' \cos ( DAa'' - 90^{\circ} ) = Aa' \sin DAa'' = Aa' \sin a ,
q'= Bb'\cos b'Bb'' = Bb'\cos (90^{\circ}-b) = Bb'\sin b&lt;/math&gt;; et consequenter
&lt;math&gt;\frac{p'}{q'}= \frac{Aa' \sin a}{
Bb' \sin b}=
\frac{AD \sin a}{BD \sin b} = \frac{x \sin a}{(h - x ) \sin b} =
\frac{Q}{P} .&lt;/math&gt;

Nihil sunt aliud &lt;math&gt;Aa'&lt;/math&gt; et &lt;math&gt;Bb'&lt;/math&gt; nisi spatiola tempusculo infinitesimo circa immobile punctum &lt;math&gt;D&lt;/math&gt; simul describenda ab &lt;math&gt;A&lt;/math&gt;
et &lt;math&gt;B&lt;/math&gt; in hypothesi turbati aequilibrii; quibus punctis &lt;math&gt;A&lt;/math&gt; et &lt;math&gt;B&lt;/math&gt;
applicantur vires &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt;: exhibent &lt;math&gt;p', q'&lt;/math&gt; illorum spatiolorum projectiones super ipsarum virium directionibus. Vires igitur &lt;math&gt;P, Q&lt;/math&gt; sese mutuo librantes circa &lt;math&gt;D&lt;/math&gt; erunt reciproce ut eae projectiones.

2º. Etiam sic : triangula &lt;math&gt;Aa'a'', DAh&lt;/math&gt; , itemque
&lt;math&gt;Bb'b'', DBh'&lt;/math&gt; habent latera sibi respective perpendicularia ; igitur &lt;math&gt;\frac {DA} {Aa'} = \frac{p}{p'} , \frac{DB} {Bb'} = \frac{q}{q'}&lt;/math&gt;.

Denotet &lt;math&gt;i&lt;/math&gt; valorem rationum aequalium &lt;math&gt; \frac{DA}{Aa'} , \frac {DB}{Bb'}   &lt;/math&gt;, projectio insuper &lt;math&gt;p'&lt;/math&gt; computata in ipsa directione respondentis potentiae &lt;math&gt;P&lt;/math&gt; censeatur positiva; projectio vero &lt;math&gt;q'&lt;/math&gt; computata in directione contraria illi , quam obtinet respondens  potentia &lt;math&gt;Q&lt;/math&gt; , censeatur negativa: erunt
&lt;math&gt;p = ip' , q = -
iq'&lt;/math&gt; ;
propterea
&lt;math&gt; \frac QP = \frac pq = -\frac{ip'}{iq'}= -\frac{p'}{q'}
, Pp' + Qq' = 0 .&lt;/math&gt; 

Huc spectat principium velocitatum &lt;u&gt;virtualium&lt;/u&gt;.

3º. Ex quovis puncto (&lt;math&gt;  M&lt;/math&gt; ) sive intra , sive extra
angulum &lt;math&gt; ACB&lt;/math&gt; , duc perpendicula &lt;math&gt; p'', q'' , r''&lt;/math&gt;  ad &lt;math&gt; P, Q, R&lt;/math&gt;  ; duc
quoque ab (&lt;math&gt;  M&lt;/math&gt; ) ad &lt;math&gt; C &lt;/math&gt; rectam ( &lt;math&gt; MC = c &lt;/math&gt; ), cui normaliter
insistat alia recta (&lt;math&gt;  E E' &lt;/math&gt; ) transiens per &lt;math&gt; C &lt;/math&gt;: singulis &lt;math&gt; P , Q,
R &lt;/math&gt; resolutis in duas , alteram juxta (&lt;math&gt;  CM &lt;/math&gt; ) , alteram juxta
( &lt;math&gt; EE'&lt;/math&gt;  ), expriment
&lt;math&gt; P\frac{p''}{c},Q\frac{q''}{c},R\frac{r''}{c} &lt;/math&gt; 
componentes juxta (&lt;math&gt;  EE'&lt;/math&gt;  ) . Quoad (&lt;math&gt;M&lt;/math&gt;) situm extra angulum &lt;math&gt;ACB&lt;/math&gt;, primae duae erant conspirantes; quoad (&lt;math&gt;M &lt;/math&gt;)
situm intra &lt;math&gt;ACB&lt;/math&gt; erunt contrariae : cum igitur &lt;math&gt;R&lt;/math&gt; resultet ex &lt;math&gt;P&lt;/math&gt;
et &lt;math&gt;Q&lt;/math&gt;, prodibit ( 9. 4° ) in primo casu
&lt;math&gt; P\frac{p''}{c}+Q\frac{q''}{c}=R\frac{r''}{c} &lt;/math&gt; et consequenter &lt;math&gt;Pp'' + Qq'' = Rr''&lt;/math&gt;,
in secundo.

&lt;math&gt; \pm(P\frac{p''}{c}-Q\frac{q''}{c}) = R\frac{r''}{c} &lt;/math&gt;, ideoque &lt;math&gt; \pm(Pp''-Qq'') = Rr'', &lt;/math&gt; sumptis signis vel superioribus , vel inferioribus , prout 
&lt;math&gt; P\frac{p''}{c} &gt;  &lt;/math&gt; vel &lt;math&gt; &lt;Q\frac{q''}{c} &lt;/math&gt;: rectangula &lt;math&gt; Pp'',Qq'', Rr'' &lt;/math&gt;  dicuntur momenta virium &lt;math&gt;P, Q, R&lt;/math&gt; quoad punctum (&lt;math&gt;M&lt;/math&gt;). Hinc stabilitur
illud: momentum resultantis &lt;math&gt;R&lt;/math&gt; aequatur summae ex momentis componentium &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; si &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; in eamdem plagam circa (&lt;math&gt;M&lt;/math&gt;) nituntur movere puncta , quibus applicitae
sunt; aequatur differentiae si nituntur movere in plagas
contrarias.

4° . Idipsum facile extenditur ad quemvis numerum
virium &lt;math&gt;P, Q, S, V, U&lt;/math&gt;, ... in dato plano jacentium : fac
v. gr. ut ternae &lt;math&gt;P, Q, S&lt;/math&gt;, in unam eamdemque plagam circa ( &lt;math&gt;M&lt;/math&gt; ) nitantur movere puncta, ad quae sunt applicitae; caeterae vero &lt;math&gt;V, U&lt;/math&gt;, ... in plagam contrariam ; sitque &lt;math&gt;L&lt;/math&gt; resultans ex &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt;; &lt;math&gt;N&lt;/math&gt; resultans ex &lt;math&gt;L&lt;/math&gt; et &lt;math&gt;S&lt;/math&gt;, ac proinde ex
&lt;math&gt;P, Q, S&lt;/math&gt;; &lt;math&gt;O&lt;/math&gt; resultans ex reliquis &lt;math&gt;V, U&lt;/math&gt;. . . Erurt
&lt;math&gt;Ll''= Pp'' + Qq'', Nn'' = Ll'' +Ss''&lt;/math&gt; ;
et consequenter
&lt;math&gt;Nn'' = Pp'' + Qq'' + Ss''&lt;/math&gt; :
simili modo obtinetur
&lt;math&gt;Oo'' = Vv'' + Uu''+&lt;/math&gt; .
Iam si &lt;math&gt;R&lt;/math&gt; exhibet resultantem ex &lt;math&gt;N&lt;/math&gt; et &lt;math&gt;O&lt;/math&gt; , ideoque ex &lt;math&gt;P,
Q, S, V , U &lt;/math&gt;, ... ; cum sit 
fist the
&lt;math&gt;Rr'' = \pm ( Nn'' - Oo'' ) &lt;/math&gt;,
erit quoque
&lt;math&gt;Rr'' = ( Pp'' + Qq'' + Ss'' - Vv'' - Uu'' - ... ) .&lt;/math&gt;

5º. Fac ut &lt;math&gt;R&lt;/math&gt; transeat per (&lt;math&gt;M&lt;/math&gt;) ; erit &lt;math&gt;r'' = 0&lt;/math&gt;: propterea
&lt;math&gt;Pp'' + Qq'' + Ss'' - Vv'' - Uu'' - ... = 0&lt;/math&gt; ;
viriumque systema consistet in aequilibrio circa immobile
punctum (&lt;math&gt;M&lt;/math&gt;) . Vocatur (&lt;math&gt;M&lt;/math&gt;) centrum momentorum.

6º. Habemus ( 2 )
&lt;math&gt;p'' = ip' , q'' = iq' , s'' = is' , v'' = -iv', u'' = - iu', ...&lt;/math&gt;
Traducetur igitur aequatio ( 5°) ad
&lt;math&gt;Pp' + Qq' + Ss' + Vv' + Uu' + ... = 0&lt;/math&gt;

7° Vires &lt;math&gt;AH, BF&lt;/math&gt; haud jacentes in eodem plano nequeunt ad unicam vim aequipollentem traduci. Si enim tradu
ad aequipollentem &lt;math&gt;DL&lt;/math&gt;, poterit etiam ex quodam
istius puncto ad quoddam punctum componentis v . gr.
&lt;math&gt;BF&lt;/math&gt; duci recta linea haud occurrens alteri componenti &lt;math&gt;AH&lt;/math&gt;:
fac ut haec recta linea evadat immobilis ; elisa &lt;math&gt;DL&lt;/math&gt;, emerget aequilibrium; sed elisa quoque &lt;math&gt;BF&lt;/math&gt;, et salva &lt;math&gt;AH&lt;/math&gt;, ex hac ultima emerget motus. In ea ergo qua sumus hypothesi de
traductione virium &lt;math&gt;AH, BF&lt;/math&gt; ad unicam &lt;math&gt;DL&lt;/math&gt; obtinebit simul
aequilibrium et motus in eodem systemate: quod nequit esse; ideoque etc. ... 

8°. Patet solidum liberumque corpus haud consistere
in aequilibrio, nisi binae aequipollentes ( 8. 10° ) vires ,
ad quas traducuntur vires omnes corpus ipsum sollicitantes , sint aequales, contrariae, jaceantque in directum .

9°. Patet quoque solidum corpus, mobile dumtaxat
circa punctum fixum , consistere in aequilibrio, si eae binae
vires aequipollentes et jaceant in eodem plano ( 7º) , et suppeditent resultantem , quae transeat per punctum illud .

10°. Solidum corpus ponatur mobile dumtaxat circa
rectam fixam &lt;math&gt;AZ&lt;/math&gt; ( fig.5 ), sintque &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; binae aequipollentes
vires, ad quas traducuntur ( 9. 10° ) vires omnes corpus ipsum sollicitantes. Duc planum &lt;math&gt;XOY&lt;/math&gt; et normaliter insistens
rectae &lt;math&gt;AZ&lt;/math&gt;, et secans in punctis v. gr. &lt;math&gt;B, C&lt;/math&gt; directiones virium &lt;math&gt;P ( = BB' ), Q ( = CC' )&lt;/math&gt;: poterit &lt;math&gt;P&lt;/math&gt; resolvi in duas , alteram &lt;math&gt;BB'''&lt;/math&gt; perpendicularem plano &lt;math&gt;XO&lt;/math&gt;Y , alteram &lt;math&gt;BB''&lt;/math&gt; jacentem in ipso &lt;math&gt;XOY&lt;/math&gt;; similiter &lt;math&gt;Q&lt;/math&gt; poterit resolvi in duas ,
alteram &lt;math&gt;CC'''&lt;/math&gt; perpendicularem eidem &lt;math&gt;XOY&lt;/math&gt;, alteram &lt;math&gt;CC''&lt;/math&gt; in
eo jacentem . Binae &lt;math&gt;BB''', CC'''&lt;/math&gt;, utpote parallelae ad rectam fixam &lt;math&gt;AZ&lt;/math&gt;, peribunt elisae : in ea igitur qua sumus hypothesi
haud consistet solidum corpus in aequilibrio, nisi resultans
ex &lt;math&gt;BB'' , CC''&lt;/math&gt; transeat per aliquod punctum &lt;math&gt;O&lt;/math&gt; rectae fixae
&lt;math&gt;AZ&lt;/math&gt; ; et consequenter ( 9. 2° ) , ductis ex &lt;math&gt;O&lt;/math&gt; in istas vires
perpendicalis &lt;math&gt;b, c&lt;/math&gt;, nisi valeat aequatio

&lt;math&gt;\frac{b}{c} = \frac{CC''}{BB''} &lt;/math&gt;:

producta ex &lt;math&gt;b&lt;/math&gt; in &lt;math&gt;BB''&lt;/math&gt; et ex &lt;math&gt;c&lt;/math&gt; in &lt;math&gt;CC''&lt;/math&gt; dicuntur momenta virium &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; quoad &lt;math&gt;AZ&lt;/math&gt; . Si &lt;math&gt;P&lt;/math&gt; v. gr. , applicita ad punctum
&lt;math&gt;B'&lt;/math&gt;, est parallela plano &lt;math&gt;XOY&lt;/math&gt;, applicabuntur ad &lt;math&gt;B&lt;/math&gt; duae quaelibet vires &lt;math&gt;H &lt;/math&gt; et &lt;math&gt;- H&lt;/math&gt; aequales, contrariae et parallelae axi &lt;math&gt;AZ&lt;/math&gt;; tum una ex iis v. gr. &lt;math&gt;H&lt;/math&gt; componetur cum &lt;math&gt;P&lt;/math&gt; : vis inde
resultans poterit transferri in punctum v. gr. &lt;math&gt;B&lt;/math&gt; plani &lt;math&gt;XOY&lt;/math&gt;,
ibique resolvi in binas, alteram &lt;math&gt;BB''' ( = H )&lt;/math&gt; parallelam rectae &lt;math&gt;AZ&lt;/math&gt;, alteram &lt;math&gt;BB'' ( = P )&lt;/math&gt; jacentem in &lt;math&gt;XOY&lt;/math&gt;; eritque
&lt;math&gt;b. BB ' ( = b. P )&lt;/math&gt; momentum vis &lt;math&gt;P&lt;/math&gt; quoad &lt;math&gt;AZ&lt;/math&gt;. Quisque autem videt , si per &lt;math&gt;B '&lt;/math&gt; ducitur planum parallelum plano &lt;math&gt;XOY&lt;/math&gt;,
et ex pancto, ubi istud novum planum secat rectam &lt;math&gt;AZ&lt;/math&gt;, demittitur perpendiculum in vim &lt;math&gt;P&lt;/math&gt; applicitam ad &lt;math&gt;B '&lt;/math&gt;, ejusmodi
perpendiculum nihil fore aliud nisi &lt;math&gt;b&lt;/math&gt;; ita ut, sive momen
tum sumatur apud planum &lt;math&gt;XOY&lt;/math&gt;, sive apud illud alterum
planum parallelum ipsi &lt;math&gt;XOY&lt;/math&gt;, perinde sit.

[[11|11]]. Fac ut vis ( 10) &lt;math&gt;BF&lt;/math&gt; (fig. 4) revolvatur circa punctum &lt;math&gt;B&lt;/math&gt;,
donec evadat parallela vi &lt;math&gt;AH&lt;/math&gt;; erit &lt;math&gt;a + b = 180^{\circ}&lt;/math&gt;, ideo
que &lt;math&gt;\sin b = \sin (180^{\circ} - a ) = \sin a&lt;/math&gt; si vires ad eamdem
plagam obvertantur ; &lt;math&gt;a + b = 360^{\circ}&lt;/math&gt;, ideoque &lt;math&gt;\sin b =
\sin ( 360^{\circ} - a ) = - \sin a &lt;/math&gt; si ad contrarias plagas. In primo
igitur casu exsistent.

&lt;math&gt;\frac{x}{h-x} = \frac{Q}{P}, x=  \frac{hQ}{P+Q}, R = P + Q , \cos \alpha =\frac{P+Q}{R}=1.&lt;/math&gt;

In secundo

&lt;math&gt;\frac{x}{h-x} = -\frac{Q}{P}, x=  \frac{hQ}{Q-P}, R = \pm(P - Q) , \cos \alpha =\frac{P-Q}{R}=\pm 1.&lt;/math&gt;

valet signum superius ubi &lt;math&gt; P &gt; Q&lt;/math&gt;, inſerius ubi &lt;math&gt;P &lt; Q&lt;/math&gt;;
siquidem &lt;math&gt;P, Q, R&lt;/math&gt; denotant hic virium dumtaxat intensitates.
Inferimus illud; resultans ex duabus parallelis viribus adaequat istarum vel summam, vel differentiam , prout
vel ambae conspirant in eamdem plagam, vel altera in unam
et altera in contrariam plagam; ipsis insuper componentibus
viribus est parallela , et ad eam plagam semper obversa ,
quam respicit major ex componentibus illis ; transit denique
per ejusmodi punctum in directione &lt;math&gt;AB&lt;/math&gt;, quod distet a
punctis applicationis componentium in reciproca earum ratione : istud punctum appellari solet centrum virium parallelarum ; estque invariabile, modo et respectiva virium
positio et ipsarum ratio non mutentur.
Si &lt;math&gt;P = Q&lt;/math&gt;, in secundo casu nulla exsistet resultans.
Non est enim ratio in ea qua sumus hypothesi cur ad
plagam unius potius componentis quam ad alterius componentis plagam sese dirigat resultans. Formulae praebent &lt;math&gt;x= \infty, R =0.&lt;/math&gt;

Etsi vires &lt;math&gt;AH&lt;/math&gt; et &lt;math&gt;BF&lt;/math&gt; (fig.6) parallelae, aequales et contrariae nequeunt librari unica vi , utpote omni resultante destitutae; librabuntur nihilominus duabus aliis viribus &lt;math&gt;AH'&lt;/math&gt; et &lt;math&gt;BF'&lt;/math&gt; parallelis, aequalibus, contrariis, et in plano &lt;math&gt;HABF&lt;/math&gt; iacentibus, dummodo ductis ex &lt;math&gt;A&lt;/math&gt; in &lt;math&gt;BF BF'&lt;/math&gt;
perpendiculis &lt;math&gt;AO&lt;/math&gt; et &lt;math&gt;AO'&lt;/math&gt;, exsistat
&lt;math&gt;BF. AO=BF'. AO'&lt;/math&gt;:
tunc enim , ductis ex &lt;math&gt;B&lt;/math&gt; in &lt;math&gt;AH&lt;/math&gt; et &lt;math&gt;AH'&lt;/math&gt; perpendiculis &lt;math&gt;BC&lt;/math&gt; et
&lt;math&gt;BC'&lt;/math&gt;, ob &lt;math&gt;BF =  AH , BF' = AH' , AO = BC , AO' = BC'&lt;/math&gt;
erit quoque
&lt;math&gt;AH. BC=AH'. BC'&lt;/math&gt;;
et consequenter ( 9. 2°) resultans ex &lt;math&gt;AH&lt;/math&gt; et &lt;math&gt;AH'&lt;/math&gt; sese diriget a puncto &lt;math&gt;A&lt;/math&gt; ad punctum &lt;math&gt;B&lt;/math&gt;, simulque resultans ex &lt;math&gt;BF&lt;/math&gt; et
&lt;math&gt;BF'&lt;/math&gt; sese diriget a puncto &lt;math&gt;B&lt;/math&gt; ad punctum &lt;math&gt;A&lt;/math&gt; ; istiusmodi praeterea resultantes sunt manifeste aequales: iccirco etc. ... Systema itaque virium &lt;math&gt;AH', AF'&lt;/math&gt; aequipollebit systemati virium
&lt;math&gt;AH , AF&lt;/math&gt; ; poteritque alterum ( mutatis ejus directionibus
in contrarias partes ) alteri substitui. Consequitur posse binas vires parallelas, aequales et contrarias transferri ab
una positione ad alteram in proprio ipsarum plano, variata simul virium et magnitudine , et directione ; modo tamen productum ex communi earum valore in mutuam distantiam maneat constans.

[[12|12]]. Sint nunc plures vires parallelae &lt;math&gt;P, P ', P ''&lt;/math&gt;, ... variis
solidi corporis punctis applicitae , quarum aliae conspirent
in unam plagam , aliae in plagam contrariam . Componendo &lt;math&gt;P&lt;/math&gt; v . gr. et &lt;math&gt;P'&lt;/math&gt; in unicam &lt;math&gt;R '&lt;/math&gt;, &lt;math&gt;R'&lt;/math&gt; et &lt;math&gt;P'&lt;/math&gt; in unicam &lt;math&gt;R''&lt;/math&gt; , &lt;math&gt;R''&lt;/math&gt; et &lt;math&gt;P'''&lt;/math&gt;
in unicam &lt;math&gt;R''' &lt;/math&gt;, etc. , ... facile devenies ( 11 ) ad illud : resultans
&lt;math&gt;R&lt;/math&gt; ex pluribus viribus parallelis adaequat differentiam inter
summam conspirantium in unam plagam et summam conspirantium in plagam contrariam ; ipsis insuper componentibus viribus est parallela , et ad eam plagam obvertitur , quam
respicit major ex illis summis .

Hinc si vires tendentes in unam plagam censentur positivae , tendentes vero in plagam contrariam negativae , obtinebit aequatio
&lt;math&gt;R = P + P' + P'' + ... (a )&lt;/math&gt;.
Ad haec : denotantibus (fig .7) &lt;math&gt;A, B, D &lt;/math&gt;, ... puncta , quibus applicantur parallelae vires &lt;math&gt;P , P ', P''&lt;/math&gt;, ... , et &lt;math&gt;AB , BD &lt;/math&gt;. ..
rigidas rectas jungentes puncta illa , cum transeant &lt;math&gt;R ', R '' &lt;/math&gt;, ...
per ejusmodi puncta &lt;math&gt;K , H &lt;/math&gt;, ... , quorum positiones sive in
rectis &lt;math&gt;AB , KD &lt;/math&gt;, ... sive in earum prolongationibus unice pendent a conditionibus
&lt;math&gt;P ' :R'= AK :AB , P'' : R'' = HK : KD,&lt;/math&gt; etc. ... ,

seu

&lt;math&gt;P: P'+P= AK : AB , P'' : P + P' + P'' = HK : KD&lt;/math&gt;, etc. ... ,

devenietur etiam ad illud : in systemate parallelarum viriam
habetur constans et immutabile centrum , per quod semper transit resultans &lt;math&gt;R&lt;/math&gt; , quacumque ceteroqui ratione componentes vires volvantur circa puncta quibus applicitae sunt ,
modo et maneant parallelae , et applicitae iisdem punctis in
iisdem respective distantiis.

[[13|13]]. Ducto quolibet plano &lt;math&gt;MQ&lt;/math&gt;, demittantur in illud ex
punctis &lt;math&gt;A , B , D,&lt;/math&gt; ... perpendicula &lt;math&gt;AM ( =z) , BN ( = z; ) , DQ
( = z''), ...&lt;/math&gt; ; sive ( 12) &lt;math&gt;K , H &lt;/math&gt;, ... sint in rectis &lt;math&gt;AB , KD &lt;/math&gt;, ... .
sive in earum prolongationibus , demittantur quoque in
idem &lt;math&gt;MQ&lt;/math&gt; ex istis punctis perpendicula &lt;math&gt;KL , HO &lt;/math&gt;, ... ; per
ipsa &lt;math&gt;K , H &lt;/math&gt;, ... agantur rectae &lt;math&gt;RS , TU &lt;/math&gt;, ... , prima rectae
MN parallela et perpendiculis &lt;math&gt;AM , BN&lt;/math&gt; occurrens in &lt;math&gt;R , S &lt;/math&gt;,
secunda rectae &lt;math&gt;LQ&lt;/math&gt; parallela et perpendiculis &lt;math&gt;KL , DQ&lt;/math&gt; occurrens in &lt;math&gt;T , U &lt;/math&gt;, etc ... Erunt
&lt;math&gt;AR = MR - AM = KL - z, BS = BN - NS =z' - KL,
DU=UQ-DQ=HO-z'', KT = KL - LT = KL - HO &lt;/math&gt;; etc ....
Jamvero ( 11 )
&lt;math&gt;BS:AR = BK :AK = P : P' ,DU :KT = DH :HK = P + P':P''&lt;/math&gt;,etc ...,
ideoque
&lt;math&gt;AR.P = BS.P', DU.P'' = KT (P + P'), &lt;/math&gt;etc....
Igitur
&lt;math&gt;(KL- z) P = (z' -KL )P',(HO- z'') P'' = (KL-HO)(P + P'),&lt;/math&gt;etc....
unde
&lt;math&gt;KL (P + P') = zP + z'P', HO (P + P + P'' ) = KL (P + P') +
z'' P '' = zP + z' P' +z'' P'',&lt;/math&gt; etc.
seu
&lt;math&gt;KL. R ' = zP + z' P', HO. R'' = zP + z'P' + z''P'' , &lt;/math&gt;etc....

Generatim exhibente &lt;math&gt;z_{\mathrm I}&lt;/math&gt;, perpendiculum ex centro omnium datarum virium parallelarum ductum in &lt;math&gt;MQ&lt;/math&gt; , habebimus

&lt;math&gt;z_{\mathrm I} R = zP + z' P' + z'' P'' + z''' P ''' + ... :&lt;/math&gt;
rectangula &lt;math&gt;z_{\mathrm I} R , zP&lt;/math&gt;, dicuntur momenta virium &lt;math&gt;R , P&lt;/math&gt;, ...
quoad plapum &lt;math&gt;MQ&lt;/math&gt;.

Haec notentur: 1° Etsi non omnia puncta , quibus applicantur parallelae vires &lt;math&gt;P , P', P'' &lt;/math&gt;... sita sunt supra planum &lt;math&gt;MQ&lt;/math&gt;
adhuc tamen algebraica summa rectangulorum sub &lt;math&gt;P , P'&lt;/math&gt; ... et
respondentibus perpendiculis ductis in &lt;math&gt;MQ&lt;/math&gt; ex punctis illis '''adaequabit''' rectangulum sub resultante &lt;math&gt;R&lt;/math&gt; et perpendiculo ducto ex centro ipsarum &lt;math&gt;P, P' , &lt;/math&gt;... in idem &lt;math&gt;MQ&lt;/math&gt;; moto enim &lt;math&gt;MQ&lt;/math&gt; versus ea puncta ita , ut maneat sibi parallelum , atque a primitiva positione recedat intervallo &lt;math&gt;h&lt;/math&gt; , si nova perpendicula exhibentur per &lt;math&gt;k, k', k '', ... k_{\mathrm I}&lt;/math&gt; erunt &lt;math display=''inline''&gt;k = z - h , k' = z'- h , k'' = z'' - h, ...  k_{\mathrm I} = z_{\mathrm I} - h &lt;/math&gt;; hinc &lt;math&gt;(k_{\mathrm I} +h) R = (k + h) P + ( k' + h) P' + (k'' + h ) P'' + &lt;/math&gt;... 

est autem ( 12.''a'') &lt;math&gt;hR =h (P + P' + P'' + ...) = hP + hP' + hP'' + ...&lt;/math&gt;;

igitur

&lt;math&gt;k_{\mathrm I} R = kP +k'P' + k'' P'' + ... &lt;/math&gt;

ubi &lt;math&gt;k, k ', k'', ... k_{\mathrm I}&lt;/math&gt; possunt esse vel positiva , vel negativa.

2° Praeter &lt;math&gt;MQ&lt;/math&gt; seu &lt;math&gt;XOY&lt;/math&gt; ( Fig.8 ) concipiantur duo alia plana &lt;math&gt;XOZ , YOZ&lt;/math&gt;; quod autem in ordine ad &lt;math&gt;XOY&lt;/math&gt; est, sit &lt;math&gt;z, z',... z_{\mathrm I} &lt;/math&gt;, sit &lt;math&gt;x, x',... x_{\mathrm I} &lt;/math&gt; in ordine ad &lt;math&gt;YOZ &lt;/math&gt;, et &lt;math&gt;y, y',... y_{\mathrm I} &lt;/math&gt; in ordine ad &lt;math&gt;XOZ&lt;/math&gt;; qua ratione assequuti sumus
&lt;math&gt;z_{\mathrm I}R=zP+z'P'+z''P'' + ...,&lt;/math&gt;
eadem assequemur (a')
&lt;math&gt;x_{\mathrm I}R=xP+x'P'+x''P'' + ...
y_{\mathrm I}R=yP+y'P'+y''P'' + ...&lt;/math&gt;

3° Si compendii causa per  &lt;math&gt;\Sigma P &lt;/math&gt; exprimitur summa
potentiarum &lt;math&gt;P, P', P'', &lt;/math&gt;
et per &lt;math&gt;\Sigma_x P, \Sigma_y  P, \Sigma_z P &lt;/math&gt; designantur summae rectangulorum sub potentiis et respectivis 
perpendiculis , formulae ( a' ) scribi poterunt in hunc modum ( 12. ''a'')
&lt;math&gt;x_{\mathrm I}\Sigma P = \Sigma_x P, y_{\mathrm I}\Sigma P = \Sigma_y P ,z_{\mathrm I}\Sigma P = \Sigma_z P, &lt;/math&gt;
unde
&lt;math&gt;x_{\mathrm I} = \frac{\Sigma_x P}{ \Sigma P} , y_{\mathrm I}= \frac{\Sigma_y P}{ \Sigma P},z_{\mathrm I}= \frac{\Sigma_z P}{ \Sigma P} &lt;/math&gt;
In hypothesi planorum &lt;math&gt;XOY , XOZ , YOZ&lt;/math&gt; orthogonalium ,
&lt;math&gt;x_{\mathrm I}, y_{\mathrm I}&lt;/math&gt; et &lt;math&gt;z_{\mathrm I}&lt;/math&gt;, erunt orthogonales coordinatae , quibus determinatur positio centri parallelarum virium .

4.° Aequatio
P + P + P + ... ... = o ( a &lt;nowiki&gt;''&lt;/nowiki&gt; )&lt;nowiki&gt;''&lt;/nowiki&gt;
manifeste denotat unam quamvis ex datis potentiis v. gr. P
esse aequalem et contrariam resultanti R, ex reliquis omni
bus P' , P &lt;nowiki&gt;''&lt;/nowiki&gt; , ... Ponamus XOY perpendiculare , et XOZ , YOZ&lt;nowiki&gt;''&lt;/nowiki&gt;
parallela directioni potentiarum ; in hac hypothesi erunt
P et R, directe contrariae si perpendicula x et y spectantia
ad punctum , cui applicalur P , spectent ambo ad centrum
quoque virium p ', P &lt;nowiki&gt;''&lt;/nowiki&gt;, ... , si nempe habeantur&lt;nowiki&gt;''&lt;/nowiki&gt;
x R , = x'P' + x &lt;nowiki&gt;''&lt;/nowiki&gt; P t ... ,&lt;nowiki&gt;''&lt;/nowiki&gt;
y R, =ÝP' +y&lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; + .
seu , ob R,
x P + x' P ' + x &lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; +
yP + ' P ' + y &lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; +
-P
= 0,
0; }(cm

5. ° Quibus positis , stabilitur illud : parallelarum
virium systema consistit in aequilibrio sub duabus conditio
nibus simul explendis ; altera est , ut evanescat earum sum
ma : altera ut evanescat summa ex earum momentis in ordi
ne ad duo plana ipsis viribus parallela. Si parallelae vires
inveniuntur omnes in eodem plano, secunda conditio jam de
18
tur summae rectangulorum sub potentiis et reSpectivis perpen-
diculis, formulae (a') scribi poterunt in hunc modum (12. a)
a:, EP :ZxP,y,ZxP: ZJP, z.l 2P:ZzP,
unde u
∙∙∙ zxp ∙∙ \sum∫ M) (0 )
∙−− ⋅ −\sum−⇂⋅−↗∫≖ \sum⇂≀ .z,--—— ZP
ln hypothesi planorum XOT, XOZ , TOZ orthogonalium ,
x, ,y, , et 2! erunt orthogonales coordinatae, quibus deter-
minatur positio centri parallelarum Vtrium.
43 Aequatio
P gr ≖⋡⋅−⊦∙∙⋅−−∙∶∘ (a&lt;nowiki&gt;'''&lt;/nowiki&gt;)&lt;nowiki&gt;'''&lt;/nowiki&gt;
manifeste denotat uuam quamvis-ex datis potentiis v. gr. P
esse aequalem et contrariam resultanti R, ex reliquis omni-
bus P', P&lt;nowiki&gt;''&lt;/nowiki&gt;, Ponamus XOV perpendiculare , et XOZ ,&lt;nowiki&gt;''&lt;/nowiki&gt;
ïOZ
parallela directioni potentiarum; in hac hypothesi erunt
P et B[ directe contrariae si perpendicula x et y spectantia
ad punctum , cui applicatur P ,
spectent ambo ad centrum
quoque virium P',P&lt;nowiki&gt;''&lt;/nowiki&gt;, , si nempe habeantur&lt;nowiki&gt;''&lt;/nowiki&gt;

&lt;nowiki&gt;::&lt;/nowiki&gt; Bl :x'F—I-x&lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; ⊣−∙∙∙∙ Ja, ∶−−∫∣⊉≀−⊢∜∣∣≖≻∥−⊢∙∙∙∙

seu,ob B' :—P,
xP—- x'P'-- x&lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; −∙∙ :o,
yp ——y' PI ...—7&lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; −∙∙ ∙∙∙ :0' )
(a&lt;nowiki&gt;''&lt;/nowiki&gt;)
5.&lt;nowiki&gt;''&lt;/nowiki&gt; Quibus positis
, stabilitur illud : parallelarum
virium systema consistit in aequilibrio sub duabus conditio-
nibus simul explendis; altera est
, ut evanescat earum sum-
ma :altera ut evanescat summa ex earum momentis in ordi-
ne ad duo plana ipsis viribus parallela. Si parallelae vires
inveniuntur omnes in eodem plano, secunda conditio iam de19
9
se explebitur quoad istud planum , satisque erit ut explea
tur quoad aliud tantummodo planum .
6. Etsi vires P, P' , P &lt;nowiki&gt;''&lt;/nowiki&gt;, ... non sunt parallelae , pos
sunt tamen reduci ad terna ejusmodi systemata , quorum pri.
mum coalescat ex viribus parallelis axi OZ , secundum ex
viribus jacentibus in plano XOY simulque parallelis axi OY ,
tertium ex viribus agentibus juxta axem OX. 

Ut demonstretur assertio , resolve unam quamvis ex datis viribus v. gr.

P applicitam puncto A in tres X, Y, Z, respective parallelas
axibus Ox, OY, OZ; ad punctum A applica duas vires H et
- H aequales , contrarias , et parallelas axi OZ ; compone
X ( = AC ) et H sese dirigentem juxta AE , sitque AB dire
ctio resultantis ; produc BA donec in N occurrat plano XOY ;
transfer in N resultantem illam , et sic translatam resolve in
binas , alteram parallelam axi OX , alteram axi OZ ; prodi
bunt componentes X ( = NC = AC ) , H ( = ND ) , qua
rum primam transfer in C ut sit C'C' ( = NC' ) = X; ad C
applica binas vires K et — K aequales , 'contrarias et pa
rallelas axi OY ; compone X ( = .CC ') et K sese dirigen
tem juxia C'F , sitque C'L directio resultantis ; produc LC
donec in V occurrat axi OX ; transfer in V novam istam re
sultantem , et sic translatam resolve in binas , alteram juxta
ox , alteram parallelam axi OY ; emergent componentes
X ( = VV' = CC&lt;nowiki&gt;''&lt;/nowiki&gt; ) , K ( = VF '): compone nunc Y et - H ;
produc directionem resultantis donec rectae C' F occurrat
v . gr. in N ' ; hanc resultantem transfer in N ' , et sic traus
latam resolve in duas , alteram parallelam axi OY , alteram
axi OZ ; exurgent componentes Y et -H applicitae puucto N:
hoc pacto vi P poterunt substitui sex vires
Z, H, — H applicitae punctis A, N, N' et parallelae axi Oz,
K, Y - K applicitae punctis V, C' et parallelae axi OY ,
X applicita puncto V et agens juxta OX .
Consimiles operationes cum possint instaurari quoad P', P ” ...
non pluribus opus est , at pateat veritas assertionis .
19
se explebitur quoad istud planum , satisque erit ut explea-
tur quoad aliud tantummodo planum .
6.o Etsi vires P, P', P&lt;nowiki&gt;''&lt;/nowiki&gt;, non sunt parallelae ,pos-
sunt tamen reduci ad terna eiusmodi systemata
, quorum pri-
mum coalescat ex viribus parallelis axi OZ , secundum ex
viribus jacentibus in plano XOT simulque parallelis axi Oï
,
tertium ex viribus agentibus juxta axem OX. Ut demon-
stretur assertio , resolve unam quamvis ex datis viribus v. gr.
P applicitam puncto A in tres X, T, Z, respective parallelas
axibus OX, OT, OZ; ad punctum A applica duas vires H et
∙∙∙ H aequales , contrarias ,
et parallelas axi OZ ; compone
X (: AC) et H sese dirigentem iuxta AE .sitque AB dire-
ctio resultantis; produc BA donec in N occurrat planc XOT;
transfer in N resultantem illam
,
et sic translatam resolve in
binas , alteram parallelam axi OX , alteram axi OZ; prodi-
bunt componentes X (: NC':AC )
, H (: ND) , qua-
rum primam transfer in C' ut sit C'C&lt;nowiki&gt;''&lt;/nowiki&gt; (: NC' :) X; ad C'
applica binas vires K et —K aequales ,
'contrarias et parallelas axi OV; compone X (:.C' C&lt;nowiki&gt;''&lt;/nowiki&gt;) et K sese dirigen-&lt;nowiki&gt;''&lt;/nowiki&gt;
tem juxt'a C'F , sitque C'L directio resultantis ; produc LC'
donec in V occurrat axi OX ; transfer in V novam istam re-
sultantem ,
et sic translatam resolve in binas , alteram juxta
OX, alteram parallelam axi OV ; emergent componentes
X (:VV':C' C&lt;nowiki&gt;''&lt;/nowiki&gt;) ,K (:VF'): compone nunc V et —H;
produc directionem resultantis donec rectae C' F occurrat
v. gr. in N'; hanc resultantem transfer inN' , et sic traus-
latam resolve' tn duas
, alteram parallelam axi OV, alteram
axi OZ ; exurgeut componentes ?et —H applicitae puncto N':
hoc pacto vi P poterunt substitui sex vires
Z,,H — H applicitae punctis A, N, Net parallelae axi OZ,
K, ï— K applicitae punctis V, C' et parallelae axi OV,
X applicita puncto V et agens juxta OX.
Consimiles operationes cum possint instaurari quoadP' ,P&lt;nowiki&gt;''&lt;/nowiki&gt;,... non pluribus opus est
,
ut pateat veritas assertionis.20
7. Axes OX , OY, OZ sumantur orthogonales ; erit
H : X = ND : NC' NC zX Z :
H ,
et consequenter perpendicula ducta ex N in plana YOZ , XOZ
exprimentur per
2X
H
g ;
erit quoque
H : Y = AC ' : C'N ' = 2 : C'N' =
2Y
H
ac proinde perpendicula ducta ex N' in eadem plana YOZ ,
XOZ exprimentur per
x18+1;
insuper
Vi : Ci = VV' : VF' , seu x - OV : y = X , K ,
ex qua eruitur perpendiculum ductum ex Vin planum YOZ,
nempe
OV =
y X
K
8 . '* Quod in ordine ad Pest X, Y, Z, H, K, sit X ', Y ,
Z ', H , K ' in ordine ad P ', sit X ”, Y &lt;nowiki&gt;''&lt;/nowiki&gt;, Z&lt;nowiki&gt;''&lt;/nowiki&gt;, H ” , K &lt;nowiki&gt;''&lt;/nowiki&gt; in or&lt;nowiki&gt;''&lt;/nowiki&gt;
dine ad P, etc. ... Systema ( 6&lt;nowiki&gt;''&lt;/nowiki&gt;) virium parallelarum axi OZ
consistet in aequilibrio sub tribus istis conditionibns ( 59)
2 + Z ' + Z &lt;nowiki&gt;''&lt;/nowiki&gt; +... + H + HP + H &lt;nowiki&gt;''&lt;/nowiki&gt; + .- H - H²- H &lt;nowiki&gt;''&lt;/nowiki&gt; -... = 0 ,
x2+x+2 + .. + ( x -7 ) +la
ZX
H
- ) H +
'
x H - X'H '-...
20
7
∙∘∙ Axes OX, 07, OZ sumantur orthogonales ;erit
H:X:ND:NC': -Nc': f—X
...-7
et consequenter perpendicula ducta ex N in plana ïOZ, XOZ
exprimentur per
zX x——s.7-i
eritquoque
H. r:.tcx ea:: aut: 2?
—, H
ac proinde perpendicula ducta ex N' in eadem plana TOZ,
XOZ exprimentur per
T
xsf'l'ïïi—i insuper
Vi:C'i:VV':VF',senx—OV:J:X,K. ↴
ex qua eruitur perpendiculum ductum ex Vin planum TOZ, )-
nempe )
∘∇∶∙≖−⋅\sum⋮∙ K
8. 01: Quod in ordine adPestX, T, Z, H, K, sit X'.ï',
Z', H', K' in ordine ad P', sit X&lt;nowiki&gt;''&lt;/nowiki&gt;, T', Z&lt;nowiki&gt;''&lt;/nowiki&gt;, H&lt;nowiki&gt;''&lt;/nowiki&gt;, K&lt;nowiki&gt;'''&lt;/nowiki&gt;m or-
dine ad P&lt;nowiki&gt;''&lt;/nowiki&gt; ,
etc.. «Systema (60) virium parallelamm axi OZ
consistet in aequilibrio sub tribus istis conditionibus (50)
z −⊦ ⊠∣⊣−≀∥⊹∙∙∙−⊦∐⊣−∐∣⊣−∐∥−⊦ ∙∙⋅− ⊟∙↧∓∣∙⊟∥∙∙∙∙ :
xZ-l—x'ZH—&lt;nowiki&gt;''&lt;/nowiki&gt;xl-(x- fl—iï' H—1-(x'
-
)&lt;nowiki&gt;''&lt;/nowiki&gt;IX, H'—)- ..

&lt;nowiki&gt;:&lt;/nowiki&gt; r H—x'H'-.. .
&lt;nowiki&gt;:&lt;/nowiki&gt; o.21

y2 +y2 + ... + 38+y'! '+ ..- ( o + #) : -
(-+ -+* ) r -...--.
seu
2 + 2 + Z&lt;nowiki&gt;''&lt;/nowiki&gt; + ... = 0 ,
x2–2x + x2–5x' + x Z&lt;nowiki&gt;''&lt;/nowiki&gt; _z&lt;nowiki&gt;''&lt;/nowiki&gt; X &lt;nowiki&gt;''&lt;/nowiki&gt; + ... = o,
y2 - zY + y'Z' — zY + y&lt;nowiki&gt;''&lt;/nowiki&gt; Z&lt;nowiki&gt;''&lt;/nowiki&gt; —z&lt;nowiki&gt;''&lt;/nowiki&gt; Y&lt;nowiki&gt;''&lt;/nowiki&gt; + ... :. =0.
360&lt;nowiki&gt;''&lt;/nowiki&gt;

Systema (69) coalescens ex viribus jacentibus in plano XOY
simulque parallelis axi OY consistet in aequilibrio 

sub duabus istis conditionibus ( 5° ) . 

Y - K + Y - K + Y&lt;nowiki&gt;''&lt;/nowiki&gt; _K&lt;nowiki&gt;''&lt;/nowiki&gt; + . + K + K + K + ... = a,
2{Y -K)+7 (9 –K)+- + (3 - X) +(37 )K +
seu
Y + Y + Y &lt;nowiki&gt;''&lt;/nowiki&gt; + ... = 0,
xY4yX + x'Y' — y'X ' + x &lt;nowiki&gt;''&lt;/nowiki&gt; Y&lt;nowiki&gt;''&lt;/nowiki&gt; -- y&lt;nowiki&gt;''&lt;/nowiki&gt; X &lt;nowiki&gt;''&lt;/nowiki&gt; +... =0.
0.}10&lt;nowiki&gt;''&lt;/nowiki&gt;)
Systema ( 6°) virium agentium juxta OX 

consistet in aequilibrio sub ista tantum conditione

X + X+X&lt;nowiki&gt;''&lt;/nowiki&gt;+... = o ( a &lt;nowiki&gt;''&lt;/nowiki&gt; &lt;nowiki&gt;''&lt;/nowiki&gt; ).
Inferimus solidum liberumque corpus viribus P , P' ,
P&lt;nowiki&gt;''&lt;/nowiki&gt; , ... sollicitatum haud mansurum in aequilibrio, nisi ex
pletis conditionibus ( a' ) , ( a &lt;nowiki&gt;''&lt;/nowiki&gt; ) , ( a &lt;nowiki&gt;''&lt;/nowiki&gt; &lt;nowiki&gt;''&lt;/nowiki&gt; ); quas ita scri
bimus ( 30 )
21
⊺∄⊹↗⋅∄∣−⊢∙∙∙⊹∫∐⊹∫∣∐∣⊹∙⋅∙− (
∫⊹.äï.) H —
(y'-]- ⋮⋅≨⋚∣⇀∙≻ H'—. ..

&lt;nowiki&gt;:&lt;/nowiki&gt; o,

seu
∅⊣−∅∣⊣−⊈∥⊹∙∙∙∶∘∙ ;
(a') xZ—zX-I-x'Z'— z'X'-l-x&lt;nowiki&gt;''&lt;/nowiki&gt;Z&lt;nowiki&gt;''&lt;/nowiki&gt;— z'X&lt;nowiki&gt;''&lt;/nowiki&gt;—-]-. ..

&lt;nowiki&gt;:&lt;/nowiki&gt; o,

yz -— zV-l—J'Z'—z'ï'—I- y&lt;nowiki&gt;''&lt;/nowiki&gt;Z&lt;nowiki&gt;''&lt;/nowiki&gt;—z&lt;nowiki&gt;''&lt;/nowiki&gt;ï&lt;nowiki&gt;''&lt;/nowiki&gt;-l— ..

&lt;nowiki&gt;:&lt;/nowiki&gt;

.
0.
Systema (60) coalescens ex viribus jacentibus in plano XOV
simulque parallelis axi OV consistet in aequilibrio sub dua-
bus istis conditionibus ( 5o )-
r—x-t-x'—x'—l-1z&lt;nowiki&gt;''&lt;/nowiki&gt;—xq-.. —[-K-]-K'-)-K&lt;nowiki&gt;''&lt;/nowiki&gt;—]—. ..

&lt;nowiki&gt;:&lt;/nowiki&gt;

0,
I
'
X
&lt;nowiki&gt;!
IX ∣
..
ï—KH—x (r—x ⊢⊢⋅∙∙−⊢ xli?) x-l-(x ïk.-')K ∙⊦∙∙≔∶⋅∘⋅
seu .
y—I—T-I- ï''&lt;/nowiki&gt;—l— ..

&lt;nowiki&gt;:&lt;/nowiki&gt;

.
0, '
h
0&lt;nowiki&gt;''&lt;/nowiki&gt;)
xï—yX-l—x'ïL-y'X' x&lt;nowiki&gt;''&lt;/nowiki&gt;ï&lt;nowiki&gt;''&lt;/nowiki&gt;-y&lt;nowiki&gt;''&lt;/nowiki&gt; &lt;nowiki&gt;''&lt;/nowiki&gt;
∙⊦∙∙∙∶−−∙ ∙
Systema (60) virium agentium iuxta OX consistet in ae-
quilibrio sub ista tantum conditione '
,x-t—X'-l-X&lt;nowiki&gt;''&lt;/nowiki&gt;-1-...:o (a&lt;nowiki&gt;'''&lt;/nowiki&gt;).
Inferimus solidum liberumque corpus viribus P, P',
P&lt;nowiki&gt;''&lt;/nowiki&gt;, .. . sollicitatum haud mansurum in aequilibrio, nisi ex-
pletis conditionibus (a' )
,
( a&lt;nowiki&gt;''&lt;/nowiki&gt; )
,
(av&lt;nowiki&gt;''&lt;/nowiki&gt; ); quas ita scri-
bimus ( 3&lt;nowiki&gt;''&lt;/nowiki&gt;)22
EX
=
0
, EY
=
0
, E2
=
0
,
}
(
a
&lt;nowiki&gt;''&lt;/nowiki&gt;
)
2
( zYX)
= 0,2
( x2–2X
)
= 0,2
(x2
– zY
)
=
0..
9
'

&lt;nowiki&gt;#&lt;/nowiki&gt; Denotet

R
' resultantem ex viribus primi syste
matis
(
6
°
)
,
R
&lt;nowiki&gt;''&lt;/nowiki&gt; ex viribus secundi
,
R
&lt;nowiki&gt;''&lt;/nowiki&gt; ex viribus tertii&lt;nowiki&gt;''&lt;/nowiki&gt;

&lt;nowiki&gt;;&lt;/nowiki&gt;

erunt
( 12

&lt;nowiki&gt;:&lt;/nowiki&gt;

a
)
R
= EZ
,
R
= EY
,
R
&lt;nowiki&gt;''&lt;/nowiki&gt;
= EX
.
Recta
, in qua agit
R
&lt;nowiki&gt;''&lt;/nowiki&gt;
, occurrit axi OX sub angulo recto

&lt;nowiki&gt;;&lt;/nowiki&gt;

et disignante
r
&lt;nowiki&gt;''&lt;/nowiki&gt; distantiam inter
O et punctum occursus
erit
( 2º
.
7
°
)
.
r
“
R
”
=
x
(Y
—
K
)
+
x'(
Y
’
–
K
”
+ ...
XK
(
s
–
&lt;nowiki&gt;''&lt;/nowiki&gt;
)
k
'
+ ...,ideoque
?&lt;nowiki&gt;''&lt;/nowiki&gt;=
EfxY
-yX
)
.
R
&lt;nowiki&gt;''&lt;/nowiki&gt;
tra
potest
R
'
&lt;nowiki&gt;''&lt;/nowiki&gt; transferri in illud punctum occursus sicque
componi cum
R
&lt;nowiki&gt;''&lt;/nowiki&gt; ut inde obtineatur resultans VR
&lt;nowiki&gt;''&lt;/nowiki&gt;
2
+
R
&lt;nowiki&gt;''&lt;/nowiki&gt;
3.
Iterum
( 9. 9º
. 10
°
.
) patet ergo vires
P
P
'
,
P
'
,
, ...
duci vel ad ternas
, vel ad binas aequipollentes
.
10.
°

&lt;nowiki&gt;#&lt;/nowiki&gt; Recta

, in qua agit
R
'
, occurrit normaliter plano
XOY

&lt;nowiki&gt;;&lt;/nowiki&gt; et designantibus

a
'
,
b
' coordinatas istius occursus
,
erunt
( 2º
. 7º
.
)
a
'
$
(xZ
- zX
)
R
'
6
Egy
Z
-
Y
) 1 R
Occurrent sibi mutuo R’et VR
”
?
+
R
&lt;nowiki&gt;''&lt;/nowiki&gt;
2, ac proinde
jacebunt in eodem plano
, quotiescumque
a
' et
b
' recident
in duas quasvis ex coordinatis illius rectae in qua agit
VR'
2
+
R
'&lt;nowiki&gt;'''&lt;/nowiki&gt;
2

&lt;nowiki&gt;;&lt;/nowiki&gt; propterea

22
ZX:0,Zï:o,ZZ:o,

&lt;nowiki&gt;;&lt;/nowiki&gt; (aVIII)

\sum (xï—ïyX):o.Z(xZ—zX):0,2(yZ—zï): 0.
9. 01: Denotet B' resultantem ex viribus primi syste-
matis '(60 ), B&lt;nowiki&gt;''&lt;/nowiki&gt; ex viribus secundi , B&lt;nowiki&gt;'''&lt;/nowiki&gt; ex viribus tertii;
erunt
( 12. a)
R,:Z Z, B&lt;nowiki&gt;''&lt;/nowiki&gt;:Zï, R&lt;nowiki&gt;'''&lt;/nowiki&gt;:ZX.
Recta, in qua agit R&lt;nowiki&gt;''&lt;/nowiki&gt;, occurrit axi OX sub angulo recto

&lt;nowiki&gt;;&lt;/nowiki&gt;

et disignante r&lt;nowiki&gt;''&lt;/nowiki&gt; distantiam inter
0 et punctum occursus,
ertt∙
(
2
∘
∘
. 7. ).
r&lt;nowiki&gt;''&lt;/nowiki&gt;R&lt;nowiki&gt;''&lt;/nowiki&gt;:x(ï—K)—)—x'(ï'—K')—-)—...-)- (x... JKX.)
K −⊦
(x'-— 2274.) K' −∙⊢∙
∙ .,ideoque r&lt;nowiki&gt;''&lt;/nowiki&gt;-— xwy-FK)

&lt;nowiki&gt;:&lt;/nowiki&gt;

potest B&lt;nowiki&gt;'''&lt;/nowiki&gt; transferri in illud punctum occursus
, sicque
componi cum B&lt;nowiki&gt;''&lt;/nowiki&gt; ut inde obtineatur resultans l/B&lt;nowiki&gt;''&lt;/nowiki&gt;3-l—B'&lt;nowiki&gt;'''&lt;/nowiki&gt;. Iterum (9. 90.100.) patet ergo vires P P', P', , .. .
tra-
duci vel ad ternas, vel ad binas aequipollentes.
↿∘∙∘⋕ Recta, in qua agit B', occurrit normaliter plano
XOT; et designantibus a', b' coordinatas istius occursus,
erunt (20. 70.)
↙⋮∣∙− X(xZ—zX) b' ∙∙∙
\sum (yZ—zï)
B'
' R'
⋅
Occurrent sibi mutuo B' et l/B&lt;nowiki&gt;''&lt;/nowiki&gt;2-)-B&lt;nowiki&gt;'''&lt;/nowiki&gt;2, ac proinde
iacebunt in eodem plano, quotiescumque a' et b' recident
in duas quasvis ex coordinatis illius rectae in qua agit
⇂∕ B&lt;nowiki&gt;''&lt;/nowiki&gt;2-I-B&lt;nowiki&gt;'''&lt;/nowiki&gt;2; propterea23
a ' - p &lt;nowiki&gt;''&lt;/nowiki&gt; : 6 = R : R &lt;nowiki&gt;''&lt;/nowiki&gt;
et consequenter
b' R' + ( r &lt;nowiki&gt;''&lt;/nowiki&gt; – a ' ) R &lt;nowiki&gt;''&lt;/nowiki&gt; = 0 ;
quae , adhibitis substitutionibus, traducitur ad
EXE(yZ — ZY) + EYXzX— « Z ) + EZE (xY yX ) = 0.
Sub hac ilaque conditione occurrent sibi mutuo vires
R' , V R &lt;nowiki&gt;''&lt;/nowiki&gt;2+ R &lt;nowiki&gt;''&lt;/nowiki&gt; ), dabuntque resultantem
VR2+ R &lt;nowiki&gt;''&lt;/nowiki&gt;2 + R &lt;nowiki&gt;'''&lt;/nowiki&gt; a = V (EX)2 + (PY )2+ ( EZ )2.
11 . '* Si nequeunt vires alium gignere motum ni
si circa immobilem axem Oz , quisque videt aequilibrii
conditiones redactum iri ad unicam r ' = 0 , seu ad quar
tam ( a &lt;nowiki&gt;''&lt;/nowiki&gt; ), Ad haec si nequeunt vires alium gignere mo
tum nisi circa immobile punctum 0 , redigentur aequili
brii conditiones ad r&lt;nowiki&gt;''&lt;/nowiki&gt; = 0 , a' = 0,6 = 0 , seu ad quar
tam , quintam et sextam ( a )
12. '* Fac ut duo solida corpora A et B ( Fig. 9) ,
alterum viribus P , P , P &lt;nowiki&gt;''&lt;/nowiki&gt;... sollicitatum , alterum viri
bus Q , , Q &lt;nowiki&gt;''&lt;/nowiki&gt; , ... , sese invicem aeque premendo apud da
lum mutui contactus punctum C maneant in aequilibrio ;
quaeritur istiusmodi pressionis magnitudo w. Duc per C pla
num tangens DD' , cui normaliter insistat recta ECE': de
notent fig, h coordinatas puncti C ; a , á , a &lt;nowiki&gt;''&lt;/nowiki&gt; angulos
interceptos recta CE axibusque orthogonalibus OX , OY ,
OZ ; et quod in ordine ad P' , P' , P &lt;nowiki&gt;''&lt;/nowiki&gt; , ... est X, Y , Z,
á , : , X , Y , Z , X ', . . . sit a , b , c , a , ... A , B ,
C , A ', ... in ordine ad C , Q , ... Pressio agens
versus E resolvetur in ternas
23
a'—- r&lt;nowiki&gt;''&lt;/nowiki&gt;: 6':a&lt;nowiki&gt;'''&lt;/nowiki&gt;: a&lt;nowiki&gt;''&lt;/nowiki&gt;
et consequenter
↘∙∙
b' B&lt;nowiki&gt;'''&lt;/nowiki&gt;—i— ( r&lt;nowiki&gt;''&lt;/nowiki&gt;-—a') B&lt;nowiki&gt;''&lt;/nowiki&gt;: 0

&lt;nowiki&gt;;&lt;/nowiki&gt;

quae , adhibitis substitutionibus, traducitur ad
ZXZUZ—zTH-ZïXzX—xZH-ZZZ (a.-T —JX):o.
Sub hac itaque conditione occurrent sibi mutuo vires
B', l/ B&lt;nowiki&gt;''&lt;/nowiki&gt;2-)- B&lt;nowiki&gt;'''&lt;/nowiki&gt;, dabuntque resultantem
⇂∕↓↖⋅≖−⊦↓⊰⋅⋅≖−⊢∐⋯≖∶ ⇂∕ (mun-)- (zx) ≕⊣−≺ \sum∣∠≻≖∙
11.&lt;nowiki&gt;''&lt;/nowiki&gt;; Si nequeunt vires alium gignere motnm ni-
si circa immobilem axem OZ, quisque videt aequilibrii
conditiones redactum iri ad unicam r&lt;nowiki&gt;''&lt;/nowiki&gt; :o
, seu ad quar-
tam (a'&lt;nowiki&gt;'''&lt;/nowiki&gt; ). Ad haec si nequeunt vires alium gignere mo-
tum nisi circa immobile punctum 0 , redigentur aequili-
brii conditiones ad r&lt;nowiki&gt;''&lt;/nowiki&gt;:0, a':o, b':o, seu ad quar-
tam, quintam et sextam ( am&lt;nowiki&gt;''&lt;/nowiki&gt;)
12.&lt;nowiki&gt;''&lt;/nowiki&gt;: Fac ut duo solida corpora A et B (Fig. 9),
alterum viribus P , P', P&lt;nowiki&gt;''&lt;/nowiki&gt;. .. sollicitatum , alterum viri-
bus Q, Q' , Q&lt;nowiki&gt;''&lt;/nowiki&gt;, .. ., sese invicem aeque premendo apud da-
tum mutui contactus punctum C maneant in aequilibrio;
quaeritur istiusmodi pressionis magnitudo 'a'. Duc per C pla-
num tangens DD', cui normaliter insistet recta ECE': de-
notent f, g ,
]: coordinatas puncti C; at, a', a&lt;nowiki&gt;''&lt;/nowiki&gt; angulos
interceptus recta CE axibusque orthogonalibns OX, Of
, OZ; et quod in ordine ad P'
, P', P&lt;nowiki&gt;''&lt;/nowiki&gt;,
...
est a:, 7, z,
x', . . X,ï, Z, X',. . . sita,b, c, a,... A,. B,
C, A', . . .
in ordine ad Q', Q, . . . Pressio :: agens
versus E resolvetur in ternas24
cosa , cose , a cos &lt;nowiki&gt;''&lt;/nowiki&gt; ,
agens vero versus E resolvetur in ternas
w cos ( 180 ° - « ) = - COS Q, a cos ( 180 ° - = - a coseć,
cos ( 180º – Ø&lt;nowiki&gt;''&lt;/nowiki&gt; ) W cos a :
in primo casu w librat ex hypothesi vires P, P,
in secundo vires Q, C, ... Igitur
EX +w cosa = 0, Erto cosá = 0 , xZ + w cosa &lt;nowiki&gt;''&lt;/nowiki&gt; = 0 ,
Σ Α W cosa = 0 ,
EB - cosa = 0,8C — a cos &lt;nowiki&gt;''&lt;/nowiki&gt; = 0 , E ( «Y -y X ) +
W ( f cos ' - g cosc) =0 ,
ElxZ - 2X ) + o ( f cosc &lt;nowiki&gt;''&lt;/nowiki&gt; — h cosc ) =0 , Ely2 -zY) +
wig cosa &lt;nowiki&gt;''&lt;/nowiki&gt; -hcosé ) =0 ,
(aB - 6A ) - ( fcos - g cos ) = 0 ,E (aC - A )
a ( f cosa &lt;nowiki&gt;''&lt;/nowiki&gt; -hcosa) = 0 , E (6C - cB ) - ( g cosa &lt;nowiki&gt;''&lt;/nowiki&gt; - h cosa') = 0 .
Eliminata , prodibunt undecim aequationes , inde
pendentes ab ipsa a , inter quantitates datas ; quibus ae
quationibus expletis, habebitur aequilibrium , poteritque ab
una quavis ex duodecim praecedentibus erui valor u .
13.0# Solidum corpus sollicitatum viribus , P P ',
P &lt;nowiki&gt;''&lt;/nowiki&gt; , ... delineatur duobus punctis fixis , sumptis in axe
v. gr. OZ ; sic facile determinabuntur pressiones M, N , L
et M ', N ', L' exercitae in puncta illa juxta coordinatos a.
xes Ox , OY, OZ. Exprimant m, n , l coordinatas unius
ex duobus panciis , et m ', ní, ľ coordinatas alterius. Quo
uiam spectari debent
24
a: cosa, a cosa', wcosac' ,
agens vero versus E' resolvetur in ternas
m cos(1800—a): — arcus a, acos (1800—at'):—w cosa',
a cos ( 180o -— ac&lt;nowiki&gt;''&lt;/nowiki&gt;) ∶≖ −meos ac&lt;nowiki&gt;''&lt;/nowiki&gt;:
in primo casu ut librat ex bypOthesi vires P, P', .
∙ ∙
,
in secundo vires Q, Q', . . .
Igitur
2X —l—w cosa::o, Zy-l—a cosa':o,ZZ—l-ar cosa:&lt;nowiki&gt;''&lt;/nowiki&gt;:o,
EA — z: cosa: :0,
2B —a cosa':o,ZC—z.ïcos at&lt;nowiki&gt;''&lt;/nowiki&gt;:o, Z (xï —7 X) −−∣−
15 (fcosa&lt;nowiki&gt;''&lt;/nowiki&gt;—-g cos ac) :0,
2( a:Z—zX)-I—w(fcosa&lt;nowiki&gt;''&lt;/nowiki&gt;—hcosa):o, XOZ—z?) −⊦
w(g cosa&lt;nowiki&gt;''&lt;/nowiki&gt;--hcosat'):o,
E( aB—bA) —w(fcos a'—gcosa):o,2 (aC—cA)—-
a(fcosa&lt;nowiki&gt;''&lt;/nowiki&gt;-h cos a):o,2 (bC-cB) -zz(gcos a&lt;nowiki&gt;''&lt;/nowiki&gt;- hcosac'):o.
Eliminata a, prodibunt undecim aequationes, inde-
pendentes ab ipsa a'
,
inter quantitates datas.; quibus ae-
quationibus expletis, habebitur aequilibrium, poteritque ab
una quavis ex duodecim praecedentibus erui valor a.
1394: Solidum corpus sollicitatum viribus, P P',
P&lt;nowiki&gt;''&lt;/nowiki&gt;, . . . detineatur duobus punctis fixis, sumptis, in axe
v. gr. OZ; sic facile determinabantur pressiones M, N, L
et M', N', L' exercitae in puncta illa juxta coordinatos a-
xes OX, DV, 02. Exprimant m, n, !coordinatas unius
ex duobus punctis, et m', n'. [ coordinatas alterius. Quo-
niam spectari debent25
M, N , -L, — M ', - N - L'
tanquam vires , quibus librantur caeterae P , P , P' ... ,
ac insuper m = 0 , n = o , m' =0 , n = 0 , necnon ( 110. )
(xY - yX ) = 0 : iccirco ( 8º. a &lt;nowiki&gt;''&lt;/nowiki&gt; )
EX - M - M ' = 0 , EY -N - N = 0,8Z -L - L ' = 0 ,
{ ( xZ - 2X ) + 2M + l'M' = 0 , E ( yZ – zY) +IN +
Ľ N' = 0 ;
quarum tertia nos edocet axem OZ premi vi XZ in dire
ctione z , reliquae vero suppeditant M , M' , N , N' . Si P ,
P ' , P &lt;nowiki&gt;''&lt;/nowiki&gt; , ... evadunt parallelae axi OZ, et se dirigunt ad
plagam negativam , erunt ( 12: 13, 2° ) ,
EX = 0 , EY = 0 , EZ = P - P - P ' -... = - R,
ExZ- X ) = - xP - X'P' - &lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; --... X, R,
(y2 — zY) = - ype ' P' y &lt;nowiki&gt;''&lt;/nowiki&gt; P &lt;nowiki&gt;''&lt;/nowiki&gt; —... = - y . R;
hic denotant P, P ', P &lt;nowiki&gt;''&lt;/nowiki&gt; , ... virium duntaxat intensitates. Quare
M + M ' = 0 , N + N = 0 , L + L + R = 0,2M +
I'M – x, R = 0 , 2N + IN - Y , R = 0 ;
unde
M = -M'
1, R
1 - T '
N = -N y R ,
, L
L +
+ LEL' = - R.
3
25
—M,-FNg-Lg—M'g—N' '..L,
tanquam vires
, quibus librantur caeterae P, P', P&lt;nowiki&gt;''&lt;/nowiki&gt;. ..,
ac insuper m::o, &lt;nowiki&gt;''&lt;/nowiki&gt;:D, in'-:(), &lt;nowiki&gt;'''&lt;/nowiki&gt; ∶−−⋅ o, necngn ( 110.)
Xxï—yX :) o:
iccirco( 80. a'&lt;nowiki&gt;'''&lt;/nowiki&gt; )
EX—M—M':o,2ï-—N-—N' :o,ZZ—L-—L' :0,
Si xZ—zX)—I-lM-I— l'M':o,Z(yZ—-zï)—l-IN-l-
&lt;nowiki&gt;!' N':o;
quarum tertia nos edocet axem OZ premi vi ZZ in dire-
ctione&lt;/nowiki&gt; :, reliquae vero suppeditant M, M' , N ,N'. Si P,
P'
,P&lt;nowiki&gt;''&lt;/nowiki&gt; ,... evadunt parallelae axi OZ, et se dirigunt ad
plagam negativam, erunt (12: 13. 20 ),
ZX:o, Zïzo,zz :P—P'-—P&lt;nowiki&gt;''&lt;/nowiki&gt;-—. ..

&lt;nowiki&gt;:&lt;/nowiki&gt; — R,

XxZ—QX):—xP -x'P'
-— x&lt;nowiki&gt;''&lt;/nowiki&gt;P&lt;nowiki&gt;''&lt;/nowiki&gt; —-. .

&lt;nowiki&gt;:&lt;/nowiki&gt;

. —a:. B,.
2(yz ∙∙∙ zï):—yP—— r' P' —.7&lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; ∙∙∙ ∙ ∙ ∙ ∶−∙ ∙−−∫∎ R;
bic denotant P, P', P&lt;nowiki&gt;''&lt;/nowiki&gt;, ... virium duntaxat intensitates. Quare
∐−⊦∐∣∶∘∙∾⊣−∐∙−−∶∘∙ ↧⋅−↽↧⋅∙−↽≖↸≓∘∙≀∐⊣⊸
I'M' — x,R:o, lN-i-l'N'—-y,R:o;
nnde
M: x,B -—M':—
---—- r—z '
Nz—N' :-—l',y'—-—R—2,L-I—L':—R- 326
14. Ex dictis de virium aequilibrio deduci possunt
aequilibrii leges in Machinis. Sic v. gr. in vecte poten
tia librabit resistentiam seu pondus, quotiescumque ( sumptis
( 10. 10 ) momentis quoad axem immobilem, circa quem po
test vectis moveri ) momentum potentiae aequatur momento
resistentiae.Idipsum obtinet quoad Axem in peritrochio ; idi
psum quoad trochleam fixam . Potentia et resisteutia istis
machinis applicantur in directione parallela planis perpen
dicularibus axi immobili; perinde igitur ( 10. 10 ° ) erit si
ve in eorum uno sive in altero accipiantur momenta ;
poteritque vectis repraesentari per lineam mobilem circa
punctum fixum , quod dicitur fulcrum , hypomoclion : axis
in peritrochio per circulares projectiones rotae ac cylin
dri in uno quovis ex dictis planis , mobiles circa com
mune immobile centrum : trochlea fixa per circulum ro
tatilem circa suum centrum , cujus circuli radius sit ipse
trochleae radius; verum quia in trochlea fixa nihil sunt
aliud perpendicula momentorum propria nisi trochleae
radii, ducti ad puncta ubi funis desinit ipsam trochleam
tangere, ideo erunt aequalia , et consequenter aequilibrium
in trochlea fixa importat potentiae ac resistentiae ae
qualitatem.
Ad trochleam mobilem quod spectat , sit P (Fig. 10)
pondus sustinendum a potentia Q : quoniam in casu aequi
librii , P et Q praebeant oportet resultantem R transeuntem
per punctum fixum 0, iccirco ( 9. 10. ) Q : P = sin \beta : sin a
= sin i : sin 2i = cos x : sin 2x = cos x : 2sin x cos
1 : 2 sin ; ac proinde
P
Q
2 sin s
Posuimus angulum OaQ dividi aequaliter directione
ponderis P : id vero facile intelligemus animadvertendo , si
filum OaQ fixum in 0 et Q , tenditur vi applicita puncto
26

14. Ex dictis de virium aequilibrio deduci possunt
aequilibrii leges in Machinis. Sic v. gr. in vecte poten-
tia librabit resistentiam seu pondus, quotiescumque ( sumptis
(10. 100) momentis quoad axem immobilem, circa quem po-
test vectis moveri ) momentum potentiae aequatur momento
resistentix-Idipsum obtinet quoad Axem in peritrochio ; idi-
psum quoad trocbleam lixam. Potentia et resistentia istis
machinis applicantur in directione parallela planis perpen-
dicularibus axi immobili; perinde igitur( 10. 100) erit si-
ve in eorum- uno sive in altero accipiantur momenta;
poteritque vectis repraesentari per lineam mobilem circa
punctum fixum, quod dicitur fulcrum, hypomoclion: axis
in peritrochio per circulares proiectiones rotae ac cylin-
dri in uno quovis ex dictis planis, mobiles circa com-
mune immobile centrum: trochlea lixa per circulum ro-
tatilem circa suum centrum,cuius circuli radius sit ipse
trochleae radius; verum quia in trochlea fixa nihil sunt
aliud perpendicula momentorum propria nisi trochleae
radii, ducti ad puncta ubi funis desinit ipsam trocbleam
tangere, ideo erunt aequalia ,
et consequenter aequilibrium
in trochlea fixa importat potentiae ac resistentiae ae-
qualitatem.
Ad trocbleam mobilem quod spectat , sit P (Fig. 10)
pondus sustinendum a potentia Q: quoniam in casu aequi-
librii , P et Q praebeant oportet resultantem R transeuntem
per punctum fixum 0, iccirco (9. 10.) Q: P:sin 13: sin «

&lt;nowiki&gt;:&lt;/nowiki&gt; sin i :sin 2i:cos x : sin Zx:cos x: Zsinxcosx
&lt;nowiki&gt;:&lt;/nowiki&gt; 1: 2 sin a:; ac proinde

P Q&lt;nowiki&gt;''&lt;/nowiki&gt;üü'
Posuimus angulum OaQ dividi aequaliter directione
ponderis P: id vero facile intelligemus animadvertendo, si
iilum OaQ fixum in 0et Q ,
tenditur vi applicita puncto
∙∙∙ '. 'una-
,.. ↙∙∙∎⋅−27
a libere excurrenti juxta ipsum Oal , punctum a necessa
rio permansurum in perimetro ellipseos , cujas foci O et Q;
ideoque in casu aequilibrii vim illam fore perimetro elli
pseos normalem ; quod certe importat praedictam anguli
OaQ divisionem . Vide n. 55. 13 .
Etiam sic : cum in casu aequilibrii funis ubique ma
neat aeque distentus , pondus P librabit resultantem R'
ex duabus viribus aequalibus Q et concurrentibus apud
punctum a ; et quoniam R' aequaliter dividit angulum Oal ,
idipsum dicendum erit de ponderis directione. Jamvero
R ' ( = P2) = Q + + Q2 + 2QQ cos 2i =2Q ( 1 +cos 2i) =
4 Q* cos 2i = 4 Q* sinºx :
rursus igitur
P
-
2sin x
angulo x = 90° respondebit minimal ; erit Q = P
2
si x = 30° ; vergente x ab 30° ad 09 , verget Q ab P ad co .
15. Vectis primi generis nuncupatur , si fulcrum sit
inter potentiam et pondus ; dicitur secundi generis si pon
dus sit inter fulcrum et potentiam ; denique si potentin me.
dium locum teneat inter fulcrum et pondus , vectis tertii ge
neris vocatur. Hinc vectes primi et secundi generis poten
tiam juvant , quatenus eo minor requiritur potentia ad da
tum pondus sustinendum , quo major est potentiae distantia
a fulcro relate ad ponderis distantiam ab eodem fulcro ;
adeo ut quodvis pondus utcumque ingens possit vectis ope
a quacumque potentia utcumque exigua sustineri : quod cum
bene nosset Archimedes , illud dixisse fertur Hieroni Regi ..
dic ubi consistam , coelum , terramque movebo ,, : vectis au
tem tertii generis potentiam non juvat , quia in hoc vecte
potentia minus distat a fulcro quam resistentia seu pondus.
27

&lt;nowiki&gt;::&lt;/nowiki&gt; libere excurrenti juxta ipsum OaQ , punctum :: necessa-

rio permansurum in perimetro ellipseos, cuius foci O et Q;
ideoque in' casu aequilibrii vim illam fore perimetro elli-
pseos normalem; quod certe importat praedictam anguli
OaQ divisionem . Vide n. 55. 13.0
Etiam sic :cum in casu aequilibrii funis ubique ma-
neat aeque distentus , pondus P librabit resultantem R'
ex duabus viribus aequalibus Q et Q concurrentibus apud
punctum a; et quoniam R' aequaliter dividit angulum OaQ,
idipsum dicendum erit de ponderis directione. Iamvero
a'
∙≺⇌−− re:? -1-Q*-l—2QQcos2i:2Q'(1-l-cva 20:
4Q' cos 3i:4Q3 sin'x:
rursus igitur
P
Q— 2sinx,
angulo x:900 respondebit minima Q

&lt;nowiki&gt;:&lt;/nowiki&gt; ä; erit Q:P

si a: 300 ,- vergente :: ab 300 ad 00
,
verget Qab P ad 00
.

15. Vectis primi generis nuncupatur, si fulcrum sit
inter potentiam et pondus; dicitur secundi generis si pon-
dus sit inter fulcrum et potentiam ;denique si potentia me-
dium locum teneat inter fulcrum et pondus , vectis tertii ge-
neris vocatur. Hinc vectes primi et secundi generis poten-
tiam iuvant, quatenus eo, minor requiritur potentia ad d'a-
tum pondus sustinendum , quo maior est potentiae distantia
a fulcro relate ad ponderis distantiam ab eodem fulcro;
adeo ut quodvis pondus utcumque ingens possit vectis ope
a quacumque potentia utcumque exigua sustineri :quod cum
bene nosset Archimedes , illnd dixisse fertur Hieroni Regi ,,
dic ubi consistam ,coelum ,terramque movebo ,, :vectis an-
tem tertii generis potentiam non juvat , quia in hoc vecte
potentia minus distat a fulcro quam resistentia seu pondus.28
Ex indicata vectis theoria redditur ratio innumerabi
liam effectuum quos quotidie cernimus fieri ; ac primo qui
dem quicumque ex lapidicinis extrahunt lapides utuntur ut
plurimum vecte ferreo : quoties autein multum resistit la
pis sive propter magnitudinem sive quod nimis firmiter aliis
adhaereat , tunc hypomoclion quam proxime ponderi admo
vent , ut facilius moveant , quod vulgo dicitur ,, dar la
leva ,, . Pro hypomoclio antem utuntur quovis sustentaculo
v . gr. lapide ; saepe etiam cum duo lapides ab invicem
sejungendi sunt , unus respectu alterius habet rationem hy
pomoclii . Hic notandum est maxillas quoque esse vectes
secundi generis quum cibi dentibus molaribus comminuen
di traduntur . Secundo : si avellendus est clavus ope mal
lei , quanto clavus , qui ponderis vicem obtinet , propior
fuerit hypomoclio , eo facilius educetur ; unde cum jam tan
tisper eductus est , ita ut extremitas mallei nequeat am
plius insistere subjectae tabulae aut parieti e quo est dedu
cendus , solemus aliud corpus interserere ut quam minima
sit distantia. Tertio : in forcipibus quoque duplex est vectis
primi generis , quorum unum est commune hypomoclion ,
clavus nempe circa quem uterque ramus volvitur , eoque va
lidius stringetur corpus quo rami , qua parte secant , brevio
res , qua parte vero applicatur potentia seu manus , longiores
erunt . Quarto : cum portas aperimus aut claudimus , eo
facilius id praestamas , quo longius a cardinibus eas impel
Iimus , nempe janua est vectis secundi generis , cujas hy
pomoclion sunt cardines. Quinto : remorum motu cymba
promovetur , quia remi sunt vectes secundi generis , quo
rum bypomoclion est aqua , cymba est pondus seu resi
stentia , manus hominis sunt potentia applicata : hinc quo
magis ab aqua remotae sunt manus quam punctum cym
bae , cui remi insistunt , eo majus est potentiae momen
ium. Sexto : ex his etiam intelligitur cur difficillima sit
bacali oblongi elevatio si per extremitatem accipiatur , el
cur quo longior fuerit ipse baculus , eo facilius curvetur
aut frangatur.
28
Ex indicata vectis theoria redditur ratio innumerabi-
lium efi'ectuum quos quotidie cernimus iieri ; ac primo qui-
dem quicumque ex lapidicinis extrahunt lapides utuntur ut
plurimum vecte ferreo :quoties autem multum resistit la-
pis sive prOpter magnitudinem sive quod nimis firmiter aliis
adhaereat ,
tuuc hypomoclion quam proxime ponderi admo-
vent
,
ut facilius moveant, quod vulgo dicitur
,, der in
leva ,, . Pro hypomocliol autem utuutur quovis sustentaculo
v. gr. lapide; saepe etiam cum duo lapides ab invicem
sejungendi sunt
, unus respectu alterius habet rationem hy-
pomoclii. Hic notandum est maxillas quoque esse vectes
secundi generis quum cibi dentibus molaribus comminuen-
di traduntur. Secundo: si avellendus est clavus ope mal-
lei, quanto clavus, qui ponderis vicem obtinet, propior
fuerit hypomoclio , eo facilius educetur ;unde cum iam tan-
tisper eductus est, ita ut extremitas mallei nequeat am-
plius insistere subjectae tabulae aut parieti e quo est dedu-
cendus , solemus aliud corpus interserere ut quam minima
sit distantia. Tertio :in forcipibus quoque duplex est vectis
primi generis, quorum unum est commune hypomoclion,
clavus nempe circa quem uterque ramus volvitur, eoque va-
lidius stringetur corpus quo rami
, qua parte secant
, brevio-
res, qua parte vero applicatur potentia seu manus ,
longiores
erunt. Quarto: cum portas aperimus aut claudimus , eo
facilius id praestamus
, quo longius a cardinibus eas impel-
limus , nempe janua est vectis secundi generis , cujus hy-
pomoclion sunt cardines. Quinto : remorum motu cymba
promovetur
, quia remi sunt vectes secundi generis , quo-
rum bypomoclion est aqua, cymba est pondus seu resi-
stentia , manus hominis sunt potentia applicata: hinc quo
magis ab aqua remotae sunt manus quam punctum cym-
hae, cui remi insistunt , eo majus est potentiae momen-
tum. Sexto : ex his etiam intelligitur cur difficillima sit
baculi oblongi elevatio si per extremitatem accipiatur ,
et
cur quo longior fuerit ipse baculus, eo facilius curvetur
aut frangatur.29
16. Supponantur nunc plures quotcumque vectes ita
dispositi , ut potentia Q in 0 ( Fig. 11 ) magis , puta decu
plo distet a fulcro A quam resistentia in L , quae simili
ter magis distet , puta noncuplo a fulcro C quam resisten
tia in K , quae rursus magis distet a fulcro D puta quin
tuplo quam resistentia in E , et haec similiter magis puta
quadruplo distet a fulcro G quam resistentia in F , haec
denique duplo magis distet a fulcro H quam pondus P in B.
Si res ita se habet , atque insuper potentia et pondus di
rectiones habeant perpendiculares ad respectivos vectes
factis AO = a , CL = a ', DK = a &lt;nowiki&gt;''&lt;/nowiki&gt; , GE = a &lt;nowiki&gt;''&lt;/nowiki&gt; , HF - a &lt;nowiki&gt;''&lt;/nowiki&gt; ,
AL = 6, CK = b' , DE = 6&lt;nowiki&gt;''&lt;/nowiki&gt; ,GF = 6 &lt;nowiki&gt;''&lt;/nowiki&gt; , HB = 6 &lt;nowiki&gt;''&lt;/nowiki&gt;
b &lt;nowiki&gt;''&lt;/nowiki&gt; , erunt
in casu aequilibrii,
L. 6 E. 6 &lt;nowiki&gt;''&lt;/nowiki&gt;
Q
F.6&lt;nowiki&gt;''&lt;/nowiki&gt; &lt;nowiki&gt;''&lt;/nowiki&gt;
il
K =
Kiba,K
E F
P. &lt;nowiki&gt;''&lt;/nowiki&gt;

&lt;nowiki&gt;;&lt;/nowiki&gt;

a
a&lt;nowiki&gt;''&lt;/nowiki&gt;
a '
IV
ex quarum multiplicatione prodibit
b 6'6&lt;nowiki&gt;''&lt;/nowiki&gt; 6 &lt;nowiki&gt;''&lt;/nowiki&gt; 8 &lt;nowiki&gt;''&lt;/nowiki&gt; P
Q
α α' α
P
a &lt;nowiki&gt;''&lt;/nowiki&gt; a &lt;nowiki&gt;''&lt;/nowiki&gt; 3600&lt;nowiki&gt;''&lt;/nowiki&gt;
Quisque videt haec applicari systemati cuicumque rotarum
dentatarum.
Supponantur quoque plures trochleae mobiles v.gr. tres
(Fig. 12) ; erunt ( 14)
Q
L
2 sin r &lt;nowiki&gt;''&lt;/nowiki&gt;
K р
LE 2 sin ac '
&gt; K = ;
2 sin x
et consequenter
Q =
P
23 sin x sin a ' sipx&lt;nowiki&gt;''&lt;/nowiki&gt;
Quod si detur systema coalescens ex trochleis fixis
v. gr , C , C' , C &lt;nowiki&gt;''&lt;/nowiki&gt;, C &lt;nowiki&gt;'''&lt;/nowiki&gt; ( Fig . 13 ) et ex mobilibus F, E, K
29
16. Supponantur nunc plures quotcumque vectes ita
dispositi , ut potentia Q in O (Fig. 11 ) magis , puta decu.
plo distet a fulcro A quam resistentia in L , quae simili-
ter magis distet
, puta noncuplo a fulcro C quam resisten-
tia in K, quae rursus magis distet a fulcro D piita quin-
tuplo quam resistentia in E, et haec similiter magis puta
quadruplo distet a fulcro G quam resistentia in F, haec
denique duplo magis distet a fulcro H quam pondus P in B.
Si res ita se habet , atque insuper potentia et pondus di-
rectiones habeant perpendiculares ad respectivos vectes
,
factis AO:a,CL :a'
, DK:
a&lt;nowiki&gt;''&lt;/nowiki&gt;, GE :a&lt;nowiki&gt;'''&lt;/nowiki&gt;,HF :a&lt;nowiki&gt;''&lt;/nowiki&gt;,
AL:&amp;, CK:6', DE :6&lt;nowiki&gt;''&lt;/nowiki&gt;, GF:b&lt;nowiki&gt;'''&lt;/nowiki&gt;, HB:ö&lt;nowiki&gt;''&lt;/nowiki&gt;
, erunt
in casu aequilibrii,
'
'
'. ∙
'&lt;nowiki&gt;'''&lt;/nowiki&gt; Q—qy'b,L—K'£.,K:E'f ∙ !' ,E—Eb ,F—Pf ;
a a a a a
ex quarum multiplicatione prodibit
Q 6 b' 1)&lt;nowiki&gt;''&lt;/nowiki&gt; b&lt;nowiki&gt;'''&lt;/nowiki&gt; 6&lt;nowiki&gt;''&lt;/nowiki&gt;P P ⇠
a
.: ∙
as an aut alv 3600
Quisque Videt baec applicari systemati cuicumque rotarum
dentatarum.
. Su pponantur quoque plures trochleae mobiles v. gr. tres
(Fig. 12) ; erunt (14). ⋅
et consequenter
Q.... 23 sinu: sinx' sin x&lt;nowiki&gt;''&lt;/nowiki&gt;
Quod si detur systema coalescens ex trochleis fixis
&lt;nowiki&gt;''&lt;/nowiki&gt;
gr, C ∙∁⋅∣ C&lt;nowiki&gt;''&lt;/nowiki&gt;. 0&lt;nowiki&gt;''&lt;/nowiki&gt; (Fig. 13) et ex mobilibus F, E, K30
uno eodemque fane conjunctis ; quoniam , librato systemate ,
funis ubique manet aeque tensus , ideo Q : Q = Q &lt;nowiki&gt;''&lt;/nowiki&gt;
Q &lt;nowiki&gt;''&lt;/nowiki&gt; = Q &lt;nowiki&gt;''&lt;/nowiki&gt; = Q = Q &lt;nowiki&gt;''&lt;/nowiki&gt; = Q &lt;nowiki&gt;''&lt;/nowiki&gt; &lt;nowiki&gt;''&lt;/nowiki&gt; . Jamvero
F E K
Q Q '&lt;nowiki&gt;'''&lt;/nowiki&gt; QP 2sin x' ' 2 sin x
2 sin 2
et consequenter
F = 2Q ' sin &lt;nowiki&gt;''&lt;/nowiki&gt; 2Q sin x &lt;nowiki&gt;''&lt;/nowiki&gt; , E = 2Q sin ü&lt;nowiki&gt;''&lt;/nowiki&gt; , K =
2Q sin x ;
cum igitur sint
L = Q&lt;nowiki&gt;''&lt;/nowiki&gt; &lt;nowiki&gt;''&lt;/nowiki&gt; , F +E + K +L = P ,
iccirco
2 Q sin x &lt;nowiki&gt;''&lt;/nowiki&gt; + 2 Q sin x' + 2 Q sin x +Q = P :
unde
P
Q =
1 +2 (sin x +sin x ' + sin x &lt;nowiki&gt;''&lt;/nowiki&gt; )
Fac demum ut puncta materialia K , K ', K &lt;nowiki&gt;''&lt;/nowiki&gt; , K '&lt;nowiki&gt;'''&lt;/nowiki&gt;,
( fig. 14 ) jungantur Glis K K' , K'K &lt;nowiki&gt;''&lt;/nowiki&gt; determinatae
quidem longitudinis, sed mobilibus circa K , K &lt;nowiki&gt;''&lt;/nowiki&gt; . Si pun
cta illa sollicitantur viribus Q , Q , Q &lt;nowiki&gt;''&lt;/nowiki&gt; , Q &lt;nowiki&gt;'''&lt;/nowiki&gt; , ad aequi
librium haec manifeste requirentur : potentia Q in di
rectione K'K tendens ab K' versus K ; resultans R' ex
Q et Q' in directione K &lt;nowiki&gt;''&lt;/nowiki&gt; K ' tendens ab K &lt;nowiki&gt;''&lt;/nowiki&gt; versus K' ; re
sultans R &lt;nowiki&gt;''&lt;/nowiki&gt; ex R' et Q &lt;nowiki&gt;''&lt;/nowiki&gt; in directione K &lt;nowiki&gt;''&lt;/nowiki&gt; K &lt;nowiki&gt;''&lt;/nowiki&gt; tendens ab K&lt;nowiki&gt;''&lt;/nowiki&gt; &lt;nowiki&gt;''&lt;/nowiki&gt; '
versus K &lt;nowiki&gt;''&lt;/nowiki&gt; ; potentia Q &lt;nowiki&gt;'''&lt;/nowiki&gt; in directione K &lt;nowiki&gt;''&lt;/nowiki&gt; K' ' ' tendens ab K &lt;nowiki&gt;''&lt;/nowiki&gt;
versus K' ' ' : demum ipsa Q's aequalis resultanti R &lt;nowiki&gt;''&lt;/nowiki&gt; .

&lt;nowiki&gt;*&lt;/nowiki&gt; Denotantibus X , Y , Z componentes coordi

natis orthogonalibusque axibus parallelas , in quas resolvi
tur Q, erunt
30
uno eodemque fune coniunctis; quoniam .
librato systemate,
funis ubique manet— aeque tensus
, ideo, Q:Q' ∶⋅−−−−∙ Q&lt;nowiki&gt;''&lt;/nowiki&gt; ∙∙∙−∙∶
Qu::
le:
Qv :va :Qvu ∙
Iamvero
F ∙∙∙ E v K
Q −⇀⋅⋅ 2 SQ..— sin m' 2 sinx
Q'—
−⋅ Zsin x&lt;nowiki&gt;''&lt;/nowiki&gt; ∙
et consequenter
F:
2Q'Isin a:&lt;nowiki&gt;''&lt;/nowiki&gt; ZQ sin x&lt;nowiki&gt;''&lt;/nowiki&gt;, E:2Q sin x', K:
2Q sinx; ⋅
cum igitur sint
LSva'sF4-E—FK—FL2P,
iccirco—
2Qsinx&lt;nowiki&gt;''&lt;/nowiki&gt;—I-2Qsinx'—]-2Qsinx—l-AQ:P:
nnde
P
1—l-2 (sinx-l—sin x' ∙−⊢ sin x&lt;nowiki&gt;''&lt;/nowiki&gt;)
.
Fac demum nt puncta materialia K,K' ,K&lt;nowiki&gt;''&lt;/nowiki&gt;, K&lt;nowiki&gt;'''&lt;/nowiki&gt;, ..:
(Gg. 14 ) iungantur filis K K', K' K&lt;nowiki&gt;''&lt;/nowiki&gt; , ... determinatae
quidem longitudinis, sed mobilibus circa K', K&lt;nowiki&gt;''&lt;/nowiki&gt;. Si pun-
cta illa sollicitantur viribus Q, Q' , Q&lt;nowiki&gt;''&lt;/nowiki&gt; , Q&lt;nowiki&gt;''&lt;/nowiki&gt; , ad aequi-
librium haec manifeste requirentur: potentia Q in di-
rectione K'K tendens ab K' versus K; resultans R' ex
Q et Q' in directione K&lt;nowiki&gt;''&lt;/nowiki&gt;K' tendens ab K&lt;nowiki&gt;''&lt;/nowiki&gt; versus K'; re-
sultans R&lt;nowiki&gt;''&lt;/nowiki&gt; ex B' et Q&lt;nowiki&gt;''&lt;/nowiki&gt; in directione K&lt;nowiki&gt;'''&lt;/nowiki&gt;K&lt;nowiki&gt;''&lt;/nowiki&gt; tendens .ab K&lt;nowiki&gt;''&lt;/nowiki&gt;
versus K&lt;nowiki&gt;''&lt;/nowiki&gt;; potentia Q&lt;nowiki&gt;'''&lt;/nowiki&gt; in directione K&lt;nowiki&gt;''&lt;/nowiki&gt; K&lt;nowiki&gt;'''&lt;/nowiki&gt; tendens ab K&lt;nowiki&gt;''&lt;/nowiki&gt;
versus K&lt;nowiki&gt;''&lt;/nowiki&gt;: demum ipsa Q&lt;nowiki&gt;'''&lt;/nowiki&gt; aequalis resultanti R&lt;nowiki&gt;''&lt;/nowiki&gt;.
&amp; Denotantibus X , T, Z componentes coordi-
natis orthogonalibusque axibus parallelas, in quas resolvi-
⋅ tur Q, erunt
Q;:31
X Y
ē
z
Q
cosinus angulorum , quos cum iis axibus intercipit l; de
notantibus insuper 2 , y , z coordinatas puncti K , et x' ,
j ', z coordinatas puncti K' , erunt
2x yay 22
KKKK KK
cosinus angulorum, quos cum ipsis axibus efficit K'K ; ob
tinebit itaque primum ex requisitis ad aequilibrium, quoties
cumque fuerint
X XX Y
DKKKK .
yg Z
KÖK &gt;
K’K &lt;nowiki&gt;''&lt;/nowiki&gt;
seu
X Y Z
(h ) .
Quod in ordine ad Q est X , Y , Z , sit X', Y ', Z ' in or
dine ad Q ' : si resolvitur l' in ternas coordinalis axibus
parallelas, eae erunt ( 9. 40. )
x + X ' , Y + Y ' , 2 + Z ';
hinc designantibus a&lt;nowiki&gt;''&lt;/nowiki&gt;, y ', z &lt;nowiki&gt;''&lt;/nowiki&gt; coordinatas puncti K&lt;nowiki&gt;''&lt;/nowiki&gt; , ob
tinebit secundum ex requisitis ad aequilibrium , ubi fuerint
X + X __ * ' - &lt;nowiki&gt;''&lt;/nowiki&gt; Y + Y_y_y&lt;nowiki&gt;''&lt;/nowiki&gt; 2 + 2_z'- &lt;nowiki&gt;''&lt;/nowiki&gt;
R ? KK R' K ” K R K&lt;nowiki&gt;''&lt;/nowiki&gt;K&lt;nowiki&gt;'''&lt;/nowiki&gt;
.
seu
X + * _ * + Y_2_Z
x - x yay 22
( h ').
31
X ? Z
Q Q Q
cosinus angulorum, quos cum iis axibus intercipit Q; de-
notantibus insuper a: , y , :: coordinatas puncti K,, et x',
y', s' coordinatas puncti K' ,
erunt
⋅⇂⋅−−⋅⊴⇂∙∣ .7-7&lt;nowiki&gt;''&lt;/nowiki&gt; z—z'
K'K , K'K . K'K
cosinus angulorum, quos cum ipsis axibus efficit K'K: ob-
tinebit itaque primum ex requisitis ad aequilibrium, quoties-
cumque fuerint '
≟−−−⋅−∝−−≄∣ it.s,-ï Z
Q K'K&lt;nowiki&gt;''&lt;/nowiki&gt; 'Q K'K &lt;nowiki&gt;''&lt;/nowiki&gt;G'ka' ↽−≖∙⊍↼∙≕∣
seu gx z
r—x' y—y' x—z'
Quod in ordine ad Q est X , T, Z , sit X', ï', Z' in or-
dine ad Q':si resolvitur Q' in ternas coordinatis axibus
parallelas, eae erunt (9. 40.)
X—FX' , T—Fï' , Z—l-Z' ; ↽
hinc designantibus z', y&lt;nowiki&gt;''&lt;/nowiki&gt;, :&lt;nowiki&gt;''&lt;/nowiki&gt; coordinatas puncti K&lt;nowiki&gt;''&lt;/nowiki&gt;, ob-
tinebit secundum ex requisitis ad aequilibrium , ubi fuerint ⋅
X—l—X' x'-x&lt;nowiki&gt;''&lt;/nowiki&gt; T—l-Tl—TI—j&lt;nowiki&gt;'''&lt;/nowiki&gt; ∅⊣−⊈∣↼↼≂∣∙ z&lt;nowiki&gt;''&lt;/nowiki&gt;
B' ⋅⋅⇀∣⋦∣∣↓⊊∣ ∙ nf- KI/KT '-T—KHK' '—
....t ∙⇁−⋅∣ ↖↽∙∣ ∣
X X T
T—Z-Z.(h).
/ II I I/32
non pluribus opus est ut intelligamus quod, expleta
X + X + X &lt;nowiki&gt;''&lt;/nowiki&gt; _Y + Y + Y &lt;nowiki&gt;''&lt;/nowiki&gt; _Z + Z + Z &lt;nowiki&gt;''&lt;/nowiki&gt;
x ' - 0 &lt;nowiki&gt;''&lt;/nowiki&gt; g'my &lt;nowiki&gt;''&lt;/nowiki&gt; z &lt;nowiki&gt;'''&lt;/nowiki&gt; - &lt;nowiki&gt;''&lt;/nowiki&gt;
( h&lt;nowiki&gt;''&lt;/nowiki&gt; ) ,&lt;nowiki&gt;''&lt;/nowiki&gt;
obtinebit tertium ex requisitis illis ; componentes X” , Y&lt;nowiki&gt;''&lt;/nowiki&gt; ,
Z&lt;nowiki&gt;''&lt;/nowiki&gt; spectant ad vim Q &lt;nowiki&gt;''&lt;/nowiki&gt;, coordinatae z ' ', y, pun.
clum K &lt;nowiki&gt;'''&lt;/nowiki&gt; . Designantibus demum X '&lt;nowiki&gt;'''&lt;/nowiki&gt; , Y
Y ' &lt;nowiki&gt;''&lt;/nowiki&gt;,
&lt;nowiki&gt;''&lt;/nowiki&gt; , Z &lt;nowiki&gt;''&lt;/nowiki&gt;
componen
tes in ordine ad Q&lt;nowiki&gt;''&lt;/nowiki&gt; , expletisque
X + X + X &lt;nowiki&gt;''&lt;/nowiki&gt; + X &lt;nowiki&gt;''&lt;/nowiki&gt; = 0 ,
Y + r' + &lt;nowiki&gt;''&lt;/nowiki&gt; + I&lt;nowiki&gt;''&lt;/nowiki&gt; = 0 ,
2 + 2 +2&lt;nowiki&gt;''&lt;/nowiki&gt; + Z &lt;nowiki&gt;''&lt;/nowiki&gt; = 0 ,
( h &lt;nowiki&gt;''&lt;/nowiki&gt; )
manifeste obtinebit quartum simulque quintum ex requisi
tis ad aequilibrium. Sub novem igitur distinctis condi
tionibus propositum quatuor virium systema consistet in
aequilibrio: si quinque darentur vires , undecim prodirent
conditiones; generatim 2 n + 1 conditiones quoad n vires.
Collatis primis ac secundis membris formularum ( h) , (h') ,
( h&lt;nowiki&gt;''&lt;/nowiki&gt;) , emergent
Y ( 2 - x ) – X (y - ) = 0 ,
( Y + Y') (a' - &lt;nowiki&gt;''&lt;/nowiki&gt; ) – ( X + X ') ( 7'- , ' ) = 0 ,
( X + Y' + Y &lt;nowiki&gt;''&lt;/nowiki&gt;) ( ' &lt; &lt;nowiki&gt;''&lt;/nowiki&gt; ) — ( X + x ' + X &lt;nowiki&gt;''&lt;/nowiki&gt;) (y &lt;nowiki&gt;''&lt;/nowiki&gt; , ' ') = 0;&lt;nowiki&gt;''&lt;/nowiki&gt;
quarum summa praebet
xY_yXfwY — y'X ' + x &lt;nowiki&gt;''&lt;/nowiki&gt; Y &lt;nowiki&gt;''&lt;/nowiki&gt; —y &lt;nowiki&gt;''&lt;/nowiki&gt; X &lt;nowiki&gt;''&lt;/nowiki&gt; + &lt;nowiki&gt;''&lt;/nowiki&gt; ( X + X' + x &lt;nowiki&gt;''&lt;/nowiki&gt;) —
x &lt;nowiki&gt;''&lt;/nowiki&gt; ( Y + Y ' + Y &lt;nowiki&gt;''&lt;/nowiki&gt; ) = 0 ,
∃⊈∙
non pluribus Opus est ut intelligamus quod, expleta
x-1-xq-x'Q—v-1-rq-rff—z-i-zq-z&lt;nowiki&gt;''&lt;/nowiki&gt; W,),
xli—xlli J/l ∙∙⇁ 7&lt;nowiki&gt;''&lt;/nowiki&gt;, z&lt;nowiki&gt;''&lt;/nowiki&gt;—z&lt;nowiki&gt;'''&lt;/nowiki&gt;
obtinebit tertium ex requisitis illis; componentes X&lt;nowiki&gt;''&lt;/nowiki&gt;, ?&lt;nowiki&gt;''&lt;/nowiki&gt;
,
⋅
Z&lt;nowiki&gt;''&lt;/nowiki&gt; spectant ad vim Q&lt;nowiki&gt;''&lt;/nowiki&gt;, coordinatae x&lt;nowiki&gt;'''&lt;/nowiki&gt;. &lt;nowiki&gt;''&lt;/nowiki&gt;, z&lt;nowiki&gt;'''&lt;/nowiki&gt; ad pun-
ctum K&lt;nowiki&gt;''&lt;/nowiki&gt; . Designentibus demnm X&lt;nowiki&gt;'''&lt;/nowiki&gt;, ï&lt;nowiki&gt;'''&lt;/nowiki&gt;
, Z&lt;nowiki&gt;'''&lt;/nowiki&gt; componen-
tes in ordine ad Q&lt;nowiki&gt;'''&lt;/nowiki&gt; , expletisque
\sum∙⊦\sum∣∙⊢\sum∦⊹\sum∣∥∶∶∘∙
T .l-T-l-TII—l- III,: 0 , (hi/I)
Z-i-ZIä-le-l—ZIflzo'
manifeste obtinebit quartum simulque quintum ex requisi-
tis ad aequilibrium: Sub novem igitur distinctis condi-*
tionibus propositum quatuor virium systema consistet in
aequilibrio: si quinque darentur vires, undecim prodirent
conditiones; generatim 2 n ⊣− ↿conditiones quoad :: vires.
Collatis primis ac secundis membris formularum (I:), (b'),
U;&lt;nowiki&gt;''&lt;/nowiki&gt; )
, emergent
?(x—x') —X (?'—?') −∙−−−∘ ∙
(
ï—l— !' )(x' ∙∙∙ x&lt;nowiki&gt;''&lt;/nowiki&gt;)—( X-l-X') (r'—y&lt;nowiki&gt;''&lt;/nowiki&gt;) :o
,
(HF-IJ&lt;nowiki&gt;''&lt;/nowiki&gt;) (x&lt;nowiki&gt;''&lt;/nowiki&gt;-— ∣∣∣≻⊣≖≖−⊦\sum∣−⊦\sum∥≻ (y'—y&lt;nowiki&gt;'''&lt;/nowiki&gt;)
−−− .;
quarum summa praebet
xy-Jx-Jlïl—y/X/ :&lt;nowiki&gt;''&lt;/nowiki&gt; ï&lt;nowiki&gt;''&lt;/nowiki&gt;—y&lt;nowiki&gt;''&lt;/nowiki&gt;X&lt;nowiki&gt;''&lt;/nowiki&gt;—l—y&lt;nowiki&gt;'''&lt;/nowiki&gt;(X XLI-X&lt;nowiki&gt;''&lt;/nowiki&gt;) ∙−⋅
↕∣∣∣≼↕⊹⊺∣⊹↕∥≻ :0 ,33
seu , ob primam et secundam ( hm) ,
-Y yXTY'y'x + x'Y&lt;nowiki&gt;''&lt;/nowiki&gt; _7 / X &lt;nowiki&gt;''&lt;/nowiki&gt; + x &lt;nowiki&gt;''&lt;/nowiki&gt; I &lt;nowiki&gt;''&lt;/nowiki&gt; —7&lt;nowiki&gt;'''&lt;/nowiki&gt;X &lt;nowiki&gt;'''''&lt;/nowiki&gt;
Simili modo collatis primis ac tertiis membris ipsarum ( h) ,
( h') , ( h&lt;nowiki&gt;''&lt;/nowiki&gt; ), attentisque prima ac tertia ( h '&lt;nowiki&gt;'''&lt;/nowiki&gt;) ; itemque col
latis secundis ac tertiis membris earumdem ( h ) , ( h ) , ( h&lt;nowiki&gt;''&lt;/nowiki&gt; ) ,&lt;nowiki&gt;''&lt;/nowiki&gt;
attentisque secunda ac tertia ( h &lt;nowiki&gt;'''&lt;/nowiki&gt;) , assequemur&lt;nowiki&gt;'''&lt;/nowiki&gt;
xZ - 2X + « Z_z'X' + x'Z&lt;nowiki&gt;''&lt;/nowiki&gt; _z&lt;nowiki&gt;''&lt;/nowiki&gt;X &lt;nowiki&gt;''&lt;/nowiki&gt; Tx &lt;nowiki&gt;''&lt;/nowiki&gt; Z '&lt;nowiki&gt;'''&lt;/nowiki&gt; —Z'y&lt;nowiki&gt;''&lt;/nowiki&gt; = 0 ,&lt;nowiki&gt;''&lt;/nowiki&gt;
32—3Y + y^2?–49 + &lt;nowiki&gt;''&lt;/nowiki&gt;Z&lt;nowiki&gt;''&lt;/nowiki&gt; &gt;&lt;nowiki&gt;''&lt;/nowiki&gt;Y &lt;nowiki&gt;''&lt;/nowiki&gt; +y &lt;nowiki&gt;''&lt;/nowiki&gt; Z&lt;nowiki&gt;''&lt;/nowiki&gt; — ;&lt;nowiki&gt;''&lt;/nowiki&gt; Y&lt;nowiki&gt;''&lt;/nowiki&gt; = 0 .
Conditiones videlicet aequilibrii ( 13. 8º. ) quoad systema
punctorum lineis rigidis inter se firmiter connexorum in
cluduntur in conditionibus aequilibrii quoad propositum
systema habens formam variabilem .

=== De centro gravitatis. ===
[[17|17]]. Constat experimentis corpora jugiter sic tendere, seu gravitare in tellurem, ut sibi commissa descendant verticaliter in eius superficiem, gravitas ergo, seu vis unde provenit iste verticalis descensus, eatenus haberi poterit pro sibi ad sensum parallela, quatenus licebit superficiem illam habere pro physice plana: constat insuper experimentis omnia quaevis corpora eodem tempore idem spatium verticaliter in vacuo percurrere, idest aequali velocitate ex aequali altitudine perpendiculariter ad horizontem descendere. Inde sequitur vires gravitatis in diversis corporibus esse illorum massis proportionales, et corpus quodlibet spectari posse tanquam aggregatum materialium graviumque particularum, quae gaudeant parallelarum virium proprietatibus: centrum virium parallelarum (12) in casu dicitur centrum gravitatis. Resultans ex omnibus gravitatis viribus, quae vigent in corporis particulis, vocatur corporis pondus; transit constanter per gravitatis centrum, et directionem obtinet horizonti perpendicularem.

Porro si massula indefinite parva &lt;math&gt;\nu&lt;/math&gt; apud datum corporis punctum dividitur per respondens volumen &lt;math&gt;\beta&lt;/math&gt;, ratio &lt;math&gt;\frac{\nu}{\beta} (= \mu ) &lt;/math&gt; vocatur corporis densitas apud illud punctum; diciturque corpus vel homogeneum, vel heterogeneum prout &lt;math&gt;\mu&lt;/math&gt; apud singula corporis puncta est vel eadem, vel diversa; in corporibus homogeneis ratio &lt;math&gt;= \mu&lt;/math&gt;  est eadem ac ratio inter totalem corporis massam et ejus totale volumen; pondusculum massulae &lt;math&gt;= \nu&lt;/math&gt;, utpote proportionale ipsi &lt;math&gt;= \mu&lt;/math&gt;, exprimitur per &lt;math&gt;= \mu&lt;/math&gt; ductam in quandam constantem &lt;math&gt;c&lt;/math&gt;; ratio &lt;math&gt;\frac{c \nu}{\beta} (= c \mu ) &lt;/math&gt;  appellatur specifica corporis gravitas apud praefatum punctum; estque densitati proportionalis.

[[18|18]]. Notetur illud: etsi corpus gravitate sua jugiter sollicitatur deorsum; hoc tamen non officit quominus adhuc (2) dicatur corpus de se et natura sua indifferens ad quietem vel motum. Gravitas enim est dumtaxat vel aliquid extrinsecum corpori, vel illi intrinsecus additum, non autem aliquid eidem essentiale. Patet, quia vel nomine gravitatis intelligitur vis quaedam, qua corpora versus terram urgentur, vel vis qua tendunt ad determinatam quamdam spatii immobilis partem. Non hoc secundum, quia eo ipso casus purus admitteretur contra principium rationis sufficientis, cum nulla appareat ratio cur mobile ad hanc potius partem ferri debeat quam ad illam, cum spatium ubique sit homogeneum; ergo primum erit dicendum: sed si ita est, certe gravitas non est corporibus essentialis; nulli enim corpori essentiale est ut sibi caetera coexistant, ac proinde unum potest existere quin existant caetera, et consequenter etiam quin existat terra.

[[19|19]]. Dato centro gravitatis corporis, facile definitur utrum corpus in dato situ extra lapsus periculum constitui possit. Nam ex eo centro demissa ad planum horizontale recta perpendiculari, quae vocatur linea directionis, si haec intra basim cadat, corpus extra lapsus periculum erit positum, secus ruet in eam partem in quam perpendicularis recta dirigitur. Hinc patet ratio cur turres aliquae &lt;u&gt;inclinatae&lt;/u&gt; non cadant, ut sunt Bononiensis, Pisana etc: linea scilicet directionis extra ipsarum basim non excurrit. Hinc etiam valde pingues, et qui magnum aliquod onus brachiis complectuntur, retrorsum; gibbosi autem et bajuli antrorsum; qui dextra pondus aliquod sustinent, sinistrorsum; qui vero sinistra, dextrorsum &lt;u&gt;inflectuntur&lt;/u&gt;. Per hanc scilicet declinationem efficiunt ut linea directionis transeat per spatium, quod inter pedes continetur; quod spatium est basis corporis humani. Eamdem ob caussam si quis velit ex. gr. dextero pede stare, crus &lt;u&gt;inclinat&lt;/u&gt; paullulum dexteram partem versus, nec diu haerere potest in eo statu , quia cum basis totius corporis sit unus dumtaxat pes, linea directiouis facile potest basis tam anguslae limites praetergredi. His autem corporis nostri flexibus ac librationibus ita ab infantia assuevimus usu continuo ut nec advertentes recto illas ordine peragamus. Patet hinc denique cur aves uni pedi insistentes dormire solent capite sub ala recondito; id nempe faciunt ut linea directionis intra pedis cui insistunt latitudinem servetur.

[[20|20]]. Centrum gravitatis inveniri potest vel ratione mechanica, vel ratione, algebraica. Ad primam quod attinet, si corpus aliquod filo suspendas, volvetur converteturque donec in aequilibrio tandem consistat, et filum ad terrae superficiem perpendiculariter dirigatur. In hac perpendiculari, quae est linea directionis per quam centrum gravitatis corporis tendit, erit centrum ipsum. Iam notetur linea a filo perpendiculari in corpore designata, rursusque ex alio puncto suspendatur corpus, et facto aequilibrio linea perpendicularis pariter notetur. In communi duarum linearum intersectione reperietur quaesitum centrum. Ratio algebraica desumitur ex dictis ( 13.2.º''a''&quot; ): sumantur nempe vires proportionales massis &lt;math&gt;m, m' , m''&lt;/math&gt;, ..... punctorum, quibus applicitae sunt; hoc pacto, ad positionem centri gravitatis determinandam exsistent
&lt;math display=&quot;block&quot;&gt;x_{\mathrm I}=\frac{\sum m x}{\sum m}, y_{\mathrm I}=\frac{\sum m y}{\sum m}, z_{\mathrm I}=\frac{\sum m z}{\sum m}  (b) &lt;/math&gt;Si corpus intelligitur divisum in varias portiones dimensionis finitae , et earum massae denotantur per  &lt;math&gt;m, m' , m''&lt;/math&gt;, adhuc valebunt formulae (b); nihilque aliud erunt &lt;math&gt;x , y , z ,x' y ',z',x''&lt;/math&gt;, ... nisi coordinatae centrorum gravitatis illarum portionum. Si corpus ponitur insuper homogeneum quoad omnes partes, erunt massae ut respondentia volumina; poteruntque haec illis substitui in formulis (''b'') : quisque videt coordinatas &lt;math&gt;x_{\mathrm I}, y_{\mathrm I}, z_{\mathrm I}&lt;/math&gt;, ex (''b'') haud pendere ab intensitate gravitatis. Caeterum plures sunt casus, in quibus centrum gravitatis absque formularum subsidio immediate cognoscitur. Sic in linea recta centrum gravitatis est medium ipsius rectae punctum: in parallelogrammo punctum, ubi binae diagonales se mutuo secant: in circulo centrum figurae: in cylindro habente bases parallelas punctum medium axeos:
in parallelepipedo punctum, ubi quatuor diagonales se mutuo secant: in sphaera ipsum magnitudinis centrum.

In triangulo centrum gravitatis est punctum illud, ubi sese invicem secant rectae lineae, quae a duobus trianguli verticibus ducuntur ad puncta media laterum oppositorum: cum enim &lt;math&gt;AD&lt;/math&gt; (Fig. 15) dividat aequaliter rectas omnes lateri &lt;math&gt;BC&lt;/math&gt; parallelas, et &lt;math&gt;BE&lt;/math&gt; rectas omnes lateri &lt;math&gt;AC&lt;/math&gt; parallelas, reperietur centrum gravitatis areae triangularis tam in &lt;math&gt;AD&lt;/math&gt; quam in &lt;math&gt;BE&lt;/math&gt;; ideoque erit in &lt;math&gt;H&lt;/math&gt;. Jamvero ducta &lt;math&gt;DE&lt;/math&gt;, ea exsistet parallela lateri &lt;math&gt;AB&lt;/math&gt;; et consequenter triangula &lt;math&gt;ABH , DEH&lt;/math&gt; erunt similia; hinc&lt;math display=&quot;block&quot;&gt;\frac{DE}{AB}=\frac{DH}{AH}&lt;/math&gt;sed, ob &lt;math&gt;CE = \frac12  AC&lt;/math&gt; et &lt;math&gt;CE = \frac12 CD = BC&lt;/math&gt;, est DE = &lt;math&gt;CE = \frac12 AB&lt;/math&gt;; igitur &lt;math&gt;DH = \frac12  AH&lt;/math&gt;; ac proinde &lt;math&gt;DH = \frac12 AD&lt;/math&gt;; et &lt;math&gt;AH = \frac23 AD&lt;/math&gt;.

In pyramide triangulari &lt;math&gt;ABCO&lt;/math&gt; (Fig. 16) erit &lt;math&gt;G&lt;/math&gt; centrum gravitatis; ubi nempe se mutuo secant binae rectae &lt;math&gt;OH , CK&lt;/math&gt;, quae ex &lt;math&gt;O&lt;/math&gt; et &lt;math&gt;C&lt;/math&gt; ducuntur ad centra gravitatis
&lt;math&gt;H&lt;/math&gt; et &lt;math&gt;K&lt;/math&gt; triangulorum &lt;math&gt;ABC , ABO&lt;/math&gt;. Secetur enim pyramis,

1.º planis parallelis triangulo &lt;math&gt;ABC&lt;/math&gt;, 

2.º planis parallelis triangulo &lt;math&gt;ABO&lt;/math&gt;; transibit &lt;math&gt;OH&lt;/math&gt; per centra gravitatis omnium illarum sectionum triangularium; transibit &lt;math&gt;CK&lt;/math&gt; per centra gravitatis omnium harum. Ergo pyramis habebit suum gravitatis centrum tam in &lt;math&gt;OH&lt;/math&gt; quam in &lt;math&gt;CK&lt;/math&gt;, et consequenter in &lt;math&gt;G&lt;/math&gt;. Ducatur nunc &lt;math&gt;HK&lt;/math&gt;; erit &lt;math&gt;HK&lt;/math&gt; parallela rectae &lt;math&gt;CO&lt;/math&gt;, et triangula similia &lt;math&gt;HKG , CGO&lt;/math&gt; praebebunt
&lt;math&gt;\frac{HK}{CO}=\frac{HG}{OG}.&lt;/math&gt;
Sed, ob &lt;math&gt;MH =\frac13 CM&lt;/math&gt; et &lt;math&gt;MK = \frac13 OM&lt;/math&gt;, est &lt;math&gt;HK = \frac13 OC&lt;/math&gt;;
ideoque &lt;math&gt;HG =\frac13 OG&lt;/math&gt;; igitur &lt;math&gt;HG = \frac14 OH&lt;/math&gt;, et &lt;math&gt;OG = \frac34 OH&lt;/math&gt;.

=== De corporum collisione ===
[[21|21]]. Quaestio de corporum collisione eo redit, ut datis velocitatibus ante collisionem, determinentur velocitates post collisionem. Corpora sese collidentia assumimus sphaerica, et in singulis stratis concentricis homogenea; in quibus proinde corporibus centrum gravitatis erit ipsum magnitudinis centrum. Corporum sese collidentium centra vel moventur in eadem recta, vel in diversis rectis; in primo casu collisio dicitur normalis, in secundo obliqua.
[[Fasciculus:Inelastischer stoß.gif|thumb]]
[[22]]. Invenire velocitatem &lt;math&gt;v''&lt;/math&gt;, quam habent duo data corpora non elastica post normalem collisionem, datis eorum
velocitatibus &lt;math&gt;v'&lt;/math&gt; et &lt;math&gt;v&lt;/math&gt; ante collisionem. Dicantur &lt;math&gt;m', m&lt;/math&gt; corporum massae; erunt &lt;math&gt;mv , m'v'&lt;/math&gt; quantitates motus ante collisionem: eatenus corpus subsequens agit in antecedens quatenus hoc lentius illo movetur, adeo ut perseveret actio donec ad aequalitatem velocitatis deveniatur; unde velocitas &lt;math&gt;v''&lt;/math&gt; post collisionem erit communis, et aequalis in utroque: summa praeterea quantitatum motus est eadem ante et post collisionem; velocitas autem obtinetur dividendo quantitatem motus per massam. Ergo demum&lt;math display=&quot;block&quot;&gt; v'' =\frac{mv + m'v'}{m + m'}&lt;/math&gt;

Haec observentur: 

1.° &lt;math&gt; v'' - v &lt;/math&gt; exprimit quantum velocitatis acquisierit corpus antecedens, quod ponimus esse
&lt;math&gt;m&lt;/math&gt;; et &lt;math&gt; v' - v'' &lt;/math&gt; quantum amiserit impellens &lt;math&gt;m'&lt;/math&gt;. 

2.° consideranda erit pro lubito alterutra velocitas tamquam negativa, si corpora ex oppositis plagis adveniunt; hinc in formulis ubicumque ea inveniatur, signo contrario erit adhibenda - Sic v. gr. si massae &lt;math&gt;m'&lt;/math&gt; directio habeatur pro positiva, sumenda erit &lt;math&gt;v&lt;/math&gt; negative, ac proinde &lt;math&gt; v'' =\frac{m'v'- mv}{m + m'}&lt;/math&gt;.

3.° ponetur &lt;math&gt;v = 0&lt;/math&gt;, si corpus impellendum &lt;math&gt;m&lt;/math&gt; quiescit; erit &lt;math&gt; v'' =\frac{m'v'}{m + m'}&lt;/math&gt;: hinc &lt;math&gt;v''&lt;/math&gt; ferme evanescet si massa &lt;math&gt;m&lt;/math&gt; sit physice infinita respectu &lt;math&gt;m'&lt;/math&gt;. 

4.º numquam habebitur perfecta quies post collisionem si &lt;math&gt;m&lt;/math&gt; et &lt;math&gt;m'&lt;/math&gt; in easdem partes oppositas, et velocitates sint reciproce ut massae, tunc &lt;math&gt; v'' = 0&lt;/math&gt;, et consequenter habebitur perfecta quies.

23. Invenire velocitates corporum perfecte elasticorum post normalem collisionem, datis velocitatibus &lt;math&gt;v', v&lt;/math&gt; ante collisionem. Perspicuum est hujusmodi corpora sequi leges non elasticorum toto compressionis tempore, tum restitutionis tempore effectum hunc instaurari. Ergo facta partium restitutione inveniri debet in corpore impulso dupla velocitatis acquisitio; dupla vero celeritatis amissio in impellente. Itaque si dicantur r' ' et &quot; velocitates corporis im
pellentis et corporis impulsi post factam restitutionem ,
erunt ( 22)
u &quot; = V - 2 ( 0--0&quot; ) = v - 2
my + ms
mtm
2 mv tv (m ' — m)
( 9 ),
m + m
(
1
vi &quot; = 0 + 2 (0 &quot; ~ v ) = 2 + 2 ( -v)
mv + m's
m + m
2 m ' ú tu (m - m ')
(9) .
mtm

24. Haec ex formulis (9) et (q' ) deducuntur .

1.• Si massae sunt aequales , elastica corpora post colli
sionem movebuntur .facta velocitatum permutatione, Nam
moveantur primo in eamdem plagam ; propter m = m' , for
mula (9) abit in
2 m v'
et ( 9 ') in 3,10 v' ; ergo etc.
Rursus praeter m = m ' habeatur etiam v = 0 , hoc est cor
2 mo
pus percussum quiescat; erit v = 0 , et v ' . = V ' ;
corpus nempe percutiens post collisionem quiescet , et per
2 mv
2 m 2 m
2 m
1
39
moveantur, vel* alterutra solum quiescat :quod si collisio
liat ad partes oppositas ,
et velocitates sint reciproce ut mas-
sae, tunc v&quot;:o ,
et consequenter habebitur perfecta quies.

23. Invenire velocitates corporum perfecte elasticorum
post normalcm collisionem, datis velocitatibus v', 0 ante col-
lisionem. Perspicuum est huiusmodi corpora sequi leges non
elasticorum tOto compressionis tempore, tum restitutionis
tempore effectum hunc instaurari. Ergo facta partium re-
stitutione inveniri debet in corpore impulso dupla velo-
citatis acquisitio; dupla vero celeritatis amissio in impel-
lente. ltaque si dicantur v'&quot; et v&quot; velocitates corporis im-
pellentis et corporis impulsi post factam restitutionem ,
erunt (22)
'
'
um:-D' --2 ('n'—v&quot;) :'--2 (,; ∙−−−−−−−−⋯⇂↓−⊢⋯∣∣↗ m −−⊢ m )
..2 mv −⊢∣v (m' −∙∙ m)
'm ∙−∙∙ m' ∓∎∎∎∎∎∎ (9):
W:w—l-2(v&quot;—v):v-l—2 Maii:-31; -v)
-&quot;2mv—l-v(m-—m) (qr).
m-l-m' J—
24. Haec ex formulis (q) et (q') deducuntur.
↿∙∘ Si massae sunt aequales, elastica corpora post colli-
sionem movebuntur.fdcta velocitatum permutatione, Nam
moveantur primo in eamdem plagam; propter m:m', for-
mula (q) abit in
2 mv ' ∙
2
'&quot;
'v'&quot;::«v p. . ,
et (q ) 111 10&quot;: :v ; ergo etc.
a m
'
-
Rursus praeter m:m' habeatur etiam :::o , hoc est cor-
- -
∣∣∣ tv 2 m &quot;( pus percussum qutescat; er1t a::o, et a:::
'v ; m
corpus nempe percutiens post collisionem quiescet , et per-40
cussum movebitur velocitate , quam percutiens habebat ante
collisionem . Demum sibi mutuo occurrant : ubicumque ergo
invenitur v , sumenda erit negative ; qua mutatione facta ,
habebuntur
2 mv 2 mv'
2 m
v, et viv v' .
2 m
Jam vides mutationes velocitatum exhiberi per ipsas litte
ras , et ubi debeat etiam mutari directio , regressus expri
mitur per mutationem signorum.
2.• Si statuatur series corporum perfecte elasticorum , ae
qualium , se mutuo tangentium , et quorum centra unam
rectam constituant , in primum autem quacumque velocitate
incidat alterum corpus aequale , movebitur tantummodo cor
pus ultimum , quiescentibus omnibus aliis . Quod si statua
tur series corporum habentium massas in progressione geo
m3,
metrica m' , m, ... ; et caeteris quiescentibus, pri
mum m' incidat in secundum velocitate v' , expriment
m2
m ? m
2 m
v' .
m +m (m *:)*,~(m2 I )
m
velocitates excitatas a primo in secundo , a secundo in ter
tio , a tertio in quarto etc. Denotante igitur n numerum cor
porum , movebitur ultimum velocitate
2 m'
N- 1
I
Cena ntmi
).
3. Quotiescumque corpus impellens minus erit corpore
impacto quiescente , toties illud regredietur , uti patet ex
formula ( 9 ) , posita v = o et m &gt; m' . Quod si m =
et v =0 , prodibit v'' = -1 , nimirum si globus minor''
∢⋅∘cussum movebitur velocitate ,quam percutiens habebat ante
collisionem. Demum sibi mutuo occurrant :ubicumque ergo
invenitur :: , sumenda erit negative; qua mntatione facta.,
habebuntur
2mv &quot; 2mv'

: —-v,et'v :

2m ∙∙−−−∙⋅∙≀≀∙ 2m
Iam vides mutationes velocitatum exhiberi per ipsas litte-
ras , et'ubi debeat etiam mutari directio , regressus expri-
mttur per mutat1onem signorum.
2.&quot; Si statuatur series corporum perfecte elasticorum, ae-
qualium, se mutuo tangentium, et quorum centra unam
rectam constituant
,
in primum autem quacumque velocitate
incidat alterum corpus aequale , movebitur tantummodo cor-
pus ultimum
, quiescentibus omnibus aliis. Quod si statua-
tur series corporum habentium massas in progressione geo-
∙ m! ma. metrtca m', m, ∙ ∙ ∙ ∙ &quot;7, , m .; et caeterts qmescenttbus , prt-
mum m' incidat in secundum velocitate v', expriment
v,2m' ⋅∙ 2m' :
,( 2m' 3
m—l—m&quot; 'v (m—l—m')&quot;, m-l-m'
velocitates excitatas a primo in secundo , a secundo in ter-
tio
, a tertio in quarto etc. Deuotante igitur n numerum cor-
porum , movebitur ultimum
⋅⋅⋮ velocitate ea&quot;
3.&quot; Quotiescumque corpus impellens minus erit corpore
impacto quiescente ,
toties illud regredietur , uti patet ex
formula (q) , posita v :o et m m'
. Quod si ut:ea
et 9

: o

, prodibit v'&quot;:
-— v' , nimirnm si globus minor41
incurrat in globum immensae massae quiescentem , resiliet
cum velocitate eadem , cum qua advenerat .
4.• Si duo corpora elastica occurrant sibi velocita
tibus v , v ', quae massis m, m ' reciproce sint proportionales ,
eadem qua venerant velocitate ambo resilient. Etenim cum
ex oppositis partibus corpora congrediantur , ac praeterea
m : m ' :: v ' : v , in formulis ( 9) , ( 9' ) sumenda erit » negative ,
et ponendum mv = m' '; quibus peractis , obtinebitur
v ' &quot; = &gt; &quot; (m + m )
et viv=v
Im + m
no-tm
Imtin
-- 5. ° Ex ipsis ( 9) et ( 9' ) eruitur m'y's mula
m 'ustomus: factum ex massa in quadratum respondentis
velocitatis dicitur vis viva ; hinc in corporibus perfecte
elasticis summa virium vivarum permanet eadem ante et post
collisiopem .

25. Formulae ( 9) , (9 ') aptari possunt etiam corporibus,
imperfecte elasticis , modo quantitatibus
2(v— mm Imus)
my tms
mtm
et 2
( mahu
my + mv
m + m
--)
substituantur
(n+ m ( = m **)e (1+- ( Inv—-).
denotante r rationem inter vim , qua partes sese resti
tuunt , et vim comprimentem. Quantitas r experimentis de
terminanda est in singulis corporum speciebus : fac ut m
quiescat , sitque co ; erit post collisionem '&quot; = -ru':
unde , cognita velocitate v' ., qua m ' offendit in m , et velo
citate negativa v'' , qua post impactum resilit , habebitur''
-
4
⋅↣
' 41
incurrat in globum immensae massae quiescentem ,
resiliat
cum velocitate eadem, cum qua advenerat.
49 Si duo corpora elastica occurrant sibi velocita-
tibus v, v', quae massis m, m' reciproce sint proportionales ,
eadem qua venerant velocitate ambo resilient. Etenim cum
ex oppositis partibus corpora congrediantur , ac .praeterea
ut :m'::v':
9, in formulis (q) ,(q' ) sumenda erit .9 negative ,
et ponendum m 9:m' v'; quibus peractis , obtinebitur
v'&quot;:
— v' (Z.—lm,) :-v'. et v&quot;:v (m )

:

v.
∙
∙⊢⋯⋅ ⋯∙−⊦⋯
∂∙∘ Ex ipsis (q)et.(q') eruitur m' v'&quot;3-l- mv&quot;::
m'∎∣∣≖ -l-m vi: factum ex massa in quadratum respondentis
velocitatis dicitur vis viva; hinc in corporibus perfecte
elasticis summa virium vivarum permanet eadem ante et post
collisionem. . 

25. Formulae (q), (q')'aptari possunt etiam corporibus ,
imperfecte elasticis , modo quantitatibus .
2(v,—nw—-mlv) et/2 (mv-l-mlv m-l-m —v) mm
substituantur (1 44) (v'. m.,-(.m'v') et≰↿⊹↗⋝⋅≼⋯⇂≩−−⋯⋅∣⇂≀∣ ∙∙∙∙∙ v) ∙
∦⇂⊣−⋯≳ m—l—m
denotante r rationem inter vim
, qua partes sese resti-
tuunt , et vim comprimentem. Quantitas r experimentis de-
terminanda est in singulis corpürum speciebus :fac ut 11:
quiescat , sitque :co ; erit post collisionem v'&quot;:
-
r v':
unde , cognita velocitate v' ., qua m' olfendit in m, et velo-
citate negativa v'&quot;
, ua post impactum resilit, habebitur '42

26. Ad collisionem obliquam quod pertinet , si corpora sibi mutuo occurrunt directionibus convergentibus bm , b'm
( Fig.17 ) et velocitatibus expressis per easdem rectas bm ,b'm ', resolvantur bm , b'm ', altera in duas by, ba, altera in duas b'y ', b'a', ita ut by, b'y' existant normales , ba vero et bá parallelae sint rectae m m corporum centra jungenti. Quoniam componentes b y , b'y' parallelae sunt tangenti TT ductae per punctum contactus, ab ipsis nullo pacto pendebit collisio, nullamque in collisione subibunt mutationem . Cor
pora igitur sese collident velocitatibus ba = ym, b'a' =
y'm '. Inventis itaque ( 23 ) v &quot; , et v '' , sumptisque ex. gr.'' mf = y &quot; , mi = &quot; in recta y r', et ductis mv = by , m'ú
= bóý , si complentur parallelogramma fv, iv', exprimentur per diagonales mf, m'i' tum velocitates , tum directiones corporum post collisionem. Haec autem ex modo dictis facile colliguntur; 

1.º Si globus minime elasticus iacidit oblique in planum immobile, progredietur secundum directionem plani cum velocitate m'v ' ( = a'm '), quae ad velocitatem priorem b'm ' erit ut sinus anguli incidentiae b'm'y' ad radium. 

2.º si globus fuerit perfecte elasticus, resiliet per m'z efficiendo angulum reflexionis z míy aequalem angulo incidentiae b'm'ý . 

3.° quod si globus incidens sit imperfecte elasticus, resiliet ad angu
lom i'm'y ', cujus cotangens ad cotangentem anguli inciden
liae b'm'y ' ut r : 1 .

=== De motu rectilineo utcumque vario.=== 
27. Nonnulla hic praemittimus ex analysi infinitesimali.
1.o Quantitas iniinitesima a: (minor videlicet qua-
cumque data utcumque parva) censeatur esse primi ordinis ;
«2 erit inlinitesima secundi ordinis; «3 iniiuitesima tertii; etc.
2.&quot; Inlinitesima a) dicetur esse primi ordinis si ra-
∙ G) ∙ ∙ ∙
a tno ∙
.. valorem habet (imtum
, secund1 s1 ∙−− valorem obtinet
ac «:43
similiter finitum , atque ita porro . Denotante generatim k
valorem illum finitum , poterit infinitesima quantitas ordinis
msimi exhiberi per
w kam
3. Sumptis aliis valoribus finitis k,; ka, ... km ,
habentur pro aequalibus
kmetkam tk , an- tkzam- ² + ... + kmiat kma
km_ ,a et kam + kamer + ... + km_, &amp; ,
kmed k * et kam + kamer t . tkm -rQ ?.
etc .... ;
admittuntur nimirum aequationes
kam tka&quot;-t ... tkm , at km
km
kam +k ,am -s +... +kimeza? + kmail
km ,
51
etc.
quatenus differentia inter utrumque membrum est minor
quacumque data quantitate alcunique parva. Huc spectat
illud : quantitates infinitesimae , quaecumque eae sint, et quo
rumcumque ordinum , absque ullo errore negliguntur prae
quantitate finita : itemque infinitesimae quantitates altiorum
ordinum absque ullo pariter errore negliguntur prae infini
tesima quantitate inferioris ordinis.
4.0 Quantitates infinitae ( majores videlicet qua
cumque data utcamque magna) cum possint exprimi person
43
similiter tinitum
, atque ita porro . Deuotante generatim k
valorem illnm finitum
, poterit intinitesima quantitas ordinis
msimi exhiberi per
m::ka&quot;
3.&quot; Sumptis aliis valoribus finitis k,
,
It,,
... k,&quot;,
habentur pro aequalibus

: , et kat'-I-k, ∝⋅⋅−≖⊣−≀∣≖∝∙−⋅≖−⊦ −⊦↗⊏⋅∙∙⋅∝−⊦∣⊏⋅⋅∙

∄⊄⋅⋅∙≖∘≖ et ka&quot; ⊣− 1, ∘⋍⋅∙∙⋅⋅⊳⋅ ⊣− ⊣− r.,, a: -
It,... «* et kat&quot; −⊦ kp?&quot; -]— ∙∙∙−∣⋅⋅ km., æ.
etc-eoo ;
.
admittuntur nimirum aequationes ⋅ '
↗⊄⊧∘↙⋅⊣−∣∁⋅⊶⋅∙−⋅⊣−∙∙∙∣−⊦↗≂⋅∙∙⋅⊄−⊦↳
↿ ⋅
km .
l
kan-l-Ic,ac&quot;&quot;-l--. &quot;'l-kaum&quot;,
hngua −⊦⋠⋅∙∙∙∸⇂⇉⊄∙−⋡↿ ⋅ «
⋅
etc. ∙ ∙ , . .
.. ⋮∙ ;
,-
∣

;

quatenus differentia, inter utrumque membrum'est'minor
quacumque data quantitate utcumque parva. Huc. &quot;spectat
illud :quantitates inünitesimae, quaecumque eae sint. et quo- ∙
rumcumque ordinum
, absque ullo errore negliguntur prae
quantitate finita :itemque infinitesimae quantitates altiorum
ordinum absque ullo pariter errore negliguntur prae ⋮≖≖∅≖∙≖∃∙∙ tesima quantitate inferioris ordinis.
∙∡∙∘ Quantitates infinitae (majores videlicet qua-
cumque data utcumque magna) cum possint exprimi per-;,44
tribuentur et ipsae in varios ordines ; illudque facile stabi
lietur : quotcumque finitae quantitates tuto : negliguntur
prae quantitate infinita ; quantitatesque infinitae ordinum in
feriorum tuto etiam negliguntur prae quantitate infinita altio
ris ordinis. Facto enim \beta . , et designantibus a, b,c, ... , 9
valores finitos , habebitur
1
. 0
a \betam + bBm - tom-&gt; + ... +9\betato
1
EL
-la + bw twat..tqomat ww .
5.- Si variabiles x, y sunt inter se per certam quam
dam relationem ita connexae ut data v. g. X , inde possit
valor y determinari , y vocatur functio quantitatis x ; ipsa
vero x dicitur independens. Si relatio inter x et y expri
mitur aequatione minime resoluta quoad functionem y habi
tam pro incognita , y appellatur functio implicita ; quod si
valor y detur expressus immediate per independentem x,
vel talis obtineatur per aequationis resolutionem , y dicitur
functio explicita. In aequatione v, g. yo -2xy + m2 =0
y functio implicita quantitatis variabilis x ; at facta re
solutione , evadet y functio explicita ipsius x , duplicemque
habebit valorem , scilicet y = x + Vx? m2 , Functio
nes explicitae quantitatis x designari solent in hunc modum
est
-
F ( x) , f ( x) , ..
6.0 Differenziale dx quantitatis x est incrementum
infinitesimum , quod ipsi x adscribitur : differentiale vero dy
functionis y = f (x ) est respondens incrementum f ( x + dx)
- f (x ) .quod ob variatam x recipit in se functio illa : pro
ponantur v. gr. invenienda differentialia functionum
44
tribuentur et ipsae in varios ordines ; illudque-facile stabi-
lietur :quotcumque finitae quantitates tuto.: negliguntur
prae quantitate infinita; quantitatesque infinitae Ordinum in-
feriorum tuto etiam negliguntur prae quantitate infinita altio-
ris ordinis. Facto enim þ:
S;, et designantibus a,b,c, ..., q
valores linitos , habebitur
∘∣⊰⋅∙⊣−∂↙⊰⋅⋅∙⋅≖−⊦∘≀⊰⊶−≖ −⊦⋅⋅⋅ ⊣−⊄∣⊰−⊦ .
∸−−
te». &quot;(a ∙⊸⊦bæ—l— ccc&quot; −⊦∙∙ .-l—qm&quot;&quot;' ∎∙−∣− r m'&quot;).
5.&quot; Si variabiles æ,y sunt inter se per certam quam-
dam relatidnem ita connexae, ut data v. g. a:
,
inde possit
valor ]determinari ,;
vocatur functio quantitatis se: ipsa
vero «: dicitur independens. Si relatio intern- et y expri-
mitur aequatione minime resoluta quoad functionem ]habi-
tam pro incognita ,
]appellatur functio implicita ; quod si
valor y detur expressus immediate per independentem :,
vel talis obtineatur per aequationis resolutionem ,
]dicitur
functio explicita. ln aeqnatione v,- g. ;&quot; -—,2ay −⊢ ⇑∙∅:o
est 7 functio implicita quantitatis variabilis z'; at facta re-
solutione
, evadet ]
functio explicita ipsius a:, duplicemque
habebit valorem, scilicet 7—:a:∶⊨ ⇂⋅∕⋅↕∙≖ −− m'. anctio-
nes explicitae quantitatis a: designari solent 1n hunc modum
,F (x), f(x),... ⋅∙
6.&quot;Dill'erentiale dx quantitatis x est incrementum
infinitesimum , quod ipsi :: adseribitnr :differentiale 'vero dy
functionis y:--f (x) est respondens incrementum f
(x dx)
—f
(a:)
, quod ob variatum se recipit in se functio illa: pro-
ponantur v. gr. inVenienda difi'erentialia functionum45
at +6,9 +0,24+ Cisin x + C , cos x+c ,
tang x + C, log x + C , a ' tc ,
ubi a et C sunt quantitates constantes.
Erit I. dy = [ alx + dx ) + ] - [ax + C ] = adx.
a
II.dy = [ f'da+ c ]- [* + c]atda X
adot
adx
x2 + xdx
.
III.dy = [ ( x + dx)* + C]-[x4 +C]=ax“-'dx + 29,
a'a- 1 )
24-2d.22
t . ax-' dx .
IV.dy = [ sin ( x + dx ) + c ]- [sinx + C ) = sin ( x + dx ) — sinx
2 cos 2xdx)sinh dz = 2cosx sind = cos xdx .
V.dy = [cos ( c + dº + C ]- [cos.FC ] = cos ( c - day -cosx =
2 sin - (2x +dx)sin __ (x -x -dx) = sin xdx.
VI.dy = [tang ( xtdx )+c] - [langat.C ]=
sin ( x + 2x)
cos(x + dx )
sinx cosxsin (xtdx)-sinxcos( x + dx ) sin ( x + dx - x )
cos2 cosx cos ( cdc )
cos2x
45
a'−⊦∁∙−⋮∙−∙⊹ C, æa-l- C,'sin a: ∙⋅⊢ C,cos æ-l-C,
tang æ-l—C, logæ-l— C, ar-l-C,
ubi a et C suntquantitates constantes.
Erit l. dy: [a( æ-l—dx) ——C ]— [ux—I»- C ]:adæ.
∥⋅↙∣↗↗⊣⋤⋮⋅−−⊦ (i]—[?&quot; C]— jd,— :—
∥∣∙↙∄∫⇋∶∐↕⊹≴≀↕≻⋅⊹∁⊐∙⋢∞⋅⊹∁∃≕∞≕∙∣↙≀↝⋍⊹↽∘↙↙⊑⋅∣≱↶∶⊄−≖∠≀↓⋅≖
&quot;I- ∙ ∙∼ ∙ :aæ&quot;-' dx.
IV. a];:[sin(æ-l-dæHCI-[sinæ-l-C] :sin(æ-l—dæ)— aina:

: 2cos—;..(Zæ-l-dæ)sin-—;. dæ: a cosa: sin-;— dx:cos ædæ .

V.dJ:[cos(æ-l—dæ)-l-C]- [cosa: −⊢∁↥ ∙−−− cos (æ—l—dæþcosæ:
2sin :(ZPFdrþin-i—(æ-x-dx):— sin ædæ.
Vl.dy::[tang (
.z—l-dæ ≻−∣−∁⋮∣ ∙
⇂⊏∄∐⊰⊅⇥∙∁⊐ —8lll(æ..ll-rlæ) cos(æ—-dæ)
aina: cosæsin(æ-l-dæ)-sinæcos(H—dx)— sin(.i—l—dx-æ)
cosa: cosæ cos( æ
⊹∠≀∙↧⋅ ) coszx ⇁⇁−∙↼46
dx
cos2 x
VII.dy= [log(x-tdx)+ c]-[logo +C ]= log ( + )
dit 15 ( 1 +4x)dx _d2log [2 + } (1- dot) +
23 (1-4 )(1-2dt)+... ]
det 103 ( 2 + + 43 + 234 + ...)
dxlog [ 2 , 718281828
dx
] ;
sumptisque logarithmis quoad basim 2 , 718281828
dx dy X
istiusmodi basis solet exprimi per e.
VIII.dy = [a ++dx + C ] - [ a * + C ] = a *+ x_qt = da? =
a* d log (a *) = a * d [ x log (a )] = a * log (a ) dx.
70. Quantitas constans C, quaecumque ea sit, non in
venitur in differentialibus: idemque proveniet differentiale
sive differentietur v. g. sin x + C, sive sin x.
dy 8º. In primo exemplo habemus a,
dz
cundo axe- ', in dy quarto dx dx
- in se
dy
a
in tertio dy dx
x2
46
da:
∙∙∙∎∙∙↼⇁∙−⇁
∞∘⇄∙⋍∙∙
∇∥∙ ↙≀↨↶−−−∏∘⊰≺⊿↾∶∙∔⋞≀∙↕≻−⊦∁∃⋅⊏∣∘∷∞⊣−∁↿⇌ log (
↿
.? −⊢↙∙⇣⋮⋮⋟⋮
dx —log (HE ↙↿−⋤⋅∶∙↙≀−≟−∅∣∘⊰∣∶≆⊣⊸≑−≺↿− ff):
⋮⇡↽≐≺↿−⋛≣≤ (fi-:): --]—
da: ↿ ↿
⊺⊅−∣∘⊰≺⊈⋅∙⊦−≆−⊣−≐−∙≡ 2..3
∢⊯∎⊦∙∙∙ '):
≦−↕∣∘∥⋣∙ 718281828 ...
];
sumptisque logaritbmis quoad basim 2, 718281828 ..., .
(II:—;:
istiusmodi basis solet exprimi per e.
Vlll'dy :[a&quot;dx—I—C ]—[ax ∙−⋅⋅⊢∁⋮∣∶∅≖↤≖− ar: daJr:

: a'd log (a'):axd [æ log (a)]:axlog (a) dt.

⋅70. Quantitas constans C,quaecumque ea sit, non in-
venitur iu differentialibus: idemqne proveniet dili'erentiale
sive dilferentietur v. g. sinæ ∙−∣⋅− C, sive siuæ.
80. In primo exemplo habemus ?: a, in se-
x
d d cundo a- . J
⊋−⋮⋮⋮ :— &amp;, 1n tertio 23:01: ', 111 quarto 71:47
in COST
in octavo
dy dy cost , in quinto sin x, in sexto dx dx
1
septimo di die= a * loga . Quisque videt
dy fore generatim novam functionem variabilis z : si ea
denotatur per f(x) , erit
2
dx
de = f( ), et dy = f ( z )dx .
Functio f '(x ) appellari solet derivata ex primitiva f( x) :
caeterum dx, seu differentiale variabilis independentis x,
spectatur quidem ut quantitas infinite parva ; sed simul con
stans atque arbitraria,
9º. Ex ivº, vº, et viº exemplo habemus
d sinx d sinx da dx :
dcosx
COS X V
sinx 1 - sinar
dcosx
77- cosa a '
dx = cosa x d tang x =
d tanga
sec2 x
dtang x
1 + langa x
Aequationes istae in hunc modum scribi possunt
dz dz
darc (sin = z ) = darc(cos = 2 ) =
V1 - Z
V 1-22
dz darc ( tang = 2 )
1 + z2
47
cosa: ,
in quinto ?; −∙−∙−∙ -—sinx, in sexto ⋛⋚∶≎∙⊂≐⊭−∙ in
septimo g

:

-.::— ,
in octavo :::-ï :axloga. Quisque videt
217— fore generatim novam functionem variabilis :: si ea

denotatur per f(æ) , erit
ngþ), et 47 :f(æ)dæ .
l
Functio f(æ) appellari solet derivata ex primitiva f(æ):
caeterum dx, seu differentiale variabilis independentis x,
spectatur quidem ut quantitas infinite parva; sed simul con-
stans atque arbitraria.
go, Ex &quot;o, vo, et vi&quot; exemplo habemus
dsinæ dsinæ dccsæ dar.... ∶−−−−−⇀⋅−⇁−⋅ :
.

: ⋅ cos 3&quot; l/ 1

-
sm'æ '&quot;&quot;:
&amp;
, dx:cos3xd tangæ:
deosæ
Vi-oos'x
dtangx ∙∙∙ dtangx
secaæ 1—i—tang3x
Aequationes istae in hunc modum scribi possunt
'
d
dare (sin— !.):sz ,darc(cos——z)-—- V
132 ,
-zz ∙ 2
darc(tang:z): T'↶−≀≘≖−−?'48
10 ° . Sicuti ex y = f( x) obtinuimus ( 8&quot; ) dy
f ( x )dx, sic ex hac obtinebimus ddy = f' ( x) dxdx
f '(x )dx?, ex qua rursus dddy = f &quot; (x )dx dx ' = f &quot; (x ) dr },
atque ita porro ; denotant fif &quot;, ... novas functiones
variabilis independentis x. Itaque si compendii causa e
xhibentur ddy, dddy, ... per dy, dy,.. , profluent
d
d’y = f '( x ) dx?,dy = f &quot; ( x ) dx3,
dy= f (x) ,
da²
d3y
= f'&quot; (r ) , ... : d.x3
assumpta v.gr.y = x ^, erunt f( x) = x ^ , f ( x ) = axa if '( ')
=a ( a - 1 ) 219-2, f ( x ) = a (a - 1 ) ( a - 2 ) x4-3, .
Differentialia dy , dºy , dy ,. . , itemque functiones deri
vatae f (x ), f ' (x) , f &quot; (x ), ... dicuntur primi, secundi, ter
tii , ... ordinis respectu functionis primitivae y = f (x ).
11 ° . Quemadmodum data functione possunt quaeri
ejus differentialia , ita vicissim dato differentiali quaeri po
test functio unde illud promanal. Sint F (x ), f (x ) ejusmo
di functiones variabilis x , ut exsistat F' ( x) =f( x) : quan
titas F ( x) + C vocatur integrale indefinitum differentia
lis f ( ) dx, designaturque praefigendo litteram ſipsi differen
tiali , ut scribatur
ſf(x) dx F ( x) + C ;
exprimit C quantitatem ( 7 &quot; ) constantem atque arbitrariam.
12° . Formula f ( x )dx ita sese aliquando exhibet,
ut statim appareat eam esse differentiale cujusdam da
tae functionis ; tunc vero in promptu est integrale: atque
hoc pacto habemus ( 6º . 9° )
f (a + 1)x*dx ******+. C,unde fredr =
xati
atito
48
100. Sicuti ex 7:f(x) obtinuimus (80) d] ∶∙∙−−⋅
f
(æ)d.r, sic ex bac obtinebimus ddj :
f
'(æ) dædæ :
f'(x)dæ', ex qua rursus dild]:f&quot;(æ)dx dæ' :f'&quot;(æ) dx3,
atque ita porro; denotant f,f &quot;,
... novas functiones
variabilis independentis æ. Itaque si compendii causa e-
xhibentur ddy, dddy, ... per dy, d37 ,. ., profluent
dïy
&amp;? d')&quot; :f'(x) da.&quot;,dfly :f&quot; (æ) das-3, ...,

: f'(æ),

113!
dæ3 :f'&quot;(.r),...:
∙∙⋅⋅∙⋅
assumpta v. gr.y:æ&quot;, erunt f
(æ):æ',f' (x):ax&quot;',f'(a-)

:: a(a — 1) x&quot;&quot;,f&quot; (x):a(a—1)(a—-2)x&quot;3, ....

Difаerentialia dy, diy, d3y,. . , itemque functiones derivatae f(x), f'(x) , f&quot;(æ) , ... dicuntur primi, secundi, tertii, ...ordinis respectu functionis primitivae y:f(x).
110. Quemadmodum data functione possunt quaeri
eius differentialia, ita vicissim dato differentiali quaeri po-
test functio unde illud promanat. SintF(x),f(æ) ejusmo-
di functiones variabilis x, ut exsistat F'(æ):f(x): quan-
titas F(x) −−∣− C vocatur integrale indefinitum differentia-
lis f
(.r) dar, designaturque praefigendo litteram ]ipsi differen-
tiali, ut scribatur
ff(æ)dæ:-—F(æ)—1-C;
exprimit C quantitatem (70) constantem atque arbitrariam.
120. Formula f(x)dx ita sese aliquando exhibet,
ut statim appareat eam esse dili'erentiale cujusdam da-
tae functionis; tunc vero in promptu est integi—ale: atque
hoc pacto habemus (60. 90)
a
&amp;
a-l-l C :: xtt-H f(a—1-1)æ dx:x ∙⋅∣− ,undefæ dx: ∉⊋∙∙⊦∙∙∙∓ ⊹∁⊒ï49
QCx
ſalog/a)d(c== q** + C, unde ſe*dx =clogiastc

;

dx
S
= arc ( sin = x ) +C ;
V 1 - 22
Sa
dx
1 + x2
= arc ( tang = x ) + C.
130. Interdum formula f (x )dz, de cujus integra
tione non constat , per quasdam substitationes transfor
matur in aliam , cujus integrale illico cognoscitur. Sic.
v . gr. positis ax = 2 , - = z ,assequimur a
dx dz 1
Si Salita
=
14a²x² arc ( tang == z) + C =
1
arc ( tang = ax ) + C ,
Sa
dix
22 ta
1
Sat
dz
a (1 + z2)
arc ( tang = 2 ) + c = a
-a arc tang * + c,
-Svador - Svado --Svet
( cos = ) + c
arc ( cos = z ) + c
= arc
fa&quot;log(a)d(cæ):a&quot; —]—C,undefa&quot; dx: -ac
dx
- ⇂∕↿∙−⋅⋥∎⊑ :arc (sin :x) −⊢∁≂
f 1112 :arc( tang:x )-l-C.
130. Interdum formula f(æ)dx, de cujus integra-
tione non constat, per quasdam substitutiones transfor-
matur in aliam, cuius integrale illico ougnoscitur. Sic.
.. æ . '. gl'. POSIUS nær-z.;— Zoasaequlmur
dæ ⇀∙∙− dz 1 — ∙−−
fl'l'a'æ' a(1-I-za) a
&quot;c (tang—z)-[-C—..
dx
xï-l-a
dz 1
faU-l—z') − a arc(tangzz)-I-C:
↿
—a.arc (tang :ax)-[- C,] ——
.—
—1-arc( tang :
−⋅⋮− ⋟⊹ C,
et
f
dx ]
adz ]
dz ∙∙∙
[fas-xa ∣∕ ∅≖∙∅≖≖≖ −⇀ ∣∕↿ -zz
arc(cos:z)-1-C :
arc(cos :?) ⊹∁∙50
140. In integrali indefinito ( 11 °) adhibeantur suc
cessive pro x peculiares valores xo, x n , ac dein ab F ( zn )
+ C subtrahatur F ( x ) +C ut , eliminata C , prodeat
F (xn) - F ( xo) : ejusmodi differentia vocatur integrale de
finitum differentialis f (x ) dx , sumptum videlicet ab x =
а &quot;
x
Xo
xh
ſ p(x)dx = F(wow )— F( xo ) .
Xo
Hinc v. gr.
a
dx
jederati
7T
a
o
Variato altero ex binis limitibus v. gr. x ny variabit
ipsum quoque integrale ; et adhibita x pro xmo erit
X
ſ f(x)dx= F ( x) — F ( xo ) :
Xo
habebitur videlicet integrale illud , quod incipit ab xo ,
quodque evanescit facto x = x,: et quoniam
aff(x)dx = d [F(x)-F(xs)] =dF( x) =f ( x) dx ;
X.
iccirco
X
S SP(x)dx = Sp«x ) dx + c .
X.
15 °. Sit arcus infinitesimus ABEH ( Fig. 18 ) , et
in eo chordae infinitesimae AB , BE , EH , quarum prima
50
140. In integrali indefinito (110) adhibeantur suc-
cessive pro x peculiares valores xo, x,, , ac dein ab F (x,)
−⋅∣− C subtrahatur F(xo)-I-C ut, eliminata C, prodeat
F(x,,)— F (x,): eiusmodi differentia vocatur integrale de-
finitum differentialis f(x)dx, sumptum videlicet ab x:
x&quot;
x, ad x:æ, ,designaturque per [f(x) dx, ut scribatur
æo
xn
]
f(æ)dx :P(æ.) — P(æ. )-
æo Hinc v. gr.
[ a
fa,-' dx: ..-.-..-—1 J'EL :
-E- a-l—1 ' xï-l-aa a.
0 0
Variato altero ex binis limitibus v. gr. x,, variabit
ipsum quoque integrale; et adhibita x pro x,, erit
.?
faa-w.r: ∌⇁≺∙↿∶≻−−∙≖⊸⇁≺∞∘≻≃⋍∙
æo
habebitur videlicet integrale illud, quod incipit ab x.,
,
quodque evanescit facto x: x,: et quoniam
&amp;?
df/(x)dæ:d[F(x)-F(æ.) ]:dF(æ) :f(æ) dx;
xo
iccirco
fff(-1')dx:ff( x
)
dxH—FC.
↿⋅⇂⋝∘⋅ x.,
Sit arcus iniinitesimus ABEH( Fig. 18 ), et
in eo chordae infinitesimae AB, BE, EH, quarum prima51
SUC:
ac tertia producantur donec concurrant in D. Quoniam an
guli DBE, DEB sunt infinitesimi, angulus quoque ODE
erit infinitesimus: designetur iste angulus per i , et fiant odest
e de
BD =
es
BE = c , DE b ;
habebimus
lur
62 =a +62 – 2ab cos ( 180° -1i ) = a + b2 +
2 ab cos i = a : + 62 + 2 ab- 2ab + 2 abcosi = (a + b )2
2 ab ( 1 — cosi ) =( a + b )2 – 4ab sin ’ şi ,
unde

: 1

4ab
sin _ i = 1
(a + b ) ( a + b )2
ariabi
et consequenter
[1 - ( +5)*] sinº in
= - = [ - (-3 ) ]su'_

: [&quot; - )*]*sist i -....

2 + b
ban
Differentia nimirum inter unitatem et rationem c ad
a + b consistet in terminis duntaxat infinitesimis , quorum
ordines excedunt omnes ordinem primum .
16º. Idipsum a fortiori dicendum de differentia
inter unitatem et rationem ipsius c ad subtensum arcum
BmE ; siquidem BmE &lt;a + b et &gt; c. Inde fit ut et ar
cus infinite parvus censealur aequalis respondenti chor
dae , et curva quaevis spectetur tamquam polygonum coa
lescens ex laterculis infinitesimis numero infinitis, et isto.
rum laterculorum prolongationes habeantur pro totidem
tangentibus apud varia curvae puncta.
rini
⋅ 500
(I.]
0an
ede-
lur
anali
bf&quot;
Lr;
(im!
51
ac tertia producantur donec concurrant in D. Quoniam an-
guli DBE, DEB sunt infinitesimi, angulus quoque ODE
erit infinitesimus: designetur iste angulus per i, et fiant
BD—fd, BE:c, DE:6;
habebimus
ea :a: ⊹∂≖ —2abcos(180'-'—-i) :03 ∙−⊦ 63 −∣−
Zabensiz—maa-i-ba -l-Zab—Zab-l-2abcosi :(cs-Fb):—
Zab,(1—- cosi):( a --[-b):
-
4absin* −≧−≀⋅∙
nnde
c*— 406 (a- -b)' ∙∙∎∙∙∙∙∙∶↿∙∙∙ . (a -l-b)3 sin ∙⋮−∎
a—ö a .

: ,

.
[1 (—r—b)]sm;-h et consequenter
c
⋍↿∙−−∶∙−∣∶↿ −≺∅≆≴≻≖∃ aina-Li—
a b a
∸⋇⋅∣∶↿ 3
−≺⋮−⋮−−⇣∙≑≻≏∃≏∘⋮∎≖∣⇩ ..;-i ∙−− ∙ ∙ ∙ ∙
DiEerentia nimirum inter unitatem et rationem c ad
a −⊦ &amp; consistet in terminis duntaxat inünitesimis, quorum
ordines excedunt omnes ordinem primum.
160. Idipsum a fortiori dicendum de differentia
inter unitatem et rationem ipsius (: ad subtensam arcum BmE ; siquidem BmE a −↿− 6 et 0. Inde Et ut et ar-
cus infinite parvus censeatur aequalis respondenti chor-
dae, et curva quaevis spectetur tamquam polygonum coa-
lescens ex laterculis infinitesimis numero infinitis, et isto-
rum laterculorum prolongationes habeantur pro totidem'
tangentibus apud varia curvae puncta.52
17º. Fac ut aequatio y f( x) pertineat ad cor
vam ABD ( Fig. 19 ) et sumptis coordinatis orthogonali
bus, sit abscissa OG = x, ordinata CB = y , infinitesimum
abscissae incrementum CC = dx : ducta per C' alia ordinata
C'B' , et per B lineola recta Bm parallela axi abscissarum
OX, erunt B'm = dy , Bm = CC = dx. Pone tangentem
BE occurrere abscissarum axi in E , normalem vero BH in H;
triangula rectangula et similia BEC , B'Bm , BCH dabunt
ydy

:

tang E
-
tang B'Bm dy, ce = ydx CH dx dy dx
CE dicitur subtangens, CH subnormalis.
18º. Ob auctam x area curvilinea BCa'a recipit
incrementum infinitesimum BB'C'C; est autem
BB'C'C =
dx
(rty + dy ) =
dxdy ydx +
2

&lt;math&gt;= ydx + f (x)dx =&lt;/math&gt;
ydr:
2
facta igitur Oa' = xo , erit
BCa'a- j^ydx = ${( )dx Xo Xo
Area BCa'a manifeste traduci polest ad rectangularem a
ream sub ejusmodi lateribus , quorum alterum sit differen
alterum vero ordinata quaedam ym media in
ter ordinatam aa' respondentem abscissae xo et ordina
tam BC respondentcm abscissae x : propterea
tia c Xo ,
X
ſ ydx = ( x - X . \ 'm , seu S f (x )dx = ( x - x . ) f ( xm ) .
X. Xo
Eadem area BCa'a spectari potest veluti summa ex infini
tis numero infinitesimis areolis rectangularibus
52
170. Fac ut aequatioy :
f
(et) pertineat ad cor--
vam ABD( Fig. 19) et sumptis coordinatis orthogonali—
bus, sit abscissa OG:x. ordinata CB: , infinitesimum

abscissae incrementum CC':dx :ducta per 0alia ordinata
C'B', et per B lineola rectaBm parallela axi abscissarum
OX, erunt B'm:dy
, Bm:CC':dx. Pone tangentem
BF. occurrere abscissarum axi in E, normalem vero BH in H;
triangula rectangula et similia BEC , B'Bm, BCH dabunt
J—— , tang E: tang B'Bm :
.. £,CF—Jjæ (31:731: '
]
.L'
CE dicitur subtangens, CH subnormalis.
180. Ob auctam x area curvilinea BCa'a recipit
incrementum infinitas-imum BB'C'C; est autem
⊞∍⋅∁∙∁−−−−↙⋚∁≺⊺ −⊢∫ ⊣−↙≀∫ )
∙−−∶
ydx −⊢ ∂⋅⋅↕−⋮↨−↗− :ydx-l— [figi-£ :ydx:
facta igitur Oa':x.,
, erit
x x
BCa'a:fydx :ff(x)dx.
xo xo
Area BCa'a manifeste traduci potest ad rectangularem a-
ream sub eiusmodi lateribus , quorum alterum sit differen-
tia x —-xo , alterum vero ordinata quaedamym media in-
ter ordinatam aa' respondentem abscissae an. et ordina-
tam BC respondentem abscissae x: propterea
x ⋅ x
fydx: (x -e-x., ly,&quot;, seuff(x)dx:(x—- x.,)f(x,,, ).
x., . xo
Eadem area BCa'a spectari potest veluti summa ex infini-
tis numero inlinitesimis areolis rectangularibus53
f ( x ) dx , f ( x +dx ) dx , COP
f ( xo +2dx ) dx f ( x — dx ) dx ;
nali
imum
Binala
.
sarum ubi nibil sunt aliud f (xo) , f( x + dx ), f (xo + 2dx), ...
nisi ordinatae respondentes abscissis xo , xo + dx , to +
2 dx ,
Quare
entem
in Hi;
bunt C
ſ f(x).lx = f(x )dx + f( xo + dx)dx +
Y :
Xo
fl xo + 2 dx )dc + .
+ f ( x -dx )dx.
recipé 19º. Ponatur arcus aB = s , ejusque incrementum
infinitesimum BB' = ds; quoniam BB'2 = Bm2 + B'ma, erit
2
ds = dx= + dy ,ideoque s= V dx=+dya =
X.
jäevitro
Xo
Tema
iffere
dia is
ordin
200. Circulus habens communia cum curva CC
( Fig. 18 ) duo proxima latcrcula v. gr. AB et BE, dici
tur osculator: sit O centrum istius circuli, BO ( r) ra
dius, OʻK et O'K' perpendicula ex O ducta in AB et BE ,
i angulus OBE , ds' et ds infinitesimi arcus laterculis AB
et BE subtensi, alter spectans ad circulum osculatorem , al
ler ad curyam CC' . Quadrilaterum KOʻKB praebet angu
lum KO'K ' = 180° — KBK' ; sed KBK' = 180°-OBE =
180° -1 ; igitur KOK' = , et consequenter ds' =
r( KOK' ) = ri' . Est autem ( 16 ° ) ds' = ds : propterea
infini
mali-
imum
linat:
arua
entem
Liuii; ↽
bum
⇟⇁∙∎↘⊰
..
recipi
rem ?
illerä
dia i?
orzlïm'
inüw'
53
f(xo)dx,f(xo-I-dx)dx,
f(xo—l—2dx)dx,. . ..f(x—dx)dx;
ubi nihil sunt. aliud f
(..-.,)
,
f(xo—l-dx), f(æQ—l-2dæ). .. . nisi ordinatae respondentes abscissis xo , xo -l-.dx , xo −∣−
de, . .. . Quare
æ
J. f(x)dx :f(xo)dx −⊢∙∣≼ xo-l-dx )dx ∙−∣−
∙↾≀⋅⋅∘
f(xo-l-2dx)dx −⊦ ∙ ∙
..
-I-f(æ-dx)dx.
190. Ponatur arcus aB: :. ejusque incrementum
iniinitesimum BB':ds; quoniam BB'a :Bma—l-B'ma ∙ erit
x
d:&quot;:dxï-l-dyïddeoque s:f V de-l-dy ∶−∙⋅−∙
æo
x
⋅∣∙↙≢∙↿∶⇂∕↿∙∙⊢∣⇃≖≼⋅≖⋅⋟∙
. xn ⋮⋅⋅
200. Circulus habens communia cum curva CG'
( Fig. 18 ) duo proxima latercula v. gr. AB et BE, dici-
tur osculator: sit 0' centrum istius circuli, BO' (:) ra-
dius, O'K et O'K' perpendicula ex 0' ducta in AB et BE,

: &quot; angulus OBE, 'ds' etïds-iniinitesimi arcus later-culis AB

et BE subtensi, alter spectans ad circulum osculatorem, al-
ter ad curvam CC'. Quadrilaterum KO'K'B praebet angu-
lum KO'K':1800 −− KBK'; sed KBK':1800—OBE:
180o — i' ; igitur KOK':i'
,
et consequenter ds&quot;:
r( KOK') :ri'. Est autem ( 16o ) ds':ds: propterea54
ds
21.• Curva CC' sit plana ; exhibeaturque per y =
f (x ), sumptis abscissis x in RX ( Fig. 20) . Erit i = Q
a = - (-a) = - dx , ideoque ( 170)
ds
ds
d x darc ( tang dy -
dx
)
Jamvero (90 )
dy
darc ( tang )
a dy
dr
dy²
dx²
df ( 30)
1 + f ? (x )
f (x ) dx ;
1 + f ? ( x )
dx
igitur
[1 + F2(x) ] }
f &quot; ( 3)
22.• Si ordinata y in curva y =f ( x) fit alicubi
maxima vel minima, exhibeaturque respondens abscissa per
Xn , quisque videt angulum interceptum tangente geometrica
et positivo abscissarum axe fore acutum vel obtusum pro
ut punctum contactus habuerit abscissam x &lt; vel &gt; xn in
casu maximi , &gt; vel &lt; x , in casu minimi , fore autem in
utroque casu = o ubi punctum contactus habuerit abscissam
x = x , Inferimus illud ( 8º. 170) : functio f (xn) est maxi
ma quotiescumque
f ( x) &lt; o quoad x = x + w ( denotat a quantitatem
infinite parvam &gt;0 ) , et f ( 2) &gt; o quoad x = xn - W ;
est minima quotiescumque
54
21 ∙∘ Curva CC' sit plana ;exhibeaturque per :7
f
(x), sumptis abscissis x
in RX (Fig. 20). Erit :&quot; a— a':
—(a'- a):
— dx , ideoque (170)
ds- ds
↗−− dx— dy
darc(tang:ä-;
Iamvero (90)
- si! darc(tang:i-'r .— dx ∙− ↙≀∣↬≺∙↿∶∟∙∙− f (adde;
dx −−↿ ,dJ' 1-t-f'(æ) l*f'ix)
dx'
igitur
3
[1 ⊣∙↾↔≖ (æ) ]
∶⊸∙ f&quot; r— (æ)
22.0 Si ordinata ;- in curva ;-:[(x) (it alicubi
maxima vel minima, exhibeaturque respondens abscissa per
x,, , quisque videt angulum interceptum tangente geometrica
et positivo abscissarum axe fore acutum vel obtusum pro-
ut punctum contactus habuerit abscissam x(vel )x,, in
casu maximi ,
)vel (x,, in casu minimi , fore autem in
utroque casu:
0 ubi punctum contactus habuerit abscissum
x:x,. Inferimus illud ( 80. 170) :functio f
(x,) est maxi-
ma quotiescumque
f
(x) (o quoad x :. x,, ↼⊢ co (denotat a quantitatem
infinite parvam )o )
.
et f' (x) )o quoad x :x. — a) ;
est minima quotiescumque55
f (x ) &lt; o quoad x = x, — W, et f ( x ) &gt; o quoad x =
X'n tw ;
valores X c.quibus respondet maxima vel minima f( xr ), quae
rendi sunt inter radices aequationis
p' ( x)
In
Si f ( x) maneret aut constanter negativa , aut constan
ter positiva, dum x versatur in viciniis x m , certe f ( x ) neque
maxima esset , neque minima .
Ad haec : quoad casum maximi, crescente x in viciniis
decrescit f' ( oc) , decrescente x decrescit f ( x) ; ideoque
df ( x)
&lt; 0 , seu f&quot; ( 30 ) &lt;0 . Quoad casum vero minimi ,
dx
crescente x crescit f (x) , decrescente x decrescit f ' (x ), et
af' ( x)
consequenter &gt; o seu f ( x) &gt;o .
23. Functiones plurium variabilium independen
tium x , 2 , u , ... designantur in hunc modum
dx
F ( x, 2, Ú, ... ) _f ( x, 2, U, ... ) , ...
Ponatur j = f (x ,2 , 9-9.) : si quaevis una ex quan
titatibus x, 2, u, spectetur uti variabilis et habeantur cae
terae pro constantibus , poterunt differentialia functionis u
eodem manifeste modo determinari ac differentialia functio
num quae ab unica pendent variabili. Ejusmodi differentialia
dicuntur partialia , ipsaque sic exhiberi queunt ,
ut
det , draf . d. , dal , ...
denotent differentialia functionis fe , primi , secundi ... ordi
nis quoad x , quoad 2 , ... Ad partiales functiones derivatas
quod pertinet , eae poterunt sic exprimi , ut per
55
f(x)(oquoad x:x,,—-o),etf (x))o quoad x:
a'.—FG);
valores x,,,quibus respondet maxima vel minima f
(x,,)
, quae-
rendi sunt inter radices aequationis
,'(æ)::00
Si f
(x) maneret aut constanter negativa , aut constan-
ter positiva,-dum x versatur in viciniis xn, certe f
(x,) neque
maxima esset
, neque minima.
Ad haec :quoad casum maximi, crescente x in viciniis
x,, decrescit ]&quot; (x) ,decrescente x decrescit f' (x); ideoque
df (x)
dx 0
.- seu f&quot; (x) (o. Quoad casum vero minimi ,
crescente x crescit f
(x) , decrescente x decrescit f' (x) ,
et
consequenter (IS .(ræ) )o seu f&quot; (x) )o.
23.&quot; Functiones plurium variabilium independen-
tium x ,z , u, designantur in hunc modum
F (x, :, ti, ...) ,f( x, :, u, ...
)
,
Ponatur p.:
f
(x, :, a.,.,.) :si quaevis una ex quan-
titatibus x, z,u. spectetur uti variabilis et habeantur cae-
terae pro constantibus
, poterunt differentialia functionis p. ↴
eodem manifeste modo determinari ac differentialia functio-
num quae ab unicapendent variabili. Eiusmodi dill'ereutialia
dicuntur partialia , ipsaque sic exhiberi queunt , ut
dxld-1 dxaPQ'&quot; ∂∷⊬∙∠↨≖≖⊬∙∙∙∙ 
denotent differentialia functionis 9. ,primi , secundi ordi-
nis quoad x , quoad :, Ad partiales functiones derivatas
quod pertinet , eae poterunt sic exprimi ,
ut per56
doll darf
dx dx2
dou , dazle
dedz dza
vel
per
fx(X , Z, Up... ) , f&quot; , (3 , 2, U, ... ) , . f : (3 , 2, U, ... ) ,
fo( %, 2 , Wo...) , ...
designentur functiones , primi , secundi ... ordinis derivatae
ex M = f ( x , % , U. ... ) quoad x , quoad 2 , ... Plerumque
tamen in his derivatis functionibus exprimendis detrahuntor ,
compendii causa , litterae d signa x , % , U 7 .** , et pro
dal d , ² l
dx dx2 d,I
d², M
dz dz
adhibentur
du del i
dx dx2
du dele
dz dz ?
9
24º . Totale functionis pe differentiale due ( quum
nempe x
spectantur omnes ut simul variabiles )
eruitur ex partialibus dx f , d , f , dul , ... ; sunt enim
% , U ,
f ( x + dx, 2, 1, ... ) - f ( x , ,U, ...) = fx ( x ,2 ,4, ... ) dx,
f ( x + dx, atdz, u, ... ) -f( x + dx, 2, U, ... ) =
f: ( x + dx, 2, u, ...) dz = f : ( x, z, u, ... ) dz,
f ( x + dx, atdz, utdu, ... ) — ( x + dx, atdz, u , ...) —
f'u ( x + dx, z + dz, il ... ) du = f ( x, 2, u, ... ) du, etc... ,
)
1
.—
56
≀≀≖≀∸ −−↙≀⇄↕≴∸ .⋅≤≀−⊦∸
−∙∙ −−−⊓≀⇄≖≴∸ dx dx&quot; , dz dza '
vel per
fx(x, :, uh&quot;) ,
f&quot;: (x, :, u, a&quot;) , ∙∙∙ f, (x' z' u, a.) ∙
f',(x, :, u, ...)
,
designentur functiones , primi , secundi .. ordinis derivatae
ex &quot;. f
(æ.:, u.
...
) quoad x, quoad :∙ ∙∙∙ Plerumque
tamen in his derivatis functionibus exprimendis detrahantur,
compendii causa ,
litterae d signa :, a, «,... ,
et pro
de- dx'P- (!sz (I,,[L da:
∙ m ⋅⋅⋅⊤ ⋅−∂⋅⊒⋅−
adhibentur
≴≀−⋅≖∸− ↙≀≖⋅⊀↓ de dw
dæ'dx' ,' de, dza
240. Totale functionis p. dilferentiale dp. (qumn
nempe x , z , n , spectantur omnes ut simul variabiles )
eruitur ex partialibus d, (1.
,d, pt
, d,, p. , ; sunt enim
fl xhi—dx, 39 ut ...) ↼f( æ, :, u, ...):
f, (æ, :, ., ...) dx,
f(æ-l—dæ, t—l—rlz, u,...) — f(x-[r-dx, :, n, ...):
f: (xä-dx, :, &quot;, mida: f, (x, s, u, ...) dz,
f( æ-I- dx, z-l—dz, u-l-du, ...) ∙− ≼∙↧∙⋅∙∣−↙∣∙↧⋅∙ z—i-dz, u, ...) ::
f,&quot; (x-i—dæi z-l-dzo nus) du:f,, (æ, Z, &quot;, ,,.) du, etc-0-
'57
quarum summa praebet
p ( x + dx, atdz, utdu , ...) — f ( x, z , l , ... ) =
fr ( x, 2, U, ...)dx + f : (x , 2, u, ...)dz + f'u ( x,
Z, U, ... ) dut ... , seu dų = d .; + d ,l + d.le + ...
25.• Potest etiam functio pe differentiari successi
ve quoad binas, lernas , ... variabiles v . gr. quoad x, z, quoad
X , 2, u ; etc. ... Id genus partialia secundi , tertii , ... ordinis
differentialia designari queunt per
d, dx M , d , d , dal , ...
sive autem functio u prius differentietur v . gr. quoad x
deinde quoad z , sive prius quoad z , deinde quoad x ,
paallulum attendenti patebit idem in utroque casu pro
venturum differentiale .
26. Detur nunc differentialis aequatio primi ordinis
dy - cydx f ( x ) dx ;
facta y = zu, et adhibita substitutione, emerget zdu + ud:
czudx = f ( x) dx . Pone udz – czudr = 0 ; habebis
dz
= cdx , log ( x ) = cx = cx log ( e) = log ( eⓇx ) ; unde 7
z &gt; eºx :
in ea qua sumus hypothesi
zdu = f(x) dx ;
igilur du = f ( x ) dx
f (x) dx , u Sf (x)dx + G ; et 7 es
ex
1
5
quarum summa praebet
f(x-f—dx, z-l—dz, (kl—du,...) —f(x,'z, u, ...):
f: (æs 31 ut ⋅∙ ')dæ—I—fg (æ, :, II,. ..)dz—l—f'u (æ,
.z, u, .,.) du-l—n., seu dy.— −∙∙ d,.p. :i- dyp. ∙−⊦ d,); ∙∣−∙∙∙
25. ∘ Potest etiam functio p. diil'erentiari successi-
vequoad binas, ternas... .variabiles v. gr. quoad x,z, quoad
æ, :, u; etc.
... Id genus partialia secundi ,
tertii,... ordinis
diii'erentialia designari queunt per
ds dxp'adudadxp-vmi
sive autem functio p. prius differentietur 11. gr. quoad ac
deinde quoad :, sive prius quoad z, deinde quoad x ,
paullulum attendenti patebit idem in utroque casu p1o-
venturum differentiale.
26.&quot; Detur nunc differentialis aequatio primi ordinis
,dy— cydx :f(x) dx;
facta ]: zu, et adhibita substitutione, emerget zdu −⋅⊢ ud:
— czudx :
f
(x) dx
. Pone udz —. czud-r :o ; habebis
dz
Z
∙−−− ∖∙∘⊄≀⋅⋍∙⋅ ,
log (z):cx:cx log (e):
log (e&quot;); unde
∙−−− cx
, z....e :
«
in ea qua sumus hypothesi
zdu :f(x) dx; ∙ ∙
'
igitur du:
,
(x) dx *fbl'c) dx :
&quot;:M—i—C; et 2 0 .: et.:
5 d58
consequenter
y = eriſ f x)dx = C ] :
integratio videlicet dalae aequationis differentialis traducitur
ad integrationem functionis f (x ) dx Porro absoluta aequa er
tionum differentialium integratio eo redit , ut quae relatio
inter quantitatem et quantitatem per eas exprimitur , aequi
valenter exprimatur per aequationes differentialibus liberalas.
27 .. Si dalur differentialis aequatio secundi or
dinis
day dy ta dx +
0 ,
dxt
by:
designantibus k et k' radices aequationis 32 taz +b =0 ,
traducelur illius integratio ad integrationem binarum pri
mi ordinis
dy ' dy - ky ' = 0 ,
dx dx
siquidem , eliminata y' , prodibit
- ky = y ' ;
a
dy
-ky)
dx dy
dx – k G - hy) == 0 ;
quae , ob k tok = -a et kk' = b , recidit in datam. Jam
vero ( 260 )
dx y ' = Cetry = e ** C
elix

:

ergo
y = ek's
es [ foe-tyde +c ] - [ * +c ]=
Ceks + C'ek's .
58
consequenter
yzccxiffiæidæzcl: Bex
integratio videlicet datae aequationis differen'tialis traducitur
f
(x) dx ad integrationem functionis
ac:
. Porro absoluta aequa-
tionum differentialium integratio: eo redit ,
ut quae relatio
inter quantitatem et quantitatem per eas exprimitur , aequi-
valenter exprimatur per aequationes differentialibus liberatas.
27.0 Si datur differentialis aequatio secundi or-
diuis
?;: 437 dr −⊦ &quot;ï; −⊢ b,! −−∶ ∘⋅
designantibus I: et k' radices aequationis z' −⊸⊢ az —]-b:0.
traducetur illius integratio ad integrationem binarum pri-
mi ordinis ⋅
alt&quot; ∙⋅− df
↙↙−↜↕∶∎∎−∎↗⋮∫−−∘∣∠≀↜↿∶−↻ ⋅⋅⋅⋅∙∙⋅−−−−−↗ '
siqnidem , eliminata y'
, prodibit
d d ∠−− '
(dx kf) A(g—F):0.
d.; ⋅ dx ] '
quae ∙Ob k −⋅⊢ ∣⊏∎:
— a et kk':b, recidit in datam. Jam-
vero (260)
.)&quot;: Ce&quot;'.y:e*&quot;[ ∫−⋅≤∎⇂−⊺∶⋮⇆∙⊹∁∙ ]

:

ek'x
ergo
∙∙∙ ': ∙∙ ∙ r ∙ rr Ceu—H).:
..7— e*. [Ca,/460 &amp;) dx—l—C] ∙−−∶ e* k—k. *C]:
- Ce&quot; ∙⋅∣−∁∎ e*&quot; .59
28.- Si daretur d²y dxata
dr. + by = f(x),tra
duceretur integratio ad integrationem binarum dy'
-ky'
da
P(z) Tipo - Ky = y'; sicque prodirent ( 260)
[Sl + c]
e** [ S ,* + c ]
y' = etxe
k et k 'sunt , ut supra, radices aequationis z2 taz + b = 0.
29.• Resumentes functionem f ( x ) , ponamus
f(x) = a, tax taqx? +R3 2 :3 + 04x4 +

:

exsurgent
f ( x) = a + 2a , x + 3az x2 + 404 x3 + ... , f&quot; ( x)
2a + 3.2a3 x + 4.394 va t ... if&quot; ( 0) = 3.2a3 +
4.3.204 x +
Facto x = 0 , emergent ao f (o) , a, =: f ( 0) , a,
i f' (o) , az =-3f&quot; (o), etc.... Hinc
etc...
f(x) = f(0)+xf ( 0) + 1&quot; (0) + &quot;(o) + ...
Sint v. gr. f (x) = e*. f (x) sinx , f (x) = cosx : quoad
primam f (o ) = 1, f (0 ) = 1 , f ' (o) = 1,8 &quot; (0 ) = 1 , etc...;
quoad secundam , f (o) = 0 , f ( 0) = 1 , p (o ) = 0 ,
fr (0 ) • , 1 , f (0) = 0, f ( 0) = 1 , etc...; quoad ter
59
. dfy dy 28.0 St daretur . (: d −⊦∙ 6]:
f
(x) ,.tra-
dxa x
duceretur integratio ad integrationem binarum Si.-..;. ':
dx '7
f
(a:), £
—)(]:
!' &lt;nowiki&gt;; sicque prodirent (260) www-rc]&lt;/nowiki&gt;
730, reli]dx
11 et k' sunt
,
ut supra ,
radices aequationis z' -l—az—-I—-b :o.
29.&quot; Resumentes functionem f
(x) , ponamus
f(x):ao—I&quot;alx −⊦∁≖∙↕≖∙−∣⋅−↷∍ ∷∙⋅∍⊣−∦∣∙∙≂↙∣−⊦∙∙∙⇋
exsurgent
f(x):a, -l-Za,x-l-3a3x3 404 ∞∍−⊢∙∙⋅ ,f&quot; (x):
2a3-1l-3. Zaax-i— 4. 3a(,x2 -[-...,f&quot;(x) :3.2a3—[—
4. 3. 244x—l-..,
,etc...
Facto x:a, emergent a,:
f(o) , a,:
f
(a) , a,:

;

f. (0) , 03 3— f&quot; (0) ' :.
etc-00. Hinc 3 '
f(x): f(0)-l-xf(0)-i-—-f'(0)'l&quot;——f (o)-b&quot;-
Sint v. gr. f(x):ex.f(x) :sinx ,f
(x):
coax :quoad
primam f(o):1, f
(0):1, f' 'o):1,f&quot; (o):1, etc...;
quoad secundam , f(o):0, f(o) :1 , f&quot; (a):0 ,
f&quot; (Ol—&quot;'
-— ∙−− 1, f' (0):0, f' (0):1, etc...; quoad ter-60
23
tiam , f (o ) = 1 , f ( 0) = 0.8&quot; ( ) 1,8 &quot; ( 0 ) = 0 ,
f (o ) = 1 , ' (o) = 0 , p (0 ) 1 , etc... ; ideoque
x2 24 3 e* : 1 tox +*+ +
sin u = r
2.3.4 2.3
x2 8: 4
cos = 1
2.3.4.5 2
+
2.3.4 2.3.4.5.6
1+
ar5
x6
1.5.0 + ...
30.• Adhibita xV - 1 pro x in istarum prima,
emerget
= 1
x2
e **vi .x4
+
2.3.4
Xc6
2 2.3.4.5.6 +
r3 xc5
+
2.3 2.3.4.5 -.)v = 7.
ܪ
unde , ob secundam ac tertiam ,
e #xVST = cos x + V1 sio x .

28. Fac nunc ut punctum materiale vi qualibet continua
sollicitetur ad motum rectilineum: sit »» velocitas puncti
in fine temporis t,,.s spatium percursum , et ds spatiolum
percurrendam subsequente tempusculo dt. Perinde spectari
poterit ds ac si motu uniformi conficeretur , sola nimirum
velocitate praeconcepta 1); siquidem nova velocitas dv, quae
labente d:accedit materiali puncto, utpote infinitesima. ne-
gligenda.Hi11c (1 )
s
[
∥
Motus rectilineus puncti materialis iugiter sollicitati vi
constanter eadem, dicitur uniformiter varius. Per ep desi-61
/
gnetur velocitas, quam vis constanter eadem gignit intra
tempus 1 , erit qe velocitas ( 6 ) genita intra tempus t :
propterea denotante vo velocitatem initialem , qua nimirum
donatur materiale punctum quum t = 0, existet
v =v, +9
ds Hinc dt votoe : fac ut tempori 1 = o respondeat
So ; habebis
s -8 = v. i + 902 ?

;

2
1
et eliminato t ,
v2— v.2 = 29 / s - s . ) :
positis v ,30,
0, erunt
V = pt , s =
- Det
, v?= 205 ,
o dicitur vis acceleratrix : el designante m
massam puno
cti materialis, m q appellatur vis motrix : insuper spatium s in
aequatione ultima vocatur allitudo debila velocitati v.
Ad motum rectilineum utcumque varium quod spe
ctat , nomine vis , acceleratricis apud terminum spatii per
carsi s nihil aliud intelligitur nisi velocitas q , quam gi.
gneret vis conversa in constantem, constantique energia
qua inibi pollet , agens loto tempore 1. lamvero exhibet
do numerum tempusculorum , ex quibus coalescit tempus
1 ; ergo velocitas illa exprimetar per dv; nimirum
1
61
gnetur velocitas, quam vis constanter eadem gignit intra
tempus 1, erit got velocitas (6) genita intra tempus ::
propterea denotante 'Uo velocitatem initialem, qua nimirum
donatur materiale punctum quum :

: o, existet

v:v, ⊣∙− cp :.
Hincd ∙−∙−:v.,-l—got: fac ut tempori :

: o respondeat

,,- s,;habebis
(2
⋅⇟−⋅⋅≖∘∶∶
'&quot;o t&quot;l&quot; 92&quot;;
et eliminato t,
vï—vo*:2?( s—s, ):
positis v,: o ,r,: 0, erunt
v:g0t, :: gt: ,
v': 291,
q; dicitur vis acceleratrix: et designante m massam pun-
cti materialis, m ? appellatur vis motrix: insuper spatium .: in
aeqnatione ultima vocatur altitudo debita velocitati 9.
Ad motum rectilineum utcumque varium quod spe-
ctat
, nomine visacceleratricis apud terminum spatii per-
cursi :nihil aliud intelligitur nisi velocitas ga, quam gi-
gneret vis conversa in constantem, constantique Aenergia ,
qua inibi pollet, agens toto tempore 1. Iamvero exhibet
−↿−∙ numerum tempuscu'lorum, ex'quibus coalescit tempus
dt
1 .: ergo velocitas illa exprimatur per Tit-dv; nimirum62
dv
dt .
et quia dy
d's d
dt ; idcirco erit quoque
dès
d12
habetur pro variabili atque independente quantitate.
29. Fac v. gr. ut materiale punctum sollicitetur versus
datum centrum vi acceleratrice, quae distantiae z' ab eo cen
tro exsistat proportionalis , ut , denotante C ' quantitatem
constantem , habeamus q =C'z' ; sit z, initialis distantia ,
ibique vo =0 , t =0 ; sit insuper v ' velocitas in distantia
z' : erit ( 28 )
v = d (20-3')
dc
dz
dt
du'
ideoque C'z' =
dt
v'dz'
dzi
Hinc 19. Cʻz'dz' = -v'dv'; ex cujus integratione prodit
C- W'2
C'z'2 =C -2'2 , z = ve C'
facto z' =0, erit v' velocitas punci materialisió centro virium ;
exprimit igitur C hujusce velocitatis quadratum : quod si fiat
z' =2 . , erit ex hypothesi v = v = o, ideoque
VT= 2.VC ;
velocitas nempe puncti materialis in centro virium est ut ipsa
initialis distancia zo.
62
0:32- ' dt -'
et quia xlv::! g:- ; idcirco erit quoque
d3s
(:):
d::
'
habetur :pro variabili atque independente quantitate.

29. Fac v. gr. ut materiale punctum sollicitetur versus
datum centrum vi acceleratrice, quae distantiae z' ab eo cen-
tro exsistat proportionalis , ut, denotante C&quot; quantitatem
constantem , habeamus q) :C'z'; sit zoïinitialis distantia ,
ibique v.:o, t:o; sit insuper v' velocitas in distantia
z' : erit ( 28 )
d(zo-z')-——dz' .d ∙
∙−− dp'— v'dz' d: d:
&quot;
eoque c.. d. dz' '
I,.....
Hinc ↿∘∙ C'z'dz':
—- v'dv'; ex cuius integratione prodit
C— 'v'ï cause—w.r: ⇂∕−∁∼⊤−⋮
facto z':o,erit v' velocitas puncti materialisin centro virium;
exprimit igitur(] hujusce velocitatis quadratum: quod si fiat
z':z,, erit ex hypothesi v':v,;—..-o, ideoque
⇂∕⋜⋮∶−−⊸−≖∘⇂∕−∁−⋮≂
velocitas nempe puncti materialis in centro virium est ut ipsa
initialis distantia z..63
2.º du
1 Tc di=
C'zi v CV C -via VC
Vic С
suinptisque integralibus ,
i = C &quot; +
ve are (sin = vo ):
v = o quando i = 0 , proinde
Vc are ( sin = o), exquav = VC sinero.
3º. Cum in centro virium sit v = VC, erit ibi
1 = sint y C , et consequenter t = Inferimus pun
n
2V0
a
1
ctum materiale eodem semper tempore quacumque 2VC
distantia z . perventurum ad centrum illud .
4º. Si materiale punctum movetur vi accelera
trice, quae distantiae a dato centro sit proportionalis , sic
absque formularum subsidio polest ostendi eodem semper
tempore punctum ipsum peryenturum ad centrum illud :
concipiantur duo puncta, quorum primum triplo magis i
nitio molus distet a centro quam secundum : quoniam ex
hypothesi vis est proportionalis distantiae a centro, erit vis
primi triplo major quam secundi , ideoque triplam velo
citatem primo tempuscalo illud acquiret, et triplum spa
lium percurret; quare etiam tripla ibidem residua erit di
stantia. Igitur et secundo tempusculo triplam velocitatem
novam acquiret, et triplum spatiolum tum praecedente, imm
⊖⊰∆
2.-
−≀≀∙↗⋅ ∙∙∙ dv' '
dv' 1 C'z' yel/CT?&quot;— ⇂∕⋅∁⋅ ⇂∕↿−−−∙∙−−∙∽⋅∙∑−⋅ :
sumptisque integralibus ,
1
(
. v' )
are sm −−− ;
C' y'C
v':o quando :: o , proinde

: z ∁∙∙−⊢ ⇂∕

!
(z.—1.-.— arc (
sin: v
), ex qua ≸↗⋅∶∶⇂∕ ⇂∕∁ ⇂∕∁ sint;/CZ
30. Cum in centro virium sit v': l/C,erit ibi
'io
. n ↿∶∶ . sunl/C, et consequenter :: ï— . Infenmus pun-
ctum materiale eodem semper tempore a quacumque 21/ C
distantia za perventurum ad centrum illud.
40. Si materiale punctum movetur vi accelera-
trice, quae distantiae a dato centro sit propmtionalis, sic
absque formula1um subsidio potest ostendi eodem semper
tempore punctum ipsum perventurum ad centrum illud:
concipiantur duo puncta, quorum primum triplo magis i-
nitio motus distet a centro quam secundum: quoniam ex
hypothesi vis est proportionalis distantiae a centro, erit vis
primi triplo maior quam secundi, ideoque triplam velo-
citatem primo tempusculo illud nequiret, et triplum spa-
tium percurret; quare etiam tripla ibidem residua erit di-
stantia. lgitur et secundo tempusculo triplam velocitatem
novam acquiret, et triplum spatiolum tum praecedente, tam64
nova vi et velocitate percurret: unde consequitur ut tripla
pariter sit lota velocitas jam acquisita , triplum totum spa
tium percursum, tripla distantia residua. Propterea et no
vo tempusculo tripla erit nova velocitas acquisita , tri
plum spatium novum percursum , tripla nova distantia; atque
ita porro. Patet igitur post tempus quodvis distantiam
primi fore triplam distantiae secundi, ac proinde imminu
ta in infinitum ac demum evanescente hujus secundi di
stantia, illius quoque primi distantiam in infinitum immi
nui ac simul evanescere: haud poterit ergo secundum pun.
clum ad centrum pervenire, quin simul cum secundo ipso
primum punctum perveniat. Hoc tantummodo discrimen e
rit, quod primum eo deveniet velocitate tripla secundi ; ex
quo manifeste consequitur , quod si primum illud punctum
ex centro cum illa tripla velocitate projicitur , debebit ad
triplam distantiam pervenire; nam vis in recessu velocita
tem codem ordine extinguit , quo generat in accesso. Por
ro quod diximus de ratione tripla , patet generatim conve
nire rationi cuicumque ; nimirum in quacumque propor
tione fuerit distantia prini punci major quam secundi ,
eodem tempore semper ambo ad centrum devenient cum
velocitalibus , quae distantiis initio habitis sint proportio
nales; et si inde discedant cum velocitatibus quibuscumque,
pervenient eodem pariter tempore ad distantias ipsis ve
locitatibus proportionales.
5º. Dicatur tempus quo materiale punctum it ac
redit uude primo discessit; erit ( 3º. )
471 276 271 0
276 VC
Quare ( 1º, 2º. )
2750 C ,
6
0 2751
0
G
220
210
VO TT
z = VC
COS 277
64
nova vi et velocitate percurret: tinde consequitur ut tripla
pariter sit tota velocitas iam acquisita, triplum totum spa-
tium percursum, tripla distantia, residua. Propterea et no-
vo tempusculo tripla erit nova velocitas acquisita, tri-
plum spatium novum percursum, tripla nova distantia; atque
ita porro. Patet igitur post tempus quodvis distantiam
primi fore triplam distantiae secundi, ac proinde imminu-
ta in infinitum ac demum evanescente huius secundi di-
stantia, illius quoque primi distantiam in infinitum immi-
nui ac simul evanescere: haud poterit ergo secundum pun-
ctum ad centrum pervenire, quin simul cum secundo ipso
primum punctum perveniat. Hoc tantummodo discrimen e-
rit, quod primum eo deveniet velocitate tripla secundi; ex
quo manifeste consequitur, quod si primum illud punctum
ex centro cum illa tripla velocitate projicitur, debebit ad
triplam distantiam pervenire; nam vis in recessu velocita-
tem eodem ordine extinguit , quo generat in accessu. Por-
ro quod diximus de ratione tripla, 'patet generatim conve-
nire rationi cuicumque; nimirum in quacumque propor-
tione fuerit distantia prinii puncti maior quam secundi,
eodem tempore semper ambo ad centrum devenient cum
velocitatibus, quae distantiis initio habitis sint proportio-
nales; et si inde discedant cum velocitatibus quibuscumque,
pervenient eodem pariter tempore ad distantias ipsis ve-
locitatibus proportionales.
50. Dicatur 9 tempus quo materiale punctum it ac
redit uude primo discessit; erit (30.)

:

∡⊺≖∙∙∙∙∙⊸ 211 ,—21t
9
⊋⇂↗⇠∁⋮↼−⇀∣−∕−⋐⇀∶⋅−∙⋅⇂∕∁−⇀⊺⋅∙ Quare (10. 20.)
∙∙∙⊇⇂∕∁ 9
220 ∙∙∙⊓∙ ⇂∕∐⋅ ∶∶↼⋤⋮−⇂∕∁∙≀≀↗⋅∶⇂∕∁⊱⋮∥ −−−−⊖−⋅ ⊋⋯⋅ ⋅− 9 271! z': l/C -—-co −−−−∙
271 s 965

=== De verticali gravium descensu atque ascensu. ===
[[30|30]]. Si gravitas aequaliter semper ad sensum corpora decidentia sollicitare intelligitur, motus erit uniformiter varius (28): positis igitur &lt;math&gt;v_0=0,s_0=0&lt;/math&gt;, et denotante &lt;math&gt;g&lt;/math&gt; vim acceleratricem ex gravitate, in ea qua sumas hypothesi determinabitur motus per formulas

&lt;math&gt;v =gt,s=gt^2/2, v^2 =2gs (b) , &lt;/math&gt;

legibusque sequentibus subjicietar.

1&lt;sup&gt;a&lt;/sup&gt;. Spatium &lt;math&gt;s&lt;/math&gt; percursum intra tempus &lt;math&gt;t&lt;/math&gt; est dimidia pars illus spatii &lt;math&gt;s'&lt;/math&gt;, quod percurreretur si grave aequali tempore pergeret moveri uniformiter cum velocitate &lt;math&gt;v&lt;/math&gt; in
fine temporis &lt;math&gt;t&lt;/math&gt; acquisita; nam (1)

&lt;math&gt;s' = tv = tgt = gt^2 = 2s.&lt;/math&gt;

2&lt;sup&gt;a&lt;/sup&gt;. Spatia totalia a gravibus libere decidentibus percursa,
sunt ut quadrata temporum quibus eadem spatia conficiuntur: item ut quadrata velocitatum tempore descensus acquisitarum  Nam

&lt;math&gt;s=gt^2/2=\frac{v^2}{2g}.&lt;/math&gt;

3&lt;sup&gt;a&lt;/sup&gt;. Spatia a gravibus libere decidentibus percorsa
aequalibus et successivis temporibus sequuntur progressio
numerorum imparium 1,3,5,7, ... ; assumpto enim
&lt;math&gt;t = 1,2,3,4 &lt;/math&gt;, ... spatia illa exprimentur per &lt;math&gt;\frac{g}{2},
\frac{4g-g}{2},\frac{9g - 4g}{2},\frac{16g-9g}{2}, \mathrm{seu}\,
\frac{3g}{2}, \frac{5g}{2}, \frac{7g}{2}.
&lt;/math&gt; 

Hae leges experientiae cum sin &lt;math&gt;t&lt;/math&gt; consentaneae, hypothesis gravitatis aequaliter semper ad sensum agentis prope
telluris superficiem existimanda est naturae conveniens: et
quoniam experimentis saepe iteratis apud nostras regiones compertum est, grave sibi relictum percurrere pedes
15, 0915 ... intervallo unius minuti secundi, erit

&lt;math&gt;g = \frac{2s}{t^2} = 2\times 15,0915 ... = 30,183 ... &lt;/math&gt;&lt;ref&gt;9,78:30,183=0,324 m/pes&lt;/ref&gt;

Eam nimirum velocitatem gravitas valet mobili communicare intervallo unius secundi, qua si mobile pergeret uniformiter moveri, absolveret singulis secundis pedes 30,2 circiter.

Deprehenderunt quidem Physici gravitatem esse diversam tum ad diversas supra terrestrem superficiem altitudines, tum ad diversas ab aequatore terrestri distantias: verum ejusmodi variationes in corporum gravitate haud fiunt sensibiles nisi sub differentiis admodum grandibus sive inter altitudines illas, sive inter illas distantias; propterea absque sensibili errore contemni poterunt in ordine ad singula corpora terrestria, quae ut plurimum veniunt consideranda.
Si retenta &lt;math&gt;s_0=0&lt;/math&gt;, ponitur &lt;math&gt;v = a&lt;/math&gt;, exsurgent (28) &lt;math display=&quot;block&quot;&gt;v=a+gt, s = at + gt^2/2, v^2-a^2 = 2gs (b').&lt;/math&gt;

[[31|31]]. Assumpta &lt;math&gt;g&lt;0&lt;/math&gt; in (b'), prodibunt&lt;math display=&quot;block&quot;&gt;v = a-gt, s = at - gt^2/2, a^2-v^2 = 2gs (b'');&lt;/math&gt;

quae formulae manifeste determinant verticalem gravium ascensum. Facta &lt;math&gt;v=0&lt;/math&gt; in tertia ac prima (b&quot;), emergent
&lt;math&gt;
s=\frac{a^2}{2g}, 
t= \frac{a}{g} (b'''),
&lt;/math&gt;
maxima nempe altitudo ad quam ascendit grave, tempusque respondens.

Obiter hic notamus illud: Si datur ejusmodi potentia &lt;math&gt;R&lt;/math&gt;, quae agendo ad modum vis instantaneae valeat massae &lt;math&gt;M'&lt;/math&gt; communicare velocitatem &lt;math&gt;a&lt;/math&gt;, ut sit (6) &lt;math&gt;R= M'a&lt;/math&gt;, ipsa &lt;math&gt;R&lt;/math&gt; agendo ad modum vis continuae per gradus infinitesimos poterit ponderosam massam &lt;math&gt;M&lt;/math&gt; sustentare libratam per totum tempus &lt;math&gt;t = \frac{M'a}{Mg}&lt;/math&gt;

Cum enim singulis tempusculis infinitesimis &lt;math&gt;dt&lt;/math&gt; gignat
gravitas in massa &lt;math&gt;M&lt;/math&gt; quantitatem motus &lt;math&gt;Mgdt&lt;/math&gt;, certe singulis &lt;math&gt;dt&lt;/math&gt; debebit &lt;math&gt;R&lt;/math&gt; ad librandam &lt;math&gt;M&lt;/math&gt; exerere actionem infinite parvam &lt;math&gt;=Mgdt&lt;/math&gt;; proinde totalis actio respondens toti &lt;math&gt;t&lt;/math&gt; erit &lt;math&gt;\int Mgdt = Mgt&lt;/math&gt;: igitur &lt;math&gt;Mgt=M'a&lt;/math&gt;; ideoque etc.

Quisque nunc videt posse vim &lt;math&gt;R&lt;/math&gt; exhiberi non solum per
&lt;math&gt;M'a&lt;/math&gt;, sed etiam per &lt;math&gt;Mgt&lt;/math&gt;.
[[Fasciculus:Atwoods machine.png|thumb]]
[[32]]. Ad motum gravium determinandum in machina Atwoodi, sint &lt;math&gt;m&lt;/math&gt; et &lt;math&gt;m +m'&lt;/math&gt; massae filo appensae: quisque videt motricem systematis vim exhiberi per &lt;math&gt;
g ( m +m' ) - gm =gm'&lt;/math&gt;; unde profluit vis acceleratrix &lt;math&gt;g\frac{m'}{2m + m'}&lt;/math&gt; substituenda loco &lt;math&gt;g&lt;/math&gt; in formulis (b). Quoniam vis ista potest pro lubito attenuari, sequitur in Machina Alwoodi posse motus velocitatem imminui quantum libuerit; quod maxime conducit et ad accuratius definienda spatia percursa, et ad aeris resistentiam tuto negligendam. Sicuti enim corpus, quod movetur in medio aliquo materiali, agit in ipsum medium, ejus particulas expellendo, exerceturque corporis actio juxta motus directionem, ita medii particulae juxta contrariam directionem reagunt (7) in corpus atque resistunt; inde oritur quidem imminutio virium in corpore, sed major vel minor, prout major vel minor velocitas communicatur medio expellendo; et consequenter prout major vel minor est velocitas corporis expellentis.

[[33|33]]. Haec notamus circa gravium motum in medio resistente.

1º. Constat gravia decidentia in pleno homogeneo motum suum vi gravitatis sic accelerare ut paullatim evadat proxime et sensibiliter uniformis. Dum nempe corpus initio movetur, primumque velocitatis gradum acquirit, aliquam hujus gradus jacturam pati debet ex opposita medii resistentia. Sed quia velocitas corporis in progressu semper augetur, multo magis augeri etiam debet medii resistentia; siquidem major corporis velocitas non solum importat ut major quoque velocitas communicetur singulis particulis removendis, sed praeterea ut major quoque resistentis materiae quantitas dato tempore dimoveatur. Ergo velocitatis gradus semper magis imminuetur: unde fit quod velocitas corporis ad valorem constantem propius semper accedat, ejusque motus paullatim evadat proxime et sensibiliter uniformis.
[[Fasciculus:Atwood.svg|thumb]]
2º. Medii resistentia cum tota exerceatur contra corporis superficiem, vis motrix inde resultans haud pendebit ab ipsius corporis massa, eritque eadem utcumque sub eadem et forma, et amplitudine superficiei, crescat vel decrescat massa: non sic dicendum de respondente vi acceleratrice, quae cum obtineatur dividendo vim motricem per corporis massam, permanente et forma, et amplitudine superficiei, erit reciproce ut ipsa massa. Hinc patet cur, caeteris paribus, quo major est massa corporis in medio resistente decidentis, eo etiam rapidior sit motus finalis.

3º. Si concipimus planum variis resistentis medii stratis normaliter occurrens velocitate &lt;math&gt;v&lt;/math&gt;, ponimusque et plani actionem in medii particulas intra singula tempuscula infinitesima sese protendere ad respondentia duntaxat strata dimovenda, et haec eadem strata illico sic dimoveri ut statim atque dimota sunt nullam praeterea actionem sive immediatam, sive medialam exerceant in dimovens planum; expressa per &lt;math&gt;ds&lt;/math&gt; crassitudine strati dimovendi intra tempusculum &lt;math&gt;dt&lt;/math&gt;, per &lt;math&gt;\mu&lt;/math&gt; densitate medii, et per &lt;math&gt;A&lt;/math&gt; area dimoventis plani, orietur inde (28) resistentia &lt;math&gt;A\mu v ds \frac{1}{dt}&lt;/math&gt; seu &lt;math&gt;A \mu v^2&lt;/math&gt;. Duplicatur resistentia in casu medii elastici (23).

4°* Si vis acceleratrix ex medii resistentia assumitur proportionalis quadrato velocitatis, ut denotante &lt;math&gt;\mathrm{k}&lt;/math&gt; quantitatem constantem (experimentis determinandam), exhiberi possit vis illa per  &lt;math&gt;g\frac{v^2}{\mathrm{k}^2}&lt;/math&gt; gravia descendentia sollicitabuntur vi acceleratrice &lt;math&gt;g-g\frac{v^2}{\mathrm{k}^2}&lt;/math&gt; ascendentia vi acceleratrice  &lt;math&gt;-\left(g+g\frac{v^2}{\mathrm{k}^2}\right)&lt;/math&gt;: proinde (28) quoad gravium descensum
 &lt;math&gt;\frac{dv}{dt}=g-g\frac{v^2}{\mathrm{k}^2}&lt;/math&gt; 
quod ascensum
 &lt;math&gt;\frac{dv}{dt}=-g-g\frac{v^2}{\mathrm{k}^2}&lt;/math&gt; 

5°* In primo casu 
 &lt;math&gt;gdt=\frac{{\mathrm{k}^2}dv}{{\mathrm{k}^2}-v^2}=\frac{\mathrm{k}}{2}\left(\frac{dv}{\mathrm{k}+v} + \frac{dv}{\mathrm{k}-v} \right)&lt;/math&gt; 
sumptisque integralibus:(27.6 °) in hypothesi velocitatis &lt;math&gt;v_0=0&lt;/math&gt;,
 &lt;math&gt;gt=\frac{\mathrm{k}}{2}\ln\left(\frac{\mathrm{k}+v}{\mathrm{k}-v} \right)&lt;/math&gt; 
unde 
 &lt;math&gt;e^{\frac{ngt}{\mathrm{k}}}=\frac{\mathrm{k}+v}{\mathrm{k}-v}&lt;/math&gt;
Primum membrum est necessario &lt;math&gt;&gt;0&lt;/math&gt;; ergo et secundum: crescente igitur &lt;math&gt;t&lt;/math&gt; crescet quidem &lt;math&gt;v&lt;/math&gt;; ita tamen ut nunquam fiat &lt;math&gt;v &gt; k&lt;/math&gt;: quod consentit cum dictis (10). Ad haec : quoniam (28)
&lt;math&gt;
dt=\frac{ds}{v}&lt;/math&gt; erit &lt;math&gt;gds=\frac{{\mathrm{k}^2}vdv}{{\mathrm{k}^2}-v^2}&lt;/math&gt; quam integrantes assequemur &lt;math&gt;gs = C - \mathrm{k}^2\ln(\mathrm{k}^2 -v^2)&lt;/math&gt;: in initio motus ex hypothesi &lt;math&gt;v =0 , s =0&lt;/math&gt;, ac proinde &lt;math&gt;C = \frac{\mathrm{k}^2}{2}\ln \mathrm{k}^2&lt;/math&gt;; igitur
 &lt;math&gt;gs= \frac{\mathrm{k}^2}{2}\ln\frac{{\mathrm{k}^2}}{{\mathrm{k}^2}-v^2}&lt;/math&gt;
6°* In secundo casu
 &lt;math&gt;gdt=\frac{{\mathrm{k}^2}dv}{{\mathrm{k}^2}+v^2}&lt;/math&gt;
ideoque (27.13°)
 &lt;math&gt;gt = C - \mathrm{k}\arctan(\frac{v}{\mathrm{k}})&lt;/math&gt;
tempori &lt;math&gt;t = 0&lt;/math&gt; respondet &lt;math&gt;v =v_0&lt;/math&gt;, et consequenter &lt;math&gt;C
= \mathrm{k}\arctan(\frac{v_0}{\mathrm{k}})&lt;/math&gt;; igitur (tang- ):
5l = k [ arc(tang = :) - arc (ranga ) ] .
ds Ad haec : ob de habemus


s
V71
ndum :
gds =
kavdv ; propterea gs = C—
kat va -- 105 (1º + vw).
log ( Kº +w.), ce
ka In initio motus s = 0 , v = Vo;hinc CF
2
gs =
log k2+0.2
katua 2
Facta v = 0 , prodibunt
k2 proind k
log
ktve t
2g 8 are (tang = ).

maxima videlicet altitudo ad quam in medio resistente ascendit grave, tempasque respondens.

7º. Fac ut , exhibente YM ( Fig. 17) directionem
normalem stratis TT ''medii resistentis , planum A oblique''
occurrat stratis ipsis sub angulo BMY ( =\beta ) . Recta bc
parallela rectae YM repraesentet velocitatem v , qua move
tur A : resoluta bc in Kc perpendicularem et BK parallelam
plano A , exprimet Aje . KC2 resistentiam medii ; et quo
niam KC bc . sin Kbc = vsin \beta , iccirco resistentia ista
Ajwa , sin a\beta .
J =

=== De gravium descensu atque ascensu per plana inclinata; de attritu; deque cochlea, et cuneo.===
[[Fasciculus:Free body.svg|thumb|Planum inclinatum]]
[[34]]. Super plano ad horizontem &lt;u&gt;inclinato&lt;/u&gt; collocetur corpus quod habeat centrum gravitatis in &lt;math&gt;G&lt;/math&gt; (Fig. 21) et massam &lt;math&gt;M&lt;/math&gt;; ex &lt;math&gt;G&lt;/math&gt; horizontem demittatur perpendiculum &lt;math&gt;GH&lt;/math&gt;; et ex &lt;math&gt;H&lt;/math&gt; ducatur alterum perpendiculum &lt;math&gt;HB&lt;/math&gt; in communem plani horizontalis et plani inclinati intersectionem; vis motrix ex corporis pondere jacebit in plano perpendiculornm &lt;math&gt;GH , HB&lt;/math&gt;; demisso enim ex &lt;math&gt;G&lt;/math&gt; perpendiculo &lt;math&gt;Gi&lt;/math&gt; in planum inclinatum, vis illa invenietur in plano &lt;math&gt;iGH&lt;/math&gt; normaliter insistente intersectioni plani inclinati et plani horizontalis; quod planum &lt;math&gt;iGH&lt;/math&gt; manifeste recidit in planum perpendiculorum &lt;math&gt;GH , HB&lt;/math&gt;. Sit &lt;math&gt;AB&lt;/math&gt; communis intersectio istius plani et plani inclinati; &lt;math&gt;AC&lt;/math&gt; perpendiculum ex &lt;math&gt;A&lt;/math&gt; demissum in &lt;math&gt;BH ... ; c&lt;/math&gt; angulus &lt;math&gt;ABC&lt;/math&gt;: recta &lt;math&gt;AB&lt;/math&gt; vocatur longitudo plani inclinati, &lt;math&gt;AC&lt;/math&gt; altitudo, &lt;math&gt;c&lt;/math&gt; &lt;u&gt;angulus inclinationis&lt;/u&gt;. Vim motricem per &lt;math&gt;GK&lt;/math&gt; repraesentatam resolve in duas &lt;math&gt;Gi , Gh&lt;/math&gt;, quarum altera sit perpendicularis, altera parallela rectae &lt;math&gt;AB&lt;/math&gt;; erunt

&lt;math&gt;Gi = gM \cos c , Gh = gM \sin c&lt;/math&gt;.

Cadat &lt;math&gt;Gi&lt;/math&gt; intra corporis basim; elisa &lt;math&gt;Gi&lt;/math&gt; a resistentia plani inclinati, gignetur motus a sola &lt;math&gt;Gh&lt;/math&gt;; quae cum maneat constanter eadem, non alium pariet motum nisi uniformiter varium. His positis, ad determinandum gravium motum per plana inclinata satis erit in (6,6' . 30) et in ( 6 &quot; . 31 ) substituere &lt;math&gt;g \sin c&lt;/math&gt; pro &lt;math&gt;g&lt;/math&gt;: denotantibus itaque &lt;math&gt;\theta&lt;/math&gt; tempus, &lt;math&gt;u&lt;/math&gt; velocitatem, et &lt;math&gt;z&lt;/math&gt; spatium, erunt quoad gravium descensum per plana inclinata u = g 9 sin c, z = * gga sin c, u = 2gz sin c ( 6 &quot; )
si tempori 0 = o respondent u = 0,2 = 0 ; et
u = u + go sinc,z = altiglasin c,u ? —a? = 2g zsin c (6 )
si tempori 0 =o respondent u = a, z = 0 : quoad ascen
sum vero
u =a -g6 sinc, z =a9— 1 g 2sinc, a ?—u? = 2gz sin c (65 &quot; )
&lt;u&gt;Componens&lt;/u&gt; &lt;math&gt;Gi&lt;/math&gt; exhibet pressionem, quam exercet grave contra planum inclinatum .

et :spatium , eruntxquoad gravium descen-
sum per plana inclinata
u:g93inc, :

: äggï sinc, 113:2gz sinc (ö&quot;)

si tempori 6:o respondent u:o,:

: o; et

u:u-1-g 9ainc.z:a G—i-äggasin c,u'—a*:Zgzsin c(b')
si tempori 9:o respondent u:a, s:o :quoad ascen-
∙ sum vero
u :::—gg sinc, :349— äggaslncaaa—uzzzgz Sine (b'-l)&quot; r Componeus Gi exhibet pressionem, quam exercet grave
contra planum inclinatum .73

35. Comparantes ( 6 ' ' ) cum (6 ) haec facile stabiliemus.
1. ' Si licals t . erunt i GH
plaui 1 : sin c , s : 2 = 1 : sin c ;
pla
inter
cula
Br
noguls
si duo nempe gravia eodem tempore delabuntur, alterum
verticaliter , alterum per planum inclinatum AB , tam ve
locitates v , u ab ipsis gravibus in descensu perpendiculari
et obliquo acquisitae , quam spatia s , z descripta , erunt
ut longitudo plani ad ejus altitudinem .
2. ° Hinc ubi ex puncto C concursus rectae verti
calis com horizontali ducatur perpendiculam CE ad plani
inclinati lougitudinem AB , grave percurret lapsu obliquo
spatium = AE eo tempore , quo percurreret lapsu verticali
totam altitudinem AC ; nam AC : AE - AB : AC.
3. ° Inde sequitur chordas omnes circuli ad supre
mam , vel infimam diametri verticalis extremitalem pertin
gentes describi eodem tempore ; eo nimirum , quo descri
beretur ipsa circali diameter.
4. ° Velocitates u , v gravium in plano ioclinato
et in recta verticali aequales sunt si gravia ex punctis aeque
altis ad eamdem rectam horizontalem pervenerint : in ea
sumus hypothesi est s = zsinc , ac proinde
a pla
ifors
16 :3
cempo
enim qua

: u2 V =U .

5.° Tempus descensus per longitudinem plani in
clinati ad tempus descensus per altitudinem est ut ipsa
longitudo ad altitudinem : nam in casu ( 4º ) u = v ; ideoque
SIDEN
g9 sin gt , et 0 : t 1 : sin c .

36. Sint nunc plura plana sibi contigua ( fig. 22. * )
diversimode ad horizontem inclinata . Si grave ab AB
transit ad planum BD , in eo transitu non retinebit in
initio plani BD totam velocitatem , quam habebat in
fine plani AB. Si enim concipitur recta AC perpendi
6 et ?
73
35. Comparantes (b &quot;') cum (6) haec facile stabiliemus.
1.&quot; Si 9:t . erunt
v:u:1:sinc,s:z:1:sinc;'
si duo nempe gravia eodem tempore delebuntur, alterum
verticaliter , alterum per planum inclinatum AB, tam ve-
locitates v ,
1: ab ipsis gravibus in descensu perpendiculari
et obliquo acquisitae , quam spatia s
, z descripta , erunt
ut longitudo plani ad ejus altitudinem .
2? Hinc ubi ex puncto G concursus rectae verti-
calis cum horizontali ducatur perpendiculum CE ad plani
inclinati longitudinem AB, grave percurret lapsu obliquo
spatium :AE eo tempore , quo percurreret lapsu verticali
totam altitudinem AC; uam AC :AE :- AB :AC.
3.&quot; Inde sequitur chordas omnes circuli ad supre-
mam , vel infimam diametri verticalis extremitatem pertin-
gentes describi eodem tempore; eo nimirum , quo descri-
beretur ipsa circnli diameter.
. 4.&quot; Velocitates 11 .'v gravium in plano inclinato
et in recta verticali aequales sunt si gravia ex punctis aeque
altis ad eamdem rectam horizontalem pervenerint: in ea
enim qua sumus hypothesi est s:zsinc
, ac proinde
v&quot;:
uz , v :u .
'
5.&quot; Tempus descensus per longitudinem plani in-
clinati ad tempus descensus per altitudinem est ut ipsa
longitudo ad altitudinem: nam in casu (40) u:v ; ideoque
g95inc:gt,et-9:t:1:sinc.
36. Sint nunc plura plana sibi contigua (fig. 22.')
diversimode ad horizontem inclinata. Si grave ab AB
transit ad planum BD, in eo transitu non retinebit in
initio plani BD totam velocitatem, quam habebat -in
fine plani AB. Si enim concipitur recta AC perpendi-
6
- .... ↹∙∙∙↽∙⊾ −↿−⇀⋅⋅⋅⋅↽∙⋅↽ f.:-.. ∙−←−−− ↘−∼∙⋅ ,. ∙∙⋅∙∙∙⇁ . ∙∙
'1
cularis plano BD producto
, et velocitas in fine plani ha
bens directionem AB concipitur resoluta in duas AC
,
CB

; illa prior AC

a novo plano BD elidetur
, utpote quae
tota insumitar in eo normaliter percutiendo
, ac seclusa
0
mois elasticitatis consideratione
, sola altera CB urgebit cor
pus per novum planum BD
, eritque velocitas prior ad no
vam
, qua nempe ingreditur novum planum ut AB

: CB sive ut

radius ad cosinum anguli ABC
, et prior velocitas ad amissam
erit ut radius ad sinum versum ejusdem anguli ABC

; cum

nempe
, si centro
B et radio BA describatur semicirculus
EAE
'
, sit velocitas prior ad amissam ul AB

: CE

.
Erraverunt igitur qui banc velocitatis jacturam minime
considerantes falsum hoc theorema confecerunt,, Ex aliitu
dine quacumque descendens grave per quotlibet ac quaeli
bet plana AB
, BD sibi contigua utcumque inclinata eamdem
in puncto infimo
D velocitatem acquiret ac cadendo perpen
diculariter ex eorum omnium altitudine,, Erit tamen veris
simum theorema si non ad plana contigua quaecumque scd
ad curvas, quae ex infinitis numero rectis lineis et infinite
parvis
( 27. 16
°
) coalescere intelliguntur
, applicetur et poterit verissime sic enunciari ,, Quodlibet grave ex quacum
que altitudine cadens supra superficiem curvam quamcum
que
, eamdem in puncto infimo velocitatem acquiret ac ca
dendo perpendiculariter ex ipsius curvae altitudine ,, Ratio
est quia velocitatum jactura in transitu de uno in aliud
planum
, decrescente angulo quem continet planum alterum
AB cum altero DB producto
, decrescit siquidem decrescente angulo ABC decrescet sinus versus CE repraesentans
velocitatem amissam
. Quare faclo infinite parvo angulo ABC
,
uti contingit in curvis
, velocitas quoque amissa fiet infinite
parva
, ac proinde grave ingredietur planum BD cum ve
locitate acquisita in descensu per planum AB
. Porro sinus
versus CE
' ita decrescit ut, facto infinite parvo primi or
dinis angulo ABC
, ipse CE
' evadat infinitesimus secundi or
dinis

; nam EC
: AC

= AC

: CE

'.
74
cularis plano BD producto
,
et velocitas in fine plani ha-
bens directionem AB concipitur resoluta in duas AC
,
CB; illa prior AC :: novo plano BD elidetur, utpote quae
tota insumitur in eo normaliter percutiendo, ac seclusao-
mnis elasticitatis consideratione, sola altera CB urgebit cor-
pus per novum planum BD, eritque veloeitas prior ad no-
vam, qua nempe ingreditur novum planum ut AB:CB sive ut
radius ad cosinum anguli ABC, et prior velocitas ad amissam
erit ut radius ad sinum versum ejusdem anguli ABC; cum
nempe, si centro
B et radio BA describatur semicirculus
EAE', sit velocitas prior ad amissam ut AB: CE'.
Erraverunt igitur qui hanc velocitatis jacturam minime
considerantes falsum hoc theorema coufecerunt,, Ex altitu-
dine qnacumque descendens grave per quotlibet ac quaeli-
bet plana AB, BD sibi contigua utcumque inclinata eamdem
in puncto infimo
D velocitatem acquirat ac cadendo perpen-
diculariter ex eorum omnium altitudine,, Erit tamen veris-
simum theorema si non ad plana contigua quaecumque sed
ad curvas, quae ex infinitis numero rectis lineis-et infinite
parvis (27. 16&quot;) coalescere intelliguntur, applicetnr; et po-
terit verissime sic enunciari ,, Quodlibet grave ex quacum-
que altitudine cadens supra superficiem curvam quamcum—
que, eamdem in puncto infimo velocitatem acquiret ac ca-
dendo pan-pendiculariter ex ipsius curvae altitudine ,, Ratio
est quia velocitatum jactura in transitu de uno in aliud
planum, decrescente angulo quem continet planum alterum
AB cum altero DB producto, decrescit; siquidem decrescen-
te angulo ABC decrescet sinus versus CE' repraesentans
velocitatem amissam. Quare facto infinite parvo angulo ABC,
nti contingit in curvis, velocitas quoque amissa fiet infinite
parva, ac proinde grave ingredietur planum BD cum ve-
locitate acquisita in descensu, per planum AB. Porro sinus
versus CE' ita decrescit ut, facto infinite parvo primi or-
dinis angulo ABC, ipse CE' evadat infiuitesimus secundi or-
diuis; nam EC: AC:AC: CE'.75
1

37. Hactenus nullam habuimus rationem attritus , seu
resistentiae ex asperitate superficierum : prominentes nem
pe unius superficiei denticuli foveas alterius ingrediun
tur ; sicque haud poterit una superficies alteri superposita
promoveri, nisi ipsi denticuli vel frangantur, vel inflectan
tur, vel , saperiori superficie identidem elevata parumper ,
e foveolis egrediantur: possunt quidem denticuli ita poli
tione imminui , ut sensum inermem effugiant, sed penitus
tolli nequeunt.Statue corpus super plano horizontali ; tum pla
num istud eousque sensim inclina , donec sub quodam angulo
c=c'corpus tantum non incipiat descendere, incipiat vero cre
scente utcumque parum c ultra c' . Attritus respondens angulo
c = c dicatur f: quoniam f accurate librat vim gM sinc'
erit f =g Msinc' ; hinc si per r exprimitur ratio attritus f ad
pressionem gM cosc' ut sit fer. GM cosc ', habebitur
.
r.gM cosc = gM sinc' ,
ideoque
r = tang c' .
0
5
Permanente qualitate massae M, itemque politionis gra
du , constat experimentis quod permanet quoque angulus c' ,
et consequenter ratio r, licet quantitas ipsius M augeatur,
vel minuatur. Inde sequitur attritum f, caeteris paribus, fo
re proportionalem pressioni r.gM cosc' .
Si ponimus attritum adhuc pressioni proportionalem
quum angulus c superat angulum c'; ad habendam ratio
nem attritus in motu gravium per plana inclinata , pro
gsinc substituetur g sin c - rg cosc in ( b ), et gsinc +
rg cosc in ( 6 &quot; ); caeterum in casu motus videtur f non a so
la pressione , sed a corporis quoque velocitate haud pa
rum pendere.
Haec subjungimus.
&quot;
75

37. Hactenus nullam habuimus rationem attritus, seu
resistentiae ex asperitate superficierum :prominentes nem-
pe unius superficiei denticuli foveas' alterius ingrediun-
tur ; sicque haud poterit una superficies alteri superposita-
promoveri, nisi ipsi denticuli vel frangantur, vel mflectan-
tur, vel, superiori superficie identidem elevata parumper ,
e foveolis egrediantur: possunt quidem denticuli ita poli-
tione imminui, ut sensum inermem effugiam, sed penitus
tolli nequeunt.Statue corpus super plano horizontali; tum pla-
num istud eousque sensim inclina
, donec sub quodam angulo
c:c' corpus tantum non incipiat descendere, incipiat vero cre-
scente utcumque parum c ultra c'. Attritus respondens angulo
c:c' dicatur f: quoniam f
accurate librat vim nginc'
erit f

: g Msinc'; hinc si perr exprimitur ratio attritus f

ad
pressionem gM cosc' ut sit:r. gM cosc', habebitur
r. gM cosc': gM sinc' ,
ideoque
r:tang c'
.
Permanente qualitate massae M, itemque politionis gra-
du, constat experimentis quod permanet quoque angulus c',
et consequenter ratio r, licet quantitas ipsius M augeatur,

: vel minuatur. Inde sequitur-attritum f,'caeteris paribus, fo-

1e proportionalem pressioni ngM cosc'.
Si ponimus attritum adhuc pressioni proportionalem
↴⋅ quum angulus c superat angulum ∁∙∍ ad habendam ratio-
lnem attritus in motu gravium per plana inclinata
, pro
igsinc substituetur gsinc—rgcosc in (b' ), et gsinc −∣−
' rgcosc tn ( b &quot;); caeterum in casu motus videtur fnon a so-
lla pressione, sed a corporis quoque velocitate haud pa-
rum pendere.
Haec subjungimus.76
1º . Si corpus in plano inclinato constitutum li
brandum sit potential applicita ( Fig. 21 ) puncto G, quae
potentia et sollicitat ad ascensum, et efficit angulum &amp; cum
AB, gignitque propterea pressionem Qsind, satis erit ut re
sultans ex viribus Q et M ( g sinc F rg cosc ) Fr (sin
exsistat ipsi plano perpendicularis , sese videlicet diri
gat juxta Gi: continet autem Q cum Gi angulum 900
An et vis Mg ( sinc F r cosc ) FrQsinc angulum
cum eadem Gi. Igitur ( 9.10 )
= 90
Q: Mg( sincar cosc ) FrQsing = sin 90 ° ; sin ( 90 °
a ) = 1 : cosa ;
ideoque
sinc Frcosc OSCMS
Q
cos a Es since
secun Sumpio superiori signo, nequit Q esse minor
do membro quin corpus descendat; sumplo inferiori si
gno, nequit Q esse major secundo membro quin corpus
ascendat; perstabit aequilibrium intra limites
sinc - rcosc sinc torcose Mg, el &lt;
cosa + rsing
Mg.
cosu - osinc
2º. In hypothesi nullius attritus erit r = 0 ; et
consequenter
sin c
Q
Mg COSU.
3º. Si Q est insuper parallela horizontali BC, e
rit a = c ; ideoque
76
1&quot;. Si corpus in plano inclinato constitutum li-
brandum sit potentia Q applicita ( Fig'. 21) puncto G, quae
potentia et sollicitat ad ascensum, et eilicit anguluma cum
AB, gignitque propterea pressionem Qsinac, satis erit ut re-
sultans ex viribus Q et M (gsinc :rgcosc ):F r Qsin a
exsistat ipsi plano perpendicularis , sese videlicet diri-
gat juxta Gi: continet autem Q cum Gi angulum :90&quot;—
a, et vis Mg ( sine: rcosc) :rQsina angulum :90&quot;
cum eadem Gi. Igitur ( 9. 1&quot; )
Q: Mgüincqzr 0050 ):t:rQsinat:sin 90&quot; :sin ( 90&quot;—
at:) 1:cosa:;
ideoque
sinc ∓r cosc
−∙∙ Mo
cos a:
r siua
Sumpto superiori signo, nequit Q esse minor secun-
do membro quin corpus descendat; sumpto inferiori si-
gno, nequit Q esse maior secundo membro quia corpus
ascendat; perstabit aequilibrium intra limites
sinc—rcosc sinc rrnsr Q)...— Mg, et Q(
−⊢ Mg.
cosa rsiuat cosa—rsiua
2&quot;. In hypothesi nullius attritus erit r: o ; et
consequenter
sin 0 M
g. szz cosa:
3&quot;. Si Q est insuper parallela horizontali BC, e-
rit at:c ; ideoque77
sipc :
Mg COSC
potentia videlicet ad pondus ut plani altitudo AC ad hori
zontaleon BC. Hinc facile deducuntur aequilibrii leges in
cochlea et cuneo.
4º. Cum cochlea non sit nisi planum inclina
tum ABC, quod circum cylindruni ducitur; dum vero co
chlea agit , potentia sit rectae lineae BC parallela, erit
potentia ad pondus seu resistentiam superandam , ut al
titudo plani seu helicam distantia ( =h ) ad basim plani
seu cylindri peripheriam ( = k ). Hinc
Q
hP ;
k
quae formula supponit distantiam inter cylindri axem et pun .
ctum cui applicatur potentia , esse ipsius cylindri radium
( = m ) : quod si distantia illa fiat alia ab r', et exhibea
tur per R' ; denotante e potentiam respondentem novae
distantiac, exsistet
mi?
R'
ac proinde Q
-
hP
R '
LP
25R .
k
In ordine ad cochleam infinitam , dicatur A radius ma
joris rotae , a radius minoris , et P' pondus seu poten
tia apud dentes ipsius rotae majoris; erunt
ар P =
Q
api A
hP
27.R '
ideoque
Q =
h a P
21AR'
77
Q sine. NT: ⋅⇀ SE.—.'
potentia videlicet ad pondus ut plani altitudo AC ad hori-
zoutalem BC. Hinc facile deducuntur aequilibrii leges in
cochlea et cuneo.
4&quot;. Cum cochlea non sit nisi planum inclina-
tum ABC, quod circum cylindrum ducitur; dum vero eo-
chlea agit, potentia sit rectae lineae BC parallela, erit
potentia ad pondus seu resistentiam superandam, ut al-
titudo plani seu helicum distantia ( :h)ad basim plani
seu cylindri peripheriam :( k). Hinc
Qz—k-i
quae formula supponit distantiam inter cylindri axem et pun-
ctum cui applicatur potentia, esse ipsius cylindri radium
(: r' ): quod si distantia illa fiat alia ab r', et exhibea-
tur per B'; denotante Q' potentiam respondentem novae
distantiae, exsistet
Q'—r' ∙∙ ∙∙∙∣≖∣⊃⋅↿⋅⋅∙−∣≀∌ ∙≺⋮−−−−∙↓⊤∙ ac ptomde QI—B— . ∣∙⋮−−−−∙ ⊋∙⋮⋮⋅⋮↸↽∙
In ordine ad cochleam infinitam, dicatur A radius ma-
ioris rotae
, a radius minoris , et P' pondus seu poten-
tia apud dentes ipsius rotae maioris; erunt
aP , hP'
P::ï'Q—an' '
ide ue
Oq haP78
1
5 ° Cuneus spectari potest tamquam coalescens ex
duobus planis inclinatis ut ABC, tum ob figuram suam ,
tum quia idem est sive pondus per planum inclinatum
trahatur sursum , sive planum sub pondere promoveatur.
Agit autem potentia in cuneo juxta CB; quoad igitur u
1
nam cunei partem ABC respondens potentia Qerit ad
m
1
respondentem resistentiam P ut AC ( = D ) , sen di
midia cunei crassities ad BC ( = H ) , idest ad altitudinem
1
Q
1
ad respondentem resistentiam P P erit pariter ut
į D ad H. Igitur
m
LQ.H - 1P.HD, Q (m - 1 ) .
A
m2 m m
P (m - 1 )
mi '
· D ;
quibus aequationibus in summam collectis ,
Q. H = P. , D ,
et consequenter
D
H totalis videlicet potentia Q ad totalem resistentiam P ut
dimidia cunei crassities D ad ejus altitudinem H ; mo
do tamen exerceatur resistentia normaliter ad H.
6º . Si in cochlea v . gr. considerandus esset at
tritus , haberetur ( 10.40.) ,
1
!
1
5&quot; Cuneus spectari potest tamquam coalescens ex
duobus planis inclinatis ut ABC, tum ob figuram suam,
tum quia idem est sive pondus per planum inclinatum
trahatur sursum, sive planum sub pondere promoveatur.
Agit autem potentia in cuneo iuxta CB; quoad igitur u-
. . 1 nam cune1 partem ABC . respondens potentta —Qer1t ad
.
'
m
respondentem resistentiam −↿−∙∶ P ut AC (: äD ), seu di-
↾
m
midia cunei crassities ad BC (: H ), idest ad altitudinem
. . 1 cunei. Quoad alteram partem respondens poteutta Q—- −− Q
m
. . 1 ad . . respondentem rc51stent1am P
-— −−∙ P er1t partter ut
171
&amp; D ad H. Igitur
D, Q—(m-1).H: −↿−↽≺≀∙∥∶∶∙−↿∙−↕⊃∙ ;.
m m m
P
∙∙
-
(m'1) -äD;
,- ut
quibus aequationibus in summam collectis,
QaHzpaL'D, et consequenter
≟≺−≀∙∙− :D
.
P −⋅⋅ H ⋅
totalis videlicet potentia Q ad totalem resistentiam P ut
dimidia cunei crassities &amp; D ad ejus altitudinem H; mo-
do tamen exerceatur resistentia normaliter ad H.
6&quot;. Si in cochlea v. gr. considerandus esset at-
tritus , haberetur (10. 4&quot;.),
≁−−−−∎⋅−−
−−⋅⋅...-—79
sinc FrcoscP =
cosc trsinc
h = 2 te r'r P ;
h Erk
P
k trh 2 trh
ideoque
Q
Qr Pr' h = 27r's
R ? -R 2 r'trh
0
70. Veniat quoque considerandus attritus in ae- ,
quilibrio corporis AB ( Fig. 23: 24 ) , quod ad rolatilem
motum circa fixum cylindrum sollicitatur vi Rjacente in
plano, cui normaliter insistit axis cylindri.
Ac primo cylindrus impleat accurate circularem cor
poris aperturam DE ( Fig. 23 ) , in quam inseritur: per cy
lindri centrum O duc rectam OEE' parallelam vi R , et
pancto E corporis AB applica duas . vires Q ', Q' aequa
les eidem R, et contrarias, alteram nempe tendentem, ab
E versus E' , alteram ab E versus O; vi R licebit substi
tuere systema virium R , Q ', Q &quot; : et cum possint absque sy
stematis turbatione sic transferri ( 11 ) R et l ' ut aequi
distent ab O, eae nitentur dumtaxat gignere motum ro
tatilem circa cylindrum quin ullam pariant pressionem a
pud ipsius cylindri superficiem ; pressio igitur in hanc su
perficiem redigetur ad solam
୧ = R , ideoque f = Rr.
Attritus fest vis tangentialis respectu superficiei cylin
dricae; hinc denotante a radium OE cylindri , et p per
pendiculum Oi ex O ductum in directionem potentiae R,
ad aequilibrium satis erit, ut exsistat ( 9. 2° )
R 1
2
р
Rr
. 79
Q-—sinc:r:rc.oscP 11:er P—h:t:2nr'rp
cosczbrsmc R::brh 2nr':t:rh ,
ideoque
—Qr' Pr' II::ZRr'r
a' &quot;B' 'an'äzrh Q! '
70. Veniat quoque considerandus attritus in ae- ↗
qnilibrio corporis AB ( Fig. 23: 24 ), quod ad rotatilem
motnm circa fixum cylindrum sollicitatur vi R iacente in
plano, cui normaliter insistit axis cylindri.
Ac primo cylindrus impleat accurate circularem ocr-
poris aperturam DE (Fig. 23), in quam inseritur: per cy- ⋅
lindri centrum O duc rectam OEE' parallelam vi B, et
pnncto E corporis AB applica duas, vires Q', Q&quot; aequa-
les eidem R, et contrarias, alteram nempe tendentem, ab
E versns E', alteram ab E versus O; vi R licebit substi-
tuere system virium R, Q', Q&quot;: et cum possint absque sy—
stematis turbatione sic transferri (11) B et Q'0ut aequi-
distent ab 0, eae uitentur dumtaxat gignere motum ro-
tatilem circa cylindrum quin ullam pariant pressionem a-
pud ipsius cylindri superficiem; pressio igitur in hanc su-
perficiem redigetur ad solam
Q&quot; −∙∙−− R, ideoque f: R r.
Attritus fest vis tangentialis respectu superficiei cylin-
dricae; hinc denotante a radium OE cylindri, et p per-
pendiculum Oi ex Oductum in directionem potentiae Pt,
ad aequilibrium satis erit, ut exsistat ( 9. 20)80
et consequenter
P
facto p &gt; ar , disrumpetur aequilibrium ; facto p &lt; ar ,
subsistet .
Ponatur secundo circularis apertura corporis baud
impleri accurate cylindro ( Fig.24) : vis R manifeste trans
ibit per contactum E cylindri et corporis AB . Resolve R
in duas EF, et ED' , quarum altera transeat per centrum 0 ,
altera tangat cylindrum : per EF exprimetur pressio ; ac
proinde
f = r.EF .
Obtinebit igitur aequilibrium quotiescumque
ED ' &lt; r. EF ,
vel ED' = r.EF :
cum autem ( 9. 1. ° ) .
ED' : R = sin FER ; sin D'EF = sin FER : 1 ,
EF : R = sin D'ER ; sin D'EF = cos FER : 1 ,
cumque ducto perpendiculo Oi ex O in ER ,
Oi Ei voa ?
OE sin FER Р cos FER 22 -
p2
a OE
iccirco praefatac aequilibrii conditiones vertentur in
Rp Rr Vap2 Rp a
Rr Va - p ?
a a
quae traducuntur ad
80
et consequenter
&quot;' p :: ar :
facto p ar, disrumpetur aequilibrium; facto p ar
,
subsistet .
l
Ponatur secundo circularis apertura corporis baud
impleri accurate cylindro (Fig.24): vis B manifeste trans-
ibit per contactum E cylindri et corporis AB
. Resolve B
in duas EF, et ED' , quarum altera transeat per centrum O,
altera tangat cylindrum: per EF exprimetur pressio; ac
proinde
f

: r. EF.

Obtinebit igitur aequilibrium quotiescumque
ED' (r. EF , vel ED' −−∶ r. EF :.
cum autem (9. 1.0).
.'
ED': R ::sin FER : sin D'EF :sin FER : 1 ,
EF fii ∙−−∶ sin D'ER; sin D'EF: cos FER : 1,
cumque ducto perpendiculo Oi ex 0 in EB
.
Oi p Et. ⇂∣ (13 ∙−− :;
' :∙−−− :... ∙ FER ↽− −∙ p
sin FER 08 a 005 OF. a ,
iccirco praefatae aequilibrii conditiones vertentur in
n,,(RrI/aa—pz ↧≹∣↗∙∙∙↧≹≀⋅ Wiz—pa
7.&quot; −−−−↴∶∎−−∙−↙≀∎ ⇀⇀ a '
quae traducuntur ad
⇁−∙↱⇁≓≓81 1
ar 2
p &lt;
р
1 + 12 vit ?
8.• Si ponitur R nihil esse aliud nisi resultans
ex datis viribus P' , Pi ad puncta data v . gr. A , B appli
citis , innotescet R ex dictis ( 10 ) , itemque p. ex ( 10.3° ) .
Sic habetur ratio attritus in vecte : caeterum in machinis
praeter resistentiam ex attritu spectanda etiam est resi
stentia ex funibus . Hi enim inflexioni suae resistunt quum
cylindris vel trochleis circumvolvuntur; et quidem eo ma
gis , quo majori pondere tenduntur , quo insuper crassio
res sunt , et quo minor fuerit trochleae, aut cylindri radius.

=== De motu gravium oblique projectorum.===
[[Fasciculus:Ferde hajitas2.svg|thumb]]
[[38]]. Grave &lt;math&gt;M&lt;/math&gt; (Fig. 25) juxta directionem MG velocitate &lt;math&gt;v_0&lt;/math&gt; projectum urgebitur duplici motu, altero aequabili per &lt;math&gt;MG&lt;/math&gt; ex impetu recepto, altero (nihil est aliud nisi motus relativus mobilis &lt;math&gt;M&lt;/math&gt; quoad ipsum &lt;math&gt;M&lt;/math&gt; iens per &lt;math&gt;MG&lt;/math&gt; sola &lt;math&gt;v_0&lt;/math&gt;) uniformiter accelerato gravitatis proprio per rectam verticalem &lt;math&gt;MR&lt;/math&gt;, vel ipsi &lt;math&gt;MR&lt;/math&gt; parallelam. Sit &lt;math&gt;S&lt;/math&gt; spatium quod cumque &lt;math&gt;MC&lt;/math&gt; primo illo aequabili motu seorsim sumpto percursum, &lt;math&gt;t&lt;/math&gt; tempus impensum ad ejusmodi spatium percurrendum, sitque &lt;math&gt;s&lt;/math&gt; spatium &lt;math&gt;MF&lt;/math&gt; pari tempore percursum secundo motu item seorsim sumpto. Completo parallelogrammo &lt;math&gt;MFQC&lt;/math&gt;, in fine temporis &lt;math&gt;t&lt;/math&gt; grave erit (5) in &lt;math&gt;Q&lt;/math&gt;;
et quia (1:30)
&lt;math&gt;S = v_0 t , s = \frac{gt^2}{2},&lt;/math&gt;
eliminato &lt;math&gt;t&lt;/math&gt;, existet
&lt;math&gt;S^2 = \frac{2 v_0^2}{g}s &lt;/math&gt;
aequatio ad curvam (dicitur parabola) in qua defertur grave. Si altitudo debita (28) velocitati &lt;math&gt;v_0&lt;/math&gt;, dicatur &lt;math&gt;\mathrm{A}&lt;/math&gt;, erit
&lt;math&gt;v_0^2 = 2g\mathrm{A}&lt;/math&gt;, et aequatio transformabitur in
&lt;math&gt;S^2 = 4 As ( c)&lt;/math&gt;.

[[39|39]]. Denotet x horizontalem rectam MK , y vertica
lem KQ , et h angulum CMK ; erunt x = S cosh , y =
CK - CQ = S sin h -5 ; unde
X
X
S =
cosh . sinh :
cosh
quibus valoribus substitutis in (c) , prodibit
x2 rcsinh
4 A
CO -Y) ,
et consequenter cos2 h
cos h
y =xtang h 1 + tang k
4 A
x2 ( c' ) .

[[40|40]]. Haec facile nunc stabiliuntur. 1º facta y = 0 ,
proveniet amplitudo jactus
4 Atangah
1 + tang h
4Asinhcosh = 2 Asin2h.
2.º Inde sequitur maximam jaclus amplitudinem
haberi sub angulo h = 45°.

3. ° Si quaeritur angulus h , sub quo proiicien
dum est grave ut offendat in datum scopum , cujus nempe
dantur coordinatae x et y , erit
2A + V 4A2-4 Ay - x2
tangh
82
aequatio ad curvam (dicitur parabola) in qua defertur
grave. Si altitudo debita (28) velocitati v, dicatur A, erit
vio:2gA, et aequatio trausformabitur in
S':4As (c) Esistente igitur 4 A 2—4 Ay-x &gt;0 , poterit sub duplici
angulo projici grave ut datum -scopum attingat : attinget
autem in fine temporis ( 38 : 39 )
S
ts Vo .
Vo cos h

4.0 in ( c ) pone
2 Atangh ta ;
1 +tangah
babebis
A tangah
1+ tang2h ya
1 + tang 2h W?
4A
( c ' ' ) .
Iam vero maxima y ( dicitur altitudo jactus ) manifeste re
spondet valori w = 0 ; altitudo igitur jactus exhibebitur
per
A tangah seu A sinh.
1 +tangah

5º . Ex eadem ( c &quot; ) quisque colligit parabolam , in qua
defertur grave, dividi a maxima y in duas aequales simi
lesque partes : extremitas maximae y vocatur vertex pa
rabolae; ipsa vero maxima y indefinite producta juxla gra
vitatis directionem appellatur axis parabolae.
[[Fasciculus:Ferde hajitas7.svg|thumb]]
6º Si angulus h fit &lt; o, ut initialis directio cadat iтfra horizontalem rectam ML, jactus amplitudo x (1°) ex &gt; fiet &lt; 0; jactus vero altitudo y ( 40 ) permanebit &gt;o. Quod si fuerit h = o, ut initialis directio recidat in rectam horizontalem ML, nulla erit amplitudo jacеus, nullaque ejus altitudo.

7º. Demittatur perpendiculum QP ex puncto Q
parabolae in axem NI ... , sintque NP = x', Q P =y'; erunt ( 1º . 4º . ) x MI — QP = 2 A sinh con -y'
y=NI — NP = A sin’h— x' : quibus valoribus substitutis
in ( c' : 39 ) , proveniet
y2= 4 A x' cosah
aequatio ad parabolam M N L inter x' ety' computatas a
vertice ; quantitas 4 A cos’h dicitur parameter parabolae ;
quod si in axe sumatur punctum H ita , ut ejus distantia
a vertice sit quarta parametri pars seu A cos ?h , habebitur
punctum illud , quod appellatur parabolae focus.

[[41]]. Cum ad curvam parabolicam describendam, corporis motus, qui fit secundum lineam projectionis, debeat esse aequabilis, qui vero fit secundum lineam verticalem, debeat esse uniformiter acceleratus, cumque hujusmodi certe neuter esse possit si medium utrique motui resistat, iccirco nonnisi in vacuo motus corporis oblique projecti fieri potest per curvam, quae sit perfecte parabolica. In medio resistente curva minus late patet, minusque assurgit quam in vacuo; duobus insuper cruribus dissimilibus &lt;math&gt;AN, NL&lt;/math&gt; (Fig. 26) componitur, quorum descendens &lt;math&gt;NL&lt;/math&gt; ad rectam quamdam &lt;math&gt;FE&lt;/math&gt; ut asymptotum accedit in infinitum, quin unquam congruant. Etenim resoluta projectionis velocitate in duas, alteram verticalem, alteram horizontalem, verticalis tum ab aeris resistentia, tum a gravitate usque ad punctum &lt;math&gt;N&lt;/math&gt; minuetur: propterea punctum &lt;math&gt;N&lt;/math&gt; minus assurget quam in vacuo: postquam grave ad &lt;math&gt;N&lt;/math&gt; pervenerit, descendet ob gravitatis vim damna ex medii resistentia reparantem, et hujusmodi descensus fiet motu verticali ad motum aequabilem (33) semper accedente. At horizontalis velocitas minuitur perpetuo, nulla interim vi iacturam reparante, atque inde fit ut recessus horizontalis a recta verticali &lt;math&gt;NP&lt;/math&gt; certum limitem non praetergrediatur, quem curva habet pro asymptoto. Haec contingunt potissimum corporibus ingenti velocitate in aere projectis.
=== De generalibus quibusdam proprietatibus motus curvilinei, orti a viribus, quarum una determinat materiale punctum ad motum aequabilem, altera ipsi materiali puncto est continue applicata.=== 
[[42|42]]. Concipiamus secundam vim agere solum in initiis quorundam tempusculorum, ac tantam velocitatem unico impulsu valido producere, quantam vis perpetuo agens producit toto illo tempusculo, ut deinde inminuta magnitudine tempusculorum in infinitum, habeatur linea curva orta ex continua vis actione.  

Projecto puncto materiali cum velocitate CB (Fig. 27) simulque illi impressa velocitate CA, abiret punctum per diagonalem CO parallelogrammi AOBC et esset in fine primi tempusculi in O cum determinatione describendi altero aequali tempusculo rectam OL = OC, eique in directum jacenlem. Si hic iterum illi imprimeretur alia velocitas OF, completo parallelogrammo FILO , incederet per diagonalem OI, essetque in fine secundi tempusculi in I cum determinatione describendi tertio tempusculo aequali rectam IM = 10, eique in directum jacentem. Sed ob impressam hic quoque aliam velocitatem IV abiret per novam parallelogrammi diagonalem IH, atque ita porro. Fieret ergo in ejusmodi hypothesi vis agentis per intervalla tempusculorum ut materiale punctum describeret polygonum COIHN etc, cujus latera certam magnitudinem et positionem haberent, definita nempe a directione virium et a ratione velocitatum, quas initio cujusvis tempusculi mobile obtineret. Hinc pro diversis virium ila agentium ordinibus numero infinitis infinita considerari possunt ejusmodi polygona, quorum alia in se ipsa redirent, desinente ultimo latere in puncto C ubi primum inceperat; alia abirent in infinitum. Concipiamus jam numerum tempusculorum augeri, et simul eorum magnitudinem imminui in infinitum, vitum magnitudine tum directione vel constantes manere, vel variare certa quadam lege ad continuam quamdam variationis rationem accedente in infinitum. Augebitur in infinitum numerus laterum polygoni determinato tempore descripti, imminutis interea in infinitum angulis, quos efficit quodlibet latus praecedens cum consequente: cum enim LI debeatur impulsui, qui initio tempusculi 0 eam
velocitatem producere concipitur, quam produceret vis to
to tempusculo agens, cumque per tempusculum infinitesimum vis ista habenda sit pro constante, existet ( 28: 30. 14. ) LI =
092; ideoque ob o finitam, et quadratum 62 infinitesimum secundi ordinis, erit etiam LI infinitesima ordinis secundi, sed OL est infinitesima ordinis primi, utpote quae tempusculo O describitur cum velocitate finita; ergo angulus LOI erit ivfinitesimus: atque eodem pacto demonstrantur infinitesimi anguli MIH , K'HN , etc. Hinc polygonum ad curvam continuam semper magis accedet; et ubi demum continua habealur actio vis, et continuae cuidam legi subjiciantur directio ipsius et magnitudo, obtinebitur curva continua cavam sui partem versus eam plagam obvertens, in quam tendunt vires.

43. Abeunte polygono in curvam , rectae CL , OM' , IH ', HK , etc abeunt in tangentes apud puncta C, O, I, H , etc. Ubi ergo in aliquo curvae puncto vis desinat agere,,
excurret mobile per tangentem apud illud punctum.

44. Sit IM (fig, 28 ) spatiolum quod tempusculo 9
mobile percurreret sola velocitate praeconcepta, et IV
spatiolum respondens vi agenti unico impnlsui valido ; ita
ut existat (42) IV ::99&quot;. Completo parallelogrammo, positis-
que lM:P ,
lH:B, et angulo MIV :i, erit (9. 3.&quot; )
∶∶ Vra-Hæ os −⊢⋅∠⇂⊃⊊↶⊖⋍ cos :.87
Evolvatur quantitas radicalis in seriem : proveniet
R = P + q9 cos i , unde R - P = º02cosi ,
neglectis infinitesimis altioris ordinis. Sit v' velocitas , qua
mobile percurrit laterculum R; erit R = v'0 : sit etiam v
velocitas , qua mobile percurreret laterculum P. si non
adesset impulsus validas in I ; erit P =v @ : hinc R --
P = vv( ) 0 .; et consequenter
v ' - v = q Ocosi.
Ex hac aequatione patet v— esse quantitatem in
finitesimam primi ordinis , positivam vel negativam prout
i &lt;vel &gt; 90° , esse autem =0 si i 90° . Inferimus il
lud : ubi tempore finito angulus , quem efformat vis ac
celeratrix cum directione tangentis , fuerit semper aculus,
acquiret mobile incrementum velocitatis finitum ; si sem
per obtusus , patietur decrementum finitum ; si semper re
lus , velocitas manebit constans.

45. In motu quovis curvilineo quadratum velocitatis aequat vim acceleratricem ductam in dimidium chordae, quae ex ejus directione abscinditur a circulo osculatore. Denotet enim a lineolam infinitesimam IM (Fig. 29) ut sito et consequenter IV = 902
cipiatur circulus , qui transiens per tria puncta 0 , I ,
H ( fig . 27. 29. ) habeat centrum in G , quique erit
circulus osculator apud curvae punctum O ; producantur
IV , MH donec occurrant peripheriae in G &quot; , G '' ; et ex''
G ducatur perpendiculum GGʻad chordam IG &quot; : erunt
IG &quot;
MG &quot; = IG &quot; = ICE Est autem MH . MG ' &quot; : MI. MO;
2
ergo MH . 21Gʻ = MI.MO = MI . 2MI , seu 21G'
2x2. Hinc v2 = . IGʻ ; ideoquc etc. Porro angulus IGG' =
2 Oxa

; con

.
px ?
22
87
Evolvatur quantitas radicalis in seriem : proveniet
B:P −⊢ o9zcos i
, unde B—P:cp92cosi ,
neglectis infiuitesimis altioris ordinis. Sit 'v' velocitas , qua mobile percurrit laterculum R; erit R: 0'9: sit etiam »
velocitas , qua mobile percurreret laterculum P. si non
adesset impulsus validus in I, erit P:-v 9: 'hinc R
-— P:(v'--v)9; et consequenter
v'—v:cp9cosi .
Ex hac aeqnatione patet 'o'—v esse quantitatem in-
fiuitesimam primi ordinis
, positivam vel negativam prout
i(vel 90&quot; , esse autem :0 si 1&quot;: 90&quot;. Inferimus il-
lnd : ubi tempore finito angulus, quem efformat vis ac-
celeratrir cum directione tangentis ,
fuerit semper acutus,
acquiret mobile incrementum velocitatis finitum; si sem-
per obtusus, patietur decrementum finitum; si semper re-
ctus, velocitas manebit constans.

45. In motu quovis curvilineo quadratum velocitatis
aequat vim acceleratricem ductam in dimidium chordae,
quae ex eius directione abscinditur a circulo. osculatore.
Denotet enim a lineolam infiuitesimam IM (fig. 29. )
gox- ; con-
92
cipiatur circulus, qui transiens per tria puncta 0, I,
II (fig. 27. 29..) habeat centrum in G, quique erit
circulus osculator apud curvae punctum 0; producantur
IV, MH donec occurrant peripheriae in G&quot;, G'&quot;; et ex
G ducatur perpendiculum GG' ad chordam IG&quot;: erunt
MG&quot;':IG&quot;, −−−∙−↧∁⇀−− ⇀∸−↧−⊊≩−⋅∎−∙ Est autemMH. MG'&quot;:MI. MO;
ut sit :9 i,
et consequenter IV:
99&quot;:
'»
ergoMH.21G':MI.-:MO MI. 2Ml.seu—-— &quot;&quot;&quot;&quot; ,210':
'v&quot;
.Hiuc v&quot;: 39. lG' ; ideoque etc. Porro angulus IGG'— −∙∙
−∙↼⇀−−
. −↼∙⋅⋅∙∙⋅↼−∎∣ −↼
∙∙∙88
90 ° -GIGʻ = 900 (MIV - MIG ) = 90 ' - ( i - 90 °) = 180 °-i ;
proinde , denotante r radium GI , erit IG ' = rsin IGG' =
rsini , et consequenter
va = grsini ( b ) .

:

46. Haec vera sunt de omni virium genere. Sint nunc
vires acceleratrices directae ' ad centrum datum : in casu,
curva ColH .... ( fig . 27. ) jacebit in plano transeunte
per rectam projectionis et per centrum virium ; quod fa
cile intelligitur , cum in eodem plano agant vires omnes.
Viribus ad punctum aliquod S tendentibus , radius
vector ( est recta , quae ab S ducitur ad mobile ) descri .
bet areas circa idem punctum temporibus proportionales ,
et viceversa.
Quod spectat ad primam assertionis partem , assum
ptis tempusculis aequalibus , et ducta recta SL conside .
rentur triangula SCO , SOL , SOI : est SCO = SOL ,
cum sivt super bases CO , OL aequales ob aequali
tatem tempusculorum , eamdemque habeant altitudinem
est etiam SOL = SOI , quia insistunt ambo eidem basi
SO, et sunt inter easdem parallelas SO , LI : ergo SCO
SOI. Eodem modo ostenditur triangula SOI , SIH aequa
lia esse eidem SIM , et proinde aequalia esse inter se ,
et similiter omnes areas sequentium triangulorum aequa
les esse inter se et cum areis praecedentibus. Quare cum
temporibus finitis quibuscumque contineantur numeri tem
pusculorum aequalium ipsis temporibus proportionales, areae
terminatae polygoni perimetro et rectis ad centrum virium
pertingentibus , hoc est compositae a lot areolis triangu
lorum aequalium quot tempuscula respondent illis tem
poribus , quibus perimetri partes describuntur , erunt
ipsis temporibus proportionales . Cum autem id locum ha
beat quomodocumque augeatur numerus tempusculorum,
eorumque magnitudo imminuatur , consequens est ut, ubi
⇤
88
⊖∘∘∙∁≖↧∁↾⋅ :soc—(MIV—MIG) :90&quot;—(i—gO&quot;):180&quot;—i ;
proinde , denotante r radium GI, erit IG':rsin IGG':
rsini , et consequenter
-v&quot;:g9rsini (6).

46. Haec vera sunt de omni virium genere. Sint nunc
vires acceleratrices directae'ad centrum datum: in casu,
curva COIH
.. ..
(Gg. 27. ) jacebit in plano transeunte
per rectam projectionis et per centrum virium; quod fa-
cile intelligitur , cum in eodem plano agant vires omnes.
Viribus ad punctum aliquod S tendentibus, radius
vector (est recta
, quae ab 5 ducitur ad mobile ) descri-
bet areas circa idem punctum temporibus proportionales,
et viceversa.
Quod spectat ad primam assertionis partem, assum-
ptis tempusculis aequalibus, et ducta recta SL conside-
rentur triangula SCO, SOL , SOI: est SCO:SOL,
cum sint super bases CO, OL aequales ob aequali-
tatem tempusculorum, eamdemque habeant altitudinem:
est etiam SOL :SOI . 'quia insistunt ambo eidem basi
50, et sunt inter easdem parallelas SO, LI : ergo 500:-
SOI. Eodem modo ostenditur triangula SOI , SIH aequa-
lia esse eidem SIM ,
et proinde aequalia esse inter se ,
et similiter omnes areas sequentium triangulorum aequa-
les esse inter se et cum areis praecedentibus. Quare cum
temporibus Gnitis quibuscumque contineantur numeri tem-
pusculorum aequalium ipsis temporibus proportionales, areae
terminatae polygoni perimetro et rectis ad centrum virium
pertingentibus , hoc est compositae a tot areolis triangu-
lorum aequalium quot tempuscula respondent illis tem-
poribus
, quibus perimetri partes describuntur ,
erunt
ipsis temporibus proportionales. Cum autem id locum ha-
beat quomodocumque augeatur numerus tempusculorum,
eorumque magnitudo imminuatur , consequens est ut, ubi89
demum polygonum abit iu curvam continuam , areae ter
minatae arcu curvilineo et rectis ad centrum virium ten
dentibus sint itidem temporibus proportionales.
Ad secundam assertionis partem quod spectat , sint
areae SCO, SOI, aequalibus temporibus confectae , omnino
aequales. Quoniam producta CO in L ita , ut existat OL =
CO, est triangulum SOL = SCO, idcirco SOL =SOI; sed
baec duo triangula habent basim communem SO ; erunt
igitur inter easdem parallelas, ideoque IL erit parallela re
ctae So. Ducatur IF parallela ad OL; motus per Ol com
ponetur ex duobus per OL et OF , quorum prior cum
oriatur a determinatione motum praecedentem continuandi
per C O , certe posterior a vi acceleratrice generabitur ,
quae propterea dirigitur ad centrum S.

47. Velocitas qua pollet mobile in eadem curva , est
reciproce proportionalis perpendiculo e centro virium du
cto in tangentem . Velocitas enim mobilis in quovis latere
polygoni est ut ipsum latus ob aequalia tempuscula , quibus
unumquodque latus percurri supponimus : est autem unum :
quodque ejusmodi latus reciproce ut perpendiculum quod
ex centro virium ducitur in latus ipsum ; siquidem id
perpendiculum habent pro altitudine triangula illa exigua
polygoni , si hujus latera pro eorumdem trianguloruin basi
bus assumantur ; ea insuper triangula sunt aequalia , et in
triangulis aequalibus debent bases esse in ratione recipro
ca altitudinum : est igitur ea velocitas reciproce ut per
pendiculum ductum ex centro virium in latera polygoni.
Sed abeunte polygono in curvam continuam , directiones la
teruın abeunt in tangentes ; ergo velocitas mobilis in quo
vis curvae puncto erit reciproce ut perpendiculum ex cen
tro virium in langentem demissum.

48. Denotet a areolam NSZ , et g perpendiculum SE
ductum ex centro S in laterculum NZ ; describetur NZ ve
NZ 2a ; siquidem NZ.SE=2NSZ: hinc ( 45 ) o
locitate v=
90
7
89
demum polygonum abit iu curvam continuam
, areae ter-
minatae arcu curvilineo et rectis ad centrum virium ten-
dentibus sint itidem temporibus proportionales.
Ad secundam assertionis partem quod spectat, sint
areae SCO, SOI, aequalibus temporibus confectae, omnino
aequales. Quoniam producta CO in L ita, ut existat OL:
CO, est triangulum SOL:SCO, idcirco SOL:SOI; sed .
haec duo triangula habent basim communem SO.; erunt
igitur inter easdem parallelas, ideoque IL erit parallela re-
ctae SO. Ducatur lF parallela ad OL; motus per OI com-
ponetur ex duobus per OL et OF, quorum prior cum
oriatur a determinatione motum praecedentem coutinuaudi
per C 0, certe posterior a vi acceleratrice generabitur ,
quae propterea dirigitur ad centrum S.

49. Quoniam radius vector , juxta quem agit vis con
tinua , potest assumi ut sibi parallelus per tempusculum
quodvis infinitesimum 0 , ipsaque vis ut constans per to
tum illud tempusculum ; ideo si mobile K incedens cur
vam CX ( fig. 30 ) viribus ad centrum S tendentibus de
scribit arcum infinitesimum HN labente , ductis SH ,
SN , et producto SN donec occurrat in H' tangenti HH &quot; ,
lineola recta H'N repraesentabit motum relativum mobi
lis K quoad ipsum Kieps per HH' sola vi praeconcepta in
H. Igitur cum motus iste relativus sit unice repelendus
( 5 ) a vi continuata per tempusculum e , exsistet
H'N son (6&quot;).

50. Haec subiungimus . 

1.&quot; Sive vires tendant ad
centrum datum
, sive non; denotantibus any, :coordinatas
puncti materialis in fine temporis t
, profecto x ,r,:peu-
debunt ab ipso :; erunt videlicet æ, y, :functiones tem-
peris :, ut scribi possit
.
——-—————.——-—-——-——.—.——...———..—91
= f ( ) , y = fi ( ) , z = 12

2. • Si vocatur s arcus a materiali puncto percursus
tempore t, w velocitas ejusdem puncti in fine ipsius t , pe
rinde spectari poterit ds ac si motu uniformi conficeretur ,
sola nimirum velocitate praeconcepta v ; siquidem nova
velocitas, dv , quae labente dt accedit materiali puncto ,
est infinitesima . Propterea hic quoque ( 28 )
ds
dt

3.º Resoluta vi o in duas , quarum altera sese
dirigat juxta tangentem , altera juxta respondentem nor
malem , erit ( 44) prima
des o cos i duษ
dc &gt;
dta
secunda (45 )
2² ♡ sini
ds²
r rdta

4.°# Incedente puncto materiali K per arcum s , mo
vebuntur motu rectilineo projectiones K' , K &quot;, K '' ipsius K in''
coordinatis orthogonalibusque axibus OX , OY, OZ ( Fig.5 ) ,
eruntque ( 28 )
dx dy dz
dt dt dt &gt;
earum velocitates in fine temporis : , quum nempe K ha
ds bet ( 2 ) velocitatem Vi acceleratrice dc
K , resoluta in ternas P ', P &quot; , D' ' ' iisdem axibus parallelas,
. , qua sollicitatur
∙ 91-
x:f(t)-J:fx(t)o 2:130)-
2.&quot; Si vocatur .: arcus a materiali puncto percnrsus
tempore :, v velocitas eiusdem puncti in fine ipsius t, pe-
rinde spectari poterit ds ac si motu uniformi couGeeretnr ,
sola nimirum velocitate praeconcepta v, ∙ siquidem nova
velocitas dv , quae labente dt accedit materiali puncto ,
est infinitesima
. Propterea hic quoque (28)
ds
Pr.—...... dt
3.&quot; Besoluta vi 9 in duas , quarum altera sese
dirigat juxta tangentem , altera juxta respondentem nor-
malem
, erit (44) prima
'n'—'v—d'v-Sd&quot;: cpcost—p ,9 de dt&quot;
secunda (45) ⋅
'
∙⋅ &quot;
'
∙ . ,,,a d;:
cpsmr— r — rdt&quot;'
4.&quot;e Incedente puncto materialiK per arcum :, mo-
vebuntur motu rectilineo projectiones K', K&quot;, K'&quot; ipsius Km
coordinatis orthogonalibusque axibus OX, Oï. OZ (Frg- 5) :
eruntque (28)
'de: (I)-' dz
dt
'
dt '
dt
earum velocitates in Gne temporis :, quum nempe K ha-
bet (2&quot;) velocitatem? .Vi acceleratrice , qua sollicitatur

: - -

K
,
resoluta in ternas P', P&quot;, P'&quot; iisdem axibus-parallelas,92
motus projectionis K' nihil erit aliud nisi motus rela
tivus puncti K quoad ipsum K sollicitatum viribus dum
dx taxat P &quot; , P ''' ; proinde velocitas debelur soli P' ex dt'''
dr ternis P' , P &quot; , P &quot; ; simili ratione ostenditur. deberi soli dt
dz P &quot; ex ternis P' ,P &quot; , P , et soli P&quot; ' ' ex iis 'componenti
dt
bus . Hinc ( 28 )
adx ddy adz
de de dt
P' ,
P &quot; , = P &quot; , dt de dt
seu
dex day daz
dt2 P'
dia
P &quot; ,
di?
= P &quot; .

5. °* Si punctum materiale incedit curvam plagam,
sumptis axibus v. gr . OX , OY in plano curvae , habebuntur
tantummodo
der day
de² P ' ,
dia
= P &quot; .
Fac v. gr. ut vis acceleratrix o sit parallela axi OY , ita
lamen ut sese dirigat ad plagam ordinatae y negativae : erunt
P = 0 , P :
ideoque
d2x
dla 0,
dy
di ?
Istarum prima suppeditat
I
92
motns projectionis K' nihil erit aliud nisi motus rela-
tivus puucti K quoad ipsnm K sollicitatam viribus dum-
taxat P&quot;, P&quot;'; proinde velocitas .j—f. debetur soli P' ex
ternis P', P&quot;, P'&quot; ; simili ratione ostenditur-(g.; deberi soli
&quot;
∙ ∙∙∙ dz ∙ n ∙∙ P ex ternis P', P&quot; ∙ ∙
, P , et −− soh P' ex 11s'componeut1-
dc
bus . Hinc (28)
' ddf ddZ ddi dt dt dt
∙−−− −−∶ '. ∙−−−:
P. −∙∙:
dt dt '&quot;
'
de P '
seu
'
' ⊒ ∙
' '
d3æ (137 d&quot;:
dt&quot; −−−∶ P
'
∙−− ∙−−− ∙−:
P
'di&quot; P '
dt&quot;

5.&quot;; Si punctum materiale incedit curvam planam,
Sumptis axibus v. gr. OX, O? in plano curvae , habebuntur
tantummodo .
⋅ ⋅
dzæ - d
dc&quot; :&quot;)&quot;Zïz'
Fac v. gr. ut vis acceleratrix q; sit parallela axi Oï
,
ita
tamen ut sese dirigat ad plagam ordinatae] negativae :erunt
ideoque
dh: ∙∙ d'] ∙∙∙ ∙∙
dt&quot; —0' −↲⋅≀⋅⇀≖− ?
Istarum prima suppeditat93
dx
dt C , x =Ct +C' ;
secunda, in hypothesi o constantis , praebet
dy ota
dt ot + C &quot;, y = 2 +0&quot; 4 + C &quot; :
eliminato t ,
y
y = c&quot; + * (** ) (* = ) .
Habes itaque, in ea qua sumus hypothesi , coordina
tas x ety expressas ( 10) per t; habes insuper aequatio
nem ad curvam, quam describit materiale punctum : re
stat ut constantes arbitrarias C, C' , C &quot;, C '&quot; determinemus.
Ponatur materiale punctum ex ipsa coordinatarum origi
ne O projici cum velocitate Yo juxta rectam inclinatam
ad OX sub angulo h: resoluta v. in' binas, alteram paral
lelam axi Ox, alteram parallelam axi OY, erit illa = v , cosh,
haec Vo sinh: initio motus obtinent simul
t = 0 , x = y = 0 ,
dx
dt = v , cosh,
dy
dt = V , sinh ;
igitur
C = Vocosh , C = 0.C &quot; = V . sinh , C = 0 ;
et consequenter
012
x = vol cosh ,y = v , sinh -
csinh
cosh
gx2
2v.cosh
93
dr
.
E—:C, æ:Ct-l-C,
secunda, in hypothesi ? constantis , praebet
' d ∙
'

: &quot; 73: :,n—j-c'.7:— ∙≌⇉−−−⊦∁∥≀−⊦ ∁⋯≖ eliminato t ,

∜−⋅−−−≺⋮⋅⋅⋅⊹∁∣⋅ (&quot;€&quot;)— −≣−≺∙≄ ;
&quot;):
.
Habes itaque, in ea qua sumus hypothesi, cbordina-
tas æ ety expressas (1&quot;) per :; habes insuper aequatio-
nem ad curvam, quam describit materiale 'punctum: re-
stat ut constantes arbitrarias C, 0, C&quot;, C'&quot; determinemus.
Ponatur materiale punctum ex ipsa coordinatarum origi-
ne O projici cum velocitate vo juxta rectam inclinatam
ad OX sub angulo h: resoluta v., in' binas., alteram paral-
lelam axi OX, alteram parallelam axi Oï, erit illa:vo cos 11,
haec:vo sin/1: initio motus obtinent simul
da: dy .
t.:o,x:o,y:o, ï

: vo cosh, ?::v., smh;

igitur
C: vo cosh , C': o .C&quot;: vo sin]: ,C&quot; ':o;
et consequenter '
cpt&quot; æsiuh (pa-&quot;
2 &quot;7— cos/1 -21Jo&quot;cos&quot;lt :

: votcOsIt,y:vosiult—94

x tangh -
9 1+ tangah
2 v2.
22.
Recole quae diximus ( 39).

6°# Fac nunc ut, permanentibus caeteris ( 5º. ) , pun
clum materiale moveatur in medio resistente: poterit vis ac
celeratrix ex resistentia medii exprimi ( 32. 33 ) generatim
per f (v ) ; per functionem videlicet velocitatis v
tem , decrescentem , evanescentem simul cum v Sit \beta an
gulus interceptus directione motus et ordinatarum axe OY ;
erunt ( 32 ) P' f (w) sin \beta , P &quot; = -- flv) cos \beta ;
ideoque
crescen
dar d²y : - flv )sin\beta , = -9 - flu) cos\beta ( c ) :
dt2 dla
insuper ( 40)
dx
dt
dy v sin\beta ,
dt = v cos\beta (c' )
quae differentiatae suppeditant
d22 dy d\beta dy do d\beta
dt sin\beta tvcos\beta dt dt2 dt cos\beta — v sin\beta ordt dt2 .
Ergo
dv
sin \beta + y cos \beta
d\beta
dt dt : -f (v )sin\beta,
do
de
d3 cos \beta-usin \beta 0 - f v ) cos\beta: dt
94
x tangh .::
t—ïngïhæt
Recole quae diximus (39).

604: Fac nunc ut, permanentibus caeteris (50.),ptm-
ctum materiale moveatur in medio resistente: poterit vis ac-
⋅ celeratrix ex resistentia medii exprimi (32. 33) generatim
per f(v); per functionem videlicet velocitatis v crescen-
tem
, decrescentem
,
evanescentem simul cum 0Sit B an-
gulus interceptus directione motus et ordinatarum axe Oï;
erunt (32) P':
-
f(v) siuþ ∙P&quot;: −− ?
−f(P) 008 p;
ideoque
d'æ
dt: :—ftv)sinþ,d —:— —f(v) cosþ (c):
insuper '(40)
da: . d .
'at—:&quot;.lnþO £: &quot;waþ (0)
quae diB'erentiatae snppeditant
dzæ
−↙⊼≖−−∶−⋇⋮∐⇪ ⊣−∙≀∘∞⇪⊼ d'B. dz :d—ïcosþ— —vsinþ dþ
dt
Ergo
——sin,8 −⋅⊢ vcos 5—d—-5 −∙−−−∙ —-f(v)sin,8,
dv
Ft— cosþ—vsin B (35—:— ep —f(v) cosþ:95
istarum primam multiplica per sin\beta , secundam per cos\beta,
tum collige in summam; eamdem primam multiplica per
cos\beta , et secundam per sin\beta , cum subtrahe; habebis
dy d\beta
+ fv) =– pcos\beta,
=
Psins
(c' ) .
dc dt
Quibus positis, haec stabilientur: cum nequeat \beta fie
ri &gt; 180° ( siquidem in transitu . per 180° vires omnes e
vaderent verticales, motusque permaneret verticalis ) , cum
que p etv existant perseveranter &gt; 0, ob secundam ( c &quot; ) erit
d\beta constanter 0 ; proinde crescente e crescet semper an dt
gulus \beta accedendo ad quemdam limitem B.
In hypothesi anguli initialis \beta. (=90° - h)&lt;90°, per
get o cos \beta per aliquod tempus esse &gt; o : sed flv ) &gt; 0 ; i
gitur , ob primam ( c''), per totum illud tempus erit de''
et consequenter crescente t decrescet v.
Prima ( c&quot; ) differentiata praebet
du
&lt; o .
d2v dv d\beta gsin\beta ;
dt -
dea + au f '(o )
seu , attenta secunda ( d ''),''
dev dy

dia
+ áf ( ) = q *sin- B
dv
facta igitur dt , emerget
dev oʻsina&gt; o.
dt
95
istarum primam multiplica per sin 13, secundam per cosþ, ⋅
tum collige in summam; eamdem primam multiplica per
cosþ ,
et secundam per. siuþ ,t'um subtrahe; habebis
∙ d d ∙
⊋⋮∙∙⊣−∣↻⇝⇌− ws?- ∙⊺∙↙↙⋛−∶∶∲−−−∘∎⋮∙∂ (a&quot;)-
Quibus positis, haec stabilientur: cum nequeat. þ Ge-
ri )180o ( siquidem in transitu.per 1800 vires omnes e-
vaderent verticales, motusque permaneret verticalis ), cum-
que (p et v existant perseveranter o, ob secundam (e&quot;) erit ↭ ∣
d ∙ ⋅
constanter £
)a; promde crescente :
crescet semper an-
gulus þ accedendo ad quemdam limitem B.
In hypothesi anguli initialis B., (:::90() -H( 90&quot;,per-
get ? .cosp per aliquod tempus esse ∘:sed iv))o'; i-
gitur
, ob primam (e&quot; ), per totum illud tempus erit ⋚∶≺∘∙ et consequenter crescente :decrescet 0.
Prima (e&quot;) differentiam praebet
⋣≖−⊦↙↨−⋛∣≼⋅⇝⇌≡≴∊∹∾⋅≖⋅∣⋮⇋ dav d d . ∖∖ seu, attenta secunda (c' '),
d'v dv ∙∙∙ ∳≖∘⋮∐≏∆⊙ ∙
⊄⋮⋮⋝⊹⊋−∑ f(V)-— v '
. .
dv
facta igitur 22 :o
, emerget
dav cp'sinïþ
üt: ⇀−− v
)0-96
Inferimus ( 27. 22°. ) velocitatem v haud exsistere capacem
maximi: poterit quidem esse capax minimi ; ita tamen , ut
mutato decremento in incrementum, hoc neque vertatur ite
rum in decrementum, neque certum quemdam limitem E
praetergrediatur. Primum patet ex eo quod , posita conver
sione incrementi in decrementum, jam obtineret maximum:
secundum ex eo quod, aucta indefinite v, augeretur indefi
dv nite flv ), simulque foret &gt;0 ; id vero adversatur pri dt
mae ( 6' ) .
Ex ( c &quot;) eruuntur binae
20 21
V2-01 ſię cos$ + fvde,B2- B;= Sosiu\beta dt ;
t t
exprimunt N,, V, velocitates , item B , B, angulos limitibus t,
2t respondentes. Fac
o cos\beta + v ) = f (t) ,
psins = fa (t) :
habebis ( 27. 18º. )
V; - v.--tfittat) • B. - = falttal) ;
exprimunt a et a numeros &gt; o et &lt; 1. Sed crescente t in
definite , vergit fi (t) ad q cosB + f (E ),et fu( t) ad qsinB
E
ac proinde
2 - -V2
limes quantitatis cos B + F( E ) ,
3. - 22 O
limesque quantitatis
sinB
E
96
∙
Inferimus (27. 220.) velocitatem v haud exsistere capacem
maximi: poterit quidem esse capax minimi; ita tamen, ut
mutato decremento in incrementum,hoc neque vertatur ite-
rum in decrementum,- neque certum quemdam limitem E
praetergrediatur. Primum patet ex eo quod, posita conver-
sione incrementi in decrementum, iam obtineret maximum:
secundum ex eo quod, aucta indefinite v, augeretur indefi-
nite f(v). simulque foret-(£)o, ∙ id vero adversatur pri-
mae (c&quot; ). ∙
Ex (e&quot;) eruuntur binae
2t
∙↗−⇂↗≖ −−−∙−− ∙∣ ( ? cosþ-l- fwndz, ↾⊖≖−,B— fra-018 de;
exprimunt v, , v, velocitates,' 1tem (i,, ,H, angulos limitibus !,
2t respondentes. Fac
9) cosB ^v):fd!) ∙∲≊∣∶∁ :
fam
habebis (27. 180.)
'Ur—vzzf— tf1(t&quot;l&quot;at) ∙⇪≖−−⇪≃∶∶⊀≖↸≖⊣−⊄⋅∁⋟⋮
exprimunt a: et «' numeros )b et ↿∙ Sed crescente :in-
≺↿⊜∊⊓⋮⇂∊∙ &quot;&quot;sit fxw ad 90053 —I-f(E).et rm ad ?&quot;&quot;B-
ac proinde
2! -—v

limes quantitatis :

Bos B4-f(E) ,x—Bz ↽− wir-B : .
limes ue uantitatis . q q E97
quoniam igitur
VI - V2 lim.
B - \beta ,
0 lim t t
erunt
Ø cos B + f(E)= 0 ; sin B
E
et consequenter
B = 180° , f(E )= .
Ex istarum prima inferimus motum materialis puncti ver
gere ad rectilineum verticalemque motum; e secunda ( viri
bus p et medii resistentis sese in limite elidentibus, utpo
te aequalibus et contrariis ) ad motum uniformem , proce
dentem videlicet a sola vi praeconcepta.
Divide primam ( c&quot; ) per secundam (c&quot;) : proveniet
dx
d\beta
sie
X-X
B-Brvm?;
iccirco ( 27. 18º. )
i\beta
Spa\beta
Q Q
Bm
exprimit um valorem medium velocitatis v. Haud praeter
greditur ' ' m certum quemdam valorem finitum ; insuper ver
git \beta ad B= 180° : ergo neque x praetergredietur finitum
valorem; ideo que materiale punctum incedet curvam prae
ditam asymptoto verticali. Recole, quae diximus nº. 41 .
Posita ( 33. 4º. ) flv )
formulae ( c) evadent k?
qua 1
quoniam igitur
&quot;r'—Va lim. :o, lim Bi—Ba :.0,
erunt
?
∘∞↿∃⊣−⊀≺≖∙∶⊢− 0
∙
∲≕⋮⋮∶⊔∄−∙−− −−∘⊰
et consequenter
3:180&quot; ,f(E):9.
Ex istarum prima inferimus motum materialis puncti ver-
97
gere ad rectilineum verticalemque motum; esecunda(viri-
bus 91 et medii resistentis sese in limite elidentibus, utpo-
te aequalibus et contrariis ) ad motum uniformem, proce-
dentem videlicet a sola vi praeconcepta.
Divide primam (c') per secundam (e&quot;) :proveniet
iccirco ( 27. 180.)
exprimit v,, valorem medium velocitatis
,,,
Haud praeter-
greditur v,, certum quemdam valorem finitum; insuper ver-
git B ad B: 1800: ergo neque æ praetergrediatur finitum
valorem; ideoque materiale punctum incedet curvam prae-
ditam asymptoto verticali. Recole, quae diximus n&quot;. 

41.
Posita ( 33. 40.) f(v):SE,-2 , formulae (c) evadent
.k?98
dar
di ?
-sing,
day
dla 9
qua
cos\beta : ka
sed haec hactenus.
7º. Intelligantur per coordinatarum orthogonalium
originem O ( Fig. 5 ) duci binae rectae 8,0&quot; intercipien
tes angulum a : earum extremitatibus junctis recta d '&quot;, erit
cosa =
02 +02.02
28 &quot;
Extremitas rectae , habeat coordinatas a ', y, z ', rectae au
tem o coordinatas x &quot;, 1 &quot; , 2 &quot; : paullulum attendenti pate
bit fore
õ = x's + y + 2,0% = &lt; &quot; + ya + z'2 ,
d's = (x - x &quot; )2 + 6 - y &quot; )2+ (z'- z&quot; )?;
adhibitis substitutionibus ,
cosa =
x' x &quot; ta'y &quot; tz'z&quot;
8o&quot;
Sint a' , b' , c' , anguli, quos Ở facit cum axibus OX, OY ,
OZ ; et a &quot; , 1 &quot; , c&quot; anguli quos d &quot; facit cum iisdem axi
bus: erunt 1
x' = cosa' , y ' = ' cos b ', z ' = ' cosc'
x &quot; = &quot; cosa &quot; , y &quot; = 0 &quot; cosb &quot;, z&quot; = 0&quot; cosc&quot; ;
rursusque adhibitis substitutionibus,
98
−∙−≂− −≌≝≖⋅ ∙ 9
9008?-
sed haec hactenus.
70. Intelligentnr per eoordinatarnm orthogonalium
originem O ( Fig. 5 ) duci binae rectae d', d&quot; intercipien-
tes angulum a: earum extremitatibusjunctis recta ö&quot;, erit
ö&quot;:
eo&quot; −⊦∂∣∣∶∎−∂≀∥≖ ⋅−∎ 26' a&quot; '
Extremitas rectae ö' habeat coordinatas 0:231, z', rectae an-
tem d&quot;coordinatas x&quot; , y&quot;, z&quot;: paullulum attendenti pate-
bit fore ⋅
∂∣≏−−∶∞↾≖−⋅⊦∙↗∣≖−⊢≖↾⋩∙ ∂∣⋅≖∙∸⋅∞↾∎≖⊹∕∣≖−∣−≖∥≖ ,
3' ⋅≖−−−−≺∙⊅∣∙∞⋅∣⋟≖⊣⊣∙↗∣⋅∫∎⋅ )'—l-(z'-z&quot; ),:
adhibitis substitutionibns ,
∙−− æ; æ&quot;——)&quot;)'&quot;—l-Z' zn
cosa ∶⋅↳ a, 6&quot;
Sint a', 6', c', anguli, quos 6' facit cum axibus OX, Oï.
OZ; et a&quot;, b&quot;, e&quot; anguli quos 6&quot; facit cum iisdem axi-
bus: erunt
x':d' cosa' ,y*zzd&quot; cosb', z':ö' cosc'
æ&quot;: ö&quot;cosa&quot;, y'': ö&quot; cosb&quot;, 2&quot;: d&quot; cosc&quot;;''
rursusque adhibitis substitutionibus,
−∙∙⋅∙−⋅−−⋅99
cosa = cosa' cosa&quot; -- cosb' cosb&quot; + cose'cosc &quot; .

* His positis, fac ut vis acceleratrix o sese constanter dirigat ad centrum datum : constituta in eo coordinatarum ori gine O, erunt sle D 5.5 cosinus angulorum , quos cum axibus coordinatis efficit ra dius vector D; et P P &quot; P '&quot; P cosinus angulorum , quos cum iisdem axibus efficit . Pro pterea P X op + . $ . Þ==1 , sumpto vel superiore, vel inferiore signo , prout o nititur vel adducere materiale punctum ad centrum illud, vel ab ipso distrahere: in primo enim casu q et D faciunt angu lum a = 180° , in secundo angulum a = 0. Inde profluit ( 49) d2x Ide² dy v + D dia D daz dt2 8.• * Sumptis axibus OX, OY in plano ( 46) cur vae , quam incedit materiale punctum , erit der Q =F Ndt² on the + 5) . 99 cosa:eosa'cosa&quot;-]-cosb' cos6&quot;-1-cose' cosa&quot;. ∙His positis, fac ut vis acceleratrix (p sese constanter di- rigat ad centrum datum: constituta in eo coordinatarum ori- gine 0, erunt æLz D'D'D cosinus angulorum, quos cum axibus coordinatis edicit ra- dius vector D; et P' P&quot; P'&quot; r ' a ' ? cosinus angulorum, quos cum iisdem axibus ellicit ep. Pro- ? se P&quot; 7 p--- ∙∙∙ ∙−−− ' D—&quot;'ï&quot;10 ? sumpto vel superiore, vel inferiore signo, prout ep nititur vel adducere materiale punctum ad centrum illud, vel ab ipso distrahere: in primo enim casu ? et D faciunt angu- lum a:1800, in secundo angulnm a −−−− 0. Inde profluit (40) ≕∙∙∙∙∙∙ dia: :: d'y )- ? D) *(dz : &quot;D'l'dcz ∐↼⊦↲⋮−−≟ ⋅⋅− ⋅ 8.0 «: Sumptis axibus OX, 0? in plano (46) cur- vae , quam incedit materiale punctum , erit100 Ad exprimendamo per coordinatas polares , exhi beat 180°-W angulum interceplum radio vectore D et axe OX ; erunt De = x ? tys , x= - Dcosw , j = Dsina . Prima semel iterumque differentiata dat dDP + Dd D = xd x + ydży + dx2 + dy? ; secunda et tertia praebent dx = Dsiow cosw - coswdD . dy = Dcos wdw tsinwdD , ideoque dsa = dx2 + dyr= D -dw2+ dD2 , Hinc 2 der dia dy a D + dla D d - D dea D 2) ܪ . ac proinde la pa (d- D dla 0 ( ) ). Ad haec : P P &quot; = P P &quot; unde D àla D y et consequenter 1 1 1 100 Ad exprimendam (p per coordinatas polares, exhi- ' beat 1800—0 angulum interceptum radio vectore D et axe OX ; erunt Dï':a:3--l-)'2 , x: — Dcosw .szsinm. Prima semel iterumque differentiam dat dDL-l-DdzDzædïæ-l-yd'y-þdæï-l-dyz .; secunda et tertia praebent dæ:Dsinm cos co —cos ad D. dy:Dcos ædwf-sinædD, ideoque d.,- ∙−−− dx: −⊦ dyaznadæ-l—doa . Hinc dïæ a: dfy ] (PL) ? Dei?-),. ∎⊃⊣−≺∄↙⇄ ⋅∎⊃−−⇤↲⋍≖ dt &quot; ac proinde dzD (deo)!) ∙−−∶ −− D — ? ∓ ∙ (aua dt Ad haec : P' a: P&quot; ⋅∙∙∙∙ )» P ∙∙∙ P .;. :ï,?—q:.ü.,unde-; 7- et con sequenter ∙∙∙∎∙∎⋅∎−⋅101 • dx yd dt rady FO : de quam integrantes assequemur dr V dc dy dt C , seu ydx - xdy = Cdt. Est autem ydxxdy = Dsinud(-Dcosw ) + Dcosad( Dsinw ) Dºdw , propterea с dwla CdtD - da da de ( ) = C2 D D4 insuper AD Code : d d - D dla de dt dD da dt dt . ( dD C do D2 dt 1 . ( D ( ( d da da) da C2 D d d dw ,!... Hit C = as dt aan zoals da ? Coil 100 dwudt da . Da aby boxe parutis 1 C2 D D2 dw² Quare J 101 quam integrantes assequemnr da: dy,&quot; ⋅ ∙∙∙∙ ∙≯≀∙⊋∙↕−− ∙∙∷⊋∙⋮−−− C, seu ydæ—Jt'dj—Cdt- Est autem ydx—ædy:Dsinæd(—DcosmH—Dcosæd(Dsinæ) : D'daii , propterea ∙−− dai—C ∙ de) 3—01 ∙ ∁↙≀⇞−−∐⇟⊄∄∾⋅∙⋅⊋⊼−−−∐−≖⋅∙ (a)—&quot;1373&quot; insuper di? d(dD. 49) d(iD ∙⊆− ≀∄⋅∣⊃∙∙ d ∙∙∙⋅∃⊂∙−⊃⋅ 71? ∙− do) ne). dt'- d; ⋅ d:; dï- ⋪∙−⋅−⊳ ↿ ⋅ ⋯↿ 41 d(ï) d D d(B) 1101 ...-2 d, ⋅ da) ∙−↽∁⋅ ↪↼⋅−↽−⇁∁ ↜⊒⋅∶≥⇀⋍−−⋅−≤⋮∶ a d: dmwdt;,, nad-'O) . ' f&quot; &quot; c: ∐≖⋅↙∄∘−∎⊃−dasz102 D + ) ( 61 ). D2 dw² Fac v. gr. ut, viribus ad datum centrum tendentibus, materiale punctum incedat curvam (dicitur spiralis loga rithmica ) repraesentatam per Draw Habebis ( 27. 6.° ) . el = 2 11름 loga dw ,de 1 D% log ? a dway log.'a ; iccirco go CP D2 a log-a + b) ( logo a+ 1 ) . vis nempe acceleratrix erit in ratione reciproca triplicata distantiarum a dato centro.

9. # . Ad constantem C quod spectat, ex coordina
larum origine 0.(Fig . 19 ) intelligantur duci bini radii ve
clores, alter ad punctum datum a habens coordinatas xo,
Yo, alter ad punctum quodvis B habens coordinatas x, y;
sitque A area sectoris terminati arcu aB et radiis illis, A
area aBca' : erit
A = A +
Xoyo
2
xy
2
&gt;
⇀↿∘⊋∙
du-
−−∶⊨ C, ...—l.).. ..,— (6) ..... i)? da: D !'
Fac 9. gr. nt, viribus ad datum centrum tendentibus,
materiale punctum incedet curvam (dicitur spiralis loga-
rithmica ) repraesentatam per
D:
ac .
Habebis ( 27. 6.&quot; )
1 -ao 1 1
T)- :a ∙↙≀−∣⋝−∙−−−−−− logadæ,d'ï-—
.. ∠≀≖−∣↿⋝∙ ..
a logia dei:-, :logæa ,

dm&quot;
iccirco
1»ng (
−∾∙∣∘⊰⋅∅⊣−∎↿⋥≻−−−∌⋮ ——(l0gi ∅−⊦
1):
vis nempe acceleratrix erit in ratione reciproca triplicata
distantiarum a dato centro.
9011. Ad constantem C quod spectat, ex coordina-
tarum origine O-(Fig. 19) intelligantur duci bini radiive-
ctores, 'alter ad punctum datum et habens coordinatas xo,
yo, alter ad punctum quodvis B habens coordinatas x, y;
sitque A area sectoris terminati arcu aB et radiis illis, A'
area cha': erit '
A—A' : æozïo ?;103
ideoque ( 27. 18° )
dA = dA - d
xy ydxxdy
2
denotante praeterea i angulum interceptum tangente in B
et respondente radio vectore D, est ( 48)
D sinids dA 2
Igitar
ydx = xdy = Dsini ds,
et consequenter ( 70 )
C dt = D sini ds ;
unde ( 20 )
ds
C = D sini - Du sini .
dt
Caetero quantitatem Dvsini esse eamdem ubicumqe suae cur
vae sit materiale punctum, liquet ex dictis ( 47 ) .

10 ° # Habemus ( 2º. 8° )
ds
de²
DP dw2 + dD CP
dc2 (Da dwa + dD&quot;)p4
dwa
dD 2
D2
[ 11
+ 9 seu
de
103
ideoque ( 27. 18o .)
dA ::dA'- J
222 −−−∫↙≀∞−⋍≀↨≧↩ −∫∂∞−≨⋅⇣⇃⋮↙≀∫ ;
denotante praeterea t' angulum interceptum tangente in B
et respondente radio vectore D, est (48)
D sini ds
2 ∙
(IA:
lgitur
ydx':xdy :Dsint' ds,
et consequenter ( 70 )
Gde:D sini ds ;
unde ( 20)
CZDsini £:Dvsiü. dt
Caetera quantitatem Dvsini esse eamdem ubicumqe suae cur-
vae sit materiale punctum, liquet ex dictis (47).
100a Habemus ( 20. 8&quot;)
2
(Isa 02 dGP—xl-JD':(02 da,-1- dDz) c,
.
— —∙∙∙
−⋅ dt&quot; ⋅⋅−⋅ dt: D4 dc.-13:
∁≖
dDa
äirl—(5)], seu
de?104
v2 = C2 -- [ + (3 ]
( m) .
11. # Quemadmoduni , data linea quam incedit
materiale punctum , innotescit q ; sic vicissim , data op , po
terit sciri linea per quam movetur materiale punctum
Denolante B quantitatem constantem et n numerum inte
B
grum , sit v . gr. g = ; erit ( 7° 6. )
D &quot;
B CP D
+ )

;

dwa
1 B
quae ,
facto D = 1
D' et
et og h , vertetur in C2
d2 D'
h D ' r-2 = +
diwa +D) .
Chaton
Haec multiplicata per 2dD ' suppeditat
E12dD' dD d
dw da +
2D'dDdD' ) -2-2 h D'n -2 d D' = 0 ;
sumptisque integralibus ,
= [CD)* + D ] - 2,0-4C = 0;
unde
dw
(6,2
dD'
2h Dina quoad o adducentem
D'2 į ad centrum ,
'? – C ).
∠⊢⋅⋅ ↿
' 2 2
1 D∶∁ Exi-(a)] (&quot;**-

↿↿∙∘∙ Quemadmodum, data linea quam incedit
materiale punctum
,
innotescit ?; sic vicissim , data ep , po-
terit sciri linea per quam movetur materiale punctum
.
Deuotante B quantitatem constantem
, et 11 numerum inte-
grum, sit v. gr. ep:DT; erit ( 70 6.)
l
'l
3— B 02 (...d D.. 57.— 25 .'.)* da: D
⋅ ↿ B ∙
quae, fama-:D et——⋜⋮−:h,vertetur tn

* D' ≀≖≖≖⋅⋅−≖⇌⇀−⊻≐≺∡∽≖ −⊦∘∙≻⋅

Haec multiplicata per 2dD' suppeditat
∶⊨≺∶≳↙⊋≞⇗∠∄−−⊣− 2D' dD')—2h D&quot;'2dD':——o ;
sumptisque integralibus ,
465)&quot; HB ]-—'— ∣⊃⋅⋅−⋅∙−⊦∁⋅−−−∘≅ n—l
unde
da: dD'
2]; quoad ?adducentem
TDV&quot; —-D'3 —C')5 ad centrum,105
dD'
dw 2h quoad o distrahentem
(0 –
a centro :
n =; D**?— D» ) *
quarum integratio praebebit relationem inter w et D' , ideo
que inter coordinatas polares w et D lineae quaesitae .
12.°* In istarum aequationum prima sume v . gr.
n = 2 ; ea sic poterit scribi
D'
doma
V
ha- C
da
h D'
h2_C Hinc
w = C &quot; + arc cos =
h - D'
VhC
cos (6-C' ' ) ;
et restitutis valoribus h , D' ,
D =
C2
B - 1 B2 – C4 C cos (W – C&quot;) ·
Pone
C2 C
= B (1 + €),
=B' ( 1 —E) ,
B-HVB2_C4 C B - V B2 - C4C
quae in summam collectae praebent
B
CPC B '
,
invicem multiplicatae suppeditant --
8
105
&amp;
dm: dD
quoad p distrahentem
(C' - 36- D'&quot;&quot;' −−∙ ∎⊃∎∌≻≩⋅ a centro :
⇀
n—1
quarum integratio praebebit relationem inter 61 et D' ,
ideo-
que inter coordinatas polares &amp;) et D lineae quaesitae .
12.&quot;; In istarum aequationum prima sume v. gr.
n:2 ; ea sic poterit scribi

: .-n

'
↶⋮≼⇂∕−−⊮−∁∙⋟
dæ:- ⇂∕↿
Hinc −≺⊓⋅≻≖∙∣≖≖−∁⋅
G):C&quot;-l— :COS(GO—C&quot;)i arC(cos ∙−∙−−−−
h—D' h—D'
⋅⇂∕∣−≖−−−⋯≖−⇀∁∙ ⇂∕∣≖≖∙−∁∙ '&quot;
et restitutis valbribus I: , D',
B—l/Bz—Clt C' cos (co—C&quot;)
D
Pone
C2 02 −−−−− −−−−−−↧≉⋅↿
). −−⋅⊨
−−−−−−∶ —B'(1—e). B—l/Ba—cac- ≺⊹⋮ ∌−⊢⇂∕∌≖−∁↙∣∁∣
quae in summam collectae praebent
c:c' B -—-−⋅⋅
. B',
invicem multiplicatae suppeditant106
&lt;= B' ? ( 1 —-z );
habebis
1 C2 C' = B
B'2 ( 1
B' ( 1 — 52)
Propterea
D =
B' ( 1 - 2)
E cosWC( '')''
(62) .
1
13.0* Potest C' esse vel &gt; 0 , vel &lt; o , vel == 0;
in primo casu erit B ' &gt; o et € &lt; 1 ; in secundo B' &lt;o et
&gt; 1 ; in tertio B ' = et z = 1. Primum ac secundum
casum alibi considerabimus .
14. * Ad tertium quod pertinet , exhibeat NI...
(Fig . 25) axem parabolae ( 40. 5.º 7.º ) ; sintque NO ( 3x)
et 00' ( =y) orthogonales coordinatae : designante 2p pa
ramelrum , exsistet
ya = 2px .
Substituto x' + ip pro x , transferetur coordinatarum
origo in focum H , eritque quoad novam originem H
ya = 2px' +p .
Duc radium HO =D) ; habebis
NHO x'
--- D cos w , y = D sin w ;
et consequenter
D2 sin ’ w = p - 2pDcosw .
Spectatur autem D ut quantitas constanter positiva ; proinde
106
↿ 'a a .
&quot;ö'.:B (1—£)1
habebis
∙∙∙ ↿ ⋅ ∙∙∙ Ca
B'3(1 -
a&quot;) ' B' (1—5') ⋅
Propterea
B' (1 — a')
D −∙− (b,) . 1— :cos (co— C&quot;)
1391» Potest C' esse vel≻∘ ∙ vel (o , vel:o;
in primo casu erit B' o et e ↿;in secundo B' (0 et s
↿; in tertio B':eo et e:1 . Primum ac secundum
⋅ casum alibi considerabimus .
145): Ad tertium quod pertinet , exhibeat Nl..
.
(Fig.25) axem parabolae (40. 5.&quot; 79); sintque NO (:.r)
et 00' (: y) orthogonales coordinatae :designante 2p pa-
rametrum
, exsistet
y':2pæ .
Substituto x' −⊦ ∙⇡∙↼ ;) pro æ .
transferetur coordinatarum
origo in focum H , eritque quoad novam originem H
7&quot; ⇌ 2pæ' ⊣− r'-
D'uc- radium HO' (:D) ; habebis
NHO':61, uf:—D cos æ,y:D sin(-);
et consequenter
D2 sin2 01:p' −∙∙ 2pDcos co .
Spectatur autem D ut quantitas constanter positiva; proinde107
DE P cosa
+ V
V pa pacos w_P(1 ~ cos )
sin? W sin? W sin4 w sin' w
Sed sin? w = 1 - Cos w = (1 — cosa) (1 + cosw ) : igitur
P
D =
1 +cosa (63) .
Designata nimirum quantitate B '(1 - 6 ) per P , et assumpta
C &quot; = 180° , recidet (62) in (63) ; unde consequitur illud :
iribus ad centrum datum tendentibus in ratione reciproca
duplicala distantiarum ab ipso centro , poterit materiale
punctum describere parabolam habentem suum focnm in
centro illo .
15.0# Quoad parabolam ( 14º. ),
(* )
sinaw 1 1 +cosw cosa a
COS
pa da P р
2 1 2 CM 1 1
D
O
Hinc ( 90.m) va - .
р
D
P
D
Sit E altitudo debita velocitati v ; erit ( 12º. 14º. )
2C E 2C?
v2 = 20E =
2BE
D2
E
B
D2 ' ( 1 -62 ) D2 p
et consequenter
2C2 E 2C 1 E
D

; unde D2 D

.
р P
Inferimus illud : si in distantia D a centro virium proji .
citur materiale punctum , haud describetur parabola nisi
107
D:∙∙∙ ;) cosa) ∙∙∙⊦ Vpa .l.-paene: c.)—p(l—cosï
≖⋮∐⇄
∙
a) s1na a) sint! ea sin' 6)
Sed sinit.):1−cosa a:(1— eos a)) ('l-l- cos a) :igitur
∼ P
D:1—i-cosm ∅⋮⋝⋅ ⋅
Designata nimirnm quantitate B'(1-- 6&quot;) per p , et assumpta
&quot;:180o , recidet (b,) in (63) ; unde consequitur illud :
viribus ad centrum datum tendentibus in ratione reciproca
duplicata distantiarum ab ipso centro
, poterit materiale
punctum describere parabolam habentem suum focum in
centro illo .
1530 Quoad parabolam (Mc,),
∙
∙−−− — ∙−−−
d'
⋅
( D) sin'm 1—cosaæ—1—l-cosæ 1—cosa1
df&quot; P&quot; ?' p P
2 ↿ ↿ ∙ 2
c- 1

;

∙ D ∙∙∙ DQ. Hlnc (90.m) 02: ∙∙∙∎∎∙ ∙
ö
∙
Sit Ealtitudo debita velocitati «a; eri; (1241. 14o.)
2311: 20» E
∙∙∙∶≿∁∶
E
ng—ZQE— Da —B'('l—-£3) ∙ [P p . 02 ,
et consequenter
zcn E—zc: '-dE-1
p.Da—p.D,uneD—. .
Inferimus illud :. si in distantia D a centro virium proii-
citnr materiale punctum , baud describetur parabola nisi108
eidem distantiae D fuerit aequalis altitudo illa , per quam
mobile vi acceleratrice vigente in puncto projectionis ca
dendo motu uniformiter accelerato acquireret velocitatem
ipsius projectionis.

51. Hactenus de motu curvilineo libero, quum nempe nihil obstat quominus mobile obtemperet viribus; fac nunc ut materiale punctudi, cujus massa = m, moveatur motu impedito, sollicitatum videlicet vi acceleratrice q adstringatur moveri vel in data superficie vel in data linea curva. Quoniam ejusmodi superficies et linea nihil praestant aliud nisi exercere in puncto materiali resistentiam m ç sibi perpendicularem, ideo motus perinde fiet ac si punctum materiale esset liberum viribusque acceleratricibus et d', seu quod eodem redit viq &quot; inde resultanti libere obtemperaret.

Pone quod motus impeditus in data linea debeatur
unice vi praeconceplae et vi gp' ut sit 9
habebis
q &quot; = 0 ; i = 90 °; et consequenter ( 45. b)
0 :
2,2
( 6' ' ' ) ;
my?
Precisa nimirum q , exprimet ( 28 ) pressio
nem exercitam a puncto materiali in lineam illam , atque
huc spectat vis centrifuga ; pressio videlicet a puncto ma
teriali exercita in eam lineam , orta e sola inertia ad prae
seulem velocitatis siatum contracta.
Ad haec : in eadem hypothesi vis acceleratricis ♡
facile colligitur ex dictis ( 36) motum impeditum fore u
niformem .
!
108
eidem distantiae D fuerit aequalis altitudo illa , per quam
mobile vi acceleratrice vigente in puncto projectionis ca-
dendo motn uniformiter accelerato acquireret velocitatem
ipsius proiectionis.

===De vi acceleratrice in motu circulari, existente centro virium in centro circuli.===

52. Ex demonstratis (47) patet istiusmodi motum esse uniformem. Sit R radius circuli, per cujus peripheriam incedit mobile: in ( b: 45 ) erant r = R, i = 90° ; in ( b' : 48) vero D =9 = r = R; et denotante A lotam circuli aream, T tempus periodicum, quo nempe mobile conficit integram circuli peripheriam, in eadem ( 8' ) erunt quoque A = n R?, = T. Hinc ex ( 6)
1
RO
et ex ( 6 ) ( c )
4 762 R
T2

53. Haec facile punc stabiliuntur. 

1º. mobile velocitate quadam projectum in distantia R a centro virium von describet circularem curvam nisi velocitas illa tanta sit quantam mobile ipsum acquireret cadendo per { R motu uniformiter accelerato et vi acceleratrice, quae viget in projectionis puncto; siquidem prima (c) suppeditat v = 2 0.4 R.

2º. In circularibus peripheriis eodem tempore
descriptis vires acceleratrices sunt ut respondentes radii: patet ex secunda (c).

3º. Ex eadem secunda (c) inferimus vires acceleratrices fore in ratione reciproca duplicata radiorum quotiescumque quadrata temporum periodicorum fuerint ut radiorum cubi.

54. Obiter haec notamus. 

1º. Ex circulari telluris rotatione circa suum axem oritur vis centrifuga (51) in materialibus punctis tam apud aequatorem quam apud circulos aequatori parallelos, generatim expressa per &lt;math&gt;m\varphi'=\frac{mv^2}{R};&lt;/math&gt; et quia rotatio illa fit motu uniformi, ideo&lt;math display=&quot;block&quot;&gt;v=\frac{2\pi}{T}\,\mathrm{ 
et}\, \varphi'=\frac{4\pi^2 R}{T^2}
&lt;/math&gt;Tempus periodicum &lt;math&gt;T&lt;/math&gt; est ubique idem; &lt;math&gt;R&lt;/math&gt; vero decrescit ab aequatore ad polos; in eadem ergo ratione ab aequatore ad polos descrescet vis centrifuga. 

2º. Exhibeat R , radium aequatoris terrestris (Fig. 31) et a geographicam latitudinem, cui respondet circulus aequatori parallelus habens radium R, erit R =R cosa , et consequenter R , cosa
T2
Resoluta q' in duas, quarum altera sit verticalis, altera horizontalis, existet illa 402R , cosa
D'cosa=
T2
et quoniam q' cosa est vis contraria gravitati, inferimus gravitatem imminui magis semper a polis ad aequatorem, ejusque decrementum fore ut quadratum ex cosinu latitudinis, spectata videlicet tellure instar sphaerae.

3º. Exprimat s altitudinem debitam velocitati rotationis; erit ( 30) 2gs = v ?, ideoque ( 10 ) 2gs = q R, et
consequenter
8
solia
R
.
2s
110
mg': mv&quot;R; )
et quia rotatio illa Et motu uniformi, ideo
27rR et ∙∙∙∙∙ ∢∏≃∣≹
T ' ?&quot; Ta
.-

ecosa: Tat quoniam cp' cosa: est vis contraria gravitati, inferimus gravi-
tatemimminui magis semper a polis ad aequatorem, ejusque
decrementum fore ut quadratum ex cosinu latitudinis ,
spectata videlicet tellure instar sphaerae.

111
Hinc innotescit ratio inter gravitatem et vim centri
fugam : sic apud aequatorem invenitur
8
R,
= 288 circiter;
2s1
inde sequitur quod gravitas sub aequatore in hypothesi tel
luris immotae esset == 1880' + q = 289 .
=== De vi acceleratrice in motu elliptico, existente centro virium in foco ellipsis. ===
55. Haec praemittimus: 

1 °. si ex puncto quovis M (Fig. 32) ducuntur duae rectae MN, MS tangentes sphaeram SN .. , erit MN = MS: ductis enim ex centro C radiis CN, CS ad contactus puncta N et S; itemque CM ad punctum M, triangula CMN, CMS rectangula in N et S habebunt latus CM commune, latera vero CN , CS aequalia; ideoque etc.

2°. Si per tangentes MN , MS ducuntur plana tangentia NMT , SMT ad sphaeram SN .... sese muluose.
cantia juxta rectam MT, angulus NMT aequalis erit an
gulo SMT: nam ex C , N , S ad punctum v . gr. T rectae MT, ductis CT , NT , ST, quoniam NT et ST jacent in planis tangentibus NMT , SMT , iccirco in triangulis CTN , CT'S anguli CNT, CST erunt recti; latera in.
super CN CS sunt aequalia , et CT commune: proinde
NT = ST. Triangala igitur MNT, MST exsistent ( 1 ° ) invicem aequilatera; ideoque etc.

3º. Si denotat p projectionem lineae rectae l in
plano quovis , et a angulum , quem efficit I cum eo plano , erit&lt;math display=&quot;block&quot;&gt;p = l\cos\alpha&lt;/math&gt;: patet ex Trigonometria.
4º. Si denotat P projectionem mn (Fig. 33) areae planae cd ( = A ) in plano quovis gr , et i angulum ,
quem efficit A cum gr , erit,
P = A cosi .
Ducatur enim planum mg parallelum areae A, in quod
demittatur ex d perpendiculum dK ( = x ) ; ducantor quo
que plana gh , de parallela plano qr; ponaturque dg = y .
Quisque videt prisma vel cylindrum md aequari prismati
vel cylindro ge: propterea Ax Py ; unde P A ; est
autem -
sindgK = cosi ; igitur etc. yу

5º. Secetür cylindrus rectus aB ( Fig. 34 ) plano
ad circularem ipsius cylindri basim inclinato; sectio AMBL
dicitur ellipsis ; punctum C, ubi planum secans occurrit axi
cylindri, vocatur ellipseos centrum; punctumque S, ubi se
crio illa tangit sphaeram sambl cylindro inscriptam , appel
latur ellipseos focus; pro cylindri base sumimus circuluin trans
euntem per centrum c sphaerae inscriptae; inde fit, ut ba
seos peripheria sit linea contactuum superficiei sphaericae
et superficiei cylindricae.

6º. Si per C ducitur linea quaevis recta LM ter
minata ad ellipseos perimetrum , ejus projectio in cylindri
base erit ipsius baseos diameter lm , ita at lc sit projectio
portionis LC, et mc projectio portionis MC. Sed lc mc ;
ergo ( 30 ) LC = MG: lineae videlicet rectae transeuntes per
ellipseos centrum , et ad ellipseos perimetrum terminatae ,
dividuntur omnes bifariam in eodem centro.

7º. Per extrema puncta 1 et m diametri lm du
ctis ad circularem cylindri basim tangentibus lh et mt ,
hae utpote perpendiculares ipsi lm erunt parallelae; rectae
quoque IL , mM utpote cylindri basi perpendiculares, erunt
parallelae; ergo plana hll , ImM cylindricam superficiem
112
40. Si denotat P proiectionem mn (Fig. 33 ) a-
reae planae cd:( A ) in plano quovis qr ,
et t' angulum ,
quem eliicit A cum qr', erit,
P:A cost'.
Ducatur enim planum mg parallelum areae A, in quod
demittatur ex d. perpendiculnde ( −−∶ æ ); ducantur quo-
que plana gh, de parallela plano qr; ponaturque liga:-7.
Quisque videt prisma vel cylindrum md aequari prismati
vel cylindro ge: propterea Aa: −−∶ Py: unde P:
.i.-A; est
autem −⋅↕⇣∙ ∶−− siudgK :cosi; igitur etc.
.7
20. Secet'ur cylindrus rectus aB (Fig. 34 )plano
ad circularem ipsius cylindri basim inclinato; sectio AMBL
dicitur ellipsis; punctum C, ubi planum secans occurrit axi
cylindri, vocatur ellipseos centrum; punctumque S, ubi se-
ctio illa tangit sphaeram sambl cylindro inscriptam, appel-
latur ellipseos focus; pro cylindri base sumimus circulum trans-
euntem per centrum c sphaerae inscriptae; inde fit, ut ba-
seos peripheria sit linea contactuum superficiei sphaericae
et superficiei cylindricae.
60. Si per C ducitur linea quaevis recta LM ter—
minata ad ellipseos perimetrum, ejus proiectio in cylindri
base erit ipsins baseos diameter lm, ita ut lc sit projectio
portionis LC, et me projectio portionis MC. Sed lc :: mc;
ergo (30) LC: MC: lineae videlicet rectae transeuntes per
ellipseos centrum
.
et ad ellipseos perimetrum terminatae.
dividuntur omnes bifariam in eodem centro.

113
tangentia existent parallela inter se; et couscquenter inter
sectiones quoque LH, MT istorum planorum cum elipseos
plano exsistent parallelae. Liquet autein intersectiones il
las esse tangentes elipseos in L et M; ellipseos igitur lan
gentes ductae per extrema puocta cujusvis rectae, quae trans
eat per centrum , quaeque terminetur ad curvae perime
trum, erunt inter se parallelae.
Recta LM secat bifariam ( 3º ) chordas omnes paral
lelas tangentibus LH , MT; ejusmodi enim chordarum pro
jectiones nibil sunt aliud nisi circularis baseos chordae pa
rallelae tangentibus lh, mi, atque ideo perpendiculares dia
metro lm , a qua proinde secantur bifariam : inde fit , ut
LM dicatur ellipseos diameter.

8º. Ex M ad focum S ducatur MS; rectae MS ,Mm tangent ( 50 ) sphaeram, altera in S , aliera in punctum lineae contactuum superficiei cylindricae et superficiei sphaericae: ergo ( 19. ) MS = Mm. Simili modo, ex L ad S du
cta LS, erit LS = LI.

9º. Plana TMS, MMT et transeunt per rectas MS,
Mm tangentes sphaeram , et sphaeram tangunt, et sese mutuo
secanı juxta MT; ergo ( 2º )anguli TMm, TMS erunt aequales :
simili ratione ostenditur angulos IILS esse aequales.

10º. Denotet a rectam Cc jungentem centra Cet c: trapezium LMml suppeditat Ll +Mm 2a ; igirur i 80 )
SL + SM 2a . Variala utcumquc positione diametri LM ,
non ideo variabit recta Cc , sed mavebit cousians in ea
dem ellipsi ; ergo summa rectarum SL et SM, quae in ea
dem ellipsi ducuntur a foco ad extrema puncta cujuscum
que diametri LM, erit quantitas constans. Ad haec: rectae
SL, SM efficiunt cum tangentibus LH , MT avgulos aequa
les SLH, SMT; cum enim LH et MTsint parallelae ( 7 °) ,
itemque Ll et Mm parallelae , angulus HLL aequalis erit
angulo TMm; proinde ( 99) etc.

11º. Revolvatur diameter LM donec transeat per focum S, sicque evadal AB: rccidet SL in SA, et SM in
113
tangentia existent parallela inter se; et consequenter inter-
sectiones quoque LH, MT istorum planorum cum elipseos
plano exsistent parallelae. Liquet autem intersectiones il-
las esse tangentes elipseos in L et M; ellipseos igitur tan-
gentes ductae per extrema puncta cuiusvis rectae, quae trans-
eat per centrum, quaeque terminetur ad curvae perime-
trum, erunt inter se parallelae.
Recta LM secat bifariam (30) chordas omnes parallelas tangentibus LH, MT; ejusmodi enim chordarum pro-
jectiones nihil sunt aliud nisi circularis baseos chordae pa-
rallelae tangentibus lh, mt, atque ideo perpendiculares dia-
metro lm, a qua proinde secantur bifariam: inde fit, ut
LM dicaturo ellipseos diameter.
.Ex M ad focum S ducatur MS; rectae MS
. Mm tangeiit (50) sphaeram, altera in S, altera in puncto
m lineae contactuum superficiei cylindricae et superficici sphae-
ricae: ergo (10. ) MS :Mm. Simili modo, ex L ad S du-
cta LS, erit LS:LI.
90. Plana TMS, mMT et transeuntper rectas MS,
Mm tangentes sphaeram, et sphaeram tangunt, et sese mutuo
secant iuxta MT; ergo (2&quot;)anguli TMm, 'I'MS erunt aequales:
simili ratione ostenditur angnlos HLS esse aequales.

12.• Aa est minimum , Bb est maximum omnium
perpendiculorum Ll , Mm , ... quae ex perimetro ellipseos
demittuntur in cylindri basim ; ergo ( 89) SA erit minima ,
SB erit maxima omnium rectarum , quae ex foco S du
cuntur ad ipsam ellipseos perimetrum .

13.• Punctum S' ita determinatum in axe trans
verso AB , ut sit CS' = CS , dicitur alter ellipseos focus.
Jam si ex S' ad M et L ducuntur rectae S'M et S'L , quo
niam SC = S'C et ( 69) LC = MC , iccirco SL et SM
erunt aequales et parallelae ; igitur ( 109)
SL + SM SM + SM = SL + SL = 2a .
Praeterea angulus SLH aequatur angulo SMR ; ergo
( 10 °.) angulus SMT aequabitur angulo SMR.

14°. Producatur MS donec tangenti LH occurrat
in H , erit ( 30. ) angulus LHS aequalis angulo SMT. Sed
( 109. ) SMT = SLH ; ergoò LHS == SLH , ideoque SL=SH:
hinc ( 13. ) HM = 2a .
56. His praemissis venio cum D &quot; o Arpere ad quaestio
nem propositam de invenienda vi acceleratrice o in motu
elliptico , exsistente centro virium in ellipseos foco S. Conci
piantur duo radii vectores SM , SN intercipientes angulum
inGnitesimum MSN , et producatur SN donec occurrat
tangenti TM ... in R ; erit ( 49 , 6 &quot; )
Q
2 NR
62
Binae NR , MH babendae sunt pro parallelis , eruntque
114
SB; ideoque (100) AB:Za. Quoad alias positiones diame-
tri LM habetur semper LM (SL ∙−⊢ SM, et consequen-
ter (100) LM 2a; igitur AB est omnium diametrorum
maxima: AB dicitur axis transversus ellipseos; diameter per-
pendicularis axi transverso dicitur axis conjugatus.

140. Producatur MS donec tangenti LH occurrat
in H , erit (70.) angulus LHS aequalis angulo SMT. Sed
(loo-) SMT:SLH ; ergö LHS:SLH ,
ideoque SL:SH:
hinc (139) HM:20.
56. His praemissis venio cum D'&quot; Atnpere ad quaestio-
nem propositam de invenienda vi acceleratrice ep in motu
elliptico , exsistente centro virium in ellipseos foco S. Conci-
piantur duo radii vectores SM , SN intercipientes angulum
infiuitesimam MSN , et producatur SN donec occurrat
tangenti TM
...
in R; erit (49. b&quot;)
2NR
∙∙∙⇀−−∙ −−⇀∙∙62
Binae NR , MH habendae sunt pro parallelis , eruntque115
proinde ( 55. 3. ) ut respondentes projectiones nr , mh in
cylindri base : hinc ( 55. 14º.)
nr . MH NR =
nr 2a
mh mh
Sit T tempus periodicum , quo nempe materiale pun
ctum totam percurrit ellipticam orbitam ; erit ( 46) ellipseos
area ad aream MSN ut Tad 0 : istae areae sunt ut re
spondentes projectiones ( 55. 4º. ) in cylindri basi , nimirum
ut ipsa cylindri basis ambll =
mila et area msn : ad
haec ; demisso perpendiculo st ex s io tangentem mt ,
erit
msn = j st , mr = 1 st (nr . mg) : quare
( mza)
712 14
ml
16 T2
2
SC
nir , mg
et consequenter
mi ml 62
T2 .
nr T2 2
st mg
Triangula mlh , mlg sunt rectangula , alterum in l ,
alterum in g ; habent insuper communem angulum in m :
iccirco
ml&quot;
= mh .
mg
Anguli mhl et hmt sunt ( 55. 7. &quot; ) aequales ; propterea
triangula mlh , stm rectangula in l ac o dabunt (55.30. 14º.)
115
proinde (55. 39) ut respondentes proiectiones nr, mi: in
cylindri base : hinc (55. 140.)
nr . MH nr

2
−∙− ∙ NR mh (: mh
Sit T tempus periodicum, quo nempe materiale pun-
ctum totam percurrit ellipticam orbitam; erit (46) ellipseos
area ad aream MSN ut T ad 9: istae areae sunt ut re?
spondentes projectiones (55. 40.) in cylindri basi , nimirum
∙ ∙ ∙ ∙ ↿≖ −∎⋅
↴ ut ipsa cylindri basis ambl(:-Z - ml&quot;) et aram nim: ad
haec ; demisso perpendiculo st ex .: in tangentem mt
, erit
mm:&amp;st,mr:äst(nr.mg)iï:quare
l ml ml: 62
nr ∙−−− ∙∙ T! :,- '
—-- '
&quot;rf&quot;; '
st2 mg
Triangula mih , mlg sunt rectangula , alterum in I,
alterum in g; habent insuper communem angulum in m :
iccirco '
tl,
— z'mll.
Anguli mi:! et hmt sunt (55. 73) aequales; propterea
triangula mllt ,
stm rectangula in 1ac :dabunt (55 . 30. 140.)116
Im mh MH 2a
SC si SM SM
Non pluribus opus est , ut assequamur
47' a3 1
( h) ;
T2 SM
vim nempe acceleratricem in ratione reciproca duplicata
radii vectoris .
Quoad aliam ellipsim
4 R² a ,
1
T ;
i
S, MI 2
hinc si
1 1
a3
T2
a ,
T

: erit op :
:

2 SM 2
S, M ,
Si nempe in diversis ellipsibus quadrala temporum pe
riodicorum sunt ut cubi semiaxium transversorum , vires
acceleratrices tendentes ad respectivos focos erunt in sola
ratione reciproca duplicata respondentium radiorum ve
ctorum .

57. Haec subjungimus .

1.º Fiat CS
CA
CS seu a
€ ; numerus &amp; K1 ) dici
lur excentricitas : ex L in axem transversum ducatur per
pendiculam Li , et ponantur Ci = x , Li = r ; erunt
SL = y2 + ( x — $ a) 2, S'L ' =y2 + ( x + ε a) 2 ,
et consequenter ( 55 , 13º. )
↿16
lm mh MH 212
∙−−∙∙−−− −∙∙sm SM SM .

Non pluribus opus est, ut assequamur
47:303 1
−− ∙∙∙ lt ; ? Ta sit-r, (
vimnempe acceleratrieem in ratione reciproca duplicata
radii veetoris.
⋅ Quoad ≘∣⋮∘⊡↾ ellipsim
∙− 4 123 a,3 1
ut ?! Tla 5! M :
hinc si .?- gz.
. −↿− ↿
⊽↓⊽∶⊺∣≖∙∁≖∣⇂∲∙∲∎⇌⇋⊤⊡∶
Si nempe in diversis ellipsibus quadrata temporum pe-
riodicorum sunt ut cubi semiaxium transversorum, vires
acceleratrices tendentes ad respectivos focos erunt in sola
ratione reciproca duplicata respondentium radiorum ve-
ctorum
.

57. Haec subjungimus.
CS 

↿∙∘ Fiat äseuï :8: numerus :
((1) dici-
tur excentricitas : ex L in axem transversum ducatur per-
pendiculum Li
,
et ponantur Ci:æ , Li:7; erunt
ST.&quot; :y2 −⊢ ≼⋅⊅−−∙⋮∠≖≽⇄⋮ ST]? :]! −⊢ ≼⋅≈⋅−⊢⋮∘≻≖ ,
et consequenter (55.130.)117
Vym + (x - ea) + V y2 + (x + ea )
2a ; !
unde
ye + ( x – sa )2 + 2V 99 + (2 - a) Vya + (xta)
ty: + (x + a ) = 4aº ;
ac propterea
V12 + (x - a)2 V y2 + (x +-a)? = 2a? —yox? - ?o ?
ex qua obtinetur
ya = (1-2) (a? – x2) ( o) ;
aequatio ad ellipsim inter x et y computatas a centro C.

2. ° Facta x = o in ( o ) , valor y inde proveniens
nihil erit aliud nisi valor semiaxis conjugati ( 110.) : hinc ,
denotante 6 istiusmodi semiaxem , exsistet
2
62 CS seu ( 10.) 1 -
62 ideoque CS' =a2-6.

al'
a a2
Inferimus distantiam inter focum et punctum illud ,
in quo semiaxis conjugatus occurrit ellipseos perimetro ,
acqnari semiaxi transverso .
39. Loco x substituatur a - ain (o) : emerget
y2 = ((1 — 82 ) ((2ax - x2 ) ( 0' ) ;
aequatio ad ellipsim inter x et y computatas a vertice A.
Jam vergente e ad 1 , simulque crescente a indefinite ver
117
Vr-l—(æ—eaP-l- l/Ja-l—(æ—l-eaPr-h? ⇥
'
nnde
y' −⊦ (æ −∙∙ id? −⊢ 21/7' ∓−⋅⋜∞∶∽≻∙ Vy' −⊢≺∙↿⊏⊹∽⋟≖
−⊦↗≖ −⊦ (..-'.]. ..). ∶−− ta:
.
EC propterea
Vm VW:2(:* —y2—æ2—s*a'
ex qua obtinetur
]&quot; −−−−−− ≺↿∙−∊≖≻ (a' --.r*) (a):
aequatio ad ellipsim inter se et] computatas a centro C.
2.(, Facta a: o in (a) , valor ]inde proveniens
nihil erit aliud nisi valor semiaxis coniugati (HO.) :hinc ,
denotante b istiusmodi semiaxem , exsistet
—2 b' CS &amp;&quot; ∙
..... 1−−∊≖−∙−∶ 23, seu (1 0,)1 ...—a—z- ;; ;1deoque CSa −−∶∅⇄∙− ∂≖⋅
Inferimus distantiam inter focum et punctum illud,
in quo semiaxis conjugatus occurrit ellipseos perimetro,
aequari semiaxi transverso
.

30. Loco a: substituatur a— a: in (0) :emerget

(1—82) (2aæ—æ2) (0') :
aequatio ad ellipsim inter se et y computatas a vertice A .
Jam vergente P. ad 1
, simulque crescente a indefinite ver-118
gat 2 (1 — ?) a ad limitem quemdam finitum B : aequatio
( 0 &quot; ) verget ad
yö = B x (o &quot; ) ,
et consequenter , precedente foco S' indefinite a vertice A ,
ellipsis repraesentata per (o' ) ad parabolam repraesentatam
( 40.70. ) per (o &quot; ) .
Inferimus illud : si a quovis parabolae puncto du
cuntur binae rectae altera ad focum , altera axi paral
lela , eae cum tangente per idem punctum ducta aequa
les ( 55. 130. ) hinc inde continebunt angulos.

4.• Pone conjugatum ellipseos axem fieri imagi
narium ; adhibe nempe 26V - 1 pro 26 : fiet
22 1-62 = , ideoque e &gt; 1 .
Q2
Aequatio nimirum ( 0) novam curvam repraesentabit,
quae dicitur hyperbola , quaeque secat abscissarum axem
in distantiis CA ( = a ) et CB ( =-a) ab G ; inde in infi .
nitum excurrit cum quatuor ramis ab axe illo magis sem
per recedentibus , quorum bini respiciunt partem posi
tivam , bini negativam , habet insuper centrum in C ,
focos in 0 et O' , exsistente CO = CO ' = ɛa .

5. ° * In aequatione ( o) substitue x' + sa pro x;
habebis
ya=( 1—62) ( a2 -x'tea) )
ad ellipsim vel hyperbolam prout &lt;&lt; vel &gt; 1 , exsisten
te coordinatarum origine in respectivo foco S vel 0. As
sumptis nunc ( 7.9 )
x = Dcosw , y = Dsina ,
118
gat 2(t-—£*)a ad limitem quemdam finitum B :aequatio
(a') verget ad
J'2Bæ ⋅ (a&quot;).
et consequenter
, recedente foco S' indefinite a vertice A ,
ellipsis repraesentata per (a') ad parabolam repraesentatam
(40. 70.) per (a&quot;)
.
Inferimus illud: si a quovis parabolae pnncto du-
cuntur binae rectae altera ad focum, altera axi paral-
lela
, eae cum tangente per idem punctum ducta aequa-
les (55.130.) hinc inde continebunt angulos.

4. 0 Pone coniugatum ellipseos axem fieri imagi-
narium; adhibe nempe ⊋∂⇂∕∙−−−−↿ pro 26 :iie't
↿∟∊≖−−∶−⋮⋮ ideoque : ↿∙
∙
Aequatio nimirum (o) novam curvam repraesentabit,
quae dicitur hyperbola , quaeque secat abscissarum axem
in distantiis CA (:a) et CB (:—a) ab C; inde in inli-
nitum excurrit cum quatuor ramis ab axe illa magis sem-
per recedentibus , quorum bini respiciunt partem posi-
tivam, bini negativam, habet insuper centrum in C,
focos in O et O' , exsistente CO:CO':sa.

5.0 11 In aeqnatione (o) substitue x' −∣− Sa pro a:;
habebis ↼
⋅
J*-——(1—8*)(a—(x'-l—w)2)
ad ellipsim vel hyperbolam prout :( vel)1
, exsisten-
te coordinatarum origine in respectivo foco S vel 0. As-
sumptis nunc (7?)
x': -- DCOSGJ
,yzDsinm ,119
erit
Dasin 6) = (1-2)( a ) - (ea - Dcosa)) &quot;) ;
quae traducitur ad
Da 2 ea ( 1-2) cosa a ' (1-2)
D =
1-6 cos26 1 - &amp; cosa
unde c
D :
a (1-2) ( ECOSW +1 ) .
18? cos26 1
Habetur D pro positiva quantitate ; sumpto itaque su
periore signo quoad &lt;&lt; 1 , emerget in ordine ad elli
psim
D
al 1-52) ( 1 t-scosa )
( 1 +acosw) ( 1 -ecosw)
a ( 1-2)
1 -ECOSW ( h) ;
sumpto inferiore signo quoad &gt;1 , prodibit in ordine ad
hyperbolam
a (1-2) ( ECOSW - 1 ) a (621)
D =
( 1 + scos ) (1 - Cosw ) 1 tecosw
(h' )
Non pluribus opus est ut intelligamus in primo ex ca
sibus alibi ( 50. 13.° 14. ) consideratis descriptum iri
ellipsim , in secundo hyperbolam , exsistente focorum al
tero in centro virium : quoad ellipsim , B= a; quoad hy
perbolam, B' = - a.
6. # Ex ( h)
119
erit
Didone-:( 1—s*)(a'—-(ea—chsæ)3) ;
quae traducitur ad
25a(1—s*) cos 6) D∙∙− a'( 1
Da −∙∙ −∊≖≱≖ .
1—szcos2ca 1—e*cos*c.1
unde
∙∙∙ ⇩≺↿∙−⋮⇄⋟ (scusa) :bt)
1----ea cosa:»
D
1
Habetur D pro positiva quantitate; sumpto itaque su-
periore signo quoad e(1
, emerget in ordine ad elli-
psim '
3( l—sï) (1—l—scosa1) —a(1 —e')
D—(l—l-Ecosw) (1—äcosm) 1—scosc1
(71) :
sumpto inferiore signo quoad s)1
, prodibit in ordine ad
hyperbolam
∙∙ -a(1—e*)(scosca——1) —a(sï—1)
(1—I—scosa1) (1—scosm). 1—I—ecosct
Non pluribus Opus est ut intelligamus in primo ex ea-
sibus alibi (50. 13.014.0) consideratis descriptum iri
ellipsim ,
in secundo byperbolam , exsistente iocorum al-
tero in centro virium :quoad ellipsim, B:; quoad hy-
perbolam, B': — a.
69
.
Ex (h)
∙−− .n..- ∙∙ -&quot; ∙∙∙∙∙∙∙−⋅↖∙∙∙− '.120
1 2
a ( 1-2)
sasin '
ECOSA= 1
€2-82cos ?
Ꭰ . dw al( 1 - E22 a-(1-6 )
a (1— $ 2) 2
( 1 - &quot;)
a (1452)
2 1 1 1
a (14 € 2 ) D
bi a - 1—62)
D2
proinde ( 50. 9.º )
02 2C2
a (1-2)
G- ) ( h &quot; ).
Ex ( h' )
€2sin ?
ECOS W =
a( 82-1 )
D
( a2( 1–82) 2
–1).
€ 2 .
a ( 821)
&amp; 2 - cos26 D
42( 1-2)
2 1 .
a (21) D
a’( 1-62) 2
1 1
a2 ( 2-1) Da,
ideoque ( 50. 10.)
V2
2C2
a 2-1 ( + za) ( 17&quot;).
120
∣a(1-52) d 0 eisinïæ sï-sïcosza)
o ' −⋅
' ⇀−
∙∙∙∙−∙ ∈∁∘⊱∞∶ ↿∙− −∙∙
czu-e*)a czu-ez):
proinde ( 50. 99 )
vï— 202 (
1 '1 )
h&quot;
∙ (tU—83) D 20 ( ).
Ex (h')

8003 6) a(83—1) ;
(ï) ∙∙∙ £2sin26)
−− '
dcc D aï(1—82)2
. &amp;:
e* (cuï—1) 1)2 −− ∊≖∁∘⊱≃∾∙∙∙ D
?
2 1
uzu—w?)a t
czu—ez? a(e*—-1) D
↿ ↿ ∙
↙≖≖≺∊≖−↿⋟ ⋅−⋅ ⋅↧⋅⊃−≖∙ '
ideoque (50. 100.)
202 1 1
)
,,, ↗⇩≕−− an:—1 )(D 'l'ïiz (h)121
Sit E altitudo debita velocitati v; erit ( 50. 12º. )
2BE
v=2qE=
Da
2C E
B (1-82) D2
Igitur in ellipsi
1 E
1
B ' D (ó -za),
2
seu ( 50)
olt
E
D
D
2a ( h &quot; );
in hyperbola
1
B'
E
Da - ( + za)
seu ( 5 )
E = 1 + (tha&quot;)
Ex (h &quot; ) et ( h ) consequitur, si in distantia D a cen
tro virium projicitur materiale punctum, haud descriptom
iri ellipsim vel hyperbolam nisi respectu ejusdem distan
tiae D fuerit minor vel major altitudo illa , per quam mo
bile vi acceleratrice vigente in puncto projectionis cadendo
molu uniformiter accelerato acquireret velocitatem ipsius
projectionis.
7 ° * Quoad ellipsim ( 50 , h. 6° )
9
∙
121
Sit E altitudo debita velocitati v.; erit (a 50. 12'.)
2BE— zcn E
D: B'(1-e*) ⋅
ï; ⊍≖∶∃∲⊡∶−∙− ⋅
Igitur in ellipsi
1'Efn'1 1&quot; 1) B&quot;Dï—-a(o za'
seu (50)
in hyperbola
seu (50)
E -D
,
⋮−⇂∃⇌−−⋅⊳⊣−⋅⇄−∅⋅⋅≺≀⋅⋟⋅ ∙ l
Ex (II&quot;) et (h') consequitur, si in distantia D a cen-
tro virium proiicitur materiale punctum, baud descriptum
iri ellipsim vel hyperbolam 'nisi respectu eiusdem distan-
tiae D fuerit minor vel major altitudo illa, per quam mo-
bile vi acceleratrice vigente in þuncto projectionis cadendo
motu uniformiter accelerato acquireret velocitatem ipsius
proiectionis. -
70t Quoad ellipsim (50. I:. 60)122
7
a 옘 E COSQ ) 1
dw² a ( 1-2) Q ( 1-22) - 5
hinc ( 50. 8º. b .)
go
Ca
a ( 1-62 )
1
Da
areo ds D sinids
Est ( 50. 9º . ) C =D sini.

; exhibet

dt 2
lam a radio vectore D descriptam tempusculo de : deno
tante igitur A totam ellipseos aream, T tempus periodi
cum, habebitur
ds C = D sini dt
2A
T
Est ( 27. 18º. )
a
A = 2V 1-* [Vaº-x:dx ;
exprimit 2 | Va?-xă de circularem aream , cujus radius
= a , et consequenter 1
A = Tla ? VT-
Propterea
1
C2 4 A2
T2
4772 24 (1-2)
T2
42 a3
et p =
Ta
0
9 D2
122
≖↿ ⋅ *

:

cosa) 1 1
dm&quot; −⇩≼↿∙∊≖⋗⊽ ⋅⋅∙↽∙↰↿∙∊≖≽∙ D '
liinc ( 50. 80. b,.)
∙−− ∁∙ ↿
,? ↼⇀ (tU-e&quot;)- ⋅
⋅∎⋝≖⋅∙
Est (50. 90.) C :D ciuili-f.; exhibet Egit-If. areo-
£
iam a radio vectore D descriptam tempusculo dt: deno-
tante igitur A totam ellipseos aream, T tempus periodi-
cum, habebitur &quot;
⋅
∙ ∙ ds 2A
C—DSID! 'a'ï—T ∙
Est (27. me.) fZl/l-Ez l/a'-æ' dx;

∽ ∘
exprimit Zf Vaz- ac2 dx circularem aream , cujus radius
o
∙−∙−−−∙− a , et consequenter
A −−∶↿∽≖ ⇂∕↿ ∙a&quot; ∙
Propterea
4A3 47taa4(1-s*) 41:303 1 ∙
Ta&quot; TTL ∅∘⊔⊢−
'1'» C*.—. 'ne'123
prorsus ut supra ( 56).

8º. Obiter notamus illud: si materiale punctum
motu elliptico movetur viribus ad ellipseos centrum len
dentibus, eae erunt in ratione directa distantiarum ab ipso
centro . Assertionis demonstratio eruitur ex dictis ( 56) : sint
enim duo radii vectores CM ', CN' sub angulo infinitesimo
M'ON' , et producatur CN' donec occurrat tangenti M'T
in R' ; erit ( 49. 6' ' )
2N'R'
♡ 02
binae N'R' , M'C censendae sunt parallelae; proinde ( 55.3º. )
m'c : n'r' = M'C : N'R'
M'C . n'r
m'c
area insuper ellipseos ad areolam M'ON' ut tempus pe
riodicum T ad tempusculum 6 ; quae areae cum sint ( 55.4º. )
ut respondentes projectiones in cylindri basi , nimirum
ut ipsa cylindri basis ambl ( = 76. cm ' ) et areola
cm'.r'm' cm '
m'cn'
V r'n'. 2 cm ) , iccirco 2 2 m'
cm'
r'n ' . 2cm
4 02 unde r'n'
762.
cm
272. cm' ; 1 4 T2 T2

et consequenter
M'C . 27. cm'
T2
N'R'
cm'
Ta 272. M'C..
--
123
prorsus ut supra (56).
80. Obiter notamus illud: si materiale punctum
motu elliptico movetur viribus ad ellipseos centrum ,ten-
dentibus, eae erunt in ratione directa distantiarum ab ipso
centro. Assertionis demonstratio eruitur ex dictis (56): sint
enim duo radii vectores CM', CN' sub angulo infinitesimo
M'CN' ,
et producatur CN'donec occurrat tangenti M'T'
in B'; erit (49. b&quot;)
2N'R'
? ∶−⋅− ⊖≖ binae NZR', M'C ceusendae sunt parallelae;proinde (55.30.)
m'c:n'r':M'C:
⋅ 'C. &quot; N'R'-— M nr :
m'c
area insuper ellipseos ad areolam M'CN' ut tempus pe-
riodicum T ad tempusculum 9; quae areae cum sint (55.4&quot;.)
ut respondentes proiectiones in cylindri basi , nimirum
ut ipsa cylindri basis amb! (: Tt. 27:23 et areola
, .
cm'.r'm' cm ∙
∙ ∙
2
...—3
CI&quot;. I o
,
—— - n .Zcm - 4 9: . .
∙ 9: a , .
32. cmlb :T2;undern :::-'F. 212. em,
et consequenter
9:
MC. ∙⊤↓⋅↴∙⋮−⋅∙ ⇄∏≖ cm

NR −− ∙ ∙−−− -;'21t'.M'C.
cm124
Propterea
. M'C :
vis nempe acceleratrix Q directe ut distantia M'C ab el
lipseos centro

* Etiam sic : in ( o. 1º. ) fac

X
Dcosw y = Dsinw ;
prodibil aequatio inter coordinatas polares ab ellipseos cen
tro computatas, nimirum
av182
Dsin ? w = (1-2) (a² - D2cos w ), unde D= V 1-8? cos26
Hinc
at 2
d:2 av1 (via1-2003 (1-8? cosaw ) V 1-2coscosti
D3 1
a* ( 1-2) D
.
ac proinde ( 50. 8º. 3 , )
CP
a4 ( 1482 )
D :
quae ad superiorem expressionem traducitur; nam ( 70. )
4724 (1-2)
C2 =
4A2
T2 T2
124
Propterea
4 ita
?

:

0132 ∙ M'C;
vis nempe acceleratrix go directe ut distantia MC ab el-
lipseos eentro.
&amp; Etiam sic: in (0. 10.) fac'
∶−∙−− -Doosa) ,y
−−−−− Dsinæ;
prodibit aequatio inter coordinatas polares ab ellipseos cen-
tro computatas, nimirum
al/1— ei
Dsin2 a): ( 1—53) (aa.-ul)2 cosm), unde D—
Hinc
([21 «
⋅
') ? cos-36) sium
113&quot; (zl/1—ea ⇂∕ ↿−⋅⋅∊≖∞≘≖∾ ≼↿∙∊≖∘∘≘≖∾⋟⇂∕↿−⋅⋮∅∾∙≖∞≻ ∙∙∙ D3 1
−− a4(1—£2)—ï '
' ⋅ ac proinde ( 50. 823, )
Ca
?
∙−− a4(1—-s2 )
quae ad superiorem expressionem traducitur; nam (72)
Ca— 4A' ∙∙∙ 4n304(1—£2l
T2 T:
⇂∕⋅↿ ∙⊽∊≖∞⊱≖∾ .125

=== De motu relativo punctorum materialium, tendentium in se mutuo viribus acceleratricibus quae sint directe ut massae in quas tenditur, et reciproce ut &lt;u&gt;quadrata&lt;/u&gt; respondentium distantiarum.===
58.* Sint m, m ', m , ... punctorum massae; a, b, c coordinatae orthogonales puncti m in ordine ad axes OX, OY, OZ (Fig. 8); x ', y', z' , x &quot; , y &quot;, z &quot; , x '&quot; , ... Coordinatae reliquorum punctorum in ordine ad novos axes et parallelos axibus Ox, OY, OZ, et habentes originem in m. Factis compendii causa ( 50. 7.0)
x ' ty's tz's =k ?, x &quot; ty's t-z&quot; = k &quot; , etc ...
erunt ( 50. 4.0) quoad motum puncti m
de a m ' x' m' ' Qc &quot; d²b m'
.
g' , m &quot; g
+
k'
&quot; &quot; ) dc2 k2 k' k &quot; 2 hit
d12 ka kita
d2c
m' z'
k'
m &quot;
k' ' ?
.
dc2 ti to...,
seu
d'a d26
dc2
m'x m'z '
Σ
k'3
niy'
Σ
dc
dca &gt; ( o ) .
dt2 k'3
Nunc quod spectat ad aliud punctum v . gr. mi' , pone ( 50.70. )
(.x &quot; —X')2 +6 &quot; -Y')2 + (z&quot; -z&quot;) = 002 ,
( z&quot; &quot; ' —x' ) 2 + 6 — ')2+ ( z' — z &quot;)2 = ' ' , etc... ;
exhibebunt


126
t ... The **** +
en +++
m &quot; yy'
+
d'a + ... ,
+ ..
vires acceleratrices ab m &quot; , m ' exercitas in m' , no
visque axibus parallelas : denotant
ac m
j'
k'a k'
. C
k'a ' ki k'2 k'
vires acceleratrices ab m exercitas in m' , iisdemque novis
axibus parallelas ; sunt insuper ata , bty' , cta' coor
dinatae puncti m' in ordine ad axes OX,OY,02; facto igitur
m &quot; m '&quot;
+ .. =
assequemur quoad motum puncti m'
20 dQ d'a+x' )
dta
mx' d2(6 + y ')
k3 dla
my'
k'3 dx '
dy '
dQ mz' dºlc + z )
dia dzi k'3
d²a d2b
Substitutis valoribus
dac
ex ( 0 ) , prodibunt dca dla dt2
daxi dl mx' m'r'
dxc '
.
day' d my'
dc2 dy
m'y'
Σ
dea k'3
k3
k3 k'3
126
m&quot; .v&quot;--.r' a&quot;; 7 '—:7' F
∙⋅⊱∷−∎∙−⊦∂⋅≖ a ⊣−∙⋅∙∙
vires acceleratrices ab m&quot;, m'&quot;
, ,..
exercitas in m'
, no-
visque axibus parallelas: denotant
ut se' m y' m :
&quot;F' la&quot;—k&quot; k&quot;&quot;'1?'-&quot;£'
vires acceleratrices ab m exercitae in m'
,
iisdemque novis
axibus parallelas ; sunt insuper a-l-z' , (Hl-y' , e—l—e' coor-
dinatae puncti m' in ordine ad axes 0X,OT,OZ; facto igitur
&quot;
m
m m 37 −∂∙−⋅⋅ −⊦ −−∶
9-
assequhmur quoad motum puncti m'
d'(a-[-æ') ∙∙∙ dQ mx' d3(b-l-y') ∙∙∙ dQ my . d,. dx&quot; k-a de dy' k'3
(P(e-l-z') .... di) me'
dt' dz' k'3
∙ ∙ ∙ dia d'b die ∙
∙⊱∎≖∣⋯⋅⋯∎⋯ valonbus dt&quot; ∙ dt' ∙ dt? ex (0) , prodibunt
g'æ/ dQ -mæ' zm'x' d'y' dQ my' zmiy'
dt' dæ' 163 It'3
'
dt2 dj'- k'3- It'3127
daa' d2 mz'
K'3
m'z'
Σ
dta dzi k3
formulae determinantes motum relatiyum puncti m' quoad
punctum m . Quoniam
00 mx' m'x
k'3
mtm x +
k'3 dx ' k'3
zel
2 X
m &quot;
come
-ac '
813 -) +mi&quot;
xc
k3
V3 k'3) +... ,
dQ
,
- - monte +
-&quot; * 7- ) + m.A-A ) +...
en e -maile +
) &quot; V +
d2 mz'
-Σ dzi k3
m &quot;
tom &quot;
t ... ;
03 k3
hinc facto
R = m &quot;
.6. – +) +
(5--**&quot; +jx +e*e&quot;)+ - ( &quot; ),
m &quot;
formulae ( 0' ) vertentur in
127
∙⇌⋅⋮⋅⋮⋅≕↙⇣≴⋅≖−−−∶↗−⋅⋮−−− ∑∶≀−≖⇣ (.,-,,
dt: dzï lt'3 k'3
formulae determinantes. motum relativum puncti m' quoad
punctum m . Quoniam
dQ ⋯⋅∙∙∙∑∽∙∙↼∙⋅⋅≈∙↾−∙ m—I-m' .
⊋⊑⋅∙⋅−⋅⊼∙∶⊤∣ k'3 −−−⋅∎−∎ ↗⊏∙⋮∣ æ
III I'll
.. ∞⋅⋅−−⋅↕∙⇗ æ&quot; ,,, æ —x' x ≺−−⊽⋮−−−⋅−↗⋮⇁⋮⋮−≻ ⊹≖⊷ ⋯⋯≻⋅⊢ df
———— —— k'3 ∙−− 72— k&quot;3
&quot; yn ∙ yl! &quot; yon—70 ..- 70&quot;.
&quot;' (
a&quot;? ≀⊏⊤∍≻−⊦∽ ↾≺↴↼⋮⋅−∣⋮∎∎ ≀∎⊄−⋅∣∎⋅⋮≻∎⊦⋅⋅⋅∙
(19 Mi z m'x' m—l-m' zo ∙⊦
dQ my' Z mfy' m-l-m'y. ∙∙⊦
⊋∎≖∎⋅∎∎∎∎ k'3 15&quot;— ⋅∎∎∎ ↗⊏⋅⋮⇂ a'.—Z&quot; z&quot; &quot;' zIIO—zt all-l .
&quot;'&quot; ea &quot;&quot;17'5) &quot;'&quot;&quot;
«W &quot;:?75) ⊹⋅⋅⋅⋅ hinc facto
1 æoæn ⊣∙∙ o n

: , z'&quot; B: m&quot; (y'—W) ∙∙∣∎∙

1 me xlv ' '&quot; zl zh, &quot;
mm (öt—or— J—æO—ïä—L) ⊣∎∙∙∙∙ (O 2, .
formulae (o') vertentur in128
dax
de2
m -tm '
+ x's K'3
dR day '
dx ' ' de
mtm
+
k'3
g
mtm
dR daz' dR
dy' '
dit de k'3
dz
Porro , cum habeamus
ka + k &quot;? – 02
x ' x &quot; ty'y &quot; + =' z&quot; =
2
k'2 + k ' ' ?
d''2''
x' x'&quot; ty'g '&quot; +z'z' &quot; etc... ;
2
poterit (o&quot; ' ) scribi etiam in hunc modum
(
R = m k'o + k&quot; — 0° ) +
22
in '&quot; . k'2 + k '''2'''
2k &quot; 2 3* 2) + ... ( o &quot; ) .

59 * Fac at systema reducatur ad duo tantum pun
eta m et m' ; habebis R = 0 , et consequenter
der mm x
+
k'2 k'
day' mtm +
dia K2 K
&gt;
dta

* 3&quot;. d2 z' mtm

dt2 +
k'2 k
Relativus videlicet motus puncti mi quoad m proveniet
m +m: (50. 4. 20. ) a vi acceleratrice tendente ad m : pro.
k' ?
⋯⊣−⋯⋮↨↾ '
(0 ∙∎∣).
klö ,d—l;
dR dïz' m—I—m'z, dR ∙
«(y' dc2 k'3 dz'
Porro, cum habeamus
&quot; k': k&quot;: ∙∙∙ ∝↭⊹⊔↤⇥⋠−−⊦⊇ ∂∣∣≖ ⋅
-k'jl −−⊢ k'&quot;a — ö&quot;&quot; x'M* x'&quot; &quot;' z'e &quot;&quot;— 2 ∙ etc... :
poterit (o&quot;) scribi etiam in hunc modum
!, k.: kn; −∙− ux.,, ∂∜≖
.).
21./&quot;a
↿ ⋅ ra −⊦∣⊏⋯≖ −⋅∂∣∙⋅≖⋅≻ ..
∙−∂⋅−∣⋅∣∣ . ka2&quot; .l.-&quot;' (0 )- mllt
59; Fac 'ut systema 'reducaturad duo 'tantum pun-
cta m et m' ;habebis R ∙−−∶ ∘,
et consequenter
d'x' m di' '
'
−⊦⋯P',—mi −⊢⋯⋅−⊣⋤↾⋮⋡−∙≛ −−−−−∘∙ (it—T k'3 k dca k' k'
da z' ∙ ⋯⊣−⋯∣ .'
d:: [ kl; '
kl :::-&quot;'o.
Relativus videlicet motus puncti m' quoad m proveniet
(50 . 40 . 70.) a vi acceleratrice mt;&quot;. tendente ad m :pro.129
&gt; 7
pterea ( 50.13º . 140.57.50. ) describet m' motu relativo
vel parabolain , vel ellipsim , vel hyperbolam , existente foco in m .
dR dR dR 60# Secunda membra formularum
dx' ' dy' ' dz
( o &quot; ) exhibent ( 50 , 4.:) vires turbantes relativum motum
puncti m' determinatum per formulas (o &quot;) . Hinc si membra
illa manent constanter tenuissima , ita ut (o ' ') et ( o&quot;) dif
ferant in terminis exiguissimis , turbationes quoque inde
provenientes manebunt tenuissimae ; lineaque ab m descri
pla circa m poterit adhuc spectari tanquam vel parabolica ,
vel elliptica , vel hyperbolica ; ita tamen , ut gaudeat ele mentis continue mulatis .

61 * Datis tribus punctis m , m ' , m &quot; ( Fig. 35 ) ,
demissoque ex m' in mm &quot; perpendiculo m'A , sint
x' = mA , y' = m'A , X &quot; = mm &quot; , z' = 0, y = 0, z &quot; = 0.
Erit ( 58)
a' x
1
R m'
(-- = m &quot;
k3
ha( x&quot; —x'to) 2ty'a )
unde prodeunt vires distrahentes m' ab m juxta directiones
x' et y' , nimirum
dR x &quot; — x 1
dx = m &quot;
[(x&quot; — x'ja traj
. DR
dy '
m &quot;
[(x“ — x'ja + y'a ] }
Denotet h angulum m'mm &quot; , et D distantiam mm' ; erunt
x ' = D cos h , y = Dsinh , et consequenter
129
pterea (50 . 130. 14&quot; . 57 . ö&quot;.) describet m' motn relativo
vel parabolam , vel ellipsim, vel hyperbolam , existente
foco in m .
dR dR dR
dæ' , d)&quot; , dz'
(o&quot;') exhibent (50 . 40:) vires turbantes relativum motum
puncti m' determinatum per formulas (a')
.Hinc si membra
illa manent constanter tenuissima ,
ita ut (o&quot;') et (a') dif-
ferant in terminis exiguissimis , turbationes quoque inde
provenientes manebunt tenuissimae ; lineaqne ab m' descri-
pta circa n; poterit adhuc spectari tanquam vel parabolica ,
vel elliptica, vel hyperbolica; ita tamen ,
ut gaudeat ele-
mentis continue mutatis .

130
[ ( x &quot; —x ) 2 + y'r] - = ( x &quot; 2—2D.x&quot; cosh + Daj - -
mi [1+ (13–2cori)]- *

* &quot; [ - P2-2.5k)+3 . ) 6–2 cos )&quot;

- 2.7 CM) 6-2005 )'+ ]
Propterea , si D est ita parva prae aut possint omitui
termini includentes factorem exsistet
(2 )&quot;,
[cº= 2') + s] =
1
+3 D cos h
73 &quot;4
ac proinde
dR
dx = m '
3 D cos h 3 D2 cos2 h
x''3''
=
2 2:14
X

m
* +357 Dosh )
( -Dout
mi ( Doco 4-3C )*.)
China + 202 )=
2 m D cos h
dR 3 DP sin h cos h
dy.
m &quot;
24
130
3
∐∙↧⋅∥−−∙↧⋅⋅⋟≖⊹∫⋅≖⋮∣ −'i;:[£&amp;—2th cosh-l— D'] −⋅⋮∎∎ :
∙∙ 3
æ&quot; :
[130 2, äl—Zcosh)]— 7:
.'! 30 D 3.5 D): D
) a': ∣∶↿⋅−⋮⊸⋅⊋∙−⊤≺⋤−∣ 2—-cosh)-i—m (;&quot; (;,—2005&quot;
−∶≣⋅−≣−−∶≟≺⋚⋛∥−≻ ≺−⋅⋅⋅⋛⋮⊽∙−⋮∞≖↗⋮≻⋮⊣−⋅∙∙∃∙
PrOpterea, si D est ita parva prae æ&quot; ut possint omitti
. . . termini D . . includentes factorem (F) ,
exsistet
[(. ' BD-io-sh. «J)- −⊦∂∣≖∃−⋮⋅↼−↿ −−∽ ↽⊦−−−−−−
ac proinde
dR ,, a.,—a.&quot; æ&quot;—æ' 1
ævo, a'./3&quot; −⊦ [[[. ∎∎∎∎∎ II
a: .r.-3 ,,(1 Dcosh BDcosh— BD'cosïh 1)
., 2 D cOs '! (D : cos: !: 2m&quot; Dcos]:
m -—-3 −−
∙ ,, −− ( æ... .) .: .r., a
−∉⋮≹∙∙∙ ∣≺∐∘⋮∐∣⇂⊹∍∘∙∙⋮∐∣≖∞≖∣≖ −∙∙
dy —--—m ∙↿∙∙∦⋮ æ&quot;], )—131
sin h cos m&quot;
-
D sin h
3 +3 m&quot; D sin h
m 

62. Bonum erit alia ratione nonnulla hic stabilire
circa vires in praefato motu relativo . 

↿∙∘ Sint duo puncta
T , P (Fig. 36.)
, quorum massae m, m', distantia vero TP
(: k');et &quot;veniat determinanda vis acceleratrix in motu
relativo puncti P quoad T . Ex hypothesi P tendit in T vi
I
acceleratrice . m . . m

; — ;et T in P v
: acceleratrice ∙−−∣−−≖ sive au.

] 3
I
tem T sollicitetur .. vn. m . m .
−− ∣−⋮∣−≖− et Pv: 17; , sive T quiescat et P
I
sollicitetur vili—ïm]— &amp;, idem in utroque casu (5) habetur
motus relativus puncti P quoad T; vis ergo acceleratrix
in istiusmodi motu erit

2.&quot; Praeter P , T detur et tertium punctum S ,
cuius massa m&quot; , ut determinentur vires iude provenien-
tes, quibus turbatur motus relativus puncti PquoadT
ortus ex vi (0) . Ducta ST
, completoque parallelogrammo
.. STPP' , exhibeat diagonalis SP (: 8&quot;) vim g.: , qua sol-
licitatur P versus S:resolvatur vis ista in duas, quarum al-
tera (: ?') sese dirigat iuxta PT , altera (:f) iuxta PP';
exhibebitur illa (8) per parallelogrammi latus PT (: k')
.
haec per latus PP':ST (: k&quot;); eritque '
m&quot; .,

; n ⇀
: m&quot; IC, '

m&quot; k, ≒≀−∣−⋮∙⊊≱⋅∙∣⇆∶∂ ∶∣⊄∙ ]; ,unde 93:77sz &quot;3 ⋅132
m' '
m &quot;
Sollicitatur T versus S vi ; et attentis f et i motus k''2''
relativus puncti P quoad T eodem prorsus modo fiet ( 5 )
sive T quiescat et P sollicitetur vi f
m '
sive T sollicite k'2
m &quot;
tur vi et P vi f. Propterea vires provenientes ex S ,
et perturbantes motum relativum puncti P quoad T , al
tera juxta PT altera juxta PP' parallelam rectae ST , ex
primentur per
k &quot; 2
ø=73
m &quot; k
&quot;
g = f mi&quot;
&quot; Cess )
( c' ) . k's
3.° Ex puncto S demittatur perpendiculum SS'
( =i) in planum curvae , quam describit P motu relati
vo quoad T; ab S ad T ducatur recta ST ( =n ) , sitque
angulus STP = a : vis q&quot; agens juxta directionem paralle.
lam rectae ST resolvetur in duas, quarum altera q&quot;cosSTS
seu q &quot; . ! existet parallela rectae ST in plano cur
vae , altera q &quot; sinSTS' seu o&quot; . perpendicularis eidem pla k &quot;
resolvetur in duas quarum altera o &quot;
no: rursus onk
cos a
aget in curvae plano juxta TP , altera om. sina in eo k &quot;
dem plano normaliter ad TP. His positis , quisque in
telligit vires perturbantes motum relativum puncti P exhiberi
posse per
132
SollicitaturT versus S vi &quot;'

; et attentis f et −∥↼↕−∙ -, motus 1.&quot;» 1.&quot;

relativus puncti Pquoad T eodem prorsus modo fiet (5)
sive T quiescat et P sollicitetur vi f— 'I—N. ,
∣∣≖ sive T sollicite/-
et P vi f. Propterea vires provenientes ex S ,
∙ m tur '! k&quot;-
et perturbant'es motum relativum puncti P quoad T , al-
tera juxta PT altera juxta PP' parallelam rectae ST, ex-
primentur per
' mllko &quot; &quot;zl! &quot; kl! 1 '
Pf&quot;??- ⊕−−⇌↾−−⊺⋇⊽≏∶⋯ (Fa-&quot;' ia&quot;) &quot;'

3.&quot; Ex puncto S demittatur perpendiculum SS'
(::t') in planum curvae , quam describit P motu relati-
vo quoad T; ab S' ad T ducatur recta S'T (::n)
, sitque
angulus S'TPr-at: vis 9&quot; agens juxta directionem paralle-
lam rectae ST resolvetur in duas, quarum altera 9&quot;cosST5'
seu 9&quot;? existet parallela rectae S'T in plano cur-
vae, altera 9&quot;sinSTS' seu q;&quot;.grperpendicularis eidem pla-
no: rursus ?&quot;]?- resolvetur in duas quarum altera ⊄∙⊅⋅⋅∙⋮∙− eos ::
aget in curvae plano iuxta TP , altera ?&quot;)—;.sinat in eo-
dem plano normaliter ad TP. His positis, quisque. in? telligit vires perturbautes motum relativum puncti P exhiberi
posse per133
COS Q = cosa ,
9 =porn o--&quot; (* - ) .com
Pa = e&quot; sin æ =
= m
m &quot; (- ) snæ ,
93 = ml - )
( c )
i

;

9 , et Q2 agentes in curvae seu orbitae plano ipsam orbi
tam turbant ; 93 perpendicularis plano orbitae turbat
ipsius plani positionem .
4. ° Pone S , T, P esse constanter in uno eo
demque plario ; erunt i = 0 , n=k&quot;, a=S'TP=STP(=h) :
proinde
PI
m &quot;
8'3 -m&quot;( )cosh ,
&quot; sink ,
}
( cm)
Q2 , 93 = 0 .
Pone insuper ST, SP ita magnas prae TP ut , ex P du
clo perpendiculo PQ in ST, assumi possit absque sensi
bili errore SP=SQ , nimirum d&quot; = k&quot; -kcosh ; erit
1
js =(k“&quot; —k'cosh)-3 = 13 +
3k'cosh
+
Hinc proxime
m &quot;
m'k ( 1-3cos'h= ( 1 +3cos2h) ,
k &quot; 3 2K3
( c&quot; )
3m''K'sinhcosh 3m'k'sin2h''
92 k&quot;3
2k'3
133

: n&quot; muli, &quot; k&quot; ! n 913? —Q ?

eos a:
ïïï —m 673 k,,a k,, 0082,
.&quot;
k ⋅
,
93——9 &quot;,;— Blna :m &quot;(ä-3- It.—741) ,——,- sin a ,- (c)
]. LII-. [
i
93:907?sz −−⋮ ⇁≖⊼↗ ;
4). et (p, agentes in curvae seu orbitae plano ipsam orhi-
tam turbant; (pg perpendicularis plano orbitae turbat
ipsius plani positionem.
4.&quot; Pone S, T, P esse constanter in uno eo-
demque platfo; erunt i:o, n.:k&quot;, a:S'TP:STP(:h):
proinde
mrlk' &quot; kn
'
(Pr −−∶ 7873- −−⋅ m
⊱∣−⊵∙−− F,.)COSII,
(e&quot;)
?::m&quot;≣∶⋅⋅⋮∙−⋅ -—k,,,)smh , 93:30.
Pone insuper ST, SP ita magnas prae TP ut. ex P du;
cto perpendiculo PQ in ST, assumi possit absque sensi-
hili errore SP:SQ , nimirum d&quot;:k&quot;—k'cosh ; erit
1
I,
, −∙∙ BkCOBh ∙≦↜−∽⋮⋅−−−−−≺↗⊏ —kcosh) ∍≔−−↼−−⊺⋮−−∣−−−−
k&quot;, −⊦
,
∙∙
Hinc proxime
∙ ?: 2773— (1—3cosïh):— (1-1—3c092h) ,
z—kHS .
(e&quot;)
—3m' &quot;ksinhcosh —3m &quot;k sinZh134
5,9 Fac ut orbita puncti P sit circularis , ipsum .
que P moveatur ad partes N : sive spectentur formulae
( 6 ') , sive (6 &quot;) , sive ( c &quot; ), aget 92 juxta orbitae tangen
tem contra motus directionem : ejus proinde valori erit
praefigendum signum negativum.
=== De pendulis; deque gravium descensu per arcus cycloidales. ===
[[Fasciculus:Simple pendulum generalized coordinates.svg|thumb|Pendulum]]
[[Fasciculus:Pendulum simplicium.svg|thumb]]
63. Pendulum constat filo tenui secundum alteram sui extremitatem fixo, quod tamquam linea recta et gravitatis expers concipitur, ex quo suspensum punctum ponderosum a directione verticali dimotum potest huc et illuc circum punctum illud alterum extremum fixum in motum circinationis per arcum excurrere. Excursio penduli ab uno arcus, quem describit, extremo &lt;math&gt;C&lt;/math&gt; (Fig. 37) ad aliud extremum &lt;math&gt;D&lt;/math&gt; dicitur &lt;u&gt;vibratio&lt;/u&gt; seu &lt;u&gt;oscillatio&lt;/u&gt;: accessus ad verticalem directionem ex &lt;math&gt;C&lt;/math&gt; in punctum infimum &lt;math&gt;B&lt;/math&gt;, vel recessus ex &lt;math&gt;B&lt;/math&gt; in &lt;math&gt;D,&lt;/math&gt; dicitur semivibratio. Si unicum ponderosum punctum pendeat e filo, pendulum dicitur simplex, si plura in diversa a suspensionis puncto distantia pendeant, dicitur compositum.  
[[Fasciculus:Pendulo simples.jpg|thumb]]
Illud facile quisque intelligit, pendulum &lt;math&gt;AB&lt;/math&gt; circa punctum fixum &lt;math&gt;A&lt;/math&gt; eodem motu arcum circuli &lt;math&gt;CBD&lt;/math&gt; descripturum ac si, sublato filo, in superficie sphaerica perfecte dura et levigata punctum ponderosum moveretur motu impedito. Sicut enim adducto puncto illo ad praedictae superficiei punctum &lt;math&gt;C&lt;/math&gt;, et exinde demisso, gravitas &lt;math&gt;CT&lt;/math&gt; horizonti perpendicularis &lt;u&gt;resolveretur&lt;/u&gt; in duas vires, quarum altera &lt;math&gt;CE&lt;/math&gt; ad tangentem &lt;math&gt;CG&lt;/math&gt; normalis insumeretur in premenda superficie, altera expressa ab ipsa &lt;math&gt;CG&lt;/math&gt; sollicitaret punctum ponderosum ad motum per tangentem infinite parvam, ac deinde per aliam atque aliam subsequentem, et sic deinceps per reliquas omnes numero infinitas et infinite parvas tangentes, quibus constare arcus descriplus concipitur; ita a filo resolvetar gravitas eodem prorsus modo , nempe partim in trahendo filo insumpta, partiin ad singulas arcus circularis tangentes infinite parvas subinde determinata, qua deducetur pendulum per arcum circularem motu omnino simili, subeunte filo &lt;math&gt;AG&lt;/math&gt; vices curvilineae superficiei: hinc sicuti punctum illud ponderosum propter suam gravitatem, postquam descendisset ex &lt;math&gt;C&lt;/math&gt; in &lt;math&gt;B&lt;/math&gt;, cogeretur ascendere ex &lt;math&gt;B&lt;/math&gt; versus &lt;math&gt;D&lt;/math&gt;, ita ob rationem similem pendulum post descensum ex &lt;math&gt;C&lt;/math&gt; in &lt;math&gt;B&lt;/math&gt; ascendet ex &lt;math&gt;B&lt;/math&gt; versus &lt;math&gt;D&lt;/math&gt;. Rursus quemadmodum ponderosum punctum in praedicta superficie ascendere inciperet per arcum &lt;math&gt;BD&lt;/math&gt; cum eadem velocitate, quam acquisivisset in puncto infimo &lt;math&gt;B&lt;/math&gt;, et ideo ad eamdem altitudinem, ex qua descendisset, perveniret, nempe usque in &lt;math&gt;D&lt;/math&gt;, ubi extincta omni velocitate, iterum gravitate sua inciperet descendere, et in puncto &lt;math&gt;B&lt;/math&gt; priori velocitate rursus acquisita, cum ea ascenderet iterum in &lt;math&gt;C&lt;/math&gt;, atque ita porro ascendendo et descendendo perpetuas et aequalęs in peripheria &lt;math&gt;CBD&lt;/math&gt; excursiones perficeret, ita ob eamdem rationem penduli oscillationes aequales essent natura sua et perpetuo duraturae, nisi ab aeris &lt;u&gt;resistentia&lt;/u&gt; et &lt;u&gt;frictione&lt;/u&gt; aliqua circa sustentationis punctum &lt;math&gt;A&lt;/math&gt; inaequales primo redderentur, ac denique extinguereatur; adimentibus scilicet ejusmodi causis in singulis oscillationibus aliquid de illa velocitate, quae producitur a gravitate.

64. Velocitates &lt;math&gt;v&lt;/math&gt; et &lt;math&gt;v'&lt;/math&gt; in puncto infimo B acquisitae a gravibus per arcus &lt;math&gt;CB, C'B&lt;/math&gt; descendentibus sunt ut ipsorum arcuum chordae. 

Per &lt;math&gt;B&lt;/math&gt; concipiamus duci tangentem et in eam ex &lt;math&gt;C&lt;/math&gt; et &lt;math&gt;C'&lt;/math&gt; demitti perpendicula &lt;math&gt;z&lt;/math&gt; et &lt;math&gt;z'&lt;/math&gt;: denotante &lt;math&gt;r&lt;/math&gt; radium &lt;math&gt;AB&lt;/math&gt; et denotantibus &lt;math&gt;k, k'&lt;/math&gt; arcus quoad radium 1 similes ''arcubus'' &lt;math&gt;CB, CB'&lt;/math&gt;, erunt 

&lt;math&gt;z = r ( 1 - \cos k ) , z' = r (1 - \cos k' ) ; &lt;/math&gt; 

et quoniam (30: 36)

&lt;math&gt;v^2 = 2gz, v'^2= 2gz';&lt;/math&gt;

propterea

&lt;math&gt;v: v' = \sqrt{2gr (1 - \cos k )} : \sqrt{2gr (1 - \cos k')} = \sin \frac{k}{2} : \sin \frac{k'}{2};&lt;/math&gt;

ideoque etc.

65. Pendulum, quod incipit descendere ex &lt;math&gt;C&lt;/math&gt;, percurrat arcum &lt;math&gt;CM&lt;/math&gt; tempore &lt;math&gt;t&lt;/math&gt;; sitque &lt;math&gt;\alpha&lt;/math&gt; arcus quoad radium 1 similis arcui &lt;math&gt;BM&lt;/math&gt;: erunt &lt;math&gt;CB =rk, BM = r\alpha&lt;/math&gt;; et designante &lt;math&gt;u&lt;/math&gt;velocitatem in puncto &lt;math&gt;M&lt;/math&gt;, exsistet

&lt;math&gt;u = - 2gr (\cos\alpha - \cos k ) = 4gr \sin\frac{k+\alpha}{2} \sin \frac{k-\alpha}{2}.&lt;/math&gt;

Si arcus &lt;math&gt;k&lt;/math&gt; est ita exiguus, ut possit absque sensibili errore substitui respondenti sinui, habebimus
&lt;math&gt;u^2 = gr(k^2 -\alpha^2),&lt;/math&gt; et consequenter (28)

&lt;math&gt;\frac{ds^2}{dt^2} = gr(k^2 -\alpha^2),&lt;/math&gt; 

unde

&lt;math&gt;dt = \frac{ds}{\sqrt{gr(k^2 -\alpha^2)}} = \frac{r\beta}{\sqrt{gr(k^2 -\alpha^2)}}= \frac{\beta}{\sqrt{\frac gr (k^2 -\alpha^2)}};&lt;/math&gt; 

&lt;math&gt;\beta&lt;/math&gt; est arcus quoad radium 1 similis arcui infinitesimo Mm
( = ds ). Nunc centro H ( Fig. 38) et radio HD ( = k) describe circulum DED' ; sume HN : Ν »
B; duc perpen
dicula Ne, ne super HD: et Ey parallelam radio HD. Trian
gula similia HEN, Eey rectangula in N, y praebent

∙∙∙⇀ ,4þf - ⇀∙⋅∙∎∙ .. ⊸∙⋅⋅⋅∙∎∎∣∙ 4.-
∙−
..137
Ey: EN = Ee: HE, seu B: V R2-42 = Ee: k :
hinc
B
Ee 8
dt ;
V R2-42 k
et consequenter
Ee dt
kV
tempusculum videlicet dt impensum ad percurrendum
seu Nn, obtinetur dividendo respondentem arcum Ee per
kV § . Inferimus tempus t impensum ad percurren
dum ka seu ND, obtineri dividendo respondentem ar
cum ED per kV ; nimirum
ED
자
름
k
Quare
VED –are(com);
ideoque
Vare(cový = ) &lt;( a ) .
10
137
Ey:EN :Ee: HE, seu ,8: V kï-aï: Ee: h:
hinc
∙
!
Ee .... B ∙∸−⋅⋅ :: Vii. dt ;
VIR-æ ]: '
r
et consequenter
⋅↙≀↥∶∎∙−−⋮∶∶⇣∶⋮ *Ve-
tempusculum videlicet dt impensum ad percurrendam þ
seu Nn, obtinetur dividendo respondentem arcum Ee per
]; Vi .
Inferimus, tempus :
impensum ad percurren-
'.
dum ]:-
ut seu ND, obtineri dividendo respondentem arcum ED per kI/ £.; nimirum
r
t ∙∙∙ ED
l.V-f,- .
Quare ∙
.yz... −∙−⊡∍ ......
«
.
rf- k —- (eos:.k),
ideoque
≀⇌⇂∕∑− arc (eo: &amp;) (a) ∙
5
10138
Iam vero in puncto infimo B (Fig. 37) exsistit a = 0 ;
erit igitur tempus semioscillationis
TT
ti
V
2
8
tempus integrae oscillationis ( a ' ).
t2 =
V
quas formulas cum non ingrediatur initialis angulus k, patet oscillationes ejusdem penduli, vel plurium pendulorum aequalis longitudinis r per arcus satis exiguos utcumque ceteroquin inaequales, fore ad sensum isochronas seu aequi
diuturnas.
Idipsum facile demonstratur hac alia ratione: angulus GCT
= 90° BAC; hinc vis acceleratrix CG , ex qua sola repetendus est penduli descensus, exhibebitur per gsing: in hypothesi nimirum arcuum satis exiguorum spectari poterit CG tamquam proportionalis distantiae a puncto infimo B, computatae in arcu BC. Ergo ( 29. 4°) etc.... Etiam sic: est ds = d rík - a ) rda ;
et consequenter
rda da
dt
V rg (k -u?)
-Vivok²-u?
factaque integratione ( 27. 13º. 14° ) prius ab a kad
a =0 , dein ab a = k ad a = -k, emergent binae (a' ) .

66. Haec notentur: 

1º: secunda ( a' ) dat
77 r
8 ( a '');''
ta
atque inde innotescit gravitas g.
138
Iam vero in puncto intimo B (Fig. 37) exsistit «:o;
erit igitur tempns semioscillationis. &quot;
∙−∣
⋍⋅∶−−−−⇄⋅−∣∕−≦−∙
tempus integrae oscillationis (,,-)
−∣−∙−
ta:T! −∙− ∣∕ :
,
quas formulas cum non ingrediatur initialis angulus k, patet oscillationes ejusdem penduli, vel plurium pendulorum
aequalis longitudinis :- per arcus satis exiguos utcumque
ceteroquin inaequales, fore ad sensum isochronas seu aequi-
diuturnas-
Idipsum facile demonstratur hac alia ratione: angulus GCT

: 900— BAC; hinc vis acceleratrix CG, ex qua sola re-

petendus est penduli descensus, exhibebitur per gsinat: in
hypothesi nimirum arcuum satis exiguorum spectari poterit
XCG tamquam proportionalis distantiae a puncto infimo B.
computatae in arcu BC. Ergo (29. 40) etc.
...
Etiam sic: est ds:dr(k— a):
-— rda;
et consequenter
rdat dat
dt −∙− ↵
−− '&quot;
VrgUe-æ .? sz-az :
factaque integratione (27. 130. 140 ) prius ab et: I: ad
«:o, dein 'ab a: ∙−−− I, ad ac ∶−−⋅ —-k, emergent binae (a')-

2º. Etsi ponderosa diversae materiei puncta permissa sunt oscillare, attamen idem semper prodiit valor g in eodem terrae loco: rursus ( 17 ) igitur devenitur ad proportionalitatem inter corporum massas et respondentes gravitatis vires.

3º. Constat observationibus longitudinem penduli
simplicis oscillationem absolventis intra mioutum secundum
eo esse minorem, quo magis ad aequatorem acceditur: quoniam ergo, haud variato tz, gravitas est ut longitudo illa, minuetur gravitas a polo ad aequatorem usque ( 30) .

4º. Apud nostras regiones praefata penduli longitudo cum sit = 3ped opol glin, 38 = 440lin, 38, factis
in ( a &quot; ) tz = 1 &quot;, r = 440lin , 38, prodibit respondens gravitatis valor g = 30ped , 183 alibi (30) indicatus.
[[Fasciculus:Double-Pendulum.svg|thumb]]
67. Quod spectat ad pendulum compositum concipiamus (Fig. 39) puncta ponderosa B, B. , B2 , . . filo appensa: invicem disjuncta conficerent haec puncta temporibus inaequalibus oscillationes suas; punctum nempe B, citius  (66) quam B, punctum B, citius quam B, etc: invicem ergo conjuncta agent ita in se mutuo, ut quae, minus distant a puncto suspensionis A retardentur ab iis quae magis distant, et quae magis distant a suspensionis puncto accelerentur ab iis quae minus distant: fiet propterea oscillatio penduli compositi tempore quodam medio inter minimum ac maximum praedictorum temporum inaequalium. Hinc sequitur fore in AB punctum quoddam B.,m suas conficiens oscillationes perinde ac esset solitarium, nulloque nexu caeteris punctis uui
retor: Bm dicitur centrum oscillationis, cujus centri distantia a puncto suspensionis est longitudo penduli simplicis suas perficientis oscillationes eodem tempore ac pendulum compositum. Inferimus oscillationes pendali compositi, et ipsas fore isochronas; modo tamen exsistant satis exiguae.

68*. Facile intelligimus ( 50. 3º. 6° : 66 ) motum penduli simplicis in medio resistente determinari generatim per aequationem

140
das
di?
= gsing -f(v ) .
derka( )
di Ob dc2
dra
dt2 et ( 27. 29º . ) sing
23
2.3 +
aequatio illa eyadit
creat + s ( « - +...)-fo) =
Pone fv) = cv ; et angulum a ita perseveranter exiguum,
ut ejus tertia potentia absque sensibili errore negligi pos
sit : habebis
da с
+
dla
ola
0 .
ds Est autem v =
dt
drak -a)
dt
da
dt ; igitur
d2C da
dt2 to dt + baro:
quam integrantes in hypothesi c constantis assequemur( 27.270. )
... [ :V546.-V21
( 6),
In experimentis, quae pendulorum ope solent institui ,
r est multo minor quam g; item densitas penduli, et con
sequenter ( 33 ) c fractio admodum parva. Fac ergo
140
tiis ∙
de'-* :gsmat —f(v)
d': d2r(k—a) (lioc
↽⇁−−− −∙−
' Ob daz dtz 27. 290. .:
rdtï
, et ( ) stna
'
a3 . '11 d'
rdzatdta-l-g(at—-— ...):fþv) :0.
Pone f(v): cv; et angulum ac ita perseveranter exiguum,
ut ejus tertia potentia absque sensibili errore negligi pos-
sit: habebis
dza g c
dia—FTa—Tv—o
.Et :
−−∠≀≖∙∙↙≀↗≺∣≖⋅⋅∝≻∙− ∎∠≀∘⊏∙⋮ ⋯⋅⋅ s auem'v— dt— dt rd , g1
quam integrantes in hypothesi c constantis assequemur(2 7.2 70.)
&quot; til &quot;'i-i.]c) —l ! —-..—.
2:32 [CG 4
r—l—C'f—B
ln expetimentis, quae pendulorum Ope solent institui,
r est multo minor quam g; item densitas penduli, et con-
sequenter (33) c fractio admodum parva. Fac ergo141
VS
ut sit VA = iVT ;
4 .
vertetur ( 6) in
ti V = 1 -til
+C'e 2 -
&quot; ]
U = e
seu ( 27. 30° )
3 [
e
] ;
C &quot; sin it +C ' ' cosit
unde
cosit
data o - [( c'i - )
(c&quot;: + * ) ainit ]
da
In joitio motus t=0 , a= k , 0 ;
di
propterea C &quot; &quot; k, C '
ck
21 ; et .
= ke - - [
sin it + cosit ] . )
2i
( 6 )
dan
dt = -ke- Ź [ ita sinic.
141
'
. . ∙
c' a . .
ä-ï': - utsit V—--€- :::tl/ −−↿ ∙
r 4 4 :-
vertetur (b) in
−−−∘⋮−≀ — tiV—1 -til/—1
at:6 2
)
C'0 v
380 ( 270 300 )
↴
c ↼
a:
∘−−≖− '[C&quot; sinit ⊣−∁⋯ cosit] ;
unde
∘⋮ C'&quot;
↙⋮⋮∶∶ .
∘∙⋅−
; t [( C'i— 20) cosa
: -— dt ∙

Cnc . .
(
0&quot; i −⊢ —2 )
sunt ].
dat ∙∙∙ ∙
In initio motus t—:o
, at:— ]:
, &quot;dt −∙− 0-
∙∙∙ k
': propterea C Ck et −−−∙−− ∙
∙−− 21' ∙142
с
Ex.Vihabemus zi 11
ll
Hlacin
c2
V C
it =-V
rc?
4g &gt;
2V EV
rc2 1
48
1
i +
VE ;factoigitur V
cr2 =c,
43
&quot; Vi ro2
4g
binae ( 6 ) sic poterunt exprimi
a = ke

* IVE Vētowi.V ]

1.)
(6 )
dm-- .- iv E sinórV.
In fine cujusque oscillationis est da
dt = 0; proinde, ob
= 0: inferimus in fine primae secundam (3&quot;),since V
12 풍VV
2.V
oscillationis fore t = &gt; in fine secundae
8
271 376 in fine tertiae t =T
8
8
gulae itaque oscillationes absolventur aequali tempore
E ,
in
V
, elc.. .. ; sin . с
g. ∘ Ex ::i habemus 21. ∶∶ − c —- .
∙
∙−−−−−−−
I/ ⇂∕−−−⋅↿rc
c ⋅ :! −⋮↓−≔⋤⋅ a
cr&quot; ,
. CZ .g 1;factoigilur V1—-- :c, ↓⊣− ' ⋅ −−∶ ⋅
binae ( b') sic poterunt exprimi
£(sz 2 inc't cosc' g
c[hc V—s Vg ∙−⊢ ::ll/:]
(ö&quot;)
(E;—..., &quot;&quot;T-V—sinc't 5-
dt :-
/
In fine cuiusque oscillationis est ≤−∝⋮∶∶∘⋮ proinde, ob
secundum (b&quot;), sinc': Vi:o: inferimus in fine primae
!'
1:
osc1llat10n1s ∙ ∙ ∙ r ∙ fore :

:

−⋮⇆⊤ V—3,
in fine secundae
271 .
⋅⊤ −∙
,
in fine tertiae :
−−−−−∣−
−− ,etc.. -sm-
gulaec itaque goscillationes absolventur aeqnali tempore143
و
=
ا
( 6 '' ) .''
8
In primo substitutis valoribus 0, 20 , 30 , ... no
pro t , emerge
Qu - ke 2 A2 = ke -
&gt;
929
as= -ke- 30
Q. = 1–1 y&quot; ke – no
hinc successivarum oscillationum amplitudines
0
2
음
k + ke ke - 9 +ke - 2
-ke 39 -
2/2
20
the
ke
seu
1(1 +-2), ( +-3, -2,

* (170-99 . -Ź- 42329......

Decrescunt igitur amplitudines istae in progressione
geometrica : quod cum confirmatum sil experimentis pen
dulorum in aere oscillantium per arcus satis exiguos, haud
majores v . g. tertia parte unius gradus, licebit quoad e
jusmodi oscillationes assumere aeris resistentiam tanquam
proportionalein simplici velocitati.
143
n −−−∽−∣∕∙⊂− (B&quot;)- 0

6
In prim: (. ⋮⋅ substitutis valoribus 9, 29 , 39 , ... 719
pro :, emerge f
⋅∙⇁− -
∙
1- 0
29 ⋅ C
a;:— ke 2 ,agzke 2 ,a3z—ke 2 39
C
119 a.::(—1)&quot; ke −⋮⋅ :
hinc successivarum oscillationum amplitudines
.
c ⋅∙⋮∙∙ ...—£
k-l—Re— i.e.]:e— 294-ke ∙
229
&quot;
ke −⋅⋮⋅⋮∂−⊦∣∥⋝ 2
39,. ;
seu
.- e ∘
9
−−∘ &amp;
Decrescunt igitur amplitudines istae in progressione
geometrica : quod cum confirmatum sit experimentis pen-
dulorum in aere oscillantium per arcus satis exiguos, haud
maiores v. g. tertia parte unius gradus, licebit quoad e-
jusmodi oscillationes assumere aeris resistentiam tanqnam
proportionalem simplici velocitati.144
Experientia insuper docet decrementum illud gradu
admodum lento procedere : sic D. Borda expertas est non
nisi post 1800 oscillationes valorem en converti in k .
Hoc posito, existet
2
1
1800c7V
.18000 e 2
2 2c seu ob (6 ' ') e
3اندبه
et consequenler
1800CTE
= c'log. 13 == c (o , 40546) . 2
( 1800)? c ?72. &quot;
= 1 - c '? , ideoque 8
4g
ro2 Sed
4
= (1800 ) 2772 /1— 2) ; igitur ( 1800)277 ?(1 — c'2)
= c (0 , 40546 ) ; unde
c' ?
( 1800 )2772 1
(1800 )222 I.(0,40546 )
&gt; et e
V
+18007
1+
0,4054612
180076
1
quam proxime.
gua
Si ( 33 , 4.&quot; )
poneretur f (v )
terminandum exsisteret
ad penduli motum de
144 *
Experientia insuper docet decrementum illud gradu
admodum lento procedam : sic D. Borda expertus est non-
. . , 1118! ∙ ⋅ ∙ ∙ 2 post 1800 osmllattones valorem ac,, com-cru mes-k.
Ilnc posito, existet
'
u
1eöocn VL
..
'
5.
20'
≀∙ ⋅
.
...—18009 2
e 2 −−∙−−− −∙− , seu 01) (b&quot;') e
⊏⋅⋮− ,
. 5 ⇁
3
(!l. 000 sequenter

: .—c'(o ∙⇁ '40546) b. &quot; .

1800072l/L .
2 g :c'lo
'.
(1800≻∖⋅:0:712.— g
PC: '
∙
'
Sed −−−−−↿ ---c2
, 1deoque 45
⇌≺↿⋅∂∘⊙≻≃⊺≖≖≺↿−⋅≺∶∣≏≻⇋ igitur ≺↿∂⊙∘≻⇄∏≖≺↿−∘∣≖≻

: c'5(o, 4054673; unde

(1800)??? - -, 1
, ↼−−
≺↿ et -c':
≨∃⊙∘⋟≖⇃∙≖⋍⊣−⋅⊏∘∙∠↥∘⋦∢⊖≻ ↼↼
'3
Vl—l&quot;(7'g55; o.4(l546)z
≖≖−−−↿ quam proxime.
↓
2 '
51 (33. 41.&quot;) poneretur f(v) ∶−−⊸∙⋮⊥⋮− , ad penduli motum de-
02
termiuandum exsisteret145
des
dt?
gsing gu2 d ?
seu de2 + sine —
8r /dala
2
= 0 .
c²lde
Haec prias multiplicata per 2du , ac dein integrata suppe
ditat
' dala Idala
ca ldt
2g
COS O
seu facto Slaa) dx = y , ideoque Coupe ( )
dy
da
dy
28
2gr COS Q
da y = 0 ;
cojas integratio traducitur ( 27. 26 °.) ad integrationem fun
ctionis
2g cosada
2gra
c2 re
Jamvero , facto compendii causa 2gr
= m , habemus c2
dem sina ) coso, da Se
ma
-mu m e sing da ,
d ( e-ma cosa )
-ma sina da - me COSQ da :
igitur
ſe-ma cosa da e -ma sina tm se-ma siac da ,
ſe-ma sina du = me-ma cosa m ſe-me. cosa da ;
ex quibus
145
d's— ∙ g.&quot;: dia g ⋅⋅ . gr
äzäfgsma— Z;- , sendt: da)!— -[-r sma 02 22 —--0.
Haec prius multiplicata per Zda , ac dein integram suppe-
ditat -
(&amp;)2— dt ∙−⊋∊∁∘≘∝−⋅⋮⋚⊆∫≺≦− r
f:) ↙≀⊄∶∘∙∙∙
dat ∙∙∙ ∙ da: a... 47&quot;
seu facto f(ä—t) fia —J—, ,
1deoque (22) −∙− ä; ,
≝⊻−≟≝∁∘⊱⊄≉≣≝⋅∫∶∘⇋ . dat
cuius integratio traducitur (27- 262) ad integrationem fun-
ctionis
Zg cosadat ∙
⋅
2grat .
∘≖ . re :
∙
2 r
'
Iamvero ,facto compendii causa −⋚−⋅:m ,habemus
02
,! (.;-&quot;&quot;&quot; since ):e'ma cosa: d-a −∙∙ m e'm' sinat da
,
d(e'macosat):-e'm ∋⋮∘⊄↙∄∝−⋅⊪∘⋅⋯ &quot;cosada:
igitur
fe'm. cosa da::
efm sinat—lfm fe'm sinat dat
,
fe'm sinat daz ⋅⋅−−− −−∶ e'm cosa: — m fe'm- cosa daz ,-
ex quibus146
ſen-Ma cosa da e -ma sina — те cosa -m2ſe-mecosadu 7,
et consequenter
2g
Sce-ma cosa da
2g ( e -ma since me- mu cosa)
r ( 1 + m2)
Erit itaque (27. 26 °.)
y = Cema +
2g ( sing - m cosa )
r ( 1 +ma )
ex qua differentiata quoad &amp; cum emergat
dy
da
Cmema + 2g (cosx + m sina )
r ( 1 +m2)
restituto valore dy
da habebimus
ca
dal 2g (cosa + m sina)
= - Cmema +
r ( 1 +m2 )
da
In initio motus a = k , = 0 ; hinc dt
Cm 2g ( cosk + m sink) e-mk
r ( 1 + m2)
propterea
-m (k - a )
Cate)
dal 2
ldt
29
r ( 1 +m2)
cosa + msina - cosk + msink)e ( h).
Facto a = o in ( h) , prodibit inde velocitas penduli in pun
cto infimo B ( Fig. 37.) : ascendet pendulum cum velocitate
146
fe'm&quot; cosa da: e'm&quot; sinat —me*mcosoc −∙∙ ⋯≖∫∘⋅⋅⊪∞∽∡∠≀∝ ∙
et consequenter
2g &quot; 2g (e-&quot;W- sinat −∙∙ rne-ma cosa)
∙−∙∣ (.'-'&quot;&quot; cosa dat:
⋅
'
r(1 −⊢ ⋯⋅≀≽ Erit itaque (27 . 260.)
2g (sinat —m cosa)
y.:Cama'i' r(1-l—m') :
ex qua differentiam quoad a: cum emergat
dy ∙− M Zg (com-[- m 5213.)-
da :Cme r (1 —f-m3) '
∙
d
' ∙ resututo valore 1, babebmus ⋅
de:
((!—S :Cma'&quot; &quot;I&quot; 25 (cosa: ∙⊢ 11: sind:? ∙
.. dt r(1-l-m3)
∙ ∙ ∙
'
da ∙
In 1n1t1o motus a:k −∙− −−−−∙ o ; bmc 'dt
2g (cosk −↿− m sinit) er:-&quot;&quot;* ∙
Cm: r(1—l-m2) ..
propterea
(de!)2 28 &quot;[COSa-Hnsina—(cosk-i'msïnk) e-m(k-a)] 32 :r(1—i—m2) (73)-
Facto a:o in (I:), prodibit inde velocitas penduli in pun-
cto infimo B (Fig. 37.) :ascendet pendulum cum velocitate147
ista versus D , conficietque arcum , cui respondebit — Q,; et
quoniam in extremo puncto illius arcus extinguitur tota ve
locitas , iccirco
COS - m sina, · ( cosk + m sink) e -m (4+ 1) = 0 ,
seu
(cosa,
m sing , emai (cosk +m sink) e-m * = 0
( h ' ).
...
mk .
maa ,
Sunt ( 27. 29.° ) emas = 1 + m « . + +
2
mak ?
=1 -mk+ -... ; est insuper m fractio admodum
parva ( 33) : neglectis igitur terminis , ubi invenitur mº ,
traducetur ( h ) ad
2
&gt;
cos@g - m (sina, cosax) = coskt m (sinkkcosk ) (h &quot; ).
Denolante o differentiam inter valores a, et k ut sit Q= k -0 ,
certe ð erit fractio tenuissima : hinc substituto k- loco
Qy in ( &quot; ) , sumpto 1 pro cosd et à pro sind , missisque
öz et mo , assequemur
2m
Osink = 2m ( sink - kcosk) , d = 0
sink (sipk- kcosk ) ;
unde
Uy= h
2m
(sink-kcosk) .
sin k
Si popimus k ita iguum , ut ejus quarta potentia prae
termitti possit , obtinebimus (27. 29.° )
147
ista versus D, &quot;conficietque arcum , cui respondebit — at,; et
quoniam in extremo puncto illius arcus extinguitur tota ve-
locitas
,
iccirco
111 cosa:, −∙∙ m sinatl — (cosk −∣− m sink) e&quot; (b'-3 1): o ,
seu
(cosa, ∙− 11: sind,) a'&quot;! — (cosk —-msink)e&quot;&quot;'* :: o (b').
Sunt (27.29.0) erat: mna? ↿−⊦⋯⊄≖−⊦
2 −⊦ ∙ ∙ ∙
∙ ,∙⋯⋆
⋯≖⇂∙∙≖ ⋅ ∙ ∙ ⋅

: i —mk—l—-—2—-— ...; est 1nsuper m fract1o admodum ∙

parva (33): neglectis igitur terminis ,.
ubi invenitur m',
traducetur (h') ad
tuom,—m (si na,—aleam,:cosk—l-müi nk—kcosk) (h&quot;).
Deuotante ö differentiam inter valores a, et k ut sit ac,: k-ö.
certe d erit fractio tenuissima : hinc substituto k—ö loco
a, in (II&quot;), sumpto 1 pro cosd et 6 pro sind
. missisque
d' et md , asscquemur
ösinkz2m (siuk—kcosk) , ö: ET- (sink—kcosk);
. sink
-
nnde
2m
sin lt at:-k— (siïnk—kcoslc) .
Si ponimus !: ita exiguum , ut eius quarta potentia prae-
termitti possit , obtinebimus (27. 293)148
2m 2m
- (sink - koska gink k
21 k2 1
2.3
2m 2m ka ( 1
k2
2.3 h2 ,
3
ac proinde
Q = k
2m
3
k2 :
quemadmodum valor a, deducitur ex k , sic ,yalor d, ex
valor az ex la , atque ila porro ; erunt nempe
2m Aa =
2.1 az ?; 3=0,- 313, etc...
| Patet illud ; si vis acceleratrix ex medij resistentia sumitur proportionalis quadrato velocitatis, haud subsistet superior lex, experimentis confirmata, de oscillationum amplitudinibus in progressione geometrica decrescentibus.
[[Fasciculus:Cycloid f.gif|thumb]]
[[Fasciculus:Cycloid03d.svg|thumb]]
[[Fasciculus:Cycloide InfinimentPetits.svg|thumb]]
69. ° * Aliquid subjungimus de gravium descensu per arcus cycloidales. Circulus A'D ( Fig. 40 ) tangens rectam A”E in A&quot; revolvatur super ipsa A”E ita, ut eam pergat semper tangere. Punctum A&quot; circuli regredietur ab A&quot; in E, lineamque curvam describet, quae appellatur cyclois: circulus ille mobilis vocatur cycloidis genitor, recta A ” E basis, diameter AB perpendicularis mediae basi dicitur axis, punctum A vertex; patet autem quemvis circuli genitoris arcum B’A' aequari rectae A'B, quae intercipitur duobus punctis A &quot; et B', in quibus extre ma puncta ipsius arcus tanguntur ab A'E; et totam basim AE aequari peripheriae circuli genitoris. Ducantur
148
2
2,&quot; (siuk—kcosk)-— &quot;' .↗⋮∍ mnk ]: (
kz 3 2.) ⋅
ac proinde
2m
'k 3k
quemadmodum valor a; deducitur ex 1: , sic.valor ag, ex
'a, , valor 013 ex ac, , atque ita porro; erunt nempe
a—a—ïaa' a—a—zma' etc - 2—1 3
1
, 3—3 T;, ∙Ducantur149
jam ex cycloidis puncto v. g . A' perpendicula A'rl= y )
et A'C , alterum in basim AE, alterum in axem AB ;
sit A'r = x ; diameter circuli genitoris dicatur 2a; exhibea
turque per &amp; arcus quoad radium -- = 1 similis arcui A'B' .
Erunt
x = A'B - B'r - A'B ' - AM = a5 - asins ,
y = B'M = asin.v.zza( 1 - сoss) .
ex istarum prima assequimur
dx = ads - acoss ds = a ( 1 - cos )de ;
et dividendo per secundam.
dc
de .
y
Est autem arc sin=
are(sin = AMM))—are (sin V Zay —ya )
a
IV2ay - y2
et consequenter de 2
2ay - y
aa
dyZay - y ?
dy
V2ay - y2 ; ergo
2a
dy = dx V?
(at ) ;
y
149
iam ex cycloidis puncto v. g. A' perpendicula A'r(:y) ⋅
et A'C, alterum in basim A&quot;E. alterum in axem .AB;
sit A&quot;r:æ; diameter circuli genitoris dicatur 20; exhibea-
turque per &amp; arcus quoad radium −∶∙−↿ similis arcui A'B'.
Erunt
æ:A&quot;B'——B'r—-A&quot;B'—-A'M:ae-—asins ,
y:B'M:asin.v.:a(1—coss) ∙
ex istarum prima assequimur
dæ:ada—acoss de:-au -cose)d£ ;
et dividendo per secundam.
d
-—æ-- :de ∙ 7,
Est autem :arc(sin: M):arc (sin ∙−−∶ ∣∕⊋∅∫−↗↾≖
a a ),
et consequenter de:
.a— ∙ a
): ∣∕ ↿−− 5351-
02
dl/Zay—yz df
a—y VZaJ—yi 'ergo150
aequatio differentialis ad cycloidem , computatis coordina
tis a baseos initio A &quot;, Quod si computentur a vertice
A , ut novae coordinatae sint AC ( = x ') , et A'C ( =y' ) ,
cum habeamus x = an — y , y = 2a — x', prodibit
-dx'adyV
x'
2a- , seu dy = dxV
2a - x
xช่ (a &quot;) .
Nunc ad gravium descensum quod pertinet per ar.
cum quemvis cycloidalem , cujus vertex in puncto inſimo
B ( Fig. 37), sit C initialis positio puncli ponderosi , quum
nempe t =0 et v = v = 0 , M positio in fine temporis 1;
quibus positionibus respondeant altitudines c et ac' supra
horizontalem rectam transeuntem per B , ut in Mha
beatur v = V 2g(c-x') : denotantibus h , se s' cycloidales
arcus CB , CM , BMBM ,, erit erit dsds== dhd (h -- ss'')) = - ds'; unde''
ds
di
ds'
dt = V2g(c -x '), ex qua obtinelur
ds'
dt
V 28(c — x ')
Formula ( a&quot; ) praebet ( 27. 19.0)
2a -x
do = Vdx =+ody&quot;a= dx V17 = dx '

;

x
hinc
da - c
a
dt dx'
V. 8 V cx' — x'2 -Va
GVFECITATE
150
aequatio diti'erentialis ad cycloidem . computatis coordina-
tis a baseos initio A&quot; Quod si computentur a vertice
A , ut novae coOrdinatae siut AC (:æ') , et A'C (:y'),
cum habeamus x:an'—-7, 1:20—æ', prodibit
I
Za—æ ∙−−−
, seu df:dx I/ æ, (a') .
Nunc ad gravium descensum quod pertinet per ar-
cum quemvis cycloidalem . cujus vertex in puncto infimo
B (Fig. 37), sit C initialis positio puncti ponderosi, quum
nempe t:o et ⇂↾−−∙∶⇂↗∘∶∶∘ , M positio in fine temporis :;
quibus positionibus respondeant altitudines c et æ'supra
horizontalem rectam transeuntem per B , ut in M ba-
beatur :»:V Zg(c-x ':) denotantibus ,: .
s, s' cycloidales
arcus CB, CM, BM , erit ds:d(h—s'):—ds'; uude
ds di' .
v −− dt— dt —l/2g(c-æ'), ex qua obtmetur
ds'
dc ∶−∙−− − ⇂∕−−−−−
⇄∊≺∁−−⋅↿⊏⋅ )
Formula (a') praebet (27. 19!)
d.;— −−∙ ⇂∕∎∎−−∎∎−∎∎∎ ↙↙∙≖⇌ ≖⊣⊸↙↿∫−
hinc
(1&quot;.—— a flx' ;.
(ll:
∙−∙ V
∙−− ∙−− V
a .; 20 .
l
—— g J/Fæ—æ'z ∙ g
∣∕↿ ∙↕⋅∎⋅≩∁≻≖⊽151
sumptisque integralibus ( 27. 9,9 ) ,
= c +Vare (co== **)

;

in positione initiali est t=0, simulque x' =c; igitur C = o, et
Vore (rosa ).
Facta x=0, prodibit tempus descensus usque ad punctum
infimum B, nimirum
11 =T
VO
ubi cum non inveniatur c , patet , ex quocumque cycloi
dis puncto demittatur grave, eodem semper tempore per
venturum ad B. Hanc cycloidis proprietatem posteaquam
detexit Hugenius , cycloidem ad pendolum adhibere cae
pit : quod qua ratione fieri possit , ostendit in parte 3. “
Horologii oscillatorii.

=== De attractione corporum in hypothesi attractionis agentis in ratione directa massarum, et in reciproca duplicata distantiarum.===
70. Pyramis AH (Fig. 41) habens basim GH infinitesimam secetur superficie sphaerica, cujus centrum in A , et radius AZ ( = r ); sit Ky = B ) projectio intersectionis VZ ( = ) in plano AB; supra basim Ky erigatur prisma KyE altitudinis CH ( =x): exprimet KE AZ sumptisque integralibus (27. 93) ,
↥⋅∶∁⊹ l
V— a1c (eo: 200) ;
in positione initiali est t:o. simulque æ':c; igitur (l:-o, et

* x −−∶−∁

Facta x':o. prodibit tempus descensus usque ad punctum
infimum B, nimirum
a
II:&quot; V— :
g .
ubi cum non inveniatur c , patet, ex quocumque cycloidis puncto demittatur grave, eodem semper tempore perventurum ad B. Hanc cycloidis prOprietatem posteaquam detexit Hugenius, cycloidem ad pendulum adhibete caepit: quod qua ratione fieri possit, ostendit in parte 3.'
Horologii oscillatorii. ⋅

lit-F.
AZZ152
vim attractivam ( p) segmenti CG in punctum A juxta directionem perpendicularem plano AB. Intelligatur enim segmentum CG secari sphaericis
superficiebus numero infinitis , quarum centrum in A , et
r2 , 7* 3 sintque eoz , A2 , A3 intersectionum areae. Erit
radii rs .
din 23

; 2

2 r2 3 p2
vis nempe attractiva cujuscumque areolae Qy , da , .
aequabit vim attractivam areolae A. Ex punctis Z, C,
ducantur in AB ... perpendicula Zy ( =n) , CB ( = n ) ...:
singulis viribus resolutis in duas, quarum altera sit paral
lela , altera perpendicularis plano AB , componentes per
pendiculares repraesentabuntur per
ni na n
2
ri ra
et quia
ni n2 n
72
iccirco
li ni 0.2 п,
=
a n
.
t'i p22 ra
his positis , quisque videt fore
n
f
152 .
vim attractivam (:f) segmenti CG in punctum A juxta
directionem perpendicularem plano AB.
Intelligatur enim segmentum CG secari sphaericis
superficiebus numero infinitis. quarum centrum in A, et
radii r; , r,. rg
.... sintque at,
, at,, ata ...
inter-—
sectionum areae. Erit
«! a; «3 ∙
. a
r,: fa, rna ra.
∙
'
vis nempe attractiva cujuscumque areolae at,, at,, ∙∙∙
aequabit vim attractivam areolae at. Ex punctis Z, C,
...
ducantur in AB
... perpendicula Zf (:n)
, CB (:m) ...:
singulis viribus resolutis in duas, quarum altera sit paralf
lela, altera perpendicularis plano AB , componentes per-
pendiculares repraesentabantur per
a! &quot;[ a:; &quot;a a n
. , ∙ , ∙ ∙ . ∙−−− .—,. r,2 r, rf r, :-a r
et quia
—:
&quot;! &quot;2 n
∙−− ∙ ∙ ∙ :∙
—;
r, r, r
iccirco
«! &quot;r a: n, a n
∙
∙−−∎ ∙ ∙ ∙ ∙:∙∙−∎ ∙ ∙∙∙∙ :
rl: rl ,.22 &quot;, &quot;, r153
Jamvero (55.4. )
\beta
= cosyZA 3
igitur &gt;
n
ela
B
sli
oli
a
et consequenter
Bx
r?
KyE f
AZ

71. Singula corporis cuiuscumque KGDH (Fig. 42) puncta trahant punctum C positione datum. Centro C et
radio quolibet CM describatur sphaera MBN; in eius
superficiem incurrat in A recta quaelibet CG permeans
corpus KGDH iuxta DG ; demittatur ex A perpendi-
culum AQ supra planum MCN; capiatur in- AQ pars
TQ aequalis segmento DG intra corpus KGDH demerso;
quod si plura fuerint huiusmodi segmenta, pars in per-
pendiculo accepta sequetur omnium summae, Si per GM?
dividitur solidum ïTXV, quod continetur plano MCN
et superficie ab omnibus punctis T determinata , expri-
met quotus vim, qua totum corpus KGDH trahit punctum
C perpendiculariter ad planum MCN.
Prodeant enim ex C infinitae numero pyramides, qua-
rum segmenta DG impleant totum corpus KGDH; pote-
runt totidem respondentia (69) prismata TQ concipi , quae
totum solidum ïTXV impleant; ergo etc.
Quoniam vires omnes sollicitantes punctum C
possunt traduci ad ternas
, quarum directiones congruant
cum tribus rectis se mutuo ad angulum rectum secantibus in ipso C; ternae vero istiusmodi vires in unam com-
↿↿154
positae dant resultantem ex illis omnibus , inde fit quod
ubi determinentur (70) ternae vires corporis KGDH re
spective perpendiculares tribus planis orthogonalibus per
punctum C traseuntibus , eae in unam contractae suppedi
tabunt et directionem , et intensitatem illius vis , quae re
sultat ex omnibus viribus punctorum constituentium cor
pus ipsum KGDH.
Si punctum C intra - corpus trahens collocaretur
accipienda esset TQ aequalis differentiae inter distantias
ipsius C ab extremis D et G rectae DG transeuntis per
C. Hinc si C fuerit situm intra ejusmodi crustae cavita
tem , ut per C ducta quavis recta , aequales hinc inde par
les illius rectae intra crustae crassitiem intercipiantur, eva
nescentibus omnibus TQ , evanescet etiam omnis vis pla
no cuicumque perpendicularis , et punctum C in aequi.
librio consister.

72. • * Coordinatarum originem O constitue in
quovis corporis puncto ; sin que x, y, z coordinatae pun
cli altrahentis ; a , b , c coordinatae puncti allracti ; '
distantia inter punctum attrahens et punctum altractum :
expriment
b - r
CZ
ba 몇
7
A'
A
cosinus angulorum , quos a continet cum axibus coor
dinatis OX , OY, OZ. Quare denotantibus Hc , H,, H,
componentes iisdem axibus parallelas , in quas rosolvitur
attrahens totius corporis vis H , et dm elementum massae,
eront
H,
- Som dm , 1 , = Sabah dm
(o)
H , Set dm :
A3
154
positae dant resultantem ex illis omnibus; inde Et quod
ubi determinentur (70) ternae vires corporis KGDH re-
spective perpendiculares tribus planis orthOgonalibus per
punctum C traseuntibus, eae in unam contractae suppedi-
tabunt et directionem, et intensitatem illius vis, quae re-
sultat ex omnibus viribus punctorum constituentium cor-
pus ipsum KGDH. ↴
Si punctum C intra -corpus trahens collocaretur ,
accipienda esset TQ aequalis differentiae inter distantias
ipsius C ab extremis D et G rectae DG transeuntis per
C. Hinc si C fuerit situm intra eiusmodi crustae cavita-
tem
, ut per C ducta quavis recta, aequales hinc inde par-
tes illius rectae intra crustae crassitiem intercipiantur, eva-
nescentibus omnibus TQ, evanescet etiam omnis vis pla-
no cuicumque perpendicularis. et punctum C in aequi-
librio consistet. ⋅
⊽∑∙∘∙ Coordinatarum originem O constitue in
quovis corporis puncto; sintque x, ],:coordinatae pun-
cti attrahentis; a, b ,&quot;c coordinatae puncti attracti; A'
distantia inter punctum. attrahens et punctum attractum:
expriment
⊄≖∙∙−−∙∙−−∙⋮≖ b—gr c—z
∆∣∙
'
∆∣
'
∆∣
cosinus angulorum, quos ∆⋅ continet cum. axibus coor-
dinatis OX , Oï, .OZ. Quare denotantibus H, , H, , H,
componentes iisdem axibus parallelas, in quas rosolvitur
attrahens totius corporis vis H ∙ et dm elementum massae,
erunt ⋅
a—æ &quot;bf—7 ⋅ ∏≖∶∶∫ ∆∣⊰ dm,
⊟⋮⇌−∽∙∣∙−⊒↙∙⇁⋮⇀≀≀≀↿∙ .-
(0)
,:szjä-äfdm:155
integralia se se protendunt ad totam corporis massam M.
Pone
Q = Sam
( o ')
habes quidem
A2 = (a — x )&quot; + ( my) + cz( )&quot; ;
sed quia integrationis limites non pendent ab a , b , c ,
ideo ex prima ( o' ) erues
dQ
da ſ
dm ,
dQ
db
den
ES ,
do dm
dQ
dc -dm ;
da
secunda vero (o' ) praebet /
a 영 1 dA a -X a
A' ? da
b
da 4'3
db A'3엷
slot
dc 4'3
traducentar itaque ( o ) ad
H=
dQ
da H ,
do
db H,
dQ
dc ( o &quot; ) ,
componentesque H , H ,, H, pendebunt ab unico integrali
l. Fiat
a : + 62 + c = A2 ,
155
integi-alia se se protendunt ad totam. corporis massam M.,,
Pone
≺≀⇌⇀⋅ dm −∙−≃
habes quidem (O,)
∆≏⇋−∙≺∅∙−∞≻⋍⊣−≼≀↗−∫≻≔⊣⊣∘∙−≖≻⋅ ;
sed quia integrationis limites non pendent ab a, b , c ,
ideo ex prima (a') erues
⋛≣∎∶∆∼∣↙≟ dm' ∎−⋅∫↲∂↙≀⋯⋅−−−−∫ &quot;'&quot;-('m-
secunda vero (a') praebet
↿ ↿
⊄∄−−−↽
∆∙ ↿ siA—, a—æ (LA-7 b—r
de'&quot; A'da A'3 ↞∙ ∠≀∣⊃−−⋅−−−−∆⋅≀∙ '
↙≀∙∙↿⊽ ac.
.... de −⋅⋅ ∆∣∍
traducentur itaque (a) ad
dQ dQ dQ−⊋⋤−∙ Hic—2? ∙ Ha ≔−⋅⊋⊂∙∙− (O&quot;) 1

componentesque H, , H,, H, pendebunt ab unico integrali
Q. Fiat ⋅
∁≖⊹∂≖−⊦∁∶≖−−∆≖ ∙156
ut secunda ( o ' ) scribi possit in hunc modum
A ' = 12—2(axtby tcz) + wa + ya + z2 ;
erit
1
-
[ 12—2( ax + by + cz) + xa + ya + za = +
+
2(ax + by + cz) xtya taza)
+
243
12(ax + by + cz)2- [ 12 (ax + by + cz )-3(x2 + ya + za) ][ x ? tye + z&quot;)
845
+
.
unde, ob prinam ( o' ) ,
m
Q

* ++ ſ(ax + by+ ca)dın

25 /(x +y +z&quot;)du + z flar+ by +czydom.co&quot;.
Sit coordinatarum origo in centro gravitatis massae at
trahentis; erit ( 20. b )
1
43
Slax ( ax + by + cz) dni = ta fædm +
bſydm + ſzam ] = 0 ;
ideoque vertetur ( o '&quot;') in
156
ut secnnda (o') scribi possit in hunc modum
A&quot;::A3—2(aæ—-l-bj-l—cz)*æï-þyl-l-z' ;
erit
−↙∃≃∙⋅⋅ −−−−−
[∆⋅−≆≺∾↼⊦∂∫⊣⊸≉≻⊣−↕⋅⇀⊦∫⋅−⊦≖≖ ⊐⋅− *
↽−⇌−↿≴↸ ⋍≺∅↕⊹≀↗∫⊣⊸∡≻−≺↕∙⊣⇀∫⋅⊹≖≖≻
2A3
12(aæ-l—b.7—l-Cz)'-[1 ⊋≼↙⇂∙↿∙∙⊹⊘↾⊣⊸∅⊢∃≼∞≖⊹∫≖⊹≂≖≻∃ ∣⋮∙∙∁≴⊣↰↾⊣∎≖∶∣∙
8A5 '
'
-I-..;
unde, ob primam (a')-
. 1
1
Q ∙∙∙ Z
∙∣∙− A3]
(aæ-l-bJ-l-czkim—
1
- 3 &quot;
555] (x'-l'f' ∙⊦≖≖≱ dnl-l— üïf(aæ'l'lïï'*l'cz)'dm-n-(0 ').
Sit coordinatarnm origo in centro gravitatis massae
&quot;'
trahentis; erit ( 20. b)
. ↿ ∆↿−⋮∫≺∘∞⊣−≀≀↗−⊢∞≻↙≀⋯∶ 33— ta xdm-i-
bjïydm ⊣− rfzdm ]: 0

:

ideoque vertetur (o&quot;') in157
M 1
Q
Δ 243f\ x3 + y* +32) dmt
3
245 (ax + by + cz)-dm-, .. ( 0 &quot; &quot;).
ca
73. Corpus KGDH Sit sphaericum , ejusque centrum
in puncto extremo B radii CB (Fig. 43) inveniatur ; ipsi corpori occurrat QA in T. et Q '; ducto perpendiculo BE
supra CA , triangula rectangula AQC, BEC propter latus
AC=CB , et angulum QAC=BCA , erunt aequalia , adeo
que QC=BE ; chordae nimirum SD,CT aequidistabunt a
centro B; erunt itaque inter se aequales , ac proinde
OʻT ( Fig. 43 ) , aequabit QT (Fig. 42): quod cum ubique contingat, erit area KGDH (Fig. 43) sic .aequalis
areae XYC (Fig. 42) , ut solida genita ab his areis cir
suos axes revolutis aequalia sint inter se. Vim proinde , qua punctum C tendit in sphaeram KGTH ( Fig. 43 ) exprimet ipsa sphaera divisa per CM (=CB) seu per
quadratum distantiae puncti C ab ' ipsius sphaerae centro; siquidem aliae duae componentes (71) evanescunt:
Sed si sphaera ita condensaretur , ut coiret in centrum ,
eodem prorsus modo exprimeretur ejus attractiva vis; ergo punctum extra sphaeram situm eadem omnino ratio
ne in ipsam tendit , ac si omnia sphaerae puncta in cen
tro compenetrarentur.
Haec vera sunt , licet corpus non sit omnino ho
mogeneum , modo tamen sint ubique bomogeneae ejus par
tes a centro aequidistantes ; quod notandum etiam in se
quenti assertione.


73. Corpus KGDH Sit Sphaericum
. eiusque centrum
in puncto extremo B radii CB (F ig, 43) inveniatur; ipsi
corpori occur1at QA in T
et Q'; ducto perpendiculo BE
snpra CA , triangula rectangula AQC, BEC propter latus
AC;-:: CB, et angulum QAC:BCA ,
erunt aequalia, adeo-
que QC—BE- , chordae nimirum SD Q' T aequidistabunt a
«centro B; erunt 'itaque inter se aequales , ac proinde
Q'T (Fig. 43) aequabit QT (Fig. 42): quod cum ubi-
qne contingat, erit area KGDH (Fig. 43 ) sic .aequalis
areae XTC (Fig. 42) , ut solida genita ab his areis cir-
ca suos axes revolutis aequaha sint inter se. Vim proin-
de ,qua punctum Ctendit in sphaeram KGTH (F 1g 43)
exprimet ipsa sphaera divisa per ∁∾∙≖ (:CB') seu per
quadratum distantiae puncti. C ab ipsius sphaerae een-
tro ; siquidem aliae duaeïcomponentes (71) evanescunt,:
Sed si sphaera ita, condensantur,, ut coiret in centrum,
eodem prorsus modo exprimeretur eius attractiva vis; er-
go punctum extra sphaeram situm eadem omnino ratio-
ne in ipsam tendit
, ac si omnia sphaerae puncta in cen-
tro compenetrarentur. ⋅ ⋅
Haec vera sunt
,
lieet corpus non sit omnino ho-
mogeneam, modo tamen sint ubique homogeneae eius par-
tes a centro aequidistantes; quod notandum etiam in se-
quenti assertione.

74. Si punctum materiae locetur intra crustam sphaericam, sive intra orbem sphaericum intus cavum terminatum binis superficiebus sphaericis concentricis, id punctum, destructis viribus consistet in aequilibrio. Sint ( Fig. 44) NEQ, MFP superficies illae concentricae , punctum vero materiae sit O. Ducta per 0 quavis chorda MNEF, et ex centro K demisso perpendiculo KC supra ipsam chordam, erunt CM=CF, CN=CE;
igitur MN = EF , ac proinde ( 72 ) etc:

75. Ex dictis ( 73. 74 ) sequitur: 

1º . punctum in superficie duarum sphaerarum positum gravitare in ipsas sphaeras in ratione radiorum directa: nam sphaerae sunt ut radiorum cubi, quibus per eorumdem quadrata divisis, prodeunt radii simplices: 

2° . gravitatem puncti intra globum homogeneum pergentis a superficie ad centrum decrescere in ratione directa distantiae a centro ipso. 

76. Haec notentur. 1º. materiale punctum valde distans a corpore attrahente, utcumque se habeat forma corporis, ea proxime ratione tendit in ipsum corpus, qua tenderet si corporis partes in centro gravitatis compenetrarentur; patet tum ex dictis (12: 20) , tum ex eo quod in casu vires attrahentes punctorum constituentium corpus considerari possint tamquani proxime parallelae et proportionales ipsorum punctorum massis.

* Patet etiam ex ( 0 &quot; . 01 .: 72 ) ; nam si A est ila na

gna , ut, retento primo termino in ( o &quot; ), possint caeteri
praetermitti absque sensibili errore , sicque habeatur
M
Q
exsistent
M C
H
M
A2 ., HH ,
M 3
42 : A
H
ac proinde
M
H
ViH + H ,* + H , +
158
Sint (Fig. 44) NEQ, MF P supetticies illae concen.
tricae, punctum vero materiae sit 0. Ducta per 0 qua-
vis chorda MNEF , et ex centro K demisso perpendicu-
lo KC supra ipsam chordam, erunt CMr—CF, CNzCE;
igitur MN— EF
, ac proinde (72) etc:

75. Ex dictis (73. 74) sequitur: 10. punctum in su-
perficie duarum sphaerarum positum gravitare in ipsas sphae-
'ras in ratione radiorum directa: nam sphaerae sunt ut ra-
diorum cubi, quibus per eorumdem quadrata divisis, pro-
deunt radii simplices: 20. gravitatem puncti intra globum
homogeneum pergentis a supe1ticie ad centrum decrescere in ratione directa distantiae a centro ipso.

76. Haec notentur. 10. materiale punctum valde di-
stans a corpore attrahente, utcumque se habeat forma cor-
'poris, ea proxime ratione tendit in ipsum corpus , qua
tenderet si corporis partes in centro gravitatis comPe-
netrarentur', patet tum ex dictis (12: 20), tum exeo quod
ih casu vires ,attrahentes punctorum constituentium cor-
pus considerari possint tamquam proxime parallelae et pro-
portionales ipsorum punctoruin massis.
' ea Patet etiam ex (a&quot; . o&quot; 72); nam si∆∙ est tta ma-
gna , ut, retento primo terminogin (o'f ), possint. caeteri praetermitti 'absque &quot;sensibili 'et-rore , sicque habeatur
.
'
. - «. ∙
⋅
r ↾
1' I
.
. M. 11&quot;
≺≀⇌⋅⊼−↿⋅ ⋅
exsistent '1-
- --M 0
'M 6
&quot;' M
∣⋅
∏⋍−⋅−− −−∶∙−−⋅∙−− ...—...; .' AaA'H' ArA'H': ∣⋅∙↘∆∙
ac proinde
−−−∙∙−−−−−−∙∙∙∙−−∙ M
H:: l/Hil'i'nya'i&quot; He's-A&quot;?159
2º. Non pluribus opus est , ut stabiliatur illud: u
bi dimensiones corporum quorumcumque se matuo attra
hentium in ratione directa massarum, et reciproca duplicata
distantiarum sint admodum exiguae prae distantiis, quibus
ipsa corpora disjunguntur, eorum alterum tendet in alterum
perinde ac si essent 'ambo in suis gravitatis centris compe
netrata . Dicantur enim M , M' massae duorum ejusmodi
corporum , m, massa cujuslibet puncti spectantis ad M , et
A distantia inter m, ac centrum gravitatis massae M ; ex
Mm ,
primet vim attractionis motricem ( 28) , qua m, len.
dit in M, simulque ( 7 ) vim attractionis motricem, qua M tendit
in ma; ideoque merit vis attractionis acceleratrix, qua
M tendit in mo . Atqui hoc pacto M tenderet in mo, si to
la massa M compenetraretur in suo gravitatis centro ; er
go M revera tendit in mi, id est in singula puncta mas
sae M' , perinde ac si tota M foret in suo gravitatis cen
tro compenetrata: cumque ob paritatem rationis idem con
tingat massae M' quoad M , jam patet veritas assertionis.

3º. Quoad sphaerica corpora, quorum partes aequidistantes a suis centris sans homogeneae, obtinet assertio, utcumque caeteroqui se habeat intercedens distantia.

=== De gravitatione universali ===
[[77|77]]. Quae de coelestium corporum motibus, ex astronomicis observationibus hic subjicimus, ad ipsorum gravitatis centra respiciunt.

1º. Areae, quas circa solem describit radius vector uniuscujusque planetae sunt respondentibus temporibus proportionales: idipsum obtinet quoad areas descriptas a radio vectore uniuscujusque satellitis seu planetae secundarii circa suum planetam primarium.

2º. Convertuntur planetae circa solem in orbitis ellipticis ita, ut singularam ellipsium alterum focum occupet sol: convertuntur planetae secundarii circa suos planetas primarios in orbitis ellipticis ita, ut istarum focum occupet respectivus planeta primarius.

3º. Quadrala temporum periodicorum sunt in diversis planetis ut cubi semiaxium transversorum: idipsum obtinet quoad diversos satellites circa respondentem planetam primarium.

[[78|78]]. Hinc 

1º. planetae urgentur vi acceleratrice &lt;u&gt;tendente in solem&lt;/u&gt;; itidem satellites urgentur vi acceleratrice &lt;u&gt;ad respectivos planetas primarios tendente&lt;/u&gt;: plauetae, nimirum gravitant in solem, satellites vero in planetas, quibus adhaerent.

2º. Unusquisque planetarum (56) urgetur in solem vi gravitatis, quae sequitur rationem reciprocam duplicatam distantiarum ab ipso sole: idem dicendum de unoquoque satellite in ordine ad suum planetam primarium .

3º. Collatis inter se viribus acceleratricibus, quibus diversi planetae urgentur in solem, eae erunt (56) in sola ratione reciproca duplicata distantiarum a sole ipso; praecisa igitur projectionis vi, si diversi planetae in aequalibus a sole distantiis constituerentur, aequali tempore in eum descenderent. Idem obtinet in satellitibus quoad respectivos planetas primarios. 

[[79|79]]. Planetae secundarii una cum primariis, quibus adhaerent, in solem urgentur eadem gravitatis lege. Nam corpus omne, quod circa corpas alterum utcumque motum describit areas temporibus proportionales, urgelur duplici vi, altera tendente ad corpus illud utcumque motum, altera utriusque communi (5:46): cum igitur planetae primarii gravitent in solem, cumque planetae secundarii circa suos primarios describant areas temporibus proportionales; propterea etc.

[[80|80]]. Gravitant in se mutuo corpora omnia, ex quibus coalescit planeticum systema. Planetae siquidem omnes cum primarii tum secundarii vi gravitatis urgentur in solem; ergo sol in planetas omnes vi ejusdem gravitatis (7) urgetur: atque hoc argumento ostendes terram gravitare in lunam (id confirmant phoenomena marini aestus) caeterosque planetas primarios in suos satellites. Quod autem planeta quilibet in alium quemvis gravitet, satis e sola comprobaretur analogia, etiamsi nulli essent effectus, ex quibus haec gravitatio immediate detegi posset. Sed ejusmodi effectus non desunt: perturbationes videlicet, quae in recensitis motibus (77) observantur, quaeque per mutuam coelestium corporum gravitatem optime determinantur (62*60). Sic cum lunae motum ad regularis calculi normam ex observationibus exigere se posse Astronomi desperarent, tandem postquam ejusdem perturbationes ex mutua corporum coelestium gravitatione investigare coeperunt, tabulas lunariam motuum potuerunt conficere, quarum tantus est cum coelo consensus, quantum sperare ex observationibus nemo potest.

[[81|81]]. Praecisis perturbationum causis , urgebitar luna in tellurem vi acceleratrice (56):&lt;math display=&quot;block&quot;&gt;
\varphi=\frac{4\pi^2 a'^3}{T^2}\frac{1}{\Delta^2};&lt;/math&gt;


denotat &lt;math&gt;T&lt;/math&gt; tempus periodicum = dieb. 27 , 322 = minut. secund. 60&lt;sup&gt;2&lt;/sup&gt;. 24. 27 , 322; &lt;math&gt;a'&lt;/math&gt; semiaxem transversum orbitae lunaris, &lt;math&gt;\Delta&lt;/math&gt; radium vectorem ipsius orbitae. Iamvero mediocris radius terrestris = 16931100&lt;ref&gt;Error in originale&lt;/ref&gt; ped., mediocris parallaxis lunaris 57' + 11&quot;, unde &lt;math&gt;a' =\frac{16931100}{\sin(57' + 11'')}&lt;/math&gt;facto  igitur &lt;math&gt;\Delta = 19631100&lt;/math&gt;, gravitatis vis qua luna urgetur in terram evadet in ipsius terrae superficie
&lt;math display=&quot;block&quot;&gt;blah blah blah&lt;/math&gt;qui valor cum sit proxime 30,2 ped., inferimus gravitatem qua luna urgetur in terram nihil esse aliud nisi gravitatem ipsam terrestrem imminutam in ratione reciproca duplicata lunaris distantiae a terrae centro.

[[82|82]]. Vis gravitatis, qua lapis v . gr. urgetur in terram, est (80) ejusdem speciei cum illa gravitatis vi, qua corpora mundani systematis in se mutuo tendunt; ergo idem in utraque erit agendi modus. Atqui vis qua totus lapis urgetur in terram resultat ex viribus, quibus singulae lapidis particulae in eamdem nituntur; igitur et vires, quibus corpora mundani systematis gravitant in se mutuo, resultant ex viribus, per quas singulae ipsorum, particulae se mutuo petunt. His positis, stabilietur illud: gravitas ita materiam afficit, ut singulae ejus particulae in alias omnes et singulas gravitent in ratione directa massarum, ad quas tenditur, et reciproca duplicata distantiarum alterius ab altera. Recole quae diximus (76.2º.3º); etenim coelestia corpora et habent dimensiones admodum exiguas prae mutuis distantiis, et induunt formam prope sphaericam.

[[83|83]]. Bonum erit nonulla hic annotare. 

1º. designantibus &lt;math&gt;M&lt;/math&gt; et &lt;math&gt;m&lt;/math&gt; solarem et planeticam massam, ex dictis (56.k, 62.c) eruitur&lt;math display=&quot;block&quot;&gt;
M + m =\frac{4 \pi^2 a^3}{T^2}
&lt;/math&gt;ratio igitur inter cubum semiaxis transversi et quadratum temporis periodici, utpote pendens a massa planetica, nequit esse accurate constans quoad diversas planetarum massas. Atqui tamen ex astronomicis observationibus infertur rationem illam, sin minus accurate, certe esse quamproxime constantem: concludendum itaque planetarum massas admodum exiguas esse, ubi comparentur cum massa solis.
2.º Eodem modo ostenditur, si lunam excipis, satellitum massas fore et ipsas valde parvas prae massis planetarum, quibus adhaerent.

3.° Quae quantitates sunt designatae per &lt;math&gt;m, 
a , T&lt;/math&gt; quoad planetam , eae designentur per &lt;math&gt;m' , a' , T'&lt;/math&gt; quoad satellitem; erit&lt;math display=&quot;block&quot;&gt;
m + m'=\frac{4 \pi^2 a'^3}{T'^2}&lt;/math&gt;Hinc (1°)&lt;math display=&quot;block&quot;&gt;
\frac{m + m'}{M + m}=\frac{T^2}{T'^2}\frac{a'^3}{a^3}
&lt;/math&gt;praetermissa ( 19. 20. ) &lt;math&gt;m'&lt;/math&gt; in numeratore primi membri, itemque m in denominatore , et facta M = 1 , prodibit.
T2
TO
a's
i
quae formula suppeditat rationem inter solarem massam
habitam pro unitate , et massas planetarum ( tellurem ex
cipe) , qui satellitibus stipantür.


4.° Quod spectat ad tellurem consideratam in
star sphaerae habentis radium R , et massam m , sit &amp; gravi .
tas prope ejus superficiem , erit (73) 8 =R. , ideoque (10. )
M +m
4 712 a3
&amp; R2T
et praetermissa ( 19. ) m in numeratore primi membri , factaque solari massa M = 1 , emerget
163
2! Eodem modo ostenditur ,
si lunam excipis,
satellitum massas fore et ipsas valde parvas prae massis planetarum, quibus adhaerent .
-l ⋅ 

:

41:2003 -
∙∙ ↶↿ m-4—m' £S.-&quot;IW.,
Hï'n'c (10)
⇀
..
.m -- in'. T., a'-3-- - M,-—-.nsl ≔−⋅⋅∙−∙− ∙ ∙∙∙ : T'a ::
&quot;
praetermissa (.10 20. ) tu' in numeratore primi membri ,
itemque ut in denominatore ,
et facta M— −∙− ↿, prodibit.
⋅ 1 ↴− ⋅↧⇁≖ -a ∙
−
⋯∙↽↽⊽ .
..;3, .'
∙⊾⊺⋅⋅
quae? .fottmule- suppeditat rationem -
intcr. solarem^ massam
habitam pro unitate ,
et massas planetarum (tellurem ex-
cipe) , qui satellitibus .stipantur.
'i-o Quæ-Spectat ad tellurem' consideratam in-
star sphaerae habentis radium R , et massam 111, sit 3gravi-
tas prope eius superficiem ,
erit (73) gr ≖⋅⇁⋅∙−⋮↾−⋮↾− , ideoque (10.)
IUI—tm,— 4 11:303.
,
gRQTi'
et praetermissa (10.) »: in numeratore primi membri, f.-
ctaque solari massa M ∙−⇁−−−∙ ↿, emerget ⋅164
&amp; R2T2
4 Ti ? a3
PE

5° Media telluris densitas ( = M) determinari potest ex penduli aberratione. Sit CB (Fig. 45) pendulum; a longitudo rectae CB, quae nec distendi possit nec inflecti; S centrum massae sphaericae ( = m' ) ad se trahentis punctum ponderosum B , r radius , M densitas; b recta CS; CD posilio penduli digressi a recta verticali CB;
&amp; angulus BCD; h angulus BCS; k recta SD: centro insuper C et radio CB intelligatur describi circularis arcus
BD , et per D duci tangens nn' . In D sollicitatur materiale
m'
punctum viribus acceleratricibus g et altera juxta ver ka
ticalem DD' , altera juxta rectam DS ; anguli
SU
1
D'Dn = CnD = 90 ° — E , SDn ' = # (CDS - 90° ) ,
to
et consequenter
b sin (h — 5)
cosD'Dn sins , cosSDn ' = sinCDS =
k
Vires igitur motrices respondentes praefatis viribus ac
celeratricibus sese librant in D quotiescumque fuerit
que
bas
bit
ha m'
23 gsine b sin ( h - €) .
sed
Pone longitudinem penduli ita exiguam prae distantia CS ,
ut absque sensibili errore sumi possit k = b ; traduce
tur aequilibrii conditio ad
gb2 sin é = m ' sin (h - ) ;
m
et substitutis ( 4º . ) valoribus 8 T RH, m
R? 3
4
90
paris 75 1'3 je , prodibit
164
g R*T3
m— 4 723 (13
50 Media telluris densitas (:: p.) determinari po-
test ex penduli aberratione . Sit CB (Fig. 45) pendulum longitudo rectae CB, quae nec disteudi possit nec inflecti; S centrum massae sphaericae (: m') ad se trahentis
punctum ponderosum B ,
r' radius
, pf densitas ; 6 recta
CS; CD positio penduli digressi a recta verticali CB;
a angulus BCD :
]: angulus BCS: k recta SD: centro in-
super C et radio CB intelligatur describi circularis arcus
BD , et per D duci tangens nn' . In D sollicitatur materiale
punctum viribus acceleratricibus 3 et ?; , altera juxta ver-
ticalem DD', altera juxta rectam DS : anguli
ix)-D'.. ∙∁∥↧⊃ :. 90o −−∙ e
, sna':∶↿≐ (CDS — 900)
.
et ⋅ consequenter
bsinUt—s) ——k .
Vires igitur motrices respondentes praefatis viribus ac-
celeratricibus sese librant in D quotiescumque fuerit
! r ' ,
!'
cosD'Dn :: sins, cosSDn' ∶∙ sinCDS −−−−⋅−
gsins:
£;- b sin( h — £) .
liane. longitudinem penduli ita exiguam prae distantia CS ,
ut absque sensibili errore sumi possit 1::6; traduce-
tur aequilibt'ii conditio ad
gbï sin ; −−∶ m' sin-(h — a) ;
et substitutis (40.) valoribus g: €; ∶∶ £- 11 R p., m' −∙−∙−−
4
,, ,
'
∙⋅ ; &quot; . .
⋅
∙⋅⋮↿∏⋅ p. , prod1h1t
!
'
I
151
sit
lla165
1
b- Rue sin
{
= 13
M
' sio
(
h
-
E)

unde
i
p3
y sin
h rº
( sinh
– coshtang :)
1
lang
E
=
Ruba tospicosh ji Rba lang

Permanentibus
r
' et
'
, valores
b
=
r
' et
h
= 90
° manife
ste suppeditant maximam penduli aberrationem
&amp;
, ut quoad
istiusmodi aberrationem sint
Se
re tang R pe
3/3
Rtang
s
Densitas fl
, prout colligitur ex aberratione penduli
, cen
setur quater vel quinquies major quam densitas aquae. 6. ** Eadem u determinatur etiam experimentis in
stitutis in libra torsionis
. Sit
( Fig
. 9.
) HH
(
=2a
'
) posi
tio vectis horizontaliter librati

;

E punctum medium, in quo vectis appenditur filo metallico verticali HA circulus horizontalis centro
E et radio EH
=
a
')

; SS

( 26
)
recta similiter horizontalis
, transiens per
E
, ibique secta
bifariam

; sint

S et
S centra sphaerarum inter se aequa
liam et quoad volumen
, et quoad massam
(
=
m
)
, ad se
trahentium massulas sphaericas
m
' et
m
&quot; inter se pariter
aequales
, quarum centra in
H et
H
'
. Movebitur vectis
circa punctum
E immotum
, motusque iste repetendus ab
attrahente sphaerarum vi juxta circuli tangentem
, cui vi
jugiter adversatur vis lorsionis ex filo metallico verticali

et quoniam corpuscula
m
' et
m
&quot; eodem prorsus donantur
motu circa
E
, satis erit alterum dumtaxat
v
. gr
.
m
' con
siderare
. Dicatur itaque
h datus angulus HES

;

&amp; angulus
,
quem in fine temporis
i continet vectis cum initiali po
sitione EH

;

k distantia inter
S et
m
' in five ipsius
t

: sol

licitabitur
m
' juxla circuli tangentem vi attractiva
!
)- a
165
63 Bpain :::/3 (fimul—15);
unde
tan E' ∙∙∙⋅ r'3 pf sin 11 p. '
r'3 (sinit −⋅ coshtang &amp;)
∙
g −⇀ nubi-l- r'3 picas/1
.pf ⇀−− lib2 teng
&amp; .
Permanentibus r' et p!
, valores
b

: r' et 11
: 900 manife-

ste suppeditant maximam penduli aberrationem

:

, ut quoad
istiusmodi aberratiouem sint
tangi—£&quot;.! P',—.
r −⇁∙
p p. Btang
&amp; ,
Densitas p., proutcolligitur ex aberratione penduli.
, cen-
setur quater vel quinquies maior quam densitas aquae.

6?- Eadem p. determinatur etiam experimentis in-
stitutis in libra torsionis
. Sit (Fig.'9*.) HH' (:Za') posi-
tio vectis horizontaliter librati;
E punctum medium
,
in
quo vectis appenditur (ilo, metallico verticali; HA circu-
lus horizontalis centro
E et radio EH (:a'); SS' (:26)
recta similiter horizontalis
,
transiens per
E
,
ibique secta
bifariam

; sint

S et S' centra sphaerarum inter se aequa-
lium et quoad volumen
,
et quoad massam (:m)
, ad se
trahentium massulas sphaericas m' et 111&quot; inter se pariter
aequales
, quarum centra in
H et H' ., Movebitur vectis
circa punctum
E immotum
, motusque iste repetendus ab
attraheute sphaerarum vi juxta circuli tangentem
, cui vi
jugiter adversatur vis torsionis ex filo metallico verticali

:

et quoniam corpuscula tu' et m&quot; eodem prorsus douantur
motu circa
E ,,satis erit alterum dumtaxat v. gr. m' con-
siderare
. Dicatur itaque ]: datus angulus HES

;

&amp; angulus
,
quem in fine temporis

: continet vectis cum initiali po-

sitione EH;
k distantia inter
5 et m' in line ipsius
t

: sol-

licitabitur m' iutta circuli tangentem vi attractiva-166
b
hak sin (h — e), eritque kº = a's - 2a'b cos (h --- e) + 6+;
experimenta insuper praebent vim torsionis proportionalem
angulo e , et consequenter expressam per ce : quoniam igi
tur labente e describit m' arcum ás , iccirco ( 50.3º. )
áre mb
sin (h -€)
dta k3
3
1
&lt;u&gt;aequatio ad motum&lt;/u&gt; corpusculi m' . Ob angulum &amp; valde exi
gaum , sin (h-5~s)) = sin h - e cos h , k - = [ a's -2a'b ( cosh
+sinh) + ] := k +. 2a'besio h ]
R3
3a'b esinh
+
ubi denotat k , valorem k respondentem initio k.
molus , quum nempe E = 0 ; proinde
sin (h-E ) sinh
23
E COS h 3abe sin’h
+
k.
k.
sinh
k.
k .
£ k cosh 3a'be sinh sinh
+
kb.
K.
her )
[ (a's tabo) cos h
k . k5.
sinh 2a'b cosah — 3a' b sin : h]
[laat69)cosh
- 2db -a'b sinä h] :
et factis compendii causa
mo
[ la'a + b ) cosh — 2a'b – a'b sinºh] +c = g' ,
166
m 1)
.
'
.
F.
∙∣⋮∙ nuUi—s) ,entque ka: a'a— 2a'b cos (71—5)-1—63;
experimenta insuper praebent vim torsionis proportionalem
angulo :,
et consequenter expressam per et: quoniam igi-
tur labente :describit m' arcum a's, iccirco (50.?)0.)
,d'38 mb ∙
aa;-a.- Fama—Q—ct
aequatio ad motum corpusculi m'
.Ob angulum :valde exi-
guum ,
sin (It—s) :sin h—s cos 11, k'3: [a&quot; −∙∙ Za'b (cos):
3 . 3
-i-ssinh) ⊣−∂∙∎∙∣− 'a': [le, −∙∙ Za'besin H's—:i?
⊣⋅− a.:-s;&quot;: h, ubi denotat k, valorem k respondentem initio
motus
, quum nempe s.: o ; proinde
sin (I;—s) sin 11 :
cbs Il 30'68 sin'h sin ls
∙−−−⋅∙≖∙−−∙− ∙∙∙⋅ ks' &quot;&quot;&quot;&quot; k3, −∎⋮∣⋮∍∙ fl&quot;, P, P,
sit' eos]: l Bez-'besin'h sinh : .

*

5,
w
cos-1. −−⋅ adb sin-t.] :
822&quot; −⋅⊼⋮−⊏≺∘⋅⋅−⊦∂⋅↗ coslt
∙∙∙⋅ ⊋∘∣∂∙∙− a'b sin' &amp;] :.
et factis compendii causa
iii- [(.-a ⊣⇁ 61) cos h—w—
a'6 .i..- h]
∙⊦≖⋅∸ −−∶
z'-167
mô sinh
wg'
23.
aequatio ad motum corpusculi m ' vertetur in
do e
a' ó (0) -- s ) ;
de²
ex cujus integratione ( 27. 28º. )
(9)*va
' @
'ri
{ = w + Ce + Ce
Sunt autem ( 27.300. )
.
va
cos (9 )* + v = on e(2)
, - ) vi
cose ( ) -va sine ( 2)
propterea
szaf1CTC&quot;) cos( ) +(c —c )V= sine ( :
sumptisque C +C' =C.cos C,, C — C = CV -ī sincs,
= - + 6, co [4 ( 4 ) + c ]
Minima vectis declinatio , í = u - C , ab aequilibrii positio
167
mb sinh
−−∣−∣⋮−≣∶−−≂−↩∾⊰
aequatio admotum c0rpusculi m' vertetur-in
,d'l
(: -d—t;:::g (6)—S):
ex cuius integratione (27 . 280.)
⋅⋅ .

*

.
.'.t ⋅↴∶
≖−↽−−−⇀↠∾−⊦∁∊ :(?) [l:—[— 08 &quot;(ä-') V
..
Sunt autem (27 .300.)
.;.
propterea &quot;
∙⋐⋅∶∶⊙−⊦≼∁−⊢∁⋅≱ ∾≘↙≺−⋚⊑⋅≻⋚⋅⋅−⊢ ((i—C' )
(V:; sint (?);;
sumptisque ∁−⊢ ∁∙∶∁≖∞∙ c, , ∁−∁∽−↽⇌∁≖⇂∕∙⊺∘⋮∥∁≖∙
⋅⋅ ∸ g' ?
≘−∙−−∙∾−⊦∁∙∞∘∣↣⋮≀ ≼⋮⇉⋟ −⊦∁≖∃∙
Minima-vectis declinatio ,
(:o)—G. ab aequilibrii pocitio-168
ue H'H respondet valori ( ) + Cs = ( 2n – 13 ;ma
= + c = 212nt : determinatis
itaque per observationem i'et s&quot; , eruetur inde
te&quot;
et ducta 00 ita , ut sit angulus HEO = w , perget vectis
moveri instar pendali horizontalis circum EO , impendet
que tempus tz = &quot; - t ad integram conficiendam oscilla
tionem , nimirum
ty=T
VAg'
Sit nunc a longitudo penduli simplicis ( 66) , quod intra
idem tempus t absolvit oscillationes suas : cum habeamus
ty = TO Van
a
erit 8 et denotante si densitatem sphaerae m ,

a'
r radium , substitutisque valoribus ( 5.0 )
4
пRр .8 3
mbsinh
wk3
471p3 M'bsinh
3wk. 3
proveuica
Ruwk3 р
p3M'bsinha
unde ar3bsinh
a'Rwk.3 1
Densitas pe sic determinata censetur esse ad
tatem ut 5,48 : 1 .
aquae densi
168
ne H'H respondet valori t'(g——,)ä -I-C,:(2n-1)1r; ma-
xima ⋮∣↾∶∶∾⊹∁≖ valori :
.(gwik) ⊹∁≖:2mr: determinatis
itaque per observationem eet :&quot; , eruetur inde
−−≘⋮−⊢⋮⋅∣∙ −− ⇄ ∙
et ducta O'O ita
,
ut sit angulus HEOzzæ, perget. vectis
moveri instar penduli horizontalis circum EO, impendet-
que tempus :::-:i &quot;—t' ad integram conGciendam oscilla-
tionem , nimirum
a'&quot; Vf- ∙

5
Sit nunc a longitudo penduli simplicis (66), quod intra
idem tempus :, absolvit oscillationes suas: cum habeamus
(3:11 V? .
. g a '
∙
erit?-;? ; et denotante p. denutatem sphaerae m ,

: - radium

. substitutisque valoribus (59)
∙∙∙⋅ ∙≤∙ nR ,— mbsinh— 4nr39'bsinh ∙ ∙
o ∙−−− 3 p. ,g 01:03 30 1:03 ,
provenit...
Rpali-03 a p. arabsinh
r3p'bsinh—c7 '
unde ∙∥∙∽ a'Rmkoï'l '
Densitas p. sic determinata censetur esse ad aquae densi-
tatem ut 5,48:1.169

7. ** Ex mariui aestus phoenomeno deduci pol
est ratio inter massam lunarem m &quot; et terrestrem m. Sit
m' ( Fig. 35 ) quodvis terrae punctum ; lunares vires distrah
entes punctum mé juxta mm&quot; et Am exprimuntur ( 62) per
2m &quot; Dcosh m&quot;Dsinh
(0) ,
( 0' ) :. x'3
X :' 3
quod in ordine ad lunam est h , x&quot; , in ordine ad so
lem sit H , X &quot; ; prodibunt consimiles vires solares
2MDcosH MDsind
X &quot; 3
( a ) , ( a ').
X'3
In casu angulorum h et H aequalium habemus
( 0) m&quot;X &quot;3
(a)
-- ( 0 )
M.2'3 ( a' )
caeteris vero paribus , ratio inter lunares et solares vi
res est eadem ac ratio inter respondentes aquarum ela
tiones ; denotante igitur p hanc secundam rationem , erit
X3
M
р
m '
M
m'
unde m
= P
3
x &quot;' 3
X &quot; 3
Observationes praebent p = 2 , 35333 : vide mechan, coel.
vol , 5. pag . 206.
Aliquid notatur de motu punctorum materialium
utcumque inter se connexorum .

84.* Vires motrices P, P , P&quot; , ... sollicitantes
istiusmodi punctorum massas m ,
mi ,
m' 0 resol
12
169
7.&quot; Ex marini aestus phaenomeno deduci pot-
est 'ratio inter massam lunarem m&quot; et terrestrem m. Sit
m' (Fig. 35) quodvis terrae punctum .; lunares vires distrah-
entes punctum m juxta mm&quot; et Am' exprimuntur (62) per
2m&quot;Dcosh m&quot;Dsinh (0) ∙ ∙−−∙↕∙−∙⋅∃−−⋅ (O,)
xara ⋅
quod in ordine ad lunam est h , a:&quot; ,
in ordine ad so-
lem sit H , X&quot;; prodibunt consimiles vires solares
2MDcosH' ⋅ MDsinH
W (a), ∙∎∎ Xl/3 (a').
In casu angulorum I; et H aequalium habemus
(o) -m&quot;X&quot;3l-(o') ∙
(a) Mx&quot;3 X(a') ⋅
caeteris vero paribus
,
ratio inter lunares et solares vi-
res est eadem ac ratio inter respondentes aquarum ela-
tiones ; denotante igitur p hanc secundam rationem
, erit
X&quot;3 d m&quot; M a:&quot;3
∙ &quot;&quot;;- un B −−∶ ∙ ,. x' 3
' m ? m X'3 31 ≊⋅∙ p:
Observationes praebent p——:2 , 35333 : avide machen. coel.
vol. 5. pag. 206.
Aliquid notatur de motu punctorum materialium
utcumque inter se conus-xarum.

843: Vires motrices P, P&quot;, P&quot;, ... sollicitantes
istiusmodi punctorum massas m , m'
, m&quot; , ...
resol-
12170
vantur singulae in ternas coordinatis axibus OX , OY ,
OZ ( Fig. 8 ) parallelas ; designentur per X , Y , Z, X',
Y , Z ' , X &quot; , . . componentes inde ortae ; sintque x, y,
3 , x ', y ', z' , x &quot; • punctorum coordinatae responden
tes temporit , ut ( 50, 1.º ) per
x == f (1 ), y = f(t), 2 = F (t), ' = fi (t), y = fz(t ), == F ,(t)
x ' = fale) ,y&quot; =f(e), z&quot; = F.(6),7: &quot; = f5e) , ... ) co)
exhibeantur aequationes ad actuales molus ; ad eos nem
pe motus , quos reapse concipiunt massae m , m' , m &quot; ,
ob actiones virium P , P' , P &quot;, ... Quoniam materialia
puncta , etsi mutuis nexibus liberala , viribusque ( 50. 4.0 )
dºx dz m
dạy
de² m
d²x
dla m . dia dla dea
d2z '
dca dc ?
sollicitala , adhuc tamen conciperent motus ( 0 ) ; ideo ,
attentis nexibus , consistent in aequilibrio vires
dez X - m
d2x
de2 Ymdạy
di?
2m X'
der'
di2 &gt;
7
dt2
Y - m
d²ý
dt2
daz'
Z ' - m '?
dt²
X &quot; -m.dºx
&quot;:
.
dt2
Conditiones ( a &quot; 13. 8. ) includuntur in conditionibus
aequilibrii quoad liberum punctorum utcumque connexo
um systema : liquet enim varians systema, semel libra
tum , adhuc permansurum in aequilibrio , etsi ejus pun
cla rigidis lineis immutabiliter connectuntur. Propterea
170
vantur singulae in ternas coordinatis axibus OX, Oï
,
OZ ( Fig. 8 ) parallelas; designentur per X, ? ,- Z, X',
1&quot;, Z' , X&quot; , . . . componentes inde ortae; sintque x,],

: , x', y', z', æ&quot; , ... punctorum coordinatae responden-

tes tempori t
, ut (50. 19) per
x:f(t), 7:112) ,z:F(t),.r ':f,(t),y ':f (t), z':F,(t),
(0)
x&quot;:-f.(t)
.
y&quot;:f.(t)
.
(&quot;zl-&quot;.m. m&quot;':--f3(t)
.
∙ ∙ ∙
exbibeantur aequationes ad actuales motus; ad eos nem-
pe motus
, quos reapse concipiunt massae m , m', m&quot;
,
ob actiones virium P, P', P&quot;, . . . Quoniam materialia
puncta , etsi mutuis nexibus liberata, viribusque (50. 49)
da.. mdzy de.
⋯∽∣↙≀≄∙↿∶ ∙↙∄≖∫⋅
&quot;B'—'— m-——- m— d,.
' md:2 '. d?
'md? ' dta '
,dzz, ad:-I?&quot; m d£2 '.. m dt:- '
∙ ∙ ∙
sollicitata, adhuc tamen conciperent motus (a); ideo ,
attentis nexibus, consistent in aequilibrio vires
(P:: (P] daz ,dzæ'
X
⇁∎−−∙ —p ⋅⋅⇁ ∙∙∙∙ ∙−−− '&quot;sz '
?
&quot;'d'T'z' Z &quot;&quot;da: X &quot;'de '
≀∠⋮⊺ &quot;rad :
&quot;
rnnndaæ ï'-——-—m '
...
dtï' ,..—z ⋯∠↙⊤ 'X— dtz
'
Conditiones (a&quot;f' 13. 8.0) includuntur in conditionibus
aequilibrii quoad liberum punctorum utcumque connexa-
rum systema :liquet enim varians systema, semel libra-
tnm, adhuc permansurum in aequilibrio, etsi eius pun-
cta rigidis lineis immutabiliter connectuntur. Propterea171
(xam )=o, 3(1— )= 0, $ (2- -o,
=[+ (rad ) – (xrm ) ] = o,

* [&gt; (2 mm ) -- ( - ) ]

&lt;math&gt; [ - (-) - = ( x ) ] &lt;/math&gt;
0
seu
daz ΣΖ - Ση
de² EX = sme , Y =sme &gt; ( 0 )
$( wYyX) =Em ( -e
)
3(y2=-1)= sm (voeding - :)
Eml
1
Z
day
de2 (0')
(2X == Z ) = 2m Z
dax
dta -
daz
dea :
formulae (o ') spectant ad translativum punctorum motum,
prima juxta OX , secunda juxta OY , tertia juxta OZ ;
formulae ( o“) ad rotatilem punctorum motum , prima cir
ca OZ , secunda circa OX , tertia circa OY ; eaedem ve
ro (o &quot; ) simul , ad punctorum motum circa fixam coor
dinatarum originem .
Haec facile nunc stabiliuntur. 1 . ** Habemus (20. 6.)
seu
æ ,dþ- ∙−−
∠∄≖⋍ ,
day d3æ ?
−∙∙ − ..(æy—yX) Em (
«: Tt&quot; ]—dt3 )
.
- ∑ dan: daz (zX—æZ): Zm( :217; — æ
—) :
formulae (c')/spectant ad translativum punctOrum motum,
prima juxta OX , secnnda juxta Oï, tertia juxta OZ;
formulae (a&quot;) ad rotatilem punctorum motum, prima' cir-
ca OZ , secunda eirca OX , tertia circa Oï ; eaedem ve-
ro (o&quot;) simul , ad pnnctorum motum circa fixam coor-
dinatarum originem.
Haec facile nunc stabiliuntur. ↿∙∘⋇ Habemus (20. b.)172
dar
Em dea dex,
dla
day Σm.
dt2
Em
daz,
dc2
da ,
Em dla
Em &gt;
Em: dt?
Hinc &gt;, ob (o' ) ,
der ΣΧ daz,
de
ΣΥ
Em ' dt2
ΣΖ
Σm (o' ' ' ) : Am dla
molo videlicet systemate punctorum m , m' , m &quot; ,
perinde ( 50. 4. ) movebitur gravitatis centrum ac si , co
euntibus punctis in ipsum centrum , applicarentur centro
eaedem vires P, P , P &quot; , ... cum iisdem directioni
bus , quibus puncta illa sollicitantur .
2 . '* Fac ut vires nihil sint aliud nisi punclo
rum actiones mutuae : denotante A actionem puncti v.
gr. m in aliud quodvis v . gr. m' , et A' actionem puncti m'
in m , erit ( 7 ) A=A' ; et expressa per D distantia inter
utrumque punctum , resolvetur A' in ternas coordinatis
axibus parallelas
x' #A D
TA
EA ;
ilem A in ternas iisdem axibus parallelas ( o'r )
ŁA to ,
EA D po', -A D
sumpto superiori signo si A , A' sunt vires attrahentes,
inferiori si repellentes . Quare EX =0, EY=0 , &amp;Z=0, et
consequenter
dér,
=0,
adi ?
day1
di?
dz,
-0,
=O ;
di?
in ea scilicet qua sumus hypothesi nullis viribus acce
↙≀⊴⋅↕⋮ d'), inuia—z- '
d'æ, dta (if/y. md:2 dïz, dta
dtz— Em 'dt: Em '
du ïm
Hinc , 06 (o') ,
d'æ, EX (Ph- Zï diru— ZZ
dF—Zm' dt&quot; &quot;Zm dt2 −∑⋯ (0 ):
moto videlicet systemate punctorum m , m' , m , . . ,
perinde (50. .f.&quot;) movebitur gravitatis centrum ac si, eo-
euntibus pnnctis in ipsum centrum
, applicarentur centro
eaedem vires P, P', P&quot; , . . . cum iisdem directioni-
bus ∙ quibus puncta illa sollicitantur.
294: Fac ut vires nihil sint aliud nisi puncto-
rum actiones mutuae :denotante A- actionem puncti v.
gr.m in aliud quodvis v. gr. m', et A' actionem puncti m'
in m, erit (7) A:A'; et expressa per D distantia inter
utrumque punctum, resolvetur A' in ternas coerdinatis
axibus parallelas
3.&quot;—æ ∙−⇠ 7—7 ...-,: z—z .
drA U , A D A D
itcm A in ternas iisdem axibus parallelas (o&quot;)
æ—x' J—y' z—z' ∙
sumpto superiori signo si A, A' sunt vires attrahentes,
inferiori si repellentes. Quare XX :0, ∑∟ -o, ZZ:o, et
consequenter
in ea scilicet qua sumus hypothesi nullis viribus acce-173
leratricibus agetur gravitatis centrum , nulloque ob mutuas
panctorum actiones afficietur motu. Huc spectat princi
pium de conservatione centri gravitatis.
3.°# Super planis XOY, YOZ, XOZ fiant proje
ctiones a, b, c, a' , b' , c' , a &quot; , 6 &quot;, c &quot; , a ' , , . . arearum
descriptarum a radiis vectoribus punctorum m, m' , m&quot;,
computatis radiis ab origine coordinatarum : erunt ( 50. 8. )
xdy — ydx Σmda = Σm ydz - zdy και Σmdb = Σm 2 2
zdxxdz Σmdc = Ση 2 unde
daa 2Em -Σm α2
dta dt
22m d2b
dta
daz
у

;

= Em (:
dla e )

: )

dec dex dez 2Σm -Σm dt2 sm
(
20
de?
et consequenter ( o &quot; )
d'a 22m dt² 8(xY4yX), 28mmdla = Eby2 — zY),
( 0 )
dac 2Em =E( zX-xZ) .
dta
4.0 # Si vires consistunt in mutuis punctorum actio
nibus , erunt ( 2.º o &quot; )
173
leratricibns agetur gravitatis centrum
, nulloque ob mutuas
punctorum actiones afficietur motn. Huc spectat princi-
pium de conservatione centri gravitatis.
3.&quot;; Super planis XOï, ïOZ, XOZ fiant proie-
ctiones a, b, c, a', b'. c', a&quot;, b&quot;, c' ', a'&quot;, ∙ ∙ ∙ arearum
descriptarum a radiis vectoribus punctorum m, m', m&quot; , .. .
computatis radiis ab origine coordinatarum : erunt (50. 8.&quot;)
∑⋅↾⋅⊿↙↓⋅−−≔−∑⋯⊔−−−−∫−≌∙∑⋯↲≀⊨∑⋯⇅−−−≖−≗↶∙ d d d—d
2 ∙ 2
Emma.—zn. fix—?? ,
unde
22m ⋛∙∶−≧∶−−⋅∑⋯≺⊰≵ :::-£v:— 7
id?-:?) ∙ ⋮⋯∶⊜≀≀∶≖∂−↽−−≖⋅⋯⋅↗≺ :: −− ::z ⇋
d'c— d:.r ædaz
et consequenter (o&quot;)
ZZm −⋛⊴⋮↥⋮−⇌∑≺∞⊺−∜∑⋟ ,
ZZmäï—b- :Zþ'Z—zï),
d (a')
220: ⋅⊋≖−∶∶ :2(zX—-æZ).
494: Si vires consistunt in mutuis punctorum actio-
nibus, erunt (2.o o&quot;)174
8 (xY - 7X ) = 0 , (yz - zY) = 0 ; $ (zX -- XZ) = 0;
ideoque
dra dc Emdl2
d2b Σm dc2 Emadt²
}
et computatis areis ab initio temporis t ,
Ema = Ct , Emb = Ct, Emc = C &quot; ! (0 ) :
huc special principium de conservatione arearum. Formu
lae ( o &quot; ) adhuc obstinent , etsi in systemate invenitur pun
ctum fixum , modo tamen in pancto illo collocetur origo
coordinatarum : siquidem vigent in casua equationes ( o&quot; ' ' ) ,
unde profluunt ( o &quot; ).
5.0 * Si arcus s refertur ad tres axes orthogona
les , ejus incrementum infinitesimum ds poterit spectari
tanquam diagonalis parallelepipedi rectanguli sub later
culis dx , dy , dz : hinc.
dsa = dx2 + dyz-tdz?, et consequenter ( 50, 2.0 )
v2= dx2+ dyatdz
dla
Erit itaque
Emvdv = Em
der
de²
d'I
ayt
ar
- .
ac proinde (0 )
Emvdv = E (Xdx + ydy + Zdz) ( o&quot; &quot; ' ) .
Fac ut E (Xdx + Ydy + Zdz) exsistat differentiale exactum ,
!
174 .
£(xï—77X):o , ZUZ—zï): :X(zX—xZ):o;
ideoque
d'a 'dzb (130
dt2
?
et computatis areis ab initio temporis :,
2ma:Ct , 2mb:C't, ch:C&quot;t (o&quot;):
huc spectat principium de conservatione arearum. Formu-
lae (ov') adbuc obstinent
, etsi in systemate invenitur pun-
ctum fixum, modo tamen in puncto illo collocetur origo
coordinatarum: siquidem vigent in casua equationes (o&quot;'),
unde profluunt (o&quot;).
594: Si arcus :refertur ad tres axes orthogona-
les, eius incrementum infiuitesimam ds poterit spectari
tanquam diagonalis parallelepipedi rectanguli sub later-
culis dx
, dy , ds :hinc.
ds':dæï-l-df3-l-dz3 ,
et consequenter (50, 2?)
daga—.dyzudzz
dt2
⋅
02.—
Erit itaque
da a :
Zmpdu:2m(d£fdx : (iyd): ↿ d
zdz) .
ac proinde (o') ∙
vadv :!(de ⊣− ⊺⊄↴⋅⊺∫ ⊣−∅∠∄≂≻⋅⋅
(o&quot;').
Fac. ut XXdæ—fïdJ—l—Zdz) exsistat differentiale exactum.175
prodeat nimirum ex differentiatione cujusdam functionis
F (x , y , z, x ', y , z, x &quot; , ... ) ; habebis
Em (u2 — V.2) = 2F (x ,y ,z,x',...) —2 F (xo,9o , zo , x '. , ... ) ;
quantitates v. , xo, Yo, Zo, x'o, ... respondent initio mo
tus. Consequitur, quod, redeuntibus iisdem coordinatis, ea
dem quoque redibit summa virium vivarum : huc spectat
principium de virium vivarum conservatione.

6. °* Denotent &lt;math&gt;h, i, k , h , i , k ' , h '' ...&lt;/math&gt; coordinatas punctorum &lt;math&gt;m, m', m''&lt;/math&gt; in ordine ad novos axes, qui et paralleli sint axibus &lt;math&gt;OX , OY , OZ ,&lt;/math&gt; et originem habeant in communi gravitatis centro; erunt
x = xrth , y = yiti , z = zetk , x' =xith' ,
y = yiti, z= z+k ', w &quot; = xrth &quot; , ... ;
quibus valoribus substitatis in ( o &quot; ) , attentisque aequatio
nibus ( 20)
dah deh , dai dai Σm Σm=0, Σ . Σm=0
dcz dta des dt2 =
dek Σm-
dt
dakı Em=0
dc2
1
nec non aequationibus ( o &quot;&quot; ), prodibunt
dai dah
2 ( XiY) = Em ( h TI dea dt2
E ( iz - kY) = Em
(ala dih),
(
- ) com
(akone )
( o &quot; )
dah
ElkX_hZ) = Em ( k dt2
175
prodeat nimirum ex differentiatione cujusdam functionis
F(x,y, :, x', y', z', æ&quot;
, ..
.) ; habebis
M(æ—voz):2F(æ,7,z,æ',..-) -2 P(æo,yo , zo , x', , ..
.);
quantitates v,
, æ., y,,zo, æ'o,
... respondent initio mo-
tns. Consequitur, quod, redeuntibus iisdem coordinatis, ea-
dem quoque redibit summa virium vivarum: liuc spectat
principium de virium vivarum conservatione.-
604: Denotent h, i, k, h', i', A', I:&quot; .. . coordi-
natas punctorum m, m', m&quot; , . ..
in ordine ad novos
axes , qui et paralleli sint axibus OX, Oï
, OZ
,
et ori-
ginem habeant in communi gravitatis centro; erunt
r—æl-i-h ,.szl-ï-i !≖∶∅∎⊹∣⊂ 'x':xx-Fll' ,
f:.yg-I-i', z':z,-l—k', x&quot;:æ,-l-h&quot;, ...;
quibus valoribus substitutis in (a&quot;)
, attentisque aequatio-
nibus (20)
d'h dïb, dii dïi,
EMzzï—an—O, zmcïS—dtï Zm--o ∙
d']: dïk,
ZmäF—äz; Zm—o ∙
nec non aequationibus (o&quot;'), prodibunt
E(hX—-iï):2m(hää—ci £b) ∙ dr2
⋅ ∙ dq: ti*i z
(iz—mzn. (.
&amp;? −:. $) ∙
(o....)
d.,, dal. ∑≺⋌⊔∅≻⇌∑⋯≺∣≂−∂−↙⋮−−∣⋅⋮⋮↙⇆⋟∙ .176
Formulae (o &quot; ) se habent ad commune gravitatis cen
trum prorsus ut formulae ( o &quot; ) ad fixam coordinatarum x ,
J, 2, x' , ... originem 0 , respiciuntque relativum syste
matis motum quoad ipsum gravitatis centrum:

7. • * Relativus rigidi liberique systematis motus
quoad gravitatis centrum determinatur per ( o &quot; &quot; &quot; ) ; motus
vero ipsius centri per ( o' ' ' ) Ad haec : si resultans ex omni
bus viribus systemati rigido applicitis transit per gravi
tatis centrum , nullus inde orietur relativus systematis mo
tas quoad ipsum centrum : etenim quoad istiusmodi mo
tum similiter procedet res ac si resultans illa exerceretur
contra punclum fixum ( 6.&quot; ). Eadem de causa , accedentibus
novis viribus , relativus systematis motus quoad gravitatis
centrum nullo pacto turbabitur , ubi eae resultantem sup
peditent transeuntem per centrum illud.
85.&amp; Pauca subjungentes de motu rigidi systematis cir
ca axem fixum praemittimus illud: praeter orthogonales axes   &lt;math&gt;OX , OY , OZ ,&lt;/math&gt;   ( Fig. 9 ) sint alii tres axes similiter orthogonales On, Op, Oq, quibuscum ii angulos efficiant designatos per ( xn) , (xp ) , ( aq) , (yn) , ( yp ) , (99) , (zn) ,
(zp ), ( z9 ) . Si panctum E, quod referebatur ad axes OX,
OY, OZ , referendum sit ad axes On, Op , Og , quaeri
tur relatio inter veteres coordinatas x , y
nip , q. Ponatur OE = a, et per (ax ), (ay ) , (az), ( an ) ,
(ap) , ( aq) exhibeantur anguli , quos OE facit com axibus
OX , OY , OZ , On , Op , Oq: erunt ( 50. 6º . )
ma
z et novas
cos (ax) =cos (an) cos ( xn) +cos ( ap) cos (xp) +
cos (aq) cos (xq) , cos (ay ) =cos ( an ) cos (yn ) +
cos (ap) cos (yp) + cos (aq) cos (79) , cos (az)
eos ( an) cos ( zn) * cos(ap) cos (zp) + cos ( aq ) cos ( 29) .
176
Formulae (o&quot;&quot;) se habent ad commune gravitatis cen-
trum prorsus ut formulae (a&quot;) ad fixam coordinatarum æ,
y, 2, æ',
.. . originem O, respiciantque relativum syste-
matis motum quoad ipsum gravitatis centrum:
73»: Relativus rigidi liberique systematis motus
quoad gravitatis centrum determinatur per (o&quot;&quot;); motus
vero ipsius centri per (o&quot;') Ad haec: si resultans ex omni-
bns viribus systemati rigido applicitis transit per gravi-
tatis centrum
. nullus inde orietur relativus systematis mo-
tus quoad ipsum centrum : etenim quoad istiusmodi mo-
tum similiter procedet res ac si resultans illa exerceretur
contra punctum fixnm (S.&quot;). Eadem de causa , accedentibus
novis viribus
, relativus systematis motns quoad gravitatis
centrum nullo pacto turbabitur , ubi eae resultantem suppeditent transeuntem per centrum illud. '
853 Pauca subiungentes de motu rigidi systematis cir-
ca axem fixum praemittimus illud: praeter orthogonales a-
xes OX, Of, OZ (Fig. 9) sint alii tres axes similiter or-
thogonales On, Op, Oq, quibuscum ii angnlos efficiant de-
signatos Per (æ&quot;)s (æpl- (xq) :(f&quot;) , 07)» (f?) :(znls
(zp), (zq). Si punctum E, quod referebatur ad axes OX,
OV, OZ, referendum sit ad axes On, Op, Oq , quaeri-
tur relatio inter veteres coordinatas æ , y , :. et novas
n , p , q. Ponatur OE :a, et per (aæ)
,
(ay) ,(az), (an),
(ap), (aq) exhibeantur anguli, quos OE facit cum axibus
OX , Oï
, OZ , On. , Op , Oq: erunt ( 50. 60.)
cos (aæ):cos (an) cos (xn) —-[-cos (ap) cos (æp) −⊢ ⋅
cos (aq)cos (xq) , cos (ay) :cos (an) cos (yn) −∙⊢
cos (ap) cos (yp) ∙−⊢ eos (aq) cos (rq) ∙
005 (az) −−∶
eos (an) cos (zn) —,l-cos(ap)cos(zp)-—- eos (aq) cos (zq).
3751177
Sed cos (ax ) =
a , cos(ay) = cos(az ) = a
cos (an) =
, cos ( ap ) = .. cos (aq ) =
= 9
a
adhibitis igitur substitutionibus , provenient
x = ncos( an) + pcos (xp) + qcos(xq) ,
y = ncos(yn) + pcos (yp ) + acos(yq) ,
x = ncos( zn ) + pcos(zp) + qcos(zq) ;
formulae praebentes quaesitam relationem .
Nunc 1 . ** Sit OX rotationis axis, datumque systema
tis punctum reperiatur constanter in plano YOZ: si per OX
et per punctum illud ducitur planum occurrens plano YOZ,
satis erit determinare situm intersectionis istorum plano
rum ut innotescat systematis positio. Concipiamus itaque
novos axes orthogonales On , Op , Oq sic constitutos , ut
firmiter adhaereant systemati , primusque incidat in OX ,
tertius in intersectionem illam ; erunt
y= pcos(zq ) + qsiu(z9) , z =qcos (29) — psin(zg) :
adhibita substitutione in secundo membro secundae ( o &quot; .84)
animadvertendo quod variato e non ideo variant novae co
ordinatae , factoque
2 m ( p2 +9 ) = B ,
proveniet
d ' (29 )
di2 - $ (72—28 )
(o'r) :
∙∙∙⋅ 177
&amp; Sed ∾⋇≺⊄∣∙↿∶≻∶−−⊶⋚ ,cos (ay): a , c08(az):ä— .
']
...
(an):
⋮⋮−∙ costam: g.... (aq) ⇌⋅−− −↙⋅↓−
.
adhibitis igitur substitutionibus , provenient
x:ncos(æn) -l-pcoa (æp) ⊣− 9005(-qu :
)»:ncosU'n) pcos (ïp) −∣⋅− ⊄∾≘∩⊄⋟ '

: &quot;cos(zn) −⊢ pcos(zp) -I-— qcos(zq) :

famulae praebentes (quaesitam relationem.
Nune ↿∙∘∙ Sit OX rotationis axis, datumque systema-
tis punctum reperiatur constanter in plano ïOZ: si per OX
et per punctum illud ducitur planum occurrens plano ïOZ,
satis erit determinare situm intersectionis istorum plano-
rum ut innotescat systematis positio. Concipiamus itaque
novos axes orthogonales'.0n, Op , Oq sic constitutos, ut
firmiter adhaereant systemati, primusque incidat in OX ,
tertius in intersectionem illam; erunt
7: pcos(zq ) ⊣−⊄⊗∃∥≺∅⊄⋟ :
3 −−∶ quos(zq) --
psiu(zq) :
adhibita substitutione in secundo membro secundae (o&quot;.84)
animadvertendo quod variato :non ideo variant novae co-
ordinatae, factoque
. Zm(p'-l-q3)-——-B.
proveniet
(P(zq)
- dt2
z.. 1
-B— ZUZ—zï) (o&quot;):178
d (29 )
velocitas ( 50. 2º BE . ) respondet radio 1 , diciturque dla
velocitas angularis: binomia patq , p'? + 92.. nihil sunt aliud
nisi quadrata perpendiculorum ex m , m' , ... in axem On de
missorum ; summa productorum ex massis m , m' ... in
quadrata respondentium perpendiculorum , seu m (pa+92) +
m ' ( p2t 92) + . . . vocatur momentum inertiae systema
tis m , m' , .... quod axem On .
2. °# Ponamus vires acceleratrices consistere in so
la gravitate g, axesque Ox , OY jacere in horizontali pla
no: erunt Y = 0, Z
8 , et consequenter
7
de( 29 )
1 1
dla B
&amp; Emy =
1
g Em [ p cos ( zq) +qsiu ( zq ) ] E
&amp; [cos(zq) . Emp + sin ( zq) . Emq).
Fac ut illud systematis punctum , quod posuimus ( 10.)
reperiri constanter in axe Oq, sit gravitatis centrum; ex
sistent ( 20)
Σmp = p,Σm = 0 , Σmg = qΣm :
proinde
d? ( )
- 1/3
sin ( zqı ) . Em ;
dt?
B
891
quae prius multiplicata per 2d( 29, ) , ac dein integrata
praebebit
[da ] = - 69.cos/ 291). Em + c
x
178
. ∘
'
d(zq) velocitas ( 50. 2 . ..
.
) 7:2- respondet radio1,d1c1turque
velocitas angularis: binomia phi-qi, p'H—q'æ. nihil sunt aliud
nisi quadrata perpendiculorum ex m, tu',... in axem On de-
missorum ; snmma productorum ex massis m , m'
...
in
quadrata respondentium perpendiculorum, seu m (pi-I—q'H-
m' (p'ï-l- q'3)-i- ....
vocatur momentum inertiae systema-
.tis m, m',
.... quod axem On.
2.0a Ponamus vires acceleratrices consistere in so-
la gravitate g, axesque OX , Oï jacere in horizontali pla-
no: erunt T:o, Z:
-— g ,et consequenter
d3(z ) 1 ↿ ∙
de? ∶−∙−−↕≣− g Em]: —B—g2m[pcos(zq)—l-qstn(zq)]
1 - .
−−−−− -B- g[cos(zq). Zmp −↘∟ stn (zq). qu].
Fac ut illud systematis punctum ,quod posuimus (10.)
reperiri constanter in axe Oq, sit gravitatis centrum; ex-
BlStent (20) ∣
Zmp :plzm:o , qu :q12m :
proinde*
dï ↿ ∙)— B gq, sm (sq,). Em;

quae prius multiplicata per 2d( sq, )
, ac dein integrata
praebebit
. d
Z [ 2
2 .
[ld-g-l ∸∶−∙∙∙⋅ ∙∙∙ ï
gqx 008(zq1). Zm ⊹∁ ∙
iis179
Exsistentibus in initio motus d (291)
= uo et ( 291) = a , erit du
2
C = uo% + B 69 , cosa. Em :
propterea
d (290) 72
=u' . +
dt 2/3
891 [ cosa - cos ( 291) ] Em (o' ) .
Huc spectat theoria penduli compositi.
3.•* Intelligantur m , m' , m &quot; , .... coire in u
nicum punctum annexum axi horizontali Ox ope rectae r;
exsurget pendulum simplex : in casu
p = p = p = ... = 0 , q = 9 = 9 &quot; = ... = 9 = r ,
B = 2m(p + g ”) = 2n ;
et consequenter quoad pendulum simplex
d ( 292) 7
2
[Company *== +
s [ cos a - cos ( 291) ] (o &quot; ).
2
4.0# Facto 8
2
B
89. EmEm , proveniet
B
9 , £ m col) ;
longitudo videlicet penduli simplicis , quod suas perficit
oscillationes eodem tempore ac pendulum compositum . Re
cole quae diximus (67 ) .
5.° * Pone nullas esse vires acceleratrices i
erit ( 1. ° 0 ' )
179
d(zq !)
dc
Exsistentibus in initio motus :u. et (zq.):a, erit
.
2 C:u.,2 −⊦⋅ ïgq, cosa. Em:
prapterea
d(dZQtli:u⋅−⊢ ∙−⋛−∊⊄∙ [cosa — cos (zq.)]2m (a').
Huc spectat theoria penduli compositi.
39»: Intelligantur m , m', m&quot;, ... . . coire in u-
nicum punctum annexum axi horizontali OX Ope rectae r;
exsurget pendulum simplex: in casu
p::p'::p&quot;: ∙ ∙ ∙ −−∙−−∶∘ , qzq'2q&quot;: ∙ ∙ ∙
∶⊄∎∶↿∙ ,
&quot;B::ZmQF-l-qa) claim ;
et consequenter quoad pendulum simplex
↙≀≺≦≦∣≖⋝⊺−−⋅↙∘≖ ; f ,.
(....
...
(..., ].
2 2
4.0a Facto ∙∓− g :ïgq, Em
, proveniet
B .
r'.&quot;—∙−∙∙ q,2m
Om) ;
longitudo videlicet penduli simplicis, quod suas perficit
oscillationes eodem tempore ac pendulum compositum. Re-
cole quae diximus (67).
5.
.,
Pone nullas esse vires acceleratrices: ,
erit (1. ∘ o)180
dº(aq )
dia
d( 24)
unde velocitas angolaris u =
dc = const. = u , .
1
1 Motus igitur exsistet uniformis , eritque velocitas angu
laris ad velocitatem puncti v . gr. m ut 1 ad radium cir
culi descripti ab ipso m , seu
1
u : v =1 : V patqz , ac proinde v = u ? (patoga )
quoad illud itaque punctum obtinebit vis contrifuga expres
sa ( 51 ) per
= u’m V pat92 .
V pr + q2
1
vam 2
Resolvatur haec vis in ternas coordinatis axibus On, Op,
Og parallelas ; prodibunt 1
+
9
р
0 , u²mV p2tga .
V p²ta?
wimb p'tgo.
Foto &gt;
seu
0
0 , ump , u'mg :
1
quoad totum ergo systema habebuntur 2
0 , użEmp , u’Emq ;
ideoque orietur pressio in axem OX. Prima membra formu
larum ( a : 13. 8.° ) in casu fiunt
0 , użEmp, użEmq , użEmnp , użEmng , u’Em (pa - pa ) :
!
hinc ubi fuerint 1
1
Emp= 0 , Emg = 0 , Emnp = 0, Emng = 0 ( o'r) ,
180
(l*(z'q)
d?
d(zq)
dc

: o

, unde velocitas angularis ::: :const.-zuo,
Motus igitur exsistet uniformis , eritque velocitas angu-
laris ad velocitatem puncti v. gr. m ut 1 ad radium cir-
culi descripti ab ipso m, seu
∣ ...—....
a: p −−−−−↿ :Vpl-I—qa , ac proinde V::u' (pH-q2 )
quoad illud itaque punctum obtinebit vis centrifuga expres-
sa (51) per
vam
l/P'&quot;l'qa −−∶
&quot;&quot;&quot; Vlf-*?&quot;
-
Resolvatur haec vis in ternas&quot; coordinatis axibus On. Op,
Oq parallelas; prodibunt
seu
0, uïrnp
,
'u'mq :
quoad totum ergo systema habebuntur
o , u'Zmp , u'qu;
ideoque orietur pressio in axem OX. Prima membra formu-
larum (a'm : 13. 8.&quot;) in casu fiunt
o , ti*Zmp, uazmq , u'Zmnp , u'Zmnq , u'Zmþq—pq) :
hinc ubi fuerint
Zmp:o , M.,—:a, Zmnpzo, zmnqzo (atur)
'181
1
vires centrifugae se muluo librabunt independenter ab axe
Ox
, nullamque iste axis patietur pressionem
. Prima et se
cunda
(oh
) important
( 20. 6.
) transitum axeos On seu
OX per gravitatis centrum tertia vero et quarta important peculiarem quandam axiuin On
, Op, Oq positionem
relate ad punctorum
m
,
m
'
,
m
&quot; systema
. Porro si On
,
Op
, Oq ita sunt positi, ut suppeditent Emnp
=
0
, Emng
=
0
,
Empq
=
o
, appellari solent principales systematis axes in
ordine ad originem
itidem quae momenta ad eos referuntur
, et ipsa dicuntur principalia inertiae momenta
. Ex
pletis tertia et quarta
(
o
&quot;
&quot;&quot;
)
, non autem prima et secun
da
, ex omnibus viribus centrifugis resultabit
( 13. 9.0 10.9
)
vis premens rolationis axem in O. 6. '* Superiores formulae manifeste applicantur
continuo materialium punctorum systemati mutando
&amp; in
ſ et
m in dm
, integrationemque protendendo ad totam
systematis massam
.
7.9 Saepe videmus corpora impulsu aliquo loca
liter mota affici simul rotationis motu

: etiam praecisis

,
quae diximus
( 84
)
, sic ostendi potest motum istum oriri ex
eo quod impulsus non habeat directionem transeuntem per
centra gravitatis corporum
. Sic
G gravitatis centrum cor
poris MM
'
( Fig
. 46
)
, et AZ vis corpori cominunicata
..
Ducatur per
G ad AZL perpendiculum GL dividalur

bifariam AZ in
C
, et resolvatur CA in AD per
G tran
seantem
, et in AB normalem rectae AZ producatur AG

donec GF aequet GA intelligatur AD applicita ad punclum F , sitque FK = AD resolvatur FK in FH parallelam et FI perpendicularem rectae LGN

: quibus posi

tis
, substituti poteront vi AZ quatuor vires CZ
, AB
, FI
,
FH
. Jamvero CZ
, FI utpote aequales
, parallelae et ad
eamdem plagam tendentes contrahuntur in unam reprae
sentatam per GE
( 11
)
=GZ
+Fl
=AZ
, transeuntem per
G
, eidemque AZ parallelam proinde movebitur centrum

G non secus ac vis AZ ipsi esset applicata
. At duae aliae
↿∂⋅↿
vires centrifugae se mutuo librabunt independenter ab axe
OX, nullamque iste axis patietur pressionem. Prima et se-
cunda (o&quot;&quot;) important (20. b.) transitum axeos On seu
OX per gravitatis centrum: tertia vero et quarta impor-
tant peculiarem quandam axium On
, Op, Oq positionem
relate. ad punctorum m, m', m&quot; ,... systema. Porro si On,
Op, Oq ita sunt positi, ut suppeditent Zmnp:o, Zmnq:o,
Zmpq:o, appellari solent principales systematis axes in
ordine ad originem
O

itidem quae momenta ad eos refe-
runtur, et ipsa dicuntur principalia inertiae momenta. Expletis tertia et quarta (on&quot;), non autem prima et secunda, ex omnibus viribus centrifugis resultabit (13. 9310!)
vis premens rotationis axem in O.

63. Superiores formulae manifeste applicantur
continuo materialium punctorum systemati mutando
2 in
]et as in dm
,
integrationemque protendendo ad totam
systematis massam.

7.&quot; Saepe videmus corpora impulsu aliquo loca-
liter mota aflici simul rotationis motu: etiam praecisis,
quae diximus (84), sic ostendi potest motum istum oriri ex
eo quod impulsus non habeat directionem transeuntem per
centra gravitatis corporum. Sit
G gravitatis centrum cor-
poris MM' (Fig. 46)
, et AZ vis corpori communicata.
Ducatur per
G ad AZL perpendiculum GL; dividatur
bifariam AZ in C, et resolvatur CA in AD per
G tran-
seuntem, et in AB normalem rectae AZ; producatur AG
donec GF aequet GA; intelligatur AD applicita ad pun-
ctum
F
, sitque FK: A

resolvatur FK in FH paral-
lelam et FI perpendicularem rectae LGN

: quibus posi-

tis, substituti poterunt vi AZ quatuor vires CZ,AB, FI
,
FH. Iamvero CZ, FI utpote aequales
, parallelae et ad
eamdem plagam tendentes contrahuntur in unam reprae-
sentatam per GE (11):GZ—-FI:AZ, transeuntem per
G, eidemque AZ parallelam proinde movebitur centrum

0 non secus ac vis AZ ipsi esset applicata. At duae aliae182
.AB, FH utpote aequales , parallelae , et ad contrarias par-
tes tendentes , nequeunt gravitatis centrum e suo loco di-
movere : spectatis itaqne istiusmodi viribus, immobile eqn-
sisteret gravitatis centrum; sed eae sese mutuo non de-
struunt, cum e diametro non opponantur. Aliud ergo praestare non poterunt nisi corporis rotationem circa gravitatis centrum. Rotationis motus incipit circa reetam aliquam seu axem, et quoniam in omnes corporis particulas ex rotatione inducitur vis centrifuga; hinc si vires centrifugae inde ortae aequilibrantur circa rectam illam, invariabilis exsistet rotationis axis, defereturque per spatium sibimet semper parallelas; secus, mutabitur indesinenter rotationis axis donec ad aequilibrium deveniatur.

=== De fluidorum corporum aequilibrio. ===
86. Fluida corpora spectamus veluti materialiam punctorum congeries; quae puncta, utpote invicem independentia, vel minimo cedunt impulsui. In massa fluida undique librata sume punctum quodvis [exhibemus per &lt;math&gt;( x , y , z )&lt;/math&gt;, denotantibus &lt;math&gt;x, y, z&lt;/math&gt; ejus coordinatas] sollicitatum vi acceleratrice &lt;math&gt;\varphi&lt;/math&gt; praebente componentes &lt;math&gt;X, Y, Z&lt;/math&gt; coordinatis axibus &lt;math&gt;OX, OY, OZ&lt;/math&gt;,  parallelas et per punctum illud fac ut transeat superficies &lt;math&gt;k&lt;/math&gt; plana, rigida atque infinitesima: consistet &lt;math&gt;k&lt;/math&gt; in aequilibrio; et consequenter pressiones hinc et illinc exercitae in &lt;math&gt;k&lt;/math&gt; ab circumpositis massae fluidae stratis, erunt vires aequales et directe contrariae, simulque normales ipsi &lt;math&gt;k&lt;/math&gt;. Ejusmodi pressionum alteram repraesenta per &lt;math&gt;\varpi k&lt;/math&gt;; ratio &lt;math&gt;\frac{\varpi k}{k}(= \varpi)&lt;/math&gt; dicitur pressio hydrostatica exercita &lt;math&gt;k&lt;/math&gt; apud punctum &lt;math&gt;( x, y , z )&lt;/math&gt; contra aream ( = 1 ) sumptam in plano superficiei &lt;math&gt;k&lt;/math&gt;. In eadem massa fluida fac ut per punctum alterum &lt;math&gt;( x_0, y , z )&lt;/math&gt; transeat talis superficies &lt;math&gt;k_0&lt;/math&gt; plana, rigida et infinitesima, quae communem habeat projectionem cum superficie &lt;math&gt;k&lt;/math&gt; in plano &lt;math&gt;YOZ&lt;/math&gt;; voca &lt;math&gt;h&lt;/math&gt; projectionem illam, et &lt;math&gt;\varpi_0&lt;/math&gt;, hydrostaticam pressionem apud punctum &lt;math&gt;(x_0, y , z)&lt;/math&gt; contra aream ( =1 ) sumptam in plano areae &lt;math&gt;k_0&lt;/math&gt;. Massa fluida adhuc perget esse librata, etsi in qualibet ejus portione intelliguntur puncta rigidis lineolis firmiter connecti, seu, quod eodem redit, etsi quaelibet ejus portio fit solida: ponatur id contingere portioni cylindricae habenti rectam parallelam axi &lt;math&gt;OX&lt;/math&gt; pro generatrice, et &lt;math&gt;k , k_0&lt;/math&gt; pro basibus; denotet &lt;math&gt;\mu&lt;/math&gt; densitatem massae fluidae apud punctum &lt;math&gt;(x , y , z)&lt;/math&gt;; sitque &lt;math&gt;x &gt; x_0&lt;/math&gt; Exprimetur per&lt;math display=&quot;block&quot;&gt;h\int_{x_0}^x \mu X dx
&lt;/math&gt;summa ex viribus motricibus, quibus juxta &lt;math&gt;OX&lt;/math&gt; sollicitantur puncta illius portionis; exprimenlur praeterea per  &lt;math display=&quot;block&quot;&gt;\frac{h}{k_0}\varpi k_0, -\frac{h}{k}\varpi k &lt;/math&gt;pressiones exercitae juxta eumdem OX , altera in basim ko,altera in basim k quod spectat ad pressiones contra lateralis superficiei puncta, eae utpote normales generatrici rectae nullas dabunt componentes axi OX parallelas. Quia igitur solidata portio perseverat in aequilibrio, iccirco

&lt;math display=&quot;block&quot;&gt;h\int_{x_0}^x \mu X dx + h\varpi_0 - h\varpi = 0, \, \mathrm{unde}\,   \varpi = \varpi_0 + \int_{x_0}^x \mu X dx .
&lt;/math&gt;

Haud mutata positione superficiei &lt;math&gt;k_0&lt;/math&gt;, revolvatur utcumque superficies &lt;math&gt;k&lt;/math&gt; circa punctum &lt;math&gt;(x , y ,z)&lt;/math&gt;: permanebit secundum membrum ultimae aequationis; ergo et primum. Quare perseverabit in eodem valore hydrostatica pressio quoad omnia plana per punctum illud utcumque ducta: huc spectat principium de aequalitate pressionis. Consequitur, si recta generatrix sumitur parallela, prius axi &lt;math&gt;OY&lt;/math&gt;, deinde axi &lt;math&gt;OZ&lt;/math&gt;, denotantibus &lt;math&gt;\varpi_0',\varpi_0''&lt;/math&gt; hydrostaticas pressiones apud puncta &lt;math&gt;( x , y_0, z) , (x , y , z_0 )&lt;/math&gt;, fore etiam&lt;math display=&quot;block&quot;&gt;
\varpi = \varpi_0' + \int_{y_0}^y \mu Y dy , \varpi  = \varpi_0''+ \int_{z_0}^z \mu Z dz &lt;/math&gt;Terni valores &lt;math&gt;\varphi&lt;/math&gt; differentiati, primus quoad &lt;math&gt;x&lt;/math&gt;,  secundus quoad &lt;math&gt;y&lt;/math&gt;, tertius quoad &lt;math&gt;z&lt;/math&gt;, praebent
&lt;math display=&quot;block&quot;&gt;
\frac{d\varpi}{dx} = \mu X,  \frac{d\varpi}{dy} = \mu Y, \frac{d\varpi}{dz} = \mu Z. (o) &lt;/math&gt;et consequenter (27.24º)
&lt;math display=&quot;block&quot;&gt;
d\varpi = \mu ( Xdx + Ydy + Zdz). ( o' ) &lt;/math&gt; Itaque conditiones requisitae ad massae fluidae aequilibrium eo redeunt ut exsistat ejusmodi functio &lt;math&gt;\varpi&lt;/math&gt; variabilium &lt;math&gt;x, y, z&lt;/math&gt;, quae expleat sive ternas (o), sive unicam (o').

87. Haec notentur. 

1º. Si fluidum continetur vase undique clauso satisque firmo, utcumque se habeat valor &lt;math&gt;
\varpi &lt;/math&gt; ex (o') quoad superficiem fluidi, is constanter aequivalebit reactioni ex vasis lateribus: at si fluidi superficies sit libera, externisque subjecta pressionibus, ad aequilibrium explenda insuper erit (o') per talem valorem &lt;math&gt;
\varpi &lt;/math&gt;, qui in singulis liberae superficiei punctis aequivaleat respondenti pressioni externae.

2º. Hinc si pressio externa vel ponitur &lt;math&gt;=0&lt;/math&gt; vel ubique eadem, erit &lt;math&gt;d\varpi = 0&lt;/math&gt; quoad superficiem fluidi librati, ideoque
&lt;math display=&quot;block&quot;&gt;Xdx + Ydy + Zdz  = O (o''). &lt;/math&gt;

3º. Traduci potest (o&quot;) ad&lt;math display=&quot;block&quot;&gt;\frac{X}{\varphi} \frac{dx}{ds}+\frac{Y }{\varphi} \frac{dy}{ds} + \frac{Z}{\varphi}\frac{dz}{ds} = 0&lt;/math&gt;exprimunt &lt;math&gt;X/\varphi, Y/\varphi, Z/\varphi&lt;/math&gt; cosinus angulorum, quos efficit vis acceleratrix &lt;math&gt;\varphi&lt;/math&gt; cum axibus coordinatis &lt;math&gt;OX, OY, OZ&lt;/math&gt;; denotant &lt;math&gt;\frac{dx}{ds}, \frac{dy}{ds},\frac{dz}{ds}&lt;/math&gt; cosinus angulorum, quos recta tangens arcum &lt;math&gt;s&lt;/math&gt; apud ejus extremum facit cum iisdem axibus: inferimus (50. 6.) vim &lt;math&gt;\varphi&lt;/math&gt; intercipere angulum = 90° cum rectis omnibus tangentibus ubivis superficiem vel nullo pacto, vel aeque pressam; ac proinde &lt;math&gt;\varphi&lt;/math&gt; sese dirigere normaliter ad istiusmodi superficiem.

4.º Integrata (o&quot;), si constanti arbitrariaeque quantitati tribuuntur alii atque alii valores, emergent aliae atque aliae aequationes, quibus totidem respondebunt distinctae superficies aeque pressae.

5.°* In hypothesi &lt;math&gt;\varphi&lt;/math&gt; tendentis ad punctum fixum, constitue ibi coordinatarum originem: denotante &lt;math&gt;D&lt;/math&gt; distantiam inter punctum illud et &lt;math&gt;( x , y , z)&lt;/math&gt;, erunt (50. 6º)&lt;math display=&quot;block&quot;&gt;X = -\varphi \frac x D, Y= -\varphi \frac y D, Z= -\varphi \frac z D&lt;/math&gt;hinc&lt;math display=&quot;block&quot;&gt;X dx + Ydy + Zdr = - \frac \varphi D (xdx + ydy + zdz).&lt;/math&gt;Est insuper &lt;math&gt;x^2 + y^2 + z^2 = D^2 &lt;/math&gt;, unde &lt;math&gt;xdx + ydy + zdz = DdD;&lt;/math&gt; et consequenter&lt;math display=&quot;block&quot;&gt;Xdx +Ydy + Zdz = -\varphi dD.&lt;/math&gt;In ordine igitur ad superficiem aeque pressam exsistet &lt;math&gt;dD = 0&lt;/math&gt;: propterea &lt;math&gt;D = C&lt;/math&gt;; ex qua &lt;math&gt;x^2 + y^2 + z^2 = C^2 &lt;/math&gt;: massa videlicet fluida atque librata induet sphaericam formam.


6. Quoad fluidum elasticitate pollens, constat experimentis densitatem &lt;math&gt;\mu&lt;/math&gt;, permanente temperie, esse proportionalem respondenti pressioni &lt;math&gt;
\varpi &lt;/math&gt;, nimirum&lt;math display=&quot;block&quot;&gt;\mu = \theta \varpi: (o''')&lt;/math&gt;Eliminata  &lt;math&gt;\mu&lt;/math&gt; ab (o') et (o''&quot;''), proveniet&lt;math display=&quot;block&quot;&gt;\frac{d\varpi}{\varpi}=\theta(Xdx + Ydy + Zdz);&lt;/math&gt;et facto &lt;math&gt;Xdx + Ydy + Zdz = df (x,y,z)&lt;/math&gt;, erit:&lt;math display=&quot;block&quot;&gt;\ln \varpi = \int \theta df + \ln C = \ln (e^{\int \theta df}) + \ln C= \ln (C e^{\int \theta df})&lt;/math&gt;hinc&lt;math display=&quot;block&quot;&gt;\varpi = C e^{\int \theta df}, \mu = C \theta e^{\int \theta df}&lt;/math&gt;coefficiens &lt;math&gt;\theta&lt;/math&gt; pendet a temperie vigente apud &lt;math&gt;(x , y , z)&lt;/math&gt;. Inferimus aequilibrii statum in fluido elastico importare temperiem vel ubique eamdem, vel talem ut sit functio quantitatis &lt;math&gt;f&lt;/math&gt;. Haec insuper quantitas est (2º, 4º) constans in unaquaque superficie aeque pressa; idipsum ergo dicendum de temperie. 


7.º Constat etiam experimentis fluidum elasticitate pollens ita contrahi vel expandi, imminuta vel aucta temperie ac permanente pressione &lt;math&gt;
\varpi' &lt;/math&gt; ut ejus volumen &lt;math&gt;
V &lt;/math&gt;minuatur vel augeatur partibus 0,00375 pro singulis gradibus thermometri centigradi; inde fit, ut posito 0,00375 = &lt;math&gt;
a &lt;/math&gt;, et aucta temperie gradibus &lt;math&gt;n&lt;/math&gt; ultra &lt;math&gt;0^\circ \mathrm{C}&lt;/math&gt; , volumen  &lt;math&gt;V&lt;/math&gt; evadet &lt;math&gt;V ( 1 + an )&lt;/math&gt;; propterea, designantibus  &lt;math&gt;\mu_0&lt;/math&gt;  et   &lt;math&gt;\mu_1&lt;/math&gt;  respondentes densitates, erit &lt;math&gt;\frac{\mu_1}{\mu_0}=\frac{1}{1+an}.&lt;/math&gt; Nunc, permanente temperie &lt;math&gt;n&lt;/math&gt;, crescat pressio ab &lt;math&gt;
\varpi' &lt;/math&gt; ad  &lt;math&gt;
\varpi &lt;/math&gt;;  denotante &lt;math&gt;\mu&lt;/math&gt;  respondentem densitatem, erit  (1º) &lt;math&gt;\frac{\varpi}{\varpi'}=\frac{\mu}{\mu_1},&lt;/math&gt; quocirca&lt;math display=&quot;block&quot;&gt;\varpi = \frac{\varpi'\mu}{\mu_1} =  \frac{\varpi'}{\mu_0} \mu ( 1 + an )&lt;/math&gt;; et facto &lt;math&gt; \frac{\varpi'}{\mu_0} =i&lt;/math&gt;, &lt;math&gt;\varpi = i \mu ( 1 + an ) (o^{(iv)})&lt;/math&gt;.

=== De gravium homogeneorumque liquidorum aequilibrio. ===
88. Planum &lt;math&gt;XOY&lt;/math&gt; sit horizontale, axisque &lt;math&gt;OZ&lt;/math&gt; (Fig. 47) vergat deorsum juxta directionem gravitatis &lt;math&gt;g&lt;/math&gt;; erunt &lt;math&gt;X=0, Y=0, Z = g&lt;/math&gt;: proinde (86. 6), &lt;math display=&quot;block&quot;&gt;d\varpi = g \mu dz ( 0^{v} )&lt;/math&gt;Si pressio externa ponitur vel = 0, vel ubique eadem, erit &lt;math&gt;d\varpi = 0&lt;/math&gt; quoad librati fluidi superficiem, ideoque &lt;math&gt;dz = 0&lt;/math&gt;, et &lt;math&gt;z = Const&lt;/math&gt;: superficies nempe illa existet plana atque horizontalis. 

Pone &lt;math&gt;\mu&lt;/math&gt;  constantem; ex (0&lt;sup&gt;v&lt;/sup&gt;) habebis

&lt;math&gt;\varpi = g \mu z + C_1&lt;/math&gt;,

In fluidi superficie aeque pressa constitue planum horizontale &lt;math&gt;XOY&lt;/math&gt;: quoad eam erit &lt;math&gt;z = 0&lt;/math&gt;; nihilque aliud denotabit &lt;math&gt;C_1&lt;/math&gt; nisi externam pressionem in aream ( = 1 ) quaquaversus per fluidum aequaliter diffusam.

Haec facile nunc stabiliuntur circa pressiones gravium homogeneorumque liquidorum intra vasa in aequilibrio consistentium. 
[[Fasciculus:Hydrostatic-pressure.svg|thumb]]
1º. Si per &lt;math&gt;\Pi&lt;/math&gt; designatur pressio in horizontalem aream &lt;math&gt;A&lt;/math&gt; demersam ad profunditatem &lt;math&gt;z&lt;/math&gt;, exsistet &lt;math&gt;\Pi = A \varpi = A (g\mu z + C_1 ) .&lt;/math&gt;

2º. Si &lt;math&gt;C_1 = 0&lt;/math&gt;, aequivalebit &lt;math&gt;\Pi&lt;/math&gt; ponderi prismatis, cujus basis est &lt;math&gt;A&lt;/math&gt;, altitudo &lt;math&gt;z&lt;/math&gt;, densitas vero eadem ac densitas liquidi.

3º. Exhibente &lt;math&gt;A&lt;/math&gt; horizontalem vasis fundum, ideoque &lt;math&gt;z&lt;/math&gt; altitudinem vasis; quoniam &lt;math&gt;\Pi&lt;/math&gt; nullatenus pendet a vasis figura, iccirco permanentibus &lt;math&gt;A&lt;/math&gt; et eadem perstabit liquidi pressio in horizontalem fundum, utcumque de caetero varient figura et capacitas vasis.

4º. Area &lt;math&gt;A&lt;/math&gt; sit oblique intra liquidum utcumque demersa: divide &lt;math&gt;A&lt;/math&gt; in areolas infinitesimas &lt;math&gt;a , a ', a'' &lt;/math&gt; quarum distantiae ab extima liquidi superficie designentur per &lt;math&gt;z' , z''...;&lt;/math&gt; denotante &lt;math&gt;\Pi'&lt;/math&gt; totalem pressionem, et &lt;math&gt;z_1&lt;/math&gt; perpendiculum ductum ex centro gravitatis areae &lt;math&gt;A&lt;/math&gt; in planum &lt;math&gt;XOY&lt;/math&gt;; erit (20) 

&lt;math&gt;\Pi' = a(g\mu z + C_1) + a'(g\mu z'+ C_1) +... = g\mu(az + a'z' + ...) + C_1( a + a' + ... ) = g\mu z_1 A + C_1 A = A (g\mu z_1 + C_1 )&lt;/math&gt;. 

Hinc si centrum gravitatis manet ad eamdem profunditatem demersum, haud variabit  &lt;math&gt;\Pi'&lt;/math&gt;, utcumque circa illud revolvatur area demersa: potest A repraesentare quamlibet rectilineam portionem internae superficiei vasis.

Ad haec: coordinatae ( 13. 3º. )

&lt;math&gt;b=\frac{\sum ax (g\mu z + C_1)}{ \sum a(g\mu z + C_1)}; b' = \frac{\sum ay (g\mu z + C_1)}{ \sum a(g\mu z + C_1)}, b'' = \frac{\sum az (g\mu z + C_1)}{ \sum a(g\mu z + C_1)} &lt;/math&gt; seu (20) 

&lt;math&gt;b =   \frac{g\mu \sum ax z + C_1 A x_1 }{ A(g\mu z_1 + C_1) }; 
b' =  \frac{g\mu \sum ay z + C_1 A y_1 }{  A(g\mu z_1 + C_1)}, 
b'' = \frac{g\mu \sum a  z^2 + C_1 A z_1 }{ A(g\mu z_1 + C_1)} &lt;/math&gt;

respondent illi puncto areae &lt;math&gt;A&lt;/math&gt;, per quod transit resultans ex parallelis viribus &lt;math&gt;a(g\mu z + C_1), a'(g\mu z'+ C_1), a''(g\mu z''+ C_1)...&lt;/math&gt;; istiusmodi punctum dicitur centrum pressionis.
[[Fasciculus:PolydirectionalPressure.svg|thumb]]
5º . Veniat considerandum solidum liquido immersum: sume apud punctum &lt;math&gt;( x , y , z )&lt;/math&gt; in solidi superficie areolam infinitesimam &lt;math&gt;k&lt;/math&gt; , et apud puncta &lt;math&gt;( x_0, y, z ) , (x , y_0, z ) , ( x , y , z_0 )&lt;/math&gt;in eadem solidi superficie areolae &lt;math&gt;k_0, k'_0, k''_0&lt;/math&gt;, sitque &lt;math&gt;h&lt;/math&gt; projectio areolae &lt;math&gt;k_0&lt;/math&gt; in plano &lt;math&gt;YOZ&lt;/math&gt;, &lt;math&gt;h'&lt;/math&gt; projectio areolae &lt;math&gt;k'_0&lt;/math&gt; in plano &lt;math&gt;XOZ, h''&lt;/math&gt;projectio areolae &lt;math&gt;k''_0&lt;/math&gt; in plano &lt;math&gt;XOY&lt;/math&gt;; congruant vero &lt;math&gt;h, h' , h''&lt;/math&gt; cum projectionibus areolae &lt;math&gt;k&lt;/math&gt; in iisdem planis: per &lt;math&gt;k(g\mu z + C_1), k_0(g\mu z + C_1),k'_0(g\mu z + C_1), k''_0(g\mu z + C_1),&lt;/math&gt;exprimentur pressiones normaliter exercitae in areolas  &lt;math&gt;k, k_0, k'_0, k''_0&lt;/math&gt;; ejusmodi pressionum prima resolvitur in &lt;math&gt;\frac{h}{k}\cdot k(g\mu z + C_1), \frac{h'}{k}\cdot k(g\mu z + C_1),\frac{h''}{k}\cdot k(g\mu z + C_1),&lt;/math&gt;&lt;ref&gt;Figura deest ergo clare non est si aequatio est recte stripta &lt;/ref&gt; parallelas rectis &lt;math&gt;OX , OY , OZ&lt;/math&gt;; secunda praebet componentem

&lt;math&gt;-\frac{h}{k_0}\cdot k_0(g\mu z + C_1)&lt;/math&gt;

parallelam rectae OX, tertia dat componentem

&lt;math&gt;-\frac{h'}{k'_0}\cdot k'_0(g\mu z + C_1)&lt;/math&gt;

parallelam rectae OY; quarta suppeditat componentem

&lt;math&gt;-\frac{h''}{k''_0}\cdot k''_0(g\mu z_0 + C_1),&lt;/math&gt;

parallelam rectae &lt;math&gt;OZ&lt;/math&gt;. His positis, quisque videt areolam &lt;math&gt;k&lt;/math&gt;, elisis componentibus horizontalibus, urgeri sursum verticali pressione&lt;math display=&quot;block&quot;&gt;h'' g \mu ( z - z_0 )&lt;/math&gt;totum igitur demersum solidum ad verticalem ascensum sollicitatur parallelis viribus praebentibus resultantem, quae aequivalet ponderi liquidi expulsi, et transit per punctum illud, ubi erat gravitatis centrum ipsius liquidi expulsi. Itaque si &lt;math&gt;V'&lt;/math&gt; et &lt;math&gt;\mu'&lt;/math&gt; exhibent volumen et densitatem solidi liquido immersi, &lt;math&gt;V&lt;/math&gt; volumen liquidi espulsi; pondus, quod superest solido, exprimelur per &lt;math&gt;g( V'\mu' - V\mu )&lt;/math&gt;: in solidis heterogeneis designat &lt;math&gt;\mu'&lt;/math&gt; densitatem mediam.

89. Sit 1º &lt;math&gt;\mu' &gt; \mu &lt;/math&gt; cum nequeat esse &lt;math&gt;V &gt; V '&lt;/math&gt;, erit semper &lt;math&gt;V'\mu' - V\mu &gt;0&lt;/math&gt;; tamdiu igitur descendet solidum, ubicumque in liquido collocetur, donec aliquod offendat obstaculum, cui adstringatur adhaerere. Si collocatur in liquidi superficie; statim atque totum fuerit demersum, exsistet &lt;math&gt;V = V';&lt;/math&gt; et consequenter perget solidum moveri vi acceleratrice &lt;math&gt;\frac{gV ' ( \mu' - \mu )}{V'\mu'}&lt;/math&gt; seu &lt;math&gt;g\left( 1 - \frac{\mu }{\mu'}\right)&lt;/math&gt;

Ab exploratis solidi ponderibus P et P' in vacuo et in li
quido elici potest ratio inter u et l ; siquidem
P = gV ' ', P = 8 ! V' ' — Vp ) , et V = V :
propterea
P í M
Hop
unde р
P
P - P
[[Fasciculus:EB1911 Hydromechanics - Fig. 3.jpg|thumb]]
Sit 2º. M '= H: tamdiu V'u ' - Vl &gt; o quamdiu &lt;math&gt;V' &gt; V&lt;/math&gt;; solidum nempe collocatum in superficie liquidi eo usque descendet, donec totum demergatur; quod ubi contigerit, evanescente V' M' — Vp , consisteret in aequilibrio nisi urgeretur adhuc vi acquisita descendendo ante et aequivalet ponderi liquidi expulsi, et transit per punctum illud, ubi erat gravitatis centrum ipsius liquidi expulsi. ltaque si V'et p! exhibent volumen et' densitatem
solidi liquido immersi, V volumen liquidi expulsi; pondus,
quod superest solido, exprimetur per
g( V'p.'—Vp.) :
in solidis heterogeneis designat p! densitatem mediam. ⋅
89. Sit. 1041!) p.: cum nequeat esseV) V', erit sem-
per V' pf ∙−− Vp.) o; tamdiu igitur descendet solidum, ubi-'
cumque in liquido collocetur, donec aliquod offendat ob-
staculum
, cui adstringatur adhaerere. Si collocatur in li-
quidi superficie; statim atque totum fuerit demersum, ex-
sistet V:V'; et consequenter perget solidum moveri vi
acceleratrice '
sv. ∣≺⊮∸⋮⋅⋅−⋅∟∸≻ ∘ −.r. v'F-I , .seu :,(1 l*') .
Ab exploratis solidi ponderibus P et'P' in vacuo et in li-
quido elici potest ratio inter p! et p.; siquidem
≖∙⊃−∙−⇀−∊⋁∙⊬↼∙∙ P',—.: g( vir—v,. ), .xv.-: V':
prOpterea
.
P p! p!— P
P' −−−⊬∙∙⊬∙ uude F- P-P'
.
Sit 20. pl: p.: tandiu V'pf -— VP) o quamdiu
V&quot; V ; solidum nempe collocatum in superficie liquidi
eo .usque descendet,, donec totum demergatur; quod ubi
contigerit, evanescentev p! —Vp. , consisteret in aequi-,-
librio nisi urgeretur adbuc vi acquisita descendendo ante192
1
V'de VM
1
1
totalem immersionem ; ad aequilibrium praeterea deberent
gravitatis centra solidi et liquidi expulsi esse in eadem re
cia verticali,

Sit 3º. p &lt; l tandiu . Vil – Ve &lt; o quandiu
V &gt; ; et facto V , erit Vų – VH = 0.
Solidum igitur collocatum intra liquidum ascendet ad li
quidi superficiem ; situm in ipsa superficie supernatabit ;
eritque portio demersa V ad volumen integrum V' ut j ': fl.
Innatantis solidi aequilibrium requirii insuper ut in eadem
recta verticali inveniantur gravitatis centra ipsius solidi et
liquidi quod expellitur.
Itaque positio aequilibrii quoad solidum homogeneum
liquido insideas determinabitur si plano ita secetur soli
dum, ut et alterius segmenti volumen sit ad solidi volu
men ia data ratione pe': fhy et haec volumina habeant sua
gravitatis centra in eadem recta , quae normaliter insistat
plano secanti: rem declaramus exemplo. Determinanda sit
positio aequilibrii in prismate recto ac triangulari , quod
ita demergitur ut et ejus bases maneant verticales, et u
na ex tribus faciebns v. g. BC ( Fig 48 ) exsistat cota ex
tra liquidum.
Quisque videt directionem plani secantis non pende
re a mutua basium distantia, satisque esse ut determine
tur intersectio De illius plani et baseos v . g. ABC. Exhi.
beant a ', a“ latera AB, AC dati trianguli ABC , et a', w &quot;
latera incognita AD, AE crianguli ADE : triangulares areae
ABC, ADE exprimentur per
3
i a'a ' sin A , Law&quot; sin A.
Sed area ABC est ad aream ADE ut integrum prisma ad
prismaticam portionem demersam ; igitur
le
IWW 'sin A: į a' a &quot; sin A = fe':J.,Was&quot;
P.
-a'a' ( k) .
192
totalem immersionem; ad aequilibrium praeterea deberent
gravitatis centra solidi et liquidi expulsi esse in eadem re-
cta verticali. ⋅
'
Sit 3&quot;. p! p. : tandiu. V'pl ∙− Vp.( o quandiu
V P- ;et factoV:V V) P- ,eritV'pf—Vp.:o.
Solidum igitur collocatum intra liquidum ascendet ad li-
quidi superficiem; situm in ipsa superficie superuatabit;
eritque portio demersa V ad volumen integrum V' ut pf: p..
Iunatantis solidi aequilibrium requirit insuper utin eadem
recta verticali inveniantur gravitatis centra ipsius solidi et
liquidi quod expellitur.
ltaque positio aequilibrii quoad solidum homogeneum
liquido insidens determinabitur si plano ita secetur soli-
dum, ut et alterius segmeuti volumen sit ad solidi volu-
men iu data ratione an., et haec volumina habeant sua
gravitatis centra in eadem recta, quae normaliter insistat
plano secanti: rem declaramus exemplo. Determinauda sit
positio aequilibrii in prismate recto ac triangulari, quod
ita demergitur ut et eius bases maneant verticales, et u- ⋅
na ex tribus faciebus v. g. BC ( Fig 48) exsistat tota ex—
tra liquidum.
Quisque videt directionem plani secantis nou pende-
re a mutua basium distantia, satisque esse ut determine-
tur intersectio DE illius plani et baseos v. g. ABC. Exhi-
beant a', a&quot; latera AB. AC dati trianguli ABC, et m', a)&quot;
latera incognita AD, AE trianguli ADE :triangulares areae
ABC, ADE exprimeutur per
∙∙⋅∙ äaa smA I &quot;-
, ämæstu A.
Sed area ABC est ad aream ADE ut integrum prisma ad
prismaticam portionem demersam: igitur
P:.
in' d'siu A: ;a' a&quot; sin A∶∶∶ [1]: .n., 'n' a&quot;:—-—a'a&quot; (k) .193

pla AM
Ž
AH '
Nunc secto bifariam in H latere BC, ducatur AH; sum
2
3
AH , centrum gravitatis trianguli ABC e
rit in M: simili modo, secto bifariam in H ' latere DE, sum
2
ptaque AN = AH', erit N centrum gravitatis trianguli 3
AM AN
ADE. Quia igitur ideo MN et HH' erunt АН
inter se parallelae: sed in casu aequilibrii recta MN, jun
gens gravitatis centra M et N , est perpendicularis rectae
DE ; ergo et HH' erit perpendicularis ipsi DE . Hinc DH=
HE: vicissim si DH =HE, erit HH' ac proinde MN per
pendicularis rectae DE; conditio nimirum necessaria ac
sufficiens ut recta jungens gravitatis centra M et N sit per
pendicularis rectae DE redigetur ad mutuam aequalitatem
rectarum DH, HE. Quibus positis , denotent B, Borangulos
DAH, BAH, et b rectam AH; triangula ADH, AHE dabunt
DA’ = w2762—2wbcos B ,HE' = w &quot; 2 + 62—20 &quot; bcoss *:
propterea
w2 -2bw' cos B = &quot;? - 26w &quot; cos \beta &quot; (k' ) .
Ex duabus ( k) et ( k ' ) eruentur a eta' , uude innotescit
positio intersectionis DE.
Quod si determinanda esset positio aequilibrii in eo
dem prismate quum ita demergitur ut puncta B et C ma
neant infra liquidi superficiem DE, foret
area BCDE : aream ABC = h' : u ': fb,
ideoque
ABC - BCDE ( ADE ): ABC Hope': fl , seu
−−∙≔∎⊾↼−−⇀
193
' Nune secto bifariam in H latere BC, ducatur AH; sum-
pta AM:
∙⋛−⋅ AH, centrum gravitatis trianguli ABC e-
rit in M: simili modo, secto bifariam in H' latere DE, sum-
ptaque AN:
∙−−≣−− AH', erit N centrum gravitatis trianguli
ADE. Quia igitur illi: −∙∙ 23,
inter se parallelae: sed in casu aequilibrii recta MN,iuu- ⋅
gens gravitatis centra M et N , est perpendicularis rectae
DE; ergo et HH' erit perpendicularis ipsi DE. Hinc DEI:-.'
HE: vicissim si DH :HE, erit HH' ac proinde MN per-
pendicularis rectae DE; conditio nimirum necessaria ac
sullicieus ut recta iungens gravitatis centra M et N sit per-
pendicularis rectae DE redigatur ad mutuam aequalitatem
rectarum DH, HE. Quibus positis, denotent B', B&quot;angulos
DAH, BAH, et brectam AH; triangula ADH,AHE dabunt
,
ideo MN et HH' erunt
BB': 'i—l-b' —29'6 cos B', B—Ea ⇌∾∣⋅≖−⊢ &amp;&quot; —20&quot;bcosB&quot;:
propterea
a)&quot; ---260' cos B': si&quot;! -
266)&quot; 'cos B&quot; (k' ).
Ex duabus (I:) et (k') erucutur a' et et&quot;, unde innotescit
positio intersectionis DE.
Quod si determinanda esset positio aequilibrii in eo-
dem prismate quum ita demergitur ut puncta B et C ma-
neant infra liquidi superficiem DE, foret
area BCDE: aream ABC −∙∶−− pl: p.': p.,
ideoque
ABC −∙− BCDE (: ADE ): ABC :p.- p.': p., seu194
Ww&quot; sin A : 1 a'a&quot; sin A = M - pe : plo
et consequenter
s'avº = ( 1- )« a”(k&quot;).
Ad haec : centrum gravitatis trianguli ABC invenilor
in recta jungente centra gravitatis portionum ADE ac BCDE.
Sed ABC et BCDE habent sua gravitatis centra in recta
perpendiculariter insistente rectae DE : adhuc igitur MN
erit perpendicularis ipsi DE ; rursusque prodibit (k' ) : eru
entur videlicet in casu w' et w &quot; ex binis ( k' ) et (k &quot; ) .

90. Determinata aequilibrii positione, restat videndum utrum aequilibrium sit stabile nec ne. Pone v. gr. innatans solidum esse tale, ut secari possit plano verticali AB ( Fig. 49. ) in duas partes omnino symmetricas tum quoad formam, tum quoad densitatem, et in casu aequilibrii sit HK intersectio plani AB et horizontalis plani repraesentantis superficiem liquidi: gravitatis centra M et N innatantis solidi et ejecti liquidi invenientur ambo in plano AB super eadem verticali CD; si solidum est homogeneum exsistet N subter M; si heterogeneum, poterit M esse vel subter N vel supra. Fac ut aliquantulo revolvatur solidum circa axem perpendicularem plano AB, sicque removeatur ab aequilibrii positione; ita tamen ut, exhibente H'K ' (Fig. 50) novam intersectionem plani AB et horizontalis plani repraesentantis superficiem liquidi, segmentum solidi respondens angulo K i K' aequetur constanter segmento quod respondet angulo H i H' ; hoc pacto haud variato ejecti liquidi volumine, permanebit ( 89.30. )
gV'p ' = gVd : proinde solidum absque initiali velocitate sibi
commissum movebitur ( 84 ) circa centrum M immotum.
Jam si ex puncto N' , ubi , amoto solido ab aequilibrii
positione , situm est gravitatis centram liquidi expulsi , du
194
&amp; o'o'f aiu A:) a'a&quot; sin A:p—p:p., /
et consequenter
Ad haec :centrum gravitatis trianguli ABC invenitur
in recta iungente centra gravitatis porticuum ADE ac BCDE.
Sed ABC et BCDE habent sua gravitatis centra in recta
perpendiculariter insistente rectae DE: adhuc&quot; igitur MN
erit perpendicularis ipsi DE; rursusque prodibit (k') :eru-
entur videlicet in casu a' et a)&quot; ex binis (k') et (Is&quot;) . ducatur verticalis recta N'R occurrens rectae CD in R , oc
cursus iste vel fiet supra M , vel infra , vel in ipso M :
in primo casu vis g Vlagens sursum juxta N'R manife
ste nitetur ut CD resumat verticalem positionem, et conse
quenter aequilibrium erit stabile ; in secundo ipsa gVp.
nitetur ut CD magis recedat a verticali positione , ideoque
aequilibrium instabile ; in tertio aequilibrium adhuc ob
tinebit quoad novam positionem .

=== De gravium liquidorum aequilibrio in vasis communicantibus. ===
[[Fasciculus:Communicating vessels.svg|thumb]]
91. Vasa communicantia dicuntur illa, quae ita sunt inter se conjuncta ut ex altero in alterum pateat aditus fluido. In altero contineatur fluidum homogeneum, cujus densitas &lt;math&gt;\mu&lt;/math&gt;; in altero fluidum pariler homogeneum cujus densitas &lt;math&gt;\mu'&lt;/math&gt;; siatque &lt;math&gt;z&lt;/math&gt; et &lt;math&gt;z+ z'&lt;/math&gt; distantiae inter punctum quodvis superficiei communis utrique fluido ac extimas fluidorum superficies. Fluidis se mutuo librantibus, exsistet (88)

&lt;math&gt;g\mu z + C_1 = g\mu ( z + z' ) +C_2.&lt;/math&gt;
[[Fasciculus:11 hidrostatica de 61 a 70.jpg|thumb]]
92. Haec facile nunc stabiliuntur. 

1.º Si vasis communicantibus idem continetur liquidum, ut sit &lt;math&gt;\mu = \mu '&lt;/math&gt;, erit &lt;math&gt;g \mu z = g \mu' z&lt;/math&gt; ideoque &lt;math&gt;z' = \frac{C_1 - C_2}{g \mu'};&lt;/math&gt; emerget ergo &lt;math&gt;z' = 0&lt;/math&gt; vel &lt;math&gt;z' &gt; 0&lt;/math&gt;, prout &lt;math&gt;C_1 = C_2 &lt;/math&gt;vel &lt;math&gt;C_1 &gt; C_2&lt;/math&gt;: in ea videlicet qua sumus hypothesi liquidum sub externis aequalibusque pressionibus manebit in utroque vase aeque altum, sub externis vero inaequalibusque pressionibus altias apud eam partem assurget ubi minor exercetur pressio. Inde profluit explicatio variorum effectuum; cujusmodi sunt hydrargyrum in barometro suspensum, aqua elevata in siphone, in antliis etc.... Sic v. gr. quoad antlias adspirantes, dum attollitur embolus ex &lt;math&gt;H'H''&lt;/math&gt; in &lt;math&gt;HI&lt;/math&gt; (Fig. 51), aer in tubo &lt;math&gt;HB'&lt;/math&gt; confestim fit rarior, et consequenter externus aer densior aquam in receptaculo vel puteo contentam cogit in tubum ascendere usque ad altitudinem v. gr. &lt;math&gt;A' B'&lt;/math&gt;: quam ob causam descendet aqua in receptaculo ab &lt;math&gt;AE&lt;/math&gt; in &lt;math&gt;ii'&lt;/math&gt;. Jam datis &lt;math&gt;H'Q ( = a ')., EQ ( = a '' ) , HH' ( = b) ,&lt;/math&gt;itemque horizontalibus receptaculi, ac tuborum &lt;math&gt;BQFD', FQA'B'&lt;/math&gt; sectionibus &lt;math&gt;\omega, \omega' , \omega ''&lt;/math&gt;, si debeat inveniri altitudo &lt;math&gt;AA'&lt;/math&gt;, pone &lt;math&gt;AA' = \beta&lt;/math&gt; et &lt;math&gt;Ai = \beta'&lt;/math&gt;: densitates aeris interni ante et post emboli elationem sunt in ratione reciproca voluminum &lt;math display=&quot;block&quot;&gt;a'\omega' + a'' \omega'' , a'\omega' + a'' \omega'' + b\omega' - \beta \omega'';&lt;/math&gt;ideoque (87. 6º) in eadem ratione erunt pressiones a et ;
hinc
( a'w ' ta'w ') a
(a' +6) + (a&quot; – 3 ) cs&quot; |
designante m aquae densitatem , aqua elevata supra ii ' exer
cebit ( 88) pressionem
a = gm (B + B ) .
Cum igitor a' to = 5W , cumque Bw &quot;' = f'w , iccirco
( a'w' + aa'') as''
(a + b ) w + la &quot; -B, w 

sia ponitur parvitatis contemnendae prae w , erit
( a'w' ta'a ') as tgms = a .
( a ' + 6) + ( a &quot; -B) w &quot;
196
sunt hydrargyrum iu barometro suspensum , aqua elevata
in siphone ,
in antliis etc.... Sic v. gr. quoad antlias ad- ∙
spirantes , dum attollitur embolus ex H'H&quot; iu Hl (Fig.51.),
aer in tubo HB' confestim Et rarior , et consequenter ex-
ternus aer densior aquam in receptaculo velputeo conten-
tam cogit iu tubum ascendere usque ad altitudinem v. gr.
A' B' : quam ob causam descendet aqua in receptaculo ab
AE in ii'. Jam datis H'Q (: a')... EQ (: a&quot;)
, HH'(:—...- 6),
itemque horizontalibus receptaculi , ac tuberum BQFD' ,
FQA'B' sectionibus a), m', ei&quot;, si debeat inveniri altitudo AA',
pone AA':B et Ai:B' :densitates aeris interni ante et
post emboli elationem sunt in ratione reciproca voluminum
a'm' −↿− a&quot;a)&quot;
, a'æ' a&quot;o)&quot; −∣⋅− ∂∾⋅−Ba)&quot; ;
ideoque (87. 60.) in eadem ratione erunt pressiones a et se' ;
hinc
(a'ai' −⋅∣− d'un&quot;) ur −⇀⋅ (a'-1-b)m'-1-(a&quot;—B)m&quot;
designante m aquae densitatem ,,aqua elevata supra ii' exer-
cebit (88) pressionem
0&quot;: sm (49 ⊣− B')-
Cum igitur a' -l-—a&quot; :0, cumque Ba)&quot;:B'm ,
iccirco
[
U'
(a'æ' ⊣∙⋅ d'ai&quot;) ar −⊢⊣−⊣−⊰⋯≺↿⊣−∾−↜∶≻∣∃⇌≔⇌
si a)&quot; ponitur parvitatis contemnendae prae a)
, erit
(a'æ' −⋅⊢ J'ai&quot;) :: l ∙−−
(a'-l—b) ∾∣∙∙⊢ (avl—þ) 0)&quot; l gmB—a-197
la eadem hypothesi , post iteratos descensus atque ascen
sus , restituto embolo ab altitudine minima H'H ' ad maxi
mam HI , pertingat aqua ad inferiorem superficiem mem
branae G ; descendente rursus embolo et denotante k alti
tudinem aquae de se librantis atmosphaericam pressionem ,
elevabitur membrana D , ac proinde rarescente adhuc aere
in consequenti emboli ascensu , aqua pergei assurgere quo
tiescumque fuerit
EQ . HQ &lt; k (HH') .
Ut enim elevetur membrana D , debei elasticitas k' aeris
HF sic augeri quum aer HF traducitur ad volumen H'F , ut
elasticitas k' ' densati aeris H'F superet elasticitatem k aeris
atmosphaerici embolo superincumbentis . Est aulem ( 87.1º . ) .
k : k &quot; = HQ : HQ ;
vis insuper elastica k' unita ponderi aquae suspens ae EQ
librat pressionem aeris atmosphaerici , nimirum
h' + EQ = k ;
et consequenter
k &quot;
k' (HQ)
H'Q
(k- EQ) (HQ)
H'Q
Igitur
( k — EQ) ( HQ)
&gt; k ; ac proinde etc. ... H'Q

2. Tubus cylindricus longitudinis h , et in una
sui extremitate clausus , impleatur hydrargyro usque ad
197
in eadem hypothesi , post iteratus descensus atque&quot; ascen-
sus , restituto embolo ab altitudine minima H'H&quot; ad maxi-
mam Hl , pertingat aqua ad inferiorem superficiem mem-
branae G; descendente rursus embolo et denotante ]: alti-
tudinem aquae de se librantis atmosphaericam pressionem ,
elevabitur membrana D, ac proinde rarescente adhuc aere
in consequenti emboli ascensu , aqua perget assurgere quo-
tiescumque fuerit
EQ
. HQ h(HH').
Ut enim elevetur membrana D , dabei elasticitas k' aeris
HF sic augeri quum aer HF traducitur ad volumen H'F , ut
elasticitas k&quot;densati aeris H'F superet elasticitatem k aeris
atmosphaerici embolo superiucumbentis .Est autem (87.10.).
k': k&quot;:H'Q :HQ ;
vis insuper elastica k' unita punderi aquae suspensae EQ
librat pressionem aeris atmosphaerici, nimirum
k&quot; -]— EQ:k ;
et consequenter
- k&quot; k' (HQ) −∙∙≺∣⊂− EQ) (HQ). ↼−− l'l'Q HQ
igitur
de −−⋅⋅ EQ) (&quot;Q)
H'Q k; ac proinde etc.
..

2.&quot; Tubus cylindricus longitudinis A, et in una
sui extremitate clausus ,
'impleatur hydrargyro usque ad198
altitudinem hoh , cum digito ad orificium in altera parte
exsistens apposito invertatur tubus ; et remolo digito , col
locetur hoc ipsum orificium in superficie hydrargyri sta
gnantis intra aliquod vas. Ascendet aer l' ad supremam in
versi tubi partem ; augescet h , et fiet = h &quot; . Jam vero ad
inveniendam h &quot; denotante k' altitadinem hydrargyri libran
tis atmosphaericam pressionem , satis erit animadvertere
h'ki
quod exhibet altitudinem hydrargyri librantis rarefa h &quot;
clum aerem h ' ; unde
hk
h - h '
+
To
k ;
ac propterea
h &quot; = h - k' = V Th — kj» + 4hºk
2
signum inferius non pertinet ad praesens problema .
lu formula ( 10)
C, C,
Sle'
Spkk
sunt C, = gu'k ' , C,
z'
'

3º. Pone ple , pe inaequales , et C, = C2 ; habebis
p.z = p ( = + z ) ,
unde
2 : 3+ = M ' : pe ;
diversorum nempe liquidorum altitudines z et ztz' in vasis
communicantibus erunt reciproce ut ipsorum liquidorum
densitates.
198
altitudinem h—h' ,
tum digito ad orificium in altera parte
exsistens apposito invertatur tubus ; et remoto digito , col-
locetur hoc ipsum orificium in superficie hydrargyri sta-
gnantis intra aliquod vas. Ascendet aer b' ad supremam iu-
versi tubi partem ; augescet h' ,
et fiet:h&quot;. Jam vero ad
inveniendam h&quot; denotante k' altitudinim hydrargyri libran-
tis atmos'phaericam pressionem , satis erit animadvertere
quod exhibet (i£—, altitudinem hydrargyri librantis rarefa-
ctum sereni I:&quot; ; unde
h—h&quot; ∙−⊢ h—Ij—L: k' ;
ac propterea
1.&quot; −∣∙ ∣⋅−∣⊏⋅∶⊨∣∕⇀≺∣≖−∣≂⊤≻⋅⊣−⊓≖⋅∣⊏⋮∙ , z
signum inferius non pertinet ad praesens problema.
lu formula (10)
sunt C, :gpjk' , Ca.-:. ∙−−−−

diversorum nempe liquidorum altitudines :
et &lt;math&gt;z+z'&lt;/math&gt; in vasis
communicantibus erunt reciproce ut ipsorum liquidorum densitates.

=== De gravium elasticorumque fluidorum aequilibrio; necnon de altitudinibus dimetiendis ope barometri, et de pondere ac densitate vaporum. ===
93. Binae (o&lt;sup&gt;iii&lt;/sup&gt; 87.), (o&lt;sup&gt;v&lt;/sup&gt; 88.) dant &lt;math display=&quot;block&quot;&gt;\frac{d\varpi}{ \varpi} = g\theta dz;&lt;/math&gt;binae (o&lt;sup&gt;iii&lt;/sup&gt;), (o&lt;sup&gt;iv&lt;/sup&gt;  87) praebent&lt;math display=&quot;block&quot;&gt;\theta=\frac{\mu}{\varpi}=\frac{1}{ i(1 + an )}&lt;/math&gt;propterea&lt;math display=&quot;block&quot;&gt;\frac{d\varpi}{ \varpi} = \frac{gdz}{ i(1 +an )} ( b ).&lt;/math&gt; Assumptis autem logarithmis quoad basim 10,&lt;math display=&quot;block&quot;&gt;d\log_{10}(\varpi)= \frac{d\varpi}{\varpi} \log_{10}[2,718281828 ] = 0 , 4342945 \frac{d\varpi}{\varpi}&lt;/math&gt;ideoque
dos dL(W)
0,4342945
Designante igitur pressionem apud punctum ( x , 0)
in hypothesi temperiei constantis formula (6) suppeditabit.
L - Llw ')
0,4342945 ( 6' ) .
i ( 1 + an )

94. Quoad punctum ( x , y ; '-— z) supra horizontale pla
num XOY ( Fig. 47 ) , aequatio ( 6' ) suppeditat
LULLG )
0.4342945
82
i (1tan)

et inde infertur valor z dimetiendae altitudinis supra XOY
sic expressus
i ( 1 + an )
L
0,4342945g ( 6 '') .''
Haec observentur: 1. ° sub temperie = 0 , et barometrica
hydrargyri elatione =2,33958 ped. apud geographicam lati
tudinem = 48° 50' 14 &quot;, ubi gravitas 30,1959 ped. ,
Biot et Arrago invenerunt densitatem hydrargyri esse ad
aeris densitatem po ut 10467 : 1 ; inde habemus respon
dentem pressionem ( 88 )
Ww=( 30,1959) ( 10467 floo ) ( 2,33958) ,
ideoque
wo -
( 30,1959 ) ( 10467) ( 2,33958)
ро
( 30,1959 ) ( 24488, 38386) =739448, 790198174.

2.o Eo minorem experimur temperiem , quo ma-
gis supra terrestrem superficiem assurgimus ,
at, igno&quot;-
mus qua lege liat ejusmodi imminutio; designantibus ?'
et ': temperies in intimo ac supremo puncto dimetiendae
altitudinis z, solet assumi
.- 'r'-l—r
'
&quot;: 2• 201
poniturque ista temperies media constanter vigere per to
tam 2.
1

3. Singulis gradibus imminutae temperiei respon
det hydrargyri condensatio = ; igitur si M et M
5550
exhibent densitates bydrargyri sub temperiebus t' ; ac to
DY in infimo ac supremo puncto altitudinis , erit
t'
M

: 1 = M' : M, unde M

5550 I' ,-1,
1
5550
rica
ati.
ed.,
e ad
00
Temperies hydrargyri tubo barometrico inclusi nonnisi
post aliquod tempus ad aequalitatem reducitur cum aeris
circumstantis temperie , hinc t'i et t, solent definiri sub
sidio thermometri , quod ad barometrum ipsum adnecti
tar ; aliae vero t ' et determinantur ope thermometri ,
quod cum barometro non communicat.

4.0 Si l' et h exprimunt barometricas altitudi
nes apud infimum et supremum punctum altitudinis
erunt ( 88 )
h' =gM'h , =
&amp;M'h
t',
1
5550
ideoque ma'
/
Jora
us ?
endae

: - * (

I' , - 1 ]
5550

5. ° experimentis pendulorum subsidio institutis
14
'201
∣∙ poniturque ista temperies media constanter vigere per to-
tam :. '
&amp;&quot; Singulis gradibus imminutae temperiei respon-
det hydrargyri condensatio: 5150; igitur si M' et M

; )

exhibent densitates bydrargyri sub temperiebus 'r', ac 't',
ll
⋅
in infimo ac supremo puncto altitudinis , erit
———- f.:—T! M' :1: ': ∙∙∙⋅ J
5550 M M, uudeM T',—Tx
'
5550&quot;
ric-a Temperies hydrargyri tubo -barometrico inclusi nonnisi
all' post aliquod tempus ad amnalitatem reducitur cum aeria
ed.. circumstantis temperie , hinc 'r', et 1.&quot;, isolent definiri sub-
sad
ron-
sidio tbermometri , quod ad barometrnm ipsum adnecti-
tur; aliae vero 1&quot; et ': determinantur ope tbermometri ,
quod cum barometro non communicat.

4.&quot; Si b' et lt exprimunt barometricas altitudi-
nes apud infimum et supremum punctum altitudinis z ,
erunt (88)
.: M'h ∙∣≖∙ ∙∙ g ∙
a *gM .m. fr.—ff: ,
∎∎− 5550
mr ideoque
nora-
0st a. ∙∙∙ h' 1 T.]- T; .»pdæ ⊺≖−−−∣≖ (&quot;5550)'
5.0 experimentis pendulorum subsidio institutis
14
- x'!
.202
probatum est , si gi est gravitas apud geographicam la
titudinem = 45 , apud aliam latitudinem å fore
g = g . (1-0,002589cos22 ) ;
erit igitur ( 1 )
30,1959 = g1 [1–0,002588 cos2 (48° 50'14'') ]''
ac proinde
30,1959 ( 1–0,002588 cos 22 )
1 -0,002588 cos2 (48 ° 50'14 &quot; )
.
6.° Quibus positis , formula ( 6 &quot; , 94) traducetur ad
24488,38( 1–0,002588cos2 [48° 50'14*]/ (1 +0,00395+7 )
X
1-0,002588cos22
L
CO
I '
1
5550
0,4342945
-) ]
ped.
e ,
formu 95. # Sumptis logarithmis quoad basim
la ( 6 &quot; , 94 ) evadet
i (1 + an ) .
,( );
upde
et consequenter ( 87. 7. ) H =
82
e iſitan )
202
probatum est
.
si g. est gravitas apud geographicum la-
titudinem;—4

5. ∘ apud aliam latitudinem ). fore
gzg, (1—0,002588cos2)t) :
erit igitur (10)
30,, 95gzg,[1—o,002588 cosz (48-50'1 4&quot;)1
ac proinde
∙− 30,1959 (1—0,002588 cos zx)
5 1-0,002588 cos2 (4so50'14&quot;) '
6.&quot; Quibus positis, formula (E&quot;, 94) traducatur ad
24488,38(1—0.002588cosz[4so50'14&quot;])(1'-1-o,003757 BH) s—
⇁⋅⊤
' - X '1—0,002588cos2').
h. r.!—T! )]
L I: &quot;( ↿−∎∎ 5550
o,4342945 pcd.
95-0 Sumptis logarithmis quoad basim e ,
formu-
la (6&quot;. 94 ) evadet

: i(1tan)L (jul-) ;

unde
a,
ex --
z
.
et consequenter (87. 7. ) p,:
e i(t-l—an)203
1
g?
i ( 1 + an) e il1 +an)
Denotent V'et i volumen et densitatem corporis aere
demersi , ipsoque aere specifice levioris : urgebitur corpus
ad verticalem ascensum vi acceleratrice
8 (V'4 — V'x ')
Vph Gelee Me gz
if1tan)
i ( 1 + an) e
Facile intelligimus , si denotat densitatem mediam glo
bi aereostatici , verticalem ascensum ipsius globi determi
natum iri per
daz
de2
8
)
(6 '' )''
.
i (1 + an ) e
i(1 + an )
Multiplica (6 &quot; ' ) per 2dz, et sume integralia; habebis ( 27. 12.9)
gz
dz2 i(1 + an)
dla с
2g role
re' f 8 + ks)
In hypothesi velocitatis initialis = o erunt simul z=0
20

o , ideoque C

Hinc do
dz et
20 82 dza
de ² (
1 e
i (1 + an ) — 2g2 (6 &quot;) .
hey 252 &quot;
203
I
3
gz
,
i (1—l—an) e i(1—l-an)
Denotent V' et pf volumen et densitatem corporis aere
demersi, ipsoque aere specifice levioris :urgebitur corpus
ad verticalem ascensum vi acceleratrice
V' —-V' ')
,
'a' , gt P
&quot;
—g,(p p.) g( —H)- VP ⊬ M sz
i (1-l—an) e
t(t-i-an)
Facile intelligimus, si denotat p! densitatem mediam glo-
bi aereostatici, verticalem ascensum ipsius globi determi-
natum iri per
I
daz g es'
dt: p.(
Multiplica (6&quot;') per 2dz, et sume integralis; habebis (27. 12.&quot;)
..
52
dzz— c zg a t(1—l—an) '] '
&amp;f— —F g
'&quot;')'
ln hypothesi velocitatis initialis :
0
erunt simul 220
(12. 20! et ⊼⋅−−−∶∘ ,
ideoque C..:
F.]iinc
d:: 25, ∙−− ...—gj— ⋅ '['
(7:3—?( 1 — 8 :
(l*'-an)) .'.2gz (6 ).204
cto
ex cujus integratione innotescet relatio inter z ac t . Fa
dez
=o, formula ( 6 ' ' ' ) suppeditabit altitudinem 2, apud dia
dz
quam exsistet f = M ; et facto = 0 , formula ( 6 &quot; ) praebe dt
bit maximam globi elationem z.

96. Quod pertinet ad pondus ac densitatem vaporum, haec subjungimus: 

1.0 Si vase undique clauso continetur satis liquidi, ut inde sese possit evolvere tantum vaporis, quantum postulat capacitas vasis, quantitas vaporis sese evolventis pertinget ad quoddam maximum unice pendens a vigente temperie: qua videlicet permanente, istud maxinium perstabit idem aut vas exsistat vacuum ab aere, aut aerem contineat, vel quodvis aliud gas ulcum
que densatum vel rarefactum: sive autem obtinuerit illud maximum, sive non, tantundem augebitur vis elastica sicci aeris, vel gas inclusi, quanta est elasticitas evoluti vaporis.

2.° Si vapor aqueus seorsum spectatus posset sub data temperie, quin ad liquidam formam redigeretur, eam librare pressionem ā, quam sub eadem temperie librat siccus aer, ex Gay-Lussac foret densitas té aquei vaporis ad sicci aeris densitatem / ut 10 : 16 , ideoque
M=
104
16

3.• Permanente temperie , fac ut aqueus vapor seor
sum consideratus libret reipsa pressionem Wri si vaporis
densitas vocatur Hiss erit ( 87 : 1. )
10

: @ = ht ' i theo unde pos = 16 Mo ;

et denotantibus P ac P, pondera aeris ac vaporis sub ae
quali volumine ,
204
ex cuius integratione innotescet relatio inter z ac :. Fa-
dzz . . . cto 27; ::o, formula (F&quot;) suppedttabtt altitudinem :, apud
quam exsistet p.:pl; et facto 5; :o, formula (ö&quot;) praehe-
bit maximam globi elationem :.

96. Quod pertinet ad pondus ac densitatem vaporum,
haec subjungimus: 1.&quot; Si vase undique clauso continetur
.satis liquidi, ut inde sese possit evolvere tantum vapo-
ris
, quantum postulat capacitas vasis , quantitas vaporis
sese evolventis pertingat ad quoddam maximum unice pen-
dens a vigente temperie :qua videlicet permanente
,
istud
maximum perstabit idem aut vas exsistat vacuum ab ae-
re, aut aerem contineat
, vel quodvis aliud gas utcum-
que densatum vel rarefactum : sive autem obtinuerit illud
maximum, sive non, tantundem augebitur vis elastica sicci
aeris, vel gas inclusi, quanta est elasticitas evoluti vaporis.
2.&quot; Si vapor aqueus seorsum spectatus posset sub da-
ta temperie , quin ad liquidam formam redigeretur , eam
librare pressionem 0, quam sub eadem temperie librat
siccus aer, ex Gay- Lussac foret densitas p! aquei va-
poris ad sicci aeris densitatemlp. ut 10 : 16, ideoque

4. • Nunc ex aqueo vapore librante pressionem ,
et ex aere sicco emergat volumen V aeris vaporosi librantis
pressionem , et habentis densitatem &amp; ; istiusmodi aeris
massa erit Vs; aer siccus in aere vaporoso contentus utpote
librans pressionem ( 1.9 ) a— , pollebit ( 87 : 1. ° ) den
( - )
sitate Quoniam igitur ( 39) vapor aequeus in
,
to
10 WI
16 W
aere vaporoso pariter contentus pollet densitate pi
propterea ad
Ve = y (0 ) tv
10 i
16 W
por
ICCI
ris
da unde
bra €
( ---+ -s)= (:-)
1 &quot;
sic v. gr. in ordine ad aerem maxime vaporosum sub
temperie =0 , et barometrica hydrargyri altitudine
2,33958 ped. , quoniam maxima pressio librata ab aqueo
vapore sub temperie = 0 respondet barometricae altitu
dini =0,015638 ped ., erunt ( 95. 1.° )
g = W = ( 10467No) ( 2,33958 )g , w = (10467 /lo) ( 0,015638 ) g;
ac proinde designante eo respondentem valorem €,
seor
pors

; 3

2,33958 0,015638
lo 8
Eo = Too
bi
Wo :)-- (*
2,33958
=0,997495po.
205
'
—Pp.,— 10 ut,
⊬∙⊬≖≔⊉∙⊅∎∙⊉∎ F 16.;—P.
,
∙ 2,33958 — 3- .0,015638 ⇌−⇀
−⋮⊥−∘⇠ −−∃↾− ).. 8
a'., ∘ a m ↼⊣∸∘ 2,33958

: o,997495p.o.

l—xu .206
Hinc
E. 0 , 997495 ;
Ho
ratio videlicet inter densitatem aeris maxime vaporosi et
densitatem sicci aeris in ea qua sumus temperiei ac pres
sionis hypothesi.

5. • Valor i jam inventus ( 94. 1. ° ) spectat ad ae
rem siccum ; quoad aerem v. gr. maxime vaporosum erit
T. .
0,997495 flo
( 30,1959 ) ( 10467 ) ( 2,33958 )
Eo 0,997495

6. Obiter notamus illud : aquam sub satis alta
praesertim temperie in vapores versam conari sese qua
quaversus incredibili vi expandere indubia evincunt expe
rimenta. Hinc usus aquei vaporis in movendis machinis :
certo quodam tuborum valvularumque artificio vapor ex
caldario introducitur in antliam , ita , ut antliae cavitates ,
alteram infra embolum , alteram supra embolum , vicissim
obtineat, vicissimque frigidae suffusione ad pristinam redeat
Jiquiditatis conditionem ; vapor inferiorem cavitatem obtinens,
attollit embolum ; superiorem, deprimit ; embolus adnexus
est alteri ex duabus cujuspiam vectis extremitatibus ; qui
vectis altera sui extremitate vel immediate vel instrumen.
torum apte conjunctorum subsidio motum communicat rotis ,
malleis , elc.... ; prout nempe importat machinae
movendae natura.

Hinc
∙⇣∘−−−∶ o,997495;
p..
ratio videlicet inter densitatem aeris maxime vaporosi et
densitatem sicci aeris in ea qua sumus temperiei ac pres-
sionis hypothesi. ,
'
5.&quot; Valor i iam iuventus (94. 1.&quot;) spectat ad ae-
rem siccum; quoad aerem v. gr. maxime vaporosum erit
↿≖∘∙∙ ar, —(3o,1959) (10467) (2.33958)
s, o,997495 ⊬∘ o,997495 '
i..—

=== De aqua egrediente per angustum foramen e vasis verticalibus sive cylindricis, sive prismaticis.===

97. Haec praemittimus ex pluries iteratis experimentis 
[[Fasciculus:Aqua egrediens.png|thumb]]
1.° Minuta corpuscula disseminata per descendentem aquam verticaliter descendunt commuui ad sensum velocitate usque ad horizontalem &lt;math&gt;HH'&lt;/math&gt; (Fig. 52), cujus distantia ab orificio &lt;math&gt;hh'&lt;/math&gt; aequat triplum radiurn ipsius &lt;math&gt;hh'&lt;/math&gt;; tum cursum flectentia, perque lineas curvas incedentia conspirant versus orificium. Aqueae igitur particulae verticaliter descendunt usque ad &lt;math&gt;HH'&lt;/math&gt;; formaturque ab &lt;math&gt;HH'&lt;/math&gt; ad &lt;math&gt;hh'&lt;/math&gt;conoides aquea &lt;math&gt;Hhh'H'&lt;/math&gt;, quiescentibus portiunculis lateralibus &lt;math&gt;B.B'&lt;/math&gt;.

2.° Adhuc obtinent et verticalis particularum descensus, et earum conspiratio ad formandam conoidem, etsi orificium aperitur in latere vasis.

3.° Aqua ex aperto orificio verticaliter saliens assurgit ad supremam fere prementis aquae superficiem.

98. Denotet &lt;math&gt;\omega&lt;/math&gt; velocitalem aquae egredientis ex orificio &lt;math&gt;hh'&lt;/math&gt;, et &lt;math&gt;z&lt;/math&gt; altitudinem prementis aquae supra orificium, erit proxime (30:31)&lt;math display=&quot;block&quot;&gt;
\omega=\sqrt{2gz}(k) . &lt;/math&gt;Ad haec; si &lt;math&gt;\alpha&lt;/math&gt; denotat horizontalem vasis basim, &lt;math&gt;a&lt;/math&gt; orificium &lt;math&gt;hh'&lt;/math&gt;, &lt;math&gt;v&lt;/math&gt; velocitatem particularum ex quibus coalescit suprema aquae superficies, erit &lt;math&gt;
\alpha v dt= a.\omega dt,&lt;/math&gt; unde &lt;math&gt;w= vi&lt;/math&gt; et facta asna,
a
imen
roting
w = nv ( k' ) .
Hinc (27 )
asis
dz ndy
do 2gz et dt
V28%
ideoque designante 2, initialem valorem » , quum nempe t = 0 ,
inted
ft?
ae-
erit
talil
men&quot;
aul?&quot;
207

98. Denotet &amp;) velocitatem aquae egredientis ex ori-
ficio hls', et : altitudinem prementis aquae supra orifi-
cium , erit proxime (30 :31 )
a) −−∶ Vig.; (k).
Ad haec :
si a denotat horizontalem vasis basim , a ori-
licium hh' , :: velocitatem particularum , ex quibus coa-
lescit suprema aquae superficies, erit
ac «.vdtzamdt, unde a): −⇀ v :
et facta «scita,
a
mzn-v (k').
Hinc (27)
ds ⇂∕ nds
— :: ∙−−−∶ 2 :∙∙∙ ∙ dt ga ,
et dt V—zgs .
ideoque designante s., initialem valorem :, quum nem-
Pe ∁−−−−∘⇟208
i - V7(
..- , ) ( &quot; ) .

99. Sit \beta volumen aquae tempore t egredientis ex
orificio a ; erit ( 98. k . k &quot; )
233= a.orde=a(28)* . * de= a/ 2018 ( 3 - V . Jde
Propterea
B =a/25)*(*.* -VERSI-)
( k' ' ) .

100. Assumpta z = o in ( k &quot;. 98 ) , prodibit tempus
O , quo vas lotum evacuatur ; nimirum
11/ 를
2n 0—
V 29
( k &quot; ) .
In ( k &quot;) et ( K '') substitae valorem molè ex ( k ' ) ; habebis''
2n 을
21
5 B
ag
V28
2n (25–2-ce). ( &quot;) .

101. Ex (k &quot; ) sequitur illud : si duo vasa habuerint
et altitudines zo , zo, el orificia a , a' aequalia , tempo
ra 0 , 0 quibus deplentur , erunt in ratione basium a,a' ,
siquidem
2n 2n '
á
0 : 0 =
V 29

: z ' .

V 28
--- N : n ' = . Q : a' :
a '
208
≖−−−−⇁ 21( soi—1 ii) (li&quot;)
-
l/Zg
99. Sit þ volumen aquae tempore :egredientis ex
orificio a ; erit (98. I:. k&quot;)
.l. s
'
s i— dþzaüdtza (25? s
'dt:a(2g)ir&quot; (
zog— &amp;
t
)dt. ⇂

Propterea
,

100. Assumpta ≖∙∶−−− ∘
in (Is&quot;. 98 )
. prodibit tempus
9 , quo vas totum evacuatur; nimirum
9: 2: ∣∙∘⋚ (z.-&quot;) ⋅
l/Zg
In (It-&quot;) et (I.-&quot;') substitue valorem s.,ïli ex (Is&quot;); habebis

:: 6— 2n 3,- ag( .

⋅−
2-
. — ⋅ 20 t): wg I.&quot;. (3 ,&quot; ( )
101. Ex (It-&quot;) sequitur illud: si duo vasa habuerint
et altitudines s.
, a',, et orificia a
.
a' aequalia. tempo-
ra 9. 9' quibus deplcntur ,
erunt in ratione basium a.d.
siquidem
∙ 2n 2n'

: 9:∶−− zo : ...z'o zn:n'— :—,-—a:a':

l/Zg a a· 209

102. Quantitates aquarum successivis et aequalibus tem
poribus effluentium decrescunt secundum numeros impa-
res ordine retrogrado sumptos. Assertio facile ostenditur
e secunda ( k ) , facto successive t=1,2,3,4, • ; nam
quantitates illae prodibunt expressae per
ag
2n (29-1 ) , L ( 49-4 ) – ( 29-11 , P.(69-9)– ( 49-4) ,
Se
ag ( 80-16)
2n
ag ( 60-9) ... , seu 2n
ag
2n (20-1 ) , 29–3 ) , (29-5 ) , (29–7), - ;
ideoque etc... Idipsum eruitur ex (k &quot; ) et ex prima (k&quot; ) ;
denotantibus enim 21 , 22, 23 , ... valores z respondentes tem
poribus 1 , 2, 3, ... eae praebebunt
&amp; 02 , 2, 3
(0-1) 2,225 2n? 2n2 , =-2,(0-2) », 23 =
S
(0-3 ) , ... 29-1 2n2
8
2n2 ;
unde
6
( 29-1 ) , 21-22 2n?
8
( 29-3 ) , Zz- 23 =
2n2
8
2n2 (26-5) , ... 29-3 8
2n?
et consequenter etc...
Hinc si dividendum sit vas in partes successivis dati tem
'209
102. Quantitates aquarum successivis et aequalibus tem--
poribus ellluentium .decrescunt secundum numeros impa-
res ordine retrogrado sumptos. Assertio facile ostenditur
e secunda (k')
.
facto successive t::1,2,3.4, .; nam
qnantitates illae prodibunt expressae per
Zn (29 'l), 2&quot; (49 4) 2&quot; (29 1) , 2n(69 9) g(49 4) ,
a—g - .. &quot;£ -
∙∙ 2&quot;(89 16) 2&quot; (69 9) . , seu
ag
- es -
es -
∙−−
7:091). 2&quot; (29 3), 2809 5)sa g(29- &quot;711&quot;;
ideoque etc... Idipsum eruitur ex (In&quot;) et exprime (k');
denotantibus enim sus,, &amp;
.... valores :respondentes tem-
poribus 1,2, 3, ... eae praebebunt
'
z.,:
⋚−⊯≖ ⊖⋅∙ z. ↼−− ⊋⋅⋚⇆≺⊖⋅↿ ):,
≖≖−−−∶⇄−⋚⊑≺⊖∙⊋≻≖∣ za⇌∎
- 2 -' ∙−−− i.— ↿∠∏−−≖⋅ i(ä 3) zo Zn' '
↴
uude ⋅
2, ∙z. ∙−−−⋮⋚≔ (29-1),z,-z, −−∶ Zif-;, (za-3), 22-33 −∙−−∙∸−
s- - ...g. .
ZI€3(29 5), ∙∙∙ Zo-x —-2na ,
et consequenter etc..
Hinc si dividendum sit vas in partes successivis dati tem-210
1 poris a unitatibus vacuandas , determinata altima 20-1
ceterae usque ad primanı erunt
320-7.526-4,72 6-7** (29-3 )z 0-10 ( 26-1 ) 0-1 .
1
1
Liquet autem fore
2:6-1 + 326-1 + 520.4 + 720-1 + . + 20-3)26-17
0
(29-1930_1 = {1 + 29-1)
o
2
0-1 = 622
6-41 1
d
.

103. Tria subjungimus, quae certissimis constant experimentis. 

1º. Vena aquae exilientis a foramine aperto in pertenui lamina magis semper contrahitur usque ad ejusmodi distantiam ab orificio, quae vix aequat ipsius orificii radium; estque venae maxime contractae area cc' ad orificii aream ut 5 : 8 circiter. Istius contractionis ratio ex eo desumenda videtur quod aqueae particulae etiam paullo extra vas retinent obliquos convergentesque motus, quibus orificium subierunt.

2.• Tanta effluit aqua intra datum tempus ex fo
ramine aperto in pertenui lamina , quantam suppeditat for
5
mula ( k &quot; ) , modo tamen pro a substituamus 8

3.º Aptatis orificio exterius tubis cylindricis, co
nicis etc., pro varietate tuborum variae habebuntur quan
titates aquae dato tempore exilientis.

104. Haec notentur 

1º. Acceleratio , per quam velocitas aquae admodum exigua usque ad HH' mutatur in finalein satisque grandem effluxus velocitatem, tota manifeste perficitur ab HH ad cc' intra spatium interceptum conoide ac vena contracta, ubi nempe descendentium stratorum amplitudines citissime decrescunt. Vas ergo ABB'A '
a.
210
poris 9 unitatibus vacuaudas
.
determinata ultima &quot;9-1 ,
ceterae usque ad primam erunt&quot;
3z9-1, 529-1,7z ⊖∙↿∙∙∙ (29-3): ∂∙↿ '
(29-1)z 9-1 ∙
Liquet autem fore
ze, ↿
∎∎⊢∍∅∂∙↿⊣−⋮≖∂ ∙↿ ⊣−⋅∄≖∂∙↿∙⊢∙∙∙↤⊋∂∙⊰≻∅∂∙↿−⊢
. 9 ∙∙∙ :
(29-1)z9-1:(1-1—29-1)ïz ⊖∙↿ —9 294 -
7

1 '.
er
perto
ejus
citci
ori.
spectandum erit tamquam terminatum tubo Hcc'll' ad se
ctionem HH ' aptato.

2.0 Motus aquae defluentis in regularibus alveis
traduci potest ad motum aquae prosilientis ex angustis
vasorum orificiis. Concipe regularem alveun secari plano
verticali ; in plano isto insculpi plura foramina , ex qui
bus effluat aqua ; iisque nova addi foramina ut indefinite
crescat foraminum numerus , totaquc sectio veluti unicum
efficiat foramen infinitis numero foraminibus' coalescens :
cum e singulis foraminibus aqua debeat effluere veloci
tate illa , qua et erumperet e vase ad eamdem altitudi
nem pleno , et , sublato plano , Queret in eodem sectionis
loco , idem ferme erit casus aquae defluentis per alveum
et aquae prosilieatis e vase ad eamdem altitudinem pleno.

3. • Si in regulari atque horizontali alveo mo
vetur inferior aqua ob superioris aquae pressionem , nec
directionum obliquitate , et fundi laterumque resistentia
turbatur conceptus motus , apud particulam quamvis de
notante i altitudinem superincumbentis aquae , exprimet
V 2gi particulae velocitatem.

4.° Quod si regularis alveus ad horizontem ex
sistat inclinatus , sitque m altitudo debita velocitati apud
supremam aquae superficiem , cum haec velocitas ( levio
ribus corporibus aquae injectis determinari potest ) utpote
orta ab inclinatione alvei debeat aquae omni esse
munis , exbibebit V 28 (i + m ) particulae velocitatem .

5.° Hinc poterit in utroque casu definiri quan
titas V aquarum intra datum tempus t defluentium apud
quamlibet regularis alvei sectionem ; sic v. gr. in hypo
thesi rectangularis sectionis habentis latitudinem r , erit
in primo casu
i
2tri.
V = tr įdi
3
no es
دالاق
uibus
fo
o for
com
S, CO
paano
relo
in 6
Dani
?ptom
Vžg
I stra
B!!
in secundo
lCP
perta
ejus-
filicii
, ori-
iot!
.aullo
uibui
⊊∣∝⊦
luan-
«de
new&quot;
ipua
! slfl'
BN
↗−
⋅ 211
spectandum erit tamquam terminatum tubo Hcc'll' ad se-
ctionem HH' aptato.
2.&quot; Motus aquae defluentis in regularibus alveis
traduci potest ad motum aquae prosilientis ex angustis
vasorum orificiis. Concipe regularem alveum secari plano
verticali .; in plano isto insculpi plura foramina , ex qui-
bus effluat aqua; iisque nova addi foramina ut indefinite
crescat foraminum numerus , totaque sectio veluti unicum
efficiat foramen inlinitis numero foraminibus' coalescens :
cum e singulis foraminibus aqua debeat eflluere veloci-
tate illa, qua et erumperet e vase ad eamdem altitudi-
nem pleno , et
, sublato plano ,
(lueret in eodem sectionis
loco, idem ferme erit casus aquae defluentis per alveum
et aquae prosilientis e vase ad eamdem altitudinem pleno.

105. Auctores non pauci tractantes de motu li
quidorum ex apertis luminibus effluentium , illud usorpare
solent tanquam principium , quod nempe unumquodque li
quidi in vase quolibet descendentis tenuissimum et hori
zontale stratum coalescat iisdem constanter particulis com
muni , eaque tantum verticali , velocitate donatis . Deno
tante v verticalem velocitalem , qua pollet in fine tempo
ris i quodvis massae liquidae punctum ( x, y, z ) sollicita
tum gravitate g , vis acceleratris de se valens producere
dy
actualem motum exprimetur ( 28) per : et qaoniam ,
dt
praecisis etiam mutuis punctorum pressionibns , adhuc ta du men vis de gigneret actualem motum ; ideo , attentis pres
sionibus , consistet in aequilibrio punctum (x , y , z) solli
du citatum vi g Propterea ( 88 )
dt
do
dz
dvi
dt (kº ) .
Attenta insuper liquidi continuitate ( liquidum ponitur in
capax compressionis ) ; sequitur , si A designat amplitudi
nem cujusvis strati horizontalis , fore ( 98)
viw = a : A , unde v =
Ä ( * &quot; &quot; ) ;
w est functio temporis t ; A distantiae ; ab XOY : sequi
212
∙∙∙ i. &amp; 3
VZU'l/ng (i-l—m) di: Z',..l/Zg g'[(10 m)⇣⇥≖∶∣⋅ a
denotat i., sectionis altitudinem .

1054: Auctores non pauci tractantes de motu li-
quidarum ex apertis luminibus ellluentium. illud usurpare
solent tanquam prineipium . quod nempe unumquodque li-
quidi in vase quolibet descendentis tenuissimum et hori-
zontale stratum coalescat iisdem constanter particulis com-
muni, eaque tantum verticali .
velocitate dona-tis . Deuo-
tante v verticalem velocitatem
, qua pollet in fine tempo-
ris : quodvis massae liquidae punctum (.r.-7, :) sollicita-
tum gravitate g. vis acceleratrix de se valens-producere
. dv ⋅
⋅ ∙
actualem motum exprimetur (28) per .d—t : et quoniam
.
praecisis etiam mutuis punctorum pressionibus , adhuc ta-
dv
men vis —d—£ gigneret actualem motu-m; ideo , attentis pres-
sionibus
.
consistet in aequilibrio 'punctum (.r.-y. :) solli-
citatnm vi g—⋛⋮ ∙ PrOpterea (88)
der . ⋅ dv
'
z,; ∙−−∶ P- (
−−⋅ (17) ('i ')
-
Atteuta insuper liquidi continuitate (liquidum ponitur in-
capax compressionis ) ; sequitur . si A designat amplitudi-
nem cuiusvis strati horizontalis .
fore (98)
psa-ca: A, lel). ,
undevzr-a— A (
cc est functio temporis :; A distantiae :ab XOï: sequi-213
1
i
tur quoque supremam descendentis liquidi superficiem ma
nere horizontalem . Ex kl( ) habemus
dv a da
a do
dt
aw dA dx
A2 dz dc
-
A dt A do
.
aw dA
a da a’w2 dA
A2 dz A di A3 da
iccirco formula (K ™ ) traducelur ad
do dz dz =+ (sds - au de):
1 sumptisque integralibus quoad % ,
==C+u(sma )
Zo
ic
exprimit 200 distantiam inter XOY et supremam liquidi
superficiem A , Denotante w, pressionem v . gr. atmosphae.
ricam in superficiem illam , assequimur
Two
-= C+4 (** 241,3.
)
unde C=0. – (
( 50-100)
.
li propterea
-=o +15(2-)-avenit SA- G --&gt;) ( 47 ).
Zu
je Apud orificium
213
tur quoque supremam descendentis liquidi superficiem ma-
nere horizontalem. Ex (F&quot;) habemus
dv a de.) am dA d: a dm
.dt—A dt A*dz dc-TA dc
as) ubi a da) ama dA ∙
Aza.&quot; Ad: A3 d.'
iccirco formula (Is&quot;) traducetur ad
' dar − das d:. am: dA
- )
⋅⊋−⋮∁≀∅−−∙↱∙≺⊰∠≀∅−−∅∙∣⊺⋮⊺−⊢ A3 d: dz ,
,. sumptisque integralibus quoad s,
⋍≖⇌∁−⊦⊬≺≊≴−∘≤≀≜∫≖≤≀⋮− −
.-)
,.
dt A 2112
zo
.i-
eXprimit s., distantiam inter XOï et supremam liquidi
superficiem A.,. Deuotante wo pressionem v. gr. atmosphae-
ricam in superficiem illam
, assequimur
2 2 2 2
≔∘−−−⋅∁−⊦⇤∸≼∊≴∘−≦⋏∘∶≕≻ ⋅
....
∁−−⇌≖≖∘
..
(g.... &quot;.? ):
[&quot; propterea
d ad:. 2
1 1
w:eod—Pgu-uþauä A a&quot; P:) (A*—. :) (k&quot;&quot;).
zo

; . Apud orificium214

1
Wo
A2 a
designantibus insuper b et i distantias ipsius orificii ab XOY et ab A. ,
m=b , 2 = b - i , 1 - % = i :
facto igitur
b
dz
A
biI !
erit ibi
mode,
gi - sa- (1-4 ) = (A &quot; .
Quoad (k &quot; &quot;&quot; ) et ( k &quot; ) notamus haec tria. 1.0# Si a est
parvitatis contemnendae , ex (k &quot; ) profluet
a = w.tugis mo ) ,
ut in casu liquidi aequilibrari (88) ; ex (k' ) vero emerget
V2gi ,
quae formula recidit in formulam ( k) .
2.0* Si , affluente novo liquido, eadem servatur
in vase altitudo liquoris , quantitates i, A., B exsistent con
stanles ac datae ; et facto
a2 1 h
A.2
↿ 1
≖⋝−−≖≖⋅∙ ∙ :::—:::;
designantibus insuper &amp; et t' distantias ipsius oriücii ab
XOT et ab A.,.
526 , sozb—t' .
s—sozi:
facto igitur
erit ibi
ad!» c.)&quot;( a2
g' Bdc 2.↿∎∎∎⊼∘⊑≻∶∘ (kl
Quoad (k&quot;&quot;) et (Is&quot;) n0tamus haec tria. ↿∙∘∙ Si a est
parvitatis contemnendae
, ex (k&quot;&quot;) profluet
Uzwoillg (z'—*o) )
ut in casu liquidi aequilibrati (88) ; ex ('tu) vero emerget
∾⋅⇌ vra-u
quae formula recidit in formulam' (lt).
2.0a Si
. affluente novo liquido. eadem servatur
in vase altitudo liquoris. quantitates i. A.,. B exsistent con-
stantes ac datae ; et facto
emuli
a ad
Qiiia215
formula ( k &quot; ) praebebit
h
d
2a Bdt V
2gi
2ada
2gi - hwa hV
2gi
h2 .62
2gi
d h
h
d
a V
2gi
V
2gi
hv
2gil
1+
v
2gi
+

: ) h h
;

-
Vzgi
unde , sumptis bogarithmis quoad basim
a
Bta
log
hy
2gi
V 2gi + hw
V
2gi
ha
non additar constans et arbitraria quantitas utpote =0
siquidem tempori t =o respondet w =o. Ex ista aequatio
ne emergit
Bhty2gi
V2gi(1
a
h
1 + e
Bhiv
2gi
a
inferimus , elapso brevi quodam tempore t, fieri ad sensum
1
V 2gii
itemque
21 5
formula (li&quot;) praebebit
d—L
.,
Bdt— iuda −− 21. V? —
⋣∊∙−⋅∣⇂≖∾≖ hl/Zgi IP
1——c.)2
Zgi
d
a ita di;—00 —( ⇂∕2gi ∣
l/2gi ) h
'⋅⊾ ⋅
l/Z-g—l ↿∙∙∙ .b— 6)
,
⇂∕⇄∃−⋮⋅∾ Vzgi
unde
, sumptis bgarithmis quoad basim e ,
'Bt: ]: a— log iii—E? :
l/th' Vzgi −− hæ
non additur constans et arbitraria quantitas utpote ∙−−−∘ ,
siquidem tempori tzo respondet 6) 30. Ex ista aequatio-
ne emergit
⋅≖∃∣≖≀⇂∕⋝⋮⋜
&amp;
l/2-g-i1—e— ::
∎∎⇀ h B'm/223-
1—l—e— a
inferimus
. elapso'brevi quodam tempore :. fieri ad sensum
1 − −−−−−−↗↓−∎∕∑∊≀⋅⊰
itemque216
= w.tuzia - 2 .) – paga?i( 1 hot G1 - ),-- VE
3.•* Si vas consistit in verticali cylindro , vel pri
smate , A erit constans , et A.=A ; insuper
dz 7-zo 1
A A B
ic
zo

=== Aliquid subjungitur circa generalem theoriam motus corporum fluidorum. ===
106.* Velocitas v, qua pollet if fine temporis !
quodvis massae fluidae punctum ( x, y, z) sollicitatum (86)
vi acceleratrice Q , resolvatur in ternas v' , w &quot; , 1 &quot;&quot; coor
dinalis axibus Ox, OY, OZ parallelas ; erunt ( 29 ) dý ,
1
dy' ' vires iisdem axibus parallelae , in dt dt
quas resolvitur' vis acceleratrix q' valens de se produ
cere actualem motum. Quoniam , etsi praecisis pun
cloruni mutuis pressionibus , adhuc tamen gignit actualem
motum ; ideo , attentis pressionibus , consistet in aequi
librio punctum ( 2, y, z ) sollicitatum viribus X
, ) – , dv',
1
Y dy&quot; , 2 dy'&quot;'; ac proinde ( 86. o )
di
1
dt
de = - ( x – do ). -- ( v- à dv&quot; ) , )
de u ( 2-2 &quot; )..
( 6)
216 .
↴
a':' 1 ↿ a Vii—g'
∙
szo-l-pg(2'.—20) uia (Aa Ag) ,
VS—K ∙
T
∙
∶⊰∙∘∙ Si vas consistit in verticali cylindro. vel pri-
smate
, A erit constans, et A.,:zA; insuper

:

Zo
Aliquid subjungitur circa generalem theoriam motus
corporum [[ uidorum.

106: Velocitas 0, qua pollet i! fine temporis :
quodvis massae fluidae punctum ( æ, y, z) sollicitatum (86)
vi acceleratrice ?, resolvatur in ternas v'. 9&quot;
.
v'&quot; coor-
dinatis axibus OK, OT. OZ parallelas ; erunt (29) ;llg-dvl ,
,, 1 ∙∙∙ ∙ ∙∙ ∙ ∙
(Z— dv
, &amp;? dv vires iisdem axibus parallelae ,
tn
quas resolvitur vis acceleratrix ?' valens de se produ-
cere actualem motum. Quoniam (a' .
etsi praecisis pun-
ctorum mutuis pressionibus, adhuc tamen gignit actualem
motum; ideo , attentis pressionibus , consistet in aequi-
librio punctum (z, y, :
) sollicitatam viribus X — −↿− dv'.
dc
y— —dv&quot;,Z—- dc dv' ; ac promde (86. o)217
1
dt ,
dy
du
dx
axt
du' !
(du ?
Habitis v ', &quot;.0&quot;, pro functionibus variabilium x, y, z, t,
exsistent ( 27. 24.0)
dy' du dv du du' dx + dy +
dzt.
dix dz dc
dur dul
dy + dz +
dt,
dy dz de
du du dy't dy + dz + dt ,
dx dy dz de
du du=
dr
seu ,
ob dx v dt , dy udt , dz udt ( 27 ) ,
dv dú dv ' dun du'
dt ,
dx dy dz de
dur dvd dv du'a + dt ,
dy dz de
( 6 ).
dy'll dy dy &quot;
\dx dy dz
dat
di axt du.
du
dyt
dzt dz dt dy
dl ;
at
will +
leo
lesin
' du'
\dx vt
alt
- de
dy ut
21&quot; + .!&quot;
to
at
dt ,
ide
dx du du.
vt
de ede :
w
itot
dy
dz
formulaeque (6) vertentur in
15
1
1
217
Habitis v'. v&quot; . v'&quot;, p. pro functionibus variabilium x,]. z,t,
∙ exsistent (27. 243) '
/
dVr-äï-I-dæ-f-g dy—l—d 7; v/dz—l-dïvt-dth
.
I, I/
dp'p&quot;: &quot;
≤⋮∙−≤⋅⇗↙↙∠∞∙⋅∣− dv ∙−∙↙∣∫⊣−↙≀↥≟ dz-l— ii)—dt,
III '&quot;
dv'&quot;: dv Tdæ—i—djr dv/Il d-v'&quot; dy −⊸⊢−−⋅ zdz—l- -^——--dt,
d d ⊬−− ⊋⊥∸↙≢↕ ↙∄↕≤−⊦ Hari- ⋮⋮↙∄≖⊣−−↙−∣≛⋮∠≀≀⊰
seu , Ob dx:
'v' dt .ei)-':«:;&quot;dt, dz 37)/&quot;dt (2&quot;..
dv! ∣∙∙− d'&quot;, v&quot; ∣∣ (if—,) dv—(ïr-v-l-ï P-l-d—z-IVI-ï-dï— dt,
&quot; &quot;
vl] II
V&quot;'—:(d—; V, &quot;J—d gr.&quot;- .v/j-l—g-z- will-i-ïi'l;-—-)d[,
(V).
dv ∣−− dvlll dui/I
−−−−⊋⋤−≼ .
[v;/1.&quot;, dv'&quot;
,
∣∣
), 1
∎∎⊢∎∎∎−∎−∎ dz 'l&quot; dt —)dt ⋅∣⊹ '
∙−− dP'. ,v/ !
dlu' ut dp' ∣∣∣ dp') ∙
dy. (dæ'v [
dy&quot; ⋅−↱⋅⋅∓⇂≀ −↽⊋−∁⋅− dt.
formulaeque (6) vertentur in
15218
dos deild dy&quot;
dx
dv'
dx dy
dz che si (
(v
do ( 6&quot;) v'
do'
dy
v
du
dz w
dur
dy
dx de - ) ,
dy ')
do dv'

: -(2

v'
dy't
dy
v
du Win
dz dx dz de
107 #. Quae portiuncula infinitesima massae fluidae a
pud punctum ( x , 3 , 2 ) sub volumine V in fine tem
poris i exprimitur per V , eadem sub volumine V+dV
in fine temporis + dt ad punctam aliud translata expri
metur per ( V+dV ) ( pe + du ); ideoque
V = V + dV) (v + dpl)= Vu + udV + Vdp. + dp.dV ,
et consequenter, misso dudv,
Vdp. + pdV= ( 6 &quot; ).
Sumatur V = dxdydz, aequale nimirum parallelepipedo
rectangulo AF ( Fig. 47. ) sub laterculis AD( =dx) , AB( =
dy ) , AH = dz); punctaque A , B, C , D , H , M , F , E po
nantur transferri tempusculo de ad A ' , B , C , D , H' , M',
F' , E , ut sit V + DV = A'F'. Transferetur A in A '
velocitatibus d' , 0 , 2, juxta coordinatos axes ,
runtque
e
x + v'dt, y tv&quot; dt , z tudt
coordinatac puncti A': designatis v ', u ' &quot; per
d7 dx d] dz : '
,
dm' dv&quot;' −∙∙: Z- ∣- dv'&quot; du''' dv'&quot; ∣∣'''
∙∙∙ ∙−−⋁∣∣∣∙− —) ∙
dz F
(
da: v dy 'v dz dt
107-. Quae portiuncula infinitesima massae fluidae a-
pud punctum (æ
,
I,: ) sub volumine V in fine tem-
poris : exprimitur per VP-o eadem sub volumine V-l-dV
in fine temporis t −⊢ dt ad punctum aliud transl'ata expri-
metur per ( V-l—dV) ( p. dy. ); ideoque
Virsz-l-dV) (p.-l—dp.) ∙−−∶ ⋁∣↓∙⊹ ⊦∙∠≀∇−⊢ ∇⊂∣≴⊥∙⋅⊢ dde .
et consequenter, misso dpdV,
. Vdp. −⊦ ⊬↙∣∇−−∶⋄
('b'&quot;).
Sumatur Vzdædydz, aequale nimirum parallelepipedo
rectangulo AF (Fig. 47.) sub laterculis AD(-:-:dæ) , AB(-—--
dy ), AH(-:dz); punctaque A, B, C, D, H, M , F , E po-
nantur transferri tempusculo dt ad A' , B' , C' , D' , H' , M',
F', E' , ut sit V −↿− dV −−∶ A'F'. Transferetur A in A'
velocitatibus v'. a:&quot; ,
∙⇂∙∥⋅ juxta coordinatas axes , e-
runtque ∕∕∕ '
..: ⊣−⋁∣↙∣⊀ ∙
]−⊦ wa: , z −⊢ war:
coordinatae puncti A': designatis v', v&quot;, «a'/' per219
fi( x , y , %, t ) , fa(x , y , z , 1), 13 (x , y , z , t) ,
expriment
fi (x , y , z + d2, e) ,fz(x , y , z + dz, t ), f3( x , y, z + dz, t)
velocitates coordinatis axibas parallelas puncti H euntis in
H '; et cum babeamus ( 27. 24.)
filx9,2 + dz,t) = f (x , y ,z, e)7df1(x,y,z,e)dz = uti
du
dz dz ,
dz
e ao
em dy ”
fa (x , y, 2 + dz, t ) = 0&quot; +
dz ,
dz
spri.
f3( x , y , z + dz, t ) = 0 !!! allt dv !
dz,
dz
IV ,
coordinatae puncti H'erunt
X
+ (v + da )dt,y + (&quot;* + de )de,

: +de+ (** + adaptada dt:

pipedo
AB =
E po inferimus, missis infinitesimis tertii ordinis, fore ( 50. 6º. )
1 , M.
in 4
5 ,
ee A'H' =
[ledesdeu + )de de +
) ]=d =+
dy &quot;
-dz dt dz dt .
dz
Motus puncti Cin C'juxta coordinatos axes fiet velocitatibus
!
15.
C ∎∙
em-
[pl'l'
IV.
. 219
fuci-'s], 3! 1), fa(æs)'s 3! t) ∙ ⊀∍≺⋅↕∎∙∫↿≖∙ :),
expriment
ftlæsfaz-l'dzs 1) ,falæoys z 'l'dz; t)sf3(æs ïs ≖∙∙∣− d:, 1)
velocitates coordinatis axibus parallelas puncti Hieuntis in
H'; et cum habeamus (27. 240.)
dfx(æJ,z,t)d fax-a',: 4—dz,t):f.(æ,y,z, :
dz dz—v ↾−⊦↙↙∙⋚∙ —dz,
&quot;
falæsïsz (I:-',! :):vlf'i'd'ä- dz:
d'UIII fave,], z-l-dz,t ): ⇝∣∣∣−⊢ —zdz.
coordinatae puncti H' erunt ' &quot;'
ahi-(» ∣⊣− −↲≖≻≳≀∙∫−⊦≼↙∣⊣−≝∂≖⋟↙∦⋅

: ↽⊦ dz −⊦ (W.;- ↙⋛↙−≖

d: )dc.
inferimus, missis infinitesimis tertii ordinis, fore (50. 60-)
A'H': RSTV) dz-dz −⊦≺⋅⋮↷⋛↗−−−≖−∥≖⋟↙≀≖≏
d:: −⊦
d.,/II dv!&quot; -
(d:-[- d: dzdi )]ä :dz-F—dzdt-
Motus puncti Cm∁∣ juxta coordinatas axes fiet velocitatibus220
falar + dx , y + dy, zil ) = fi(x , y , 2,1 ) +
afı( 8•
7,5,6)
det dfi (x , y , 2,1) du
dy
dy
dic = tIdxt dx
falx + dx , y tdy, 2, 1 ) = &quot;&quot; + -dat dx
du &quot;
dy ,
dy
dumi
dy !!!
ON +
f3(x + dx ,y + dy , z , 1 ) = &quot;&quot; dat dx dy ;
dy
inde prodeunt coordinatae puncti C
du
d )
dy x + dx + (v + ad det )dt
y + dy + ( ** + na tempat day ) di,

: + ( v&quot; + data darym dy de :

motus puncti F in F ' juxta coordinatos axes fiet velocitatibus
du'
filxtdxy + dy,z + d2,2)= x + xdx + dydy +
du
dz,
dz
dy&quot; du&quot;
falxtdx,y + dy,atdz,t)= &quot; + de + dy dy
du &quot;
da dz,
du &quot;
du
f3( x + dx, y + dy; z + dz,t) = 1 &quot;'
-dxt
dy&quot;
dy dy +
dz;
dx dz
inde exsurgunt coordinatae puncti F
220
fdæ'i'dng—i-dy- ≖∙⋅↕⋟⇌∣≖≺∝⇟∫∙ ze t)&quot;,-
df,(æ.j,z,t) dfl(æsyszvt)
dw'd d &quot; ≀∂≖≺∙↿⊏−≱⊢↙↕↡∫↽⊢∂∫⋅ ≖↿−⊸⋅⋅⇂∙∥⊣−↽⊋− &quot;L &quot;ad; df '
f3(æ-l—dæ,y—]-dy,z, :
)-— v'∣∣⊹−⇁∙
inde prodeunt coordinatae puncti C'
∙↴⊲−⊢∠∄∸≀∶⊣− (⊣−⋅≦⋮∠↴↧⋅↕⊣⇀−− ↙∄⋤↙↿∫⋟≴↙≀⋅
∫∔∂⋮∫⊹ (
∣∣⊣↼ d,,⇡⋮≀−−⋅∶≴←⊦≤−⋚−∥⇩≀ wa.)
vll/ du ≖↽⊦≺⊛ ∣∣∣
w&quot;'-l--d—-æ dr—l— df )dz:
motus puncti Fm F'luxta coordinatas axes Eet velocitatibus
∙ . . . '
' d ,.
fia—W;? ∂∫∙≖−−∠⇣∙≀⋟∶⋁⊹⊼∶↙⊩⊦≣↗ −⋤−∶⊔⊹ −:dz, dv&quot; dv&quot; dv&quot;
ta(æ-i-er-l-dr, a—l—dzn): ⊎∣∣⊣⋅∙⋣∂≛−⊦ df dy ! dz ds,
d&quot; III d..&quot; f3(æ—]—dx, ⊹↙∄∙↗∙≖−⊢∣≂∙ ():—Ju&quot; i-i-d; dælL dr cir—]— —-dz;
inde exsurguut coordinatae puncti F'221
dyn
= da + ( + van de tener tous de Jdeo
s + d3 + ( v +adar an nas tudi nadia )dt,
s + de + ( * + dpt dathetn dy + advan die Jde:
1
inferimus, missis infinitesimis tertii ordinis, fore
CF = [ 'de de + oem )deº de +
( de + de &quot;de de ))]]* = da +
dy&quot;
de dt.
dz
Ad motum puncti B in B ', computatum in coordinatis axi
bus, spectant velocitates
f( x, y +dy, z, t ) , falx , yt dy, z, t), f3(x ,y + dy, z, t ) ;
ad con similem vero motum puncti M in M' velocitates tatibus
fi(x ,y + dy, z + dz, t) , 82(x , y + dy ,ztdz, t ) ,
th
dan
f3(x , y + dy , z + dz , t ) :
dy&quot;.
propterea coordinatae puncti B ’ desi
dz
+ (x + dy dy )de , y + dy + (.* + dar dy ) dt,
du&quot;,
dy
7
+ (*&quot;'+ dydy hdi:
7;
(221'
I
æ—l—dæ-l-(tb -]-d −∙⋮dx-j—d −−∣vlddy-l— ——dz )dt,
,,
'
dv&quot; . dv&quot; y-l-dJ-l-( −⊢−− da.−−∥↙≀↓⊣− ⊒∫−∠≀∫−⊢−− ↙≀≖≻∠≀∁∙
&quot; dv&quot;: z-l—ds-l—(&quot; ⊣−≦−≦⊥∅≀∝−⊦↙∄ dyd ∣
↙↙≖ d: )dt:
unferimus, missis infinitesimis tertii ordinis, fore
∙∙∙⋅ dv) ∙ '
(du&quot;ïd) ∙ es'—[(? d: a: &amp;
∠≀∥≀⋍⋅−⊦
≺∁≀≖−⊢⋛≖ ——dzdt )]; :dz—l-Q—ds dt.
d:.
Ad motnm puncti B in B', computatnm in coordinatis axi-
bus, spectant velocitates
I.i-ïs;)&quot; ⊣∙∙ dy. 39 i)
'
f2(æay—l— d]: 2. t)af3(-'rsy—l—dft Z, !) 3
ad consimilem vero motum &quot;puncti M in M' velocitates
↿∎≺∞∙∙↗↾⊣−∠∄∫∙≖⊣− dzs t) sf2(æ sy'l—fi'r, ≖−⊦∠∄≖ ∙ :) ∙
fam ,y-l-dy.a-1- a.:):
propterea coordinatae puncti B'
æ-i- (&quot;'-l- ⋛⋚∠∄∫≻↙∄∁ ,J—l-dy—l-('
≻≖≀⋅≂⋮ &quot;j,-l— 72:41)!&quot;-
z −∣⋅− (vm-I— dv
dy222
coordinatae puncti M
++ (1 - en deJdt,
y + dy + (** + disa dy + &quot;deJdi.
z + dz + (** +
(** + + en in diehele
hinc
B'M dz +
du
dzdt .
dz
Ad motum puncti D in D ', computatum in coordinatis a
xibus, pertinent velocitates
fi ( x + dx, y , z , 1 ) , fa( x + dx,y ,z, 1), f3(x + dx, y , z ,t ) ;
ad consimilem autem motum puncti E in E' velocitates
filx + dr, y, z + dz, t ) , f (x + dx, y , z + dz , t ) ,
f (x + dx ,y , z + dz , t ):
proinde coordinatae puncti D'
de&quot;

* + dx + (ut ea adx)de,y + ( * + dxdx )dt,

++ (- + de -dx)dici
222
coordinatae puncti M'
x-l—(tf— ⋛≶↙≀∫⊣− ——dz)dt,
maH-( ⇂≀⋅⋛−−⊦ pri—4449
≖−⊦↶≀≖↼⊦≼ ⋮⋅∠⋛⋮−⊣− MH-;'dzdu)
hinc
,B'M'-— ∙−− tis-l- -—-dzdt.
Ad motnm puncti D in D', computatum in coordinatis a-
xibus, pertinent velocitates
ru(æ ∙−⊦ ciæ,], zit)1fa(æ &quot;l'dæofszo t) sf3(x&quot;'i&quot;dæs)'; 2; 1);
ad consimilem autem motum puncti E in E' velocitates
fdx—i-dæqæz-l-dz, t ) 'fa(x-l-dx,y , z—l—dz .
t ).
B(æ-j-dæ ,y, s--]— dz .: ):.
proinde coordinatae puncti D'
∝⋅⊦⊄↿↕∸−⊦≼↩∙−⊦ ——dæ)dc ,y—l—(v' −⊦≤−−⋮⋅⊑⋅↙≀⋅⊐∁⋟↲↥∙

: ∙⋅⊢ ≺∙∽⋯∙⊢ ⋛⋮≽∙⋮⇣↿∙↕≻≺≀∷223

puncti autem E
drt
s + dx +((uv + des de +die dz)dt ,y+ (** + de la
de &quot;a )de, a (* &quot; + dz) dt;
2 + dz to dv&quot;&quot; dy&quot; ,
det da dz
et consequenter
D'E' = dz to
du&quot;.
dz dzdt .
Itaque
A'H ' = C'F' = B'M ' = D'E' = dz +
du 
dzdt :
đz
simili modo eruuntur
AD = B'C ' FM H'E di = dx +
dx
dxdt ,
1t ) ;
A'B' C'D FE H'M ' =
dú'
dy + dydt.
dy es
thi
Ex laterculorum aequalitate manifeste consequitur eorum
parallelismus ; eritque A'F' parallelepipedum obliquangu
lum ; ita tamen , ut ejus anguli infinities parum diffe
rant ab angulis rectis parallelepipedi rectanguli AF ; quan
doquidem AF nonnisi tempusculo infinitesimo transfer
tur in A'F ' . Nunc ex H ' v . gr. due perpendiculum Ha in
areolam A'B'C'D ; erit
A'F ' = H'a . A'B'C'D' = H'a . A'B ' . A'D' sio B'A'D '
A'H ' . A'B' . A'D' sin B'A'D' sip H'A'a :
d:.
223
puncti autem E'
dv' '
dv' )
.. da: −⊢ dz z .7' 41- dæ—t- ∙⋅∎∙⋅−⊢ du:-1- (v' ∙∙⊢ da:
dv&quot; dv'&quot; dv&quot;' )
−∙ dz —-d.r —-d d ;
dz)dt ,
z-t-dz-tï 'v. −⋅⊢ da: ∙−⊢ ds :
f
et consequenter
⋅ dv&quot;:
D'E'c: dz −⊢ 71&quot;—2. dzdt .
Itaque
llo
'
'v
A'H':C'F' −−∶ B'M' ∙−−∶ D'E' :: d: ⊣⋅− ∙−≀⋮⋅≖−↙≀⋍⊄∄↥ :
simili modo eruuntur
A'n' :: B'C' −−∶ F'M' −−∶ H'E':dæ −⊢↙≟⋛ dædt
.
A'B'r: C'D' ::F'E': H'M' ∙⋅−−−∸ dy ⊣⋅−∙≣⊥⋅ ↙≀∙↨↾∠∄≀⋅∙ ]
Ex latel-culorum aequalitate manifeste consequitur eorum
parallelismus; eritque A'F' parallelepipcdum obliquangu—
lum; ita tamen , ut eius anguli infinities parum diffe-
rant ab angulis rectis parallelepipedi rectanguli AF ; quan-
doquidem AF nonnisi tempusculo inünitesimo transfer-
tur in A'F' . Nunc ex H' v. gr. due perpendiculum H'a in
areolam A'B'C'D' ; erit ∙ ∼
A'F' ∙−−− H'a . A'B'C'D' ::
'
'a . A'B' . A'D' sin B'A'D' ∶−−⋅≖
A'H' . A'B' . A'D' sin B'A'D' sin H'A'a :224
denotantibus w et w'angulos infinitesimos , poterunt anguli
B'A'D ' , H'A'a repraesentari per 90º + w , 90 ° +6 ; iccirco
sin B'A'D' sin H'A'a = sin (90 ° + w ) sin ( 90° + W' ) =
62 614 w'4 coswcos6= ( 1 -... ) ( 1 ...) . 2 2.3.4 2 2.3.4
Quare , missis infinitesimis quinti ordinis ,
dv &quot; A'F ' = (dx +
dv' du &quot;
dxdt) (dy +
dx dzdt) x
dy dydı) (dz + az
wa
du
( - -
(1-7 • det dy&quot;
• det du&quot;
dt) dxdydz ;
2 dy
dz
ideoque
dV = AF - V =
-
(dv
dx- det
-det
dy
di) dxdydz.
His positis , vertelur ( 6 '' ) in''
dxdydzdje ele
dv
\dx
dtot dt +
dv&quot;
dy
dz di )dxdyds= 0,
seu ( 106.6' )
dje du ut u'+
dr dz
djelo
du +
dy
dt
dvi dv&quot; du &quot; .
dy + demon dz ) =
(619) .
dx

108. * Si massa fluida est incapax compressionis,
unaquaeque particula immutabilem habebit densitatem
eritque du = o : proinde ( 106 , 6')
224
denotantibus eo et tu'-angulos inünitesimos , poterunt anguli
B'A'D' , H'A'a repraesentari per 900-l-cu , 900-l-co'; iccirco
sin B'A'D' sin H'A'a −−∶⋅ sin (90&quot;-FG) sin (900 ⊣−∙ w') ∶∸⋅
. ↿ a): 034 ↿ tu' te'-': )
cosmcosw—( ∙−⋅⋍−−⊢≳∙∙⋝⋅∕⇂∙−−⋯⋟≺ −−∙⋮∙ m—m .
Quare , missis infinitesimis quinti ordinis
,
I d'u' dr&quot; d'v &quot;
'
'
∙−−−
− −
'
AF ∙−− ∙−− (dx—t- dædxdt,(dy-t—- d] afydt) (dz-t- az dzdr) )(
a 'a .'
&quot; '&quot; (1 &quot;2' −−∘≩≻∙−− ≺↿⊹⋛⊰↲↥⊹≘⋚∂∁⊹↙≩⊤∶∂∊≻∂∅∂∫∂∥
ideoque
(lv-.:A'F' v
(
vd: : dv dc.-92:11) dædyde .
dæ dy dz
His positis , vertetur (b&quot;') in
dt&quot; I),, vl'l dæd.) dzdp-t— &quot;(c'ïx dc ⊣−∙ 217 dc ⊣−∙ 72- dt)dædydz ::o,
seu (106 . b')
&quot;&quot;
↿≀∙⊣⋅− d&quot; ∣∣⊣− a'&quot; &quot;-4-'-'—'-' −⊢

: 17- 2?&quot; f??&quot; dt

d'v- ⋅⋅⊢ dv&quot; dv'&quot; ⋅
⋅⋅∙∙− o
F- b,; ( ⊣⋅−⋮⋤ )
—- ( ).
108.s Si massa fluida est incapax compressionis,
unaquaeque particula immutabilem liabebit densitatem ,
eritque dy.:o : proinde (106
. b')225
de
vt
die du du v &quot; + 2 &quot; +
=
0

;

dy dx dz de
et consequenter
( 107.b
)
(
6
)
dv dy
d.x
+
+
dy dz
Formulae
(
6
&quot;
)
,
(69) suppeditant incognitas
a
,
l
,
v
,
v
&quot;,
v
&quot;
.
expressas per
x
,
y
,
z
,
t

; obtentis autem

v
'
,
v
&quot;
,
?,
&quot; per xy
, zat
,
eruentur
x
,
y
,
z per
t ex formulis
dy dx dz
dc
-
”
dc
ru!!!
dt
Si massa fluida incapas compressionis est insuper ho
mogenea
, prima
(
6
&quot;
) fiet idemtica
, satisque erunt
(
6
&quot;
) et
secunda
(
6
&quot;
)
.ad incognitas
,
u
'
,
v
&quot;
v
'&quot; determinandas
. De
mum si massa fluida pollet elasticitate
, formalis
(
6
&quot;
) et
(61
)
jungenda erit formula
(
o
&quot;
. 87.6
'
)
.

=== De tubis capillaribus. ===
[[Fasciculus:Capillarity.svg|thumb|Capillares]]
109. Etsi liquidum homogeneam in vasis communicantibus (92.1º.) manet aeque altum, iu tubis tamen vitreis admodum angustis (dicuntur capillares) utrinque apertis, et altera extremitate demersis aquae vel hydrargyro, cernimus aquam suprema superficie concava terminatam ascendere supra horizoutalem circumambientis liquidi superficiem, hydrargyrum vero suprema superficie convexa terminatum descendere infra horizontalem circumdantis liquidi superficiem: ad istius modi ascensum descensumque explicandum, haec animadvertimus.

1º. ln phaenomenis gravium liquidorum expendendis gravitatem considerantes haud habuimus rationem sive virium quibus liquidi particulae se mutuo petunt, sive virium quibus vasorum materies ad se trahit particulas illas. Porro materiales particulae duplici pollent vi attractiva; altera se prodit utcumque crescant distantiae, sequiturque (82) rationem reciprocam duplicatam distantiarum; altera se prodit dumtaxat in contactu vel quamproxime contactum, sequiturque rationem quamdam distantiarum nondum compertam. Ubi sermo est de liquorum aequilibrio, possumus ab attractione primi generis absque sensibili errore praescindere: ad attractionem secundi generis quod pertinet; cum in contactu exsistat validissima, inde fit ut suprema liquidi superficies prope vasorum latera induat figuram curvam, modo concavarn, modo convexam, et nonnisi ad aliquam ab ipsis lateribus distantiam dici queat physice horizontalis. Exhibeat TT' (Fig. 53) verticalem tubum v. gr. vitreum, utrinque apertum, et infra horizontalem liquidi superficiem partim demersum; O centrum circularis areae tubo interceptae apud eam superficiem; A particulam liquidi in area ista sub actionem vilreae particulae R; OX rectam transeuntem per A; OY horizontalem rectam perpendiculariter insistentem rectae OX; OZ verticalem rectam. Si denotat vim qua A tendit in R, designatis per h, k, i cosinibus angulorum quos AR facit cum ox, oy,
OZ, resolvetur in ternas
ph , pk , ọ
iisdem OX , OY , OZ parallelas: ex R in planum XOY
ducatur perpendiculum Rp , producaturque in R' donec fiat
R'p = Rp ; teadet A in R' vi aequipollente ternis
ch , pk , - oi : demissis perpendiculis ex R , R' in planum Xoz , iisque productis donec productiones aequentur ipsis perpendicu
226
rium quibus liquidi particulae se mutuo petunt, sive virium quibus vasorum materies ad se trahitparticulas illas. 

manifeste determinabuutur in tubo duo puncta , quorum vires dabunt componentes
gh , - ok , pi ,
sh , - ok , - qi :
in ferimus particulam A , elisis componentibus parallelis
rectae OY , itemque componentibus parallelis rectae OZ ,
sollicitatum iri juxta AX vi
4Σ φh
proveniente ex tubi materia. In OX sume Ab = Aa ; duc
verticalem bb' ; et quod in ordine ad tubi materiam est q,
in ordine ad liquidi materiam sit q' : quisque intelligit par
ticulam An elisis componentibus horizontalibus, trahi ver
ticaliter deorsum vi
4 Epi
promanante ex liquido intercepto superficie cylindrica , quam
general recta bb' dum sibi constanter parallela movetur in
peripheria circuli habentis radium Ab. Liquidum ultra
superficiem cylindricam praebet vires
.
- 2 Eph , 2 &quot;pi,
alteram horizontaliter agentem juxta XO , alteram verti
caliter deorsum. Omnes itaque vires sollicitantes particulam
A traducentur ad horizontalem
4Eph -2Ep'h = 2 [2Eoh —Eph] ,
et ad verticalem
45' pit 23&quot; ' i.
227
lis, manifeste determinabuutur in tubo duo puncta. quo-
rnm vires dabunt componentes
9ht—9k09i'
-ph.-—9k,—qn':
inferimus particulam A .
elisis componentibus parallelis
rectae 0?
,
itemque componentibus parallelis rectae OZ
,
sollicitatumeiri juxta AX vi
4297:
proveniente ex tubi materia. In OX sume Ab: Aa; duc
verticalem 65; et quod in ordine ad tubi materiam est p,
in ordine, ad liquidi materiam sit go' :quisque intelligit par-
ticulam A. elisis componentibus horizontalibus, trahi ver-
ticaliter deorsum vi ↽
42'9'i.
promanante ex liquido intercepto superficie cylindrica, quam
generat recta bb' dum sibi constanter parallela movetur in
peripheria circuli habentis radium Ab. Liquidum ultra
superficiem cylindricam praebet vires
-— 2
297: , 2E'p'i ,
alteram horizontaliter agentem juxta KO, alteram verti-
caliter deorsum. Omnes itaque vires sollicitantes particulam
A traducentur ad horizontalem
4ng −− ⇄∑∲∣∣∣:2929]: −∙− ∑⊈⊅⋅∣⋅⊐ .
et ad verticalem (f)
42: p'i-l- 22&quot;qa' i.228
Potest 2Eph -Eph esse aut &gt; o , velo, vel = 0: in primo
casu vis aequipollens et gravitati , et binis (f ) , deviabit a di
rectione verticali faciendo angulum acutum cum AX; et
quia ( 83.3º. ) vis illa debet normaliter sese dirigere ad libra
tam liquidi superficiem , ideo suprema liquidi superficies in
duet curvam concavamque figuram : in secundo casu vis
aequipollens et gravitati, et binis (f), deviabit quidem a ver
ticali directione, sed faciendo angulum oblusum cum AX ;
propterea ( 87. 3 • ) suprema liquidi superficies induet curyam
convexamque figuram : in tertio denique casu ex duabus (8)
remanebit sola verticalis, et consequenter suprema liquidi
superficies erit plana atque horizontalis.

2º . Massae liquidae OS , OS' (Fig. 54 ) ejusdem
naturae, planisque superficiebus OP , OʻP ' terminatae, ae
qualiter trahunt exilissimas columnellas liquidas AR , A'R'
perpendiculariter superficiebus ipsis insistentes; nisi tamen
columella AR intra massam OS trahitur deorsum , columel
la vero A'R' extra massam O'S' trahitur sursum. Intelligan
tur enim centris A et A ', radiisque aequalibus AB et A'B ',
ultra quos sensibilis attractio liquidi non protenditur, des
cribi hemisphaeria BCD , B'C'D' : si vires , quibus hemi
sphaeria agunt in particulas A, A ', resolvuntur in binas, alte
ram horizontalem , alteram verticalem; elisis horizontalibus,
remanebunt verticales, quarum summa est eadem utrinque
cum hoc tantum discrimine quod A trahitur deorsum et A
sursum. ln columellis sume nunc duo alia puncta E, Eʻae
quidistantia ab A , A' , radiisque aequalibus EL, E'L ' ( = AB)
describe segmenta sphaerica FML, F'M'L' : accepla EV=EA,
ductoque plano horizontali HK, vires ex QHKN , QFMN
in punctum E utpote aequales et contrariae se mutuo de
struent , ipsumque E solo segmento HLK deorsum trahe
tur : vis ex HLK deorsum sollicitans particulam E ae
quatur vi ex F'L'M ' sursum trahenti particulam E'; siqui
dem haec segmenta sunt aequalia et similiter posita ad
contrarias partes quoad E et E '. Cum igitur idem redeat
228
Potest 229h—29'h esse aut) a. vel( a, vel:o: in primo
casu vis aequipollens et gravitati, et binis ([ ), deviabit a di-
rectione verticali faciendo angulum acntum- cum AK; et
quia ( 87. 30.) vis illa debet normaliter sese dirigere ad libra-
tam liquidi superficiem, ideo suprema liquidi superficies in-
duet curvam concavamque figuram: in secundo casu vis '
aequipollens et gravitati, et binis (f), deviabit quidema ver-
ticali directione, sed faciendo angulum obtusum cum ax,
propterea (87. 3&quot;-) suprema liquidi superficies induet curvam
convexamque figuram :in tertio denique casu ex duabus (f)
remanebit sola verticalis, et consequenter suprema liquidi
supedicies erit plana atque horizontalis.
2&quot;. Massae liquidae OS , US' (Fig. 54) eiusdem
naturae, planisque superficiebus OP
, O'P' terminatae, ae-
qualiter trahunt exilissimas columellas liquidas AR , A'R'
perpendiculariter superficiebus ipsis insistentes; nisi tamen
columella AR intra massam OS trahitur deorsum. columel-
la vero A'R' extra massam O'S'trabitur sursum. Intelligan-
tur enim centris A et A', radiisque aequalibus AB et A'B',
ultra quos sensibilis attractio liquidi non protenditur,des-
cribi hemisphaeria BCD , B'C'D' : si vires , quibus hemi-
sphaeria aguntin particulas A, A', resolvuntur in binas, alte-
ram horizontalem , alteram verticalem;elisis horisontalibus,
remanebunt verticales, quarum summa est eadem utrinque
cum hoc tantum discrimine quod A trahitur deorsum et A'
sursum. ln columellis sume nunc duo alia puncta E, E'ae-
quidistantia ab A , A', radiisque aequalibus EL, E'L' (::AB)
describe segmenta sphaerica FML, F'M'L': accepta EVzEA,
doctoque plano horizontali HK, vires ex QHKN , QFMN
in punctum E utpote aequales et contrariae se mutuo de-
struent
,
ipsumque E solo segmenta HLK deorsum trahe-
tnr : vis ex HLK deorsum sollicitans particulam E ae-
quatur vi ex F'L'M' sursum trahenti particulam E'; siqui-
dem haec segmenta sunt aequalia et similiter posita ad
contrarias partes quoad E et E'. Cum igitur idem redeat229
cimo
di.

; et

bra
sin
vis
Ver
LAX;
ryam
argumentum de caeteris particulis inter A et C , necnon
inter A ' et C ' ( ponimus A'C &quot; — A'C' ) , cumque particulae
infra C viribus contrariis et aequalibus urgeantur, infra C
sensibili non subjiciantur actioni, jam patet etc
In eodem liquido vis, qua deorsum vel sursum colamella
trahitur, constans est; eam in sequentibus exhibebimus per K.

3º. Fac ut massa liquida BAB'QQ (Fig. 55) , quae intercipitur superficie sphaerica BAB' et plano tangente
QQ, trahat externam columellam liquidanı AR perpendicula
riter insistentem plano tangenti apud contactum A : quo
niam BAB O'Q gignitur rotatione areae ABQ circa ra
dium AO, perinde erit sive spectetur attractio massae BAB'Q'Q,
sive spectentur attractiones infinitarum numero superficierum
cylindricarum , quae in ea rotatione gignuntur a perpendicu
lis DC , D'C' , ... demissis ex punctis D , D ' . circu
laris arcus BA in rectam QA. Exprimant p , pi ... per
pendicula DC , D'C', . . ; 9.9 , .. perpendiculorum di
stantias AC , AC' . .. ab A computatas in AQ; sitque r
sphaericae superficiei radius OA: ob magnam lineolarum
p , p ', . . . tenuitatem prae q, , ..
quidi
Eden
A'R'
ameo
amel
gaD
AB,
des
erunt
lemi
aleo
р
92
2r ,pa2r
libus
Bogu
et A
et consequenter praefatae superficies cylindricae exhibe buntur per
-AB
EL
MY
2πη 비유
Toq3
,2πα g's Tig'3
seu 9
dem 2r 2r
cabe
ae
aqui.
Atqui ob eamdem illam tenuitatem puncta uniuscujusque su :
perficiei cylindricae haberi possunt pro aequidistantibus ab
unoquoque columellae puncto: designantibus igitur o, o, ..
quantitates pendentes et a certa quadam distantiarum lege,
deal
sit
ac-
qui'
de:!
229
argumentum de caeteris particulis inter A et C- , necnon
inter A' et C' (ponimus A'C&quot; :: A&quot;C), cumque particulae
infra C viribus contrariis et aequalibus urgeantur, infraC&quot;
sensibili non subjiciantur actioni, iam patet etc
..... ∙
.In eodem liquido vis, qua deorsum vel sursum colnmella
trahitur, constans est; eam in sequentibus exhibebimus per K.
30. Fac ut massa liquida BAB'Q'Q (Fig. 55), quae
intercipitur superficie sphaerica BAB' et plano tangente
QQ',trahat externam columellam liquidam AR perpendicula-
riter insistentem plano tangenti apud contactum A*: quo-
niam BAB'Q'Q gignitur rotatione areae ABQ circa ra-
dium AO, perinde erit sive spectetur attractio massae BAB'Q'Q,
sive spectentur attractiones infinitarum numero superficierum
cylindricarum, quae in ea rotatione gignuntura perpendicu-
lis DC , D'C', . . . demissis ex punctis D, D' . . . circu-
laris arcus BA in rectam QA. Exprimant p , p', ... per-
pendicula DC, D'C',. :
.; q, q' , .. perpendiculorum di-
stantias AC, A.C' .. ab A computatas in AQ, sitque :-
sphaericae superficiei radius OA: ob magnam lineolarum
p, p, . ..
tenuitatem prae q, q', ...,
erunt
9' ∣
vf:
?&quot; 2r'p— 2r'...'l
et consequenter praefatae superücies cylindricae exhibe-
buntur per
' q,! &quot;03 &quot;q '3
, ∙ ∙ ∙
'
seu , 21) r r
q?
2:- 2nq ,Zitq ,...
Atqui ob eamdem illam tenuitatem puncta uniuscuiusque su.-
perficiei cylindricae haberi possunt pro aequidistantibus ab
unoquoque columellae puncto: designantibus igitur &amp;, ö'. ..
qnantitates pendentes et a certa quadam distantiarum lege,.
Kn.-&quot;M— . ⇀ ⋅−−∙⇀∙⋅↼ ⋅⋅−↪∎⋅⊾ −−↼↼∎↼ ↽− ↼−⋅−⋅−⇀−⇀−⋅∙∎∙∙↼ −−↼
↰⋅−↽⋅
- −⋍⇂∙⋅−230
et a liquidi densitate, et a cosinibus angulorum quos cum
AO faciunt rectae ab attrahentibus superficierum punctis
ductae ad attracta columellae puncta, eae superficies colu
mellam sursum verticaliter trahent viribus
Teq38 Tog'38
totaque massa BAB'D'Q columellam AR sursum verticali
ter trahet vi
1938 +7.9'38' +
.
Si concipitur altera massa liquida PAP'OʻQ intercepta pla
no QQ et nova superficie sphaerica PAP, cujus radius
O'A = p , simili ratione ostendetur vim ex PAP'Q'Q fore
παδ+πα35 '+ .
.
Vires itaque istae erunt ut
- Eq: 8 : &quot;5q?: = &gt;
erunt nempe reciproce at sphaericarum superficierum ra
dii. Hinc designante H opportunam quantitatem constan
tem , exprimet
H
vim, qua sursum verticaliter massa liquida BAB'Q'Q trahit
.columellam AR : caeterum quisque videt fore H = 12q30.
230
et a liquidi densitate, et a cusinibus angulum quos cum ⋅
AO faciunt rectae ab attrahentibns superficierum punctis
ductae ad attracta columellae pnncta, eae superficies colu-
mellam sursum verticaliter trahent viribus
nq3d th'3ö&quot;
,.... r :-
totaque massa BAB'Q'Q columellam AR sursum verticali-
tcr trahet vi
∏⊄∍∂−⊢∏⊄⋅∃∂⋅⊹ ∙ ∙ ∙ ∙
.
r .
Si concipitur altera massa liquida PAP'Q'Qintcrcepta pla-
no QQ' et nova superficie sphaerica PAP', cuius radius) O'A-z r' , simili ratione ostendetur vim ex PAP'Q'Q fore
12:738 −−⊢ ∏⊄≖∃∂∣⋅−⊢ ∙ ∙ ∙ ∙
r
Vires itaque istae erunt ut
7! :: ↿ ↿
r Zq d . —r,2q ∙−−≀∙ −∙⋮∙∙ ,
erunt nempe reciproce ut sphaericarum superficierum ra-
dii. Hinc designante H Opportunam quantitatem constan-
tem
, exprimet
H
∙∙−∙−
'
vim, qua sursum verticaliter massa liquida BAB'Q'Q trahit
.columellam'AR : caeterum quisque videt fore H::an36.231

: cum e.

tis r
r
H

4. Quantitas K major est quam
nim K exprimat vim ( 2.0) , qua sursum trahitur columella
H
AR a massa liquida LFAF'L' , exprimet K vim
qua sursum trahitur AR a segmento sphaerico MBAB '.
H
Id vero importat K &gt; o ; ergo etc.

5.0 Massa liquida BAB'E'E terminetur superficie
concavo -spherica BAB' : ducto per A plano tangente QQ ,
sollicitabitur columella AR deorsum ( 2.9 ) vi K ex EFAF'E' ,
H
sarsum (3.9) vi ex BAB'F'F ; tota igitur BAB'E'E trahet
deorsum columellam AR vi (4.0)
.
bio
3
13
н
. K
IR
in i, i' , ... ,

6.• Superficies sphaerica NAN' habens radium
O'A = 0A tangatur plano QQ in A ; columella AR ae
que trahetur sursum a massa liquida NAN'Q'Q ac trabi
lur a massa BAB'Q'Q : patet ( 3. ) çum ex eo quod, pro
ductis DC , D'C' , ... donec occurrant arcui circulari AN
exsistunt DC=Ci; D'C' =C'i, ... ; tum ex
eo quod Ci , Ci', ... , sunt tenuissimae prae AC, AC,
si qua pars columellae non trahitur sursum
sit tenuissima prae reliqua parte sursum altracta .

7.º Columella igitur AR magis trahetur deor
sum ab EE'N'AN quam ab EE'F'AF ; excessusque unius
H
attractionis supra alteram erit . Propterea massa liqui
da desinens in superficiem convexo- sphaericam NAN' traliet
deorsum columellam AR vi
ita ut ea
1
K +
1
i.
231
H 4.&quot; Quantitas K major est quam —-: cum e-
- r
nim K exprimat-vim (29)
. qua sursum trahitur columella
AR a massa liqui/da LFAF'L', exprimet K —E r
vim ,
qua sursum trahitur AR a segmento sphaerico MBAB'.
Id veroinrportat'K—g- ≻∘ ∙∙∙ ergo etc.
. . .
59 Massa liquida BAB'E'E terminetur superficie
concavo-spherica BAB' : ducto per A plano tangente QQ',
sollicitabitur columella AR deorsum (2.0) vi K ex EFAF'E',
sursum (3.&quot;) vi!-.;l ex BAB'F'F; tota igitur BAB'E'E trabet
deorsum columellam AR vi (4.0)
∙
K—ll'a r
6.0 Superficies sphaerica NAN' habens radium
()&quot;AzOA tangatur plano QQ' in A; columella AB ae-
que trahetur sursum a massa liquida NAN'Q'Q ac trabi-
tnr a massa BAB'Q'Q: patet (39) tum ex eo quod, pro—
ductis DC, D'C' , ... donec occurrant arcui circulari AN
in i, s&quot;.
..., exsistunt DCxCi; D'C'..-::C't&quot;, ...: tum ex
eo quod Ci, C'i',
. . . ,
sunt tenuissimae prae AC. AC',
ita ut si qua pars columellae non trahitur sursum , ea
sit tenuissima prae reliqua parte sursum attracta.
7.o Columella igitur AR magis trahetur deor-
sum ab EE'N'AN quam ab EE'F'AF; eicessusque unius
H
attractionis snpra alteram erit -— . Propterea massa liqui-
⋅ r
da desinens in superficiem convexo-sphaericam NAN' trahet
deorsum columellam AR vi
n
∣≺⊣−−−⊑−⋅∙232

8.º Pone superficiem BAB' neque esse sphaericam
,
neque gigni rotatione ullius curvae circa AO

; secla BAB

planis transeuntibus per A0
, curvilineae sectiones apud
contaclum
A gaudebunt inaequalibus osculi radiis

; quos

inter
( demonstrationem suo tempore videre erit in parte 3.4 nostrorum elementorum matheseos 0. 118
) bi
ni reperiunlur
, alter minimus
(
=
r
)
, alter maxi
mus
(
=
r
'
), pertinentes ad binas sectiones sub angulo re
cto invicem constitutas
. Iam
, in ea qua sumus hypothe
si
, hoc pacto determinabitur visex BAB'Q'Q sursum
verticaliter trahens columellam AR
. Intelligatur coalesce
re BAB'Q'Q ex infinitis numero superficiebus cylindri
cis normaliter insistentibus plano tangenti QQ
'

:

' unaquae
que superficies cylindrica non eamdem habebit ubique
altitudinem

; sed apud bina puncta

e diametro opposita
,
quibus nempe maximus respondet circựlus osculator
, al
titudo erit minima

; apud bina puncta

e diametro pari
ter opposita
, perque gradus 90 ab illis primis sejuncta
,
quibus videlicet minimus respondet circulus osculator
,
altitudo erit maxima

: apud intermedia puncta altitudines

interjacebunt minimam maximamque
. Quapropter evoluta
superficie cylindrica super aliquo plano
, ea poterit reprae
sentari per aream QNN
&quot;
Q
&quot;
( Fig
. 56
)

; NN

&quot; aequatur
basi superficiei cylindricae

; QN et

Q
&quot;
N
&quot; simul cum Q'N
'
exhibent altitndines minimas

; Fu et F'u

' altitudines ma
ximas hinc Nu

= uN
'
= N'u
'
= u'N
&quot;
. ob perexiguum ba
seos cylindricae radium poterunt QF
, Q'F
, Q'F
'
,
(
&quot;
F
'
haberi pro lineis rectis

; eritque

1
QN
+Fu NN &quot; QʻF'Q'FQ = 1NuFQ = 4 Nu 2 NN ” QN + F4 2
232
8.&quot; Pone superficiem BAB' neque esse sphaericam,
neque gigni rotatione ullius curvae circa AO; secta BAB'
planis transeuntibus per AO, curvilineae sectiones apud
contactum
A gaudebunt inaequalibus osculi radiis quos

inter (demonstrationem suo tempore videre erit in par-
te 3.*' nostrorum elementorum matheseos n. 118) bi-s
ni reperiuntur, alter minimus
( ::r)
, alter maxi-
mus (:r'), pertinentes ad binas sectiones sub angulo re-
cto invicem constitutas. Iam
,
in ea qua sumus hypothe-
si, hoc pacto determinabitur vis et BAB'Q'Q sursum
verticaliter trahens columellam AR. Intelligatur coalesce-
re BABHQQ ex infinitis numero superficiebus cylindri-
cis normaliter insisteutibus plano tangenti QQ' :'unaquae-
que superficies cylindrica- non eamdem habebit ubique
altitudinem; sed apud bina puncta
e diametro opposita,
quibus nempe maximus respondet circulus osculator
, al-
titudo erit minima; apud bina puncta
c diametro pari-
ter opposita, perque gradus 90 ab illis &quot;primis seiuncta,
quibus videlicet minimus respondet circulus osculator
,
altitudo erit maxima: apud intermedia puncta altitudines
interjacebunt minimam maximamque. Quapropter evoluta
superficie cylindrica spper aliquo plano
, ea poterit reprae-
sentari per aream QNN&quot;Q&quot; (Fig. 56); NN&quot; aequatur
basi snperficiei cylindricae QN et Q&quot;N' simul cum Q'N'

exhibent altitudines minimas -; Fa et F'u' altitudines ma-
ximas; biuc NucuN'zN'u'2u'N&quot;
.
ob perexiguum ba-
seos cylindricae radium poterunt QF
, Q'F, Q'F ', Q&quot;F'
habcri pro lineis rectis; eritque
NN&quot;Q'F'Q'FQ::4NuFQ

:

4 Nu 'QN'zl-F&quot;

:

∙ NN&quot; QNj'F&quot;
.233
ericam,
a BAB
es apud
i quoi
Retentis igitur denominationibus ( 3.9 ) , superficies cylin
dricae , ex quibus intelligitur coalescere massa BAB'Q'Q
( Fig. 55 ) , erunt
92 Lo pár.
+
92 q /2
+
2r
2r' 2r'
2πα ,
2r
2tq'
&gt;
seu
mari 2 2
alore
ypothe
Sursa mga (: + ?). mg ( + ),...
Dalesce
linde
naquat
ubige
pposila,
et consequenter massa BAB'Q'Q desinens in superficiem
BAB' utcumque concavam trahet verticaliter sursum co
lumellam AR vi
or , al
πα
2
o pari ?( + 1) + &quot;, ( +3)x + ...
Atqui ( 3.9 ) 7.938 + Teq'38 ' + .... = H :
Ejuncta,
ulator ,
Studios
evolu
reprat
equatur
exprimetur ergo vis illa per
16+)
les m2
um bio
1 07
9. Sume Q '&quot;'N '&quot; et F &quot; u &quot; ( Fig. 56 ) aequidi
stantes ab QN et Fu : erunt Q &quot; N &quot;&quot; , F &quot; u &quot; duae ex al
titudinibus intermediis ( 8. ) respondentes duabus sectio
nibus curvilineis ad angulum rectum invicem constitutis.
Radii circulorum osculantium sectiones istas apud A ( Fig.
55 ) designentar per po&quot; et p' : erit ( Fig. 56 )
9'2
Q&quot; N &quot; +F &quot; u &quot; =
92
2r &quot;.
qo +
27 
+
g'?
2r . ' 2r' '
1
16
233
Retentis' igitur denominationibus (3.&quot;) , superficies cylin-
dricae, ex quibus intelligitur coalescere massa BAB'Q'Q
(Fig. 55 )
,
erunt
/
£ fl&quot; q&quot; a&quot;
2r .l-Zr ∙ 2r
21tq 2 , 27tq .l-Zr' 2 , ∙ , seu
↔⇍≖↙≀⋮↿≺ ,
') &quot;rv ')
⋅⋅
2 rii-&quot; 2
≀∙⊣−∣⋅⋅∅⋅⋅⋅⋅
et consequenter massa BAB'Q'Q desinens in superficiem
BAB' utcumque concavam trahet verticaliter sursum co-
lumellam AR vi
&quot;f(ï-Fl?) ∂⊹∙⋮≖−≣−⋅⋮−≺−∶−−⊦∙≙≻∂↝⊹∙ ∙∙
r
Atqui (3.0) nq3ö -l-7tq'3d' ∙−⊦ ∙ ∙ ∙:∙ H:
exprimetur ergo vis illa per
⋅≣⋅≺⊥⊣−−↿⊺≻ ⋅

: - r

9.0 Sume Q'&quot;&quot;'N et F&quot;u&quot; (Fig. 56) aequidi-
stantes ab QN et Fu : erunt Q'&quot;N&quot;', F&quot;u&quot; duae ex al-
titudinibus intermediis (89) respondentes duabus sectio- .
nibus curvilineis ad angulum ≖⋅∁∁⋯⊞∙ invicem constitutis.
Radii circulorum osculantium sectiones istas apud A ( Fig.
55 ) designentur per r&quot; et r'&quot; : erit (Fig. 56)
0 (
ns '
lr Jr ' ∎∙ lr.
'
a a 'a' ': ≺≀⋅⋅∙↓↜⇃∙⋅∙−⊦∣∂⇁⇈≀↓∙∙∶∶−∙↙−⇃−⋅− ⊄
⋞∣ −⊦⋞∣∙∙234
est autem
mane cc
Q &quot; N '&quot; + F &quot; u&quot; = QN + Fu =
92
2r
9'2
2r +
2r etc.:
1
igitur
+++++
110
et consequenter
16 + *) = &quot; 6 + - ): under
10.º Si superficies concava BAB' ( Fig. 55 ) gi
gnitur rotatione alicujus curyae circa OA , fiet
Cor
ace
r = r = r ' = r '&quot;
I supe
ac proinde vis ex BAB'Q'Q sursum verticaliter trahens
columellam AR erit ban,
zebic
H
reciproce nimirum ut radius osculi apud A.
11.• Facile nunc intelligimus attractionem mas
sae liquidae BAB'E'E, terminatae superficie utcumque con
caya BAB' , in columellam AR fore
K - &quot; ( + -) .velK6+ );
234
est autem
↾∣∣ ⇌∎
!
'2 :
NI&quot; F&quot; &quot;: −∙ Q. ∙−⊦ u QN—l—Fu q2r ↿ q 0
L
I 2r' , 2r ∙−⊦ 2r' ,etc.t
igitur
1 ↿− ↿ ↿
i- ∣
rr ∣⋅⊤⋅ .'&quot;? '
et consequenter
H 1 ↿ H
∎↿ ↿ ⋅
−⋮−≺−≀∶⇀⊦ r' )— 2
(r&quot; hl-r'&quot;) ⋅∙
10.0 Si superficies con-cava BAB' (Fig. 55 ) gi.
gnitur rotatione alicujus curvae circa OA ,
fiet
I ';
∣←−∶≀∙ :r :r '&quot; ;
ac proinde vis ex BAB'Q'Q sursum verticaliter trahens
columellam AR erit
'H-
&quot;,
reciproce nimirum ut radius osculi apud A.

11.(, Facile nunc intelligimus attractionem mas-
sae liquidae BAB'E'E, terminatae superficie utcumque con-
cava BAB' ,
in columellam AR fore
H
1,1) H(1,1 ∙
K
2(r '7 'velK 2r&quot; 'r'&quot;)'
ïfbiu235
massae vero liquidae NAN'E'E, terminatae superficie utcum
que convexa NAN' , in ipsam AR fore
K + 6 + -) .
ved K + &quot;6+ ) :
fiet =r =r&quot; = r&quot; in casu superficiei genitae rotatione li
neae curvae circa OA.

110. His declaratis , venio ad ascensum descensum que liquorum in tubis capillaribus.

1.° Aqua in tubis vitreis diversae capacitatis ad diversas ascendit altitudines, quae sunt reciproce ut tuborum diametri: idipsum contingit oleo, spiritui vini, etc..; ascendentesque liquores terminantur superne concava superficie. Concavae superficiei causam adsignavimus (109 1.): ad ascensum quod pertinet, sit QQ ( Fig. 57 ) suprema superficies aquae circumambientis tubum LE , et I A'B' concava superficies aquae intra tubum; sint insuper A'R, VR' binae columellae verticales, altera intra, altera extra tubum, quas columellas jungat horizontalis columella RR'. Urgebitur A'R deorsum gravitate simulque vi (109. 11. ° )
K - 16 + );
urgebitur VR' deorsum gravitate simulque vi ( 109. 2.° )
K :.
cum igitur
Н
K &gt; K
(
2 + ),
235
massae vero liquidae NAN'E'E, terminatae superficie utcum-
que convexa NAN' ,
in'ipsam 'AR fore
H 1 1 H 1 1 K—(-2(,, ∣∣⋅ .).ve1K( ,(,.,'. ...-)
r r
fiet r:r':r&quot;—-::r&quot;' in casu snperficiei genitae rotatione li-
neae curvae circa OA&quot;.

110. His declaratis, venio ad ascensum descensumque liquorum in tubis capillaribus.

1.&quot; Aqua in tubis vitreis diversae capacitatis ad diversas ascendit altitudines, quae sunt reciproce ut tuborum diametri: idipsum contingit oleo, spiritui vini, etc..; ascendentesque liquores terminantur superne concava superficie. 

Concavae superficiei causam adsignavimus (10913): ↙
ad ascensum quod pertinet , sit QQ' (Fig. 57) supre-
ma superficies aquae circumambientis tubum LE , et I'A'B'
concava superficies aquae intra tubum; sint insuper A'R,
VR' binae columellae verticales, altera intra, altera extra
tabum, quas columellas iungat horizontalis columella RR'.
Urgebitur A'R deorsum gravitate simulque vi ( 109. 11.&quot;)
H1711,
K
2(rlr1)a
urgebitur VR' deorsum gravitate simulque vi ( 109. Z.&quot;)
∕
cum igitur236
haud poterunt A'R , VR' consistere in aequilibrio nisi A'R
ascendat supra QQ . Denotet z altitudinem AA , ad quam ascendit columella A'R supra QQ'; sitque c gravitas specifica liquoris: fiet eousque columellae ascensio donec habeatur
H
K = K
-16 + )+c=;unde == 2c ( + ).

Vires ex materia tubi eas tantum liquidi particulas afficiant, quae ad internam ipsius tubi superficiem maxime accedunt; iccirco liquidum perinde trahetur, a tubo ac si interna superficies esset plana: permanente igitur tubi ac liquoris qualitate, etsi variat tubi diameter, eodem tamen pacto trahentur liquoris particulae versus tubum; quae attractio quia cum constanti attractione liquoris erga seipsum consociatur, extrema latercula curvae BAB' aeque inclinabuntur ubilibet ad internam tuborum superficiem. Arcus itaque omnes BAB' erunt similes in diversis tubis, ipsorumque arcuum rotatione circa tubi axem gignetur superficies concava superne terminans liquorem : bini videlicet
r et r' exsistent aequales, simulque tuborum diametris pro
portionales; ideoque altitudo z reciproce ut eae diametri.

2. • Hydrargyrum in tubis vitreis descendit in
fra circumambientis hydrargyri superficiem QQ ad ejus
modi altitudines , quae sunt tuborum diametris recipro
ce proportionales ; descendensque liquidum terminatur su
perne convexa superficie NOM.
Convexitatis causam adsignavimus ( 109. 1.0 ) : ad de
scensnm quod spectat , columellarum OR et VR' altera
sollicitatur deorsum gravitate simulque vi ( 109. 11.° )
0
{
K + C + ) ,
236
haud poterunt A'R, VR' consistere in aequilibrio nisi A'R
ascendat supra QQ'. Denotet :altitudinem AA' , ad quam
ascendit columella A'R supra QQ'; sitque c gravitas spe-
cifica liquoris: fiet eousque columellae ascensio donec ha-
beatur
H ↿ ↿ H 1 ↿
∣≮≓⋅∶∶∣⊊∙− −∙≨∙⋖−−∙−⊢⊤≻−⊢∶≖∙ under—' 2c(r ≓≀∙∙≻ .
r .
!
Vires ex materia tubi eas tantum liquidi particulas af-
ficiunt, quae ad internam ipsius tubi superficiem maxi-
me accedunt; iccirco liquidum perinde trahetur, a tuba ac si
interna superficies esset plana : permanente igitur tubi ite
liquoris qualitate, etsi variat tubi diameter, eadem tamen
pacto trahentur liquoris particulae versus tubum; quae
attractio quia cum constanti attractione liquoris erga se-
ipsum consociatnr, extrema latercula curvae BAB' aeque
incl'uabuntur ubilibet ad internam tubarum superficiem.
Arcus itaque omnes BAB' erunt similes in diversis tubis,
ipsorumque arcuum rotatione circa tubi axem gignetursuper-
iicies concava superne terminans liquorem : bini videlicet
r et r' exsistent aequales, simulque tubarum diametris pro-
portionales; ideoque altitudo :reciproce ut eae diametri.

2.0 Hydrargyrum in tubis vitreis descendit in-
fra circumambientis hydrargyri superficiem QQ' ad eius-
modi altitudines , quae sunt tubarum diametris recipro-
ce proportionales; descendensque liquidum terminatur su-
perne convexa superficie NOM.
Convexitatis causam adsignavimus (109. 1.(, ) : ad de-
scensnm quod spectat, columellarum OR et VR' altera
sollicitatur deorsum gravitate simulque vi (109. 11.&quot;)
Hi 1
K&quot;.'a(1&quot;'r&quot;)'
I:
'P.?237
altera sollicitatur deorsum gravitate simulque vi ( 109. 2.0)
K :
cum igitur
K &lt; K + -( + )
haud 'poterunt OR et VR' sese librarè nisi OR descen
dat infra QQ. Designet é altitudinem AO , ad quam
deprimitur columella OR infra QQ' ; sitque c' gravitas
specifica hydrargyri : eousque fiet columellae depressio do
nec habeatur,
H
K=K + ( + )- c'z' , unde z' = 2c'
(
+ &gt;&gt;).
Ut supra ( 1. ) ostenditur binos r , r' fore aequales, simul
que proportionales tuborum diametris ; iccirco etc.

111. Nonnulla subjungimas, quorum ratio desumitur ex animadversionibus (109).

1.º Duae laminae vitreae et parallelae PP ', SS ' demergantur verticaliter in aquam, earumque mutua distantia aequetur diametro tubi capillaris LE: suprema aquae superficies B &quot; A &quot; B &quot; inter laminas evadet concava instar canalis horizontalis; altitudo vero A'al= x ), ad quam attollitur aqua, erit duplo minor altitudine ad quam attollitur in tubo LE.: Infima superficiei B &quot; A &quot; B '&quot;' puncta jacent omnia inea
dem recta A'A'&quot;: secetur B &quot; A &quot; B &quot; &quot; plano perpendiculari ad A &quot; A &quot; '; sectio erit ubilibet arcus arcui BA'B' similis et
aequalis : istorun arcuum radius osculi apud puncta infima
dicatur r ; in tubo LE erit p = r , in laminis r= -0 . Colamellarum igitur A'R , VR aequilibrium praebebit ..
237
altera sollicitatur deorsum gravitate simulque vi (109. 23)
K :
cum igitur
X(K '
H( ↿
r ral,).r
baud' poterunt OR et VR' sese librare nisi OR descen-
dat infra -'QQ. Designet z' altitudinem AO,- ad quam
deprimitur columella OR infra QQ'; sitque c' gravitas
specifica hydrargyri: eousque fiet columellae depressio do-
nec habeatur.
1
∙
↿
'
∣∟−∣≖⊹−−⊸ 2(-—--]——-)-—c&quot;,z undez'—.2[:7(1 [
r').
Ut supra (1 .&quot;) ostenditur binos r, 'r' fore aequales, simul-
que praportionales tubarum diametris; iccirco etc.
,.,11.1 Nonnulla subjungimtts ,- quorum ratio desu-mitur ex animadversionibus (109).↿∙∘ ∐∎≖≔∘∙ laminae vitreae et parallelae PP', SS'demergantur verticaliter in aquam, earumque mutua distantiasequetur diametro tubi capillaris LE : suprema aquae super-ficies B&quot;A&quot;B&quot;' inter laminas evadet concava instar canalishorizontalis; altitudo vero A&quot;a(:.r) , ad quam attollituraqua , erit duplo minor altitudine , ad quam attolliturintnhoIfE.- ⋅∶∶⊸∙Infima superficiei B&quot;A&quot;B&quot;' puncta iacent omnia infen-dem recta A&quot;A'&quot;: secetur B&quot;A&quot;B&quot;' plano perpendiculariad A&quot;A&quot;'; sectio erit ubilibet arcus arcui BA'B' similis etaequalis: istorum arcuum radius osculi apud puncta infimadicatur r; in tuba LE erit r'zzr, in laminis r'zoo; Cos-lumellarnm igitur A'R , VR' aequilibrium praebebit't-dïff2382c ( + ) &quot; ;Tetscolumellarum autem A ” R &quot; , VR aequilibrium suppeditabitH101 IK -K - I ( + ) +re , f =.2c1Hinc xai 2ż z ; ideoque etc.....2.0 Laminae PP ' , SS , sibi commissae ad sematuo accedunt.Sit P&quot; punctum quodvis laminae PP: infra QQ adprofunditatem Alla &quot; : columella verticalis Alla&quot; transmittetpuncto P vim ( 1.0 ) .K- ( + ) +ostan&quot;) = KH+2r Tersus1 .C2c 5+c. a a'&quot; = K +0. aa!&quot;dicenndnetfenotaversus Qt : attenta columella horizontali . a &quot; ' P &quot; , urgebi amelltur P vi seu pressione externa + traiKversus Q't': colamella verticalis V'a transmittet puncto P &quot;vimK+c.aa &quot; &quot;versus Qi' : attenta columella horizontali a'P ' , solicitabitur P &quot; vi seu pressione interna 16TSU238H(t 1) H 1z— ⇂ ∙−−− ∙⋅ ;20 r !' C !'columellarum autem A&quot;R&quot;', VR' aequilibrium suppeditabit!H 1 1 H 1− ↿ ∣ ⊫−∙− −⋅⋅ ∙!cx . ∙∖Hlnc ∙ æ;—2—z;l ∙ ⋅ 1deoqueetc......⋅⋅ n .'2.o Laminae 'PP. SS', sibi commissae ad sea') 'At mutuo aeeedunt. )Sit P&quot; punctum quodvis laminae PP' infra QQ' adprofunditatem A&quot;a &quot;: columella verticalis A&quot;a transmittetpuncto P&quot; vim (1.). &amp; ↽(⋅. H 1 1⋅ &quot; HK(cæ—-—aa&quot;' :K −−−∙−−∙ 2 ≀⋅∙⋮∙∙∞ .'. (M' . 2r. . '&quot;1ciii .-—--)-c.aa :::K'I—ILmaa'&quot;.⋅.. .: '&quot;.versuth : attenta columella horizantali- a.'&quot;P&quot; , urgebi-tur P&quot; vi seu pressione externa,.. ⋅∙ ⋅('. a∙ t '. '(1
..

infima
dicatur r; in tuba LE erit r'zzr, in laminis r'zoo; Cos-
lumellarnm igitur A'R , VR' aequilibrium praebebit't-dïff238
2c ( + ) &quot; ;
Tets
columellarum autem A ” R &quot; , VR aequilibrium suppeditabit
H
101 I
K -K - I ( + ) +re , f =
.
2c
1
Hinc xai 2ż z ; ideoque etc.....
2.0 Laminae PP ' , SS , sibi commissae ad se
matuo accedunt.
Sit P&quot; punctum quodvis laminae PP: infra QQ ad
profunditatem Alla &quot; : columella verticalis Alla&quot; transmittet
puncto P vim ( 1.0 ) .
K
- ( + ) +ostan&quot;) = K
H
+
2r Tersus
1 .
C
2c 5+c. a a'&quot; = K +0. aa!&quot;
dicen
ndnet
fenota
versus Qt : attenta columella horizontali . a &quot; ' P &quot; , urgebi amell
tur P vi seu pressione externa + trai
K
versus Q't': colamella verticalis V'a transmittet puncto P &quot;
vim
K+c.aa &quot; &quot;
versus Qi' : attenta columella horizontali a'P ' , solicita
bitur P &quot; vi seu pressione interna 16TSU
238
H(t 1) H 1
z— ⇂ ∙−−− ∙⋅ ;
20 r !' C !'
columellarum autem A&quot;R&quot;', VR' aequilibrium suppeditabit
!
H 1 1 H 1
− ↿ ∣ ⊫−∙− −⋅⋅ ∙
!
cx . ∙∖
Hlnc ∙ æ;—2—z;l ∙ ⋅ 1deoqueetc......
⋅⋅ n .
'2.o Laminae 'PP. SS', sibi commissae ad se
a
') 'At mutuo aeeedunt. )
Sit P&quot; punctum quodvis laminae PP' infra QQ' ad
profunditatem A&quot;a &quot;: columella verticalis A&quot;a transmittet
puncto P&quot; vim (1.). &amp; ↽
(
⋅
. H 1 1
⋅ &quot; H
K
(
cæ—-—aa&quot;' :K −−−∙−−∙ 2 ≀⋅∙⋮∙∙∞ .'. (M' . 2r
. . '
&quot;
1
ciii .-—--)-c.aa :::K'I—ILmaa'&quot;
.
⋅
.
. .: '
&quot;
.versuth : attenta columella horizantali- a.'&quot;P&quot; , urgebi-
tur P&quot; vi seu pressione externa
,
.. ⋅∙ ⋅
('. a
∙ t '. '
(1
..

: - ∎∙ 3 ' ∙

.
versus Q't': columella verticalis V'a' transmittat puncto P&quot;
visu ⋅⋅ ' ⇀
'
'
'
⋅
⋅⋅
K—l—caa ?
.
versus Q't': attenta columella horizontali a'P&quot;
, sollicita-
bitur P&quot; vi seu :pressione interna &quot;
'
'
-
⇂⇣ r 'a'
ms
1111:
&quot;But
&quot;fin239
K
versas Qt : erit igitur p &quot; aeque sollicitatum hinc et illinc,
nullusque motus gignetur ab istis viribus.
Sit P' &quot; ' punctum ipsius PP' inter QQ et A &quot; A '' : ver''
ricalis columella A &quot; a &quot; transmittet puncto P &quot; ' vim
к
-16 + ) + ( 6 – aa &quot;) = K -
H
+
2r
H
S c.aa&quot; = K- c.aa' '
2r
versus Qt : ob columellam horizontalem P '&quot; a &quot; urgebitur
P &quot; &quot; vi seu pressione externa
K
versus Qt : cum igitur K &gt; K - c.aa &quot; , nitelur &quot; mo
veri ad plagam (t' .
Ascendet aliquantulum aqua externa prope laminam ÞP
induetque ( 109. 1. ) figuram concavam ee'e ' ; propterea,
denotante &amp; radium osculi apud punctum v . gr. e' , co
lumella e'b normaliter insistens superficiei curvae ee'e&quot; in
e' transmittet puncto v. gr. b vim
H
K
29 ( +4) = K 2 €
versus Qt’ : attenta columella horizontali e'b ', urgebi
tur b ' vi seu pressione interna
K
versus Qt : ex aqua éb'é' proveniet in bi vis
∙ 239
K
Versus Qt: erit igitur P&quot; aeque sollicitatum hinc et illinc,
nullusque motus gignetur ab istis .viribus. Sit P'&quot; punctum ipsius PP' inter QQ' et A&quot;A&quot;': ver-
ticalis columella A&quot; a&quot; transmittet puncto P&quot;'v vim
K
--—-(—1--—l--—) —[—c(x—-aa&quot; ∙−−∶ ∣⊂−∙≗ ∙⋅⊢
H
—— c.aa&quot;
−−−−− K— c.aa&quot; 2r ∕
versus Qt'
- ob columellam horizontalem P&quot;'a &quot;
P'&quot; vi sen pressione externa
&quot;—
urgebitur
K
versus Q't'. ∙ cum igitur K)K—c.aa&quot; , uitetur P&quot;' mo-
veri ad plagam Q't' .
Ascendet aliquantulum aqua externa prope laminam PP' induetque (109.1.?) figuram concavam ee'e&quot; ; propterea,
denotante &amp; radium osculi apud punctum v. gr. e'
, co-
lumella e'b normaliter insistens superficiei curvae ee'e&quot; in
e' transmittet puncto v. gr. b' vim
H 1 1 H K—
⊸≳↽≼−⋮⇀⊣−∘−∘−⊢ ≖⊊−−⊸⇄−∊
versus Q't'- attenta columella horizontali e',b' urgebi-
tur b' vi seu pressione interna '-
. K
8
versus Qt: ex aqua e'b'e&quot; proveniet in b' vis240
c.e&quot; 6
versus Qit' : columella verticalis A &quot; 6 &quot; transmitiet puncto
b' yim
H
K
25
+ c . A &quot; 6 &quot;
versus Qt : attenta columella horizontali b'b' impelletur 6
vi seu pressione esterna
K
versus Qt . Est
H
2r = cx = C . A'a ;
librato insuper liquido , pressiones apud V' et é' sunt ae
quales , et consequenter
K = K
H
28 to.e&quot; 6 ,
H
= c.eb' ;
2 €
detractisque proinde viribus versus Qt ex viribus versus
Q'ť , emerget
H н
K
2e-K + c.e^ 6—K+
-c.A &quot;b&quot; + K = c.eb - c.A &quot;6&quot; +
H H
2r 2 €
c (e&quot; b' — A &quot; 6&quot; + 1&quot; a - c'b') = c.b &quot;a &gt; o :
sollicitabitur ergo b' vi c.6 &quot; a versus Q't' .
Veniat denique spectandum in lamina PP punctum p '
inter A &quot; A &quot; et B &quot; B &quot; : sit P'a&quot; columella horizontalis ; a'i
columella perpendicularis superficiei curvae B’A &quot; B &quot; &quot; apud
a &quot; ; dicaturque é radius osculi in a ' ' . Transmittet a'i
puncto Ph vim
240
c . e&quot;b'
versus Q't' :columella verticalis A&quot;b&quot; transmittet puncto
b' vim
. H &quot;&quot;
K—Z—i—c'Ab
versus Qt :attenta columella horizontali b'b&quot; impelletur b'
vi seu pressione externa
↴
K
versus Q't'. Est
H
∙∠−≀∙−∶∶∘∙↿∽⋅∶∘∙∆∎∣∅
librato insuper liquido , pressiones apud V' et a' sunt ac-
quales , et consequenter
-——-K——'l&quot;c. e&quot;6'. Eli-:o- e&quot;b':
detractisque proinde viribus versus Qt ex viribus versus
Q&quot;: , emerget ⋅
K—is'&quot;. -K—]—
.- .∘∣∙∣↗∣−−⋅↧≮−⊦ ≛↿−⊑∙⊸∙∆∦∣⊃⋅∣−⊢↧⊊∶⊸⋅⊜∣⋅∣⊃⋅−∘⋅∆⋅⋅≀≀⋅∣−⊦
H H &quot;
.!
.,
h 1
∙−−
&quot;
.
ii.—22&quot;— ..—c(&quot;eb'- Ab -I-Aa-c.b)—-c.b a)o.
sollicit'abitur ergo b' vi c.b&quot;a vcrsus Q't' .
Veniat denique Spectandum in lamina PP' punctum P&quot;
inter A&quot;A&quot;' et B&quot;B&quot; : sit P&quot;'a&quot; columella horizontalis; a&quot;i
columella perpendicularis snperficiei curvae B&quot;A&quot;B&quot;' apud
a&quot; ; diceturque e' radius osculi in a&quot;. Transmittet a&quot;:
puncto P&quot; vim.241
K
H
2 €
.
versus Qt : ex liquido superincumbente proveniet in p ' ' vis
B ' P &quot;
versus Qc : attenta P &quot; a &quot; urgebitur p &quot; vi seu pressione
externa
K
versus Qit' : librato liquido , pressiones apud a &quot; et A ” sunt
aequales ; proinde ducta horizontali Allu ,
H H
K tc.P&quot; u = K = K- C. A'a ,
2 € 2r
H
= c ( P''u + A &quot; u ) = c.P'' ' :
26
detractis igitur primis duabus viribus ex tertia , assequemur
H
K - K +
c . B ' piv H
-C.B&quot; piv
2€ 22'
c ( Piu' B ' p ' ) &gt; 0.
Lamina itaque PP' movebitur versus Q'C' : eadem de causa
movebitur lamina SS' versus Qt ; ideoque etc...

3. Si aquae substituitur hydrargyrum , supre
ma liquidi superficies inter laminas PP' , SS ' fiet con
vexa instar horizontalis inversique canalis ; deprimetur li
quidam ad altitudinem duplo minorem quam in tubo LE ;
ipsae insuper laminae adhuc ad se mutuo accedent . Haec
explicantur simili ratione ac ( 10, 20. )..
241
versus Qt :ex liquido superincumbente proveniet in P&quot; vis
&amp; ∙ B1v Ptv
versus Qt : attenta P&quot; a&quot;
externa
urgebitur P&quot; vi seu pressione
K
versus Q't' :librato liquido . pressiones apud a&quot; et A&quot; sunt
aequales .; proinde ducta horizontali A&quot;u,
H &quot;
∙∙∙ H ∙∙
↿⊂−∙−∙⋮≳−⋮∙−⊢∘∙↧∙ ∥⋅−−↿⊊∙−⋮≳≀∙−−∙−−↧⊊−∁∙ A a
.
H ,,
⊓∣
27-—-c(P&quot;u-l—Au)——-—c.P u:
detractis igitur primis duabus viribus ex tertia , assequemur
⊏−≖⊂−⊢⋮−⋮∶∶−∘∙∌∏ ↕⊃≖⊽∶∶∶ IST—c .B&quot; Piv:
c (P&quot;'u' — Blv P&quot;) ∘∙
Lumina itaque PP' movebitur versus Q't' : eadem de causa
movebitur lamina SS' versus Qt ; ideoque etc...
3.&quot; Si aquae substituitur hydrargyrum , supre-
ma liquidi superficies inter laminas PP' , SS' fiet con-
vexa instar horizontalis inversique canalis; deprimetur li-
quidam ad altitudinem duplo minorem quam in tubo LE;
ipsae insuper laminae adhuc ad se mutuo accedent. Haec
explicantur simili ratione ac (10. 20.)
.242

4. • Super vitream laminam horizontalem AA'B'B
( Fig . 58 ) affunde gattam olei terebinthini mm ' ; tum al
teram laminam vitream A &quot; A'B'B &quot; priori AA'B'B impone
sub angulo sic exiguo , ut imposita lamina gutlam le
viter attingat ; conspicies mm' , instar trochleae , termi
natam quodam canaliculo ; qui canaliculus plano hori
zontali sectus dabit curvilineam convexamque sectionem
plano verticali sectus curvilineam concavamque sectionem .
Radius convexitatis ( € ) manet proxime idem in
punctis m et m' e diametro oppositis ; radius vero con
cavitatis ( = r' ) in puncto m' magis accedente ad A'B'
minor erit quam radius concavitatis ( = r) in puncto m
minus accedente ad ipsam A'B ' . Spectantes columellam
mam ' perpendicularem rectae A'B' , quoniam r et é ' apud m
obverluntur ad plagas contrarias , itemque r et ê &quot; apud m' ,
facile intelligemus ( 109. 110. ) sollicitatum iri mam ver
sus A'B' vi
H
K
2
simulque versus AB vi
K
16-&gt;).
Cum igitur &gt; m , prima vis erit major quam secunda ;
columellaque mam , et una cum mam' tola gutta mo
vebitur versus A'B' motu accelerato : idipsum contingit
guttaeaqueae . At si ejusmodi guttis substituatur gutta
hydrargyri , haec movebitur versus AB ; ratio est quia
gutta hydrargyri tam in sectione horizontali quam in
verticali praebet curvam convexam , radiusque novae con
vexitatis in m superat radium novae convexitatis in m .

5.° Capillaris tubus in aquam QQtt ( Fig. 57. ) demergatur; tum, apposito digito ad orificium inferius extrahatur : remoto digito , aqua jam elevata eflaet ali
quantulum ex orificio illo , ibique demum haerebit sus
pensa in guttam rotundam conformata ; residuae vero
aquae altitudo in tubo extracto invenitur major quam
altitudo ( 110, 1º.)
H
Z =
16 + 3) = * ( + ) cr
supra QQ in tubo demerso.
Exprimant et w , altera radium convexitatis apud infi
mam aquae superficiem in tubo extracto ,
altera ipsius
aquae altitudinem : ex aqueae columnae aequilibrio pro
fluit&quot; ( 110. 1 ° 2°. )
H H
K
+++
www =
= ktö : +
H
co ;
cr
ideoque w &gt; z . Si aquae substituitur hydrargyrum , tam
suprema quam infima superficies liquidi exsistet conve
x2 ; ex aequilibrio igitur hydrargyri in tubo extracto
emerget ( 110. 20. )
H H
H
K + tow == Kt H
co CI
et consequenter a = 0 si r = 0 .

112. Quae diximus de liquorum ascensu tubulis vitreis, applicari possunt ascensui liquorum in tenuibus cujuscumque materiei tabulis: hinc patet cur liquida ascendendo imbuant spongias, saccharum, ellychnia etc: cur succus inserviens plantarum vegetationi sursum ex terra eluctetur; etc... Istiusmodi corpora vel constant exilissimis fibris, in quibus tanquam in totidem capillaribus tubis ascendit liquidum, vel innumeros habent angustos meatus vicem tubulorum varie flexorum supplentes. 

Caeterum methodo inhaerentes, qua D. Pessuti LaPlacianam theoriam ad faciliorem formam traduxit, capillarium luborum phoenomena explicavimus in hypothesi liquidorum eamdem usque ad extimas omuino superficies obtinentium densitatem: non enim nobis in animo est vel leviter attingere novam theoriam, quam de actione capillari anno 1831 edidit D. Poisson.

== ACUSTICAE PRINCIPIA ==

=== Notiones preambulae ===
[[113|113]]. Acustica agit de sono: non defuerunt, qui sonum consistere putabant in efluviorum a soporo corpore vibratorum motu quae efluvia ex affrictu, vel contusione sonori corporis ejaculantur atque huic affinis est alia quaedam sententia, quod contusione illa vel affrictu particulae aeris purioris in eo corpore absconditi, vel ipsum circumdantis, expellantur et ad aures usque excurrant. Verum experimento machinae pneumaticae compertum est, quod incluso tintinnabulo vel horologio horas personante in recipiente, ubi aer incipit exhauriri, incipit sonus minui; ubi autem totus exhaustus est aer, nihil jam soni auditur, utcumque pergat tintinnabulum concuti, aut horologium pulsibus affici. Hoc probat sonum non consistere in effluviis a sonoro corpore vibrati cur enim non emittuntur amplius, aut ad aures non permeant, cum imo liberius ob minora obstacula deberent? &lt;u&gt;Ad majorem rei evidentiam&lt;/u&gt; ita hoc experimentum instituitur horologium in vitro aere pleno ac probe clauso reponitur, ne aer scilicet inde possit exhauriri tum in recipiente pneumatico collocatur, atque ex hoc educentes aerem animadvertimus sonum nullum audiri. Machina horaria aere circumsepta est ergo nullimode suspicari licet aliquid deesse circa ipsum corpus sonorum quominus sonus exaudiatur. 

Dicendum potius non audiri sonum propter defectum aeris intermedii inter utrumque corpus. Porro corpus cum resonat, motu tremulo atque &lt;u&gt;oscillatorio&lt;/u&gt; minimarum partium afficitur singulis autem oscillationibus aer corpus tremulam circumdans concutitur, similesque recipit vibrationes, quas in ulteriores particulas aereas pariter defert nisi quod impulsus in circumfusum aerem delapsi atque auditus organum afficientes eo minores ac debiliores fiunt quo magis a fonte recedunt. Enimvero corpora, quae sonora dicantur, tunc sonum excitant quando ita agitantur, ut illorum partes tremulo ac vibratorio satisque rapido concutiantur motu: sic campanae malleo percussae, ratione elasticitatis concipiunt tremoris motum, qui major fit postquam vehementius ac diutius agitatae fuerint: instrumenta musica, dum illorum fides agitantur, simili pariter tremore concutiuntur; hinc est quod chartae frustula resonanti corpori imposita subsultare cernuntur. His positis certum est tremorem similem communicari debere aeri immediate ambienti, et deinde tremorem late &lt;u&gt;diffundi&lt;/u&gt; per &lt;u&gt;aereas particulas&lt;/u&gt;; nam particulae aeris sonoro corpori proximae illius impulsu comprimuntur; et cum sint elasticae , post &lt;u&gt;compressionem&lt;/u&gt; &lt;u&gt;dilatantur&lt;/u&gt;, aliasque sibi proximas urgent; atque hoc pacto et illae vibrant, et longe lateque in particulas aereas similis vibratio omni ex parte discurrit. Hinc pulvisculi aeri innatantes, qui radio solis in obscurum conclave intromisso conspicui sunt, agitari videntur si sonus proxime intendatur: pulsato prope stagnantem aquam tympano, validius et crispari, et subsultare aqua pariter cernitur.

Haec notentur. 1º . Vis acceleratrix &lt;math&gt;\varphi&lt;/math&gt; in vibrante particula resonantis corporis ita pendet a spatiolo &lt;math&gt;z'&lt;/math&gt; quod particulae superest excurrendum usque ad nativam aequilibrii positionem, ut crescente, decrescente , vel evanescente &lt;math&gt;z'&lt;/math&gt; crescat simul, decrescat, vel evanescat; propterea&lt;math display=&quot;block&quot;&gt;\varphi = C'z' + C'' z'^2 + C''' z'^3 + ...;&lt;/math&gt;et quia particularum excursiones exsistunt exiguissimae, erit,&lt;math display=&quot;block&quot;&gt;\varphi = C'z':&lt;/math&gt;vis nempe acceleratrix assumi potest proportionalis spatiolo &lt;math&gt;z'&lt;/math&gt;. 

2º. Non pluribus opus est ut intelligamus (29.4º) vibrationes omnes, sive majores, sive minores ejusdem particulae fore aequidiuturnas.

[[114|114]]. Progignitur quoque sonus ab aere vehementer compresso seseque statim restituente: etenim propter impetum in restitutione conceptum ad majorem, quam in statu naturali occupabat, extensionem perveniet; ac proinde cogetur se rursus contrahere, minusque naturali spatio tenere. His autem successivis contractionibus et expansionibus in reliquo aere pulsus &lt;u&gt;excitantur&lt;/u&gt;: sic producitur sonus v. gr. virgae aerem celerrime perstringentis: simili modo qui in tibiam insufflat, sonum gignit; dum nempe per tubi orificium aer insufflatione intromittitur, ille, qui continebatur in tubo, necessario secundum longitudinem comprimitur; unde fit ut is iterum expandatur, tum denuo coarctetur; atque hoc pacto, quamdiu perseverat inflatio, perficiantur oscillationes, hisque sonus progignatur. Certe si aerea columna tubo &lt;u&gt;inclusa&lt;/u&gt; non afficiatur nisi motu totius, sonus minime obtinebitur; utcumque vero &lt;u&gt;excitentur vibrationes&lt;/u&gt;; ut &lt;u&gt;perceptibilem&lt;/u&gt; sonum edant, earum numerus intra minutum secundum non debet praetergredi quosdam certos limites, videlicet 6 circiter et amplius 24000; uti compertum est experimentis D&lt;sup&gt;''ni''&lt;/sup&gt; Savart.

[[115|115]]. Saepe contingit nos voce elatiori quibusdam in locis loquentes, aliquo tempore postquam siluimus repente audire rursus verba a nobis antea prolata; atque haec est illa echo, de qua plura fabulantur poetae. Philosophi in hoc conveniunt, quod echo sit motus reflexus aeris, qui &lt;u&gt;motu ondulatorio&lt;/u&gt; affectus obici incurrens resilit consimili motu, et rursum aures nostras afficiens nos determinat ad eumdem sonum audiendum, quem antea audivimus: ut autem effectus iste contingat, necesse est aliquanto longius a loquente obicem existere. Ratio est quia si &lt;u&gt;obex&lt;/u&gt; proximior fuerit, sonus reflexus efficiet in auribus impressionem suam antequam impressio soni directi defecerit; tunc vero non poterit secunda impressio a prima discerni. Aliquando semel tantum, aliquando saepius eadem vox per reflexionem auditur: primam contingit quando ab unico loco vox collecta rejicitur, vel a pluribus, sed ad eamdem distantiam: secundum quando vox in pluribus locis ad diversas distantias collecta revertitur ad aures sensibili successione. Hinc intelligitur quare in vallibus, quas undique colles cingunt, echo saepius iteretur.

[[116|116]]. Non solus aer est &lt;u&gt;medium&lt;/u&gt; idoneum transmissioni sonorum: nam per alia quoque elastica fluida propagatur sonus. Vapores ipsi, in quos aqua, spiritus vini etc. attenuantur, sonum transmittunt; etenim si recipiens pneumaticum aere atmosphaerico evacuetur, tum aliquo ex dictis fluidis repleatur, sonus campanae vel horologii adhuc bene audietur: quin et liquores, aqua v. gr. sonum non intercipiunt, sed ipsum debilitatum licet propagant; qui enim intra aquam sunt, audiunt sonos extra aquam editos; et qui extra aquam sunt, audiunt sonos editos intra aquam. Tandem etiam corpora solida deferunt sonos ad ingentes distantias: celebre est apud milites ita terram excavare donec strato alicui bene solido aurem applicare possint, ut ex reboatu agnoscant adventum hostilis legionis, praesertim equitatus; huic strato non raro tympanum imponunt, atque levia corpora tympano imposita ex sonoris tremoribus subsultant.

=== De intensitate soni; deque ejus gravitate, et acutie. ===

[[117|117]]. &lt;u&gt;Intensitas&lt;/u&gt; major vel minor soni importat majorem vel minorem ejusdem soni vim ad sensationem excitandam, quae proinde in intensiore sono vehementior est, ita ut aures prae violentia laedat aliquando; in remissiore ita debilis, ut vix aliquando audiatur. Iamvero evidens est quod quo plures sunt partes sive in corpore sonoro, sive in aere simul oscillantes, eo plus motus atque activitatis, caeteris paribus, habent; ac proinde vehementius organum auditus pulsare poterunt: quo singularum partium itus et reditus major est, seu quo fortius singulae particulae comprimuntur et restituantur in unaquaque oscillatione sive in corpore sonoro, sive deinde in aere, fortiori item impressione aptae erunt organum auditus afficere. Contra, quo pauciores partes sonori corporis oscillant, eo minus communicabitur motus particulis aeris, et consequenter ab his minus afficietor auditus organum: quo singulae sonori corporis partes unamquamque oscillationem minorem habent, eo minorem item oscillationem in aeris particulis producent, ac proinde impressione minus valida auditus organum concutient. Quod ratione perspectum est, &lt;u&gt;experientia quoque confirmatur&lt;/u&gt;; et quod ad sonum, quem vocant primitivum, attinet, corpora densiora, caeteris paribus, magis sopora sunt quam quae ex opposito; atqui hoc nonnisi quia plures particulae in his oscillant simul; ergo ex numero particularum oscillantium sonus major vel minor pendet. Rursum inter corpora aeque densa, atqae elastica, quod validius percutitur validiorem profecto sonum excitat et &lt;u&gt;magnitudo&lt;/u&gt; soni &lt;u&gt;magnitudini&lt;/u&gt; percussionis est proportionalis: undenam hoc repetendum est nisi ex eo quod validior percussio fortias comprimit atque oscillare vehementius cogit particulas elasticas? Quoad derivatum sonum res constat experimento machinae pneumaticae (113): cum enim exhauriri aer incipit, sonus incipit imminui; atqui hoc est quia  aeris quantitas in excipulo imminuitur; et cum rarior evadat aer, minus valide comprimi et restitui ejus particulae debent; neque enim ulla alia &lt;u&gt;probabilis&lt;/u&gt; causa est. Condensando insuper aerem in eodem excipulo ultra &lt;u&gt;statum ordinarium&lt;/u&gt;, quem tenet in almosphaera, compertum est quod condensatus aer sopam reddit intensiorem; atque hoc quidem ita, ut intensitatis augmentum proportionem servet cum augmento condensationis. Franciscus M. Zannotti diligentius rem exploravit: aerem inclusum vase calefecit; quo pacto aeris &lt;u&gt;elasticitatem&lt;/u&gt; auxit, &lt;u&gt;densitate&lt;/u&gt; eadem servata, cum nullus permitteretur aeri exitas; et tunc sonus intendebatur, At rima aliqua in vase relicta, per quam aer posset erumpere, tum igne admoto, sonus multo minor visus est quam antea fuerat. Cum igitur, permanente aeris elasticitate, non idem permanserit sonus, rursus patet quod soni intensitas non solum ab elasticitate, et consequenter a magnitudine vibrationum, sed a densitate, id est a numero particularum vibrantium dependet. Nec arte solum ex rarefacto vel condensato aere intensitas soni mutata deprehenditur , sed naturali etiam aeris rarefacti vel condensati constituțione idem evenit: hinc in altissimis montibus, ubi aer rarior est, ac proinde minus elasticus, sonus multo est remissior quam in planitie , ubi condensatione atque elasticitate pollet majori.

118. Ex his explicantur sequentia circa soni intensitatem. 

1º. In aperto aere sonus calore minuitur, in clauso vero calore augetur: apertus enim aer, ubi calore afficitur, sese continuo dilatat, adeoque ejus intensitas minuitur, quin &lt;u&gt;elasticitas&lt;/u&gt; augeri debeat; quia nempe habet quo se rarefactus recipiat; ergo minor numerus particularum oscillat, adeoque remissior sonus. Contra, si aer undique clausus est, cum densitas eadem manere debeat, elasticitas autem ex calore crescat, idem erit particularum numerus, sed singularum oscillatio propter auctam elasticitatem augebitur; ergo intensior sonus.

2°. Sunt qui dicunt, aestate sonum intensiorem esse, caeteris paribus, quam hyeme; alii contra opponunt, quod hyeme intensior sit sonus quam aestate. Si in re incerta quoad factum et ex circumstantiarum varietate adeo varia ut fortasse determinari non possit , si inquam ratio reddenda esset, ajendum sonum aestate imminui debere, quia aer terram ambiens calore rarefactus minori densitate pollet, ac proinde minor erit numerus particularum oscillantium. Cum autem ex calore elasticitas crescat , hoc capite augeri debet sonus , cum nempe singularum particularum oscillationes validiores debeant. Videndum igitur quid praevaleat; et juxta vel densitatem hyeine praevalentem imminutioni elasticitatis, vel elasticitatem praevalentem aestate imminutioni densitatis, qui effectus sequi debeat.

3º. Hinc etiam explicant nonnulli cur nocte, caeteris paribus, soni majores sint quam interdiu; quia nempe densior est per noctem aer ob calorem minorem; at hujus rei explicatio verior est, quod per noctem, cessante ea aeris commotione quae per diem habetur ex multiplici strepitu, magis aptus sit aer ad soni vibrationes concipiendas et deferendas, organumque auditus nulla alia sensatione percussum aptius sit ad peculiarem aliquem sonum exaudiendum.

119. Discrimen inter gravem et acutum in sono importare profecto debet diversitatem aliquam in motu aeris, quo afficitur organum auditus, atque adeo in motu sonori corporis ex quo in aere motus hujusmodi derivatur; nam cum sensatio sonii ex impressione organi auditorii oriatur, at omnis alia sensatio ex impressione organi proportionati, et impressio ista per motum aeris ad organum appellentis fiat, profecto diversa impressio, quae a sono gravi atque acuto fit, diversum motum exigit tum in aere ex quo immediate producitur, tum in corpore sonoro a quo mediate progignitur; atqui ista diversitas non ex validiori vibratione seu oscillatione majori provenit; ex hac enim quantitas sive intensitas soni (117), non autem qualitas seu tonus procedit; ergo diversitas ista in celeriori seu crebriori vibratione partium aeris, et consequenter sonori corporis, derivanda videtur. Ratio consequentiae est, quia non alia diversitas saltem probabilior in oscillatione partium concipi potest quam, ut haec sit vel major ut scilicet quisque itus et reditus spatium majus percurrat, vel quod sit celerior ut scilicet eodem tempore plures situs ac reditus habeantur. Ergo cum ex primo capite discrimen acuti et gravis repeti nequeat, nihil afferri probabilius potest: quam celeritas oscillationum, quae certe in satione diversitatem afferre debet. 

Quoniam vero in rebus physicis natura explorari maxime debet experimentis atque observationibus, ita prosequor. Constat in chordis musicis, eas quae vel breviores sunt, vel magis tensae , vel minoris diametri (nam ex hoc triplici capite diversitas tonorum habetur in fidibus) acutius resonare; contra graviorem sonum reddere eas, quae longiores sunt, vel minus tensae vel majoris diametri: atqui chordae breviores vel magis tensae etc. percussae, plures numero vibrationes pari temporis intervallo producunt, pauciores aliae; hoc patet ex ipso sensuum testimonio: ergo sonus acutus habetur in chordis, quae frequentius dato tempore oscillant; gravis autem etc. In ea insuper proportione, in qua frequentiores aut rariores sunt vibrationes chordae musicae, est etiam magis vel minus acutus sonus: ergo frequentior aut rarior vibratio omnino connexionem habet cum tono per chordam musicam producto; pendetque tonus ex illa &lt;u&gt;frequentia&lt;/u&gt; aut raritate vibrationum, tamquam effectus a sua causa. Quod dictum est de chordis musicis, valet etiam in campanis et pocalis vitreis, aliisque id genus sonoris corporibus; haec enim percussa figuram rotundam in ovalem mutant, eorumque proinde fibrae eundo et redeundo oscillare debent, atque ex hac oscillatione sonus oriri colligitur; ut autem gravior vel acutior est sonus corporis, ita in figura immutatio et restitutio seu fibrarum ítus ac reditus rariores sunt aut crebriores. Porro si id in sonoro corpore contingit, ut gravior sonus obtineatur quando minor vibrationum numerus habetur in corpore, jam tunc in aere quoque minor vibrationum numerus haberi dicendus est: siquidem tot numero vibrationes dato temporis intervallo producuntur iu aereis particulis a tremulo motu corporis resonantis, quot ab ipsius corporis sonori fibris seu particulis eodem tempore peraguntur; et vice versa quot in aere gigni ac propagari vibrationes constat, totidem in ipso corpore resonante produci dicendum est.

120. Dum plurium corporum sonus ita temperatur ut gratus sit auribus, dicitur consonantia seu concentus, si ingratum  sonum produxerint, appellamus dissonantiam: in sonis ita temperandis ut sint jucundi, ars musica versatur. Tonus musicus seu consonantia pendet ex eo quod certo tempore certus vibrationum numerus a pluribus sonoris corporibus peragatur, et particulis aereis communicetur. Si duo vel plura corpora sonora intra idem tempus vibrationem absolverint , consonantia est omnium perfectissima , et sonus dicitur unisonus ; si eodem tempore unum corpus unam , aliud duas vibrationes expleat, consonantia haec dicitur octava: ita appellatur ex eo quod per quandam tonorum seriem ascendendo hic tonus a musicis octavo loco constituitur. Si eo tempore quo unum duas vibrationes, aliud tres absolvat, adeoque secunda unius cum tertia alterius concurrat, dicitur quinta: si eo tempore quo unum tres, aliud quatuor vibrationes conficiat, quarta nuncupatur; atque istae sunt consonantiae illae, quas Pythagoras advertisse traditur, dum quinque fabri malleis ferreis massam ferream contunderent. Consonantiae istae in vibrationibus chordarum inventae sunt ; imo etiam alii successu temporis consonantiae gradus additi , quos diligenter musicae scriptores explicant. Si videlicet numeri vibrationum , quas dato tempore chordae  musicae efficiunt , sunt ut &lt;math&gt;1 , \frac9 8, \frac5 4, \frac4 3, \frac 3 2 ,\frac5 3,  \frac{15}8, 2&lt;/math&gt; chordae illae edent notissimos tonos ''do, re , mi , fa , sol , la , si , do'': constat experimentis saepissime iteratis; etenim chordae homogeneae , aeque crassae , eodemque pondere tensae , quarum longitudines sint uti &lt;math&gt;1 , \frac89, \frac 4 5, \frac 3 4, \frac 2 3, \frac 3 5,  \frac8{15}, \frac12&lt;/math&gt;praefatos tonos edunt. 

Haec subjungimus circa exiguissimas chordarum vibrationes. 

1°. Chorda homogenea &lt;math&gt;AB&lt;/math&gt; (Fig. 59) uniformiter crassa ubique tensa aequaliter, punctisque &lt;math&gt;A&lt;/math&gt; et &lt;math&gt;B&lt;/math&gt; fixa, traducatur ad datam formam curvilineam &lt;math&gt;AC''B&lt;/math&gt;; tum sibi relinquatur: pro quovis temporis momento determinanda proponitur curva &lt;math&gt;AC'''B&lt;/math&gt;, in quam abit chorda. Sint &lt;math&gt;AO ( =x)&lt;/math&gt; et &lt;math&gt;S'O ( = y )&lt;/math&gt; coordinatae orthogonales; &lt;math&gt;h&lt;/math&gt; longitudo chordae &lt;math&gt;AB&lt;/math&gt;; &lt;math&gt;M&lt;/math&gt; massa; &lt;math&gt;\theta&lt;/math&gt; tensio: in ea qua sumus exiguissimarum vibrationum hypothesi, maxima chordae elongatio ab aequilibrii positione cum sit ferme insensibilis, haec obtinebunt quamproxime. Primo: apud quodvis chordae vibrantis punctum Seadem vigebit constanter tensio &lt;math&gt;\theta&lt;/math&gt;. Secundo: movebitur &lt;math&gt;S&lt;/math&gt; juxta directionem &lt;math&gt;SO&lt;/math&gt; respondentis ordinatae. Tertio: denotante a angulum tenuissimum &lt;math&gt;S'EA&lt;/math&gt; interceptum tangente &lt;math&gt;S'E&lt;/math&gt; et abscissarum axe &lt;math&gt;AB&lt;/math&gt;, erunt &lt;math&gt;\alpha = \sin \alpha = \tan\alpha ;\, \cos \alpha =1&lt;/math&gt;. Quoniam exercetur &lt;math&gt;\theta&lt;/math&gt; juxta vibrantis chordae longitudinem; sumptis arcubus infinitesimis &lt;math&gt;S'i , Si&lt;/math&gt;, denotabunt &lt;math&gt;S'i\, \mathrm{et}\, S'i'&lt;/math&gt; directiones tensionum apud &lt;math&gt;S'&lt;/math&gt;: resolvatur tensio juxta &lt;math&gt;Si&lt;/math&gt; in duas, quarum altera existat parallela rectae &lt;math&gt;AB&lt;/math&gt;, altera perpendicularis eidem &lt;math&gt;AB&lt;/math&gt;; et idipsum fiat quoad tensionem juxta &lt;math&gt;S'i'&lt;/math&gt;. Componentes parallelae axi &lt;math&gt;AB&lt;/math&gt; se mutuo destruent; componentes vero perpendiculares ipsi &lt;math&gt;AB&lt;/math&gt; exprimentur per &lt;math&gt;\theta\sin\alpha&lt;/math&gt; versus &lt;math&gt;O&lt;/math&gt;, et Osini atda ) versus S , seu per Ox et Oatd « ). Superest igitur vis
- Oda
gignens motum juxta SO : differentiale da sumendum quoad
x tantum, utpote denotans variationem anguli a in eadem curva AC &quot; B. Quisque videt --Oda esse vim motricem, cujusmodi est tensio  &lt;math&gt;\theta&lt;/math&gt;: propterea, designante dm elementum massae , exprimetur per
Oda
dm
∶≀≤↓⇟≓
miter crassa, ubique tensa aequaliter, punctisque A et B fixa, traducatur ad datam formam curvilineam AC&quot;B;
tum sibi relinquatur: pro quovis temporis momento de-
terminanda proponitur curva AC&quot;'B, in quam abit chorda. Siut AO (:æ) et S'O (::y) coordinatae orthogonales; '
h longitudo chordae AB; M massa;9 tensio: in ea qua
sumus exiguissimarnm vibratiouum hypothesi, maxima chordae elongatio ab aequilibrii positione cum- sit ferme in-
sensibilis , haee obtinebunt quamproxime. Primo: apud
quodvis chordae «vibrantis punctum S' eadem vigebit constanter tensio 9. Secundo: movebitur S' inxta directio-
nem SO respondentis ordinatae. Tertio :denotante et an-
gulnm tenuissimum S'EA interceptum tangente S*E et
abscissarum axe AB, erunt ut :sina −∙−−−− tangat ; cos at −−−∶↿∙
Quoniam exercetur 9 iuxta vibrantis chordae longitudinem :
[snmptis arcubus infinitesimi: S'i
, S'i', denotabunt S'i et S'i'
directiones tensionum apud S': resolvatur tensio iuxta Si
in duas
, quarum altera existat parallela rectae AB, alte-
ra perpendicularis eidem AB; et idipsum fiat quoad ten-
sionem iuxta S'i'. Componentes parallelae axi AB se mu-
tuo destruent ; componentes vero perpendiculares ipsi AB
exprimentur per Osina versus O, et 9sin( at-l-dat) ver-
sus S , seu per 90: et B(a-I—dat). Superest igitur vis
—9dat
gignens motum juxta SOI: dili'erentiale dat sumendum quoad, utpote denotans variationem anguli &amp; in ea-
dem cnrva AC'&quot;B. Quisque videt —-9dat esse vim motri-
cem , cuiusmodi est tensio 9: prapterea
, designante dm
elementum massae , exprimatur per
Gala
&quot;2711-255
respondens vis acceleratrix. Ob uniformem chordae cras
sitiem ,
dx
h
dm
M
Mar
ideoque dm =
h

;

insuper
a = tang a =
dy
dx ;
sumptisque differentialibus quoad x ,
da dany
dx dx²
Facto itaque compendii causa
on
M
superior expressio vis acceleratricis traducetur ad
d²y C2 dx²
unde ( 28 )
day
da(SS)
dia
d ” (SO - SO )
dc2
d²y
dx² de²
seu
1
255
respondens vis acceleratrix. Ob uniformem chordae cras-
sitiem ,
, ideoque dni ∙∙∶−− —-—de ;
9.&quot;M :.
da: ∙∙∙⋅ h −∙−
insuper
at:—— tangat-agi ;
sumptisque diiferentialibus quoad a:
,
data:-dv dx dx: ⋅
Facto itaque compendii causa
911
∙∙−∙∙↽−∙∶∘∙ ∙ M
superior expressio vis acceleratricis traducetur ad
da
⋅−∘⋅⊒≀−⋛−⋮⋅ ∙
nnde (28 )
,↶≀≖↗∙∙ irss; -dz(so-s'b) ∙∙∙ a.,
c dx&quot; dt: d? d;:
— '
sen256
day c2 day
dia (a) .
dx2
Formula ( a) suppeditat quaesitam problematis solutionem.
2. • Fac ut vis acceleratrix sit ut 1 , nimirum
day C'y ; erit dta
der
· c?
day
dx2 C'y
1 seu tör
so . dra
Inde habemus ( 27 , 27.0 )
VVT
y=CC + C, e
с - CV-4
evanescente X , evanescit et y ; hinc C = -C, , et con
sequenter ( 27. 30. )

* V0V1

y = C , [e
- *70V1
1 =
2011'sin rc=2C1V= 1sin
2 VMCMC
h9
facta x = h , evanescet y ; proinde
sin k V MC
ik V
MC'
ho TT C' =
OTE2
LM .
ho
9
ordinata CC &quot; respondens abscissae AC ( **)
256 ∙
da,, ⋅
−−↙⋮∎⋅≒↿∣
dx? :d—t; (a) . .
jl
C:
Formula (a) suppeditat quaesitam problematis solutionem. *
2.&quot; Fac ut vis acceleratrix sit uty
, nimirum '
lude habemus (27. 27.?)
f.. t/CV −−↿
-..-z ∁∣⇂∕∶⋅↿⋅
yiL-3010 c
⊣⋅∙ O; 6 - c :
evanescente a:, evanescit et )&quot;
, hinc C,: ---Cl , et con-
sequenter (27. 30. ∘ )
⋅−∙⊽⋮⋅∣∕∁⋅⇂∕∶↿ ... ⋅⋮∸−⇂∕∁⋅⇂∕∶↿
c c
y-—-—C,[e -—e ]∶∶
2C1V —-1 'sin −⋅⋮−− ∣∕∎∁⋅∶⊋∁∎≖∣∕∶−↴ sinx 9:709. :
facta x:h , evanescet y; proinde
. ⋅∙∥⋅∪⋅∙∙ VH?&quot; 97:-
&quot;[III 119 ∙−−−∘∙≀∙ WC;&quot;, (:::-IIM:
,
ordinata CC'&quot; respondens abscissae AC (;.-ä 11) ex-
'257
hibeatur per y ' , erit

;; = - 20, vt in . V MC =20,V = tsin.V MORE

TT
2C,V -1 sin î
2C, V 31 .
Propterea
2 = sin -
77 ()a' ) ;
aequatio ad curvam AC''B.''
3.° Per ty denotetur tempus unius semivibra
tionis ; erit ( 29. 3.° )
TT
1,5
2V C
VhM

;

0
et consequenter tempus unius vibrationis
hM ta VRM
Ad haec : designante n numerum vibrationum , quae ip
tra temporis unitatem absolvuntur , exsistet
1
V
TANTE
12
In hypothesi chordae cylindricae habentis radium
r el densitatem , erit M = fErPhò ; ideoque
257
hibeatur per J&quot;, erit
. -—- .
': MC' −−
∙ h M9112
r;.—20. ⇂∕⋅−↿ sm ∙− ∙−−−⇌⊋∁≖⇂∕⋅−∙↿ 810 2— IPGM :: 2 116
∙−− 7! −∙∙ ≢∁⋅↾∕∙−↿ sin ∙−⇇∋∙− −−−∙⇌ my.—1 .
Propterea
yzy' sin :; 71 (a') ;
aequatio ad curvam AC'&quot;B,
3.0 Per t. denotetur tempus unius semivibra—
tionis; erit ( 29. 33 )
et consequenter tempus unius vibrationis
.:Vg.
Ad haec: designante n uumerum vibratiouum, quae in-
tra temporis unitatem absolvuntur , exsistet
In hypothesi chordae cylindricae habentis radium
r et densitatem 8 , erit Mr.-:Ttrïhö ; ideoque558
13=rkVis, n - EVO
4. • Facta Osy. , velocitas puncti S in fine
temporis ( erit ( 29. 1.° 2.° )
v = y.Vī sine VC -Yosin
hinc ( 29. 1. ° )
=V
9. C - 02
C
yo V1 - sin’LVA
yo coseV C sy= . COS cos r.
Simili modo , facta CC“ =jo , velocitas pancti C in
fine temporis ( erit
7 t
yo sin Ti; simulquey'= y's cososeme- T ;
ta
et aequatio (a' ) ad chordam vibrantem poterit scribi in
hunc modum
y = yo com-A sin
C
-TT h
( á ').
5.° Si abscissae x in ( a'') substiluitur vel anh''
vel ( 2n + 1) h, prodibit y = o quotiescumque n aut erit =0,
aut erit quivis numerus integer : binae videlicet
558
9
≄≖−⊣⋅↗≖⇂∕∂ ∙
∙≖⋮−−⇀⊑⋅−−≀≖ '?Eä'
4.0 Facta OS;-:]. , velocitas puncti S in line
temporis : erit (29. 1.&quot; 2.')
v': J/ö' sint t/&quot;ä ∶∶−∶−≖−∫∘ sin-;- tt
!- 8
hinc (29. 1.&quot;-)
∙⊺∶−−−∙∙⇂∕⋅↗⇗ (S'—v ∙−−− J. l/1—s1n'q/ Q'::
j'. costV C' :y, cos ∙⋮− 11.
2
Simili modo
,
facta CC
∙−−∶ y'.. , velocitas puncti C&quot; in
fine temporis :erit
' n
s
∙ t ∙
' ' :
si:—y., s1n——1r;stmnlquey-:yocos—1t ;
t

: , :,

et aequatio (a') ad chordam vibrantem poterit scribi in
hunc modum
∙−− '
cst nsinæn ⋅ (a&quot;) J—yo O.t2 h .
5.0 Si abscissae a: in (a&quot;) substituitur vel an]:
vel (a'n-l-nh, prodibit yzo quotiescumque 11 aut erit 20,
aut erit quivis numerus integer: binae videlicet
Je!259
x = 2nh ,
( 2n+1 ) h
spectabunt ad quiescentia chordae vibrantis puncta. In
ferimus illud : chorda AB produci potest ultra limites A
et B quin puncta A et B per iteratas chordae vibra
tiones a statu quietis dimoveantur , etsi puncta illa poo
nuntur de se mobilia ; modo tamen AB in eamdem ac
antea conformelur initialem curvam , eidemque subjiciatur
tensioni : imo sumpta BH = HH' = =h , ita vibra
tiones suas conficiet chorda ABHH ' .
ut puncta A,
B , H , H ', ... in quiete persistant. Ad istiusmodi vi
brantis chordae figuram quod pertinet , sit v . gr. HD
HD = AO = x ; erunt AD = AH -HD = 2h - x , AD = 2h + x :
in la &quot; ) substitue prius 2h-x , deinde 2hta loco x ;
provenient ordinatae yı ety respondentes punctis D et D ',
nimirum
visy.cos Ti sin ( 2
a sin 16 는 (2-m ) R = my'o cos t2
sa= com sin ( 2+ )n = foco na sio ža .
Igitur y = -y, ya= y : ordinatae scilicet y , y , sunt
aequales , et ad eamdem plagam obversae ; ordinatae ve
ro y , y sunt quidem aequales , sed obversae ad con
trarias plagas. Chorda itaque dividitur in partes alterna
tim vibrantes supra et infra rectam AH'.
6.** Quoad (a) generatim spectatam ; denotanti
bus f et F binas functiones arbitrarias , satis hic erit
animadvertere eam expletum iri per
y =-flatct) + F ( x - ct) (a ' ' ) ;
siquidem
259
a.:2n71 ∙ :r (2n—l-1)I;
spectabunt ad quiescentia chordae vibrantis puncta. ln-
ferimus illnd: chorda AB produci potest ultra limites A
et B quin puncta, A et B per iteratas chordae vibra-
tiones a statu quietis dimoveantnr, etsi pnncta illa po-
nantur de se mobilia; modo tamen AB in eamdem ac
antea conformetur initialem curvam, eidemque subjiciatnr
tensioni: imo sumpta BH −−−−− HH' ::
...
zh
,
ita vibra-
tiones suas conficiet chorda ABHH' . .. ,
ut puncta A,
B , H , H', ...
in quiete persistant. Ad istiusmodi vi-
brantis chordae figuram quod pertinet ,
sit v. gr. Hl):
HD'zAOsæ; erunt AD;:AH-HDzah-æ
, AD':.2h-l—æ :
in (a&quot;) substitue prins alz—a:, deinde Zh—l-æ loco :;
provenient ordinatae y. etj, respondentes-punctis D et D',
nimirnm
'
tnsin(2 −⋅⋮≻↿∎∎∶ 'cos tu' æ
fac:-Tou.&quot; (: 'l &quot;70 :: Olli-i:&quot;.
, s ∙ æ ,
t
. æ
Jar—jre cos-1t am (2 −⋅⊢ --)1t :yocos —-1t sm --1t
.
- :, h : 11
Igitur y. ∙∶−−−∙ −∫∙ ∙↗≀≏−−−−−⋮↗↟⇌ ordinatae scilicet y , ;, sunt
aeciuales ,
et ad eamdem plagam obversae; ordinatae ve-
ro y, y, sunt quidem aequales, sed obversae ad cou-
trarias plagas. Chorda itaque dividitur in partes alterna-
tim vibrantes supra et infra rectam AH'.
6
∙∘∙ Quoad (a) generatim spectatam; denotanti-
bus f
et E binas functiones arbitrarias
,
satis hic erit
animadvertere eam expletum iri per
Fnæ-l-ct) −∣⋅− P(æ—ct) (if&quot;) ;
siquidem'260
dº[fix + c ) + F(x – ct) ]_da[fixtet) + F (x – ct )] .
do[ ) dt2
7 . ** Velocilas puncli S in fine temporis i prodit
expressa ( 28) per
dOS - OS')
dt
dy
dt [flatct)-F&quot;(x – ct)]:
initio motus , quum nempe t = 0 , est v=0 ; iccirco
c [ f '( x )-F'(x )] = 0, $' ( x)=F&quot; ( x) , et f (x ) = F (x ) ;
aequationes igitur determinantes et curvam ASB , et ve
locitatem traducentur ad
y = f(x + ct) + f(xớctct), v '= -c[ f '( x + chf'( x - ct) ] .
Facto t = 0 , istarum prima praebebit
y = 2f \x ) ,
aequationem videlicet ad curvam datam ACSB : ex hac
itaque curva pendet natura functionis f. Caeterum , ge
neralem de integratione differentialium partialiumque ae
quationum doctrinam suo tempore videre erit in parte 3.4
nostrorum elementorum Matheseos n. 200 , 201 ,
121. Si chorda instrumenti musici percutiatur , et pro
pe adsit instrumentum aliud , in quo chorda sit ad aniso
num cum priore tensa , baec alterius instrumenti chorda
sensim tremere incipiet , et undulationes sensim majores
concipiendo ad sonum ipsa quoque excitabitur eumdem to
num reddendo quem prior illa chorda percussa reddit .
Jam vero si ad hujus rei rationem attendas, plana erit juxta
theoriam traditam : sicut enim pendulum quiescens etiam
minimo impulsu , uti est qui ex insufflatione procedit , ad
260
⊄≀⇟⊏∣≺↕⊣⊸↥≻ −⊢ Für—ct) ],. «l*[m—l-aH-Fw—ctü.
dt:
(
7. ∙∙∙ Velocitas puncti S'111 fine temporis :prodit
expressa (28) per
os.-os d
v. ∙∙∙⋅ &amp; dt ):... .... :]? ∶−∙−− —c[f(æ-l-ct)—-F'(:r—ct)]:
initio motus , qunm nempe t::o
,est ⇂↓−∙−−−∘⋅ , iccirco
c[f (æ)—-F' (x)] ∙−−− o, f(xrr—F' (x)
, etfix):F(æ)' ,
aequationes igitur determinantes et curvam AS'B ,
et ve—
locitatem v' traducentur ad
y-fþ—I—ct) *Aæ—ct) , ∙≀⋅−−− ∙−−− ∙−− c[f '(æ-i-ctF-f'w—ct) ].
Facto t--—-o
, istarum prima praebebit
F2nx) :
aequationem videlicet: ad curvam datam AG&quot;B : ex haei
itaqua curva pendet natura functionis f. Caeternm, ge-
neralem de integratione differentialium partialinmqne ae-
qnationnm doctrinam suo tempore videre erit in parte 3.&quot;
nostrorum elementorum Matheseos n. 200, 201, .-. .
.-

121. Si chorda instrumenti musici percutiatur, et prope adsit instrumentum aliud, in quo chorda sit ad unisonum cum, priore tensa, haec alterius instrumenti chorda sensim tremere incipiet, et undulationes sensim maiores concipiendo ad sonum ipsa quoque excitabitur eumdem tonum reddendo quem prior illa chorda percussa reddit. 

Jam vero si ad huius rei rationem attendas, plana erit iuxta theoriam traditam: sicut enim pendulum quiescens etiam minimo impulsu , uti est qui ex iusufilatione procedit , ad motum oscillatorium minimum primo concitabitur , et si in
suflationem saepius repetas , poteris sensim oscillationes
majores , ac majores perficere (tunc tamen id fiel quando novi isti impulsus certa periodo, parique intervallo habeantur; si enim pendulum contra insufflantem venit, insufflantes rursum potius motum impediemus quam adjuvabimus, atque idet' finita una oscillatione debet opportune rur
sus alius impulsus addi , sicque ubi oscillationes penduli
et novi impulsus certa periodo sibi respondeant , effectus
habebitur) ita in chorda praefata evenit ut trémitns con
cipiatur et augeatur ; donec excitetur sonus ; quia nempe
Oscillationes unius chordae consentiunt cum oscillationibus
ad quas altera determinabilis est , iccirco ex repetitis chor
dae percussae uşdulationibus , quae sunt isochronae undulationibus alterius , obtinebitur ut hae augeantar donec so
nus excitetur in chorda etiam plectro minime percussa. Ex hac doctrina infero: ergo in utraque chorda oscillationes sunt pares numero; ergo cum tonus ab utraque redditus idem sit, tonus igitur a numero vibrationum hujusmodi pendet. Ad magis declarandam traditam doctrinam de acutie et gravitate sonorum, utile erit non nulla hic explicanda proponere circa chordas vibrantes. 

Ac 1º. quamvis chordae non sint unisonae, attamen una percussa, alia sonum edit, si modo tensae sint ad octavam, aut alias quasdam habeant armonicas proportiones. 

2º. Si duae chordae tensae sint ad octavam, et pulsetur chorda longior; quae dimidia ejus est, reddet tonum sui proprium, scilicet octavam acutam; at si pulsetur chorda brevior, excitabitur in longiore tonus non sui proprius, scilicet ad octavam gravem, sed tonus chordae brevioris. 

3º, Refert Sauverius hoc phoenomenon: chorda longa 5 ped. percutiatar, et notetur tonus;
tum ad distantiam unius pedis ponatur supra chordam le
ve aliquod obstaculum velati plumae frustulum , quod ta
men non impediat molus communicationem : si quinta haec
1
1
1
261
motum oscillatorinm minimum primo concitabitur ,
et si in-
snæationem saepius repetas, poteris sensimf oscillationes
maiores , ac maiores perficere (tunc tamen id fiet quando
novi isti impulsus certa periodo», parique intervallo babe-
autur; si enim pendulum contra'insumantem venit ,
insuf-
Hantes rursum Potius motum impediemus quam' adiuvabi-
mns , atque idet-' finita nna'osci'llatione dehet opportune rur-
sns alius' impulsnsgaddi, sicque ubi oscillationes penduli
et novi impulsus certa periodo sibi respondeant , effectus
habebitur) ita in chorda praefata evenit ut tremitus con-
cipiatur et augeatur, donec excitetur sonus; quia nempe
oscillationes unins chordae consentiunt cum oscillationibus
ad quas altera determinabilia est
,
iccirco ex repetitis chor-
dae percussae undulationibus , quae sunt isochronae undu-
lationibus ulterius-, obtinebitur ut hac augeantur donec ac-
nos excitetur in chorda: etiam plectro minime percussa. Ex
hac&quot; doctrina infero: ergo in utraque chorda 'oscillationes
sunt pares numero; &quot;ergo cum tonus ab utraque redditus
idem sit, to'nus igitur a numero vibratiouum hujusmodi
pendet. , - - -
'
'
'
Ad magis declarandam- traditam doctrinam de acutie
et gravitate sonorum, utile erit non nulla hic &quot;explicanda
prcponere circa chordas vibrantes.

q'uod ta-
men non impediat motus communicationem: si quinta haec
.*262
P
-chordae pars pulselur, tongm efficiet proprium chordae d
- nias pedis; hunc autem ipsum tonum reddit etiam reliqua
chordae pars, etsi quadrupla. Rursum si obstaculum pona
tur post duos pedes , eveniet ut pars brevior citius oscil
let, et longioris motum perturbet; subinde utraque chor
dae pars ita , sese componet; ut vibrationes eodem tempo
re compleat: tunc vero tonus reddetur neq w proprius chor
dae duorum pedum , neque trium, sed proprius chordae u
nius pedis.
Ad primum quod attinet , quoties duae chordae len
sae sunt ad octavam, jam vibrationi unius chordae ,respon
dent duae vibrationes alterius; ergo quamvis singulae , O
scillationes non conveniant, adeoque tremitus aeris non re
novet impulsum in alia chorda post singulas ejusdem oscil
lationes, renovari tamen potest impulsus hic post binas ;
eo ipso poterit chorda ad octavam tensa , etsi difficilius ,
ad oscillandum determinari ex alterins oscillationibus. Idem
valet de aliis chordis quae eam habent proportionem ut
oscillationes recurrere possint post aliquem ipsarum nu
merum: ac proinde illae, quae vel ejusmodi recursum non
admittuut , vel quarum recursus majorem postulat' quam
par est vibrationum numerum, non ita invicem ad reso
nandum poterunt determinari.
Ad secundum : quod chorda brevior resonans ad pulsa
tionem longioris reddat tonum sui proprium , cohaeret cum
doctrina jam tradita : quod autem chorda longior reddat
Lonum proprium chordae brevioris non officit; etenim si
chorda sit dupla , quasi in duas dividetur, neque tota oscil
Jabit ( 120..5º. ) per modum unius, sed habens in medio
punclum quiescens, seu nodúm, oscillabit seorsim in sin
gulis dimidiis partibus, ac si, scamould adjecto , bifariam
arte divisa esset ; , et si chorda ' triplo sit longior , ia - tres
partes aequales dividetur: quo posito , nil mirum quod chor
da dupla non sui proprium tonum , sed tonum subduplae
reddat, et tripla sonum subtriplae.
ic
0
at
LE
262
chordae pars pulsetur, tonum efficiet proprium chordae n-
onius&quot; pedis; hunc autem ipsum tonum reddit etiam reliqua
chordae pars,; etsi quadrupla. Rursum siobstacnlum pona-
tur post duos pedes'. eveniet ut pars brevior. citius oscil-
.let,. et, longioris motnm perturbet; subinde utraque-. chor-
dae pars ita sese componet&quot;, ut vibrationes eodem tempo-
re compleat: tunc vero tonus reddetur neq ,.: proprius chor-
dae duorum pedum, neque trium, sed proprius [chordae u-
,niuspedis. ↴
⋅⋅∶⇟⋡∢⋅ ∙

Ad primum quod attinet, quoties duae chordae tensae sunt ad octavam, iam vibratioui unins chordae respondent duae vibrationes alterius; ergo quamvis singulae oscillatioueslnon conveniant adeoque tremitus aeris non renovet impulsum in alia chordae post singulas ejusdem oscillationes, renovari tamen potest impulsus hic post binas; eo ipso poterit chorda ad octavam tensa, etsi difficilius, ad oscillandum determinari. ex&quot; alterius. oscillationibus. Idem
.valet de aliis chordis-anae-eam habent prOportionem ut
oscillationes recurrere possint post- aliquem - ipsarnm nn-
merum: ac proinde illae, quae vel ejusmodi recursum non
admittunt
,vel quarum recursus majoreni- pastulat' quam
par est vibratiouum numerum, non ita invicem a'd reso-
.nandum poterunt determinari.

Ad secundum: quod chorda brevior resonans ad pulsationem longioris reddat tonum sui pmprium cohaeret cum doctrina iam tradita: quod autem chorda longior reddattonum - proprium  chordae brevioris non officit; (etenim-si
chorda sit dupla, quasi in &quot;duas dividetur,- neque- tota oscil-
Jabit (120. 50.) per modum unins, bed habens in medio
punctum quiescens, s'eu nodnm, oscillabit seorsim in sin-
gulis dimidiis partibus, ac si, scamnnlb adiecto , bifariam
arte .'divisa esset;. et: si chorda' triplo sit longior, in- tres
partes aequales dividetur: quo posito, nil mirum-quod chor-
da dupla non aui proprium tionnm, sed tonum'subduplae
reddat, et tripla sonum subtriplae.
n-rts lar-Q .-263

Ad tertium: idem Sauverius hanc in Academia Pari
siensi explicationem attulit . Dum chorda nullo obstaculo
apposito pulsatur, vibrationes efficit toti suae longitudini
proportionales: at dum leve illud obstaculum apponitur post
pedem unum , undulatio totalis chordae dividitur ; prima
enim pars chordae , utpote quinta chordae totius ,
quinquies citius oscillare debet quam oscillaret integra
chorda : sic citius oscillando abripiet partem sibi proxi
mam in vibrationes aequales ; secunda pars tertiam, atque
ita singulae quinque partes seorsum oscillationes pera
geat. Alterum vero, quod magis est admirabile, ila ab eo
dem auctore explicatur ; pars brevior chordae, scilicet duo
rum pedum, citius oscillans quam reliqua , secum abripit
per sui motus communicationem partem sibi similem, nem
pe duorum pedum; in quinto autem pede oscillationes e
tiam communicantur, quae cum esse debeant longitudini
proportionales, duplo crebrius oscillabit extrema haec chor
dae pars quam reliquae; proinde ista sibi proximam unius
pedis partem trahet ad analogas oscillationes , et secunda
tertiam atque ita de reliquis , donec in hoc etiam casu quin
que chordae partes oscillent juxta longitudinem propriam ,
et consequenter sonum reddant respondentem longitudini
upius pedis.

122. Quaeri potest quomodo sonus trans obicem queat communicari ita, ut tonus proprius sonori corporis permaneat; nam fibrae, seu partes elasticae obicis puta parietis aut cancelli vitrei, ad motum concitatae vel sui proprium tonum reddere debent, vel si dissimiles sint, plurium tonorum mixturam, quod non accidit. Respondeo nullam esse difficultatem, si immediate per aerem soni propagatio habeatur, etiam intermedio exsistente obice. Quod si per obicem sonus diffunditur, in ipso admitti possunt partes aptae diversos sonos reddere aerique transposito communicare; atque ita, ut ille sensibilis sit trans obicem tonus, qui a partibus analogam oscillationem habentibus cum sonoro corpore communicatur. Forte etiam dici potest, quod si fibrae non habentur aptae eum tonum reddere, dividantur, ut in chorda non unisona contingit, adeo ut idem tonus transmitti possit.

123. Quoniam de tonis, ex quibus qualitas soni denominatur, egimus; quaerendum esset unde asperitas aut lenitas, quae pariter ad qualitatem quamdam soni pertinet, proficiscatur. Animadverte sonum quemcumque non esse simplicem, sed compositum e sono plurimarum sonori corporis partium: sic chorda musica percussa non simplicem edit sonum, sed quemdam veluti concentum edicit , qui a
peritioribus musicis probe dignoscitur; in quo tamen cum
fortior tonus praevaleat, alios minores obruit : coexsistunt
videlicet in chorda sonora, et generatim in quovis particu-
larum s'ystemate, variae exiguarum oscillationum species.
Imo vero non tantum sonorum ipsum corpus attendendum
est plerumque v. gr.-chorda musica, sed instrumentum i-
psum cui chorda adhaeret: variae insuper reflexiones ani-
madverti debent, quibus aer ad aurem deveniens diversas
subit modificationes. Itaque si vibrationes partium sonori
corporis sint bomologae, sonus lenis erit; si contra, asper:
atque hinc aspere sonant chordae inaequales in materia ,
crassitie etc; item ex reflexione aequabili atque uniformi
sive instrumenti, cui chorda adhaeret, sive circumstantium
corporum, lenitas soni orietur, asperitas ex opposito. 'Bo-
num erit observare quod chorda musica vehementius quam
par est distraCta stridet; quia videlicet valde percussa non
eam' servat legem quam in moderatis percussionibus obti-
net ut sub eodem tempore oscillationes suas sive majores,
sive minores dbsolvat; sed continget ut tempora oscillationum
inordinate mutentur, stridorque pro tono solito erumpat.

124. Haec notentur 

1º. chordarum vibrationes hactenus consideratae, dicuntur transversae: quae nimirum- obtinen-
tur chordam percutiendo in directione ad ejus axem perpendiculari: quod si atteratur chorda in directione ad e
jus axem parallela, adhuc sodos edet, sed , caeteris pari
bus, multo acutiores quam qui ex vibrationibus transver
sis progignuntur ; idque ex eo repetendum esse videtur
quod elasticitas propria chordae in vibrationibus longitudi
nalibus validior sit quam in transversis.

2.° Ubi in longitudinalibus vibrationibus chorda
rum obtineant &lt;u&gt;nodi&lt;/u&gt; , molus ita fiet ut partes hinc illinc cira
ca podum quemlibet positae simul ad ipsum nodum accedant, simulque alternatim recedant.

3º. Corpus omne, dum resonat, dividitur in plu
res partes vibrantes invicem ' separatas lineis , quae vocan
tur &lt;u&gt;nodales&lt;/u&gt;, quaeque oculis subjiciuntur spargendo per su
perficiem corporis minutissima arenae grana: haec enim su :
per lineis illis acervari observantur. Nodales propterea li
neae modo' sunt rectae, modo curyae, modo ex rectis si
mul et curvis coalescunt.

4.º  Malála nodalium linearum figura, plerumque mutatur et sonus; semper autem acutior vel gravior evadet sonus, prout corporis superficies in majores vel minores numero parles vibrantes dividetur ab ipsis noda libus lineis.

5.° Laminae rigidae ex ferro, vitro etc. in transversis vibrationibus absolvendis sequuntur leges alias ab illis, quas sequuntur chordae.

=== De directa soni propagatione per aerem. ===
125. Experientia nos edocet quod in iisdem circumstantiis sonus aequabili velocitate in toto decursu devehiеur; atque omnes soni , sive intensi , sive remissi , sive graves, sive acuti eadem velocitate diffunduntur.  

Nam 1.º Academici Florentini ad percurrendam distantiam unius milliaris sonum tormenti bellici impendisse quinque secundorum tempus experti sunt, ejusdem vero tormenti sonum ad conficiendum dimidium milliare impendisse dimidium tempus testantur aequabili nimirum velocitate perrexit sonus. Derhamus saepius repetitis experimentis idipsum invenit, adeo ut ab uno ad duodecim milliaria sumens intervalla invenerit aequale spatium aequali tempore in quavis a sonoro corpore distantia confici. 

2.º Prope sonorum corpus intensior est sonus, remissior in majore a sonoro corpore distantia atqui tam prope quam procul a sonoro corpore aequali velocitate pergit sonus ergo tam intensus, quam remissus etc. Hoc ipsum institutis ad id experimentis etiam constat Gassendus sclopeti et tormenti bellici fragorem eodem tempore pervenisse affirmat, cum eodem tempore exploderentur. Florentini et Derhamus in diversi generis tormentis idipsum evenisse notant itemque tormenti bellici minoris et mallei fragorem idem unius milliaris intervallum confecisse eodem tempore. Certum est ergo tam intensum, quam remissum etc. Huc spectat quod Derhamus quoque notat post Florentinos, scilicet eodem tempore sonum ad aures pervenire sive tormentum ad observatorem convertatur, sive ad contrariam plagam videtur enim intensior in eam partem, in quam tormentum dirigitur, esse debere sonus.

3.° In concentu sive ex instrumentorum pulsatione, si malleorum ictibus etiam ad satis notabilem distantiam dignoscitur tonorum successio eo praecise ordine, quo ictus varios tonos producentes habentur successive, et quidem sine sensibili temporis mora atqui si toni diversi non eadem propagarentur velocitate, jam qui toni successive habentur, non successive atque ordine illo ad aures venirent ergo etc. Erit fortasse qui quaerat qua ratione fieri possit ut sonus in quavis distantia, sive intensus, sive remissus, uniformiter &lt;u&gt;propagetur&lt;/u&gt;. Respondeo: eadem materiae quantitas eodem tempore, tum ex vi majore, tum ex minore, undulare potest ergo eadem aeris portio, seu &lt;u&gt;unda ejusdem latitudinis&lt;/u&gt;, eodem tempore potest undulationem perficere, sive ex majori, sive ex minori vi impellente. Antecedens est evidens; pendulum enim idem , adeoque eadem massa , eodem tempore oscillationes peragit sive magis , sive minus impellatur ad oscillandum: ergo a pari eadem aeris quantitas oscillare potest sub eodem tempore sive ex majori , sive ex minori impulsu. Sed si eadem aeris quantitas aequali tempore potest comprimi et restitui , jam eodem tempore potest sonus, ad datam distantiam pervenire , sive intensior , sive remissior: haec minor est evidens; si enim eadem est &lt;u&gt;latitudo undae&lt;/u&gt; , idemque tempus, jam eodem intervallo temporis spatium datum a sono conficietur; ergo sive intensus sit , sive remissus , seu vi majori aut minori aereae undae propellantor, eadem esse potest soni velocitas.

Quid ergo provenit ex hoc quod in sono intensiore vis major aerem impellat? Nempe quod ejusdem latitudinis unda, licet eodem tempore conficiatur , compressionem tamen ac restitutionem patiatur validiorem , vel languidiorem; sicut in pendulo accidit , quod eodem tempore oscillans ex impulsione maiori oscillationem concipit magis validam , et minus ex vi minori. Atqui hoc idem praestat minorem intensitatem , non autem minorem soni velocitatem . Ostendo: intensitas soni pendet a vi , qua in organum appellunt aeris particulae ; ergo si vi majore condensantur , et restituuntur , intensiorem efficient soni sensationem; at velocitas ex dictis pendet a latitudine undae, et tempore quo perficitur: neque latitudo immutatur , neque tempus; ergo non mutatur velocitas. Quod autem neque latitudo , neque tempus mutetur , ita probari potest. Latitudo enim undae , seu aeris quantitas ad oscillandum per modum unius determinata , ea esse debet quae potest obtemperare vibrationibus sonori corporis , a quo unda producitur , quaeque potest oscillationes suas eodem tempore complere quo sonorum corpus oscillationes suas perficit: ergo latitudo undae proportionari debet tempori quo sonorum corpus perficit vibrationes suas. Atqui sive intensior , sive remissior sit sonus, tempus quo sonorum corpus vibrationes suas complet , est ( 113. 2.°) semper idem; ergo item latitudo undae aereae eadem esse semper debet. Idem probat simul, quod sicut eadem latitudo, ita idem esse debet tempus quo unda perficitur. Et sane si tempus mutaretur , deberet quoque mutari tonus: atqui idem manet tonus in quacumque distantia a sonoro corpore , et quidem sive corpus resonet intensius , sive remissius; ergo etc.   

Hinc dum de sono agitur duplex in motu undae aereae velocitas distinguenda est: altera importat tempus quo unda conficitur , seu quo segmentum aeris datae latitudinis oscillat ; altera importat motum particularum aerearum itum et reditum perficientium in ejusdem undae efformatione. 

Quaeri hic potest in quanam ratione intensitas soni minuatur in progressu . Reponunt communiter quod intensitas soni est in ratione duplicata distantiarum inversa a centro soni : rationem afferunt , quia sonus quantum est de se aequabiliter undequaque diffunditur in modum sphaerae. Atqui ex hac aequabili in modum sphaerae diffusione sequitur decrementum in ratione praedicta ; nam si ita diffunditur , debet in ea proportione intensive decrescere , qua extensive augetur , sea qua latius materia , cui communicatur motus , sese expandit ; sed hujusmodi extensionis augmentum est in ratione duplicata distantiarum ; hanc enim rationem sequuntur sphaericae superficies : ergo etc... . 

126. Sit c velocitas , qua propagatur sonus ; &lt;math&gt;\Delta&lt;/math&gt; distantia inter vibrantem sonori corporis particulam et particulam aeream : exprimet tempus a sono impensum ad percurrendam distantiam &lt;math&gt;\Delta&lt;/math&gt; ; motusque particulae vibrantis nonnisi post tempus I = pertinget ad aeream particulam: propterea substituto 2— —c- 'a duabus ulti-
mis formulis(29. 5.&quot;), si :
∙−−≜−⋅ incipit ab 0
, ultraque
progreditur, determinabitur aereae particulae motus per»
∙ 271: A , 9 211 .A.
∦⋅⇋↙∁∘∎∐∙−⊖−⋅≺∁−−∘−−≻ ,
szC-Z-n' 008
! j(t—z).
F30i20,1t,2,3,4,-...,act—-e—:i9,tln- c
de habes A:c(t—-i9): erit
⇂↓∣∶⇂∕∁ sin 21'12:o.
Sumptis ergo distantiis Azct, c(t—G), c(t—29), c(t—BG), ...,
uulla velocitas v' ibi invenietur : aer proinde in locisi il-
lis omnino quiescet quando desinit tempus :; eritque n—
sque ad Ar.-ct in plureswundas distinctum similes et aequa-
les ; quarum communis latitudo ::09 ; numerus vero

:

∆−∘−⊖− ⋅
∆
Quantitas l/C sin −⋛≖−≺ t — A;) manet positiva ab
t — 30- :::-id ad :
2 — (i ] &amp;) 6 ; manet-negativa
A . A .
⋅ ∙ ∙ ∙
ab t—-—c- :::(12-l-ä)9 ad t— -c—-.-:(1-l-1)9. Ertt 1g1-
tur v' positiva inter A———-0(t—i9) et A −−∶ c [t—(i—i—ä— )9];
erit negativa inter Ach-t—(i—i-ä-W] et A:c[t—(i-l—1)9].
in tribus hisce distantiis est praeterea v':o. Ergo quae-
libet ex dictis undis constat duabus partibus aequalibus ;
recedit aereum fluidum ab oscillante sonori corporis par-
ticula in anteriora parte, accedit in posteriore; quiescit stra-270
tum medium ; maxima viget aerearum particularum velo
citas in medio semiundae anterioris ; maxima item in me
die semiundae posterioris.

127. Soni velocitas augetur a vento secundo, minui
tur ab adverso. Derhamus videns ab aliis affirmari nullam
mutationem afferri a ventis circa soni velocitatem , hanc
rem statuit explorare ita exacte et diu , ut ambigendi lo
cus omnis tolleretur. Ad hoc autem summa ipse fruens
opportunitate experimenta habebat omnino in promptu .
Nam cum ex arce Blancheath , ubi tyrones rei tormenta
riae exercebantur , saepe exploderentur tormenta bellica ,
ipse e sua Ecclesia in agro Upminsther ad 13 milliaria
distante flammam advertere poterat ; animadvertit autem
optimo usus chronometro non semel aut iterum , sed
triennio integro. Porro ex tabula , quam observationum
suarum confecit, quaeque habetur in Transactionibus An
glicanis, et a Masschembroekio descripta fuit in suis com
mentariis ad lentamina Florentinorum , constat quod so
ni velocitas inter tempus quo ventus favens spirabat ,
et contra venius sono adversus erat, cum scilicet in utro
que casu yentus validus admodum esset , discrepat un
decim semisecundis circiter in praedicto intervallo. Ergo
experimentis hisce insistendo dicendum augeri secundo ven
to soni velocitatem , imminui autem etc.
Derhami observationibus consentiunt observationes Aca
demicorum Parisiensium , qui anno 1738 exploraturi ve
locitatem soni jussu Regiae Academiae pariter testantur
non eandem esse adverso ac secundo vento velocitatem
qua propagatur. Rationis momentum experientiae suffra
gatur : nam ventus transfert loco aerem ; ergo undas so
noras ad oscillationem a sonoro corpore impulsas trans
fert ; ergo tantum accelerari debet propagatio soni , quan
tum aeris sonori translatio ratione venti importat.
Opporluna est comparatio circulorum in aqua exci
latorum ope lapilli decidentis : si enim aqua non sit sta
270
tum medium; maxima viget aerearnm particularum velo-
citas in medio semiundae anterioris; maxima item in me-
die semiundae posterioris.

271
SUS asserue
gnans sed fluens aequabili motu ; jam dum post lapidis
descensum circuli successive efformantur , lota ipsa aqua,
in qua efformantur circuli , localiter transfertur ; ergo
circuli appellent ad datum locum citius in eam partem
versus quam defluit aqua quam ad alteram in eadem
distantia ex opposita parte : ita paritate rationis in sono.
Iis , quae . modo diximus , objici possunt experimenta
Cl. virorum qui de velocitate soni nihil immutari pror
sive secundus sit , sive adversus ventus
runt. Gassendus enim , et Mersennus id sibi accidisse te
stantar ; et Academici Florentini , collocatis observatori
bus inter se duo milliaria distantibus , dum ventus spi
raret , asserunt tormenti bellici , quod medio illo inter
vallo situm erat , fragorem pervenisse eodem tempore ad
utrosque , etsi aliis faveret ventus , aliis esset contrarius.
At cum experimenta recte instituta aliis item recte
institutis nequeant esse contraria , yidendum quaenam
praevaleant. ' Florentini , quorum tentamen prae caeteris
in medium afferri solet , unica nocte , et in distantia pau
corum milliarium experimentum instituerunt . Derhamus
triennio experimenta iteravit , et in 13 milliarium distan
tia ; haec autem distantia in experimentis Derhami eadem
erat semper , a sua scilicet Ecclesia ad arcem ; in ten
tamine Florentinorum tormentum constitutum est medio
inter duos observatores intervallo ; quod intervallum utrin
que aequale asseritur , sed fortasse non ita esse potuit.
Ergo apparet quam recte Derhami observationes antepo
nantur observationibus Florentinorum , atque eodem jure
Gassendi et Mersenni . Reipsa etsi experimento quodam
Musschembroekius quoque deprehendere sibi visus fuerit
aequalem velocitatem tam secundo quam adverso spirante
vento , tamen Derhamo assentitur , et Florentinis quo
rum sagacitatem saepe alibi commendat , minime in hoc
adstipulatur.
Obiter hic notamus quod juxta auctores ferme omnes
etiam intensitatem sąni auget ventus secundas , et minuit
.
1
271-
gnans 'sed fluens aequabili motn; jam dum post lapidis
descensum circuli successive eil'ormantur ,
tota ipsa aqua,
in qua eB'ormantur circuli ,
localiter transfertur; ergo
circuli appellent ad datum locnm citius in eam partem
versus quam defluit aqua quam ad alteram in eadem
distantia ex opposita parte: ita paritate rationis in sono.
Iis
, quae-modo diximus, objici possunt experimenta
Cl. virorum qui de velocitate soni nihil immutari pror-
sus , sive secundus sit , sive adversus ventus
, asserue-
runt. Gassendus enim-, et Mersennus id sibi accidisse te-
stantur; et Academici Florentini, collocatis observatori-
bus inter se duo milliaria distantibus , dum ventus spi-
raret
, asserunt tormenti bellici , quod medio illo inter-
vallo situm erat
,
fragorem pervenisse eodem tempore ad
utrosque, etsi aliis faveret ventus
, aliis esset contrarius.
At cum experimenta recte instituta aliis item recte
institutis nequeant esse contraria , videndum quaenam
praevaleant.x Florentini , quorum tentamen prae caeteris
in medium afferri solet , unica nocte
.,
et in distantia pau-
corum milliarium experimentum instituerunt. Derhamus
triennio experimenta iteravit, et in 13 milliarium distan-
tia; haec autem distantia in experimentis Derhami eadem
erat semper, a sua scilicet Ecclesia ad arcem; in ten-
tamine Florentinorum tormentum constitutum est medio
inter duos observatores intervallo; quod intervallum utrin-
que aequale asseritur , sed fortasse non ita esse potuit.
Ergo apparet quam recte Derhami observationes antepo-
nantur observationibus Florentinorum , atque eodem iure
Gassendi et Mersenni . Reipsa etsi experimento quodam
Musschembroekius quoque deprehendere sibixvisus fuerit
aequalem velocitatem tam secundo quam adverso spirante
vento, tamen Derbamo assentitur , et Florentinis , quo-
rum sagacitatem saepe alibi commendat, minime in hoc
adstipulatur.
Obiter hic notamus quod iuxta auctores ferme omnes
etiam intensitatem soni auget ventus secundus
,
et minuit
.272
1
P
TE
8
ta
11
11
adversus. Hoc , ajunt , experientia vulgari notum est : si
quidem campanae sonus , aut tormenti explosi fragor multo
melius auditur si conspiret in eam partem ventus quan
si contrarius sit ; et saepe ad aliquam distantiam auditar
ope venti secundi, ad quam , cum ventus est adversas ,
minime audiri potest : auget ergo ventus soni intensita
tem. Ratio quoque idipsum suadet : nam vencus secundus
undas sonoras transfert ; ergo ad aurem appellent undae
sonorae , quae a centro minus remotae erunt , adeoque
intensiorem deyehunt sonum .

128. Ad soni velocitatem determinandam multa in
stituta sunt experimenta , quae tamen non satis conve
niunt : experimenta instituta ab Academicis Parisiensibus
anno 1738 praebuerunt soni velocitatem , seu spatium minuto
secundo a sono percursum = 172 , 56 hexap. = 336 , 32
metr. Apud Madras in India orientali D. Goldingham ex
perimentis per annum integrum multoties repetitis ( Annal.
de Plays . et de Chim . tom. 23. pag. 12 ) exploravit soni ve
locitatem : prodiit mediocris velocitas 1134 , 33 ped.
Britan . = 345 , 74 metr. Varias hujusmodi mensuras vi
dere est in tabella , quam protulere DD Moll , Van-Beek
etc. ( Bibliotheque universelle tom. 30) : qui Auctores opus
definiendae velocitatis soni susceperunt anno 1823 , perfe
ceruntque in Hollandia , assumpto ad observationes eo spa
lio , quod Zevenboompies et Koolijesberg interjacet. Ten
tamiva sumpta die 28 Junii praebuerunt soni velocitatem
339 , 34 metr.
Hujus diversitatis plures esse possunt rationes : ac 19. In
strumenti aut attentionis exquisitae ad instrumentum deſe
ctus ; cum enim flamma attendi debeat simulque penduli
oscillatio , jam facile est ut vibratio aliqua initio non nu
meretur. 2. Spatium exiguum ab aliquibus assumptum ;
minimus enim error facilius est contemaibilis , si ingens
intermediet spatium. 3.° Venti qui aut retardant , aut ac
celerant souum . llaec variationis causa attenuari potest , ac
PL
M
ti
.
0
272 '
adversus. Hoc , aiunt , experientia vulgari notum est: si-
quidem campanae sonus ,
aut tormenti explosi fragor multe
melius - auditur si couspiret in eam partem ventus quam
'si comrarius sit :
et saepe ad aliquam distantiam auditur
Ope venti secundi, ad quam, cum ventus est adversus,
minime audiri potest: auget ergo ventus soni intensita-
tem. Batio quoque idipsum suadet: nam ventus secundas
undas sonoras transfert; ergo ad aurem appellent undae
sonorae , quae a centro minus remotae erunt, adeoque
intensiorem devehunt sonum.
128. Ad soni velocitatem determinandam multa in-
stituta sunt experimenta , quae tamen non 'satis conve-
niunt : experimenta ⋅ instituta ab Academicis Parisiensibus
anno 1738 praebuerunt soni velocitatem
, seu spatium minuto
secundo a sono percursum ∙−−∶ 172 , 56 hexap. −−∶ 336, 32
metr. Apud Madras in India Orientali D. Goldingham ex-
perimentis per annum integrum multoties repetitis (Annal.
de Phys. et de Chim. tom. 23. pag. 12) exploravit soni ve-
locitatem :prodiit mediocris velocitas :: 1134
, 33 ped.
Britan. −−∙− 345 , 74 metr. Varias hujusmodi mensuras vi-
dere est* in tabella , quam protulere DD Moll
, Van-Beelt
etc. ( Bibliotheque unive'rselle tom. 30) :qui Auctores opus
definiendae velocitatis soni susceperunt anno 1823
, perfe-
ceruntque in Hollandia , assumpto ad observationes eo spa-
tio , quod Zevenboompics et Kooltjesberg interiacet. Ten-
tamiua sumpta die 28 Junii praebuerunt soni velocitatem
−∸−⇁∙ 339 , 34 metr.
Huius diversitatis plures esse possunt rationes: ac ↿∘∙ In-
strumenti aut attentionis exquisitae ad instrumentum defe-
ctus; cum enim flamma attendi debeat simulque penduli
oscillatio , iam facile est ut vibratio aliqua initio non nu-
meretur. 2.*' Spatium exiguum ab aliquibus assumptum;
minimus enim error facilius est contemnibilis , si ingens
intermediet spatium. 39 Venti qui aut retardant ,
aut ac-
celerent sonum. llaec variationis causa attenuari potest
, ac
J
.
maälzz—äwæ-EL'T-aa
&amp;.273
ferme destrui , si collocatis tormentis bellicis in utraque
extremitate A et B illius distantiae , quam debet sonus
percurrere , tormenta ipsa eodem temporis momento ex
plodantur ; tunc enim si determinetur velocitas , qua per
venit sonus ex A in B , itemque velocitas qua pervenit ex
B in A , harum velocitatum semisumma erit velocitas illa ,
qua propagaretur sonus in aere tranquillo. 4.º Animadvertit
Musschembroekius quod cum sonus non in instanti audia
tur , sed initio minus , subinde organum aliquanto vehe
mentius percellat, hinc quidam ad initium , alii ad progres
sum sensationem soni potuerunt animadvertere , atque inde
inter se discrepare. 5.9 Varia atmosphaerae temperies.
Determinatio velocitatis , qua sonus propagatur , utilis
est nautis ut agnoscant quantum littus , aut alia navis
distet ; militibus ut quantam oppugnata urbs distet ; geo
graphis item ut quantum inter duo loca , praecipue cum
intervallum hexapeda metiri non licet , intersit . Etenim nu
merando minuta secunda ab erumpente flamma ad usque
audiendum sonum tormenti bellici , distantia loci colligi
potest . Quod si ea sit locorum constitutio ut flamma videri
non possit,o res ita supplenda est , ut cum ad aurem per
venit souüs , exploso statim alio tormento bellico , alter hic
sonus ad primum observatorem perveniat : si hic nume
ravit minuta secunda ab eo puncto , quo explosit suum
tormentum usque ad punctum quo audivit sonum alterius
tormenti , et haec minuta bifariam dividantur , habebitur
tempus propagationis soni inter duo illa loca : ita etiam nu
bis distantiam aliqui metiri docent , numerando scilicet mi
nuta secunda , quae inter fulgur emicans et auditionem to
nitrus intersunt .

129.# Nonnulla subjicimus ex theoria fluidorum
( 106. 107 ) ad soni propagationem applicata ; ita tamen ,
ut ad aeris gravitatem minime attendamus , libratu mque
aerem spectemus tanquam elasticum fluidum eadem ubique
pollens et densitate p' , et pressione a' , et temperie n.
273
ferme destrui
.
si collocatis tormentis bellicis in utraque
extremitate A et B illius distantiae . quam debet sonus
percurrere ,
tormenta ipsa eodem temporis momento explodantur; tunc enim si determinetur velocitas , qua pervenit sonus ex A in B , itemque velocitas qua pervenit;:
B in A , harum velocitatum semisumma erit velocitas illa,
qua propagaretur sonus in aere tranquillo. 49 Animadvertit
Musschembroeltius quod cum sonus non in instanti audia-
tur, sed initio minus , subinde organum aliquanto vebe-
mentius percellat, hinc quidam ad initium , alii ad progres-
sum sensationem soni potuerunt animadvertere , atque inde
inter se discrepare. 5.o Varia atmosphaerae temperies.
Determinatio velocitatis
, qua sonus propagatur
, utilis
est nautis ut agnoscant quantum littus ', aut alia navis
distet: militibus ut quantum oppugnata urbs distet; geographis item ut quantum inter duo loca , praecipue cum
intervallum bexapeda metiri non licet ,
intersit. Etenim nu-
merando minuta-secunda ab erumpente flamma ad usque
audiendum sonum tormenti bellici
, distantia loci colligi
potest . Quod si ea sit locorum constitutio ut flamma videri
non possit A, res ita supplenda est
,
ut cum ad aurem per-
venit somä , exploso statim alio tormento bellico
, alter bie
sonus ad primum observatorem perveniat : si bic numeravit minuta secunda ab eo puncto
, quo explosiot suum
tormentum usque ad punctum quo audivit sonum alterius
tormenti , et haec minuta bifariam dividantur , habebitur
tempus prcpagationis soni inter duo illa loca :ita etiam nu-
bis distantiam aliqui metiri docent , numerando-scilicet minuta secunda , quae inter fulgur emicans et auditionem to-
nitrus intersunt.
1294 Nonnulla snbiicimus ex theoria fluidorum
(106 . 107) ad soni propagationem applicata ; ita tamen ,
ut ad aeris gravitatem minime attendamus
,
libratumque
aerem spectemus tanquam elasticum fluidum eadem ubique
pollens et densitate ≀⊥⋅ ,
et pressione a: ,-et tmperie n.
..—274
10 Fac ut concutiantur librati aeris particulae
comprehensae sphaerico spatiolo habente radianı = (y , et
centrum in coordinatarum origine 0 ; talem vero patiantur
in densitate variationem , et velocitatem recipiant juxta re
spondentes radios vectores a , ut utraque exsistat admodum
exigua , et altera queat repraesentari per f ( ) , altera per
f ( Q) , evanescentibus fg , f quoad a = o et « &gt; « ,: sit r
distantia puncti ( x , y , z) ab 0 , ut obtineant
i
x2 + y2 + z = p2 xdx + ydy + zdz = rdr ,
Propagato motu per reliquum fluidum ; quoniam v' , v &quot; , 20 &quot;
sunt constanter parvissimae , et ad fluidi gravitatem minime
attendimus , iccirco formulae ( 6 &quot; , 106 ) , missis terminis
exiguissimis secundi ordinis , factisque X = 0 , Y = o, Z=0,
dabunt quoad punctum ( aco y, z)
1 do dui 1 do dv &quot; 1 das
de
dv'&quot;
dt &gt;
M dx dt I dy to da
et consequenter
lo
I can do to edip dy+
dz dz
du dvi'
dy + dt dt (©) .
Jam vero
dic dir -dx do
dy
do dr
dr dx &amp; ar ፊ dydy 9
dy
dosdz dz
da dr
dr de dz 2
.
274
↿∘∙≖∎⊀ Fac' ut concutiuntur Iibrati aeris particulae
comprehensae sphaerico spatiolo habente radium −−−−≖ a, ,
'et
centrum in coordinatarum origine 0; talem vero patiantur -
iu densitate variationem , et velocitatem recipiant iuxta re-
spondentes radios vectores &amp;!
, ut utraque exsistat admodum
exigua, et altera queat repraesentari per f! (a) , altera per
f (a) , evanescentibus !; ,
f quoad ac −−∶ ∘
et a) 0:' :. sit :-
distantia puncti (æ
,y, :) ab 0, ut obtineant
x' ∙−⊢∫∙⊣−≖≏∶−−≀∙≖ , ædx-i—ydy-t-zdzzzrdr,
PrOpag'ato .motu per reliquum Huidum ; quoniam v', 11&quot;, v'&quot;
sunt constanter parvissimae ,
et ad fluidi gravitatem minime
attendimus ,
iccirco formulae (ö&quot; . 106 ), missis terminis
exiguissimis secundi ordinis
,factisque X::o, ïze, Zzo,
dabunt quoad punctum (æ, y, :)
1 da der ↿⊄∄↑≖⋅∙∙∙∙ dv&quot; 1
ftdæ— dt'pdj&quot; dt'p.
et consequenter
1 der da der − − ... −−⋍≀ )
↽−
⊬
(dxdx—i- dy d),—i- dz :.
eiu' dv&quot; dv'&quot; )
∙
—- de—k-äuy—F—ät— d: (i)-
Jam vero
(la-: (lux dr dar das dr
,
(Erit: ∙−−−−∙⊋−∣∙∙ (?;-lld? ïydj :
z; gd)»- ,
dadz—dw drdz ,275
ac proinde
du do dos
dx + dy +
dz =
dx dz 1
do Idr dr dr
• dxt
dr dy
do-dr
) = dr
v , &quot; =
insuper v ' un
v , ideoque
dv' dv &quot; dv '' d (v'da tudytou '&quot; dz) dx +''
-dyti dz =
dt dc dc dt
dfædx + ydy + zdz
d (vdr)
dc dt
traducetur igitur ( i) ad
1 do dr d ( vdr)
dt -
( i ) .
f . dr
Ponentes
dQ u'dx + u'dy + v &quot;da = dQ ,ut sint v'=
dx 10&quot; :
dQ
dy
2011 !
dQ
dz
assequimur
dQ d
To
d
Come)
vdr d (vr)
dr,
dc dr -
' de
dr dr :
dr dt
vertelur itaque ( i ) in
275
ac proinde
Heia-4- — ;d; -]-d 2; ad;:
≤↾−⋮⋅↾ ïta.-.- −∙⋅⊄∄↗∙⋅∹−∙≦− ∙⋅−⋤−↙∄⇝⇌∶−∙⋡−−↙≀≀∙ ;
'
∙−−−⋅⋮∙⋅ ∙−− £ &quot;zl. lnsuPero—rv,-v' '.—r-v,-v&quot; rc:,ideoque
dv' *d-v&quot; ...-'de: &quot;' d(v'dx -1-v&quot;dy—1-v'&quot;da)
ïdïdæ'l' dc ↙↡↗⋅⋅⊢ ⋅⊋−⋮⋅∂≖ dt ∙−−−⋅
d
(ædæ A-ydy ∙−⊢ zdz 0)

: - d(wdr) ∙↗

d: ∙
&quot;' dt
traducetur igitur (i) ad
1 du! ∙∙∙ d(vdr) .,dr ∙−−− −− dt (( ).

Ponentes
u m ∙
l
∙
'o v'dx-t-m dy—t—v ds:dQ,ut sint d
∙↗∶⋛−≣−∙⇝ :::-g. *v
⋅⋅∙−−−∶∙−↿⋚≳−∙ '
assequimur
⋅
dQ d(vr) d —&quot;'Q) d (....) — (dr ∙∙∙⋅ ⋅ dt
.
⇀
mi'-&quot;ïd&quot; d: −⇀ dt 4&quot; ∙− dr 4&quot;
vertetur itaque (t&quot;) in276
1 do Cena
( i &quot; ) .
hdr dr
Pertingente motu ad punctum (x , y , z) , crescit ibi librati
aeris densitas M , et evadit
l = h' ( 1 + $) ;
augetur aliquantulum etiam temperies n in ipso condensa
tionis actu , fitque ntv : pressio , quae ob auctam den
sitatem evaderet a' ( 1 + 8) , augescit adhuc propter incre
mentum v ; et cum v pendeat ab € , novum pressionis in
crementum pendebit rursus ab z , eritque ob incremento
rum tenuitatem ipsi &amp; ad sensum proportionale ; iccirco ,
praetermisso é , emerget pressio ex duplici capite aucta
m = (1 + 5) (1 - +-AE) w [1+ (1 + A ) £] .
Poterit ergo ( i&quot; ) sic scribi
1
ale
de de .
( 1 + A
M
17 € dr dr &gt;
seu
dt
is 13
( 1 +A)
dL ( 1 + -E)
dr dr
Hinc
Bis
( 1 - +- A ) L ( 1 + E)
dQ
dt
276
Pertingente motu ad punctum (a:, y, :)
, crescit ibi librati
aeris densitas p! ∙ et evadit
it:-&quot;a' ≺↿−⊦⋮≻⋮
augetur aliquantulum etiam temperies 1: in ipso condensa-
tionis .actu
,
(itque n—l— »: pressio , quae ob auctam den-
sitatem evaderet m' (1 ∙−⊦ e) , augescit adhuc propter incre-
mentum »; et cum 9 pendeat ab a , novum pressionis in-
crementum pendebit rursus ab a , eritque ob incremento-
rum tenuitatem ipsi a ad sensum proportionale: iccirco
,
praetermisso ?
, emerget pressio ex duplici capite aucta
a:d(1-—1—s)(1-1-Aa) −−∶ a'[1-t-(1-t-A)s] .
Poterit ergo (i&quot;) sic scribi ∙
' d(ig) a' 1 de dt −− ↿⊣⇁∆∼ ∙−− — pii '
↿⊣−∙∊ dr dr
'
seu dc.-112) . :
p.
liinc
' d
⋮⋝−∽ ≺↿⊣⇁⋀≻↧∙≺↿⊣−⊽∊≻↽−∙−−−∙− 3- - p. dt277
est autem ( 27.29º. )
?
L ( 1 + E) = E
+ - +
Propterea , facto ( 1 + A ) = C ,
A dQ
&quot; .
ca do
Ad haec :
dv ' dy&quot;
dx
dur
dz
d’Q
dx² +
d’Q
dy ?
+
d2Q
dz² ;
dy
formula igitur ( 619. 107) , substituto p. ( 1 + €) loco fe , mis
sis terminis exiguissimis secundi ordinis, atque attenta ( i'''),'''
praebebit
d2Q daQ
dea = ca
e
d Q
dy ?
det
d2Q
da ?
( it ) ;
\ dx²
et quoniam
dQ dQdr dQ y
dr dx
dQ dQ x
dQz
dr
of
dQ_dQ
dxi
dz dr dy dr
unde
d’Q
dx²
daQ xa
dra 2
dQy? +z2 d2Q
dr p3 dy?
d’Qys , dQ x2 + z3
dr ra dr p3
d'Q
d22
d2Q 22
dr.2 p2
dQ x2 +y2
dr 产 产
277
est autem (27 .290.)
e* 53 si
1 :−∙− ∙−−− Ou: ∙∙ ↥⋅≺−⊢∊≻ s ⇄⊣⋅∙∃ 4(.,, :
Proptereü , factO :
(1 :A) −∙− c,,
1 dQ
ca dt
Ad haec :
d'v' dv&quot; ⊣⇀ ↙∣⊛∣∦ ↙≀≏⊄⊋ sz dïQ .
da: d] dz −⇀⋅ dx: d),: d:&quot; a
formula igitur (ö&quot; . 107) , substituto p: (ii-145) loco p. , mis-
sis terminis exiguissimis secundi ordinis, atque attenta (zw'),
' ∙ ?' praebebit
sz sz daQ sz .&quot;
.
(.i—t;. ⇀−− c&quot; da,-3 ∙−⊦ ∠∄∫≖ .* dzg) (: la.
et quoniam ⋅
'
∙ '
dQ dQ dr-—dQæ dQ—JQJ, iq—æi
dæ' drdæ dr-r'äy dr r'dz—drr'
unde ( ⋅ ⇁ ⋅ ,
dj—æ i,*deail'zz dag—dïQlyiA-iQxa—an dat.:—dr: rr: dr &quot;3 ,dyl d'.) rg ∙ &amp; r3
dQdeina *igæ'ä-J' .
d:&quot; dr: ra dr 'a &quot;.278
ideo traducetur ( i&quot; ) ad
d’Q
dia coloro
d-Q
( dra
2 dQ
r Thedrbest seu da (rQ )
dla ca d ( )
dra
Ex (i) habemus ( 120. 6º. )
Q = -- [80+ c ) + F(r — ct)] ;
et consequenter
dQ 1
dr [ f'ir tt) + F' ( r - ct )]
) — ]
( i&quot; )
Ar + c ) + F (r - ce}]
- [f(r + c )—– F&quot;( – ce)].
1 dQ
c2 dt
Ad f et F determinandas , sume t=0 ; habebis
f(a ) f (a ) : . E =
proinde
a&gt; f( x ) = af ( a ) + aF'( a ) f « ) - F( a ) ,
- caf( a ) = f ( ) – F' ( « ) .
Pone
fa) +F(a) = w , fra ) — F( X) = w ;
erunt
.
278
ideo traducetur (i&quot; ) ad
432—
. «PQ-,. 2
sit'—c &amp;? 747↲≺≀≻∙⊷≖∂↿≺↾≬⋗−≖∙↲≖≺↗≺≀≻ de'—' dn
Ex (.&quot;) habemus 120. 60.)
- 1
Q ∙−−− ;- [f(r.:i- ct) −⊦ F(r— et)] :
et consequenter−∙∙ :? ∙−−⋮∙ [f'(r-l-ct)-]-—F'(r-—ct)]

—--—1r;-[f(r-]-ct)-]—F(Qr—ct)] ,
1 dQ— ↿
s —
⊑
ca d: ;S.-[f(r-t-cn— F'(r— cs )].
Ad f
et F determinandus, sume :::-o; habebis
w:f(a)
, s:f,(a):
proinde ⋅
æf(a):af(a)—1—al-&quot;(a) -—f( cc)—P(a),
—eaf,(a)——:f(a)—F'(a).
Pone
fe) -t-F(a) :::.)
,f(a)- F(ac) ∶−∙−∾⋅
erunt
0&quot;)279
d @ = f( ) + F&quot;(x)= f(a) —F(«) da = f( x )da ;
dw ' = [f ( ) — F ' ( ) ] da = -ca f ( ) da ;
unde
a fixdx , w ==
cfafica) da :
hae suppeditant
f(Q )
w -two
2
1
2 frazda - of facada,
F(x)= afscada + ; fafceda;
ideoque ( iº )
f(x)= ff( )fat a pascafica),
Standa+ af )+ caf,ca).
F ( a )
2
20# Secunda membra (2011) evanescunt quoad a &gt;
Az ; ut igitur functiones
flrtct) , f'(x + ct) , Fr — ct ) , F &quot; (r — ct)
sint aliquae , non debet r ct esse &gt; &amp; : atqui in ordi .
ne ad fluidi particulas ultra Qi , cum e sit quantitas posi
tiva, est semper s + ct &gt; As ; ad has ergo particulas quod
attinet, erunt constanter
279
d(a-i)— af(a)—t-aF'(ac) —f(a) --F(a) dae— f(a)da;
« - æ
dar.-:. [f(az) —- F' (et)] da :: — cat &amp; (et) da;
uude '
∙∾−−−∶∶∝ «a)daz , Q':-irc af,(a)daz:
bae— suppeditant
aH—a' 1 - 1
f(a)-— 2 ∙−− ⋣∙ ⊄∫⇟↸∝⋟↙≀∝−−−⋮−−∘∫∝ f,(a)da,
c.)—of 1 1
Hall- 2
&quot;*.2 «li(alda—r—ïcfafdaW-ï &amp;
ideoque (i'&quot;)
1 1 1
f(a): -2- f(a') fat—1- ä-a ((a)—ïm f,(a) ,
↿ 1 1
F'(a) ∶−∙−−∙ ∙⋮⋅≳−∫∫≼∝⋝↙≀∘⊢⊢ -2-af(a)-1- -2—- caf,(a).
2011 Secunda membra (im) evanescunt quoad ac)
«,.; ut igitur functiones ⇀
f(r-t-ct) , f(f-Jf- ct)
, F(r -
ct )
,F'(r
-
et)
sint aliquae , non debet r

: b et esse )a, :atqui in ordine ad fluidi particulas ultra et, , cum t sit quantitas posi- tiva, est semper :- −⊢ ct a, ; ad has ergo particulas quod attinet, erunt constanter280 fir + ct ) = 0 , f ( r + c ) = 0 ; et consequenter -F(r —c)F( r -ce ) , 6 = 1.- F &quot; (r — ce) (**** ) . 30 Aereae particulae respondentes radio vectorir non incipiunt moveri nisi quum tempus sic increvit , ut habeatur rct = ly , seu r = ctt cy : inferimus sonum propagatum iri uniformiter velocitate V ( 11 + A ) Quod spectat ad numerum A, habemus (87. 70. ) a = im [1 + a (n + v)] = im '(1 + E)[ 1-+ an + ») ] , itemque ( 10.) 5 '[1+ (1 + A )ɛ] =; if' ( 1+ an) [1 + ( 1 + A ) ]: hinc i '(1 + E)[ 1-+-ant-v) ] = iu'1 + an ) [1+ ( 1 + A )ɛ ]; ex qua eruitur av A av( 17) El 1 + an ) $ ( 1 + an ) Ponamus vase aliquo accurate obserato aerem conti neri ejusdem densitatis pé ac temperiei n cum aere exter• no; sitque h altitudo barometrica utrique communis : con . ⋀≀∙⊣∙∙∘⊔≔≖∘∙⊓≀⋅⇀⊢∝⋟∶∘⋮ et consequenter 1 1 ⇀ ↿ ∙ −⋅−−−− —F'(r-ct)-—;F( r—ct). : ⋅−−−− -—F'(r—ct)(t&quot;&quot;). r r cr 3&quot;:- Aereae particulae respondentes radio vectori r non 1nc1piunt moveri nisi quum tempus sic increvit, ut babeatur r— ct:ac, , seu ::- ct ∙⊦∙ at, :inferimus sonum prcpagatum iri uniformiter velocitate 'c: Vä- (1—1-A) (i&quot;) - Quod spectat ad numerum A, habemns (87. 70.) ∙ saiw-1-a(n-1-v)]:zp'u-u-e)[1-.-a(nM)]. itemque (10.) 6 −∙−−−⊤ w'[1—t-(1 a—A)e] :; ip'U—t— an)[1 ∙−⊦⋅ (1 ∙⊢ A)e]: bino - ⋅ ↴ 's ip'(1-1-s)[1-1-a(n-1-v)] ∶−− ≀⋅⊬⋅≺↿−⊢⊄⋯≻⊏↿⊣−≺↿−⊢∆≻∊⊐⋮ ex qua eruitur ↼ . cru-H:) av ∙−−− −∙∙ e(1-1-an) s(1-1-an) Ponamus vase aliquo accurate obserat'o aerem conti-- neri eiusdem densitatis pf ac temperiei iz cum aere exter- no; sitque !: altitudo barometrica utrique communis: con-281 1 11 cipiatur extrahi e vase aliquantulum inclusi aeris, vel qui erat inclusus aliquantulo magis comprimi , et denotet d'1 Fé) densitatem , h' altitudinem barometricam, postquam aer in tra vas ad pristinam redierit temperiem n. Tum constituta parumper communicatione cum externo aere, donec nimirum redigaturad h, mutationem quandam suscipiet lam p' ( 13) quam n; et illa quidem transformabitur in u'1 *8' ) (18&quot; ), haec autem in ny. Sed cum v' brevi evanescat, et so la n supersit quin variet MIFÉ' ) (1 # &quot; ) , mutabitur iterum h et evadet h &quot; . Alteram instituendi experimenti rationem sequuti sunt Desormes et Clement , alteram Gay - Lussac et Welter: inspiciatur sequens tabella . torie pun Desormes et Clement. n = 12 , 5 , heo” , 7665 , h - hs o ” , 01381 , 11 h - h&quot; = 0 , 003611 ; 2 &quot; hi sese restituit ad h intra tempus &lt; &lt; 5 Gay - Lussac et Welter. n = 13° , h = omom,, 757 757 ,, hh -- hh : = 0 &quot; , 0163644 , h &quot; - h = 0 , 0044409 ; q &quot; h sese restituit ad h intra tempus 6 Iam vero, depolante D densitatem hydrargyri , sunt conti erter Dgh = ip (176) (1 + an ) , 000 19 villi] 1 !' torir L, 111 )num conti- erit?' con- 281 cipiatur extrabi ei vase aliquantulum inclusi aeris,'*vel qui eratinclusus aliquantulo magis comprimi, et denotet p.'(1::1:s') densitatem, h' altitudinem barometricam, postquam aer in- tra vas ad pristinam redierit temperiem 11. Tum constituta parumper communicatione cum externo aere, donec nimirum h' redigatur ad h, mutationem quamdam suscipiet tam (if( quam ∎∶∙∶∔⋅∶∊⋅⋟ .: et illa quidem transformabitur 111 p.'(1.-.;:s')(1:£ e&quot;), haec autem in :::». Sed cum v' brevi evanescat, et so- la n supersit quin variat p.'(1:1: a' ) (1 :1: e&quot; ) , mutabitur iterum I: et evadet h&quot;. Alteram instituendi experimenti rationem sequuti sunt Desormes et Clement , alteram Gay- Lussac et Welter: inspiciatur sequens tabella. Desormes et Clement. 11:12&quot;, 5 , h:o&quot;,i7665 , 11 -h': 0&quot;, 01381 , h — h&quot;:o&quot;, 003611 ; h' sese restituit ad h intra tempus ≺∙−≣− ∙ Gay - Lussac et Welter. ' n:130, h:o'&quot; , 757 ,h'-— h:d&quot;,0163644 , h&quot; — h:o'&quot;, 0044409 ; 1&quot; h' sese restituit ad h intra tempus (—6—- . hm vero, denotante D densitatem bydrargyri , sunt Dgh':i;t'(1q:e')(1—t-an), ∎⊨∎ 'i i ! 19282 Dgh = id'l 176) ( 1 #t&quot; ) ( 1 + a nv( ) ) , $ Dgh &quot; = id'l 176 ) ( 1 + &quot;) ( 1 + an ) : hinc h &quot; = 1 &amp; € &quot; , h h &quot; 1+ anty') 1+ an = 1 + R 1 + an h &quot; 'h αν&quot;' h hh&quot; h&quot;

1 tan
ideoque }
ań
= ( 1 - an )
h hh&quot;
h &quot; 7 &quot; -h*
Substitatis valoribus ex Gay - Lussac et Welter ,
αν
€&quot; ( 1 + an)
=0, 3785020934 :
1
R
et quoniam iste numerus neque ex temperie neque ex pres
sione pendere videtur, iccirco poterit generatim assumi 1
A= 0, 3785020934 ;
sicque soni velocitas prodibit expressa per ( 94. 1 ° )
V
1 , 3785020934
to
fe
-V 1,3785020934i(1+ an) = 1009 , 614V1+ an (i&quot; ).
282
1131. −−−−⋅⋅⋅⊬∣≺↿∓⋮⋮≻ ≺↿ :::&quot; ) ( ↿
....
(a:-:») )
.
Dgh&quot;:ip.'(1q:s-' ) (1:t:€&quot;) (1 qum):
tibine
−≸⋮−⋤⋅−⋅
≕↿
∙∙⋅⊧∙≘⋅⋅ ,
∣∣⋮∙⋅ −−⋅↿−⊦⋅↿∘∙≦↾∙≔⋮∙⋓⇗≱ −−⋅↿∙−⋅⊦−∙↿−−∙⋮⋮≔−∙
zh :&quot; ∙∙∙ l:&quot; --l:' ,.4, .av' −∣∎ −∦∣⇂∙∙ ;
↙
h' 1 −⊢⋅⋯∎ &amp;
ideoque
av' h' b—h&quot;
e&quot;(1-t-an)— h&quot; h&quot;—-h'
⋅
Substitutis valoribus ex Gay-Lussac et Welter
,
av'
et quoniam iste. numerus neque ex temperieneque ex pres-
sione pendere videtur, iccirco poterit generatim assumi
sicque soni velocitas prodibit expressa per (94. 10)
I
c ∸−−−⇀ ↿∙ 3785020934 1;—
wjt—lV1, 3785020934 ⋅⋅≺↿∙⊢ an) ∙−−∶ 1009,- 614 ∣∕↿⊣⇀∘≀∎ tc&quot;)- H283

Si attendenda est quoque bygrometrica aeris constitutio, de
notante 6, pressionem libratam ab aqueo vapore , pro ui'
substituendum erit ( 96. 4º. )
1
seu
i( 1+ an)
exsistet nempe
V
11
w' il 1 to an )
1 , 3785020934 3
--8
W1
009 , 614
V
8 ã' (1+ an )
80-30 ,
(i &quot; ) .
In soni velocitatem diligentissime inquisiverunt an
no 1822 DD. Arago, Prony , Mathieu , Bouvard, Humboldt
et Gay - Lussac: distantia, ad quam observationes de cor
ruscatione flammae et fragore instituebantur in explosionibus
Lormenti bellici, ea fuit quae Monthlery et Villejuif inter
jacet ; velocitas inde deducta, seu spatium iolra 1&quot; a so
no percursum, 89 Erat autemn =15°, 9; unde
Vitan = 1 , 029 : dabit igitur formula ( it )
340metr.
103gped
.
893 metr .
337 , 432 . &gt;
Hygrometricam quoque aeris constitutionem notarunt
Auctores Cl . Sub mediocri videlicet altitudine barometrica
metr .
0 76 index hygrometri, quod vocant a capello, o
slendebat grad . 72 : in hac vero hygrometrica aeris consti
lutione, et sub temperie 15° , 9 ,pressioni , respóndet ba
metr.
rometrica aliiludo 0 00679; hinc
283
Si attendenda est quoque bygrometrica aeris constitutio, de-
notante u', pressionem libratam ab aqueo vapore , pro pf
substituendum erit (96. 40.)
exsistet nempe
' ∙
1
T
∘⋅−−− ∣∕ 1, 3785020934 &quot;'
'(
a, ↼⋅⊢ s
'&quot;&quot;- ..
8
1009 614⇂∕ afuit—13:111) (i&quot;)-
In soni velocitatem diligentissime inquisiverunt au-
no 1822 00. Arago, Prony, Mathieu, Bouvard, Humboldt
et Gay-Lussac: distantia, ad quam observationes de cor-
ruscatione dammae et fragore instituebantur-in explosionibus
tormenti bellici, ea fuit quae Montblery et Villejuif inter-
iacet :
velocitas iude deducta, seu spatium intra 1&quot; a so-
no percursum, :340'm&quot; ,89 Erat autemn:150,9; unde
l/1-t-an :1, 029: dabit igitur formula (ix)
0:1038ped' ,
893 :..- 337'&quot;&quot;' ,432.
Hygrometricam quoque aeris constitutionem natarunt
Auctores Cl. Sub mediocri videlicet altitudine barometrica
Gum. , 76 index bygrometri, quod vocant :: capella, o-
stendebat grad. 72:' m hac vero hygrometrica aeris consti-
tutione, et sub temperie 150, 9 ,pressioni a', respöndet ba-
rometrica altitudo Omm ,00679; binc284
v
80
8w' 30,
=1,002 ;
et consequenter ex (3 &quot; ) eruetur
1040ped ., 97 = 338metr .
11 .
Consensus itaque experientiam inter et expositam theo .
riam tantus invenitur , ut major profecto desiderari non
debeat in praesenti argumento : difficile admodum est in id
genus observationibus ventorum vim prorsus eludere, alias
que causas declinare quae huic consensui multipliciter no
cere possunt : mirum deinde quantum ardua res sit va
lorem A experimentis accurate determinare.

4. °* Evanescunt secunda membra (iº !! ) etiam quoad
a = o : in distantia igitur r evanescent &amp; , v statim atque,
labente tempore , eo devenitur ut sit rect = o . Quia er
go in distantia illa incipiunt , v esse aliquae quum
rct = lg, sequitur motum in distantia illa minime du
raturum ultra tempus Eaedem itaque &amp; , v evanescent
in distantia r- , statim atque incipiunt esse aliquae in di
stantia r : propterea non cientur una nisi particulae con
stituentes stratum crassiliei

5.° Velocitas v duabus ( 2.º į&quot; ) constat partibus , quarum altera sequitur rationem reciprocam distan
tiae a centro unde promanat sonus , altera rationem reciprocam duplicatam ejusdem distantiae: functiones praeterea F, Fmanent constanter parvolae. Quia igitur im
pulsio in datum obicem facta pendet a velocitate v , patet , quo longius propagatur sonus , eo magis ipsum debilitatum audiri. Quum sonus ad modicam pervenerit distantiam, licebit secundam illam partem negligere; eritque 

Inferimus illud: si impulsio in obicem facta quadrato ve.
locitatis v sumitur proportioualis , rationem duplicatam di
stantiarum sequetur soni debilitatio ( 125 ) .


6.°* Fac ut librati aeris particulae concutiantur
una circum plura puncta O , 0 &quot; , ... ; quorum distan
tiae ab ( x , y, z ) exhibeantur per r' , o&quot; .... ; ipsis.
que O' , 0 ' , ... , tanquam originibu's respondeant sua
axium systemata parallela systemati habeati originem O.
Quoniam novae coordinatae s ', x &quot;, ...5,0 &quot; , ... é , z &quot; ..
constantibus quantitatibus differunt ab x, y, z ; ideo
dr ' dr dr &quot; ar 
dx doc ' F ' da d.x &quot;
!
y' g &quot;
dr'
dy
&gt;
dy '
p&quot;
dr
dy &quot;
dr
dz&quot;
dr dr
dy
dr'i
dz
2
dz dz'
el consequenter
dQ _dQ dr dQdr&quot;
tar
dx
+ .. dx dr' dx
dQ x'
dQ y
dr +
dQxt&quot;
dr'' gli t....''
dQ_ dQ y
+
dy dr p ' dr to.
dQ
dz
dQ á
dr '
+
dQ
di &quot;
.
. ilemque
to
d²Q x 2 dQ 7/ 2+22
dx² dr'a g'a + +
285
1 '
r
inferimus illud : si impulsio in obicem facta quadrato ve-
locitatis v sumitur proportionalis, ratiunem duplicatam di-
stantiarum sequetur soni debilitatio (125).
691» Fac ut librati aeris particulae concutientur
una circum 'plura puncta O', 0&quot;, ... ; quorum distan-
tiae .ab (æ, y, :) exbibeantur per r'
, r&quot; . ...; ipsis-
que O'
, O&quot;, ,
tanquam originibus respondeant sua
axium systemata parallela systemati habenti originem 0.
Quoniam novae coordinatae se', a:&quot;,
...y' ,y&quot;, ... z', :&quot;
..
constantibus quantitatibus differunt ab a:, y, :;
ideo
dr' dr' æ' dr&quot; ∙∙∙ dr&quot; ∙∙∙⋅ æ&quot; ⇀
dx daf—r dx' dr&quot; ≀⋅∎⋅↬⋅⋅⋅
et consequenter
dQ 'der- −⊦↙≀≺≀∂∙↾∙∙⋅ du:- dr'dæ dr&quot; da:
dQ æ' dQ æ&quot; &quot;
dQ ∙−−↙≀≺⊇∙⊺ d.QJ
&quot;?;/7 21.-717 −⊢∙∙ 4&quot;?!— −−⊣∎∎∙∙
': &quot;dy— dr'r'
.dQ —dQ f:: dQ ∙⋮↾∙⋅ ∙−⊦ '
dz —dr' l&quot;-1 dr&quot; r&quot; ⋅ ..
itemque
'PQ −− ↨≖≬x&quot; dQ ∟∣≖⊣−≖∙∙∣∷ .
⋅
'
dæ' dr'3 :&quot;2 −⊦⋅−−(Ti—' −−⋅∣⋅∙−⋅↾⊰ ..,-286
daQ x2 dQ &quot; 272&quot;
+
dr''2 p &quot; 2 dri p/13 + ... ,''
da d’Qy'a
dya drar'a tari
dQ x2+22 +
p3
d'Q.7 &quot;?, dQ x&quot; : + z'2 + ti. dr&quot; ' a p &quot; 2
lo:
dri
d2Q
ddza
daQ z'2
dr2 p'2
dQ x's + y'2 dQ 242
dr' 3
+
dril2 pll2 +
dQ x2+ y'a
ti ..
Adhibitis substitutionibus in ( i ' ', 1.0 ) ,
d'Q
de2
( d - Q
= c2 Adr'a +
2 dQ d2Q 2 dQ
z dr + dra +pdr&quot; + ... )

;

ex cujus forma intelligimus fore
Q = [filr'tou + F (r — ct)]+
[far&quot; +41++ F.(r&quot; ct) ] + . ( * &quot; ).

Nunc facile stabilitur illud : in hypothesi plurium concus
sionum simultanearum , ubi eae ad punctum ( x , y , z )
eodem temporis momento una perlingant , numerus e ni
hil erit aliud nisi summa consimilium numerorum re
spondentium iisdem concussionibus seorsum spectatis ; si
quidem ( 1.0 ) .
286
(PQ ∙⋅⇂⋅∥∙ ↿ dQ
dr&quot;: ∙↗≀∦≖∙⊦≖∥≖⊹ r'&quot; ∣
dr&quot; r&quot;3 ⋅⋅
'
'
- «PQ— d'Q 7&quot; ∙⊦↙∄≺≀∙−−−−∙−−−−−⊦⋅ æ'2-l-z'2
d]:—
d'Q )&quot; dQ æ'ä-l—z&quot;; .
dr&quot; ≀⋅∥⋮⊹↲≀⋅∙∣∣ r' '34- ⊣− ⋅⋅. '
d-Q Adeo ∷⋅≖ ∣dQ ⊴↾∶∣≖−⊦∜∣⋅ æno z.&quot;:
dza 'di'/3 r'&quot;
'
dr' r'3 dr&quot;3 r&quot;'
dQ ∙⋅≖∥≖⊣−∜∥≕ ↿
dr&quot; rl'3. 'l .
.. ∙ ∙
Adbibitis substitutionibus in (i&quot;. 1.&quot;).
duo (PQ 2 dQ 2 dQ ∙
∙−−− ∘≺↙↙↾∣≏−⊣−≀⋅∣ ∡≔∣∙⊦≤∶−−⊽− ⋖⋮≀≕−⊽∣−⊋−↾⊽⊣−⋯≻
ex cuius forma intelligimus fore
Q ∶−∎⋅
⋅↗⋮⊤⋅∐⋩≖≺⋅∦⊣⊸⊩⊢−∶⋮∙− ∇≖≪↗⋅⊣⊸≀⊢⊢ F,(r'—-ct)]-l—
F.(r⋅⋅⋯ ]-l—-
Nunc facile stabilitur illud :, in hypothesi plurium concus-
sionnm simultanearum , ubi eae ad punctum. (a: ,
J
,

: )

eodem temporis momento una pertingant, numerus :ni-
hil erit aliud nisi summa consimilium numerorum re-
spondentium iisdem concussionibus seorsum spectatis; si-
quidem (1.&quot;).287
DP zo al
- F&quot;
['r( + ce)-F'(x'ct)
[facr &quot;+ c8)— F'xr&quot; —cr) ] - ...
Insuper
DP
dQx' dQx&quot;
+ t . dx dr ' dr&quot; r &quot; + ...
G [r« tch+F&quot; ret) ]– 16 +6 +
F.( c )]) + ( - &quot;+e +F',(==ci)] –
[for&quot;tor)+F60—60)) + ....
vº dQ
dy
dQ r' dl go &quot;
+ + .. dr ' r ' + dr &quot; r &quot;
G - triktet)tF'(x - ce )]= i [ fim'tot +
Fa(r = -1)]) + ( -186 *408)+ F&quot;(&quot; –ce)] -
wraca&quot; terhFall -ct)]) + .
287
⋅⇌⊐ ∙−−≕↿−∙⋅ ?,?&quot;-- --',..'[f .
(r -!-c:)—F'.(r -—-c:)1—-
⋅≺∽⋅−∎↿⋅⊤∶∁↿⋮⊅⋍∣⋅∥⊹⋯∙− ↧⋅⋅∣∙≺≀∙∙⋅∙⊳∙−∙∘≀≻ ]— ....
lnsuper
&quot;- «me ⋅≄⋅ .... −−∶ .. ↼−−⋍⋜⊑⋅∙−−⋅∡⋰−∙⋅⊤−⊢⊿−≀⋅−∙⇉↗⊷ −⊦ ∙
⋅⋅
⋅
&quot;
, 1! ' '
⋅∎ ≺∎≙∶∎↾⋅⊀∎≺∣⋅⋅−∣∎⊸∘⊣−∏⋅∎ (' -—0t )]-— ∙≀−∙⋅∙−∙−∣⋅∫∎≼∣∙∎∙⊦⊸↥⊢∣−
,
' ⋅ æ' .
⋅ ↿ ⋅
⋅ ⋅
'
∙
Fl(r &quot;'&quot;'ct)
&quot;
])&quot;T'f'l'(—: [f,(r'Lï-CO—l-F', ∎⋅ ( r—ct )] ∙∙∙⋅
r
∙
∙≖−⋅≟⊑ ⊏∣≖≺↗⋅⋅−∣⊸≀≻⊣−↿⋮⋅≖≺⋅∙⋅⋅∙−−∝≻∃ )£f.-.- −⊦ ∙
⋅∙∙
. ∂≺≀∙∙−⋅∠≀≖≀∜⋅
lu.—dy dr' r' ä-l-dr&quot;— r&quot;∶∣⋅−⊦∎⋅ .. 1
(£.- ⊏⊀⋅≺≀⊤∙∙⇀⊸⋅≀⊢⊢≖⋅⋅∙⋅≺ r'-ct )1: ;; [f.(f-l-cu-l-
F.(r'— et)] ≻∙∜−⋮−⊦r ≺↿⊤↕∣∣≖≺↗⊓⊣⊸⋍⋝⊣−⇂⋅⇁⋅⋅≺∣∙⊷∙−∘≀∏ :- - ⋅
⋝∙≟≟∁∣≖≼↾∣⋅⊹≕⊢∣⋅⇁≖≺∙⋅≀∙∙⊸∁∏ ⋟∑≖∙⊤⋅⋮− −⊦∢ ∙ ∙ ∙288
dQ
dz
dQz
dr
dQz&quot;
t .
dr '' r &quot;''
( -fr.( tre)+F ;(r = cr)] - Pfalriteest
Fu
F.(x*—- )]) + ( far&quot;+c)+ F &quot;–ce)] -
pen
na[ far tcent Ffrº-cr)]) + ...;
UI
De
go
inferimus velocitatem v debitam simultaneis concussioni
bus circum 0,0 eodem temporis momento ad
punctum ( x , y , z ) una pertingentibus nihil fore aliud
nisi resultantem ex velocitatibus , quae debentur iisdem
concussionibus seorsim spectatis : atque hinc facile intel
ligimus cur, pluribus corporibus simul resonantibus , inter
oscillationes in aere excitalas non habeatur confusio ,
omnesque diversi soni inde orti ad aures distincte per
veniant. Huc spectat principium de superpositione exiguo
rum motuum.
7.04 Redeuntes ad unicam concussionem in 0 ,
ponamus aerem contineri tubo cylindrico , cujus axis ox,
motumque particularum esse ipsi OX parallelum : erunt
v&quot; = 0, v&quot; = 0; propterea formula ( i&quot; ) evadet
d2Q daQ
de unde Q = f ( x + .ct) + F ( x - ct ) ; ረder2 1
et consequenter
288
... dQ- JQ :
&quot;'l-(,Q' z'—dz −−↲≀⋅⋅ r' dr&quot; r'

ll ≼⋅≟≑⊏∣∣∙≺↗⋅⊣⇥≻⊣−≖⋅⇁⋅≖≺↗⋅⋅−⋅∘≀∏ ∙−⋅⋅⋮−⋅⋮⋅⇆⋅⋅⊔≖⋅∊⋅↾⋅⊣↽⊸≀⊢⊦ ∙
,
⋅ mo*—cn] )f— ⊣−≺⊽⊏ ↑∼≖≼↗⋅∙−⊦∘↥≻−⊦ ≖∸⋅∙≖≺↗∙∙−∘≀∏ −
⋮∙−⊦∘≀≻⊹↧⊸⇁≖≺≀∙↝−−∘≀∏⋟−⋮⊽ −↿− ∙ ∙
⋅
inferimus velocitatem v debitam simultaneis concussioni-
bus circum O'. 0&quot;
, ... eodem temporis momento ad
punctum ( x . y ,

: ) una pertingentibus nihil fore aliud nisi resultantem ex velocitatibus , quae debentur iisdem concussionibus seorsim spectatis: atque hinc facile intel- ligimus cur, pluribus corporibus simul resonantibus . inter oscillationes in aere excitatas non babeatur confusio. omnesque diversi soni inde orti ad aures distincte per- veniant. Huc spectat principium de superpositione exiguo- rum motuum. 7.&quot; Redeuntes ad unicam concussionem in 0, ponamus aerem contineri tubo cylindrico, cuius axis OX. motumque particularum esse ipsi OX parallelum: erunt 0' '::--0. 0&quot; ':o; propterea formula (i&quot;) evadet ⋅ 32? —-c £?. unde Q—−−⋅∣↗≼∶∁−∔⊸⇂⋟ -l-F(x—-ct); et consequenter289 dQ 1 dQ dx = pilatot) + F '( x - 1), E = - ca do [fotot) – F (x – ct )] . Functiones f et F absque ulla difficultate determinantur: sunt enim ( 1.9). f( x) = f(@ + F'(Q ), - cf:(Q ) = f (a) F '( ) ; ideoque f'( X) = f (Q )-cfi(Q ) 2 f(@ t-of ( ) F (a ) = 2 Ultimae ac penultimae aequationis secunda membra eva nescisnt statim ac a fil &gt;Oto : erit itaque f ( t ) = 0 quoad -aereas particulas ultra azi proinde quoad ejusmodi particulas F ' ( x-ct ) . Hinc sequitur souum adhuc ( 3. ) propagatum iri uniformiter velocitate се V 11 + 4 ). • De reflexa soni propagatione per aerem .
: 130. Cam in directa propagatione sonoras aer offen dit obicem aptum, reflectitur; hinc echo ( 115 ) progignitur; assertio sic probatur . Constat quod corpus in motu positum , si in obstacu lum incidit , quod elasticum sit , vel durum , et corpus ipsum ⋅ 289 v −∸−≖ B:] ')(æ-l-ct -l-F'(æ—ct), a:.— — ∙−∙−−−⋅⋅∶ ⋅↿ ∙

: - [f(ar-l—ct)

-
F'(x—-ct)] .
Functiones f
et P absque ulla didicu-l'tate determinantur:
sunt enim (1.&quot;). ⋅
⊞≀∝≻−−−↿≺⊄⊢⊦⋮⇁≀≺∝≻∙ ∙− cf.(a)-—:f(a1—F'(a) :
ideoque f(ao ⇌≖
aa)-zcnm) ∙ Ha): Karl-faa) ∙
Ultimae ac penultimae aequationis secunda membra eva-
nescunt statim ac « Et )a.. : erit itaque fur-H:):o
quoad aereas particulas ultra «.'
,proinde quoad eiusmodi
particulas ⋅-cs :: F' (a:—ct ).

Hinc sequitur sonum adbuc (3.&quot;) prcpagatum iri unifor-
miter velocitate
C::V-z-I—(i-I—A). ' De reflexa soni prcpagatione per aerem ∙⋅

130. Cum in directa propagatione sonoras aer oü'en-
dit obicem aptum, reflectitur: binc echo (115) progiguitur: assertio sic probatur.  

Constat quod corpus in motu positum, si in obstaculum incidit, quod elasticum sit, vel durum, et corpus ipsum impingens elasticilate gaudet, debet molus directionem mutare ac reflecti: ergo aer, elasticus cum sit, ubi in obstaculum offendit, quod vel elasticum sit, vel certe non molle, reflecti debet; undae videlicet aereae, quae ex sonoro corpore progignantur ac propagantur directe, debent obicem offendendo regredi, sonumque reflexum progignere. Exemplo circulorum in aqua ex injecto lapide excitatorum res oculis subjicitur: circuli enim isti ubi ad ripam appellunt, reflectuntur inde eo ordine, quo appulerunt . Aliter sic: ejusdem naturae est echo cum sono ipso directo; obtinet enim utrinque sonus eodem generatim tono, iisdemque affectionibus praeditus; ergo echo gigni debet eodem modo quo sonus directas: atqui hic per undas aereas successive a sonori corporis motu genitas procreatur; ergo per similes undas etc. Hinc in aperta planitie, ubi nullas est obex, sono directo minime Echo respondet. Cohaeret doctrina com Echo phoenomenis. Nam 

1° redit reflexa vox duplo temporis intervallo: ab experientia doctus sum, inquit Derhamus, Echo redire duplo intervallo, quo vox primaria ad objectum phonocanticum pertingebat; scilicet tempus requiritur ut ad obicem vox primaria deveniat, et rursum tantumdem temporis exigitur ut reflexa ab obice redeat ad loquentem. 

2º. Remissior plerumque est Echo quam vox directa audiri soleat; aliquando tamen intensius resonat Echo quam sonus directus audiatur. Ralio primi est: cum soni intensitas decrescat pro aucta distuntia a sonoro corpore, jam decrescit sonus ad obicem pergens; inde autem regrediens, et novas undas progignens, iterum decrescere
debet intensitas: ratio secundi, quia si obstaculum concavum
sit, plures colligere poterit radios phonicos , quos unitos simul in uno loco regerat. 

3º. Aliquando ( 115 ) seinel vox reflectitur, aliquando saepius: prima dicitur Echo monophona,
altera polyphona. Si enim obstaculum unicum sit , jam nonnisi
semel potest vocem remittere; contra saepius remittitur duplici ex causa. Prima est cum iu variis distantiis plura habentar
290
impingens elasticitate gaudet , debet motus directionem mu-
tare ac reflecti :ergo aer , elasticus cum sit ∙ ubi in ob-
staculum offendit
, quod vel elasticum sit, vel certe non
molle
,
reflecti debet; undae videlicet aereae, quae ex so-
noro corpore prOgignuntnr ac propagantur directe , debent
obicem oll'endendo regredi , sonumque reflexum progignere . 

3&quot;. Aliquando (1 15) semel vox refle-
ctitur,aliquando saepius: prima dicitur Echo monopbona,
altera polyphona. Si enim obstaculum unicum sit. iam nonnisi
semel potest vocem remittere; contra saepius remittitur dupli-
ci ex causa. Prima ut cum iu variis distantiis plura habentur291
obstacula: altera causa est, cum duo sunt obices e regione col
locati; vox enim ex uno reflexa in alterum incidit, atque ex
hoc iterum reflexa iucidit in primum, et ita porro. Apud
veteres memoratur Olympiae porticus ; quam eplaphonam
dicebant, quod septies eamdem vocem redderet , ut tradit
Plinius. Prope Mediolanum celebris est Echo in palatio
Simonetta, in quo ope duorum parietum parallelorum fra
gor minoris fistulae bellicae vicies, et aliquando tricies re
petitur teste Schoto. 

40. Echo saepius unam tantum syllabam, aliquando plures refert: echo monosyllaba prima, et polysyllaba altera dicitur: habentur loca, ex quibus integer versus hexameter repetitur. Ea nempe est obicis ( 115) distantia, ut sonus reflexus primarum syllabarum tunc demumad aures regrediendo perveniat quando vocis directae impressio jam desinit; ac tunc sonus primae syllabae, qui opportune regreditur jam expleto versu, poterit esse sepsibilis, itemque aliarum successive. 

5º. Echo redditur aliquando a silyis; imo etiam a campis sulco exasperatis et a planitie cespitibus ac virgultis inspersa reverberalur vox; reverberari autem a sulcis ac cespitibus animadvertit Kircherus, quia quando sulci eversi, ac virgulta praecisa fuerunt Echo nulla reddebatur: talis nempe esse potest irre
gularis partium reflectentium dispositio, ut etiamsi plures ra
dii phonici dispergantur, non pauci tamen in eumdem lo
cum collineent.

131. Reflexio soni fil ad angulos incidentiae et refle
xionis aequales : quod sic explicamus . Sit AB ( Fig. 60. ) fir
ma , planaque superficies ; KCK' recta perpendicularis su
perficiei AB ; K centrum sonorum , ex quo propagantur
sphaericae undae CDD', C'EE', etc... appellentes ad AB in C,
C'ete ... Fiet soni reflexio in C, C , ..; ethabitis C , C ... pro noris
secundariarum undarum centris , ipsae secundariae undae
remittentur cum eadem principalis undae velocitate. Pro
grediente unda principali ab CDD' usque ad BB ' , unda
manans ex C progredietur ab C usque ad Q ; repraesenta
291 obstacula: altera-causa est, cum duo sunt obices e regione col-
locati; vox enim ex uno reflexa in alterum incidit, atque ex
hoc iterum reflexa incidit in primum, et ita porro. Apud
veteres memoratur Olympiae porticus; quam eptaphonam
dicebant, quod septies eamdem vocem redderet, ut tradit
Plinius. PrOpe Mediolanum celebris est Echo in palatio
Simonetta, in quo ope duorum parietum parallelorum fra-
gor minoris fistulae bellicae vicies, et aliquando tricies re-
petitur teste Scboto. 40. Echo saepius unam tantum syl-
labam, aliquando plures refert: echo monosyllaba prima, et
polysyllaba altera dicitur: habentur loca, ex quibus integer
versus hexameter repetitur. Ea nempe est obicis (115) di-
stantia, nt sonus reflexus primarum syllabarum tunc demum
ad aures regredieodo perveniat quando vocis directae im-
pressio- iam desinit; ac tunc sonus primae syllabae, qui op-
portune regreditur iam expleto versu, poterit esse sensi-
bilis, itemque aliarum successive. 50. Echo redditur ali-
quando &amp;
silvis; iuno etiam a campis sulco exasperatis et a
planitie cespitibus ac virgultis inspersa reverberatur vox ;
reverberari autem a sulcis ac cespitibus animadvertit Kir-
cberus. quia quando sulci eversi, ac virgulta praecisa fue-
runt Echo nulla reddebatur: talis nempe esse potest irre-
gularis partium reflectendum dispositio, ut etiamsi plures ra-
dii phonici dispergentur, non pauci tamen in eumdem lo-
cum collineent.

131. Beflexio soni fit ad angulos incidentiae et refle-
xionis aequales: quod sic explicamus .Sit AB (Fig. 60.) Gr-
ma, plenaque superficies; KCK' recta perpendicularis su-
perficiei AB ; K centrum sonorum , ex quo propagantur
sphaericae undae CDD', C'EE', etc... appellentes ad AB in C,
B' etc... Fiet soni reflexio in C, C',..; et habitis C,C'... pro novis
secundariarum undarum centris, ipsae secundariae undae
remittentur cum eadem principalis undae velocitate. Pro-x grcdieute unda principali ab CDD' usque ad BB' , unda
manans ex C prOgredietur ab C usqæ ad Q; repraesenta-292
biturque hemisphaerio , cujas semidiameter CQ = D'B ' :
item progrediente unda principali ab C'EE usque ad BB” ,
unda manans ex C' progredietur ab C usque ad C &quot; : re
praesentabiturque hemisphaerio , cujus semidiameter CC&quot;
E'B' ; alque ita porro. Inferimus , si concipitur superficies
curva AQC &quot; B tangens omnia haec hemisphaeria in Q, C &quot; ....,
in ea fore puncta illa , quae a secundariis andis reflexis
attinguntur eodem instanti , quaeque tunc incipient concuti
quum principalis unda pervenerit ad BB' ; exhibebit nimi
rum AQC&quot;B superficiem undae reflexae; et quoniam productis
infra AB superficiebus BB' , Qa, C'a ' , ..., recta KA' exsistit per:
pendicularis ad primam et secundam, recta KH ad primam et
tertiam , etc ... ; ac proinde sphaerica superficies B'BA'A tan
git sphaericas superficies QaA' , C'a'H , . ; sequitur super
ficiem AQB undae reflexae fore sphaericam , ejusque cen
trum in K , et semidiametrum K'Q = KA' . Jamvero quem
admodum auris collocata v. gr. in C deprehendit sonum
directum venire juxta KC' perpendicularem undae incidenti ,
sic auris in C' deprehendet sonum reflexum venire juxta
K'C &quot; perpendicularem updae reflexae ; et cum , ob mutuum
sphaericarum superficierum AQB , C'a'H contactum , recta
K'C &quot; transeat per C' ; cumque , ob latus KC = K'C , et
latus CC commune , triangula rectangula KCC , KCC' dent
angulum KCC aequalem angulo K'C'C , erit angulus KCC
angulo C &quot; CB ; ideoque angulus incidentiae aequalis an
gulo reflexionis .
Sit nunc firma curvilineaque superficies AB ( Fig 61. ) ,
in quam incidant undae CE , HE&quot; , ... BB' propagatae ex
centro sonoro K ; si centris C , H , ... describuntur sphae
rae , quarum semidiametri ( KB-KC' ) , ( KB - KH ) , ... ,
aereae particulae sitae in superficie BD tangente sphaeras
istas incipient simul affici motu reflexo stalia ab adventu
undac ex K in B. Erit igitur BD superficies undae refle
xae : quam superficiem pon esse sphaericam nemo est qui
non videat. Fac ut puncta C , H sint inter se infinite vi
292
biturqne hemisphaerio , cuins semidiameter CQ ∶⋅−⋅ D'B' :
itcm progrediente unda principali ab C'EE' usque ad BB',
unda manans ex 0 progredietur ab C' usque ad C&quot;; re-
praesentabitnrque hemisphaerio
.cnius semidiameter C'C' ::
E'B' ; atque ita porro. Inferimus ,si concipitur superficies
curva AQC&quot;B tangens omnia haec hemisphaeria in Q, C&quot;,...,
in ea fore puncta illa , quae a secundariis undis reflexis
attinguntur eodem instanti , quaeque tunc incipient concuti -
quum principalis unda pervenerit ad BB' ;exhibebit nimi-
rum AQC&quot;B superficiem undae reflexae; et quoniam productis
infra AB superficiebus BB',Qa, C'a',..., recta KA' exsistit per-
pendicularis ad primam et secundam, recta KH ad primam et
tertiam
,
etc... ; ac proinde sphaerica superficies B'BA'A tan-
git sphaericas superficies QaA', C&quot;a'H .∙∙∙ ;sequitur super-
ficiem AQB undae reflexae fore sphaericam, eiusque cen-
trum in K', et semidiametrum K'Q ∶⋅−∙⋅ KA'. Iamvero qnem-
admodum auris collocata v. gr. in C' deprehendit sonum
directum venire iuxta KC' perpendicularem nudae incidenti ,
sic auris in C&quot; deprehendet sonum reflexum venire iuxta
K'C&quot; perpendicularem undae reflexae ; et cum , ob mutuum
sphaericarum superficierum AQB , C&quot;a'H contactum
,
recta
K'C&quot; transeat per C'; cumque , ob latus KC −∙−∸− K'C , et
latus CC' commune ,
triangula rectangula KCC', K'CC' dent
angulum KC'C aequalem angulO'K'C'C , erit angulus KC'C

: angulo C&quot;CB; ideoque angulus incidentiae aequalis angulo reflexionis . Sit nunc firma curvilineaqne superficies AB (Fig GI.), in quam incidant undae C'E' , HE&quot; ,... BB' propagatae ex centro sonoro K; si centris C' , H , ... describantur sphae- rae , quarum semidiametri ( KB—KC') , (KB—KH) , , aereae particulae sitae in superficie BD tangente sphaeras istas incipient simul affici motu reflexo statim ab adventu undae ex K in B. Erit igitur BD superficies undae refle- xae : quam superficiem non esse Sphaericam nemo est qui non videat. Fac ut puncta C' , H sint inter se infinite vi-293 cina , sintque C'C &quot; , HQ normales ad BD : ex H ductis per pendiculis Ha , Ha' in KC , C'C &quot; , erit Ca ' = CC &quot;—HQ = KB - KC ) - (KB - KH ) = KH - KC = Ca. Quoniam igitur triangula rectangula Cal , Ca'H habent latera aequalia C'a , C'a ', latusque C'H commuue , habebunt ae quales angulos ac'h , a'C'H : hinc sequitur , etsi unda re flexa non est sphaerica , adhuc tamen reflexionis angulum fore aequalem angulo incidentiae.

132. * Haec deducimus ex ( 129) in ordine ad aereum
fluidum concussum in K ( Fig. 60 ) , planoque fixo AB ter
minatum. 1 °* Sumpta x in KC normaliter ad AB, peribit
apud AB tota componens v' ; erit nempe ( 129. 10. )
dQ
dxdo O ( a )
quoad x = KC ( = h ). ProducaturKC donec KC = KC;
radius vector r' computetur ab K' ; et x ab eodem K' in
K'C ; explebitur (a) per
Q = --[Pr + c ) + F(ra) ] +
[fri + ce ) + F(x – ċe)]
( a ) ;
siquidem quoad puncta sita in AB
dQ dQ r=r' ,
dr x = h , it's - h ,
dr
dris
dx dx
Determinatis praeterea f et F ex ( i&quot; &quot; . 129. 10. ) , re
praesentabit ( a' ) initialem fluidi statum: quoniain igitur ( a' )
a— 293
cina , sintque C',C&quot; HQ normales ad BD: ex H ductis per-
pendiculis Ha ,
[in' in KC', 0C.&quot; , erit
∁≮≖⇌∁⋅∙∁ ∙∶−⇀−∐≺≀ (KB'—-KC';-(KB-—Kll)—-KH-—KC':C a.
Quoniam igitur triangula rectangulaC aH, C:: 'H habent latera
aequalia C'a
, C'a', latusque C'H commune , habebant ae-
quales angulos aC',H a'C' H hinc sequitur , etsi unda re-
flexa non est sphaerica , adhuc tamen reflexionis angulum
fore aequalem angulo incidentiae.
1324: Haec deducimus ex (129) in ordine ad aereum
fluidum concussam in K (Fig. 60), planoque fixo AB ter-
minutum. ↿∘∙ Sumpta æ in KC normaliter ad AB, peribit
apud AB tota componens v'; erit nempe (129. 10.)
19.
da: :0 (a)
quoad .c— KC (: It ). Producatur KC donec K' C:: KC:
radius vector r' computetur ab K'; et .r' ab eodem K' in
K'C; explebitur (a) per
≬⇌−⋮−∥↸≀∙⊣−∘≖⋮⋟⊣−⊏⋅⇁≺↗⋅−⋅−∘∩⊐−⊦
↿
−≀−∙−∙⋅−∣⋮⋀≀∙⋅−∣⋅−∘≀⊅⊹⊞↱⋅−−⊄⋮↕∙⋟⋅∙∣ (a');
siquidem quoad puncta sita in AB
∙∙ dQ dQ ∙∙∙
↙≀↾∙∙∙⊲ dr'
⋅⋅−—&quot;'-27—27 ***-&quot; ∙⋅↕−⇀−∣⋅∙⋅↴∙⋮⋮⊒−− 2;-
Determinatis praeterea f
et F ex (i'&quot;. 129. 10.), re-
praesentabit (a') initialem fluidi statum: quoniam igitur (a')291 !
1
1
1
1
1
et satisfacit conditioni ( a ) , et exprimit initialem fluidi statum,
poterunt per ( a' ) definiri, quae spectant ad motus propaga
tionem, attento obstaculo AB.
2º . * Punctum C &quot; , ad quod pertinent radii vecto
res r et r seu KC&quot; et K'C &quot; , perinde motum concipiet ac
si ( 129. 6. &quot; ) , sublato plano AB, fluidum eadem omnino
ratione concuteretur simul circa duo puncta K et K' . Per
tinget itaque ( 229. 4º. ) concussio ad C &quot;, primum in fine
temporis deinde in fine temporis : hinc bi
ni successive motus in C &quot; , alter directus, alter reflexus ; et
quia secunda concussio non pervenit ad C &quot; nisi quum tempus
sic invrevit, ut habeatur r = ct + a,, iccirco eadem velo
citate c regredietur motus, qua incedebat antequam in obi
cem impiogeret. Ad haec : cum anguli KC'C, K'C'C sint
aequales, rursus ( 131 ) patet sonum illisum obici AB re
gressurum efficiendo angulum reflexionis aequalem angulo
incidentiae.
C. с
De instrumentis pneumaticis.

133. In instrumentis pneumaticis soni genesis repe
tenda non est saltem praecipue ex oscillatione partium so
lidarum ipsius instrumenti. Etenim si in hisce instrumentis
dicatur soous creari eodem modo ac in instrumentis per
cussione resonantibus, jam sonus ipse connexionem haberet
maximam cum materia qua instrumentum compactum est ,
nec non cum ejusdem crassitie; quod tum ratione verissi
mum apparet , lum etiam constat ex vi paritatis. Ratione
quidem: nam in instrumentis ex diversa materia compactis,
quorum proinde particulae non aeque elasticae sunt, et ad mo.
cum oscillatorium non aeque aptae, non eodem modo tremu.
lus ille motus per insufflationem excitari debet ; quo vero
crassius instrumentum est, eo in plures continuas partes 0
294
et satisfacit conditioni (a), et exprimit initialem fluidi statum,
poterunt per (a') definiri, quae spectant ad motus prcpaga-
tionem, attento obstaculo AB.
20.a Punctum C&quot;, ad quod pertinent radii vecto-
res r et r' seu KC&quot; et K'C&quot;, perinde motum concipiet ac
si (129. 6.0), sublato plano AB, fluidum eadem omnino
ratione concuteretur simul circa duo puncta K et K'. Per-
tinget itaque (229. 40.) concussio ad C&quot;, primum in fine
tempons c
, deinde in fine temporis c : hinc bi-
ni successive motus in C&quot;, alter directus, alter reflexus; et
quia secunda concussio non pervenit ad C&quot; nisi quam tempus
sic iuvrevit, ut habeatur r': ct ⊣−∙ a. ,
iccirco eadem velo-
citate c regredietur motus, qua incedebat antequam in obi-
cem impingeret. Ad haec : cum anguli KC'C, K'C'C sint
aequales, rursus (131) patet sonum illisum obici AB re-
gressurum efficiendo angulum reflexionis aequalem angulo
incidentiae.
∙ r—al r'e—a
De instrumentis pneumatict's.

133. In instrumentis pneumaticis soni genesis repe-
tenda non est saltem praecipue ex oscillatione partium so-
lidaram ipsius instrumenti. Etenim si in hisce instrumentis
dicatur sonus creari eodem modo ac in instrumentis per-
cussione resonantibus, jam sonus ipse connexionem haberet
mammam cum materia qua instrumentum compactum est,
nec non cum eiusdem crassitie; quod tum ratione verissi-
mum apparet ,
tum etiam constat ex vi paritatis. Ratione
quidem: nam in instrumentis ex diversa materia compactis,
quorum proinde particulae non aeque elasticae sunt, et ad mo-
tum oscillatorium non aeque aptae, non eodem modo tremu-
lus ille motus per insufilationem excitari debet ; quo vero
crassius instrumentum est, eo in plures continuas partes o-295
scillatorius molus dispesci debet. Vi paritatis autem : nam
reipsa instrumenta, quae percussione sopant, pro materiae di
versitate etiam in pari crassitudine diversimode contremiscunt;
et si ejusdem sint materiae, pro diversa crassitie diversum
item sonum edunt. Ergo sonus in instrumentis pneumaticis
maximam connexionem etc. si in his soni genesis etc. Atqui
hoc est falsum : in tibiis enim cylindricis ejusdem longitu
dinis idem habetur sonus aut fere idem , nullo respectu
habito ad materiam aut crassitiem ipsius instrumenti , ut
constat experimentis; totumque artificium pro varietate to
norum pendet ex instrumenti variata longitudine: propte
rea etc.
Quonam igitur pacto in hujusmodi instrumentis erit
soni genesis explicanda ? Eo videlicet , quem indicavimus
(114.). In interna instrumenti capacitate aeris columna includitur, quae externi aeris pressione urgetur: dum igitur
per exiguum orificium fistulae alius aer insufflatione intro
mittitur, aer ille inclusus condensari debet , atque urgeri
contra aerem externum; externus autem vi suae pressionis
resistere; et quum aeris interni aucta pressio vincit, hic se
dilatando condensabit externum aerem , repelletque; externus
aer ita densatus ut ejus elasticitas praevaleat, se restituendo
rursum comprimet internum aerem : in columna videlicet illa
fiei compressio et restitutio, sicque in aeris particulis oscillato
rius motus excitabitur; qui motus communicabitur externo
aeri contiguo, et ad aures perveniet. Aer itaque secundum lon
gitudinem fistulae se habet instar chordae peragentis longita
dinales vibrationes: quo tibia et consequenter columna aerea
longior est, eo etiam longiores undae efformantur; longius
que erit tempus compressionis et restitutionis , ac proinde
Lonus gravior. Hinc in instrumentis, quae secundum longi
Ludinem sunt foraminibus instructa, modo hoc et modo il
lud foramen aperiendo, sublato digito, varii obtinentur to
ni; siquidem externum aerem sic admittendo , modo ma
jorem et modo minorem columnae aereae longitudinem ha
295
scillatdrius motus dispesci debet. Vi paritatis autem: nam
reipsa instrumenta, quae percussione sonant, pro materiae di-
versitate etiam in pari crassitudine diversimode contremiscunt;
et si ejusdem sint materiae, pro 'diversa crassitie diversum
item sonum edunt. Ergo sonus in iustrumentispneumaticis
maximam connexionem etc. si in his soni genesis etc. Atqui
'
hoc est falsum: in tibiis enim cylindricis ejusdem longitu-
dinis idem habetar sonus aut fere idem , nullo respectn
habito ad materiam aut crassitiem ipsius instrumenti ,
ut
constat experimentis; totumque artificium pro varietate to-
norum pendet ex instrumenti variata longitudine: propte-
rea etc.
Quonam igitur pacto in hujusmodi instrumentis erit
soni genesis explicanda ? Eo videlicet
, quem iudicavimus
(114.).ln interna instrumenti-capacitate aeris columna in-
cluditur, quae externi aeris pressione urgetur: dum igitur
per exiguum orificium fistulae alius aer iusufflatione intro-
mittitur, aer ille inclusus condensari debet, atque urgeri
contra aerem externum; externus autem vi suae pressionis
resistere; et quam aeris interni aucta pressio vincit, hic se
dilatando condensabit externum aerem, repelletque; externus
aer ita densatus ut ejus elasticitas praevaleat, se restituendo
rursum comprimet internum aerem: in columna videlicet illa
fiet compressio et restitutio, sicque in aeris particulis oscillato-
rius motus excitabitur; qui motus communicabitur externo
aeri contiguo, et ad aures perveniet. Aer itaque secundum lon-
gitudinem fistulae se habet instar chordae peragentia longitu-
dinales vibrationes: quo tibia et consequenter columna aerea
longior est, eo etiam longiores undae efi'ormantur; longius-
que erit tempus' compressionis et restitutionis , ac proinde
tonus gravior. Hinc in instrumentis, quae secundum lougi-
tudinem sunt foraminibus instructa, modo hoc et modo il-
lud foramen aperiendo, sublato digito, varii obtineatur to-
ni; siquidem externum aerem sic admittendo , modo ma-
iorem et modo minorem columnae aereae longitudinem ha-296
benius. Ita in chordis, pro majori chordae longitudine gra
vior est tonus, acutior pro minori; et digitis comprimendo
camdem chordam, ut evadat plus aut minus longa , varios
assequimur tonos .
Dixi soni genesim repetendam non esse saltem prae
cipue ex oscillatione solidarum partium etc. Etsi enim ex
materia instrumenti non habetur varietas quoad soni qua
litatem , aut valde notabilem intensitatem ; varietas tamen
habelur quoad meliorem aliquam resonantiam; idque ex eo
desumendum videtur quod aer inclusus pro diversitate cor
poris includentis melius aut minus bene oscillare potest ;
magis nimirum aut minus impeditus adhaesione ad ipsum
corpus et scabritie aliqua. Ad haec; si instrumentum pneu
maticum sit compactum ex materia non resistente, quale v.g.
esset instrumentum membranaceum , tunc per vibrationes
columnae aereae excitari poterit sensibilis motus oscillato
rius in partibus instrumenti; quae partes vicissim in aerem
vibrantem reagendo valebunt sonum ipsum modificari etiam
quoad tonum; et quidem plurimum si instrumentum sit val
de breve, quemadmodum expertus est D. Savarı; adeo ut
brevi tubo membranaceo obtineri possil magna varietas
lonorum , qui eo graviores erunt quo minus tenditur mem
brana.

134. Haec proponimus explicanda circa instrumenta
pneumatica. 

1º. Aperto aliquo foramine ex. gr. tertio, cae
lerisque clausis, ac deinde aperto alio puta quinto , variat
lonus: at si ' per aperitionem tertii inducitur communicatio
interni aeris cum externo, nonne ex dictis ( 133 ) audiri de
beret idem tonus sive apertum sive clausum sit quintam
foramen ? 

2º. Sola inflationis intensione mutantur toni , e
tiam servata eadem internae columnae longitudine 

3º. In
canna organi ejusdem diametri superius clausa, si subdupla
sit longitudo , idem redditur tonus qui obtinetur ex can
na superius aperta, et longitudinis duplae.
Ad 1. Cum varia in instrumento pneumatico fora
mina aperiuntur, variae interni aeris columnae communi
296
hemas. ita in chordis, pro maiori chordae longitudine g'ra-
vior est tonus, acutior pro minori; et digitis comprimendo
eamdem chordam, ut evadat plus aut minus longa , varios
assequimur tonos.
Dixi soni geneaim repetendam non esse saltem prae-
cipue ex oscillatione solidarum partium ctc. Etsi enim ex
materia instrumenti non habetur varietas quoad soni qua-
litatem , aut valde notabilem intensitatem ; varietas tamen
habetur quoad meliorem aliquam resonantiam; idque exeo
desumendum videtur quod aer inclusus pro diversitate cor-
poris incladentis melius aut minus bene oscillare potest;
magis nimirum aut minus impeditus adbaesione ad ipsum
corpus et scabritia aliqua. Ad haec; si instrumentum pneu-
maticum sit compactum ex materia non resistente, quale v.g.
esset instrumentum membranaceum ,
tunc per vibrationes
columnae aerea'e excitari poterit sensibilis motus oscillato-
rius in partibus instrumenti; quae partes vicissim in aerem
vibrantem reagendo valebunt sonum ipsum modificari etiam
quoad tonum; et quidem plurimum si instrumentum sit val-
de breve, quemadmodum expertus est D. Savart; adeo ut
brevi tubo membranacea obtineri possit magna varietas
tonorum, qui eo graviores erunt quo minus tenditur mem-
liraua.

134. Haec proponimus explicanda cirea instrumenta
pneumatica. 10. Aperto aliquo foramine ex. gr. tertio, cae-
terisque clausis, ac deinde aperto alio puta quinto. variat
tonus: at si' per aperitionem tertii inducitur communicatio
interni aeris cum externo, nonne ex dictis (133) audiri de-
beret idem tonus sive apertum sive clausum sit quintum
foramen? 20. Sola inflationis intensione, mutantur toni. e-
tiam servata eadem internae columnae longitudine 3&quot;. In
canna organi eiusdem diametri superius clausa, si subdupla
sit longitudo, idem redditur tonus qui obtinetur ex can-
na superius aperta, et longitudinis duplae.
Ad 1.&quot; Cum varia in instrumento pneumatico forf- mina aperiuntur, variae interni aeris columnae communi-297
cantes çum aere externo excitantur; non ita tamen commu
nicantes, ut simul non etiam inter se communicent; ergo
looi variare per plurium foraminum aperitionem debent ,
etsi exquisitam ejus rei rationem aegerrime quis reddere possit.
Ad 2.&quot; Ut per vehementiorem percussionem in chor
da instrumenti fidicularis contingit ut ea resonet ad oclavam,
ita in columna aerea per variam inflationis intensionem con
tingit ut tonus mutetur; et sicut certum est in chorda mu
sica quod ea tunc dividitur in duas partes separatim oscil
lantes, ita eadem asserenda est fieri divisio et oscillatio in
columna aerea sub tempore, quod sił proportionale tono
quem reddit. Hinc deducitur explicatio saltus ut ajunt tu
bae v. gr. ad octayam: cam paulo vehementius inspiralur tu
ba, cogitur aer ad celeriorem motum , quem tamen colu
mnae aereae jam vibrantes , utpote nimis longae, praesta
re non possunt. Dividitur igitur columna per medium ita ,
ut duo nova segmenta aeris aequalia suas vibrationes separatim
peragant. Ubi vero saltus non sit ad octavam , alia divi
sio fieri dicenda est .
Ad 3.&quot; Ia medio cannae duplae efformátur nodus ,
habetur aereum stratum quiescens , quemadmodum
habetur in orificio clauso cannae subduplae ; adeoque ea
dem undae aereae longitudo in utraque canna , idemque
proinde tonus .

135.* Sit tubus cylindricus determinatae longitudinis 1,
firmiter obseratus apud alterum orificium , aperius apnd al
terum : aequilibrium aereae columnae inclosae ita turbari
pono, ut qui aer orificio aperto ( ubi initium distantiae x
consliluo ) respondet, nullam densitatis variationem subeat,
et qui orificio clauso, nullatenus moveatur.Functiones ( 129.7 °) .
f, fx , ac proinde f , F ' tanquam datas assumo ab x = 0
ad x = l. E statu aeris apud extremitates tubi habemus
= o si x = 0, v = 0 si x = l; hinc
seu
fl + 1) + F'll — cl) = 0 ( 0 ) ,
20
297
can'tcs cum aere externo excitantur; non ita tamen cdmmu-
nicantes, ut simul non etiam inter se communicent; ergo
toni variare per plurium foraminum aperitionem debent
,
etsi exquisitam eius rei rationem aegerrime quis reddere possit.
Ad 2.&quot; Ut per vehementiorem percussionem in chor-
da instrumenti fidicularis cdntingit ut ea resonet ad octavam,
ita in columna aerea per variam inflationis intensionem cou-
ting'it nt tonus mutetur; et sicut certum est in chorda mu-
sica quod ea tunc dividitur in duas partes separatim oscil-
lantes, ita eadem assereuda est fieri divisio-et oscillatio in
columna aerea sub tempore, quod sit proportionale tono
quem reddit. Hinc deducitur explicatio saltus ut aiunt tu-
bae v. gr. ad octavam: cum paulo vehementius inspiratur tu-
ba, cOgitur aer ad celeriorem motum, quem tamen colu-
mnae aereae iam vibrantes, utpote nimis longae, praesta-
re non possunt. Dividitur igitur columna per medium ita,
ut duo nova segmenta aeris aequalia suas vibrationes separatim
peragant. Ubi vero saltus non sit ad octavam, alia divi-
sio fieri dicenda est, ⋅ '
Ad 3.&quot; In medio cannae duplae eEorm'atur nodus ,
seu habetur aereum stratum quiescens, quemadmodum
habetur in orificio clauso cannae subduplae: adeoque ea-
dem uudae aereae longitudo in utraque canna
,
idemque
proinde tonus.

135 Sit tubus cylindricns determinatae longitudinis !,
firmiter obseratus apud alterum orificium, apertas apud al-
tequm : aequilibrium aereae columnae inclusae ita turbari
pono, ut qui aer orificio aperto ( ubi initium distantiae .a-
constituo ) respondet, nullam densitatis variationem subeat,
et quiorificio clauso, nullatenus moveatur.F unctiones (129.7&quot;).
f, f. , ac proinde f, F' tanquam datas assume ab a: 30
ad .r.-zl. E statu aeris apud extremitates tubi habemus

:: osi æzo,v:——osiæ:-:l; hinc

≀≖∣⋅∶≀−∙⊢≀∶∠⊢⊢ F'(l—c1):o ( o )
.
20298
Fll — ct) - f ( c ) = 0 ( o' ) .
In (0 ) substituatur ct +1- x in locum ct : prodibit
f (21 + ci - x) = - F '(x - 1) ( 0&quot; ) ;
unde
= f'( x + cl) – f'( 21+ ct - x ) ,
c = -f(x + ct) -f(21 +ct - x) :
( o ' ')
in ( o &quot; ) fiat x = 0 ; erit ob ( o')
f (c + 2) = -F ( - ct) = -f (c ) (0&quot; &quot; );
subrogato ct +21 in locum ct, habebitur
f '( c +4 ) = -f( ct + 2 ) = f(t ) (o ');
denique si in ( 0 &quot;) ponitur ci=0, emerget
f( 21 — x ) = - F ( x ) ( 0 &quot; ) .
.
Aequationes ( o' : 0' ! ) satis sunt, ut functionem f con
siderare possimus veluti datam quoad omnes positivos ya.
lores quantitatis variabilis , ad quam respicit ipsa functio.
Etenim e statu initiali atque arbitrario aereae columnae ad
motum incitatae , data est F' ab x = o ad x = l ; ergo
ob (o &quot; ) data erit fab x = l ad x = 21 : ex eodem sta
tu jam dala erat fab x = o ad x = l ; ergo dabiturf
ab x = o ad x = 21. Aequatio autem ( o&quot;) rem evidentissime
absolvit quoad caeteros quantitatis variabilis positivos va
lores. Ergo etc.
298
F'(—ct)——f(ct):-to (c').
In (o) substituetur et −⋅⊢≀ —- a: in locum et: prodibit
f(2l −∣⋅− ct — a:) ∶−∙−− F'(x—ct) (o&quot; );
uude
v:f(æ-i-ct)—f(2l—l—ct—æ).
etc:—f(x-i-cz)—f(2l -i-ct—æ):
111 (o&quot;) fiat m::- o; erit Ob (0')
(o&quot;')
f(cs −↿− 21) ∶−∙− −F'( —ct) −−∶ —f(ct) (o&quot;):
subrogato et —-[-21 in locum et, habebitur
f(cz ↽⊢ 4!) ∙−−− —-f(ct −⊦ 21) ↽↼−−⋅≖ f(ct) (a';
denique si in (a'—') ponitur ctzo, emerget
f(ZI—æ) z—F'Lr) (o&quot;).
Aequationes (a': a'!) satis sunt, ut functionem f coa-
siderare possimus veluti datam quoad omnes positivos ,va-
lores quantitatis variabilis, ad quamrespicit ipsa functio.
Etenim e statu initiali atque arbitrario aereae columnae ad
motum incitatae
, data est F' ab a: a ad w:! : ergo
ob (o&quot;) data erit f
ab æ-—-:l ad se:21 :ex eodem sta-
tu iam data erat f' ab a: :: o ada::1; ergo dabitur ['
ab a: :: 0 ad se:21. Aequatio autem (o&quot;) rem evidentissime
absolvit quoad caeteros quantitatis variabilis positivos va-
lores. Ergo etc.299
Quoniam ab x = o ad x = 21 dependet f' ab ini
tiali atque arbitrario statu aereae columnae ; poterit igitur
sic assumi inter illos limites , ut facto i = 1,3,5,7 ,..., sit
21
f'(c + % -f(cc + = p&quot;(ce) ( 0 &quot; &quot; );
numeri pares = 2, 4, 6 ... debent excludi ob aequatio
dem (o &quot; ). Instaurantur ergo iidem functionis f valores
42 quotiescumque tempus t evadit it ; sed ( o &quot; ) a functio ic
ne f unice dependent v, E. Columna igitur aerea in eum
dem restituitur statum per aequalia intervalla, suasque com
41
plet oscillationes intra tempus ; quarum propterea nu
merus intra q ' erit
ic
ic
41
136.. Evanescet (135.0 '&quot;') velocitas v ubi fuerit f ( x + cos
= f (21 + ct - x ) ; evanescet e si f'( x + ct) - f (21+
ci - x ). Primum contingit ( 135 : 0 &quot; ) quando (22 +ic - x )
41'2 - ( x + cl) seu 21 2x ; secundum quando 21—
21&quot; i 1
2x= • Hinc 1º. facto i = 0 , 1 , 2 , 3 .. i
scet aer in distantiis
2 , quie
lli - 21 )
X
2º . Facto i&quot; =1 , 3 , 5 .... 1 ; movebitur aer in
locis
299
Quoniam ab a: a ad a: 2! dependet ;&quot; ab ini-
tiali atque arbitrario statu aereae columnae ; poterit igitur
sic assumi inter'illos limites, ut facto i:1,3,5,7,...., sit
41 21 ∣
'&quot;
∣⇃≺∁⊢⊢ ∙
−−⋮∙⊣∶∶≕−∣↙≼⋄⊢⊢ 7): f
(ct) (0 ):
numeri pares r':2, 4, 6 ...
debent excludi ob aequatio-
nem (o&quot;). Instaurantur ergo iidem functionis f' valores ,
quotiescumque tempus :
evadit t −∣∙⋅ &amp; sed (o&quot;')a functio-ne )&quot; unice dependent v, :. Columne igitur aerea in eum--
dem restituitur statum per aequalia intervalla, suasque com-
. . . plet 4! osc1llat1ones intra tempus ∙≀⋅−≔∙ ; quarum propterea nu-
merus intra 1&quot; erit
t'

136. a Evanescet (135.o&quot;')velocitas :: ubi fuerit f' (æ—l—ct

f(ZH— ct —- æ ); evanescet :si f(x −∣⋅− et):
— f(ZI-i-
ct ∙−− a: ). Primum contingit (135: a'&quot;) quando (21—1—4 cs --' a:)
41&quot;! −∙− ( æ ⋅−⊢ ct) seu 2! -- 21: −∙−−−−− T;secundum quando 21—
2
∙∣∣ ∙−↿ .
2x: −⋮∙−↨ ∙Hinc ↿∘∙ facto 1&quot;: a, 1, 2, 3 ....
'T, qu1e-
scet. aer in distantiis' -
∙∙∙ [( t' — 21&quot;)
æ
!
20. Facto 1'&quot; :1, 3, 5 ....
i; movebitur aer in
locis300
llimi)
quin tamen ullam patiatur densitatis variationem, Aper
tis itaque foraminibus in hisce postremis locis , nullo pa
cto sonus mutari debet ; quod experientiae consonum re
peritur: imo non mutabitur sonus, licet lubo abscindatur
pars 1- x , quae ultra locum x ad fundum usque pro
tenditur. Atqui pars reliqua nihil aliud est nisi tubus
in utraque patens extremitate: ergo si de hujusnuodi cubis
sermo sit, posita e = o apud unum orificium erit quo
que apnd alterum { =0.
137. # In tubis itaque cylindricis, quorum ambo ori
ficia libera omnino sunt, habetur ( 129.7 .)

#

Fl—ct) — 9 (2+ cl ) = 0, F1 — ct) - f'(C ) = 0.
Hinc facile deducuntur ( 135 ) sequentes aequationes
f (21 + cix) = F ' x - ct), v = P ( x + ct) + P (21 + c1 - ),
c : = f ( 21 + (1 - x ) — f (x + c ! ), f (ce + 21 = F (-1)= f (c ),
f'(21 — * ) = F ( x ).
Quia vero ab x = o ad x = 21 rursus dependet p
ab initiali atque arbitrario statu 'aereae columnae , ic
circo poterit etiam asseri sequens aequatio.
f (ce + *-) = f ( c )
in praesenti est i = 1. 2, 3, 4 ......
æs. lu.—'r')
1
,
quia tamen ullam patiatur densitatis variationem. Aper-
tis itaque foraminibus in hisce postremis locis, nullo pa-
cto sonus mutari debet; quod experientiae consonumre-
peritur: imo non mutabitur sonus, licet tubo abscindatur
pars l—æ, quae ultra ↙
locum a: ad fundum usque pro-
tenditur. Atqui pars reliqua nihil aliud est nisi tubas
in utraque patens extremitate: ergo si de huiusmodi tubis
sermo sit, posita :

: o apud/uuum oriücium erit quo-

que apud alterum::o.

137.a In tubis itaque cylindricis, quorum ambo ori-
licia libera omnino sunt, habetur (129. 70.)
F'U—ct) -
f(l −↿− ct) ∙−−−−∙∙ o,F'( -
ct) --f(ct): 0.
Hinc facile deducuntur (135) sequentes aequationes
f(21 -l-ct -æ):F'(æ—-ct), v ::f'(æ ⊣− et)-t— f(ZI-i-ct—x).
es:/(21 -t-ct —x)-— f(æ-t—ct), f(ct-t-Zl):F'(—cc):f(c1),
f(2l — a: ): F' (æ)-
Quia vero ab a: 0 ad ..r:21 rursus dependet f
ab initiali atque arbitraria statu 'aereae columnae
,
ic-
circo poterit etiam asseri sequens aequatio.
f(ct—i- -—2'-£-) :f'(ct )

:

in praesenti est i: 1. 2, 3, 4 ......
≡⊲∙⋅⇀≣∎ lJ-r

: 301

22
Iterat ergo aerea columna per aequalia intervalla ic
oscillationes suas , quarum proinde numerus intra 1 &quot; erit
ic n =
21
Haud immoror inquisitioni distantiarum , ubi a
er vel quiescit, vel nativam retinet densitatem : hujusmo.
di namque investigatio similiter perficitur ac in Lubis,
quorum unum orificium apertum est . Satius forsan e
rit adnotare quod, facto i = 1 , exhibet ( 137 ) aequatio n '
relationem inter principalem tonum n ', redditum ab
elastico fluido intra tubum oscillante, et velocitatem c qua
sonus incederet si per ipsum fluidum propagaretur. Hinc
patet quomodo experimentis indagari possit velocitas c in
aliis elasticis flaidis ab aere atmosphaerico diversis : ex
tentaminibus Van - Rees, Frammeyer, et Moll prodiit so
ni velocitas sub temperie 10.° C
21
io gas oxigenio 3,7m, 9 :
bydrogenio 1233 , 3 ,
nitrogenio .
.
339 .
oxido nitrico 317 , 4 ,
acido salphuroso 229 , 2 ,
acido carbonico 370 , 7 ,
. .
suboxido carbonico . . 341,1
etc. etc.
301
. . 2! [terat ergo aerea columna per sequsl1a1ntervalla ∙∙∙∙⋮∙⋅−
oscillationes suas ∙ quarum ⋅proinde numerus intra 1&quot; crit
'
IC
Haud immoror inquisitioni distantiarum , ubi a-
er vel quiescit, vel nativam retinet densitatem: huiusmo-
di namque investigatio similiter perficitur ac in tubis,
quorum unum orificium apertum est. Satius forsan e-
rit adnotare quod, facto 1':1,exhibet (157) aequatio n'
0
∙−−∶ -2-l- relationem inter principalem tonum n', redditum ab
clastico flaido intra tubam oscillante, et velocitatem e qua
sonus incederet si per ipsum fluidum prcpagaretar. Hinc
patet quomodo experimentis indagari possit velocitasc in
aliis elasticis fluidis ab aere atmosphaerico diversis: ex
tentaminibas Van— Bees, Frammeyer, et Moll prodiit so-
ni velocitas sub temperie 10.0C
in gas oxigenio . . . . . . 317',g,
hydrogenio . . . . . 1233,3,
nitrogenio. . . . . . 339 . ∙
oxido uitrico . . . . 317 ,4
acido sulphuroso . . -
229 , 2
,
acido carbonico . . . . 370 , 7 ,
suboxido carbonico . . . 341 ,
1
,
etc. etc.302

138. Si tubus proponilur utrinque obseratus , quis
que videt fore v = o apud ambas extremitates; unde (129.7°)
f (c ) + F ( -ct)= 0,flfct) + F (l ct) = 0,
quarum ope determinatur motus inclusi aeris,
De propagatione soni per liquida , et per solida corpora.
139.* Quod spectat ad liquida corpora , in comperlo
est aquam v. g. contrahi perparum posse atque restitui
in suis partibus : itaque qua ratione turbatum posuimus
( 129..1 . ° ) aequilibrium , eadem in praesenti imaginemur
turbari . Propagato motu , densitas ré aquae libratae ver
tetur in je = pili + :) apud (2. , y , z ) ; et pressio o'
in w= '+Ae ; exprimit A numerum experimentis deter
minandum. Sumptis hic quoque X=0, Y=0, Z=o, et ra
tiocinando ut in citato n. ° assequemur
d
dQ
1 do dt (
dQ
dt
dr
A dL {1+ :)
dr seu
р. dr pi
A
tum facto c ” , perveniemus ad formulas (i' '.;&quot; . .
ji
į &quot; : 129, 1.0 ) . Non pluribus opus est ut intelligamus
( 129. 2.° 3,0 ) sonum per aquam diffundi aequabiliter ve
locitate.
VA
Numerus A potest determinari ex parvula contractione ,
quam juxta longitudinem à ( haud variata diametro ) pa
302
1381: Si tnbus praponitnr utrinqne obseratus , quis-
que videt fore v:o apud ambas extremitates; unde (129.?)
f(ctH-FX --ct):o,f7(l—i-ct)-i-F'(l—ct):a,
quarum Ope determinatur motus inclusi aeris.
De propagatione soni per liquida, et per solida corpora.
139:- Quod spectat ad liquida corpora, in comperto
est aquam v. g. contrahi perparum posse atque restitui
in suis partibus :itaqne qua ratione turbatum posuimus
( 129. ⋅↿∙∘ ) aequilibrium, eadem in praesenti imaginemur
turbari. Pr0pagato motn, densitas pf aquae libratae ver-
tetur in þ.:yJU—I—s) apud (.x.-,,] ,
z) :
et pressio a'
in ≔≖⇌≖⋝∣↰∟⋀⋮⋅⋮ eXprimit A numerum experimentis deter-
minandum. Sumptis hic quoque X:0, ↧↗−−−−⋅∘∙ Z:o, et ru-
tiocinando ut in citato 11.0 assequemur
d
dQ) ↼ 1 de ∙− dQ) (Et? A; JLu-Jr-s) ∙− &quot;( dt ∙
a d?
.
dr ' se.. a' d.- 4.-
A -
∙∣∣ -1 tum facto ?

: cz, perveniamus ad formulas (:

'.
t
'.
i'
i&quot;: 129, ↿∙∘ )
. Non pluribus opus est ut intelligamus
( 129. 2.0 3.&quot;) sonum per aquam diffundi aequabiliter ve-
locitate. ⋅
.: ⇂∕⋅−⋮∶⇡∣−⋅
Numerus A potest determinari'ex parvula contractione f:,
quam iuxta longitudinem l (haud variata diametro) pa-303
tilur columna aquea ob incrementum 5. superadditum
pressioni o '. Nam 1 : 1 - B = + ): , ideoque &lt; =
B \beta

: ' sed o=u'two=a' +As, igitur スー B 2

6.
A :
σολ
E \beta
In hypothesi pressionis . = 0 &quot; , 76) g, ac temperiei n=
10.• C, experimenta Dni Canton suppeditant B = 0,000046 ),
inter quem valorem et quos invenerunt DD . Parkins et
Oersted , nimirum
B=0,0000452 , B=0,0000482 ,
parvula est differentia. Ponatur hydrargyri densitas 1 ;
erit proxime u'= . : assumpta igitur g=9m, 8088,
13,5819
1
emerget
c=1483&quot; , 59.
Sonus videlicet propagatur per aquam plus quadruplo ce
lerius quam per aerem. D. Beudant dicit in hac se fuis
se sententia , ut e suis experimentis in mari institutis ta
lem deduceret soni velocitatem , quae 1500m saltem aequaret.
140.* Quisque videt soni velocitatem eadem ratione
posse in caeteris corporibus , sive liquidis , sive solidis ,
determinari , modo eorum partes contrahi perparum queant
atque restitui . Sic , manente 5 . = (0,76 ) 8 , obtinuit idem
ipse Canton hydrargyri contractionem B = 0,0000032 : as
sumpla igitur u = 1 , erit
c = 1576m , 35
1
303
titur columna aquea' ob incrementum m superadditum
pressioni w'- Nam 71: l—þ:p.'(1-l-s): p!
,
ideoque ::
P −⇀ 13 ⇤
⋅
m−⊸T;'sed ∏∙−−∶∏∎∙⊦∏∘∶−−∸⋅∄≖⋅−⊦∆⋮∙ igitur
'
A ∙− a'., ∙∙∙ wo).
8
5
In hypothesi pressionis uro :( o'&quot;, 76) g, ac temperiei :::,
10.(, C. experimenta Dni Cauton suppeditant þ:0,0000461,
inter quem valorem et quos invenerunt DD. Parltins et
Oersted , nimirum
,ezo,oooo45) ∙ þ:0,000048).
.
parvula est differentia. Ponatur hydrargyri densitas :1;
↿
15.5819
erit proxime pf: : assumpta igitur g:9&quot;?,,8088,
emerget ⋅
c:1483&quot;'
. 59.
Sonus videlicet prcpagatur per aquam plus quadruplo cc-
lerius quam per aerem. D. Bendant dicit in hac se fuis-
se seateutia , ut e suis exPerimentis in mari institutis ta-
lem deduceret soni velocitatem, quae 1500&quot; saltem aequaret.
1403 Quisque videt soni velocitatem eadem ratione
posse in caeteris corporibus
.
sive liquidis , sive soli-dis ,
determinari , modo eorum partes contrahi perparum queant-
atque restitui. Sic, manente wo:(0,76) g , obtinuit idem
ipse Canton hydrargyri contractionem [5:0,0000037t : as-
sumpta igitur pf:1 , erit
0:1576,&quot; ∙ 35304
velocitas , qua per hydrargyrom diffunditur sonus. Ante
quam usum contractionis \beta animadverteret Laplace ad de finiendam soni velocitatem per liquida et solida corpora ,
exhibuerat Chladni in sua Acustica aliam methodum
sane ingeniosam , ejusdem velocitatis investigandae in cor
poribus solidis.
141. # Innititur ista methodus analogiae, quam norunt
Physici inter oscillationes aeris in tubo cylindrico apud
ambas extremitales aperto et longitudinales oscillationes
virgae rigidae , cujus ambo extrema omnino libera sint.
Exprimat enimvero l oscillantis virgae longitudinem ; n'
principalem tonum , quem edit resonans virga ; c' quae
sitam velocitatem . Erit ( 137 )
n &quot; ; unde n ' : n &quot; = 0 : c' , ' =
21
Iam si velocitas soni per aerem repraesenterar per &quot; , ex
perimenta D.ni Chladoi praebent soni velocitatem c
per stannum . 717 를
per argentum
per cuprum .
12
per ferrum et vitrum ... 17
per varia lignorum genera 11 ad 17 , . .
Ad explorandam soni velocitatem per ferruin fusionis , in
promptu habebat D. Biot 376 tubos ex hoc metallo com .
pactos ; quibus singulis mediocris erat longitudo duorum
304
velocitas , qua per hydrargyrnm diffunditur sonus. Ante-
quam usum contractionis þ animadverteret Laplace ad de-
finiendum soni velocitatem per liquida et solida corpora ,
cxbibuerat Chladni in sua Acustica aliam methodum,
sane ingeniosam .
ejusdem velocitatis investigandae in cor-
poribus solidis.
1414: lnnititur ista methodus analogiae, quam norunt
Physici inter oscillationes aeris in tabo cylindrico apud
ambas extremitates aperto et lougitudiuales' oscillationes
virgae rigidae, cuius ambo extrema omnino libera sint.
Exprimat enimvero! oscillantis virgae longitudinem ; n&quot;
principalem tonum
, quem edit resonans virga ; c' quae-
sitam velocitatem. Erit (137) '
c, '
nn 11&quot;:-2-i-;nuden:n&quot;:c:c', c':c-—J.
»
Iam si velocitas soni per aerem repraesentetur per 1, ex-
perimenta D.!d Chladni praebent soni velocitatem c'
per stannum . ∙ ∙ ∙ ∙
, 7
vet--
per argentnm . . . . . . 9 ,
per cuprum . . . . .
.. . 12 ,
per ferrum et vitrum . . .
17
.
per varia lignorum genera . .
11 ad 17,
Ad explorandam soni velocitatem per ferrum' fusionis ,
in
promptu habebat. D. Biot 376 tubos ex hoc metallo com'
pactos ; quibus singulis mediocris erat longitudo duorum

* a305

metr. cum partibus millesimis 515. Sumptis experimentis,
prodiit soni velocitas 104 ; nisi quod jungebantur ii tu
bi ope plumbi, quod aliquanto sonum retardare videtur.

=== De vocis humanae origine. ===
142. Vocis humanae organum etsi considerari maxi
me debet tamquam instrumentum pneumaticum ftexili
et elastica materia ex parte compactum , non tamen ita
est ut cum instrumentis etiam fidicularibus aliquam non
babeat analogiam. Quod ut melius intelligatur , nonnulla
ex anatomicis sunt hic afferenda.
Palmo est viscus respirationi inserviens: in duas par
tes distinguitur , dexteram et sinistram , et duo magni lo
bi dicuntur , etsi quivis ex his duobus dividitur mino
ribus aliis. Substantia constat molli , spongiosa , rara et
vessiculosa ita ut ad aerem excipiendum aptissimus sit :
motu ergo dilatationis aere impletur , et constrictionis
motu eundem expellit ; atque aer ita expulsus primo per
multiplices canaliculos lobis interspereos , qui bronchia
dicuntur ; tum per duos ex utroque lobo emergentes ; de.
mum per ampliorem canalem emergit , qui ex praefa
tis duobus in unum conjunctis coalescit. Hic canalis seu tubus
ad oris usque radices ascendens trachea seu aspera arteria
nuncupatur ; in summitate asperae arteriae brevis canaliculus
habetur , qui larynx dicitur , cujus summitatem facit rima
ovalis a duabus membranis horizontaliter jacentibus relicta ;
quae rima glottis dicitur : atque huic superposita est epi
glottis ; tenuis scilicet et mobilis cartilago glottidem te
gens , quae ad hoc praecipue statuta esse videtur, ut dum
aliquid deglutimus , quidquam cibi aut potus in asperam
arteriam minime irruat, sed per contiguum canalem, qui
exophagus dicitur, et cujus orificium pharynx vocatur , de
more in stomachum demittatur. Itaque lobi pulmonis instar
}
305
metr. cum partibus millesimis 515. Sumptis experimentis.
prodiit soni velocitas 10;- ; nisi quod iungebantur ii ftu-
bi »ope plumbi, quod aliquanto sonum retardare videtur.
De vocis humanae origine.


142.1Vocis humanae organum etsi considerari maxi-
me debet tamquam instrumentum pneumaticum fiexili
et elastica materia ex parte compactam, non tamen ita
est ut cum instrumentis etiam fidicularibus aliquam non
habeat analogiam. Quod ut melius intelligatur, nonnulla
ex anatomicis sunt hic aderenda.
⋅ Palmo est viscus respirationi inserviens: in duas par-
tcc distinguitur , dexteram et sinistram ,
et duo magni lo-
bi dicuntur , etsi quivis ex his duobus dividitur mino-
ribus aliis. Substantia constat molli , spongiosa , rara et
vesaiculosa ita ut ad aerem excipiendum aptissimus sit:
motu ergo dilatationis aere impletur , et constrictioais
motu eumdem expellit; atque aer ita expulsus primo per
multiplices canaliculos' lobis interspereos , qui bronchia
dicuntur; tum per duos ex utroque lobo emergentes :dc-
mum per ampliorem canalem emergit , qui ex praefa-
tis duobus in unum coniunctis coalescit. Hic canalis seu tubas
ad oris usqne radices ascendens tracbea seu aspera arteria
nuncupatur; in summitate asperae arteriae brevis canaliculus
habetur , qui laryux dicitur, cuius summitatem facit rima
ovalis a duabus membranis horizontaliter jacentibus relicta;
quae rima glottis dicitur : atque huic superposita est epi-
glottis; tenuis scilicet et mobilis cartilago glottidem te-
gens, quae ad hoc praecipue statuta esse videtur, ut dum
aliquid deglutimus, quidquam cibi aut potus in asperam
arteriam minime irruat, sed per contiguum canalem, qui
cxophagus dicitur, et cuius orificium pharynx vocatur
, de
more in stomachum demittatur. Itaque lobi pulmonis instar306
re cur com
follium aerem excipiunt, cum compressi illum emittunt per
asperam arteriam : aer ita expulsus per asperam arteriam
in arctiorem laryngis canalem irroit , atque ita ex am
pliori in angustius spatium redactus compressionem pati
debet , oscillatoriumque motum concipere. Sed quia la
rynx flexili et elastica materia compingitur, iccirco ( 133)
ad motum tremulum ab aere vibrante excitabitur. Deinde
vero in eumdem aerem diversimode reagendo, prout magis
vel minus erit tensa , ejus Oscillationes diversimode quoque
modificabitur.
Obiter notamus antiquos et cum iis Galenum male
organum vocis humanae in trachea constituisse ; quam
arbitrabantur vices gerere tubi, per quem aer ad sonum
jam excitatus excurrit. Refelles hanc opinionem consi
derans aerem qui tracheam ascendit , libere ascende
et liberius habere spatium ; unde non est
primi debeat et oscillatorium motum habere : cum autem
glottis sit multo arctior quam trachea , per glottidem
transiens habet quidem unde comprimi possit. Haec de
voce humana : ad vocem enim quod spectat quorumdam
animalium , uti sunt multae aves ; hae cum etiam exse
cto collo , sola ventris compressione sonum edant , in his
utique trachea concurrit ad sonum ipsum modificandum .
Sed nil hinc eruitur contra jam dicta: in istis namque avi
bus trachea habetur supra glottidem , seu gloutis esse obser
vatur non ad summitatem , sed infra tracheam ; contra
ac est in homine , et plerisque aliis animalibus.
In monumentis Academiae Parisiensis ad an. 1741
observat Ferreinius intra laryngem duas haberi fibras ad
labrum glottidis ; quae fibrae ex impetu aeris per angu
stiorem laryngis canaliculum irrumpentis ad tremitum con
citantur , atque hoc tremitu resonant , quemadmodum in
fidibus contingit ; unde dictum est vocis humanae organum
analogiam habere aliquam ad instrumenta fidicularia . Sum
psit ille plures laryages cum sua glottide ; dunque insuf
306
folliam aerem excipiunt. tam compressi illam emittunt per
asperam arteriam: aer ita expulsus per asperam arteriam
in arctiorem laryngis canalem irruit ,. atque ita 'ex am-
pliori in angustias spatium redactus compressionem pati
debet, oscillatoriamque motum concipere. Sed quia la-
rynx flexili et elastica materia compingitur. iccirco (133)
ad motum tremulum ab aere vibrante excitabitur. Deinde
vero in eamdem aerem diversimode reagendo, prout magis
vel minus erit tensa, eius oscillationes diversimode quoque
modificabitur. ∙
Obiter notamus antiquos ,et cum iis Galenum male
organum vocis humanae in trachea constituisse; quam
arbitrabantur vices gerere tubi, per quem aer ad sonum
iam excitatus excurrit. Refelles hanc opinionem consi-
derans aerem. qui tracheam ascendit
,
libere ascende-
re,'et liberius habere spatium ; unde non est cur ,com-
primi debeat et oscillatorium motum habere: cum autem
glottis sit multo arctior quam trachea , per glottidem
transiens habet quidem unde comprimi possit. Haec de
voce humana : ad vocem enim quod spectat quorumdam
animalium , uti, sunt multae aves; hac cum etiam exse-
cto collo, sola ventris compressione sonum edant, in his
utique-trachea concurrit ad sonum ipsum modificandam.
Sed nil hinc eruitur contra iam dicta: in istis namque avi-
bus tracbea habetur supra glottidem , sen glottis esse obser-
vatur non ad summitatem, (sed infra tracheam ; contra
ac est in homine , et plerisque aliis animalibus. .
In. monumentis Academiae Parisiensis ad an. 1741
observat .Ferreinius intra laryngem duas haberi, fibras ad
labrum glottidis ; quae fibrae ex impetu aeris per angu-
stiorem laryugis canaliculata irrumpentis ad tremitum.con-
citantur, atque hoc tremitu resonant
, quemadmodum in
fidibus contingit; unde dictum est .vocis humanae organum
analogiam habere aliquam ad instrumenta fidicularia. Sum-
psit illa plures' larynges cum sua glottidc; dumque insuf-307
Aando sonus vocis animalis excitabatur, microscopio Gibras
praedictas inspiciendo tremor et vibratio in iisdem cerne
batur , prout in chordis musicis habetur dum resonant.
Atqui eo ipso ex earum tremitu ab irruente aere tamquam
a vi percutiente excitato sonus gigni debet, vel jam geni
tus modificationem quamdam recipere.
Rursus , sicut in chordis musicis contingit ' ut chorda
brevior det sonum acutiorem , graviorem longior : ita ani
madvertendum hic fuit an fibrarum illarum major minor
ve longitudo toni mutationem induceret. Compertum au
tem est quod , impedita illarum fibrarum parte ne tre
meret , tonus prodibat acutior.
Sumpsit etiam larynges bovis , canis, aliorumque ani
malium , deinde insufflando excitabatur mugitus bovis , et
conformis aliis animalibus sonus. Movendo autem musculos
ita , ut traherentur et distenderentur fibrae, excitabantur
mutationes soni , quae haberi solent in varia horum ani
malium voce.
Notetur illud : cum tensio vel remissio fibrarum glot
tidis et cartilagineae substantiue , qua larynx constat , ab
eodem musculo dependeat , ut notat Savart , consequitur
una cum fibris illis etiam laryngem tendi vel remitti.
Laxatis fibris, orificium glottidis ampliatur , et sonus pro
dit gravior ; tensis vero , orificium restringitur, et sonus
evadit acutior , ut in canibus observavit D. Magendie.

143. Si vox in larynge et glottide tanquam in proprio
organo fit ; quid ergo, inquies, os atque ejus partes con
ferunt ad formationem vocis ?
Respondeo oris cavitatem , linguam, dentes, labia con
currere ad modificationem perfectionemque vocis ; quae
larynge et glottide incipit quidem , sed non omnimode
ibi perficitur : nam quod in illis partibus sufficiens habea.
tur organum quin prorsus necessaria sint oris et linguae or
gana ad exhibendum aliquo modo sonum animalis pro
prium , apparet ex eo quod grues abscisso
in
et anseres 1
307
flando sonus vocis animalis excitabatur, microscupio fibras
praedictas inspiciendo tremor et vibratio in iisdem cerne-
batur, prout in chordis musicis habetur dum resonant.
Atqui eo ipso ex earum trcmitu ab irruente aere tamquam
a vi percutiente excitato sonus gigni debet. vel iam geni-
tus modificationem quamdam recipere.
Rursus , sicut in chordis musicis contingit 'ut chorda
brevior det sonum acutiorem, graviorem longior :
ita ani-
madvertendum hic fuit an librarum illarum maior minor-
ve longitudo toni mutationem induceret. Compertum au-
tem est quod ,
impedita illarum fibrarum parte ac tre-
meret. tonus prodibat acutior.
Sumpsit etiam larynges bovis , canis. aliorumque sni-
malium, deinde insufflaudo excitabatur mugitus bovis ,
et
conformis aliis animalibus sonus. Movendo autem musculos
ita, ut traherentur et distenderentur fibrae, excitabantur
'mutationes soni, quae haberi solent in varia horum ani-
malium voce.
Notetur illud: cum tensio vel remissio librarum glot—
tidis et cartilagineae substantiae, qua larynx constat , ab
eodem musculo dependeat , ut notat Savart, consequitur
una cum fibris illis etiam laryngem tendi vel remitti.
Laxatis' fibris, ,orificiu-m glottidis ampliatur, et sonus pro-
dit gravior; tensis vero , orificium restringitur. et sonus
evadit acutior, ut in canibus observavit D. Magendie.

143. Si vox in larynge et glottide tanquam in proprio
organo fit; quid ergo, inquies, os atque eius partes cou-
ferunt ad formationem vocis?
Respondeo oris cavitatem. linguam, dentes. labia con-
currere ad modificationem perfectionemque'vocis; (quae in
⇁
larynge et glottide incipit ⋅ quidem , sed non omnimode
⋅⋅
ibi perficitur: nam quod in illis partibus sufficiens habea-
tur organum qain prorsus necessaria sint oris et linguae or-
,gana ad exhibendum aliquo modo sonum animalis pro-
prium ,,apparet ex eo quod grues et anseres , abscisso308
capite , ex ventris compressione sonos edere possint iis si
miles, quos viventes edebant. Ad modificationem igitur per
fectionemque vocis in larynge et glottide inchoatae caete
ra concurrunt : neque haec modificatio in mera reflexione
consistit, sed in resonantia proportionata tono soni a
glottide emissi .
Ad articulatarum vocum formationem quod attinet ,
ea praecipue a mota linguae et labiorum repeti solet :
inter caeteros P. Fabri diligenter expendit quo pacto lin
gua et labia componantur ad cujusque syllabae efforma
tionem .

144. Dices: potest sonus excitari aerem expellendo
per angustius spatium ; atque ita sibilus per labiorum com
pressionem excitatur. Ergo dicendum videtur quod ex 90
la emissione aeris per angustius glottidis spatium vox effor
inari possit quin confugiamus ad tremitum laryngis et
fibrarum glottidis ; qui tremitus effectus erit soni quin in
sonum ipsum influat.
Respondeo : etsi sonus aliquis obtineri praecise pos
sit per hoc quod ex ampliore in angustius spatium aer
cogatar transire ; attamen quae hactenus diximus suadent
tremitum laryngis et fibrarum ad vocis formationem con .
currere; attenta praecipue varietate maxima , quae in vo
cis modificatione habetur. Novimus enim et singulos ho
mines modificari quam maxime vocem , et in diversis ho
minibus quam maxime diversum esse vocis sonum . Iam ve
ro cum habeatur sibilus per solam labiorum compressio
nem , inde expulso violenter aere , exigua est hujusmodi
soni diversitas; et omnes fere homines eumdem sonum ef
ficiunt , licet in diversa intensione : ergo cum contra in
voce diversitas sit maxima et proprius cuique sit homini so
nus , ad diversam fibrarum et laryngis materiam ac ten
sionem recurrendum potius videtur. Scio equidem ab in
strumentis pneumaticis etiam materia resistente compactis
et lingula instructis magnam edi posse varietatem sono
308
capite , ex ventris compressione sonos edere possint iis si-
miles, quos viventes edebant. Ad modificationem igitur per-
fectionemque vocis in laryuge et glottide inchoatae caete-
ra concurrunt: neque haec modificatio in mera reflexione
consistit, sed in resonantia prOportionata tono soni a
glottide emissi.
Ad articulatarum vocum formationem quod attinet ,
ea praecipue a motu linguae et labiorum repeti solet:
inter caeteros P. Fabri diligenter expendit quo pacto lin-
gua et labia componantur ad cuiusque syllabae efforma-
tionem.

144. Dices: potest sonus excitari aerem eXpellendo
per angustius spatium : atque ita sibilus per labiorum com-
pressionem excitatur. Ergo dicendum videtur quod ex so-
la emissione aeris per angustius glottidis spatium vox effor-
mari possit' quin confugiamus ad tremitum laryngis et
fibrarnm glottidis; qui tremitus effectus erit soni quin in'
sonum ipsum influat.
Respondeo: etsi sonus aliquis obtineri praecise pos-
sit per hoc quod ex ampliore in angustius spatium aer
cogatur transire; attamen quae hactenus diximus suadent
tremitum laryngis et librarum ad vocis formationem cou-
1:11rrere; attenta praecipue varietate maxima, quae in vo-
cis modificatione habetur. Novimus enim et singulos bo-
mines modificari quam maxime vocem, et in diversis ho-
minibus quam maxime diversum esse vocis sonum. Iam ve-
ro cum habeatur sibilus per solam labiorum compressio-
nem ,
inde expulso violenter aere , exigua est huiusmodi
soni diversitas; et omnes fere homines eumdem sonum ef-
ficiunt
,
licet in diversa intensione : ergo cum contra in
voce diversitas sit maxima et proprius cuique sit homini so-
nus , ad diversam librarum et laryngis materiam ac ten-
sionem recurrendum potius videtur. Scio equidem ab in-
strumentis pneumaticis etiam materia resistente compactis
et lingula instructis magnam edi posse varietatem sono-309
recessum
rum , atque ad instrumenta ista referri organum vocis ab
auctoribus non paucis. Verum non video quomodo glotti
dis fibrae se habeant ad vocis organum perinde ac lin
gula : si non ita haec movetur , ut epistomium alterne
aperiatur claudaturque ; licet ea citissime oscillet , nullus
inde prodibit sensibilis sonus . Iam vero glottidis fibrae
non sic oscillant , ut per mutuum accessum et
alterne claudatur aperiaturque ipsius glottidis foramen .
In glottidis fibris aeris irrumpentis impetu ad tremitum
concitalis auctores aliqui cum Ferreinio organum vocis ma
xime constituunt , illudque ad instrumenta fidicularia po
tissime revocant , minime considerantes quod hujusmodi fi
brae careant ea longitudine et crassitie , quae necessaria
esset ad graves atque ingentes humanae vocis tonos effi
ciendos,

145. Quaeres 

1.º qui sit defectus, ob quem mutis loquela deest. Cum plerique muti sint, quia sunt a nativitate surdi, quique proinde cum non possint alios loquentes audire ne loqui quidem discunt unquam; attamen sunt qui auditu pollent, at loquendi facultate destituuntur. Vitium in his multiplex esse potest; aut ex humorum nimietate et crassitie; aut ex fibrarum inelasticitate, qua etiam fit ut, timore insolito obrigescentibus fibris, vox impediatur in iis qui caeterum muti non sunt; vel ex nimia linguae turgescentia; vel alio vitio: adeoque non desunt exempla mutorum arte medica, aut etiam solius naturae auxilio loquelam adipiscentium. 

2.º Cur aves aliquae humanam vocem aemulentur, pleraeque non item. In psitlacis diligenter rem inspexit Kircherus, atque animadvertit pro corporis quantitate os satis magnum et excavatum, maxillas turgentes, linguam maxime flexilem, et rostrum superius contra indolem aliarum avium mobile; unde bruta pro majore vel minore aptitudine ad oris dilatationem, flexilitatem linguae, labiorum, vel rostri modificationem apta erunt ad sovum humanae vocis imitandum. Picae io
309
rum , atque ad instrumenta ista referri organum, vocis. ab
auctoribus non paucis. Verum non video quomodo glottidis fibrae se habeant ad vocis organum perinde ac lingula: si non ita haec movetur, ut epistomium alterne aperiatur claudaturque; licet ea citissime oscillet, nullus inde prodibit sensibilis sonus. Iam vero glottidis fibrae non sic oscillant, ut per mutuum accessum et recessum alterne claudatur aperiaturque ipsius glottidis foramen. In glottidis fibris aeris irrumpentis impetu ad tremitum concitatis auctores aliqui cum Ferreinio organum vocis maxime constituunt, illudque ad instrumenta fidicularia potissime revocant, minime considerantes quod huiusmodi fibrae careant ea longitudine et crassitie, quae necessaria esset ad graves atque ingentes humanae vocis tonos efiiciendos. 

145. Quaeres ↿∙∘ qui sit defectus , ob quem mutis
loquela deest. Cum plerique muti sint, quia sunt a nati-
vitate surdi
, quique proinde cum non possint alios loquen-
tes audire ne loqui quidem discunt unquam; attamen sunt
qui auditu pollent, at loquendi facultate destituuntur. Vitium
in his multiplex esse potest: aut ex humOrum nimietate
et crassitie; aut ex fibrarum inelasticitate , qua etiam fit
ut
,
timore insolito obrigescentibus fibris
, vox impedia-
'tur in iis qui caeterum muti non sunt; vel ex nimia lin-
guae turgescentia; vel alio vitio: adeoque non desunt exem-
pla mutorum arte medica
,
aut etiam solius naturae auxi-
lio loquelam adipiscentium. 2.(' Cur aves aliquae humanam
vocem aemulentur , pleraeque non item. In psittacis dili-
genter rem inspexit Kircherus , atque animadvertit pro
corporis quantitate os satis magnum et excavatum, maxil-
las turgentes, linguam maxime flexilem , et rostrum su-
perius contra indolem aliarum avium mobile; unde bru-
te pro majore vel minore aptitudine ad oris dilatationem
,
⋅ Hexilitaïem linguae ,
labiorum , vel rostri modificationem
apta'erunt ad sonum humanae vocis imitandum. Picae- iu-310
a
ter caeteras aves , et corvi antiquitus etiam ad voces hu
manas formandas instituebantur. 

3. ° An verum sit quod vox ita procreari possit ut infra laryngem genita videatur ,
ideoque sonus audiatur non tanquam ex ore , sed tanquam
ex alvo veniens. Ita de facto Pythonissae , quas Ethnicorum
historiae , et sacra scriptura etiam memorat, loquebantur:
aliquos tamen sine ulla suspicione daemoniaci operis ven
triloquos esse exemplis pluribus compertum est. Sicut enim
loquitur aerem expellendo , qui in larynge et glottide vo
cem excitat ; ita fieri potest ut aerem ore ac naribus at
lrahendo in gloutide item parem molum excitemus , sicque
non ex ore sed infra laryngem vox orta videatur,
प
be
AL

=== De auditus organo. ===
146. Externa auris pars palula est; et ex cartilagine intus concava atque elastica constat; quae in concham
sea cavitatem referentem conchae figuram desinit. Inser
vit ad colligendas uudas soni : hinc quasi natura duce qui
minus acuto pollet auditu , aut ad vocein nimis e lon
ginquo attendit , manum ad aures applicat , ut eo pacto
plures colligat aeris undas. Externa haec auris pars , quae
auricula simpliciter dicitur , musculis adornatur , quorum
ope sunt aliqui homines qui auriculam ad libitum mo
vent ; oves autem , equi et bruta alia multo facilius :
adnotant nonnulli Analomici ila necessariam esse exter
banc partem ut sonorus lenius allabatur
in internas cavitates, ut nonnisi confusa et quasi cum
inurmure fluentis aquae audiant ii, quibus auriculae abscis
sau sint. Animadvertendum tamen reptilia et aves hoc ex
lerno adminiculo carere.
Ad fundum conchae incipit meatus auditorius , qui est
canaliculus aliquanto tortuosus ; et ex majori latitudine in
minorem paullatim coarctator. Ita factum notat Val
9
nam aer 1
1
310.
ter caeteras aves , 'et corvi antiquitus etiam ad voces hn-
manas formandasinstituebantur. 3.0 An verum sit quod
'vox ita proci-cari possit ut infra laryngem genita videatur,
ideoque sonus audiatur non tanquam ex ore , sed tanquam
ex alvo veniens. Ita de facto Pythonissae , quas Ethnicorum
historiae ,
et sacra scriptura etiam memorat, loquebantur:
aliquos tamen sine ulla suspicione daemoniaci operis ven-
triloquos esse exemplis pluribus compertum est. Sicut enim
loquitur aerem expellendo , qui in larynge et glottide vo-
cem excitat; ita fieri potest ut aerem ore ac naribus at-
trahendo in glottide item parem motum excitemus, sicque
non ex ore sed infra laryngem vox orta videatur,
De auditu: organo.
146. Externa auris pars patula est, et ex cartilagi-
ne iutus concava atque elastica constat; quae in concham
sen cavitatem referentem conchae figuram desinit. Inser-
vit,ad colligendas undas soni: hinc qnasi natura duce qui
minus acuto pollet auditu
,
aut ad vocem nimis e lon-
giuquo attendit, manum ad aures applicat , ut eo pacto
plures colligat aeris undas. Externa haec auris pars, quae
auricula simpliciter dicitur
, musculis adornatur , quorum
0pe sunt aliqui homines qui auriculam ad Hibitum mo-
vent; oves autem
, equi et bruta alia multo facilius :
adnotaut nonnulli Anatomici itaqnecessariam esSe exter-
nam lianc partem ut aer sonorus lenius allahatur
in internas cavitates, ut nonnisi confusa et quasi- cum
murmure fluentis aquae audiant ii, quibus auriculae abscis-
sae sint. Animadverteudum tamen reptilia et aves hoc ex-
terno adminiculo carere.
Ad fundum conchae incipit meatus auditorins , qui est
canaliculus aliquanto tortuosus; et ex maiori latitudine in
minorem paullatim coarctatur. Ita factum notat Val-311
sa
salva at sonus intendatur magis , sicuti in recurvis lubis
a surdastris adhiberi solitis intenditur ; alii potius ad im
minuendum aeris impetum , ne in auris interiora fortius
impellat , has tortuositates in organo auditus a natura in
stilutas putant. In auditorio meatu humor quidam amarus
ac viscosus habetur , seu cerumen ; exsudat e glandulis
quas sebaceas vocant , et institutum est ut minima ani
malcula ab ingressu ad interiora auris arceantur .
Ad finem hujus canalis habetur membrana tympani:
haec membrana tenuissima , sicca , pellucida et valde ela
stica , obtensa est annullo , qui tamen totum circuitum
non complet ; et fere ad similitudinem pellis tympani mi
litaris cavitatem interiorem superambit : non est recte exten
sed curva nonnihil ; coacava scilicet respectu auris
externae , convexa ad partes internas . Fuit acerrima quae
stio , an membrana tympani omnem communicationem in
ter externam internamve aurem excludat , an contra per
via sit aeri externo. Argumentum pro communicatione va
lidum est , quod aliqui fumum ore exceptum per aurem
emittunt ; neque id semper imposturae vertendam est ,
ut compertum fuisse Nolletus ait a viro , cni Academia
regia jussum fecerat facti veritatem explorare. Argumen
tum contra communicationem est , quod Valsava , immis
so in aurem internam hydrargyro , quantumvis excute .
retur , nihil unquam per externam aurem defluxit ; quam
quam reponere soleat qui communicationem tuetur , quod
in cadaveris organo non est necesse partium structuram sal
vari.
Post pellem tympani habetur cavitas aere plena , quae
capsula dicitur , quaeque cum membrana praedicta tym
panum constituit. In hac sunt quatuor ossicula quae ap
pellantur malleus , incus , os orbiculare , stapia. Alii tria
tantum numerant , omisso osse orbiculari , vel quia, cum
ompium humani corporis ossiam minimum sit , adeo ut
non superet dimidium grani millii , animadversionem fu
31↿⋮ salva ut sonus intendatur magis , sicuti in recurvis tubis
'a snrdastris adhiberi solitis intenditur; alii potius ad im-
miuuendum aeris impetum , ne in auris interiora fortius
impellat, has tortuositates in organo auditus a natura in-
stitutas putant. In auditorio meatu humor quidam amarus
ac viscosus habetur , seu cerumen; exsudat e glandulis,
quas sebaceas vocant , et institutum est ut minima ani-
malcula ab ingressu ad interiora auris arceantur.
Ad finem hujus canalis habetur membrana tympani:
haec membrana tenuissima , sicca , pellucida et valde ela-
stica
, obtensa est annnllo , qui tamen totnm circuitum
non complet; et fere ad similitudinem pellis tympani «mi—
litaris cavitatem interiorem superambit: non est recte exten-
sa , sed 'curva nonnihil : concava scilicet respectu auris.
externae
, convexa ad partes internas. Fuit acerrima qnae-
stio , an membrana tympani omnem communicationem in-
ter externam internamve aurem excludat , an contra -per-
via sit aeri externo. Argumentum pro-communicatione va-
lidum est
, quod aliqui fumum ore exceptum per aurem
emittunt; neque id semper imposturae vertendam est,
ut compertum fuisse 'Nolletus ait a- viro, cni Academia
regia iussum fecerat facti veritatem explorare. Argumen-
tum eontra communicationem est
, quod Valsava ,
immis-
so in aurem internam hydrargyro , quantumvis excute-
retur, nihil unquam per externam aurem defluxit; quam-
quam reponere soleat qui communicationem tuetur
, quod
in cadaveris organo non est necesse partinm structuram sal-
var]. '
Post pellem tympani habetur- cavitas aere plena , quae
capsula dicitur
, quaeque eum membrana praedicta tym-
panum constituit. In hae sunt quatuor ossicula quae ap-
pellantur mallens
.
incus , os orbiculare
, stapia. Alii tria
tantum numerant
, omisso osse orbiculari, vel quia, cum
omnium humani uerporis ossium minimum sit, adeo ut
non superet dimidium grani millii
, animadversionem fu-312
1
1
gerit : vel quia ita adhaeret slapiae et incudi , at cum al
tero ex his confundi potuerit, Circa haec ossicula nolan
dum , quod ejusdem magnitudinis sint tam in infante quam
in adulto viro contra aliarum omnium corporis partium in- .
dolem , quae aetatis progressu augentur. Hoc ideo factum
esse docet Valsalva , ne augmento partium auditui inservien
tium alia sit sonorum ratio adulla aetate ac fuit ab ini
tio ; et ideas gravis atque acuti quas pueri imbibimus, ma
tare aetate proficiente cogamur.
In tympani cavitate habetur canalis quidam seu lu
ba Eustachiana dicta ab ipsius inventore : per hanc tu
bam ab interna auris cavitate obtinet communicatio cum
cavitate oris; desinit enim ista tuba ad radices uvae; quem
prope locum etiam interiora nasi communicant : hujus tu
bae ope fit , ut sonus ex oris cavitate auri communicetur,
ideoque qui dentibus stringit corpus resonans sobum au.
dit etiam auribus impeditis ; et surdastri hiante ore so
nos excipere solent , ut tali pacto juvelur melius auditio.
Praeter foramen ex quo tuba Eustachiana procedit ,
duo alia babentur in tympani cavitate foramina , quorum
unum dicitur fenestra ovalis , allerum fenestra rotunda.
Feuestra ovalis basi slapiae occluditur, rotunda solo mem
branulae tegumento obtegitur. .
Ex tympani cavitate per duo praedicta foramina , ovale
scilicet ac rotundum , itur in labyriothum , qui - est inte
rior alia cavitas in osse petroso ulterius excavata , et quo
dam liquido plena : in hac tres partes distingui solent ;
prima est vestibulum labyrinthi ; secunda constat tribus
ossiculis semicircularibus , quibus nomen labyrinthi pecu
liarins aliqui tribuunt ; tertia est cochlea seu limax, quae
ex osse constat in cochleae modum conlorto duos gyros
cum dimidio faciente. Elsi cochlea unus canalis videri
possit , est lameu revera duplex : dividitur enim secun
dum longitudinem medio segmento , parim osseo , partim
membranaceo , quod dicitur lamina spiralis. Cochlea in
1
1
1
i
1
1
1
•
2
0
1
1
1
.
312
gerit: vel quia ita adhaeret stapiae et incudi , at cum al-
tero ex his confundi potuerit. Circa haec ossicula notan-
dum , quod eiusdem magnitudinis sint tam in infante quam
in adulto viro contra aliarum omnium corporis partium in--
dolem, quae aetatis progressu augentur. Hoc ideo factum
esse docet Valsalva, ne augmento partium auditui inservien-
tium alia sit sonorum ratio adulta aetate ac fuit ab iui-
tio; et ideas gravis atque acuti quas pueri imbibitüus, mu-
tare aetate proficiente cogamur.
ln tympani cavitate habetur canalis quidam seu tu-
ba Eustachiaua dicta ab ipsius inventore: per hanc tu-
bam ab interna auris cavitate obtinet communicatio cum
cavitate oris; desinit enim ista tuba ad radices uvae; quem
prope locum etiam interiora nasi communicant :-huius tu-
bae upe fit
, ut sonus ex oris cavitate auri .communicetur,
ideoque qui dentibus stringit corpus resonans sonum su-
dit etiam auribus impeditis ; et surdastri hiante ore so-
nos excipere solent , ut tali pacto iuvetur melius auditio-
Praeter foramen ex quo tuba Eustachiana procedit,
duo alia habentur in tympani cavitate foramina , quorum
unum dicitur fenestra ovalis, alterum fenestra rotunda.
Feuestra ovalis basi stapiae occluditur, rotunda solo mem-
branulae tegumento obtegitur. .
Ex tympani cavitate per duo praedicta foramina
, ovsle
scilicet ac rotundum ,
itur in labyrinthum
, qui-est inte-
- rior alia cavitas in esse petroso ulterius excavata
,
et quo-
dam liquido plena: in hac,tres partes distingui solent;
prima est vestibulum labyrinthi ; secunda constat tribus
ossiculis semicircularibus , quibus nomen labyrinthi pecu-
liarius aliqui tribuunt; tertia est cochlea seu limax, quae
ex osse constat in cochleae modum contorto duos gyros
cum dimidio. faciente. Etsi cochlea unus canalis videri
possit ,
est tamen revera duplex: dividitur enim secun-
dum longitudinem medio segmento, partim osseo , partim
membranacea , quod dicitur lamina spiralis. Cochlea in313
avibus deest , si vera refert Boyle ; at ipsemet notat de
fectum hunc suppleri per cavitatem oblongam instar sacci.
Plures rami nervei per foramina perexigua ex eo ,
qui dicitur uervus auditorius , propagati per totam fere
aurem distribuuntur : in labyrinthum per quinque fora
mina ingrediuntur Gibrae nerveae , et ejus cavitatem inves
tiunt ; sed de nervis totum auris organum permeantibus
non vacat disserere. Id unum adnoto ex Mairano , spira
lem laminam fibrillis ita instructam esse ut quemadmo
dum ipsa ascendens ad cochleae apicem . semper angustior
fit , ita fibrae ipsae breviores semper evadunt.
147. Quaenam vero ex hactenus descriptis auris par
tibus pro praecipuo atque immediato auditionis organo sta
tueuda est ?. Aliqui membranam tympani assignarunt : at
experientia constat quod hac membrana lacerata vel erepta
adhuc manet auditio aliqua. Alii inepte in ossiculis intra
capsulam contentis organom auditus statuerunt , et sonum
ab anima immediate perceptibilem ex horum collisione
nasci affirmarunt. Praeter quam quod solus malleus in a
nimantibus quibusdam habeatur, falsum omnino est collidi
invicem haec ossicula atque inde sonum creari : adnexum
enim est caput mallei firmiter corpori incudis , et hujus
processus alter stapiae; adeoque cum aer exterous tympa
ni membranam impellit, omnia per modum unius intromit
tuntur et conjuncta simul sese restituunt ad locum pristi
num. Magis autem absona est illorum sententia , qui in
aere per capsam et labyrinthum existente auditus organum
collocabant: aerem hunc, quem innatum, insitum, vernacu
lum, implantatum dicebant, animatum statuere non vere
bantur. Communiter nunc auditus organum collocatur in
fibrillis nerveis per aurem internam, ac praesertim intra co
chleam disseminatis. Tremores itaque a corpore
excitati communicantur membranae tympani; tum per aea
rem in tympano existentem , nec non per ossiculorum se
riem, ad parietes asque labyrinthi et praecipue ad dupli
Sonoro
21
313
avibus deest , si vera refert Boyle ;
'at ipsemet notat de-
fectum hunc suppleri per cavitatem oblongam instar sacci.
Plures rami nervei per foramina perexigua ex eo ,- qui dicitur nervus auditorius, prcpagati per totam fere
aurem distribuuntur: in labyrinthum per quinque fora-
mina ingrediuntur fibrae nerveae, et eius cavitatem inves-
tiunt; sed de nervis totum auris organum permeantibus
non vacat disserere. Id unum adnoto ex Mairano , spira-
lem laminam fibrillis ita instructam esse ut quemadmo-
dum ipse ascendens ad cochleae apicem- semper angustior
fit
,
ita fibrae ipsae breviores semper evadunt.

147. Quaenam vero ex hactenus descriptis auris par-
tibus pro praecipuo atque immediatoauditionis organo sta-
tuenda est ?. Aliqui membranam tympani assignarent : at
experientia constat quod hac membrana lacerata vel erepta
adhuc manet auditio aliqua. Alii inepte in ossiculis intra
capsulam contentis organum auditus statuerunt
,
et sonum
ab anima- immediate perceptibilem ex horum collisione
nasci affirmarunt. Praeter quam quod solus malleus in a-
nimantibus quibusdam habeatur, falsum omnino est collidi
invicem haec ossicula atque inde sonum creari: adnexum
enim est caput mallei firmiter corpori incudis ,
et huius
processus alter stapiae; adeoque cum aer externus tympa-
ni membranam impellit, omnia per modum uniua intromit-
tuntnr et coniuncta simul sese restituunt ad, locum pristi-
num. Magis autem absona est illorum sententia , qui in..
aere per capsam et labyrinthum existente auditus organum
collocabant: aerem hunc, quem innatum, insitum, vernacu-
lum, implantatum dicebant, animatum statuere non vere-
bantur. Communiter nunc auditus organum collocatur in
fibrillis nerveis per aurem internam, ac praesertim intra co-
chleam disseminatis. Tremores itaque a sonoro corpore
excitati commnnicantur membranae tympani; tum per aes-.
rem in tympano existentem, nec non per ossiculorum se-
riem, ad parietes usque labyrinthi et praecipue ad dupli-
21
is314
cem fenestram , ovalem ac rolundam , transmissi deducuntur
ad liquidum cavitate labyrinthi contenlum ; inde vero ad fi
brillas nerveas praedictas, atque ad nervum ipsum audito
rium: unde fit, ut ex lege commercii anima ad sensationem
soni determinetur. Animadvertit Mairanus, quod sicut in
instrumentis quibusdam musicis binae et binae chordae pro
tonis singulis disponuntur, ita in lamina spirali nerveae fi
brillae dispositae sint : ex quo infert huc potius spectare
organum quam ad alias auris internas partes.
148. Quaeres 1º. Cur quibusdam grata , aliis pene ni
hil, aut etiam molesta sit harmonia. Alibi ( 121 ) dictumn
est chordam upisonam facile ad tremitum concitari: aliam
item , sed difficilius prout majorem minoremve cum chor
da percussa harmonicam proportionem habet. Alert Kir
cherus aliud experimentum , quod ad rem aptum est etiam
magis : experimentum ita se habet. Quinque sumantur
scyphi vitrei.ejusdem magnitudinis et capacitatis , et unus
quidem liquore impleatur, qui acquavite dicitur; alter vi
no ; tertius aqua puriori; quartus aqua communi ; quintus
crassiori aqua: tum ora cujusque poculi confricando sonus
quam fieri potest acntissimus excitetur. In primo quidem
• scypho spiritus ille maxime subsultabit; vinum moderatam su
bibit concitationem ; adhuc moderatior erit molus purio
ris aquae, et ita porro . Ex his intelligitur quomodo variae
in variis hominibus partium corporis commotiones orian
tur e sono. Cum autem animi molus, in quibus voluptas
consistit vel molestia , pendeant ex partium corporis affe
ctionibus; iis gratissima accidere poterit harmonia, quibus
ea solidorum ac fluidorum constitutio est , ut in iisdem com
motio consequatur impressionem factam in organo auditus
satis . vivida et animi moribus cum voluptate conjunctis ex
citandis apta: ii erunt ad harmoniam indifferentes, in qui
bus impressionem factam in organo auditus vix ulla con
sequitur alteratio solidarum fuidarumve corporis partium
quae pariat animi motus vel consonos, vel incongruos: iis
314
cem fenestram, ovalem ac rotundam, transmissi deducuntur
ad liquidum cavitate labyrinthi contentum; inde vero ad fi-
brillas nerveas praedictas, atque ad nervum ipsum audito-
rium: nnde fit, ut ex lege commerciianima ad sensationem
aoni determinetur. Animadvertit Mairanus, quod sicut in
instrumentis quibusdam musicis binae et binae chordae pro
tonis singulis disponuntur, ita ip lamina spirali nerveae fi-
brillae dispositae sint: ex quo infert huc potius spectare
organum quam ad alias auris internas partes.
148. Quaeres 1&quot;. Cur quibusdam grata, aliis pene ni-
hil, aut etiam molesta sit harmonia. Alibi (121 ) dictum
est chordam unisonam facile ad tremitum concitari: aliam
item, sed difficilius prout majorem minoremve cum chor-
da percussa harmonicum proportionem habet. Affert Kir-
cherus aliud experimentum, quod .ad rem aptum est etiam
magis : experimentum ita se habet. Quinque sumantur
scyphi vitrei-ejusdem magnitudinis et capacitatis, et unus
quidem liqum'e impleatur, qui acquavite dicitur; alter vi-
no; tertius aqua PUI'lOl'i; quartus aqua communi; quintus
crassiori aqua: tum ora cujusque poculi confricando sonus
quam fieri potest acutissimus excitetur. In primo quidem
scypho spiritus ille maxime subsultahit; vinum moderatam su-
bibit concitationem; adhuc moderatior erit motus purio-
ris aquae, et ita porro. Ex his intelligitur quomodo variae
in variis hominibus partium corporis commotiones orian-
tur e sono. Cum autem animi motus, in quibus voluptas
consistit vel molestia, pendeant ex partium corporis affe-
ctionibus; iis gratissima accidere poterit harmonia, quibus
easolidorum ac fluidorum constitutio est, ut in iisdem com-
motio consequatur impressionem factam in organo auditus
satis.vivida et animi motibus cum voluptate conjunctis ex-
citandis apta: ii erunt ad harmoniam indifferentes. tu qui-
bus impressionem factam in Organo auditus vix ulla con-
sequitur alteratio solidarum fluidarumve corporis partium ∙
quae pariat animi motus vel consouos, vel incongruos: iis1
1
315
denique molestia etiam accidet, quibus ex impressione ner
vorum acusticorum contingat incongrua motuum alteratio
in partibus corporis ad pracfatos animi molus inservienti
bus: quo fit etiam mechanice ut alii aliis sonorum gene
ribus vel delectentur magis, vel contra. Hanc tamen me
chanicam causam non arbitror esse sufficientem atque adae
quatam: admittenda est causa ex rationali natura hominis
pendens. Consistit harmonia in proportione illa , quam so
ni habent inter se ; unde fit ut in organo auditus vibra
tiones diversi generis, aliae frequentiores, aliaė tardio
res efficiantur: dum vibrationes istae organum anditus af
ficiunt, mens easdein comparat inter se, earumque propor
tionem animadvertit : si haec proportio ejusmodi sit ut fa
cile possit a mente percipi, et vibrationes facile comparari
queant, ex hoc gaudet mens; si confusa et inordinata vi
brationum sit comparatio , neque has mens facile con
ferre inter se potest, obruelur taedio: et quia imperi
tas in musica facilius comparat simpliciores consonantias quam
magis compositas, ideo musica planissima vulgo arridet ; qui
vero periti sunt et copiosioribus compositiouibus assueti, vix
patiuntur musicam nimis simplicem: item cum ex consue
tadine pendeat ut aliquas harmonicas proportiones faci
lius mens assequatur quam alias ; inde oritur at volu
ptas ex eo musices genere major sit, cui quis sit assue
tus; adeo ut ex hoc capite diversis nationibus diversae
placeant musicae species. Porro ab exercitatione etiam
profluit ut gratior accidat musica , etiam qua ex parte
mechanice voluptatem parit; ex assuetudine enim in fi
brillas nerveas docilitas inducitur ad recipiendas facilius
impressiones harmonicas.
2º. Cur duabus auribus unus idemque sonus au
diatur. Communis responsio est hujusmodi : cum in utra.
que aure creetur simillima impressio; non duplicem , sed
voam sensationem ab anima haber¡ necesse est. Qua in
re scite animadvertit Valsalva , summa industria provisum
315
denique molestia etiam accidet, quibus ex impressione ner-
vorum acnsticorum contingat incongrua motuum alteratio
in partibus corporis ad praefatos animi motus inservienti-
bus: quo fit etiam mechanica ut alii aliis sonorum gene-
ribus vel delectentur magis, 'vel contra. Hanc tamen me-
chanicam causam non arbitror esse sufficientem atque adae-i
quatam: admittenda est causa ex rationali natura hominis
pendens. Consistit harmonia in praportione illa, quam so-
ni habent inter se; unde fit ut in organo auditusvibraP-
tiones diversi generis, aliae frequentiores, aliae tardio-
res efficiantur: dum vibrationes istae organum auditus af-
Hciunt, mens easdem comparat inter se, earumque propor-
tionem animadvertit: si haec proportio ejusmodi sit ut fa-
cile possit a mente percipi, et vibrationes facile comparari
queant, ex hoc gaudet mens; si confusa et inordinata vi-
bratiouum sit comparatio , neque has mens facile con-
ferre inter se potest, obruetur taedio: et quia imperi-
tus in musica facilius comparat simpliciores consonantias quam
magis compositas, ideo musica planissima vulgo arridet ; qui
vero periti sunt et c0piosioribus compositionibus assueti, vix
patiuntur musicam nimis simplicem: item cum ex consue-,
tudine pendeat ut aliquas harmonicas preportiones faci-
lius mens assequetur quam alias; inde oritur ut volu-
ptas ex eo mus1ces genere major sit, cui quis sit assue-
tus; adeo ut ex hoc capite diversis nationibus diversae
placeant musicae species. Porro ab exercitatione etiam
profluit ut gratior accidat musica, etiam qua ex parte
mechanica voluptatem parit; ex assuetudine enim in fi-
brillas nerveas docilitas inducitur ad recipiendas facilius
impressiones harmonicas. ⋅
20. Cur duabus auribus unus idemque sonus au-
diatur. Communis responsio est huiusmodi: cum in utra-
que aure creetur simillima impressio; non duplicem, sed,
unam sensationem ab anima haberi necesse est. Qua in
re scite animadvertit Valsalva, summa industria provisum316
fuisse a natura ut in utraque aure quam simillima es
sent organa omnia ; adeo ut, dum in diversis hominibus
structura partium nonnihil variat, in eodem tamen bomi
mine nulla prorsus sit utriusque auris vel minima variatio .
Notetur illud : quemadmodum eadem chorda varios
tonos varia tensione edere potest, ita membrana tympani
lenditur diversimode ut variis tonis aple accomodetur ;
eapropter manubrium mallei eidem adnexum est, et ba
sis stapiae eodem modo membranae fenestrae ovalis: ten
sio autem et relaxatio membranae, nobis insciis , potest na
turaliter fieri. Itaque, ut primae vibrationes membranam
feriunt, admonita natura organum opportuna tensione aptat
ut respondens tonus in successivis vibrationibus fiat ma
gis vel minus sensibilis.
316
fuisse a natura ut in utraque aure quam simillima es-
sent organa omnia; adeo ut, dum in diversis hominibus
structura partium nonnihil variat, in eodem tamen homi-
miue nulla prorsus sit utriusque auris vel minima variatio.
Notetur illud: quemadmodum eadem chorda varios
tonos varia tensione edere potest, ita membrana tympani
tenditur diversimode ut variis tonis apte accomodetur;
eapropter manubrium mallei eidem adnexum est, et ba-
sis stapiae eodem modo membranae fenestrae ovalis: ten-
sio autem et relaxatio membranae, nobis insciis,potest na-
turaliter fieri. Itaque, ut primae vibrationes membranam
feriunt, admonita natura organum opportuna tensione aptat
ut respondens tonus in successivis vibrationibus fiat ma-
gis vel minus sensibilis.INDEX
RERUM QUAE IN PRIMO VOLUMINE CONTINENTUR.
MECHANICA E PRINCIPIA
Notiones praeambulae. pag. 1 .
Molus uniformis et varius : velocitas et quantitas mo
tas in motu uniformi. num . 1 .
Corporum indifferentia ad motum et ad quietem: quid
vires : quid earum aequilibrium ; et quomodo repraesen
tentur sive per lineas rectas, sive per numeros . n. 2, 3, 4.
Principiom motus • relativi : vires sunt ut quantitates
motus , n. 5, 6 .
Principium actionis et reactionis : mutatio status in
corpore haud repente gignitur a viribus extrinsecis , sed
per gradus indefinitae attenuationis capaces: quid vires in
stanlaneae et continuae. n. 7 .
De virium compositione et resolutione , deque earum
momentis et aequilibrio : aliquid quoque notatur de
vecte, axe in peritrochio, trochlca etc. pag. 6.
Compositio virium materiali puncto applicatarum: ae
quilibrium: varia circa virium resolutionem .. n. 8. 9. 10.
⋅ N D EX
RERUM QUAE IN ramo VOLUMINE CONTINENTUR.
' MECHANICAE PRINCIPIA
∙
W
⋅∙
Nott'ones praeambulae. pag. 1.
Motus uniformis et varius: velocitas et quantitas mo-
tus in motu uniformi. . . . . . . . . . num-1.
Corporum indifferentia ad motum et ad quietem: quid
vires: quid earum aequilibrium; et quomodo repraesen-
tentur sive per lineas rectas, sive per numeros. n.2, 3, 4.
. Principium motus 'relativi: vires sunt ut quantitates.
motus. ∙ ∙ ∙ ∙↴ ∙ ∙
.' ∙ ∙ ∙ ∙ ∙ ∙ n. 5, 6.
Principium actionis et reactionis: mutatio status in
corpore haud repente gignitur a viribus extrinsecis ,
sed
per gradus indefinitae attenuationis capaces: quid vires in-
stantaneae et-continuae.
. . . . . . . . . n. 7.»
De virium compositione et resolutione , deque earum
momentis et aequilibrio : aliquid quoque notatur de
vecte, axe in peritrochio, trochlea etc. pag. 6.
Compositio virium materiali puncto applicatarum: ae-
quilibrium: varia circa virium resolutionem.. n. 8. 9. 10.318
Compositio duarum virium extremis rectae rigidae punctis
applicatarum, et in eodem plano jacentium: aequilibrium circa
immobile punctum: principiam velocitatum virtualium in ordi
ne ad istiusmodi vires: momenta virium quoad punctum ( M) :
momentum resultantis aequatur summae ex momentis com
ponentium si hae in eamdem plagam circa ( M ) nituntur
movere puncta , quibus applicitae sunt; aequatur differentiae
si nituntur movere in plagas contrarias. n. 10. 10. 20.30,
Haec ipsa extenduntur ad quemvis numerum virium in
dato plano jacentium.
n. 10. 4º , 5º , 6º.
Binae vires haud jacentes in eodem plano nequeunt
ad unicam vim aequipollentem traduci : vires in aequili
brio constitutae; sollicitantesque vel solidum liberumque cor
pus, vel solidam corpus mobile duntaxat circa punctum fi
xum, vel solidum corpus mobile tantummodo circa asem
fixum : momenta quoad axem . n. 10: 70. ... 10 °.
Vires parallelae: vis inde resaltans: earum centrum :
momenta quoad planum: respondens theorema n . 11 , 12 ,
13. 10. 2º. 3º.
Parallelarum virium systema consistit in aequilibrio sub
duabus conditionibus simul explendis; altera est ut evane
scat earum summa; altera ut evanescat summa ex earum
momentis in ordine ad duo plana ipsis viribus paral
lela . n. 13. 4º. 5º.
.
Etsi vires non sunt parallelae, possunt tamen rednci
ad terna ejusmodi systemata, quorum primum coalescat ex
viribus parallelis axi OZ, secundum ex viribus jacentibus
in plano XOY simulque parallelis axi Qy, tertium ex vi
ribus agentibus juxta axem OX: aequilibrii conditiones quoad
systemata rigida viribus minime parallelis sollicitata; 1º , in
318
Compositio duarum virium extremis rectae rigidae punctis
applicatarum,etin eodem plano jacentium: aequilibrium circa
immobile punctum: princi pium velocitatum virtualium in ordi-
ne ad istiusmodi vires: momenta virium quoad punctum (M):
momentum resultantis aequatur summae ex momentis com-
ponentium si hae in eamdem plagam circa (M ) nituhtur
movere puncta, quibus applicitae sunt; aequatur differentiae
si nituntur movere in plagas contrarias. n. 10. 10. 2230.
∙ Haec ipsa extenduntur ad quemvis numerum virium in
dato plano jacentium.
. . . . . . n. 10. 40. 50. 6&quot;.
Binae vires haud jacentes in eodem plano nequeunt
ad unicam vim aequipollentem traduci : vires in aequili-
brio constitutae; sollicitantesque vel solidum liberumque cor-
pus, vel solidum corpus mobile duntaxat circa punctum fi-
xum, vel solidum corpus mobile tantummodo circa axem
fixum: momenta quoad axem. .'
. . n. 10: 70. 10&quot;-
Vires parallelae: vis inde resultans: earum centrum:
momenta quoad planum: respondens theorema &quot;. 11, 121
13. 10. 20. 30.
Parallelarum virium systema consistit in aequilibrio sub
duabus conditionibus simul explendis; altera est ut evane-
scat earum summa; altera ut evanescat summa ex earum
momentis in ordine ad duo plana ipsis viribus paral-
lela.
. . . . . . . . . . . . . n.13.4&quot;.5'-
Etsi vires non sunt parallelae, possunt tamen -reduci
ad terna eiusmodi systemata. quorum primum coalescat ex
viribus parallelis axi OZ, secundum ex viribus jacentibus
in plano XOV simulque parallelis axi QT, tertium ex vi-
ribus agentibus juxta axem OX: aequilibrii conditiones quoad
systemata rigida viribus minime parallelis sollicitata;1&quot;.in319
hypothesi systematis liberi; 2 °. in hypothesi systematis de
tenti puncto fixo; 3º . in hypothesi systematis detenti axe
fixo: conditio explenda ut plures vires minime parallelae
traduci possint ad unicam aequipollentem . n. 13. 6 ... 11º.
Duo solida corpora , datis viribus sollicitata , sese in
vicem aeque premendo apud datum mutui contaclus pan
ctum manent in aequilibrio : determinatur istiusmodi pres
sionis magnitudo. n. 13. 12 .
Solidum corpus , datis viribus sollicitatum, detinetur
duobus punctis fixis, sumptis in axe v. gr. OZ: determi
nantur pressiones exercitae in puncta illa juxta coordi
nalos axes OX, OY, OZ. n. 13. 13 .
Exempla aequilibrii in quibusdam machinis, praeci
so attritu : aequilibrium punctoruni materialium juncto
rum flis determinatae quidem longitudinis sed mobili
bas circa data puncta. n. 14. 15. 16 .
De centro gravitatis. pag. 33.
Varia de vi gravitatis, de corporum densitate , deque
specifica eorum gravitate: linea directionis. n . 17 , 18 , 19.
Generales formulae determinantes centrum gravita
tis: inveniri potest ratione mechanica: peculiari metho
do determinalur in triangulo et pyramide triangulari, n. 20.
De corporum collisione. pag . 37 .
Normalis collisio : 1º. corporum non elasticorum : 2 ° .
corporum perfecte elasticorum : 3º . corporum imperfe
cte elasticorum . n. 21 , 22, ... 25 .
319
hypothesi systematis liberi: &quot;. in hypothesi systematis de-
⋅ teuti puncto fixo: 30. in hypothesi systematis detenti axe
fixo: conditio explenda ut plures vires minime parallelae
traduci possint ad unicam aequipollentem. n. 13. 60... 1'l0.
Duo solida corpora, datis viribus sollicitata, sese in-
vicem aeque premendo apud datum mutui contactus pun-
ctum manent in aequilibrio: determinatur istiusmodi pres-
sionis magnitudo. . . . . . . . . . n. 13. 120.
Solidum corpus . datis viribus sollicitatum. detinetur
duobus punctis fixis, sumptis in axe v- gr. OZ: determi-
nantur pressiones exercitae in puncta illa iuxta coordi-'
natos axes OX, 0ï,OZ. . . . ∎∙ ∙ ∙ n.13.130.
Exempla aequilibrii. in quibusdam machinis, praeci-
so attritu : aequilibrium punctorum materialium iuncto-
rum filis determinatae quidem longitudinis sed mobili-
bus circa data puncta.
. . . . . . n.14.15.'16.
De centro gravitatis. pag. 33.
Varia de vi gravitatis, de corporum densitate.deque
specifica eorum gravitate: linea directionis. n. 17, 18, 19.
Geuerales formulae determinantes centrum gravita-
tis: inveniri potest ratione mechanica: peculiari metho-
do determinatur in triangulo et pyramide triangulari. n. 20-
Dä corporum collisione. pag. 37-
Normalis collisio: 10. corporum non elasticorum: 2&quot;.
corporum perfecte elasticorum : 3&quot;. corporum imperfe-
cte elasticorum. . .
-
. . . . . n.21,22,...25.320
Obliqua eorumdem corporum collisio. n . 26.
De motu rectilineo utcumque vario. pag. 42
Praemittantur nonnulla ex analysi infinitesimali, e
jusque ad res geometricas applicatione. n. 27. 10.2 ... 300.
Formulae spectantes ad motum rectilineum utcumque
varium : formulae quoad motum rectilineum uniformiter
varium: vis acceleratrix : vis motrix. n. 28.
Formulae pertinentes ad motum rectilineum utcum
que varium applicantur ad materiale punctum sollicita
tum vi acceleratrice, quae sit distantiae a dato centro pro
portionalis. n. 29.

De verticali gravium descensu atque ascensu . pag.
65.
Formulae, legesque huc spectantes: motus gravium in
machina Atwoodi. n . 30, 31 , 32 .
Quid si gravium descensus vel ascensus fiat in me
dio resistente sub ea conditione, ut resistentia medii sit pro .
portionalis quadrato velocitatis. n. 33.
De gravium descensu per plana inclinala ; de attritu ;
deque cochlea, et cuneo. pag. 71.
Formulae determinantes descensum gravium per plana
inclinata: gravium descensus per plana inclinata compara
tur cum verticali eorum descensu. n . 34, 35.
320
⋅∡ Obliqua eorumdem corporum collisio. . ∙⋅ n. 26.
De motu rectilineo utcumque uario. pag. 42
Praemittuntur nonnulla ex analysi infiuitesimali, e-
iusque ad res geometricas applicatione. n. 27. 10. 2&quot;....300.
Formulae spectantes ad motum rectilineum utcumque
varium: formulae quoad mo'tum rectilineum uniformiter
varium: vis acceleratrix: vis motrix. . . . . . n. 28.
Formulae pertinentes ad motum rectilineum utcum-
que varium applicantur ad materiale punctum sollicita-
tnm vi acceleratrice. quae sit distantiae a dato centro pro-
portionalis. ............n.ag.
!
De verticali gravium descensu atque ascensu. pag. 65.
Formulae, legesque huc spectantes: motus gravium in
machina Atwoodi. . . . . . . . . n. 30,31,32-
Quid si gravium descensus vel ascensus liat in me-
dio resistente sub ea conditione, ut resistentia medii sit pro-
portionalis quadrato velocitatis. . . . . . . n. 33-
De gravium descensu per plana inclinata; de attritu; ⇥
deque cochlea, et cuneo. ∙ pag. 71.
Formulae determinantes descensum gravium per plana
inclinata: gravium descensus per plana inclinata compara-
tur cum verticali eorum descensu. . . . n. 34, 35-321
Gravium descensus per plura plana inclinata sibi con
rigua . n. 36.
non.
Unde orialur attritus , caeteris paribus , est proportio
nalis pressioni : quomodo habeatur ratio attritus in motu
gravium per plana inclinata : grave in plano inclinato li
brandum potentia aliqua, sive habeatur ratio attritus , sive
n. 37. 10. 20 30
Aequilibrii leges in cochlea, et cuneo. n. 37. 4º. 5º.
Spectatur attritus in aequilibrio cochleae: itemque in
aequilibrio corporis ad rotatilem motum circa fixum cy
lindrum sollicitati : in machinis praeter resistentiam ex at
tritu spectanda etiam est resistentia ex funibus n. 37.6º.70.8° .
De motu gravium oblique projectorum . pag . 81 ,
Aequatio ad curvam, quam describunt gravia oblique
projecta; istiusmodi curva dicitur parabola. n. 38, 39.
Amplitudo jactus: maxima jactus amplitudo habetur sub
angulo projectionis semirecto: sub quo angulo projiciendum sit
grave ut offendat in datum scopum : altitudo jactus : ali
quid subjungitur de proprietatibus praefatae curvae. n. 40. 1º.
2 ° .... 70
Quid si gravia oblique projiciantur in medio resi
n. 41 .
stente.
De generalibus quibusdam proprietatibus motus curvili
nei, orti a viribus quarum una determinat materiale
punctum ad motum uniformem , altera ipsi materia
li puncto est continue applicata .
.
pag. 85.
Ubi in aliquo curvae puncto vis acceleratrix desinat
agere, excurret mobile per tangentem în punclo illo : ubi
321
Gnavium descensus-per plura plana inclinata sibi con-
ligua...............n.36.
Unde oriatur attritus. caeteris paribus, est proportio-
nalis pressioni: quomodo habeatur ratio attritus in motu
gravium per plana inclinata: grave in plano inclinato li-
brandum potentia aliqua, sive habeatur ratio attritus, sive
non. . , . . . . . . . . . . n. 37.10.2030.
Aequilibrii leges in cochlea, et cuneo. n. 37. 40. 50.
Spectatur attritus in aequilibrio cochleae: itemque in
aequilibrio corporis ad rotatilem motum circa fixum cy-
lindrum sollicitati: in machinis praeter resistentiam ex et-
tritu spectanda etiam est resistentia ex funibus n. 37.60.70.80.
De motu gravium oblique projectorum: pag. 81,
∙ Aequatio ad curvam, quam describunt gravia oblique
proiecta; istiusmodi curva dicitur parabola. . n. 38. 39.
Amplitudo iactus: maxima jactus amplitudo habetur sub
angulo projectiouis semirecto: sub quo angulo proiiciendum sit
grave ut offendat in datum scopum : altitudo jactus: ali-
quid subiungitur de proprietatibus praefatae curvae. n. 40. 10.
20 .... 70.
Quid si gravia oblique projiciantnr in medio resi-
stente. ↖∙∙∙∙∙∙∙∙∙∙⋅∙∙∙∥∙∡∎∙
De generalibus quibusdam praprietatibus motus curvili-
'nei, orti a viribus quarum una determinat materiale
punctum ad motum uniformem. altera ipsi materia-
li puncto est continue applicata.
. . . . pag. 85-
Ubi in aliquo curvae puncto vis acceleratrix desinat
agere, excurret mobile per tangentem in puncto illo: ubi322
tempore finito angulus, quem efformat vis acceleratrix cum
directione tangentis , fuerit semper acutus, acquiret mo
bile incrementum velocitatis finitum ; si semper obtusus,
patietur decrementum finitum ; si semper rectus , veloci
tas manebit constans: quadratum velocitatis adaequat vim
acceleratricem ductam in dimidium chordae, quae ex ejus
directione abscinditur ab osculatore circulo. n . 42.... 45.
Haec vera sunt de omnium virium genere; ponantur
vires acceleratrices ad centrum datum directae : jacebit cur
va in plano transeunte per rectam projectionis et per cen
trum virium: radius vector describet areas circa virium cen
trum temporibus proportionales: viceversa si radius ve
ctor describit areas circa punctum aliquod temporibus pro
portionales, vis acceleratrix erit constanter directa ad pun
ctum illud: velocitas, qua pollet mobile in eadem curva ,
exsistit reciproce proportionalis perpendiculo ducto a centro
virium in tangentem: vis acceleratrix in quovis curvae pun:
cto est directe ul radius vector, et reciproce ut factum
ex osculi radio in cnbum praefati perpendiculi : si ultra
punctum contactus sumitur arcus infinitesimus, a materiali
puncto describendus subsequente tempusculo, radiusque ve
ctor pertingens ad hujus arcus extremitatem producitur
donec occurrat tangenti, vis acceleralrix in contactus pun
cto erit directe ut pars radii vectoris producti intercepta
ac tangente , et reciproce ut quadratum tempuscu
li .
arcu
n. 46,
49.
Sive vires tendant ad centrum datum, sive non ; coor
dinatae puncti materialis in fine temporis e spectandae sunt
tanquam functiones ipsius t : formulae respicientes et veloci
tatem in quolibet curyae puncto, et binas componentes, al
teram juxta tangentem , alteram juxta normalem , in quas
resolvitur yis acceleratrix. n, 50. 10. 2º . 3º.
.
322
tempore linito angulus, quem etl'ormat- vis acceleratrix cum
directione tangentis ,
fuerit semper acutus, acquirat mo-
bile incrementum velocitatis Gnitum; si semper obtusus,
patietur decrementum (initum: si semper rectus, veloci-
tas mauebit constans: quadratum velocitatis adaequat vim.
acceleratricem ductam in dimidium chordae, quae ex eius
directione abscinditur ab osculatore circulo. n. 42.... 45.
Haec vera sunt de omnium virium genere; ponantur
vires acceleratrices ad centrum datum directae: iacebit cur-
va in plano transeunte per rectam projectiouis et per cen-
trum virium: radius vector describet areas circa virium cen-
trum temporibus proportionales: viceversa si radius ve-
ctor describit areas circa punctum aliquod temporibus pro-
portionales, vis acceleratrix erit constanter directa ad pun-
ctum illud: velocitas, qua pollet mobile in eadem curva
.
exsistit reciproce proportionalis perpendiculo ducto a centro
virium in tangentem: vis acceleratrix in quovis curvae pung-
cto est directe ut radius vector, et reciproce ut factum
ex osculi radio iu cubum praefati perpendiculi: si ultra
punctum contactus sumitur arcusiufiuitesimus, a materiali
puncto describendus subsequente tempusculo, radiosque ve-
ctor pertingens-ad huius arcus extremitatem producitur
donec occurrat tangenti, vis acceleratrix in contactus pun-
cto erit directe ut pars radii vectoris producti intercepta
arcu ac tangente ,
et reciproce ut quadratum tempuscu-
li. ∙∙∙∙∙⋅∙∙∙ ∙ ∙∙ ..n.46,...49-
Sive vires tendant ad centrum datum, sive non; coor-
dinatae puncti materialis in fine temporis t spectandae sunt
tanquam functiones ipsius :: formulae respicientes et veloci-
tatem in quolibet curvae puncto, et binas componentes, al-
teram juxta tangentem, alteram juxta normalcm. in. qu&quot;.
resolvitur vis acceleratrix. . . . . . n. 50.1'-2&quot;- 30,323
Resolata vi acceleratrice in ternas componentes axi
bus coordinatis parallelas, stabiliuntur formulae huc per
tinentes: applicantur formulae ad duas quaestiones, quarum al
tera respicit gravia oblique projecta in vacuo, altera respicit
gravia oblique projecta in medio resistente. n. 50. 4º. 5º . 6º.
Quomodo vis acceleratrix directa ad centrum expri
matur generatim per coordinatas polares : quomodo, data vi
acceleratrice directa ad centrum , inveniri possit aequatio
inter coordinatas polares ad lineam per quam movetur ma
teriale punclum: exemplum desumptum a vi acceleratrice ,
quae sit reciproce ut quadratum radii vectoris: sub hac le
ge poterit materiale punctum describere parabolam haben
tem suum focum in centro virium:
quaenam velocitas pro
jectionis ad id sit necessaria. n. 50. 7º. 8º... 15 °
Motus curvilineus impeditus : vis centrifuga. n. 51 .
De vi acceleratrice in motu circulari, existente
centro virium in centro circuli. pag . 109,
Istiusmodi motus ' est uniformis: vis acceleratrix obti
netur dividendo quadratum velocitatis per curvae circularis
radium: varia inde inferuntur et quoad projectionis velo citatem necessariam ad describendam cicularem curvam , et
quoad vires acceleratrices in diversis peripheriis circula
ribus. n. 52 , 53.
Vis centrifuga orta ex circulari telluris rotatione cir
ca suum axem : qua ratione decrescat ab aequatore ad po
los: qua ratione vis centrifuga imminuat gravitatem a po
lis ad aequatoren . n. 54.
323
Resoluta vi acceleratrice in ternas componentes axi-
bus coordiuatis parallelas, stabiliuntur formulae huc per-
tinentes: applicantur formulae ad duas quaestiones, quarum al-
tera respicitgravia oblique projecta in vacuo, altera respicit
gravia oblique proiecta in medio resistente. n. 50. 40. 50. 60.
Quomodo vis acceleratrix directa ad centrum lexpri- matur generatim per coordinatas polares: quomodo, data vi
acceleratrice directa ad centrum
.
inveniri possit aequatio
inter coordinatas polares ad lineam per quam movetur ma-
teriale punctum: exemplum desumptum a vi acceleratrice,
quae sit reciproce'ut quadratum radii vectoris: sub hac le-
ge poterit materiale punctum describere parabolam haben-
tem suum focum in centro virium: quaenam velocitas pro-
fectionis ad id sit necessaria. '
. . n. 50. 7&quot;. 80...150.
Motus curvilineus impeditus: vis centrifuga. n. St.
De vi acceleratrice in motu circulari, existente
centro m'rium' in centro circuli . pag. 109.
Istiusmodi motus 'est uniformis: vis acceleratrix obti-
netur dividendo quadratum velocitatis per curvae circularis
radium: varia iude inferuntur et quoad proiectionis velo-
citatem necessariam ad&quot; describendam cicularem curvam, et
quoad vires acceleratrices in diversis peripheriis circula-
ribus-.............n.52,53.
.
Vis centrifuga orta ex circulari telluris rotatione cir-
ca suum axem: qua ratione decrescat ab aequatore ad po-
los: qua ratione vis centrifuga imminuat gravitatem a po-
lis ad aequatorem.
. . .. . . . . ∙∙ ∙ n. 54-324
De vi acceleratrice in motu elliptico, existente
centro virium in foco ellipsis pag. 111,
Varia praemittuntur et circa rectas ita ductas ex puncto
quovis, ut tangant datam sphaeram ; et circa plaua tangentia
ducta per ejusmodi rectas ; et circa rectarum , arearumque
planarum projectiones in plano quolibet ; sed praecipue
circa ellipsim. n. 55. 1º, 2º ...14 °. . .
Quibus praemissis, demonstratur illud : existente cen
tro virium in foco ellipseos , vis acceleratrix in motu el
liptico est reciproce ut quadratum radii vectoris : quid in
duabus ellipsibus si quadrata temporum periodicorum sint ut cubi semiaxiun transversorum . n. 56.
Paucis subjunctis de ellipsi , parabola , et hyperbo
la, demonstratur quod, agentibus viribus in ratione reci
proca duplicata distantiarum a dato centro, praeter para
bolam poterit quoque mobile describere vel ellipsim
vel hyperbolam, existente focorum altero in centro virium:
quaenam projectionis velocitas requiratur ad ellipsim de
scribendam , quaenam ad hyperbolam. n, 67. 1.2.7 .
Obiter de lege virium in motu elliptico, ubi eae ten
dant ad ellipseos centrum . n. 57 , 8 .
De motu relativo punctorum materialium , tendentium
in se mutuo viribus acceleratricibus quae sint di
recte ut massae in quas tenditur, et reciproce ut qua
drata respondentium distantiarum . pag. 125.
Generales ad istiusmodi motum aequationes differen
tiales. n, 58,
324
- De ui acceleratrice in motu elliptica. existente
centro virium in foco ellipsis pag. 111.
Varia praemittuntur et circa rectas ita ductas ex puncto
quovis, ut tangant datam sphaeram; et circa plana tangentia
ducta per ejusmodi rectas; et circa rectarum, arearumque
planarum proiectiones in plano quolibet ; sed praecipue
circa ellipsim. . . . . . . . . n.55.10. 20 ...140.
Quibus praemissis, demonstratur illud: existente cen-
tro virium in foco ellipseos , vis acceleratrix in motu el-
liptico est reciproce ut quadratum radii vectoris: quid in
duabus ellipsibus si quadrata temporum periodicorum sint
ut cubi semiaxium transversorum . . . . . . n- 56.
Paucis subjunctis de ellipsi, parabola ,
et hyperbo-
la, demonstratur quod, agentibus viribus in ratione reci-
proca duplicata distantiarum a dato centro, praeter para-
bolam poterit quoque mobile describere vel ellipsim ,
vel hyperbolam, existente focorum altero in centro virium:
quaenam proiectiouis velocitas requiratur ad ellipsim de-
scribendam, quaenam ad hyperbolam. . n. 57. ↿∘∙ ⋍∘∙∙∙ 70.
Obiter de lege virium in motu elliptica, ubi eae ten- dant ad ellipseos centrum.
. . . . . . n. 57. 8&quot;.
De motu relativo punctorum &quot;materialium
,
tendentium
in se mutuo viribus acceleratricibus quae sint di-
recte ut massae in quas tenditur, et reciproce ut quab
drata respondentium distantiarum. pag. 125.
Gener-ales ad istiusmodi motum aequationes dideren-
tiüles- ∙∎∎ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ &quot;a 58.325
Spectantur duo tantum materialia puncta: vires per
turbantes ex reliquis punctis. n. 59, ... 62.
De pendulis ; deque gravium descensu per
arcus cycloidales. pag . 134.
Quid pendulum simplex ; quid compositum : vires
gignentes motum penduli simplicis n . 63.
Velocitates in puncto infimo acquistae a gravibus per
inaequales ejusdem circuli arcus descendentibus sunt ut
ipsorum arcuum chordae .
. n. 64.
Oscillationes penduli simplicis per arcus satis exi
guos , ulcumque ceteroquin inaequales , sunt ad sensum
isochronae seu aequidiuturnae : quid ex doctrina penduli
simplicis circa terrestrem gravitatem n. 65 , 66.
Centrum oscillationis in pendulo composito : etiam
oscillationes penduli compositi suņt isochronae, modo ta
men existant satis exiguae .
n. 67.
Oscillationes penduli simplicis in medio resistente :
primo in hypothesi resistentiae proportionalis simplici ve
locitati; deinde in hypothesi resistentiae proportionalis qua
drato velocitatis .
n. 68.
n . 69 .
Paucis praemissis de cycloide, demonstratur illud : ex
quocumque cycloidis puncto demittatur grave , eodem sem
per tempore perveniet ad punctum infimum
De attractione corporum in hypothesi attractionis agentis
in ratione directa massarum , et in reciproca
duplicata distantiarum .
Attractio corporum quorumcumque in materiale pun
clum situm sive extra corpus attrahens, sive intra. n . 70,71,72.
pag . 151 .
325
Spectautur duo tantum materialia puncta: vires per-
turbantes ex reliquis punctis. . . . . . n. 59....62.
De pendulis; deque gravium descensu per
arcus cycloidales. pag. 134.
Quid pendulum simplex; quid compositum : vires
gignentes motum penduli simplicis
. . . . . n. 63.
Velocitates in puncto intimo acquistae a gravibus per
inaequales ejusdem circuli arcus descendentibus sunt ut
ipsorum arcuum chordae
. . . . . .- . . n. 64.
Oscillationes penduli simplicis per arcus satis exi-
guos, utcumque ceteroquin iuaequales ,
sunt ad sensum
isochrouae seu aequidiuturuae: quid ex doctrina penduli
simplicis circa terrestrem gravitatem . . . n. 65
, 66.
Centrum oscillationis in pendulo composito: etiam
oscillationes penduli compositi sunt isochrouae, modo ta-
men existant satis exiguae . . . . . . . . n. 67.
Oscillationes penduli simplicis in medio resistente:
primo in hypothesi resistentiae proportionalis simplici ve-
locitati; deinde in hypothesi resistentiae proportionalis qua-
drato velocitatis
. . . .. . . . . . . n. 68.
Paucis praemissis de cycloide, demonstratur illud : ex
quocumque cycloidis puncto demittatur grave , eodem sem-
per tempore perveniet ad punctum infimum
. . n.. 69.
De attractione corporum in hypothesi attractionis agentis
in ratione directa massarum, et in reciproca
duplicata distantiarum. pag. 151 .
Attractio corporum quorumcumque in materiale pun-
ctum situm sive extra corpus attrahens, sireintrafn. 70,71,72.326
Expediuntur quae pertinent ad attractionem corpo
rum sphaericorum in punctum materiale n. 73,74,75.
Materiale punctum valde distans a corpore attrahente,
utcumque se habeat forma corporis, ea proxime ratione tendit
in ipsum corpus, qua tenderet si corporis partes in centro
gravitatis compenetrarentur: ubi dimensiones corporum quo
rumcuinque se mutuo attrahentium sint admodum exiguae
prae distantiis , quibus ipsa corpora disjunguntur , eorum
alterum tendet in alterum perinde ac si essent ambo in
suis gravitatis centris compenetrata ; haec assertio quoad
sphaerica corpora valet utcumque se habeat intercedens
distantia n. 76.
De gravitatione universali. pag. 159.
Ex mutua coelestium corporum gravitatione collata
cum terrestrii gravitate inferimus illud : gravitas ita ma
teriam afficit , ut singulae ejus particulae in alias omnes
et singulas gravitent in ratione directa massarum ad quas
tenditur , et reciproca duplicata distantiarum alterius ab
altera n . 77 , ...82.
.
Aliquid circa solarem et planeticas massas... n.83.10... 4.
Media telluris densitas determinata ex penduli aber
ratione ; itemque experimentis institutis in libra
siouis n. 83. 5. 6.°
tor
Quomodo ex marini aestus phoenomeno deduci pos
sit ratio inter lunarem ac terrestrem massam . n. 83.7 . °
326
Expediuntur quae pertinent ad attractionem corpo-
rum sphaericorum iu punctum materiale
. n. 73,74,75.
Materiale punctum- valde distans a corpore attraheute,
utcumque se habeat forma corporis, ea proxime ratione tendit
in ipsum corpus, qua tenderet si corporis partes in centro
gravitatis compenetrarentur: ubi dimensiones corporum quo-
rumcumque se mutuo attrahentium sint admodum exiguae
prae distantiis, quibus ipsa corpora disiunguntur , eorum
alterum tendet in alterum perinde ac si essent ambo in
suis gravitatis centris compenetrata ; haec assertio quoad
sphaerica corpora valet utcumque se habeat intercedens
distantia ...............n76
De gravitatione universali. pag. 159.
Ex mutua coelestium corporum gravitatione collata
cum terrestrii gravitate.iuferimus illud: gravitas ita'ma-
teriam allicit, ut singulae eius particulae in alias omnes
et singulas graviteut in ratione directa massarum ad quas
tenditur, et reciproca duplicata distantiarum alterius ab
altera . . . . . . . . . ∎∙ ∙ ∙ n. 77,...82.
Aliquid circa solarem et plaueticas massas...n.83.10...4.'
Media telluris densitas determinata ex penduli aber-
ratione : itemque experimentis institutis in libra tor-
Sioni. ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙∙ ∙ ∙ n. 830 5-0 S..
Quomodo ex marini aestus phoenomeuo deduci pos-
sit ratio inter lunarem ac terrestrem massam. n. 83. 7!327
Aliquid notatur de motu punctorum materialium
utcumque inter se connexorum pag. 169.
Formulae spectantes et ad translativum punctorum
motum juxta coordinatos axes, et ad rotatilem eorum mo
tum circum axes ipsos n. 84.
Moto punctorum systemate, perinde movebitur com
mune gravitatis centrum ac si , coeuntibus punctis in
ipsum centrum , applicarentur centro eaedem vires cum
iisdem directionibus , quibus pancta sollicitantur. n. 84.1.6
Principium de conservatione centri gravitatis : item
de conservatione arearum : necnon de conservatione vi
rium vivarum n. 84. 2.0 ... 5 ..
Relativus rigidi liberique systematis motus quoad
gravitatis centrum n. 84. 6. ° 7.0
Motus rigidi systematis circa axem fixum ; quibus
cuinque caeteroquin viribus acceleratricibus sollicitetur sy
stema : quid si vires acceleratrices consistant in sola gra
vitate ; huc spectat theoria penduli compositi : longitudo
penduli simplicis , quod suas perficit oscillationes eodem
tempore ac pendulum compositum : quid si nullae sint
vires acceleratrices : inertiae momenta quoad axem
principales systematis axes : principalia inertiae momen
n. 85. 1.° 2.° ... 7.0
.
ta .
De fluidorum corporum aequilibrio pag. 182.
Ex perfecta mobilitate , qua ponuntur gaudere flui
dorum corporum particulae , ostenditur principium de
aequalitate pressionis , atque inde eruuntur conditiones
requisitae ad aequilibrium cujusvis massae fluidae. n. 86.
327
Aliquid notatur de motu punctorum materialium
utcumque inter se connexorum pag. 169.
Formulae spectantes et ad translativum punctorum
motum iuxta coordinatos axes, et ad rotatilem eorum mo-
tum circum axes ipsos . . . . . . . . . n. 84.
Moto punctorum, systemate, perinde movebitur com-
mune gravitatis centrum ac si
, coeuntibus punctis in
ipsum centrum, applicarentur centro eaedem vires cum
iisdem directionibus , quibus puncta sollicitantur. n. 84.1.'
Principium de conservatione centri gravitatis: item
de conservatione arearum : necnon de conservatione vi-
rium vivarum . . . .
'. ⋅∙ ∙ ∙ n. 84. Z.&quot;...Sæ
Belativus rigidi liberique systematis motus quoad
gravitatis centrum
. . . . . . . . n.84. 6.&quot; 79
Motus rigidi systematis circa axem fixum .: quibus-
cumque caeteroquin viribus acceleratricibus sollicitetur sy-
stema :quid si vires acceleratrices consistant in sola gra-
vitate; huc spectat theoria penduli compositi: longitudo
penduli simplicis
, quod suas perficit oscillationes eodem
tempore ac pendulum compositum .: quid si nullae sint
vires acceleratrices : inertiae momenta quoad axem :
principales systematis axes : principalia inertiae momen-
ta.
. . . . . . . . . . n.85.1.0 Z.&quot;... 7.&quot;
De fluidorum corporum aequilibrio pag. 182.
Ex perfecta 'mobilitate
. qua ponuntur gaudere Hui-
dorum corporum particulae ,, ostenditur principium de
aequalitate pressionis , atque inde eruuntur conditiones
requisitae ad aequilibrium cujusvis massae Huidae. n. 86.328
Quid notandum circa superficiem massae fluidae li
bratae n. 87, 1. ° 2.° ... 5 .
Quid circa fluidum elasticitate pollens, ni 87. 6.0 7 .
De gravium homogeneorumque liquidorum
aequilibrio. pag. 187.
Liquida constituta in aequilibrio intra vasa : pres.
siones in areas sive horizontaliter , sive oblique demer
sas : centrum pressionis .
n. 98. 1. ° . , . 4.0
Solida liquidis immersa : aequilibrii positiones quoad
solidum liquido insidens n. 88.5 . , 89. 1.° 2. ° 3.°
Utrum aequilibrium sit stabile , nec ne. n. 90.
.
De gravium liquidorum aequilibrio in vasis
communicantibus.
pag. 195.
Quid si vasis communicantibus idem contineatur li
quidum : explicatio variorum effectuum ; antliae adspi
ranles , etc n. 91 , 92. 1.° 2.°
Quid si diversa contineantur liquida.
. n. 92. 3.0
De gravium elasticorumque fluidorum aequilibrio nec
non de altitudinibus dimetiendis ope barometri, et de
pondere ac densitate vaporum . pag. 199.
Conditio aequilibrii expressa per aequationem dif
ferentialem : perficitur integratio in hypothesi temperiei
constantis n. 93.
Inde eruitur formula inserviens ad altitudines di
328
Quid notandum circa superüciemi massae liuidae li-
bratae
. . . . . . . . . n. 87.1.02.&quot;...5.0
Quid circa fluidum elasticitate pollens. n: 87. 6.&quot; 7.&quot;
-De gravium homogeneorumque liquidarum
aequilibrio. pag. 187.
Liquida constituta in aequilibrio intra vasa: pres- ⋅
tiones in areas sive horizontaliter , sive oblique demer-
sas: centrum pressionis . . . . . n. 88. 1.&quot;
..,.
4.&quot;
∙
!'
Solida liquidis immersa : aequilibrii positiones quoad
solidum liquido insidens . . n. 88.5.&quot;, 89. 1.&quot; 2.0 3.&quot;
Utrum aequilibrium sit stabile, nec ne. . . n. 90.
De gravium liquidarum aequilibrio in 'vasis
communicantibus. pag. 195.
Quid si vasis communicantibus idem continaptur li-
quidum: explicatio variorum effectuum : antliae adspi-
TODIBBQ etc ∙ ∙ ∙ ∙ ∙
↼
∙ ∙ a ∙ &quot;.
5 91. 92. 1.02.0
Quid si diversa cbntiueantur liquida. . . n. 92. 3.&quot;
De gravium elasticorumque fluidorum aequilibrio nec
non de altitudinibus dimetiendis ope barometri, et de
pondere ac densitate. naporum. pag. 199.
Conditio aequilibrii expressa per aequationem dif-
ferentialem : perficitur integratio in hypothesi temperiei
constantis . . . .
.'&quot;.
. . ⋅ n. 93.
Inde eruitur formula inservieus ad altitudines di-329
metiendas ope barometri : varia observantur pro commo
diori formulae usu n. 94. 1. ° 2.° ... 6.•
Verticalis ascensus globi aereostatici : maxima glo
bi elatio . n. 95.
Maxima quantitas vaporis sese evolventis in vase un
dique clauso : vis elastica sicci aeris aucta ob evolu
tum vaporem : ratio inter densitatem aquei vaporis ac
densitatem sicci aeris sub eadem temperie eademque pres
sione : ratio inter eorum densitates ac pondera sub ea
dem temperie et diversis pressionibus : densitas aeris va
porosi librantis datam pressionem sub temperie data. n. 96.1 . °
2 .
Usus aquei vaporis in movendis machinis. n . 99.6. •
De aqua egrediente per angustum foramen
e vasis verticalibus sive cylindricis,
sive prismaticis. pag. 206 .
Nonnulla praemittuntur ex pluries iteratis experimen
tis .
n . 97.
Quaenam velocitas aquae egredientis: tempus impen
sum in descensu usque ad quamlibet altitudinem datam . n.98.
Quantitas aquae dato tempore egredientis : tempus
quo vas totum evacualur n. 99, 100,
Ratio inter tempora , quibus deplentur duo vasa ha
bentia et altitudines et orificia aequalia : quantitales aqua
rum successivis ' et aequalibus temporibus ex vasis ori
ficio efluentium : divisio vasorum in partes successivis dati
temporis unitatibus vacuandas n. 101 , 102.
22
' 329
metiendus ope barometri : varia observantur pro commo-
diori formulae usu . . . . . n. 94. 1..) .&quot;
... 6.&quot;
Verticalis ascensus globi aereostatici : maxima glo-
bi elatio. ∙ ∙ ∙ ∙ ∙ ' ∙ ∙ ∙ ∙ ∙ ∙
∙∎∎ ∙ n- 950
-Maxima quantitas vaporis sese evolventis in vase uu-
dique clauso : vis elastica sicci aeris aucta&quot; ob evolu-
tum vaporem : ratio inter densitatem aquei vaporis ac
densitatem sicci aeris sub eadem temperie eademque pres-
sione: ratio inter eorum densitates ac pondera sub ea-
dem temperie' et diva-sis- pressionibus: densitas aeris va-
porosi librantis datam pressionem sub temperie data. n. 961.&quot;
2.0
...
5.&quot;
Usus aquei vaporis in movendis machinis. n. 99. 6.&quot;
De aqua egrediente per angustum foramen
e vasis «verticalibus sive cylindricis,
sive prismaticis. pag. 206.
Nonnulla praemittuntur ex pluries iteratis experimen-
tis ∙ ∙ ∙ ∙ ∙ ⋅∙⋅ ∙ ∙ . ∙ ∙ ∙ ∙∎∎ ∙ ∙ ∙ &quot;o 970
Quaenam velocitas aquae egredientis: tempus impen-
sum in descensu usque ad quamlibet altitudinem datam. n.98.
,. Quantitas aquae dato tempore egredientis: tempus
quo vas tatum evacuatur . . . . . . n. 99,100.
Ratio inter tempora, quibus deplentur duo vasa ha-
bentia et altitudines 'et oriiicia aequalia : quantitates aqua-
rum successivis' et aequalibus temporibus ex vasis ori-
iicio efluentium: divisio vasorum in partes successivis dati
temporis unitatibus vacuandas . . . ∙∙ n. 101, 102.
22330
Contractio venae aqueae n. 103.
Ubinam perficiatur acceleratio , per quam velocitas
aquae descendentis admodum exigua mutatur in finalem
satisque grandem effluxus velocitatem . n . 104. 1.0
Quomodo motus aquae defluentis in regularibus al
veis traduci possit ad motum aquae prosilientis ex an
gustis vasorum orificiis n. 104. 2.• , ..5.
Illud cum Auctoribus non paucis assumitur tanquam
principium , quod nempe unumquodque liquidi in vase
quolibet descendentis tenuissimum et horizontale stra
tum coalescat iisdem constanter particulis communi , ea
que tantum verticali , velocitale donatis ; inde vero eruun
tur , quae pertinent ad ipsius liquidi motum n . 105.
Aliquid subjungiur circa generalem theoriam
motus corporum fluidorum . pag. 216.
Aequationes ad istiusmodi motum et quum massa
fluida homogenea vel heterogenea est incapax compressio
nis, et quum massa fluida pollet elasticitate. n. 106,107,108.
De tubis capillaribus. pag. 225.
sol Vires ex materia tubi , et ex materia liquidi ,
licitantes datam ipsius liquidi particulam : attentis viri
bus istis , suprema liquidi superficies vel induet curvam
concavamque figuram , vel curvam convexamque , vel ma
nebit, plana atque horizontalis, n. 109,1.9
330
Contractio venae aqueae . . . . -. . n. 103.
Ubinam perficiatur acceleratio, per quam velocitas
aquae descendentis admodum exigua mutatur in finalem
satisque grandem effluxus velocitatem. .
-
.n. 104. 1.&quot;
Quomodo motus aquae defluentis in regularibus al-
veis traduci possit ad motum aquae prosilientis ex au-
gustis vasorum orificiis
. . . . . n. 104. Z.&quot;. ..5.&quot;
Illud cum Auctoribus non paucis assumitur tanquam
principium, quod nempe unumquodque liquidi in vase
quolibet descendentis tenuissimum et horizontale stra-
tum coalescat iisdem constanter particulis communi , ea-
que tantum verticali, velocitate donatis :
inde vero eruun-
tur, qnae pertinent ad ipsius liquidi motum
. ∙⋅ n. 105-
Aliquid subjungiur circa generalem theoriam
motus corporum fluidorum. pag. 215-
Aequationes ad istiusmodi motum et quum massa
fluida homogenea vel heterogenea est incapax compressio-
nis, et quum massa fluida pollet elasticitate. n. 106,107,108.
De tubis capillaribus. pag. 225.
Vires ex materia tubi
,
et ex materia liquidi .
sol-
licitantes datam ipsius liquidi particulam: attentis viri-
bus istis , suprema liquidi superficies vel induet curvam
concavamque figuram , vel curvam couvexamque , vel ma-
nebit, plana atque horizontalis, ,. . . .. . n. 109.1.&quot;331
Quam attractionem exerceat massa liquida , cujus su
prema superficies est plana , in columellam liquidam per
pendiculariter illi superficiei planae insistentem . n. 109.2 .
Quam attractionem exerceat massa liquida , cujus su.
prema superficies est vel concavo -sphaerica vel convexo
sphaerica , in columellam liquidam perpendiculariter in
sistentem plano tangenti , dactó vel per punctum infimum
superficiei concavo-sphaericae vel per supremum superfi
ciei convexo -sphaericae. n. 109. 3.° 4.0 ... 70
Quid si massa liquida terminetur superficie concaya
vel convexa , quae non sit sphaerica. n. 109. 8.° ... 11.º
His declaratis , explicamus ascensum descensumque
liquorum in lubis capillaribus n. 110,
.
Nonnalla subjunguntur , quorum ratio desumitur ab
actione capillari . n. 111. 1.° 2.° ... 5 ° , 112
)
ACUSTICAE PRINCIPIA
Notiones praeambulae.
1
pag . 245.
Corpora, quae sonora dicuntur tunc sonum exci
tant quando ita agitantur , ut illorum partes tremulo ac
vibratorio satisque rapido concutiantur motu ; qui motus
communicatus aeri ambienti , et late diffusus afficit orga
nym auditus: vis acceleratrix in vibrante particula resonan
tis corporis.
. n . 113. 10. 20.
331
Quam attractiduem exerceat massa liquida , cuius su-
prema superficies est plana ,
in columellam liquidam per-
pendiculariter illi superficiei planae insistentem. n. 1092.&quot;
Quam attractionem exerceat massa liquida , cuius su-
prema- superficies est vel concavo-sphaerica vel convexo-
sphaerica, in columellam liquidam peu-pendiculariter in-
sistentem plano tangenti , dnctö vel per punctum infimum
superficiei concavo-sphaericae vel per supremum superfi-
ciei convexo-sphaericae. . . . n. 109. 3.&quot; 4.&quot;
.. . 7.&quot;
Quid si massa liquida terminetur superficie concava
vel convexa, quae non sit sphaerica. n. 109. 8.&quot;. ..
11.&quot;
His declaratis , explicamus ascensum descensumque
liquorum iu .tubis capillaribus . . . . . . n. 110.
Nonnulla subjunguntur ∙ quorum -ratio desumitur ab
actione capillari. . . . . n. 111. 1.&quot; 2.&quot;
. . . 5&quot;,112
AOUSTIGAE
W
PRINCIPIA
Notiones praeambulae. ∣
pag. 245.
Corpora, quae sonora dicuntur
,
tunc sonum exci-
tant quando ita agitantur, ut illorum partes tremulo ac
vibratorio satisque rapido concutiuntur motu; qui motus
communicatus aeri ambienti, et late diffusus afficit orga-
num auditus: vis acceleratrix in vibrante particula resonan-
tis corporis. . . . . . . '. . . . n. 113.1&quot;. 2&quot;.332
Progignitur quoque sonus ab aere vehementer compres
so , seseque statim restituente , n. 114.
.
Soni reflexio; inde echo. n . 115 .
Non solus aer est medium ideoneum transmissioni
sonorum. n. 116.
De intensitate soni; deque ejus gravitate,
et acutie .
pag. 248.
Sonus intensior ex eo gignitur quod in sonoro cor
pore plures ejusdem partes simul oscillant, et majus spa
tium singulis oscillationibus dato tempusculo percurrunt;
atque ita in aere ex numero item et majori oscillatione
partium aeris intensitas soni dependet ; remissior autem
sonus ex opposito.
n. 117.
Nonnulla explicantur circa soni intensitatem. n . 118.
ex Soni gravis et acuti discrimen repetendum est
numero vibrationum in partibus sonori corporis, ita ut
sonus gravior oriatur ex minus frequentibus vibrationibus so
nori corporis, ex crebrioribus contra sonus acutus ; idem
que de oscillationibus aeris in sono derivato. n. 119.
Quid consonantia , et quid dissonantia: varii conso
nantiae gradus: theoria chordaram vibrantium in hypothe
si vibrationum admodum exiguarum. n. 120. 1 ” 2 ”... 7 .
Varia proponuntur explicanda circa chordas vibran
tes . n. 121 .
332
Progignitur quoque sonus ab aere vdhemeuter compres-
so, seseque statim restituente. . . . . . . n. 114.
Soni reflexio; inde echo. . . . . . . n. 115.
Non solus aer est medium ideoneum transmissioni
SODOmm. ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ &quot;. 116.
De intensitate soni; deque eius gravitate,
et acutie. pag. 248.
Sonu's intensior ex eo gignitur quod in sonoro cor-
pore plures eiusdem partes simul oscillaut, et maius spa-
tium singulis oscillationibus dato tempusculo percurrunt:
atque ita in aere ex numero item et maiori oscillatione
partium aeris intensitas soni dependet; remissior autem
sonus ex opposito. . . . . . . . . . ,n. 117.
Nonnulla explicantur circa soni intensitatem.
. .n. 118.
Soni' gravis et acuti discrimen repetendum est ex
numero vibratiouum in partibus sonori corporis, ita ut
sonus gravior oriatur ex minus frequentibus vibrationibus so-
nori corporis, ex crebrioribus contra sonus acutus,- idem-
que de oscillationibus aeris in sono derivato. . n. 119.
Quid consonantia, et quid dissonantia: varii conso-
nantiae gradus: theoria chordarum vibrantium in hypothe-
si vibrationum admodum exiguarum. n- 120. 1&quot; 2&quot;... 7&quot;.
Varia proponuntur explicanda circa chordas vibran-
tes. ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙
Q &quot;. 1210333
Quomodo sonus trans obicem possit communicari ita,
ut tonus proprius sonori corporis permaneat. n. 122.
Unde asperitas aut lenitas soni proficiscatur. n. 123.
Transversae et longitudinales chordarum vibratio
nes: nodi in chordis vibrantibus: lineae nodales in super
ficiebus corporum resonantium : vibrationes laminarum ri
gidarum .
n. 124 .
De directa soni propagatione per aerem . pag. 265.
In iisdem circumstantiis sonus aequabili velocitate
in toto decursu devehitur; omnesque soni sive intensi ,
sive remissi, sive graves, sive acuti eadem velocitate dif
funduntur: qua ratione intensitas soni minuatur in pro
gressu . n. 125,
Undae sonorae constitutio. n. 126,
Soni et velocitas, et intensitas augetur a vento se
cundo, minuitur ab adverso . n . 127.
Experimenta instituta ad soni velocitatem determi
nandam; quae tamen experimenta non satis conveniunt :
hujus diversitatis rationes : quaenam utilitas ex determi
natione velocitatis qua propagatur sonus. .
n. 128
Generalis de fluidorum motu theoria applicatur ad
soni propagationem : soni velocitas eruta ex applicatione
theoriae comparatur cum velocitate quam praebent ex
perimenta .
n. 129. 10. 2º. 3º.
Crassities aerei strati, in quo particulae cientur una :
si impulsio in obicem facta quadrato velocitatis sumitur
-
22&quot;
333
Quqmodo sonus trans obicem possit communicari ita,
ut tonus proprius sonori corporis permaneat.
. n. 122.
Unde asperitas aut leuitas soni proficiscatur. n. 123.
Trausversae et longitudinales chordarum vibratio,-
nes: nodi in chordis vibrantibns: lineae nodales in super-'-
iiciebus corporum resonantium: vibrationes laminarum ri-
gidarum...........;..n.124.
De directa soni propagatione per aerem. pag. 265.
In iisdem circumstantiis sonus aequabili velocitate
in toto decursu devehitur; omnesque soni sive intensi ,
sive remissi, sive graves, sive acuti eadem velocitate dif-
fuuduntur: qua ratione intensitas soni minuatur in pro-
gressu..............-n.125.
Undae sonorae constitutio. . . . . . . n.126,
Soni et velocitas, et intensitas augetur a vento se-
eundi), mall!!! EI) adverw. ∙ ∙ ∙ ∙ ∙ ∙ n- 127.
Experimenta instituta ad soni velocitatem determi-
nandam; quae tamen experimenta non satis conveniunt:
hujus diversitatis rationes: quaenam utilitas ex determi-
natione velocitatis qua prOpagatur sonus. . . n. 128.
Generalis de fluidorum motu theoria applicatur ad
soni propagationem: soni velocitas eruta ex applicatione
theoriae comparatur cum velocitate quam praebent ex-
perimenta . . . . . . . . . . n.129.1o.2&quot;.3&quot;.
Crassities aerei strati, in quo particulae cientur uua:
si impulsio iu obicem facta quadrato velocitatis sumitur
22'334
proportionalis, rationem duplicatam distantiarum .sequetur
soni debilitatio.
n. 129. 4. 5 .
Cur pluribus corporibus simul resonantibus , inter
oscillationes in aere excitatas non habeatur confusio , omnes
que diversi soni inde orti ad aures distincte perveniant:
huc spectat principium de superpositione exiguorum mo
tuum. n. 129. 6.
Propagatio soni in cubis cylindricis indefinitae lon
gitudinis. n. 129.7 .
J
De reflexa soni propagatione per aerem pag. 289.
Cum in directa propagatione sonorus aer . offendit o
bicem aptum , reflectitur : varia ad echo spectantia ex
plicantur. n. 130.
Reflexio soni fit ad angulos incidentiae et reflexionis
aequales; regrediturque sonus eadem velocitate qua incedebat
antequam in obicem impingeret. n . 131 , 132, 1º. 2º
De instrumentis pneumaticis. pag. 294.
In instrumentis pneumaticis soni genesis repetenda non
est saltem praecipue ex oscillatione partium solidarum i
psius instrumenti : quo pacto sit explicanda : aer secun
dum fistulae longitudinem se habet instar chordae peragen
tis longitudinales vibrationes: etsi ex materia instrumenti
non habetur varietas quoad soni qualitatem, aut valde uota
bilem intensitatem ; varietas tamen habetur quoad meliorem
334
proportionalis, rationem duplicatam distantiarum .sequetur
soni dehilitatio. . . . . . . . ∙ ∙ n.129.40.50.
Cur pluribus corporibus simul resonantibns ,
inter
oscillationes in aere excitatas non habeatur confusio,omues-
que diversi soni inde orti ad aures distincte perveniant:
huc spectat principium de superpositione exiguorum mo-
tuum. ∙⋅∙ '.
. . . . . . . . . . n.129.6&quot;.
Propagatioi soni in. tubis cylindricis indefinitae lou-
gitudinisa, ∙ ∙ ∙ . . . . . .. . . n.129.7&quot;.
]
De refleæa soni propagatione per aerem pag. 289.
!. ∙ . .
Cum indirecta prOpagatione sonorus aer .oii'eudit o-
bicem aptum, reflectitur : varia ad echo spectantia ex-
Plimnturo. ∙∙ ∙∙ ∙∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ n. 1300
. Reflexio soni iit ad angulos incidentiae et reflexionis
aequales: regrediturque sonus eadem velocitate qua incedebat
antequam in obicem impingeret. . . .&quot;' 131, 132.1&quot;.2&quot;.
'pul '
De instrumentis pneumaticis. pag. 294.
In instrumentis pneumaticis soni genesis repetenda non
est saltem praecipue ex oscillatione partium solidarum i-
psius instrumenti: quo pacto sit explicanda: aer secun-
dum fistulae longitudinem se habet instar chordae peragen-
tis lougitudinales vibrationes: etsi ex materia instrumenti
non habetur varietas quoad soni qualitatem, aut valde uota-
bilem intensitatem; varietas tamen habetur quoad meliorem335
aliquam resonantiam : quid si intrumentum pneumaticum sit
compactum ex materia non resistente , quale v. g. esset in
strumentum membranaceum .. . n. 133.
Nonnulla proponuntur explicanda circa instrumenta
pneumatica. n . 134.
Tremulus aeris motus in tubis cylindricis determinatae
longitudinis :
1º. Quum tubus est firmiter obseratus apud alterum
orificium simulque apertus apud alterum n. 135, 136.
2°. Quum tubus est patens in utraque extremitate: in
de eruitur ratio investigandi velocitatem , qua propagatur
sonus in aliis fluidis elasticis diversis ab aere. n. 137.
3 °. Quum tubus est utrinque obseratus. n. 138.
De propagatione soni per liquida, et per solida
corpora . pag. 302.
Formulae huc spectantes: parvula contractio aquae et
hydrargiri ob auctam pressionem: usus istius contractionis in
determinanda velocitate soni per haec duo liquida. n . 139,140 .
Analogia inter oscillationes aeris in tubo cylindrico a
pud ambas extremitates aperto et longitudinales oscillationes
virgae rigidae suppeditat peculiarem methodum investigandi
velocitatem propagationis per solida corpora. n. 141 .
De vocis humanae origine. pag. 305.
Nonnulla ex anatomicis praemittuntur; quibus praemis
sis , stabilitur illud : vocis humanae organum etsi conside
rari maxime debeat tanquam instrumentum pneumaticum
335
aliquam resonantiam: quid si intrumentnm pneumaticum sit
compactum ex materia nou'reaistente, quale v.. g. esset in-
strumentum membranaceum. .. .. .. .. .. .. . n. 133.
Nonnulla proponuntur explicanda circa instrumenta
pneumatica. .. . .. .. .. .. .. .. .. .. .. ..
.n. 134.
Tremulus aeris motus'iu tubis cylindricis determinatae
longitudinis :
⇝
↿∘∙ ⊄⊇⇂⋯⋯∙⋯∣⋯∘⋅⊖⊱⇂ firmiter ohseratus apud alternm
orificium simulque apertus apud alterum . n. 135,136.
20. Quum tuhus est patens in utraque extremitate: in-
de eruitur ratio investigandi velocitatem
, qua propagatur
sonus in aliis fluidis elasticis diversis ab aere. n. 137.
(
3&quot;. Quum tubus est utrinque ohseratus. . n. 138.
i '
⋅
⋅ ↼
De prapagau'one soni per liquida, ettper &quot;solida ⊳∣ corpora. pag. 302.
.Fornrnlae huc spectantes: parvula contractio ailuae et
hydrargiri ob auctam pressionem: ususistius contractionis in
determinanda velocitate soni per haec duo liquida.ia.139, 140.
Analogia inter oscillationes aeris in tuho cylindrico a-
pud ambas extremitates aperto et lougitudinales oscillationes
virgae rigidae suppeditat peculiarem methodum investigandi
velocitatem propagatiouis per solida corpora. . n. 141.
De 'vocis humanae origine. pag. 305.
Nonuulla ex anatomicis praemittuntur; quibus praemis-
sis, stahilitur illud: vocis humanae organum etsi conside-
rari maxime debeat tanquam instrumentum pneumaticum
∩336
flexili et elastica materia ex parte compactum , non tamen ita
est ut cum instrumentis fidicularibus aliquam non habeat
analogiam . n. 142.
Quid, os atque ejus partes conferant ad formationem
vocis.
n. 143.
Variae refellantur sententiae de humanae vocis ori
gine; variaeque circa vocem humanam proponuntur quae
stiones. n . 144 , 145 .

De auditus organo . pag. 310.
Auris descriptio.
n. 146.
Quaenam ex auris partibus pro praecipuo atque im
mediato auditionis organo statuenda sit. n. 147.
Cur quibusdam grata, aliis pene nihil, aut etiam mo lesta sit harmonia ,
n. 148. 19.
Cur daabus auribus unus idemque sonus audiatur n.148.2 °.
1
336 '
Bexiliot'elgstica materia ex parte compactum, non tamen ita
eat ut cum, instrumentis iidicularibus aliquam non habeat
malogihmoo-o-0 ∙⋅∙∙∙⋅∙ ∙ ∙ ∙ ∙ ∙ ∙
&quot;0142.
Quid, os atque eius partes conferant ad formationem
'owa ∙∙∙ ∙ ∙ ∙ ∙ ∙ ∙∙∙∙∙ ∙ ∙
&quot;.1430
Variae refelluntur sententiae de humanae vocis ori-
gine, variaeque circa vocem humanam proponuntur quae-
'none'- ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ &quot;- 14401450
De auditus organo. pag. 310.
Auriadeacriptio. . . . . . . . . . n.146.
Quaenam ex auris partibus pro praecipuo atque im-
mediato auditionia organo statuenda- sit.
. . . n. 147.
Cur quibusdam grata, aliis pene nihil, aut etiam mo-
lestasit harmonia. ∙ ∙ ∙
. . . . . . n. 148. ∎∘∙
Cur duabus auribus unus idemque sonus audiatur n.148.2'.ERRATA CORRIGE
pag. lin .
1. 4. saepae
1. 5. decresit
4. 28. istanti
5. 13. rive
7. 29. poductis
8. 14. sin a
14 29. AH'.BC
15. 3. AF
saepe
decrescit
instanti .
siye .
prodactis
sin a .
AH'.BC
BF'
BF .
Y.
4. AF
24, 7. Sy
50. 6 et 7.
S * S

*

17. ſsfla)dx Sfaxdx.
52. 14.
f (x )dx f '( x )d.x2
2
2
56. 18.- ( tdx ,z + dz,u,...) -f(xtdx , z + de, u, ...).
dull .
Sfx )dx 22. ( x) dx
eck
58. 1.-C
+0 .
57. 4. del
1
Sfaydar
62. 3. W
v'dz'
11. dzi
63. 8. sint va
69. 12. quod
... 17.
v'du'
da
sintVC .
quoad
ngt 2gt
70. 7.- 7 . kalog(k2—12)
.
.log(k ?-- ).
1
-
1
!
ERBATA CORRIGE
pag. lin.
&quot;. 4. saepae saepe .
1. 5. decresit decrescit
4. 28. istanti instanti
.
5. 13. rive sive
.
7. 29. poductis productis .
8. 14. aina sin at
.
14 29. AH'.BC AH'.BC'
.
15. 3. AF' BF' .
⋅
∙ ∙ ∙ 4. AF BF
. 24, 7. &lt;nowiki&gt;:] z?&lt;/nowiki&gt;
.
50. 6et 7. f:&quot; f:&quot;
... 17. JfftæMx [f(xkiæ .
52. 14. figit&quot; f'(æ2)dæ* .
56. 18.—(æ-]-dæ,z-l—dz,u,...) --f(a—-[-dx , z—l-dz, u, -
.
)- - 57. 4. d,,p. dup.
.
.. .
22. 111-2635 f(ældæ
.
803 803
58. 1.-:.-C —]—G .
62. 3. 9) p
↿↿∙∙∙ v'dz' til—tf
.
d:, dz'
.
63. 8. siun/C sint;/C .
69. 12. quod ' quoad .
⋅ n : 2 c
...
17. ∘−⋚∟ ∉−≓∙∙
k: 70. 7. −∙∙ 2 2
∣⊂≄∣∘⊰≼∣∁≖−⋁≖⋟ −−⋅⊋−⋅ lOg(k —P )
--
. ∙∙∽∙∙⋅ −∙− ↼∙ -
∙−⊣ERRATA CORRIGE
pag. lin .
kdv Ka dy
70. 12 .
katus kype &quot;
71. 13 et 14. KC Kc
72. 23. u = ułgosinc u = a + g9 sinc .
75. 23. pressioni r.gMcosc' pressioni gMcosc' .
87. 2. Denotet enim a Denotet enim x .
IG &quot; IG &quot;
27. = IC &quot; : IC =
2 2
110. 9. R = Rcosa R = R , cosa
111. 5. 1880 to 288q'to .
da dala 146. 8 .
idt
148. 12. 69.º* 69. *
149. 6. x = A'B' - B'r - A'B ' - A'M x=A''B'-B'r=A'B'-A'M .''
x' ? c
x
151. 2 .
ic
(de)
Centre Ide
i
152. 78 et 20. r2
153. 22. (69)
154. 17.
72.°*
157. 8. SD
161. 26. 16931100
193. 23. u : M ' : fle ..
205. 7. aequeus
208. 14. aia
r2
( 70 ) .
72.*
GD .
19631100 .
Me : No :
aqueus ,
Q:.
i
3
ERRATA CORRIGE
∙∙∙ ∙∙∙∄≾≖∠≀⇂↗ *
kæ-I—uz ⋅
Kc
uza-l—gg sinc .
23. pressioni ngMcosc' pressioni gMcosc' .
pag. lin
70. 12:
liti—v- kZ.-v2
71. 13 et 14. KC
72. 23. uzu-l—gasinc '
75.
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. 27.:IC :::—2— . 10: -—2-— .
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.
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208.1.£. a:a' «:a' .
Denotet enim æ .</text>
      <sha1>bysld27ncvn69ltjno852vkjsd1plim</sha1>
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      <text bytes="724882" xml:space="preserve">== PRAEFATIO ==
Rerum naturalium ordinem considerare, Deumque in iis mirifice operantem intueri, proprium est verae sapientiae, quam Philosophia profitetur. Haec scientia, quae dicitur Physica , inter scientias homine dignissimas. atque inter praecipua Dei dona jure commendatur: ecquid enim potest esse praestantius aut utilius quam divinae sapientiae opera, Deumque ipsum suas in natura perfectiones ostentantem contemplari? An quod Deus omnipotentia sua non judicavit indignum in iis quae creavit , quod in iis quae regit et gubernat attentione sua dignatur Providentia Dei, hoc nos meditari supervacaneum atque otiosum iudicabimus? Otiosam illam dicerem Physicam, quae ita immoraretur in Operis consideratione, ut opificis non perpetue suspicere! industriam: caecus est, qui Deum non videt in natura ejusque providentiam ac sapientiam non admiratur. Similem illum dixerim homini, qui librum ob Oculos apertum tenens characterum elegantiam contemplatur, numerat verba; sensum non penetrat. ⋅Neque vero minus utilis Naturae cognitio ministris Ecclesiae quam caeteris hominibus existimanda est: imo et hanc ipsis maxime necessariam duxerim hoc praesertim tempore cum homines vano inflati doctrinae apparatu scientias pro viribus adversus Religionem convertant , et Phyicam praecipue revelationi satagant opponere , vereque Opponi non desinant clamare eoram ignaris. Cum igitur se linguae impiae in injuriam Religionis armant, pudeat hominem Religionis amantem, et eo charactere insignitum qui ipsum Religionis statuat defensorem, aut turpiter obmutescere, aut Religionem. male defensam hominibus impiis vanum jactantibus triumphum, et ministrorum ignorantiam in Religionis opprobrium vertentium, deridendam proponere. Quod si nihil a viro ecclesiastico quaereretur aliud in Physica quam honesta mentis recreatio, justaque serii laboris intermissio, quid potest esse homini laboribus sacris defuncto aut jucundius aut dignius quam otium inutile, ac saepae periculosum, otio erudito et physico commutare? Quam multa offeret naturae spectaculum , ipsiusque arcanorum inquisitio, quae laudabilem nutriant curiositatem ,utiles praebeant considerationes, suaves atque innoxias pariant delicias! Etenim nullam esse in Philosophia partem quae majori voluptate quam naturae contemplatio animum compleat, Tullii auctoritate facile possem confirmare. Haec nihilominus voluptas non sine studio ac labore comparari potest: sapientissimus enim rerum omnium creator et rector Deus, quamvis sensuum nostrorum perceptioni corpora haec sensibilia subjecerit, illorum tamen naturam et vim mira quadam sepsit caligine, ut quicumque ad eam penitus scrutandam accedunt, in media luce densis veluti tenebris repente se obrui sentiant; quas tenebras etiamsi non ita discutere possimus ut claram ac certam rerum omnium scientiam assequamur, attamen si nos studii, diligentiae ac laboris non piguerit, ita tenebras attenuari experiemur ut multarum rerum certam cognitionem , plurimarum admodum probabilem obtineamus . Ad occulta Naturae arcana inquirenda duae sunt viae, quas eximii ingenii vir Franciscus Baconus de Verulanio notavit in novo scientiarum organo lib . 1. aphor, 19. Prima, qua a sensu et particularibus incipientes advolamus ad axiomata maxime generalia; atque ex iis principiis, eorumque immota veritate judicamus et invenimus axiomata media . Altera a sensu et particularibus excitat axiomata ascendendo continenter et gradatim , ut ultimo loco perveniatur ad ma
xime generalia. Primam viam plures arripuerunt, qui conjecturas non admodum graves sequuti , atque experientia
non satis accurata innixi generalia axiomata nimia festina
tione constituerunt , iisque naturalium causarum et effe
ctyum vim omnem contineri voluerunt; atque in iis tuen
∼∣⋁in Physica quam honesta mentis recreatio, iustaque serii
laboris intermissio, quid potest esse homini laboribus sacris defuncto aut iucundius aut dignius quam Otium inutile,
ac saepae periculosum, Otioterudito et physico commutare?
Quam multa offeret naturae speCtaculum, ipsiusque arca-
norum inquisitio, quae laudabilem nutriant curiositatem,
utiles praebeant considerationes, suaves atque innoxias pariant delicias! Etenim nullam esse in Philosophia partem
quae majoriivoluptate quam naturae contemplatio animum
compleat, Tullii auctoritate facile possem confirmare.
Haec nihilominus voluptas non' sine studio. ac labore
Comparari potest: sapientissimus enim rerum omnium creator
et rector Deus, quamvis sensuum nostrorum perceptioni
corpora haec sensibilia subiecerit, illorum tamen naturam
et vim miraaquadam sepsit caligine, ut quicumque ad eam
penitus scrutantium accedunt, in media luce densis veluti
tenebris repente se obrui sentiant; quas tenebras etiamsi
non ita discutere possimus ut claram ac certam rerum o-
mnium scientiam assequamur, attamen si nos studii. dili-gentiae ac laboris non piguerit, ita tenebras attenuari ex-
periemur ut multarum rerum certam-cognitionem , pluri-
marum admodum probabileur Obtineamus. Ad Occulta Naturae arcana inquirenda duae sunt viae, quas eximii inge-
nii vir Franciscus Baconus de,.Verulamio notavit&quot; in novo
scientiarum organo lib. ∎∙ aphor. 19. Prima, qua a sensu et particularibus incipientes advolamus.ad axiomata-; mas-
xime generalia; atque ex iis principiis, eorumque-[immota
veritate iudicamus et invenimus axiomata 'media. :Altera'a
sensu et particularibus excitat axiomata ascendendo contio
nenter et gradatim, ut ultimo loco perveniatur adfusa-i-
xime generalis. Primam viam plures arripueruut, qui' con-
iecturas non admodum graves,,s'equuti , atque experientia
non satis accurata innixi generalia axiomata nimia festina-
tione constituerunt,, iisque naturalium causarum et eil'e-
ctuum vim omnem contineri voluerunt; atque in iis tuen-dis totam ingenii aciem intendentes inciderunt in perver
sam philosophandi rationem , adeo ut rerum universitatem
commenti sint omnino aliam ac éa est. Altera aliis placuit
via, qui rerum naturam in rebus ipsis longa observatione
atque accurata experientia quaerendam esse statuerunt; isti
effectuum et causarum naturalium indolem singillatim in
quirere coeperunt, corporum texturám intimam , configu
rationem, motum scrutati sunt; atque ex his; aliisque in
numeris diligentissime observatis Naturae leges deduxerunt,
verasque rerum causas deprehenderunt. ' Hoc pacto plura
nostris temporibus certissima sunt , quae olim ignoraban
tur : alia probabili conjectura assecuti sumus : adhuc ta
men non pauca restant ambigua et involuta ; sed non de
erunt fortasse, qui aliquando novas in lucem eruant veritates.
Notandum vero Naturam ubique mathematicam esse ,
eamque velle absque Mathesi expiscari perinde fore, ait Gul
lielminus , ac sine cruribus ambulare. Porro tota Naturae
compago soliditate constal geometrica, resque physica rei geo
metricae unitur mystico quodam nexu, quem soli mathe
maticae Analysi datum est reserare: Analyseos ductu ex ob
servationibus et experimentis ab iis quae in promptu sunt
ad reconditiora deferimur, et ab exteriori venustate ad in
ternos naturae sinus. Observationes quidem virium existentiam demonstrant, sed proprium est Analyseos pate
facere virium leges, curvas a corporibus in gyrum actis
descriptas, circumvolutionum ac motuum inaequalitates et
aberrationes; quae omnia aspectabilem hanc Mundi ma
chinam maxime illustrant . Quid ab Analyseos indole magis abhorrere videbatur quam electricorum phoenomenorum congeries, cum et innumera prope sint, et innumeris obnoxia conditionibus? Ad electricas tamen vires expendendas accessit Analysis , earumque non paucos effectus leges
que aequationibus definivit.
Ut Tyronum , qui physicis praelectionibus in Romano
Soc. Jesu Collegio dant operam, commodo utilitatique ser
'
dis totam ingenii aciem intendentes inciderunt 'inrïperwe'r.»
sam philosophandi rationem, adeo ut rerum.:nniversitatem
commenti sint omnino aliam-ac ea est. .Altera aliis placuit
via, qui rerum naturamin: rebus ipsis longa-ObservatiOne
atque- accurata - experientia quaerendam, 'esse' statuerunt: :.i'sd
effectuum. ïet; causarum. naturalium 'indolem tsin'gillat'im in—
quirere coeperunt, corporumf-textuttam--imimdmf, configu-
rationem, motum scrutati sunt; atque ex his, aliisque-.in-
numeris diligentissime observatis Naturae leges deduxerunt,
verasque rerum causas deprehenderunt. 'Hoc pacto plura
nostris tempOribus certissima sunt, quae Olim ignoraban-
tur: alia probabili coniectura assecuti sumus : adhuc ta-
men non pauca restant— ambigua et.-involuta; sed non de-
erunt fortasse, qui aliquando novas in lucem eruant veritates.
Notandum vero Naturam ubique mathematicam esse,
eamque velle absque Mathesi expiscari perinde fore, ait Gul-
lielminus, ac sine cruribus ambulare. Porro tota Naturae
compago soliditate constat geometrica, resque physica rei geo-
metricae unitur mystico quodam nexu, quem soli mathe-
maticae Analysi datum est reserare: Analyseos ductu ex Ob-
servationibus et experimentis ab iis quae in promptu sunt
ad reconditiora deferimur, et ab exteriori venustate ad in-
ternqs naturae sinus. Observationes quidem virium exi-
stentiam demonstrant, sed prOprium est Analyseos pate-
facere virium leges, curvas a corporibus in gyrum actis
descriptas, circumvolutionum se motuum inaequalitates et
aberrationes; quae omnia aspectabilem hanc Mundi mn-
chinam maxime illustrant. Quid ab Analyseos indole ma-
gis abhorrere videbatur quam electricorum phoenomenorum
congeries, cum et innumera prope sint, et innumeris Ob-
noxia conditionibus? Ad electricas tamen vires eXpenden-
das accessit Analysis, earumque non paucos eil'ectus' leges-
que aequationibus definivit.
Ut Tyronum, qui physicis praelectionibus in Romano
Soc. Iesu Collegio dant operam, commodo utilitatique ser-VI
viam , in lucem edo illas Physicae partes , quae intimiori
vinculo nectuntur mathesi, quarumque explicandarum cu
ra mihi est demandata. A Mechanica exordior ; siquidem
reliquarum est veluti basis et fundamentum : caeterum sic
Physicae mathematicae tractationem concinnavi, ut quae aste
risco inveniuntur signata, possint ab iis Tyronibus omitti,
qui primo duntaxat Philosophiae anno mathematicis insti
tutionibus studuerunt.
VI
viam
,
in lucem edo illas Physicae partes
, quae intimiori
vinculo nectuntur mathesi, quarumque explicandarum cu-
ra mihi est demandata. A Mechanica exordiar.; siquidem
reliquarum est veluti basiset fundamentum: caeterum sic
Physicae mathematicae tractationem concinnavi, ut quae aste-
risco inveniuntur signata, pOssint ab iis Tyron'ibus omitti,
qui primo duntaxat Philosophiae anno mathematicis insti-
tutionibus studuerunt.

== MECHANICAE PRINCIPIA ==

=== Notiones Praeambulae ===
[[1|1]]. Moto puncto materiali, si ratio inter numericos spatii percursi ac respondentis temporis valores &lt;math&gt;s&lt;/math&gt; ac &lt;math&gt;t&lt;/math&gt; permanet eadem, motus dicitur uniformis; quod si ratio illa jugiter mutetur, motus dicitur varius, acceleratus nempe vel
retardatus, prout crescente
e crescit vel decrescit ipsa &lt;math&gt;\frac s t &lt;/math&gt;: porro motus rectilineus atque uniformis est simplicissimus
omnium motuum, quorum exsistit capax punctum materiale. In &lt;u&gt;motu uniformi&lt;/u&gt; ratio &lt;math&gt;\frac s t &lt;/math&gt; dicitur &lt;u&gt;velocitas&lt;/u&gt;; qua designata per &lt;math&gt;v&lt;/math&gt;, erit

&lt;math&gt;v = \frac s t .&lt;/math&gt;

Quoad punctum materiale, cujus massa seu quantitas materiae (&lt;math&gt;=m&lt;/math&gt;), et velocitas &lt;math&gt;=v&lt;/math&gt;, factum &lt;math&gt;mv&lt;/math&gt; appellatur quantitas motus.

[[2|2]]. Corpus de se est indifferens ad motam et ad quietem. Haec indifferentia sic probari potest ex natura loci: nequit corpus de se postulare at localiter moveatur nisi exigat natura sua non esse in loco ubi est, et locum in quo non est occupare; contra nequit corpus de se quietem
exigere nisi exigat natura sua esse potius in loco ubi est quam in loco quem occuparet si moveretur. Neutrum vero ex natura sua exigit corpus; cum enim omnia loca sint
ejusdem rationis, jam nulla datur ratio cur corporea substantia exigat esse potius uno in loco quam in alio: propterea etc.

[[3|3]]. Quae causae motum gignunt, accelerant, retardant, detorquent, eae vocantur potentiae seu vires. Plures potentiae corpori aut corporum systemati applicitae sese ita possunt impedire, ut nullus inde oriatur motus; tunc vero potentiae dicuntur constitutae in aequilibrio. Fac ut duae vires punctum materiale sollicitent in partes contrarias; si eae sunt in aequilibrio, dicentur aequales: pone duas, tres etc. . . : . ex ejusmodi viribus aequalibus applicari puncto materiali ita , ut in unam eamdemque rectam conspirent; inde habebis vim duplam , triplam etc. . . . Poterunt nempe vires omnes exprimi per numeros ; et consequenter repraesentari per lineas rectas istis numeris proportionales, quarum directiones cum ipsarum virium directionibus congruant. Mechanica tota est in aequilibrii ac
motus doctrina consideranda.

[[4|4]]. Finge tibi globum &lt;math&gt;A&lt;/math&gt; quiescentem e filo pendulum,
in quem impingat globus &lt;math&gt;B&lt;/math&gt; cum certo quodam velocitatis
gradu. Si nullam motui resistentiam afferret globus &lt;math&gt;A&lt;/math&gt;, eadem velocitate pergeret moveri &lt;math&gt;B&lt;/math&gt;, qua movebatur antea , secum pertrahendo globum &lt;math&gt;A&lt;/math&gt;: cur enim minueretur motus in &lt;math&gt;B&lt;/math&gt;, cum globus &lt;math&gt;A&lt;/math&gt; nihil obstaret illius motui , et ipse loco suo facile cederet? Iamvero si experientiam consulimus, multo aliter rem evenire comperiemus: cedit quidem loco suo globus &lt;math&gt;A&lt;/math&gt;, sed non sine detrimento motus
in &lt;math&gt;B&lt;/math&gt;, eoque majori quo majorem globus &lt;math&gt;A&lt;/math&gt; opponit massam impellenti se globo &lt;math&gt;B&lt;/math&gt;. Resistere igitur motui , status
que mutationi obniti concipitur &lt;math&gt;A&lt;/math&gt;, acquisitumque motum
resistentia sua destruere in &lt;math&gt;B&lt;/math&gt;. Motus habetur tamquam vis
activae effectus; quod autem vis activae effectum destruit,
potest et ipsum verae vis nomen accipere. In ipsis etiam
corporibus motis sese prodit ejusmodi vis: corpus certo quodam velocitatis gradu donatum, eumdem servabit nisi quem
inveniat obicem , nec ullum sui motus augmentum patietur nisi cum vis alienae in ipsum agentis detrimento; haud
aliter ac restitit primo motui dum quiesceret; ipso in motu resistit majori motui.

Non ergo praefata indifferentia sita est in non renitentia ad motum ex quiete, vel in non renitentia ad quietem ex mota, sed in eo quod corpus de se non magis ad
motum quam ad quietem tendat, nec magis resistat quieti
si fuerit in motu quam molui renitatur si fuerit in quiete.
Quoniam igitur ab ipsa materia nequit oriri ulla de
terminatio ( huc pertinet materiae inertia ) ad novum statum sive quietis, sive motas; profecto deficiente causa quae
materiale punctum determinet ad hunc potius quam ad
illum novum statum, punctum ipsum si in quiete sit quiescet semper, si ad motum semel fuit excitatum perget moveri cum eadem perpetuo velocitate et directione: porro
motus directio est recta linea, quam mobile aut describit,
aut describere nititur; primum obtinet in motibus rectilineis, secundum in curvilineis. 

[[5|5]]. Duo puncta materialia &lt;math&gt;H&lt;/math&gt; et &lt;math&gt;K&lt;/math&gt; ( fig 1. ) eamdem
massam habentia, eamdemque lineam communi vi &lt;math&gt;P&lt;/math&gt; incedentia, haud dubie conjunctim procedent: verum ubi puncto &lt;math&gt;K&lt;/math&gt; praeter &lt;math&gt;P&lt;/math&gt; applicetur et vis &lt;math&gt;Q&lt;/math&gt;, disjungetur illico &lt;math&gt;K&lt;/math&gt;
ab &lt;math&gt;H&lt;/math&gt;, et observator constitutus in &lt;math&gt;H&lt;/math&gt; deprehendet: motum puncti &lt;math&gt;K&lt;/math&gt; perinde ac deprehenderet si &lt;math&gt;H&lt;/math&gt; quiesceret et &lt;math&gt;K&lt;/math&gt;
moveretur sola &lt;math&gt;Q&lt;/math&gt;: sive nimirum ponatur &lt;math&gt;H&lt;/math&gt; moveri vi &lt;math&gt;P&lt;/math&gt;
et &lt;math&gt;K&lt;/math&gt; viribus &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt;, sive &lt;math&gt;H&lt;/math&gt; quiescere et &lt;math&gt;K&lt;/math&gt; moveri unica
&lt;math&gt;Q&lt;/math&gt;, idem in utroque casu, experientia teste , prodibit motus puncti &lt;math&gt;K&lt;/math&gt; quoad &lt;math&gt;H&lt;/math&gt;: huc spectat principium motus
relativi . Jamvero in secundo casu motus relativus soli &lt;math&gt;Q&lt;/math&gt; est
manifeste, adscribendus; idipsum ergo dicendum et in primo. Effectus videlicet a nova vi &lt;math&gt;Q&lt;/math&gt; genitus in puncto materiali &lt;math&gt;K&lt;/math&gt; idem est utcumque caeteroqui se habeat praecedens status ipsius &lt;math&gt;K&lt;/math&gt;: quod consequi videtur ex materiei
inertia. Etenim si variato statu praecedente variaret effectus ille, non aeque se haberet materia ad status omnes ,
punctumque materiale sibi commissum rediret tandem in
statum illum , ad quem magis tendit; sicque ab ipsa materia oriretur determinatio ad novum statum.

[[6|6]]. Exhibeant
&lt;math&gt;v&lt;/math&gt; et
&lt;math&gt;v^\prime&lt;/math&gt; velocitates, quas gignunt vires
&lt;math&gt;P&lt;/math&gt; et
&lt;math&gt;Q&lt;/math&gt;, sitque
&lt;math&gt;u&lt;/math&gt; velocitas
, quam generat vis assumpta pro
communi mensura (3) ipsarum
&lt;math&gt;P&lt;/math&gt; et
&lt;math&gt;Q&lt;/math&gt;; erunt (5)
&lt;math&gt;v = Pu,
v^\prime = Qu&lt;/math&gt;, unde:
&lt;math&gt;v:v^\prime=Pu: Qu=P: Q&lt;/math&gt;.
Permanente videlicet massa, vires erunt ut simplices velocitates: et quoniam permanente velocitate et variata massa, vis est ut massa ipsa; inferimus vires esse ut motus quantitates.

[[7|7]]. Dixi ([[4]]) tantam motus quantitatem excitari in globo
&lt;math&gt;A&lt;/math&gt; quantam ipse
&lt;math&gt;A&lt;/math&gt; resistendo destruit in globo
&lt;math&gt;B&lt;/math&gt;: atque huc spectat illud de actione et reactione principium,
quod sic enunciari solet &quot;actioni contraria semper et aequalis est reactio, sive duorum corporum actiones in se mutuo semper sunt aequales, et in contrarias partes diriguntur&quot;. Huic autem principio locus est in rerum natura sive corpora in contactu agant in se mutuo, sive dissitis e locis sese invicem ad status mutationem quocumque modo determinent.

Notetur illud: cum corpus omne obnitatur semper sui statos mutationi, inferimus ipsam status mutationem haud repente gigni a viribus extrinsecis, sed per gradus indefinitae attenuationis capaces: secus enim dicendum foret inesse materiei vim quamdam infinitam. Siquidem in hypothesi finitae mutationis instantaneae materies valeret opponere resistentiam finitam, labente tempusculo infinite quod nequit admitti. Verum quia vires quaedam tam cito gignunt mutationem status, ut eam in istanti videantur absolvere; inde fit ut vires dividi soleant in instantaneas, et continuas.

=== De virium compositione et resolutione, deque earum momentis et aequilibrio: aliquid quoque notatur de vecte, axe in peritrochio , trochlea etc. . . . ===
[[8|8]]. Fac ut per communem vim &lt;math&gt;P&lt;/math&gt; puncta &lt;math&gt;H&lt;/math&gt; et &lt;math&gt;K&lt;/math&gt; (fig. 2.)
determinentur ambo ad percurrendam motu uniformi rectam lineam &lt;math&gt;AB&lt;/math&gt; intra tempus &lt;math&gt;t&lt;/math&gt; , per &lt;math&gt;Q&lt;/math&gt; vero determinetur
&lt;math&gt;K&lt;/math&gt; ad percurrendam motu pariter uniformi rectam lineam 
&lt;math&gt;AD&lt;/math&gt; intra idem tempus &lt;math&gt;t&lt;/math&gt; ; et comple parallelogrammum &lt;math&gt;BD&lt;/math&gt;.
Ex principio motus relativi punctum &lt;math&gt;K&lt;/math&gt; in fine temporis &lt;math&gt;t&lt;/math&gt;
reperietur in &lt;math&gt;C&lt;/math&gt; ; ac proinde intra tempus &lt;math&gt;t&lt;/math&gt; percurret motu
uniformi diagonalem &lt;math&gt;AC&lt;/math&gt; : idem nimirum existet motus sive mobile feratur per diagonalem &lt;math&gt;AC&lt;/math&gt; velocitate
&lt;math&gt;\frac{AC}{t}&lt;/math&gt; ex vi 
unica impressa &lt;math&gt;R&lt;/math&gt;, sive conjunctis viribus &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; impellatur per latera &lt;math&gt;AB&lt;/math&gt; et &lt;math&gt;AD&lt;/math&gt; velocitatibus &lt;math&gt;\frac{AB}{t}&lt;/math&gt; et &lt;math&gt;\frac{AD}{t}&lt;/math&gt;; eritque (6)

&lt;math&gt;
R : P : Q : =AC: AB: AD.
&lt;/math&gt;

Hinc pro duabus viribus &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; poterit, substitui vis
&lt;math&gt;R&lt;/math&gt;; quae substitutio dicitur virium compositio : et viceversa
pro &lt;math&gt;R&lt;/math&gt; poterunt substitui duae &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt;; quae substitutio dicitur virium resolutio : &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; vocantur componentes, &lt;math&gt;R&lt;/math&gt;
resultans, vel etiam composita.

[[9|9]]. Haec notentur. 
1º. ex tribus &lt;math&gt;R&lt;/math&gt; , &lt;math&gt;P&lt;/math&gt; , &lt;math&gt;Q&lt;/math&gt; unaquae
vis potest repraesentari per sinum anguli, qui sub aliarum
directionibus continetur ; nam
&lt;math&gt;
R : P : Q = AC : DC: AD = \sin BAD : \sin CAD : \sin BAC .
&lt;/math&gt;

2°. Hinc &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; sunt reciproce ut perpendicula ,
quae a puncto quolibet resultantis &lt;math&gt;R&lt;/math&gt; ducuntur ad ipsarum
&lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; directiones .

3º. Denotante &lt;math&gt;i&lt;/math&gt; angulum interceptum directionibus
virium &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt;, triangulum &lt;math&gt;A C D&lt;/math&gt; praebebit

&lt;math&gt;
RP = P^2 + Q^2 - 2PQ \cos(180^{\circ} - i) = P^2 + Q^2 + 2PQ \cos i.
&lt;/math&gt;

4°. Si punctum &lt;math&gt;K&lt;/math&gt; ( fig. 3. ) urgetur viribus &lt;math&gt;KA, KB, KC, KD&lt;/math&gt; etc. . . , ducantur autem &lt;math&gt;Aa&lt;/math&gt; parallela et
&lt;math&gt;= KB&lt;/math&gt; , &lt;math&gt;Aa'&lt;/math&gt; &lt;math&gt;Aa''&lt;/math&gt; parallela et &lt;math&gt;= KC&lt;/math&gt; , &lt;math&gt;a'' a''' &lt;/math&gt; parallela et&lt;math&gt; =
KD&lt;/math&gt; , etc. vis cunctis aequivalens exhibebitur manifeste per lineam rectam &lt;math&gt;Ka'''&lt;/math&gt;, quae jungit punctum &lt;math&gt;K&lt;/math&gt;
et extremitatem &lt;math&gt;a'''&lt;/math&gt; ultimae &lt;math&gt;a''a'''&lt;/math&gt; . Porro linearum rectarum aequalium et parallelarum projectiones sive in recta
quavis &lt;math&gt;EE'&lt;/math&gt;, sive in plano quovis , sunt aequales et parallelae: hinc virium &lt;math&gt;KA, KB, KC, KD&lt;/math&gt;, etc. . . projectiones in recta &lt;math&gt;EE'&lt;/math&gt; simul sumptae aequabuntur projectionibus rectarum &lt;math&gt;KA, Aa', a'a'', a'' a'''&lt;/math&gt; etc. , in eadem &lt;math&gt;EE'&lt;/math&gt; simul pariter sumptis. Harum vero projectionum summa nihil est aliud nisi projectio resultantis &lt;math&gt;Ka'''&lt;/math&gt; : igitur projectio resultantis aequabitur projectionibus componentium &lt;math&gt;KA, KB, KC, KD&lt;/math&gt;, etc. , in summam contractis , si modo habeatur ratio signorum, ut censeantur negativae, quae vergunt v. gr. ad &lt;math&gt;E&lt;/math&gt;, habitis pro positivis, quae versus &lt;math&gt;E'&lt;/math&gt; se dirigunt.

5°. In hypothesi trium duntaxat virium &lt;math&gt;KA, KB , KC&lt;/math&gt;, quisque videt aequipollentem vim repraesentatum iri
per diagonalem parallelepipedi sub lateribis &lt;math&gt;KA, KB, KC&lt;/math&gt;.

6°. Si punctum &lt;math&gt;K&lt;/math&gt; urgetur vi &lt;math&gt;Ka'''&lt;/math&gt;, constructo ad
libitum polygono &lt;math&gt;a''' a'' ... K&lt;/math&gt;, ductaque &lt;math&gt;KD&lt;/math&gt; parallela et &lt;math&gt;=a''' a''&lt;/math&gt; , &lt;math&gt;KC&lt;/math&gt; parallela et &lt;math&gt;= a'' a'&lt;/math&gt;, &lt;math&gt;KB&lt;/math&gt; parallela et &lt;math&gt;= a' A&lt;/math&gt; etc.
resolvetur &lt;math&gt;Ka'''&lt;/math&gt; in &lt;math&gt;KD, KC, KB&lt;/math&gt;, etc ....

7°. Ad resolvendam &lt;math&gt;Ka'''&lt;/math&gt; in ternas sese dirigentes
juxta datas rectas &lt;math&gt;KB, KC, KD&lt;/math&gt;, satis erit per &lt;math&gt;a'''&lt;/math&gt; ducere
tria plana parallela planis &lt;math&gt;BKC, CKD, BKD&lt;/math&gt;; hoc pacto
exsurget parallelepipedum , cujus latera apud &lt;math&gt;K&lt;/math&gt; exhibebunt
( 5°) quaesitas vires componentes.

8°.  Puncta &lt;math&gt;B, C, D, K&lt;/math&gt;, ponantur inter se rigidis
lineis connexa: manentibus virium directionibus, si ternae
componentes intelliguntur applicitae punctis &lt;math&gt;B, C, D&lt;/math&gt;, adhuc
iis manifeste aequipollebit &lt;math&gt;Ka'''&lt;/math&gt; . Inferimus vim quamvis
&lt;math&gt;Ka'''&lt;/math&gt; resolvi posse in ternas, quae et sint applicitae tribus punctis ad libitum sumptis ( si sumuntur ita , ut in
eorum plano inveniatur etiam punctum &lt;math&gt;K&lt;/math&gt;, non debebit
&lt;math&gt;Ka'''&lt;/math&gt; esse extra id planum ) et sese dirigant juxta rectas ab istiusmodi punctis ductas ad punctum &lt;math&gt;K&lt;/math&gt; , cui applicatur ipsa &lt;math&gt;Ka'''&lt;/math&gt;.

9º. Dato systemate punctorum materialium rigidis
lineis inter se firmiter connexorum ( huc spectat corpus solidum ) respondentibusque viribus sollicitatorum; quia possunt (8º. ) singulae vires resolvi in cernas applicitas tribus
punctis &lt;math&gt;A , B, C&lt;/math&gt; ad libitum sumptis, poterunt ( 4°) omnes
traduci ad aequipollens trium virium systema.

10° . Per unam ex hisce tribus viribus duc planum , quod secet reliquas duas : vis , per quam ducitur planum , poterit resolvi ( 4° ) in binas , applicitas intersectionum punctis. Inde fit, ut vires omnes solidum corpus sollicitantes
traduci etiam possint ad aequipollens duarum virium systema.

[[10|10]]. Facile est determinare quandonam plures potentiae eidem puncto applicitae in aequilibrio permaneant.
Binas potentias pro lubito sumptas compone, et pro illis
aequipollentem substitue , atque id iterato donec ad duas
devenias. Si hae directe contrariae et aequales inveniuntur, constabit omnes potentias in aequilibrio consistere .

Facile etiam intelliges quanam ratione inveniri possit potentia duabus &lt;math&gt;AH, BF&lt;/math&gt; ( fig. 4. ) in eodem plano jacentibus,
rectaeque rigidae &lt;math&gt;AB&lt;/math&gt; applicatis aequivalens, et aequilibrium
obtineri; productis (?) enim directionibus &lt;math&gt;AH, BF&lt;/math&gt; donec concurrant
in &lt;math&gt;C&lt;/math&gt;, transferantur potentiae in punctum &lt;math&gt;C&lt;/math&gt;. Sumptis in earum directionibus &lt;math&gt;CH' = AH&lt;/math&gt;, et &lt;math&gt;CF' = BF&lt;/math&gt;, istae componantur. Facto parallelogrammo &lt;math&gt;CF'KH'&lt;/math&gt;, cujus diameter &lt;math&gt;CK&lt;/math&gt; equivalentem vim repraesentabit, haec producatur donec concurrat in &lt;math&gt;D&lt;/math&gt; cum &lt;math&gt;AB&lt;/math&gt;; perspicuum est potentiam &lt;math&gt;KC&lt;/math&gt; translatam in &lt;math&gt;DL&lt;/math&gt; et rectae &lt;math&gt;AB&lt;/math&gt; applicitam in D aequipollere duabus &lt;math&gt;AH , BF&lt;/math&gt;. Quare si &lt;math&gt;AB&lt;/math&gt; in puncto &lt;math&gt;D&lt;/math&gt; sustentetur, potentiae &lt;math&gt;AH, BF&lt;/math&gt; in aequilibrio quiescent; et constabit quam potentiam exerceat punctum &lt;math&gt;D&lt;/math&gt;, nimirum aequalem et oppositam potentiae aequivalenti &lt;math&gt;DL&lt;/math&gt;. Ad positionem puncti &lt;math&gt;D&lt;/math&gt; quod pertinet, concipiamus ex eo duci duo perpendicula &lt;math&gt;p&lt;/math&gt; et &lt;math&gt;q&lt;/math&gt; , alterum in &lt;math&gt;AH&lt;/math&gt; , alterum
in &lt;math&gt;BF&lt;/math&gt; ; sintque &lt;math&gt;AH = P , BF = Q&lt;/math&gt;, longitudo &lt;math&gt;AB = h ,
AD = x&lt;/math&gt;, angulus &lt;math&gt;BAC =a&lt;/math&gt;, angulus &lt;math&gt;ABC = b&lt;/math&gt; : erunt &lt;math&gt;p =
x \sin a, q = ( h- x ) \sin b &lt;/math&gt;, ideoque &lt;math&gt;\frac p q = \frac{x \sin a}{(h - x ) \sin b}  &lt;/math&gt;
Sed( 9.2º )
&lt;math&gt;\frac p q = \frac Q P &lt;/math&gt;;
igitur
&lt;math&gt; \frac Q P = 
\frac{x \sin a}{
(h- x ) \sin b} &lt;/math&gt;, unde &lt;math&gt; \frac{x }{
(h- x ) } = \frac{Q \sin b}{P \sin a }.   &lt;/math&gt;

Quod spectat ad angulum interceptum resultante &lt;math&gt;CK&lt;/math&gt; et
data recta &lt;math&gt;AC&lt;/math&gt; , is dicatur &lt;math&gt;\alpha&lt;/math&gt; : erit ( 9. 1º )
&lt;math&gt;P : Q = \sin BCD : \sin ACD= \sin ( 180^{\circ}- a - b- \alpha) : \sin \alpha&lt;/math&gt;,
unde

&lt;math&gt;\tan \alpha =\frac{
Q \sin ( a + b )}{
P - Q \cos ( a + b )}.&lt;/math&gt;

Quod vero spectat ad resultantem
&lt;math&gt;CK ( = R )&lt;/math&gt; , habemus ( 9. 3º )
&lt;math&gt;R^2 = P^2 + Q^2 - 2P Q\cos ( a + b )&lt;/math&gt;.

Penultima formula traduci potest ad

&lt;math&gt;\cos \alpha = \frac{P - Q \cos ( a + b )}{
R}.   &lt;/math&gt;

Haec subjungimus. 1º. Recta &lt;math&gt;AB&lt;/math&gt; rotetur circa &lt;math&gt;D&lt;/math&gt;, ut ejus extrema puncta &lt;math&gt;A&lt;/math&gt; et &lt;math&gt;B&lt;/math&gt; eodem tempusculo infinitesimo
describant circulares arcus infinitesimos &lt;math&gt;Aa', Bb'&lt;/math&gt;; ex &lt;math&gt;a'&lt;/math&gt; et &lt;math&gt;b'&lt;/math&gt;
duc perpendicula &lt;math&gt;a'a'', b'b''&lt;/math&gt; in directiones virium &lt;math&gt;AH ,
BF&lt;/math&gt; ; sintque &lt;math&gt;Aa'' = p' , Bb'' = q'&lt;/math&gt;: erunt &lt;math&gt;p' = Aa' \cos a'Aa''
= Aa' \cos ( DAa'' - 90^{\circ} ) = Aa' \sin DAa'' = Aa' \sin a ,
q'= Bb'\cos b'Bb'' = Bb'\cos (90^{\circ}-b) = Bb'\sin b&lt;/math&gt;; et consequenter
&lt;math&gt;\frac{p'}{q'}= \frac{Aa' \sin a}{
Bb' \sin b}=
\frac{AD \sin a}{BD \sin b} = \frac{x \sin a}{(h - x ) \sin b} =
\frac{Q}{P} .&lt;/math&gt;

Nihil sunt aliud &lt;math&gt;Aa'&lt;/math&gt; et &lt;math&gt;Bb'&lt;/math&gt; nisi spatiola tempusculo infinitesimo circa immobile punctum &lt;math&gt;D&lt;/math&gt; simul describenda ab &lt;math&gt;A&lt;/math&gt;
et &lt;math&gt;B&lt;/math&gt; in hypothesi turbati aequilibrii; quibus punctis &lt;math&gt;A&lt;/math&gt; et &lt;math&gt;B&lt;/math&gt;
applicantur vires &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt;: exhibent &lt;math&gt;p', q'&lt;/math&gt; illorum spatiolorum projectiones super ipsarum virium directionibus. Vires igitur &lt;math&gt;P, Q&lt;/math&gt; sese mutuo librantes circa &lt;math&gt;D&lt;/math&gt; erunt reciproce ut eae projectiones.

2º. Etiam sic : triangula &lt;math&gt;Aa'a'', DAh&lt;/math&gt; , itemque
&lt;math&gt;Bb'b'', DBh'&lt;/math&gt; habent latera sibi respective perpendicularia ; igitur &lt;math&gt;\frac {DA} {Aa'} = \frac{p}{p'} , \frac{DB} {Bb'} = \frac{q}{q'}&lt;/math&gt;.

Denotet &lt;math&gt;i&lt;/math&gt; valorem rationum aequalium &lt;math&gt; \frac{DA}{Aa'} , \frac {DB}{Bb'}   &lt;/math&gt;, projectio insuper &lt;math&gt;p'&lt;/math&gt; computata in ipsa directione respondentis potentiae &lt;math&gt;P&lt;/math&gt; censeatur positiva; projectio vero &lt;math&gt;q'&lt;/math&gt; computata in directione contraria illi , quam obtinet respondens  potentia &lt;math&gt;Q&lt;/math&gt; , censeatur negativa: erunt
&lt;math&gt;p = ip' , q = -
iq'&lt;/math&gt; ;
propterea
&lt;math&gt; \frac QP = \frac pq = -\frac{ip'}{iq'}= -\frac{p'}{q'}
, Pp' + Qq' = 0 .&lt;/math&gt; 

Huc spectat principium velocitatum &lt;u&gt;virtualium&lt;/u&gt;.

3º. Ex quovis puncto (&lt;math&gt;  M&lt;/math&gt; ) sive intra , sive extra
angulum &lt;math&gt; ACB&lt;/math&gt; , duc perpendicula &lt;math&gt; p'', q'' , r''&lt;/math&gt;  ad &lt;math&gt; P, Q, R&lt;/math&gt;  ; duc
quoque ab (&lt;math&gt;  M&lt;/math&gt; ) ad &lt;math&gt; C &lt;/math&gt; rectam ( &lt;math&gt; MC = c &lt;/math&gt; ), cui normaliter
insistat alia recta (&lt;math&gt;  E E' &lt;/math&gt; ) transiens per &lt;math&gt; C &lt;/math&gt;: singulis &lt;math&gt; P , Q,
R &lt;/math&gt; resolutis in duas , alteram juxta (&lt;math&gt;  CM &lt;/math&gt; ) , alteram juxta
( &lt;math&gt; EE'&lt;/math&gt;  ), expriment
&lt;math&gt; P\frac{p''}{c},Q\frac{q''}{c},R\frac{r''}{c} &lt;/math&gt; 
componentes juxta (&lt;math&gt;  EE'&lt;/math&gt;  ) . Quoad (&lt;math&gt;M&lt;/math&gt;) situm extra angulum &lt;math&gt;ACB&lt;/math&gt;, primae duae erant conspirantes; quoad (&lt;math&gt;M &lt;/math&gt;)
situm intra &lt;math&gt;ACB&lt;/math&gt; erunt contrariae : cum igitur &lt;math&gt;R&lt;/math&gt; resultet ex &lt;math&gt;P&lt;/math&gt;
et &lt;math&gt;Q&lt;/math&gt;, prodibit ( 9. 4° ) in primo casu
&lt;math&gt; P\frac{p''}{c}+Q\frac{q''}{c}=R\frac{r''}{c} &lt;/math&gt; et consequenter &lt;math&gt;Pp'' + Qq'' = Rr''&lt;/math&gt;,
in secundo.

&lt;math&gt; \pm(P\frac{p''}{c}-Q\frac{q''}{c}) = R\frac{r''}{c} &lt;/math&gt;, ideoque &lt;math&gt; \pm(Pp''-Qq'') = Rr'', &lt;/math&gt; sumptis signis vel superioribus , vel inferioribus , prout 
&lt;math&gt; P\frac{p''}{c} &gt;  &lt;/math&gt; vel &lt;math&gt; &lt;Q\frac{q''}{c} &lt;/math&gt;: rectangula &lt;math&gt; Pp'',Qq'', Rr'' &lt;/math&gt;  dicuntur momenta virium &lt;math&gt;P, Q, R&lt;/math&gt; quoad punctum (&lt;math&gt;M&lt;/math&gt;). Hinc stabilitur
illud: momentum resultantis &lt;math&gt;R&lt;/math&gt; aequatur summae ex momentis componentium &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; si &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; in eamdem plagam circa (&lt;math&gt;M&lt;/math&gt;) nituntur movere puncta , quibus applicitae
sunt; aequatur differentiae si nituntur movere in plagas
contrarias.

4° . Idipsum facile extenditur ad quemvis numerum
virium &lt;math&gt;P, Q, S, V, U&lt;/math&gt;, ... in dato plano jacentium : fac
v. gr. ut ternae &lt;math&gt;P, Q, S&lt;/math&gt;, in unam eamdemque plagam circa ( &lt;math&gt;M&lt;/math&gt; ) nitantur movere puncta, ad quae sunt applicitae; caeterae vero &lt;math&gt;V, U&lt;/math&gt;, ... in plagam contrariam ; sitque &lt;math&gt;L&lt;/math&gt; resultans ex &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt;; &lt;math&gt;N&lt;/math&gt; resultans ex &lt;math&gt;L&lt;/math&gt; et &lt;math&gt;S&lt;/math&gt;, ac proinde ex
&lt;math&gt;P, Q, S&lt;/math&gt;; &lt;math&gt;O&lt;/math&gt; resultans ex reliquis &lt;math&gt;V, U&lt;/math&gt;. . . Erurt
&lt;math&gt;Ll''= Pp'' + Qq'', Nn'' = Ll'' +Ss''&lt;/math&gt; ;
et consequenter
&lt;math&gt;Nn'' = Pp'' + Qq'' + Ss''&lt;/math&gt; :
simili modo obtinetur
&lt;math&gt;Oo'' = Vv'' + Uu''+&lt;/math&gt; .
Iam si &lt;math&gt;R&lt;/math&gt; exhibet resultantem ex &lt;math&gt;N&lt;/math&gt; et &lt;math&gt;O&lt;/math&gt; , ideoque ex &lt;math&gt;P,
Q, S, V , U &lt;/math&gt;, ... ; cum sit 
fist the
&lt;math&gt;Rr'' = \pm ( Nn'' - Oo'' ) &lt;/math&gt;,
erit quoque
&lt;math&gt;Rr'' = ( Pp'' + Qq'' + Ss'' - Vv'' - Uu'' - ... ) .&lt;/math&gt;

5º. Fac ut &lt;math&gt;R&lt;/math&gt; transeat per (&lt;math&gt;M&lt;/math&gt;) ; erit &lt;math&gt;r'' = 0&lt;/math&gt;: propterea
&lt;math&gt;Pp'' + Qq'' + Ss'' - Vv'' - Uu'' - ... = 0&lt;/math&gt; ;
viriumque systema consistet in aequilibrio circa immobile
punctum (&lt;math&gt;M&lt;/math&gt;) . Vocatur (&lt;math&gt;M&lt;/math&gt;) centrum momentorum.

6º. Habemus ( 2 )
&lt;math&gt;p'' = ip' , q'' = iq' , s'' = is' , v'' = -iv', u'' = - iu', ...&lt;/math&gt;
Traducetur igitur aequatio ( 5°) ad
&lt;math&gt;Pp' + Qq' + Ss' + Vv' + Uu' + ... = 0&lt;/math&gt;

7° Vires &lt;math&gt;AH, BF&lt;/math&gt; haud jacentes in eodem plano nequeunt ad unicam vim aequipollentem traduci. Si enim tradu
ad aequipollentem &lt;math&gt;DL&lt;/math&gt;, poterit etiam ex quodam
istius puncto ad quoddam punctum componentis v . gr.
&lt;math&gt;BF&lt;/math&gt; duci recta linea haud occurrens alteri componenti &lt;math&gt;AH&lt;/math&gt;:
fac ut haec recta linea evadat immobilis ; elisa &lt;math&gt;DL&lt;/math&gt;, emerget aequilibrium; sed elisa quoque &lt;math&gt;BF&lt;/math&gt;, et salva &lt;math&gt;AH&lt;/math&gt;, ex hac ultima emerget motus. In ea ergo qua sumus hypothesi de
traductione virium &lt;math&gt;AH, BF&lt;/math&gt; ad unicam &lt;math&gt;DL&lt;/math&gt; obtinebit simul
aequilibrium et motus in eodem systemate: quod nequit esse; ideoque etc. ... 

8°. Patet solidum liberumque corpus haud consistere
in aequilibrio, nisi binae aequipollentes ( 8. 10° ) vires ,
ad quas traducuntur vires omnes corpus ipsum sollicitantes , sint aequales, contrariae, jaceantque in directum .

9°. Patet quoque solidum corpus, mobile dumtaxat
circa punctum fixum , consistere in aequilibrio, si eae binae
vires aequipollentes et jaceant in eodem plano ( 7º) , et suppeditent resultantem , quae transeat per punctum illud .

10°. Solidum corpus ponatur mobile dumtaxat circa
rectam fixam &lt;math&gt;AZ&lt;/math&gt; ( fig.5 ), sintque &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; binae aequipollentes
vires, ad quas traducuntur ( 9. 10° ) vires omnes corpus ipsum sollicitantes. Duc planum &lt;math&gt;XOY&lt;/math&gt; et normaliter insistens
rectae &lt;math&gt;AZ&lt;/math&gt;, et secans in punctis v. gr. &lt;math&gt;B, C&lt;/math&gt; directiones virium &lt;math&gt;P ( = BB' ), Q ( = CC' )&lt;/math&gt;: poterit &lt;math&gt;P&lt;/math&gt; resolvi in duas , alteram &lt;math&gt;BB'''&lt;/math&gt; perpendicularem plano &lt;math&gt;XO&lt;/math&gt;Y , alteram &lt;math&gt;BB''&lt;/math&gt; jacentem in ipso &lt;math&gt;XOY&lt;/math&gt;; similiter &lt;math&gt;Q&lt;/math&gt; poterit resolvi in duas ,
alteram &lt;math&gt;CC'''&lt;/math&gt; perpendicularem eidem &lt;math&gt;XOY&lt;/math&gt;, alteram &lt;math&gt;CC''&lt;/math&gt; in
eo jacentem . Binae &lt;math&gt;BB''', CC'''&lt;/math&gt;, utpote parallelae ad rectam fixam &lt;math&gt;AZ&lt;/math&gt;, peribunt elisae : in ea igitur qua sumus hypothesi
haud consistet solidum corpus in aequilibrio, nisi resultans
ex &lt;math&gt;BB'' , CC''&lt;/math&gt; transeat per aliquod punctum &lt;math&gt;O&lt;/math&gt; rectae fixae
&lt;math&gt;AZ&lt;/math&gt; ; et consequenter ( 9. 2° ) , ductis ex &lt;math&gt;O&lt;/math&gt; in istas vires
perpendicalis &lt;math&gt;b, c&lt;/math&gt;, nisi valeat aequatio

&lt;math&gt;\frac{b}{c} = \frac{CC''}{BB''} &lt;/math&gt;:

producta ex &lt;math&gt;b&lt;/math&gt; in &lt;math&gt;BB''&lt;/math&gt; et ex &lt;math&gt;c&lt;/math&gt; in &lt;math&gt;CC''&lt;/math&gt; dicuntur momenta virium &lt;math&gt;P&lt;/math&gt; et &lt;math&gt;Q&lt;/math&gt; quoad &lt;math&gt;AZ&lt;/math&gt; . Si &lt;math&gt;P&lt;/math&gt; v. gr. , applicita ad punctum
&lt;math&gt;B'&lt;/math&gt;, est parallela plano &lt;math&gt;XOY&lt;/math&gt;, applicabuntur ad &lt;math&gt;B&lt;/math&gt; duae quaelibet vires &lt;math&gt;H &lt;/math&gt; et &lt;math&gt;- H&lt;/math&gt; aequales, contrariae et parallelae axi &lt;math&gt;AZ&lt;/math&gt;; tum una ex iis v. gr. &lt;math&gt;H&lt;/math&gt; componetur cum &lt;math&gt;P&lt;/math&gt; : vis inde
resultans poterit transferri in punctum v. gr. &lt;math&gt;B&lt;/math&gt; plani &lt;math&gt;XOY&lt;/math&gt;,
ibique resolvi in binas, alteram &lt;math&gt;BB''' ( = H )&lt;/math&gt; parallelam rectae &lt;math&gt;AZ&lt;/math&gt;, alteram &lt;math&gt;BB'' ( = P )&lt;/math&gt; jacentem in &lt;math&gt;XOY&lt;/math&gt;; eritque
&lt;math&gt;b. BB ' ( = b. P )&lt;/math&gt; momentum vis &lt;math&gt;P&lt;/math&gt; quoad &lt;math&gt;AZ&lt;/math&gt;. Quisque autem videt , si per &lt;math&gt;B '&lt;/math&gt; ducitur planum parallelum plano &lt;math&gt;XOY&lt;/math&gt;,
et ex pancto, ubi istud novum planum secat rectam &lt;math&gt;AZ&lt;/math&gt;, demittitur perpendiculum in vim &lt;math&gt;P&lt;/math&gt; applicitam ad &lt;math&gt;B '&lt;/math&gt;, ejusmodi
perpendiculum nihil fore aliud nisi &lt;math&gt;b&lt;/math&gt;; ita ut, sive momen
tum sumatur apud planum &lt;math&gt;XOY&lt;/math&gt;, sive apud illud alterum
planum parallelum ipsi &lt;math&gt;XOY&lt;/math&gt;, perinde sit.

[[11|11]]. Fac ut vis ( 10) &lt;math&gt;BF&lt;/math&gt; (fig. 4) revolvatur circa punctum &lt;math&gt;B&lt;/math&gt;,
donec evadat parallela vi &lt;math&gt;AH&lt;/math&gt;; erit &lt;math&gt;a + b = 180^{\circ}&lt;/math&gt;, ideo
que &lt;math&gt;\sin b = \sin (180^{\circ} - a ) = \sin a&lt;/math&gt; si vires ad eamdem
plagam obvertantur ; &lt;math&gt;a + b = 360^{\circ}&lt;/math&gt;, ideoque &lt;math&gt;\sin b =
\sin ( 360^{\circ} - a ) = - \sin a &lt;/math&gt; si ad contrarias plagas. In primo
igitur casu exsistent.

&lt;math&gt;\frac{x}{h-x} = \frac{Q}{P}, x=  \frac{hQ}{P+Q}, R = P + Q , \cos \alpha =\frac{P+Q}{R}=1.&lt;/math&gt;

In secundo

&lt;math&gt;\frac{x}{h-x} = -\frac{Q}{P}, x=  \frac{hQ}{Q-P}, R = \pm(P - Q) , \cos \alpha =\frac{P-Q}{R}=\pm 1.&lt;/math&gt;

valet signum superius ubi &lt;math&gt; P &gt; Q&lt;/math&gt;, inſerius ubi &lt;math&gt;P &lt; Q&lt;/math&gt;;
siquidem &lt;math&gt;P, Q, R&lt;/math&gt; denotant hic virium dumtaxat intensitates.
Inferimus illud; resultans ex duabus parallelis viribus adaequat istarum vel summam, vel differentiam , prout
vel ambae conspirant in eamdem plagam, vel altera in unam
et altera in contrariam plagam; ipsis insuper componentibus
viribus est parallela , et ad eam plagam semper obversa ,
quam respicit major ex componentibus illis ; transit denique
per ejusmodi punctum in directione &lt;math&gt;AB&lt;/math&gt;, quod distet a
punctis applicationis componentium in reciproca earum ratione : istud punctum appellari solet centrum virium parallelarum ; estque invariabile, modo et respectiva virium
positio et ipsarum ratio non mutentur.
Si &lt;math&gt;P = Q&lt;/math&gt;, in secundo casu nulla exsistet resultans.
Non est enim ratio in ea qua sumus hypothesi cur ad
plagam unius potius componentis quam ad alterius componentis plagam sese dirigat resultans. Formulae praebent &lt;math&gt;x= \infty, R =0.&lt;/math&gt;

Etsi vires &lt;math&gt;AH&lt;/math&gt; et &lt;math&gt;BF&lt;/math&gt; (fig.6) parallelae, aequales et contrariae nequeunt librari unica vi , utpote omni resultante destitutae; librabuntur nihilominus duabus aliis viribus &lt;math&gt;AH'&lt;/math&gt; et &lt;math&gt;BF'&lt;/math&gt; parallelis, aequalibus, contrariis, et in plano &lt;math&gt;HABF&lt;/math&gt; iacentibus, dummodo ductis ex &lt;math&gt;A&lt;/math&gt; in &lt;math&gt;BF BF'&lt;/math&gt;
perpendiculis &lt;math&gt;AO&lt;/math&gt; et &lt;math&gt;AO'&lt;/math&gt;, exsistat
&lt;math&gt;BF. AO=BF'. AO'&lt;/math&gt;:
tunc enim , ductis ex &lt;math&gt;B&lt;/math&gt; in &lt;math&gt;AH&lt;/math&gt; et &lt;math&gt;AH'&lt;/math&gt; perpendiculis &lt;math&gt;BC&lt;/math&gt; et
&lt;math&gt;BC'&lt;/math&gt;, ob &lt;math&gt;BF =  AH , BF' = AH' , AO = BC , AO' = BC'&lt;/math&gt;
erit quoque
&lt;math&gt;AH. BC=AH'. BC'&lt;/math&gt;;
et consequenter ( 9. 2°) resultans ex &lt;math&gt;AH&lt;/math&gt; et &lt;math&gt;AH'&lt;/math&gt; sese diriget a puncto &lt;math&gt;A&lt;/math&gt; ad punctum &lt;math&gt;B&lt;/math&gt;, simulque resultans ex &lt;math&gt;BF&lt;/math&gt; et
&lt;math&gt;BF'&lt;/math&gt; sese diriget a puncto &lt;math&gt;B&lt;/math&gt; ad punctum &lt;math&gt;A&lt;/math&gt; ; istiusmodi praeterea resultantes sunt manifeste aequales: iccirco etc. ... Systema itaque virium &lt;math&gt;AH', AF'&lt;/math&gt; aequipollebit systemati virium
&lt;math&gt;AH , AF&lt;/math&gt; ; poteritque alterum ( mutatis ejus directionibus
in contrarias partes ) alteri substitui. Consequitur posse binas vires parallelas, aequales et contrarias transferri ab
una positione ad alteram in proprio ipsarum plano, variata simul virium et magnitudine , et directione ; modo tamen productum ex communi earum valore in mutuam distantiam maneat constans.

[[12|12]]. Sint nunc plures vires parallelae &lt;math&gt;P, P ', P ''&lt;/math&gt;, ... variis
solidi corporis punctis applicitae , quarum aliae conspirent
in unam plagam , aliae in plagam contrariam . Componendo &lt;math&gt;P&lt;/math&gt; v . gr. et &lt;math&gt;P'&lt;/math&gt; in unicam &lt;math&gt;R '&lt;/math&gt;, &lt;math&gt;R'&lt;/math&gt; et &lt;math&gt;P'&lt;/math&gt; in unicam &lt;math&gt;R''&lt;/math&gt; , &lt;math&gt;R''&lt;/math&gt; et &lt;math&gt;P'''&lt;/math&gt;
in unicam &lt;math&gt;R''' &lt;/math&gt;, etc. , ... facile devenies ( 11 ) ad illud : resultans
&lt;math&gt;R&lt;/math&gt; ex pluribus viribus parallelis adaequat differentiam inter
summam conspirantium in unam plagam et summam conspirantium in plagam contrariam ; ipsis insuper componentibus viribus est parallela , et ad eam plagam obvertitur , quam
respicit major ex illis summis .

Hinc si vires tendentes in unam plagam censentur positivae , tendentes vero in plagam contrariam negativae , obtinebit aequatio
&lt;math&gt;R = P + P' + P'' + ... (a )&lt;/math&gt;.
Ad haec : denotantibus (fig .7) &lt;math&gt;A, B, D &lt;/math&gt;, ... puncta , quibus applicantur parallelae vires &lt;math&gt;P , P ', P''&lt;/math&gt;, ... , et &lt;math&gt;AB , BD &lt;/math&gt;. ..
rigidas rectas jungentes puncta illa , cum transeant &lt;math&gt;R ', R '' &lt;/math&gt;, ...
per ejusmodi puncta &lt;math&gt;K , H &lt;/math&gt;, ... , quorum positiones sive in
rectis &lt;math&gt;AB , KD &lt;/math&gt;, ... sive in earum prolongationibus unice pendent a conditionibus
&lt;math&gt;P ' :R'= AK :AB , P'' : R'' = HK : KD,&lt;/math&gt; etc. ... ,

seu

&lt;math&gt;P: P'+P= AK : AB , P'' : P + P' + P'' = HK : KD&lt;/math&gt;, etc. ... ,

devenietur etiam ad illud : in systemate parallelarum viriam
habetur constans et immutabile centrum , per quod semper transit resultans &lt;math&gt;R&lt;/math&gt; , quacumque ceteroqui ratione componentes vires volvantur circa puncta quibus applicitae sunt ,
modo et maneant parallelae , et applicitae iisdem punctis in
iisdem respective distantiis.

[[13|13]]. Ducto quolibet plano &lt;math&gt;MQ&lt;/math&gt;, demittantur in illud ex
punctis &lt;math&gt;A , B , D,&lt;/math&gt; ... perpendicula &lt;math&gt;AM ( =z) , BN ( = z; ) , DQ
( = z''), ...&lt;/math&gt; ; sive ( 12) &lt;math&gt;K , H &lt;/math&gt;, ... sint in rectis &lt;math&gt;AB , KD &lt;/math&gt;, ... .
sive in earum prolongationibus , demittantur quoque in
idem &lt;math&gt;MQ&lt;/math&gt; ex istis punctis perpendicula &lt;math&gt;KL , HO &lt;/math&gt;, ... ; per
ipsa &lt;math&gt;K , H &lt;/math&gt;, ... agantur rectae &lt;math&gt;RS , TU &lt;/math&gt;, ... , prima rectae
MN parallela et perpendiculis &lt;math&gt;AM , BN&lt;/math&gt; occurrens in &lt;math&gt;R , S &lt;/math&gt;,
secunda rectae &lt;math&gt;LQ&lt;/math&gt; parallela et perpendiculis &lt;math&gt;KL , DQ&lt;/math&gt; occurrens in &lt;math&gt;T , U &lt;/math&gt;, etc ... Erunt
&lt;math&gt;AR = MR - AM = KL - z, BS = BN - NS =z' - KL,
DU=UQ-DQ=HO-z'', KT = KL - LT = KL - HO &lt;/math&gt;; etc ....
Jamvero ( 11 )
&lt;math&gt;BS:AR = BK :AK = P : P' ,DU :KT = DH :HK = P + P':P''&lt;/math&gt;,etc ...,
ideoque
&lt;math&gt;AR.P = BS.P', DU.P'' = KT (P + P'), &lt;/math&gt;etc....
Igitur
&lt;math&gt;(KL- z) P = (z' -KL )P',(HO- z'') P'' = (KL-HO)(P + P'),&lt;/math&gt;etc....
unde
&lt;math&gt;KL (P + P') = zP + z'P', HO (P + P + P'' ) = KL (P + P') +
z'' P '' = zP + z' P' +z'' P'',&lt;/math&gt; etc.
seu
&lt;math&gt;KL. R ' = zP + z' P', HO. R'' = zP + z'P' + z''P'' , &lt;/math&gt;etc....

Generatim exhibente &lt;math&gt;z_{\mathrm I}&lt;/math&gt;, perpendiculum ex centro omnium datarum virium parallelarum ductum in &lt;math&gt;MQ&lt;/math&gt; , habebimus

&lt;math&gt;z_{\mathrm I} R = zP + z' P' + z'' P'' + z''' P ''' + ... :&lt;/math&gt;
rectangula &lt;math&gt;z_{\mathrm I} R , zP&lt;/math&gt;, dicuntur momenta virium &lt;math&gt;R , P&lt;/math&gt;, ...
quoad plapum &lt;math&gt;MQ&lt;/math&gt;.

Haec notentur: 1° Etsi non omnia puncta , quibus applicantur parallelae vires &lt;math&gt;P , P', P'' &lt;/math&gt;... sita sunt supra planum &lt;math&gt;MQ&lt;/math&gt;
adhuc tamen algebraica summa rectangulorum sub &lt;math&gt;P , P'&lt;/math&gt; ... et
respondentibus perpendiculis ductis in &lt;math&gt;MQ&lt;/math&gt; ex punctis illis '''adaequabit''' rectangulum sub resultante &lt;math&gt;R&lt;/math&gt; et perpendiculo ducto ex centro ipsarum &lt;math&gt;P, P' , &lt;/math&gt;... in idem &lt;math&gt;MQ&lt;/math&gt;; moto enim &lt;math&gt;MQ&lt;/math&gt; versus ea puncta ita , ut maneat sibi parallelum , atque a primitiva positione recedat intervallo &lt;math&gt;h&lt;/math&gt; , si nova perpendicula exhibentur per &lt;math&gt;k, k', k '', ... k_{\mathrm I}&lt;/math&gt; erunt &lt;math display=''inline''&gt;k = z - h , k' = z'- h , k'' = z'' - h, ...  k_{\mathrm I} = z_{\mathrm I} - h &lt;/math&gt;; hinc &lt;math&gt;(k_{\mathrm I} +h) R = (k + h) P + ( k' + h) P' + (k'' + h ) P'' + &lt;/math&gt;... 

est autem ( 12.''a'') &lt;math&gt;hR =h (P + P' + P'' + ...) = hP + hP' + hP'' + ...&lt;/math&gt;;

igitur

&lt;math&gt;k_{\mathrm I} R = kP +k'P' + k'' P'' + ... &lt;/math&gt;

ubi &lt;math&gt;k, k ', k'', ... k_{\mathrm I}&lt;/math&gt; possunt esse vel positiva , vel negativa.

2° Praeter &lt;math&gt;MQ&lt;/math&gt; seu &lt;math&gt;XOY&lt;/math&gt; ( Fig.8 ) concipiantur duo alia plana &lt;math&gt;XOZ , YOZ&lt;/math&gt;; quod autem in ordine ad &lt;math&gt;XOY&lt;/math&gt; est, sit &lt;math&gt;z, z',... z_{\mathrm I} &lt;/math&gt;, sit &lt;math&gt;x, x',... x_{\mathrm I} &lt;/math&gt; in ordine ad &lt;math&gt;YOZ &lt;/math&gt;, et &lt;math&gt;y, y',... y_{\mathrm I} &lt;/math&gt; in ordine ad &lt;math&gt;XOZ&lt;/math&gt;; qua ratione assequuti sumus
&lt;math&gt;z_{\mathrm I}R=zP+z'P'+z''P'' + ...,&lt;/math&gt;
eadem assequemur (a')
&lt;math&gt;x_{\mathrm I}R=xP+x'P'+x''P'' + ...
y_{\mathrm I}R=yP+y'P'+y''P'' + ...&lt;/math&gt;

3° Si compendii causa per  &lt;math&gt;\Sigma P &lt;/math&gt; exprimitur summa
potentiarum &lt;math&gt;P, P', P'', &lt;/math&gt;
et per &lt;math&gt;\Sigma_x P, \Sigma_y  P, \Sigma_z P &lt;/math&gt; designantur summae rectangulorum sub potentiis et respectivis 
perpendiculis , formulae ( a' ) scribi poterunt in hunc modum ( 12. ''a'')
&lt;math&gt;x_{\mathrm I}\Sigma P = \Sigma_x P, y_{\mathrm I}\Sigma P = \Sigma_y P ,z_{\mathrm I}\Sigma P = \Sigma_z P, &lt;/math&gt;
unde
&lt;math&gt;x_{\mathrm I} = \frac{\Sigma_x P}{ \Sigma P} , y_{\mathrm I}= \frac{\Sigma_y P}{ \Sigma P},z_{\mathrm I}= \frac{\Sigma_z P}{ \Sigma P} &lt;/math&gt;
In hypothesi planorum &lt;math&gt;XOY , XOZ , YOZ&lt;/math&gt; orthogonalium ,
&lt;math&gt;x_{\mathrm I}, y_{\mathrm I}&lt;/math&gt; et &lt;math&gt;z_{\mathrm I}&lt;/math&gt;, erunt orthogonales coordinatae , quibus determinatur positio centri parallelarum virium .

4.° Aequatio
P + P + P + ... ... = o ( a &lt;nowiki&gt;''&lt;/nowiki&gt; )&lt;nowiki&gt;''&lt;/nowiki&gt;
manifeste denotat unam quamvis ex datis potentiis v. gr. P
esse aequalem et contrariam resultanti R, ex reliquis omni
bus P' , P &lt;nowiki&gt;''&lt;/nowiki&gt; , ... Ponamus XOY perpendiculare , et XOZ , YOZ&lt;nowiki&gt;''&lt;/nowiki&gt;
parallela directioni potentiarum ; in hac hypothesi erunt
P et R, directe contrariae si perpendicula x et y spectantia
ad punctum , cui applicalur P , spectent ambo ad centrum
quoque virium p ', P &lt;nowiki&gt;''&lt;/nowiki&gt;, ... , si nempe habeantur&lt;nowiki&gt;''&lt;/nowiki&gt;
x R , = x'P' + x &lt;nowiki&gt;''&lt;/nowiki&gt; P t ... ,&lt;nowiki&gt;''&lt;/nowiki&gt;
y R, =ÝP' +y&lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; + .
seu , ob R,
x P + x' P ' + x &lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; +
yP + ' P ' + y &lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; +
-P
= 0,
0; }(cm

5. ° Quibus positis , stabilitur illud : parallelarum
virium systema consistit in aequilibrio sub duabus conditio
nibus simul explendis ; altera est , ut evanescat earum sum
ma : altera ut evanescat summa ex earum momentis in ordi
ne ad duo plana ipsis viribus parallela. Si parallelae vires
inveniuntur omnes in eodem plano, secunda conditio jam de
18
tur summae rectangulorum sub potentiis et reSpectivis perpen-
diculis, formulae (a') scribi poterunt in hunc modum (12. a)
a:, EP :ZxP,y,ZxP: ZJP, z.l 2P:ZzP,
unde u
∙∙∙ zxp ∙∙ \sum∫ M) (0 )
∙−− ⋅ −\sum−⇂⋅−↗∫≖ \sum⇂≀ .z,--—— ZP
ln hypothesi planorum XOT, XOZ , TOZ orthogonalium ,
x, ,y, , et 2! erunt orthogonales coordinatae, quibus deter-
minatur positio centri parallelarum Vtrium.
43 Aequatio
P gr ≖⋡⋅−⊦∙∙⋅−−∙∶∘ (a&lt;nowiki&gt;'''&lt;/nowiki&gt;)&lt;nowiki&gt;'''&lt;/nowiki&gt;
manifeste denotat uuam quamvis-ex datis potentiis v. gr. P
esse aequalem et contrariam resultanti R, ex reliquis omni-
bus P', P&lt;nowiki&gt;''&lt;/nowiki&gt;, Ponamus XOV perpendiculare , et XOZ ,&lt;nowiki&gt;''&lt;/nowiki&gt;
ïOZ
parallela directioni potentiarum; in hac hypothesi erunt
P et B[ directe contrariae si perpendicula x et y spectantia
ad punctum , cui applicatur P ,
spectent ambo ad centrum
quoque virium P',P&lt;nowiki&gt;''&lt;/nowiki&gt;, , si nempe habeantur&lt;nowiki&gt;''&lt;/nowiki&gt;

&lt;nowiki&gt;::&lt;/nowiki&gt; Bl :x'F—I-x&lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; ⊣−∙∙∙∙ Ja, ∶−−∫∣⊉≀−⊢∜∣∣≖≻∥−⊢∙∙∙∙

seu,ob B' :—P,
xP—- x'P'-- x&lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; −∙∙ :o,
yp ——y' PI ...—7&lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; −∙∙ ∙∙∙ :0' )
(a&lt;nowiki&gt;''&lt;/nowiki&gt;)
5.&lt;nowiki&gt;''&lt;/nowiki&gt; Quibus positis
, stabilitur illud : parallelarum
virium systema consistit in aequilibrio sub duabus conditio-
nibus simul explendis; altera est
, ut evanescat earum sum-
ma :altera ut evanescat summa ex earum momentis in ordi-
ne ad duo plana ipsis viribus parallela. Si parallelae vires
inveniuntur omnes in eodem plano, secunda conditio iam de19
9
se explebitur quoad istud planum , satisque erit ut explea
tur quoad aliud tantummodo planum .
6. Etsi vires P, P' , P &lt;nowiki&gt;''&lt;/nowiki&gt;, ... non sunt parallelae , pos
sunt tamen reduci ad terna ejusmodi systemata , quorum pri.
mum coalescat ex viribus parallelis axi OZ , secundum ex
viribus jacentibus in plano XOY simulque parallelis axi OY ,
tertium ex viribus agentibus juxta axem OX. 

Ut demonstretur assertio , resolve unam quamvis ex datis viribus v. gr.

P applicitam puncto A in tres X, Y, Z, respective parallelas
axibus Ox, OY, OZ; ad punctum A applica duas vires H et
- H aequales , contrarias , et parallelas axi OZ ; compone
X ( = AC ) et H sese dirigentem juxta AE , sitque AB dire
ctio resultantis ; produc BA donec in N occurrat plano XOY ;
transfer in N resultantem illam , et sic translatam resolve in
binas , alteram parallelam axi OX , alteram axi OZ ; prodi
bunt componentes X ( = NC = AC ) , H ( = ND ) , qua
rum primam transfer in C ut sit C'C' ( = NC' ) = X; ad C
applica binas vires K et — K aequales , 'contrarias et pa
rallelas axi OY ; compone X ( = .CC ') et K sese dirigen
tem juxia C'F , sitque C'L directio resultantis ; produc LC
donec in V occurrat axi OX ; transfer in V novam istam re
sultantem , et sic translatam resolve in binas , alteram juxta
ox , alteram parallelam axi OY ; emergent componentes
X ( = VV' = CC&lt;nowiki&gt;''&lt;/nowiki&gt; ) , K ( = VF '): compone nunc Y et - H ;
produc directionem resultantis donec rectae C' F occurrat
v . gr. in N ' ; hanc resultantem transfer in N ' , et sic traus
latam resolve in duas , alteram parallelam axi OY , alteram
axi OZ ; exurgent componentes Y et -H applicitae puucto N:
hoc pacto vi P poterunt substitui sex vires
Z, H, — H applicitae punctis A, N, N' et parallelae axi Oz,
K, Y - K applicitae punctis V, C' et parallelae axi OY ,
X applicita puncto V et agens juxta OX .
Consimiles operationes cum possint instaurari quoad P', P ” ...
non pluribus opus est , at pateat veritas assertionis .
19
se explebitur quoad istud planum , satisque erit ut explea-
tur quoad aliud tantummodo planum .
6.o Etsi vires P, P', P&lt;nowiki&gt;''&lt;/nowiki&gt;, non sunt parallelae ,pos-
sunt tamen reduci ad terna eiusmodi systemata
, quorum pri-
mum coalescat ex viribus parallelis axi OZ , secundum ex
viribus jacentibus in plano XOT simulque parallelis axi Oï
,
tertium ex viribus agentibus juxta axem OX. Ut demon-
stretur assertio , resolve unam quamvis ex datis viribus v. gr.
P applicitam puncto A in tres X, T, Z, respective parallelas
axibus OX, OT, OZ; ad punctum A applica duas vires H et
∙∙∙ H aequales , contrarias ,
et parallelas axi OZ ; compone
X (: AC) et H sese dirigentem iuxta AE .sitque AB dire-
ctio resultantis; produc BA donec in N occurrat planc XOT;
transfer in N resultantem illam
,
et sic translatam resolve in
binas , alteram parallelam axi OX , alteram axi OZ; prodi-
bunt componentes X (: NC':AC )
, H (: ND) , qua-
rum primam transfer in C' ut sit C'C&lt;nowiki&gt;''&lt;/nowiki&gt; (: NC' :) X; ad C'
applica binas vires K et —K aequales ,
'contrarias et parallelas axi OV; compone X (:.C' C&lt;nowiki&gt;''&lt;/nowiki&gt;) et K sese dirigen-&lt;nowiki&gt;''&lt;/nowiki&gt;
tem juxt'a C'F , sitque C'L directio resultantis ; produc LC'
donec in V occurrat axi OX ; transfer in V novam istam re-
sultantem ,
et sic translatam resolve in binas , alteram juxta
OX, alteram parallelam axi OV ; emergent componentes
X (:VV':C' C&lt;nowiki&gt;''&lt;/nowiki&gt;) ,K (:VF'): compone nunc V et —H;
produc directionem resultantis donec rectae C' F occurrat
v. gr. in N'; hanc resultantem transfer inN' , et sic traus-
latam resolve' tn duas
, alteram parallelam axi OV, alteram
axi OZ ; exurgeut componentes ?et —H applicitae puncto N':
hoc pacto vi P poterunt substitui sex vires
Z,,H — H applicitae punctis A, N, Net parallelae axi OZ,
K, ï— K applicitae punctis V, C' et parallelae axi OV,
X applicita puncto V et agens juxta OX.
Consimiles operationes cum possint instaurari quoadP' ,P&lt;nowiki&gt;''&lt;/nowiki&gt;,... non pluribus opus est
,
ut pateat veritas assertionis.20
7. Axes OX , OY, OZ sumantur orthogonales ; erit
H : X = ND : NC' NC zX Z :
H ,
et consequenter perpendicula ducta ex N in plana YOZ , XOZ
exprimentur per
2X
H
g ;
erit quoque
H : Y = AC ' : C'N ' = 2 : C'N' =
2Y
H
ac proinde perpendicula ducta ex N' in eadem plana YOZ ,
XOZ exprimentur per
x18+1;
insuper
Vi : Ci = VV' : VF' , seu x - OV : y = X , K ,
ex qua eruitur perpendiculum ductum ex Vin planum YOZ,
nempe
OV =
y X
K
8 . '* Quod in ordine ad Pest X, Y, Z, H, K, sit X ', Y ,
Z ', H , K ' in ordine ad P ', sit X ”, Y &lt;nowiki&gt;''&lt;/nowiki&gt;, Z&lt;nowiki&gt;''&lt;/nowiki&gt;, H ” , K &lt;nowiki&gt;''&lt;/nowiki&gt; in or&lt;nowiki&gt;''&lt;/nowiki&gt;
dine ad P, etc. ... Systema ( 6&lt;nowiki&gt;''&lt;/nowiki&gt;) virium parallelarum axi OZ
consistet in aequilibrio sub tribus istis conditionibns ( 59)
2 + Z ' + Z &lt;nowiki&gt;''&lt;/nowiki&gt; +... + H + HP + H &lt;nowiki&gt;''&lt;/nowiki&gt; + .- H - H²- H &lt;nowiki&gt;''&lt;/nowiki&gt; -... = 0 ,
x2+x+2 + .. + ( x -7 ) +la
ZX
H
- ) H +
'
x H - X'H '-...
20
7
∙∘∙ Axes OX, 07, OZ sumantur orthogonales ;erit
H:X:ND:NC': -Nc': f—X
...-7
et consequenter perpendicula ducta ex N in plana ïOZ, XOZ
exprimentur per
zX x——s.7-i
eritquoque
H. r:.tcx ea:: aut: 2?
—, H
ac proinde perpendicula ducta ex N' in eadem plana TOZ,
XOZ exprimentur per
T
xsf'l'ïïi—i insuper
Vi:C'i:VV':VF',senx—OV:J:X,K. ↴
ex qua eruitur perpendiculum ductum ex Vin planum TOZ, )-
nempe )
∘∇∶∙≖−⋅\sum⋮∙ K
8. 01: Quod in ordine adPestX, T, Z, H, K, sit X'.ï',
Z', H', K' in ordine ad P', sit X&lt;nowiki&gt;''&lt;/nowiki&gt;, T', Z&lt;nowiki&gt;''&lt;/nowiki&gt;, H&lt;nowiki&gt;''&lt;/nowiki&gt;, K&lt;nowiki&gt;'''&lt;/nowiki&gt;m or-
dine ad P&lt;nowiki&gt;''&lt;/nowiki&gt; ,
etc.. «Systema (60) virium parallelamm axi OZ
consistet in aequilibrio sub tribus istis conditionibus (50)
z −⊦ ⊠∣⊣−≀∥⊹∙∙∙−⊦∐⊣−∐∣⊣−∐∥−⊦ ∙∙⋅− ⊟∙↧∓∣∙⊟∥∙∙∙∙ :
xZ-l—x'ZH—&lt;nowiki&gt;''&lt;/nowiki&gt;xl-(x- fl—iï' H—1-(x'
-
)&lt;nowiki&gt;''&lt;/nowiki&gt;IX, H'—)- ..

&lt;nowiki&gt;:&lt;/nowiki&gt; r H—x'H'-.. .
&lt;nowiki&gt;:&lt;/nowiki&gt; o.21

y2 +y2 + ... + 38+y'! '+ ..- ( o + #) : -
(-+ -+* ) r -...--.
seu
2 + 2 + Z&lt;nowiki&gt;''&lt;/nowiki&gt; + ... = 0 ,
x2–2x + x2–5x' + x Z&lt;nowiki&gt;''&lt;/nowiki&gt; _z&lt;nowiki&gt;''&lt;/nowiki&gt; X &lt;nowiki&gt;''&lt;/nowiki&gt; + ... = o,
y2 - zY + y'Z' — zY + y&lt;nowiki&gt;''&lt;/nowiki&gt; Z&lt;nowiki&gt;''&lt;/nowiki&gt; —z&lt;nowiki&gt;''&lt;/nowiki&gt; Y&lt;nowiki&gt;''&lt;/nowiki&gt; + ... :. =0.
360&lt;nowiki&gt;''&lt;/nowiki&gt;

Systema (69) coalescens ex viribus jacentibus in plano XOY
simulque parallelis axi OY consistet in aequilibrio 

sub duabus istis conditionibus ( 5° ) . 

Y - K + Y - K + Y&lt;nowiki&gt;''&lt;/nowiki&gt; _K&lt;nowiki&gt;''&lt;/nowiki&gt; + . + K + K + K + ... = a,
2{Y -K)+7 (9 –K)+- + (3 - X) +(37 )K +
seu
Y + Y + Y &lt;nowiki&gt;''&lt;/nowiki&gt; + ... = 0,
xY4yX + x'Y' — y'X ' + x &lt;nowiki&gt;''&lt;/nowiki&gt; Y&lt;nowiki&gt;''&lt;/nowiki&gt; -- y&lt;nowiki&gt;''&lt;/nowiki&gt; X &lt;nowiki&gt;''&lt;/nowiki&gt; +... =0.
0.}10&lt;nowiki&gt;''&lt;/nowiki&gt;)
Systema ( 6°) virium agentium juxta OX 

consistet in aequilibrio sub ista tantum conditione

X + X+X&lt;nowiki&gt;''&lt;/nowiki&gt;+... = o ( a &lt;nowiki&gt;''&lt;/nowiki&gt; &lt;nowiki&gt;''&lt;/nowiki&gt; ).
Inferimus solidum liberumque corpus viribus P , P' ,
P&lt;nowiki&gt;''&lt;/nowiki&gt; , ... sollicitatum haud mansurum in aequilibrio, nisi ex
pletis conditionibus ( a' ) , ( a &lt;nowiki&gt;''&lt;/nowiki&gt; ) , ( a &lt;nowiki&gt;''&lt;/nowiki&gt; &lt;nowiki&gt;''&lt;/nowiki&gt; ); quas ita scri
bimus ( 30 )
21
⊺∄⊹↗⋅∄∣−⊢∙∙∙⊹∫∐⊹∫∣∐∣⊹∙⋅∙− (
∫⊹.äï.) H —
(y'-]- ⋮⋅≨⋚∣⇀∙≻ H'—. ..

&lt;nowiki&gt;:&lt;/nowiki&gt; o,

seu
∅⊣−∅∣⊣−⊈∥⊹∙∙∙∶∘∙ ;
(a') xZ—zX-I-x'Z'— z'X'-l-x&lt;nowiki&gt;''&lt;/nowiki&gt;Z&lt;nowiki&gt;''&lt;/nowiki&gt;— z'X&lt;nowiki&gt;''&lt;/nowiki&gt;—-]-. ..

&lt;nowiki&gt;:&lt;/nowiki&gt; o,

yz -— zV-l—J'Z'—z'ï'—I- y&lt;nowiki&gt;''&lt;/nowiki&gt;Z&lt;nowiki&gt;''&lt;/nowiki&gt;—z&lt;nowiki&gt;''&lt;/nowiki&gt;ï&lt;nowiki&gt;''&lt;/nowiki&gt;-l— ..

&lt;nowiki&gt;:&lt;/nowiki&gt;

.
0.
Systema (60) coalescens ex viribus jacentibus in plano XOV
simulque parallelis axi OV consistet in aequilibrio sub dua-
bus istis conditionibus ( 5o )-
r—x-t-x'—x'—l-1z&lt;nowiki&gt;''&lt;/nowiki&gt;—xq-.. —[-K-]-K'-)-K&lt;nowiki&gt;''&lt;/nowiki&gt;—]—. ..

&lt;nowiki&gt;:&lt;/nowiki&gt;

0,
I
'
X
&lt;nowiki&gt;!
IX ∣
..
ï—KH—x (r—x ⊢⊢⋅∙∙−⊢ xli?) x-l-(x ïk.-')K ∙⊦∙∙≔∶⋅∘⋅
seu .
y—I—T-I- ï''&lt;/nowiki&gt;—l— ..

&lt;nowiki&gt;:&lt;/nowiki&gt;

.
0, '
h
0&lt;nowiki&gt;''&lt;/nowiki&gt;)
xï—yX-l—x'ïL-y'X' x&lt;nowiki&gt;''&lt;/nowiki&gt;ï&lt;nowiki&gt;''&lt;/nowiki&gt;-y&lt;nowiki&gt;''&lt;/nowiki&gt; &lt;nowiki&gt;''&lt;/nowiki&gt;
∙⊦∙∙∙∶−−∙ ∙
Systema (60) virium agentium iuxta OX consistet in ae-
quilibrio sub ista tantum conditione '
,x-t—X'-l-X&lt;nowiki&gt;''&lt;/nowiki&gt;-1-...:o (a&lt;nowiki&gt;'''&lt;/nowiki&gt;).
Inferimus solidum liberumque corpus viribus P, P',
P&lt;nowiki&gt;''&lt;/nowiki&gt;, .. . sollicitatum haud mansurum in aequilibrio, nisi ex-
pletis conditionibus (a' )
,
( a&lt;nowiki&gt;''&lt;/nowiki&gt; )
,
(av&lt;nowiki&gt;''&lt;/nowiki&gt; ); quas ita scri-
bimus ( 3&lt;nowiki&gt;''&lt;/nowiki&gt;)22
EX
=
0
, EY
=
0
, E2
=
0
,
}
(
a
&lt;nowiki&gt;''&lt;/nowiki&gt;
)
2
( zYX)
= 0,2
( x2–2X
)
= 0,2
(x2
– zY
)
=
0..
9
'

&lt;nowiki&gt;#&lt;/nowiki&gt; Denotet

R
' resultantem ex viribus primi syste
matis
(
6
°
)
,
R
&lt;nowiki&gt;''&lt;/nowiki&gt; ex viribus secundi
,
R
&lt;nowiki&gt;''&lt;/nowiki&gt; ex viribus tertii&lt;nowiki&gt;''&lt;/nowiki&gt;

&lt;nowiki&gt;;&lt;/nowiki&gt;

erunt
( 12

&lt;nowiki&gt;:&lt;/nowiki&gt;

a
)
R
= EZ
,
R
= EY
,
R
&lt;nowiki&gt;''&lt;/nowiki&gt;
= EX
.
Recta
, in qua agit
R
&lt;nowiki&gt;''&lt;/nowiki&gt;
, occurrit axi OX sub angulo recto

&lt;nowiki&gt;;&lt;/nowiki&gt;

et disignante
r
&lt;nowiki&gt;''&lt;/nowiki&gt; distantiam inter
O et punctum occursus
erit
( 2º
.
7
°
)
.
r
“
R
”
=
x
(Y
—
K
)
+
x'(
Y
’
–
K
”
+ ...
XK
(
s
–
&lt;nowiki&gt;''&lt;/nowiki&gt;
)
k
'
+ ...,ideoque
?&lt;nowiki&gt;''&lt;/nowiki&gt;=
EfxY
-yX
)
.
R
&lt;nowiki&gt;''&lt;/nowiki&gt;
tra
potest
R
'
&lt;nowiki&gt;''&lt;/nowiki&gt; transferri in illud punctum occursus sicque
componi cum
R
&lt;nowiki&gt;''&lt;/nowiki&gt; ut inde obtineatur resultans VR
&lt;nowiki&gt;''&lt;/nowiki&gt;
2
+
R
&lt;nowiki&gt;''&lt;/nowiki&gt;
3.
Iterum
( 9. 9º
. 10
°
.
) patet ergo vires
P
P
'
,
P
'
,
, ...
duci vel ad ternas
, vel ad binas aequipollentes
.
10.
°

&lt;nowiki&gt;#&lt;/nowiki&gt; Recta

, in qua agit
R
'
, occurrit normaliter plano
XOY

&lt;nowiki&gt;;&lt;/nowiki&gt; et designantibus

a
'
,
b
' coordinatas istius occursus
,
erunt
( 2º
. 7º
.
)
a
'
$
(xZ
- zX
)
R
'
6
Egy
Z
-
Y
) 1 R
Occurrent sibi mutuo R’et VR
”
?
+
R
&lt;nowiki&gt;''&lt;/nowiki&gt;
2, ac proinde
jacebunt in eodem plano
, quotiescumque
a
' et
b
' recident
in duas quasvis ex coordinatis illius rectae in qua agit
VR'
2
+
R
'&lt;nowiki&gt;'''&lt;/nowiki&gt;
2

&lt;nowiki&gt;;&lt;/nowiki&gt; propterea

22
ZX:0,Zï:o,ZZ:o,

&lt;nowiki&gt;;&lt;/nowiki&gt; (aVIII)

\sum (xï—ïyX):o.Z(xZ—zX):0,2(yZ—zï): 0.
9. 01: Denotet B' resultantem ex viribus primi syste-
matis '(60 ), B&lt;nowiki&gt;''&lt;/nowiki&gt; ex viribus secundi , B&lt;nowiki&gt;'''&lt;/nowiki&gt; ex viribus tertii;
erunt
( 12. a)
R,:Z Z, B&lt;nowiki&gt;''&lt;/nowiki&gt;:Zï, R&lt;nowiki&gt;'''&lt;/nowiki&gt;:ZX.
Recta, in qua agit R&lt;nowiki&gt;''&lt;/nowiki&gt;, occurrit axi OX sub angulo recto

&lt;nowiki&gt;;&lt;/nowiki&gt;

et disignante r&lt;nowiki&gt;''&lt;/nowiki&gt; distantiam inter
0 et punctum occursus,
ertt∙
(
2
∘
∘
. 7. ).
r&lt;nowiki&gt;''&lt;/nowiki&gt;R&lt;nowiki&gt;''&lt;/nowiki&gt;:x(ï—K)—)—x'(ï'—K')—-)—...-)- (x... JKX.)
K −⊦
(x'-— 2274.) K' −∙⊢∙
∙ .,ideoque r&lt;nowiki&gt;''&lt;/nowiki&gt;-— xwy-FK)

&lt;nowiki&gt;:&lt;/nowiki&gt;

potest B&lt;nowiki&gt;'''&lt;/nowiki&gt; transferri in illud punctum occursus
, sicque
componi cum B&lt;nowiki&gt;''&lt;/nowiki&gt; ut inde obtineatur resultans l/B&lt;nowiki&gt;''&lt;/nowiki&gt;3-l—B'&lt;nowiki&gt;'''&lt;/nowiki&gt;. Iterum (9. 90.100.) patet ergo vires P P', P', , .. .
tra-
duci vel ad ternas, vel ad binas aequipollentes.
↿∘∙∘⋕ Recta, in qua agit B', occurrit normaliter plano
XOT; et designantibus a', b' coordinatas istius occursus,
erunt (20. 70.)
↙⋮∣∙− X(xZ—zX) b' ∙∙∙
\sum (yZ—zï)
B'
' R'
⋅
Occurrent sibi mutuo B' et l/B&lt;nowiki&gt;''&lt;/nowiki&gt;2-)-B&lt;nowiki&gt;'''&lt;/nowiki&gt;2, ac proinde
iacebunt in eodem plano, quotiescumque a' et b' recident
in duas quasvis ex coordinatis illius rectae in qua agit
⇂∕ B&lt;nowiki&gt;''&lt;/nowiki&gt;2-I-B&lt;nowiki&gt;'''&lt;/nowiki&gt;2; propterea23
a ' - p &lt;nowiki&gt;''&lt;/nowiki&gt; : 6 = R : R &lt;nowiki&gt;''&lt;/nowiki&gt;
et consequenter
b' R' + ( r &lt;nowiki&gt;''&lt;/nowiki&gt; – a ' ) R &lt;nowiki&gt;''&lt;/nowiki&gt; = 0 ;
quae , adhibitis substitutionibus, traducitur ad
EXE(yZ — ZY) + EYXzX— « Z ) + EZE (xY yX ) = 0.
Sub hac ilaque conditione occurrent sibi mutuo vires
R' , V R &lt;nowiki&gt;''&lt;/nowiki&gt;2+ R &lt;nowiki&gt;''&lt;/nowiki&gt; ), dabuntque resultantem
VR2+ R &lt;nowiki&gt;''&lt;/nowiki&gt;2 + R &lt;nowiki&gt;'''&lt;/nowiki&gt; a = V (EX)2 + (PY )2+ ( EZ )2.
11 . '* Si nequeunt vires alium gignere motum ni
si circa immobilem axem Oz , quisque videt aequilibrii
conditiones redactum iri ad unicam r ' = 0 , seu ad quar
tam ( a &lt;nowiki&gt;''&lt;/nowiki&gt; ), Ad haec si nequeunt vires alium gignere mo
tum nisi circa immobile punctum 0 , redigentur aequili
brii conditiones ad r&lt;nowiki&gt;''&lt;/nowiki&gt; = 0 , a' = 0,6 = 0 , seu ad quar
tam , quintam et sextam ( a )
12. '* Fac ut duo solida corpora A et B ( Fig. 9) ,
alterum viribus P , P , P &lt;nowiki&gt;''&lt;/nowiki&gt;... sollicitatum , alterum viri
bus Q , , Q &lt;nowiki&gt;''&lt;/nowiki&gt; , ... , sese invicem aeque premendo apud da
lum mutui contactus punctum C maneant in aequilibrio ;
quaeritur istiusmodi pressionis magnitudo w. Duc per C pla
num tangens DD' , cui normaliter insistat recta ECE': de
notent fig, h coordinatas puncti C ; a , á , a &lt;nowiki&gt;''&lt;/nowiki&gt; angulos
interceptos recta CE axibusque orthogonalibus OX , OY ,
OZ ; et quod in ordine ad P' , P' , P &lt;nowiki&gt;''&lt;/nowiki&gt; , ... est X, Y , Z,
á , : , X , Y , Z , X ', . . . sit a , b , c , a , ... A , B ,
C , A ', ... in ordine ad C , Q , ... Pressio agens
versus E resolvetur in ternas
23
a'—- r&lt;nowiki&gt;''&lt;/nowiki&gt;: 6':a&lt;nowiki&gt;'''&lt;/nowiki&gt;: a&lt;nowiki&gt;''&lt;/nowiki&gt;
et consequenter
↘∙∙
b' B&lt;nowiki&gt;'''&lt;/nowiki&gt;—i— ( r&lt;nowiki&gt;''&lt;/nowiki&gt;-—a') B&lt;nowiki&gt;''&lt;/nowiki&gt;: 0

&lt;nowiki&gt;;&lt;/nowiki&gt;

quae , adhibitis substitutionibus, traducitur ad
ZXZUZ—zTH-ZïXzX—xZH-ZZZ (a.-T —JX):o.
Sub hac itaque conditione occurrent sibi mutuo vires
B', l/ B&lt;nowiki&gt;''&lt;/nowiki&gt;2-)- B&lt;nowiki&gt;'''&lt;/nowiki&gt;, dabuntque resultantem
⇂∕↓↖⋅≖−⊦↓⊰⋅⋅≖−⊢∐⋯≖∶ ⇂∕ (mun-)- (zx) ≕⊣−≺ \sum∣∠≻≖∙
11.&lt;nowiki&gt;''&lt;/nowiki&gt;; Si nequeunt vires alium gignere motnm ni-
si circa immobilem axem OZ, quisque videt aequilibrii
conditiones redactum iri ad unicam r&lt;nowiki&gt;''&lt;/nowiki&gt; :o
, seu ad quar-
tam (a'&lt;nowiki&gt;'''&lt;/nowiki&gt; ). Ad haec si nequeunt vires alium gignere mo-
tum nisi circa immobile punctum 0 , redigentur aequili-
brii conditiones ad r&lt;nowiki&gt;''&lt;/nowiki&gt;:0, a':o, b':o, seu ad quar-
tam, quintam et sextam ( am&lt;nowiki&gt;''&lt;/nowiki&gt;)
12.&lt;nowiki&gt;''&lt;/nowiki&gt;: Fac ut duo solida corpora A et B (Fig. 9),
alterum viribus P , P', P&lt;nowiki&gt;''&lt;/nowiki&gt;. .. sollicitatum , alterum viri-
bus Q, Q' , Q&lt;nowiki&gt;''&lt;/nowiki&gt;, .. ., sese invicem aeque premendo apud da-
tum mutui contactus punctum C maneant in aequilibrio;
quaeritur istiusmodi pressionis magnitudo 'a'. Duc per C pla-
num tangens DD', cui normaliter insistet recta ECE': de-
notent f, g ,
]: coordinatas puncti C; at, a', a&lt;nowiki&gt;''&lt;/nowiki&gt; angulos
interceptus recta CE axibusque orthogonalibns OX, Of
, OZ; et quod in ordine ad P'
, P', P&lt;nowiki&gt;''&lt;/nowiki&gt;,
...
est a:, 7, z,
x', . . X,ï, Z, X',. . . sita,b, c, a,... A,. B,
C, A', . . .
in ordine ad Q', Q, . . . Pressio :: agens
versus E resolvetur in ternas24
cosa , cose , a cos &lt;nowiki&gt;''&lt;/nowiki&gt; ,
agens vero versus E resolvetur in ternas
w cos ( 180 ° - « ) = - COS Q, a cos ( 180 ° - = - a coseć,
cos ( 180º – Ø&lt;nowiki&gt;''&lt;/nowiki&gt; ) W cos a :
in primo casu w librat ex hypothesi vires P, P,
in secundo vires Q, C, ... Igitur
EX +w cosa = 0, Erto cosá = 0 , xZ + w cosa &lt;nowiki&gt;''&lt;/nowiki&gt; = 0 ,
Σ Α W cosa = 0 ,
EB - cosa = 0,8C — a cos &lt;nowiki&gt;''&lt;/nowiki&gt; = 0 , E ( «Y -y X ) +
W ( f cos ' - g cosc) =0 ,
ElxZ - 2X ) + o ( f cosc &lt;nowiki&gt;''&lt;/nowiki&gt; — h cosc ) =0 , Ely2 -zY) +
wig cosa &lt;nowiki&gt;''&lt;/nowiki&gt; -hcosé ) =0 ,
(aB - 6A ) - ( fcos - g cos ) = 0 ,E (aC - A )
a ( f cosa &lt;nowiki&gt;''&lt;/nowiki&gt; -hcosa) = 0 , E (6C - cB ) - ( g cosa &lt;nowiki&gt;''&lt;/nowiki&gt; - h cosa') = 0 .
Eliminata , prodibunt undecim aequationes , inde
pendentes ab ipsa a , inter quantitates datas ; quibus ae
quationibus expletis, habebitur aequilibrium , poteritque ab
una quavis ex duodecim praecedentibus erui valor u .
13.0# Solidum corpus sollicitatum viribus , P P ',
P &lt;nowiki&gt;''&lt;/nowiki&gt; , ... delineatur duobus punctis fixis , sumptis in axe
v. gr. OZ ; sic facile determinabuntur pressiones M, N , L
et M ', N ', L' exercitae in puncta illa juxta coordinatos a.
xes Ox , OY, OZ. Exprimant m, n , l coordinatas unius
ex duobus panciis , et m ', ní, ľ coordinatas alterius. Quo
uiam spectari debent
24
a: cosa, a cosa', wcosac' ,
agens vero versus E' resolvetur in ternas
m cos(1800—a): — arcus a, acos (1800—at'):—w cosa',
a cos ( 180o -— ac&lt;nowiki&gt;''&lt;/nowiki&gt;) ∶≖ −meos ac&lt;nowiki&gt;''&lt;/nowiki&gt;:
in primo casu ut librat ex bypOthesi vires P, P', .
∙ ∙
,
in secundo vires Q, Q', . . .
Igitur
2X —l—w cosa::o, Zy-l—a cosa':o,ZZ—l-ar cosa:&lt;nowiki&gt;''&lt;/nowiki&gt;:o,
EA — z: cosa: :0,
2B —a cosa':o,ZC—z.ïcos at&lt;nowiki&gt;''&lt;/nowiki&gt;:o, Z (xï —7 X) −−∣−
15 (fcosa&lt;nowiki&gt;''&lt;/nowiki&gt;—-g cos ac) :0,
2( a:Z—zX)-I—w(fcosa&lt;nowiki&gt;''&lt;/nowiki&gt;—hcosa):o, XOZ—z?) −⊦
w(g cosa&lt;nowiki&gt;''&lt;/nowiki&gt;--hcosat'):o,
E( aB—bA) —w(fcos a'—gcosa):o,2 (aC—cA)—-
a(fcosa&lt;nowiki&gt;''&lt;/nowiki&gt;-h cos a):o,2 (bC-cB) -zz(gcos a&lt;nowiki&gt;''&lt;/nowiki&gt;- hcosac'):o.
Eliminata a, prodibunt undecim aequationes, inde-
pendentes ab ipsa a'
,
inter quantitates datas.; quibus ae-
quationibus expletis, habebitur aequilibrium, poteritque ab
una quavis ex duodecim praecedentibus erui valor a.
1394: Solidum corpus sollicitatum viribus, P P',
P&lt;nowiki&gt;''&lt;/nowiki&gt;, . . . detineatur duobus punctis fixis, sumptis, in axe
v. gr. OZ; sic facile determinabantur pressiones M, N, L
et M', N', L' exercitae in puncta illa juxta coordinatos a-
xes OX, DV, 02. Exprimant m, n, !coordinatas unius
ex duobus punctis, et m', n'. [ coordinatas alterius. Quo-
niam spectari debent25
M, N , -L, — M ', - N - L'
tanquam vires , quibus librantur caeterae P , P , P' ... ,
ac insuper m = 0 , n = o , m' =0 , n = 0 , necnon ( 110. )
(xY - yX ) = 0 : iccirco ( 8º. a &lt;nowiki&gt;''&lt;/nowiki&gt; )
EX - M - M ' = 0 , EY -N - N = 0,8Z -L - L ' = 0 ,
{ ( xZ - 2X ) + 2M + l'M' = 0 , E ( yZ – zY) +IN +
Ľ N' = 0 ;
quarum tertia nos edocet axem OZ premi vi XZ in dire
ctione z , reliquae vero suppeditant M , M' , N , N' . Si P ,
P ' , P &lt;nowiki&gt;''&lt;/nowiki&gt; , ... evadunt parallelae axi OZ, et se dirigunt ad
plagam negativam , erunt ( 12: 13, 2° ) ,
EX = 0 , EY = 0 , EZ = P - P - P ' -... = - R,
ExZ- X ) = - xP - X'P' - &lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; --... X, R,
(y2 — zY) = - ype ' P' y &lt;nowiki&gt;''&lt;/nowiki&gt; P &lt;nowiki&gt;''&lt;/nowiki&gt; —... = - y . R;
hic denotant P, P ', P &lt;nowiki&gt;''&lt;/nowiki&gt; , ... virium duntaxat intensitates. Quare
M + M ' = 0 , N + N = 0 , L + L + R = 0,2M +
I'M – x, R = 0 , 2N + IN - Y , R = 0 ;
unde
M = -M'
1, R
1 - T '
N = -N y R ,
, L
L +
+ LEL' = - R.
3
25
—M,-FNg-Lg—M'g—N' '..L,
tanquam vires
, quibus librantur caeterae P, P', P&lt;nowiki&gt;''&lt;/nowiki&gt;. ..,
ac insuper m::o, &lt;nowiki&gt;''&lt;/nowiki&gt;:D, in'-:(), &lt;nowiki&gt;'''&lt;/nowiki&gt; ∶−−⋅ o, necngn ( 110.)
Xxï—yX :) o:
iccirco( 80. a'&lt;nowiki&gt;'''&lt;/nowiki&gt; )
EX—M—M':o,2ï-—N-—N' :o,ZZ—L-—L' :0,
Si xZ—zX)—I-lM-I— l'M':o,Z(yZ—-zï)—l-IN-l-
&lt;nowiki&gt;!' N':o;
quarum tertia nos edocet axem OZ premi vi ZZ in dire-
ctione&lt;/nowiki&gt; :, reliquae vero suppeditant M, M' , N ,N'. Si P,
P'
,P&lt;nowiki&gt;''&lt;/nowiki&gt; ,... evadunt parallelae axi OZ, et se dirigunt ad
plagam negativam, erunt (12: 13. 20 ),
ZX:o, Zïzo,zz :P—P'-—P&lt;nowiki&gt;''&lt;/nowiki&gt;-—. ..

&lt;nowiki&gt;:&lt;/nowiki&gt; — R,

XxZ—QX):—xP -x'P'
-— x&lt;nowiki&gt;''&lt;/nowiki&gt;P&lt;nowiki&gt;''&lt;/nowiki&gt; —-. .

&lt;nowiki&gt;:&lt;/nowiki&gt;

. —a:. B,.
2(yz ∙∙∙ zï):—yP—— r' P' —.7&lt;nowiki&gt;''&lt;/nowiki&gt; P&lt;nowiki&gt;''&lt;/nowiki&gt; ∙∙∙ ∙ ∙ ∙ ∶−∙ ∙−−∫∎ R;
bic denotant P, P', P&lt;nowiki&gt;''&lt;/nowiki&gt;, ... virium duntaxat intensitates. Quare
∐−⊦∐∣∶∘∙∾⊣−∐∙−−∶∘∙ ↧⋅−↽↧⋅∙−↽≖↸≓∘∙≀∐⊣⊸
I'M' — x,R:o, lN-i-l'N'—-y,R:o;
nnde
M: x,B -—M':—
---—- r—z '
Nz—N' :-—l',y'—-—R—2,L-I—L':—R- 326
14. Ex dictis de virium aequilibrio deduci possunt
aequilibrii leges in Machinis. Sic v. gr. in vecte poten
tia librabit resistentiam seu pondus, quotiescumque ( sumptis
( 10. 10 ) momentis quoad axem immobilem, circa quem po
test vectis moveri ) momentum potentiae aequatur momento
resistentiae.Idipsum obtinet quoad Axem in peritrochio ; idi
psum quoad trochleam fixam . Potentia et resisteutia istis
machinis applicantur in directione parallela planis perpen
dicularibus axi immobili; perinde igitur ( 10. 10 ° ) erit si
ve in eorum uno sive in altero accipiantur momenta ;
poteritque vectis repraesentari per lineam mobilem circa
punctum fixum , quod dicitur fulcrum , hypomoclion : axis
in peritrochio per circulares projectiones rotae ac cylin
dri in uno quovis ex dictis planis , mobiles circa com
mune immobile centrum : trochlea fixa per circulum ro
tatilem circa suum centrum , cujus circuli radius sit ipse
trochleae radius; verum quia in trochlea fixa nihil sunt
aliud perpendicula momentorum propria nisi trochleae
radii, ducti ad puncta ubi funis desinit ipsam trochleam
tangere, ideo erunt aequalia , et consequenter aequilibrium
in trochlea fixa importat potentiae ac resistentiae ae
qualitatem.
Ad trochleam mobilem quod spectat , sit P (Fig. 10)
pondus sustinendum a potentia Q : quoniam in casu aequi
librii , P et Q praebeant oportet resultantem R transeuntem
per punctum fixum 0, iccirco ( 9. 10. ) Q : P = sin \beta : sin a
= sin i : sin 2i = cos x : sin 2x = cos x : 2sin x cos
1 : 2 sin ; ac proinde
P
Q
2 sin s
Posuimus angulum OaQ dividi aequaliter directione
ponderis P : id vero facile intelligemus animadvertendo , si
filum OaQ fixum in 0 et Q , tenditur vi applicita puncto
26

14. Ex dictis de virium aequilibrio deduci possunt
aequilibrii leges in Machinis. Sic v. gr. in vecte poten-
tia librabit resistentiam seu pondus, quotiescumque ( sumptis
(10. 100) momentis quoad axem immobilem, circa quem po-
test vectis moveri ) momentum potentiae aequatur momento
resistentix-Idipsum obtinet quoad Axem in peritrochio ; idi-
psum quoad trocbleam lixam. Potentia et resistentia istis
machinis applicantur in directione parallela planis perpen-
dicularibus axi immobili; perinde igitur( 10. 100) erit si-
ve in eorum- uno sive in altero accipiantur momenta;
poteritque vectis repraesentari per lineam mobilem circa
punctum fixum, quod dicitur fulcrum, hypomoclion: axis
in peritrochio per circulares proiectiones rotae ac cylin-
dri in uno quovis ex dictis planis, mobiles circa com-
mune immobile centrum: trochlea lixa per circulum ro-
tatilem circa suum centrum,cuius circuli radius sit ipse
trochleae radius; verum quia in trochlea fixa nihil sunt
aliud perpendicula momentorum propria nisi trochleae
radii, ducti ad puncta ubi funis desinit ipsam trocbleam
tangere, ideo erunt aequalia ,
et consequenter aequilibrium
in trochlea fixa importat potentiae ac resistentiae ae-
qualitatem.
Ad trocbleam mobilem quod spectat , sit P (Fig. 10)
pondus sustinendum a potentia Q: quoniam in casu aequi-
librii , P et Q praebeant oportet resultantem R transeuntem
per punctum fixum 0, iccirco (9. 10.) Q: P:sin 13: sin «

&lt;nowiki&gt;:&lt;/nowiki&gt; sin i :sin 2i:cos x : sin Zx:cos x: Zsinxcosx
&lt;nowiki&gt;:&lt;/nowiki&gt; 1: 2 sin a:; ac proinde

P Q&lt;nowiki&gt;''&lt;/nowiki&gt;üü'
Posuimus angulum OaQ dividi aequaliter directione
ponderis P: id vero facile intelligemus animadvertendo, si
iilum OaQ fixum in 0et Q ,
tenditur vi applicita puncto
∙∙∙ '. 'una-
,.. ↙∙∙∎⋅−27
a libere excurrenti juxta ipsum Oal , punctum a necessa
rio permansurum in perimetro ellipseos , cujas foci O et Q;
ideoque in casu aequilibrii vim illam fore perimetro elli
pseos normalem ; quod certe importat praedictam anguli
OaQ divisionem . Vide n. 55. 13 .
Etiam sic : cum in casu aequilibrii funis ubique ma
neat aeque distentus , pondus P librabit resultantem R'
ex duabus viribus aequalibus Q et concurrentibus apud
punctum a ; et quoniam R' aequaliter dividit angulum Oal ,
idipsum dicendum erit de ponderis directione. Jamvero
R ' ( = P2) = Q + + Q2 + 2QQ cos 2i =2Q ( 1 +cos 2i) =
4 Q* cos 2i = 4 Q* sinºx :
rursus igitur
P
-
2sin x
angulo x = 90° respondebit minimal ; erit Q = P
2
si x = 30° ; vergente x ab 30° ad 09 , verget Q ab P ad co .
15. Vectis primi generis nuncupatur , si fulcrum sit
inter potentiam et pondus ; dicitur secundi generis si pon
dus sit inter fulcrum et potentiam ; denique si potentin me.
dium locum teneat inter fulcrum et pondus , vectis tertii ge
neris vocatur. Hinc vectes primi et secundi generis poten
tiam juvant , quatenus eo minor requiritur potentia ad da
tum pondus sustinendum , quo major est potentiae distantia
a fulcro relate ad ponderis distantiam ab eodem fulcro ;
adeo ut quodvis pondus utcumque ingens possit vectis ope
a quacumque potentia utcumque exigua sustineri : quod cum
bene nosset Archimedes , illud dixisse fertur Hieroni Regi ..
dic ubi consistam , coelum , terramque movebo ,, : vectis au
tem tertii generis potentiam non juvat , quia in hoc vecte
potentia minus distat a fulcro quam resistentia seu pondus.
27

&lt;nowiki&gt;::&lt;/nowiki&gt; libere excurrenti juxta ipsum OaQ , punctum :: necessa-

rio permansurum in perimetro ellipseos, cuius foci O et Q;
ideoque in' casu aequilibrii vim illam fore perimetro elli-
pseos normalem; quod certe importat praedictam anguli
OaQ divisionem . Vide n. 55. 13.0
Etiam sic :cum in casu aequilibrii funis ubique ma-
neat aeque distentus , pondus P librabit resultantem R'
ex duabus viribus aequalibus Q et Q concurrentibus apud
punctum a; et quoniam R' aequaliter dividit angulum OaQ,
idipsum dicendum erit de ponderis directione. Iamvero
a'
∙≺⇌−− re:? -1-Q*-l—2QQcos2i:2Q'(1-l-cva 20:
4Q' cos 3i:4Q3 sin'x:
rursus igitur
P
Q— 2sinx,
angulo x:900 respondebit minima Q

&lt;nowiki&gt;:&lt;/nowiki&gt; ä; erit Q:P

si a: 300 ,- vergente :: ab 300 ad 00
,
verget Qab P ad 00
.

15. Vectis primi generis nuncupatur, si fulcrum sit
inter potentiam et pondus; dicitur secundi generis si pon-
dus sit inter fulcrum et potentiam ;denique si potentia me-
dium locum teneat inter fulcrum et pondus , vectis tertii ge-
neris vocatur. Hinc vectes primi et secundi generis poten-
tiam iuvant, quatenus eo, minor requiritur potentia ad d'a-
tum pondus sustinendum , quo maior est potentiae distantia
a fulcro relate ad ponderis distantiam ab eodem fulcro;
adeo ut quodvis pondus utcumque ingens possit vectis ope
a quacumque potentia utcumque exigua sustineri :quod cum
bene nosset Archimedes , illnd dixisse fertur Hieroni Regi ,,
dic ubi consistam ,coelum ,terramque movebo ,, :vectis an-
tem tertii generis potentiam non juvat , quia in hoc vecte
potentia minus distat a fulcro quam resistentia seu pondus.28
Ex indicata vectis theoria redditur ratio innumerabi
liam effectuum quos quotidie cernimus fieri ; ac primo qui
dem quicumque ex lapidicinis extrahunt lapides utuntur ut
plurimum vecte ferreo : quoties autein multum resistit la
pis sive propter magnitudinem sive quod nimis firmiter aliis
adhaereat , tunc hypomoclion quam proxime ponderi admo
vent , ut facilius moveant , quod vulgo dicitur ,, dar la
leva ,, . Pro hypomoclio antem utuntur quovis sustentaculo
v . gr. lapide ; saepe etiam cum duo lapides ab invicem
sejungendi sunt , unus respectu alterius habet rationem hy
pomoclii . Hic notandum est maxillas quoque esse vectes
secundi generis quum cibi dentibus molaribus comminuen
di traduntur . Secundo : si avellendus est clavus ope mal
lei , quanto clavus , qui ponderis vicem obtinet , propior
fuerit hypomoclio , eo facilius educetur ; unde cum jam tan
tisper eductus est , ita ut extremitas mallei nequeat am
plius insistere subjectae tabulae aut parieti e quo est dedu
cendus , solemus aliud corpus interserere ut quam minima
sit distantia. Tertio : in forcipibus quoque duplex est vectis
primi generis , quorum unum est commune hypomoclion ,
clavus nempe circa quem uterque ramus volvitur , eoque va
lidius stringetur corpus quo rami , qua parte secant , brevio
res , qua parte vero applicatur potentia seu manus , longiores
erunt . Quarto : cum portas aperimus aut claudimus , eo
facilius id praestamas , quo longius a cardinibus eas impel
Iimus , nempe janua est vectis secundi generis , cujas hy
pomoclion sunt cardines. Quinto : remorum motu cymba
promovetur , quia remi sunt vectes secundi generis , quo
rum bypomoclion est aqua , cymba est pondus seu resi
stentia , manus hominis sunt potentia applicata : hinc quo
magis ab aqua remotae sunt manus quam punctum cym
bae , cui remi insistunt , eo majus est potentiae momen
ium. Sexto : ex his etiam intelligitur cur difficillima sit
bacali oblongi elevatio si per extremitatem accipiatur , el
cur quo longior fuerit ipse baculus , eo facilius curvetur
aut frangatur.
28
Ex indicata vectis theoria redditur ratio innumerabi-
lium efi'ectuum quos quotidie cernimus iieri ; ac primo qui-
dem quicumque ex lapidicinis extrahunt lapides utuntur ut
plurimum vecte ferreo :quoties autem multum resistit la-
pis sive prOpter magnitudinem sive quod nimis firmiter aliis
adhaereat ,
tuuc hypomoclion quam proxime ponderi admo-
vent
,
ut facilius moveant, quod vulgo dicitur
,, der in
leva ,, . Pro hypomocliol autem utuutur quovis sustentaculo
v. gr. lapide; saepe etiam cum duo lapides ab invicem
sejungendi sunt
, unus respectu alterius habet rationem hy-
pomoclii. Hic notandum est maxillas quoque esse vectes
secundi generis quum cibi dentibus molaribus comminuen-
di traduntur. Secundo: si avellendus est clavus ope mal-
lei, quanto clavus, qui ponderis vicem obtinet, propior
fuerit hypomoclio , eo facilius educetur ;unde cum iam tan-
tisper eductus est, ita ut extremitas mallei nequeat am-
plius insistere subjectae tabulae aut parieti e quo est dedu-
cendus , solemus aliud corpus interserere ut quam minima
sit distantia. Tertio :in forcipibus quoque duplex est vectis
primi generis, quorum unum est commune hypomoclion,
clavus nempe circa quem uterque ramus volvitur, eoque va-
lidius stringetur corpus quo rami
, qua parte secant
, brevio-
res, qua parte vero applicatur potentia seu manus ,
longiores
erunt. Quarto: cum portas aperimus aut claudimus , eo
facilius id praestamus
, quo longius a cardinibus eas impel-
limus , nempe janua est vectis secundi generis , cujus hy-
pomoclion sunt cardines. Quinto : remorum motu cymba
promovetur
, quia remi sunt vectes secundi generis , quo-
rum bypomoclion est aqua, cymba est pondus seu resi-
stentia , manus hominis sunt potentia applicata: hinc quo
magis ab aqua remotae sunt manus quam punctum cym-
hae, cui remi insistunt , eo majus est potentiae momen-
tum. Sexto : ex his etiam intelligitur cur difficillima sit
baculi oblongi elevatio si per extremitatem accipiatur ,
et
cur quo longior fuerit ipse baculus, eo facilius curvetur
aut frangatur.29
16. Supponantur nunc plures quotcumque vectes ita
dispositi , ut potentia Q in 0 ( Fig. 11 ) magis , puta decu
plo distet a fulcro A quam resistentia in L , quae simili
ter magis distet , puta noncuplo a fulcro C quam resisten
tia in K , quae rursus magis distet a fulcro D puta quin
tuplo quam resistentia in E , et haec similiter magis puta
quadruplo distet a fulcro G quam resistentia in F , haec
denique duplo magis distet a fulcro H quam pondus P in B.
Si res ita se habet , atque insuper potentia et pondus di
rectiones habeant perpendiculares ad respectivos vectes
factis AO = a , CL = a ', DK = a &lt;nowiki&gt;''&lt;/nowiki&gt; , GE = a &lt;nowiki&gt;''&lt;/nowiki&gt; , HF - a &lt;nowiki&gt;''&lt;/nowiki&gt; ,
AL = 6, CK = b' , DE = 6&lt;nowiki&gt;''&lt;/nowiki&gt; ,GF = 6 &lt;nowiki&gt;''&lt;/nowiki&gt; , HB = 6 &lt;nowiki&gt;''&lt;/nowiki&gt;
b &lt;nowiki&gt;''&lt;/nowiki&gt; , erunt
in casu aequilibrii,
L. 6 E. 6 &lt;nowiki&gt;''&lt;/nowiki&gt;
Q
F.6&lt;nowiki&gt;''&lt;/nowiki&gt; &lt;nowiki&gt;''&lt;/nowiki&gt;
il
K =
Kiba,K
E F
P. &lt;nowiki&gt;''&lt;/nowiki&gt;

&lt;nowiki&gt;;&lt;/nowiki&gt;

a
a&lt;nowiki&gt;''&lt;/nowiki&gt;
a '
IV
ex quarum multiplicatione prodibit
b 6'6&lt;nowiki&gt;''&lt;/nowiki&gt; 6 &lt;nowiki&gt;''&lt;/nowiki&gt; 8 &lt;nowiki&gt;''&lt;/nowiki&gt; P
Q
α α' α
P
a &lt;nowiki&gt;''&lt;/nowiki&gt; a &lt;nowiki&gt;''&lt;/nowiki&gt; 3600&lt;nowiki&gt;''&lt;/nowiki&gt;
Quisque videt haec applicari systemati cuicumque rotarum
dentatarum.
Supponantur quoque plures trochleae mobiles v.gr. tres
(Fig. 12) ; erunt ( 14)
Q
L
2 sin r &lt;nowiki&gt;''&lt;/nowiki&gt;
K р
LE 2 sin ac '
&gt; K = ;
2 sin x
et consequenter
Q =
P
23 sin x sin a ' sipx&lt;nowiki&gt;''&lt;/nowiki&gt;
Quod si detur systema coalescens ex trochleis fixis
v. gr , C , C' , C &lt;nowiki&gt;''&lt;/nowiki&gt;, C &lt;nowiki&gt;'''&lt;/nowiki&gt; ( Fig . 13 ) et ex mobilibus F, E, K
29
16. Supponantur nunc plures quotcumque vectes ita
dispositi , ut potentia Q in O (Fig. 11 ) magis , puta decu.
plo distet a fulcro A quam resistentia in L , quae simili-
ter magis distet
, puta noncuplo a fulcro C quam resisten-
tia in K, quae rursus magis distet a fulcro D piita quin-
tuplo quam resistentia in E, et haec similiter magis puta
quadruplo distet a fulcro G quam resistentia in F, haec
denique duplo magis distet a fulcro H quam pondus P in B.
Si res ita se habet , atque insuper potentia et pondus di-
rectiones habeant perpendiculares ad respectivos vectes
,
factis AO:a,CL :a'
, DK:
a&lt;nowiki&gt;''&lt;/nowiki&gt;, GE :a&lt;nowiki&gt;'''&lt;/nowiki&gt;,HF :a&lt;nowiki&gt;''&lt;/nowiki&gt;,
AL:&amp;, CK:6', DE :6&lt;nowiki&gt;''&lt;/nowiki&gt;, GF:b&lt;nowiki&gt;'''&lt;/nowiki&gt;, HB:ö&lt;nowiki&gt;''&lt;/nowiki&gt;
, erunt
in casu aequilibrii,
'
'
'. ∙
'&lt;nowiki&gt;'''&lt;/nowiki&gt; Q—qy'b,L—K'£.,K:E'f ∙ !' ,E—Eb ,F—Pf ;
a a a a a
ex quarum multiplicatione prodibit
Q 6 b' 1)&lt;nowiki&gt;''&lt;/nowiki&gt; b&lt;nowiki&gt;'''&lt;/nowiki&gt; 6&lt;nowiki&gt;''&lt;/nowiki&gt;P P ⇠
a
.: ∙
as an aut alv 3600
Quisque Videt baec applicari systemati cuicumque rotarum
dentatarum.
. Su pponantur quoque plures trochleae mobiles v. gr. tres
(Fig. 12) ; erunt (14). ⋅
et consequenter
Q.... 23 sinu: sinx' sin x&lt;nowiki&gt;''&lt;/nowiki&gt;
Quod si detur systema coalescens ex trochleis fixis
&lt;nowiki&gt;''&lt;/nowiki&gt;
gr, C ∙∁⋅∣ C&lt;nowiki&gt;''&lt;/nowiki&gt;. 0&lt;nowiki&gt;''&lt;/nowiki&gt; (Fig. 13) et ex mobilibus F, E, K30
uno eodemque fane conjunctis ; quoniam , librato systemate ,
funis ubique manet aeque tensus , ideo Q : Q = Q &lt;nowiki&gt;''&lt;/nowiki&gt;
Q &lt;nowiki&gt;''&lt;/nowiki&gt; = Q &lt;nowiki&gt;''&lt;/nowiki&gt; = Q = Q &lt;nowiki&gt;''&lt;/nowiki&gt; = Q &lt;nowiki&gt;''&lt;/nowiki&gt; &lt;nowiki&gt;''&lt;/nowiki&gt; . Jamvero
F E K
Q Q '&lt;nowiki&gt;'''&lt;/nowiki&gt; QP 2sin x' ' 2 sin x
2 sin 2
et consequenter
F = 2Q ' sin &lt;nowiki&gt;''&lt;/nowiki&gt; 2Q sin x &lt;nowiki&gt;''&lt;/nowiki&gt; , E = 2Q sin ü&lt;nowiki&gt;''&lt;/nowiki&gt; , K =
2Q sin x ;
cum igitur sint
L = Q&lt;nowiki&gt;''&lt;/nowiki&gt; &lt;nowiki&gt;''&lt;/nowiki&gt; , F +E + K +L = P ,
iccirco
2 Q sin x &lt;nowiki&gt;''&lt;/nowiki&gt; + 2 Q sin x' + 2 Q sin x +Q = P :
unde
P
Q =
1 +2 (sin x +sin x ' + sin x &lt;nowiki&gt;''&lt;/nowiki&gt; )
Fac demum ut puncta materialia K , K ', K &lt;nowiki&gt;''&lt;/nowiki&gt; , K '&lt;nowiki&gt;'''&lt;/nowiki&gt;,
( fig. 14 ) jungantur Glis K K' , K'K &lt;nowiki&gt;''&lt;/nowiki&gt; determinatae
quidem longitudinis, sed mobilibus circa K , K &lt;nowiki&gt;''&lt;/nowiki&gt; . Si pun
cta illa sollicitantur viribus Q , Q , Q &lt;nowiki&gt;''&lt;/nowiki&gt; , Q &lt;nowiki&gt;'''&lt;/nowiki&gt; , ad aequi
librium haec manifeste requirentur : potentia Q in di
rectione K'K tendens ab K' versus K ; resultans R' ex
Q et Q' in directione K &lt;nowiki&gt;''&lt;/nowiki&gt; K ' tendens ab K &lt;nowiki&gt;''&lt;/nowiki&gt; versus K' ; re
sultans R &lt;nowiki&gt;''&lt;/nowiki&gt; ex R' et Q &lt;nowiki&gt;''&lt;/nowiki&gt; in directione K &lt;nowiki&gt;''&lt;/nowiki&gt; K &lt;nowiki&gt;''&lt;/nowiki&gt; tendens ab K&lt;nowiki&gt;''&lt;/nowiki&gt; &lt;nowiki&gt;''&lt;/nowiki&gt; '
versus K &lt;nowiki&gt;''&lt;/nowiki&gt; ; potentia Q &lt;nowiki&gt;'''&lt;/nowiki&gt; in directione K &lt;nowiki&gt;''&lt;/nowiki&gt; K' ' ' tendens ab K &lt;nowiki&gt;''&lt;/nowiki&gt;
versus K' ' ' : demum ipsa Q's aequalis resultanti R &lt;nowiki&gt;''&lt;/nowiki&gt; .

&lt;nowiki&gt;*&lt;/nowiki&gt; Denotantibus X , Y , Z componentes coordi

natis orthogonalibusque axibus parallelas , in quas resolvi
tur Q, erunt
30
uno eodemque fune coniunctis; quoniam .
librato systemate,
funis ubique manet— aeque tensus
, ideo, Q:Q' ∶⋅−−−−∙ Q&lt;nowiki&gt;''&lt;/nowiki&gt; ∙∙∙−∙∶
Qu::
le:
Qv :va :Qvu ∙
Iamvero
F ∙∙∙ E v K
Q −⇀⋅⋅ 2 SQ..— sin m' 2 sinx
Q'—
−⋅ Zsin x&lt;nowiki&gt;''&lt;/nowiki&gt; ∙
et consequenter
F:
2Q'Isin a:&lt;nowiki&gt;''&lt;/nowiki&gt; ZQ sin x&lt;nowiki&gt;''&lt;/nowiki&gt;, E:2Q sin x', K:
2Q sinx; ⋅
cum igitur sint
LSva'sF4-E—FK—FL2P,
iccirco—
2Qsinx&lt;nowiki&gt;''&lt;/nowiki&gt;—I-2Qsinx'—]-2Qsinx—l-AQ:P:
nnde
P
1—l-2 (sinx-l—sin x' ∙−⊢ sin x&lt;nowiki&gt;''&lt;/nowiki&gt;)
.
Fac demum nt puncta materialia K,K' ,K&lt;nowiki&gt;''&lt;/nowiki&gt;, K&lt;nowiki&gt;'''&lt;/nowiki&gt;, ..:
(Gg. 14 ) iungantur filis K K', K' K&lt;nowiki&gt;''&lt;/nowiki&gt; , ... determinatae
quidem longitudinis, sed mobilibus circa K', K&lt;nowiki&gt;''&lt;/nowiki&gt;. Si pun-
cta illa sollicitantur viribus Q, Q' , Q&lt;nowiki&gt;''&lt;/nowiki&gt; , Q&lt;nowiki&gt;''&lt;/nowiki&gt; , ad aequi-
librium haec manifeste requirentur: potentia Q in di-
rectione K'K tendens ab K' versus K; resultans R' ex
Q et Q' in directione K&lt;nowiki&gt;''&lt;/nowiki&gt;K' tendens ab K&lt;nowiki&gt;''&lt;/nowiki&gt; versus K'; re-
sultans R&lt;nowiki&gt;''&lt;/nowiki&gt; ex B' et Q&lt;nowiki&gt;''&lt;/nowiki&gt; in directione K&lt;nowiki&gt;'''&lt;/nowiki&gt;K&lt;nowiki&gt;''&lt;/nowiki&gt; tendens .ab K&lt;nowiki&gt;''&lt;/nowiki&gt;
versus K&lt;nowiki&gt;''&lt;/nowiki&gt;; potentia Q&lt;nowiki&gt;'''&lt;/nowiki&gt; in directione K&lt;nowiki&gt;''&lt;/nowiki&gt; K&lt;nowiki&gt;'''&lt;/nowiki&gt; tendens ab K&lt;nowiki&gt;''&lt;/nowiki&gt;
versus K&lt;nowiki&gt;''&lt;/nowiki&gt;: demum ipsa Q&lt;nowiki&gt;'''&lt;/nowiki&gt; aequalis resultanti R&lt;nowiki&gt;''&lt;/nowiki&gt;.
&amp; Denotantibus X , T, Z componentes coordi-
natis orthogonalibusque axibus parallelas, in quas resolvi-
⋅ tur Q, erunt
Q;:31
X Y
ē
z
Q
cosinus angulorum , quos cum iis axibus intercipit l; de
notantibus insuper 2 , y , z coordinatas puncti K , et x' ,
j ', z coordinatas puncti K' , erunt
2x yay 22
KKKK KK
cosinus angulorum, quos cum ipsis axibus efficit K'K ; ob
tinebit itaque primum ex requisitis ad aequilibrium, quoties
cumque fuerint
X XX Y
DKKKK .
yg Z
KÖK &gt;
K’K &lt;nowiki&gt;''&lt;/nowiki&gt;
seu
X Y Z
(h ) .
Quod in ordine ad Q est X , Y , Z , sit X', Y ', Z ' in or
dine ad Q ' : si resolvitur l' in ternas coordinalis axibus
parallelas, eae erunt ( 9. 40. )
x + X ' , Y + Y ' , 2 + Z ';
hinc designantibus a&lt;nowiki&gt;''&lt;/nowiki&gt;, y ', z &lt;nowiki&gt;''&lt;/nowiki&gt; coordinatas puncti K&lt;nowiki&gt;''&lt;/nowiki&gt; , ob
tinebit secundum ex requisitis ad aequilibrium , ubi fuerint
X + X __ * ' - &lt;nowiki&gt;''&lt;/nowiki&gt; Y + Y_y_y&lt;nowiki&gt;''&lt;/nowiki&gt; 2 + 2_z'- &lt;nowiki&gt;''&lt;/nowiki&gt;
R ? KK R' K ” K R K&lt;nowiki&gt;''&lt;/nowiki&gt;K&lt;nowiki&gt;'''&lt;/nowiki&gt;
.
seu
X + * _ * + Y_2_Z
x - x yay 22
( h ').
31
X ? Z
Q Q Q
cosinus angulorum, quos cum iis axibus intercipit Q; de-
notantibus insuper a: , y , :: coordinatas puncti K,, et x',
y', s' coordinatas puncti K' ,
erunt
⋅⇂⋅−−⋅⊴⇂∙∣ .7-7&lt;nowiki&gt;''&lt;/nowiki&gt; z—z'
K'K , K'K . K'K
cosinus angulorum, quos cum ipsis axibus efficit K'K: ob-
tinebit itaque primum ex requisitis ad aequilibrium, quoties-
cumque fuerint '
≟−−−⋅−∝−−≄∣ it.s,-ï Z
Q K'K&lt;nowiki&gt;''&lt;/nowiki&gt; 'Q K'K &lt;nowiki&gt;''&lt;/nowiki&gt;G'ka' ↽−≖∙⊍↼∙≕∣
seu gx z
r—x' y—y' x—z'
Quod in ordine ad Q est X , T, Z , sit X', ï', Z' in or-
dine ad Q':si resolvitur Q' in ternas coordinatis axibus
parallelas, eae erunt (9. 40.)
X—FX' , T—Fï' , Z—l-Z' ; ↽
hinc designantibus z', y&lt;nowiki&gt;''&lt;/nowiki&gt;, :&lt;nowiki&gt;''&lt;/nowiki&gt; coordinatas puncti K&lt;nowiki&gt;''&lt;/nowiki&gt;, ob-
tinebit secundum ex requisitis ad aequilibrium , ubi fuerint ⋅
X—l—X' x'-x&lt;nowiki&gt;''&lt;/nowiki&gt; T—l-Tl—TI—j&lt;nowiki&gt;'''&lt;/nowiki&gt; ∅⊣−⊈∣↼↼≂∣∙ z&lt;nowiki&gt;''&lt;/nowiki&gt;
B' ⋅⋅⇀∣⋦∣∣↓⊊∣ ∙ nf- KI/KT '-T—KHK' '—
....t ∙⇁−⋅∣ ↖↽∙∣ ∣
X X T
T—Z-Z.(h).
/ II I I/32
non pluribus opus est ut intelligamus quod, expleta
X + X + X &lt;nowiki&gt;''&lt;/nowiki&gt; _Y + Y + Y &lt;nowiki&gt;''&lt;/nowiki&gt; _Z + Z + Z &lt;nowiki&gt;''&lt;/nowiki&gt;
x ' - 0 &lt;nowiki&gt;''&lt;/nowiki&gt; g'my &lt;nowiki&gt;''&lt;/nowiki&gt; z &lt;nowiki&gt;'''&lt;/nowiki&gt; - &lt;nowiki&gt;''&lt;/nowiki&gt;
( h&lt;nowiki&gt;''&lt;/nowiki&gt; ) ,&lt;nowiki&gt;''&lt;/nowiki&gt;
obtinebit tertium ex requisitis illis ; componentes X” , Y&lt;nowiki&gt;''&lt;/nowiki&gt; ,
Z&lt;nowiki&gt;''&lt;/nowiki&gt; spectant ad vim Q &lt;nowiki&gt;''&lt;/nowiki&gt;, coordinatae z ' ', y, pun.
clum K &lt;nowiki&gt;'''&lt;/nowiki&gt; . Designantibus demum X '&lt;nowiki&gt;'''&lt;/nowiki&gt; , Y
Y ' &lt;nowiki&gt;''&lt;/nowiki&gt;,
&lt;nowiki&gt;''&lt;/nowiki&gt; , Z &lt;nowiki&gt;''&lt;/nowiki&gt;
componen
tes in ordine ad Q&lt;nowiki&gt;''&lt;/nowiki&gt; , expletisque
X + X + X &lt;nowiki&gt;''&lt;/nowiki&gt; + X &lt;nowiki&gt;''&lt;/nowiki&gt; = 0 ,
Y + r' + &lt;nowiki&gt;''&lt;/nowiki&gt; + I&lt;nowiki&gt;''&lt;/nowiki&gt; = 0 ,
2 + 2 +2&lt;nowiki&gt;''&lt;/nowiki&gt; + Z &lt;nowiki&gt;''&lt;/nowiki&gt; = 0 ,
( h &lt;nowiki&gt;''&lt;/nowiki&gt; )
manifeste obtinebit quartum simulque quintum ex requisi
tis ad aequilibrium. Sub novem igitur distinctis condi
tionibus propositum quatuor virium systema consistet in
aequilibrio: si quinque darentur vires , undecim prodirent
conditiones; generatim 2 n + 1 conditiones quoad n vires.
Collatis primis ac secundis membris formularum ( h) , (h') ,
( h&lt;nowiki&gt;''&lt;/nowiki&gt;) , emergent
Y ( 2 - x ) – X (y - ) = 0 ,
( Y + Y') (a' - &lt;nowiki&gt;''&lt;/nowiki&gt; ) – ( X + X ') ( 7'- , ' ) = 0 ,
( X + Y' + Y &lt;nowiki&gt;''&lt;/nowiki&gt;) ( ' &lt; &lt;nowiki&gt;''&lt;/nowiki&gt; ) — ( X + x ' + X &lt;nowiki&gt;''&lt;/nowiki&gt;) (y &lt;nowiki&gt;''&lt;/nowiki&gt; , ' ') = 0;&lt;nowiki&gt;''&lt;/nowiki&gt;
quarum summa praebet
xY_yXfwY — y'X ' + x &lt;nowiki&gt;''&lt;/nowiki&gt; Y &lt;nowiki&gt;''&lt;/nowiki&gt; —y &lt;nowiki&gt;''&lt;/nowiki&gt; X &lt;nowiki&gt;''&lt;/nowiki&gt; + &lt;nowiki&gt;''&lt;/nowiki&gt; ( X + X' + x &lt;nowiki&gt;''&lt;/nowiki&gt;) —
x &lt;nowiki&gt;''&lt;/nowiki&gt; ( Y + Y ' + Y &lt;nowiki&gt;''&lt;/nowiki&gt; ) = 0 ,
∃⊈∙
non pluribus Opus est ut intelligamus quod, expleta
x-1-xq-x'Q—v-1-rq-rff—z-i-zq-z&lt;nowiki&gt;''&lt;/nowiki&gt; W,),
xli—xlli J/l ∙∙⇁ 7&lt;nowiki&gt;''&lt;/nowiki&gt;, z&lt;nowiki&gt;''&lt;/nowiki&gt;—z&lt;nowiki&gt;'''&lt;/nowiki&gt;
obtinebit tertium ex requisitis illis; componentes X&lt;nowiki&gt;''&lt;/nowiki&gt;, ?&lt;nowiki&gt;''&lt;/nowiki&gt;
,
⋅
Z&lt;nowiki&gt;''&lt;/nowiki&gt; spectant ad vim Q&lt;nowiki&gt;''&lt;/nowiki&gt;, coordinatae x&lt;nowiki&gt;'''&lt;/nowiki&gt;. &lt;nowiki&gt;''&lt;/nowiki&gt;, z&lt;nowiki&gt;'''&lt;/nowiki&gt; ad pun-
ctum K&lt;nowiki&gt;''&lt;/nowiki&gt; . Designentibus demnm X&lt;nowiki&gt;'''&lt;/nowiki&gt;, ï&lt;nowiki&gt;'''&lt;/nowiki&gt;
, Z&lt;nowiki&gt;'''&lt;/nowiki&gt; componen-
tes in ordine ad Q&lt;nowiki&gt;'''&lt;/nowiki&gt; , expletisque
\sum∙⊦\sum∣∙⊢\sum∦⊹\sum∣∥∶∶∘∙
T .l-T-l-TII—l- III,: 0 , (hi/I)
Z-i-ZIä-le-l—ZIflzo'
manifeste obtinebit quartum simulque quintum ex requisi-
tis ad aequilibrium: Sub novem igitur distinctis condi-*
tionibus propositum quatuor virium systema consistet in
aequilibrio: si quinque darentur vires, undecim prodirent
conditiones; generatim 2 n ⊣− ↿conditiones quoad :: vires.
Collatis primis ac secundis membris formularum (I:), (b'),
U;&lt;nowiki&gt;''&lt;/nowiki&gt; )
, emergent
?(x—x') —X (?'—?') −∙−−−∘ ∙
(
ï—l— !' )(x' ∙∙∙ x&lt;nowiki&gt;''&lt;/nowiki&gt;)—( X-l-X') (r'—y&lt;nowiki&gt;''&lt;/nowiki&gt;) :o
,
(HF-IJ&lt;nowiki&gt;''&lt;/nowiki&gt;) (x&lt;nowiki&gt;''&lt;/nowiki&gt;-— ∣∣∣≻⊣≖≖−⊦\sum∣−⊦\sum∥≻ (y'—y&lt;nowiki&gt;'''&lt;/nowiki&gt;)
−−− .;
quarum summa praebet
xy-Jx-Jlïl—y/X/ :&lt;nowiki&gt;''&lt;/nowiki&gt; ï&lt;nowiki&gt;''&lt;/nowiki&gt;—y&lt;nowiki&gt;''&lt;/nowiki&gt;X&lt;nowiki&gt;''&lt;/nowiki&gt;—l—y&lt;nowiki&gt;'''&lt;/nowiki&gt;(X XLI-X&lt;nowiki&gt;''&lt;/nowiki&gt;) ∙−⋅
↕∣∣∣≼↕⊹⊺∣⊹↕∥≻ :0 ,33
seu , ob primam et secundam ( hm) ,
-Y yXTY'y'x + x'Y&lt;nowiki&gt;''&lt;/nowiki&gt; _7 / X &lt;nowiki&gt;''&lt;/nowiki&gt; + x &lt;nowiki&gt;''&lt;/nowiki&gt; I &lt;nowiki&gt;''&lt;/nowiki&gt; —7&lt;nowiki&gt;'''&lt;/nowiki&gt;X &lt;nowiki&gt;'''''&lt;/nowiki&gt;
Simili modo collatis primis ac tertiis membris ipsarum ( h) ,
( h') , ( h&lt;nowiki&gt;''&lt;/nowiki&gt; ), attentisque prima ac tertia ( h '&lt;nowiki&gt;'''&lt;/nowiki&gt;) ; itemque col
latis secundis ac tertiis membris earumdem ( h ) , ( h ) , ( h&lt;nowiki&gt;''&lt;/nowiki&gt; ) ,&lt;nowiki&gt;''&lt;/nowiki&gt;
attentisque secunda ac tertia ( h &lt;nowiki&gt;'''&lt;/nowiki&gt;) , assequemur&lt;nowiki&gt;'''&lt;/nowiki&gt;
xZ - 2X + « Z_z'X' + x'Z&lt;nowiki&gt;''&lt;/nowiki&gt; _z&lt;nowiki&gt;''&lt;/nowiki&gt;X &lt;nowiki&gt;''&lt;/nowiki&gt; Tx &lt;nowiki&gt;''&lt;/nowiki&gt; Z '&lt;nowiki&gt;'''&lt;/nowiki&gt; —Z'y&lt;nowiki&gt;''&lt;/nowiki&gt; = 0 ,&lt;nowiki&gt;''&lt;/nowiki&gt;
32—3Y + y^2?–49 + &lt;nowiki&gt;''&lt;/nowiki&gt;Z&lt;nowiki&gt;''&lt;/nowiki&gt; &gt;&lt;nowiki&gt;''&lt;/nowiki&gt;Y &lt;nowiki&gt;''&lt;/nowiki&gt; +y &lt;nowiki&gt;''&lt;/nowiki&gt; Z&lt;nowiki&gt;''&lt;/nowiki&gt; — ;&lt;nowiki&gt;''&lt;/nowiki&gt; Y&lt;nowiki&gt;''&lt;/nowiki&gt; = 0 .
Conditiones videlicet aequilibrii ( 13. 8º. ) quoad systema
punctorum lineis rigidis inter se firmiter connexorum in
cluduntur in conditionibus aequilibrii quoad propositum
systema habens formam variabilem .

=== De centro gravitatis. ===
[[17|17]]. Constat experimentis corpora jugiter sic tendere, seu gravitare in tellurem, ut sibi commissa descendant verticaliter in eius superficiem, gravitas ergo, seu vis unde provenit iste verticalis descensus, eatenus haberi poterit pro sibi ad sensum parallela, quatenus licebit superficiem illam habere pro physice plana: constat insuper experimentis omnia quaevis corpora eodem tempore idem spatium verticaliter in vacuo percurrere, idest aequali velocitate ex aequali altitudine perpendiculariter ad horizontem descendere. Inde sequitur vires gravitatis in diversis corporibus esse illorum massis proportionales, et corpus quodlibet spectari posse tanquam aggregatum materialium graviumque particularum, quae gaudeant parallelarum virium proprietatibus: centrum virium parallelarum (12) in casu dicitur centrum gravitatis. Resultans ex omnibus gravitatis viribus, quae vigent in corporis particulis, vocatur corporis pondus; transit constanter per gravitatis centrum, et directionem obtinet horizonti perpendicularem.

Porro si massula indefinite parva &lt;math&gt;\nu&lt;/math&gt; apud datum corporis punctum dividitur per respondens volumen &lt;math&gt;\beta&lt;/math&gt;, ratio &lt;math&gt;\frac{\nu}{\beta} (= \mu ) &lt;/math&gt; vocatur corporis densitas apud illud punctum; diciturque corpus vel homogeneum, vel heterogeneum prout &lt;math&gt;\mu&lt;/math&gt; apud singula corporis puncta est vel eadem, vel diversa; in corporibus homogeneis ratio &lt;math&gt;= \mu&lt;/math&gt;  est eadem ac ratio inter totalem corporis massam et ejus totale volumen; pondusculum massulae &lt;math&gt;= \nu&lt;/math&gt;, utpote proportionale ipsi &lt;math&gt;= \mu&lt;/math&gt;, exprimitur per &lt;math&gt;= \mu&lt;/math&gt; ductam in quandam constantem &lt;math&gt;c&lt;/math&gt;; ratio &lt;math&gt;\frac{c \nu}{\beta} (= c \mu ) &lt;/math&gt;  appellatur specifica corporis gravitas apud praefatum punctum; estque densitati proportionalis.

[[18|18]]. Notetur illud: etsi corpus gravitate sua jugiter sollicitatur deorsum; hoc tamen non officit quominus adhuc (2) dicatur corpus de se et natura sua indifferens ad quietem vel motum. Gravitas enim est dumtaxat vel aliquid extrinsecum corpori, vel illi intrinsecus additum, non autem aliquid eidem essentiale. Patet, quia vel nomine gravitatis intelligitur vis quaedam, qua corpora versus terram urgentur, vel vis qua tendunt ad determinatam quamdam spatii immobilis partem. Non hoc secundum, quia eo ipso casus purus admitteretur contra principium rationis sufficientis, cum nulla appareat ratio cur mobile ad hanc potius partem ferri debeat quam ad illam, cum spatium ubique sit homogeneum; ergo primum erit dicendum: sed si ita est, certe gravitas non est corporibus essentialis; nulli enim corpori essentiale est ut sibi caetera coexistant, ac proinde unum potest existere quin existant caetera, et consequenter etiam quin existat terra.

[[19|19]]. Dato centro gravitatis corporis, facile definitur utrum corpus in dato situ extra lapsus periculum constitui possit. Nam ex eo centro demissa ad planum horizontale recta perpendiculari, quae vocatur linea directionis, si haec intra basim cadat, corpus extra lapsus periculum erit positum, secus ruet in eam partem in quam perpendicularis recta dirigitur. Hinc patet ratio cur turres aliquae &lt;u&gt;inclinatae&lt;/u&gt; non cadant, ut sunt Bononiensis, Pisana etc: linea scilicet directionis extra ipsarum basim non excurrit. Hinc etiam valde pingues, et qui magnum aliquod onus brachiis complectuntur, retrorsum; gibbosi autem et bajuli antrorsum; qui dextra pondus aliquod sustinent, sinistrorsum; qui vero sinistra, dextrorsum &lt;u&gt;inflectuntur&lt;/u&gt;. Per hanc scilicet declinationem efficiunt ut linea directionis transeat per spatium, quod inter pedes continetur; quod spatium est basis corporis humani. Eamdem ob caussam si quis velit ex. gr. dextero pede stare, crus &lt;u&gt;inclinat&lt;/u&gt; paullulum dexteram partem versus, nec diu haerere potest in eo statu , quia cum basis totius corporis sit unus dumtaxat pes, linea directiouis facile potest basis tam anguslae limites praetergredi. His autem corporis nostri flexibus ac librationibus ita ab infantia assuevimus usu continuo ut nec advertentes recto illas ordine peragamus. Patet hinc denique cur aves uni pedi insistentes dormire solent capite sub ala recondito; id nempe faciunt ut linea directionis intra pedis cui insistunt latitudinem servetur.

[[20|20]]. Centrum gravitatis inveniri potest vel ratione mechanica, vel ratione, algebraica. Ad primam quod attinet, si corpus aliquod filo suspendas, volvetur converteturque donec in aequilibrio tandem consistat, et filum ad terrae superficiem perpendiculariter dirigatur. In hac perpendiculari, quae est linea directionis per quam centrum gravitatis corporis tendit, erit centrum ipsum. Iam notetur linea a filo perpendiculari in corpore designata, rursusque ex alio puncto suspendatur corpus, et facto aequilibrio linea perpendicularis pariter notetur. In communi duarum linearum intersectione reperietur quaesitum centrum. Ratio algebraica desumitur ex dictis ( 13.2.º''a''&quot; ): sumantur nempe vires proportionales massis &lt;math&gt;m, m' , m''&lt;/math&gt;, ..... punctorum, quibus applicitae sunt; hoc pacto, ad positionem centri gravitatis determinandam exsistent
&lt;math display=&quot;block&quot;&gt;x_{\mathrm I}=\frac{\sum m x}{\sum m}, y_{\mathrm I}=\frac{\sum m y}{\sum m}, z_{\mathrm I}=\frac{\sum m z}{\sum m}  (b) &lt;/math&gt;Si corpus intelligitur divisum in varias portiones dimensionis finitae , et earum massae denotantur per  &lt;math&gt;m, m' , m''&lt;/math&gt;, adhuc valebunt formulae (b); nihilque aliud erunt &lt;math&gt;x , y , z ,x' y ',z',x''&lt;/math&gt;, ... nisi coordinatae centrorum gravitatis illarum portionum. Si corpus ponitur insuper homogeneum quoad omnes partes, erunt massae ut respondentia volumina; poteruntque haec illis substitui in formulis (''b'') : quisque videt coordinatas &lt;math&gt;x_{\mathrm I}, y_{\mathrm I}, z_{\mathrm I}&lt;/math&gt;, ex (''b'') haud pendere ab intensitate gravitatis. Caeterum plures sunt casus, in quibus centrum gravitatis absque formularum subsidio immediate cognoscitur. Sic in linea recta centrum gravitatis est medium ipsius rectae punctum: in parallelogrammo punctum, ubi binae diagonales se mutuo secant: in circulo centrum figurae: in cylindro habente bases parallelas punctum medium axeos:
in parallelepipedo punctum, ubi quatuor diagonales se mutuo secant: in sphaera ipsum magnitudinis centrum.

In triangulo centrum gravitatis est punctum illud, ubi sese invicem secant rectae lineae, quae a duobus trianguli verticibus ducuntur ad puncta media laterum oppositorum: cum enim &lt;math&gt;AD&lt;/math&gt; (Fig. 15) dividat aequaliter rectas omnes lateri &lt;math&gt;BC&lt;/math&gt; parallelas, et &lt;math&gt;BE&lt;/math&gt; rectas omnes lateri &lt;math&gt;AC&lt;/math&gt; parallelas, reperietur centrum gravitatis areae triangularis tam in &lt;math&gt;AD&lt;/math&gt; quam in &lt;math&gt;BE&lt;/math&gt;; ideoque erit in &lt;math&gt;H&lt;/math&gt;. Jamvero ducta &lt;math&gt;DE&lt;/math&gt;, ea exsistet parallela lateri &lt;math&gt;AB&lt;/math&gt;; et consequenter triangula &lt;math&gt;ABH , DEH&lt;/math&gt; erunt similia; hinc&lt;math display=&quot;block&quot;&gt;\frac{DE}{AB}=\frac{DH}{AH}&lt;/math&gt;sed, ob &lt;math&gt;CE = \frac12  AC&lt;/math&gt; et &lt;math&gt;CE = \frac12 CD = BC&lt;/math&gt;, est DE = &lt;math&gt;CE = \frac12 AB&lt;/math&gt;; igitur &lt;math&gt;DH = \frac12  AH&lt;/math&gt;; ac proinde &lt;math&gt;DH = \frac12 AD&lt;/math&gt;; et &lt;math&gt;AH = \frac23 AD&lt;/math&gt;.

In pyramide triangulari &lt;math&gt;ABCO&lt;/math&gt; (Fig. 16) erit &lt;math&gt;G&lt;/math&gt; centrum gravitatis; ubi nempe se mutuo secant binae rectae &lt;math&gt;OH , CK&lt;/math&gt;, quae ex &lt;math&gt;O&lt;/math&gt; et &lt;math&gt;C&lt;/math&gt; ducuntur ad centra gravitatis
&lt;math&gt;H&lt;/math&gt; et &lt;math&gt;K&lt;/math&gt; triangulorum &lt;math&gt;ABC , ABO&lt;/math&gt;. Secetur enim pyramis,

1.º planis parallelis triangulo &lt;math&gt;ABC&lt;/math&gt;, 

2.º planis parallelis triangulo &lt;math&gt;ABO&lt;/math&gt;; transibit &lt;math&gt;OH&lt;/math&gt; per centra gravitatis omnium illarum sectionum triangularium; transibit &lt;math&gt;CK&lt;/math&gt; per centra gravitatis omnium harum. Ergo pyramis habebit suum gravitatis centrum tam in &lt;math&gt;OH&lt;/math&gt; quam in &lt;math&gt;CK&lt;/math&gt;, et consequenter in &lt;math&gt;G&lt;/math&gt;. Ducatur nunc &lt;math&gt;HK&lt;/math&gt;; erit &lt;math&gt;HK&lt;/math&gt; parallela rectae &lt;math&gt;CO&lt;/math&gt;, et triangula similia &lt;math&gt;HKG , CGO&lt;/math&gt; praebebunt
&lt;math&gt;\frac{HK}{CO}=\frac{HG}{OG}.&lt;/math&gt;
Sed, ob &lt;math&gt;MH =\frac13 CM&lt;/math&gt; et &lt;math&gt;MK = \frac13 OM&lt;/math&gt;, est &lt;math&gt;HK = \frac13 OC&lt;/math&gt;;
ideoque &lt;math&gt;HG =\frac13 OG&lt;/math&gt;; igitur &lt;math&gt;HG = \frac14 OH&lt;/math&gt;, et &lt;math&gt;OG = \frac34 OH&lt;/math&gt;.

=== De corporum collisione ===
[[21|21]]. Quaestio de corporum collisione eo redit, ut datis velocitatibus ante collisionem, determinentur velocitates post collisionem. Corpora sese collidentia assumimus sphaerica, et in singulis stratis concentricis homogenea; in quibus proinde corporibus centrum gravitatis erit ipsum magnitudinis centrum. Corporum sese collidentium centra vel moventur in eadem recta, vel in diversis rectis; in primo casu collisio dicitur normalis, in secundo obliqua.
[[Fasciculus:Inelastischer stoß.gif|thumb]]
[[22]]. Invenire velocitatem &lt;math&gt;v''&lt;/math&gt;, quam habent duo data corpora non elastica post normalem collisionem, datis eorum
velocitatibus &lt;math&gt;v'&lt;/math&gt; et &lt;math&gt;v&lt;/math&gt; ante collisionem. Dicantur &lt;math&gt;m', m&lt;/math&gt; corporum massae; erunt &lt;math&gt;mv , m'v'&lt;/math&gt; quantitates motus ante collisionem: eatenus corpus subsequens agit in antecedens quatenus hoc lentius illo movetur, adeo ut perseveret actio donec ad aequalitatem velocitatis deveniatur; unde velocitas &lt;math&gt;v''&lt;/math&gt; post collisionem erit communis, et aequalis in utroque: summa praeterea quantitatum motus est eadem ante et post collisionem; velocitas autem obtinetur dividendo quantitatem motus per massam. Ergo demum&lt;math display=&quot;block&quot;&gt; v'' =\frac{mv + m'v'}{m + m'}&lt;/math&gt;

Haec observentur: 

1.° &lt;math&gt; v'' - v &lt;/math&gt; exprimit quantum velocitatis acquisierit corpus antecedens, quod ponimus esse
&lt;math&gt;m&lt;/math&gt;; et &lt;math&gt; v' - v'' &lt;/math&gt; quantum amiserit impellens &lt;math&gt;m'&lt;/math&gt;. 

2.° consideranda erit pro lubito alterutra velocitas tamquam negativa, si corpora ex oppositis plagis adveniunt; hinc in formulis ubicumque ea inveniatur, signo contrario erit adhibenda - Sic v. gr. si massae &lt;math&gt;m'&lt;/math&gt; directio habeatur pro positiva, sumenda erit &lt;math&gt;v&lt;/math&gt; negative, ac proinde &lt;math&gt; v'' =\frac{m'v'- mv}{m + m'}&lt;/math&gt;.

3.° ponetur &lt;math&gt;v = 0&lt;/math&gt;, si corpus impellendum &lt;math&gt;m&lt;/math&gt; quiescit; erit &lt;math&gt; v'' =\frac{m'v'}{m + m'}&lt;/math&gt;: hinc &lt;math&gt;v''&lt;/math&gt; ferme evanescet si massa &lt;math&gt;m&lt;/math&gt; sit physice infinita respectu &lt;math&gt;m'&lt;/math&gt;. 

4.º numquam habebitur perfecta quies post collisionem si &lt;math&gt;m&lt;/math&gt; et &lt;math&gt;m'&lt;/math&gt; in easdem partes oppositas, et velocitates sint reciproce ut massae, tunc &lt;math&gt; v'' = 0&lt;/math&gt;, et consequenter habebitur perfecta quies.

23. Invenire velocitates corporum perfecte elasticorum post normalem collisionem, datis velocitatibus &lt;math&gt;v', v&lt;/math&gt; ante collisionem. Perspicuum est hujusmodi corpora sequi leges non elasticorum toto compressionis tempore, tum restitutionis tempore effectum hunc instaurari. Ergo facta partium restitutione inveniri debet in corpore impulso dupla velocitatis acquisitio; dupla vero celeritatis amissio in impellente. Itaque si dicantur r' ' et &quot; velocitates corporis im
pellentis et corporis impulsi post factam restitutionem ,
erunt ( 22)
u &quot; = V - 2 ( 0--0&quot; ) = v - 2
my + ms
mtm
2 mv tv (m ' — m)
( 9 ),
m + m
(
1
vi &quot; = 0 + 2 (0 &quot; ~ v ) = 2 + 2 ( -v)
mv + m's
m + m
2 m ' ú tu (m - m ')
(9) .
mtm

24. Haec ex formulis (9) et (q' ) deducuntur .

1.• Si massae sunt aequales , elastica corpora post colli
sionem movebuntur .facta velocitatum permutatione, Nam
moveantur primo in eamdem plagam ; propter m = m' , for
mula (9) abit in
2 m v'
et ( 9 ') in 3,10 v' ; ergo etc.
Rursus praeter m = m ' habeatur etiam v = 0 , hoc est cor
2 mo
pus percussum quiescat; erit v = 0 , et v ' . = V ' ;
corpus nempe percutiens post collisionem quiescet , et per
2 mv
2 m 2 m
2 m
1
39
moveantur, vel* alterutra solum quiescat :quod si collisio
liat ad partes oppositas ,
et velocitates sint reciproce ut mas-
sae, tunc v&quot;:o ,
et consequenter habebitur perfecta quies.

23. Invenire velocitates corporum perfecte elasticorum
post normalcm collisionem, datis velocitatibus v', 0 ante col-
lisionem. Perspicuum est huiusmodi corpora sequi leges non
elasticorum tOto compressionis tempore, tum restitutionis
tempore effectum hunc instaurari. Ergo facta partium re-
stitutione inveniri debet in corpore impulso dupla velo-
citatis acquisitio; dupla vero celeritatis amissio in impel-
lente. ltaque si dicantur v'&quot; et v&quot; velocitates corporis im-
pellentis et corporis impulsi post factam restitutionem ,
erunt (22)
'
'
um:-D' --2 ('n'—v&quot;) :'--2 (,; ∙−−−−−−−−⋯⇂↓−⊢⋯∣∣↗ m −−⊢ m )
..2 mv −⊢∣v (m' −∙∙ m)
'm ∙−∙∙ m' ∓∎∎∎∎∎∎ (9):
W:w—l-2(v&quot;—v):v-l—2 Maii:-31; -v)
-&quot;2mv—l-v(m-—m) (qr).
m-l-m' J—
24. Haec ex formulis (q) et (q') deducuntur.
↿∙∘ Si massae sunt aequales, elastica corpora post colli-
sionem movebuntur.fdcta velocitatum permutatione, Nam
moveantur primo in eamdem plagam; propter m:m', for-
mula (q) abit in
2 mv ' ∙
2
'&quot;
'v'&quot;::«v p. . ,
et (q ) 111 10&quot;: :v ; ergo etc.
a m
'
-
Rursus praeter m:m' habeatur etiam :::o , hoc est cor-
- -
∣∣∣ tv 2 m &quot;( pus percussum qutescat; er1t a::o, et a:::
'v ; m
corpus nempe percutiens post collisionem quiescet , et per-40
cussum movebitur velocitate , quam percutiens habebat ante
collisionem . Demum sibi mutuo occurrant : ubicumque ergo
invenitur v , sumenda erit negative ; qua mutatione facta ,
habebuntur
2 mv 2 mv'
2 m
v, et viv v' .
2 m
Jam vides mutationes velocitatum exhiberi per ipsas litte
ras , et ubi debeat etiam mutari directio , regressus expri
mitur per mutationem signorum.
2.• Si statuatur series corporum perfecte elasticorum , ae
qualium , se mutuo tangentium , et quorum centra unam
rectam constituant , in primum autem quacumque velocitate
incidat alterum corpus aequale , movebitur tantummodo cor
pus ultimum , quiescentibus omnibus aliis . Quod si statua
tur series corporum habentium massas in progressione geo
m3,
metrica m' , m, ... ; et caeteris quiescentibus, pri
mum m' incidat in secundum velocitate v' , expriment
m2
m ? m
2 m
v' .
m +m (m *:)*,~(m2 I )
m
velocitates excitatas a primo in secundo , a secundo in ter
tio , a tertio in quarto etc. Denotante igitur n numerum cor
porum , movebitur ultimum velocitate
2 m'
N- 1
I
Cena ntmi
).
3. Quotiescumque corpus impellens minus erit corpore
impacto quiescente , toties illud regredietur , uti patet ex
formula ( 9 ) , posita v = o et m &gt; m' . Quod si m =
et v =0 , prodibit v'' = -1 , nimirum si globus minor''
∢⋅∘cussum movebitur velocitate ,quam percutiens habebat ante
collisionem. Demum sibi mutuo occurrant :ubicumque ergo
invenitur :: , sumenda erit negative; qua mntatione facta.,
habebuntur
2mv &quot; 2mv'

: —-v,et'v :

2m ∙∙−−−∙⋅∙≀≀∙ 2m
Iam vides mutationes velocitatum exhiberi per ipsas litte-
ras , et'ubi debeat etiam mutari directio , regressus expri-
mttur per mutat1onem signorum.
2.&quot; Si statuatur series corporum perfecte elasticorum, ae-
qualium, se mutuo tangentium, et quorum centra unam
rectam constituant
,
in primum autem quacumque velocitate
incidat alterum corpus aequale , movebitur tantummodo cor-
pus ultimum
, quiescentibus omnibus aliis. Quod si statua-
tur series corporum habentium massas in progressione geo-
∙ m! ma. metrtca m', m, ∙ ∙ ∙ ∙ &quot;7, , m .; et caeterts qmescenttbus , prt-
mum m' incidat in secundum velocitate v', expriment
v,2m' ⋅∙ 2m' :
,( 2m' 3
m—l—m&quot; 'v (m—l—m')&quot;, m-l-m'
velocitates excitatas a primo in secundo , a secundo in ter-
tio
, a tertio in quarto etc. Deuotante igitur n numerum cor-
porum , movebitur ultimum
⋅⋅⋮ velocitate ea&quot;
3.&quot; Quotiescumque corpus impellens minus erit corpore
impacto quiescente ,
toties illud regredietur , uti patet ex
formula (q) , posita v :o et m m'
. Quod si ut:ea
et 9

: o

, prodibit v'&quot;:
-— v' , nimirnm si globus minor41
incurrat in globum immensae massae quiescentem , resiliet
cum velocitate eadem , cum qua advenerat .
4.• Si duo corpora elastica occurrant sibi velocita
tibus v , v ', quae massis m, m ' reciproce sint proportionales ,
eadem qua venerant velocitate ambo resilient. Etenim cum
ex oppositis partibus corpora congrediantur , ac praeterea
m : m ' :: v ' : v , in formulis ( 9) , ( 9' ) sumenda erit » negative ,
et ponendum mv = m' '; quibus peractis , obtinebitur
v ' &quot; = &gt; &quot; (m + m )
et viv=v
Im + m
no-tm
Imtin
-- 5. ° Ex ipsis ( 9) et ( 9' ) eruitur m'y's mula
m 'ustomus: factum ex massa in quadratum respondentis
velocitatis dicitur vis viva ; hinc in corporibus perfecte
elasticis summa virium vivarum permanet eadem ante et post
collisiopem .

25. Formulae ( 9) , (9 ') aptari possunt etiam corporibus,
imperfecte elasticis , modo quantitatibus
2(v— mm Imus)
my tms
mtm
et 2
( mahu
my + mv
m + m
--)
substituantur
(n+ m ( = m **)e (1+- ( Inv—-).
denotante r rationem inter vim , qua partes sese resti
tuunt , et vim comprimentem. Quantitas r experimentis de
terminanda est in singulis corporum speciebus : fac ut m
quiescat , sitque co ; erit post collisionem '&quot; = -ru':
unde , cognita velocitate v' ., qua m ' offendit in m , et velo
citate negativa v'' , qua post impactum resilit , habebitur''
-
4
⋅↣
' 41
incurrat in globum immensae massae quiescentem ,
resiliat
cum velocitate eadem, cum qua advenerat.
49 Si duo corpora elastica occurrant sibi velocita-
tibus v, v', quae massis m, m' reciproce sint proportionales ,
eadem qua venerant velocitate ambo resilient. Etenim cum
ex oppositis partibus corpora congrediantur , ac .praeterea
ut :m'::v':
9, in formulis (q) ,(q' ) sumenda erit .9 negative ,
et ponendum m 9:m' v'; quibus peractis , obtinebitur
v'&quot;:
— v' (Z.—lm,) :-v'. et v&quot;:v (m )

:

v.
∙
∙⊢⋯⋅ ⋯∙−⊦⋯
∂∙∘ Ex ipsis (q)et.(q') eruitur m' v'&quot;3-l- mv&quot;::
m'∎∣∣≖ -l-m vi: factum ex massa in quadratum respondentis
velocitatis dicitur vis viva; hinc in corporibus perfecte
elasticis summa virium vivarum permanet eadem ante et post
collisionem. . 

25. Formulae (q), (q')'aptari possunt etiam corporibus ,
imperfecte elasticis , modo quantitatibus .
2(v,—nw—-mlv) et/2 (mv-l-mlv m-l-m —v) mm
substituantur (1 44) (v'. m.,-(.m'v') et≰↿⊹↗⋝⋅≼⋯⇂≩−−⋯⋅∣⇂≀∣ ∙∙∙∙∙ v) ∙
∦⇂⊣−⋯≳ m—l—m
denotante r rationem inter vim
, qua partes sese resti-
tuunt , et vim comprimentem. Quantitas r experimentis de-
terminanda est in singulis corpürum speciebus :fac ut 11:
quiescat , sitque :co ; erit post collisionem v'&quot;:
-
r v':
unde , cognita velocitate v' ., qua m' olfendit in m, et velo-
citate negativa v'&quot;
, ua post impactum resilit, habebitur '42

26. Ad collisionem obliquam quod pertinet , si corpora sibi mutuo occurrunt directionibus convergentibus bm , b'm
( Fig.17 ) et velocitatibus expressis per easdem rectas bm ,b'm ', resolvantur bm , b'm ', altera in duas by, ba, altera in duas b'y ', b'a', ita ut by, b'y' existant normales , ba vero et bá parallelae sint rectae m m corporum centra jungenti. Quoniam componentes b y , b'y' parallelae sunt tangenti TT ductae per punctum contactus, ab ipsis nullo pacto pendebit collisio, nullamque in collisione subibunt mutationem . Cor
pora igitur sese collident velocitatibus ba = ym, b'a' =
y'm '. Inventis itaque ( 23 ) v &quot; , et v '' , sumptisque ex. gr.'' mf = y &quot; , mi = &quot; in recta y r', et ductis mv = by , m'ú
= bóý , si complentur parallelogramma fv, iv', exprimentur per diagonales mf, m'i' tum velocitates , tum directiones corporum post collisionem. Haec autem ex modo dictis facile colliguntur; 

1.º Si globus minime elasticus iacidit oblique in planum immobile, progredietur secundum directionem plani cum velocitate m'v ' ( = a'm '), quae ad velocitatem priorem b'm ' erit ut sinus anguli incidentiae b'm'y' ad radium. 

2.º si globus fuerit perfecte elasticus, resiliet per m'z efficiendo angulum reflexionis z míy aequalem angulo incidentiae b'm'ý . 

3.° quod si globus incidens sit imperfecte elasticus, resiliet ad angu
lom i'm'y ', cujus cotangens ad cotangentem anguli inciden
liae b'm'y ' ut r : 1 .

=== De motu rectilineo utcumque vario.=== 
27. Nonnulla hic praemittimus ex analysi infinitesimali.
1.o Quantitas iniinitesima a: (minor videlicet qua-
cumque data utcumque parva) censeatur esse primi ordinis ;
«2 erit inlinitesima secundi ordinis; «3 iniiuitesima tertii; etc.
2.&quot; Inlinitesima a) dicetur esse primi ordinis si ra-
∙ G) ∙ ∙ ∙
a tno ∙
.. valorem habet (imtum
, secund1 s1 ∙−− valorem obtinet
ac «:43
similiter finitum , atque ita porro . Denotante generatim k
valorem illum finitum , poterit infinitesima quantitas ordinis
msimi exhiberi per
w kam
3. Sumptis aliis valoribus finitis k,; ka, ... km ,
habentur pro aequalibus
kmetkam tk , an- tkzam- ² + ... + kmiat kma
km_ ,a et kam + kamer + ... + km_, &amp; ,
kmed k * et kam + kamer t . tkm -rQ ?.
etc .... ;
admittuntur nimirum aequationes
kam tka&quot;-t ... tkm , at km
km
kam +k ,am -s +... +kimeza? + kmail
km ,
51
etc.
quatenus differentia inter utrumque membrum est minor
quacumque data quantitate alcunique parva. Huc spectat
illud : quantitates infinitesimae , quaecumque eae sint, et quo
rumcumque ordinum , absque ullo errore negliguntur prae
quantitate finita : itemque infinitesimae quantitates altiorum
ordinum absque ullo pariter errore negliguntur prae infini
tesima quantitate inferioris ordinis.
4.0 Quantitates infinitae ( majores videlicet qua
cumque data utcamque magna) cum possint exprimi person
43
similiter tinitum
, atque ita porro . Deuotante generatim k
valorem illnm finitum
, poterit intinitesima quantitas ordinis
msimi exhiberi per
m::ka&quot;
3.&quot; Sumptis aliis valoribus finitis k,
,
It,,
... k,&quot;,
habentur pro aequalibus

: , et kat'-I-k, ∝⋅⋅−≖⊣−≀∣≖∝∙−⋅≖−⊦ −⊦↗⊏⋅∙∙⋅∝−⊦∣⊏⋅⋅∙

∄⊄⋅⋅∙≖∘≖ et ka&quot; ⊣− 1, ∘⋍⋅∙∙⋅⋅⊳⋅ ⊣− ⊣− r.,, a: -
It,... «* et kat&quot; −⊦ kp?&quot; -]— ∙∙∙−∣⋅⋅ km., æ.
etc-eoo ;
.
admittuntur nimirum aequationes ⋅ '
↗⊄⊧∘↙⋅⊣−∣∁⋅⊶⋅∙−⋅⊣−∙∙∙∣−⊦↗≂⋅∙∙⋅⊄−⊦↳
↿ ⋅
km .
l
kan-l-Ic,ac&quot;&quot;-l--. &quot;'l-kaum&quot;,
hngua −⊦⋠⋅∙∙∙∸⇂⇉⊄∙−⋡↿ ⋅ «
⋅
etc. ∙ ∙ , . .
.. ⋮∙ ;
,-
∣

;

quatenus differentia, inter utrumque membrum'est'minor
quacumque data quantitate utcumque parva. Huc. &quot;spectat
illud :quantitates inünitesimae, quaecumque eae sint. et quo- ∙
rumcumque ordinum
, absque ullo errore negliguntur prae
quantitate finita :itemque infinitesimae quantitates altiorum
ordinum absque ullo pariter errore negliguntur prae ⋮≖≖∅≖∙≖∃∙∙ tesima quantitate inferioris ordinis.
∙∡∙∘ Quantitates infinitae (majores videlicet qua-
cumque data utcumque magna) cum possint exprimi per-;,44
tribuentur et ipsae in varios ordines ; illudque facile stabi
lietur : quotcumque finitae quantitates tuto : negliguntur
prae quantitate infinita ; quantitatesque infinitae ordinum in
feriorum tuto etiam negliguntur prae quantitate infinita altio
ris ordinis. Facto enim \beta . , et designantibus a, b,c, ... , 9
valores finitos , habebitur
1
. 0
a \betam + bBm - tom-&gt; + ... +9\betato
1
EL
-la + bw twat..tqomat ww .
5.- Si variabiles x, y sunt inter se per certam quam
dam relationem ita connexae ut data v. g. X , inde possit
valor y determinari , y vocatur functio quantitatis x ; ipsa
vero x dicitur independens. Si relatio inter x et y expri
mitur aequatione minime resoluta quoad functionem y habi
tam pro incognita , y appellatur functio implicita ; quod si
valor y detur expressus immediate per independentem x,
vel talis obtineatur per aequationis resolutionem , y dicitur
functio explicita. In aequatione v, g. yo -2xy + m2 =0
y functio implicita quantitatis variabilis x ; at facta re
solutione , evadet y functio explicita ipsius x , duplicemque
habebit valorem , scilicet y = x + Vx? m2 , Functio
nes explicitae quantitatis x designari solent in hunc modum
est
-
F ( x) , f ( x) , ..
6.0 Differenziale dx quantitatis x est incrementum
infinitesimum , quod ipsi x adscribitur : differentiale vero dy
functionis y = f (x ) est respondens incrementum f ( x + dx)
- f (x ) .quod ob variatam x recipit in se functio illa : pro
ponantur v. gr. invenienda differentialia functionum
44
tribuentur et ipsae in varios ordines ; illudque-facile stabi-
lietur :quotcumque finitae quantitates tuto.: negliguntur
prae quantitate infinita; quantitatesque infinitae Ordinum in-
feriorum tuto etiam negliguntur prae quantitate infinita altio-
ris ordinis. Facto enim þ:
S;, et designantibus a,b,c, ..., q
valores linitos , habebitur
∘∣⊰⋅∙⊣−∂↙⊰⋅⋅∙⋅≖−⊦∘≀⊰⊶−≖ −⊦⋅⋅⋅ ⊣−⊄∣⊰−⊦ .
∸−−
te». &quot;(a ∙⊸⊦bæ—l— ccc&quot; −⊦∙∙ .-l—qm&quot;&quot;' ∎∙−∣− r m'&quot;).
5.&quot; Si variabiles æ,y sunt inter se per certam quam-
dam relatidnem ita connexae, ut data v. g. a:
,
inde possit
valor ]determinari ,;
vocatur functio quantitatis se: ipsa
vero «: dicitur independens. Si relatio intern- et y expri-
mitur aequatione minime resoluta quoad functionem ]habi-
tam pro incognita ,
]appellatur functio implicita ; quod si
valor y detur expressus immediate per independentem :,
vel talis obtineatur per aequationis resolutionem ,
]dicitur
functio explicita. ln aeqnatione v,- g. ;&quot; -—,2ay −⊢ ⇑∙∅:o
est 7 functio implicita quantitatis variabilis z'; at facta re-
solutione
, evadet ]
functio explicita ipsius a:, duplicemque
habebit valorem, scilicet 7—:a:∶⊨ ⇂⋅∕⋅↕∙≖ −− m'. anctio-
nes explicitae quantitatis a: designari solent 1n hunc modum
,F (x), f(x),... ⋅∙
6.&quot;Dill'erentiale dx quantitatis x est incrementum
infinitesimum , quod ipsi :: adseribitnr :differentiale 'vero dy
functionis y:--f (x) est respondens incrementum f
(x dx)
—f
(a:)
, quod ob variatum se recipit in se functio illa: pro-
ponantur v. gr. inVenienda difi'erentialia functionum45
at +6,9 +0,24+ Cisin x + C , cos x+c ,
tang x + C, log x + C , a ' tc ,
ubi a et C sunt quantitates constantes.
Erit I. dy = [ alx + dx ) + ] - [ax + C ] = adx.
a
II.dy = [ f'da+ c ]- [* + c]atda X
adot
adx
x2 + xdx
.
III.dy = [ ( x + dx)* + C]-[x4 +C]=ax“-'dx + 29,
a'a- 1 )
24-2d.22
t . ax-' dx .
IV.dy = [ sin ( x + dx ) + c ]- [sinx + C ) = sin ( x + dx ) — sinx
2 cos 2xdx)sinh dz = 2cosx sind = cos xdx .
V.dy = [cos ( c + dº + C ]- [cos.FC ] = cos ( c - day -cosx =
2 sin - (2x +dx)sin __ (x -x -dx) = sin xdx.
VI.dy = [tang ( xtdx )+c] - [langat.C ]=
sin ( x + 2x)
cos(x + dx )
sinx cosxsin (xtdx)-sinxcos( x + dx ) sin ( x + dx - x )
cos2 cosx cos ( cdc )
cos2x
45
a'−⊦∁∙−⋮∙−∙⊹ C, æa-l- C,'sin a: ∙⋅⊢ C,cos æ-l-C,
tang æ-l—C, logæ-l— C, ar-l-C,
ubi a et C suntquantitates constantes.
Erit l. dy: [a( æ-l—dx) ——C ]— [ux—I»- C ]:adæ.
∥⋅↙∣↗↗⊣⋤⋮⋅−−⊦ (i]—[?&quot; C]— jd,— :—
∥∣∙↙∄∫⇋∶∐↕⊹≴≀↕≻⋅⊹∁⊐∙⋢∞⋅⊹∁∃≕∞≕∙∣↙≀↝⋍⊹↽∘↙↙⊑⋅∣≱↶∶⊄−≖∠≀↓⋅≖
&quot;I- ∙ ∙∼ ∙ :aæ&quot;-' dx.
IV. a];:[sin(æ-l-dæHCI-[sinæ-l-C] :sin(æ-l—dæ)— aina:

: 2cos—;..(Zæ-l-dæ)sin-—;. dæ: a cosa: sin-;— dx:cos ædæ .

V.dJ:[cos(æ-l—dæ)-l-C]- [cosa: −⊢∁↥ ∙−−− cos (æ—l—dæþcosæ:
2sin :(ZPFdrþin-i—(æ-x-dx):— sin ædæ.
Vl.dy::[tang (
.z—l-dæ ≻−∣−∁⋮∣ ∙
⇂⊏∄∐⊰⊅⇥∙∁⊐ —8lll(æ..ll-rlæ) cos(æ—-dæ)
aina: cosæsin(æ-l-dæ)-sinæcos(H—dx)— sin(.i—l—dx-æ)
cosa: cosæ cos( æ
⊹∠≀∙↧⋅ ) coszx ⇁⇁−∙↼46
dx
cos2 x
VII.dy= [log(x-tdx)+ c]-[logo +C ]= log ( + )
dit 15 ( 1 +4x)dx _d2log [2 + } (1- dot) +
23 (1-4 )(1-2dt)+... ]
det 103 ( 2 + + 43 + 234 + ...)
dxlog [ 2 , 718281828
dx
] ;
sumptisque logarithmis quoad basim 2 , 718281828
dx dy X
istiusmodi basis solet exprimi per e.
VIII.dy = [a ++dx + C ] - [ a * + C ] = a *+ x_qt = da? =
a* d log (a *) = a * d [ x log (a )] = a * log (a ) dx.
70. Quantitas constans C, quaecumque ea sit, non in
venitur in differentialibus: idemque proveniet differentiale
sive differentietur v. g. sin x + C, sive sin x.
dy 8º. In primo exemplo habemus a,
dz
cundo axe- ', in dy quarto dx dx
- in se
dy
a
in tertio dy dx
x2
46
da:
∙∙∙∎∙∙↼⇁∙−⇁
∞∘⇄∙⋍∙∙
∇∥∙ ↙≀↨↶−−−∏∘⊰≺⊿↾∶∙∔⋞≀∙↕≻−⊦∁∃⋅⊏∣∘∷∞⊣−∁↿⇌ log (
↿
.? −⊢↙∙⇣⋮⋮⋟⋮
dx —log (HE ↙↿−⋤⋅∶∙↙≀−≟−∅∣∘⊰∣∶≆⊣⊸≑−≺↿− ff):
⋮⇡↽≐≺↿−⋛≣≤ (fi-:): --]—
da: ↿ ↿
⊺⊅−∣∘⊰≺⊈⋅∙⊦−≆−⊣−≐−∙≡ 2..3
∢⊯∎⊦∙∙∙ '):
≦−↕∣∘∥⋣∙ 718281828 ...
];
sumptisque logaritbmis quoad basim 2, 718281828 ..., .
(II:—;:
istiusmodi basis solet exprimi per e.
Vlll'dy :[a&quot;dx—I—C ]—[ax ∙−⋅⋅⊢∁⋮∣∶∅≖↤≖− ar: daJr:

: a'd log (a'):axd [æ log (a)]:axlog (a) dt.

⋅70. Quantitas constans C,quaecumque ea sit, non in-
venitur iu differentialibus: idemqne proveniet dili'erentiale
sive dilferentietur v. g. sinæ ∙−∣⋅− C, sive siuæ.
80. In primo exemplo habemus ?: a, in se-
x
d d cundo a- . J
⊋−⋮⋮⋮ :— &amp;, 1n tertio 23:01: ', 111 quarto 71:47
in COST
in octavo
dy dy cost , in quinto sin x, in sexto dx dx
1
septimo di die= a * loga . Quisque videt
dy fore generatim novam functionem variabilis z : si ea
denotatur per f(x) , erit
2
dx
de = f( ), et dy = f ( z )dx .
Functio f '(x ) appellari solet derivata ex primitiva f( x) :
caeterum dx, seu differentiale variabilis independentis x,
spectatur quidem ut quantitas infinite parva ; sed simul con
stans atque arbitraria,
9º. Ex ivº, vº, et viº exemplo habemus
d sinx d sinx da dx :
dcosx
COS X V
sinx 1 - sinar
dcosx
77- cosa a '
dx = cosa x d tang x =
d tanga
sec2 x
dtang x
1 + langa x
Aequationes istae in hunc modum scribi possunt
dz dz
darc (sin = z ) = darc(cos = 2 ) =
V1 - Z
V 1-22
dz darc ( tang = 2 )
1 + z2
47
cosa: ,
in quinto ?; −∙−∙−∙ -—sinx, in sexto ⋛⋚∶≎∙⊂≐⊭−∙ in
septimo g

:

-.::— ,
in octavo :::-ï :axloga. Quisque videt
217— fore generatim novam functionem variabilis :: si ea

denotatur per f(æ) , erit
ngþ), et 47 :f(æ)dæ .
l
Functio f(æ) appellari solet derivata ex primitiva f(æ):
caeterum dx, seu differentiale variabilis independentis x,
spectatur quidem ut quantitas infinite parva; sed simul con-
stans atque arbitraria.
go, Ex &quot;o, vo, et vi&quot; exemplo habemus
dsinæ dsinæ dccsæ dar.... ∶−−−−−⇀⋅−⇁−⋅ :
.

: ⋅ cos 3&quot; l/ 1

-
sm'æ '&quot;&quot;:
&amp;
, dx:cos3xd tangæ:
deosæ
Vi-oos'x
dtangx ∙∙∙ dtangx
secaæ 1—i—tang3x
Aequationes istae in hunc modum scribi possunt
'
d
dare (sin— !.):sz ,darc(cos——z)-—- V
132 ,
-zz ∙ 2
darc(tang:z): T'↶−≀≘≖−−?'48
10 ° . Sicuti ex y = f( x) obtinuimus ( 8&quot; ) dy
f ( x )dx, sic ex hac obtinebimus ddy = f' ( x) dxdx
f '(x )dx?, ex qua rursus dddy = f &quot; (x )dx dx ' = f &quot; (x ) dr },
atque ita porro ; denotant fif &quot;, ... novas functiones
variabilis independentis x. Itaque si compendii causa e
xhibentur ddy, dddy, ... per dy, dy,.. , profluent
d
d’y = f '( x ) dx?,dy = f &quot; ( x ) dx3,
dy= f (x) ,
da²
d3y
= f'&quot; (r ) , ... : d.x3
assumpta v.gr.y = x ^, erunt f( x) = x ^ , f ( x ) = axa if '( ')
=a ( a - 1 ) 219-2, f ( x ) = a (a - 1 ) ( a - 2 ) x4-3, .
Differentialia dy , dºy , dy ,. . , itemque functiones deri
vatae f (x ), f ' (x) , f &quot; (x ), ... dicuntur primi, secundi, ter
tii , ... ordinis respectu functionis primitivae y = f (x ).
11 ° . Quemadmodum data functione possunt quaeri
ejus differentialia , ita vicissim dato differentiali quaeri po
test functio unde illud promanal. Sint F (x ), f (x ) ejusmo
di functiones variabilis x , ut exsistat F' ( x) =f( x) : quan
titas F ( x) + C vocatur integrale indefinitum differentia
lis f ( ) dx, designaturque praefigendo litteram ſipsi differen
tiali , ut scribatur
ſf(x) dx F ( x) + C ;
exprimit C quantitatem ( 7 &quot; ) constantem atque arbitrariam.
12° . Formula f ( x )dx ita sese aliquando exhibet,
ut statim appareat eam esse differentiale cujusdam da
tae functionis ; tunc vero in promptu est integrale: atque
hoc pacto habemus ( 6º . 9° )
f (a + 1)x*dx ******+. C,unde fredr =
xati
atito
48
100. Sicuti ex 7:f(x) obtinuimus (80) d] ∶∙∙−−⋅
f
(æ)d.r, sic ex bac obtinebimus ddj :
f
'(æ) dædæ :
f'(x)dæ', ex qua rursus dild]:f&quot;(æ)dx dæ' :f'&quot;(æ) dx3,
atque ita porro; denotant f,f &quot;,
... novas functiones
variabilis independentis æ. Itaque si compendii causa e-
xhibentur ddy, dddy, ... per dy, d37 ,. ., profluent
dïy
&amp;? d')&quot; :f'(x) da.&quot;,dfly :f&quot; (æ) das-3, ...,

: f'(æ),

113!
dæ3 :f'&quot;(.r),...:
∙∙⋅⋅∙⋅
assumpta v. gr.y:æ&quot;, erunt f
(æ):æ',f' (x):ax&quot;',f'(a-)

:: a(a — 1) x&quot;&quot;,f&quot; (x):a(a—1)(a—-2)x&quot;3, ....

Difаerentialia dy, diy, d3y,. . , itemque functiones derivatae f(x), f'(x) , f&quot;(æ) , ... dicuntur primi, secundi, tertii, ...ordinis respectu functionis primitivae y:f(x).
110. Quemadmodum data functione possunt quaeri
eius differentialia, ita vicissim dato differentiali quaeri po-
test functio unde illud promanat. SintF(x),f(æ) ejusmo-
di functiones variabilis x, ut exsistat F'(æ):f(x): quan-
titas F(x) −−∣− C vocatur integrale indefinitum differentia-
lis f
(.r) dar, designaturque praefigendo litteram ]ipsi differen-
tiali, ut scribatur
ff(æ)dæ:-—F(æ)—1-C;
exprimit C quantitatem (70) constantem atque arbitrariam.
120. Formula f(x)dx ita sese aliquando exhibet,
ut statim appareat eam esse dili'erentiale cujusdam da-
tae functionis; tunc vero in promptu est integi—ale: atque
hoc pacto habemus (60. 90)
a
&amp;
a-l-l C :: xtt-H f(a—1-1)æ dx:x ∙⋅∣− ,undefæ dx: ∉⊋∙∙⊦∙∙∙∓ ⊹∁⊒ï49
QCx
ſalog/a)d(c== q** + C, unde ſe*dx =clogiastc

;

dx
S
= arc ( sin = x ) +C ;
V 1 - 22
Sa
dx
1 + x2
= arc ( tang = x ) + C.
130. Interdum formula f (x )dz, de cujus integra
tione non constat , per quasdam substitationes transfor
matur in aliam , cujus integrale illico cognoscitur. Sic.
v . gr. positis ax = 2 , - = z ,assequimur a
dx dz 1
Si Salita
=
14a²x² arc ( tang == z) + C =
1
arc ( tang = ax ) + C ,
Sa
dix
22 ta
1
Sat
dz
a (1 + z2)
arc ( tang = 2 ) + c = a
-a arc tang * + c,
-Svador - Svado --Svet
( cos = ) + c
arc ( cos = z ) + c
= arc
fa&quot;log(a)d(cæ):a&quot; —]—C,undefa&quot; dx: -ac
dx
- ⇂∕↿∙−⋅⋥∎⊑ :arc (sin :x) −⊢∁≂
f 1112 :arc( tang:x )-l-C.
130. Interdum formula f(æ)dx, de cujus integra-
tione non constat, per quasdam substitutiones transfor-
matur in aliam, cuius integrale illico ougnoscitur. Sic.
.. æ . '. gl'. POSIUS nær-z.;— Zoasaequlmur
dæ ⇀∙∙− dz 1 — ∙−−
fl'l'a'æ' a(1-I-za) a
&quot;c (tang—z)-[-C—..
dx
xï-l-a
dz 1
faU-l—z') − a arc(tangzz)-I-C:
↿
—a.arc (tang :ax)-[- C,] ——
.—
—1-arc( tang :
−⋅⋮− ⋟⊹ C,
et
f
dx ]
adz ]
dz ∙∙∙
[fas-xa ∣∕ ∅≖∙∅≖≖≖ −⇀ ∣∕↿ -zz
arc(cos:z)-1-C :
arc(cos :?) ⊹∁∙50
140. In integrali indefinito ( 11 °) adhibeantur suc
cessive pro x peculiares valores xo, x n , ac dein ab F ( zn )
+ C subtrahatur F ( x ) +C ut , eliminata C , prodeat
F (xn) - F ( xo) : ejusmodi differentia vocatur integrale de
finitum differentialis f (x ) dx , sumptum videlicet ab x =
а &quot;
x
Xo
xh
ſ p(x)dx = F(wow )— F( xo ) .
Xo
Hinc v. gr.
a
dx
jederati
7T
a
o
Variato altero ex binis limitibus v. gr. x ny variabit
ipsum quoque integrale ; et adhibita x pro xmo erit
X
ſ f(x)dx= F ( x) — F ( xo ) :
Xo
habebitur videlicet integrale illud , quod incipit ab xo ,
quodque evanescit facto x = x,: et quoniam
aff(x)dx = d [F(x)-F(xs)] =dF( x) =f ( x) dx ;
X.
iccirco
X
S SP(x)dx = Sp«x ) dx + c .
X.
15 °. Sit arcus infinitesimus ABEH ( Fig. 18 ) , et
in eo chordae infinitesimae AB , BE , EH , quarum prima
50
140. In integrali indefinito (110) adhibeantur suc-
cessive pro x peculiares valores xo, x,, , ac dein ab F (x,)
−⋅∣− C subtrahatur F(xo)-I-C ut, eliminata C, prodeat
F(x,,)— F (x,): eiusmodi differentia vocatur integrale de-
finitum differentialis f(x)dx, sumptum videlicet ab x:
x&quot;
x, ad x:æ, ,designaturque per [f(x) dx, ut scribatur
æo
xn
]
f(æ)dx :P(æ.) — P(æ. )-
æo Hinc v. gr.
[ a
fa,-' dx: ..-.-..-—1 J'EL :
-E- a-l—1 ' xï-l-aa a.
0 0
Variato altero ex binis limitibus v. gr. x,, variabit
ipsum quoque integrale; et adhibita x pro x,, erit
.?
faa-w.r: ∌⇁≺∙↿∶≻−−∙≖⊸⇁≺∞∘≻≃⋍∙
æo
habebitur videlicet integrale illud, quod incipit ab x.,
,
quodque evanescit facto x: x,: et quoniam
&amp;?
df/(x)dæ:d[F(x)-F(æ.) ]:dF(æ) :f(æ) dx;
xo
iccirco
fff(-1')dx:ff( x
)
dxH—FC.
↿⋅⇂⋝∘⋅ x.,
Sit arcus iniinitesimus ABEH( Fig. 18 ), et
in eo chordae infinitesimae AB, BE, EH, quarum prima51
SUC:
ac tertia producantur donec concurrant in D. Quoniam an
guli DBE, DEB sunt infinitesimi, angulus quoque ODE
erit infinitesimus: designetur iste angulus per i , et fiant odest
e de
BD =
es
BE = c , DE b ;
habebimus
lur
62 =a +62 – 2ab cos ( 180° -1i ) = a + b2 +
2 ab cos i = a : + 62 + 2 ab- 2ab + 2 abcosi = (a + b )2
2 ab ( 1 — cosi ) =( a + b )2 – 4ab sin ’ şi ,
unde

: 1

4ab
sin _ i = 1
(a + b ) ( a + b )2
ariabi
et consequenter
[1 - ( +5)*] sinº in
= - = [ - (-3 ) ]su'_

: [&quot; - )*]*sist i -....

2 + b
ban
Differentia nimirum inter unitatem et rationem c ad
a + b consistet in terminis duntaxat infinitesimis , quorum
ordines excedunt omnes ordinem primum .
16º. Idipsum a fortiori dicendum de differentia
inter unitatem et rationem ipsius c ad subtensum arcum
BmE ; siquidem BmE &lt;a + b et &gt; c. Inde fit ut et ar
cus infinite parvus censealur aequalis respondenti chor
dae , et curva quaevis spectetur tamquam polygonum coa
lescens ex laterculis infinitesimis numero infinitis, et isto.
rum laterculorum prolongationes habeantur pro totidem
tangentibus apud varia curvae puncta.
rini
⋅ 500
(I.]
0an
ede-
lur
anali
bf&quot;
Lr;
(im!
51
ac tertia producantur donec concurrant in D. Quoniam an-
guli DBE, DEB sunt infinitesimi, angulus quoque ODE
erit infinitesimus: designetur iste angulus per i, et fiant
BD—fd, BE:c, DE:6;
habebimus
ea :a: ⊹∂≖ —2abcos(180'-'—-i) :03 ∙−⊦ 63 −∣−
Zabensiz—maa-i-ba -l-Zab—Zab-l-2abcosi :(cs-Fb):—
Zab,(1—- cosi):( a --[-b):
-
4absin* −≧−≀⋅∙
nnde
c*— 406 (a- -b)' ∙∙∎∙∙∙∙∙∶↿∙∙∙ . (a -l-b)3 sin ∙⋮−∎
a—ö a .

: ,

.
[1 (—r—b)]sm;-h et consequenter
c
⋍↿∙−−∶∙−∣∶↿ −≺∅≆≴≻≖∃ aina-Li—
a b a
∸⋇⋅∣∶↿ 3
−≺⋮−⋮−−⇣∙≑≻≏∃≏∘⋮∎≖∣⇩ ..;-i ∙−− ∙ ∙ ∙ ∙
DiEerentia nimirum inter unitatem et rationem c ad
a −⊦ &amp; consistet in terminis duntaxat inünitesimis, quorum
ordines excedunt omnes ordinem primum.
160. Idipsum a fortiori dicendum de differentia
inter unitatem et rationem ipsius (: ad subtensam arcum BmE ; siquidem BmE a −↿− 6 et 0. Inde Et ut et ar-
cus infinite parvus censeatur aequalis respondenti chor-
dae, et curva quaevis spectetur tamquam polygonum coa-
lescens ex laterculis infinitesimis numero infinitis, et isto-
rum laterculorum prolongationes habeantur pro totidem'
tangentibus apud varia curvae puncta.52
17º. Fac ut aequatio y f( x) pertineat ad cor
vam ABD ( Fig. 19 ) et sumptis coordinatis orthogonali
bus, sit abscissa OG = x, ordinata CB = y , infinitesimum
abscissae incrementum CC = dx : ducta per C' alia ordinata
C'B' , et per B lineola recta Bm parallela axi abscissarum
OX, erunt B'm = dy , Bm = CC = dx. Pone tangentem
BE occurrere abscissarum axi in E , normalem vero BH in H;
triangula rectangula et similia BEC , B'Bm , BCH dabunt
ydy

:

tang E
-
tang B'Bm dy, ce = ydx CH dx dy dx
CE dicitur subtangens, CH subnormalis.
18º. Ob auctam x area curvilinea BCa'a recipit
incrementum infinitesimum BB'C'C; est autem
BB'C'C =
dx
(rty + dy ) =
dxdy ydx +
2

&lt;math&gt;= ydx + f (x)dx =&lt;/math&gt;
ydr:
2
facta igitur Oa' = xo , erit
BCa'a- j^ydx = ${( )dx Xo Xo
Area BCa'a manifeste traduci polest ad rectangularem a
ream sub ejusmodi lateribus , quorum alterum sit differen
alterum vero ordinata quaedam ym media in
ter ordinatam aa' respondentem abscissae xo et ordina
tam BC respondentcm abscissae x : propterea
tia c Xo ,
X
ſ ydx = ( x - X . \ 'm , seu S f (x )dx = ( x - x . ) f ( xm ) .
X. Xo
Eadem area BCa'a spectari potest veluti summa ex infini
tis numero infinitesimis areolis rectangularibus
52
170. Fac ut aequatioy :
f
(et) pertineat ad cor--
vam ABD( Fig. 19) et sumptis coordinatis orthogonali—
bus, sit abscissa OG:x. ordinata CB: , infinitesimum

abscissae incrementum CC':dx :ducta per 0alia ordinata
C'B', et per B lineola rectaBm parallela axi abscissarum
OX, erunt B'm:dy
, Bm:CC':dx. Pone tangentem
BF. occurrere abscissarum axi in E, normalem vero BH in H;
triangula rectangula et similia BEC , B'Bm, BCH dabunt
J—— , tang E: tang B'Bm :
.. £,CF—Jjæ (31:731: '
]
.L'
CE dicitur subtangens, CH subnormalis.
180. Ob auctam x area curvilinea BCa'a recipit
incrementum infinitas-imum BB'C'C; est autem
⊞∍⋅∁∙∁−−−−↙⋚∁≺⊺ −⊢∫ ⊣−↙≀∫ )
∙−−∶
ydx −⊢ ∂⋅⋅↕−⋮↨−↗− :ydx-l— [figi-£ :ydx:
facta igitur Oa':x.,
, erit
x x
BCa'a:fydx :ff(x)dx.
xo xo
Area BCa'a manifeste traduci potest ad rectangularem a-
ream sub eiusmodi lateribus , quorum alterum sit differen-
tia x —-xo , alterum vero ordinata quaedamym media in-
ter ordinatam aa' respondentem abscissae an. et ordina-
tam BC respondentem abscissae x: propterea
x ⋅ x
fydx: (x -e-x., ly,&quot;, seuff(x)dx:(x—- x.,)f(x,,, ).
x., . xo
Eadem area BCa'a spectari potest veluti summa ex infini-
tis numero inlinitesimis areolis rectangularibus53
f ( x ) dx , f ( x +dx ) dx , COP
f ( xo +2dx ) dx f ( x — dx ) dx ;
nali
imum
Binala
.
sarum ubi nibil sunt aliud f (xo) , f( x + dx ), f (xo + 2dx), ...
nisi ordinatae respondentes abscissis xo , xo + dx , to +
2 dx ,
Quare
entem
in Hi;
bunt C
ſ f(x).lx = f(x )dx + f( xo + dx)dx +
Y :
Xo
fl xo + 2 dx )dc + .
+ f ( x -dx )dx.
recipé 19º. Ponatur arcus aB = s , ejusque incrementum
infinitesimum BB' = ds; quoniam BB'2 = Bm2 + B'ma, erit
2
ds = dx= + dy ,ideoque s= V dx=+dya =
X.
jäevitro
Xo
Tema
iffere
dia is
ordin
200. Circulus habens communia cum curva CC
( Fig. 18 ) duo proxima latcrcula v. gr. AB et BE, dici
tur osculator: sit O centrum istius circuli, BO ( r) ra
dius, OʻK et O'K' perpendicula ex O ducta in AB et BE ,
i angulus OBE , ds' et ds infinitesimi arcus laterculis AB
et BE subtensi, alter spectans ad circulum osculatorem , al
ler ad curyam CC' . Quadrilaterum KOʻKB praebet angu
lum KO'K ' = 180° — KBK' ; sed KBK' = 180°-OBE =
180° -1 ; igitur KOK' = , et consequenter ds' =
r( KOK' ) = ri' . Est autem ( 16 ° ) ds' = ds : propterea
infini
mali-
imum
linat:
arua
entem
Liuii; ↽
bum
⇟⇁∙∎↘⊰
..
recipi
rem ?
illerä
dia i?
orzlïm'
inüw'
53
f(xo)dx,f(xo-I-dx)dx,
f(xo—l—2dx)dx,. . ..f(x—dx)dx;
ubi nihil sunt. aliud f
(..-.,)
,
f(xo—l-dx), f(æQ—l-2dæ). .. . nisi ordinatae respondentes abscissis xo , xo -l-.dx , xo −∣−
de, . .. . Quare
æ
J. f(x)dx :f(xo)dx −⊢∙∣≼ xo-l-dx )dx ∙−∣−
∙↾≀⋅⋅∘
f(xo-l-2dx)dx −⊦ ∙ ∙
..
-I-f(æ-dx)dx.
190. Ponatur arcus aB: :. ejusque incrementum
iniinitesimum BB':ds; quoniam BB'a :Bma—l-B'ma ∙ erit
x
d:&quot;:dxï-l-dyïddeoque s:f V de-l-dy ∶−∙⋅−∙
æo
x
⋅∣∙↙≢∙↿∶⇂∕↿∙∙⊢∣⇃≖≼⋅≖⋅⋟∙
. xn ⋮⋅⋅
200. Circulus habens communia cum curva CG'
( Fig. 18 ) duo proxima latercula v. gr. AB et BE, dici-
tur osculator: sit 0' centrum istius circuli, BO' (:) ra-
dius, O'K et O'K' perpendicula ex 0' ducta in AB et BE,

: &quot; angulus OBE, 'ds' etïds-iniinitesimi arcus later-culis AB

et BE subtensi, alter spectans ad circulum osculatorem, al-
ter ad curvam CC'. Quadrilaterum KO'K'B praebet angu-
lum KO'K':1800 −− KBK'; sed KBK':1800—OBE:
180o — i' ; igitur KOK':i'
,
et consequenter ds&quot;:
r( KOK') :ri'. Est autem ( 16o ) ds':ds: propterea54
ds
21.• Curva CC' sit plana ; exhibeaturque per y =
f (x ), sumptis abscissis x in RX ( Fig. 20) . Erit i = Q
a = - (-a) = - dx , ideoque ( 170)
ds
ds
d x darc ( tang dy -
dx
)
Jamvero (90 )
dy
darc ( tang )
a dy
dr
dy²
dx²
df ( 30)
1 + f ? (x )
f (x ) dx ;
1 + f ? ( x )
dx
igitur
[1 + F2(x) ] }
f &quot; ( 3)
22.• Si ordinata y in curva y =f ( x) fit alicubi
maxima vel minima, exhibeaturque respondens abscissa per
Xn , quisque videt angulum interceptum tangente geometrica
et positivo abscissarum axe fore acutum vel obtusum pro
ut punctum contactus habuerit abscissam x &lt; vel &gt; xn in
casu maximi , &gt; vel &lt; x , in casu minimi , fore autem in
utroque casu = o ubi punctum contactus habuerit abscissam
x = x , Inferimus illud ( 8º. 170) : functio f (xn) est maxi
ma quotiescumque
f ( x) &lt; o quoad x = x + w ( denotat a quantitatem
infinite parvam &gt;0 ) , et f ( 2) &gt; o quoad x = xn - W ;
est minima quotiescumque
54
21 ∙∘ Curva CC' sit plana ;exhibeaturque per :7
f
(x), sumptis abscissis x
in RX (Fig. 20). Erit :&quot; a— a':
—(a'- a):
— dx , ideoque (170)
ds- ds
↗−− dx— dy
darc(tang:ä-;
Iamvero (90)
- si! darc(tang:i-'r .— dx ∙− ↙≀∣↬≺∙↿∶∟∙∙− f (adde;
dx −−↿ ,dJ' 1-t-f'(æ) l*f'ix)
dx'
igitur
3
[1 ⊣∙↾↔≖ (æ) ]
∶⊸∙ f&quot; r— (æ)
22.0 Si ordinata ;- in curva ;-:[(x) (it alicubi
maxima vel minima, exhibeaturque respondens abscissa per
x,, , quisque videt angulum interceptum tangente geometrica
et positivo abscissarum axe fore acutum vel obtusum pro-
ut punctum contactus habuerit abscissam x(vel )x,, in
casu maximi ,
)vel (x,, in casu minimi , fore autem in
utroque casu:
0 ubi punctum contactus habuerit abscissum
x:x,. Inferimus illud ( 80. 170) :functio f
(x,) est maxi-
ma quotiescumque
f
(x) (o quoad x :. x,, ↼⊢ co (denotat a quantitatem
infinite parvam )o )
.
et f' (x) )o quoad x :x. — a) ;
est minima quotiescumque55
f (x ) &lt; o quoad x = x, — W, et f ( x ) &gt; o quoad x =
X'n tw ;
valores X c.quibus respondet maxima vel minima f( xr ), quae
rendi sunt inter radices aequationis
p' ( x)
In
Si f ( x) maneret aut constanter negativa , aut constan
ter positiva, dum x versatur in viciniis x m , certe f ( x ) neque
maxima esset , neque minima .
Ad haec : quoad casum maximi, crescente x in viciniis
decrescit f' ( oc) , decrescente x decrescit f ( x) ; ideoque
df ( x)
&lt; 0 , seu f&quot; ( 30 ) &lt;0 . Quoad casum vero minimi ,
dx
crescente x crescit f (x) , decrescente x decrescit f ' (x ), et
af' ( x)
consequenter &gt; o seu f ( x) &gt;o .
23. Functiones plurium variabilium independen
tium x , 2 , u , ... designantur in hunc modum
dx
F ( x, 2, Ú, ... ) _f ( x, 2, U, ... ) , ...
Ponatur j = f (x ,2 , 9-9.) : si quaevis una ex quan
titatibus x, 2, u, spectetur uti variabilis et habeantur cae
terae pro constantibus , poterunt differentialia functionis u
eodem manifeste modo determinari ac differentialia functio
num quae ab unica pendent variabili. Ejusmodi differentialia
dicuntur partialia , ipsaque sic exhiberi queunt ,
ut
det , draf . d. , dal , ...
denotent differentialia functionis fe , primi , secundi ... ordi
nis quoad x , quoad 2 , ... Ad partiales functiones derivatas
quod pertinet , eae poterunt sic exprimi , ut per
55
f(x)(oquoad x:x,,—-o),etf (x))o quoad x:
a'.—FG);
valores x,,,quibus respondet maxima vel minima f
(x,,)
, quae-
rendi sunt inter radices aequationis
,'(æ)::00
Si f
(x) maneret aut constanter negativa , aut constan-
ter positiva,-dum x versatur in viciniis xn, certe f
(x,) neque
maxima esset
, neque minima.
Ad haec :quoad casum maximi, crescente x in viciniis
x,, decrescit ]&quot; (x) ,decrescente x decrescit f' (x); ideoque
df (x)
dx 0
.- seu f&quot; (x) (o. Quoad casum vero minimi ,
crescente x crescit f
(x) , decrescente x decrescit f' (x) ,
et
consequenter (IS .(ræ) )o seu f&quot; (x) )o.
23.&quot; Functiones plurium variabilium independen-
tium x ,z , u, designantur in hunc modum
F (x, :, ti, ...) ,f( x, :, u, ...
)
,
Ponatur p.:
f
(x, :, a.,.,.) :si quaevis una ex quan-
titatibus x, z,u. spectetur uti variabilis et habeantur cae-
terae pro constantibus
, poterunt differentialia functionis p. ↴
eodem manifeste modo determinari ac differentialia functio-
num quae ab unicapendent variabili. Eiusmodi dill'ereutialia
dicuntur partialia , ipsaque sic exhiberi queunt , ut
dxld-1 dxaPQ'&quot; ∂∷⊬∙∠↨≖≖⊬∙∙∙∙ 
denotent differentialia functionis 9. ,primi , secundi ordi-
nis quoad x , quoad :, Ad partiales functiones derivatas
quod pertinet , eae poterunt sic exprimi ,
ut per56
doll darf
dx dx2
dou , dazle
dedz dza
vel
per
fx(X , Z, Up... ) , f&quot; , (3 , 2, U, ... ) , . f : (3 , 2, U, ... ) ,
fo( %, 2 , Wo...) , ...
designentur functiones , primi , secundi ... ordinis derivatae
ex M = f ( x , % , U. ... ) quoad x , quoad 2 , ... Plerumque
tamen in his derivatis functionibus exprimendis detrahuntor ,
compendii causa , litterae d signa x , % , U 7 .** , et pro
dal d , ² l
dx dx2 d,I
d², M
dz dz
adhibentur
du del i
dx dx2
du dele
dz dz ?
9
24º . Totale functionis pe differentiale due ( quum
nempe x
spectantur omnes ut simul variabiles )
eruitur ex partialibus dx f , d , f , dul , ... ; sunt enim
% , U ,
f ( x + dx, 2, 1, ... ) - f ( x , ,U, ...) = fx ( x ,2 ,4, ... ) dx,
f ( x + dx, atdz, u, ... ) -f( x + dx, 2, U, ... ) =
f: ( x + dx, 2, u, ...) dz = f : ( x, z, u, ... ) dz,
f ( x + dx, atdz, utdu, ... ) — ( x + dx, atdz, u , ...) —
f'u ( x + dx, z + dz, il ... ) du = f ( x, 2, u, ... ) du, etc... ,
)
1
.—
56
≀≀≖≀∸ −−↙≀⇄↕≴∸ .⋅≤≀−⊦∸
−∙∙ −−−⊓≀⇄≖≴∸ dx dx&quot; , dz dza '
vel per
fx(x, :, uh&quot;) ,
f&quot;: (x, :, u, a&quot;) , ∙∙∙ f, (x' z' u, a.) ∙
f',(x, :, u, ...)
,
designentur functiones , primi , secundi .. ordinis derivatae
ex &quot;. f
(æ.:, u.
...
) quoad x, quoad :∙ ∙∙∙ Plerumque
tamen in his derivatis functionibus exprimendis detrahantur,
compendii causa ,
litterae d signa :, a, «,... ,
et pro
de- dx'P- (!sz (I,,[L da:
∙ m ⋅⋅⋅⊤ ⋅−∂⋅⊒⋅−
adhibentur
≴≀−⋅≖∸− ↙≀≖⋅⊀↓ de dw
dæ'dx' ,' de, dza
240. Totale functionis p. dilferentiale dp. (qumn
nempe x , z , n , spectantur omnes ut simul variabiles )
eruitur ex partialibus d, (1.
,d, pt
, d,, p. , ; sunt enim
fl xhi—dx, 39 ut ...) ↼f( æ, :, u, ...):
f, (æ, :, ., ...) dx,
f(æ-l—dæ, t—l—rlz, u,...) — f(x-[r-dx, :, n, ...):
f: (xä-dx, :, &quot;, mida: f, (x, s, u, ...) dz,
f( æ-I- dx, z-l—dz, u-l-du, ...) ∙− ≼∙↧∙⋅∙∣−↙∣∙↧⋅∙ z—i-dz, u, ...) ::
f,&quot; (x-i—dæi z-l-dzo nus) du:f,, (æ, Z, &quot;, ,,.) du, etc-0-
'57
quarum summa praebet
p ( x + dx, atdz, utdu , ...) — f ( x, z , l , ... ) =
fr ( x, 2, U, ...)dx + f : (x , 2, u, ...)dz + f'u ( x,
Z, U, ... ) dut ... , seu dų = d .; + d ,l + d.le + ...
25.• Potest etiam functio pe differentiari successi
ve quoad binas, lernas , ... variabiles v . gr. quoad x, z, quoad
X , 2, u ; etc. ... Id genus partialia secundi , tertii , ... ordinis
differentialia designari queunt per
d, dx M , d , d , dal , ...
sive autem functio u prius differentietur v . gr. quoad x
deinde quoad z , sive prius quoad z , deinde quoad x ,
paallulum attendenti patebit idem in utroque casu pro
venturum differentiale .
26. Detur nunc differentialis aequatio primi ordinis
dy - cydx f ( x ) dx ;
facta y = zu, et adhibita substitutione, emerget zdu + ud:
czudx = f ( x) dx . Pone udz – czudr = 0 ; habebis
dz
= cdx , log ( x ) = cx = cx log ( e) = log ( eⓇx ) ; unde 7
z &gt; eºx :
in ea qua sumus hypothesi
zdu = f(x) dx ;
igilur du = f ( x ) dx
f (x) dx , u Sf (x)dx + G ; et 7 es
ex
1
5
quarum summa praebet
f(x-f—dx, z-l—dz, (kl—du,...) —f(x,'z, u, ...):
f: (æs 31 ut ⋅∙ ')dæ—I—fg (æ, :, II,. ..)dz—l—f'u (æ,
.z, u, .,.) du-l—n., seu dy.— −∙∙ d,.p. :i- dyp. ∙−⊦ d,); ∙∣−∙∙∙
25. ∘ Potest etiam functio p. diil'erentiari successi-
vequoad binas, ternas... .variabiles v. gr. quoad x,z, quoad
æ, :, u; etc.
... Id genus partialia secundi ,
tertii,... ordinis
diii'erentialia designari queunt per
ds dxp'adudadxp-vmi
sive autem functio p. prius differentietur 11. gr. quoad ac
deinde quoad :, sive prius quoad z, deinde quoad x ,
paullulum attendenti patebit idem in utroque casu p1o-
venturum differentiale.
26.&quot; Detur nunc differentialis aequatio primi ordinis
,dy— cydx :f(x) dx;
facta ]: zu, et adhibita substitutione, emerget zdu −⋅⊢ ud:
— czudx :
f
(x) dx
. Pone udz —. czud-r :o ; habebis
dz
Z
∙−−− ∖∙∘⊄≀⋅⋍∙⋅ ,
log (z):cx:cx log (e):
log (e&quot;); unde
∙−−− cx
, z....e :
«
in ea qua sumus hypothesi
zdu :f(x) dx; ∙ ∙
'
igitur du:
,
(x) dx *fbl'c) dx :
&quot;:M—i—C; et 2 0 .: et.:
5 d58
consequenter
y = eriſ f x)dx = C ] :
integratio videlicet dalae aequationis differentialis traducitur
ad integrationem functionis f (x ) dx Porro absoluta aequa er
tionum differentialium integratio eo redit , ut quae relatio
inter quantitatem et quantitatem per eas exprimitur , aequi
valenter exprimatur per aequationes differentialibus liberalas.
27 .. Si dalur differentialis aequatio secundi or
dinis
day dy ta dx +
0 ,
dxt
by:
designantibus k et k' radices aequationis 32 taz +b =0 ,
traducelur illius integratio ad integrationem binarum pri
mi ordinis
dy ' dy - ky ' = 0 ,
dx dx
siquidem , eliminata y' , prodibit
- ky = y ' ;
a
dy
-ky)
dx dy
dx – k G - hy) == 0 ;
quae , ob k tok = -a et kk' = b , recidit in datam. Jam
vero ( 260 )
dx y ' = Cetry = e ** C
elix

:

ergo
y = ek's
es [ foe-tyde +c ] - [ * +c ]=
Ceks + C'ek's .
58
consequenter
yzccxiffiæidæzcl: Bex
integratio videlicet datae aequationis differen'tialis traducitur
f
(x) dx ad integrationem functionis
ac:
. Porro absoluta aequa-
tionum differentialium integratio: eo redit ,
ut quae relatio
inter quantitatem et quantitatem per eas exprimitur , aequi-
valenter exprimatur per aequationes differentialibus liberatas.
27.0 Si datur differentialis aequatio secundi or-
diuis
?;: 437 dr −⊦ &quot;ï; −⊢ b,! −−∶ ∘⋅
designantibus I: et k' radices aequationis z' −⊸⊢ az —]-b:0.
traducetur illius integratio ad integrationem binarum pri-
mi ordinis ⋅
alt&quot; ∙⋅− df
↙↙−↜↕∶∎∎−∎↗⋮∫−−∘∣∠≀↜↿∶−↻ ⋅⋅⋅⋅∙∙⋅−−−−−↗ '
siqnidem , eliminata y'
, prodibit
d d ∠−− '
(dx kf) A(g—F):0.
d.; ⋅ dx ] '
quae ∙Ob k −⋅⊢ ∣⊏∎:
— a et kk':b, recidit in datam. Jam-
vero (260)
.)&quot;: Ce&quot;'.y:e*&quot;[ ∫−⋅≤∎⇂−⊺∶⋮⇆∙⊹∁∙ ]

:

ek'x
ergo
∙∙∙ ': ∙∙ ∙ r ∙ rr Ceu—H).:
..7— e*. [Ca,/460 &amp;) dx—l—C] ∙−−∶ e* k—k. *C]:
- Ce&quot; ∙⋅∣−∁∎ e*&quot; .59
28.- Si daretur d²y dxata
dr. + by = f(x),tra
duceretur integratio ad integrationem binarum dy'
-ky'
da
P(z) Tipo - Ky = y'; sicque prodirent ( 260)
[Sl + c]
e** [ S ,* + c ]
y' = etxe
k et k 'sunt , ut supra, radices aequationis z2 taz + b = 0.
29.• Resumentes functionem f ( x ) , ponamus
f(x) = a, tax taqx? +R3 2 :3 + 04x4 +

:

exsurgent
f ( x) = a + 2a , x + 3az x2 + 404 x3 + ... , f&quot; ( x)
2a + 3.2a3 x + 4.394 va t ... if&quot; ( 0) = 3.2a3 +
4.3.204 x +
Facto x = 0 , emergent ao f (o) , a, =: f ( 0) , a,
i f' (o) , az =-3f&quot; (o), etc.... Hinc
etc...
f(x) = f(0)+xf ( 0) + 1&quot; (0) + &quot;(o) + ...
Sint v. gr. f (x) = e*. f (x) sinx , f (x) = cosx : quoad
primam f (o ) = 1, f (0 ) = 1 , f ' (o) = 1,8 &quot; (0 ) = 1 , etc...;
quoad secundam , f (o) = 0 , f ( 0) = 1 , p (o ) = 0 ,
fr (0 ) • , 1 , f (0) = 0, f ( 0) = 1 , etc...; quoad ter
59
. dfy dy 28.0 St daretur . (: d −⊦∙ 6]:
f
(x) ,.tra-
dxa x
duceretur integratio ad integrationem binarum Si.-..;. ':
dx '7
f
(a:), £
—)(]:
!' &lt;nowiki&gt;; sicque prodirent (260) www-rc]&lt;/nowiki&gt;
730, reli]dx
11 et k' sunt
,
ut supra ,
radices aequationis z' -l—az—-I—-b :o.
29.&quot; Resumentes functionem f
(x) , ponamus
f(x):ao—I&quot;alx −⊦∁≖∙↕≖∙−∣⋅−↷∍ ∷∙⋅∍⊣−∦∣∙∙≂↙∣−⊦∙∙∙⇋
exsurgent
f(x):a, -l-Za,x-l-3a3x3 404 ∞∍−⊢∙∙⋅ ,f&quot; (x):
2a3-1l-3. Zaax-i— 4. 3a(,x2 -[-...,f&quot;(x) :3.2a3—[—
4. 3. 244x—l-..,
,etc...
Facto x:a, emergent a,:
f(o) , a,:
f
(a) , a,:

;

f. (0) , 03 3— f&quot; (0) ' :.
etc-00. Hinc 3 '
f(x): f(0)-l-xf(0)-i-—-f'(0)'l&quot;——f (o)-b&quot;-
Sint v. gr. f(x):ex.f(x) :sinx ,f
(x):
coax :quoad
primam f(o):1, f
(0):1, f' 'o):1,f&quot; (o):1, etc...;
quoad secundam , f(o):0, f(o) :1 , f&quot; (a):0 ,
f&quot; (Ol—&quot;'
-— ∙−− 1, f' (0):0, f' (0):1, etc...; quoad ter-60
23
tiam , f (o ) = 1 , f ( 0) = 0.8&quot; ( ) 1,8 &quot; ( 0 ) = 0 ,
f (o ) = 1 , ' (o) = 0 , p (0 ) 1 , etc... ; ideoque
x2 24 3 e* : 1 tox +*+ +
sin u = r
2.3.4 2.3
x2 8: 4
cos = 1
2.3.4.5 2
+
2.3.4 2.3.4.5.6
1+
ar5
x6
1.5.0 + ...
30.• Adhibita xV - 1 pro x in istarum prima,
emerget
= 1
x2
e **vi .x4
+
2.3.4
Xc6
2 2.3.4.5.6 +
r3 xc5
+
2.3 2.3.4.5 -.)v = 7.
ܪ
unde , ob secundam ac tertiam ,
e #xVST = cos x + V1 sio x .

28. Fac nunc ut punctum materiale vi qualibet continua
sollicitetur ad motum rectilineum: sit »» velocitas puncti
in fine temporis t,,.s spatium percursum , et ds spatiolum
percurrendam subsequente tempusculo dt. Perinde spectari
poterit ds ac si motu uniformi conficeretur , sola nimirum
velocitate praeconcepta 1); siquidem nova velocitas dv, quae
labente d:accedit materiali puncto, utpote infinitesima. ne-
gligenda.Hi11c (1 )
s
[
∥
Motus rectilineus puncti materialis iugiter sollicitati vi
constanter eadem, dicitur uniformiter varius. Per ep desi-61
/
gnetur velocitas, quam vis constanter eadem gignit intra
tempus 1 , erit qe velocitas ( 6 ) genita intra tempus t :
propterea denotante vo velocitatem initialem , qua nimirum
donatur materiale punctum quum t = 0, existet
v =v, +9
ds Hinc dt votoe : fac ut tempori 1 = o respondeat
So ; habebis
s -8 = v. i + 902 ?

;

2
1
et eliminato t ,
v2— v.2 = 29 / s - s . ) :
positis v ,30,
0, erunt
V = pt , s =
- Det
, v?= 205 ,
o dicitur vis acceleratrix : el designante m
massam puno
cti materialis, m q appellatur vis motrix : insuper spatium s in
aequatione ultima vocatur allitudo debila velocitati v.
Ad motum rectilineum utcumque varium quod spe
ctat , nomine vis , acceleratricis apud terminum spatii per
carsi s nihil aliud intelligitur nisi velocitas q , quam gi.
gneret vis conversa in constantem, constantique energia
qua inibi pollet , agens loto tempore 1. lamvero exhibet
do numerum tempusculorum , ex quibus coalescit tempus
1 ; ergo velocitas illa exprimetar per dv; nimirum
1
61
gnetur velocitas, quam vis constanter eadem gignit intra
tempus 1, erit got velocitas (6) genita intra tempus ::
propterea denotante 'Uo velocitatem initialem, qua nimirum
donatur materiale punctum quum :

: o, existet

v:v, ⊣∙− cp :.
Hincd ∙−∙−:v.,-l—got: fac ut tempori :

: o respondeat

,,- s,;habebis
(2
⋅⇟−⋅⋅≖∘∶∶
'&quot;o t&quot;l&quot; 92&quot;;
et eliminato t,
vï—vo*:2?( s—s, ):
positis v,: o ,r,: 0, erunt
v:g0t, :: gt: ,
v': 291,
q; dicitur vis acceleratrix: et designante m massam pun-
cti materialis, m ? appellatur vis motrix: insuper spatium .: in
aeqnatione ultima vocatur altitudo debita velocitati 9.
Ad motum rectilineum utcumque varium quod spe-
ctat
, nomine visacceleratricis apud terminum spatii per-
cursi :nihil aliud intelligitur nisi velocitas ga, quam gi-
gneret vis conversa in constantem, constantique Aenergia ,
qua inibi pollet, agens toto tempore 1. Iamvero exhibet
−↿−∙ numerum tempuscu'lorum, ex'quibus coalescit tempus
dt
1 .: ergo velocitas illa exprimatur per Tit-dv; nimirum62
dv
dt .
et quia dy
d's d
dt ; idcirco erit quoque
dès
d12
habetur pro variabili atque independente quantitate.
29. Fac v. gr. ut materiale punctum sollicitetur versus
datum centrum vi acceleratrice, quae distantiae z' ab eo cen
tro exsistat proportionalis , ut , denotante C ' quantitatem
constantem , habeamus q =C'z' ; sit z, initialis distantia ,
ibique vo =0 , t =0 ; sit insuper v ' velocitas in distantia
z' : erit ( 28 )
v = d (20-3')
dc
dz
dt
du'
ideoque C'z' =
dt
v'dz'
dzi
Hinc 19. Cʻz'dz' = -v'dv'; ex cujus integratione prodit
C- W'2
C'z'2 =C -2'2 , z = ve C'
facto z' =0, erit v' velocitas punci materialisió centro virium ;
exprimit igitur C hujusce velocitatis quadratum : quod si fiat
z' =2 . , erit ex hypothesi v = v = o, ideoque
VT= 2.VC ;
velocitas nempe puncti materialis in centro virium est ut ipsa
initialis distancia zo.
62
0:32- ' dt -'
et quia xlv::! g:- ; idcirco erit quoque
d3s
(:):
d::
'
habetur :pro variabili atque independente quantitate.

29. Fac v. gr. ut materiale punctum sollicitetur versus
datum centrum vi acceleratrice, quae distantiae z' ab eo cen-
tro exsistat proportionalis , ut, denotante C&quot; quantitatem
constantem , habeamus q) :C'z'; sit zoïinitialis distantia ,
ibique v.:o, t:o; sit insuper v' velocitas in distantia
z' : erit ( 28 )
d(zo-z')-——dz' .d ∙
∙−− dp'— v'dz' d: d:
&quot;
eoque c.. d. dz' '
I,.....
Hinc ↿∘∙ C'z'dz':
—- v'dv'; ex cuius integratione prodit
C— 'v'ï cause—w.r: ⇂∕−∁∼⊤−⋮
facto z':o,erit v' velocitas puncti materialisin centro virium;
exprimit igitur(] hujusce velocitatis quadratum: quod si fiat
z':z,, erit ex hypothesi v':v,;—..-o, ideoque
⇂∕⋜⋮∶−−⊸−≖∘⇂∕−∁−⋮≂
velocitas nempe puncti materialis in centro virium est ut ipsa
initialis distantia z..63
2.º du
1 Tc di=
C'zi v CV C -via VC
Vic С
suinptisque integralibus ,
i = C &quot; +
ve are (sin = vo ):
v = o quando i = 0 , proinde
Vc are ( sin = o), exquav = VC sinero.
3º. Cum in centro virium sit v = VC, erit ibi
1 = sint y C , et consequenter t = Inferimus pun
n
2V0
a
1
ctum materiale eodem semper tempore quacumque 2VC
distantia z . perventurum ad centrum illud .
4º. Si materiale punctum movetur vi accelera
trice, quae distantiae a dato centro sit proportionalis , sic
absque formularum subsidio polest ostendi eodem semper
tempore punctum ipsum peryenturum ad centrum illud :
concipiantur duo puncta, quorum primum triplo magis i
nitio molus distet a centro quam secundum : quoniam ex
hypothesi vis est proportionalis distantiae a centro, erit vis
primi triplo major quam secundi , ideoque triplam velo
citatem primo tempuscalo illud acquiret, et triplum spa
lium percurret; quare etiam tripla ibidem residua erit di
stantia. Igitur et secundo tempusculo triplam velocitatem
novam acquiret, et triplum spatiolum tum praecedente, imm
⊖⊰∆
2.-
−≀≀∙↗⋅ ∙∙∙ dv' '
dv' 1 C'z' yel/CT?&quot;— ⇂∕⋅∁⋅ ⇂∕↿−−−∙∙−−∙∽⋅∙∑−⋅ :
sumptisque integralibus ,
1
(
. v' )
are sm −−− ;
C' y'C
v':o quando :: o , proinde

: z ∁∙∙−⊢ ⇂∕

!
(z.—1.-.— arc (
sin: v
), ex qua ≸↗⋅∶∶⇂∕ ⇂∕∁ ⇂∕∁ sint;/CZ
30. Cum in centro virium sit v': l/C,erit ibi
'io
. n ↿∶∶ . sunl/C, et consequenter :: ï— . Infenmus pun-
ctum materiale eodem semper tempore a quacumque 21/ C
distantia za perventurum ad centrum illud.
40. Si materiale punctum movetur vi accelera-
trice, quae distantiae a dato centro sit propmtionalis, sic
absque formula1um subsidio potest ostendi eodem semper
tempore punctum ipsum perventurum ad centrum illud:
concipiantur duo puncta, quorum primum triplo magis i-
nitio motus distet a centro quam secundum: quoniam ex
hypothesi vis est proportionalis distantiae a centro, erit vis
primi triplo maior quam secundi, ideoque triplam velo-
citatem primo tempusculo illud nequiret, et triplum spa-
tium percurret; quare etiam tripla ibidem residua erit di-
stantia. lgitur et secundo tempusculo triplam velocitatem
novam acquiret, et triplum spatiolum tum praecedente, tam64
nova vi et velocitate percurret: unde consequitur ut tripla
pariter sit lota velocitas jam acquisita , triplum totum spa
tium percursum, tripla distantia residua. Propterea et no
vo tempusculo tripla erit nova velocitas acquisita , tri
plum spatium novum percursum , tripla nova distantia; atque
ita porro. Patet igitur post tempus quodvis distantiam
primi fore triplam distantiae secundi, ac proinde imminu
ta in infinitum ac demum evanescente hujus secundi di
stantia, illius quoque primi distantiam in infinitum immi
nui ac simul evanescere: haud poterit ergo secundum pun.
clum ad centrum pervenire, quin simul cum secundo ipso
primum punctum perveniat. Hoc tantummodo discrimen e
rit, quod primum eo deveniet velocitate tripla secundi ; ex
quo manifeste consequitur , quod si primum illud punctum
ex centro cum illa tripla velocitate projicitur , debebit ad
triplam distantiam pervenire; nam vis in recessu velocita
tem codem ordine extinguit , quo generat in accesso. Por
ro quod diximus de ratione tripla , patet generatim conve
nire rationi cuicumque ; nimirum in quacumque propor
tione fuerit distantia prini punci major quam secundi ,
eodem tempore semper ambo ad centrum devenient cum
velocitalibus , quae distantiis initio habitis sint proportio
nales; et si inde discedant cum velocitatibus quibuscumque,
pervenient eodem pariter tempore ad distantias ipsis ve
locitatibus proportionales.
5º. Dicatur tempus quo materiale punctum it ac
redit uude primo discessit; erit ( 3º. )
471 276 271 0
276 VC
Quare ( 1º, 2º. )
2750 C ,
6
0 2751
0
G
220
210
VO TT
z = VC
COS 277
64
nova vi et velocitate percurret: tinde consequitur ut tripla
pariter sit tota velocitas iam acquisita, triplum totum spa-
tium percursum, tripla distantia, residua. Propterea et no-
vo tempusculo tripla erit nova velocitas acquisita, tri-
plum spatium novum percursum, tripla nova distantia; atque
ita porro. Patet igitur post tempus quodvis distantiam
primi fore triplam distantiae secundi, ac proinde imminu-
ta in infinitum ac demum evanescente huius secundi di-
stantia, illius quoque primi distantiam in infinitum immi-
nui ac simul evanescere: haud poterit ergo secundum pun-
ctum ad centrum pervenire, quin simul cum secundo ipso
primum punctum perveniat. Hoc tantummodo discrimen e-
rit, quod primum eo deveniet velocitate tripla secundi; ex
quo manifeste consequitur, quod si primum illud punctum
ex centro cum illa tripla velocitate projicitur, debebit ad
triplam distantiam pervenire; nam vis in recessu velocita-
tem eodem ordine extinguit , quo generat in accessu. Por-
ro quod diximus de ratione tripla, 'patet generatim conve-
nire rationi cuicumque; nimirum in quacumque propor-
tione fuerit distantia prinii puncti maior quam secundi,
eodem tempore semper ambo ad centrum devenient cum
velocitatibus, quae distantiis initio habitis sint proportio-
nales; et si inde discedant cum velocitatibus quibuscumque,
pervenient eodem pariter tempore ad distantias ipsis ve-
locitatibus proportionales.
50. Dicatur 9 tempus quo materiale punctum it ac
redit uude primo discessit; erit (30.)

:

∡⊺≖∙∙∙∙∙⊸ 211 ,—21t
9
⊋⇂↗⇠∁⋮↼−⇀∣−∕−⋐⇀∶⋅−∙⋅⇂∕∁−⇀⊺⋅∙ Quare (10. 20.)
∙∙∙⊇⇂∕∁ 9
220 ∙∙∙⊓∙ ⇂∕∐⋅ ∶∶↼⋤⋮−⇂∕∁∙≀≀↗⋅∶⇂∕∁⊱⋮∥ −−−−⊖−⋅ ⊋⋯⋅ ⋅− 9 271! z': l/C -—-co −−−−∙
271 s 965

=== De verticali gravium descensu atque ascensu. ===
[[30|30]]. Si gravitas aequaliter semper ad sensum corpora decidentia sollicitare intelligitur, motus erit uniformiter varius (28): positis igitur &lt;math&gt;v_0=0,s_0=0&lt;/math&gt;, et denotante &lt;math&gt;g&lt;/math&gt; vim acceleratricem ex gravitate, in ea qua sumas hypothesi determinabitur motus per formulas

&lt;math&gt;v =gt,s=gt^2/2, v^2 =2gs (b) , &lt;/math&gt;

legibusque sequentibus subjicietar.

1&lt;sup&gt;a&lt;/sup&gt;. Spatium &lt;math&gt;s&lt;/math&gt; percursum intra tempus &lt;math&gt;t&lt;/math&gt; est dimidia pars illus spatii &lt;math&gt;s'&lt;/math&gt;, quod percurreretur si grave aequali tempore pergeret moveri uniformiter cum velocitate &lt;math&gt;v&lt;/math&gt; in
fine temporis &lt;math&gt;t&lt;/math&gt; acquisita; nam (1)

&lt;math&gt;s' = tv = tgt = gt^2 = 2s.&lt;/math&gt;

2&lt;sup&gt;a&lt;/sup&gt;. Spatia totalia a gravibus libere decidentibus percursa,
sunt ut quadrata temporum quibus eadem spatia conficiuntur: item ut quadrata velocitatum tempore descensus acquisitarum  Nam

&lt;math&gt;s=gt^2/2=\frac{v^2}{2g}.&lt;/math&gt;

3&lt;sup&gt;a&lt;/sup&gt;. Spatia a gravibus libere decidentibus percorsa
aequalibus et successivis temporibus sequuntur progressio
numerorum imparium 1,3,5,7, ... ; assumpto enim
&lt;math&gt;t = 1,2,3,4 &lt;/math&gt;, ... spatia illa exprimentur per &lt;math&gt;\frac{g}{2},
\frac{4g-g}{2},\frac{9g - 4g}{2},\frac{16g-9g}{2}, \mathrm{seu}\,
\frac{3g}{2}, \frac{5g}{2}, \frac{7g}{2}.
&lt;/math&gt; 

Hae leges experientiae cum sin &lt;math&gt;t&lt;/math&gt; consentaneae, hypothesis gravitatis aequaliter semper ad sensum agentis prope
telluris superficiem existimanda est naturae conveniens: et
quoniam experimentis saepe iteratis apud nostras regiones compertum est, grave sibi relictum percurrere pedes
15, 0915 ... intervallo unius minuti secundi, erit

&lt;math&gt;g = \frac{2s}{t^2} = 2\times 15,0915 ... = 30,183 ... &lt;/math&gt;&lt;ref&gt;9,78:30,183=0,324 m/pes&lt;/ref&gt;

Eam nimirum velocitatem gravitas valet mobili communicare intervallo unius secundi, qua si mobile pergeret uniformiter moveri, absolveret singulis secundis pedes 30,2 circiter.

Deprehenderunt quidem Physici gravitatem esse diversam tum ad diversas supra terrestrem superficiem altitudines, tum ad diversas ab aequatore terrestri distantias: verum ejusmodi variationes in corporum gravitate haud fiunt sensibiles nisi sub differentiis admodum grandibus sive inter altitudines illas, sive inter illas distantias; propterea absque sensibili errore contemni poterunt in ordine ad singula corpora terrestria, quae ut plurimum veniunt consideranda.
Si retenta &lt;math&gt;s_0=0&lt;/math&gt;, ponitur &lt;math&gt;v = a&lt;/math&gt;, exsurgent (28) &lt;math display=&quot;block&quot;&gt;v=a+gt, s = at + gt^2/2, v^2-a^2 = 2gs (b').&lt;/math&gt;

[[31|31]]. Assumpta &lt;math&gt;g&lt;0&lt;/math&gt; in (b'), prodibunt&lt;math display=&quot;block&quot;&gt;v = a-gt, s = at - gt^2/2, a^2-v^2 = 2gs (b'');&lt;/math&gt;

quae formulae manifeste determinant verticalem gravium ascensum. Facta &lt;math&gt;v=0&lt;/math&gt; in tertia ac prima (b&quot;), emergent
&lt;math&gt;
s=\frac{a^2}{2g}, 
t= \frac{a}{g} (b'''),
&lt;/math&gt;
maxima nempe altitudo ad quam ascendit grave, tempusque respondens.

Obiter hic notamus illud: Si datur ejusmodi potentia &lt;math&gt;R&lt;/math&gt;, quae agendo ad modum vis instantaneae valeat massae &lt;math&gt;M'&lt;/math&gt; communicare velocitatem &lt;math&gt;a&lt;/math&gt;, ut sit (6) &lt;math&gt;R= M'a&lt;/math&gt;, ipsa &lt;math&gt;R&lt;/math&gt; agendo ad modum vis continuae per gradus infinitesimos poterit ponderosam massam &lt;math&gt;M&lt;/math&gt; sustentare libratam per totum tempus &lt;math&gt;t = \frac{M'a}{Mg}&lt;/math&gt;

Cum enim singulis tempusculis infinitesimis &lt;math&gt;dt&lt;/math&gt; gignat
gravitas in massa &lt;math&gt;M&lt;/math&gt; quantitatem motus &lt;math&gt;Mgdt&lt;/math&gt;, certe singulis &lt;math&gt;dt&lt;/math&gt; debebit &lt;math&gt;R&lt;/math&gt; ad librandam &lt;math&gt;M&lt;/math&gt; exerere actionem infinite parvam &lt;math&gt;=Mgdt&lt;/math&gt;; proinde totalis actio respondens toti &lt;math&gt;t&lt;/math&gt; erit &lt;math&gt;\int Mgdt = Mgt&lt;/math&gt;: igitur &lt;math&gt;Mgt=M'a&lt;/math&gt;; ideoque etc.

Quisque nunc videt posse vim &lt;math&gt;R&lt;/math&gt; exhiberi non solum per
&lt;math&gt;M'a&lt;/math&gt;, sed etiam per &lt;math&gt;Mgt&lt;/math&gt;.
[[Fasciculus:Atwoods machine.png|thumb]]
[[32]]. Ad motum gravium determinandum in machina Atwoodi, sint &lt;math&gt;m&lt;/math&gt; et &lt;math&gt;m +m'&lt;/math&gt; massae filo appensae: quisque videt motricem systematis vim exhiberi per &lt;math&gt;
g ( m +m' ) - gm =gm'&lt;/math&gt;; unde profluit vis acceleratrix &lt;math&gt;g\frac{m'}{2m + m'}&lt;/math&gt; substituenda loco &lt;math&gt;g&lt;/math&gt; in formulis (b). Quoniam vis ista potest pro lubito attenuari, sequitur in Machina Alwoodi posse motus velocitatem imminui quantum libuerit; quod maxime conducit et ad accuratius definienda spatia percursa, et ad aeris resistentiam tuto negligendam. Sicuti enim corpus, quod movetur in medio aliquo materiali, agit in ipsum medium, ejus particulas expellendo, exerceturque corporis actio juxta motus directionem, ita medii particulae juxta contrariam directionem reagunt (7) in corpus atque resistunt; inde oritur quidem imminutio virium in corpore, sed major vel minor, prout major vel minor velocitas communicatur medio expellendo; et consequenter prout major vel minor est velocitas corporis expellentis.

[[33|33]]. Haec notamus circa gravium motum in medio resistente.

1º. Constat gravia decidentia in pleno homogeneo motum suum vi gravitatis sic accelerare ut paullatim evadat proxime et sensibiliter uniformis. Dum nempe corpus initio movetur, primumque velocitatis gradum acquirit, aliquam hujus gradus jacturam pati debet ex opposita medii resistentia. Sed quia velocitas corporis in progressu semper augetur, multo magis augeri etiam debet medii resistentia; siquidem major corporis velocitas non solum importat ut major quoque velocitas communicetur singulis particulis removendis, sed praeterea ut major quoque resistentis materiae quantitas dato tempore dimoveatur. Ergo velocitatis gradus semper magis imminuetur: unde fit quod velocitas corporis ad valorem constantem propius semper accedat, ejusque motus paullatim evadat proxime et sensibiliter uniformis.
[[Fasciculus:Atwood.svg|thumb]]
2º. Medii resistentia cum tota exerceatur contra corporis superficiem, vis motrix inde resultans haud pendebit ab ipsius corporis massa, eritque eadem utcumque sub eadem et forma, et amplitudine superficiei, crescat vel decrescat massa: non sic dicendum de respondente vi acceleratrice, quae cum obtineatur dividendo vim motricem per corporis massam, permanente et forma, et amplitudine superficiei, erit reciproce ut ipsa massa. Hinc patet cur, caeteris paribus, quo major est massa corporis in medio resistente decidentis, eo etiam rapidior sit motus finalis.

3º. Si concipimus planum variis resistentis medii stratis normaliter occurrens velocitate &lt;math&gt;v&lt;/math&gt;, ponimusque et plani actionem in medii particulas intra singula tempuscula infinitesima sese protendere ad respondentia duntaxat strata dimovenda, et haec eadem strata illico sic dimoveri ut statim atque dimota sunt nullam praeterea actionem sive immediatam, sive medialam exerceant in dimovens planum; expressa per &lt;math&gt;ds&lt;/math&gt; crassitudine strati dimovendi intra tempusculum &lt;math&gt;dt&lt;/math&gt;, per &lt;math&gt;\mu&lt;/math&gt; densitate medii, et per &lt;math&gt;A&lt;/math&gt; area dimoventis plani, orietur inde (28) resistentia &lt;math&gt;A\mu v ds \frac{1}{dt}&lt;/math&gt; seu &lt;math&gt;A \mu v^2&lt;/math&gt;. Duplicatur resistentia in casu medii elastici (23).

4°* Si vis acceleratrix ex medii resistentia assumitur proportionalis quadrato velocitatis, ut denotante &lt;math&gt;\mathrm{k}&lt;/math&gt; quantitatem constantem (experimentis determinandam), exhiberi possit vis illa per  &lt;math&gt;g\frac{v^2}{\mathrm{k}^2}&lt;/math&gt; gravia descendentia sollicitabuntur vi acceleratrice &lt;math&gt;g-g\frac{v^2}{\mathrm{k}^2}&lt;/math&gt; ascendentia vi acceleratrice  &lt;math&gt;-\left(g+g\frac{v^2}{\mathrm{k}^2}\right)&lt;/math&gt;: proinde (28) quoad gravium descensum
 &lt;math&gt;\frac{dv}{dt}=g-g\frac{v^2}{\mathrm{k}^2}&lt;/math&gt; 
quod ascensum
 &lt;math&gt;\frac{dv}{dt}=-g-g\frac{v^2}{\mathrm{k}^2}&lt;/math&gt; 

5°* In primo casu 
 &lt;math&gt;gdt=\frac{{\mathrm{k}^2}dv}{{\mathrm{k}^2}-v^2}=\frac{\mathrm{k}}{2}\left(\frac{dv}{\mathrm{k}+v} + \frac{dv}{\mathrm{k}-v} \right)&lt;/math&gt; 
sumptisque integralibus:(27.6 °) in hypothesi velocitatis &lt;math&gt;v_0=0&lt;/math&gt;,
 &lt;math&gt;gt=\frac{\mathrm{k}}{2}\ln\left(\frac{\mathrm{k}+v}{\mathrm{k}-v} \right)&lt;/math&gt; 
unde 
 &lt;math&gt;e^{\frac{ngt}{\mathrm{k}}}=\frac{\mathrm{k}+v}{\mathrm{k}-v}&lt;/math&gt;
Primum membrum est necessario &lt;math&gt;&gt;0&lt;/math&gt;; ergo et secundum: crescente igitur &lt;math&gt;t&lt;/math&gt; crescet quidem &lt;math&gt;v&lt;/math&gt;; ita tamen ut nunquam fiat &lt;math&gt;v &gt; k&lt;/math&gt;: quod consentit cum dictis (10). Ad haec : quoniam (28)
&lt;math&gt;
dt=\frac{ds}{v}&lt;/math&gt; erit &lt;math&gt;gds=\frac{{\mathrm{k}^2}vdv}{{\mathrm{k}^2}-v^2}&lt;/math&gt; quam integrantes assequemur &lt;math&gt;gs = C - \mathrm{k}^2\ln(\mathrm{k}^2 -v^2)&lt;/math&gt;: in initio motus ex hypothesi &lt;math&gt;v =0 , s =0&lt;/math&gt;, ac proinde &lt;math&gt;C = \frac{\mathrm{k}^2}{2}\ln \mathrm{k}^2&lt;/math&gt;; igitur
 &lt;math&gt;gs= \frac{\mathrm{k}^2}{2}\ln\frac{{\mathrm{k}^2}}{{\mathrm{k}^2}-v^2}&lt;/math&gt;
6°* In secundo casu
 &lt;math&gt;gdt=\frac{{\mathrm{k}^2}dv}{{\mathrm{k}^2}+v^2}&lt;/math&gt;
ideoque (27.13°)
 &lt;math&gt;gt = C - \mathrm{k}\arctan(\frac{v}{\mathrm{k}})&lt;/math&gt;
tempori &lt;math&gt;t = 0&lt;/math&gt; respondet &lt;math&gt;v =v_0&lt;/math&gt;, et consequenter &lt;math&gt;C
= \mathrm{k}\arctan(\frac{v_0}{\mathrm{k}})&lt;/math&gt;; igitur (tang- ):
5l = k [ arc(tang = :) - arc (ranga ) ] .
ds Ad haec : ob de habemus


s
V71
ndum :
gds =
kavdv ; propterea gs = C—
kat va -- 105 (1º + vw).
log ( Kº +w.), ce
ka In initio motus s = 0 , v = Vo;hinc CF
2
gs =
log k2+0.2
katua 2
Facta v = 0 , prodibunt
k2 proind k
log
ktve t
2g 8 are (tang = ).

maxima videlicet altitudo ad quam in medio resistente ascendit grave, tempasque respondens.

7º. Fac ut , exhibente YM ( Fig. 17) directionem
normalem stratis TT ''medii resistentis , planum A oblique''
occurrat stratis ipsis sub angulo BMY ( =\beta ) . Recta bc
parallela rectae YM repraesentet velocitatem v , qua move
tur A : resoluta bc in Kc perpendicularem et BK parallelam
plano A , exprimet Aje . KC2 resistentiam medii ; et quo
niam KC bc . sin Kbc = vsin \beta , iccirco resistentia ista
Ajwa , sin a\beta .
J =

=== De gravium descensu atque ascensu per plana inclinata; de attritu; deque cochlea, et cuneo.===
[[Fasciculus:Free body.svg|thumb|Planum inclinatum]]
[[34]]. Super plano ad horizontem &lt;u&gt;inclinato&lt;/u&gt; collocetur corpus quod habeat centrum gravitatis in &lt;math&gt;G&lt;/math&gt; (Fig. 21) et massam &lt;math&gt;M&lt;/math&gt;; ex &lt;math&gt;G&lt;/math&gt; horizontem demittatur perpendiculum &lt;math&gt;GH&lt;/math&gt;; et ex &lt;math&gt;H&lt;/math&gt; ducatur alterum perpendiculum &lt;math&gt;HB&lt;/math&gt; in communem plani horizontalis et plani inclinati intersectionem; vis motrix ex corporis pondere jacebit in plano perpendiculornm &lt;math&gt;GH , HB&lt;/math&gt;; demisso enim ex &lt;math&gt;G&lt;/math&gt; perpendiculo &lt;math&gt;Gi&lt;/math&gt; in planum inclinatum, vis illa invenietur in plano &lt;math&gt;iGH&lt;/math&gt; normaliter insistente intersectioni plani inclinati et plani horizontalis; quod planum &lt;math&gt;iGH&lt;/math&gt; manifeste recidit in planum perpendiculorum &lt;math&gt;GH , HB&lt;/math&gt;. Sit &lt;math&gt;AB&lt;/math&gt; communis intersectio istius plani et plani inclinati; &lt;math&gt;AC&lt;/math&gt; perpendiculum ex &lt;math&gt;A&lt;/math&gt; demissum in &lt;math&gt;BH ... ; c&lt;/math&gt; angulus &lt;math&gt;ABC&lt;/math&gt;: recta &lt;math&gt;AB&lt;/math&gt; vocatur longitudo plani inclinati, &lt;math&gt;AC&lt;/math&gt; altitudo, &lt;math&gt;c&lt;/math&gt; &lt;u&gt;angulus inclinationis&lt;/u&gt;. Vim motricem per &lt;math&gt;GK&lt;/math&gt; repraesentatam resolve in duas &lt;math&gt;Gi , Gh&lt;/math&gt;, quarum altera sit perpendicularis, altera parallela rectae &lt;math&gt;AB&lt;/math&gt;; erunt

&lt;math&gt;Gi = gM \cos c , Gh = gM \sin c&lt;/math&gt;.

Cadat &lt;math&gt;Gi&lt;/math&gt; intra corporis basim; elisa &lt;math&gt;Gi&lt;/math&gt; a resistentia plani inclinati, gignetur motus a sola &lt;math&gt;Gh&lt;/math&gt;; quae cum maneat constanter eadem, non alium pariet motum nisi uniformiter varium. His positis, ad determinandum gravium motum per plana inclinata satis erit in (6,6' . 30) et in ( 6 &quot; . 31 ) substituere &lt;math&gt;g \sin c&lt;/math&gt; pro &lt;math&gt;g&lt;/math&gt;: denotantibus itaque &lt;math&gt;\theta&lt;/math&gt; tempus, &lt;math&gt;u&lt;/math&gt; velocitatem, et &lt;math&gt;z&lt;/math&gt; spatium, erunt quoad gravium descensum per plana inclinata u = g 9 sin c, z = * gga sin c, u = 2gz sin c ( 6 &quot; )
si tempori 0 = o respondent u = 0,2 = 0 ; et
u = u + go sinc,z = altiglasin c,u ? —a? = 2g zsin c (6 )
si tempori 0 =o respondent u = a, z = 0 : quoad ascen
sum vero
u =a -g6 sinc, z =a9— 1 g 2sinc, a ?—u? = 2gz sin c (65 &quot; )
&lt;u&gt;Componens&lt;/u&gt; &lt;math&gt;Gi&lt;/math&gt; exhibet pressionem, quam exercet grave contra planum inclinatum .

et :spatium , eruntxquoad gravium descen-
sum per plana inclinata
u:g93inc, :

: äggï sinc, 113:2gz sinc (ö&quot;)

si tempori 6:o respondent u:o,:

: o; et

u:u-1-g 9ainc.z:a G—i-äggasin c,u'—a*:Zgzsin c(b')
si tempori 9:o respondent u:a, s:o :quoad ascen-
∙ sum vero
u :::—gg sinc, :349— äggaslncaaa—uzzzgz Sine (b'-l)&quot; r Componeus Gi exhibet pressionem, quam exercet grave
contra planum inclinatum .73

35. Comparantes ( 6 ' ' ) cum (6 ) haec facile stabiliemus.
1. ' Si licals t . erunt i GH
plaui 1 : sin c , s : 2 = 1 : sin c ;
pla
inter
cula
Br
noguls
si duo nempe gravia eodem tempore delabuntur, alterum
verticaliter , alterum per planum inclinatum AB , tam ve
locitates v , u ab ipsis gravibus in descensu perpendiculari
et obliquo acquisitae , quam spatia s , z descripta , erunt
ut longitudo plani ad ejus altitudinem .
2. ° Hinc ubi ex puncto C concursus rectae verti
calis com horizontali ducatur perpendiculam CE ad plani
inclinati lougitudinem AB , grave percurret lapsu obliquo
spatium = AE eo tempore , quo percurreret lapsu verticali
totam altitudinem AC ; nam AC : AE - AB : AC.
3. ° Inde sequitur chordas omnes circuli ad supre
mam , vel infimam diametri verticalis extremitalem pertin
gentes describi eodem tempore ; eo nimirum , quo descri
beretur ipsa circali diameter.
4. ° Velocitates u , v gravium in plano ioclinato
et in recta verticali aequales sunt si gravia ex punctis aeque
altis ad eamdem rectam horizontalem pervenerint : in ea
sumus hypothesi est s = zsinc , ac proinde
a pla
ifors
16 :3
cempo
enim qua

: u2 V =U .

5.° Tempus descensus per longitudinem plani in
clinati ad tempus descensus per altitudinem est ut ipsa
longitudo ad altitudinem : nam in casu ( 4º ) u = v ; ideoque
SIDEN
g9 sin gt , et 0 : t 1 : sin c .

36. Sint nunc plura plana sibi contigua ( fig. 22. * )
diversimode ad horizontem inclinata . Si grave ab AB
transit ad planum BD , in eo transitu non retinebit in
initio plani BD totam velocitatem , quam habebat in
fine plani AB. Si enim concipitur recta AC perpendi
6 et ?
73
35. Comparantes (b &quot;') cum (6) haec facile stabiliemus.
1.&quot; Si 9:t . erunt
v:u:1:sinc,s:z:1:sinc;'
si duo nempe gravia eodem tempore delebuntur, alterum
verticaliter , alterum per planum inclinatum AB, tam ve-
locitates v ,
1: ab ipsis gravibus in descensu perpendiculari
et obliquo acquisitae , quam spatia s
, z descripta , erunt
ut longitudo plani ad ejus altitudinem .
2? Hinc ubi ex puncto G concursus rectae verti-
calis cum horizontali ducatur perpendiculum CE ad plani
inclinati longitudinem AB, grave percurret lapsu obliquo
spatium :AE eo tempore , quo percurreret lapsu verticali
totam altitudinem AC; uam AC :AE :- AB :AC.
3.&quot; Inde sequitur chordas omnes circuli ad supre-
mam , vel infimam diametri verticalis extremitatem pertin-
gentes describi eodem tempore; eo nimirum , quo descri-
beretur ipsa circnli diameter.
. 4.&quot; Velocitates 11 .'v gravium in plano inclinato
et in recta verticali aequales sunt si gravia ex punctis aeque
altis ad eamdem rectam horizontalem pervenerint: in ea
enim qua sumus hypothesi est s:zsinc
, ac proinde
v&quot;:
uz , v :u .
'
5.&quot; Tempus descensus per longitudinem plani in-
clinati ad tempus descensus per altitudinem est ut ipsa
longitudo ad altitudinem: nam in casu (40) u:v ; ideoque
g95inc:gt,et-9:t:1:sinc.
36. Sint nunc plura plana sibi contigua (fig. 22.')
diversimode ad horizontem inclinata. Si grave ab AB
transit ad planum BD, in eo transitu non retinebit in
initio plani BD totam velocitatem, quam habebat -in
fine plani AB. Si enim concipitur recta AC perpendi-
6
- .... ↹∙∙∙↽∙⊾ −↿−⇀⋅⋅⋅⋅↽∙⋅↽ f.:-.. ∙−←−−− ↘−∼∙⋅ ,. ∙∙⋅∙∙∙⇁ . ∙∙
'1
cularis plano BD producto
, et velocitas in fine plani ha
bens directionem AB concipitur resoluta in duas AC
,
CB

; illa prior AC

a novo plano BD elidetur
, utpote quae
tota insumitar in eo normaliter percutiendo
, ac seclusa
0
mois elasticitatis consideratione
, sola altera CB urgebit cor
pus per novum planum BD
, eritque velocitas prior ad no
vam
, qua nempe ingreditur novum planum ut AB

: CB sive ut

radius ad cosinum anguli ABC
, et prior velocitas ad amissam
erit ut radius ad sinum versum ejusdem anguli ABC

; cum

nempe
, si centro
B et radio BA describatur semicirculus
EAE
'
, sit velocitas prior ad amissam ul AB

: CE

.
Erraverunt igitur qui banc velocitatis jacturam minime
considerantes falsum hoc theorema confecerunt,, Ex aliitu
dine quacumque descendens grave per quotlibet ac quaeli
bet plana AB
, BD sibi contigua utcumque inclinata eamdem
in puncto infimo
D velocitatem acquiret ac cadendo perpen
diculariter ex eorum omnium altitudine,, Erit tamen veris
simum theorema si non ad plana contigua quaecumque scd
ad curvas, quae ex infinitis numero rectis lineis et infinite
parvis
( 27. 16
°
) coalescere intelliguntur
, applicetur et poterit verissime sic enunciari ,, Quodlibet grave ex quacum
que altitudine cadens supra superficiem curvam quamcum
que
, eamdem in puncto infimo velocitatem acquiret ac ca
dendo perpendiculariter ex ipsius curvae altitudine ,, Ratio
est quia velocitatum jactura in transitu de uno in aliud
planum
, decrescente angulo quem continet planum alterum
AB cum altero DB producto
, decrescit siquidem decrescente angulo ABC decrescet sinus versus CE repraesentans
velocitatem amissam
. Quare faclo infinite parvo angulo ABC
,
uti contingit in curvis
, velocitas quoque amissa fiet infinite
parva
, ac proinde grave ingredietur planum BD cum ve
locitate acquisita in descensu per planum AB
. Porro sinus
versus CE
' ita decrescit ut, facto infinite parvo primi or
dinis angulo ABC
, ipse CE
' evadat infinitesimus secundi or
dinis

; nam EC
: AC

= AC

: CE

'.
74
cularis plano BD producto
,
et velocitas in fine plani ha-
bens directionem AB concipitur resoluta in duas AC
,
CB; illa prior AC :: novo plano BD elidetur, utpote quae
tota insumitur in eo normaliter percutiendo, ac seclusao-
mnis elasticitatis consideratione, sola altera CB urgebit cor-
pus per novum planum BD, eritque veloeitas prior ad no-
vam, qua nempe ingreditur novum planum ut AB:CB sive ut
radius ad cosinum anguli ABC, et prior velocitas ad amissam
erit ut radius ad sinum versum ejusdem anguli ABC; cum
nempe, si centro
B et radio BA describatur semicirculus
EAE', sit velocitas prior ad amissam ut AB: CE'.
Erraverunt igitur qui hanc velocitatis jacturam minime
considerantes falsum hoc theorema coufecerunt,, Ex altitu-
dine qnacumque descendens grave per quotlibet ac quaeli-
bet plana AB, BD sibi contigua utcumque inclinata eamdem
in puncto infimo
D velocitatem acquirat ac cadendo perpen-
diculariter ex eorum omnium altitudine,, Erit tamen veris-
simum theorema si non ad plana contigua quaecumque sed
ad curvas, quae ex infinitis numero rectis lineis-et infinite
parvis (27. 16&quot;) coalescere intelliguntur, applicetnr; et po-
terit verissime sic enunciari ,, Quodlibet grave ex quacum-
que altitudine cadens supra superficiem curvam quamcum—
que, eamdem in puncto infimo velocitatem acquiret ac ca-
dendo pan-pendiculariter ex ipsius curvae altitudine ,, Ratio
est quia velocitatum jactura in transitu de uno in aliud
planum, decrescente angulo quem continet planum alterum
AB cum altero DB producto, decrescit; siquidem decrescen-
te angulo ABC decrescet sinus versus CE' repraesentans
velocitatem amissam. Quare facto infinite parvo angulo ABC,
nti contingit in curvis, velocitas quoque amissa fiet infinite
parva, ac proinde grave ingredietur planum BD cum ve-
locitate acquisita in descensu, per planum AB. Porro sinus
versus CE' ita decrescit ut, facto infinite parvo primi or-
dinis angulo ABC, ipse CE' evadat infiuitesimus secundi or-
diuis; nam EC: AC:AC: CE'.75
1

37. Hactenus nullam habuimus rationem attritus , seu
resistentiae ex asperitate superficierum : prominentes nem
pe unius superficiei denticuli foveas alterius ingrediun
tur ; sicque haud poterit una superficies alteri superposita
promoveri, nisi ipsi denticuli vel frangantur, vel inflectan
tur, vel , saperiori superficie identidem elevata parumper ,
e foveolis egrediantur: possunt quidem denticuli ita poli
tione imminui , ut sensum inermem effugiant, sed penitus
tolli nequeunt.Statue corpus super plano horizontali ; tum pla
num istud eousque sensim inclina , donec sub quodam angulo
c=c'corpus tantum non incipiat descendere, incipiat vero cre
scente utcumque parum c ultra c' . Attritus respondens angulo
c = c dicatur f: quoniam f accurate librat vim gM sinc'
erit f =g Msinc' ; hinc si per r exprimitur ratio attritus f ad
pressionem gM cosc' ut sit fer. GM cosc ', habebitur
.
r.gM cosc = gM sinc' ,
ideoque
r = tang c' .
0
5
Permanente qualitate massae M, itemque politionis gra
du , constat experimentis quod permanet quoque angulus c' ,
et consequenter ratio r, licet quantitas ipsius M augeatur,
vel minuatur. Inde sequitur attritum f, caeteris paribus, fo
re proportionalem pressioni r.gM cosc' .
Si ponimus attritum adhuc pressioni proportionalem
quum angulus c superat angulum c'; ad habendam ratio
nem attritus in motu gravium per plana inclinata , pro
gsinc substituetur g sin c - rg cosc in ( b ), et gsinc +
rg cosc in ( 6 &quot; ); caeterum in casu motus videtur f non a so
la pressione , sed a corporis quoque velocitate haud pa
rum pendere.
Haec subjungimus.
&quot;
75

37. Hactenus nullam habuimus rationem attritus, seu
resistentiae ex asperitate superficierum :prominentes nem-
pe unius superficiei denticuli foveas' alterius ingrediun-
tur ; sicque haud poterit una superficies alteri superposita-
promoveri, nisi ipsi denticuli vel frangantur, vel mflectan-
tur, vel, superiori superficie identidem elevata parumper ,
e foveolis egrediantur: possunt quidem denticuli ita poli-
tione imminui, ut sensum inermem effugiam, sed penitus
tolli nequeunt.Statue corpus super plano horizontali; tum pla-
num istud eousque sensim inclina
, donec sub quodam angulo
c:c' corpus tantum non incipiat descendere, incipiat vero cre-
scente utcumque parum c ultra c'. Attritus respondens angulo
c:c' dicatur f: quoniam f
accurate librat vim nginc'
erit f

: g Msinc'; hinc si perr exprimitur ratio attritus f

ad
pressionem gM cosc' ut sit:r. gM cosc', habebitur
r. gM cosc': gM sinc' ,
ideoque
r:tang c'
.
Permanente qualitate massae M, itemque politionis gra-
du, constat experimentis quod permanet quoque angulus c',
et consequenter ratio r, licet quantitas ipsius M augeatur,

: vel minuatur. Inde sequitur-attritum f,'caeteris paribus, fo-

1e proportionalem pressioni ngM cosc'.
Si ponimus attritum adhuc pressioni proportionalem
↴⋅ quum angulus c superat angulum ∁∙∍ ad habendam ratio-
lnem attritus in motu gravium per plana inclinata
, pro
igsinc substituetur gsinc—rgcosc in (b' ), et gsinc −∣−
' rgcosc tn ( b &quot;); caeterum in casu motus videtur fnon a so-
lla pressione, sed a corporis quoque velocitate haud pa-
rum pendere.
Haec subjungimus.76
1º . Si corpus in plano inclinato constitutum li
brandum sit potential applicita ( Fig. 21 ) puncto G, quae
potentia et sollicitat ad ascensum, et efficit angulum &amp; cum
AB, gignitque propterea pressionem Qsind, satis erit ut re
sultans ex viribus Q et M ( g sinc F rg cosc ) Fr (sin
exsistat ipsi plano perpendicularis , sese videlicet diri
gat juxta Gi: continet autem Q cum Gi angulum 900
An et vis Mg ( sinc F r cosc ) FrQsinc angulum
cum eadem Gi. Igitur ( 9.10 )
= 90
Q: Mg( sincar cosc ) FrQsing = sin 90 ° ; sin ( 90 °
a ) = 1 : cosa ;
ideoque
sinc Frcosc OSCMS
Q
cos a Es since
secun Sumpio superiori signo, nequit Q esse minor
do membro quin corpus descendat; sumplo inferiori si
gno, nequit Q esse major secundo membro quin corpus
ascendat; perstabit aequilibrium intra limites
sinc - rcosc sinc torcose Mg, el &lt;
cosa + rsing
Mg.
cosu - osinc
2º. In hypothesi nullius attritus erit r = 0 ; et
consequenter
sin c
Q
Mg COSU.
3º. Si Q est insuper parallela horizontali BC, e
rit a = c ; ideoque
76
1&quot;. Si corpus in plano inclinato constitutum li-
brandum sit potentia Q applicita ( Fig'. 21) puncto G, quae
potentia et sollicitat ad ascensum, et eilicit anguluma cum
AB, gignitque propterea pressionem Qsinac, satis erit ut re-
sultans ex viribus Q et M (gsinc :rgcosc ):F r Qsin a
exsistat ipsi plano perpendicularis , sese videlicet diri-
gat juxta Gi: continet autem Q cum Gi angulum :90&quot;—
a, et vis Mg ( sine: rcosc) :rQsina angulum :90&quot;
cum eadem Gi. Igitur ( 9. 1&quot; )
Q: Mgüincqzr 0050 ):t:rQsinat:sin 90&quot; :sin ( 90&quot;—
at:) 1:cosa:;
ideoque
sinc ∓r cosc
−∙∙ Mo
cos a:
r siua
Sumpto superiori signo, nequit Q esse minor secun-
do membro quin corpus descendat; sumpto inferiori si-
gno, nequit Q esse maior secundo membro quia corpus
ascendat; perstabit aequilibrium intra limites
sinc—rcosc sinc rrnsr Q)...— Mg, et Q(
−⊢ Mg.
cosa rsiuat cosa—rsiua
2&quot;. In hypothesi nullius attritus erit r: o ; et
consequenter
sin 0 M
g. szz cosa:
3&quot;. Si Q est insuper parallela horizontali BC, e-
rit at:c ; ideoque77
sipc :
Mg COSC
potentia videlicet ad pondus ut plani altitudo AC ad hori
zontaleon BC. Hinc facile deducuntur aequilibrii leges in
cochlea et cuneo.
4º. Cum cochlea non sit nisi planum inclina
tum ABC, quod circum cylindruni ducitur; dum vero co
chlea agit , potentia sit rectae lineae BC parallela, erit
potentia ad pondus seu resistentiam superandam , ut al
titudo plani seu helicam distantia ( =h ) ad basim plani
seu cylindri peripheriam ( = k ). Hinc
Q
hP ;
k
quae formula supponit distantiam inter cylindri axem et pun .
ctum cui applicatur potentia , esse ipsius cylindri radium
( = m ) : quod si distantia illa fiat alia ab r', et exhibea
tur per R' ; denotante e potentiam respondentem novae
distantiac, exsistet
mi?
R'
ac proinde Q
-
hP
R '
LP
25R .
k
In ordine ad cochleam infinitam , dicatur A radius ma
joris rotae , a radius minoris , et P' pondus seu poten
tia apud dentes ipsius rotae majoris; erunt
ар P =
Q
api A
hP
27.R '
ideoque
Q =
h a P
21AR'
77
Q sine. NT: ⋅⇀ SE.—.'
potentia videlicet ad pondus ut plani altitudo AC ad hori-
zoutalem BC. Hinc facile deducuntur aequilibrii leges in
cochlea et cuneo.
4&quot;. Cum cochlea non sit nisi planum inclina-
tum ABC, quod circum cylindrum ducitur; dum vero eo-
chlea agit, potentia sit rectae lineae BC parallela, erit
potentia ad pondus seu resistentiam superandam, ut al-
titudo plani seu helicum distantia ( :h)ad basim plani
seu cylindri peripheriam :( k). Hinc
Qz—k-i
quae formula supponit distantiam inter cylindri axem et pun-
ctum cui applicatur potentia, esse ipsius cylindri radium
(: r' ): quod si distantia illa fiat alia ab r', et exhibea-
tur per B'; denotante Q' potentiam respondentem novae
distantiae, exsistet
Q'—r' ∙∙ ∙∙∙∣≖∣⊃⋅↿⋅⋅∙−∣≀∌ ∙≺⋮−−−−∙↓⊤∙ ac ptomde QI—B— . ∣∙⋮−−−−∙ ⊋∙⋮⋮⋅⋮↸↽∙
In ordine ad cochleam infinitam, dicatur A radius ma-
ioris rotae
, a radius minoris , et P' pondus seu poten-
tia apud dentes ipsius rotae maioris; erunt
aP , hP'
P::ï'Q—an' '
ide ue
Oq haP78
1
5 ° Cuneus spectari potest tamquam coalescens ex
duobus planis inclinatis ut ABC, tum ob figuram suam ,
tum quia idem est sive pondus per planum inclinatum
trahatur sursum , sive planum sub pondere promoveatur.
Agit autem potentia in cuneo juxta CB; quoad igitur u
1
nam cunei partem ABC respondens potentia Qerit ad
m
1
respondentem resistentiam P ut AC ( = D ) , sen di
midia cunei crassities ad BC ( = H ) , idest ad altitudinem
1
Q
1
ad respondentem resistentiam P P erit pariter ut
į D ad H. Igitur
m
LQ.H - 1P.HD, Q (m - 1 ) .
A
m2 m m
P (m - 1 )
mi '
· D ;
quibus aequationibus in summam collectis ,
Q. H = P. , D ,
et consequenter
D
H totalis videlicet potentia Q ad totalem resistentiam P ut
dimidia cunei crassities D ad ejus altitudinem H ; mo
do tamen exerceatur resistentia normaliter ad H.
6º . Si in cochlea v . gr. considerandus esset at
tritus , haberetur ( 10.40.) ,
1
!
1
5&quot; Cuneus spectari potest tamquam coalescens ex
duobus planis inclinatis ut ABC, tum ob figuram suam,
tum quia idem est sive pondus per planum inclinatum
trahatur sursum, sive planum sub pondere promoveatur.
Agit autem potentia in cuneo iuxta CB; quoad igitur u-
. . 1 nam cune1 partem ABC . respondens potentta —Qer1t ad
.
'
m
respondentem resistentiam −↿−∙∶ P ut AC (: äD ), seu di-
↾
m
midia cunei crassities ad BC (: H ), idest ad altitudinem
. . 1 cunei. Quoad alteram partem respondens poteutta Q—- −− Q
m
. . 1 ad . . respondentem rc51stent1am P
-— −−∙ P er1t partter ut
171
&amp; D ad H. Igitur
D, Q—(m-1).H: −↿−↽≺≀∙∥∶∶∙−↿∙−↕⊃∙ ;.
m m m
P
∙∙
-
(m'1) -äD;
,- ut
quibus aequationibus in summam collectis,
QaHzpaL'D, et consequenter
≟≺−≀∙∙− :D
.
P −⋅⋅ H ⋅
totalis videlicet potentia Q ad totalem resistentiam P ut
dimidia cunei crassities &amp; D ad ejus altitudinem H; mo-
do tamen exerceatur resistentia normaliter ad H.
6&quot;. Si in cochlea v. gr. considerandus esset at-
tritus , haberetur (10. 4&quot;.),
≁−−−−∎⋅−−
−−⋅⋅...-—79
sinc FrcoscP =
cosc trsinc
h = 2 te r'r P ;
h Erk
P
k trh 2 trh
ideoque
Q
Qr Pr' h = 27r's
R ? -R 2 r'trh
0
70. Veniat quoque considerandus attritus in ae- ,
quilibrio corporis AB ( Fig. 23: 24 ) , quod ad rolatilem
motum circa fixum cylindrum sollicitatur vi Rjacente in
plano, cui normaliter insistit axis cylindri.
Ac primo cylindrus impleat accurate circularem cor
poris aperturam DE ( Fig. 23 ) , in quam inseritur: per cy
lindri centrum O duc rectam OEE' parallelam vi R , et
pancto E corporis AB applica duas . vires Q ', Q' aequa
les eidem R, et contrarias, alteram nempe tendentem, ab
E versus E' , alteram ab E versus O; vi R licebit substi
tuere systema virium R , Q ', Q &quot; : et cum possint absque sy
stematis turbatione sic transferri ( 11 ) R et l ' ut aequi
distent ab O, eae nitentur dumtaxat gignere motum ro
tatilem circa cylindrum quin ullam pariant pressionem a
pud ipsius cylindri superficiem ; pressio igitur in hanc su
perficiem redigetur ad solam
୧ = R , ideoque f = Rr.
Attritus fest vis tangentialis respectu superficiei cylin
dricae; hinc denotante a radium OE cylindri , et p per
pendiculum Oi ex O ductum in directionem potentiae R,
ad aequilibrium satis erit, ut exsistat ( 9. 2° )
R 1
2
р
Rr
. 79
Q-—sinc:r:rc.oscP 11:er P—h:t:2nr'rp
cosczbrsmc R::brh 2nr':t:rh ,
ideoque
—Qr' Pr' II::ZRr'r
a' &quot;B' 'an'äzrh Q! '
70. Veniat quoque considerandus attritus in ae- ↗
qnilibrio corporis AB ( Fig. 23: 24 ), quod ad rotatilem
motnm circa fixum cylindrum sollicitatur vi R iacente in
plano, cui normaliter insistit axis cylindri.
Ac primo cylindrus impleat accurate circularem ocr-
poris aperturam DE (Fig. 23), in quam inseritur: per cy- ⋅
lindri centrum O duc rectam OEE' parallelam vi B, et
pnncto E corporis AB applica duas, vires Q', Q&quot; aequa-
les eidem R, et contrarias, alteram nempe tendentem, ab
E versns E', alteram ab E versus O; vi R licebit substi-
tuere system virium R, Q', Q&quot;: et cum possint absque sy—
stematis turbatione sic transferri (11) B et Q'0ut aequi-
distent ab 0, eae uitentur dumtaxat gignere motum ro-
tatilem circa cylindrum quin ullam pariant pressionem a-
pud ipsius cylindri superficiem; pressio igitur in hanc su-
perficiem redigetur ad solam
Q&quot; −∙∙−− R, ideoque f: R r.
Attritus fest vis tangentialis respectu superficiei cylin-
dricae; hinc denotante a radium OE cylindri, et p per-
pendiculum Oi ex Oductum in directionem potentiae Pt,
ad aequilibrium satis erit, ut exsistat ( 9. 20)80
et consequenter
P
facto p &gt; ar , disrumpetur aequilibrium ; facto p &lt; ar ,
subsistet .
Ponatur secundo circularis apertura corporis baud
impleri accurate cylindro ( Fig.24) : vis R manifeste trans
ibit per contactum E cylindri et corporis AB . Resolve R
in duas EF, et ED' , quarum altera transeat per centrum 0 ,
altera tangat cylindrum : per EF exprimetur pressio ; ac
proinde
f = r.EF .
Obtinebit igitur aequilibrium quotiescumque
ED ' &lt; r. EF ,
vel ED' = r.EF :
cum autem ( 9. 1. ° ) .
ED' : R = sin FER ; sin D'EF = sin FER : 1 ,
EF : R = sin D'ER ; sin D'EF = cos FER : 1 ,
cumque ducto perpendiculo Oi ex O in ER ,
Oi Ei voa ?
OE sin FER Р cos FER 22 -
p2
a OE
iccirco praefatac aequilibrii conditiones vertentur in
Rp Rr Vap2 Rp a
Rr Va - p ?
a a
quae traducuntur ad
80
et consequenter
&quot;' p :: ar :
facto p ar, disrumpetur aequilibrium; facto p ar
,
subsistet .
l
Ponatur secundo circularis apertura corporis baud
impleri accurate cylindro (Fig.24): vis B manifeste trans-
ibit per contactum E cylindri et corporis AB
. Resolve B
in duas EF, et ED' , quarum altera transeat per centrum O,
altera tangat cylindrum: per EF exprimetur pressio; ac
proinde
f

: r. EF.

Obtinebit igitur aequilibrium quotiescumque
ED' (r. EF , vel ED' −−∶ r. EF :.
cum autem (9. 1.0).
.'
ED': R ::sin FER : sin D'EF :sin FER : 1 ,
EF fii ∙−−∶ sin D'ER; sin D'EF: cos FER : 1,
cumque ducto perpendiculo Oi ex 0 in EB
.
Oi p Et. ⇂∣ (13 ∙−− :;
' :∙−−− :... ∙ FER ↽− −∙ p
sin FER 08 a 005 OF. a ,
iccirco praefatae aequilibrii conditiones vertentur in
n,,(RrI/aa—pz ↧≹∣↗∙∙∙↧≹≀⋅ Wiz—pa
7.&quot; −−−−↴∶∎−−∙−↙≀∎ ⇀⇀ a '
quae traducuntur ad
⇁−∙↱⇁≓≓81 1
ar 2
p &lt;
р
1 + 12 vit ?
8.• Si ponitur R nihil esse aliud nisi resultans
ex datis viribus P' , Pi ad puncta data v . gr. A , B appli
citis , innotescet R ex dictis ( 10 ) , itemque p. ex ( 10.3° ) .
Sic habetur ratio attritus in vecte : caeterum in machinis
praeter resistentiam ex attritu spectanda etiam est resi
stentia ex funibus . Hi enim inflexioni suae resistunt quum
cylindris vel trochleis circumvolvuntur; et quidem eo ma
gis , quo majori pondere tenduntur , quo insuper crassio
res sunt , et quo minor fuerit trochleae, aut cylindri radius.

=== De motu gravium oblique projectorum.===
[[Fasciculus:Ferde hajitas2.svg|thumb]]
[[38]]. Grave &lt;math&gt;M&lt;/math&gt; (Fig. 25) juxta directionem MG velocitate &lt;math&gt;v_0&lt;/math&gt; projectum urgebitur duplici motu, altero aequabili per &lt;math&gt;MG&lt;/math&gt; ex impetu recepto, altero (nihil est aliud nisi motus relativus mobilis &lt;math&gt;M&lt;/math&gt; quoad ipsum &lt;math&gt;M&lt;/math&gt; iens per &lt;math&gt;MG&lt;/math&gt; sola &lt;math&gt;v_0&lt;/math&gt;) uniformiter accelerato gravitatis proprio per rectam verticalem &lt;math&gt;MR&lt;/math&gt;, vel ipsi &lt;math&gt;MR&lt;/math&gt; parallelam. Sit &lt;math&gt;S&lt;/math&gt; spatium quod cumque &lt;math&gt;MC&lt;/math&gt; primo illo aequabili motu seorsim sumpto percursum, &lt;math&gt;t&lt;/math&gt; tempus impensum ad ejusmodi spatium percurrendum, sitque &lt;math&gt;s&lt;/math&gt; spatium &lt;math&gt;MF&lt;/math&gt; pari tempore percursum secundo motu item seorsim sumpto. Completo parallelogrammo &lt;math&gt;MFQC&lt;/math&gt;, in fine temporis &lt;math&gt;t&lt;/math&gt; grave erit (5) in &lt;math&gt;Q&lt;/math&gt;;
et quia (1:30)
&lt;math&gt;S = v_0 t , s = \frac{gt^2}{2},&lt;/math&gt;
eliminato &lt;math&gt;t&lt;/math&gt;, existet
&lt;math&gt;S^2 = \frac{2 v_0^2}{g}s &lt;/math&gt;
aequatio ad curvam (dicitur parabola) in qua defertur grave. Si altitudo debita (28) velocitati &lt;math&gt;v_0&lt;/math&gt;, dicatur &lt;math&gt;\mathrm{A}&lt;/math&gt;, erit
&lt;math&gt;v_0^2 = 2g\mathrm{A}&lt;/math&gt;, et aequatio transformabitur in
&lt;math&gt;S^2 = 4 As ( c)&lt;/math&gt;.

[[39|39]]. Denotet x horizontalem rectam MK , y vertica
lem KQ , et h angulum CMK ; erunt x = S cosh , y =
CK - CQ = S sin h -5 ; unde
X
X
S =
cosh . sinh :
cosh
quibus valoribus substitutis in (c) , prodibit
x2 rcsinh
4 A
CO -Y) ,
et consequenter cos2 h
cos h
y =xtang h 1 + tang k
4 A
x2 ( c' ) .

[[40|40]]. Haec facile nunc stabiliuntur. 1º facta y = 0 ,
proveniet amplitudo jactus
4 Atangah
1 + tang h
4Asinhcosh = 2 Asin2h.
2.º Inde sequitur maximam jaclus amplitudinem
haberi sub angulo h = 45°.

3. ° Si quaeritur angulus h , sub quo proiicien
dum est grave ut offendat in datum scopum , cujus nempe
dantur coordinatae x et y , erit
2A + V 4A2-4 Ay - x2
tangh
82
aequatio ad curvam (dicitur parabola) in qua defertur
grave. Si altitudo debita (28) velocitati v, dicatur A, erit
vio:2gA, et aequatio trausformabitur in
S':4As (c) Esistente igitur 4 A 2—4 Ay-x &gt;0 , poterit sub duplici
angulo projici grave ut datum -scopum attingat : attinget
autem in fine temporis ( 38 : 39 )
S
ts Vo .
Vo cos h

4.0 in ( c ) pone
2 Atangh ta ;
1 +tangah
babebis
A tangah
1+ tang2h ya
1 + tang 2h W?
4A
( c ' ' ) .
Iam vero maxima y ( dicitur altitudo jactus ) manifeste re
spondet valori w = 0 ; altitudo igitur jactus exhibebitur
per
A tangah seu A sinh.
1 +tangah

5º . Ex eadem ( c &quot; ) quisque colligit parabolam , in qua
defertur grave, dividi a maxima y in duas aequales simi
lesque partes : extremitas maximae y vocatur vertex pa
rabolae; ipsa vero maxima y indefinite producta juxla gra
vitatis directionem appellatur axis parabolae.
[[Fasciculus:Ferde hajitas7.svg|thumb]]
6º Si angulus h fit &lt; o, ut initialis directio cadat iтfra horizontalem rectam ML, jactus amplitudo x (1°) ex &gt; fiet &lt; 0; jactus vero altitudo y ( 40 ) permanebit &gt;o. Quod si fuerit h = o, ut initialis directio recidat in rectam horizontalem ML, nulla erit amplitudo jacеus, nullaque ejus altitudo.

7º. Demittatur perpendiculum QP ex puncto Q
parabolae in axem NI ... , sintque NP = x', Q P =y'; erunt ( 1º . 4º . ) x MI — QP = 2 A sinh con -y'
y=NI — NP = A sin’h— x' : quibus valoribus substitutis
in ( c' : 39 ) , proveniet
y2= 4 A x' cosah
aequatio ad parabolam M N L inter x' ety' computatas a
vertice ; quantitas 4 A cos’h dicitur parameter parabolae ;
quod si in axe sumatur punctum H ita , ut ejus distantia
a vertice sit quarta parametri pars seu A cos ?h , habebitur
punctum illud , quod appellatur parabolae focus.

[[41]]. Cum ad curvam parabolicam describendam, corporis motus, qui fit secundum lineam projectionis, debeat esse aequabilis, qui vero fit secundum lineam verticalem, debeat esse uniformiter acceleratus, cumque hujusmodi certe neuter esse possit si medium utrique motui resistat, iccirco nonnisi in vacuo motus corporis oblique projecti fieri potest per curvam, quae sit perfecte parabolica. In medio resistente curva minus late patet, minusque assurgit quam in vacuo; duobus insuper cruribus dissimilibus &lt;math&gt;AN, NL&lt;/math&gt; (Fig. 26) componitur, quorum descendens &lt;math&gt;NL&lt;/math&gt; ad rectam quamdam &lt;math&gt;FE&lt;/math&gt; ut asymptotum accedit in infinitum, quin unquam congruant. Etenim resoluta projectionis velocitate in duas, alteram verticalem, alteram horizontalem, verticalis tum ab aeris resistentia, tum a gravitate usque ad punctum &lt;math&gt;N&lt;/math&gt; minuetur: propterea punctum &lt;math&gt;N&lt;/math&gt; minus assurget quam in vacuo: postquam grave ad &lt;math&gt;N&lt;/math&gt; pervenerit, descendet ob gravitatis vim damna ex medii resistentia reparantem, et hujusmodi descensus fiet motu verticali ad motum aequabilem (33) semper accedente. At horizontalis velocitas minuitur perpetuo, nulla interim vi iacturam reparante, atque inde fit ut recessus horizontalis a recta verticali &lt;math&gt;NP&lt;/math&gt; certum limitem non praetergrediatur, quem curva habet pro asymptoto. Haec contingunt potissimum corporibus ingenti velocitate in aere projectis.
=== De generalibus quibusdam proprietatibus motus curvilinei, orti a viribus, quarum una determinat materiale punctum ad motum aequabilem, altera ipsi materiali puncto est continue applicata.=== 
[[42|42]]. Concipiamus secundam vim agere solum in initiis quorundam tempusculorum, ac tantam velocitatem unico impulsu valido producere, quantam vis perpetuo agens producit toto illo tempusculo, ut deinde inminuta magnitudine tempusculorum in infinitum, habeatur linea curva orta ex continua vis actione.  

Projecto puncto materiali cum velocitate CB (Fig. 27) simulque illi impressa velocitate CA, abiret punctum per diagonalem CO parallelogrammi AOBC et esset in fine primi tempusculi in O cum determinatione describendi altero aequali tempusculo rectam OL = OC, eique in directum jacenlem. Si hic iterum illi imprimeretur alia velocitas OF, completo parallelogrammo FILO , incederet per diagonalem OI, essetque in fine secundi tempusculi in I cum determinatione describendi tertio tempusculo aequali rectam IM = 10, eique in directum jacentem. Sed ob impressam hic quoque aliam velocitatem IV abiret per novam parallelogrammi diagonalem IH, atque ita porro. Fieret ergo in ejusmodi hypothesi vis agentis per intervalla tempusculorum ut materiale punctum describeret polygonum COIHN etc, cujus latera certam magnitudinem et positionem haberent, definita nempe a directione virium et a ratione velocitatum, quas initio cujusvis tempusculi mobile obtineret. Hinc pro diversis virium ila agentium ordinibus numero infinitis infinita considerari possunt ejusmodi polygona, quorum alia in se ipsa redirent, desinente ultimo latere in puncto C ubi primum inceperat; alia abirent in infinitum. Concipiamus jam numerum tempusculorum augeri, et simul eorum magnitudinem imminui in infinitum, vitum magnitudine tum directione vel constantes manere, vel variare certa quadam lege ad continuam quamdam variationis rationem accedente in infinitum. Augebitur in infinitum numerus laterum polygoni determinato tempore descripti, imminutis interea in infinitum angulis, quos efficit quodlibet latus praecedens cum consequente: cum enim LI debeatur impulsui, qui initio tempusculi 0 eam
velocitatem producere concipitur, quam produceret vis to
to tempusculo agens, cumque per tempusculum infinitesimum vis ista habenda sit pro constante, existet ( 28: 30. 14. ) LI =
092; ideoque ob o finitam, et quadratum 62 infinitesimum secundi ordinis, erit etiam LI infinitesima ordinis secundi, sed OL est infinitesima ordinis primi, utpote quae tempusculo O describitur cum velocitate finita; ergo angulus LOI erit ivfinitesimus: atque eodem pacto demonstrantur infinitesimi anguli MIH , K'HN , etc. Hinc polygonum ad curvam continuam semper magis accedet; et ubi demum continua habealur actio vis, et continuae cuidam legi subjiciantur directio ipsius et magnitudo, obtinebitur curva continua cavam sui partem versus eam plagam obvertens, in quam tendunt vires.

43. Abeunte polygono in curvam , rectae CL , OM' , IH ', HK , etc abeunt in tangentes apud puncta C, O, I, H , etc. Ubi ergo in aliquo curvae puncto vis desinat agere,,
excurret mobile per tangentem apud illud punctum.

44. Sit IM (fig, 28 ) spatiolum quod tempusculo 9
mobile percurreret sola velocitate praeconcepta, et IV
spatiolum respondens vi agenti unico impnlsui valido ; ita
ut existat (42) IV ::99&quot;. Completo parallelogrammo, positis-
que lM:P ,
lH:B, et angulo MIV :i, erit (9. 3.&quot; )
∶∶ Vra-Hæ os −⊢⋅∠⇂⊃⊊↶⊖⋍ cos :.87
Evolvatur quantitas radicalis in seriem : proveniet
R = P + q9 cos i , unde R - P = º02cosi ,
neglectis infinitesimis altioris ordinis. Sit v' velocitas , qua
mobile percurrit laterculum R; erit R = v'0 : sit etiam v
velocitas , qua mobile percurreret laterculum P. si non
adesset impulsus validas in I ; erit P =v @ : hinc R --
P = vv( ) 0 .; et consequenter
v ' - v = q Ocosi.
Ex hac aequatione patet v— esse quantitatem in
finitesimam primi ordinis , positivam vel negativam prout
i &lt;vel &gt; 90° , esse autem =0 si i 90° . Inferimus il
lud : ubi tempore finito angulus , quem efformat vis ac
celeratrix cum directione tangentis , fuerit semper aculus,
acquiret mobile incrementum velocitatis finitum ; si sem
per obtusus , patietur decrementum finitum ; si semper re
lus , velocitas manebit constans.

45. In motu quovis curvilineo quadratum velocitatis aequat vim acceleratricem ductam in dimidium chordae, quae ex ejus directione abscinditur a circulo osculatore. Denotet enim a lineolam infinitesimam IM (Fig. 29) ut sito et consequenter IV = 902
cipiatur circulus , qui transiens per tria puncta 0 , I ,
H ( fig . 27. 29. ) habeat centrum in G , quique erit
circulus osculator apud curvae punctum O ; producantur
IV , MH donec occurrant peripheriae in G &quot; , G '' ; et ex''
G ducatur perpendiculum GGʻad chordam IG &quot; : erunt
IG &quot;
MG &quot; = IG &quot; = ICE Est autem MH . MG ' &quot; : MI. MO;
2
ergo MH . 21Gʻ = MI.MO = MI . 2MI , seu 21G'
2x2. Hinc v2 = . IGʻ ; ideoquc etc. Porro angulus IGG' =
2 Oxa

; con

.
px ?
22
87
Evolvatur quantitas radicalis in seriem : proveniet
B:P −⊢ o9zcos i
, unde B—P:cp92cosi ,
neglectis infiuitesimis altioris ordinis. Sit 'v' velocitas , qua mobile percurrit laterculum R; erit R: 0'9: sit etiam »
velocitas , qua mobile percurreret laterculum P. si non
adesset impulsus validus in I, erit P:-v 9: 'hinc R
-— P:(v'--v)9; et consequenter
v'—v:cp9cosi .
Ex hac aeqnatione patet 'o'—v esse quantitatem in-
fiuitesimam primi ordinis
, positivam vel negativam prout
i(vel 90&quot; , esse autem :0 si 1&quot;: 90&quot;. Inferimus il-
lnd : ubi tempore finito angulus, quem efformat vis ac-
celeratrir cum directione tangentis ,
fuerit semper acutus,
acquiret mobile incrementum velocitatis finitum; si sem-
per obtusus, patietur decrementum finitum; si semper re-
ctus, velocitas manebit constans.

45. In motu quovis curvilineo quadratum velocitatis
aequat vim acceleratricem ductam in dimidium chordae,
quae ex eius directione abscinditur a circulo. osculatore.
Denotet enim a lineolam infiuitesimam IM (fig. 29. )
gox- ; con-
92
cipiatur circulus, qui transiens per tria puncta 0, I,
II (fig. 27. 29..) habeat centrum in G, quique erit
circulus osculator apud curvae punctum 0; producantur
IV, MH donec occurrant peripheriae in G&quot;, G'&quot;; et ex
G ducatur perpendiculum GG' ad chordam IG&quot;: erunt
MG&quot;':IG&quot;, −−−∙−↧∁⇀−− ⇀∸−↧−⊊≩−⋅∎−∙ Est autemMH. MG'&quot;:MI. MO;
ut sit :9 i,
et consequenter IV:
99&quot;:
'»
ergoMH.21G':MI.-:MO MI. 2Ml.seu—-— &quot;&quot;&quot;&quot; ,210':
'v&quot;
.Hiuc v&quot;: 39. lG' ; ideoque etc. Porro angulus IGG'— −∙∙
−∙↼⇀−−
. −↼∙⋅⋅∙∙⋅↼−∎∣ −↼
∙∙∙88
90 ° -GIGʻ = 900 (MIV - MIG ) = 90 ' - ( i - 90 °) = 180 °-i ;
proinde , denotante r radium GI , erit IG ' = rsin IGG' =
rsini , et consequenter
va = grsini ( b ) .

:

46. Haec vera sunt de omni virium genere. Sint nunc
vires acceleratrices directae ' ad centrum datum : in casu,
curva ColH .... ( fig . 27. ) jacebit in plano transeunte
per rectam projectionis et per centrum virium ; quod fa
cile intelligitur , cum in eodem plano agant vires omnes.
Viribus ad punctum aliquod S tendentibus , radius
vector ( est recta , quae ab S ducitur ad mobile ) descri .
bet areas circa idem punctum temporibus proportionales ,
et viceversa.
Quod spectat ad primam assertionis partem , assum
ptis tempusculis aequalibus , et ducta recta SL conside .
rentur triangula SCO , SOL , SOI : est SCO = SOL ,
cum sivt super bases CO , OL aequales ob aequali
tatem tempusculorum , eamdemque habeant altitudinem
est etiam SOL = SOI , quia insistunt ambo eidem basi
SO, et sunt inter easdem parallelas SO , LI : ergo SCO
SOI. Eodem modo ostenditur triangula SOI , SIH aequa
lia esse eidem SIM , et proinde aequalia esse inter se ,
et similiter omnes areas sequentium triangulorum aequa
les esse inter se et cum areis praecedentibus. Quare cum
temporibus finitis quibuscumque contineantur numeri tem
pusculorum aequalium ipsis temporibus proportionales, areae
terminatae polygoni perimetro et rectis ad centrum virium
pertingentibus , hoc est compositae a lot areolis triangu
lorum aequalium quot tempuscula respondent illis tem
poribus , quibus perimetri partes describuntur , erunt
ipsis temporibus proportionales . Cum autem id locum ha
beat quomodocumque augeatur numerus tempusculorum,
eorumque magnitudo imminuatur , consequens est ut, ubi
⇤
88
⊖∘∘∙∁≖↧∁↾⋅ :soc—(MIV—MIG) :90&quot;—(i—gO&quot;):180&quot;—i ;
proinde , denotante r radium GI, erit IG':rsin IGG':
rsini , et consequenter
-v&quot;:g9rsini (6).

46. Haec vera sunt de omni virium genere. Sint nunc
vires acceleratrices directae'ad centrum datum: in casu,
curva COIH
.. ..
(Gg. 27. ) jacebit in plano transeunte
per rectam projectionis et per centrum virium; quod fa-
cile intelligitur , cum in eodem plano agant vires omnes.
Viribus ad punctum aliquod S tendentibus, radius
vector (est recta
, quae ab 5 ducitur ad mobile ) descri-
bet areas circa idem punctum temporibus proportionales,
et viceversa.
Quod spectat ad primam assertionis partem, assum-
ptis tempusculis aequalibus, et ducta recta SL conside-
rentur triangula SCO, SOL , SOI: est SCO:SOL,
cum sint super bases CO, OL aequales ob aequali-
tatem tempusculorum, eamdemque habeant altitudinem:
est etiam SOL :SOI . 'quia insistunt ambo eidem basi
50, et sunt inter easdem parallelas SO, LI : ergo 500:-
SOI. Eodem modo ostenditur triangula SOI , SIH aequa-
lia esse eidem SIM ,
et proinde aequalia esse inter se ,
et similiter omnes areas sequentium triangulorum aequa-
les esse inter se et cum areis praecedentibus. Quare cum
temporibus Gnitis quibuscumque contineantur numeri tem-
pusculorum aequalium ipsis temporibus proportionales, areae
terminatae polygoni perimetro et rectis ad centrum virium
pertingentibus , hoc est compositae a tot areolis triangu-
lorum aequalium quot tempuscula respondent illis tem-
poribus
, quibus perimetri partes describuntur ,
erunt
ipsis temporibus proportionales. Cum autem id locum ha-
beat quomodocumque augeatur numerus tempusculorum,
eorumque magnitudo imminuatur , consequens est ut, ubi89
demum polygonum abit iu curvam continuam , areae ter
minatae arcu curvilineo et rectis ad centrum virium ten
dentibus sint itidem temporibus proportionales.
Ad secundam assertionis partem quod spectat , sint
areae SCO, SOI, aequalibus temporibus confectae , omnino
aequales. Quoniam producta CO in L ita , ut existat OL =
CO, est triangulum SOL = SCO, idcirco SOL =SOI; sed
baec duo triangula habent basim communem SO ; erunt
igitur inter easdem parallelas, ideoque IL erit parallela re
ctae So. Ducatur IF parallela ad OL; motus per Ol com
ponetur ex duobus per OL et OF , quorum prior cum
oriatur a determinatione motum praecedentem continuandi
per C O , certe posterior a vi acceleratrice generabitur ,
quae propterea dirigitur ad centrum S.

47. Velocitas qua pollet mobile in eadem curva , est
reciproce proportionalis perpendiculo e centro virium du
cto in tangentem . Velocitas enim mobilis in quovis latere
polygoni est ut ipsum latus ob aequalia tempuscula , quibus
unumquodque latus percurri supponimus : est autem unum :
quodque ejusmodi latus reciproce ut perpendiculum quod
ex centro virium ducitur in latus ipsum ; siquidem id
perpendiculum habent pro altitudine triangula illa exigua
polygoni , si hujus latera pro eorumdem trianguloruin basi
bus assumantur ; ea insuper triangula sunt aequalia , et in
triangulis aequalibus debent bases esse in ratione recipro
ca altitudinum : est igitur ea velocitas reciproce ut per
pendiculum ductum ex centro virium in latera polygoni.
Sed abeunte polygono in curvam continuam , directiones la
teruın abeunt in tangentes ; ergo velocitas mobilis in quo
vis curvae puncto erit reciproce ut perpendiculum ex cen
tro virium in langentem demissum.

48. Denotet a areolam NSZ , et g perpendiculum SE
ductum ex centro S in laterculum NZ ; describetur NZ ve
NZ 2a ; siquidem NZ.SE=2NSZ: hinc ( 45 ) o
locitate v=
90
7
89
demum polygonum abit iu curvam continuam
, areae ter-
minatae arcu curvilineo et rectis ad centrum virium ten-
dentibus sint itidem temporibus proportionales.
Ad secundam assertionis partem quod spectat, sint
areae SCO, SOI, aequalibus temporibus confectae, omnino
aequales. Quoniam producta CO in L ita, ut existat OL:
CO, est triangulum SOL:SCO, idcirco SOL:SOI; sed .
haec duo triangula habent basim communem SO.; erunt
igitur inter easdem parallelas, ideoque IL erit parallela re-
ctae SO. Ducatur lF parallela ad OL; motus per OI com-
ponetur ex duobus per OL et OF, quorum prior cum
oriatur a determinatione motum praecedentem coutinuaudi
per C 0, certe posterior a vi acceleratrice generabitur ,
quae propterea dirigitur ad centrum S.

49. Quoniam radius vector , juxta quem agit vis con
tinua , potest assumi ut sibi parallelus per tempusculum
quodvis infinitesimum 0 , ipsaque vis ut constans per to
tum illud tempusculum ; ideo si mobile K incedens cur
vam CX ( fig. 30 ) viribus ad centrum S tendentibus de
scribit arcum infinitesimum HN labente , ductis SH ,
SN , et producto SN donec occurrat in H' tangenti HH &quot; ,
lineola recta H'N repraesentabit motum relativum mobi
lis K quoad ipsum Kieps per HH' sola vi praeconcepta in
H. Igitur cum motus iste relativus sit unice repelendus
( 5 ) a vi continuata per tempusculum e , exsistet
H'N son (6&quot;).

50. Haec subiungimus . 

1.&quot; Sive vires tendant ad
centrum datum
, sive non; denotantibus any, :coordinatas
puncti materialis in fine temporis t
, profecto x ,r,:peu-
debunt ab ipso :; erunt videlicet æ, y, :functiones tem-
peris :, ut scribi possit
.
——-—————.——-—-——-——.—.——...———..—91
= f ( ) , y = fi ( ) , z = 12

2. • Si vocatur s arcus a materiali puncto percursus
tempore t, w velocitas ejusdem puncti in fine ipsius t , pe
rinde spectari poterit ds ac si motu uniformi conficeretur ,
sola nimirum velocitate praeconcepta v ; siquidem nova
velocitas, dv , quae labente dt accedit materiali puncto ,
est infinitesima . Propterea hic quoque ( 28 )
ds
dt

3.º Resoluta vi o in duas , quarum altera sese
dirigat juxta tangentem , altera juxta respondentem nor
malem , erit ( 44) prima
des o cos i duษ
dc &gt;
dta
secunda (45 )
2² ♡ sini
ds²
r rdta

4.°# Incedente puncto materiali K per arcum s , mo
vebuntur motu rectilineo projectiones K' , K &quot;, K '' ipsius K in''
coordinatis orthogonalibusque axibus OX , OY, OZ ( Fig.5 ) ,
eruntque ( 28 )
dx dy dz
dt dt dt &gt;
earum velocitates in fine temporis : , quum nempe K ha
ds bet ( 2 ) velocitatem Vi acceleratrice dc
K , resoluta in ternas P ', P &quot; , D' ' ' iisdem axibus parallelas,
. , qua sollicitatur
∙ 91-
x:f(t)-J:fx(t)o 2:130)-
2.&quot; Si vocatur .: arcus a materiali puncto percnrsus
tempore :, v velocitas eiusdem puncti in fine ipsius t, pe-
rinde spectari poterit ds ac si motu uniformi couGeeretnr ,
sola nimirum velocitate praeconcepta v, ∙ siquidem nova
velocitas dv , quae labente dt accedit materiali puncto ,
est infinitesima
. Propterea hic quoque (28)
ds
Pr.—...... dt
3.&quot; Besoluta vi 9 in duas , quarum altera sese
dirigat juxta tangentem , altera juxta respondentem nor-
malem
, erit (44) prima
'n'—'v—d'v-Sd&quot;: cpcost—p ,9 de dt&quot;
secunda (45) ⋅
'
∙⋅ &quot;
'
∙ . ,,,a d;:
cpsmr— r — rdt&quot;'
4.&quot;e Incedente puncto materialiK per arcum :, mo-
vebuntur motu rectilineo projectiones K', K&quot;, K'&quot; ipsius Km
coordinatis orthogonalibusque axibus OX, Oï. OZ (Frg- 5) :
eruntque (28)
'de: (I)-' dz
dt
'
dt '
dt
earum velocitates in Gne temporis :, quum nempe K ha-
bet (2&quot;) velocitatem? .Vi acceleratrice , qua sollicitatur

: - -

K
,
resoluta in ternas P', P&quot;, P'&quot; iisdem axibus-parallelas,92
motus projectionis K' nihil erit aliud nisi motus rela
tivus puncti K quoad ipsum K sollicitatum viribus dum
dx taxat P &quot; , P ''' ; proinde velocitas debelur soli P' ex dt'''
dr ternis P' , P &quot; , P &quot; ; simili ratione ostenditur. deberi soli dt
dz P &quot; ex ternis P' ,P &quot; , P , et soli P&quot; ' ' ex iis 'componenti
dt
bus . Hinc ( 28 )
adx ddy adz
de de dt
P' ,
P &quot; , = P &quot; , dt de dt
seu
dex day daz
dt2 P'
dia
P &quot; ,
di?
= P &quot; .

5. °* Si punctum materiale incedit curvam plagam,
sumptis axibus v. gr . OX , OY in plano curvae , habebuntur
tantummodo
der day
de² P ' ,
dia
= P &quot; .
Fac v. gr. ut vis acceleratrix o sit parallela axi OY , ita
lamen ut sese dirigat ad plagam ordinatae y negativae : erunt
P = 0 , P :
ideoque
d2x
dla 0,
dy
di ?
Istarum prima suppeditat
I
92
motns projectionis K' nihil erit aliud nisi motus rela-
tivus puucti K quoad ipsnm K sollicitatam viribus dum-
taxat P&quot;, P&quot;'; proinde velocitas .j—f. debetur soli P' ex
ternis P', P&quot;, P'&quot; ; simili ratione ostenditur-(g.; deberi soli
&quot;
∙ ∙∙∙ dz ∙ n ∙∙ P ex ternis P', P&quot; ∙ ∙
, P , et −− soh P' ex 11s'componeut1-
dc
bus . Hinc (28)
' ddf ddZ ddi dt dt dt
∙−−− −−∶ '. ∙−−−:
P. −∙∙:
dt dt '&quot;
'
de P '
seu
'
' ⊒ ∙
' '
d3æ (137 d&quot;:
dt&quot; −−−∶ P
'
∙−− ∙−−− ∙−:
P
'di&quot; P '
dt&quot;

5.&quot;; Si punctum materiale incedit curvam planam,
Sumptis axibus v. gr. OX, O? in plano curvae , habebuntur
tantummodo .
⋅ ⋅
dzæ - d
dc&quot; :&quot;)&quot;Zïz'
Fac v. gr. ut vis acceleratrix q; sit parallela axi Oï
,
ita
tamen ut sese dirigat ad plagam ordinatae] negativae :erunt
ideoque
dh: ∙∙ d'] ∙∙∙ ∙∙
dt&quot; —0' −↲⋅≀⋅⇀≖− ?
Istarum prima suppeditat93
dx
dt C , x =Ct +C' ;
secunda, in hypothesi o constantis , praebet
dy ota
dt ot + C &quot;, y = 2 +0&quot; 4 + C &quot; :
eliminato t ,
y
y = c&quot; + * (** ) (* = ) .
Habes itaque, in ea qua sumus hypothesi , coordina
tas x ety expressas ( 10) per t; habes insuper aequatio
nem ad curvam, quam describit materiale punctum : re
stat ut constantes arbitrarias C, C' , C &quot;, C '&quot; determinemus.
Ponatur materiale punctum ex ipsa coordinatarum origi
ne O projici cum velocitate Yo juxta rectam inclinatam
ad OX sub angulo h: resoluta v. in' binas, alteram paral
lelam axi Ox, alteram parallelam axi OY, erit illa = v , cosh,
haec Vo sinh: initio motus obtinent simul
t = 0 , x = y = 0 ,
dx
dt = v , cosh,
dy
dt = V , sinh ;
igitur
C = Vocosh , C = 0.C &quot; = V . sinh , C = 0 ;
et consequenter
012
x = vol cosh ,y = v , sinh -
csinh
cosh
gx2
2v.cosh
93
dr
.
E—:C, æ:Ct-l-C,
secunda, in hypothesi ? constantis , praebet
' d ∙
'

: &quot; 73: :,n—j-c'.7:— ∙≌⇉−−−⊦∁∥≀−⊦ ∁⋯≖ eliminato t ,

∜−⋅−−−≺⋮⋅⋅⋅⊹∁∣⋅ (&quot;€&quot;)— −≣−≺∙≄ ;
&quot;):
.
Habes itaque, in ea qua sumus hypothesi, cbordina-
tas æ ety expressas (1&quot;) per :; habes insuper aequatio-
nem ad curvam, quam describit materiale 'punctum: re-
stat ut constantes arbitrarias C, 0, C&quot;, C'&quot; determinemus.
Ponatur materiale punctum ex ipsa coordinatarum origi-
ne O projici cum velocitate vo juxta rectam inclinatam
ad OX sub angulo h: resoluta v., in' binas., alteram paral-
lelam axi OX, alteram parallelam axi Oï, erit illa:vo cos 11,
haec:vo sin/1: initio motus obtinent simul
da: dy .
t.:o,x:o,y:o, ï

: vo cosh, ?::v., smh;

igitur
C: vo cosh , C': o .C&quot;: vo sin]: ,C&quot; ':o;
et consequenter '
cpt&quot; æsiuh (pa-&quot;
2 &quot;7— cos/1 -21Jo&quot;cos&quot;lt :

: votcOsIt,y:vosiult—94

x tangh -
9 1+ tangah
2 v2.
22.
Recole quae diximus ( 39).

6°# Fac nunc ut, permanentibus caeteris ( 5º. ) , pun
clum materiale moveatur in medio resistente: poterit vis ac
celeratrix ex resistentia medii exprimi ( 32. 33 ) generatim
per f (v ) ; per functionem videlicet velocitatis v
tem , decrescentem , evanescentem simul cum v Sit \beta an
gulus interceptus directione motus et ordinatarum axe OY ;
erunt ( 32 ) P' f (w) sin \beta , P &quot; = -- flv) cos \beta ;
ideoque
crescen
dar d²y : - flv )sin\beta , = -9 - flu) cos\beta ( c ) :
dt2 dla
insuper ( 40)
dx
dt
dy v sin\beta ,
dt = v cos\beta (c' )
quae differentiatae suppeditant
d22 dy d\beta dy do d\beta
dt sin\beta tvcos\beta dt dt2 dt cos\beta — v sin\beta ordt dt2 .
Ergo
dv
sin \beta + y cos \beta
d\beta
dt dt : -f (v )sin\beta,
do
de
d3 cos \beta-usin \beta 0 - f v ) cos\beta: dt
94
x tangh .::
t—ïngïhæt
Recole quae diximus (39).

604: Fac nunc ut, permanentibus caeteris (50.),ptm-
ctum materiale moveatur in medio resistente: poterit vis ac-
⋅ celeratrix ex resistentia medii exprimi (32. 33) generatim
per f(v); per functionem videlicet velocitatis v crescen-
tem
, decrescentem
,
evanescentem simul cum 0Sit B an-
gulus interceptus directione motus et ordinatarum axe Oï;
erunt (32) P':
-
f(v) siuþ ∙P&quot;: −− ?
−f(P) 008 p;
ideoque
d'æ
dt: :—ftv)sinþ,d —:— —f(v) cosþ (c):
insuper '(40)
da: . d .
'at—:&quot;.lnþO £: &quot;waþ (0)
quae diB'erentiatae snppeditant
dzæ
−↙⊼≖−−∶−⋇⋮∐⇪ ⊣−∙≀∘∞⇪⊼ d'B. dz :d—ïcosþ— —vsinþ dþ
dt
Ergo
——sin,8 −⋅⊢ vcos 5—d—-5 −∙−−−∙ —-f(v)sin,8,
dv
Ft— cosþ—vsin B (35—:— ep —f(v) cosþ:95
istarum primam multiplica per sin\beta , secundam per cos\beta,
tum collige in summam; eamdem primam multiplica per
cos\beta , et secundam per sin\beta , cum subtrahe; habebis
dy d\beta
+ fv) =– pcos\beta,
=
Psins
(c' ) .
dc dt
Quibus positis, haec stabilientur: cum nequeat \beta fie
ri &gt; 180° ( siquidem in transitu . per 180° vires omnes e
vaderent verticales, motusque permaneret verticalis ) , cum
que p etv existant perseveranter &gt; 0, ob secundam ( c &quot; ) erit
d\beta constanter 0 ; proinde crescente e crescet semper an dt
gulus \beta accedendo ad quemdam limitem B.
In hypothesi anguli initialis \beta. (=90° - h)&lt;90°, per
get o cos \beta per aliquod tempus esse &gt; o : sed flv ) &gt; 0 ; i
gitur , ob primam ( c''), per totum illud tempus erit de''
et consequenter crescente t decrescet v.
Prima ( c&quot; ) differentiata praebet
du
&lt; o .
d2v dv d\beta gsin\beta ;
dt -
dea + au f '(o )
seu , attenta secunda ( d ''),''
dev dy

dia
+ áf ( ) = q *sin- B
dv
facta igitur dt , emerget
dev oʻsina&gt; o.
dt
95
istarum primam multiplica per sin 13, secundam per cosþ, ⋅
tum collige in summam; eamdem primam multiplica per
cosþ ,
et secundam per. siuþ ,t'um subtrahe; habebis
∙ d d ∙
⊋⋮∙∙⊣−∣↻⇝⇌− ws?- ∙⊺∙↙↙⋛−∶∶∲−−−∘∎⋮∙∂ (a&quot;)-
Quibus positis, haec stabilientur: cum nequeat. þ Ge-
ri )180o ( siquidem in transitu.per 1800 vires omnes e-
vaderent verticales, motusque permaneret verticalis ), cum-
que (p et v existant perseveranter o, ob secundam (e&quot;) erit ↭ ∣
d ∙ ⋅
constanter £
)a; promde crescente :
crescet semper an-
gulus þ accedendo ad quemdam limitem B.
In hypothesi anguli initialis B., (:::90() -H( 90&quot;,per-
get ? .cosp per aliquod tempus esse ∘:sed iv))o'; i-
gitur
, ob primam (e&quot; ), per totum illud tempus erit ⋚∶≺∘∙ et consequenter crescente :decrescet 0.
Prima (e&quot;) differentiam praebet
⋣≖−⊦↙↨−⋛∣≼⋅⇝⇌≡≴∊∹∾⋅≖⋅∣⋮⇋ dav d d . ∖∖ seu, attenta secunda (c' '),
d'v dv ∙∙∙ ∳≖∘⋮∐≏∆⊙ ∙
⊄⋮⋮⋝⊹⊋−∑ f(V)-— v '
. .
dv
facta igitur 22 :o
, emerget
dav cp'sinïþ
üt: ⇀−− v
)0-96
Inferimus ( 27. 22°. ) velocitatem v haud exsistere capacem
maximi: poterit quidem esse capax minimi ; ita tamen , ut
mutato decremento in incrementum, hoc neque vertatur ite
rum in decrementum, neque certum quemdam limitem E
praetergrediatur. Primum patet ex eo quod , posita conver
sione incrementi in decrementum, jam obtineret maximum:
secundum ex eo quod, aucta indefinite v, augeretur indefi
dv nite flv ), simulque foret &gt;0 ; id vero adversatur pri dt
mae ( 6' ) .
Ex ( c &quot;) eruuntur binae
20 21
V2-01 ſię cos$ + fvde,B2- B;= Sosiu\beta dt ;
t t
exprimunt N,, V, velocitates , item B , B, angulos limitibus t,
2t respondentes. Fac
o cos\beta + v ) = f (t) ,
psins = fa (t) :
habebis ( 27. 18º. )
V; - v.--tfittat) • B. - = falttal) ;
exprimunt a et a numeros &gt; o et &lt; 1. Sed crescente t in
definite , vergit fi (t) ad q cosB + f (E ),et fu( t) ad qsinB
E
ac proinde
2 - -V2
limes quantitatis cos B + F( E ) ,
3. - 22 O
limesque quantitatis
sinB
E
96
∙
Inferimus (27. 220.) velocitatem v haud exsistere capacem
maximi: poterit quidem esse capax minimi; ita tamen, ut
mutato decremento in incrementum,hoc neque vertatur ite-
rum in decrementum,- neque certum quemdam limitem E
praetergrediatur. Primum patet ex eo quod, posita conver-
sione incrementi in decrementum, iam obtineret maximum:
secundum ex eo quod, aucta indefinite v, augeretur indefi-
nite f(v). simulque foret-(£)o, ∙ id vero adversatur pri-
mae (c&quot; ). ∙
Ex (e&quot;) eruuntur binae
2t
∙↗−⇂↗≖ −−−∙−− ∙∣ ( ? cosþ-l- fwndz, ↾⊖≖−,B— fra-018 de;
exprimunt v, , v, velocitates,' 1tem (i,, ,H, angulos limitibus !,
2t respondentes. Fac
9) cosB ^v):fd!) ∙∲≊∣∶∁ :
fam
habebis (27. 180.)
'Ur—vzzf— tf1(t&quot;l&quot;at) ∙⇪≖−−⇪≃∶∶⊀≖↸≖⊣−⊄⋅∁⋟⋮
exprimunt a: et «' numeros )b et ↿∙ Sed crescente :in-
≺↿⊜∊⊓⋮⇂∊∙ &quot;&quot;sit fxw ad 90053 —I-f(E).et rm ad ?&quot;&quot;B-
ac proinde
2! -—v

limes quantitatis :

Bos B4-f(E) ,x—Bz ↽− wir-B : .
limes ue uantitatis . q q E97
quoniam igitur
VI - V2 lim.
B - \beta ,
0 lim t t
erunt
Ø cos B + f(E)= 0 ; sin B
E
et consequenter
B = 180° , f(E )= .
Ex istarum prima inferimus motum materialis puncti ver
gere ad rectilineum verticalemque motum; e secunda ( viri
bus p et medii resistentis sese in limite elidentibus, utpo
te aequalibus et contrariis ) ad motum uniformem , proce
dentem videlicet a sola vi praeconcepta.
Divide primam ( c&quot; ) per secundam (c&quot;) : proveniet
dx
d\beta
sie
X-X
B-Brvm?;
iccirco ( 27. 18º. )
i\beta
Spa\beta
Q Q
Bm
exprimit um valorem medium velocitatis v. Haud praeter
greditur ' ' m certum quemdam valorem finitum ; insuper ver
git \beta ad B= 180° : ergo neque x praetergredietur finitum
valorem; ideo que materiale punctum incedet curvam prae
ditam asymptoto verticali. Recole, quae diximus nº. 41 .
Posita ( 33. 4º. ) flv )
formulae ( c) evadent k?
qua 1
quoniam igitur
&quot;r'—Va lim. :o, lim Bi—Ba :.0,
erunt
?
∘∞↿∃⊣−⊀≺≖∙∶⊢− 0
∙
∲≕⋮⋮∶⊔∄−∙−− −−∘⊰
et consequenter
3:180&quot; ,f(E):9.
Ex istarum prima inferimus motum materialis puncti ver-
97
gere ad rectilineum verticalemque motum; esecunda(viri-
bus 91 et medii resistentis sese in limite elidentibus, utpo-
te aequalibus et contrariis ) ad motum uniformem, proce-
dentem videlicet a sola vi praeconcepta.
Divide primam (c') per secundam (e&quot;) :proveniet
iccirco ( 27. 180.)
exprimit v,, valorem medium velocitatis
,,,
Haud praeter-
greditur v,, certum quemdam valorem finitum; insuper ver-
git B ad B: 1800: ergo neque æ praetergrediatur finitum
valorem; ideoque materiale punctum incedet curvam prae-
ditam asymptoto verticali. Recole, quae diximus n&quot;. 

41.
Posita ( 33. 40.) f(v):SE,-2 , formulae (c) evadent
.k?98
dar
di ?
-sing,
day
dla 9
qua
cos\beta : ka
sed haec hactenus.
7º. Intelligantur per coordinatarum orthogonalium
originem O ( Fig. 5 ) duci binae rectae 8,0&quot; intercipien
tes angulum a : earum extremitatibus junctis recta d '&quot;, erit
cosa =
02 +02.02
28 &quot;
Extremitas rectae , habeat coordinatas a ', y, z ', rectae au
tem o coordinatas x &quot;, 1 &quot; , 2 &quot; : paullulum attendenti pate
bit fore
õ = x's + y + 2,0% = &lt; &quot; + ya + z'2 ,
d's = (x - x &quot; )2 + 6 - y &quot; )2+ (z'- z&quot; )?;
adhibitis substitutionibus ,
cosa =
x' x &quot; ta'y &quot; tz'z&quot;
8o&quot;
Sint a' , b' , c' , anguli, quos Ở facit cum axibus OX, OY ,
OZ ; et a &quot; , 1 &quot; , c&quot; anguli quos d &quot; facit cum iisdem axi
bus: erunt 1
x' = cosa' , y ' = ' cos b ', z ' = ' cosc'
x &quot; = &quot; cosa &quot; , y &quot; = 0 &quot; cosb &quot;, z&quot; = 0&quot; cosc&quot; ;
rursusque adhibitis substitutionibus,
98
−∙−≂− −≌≝≖⋅ ∙ 9
9008?-
sed haec hactenus.
70. Intelligentnr per eoordinatarnm orthogonalium
originem O ( Fig. 5 ) duci binae rectae d', d&quot; intercipien-
tes angulum a: earum extremitatibusjunctis recta ö&quot;, erit
ö&quot;:
eo&quot; −⊦∂∣∣∶∎−∂≀∥≖ ⋅−∎ 26' a&quot; '
Extremitas rectae ö' habeat coordinatas 0:231, z', rectae an-
tem d&quot;coordinatas x&quot; , y&quot;, z&quot;: paullulum attendenti pate-
bit fore ⋅
∂∣≏−−∶∞↾≖−⋅⊦∙↗∣≖−⊢≖↾⋩∙ ∂∣⋅≖∙∸⋅∞↾∎≖⊹∕∣≖−∣−≖∥≖ ,
3' ⋅≖−−−−≺∙⊅∣∙∞⋅∣⋟≖⊣⊣∙↗∣⋅∫∎⋅ )'—l-(z'-z&quot; ),:
adhibitis substitutionibns ,
∙−− æ; æ&quot;——)&quot;)'&quot;—l-Z' zn
cosa ∶⋅↳ a, 6&quot;
Sint a', 6', c', anguli, quos 6' facit cum axibus OX, Oï.
OZ; et a&quot;, b&quot;, e&quot; anguli quos 6&quot; facit cum iisdem axi-
bus: erunt
x':d' cosa' ,y*zzd&quot; cosb', z':ö' cosc'
æ&quot;: ö&quot;cosa&quot;, y'': ö&quot; cosb&quot;, 2&quot;: d&quot; cosc&quot;;''
rursusque adhibitis substitutionibus,
−∙∙⋅∙−⋅−−⋅99
cosa = cosa' cosa&quot; -- cosb' cosb&quot; + cose'cosc &quot; .

* His positis, fac ut vis acceleratrix o sese constanter dirigat ad centrum datum : constituta in eo coordinatarum ori gine O, erunt sle D 5.5 cosinus angulorum , quos cum axibus coordinatis efficit ra dius vector D; et P P &quot; P '&quot; P cosinus angulorum , quos cum iisdem axibus efficit . Pro pterea P X op + . $ . Þ==1 , sumpto vel superiore, vel inferiore signo , prout o nititur vel adducere materiale punctum ad centrum illud, vel ab ipso distrahere: in primo enim casu q et D faciunt angu lum a = 180° , in secundo angulum a = 0. Inde profluit ( 49) d2x Ide² dy v + D dia D daz dt2 8.• * Sumptis axibus OX, OY in plano ( 46) cur vae , quam incedit materiale punctum , erit der Q =F Ndt² on the + 5) . 99 cosa:eosa'cosa&quot;-]-cosb' cos6&quot;-1-cose' cosa&quot;. ∙His positis, fac ut vis acceleratrix (p sese constanter di- rigat ad centrum datum: constituta in eo coordinatarum ori- gine 0, erunt æLz D'D'D cosinus angulorum, quos cum axibus coordinatis edicit ra- dius vector D; et P' P&quot; P'&quot; r ' a ' ? cosinus angulorum, quos cum iisdem axibus ellicit ep. Pro- ? se P&quot; 7 p--- ∙∙∙ ∙−−− ' D—&quot;'ï&quot;10 ? sumpto vel superiore, vel inferiore signo, prout ep nititur vel adducere materiale punctum ad centrum illud, vel ab ipso distrahere: in primo enim casu ? et D faciunt angu- lum a:1800, in secundo angulnm a −−−− 0. Inde profluit (40) ≕∙∙∙∙∙∙ dia: :: d'y )- ? D) *(dz : &quot;D'l'dcz ∐↼⊦↲⋮−−≟ ⋅⋅− ⋅ 8.0 «: Sumptis axibus OX, 0? in plano (46) cur- vae , quam incedit materiale punctum , erit100 Ad exprimendamo per coordinatas polares , exhi beat 180°-W angulum interceplum radio vectore D et axe OX ; erunt De = x ? tys , x= - Dcosw , j = Dsina . Prima semel iterumque differentiata dat dDP + Dd D = xd x + ydży + dx2 + dy? ; secunda et tertia praebent dx = Dsiow cosw - coswdD . dy = Dcos wdw tsinwdD , ideoque dsa = dx2 + dyr= D -dw2+ dD2 , Hinc 2 der dia dy a D + dla D d - D dea D 2) ܪ . ac proinde la pa (d- D dla 0 ( ) ). Ad haec : P P &quot; = P P &quot; unde D àla D y et consequenter 1 1 1 100 Ad exprimendam (p per coordinatas polares, exhi- ' beat 1800—0 angulum interceptum radio vectore D et axe OX ; erunt Dï':a:3--l-)'2 , x: — Dcosw .szsinm. Prima semel iterumque differentiam dat dDL-l-DdzDzædïæ-l-yd'y-þdæï-l-dyz .; secunda et tertia praebent dæ:Dsinm cos co —cos ad D. dy:Dcos ædwf-sinædD, ideoque d.,- ∙−−− dx: −⊦ dyaznadæ-l—doa . Hinc dïæ a: dfy ] (PL) ? Dei?-),. ∎⊃⊣−≺∄↙⇄ ⋅∎⊃−−⇤↲⋍≖ dt &quot; ac proinde dzD (deo)!) ∙−−∶ −− D — ? ∓ ∙ (aua dt Ad haec : P' a: P&quot; ⋅∙∙∙∙ )» P ∙∙∙ P .;. :ï,?—q:.ü.,unde-; 7- et con sequenter ∙∙∙∎∙∎⋅∎−⋅101 • dx yd dt rady FO : de quam integrantes assequemur dr V dc dy dt C , seu ydx - xdy = Cdt. Est autem ydxxdy = Dsinud(-Dcosw ) + Dcosad( Dsinw ) Dºdw , propterea с dwla CdtD - da da de ( ) = C2 D D4 insuper AD Code : d d - D dla de dt dD da dt dt . ( dD C do D2 dt 1 . ( D ( ( d da da) da C2 D d d dw ,!... Hit C = as dt aan zoals da ? Coil 100 dwudt da . Da aby boxe parutis 1 C2 D D2 dw² Quare J 101 quam integrantes assequemnr da: dy,&quot; ⋅ ∙∙∙∙ ∙≯≀∙⊋∙↕−− ∙∙∷⊋∙⋮−−− C, seu ydæ—Jt'dj—Cdt- Est autem ydx—ædy:Dsinæd(—DcosmH—Dcosæd(Dsinæ) : D'daii , propterea ∙−− dai—C ∙ de) 3—01 ∙ ∁↙≀⇞−−∐⇟⊄∄∾⋅∙⋅⊋⊼−−−∐−≖⋅∙ (a)—&quot;1373&quot; insuper di? d(dD. 49) d(iD ∙⊆− ≀∄⋅∣⊃∙∙ d ∙∙∙⋅∃⊂∙−⊃⋅ 71? ∙− do) ne). dt'- d; ⋅ d:; dï- ⋪∙−⋅−⊳ ↿ ⋅ ⋯↿ 41 d(ï) d D d(B) 1101 ...-2 d, ⋅ da) ∙−↽∁⋅ ↪↼⋅−↽−⇁∁ ↜⊒⋅∶≥⇀⋍−−⋅−≤⋮∶ a d: dmwdt;,, nad-'O) . ' f&quot; &quot; c: ∐≖⋅↙∄∘−∎⊃−dasz102 D + ) ( 61 ). D2 dw² Fac v. gr. ut, viribus ad datum centrum tendentibus, materiale punctum incedat curvam (dicitur spiralis loga rithmica ) repraesentatam per Draw Habebis ( 27. 6.° ) . el = 2 11름 loga dw ,de 1 D% log ? a dway log.'a ; iccirco go CP D2 a log-a + b) ( logo a+ 1 ) . vis nempe acceleratrix erit in ratione reciproca triplicata distantiarum a dato centro.

9. # . Ad constantem C quod spectat, ex coordina
larum origine 0.(Fig . 19 ) intelligantur duci bini radii ve
clores, alter ad punctum datum a habens coordinatas xo,
Yo, alter ad punctum quodvis B habens coordinatas x, y;
sitque A area sectoris terminati arcu aB et radiis illis, A
area aBca' : erit
A = A +
Xoyo
2
xy
2
&gt;
⇀↿∘⊋∙
du-
−−∶⊨ C, ...—l.).. ..,— (6) ..... i)? da: D !'
Fac 9. gr. nt, viribus ad datum centrum tendentibus,
materiale punctum incedet curvam (dicitur spiralis loga-
rithmica ) repraesentatam per
D:
ac .
Habebis ( 27. 6.&quot; )
1 -ao 1 1
T)- :a ∙↙≀−∣⋝−∙−−−−−− logadæ,d'ï-—
.. ∠≀≖−∣↿⋝∙ ..
a logia dei:-, :logæa ,

dm&quot;
iccirco
1»ng (
−∾∙∣∘⊰⋅∅⊣−∎↿⋥≻−−−∌⋮ ——(l0gi ∅−⊦
1):
vis nempe acceleratrix erit in ratione reciproca triplicata
distantiarum a dato centro.
9011. Ad constantem C quod spectat, ex coordina-
tarum origine O-(Fig. 19) intelligantur duci bini radiive-
ctores, 'alter ad punctum datum et habens coordinatas xo,
yo, alter ad punctum quodvis B habens coordinatas x, y;
sitque A area sectoris terminati arcu aB et radiis illis, A'
area cha': erit '
A—A' : æozïo ?;103
ideoque ( 27. 18° )
dA = dA - d
xy ydxxdy
2
denotante praeterea i angulum interceptum tangente in B
et respondente radio vectore D, est ( 48)
D sinids dA 2
Igitar
ydx = xdy = Dsini ds,
et consequenter ( 70 )
C dt = D sini ds ;
unde ( 20 )
ds
C = D sini - Du sini .
dt
Caetero quantitatem Dvsini esse eamdem ubicumqe suae cur
vae sit materiale punctum, liquet ex dictis ( 47 ) .

10 ° # Habemus ( 2º. 8° )
ds
de²
DP dw2 + dD CP
dc2 (Da dwa + dD&quot;)p4
dwa
dD 2
D2
[ 11
+ 9 seu
de
103
ideoque ( 27. 18o .)
dA ::dA'- J
222 −−−∫↙≀∞−⋍≀↨≧↩ −∫∂∞−≨⋅⇣⇃⋮↙≀∫ ;
denotante praeterea t' angulum interceptum tangente in B
et respondente radio vectore D, est (48)
D sini ds
2 ∙
(IA:
lgitur
ydx':xdy :Dsint' ds,
et consequenter ( 70 )
Gde:D sini ds ;
unde ( 20)
CZDsini £:Dvsiü. dt
Caetera quantitatem Dvsini esse eamdem ubicumqe suae cur-
vae sit materiale punctum, liquet ex dictis (47).
100a Habemus ( 20. 8&quot;)
2
(Isa 02 dGP—xl-JD':(02 da,-1- dDz) c,
.
— —∙∙∙
−⋅ dt&quot; ⋅⋅−⋅ dt: D4 dc.-13:
∁≖
dDa
äirl—(5)], seu
de?104
v2 = C2 -- [ + (3 ]
( m) .
11. # Quemadmoduni , data linea quam incedit
materiale punctum , innotescit q ; sic vicissim , data op , po
terit sciri linea per quam movetur materiale punctum
Denolante B quantitatem constantem et n numerum inte
B
grum , sit v . gr. g = ; erit ( 7° 6. )
D &quot;
B CP D
+ )

;

dwa
1 B
quae ,
facto D = 1
D' et
et og h , vertetur in C2
d2 D'
h D ' r-2 = +
diwa +D) .
Chaton
Haec multiplicata per 2dD ' suppeditat
E12dD' dD d
dw da +
2D'dDdD' ) -2-2 h D'n -2 d D' = 0 ;
sumptisque integralibus ,
= [CD)* + D ] - 2,0-4C = 0;
unde
dw
(6,2
dD'
2h Dina quoad o adducentem
D'2 į ad centrum ,
'? – C ).
∠⊢⋅⋅ ↿
' 2 2
1 D∶∁ Exi-(a)] (&quot;**-

↿↿∙∘∙ Quemadmodum, data linea quam incedit
materiale punctum
,
innotescit ?; sic vicissim , data ep , po-
terit sciri linea per quam movetur materiale punctum
.
Deuotante B quantitatem constantem
, et 11 numerum inte-
grum, sit v. gr. ep:DT; erit ( 70 6.)
l
'l
3— B 02 (...d D.. 57.— 25 .'.)* da: D
⋅ ↿ B ∙
quae, fama-:D et——⋜⋮−:h,vertetur tn

* D' ≀≖≖≖⋅⋅−≖⇌⇀−⊻≐≺∡∽≖ −⊦∘∙≻⋅

Haec multiplicata per 2dD' suppeditat
∶⊨≺∶≳↙⊋≞⇗∠∄−−⊣− 2D' dD')—2h D&quot;'2dD':——o ;
sumptisque integralibus ,
465)&quot; HB ]-—'— ∣⊃⋅⋅−⋅∙−⊦∁⋅−−−∘≅ n—l
unde
da: dD'
2]; quoad ?adducentem
TDV&quot; —-D'3 —C')5 ad centrum,105
dD'
dw 2h quoad o distrahentem
(0 –
a centro :
n =; D**?— D» ) *
quarum integratio praebebit relationem inter w et D' , ideo
que inter coordinatas polares w et D lineae quaesitae .
12.°* In istarum aequationum prima sume v . gr.
n = 2 ; ea sic poterit scribi
D'
doma
V
ha- C
da
h D'
h2_C Hinc
w = C &quot; + arc cos =
h - D'
VhC
cos (6-C' ' ) ;
et restitutis valoribus h , D' ,
D =
C2
B - 1 B2 – C4 C cos (W – C&quot;) ·
Pone
C2 C
= B (1 + €),
=B' ( 1 —E) ,
B-HVB2_C4 C B - V B2 - C4C
quae in summam collectae praebent
B
CPC B '
,
invicem multiplicatae suppeditant --
8
105
&amp;
dm: dD
quoad p distrahentem
(C' - 36- D'&quot;&quot;' −−∙ ∎⊃∎∌≻≩⋅ a centro :
⇀
n—1
quarum integratio praebebit relationem inter 61 et D' ,
ideo-
que inter coordinatas polares &amp;) et D lineae quaesitae .
12.&quot;; In istarum aequationum prima sume v. gr.
n:2 ; ea sic poterit scribi

: .-n

'
↶⋮≼⇂∕−−⊮−∁∙⋟
dæ:- ⇂∕↿
Hinc −≺⊓⋅≻≖∙∣≖≖−∁⋅
G):C&quot;-l— :COS(GO—C&quot;)i arC(cos ∙−∙−−−−
h—D' h—D'
⋅⇂∕∣−≖−−−⋯≖−⇀∁∙ ⇂∕∣≖≖∙−∁∙ '&quot;
et restitutis valbribus I: , D',
B—l/Bz—Clt C' cos (co—C&quot;)
D
Pone
C2 02 −−−−− −−−−−−↧≉⋅↿
). −−⋅⊨
−−−−−−∶ —B'(1—e). B—l/Ba—cac- ≺⊹⋮ ∌−⊢⇂∕∌≖−∁↙∣∁∣
quae in summam collectae praebent
c:c' B -—-−⋅⋅
. B',
invicem multiplicatae suppeditant106
&lt;= B' ? ( 1 —-z );
habebis
1 C2 C' = B
B'2 ( 1
B' ( 1 — 52)
Propterea
D =
B' ( 1 - 2)
E cosWC( '')''
(62) .
1
13.0* Potest C' esse vel &gt; 0 , vel &lt; o , vel == 0;
in primo casu erit B ' &gt; o et € &lt; 1 ; in secundo B' &lt;o et
&gt; 1 ; in tertio B ' = et z = 1. Primum ac secundum
casum alibi considerabimus .
14. * Ad tertium quod pertinet , exhibeat NI...
(Fig . 25) axem parabolae ( 40. 5.º 7.º ) ; sintque NO ( 3x)
et 00' ( =y) orthogonales coordinatae : designante 2p pa
ramelrum , exsistet
ya = 2px .
Substituto x' + ip pro x , transferetur coordinatarum
origo in focum H , eritque quoad novam originem H
ya = 2px' +p .
Duc radium HO =D) ; habebis
NHO x'
--- D cos w , y = D sin w ;
et consequenter
D2 sin ’ w = p - 2pDcosw .
Spectatur autem D ut quantitas constanter positiva ; proinde
106
↿ 'a a .
&quot;ö'.:B (1—£)1
habebis
∙∙∙ ↿ ⋅ ∙∙∙ Ca
B'3(1 -
a&quot;) ' B' (1—5') ⋅
Propterea
B' (1 — a')
D −∙− (b,) . 1— :cos (co— C&quot;)
1391» Potest C' esse vel≻∘ ∙ vel (o , vel:o;
in primo casu erit B' o et e ↿;in secundo B' (0 et s
↿; in tertio B':eo et e:1 . Primum ac secundum
⋅ casum alibi considerabimus .
145): Ad tertium quod pertinet , exhibeat Nl..
.
(Fig.25) axem parabolae (40. 5.&quot; 79); sintque NO (:.r)
et 00' (: y) orthogonales coordinatae :designante 2p pa-
rametrum
, exsistet
y':2pæ .
Substituto x' −⊦ ∙⇡∙↼ ;) pro æ .
transferetur coordinatarum
origo in focum H , eritque quoad novam originem H
7&quot; ⇌ 2pæ' ⊣− r'-
D'uc- radium HO' (:D) ; habebis
NHO':61, uf:—D cos æ,y:D sin(-);
et consequenter
D2 sin2 01:p' −∙∙ 2pDcos co .
Spectatur autem D ut quantitas constanter positiva; proinde107
DE P cosa
+ V
V pa pacos w_P(1 ~ cos )
sin? W sin? W sin4 w sin' w
Sed sin? w = 1 - Cos w = (1 — cosa) (1 + cosw ) : igitur
P
D =
1 +cosa (63) .
Designata nimirum quantitate B '(1 - 6 ) per P , et assumpta
C &quot; = 180° , recidet (62) in (63) ; unde consequitur illud :
iribus ad centrum datum tendentibus in ratione reciproca
duplicala distantiarum ab ipso centro , poterit materiale
punctum describere parabolam habentem suum focnm in
centro illo .
15.0# Quoad parabolam ( 14º. ),
(* )
sinaw 1 1 +cosw cosa a
COS
pa da P р
2 1 2 CM 1 1
D
O
Hinc ( 90.m) va - .
р
D
P
D
Sit E altitudo debita velocitati v ; erit ( 12º. 14º. )
2C E 2C?
v2 = 20E =
2BE
D2
E
B
D2 ' ( 1 -62 ) D2 p
et consequenter
2C2 E 2C 1 E
D

; unde D2 D

.
р P
Inferimus illud : si in distantia D a centro virium proji .
citur materiale punctum , haud describetur parabola nisi
107
D:∙∙∙ ;) cosa) ∙∙∙⊦ Vpa .l.-paene: c.)—p(l—cosï
≖⋮∐⇄
∙
a) s1na a) sint! ea sin' 6)
Sed sinit.):1−cosa a:(1— eos a)) ('l-l- cos a) :igitur
∼ P
D:1—i-cosm ∅⋮⋝⋅ ⋅
Designata nimirnm quantitate B'(1-- 6&quot;) per p , et assumpta
&quot;:180o , recidet (b,) in (63) ; unde consequitur illud :
viribus ad centrum datum tendentibus in ratione reciproca
duplicata distantiarum ab ipso centro
, poterit materiale
punctum describere parabolam habentem suum focum in
centro illo .
1530 Quoad parabolam (Mc,),
∙
∙−−− — ∙−−−
d'
⋅
( D) sin'm 1—cosaæ—1—l-cosæ 1—cosa1
df&quot; P&quot; ?' p P
2 ↿ ↿ ∙ 2
c- 1

;

∙ D ∙∙∙ DQ. Hlnc (90.m) 02: ∙∙∙∎∎∙ ∙
ö
∙
Sit Ealtitudo debita velocitati «a; eri; (1241. 14o.)
2311: 20» E
∙∙∙∶≿∁∶
E
ng—ZQE— Da —B'('l—-£3) ∙ [P p . 02 ,
et consequenter
zcn E—zc: '-dE-1
p.Da—p.D,uneD—. .
Inferimus illud :. si in distantia D a centro virium proii-
citnr materiale punctum , baud describetur parabola nisi108
eidem distantiae D fuerit aequalis altitudo illa , per quam
mobile vi acceleratrice vigente in puncto projectionis ca
dendo motu uniformiter accelerato acquireret velocitatem
ipsius projectionis.

51. Hactenus de motu curvilineo libero, quum nempe nihil obstat quominus mobile obtemperet viribus; fac nunc ut materiale punctudi, cujus massa = m, moveatur motu impedito, sollicitatum videlicet vi acceleratrice q adstringatur moveri vel in data superficie vel in data linea curva. Quoniam ejusmodi superficies et linea nihil praestant aliud nisi exercere in puncto materiali resistentiam m ç sibi perpendicularem, ideo motus perinde fiet ac si punctum materiale esset liberum viribusque acceleratricibus et d', seu quod eodem redit viq &quot; inde resultanti libere obtemperaret.

Pone quod motus impeditus in data linea debeatur
unice vi praeconceplae et vi gp' ut sit 9
habebis
q &quot; = 0 ; i = 90 °; et consequenter ( 45. b)
0 :
2,2
( 6' ' ' ) ;
my?
Precisa nimirum q , exprimet ( 28 ) pressio
nem exercitam a puncto materiali in lineam illam , atque
huc spectat vis centrifuga ; pressio videlicet a puncto ma
teriali exercita in eam lineam , orta e sola inertia ad prae
seulem velocitatis siatum contracta.
Ad haec : in eadem hypothesi vis acceleratricis ♡
facile colligitur ex dictis ( 36) motum impeditum fore u
niformem .
!
108
eidem distantiae D fuerit aequalis altitudo illa , per quam
mobile vi acceleratrice vigente in puncto projectionis ca-
dendo motn uniformiter accelerato acquireret velocitatem
ipsius proiectionis.

===De vi acceleratrice in motu circulari, existente centro virium in centro circuli.===

52. Ex demonstratis (47) patet istiusmodi motum esse uniformem. Sit R radius circuli, per cujus peripheriam incedit mobile: in ( b: 45 ) erant r = R, i = 90° ; in ( b' : 48) vero D =9 = r = R; et denotante A lotam circuli aream, T tempus periodicum, quo nempe mobile conficit integram circuli peripheriam, in eadem ( 8' ) erunt quoque A = n R?, = T. Hinc ex ( 6)
1
RO
et ex ( 6 ) ( c )
4 762 R
T2

53. Haec facile punc stabiliuntur. 

1º. mobile velocitate quadam projectum in distantia R a centro virium von describet circularem curvam nisi velocitas illa tanta sit quantam mobile ipsum acquireret cadendo per { R motu uniformiter accelerato et vi acceleratrice, quae viget in projectionis puncto; siquidem prima (c) suppeditat v = 2 0.4 R.

2º. In circularibus peripheriis eodem tempore
descriptis vires acceleratrices sunt ut respondentes radii: patet ex secunda (c).

3º. Ex eadem secunda (c) inferimus vires acceleratrices fore in ratione reciproca duplicata radiorum quotiescumque quadrata temporum periodicorum fuerint ut radiorum cubi.

54. Obiter haec notamus. 

1º. Ex circulari telluris rotatione circa suum axem oritur vis centrifuga (51) in materialibus punctis tam apud aequatorem quam apud circulos aequatori parallelos, generatim expressa per &lt;math&gt;m\varphi'=\frac{mv^2}{R};&lt;/math&gt; et quia rotatio illa fit motu uniformi, ideo&lt;math display=&quot;block&quot;&gt;v=\frac{2\pi}{T}\,\mathrm{ 
et}\, \varphi'=\frac{4\pi^2 R}{T^2}
&lt;/math&gt;Tempus periodicum &lt;math&gt;T&lt;/math&gt; est ubique idem; &lt;math&gt;R&lt;/math&gt; vero decrescit ab aequatore ad polos; in eadem ergo ratione ab aequatore ad polos descrescet vis centrifuga. 

2º. Exhibeat R , radium aequatoris terrestris (Fig. 31) et a geographicam latitudinem, cui respondet circulus aequatori parallelus habens radium R, erit R =R cosa , et consequenter R , cosa
T2
Resoluta q' in duas, quarum altera sit verticalis, altera horizontalis, existet illa 402R , cosa
D'cosa=
T2
et quoniam q' cosa est vis contraria gravitati, inferimus gravitatem imminui magis semper a polis ad aequatorem, ejusque decrementum fore ut quadratum ex cosinu latitudinis, spectata videlicet tellure instar sphaerae.

3º. Exprimat s altitudinem debitam velocitati rotationis; erit ( 30) 2gs = v ?, ideoque ( 10 ) 2gs = q R, et
consequenter
8
solia
R
.
2s
110
mg': mv&quot;R; )
et quia rotatio illa Et motu uniformi, ideo
27rR et ∙∙∙∙∙ ∢∏≃∣≹
T ' ?&quot; Ta
.-

ecosa: Tat quoniam cp' cosa: est vis contraria gravitati, inferimus gravi-
tatemimminui magis semper a polis ad aequatorem, ejusque
decrementum fore ut quadratum ex cosinu latitudinis ,
spectata videlicet tellure instar sphaerae.

111
Hinc innotescit ratio inter gravitatem et vim centri
fugam : sic apud aequatorem invenitur
8
R,
= 288 circiter;
2s1
inde sequitur quod gravitas sub aequatore in hypothesi tel
luris immotae esset == 1880' + q = 289 .
=== De vi acceleratrice in motu elliptico, existente centro virium in foco ellipsis. ===
55. Haec praemittimus: 

1 °. si ex puncto quovis M (Fig. 32) ducuntur duae rectae MN, MS tangentes sphaeram SN .. , erit MN = MS: ductis enim ex centro C radiis CN, CS ad contactus puncta N et S; itemque CM ad punctum M, triangula CMN, CMS rectangula in N et S habebunt latus CM commune, latera vero CN , CS aequalia; ideoque etc.

2°. Si per tangentes MN , MS ducuntur plana tangentia NMT , SMT ad sphaeram SN .... sese muluose.
cantia juxta rectam MT, angulus NMT aequalis erit an
gulo SMT: nam ex C , N , S ad punctum v . gr. T rectae MT, ductis CT , NT , ST, quoniam NT et ST jacent in planis tangentibus NMT , SMT , iccirco in triangulis CTN , CT'S anguli CNT, CST erunt recti; latera in.
super CN CS sunt aequalia , et CT commune: proinde
NT = ST. Triangala igitur MNT, MST exsistent ( 1 ° ) invicem aequilatera; ideoque etc.

3º. Si denotat p projectionem lineae rectae l in
plano quovis , et a angulum , quem efficit I cum eo plano , erit&lt;math display=&quot;block&quot;&gt;p = l\cos\alpha&lt;/math&gt;: patet ex Trigonometria.
4º. Si denotat P projectionem mn (Fig. 33) areae planae cd ( = A ) in plano quovis gr , et i angulum ,
quem efficit A cum gr , erit,
P = A cosi .
Ducatur enim planum mg parallelum areae A, in quod
demittatur ex d perpendiculum dK ( = x ) ; ducantor quo
que plana gh , de parallela plano qr; ponaturque dg = y .
Quisque videt prisma vel cylindrum md aequari prismati
vel cylindro ge: propterea Ax Py ; unde P A ; est
autem -
sindgK = cosi ; igitur etc. yу

5º. Secetür cylindrus rectus aB ( Fig. 34 ) plano
ad circularem ipsius cylindri basim inclinato; sectio AMBL
dicitur ellipsis ; punctum C, ubi planum secans occurrit axi
cylindri, vocatur ellipseos centrum; punctumque S, ubi se
crio illa tangit sphaeram sambl cylindro inscriptam , appel
latur ellipseos focus; pro cylindri base sumimus circuluin trans
euntem per centrum c sphaerae inscriptae; inde fit, ut ba
seos peripheria sit linea contactuum superficiei sphaericae
et superficiei cylindricae.

6º. Si per C ducitur linea quaevis recta LM ter
minata ad ellipseos perimetrum , ejus projectio in cylindri
base erit ipsius baseos diameter lm , ita at lc sit projectio
portionis LC, et mc projectio portionis MC. Sed lc mc ;
ergo ( 30 ) LC = MG: lineae videlicet rectae transeuntes per
ellipseos centrum , et ad ellipseos perimetrum terminatae ,
dividuntur omnes bifariam in eodem centro.

7º. Per extrema puncta 1 et m diametri lm du
ctis ad circularem cylindri basim tangentibus lh et mt ,
hae utpote perpendiculares ipsi lm erunt parallelae; rectae
quoque IL , mM utpote cylindri basi perpendiculares, erunt
parallelae; ergo plana hll , ImM cylindricam superficiem
112
40. Si denotat P proiectionem mn (Fig. 33 ) a-
reae planae cd:( A ) in plano quovis qr ,
et t' angulum ,
quem eliicit A cum qr', erit,
P:A cost'.
Ducatur enim planum mg parallelum areae A, in quod
demittatur ex d. perpendiculnde ( −−∶ æ ); ducantur quo-
que plana gh, de parallela plano qr; ponaturque liga:-7.
Quisque videt prisma vel cylindrum md aequari prismati
vel cylindro ge: propterea Aa: −−∶ Py: unde P:
.i.-A; est
autem −⋅↕⇣∙ ∶−− siudgK :cosi; igitur etc.
.7
20. Secet'ur cylindrus rectus aB (Fig. 34 )plano
ad circularem ipsius cylindri basim inclinato; sectio AMBL
dicitur ellipsis; punctum C, ubi planum secans occurrit axi
cylindri, vocatur ellipseos centrum; punctumque S, ubi se-
ctio illa tangit sphaeram sambl cylindro inscriptam, appel-
latur ellipseos focus; pro cylindri base sumimus circulum trans-
euntem per centrum c sphaerae inscriptae; inde fit, ut ba-
seos peripheria sit linea contactuum superficiei sphaericae
et superficiei cylindricae.
60. Si per C ducitur linea quaevis recta LM ter—
minata ad ellipseos perimetrum, ejus proiectio in cylindri
base erit ipsins baseos diameter lm, ita ut lc sit projectio
portionis LC, et me projectio portionis MC. Sed lc :: mc;
ergo (30) LC: MC: lineae videlicet rectae transeuntes per
ellipseos centrum
.
et ad ellipseos perimetrum terminatae.
dividuntur omnes bifariam in eodem centro.

113
tangentia existent parallela inter se; et couscquenter inter
sectiones quoque LH, MT istorum planorum cum elipseos
plano exsistent parallelae. Liquet autein intersectiones il
las esse tangentes elipseos in L et M; ellipseos igitur lan
gentes ductae per extrema puocta cujusvis rectae, quae trans
eat per centrum , quaeque terminetur ad curvae perime
trum, erunt inter se parallelae.
Recta LM secat bifariam ( 3º ) chordas omnes paral
lelas tangentibus LH , MT; ejusmodi enim chordarum pro
jectiones nibil sunt aliud nisi circularis baseos chordae pa
rallelae tangentibus lh, mi, atque ideo perpendiculares dia
metro lm , a qua proinde secantur bifariam : inde fit , ut
LM dicatur ellipseos diameter.

8º. Ex M ad focum S ducatur MS; rectae MS ,Mm tangent ( 50 ) sphaeram, altera in S , aliera in punctum lineae contactuum superficiei cylindricae et superficiei sphaericae: ergo ( 19. ) MS = Mm. Simili modo, ex L ad S du
cta LS, erit LS = LI.

9º. Plana TMS, MMT et transeunt per rectas MS,
Mm tangentes sphaeram , et sphaeram tangunt, et sese mutuo
secanı juxta MT; ergo ( 2º )anguli TMm, TMS erunt aequales :
simili ratione ostenditur angulos IILS esse aequales.

10º. Denotet a rectam Cc jungentem centra Cet c: trapezium LMml suppeditat Ll +Mm 2a ; igirur i 80 )
SL + SM 2a . Variala utcumquc positione diametri LM ,
non ideo variabit recta Cc , sed mavebit cousians in ea
dem ellipsi ; ergo summa rectarum SL et SM, quae in ea
dem ellipsi ducuntur a foco ad extrema puncta cujuscum
que diametri LM, erit quantitas constans. Ad haec: rectae
SL, SM efficiunt cum tangentibus LH , MT avgulos aequa
les SLH, SMT; cum enim LH et MTsint parallelae ( 7 °) ,
itemque Ll et Mm parallelae , angulus HLL aequalis erit
angulo TMm; proinde ( 99) etc.

11º. Revolvatur diameter LM donec transeat per focum S, sicque evadal AB: rccidet SL in SA, et SM in
113
tangentia existent parallela inter se; et consequenter inter-
sectiones quoque LH, MT istorum planorum cum elipseos
plano exsistent parallelae. Liquet autem intersectiones il-
las esse tangentes elipseos in L et M; ellipseos igitur tan-
gentes ductae per extrema puncta cuiusvis rectae, quae trans-
eat per centrum, quaeque terminetur ad curvae perime-
trum, erunt inter se parallelae.
Recta LM secat bifariam (30) chordas omnes parallelas tangentibus LH, MT; ejusmodi enim chordarum pro-
jectiones nihil sunt aliud nisi circularis baseos chordae pa-
rallelae tangentibus lh, mt, atque ideo perpendiculares dia-
metro lm, a qua proinde secantur bifariam: inde fit, ut
LM dicaturo ellipseos diameter.
.Ex M ad focum S ducatur MS; rectae MS
. Mm tangeiit (50) sphaeram, altera in S, altera in puncto
m lineae contactuum superficiei cylindricae et superficici sphae-
ricae: ergo (10. ) MS :Mm. Simili modo, ex L ad S du-
cta LS, erit LS:LI.
90. Plana TMS, mMT et transeuntper rectas MS,
Mm tangentes sphaeram, et sphaeram tangunt, et sese mutuo
secant iuxta MT; ergo (2&quot;)anguli TMm, 'I'MS erunt aequales:
simili ratione ostenditur angnlos HLS esse aequales.

12.• Aa est minimum , Bb est maximum omnium
perpendiculorum Ll , Mm , ... quae ex perimetro ellipseos
demittuntur in cylindri basim ; ergo ( 89) SA erit minima ,
SB erit maxima omnium rectarum , quae ex foco S du
cuntur ad ipsam ellipseos perimetrum .

13.• Punctum S' ita determinatum in axe trans
verso AB , ut sit CS' = CS , dicitur alter ellipseos focus.
Jam si ex S' ad M et L ducuntur rectae S'M et S'L , quo
niam SC = S'C et ( 69) LC = MC , iccirco SL et SM
erunt aequales et parallelae ; igitur ( 109)
SL + SM SM + SM = SL + SL = 2a .
Praeterea angulus SLH aequatur angulo SMR ; ergo
( 10 °.) angulus SMT aequabitur angulo SMR.

14°. Producatur MS donec tangenti LH occurrat
in H , erit ( 30. ) angulus LHS aequalis angulo SMT. Sed
( 109. ) SMT = SLH ; ergoò LHS == SLH , ideoque SL=SH:
hinc ( 13. ) HM = 2a .
56. His praemissis venio cum D &quot; o Arpere ad quaestio
nem propositam de invenienda vi acceleratrice o in motu
elliptico , exsistente centro virium in ellipseos foco S. Conci
piantur duo radii vectores SM , SN intercipientes angulum
inGnitesimum MSN , et producatur SN donec occurrat
tangenti TM ... in R ; erit ( 49 , 6 &quot; )
Q
2 NR
62
Binae NR , MH babendae sunt pro parallelis , eruntque
114
SB; ideoque (100) AB:Za. Quoad alias positiones diame-
tri LM habetur semper LM (SL ∙−⊢ SM, et consequen-
ter (100) LM 2a; igitur AB est omnium diametrorum
maxima: AB dicitur axis transversus ellipseos; diameter per-
pendicularis axi transverso dicitur axis conjugatus.

140. Producatur MS donec tangenti LH occurrat
in H , erit (70.) angulus LHS aequalis angulo SMT. Sed
(loo-) SMT:SLH ; ergö LHS:SLH ,
ideoque SL:SH:
hinc (139) HM:20.
56. His praemissis venio cum D'&quot; Atnpere ad quaestio-
nem propositam de invenienda vi acceleratrice ep in motu
elliptico , exsistente centro virium in ellipseos foco S. Conci-
piantur duo radii vectores SM , SN intercipientes angulum
infiuitesimam MSN , et producatur SN donec occurrat
tangenti TM
...
in R; erit (49. b&quot;)
2NR
∙∙∙⇀−−∙ −−⇀∙∙62
Binae NR , MH habendae sunt pro parallelis , eruntque115
proinde ( 55. 3. ) ut respondentes projectiones nr , mh in
cylindri base : hinc ( 55. 14º.)
nr . MH NR =
nr 2a
mh mh
Sit T tempus periodicum , quo nempe materiale pun
ctum totam percurrit ellipticam orbitam ; erit ( 46) ellipseos
area ad aream MSN ut Tad 0 : istae areae sunt ut re
spondentes projectiones ( 55. 4º. ) in cylindri basi , nimirum
ut ipsa cylindri basis ambll =
mila et area msn : ad
haec ; demisso perpendiculo st ex s io tangentem mt ,
erit
msn = j st , mr = 1 st (nr . mg) : quare
( mza)
712 14
ml
16 T2
2
SC
nir , mg
et consequenter
mi ml 62
T2 .
nr T2 2
st mg
Triangula mlh , mlg sunt rectangula , alterum in l ,
alterum in g ; habent insuper communem angulum in m :
iccirco
ml&quot;
= mh .
mg
Anguli mhl et hmt sunt ( 55. 7. &quot; ) aequales ; propterea
triangula mlh , stm rectangula in l ac o dabunt (55.30. 14º.)
115
proinde (55. 39) ut respondentes proiectiones nr, mi: in
cylindri base : hinc (55. 140.)
nr . MH nr

2
−∙− ∙ NR mh (: mh
Sit T tempus periodicum, quo nempe materiale pun-
ctum totam percurrit ellipticam orbitam; erit (46) ellipseos
area ad aream MSN ut T ad 9: istae areae sunt ut re?
spondentes projectiones (55. 40.) in cylindri basi , nimirum
∙ ∙ ∙ ∙ ↿≖ −∎⋅
↴ ut ipsa cylindri basis ambl(:-Z - ml&quot;) et aram nim: ad
haec ; demisso perpendiculo st ex .: in tangentem mt
, erit
mm:&amp;st,mr:äst(nr.mg)iï:quare
l ml ml: 62
nr ∙−−− ∙∙ T! :,- '
—-- '
&quot;rf&quot;; '
st2 mg
Triangula mih , mlg sunt rectangula , alterum in I,
alterum in g; habent insuper communem angulum in m :
iccirco '
tl,
— z'mll.
Anguli mi:! et hmt sunt (55. 73) aequales; propterea
triangula mllt ,
stm rectangula in 1ac :dabunt (55 . 30. 140.)116
Im mh MH 2a
SC si SM SM
Non pluribus opus est , ut assequamur
47' a3 1
( h) ;
T2 SM
vim nempe acceleratricem in ratione reciproca duplicata
radii vectoris .
Quoad aliam ellipsim
4 R² a ,
1
T ;
i
S, MI 2
hinc si
1 1
a3
T2
a ,
T

: erit op :
:

2 SM 2
S, M ,
Si nempe in diversis ellipsibus quadrala temporum pe
riodicorum sunt ut cubi semiaxium transversorum , vires
acceleratrices tendentes ad respectivos focos erunt in sola
ratione reciproca duplicata respondentium radiorum ve
ctorum .

57. Haec subjungimus .

1.º Fiat CS
CA
CS seu a
€ ; numerus &amp; K1 ) dici
lur excentricitas : ex L in axem transversum ducatur per
pendiculam Li , et ponantur Ci = x , Li = r ; erunt
SL = y2 + ( x — $ a) 2, S'L ' =y2 + ( x + ε a) 2 ,
et consequenter ( 55 , 13º. )
↿16
lm mh MH 212
∙−−∙∙−−− −∙∙sm SM SM .

Non pluribus opus est, ut assequamur
47:303 1
−− ∙∙∙ lt ; ? Ta sit-r, (
vimnempe acceleratrieem in ratione reciproca duplicata
radii veetoris.
⋅ Quoad ≘∣⋮∘⊡↾ ellipsim
∙− 4 123 a,3 1
ut ?! Tla 5! M :
hinc si .?- gz.
. −↿− ↿
⊽↓⊽∶⊺∣≖∙∁≖∣⇂∲∙∲∎⇌⇋⊤⊡∶
Si nempe in diversis ellipsibus quadrata temporum pe-
riodicorum sunt ut cubi semiaxium transversorum, vires
acceleratrices tendentes ad respectivos focos erunt in sola
ratione reciproca duplicata respondentium radiorum ve-
ctorum
.

57. Haec subjungimus.
CS 

↿∙∘ Fiat äseuï :8: numerus :
((1) dici-
tur excentricitas : ex L in axem transversum ducatur per-
pendiculum Li
,
et ponantur Ci:æ , Li:7; erunt
ST.&quot; :y2 −⊢ ≼⋅⊅−−∙⋮∠≖≽⇄⋮ ST]? :]! −⊢ ≼⋅≈⋅−⊢⋮∘≻≖ ,
et consequenter (55.130.)117
Vym + (x - ea) + V y2 + (x + ea )
2a ; !
unde
ye + ( x – sa )2 + 2V 99 + (2 - a) Vya + (xta)
ty: + (x + a ) = 4aº ;
ac propterea
V12 + (x - a)2 V y2 + (x +-a)? = 2a? —yox? - ?o ?
ex qua obtinetur
ya = (1-2) (a? – x2) ( o) ;
aequatio ad ellipsim inter x et y computatas a centro C.

2. ° Facta x = o in ( o ) , valor y inde proveniens
nihil erit aliud nisi valor semiaxis conjugati ( 110.) : hinc ,
denotante 6 istiusmodi semiaxem , exsistet
2
62 CS seu ( 10.) 1 -
62 ideoque CS' =a2-6.

al'
a a2
Inferimus distantiam inter focum et punctum illud ,
in quo semiaxis conjugatus occurrit ellipseos perimetro ,
acqnari semiaxi transverso .
39. Loco x substituatur a - ain (o) : emerget
y2 = ((1 — 82 ) ((2ax - x2 ) ( 0' ) ;
aequatio ad ellipsim inter x et y computatas a vertice A.
Jam vergente e ad 1 , simulque crescente a indefinite ver
117
Vr-l—(æ—eaP-l- l/Ja-l—(æ—l-eaPr-h? ⇥
'
nnde
y' −⊦ (æ −∙∙ id? −⊢ 21/7' ∓−⋅⋜∞∶∽≻∙ Vy' −⊢≺∙↿⊏⊹∽⋟≖
−⊦↗≖ −⊦ (..-'.]. ..). ∶−− ta:
.
EC propterea
Vm VW:2(:* —y2—æ2—s*a'
ex qua obtinetur
]&quot; −−−−−− ≺↿∙−∊≖≻ (a' --.r*) (a):
aequatio ad ellipsim inter se et] computatas a centro C.
2.(, Facta a: o in (a) , valor ]inde proveniens
nihil erit aliud nisi valor semiaxis coniugati (HO.) :hinc ,
denotante b istiusmodi semiaxem , exsistet
—2 b' CS &amp;&quot; ∙
..... 1−−∊≖−∙−∶ 23, seu (1 0,)1 ...—a—z- ;; ;1deoque CSa −−∶∅⇄∙− ∂≖⋅
Inferimus distantiam inter focum et punctum illud,
in quo semiaxis conjugatus occurrit ellipseos perimetro,
aequari semiaxi transverso
.

30. Loco a: substituatur a— a: in (0) :emerget

(1—82) (2aæ—æ2) (0') :
aequatio ad ellipsim inter se et y computatas a vertice A .
Jam vergente P. ad 1
, simulque crescente a indefinite ver-118
gat 2 (1 — ?) a ad limitem quemdam finitum B : aequatio
( 0 &quot; ) verget ad
yö = B x (o &quot; ) ,
et consequenter , precedente foco S' indefinite a vertice A ,
ellipsis repraesentata per (o' ) ad parabolam repraesentatam
( 40.70. ) per (o &quot; ) .
Inferimus illud : si a quovis parabolae puncto du
cuntur binae rectae altera ad focum , altera axi paral
lela , eae cum tangente per idem punctum ducta aequa
les ( 55. 130. ) hinc inde continebunt angulos.

4.• Pone conjugatum ellipseos axem fieri imagi
narium ; adhibe nempe 26V - 1 pro 26 : fiet
22 1-62 = , ideoque e &gt; 1 .
Q2
Aequatio nimirum ( 0) novam curvam repraesentabit,
quae dicitur hyperbola , quaeque secat abscissarum axem
in distantiis CA ( = a ) et CB ( =-a) ab G ; inde in infi .
nitum excurrit cum quatuor ramis ab axe illo magis sem
per recedentibus , quorum bini respiciunt partem posi
tivam , bini negativam , habet insuper centrum in C ,
focos in 0 et O' , exsistente CO = CO ' = ɛa .

5. ° * In aequatione ( o) substitue x' + sa pro x;
habebis
ya=( 1—62) ( a2 -x'tea) )
ad ellipsim vel hyperbolam prout &lt;&lt; vel &gt; 1 , exsisten
te coordinatarum origine in respectivo foco S vel 0. As
sumptis nunc ( 7.9 )
x = Dcosw , y = Dsina ,
118
gat 2(t-—£*)a ad limitem quemdam finitum B :aequatio
(a') verget ad
J'2Bæ ⋅ (a&quot;).
et consequenter
, recedente foco S' indefinite a vertice A ,
ellipsis repraesentata per (a') ad parabolam repraesentatam
(40. 70.) per (a&quot;)
.
Inferimus illud: si a quovis parabolae pnncto du-
cuntur binae rectae altera ad focum, altera axi paral-
lela
, eae cum tangente per idem punctum ducta aequa-
les (55.130.) hinc inde continebunt angulos.

4. 0 Pone coniugatum ellipseos axem fieri imagi-
narium; adhibe nempe ⊋∂⇂∕∙−−−−↿ pro 26 :iie't
↿∟∊≖−−∶−⋮⋮ ideoque : ↿∙
∙
Aequatio nimirum (o) novam curvam repraesentabit,
quae dicitur hyperbola , quaeque secat abscissarum axem
in distantiis CA (:a) et CB (:—a) ab C; inde in inli-
nitum excurrit cum quatuor ramis ab axe illa magis sem-
per recedentibus , quorum bini respiciunt partem posi-
tivam, bini negativam, habet insuper centrum in C,
focos in O et O' , exsistente CO:CO':sa.

5.0 11 In aeqnatione (o) substitue x' −∣− Sa pro a:;
habebis ↼
⋅
J*-——(1—8*)(a—(x'-l—w)2)
ad ellipsim vel hyperbolam prout :( vel)1
, exsisten-
te coordinatarum origine in respectivo foco S vel 0. As-
sumptis nunc (7?)
x': -- DCOSGJ
,yzDsinm ,119
erit
Dasin 6) = (1-2)( a ) - (ea - Dcosa)) &quot;) ;
quae traducitur ad
Da 2 ea ( 1-2) cosa a ' (1-2)
D =
1-6 cos26 1 - &amp; cosa
unde c
D :
a (1-2) ( ECOSW +1 ) .
18? cos26 1
Habetur D pro positiva quantitate ; sumpto itaque su
periore signo quoad &lt;&lt; 1 , emerget in ordine ad elli
psim
D
al 1-52) ( 1 t-scosa )
( 1 +acosw) ( 1 -ecosw)
a ( 1-2)
1 -ECOSW ( h) ;
sumpto inferiore signo quoad &gt;1 , prodibit in ordine ad
hyperbolam
a (1-2) ( ECOSW - 1 ) a (621)
D =
( 1 + scos ) (1 - Cosw ) 1 tecosw
(h' )
Non pluribus opus est ut intelligamus in primo ex ca
sibus alibi ( 50. 13.° 14. ) consideratis descriptum iri
ellipsim , in secundo hyperbolam , exsistente focorum al
tero in centro virium : quoad ellipsim , B= a; quoad hy
perbolam, B' = - a.
6. # Ex ( h)
119
erit
Didone-:( 1—s*)(a'—-(ea—chsæ)3) ;
quae traducitur ad
25a(1—s*) cos 6) D∙∙− a'( 1
Da −∙∙ −∊≖≱≖ .
1—szcos2ca 1—e*cos*c.1
unde
∙∙∙ ⇩≺↿∙−⋮⇄⋟ (scusa) :bt)
1----ea cosa:»
D
1
Habetur D pro positiva quantitate; sumpto itaque su-
periore signo quoad e(1
, emerget in ordine ad elli-
psim '
3( l—sï) (1—l—scosa1) —a(1 —e')
D—(l—l-Ecosw) (1—äcosm) 1—scosc1
(71) :
sumpto inferiore signo quoad s)1
, prodibit in ordine ad
hyperbolam
∙∙ -a(1—e*)(scosca——1) —a(sï—1)
(1—I—scosa1) (1—scosm). 1—I—ecosct
Non pluribus Opus est ut intelligamus in primo ex ea-
sibus alibi (50. 13.014.0) consideratis descriptum iri
ellipsim ,
in secundo byperbolam , exsistente iocorum al-
tero in centro virium :quoad ellipsim, B:; quoad hy-
perbolam, B': — a.
69
.
Ex (h)
∙−− .n..- ∙∙ -&quot; ∙∙∙∙∙∙∙−⋅↖∙∙∙− '.120
1 2
a ( 1-2)
sasin '
ECOSA= 1
€2-82cos ?
Ꭰ . dw al( 1 - E22 a-(1-6 )
a (1— $ 2) 2
( 1 - &quot;)
a (1452)
2 1 1 1
a (14 € 2 ) D
bi a - 1—62)
D2
proinde ( 50. 9.º )
02 2C2
a (1-2)
G- ) ( h &quot; ).
Ex ( h' )
€2sin ?
ECOS W =
a( 82-1 )
D
( a2( 1–82) 2
–1).
€ 2 .
a ( 821)
&amp; 2 - cos26 D
42( 1-2)
2 1 .
a (21) D
a’( 1-62) 2
1 1
a2 ( 2-1) Da,
ideoque ( 50. 10.)
V2
2C2
a 2-1 ( + za) ( 17&quot;).
120
∣a(1-52) d 0 eisinïæ sï-sïcosza)
o ' −⋅
' ⇀−
∙∙∙∙−∙ ∈∁∘⊱∞∶ ↿∙− −∙∙
czu-e*)a czu-ez):
proinde ( 50. 99 )
vï— 202 (
1 '1 )
h&quot;
∙ (tU—83) D 20 ( ).
Ex (h')

8003 6) a(83—1) ;
(ï) ∙∙∙ £2sin26)
−− '
dcc D aï(1—82)2
. &amp;:
e* (cuï—1) 1)2 −− ∊≖∁∘⊱≃∾∙∙∙ D
?
2 1
uzu—w?)a t
czu—ez? a(e*—-1) D
↿ ↿ ∙
↙≖≖≺∊≖−↿⋟ ⋅−⋅ ⋅↧⋅⊃−≖∙ '
ideoque (50. 100.)
202 1 1
)
,,, ↗⇩≕−− an:—1 )(D 'l'ïiz (h)121
Sit E altitudo debita velocitati v; erit ( 50. 12º. )
2BE
v=2qE=
Da
2C E
B (1-82) D2
Igitur in ellipsi
1 E
1
B ' D (ó -za),
2
seu ( 50)
olt
E
D
D
2a ( h &quot; );
in hyperbola
1
B'
E
Da - ( + za)
seu ( 5 )
E = 1 + (tha&quot;)
Ex (h &quot; ) et ( h ) consequitur, si in distantia D a cen
tro virium projicitur materiale punctum, haud descriptom
iri ellipsim vel hyperbolam nisi respectu ejusdem distan
tiae D fuerit minor vel major altitudo illa , per quam mo
bile vi acceleratrice vigente in puncto projectionis cadendo
molu uniformiter accelerato acquireret velocitatem ipsius
projectionis.
7 ° * Quoad ellipsim ( 50 , h. 6° )
9
∙
121
Sit E altitudo debita velocitati v.; erit (a 50. 12'.)
2BE— zcn E
D: B'(1-e*) ⋅
ï; ⊍≖∶∃∲⊡∶−∙− ⋅
Igitur in ellipsi
1'Efn'1 1&quot; 1) B&quot;Dï—-a(o za'
seu (50)
in hyperbola
seu (50)
E -D
,
⋮−⇂∃⇌−−⋅⊳⊣−⋅⇄−∅⋅⋅≺≀⋅⋟⋅ ∙ l
Ex (II&quot;) et (h') consequitur, si in distantia D a cen-
tro virium proiicitur materiale punctum, baud descriptum
iri ellipsim vel hyperbolam 'nisi respectu eiusdem distan-
tiae D fuerit minor vel major altitudo illa, per quam mo-
bile vi acceleratrice vigente in þuncto projectionis cadendo
motu uniformiter accelerato acquireret velocitatem ipsius
proiectionis. -
70t Quoad ellipsim (50. I:. 60)122
7
a 옘 E COSQ ) 1
dw² a ( 1-2) Q ( 1-22) - 5
hinc ( 50. 8º. b .)
go
Ca
a ( 1-62 )
1
Da
areo ds D sinids
Est ( 50. 9º . ) C =D sini.

; exhibet

dt 2
lam a radio vectore D descriptam tempusculo de : deno
tante igitur A totam ellipseos aream, T tempus periodi
cum, habebitur
ds C = D sini dt
2A
T
Est ( 27. 18º. )
a
A = 2V 1-* [Vaº-x:dx ;
exprimit 2 | Va?-xă de circularem aream , cujus radius
= a , et consequenter 1
A = Tla ? VT-
Propterea
1
C2 4 A2
T2
4772 24 (1-2)
T2
42 a3
et p =
Ta
0
9 D2
122
≖↿ ⋅ *

:

cosa) 1 1
dm&quot; −⇩≼↿∙∊≖⋗⊽ ⋅⋅∙↽∙↰↿∙∊≖≽∙ D '
liinc ( 50. 80. b,.)
∙−− ∁∙ ↿
,? ↼⇀ (tU-e&quot;)- ⋅
⋅∎⋝≖⋅∙
Est (50. 90.) C :D ciuili-f.; exhibet Egit-If. areo-
£
iam a radio vectore D descriptam tempusculo dt: deno-
tante igitur A totam ellipseos aream, T tempus periodi-
cum, habebitur &quot;
⋅
∙ ∙ ds 2A
C—DSID! 'a'ï—T ∙
Est (27. me.) fZl/l-Ez l/a'-æ' dx;

∽ ∘
exprimit Zf Vaz- ac2 dx circularem aream , cujus radius
o
∙−∙−−−∙− a , et consequenter
A −−∶↿∽≖ ⇂∕↿ ∙a&quot; ∙
Propterea
4A3 47taa4(1-s*) 41:303 1 ∙
Ta&quot; TTL ∅∘⊔⊢−
'1'» C*.—. 'ne'123
prorsus ut supra ( 56).

8º. Obiter notamus illud: si materiale punctum
motu elliptico movetur viribus ad ellipseos centrum len
dentibus, eae erunt in ratione directa distantiarum ab ipso
centro . Assertionis demonstratio eruitur ex dictis ( 56) : sint
enim duo radii vectores CM ', CN' sub angulo infinitesimo
M'ON' , et producatur CN' donec occurrat tangenti M'T
in R' ; erit ( 49. 6' ' )
2N'R'
♡ 02
binae N'R' , M'C censendae sunt parallelae; proinde ( 55.3º. )
m'c : n'r' = M'C : N'R'
M'C . n'r
m'c
area insuper ellipseos ad areolam M'ON' ut tempus pe
riodicum T ad tempusculum 6 ; quae areae cum sint ( 55.4º. )
ut respondentes projectiones in cylindri basi , nimirum
ut ipsa cylindri basis ambl ( = 76. cm ' ) et areola
cm'.r'm' cm '
m'cn'
V r'n'. 2 cm ) , iccirco 2 2 m'
cm'
r'n ' . 2cm
4 02 unde r'n'
762.
cm
272. cm' ; 1 4 T2 T2

et consequenter
M'C . 27. cm'
T2
N'R'
cm'
Ta 272. M'C..
--
123
prorsus ut supra (56).
80. Obiter notamus illud: si materiale punctum
motu elliptico movetur viribus ad ellipseos centrum ,ten-
dentibus, eae erunt in ratione directa distantiarum ab ipso
centro. Assertionis demonstratio eruitur ex dictis (56): sint
enim duo radii vectores CM', CN' sub angulo infinitesimo
M'CN' ,
et producatur CN'donec occurrat tangenti M'T'
in B'; erit (49. b&quot;)
2N'R'
? ∶−⋅− ⊖≖ binae NZR', M'C ceusendae sunt parallelae;proinde (55.30.)
m'c:n'r':M'C:
⋅ 'C. &quot; N'R'-— M nr :
m'c
area insuper ellipseos ad areolam M'CN' ut tempus pe-
riodicum T ad tempusculum 9; quae areae cum sint (55.4&quot;.)
ut respondentes proiectiones in cylindri basi , nimirum
ut ipsa cylindri basis amb! (: Tt. 27:23 et areola
, .
cm'.r'm' cm ∙
∙ ∙
2
...—3
CI&quot;. I o
,
—— - n .Zcm - 4 9: . .
∙ 9: a , .
32. cmlb :T2;undern :::-'F. 212. em,
et consequenter
9:
MC. ∙⊤↓⋅↴∙⋮−⋅∙ ⇄∏≖ cm

NR −− ∙ ∙−−− -;'21t'.M'C.
cm124
Propterea
. M'C :
vis nempe acceleratrix Q directe ut distantia M'C ab el
lipseos centro

* Etiam sic : in ( o. 1º. ) fac

X
Dcosw y = Dsinw ;
prodibil aequatio inter coordinatas polares ab ellipseos cen
tro computatas, nimirum
av182
Dsin ? w = (1-2) (a² - D2cos w ), unde D= V 1-8? cos26
Hinc
at 2
d:2 av1 (via1-2003 (1-8? cosaw ) V 1-2coscosti
D3 1
a* ( 1-2) D
.
ac proinde ( 50. 8º. 3 , )
CP
a4 ( 1482 )
D :
quae ad superiorem expressionem traducitur; nam ( 70. )
4724 (1-2)
C2 =
4A2
T2 T2
124
Propterea
4 ita
?

:

0132 ∙ M'C;
vis nempe acceleratrix go directe ut distantia MC ab el-
lipseos eentro.
&amp; Etiam sic: in (0. 10.) fac'
∶−∙−− -Doosa) ,y
−−−−− Dsinæ;
prodibit aequatio inter coordinatas polares ab ellipseos cen-
tro computatas, nimirum
al/1— ei
Dsin2 a): ( 1—53) (aa.-ul)2 cosm), unde D—
Hinc
([21 «
⋅
') ? cos-36) sium
113&quot; (zl/1—ea ⇂∕ ↿−⋅⋅∊≖∞≘≖∾ ≼↿∙∊≖∘∘≘≖∾⋟⇂∕↿−⋅⋮∅∾∙≖∞≻ ∙∙∙ D3 1
−− a4(1—£2)—ï '
' ⋅ ac proinde ( 50. 823, )
Ca
?
∙−− a4(1—-s2 )
quae ad superiorem expressionem traducitur; nam (72)
Ca— 4A' ∙∙∙ 4n304(1—£2l
T2 T:
⇂∕⋅↿ ∙⊽∊≖∞⊱≖∾ .125

=== De motu relativo punctorum materialium, tendentium in se mutuo viribus acceleratricibus quae sint directe ut massae in quas tenditur, et reciproce ut &lt;u&gt;quadrata&lt;/u&gt; respondentium distantiarum.===
58.* Sint m, m ', m , ... punctorum massae; a, b, c coordinatae orthogonales puncti m in ordine ad axes OX, OY, OZ (Fig. 8); x ', y', z' , x &quot; , y &quot;, z &quot; , x '&quot; , ... Coordinatae reliquorum punctorum in ordine ad novos axes et parallelos axibus Ox, OY, OZ, et habentes originem in m. Factis compendii causa ( 50. 7.0)
x ' ty's tz's =k ?, x &quot; ty's t-z&quot; = k &quot; , etc ...
erunt ( 50. 4.0) quoad motum puncti m
de a m ' x' m' ' Qc &quot; d²b m'
.
g' , m &quot; g
+
k'
&quot; &quot; ) dc2 k2 k' k &quot; 2 hit
d12 ka kita
d2c
m' z'
k'
m &quot;
k' ' ?
.
dc2 ti to...,
seu
d'a d26
dc2
m'x m'z '
Σ
k'3
niy'
Σ
dc
dca &gt; ( o ) .
dt2 k'3
Nunc quod spectat ad aliud punctum v . gr. mi' , pone ( 50.70. )
(.x &quot; —X')2 +6 &quot; -Y')2 + (z&quot; -z&quot;) = 002 ,
( z&quot; &quot; ' —x' ) 2 + 6 — ')2+ ( z' — z &quot;)2 = ' ' , etc... ;
exhibebunt


126
t ... The **** +
en +++
m &quot; yy'
+
d'a + ... ,
+ ..
vires acceleratrices ab m &quot; , m ' exercitas in m' , no
visque axibus parallelas : denotant
ac m
j'
k'a k'
. C
k'a ' ki k'2 k'
vires acceleratrices ab m exercitas in m' , iisdemque novis
axibus parallelas ; sunt insuper ata , bty' , cta' coor
dinatae puncti m' in ordine ad axes OX,OY,02; facto igitur
m &quot; m '&quot;
+ .. =
assequemur quoad motum puncti m'
20 dQ d'a+x' )
dta
mx' d2(6 + y ')
k3 dla
my'
k'3 dx '
dy '
dQ mz' dºlc + z )
dia dzi k'3
d²a d2b
Substitutis valoribus
dac
ex ( 0 ) , prodibunt dca dla dt2
daxi dl mx' m'r'
dxc '
.
day' d my'
dc2 dy
m'y'
Σ
dea k'3
k3
k3 k'3
126
m&quot; .v&quot;--.r' a&quot;; 7 '—:7' F
∙⋅⊱∷−∎∙−⊦∂⋅≖ a ⊣−∙⋅∙∙
vires acceleratrices ab m&quot;, m'&quot;
, ,..
exercitas in m'
, no-
visque axibus parallelas: denotant
ut se' m y' m :
&quot;F' la&quot;—k&quot; k&quot;&quot;'1?'-&quot;£'
vires acceleratrices ab m exercitae in m'
,
iisdemque novis
axibus parallelas ; sunt insuper a-l-z' , (Hl-y' , e—l—e' coor-
dinatae puncti m' in ordine ad axes 0X,OT,OZ; facto igitur
&quot;
m
m m 37 −∂∙−⋅⋅ −⊦ −−∶
9-
assequhmur quoad motum puncti m'
d'(a-[-æ') ∙∙∙ dQ mx' d3(b-l-y') ∙∙∙ dQ my . d,. dx&quot; k-a de dy' k'3
(P(e-l-z') .... di) me'
dt' dz' k'3
∙ ∙ ∙ dia d'b die ∙
∙⊱∎≖∣⋯⋅⋯∎⋯ valonbus dt&quot; ∙ dt' ∙ dt? ex (0) , prodibunt
g'æ/ dQ -mæ' zm'x' d'y' dQ my' zmiy'
dt' dæ' 163 It'3
'
dt2 dj'- k'3- It'3127
daa' d2 mz'
K'3
m'z'
Σ
dta dzi k3
formulae determinantes motum relatiyum puncti m' quoad
punctum m . Quoniam
00 mx' m'x
k'3
mtm x +
k'3 dx ' k'3
zel
2 X
m &quot;
come
-ac '
813 -) +mi&quot;
xc
k3
V3 k'3) +... ,
dQ
,
- - monte +
-&quot; * 7- ) + m.A-A ) +...
en e -maile +
) &quot; V +
d2 mz'
-Σ dzi k3
m &quot;
tom &quot;
t ... ;
03 k3
hinc facto
R = m &quot;
.6. – +) +
(5--**&quot; +jx +e*e&quot;)+ - ( &quot; ),
m &quot;
formulae ( 0' ) vertentur in
127
∙⇌⋅⋮⋅⋮⋅≕↙⇣≴⋅≖−−−∶↗−⋅⋮−−− ∑∶≀−≖⇣ (.,-,,
dt: dzï lt'3 k'3
formulae determinantes. motum relativum puncti m' quoad
punctum m . Quoniam
dQ ⋯⋅∙∙∙∑∽∙∙↼∙⋅⋅≈∙↾−∙ m—I-m' .
⊋⊑⋅∙⋅−⋅⊼∙∶⊤∣ k'3 −−−⋅∎−∎ ↗⊏∙⋮∣ æ
III I'll
.. ∞⋅⋅−−⋅↕∙⇗ æ&quot; ,,, æ —x' x ≺−−⊽⋮−−−⋅−↗⋮⇁⋮⋮−≻ ⊹≖⊷ ⋯⋯≻⋅⊢ df
———— —— k'3 ∙−− 72— k&quot;3
&quot; yn ∙ yl! &quot; yon—70 ..- 70&quot;.
&quot;' (
a&quot;? ≀⊏⊤∍≻−⊦∽ ↾≺↴↼⋮⋅−∣⋮∎∎ ≀∎⊄−⋅∣∎⋅⋮≻∎⊦⋅⋅⋅∙
(19 Mi z m'x' m—l-m' zo ∙⊦
dQ my' Z mfy' m-l-m'y. ∙∙⊦
⊋∎≖∎⋅∎∎∎∎ k'3 15&quot;— ⋅∎∎∎ ↗⊏⋅⋮⇂ a'.—Z&quot; z&quot; &quot;' zIIO—zt all-l .
&quot;'&quot; ea &quot;&quot;17'5) &quot;'&quot;&quot;
«W &quot;:?75) ⊹⋅⋅⋅⋅ hinc facto
1 æoæn ⊣∙∙ o n

: , z'&quot; B: m&quot; (y'—W) ∙∙∣∎∙

1 me xlv ' '&quot; zl zh, &quot;
mm (öt—or— J—æO—ïä—L) ⊣∎∙∙∙∙ (O 2, .
formulae (o') vertentur in128
dax
de2
m -tm '
+ x's K'3
dR day '
dx ' ' de
mtm
+
k'3
g
mtm
dR daz' dR
dy' '
dit de k'3
dz
Porro , cum habeamus
ka + k &quot;? – 02
x ' x &quot; ty'y &quot; + =' z&quot; =
2
k'2 + k ' ' ?
d''2''
x' x'&quot; ty'g '&quot; +z'z' &quot; etc... ;
2
poterit (o&quot; ' ) scribi etiam in hunc modum
(
R = m k'o + k&quot; — 0° ) +
22
in '&quot; . k'2 + k '''2'''
2k &quot; 2 3* 2) + ... ( o &quot; ) .

59 * Fac at systema reducatur ad duo tantum pun
eta m et m' ; habebis R = 0 , et consequenter
der mm x
+
k'2 k'
day' mtm +
dia K2 K
&gt;
dta

* 3&quot;. d2 z' mtm

dt2 +
k'2 k
Relativus videlicet motus puncti mi quoad m proveniet
m +m: (50. 4. 20. ) a vi acceleratrice tendente ad m : pro.
k' ?
⋯⊣−⋯⋮↨↾ '
(0 ∙∎∣).
klö ,d—l;
dR dïz' m—I—m'z, dR ∙
«(y' dc2 k'3 dz'
Porro, cum habeamus
&quot; k': k&quot;: ∙∙∙ ∝↭⊹⊔↤⇥⋠−−⊦⊇ ∂∣∣≖ ⋅
-k'jl −−⊢ k'&quot;a — ö&quot;&quot; x'M* x'&quot; &quot;' z'e &quot;&quot;— 2 ∙ etc... :
poterit (o&quot;) scribi etiam in hunc modum
!, k.: kn; −∙− ux.,, ∂∜≖
.).
21./&quot;a
↿ ⋅ ra −⊦∣⊏⋯≖ −⋅∂∣∙⋅≖⋅≻ ..
∙−∂⋅−∣⋅∣∣ . ka2&quot; .l.-&quot;' (0 )- mllt
59; Fac 'ut systema 'reducaturad duo 'tantum pun-
cta m et m' ;habebis R ∙−−∶ ∘,
et consequenter
d'x' m di' '
'
−⊦⋯P',—mi −⊢⋯⋅−⊣⋤↾⋮⋡−∙≛ −−−−−∘∙ (it—T k'3 k dca k' k'
da z' ∙ ⋯⊣−⋯∣ .'
d:: [ kl; '
kl :::-&quot;'o.
Relativus videlicet motus puncti m' quoad m proveniet
(50 . 40 . 70.) a vi acceleratrice mt;&quot;. tendente ad m :pro.129
&gt; 7
pterea ( 50.13º . 140.57.50. ) describet m' motu relativo
vel parabolain , vel ellipsim , vel hyperbolam , existente foco in m .
dR dR dR 60# Secunda membra formularum
dx' ' dy' ' dz
( o &quot; ) exhibent ( 50 , 4.:) vires turbantes relativum motum
puncti m' determinatum per formulas (o &quot;) . Hinc si membra
illa manent constanter tenuissima , ita ut (o ' ') et ( o&quot;) dif
ferant in terminis exiguissimis , turbationes quoque inde
provenientes manebunt tenuissimae ; lineaque ab m descri
pla circa m poterit adhuc spectari tanquam vel parabolica ,
vel elliptica , vel hyperbolica ; ita tamen , ut gaudeat ele mentis continue mulatis .

61 * Datis tribus punctis m , m ' , m &quot; ( Fig. 35 ) ,
demissoque ex m' in mm &quot; perpendiculo m'A , sint
x' = mA , y' = m'A , X &quot; = mm &quot; , z' = 0, y = 0, z &quot; = 0.
Erit ( 58)
a' x
1
R m'
(-- = m &quot;
k3
ha( x&quot; —x'to) 2ty'a )
unde prodeunt vires distrahentes m' ab m juxta directiones
x' et y' , nimirum
dR x &quot; — x 1
dx = m &quot;
[(x&quot; — x'ja traj
. DR
dy '
m &quot;
[(x“ — x'ja + y'a ] }
Denotet h angulum m'mm &quot; , et D distantiam mm' ; erunt
x ' = D cos h , y = Dsinh , et consequenter
129
pterea (50 . 130. 14&quot; . 57 . ö&quot;.) describet m' motn relativo
vel parabolam , vel ellipsim, vel hyperbolam , existente
foco in m .
dR dR dR
dæ' , d)&quot; , dz'
(o&quot;') exhibent (50 . 40:) vires turbantes relativum motum
puncti m' determinatum per formulas (a')
.Hinc si membra
illa manent constanter tenuissima ,
ita ut (o&quot;') et (a') dif-
ferant in terminis exiguissimis , turbationes quoque inde
provenientes manebunt tenuissimae ; lineaqne ab m' descri-
pta circa n; poterit adhuc spectari tanquam vel parabolica ,
vel elliptica, vel hyperbolica; ita tamen ,
ut gaudeat ele-
mentis continue mutatis .

130
[ ( x &quot; —x ) 2 + y'r] - = ( x &quot; 2—2D.x&quot; cosh + Daj - -
mi [1+ (13–2cori)]- *

* &quot; [ - P2-2.5k)+3 . ) 6–2 cos )&quot;

- 2.7 CM) 6-2005 )'+ ]
Propterea , si D est ita parva prae aut possint omitui
termini includentes factorem exsistet
(2 )&quot;,
[cº= 2') + s] =
1
+3 D cos h
73 &quot;4
ac proinde
dR
dx = m '
3 D cos h 3 D2 cos2 h
x''3''
=
2 2:14
X

m
* +357 Dosh )
( -Dout
mi ( Doco 4-3C )*.)
China + 202 )=
2 m D cos h
dR 3 DP sin h cos h
dy.
m &quot;
24
130
3
∐∙↧⋅∥−−∙↧⋅⋅⋟≖⊹∫⋅≖⋮∣ −'i;:[£&amp;—2th cosh-l— D'] −⋅⋮∎∎ :
∙∙ 3
æ&quot; :
[130 2, äl—Zcosh)]— 7:
.'! 30 D 3.5 D): D
) a': ∣∶↿⋅−⋮⊸⋅⊋∙−⊤≺⋤−∣ 2—-cosh)-i—m (;&quot; (;,—2005&quot;
−∶≣⋅−≣−−∶≟≺⋚⋛∥−≻ ≺−⋅⋅⋅⋛⋮⊽∙−⋮∞≖↗⋮≻⋮⊣−⋅∙∙∃∙
PrOpterea, si D est ita parva prae æ&quot; ut possint omitti
. . . termini D . . includentes factorem (F) ,
exsistet
[(. ' BD-io-sh. «J)- −⊦∂∣≖∃−⋮⋅↼−↿ −−∽ ↽⊦−−−−−−
ac proinde
dR ,, a.,—a.&quot; æ&quot;—æ' 1
ævo, a'./3&quot; −⊦ [[[. ∎∎∎∎∎ II
a: .r.-3 ,,(1 Dcosh BDcosh— BD'cosïh 1)
., 2 D cOs '! (D : cos: !: 2m&quot; Dcos]:
m -—-3 −−
∙ ,, −− ( æ... .) .: .r., a
−∉⋮≹∙∙∙ ∣≺∐∘⋮∐∣⇂⊹∍∘∙∙⋮∐∣≖∞≖∣≖ −∙∙
dy —--—m ∙↿∙∙∦⋮ æ&quot;], )—131
sin h cos m&quot;
-
D sin h
3 +3 m&quot; D sin h
m 

62. Bonum erit alia ratione nonnulla hic stabilire
circa vires in praefato motu relativo . 

↿∙∘ Sint duo puncta
T , P (Fig. 36.)
, quorum massae m, m', distantia vero TP
(: k');et &quot;veniat determinanda vis acceleratrix in motu
relativo puncti P quoad T . Ex hypothesi P tendit in T vi
I
acceleratrice . m . . m

; — ;et T in P v
: acceleratrice ∙−−∣−−≖ sive au.

] 3
I
tem T sollicitetur .. vn. m . m .
−− ∣−⋮∣−≖− et Pv: 17; , sive T quiescat et P
I
sollicitetur vili—ïm]— &amp;, idem in utroque casu (5) habetur
motus relativus puncti P quoad T; vis ergo acceleratrix
in istiusmodi motu erit

2.&quot; Praeter P , T detur et tertium punctum S ,
cuius massa m&quot; , ut determinentur vires iude provenien-
tes, quibus turbatur motus relativus puncti PquoadT
ortus ex vi (0) . Ducta ST
, completoque parallelogrammo
.. STPP' , exhibeat diagonalis SP (: 8&quot;) vim g.: , qua sol-
licitatur P versus S:resolvatur vis ista in duas, quarum al-
tera (: ?') sese dirigat iuxta PT , altera (:f) iuxta PP';
exhibebitur illa (8) per parallelogrammi latus PT (: k')
.
haec per latus PP':ST (: k&quot;); eritque '
m&quot; .,

; n ⇀
: m&quot; IC, '

m&quot; k, ≒≀−∣−⋮∙⊊≱⋅∙∣⇆∶∂ ∶∣⊄∙ ]; ,unde 93:77sz &quot;3 ⋅132
m' '
m &quot;
Sollicitatur T versus S vi ; et attentis f et i motus k''2''
relativus puncti P quoad T eodem prorsus modo fiet ( 5 )
sive T quiescat et P sollicitetur vi f
m '
sive T sollicite k'2
m &quot;
tur vi et P vi f. Propterea vires provenientes ex S ,
et perturbantes motum relativum puncti P quoad T , al
tera juxta PT altera juxta PP' parallelam rectae ST , ex
primentur per
k &quot; 2
ø=73
m &quot; k
&quot;
g = f mi&quot;
&quot; Cess )
( c' ) . k's
3.° Ex puncto S demittatur perpendiculum SS'
( =i) in planum curvae , quam describit P motu relati
vo quoad T; ab S ad T ducatur recta ST ( =n ) , sitque
angulus STP = a : vis q&quot; agens juxta directionem paralle.
lam rectae ST resolvetur in duas, quarum altera q&quot;cosSTS
seu q &quot; . ! existet parallela rectae ST in plano cur
vae , altera q &quot; sinSTS' seu o&quot; . perpendicularis eidem pla k &quot;
resolvetur in duas quarum altera o &quot;
no: rursus onk
cos a
aget in curvae plano juxta TP , altera om. sina in eo k &quot;
dem plano normaliter ad TP. His positis , quisque in
telligit vires perturbantes motum relativum puncti P exhiberi
posse per
132
SollicitaturT versus S vi &quot;'

; et attentis f et −∥↼↕−∙ -, motus 1.&quot;» 1.&quot;

relativus puncti Pquoad T eodem prorsus modo fiet (5)
sive T quiescat et P sollicitetur vi f— 'I—N. ,
∣∣≖ sive T sollicite/-
et P vi f. Propterea vires provenientes ex S ,
∙ m tur '! k&quot;-
et perturbant'es motum relativum puncti P quoad T , al-
tera juxta PT altera juxta PP' parallelam rectae ST, ex-
primentur per
' mllko &quot; &quot;zl! &quot; kl! 1 '
Pf&quot;??- ⊕−−⇌↾−−⊺⋇⊽≏∶⋯ (Fa-&quot;' ia&quot;) &quot;'

3.&quot; Ex puncto S demittatur perpendiculum SS'
(::t') in planum curvae , quam describit P motu relati-
vo quoad T; ab S' ad T ducatur recta S'T (::n)
, sitque
angulus S'TPr-at: vis 9&quot; agens juxta directionem paralle-
lam rectae ST resolvetur in duas, quarum altera 9&quot;cosST5'
seu 9&quot;? existet parallela rectae S'T in plano cur-
vae, altera 9&quot;sinSTS' seu q;&quot;.grperpendicularis eidem pla-
no: rursus ?&quot;]?- resolvetur in duas quarum altera ⊄∙⊅⋅⋅∙⋮∙− eos ::
aget in curvae plano iuxta TP , altera ?&quot;)—;.sinat in eo-
dem plano normaliter ad TP. His positis, quisque. in? telligit vires perturbautes motum relativum puncti P exhiberi
posse per133
COS Q = cosa ,
9 =porn o--&quot; (* - ) .com
Pa = e&quot; sin æ =
= m
m &quot; (- ) snæ ,
93 = ml - )
( c )
i

;

9 , et Q2 agentes in curvae seu orbitae plano ipsam orbi
tam turbant ; 93 perpendicularis plano orbitae turbat
ipsius plani positionem .
4. ° Pone S , T, P esse constanter in uno eo
demque plario ; erunt i = 0 , n=k&quot;, a=S'TP=STP(=h) :
proinde
PI
m &quot;
8'3 -m&quot;( )cosh ,
&quot; sink ,
}
( cm)
Q2 , 93 = 0 .
Pone insuper ST, SP ita magnas prae TP ut , ex P du
clo perpendiculo PQ in ST, assumi possit absque sensi
bili errore SP=SQ , nimirum d&quot; = k&quot; -kcosh ; erit
1
js =(k“&quot; —k'cosh)-3 = 13 +
3k'cosh
+
Hinc proxime
m &quot;
m'k ( 1-3cos'h= ( 1 +3cos2h) ,
k &quot; 3 2K3
( c&quot; )
3m''K'sinhcosh 3m'k'sin2h''
92 k&quot;3
2k'3
133

: n&quot; muli, &quot; k&quot; ! n 913? —Q ?

eos a:
ïïï —m 673 k,,a k,, 0082,
.&quot;
k ⋅
,
93——9 &quot;,;— Blna :m &quot;(ä-3- It.—741) ,——,- sin a ,- (c)
]. LII-. [
i
93:907?sz −−⋮ ⇁≖⊼↗ ;
4). et (p, agentes in curvae seu orbitae plano ipsam orhi-
tam turbant; (pg perpendicularis plano orbitae turbat
ipsius plani positionem.
4.&quot; Pone S, T, P esse constanter in uno eo-
demque platfo; erunt i:o, n.:k&quot;, a:S'TP:STP(:h):
proinde
mrlk' &quot; kn
'
(Pr −−∶ 7873- −−⋅ m
⊱∣−⊵∙−− F,.)COSII,
(e&quot;)
?::m&quot;≣∶⋅⋅⋮∙−⋅ -—k,,,)smh , 93:30.
Pone insuper ST, SP ita magnas prae TP ut. ex P du;
cto perpendiculo PQ in ST, assumi possit absque sensi-
hili errore SP:SQ , nimirum d&quot;:k&quot;—k'cosh ; erit
1
I,
, −∙∙ BkCOBh ∙≦↜−∽⋮⋅−−−−−≺↗⊏ —kcosh) ∍≔−−↼−−⊺⋮−−∣−−−−
k&quot;, −⊦
,
∙∙
Hinc proxime
∙ ?: 2773— (1—3cosïh):— (1-1—3c092h) ,
z—kHS .
(e&quot;)
—3m' &quot;ksinhcosh —3m &quot;k sinZh134
5,9 Fac ut orbita puncti P sit circularis , ipsum .
que P moveatur ad partes N : sive spectentur formulae
( 6 ') , sive (6 &quot;) , sive ( c &quot; ), aget 92 juxta orbitae tangen
tem contra motus directionem : ejus proinde valori erit
praefigendum signum negativum.
=== De pendulis; deque gravium descensu per arcus cycloidales. ===
[[Fasciculus:Simple pendulum generalized coordinates.svg|thumb|Pendulum]]
[[Fasciculus:Pendulum simplicium.svg|thumb]]
63. Pendulum constat filo tenui secundum alteram sui extremitatem fixo, quod tamquam linea recta et gravitatis expers concipitur, ex quo suspensum punctum ponderosum a directione verticali dimotum potest huc et illuc circum punctum illud alterum extremum fixum in motum circinationis per arcum excurrere. Excursio penduli ab uno arcus, quem describit, extremo &lt;math&gt;C&lt;/math&gt; (Fig. 37) ad aliud extremum &lt;math&gt;D&lt;/math&gt; dicitur &lt;u&gt;vibratio&lt;/u&gt; seu &lt;u&gt;oscillatio&lt;/u&gt;: accessus ad verticalem directionem ex &lt;math&gt;C&lt;/math&gt; in punctum infimum &lt;math&gt;B&lt;/math&gt;, vel recessus ex &lt;math&gt;B&lt;/math&gt; in &lt;math&gt;D,&lt;/math&gt; dicitur semivibratio. Si unicum ponderosum punctum pendeat e filo, pendulum dicitur simplex, si plura in diversa a suspensionis puncto distantia pendeant, dicitur compositum.  
[[Fasciculus:Pendulo simples.jpg|thumb]]
Illud facile quisque intelligit, pendulum &lt;math&gt;AB&lt;/math&gt; circa punctum fixum &lt;math&gt;A&lt;/math&gt; eodem motu arcum circuli &lt;math&gt;CBD&lt;/math&gt; descripturum ac si, sublato filo, in superficie sphaerica perfecte dura et levigata punctum ponderosum moveretur motu impedito. Sicut enim adducto puncto illo ad praedictae superficiei punctum &lt;math&gt;C&lt;/math&gt;, et exinde demisso, gravitas &lt;math&gt;CT&lt;/math&gt; horizonti perpendicularis &lt;u&gt;resolveretur&lt;/u&gt; in duas vires, quarum altera &lt;math&gt;CE&lt;/math&gt; ad tangentem &lt;math&gt;CG&lt;/math&gt; normalis insumeretur in premenda superficie, altera expressa ab ipsa &lt;math&gt;CG&lt;/math&gt; sollicitaret punctum ponderosum ad motum per tangentem infinite parvam, ac deinde per aliam atque aliam subsequentem, et sic deinceps per reliquas omnes numero infinitas et infinite parvas tangentes, quibus constare arcus descriplus concipitur; ita a filo resolvetar gravitas eodem prorsus modo , nempe partim in trahendo filo insumpta, partiin ad singulas arcus circularis tangentes infinite parvas subinde determinata, qua deducetur pendulum per arcum circularem motu omnino simili, subeunte filo &lt;math&gt;AG&lt;/math&gt; vices curvilineae superficiei: hinc sicuti punctum illud ponderosum propter suam gravitatem, postquam descendisset ex &lt;math&gt;C&lt;/math&gt; in &lt;math&gt;B&lt;/math&gt;, cogeretur ascendere ex &lt;math&gt;B&lt;/math&gt; versus &lt;math&gt;D&lt;/math&gt;, ita ob rationem similem pendulum post descensum ex &lt;math&gt;C&lt;/math&gt; in &lt;math&gt;B&lt;/math&gt; ascendet ex &lt;math&gt;B&lt;/math&gt; versus &lt;math&gt;D&lt;/math&gt;. Rursus quemadmodum ponderosum punctum in praedicta superficie ascendere inciperet per arcum &lt;math&gt;BD&lt;/math&gt; cum eadem velocitate, quam acquisivisset in puncto infimo &lt;math&gt;B&lt;/math&gt;, et ideo ad eamdem altitudinem, ex qua descendisset, perveniret, nempe usque in &lt;math&gt;D&lt;/math&gt;, ubi extincta omni velocitate, iterum gravitate sua inciperet descendere, et in puncto &lt;math&gt;B&lt;/math&gt; priori velocitate rursus acquisita, cum ea ascenderet iterum in &lt;math&gt;C&lt;/math&gt;, atque ita porro ascendendo et descendendo perpetuas et aequalęs in peripheria &lt;math&gt;CBD&lt;/math&gt; excursiones perficeret, ita ob eamdem rationem penduli oscillationes aequales essent natura sua et perpetuo duraturae, nisi ab aeris &lt;u&gt;resistentia&lt;/u&gt; et &lt;u&gt;frictione&lt;/u&gt; aliqua circa sustentationis punctum &lt;math&gt;A&lt;/math&gt; inaequales primo redderentur, ac denique extinguereatur; adimentibus scilicet ejusmodi causis in singulis oscillationibus aliquid de illa velocitate, quae producitur a gravitate.

64. Velocitates &lt;math&gt;v&lt;/math&gt; et &lt;math&gt;v'&lt;/math&gt; in puncto infimo B acquisitae a gravibus per arcus &lt;math&gt;CB, C'B&lt;/math&gt; descendentibus sunt ut ipsorum arcuum chordae. 

Per &lt;math&gt;B&lt;/math&gt; concipiamus duci tangentem et in eam ex &lt;math&gt;C&lt;/math&gt; et &lt;math&gt;C'&lt;/math&gt; demitti perpendicula &lt;math&gt;z&lt;/math&gt; et &lt;math&gt;z'&lt;/math&gt;: denotante &lt;math&gt;r&lt;/math&gt; radium &lt;math&gt;AB&lt;/math&gt; et denotantibus &lt;math&gt;k, k'&lt;/math&gt; arcus quoad radium 1 similes ''arcubus'' &lt;math&gt;CB, CB'&lt;/math&gt;, erunt 

&lt;math&gt;z = r ( 1 - \cos k ) , z' = r (1 - \cos k' ) ; &lt;/math&gt; 

et quoniam (30: 36)

&lt;math&gt;v^2 = 2gz, v'^2= 2gz';&lt;/math&gt;

propterea

&lt;math&gt;v: v' = \sqrt{2gr (1 - \cos k )} : \sqrt{2gr (1 - \cos k')} = \sin \frac{k}{2} : \sin \frac{k'}{2};&lt;/math&gt;

ideoque etc.

65. Pendulum, quod incipit descendere ex &lt;math&gt;C&lt;/math&gt;, percurrat arcum &lt;math&gt;CM&lt;/math&gt; tempore &lt;math&gt;t&lt;/math&gt;; sitque &lt;math&gt;\alpha&lt;/math&gt; arcus quoad radium 1 similis arcui &lt;math&gt;BM&lt;/math&gt;: erunt &lt;math&gt;CB =rk, BM = r\alpha&lt;/math&gt;; et designante &lt;math&gt;u&lt;/math&gt;velocitatem in puncto &lt;math&gt;M&lt;/math&gt;, exsistet

&lt;math&gt;u = - 2gr (\cos\alpha - \cos k ) = 4gr \sin\frac{k+\alpha}{2} \sin \frac{k-\alpha}{2}.&lt;/math&gt;

Si arcus &lt;math&gt;k&lt;/math&gt; est ita exiguus, ut possit absque sensibili errore substitui respondenti sinui, habebimus
&lt;math&gt;u^2 = gr(k^2 -\alpha^2),&lt;/math&gt; et consequenter (28)

&lt;math&gt;\frac{ds^2}{dt^2} = gr(k^2 -\alpha^2),&lt;/math&gt; 

unde

&lt;math&gt;dt = \frac{ds}{\sqrt{gr(k^2 -\alpha^2)}} = \frac{r\beta}{\sqrt{gr(k^2 -\alpha^2)}}= \frac{\beta}{\sqrt{\frac gr (k^2 -\alpha^2)}};&lt;/math&gt; 

&lt;math&gt;\beta&lt;/math&gt; est arcus quoad radium 1 similis arcui infinitesimo Mm
( = ds ). Nunc centro H ( Fig. 38) et radio HD ( = k) describe circulum DED' ; sume HN : Ν »
B; duc perpen
dicula Ne, ne super HD: et Ey parallelam radio HD. Trian
gula similia HEN, Eey rectangula in N, y praebent

∙∙∙⇀ ,4þf - ⇀∙⋅∙∎∙ .. ⊸∙⋅⋅⋅∙∎∎∣∙ 4.-
∙−
..137
Ey: EN = Ee: HE, seu B: V R2-42 = Ee: k :
hinc
B
Ee 8
dt ;
V R2-42 k
et consequenter
Ee dt
kV
tempusculum videlicet dt impensum ad percurrendum
seu Nn, obtinetur dividendo respondentem arcum Ee per
kV § . Inferimus tempus t impensum ad percurren
dum ka seu ND, obtineri dividendo respondentem ar
cum ED per kV ; nimirum
ED
자
름
k
Quare
VED –are(com);
ideoque
Vare(cový = ) &lt;( a ) .
10
137
Ey:EN :Ee: HE, seu ,8: V kï-aï: Ee: h:
hinc
∙
!
Ee .... B ∙∸−⋅⋅ :: Vii. dt ;
VIR-æ ]: '
r
et consequenter
⋅↙≀↥∶∎∙−−⋮∶∶⇣∶⋮ *Ve-
tempusculum videlicet dt impensum ad percurrendam þ
seu Nn, obtinetur dividendo respondentem arcum Ee per
]; Vi .
Inferimus, tempus :
impensum ad percurren-
'.
dum ]:-
ut seu ND, obtineri dividendo respondentem arcum ED per kI/ £.; nimirum
r
t ∙∙∙ ED
l.V-f,- .
Quare ∙
.yz... −∙−⊡∍ ......
«
.
rf- k —- (eos:.k),
ideoque
≀⇌⇂∕∑− arc (eo: &amp;) (a) ∙
5
10138
Iam vero in puncto infimo B (Fig. 37) exsistit a = 0 ;
erit igitur tempus semioscillationis
TT
ti
V
2
8
tempus integrae oscillationis ( a ' ).
t2 =
V
quas formulas cum non ingrediatur initialis angulus k, patet oscillationes ejusdem penduli, vel plurium pendulorum aequalis longitudinis r per arcus satis exiguos utcumque ceteroquin inaequales, fore ad sensum isochronas seu aequi
diuturnas.
Idipsum facile demonstratur hac alia ratione: angulus GCT
= 90° BAC; hinc vis acceleratrix CG , ex qua sola repetendus est penduli descensus, exhibebitur per gsing: in hypothesi nimirum arcuum satis exiguorum spectari poterit CG tamquam proportionalis distantiae a puncto infimo B, computatae in arcu BC. Ergo ( 29. 4°) etc.... Etiam sic: est ds = d rík - a ) rda ;
et consequenter
rda da
dt
V rg (k -u?)
-Vivok²-u?
factaque integratione ( 27. 13º. 14° ) prius ab a kad
a =0 , dein ab a = k ad a = -k, emergent binae (a' ) .

66. Haec notentur: 

1º: secunda ( a' ) dat
77 r
8 ( a '');''
ta
atque inde innotescit gravitas g.
138
Iam vero in puncto intimo B (Fig. 37) exsistit «:o;
erit igitur tempns semioscillationis. &quot;
∙−∣
⋍⋅∶−−−−⇄⋅−∣∕−≦−∙
tempus integrae oscillationis (,,-)
−∣−∙−
ta:T! −∙− ∣∕ :
,
quas formulas cum non ingrediatur initialis angulus k, patet oscillationes ejusdem penduli, vel plurium pendulorum
aequalis longitudinis :- per arcus satis exiguos utcumque
ceteroquin inaequales, fore ad sensum isochronas seu aequi-
diuturnas-
Idipsum facile demonstratur hac alia ratione: angulus GCT

: 900— BAC; hinc vis acceleratrix CG, ex qua sola re-

petendus est penduli descensus, exhibebitur per gsinat: in
hypothesi nimirum arcuum satis exiguorum spectari poterit
XCG tamquam proportionalis distantiae a puncto infimo B.
computatae in arcu BC. Ergo (29. 40) etc.
...
Etiam sic: est ds:dr(k— a):
-— rda;
et consequenter
rdat dat
dt −∙− ↵
−− '&quot;
VrgUe-æ .? sz-az :
factaque integratione (27. 130. 140 ) prius ab et: I: ad
«:o, dein 'ab a: ∙−−− I, ad ac ∶−−⋅ —-k, emergent binae (a')-

2º. Etsi ponderosa diversae materiei puncta permissa sunt oscillare, attamen idem semper prodiit valor g in eodem terrae loco: rursus ( 17 ) igitur devenitur ad proportionalitatem inter corporum massas et respondentes gravitatis vires.

3º. Constat observationibus longitudinem penduli
simplicis oscillationem absolventis intra mioutum secundum
eo esse minorem, quo magis ad aequatorem acceditur: quoniam ergo, haud variato tz, gravitas est ut longitudo illa, minuetur gravitas a polo ad aequatorem usque ( 30) .

4º. Apud nostras regiones praefata penduli longitudo cum sit = 3ped opol glin, 38 = 440lin, 38, factis
in ( a &quot; ) tz = 1 &quot;, r = 440lin , 38, prodibit respondens gravitatis valor g = 30ped , 183 alibi (30) indicatus.
[[Fasciculus:Double-Pendulum.svg|thumb]]
67. Quod spectat ad pendulum compositum concipiamus (Fig. 39) puncta ponderosa B, B. , B2 , . . filo appensa: invicem disjuncta conficerent haec puncta temporibus inaequalibus oscillationes suas; punctum nempe B, citius  (66) quam B, punctum B, citius quam B, etc: invicem ergo conjuncta agent ita in se mutuo, ut quae, minus distant a puncto suspensionis A retardentur ab iis quae magis distant, et quae magis distant a suspensionis puncto accelerentur ab iis quae minus distant: fiet propterea oscillatio penduli compositi tempore quodam medio inter minimum ac maximum praedictorum temporum inaequalium. Hinc sequitur fore in AB punctum quoddam B.,m suas conficiens oscillationes perinde ac esset solitarium, nulloque nexu caeteris punctis uui
retor: Bm dicitur centrum oscillationis, cujus centri distantia a puncto suspensionis est longitudo penduli simplicis suas perficientis oscillationes eodem tempore ac pendulum compositum. Inferimus oscillationes pendali compositi, et ipsas fore isochronas; modo tamen exsistant satis exiguae.

68*. Facile intelligimus ( 50. 3º. 6° : 66 ) motum penduli simplicis in medio resistente determinari generatim per aequationem

140
das
di?
= gsing -f(v ) .
derka( )
di Ob dc2
dra
dt2 et ( 27. 29º . ) sing
23
2.3 +
aequatio illa eyadit
creat + s ( « - +...)-fo) =
Pone fv) = cv ; et angulum a ita perseveranter exiguum,
ut ejus tertia potentia absque sensibili errore negligi pos
sit : habebis
da с
+
dla
ola
0 .
ds Est autem v =
dt
drak -a)
dt
da
dt ; igitur
d2C da
dt2 to dt + baro:
quam integrantes in hypothesi c constantis assequemur( 27.270. )
... [ :V546.-V21
( 6),
In experimentis, quae pendulorum ope solent institui ,
r est multo minor quam g; item densitas penduli, et con
sequenter ( 33 ) c fractio admodum parva. Fac ergo
140
tiis ∙
de'-* :gsmat —f(v)
d': d2r(k—a) (lioc
↽⇁−−− −∙−
' Ob daz dtz 27. 290. .:
rdtï
, et ( ) stna
'
a3 . '11 d'
rdzatdta-l-g(at—-— ...):fþv) :0.
Pone f(v): cv; et angulum ac ita perseveranter exiguum,
ut ejus tertia potentia absque sensibili errore negligi pos-
sit: habebis
dza g c
dia—FTa—Tv—o
.Et :
−−∠≀≖∙∙↙≀↗≺∣≖⋅⋅∝≻∙− ∎∠≀∘⊏∙⋮ ⋯⋅⋅ s auem'v— dt— dt rd , g1
quam integrantes in hypothesi c constantis assequemur(2 7.2 70.)
&quot; til &quot;'i-i.]c) —l ! —-..—.
2:32 [CG 4
r—l—C'f—B
ln expetimentis, quae pendulorum Ope solent institui,
r est multo minor quam g; item densitas penduli, et con-
sequenter (33) c fractio admodum parva. Fac ergo141
VS
ut sit VA = iVT ;
4 .
vertetur ( 6) in
ti V = 1 -til
+C'e 2 -
&quot; ]
U = e
seu ( 27. 30° )
3 [
e
] ;
C &quot; sin it +C ' ' cosit
unde
cosit
data o - [( c'i - )
(c&quot;: + * ) ainit ]
da
In joitio motus t=0 , a= k , 0 ;
di
propterea C &quot; &quot; k, C '
ck
21 ; et .
= ke - - [
sin it + cosit ] . )
2i
( 6 )
dan
dt = -ke- Ź [ ita sinic.
141
'
. . ∙
c' a . .
ä-ï': - utsit V—--€- :::tl/ −−↿ ∙
r 4 4 :-
vertetur (b) in
−−−∘⋮−≀ — tiV—1 -til/—1
at:6 2
)
C'0 v
380 ( 270 300 )
↴
c ↼
a:
∘−−≖− '[C&quot; sinit ⊣−∁⋯ cosit] ;
unde
∘⋮ C'&quot;
↙⋮⋮∶∶ .
∘∙⋅−
; t [( C'i— 20) cosa
: -— dt ∙

Cnc . .
(
0&quot; i −⊢ —2 )
sunt ].
dat ∙∙∙ ∙
In initio motus t—:o
, at:— ]:
, &quot;dt −∙− 0-
∙∙∙ k
': propterea C Ck et −−−∙−− ∙
∙−− 21' ∙142
с
Ex.Vihabemus zi 11
ll
Hlacin
c2
V C
it =-V
rc?
4g &gt;
2V EV
rc2 1
48
1
i +
VE ;factoigitur V
cr2 =c,
43
&quot; Vi ro2
4g
binae ( 6 ) sic poterunt exprimi
a = ke

* IVE Vētowi.V ]

1.)
(6 )
dm-- .- iv E sinórV.
In fine cujusque oscillationis est da
dt = 0; proinde, ob
= 0: inferimus in fine primae secundam (3&quot;),since V
12 풍VV
2.V
oscillationis fore t = &gt; in fine secundae
8
271 376 in fine tertiae t =T
8
8
gulae itaque oscillationes absolventur aequali tempore
E ,
in
V
, elc.. .. ; sin . с
g. ∘ Ex ::i habemus 21. ∶∶ − c —- .
∙
∙−−−−−−−
I/ ⇂∕−−−⋅↿rc
c ⋅ :! −⋮↓−≔⋤⋅ a
cr&quot; ,
. CZ .g 1;factoigilur V1—-- :c, ↓⊣− ' ⋅ −−∶ ⋅
binae ( b') sic poterunt exprimi
£(sz 2 inc't cosc' g
c[hc V—s Vg ∙−⊢ ::ll/:]
(ö&quot;)
(E;—..., &quot;&quot;T-V—sinc't 5-
dt :-
/
In fine cuiusque oscillationis est ≤−∝⋮∶∶∘⋮ proinde, ob
secundum (b&quot;), sinc': Vi:o: inferimus in fine primae
!'
1:
osc1llat10n1s ∙ ∙ ∙ r ∙ fore :

:

−⋮⇆⊤ V—3,
in fine secundae
271 .
⋅⊤ −∙
,
in fine tertiae :
−−−−−∣−
−− ,etc.. -sm-
gulaec itaque goscillationes absolventur aeqnali tempore143
و
=
ا
( 6 '' ) .''
8
In primo substitutis valoribus 0, 20 , 30 , ... no
pro t , emerge
Qu - ke 2 A2 = ke -
&gt;
929
as= -ke- 30
Q. = 1–1 y&quot; ke – no
hinc successivarum oscillationum amplitudines
0
2
음
k + ke ke - 9 +ke - 2
-ke 39 -
2/2
20
the
ke
seu
1(1 +-2), ( +-3, -2,

* (170-99 . -Ź- 42329......

Decrescunt igitur amplitudines istae in progressione
geometrica : quod cum confirmatum sil experimentis pen
dulorum in aere oscillantium per arcus satis exiguos, haud
majores v . g. tertia parte unius gradus, licebit quoad e
jusmodi oscillationes assumere aeris resistentiam tanquam
proportionalein simplici velocitati.
143
n −−−∽−∣∕∙⊂− (B&quot;)- 0

6
In prim: (. ⋮⋅ substitutis valoribus 9, 29 , 39 , ... 719
pro :, emerge f
⋅∙⇁− -
∙
1- 0
29 ⋅ C
a;:— ke 2 ,agzke 2 ,a3z—ke 2 39
C
119 a.::(—1)&quot; ke −⋮⋅ :
hinc successivarum oscillationum amplitudines
.
c ⋅∙⋮∙∙ ...—£
k-l—Re— i.e.]:e— 294-ke ∙
229
&quot;
ke −⋅⋮⋅⋮∂−⊦∣∥⋝ 2
39,. ;
seu
.- e ∘
9
−−∘ &amp;
Decrescunt igitur amplitudines istae in progressione
geometrica : quod cum confirmatum sit experimentis pen-
dulorum in aere oscillantium per arcus satis exiguos, haud
maiores v. g. tertia parte unius gradus, licebit quoad e-
jusmodi oscillationes assumere aeris resistentiam tanqnam
proportionalem simplici velocitati.144
Experientia insuper docet decrementum illud gradu
admodum lento procedere : sic D. Borda expertas est non
nisi post 1800 oscillationes valorem en converti in k .
Hoc posito, existet
2
1
1800c7V
.18000 e 2
2 2c seu ob (6 ' ') e
3اندبه
et consequenler
1800CTE
= c'log. 13 == c (o , 40546) . 2
( 1800)? c ?72. &quot;
= 1 - c '? , ideoque 8
4g
ro2 Sed
4
= (1800 ) 2772 /1— 2) ; igitur ( 1800)277 ?(1 — c'2)
= c (0 , 40546 ) ; unde
c' ?
( 1800 )2772 1
(1800 )222 I.(0,40546 )
&gt; et e
V
+18007
1+
0,4054612
180076
1
quam proxime.
gua
Si ( 33 , 4.&quot; )
poneretur f (v )
terminandum exsisteret
ad penduli motum de
144 *
Experientia insuper docet decrementum illud gradu
admodum lento procedam : sic D. Borda expertus est non-
. . , 1118! ∙ ⋅ ∙ ∙ 2 post 1800 osmllattones valorem ac,, com-cru mes-k.
Ilnc posito, existet
'
u
1eöocn VL
..
'
5.
20'
≀∙ ⋅
.
...—18009 2
e 2 −−∙−−− −∙− , seu 01) (b&quot;') e
⊏⋅⋮− ,
. 5 ⇁
3
(!l. 000 sequenter

: .—c'(o ∙⇁ '40546) b. &quot; .

1800072l/L .
2 g :c'lo
'.
(1800≻∖⋅:0:712.— g
PC: '
∙
'
Sed −−−−−↿ ---c2
, 1deoque 45
⇌≺↿⋅∂∘⊙≻≃⊺≖≖≺↿−⋅≺∶∣≏≻⇋ igitur ≺↿∂⊙∘≻⇄∏≖≺↿−∘∣≖≻

: c'5(o, 4054673; unde

(1800)??? - -, 1
, ↼−−
≺↿ et -c':
≨∃⊙∘⋟≖⇃∙≖⋍⊣−⋅⊏∘∙∠↥∘⋦∢⊖≻ ↼↼
'3
Vl—l&quot;(7'g55; o.4(l546)z
≖≖−−−↿ quam proxime.
↓
2 '
51 (33. 41.&quot;) poneretur f(v) ∶−−⊸∙⋮⊥⋮− , ad penduli motum de-
02
termiuandum exsisteret145
des
dt?
gsing gu2 d ?
seu de2 + sine —
8r /dala
2
= 0 .
c²lde
Haec prias multiplicata per 2du , ac dein integrata suppe
ditat
' dala Idala
ca ldt
2g
COS O
seu facto Slaa) dx = y , ideoque Coupe ( )
dy
da
dy
28
2gr COS Q
da y = 0 ;
cojas integratio traducitur ( 27. 26 °.) ad integrationem fun
ctionis
2g cosada
2gra
c2 re
Jamvero , facto compendii causa 2gr
= m , habemus c2
dem sina ) coso, da Se
ma
-mu m e sing da ,
d ( e-ma cosa )
-ma sina da - me COSQ da :
igitur
ſe-ma cosa da e -ma sina tm se-ma siac da ,
ſe-ma sina du = me-ma cosa m ſe-me. cosa da ;
ex quibus
145
d's— ∙ g.&quot;: dia g ⋅⋅ . gr
äzäfgsma— Z;- , sendt: da)!— -[-r sma 02 22 —--0.
Haec prius multiplicata per Zda , ac dein integram suppe-
ditat -
(&amp;)2— dt ∙−⊋∊∁∘≘∝−⋅⋮⋚⊆∫≺≦− r
f:) ↙≀⊄∶∘∙∙∙
dat ∙∙∙ ∙ da: a... 47&quot;
seu facto f(ä—t) fia —J—, ,
1deoque (22) −∙− ä; ,
≝⊻−≟≝∁∘⊱⊄≉≣≝⋅∫∶∘⇋ . dat
cuius integratio traducitur (27- 262) ad integrationem fun-
ctionis
Zg cosadat ∙
⋅
2grat .
∘≖ . re :
∙
2 r
'
Iamvero ,facto compendii causa −⋚−⋅:m ,habemus
02
,! (.;-&quot;&quot;&quot; since ):e'ma cosa: d-a −∙∙ m e'm' sinat da
,
d(e'macosat):-e'm ∋⋮∘⊄↙∄∝−⋅⊪∘⋅⋯ &quot;cosada:
igitur
fe'm. cosa da::
efm sinat—lfm fe'm sinat dat
,
fe'm sinat daz ⋅⋅−−− −−∶ e'm cosa: — m fe'm- cosa daz ,-
ex quibus146
ſen-Ma cosa da e -ma sina — те cosa -m2ſe-mecosadu 7,
et consequenter
2g
Sce-ma cosa da
2g ( e -ma since me- mu cosa)
r ( 1 + m2)
Erit itaque (27. 26 °.)
y = Cema +
2g ( sing - m cosa )
r ( 1 +ma )
ex qua differentiata quoad &amp; cum emergat
dy
da
Cmema + 2g (cosx + m sina )
r ( 1 +m2)
restituto valore dy
da habebimus
ca
dal 2g (cosa + m sina)
= - Cmema +
r ( 1 +m2 )
da
In initio motus a = k , = 0 ; hinc dt
Cm 2g ( cosk + m sink) e-mk
r ( 1 + m2)
propterea
-m (k - a )
Cate)
dal 2
ldt
29
r ( 1 +m2)
cosa + msina - cosk + msink)e ( h).
Facto a = o in ( h) , prodibit inde velocitas penduli in pun
cto infimo B ( Fig. 37.) : ascendet pendulum cum velocitate
146
fe'm&quot; cosa da: e'm&quot; sinat —me*mcosoc −∙∙ ⋯≖∫∘⋅⋅⊪∞∽∡∠≀∝ ∙
et consequenter
2g &quot; 2g (e-&quot;W- sinat −∙∙ rne-ma cosa)
∙−∙∣ (.'-'&quot;&quot; cosa dat:
⋅
'
r(1 −⊢ ⋯⋅≀≽ Erit itaque (27 . 260.)
2g (sinat —m cosa)
y.:Cama'i' r(1-l—m') :
ex qua differentiam quoad a: cum emergat
dy ∙− M Zg (com-[- m 5213.)-
da :Cme r (1 —f-m3) '
∙
d
' ∙ resututo valore 1, babebmus ⋅
de:
((!—S :Cma'&quot; &quot;I&quot; 25 (cosa: ∙⊢ 11: sind:? ∙
.. dt r(1-l-m3)
∙ ∙ ∙
'
da ∙
In 1n1t1o motus a:k −∙− −−−−∙ o ; bmc 'dt
2g (cosk −↿− m sinit) er:-&quot;&quot;* ∙
Cm: r(1—l-m2) ..
propterea
(de!)2 28 &quot;[COSa-Hnsina—(cosk-i'msïnk) e-m(k-a)] 32 :r(1—i—m2) (73)-
Facto a:o in (I:), prodibit inde velocitas penduli in pun-
cto infimo B (Fig. 37.) :ascendet pendulum cum velocitate147
ista versus D , conficietque arcum , cui respondebit — Q,; et
quoniam in extremo puncto illius arcus extinguitur tota ve
locitas , iccirco
COS - m sina, · ( cosk + m sink) e -m (4+ 1) = 0 ,
seu
(cosa,
m sing , emai (cosk +m sink) e-m * = 0
( h ' ).
...
mk .
maa ,
Sunt ( 27. 29.° ) emas = 1 + m « . + +
2
mak ?
=1 -mk+ -... ; est insuper m fractio admodum
parva ( 33) : neglectis igitur terminis , ubi invenitur mº ,
traducetur ( h ) ad
2
&gt;
cos@g - m (sina, cosax) = coskt m (sinkkcosk ) (h &quot; ).
Denolante o differentiam inter valores a, et k ut sit Q= k -0 ,
certe ð erit fractio tenuissima : hinc substituto k- loco
Qy in ( &quot; ) , sumpto 1 pro cosd et à pro sind , missisque
öz et mo , assequemur
2m
Osink = 2m ( sink - kcosk) , d = 0
sink (sipk- kcosk ) ;
unde
Uy= h
2m
(sink-kcosk) .
sin k
Si popimus k ita iguum , ut ejus quarta potentia prae
termitti possit , obtinebimus (27. 29.° )
147
ista versus D, &quot;conficietque arcum , cui respondebit — at,; et
quoniam in extremo puncto illius arcus extinguitur tota ve-
locitas
,
iccirco
111 cosa:, −∙∙ m sinatl — (cosk −∣− m sink) e&quot; (b'-3 1): o ,
seu
(cosa, ∙− 11: sind,) a'&quot;! — (cosk —-msink)e&quot;&quot;'* :: o (b').
Sunt (27.29.0) erat: mna? ↿−⊦⋯⊄≖−⊦
2 −⊦ ∙ ∙ ∙
∙ ,∙⋯⋆
⋯≖⇂∙∙≖ ⋅ ∙ ∙ ⋅

: i —mk—l—-—2—-— ...; est 1nsuper m fract1o admodum ∙

parva (33): neglectis igitur terminis ,.
ubi invenitur m',
traducetur (h') ad
tuom,—m (si na,—aleam,:cosk—l-müi nk—kcosk) (h&quot;).
Deuotante ö differentiam inter valores a, et k ut sit ac,: k-ö.
certe d erit fractio tenuissima : hinc substituto k—ö loco
a, in (II&quot;), sumpto 1 pro cosd et 6 pro sind
. missisque
d' et md , asscquemur
ösinkz2m (siuk—kcosk) , ö: ET- (sink—kcosk);
. sink
-
nnde
2m
sin lt at:-k— (siïnk—kcoslc) .
Si ponimus !: ita exiguum , ut eius quarta potentia prae-
termitti possit , obtinebimus (27. 293)148
2m 2m
- (sink - koska gink k
21 k2 1
2.3
2m 2m ka ( 1
k2
2.3 h2 ,
3
ac proinde
Q = k
2m
3
k2 :
quemadmodum valor a, deducitur ex k , sic ,yalor d, ex
valor az ex la , atque ila porro ; erunt nempe
2m Aa =
2.1 az ?; 3=0,- 313, etc...
| Patet illud ; si vis acceleratrix ex medij resistentia sumitur proportionalis quadrato velocitatis, haud subsistet superior lex, experimentis confirmata, de oscillationum amplitudinibus in progressione geometrica decrescentibus.
[[Fasciculus:Cycloid f.gif|thumb]]
[[Fasciculus:Cycloid03d.svg|thumb]]
[[Fasciculus:Cycloide InfinimentPetits.svg|thumb]]
69. ° * Aliquid subjungimus de gravium descensu per arcus cycloidales. Circulus A'D ( Fig. 40 ) tangens rectam A”E in A&quot; revolvatur super ipsa A”E ita, ut eam pergat semper tangere. Punctum A&quot; circuli regredietur ab A&quot; in E, lineamque curvam describet, quae appellatur cyclois: circulus ille mobilis vocatur cycloidis genitor, recta A ” E basis, diameter AB perpendicularis mediae basi dicitur axis, punctum A vertex; patet autem quemvis circuli genitoris arcum B’A' aequari rectae A'B, quae intercipitur duobus punctis A &quot; et B', in quibus extre ma puncta ipsius arcus tanguntur ab A'E; et totam basim AE aequari peripheriae circuli genitoris. Ducantur
148
2
2,&quot; (siuk—kcosk)-— &quot;' .↗⋮∍ mnk ]: (
kz 3 2.) ⋅
ac proinde
2m
'k 3k
quemadmodum valor a; deducitur ex 1: , sic.valor ag, ex
'a, , valor 013 ex ac, , atque ita porro; erunt nempe
a—a—ïaa' a—a—zma' etc - 2—1 3
1
, 3—3 T;, ∙Ducantur149
jam ex cycloidis puncto v. g . A' perpendicula A'rl= y )
et A'C , alterum in basim AE, alterum in axem AB ;
sit A'r = x ; diameter circuli genitoris dicatur 2a; exhibea
turque per &amp; arcus quoad radium -- = 1 similis arcui A'B' .
Erunt
x = A'B - B'r - A'B ' - AM = a5 - asins ,
y = B'M = asin.v.zza( 1 - сoss) .
ex istarum prima assequimur
dx = ads - acoss ds = a ( 1 - cos )de ;
et dividendo per secundam.
dc
de .
y
Est autem arc sin=
are(sin = AMM))—are (sin V Zay —ya )
a
IV2ay - y2
et consequenter de 2
2ay - y
aa
dyZay - y ?
dy
V2ay - y2 ; ergo
2a
dy = dx V?
(at ) ;
y
149
iam ex cycloidis puncto v. g. A' perpendicula A'r(:y) ⋅
et A'C, alterum in basim A&quot;E. alterum in axem .AB;
sit A&quot;r:æ; diameter circuli genitoris dicatur 20; exhibea-
turque per &amp; arcus quoad radium −∶∙−↿ similis arcui A'B'.
Erunt
æ:A&quot;B'——B'r—-A&quot;B'—-A'M:ae-—asins ,
y:B'M:asin.v.:a(1—coss) ∙
ex istarum prima assequimur
dæ:ada—acoss de:-au -cose)d£ ;
et dividendo per secundam.
d
-—æ-- :de ∙ 7,
Est autem :arc(sin: M):arc (sin ∙−−∶ ∣∕⊋∅∫−↗↾≖
a a ),
et consequenter de:
.a— ∙ a
): ∣∕ ↿−− 5351-
02
dl/Zay—yz df
a—y VZaJ—yi 'ergo150
aequatio differentialis ad cycloidem , computatis coordina
tis a baseos initio A &quot;, Quod si computentur a vertice
A , ut novae coordinatae sint AC ( = x ') , et A'C ( =y' ) ,
cum habeamus x = an — y , y = 2a — x', prodibit
-dx'adyV
x'
2a- , seu dy = dxV
2a - x
xช่ (a &quot;) .
Nunc ad gravium descensum quod pertinet per ar.
cum quemvis cycloidalem , cujus vertex in puncto inſimo
B ( Fig. 37), sit C initialis positio puncli ponderosi , quum
nempe t =0 et v = v = 0 , M positio in fine temporis 1;
quibus positionibus respondeant altitudines c et ac' supra
horizontalem rectam transeuntem per B , ut in Mha
beatur v = V 2g(c-x') : denotantibus h , se s' cycloidales
arcus CB , CM , BMBM ,, erit erit dsds== dhd (h -- ss'')) = - ds'; unde''
ds
di
ds'
dt = V2g(c -x '), ex qua obtinelur
ds'
dt
V 28(c — x ')
Formula ( a&quot; ) praebet ( 27. 19.0)
2a -x
do = Vdx =+ody&quot;a= dx V17 = dx '

;

x
hinc
da - c
a
dt dx'
V. 8 V cx' — x'2 -Va
GVFECITATE
150
aequatio diti'erentialis ad cycloidem . computatis coordina-
tis a baseos initio A&quot; Quod si computentur a vertice
A , ut novae coOrdinatae siut AC (:æ') , et A'C (:y'),
cum habeamus x:an'—-7, 1:20—æ', prodibit
I
Za—æ ∙−−−
, seu df:dx I/ æ, (a') .
Nunc ad gravium descensum quod pertinet per ar-
cum quemvis cycloidalem . cujus vertex in puncto infimo
B (Fig. 37), sit C initialis positio puncti ponderosi, quum
nempe t:o et ⇂↾−−∙∶⇂↗∘∶∶∘ , M positio in fine temporis :;
quibus positionibus respondeant altitudines c et æ'supra
horizontalem rectam transeuntem per B , ut in M ba-
beatur :»:V Zg(c-x ':) denotantibus ,: .
s, s' cycloidales
arcus CB, CM, BM , erit ds:d(h—s'):—ds'; uude
ds di' .
v −− dt— dt —l/2g(c-æ'), ex qua obtmetur
ds'
dc ∶−∙−− − ⇂∕−−−−−
⇄∊≺∁−−⋅↿⊏⋅ )
Formula (a') praebet (27. 19!)
d.;— −−∙ ⇂∕∎∎−−∎∎−∎∎∎ ↙↙∙≖⇌ ≖⊣⊸↙↿∫−
hinc
(1&quot;.—— a flx' ;.
(ll:
∙−∙ V
∙−− ∙−− V
a .; 20 .
l
—— g J/Fæ—æ'z ∙ g
∣∕↿ ∙↕⋅∎⋅≩∁≻≖⊽151
sumptisque integralibus ( 27. 9,9 ) ,
= c +Vare (co== **)

;

in positione initiali est t=0, simulque x' =c; igitur C = o, et
Vore (rosa ).
Facta x=0, prodibit tempus descensus usque ad punctum
infimum B, nimirum
11 =T
VO
ubi cum non inveniatur c , patet , ex quocumque cycloi
dis puncto demittatur grave, eodem semper tempore per
venturum ad B. Hanc cycloidis proprietatem posteaquam
detexit Hugenius , cycloidem ad pendolum adhibere cae
pit : quod qua ratione fieri possit , ostendit in parte 3. “
Horologii oscillatorii.

=== De attractione corporum in hypothesi attractionis agentis in ratione directa massarum, et in reciproca duplicata distantiarum.===
70. Pyramis AH (Fig. 41) habens basim GH infinitesimam secetur superficie sphaerica, cujus centrum in A , et radius AZ ( = r ); sit Ky = B ) projectio intersectionis VZ ( = ) in plano AB; supra basim Ky erigatur prisma KyE altitudinis CH ( =x): exprimet KE AZ sumptisque integralibus (27. 93) ,
↥⋅∶∁⊹ l
V— a1c (eo: 200) ;
in positione initiali est t:o. simulque æ':c; igitur (l:-o, et

* x −−∶−∁

Facta x':o. prodibit tempus descensus usque ad punctum
infimum B, nimirum
a
II:&quot; V— :
g .
ubi cum non inveniatur c , patet, ex quocumque cycloidis puncto demittatur grave, eodem semper tempore perventurum ad B. Hanc cycloidis prOprietatem posteaquam detexit Hugenius, cycloidem ad pendulum adhibete caepit: quod qua ratione fieri possit, ostendit in parte 3.'
Horologii oscillatorii. ⋅

lit-F.
AZZ152
vim attractivam ( p) segmenti CG in punctum A juxta directionem perpendicularem plano AB. Intelligatur enim segmentum CG secari sphaericis
superficiebus numero infinitis , quarum centrum in A , et
r2 , 7* 3 sintque eoz , A2 , A3 intersectionum areae. Erit
radii rs .
din 23

; 2

2 r2 3 p2
vis nempe attractiva cujuscumque areolae Qy , da , .
aequabit vim attractivam areolae A. Ex punctis Z, C,
ducantur in AB ... perpendicula Zy ( =n) , CB ( = n ) ...:
singulis viribus resolutis in duas, quarum altera sit paral
lela , altera perpendicularis plano AB , componentes per
pendiculares repraesentabuntur per
ni na n
2
ri ra
et quia
ni n2 n
72
iccirco
li ni 0.2 п,
=
a n
.
t'i p22 ra
his positis , quisque videt fore
n
f
152 .
vim attractivam (:f) segmenti CG in punctum A juxta
directionem perpendicularem plano AB.
Intelligatur enim segmentum CG secari sphaericis
superficiebus numero infinitis. quarum centrum in A, et
radii r; , r,. rg
.... sintque at,
, at,, ata ...
inter-—
sectionum areae. Erit
«! a; «3 ∙
. a
r,: fa, rna ra.
∙
'
vis nempe attractiva cujuscumque areolae at,, at,, ∙∙∙
aequabit vim attractivam areolae at. Ex punctis Z, C,
...
ducantur in AB
... perpendicula Zf (:n)
, CB (:m) ...:
singulis viribus resolutis in duas, quarum altera sit paralf
lela, altera perpendicularis plano AB , componentes per-
pendiculares repraesentabantur per
a! &quot;[ a:; &quot;a a n
. , ∙ , ∙ ∙ . ∙−−− .—,. r,2 r, rf r, :-a r
et quia
—:
&quot;! &quot;2 n
∙−− ∙ ∙ ∙ :∙
—;
r, r, r
iccirco
«! &quot;r a: n, a n
∙
∙−−∎ ∙ ∙ ∙ ∙:∙∙−∎ ∙ ∙∙∙∙ :
rl: rl ,.22 &quot;, &quot;, r153
Jamvero (55.4. )
\beta
= cosyZA 3
igitur &gt;
n
ela
B
sli
oli
a
et consequenter
Bx
r?
KyE f
AZ

71. Singula corporis cuiuscumque KGDH (Fig. 42) puncta trahant punctum C positione datum. Centro C et
radio quolibet CM describatur sphaera MBN; in eius
superficiem incurrat in A recta quaelibet CG permeans
corpus KGDH iuxta DG ; demittatur ex A perpendi-
culum AQ supra planum MCN; capiatur in- AQ pars
TQ aequalis segmento DG intra corpus KGDH demerso;
quod si plura fuerint huiusmodi segmenta, pars in per-
pendiculo accepta sequetur omnium summae, Si per GM?
dividitur solidum ïTXV, quod continetur plano MCN
et superficie ab omnibus punctis T determinata , expri-
met quotus vim, qua totum corpus KGDH trahit punctum
C perpendiculariter ad planum MCN.
Prodeant enim ex C infinitae numero pyramides, qua-
rum segmenta DG impleant totum corpus KGDH; pote-
runt totidem respondentia (69) prismata TQ concipi , quae
totum solidum ïTXV impleant; ergo etc.
Quoniam vires omnes sollicitantes punctum C
possunt traduci ad ternas
, quarum directiones congruant
cum tribus rectis se mutuo ad angulum rectum secantibus in ipso C; ternae vero istiusmodi vires in unam com-
↿↿154
positae dant resultantem ex illis omnibus , inde fit quod
ubi determinentur (70) ternae vires corporis KGDH re
spective perpendiculares tribus planis orthogonalibus per
punctum C traseuntibus , eae in unam contractae suppedi
tabunt et directionem , et intensitatem illius vis , quae re
sultat ex omnibus viribus punctorum constituentium cor
pus ipsum KGDH.
Si punctum C intra - corpus trahens collocaretur
accipienda esset TQ aequalis differentiae inter distantias
ipsius C ab extremis D et G rectae DG transeuntis per
C. Hinc si C fuerit situm intra ejusmodi crustae cavita
tem , ut per C ducta quavis recta , aequales hinc inde par
les illius rectae intra crustae crassitiem intercipiantur, eva
nescentibus omnibus TQ , evanescet etiam omnis vis pla
no cuicumque perpendicularis , et punctum C in aequi.
librio consister.

72. • * Coordinatarum originem O constitue in
quovis corporis puncto ; sin que x, y, z coordinatae pun
cli altrahentis ; a , b , c coordinatae puncti allracti ; '
distantia inter punctum attrahens et punctum altractum :
expriment
b - r
CZ
ba 몇
7
A'
A
cosinus angulorum , quos a continet cum axibus coor
dinatis OX , OY, OZ. Quare denotantibus Hc , H,, H,
componentes iisdem axibus parallelas , in quas rosolvitur
attrahens totius corporis vis H , et dm elementum massae,
eront
H,
- Som dm , 1 , = Sabah dm
(o)
H , Set dm :
A3
154
positae dant resultantem ex illis omnibus; inde Et quod
ubi determinentur (70) ternae vires corporis KGDH re-
spective perpendiculares tribus planis orthOgonalibus per
punctum C traseuntibus, eae in unam contractae suppedi-
tabunt et directionem, et intensitatem illius vis, quae re-
sultat ex omnibus viribus punctorum constituentium cor-
pus ipsum KGDH. ↴
Si punctum C intra -corpus trahens collocaretur ,
accipienda esset TQ aequalis differentiae inter distantias
ipsius C ab extremis D et G rectae DG transeuntis per
C. Hinc si C fuerit situm intra eiusmodi crustae cavita-
tem
, ut per C ducta quavis recta, aequales hinc inde par-
tes illius rectae intra crustae crassitiem intercipiantur, eva-
nescentibus omnibus TQ, evanescet etiam omnis vis pla-
no cuicumque perpendicularis. et punctum C in aequi-
librio consistet. ⋅
⊽∑∙∘∙ Coordinatarum originem O constitue in
quovis corporis puncto; sintque x, ],:coordinatae pun-
cti attrahentis; a, b ,&quot;c coordinatae puncti attracti; A'
distantia inter punctum. attrahens et punctum attractum:
expriment
⊄≖∙∙−−∙∙−−∙⋮≖ b—gr c—z
∆∣∙
'
∆∣
'
∆∣
cosinus angulorum, quos ∆⋅ continet cum. axibus coor-
dinatis OX , Oï, .OZ. Quare denotantibus H, , H, , H,
componentes iisdem axibus parallelas, in quas rosolvitur
attrahens totius corporis vis H ∙ et dm elementum massae,
erunt ⋅
a—æ &quot;bf—7 ⋅ ∏≖∶∶∫ ∆∣⊰ dm,
⊟⋮⇌−∽∙∣∙−⊒↙∙⇁⋮⇀≀≀≀↿∙ .-
(0)
,:szjä-äfdm:155
integralia se se protendunt ad totam corporis massam M.
Pone
Q = Sam
( o ')
habes quidem
A2 = (a — x )&quot; + ( my) + cz( )&quot; ;
sed quia integrationis limites non pendent ab a , b , c ,
ideo ex prima ( o' ) erues
dQ
da ſ
dm ,
dQ
db
den
ES ,
do dm
dQ
dc -dm ;
da
secunda vero (o' ) praebet /
a 영 1 dA a -X a
A' ? da
b
da 4'3
db A'3엷
slot
dc 4'3
traducentar itaque ( o ) ad
H=
dQ
da H ,
do
db H,
dQ
dc ( o &quot; ) ,
componentesque H , H ,, H, pendebunt ab unico integrali
l. Fiat
a : + 62 + c = A2 ,
155
integi-alia se se protendunt ad totam. corporis massam M.,,
Pone
≺≀⇌⇀⋅ dm −∙−≃
habes quidem (O,)
∆≏⇋−∙≺∅∙−∞≻⋍⊣−≼≀↗−∫≻≔⊣⊣∘∙−≖≻⋅ ;
sed quia integrationis limites non pendent ab a, b , c ,
ideo ex prima (a') erues
⋛≣∎∶∆∼∣↙≟ dm' ∎−⋅∫↲∂↙≀⋯⋅−−−−∫ &quot;'&quot;-('m-
secunda vero (a') praebet
↿ ↿
⊄∄−−−↽
∆∙ ↿ siA—, a—æ (LA-7 b—r
de'&quot; A'da A'3 ↞∙ ∠≀∣⊃−−⋅−−−−∆⋅≀∙ '
↙≀∙∙↿⊽ ac.
.... de −⋅⋅ ∆∣∍
traducentur itaque (a) ad
dQ dQ dQ−⊋⋤−∙ Hic—2? ∙ Ha ≔−⋅⊋⊂∙∙− (O&quot;) 1

componentesque H, , H,, H, pendebunt ab unico integrali
Q. Fiat ⋅
∁≖⊹∂≖−⊦∁∶≖−−∆≖ ∙156
ut secunda ( o ' ) scribi possit in hunc modum
A ' = 12—2(axtby tcz) + wa + ya + z2 ;
erit
1
-
[ 12—2( ax + by + cz) + xa + ya + za = +
+
2(ax + by + cz) xtya taza)
+
243
12(ax + by + cz)2- [ 12 (ax + by + cz )-3(x2 + ya + za) ][ x ? tye + z&quot;)
845
+
.
unde, ob prinam ( o' ) ,
m
Q

* ++ ſ(ax + by+ ca)dın

25 /(x +y +z&quot;)du + z flar+ by +czydom.co&quot;.
Sit coordinatarum origo in centro gravitatis massae at
trahentis; erit ( 20. b )
1
43
Slax ( ax + by + cz) dni = ta fædm +
bſydm + ſzam ] = 0 ;
ideoque vertetur ( o '&quot;') in
156
ut secnnda (o') scribi possit in hunc modum
A&quot;::A3—2(aæ—-l-bj-l—cz)*æï-þyl-l-z' ;
erit
−↙∃≃∙⋅⋅ −−−−−
[∆⋅−≆≺∾↼⊦∂∫⊣⊸≉≻⊣−↕⋅⇀⊦∫⋅−⊦≖≖ ⊐⋅− *
↽−⇌−↿≴↸ ⋍≺∅↕⊹≀↗∫⊣⊸∡≻−≺↕∙⊣⇀∫⋅⊹≖≖≻
2A3
12(aæ-l—b.7—l-Cz)'-[1 ⊋≼↙⇂∙↿∙∙⊹⊘↾⊣⊸∅⊢∃≼∞≖⊹∫≖⊹≂≖≻∃ ∣⋮∙∙∁≴⊣↰↾⊣∎≖∶∣∙
8A5 '
'
-I-..;
unde, ob primam (a')-
. 1
1
Q ∙∙∙ Z
∙∣∙− A3]
(aæ-l-bJ-l-czkim—
1
- 3 &quot;
555] (x'-l'f' ∙⊦≖≖≱ dnl-l— üïf(aæ'l'lïï'*l'cz)'dm-n-(0 ').
Sit coordinatarnm origo in centro gravitatis massae
&quot;'
trahentis; erit ( 20. b)
. ↿ ∆↿−⋮∫≺∘∞⊣−≀≀↗−⊢∞≻↙≀⋯∶ 33— ta xdm-i-
bjïydm ⊣− rfzdm ]: 0

:

ideoque vertetur (o&quot;') in157
M 1
Q
Δ 243f\ x3 + y* +32) dmt
3
245 (ax + by + cz)-dm-, .. ( 0 &quot; &quot;).
ca
73. Corpus KGDH Sit sphaericum , ejusque centrum
in puncto extremo B radii CB (Fig. 43) inveniatur ; ipsi corpori occurrat QA in T. et Q '; ducto perpendiculo BE
supra CA , triangula rectangula AQC, BEC propter latus
AC=CB , et angulum QAC=BCA , erunt aequalia , adeo
que QC=BE ; chordae nimirum SD,CT aequidistabunt a
centro B; erunt itaque inter se aequales , ac proinde
OʻT ( Fig. 43 ) , aequabit QT (Fig. 42): quod cum ubique contingat, erit area KGDH (Fig. 43) sic .aequalis
areae XYC (Fig. 42) , ut solida genita ab his areis cir
suos axes revolutis aequalia sint inter se. Vim proinde , qua punctum C tendit in sphaeram KGTH ( Fig. 43 ) exprimet ipsa sphaera divisa per CM (=CB) seu per
quadratum distantiae puncti C ab ' ipsius sphaerae centro; siquidem aliae duae componentes (71) evanescunt:
Sed si sphaera ita condensaretur , ut coiret in centrum ,
eodem prorsus modo exprimeretur ejus attractiva vis; ergo punctum extra sphaeram situm eadem omnino ratio
ne in ipsam tendit , ac si omnia sphaerae puncta in cen
tro compenetrarentur.
Haec vera sunt , licet corpus non sit omnino ho
mogeneum , modo tamen sint ubique bomogeneae ejus par
tes a centro aequidistantes ; quod notandum etiam in se
quenti assertione.


73. Corpus KGDH Sit Sphaericum
. eiusque centrum
in puncto extremo B radii CB (F ig, 43) inveniatur; ipsi
corpori occur1at QA in T
et Q'; ducto perpendiculo BE
snpra CA , triangula rectangula AQC, BEC propter latus
AC;-:: CB, et angulum QAC:BCA ,
erunt aequalia, adeo-
que QC—BE- , chordae nimirum SD Q' T aequidistabunt a
«centro B; erunt 'itaque inter se aequales , ac proinde
Q'T (Fig. 43) aequabit QT (Fig. 42): quod cum ubi-
qne contingat, erit area KGDH (Fig. 43 ) sic .aequalis
areae XTC (Fig. 42) , ut solida genita ab his areis cir-
ca suos axes revolutis aequaha sint inter se. Vim proin-
de ,qua punctum Ctendit in sphaeram KGTH (F 1g 43)
exprimet ipsa sphaera divisa per ∁∾∙≖ (:CB') seu per
quadratum distantiae puncti. C ab ipsius sphaerae een-
tro ; siquidem aliae duaeïcomponentes (71) evanescunt,:
Sed si sphaera ita, condensantur,, ut coiret in centrum,
eodem prorsus modo exprimeretur eius attractiva vis; er-
go punctum extra sphaeram situm eadem omnino ratio-
ne in ipsam tendit
, ac si omnia sphaerae puncta in cen-
tro compenetrarentur. ⋅ ⋅
Haec vera sunt
,
lieet corpus non sit omnino ho-
mogeneam, modo tamen sint ubique homogeneae eius par-
tes a centro aequidistantes; quod notandum etiam in se-
quenti assertione.

74. Si punctum materiae locetur intra crustam sphaericam, sive intra orbem sphaericum intus cavum terminatum binis superficiebus sphaericis concentricis, id punctum, destructis viribus consistet in aequilibrio. Sint ( Fig. 44) NEQ, MFP superficies illae concentricae , punctum vero materiae sit O. Ducta per 0 quavis chorda MNEF, et ex centro K demisso perpendiculo KC supra ipsam chordam, erunt CM=CF, CN=CE;
igitur MN = EF , ac proinde ( 72 ) etc:

75. Ex dictis ( 73. 74 ) sequitur: 

1º . punctum in superficie duarum sphaerarum positum gravitare in ipsas sphaeras in ratione radiorum directa: nam sphaerae sunt ut radiorum cubi, quibus per eorumdem quadrata divisis, prodeunt radii simplices: 

2° . gravitatem puncti intra globum homogeneum pergentis a superficie ad centrum decrescere in ratione directa distantiae a centro ipso. 

76. Haec notentur. 1º. materiale punctum valde distans a corpore attrahente, utcumque se habeat forma corporis, ea proxime ratione tendit in ipsum corpus, qua tenderet si corporis partes in centro gravitatis compenetrarentur; patet tum ex dictis (12: 20) , tum ex eo quod in casu vires attrahentes punctorum constituentium corpus considerari possint tamquani proxime parallelae et proportionales ipsorum punctorum massis.

* Patet etiam ex ( 0 &quot; . 01 .: 72 ) ; nam si A est ila na

gna , ut, retento primo termino in ( o &quot; ), possint caeteri
praetermitti absque sensibili errore , sicque habeatur
M
Q
exsistent
M C
H
M
A2 ., HH ,
M 3
42 : A
H
ac proinde
M
H
ViH + H ,* + H , +
158
Sint (Fig. 44) NEQ, MF P supetticies illae concen.
tricae, punctum vero materiae sit 0. Ducta per 0 qua-
vis chorda MNEF , et ex centro K demisso perpendicu-
lo KC supra ipsam chordam, erunt CMr—CF, CNzCE;
igitur MN— EF
, ac proinde (72) etc:

75. Ex dictis (73. 74) sequitur: 10. punctum in su-
perficie duarum sphaerarum positum gravitare in ipsas sphae-
'ras in ratione radiorum directa: nam sphaerae sunt ut ra-
diorum cubi, quibus per eorumdem quadrata divisis, pro-
deunt radii simplices: 20. gravitatem puncti intra globum
homogeneum pergentis a supe1ticie ad centrum decrescere in ratione directa distantiae a centro ipso.

76. Haec notentur. 10. materiale punctum valde di-
stans a corpore attrahente, utcumque se habeat forma cor-
'poris, ea proxime ratione tendit in ipsum corpus , qua
tenderet si corporis partes in centro gravitatis comPe-
netrarentur', patet tum ex dictis (12: 20), tum exeo quod
ih casu vires ,attrahentes punctorum constituentium cor-
pus considerari possint tamquam proxime parallelae et pro-
portionales ipsorum punctoruin massis.
' ea Patet etiam ex (a&quot; . o&quot; 72); nam si∆∙ est tta ma-
gna , ut, retento primo terminogin (o'f ), possint. caeteri praetermitti 'absque &quot;sensibili 'et-rore , sicque habeatur
.
'
. - «. ∙
⋅
r ↾
1' I
.
. M. 11&quot;
≺≀⇌⋅⊼−↿⋅ ⋅
exsistent '1-
- --M 0
'M 6
&quot;' M
∣⋅
∏⋍−⋅−− −−∶∙−−⋅∙−− ...—...; .' AaA'H' ArA'H': ∣⋅∙↘∆∙
ac proinde
−−−∙∙−−−−−−∙∙∙∙−−∙ M
H:: l/Hil'i'nya'i&quot; He's-A&quot;?159
2º. Non pluribus opus est , ut stabiliatur illud: u
bi dimensiones corporum quorumcumque se matuo attra
hentium in ratione directa massarum, et reciproca duplicata
distantiarum sint admodum exiguae prae distantiis, quibus
ipsa corpora disjunguntur, eorum alterum tendet in alterum
perinde ac si essent 'ambo in suis gravitatis centris compe
netrata . Dicantur enim M , M' massae duorum ejusmodi
corporum , m, massa cujuslibet puncti spectantis ad M , et
A distantia inter m, ac centrum gravitatis massae M ; ex
Mm ,
primet vim attractionis motricem ( 28) , qua m, len.
dit in M, simulque ( 7 ) vim attractionis motricem, qua M tendit
in ma; ideoque merit vis attractionis acceleratrix, qua
M tendit in mo . Atqui hoc pacto M tenderet in mo, si to
la massa M compenetraretur in suo gravitatis centro ; er
go M revera tendit in mi, id est in singula puncta mas
sae M' , perinde ac si tota M foret in suo gravitatis cen
tro compenetrata: cumque ob paritatem rationis idem con
tingat massae M' quoad M , jam patet veritas assertionis.

3º. Quoad sphaerica corpora, quorum partes aequidistantes a suis centris sans homogeneae, obtinet assertio, utcumque caeteroqui se habeat intercedens distantia.

=== De gravitatione universali ===
[[77|77]]. Quae de coelestium corporum motibus, ex astronomicis observationibus hic subjicimus, ad ipsorum gravitatis centra respiciunt.

1º. Areae, quas circa solem describit radius vector uniuscujusque planetae sunt respondentibus temporibus proportionales: idipsum obtinet quoad areas descriptas a radio vectore uniuscujusque satellitis seu planetae secundarii circa suum planetam primarium.

2º. Convertuntur planetae circa solem in orbitis ellipticis ita, ut singularam ellipsium alterum focum occupet sol: convertuntur planetae secundarii circa suos planetas primarios in orbitis ellipticis ita, ut istarum focum occupet respectivus planeta primarius.

3º. Quadrala temporum periodicorum sunt in diversis planetis ut cubi semiaxium transversorum: idipsum obtinet quoad diversos satellites circa respondentem planetam primarium.

[[78|78]]. Hinc 

1º. planetae urgentur vi acceleratrice &lt;u&gt;tendente in solem&lt;/u&gt;; itidem satellites urgentur vi acceleratrice &lt;u&gt;ad respectivos planetas primarios tendente&lt;/u&gt;: plauetae, nimirum gravitant in solem, satellites vero in planetas, quibus adhaerent.

2º. Unusquisque planetarum (56) urgetur in solem vi gravitatis, quae sequitur rationem reciprocam duplicatam distantiarum ab ipso sole: idem dicendum de unoquoque satellite in ordine ad suum planetam primarium .

3º. Collatis inter se viribus acceleratricibus, quibus diversi planetae urgentur in solem, eae erunt (56) in sola ratione reciproca duplicata distantiarum a sole ipso; praecisa igitur projectionis vi, si diversi planetae in aequalibus a sole distantiis constituerentur, aequali tempore in eum descenderent. Idem obtinet in satellitibus quoad respectivos planetas primarios. 

[[79|79]]. Planetae secundarii una cum primariis, quibus adhaerent, in solem urgentur eadem gravitatis lege. Nam corpus omne, quod circa corpas alterum utcumque motum describit areas temporibus proportionales, urgelur duplici vi, altera tendente ad corpus illud utcumque motum, altera utriusque communi (5:46): cum igitur planetae primarii gravitent in solem, cumque planetae secundarii circa suos primarios describant areas temporibus proportionales; propterea etc.

[[80|80]]. Gravitant in se mutuo corpora omnia, ex quibus coalescit planeticum systema. Planetae siquidem omnes cum primarii tum secundarii vi gravitatis urgentur in solem; ergo sol in planetas omnes vi ejusdem gravitatis (7) urgetur: atque hoc argumento ostendes terram gravitare in lunam (id confirmant phoenomena marini aestus) caeterosque planetas primarios in suos satellites. Quod autem planeta quilibet in alium quemvis gravitet, satis e sola comprobaretur analogia, etiamsi nulli essent effectus, ex quibus haec gravitatio immediate detegi posset. Sed ejusmodi effectus non desunt: perturbationes videlicet, quae in recensitis motibus (77) observantur, quaeque per mutuam coelestium corporum gravitatem optime determinantur (62*60). Sic cum lunae motum ad regularis calculi normam ex observationibus exigere se posse Astronomi desperarent, tandem postquam ejusdem perturbationes ex mutua corporum coelestium gravitatione investigare coeperunt, tabulas lunariam motuum potuerunt conficere, quarum tantus est cum coelo consensus, quantum sperare ex observationibus nemo potest.

[[81|81]]. Praecisis perturbationum causis , urgebitar luna in tellurem vi acceleratrice (56):&lt;math display=&quot;block&quot;&gt;
\varphi=\frac{4\pi^2 a'^3}{T^2}\frac{1}{\Delta^2};&lt;/math&gt;


denotat &lt;math&gt;T&lt;/math&gt; tempus periodicum = dieb. 27 , 322 = minut. secund. 60&lt;sup&gt;2&lt;/sup&gt;. 24. 27 , 322; &lt;math&gt;a'&lt;/math&gt; semiaxem transversum orbitae lunaris, &lt;math&gt;\Delta&lt;/math&gt; radium vectorem ipsius orbitae. Iamvero mediocris radius terrestris = 16931100&lt;ref&gt;Error in originale&lt;/ref&gt; ped., mediocris parallaxis lunaris 57' + 11&quot;, unde &lt;math&gt;a' =\frac{16931100}{\sin(57' + 11'')}&lt;/math&gt;facto  igitur &lt;math&gt;\Delta = 19631100&lt;/math&gt;, gravitatis vis qua luna urgetur in terram evadet in ipsius terrae superficie
&lt;math display=&quot;block&quot;&gt;blah blah blah&lt;/math&gt;qui valor cum sit proxime 30,2 ped., inferimus gravitatem qua luna urgetur in terram nihil esse aliud nisi gravitatem ipsam terrestrem imminutam in ratione reciproca duplicata lunaris distantiae a terrae centro.

[[82|82]]. Vis gravitatis, qua lapis v . gr. urgetur in terram, est (80) ejusdem speciei cum illa gravitatis vi, qua corpora mundani systematis in se mutuo tendunt; ergo idem in utraque erit agendi modus. Atqui vis qua totus lapis urgetur in terram resultat ex viribus, quibus singulae lapidis particulae in eamdem nituntur; igitur et vires, quibus corpora mundani systematis gravitant in se mutuo, resultant ex viribus, per quas singulae ipsorum, particulae se mutuo petunt. His positis, stabilietur illud: gravitas ita materiam afficit, ut singulae ejus particulae in alias omnes et singulas gravitent in ratione directa massarum, ad quas tenditur, et reciproca duplicata distantiarum alterius ab altera. Recole quae diximus (76.2º.3º); etenim coelestia corpora et habent dimensiones admodum exiguas prae mutuis distantiis, et induunt formam prope sphaericam.

[[83|83]]. Bonum erit nonulla hic annotare. 

1º. designantibus &lt;math&gt;M&lt;/math&gt; et &lt;math&gt;m&lt;/math&gt; solarem et planeticam massam, ex dictis (56.k, 62.c) eruitur&lt;math display=&quot;block&quot;&gt;
M + m =\frac{4 \pi^2 a^3}{T^2}
&lt;/math&gt;ratio igitur inter cubum semiaxis transversi et quadratum temporis periodici, utpote pendens a massa planetica, nequit esse accurate constans quoad diversas planetarum massas. Atqui tamen ex astronomicis observationibus infertur rationem illam, sin minus accurate, certe esse quamproxime constantem: concludendum itaque planetarum massas admodum exiguas esse, ubi comparentur cum massa solis.
2.º Eodem modo ostenditur, si lunam excipis, satellitum massas fore et ipsas valde parvas prae massis planetarum, quibus adhaerent.

3.° Quae quantitates sunt designatae per &lt;math&gt;m, 
a , T&lt;/math&gt; quoad planetam , eae designentur per &lt;math&gt;m' , a' , T'&lt;/math&gt; quoad satellitem; erit&lt;math display=&quot;block&quot;&gt;
m + m'=\frac{4 \pi^2 a'^3}{T'^2}&lt;/math&gt;Hinc (1°)&lt;math display=&quot;block&quot;&gt;
\frac{m + m'}{M + m}=\frac{T^2}{T'^2}\frac{a'^3}{a^3}
&lt;/math&gt;praetermissa ( 19. 20. ) &lt;math&gt;m'&lt;/math&gt; in numeratore primi membri, itemque m in denominatore , et facta M = 1 , prodibit.
T2
TO
a's
i
quae formula suppeditat rationem inter solarem massam
habitam pro unitate , et massas planetarum ( tellurem ex
cipe) , qui satellitibus stipantür.


4.° Quod spectat ad tellurem consideratam in
star sphaerae habentis radium R , et massam m , sit &amp; gravi .
tas prope ejus superficiem , erit (73) 8 =R. , ideoque (10. )
M +m
4 712 a3
&amp; R2T
et praetermissa ( 19. ) m in numeratore primi membri , factaque solari massa M = 1 , emerget
163
2! Eodem modo ostenditur ,
si lunam excipis,
satellitum massas fore et ipsas valde parvas prae massis planetarum, quibus adhaerent .
-l ⋅ 

:

41:2003 -
∙∙ ↶↿ m-4—m' £S.-&quot;IW.,
Hï'n'c (10)
⇀
..
.m -- in'. T., a'-3-- - M,-—-.nsl ≔−⋅⋅∙−∙− ∙ ∙∙∙ : T'a ::
&quot;
praetermissa (.10 20. ) tu' in numeratore primi membri ,
itemque ut in denominatore ,
et facta M— −∙− ↿, prodibit.
⋅ 1 ↴− ⋅↧⇁≖ -a ∙
−
⋯∙↽↽⊽ .
..;3, .'
∙⊾⊺⋅⋅
quae? .fottmule- suppeditat rationem -
intcr. solarem^ massam
habitam pro unitate ,
et massas planetarum (tellurem ex-
cipe) , qui satellitibus .stipantur.
'i-o Quæ-Spectat ad tellurem' consideratam in-
star sphaerae habentis radium R , et massam 111, sit 3gravi-
tas prope eius superficiem ,
erit (73) gr ≖⋅⇁⋅∙−⋮↾−⋮↾− , ideoque (10.)
IUI—tm,— 4 11:303.
,
gRQTi'
et praetermissa (10.) »: in numeratore primi membri, f.-
ctaque solari massa M ∙−⇁−−−∙ ↿, emerget ⋅164
&amp; R2T2
4 Ti ? a3
PE

5° Media telluris densitas ( = M) determinari potest ex penduli aberratione. Sit CB (Fig. 45) pendulum; a longitudo rectae CB, quae nec distendi possit nec inflecti; S centrum massae sphaericae ( = m' ) ad se trahentis punctum ponderosum B , r radius , M densitas; b recta CS; CD posilio penduli digressi a recta verticali CB;
&amp; angulus BCD; h angulus BCS; k recta SD: centro insuper C et radio CB intelligatur describi circularis arcus
BD , et per D duci tangens nn' . In D sollicitatur materiale
m'
punctum viribus acceleratricibus g et altera juxta ver ka
ticalem DD' , altera juxta rectam DS ; anguli
SU
1
D'Dn = CnD = 90 ° — E , SDn ' = # (CDS - 90° ) ,
to
et consequenter
b sin (h — 5)
cosD'Dn sins , cosSDn ' = sinCDS =
k
Vires igitur motrices respondentes praefatis viribus ac
celeratricibus sese librant in D quotiescumque fuerit
que
bas
bit
ha m'
23 gsine b sin ( h - €) .
sed
Pone longitudinem penduli ita exiguam prae distantia CS ,
ut absque sensibili errore sumi possit k = b ; traduce
tur aequilibrii conditio ad
gb2 sin é = m ' sin (h - ) ;
m
et substitutis ( 4º . ) valoribus 8 T RH, m
R? 3
4
90
paris 75 1'3 je , prodibit
164
g R*T3
m— 4 723 (13
50 Media telluris densitas (:: p.) determinari po-
test ex penduli aberratione . Sit CB (Fig. 45) pendulum longitudo rectae CB, quae nec disteudi possit nec inflecti; S centrum massae sphaericae (: m') ad se trahentis
punctum ponderosum B ,
r' radius
, pf densitas ; 6 recta
CS; CD positio penduli digressi a recta verticali CB;
a angulus BCD :
]: angulus BCS: k recta SD: centro in-
super C et radio CB intelligatur describi circularis arcus
BD , et per D duci tangens nn' . In D sollicitatur materiale
punctum viribus acceleratricibus 3 et ?; , altera juxta ver-
ticalem DD', altera juxta rectam DS : anguli
ix)-D'.. ∙∁∥↧⊃ :. 90o −−∙ e
, sna':∶↿≐ (CDS — 900)
.
et ⋅ consequenter
bsinUt—s) ——k .
Vires igitur motrices respondentes praefatis viribus ac-
celeratricibus sese librant in D quotiescumque fuerit
! r ' ,
!'
cosD'Dn :: sins, cosSDn' ∶∙ sinCDS −−−−⋅−
gsins:
£;- b sin( h — £) .
liane. longitudinem penduli ita exiguam prae distantia CS ,
ut absque sensibili errore sumi possit 1::6; traduce-
tur aequilibt'ii conditio ad
gbï sin ; −−∶ m' sin-(h — a) ;
et substitutis (40.) valoribus g: €; ∶∶ £- 11 R p., m' −∙−∙−−
4
,, ,
'
∙⋅ ; &quot; . .
⋅
∙⋅⋮↿∏⋅ p. , prod1h1t
!
'
I
151
sit
lla165
1
b- Rue sin
{
= 13
M
' sio
(
h
-
E)

unde
i
p3
y sin
h rº
( sinh
– coshtang :)
1
lang
E
=
Ruba tospicosh ji Rba lang

Permanentibus
r
' et
'
, valores
b
=
r
' et
h
= 90
° manife
ste suppeditant maximam penduli aberrationem
&amp;
, ut quoad
istiusmodi aberrationem sint
Se
re tang R pe
3/3
Rtang
s
Densitas fl
, prout colligitur ex aberratione penduli
, cen
setur quater vel quinquies major quam densitas aquae. 6. ** Eadem u determinatur etiam experimentis in
stitutis in libra torsionis
. Sit
( Fig
. 9.
) HH
(
=2a
'
) posi
tio vectis horizontaliter librati

;

E punctum medium, in quo vectis appenditur filo metallico verticali HA circulus horizontalis centro
E et radio EH
=
a
')

; SS

( 26
)
recta similiter horizontalis
, transiens per
E
, ibique secta
bifariam

; sint

S et
S centra sphaerarum inter se aequa
liam et quoad volumen
, et quoad massam
(
=
m
)
, ad se
trahentium massulas sphaericas
m
' et
m
&quot; inter se pariter
aequales
, quarum centra in
H et
H
'
. Movebitur vectis
circa punctum
E immotum
, motusque iste repetendus ab
attrahente sphaerarum vi juxta circuli tangentem
, cui vi
jugiter adversatur vis lorsionis ex filo metallico verticali

et quoniam corpuscula
m
' et
m
&quot; eodem prorsus donantur
motu circa
E
, satis erit alterum dumtaxat
v
. gr
.
m
' con
siderare
. Dicatur itaque
h datus angulus HES

;

&amp; angulus
,
quem in fine temporis
i continet vectis cum initiali po
sitione EH

;

k distantia inter
S et
m
' in five ipsius
t

: sol

licitabitur
m
' juxla circuli tangentem vi attractiva
!
)- a
165
63 Bpain :::/3 (fimul—15);
unde
tan E' ∙∙∙⋅ r'3 pf sin 11 p. '
r'3 (sinit −⋅ coshtang &amp;)
∙
g −⇀ nubi-l- r'3 picas/1
.pf ⇀−− lib2 teng
&amp; .
Permanentibus r' et p!
, valores
b

: r' et 11
: 900 manife-

ste suppeditant maximam penduli aberrationem

:

, ut quoad
istiusmodi aberratiouem sint
tangi—£&quot;.! P',—.
r −⇁∙
p p. Btang
&amp; ,
Densitas p., proutcolligitur ex aberratione penduli.
, cen-
setur quater vel quinquies maior quam densitas aquae.

6?- Eadem p. determinatur etiam experimentis in-
stitutis in libra torsionis
. Sit (Fig.'9*.) HH' (:Za') posi-
tio vectis horizontaliter librati;
E punctum medium
,
in
quo vectis appenditur (ilo, metallico verticali; HA circu-
lus horizontalis centro
E et radio EH (:a'); SS' (:26)
recta similiter horizontalis
,
transiens per
E
,
ibique secta
bifariam

; sint

S et S' centra sphaerarum inter se aequa-
lium et quoad volumen
,
et quoad massam (:m)
, ad se
trahentium massulas sphaericas m' et 111&quot; inter se pariter
aequales
, quarum centra in
H et H' ., Movebitur vectis
circa punctum
E immotum
, motusque iste repetendus ab
attraheute sphaerarum vi juxta circuli tangentem
, cui vi
jugiter adversatur vis torsionis ex filo metallico verticali

:

et quoniam corpuscula tu' et m&quot; eodem prorsus douantur
motu circa
E ,,satis erit alterum dumtaxat v. gr. m' con-
siderare
. Dicatur itaque ]: datus angulus HES

;

&amp; angulus
,
quem in fine temporis

: continet vectis cum initiali po-

sitione EH;
k distantia inter
5 et m' in line ipsius
t

: sol-

licitabitur m' iutta circuli tangentem vi attractiva-166
b
hak sin (h — e), eritque kº = a's - 2a'b cos (h --- e) + 6+;
experimenta insuper praebent vim torsionis proportionalem
angulo e , et consequenter expressam per ce : quoniam igi
tur labente e describit m' arcum ás , iccirco ( 50.3º. )
áre mb
sin (h -€)
dta k3
3
1
&lt;u&gt;aequatio ad motum&lt;/u&gt; corpusculi m' . Ob angulum &amp; valde exi
gaum , sin (h-5~s)) = sin h - e cos h , k - = [ a's -2a'b ( cosh
+sinh) + ] := k +. 2a'besio h ]
R3
3a'b esinh
+
ubi denotat k , valorem k respondentem initio k.
molus , quum nempe E = 0 ; proinde
sin (h-E ) sinh
23
E COS h 3abe sin’h
+
k.
k.
sinh
k.
k .
£ k cosh 3a'be sinh sinh
+
kb.
K.
her )
[ (a's tabo) cos h
k . k5.
sinh 2a'b cosah — 3a' b sin : h]
[laat69)cosh
- 2db -a'b sinä h] :
et factis compendii causa
mo
[ la'a + b ) cosh — 2a'b – a'b sinºh] +c = g' ,
166
m 1)
.
'
.
F.
∙∣⋮∙ nuUi—s) ,entque ka: a'a— 2a'b cos (71—5)-1—63;
experimenta insuper praebent vim torsionis proportionalem
angulo :,
et consequenter expressam per et: quoniam igi-
tur labente :describit m' arcum a's, iccirco (50.?)0.)
,d'38 mb ∙
aa;-a.- Fama—Q—ct
aequatio ad motum corpusculi m'
.Ob angulum :valde exi-
guum ,
sin (It—s) :sin h—s cos 11, k'3: [a&quot; −∙∙ Za'b (cos):
3 . 3
-i-ssinh) ⊣−∂∙∎∙∣− 'a': [le, −∙∙ Za'besin H's—:i?
⊣⋅− a.:-s;&quot;: h, ubi denotat k, valorem k respondentem initio
motus
, quum nempe s.: o ; proinde
sin (I;—s) sin 11 :
cbs Il 30'68 sin'h sin ls
∙−−−⋅∙≖∙−−∙− ∙∙∙⋅ ks' &quot;&quot;&quot;&quot; k3, −∎⋮∣⋮∍∙ fl&quot;, P, P,
sit' eos]: l Bez-'besin'h sinh : .

*

5,
w
cos-1. −−⋅ adb sin-t.] :
822&quot; −⋅⊼⋮−⊏≺∘⋅⋅−⊦∂⋅↗ coslt
∙∙∙⋅ ⊋∘∣∂∙∙− a'b sin' &amp;] :.
et factis compendii causa
iii- [(.-a ⊣⇁ 61) cos h—w—
a'6 .i..- h]
∙⊦≖⋅∸ −−∶
z'-167
mô sinh
wg'
23.
aequatio ad motum corpusculi m ' vertetur in
do e
a' ó (0) -- s ) ;
de²
ex cujus integratione ( 27. 28º. )
(9)*va
' @
'ri
{ = w + Ce + Ce
Sunt autem ( 27.300. )
.
va
cos (9 )* + v = on e(2)
, - ) vi
cose ( ) -va sine ( 2)
propterea
szaf1CTC&quot;) cos( ) +(c —c )V= sine ( :
sumptisque C +C' =C.cos C,, C — C = CV -ī sincs,
= - + 6, co [4 ( 4 ) + c ]
Minima vectis declinatio , í = u - C , ab aequilibrii positio
167
mb sinh
−−∣−∣⋮−≣∶−−≂−↩∾⊰
aequatio admotum c0rpusculi m' vertetur-in
,d'l
(: -d—t;:::g (6)—S):
ex cuius integratione (27 . 280.)
⋅⋅ .

*

.
.'.t ⋅↴∶
≖−↽−−−⇀↠∾−⊦∁∊ :(?) [l:—[— 08 &quot;(ä-') V
..
Sunt autem (27 .300.)
.;.
propterea &quot;
∙⋐⋅∶∶⊙−⊦≼∁−⊢∁⋅≱ ∾≘↙≺−⋚⊑⋅≻⋚⋅⋅−⊢ ((i—C' )
(V:; sint (?);;
sumptisque ∁−⊢ ∁∙∶∁≖∞∙ c, , ∁−∁∽−↽⇌∁≖⇂∕∙⊺∘⋮∥∁≖∙
⋅⋅ ∸ g' ?
≘−∙−−∙∾−⊦∁∙∞∘∣↣⋮≀ ≼⋮⇉⋟ −⊦∁≖∃∙
Minima-vectis declinatio ,
(:o)—G. ab aequilibrii pocitio-168
ue H'H respondet valori ( ) + Cs = ( 2n – 13 ;ma
= + c = 212nt : determinatis
itaque per observationem i'et s&quot; , eruetur inde
te&quot;
et ducta 00 ita , ut sit angulus HEO = w , perget vectis
moveri instar pendali horizontalis circum EO , impendet
que tempus tz = &quot; - t ad integram conficiendam oscilla
tionem , nimirum
ty=T
VAg'
Sit nunc a longitudo penduli simplicis ( 66) , quod intra
idem tempus t absolvit oscillationes suas : cum habeamus
ty = TO Van
a
erit 8 et denotante si densitatem sphaerae m ,

a'
r radium , substitutisque valoribus ( 5.0 )
4
пRр .8 3
mbsinh
wk3
471p3 M'bsinh
3wk. 3
proveuica
Ruwk3 р
p3M'bsinha
unde ar3bsinh
a'Rwk.3 1
Densitas pe sic determinata censetur esse ad
tatem ut 5,48 : 1 .
aquae densi
168
ne H'H respondet valori t'(g——,)ä -I-C,:(2n-1)1r; ma-
xima ⋮∣↾∶∶∾⊹∁≖ valori :
.(gwik) ⊹∁≖:2mr: determinatis
itaque per observationem eet :&quot; , eruetur inde
−−≘⋮−⊢⋮⋅∣∙ −− ⇄ ∙
et ducta O'O ita
,
ut sit angulus HEOzzæ, perget. vectis
moveri instar penduli horizontalis circum EO, impendet-
que tempus :::-:i &quot;—t' ad integram conGciendam oscilla-
tionem , nimirum
a'&quot; Vf- ∙

5
Sit nunc a longitudo penduli simplicis (66), quod intra
idem tempus :, absolvit oscillationes suas: cum habeamus
(3:11 V? .
. g a '
∙
erit?-;? ; et denotante p. denutatem sphaerae m ,

: - radium

. substitutisque valoribus (59)
∙∙∙⋅ ∙≤∙ nR ,— mbsinh— 4nr39'bsinh ∙ ∙
o ∙−−− 3 p. ,g 01:03 30 1:03 ,
provenit...
Rpali-03 a p. arabsinh
r3p'bsinh—c7 '
unde ∙∥∙∽ a'Rmkoï'l '
Densitas p. sic determinata censetur esse ad aquae densi-
tatem ut 5,48:1.169

7. ** Ex mariui aestus phoenomeno deduci pol
est ratio inter massam lunarem m &quot; et terrestrem m. Sit
m' ( Fig. 35 ) quodvis terrae punctum ; lunares vires distrah
entes punctum mé juxta mm&quot; et Am exprimuntur ( 62) per
2m &quot; Dcosh m&quot;Dsinh
(0) ,
( 0' ) :. x'3
X :' 3
quod in ordine ad lunam est h , x&quot; , in ordine ad so
lem sit H , X &quot; ; prodibunt consimiles vires solares
2MDcosH MDsind
X &quot; 3
( a ) , ( a ').
X'3
In casu angulorum h et H aequalium habemus
( 0) m&quot;X &quot;3
(a)
-- ( 0 )
M.2'3 ( a' )
caeteris vero paribus , ratio inter lunares et solares vi
res est eadem ac ratio inter respondentes aquarum ela
tiones ; denotante igitur p hanc secundam rationem , erit
X3
M
р
m '
M
m'
unde m
= P
3
x &quot;' 3
X &quot; 3
Observationes praebent p = 2 , 35333 : vide mechan, coel.
vol , 5. pag . 206.
Aliquid notatur de motu punctorum materialium
utcumque inter se connexorum .

84.* Vires motrices P, P , P&quot; , ... sollicitantes
istiusmodi punctorum massas m ,
mi ,
m' 0 resol
12
169
7.&quot; Ex marini aestus phaenomeno deduci pot-
est 'ratio inter massam lunarem m&quot; et terrestrem m. Sit
m' (Fig. 35) quodvis terrae punctum .; lunares vires distrah-
entes punctum m juxta mm&quot; et Am' exprimuntur (62) per
2m&quot;Dcosh m&quot;Dsinh (0) ∙ ∙−−∙↕∙−∙⋅∃−−⋅ (O,)
xara ⋅
quod in ordine ad lunam est h , a:&quot; ,
in ordine ad so-
lem sit H , X&quot;; prodibunt consimiles vires solares
2MDcosH' ⋅ MDsinH
W (a), ∙∎∎ Xl/3 (a').
In casu angulorum I; et H aequalium habemus
(o) -m&quot;X&quot;3l-(o') ∙
(a) Mx&quot;3 X(a') ⋅
caeteris vero paribus
,
ratio inter lunares et solares vi-
res est eadem ac ratio inter respondentes aquarum ela-
tiones ; denotante igitur p hanc secundam rationem
, erit
X&quot;3 d m&quot; M a:&quot;3
∙ &quot;&quot;;- un B −−∶ ∙ ,. x' 3
' m ? m X'3 31 ≊⋅∙ p:
Observationes praebent p——:2 , 35333 : avide machen. coel.
vol. 5. pag. 206.
Aliquid notatur de motu punctorum materialium
utcumque inter se conus-xarum.

843: Vires motrices P, P&quot;, P&quot;, ... sollicitantes
istiusmodi punctorum massas m , m'
, m&quot; , ...
resol-
12170
vantur singulae in ternas coordinatis axibus OX , OY ,
OZ ( Fig. 8 ) parallelas ; designentur per X , Y , Z, X',
Y , Z ' , X &quot; , . . componentes inde ortae ; sintque x, y,
3 , x ', y ', z' , x &quot; • punctorum coordinatae responden
tes temporit , ut ( 50, 1.º ) per
x == f (1 ), y = f(t), 2 = F (t), ' = fi (t), y = fz(t ), == F ,(t)
x ' = fale) ,y&quot; =f(e), z&quot; = F.(6),7: &quot; = f5e) , ... ) co)
exhibeantur aequationes ad actuales molus ; ad eos nem
pe motus , quos reapse concipiunt massae m , m' , m &quot; ,
ob actiones virium P , P' , P &quot;, ... Quoniam materialia
puncta , etsi mutuis nexibus liberala , viribusque ( 50. 4.0 )
dºx dz m
dạy
de² m
d²x
dla m . dia dla dea
d2z '
dca dc ?
sollicitala , adhuc tamen conciperent motus ( 0 ) ; ideo ,
attentis nexibus , consistent in aequilibrio vires
dez X - m
d2x
de2 Ymdạy
di?
2m X'
der'
di2 &gt;
7
dt2
Y - m
d²ý
dt2
daz'
Z ' - m '?
dt²
X &quot; -m.dºx
&quot;:
.
dt2
Conditiones ( a &quot; 13. 8. ) includuntur in conditionibus
aequilibrii quoad liberum punctorum utcumque connexo
um systema : liquet enim varians systema, semel libra
tum , adhuc permansurum in aequilibrio , etsi ejus pun
cla rigidis lineis immutabiliter connectuntur. Propterea
170
vantur singulae in ternas coordinatis axibus OX, Oï
,
OZ ( Fig. 8 ) parallelas; designentur per X, ? ,- Z, X',
1&quot;, Z' , X&quot; , . . . componentes inde ortae; sintque x,],

: , x', y', z', æ&quot; , ... punctorum coordinatae responden-

tes tempori t
, ut (50. 19) per
x:f(t), 7:112) ,z:F(t),.r ':f,(t),y ':f (t), z':F,(t),
(0)
x&quot;:-f.(t)
.
y&quot;:f.(t)
.
(&quot;zl-&quot;.m. m&quot;':--f3(t)
.
∙ ∙ ∙
exbibeantur aequationes ad actuales motus; ad eos nem-
pe motus
, quos reapse concipiunt massae m , m', m&quot;
,
ob actiones virium P, P', P&quot;, . . . Quoniam materialia
puncta , etsi mutuis nexibus liberata, viribusque (50. 49)
da.. mdzy de.
⋯∽∣↙≀≄∙↿∶ ∙↙∄≖∫⋅
&quot;B'—'— m-——- m— d,.
' md:2 '. d?
'md? ' dta '
,dzz, ad:-I?&quot; m d£2 '.. m dt:- '
∙ ∙ ∙
sollicitata, adhuc tamen conciperent motus (a); ideo ,
attentis nexibus, consistent in aequilibrio vires
(P:: (P] daz ,dzæ'
X
⇁∎−−∙ —p ⋅⋅⇁ ∙∙∙∙ ∙−−− '&quot;sz '
?
&quot;'d'T'z' Z &quot;&quot;da: X &quot;'de '
≀∠⋮⊺ &quot;rad :
&quot;
rnnndaæ ï'-——-—m '
...
dtï' ,..—z ⋯∠↙⊤ 'X— dtz
'
Conditiones (a&quot;f' 13. 8.0) includuntur in conditionibus
aequilibrii quoad liberum punctorum utcumque connexa-
rum systema :liquet enim varians systema, semel libra-
tnm, adhuc permansurum in aequilibrio, etsi eius pun-
cta rigidis lineis immutabiliter connectuntur. Propterea171
(xam )=o, 3(1— )= 0, $ (2- -o,
=[+ (rad ) – (xrm ) ] = o,

* [&gt; (2 mm ) -- ( - ) ]

&lt;math&gt; [ - (-) - = ( x ) ] &lt;/math&gt;
0
seu
daz ΣΖ - Ση
de² EX = sme , Y =sme &gt; ( 0 )
$( wYyX) =Em ( -e
)
3(y2=-1)= sm (voeding - :)
Eml
1
Z
day
de2 (0')
(2X == Z ) = 2m Z
dax
dta -
daz
dea :
formulae (o ') spectant ad translativum punctorum motum,
prima juxta OX , secunda juxta OY , tertia juxta OZ ;
formulae ( o“) ad rotatilem punctorum motum , prima cir
ca OZ , secunda circa OX , tertia circa OY ; eaedem ve
ro (o &quot; ) simul , ad punctorum motum circa fixam coor
dinatarum originem .
Haec facile nunc stabiliuntur. 1 . ** Habemus (20. 6.)
seu
æ ,dþ- ∙−−
∠∄≖⋍ ,
day d3æ ?
−∙∙ − ..(æy—yX) Em (
«: Tt&quot; ]—dt3 )
.
- ∑ dan: daz (zX—æZ): Zm( :217; — æ
—) :
formulae (c')/spectant ad translativum punctOrum motum,
prima juxta OX , secnnda juxta Oï, tertia juxta OZ;
formulae (a&quot;) ad rotatilem punctorum motum, prima' cir-
ca OZ , secunda eirca OX , tertia circa Oï ; eaedem ve-
ro (o&quot;) simul , ad pnnctorum motum circa fixam coor-
dinatarum originem.
Haec facile nunc stabiliuntur. ↿∙∘⋇ Habemus (20. b.)172
dar
Em dea dex,
dla
day Σm.
dt2
Em
daz,
dc2
da ,
Em dla
Em &gt;
Em: dt?
Hinc &gt;, ob (o' ) ,
der ΣΧ daz,
de
ΣΥ
Em ' dt2
ΣΖ
Σm (o' ' ' ) : Am dla
molo videlicet systemate punctorum m , m' , m &quot; ,
perinde ( 50. 4. ) movebitur gravitatis centrum ac si , co
euntibus punctis in ipsum centrum , applicarentur centro
eaedem vires P, P , P &quot; , ... cum iisdem directioni
bus , quibus puncta illa sollicitantur .
2 . '* Fac ut vires nihil sint aliud nisi punclo
rum actiones mutuae : denotante A actionem puncti v.
gr. m in aliud quodvis v . gr. m' , et A' actionem puncti m'
in m , erit ( 7 ) A=A' ; et expressa per D distantia inter
utrumque punctum , resolvetur A' in ternas coordinatis
axibus parallelas
x' #A D
TA
EA ;
ilem A in ternas iisdem axibus parallelas ( o'r )
ŁA to ,
EA D po', -A D
sumpto superiori signo si A , A' sunt vires attrahentes,
inferiori si repellentes . Quare EX =0, EY=0 , &amp;Z=0, et
consequenter
dér,
=0,
adi ?
day1
di?
dz,
-0,
=O ;
di?
in ea scilicet qua sumus hypothesi nullis viribus acce
↙≀⊴⋅↕⋮ d'), inuia—z- '
d'æ, dta (if/y. md:2 dïz, dta
dtz— Em 'dt: Em '
du ïm
Hinc , 06 (o') ,
d'æ, EX (Ph- Zï diru— ZZ
dF—Zm' dt&quot; &quot;Zm dt2 −∑⋯ (0 ):
moto videlicet systemate punctorum m , m' , m , . . ,
perinde (50. .f.&quot;) movebitur gravitatis centrum ac si, eo-
euntibus pnnctis in ipsum centrum
, applicarentur centro
eaedem vires P, P', P&quot; , . . . cum iisdem directioni-
bus ∙ quibus puncta illa sollicitantur.
294: Fac ut vires nihil sint aliud nisi puncto-
rum actiones mutuae :denotante A- actionem puncti v.
gr.m in aliud quodvis v. gr. m', et A' actionem puncti m'
in m, erit (7) A:A'; et expressa per D distantia inter
utrumque punctum, resolvetur A' in ternas coerdinatis
axibus parallelas
3.&quot;—æ ∙−⇠ 7—7 ...-,: z—z .
drA U , A D A D
itcm A in ternas iisdem axibus parallelas (o&quot;)
æ—x' J—y' z—z' ∙
sumpto superiori signo si A, A' sunt vires attrahentes,
inferiori si repellentes. Quare XX :0, ∑∟ -o, ZZ:o, et
consequenter
in ea scilicet qua sumus hypothesi nullis viribus acce-173
leratricibus agetur gravitatis centrum , nulloque ob mutuas
panctorum actiones afficietur motu. Huc spectat princi
pium de conservatione centri gravitatis.
3.°# Super planis XOY, YOZ, XOZ fiant proje
ctiones a, b, c, a' , b' , c' , a &quot; , 6 &quot;, c &quot; , a ' , , . . arearum
descriptarum a radiis vectoribus punctorum m, m' , m&quot;,
computatis radiis ab origine coordinatarum : erunt ( 50. 8. )
xdy — ydx Σmda = Σm ydz - zdy και Σmdb = Σm 2 2
zdxxdz Σmdc = Ση 2 unde
daa 2Em -Σm α2
dta dt
22m d2b
dta
daz
у

;

= Em (:
dla e )

: )

dec dex dez 2Σm -Σm dt2 sm
(
20
de?
et consequenter ( o &quot; )
d'a 22m dt² 8(xY4yX), 28mmdla = Eby2 — zY),
( 0 )
dac 2Em =E( zX-xZ) .
dta
4.0 # Si vires consistunt in mutuis punctorum actio
nibus , erunt ( 2.º o &quot; )
173
leratricibns agetur gravitatis centrum
, nulloque ob mutuas
punctorum actiones afficietur motn. Huc spectat princi-
pium de conservatione centri gravitatis.
3.&quot;; Super planis XOï, ïOZ, XOZ fiant proie-
ctiones a, b, c, a', b'. c', a&quot;, b&quot;, c' ', a'&quot;, ∙ ∙ ∙ arearum
descriptarum a radiis vectoribus punctorum m, m', m&quot; , .. .
computatis radiis ab origine coordinatarum : erunt (50. 8.&quot;)
∑⋅↾⋅⊿↙↓⋅−−≔−∑⋯⊔−−−−∫−≌∙∑⋯↲≀⊨∑⋯⇅−−−≖−≗↶∙ d d d—d
2 ∙ 2
Emma.—zn. fix—?? ,
unde
22m ⋛∙∶−≧∶−−⋅∑⋯≺⊰≵ :::-£v:— 7
id?-:?) ∙ ⋮⋯∶⊜≀≀∶≖∂−↽−−≖⋅⋯⋅↗≺ :: −− ::z ⇋
d'c— d:.r ædaz
et consequenter (o&quot;)
ZZm −⋛⊴⋮↥⋮−⇌∑≺∞⊺−∜∑⋟ ,
ZZmäï—b- :Zþ'Z—zï),
d (a')
220: ⋅⊋≖−∶∶ :2(zX—-æZ).
494: Si vires consistunt in mutuis punctorum actio-
nibus, erunt (2.o o&quot;)174
8 (xY - 7X ) = 0 , (yz - zY) = 0 ; $ (zX -- XZ) = 0;
ideoque
dra dc Emdl2
d2b Σm dc2 Emadt²
}
et computatis areis ab initio temporis t ,
Ema = Ct , Emb = Ct, Emc = C &quot; ! (0 ) :
huc special principium de conservatione arearum. Formu
lae ( o &quot; ) adhuc obstinent , etsi in systemate invenitur pun
ctum fixum , modo tamen in pancto illo collocetur origo
coordinatarum : siquidem vigent in casua equationes ( o&quot; ' ' ) ,
unde profluunt ( o &quot; ).
5.0 * Si arcus s refertur ad tres axes orthogona
les , ejus incrementum infinitesimum ds poterit spectari
tanquam diagonalis parallelepipedi rectanguli sub later
culis dx , dy , dz : hinc.
dsa = dx2 + dyz-tdz?, et consequenter ( 50, 2.0 )
v2= dx2+ dyatdz
dla
Erit itaque
Emvdv = Em
der
de²
d'I
ayt
ar
- .
ac proinde (0 )
Emvdv = E (Xdx + ydy + Zdz) ( o&quot; &quot; ' ) .
Fac ut E (Xdx + Ydy + Zdz) exsistat differentiale exactum ,
!
174 .
£(xï—77X):o , ZUZ—zï): :X(zX—xZ):o;
ideoque
d'a 'dzb (130
dt2
?
et computatis areis ab initio temporis :,
2ma:Ct , 2mb:C't, ch:C&quot;t (o&quot;):
huc spectat principium de conservatione arearum. Formu-
lae (ov') adbuc obstinent
, etsi in systemate invenitur pun-
ctum fixum, modo tamen in puncto illo collocetur origo
coordinatarum: siquidem vigent in casua equationes (o&quot;'),
unde profluunt (o&quot;).
594: Si arcus :refertur ad tres axes orthogona-
les, eius incrementum infiuitesimam ds poterit spectari
tanquam diagonalis parallelepipedi rectanguli sub later-
culis dx
, dy , ds :hinc.
ds':dæï-l-df3-l-dz3 ,
et consequenter (50, 2?)
daga—.dyzudzz
dt2
⋅
02.—
Erit itaque
da a :
Zmpdu:2m(d£fdx : (iyd): ↿ d
zdz) .
ac proinde (o') ∙
vadv :!(de ⊣− ⊺⊄↴⋅⊺∫ ⊣−∅∠∄≂≻⋅⋅
(o&quot;').
Fac. ut XXdæ—fïdJ—l—Zdz) exsistat differentiale exactum.175
prodeat nimirum ex differentiatione cujusdam functionis
F (x , y , z, x ', y , z, x &quot; , ... ) ; habebis
Em (u2 — V.2) = 2F (x ,y ,z,x',...) —2 F (xo,9o , zo , x '. , ... ) ;
quantitates v. , xo, Yo, Zo, x'o, ... respondent initio mo
tus. Consequitur, quod, redeuntibus iisdem coordinatis, ea
dem quoque redibit summa virium vivarum : huc spectat
principium de virium vivarum conservatione.

6. °* Denotent &lt;math&gt;h, i, k , h , i , k ' , h '' ...&lt;/math&gt; coordinatas punctorum &lt;math&gt;m, m', m''&lt;/math&gt; in ordine ad novos axes, qui et paralleli sint axibus &lt;math&gt;OX , OY , OZ ,&lt;/math&gt; et originem habeant in communi gravitatis centro; erunt
x = xrth , y = yiti , z = zetk , x' =xith' ,
y = yiti, z= z+k ', w &quot; = xrth &quot; , ... ;
quibus valoribus substitatis in ( o &quot; ) , attentisque aequatio
nibus ( 20)
dah deh , dai dai Σm Σm=0, Σ . Σm=0
dcz dta des dt2 =
dek Σm-
dt
dakı Em=0
dc2
1
nec non aequationibus ( o &quot;&quot; ), prodibunt
dai dah
2 ( XiY) = Em ( h TI dea dt2
E ( iz - kY) = Em
(ala dih),
(
- ) com
(akone )
( o &quot; )
dah
ElkX_hZ) = Em ( k dt2
175
prodeat nimirum ex differentiatione cujusdam functionis
F(x,y, :, x', y', z', æ&quot;
, ..
.) ; habebis
M(æ—voz):2F(æ,7,z,æ',..-) -2 P(æo,yo , zo , x', , ..
.);
quantitates v,
, æ., y,,zo, æ'o,
... respondent initio mo-
tns. Consequitur, quod, redeuntibus iisdem coordinatis, ea-
dem quoque redibit summa virium vivarum: liuc spectat
principium de virium vivarum conservatione.-
604: Denotent h, i, k, h', i', A', I:&quot; .. . coordi-
natas punctorum m, m', m&quot; , . ..
in ordine ad novos
axes , qui et paralleli sint axibus OX, Oï
, OZ
,
et ori-
ginem habeant in communi gravitatis centro; erunt
r—æl-i-h ,.szl-ï-i !≖∶∅∎⊹∣⊂ 'x':xx-Fll' ,
f:.yg-I-i', z':z,-l—k', x&quot;:æ,-l-h&quot;, ...;
quibus valoribus substitutis in (a&quot;)
, attentisque aequatio-
nibus (20)
d'h dïb, dii dïi,
EMzzï—an—O, zmcïS—dtï Zm--o ∙
d']: dïk,
ZmäF—äz; Zm—o ∙
nec non aequationibus (o&quot;'), prodibunt
E(hX—-iï):2m(hää—ci £b) ∙ dr2
⋅ ∙ dq: ti*i z
(iz—mzn. (.
&amp;? −:. $) ∙
(o....)
d.,, dal. ∑≺⋌⊔∅≻⇌∑⋯≺∣≂−∂−↙⋮−−∣⋅⋮⋮↙⇆⋟∙ .176
Formulae (o &quot; ) se habent ad commune gravitatis cen
trum prorsus ut formulae ( o &quot; ) ad fixam coordinatarum x ,
J, 2, x' , ... originem 0 , respiciuntque relativum syste
matis motum quoad ipsum gravitatis centrum:

7. • * Relativus rigidi liberique systematis motus
quoad gravitatis centrum determinatur per ( o &quot; &quot; &quot; ) ; motus
vero ipsius centri per ( o' ' ' ) Ad haec : si resultans ex omni
bus viribus systemati rigido applicitis transit per gravi
tatis centrum , nullus inde orietur relativus systematis mo
tas quoad ipsum centrum : etenim quoad istiusmodi mo
tum similiter procedet res ac si resultans illa exerceretur
contra punclum fixum ( 6.&quot; ). Eadem de causa , accedentibus
novis viribus , relativus systematis motus quoad gravitatis
centrum nullo pacto turbabitur , ubi eae resultantem sup
peditent transeuntem per centrum illud.
85.&amp; Pauca subjungentes de motu rigidi systematis cir
ca axem fixum praemittimus illud: praeter orthogonales axes   &lt;math&gt;OX , OY , OZ ,&lt;/math&gt;   ( Fig. 9 ) sint alii tres axes similiter orthogonales On, Op, Oq, quibuscum ii angulos efficiant designatos per ( xn) , (xp ) , ( aq) , (yn) , ( yp ) , (99) , (zn) ,
(zp ), ( z9 ) . Si panctum E, quod referebatur ad axes OX,
OY, OZ , referendum sit ad axes On, Op , Og , quaeri
tur relatio inter veteres coordinatas x , y
nip , q. Ponatur OE = a, et per (ax ), (ay ) , (az), ( an ) ,
(ap) , ( aq) exhibeantur anguli , quos OE facit com axibus
OX , OY , OZ , On , Op , Oq: erunt ( 50. 6º . )
ma
z et novas
cos (ax) =cos (an) cos ( xn) +cos ( ap) cos (xp) +
cos (aq) cos (xq) , cos (ay ) =cos ( an ) cos (yn ) +
cos (ap) cos (yp) + cos (aq) cos (79) , cos (az)
eos ( an) cos ( zn) * cos(ap) cos (zp) + cos ( aq ) cos ( 29) .
176
Formulae (o&quot;&quot;) se habent ad commune gravitatis cen-
trum prorsus ut formulae (a&quot;) ad fixam coordinatarum æ,
y, 2, æ',
.. . originem O, respiciantque relativum syste-
matis motum quoad ipsum gravitatis centrum:
73»: Relativus rigidi liberique systematis motus
quoad gravitatis centrum determinatur per (o&quot;&quot;); motus
vero ipsius centri per (o&quot;') Ad haec: si resultans ex omni-
bns viribus systemati rigido applicitis transit per gravi-
tatis centrum
. nullus inde orietur relativus systematis mo-
tus quoad ipsum centrum : etenim quoad istiusmodi mo-
tum similiter procedet res ac si resultans illa exerceretur
contra punctum fixnm (S.&quot;). Eadem de causa , accedentibus
novis viribus
, relativus systematis motns quoad gravitatis
centrum nullo pacto turbabitur , ubi eae resultantem suppeditent transeuntem per centrum illud. '
853 Pauca subiungentes de motu rigidi systematis cir-
ca axem fixum praemittimus illud: praeter orthogonales a-
xes OX, Of, OZ (Fig. 9) sint alii tres axes similiter or-
thogonales On, Op, Oq, quibuscum ii angnlos efficiant de-
signatos Per (æ&quot;)s (æpl- (xq) :(f&quot;) , 07)» (f?) :(znls
(zp), (zq). Si punctum E, quod referebatur ad axes OX,
OV, OZ, referendum sit ad axes On, Op, Oq , quaeri-
tur relatio inter veteres coordinatas æ , y , :. et novas
n , p , q. Ponatur OE :a, et per (aæ)
,
(ay) ,(az), (an),
(ap), (aq) exhibeantur anguli, quos OE facit cum axibus
OX , Oï
, OZ , On. , Op , Oq: erunt ( 50. 60.)
cos (aæ):cos (an) cos (xn) —-[-cos (ap) cos (æp) −⊢ ⋅
cos (aq)cos (xq) , cos (ay) :cos (an) cos (yn) −∙⊢
cos (ap) cos (yp) ∙−⊢ eos (aq) cos (rq) ∙
005 (az) −−∶
eos (an) cos (zn) —,l-cos(ap)cos(zp)-—- eos (aq) cos (zq).
3751177
Sed cos (ax ) =
a , cos(ay) = cos(az ) = a
cos (an) =
, cos ( ap ) = .. cos (aq ) =
= 9
a
adhibitis igitur substitutionibus , provenient
x = ncos( an) + pcos (xp) + qcos(xq) ,
y = ncos(yn) + pcos (yp ) + acos(yq) ,
x = ncos( zn ) + pcos(zp) + qcos(zq) ;
formulae praebentes quaesitam relationem .
Nunc 1 . ** Sit OX rotationis axis, datumque systema
tis punctum reperiatur constanter in plano YOZ: si per OX
et per punctum illud ducitur planum occurrens plano YOZ,
satis erit determinare situm intersectionis istorum plano
rum ut innotescat systematis positio. Concipiamus itaque
novos axes orthogonales On , Op , Oq sic constitutos , ut
firmiter adhaereant systemati , primusque incidat in OX ,
tertius in intersectionem illam ; erunt
y= pcos(zq ) + qsiu(z9) , z =qcos (29) — psin(zg) :
adhibita substitutione in secundo membro secundae ( o &quot; .84)
animadvertendo quod variato e non ideo variant novae co
ordinatae , factoque
2 m ( p2 +9 ) = B ,
proveniet
d ' (29 )
di2 - $ (72—28 )
(o'r) :
∙∙∙⋅ 177
&amp; Sed ∾⋇≺⊄∣∙↿∶≻∶−−⊶⋚ ,cos (ay): a , c08(az):ä— .
']
...
(an):
⋮⋮−∙ costam: g.... (aq) ⇌⋅−− −↙⋅↓−
.
adhibitis igitur substitutionibus , provenient
x:ncos(æn) -l-pcoa (æp) ⊣− 9005(-qu :
)»:ncosU'n) pcos (ïp) −∣⋅− ⊄∾≘∩⊄⋟ '

: &quot;cos(zn) −⊢ pcos(zp) -I-— qcos(zq) :

famulae praebentes (quaesitam relationem.
Nune ↿∙∘∙ Sit OX rotationis axis, datumque systema-
tis punctum reperiatur constanter in plano ïOZ: si per OX
et per punctum illud ducitur planum occurrens plano ïOZ,
satis erit determinare situm intersectionis istorum plano-
rum ut innotescat systematis positio. Concipiamus itaque
novos axes orthogonales'.0n, Op , Oq sic constitutos, ut
firmiter adhaereant systemati, primusque incidat in OX ,
tertius in intersectionem illam; erunt
7: pcos(zq ) ⊣−⊄⊗∃∥≺∅⊄⋟ :
3 −−∶ quos(zq) --
psiu(zq) :
adhibita substitutione in secundo membro secundae (o&quot;.84)
animadvertendo quod variato :non ideo variant novae co-
ordinatae, factoque
. Zm(p'-l-q3)-——-B.
proveniet
(P(zq)
- dt2
z.. 1
-B— ZUZ—zï) (o&quot;):178
d (29 )
velocitas ( 50. 2º BE . ) respondet radio 1 , diciturque dla
velocitas angularis: binomia patq , p'? + 92.. nihil sunt aliud
nisi quadrata perpendiculorum ex m , m' , ... in axem On de
missorum ; summa productorum ex massis m , m' ... in
quadrata respondentium perpendiculorum , seu m (pa+92) +
m ' ( p2t 92) + . . . vocatur momentum inertiae systema
tis m , m' , .... quod axem On .
2. °# Ponamus vires acceleratrices consistere in so
la gravitate g, axesque Ox , OY jacere in horizontali pla
no: erunt Y = 0, Z
8 , et consequenter
7
de( 29 )
1 1
dla B
&amp; Emy =
1
g Em [ p cos ( zq) +qsiu ( zq ) ] E
&amp; [cos(zq) . Emp + sin ( zq) . Emq).
Fac ut illud systematis punctum , quod posuimus ( 10.)
reperiri constanter in axe Oq, sit gravitatis centrum; ex
sistent ( 20)
Σmp = p,Σm = 0 , Σmg = qΣm :
proinde
d? ( )
- 1/3
sin ( zqı ) . Em ;
dt?
B
891
quae prius multiplicata per 2d( 29, ) , ac dein integrata
praebebit
[da ] = - 69.cos/ 291). Em + c
x
178
. ∘
'
d(zq) velocitas ( 50. 2 . ..
.
) 7:2- respondet radio1,d1c1turque
velocitas angularis: binomia phi-qi, p'H—q'æ. nihil sunt aliud
nisi quadrata perpendiculorum ex m, tu',... in axem On de-
missorum ; snmma productorum ex massis m , m'
...
in
quadrata respondentium perpendiculorum, seu m (pi-I—q'H-
m' (p'ï-l- q'3)-i- ....
vocatur momentum inertiae systema-
.tis m, m',
.... quod axem On.
2.0a Ponamus vires acceleratrices consistere in so-
la gravitate g, axesque OX , Oï jacere in horizontali pla-
no: erunt T:o, Z:
-— g ,et consequenter
d3(z ) 1 ↿ ∙
de? ∶−∙−−↕≣− g Em]: —B—g2m[pcos(zq)—l-qstn(zq)]
1 - .
−−−−− -B- g[cos(zq). Zmp −↘∟ stn (zq). qu].
Fac ut illud systematis punctum ,quod posuimus (10.)
reperiri constanter in axe Oq, sit gravitatis centrum; ex-
BlStent (20) ∣
Zmp :plzm:o , qu :q12m :
proinde*
dï ↿ ∙)— B gq, sm (sq,). Em;

quae prius multiplicata per 2d( sq, )
, ac dein integrata
praebebit
. d
Z [ 2
2 .
[ld-g-l ∸∶−∙∙∙⋅ ∙∙∙ ï
gqx 008(zq1). Zm ⊹∁ ∙
iis179
Exsistentibus in initio motus d (291)
= uo et ( 291) = a , erit du
2
C = uo% + B 69 , cosa. Em :
propterea
d (290) 72
=u' . +
dt 2/3
891 [ cosa - cos ( 291) ] Em (o' ) .
Huc spectat theoria penduli compositi.
3.•* Intelligantur m , m' , m &quot; , .... coire in u
nicum punctum annexum axi horizontali Ox ope rectae r;
exsurget pendulum simplex : in casu
p = p = p = ... = 0 , q = 9 = 9 &quot; = ... = 9 = r ,
B = 2m(p + g ”) = 2n ;
et consequenter quoad pendulum simplex
d ( 292) 7
2
[Company *== +
s [ cos a - cos ( 291) ] (o &quot; ).
2
4.0# Facto 8
2
B
89. EmEm , proveniet
B
9 , £ m col) ;
longitudo videlicet penduli simplicis , quod suas perficit
oscillationes eodem tempore ac pendulum compositum . Re
cole quae diximus (67 ) .
5.° * Pone nullas esse vires acceleratrices i
erit ( 1. ° 0 ' )
179
d(zq !)
dc
Exsistentibus in initio motus :u. et (zq.):a, erit
.
2 C:u.,2 −⊦⋅ ïgq, cosa. Em:
prapterea
d(dZQtli:u⋅−⊢ ∙−⋛−∊⊄∙ [cosa — cos (zq.)]2m (a').
Huc spectat theoria penduli compositi.
39»: Intelligantur m , m', m&quot;, ... . . coire in u-
nicum punctum annexum axi horizontali OX Ope rectae r;
exsurget pendulum simplex: in casu
p::p'::p&quot;: ∙ ∙ ∙ −−∙−−∶∘ , qzq'2q&quot;: ∙ ∙ ∙
∶⊄∎∶↿∙ ,
&quot;B::ZmQF-l-qa) claim ;
et consequenter quoad pendulum simplex
↙≀≺≦≦∣≖⋝⊺−−⋅↙∘≖ ; f ,.
(....
...
(..., ].
2 2
4.0a Facto ∙∓− g :ïgq, Em
, proveniet
B .
r'.&quot;—∙−∙∙ q,2m
Om) ;
longitudo videlicet penduli simplicis, quod suas perficit
oscillationes eodem tempore ac pendulum compositum. Re-
cole quae diximus (67).
5.
.,
Pone nullas esse vires acceleratrices: ,
erit (1. ∘ o)180
dº(aq )
dia
d( 24)
unde velocitas angolaris u =
dc = const. = u , .
1
1 Motus igitur exsistet uniformis , eritque velocitas angu
laris ad velocitatem puncti v . gr. m ut 1 ad radium cir
culi descripti ab ipso m , seu
1
u : v =1 : V patqz , ac proinde v = u ? (patoga )
quoad illud itaque punctum obtinebit vis contrifuga expres
sa ( 51 ) per
= u’m V pat92 .
V pr + q2
1
vam 2
Resolvatur haec vis in ternas coordinatis axibus On, Op,
Og parallelas ; prodibunt 1
+
9
р
0 , u²mV p2tga .
V p²ta?
wimb p'tgo.
Foto &gt;
seu
0
0 , ump , u'mg :
1
quoad totum ergo systema habebuntur 2
0 , użEmp , u’Emq ;
ideoque orietur pressio in axem OX. Prima membra formu
larum ( a : 13. 8.° ) in casu fiunt
0 , użEmp, użEmq , użEmnp , użEmng , u’Em (pa - pa ) :
!
hinc ubi fuerint 1
1
Emp= 0 , Emg = 0 , Emnp = 0, Emng = 0 ( o'r) ,
180
(l*(z'q)
d?
d(zq)
dc

: o

, unde velocitas angularis ::: :const.-zuo,
Motus igitur exsistet uniformis , eritque velocitas angu-
laris ad velocitatem puncti v. gr. m ut 1 ad radium cir-
culi descripti ab ipso m, seu
∣ ...—....
a: p −−−−−↿ :Vpl-I—qa , ac proinde V::u' (pH-q2 )
quoad illud itaque punctum obtinebit vis centrifuga expres-
sa (51) per
vam
l/P'&quot;l'qa −−∶
&quot;&quot;&quot; Vlf-*?&quot;
-
Resolvatur haec vis in ternas&quot; coordinatis axibus On. Op,
Oq parallelas; prodibunt
seu
0, uïrnp
,
'u'mq :
quoad totum ergo systema habebuntur
o , u'Zmp , u'qu;
ideoque orietur pressio in axem OX. Prima membra formu-
larum (a'm : 13. 8.&quot;) in casu fiunt
o , ti*Zmp, uazmq , u'Zmnp , u'Zmnq , u'Zmþq—pq) :
hinc ubi fuerint
Zmp:o , M.,—:a, Zmnpzo, zmnqzo (atur)
'181
1
vires centrifugae se muluo librabunt independenter ab axe
Ox
, nullamque iste axis patietur pressionem
. Prima et se
cunda
(oh
) important
( 20. 6.
) transitum axeos On seu
OX per gravitatis centrum tertia vero et quarta important peculiarem quandam axiuin On
, Op, Oq positionem
relate ad punctorum
m
,
m
'
,
m
&quot; systema
. Porro si On
,
Op
, Oq ita sunt positi, ut suppeditent Emnp
=
0
, Emng
=
0
,
Empq
=
o
, appellari solent principales systematis axes in
ordine ad originem
itidem quae momenta ad eos referuntur
, et ipsa dicuntur principalia inertiae momenta
. Ex
pletis tertia et quarta
(
o
&quot;
&quot;&quot;
)
, non autem prima et secun
da
, ex omnibus viribus centrifugis resultabit
( 13. 9.0 10.9
)
vis premens rolationis axem in O. 6. '* Superiores formulae manifeste applicantur
continuo materialium punctorum systemati mutando
&amp; in
ſ et
m in dm
, integrationemque protendendo ad totam
systematis massam
.
7.9 Saepe videmus corpora impulsu aliquo loca
liter mota affici simul rotationis motu

: etiam praecisis

,
quae diximus
( 84
)
, sic ostendi potest motum istum oriri ex
eo quod impulsus non habeat directionem transeuntem per
centra gravitatis corporum
. Sic
G gravitatis centrum cor
poris MM
'
( Fig
. 46
)
, et AZ vis corpori cominunicata
..
Ducatur per
G ad AZL perpendiculum GL dividalur

bifariam AZ in
C
, et resolvatur CA in AD per
G tran
seantem
, et in AB normalem rectae AZ producatur AG

donec GF aequet GA intelligatur AD applicita ad punclum F , sitque FK = AD resolvatur FK in FH parallelam et FI perpendicularem rectae LGN

: quibus posi

tis
, substituti poteront vi AZ quatuor vires CZ
, AB
, FI
,
FH
. Jamvero CZ
, FI utpote aequales
, parallelae et ad
eamdem plagam tendentes contrahuntur in unam reprae
sentatam per GE
( 11
)
=GZ
+Fl
=AZ
, transeuntem per
G
, eidemque AZ parallelam proinde movebitur centrum

G non secus ac vis AZ ipsi esset applicata
. At duae aliae
↿∂⋅↿
vires centrifugae se mutuo librabunt independenter ab axe
OX, nullamque iste axis patietur pressionem. Prima et se-
cunda (o&quot;&quot;) important (20. b.) transitum axeos On seu
OX per gravitatis centrum: tertia vero et quarta impor-
tant peculiarem quandam axium On
, Op, Oq positionem
relate. ad punctorum m, m', m&quot; ,... systema. Porro si On,
Op, Oq ita sunt positi, ut suppeditent Zmnp:o, Zmnq:o,
Zmpq:o, appellari solent principales systematis axes in
ordine ad originem
O

itidem quae momenta ad eos refe-
runtur, et ipsa dicuntur principalia inertiae momenta. Expletis tertia et quarta (on&quot;), non autem prima et secunda, ex omnibus viribus centrifugis resultabit (13. 9310!)
vis premens rotationis axem in O.

63. Superiores formulae manifeste applicantur
continuo materialium punctorum systemati mutando
2 in
]et as in dm
,
integrationemque protendendo ad totam
systematis massam.

7.&quot; Saepe videmus corpora impulsu aliquo loca-
liter mota aflici simul rotationis motu: etiam praecisis,
quae diximus (84), sic ostendi potest motum istum oriri ex
eo quod impulsus non habeat directionem transeuntem per
centra gravitatis corporum. Sit
G gravitatis centrum cor-
poris MM' (Fig. 46)
, et AZ vis corpori communicata.
Ducatur per
G ad AZL perpendiculum GL; dividatur
bifariam AZ in C, et resolvatur CA in AD per
G tran-
seuntem, et in AB normalem rectae AZ; producatur AG
donec GF aequet GA; intelligatur AD applicita ad pun-
ctum
F
, sitque FK: A

resolvatur FK in FH paral-
lelam et FI perpendicularem rectae LGN

: quibus posi-

tis, substituti poterunt vi AZ quatuor vires CZ,AB, FI
,
FH. Iamvero CZ, FI utpote aequales
, parallelae et ad
eamdem plagam tendentes contrahuntur in unam reprae-
sentatam per GE (11):GZ—-FI:AZ, transeuntem per
G, eidemque AZ parallelam proinde movebitur centrum

0 non secus ac vis AZ ipsi esset applicata. At duae aliae182
.AB, FH utpote aequales , parallelae , et ad contrarias par-
tes tendentes , nequeunt gravitatis centrum e suo loco di-
movere : spectatis itaqne istiusmodi viribus, immobile eqn-
sisteret gravitatis centrum; sed eae sese mutuo non de-
struunt, cum e diametro non opponantur. Aliud ergo praestare non poterunt nisi corporis rotationem circa gravitatis centrum. Rotationis motus incipit circa reetam aliquam seu axem, et quoniam in omnes corporis particulas ex rotatione inducitur vis centrifuga; hinc si vires centrifugae inde ortae aequilibrantur circa rectam illam, invariabilis exsistet rotationis axis, defereturque per spatium sibimet semper parallelas; secus, mutabitur indesinenter rotationis axis donec ad aequilibrium deveniatur.

=== De fluidorum corporum aequilibrio. ===
86. Fluida corpora spectamus veluti materialiam punctorum congeries; quae puncta, utpote invicem independentia, vel minimo cedunt impulsui. In massa fluida undique librata sume punctum quodvis [exhibemus per &lt;math&gt;( x , y , z )&lt;/math&gt;, denotantibus &lt;math&gt;x, y, z&lt;/math&gt; ejus coordinatas] sollicitatum vi acceleratrice &lt;math&gt;\varphi&lt;/math&gt; praebente componentes &lt;math&gt;X, Y, Z&lt;/math&gt; coordinatis axibus &lt;math&gt;OX, OY, OZ&lt;/math&gt;,  parallelas et per punctum illud fac ut transeat superficies &lt;math&gt;k&lt;/math&gt; plana, rigida atque infinitesima: consistet &lt;math&gt;k&lt;/math&gt; in aequilibrio; et consequenter pressiones hinc et illinc exercitae in &lt;math&gt;k&lt;/math&gt; ab circumpositis massae fluidae stratis, erunt vires aequales et directe contrariae, simulque normales ipsi &lt;math&gt;k&lt;/math&gt;. Ejusmodi pressionum alteram repraesenta per &lt;math&gt;\varpi k&lt;/math&gt;; ratio &lt;math&gt;\frac{\varpi k}{k}(= \varpi)&lt;/math&gt; dicitur pressio hydrostatica exercita &lt;math&gt;k&lt;/math&gt; apud punctum &lt;math&gt;( x, y , z )&lt;/math&gt; contra aream ( = 1 ) sumptam in plano superficiei &lt;math&gt;k&lt;/math&gt;. In eadem massa fluida fac ut per punctum alterum &lt;math&gt;( x_0, y , z )&lt;/math&gt; transeat talis superficies &lt;math&gt;k_0&lt;/math&gt; plana, rigida et infinitesima, quae communem habeat projectionem cum superficie &lt;math&gt;k&lt;/math&gt; in plano &lt;math&gt;YOZ&lt;/math&gt;; voca &lt;math&gt;h&lt;/math&gt; projectionem illam, et &lt;math&gt;\varpi_0&lt;/math&gt;, hydrostaticam pressionem apud punctum &lt;math&gt;(x_0, y , z)&lt;/math&gt; contra aream ( =1 ) sumptam in plano areae &lt;math&gt;k_0&lt;/math&gt;. Massa fluida adhuc perget esse librata, etsi in qualibet ejus portione intelliguntur puncta rigidis lineolis firmiter connecti, seu, quod eodem redit, etsi quaelibet ejus portio fit solida: ponatur id contingere portioni cylindricae habenti rectam parallelam axi &lt;math&gt;OX&lt;/math&gt; pro generatrice, et &lt;math&gt;k , k_0&lt;/math&gt; pro basibus; denotet &lt;math&gt;\mu&lt;/math&gt; densitatem massae fluidae apud punctum &lt;math&gt;(x , y , z)&lt;/math&gt;; sitque &lt;math&gt;x &gt; x_0&lt;/math&gt; Exprimetur per&lt;math display=&quot;block&quot;&gt;h\int_{x_0}^x \mu X dx
&lt;/math&gt;summa ex viribus motricibus, quibus juxta &lt;math&gt;OX&lt;/math&gt; sollicitantur puncta illius portionis; exprimenlur praeterea per  &lt;math display=&quot;block&quot;&gt;\frac{h}{k_0}\varpi k_0, -\frac{h}{k}\varpi k &lt;/math&gt;pressiones exercitae juxta eumdem OX , altera in basim ko,altera in basim k quod spectat ad pressiones contra lateralis superficiei puncta, eae utpote normales generatrici rectae nullas dabunt componentes axi OX parallelas. Quia igitur solidata portio perseverat in aequilibrio, iccirco

&lt;math display=&quot;block&quot;&gt;h\int_{x_0}^x \mu X dx + h\varpi_0 - h\varpi = 0, \, \mathrm{unde}\,   \varpi = \varpi_0 + \int_{x_0}^x \mu X dx .
&lt;/math&gt;

Haud mutata positione superficiei &lt;math&gt;k_0&lt;/math&gt;, revolvatur utcumque superficies &lt;math&gt;k&lt;/math&gt; circa punctum &lt;math&gt;(x , y ,z)&lt;/math&gt;: permanebit secundum membrum ultimae aequationis; ergo et primum. Quare perseverabit in eodem valore hydrostatica pressio quoad omnia plana per punctum illud utcumque ducta: huc spectat principium de aequalitate pressionis. Consequitur, si recta generatrix sumitur parallela, prius axi &lt;math&gt;OY&lt;/math&gt;, deinde axi &lt;math&gt;OZ&lt;/math&gt;, denotantibus &lt;math&gt;\varpi_0',\varpi_0''&lt;/math&gt; hydrostaticas pressiones apud puncta &lt;math&gt;( x , y_0, z) , (x , y , z_0 )&lt;/math&gt;, fore etiam&lt;math display=&quot;block&quot;&gt;
\varpi = \varpi_0' + \int_{y_0}^y \mu Y dy , \varpi  = \varpi_0''+ \int_{z_0}^z \mu Z dz &lt;/math&gt;Terni valores &lt;math&gt;\varphi&lt;/math&gt; differentiati, primus quoad &lt;math&gt;x&lt;/math&gt;,  secundus quoad &lt;math&gt;y&lt;/math&gt;, tertius quoad &lt;math&gt;z&lt;/math&gt;, praebent
&lt;math display=&quot;block&quot;&gt;
\frac{d\varpi}{dx} = \mu X,  \frac{d\varpi}{dy} = \mu Y, \frac{d\varpi}{dz} = \mu Z. (o) &lt;/math&gt;et consequenter (27.24º)
&lt;math display=&quot;block&quot;&gt;
d\varpi = \mu ( Xdx + Ydy + Zdz). ( o' ) &lt;/math&gt; Itaque conditiones requisitae ad massae fluidae aequilibrium eo redeunt ut exsistat ejusmodi functio &lt;math&gt;\varpi&lt;/math&gt; variabilium &lt;math&gt;x, y, z&lt;/math&gt;, quae expleat sive ternas (o), sive unicam (o').

87. Haec notentur. 

1º. Si fluidum continetur vase undique clauso satisque firmo, utcumque se habeat valor &lt;math&gt;
\varpi &lt;/math&gt; ex (o') quoad superficiem fluidi, is constanter aequivalebit reactioni ex vasis lateribus: at si fluidi superficies sit libera, externisque subjecta pressionibus, ad aequilibrium explenda insuper erit (o') per talem valorem &lt;math&gt;
\varpi &lt;/math&gt;, qui in singulis liberae superficiei punctis aequivaleat respondenti pressioni externae.

2º. Hinc si pressio externa vel ponitur &lt;math&gt;=0&lt;/math&gt; vel ubique eadem, erit &lt;math&gt;d\varpi = 0&lt;/math&gt; quoad superficiem fluidi librati, ideoque
&lt;math display=&quot;block&quot;&gt;Xdx + Ydy + Zdz  = O (o''). &lt;/math&gt;

3º. Traduci potest (o&quot;) ad&lt;math display=&quot;block&quot;&gt;\frac{X}{\varphi} \frac{dx}{ds}+\frac{Y }{\varphi} \frac{dy}{ds} + \frac{Z}{\varphi}\frac{dz}{ds} = 0&lt;/math&gt;exprimunt &lt;math&gt;X/\varphi, Y/\varphi, Z/\varphi&lt;/math&gt; cosinus angulorum, quos efficit vis acceleratrix &lt;math&gt;\varphi&lt;/math&gt; cum axibus coordinatis &lt;math&gt;OX, OY, OZ&lt;/math&gt;; denotant &lt;math&gt;\frac{dx}{ds}, \frac{dy}{ds},\frac{dz}{ds}&lt;/math&gt; cosinus angulorum, quos recta tangens arcum &lt;math&gt;s&lt;/math&gt; apud ejus extremum facit cum iisdem axibus: inferimus (50. 6.) vim &lt;math&gt;\varphi&lt;/math&gt; intercipere angulum = 90° cum rectis omnibus tangentibus ubivis superficiem vel nullo pacto, vel aeque pressam; ac proinde &lt;math&gt;\varphi&lt;/math&gt; sese dirigere normaliter ad istiusmodi superficiem.

4.º Integrata (o&quot;), si constanti arbitrariaeque quantitati tribuuntur alii atque alii valores, emergent aliae atque aliae aequationes, quibus totidem respondebunt distinctae superficies aeque pressae.

5.°* In hypothesi &lt;math&gt;\varphi&lt;/math&gt; tendentis ad punctum fixum, constitue ibi coordinatarum originem: denotante &lt;math&gt;D&lt;/math&gt; distantiam inter punctum illud et &lt;math&gt;( x , y , z)&lt;/math&gt;, erunt (50. 6º)&lt;math display=&quot;block&quot;&gt;X = -\varphi \frac x D, Y= -\varphi \frac y D, Z= -\varphi \frac z D&lt;/math&gt;hinc&lt;math display=&quot;block&quot;&gt;X dx + Ydy + Zdr = - \frac \varphi D (xdx + ydy + zdz).&lt;/math&gt;Est insuper &lt;math&gt;x^2 + y^2 + z^2 = D^2 &lt;/math&gt;, unde &lt;math&gt;xdx + ydy + zdz = DdD;&lt;/math&gt; et consequenter&lt;math display=&quot;block&quot;&gt;Xdx +Ydy + Zdz = -\varphi dD.&lt;/math&gt;In ordine igitur ad superficiem aeque pressam exsistet &lt;math&gt;dD = 0&lt;/math&gt;: propterea &lt;math&gt;D = C&lt;/math&gt;; ex qua &lt;math&gt;x^2 + y^2 + z^2 = C^2 &lt;/math&gt;: massa videlicet fluida atque librata induet sphaericam formam.


6. Quoad fluidum elasticitate pollens, constat experimentis densitatem &lt;math&gt;\mu&lt;/math&gt;, permanente temperie, esse proportionalem respondenti pressioni &lt;math&gt;
\varpi &lt;/math&gt;, nimirum&lt;math display=&quot;block&quot;&gt;\mu = \theta \varpi: (o''')&lt;/math&gt;Eliminata  &lt;math&gt;\mu&lt;/math&gt; ab (o') et (o''&quot;''), proveniet&lt;math display=&quot;block&quot;&gt;\frac{d\varpi}{\varpi}=\theta(Xdx + Ydy + Zdz);&lt;/math&gt;et facto &lt;math&gt;Xdx + Ydy + Zdz = df (x,y,z)&lt;/math&gt;, erit:&lt;math display=&quot;block&quot;&gt;\ln \varpi = \int \theta df + \ln C = \ln (e^{\int \theta df}) + \ln C= \ln (C e^{\int \theta df})&lt;/math&gt;hinc&lt;math display=&quot;block&quot;&gt;\varpi = C e^{\int \theta df}, \mu = C \theta e^{\int \theta df}&lt;/math&gt;coefficiens &lt;math&gt;\theta&lt;/math&gt; pendet a temperie vigente apud &lt;math&gt;(x , y , z)&lt;/math&gt;. Inferimus aequilibrii statum in fluido elastico importare temperiem vel ubique eamdem, vel talem ut sit functio quantitatis &lt;math&gt;f&lt;/math&gt;. Haec insuper quantitas est (2º, 4º) constans in unaquaque superficie aeque pressa; idipsum ergo dicendum de temperie. 


7.º Constat etiam experimentis fluidum elasticitate pollens ita contrahi vel expandi, imminuta vel aucta temperie ac permanente pressione &lt;math&gt;
\varpi' &lt;/math&gt; ut ejus volumen &lt;math&gt;
V &lt;/math&gt;minuatur vel augeatur partibus 0,00375 pro singulis gradibus thermometri centigradi; inde fit, ut posito 0,00375 = &lt;math&gt;
a &lt;/math&gt;, et aucta temperie gradibus &lt;math&gt;n&lt;/math&gt; ultra &lt;math&gt;0^\circ \mathrm{C}&lt;/math&gt; , volumen  &lt;math&gt;V&lt;/math&gt; evadet &lt;math&gt;V ( 1 + an )&lt;/math&gt;; propterea, designantibus  &lt;math&gt;\mu_0&lt;/math&gt;  et   &lt;math&gt;\mu_1&lt;/math&gt;  respondentes densitates, erit &lt;math&gt;\frac{\mu_1}{\mu_0}=\frac{1}{1+an}.&lt;/math&gt; Nunc, permanente temperie &lt;math&gt;n&lt;/math&gt;, crescat pressio ab &lt;math&gt;
\varpi' &lt;/math&gt; ad  &lt;math&gt;
\varpi &lt;/math&gt;;  denotante &lt;math&gt;\mu&lt;/math&gt;  respondentem densitatem, erit  (1º) &lt;math&gt;\frac{\varpi}{\varpi'}=\frac{\mu}{\mu_1},&lt;/math&gt; quocirca&lt;math display=&quot;block&quot;&gt;\varpi = \frac{\varpi'\mu}{\mu_1} =  \frac{\varpi'}{\mu_0} \mu ( 1 + an )&lt;/math&gt;; et facto &lt;math&gt; \frac{\varpi'}{\mu_0} =i&lt;/math&gt;, &lt;math&gt;\varpi = i \mu ( 1 + an ) (o^{(iv)})&lt;/math&gt;.

=== De gravium homogeneorumque liquidorum aequilibrio. ===
88. Planum &lt;math&gt;XOY&lt;/math&gt; sit horizontale, axisque &lt;math&gt;OZ&lt;/math&gt; (Fig. 47) vergat deorsum juxta directionem gravitatis &lt;math&gt;g&lt;/math&gt;; erunt &lt;math&gt;X=0, Y=0, Z = g&lt;/math&gt;: proinde (86. 6), &lt;math display=&quot;block&quot;&gt;d\varpi = g \mu dz ( 0^{v} )&lt;/math&gt;Si pressio externa ponitur vel = 0, vel ubique eadem, erit &lt;math&gt;d\varpi = 0&lt;/math&gt; quoad librati fluidi superficiem, ideoque &lt;math&gt;dz = 0&lt;/math&gt;, et &lt;math&gt;z = Const&lt;/math&gt;: superficies nempe illa existet plana atque horizontalis. 

Pone &lt;math&gt;\mu&lt;/math&gt;  constantem; ex (0&lt;sup&gt;v&lt;/sup&gt;) habebis

&lt;math&gt;\varpi = g \mu z + C_1&lt;/math&gt;,

In fluidi superficie aeque pressa constitue planum horizontale &lt;math&gt;XOY&lt;/math&gt;: quoad eam erit &lt;math&gt;z = 0&lt;/math&gt;; nihilque aliud denotabit &lt;math&gt;C_1&lt;/math&gt; nisi externam pressionem in aream ( = 1 ) quaquaversus per fluidum aequaliter diffusam.

Haec facile nunc stabiliuntur circa pressiones gravium homogeneorumque liquidorum intra vasa in aequilibrio consistentium. 
[[Fasciculus:Hydrostatic-pressure.svg|thumb]]
1º. Si per &lt;math&gt;\Pi&lt;/math&gt; designatur pressio in horizontalem aream &lt;math&gt;A&lt;/math&gt; demersam ad profunditatem &lt;math&gt;z&lt;/math&gt;, exsistet &lt;math&gt;\Pi = A \varpi = A (g\mu z + C_1 ) .&lt;/math&gt;

2º. Si &lt;math&gt;C_1 = 0&lt;/math&gt;, aequivalebit &lt;math&gt;\Pi&lt;/math&gt; ponderi prismatis, cujus basis est &lt;math&gt;A&lt;/math&gt;, altitudo &lt;math&gt;z&lt;/math&gt;, densitas vero eadem ac densitas liquidi.

3º. Exhibente &lt;math&gt;A&lt;/math&gt; horizontalem vasis fundum, ideoque &lt;math&gt;z&lt;/math&gt; altitudinem vasis; quoniam &lt;math&gt;\Pi&lt;/math&gt; nullatenus pendet a vasis figura, iccirco permanentibus &lt;math&gt;A&lt;/math&gt; et eadem perstabit liquidi pressio in horizontalem fundum, utcumque de caetero varient figura et capacitas vasis.

4º. Area &lt;math&gt;A&lt;/math&gt; sit oblique intra liquidum utcumque demersa: divide &lt;math&gt;A&lt;/math&gt; in areolas infinitesimas &lt;math&gt;a , a ', a'' &lt;/math&gt; quarum distantiae ab extima liquidi superficie designentur per &lt;math&gt;z' , z''...;&lt;/math&gt; denotante &lt;math&gt;\Pi'&lt;/math&gt; totalem pressionem, et &lt;math&gt;z_1&lt;/math&gt; perpendiculum ductum ex centro gravitatis areae &lt;math&gt;A&lt;/math&gt; in planum &lt;math&gt;XOY&lt;/math&gt;; erit (20) 

&lt;math&gt;\Pi' = a(g\mu z + C_1) + a'(g\mu z'+ C_1) +... = g\mu(az + a'z' + ...) + C_1( a + a' + ... ) = g\mu z_1 A + C_1 A = A (g\mu z_1 + C_1 )&lt;/math&gt;. 

Hinc si centrum gravitatis manet ad eamdem profunditatem demersum, haud variabit  &lt;math&gt;\Pi'&lt;/math&gt;, utcumque circa illud revolvatur area demersa: potest A repraesentare quamlibet rectilineam portionem internae superficiei vasis.

Ad haec: coordinatae ( 13. 3º. )

&lt;math&gt;b=\frac{\sum ax (g\mu z + C_1)}{ \sum a(g\mu z + C_1)}; b' = \frac{\sum ay (g\mu z + C_1)}{ \sum a(g\mu z + C_1)}, b'' = \frac{\sum az (g\mu z + C_1)}{ \sum a(g\mu z + C_1)} &lt;/math&gt; seu (20) 

&lt;math&gt;b =   \frac{g\mu \sum ax z + C_1 A x_1 }{ A(g\mu z_1 + C_1) }; 
b' =  \frac{g\mu \sum ay z + C_1 A y_1 }{  A(g\mu z_1 + C_1)}, 
b'' = \frac{g\mu \sum a  z^2 + C_1 A z_1 }{ A(g\mu z_1 + C_1)} &lt;/math&gt;

respondent illi puncto areae &lt;math&gt;A&lt;/math&gt;, per quod transit resultans ex parallelis viribus &lt;math&gt;a(g\mu z + C_1), a'(g\mu z'+ C_1), a''(g\mu z''+ C_1)...&lt;/math&gt;; istiusmodi punctum dicitur centrum pressionis.
[[Fasciculus:PolydirectionalPressure.svg|thumb]]
5º . Veniat considerandum solidum liquido immersum: sume apud punctum &lt;math&gt;( x , y , z )&lt;/math&gt; in solidi superficie areolam infinitesimam &lt;math&gt;k&lt;/math&gt; , et apud puncta &lt;math&gt;( x_0, y, z ) , (x , y_0, z ) , ( x , y , z_0 )&lt;/math&gt;in eadem solidi superficie areolae &lt;math&gt;k_0, k'_0, k''_0&lt;/math&gt;, sitque &lt;math&gt;h&lt;/math&gt; projectio areolae &lt;math&gt;k_0&lt;/math&gt; in plano &lt;math&gt;YOZ&lt;/math&gt;, &lt;math&gt;h'&lt;/math&gt; projectio areolae &lt;math&gt;k'_0&lt;/math&gt; in plano &lt;math&gt;XOZ, h''&lt;/math&gt;projectio areolae &lt;math&gt;k''_0&lt;/math&gt; in plano &lt;math&gt;XOY&lt;/math&gt;; congruant vero &lt;math&gt;h, h' , h''&lt;/math&gt; cum projectionibus areolae &lt;math&gt;k&lt;/math&gt; in iisdem planis: per &lt;math&gt;k(g\mu z + C_1), k_0(g\mu z + C_1),k'_0(g\mu z + C_1), k''_0(g\mu z + C_1),&lt;/math&gt;exprimentur pressiones normaliter exercitae in areolas  &lt;math&gt;k, k_0, k'_0, k''_0&lt;/math&gt;; ejusmodi pressionum prima resolvitur in &lt;math&gt;\frac{h}{k}\cdot k(g\mu z + C_1), \frac{h'}{k}\cdot k(g\mu z + C_1),\frac{h''}{k}\cdot k(g\mu z + C_1),&lt;/math&gt;&lt;ref&gt;Figura deest ergo clare non est si aequatio est recte stripta &lt;/ref&gt; parallelas rectis &lt;math&gt;OX , OY , OZ&lt;/math&gt;; secunda praebet componentem

&lt;math&gt;-\frac{h}{k_0}\cdot k_0(g\mu z + C_1)&lt;/math&gt;

parallelam rectae OX, tertia dat componentem

&lt;math&gt;-\frac{h'}{k'_0}\cdot k'_0(g\mu z + C_1)&lt;/math&gt;

parallelam rectae OY; quarta suppeditat componentem

&lt;math&gt;-\frac{h''}{k''_0}\cdot k''_0(g\mu z_0 + C_1),&lt;/math&gt;

parallelam rectae &lt;math&gt;OZ&lt;/math&gt;. His positis, quisque videt areolam &lt;math&gt;k&lt;/math&gt;, elisis componentibus horizontalibus, urgeri sursum verticali pressione&lt;math display=&quot;block&quot;&gt;h'' g \mu ( z - z_0 )&lt;/math&gt;totum igitur demersum solidum ad verticalem ascensum sollicitatur parallelis viribus praebentibus resultantem, quae aequivalet ponderi liquidi expulsi, et transit per punctum illud, ubi erat gravitatis centrum ipsius liquidi expulsi. Itaque si &lt;math&gt;V'&lt;/math&gt; et &lt;math&gt;\mu'&lt;/math&gt; exhibent volumen et densitatem solidi liquido immersi, &lt;math&gt;V&lt;/math&gt; volumen liquidi espulsi; pondus, quod superest solido, exprimelur per &lt;math&gt;g( V'\mu' - V\mu )&lt;/math&gt;: in solidis heterogeneis designat &lt;math&gt;\mu'&lt;/math&gt; densitatem mediam.

89. Sit 1º &lt;math&gt;\mu' &gt; \mu &lt;/math&gt; cum nequeat esse &lt;math&gt;V &gt; V '&lt;/math&gt;, erit semper &lt;math&gt;V'\mu' - V\mu &gt;0&lt;/math&gt;; tamdiu igitur descendet solidum, ubicumque in liquido collocetur, donec aliquod offendat obstaculum, cui adstringatur adhaerere. Si collocatur in liquidi superficie; statim atque totum fuerit demersum, exsistet &lt;math&gt;V = V';&lt;/math&gt; et consequenter perget solidum moveri vi acceleratrice &lt;math&gt;\frac{gV ' ( \mu' - \mu )}{V'\mu'}&lt;/math&gt; seu &lt;math&gt;g\left( 1 - \frac{\mu }{\mu'}\right)&lt;/math&gt;

Ab exploratis solidi ponderibus P et P' in vacuo et in li
quido elici potest ratio inter u et l ; siquidem
P = gV ' ', P = 8 ! V' ' — Vp ) , et V = V :
propterea
P í M
Hop
unde р
P
P - P
[[Fasciculus:EB1911 Hydromechanics - Fig. 3.jpg|thumb]]
Sit 2º. M '= H: tamdiu V'u ' - Vl &gt; o quamdiu &lt;math&gt;V' &gt; V&lt;/math&gt;; solidum nempe collocatum in superficie liquidi eo usque descendet, donec totum demergatur; quod ubi contigerit, evanescente V' M' — Vp , consisteret in aequilibrio nisi urgeretur adhuc vi acquisita descendendo ante et aequivalet ponderi liquidi expulsi, et transit per punctum illud, ubi erat gravitatis centrum ipsius liquidi expulsi. ltaque si V'et p! exhibent volumen et' densitatem
solidi liquido immersi, V volumen liquidi expulsi; pondus,
quod superest solido, exprimetur per
g( V'p.'—Vp.) :
in solidis heterogeneis designat p! densitatem mediam. ⋅
89. Sit. 1041!) p.: cum nequeat esseV) V', erit sem-
per V' pf ∙−− Vp.) o; tamdiu igitur descendet solidum, ubi-'
cumque in liquido collocetur, donec aliquod offendat ob-
staculum
, cui adstringatur adhaerere. Si collocatur in li-
quidi superficie; statim atque totum fuerit demersum, ex-
sistet V:V'; et consequenter perget solidum moveri vi
acceleratrice '
sv. ∣≺⊮∸⋮⋅⋅−⋅∟∸≻ ∘ −.r. v'F-I , .seu :,(1 l*') .
Ab exploratis solidi ponderibus P et'P' in vacuo et in li-
quido elici potest ratio inter p! et p.; siquidem
≖∙⊃−∙−⇀−∊⋁∙⊬↼∙∙ P',—.: g( vir—v,. ), .xv.-: V':
prOpterea
.
P p! p!— P
P' −−−⊬∙∙⊬∙ uude F- P-P'
.
Sit 20. pl: p.: tandiu V'pf -— VP) o quamdiu
V&quot; V ; solidum nempe collocatum in superficie liquidi
eo .usque descendet,, donec totum demergatur; quod ubi
contigerit, evanescentev p! —Vp. , consisteret in aequi-,-
librio nisi urgeretur adbuc vi acquisita descendendo ante192
1
V'de VM
1
1
totalem immersionem ; ad aequilibrium praeterea deberent
gravitatis centra solidi et liquidi expulsi esse in eadem re
cia verticali,

Sit 3º. p &lt; l tandiu . Vil – Ve &lt; o quandiu
V &gt; ; et facto V , erit Vų – VH = 0.
Solidum igitur collocatum intra liquidum ascendet ad li
quidi superficiem ; situm in ipsa superficie supernatabit ;
eritque portio demersa V ad volumen integrum V' ut j ': fl.
Innatantis solidi aequilibrium requirii insuper ut in eadem
recta verticali inveniantur gravitatis centra ipsius solidi et
liquidi quod expellitur.
Itaque positio aequilibrii quoad solidum homogeneum
liquido insideas determinabitur si plano ita secetur soli
dum, ut et alterius segmenti volumen sit ad solidi volu
men ia data ratione pe': fhy et haec volumina habeant sua
gravitatis centra in eadem recta , quae normaliter insistat
plano secanti: rem declaramus exemplo. Determinanda sit
positio aequilibrii in prismate recto ac triangulari , quod
ita demergitur ut et ejus bases maneant verticales, et u
na ex tribus faciebns v. g. BC ( Fig 48 ) exsistat cota ex
tra liquidum.
Quisque videt directionem plani secantis non pende
re a mutua basium distantia, satisque esse ut determine
tur intersectio De illius plani et baseos v . g. ABC. Exhi.
beant a ', a“ latera AB, AC dati trianguli ABC , et a', w &quot;
latera incognita AD, AE crianguli ADE : triangulares areae
ABC, ADE exprimentur per
3
i a'a ' sin A , Law&quot; sin A.
Sed area ABC est ad aream ADE ut integrum prisma ad
prismaticam portionem demersam ; igitur
le
IWW 'sin A: į a' a &quot; sin A = fe':J.,Was&quot;
P.
-a'a' ( k) .
192
totalem immersionem; ad aequilibrium praeterea deberent
gravitatis centra solidi et liquidi expulsi esse in eadem re-
cta verticali. ⋅
'
Sit 3&quot;. p! p. : tandiu. V'pl ∙− Vp.( o quandiu
V P- ;et factoV:V V) P- ,eritV'pf—Vp.:o.
Solidum igitur collocatum intra liquidum ascendet ad li-
quidi superficiem; situm in ipsa superficie superuatabit;
eritque portio demersa V ad volumen integrum V' ut pf: p..
Iunatantis solidi aequilibrium requirit insuper utin eadem
recta verticali inveniantur gravitatis centra ipsius solidi et
liquidi quod expellitur.
ltaque positio aequilibrii quoad solidum homogeneum
liquido insidens determinabitur si plano ita secetur soli-
dum, ut et alterius segmeuti volumen sit ad solidi volu-
men iu data ratione an., et haec volumina habeant sua
gravitatis centra in eadem recta, quae normaliter insistat
plano secanti: rem declaramus exemplo. Determinauda sit
positio aequilibrii in prismate recto ac triangulari, quod
ita demergitur ut et eius bases maneant verticales, et u- ⋅
na ex tribus faciebus v. g. BC ( Fig 48) exsistat tota ex—
tra liquidum.
Quisque videt directionem plani secantis nou pende-
re a mutua basium distantia, satisque esse ut determine-
tur intersectio DE illius plani et baseos v. g. ABC. Exhi-
beant a', a&quot; latera AB. AC dati trianguli ABC, et m', a)&quot;
latera incognita AD, AE trianguli ADE :triangulares areae
ABC, ADE exprimeutur per
∙∙⋅∙ äaa smA I &quot;-
, ämæstu A.
Sed area ABC est ad aream ADE ut integrum prisma ad
prismaticam portionem demersam: igitur
P:.
in' d'siu A: ;a' a&quot; sin A∶∶∶ [1]: .n., 'n' a&quot;:—-—a'a&quot; (k) .193

pla AM
Ž
AH '
Nunc secto bifariam in H latere BC, ducatur AH; sum
2
3
AH , centrum gravitatis trianguli ABC e
rit in M: simili modo, secto bifariam in H ' latere DE, sum
2
ptaque AN = AH', erit N centrum gravitatis trianguli 3
AM AN
ADE. Quia igitur ideo MN et HH' erunt АН
inter se parallelae: sed in casu aequilibrii recta MN, jun
gens gravitatis centra M et N , est perpendicularis rectae
DE ; ergo et HH' erit perpendicularis ipsi DE . Hinc DH=
HE: vicissim si DH =HE, erit HH' ac proinde MN per
pendicularis rectae DE; conditio nimirum necessaria ac
sufficiens ut recta jungens gravitatis centra M et N sit per
pendicularis rectae DE redigetur ad mutuam aequalitatem
rectarum DH, HE. Quibus positis , denotent B, Borangulos
DAH, BAH, et b rectam AH; triangula ADH, AHE dabunt
DA’ = w2762—2wbcos B ,HE' = w &quot; 2 + 62—20 &quot; bcoss *:
propterea
w2 -2bw' cos B = &quot;? - 26w &quot; cos \beta &quot; (k' ) .
Ex duabus ( k) et ( k ' ) eruentur a eta' , uude innotescit
positio intersectionis DE.
Quod si determinanda esset positio aequilibrii in eo
dem prismate quum ita demergitur ut puncta B et C ma
neant infra liquidi superficiem DE, foret
area BCDE : aream ABC = h' : u ': fb,
ideoque
ABC - BCDE ( ADE ): ABC Hope': fl , seu
−−∙≔∎⊾↼−−⇀
193
' Nune secto bifariam in H latere BC, ducatur AH; sum-
pta AM:
∙⋛−⋅ AH, centrum gravitatis trianguli ABC e-
rit in M: simili modo, secto bifariam in H' latere DE, sum-
ptaque AN:
∙−−≣−− AH', erit N centrum gravitatis trianguli
ADE. Quia igitur illi: −∙∙ 23,
inter se parallelae: sed in casu aequilibrii recta MN,iuu- ⋅
gens gravitatis centra M et N , est perpendicularis rectae
DE; ergo et HH' erit perpendicularis ipsi DE. Hinc DEI:-.'
HE: vicissim si DH :HE, erit HH' ac proinde MN per-
pendicularis rectae DE; conditio nimirum necessaria ac
sullicieus ut recta iungens gravitatis centra M et N sit per-
pendicularis rectae DE redigatur ad mutuam aequalitatem
rectarum DH, HE. Quibus positis, denotent B', B&quot;angulos
DAH, BAH, et brectam AH; triangula ADH,AHE dabunt
,
ideo MN et HH' erunt
BB': 'i—l-b' —29'6 cos B', B—Ea ⇌∾∣⋅≖−⊢ &amp;&quot; —20&quot;bcosB&quot;:
propterea
a)&quot; ---260' cos B': si&quot;! -
266)&quot; 'cos B&quot; (k' ).
Ex duabus (I:) et (k') erucutur a' et et&quot;, unde innotescit
positio intersectionis DE.
Quod si determinanda esset positio aequilibrii in eo-
dem prismate quum ita demergitur ut puncta B et C ma-
neant infra liquidi superficiem DE, foret
area BCDE: aream ABC −∙∶−− pl: p.': p.,
ideoque
ABC −∙− BCDE (: ADE ): ABC :p.- p.': p., seu194
Ww&quot; sin A : 1 a'a&quot; sin A = M - pe : plo
et consequenter
s'avº = ( 1- )« a”(k&quot;).
Ad haec : centrum gravitatis trianguli ABC invenilor
in recta jungente centra gravitatis portionum ADE ac BCDE.
Sed ABC et BCDE habent sua gravitatis centra in recta
perpendiculariter insistente rectae DE : adhuc igitur MN
erit perpendicularis ipsi DE ; rursusque prodibit (k' ) : eru
entur videlicet in casu w' et w &quot; ex binis ( k' ) et (k &quot; ) .

90. Determinata aequilibrii positione, restat videndum utrum aequilibrium sit stabile nec ne. Pone v. gr. innatans solidum esse tale, ut secari possit plano verticali AB ( Fig. 49. ) in duas partes omnino symmetricas tum quoad formam, tum quoad densitatem, et in casu aequilibrii sit HK intersectio plani AB et horizontalis plani repraesentantis superficiem liquidi: gravitatis centra M et N innatantis solidi et ejecti liquidi invenientur ambo in plano AB super eadem verticali CD; si solidum est homogeneum exsistet N subter M; si heterogeneum, poterit M esse vel subter N vel supra. Fac ut aliquantulo revolvatur solidum circa axem perpendicularem plano AB, sicque removeatur ab aequilibrii positione; ita tamen ut, exhibente H'K ' (Fig. 50) novam intersectionem plani AB et horizontalis plani repraesentantis superficiem liquidi, segmentum solidi respondens angulo K i K' aequetur constanter segmento quod respondet angulo H i H' ; hoc pacto haud variato ejecti liquidi volumine, permanebit ( 89.30. )
gV'p ' = gVd : proinde solidum absque initiali velocitate sibi
commissum movebitur ( 84 ) circa centrum M immotum.
Jam si ex puncto N' , ubi , amoto solido ab aequilibrii
positione , situm est gravitatis centram liquidi expulsi , du
194
&amp; o'o'f aiu A:) a'a&quot; sin A:p—p:p., /
et consequenter
Ad haec :centrum gravitatis trianguli ABC invenitur
in recta iungente centra gravitatis porticuum ADE ac BCDE.
Sed ABC et BCDE habent sua gravitatis centra in recta
perpendiculariter insistente rectae DE: adhuc&quot; igitur MN
erit perpendicularis ipsi DE; rursusque prodibit (k') :eru-
entur videlicet in casu a' et a)&quot; ex binis (k') et (Is&quot;) . ducatur verticalis recta N'R occurrens rectae CD in R , oc
cursus iste vel fiet supra M , vel infra , vel in ipso M :
in primo casu vis g Vlagens sursum juxta N'R manife
ste nitetur ut CD resumat verticalem positionem, et conse
quenter aequilibrium erit stabile ; in secundo ipsa gVp.
nitetur ut CD magis recedat a verticali positione , ideoque
aequilibrium instabile ; in tertio aequilibrium adhuc ob
tinebit quoad novam positionem .

=== De gravium liquidorum aequilibrio in vasis communicantibus. ===
[[Fasciculus:Communicating vessels.svg|thumb]]
91. Vasa communicantia dicuntur illa, quae ita sunt inter se conjuncta ut ex altero in alterum pateat aditus fluido. In altero contineatur fluidum homogeneum, cujus densitas &lt;math&gt;\mu&lt;/math&gt;; in altero fluidum pariler homogeneum cujus densitas &lt;math&gt;\mu'&lt;/math&gt;; siatque &lt;math&gt;z&lt;/math&gt; et &lt;math&gt;z+ z'&lt;/math&gt; distantiae inter punctum quodvis superficiei communis utrique fluido ac extimas fluidorum superficies. Fluidis se mutuo librantibus, exsistet (88)

&lt;math&gt;g\mu z + C_1 = g\mu ( z + z' ) +C_2.&lt;/math&gt;
[[Fasciculus:11 hidrostatica de 61 a 70.jpg|thumb]]
92. Haec facile nunc stabiliuntur. 

1.º Si vasis communicantibus idem continetur liquidum, ut sit &lt;math&gt;\mu = \mu '&lt;/math&gt;, erit &lt;math&gt;g \mu z = g \mu' z&lt;/math&gt; ideoque &lt;math&gt;z' = \frac{C_1 - C_2}{g \mu'};&lt;/math&gt; emerget ergo &lt;math&gt;z' = 0&lt;/math&gt; vel &lt;math&gt;z' &gt; 0&lt;/math&gt;, prout &lt;math&gt;C_1 = C_2 &lt;/math&gt;vel &lt;math&gt;C_1 &gt; C_2&lt;/math&gt;: in ea videlicet qua sumus hypothesi liquidum sub externis aequalibusque pressionibus manebit in utroque vase aeque altum, sub externis vero inaequalibusque pressionibus altias apud eam partem assurget ubi minor exercetur pressio. Inde profluit explicatio variorum effectuum; cujusmodi sunt hydrargyrum in barometro suspensum, aqua elevata in siphone, in antliis etc.... Sic v. gr. quoad antlias adspirantes, dum attollitur embolus ex &lt;math&gt;H'H''&lt;/math&gt; in &lt;math&gt;HI&lt;/math&gt; (Fig. 51), aer in tubo &lt;math&gt;HB'&lt;/math&gt; confestim fit rarior, et consequenter externus aer densior aquam in receptaculo vel puteo contentam cogit in tubum ascendere usque ad altitudinem v. gr. &lt;math&gt;A' B'&lt;/math&gt;: quam ob causam descendet aqua in receptaculo ab &lt;math&gt;AE&lt;/math&gt; in &lt;math&gt;ii'&lt;/math&gt;. Jam datis &lt;math&gt;H'Q ( = a ')., EQ ( = a '' ) , HH' ( = b) ,&lt;/math&gt;itemque horizontalibus receptaculi, ac tuborum &lt;math&gt;BQFD', FQA'B'&lt;/math&gt; sectionibus &lt;math&gt;\omega, \omega' , \omega ''&lt;/math&gt;, si debeat inveniri altitudo &lt;math&gt;AA'&lt;/math&gt;, pone &lt;math&gt;AA' = \beta&lt;/math&gt; et &lt;math&gt;Ai = \beta'&lt;/math&gt;: densitates aeris interni ante et post emboli elationem sunt in ratione reciproca voluminum &lt;math display=&quot;block&quot;&gt;a'\omega' + a'' \omega'' , a'\omega' + a'' \omega'' + b\omega' - \beta \omega'';&lt;/math&gt;ideoque (87. 6º) in eadem ratione erunt pressiones a et ;
hinc
( a'w ' ta'w ') a
(a' +6) + (a&quot; – 3 ) cs&quot; |
designante m aquae densitatem , aqua elevata supra ii ' exer
cebit ( 88) pressionem
a = gm (B + B ) .
Cum igitor a' to = 5W , cumque Bw &quot;' = f'w , iccirco
( a'w' + aa'') as''
(a + b ) w + la &quot; -B, w 

sia ponitur parvitatis contemnendae prae w , erit
( a'w' ta'a ') as tgms = a .
( a ' + 6) + ( a &quot; -B) w &quot;
196
sunt hydrargyrum iu barometro suspensum , aqua elevata
in siphone ,
in antliis etc.... Sic v. gr. quoad antlias ad- ∙
spirantes , dum attollitur embolus ex H'H&quot; iu Hl (Fig.51.),
aer in tubo HB' confestim Et rarior , et consequenter ex-
ternus aer densior aquam in receptaculo velputeo conten-
tam cogit iu tubum ascendere usque ad altitudinem v. gr.
A' B' : quam ob causam descendet aqua in receptaculo ab
AE in ii'. Jam datis H'Q (: a')... EQ (: a&quot;)
, HH'(:—...- 6),
itemque horizontalibus receptaculi , ac tuberum BQFD' ,
FQA'B' sectionibus a), m', ei&quot;, si debeat inveniri altitudo AA',
pone AA':B et Ai:B' :densitates aeris interni ante et
post emboli elationem sunt in ratione reciproca voluminum
a'm' −↿− a&quot;a)&quot;
, a'æ' a&quot;o)&quot; −∣⋅− ∂∾⋅−Ba)&quot; ;
ideoque (87. 60.) in eadem ratione erunt pressiones a et se' ;
hinc
(a'ai' −⋅∣− d'un&quot;) ur −⇀⋅ (a'-1-b)m'-1-(a&quot;—B)m&quot;
designante m aquae densitatem ,,aqua elevata supra ii' exer-
cebit (88) pressionem
0&quot;: sm (49 ⊣− B')-
Cum igitur a' -l-—a&quot; :0, cumque Ba)&quot;:B'm ,
iccirco
[
U'
(a'æ' ⊣∙⋅ d'ai&quot;) ar −⊢⊣−⊣−⊰⋯≺↿⊣−∾−↜∶≻∣∃⇌≔⇌
si a)&quot; ponitur parvitatis contemnendae prae a)
, erit
(a'æ' −⋅⊢ J'ai&quot;) :: l ∙−−
(a'-l—b) ∾∣∙∙⊢ (avl—þ) 0)&quot; l gmB—a-197
la eadem hypothesi , post iteratos descensus atque ascen
sus , restituto embolo ab altitudine minima H'H ' ad maxi
mam HI , pertingat aqua ad inferiorem superficiem mem
branae G ; descendente rursus embolo et denotante k alti
tudinem aquae de se librantis atmosphaericam pressionem ,
elevabitur membrana D , ac proinde rarescente adhuc aere
in consequenti emboli ascensu , aqua pergei assurgere quo
tiescumque fuerit
EQ . HQ &lt; k (HH') .
Ut enim elevetur membrana D , debei elasticitas k' aeris
HF sic augeri quum aer HF traducitur ad volumen H'F , ut
elasticitas k' ' densati aeris H'F superet elasticitatem k aeris
atmosphaerici embolo superincumbentis . Est aulem ( 87.1º . ) .
k : k &quot; = HQ : HQ ;
vis insuper elastica k' unita ponderi aquae suspens ae EQ
librat pressionem aeris atmosphaerici , nimirum
h' + EQ = k ;
et consequenter
k &quot;
k' (HQ)
H'Q
(k- EQ) (HQ)
H'Q
Igitur
( k — EQ) ( HQ)
&gt; k ; ac proinde etc. ... H'Q

2. Tubus cylindricus longitudinis h , et in una
sui extremitate clausus , impleatur hydrargyro usque ad
197
in eadem hypothesi , post iteratus descensus atque&quot; ascen-
sus , restituto embolo ab altitudine minima H'H&quot; ad maxi-
mam Hl , pertingat aqua ad inferiorem superficiem mem-
branae G; descendente rursus embolo et denotante ]: alti-
tudinem aquae de se librantis atmosphaericam pressionem ,
elevabitur membrana D, ac proinde rarescente adhuc aere
in consequenti emboli ascensu , aqua perget assurgere quo-
tiescumque fuerit
EQ
. HQ h(HH').
Ut enim elevetur membrana D , dabei elasticitas k' aeris
HF sic augeri quum aer HF traducitur ad volumen H'F , ut
elasticitas k&quot;densati aeris H'F superet elasticitatem k aeris
atmosphaerici embolo superiucumbentis .Est autem (87.10.).
k': k&quot;:H'Q :HQ ;
vis insuper elastica k' unita punderi aquae suspensae EQ
librat pressionem aeris atmosphaerici, nimirum
k&quot; -]— EQ:k ;
et consequenter
- k&quot; k' (HQ) −∙∙≺∣⊂− EQ) (HQ). ↼−− l'l'Q HQ
igitur
de −−⋅⋅ EQ) (&quot;Q)
H'Q k; ac proinde etc.
..

2.&quot; Tubus cylindricus longitudinis A, et in una
sui extremitate clausus ,
'impleatur hydrargyro usque ad198
altitudinem hoh , cum digito ad orificium in altera parte
exsistens apposito invertatur tubus ; et remolo digito , col
locetur hoc ipsum orificium in superficie hydrargyri sta
gnantis intra aliquod vas. Ascendet aer l' ad supremam in
versi tubi partem ; augescet h , et fiet = h &quot; . Jam vero ad
inveniendam h &quot; denotante k' altitadinem hydrargyri libran
tis atmosphaericam pressionem , satis erit animadvertere
h'ki
quod exhibet altitudinem hydrargyri librantis rarefa h &quot;
clum aerem h ' ; unde
hk
h - h '
+
To
k ;
ac propterea
h &quot; = h - k' = V Th — kj» + 4hºk
2
signum inferius non pertinet ad praesens problema .
lu formula ( 10)
C, C,
Sle'
Spkk
sunt C, = gu'k ' , C,
z'
'

3º. Pone ple , pe inaequales , et C, = C2 ; habebis
p.z = p ( = + z ) ,
unde
2 : 3+ = M ' : pe ;
diversorum nempe liquidorum altitudines z et ztz' in vasis
communicantibus erunt reciproce ut ipsorum liquidorum
densitates.
198
altitudinem h—h' ,
tum digito ad orificium in altera parte
exsistens apposito invertatur tubus ; et remoto digito , col-
locetur hoc ipsum orificium in superficie hydrargyri sta-
gnantis intra aliquod vas. Ascendet aer b' ad supremam iu-
versi tubi partem ; augescet h' ,
et fiet:h&quot;. Jam vero ad
inveniendam h&quot; denotante k' altitudinim hydrargyri libran-
tis atmos'phaericam pressionem , satis erit animadvertere
quod exhibet (i£—, altitudinem hydrargyri librantis rarefa-
ctum sereni I:&quot; ; unde
h—h&quot; ∙−⊢ h—Ij—L: k' ;
ac propterea
1.&quot; −∣∙ ∣⋅−∣⊏⋅∶⊨∣∕⇀≺∣≖−∣≂⊤≻⋅⊣−⊓≖⋅∣⊏⋮∙ , z
signum inferius non pertinet ad praesens problema.
lu formula (10)
sunt C, :gpjk' , Ca.-:. ∙−−−−

diversorum nempe liquidorum altitudines :
et &lt;math&gt;z+z'&lt;/math&gt; in vasis
communicantibus erunt reciproce ut ipsorum liquidorum densitates.

=== De gravium elasticorumque fluidorum aequilibrio; necnon de altitudinibus dimetiendis ope barometri, et de pondere ac densitate vaporum. ===
93. Binae (o&lt;sup&gt;iii&lt;/sup&gt; 87.), (o&lt;sup&gt;v&lt;/sup&gt; 88.) dant &lt;math display=&quot;block&quot;&gt;\frac{d\varpi}{ \varpi} = g\theta dz;&lt;/math&gt;binae (o&lt;sup&gt;iii&lt;/sup&gt;), (o&lt;sup&gt;iv&lt;/sup&gt;  87) praebent&lt;math display=&quot;block&quot;&gt;\theta=\frac{\mu}{\varpi}=\frac{1}{ i(1 + an )}&lt;/math&gt;propterea&lt;math display=&quot;block&quot;&gt;\frac{d\varpi}{ \varpi} = \frac{gdz}{ i(1 +an )} ( b ).&lt;/math&gt; Assumptis autem logarithmis quoad basim 10,&lt;math display=&quot;block&quot;&gt;d\log_{10}(\varpi)= \frac{d\varpi}{\varpi} \log_{10}[2,718281828 ] = 0 , 4342945 \frac{d\varpi}{\varpi}&lt;/math&gt;ideoque
dos dL(W)
0,4342945
Designante igitur pressionem apud punctum ( x , 0)
in hypothesi temperiei constantis formula (6) suppeditabit.
L - Llw ')
0,4342945 ( 6' ) .
i ( 1 + an )

94. Quoad punctum ( x , y ; '-— z) supra horizontale pla
num XOY ( Fig. 47 ) , aequatio ( 6' ) suppeditat
LULLG )
0.4342945
82
i (1tan)

et inde infertur valor z dimetiendae altitudinis supra XOY
sic expressus
i ( 1 + an )
L
0,4342945g ( 6 '') .''
Haec observentur: 1. ° sub temperie = 0 , et barometrica
hydrargyri elatione =2,33958 ped. apud geographicam lati
tudinem = 48° 50' 14 &quot;, ubi gravitas 30,1959 ped. ,
Biot et Arrago invenerunt densitatem hydrargyri esse ad
aeris densitatem po ut 10467 : 1 ; inde habemus respon
dentem pressionem ( 88 )
Ww=( 30,1959) ( 10467 floo ) ( 2,33958) ,
ideoque
wo -
( 30,1959 ) ( 10467) ( 2,33958)
ро
( 30,1959 ) ( 24488, 38386) =739448, 790198174.

2.o Eo minorem experimur temperiem , quo ma-
gis supra terrestrem superficiem assurgimus ,
at, igno&quot;-
mus qua lege liat ejusmodi imminutio; designantibus ?'
et ': temperies in intimo ac supremo puncto dimetiendae
altitudinis z, solet assumi
.- 'r'-l—r
'
&quot;: 2• 201
poniturque ista temperies media constanter vigere per to
tam 2.
1

3. Singulis gradibus imminutae temperiei respon
det hydrargyri condensatio = ; igitur si M et M
5550
exhibent densitates bydrargyri sub temperiebus t' ; ac to
DY in infimo ac supremo puncto altitudinis , erit
t'
M

: 1 = M' : M, unde M

5550 I' ,-1,
1
5550
rica
ati.
ed.,
e ad
00
Temperies hydrargyri tubo barometrico inclusi nonnisi
post aliquod tempus ad aequalitatem reducitur cum aeris
circumstantis temperie , hinc t'i et t, solent definiri sub
sidio thermometri , quod ad barometrum ipsum adnecti
tar ; aliae vero t ' et determinantur ope thermometri ,
quod cum barometro non communicat.

4.0 Si l' et h exprimunt barometricas altitudi
nes apud infimum et supremum punctum altitudinis
erunt ( 88 )
h' =gM'h , =
&amp;M'h
t',
1
5550
ideoque ma'
/
Jora
us ?
endae

: - * (

I' , - 1 ]
5550

5. ° experimentis pendulorum subsidio institutis
14
'201
∣∙ poniturque ista temperies media constanter vigere per to-
tam :. '
&amp;&quot; Singulis gradibus imminutae temperiei respon-
det hydrargyri condensatio: 5150; igitur si M' et M

; )

exhibent densitates bydrargyri sub temperiebus 'r', ac 't',
ll
⋅
in infimo ac supremo puncto altitudinis , erit
———- f.:—T! M' :1: ': ∙∙∙⋅ J
5550 M M, uudeM T',—Tx
'
5550&quot;
ric-a Temperies hydrargyri tubo -barometrico inclusi nonnisi
all' post aliquod tempus ad amnalitatem reducitur cum aeria
ed.. circumstantis temperie , hinc 'r', et 1.&quot;, isolent definiri sub-
sad
ron-
sidio tbermometri , quod ad barometrnm ipsum adnecti-
tur; aliae vero 1&quot; et ': determinantur ope tbermometri ,
quod cum barometro non communicat.

4.&quot; Si b' et lt exprimunt barometricas altitudi-
nes apud infimum et supremum punctum altitudinis z ,
erunt (88)
.: M'h ∙∣≖∙ ∙∙ g ∙
a *gM .m. fr.—ff: ,
∎∎− 5550
mr ideoque
nora-
0st a. ∙∙∙ h' 1 T.]- T; .»pdæ ⊺≖−−−∣≖ (&quot;5550)'
5.0 experimentis pendulorum subsidio institutis
14
- x'!
.202
probatum est , si gi est gravitas apud geographicam la
titudinem = 45 , apud aliam latitudinem å fore
g = g . (1-0,002589cos22 ) ;
erit igitur ( 1 )
30,1959 = g1 [1–0,002588 cos2 (48° 50'14'') ]''
ac proinde
30,1959 ( 1–0,002588 cos 22 )
1 -0,002588 cos2 (48 ° 50'14 &quot; )
.
6.° Quibus positis , formula ( 6 &quot; , 94) traducetur ad
24488,38( 1–0,002588cos2 [48° 50'14*]/ (1 +0,00395+7 )
X
1-0,002588cos22
L
CO
I '
1
5550
0,4342945
-) ]
ped.
e ,
formu 95. # Sumptis logarithmis quoad basim
la ( 6 &quot; , 94 ) evadet
i (1 + an ) .
,( );
upde
et consequenter ( 87. 7. ) H =
82
e iſitan )
202
probatum est
.
si g. est gravitas apud geographicum la-
titudinem;—4

5. ∘ apud aliam latitudinem ). fore
gzg, (1—0,002588cos2)t) :
erit igitur (10)
30,, 95gzg,[1—o,002588 cosz (48-50'1 4&quot;)1
ac proinde
∙− 30,1959 (1—0,002588 cos zx)
5 1-0,002588 cos2 (4so50'14&quot;) '
6.&quot; Quibus positis, formula (E&quot;, 94) traducatur ad
24488,38(1—0.002588cosz[4so50'14&quot;])(1'-1-o,003757 BH) s—
⇁⋅⊤
' - X '1—0,002588cos2').
h. r.!—T! )]
L I: &quot;( ↿−∎∎ 5550
o,4342945 pcd.
95-0 Sumptis logarithmis quoad basim e ,
formu-
la (6&quot;. 94 ) evadet

: i(1tan)L (jul-) ;

unde
a,
ex --
z
.
et consequenter (87. 7. ) p,:
e i(t-l—an)203
1
g?
i ( 1 + an) e il1 +an)
Denotent V'et i volumen et densitatem corporis aere
demersi , ipsoque aere specifice levioris : urgebitur corpus
ad verticalem ascensum vi acceleratrice
8 (V'4 — V'x ')
Vph Gelee Me gz
if1tan)
i ( 1 + an) e
Facile intelligimus , si denotat densitatem mediam glo
bi aereostatici , verticalem ascensum ipsius globi determi
natum iri per
daz
de2
8
)
(6 '' )''
.
i (1 + an ) e
i(1 + an )
Multiplica (6 &quot; ' ) per 2dz, et sume integralia; habebis ( 27. 12.9)
gz
dz2 i(1 + an)
dla с
2g role
re' f 8 + ks)
In hypothesi velocitatis initialis = o erunt simul z=0
20

o , ideoque C

Hinc do
dz et
20 82 dza
de ² (
1 e
i (1 + an ) — 2g2 (6 &quot;) .
hey 252 &quot;
203
I
3
gz
,
i (1—l—an) e i(1—l-an)
Denotent V' et pf volumen et densitatem corporis aere
demersi, ipsoque aere specifice levioris :urgebitur corpus
ad verticalem ascensum vi acceleratrice
V' —-V' ')
,
'a' , gt P
&quot;
—g,(p p.) g( —H)- VP ⊬ M sz
i (1-l—an) e
t(t-i-an)
Facile intelligimus, si denotat p! densitatem mediam glo-
bi aereostatici, verticalem ascensum ipsius globi determi-
natum iri per
I
daz g es'
dt: p.(
Multiplica (6&quot;') per 2dz, et sume integralis; habebis (27. 12.&quot;)
..
52
dzz— c zg a t(1—l—an) '] '
&amp;f— —F g
'&quot;')'
ln hypothesi velocitatis initialis :
0
erunt simul 220
(12. 20! et ⊼⋅−−−∶∘ ,
ideoque C..:
F.]iinc
d:: 25, ∙−− ...—gj— ⋅ '['
(7:3—?( 1 — 8 :
(l*'-an)) .'.2gz (6 ).204
cto
ex cujus integratione innotescet relatio inter z ac t . Fa
dez
=o, formula ( 6 ' ' ' ) suppeditabit altitudinem 2, apud dia
dz
quam exsistet f = M ; et facto = 0 , formula ( 6 &quot; ) praebe dt
bit maximam globi elationem z.

96. Quod pertinet ad pondus ac densitatem vaporum, haec subjungimus: 

1.0 Si vase undique clauso continetur satis liquidi, ut inde sese possit evolvere tantum vaporis, quantum postulat capacitas vasis, quantitas vaporis sese evolventis pertinget ad quoddam maximum unice pendens a vigente temperie: qua videlicet permanente, istud maxinium perstabit idem aut vas exsistat vacuum ab aere, aut aerem contineat, vel quodvis aliud gas ulcum
que densatum vel rarefactum: sive autem obtinuerit illud maximum, sive non, tantundem augebitur vis elastica sicci aeris, vel gas inclusi, quanta est elasticitas evoluti vaporis.

2.° Si vapor aqueus seorsum spectatus posset sub data temperie, quin ad liquidam formam redigeretur, eam librare pressionem ā, quam sub eadem temperie librat siccus aer, ex Gay-Lussac foret densitas té aquei vaporis ad sicci aeris densitatem / ut 10 : 16 , ideoque
M=
104
16

3.• Permanente temperie , fac ut aqueus vapor seor
sum consideratus libret reipsa pressionem Wri si vaporis
densitas vocatur Hiss erit ( 87 : 1. )
10

: @ = ht ' i theo unde pos = 16 Mo ;

et denotantibus P ac P, pondera aeris ac vaporis sub ae
quali volumine ,
204
ex cuius integratione innotescet relatio inter z ac :. Fa-
dzz . . . cto 27; ::o, formula (F&quot;) suppedttabtt altitudinem :, apud
quam exsistet p.:pl; et facto 5; :o, formula (ö&quot;) praehe-
bit maximam globi elationem :.

96. Quod pertinet ad pondus ac densitatem vaporum,
haec subjungimus: 1.&quot; Si vase undique clauso continetur
.satis liquidi, ut inde sese possit evolvere tantum vapo-
ris
, quantum postulat capacitas vasis , quantitas vaporis
sese evolventis pertingat ad quoddam maximum unice pen-
dens a vigente temperie :qua videlicet permanente
,
istud
maximum perstabit idem aut vas exsistat vacuum ab ae-
re, aut aerem contineat
, vel quodvis aliud gas utcum-
que densatum vel rarefactum : sive autem obtinuerit illud
maximum, sive non, tantundem augebitur vis elastica sicci
aeris, vel gas inclusi, quanta est elasticitas evoluti vaporis.
2.&quot; Si vapor aqueus seorsum spectatus posset sub da-
ta temperie , quin ad liquidam formam redigeretur , eam
librare pressionem 0, quam sub eadem temperie librat
siccus aer, ex Gay- Lussac foret densitas p! aquei va-
poris ad sicci aeris densitatemlp. ut 10 : 16, ideoque

4. • Nunc ex aqueo vapore librante pressionem ,
et ex aere sicco emergat volumen V aeris vaporosi librantis
pressionem , et habentis densitatem &amp; ; istiusmodi aeris
massa erit Vs; aer siccus in aere vaporoso contentus utpote
librans pressionem ( 1.9 ) a— , pollebit ( 87 : 1. ° ) den
( - )
sitate Quoniam igitur ( 39) vapor aequeus in
,
to
10 WI
16 W
aere vaporoso pariter contentus pollet densitate pi
propterea ad
Ve = y (0 ) tv
10 i
16 W
por
ICCI
ris
da unde
bra €
( ---+ -s)= (:-)
1 &quot;
sic v. gr. in ordine ad aerem maxime vaporosum sub
temperie =0 , et barometrica hydrargyri altitudine
2,33958 ped. , quoniam maxima pressio librata ab aqueo
vapore sub temperie = 0 respondet barometricae altitu
dini =0,015638 ped ., erunt ( 95. 1.° )
g = W = ( 10467No) ( 2,33958 )g , w = (10467 /lo) ( 0,015638 ) g;
ac proinde designante eo respondentem valorem €,
seor
pors

; 3

2,33958 0,015638
lo 8
Eo = Too
bi
Wo :)-- (*
2,33958
=0,997495po.
205
'
—Pp.,— 10 ut,
⊬∙⊬≖≔⊉∙⊅∎∙⊉∎ F 16.;—P.
,
∙ 2,33958 — 3- .0,015638 ⇌−⇀
−⋮⊥−∘⇠ −−∃↾− ).. 8
a'., ∘ a m ↼⊣∸∘ 2,33958

: o,997495p.o.

l—xu .206
Hinc
E. 0 , 997495 ;
Ho
ratio videlicet inter densitatem aeris maxime vaporosi et
densitatem sicci aeris in ea qua sumus temperiei ac pres
sionis hypothesi.

5. • Valor i jam inventus ( 94. 1. ° ) spectat ad ae
rem siccum ; quoad aerem v. gr. maxime vaporosum erit
T. .
0,997495 flo
( 30,1959 ) ( 10467 ) ( 2,33958 )
Eo 0,997495

6. Obiter notamus illud : aquam sub satis alta
praesertim temperie in vapores versam conari sese qua
quaversus incredibili vi expandere indubia evincunt expe
rimenta. Hinc usus aquei vaporis in movendis machinis :
certo quodam tuborum valvularumque artificio vapor ex
caldario introducitur in antliam , ita , ut antliae cavitates ,
alteram infra embolum , alteram supra embolum , vicissim
obtineat, vicissimque frigidae suffusione ad pristinam redeat
Jiquiditatis conditionem ; vapor inferiorem cavitatem obtinens,
attollit embolum ; superiorem, deprimit ; embolus adnexus
est alteri ex duabus cujuspiam vectis extremitatibus ; qui
vectis altera sui extremitate vel immediate vel instrumen.
torum apte conjunctorum subsidio motum communicat rotis ,
malleis , elc.... ; prout nempe importat machinae
movendae natura.

Hinc
∙⇣∘−−−∶ o,997495;
p..
ratio videlicet inter densitatem aeris maxime vaporosi et
densitatem sicci aeris in ea qua sumus temperiei ac pres-
sionis hypothesi. ,
'
5.&quot; Valor i iam iuventus (94. 1.&quot;) spectat ad ae-
rem siccum; quoad aerem v. gr. maxime vaporosum erit
↿≖∘∙∙ ar, —(3o,1959) (10467) (2.33958)
s, o,997495 ⊬∘ o,997495 '
i..—

=== De aqua egrediente per angustum foramen e vasis verticalibus sive cylindricis, sive prismaticis.===

97. Haec praemittimus ex pluries iteratis experimentis 
[[Fasciculus:Aqua egrediens.png|thumb]]
1.° Minuta corpuscula disseminata per descendentem aquam verticaliter descendunt commuui ad sensum velocitate usque ad horizontalem &lt;math&gt;HH'&lt;/math&gt; (Fig. 52), cujus distantia ab orificio &lt;math&gt;hh'&lt;/math&gt; aequat triplum radiurn ipsius &lt;math&gt;hh'&lt;/math&gt;; tum cursum flectentia, perque lineas curvas incedentia conspirant versus orificium. Aqueae igitur particulae verticaliter descendunt usque ad &lt;math&gt;HH'&lt;/math&gt;; formaturque ab &lt;math&gt;HH'&lt;/math&gt; ad &lt;math&gt;hh'&lt;/math&gt;conoides aquea &lt;math&gt;Hhh'H'&lt;/math&gt;, quiescentibus portiunculis lateralibus &lt;math&gt;B.B'&lt;/math&gt;.

2.° Adhuc obtinent et verticalis particularum descensus, et earum conspiratio ad formandam conoidem, etsi orificium aperitur in latere vasis.

3.° Aqua ex aperto orificio verticaliter saliens assurgit ad supremam fere prementis aquae superficiem.

98. Denotet &lt;math&gt;\omega&lt;/math&gt; velocitalem aquae egredientis ex orificio &lt;math&gt;hh'&lt;/math&gt;, et &lt;math&gt;z&lt;/math&gt; altitudinem prementis aquae supra orificium, erit proxime (30:31)&lt;math display=&quot;block&quot;&gt;
\omega=\sqrt{2gz}(k) . &lt;/math&gt;Ad haec; si &lt;math&gt;\alpha&lt;/math&gt; denotat horizontalem vasis basim, &lt;math&gt;a&lt;/math&gt; orificium &lt;math&gt;hh'&lt;/math&gt;, &lt;math&gt;v&lt;/math&gt; velocitatem particularum ex quibus coalescit suprema aquae superficies, erit &lt;math&gt;
\alpha v dt= a.\omega dt,&lt;/math&gt; unde &lt;math&gt;w= vi&lt;/math&gt; et facta asna,
a
imen
roting
w = nv ( k' ) .
Hinc (27 )
asis
dz ndy
do 2gz et dt
V28%
ideoque designante 2, initialem valorem » , quum nempe t = 0 ,
inted
ft?
ae-
erit
talil
men&quot;
aul?&quot;
207

98. Denotet &amp;) velocitatem aquae egredientis ex ori-
ficio hls', et : altitudinem prementis aquae supra orifi-
cium , erit proxime (30 :31 )
a) −−∶ Vig.; (k).
Ad haec :
si a denotat horizontalem vasis basim , a ori-
licium hh' , :: velocitatem particularum , ex quibus coa-
lescit suprema aquae superficies, erit
ac «.vdtzamdt, unde a): −⇀ v :
et facta «scita,
a
mzn-v (k').
Hinc (27)
ds ⇂∕ nds
— :: ∙−−−∶ 2 :∙∙∙ ∙ dt ga ,
et dt V—zgs .
ideoque designante s., initialem valorem :, quum nem-
Pe ∁−−−−∘⇟208
i - V7(
..- , ) ( &quot; ) .

99. Sit \beta volumen aquae tempore t egredientis ex
orificio a ; erit ( 98. k . k &quot; )
233= a.orde=a(28)* . * de= a/ 2018 ( 3 - V . Jde
Propterea
B =a/25)*(*.* -VERSI-)
( k' ' ) .

100. Assumpta z = o in ( k &quot;. 98 ) , prodibit tempus
O , quo vas lotum evacuatur ; nimirum
11/ 를
2n 0—
V 29
( k &quot; ) .
In ( k &quot;) et ( K '') substitae valorem molè ex ( k ' ) ; habebis''
2n 을
21
5 B
ag
V28
2n (25–2-ce). ( &quot;) .

101. Ex (k &quot; ) sequitur illud : si duo vasa habuerint
et altitudines zo , zo, el orificia a , a' aequalia , tempo
ra 0 , 0 quibus deplentur , erunt in ratione basium a,a' ,
siquidem
2n 2n '
á
0 : 0 =
V 29

: z ' .

V 28
--- N : n ' = . Q : a' :
a '
208
≖−−−−⇁ 21( soi—1 ii) (li&quot;)
-
l/Zg
99. Sit þ volumen aquae tempore :egredientis ex
orificio a ; erit (98. I:. k&quot;)
.l. s
'
s i— dþzaüdtza (25? s
'dt:a(2g)ir&quot; (
zog— &amp;
t
)dt. ⇂

Propterea
,

100. Assumpta ≖∙∶−−− ∘
in (Is&quot;. 98 )
. prodibit tempus
9 , quo vas totum evacuatur; nimirum
9: 2: ∣∙∘⋚ (z.-&quot;) ⋅
l/Zg
In (It-&quot;) et (I.-&quot;') substitue valorem s.,ïli ex (Is&quot;); habebis

:: 6— 2n 3,- ag( .

⋅−
2-
. — ⋅ 20 t): wg I.&quot;. (3 ,&quot; ( )
101. Ex (It-&quot;) sequitur illud: si duo vasa habuerint
et altitudines s.
, a',, et orificia a
.
a' aequalia. tempo-
ra 9. 9' quibus deplcntur ,
erunt in ratione basium a.d.
siquidem
∙ 2n 2n'

: 9:∶−− zo : ...z'o zn:n'— :—,-—a:a':

l/Zg a a· 209

102. Quantitates aquarum successivis et aequalibus tem
poribus effluentium decrescunt secundum numeros impa-
res ordine retrogrado sumptos. Assertio facile ostenditur
e secunda ( k ) , facto successive t=1,2,3,4, • ; nam
quantitates illae prodibunt expressae per
ag
2n (29-1 ) , L ( 49-4 ) – ( 29-11 , P.(69-9)– ( 49-4) ,
Se
ag ( 80-16)
2n
ag ( 60-9) ... , seu 2n
ag
2n (20-1 ) , 29–3 ) , (29-5 ) , (29–7), - ;
ideoque etc... Idipsum eruitur ex (k &quot; ) et ex prima (k&quot; ) ;
denotantibus enim 21 , 22, 23 , ... valores z respondentes tem
poribus 1 , 2, 3, ... eae praebebunt
&amp; 02 , 2, 3
(0-1) 2,225 2n? 2n2 , =-2,(0-2) », 23 =
S
(0-3 ) , ... 29-1 2n2
8
2n2 ;
unde
6
( 29-1 ) , 21-22 2n?
8
( 29-3 ) , Zz- 23 =
2n2
8
2n2 (26-5) , ... 29-3 8
2n?
et consequenter etc...
Hinc si dividendum sit vas in partes successivis dati tem
'209
102. Quantitates aquarum successivis et aequalibus tem--
poribus ellluentium .decrescunt secundum numeros impa-
res ordine retrogrado sumptos. Assertio facile ostenditur
e secunda (k')
.
facto successive t::1,2,3.4, .; nam
qnantitates illae prodibunt expressae per
Zn (29 'l), 2&quot; (49 4) 2&quot; (29 1) , 2n(69 9) g(49 4) ,
a—g - .. &quot;£ -
∙∙ 2&quot;(89 16) 2&quot; (69 9) . , seu
ag
- es -
es -
∙−−
7:091). 2&quot; (29 3), 2809 5)sa g(29- &quot;711&quot;;
ideoque etc... Idipsum eruitur ex (In&quot;) et exprime (k');
denotantibus enim sus,, &amp;
.... valores :respondentes tem-
poribus 1,2, 3, ... eae praebebunt
'
z.,:
⋚−⊯≖ ⊖⋅∙ z. ↼−− ⊋⋅⋚⇆≺⊖⋅↿ ):,
≖≖−−−∶⇄−⋚⊑≺⊖∙⊋≻≖∣ za⇌∎
- 2 -' ∙−−− i.— ↿∠∏−−≖⋅ i(ä 3) zo Zn' '
↴
uude ⋅
2, ∙z. ∙−−−⋮⋚≔ (29-1),z,-z, −−∶ Zif-;, (za-3), 22-33 −∙−−∙∸−
s- - ...g. .
ZI€3(29 5), ∙∙∙ Zo-x —-2na ,
et consequenter etc..
Hinc si dividendum sit vas in partes successivis dati tem-210
1 poris a unitatibus vacuandas , determinata altima 20-1
ceterae usque ad primanı erunt
320-7.526-4,72 6-7** (29-3 )z 0-10 ( 26-1 ) 0-1 .
1
1
Liquet autem fore
2:6-1 + 326-1 + 520.4 + 720-1 + . + 20-3)26-17
0
(29-1930_1 = {1 + 29-1)
o
2
0-1 = 622
6-41 1
d
.

103. Tria subjungimus, quae certissimis constant experimentis. 

1º. Vena aquae exilientis a foramine aperto in pertenui lamina magis semper contrahitur usque ad ejusmodi distantiam ab orificio, quae vix aequat ipsius orificii radium; estque venae maxime contractae area cc' ad orificii aream ut 5 : 8 circiter. Istius contractionis ratio ex eo desumenda videtur quod aqueae particulae etiam paullo extra vas retinent obliquos convergentesque motus, quibus orificium subierunt.

2.• Tanta effluit aqua intra datum tempus ex fo
ramine aperto in pertenui lamina , quantam suppeditat for
5
mula ( k &quot; ) , modo tamen pro a substituamus 8

3.º Aptatis orificio exterius tubis cylindricis, co
nicis etc., pro varietate tuborum variae habebuntur quan
titates aquae dato tempore exilientis.

104. Haec notentur 

1º. Acceleratio , per quam velocitas aquae admodum exigua usque ad HH' mutatur in finalein satisque grandem effluxus velocitatem, tota manifeste perficitur ab HH ad cc' intra spatium interceptum conoide ac vena contracta, ubi nempe descendentium stratorum amplitudines citissime decrescunt. Vas ergo ABB'A '
a.
210
poris 9 unitatibus vacuaudas
.
determinata ultima &quot;9-1 ,
ceterae usque ad primam erunt&quot;
3z9-1, 529-1,7z ⊖∙↿∙∙∙ (29-3): ∂∙↿ '
(29-1)z 9-1 ∙
Liquet autem fore
ze, ↿
∎∎⊢∍∅∂∙↿⊣−⋮≖∂ ∙↿ ⊣−⋅∄≖∂∙↿∙⊢∙∙∙↤⊋∂∙⊰≻∅∂∙↿−⊢
. 9 ∙∙∙ :
(29-1)z9-1:(1-1—29-1)ïz ⊖∙↿ —9 294 -
7

1 '.
er
perto
ejus
citci
ori.
spectandum erit tamquam terminatum tubo Hcc'll' ad se
ctionem HH ' aptato.

2.0 Motus aquae defluentis in regularibus alveis
traduci potest ad motum aquae prosilientis ex angustis
vasorum orificiis. Concipe regularem alveun secari plano
verticali ; in plano isto insculpi plura foramina , ex qui
bus effluat aqua ; iisque nova addi foramina ut indefinite
crescat foraminum numerus , totaquc sectio veluti unicum
efficiat foramen infinitis numero foraminibus' coalescens :
cum e singulis foraminibus aqua debeat effluere veloci
tate illa , qua et erumperet e vase ad eamdem altitudi
nem pleno , et , sublato plano , Queret in eodem sectionis
loco , idem ferme erit casus aquae defluentis per alveum
et aquae prosilieatis e vase ad eamdem altitudinem pleno.

3. • Si in regulari atque horizontali alveo mo
vetur inferior aqua ob superioris aquae pressionem , nec
directionum obliquitate , et fundi laterumque resistentia
turbatur conceptus motus , apud particulam quamvis de
notante i altitudinem superincumbentis aquae , exprimet
V 2gi particulae velocitatem.

4.° Quod si regularis alveus ad horizontem ex
sistat inclinatus , sitque m altitudo debita velocitati apud
supremam aquae superficiem , cum haec velocitas ( levio
ribus corporibus aquae injectis determinari potest ) utpote
orta ab inclinatione alvei debeat aquae omni esse
munis , exbibebit V 28 (i + m ) particulae velocitatem .

5.° Hinc poterit in utroque casu definiri quan
titas V aquarum intra datum tempus t defluentium apud
quamlibet regularis alvei sectionem ; sic v. gr. in hypo
thesi rectangularis sectionis habentis latitudinem r , erit
in primo casu
i
2tri.
V = tr įdi
3
no es
دالاق
uibus
fo
o for
com
S, CO
paano
relo
in 6
Dani
?ptom
Vžg
I stra
B!!
in secundo
lCP
perta
ejus-
filicii
, ori-
iot!
.aullo
uibui
⊊∣∝⊦
luan-
«de
new&quot;
ipua
! slfl'
BN
↗−
⋅ 211
spectandum erit tamquam terminatum tubo Hcc'll' ad se-
ctionem HH' aptato.
2.&quot; Motus aquae defluentis in regularibus alveis
traduci potest ad motum aquae prosilientis ex angustis
vasorum orificiis. Concipe regularem alveum secari plano
verticali .; in plano isto insculpi plura foramina , ex qui-
bus effluat aqua; iisque nova addi foramina ut indefinite
crescat foraminum numerus , totaque sectio veluti unicum
efficiat foramen inlinitis numero foraminibus' coalescens :
cum e singulis foraminibus aqua debeat eflluere veloci-
tate illa, qua et erumperet e vase ad eamdem altitudi-
nem pleno , et
, sublato plano ,
(lueret in eodem sectionis
loco, idem ferme erit casus aquae defluentis per alveum
et aquae prosilientis e vase ad eamdem altitudinem pleno.

105. Auctores non pauci tractantes de motu li
quidorum ex apertis luminibus effluentium , illud usorpare
solent tanquam principium , quod nempe unumquodque li
quidi in vase quolibet descendentis tenuissimum et hori
zontale stratum coalescat iisdem constanter particulis com
muni , eaque tantum verticali , velocitate donatis . Deno
tante v verticalem velocitalem , qua pollet in fine tempo
ris i quodvis massae liquidae punctum ( x, y, z ) sollicita
tum gravitate g , vis acceleratris de se valens producere
dy
actualem motum exprimetur ( 28) per : et qaoniam ,
dt
praecisis etiam mutuis punctorum pressionibns , adhuc ta du men vis de gigneret actualem motum ; ideo , attentis pres
sionibus , consistet in aequilibrio punctum (x , y , z) solli
du citatum vi g Propterea ( 88 )
dt
do
dz
dvi
dt (kº ) .
Attenta insuper liquidi continuitate ( liquidum ponitur in
capax compressionis ) ; sequitur , si A designat amplitudi
nem cujusvis strati horizontalis , fore ( 98)
viw = a : A , unde v =
Ä ( * &quot; &quot; ) ;
w est functio temporis t ; A distantiae ; ab XOY : sequi
212
∙∙∙ i. &amp; 3
VZU'l/ng (i-l—m) di: Z',..l/Zg g'[(10 m)⇣⇥≖∶∣⋅ a
denotat i., sectionis altitudinem .

1054: Auctores non pauci tractantes de motu li-
quidarum ex apertis luminibus ellluentium. illud usurpare
solent tanquam prineipium . quod nempe unumquodque li-
quidi in vase quolibet descendentis tenuissimum et hori-
zontale stratum coalescat iisdem constanter particulis com-
muni, eaque tantum verticali .
velocitate dona-tis . Deuo-
tante v verticalem velocitatem
, qua pollet in fine tempo-
ris : quodvis massae liquidae punctum (.r.-7, :) sollicita-
tum gravitate g. vis acceleratrix de se valens-producere
. dv ⋅
⋅ ∙
actualem motum exprimetur (28) per .d—t : et quoniam
.
praecisis etiam mutuis punctorum pressionibus , adhuc ta-
dv
men vis —d—£ gigneret actualem motu-m; ideo , attentis pres-
sionibus
.
consistet in aequilibrio 'punctum (.r.-y. :) solli-
citatnm vi g—⋛⋮ ∙ PrOpterea (88)
der . ⋅ dv
'
z,; ∙−−∶ P- (
−−⋅ (17) ('i ')
-
Atteuta insuper liquidi continuitate (liquidum ponitur in-
capax compressionis ) ; sequitur . si A designat amplitudi-
nem cuiusvis strati horizontalis .
fore (98)
psa-ca: A, lel). ,
undevzr-a— A (
cc est functio temporis :; A distantiae :ab XOï: sequi-213
1
i
tur quoque supremam descendentis liquidi superficiem ma
nere horizontalem . Ex kl( ) habemus
dv a da
a do
dt
aw dA dx
A2 dz dc
-
A dt A do
.
aw dA
a da a’w2 dA
A2 dz A di A3 da
iccirco formula (K ™ ) traducelur ad
do dz dz =+ (sds - au de):
1 sumptisque integralibus quoad % ,
==C+u(sma )
Zo
ic
exprimit 200 distantiam inter XOY et supremam liquidi
superficiem A , Denotante w, pressionem v . gr. atmosphae.
ricam in superficiem illam , assequimur
Two
-= C+4 (** 241,3.
)
unde C=0. – (
( 50-100)
.
li propterea
-=o +15(2-)-avenit SA- G --&gt;) ( 47 ).
Zu
je Apud orificium
213
tur quoque supremam descendentis liquidi superficiem ma-
nere horizontalem. Ex (F&quot;) habemus
dv a de.) am dA d: a dm
.dt—A dt A*dz dc-TA dc
as) ubi a da) ama dA ∙
Aza.&quot; Ad: A3 d.'
iccirco formula (Is&quot;) traducetur ad
' dar − das d:. am: dA
- )
⋅⊋−⋮∁≀∅−−∙↱∙≺⊰∠≀∅−−∅∙∣⊺⋮⊺−⊢ A3 d: dz ,
,. sumptisque integralibus quoad s,
⋍≖⇌∁−⊦⊬≺≊≴−∘≤≀≜∫≖≤≀⋮− −
.-)
,.
dt A 2112
zo
.i-
eXprimit s., distantiam inter XOï et supremam liquidi
superficiem A.,. Deuotante wo pressionem v. gr. atmosphae-
ricam in superficiem illam
, assequimur
2 2 2 2
≔∘−−−⋅∁−⊦⇤∸≼∊≴∘−≦⋏∘∶≕≻ ⋅
....
∁−−⇌≖≖∘
..
(g.... &quot;.? ):
[&quot; propterea
d ad:. 2
1 1
w:eod—Pgu-uþauä A a&quot; P:) (A*—. :) (k&quot;&quot;).
zo

; . Apud orificium214

1
Wo
A2 a
designantibus insuper b et i distantias ipsius orificii ab XOY et ab A. ,
m=b , 2 = b - i , 1 - % = i :
facto igitur
b
dz
A
biI !
erit ibi
mode,
gi - sa- (1-4 ) = (A &quot; .
Quoad (k &quot; &quot;&quot; ) et ( k &quot; ) notamus haec tria. 1.0# Si a est
parvitatis contemnendae , ex (k &quot; ) profluet
a = w.tugis mo ) ,
ut in casu liquidi aequilibrari (88) ; ex (k' ) vero emerget
V2gi ,
quae formula recidit in formulam ( k) .
2.0* Si , affluente novo liquido, eadem servatur
in vase altitudo liquoris , quantitates i, A., B exsistent con
stanles ac datae ; et facto
a2 1 h
A.2
↿ 1
≖⋝−−≖≖⋅∙ ∙ :::—:::;
designantibus insuper &amp; et t' distantias ipsius oriücii ab
XOT et ab A.,.
526 , sozb—t' .
s—sozi:
facto igitur
erit ibi
ad!» c.)&quot;( a2
g' Bdc 2.↿∎∎∎⊼∘⊑≻∶∘ (kl
Quoad (k&quot;&quot;) et (Is&quot;) n0tamus haec tria. ↿∙∘∙ Si a est
parvitatis contemnendae
, ex (k&quot;&quot;) profluet
Uzwoillg (z'—*o) )
ut in casu liquidi aequilibrati (88) ; ex ('tu) vero emerget
∾⋅⇌ vra-u
quae formula recidit in formulam' (lt).
2.0a Si
. affluente novo liquido. eadem servatur
in vase altitudo liquoris. quantitates i. A.,. B exsistent con-
stantes ac datae ; et facto
emuli
a ad
Qiiia215
formula ( k &quot; ) praebebit
h
d
2a Bdt V
2gi
2ada
2gi - hwa hV
2gi
h2 .62
2gi
d h
h
d
a V
2gi
V
2gi
hv
2gil
1+
v
2gi
+

: ) h h
;

-
Vzgi
unde , sumptis bogarithmis quoad basim
a
Bta
log
hy
2gi
V 2gi + hw
V
2gi
ha
non additar constans et arbitraria quantitas utpote =0
siquidem tempori t =o respondet w =o. Ex ista aequatio
ne emergit
Bhty2gi
V2gi(1
a
h
1 + e
Bhiv
2gi
a
inferimus , elapso brevi quodam tempore t, fieri ad sensum
1
V 2gii
itemque
21 5
formula (li&quot;) praebebit
d—L
.,
Bdt— iuda −− 21. V? —
⋣∊∙−⋅∣⇂≖∾≖ hl/Zgi IP
1——c.)2
Zgi
d
a ita di;—00 —( ⇂∕2gi ∣
l/2gi ) h
'⋅⊾ ⋅
l/Z-g—l ↿∙∙∙ .b— 6)
,
⇂∕⇄∃−⋮⋅∾ Vzgi
unde
, sumptis bgarithmis quoad basim e ,
'Bt: ]: a— log iii—E? :
l/th' Vzgi −− hæ
non additur constans et arbitraria quantitas utpote ∙−−−∘ ,
siquidem tempori tzo respondet 6) 30. Ex ista aequatio-
ne emergit
⋅≖∃∣≖≀⇂∕⋝⋮⋜
&amp;
l/2-g-i1—e— ::
∎∎⇀ h B'm/223-
1—l—e— a
inferimus
. elapso'brevi quodam tempore :. fieri ad sensum
1 − −−−−−−↗↓−∎∕∑∊≀⋅⊰
itemque216
= w.tuzia - 2 .) – paga?i( 1 hot G1 - ),-- VE
3.•* Si vas consistit in verticali cylindro , vel pri
smate , A erit constans , et A.=A ; insuper
dz 7-zo 1
A A B
ic
zo

=== Aliquid subjungitur circa generalem theoriam motus corporum fluidorum. ===
106.* Velocitas v, qua pollet if fine temporis !
quodvis massae fluidae punctum ( x, y, z) sollicitatum (86)
vi acceleratrice Q , resolvatur in ternas v' , w &quot; , 1 &quot;&quot; coor
dinalis axibus Ox, OY, OZ parallelas ; erunt ( 29 ) dý ,
1
dy' ' vires iisdem axibus parallelae , in dt dt
quas resolvitur' vis acceleratrix q' valens de se produ
cere actualem motum. Quoniam , etsi praecisis pun
cloruni mutuis pressionibus , adhuc tamen gignit actualem
motum ; ideo , attentis pressionibus , consistet in aequi
librio punctum ( 2, y, z ) sollicitatum viribus X
, ) – , dv',
1
Y dy&quot; , 2 dy'&quot;'; ac proinde ( 86. o )
di
1
dt
de = - ( x – do ). -- ( v- à dv&quot; ) , )
de u ( 2-2 &quot; )..
( 6)
216 .
↴
a':' 1 ↿ a Vii—g'
∙
szo-l-pg(2'.—20) uia (Aa Ag) ,
VS—K ∙
T
∙
∶⊰∙∘∙ Si vas consistit in verticali cylindro. vel pri-
smate
, A erit constans, et A.,:zA; insuper

:

Zo
Aliquid subjungitur circa generalem theoriam motus
corporum [[ uidorum.

106: Velocitas 0, qua pollet i! fine temporis :
quodvis massae fluidae punctum ( æ, y, z) sollicitatum (86)
vi acceleratrice ?, resolvatur in ternas v'. 9&quot;
.
v'&quot; coor-
dinatis axibus OK, OT. OZ parallelas ; erunt (29) ;llg-dvl ,
,, 1 ∙∙∙ ∙ ∙∙ ∙ ∙
(Z— dv
, &amp;? dv vires iisdem axibus parallelae ,
tn
quas resolvitur vis acceleratrix ?' valens de se produ-
cere actualem motum. Quoniam (a' .
etsi praecisis pun-
ctorum mutuis pressionibus, adhuc tamen gignit actualem
motum; ideo , attentis pressionibus , consistet in aequi-
librio punctum (z, y, :
) sollicitatam viribus X — −↿− dv'.
dc
y— —dv&quot;,Z—- dc dv' ; ac promde (86. o)217
1
dt ,
dy
du
dx
axt
du' !
(du ?
Habitis v ', &quot;.0&quot;, pro functionibus variabilium x, y, z, t,
exsistent ( 27. 24.0)
dy' du dv du du' dx + dy +
dzt.
dix dz dc
dur dul
dy + dz +
dt,
dy dz de
du du dy't dy + dz + dt ,
dx dy dz de
du du=
dr
seu ,
ob dx v dt , dy udt , dz udt ( 27 ) ,
dv dú dv ' dun du'
dt ,
dx dy dz de
dur dvd dv du'a + dt ,
dy dz de
( 6 ).
dy'll dy dy &quot;
\dx dy dz
dat
di axt du.
du
dyt
dzt dz dt dy
dl ;
at
will +
leo
lesin
' du'
\dx vt
alt
- de
dy ut
21&quot; + .!&quot;
to
at
dt ,
ide
dx du du.
vt
de ede :
w
itot
dy
dz
formulaeque (6) vertentur in
15
1
1
217
Habitis v'. v&quot; . v'&quot;, p. pro functionibus variabilium x,]. z,t,
∙ exsistent (27. 243) '
/
dVr-äï-I-dæ-f-g dy—l—d 7; v/dz—l-dïvt-dth
.
I, I/
dp'p&quot;: &quot;
≤⋮∙−≤⋅⇗↙↙∠∞∙⋅∣− dv ∙−∙↙∣∫⊣−↙≀↥≟ dz-l— ii)—dt,
III '&quot;
dv'&quot;: dv Tdæ—i—djr dv/Il d-v'&quot; dy −⊸⊢−−⋅ zdz—l- -^——--dt,
d d ⊬−− ⊋⊥∸↙≢↕ ↙∄↕≤−⊦ Hari- ⋮⋮↙∄≖⊣−−↙−∣≛⋮∠≀≀⊰
seu , Ob dx:
'v' dt .ei)-':«:;&quot;dt, dz 37)/&quot;dt (2&quot;..
dv! ∣∙∙− d'&quot;, v&quot; ∣∣ (if—,) dv—(ïr-v-l-ï P-l-d—z-IVI-ï-dï— dt,
&quot; &quot;
vl] II
V&quot;'—:(d—; V, &quot;J—d gr.&quot;- .v/j-l—g-z- will-i-ïi'l;-—-)d[,
(V).
dv ∣−− dvlll dui/I
−−−−⊋⋤−≼ .
[v;/1.&quot;, dv'&quot;
,
∣∣
), 1
∎∎⊢∎∎∎−∎−∎ dz 'l&quot; dt —)dt ⋅∣⊹ '
∙−− dP'. ,v/ !
dlu' ut dp' ∣∣∣ dp') ∙
dy. (dæ'v [
dy&quot; ⋅−↱⋅⋅∓⇂≀ −↽⊋−∁⋅− dt.
formulaeque (6) vertentur in
15218
dos deild dy&quot;
dx
dv'
dx dy
dz che si (
(v
do ( 6&quot;) v'
do'
dy
v
du
dz w
dur
dy
dx de - ) ,
dy ')
do dv'

: -(2

v'
dy't
dy
v
du Win
dz dx dz de
107 #. Quae portiuncula infinitesima massae fluidae a
pud punctum ( x , 3 , 2 ) sub volumine V in fine tem
poris i exprimitur per V , eadem sub volumine V+dV
in fine temporis + dt ad punctam aliud translata expri
metur per ( V+dV ) ( pe + du ); ideoque
V = V + dV) (v + dpl)= Vu + udV + Vdp. + dp.dV ,
et consequenter, misso dudv,
Vdp. + pdV= ( 6 &quot; ).
Sumatur V = dxdydz, aequale nimirum parallelepipedo
rectangulo AF ( Fig. 47. ) sub laterculis AD( =dx) , AB( =
dy ) , AH = dz); punctaque A , B, C , D , H , M , F , E po
nantur transferri tempusculo de ad A ' , B , C , D , H' , M',
F' , E , ut sit V + DV = A'F'. Transferetur A in A '
velocitatibus d' , 0 , 2, juxta coordinatos axes ,
runtque
e
x + v'dt, y tv&quot; dt , z tudt
coordinatac puncti A': designatis v ', u ' &quot; per
d7 dx d] dz : '
,
dm' dv&quot;' −∙∙: Z- ∣- dv'&quot; du''' dv'&quot; ∣∣'''
∙∙∙ ∙−−⋁∣∣∣∙− —) ∙
dz F
(
da: v dy 'v dz dt
107-. Quae portiuncula infinitesima massae fluidae a-
pud punctum (æ
,
I,: ) sub volumine V in fine tem-
poris : exprimitur per VP-o eadem sub volumine V-l-dV
in fine temporis t −⊢ dt ad punctum aliud transl'ata expri-
metur per ( V-l—dV) ( p. dy. ); ideoque
Virsz-l-dV) (p.-l—dp.) ∙−−∶ ⋁∣↓∙⊹ ⊦∙∠≀∇−⊢ ∇⊂∣≴⊥∙⋅⊢ dde .
et consequenter, misso dpdV,
. Vdp. −⊦ ⊬↙∣∇−−∶⋄
('b'&quot;).
Sumatur Vzdædydz, aequale nimirum parallelepipedo
rectangulo AF (Fig. 47.) sub laterculis AD(-:-:dæ) , AB(-—--
dy ), AH(-:dz); punctaque A, B, C, D, H, M , F , E po-
nantur transferri tempusculo dt ad A' , B' , C' , D' , H' , M',
F', E' , ut sit V −↿− dV −−∶ A'F'. Transferetur A in A'
velocitatibus v'. a:&quot; ,
∙⇂∙∥⋅ juxta coordinatas axes , e-
runtque ∕∕∕ '
..: ⊣−⋁∣↙∣⊀ ∙
]−⊦ wa: , z −⊢ war:
coordinatae puncti A': designatis v', v&quot;, «a'/' per219
fi( x , y , %, t ) , fa(x , y , z , 1), 13 (x , y , z , t) ,
expriment
fi (x , y , z + d2, e) ,fz(x , y , z + dz, t ), f3( x , y, z + dz, t)
velocitates coordinatis axibas parallelas puncti H euntis in
H '; et cum babeamus ( 27. 24.)
filx9,2 + dz,t) = f (x , y ,z, e)7df1(x,y,z,e)dz = uti
du
dz dz ,
dz
e ao
em dy ”
fa (x , y, 2 + dz, t ) = 0&quot; +
dz ,
dz
spri.
f3( x , y , z + dz, t ) = 0 !!! allt dv !
dz,
dz
IV ,
coordinatae puncti H'erunt
X
+ (v + da )dt,y + (&quot;* + de )de,

: +de+ (** + adaptada dt:

pipedo
AB =
E po inferimus, missis infinitesimis tertii ordinis, fore ( 50. 6º. )
1 , M.
in 4
5 ,
ee A'H' =
[ledesdeu + )de de +
) ]=d =+
dy &quot;
-dz dt dz dt .
dz
Motus puncti Cin C'juxta coordinatos axes fiet velocitatibus
!
15.
C ∎∙
em-
[pl'l'
IV.
. 219
fuci-'s], 3! 1), fa(æs)'s 3! t) ∙ ⊀∍≺⋅↕∎∙∫↿≖∙ :),
expriment
ftlæsfaz-l'dzs 1) ,falæoys z 'l'dz; t)sf3(æs ïs ≖∙∙∣− d:, 1)
velocitates coordinatis axibus parallelas puncti Hieuntis in
H'; et cum habeamus (27. 240.)
dfx(æJ,z,t)d fax-a',: 4—dz,t):f.(æ,y,z, :
dz dz—v ↾−⊦↙↙∙⋚∙ —dz,
&quot;
falæsïsz (I:-',! :):vlf'i'd'ä- dz:
d'UIII fave,], z-l-dz,t ): ⇝∣∣∣−⊢ —zdz.
coordinatae puncti H' erunt ' &quot;'
ahi-(» ∣⊣− −↲≖≻≳≀∙∫−⊦≼↙∣⊣−≝∂≖⋟↙∦⋅

: ↽⊦ dz −⊦ (W.;- ↙⋛↙−≖

d: )dc.
inferimus, missis infinitesimis tertii ordinis, fore (50. 60-)
A'H': RSTV) dz-dz −⊦≺⋅⋮↷⋛↗−−−≖−∥≖⋟↙≀≖≏
d:: −⊦
d.,/II dv!&quot; -
(d:-[- d: dzdi )]ä :dz-F—dzdt-
Motus puncti Cm∁∣ juxta coordinatas axes fiet velocitatibus220
falar + dx , y + dy, zil ) = fi(x , y , 2,1 ) +
afı( 8•
7,5,6)
det dfi (x , y , 2,1) du
dy
dy
dic = tIdxt dx
falx + dx , y tdy, 2, 1 ) = &quot;&quot; + -dat dx
du &quot;
dy ,
dy
dumi
dy !!!
ON +
f3(x + dx ,y + dy , z , 1 ) = &quot;&quot; dat dx dy ;
dy
inde prodeunt coordinatae puncti C
du
d )
dy x + dx + (v + ad det )dt
y + dy + ( ** + na tempat day ) di,

: + ( v&quot; + data darym dy de :

motus puncti F in F ' juxta coordinatos axes fiet velocitatibus
du'
filxtdxy + dy,z + d2,2)= x + xdx + dydy +
du
dz,
dz
dy&quot; du&quot;
falxtdx,y + dy,atdz,t)= &quot; + de + dy dy
du &quot;
da dz,
du &quot;
du
f3( x + dx, y + dy; z + dz,t) = 1 &quot;'
-dxt
dy&quot;
dy dy +
dz;
dx dz
inde exsurgunt coordinatae puncti F
220
fdæ'i'dng—i-dy- ≖∙⋅↕⋟⇌∣≖≺∝⇟∫∙ ze t)&quot;,-
df,(æ.j,z,t) dfl(æsyszvt)
dw'd d &quot; ≀∂≖≺∙↿⊏−≱⊢↙↕↡∫↽⊢∂∫⋅ ≖↿−⊸⋅⋅⇂∙∥⊣−↽⊋− &quot;L &quot;ad; df '
f3(æ-l—dæ,y—]-dy,z, :
)-— v'∣∣⊹−⇁∙
inde prodeunt coordinatae puncti C'
∙↴⊲−⊢∠∄∸≀∶⊣− (⊣−⋅≦⋮∠↴↧⋅↕⊣⇀−− ↙∄⋤↙↿∫⋟≴↙≀⋅
∫∔∂⋮∫⊹ (
∣∣⊣↼ d,,⇡⋮≀−−⋅∶≴←⊦≤−⋚−∥⇩≀ wa.)
vll/ du ≖↽⊦≺⊛ ∣∣∣
w&quot;'-l--d—-æ dr—l— df )dz:
motus puncti Fm F'luxta coordinatas axes Eet velocitatibus
∙ . . . '
' d ,.
fia—W;? ∂∫∙≖−−∠⇣∙≀⋟∶⋁⊹⊼∶↙⊩⊦≣↗ −⋤−∶⊔⊹ −:dz, dv&quot; dv&quot; dv&quot;
ta(æ-i-er-l-dr, a—l—dzn): ⊎∣∣⊣⋅∙⋣∂≛−⊦ df dy ! dz ds,
d&quot; III d..&quot; f3(æ—]—dx, ⊹↙∄∙↗∙≖−⊢∣≂∙ ():—Ju&quot; i-i-d; dælL dr cir—]— —-dz;
inde exsurguut coordinatae puncti F'221
dyn
= da + ( + van de tener tous de Jdeo
s + d3 + ( v +adar an nas tudi nadia )dt,
s + de + ( * + dpt dathetn dy + advan die Jde:
1
inferimus, missis infinitesimis tertii ordinis, fore
CF = [ 'de de + oem )deº de +
( de + de &quot;de de ))]]* = da +
dy&quot;
de dt.
dz
Ad motum puncti B in B ', computatum in coordinatis axi
bus, spectant velocitates
f( x, y +dy, z, t ) , falx , yt dy, z, t), f3(x ,y + dy, z, t ) ;
ad con similem vero motum puncti M in M' velocitates tatibus
fi(x ,y + dy, z + dz, t) , 82(x , y + dy ,ztdz, t ) ,
th
dan
f3(x , y + dy , z + dz , t ) :
dy&quot;.
propterea coordinatae puncti B ’ desi
dz
+ (x + dy dy )de , y + dy + (.* + dar dy ) dt,
du&quot;,
dy
7
+ (*&quot;'+ dydy hdi:
7;
(221'
I
æ—l—dæ-l-(tb -]-d −∙⋮dx-j—d −−∣vlddy-l— ——dz )dt,
,,
'
dv&quot; . dv&quot; y-l-dJ-l-( −⊢−− da.−−∥↙≀↓⊣− ⊒∫−∠≀∫−⊢−− ↙≀≖≻∠≀∁∙
&quot; dv&quot;: z-l—ds-l—(&quot; ⊣−≦−≦⊥∅≀∝−⊦↙∄ dyd ∣
↙↙≖ d: )dt:
unferimus, missis infinitesimis tertii ordinis, fore
∙∙∙⋅ dv) ∙ '
(du&quot;ïd) ∙ es'—[(? d: a: &amp;
∠≀∥≀⋍⋅−⊦
≺∁≀≖−⊢⋛≖ ——dzdt )]; :dz—l-Q—ds dt.
d:.
Ad motnm puncti B in B', computatnm in coordinatis axi-
bus, spectant velocitates
I.i-ïs;)&quot; ⊣∙∙ dy. 39 i)
'
f2(æay—l— d]: 2. t)af3(-'rsy—l—dft Z, !) 3
ad consimilem vero motum &quot;puncti M in M' velocitates
↿∎≺∞∙∙↗↾⊣−∠∄∫∙≖⊣− dzs t) sf2(æ sy'l—fi'r, ≖−⊦∠∄≖ ∙ :) ∙
fam ,y-l-dy.a-1- a.:):
propterea coordinatae puncti B'
æ-i- (&quot;'-l- ⋛⋚∠∄∫≻↙∄∁ ,J—l-dy—l-('
≻≖≀⋅≂⋮ &quot;j,-l— 72:41)!&quot;-
z −∣⋅− (vm-I— dv
dy222
coordinatae puncti M
++ (1 - en deJdt,
y + dy + (** + disa dy + &quot;deJdi.
z + dz + (** +
(** + + en in diehele
hinc
B'M dz +
du
dzdt .
dz
Ad motum puncti D in D ', computatum in coordinatis a
xibus, pertinent velocitates
fi ( x + dx, y , z , 1 ) , fa( x + dx,y ,z, 1), f3(x + dx, y , z ,t ) ;
ad consimilem autem motum puncti E in E' velocitates
filx + dr, y, z + dz, t ) , f (x + dx, y , z + dz , t ) ,
f (x + dx ,y , z + dz , t ):
proinde coordinatae puncti D'
de&quot;

* + dx + (ut ea adx)de,y + ( * + dxdx )dt,

++ (- + de -dx)dici
222
coordinatae puncti M'
x-l—(tf— ⋛≶↙≀∫⊣− ——dz)dt,
maH-( ⇂≀⋅⋛−−⊦ pri—4449
≖−⊦↶≀≖↼⊦≼ ⋮⋅∠⋛⋮−⊣− MH-;'dzdu)
hinc
,B'M'-— ∙−− tis-l- -—-dzdt.
Ad motnm puncti D in D', computatum in coordinatis a-
xibus, pertinent velocitates
ru(æ ∙−⊦ ciæ,], zit)1fa(æ &quot;l'dæofszo t) sf3(x&quot;'i&quot;dæs)'; 2; 1);
ad consimilem autem motum puncti E in E' velocitates
fdx—i-dæqæz-l-dz, t ) 'fa(x-l-dx,y , z—l—dz .
t ).
B(æ-j-dæ ,y, s--]— dz .: ):.
proinde coordinatae puncti D'
∝⋅⊦⊄↿↕∸−⊦≼↩∙−⊦ ——dæ)dc ,y—l—(v' −⊦≤−−⋮⋅⊑⋅↙≀⋅⊐∁⋟↲↥∙

: ∙⋅⊢ ≺∙∽⋯∙⊢ ⋛⋮≽∙⋮⇣↿∙↕≻≺≀∷223

puncti autem E
drt
s + dx +((uv + des de +die dz)dt ,y+ (** + de la
de &quot;a )de, a (* &quot; + dz) dt;
2 + dz to dv&quot;&quot; dy&quot; ,
det da dz
et consequenter
D'E' = dz to
du&quot;.
dz dzdt .
Itaque
A'H ' = C'F' = B'M ' = D'E' = dz +
du 
dzdt :
đz
simili modo eruuntur
AD = B'C ' FM H'E di = dx +
dx
dxdt ,
1t ) ;
A'B' C'D FE H'M ' =
dú'
dy + dydt.
dy es
thi
Ex laterculorum aequalitate manifeste consequitur eorum
parallelismus ; eritque A'F' parallelepipedum obliquangu
lum ; ita tamen , ut ejus anguli infinities parum diffe
rant ab angulis rectis parallelepipedi rectanguli AF ; quan
doquidem AF nonnisi tempusculo infinitesimo transfer
tur in A'F ' . Nunc ex H ' v . gr. due perpendiculum Ha in
areolam A'B'C'D ; erit
A'F ' = H'a . A'B'C'D' = H'a . A'B ' . A'D' sio B'A'D '
A'H ' . A'B' . A'D' sin B'A'D' sip H'A'a :
d:.
223
puncti autem E'
dv' '
dv' )
.. da: −⊢ dz z .7' 41- dæ—t- ∙⋅∎∙⋅−⊢ du:-1- (v' ∙∙⊢ da:
dv&quot; dv'&quot; dv&quot;' )
−∙ dz —-d.r —-d d ;
dz)dt ,
z-t-dz-tï 'v. −⋅⊢ da: ∙−⊢ ds :
f
et consequenter
⋅ dv&quot;:
D'E'c: dz −⊢ 71&quot;—2. dzdt .
Itaque
llo
'
'v
A'H':C'F' −−∶ B'M' ∙−−∶ D'E' :: d: ⊣⋅− ∙−≀⋮⋅≖−↙≀⋍⊄∄↥ :
simili modo eruuntur
A'n' :: B'C' −−∶ F'M' −−∶ H'E':dæ −⊢↙≟⋛ dædt
.
A'B'r: C'D' ::F'E': H'M' ∙⋅−−−∸ dy ⊣⋅−∙≣⊥⋅ ↙≀∙↨↾∠∄≀⋅∙ ]
Ex latel-culorum aequalitate manifeste consequitur eorum
parallelismus; eritque A'F' parallelepipcdum obliquangu—
lum; ita tamen , ut eius anguli infinities parum diffe-
rant ab angulis rectis parallelepipedi rectanguli AF ; quan-
doquidem AF nonnisi tempusculo inünitesimo transfer-
tur in A'F' . Nunc ex H' v. gr. due perpendiculum H'a in
areolam A'B'C'D' ; erit ∙ ∼
A'F' ∙−−− H'a . A'B'C'D' ::
'
'a . A'B' . A'D' sin B'A'D' ∶−−⋅≖
A'H' . A'B' . A'D' sin B'A'D' sin H'A'a :224
denotantibus w et w'angulos infinitesimos , poterunt anguli
B'A'D ' , H'A'a repraesentari per 90º + w , 90 ° +6 ; iccirco
sin B'A'D' sin H'A'a = sin (90 ° + w ) sin ( 90° + W' ) =
62 614 w'4 coswcos6= ( 1 -... ) ( 1 ...) . 2 2.3.4 2 2.3.4
Quare , missis infinitesimis quinti ordinis ,
dv &quot; A'F ' = (dx +
dv' du &quot;
dxdt) (dy +
dx dzdt) x
dy dydı) (dz + az
wa
du
( - -
(1-7 • det dy&quot;
• det du&quot;
dt) dxdydz ;
2 dy
dz
ideoque
dV = AF - V =
-
(dv
dx- det
-det
dy
di) dxdydz.
His positis , vertelur ( 6 '' ) in''
dxdydzdje ele
dv
\dx
dtot dt +
dv&quot;
dy
dz di )dxdyds= 0,
seu ( 106.6' )
dje du ut u'+
dr dz
djelo
du +
dy
dt
dvi dv&quot; du &quot; .
dy + demon dz ) =
(619) .
dx

108. * Si massa fluida est incapax compressionis,
unaquaeque particula immutabilem habebit densitatem
eritque du = o : proinde ( 106 , 6')
224
denotantibus eo et tu'-angulos inünitesimos , poterunt anguli
B'A'D' , H'A'a repraesentari per 900-l-cu , 900-l-co'; iccirco
sin B'A'D' sin H'A'a −−∶⋅ sin (90&quot;-FG) sin (900 ⊣−∙ w') ∶∸⋅
. ↿ a): 034 ↿ tu' te'-': )
cosmcosw—( ∙−⋅⋍−−⊢≳∙∙⋝⋅∕⇂∙−−⋯⋟≺ −−∙⋮∙ m—m .
Quare , missis infinitesimis quinti ordinis
,
I d'u' dr&quot; d'v &quot;
'
'
∙−−−
− −
'
AF ∙−− ∙−− (dx—t- dædxdt,(dy-t—- d] afydt) (dz-t- az dzdr) )(
a 'a .'
&quot; '&quot; (1 &quot;2' −−∘≩≻∙−− ≺↿⊹⋛⊰↲↥⊹≘⋚∂∁⊹↙≩⊤∶∂∊≻∂∅∂∫∂∥
ideoque
(lv-.:A'F' v
(
vd: : dv dc.-92:11) dædyde .
dæ dy dz
His positis , vertetur (b&quot;') in
dt&quot; I),, vl'l dæd.) dzdp-t— &quot;(c'ïx dc ⊣−∙ 217 dc ⊣−∙ 72- dt)dædydz ::o,
seu (106 . b')
&quot;&quot;
↿≀∙⊣⋅− d&quot; ∣∣⊣− a'&quot; &quot;-4-'-'—'-' −⊢

: 17- 2?&quot; f??&quot; dt

d'v- ⋅⋅⊢ dv&quot; dv'&quot; ⋅
⋅⋅∙∙− o
F- b,; ( ⊣⋅−⋮⋤ )
—- ( ).
108.s Si massa fluida est incapax compressionis,
unaquaeque particula immutabilem liabebit densitatem ,
eritque dy.:o : proinde (106
. b')225
de
vt
die du du v &quot; + 2 &quot; +
=
0

;

dy dx dz de
et consequenter
( 107.b
)
(
6
)
dv dy
d.x
+
+
dy dz
Formulae
(
6
&quot;
)
,
(69) suppeditant incognitas
a
,
l
,
v
,
v
&quot;,
v
&quot;
.
expressas per
x
,
y
,
z
,
t

; obtentis autem

v
'
,
v
&quot;
,
?,
&quot; per xy
, zat
,
eruentur
x
,
y
,
z per
t ex formulis
dy dx dz
dc
-
”
dc
ru!!!
dt
Si massa fluida incapas compressionis est insuper ho
mogenea
, prima
(
6
&quot;
) fiet idemtica
, satisque erunt
(
6
&quot;
) et
secunda
(
6
&quot;
)
.ad incognitas
,
u
'
,
v
&quot;
v
'&quot; determinandas
. De
mum si massa fluida pollet elasticitate
, formalis
(
6
&quot;
) et
(61
)
jungenda erit formula
(
o
&quot;
. 87.6
'
)
.

=== De tubis capillaribus. ===
[[Fasciculus:Capillarity.svg|thumb|Capillares]]
109. Etsi liquidum homogeneam in vasis communicantibus (92.1º.) manet aeque altum, iu tubis tamen vitreis admodum angustis (dicuntur capillares) utrinque apertis, et altera extremitate demersis aquae vel hydrargyro, cernimus aquam suprema superficie concava terminatam ascendere supra horizoutalem circumambientis liquidi superficiem, hydrargyrum vero suprema superficie convexa terminatum descendere infra horizontalem circumdantis liquidi superficiem: ad istius modi ascensum descensumque explicandum, haec animadvertimus.

1º. ln phaenomenis gravium liquidorum expendendis gravitatem considerantes haud habuimus rationem sive virium quibus liquidi particulae se mutuo petunt, sive virium quibus vasorum materies ad se trahit particulas illas. Porro materiales particulae duplici pollent vi attractiva; altera se prodit utcumque crescant distantiae, sequiturque (82) rationem reciprocam duplicatam distantiarum; altera se prodit dumtaxat in contactu vel quamproxime contactum, sequiturque rationem quamdam distantiarum nondum compertam. Ubi sermo est de liquorum aequilibrio, possumus ab attractione primi generis absque sensibili errore praescindere: ad attractionem secundi generis quod pertinet; cum in contactu exsistat validissima, inde fit ut suprema liquidi superficies prope vasorum latera induat figuram curvam, modo concavarn, modo convexam, et nonnisi ad aliquam ab ipsis lateribus distantiam dici queat physice horizontalis. Exhibeat TT' (Fig. 53) verticalem tubum v. gr. vitreum, utrinque apertum, et infra horizontalem liquidi superficiem partim demersum; O centrum circularis areae tubo interceptae apud eam superficiem; A particulam liquidi in area ista sub actionem vilreae particulae R; OX rectam transeuntem per A; OY horizontalem rectam perpendiculariter insistentem rectae OX; OZ verticalem rectam. Si denotat vim qua A tendit in R, designatis per h, k, i cosinibus angulorum quos AR facit cum ox, oy,
OZ, resolvetur in ternas
ph , pk , ọ
iisdem OX , OY , OZ parallelas: ex R in planum XOY
ducatur perpendiculum Rp , producaturque in R' donec fiat
R'p = Rp ; teadet A in R' vi aequipollente ternis
ch , pk , - oi : demissis perpendiculis ex R , R' in planum Xoz , iisque productis donec productiones aequentur ipsis perpendicu
226
rium quibus liquidi particulae se mutuo petunt, sive virium quibus vasorum materies ad se trahitparticulas illas. 

manifeste determinabuutur in tubo duo puncta , quorum vires dabunt componentes
gh , - ok , pi ,
sh , - ok , - qi :
in ferimus particulam A , elisis componentibus parallelis
rectae OY , itemque componentibus parallelis rectae OZ ,
sollicitatum iri juxta AX vi
4Σ φh
proveniente ex tubi materia. In OX sume Ab = Aa ; duc
verticalem bb' ; et quod in ordine ad tubi materiam est q,
in ordine ad liquidi materiam sit q' : quisque intelligit par
ticulam An elisis componentibus horizontalibus, trahi ver
ticaliter deorsum vi
4 Epi
promanante ex liquido intercepto superficie cylindrica , quam
general recta bb' dum sibi constanter parallela movetur in
peripheria circuli habentis radium Ab. Liquidum ultra
superficiem cylindricam praebet vires
.
- 2 Eph , 2 &quot;pi,
alteram horizontaliter agentem juxta XO , alteram verti
caliter deorsum. Omnes itaque vires sollicitantes particulam
A traducentur ad horizontalem
4Eph -2Ep'h = 2 [2Eoh —Eph] ,
et ad verticalem
45' pit 23&quot; ' i.
227
lis, manifeste determinabuutur in tubo duo puncta. quo-
rnm vires dabunt componentes
9ht—9k09i'
-ph.-—9k,—qn':
inferimus particulam A .
elisis componentibus parallelis
rectae 0?
,
itemque componentibus parallelis rectae OZ
,
sollicitatumeiri juxta AX vi
4297:
proveniente ex tubi materia. In OX sume Ab: Aa; duc
verticalem 65; et quod in ordine ad tubi materiam est p,
in ordine, ad liquidi materiam sit go' :quisque intelligit par-
ticulam A. elisis componentibus horizontalibus, trahi ver-
ticaliter deorsum vi ↽
42'9'i.
promanante ex liquido intercepto superficie cylindrica, quam
generat recta bb' dum sibi constanter parallela movetur in
peripheria circuli habentis radium Ab. Liquidum ultra
superficiem cylindricam praebet vires
-— 2
297: , 2E'p'i ,
alteram horizontaliter agentem juxta KO, alteram verti-
caliter deorsum. Omnes itaque vires sollicitantes particulam
A traducentur ad horizontalem
4ng −− ⇄∑∲∣∣∣:2929]: −∙− ∑⊈⊅⋅∣⋅⊐ .
et ad verticalem (f)
42: p'i-l- 22&quot;qa' i.228
Potest 2Eph -Eph esse aut &gt; o , velo, vel = 0: in primo
casu vis aequipollens et gravitati , et binis (f ) , deviabit a di
rectione verticali faciendo angulum acutum cum AX; et
quia ( 83.3º. ) vis illa debet normaliter sese dirigere ad libra
tam liquidi superficiem , ideo suprema liquidi superficies in
duet curvam concavamque figuram : in secundo casu vis
aequipollens et gravitati, et binis (f), deviabit quidem a ver
ticali directione, sed faciendo angulum oblusum cum AX ;
propterea ( 87. 3 • ) suprema liquidi superficies induet curyam
convexamque figuram : in tertio denique casu ex duabus (8)
remanebit sola verticalis, et consequenter suprema liquidi
superficies erit plana atque horizontalis.

2º . Massae liquidae OS , OS' (Fig. 54 ) ejusdem
naturae, planisque superficiebus OP , OʻP ' terminatae, ae
qualiter trahunt exilissimas columnellas liquidas AR , A'R'
perpendiculariter superficiebus ipsis insistentes; nisi tamen
columella AR intra massam OS trahitur deorsum , columel
la vero A'R' extra massam O'S' trahitur sursum. Intelligan
tur enim centris A et A ', radiisque aequalibus AB et A'B ',
ultra quos sensibilis attractio liquidi non protenditur, des
cribi hemisphaeria BCD , B'C'D' : si vires , quibus hemi
sphaeria agunt in particulas A, A ', resolvuntur in binas, alte
ram horizontalem , alteram verticalem; elisis horizontalibus,
remanebunt verticales, quarum summa est eadem utrinque
cum hoc tantum discrimine quod A trahitur deorsum et A
sursum. ln columellis sume nunc duo alia puncta E, Eʻae
quidistantia ab A , A' , radiisque aequalibus EL, E'L ' ( = AB)
describe segmenta sphaerica FML, F'M'L' : accepla EV=EA,
ductoque plano horizontali HK, vires ex QHKN , QFMN
in punctum E utpote aequales et contrariae se mutuo de
struent , ipsumque E solo segmento HLK deorsum trahe
tur : vis ex HLK deorsum sollicitans particulam E ae
quatur vi ex F'L'M ' sursum trahenti particulam E'; siqui
dem haec segmenta sunt aequalia et similiter posita ad
contrarias partes quoad E et E '. Cum igitur idem redeat
228
Potest 229h—29'h esse aut) a. vel( a, vel:o: in primo
casu vis aequipollens et gravitati, et binis ([ ), deviabit a di-
rectione verticali faciendo angulum acntum- cum AK; et
quia ( 87. 30.) vis illa debet normaliter sese dirigere ad libra-
tam liquidi superficiem, ideo suprema liquidi superficies in-
duet curvam concavamque figuram: in secundo casu vis '
aequipollens et gravitati, et binis (f), deviabit quidema ver-
ticali directione, sed faciendo angulum obtusum cum ax,
propterea (87. 3&quot;-) suprema liquidi superficies induet curvam
convexamque figuram :in tertio denique casu ex duabus (f)
remanebit sola verticalis, et consequenter suprema liquidi
supedicies erit plana atque horizontalis.
2&quot;. Massae liquidae OS , US' (Fig. 54) eiusdem
naturae, planisque superficiebus OP
, O'P' terminatae, ae-
qualiter trahunt exilissimas columellas liquidas AR , A'R'
perpendiculariter superficiebus ipsis insistentes; nisi tamen
columella AR intra massam OS trahitur deorsum. columel-
la vero A'R' extra massam O'S'trabitur sursum. Intelligan-
tur enim centris A et A', radiisque aequalibus AB et A'B',
ultra quos sensibilis attractio liquidi non protenditur,des-
cribi hemisphaeria BCD , B'C'D' : si vires , quibus hemi-
sphaeria aguntin particulas A, A', resolvuntur in binas, alte-
ram horizontalem , alteram verticalem;elisis horisontalibus,
remanebunt verticales, quarum summa est eadem utrinque
cum hoc tantum discrimine quod A trahitur deorsum et A'
sursum. ln columellis sume nunc duo alia puncta E, E'ae-
quidistantia ab A , A', radiisque aequalibus EL, E'L' (::AB)
describe segmenta sphaerica FML, F'M'L': accepta EVzEA,
doctoque plano horizontali HK, vires ex QHKN , QFMN
in punctum E utpote aequales et contrariae se mutuo de-
struent
,
ipsumque E solo segmenta HLK deorsum trahe-
tnr : vis ex HLK deorsum sollicitans particulam E ae-
quatur vi ex F'L'M' sursum trahenti particulam E'; siqui-
dem haec segmenta sunt aequalia et similiter posita ad
contrarias partes quoad E et E'. Cum igitur idem redeat229
cimo
di.

; et

bra
sin
vis
Ver
LAX;
ryam
argumentum de caeteris particulis inter A et C , necnon
inter A ' et C ' ( ponimus A'C &quot; — A'C' ) , cumque particulae
infra C viribus contrariis et aequalibus urgeantur, infra C
sensibili non subjiciantur actioni, jam patet etc
In eodem liquido vis, qua deorsum vel sursum colamella
trahitur, constans est; eam in sequentibus exhibebimus per K.

3º. Fac ut massa liquida BAB'QQ (Fig. 55) , quae intercipitur superficie sphaerica BAB' et plano tangente
QQ, trahat externam columellam liquidanı AR perpendicula
riter insistentem plano tangenti apud contactum A : quo
niam BAB O'Q gignitur rotatione areae ABQ circa ra
dium AO, perinde erit sive spectetur attractio massae BAB'Q'Q,
sive spectentur attractiones infinitarum numero superficierum
cylindricarum , quae in ea rotatione gignuntur a perpendicu
lis DC , D'C' , ... demissis ex punctis D , D ' . circu
laris arcus BA in rectam QA. Exprimant p , pi ... per
pendicula DC , D'C', . . ; 9.9 , .. perpendiculorum di
stantias AC , AC' . .. ab A computatas in AQ; sitque r
sphaericae superficiei radius OA: ob magnam lineolarum
p , p ', . . . tenuitatem prae q, , ..
quidi
Eden
A'R'
ameo
amel
gaD
AB,
des
erunt
lemi
aleo
р
92
2r ,pa2r
libus
Bogu
et A
et consequenter praefatae superficies cylindricae exhibe buntur per
-AB
EL
MY
2πη 비유
Toq3
,2πα g's Tig'3
seu 9
dem 2r 2r
cabe
ae
aqui.
Atqui ob eamdem illam tenuitatem puncta uniuscujusque su :
perficiei cylindricae haberi possunt pro aequidistantibus ab
unoquoque columellae puncto: designantibus igitur o, o, ..
quantitates pendentes et a certa quadam distantiarum lege,
deal
sit
ac-
qui'
de:!
229
argumentum de caeteris particulis inter A et C- , necnon
inter A' et C' (ponimus A'C&quot; :: A&quot;C), cumque particulae
infra C viribus contrariis et aequalibus urgeantur, infraC&quot;
sensibili non subjiciantur actioni, iam patet etc
..... ∙
.In eodem liquido vis, qua deorsum vel sursum colnmella
trahitur, constans est; eam in sequentibus exhibebimus per K.
30. Fac ut massa liquida BAB'Q'Q (Fig. 55), quae
intercipitur superficie sphaerica BAB' et plano tangente
QQ',trahat externam columellam liquidam AR perpendicula-
riter insistentem plano tangenti apud contactum A*: quo-
niam BAB'Q'Q gignitur rotatione areae ABQ circa ra-
dium AO, perinde erit sive spectetur attractio massae BAB'Q'Q,
sive spectentur attractiones infinitarum numero superficierum
cylindricarum, quae in ea rotatione gignuntura perpendicu-
lis DC , D'C', . . . demissis ex punctis D, D' . . . circu-
laris arcus BA in rectam QA. Exprimant p , p', ... per-
pendicula DC, D'C',. :
.; q, q' , .. perpendiculorum di-
stantias AC, A.C' .. ab A computatas in AQ, sitque :-
sphaericae superficiei radius OA: ob magnam lineolarum
p, p, . ..
tenuitatem prae q, q', ...,
erunt
9' ∣
vf:
?&quot; 2r'p— 2r'...'l
et consequenter praefatae superücies cylindricae exhibe-
buntur per
' q,! &quot;03 &quot;q '3
, ∙ ∙ ∙
'
seu , 21) r r
q?
2:- 2nq ,Zitq ,...
Atqui ob eamdem illam tenuitatem puncta uniuscuiusque su.-
perficiei cylindricae haberi possunt pro aequidistantibus ab
unoquoque columellae puncto: designantibus igitur &amp;, ö'. ..
qnantitates pendentes et a certa quadam distantiarum lege,.
Kn.-&quot;M— . ⇀ ⋅−−∙⇀∙⋅↼ ⋅⋅−↪∎⋅⊾ −−↼↼∎↼ ↽− ↼−⋅−⋅−⇀−⇀−⋅∙∎∙∙↼ −−↼
↰⋅−↽⋅
- −⋍⇂∙⋅−230
et a liquidi densitate, et a cosinibus angulorum quos cum
AO faciunt rectae ab attrahentibus superficierum punctis
ductae ad attracta columellae puncta, eae superficies colu
mellam sursum verticaliter trahent viribus
Teq38 Tog'38
totaque massa BAB'D'Q columellam AR sursum verticali
ter trahet vi
1938 +7.9'38' +
.
Si concipitur altera massa liquida PAP'OʻQ intercepta pla
no QQ et nova superficie sphaerica PAP, cujus radius
O'A = p , simili ratione ostendetur vim ex PAP'Q'Q fore
παδ+πα35 '+ .
.
Vires itaque istae erunt ut
- Eq: 8 : &quot;5q?: = &gt;
erunt nempe reciproce at sphaericarum superficierum ra
dii. Hinc designante H opportunam quantitatem constan
tem , exprimet
H
vim, qua sursum verticaliter massa liquida BAB'Q'Q trahit
.columellam AR : caeterum quisque videt fore H = 12q30.
230
et a liquidi densitate, et a cusinibus angulum quos cum ⋅
AO faciunt rectae ab attrahentibns superficierum punctis
ductae ad attracta columellae pnncta, eae superficies colu-
mellam sursum verticaliter trahent viribus
nq3d th'3ö&quot;
,.... r :-
totaque massa BAB'Q'Q columellam AR sursum verticali-
tcr trahet vi
∏⊄∍∂−⊢∏⊄⋅∃∂⋅⊹ ∙ ∙ ∙ ∙
.
r .
Si concipitur altera massa liquida PAP'Q'Qintcrcepta pla-
no QQ' et nova superficie sphaerica PAP', cuius radius) O'A-z r' , simili ratione ostendetur vim ex PAP'Q'Q fore
12:738 −−⊢ ∏⊄≖∃∂∣⋅−⊢ ∙ ∙ ∙ ∙
r
Vires itaque istae erunt ut
7! :: ↿ ↿
r Zq d . —r,2q ∙−−≀∙ −∙⋮∙∙ ,
erunt nempe reciproce ut sphaericarum superficierum ra-
dii. Hinc designante H Opportunam quantitatem constan-
tem
, exprimet
H
∙∙−∙−
'
vim, qua sursum verticaliter massa liquida BAB'Q'Q trahit
.columellam'AR : caeterum quisque videt fore H::an36.231

: cum e.

tis r
r
H

4. Quantitas K major est quam
nim K exprimat vim ( 2.0) , qua sursum trahitur columella
H
AR a massa liquida LFAF'L' , exprimet K vim
qua sursum trahitur AR a segmento sphaerico MBAB '.
H
Id vero importat K &gt; o ; ergo etc.

5.0 Massa liquida BAB'E'E terminetur superficie
concavo -spherica BAB' : ducto per A plano tangente QQ ,
sollicitabitur columella AR deorsum ( 2.9 ) vi K ex EFAF'E' ,
H
sarsum (3.9) vi ex BAB'F'F ; tota igitur BAB'E'E trahet
deorsum columellam AR vi (4.0)
.
bio
3
13
н
. K
IR
in i, i' , ... ,

6.• Superficies sphaerica NAN' habens radium
O'A = 0A tangatur plano QQ in A ; columella AR ae
que trahetur sursum a massa liquida NAN'Q'Q ac trabi
lur a massa BAB'Q'Q : patet ( 3. ) çum ex eo quod, pro
ductis DC , D'C' , ... donec occurrant arcui circulari AN
exsistunt DC=Ci; D'C' =C'i, ... ; tum ex
eo quod Ci , Ci', ... , sunt tenuissimae prae AC, AC,
si qua pars columellae non trahitur sursum
sit tenuissima prae reliqua parte sursum altracta .

7.º Columella igitur AR magis trahetur deor
sum ab EE'N'AN quam ab EE'F'AF ; excessusque unius
H
attractionis supra alteram erit . Propterea massa liqui
da desinens in superficiem convexo- sphaericam NAN' traliet
deorsum columellam AR vi
ita ut ea
1
K +
1
i.
231
H 4.&quot; Quantitas K major est quam —-: cum e-
- r
nim K exprimat-vim (29)
. qua sursum trahitur columella
AR a massa liqui/da LFAF'L', exprimet K —E r
vim ,
qua sursum trahitur AR a segmento sphaerico MBAB'.
Id veroinrportat'K—g- ≻∘ ∙∙∙ ergo etc.
. . .
59 Massa liquida BAB'E'E terminetur superficie
concavo-spherica BAB' : ducto per A plano tangente QQ',
sollicitabitur columella AR deorsum (2.0) vi K ex EFAF'E',
sursum (3.&quot;) vi!-.;l ex BAB'F'F; tota igitur BAB'E'E trabet
deorsum columellam AR vi (4.0)
∙
K—ll'a r
6.0 Superficies sphaerica NAN' habens radium
()&quot;AzOA tangatur plano QQ' in A; columella AB ae-
que trahetur sursum a massa liquida NAN'Q'Q ac trabi-
tnr a massa BAB'Q'Q: patet (39) tum ex eo quod, pro—
ductis DC, D'C' , ... donec occurrant arcui circulari AN
in i, s&quot;.
..., exsistunt DCxCi; D'C'..-::C't&quot;, ...: tum ex
eo quod Ci, C'i',
. . . ,
sunt tenuissimae prae AC. AC',
ita ut si qua pars columellae non trahitur sursum , ea
sit tenuissima prae reliqua parte sursum attracta.
7.o Columella igitur AR magis trahetur deor-
sum ab EE'N'AN quam ab EE'F'AF; eicessusque unius
H
attractionis snpra alteram erit -— . Propterea massa liqui-
⋅ r
da desinens in superficiem convexo-sphaericam NAN' trahet
deorsum columellam AR vi
n
∣≺⊣−−−⊑−⋅∙232

8.º Pone superficiem BAB' neque esse sphaericam
,
neque gigni rotatione ullius curvae circa AO

; secla BAB

planis transeuntibus per A0
, curvilineae sectiones apud
contaclum
A gaudebunt inaequalibus osculi radiis

; quos

inter
( demonstrationem suo tempore videre erit in parte 3.4 nostrorum elementorum matheseos 0. 118
) bi
ni reperiunlur
, alter minimus
(
=
r
)
, alter maxi
mus
(
=
r
'
), pertinentes ad binas sectiones sub angulo re
cto invicem constitutas
. Iam
, in ea qua sumus hypothe
si
, hoc pacto determinabitur visex BAB'Q'Q sursum
verticaliter trahens columellam AR
. Intelligatur coalesce
re BAB'Q'Q ex infinitis numero superficiebus cylindri
cis normaliter insistentibus plano tangenti QQ
'

:

' unaquae
que superficies cylindrica non eamdem habebit ubique
altitudinem

; sed apud bina puncta

e diametro opposita
,
quibus nempe maximus respondet circựlus osculator
, al
titudo erit minima

; apud bina puncta

e diametro pari
ter opposita
, perque gradus 90 ab illis primis sejuncta
,
quibus videlicet minimus respondet circulus osculator
,
altitudo erit maxima

: apud intermedia puncta altitudines

interjacebunt minimam maximamque
. Quapropter evoluta
superficie cylindrica super aliquo plano
, ea poterit reprae
sentari per aream QNN
&quot;
Q
&quot;
( Fig
. 56
)

; NN

&quot; aequatur
basi superficiei cylindricae

; QN et

Q
&quot;
N
&quot; simul cum Q'N
'
exhibent altitndines minimas

; Fu et F'u

' altitudines ma
ximas hinc Nu

= uN
'
= N'u
'
= u'N
&quot;
. ob perexiguum ba
seos cylindricae radium poterunt QF
, Q'F
, Q'F
'
,
(
&quot;
F
'
haberi pro lineis rectis

; eritque

1
QN
+Fu NN &quot; QʻF'Q'FQ = 1NuFQ = 4 Nu 2 NN ” QN + F4 2
232
8.&quot; Pone superficiem BAB' neque esse sphaericam,
neque gigni rotatione ullius curvae circa AO; secta BAB'
planis transeuntibus per AO, curvilineae sectiones apud
contactum
A gaudebunt inaequalibus osculi radiis quos

inter (demonstrationem suo tempore videre erit in par-
te 3.*' nostrorum elementorum matheseos n. 118) bi-s
ni reperiuntur, alter minimus
( ::r)
, alter maxi-
mus (:r'), pertinentes ad binas sectiones sub angulo re-
cto invicem constitutas. Iam
,
in ea qua sumus hypothe-
si, hoc pacto determinabitur vis et BAB'Q'Q sursum
verticaliter trahens columellam AR. Intelligatur coalesce-
re BABHQQ ex infinitis numero superficiebus cylindri-
cis normaliter insisteutibus plano tangenti QQ' :'unaquae-
que superficies cylindrica- non eamdem habebit ubique
altitudinem; sed apud bina puncta
e diametro opposita,
quibus nempe maximus respondet circulus osculator
, al-
titudo erit minima; apud bina puncta
c diametro pari-
ter opposita, perque gradus 90 ab illis &quot;primis seiuncta,
quibus videlicet minimus respondet circulus osculator
,
altitudo erit maxima: apud intermedia puncta altitudines
interjacebunt minimam maximamque. Quapropter evoluta
superficie cylindrica spper aliquo plano
, ea poterit reprae-
sentari per aream QNN&quot;Q&quot; (Fig. 56); NN&quot; aequatur
basi snperficiei cylindricae QN et Q&quot;N' simul cum Q'N'

exhibent altitudines minimas -; Fa et F'u' altitudines ma-
ximas; biuc NucuN'zN'u'2u'N&quot;
.
ob perexiguum ba-
seos cylindricae radium poterunt QF
, Q'F, Q'F ', Q&quot;F'
habcri pro lineis rectis; eritque
NN&quot;Q'F'Q'FQ::4NuFQ

:

4 Nu 'QN'zl-F&quot;

:

∙ NN&quot; QNj'F&quot;
.233
ericam,
a BAB
es apud
i quoi
Retentis igitur denominationibus ( 3.9 ) , superficies cylin
dricae , ex quibus intelligitur coalescere massa BAB'Q'Q
( Fig. 55 ) , erunt
92 Lo pár.
+
92 q /2
+
2r
2r' 2r'
2πα ,
2r
2tq'
&gt;
seu
mari 2 2
alore
ypothe
Sursa mga (: + ?). mg ( + ),...
Dalesce
linde
naquat
ubige
pposila,
et consequenter massa BAB'Q'Q desinens in superficiem
BAB' utcumque concavam trahet verticaliter sursum co
lumellam AR vi
or , al
πα
2
o pari ?( + 1) + &quot;, ( +3)x + ...
Atqui ( 3.9 ) 7.938 + Teq'38 ' + .... = H :
Ejuncta,
ulator ,
Studios
evolu
reprat
equatur
exprimetur ergo vis illa per
16+)
les m2
um bio
1 07
9. Sume Q '&quot;'N '&quot; et F &quot; u &quot; ( Fig. 56 ) aequidi
stantes ab QN et Fu : erunt Q &quot; N &quot;&quot; , F &quot; u &quot; duae ex al
titudinibus intermediis ( 8. ) respondentes duabus sectio
nibus curvilineis ad angulum rectum invicem constitutis.
Radii circulorum osculantium sectiones istas apud A ( Fig.
55 ) designentar per po&quot; et p' : erit ( Fig. 56 )
9'2
Q&quot; N &quot; +F &quot; u &quot; =
92
2r &quot;.
qo +
27 
+
g'?
2r . ' 2r' '
1
16
233
Retentis' igitur denominationibus (3.&quot;) , superficies cylin-
dricae, ex quibus intelligitur coalescere massa BAB'Q'Q
(Fig. 55 )
,
erunt
/
£ fl&quot; q&quot; a&quot;
2r .l-Zr ∙ 2r
21tq 2 , 27tq .l-Zr' 2 , ∙ , seu
↔⇍≖↙≀⋮↿≺ ,
') &quot;rv ')
⋅⋅
2 rii-&quot; 2
≀∙⊣−∣⋅⋅∅⋅⋅⋅⋅
et consequenter massa BAB'Q'Q desinens in superficiem
BAB' utcumque concavam trahet verticaliter sursum co-
lumellam AR vi
&quot;f(ï-Fl?) ∂⊹∙⋮≖−≣−⋅⋮−≺−∶−−⊦∙≙≻∂↝⊹∙ ∙∙
r
Atqui (3.0) nq3ö -l-7tq'3d' ∙−⊦ ∙ ∙ ∙:∙ H:
exprimetur ergo vis illa per
⋅≣⋅≺⊥⊣−−↿⊺≻ ⋅

: - r

9.0 Sume Q'&quot;&quot;'N et F&quot;u&quot; (Fig. 56) aequidi-
stantes ab QN et Fu : erunt Q'&quot;N&quot;', F&quot;u&quot; duae ex al-
titudinibus intermediis (89) respondentes duabus sectio- .
nibus curvilineis ad angulum ≖⋅∁∁⋯⊞∙ invicem constitutis.
Radii circulorum osculantium sectiones istas apud A ( Fig.
55 ) designentur per r&quot; et r'&quot; : erit (Fig. 56)
0 (
ns '
lr Jr ' ∎∙ lr.
'
a a 'a' ': ≺≀⋅⋅∙↓↜⇃∙⋅∙−⊦∣∂⇁⇈≀↓∙∙∶∶−∙↙−⇃−⋅− ⊄
⋞∣ −⊦⋞∣∙∙234
est autem
mane cc
Q &quot; N '&quot; + F &quot; u&quot; = QN + Fu =
92
2r
9'2
2r +
2r etc.:
1
igitur
+++++
110
et consequenter
16 + *) = &quot; 6 + - ): under
10.º Si superficies concava BAB' ( Fig. 55 ) gi
gnitur rotatione alicujus curyae circa OA , fiet
Cor
ace
r = r = r ' = r '&quot;
I supe
ac proinde vis ex BAB'Q'Q sursum verticaliter trahens
columellam AR erit ban,
zebic
H
reciproce nimirum ut radius osculi apud A.
11.• Facile nunc intelligimus attractionem mas
sae liquidae BAB'E'E, terminatae superficie utcumque con
caya BAB' , in columellam AR fore
K - &quot; ( + -) .velK6+ );
234
est autem
↾∣∣ ⇌∎
!
'2 :
NI&quot; F&quot; &quot;: −∙ Q. ∙−⊦ u QN—l—Fu q2r ↿ q 0
L
I 2r' , 2r ∙−⊦ 2r' ,etc.t
igitur
1 ↿− ↿ ↿
i- ∣
rr ∣⋅⊤⋅ .'&quot;? '
et consequenter
H 1 ↿ H
∎↿ ↿ ⋅
−⋮−≺−≀∶⇀⊦ r' )— 2
(r&quot; hl-r'&quot;) ⋅∙
10.0 Si superficies con-cava BAB' (Fig. 55 ) gi.
gnitur rotatione alicujus curvae circa OA ,
fiet
I ';
∣←−∶≀∙ :r :r '&quot; ;
ac proinde vis ex BAB'Q'Q sursum verticaliter trahens
columellam AR erit
'H-
&quot;,
reciproce nimirum ut radius osculi apud A.

11.(, Facile nunc intelligimus attractionem mas-
sae liquidae BAB'E'E, terminatae superficie utcumque con-
cava BAB' ,
in columellam AR fore
H
1,1) H(1,1 ∙
K
2(r '7 'velK 2r&quot; 'r'&quot;)'
ïfbiu235
massae vero liquidae NAN'E'E, terminatae superficie utcum
que convexa NAN' , in ipsam AR fore
K + 6 + -) .
ved K + &quot;6+ ) :
fiet =r =r&quot; = r&quot; in casu superficiei genitae rotatione li
neae curvae circa OA.

110. His declaratis , venio ad ascensum descensum que liquorum in tubis capillaribus.

1.° Aqua in tubis vitreis diversae capacitatis ad diversas ascendit altitudines, quae sunt reciproce ut tuborum diametri: idipsum contingit oleo, spiritui vini, etc..; ascendentesque liquores terminantur superne concava superficie. Concavae superficiei causam adsignavimus (109 1.): ad ascensum quod pertinet, sit QQ ( Fig. 57 ) suprema superficies aquae circumambientis tubum LE , et I A'B' concava superficies aquae intra tubum; sint insuper A'R, VR' binae columellae verticales, altera intra, altera extra tubum, quas columellas jungat horizontalis columella RR'. Urgebitur A'R deorsum gravitate simulque vi (109. 11. ° )
K - 16 + );
urgebitur VR' deorsum gravitate simulque vi ( 109. 2.° )
K :.
cum igitur
Н
K &gt; K
(
2 + ),
235
massae vero liquidae NAN'E'E, terminatae superficie utcum-
que convexa NAN' ,
in'ipsam 'AR fore
H 1 1 H 1 1 K—(-2(,, ∣∣⋅ .).ve1K( ,(,.,'. ...-)
r r
fiet r:r':r&quot;—-::r&quot;' in casu snperficiei genitae rotatione li-
neae curvae circa OA&quot;.

110. His declaratis, venio ad ascensum descensumque liquorum in tubis capillaribus.

1.&quot; Aqua in tubis vitreis diversae capacitatis ad diversas ascendit altitudines, quae sunt reciproce ut tuborum diametri: idipsum contingit oleo, spiritui vini, etc..; ascendentesque liquores terminantur superne concava superficie. 

Concavae superficiei causam adsignavimus (10913): ↙
ad ascensum quod pertinet , sit QQ' (Fig. 57) supre-
ma superficies aquae circumambientis tubum LE , et I'A'B'
concava superficies aquae intra tubum; sint insuper A'R,
VR' binae columellae verticales, altera intra, altera extra
tabum, quas columellas iungat horizontalis columella RR'.
Urgebitur A'R deorsum gravitate simulque vi ( 109. 11.&quot;)
H1711,
K
2(rlr1)a
urgebitur VR' deorsum gravitate simulque vi ( 109. Z.&quot;)
∕
cum igitur236
haud poterunt A'R , VR' consistere in aequilibrio nisi A'R
ascendat supra QQ . Denotet z altitudinem AA , ad quam ascendit columella A'R supra QQ'; sitque c gravitas specifica liquoris: fiet eousque columellae ascensio donec habeatur
H
K = K
-16 + )+c=;unde == 2c ( + ).

Vires ex materia tubi eas tantum liquidi particulas afficiant, quae ad internam ipsius tubi superficiem maxime accedunt; iccirco liquidum perinde trahetur, a tubo ac si interna superficies esset plana: permanente igitur tubi ac liquoris qualitate, etsi variat tubi diameter, eodem tamen pacto trahentur liquoris particulae versus tubum; quae attractio quia cum constanti attractione liquoris erga seipsum consociatur, extrema latercula curvae BAB' aeque inclinabuntur ubilibet ad internam tuborum superficiem. Arcus itaque omnes BAB' erunt similes in diversis tubis, ipsorumque arcuum rotatione circa tubi axem gignetur superficies concava superne terminans liquorem : bini videlicet
r et r' exsistent aequales, simulque tuborum diametris pro
portionales; ideoque altitudo z reciproce ut eae diametri.

2. • Hydrargyrum in tubis vitreis descendit in
fra circumambientis hydrargyri superficiem QQ ad ejus
modi altitudines , quae sunt tuborum diametris recipro
ce proportionales ; descendensque liquidum terminatur su
perne convexa superficie NOM.
Convexitatis causam adsignavimus ( 109. 1.0 ) : ad de
scensnm quod spectat , columellarum OR et VR' altera
sollicitatur deorsum gravitate simulque vi ( 109. 11.° )
0
{
K + C + ) ,
236
haud poterunt A'R, VR' consistere in aequilibrio nisi A'R
ascendat supra QQ'. Denotet :altitudinem AA' , ad quam
ascendit columella A'R supra QQ'; sitque c gravitas spe-
cifica liquoris: fiet eousque columellae ascensio donec ha-
beatur
H ↿ ↿ H 1 ↿
∣≮≓⋅∶∶∣⊊∙− −∙≨∙⋖−−∙−⊢⊤≻−⊢∶≖∙ under—' 2c(r ≓≀∙∙≻ .
r .
!
Vires ex materia tubi eas tantum liquidi particulas af-
ficiunt, quae ad internam ipsius tubi superficiem maxi-
me accedunt; iccirco liquidum perinde trahetur, a tuba ac si
interna superficies esset plana : permanente igitur tubi ite
liquoris qualitate, etsi variat tubi diameter, eadem tamen
pacto trahentur liquoris particulae versus tubum; quae
attractio quia cum constanti attractione liquoris erga se-
ipsum consociatnr, extrema latercula curvae BAB' aeque
incl'uabuntur ubilibet ad internam tubarum superficiem.
Arcus itaque omnes BAB' erunt similes in diversis tubis,
ipsorumque arcuum rotatione circa tubi axem gignetursuper-
iicies concava superne terminans liquorem : bini videlicet
r et r' exsistent aequales, simulque tubarum diametris pro-
portionales; ideoque altitudo :reciproce ut eae diametri.

2.0 Hydrargyrum in tubis vitreis descendit in-
fra circumambientis hydrargyri superficiem QQ' ad eius-
modi altitudines , quae sunt tubarum diametris recipro-
ce proportionales; descendensque liquidum terminatur su-
perne convexa superficie NOM.
Convexitatis causam adsignavimus (109. 1.(, ) : ad de-
scensnm quod spectat, columellarum OR et VR' altera
sollicitatur deorsum gravitate simulque vi (109. 11.&quot;)
Hi 1
K&quot;.'a(1&quot;'r&quot;)'
I:
'P.?237
altera sollicitatur deorsum gravitate simulque vi ( 109. 2.0)
K :
cum igitur
K &lt; K + -( + )
haud 'poterunt OR et VR' sese librarè nisi OR descen
dat infra QQ. Designet é altitudinem AO , ad quam
deprimitur columella OR infra QQ' ; sitque c' gravitas
specifica hydrargyri : eousque fiet columellae depressio do
nec habeatur,
H
K=K + ( + )- c'z' , unde z' = 2c'
(
+ &gt;&gt;).
Ut supra ( 1. ) ostenditur binos r , r' fore aequales, simul
que proportionales tuborum diametris ; iccirco etc.

111. Nonnulla subjungimas, quorum ratio desumitur ex animadversionibus (109).

1.º Duae laminae vitreae et parallelae PP ', SS ' demergantur verticaliter in aquam, earumque mutua distantia aequetur diametro tubi capillaris LE: suprema aquae superficies B &quot; A &quot; B &quot; inter laminas evadet concava instar canalis horizontalis; altitudo vero A'al= x ), ad quam attollitur aqua, erit duplo minor altitudine ad quam attollitur in tubo LE.: Infima superficiei B &quot; A &quot; B '&quot;' puncta jacent omnia inea
dem recta A'A'&quot;: secetur B &quot; A &quot; B &quot; &quot; plano perpendiculari ad A &quot; A &quot; '; sectio erit ubilibet arcus arcui BA'B' similis et
aequalis : istorun arcuum radius osculi apud puncta infima
dicatur r ; in tubo LE erit p = r , in laminis r= -0 . Colamellarum igitur A'R , VR aequilibrium praebebit ..
237
altera sollicitatur deorsum gravitate simulque vi (109. 23)
K :
cum igitur
X(K '
H( ↿
r ral,).r
baud' poterunt OR et VR' sese librare nisi OR descen-
dat infra -'QQ. Designet z' altitudinem AO,- ad quam
deprimitur columella OR infra QQ'; sitque c' gravitas
specifica hydrargyri: eousque fiet columellae depressio do-
nec habeatur.
1
∙
↿
'
∣∟−∣≖⊹−−⊸ 2(-—--]——-)-—c&quot;,z undez'—.2[:7(1 [
r').
Ut supra (1 .&quot;) ostenditur binos r, 'r' fore aequales, simul-
que praportionales tubarum diametris; iccirco etc.
,.,11.1 Nonnulla subjungimtts ,- quorum ratio desu-mitur ex animadversionibus (109).↿∙∘ ∐∎≖≔∘∙ laminae vitreae et parallelae PP', SS'demergantur verticaliter in aquam, earumque mutua distantiasequetur diametro tubi capillaris LE : suprema aquae super-ficies B&quot;A&quot;B&quot;' inter laminas evadet concava instar canalishorizontalis; altitudo vero A&quot;a(:.r) , ad quam attollituraqua , erit duplo minor altitudine , ad quam attolliturintnhoIfE.- ⋅∶∶⊸∙Infima superficiei B&quot;A&quot;B&quot;' puncta iacent omnia infen-dem recta A&quot;A'&quot;: secetur B&quot;A&quot;B&quot;' plano perpendiculariad A&quot;A&quot;'; sectio erit ubilibet arcus arcui BA'B' similis etaequalis: istorum arcuum radius osculi apud puncta infimadicatur r; in tuba LE erit r'zzr, in laminis r'zoo; Cos-lumellarnm igitur A'R , VR' aequilibrium praebebit't-dïff2382c ( + ) &quot; ;Tetscolumellarum autem A ” R &quot; , VR aequilibrium suppeditabitH101 IK -K - I ( + ) +re , f =.2c1Hinc xai 2ż z ; ideoque etc.....2.0 Laminae PP ' , SS , sibi commissae ad sematuo accedunt.Sit P&quot; punctum quodvis laminae PP: infra QQ adprofunditatem Alla &quot; : columella verticalis Alla&quot; transmittetpuncto P vim ( 1.0 ) .K- ( + ) +ostan&quot;) = KH+2r Tersus1 .C2c 5+c. a a'&quot; = K +0. aa!&quot;dicenndnetfenotaversus Qt : attenta columella horizontali . a &quot; ' P &quot; , urgebi amelltur P vi seu pressione externa + traiKversus Q't': colamella verticalis V'a transmittet puncto P &quot;vimK+c.aa &quot; &quot;versus Qi' : attenta columella horizontali a'P ' , solicitabitur P &quot; vi seu pressione interna 16TSU238H(t 1) H 1z— ⇂ ∙−−− ∙⋅ ;20 r !' C !'columellarum autem A&quot;R&quot;', VR' aequilibrium suppeditabit!H 1 1 H 1− ↿ ∣ ⊫−∙− −⋅⋅ ∙!cx . ∙∖Hlnc ∙ æ;—2—z;l ∙ ⋅ 1deoqueetc......⋅⋅ n .'2.o Laminae 'PP. SS', sibi commissae ad sea') 'At mutuo aeeedunt. )Sit P&quot; punctum quodvis laminae PP' infra QQ' adprofunditatem A&quot;a &quot;: columella verticalis A&quot;a transmittetpuncto P&quot; vim (1.). &amp; ↽(⋅. H 1 1⋅ &quot; HK(cæ—-—aa&quot;' :K −−−∙−−∙ 2 ≀⋅∙⋮∙∙∞ .'. (M' . 2r. . '&quot;1ciii .-—--)-c.aa :::K'I—ILmaa'&quot;.⋅.. .: '&quot;.versuth : attenta columella horizantali- a.'&quot;P&quot; , urgebi-tur P&quot; vi seu pressione externa,.. ⋅∙ ⋅('. a∙ t '. '(1
..

infima
dicatur r; in tuba LE erit r'zzr, in laminis r'zoo; Cos-
lumellarnm igitur A'R , VR' aequilibrium praebebit't-dïff238
2c ( + ) &quot; ;
Tets
columellarum autem A ” R &quot; , VR aequilibrium suppeditabit
H
101 I
K -K - I ( + ) +re , f =
.
2c
1
Hinc xai 2ż z ; ideoque etc.....
2.0 Laminae PP ' , SS , sibi commissae ad se
matuo accedunt.
Sit P&quot; punctum quodvis laminae PP: infra QQ ad
profunditatem Alla &quot; : columella verticalis Alla&quot; transmittet
puncto P vim ( 1.0 ) .
K
- ( + ) +ostan&quot;) = K
H
+
2r Tersus
1 .
C
2c 5+c. a a'&quot; = K +0. aa!&quot;
dicen
ndnet
fenota
versus Qt : attenta columella horizontali . a &quot; ' P &quot; , urgebi amell
tur P vi seu pressione externa + trai
K
versus Q't': colamella verticalis V'a transmittet puncto P &quot;
vim
K+c.aa &quot; &quot;
versus Qi' : attenta columella horizontali a'P ' , solicita
bitur P &quot; vi seu pressione interna 16TSU
238
H(t 1) H 1
z— ⇂ ∙−−− ∙⋅ ;
20 r !' C !'
columellarum autem A&quot;R&quot;', VR' aequilibrium suppeditabit
!
H 1 1 H 1
− ↿ ∣ ⊫−∙− −⋅⋅ ∙
!
cx . ∙∖
Hlnc ∙ æ;—2—z;l ∙ ⋅ 1deoqueetc......
⋅⋅ n .
'2.o Laminae 'PP. SS', sibi commissae ad se
a
') 'At mutuo aeeedunt. )
Sit P&quot; punctum quodvis laminae PP' infra QQ' ad
profunditatem A&quot;a &quot;: columella verticalis A&quot;a transmittet
puncto P&quot; vim (1.). &amp; ↽
(
⋅
. H 1 1
⋅ &quot; H
K
(
cæ—-—aa&quot;' :K −−−∙−−∙ 2 ≀⋅∙⋮∙∙∞ .'. (M' . 2r
. . '
&quot;
1
ciii .-—--)-c.aa :::K'I—ILmaa'&quot;
.
⋅
.
. .: '
&quot;
.versuth : attenta columella horizantali- a.'&quot;P&quot; , urgebi-
tur P&quot; vi seu pressione externa
,
.. ⋅∙ ⋅
('. a
∙ t '. '
(1
..

: - ∎∙ 3 ' ∙

.
versus Q't': columella verticalis V'a' transmittat puncto P&quot;
visu ⋅⋅ ' ⇀
'
'
'
⋅
⋅⋅
K—l—caa ?
.
versus Q't': attenta columella horizontali a'P&quot;
, sollicita-
bitur P&quot; vi seu :pressione interna &quot;
'
'
-
⇂⇣ r 'a'
ms
1111:
&quot;But
&quot;fin239
K
versas Qt : erit igitur p &quot; aeque sollicitatum hinc et illinc,
nullusque motus gignetur ab istis viribus.
Sit P' &quot; ' punctum ipsius PP' inter QQ et A &quot; A '' : ver''
ricalis columella A &quot; a &quot; transmittet puncto P &quot; ' vim
к
-16 + ) + ( 6 – aa &quot;) = K -
H
+
2r
H
S c.aa&quot; = K- c.aa' '
2r
versus Qt : ob columellam horizontalem P '&quot; a &quot; urgebitur
P &quot; &quot; vi seu pressione externa
K
versus Qt : cum igitur K &gt; K - c.aa &quot; , nitelur &quot; mo
veri ad plagam (t' .
Ascendet aliquantulum aqua externa prope laminam ÞP
induetque ( 109. 1. ) figuram concavam ee'e ' ; propterea,
denotante &amp; radium osculi apud punctum v . gr. e' , co
lumella e'b normaliter insistens superficiei curvae ee'e&quot; in
e' transmittet puncto v. gr. b vim
H
K
29 ( +4) = K 2 €
versus Qt’ : attenta columella horizontali e'b ', urgebi
tur b ' vi seu pressione interna
K
versus Qt : ex aqua éb'é' proveniet in bi vis
∙ 239
K
Versus Qt: erit igitur P&quot; aeque sollicitatum hinc et illinc,
nullusque motus gignetur ab istis .viribus. Sit P'&quot; punctum ipsius PP' inter QQ' et A&quot;A&quot;': ver-
ticalis columella A&quot; a&quot; transmittet puncto P&quot;'v vim
K
--—-(—1--—l--—) —[—c(x—-aa&quot; ∙−−∶ ∣⊂−∙≗ ∙⋅⊢
H
—— c.aa&quot;
−−−−− K— c.aa&quot; 2r ∕
versus Qt'
- ob columellam horizontalem P&quot;'a &quot;
P'&quot; vi sen pressione externa
&quot;—
urgebitur
K
versus Q't'. ∙ cum igitur K)K—c.aa&quot; , uitetur P&quot;' mo-
veri ad plagam Q't' .
Ascendet aliquantulum aqua externa prope laminam PP' induetque (109.1.?) figuram concavam ee'e&quot; ; propterea,
denotante &amp; radium osculi apud punctum v. gr. e'
, co-
lumella e'b normaliter insistens superficiei curvae ee'e&quot; in
e' transmittet puncto v. gr. b' vim
H 1 1 H K—
⊸≳↽≼−⋮⇀⊣−∘−∘−⊢ ≖⊊−−⊸⇄−∊
versus Q't'- attenta columella horizontali e',b' urgebi-
tur b' vi seu pressione interna '-
. K
8
versus Qt: ex aqua e'b'e&quot; proveniet in b' vis240
c.e&quot; 6
versus Qit' : columella verticalis A &quot; 6 &quot; transmitiet puncto
b' yim
H
K
25
+ c . A &quot; 6 &quot;
versus Qt : attenta columella horizontali b'b' impelletur 6
vi seu pressione esterna
K
versus Qt . Est
H
2r = cx = C . A'a ;
librato insuper liquido , pressiones apud V' et é' sunt ae
quales , et consequenter
K = K
H
28 to.e&quot; 6 ,
H
= c.eb' ;
2 €
detractisque proinde viribus versus Qt ex viribus versus
Q'ť , emerget
H н
K
2e-K + c.e^ 6—K+
-c.A &quot;b&quot; + K = c.eb - c.A &quot;6&quot; +
H H
2r 2 €
c (e&quot; b' — A &quot; 6&quot; + 1&quot; a - c'b') = c.b &quot;a &gt; o :
sollicitabitur ergo b' vi c.6 &quot; a versus Q't' .
Veniat denique spectandum in lamina PP punctum p '
inter A &quot; A &quot; et B &quot; B &quot; : sit P'a&quot; columella horizontalis ; a'i
columella perpendicularis superficiei curvae B’A &quot; B &quot; &quot; apud
a &quot; ; dicaturque é radius osculi in a ' ' . Transmittet a'i
puncto Ph vim
240
c . e&quot;b'
versus Q't' :columella verticalis A&quot;b&quot; transmittet puncto
b' vim
. H &quot;&quot;
K—Z—i—c'Ab
versus Qt :attenta columella horizontali b'b&quot; impelletur b'
vi seu pressione externa
↴
K
versus Q't'. Est
H
∙∠−≀∙−∶∶∘∙↿∽⋅∶∘∙∆∎∣∅
librato insuper liquido , pressiones apud V' et a' sunt ac-
quales , et consequenter
-——-K——'l&quot;c. e&quot;6'. Eli-:o- e&quot;b':
detractisque proinde viribus versus Qt ex viribus versus
Q&quot;: , emerget ⋅
K—is'&quot;. -K—]—
.- .∘∣∙∣↗∣−−⋅↧≮−⊦ ≛↿−⊑∙⊸∙∆∦∣⊃⋅∣−⊢↧⊊∶⊸⋅⊜∣⋅∣⊃⋅−∘⋅∆⋅⋅≀≀⋅∣−⊦
H H &quot;
.!
.,
h 1
∙−−
&quot;
.
ii.—22&quot;— ..—c(&quot;eb'- Ab -I-Aa-c.b)—-c.b a)o.
sollicit'abitur ergo b' vi c.b&quot;a vcrsus Q't' .
Veniat denique Spectandum in lamina PP' punctum P&quot;
inter A&quot;A&quot;' et B&quot;B&quot; : sit P&quot;'a&quot; columella horizontalis; a&quot;i
columella perpendicularis snperficiei curvae B&quot;A&quot;B&quot;' apud
a&quot; ; diceturque e' radius osculi in a&quot;. Transmittet a&quot;:
puncto P&quot; vim.241
K
H
2 €
.
versus Qt : ex liquido superincumbente proveniet in p ' ' vis
B ' P &quot;
versus Qc : attenta P &quot; a &quot; urgebitur p &quot; vi seu pressione
externa
K
versus Qit' : librato liquido , pressiones apud a &quot; et A ” sunt
aequales ; proinde ducta horizontali Allu ,
H H
K tc.P&quot; u = K = K- C. A'a ,
2 € 2r
H
= c ( P''u + A &quot; u ) = c.P'' ' :
26
detractis igitur primis duabus viribus ex tertia , assequemur
H
K - K +
c . B ' piv H
-C.B&quot; piv
2€ 22'
c ( Piu' B ' p ' ) &gt; 0.
Lamina itaque PP' movebitur versus Q'C' : eadem de causa
movebitur lamina SS' versus Qt ; ideoque etc...

3. Si aquae substituitur hydrargyrum , supre
ma liquidi superficies inter laminas PP' , SS ' fiet con
vexa instar horizontalis inversique canalis ; deprimetur li
quidam ad altitudinem duplo minorem quam in tubo LE ;
ipsae insuper laminae adhuc ad se mutuo accedent . Haec
explicantur simili ratione ac ( 10, 20. )..
241
versus Qt :ex liquido superincumbente proveniet in P&quot; vis
&amp; ∙ B1v Ptv
versus Qt : attenta P&quot; a&quot;
externa
urgebitur P&quot; vi seu pressione
K
versus Q't' :librato liquido . pressiones apud a&quot; et A&quot; sunt
aequales .; proinde ducta horizontali A&quot;u,
H &quot;
∙∙∙ H ∙∙
↿⊂−∙−∙⋮≳−⋮∙−⊢∘∙↧∙ ∥⋅−−↿⊊∙−⋮≳≀∙−−∙−−↧⊊−∁∙ A a
.
H ,,
⊓∣
27-—-c(P&quot;u-l—Au)——-—c.P u:
detractis igitur primis duabus viribus ex tertia , assequemur
⊏−≖⊂−⊢⋮−⋮∶∶−∘∙∌∏ ↕⊃≖⊽∶∶∶ IST—c .B&quot; Piv:
c (P&quot;'u' — Blv P&quot;) ∘∙
Lumina itaque PP' movebitur versus Q't' : eadem de causa
movebitur lamina SS' versus Qt ; ideoque etc...
3.&quot; Si aquae substituitur hydrargyrum , supre-
ma liquidi superficies inter laminas PP' , SS' fiet con-
vexa instar horizontalis inversique canalis; deprimetur li-
quidam ad altitudinem duplo minorem quam in tubo LE;
ipsae insuper laminae adhuc ad se mutuo accedent. Haec
explicantur simili ratione ac (10. 20.)
.242

4. • Super vitream laminam horizontalem AA'B'B
( Fig . 58 ) affunde gattam olei terebinthini mm ' ; tum al
teram laminam vitream A &quot; A'B'B &quot; priori AA'B'B impone
sub angulo sic exiguo , ut imposita lamina gutlam le
viter attingat ; conspicies mm' , instar trochleae , termi
natam quodam canaliculo ; qui canaliculus plano hori
zontali sectus dabit curvilineam convexamque sectionem
plano verticali sectus curvilineam concavamque sectionem .
Radius convexitatis ( € ) manet proxime idem in
punctis m et m' e diametro oppositis ; radius vero con
cavitatis ( = r' ) in puncto m' magis accedente ad A'B'
minor erit quam radius concavitatis ( = r) in puncto m
minus accedente ad ipsam A'B ' . Spectantes columellam
mam ' perpendicularem rectae A'B' , quoniam r et é ' apud m
obverluntur ad plagas contrarias , itemque r et ê &quot; apud m' ,
facile intelligemus ( 109. 110. ) sollicitatum iri mam ver
sus A'B' vi
H
K
2
simulque versus AB vi
K
16-&gt;).
Cum igitur &gt; m , prima vis erit major quam secunda ;
columellaque mam , et una cum mam' tola gutta mo
vebitur versus A'B' motu accelerato : idipsum contingit
guttaeaqueae . At si ejusmodi guttis substituatur gutta
hydrargyri , haec movebitur versus AB ; ratio est quia
gutta hydrargyri tam in sectione horizontali quam in
verticali praebet curvam convexam , radiusque novae con
vexitatis in m superat radium novae convexitatis in m .

5.° Capillaris tubus in aquam QQtt ( Fig. 57. ) demergatur; tum, apposito digito ad orificium inferius extrahatur : remoto digito , aqua jam elevata eflaet ali
quantulum ex orificio illo , ibique demum haerebit sus
pensa in guttam rotundam conformata ; residuae vero
aquae altitudo in tubo extracto invenitur major quam
altitudo ( 110, 1º.)
H
Z =
16 + 3) = * ( + ) cr
supra QQ in tubo demerso.
Exprimant et w , altera radium convexitatis apud infi
mam aquae superficiem in tubo extracto ,
altera ipsius
aquae altitudinem : ex aqueae columnae aequilibrio pro
fluit&quot; ( 110. 1 ° 2°. )
H H
K
+++
www =
= ktö : +
H
co ;
cr
ideoque w &gt; z . Si aquae substituitur hydrargyrum , tam
suprema quam infima superficies liquidi exsistet conve
x2 ; ex aequilibrio igitur hydrargyri in tubo extracto
emerget ( 110. 20. )
H H
H
K + tow == Kt H
co CI
et consequenter a = 0 si r = 0 .

112. Quae diximus de liquorum ascensu tubulis vitreis, applicari possunt ascensui liquorum in tenuibus cujuscumque materiei tabulis: hinc patet cur liquida ascendendo imbuant spongias, saccharum, ellychnia etc: cur succus inserviens plantarum vegetationi sursum ex terra eluctetur; etc... Istiusmodi corpora vel constant exilissimis fibris, in quibus tanquam in totidem capillaribus tubis ascendit liquidum, vel innumeros habent angustos meatus vicem tubulorum varie flexorum supplentes. 

Caeterum methodo inhaerentes, qua D. Pessuti LaPlacianam theoriam ad faciliorem formam traduxit, capillarium luborum phoenomena explicavimus in hypothesi liquidorum eamdem usque ad extimas omuino superficies obtinentium densitatem: non enim nobis in animo est vel leviter attingere novam theoriam, quam de actione capillari anno 1831 edidit D. Poisson.

== ACUSTICAE PRINCIPIA ==

=== Notiones preambulae ===
[[113|113]]. Acustica agit de sono: non defuerunt, qui sonum consistere putabant in efluviorum a soporo corpore vibratorum motu quae efluvia ex affrictu, vel contusione sonori corporis ejaculantur atque huic affinis est alia quaedam sententia, quod contusione illa vel affrictu particulae aeris purioris in eo corpore absconditi, vel ipsum circumdantis, expellantur et ad aures usque excurrant. Verum experimento machinae pneumaticae compertum est, quod incluso tintinnabulo vel horologio horas personante in recipiente, ubi aer incipit exhauriri, incipit sonus minui; ubi autem totus exhaustus est aer, nihil jam soni auditur, utcumque pergat tintinnabulum concuti, aut horologium pulsibus affici. Hoc probat sonum non consistere in effluviis a sonoro corpore vibrati cur enim non emittuntur amplius, aut ad aures non permeant, cum imo liberius ob minora obstacula deberent? &lt;u&gt;Ad majorem rei evidentiam&lt;/u&gt; ita hoc experimentum instituitur horologium in vitro aere pleno ac probe clauso reponitur, ne aer scilicet inde possit exhauriri tum in recipiente pneumatico collocatur, atque ex hoc educentes aerem animadvertimus sonum nullum audiri. Machina horaria aere circumsepta est ergo nullimode suspicari licet aliquid deesse circa ipsum corpus sonorum quominus sonus exaudiatur. 

Dicendum potius non audiri sonum propter defectum aeris intermedii inter utrumque corpus. Porro corpus cum resonat, motu tremulo atque &lt;u&gt;oscillatorio&lt;/u&gt; minimarum partium afficitur singulis autem oscillationibus aer corpus tremulam circumdans concutitur, similesque recipit vibrationes, quas in ulteriores particulas aereas pariter defert nisi quod impulsus in circumfusum aerem delapsi atque auditus organum afficientes eo minores ac debiliores fiunt quo magis a fonte recedunt. Enimvero corpora, quae sonora dicantur, tunc sonum excitant quando ita agitantur, ut illorum partes tremulo ac vibratorio satisque rapido concutiantur motu: sic campanae malleo percussae, ratione elasticitatis concipiunt tremoris motum, qui major fit postquam vehementius ac diutius agitatae fuerint: instrumenta musica, dum illorum fides agitantur, simili pariter tremore concutiuntur; hinc est quod chartae frustula resonanti corpori imposita subsultare cernuntur. His positis certum est tremorem similem communicari debere aeri immediate ambienti, et deinde tremorem late &lt;u&gt;diffundi&lt;/u&gt; per &lt;u&gt;aereas particulas&lt;/u&gt;; nam particulae aeris sonoro corpori proximae illius impulsu comprimuntur; et cum sint elasticae , post &lt;u&gt;compressionem&lt;/u&gt; &lt;u&gt;dilatantur&lt;/u&gt;, aliasque sibi proximas urgent; atque hoc pacto et illae vibrant, et longe lateque in particulas aereas similis vibratio omni ex parte discurrit. Hinc pulvisculi aeri innatantes, qui radio solis in obscurum conclave intromisso conspicui sunt, agitari videntur si sonus proxime intendatur: pulsato prope stagnantem aquam tympano, validius et crispari, et subsultare aqua pariter cernitur.

Haec notentur. 1º . Vis acceleratrix &lt;math&gt;\varphi&lt;/math&gt; in vibrante particula resonantis corporis ita pendet a spatiolo &lt;math&gt;z'&lt;/math&gt; quod particulae superest excurrendum usque ad nativam aequilibrii positionem, ut crescente, decrescente , vel evanescente &lt;math&gt;z'&lt;/math&gt; crescat simul, decrescat, vel evanescat; propterea&lt;math display=&quot;block&quot;&gt;\varphi = C'z' + C'' z'^2 + C''' z'^3 + ...;&lt;/math&gt;et quia particularum excursiones exsistunt exiguissimae, erit,&lt;math display=&quot;block&quot;&gt;\varphi = C'z':&lt;/math&gt;vis nempe acceleratrix assumi potest proportionalis spatiolo &lt;math&gt;z'&lt;/math&gt;. 

2º. Non pluribus opus est ut intelligamus (29.4º) vibrationes omnes, sive majores, sive minores ejusdem particulae fore aequidiuturnas.

[[114|114]]. Progignitur quoque sonus ab aere vehementer compresso seseque statim restituente: etenim propter impetum in restitutione conceptum ad majorem, quam in statu naturali occupabat, extensionem perveniet; ac proinde cogetur se rursus contrahere, minusque naturali spatio tenere. His autem successivis contractionibus et expansionibus in reliquo aere pulsus &lt;u&gt;excitantur&lt;/u&gt;: sic producitur sonus v. gr. virgae aerem celerrime perstringentis: simili modo qui in tibiam insufflat, sonum gignit; dum nempe per tubi orificium aer insufflatione intromittitur, ille, qui continebatur in tubo, necessario secundum longitudinem comprimitur; unde fit ut is iterum expandatur, tum denuo coarctetur; atque hoc pacto, quamdiu perseverat inflatio, perficiantur oscillationes, hisque sonus progignatur. Certe si aerea columna tubo &lt;u&gt;inclusa&lt;/u&gt; non afficiatur nisi motu totius, sonus minime obtinebitur; utcumque vero &lt;u&gt;excitentur vibrationes&lt;/u&gt;; ut &lt;u&gt;perceptibilem&lt;/u&gt; sonum edant, earum numerus intra minutum secundum non debet praetergredi quosdam certos limites, videlicet 6 circiter et amplius 24000; uti compertum est experimentis D&lt;sup&gt;''ni''&lt;/sup&gt; Savart.

[[115|115]]. Saepe contingit nos voce elatiori quibusdam in locis loquentes, aliquo tempore postquam siluimus repente audire rursus verba a nobis antea prolata; atque haec est illa echo, de qua plura fabulantur poetae. Philosophi in hoc conveniunt, quod echo sit motus reflexus aeris, qui &lt;u&gt;motu ondulatorio&lt;/u&gt; affectus obici incurrens resilit consimili motu, et rursum aures nostras afficiens nos determinat ad eumdem sonum audiendum, quem antea audivimus: ut autem effectus iste contingat, necesse est aliquanto longius a loquente obicem existere. Ratio est quia si &lt;u&gt;obex&lt;/u&gt; proximior fuerit, sonus reflexus efficiet in auribus impressionem suam antequam impressio soni directi defecerit; tunc vero non poterit secunda impressio a prima discerni. Aliquando semel tantum, aliquando saepius eadem vox per reflexionem auditur: primam contingit quando ab unico loco vox collecta rejicitur, vel a pluribus, sed ad eamdem distantiam: secundum quando vox in pluribus locis ad diversas distantias collecta revertitur ad aures sensibili successione. Hinc intelligitur quare in vallibus, quas undique colles cingunt, echo saepius iteretur.

[[116|116]]. Non solus aer est &lt;u&gt;medium&lt;/u&gt; idoneum transmissioni sonorum: nam per alia quoque elastica fluida propagatur sonus. Vapores ipsi, in quos aqua, spiritus vini etc. attenuantur, sonum transmittunt; etenim si recipiens pneumaticum aere atmosphaerico evacuetur, tum aliquo ex dictis fluidis repleatur, sonus campanae vel horologii adhuc bene audietur: quin et liquores, aqua v. gr. sonum non intercipiunt, sed ipsum debilitatum licet propagant; qui enim intra aquam sunt, audiunt sonos extra aquam editos; et qui extra aquam sunt, audiunt sonos editos intra aquam. Tandem etiam corpora solida deferunt sonos ad ingentes distantias: celebre est apud milites ita terram excavare donec strato alicui bene solido aurem applicare possint, ut ex reboatu agnoscant adventum hostilis legionis, praesertim equitatus; huic strato non raro tympanum imponunt, atque levia corpora tympano imposita ex sonoris tremoribus subsultant.

=== De intensitate soni; deque ejus gravitate, et acutie. ===

[[117|117]]. &lt;u&gt;Intensitas&lt;/u&gt; major vel minor soni importat majorem vel minorem ejusdem soni vim ad sensationem excitandam, quae proinde in intensiore sono vehementior est, ita ut aures prae violentia laedat aliquando; in remissiore ita debilis, ut vix aliquando audiatur. Iamvero evidens est quod quo plures sunt partes sive in corpore sonoro, sive in aere simul oscillantes, eo plus motus atque activitatis, caeteris paribus, habent; ac proinde vehementius organum auditus pulsare poterunt: quo singularum partium itus et reditus major est, seu quo fortius singulae particulae comprimuntur et restituantur in unaquaque oscillatione sive in corpore sonoro, sive deinde in aere, fortiori item impressione aptae erunt organum auditus afficere. Contra, quo pauciores partes sonori corporis oscillant, eo minus communicabitur motus particulis aeris, et consequenter ab his minus afficietor auditus organum: quo singulae sonori corporis partes unamquamque oscillationem minorem habent, eo minorem item oscillationem in aeris particulis producent, ac proinde impressione minus valida auditus organum concutient. Quod ratione perspectum est, &lt;u&gt;experientia quoque confirmatur&lt;/u&gt;; et quod ad sonum, quem vocant primitivum, attinet, corpora densiora, caeteris paribus, magis sopora sunt quam quae ex opposito; atqui hoc nonnisi quia plures particulae in his oscillant simul; ergo ex numero particularum oscillantium sonus major vel minor pendet. Rursum inter corpora aeque densa, atqae elastica, quod validius percutitur validiorem profecto sonum excitat et &lt;u&gt;magnitudo&lt;/u&gt; soni &lt;u&gt;magnitudini&lt;/u&gt; percussionis est proportionalis: undenam hoc repetendum est nisi ex eo quod validior percussio fortias comprimit atque oscillare vehementius cogit particulas elasticas? Quoad derivatum sonum res constat experimento machinae pneumaticae (113): cum enim exhauriri aer incipit, sonus incipit imminui; atqui hoc est quia  aeris quantitas in excipulo imminuitur; et cum rarior evadat aer, minus valide comprimi et restitui ejus particulae debent; neque enim ulla alia &lt;u&gt;probabilis&lt;/u&gt; causa est. Condensando insuper aerem in eodem excipulo ultra &lt;u&gt;statum ordinarium&lt;/u&gt;, quem tenet in almosphaera, compertum est quod condensatus aer sopam reddit intensiorem; atque hoc quidem ita, ut intensitatis augmentum proportionem servet cum augmento condensationis. Franciscus M. Zannotti diligentius rem exploravit: aerem inclusum vase calefecit; quo pacto aeris &lt;u&gt;elasticitatem&lt;/u&gt; auxit, &lt;u&gt;densitate&lt;/u&gt; eadem servata, cum nullus permitteretur aeri exitas; et tunc sonus intendebatur, At rima aliqua in vase relicta, per quam aer posset erumpere, tum igne admoto, sonus multo minor visus est quam antea fuerat. Cum igitur, permanente aeris elasticitate, non idem permanserit sonus, rursus patet quod soni intensitas non solum ab elasticitate, et consequenter a magnitudine vibrationum, sed a densitate, id est a numero particularum vibrantium dependet. Nec arte solum ex rarefacto vel condensato aere intensitas soni mutata deprehenditur , sed naturali etiam aeris rarefacti vel condensati constituțione idem evenit: hinc in altissimis montibus, ubi aer rarior est, ac proinde minus elasticus, sonus multo est remissior quam in planitie , ubi condensatione atque elasticitate pollet majori.

118. Ex his explicantur sequentia circa soni intensitatem. 

1º. In aperto aere sonus calore minuitur, in clauso vero calore augetur: apertus enim aer, ubi calore afficitur, sese continuo dilatat, adeoque ejus intensitas minuitur, quin &lt;u&gt;elasticitas&lt;/u&gt; augeri debeat; quia nempe habet quo se rarefactus recipiat; ergo minor numerus particularum oscillat, adeoque remissior sonus. Contra, si aer undique clausus est, cum densitas eadem manere debeat, elasticitas autem ex calore crescat, idem erit particularum numerus, sed singularum oscillatio propter auctam elasticitatem augebitur; ergo intensior sonus.

2°. Sunt qui dicunt, aestate sonum intensiorem esse, caeteris paribus, quam hyeme; alii contra opponunt, quod hyeme intensior sit sonus quam aestate. Si in re incerta quoad factum et ex circumstantiarum varietate adeo varia ut fortasse determinari non possit , si inquam ratio reddenda esset, ajendum sonum aestate imminui debere, quia aer terram ambiens calore rarefactus minori densitate pollet, ac proinde minor erit numerus particularum oscillantium. Cum autem ex calore elasticitas crescat , hoc capite augeri debet sonus , cum nempe singularum particularum oscillationes validiores debeant. Videndum igitur quid praevaleat; et juxta vel densitatem hyeine praevalentem imminutioni elasticitatis, vel elasticitatem praevalentem aestate imminutioni densitatis, qui effectus sequi debeat.

3º. Hinc etiam explicant nonnulli cur nocte, caeteris paribus, soni majores sint quam interdiu; quia nempe densior est per noctem aer ob calorem minorem; at hujus rei explicatio verior est, quod per noctem, cessante ea aeris commotione quae per diem habetur ex multiplici strepitu, magis aptus sit aer ad soni vibrationes concipiendas et deferendas, organumque auditus nulla alia sensatione percussum aptius sit ad peculiarem aliquem sonum exaudiendum.

119. Discrimen inter gravem et acutum in sono importare profecto debet diversitatem aliquam in motu aeris, quo afficitur organum auditus, atque adeo in motu sonori corporis ex quo in aere motus hujusmodi derivatur; nam cum sensatio sonii ex impressione organi auditorii oriatur, at omnis alia sensatio ex impressione organi proportionati, et impressio ista per motum aeris ad organum appellentis fiat, profecto diversa impressio, quae a sono gravi atque acuto fit, diversum motum exigit tum in aere ex quo immediate producitur, tum in corpore sonoro a quo mediate progignitur; atqui ista diversitas non ex validiori vibratione seu oscillatione majori provenit; ex hac enim quantitas sive intensitas soni (117), non autem qualitas seu tonus procedit; ergo diversitas ista in celeriori seu crebriori vibratione partium aeris, et consequenter sonori corporis, derivanda videtur. Ratio consequentiae est, quia non alia diversitas saltem probabilior in oscillatione partium concipi potest quam, ut haec sit vel major ut scilicet quisque itus et reditus spatium majus percurrat, vel quod sit celerior ut scilicet eodem tempore plures situs ac reditus habeantur. Ergo cum ex primo capite discrimen acuti et gravis repeti nequeat, nihil afferri probabilius potest: quam celeritas oscillationum, quae certe in satione diversitatem afferre debet. 

Quoniam vero in rebus physicis natura explorari maxime debet experimentis atque observationibus, ita prosequor. Constat in chordis musicis, eas quae vel breviores sunt, vel magis tensae , vel minoris diametri (nam ex hoc triplici capite diversitas tonorum habetur in fidibus) acutius resonare; contra graviorem sonum reddere eas, quae longiores sunt, vel minus tensae vel majoris diametri: atqui chordae breviores vel magis tensae etc. percussae, plures numero vibrationes pari temporis intervallo producunt, pauciores aliae; hoc patet ex ipso sensuum testimonio: ergo sonus acutus habetur in chordis, quae frequentius dato tempore oscillant; gravis autem etc. In ea insuper proportione, in qua frequentiores aut rariores sunt vibrationes chordae musicae, est etiam magis vel minus acutus sonus: ergo frequentior aut rarior vibratio omnino connexionem habet cum tono per chordam musicam producto; pendetque tonus ex illa &lt;u&gt;frequentia&lt;/u&gt; aut raritate vibrationum, tamquam effectus a sua causa. Quod dictum est de chordis musicis, valet etiam in campanis et pocalis vitreis, aliisque id genus sonoris corporibus; haec enim percussa figuram rotundam in ovalem mutant, eorumque proinde fibrae eundo et redeundo oscillare debent, atque ex hac oscillatione sonus oriri colligitur; ut autem gravior vel acutior est sonus corporis, ita in figura immutatio et restitutio seu fibrarum ítus ac reditus rariores sunt aut crebriores. Porro si id in sonoro corpore contingit, ut gravior sonus obtineatur quando minor vibrationum numerus habetur in corpore, jam tunc in aere quoque minor vibrationum numerus haberi dicendus est: siquidem tot numero vibrationes dato temporis intervallo producuntur iu aereis particulis a tremulo motu corporis resonantis, quot ab ipsius corporis sonori fibris seu particulis eodem tempore peraguntur; et vice versa quot in aere gigni ac propagari vibrationes constat, totidem in ipso corpore resonante produci dicendum est.

120. Dum plurium corporum sonus ita temperatur ut gratus sit auribus, dicitur consonantia seu concentus, si ingratum  sonum produxerint, appellamus dissonantiam: in sonis ita temperandis ut sint jucundi, ars musica versatur. Tonus musicus seu consonantia pendet ex eo quod certo tempore certus vibrationum numerus a pluribus sonoris corporibus peragatur, et particulis aereis communicetur. Si duo vel plura corpora sonora intra idem tempus vibrationem absolverint , consonantia est omnium perfectissima , et sonus dicitur unisonus ; si eodem tempore unum corpus unam , aliud duas vibrationes expleat, consonantia haec dicitur octava: ita appellatur ex eo quod per quandam tonorum seriem ascendendo hic tonus a musicis octavo loco constituitur. Si eo tempore quo unum duas vibrationes, aliud tres absolvat, adeoque secunda unius cum tertia alterius concurrat, dicitur quinta: si eo tempore quo unum tres, aliud quatuor vibrationes conficiat, quarta nuncupatur; atque istae sunt consonantiae illae, quas Pythagoras advertisse traditur, dum quinque fabri malleis ferreis massam ferream contunderent. Consonantiae istae in vibrationibus chordarum inventae sunt ; imo etiam alii successu temporis consonantiae gradus additi , quos diligenter musicae scriptores explicant. Si videlicet numeri vibrationum , quas dato tempore chordae  musicae efficiunt , sunt ut &lt;math&gt;1 , \frac9 8, \frac5 4, \frac4 3, \frac 3 2 ,\frac5 3,  \frac{15}8, 2&lt;/math&gt; chordae illae edent notissimos tonos ''do, re , mi , fa , sol , la , si , do'': constat experimentis saepissime iteratis; etenim chordae homogeneae , aeque crassae , eodemque pondere tensae , quarum longitudines sint uti &lt;math&gt;1 , \frac89, \frac 4 5, \frac 3 4, \frac 2 3, \frac 3 5,  \frac8{15}, \frac12&lt;/math&gt;praefatos tonos edunt. 

Haec subjungimus circa exiguissimas chordarum vibrationes. 

1°. Chorda homogenea &lt;math&gt;AB&lt;/math&gt; (Fig. 59) uniformiter crassa ubique tensa aequaliter, punctisque &lt;math&gt;A&lt;/math&gt; et &lt;math&gt;B&lt;/math&gt; fixa, traducatur ad datam formam curvilineam &lt;math&gt;AC''B&lt;/math&gt;; tum sibi relinquatur: pro quovis temporis momento determinanda proponitur curva &lt;math&gt;AC'''B&lt;/math&gt;, in quam abit chorda. Sint &lt;math&gt;AO ( =x)&lt;/math&gt; et &lt;math&gt;S'O ( = y )&lt;/math&gt; coordinatae orthogonales; &lt;math&gt;h&lt;/math&gt; longitudo chordae &lt;math&gt;AB&lt;/math&gt;; &lt;math&gt;M&lt;/math&gt; massa; &lt;math&gt;\theta&lt;/math&gt; tensio: in ea qua sumus exiguissimarum vibrationum hypothesi, maxima chordae elongatio ab aequilibrii positione cum sit ferme insensibilis, haec obtinebunt quamproxime. Primo: apud quodvis chordae vibrantis punctum Seadem vigebit constanter tensio &lt;math&gt;\theta&lt;/math&gt;. Secundo: movebitur &lt;math&gt;S&lt;/math&gt; juxta directionem &lt;math&gt;SO&lt;/math&gt; respondentis ordinatae. Tertio: denotante a angulum tenuissimum &lt;math&gt;S'EA&lt;/math&gt; interceptum tangente &lt;math&gt;S'E&lt;/math&gt; et abscissarum axe &lt;math&gt;AB&lt;/math&gt;, erunt &lt;math&gt;\alpha = \sin \alpha = \tan\alpha ;\, \cos \alpha =1&lt;/math&gt;. Quoniam exercetur &lt;math&gt;\theta&lt;/math&gt; juxta vibrantis chordae longitudinem; sumptis arcubus infinitesimis &lt;math&gt;S'i , Si&lt;/math&gt;, denotabunt &lt;math&gt;S'i\, \mathrm{et}\, S'i'&lt;/math&gt; directiones tensionum apud &lt;math&gt;S'&lt;/math&gt;: resolvatur tensio juxta &lt;math&gt;Si&lt;/math&gt; in duas, quarum altera existat parallela rectae &lt;math&gt;AB&lt;/math&gt;, altera perpendicularis eidem &lt;math&gt;AB&lt;/math&gt;; et idipsum fiat quoad tensionem juxta &lt;math&gt;S'i'&lt;/math&gt;. Componentes parallelae axi &lt;math&gt;AB&lt;/math&gt; se mutuo destruent; componentes vero perpendiculares ipsi &lt;math&gt;AB&lt;/math&gt; exprimentur per &lt;math&gt;\theta\sin\alpha&lt;/math&gt; versus &lt;math&gt;O&lt;/math&gt;, et Osini atda ) versus S , seu per Ox et Oatd « ). Superest igitur vis
- Oda
gignens motum juxta SO : differentiale da sumendum quoad
x tantum, utpote denotans variationem anguli a in eadem curva AC &quot; B. Quisque videt --Oda esse vim motricem, cujusmodi est tensio  &lt;math&gt;\theta&lt;/math&gt;: propterea, designante dm elementum massae , exprimetur per
Oda
dm
∶≀≤↓⇟≓
miter crassa, ubique tensa aequaliter, punctisque A et B fixa, traducatur ad datam formam curvilineam AC&quot;B;
tum sibi relinquatur: pro quovis temporis momento de-
terminanda proponitur curva AC&quot;'B, in quam abit chorda. Siut AO (:æ) et S'O (::y) coordinatae orthogonales; '
h longitudo chordae AB; M massa;9 tensio: in ea qua
sumus exiguissimarnm vibratiouum hypothesi, maxima chordae elongatio ab aequilibrii positione cum- sit ferme in-
sensibilis , haee obtinebunt quamproxime. Primo: apud
quodvis chordae «vibrantis punctum S' eadem vigebit constanter tensio 9. Secundo: movebitur S' inxta directio-
nem SO respondentis ordinatae. Tertio :denotante et an-
gulnm tenuissimum S'EA interceptum tangente S*E et
abscissarum axe AB, erunt ut :sina −∙−−−− tangat ; cos at −−−∶↿∙
Quoniam exercetur 9 iuxta vibrantis chordae longitudinem :
[snmptis arcubus infinitesimi: S'i
, S'i', denotabunt S'i et S'i'
directiones tensionum apud S': resolvatur tensio iuxta Si
in duas
, quarum altera existat parallela rectae AB, alte-
ra perpendicularis eidem AB; et idipsum fiat quoad ten-
sionem iuxta S'i'. Componentes parallelae axi AB se mu-
tuo destruent ; componentes vero perpendiculares ipsi AB
exprimentur per Osina versus O, et 9sin( at-l-dat) ver-
sus S , seu per 90: et B(a-I—dat). Superest igitur vis
—9dat
gignens motum juxta SOI: dili'erentiale dat sumendum quoad, utpote denotans variationem anguli &amp; in ea-
dem cnrva AC'&quot;B. Quisque videt —-9dat esse vim motri-
cem , cuiusmodi est tensio 9: prapterea
, designante dm
elementum massae , exprimatur per
Gala
&quot;2711-255
respondens vis acceleratrix. Ob uniformem chordae cras
sitiem ,
dx
h
dm
M
Mar
ideoque dm =
h

;

insuper
a = tang a =
dy
dx ;
sumptisque differentialibus quoad x ,
da dany
dx dx²
Facto itaque compendii causa
on
M
superior expressio vis acceleratricis traducetur ad
d²y C2 dx²
unde ( 28 )
day
da(SS)
dia
d ” (SO - SO )
dc2
d²y
dx² de²
seu
1
255
respondens vis acceleratrix. Ob uniformem chordae cras-
sitiem ,
, ideoque dni ∙∙∶−− —-—de ;
9.&quot;M :.
da: ∙∙∙⋅ h −∙−
insuper
at:—— tangat-agi ;
sumptisque diiferentialibus quoad a:
,
data:-dv dx dx: ⋅
Facto itaque compendii causa
911
∙∙−∙∙↽−∙∶∘∙ ∙ M
superior expressio vis acceleratricis traducetur ad
da
⋅−∘⋅⊒≀−⋛−⋮⋅ ∙
nnde (28 )
,↶≀≖↗∙∙ irss; -dz(so-s'b) ∙∙∙ a.,
c dx&quot; dt: d? d;:
— '
sen256
day c2 day
dia (a) .
dx2
Formula ( a) suppeditat quaesitam problematis solutionem.
2. • Fac ut vis acceleratrix sit ut 1 , nimirum
day C'y ; erit dta
der
· c?
day
dx2 C'y
1 seu tör
so . dra
Inde habemus ( 27 , 27.0 )
VVT
y=CC + C, e
с - CV-4
evanescente X , evanescit et y ; hinc C = -C, , et con
sequenter ( 27. 30. )

* V0V1

y = C , [e
- *70V1
1 =
2011'sin rc=2C1V= 1sin
2 VMCMC
h9
facta x = h , evanescet y ; proinde
sin k V MC
ik V
MC'
ho TT C' =
OTE2
LM .
ho
9
ordinata CC &quot; respondens abscissae AC ( **)
256 ∙
da,, ⋅
−−↙⋮∎⋅≒↿∣
dx? :d—t; (a) . .
jl
C:
Formula (a) suppeditat quaesitam problematis solutionem. *
2.&quot; Fac ut vis acceleratrix sit uty
, nimirum '
lude habemus (27. 27.?)
f.. t/CV −−↿
-..-z ∁∣⇂∕∶⋅↿⋅
yiL-3010 c
⊣⋅∙ O; 6 - c :
evanescente a:, evanescit et )&quot;
, hinc C,: ---Cl , et con-
sequenter (27. 30. ∘ )
⋅−∙⊽⋮⋅∣∕∁⋅⇂∕∶↿ ... ⋅⋮∸−⇂∕∁⋅⇂∕∶↿
c c
y-—-—C,[e -—e ]∶∶
2C1V —-1 'sin −⋅⋮−− ∣∕∎∁⋅∶⊋∁∎≖∣∕∶−↴ sinx 9:709. :
facta x:h , evanescet y; proinde
. ⋅∙∥⋅∪⋅∙∙ VH?&quot; 97:-
&quot;[III 119 ∙−−−∘∙≀∙ WC;&quot;, (:::-IIM:
,
ordinata CC'&quot; respondens abscissae AC (;.-ä 11) ex-
'257
hibeatur per y ' , erit

;; = - 20, vt in . V MC =20,V = tsin.V MORE

TT
2C,V -1 sin î
2C, V 31 .
Propterea
2 = sin -
77 ()a' ) ;
aequatio ad curvam AC''B.''
3.° Per ty denotetur tempus unius semivibra
tionis ; erit ( 29. 3.° )
TT
1,5
2V C
VhM

;

0
et consequenter tempus unius vibrationis
hM ta VRM
Ad haec : designante n numerum vibrationum , quae ip
tra temporis unitatem absolvuntur , exsistet
1
V
TANTE
12
In hypothesi chordae cylindricae habentis radium
r el densitatem , erit M = fErPhò ; ideoque
257
hibeatur per J&quot;, erit
. -—- .
': MC' −−
∙ h M9112
r;.—20. ⇂∕⋅−↿ sm ∙− ∙−−−⇌⊋∁≖⇂∕⋅−∙↿ 810 2— IPGM :: 2 116
∙−− 7! −∙∙ ≢∁⋅↾∕∙−↿ sin ∙−⇇∋∙− −−−∙⇌ my.—1 .
Propterea
yzy' sin :; 71 (a') ;
aequatio ad curvam AC'&quot;B,
3.0 Per t. denotetur tempus unius semivibra—
tionis; erit ( 29. 33 )
et consequenter tempus unius vibrationis
.:Vg.
Ad haec: designante n uumerum vibratiouum, quae in-
tra temporis unitatem absolvuntur , exsistet
In hypothesi chordae cylindricae habentis radium
r et densitatem 8 , erit Mr.-:Ttrïhö ; ideoque558
13=rkVis, n - EVO
4. • Facta Osy. , velocitas puncti S in fine
temporis ( erit ( 29. 1.° 2.° )
v = y.Vī sine VC -Yosin
hinc ( 29. 1. ° )
=V
9. C - 02
C
yo V1 - sin’LVA
yo coseV C sy= . COS cos r.
Simili modo , facta CC“ =jo , velocitas pancti C in
fine temporis ( erit
7 t
yo sin Ti; simulquey'= y's cososeme- T ;
ta
et aequatio (a' ) ad chordam vibrantem poterit scribi in
hunc modum
y = yo com-A sin
C
-TT h
( á ').
5.° Si abscissae x in ( a'') substiluitur vel anh''
vel ( 2n + 1) h, prodibit y = o quotiescumque n aut erit =0,
aut erit quivis numerus integer : binae videlicet
558
9
≄≖−⊣⋅↗≖⇂∕∂ ∙
∙≖⋮−−⇀⊑⋅−−≀≖ '?Eä'
4.0 Facta OS;-:]. , velocitas puncti S in line
temporis : erit (29. 1.&quot; 2.')
v': J/ö' sint t/&quot;ä ∶∶−∶−≖−∫∘ sin-;- tt
!- 8
hinc (29. 1.&quot;-)
∙⊺∶−−−∙∙⇂∕⋅↗⇗ (S'—v ∙−−− J. l/1—s1n'q/ Q'::
j'. costV C' :y, cos ∙⋮− 11.
2
Simili modo
,
facta CC
∙−−∶ y'.. , velocitas puncti C&quot; in
fine temporis :erit
' n
s
∙ t ∙
' ' :
si:—y., s1n——1r;stmnlquey-:yocos—1t ;
t

: , :,

et aequatio (a') ad chordam vibrantem poterit scribi in
hunc modum
∙−− '
cst nsinæn ⋅ (a&quot;) J—yo O.t2 h .
5.0 Si abscissae a: in (a&quot;) substituitur vel an]:
vel (a'n-l-nh, prodibit yzo quotiescumque 11 aut erit 20,
aut erit quivis numerus integer: binae videlicet
Je!259
x = 2nh ,
( 2n+1 ) h
spectabunt ad quiescentia chordae vibrantis puncta. In
ferimus illud : chorda AB produci potest ultra limites A
et B quin puncta A et B per iteratas chordae vibra
tiones a statu quietis dimoveantur , etsi puncta illa poo
nuntur de se mobilia ; modo tamen AB in eamdem ac
antea conformelur initialem curvam , eidemque subjiciatur
tensioni : imo sumpta BH = HH' = =h , ita vibra
tiones suas conficiet chorda ABHH ' .
ut puncta A,
B , H , H ', ... in quiete persistant. Ad istiusmodi vi
brantis chordae figuram quod pertinet , sit v . gr. HD
HD = AO = x ; erunt AD = AH -HD = 2h - x , AD = 2h + x :
in la &quot; ) substitue prius 2h-x , deinde 2hta loco x ;
provenient ordinatae yı ety respondentes punctis D et D ',
nimirum
visy.cos Ti sin ( 2
a sin 16 는 (2-m ) R = my'o cos t2
sa= com sin ( 2+ )n = foco na sio ža .
Igitur y = -y, ya= y : ordinatae scilicet y , y , sunt
aequales , et ad eamdem plagam obversae ; ordinatae ve
ro y , y sunt quidem aequales , sed obversae ad con
trarias plagas. Chorda itaque dividitur in partes alterna
tim vibrantes supra et infra rectam AH'.
6.** Quoad (a) generatim spectatam ; denotanti
bus f et F binas functiones arbitrarias , satis hic erit
animadvertere eam expletum iri per
y =-flatct) + F ( x - ct) (a ' ' ) ;
siquidem
259
a.:2n71 ∙ :r (2n—l-1)I;
spectabunt ad quiescentia chordae vibrantis puncta. ln-
ferimus illnd: chorda AB produci potest ultra limites A
et B quin puncta, A et B per iteratas chordae vibra-
tiones a statu quietis dimoveantnr, etsi pnncta illa po-
nantur de se mobilia; modo tamen AB in eamdem ac
antea conformetur initialem curvam, eidemque subjiciatnr
tensioni: imo sumpta BH −−−−− HH' ::
...
zh
,
ita vibra-
tiones suas conficiet chorda ABHH' . .. ,
ut puncta A,
B , H , H', ...
in quiete persistant. Ad istiusmodi vi-
brantis chordae figuram quod pertinet ,
sit v. gr. Hl):
HD'zAOsæ; erunt AD;:AH-HDzah-æ
, AD':.2h-l—æ :
in (a&quot;) substitue prins alz—a:, deinde Zh—l-æ loco :;
provenient ordinatae y. etj, respondentes-punctis D et D',
nimirnm
'
tnsin(2 −⋅⋮≻↿∎∎∶ 'cos tu' æ
fac:-Tou.&quot; (: 'l &quot;70 :: Olli-i:&quot;.
, s ∙ æ ,
t
. æ
Jar—jre cos-1t am (2 −⋅⊢ --)1t :yocos —-1t sm --1t
.
- :, h : 11
Igitur y. ∙∶−−−∙ −∫∙ ∙↗≀≏−−−−−⋮↗↟⇌ ordinatae scilicet y , ;, sunt
aeciuales ,
et ad eamdem plagam obversae; ordinatae ve-
ro y, y, sunt quidem aequales, sed obversae ad cou-
trarias plagas. Chorda itaque dividitur in partes alterna-
tim vibrantes supra et infra rectam AH'.
6
∙∘∙ Quoad (a) generatim spectatam; denotanti-
bus f
et E binas functiones arbitrarias
,
satis hic erit
animadvertere eam expletum iri per
Fnæ-l-ct) −∣⋅− P(æ—ct) (if&quot;) ;
siquidem'260
dº[fix + c ) + F(x – ct) ]_da[fixtet) + F (x – ct )] .
do[ ) dt2
7 . ** Velocilas puncli S in fine temporis i prodit
expressa ( 28) per
dOS - OS')
dt
dy
dt [flatct)-F&quot;(x – ct)]:
initio motus , quum nempe t = 0 , est v=0 ; iccirco
c [ f '( x )-F'(x )] = 0, $' ( x)=F&quot; ( x) , et f (x ) = F (x ) ;
aequationes igitur determinantes et curvam ASB , et ve
locitatem traducentur ad
y = f(x + ct) + f(xớctct), v '= -c[ f '( x + chf'( x - ct) ] .
Facto t = 0 , istarum prima praebebit
y = 2f \x ) ,
aequationem videlicet ad curvam datam ACSB : ex hac
itaque curva pendet natura functionis f. Caeterum , ge
neralem de integratione differentialium partialiumque ae
quationum doctrinam suo tempore videre erit in parte 3.4
nostrorum elementorum Matheseos n. 200 , 201 ,
121. Si chorda instrumenti musici percutiatur , et pro
pe adsit instrumentum aliud , in quo chorda sit ad aniso
num cum priore tensa , baec alterius instrumenti chorda
sensim tremere incipiet , et undulationes sensim majores
concipiendo ad sonum ipsa quoque excitabitur eumdem to
num reddendo quem prior illa chorda percussa reddit .
Jam vero si ad hujus rei rationem attendas, plana erit juxta
theoriam traditam : sicut enim pendulum quiescens etiam
minimo impulsu , uti est qui ex insufflatione procedit , ad
260
⊄≀⇟⊏∣≺↕⊣⊸↥≻ −⊢ Für—ct) ],. «l*[m—l-aH-Fw—ctü.
dt:
(
7. ∙∙∙ Velocitas puncti S'111 fine temporis :prodit
expressa (28) per
os.-os d
v. ∙∙∙⋅ &amp; dt ):... .... :]? ∶−∙−− —c[f(æ-l-ct)—-F'(:r—ct)]:
initio motus , qunm nempe t::o
,est ⇂↓−∙−−−∘⋅ , iccirco
c[f (æ)—-F' (x)] ∙−−− o, f(xrr—F' (x)
, etfix):F(æ)' ,
aequationes igitur determinantes et curvam AS'B ,
et ve—
locitatem v' traducentur ad
y-fþ—I—ct) *Aæ—ct) , ∙≀⋅−−− ∙−−− ∙−− c[f '(æ-i-ctF-f'w—ct) ].
Facto t--—-o
, istarum prima praebebit
F2nx) :
aequationem videlicet: ad curvam datam AG&quot;B : ex haei
itaqua curva pendet natura functionis f. Caeternm, ge-
neralem de integratione differentialium partialinmqne ae-
qnationnm doctrinam suo tempore videre erit in parte 3.&quot;
nostrorum elementorum Matheseos n. 200, 201, .-. .
.-

121. Si chorda instrumenti musici percutiatur, et prope adsit instrumentum aliud, in quo chorda sit ad unisonum cum, priore tensa, haec alterius instrumenti chorda sensim tremere incipiet, et undulationes sensim maiores concipiendo ad sonum ipsa quoque excitabitur eumdem tonum reddendo quem prior illa chorda percussa reddit. 

Jam vero si ad huius rei rationem attendas, plana erit iuxta theoriam traditam: sicut enim pendulum quiescens etiam minimo impulsu , uti est qui ex iusufilatione procedit , ad motum oscillatorium minimum primo concitabitur , et si in
suflationem saepius repetas , poteris sensim oscillationes
majores , ac majores perficere (tunc tamen id fiel quando novi isti impulsus certa periodo, parique intervallo habeantur; si enim pendulum contra insufflantem venit, insufflantes rursum potius motum impediemus quam adjuvabimus, atque idet' finita una oscillatione debet opportune rur
sus alius impulsus addi , sicque ubi oscillationes penduli
et novi impulsus certa periodo sibi respondeant , effectus
habebitur) ita in chorda praefata evenit ut trémitns con
cipiatur et augeatur ; donec excitetur sonus ; quia nempe
Oscillationes unius chordae consentiunt cum oscillationibus
ad quas altera determinabilis est , iccirco ex repetitis chor
dae percussae uşdulationibus , quae sunt isochronae undulationibus alterius , obtinebitur ut hae augeantar donec so
nus excitetur in chorda etiam plectro minime percussa. Ex hac doctrina infero: ergo in utraque chorda oscillationes sunt pares numero; ergo cum tonus ab utraque redditus idem sit, tonus igitur a numero vibrationum hujusmodi pendet. Ad magis declarandam traditam doctrinam de acutie et gravitate sonorum, utile erit non nulla hic explicanda proponere circa chordas vibrantes. 

Ac 1º. quamvis chordae non sint unisonae, attamen una percussa, alia sonum edit, si modo tensae sint ad octavam, aut alias quasdam habeant armonicas proportiones. 

2º. Si duae chordae tensae sint ad octavam, et pulsetur chorda longior; quae dimidia ejus est, reddet tonum sui proprium, scilicet octavam acutam; at si pulsetur chorda brevior, excitabitur in longiore tonus non sui proprius, scilicet ad octavam gravem, sed tonus chordae brevioris. 

3º, Refert Sauverius hoc phoenomenon: chorda longa 5 ped. percutiatar, et notetur tonus;
tum ad distantiam unius pedis ponatur supra chordam le
ve aliquod obstaculum velati plumae frustulum , quod ta
men non impediat molus communicationem : si quinta haec
1
1
1
261
motum oscillatorinm minimum primo concitabitur ,
et si in-
snæationem saepius repetas, poteris sensimf oscillationes
maiores , ac maiores perficere (tunc tamen id fiet quando
novi isti impulsus certa periodo», parique intervallo babe-
autur; si enim pendulum contra'insumantem venit ,
insuf-
Hantes rursum Potius motum impediemus quam' adiuvabi-
mns , atque idet-' finita nna'osci'llatione dehet opportune rur-
sns alius' impulsnsgaddi, sicque ubi oscillationes penduli
et novi impulsus certa periodo sibi respondeant , effectus
habebitur) ita in chorda praefata evenit ut tremitus con-
cipiatur et augeatur, donec excitetur sonus; quia nempe
oscillationes unins chordae consentiunt cum oscillationibus
ad quas altera determinabilia est
,
iccirco ex repetitis chor-
dae percussae undulationibus , quae sunt isochronae undu-
lationibus ulterius-, obtinebitur ut hac augeantur donec ac-
nos excitetur in chorda: etiam plectro minime percussa. Ex
hac&quot; doctrina infero: ergo in utraque chorda 'oscillationes
sunt pares numero; &quot;ergo cum tonus ab utraque redditus
idem sit, to'nus igitur a numero vibratiouum hujusmodi
pendet. , - - -
'
'
'
Ad magis declarandam- traditam doctrinam de acutie
et gravitate sonorum, utile erit non nulla hic &quot;explicanda
prcponere circa chordas vibrantes.

q'uod ta-
men non impediat motus communicationem: si quinta haec
.*262
P
-chordae pars pulselur, tongm efficiet proprium chordae d
- nias pedis; hunc autem ipsum tonum reddit etiam reliqua
chordae pars, etsi quadrupla. Rursum si obstaculum pona
tur post duos pedes , eveniet ut pars brevior citius oscil
let, et longioris motum perturbet; subinde utraque chor
dae pars ita , sese componet; ut vibrationes eodem tempo
re compleat: tunc vero tonus reddetur neq w proprius chor
dae duorum pedum , neque trium, sed proprius chordae u
nius pedis.
Ad primum quod attinet , quoties duae chordae len
sae sunt ad octavam, jam vibrationi unius chordae ,respon
dent duae vibrationes alterius; ergo quamvis singulae , O
scillationes non conveniant, adeoque tremitus aeris non re
novet impulsum in alia chorda post singulas ejusdem oscil
lationes, renovari tamen potest impulsus hic post binas ;
eo ipso poterit chorda ad octavam tensa , etsi difficilius ,
ad oscillandum determinari ex alterins oscillationibus. Idem
valet de aliis chordis quae eam habent proportionem ut
oscillationes recurrere possint post aliquem ipsarum nu
merum: ac proinde illae, quae vel ejusmodi recursum non
admittuut , vel quarum recursus majorem postulat' quam
par est vibrationum numerum, non ita invicem ad reso
nandum poterunt determinari.
Ad secundum : quod chorda brevior resonans ad pulsa
tionem longioris reddat tonum sui proprium , cohaeret cum
doctrina jam tradita : quod autem chorda longior reddat
Lonum proprium chordae brevioris non officit; etenim si
chorda sit dupla , quasi in duas dividetur, neque tota oscil
Jabit ( 120..5º. ) per modum unius, sed habens in medio
punclum quiescens, seu nodúm, oscillabit seorsim in sin
gulis dimidiis partibus, ac si, scamould adjecto , bifariam
arte divisa esset ; , et si chorda ' triplo sit longior , ia - tres
partes aequales dividetur: quo posito , nil mirum quod chor
da dupla non sui proprium tonum , sed tonum subduplae
reddat, et tripla sonum subtriplae.
ic
0
at
LE
262
chordae pars pulsetur, tonum efficiet proprium chordae n-
onius&quot; pedis; hunc autem ipsum tonum reddit etiam reliqua
chordae pars,; etsi quadrupla. Rursum siobstacnlum pona-
tur post duos pedes'. eveniet ut pars brevior. citius oscil-
.let,. et, longioris motnm perturbet; subinde utraque-. chor-
dae pars ita sese componet&quot;, ut vibrationes eodem tempo-
re compleat: tunc vero tonus reddetur neq ,.: proprius chor-
dae duorum pedum, neque trium, sed proprius [chordae u-
,niuspedis. ↴
⋅⋅∶⇟⋡∢⋅ ∙

Ad primum quod attinet, quoties duae chordae tensae sunt ad octavam, iam vibratioui unins chordae respondent duae vibrationes alterius; ergo quamvis singulae oscillatioueslnon conveniant adeoque tremitus aeris non renovet impulsum in alia chordae post singulas ejusdem oscillationes, renovari tamen potest impulsus hic post binas; eo ipso poterit chorda ad octavam tensa, etsi difficilius, ad oscillandum determinari. ex&quot; alterius. oscillationibus. Idem
.valet de aliis chordis-anae-eam habent prOportionem ut
oscillationes recurrere possint post- aliquem - ipsarnm nn-
merum: ac proinde illae, quae vel ejusmodi recursum non
admittunt
,vel quarum recursus majoreni- pastulat' quam
par est vibratiouum numerum, non ita invicem a'd reso-
.nandum poterunt determinari.

Ad secundum: quod chorda brevior resonans ad pulsationem longioris reddat tonum sui pmprium cohaeret cum doctrina iam tradita: quod autem chorda longior reddattonum - proprium  chordae brevioris non officit; (etenim-si
chorda sit dupla, quasi in &quot;duas dividetur,- neque- tota oscil-
Jabit (120. 50.) per modum unins, bed habens in medio
punctum quiescens, s'eu nodnm, oscillabit seorsim in sin-
gulis dimidiis partibus, ac si, scamnnlb adiecto , bifariam
arte .'divisa esset;. et: si chorda' triplo sit longior, in- tres
partes aequales dividetur: quo posito, nil mirum-quod chor-
da dupla non aui proprium tionnm, sed tonum'subduplae
reddat, et tripla sonum subtriplae.
n-rts lar-Q .-263

Ad tertium: idem Sauverius hanc in Academia Pari
siensi explicationem attulit . Dum chorda nullo obstaculo
apposito pulsatur, vibrationes efficit toti suae longitudini
proportionales: at dum leve illud obstaculum apponitur post
pedem unum , undulatio totalis chordae dividitur ; prima
enim pars chordae , utpote quinta chordae totius ,
quinquies citius oscillare debet quam oscillaret integra
chorda : sic citius oscillando abripiet partem sibi proxi
mam in vibrationes aequales ; secunda pars tertiam, atque
ita singulae quinque partes seorsum oscillationes pera
geat. Alterum vero, quod magis est admirabile, ila ab eo
dem auctore explicatur ; pars brevior chordae, scilicet duo
rum pedum, citius oscillans quam reliqua , secum abripit
per sui motus communicationem partem sibi similem, nem
pe duorum pedum; in quinto autem pede oscillationes e
tiam communicantur, quae cum esse debeant longitudini
proportionales, duplo crebrius oscillabit extrema haec chor
dae pars quam reliquae; proinde ista sibi proximam unius
pedis partem trahet ad analogas oscillationes , et secunda
tertiam atque ita de reliquis , donec in hoc etiam casu quin
que chordae partes oscillent juxta longitudinem propriam ,
et consequenter sonum reddant respondentem longitudini
upius pedis.

122. Quaeri potest quomodo sonus trans obicem queat communicari ita, ut tonus proprius sonori corporis permaneat; nam fibrae, seu partes elasticae obicis puta parietis aut cancelli vitrei, ad motum concitatae vel sui proprium tonum reddere debent, vel si dissimiles sint, plurium tonorum mixturam, quod non accidit. Respondeo nullam esse difficultatem, si immediate per aerem soni propagatio habeatur, etiam intermedio exsistente obice. Quod si per obicem sonus diffunditur, in ipso admitti possunt partes aptae diversos sonos reddere aerique transposito communicare; atque ita, ut ille sensibilis sit trans obicem tonus, qui a partibus analogam oscillationem habentibus cum sonoro corpore communicatur. Forte etiam dici potest, quod si fibrae non habentur aptae eum tonum reddere, dividantur, ut in chorda non unisona contingit, adeo ut idem tonus transmitti possit.

123. Quoniam de tonis, ex quibus qualitas soni denominatur, egimus; quaerendum esset unde asperitas aut lenitas, quae pariter ad qualitatem quamdam soni pertinet, proficiscatur. Animadverte sonum quemcumque non esse simplicem, sed compositum e sono plurimarum sonori corporis partium: sic chorda musica percussa non simplicem edit sonum, sed quemdam veluti concentum edicit , qui a
peritioribus musicis probe dignoscitur; in quo tamen cum
fortior tonus praevaleat, alios minores obruit : coexsistunt
videlicet in chorda sonora, et generatim in quovis particu-
larum s'ystemate, variae exiguarum oscillationum species.
Imo vero non tantum sonorum ipsum corpus attendendum
est plerumque v. gr.-chorda musica, sed instrumentum i-
psum cui chorda adhaeret: variae insuper reflexiones ani-
madverti debent, quibus aer ad aurem deveniens diversas
subit modificationes. Itaque si vibrationes partium sonori
corporis sint bomologae, sonus lenis erit; si contra, asper:
atque hinc aspere sonant chordae inaequales in materia ,
crassitie etc; item ex reflexione aequabili atque uniformi
sive instrumenti, cui chorda adhaeret, sive circumstantium
corporum, lenitas soni orietur, asperitas ex opposito. 'Bo-
num erit observare quod chorda musica vehementius quam
par est distraCta stridet; quia videlicet valde percussa non
eam' servat legem quam in moderatis percussionibus obti-
net ut sub eodem tempore oscillationes suas sive majores,
sive minores dbsolvat; sed continget ut tempora oscillationum
inordinate mutentur, stridorque pro tono solito erumpat.

124. Haec notentur 

1º. chordarum vibrationes hactenus consideratae, dicuntur transversae: quae nimirum- obtinen-
tur chordam percutiendo in directione ad ejus axem perpendiculari: quod si atteratur chorda in directione ad e
jus axem parallela, adhuc sodos edet, sed , caeteris pari
bus, multo acutiores quam qui ex vibrationibus transver
sis progignuntur ; idque ex eo repetendum esse videtur
quod elasticitas propria chordae in vibrationibus longitudi
nalibus validior sit quam in transversis.

2.° Ubi in longitudinalibus vibrationibus chorda
rum obtineant &lt;u&gt;nodi&lt;/u&gt; , molus ita fiet ut partes hinc illinc cira
ca podum quemlibet positae simul ad ipsum nodum accedant, simulque alternatim recedant.

3º. Corpus omne, dum resonat, dividitur in plu
res partes vibrantes invicem ' separatas lineis , quae vocan
tur &lt;u&gt;nodales&lt;/u&gt;, quaeque oculis subjiciuntur spargendo per su
perficiem corporis minutissima arenae grana: haec enim su :
per lineis illis acervari observantur. Nodales propterea li
neae modo' sunt rectae, modo curyae, modo ex rectis si
mul et curvis coalescunt.

4.º  Malála nodalium linearum figura, plerumque mutatur et sonus; semper autem acutior vel gravior evadet sonus, prout corporis superficies in majores vel minores numero parles vibrantes dividetur ab ipsis noda libus lineis.

5.° Laminae rigidae ex ferro, vitro etc. in transversis vibrationibus absolvendis sequuntur leges alias ab illis, quas sequuntur chordae.

=== De directa soni propagatione per aerem. ===
125. Experientia nos edocet quod in iisdem circumstantiis sonus aequabili velocitate in toto decursu devehiеur; atque omnes soni , sive intensi , sive remissi , sive graves, sive acuti eadem velocitate diffunduntur.  

Nam 1.º Academici Florentini ad percurrendam distantiam unius milliaris sonum tormenti bellici impendisse quinque secundorum tempus experti sunt, ejusdem vero tormenti sonum ad conficiendum dimidium milliare impendisse dimidium tempus testantur aequabili nimirum velocitate perrexit sonus. Derhamus saepius repetitis experimentis idipsum invenit, adeo ut ab uno ad duodecim milliaria sumens intervalla invenerit aequale spatium aequali tempore in quavis a sonoro corpore distantia confici. 

2.º Prope sonorum corpus intensior est sonus, remissior in majore a sonoro corpore distantia atqui tam prope quam procul a sonoro corpore aequali velocitate pergit sonus ergo tam intensus, quam remissus etc. Hoc ipsum institutis ad id experimentis etiam constat Gassendus sclopeti et tormenti bellici fragorem eodem tempore pervenisse affirmat, cum eodem tempore exploderentur. Florentini et Derhamus in diversi generis tormentis idipsum evenisse notant itemque tormenti bellici minoris et mallei fragorem idem unius milliaris intervallum confecisse eodem tempore. Certum est ergo tam intensum, quam remissum etc. Huc spectat quod Derhamus quoque notat post Florentinos, scilicet eodem tempore sonum ad aures pervenire sive tormentum ad observatorem convertatur, sive ad contrariam plagam videtur enim intensior in eam partem, in quam tormentum dirigitur, esse debere sonus.

3.° In concentu sive ex instrumentorum pulsatione, si malleorum ictibus etiam ad satis notabilem distantiam dignoscitur tonorum successio eo praecise ordine, quo ictus varios tonos producentes habentur successive, et quidem sine sensibili temporis mora atqui si toni diversi non eadem propagarentur velocitate, jam qui toni successive habentur, non successive atque ordine illo ad aures venirent ergo etc. Erit fortasse qui quaerat qua ratione fieri possit ut sonus in quavis distantia, sive intensus, sive remissus, uniformiter &lt;u&gt;propagetur&lt;/u&gt;. Respondeo: eadem materiae quantitas eodem tempore, tum ex vi majore, tum ex minore, undulare potest ergo eadem aeris portio, seu &lt;u&gt;unda ejusdem latitudinis&lt;/u&gt;, eodem tempore potest undulationem perficere, sive ex majori, sive ex minori vi impellente. Antecedens est evidens; pendulum enim idem , adeoque eadem massa , eodem tempore oscillationes peragit sive magis , sive minus impellatur ad oscillandum: ergo a pari eadem aeris quantitas oscillare potest sub eodem tempore sive ex majori , sive ex minori impulsu. Sed si eadem aeris quantitas aequali tempore potest comprimi et restitui , jam eodem tempore potest sonus, ad datam distantiam pervenire , sive intensior , sive remissior: haec minor est evidens; si enim eadem est &lt;u&gt;latitudo undae&lt;/u&gt; , idemque tempus, jam eodem intervallo temporis spatium datum a sono conficietur; ergo sive intensus sit , sive remissus , seu vi majori aut minori aereae undae propellantor, eadem esse potest soni velocitas.

Quid ergo provenit ex hoc quod in sono intensiore vis major aerem impellat? Nempe quod ejusdem latitudinis unda, licet eodem tempore conficiatur , compressionem tamen ac restitutionem patiatur validiorem , vel languidiorem; sicut in pendulo accidit , quod eodem tempore oscillans ex impulsione maiori oscillationem concipit magis validam , et minus ex vi minori. Atqui hoc idem praestat minorem intensitatem , non autem minorem soni velocitatem . Ostendo: intensitas soni pendet a vi , qua in organum appellunt aeris particulae ; ergo si vi majore condensantur , et restituuntur , intensiorem efficient soni sensationem; at velocitas ex dictis pendet a latitudine undae, et tempore quo perficitur: neque latitudo immutatur , neque tempus; ergo non mutatur velocitas. Quod autem neque latitudo , neque tempus mutetur , ita probari potest. Latitudo enim undae , seu aeris quantitas ad oscillandum per modum unius determinata , ea esse debet quae potest obtemperare vibrationibus sonori corporis , a quo unda producitur , quaeque potest oscillationes suas eodem tempore complere quo sonorum corpus oscillationes suas perficit: ergo latitudo undae proportionari debet tempori quo sonorum corpus perficit vibrationes suas. Atqui sive intensior , sive remissior sit sonus, tempus quo sonorum corpus vibrationes suas complet , est ( 113. 2.°) semper idem; ergo item latitudo undae aereae eadem esse semper debet. Idem probat simul, quod sicut eadem latitudo, ita idem esse debet tempus quo unda perficitur. Et sane si tempus mutaretur , deberet quoque mutari tonus: atqui idem manet tonus in quacumque distantia a sonoro corpore , et quidem sive corpus resonet intensius , sive remissius; ergo etc.   

Hinc dum de sono agitur duplex in motu undae aereae velocitas distinguenda est: altera importat tempus quo unda conficitur , seu quo segmentum aeris datae latitudinis oscillat ; altera importat motum particularum aerearum itum et reditum perficientium in ejusdem undae efformatione. 

Quaeri hic potest in quanam ratione intensitas soni minuatur in progressu . Reponunt communiter quod intensitas soni est in ratione duplicata distantiarum inversa a centro soni : rationem afferunt , quia sonus quantum est de se aequabiliter undequaque diffunditur in modum sphaerae. Atqui ex hac aequabili in modum sphaerae diffusione sequitur decrementum in ratione praedicta ; nam si ita diffunditur , debet in ea proportione intensive decrescere , qua extensive augetur , sea qua latius materia , cui communicatur motus , sese expandit ; sed hujusmodi extensionis augmentum est in ratione duplicata distantiarum ; hanc enim rationem sequuntur sphaericae superficies : ergo etc... . 

126. Sit c velocitas , qua propagatur sonus ; &lt;math&gt;\Delta&lt;/math&gt; distantia inter vibrantem sonori corporis particulam et particulam aeream : exprimet tempus a sono impensum ad percurrendam distantiam &lt;math&gt;\Delta&lt;/math&gt; ; motusque particulae vibrantis nonnisi post tempus I = pertinget ad aeream particulam: propterea substituto 2— —c- 'a duabus ulti-
mis formulis(29. 5.&quot;), si :
∙−−≜−⋅ incipit ab 0
, ultraque
progreditur, determinabitur aereae particulae motus per»
∙ 271: A , 9 211 .A.
∦⋅⇋↙∁∘∎∐∙−⊖−⋅≺∁−−∘−−≻ ,
szC-Z-n' 008
! j(t—z).
F30i20,1t,2,3,4,-...,act—-e—:i9,tln- c
de habes A:c(t—-i9): erit
⇂↓∣∶⇂∕∁ sin 21'12:o.
Sumptis ergo distantiis Azct, c(t—G), c(t—29), c(t—BG), ...,
uulla velocitas v' ibi invenietur : aer proinde in locisi il-
lis omnino quiescet quando desinit tempus :; eritque n—
sque ad Ar.-ct in plureswundas distinctum similes et aequa-
les ; quarum communis latitudo ::09 ; numerus vero

:

∆−∘−⊖− ⋅
∆
Quantitas l/C sin −⋛≖−≺ t — A;) manet positiva ab
t — 30- :::-id ad :
2 — (i ] &amp;) 6 ; manet-negativa
A . A .
⋅ ∙ ∙ ∙
ab t—-—c- :::(12-l-ä)9 ad t— -c—-.-:(1-l-1)9. Ertt 1g1-
tur v' positiva inter A———-0(t—i9) et A −−∶ c [t—(i—i—ä— )9];
erit negativa inter Ach-t—(i—i-ä-W] et A:c[t—(i-l—1)9].
in tribus hisce distantiis est praeterea v':o. Ergo quae-
libet ex dictis undis constat duabus partibus aequalibus ;
recedit aereum fluidum ab oscillante sonori corporis par-
ticula in anteriora parte, accedit in posteriore; quiescit stra-270
tum medium ; maxima viget aerearum particularum velo
citas in medio semiundae anterioris ; maxima item in me
die semiundae posterioris.

127. Soni velocitas augetur a vento secundo, minui
tur ab adverso. Derhamus videns ab aliis affirmari nullam
mutationem afferri a ventis circa soni velocitatem , hanc
rem statuit explorare ita exacte et diu , ut ambigendi lo
cus omnis tolleretur. Ad hoc autem summa ipse fruens
opportunitate experimenta habebat omnino in promptu .
Nam cum ex arce Blancheath , ubi tyrones rei tormenta
riae exercebantur , saepe exploderentur tormenta bellica ,
ipse e sua Ecclesia in agro Upminsther ad 13 milliaria
distante flammam advertere poterat ; animadvertit autem
optimo usus chronometro non semel aut iterum , sed
triennio integro. Porro ex tabula , quam observationum
suarum confecit, quaeque habetur in Transactionibus An
glicanis, et a Masschembroekio descripta fuit in suis com
mentariis ad lentamina Florentinorum , constat quod so
ni velocitas inter tempus quo ventus favens spirabat ,
et contra venius sono adversus erat, cum scilicet in utro
que casu yentus validus admodum esset , discrepat un
decim semisecundis circiter in praedicto intervallo. Ergo
experimentis hisce insistendo dicendum augeri secundo ven
to soni velocitatem , imminui autem etc.
Derhami observationibus consentiunt observationes Aca
demicorum Parisiensium , qui anno 1738 exploraturi ve
locitatem soni jussu Regiae Academiae pariter testantur
non eandem esse adverso ac secundo vento velocitatem
qua propagatur. Rationis momentum experientiae suffra
gatur : nam ventus transfert loco aerem ; ergo undas so
noras ad oscillationem a sonoro corpore impulsas trans
fert ; ergo tantum accelerari debet propagatio soni , quan
tum aeris sonori translatio ratione venti importat.
Opporluna est comparatio circulorum in aqua exci
latorum ope lapilli decidentis : si enim aqua non sit sta
270
tum medium; maxima viget aerearnm particularum velo-
citas in medio semiundae anterioris; maxima item in me-
die semiundae posterioris.

271
SUS asserue
gnans sed fluens aequabili motu ; jam dum post lapidis
descensum circuli successive efformantur , lota ipsa aqua,
in qua efformantur circuli , localiter transfertur ; ergo
circuli appellent ad datum locum citius in eam partem
versus quam defluit aqua quam ad alteram in eadem
distantia ex opposita parte : ita paritate rationis in sono.
Iis , quae . modo diximus , objici possunt experimenta
Cl. virorum qui de velocitate soni nihil immutari pror
sive secundus sit , sive adversus ventus
runt. Gassendus enim , et Mersennus id sibi accidisse te
stantar ; et Academici Florentini , collocatis observatori
bus inter se duo milliaria distantibus , dum ventus spi
raret , asserunt tormenti bellici , quod medio illo inter
vallo situm erat , fragorem pervenisse eodem tempore ad
utrosque , etsi aliis faveret ventus , aliis esset contrarius.
At cum experimenta recte instituta aliis item recte
institutis nequeant esse contraria , yidendum quaenam
praevaleant. ' Florentini , quorum tentamen prae caeteris
in medium afferri solet , unica nocte , et in distantia pau
corum milliarium experimentum instituerunt . Derhamus
triennio experimenta iteravit , et in 13 milliarium distan
tia ; haec autem distantia in experimentis Derhami eadem
erat semper , a sua scilicet Ecclesia ad arcem ; in ten
tamine Florentinorum tormentum constitutum est medio
inter duos observatores intervallo ; quod intervallum utrin
que aequale asseritur , sed fortasse non ita esse potuit.
Ergo apparet quam recte Derhami observationes antepo
nantur observationibus Florentinorum , atque eodem jure
Gassendi et Mersenni . Reipsa etsi experimento quodam
Musschembroekius quoque deprehendere sibi visus fuerit
aequalem velocitatem tam secundo quam adverso spirante
vento , tamen Derhamo assentitur , et Florentinis quo
rum sagacitatem saepe alibi commendat , minime in hoc
adstipulatur.
Obiter hic notamus quod juxta auctores ferme omnes
etiam intensitatem sąni auget ventus secundas , et minuit
.
1
271-
gnans 'sed fluens aequabili motn; jam dum post lapidis
descensum circuli successive eil'ormantur ,
tota ipsa aqua,
in qua eB'ormantur circuli ,
localiter transfertur; ergo
circuli appellent ad datum locnm citius in eam partem
versus quam defluit aqua quam ad alteram in eadem
distantia ex opposita parte: ita paritate rationis in sono.
Iis
, quae-modo diximus, objici possunt experimenta
Cl. virorum qui de velocitate soni nihil immutari pror-
sus , sive secundus sit , sive adversus ventus
, asserue-
runt. Gassendus enim-, et Mersennus id sibi accidisse te-
stantur; et Academici Florentini, collocatis observatori-
bus inter se duo milliaria distantibus , dum ventus spi-
raret
, asserunt tormenti bellici , quod medio illo inter-
vallo situm erat
,
fragorem pervenisse eodem tempore ad
utrosque, etsi aliis faveret ventus
, aliis esset contrarius.
At cum experimenta recte instituta aliis item recte
institutis nequeant esse contraria , videndum quaenam
praevaleant.x Florentini , quorum tentamen prae caeteris
in medium afferri solet , unica nocte
.,
et in distantia pau-
corum milliarium experimentum instituerunt. Derhamus
triennio experimenta iteravit, et in 13 milliarium distan-
tia; haec autem distantia in experimentis Derhami eadem
erat semper, a sua scilicet Ecclesia ad arcem; in ten-
tamine Florentinorum tormentum constitutum est medio
inter duos observatores intervallo; quod intervallum utrin-
que aequale asseritur , sed fortasse non ita esse potuit.
Ergo apparet quam recte Derhami observationes antepo-
nantur observationibus Florentinorum , atque eodem iure
Gassendi et Mersenni . Reipsa etsi experimento quodam
Musschembroekius quoque deprehendere sibixvisus fuerit
aequalem velocitatem tam secundo quam adverso spirante
vento, tamen Derbamo assentitur , et Florentinis , quo-
rum sagacitatem saepe alibi commendat, minime in hoc
adstipulatur.
Obiter hic notamus quod iuxta auctores ferme omnes
etiam intensitatem soni auget ventus secundus
,
et minuit
.272
1
P
TE
8
ta
11
11
adversus. Hoc , ajunt , experientia vulgari notum est : si
quidem campanae sonus , aut tormenti explosi fragor multo
melius auditur si conspiret in eam partem ventus quan
si contrarius sit ; et saepe ad aliquam distantiam auditar
ope venti secundi, ad quam , cum ventus est adversas ,
minime audiri potest : auget ergo ventus soni intensita
tem. Ratio quoque idipsum suadet : nam vencus secundus
undas sonoras transfert ; ergo ad aurem appellent undae
sonorae , quae a centro minus remotae erunt , adeoque
intensiorem deyehunt sonum .

128. Ad soni velocitatem determinandam multa in
stituta sunt experimenta , quae tamen non satis conve
niunt : experimenta instituta ab Academicis Parisiensibus
anno 1738 praebuerunt soni velocitatem , seu spatium minuto
secundo a sono percursum = 172 , 56 hexap. = 336 , 32
metr. Apud Madras in India orientali D. Goldingham ex
perimentis per annum integrum multoties repetitis ( Annal.
de Plays . et de Chim . tom. 23. pag. 12 ) exploravit soni ve
locitatem : prodiit mediocris velocitas 1134 , 33 ped.
Britan . = 345 , 74 metr. Varias hujusmodi mensuras vi
dere est in tabella , quam protulere DD Moll , Van-Beek
etc. ( Bibliotheque universelle tom. 30) : qui Auctores opus
definiendae velocitatis soni susceperunt anno 1823 , perfe
ceruntque in Hollandia , assumpto ad observationes eo spa
lio , quod Zevenboompies et Koolijesberg interjacet. Ten
tamiva sumpta die 28 Junii praebuerunt soni velocitatem
339 , 34 metr.
Hujus diversitatis plures esse possunt rationes : ac 19. In
strumenti aut attentionis exquisitae ad instrumentum deſe
ctus ; cum enim flamma attendi debeat simulque penduli
oscillatio , jam facile est ut vibratio aliqua initio non nu
meretur. 

2. Spatium exiguum ab aliquibus assumptum ;
minimus enim error facilius est contemaibilis , si ingens
intermediet spatium. 

3.° Venti qui aut retardant , aut ac
celerant souum . llaec variationis causa attenuari potest , ac
PL
M
ti
.
0
272 '
adversus. Hoc , aiunt , experientia vulgari notum est: si-
quidem campanae sonus ,
aut tormenti explosi fragor multe
melius - auditur si couspiret in eam partem ventus quam
'si comrarius sit :
et saepe ad aliquam distantiam auditur
Ope venti secundi, ad quam, cum ventus est adversus,
minime audiri potest: auget ergo ventus soni intensita-
tem. Batio quoque idipsum suadet: nam ventus secundas
undas sonoras transfert; ergo ad aurem appellent undae
sonorae , quae a centro minus remotae erunt, adeoque
intensiorem devehunt sonum.

39 Venti qui aut retardant ,
aut ac-
celerent sonum. llaec variationis causa attenuari potest
, ac
J
.
maälzz—äwæ-EL'T-aa
&amp;.273
ferme destrui , si collocatis tormentis bellicis in utraque
extremitate A et B illius distantiae , quam debet sonus
percurrere , tormenta ipsa eodem temporis momento ex
plodantur ; tunc enim si determinetur velocitas , qua per
venit sonus ex A in B , itemque velocitas qua pervenit ex
B in A , harum velocitatum semisumma erit velocitas illa ,
qua propagaretur sonus in aere tranquillo. 

4.º Animadvertit
Musschembroekius quod cum sonus non in instanti audia
tur , sed initio minus , subinde organum aliquanto vehe
mentius percellat, hinc quidam ad initium , alii ad progres
sum sensationem soni potuerunt animadvertere , atque inde
inter se discrepare. 

5.9 Varia atmosphaerae temperies.
Determinatio velocitatis , qua sonus propagatur , utilis
est nautis ut agnoscant quantum littus , aut alia navis
distet ; militibus ut quantam oppugnata urbs distet ; geo
graphis item ut quantum inter duo loca , praecipue cum
intervallum hexapeda metiri non licet , intersit . Etenim nu
merando minuta secunda ab erumpente flamma ad usque
audiendum sonum tormenti bellici , distantia loci colligi
potest . Quod si ea sit locorum constitutio ut flamma videri
non possit,o res ita supplenda est , ut cum ad aurem per
venit souüs , exploso statim alio tormento bellico , alter hic
sonus ad primum observatorem perveniat : si hic nume
ravit minuta secunda ab eo puncto , quo explosit suum
tormentum usque ad punctum quo audivit sonum alterius
tormenti , et haec minuta bifariam dividantur , habebitur
tempus propagationis soni inter duo illa loca : ita etiam nu
bis distantiam aliqui metiri docent , numerando scilicet mi
nuta secunda , quae inter fulgur emicans et auditionem to
nitrus intersunt .

129.# Nonnulla subjicimus ex theoria fluidorum
( 106. 107 ) ad soni propagationem applicata ; ita tamen ,
ut ad aeris gravitatem minime attendamus , libratu mque
aerem spectemus tanquam elasticum fluidum eadem ubique
pollens et densitate p' , et pressione a' , et temperie n.
273
ferme destrui
.
si collocatis tormentis bellicis in utraque
extremitate A et B illius distantiae . quam debet sonus
percurrere ,
tormenta ipsa eodem temporis momento explodantur; tunc enim si determinetur velocitas , qua pervenit sonus ex A in B , itemque velocitas qua pervenit;:
B in A , harum velocitatum semisumma erit velocitas illa,
qua propagaretur sonus in aere tranquillo. 49 Animadvertit
Musschembroeltius quod cum sonus non in instanti audia-
tur, sed initio minus , subinde organum aliquanto vebe-
mentius percellat, hinc quidam ad initium , alii ad progres-
sum sensationem soni potuerunt animadvertere , atque inde
inter se discrepare. 5.o Varia atmosphaerae temperies.
Determinatio velocitatis
, qua sonus propagatur
, utilis
est nautis ut agnoscant quantum littus ', aut alia navis
distet: militibus ut quantum oppugnata urbs distet; geographis item ut quantum inter duo loca , praecipue cum
intervallum bexapeda metiri non licet ,
intersit. Etenim nu-
merando minuta-secunda ab erumpente flamma ad usque
audiendum sonum tormenti bellici
, distantia loci colligi
potest . Quod si ea sit locorum constitutio ut flamma videri
non possit A, res ita supplenda est
,
ut cum ad aurem per-
venit somä , exploso statim alio tormento bellico
, alter bie
sonus ad primum observatorem perveniat : si bic numeravit minuta secunda ab eo puncto
, quo explosiot suum
tormentum usque ad punctum quo audivit sonum alterius
tormenti , et haec minuta bifariam dividantur , habebitur
tempus prcpagationis soni inter duo illa loca :ita etiam nu-
bis distantiam aliqui metiri docent , numerando-scilicet minuta secunda , quae inter fulgur emicans et auditionem to-
nitrus intersunt.
1294 Nonnulla snbiicimus ex theoria fluidorum
(106 . 107) ad soni propagationem applicata ; ita tamen ,
ut ad aeris gravitatem minime attendamus
,
libratumque
aerem spectemus tanquam elasticum fluidum eadem ubique
pollens et densitate ≀⊥⋅ ,
et pressione a: ,-et tmperie n.
..—274
10 Fac ut concutiantur librati aeris particulae
comprehensae sphaerico spatiolo habente radianı = (y , et
centrum in coordinatarum origine 0 ; talem vero patiantur
in densitate variationem , et velocitatem recipiant juxta re
spondentes radios vectores a , ut utraque exsistat admodum
exigua , et altera queat repraesentari per f ( ) , altera per
f ( Q) , evanescentibus fg , f quoad a = o et « &gt; « ,: sit r
distantia puncti ( x , y , z) ab 0 , ut obtineant
i
x2 + y2 + z = p2 xdx + ydy + zdz = rdr ,
Propagato motu per reliquum fluidum ; quoniam v' , v &quot; , 20 &quot;
sunt constanter parvissimae , et ad fluidi gravitatem minime
attendimus , iccirco formulae ( 6 &quot; , 106 ) , missis terminis
exiguissimis secundi ordinis , factisque X = 0 , Y = o, Z=0,
dabunt quoad punctum ( aco y, z)
1 do dui 1 do dv &quot; 1 das
de
dv'&quot;
dt &gt;
M dx dt I dy to da
et consequenter
lo
I can do to edip dy+
dz dz
du dvi'
dy + dt dt (©) .
Jam vero
dic dir -dx do
dy
do dr
dr dx &amp; ar ፊ dydy 9
dy
dosdz dz
da dr
dr de dz 2
.
274
↿∘∙≖∎⊀ Fac' ut concutiuntur Iibrati aeris particulae
comprehensae sphaerico spatiolo habente radium −−−−≖ a, ,
'et
centrum in coordinatarum origine 0; talem vero patiantur -
iu densitate variationem , et velocitatem recipiant iuxta re-
spondentes radios vectores &amp;!
, ut utraque exsistat admodum
exigua, et altera queat repraesentari per f! (a) , altera per
f (a) , evanescentibus !; ,
f quoad ac −−∶ ∘
et a) 0:' :. sit :-
distantia puncti (æ
,y, :) ab 0, ut obtineant
x' ∙−⊢∫∙⊣−≖≏∶−−≀∙≖ , ædx-i—ydy-t-zdzzzrdr,
PrOpag'ato .motu per reliquum Huidum ; quoniam v', 11&quot;, v'&quot;
sunt constanter parvissimae ,
et ad fluidi gravitatem minime
attendimus ,
iccirco formulae (ö&quot; . 106 ), missis terminis
exiguissimis secundi ordinis
,factisque X::o, ïze, Zzo,
dabunt quoad punctum (æ, y, :)
1 da der ↿⊄∄↑≖⋅∙∙∙∙ dv&quot; 1
ftdæ— dt'pdj&quot; dt'p.
et consequenter
1 der da der − − ... −−⋍≀ )
↽−
⊬
(dxdx—i- dy d),—i- dz :.
eiu' dv&quot; dv'&quot; )
∙
—- de—k-äuy—F—ät— d: (i)-
Jam vero
(la-: (lux dr dar das dr
,
(Erit: ∙−−−−∙⊋−∣∙∙ (?;-lld? ïydj :
z; gd)»- ,
dadz—dw drdz ,275
ac proinde
du do dos
dx + dy +
dz =
dx dz 1
do Idr dr dr
• dxt
dr dy
do-dr
) = dr
v , &quot; =
insuper v ' un
v , ideoque
dv' dv &quot; dv '' d (v'da tudytou '&quot; dz) dx +''
-dyti dz =
dt dc dc dt
dfædx + ydy + zdz
d (vdr)
dc dt
traducetur igitur ( i) ad
1 do dr d ( vdr)
dt -
( i ) .
f . dr
Ponentes
dQ u'dx + u'dy + v &quot;da = dQ ,ut sint v'=
dx 10&quot; :
dQ
dy
2011 !
dQ
dz
assequimur
dQ d
To
d
Come)
vdr d (vr)
dr,
dc dr -
' de
dr dr :
dr dt
vertelur itaque ( i ) in
275
ac proinde
Heia-4- — ;d; -]-d 2; ad;:
≤↾−⋮⋅↾ ïta.-.- −∙⋅⊄∄↗∙⋅∹−∙≦− ∙⋅−⋤−↙∄⇝⇌∶−∙⋡−−↙≀≀∙ ;
'
∙−−−⋅⋮∙⋅ ∙−− £ &quot;zl. lnsuPero—rv,-v' '.—r-v,-v&quot; rc:,ideoque
dv' *d-v&quot; ...-'de: &quot;' d(v'dx -1-v&quot;dy—1-v'&quot;da)
ïdïdæ'l' dc ↙↡↗⋅⋅⊢ ⋅⊋−⋮⋅∂≖ dt ∙−−−⋅
d
(ædæ A-ydy ∙−⊢ zdz 0)

: - d(wdr) ∙↗

d: ∙
&quot;' dt
traducetur igitur (i) ad
1 du! ∙∙∙ d(vdr) .,dr ∙−−− −− dt (( ).

Ponentes
u m ∙
l
∙
'o v'dx-t-m dy—t—v ds:dQ,ut sint d
∙↗∶⋛−≣−∙⇝ :::-g. *v
⋅⋅∙−−−∶∙−↿⋚≳−∙ '
assequimur
⋅
dQ d(vr) d —&quot;'Q) d (....) — (dr ∙∙∙⋅ ⋅ dt
.
⇀
mi'-&quot;ïd&quot; d: −⇀ dt 4&quot; ∙− dr 4&quot;
vertetur itaque (t&quot;) in276
1 do Cena
( i &quot; ) .
hdr dr
Pertingente motu ad punctum (x , y , z) , crescit ibi librati
aeris densitas M , et evadit
l = h' ( 1 + $) ;
augetur aliquantulum etiam temperies n in ipso condensa
tionis actu , fitque ntv : pressio , quae ob auctam den
sitatem evaderet a' ( 1 + 8) , augescit adhuc propter incre
mentum v ; et cum v pendeat ab € , novum pressionis in
crementum pendebit rursus ab z , eritque ob incremento
rum tenuitatem ipsi &amp; ad sensum proportionale ; iccirco ,
praetermisso é , emerget pressio ex duplici capite aucta
m = (1 + 5) (1 - +-AE) w [1+ (1 + A ) £] .
Poterit ergo ( i&quot; ) sic scribi
1
ale
de de .
( 1 + A
M
17 € dr dr &gt;
seu
dt
is 13
( 1 +A)
dL ( 1 + -E)
dr dr
Hinc
Bis
( 1 - +- A ) L ( 1 + E)
dQ
dt
276
Pertingente motu ad punctum (a:, y, :)
, crescit ibi librati
aeris densitas p! ∙ et evadit
it:-&quot;a' ≺↿−⊦⋮≻⋮
augetur aliquantulum etiam temperies 1: in ipso condensa-
tionis .actu
,
(itque n—l— »: pressio , quae ob auctam den-
sitatem evaderet m' (1 ∙−⊦ e) , augescit adhuc propter incre-
mentum »; et cum 9 pendeat ab a , novum pressionis in-
crementum pendebit rursus ab a , eritque ob incremento-
rum tenuitatem ipsi a ad sensum proportionale: iccirco
,
praetermisso ?
, emerget pressio ex duplici capite aucta
a:d(1-—1—s)(1-1-Aa) −−∶ a'[1-t-(1-t-A)s] .
Poterit ergo (i&quot;) sic scribi ∙
' d(ig) a' 1 de dt −− ↿⊣⇁∆∼ ∙−− — pii '
↿⊣−∙∊ dr dr
'
seu dc.-112) . :
p.
liinc
' d
⋮⋝−∽ ≺↿⊣⇁⋀≻↧∙≺↿⊣−⊽∊≻↽−∙−−−∙− 3- - p. dt277
est autem ( 27.29º. )
?
L ( 1 + E) = E
+ - +
Propterea , facto ( 1 + A ) = C ,
A dQ
&quot; .
ca do
Ad haec :
dv ' dy&quot;
dx
dur
dz
d’Q
dx² +
d’Q
dy ?
+
d2Q
dz² ;
dy
formula igitur ( 619. 107) , substituto p. ( 1 + €) loco fe , mis
sis terminis exiguissimis secundi ordinis, atque attenta ( i'''),'''
praebebit
d2Q daQ
dea = ca
e
d Q
dy ?
det
d2Q
da ?
( it ) ;
\ dx²
et quoniam
dQ dQdr dQ y
dr dx
dQ dQ x
dQz
dr
of
dQ_dQ
dxi
dz dr dy dr
unde
d’Q
dx²
daQ xa
dra 2
dQy? +z2 d2Q
dr p3 dy?
d’Qys , dQ x2 + z3
dr ra dr p3
d'Q
d22
d2Q 22
dr.2 p2
dQ x2 +y2
dr 产 产
277
est autem (27 .290.)
e* 53 si
1 :−∙− ∙−−− Ou: ∙∙ ↥⋅≺−⊢∊≻ s ⇄⊣⋅∙∃ 4(.,, :
Proptereü , factO :
(1 :A) −∙− c,,
1 dQ
ca dt
Ad haec :
d'v' dv&quot; ⊣⇀ ↙∣⊛∣∦ ↙≀≏⊄⊋ sz dïQ .
da: d] dz −⇀⋅ dx: d),: d:&quot; a
formula igitur (ö&quot; . 107) , substituto p: (ii-145) loco p. , mis-
sis terminis exiguissimis secundi ordinis, atque attenta (zw'),
' ∙ ?' praebebit
sz sz daQ sz .&quot;
.
(.i—t;. ⇀−− c&quot; da,-3 ∙−⊦ ∠∄∫≖ .* dzg) (: la.
et quoniam ⋅
'
∙ '
dQ dQ dr-—dQæ dQ—JQJ, iq—æi
dæ' drdæ dr-r'äy dr r'dz—drr'
unde ( ⋅ ⇁ ⋅ ,
dj—æ i,*deail'zz dag—dïQlyiA-iQxa—an dat.:—dr: rr: dr &quot;3 ,dyl d'.) rg ∙ &amp; r3
dQdeina *igæ'ä-J' .
d:&quot; dr: ra dr 'a &quot;.278
ideo traducetur ( i&quot; ) ad
d’Q
dia coloro
d-Q
( dra
2 dQ
r Thedrbest seu da (rQ )
dla ca d ( )
dra
Ex (i) habemus ( 120. 6º. )
Q = -- [80+ c ) + F(r — ct)] ;
et consequenter
dQ 1
dr [ f'ir tt) + F' ( r - ct )]
) — ]
( i&quot; )
Ar + c ) + F (r - ce}]
- [f(r + c )—– F&quot;( – ce)].
1 dQ
c2 dt
Ad f et F determinandas , sume t=0 ; habebis
f(a ) f (a ) : . E =
proinde
a&gt; f( x ) = af ( a ) + aF'( a ) f « ) - F( a ) ,
- caf( a ) = f ( ) – F' ( « ) .
Pone
fa) +F(a) = w , fra ) — F( X) = w ;
erunt
.
278
ideo traducetur (i&quot; ) ad
432—
. «PQ-,. 2
sit'—c &amp;? 747↲≺≀≻∙⊷≖∂↿≺↾≬⋗−≖∙↲≖≺↗≺≀≻ de'—' dn
Ex (.&quot;) habemus 120. 60.)
- 1
Q ∙−−− ;- [f(r.:i- ct) −⊦ F(r— et)] :
et consequenter−∙∙ :? ∙−−⋮∙ [f'(r-l-ct)-]-—F'(r-—ct)]

—--—1r;-[f(r-]-ct)-]—F(Qr—ct)] ,
1 dQ— ↿
s —
⊑
ca d: ;S.-[f(r-t-cn— F'(r— cs )].
Ad f
et F determinandus, sume :::-o; habebis
w:f(a)
, s:f,(a):
proinde ⋅
æf(a):af(a)—1—al-&quot;(a) -—f( cc)—P(a),
—eaf,(a)——:f(a)—F'(a).
Pone
fe) -t-F(a) :::.)
,f(a)- F(ac) ∶−∙−∾⋅
erunt
0&quot;)279
d @ = f( ) + F&quot;(x)= f(a) —F(«) da = f( x )da ;
dw ' = [f ( ) — F ' ( ) ] da = -ca f ( ) da ;
unde
a fixdx , w ==
cfafica) da :
hae suppeditant
f(Q )
w -two
2
1
2 frazda - of facada,
F(x)= afscada + ; fafceda;
ideoque ( iº )
f(x)= ff( )fat a pascafica),
Standa+ af )+ caf,ca).
F ( a )
2
20# Secunda membra (2011) evanescunt quoad a &gt;
Az ; ut igitur functiones
flrtct) , f'(x + ct) , Fr — ct ) , F &quot; (r — ct)
sint aliquae , non debet r ct esse &gt; &amp; : atqui in ordi .
ne ad fluidi particulas ultra Qi , cum e sit quantitas posi
tiva, est semper s + ct &gt; As ; ad has ergo particulas quod
attinet, erunt constanter
279
d(a-i)— af(a)—t-aF'(ac) —f(a) --F(a) dae— f(a)da;
« - æ
dar.-:. [f(az) —- F' (et)] da :: — cat &amp; (et) da;
uude '
∙∾−−−∶∶∝ «a)daz , Q':-irc af,(a)daz:
bae— suppeditant
aH—a' 1 - 1
f(a)-— 2 ∙−− ⋣∙ ⊄∫⇟↸∝⋟↙≀∝−−−⋮−−∘∫∝ f,(a)da,
c.)—of 1 1
Hall- 2
&quot;*.2 «li(alda—r—ïcfafdaW-ï &amp;
ideoque (i'&quot;)
1 1 1
f(a): -2- f(a') fat—1- ä-a ((a)—ïm f,(a) ,
↿ 1 1
F'(a) ∶−∙−−∙ ∙⋮⋅≳−∫∫≼∝⋝↙≀∘⊢⊢ -2-af(a)-1- -2—- caf,(a).
2011 Secunda membra (im) evanescunt quoad ac)
«,.; ut igitur functiones ⇀
f(r-t-ct) , f(f-Jf- ct)
, F(r -
ct )
,F'(r
-
et)
sint aliquae , non debet r

: b et esse )a, :atqui in ordine ad fluidi particulas ultra et, , cum t sit quantitas posi- tiva, est semper :- −⊢ ct a, ; ad has ergo particulas quod attinet, erunt constanter280 fir + ct ) = 0 , f ( r + c ) = 0 ; et consequenter -F(r —c)F( r -ce ) , 6 = 1.- F &quot; (r — ce) (**** ) . 30 Aereae particulae respondentes radio vectorir non incipiunt moveri nisi quum tempus sic increvit , ut habeatur rct = ly , seu r = ctt cy : inferimus sonum propagatum iri uniformiter velocitate V ( 11 + A ) Quod spectat ad numerum A, habemus (87. 70. ) a = im [1 + a (n + v)] = im '(1 + E)[ 1-+ an + ») ] , itemque ( 10.) 5 '[1+ (1 + A )ɛ] =; if' ( 1+ an) [1 + ( 1 + A ) ]: hinc i '(1 + E)[ 1-+-ant-v) ] = iu'1 + an ) [1+ ( 1 + A )ɛ ]; ex qua eruitur av A av( 17) El 1 + an ) $ ( 1 + an ) Ponamus vase aliquo accurate obserato aerem conti neri ejusdem densitatis pé ac temperiei n cum aere exter• no; sitque h altitudo barometrica utrique communis : con . ⋀≀∙⊣∙∙∘⊔≔≖∘∙⊓≀⋅⇀⊢∝⋟∶∘⋮ et consequenter 1 1 ⇀ ↿ ∙ −⋅−−−− —F'(r-ct)-—;F( r—ct). : ⋅−−−− -—F'(r—ct)(t&quot;&quot;). r r cr 3&quot;:- Aereae particulae respondentes radio vectori r non 1nc1piunt moveri nisi quum tempus sic increvit, ut babeatur r— ct:ac, , seu ::- ct ∙⊦∙ at, :inferimus sonum prcpagatum iri uniformiter velocitate 'c: Vä- (1—1-A) (i&quot;) - Quod spectat ad numerum A, habemns (87. 70.) ∙ saiw-1-a(n-1-v)]:zp'u-u-e)[1-.-a(nM)]. itemque (10.) 6 −∙−−−⊤ w'[1—t-(1 a—A)e] :; ip'U—t— an)[1 ∙−⊦⋅ (1 ∙⊢ A)e]: bino - ⋅ ↴ 's ip'(1-1-s)[1-1-a(n-1-v)] ∶−− ≀⋅⊬⋅≺↿−⊢⊄⋯≻⊏↿⊣−≺↿−⊢∆≻∊⊐⋮ ex qua eruitur ↼ . cru-H:) av ∙−−− −∙∙ e(1-1-an) s(1-1-an) Ponamus vase aliquo accurate obserat'o aerem conti-- neri eiusdem densitatis pf ac temperiei iz cum aere exter- no; sitque !: altitudo barometrica utrique communis: con-281 1 11 cipiatur extrahi e vase aliquantulum inclusi aeris, vel qui erat inclusus aliquantulo magis comprimi , et denotet d'1 Fé) densitatem , h' altitudinem barometricam, postquam aer in tra vas ad pristinam redierit temperiem n. Tum constituta parumper communicatione cum externo aere, donec nimirum redigaturad h, mutationem quandam suscipiet lam p' ( 13) quam n; et illa quidem transformabitur in u'1 *8' ) (18&quot; ), haec autem in ny. Sed cum v' brevi evanescat, et so la n supersit quin variet MIFÉ' ) (1 # &quot; ) , mutabitur iterum h et evadet h &quot; . Alteram instituendi experimenti rationem sequuti sunt Desormes et Clement , alteram Gay - Lussac et Welter: inspiciatur sequens tabella . torie pun Desormes et Clement. n = 12 , 5 , heo” , 7665 , h - hs o ” , 01381 , 11 h - h&quot; = 0 , 003611 ; 2 &quot; hi sese restituit ad h intra tempus &lt; &lt; 5 Gay - Lussac et Welter. n = 13° , h = omom,, 757 757 ,, hh -- hh : = 0 &quot; , 0163644 , h &quot; - h = 0 , 0044409 ; q &quot; h sese restituit ad h intra tempus 6 Iam vero, depolante D densitatem hydrargyri , sunt conti erter Dgh = ip (176) (1 + an ) , 000 19 villi] 1 !' torir L, 111 )num conti- erit?' con- 281 cipiatur extrabi ei vase aliquantulum inclusi aeris,'*vel qui eratinclusus aliquantulo magis comprimi, et denotet p.'(1::1:s') densitatem, h' altitudinem barometricam, postquam aer in- tra vas ad pristinam redierit temperiem 11. Tum constituta parumper communicatione cum externo aere, donec nimirum h' redigatur ad h, mutationem quamdam suscipiet tam (if( quam ∎∶∙∶∔⋅∶∊⋅⋟ .: et illa quidem transformabitur 111 p.'(1.-.;:s')(1:£ e&quot;), haec autem in :::». Sed cum v' brevi evanescat, et so- la n supersit quin variat p.'(1:1: a' ) (1 :1: e&quot; ) , mutabitur iterum I: et evadet h&quot;. Alteram instituendi experimenti rationem sequuti sunt Desormes et Clement , alteram Gay- Lussac et Welter: inspiciatur sequens tabella. Desormes et Clement. 11:12&quot;, 5 , h:o&quot;,i7665 , 11 -h': 0&quot;, 01381 , h — h&quot;:o&quot;, 003611 ; h' sese restituit ad h intra tempus ≺∙−≣− ∙ Gay - Lussac et Welter. ' n:130, h:o'&quot; , 757 ,h'-— h:d&quot;,0163644 , h&quot; — h:o'&quot;, 0044409 ; 1&quot; h' sese restituit ad h intra tempus (—6—- . hm vero, denotante D densitatem bydrargyri , sunt Dgh':i;t'(1q:e')(1—t-an), ∎⊨∎ 'i i ! 19282 Dgh = id'l 176) ( 1 #t&quot; ) ( 1 + a nv( ) ) , $ Dgh &quot; = id'l 176 ) ( 1 + &quot;) ( 1 + an ) : hinc h &quot; = 1 &amp; € &quot; , h h &quot; 1+ anty') 1+ an = 1 + R 1 + an h &quot; 'h αν&quot;' h hh&quot; h&quot;

1 tan
ideoque }
ań
= ( 1 - an )
h hh&quot;
h &quot; 7 &quot; -h*
Substitatis valoribus ex Gay - Lussac et Welter ,
αν
€&quot; ( 1 + an)
=0, 3785020934 :
1
R
et quoniam iste numerus neque ex temperie neque ex pres
sione pendere videtur, iccirco poterit generatim assumi 1
A= 0, 3785020934 ;
sicque soni velocitas prodibit expressa per ( 94. 1 ° )
V
1 , 3785020934
to
fe
-V 1,3785020934i(1+ an) = 1009 , 614V1+ an (i&quot; ).
282
1131. −−−−⋅⋅⋅⊬∣≺↿∓⋮⋮≻ ≺↿ :::&quot; ) ( ↿
....
(a:-:») )
.
Dgh&quot;:ip.'(1q:s-' ) (1:t:€&quot;) (1 qum):
tibine
−≸⋮−⋤⋅−⋅
≕↿
∙∙⋅⊧∙≘⋅⋅ ,
∣∣⋮∙⋅ −−⋅↿−⊦⋅↿∘∙≦↾∙≔⋮∙⋓⇗≱ −−⋅↿∙−⋅⊦−∙↿−−∙⋮⋮≔−∙
zh :&quot; ∙∙∙ l:&quot; --l:' ,.4, .av' −∣∎ −∦∣⇂∙∙ ;
↙
h' 1 −⊢⋅⋯∎ &amp;
ideoque
av' h' b—h&quot;
e&quot;(1-t-an)— h&quot; h&quot;—-h'
⋅
Substitutis valoribus ex Gay-Lussac et Welter
,
av'
et quoniam iste. numerus neque ex temperieneque ex pres-
sione pendere videtur, iccirco poterit generatim assumi
sicque soni velocitas prodibit expressa per (94. 10)
I
c ∸−−−⇀ ↿∙ 3785020934 1;—
wjt—lV1, 3785020934 ⋅⋅≺↿∙⊢ an) ∙−−∶ 1009,- 614 ∣∕↿⊣⇀∘≀∎ tc&quot;)- H283

Si attendenda est quoque bygrometrica aeris constitutio, de
notante 6, pressionem libratam ab aqueo vapore , pro ui'
substituendum erit ( 96. 4º. )
1
seu
i( 1+ an)
exsistet nempe
V
11
w' il 1 to an )
1 , 3785020934 3
--8
W1
009 , 614
V
8 ã' (1+ an )
80-30 ,
(i &quot; ) .
In soni velocitatem diligentissime inquisiverunt an
no 1822 DD. Arago, Prony , Mathieu , Bouvard, Humboldt
et Gay - Lussac: distantia, ad quam observationes de cor
ruscatione flammae et fragore instituebantur in explosionibus
Lormenti bellici, ea fuit quae Monthlery et Villejuif inter
jacet ; velocitas inde deducta, seu spatium iolra 1&quot; a so
no percursum, 89 Erat autemn =15°, 9; unde
Vitan = 1 , 029 : dabit igitur formula ( it )
340metr.
103gped
.
893 metr .
337 , 432 . &gt;
Hygrometricam quoque aeris constitutionem notarunt
Auctores Cl . Sub mediocri videlicet altitudine barometrica
metr .
0 76 index hygrometri, quod vocant a capello, o
slendebat grad . 72 : in hac vero hygrometrica aeris consti
lutione, et sub temperie 15° , 9 ,pressioni , respóndet ba
metr.
rometrica aliiludo 0 00679; hinc
283
Si attendenda est quoque bygrometrica aeris constitutio, de-
notante u', pressionem libratam ab aqueo vapore , pro pf
substituendum erit (96. 40.)
exsistet nempe
' ∙
1
T
∘⋅−−− ∣∕ 1, 3785020934 &quot;'
'(
a, ↼⋅⊢ s
'&quot;&quot;- ..
8
1009 614⇂∕ afuit—13:111) (i&quot;)-
In soni velocitatem diligentissime inquisiverunt au-
no 1822 00. Arago, Prony, Mathieu, Bouvard, Humboldt
et Gay-Lussac: distantia, ad quam observationes de cor-
ruscatione dammae et fragore instituebantur-in explosionibus
tormenti bellici, ea fuit quae Montblery et Villejuif inter-
iacet :
velocitas iude deducta, seu spatium intra 1&quot; a so-
no percursum, :340'm&quot; ,89 Erat autemn:150,9; unde
l/1-t-an :1, 029: dabit igitur formula (ix)
0:1038ped' ,
893 :..- 337'&quot;&quot;' ,432.
Hygrometricam quoque aeris constitutionem natarunt
Auctores Cl. Sub mediocri videlicet altitudine barometrica
Gum. , 76 index bygrometri, quod vocant :: capella, o-
stendebat grad. 72:' m hac vero hygrometrica aeris consti-
tutione, et sub temperie 150, 9 ,pressioni a', respöndet ba-
rometrica altitudo Omm ,00679; binc284
v
80
8w' 30,
=1,002 ;
et consequenter ex (3 &quot; ) eruetur
1040ped ., 97 = 338metr .
11 .
Consensus itaque experientiam inter et expositam theo .
riam tantus invenitur , ut major profecto desiderari non
debeat in praesenti argumento : difficile admodum est in id
genus observationibus ventorum vim prorsus eludere, alias
que causas declinare quae huic consensui multipliciter no
cere possunt : mirum deinde quantum ardua res sit va
lorem A experimentis accurate determinare.

4. °* Evanescunt secunda membra (iº !! ) etiam quoad
a = o : in distantia igitur r evanescent &amp; , v statim atque,
labente tempore , eo devenitur ut sit rect = o . Quia er
go in distantia illa incipiunt , v esse aliquae quum
rct = lg, sequitur motum in distantia illa minime du
raturum ultra tempus Eaedem itaque &amp; , v evanescent
in distantia r- , statim atque incipiunt esse aliquae in di
stantia r : propterea non cientur una nisi particulae con
stituentes stratum crassiliei

5.° Velocitas v duabus ( 2.º į&quot; ) constat partibus , quarum altera sequitur rationem reciprocam distan
tiae a centro unde promanat sonus , altera rationem reciprocam duplicatam ejusdem distantiae: functiones praeterea F, Fmanent constanter parvolae. Quia igitur im
pulsio in datum obicem facta pendet a velocitate v , patet , quo longius propagatur sonus , eo magis ipsum debilitatum audiri. Quum sonus ad modicam pervenerit distantiam, licebit secundam illam partem negligere; eritque 

Inferimus illud: si impulsio in obicem facta quadrato ve.
locitatis v sumitur proportioualis , rationem duplicatam di
stantiarum sequetur soni debilitatio ( 125 ) .


6.°* Fac ut librati aeris particulae concutiantur
una circum plura puncta O , 0 &quot; , ... ; quorum distan
tiae ab ( x , y, z ) exhibeantur per r' , o&quot; .... ; ipsis.
que O' , 0 ' , ... , tanquam originibu's respondeant sua
axium systemata parallela systemati habeati originem O.
Quoniam novae coordinatae s ', x &quot;, ...5,0 &quot; , ... é , z &quot; ..
constantibus quantitatibus differunt ab x, y, z ; ideo
dr ' dr dr &quot; ar 
dx doc ' F ' da d.x &quot;
!
y' g &quot;
dr'
dy
&gt;
dy '
p&quot;
dr
dy &quot;
dr
dz&quot;
dr dr
dy
dr'i
dz
2
dz dz'
el consequenter
dQ _dQ dr dQdr&quot;
tar
dx
+ .. dx dr' dx
dQ x'
dQ y
dr +
dQxt&quot;
dr'' gli t....''
dQ_ dQ y
+
dy dr p ' dr to.
dQ
dz
dQ á
dr '
+
dQ
di &quot;
.
. ilemque
to
d²Q x 2 dQ 7/ 2+22
dx² dr'a g'a + +
285
1 '
r
inferimus illud : si impulsio in obicem facta quadrato ve-
locitatis v sumitur proportionalis, ratiunem duplicatam di-
stantiarum sequetur soni debilitatio (125).
691» Fac ut librati aeris particulae concutientur
una circum 'plura puncta O', 0&quot;, ... ; quorum distan-
tiae .ab (æ, y, :) exbibeantur per r'
, r&quot; . ...; ipsis-
que O'
, O&quot;, ,
tanquam originibus respondeant sua
axium systemata parallela systemati habenti originem 0.
Quoniam novae coordinatae se', a:&quot;,
...y' ,y&quot;, ... z', :&quot;
..
constantibus quantitatibus differunt ab a:, y, :;
ideo
dr' dr' æ' dr&quot; ∙∙∙ dr&quot; ∙∙∙⋅ æ&quot; ⇀
dx daf—r dx' dr&quot; ≀⋅∎⋅↬⋅⋅⋅
et consequenter
dQ 'der- −⊦↙≀≺≀∂∙↾∙∙⋅ du:- dr'dæ dr&quot; da:
dQ æ' dQ æ&quot; &quot;
dQ ∙−−↙≀≺⊇∙⊺ d.QJ
&quot;?;/7 21.-717 −⊢∙∙ 4&quot;?!— −−⊣∎∎∙∙
': &quot;dy— dr'r'
.dQ —dQ f:: dQ ∙⋮↾∙⋅ ∙−⊦ '
dz —dr' l&quot;-1 dr&quot; r&quot; ⋅ ..
itemque
'PQ −− ↨≖≬x&quot; dQ ∟∣≖⊣−≖∙∙∣∷ .
⋅
'
dæ' dr'3 :&quot;2 −⊦⋅−−(Ti—' −−⋅∣⋅∙−⋅↾⊰ ..,-286
daQ x2 dQ &quot; 272&quot;
+
dr''2 p &quot; 2 dri p/13 + ... ,''
da d’Qy'a
dya drar'a tari
dQ x2+22 +
p3
d'Q.7 &quot;?, dQ x&quot; : + z'2 + ti. dr&quot; ' a p &quot; 2
lo:
dri
d2Q
ddza
daQ z'2
dr2 p'2
dQ x's + y'2 dQ 242
dr' 3
+
dril2 pll2 +
dQ x2+ y'a
ti ..
Adhibitis substitutionibus in ( i ' ', 1.0 ) ,
d'Q
de2
( d - Q
= c2 Adr'a +
2 dQ d2Q 2 dQ
z dr + dra +pdr&quot; + ... )

;

ex cujus forma intelligimus fore
Q = [filr'tou + F (r — ct)]+
[far&quot; +41++ F.(r&quot; ct) ] + . ( * &quot; ).

Nunc facile stabilitur illud : in hypothesi plurium concus
sionum simultanearum , ubi eae ad punctum ( x , y , z )
eodem temporis momento una perlingant , numerus e ni
hil erit aliud nisi summa consimilium numerorum re
spondentium iisdem concussionibus seorsum spectatis ; si
quidem ( 1.0 ) .
286
(PQ ∙⋅⇂⋅∥∙ ↿ dQ
dr&quot;: ∙↗≀∦≖∙⊦≖∥≖⊹ r'&quot; ∣
dr&quot; r&quot;3 ⋅⋅
'
'
- «PQ— d'Q 7&quot; ∙⊦↙∄≺≀∙−−−−∙−−−−−⊦⋅ æ'2-l-z'2
d]:—
d'Q )&quot; dQ æ'ä-l—z&quot;; .
dr&quot; ≀⋅∥⋮⊹↲≀⋅∙∣∣ r' '34- ⊣− ⋅⋅. '
d-Q Adeo ∷⋅≖ ∣dQ ⊴↾∶∣≖−⊦∜∣⋅ æno z.&quot;:
dza 'di'/3 r'&quot;
'
dr' r'3 dr&quot;3 r&quot;'
dQ ∙⋅≖∥≖⊣−∜∥≕ ↿
dr&quot; rl'3. 'l .
.. ∙ ∙
Adbibitis substitutionibus in (i&quot;. 1.&quot;).
duo (PQ 2 dQ 2 dQ ∙
∙−−− ∘≺↙↙↾∣≏−⊣−≀⋅∣ ∡≔∣∙⊦≤∶−−⊽− ⋖⋮≀≕−⊽∣−⊋−↾⊽⊣−⋯≻
ex cuius forma intelligimus fore
Q ∶−∎⋅
⋅↗⋮⊤⋅∐⋩≖≺⋅∦⊣⊸⊩⊢−∶⋮∙− ∇≖≪↗⋅⊣⊸≀⊢⊢ F,(r'—-ct)]-l—
F.(r⋅⋅⋯ ]-l—-
Nunc facile stabilitur illud :, in hypothesi plurium concus-
sionnm simultanearum , ubi eae ad punctum. (a: ,
J
,

: )

eodem temporis momento una pertingant, numerus :ni-
hil erit aliud nisi summa consimilium numerorum re-
spondentium iisdem concussionibus seorsum spectatis; si-
quidem (1.&quot;).287
DP zo al
- F&quot;
['r( + ce)-F'(x'ct)
[facr &quot;+ c8)— F'xr&quot; —cr) ] - ...
Insuper
DP
dQx' dQx&quot;
+ t . dx dr ' dr&quot; r &quot; + ...
G [r« tch+F&quot; ret) ]– 16 +6 +
F.( c )]) + ( - &quot;+e +F',(==ci)] –
[for&quot;tor)+F60—60)) + ....
vº dQ
dy
dQ r' dl go &quot;
+ + .. dr ' r ' + dr &quot; r &quot;
G - triktet)tF'(x - ce )]= i [ fim'tot +
Fa(r = -1)]) + ( -186 *408)+ F&quot;(&quot; –ce)] -
wraca&quot; terhFall -ct)]) + .
287
⋅⇌⊐ ∙−−≕↿−∙⋅ ?,?&quot;-- --',..'[f .
(r -!-c:)—F'.(r -—-c:)1—-
⋅≺∽⋅−∎↿⋅⊤∶∁↿⋮⊅⋍∣⋅∥⊹⋯∙− ↧⋅⋅∣∙≺≀∙∙⋅∙⊳∙−∙∘≀≻ ]— ....
lnsuper
&quot;- «me ⋅≄⋅ .... −−∶ .. ↼−−⋍⋜⊑⋅∙−−⋅∡⋰−∙⋅⊤−⊢⊿−≀⋅−∙⇉↗⊷ −⊦ ∙
⋅⋅
⋅
&quot;
, 1! ' '
⋅∎ ≺∎≙∶∎↾⋅⊀∎≺∣⋅⋅−∣∎⊸∘⊣−∏⋅∎ (' -—0t )]-— ∙≀−∙⋅∙−∙−∣⋅∫∎≼∣∙∎∙⊦⊸↥⊢∣−
,
' ⋅ æ' .
⋅ ↿ ⋅
⋅ ⋅
'
∙
Fl(r &quot;'&quot;'ct)
&quot;
])&quot;T'f'l'(—: [f,(r'Lï-CO—l-F', ∎⋅ ( r—ct )] ∙∙∙⋅
r
∙
∙≖−⋅≟⊑ ⊏∣≖≺↗⋅⋅−∣⊸≀≻⊣−↿⋮⋅≖≺⋅∙⋅⋅∙−−∝≻∃ )£f.-.- −⊦ ∙
⋅∙∙
. ∂≺≀∙∙−⋅∠≀≖≀∜⋅
lu.—dy dr' r' ä-l-dr&quot;— r&quot;∶∣⋅−⊦∎⋅ .. 1
(£.- ⊏⊀⋅≺≀⊤∙∙⇀⊸⋅≀⊢⊢≖⋅⋅∙⋅≺ r'-ct )1: ;; [f.(f-l-cu-l-
F.(r'— et)] ≻∙∜−⋮−⊦r ≺↿⊤↕∣∣≖≺↗⊓⊣⊸⋍⋝⊣−⇂⋅⇁⋅⋅≺∣∙⊷∙−∘≀∏ :- - ⋅
⋝∙≟≟∁∣≖≼↾∣⋅⊹≕⊢∣⋅⇁≖≺∙⋅≀∙∙⊸∁∏ ⋟∑≖∙⊤⋅⋮− −⊦∢ ∙ ∙ ∙288
dQ
dz
dQz
dr
dQz&quot;
t .
dr '' r &quot;''
( -fr.( tre)+F ;(r = cr)] - Pfalriteest
Fu
F.(x*—- )]) + ( far&quot;+c)+ F &quot;–ce)] -
pen
na[ far tcent Ffrº-cr)]) + ...;
UI
De
go
inferimus velocitatem v debitam simultaneis concussioni
bus circum 0,0 eodem temporis momento ad
punctum ( x , y , z ) una pertingentibus nihil fore aliud
nisi resultantem ex velocitatibus , quae debentur iisdem
concussionibus seorsim spectatis : atque hinc facile intel
ligimus cur, pluribus corporibus simul resonantibus , inter
oscillationes in aere excitalas non habeatur confusio ,
omnesque diversi soni inde orti ad aures distincte per
veniant. Huc spectat principium de superpositione exiguo
rum motuum.
7.04 Redeuntes ad unicam concussionem in 0 ,
ponamus aerem contineri tubo cylindrico , cujus axis ox,
motumque particularum esse ipsi OX parallelum : erunt
v&quot; = 0, v&quot; = 0; propterea formula ( i&quot; ) evadet
d2Q daQ
de unde Q = f ( x + .ct) + F ( x - ct ) ; ረder2 1
et consequenter
288
... dQ- JQ :
&quot;'l-(,Q' z'—dz −−↲≀⋅⋅ r' dr&quot; r'

ll ≼⋅≟≑⊏∣∣∙≺↗⋅⊣⇥≻⊣−≖⋅⇁⋅≖≺↗⋅⋅−⋅∘≀∏ ∙−⋅⋅⋮−⋅⋮⋅⇆⋅⋅⊔≖⋅∊⋅↾⋅⊣↽⊸≀⊢⊦ ∙
,
⋅ mo*—cn] )f— ⊣−≺⊽⊏ ↑∼≖≼↗⋅∙−⊦∘↥≻−⊦ ≖∸⋅∙≖≺↗∙∙−∘≀∏ −
⋮∙−⊦∘≀≻⊹↧⊸⇁≖≺≀∙↝−−∘≀∏⋟−⋮⊽ −↿− ∙ ∙
⋅
inferimus velocitatem v debitam simultaneis concussioni-
bus circum O'. 0&quot;
, ... eodem temporis momento ad
punctum ( x . y ,

: ) una pertingentibus nihil fore aliud nisi resultantem ex velocitatibus , quae debentur iisdem concussionibus seorsim spectatis: atque hinc facile intel- ligimus cur, pluribus corporibus simul resonantibus . inter oscillationes in aere excitatas non babeatur confusio. omnesque diversi soni inde orti ad aures distincte per- veniant. Huc spectat principium de superpositione exiguo- rum motuum. 7.&quot; Redeuntes ad unicam concussionem in 0, ponamus aerem contineri tubo cylindrico, cuius axis OX. motumque particularum esse ipsi OX parallelum: erunt 0' '::--0. 0&quot; ':o; propterea formula (i&quot;) evadet ⋅ 32? —-c £?. unde Q—−−⋅∣↗≼∶∁−∔⊸⇂⋟ -l-F(x—-ct); et consequenter289 dQ 1 dQ dx = pilatot) + F '( x - 1), E = - ca do [fotot) – F (x – ct )] . Functiones f et F absque ulla difficultate determinantur: sunt enim ( 1.9). f( x) = f(@ + F'(Q ), - cf:(Q ) = f (a) F '( ) ; ideoque f'( X) = f (Q )-cfi(Q ) 2 f(@ t-of ( ) F (a ) = 2 Ultimae ac penultimae aequationis secunda membra eva nescisnt statim ac a fil &gt;Oto : erit itaque f ( t ) = 0 quoad -aereas particulas ultra azi proinde quoad ejusmodi particulas F ' ( x-ct ) . Hinc sequitur souum adhuc ( 3. ) propagatum iri uniformiter velocitate се V 11 + 4 ). • De reflexa soni propagatione per aerem .
: 130. Cam in directa propagatione sonoras aer offen dit obicem aptum, reflectitur; hinc echo ( 115 ) progignitur; assertio sic probatur . Constat quod corpus in motu positum , si in obstacu lum incidit , quod elasticum sit , vel durum , et corpus ipsum ⋅ 289 v −∸−≖ B:] ')(æ-l-ct -l-F'(æ—ct), a:.— — ∙−∙−−−⋅⋅∶ ⋅↿ ∙

: - [f(ar-l—ct)

-
F'(x—-ct)] .
Functiones f
et P absque ulla didicu-l'tate determinantur:
sunt enim (1.&quot;). ⋅
⊞≀∝≻−−−↿≺⊄⊢⊦⋮⇁≀≺∝≻∙ ∙− cf.(a)-—:f(a1—F'(a) :
ideoque f(ao ⇌≖
aa)-zcnm) ∙ Ha): Karl-faa) ∙
Ultimae ac penultimae aequationis secunda membra eva-
nescunt statim ac « Et )a.. : erit itaque fur-H:):o
quoad aereas particulas ultra «.'
,proinde quoad eiusmodi
particulas ⋅-cs :: F' (a:—ct ).

Hinc sequitur sonum adbuc (3.&quot;) prcpagatum iri unifor-
miter velocitate
C::V-z-I—(i-I—A). ' De reflexa soni prcpagatione per aerem ∙⋅

130. Cum in directa propagatione sonoras aer oü'en-
dit obicem aptum, reflectitur: binc echo (115) progiguitur: assertio sic probatur.  

Constat quod corpus in motu positum, si in obstaculum incidit, quod elasticum sit, vel durum, et corpus ipsum impingens elasticilate gaudet, debet molus directionem mutare ac reflecti: ergo aer, elasticus cum sit, ubi in obstaculum offendit, quod vel elasticum sit, vel certe non molle, reflecti debet; undae videlicet aereae, quae ex sonoro corpore progignantur ac propagantur directe, debent obicem offendendo regredi, sonumque reflexum progignere. Exemplo circulorum in aqua ex injecto lapide excitatorum res oculis subjicitur: circuli enim isti ubi ad ripam appellunt, reflectuntur inde eo ordine, quo appulerunt . Aliter sic: ejusdem naturae est echo cum sono ipso directo; obtinet enim utrinque sonus eodem generatim tono, iisdemque affectionibus praeditus; ergo echo gigni debet eodem modo quo sonus directas: atqui hic per undas aereas successive a sonori corporis motu genitas procreatur; ergo per similes undas etc. Hinc in aperta planitie, ubi nullas est obex, sono directo minime Echo respondet. Cohaeret doctrina com Echo phoenomenis. Nam 

1° redit reflexa vox duplo temporis intervallo: ab experientia doctus sum, inquit Derhamus, Echo redire duplo intervallo, quo vox primaria ad objectum phonocanticum pertingebat; scilicet tempus requiritur ut ad obicem vox primaria deveniat, et rursum tantumdem temporis exigitur ut reflexa ab obice redeat ad loquentem. 

2º. Remissior plerumque est Echo quam vox directa audiri soleat; aliquando tamen intensius resonat Echo quam sonus directus audiatur. Ralio primi est: cum soni intensitas decrescat pro aucta distuntia a sonoro corpore, jam decrescit sonus ad obicem pergens; inde autem regrediens, et novas undas progignens, iterum decrescere
debet intensitas: ratio secundi, quia si obstaculum concavum
sit, plures colligere poterit radios phonicos , quos unitos simul in uno loco regerat. 

3º. Aliquando ( 115 ) seinel vox reflectitur, aliquando saepius: prima dicitur Echo monophona,
altera polyphona. Si enim obstaculum unicum sit , jam nonnisi
semel potest vocem remittere; contra saepius remittitur duplici ex causa. Prima est cum iu variis distantiis plura habentar
290
impingens elasticitate gaudet , debet motus directionem mu-
tare ac reflecti :ergo aer , elasticus cum sit ∙ ubi in ob-
staculum offendit
, quod vel elasticum sit, vel certe non
molle
,
reflecti debet; undae videlicet aereae, quae ex so-
noro corpore prOgignuntnr ac propagantur directe , debent
obicem oll'endendo regredi , sonumque reflexum progignere . 

3&quot;. Aliquando (1 15) semel vox refle-
ctitur,aliquando saepius: prima dicitur Echo monopbona,
altera polyphona. Si enim obstaculum unicum sit. iam nonnisi
semel potest vocem remittere; contra saepius remittitur dupli-
ci ex causa. Prima ut cum iu variis distantiis plura habentur291
obstacula: altera causa est, cum duo sunt obices e regione col
locati; vox enim ex uno reflexa in alterum incidit, atque ex
hoc iterum reflexa iucidit in primum, et ita porro. Apud
veteres memoratur Olympiae porticus ; quam eplaphonam
dicebant, quod septies eamdem vocem redderet , ut tradit
Plinius. Prope Mediolanum celebris est Echo in palatio
Simonetta, in quo ope duorum parietum parallelorum fra
gor minoris fistulae bellicae vicies, et aliquando tricies re
petitur teste Schoto. 

40. Echo saepius unam tantum syllabam, aliquando plures refert: echo monosyllaba prima, et polysyllaba altera dicitur: habentur loca, ex quibus integer versus hexameter repetitur. Ea nempe est obicis ( 115) distantia, ut sonus reflexus primarum syllabarum tunc demumad aures regrediendo perveniat quando vocis directae impressio jam desinit; ac tunc sonus primae syllabae, qui opportune regreditur jam expleto versu, poterit esse sepsibilis, itemque aliarum successive. 

5º. Echo redditur aliquando a silyis; imo etiam a campis sulco exasperatis et a planitie cespitibus ac virgultis inspersa reverberalur vox; reverberari autem a sulcis ac cespitibus animadvertit Kircherus, quia quando sulci eversi, ac virgulta praecisa fuerunt Echo nulla reddebatur: talis nempe esse potest irre
gularis partium reflectentium dispositio, ut etiamsi plures ra
dii phonici dispergantur, non pauci tamen in eumdem lo
cum collineent.

131. Reflexio soni fil ad angulos incidentiae et refle
xionis aequales : quod sic explicamus . Sit AB ( Fig. 60. ) fir
ma , planaque superficies ; KCK' recta perpendicularis su
perficiei AB ; K centrum sonorum , ex quo propagantur
sphaericae undae CDD', C'EE', etc... appellentes ad AB in C,
C'ete ... Fiet soni reflexio in C, C , ..; ethabitis C , C ... pro noris
secundariarum undarum centris , ipsae secundariae undae
remittentur cum eadem principalis undae velocitate. Pro
grediente unda principali ab CDD' usque ad BB ' , unda
manans ex C progredietur ab C usque ad Q ; repraesenta
291 obstacula: altera-causa est, cum duo sunt obices e regione col-
locati; vox enim ex uno reflexa in alterum incidit, atque ex
hoc iterum reflexa incidit in primum, et ita porro. Apud
veteres memoratur Olympiae porticus; quam eptaphonam
dicebant, quod septies eamdem vocem redderet, ut tradit
Plinius. PrOpe Mediolanum celebris est Echo in palatio
Simonetta, in quo ope duorum parietum parallelorum fra-
gor minoris fistulae bellicae vicies, et aliquando tricies re-
petitur teste Scboto. 40. Echo saepius unam tantum syl-
labam, aliquando plures refert: echo monosyllaba prima, et
polysyllaba altera dicitur: habentur loca, ex quibus integer
versus hexameter repetitur. Ea nempe est obicis (115) di-
stantia, nt sonus reflexus primarum syllabarum tunc demum
ad aures regredieodo perveniat quando vocis directae im-
pressio- iam desinit; ac tunc sonus primae syllabae, qui op-
portune regreditur iam expleto versu, poterit esse sensi-
bilis, itemque aliarum successive. 50. Echo redditur ali-
quando &amp;
silvis; iuno etiam a campis sulco exasperatis et a
planitie cespitibus ac virgultis inspersa reverberatur vox ;
reverberari autem a sulcis ac cespitibus animadvertit Kir-
cberus. quia quando sulci eversi, ac virgulta praecisa fue-
runt Echo nulla reddebatur: talis nempe esse potest irre-
gularis partium reflectendum dispositio, ut etiamsi plures ra-
dii phonici dispergentur, non pauci tamen in eumdem lo-
cum collineent.

131. Beflexio soni fit ad angulos incidentiae et refle-
xionis aequales: quod sic explicamus .Sit AB (Fig. 60.) Gr-
ma, plenaque superficies; KCK' recta perpendicularis su-
perficiei AB ; K centrum sonorum , ex quo propagantur
sphaericae undae CDD', C'EE', etc... appellentes ad AB in C,
B' etc... Fiet soni reflexio in C, C',..; et habitis C,C'... pro novis
secundariarum undarum centris, ipsae secundariae undae
remittentur cum eadem principalis undae velocitate. Pro-x grcdieute unda principali ab CDD' usque ad BB' , unda
manans ex C prOgredietur ab C usqæ ad Q; repraesenta-292
biturque hemisphaerio , cujas semidiameter CQ = D'B ' :
item progrediente unda principali ab C'EE usque ad BB” ,
unda manans ex C' progredietur ab C usque ad C &quot; : re
praesentabiturque hemisphaerio , cujus semidiameter CC&quot;
E'B' ; alque ita porro. Inferimus , si concipitur superficies
curva AQC &quot; B tangens omnia haec hemisphaeria in Q, C &quot; ....,
in ea fore puncta illa , quae a secundariis andis reflexis
attinguntur eodem instanti , quaeque tunc incipient concuti
quum principalis unda pervenerit ad BB' ; exhibebit nimi
rum AQC&quot;B superficiem undae reflexae; et quoniam productis
infra AB superficiebus BB' , Qa, C'a ' , ..., recta KA' exsistit per:
pendicularis ad primam et secundam, recta KH ad primam et
tertiam , etc ... ; ac proinde sphaerica superficies B'BA'A tan
git sphaericas superficies QaA' , C'a'H , . ; sequitur super
ficiem AQB undae reflexae fore sphaericam , ejusque cen
trum in K , et semidiametrum K'Q = KA' . Jamvero quem
admodum auris collocata v. gr. in C deprehendit sonum
directum venire juxta KC' perpendicularem undae incidenti ,
sic auris in C' deprehendet sonum reflexum venire juxta
K'C &quot; perpendicularem updae reflexae ; et cum , ob mutuum
sphaericarum superficierum AQB , C'a'H contactum , recta
K'C &quot; transeat per C' ; cumque , ob latus KC = K'C , et
latus CC commune , triangula rectangula KCC , KCC' dent
angulum KCC aequalem angulo K'C'C , erit angulus KCC
angulo C &quot; CB ; ideoque angulus incidentiae aequalis an
gulo reflexionis .
Sit nunc firma curvilineaque superficies AB ( Fig 61. ) ,
in quam incidant undae CE , HE&quot; , ... BB' propagatae ex
centro sonoro K ; si centris C , H , ... describuntur sphae
rae , quarum semidiametri ( KB-KC' ) , ( KB - KH ) , ... ,
aereae particulae sitae in superficie BD tangente sphaeras
istas incipient simul affici motu reflexo stalia ab adventu
undac ex K in B. Erit igitur BD superficies undae refle
xae : quam superficiem pon esse sphaericam nemo est qui
non videat. Fac ut puncta C , H sint inter se infinite vi
292
biturqne hemisphaerio , cuins semidiameter CQ ∶⋅−⋅ D'B' :
itcm progrediente unda principali ab C'EE' usque ad BB',
unda manans ex 0 progredietur ab C' usque ad C&quot;; re-
praesentabitnrque hemisphaerio
.cnius semidiameter C'C' ::
E'B' ; atque ita porro. Inferimus ,si concipitur superficies
curva AQC&quot;B tangens omnia haec hemisphaeria in Q, C&quot;,...,
in ea fore puncta illa , quae a secundariis undis reflexis
attinguntur eodem instanti , quaeque tunc incipient concuti -
quum principalis unda pervenerit ad BB' ;exhibebit nimi-
rum AQC&quot;B superficiem undae reflexae; et quoniam productis
infra AB superficiebus BB',Qa, C'a',..., recta KA' exsistit per-
pendicularis ad primam et secundam, recta KH ad primam et
tertiam
,
etc... ; ac proinde sphaerica superficies B'BA'A tan-
git sphaericas superficies QaA', C&quot;a'H .∙∙∙ ;sequitur super-
ficiem AQB undae reflexae fore sphaericam, eiusque cen-
trum in K', et semidiametrum K'Q ∶⋅−∙⋅ KA'. Iamvero qnem-
admodum auris collocata v. gr. in C' deprehendit sonum
directum venire iuxta KC' perpendicularem nudae incidenti ,
sic auris in C&quot; deprehendet sonum reflexum venire iuxta
K'C&quot; perpendicularem undae reflexae ; et cum , ob mutuum
sphaericarum superficierum AQB , C&quot;a'H contactum
,
recta
K'C&quot; transeat per C'; cumque , ob latus KC −∙−∸− K'C , et
latus CC' commune ,
triangula rectangula KCC', K'CC' dent
angulum KC'C aequalem angulO'K'C'C , erit angulus KC'C

: angulo C&quot;CB; ideoque angulus incidentiae aequalis angulo reflexionis . Sit nunc firma curvilineaqne superficies AB (Fig GI.), in quam incidant undae C'E' , HE&quot; ,... BB' propagatae ex centro sonoro K; si centris C' , H , ... describantur sphae- rae , quarum semidiametri ( KB—KC') , (KB—KH) , , aereae particulae sitae in superficie BD tangente sphaeras istas incipient simul affici motu reflexo statim ab adventu undae ex K in B. Erit igitur BD superficies undae refle- xae : quam superficiem non esse Sphaericam nemo est qui non videat. Fac ut puncta C' , H sint inter se infinite vi-293 cina , sintque C'C &quot; , HQ normales ad BD : ex H ductis per pendiculis Ha , Ha' in KC , C'C &quot; , erit Ca ' = CC &quot;—HQ = KB - KC ) - (KB - KH ) = KH - KC = Ca. Quoniam igitur triangula rectangula Cal , Ca'H habent latera aequalia C'a , C'a ', latusque C'H commuue , habebunt ae quales angulos ac'h , a'C'H : hinc sequitur , etsi unda re flexa non est sphaerica , adhuc tamen reflexionis angulum fore aequalem angulo incidentiae.

132. * Haec deducimus ex ( 129) in ordine ad aereum
fluidum concussum in K ( Fig. 60 ) , planoque fixo AB ter
minatum. 1 °* Sumpta x in KC normaliter ad AB, peribit
apud AB tota componens v' ; erit nempe ( 129. 10. )
dQ
dxdo O ( a )
quoad x = KC ( = h ). ProducaturKC donec KC = KC;
radius vector r' computetur ab K' ; et x ab eodem K' in
K'C ; explebitur (a) per
Q = --[Pr + c ) + F(ra) ] +
[fri + ce ) + F(x – ċe)]
( a ) ;
siquidem quoad puncta sita in AB
dQ dQ r=r' ,
dr x = h , it's - h ,
dr
dris
dx dx
Determinatis praeterea f et F ex ( i&quot; &quot; . 129. 10. ) , re
praesentabit ( a' ) initialem fluidi statum: quoniain igitur ( a' )
a— 293
cina , sintque C',C&quot; HQ normales ad BD: ex H ductis per-
pendiculis Ha ,
[in' in KC', 0C.&quot; , erit
∁≮≖⇌∁⋅∙∁ ∙∶−⇀−∐≺≀ (KB'—-KC';-(KB-—Kll)—-KH-—KC':C a.
Quoniam igitur triangula rectangulaC aH, C:: 'H habent latera
aequalia C'a
, C'a', latusque C'H commune , habebant ae-
quales angulos aC',H a'C' H hinc sequitur , etsi unda re-
flexa non est sphaerica , adhuc tamen reflexionis angulum
fore aequalem angulo incidentiae.
1324: Haec deducimus ex (129) in ordine ad aereum
fluidum concussam in K (Fig. 60), planoque fixo AB ter-
minutum. ↿∘∙ Sumpta æ in KC normaliter ad AB, peribit
apud AB tota componens v'; erit nempe (129. 10.)
19.
da: :0 (a)
quoad .c— KC (: It ). Producatur KC donec K' C:: KC:
radius vector r' computetur ab K'; et .r' ab eodem K' in
K'C; explebitur (a) per
≬⇌−⋮−∥↸≀∙⊣−∘≖⋮⋟⊣−⊏⋅⇁≺↗⋅−⋅−∘∩⊐−⊦
↿
−≀−∙−∙⋅−∣⋮⋀≀∙⋅−∣⋅−∘≀⊅⊹⊞↱⋅−−⊄⋮↕∙⋟⋅∙∣ (a');
siquidem quoad puncta sita in AB
∙∙ dQ dQ ∙∙∙
↙≀↾∙∙∙⊲ dr'
⋅⋅−—&quot;'-27—27 ***-&quot; ∙⋅↕−⇀−∣⋅∙⋅↴∙⋮⋮⊒−− 2;-
Determinatis praeterea f
et F ex (i'&quot;. 129. 10.), re-
praesentabit (a') initialem fluidi statum: quoniam igitur (a')291 !
1
1
1
1
1
et satisfacit conditioni ( a ) , et exprimit initialem fluidi statum,
poterunt per ( a' ) definiri, quae spectant ad motus propaga
tionem, attento obstaculo AB.
2º . * Punctum C &quot; , ad quod pertinent radii vecto
res r et r seu KC&quot; et K'C &quot; , perinde motum concipiet ac
si ( 129. 6. &quot; ) , sublato plano AB, fluidum eadem omnino
ratione concuteretur simul circa duo puncta K et K' . Per
tinget itaque ( 229. 4º. ) concussio ad C &quot;, primum in fine
temporis deinde in fine temporis : hinc bi
ni successive motus in C &quot; , alter directus, alter reflexus ; et
quia secunda concussio non pervenit ad C &quot; nisi quum tempus
sic invrevit, ut habeatur r = ct + a,, iccirco eadem velo
citate c regredietur motus, qua incedebat antequam in obi
cem impiogeret. Ad haec : cum anguli KC'C, K'C'C sint
aequales, rursus ( 131 ) patet sonum illisum obici AB re
gressurum efficiendo angulum reflexionis aequalem angulo
incidentiae.
C. с
De instrumentis pneumaticis.

133. In instrumentis pneumaticis soni genesis repe
tenda non est saltem praecipue ex oscillatione partium so
lidarum ipsius instrumenti. Etenim si in hisce instrumentis
dicatur soous creari eodem modo ac in instrumentis per
cussione resonantibus, jam sonus ipse connexionem haberet
maximam cum materia qua instrumentum compactum est ,
nec non cum ejusdem crassitie; quod tum ratione verissi
mum apparet , lum etiam constat ex vi paritatis. Ratione
quidem: nam in instrumentis ex diversa materia compactis,
quorum proinde particulae non aeque elasticae sunt, et ad mo.
cum oscillatorium non aeque aptae, non eodem modo tremu.
lus ille motus per insufflationem excitari debet ; quo vero
crassius instrumentum est, eo in plures continuas partes 0
294
et satisfacit conditioni (a), et exprimit initialem fluidi statum,
poterunt per (a') definiri, quae spectant ad motus prcpaga-
tionem, attento obstaculo AB.
20.a Punctum C&quot;, ad quod pertinent radii vecto-
res r et r' seu KC&quot; et K'C&quot;, perinde motum concipiet ac
si (129. 6.0), sublato plano AB, fluidum eadem omnino
ratione concuteretur simul circa duo puncta K et K'. Per-
tinget itaque (229. 40.) concussio ad C&quot;, primum in fine
tempons c
, deinde in fine temporis c : hinc bi-
ni successive motus in C&quot;, alter directus, alter reflexus; et
quia secunda concussio non pervenit ad C&quot; nisi quam tempus
sic iuvrevit, ut habeatur r': ct ⊣−∙ a. ,
iccirco eadem velo-
citate c regredietur motus, qua incedebat antequam in obi-
cem impingeret. Ad haec : cum anguli KC'C, K'C'C sint
aequales, rursus (131) patet sonum illisum obici AB re-
gressurum efficiendo angulum reflexionis aequalem angulo
incidentiae.
∙ r—al r'e—a
De instrumentis pneumatict's.

133. In instrumentis pneumaticis soni genesis repe-
tenda non est saltem praecipue ex oscillatione partium so-
lidaram ipsius instrumenti. Etenim si in hisce instrumentis
dicatur sonus creari eodem modo ac in instrumentis per-
cussione resonantibus, jam sonus ipse connexionem haberet
mammam cum materia qua instrumentum compactum est,
nec non cum eiusdem crassitie; quod tum ratione verissi-
mum apparet ,
tum etiam constat ex vi paritatis. Ratione
quidem: nam in instrumentis ex diversa materia compactis,
quorum proinde particulae non aeque elasticae sunt, et ad mo-
tum oscillatorium non aeque aptae, non eodem modo tremu-
lus ille motus per insufilationem excitari debet ; quo vero
crassius instrumentum est, eo in plures continuas partes o-295
scillatorius molus dispesci debet. Vi paritatis autem : nam
reipsa instrumenta, quae percussione sopant, pro materiae di
versitate etiam in pari crassitudine diversimode contremiscunt;
et si ejusdem sint materiae, pro diversa crassitie diversum
item sonum edunt. Ergo sonus in instrumentis pneumaticis
maximam connexionem etc. si in his soni genesis etc. Atqui
hoc est falsum : in tibiis enim cylindricis ejusdem longitu
dinis idem habetur sonus aut fere idem , nullo respectu
habito ad materiam aut crassitiem ipsius instrumenti , ut
constat experimentis; totumque artificium pro varietate to
norum pendet ex instrumenti variata longitudine: propte
rea etc.
Quonam igitur pacto in hujusmodi instrumentis erit
soni genesis explicanda ? Eo videlicet , quem indicavimus
(114.). In interna instrumenti capacitate aeris columna includitur, quae externi aeris pressione urgetur: dum igitur
per exiguum orificium fistulae alius aer insufflatione intro
mittitur, aer ille inclusus condensari debet , atque urgeri
contra aerem externum; externus autem vi suae pressionis
resistere; et quum aeris interni aucta pressio vincit, hic se
dilatando condensabit externum aerem , repelletque; externus
aer ita densatus ut ejus elasticitas praevaleat, se restituendo
rursum comprimet internum aerem : in columna videlicet illa
fiei compressio et restitutio, sicque in aeris particulis oscillato
rius motus excitabitur; qui motus communicabitur externo
aeri contiguo, et ad aures perveniet. Aer itaque secundum lon
gitudinem fistulae se habet instar chordae peragentis longita
dinales vibrationes: quo tibia et consequenter columna aerea
longior est, eo etiam longiores undae efformantur; longius
que erit tempus compressionis et restitutionis , ac proinde
Lonus gravior. Hinc in instrumentis, quae secundum longi
Ludinem sunt foraminibus instructa, modo hoc et modo il
lud foramen aperiendo, sublato digito, varii obtinentur to
ni; siquidem externum aerem sic admittendo , modo ma
jorem et modo minorem columnae aereae longitudinem ha
295
scillatdrius motus dispesci debet. Vi paritatis autem: nam
reipsa instrumenta, quae percussione sonant, pro materiae di-
versitate etiam in pari crassitudine diversimode contremiscunt;
et si ejusdem sint materiae, pro 'diversa crassitie diversum
item sonum edunt. Ergo sonus in iustrumentispneumaticis
maximam connexionem etc. si in his soni genesis etc. Atqui
'
hoc est falsum: in tibiis enim cylindricis ejusdem longitu-
dinis idem habetar sonus aut fere idem , nullo respectn
habito ad materiam aut crassitiem ipsius instrumenti ,
ut
constat experimentis; totumque artificium pro varietate to-
norum pendet ex instrumenti variata longitudine: propte-
rea etc.
Quonam igitur pacto in hujusmodi instrumentis erit
soni genesis explicanda ? Eo videlicet
, quem iudicavimus
(114.).ln interna instrumenti-capacitate aeris columna in-
cluditur, quae externi aeris pressione urgetur: dum igitur
per exiguum orificium fistulae alius aer iusufflatione intro-
mittitur, aer ille inclusus condensari debet, atque urgeri
contra aerem externum; externus autem vi suae pressionis
resistere; et quam aeris interni aucta pressio vincit, hic se
dilatando condensabit externum aerem, repelletque; externus
aer ita densatus ut ejus elasticitas praevaleat, se restituendo
rursum comprimet internum aerem: in columna videlicet illa
fiet compressio et restitutio, sicque in aeris particulis oscillato-
rius motus excitabitur; qui motus communicabitur externo
aeri contiguo, et ad aures perveniet. Aer itaque secundum lon-
gitudinem fistulae se habet instar chordae peragentia longitu-
dinales vibrationes: quo tibia et consequenter columna aerea
longior est, eo etiam longiores undae efi'ormantur; longius-
que erit tempus' compressionis et restitutionis , ac proinde
tonus gravior. Hinc in instrumentis, quae secundum lougi-
tudinem sunt foraminibus instructa, modo hoc et modo il-
lud foramen aperiendo, sublato digito, varii obtineatur to-
ni; siquidem externum aerem sic admittendo , modo ma-
iorem et modo minorem columnae aereae longitudinem ha-296
benius. Ita in chordis, pro majori chordae longitudine gra
vior est tonus, acutior pro minori; et digitis comprimendo
camdem chordam, ut evadat plus aut minus longa , varios
assequimur tonos .
Dixi soni genesim repetendam non esse saltem prae
cipue ex oscillatione solidarum partium etc. Etsi enim ex
materia instrumenti non habetur varietas quoad soni qua
litatem , aut valde notabilem intensitatem ; varietas tamen
habelur quoad meliorem aliquam resonantiam; idque ex eo
desumendum videtur quod aer inclusus pro diversitate cor
poris includentis melius aut minus bene oscillare potest ;
magis nimirum aut minus impeditus adhaesione ad ipsum
corpus et scabritie aliqua. Ad haec; si instrumentum pneu
maticum sit compactum ex materia non resistente, quale v.g.
esset instrumentum membranaceum , tunc per vibrationes
columnae aereae excitari poterit sensibilis motus oscillato
rius in partibus instrumenti; quae partes vicissim in aerem
vibrantem reagendo valebunt sonum ipsum modificari etiam
quoad tonum; et quidem plurimum si instrumentum sit val
de breve, quemadmodum expertus est D. Savarı; adeo ut
brevi tubo membranaceo obtineri possil magna varietas
lonorum , qui eo graviores erunt quo minus tenditur mem
brana.

134. Haec proponimus explicanda circa instrumenta
pneumatica. 

1º. Aperto aliquo foramine ex. gr. tertio, cae
lerisque clausis, ac deinde aperto alio puta quinto , variat
lonus: at si ' per aperitionem tertii inducitur communicatio
interni aeris cum externo, nonne ex dictis ( 133 ) audiri de
beret idem tonus sive apertum sive clausum sit quintam
foramen ? 

2º. Sola inflationis intensione mutantur toni , e
tiam servata eadem internae columnae longitudine 

3º. In
canna organi ejusdem diametri superius clausa, si subdupla
sit longitudo , idem redditur tonus qui obtinetur ex can
na superius aperta, et longitudinis duplae.
Ad 1. Cum varia in instrumento pneumatico fora
mina aperiuntur, variae interni aeris columnae communi
296
hemas. ita in chordis, pro maiori chordae longitudine g'ra-
vior est tonus, acutior pro minori; et digitis comprimendo
eamdem chordam, ut evadat plus aut minus longa , varios
assequimur tonos.
Dixi soni geneaim repetendam non esse saltem prae-
cipue ex oscillatione solidarum partium ctc. Etsi enim ex
materia instrumenti non habetur varietas quoad soni qua-
litatem , aut valde notabilem intensitatem ; varietas tamen
habetur quoad meliorem aliquam resonantiam; idque exeo
desumendum videtur quod aer inclusus pro diversitate cor-
poris incladentis melius aut minus bene oscillare potest;
magis nimirum aut minus impeditus adbaesione ad ipsum
corpus et scabritia aliqua. Ad haec; si instrumentum pneu-
maticum sit compactum ex materia non resistente, quale v.g.
esset instrumentum membranaceum ,
tunc per vibrationes
columnae aerea'e excitari poterit sensibilis motus oscillato-
rius in partibus instrumenti; quae partes vicissim in aerem
vibrantem reagendo valebunt sonum ipsum modificari etiam
quoad tonum; et quidem plurimum si instrumentum sit val-
de breve, quemadmodum expertus est D. Savart; adeo ut
brevi tubo membranacea obtineri possit magna varietas
tonorum, qui eo graviores erunt quo minus tenditur mem-
liraua.

134. Haec proponimus explicanda cirea instrumenta
pneumatica. 10. Aperto aliquo foramine ex. gr. tertio, cae-
terisque clausis, ac deinde aperto alio puta quinto. variat
tonus: at si' per aperitionem tertii inducitur communicatio
interni aeris cum externo, nonne ex dictis (133) audiri de-
beret idem tonus sive apertum sive clausum sit quintum
foramen? 20. Sola inflationis intensione, mutantur toni. e-
tiam servata eadem internae columnae longitudine 3&quot;. In
canna organi eiusdem diametri superius clausa, si subdupla
sit longitudo, idem redditur tonus qui obtinetur ex can-
na superius aperta, et longitudinis duplae.
Ad 1.&quot; Cum varia in instrumento pneumatico forf- mina aperiuntur, variae interni aeris columnae communi-297
cantes çum aere externo excitantur; non ita tamen commu
nicantes, ut simul non etiam inter se communicent; ergo
looi variare per plurium foraminum aperitionem debent ,
etsi exquisitam ejus rei rationem aegerrime quis reddere possit.
Ad 2.&quot; Ut per vehementiorem percussionem in chor
da instrumenti fidicularis contingit ut ea resonet ad oclavam,
ita in columna aerea per variam inflationis intensionem con
tingit ut tonus mutetur; et sicut certum est in chorda mu
sica quod ea tunc dividitur in duas partes separatim oscil
lantes, ita eadem asserenda est fieri divisio et oscillatio in
columna aerea sub tempore, quod sił proportionale tono
quem reddit. Hinc deducitur explicatio saltus ut ajunt tu
bae v. gr. ad octayam: cam paulo vehementius inspiralur tu
ba, cogitur aer ad celeriorem motum , quem tamen colu
mnae aereae jam vibrantes , utpote nimis longae, praesta
re non possunt. Dividitur igitur columna per medium ita ,
ut duo nova segmenta aeris aequalia suas vibrationes separatim
peragant. Ubi vero saltus non sit ad octavam , alia divi
sio fieri dicenda est .
Ad 3.&quot; Ia medio cannae duplae efformátur nodus ,
habetur aereum stratum quiescens , quemadmodum
habetur in orificio clauso cannae subduplae ; adeoque ea
dem undae aereae longitudo in utraque canna , idemque
proinde tonus .

135.* Sit tubus cylindricus determinatae longitudinis 1,
firmiter obseratus apud alterum orificium , aperius apnd al
terum : aequilibrium aereae columnae inclosae ita turbari
pono, ut qui aer orificio aperto ( ubi initium distantiae x
consliluo ) respondet, nullam densitatis variationem subeat,
et qui orificio clauso, nullatenus moveatur.Functiones ( 129.7 °) .
f, fx , ac proinde f , F ' tanquam datas assumo ab x = 0
ad x = l. E statu aeris apud extremitates tubi habemus
= o si x = 0, v = 0 si x = l; hinc
seu
fl + 1) + F'll — cl) = 0 ( 0 ) ,
20
297
can'tcs cum aere externo excitantur; non ita tamen cdmmu-
nicantes, ut simul non etiam inter se communicent; ergo
toni variare per plurium foraminum aperitionem debent
,
etsi exquisitam eius rei rationem aegerrime quis reddere possit.
Ad 2.&quot; Ut per vehementiorem percussionem in chor-
da instrumenti fidicularis cdntingit ut ea resonet ad octavam,
ita in columna aerea per variam inflationis intensionem cou-
ting'it nt tonus mutetur; et sicut certum est in chorda mu-
sica quod ea tunc dividitur in duas partes separatim oscil-
lantes, ita eadem assereuda est fieri divisio-et oscillatio in
columna aerea sub tempore, quod sit proportionale tono
quem reddit. Hinc deducitur explicatio saltus ut aiunt tu-
bae v. gr. ad octavam: cum paulo vehementius inspiratur tu-
ba, cOgitur aer ad celeriorem motum, quem tamen colu-
mnae aereae iam vibrantes, utpote nimis longae, praesta-
re non possunt. Dividitur igitur columna per medium ita,
ut duo nova segmenta aeris aequalia suas vibrationes separatim
peragant. Ubi vero saltus non sit ad octavam, alia divi-
sio fieri dicenda est, ⋅ '
Ad 3.&quot; In medio cannae duplae eEorm'atur nodus ,
seu habetur aereum stratum quiescens, quemadmodum
habetur in orificio clauso cannae subduplae: adeoque ea-
dem uudae aereae longitudo in utraque canna
,
idemque
proinde tonus.

135 Sit tubus cylindricns determinatae longitudinis !,
firmiter obseratus apud alterum orificium, apertas apud al-
tequm : aequilibrium aereae columnae inclusae ita turbari
pono, ut qui aer orificio aperto ( ubi initium distantiae .a-
constituo ) respondet, nullam densitatis variationem subeat,
et quiorificio clauso, nullatenus moveatur.F unctiones (129.7&quot;).
f, f. , ac proinde f, F' tanquam datas assume ab a: 30
ad .r.-zl. E statu aeris apud extremitates tubi habemus

:: osi æzo,v:——osiæ:-:l; hinc

≀≖∣⋅∶≀−∙⊢≀∶∠⊢⊢ F'(l—c1):o ( o )
.
20298
Fll — ct) - f ( c ) = 0 ( o' ) .
In (0 ) substituatur ct +1- x in locum ct : prodibit
f (21 + ci - x) = - F '(x - 1) ( 0&quot; ) ;
unde
= f'( x + cl) – f'( 21+ ct - x ) ,
c = -f(x + ct) -f(21 +ct - x) :
( o ' ')
in ( o &quot; ) fiat x = 0 ; erit ob ( o')
f (c + 2) = -F ( - ct) = -f (c ) (0&quot; &quot; );
subrogato ct +21 in locum ct, habebitur
f '( c +4 ) = -f( ct + 2 ) = f(t ) (o ');
denique si in ( 0 &quot;) ponitur ci=0, emerget
f( 21 — x ) = - F ( x ) ( 0 &quot; ) .
.
Aequationes ( o' : 0' ! ) satis sunt, ut functionem f con
siderare possimus veluti datam quoad omnes positivos ya.
lores quantitatis variabilis , ad quam respicit ipsa functio.
Etenim e statu initiali atque arbitrario aereae columnae ad
motum incitatae , data est F' ab x = o ad x = l ; ergo
ob (o &quot; ) data erit fab x = l ad x = 21 : ex eodem sta
tu jam dala erat fab x = o ad x = l ; ergo dabiturf
ab x = o ad x = 21. Aequatio autem ( o&quot;) rem evidentissime
absolvit quoad caeteros quantitatis variabilis positivos va
lores. Ergo etc.
298
F'(—ct)——f(ct):-to (c').
In (o) substituetur et −⋅⊢≀ —- a: in locum et: prodibit
f(2l −∣⋅− ct — a:) ∶−∙−− F'(x—ct) (o&quot; );
uude
v:f(æ-i-ct)—f(2l—l—ct—æ).
etc:—f(x-i-cz)—f(2l -i-ct—æ):
111 (o&quot;) fiat m::- o; erit Ob (0')
(o&quot;')
f(cs −↿− 21) ∶−∙− −F'( —ct) −−∶ —f(ct) (o&quot;):
subrogato et —-[-21 in locum et, habebitur
f(cz ↽⊢ 4!) ∙−−− —-f(ct −⊦ 21) ↽↼−−⋅≖ f(ct) (a';
denique si in (a'—') ponitur ctzo, emerget
f(ZI—æ) z—F'Lr) (o&quot;).
Aequationes (a': a'!) satis sunt, ut functionem f coa-
siderare possimus veluti datam quoad omnes positivos ,va-
lores quantitatis variabilis, ad quamrespicit ipsa functio.
Etenim e statu initiali atque arbitrario aereae columnae ad
motum incitatae
, data est F' ab a: a ad w:! : ergo
ob (o&quot;) data erit f
ab æ-—-:l ad se:21 :ex eodem sta-
tu iam data erat f' ab a: :: o ada::1; ergo dabitur ['
ab a: :: 0 ad se:21. Aequatio autem (o&quot;) rem evidentissime
absolvit quoad caeteros quantitatis variabilis positivos va-
lores. Ergo etc.299
Quoniam ab x = o ad x = 21 dependet f' ab ini
tiali atque arbitrario statu aereae columnae ; poterit igitur
sic assumi inter illos limites , ut facto i = 1,3,5,7 ,..., sit
21
f'(c + % -f(cc + = p&quot;(ce) ( 0 &quot; &quot; );
numeri pares = 2, 4, 6 ... debent excludi ob aequatio
dem (o &quot; ). Instaurantur ergo iidem functionis f valores
42 quotiescumque tempus t evadit it ; sed ( o &quot; ) a functio ic
ne f unice dependent v, E. Columna igitur aerea in eum
dem restituitur statum per aequalia intervalla, suasque com
41
plet oscillationes intra tempus ; quarum propterea nu
merus intra q ' erit
ic
ic
41
136.. Evanescet (135.0 '&quot;') velocitas v ubi fuerit f ( x + cos
= f (21 + ct - x ) ; evanescet e si f'( x + ct) - f (21+
ci - x ). Primum contingit ( 135 : 0 &quot; ) quando (22 +ic - x )
41'2 - ( x + cl) seu 21 2x ; secundum quando 21—
21&quot; i 1
2x= • Hinc 1º. facto i = 0 , 1 , 2 , 3 .. i
scet aer in distantiis
2 , quie
lli - 21 )
X
2º . Facto i&quot; =1 , 3 , 5 .... 1 ; movebitur aer in
locis
299
Quoniam ab a: a ad a: 2! dependet ;&quot; ab ini-
tiali atque arbitrario statu aereae columnae ; poterit igitur
sic assumi inter'illos limites, ut facto i:1,3,5,7,...., sit
41 21 ∣
'&quot;
∣⇃≺∁⊢⊢ ∙
−−⋮∙⊣∶∶≕−∣↙≼⋄⊢⊢ 7): f
(ct) (0 ):
numeri pares r':2, 4, 6 ...
debent excludi ob aequatio-
nem (o&quot;). Instaurantur ergo iidem functionis f' valores ,
quotiescumque tempus :
evadit t −∣∙⋅ &amp; sed (o&quot;')a functio-ne )&quot; unice dependent v, :. Columne igitur aerea in eum--
dem restituitur statum per aequalia intervalla, suasque com-
. . . plet 4! osc1llat1ones intra tempus ∙≀⋅−≔∙ ; quarum propterea nu-
merus intra 1&quot; erit
t'

136. a Evanescet (135.o&quot;')velocitas :: ubi fuerit f' (æ—l—ct

f(ZH— ct —- æ ); evanescet :si f(x −∣⋅− et):
— f(ZI-i-
ct ∙−− a: ). Primum contingit (135: a'&quot;) quando (21—1—4 cs --' a:)
41&quot;! −∙− ( æ ⋅−⊢ ct) seu 2! -- 21: −∙−−−−− T;secundum quando 21—
2
∙∣∣ ∙−↿ .
2x: −⋮∙−↨ ∙Hinc ↿∘∙ facto 1&quot;: a, 1, 2, 3 ....
'T, qu1e-
scet. aer in distantiis' -
∙∙∙ [( t' — 21&quot;)
æ
!
20. Facto 1'&quot; :1, 3, 5 ....
i; movebitur aer in
locis300
llimi)
quin tamen ullam patiatur densitatis variationem, Aper
tis itaque foraminibus in hisce postremis locis , nullo pa
cto sonus mutari debet ; quod experientiae consonum re
peritur: imo non mutabitur sonus, licet lubo abscindatur
pars 1- x , quae ultra locum x ad fundum usque pro
tenditur. Atqui pars reliqua nihil aliud est nisi tubus
in utraque patens extremitate: ergo si de hujusnuodi cubis
sermo sit, posita e = o apud unum orificium erit quo
que apnd alterum { =0.
137. # In tubis itaque cylindricis, quorum ambo ori
ficia libera omnino sunt, habetur ( 129.7 .)

#

Fl—ct) — 9 (2+ cl ) = 0, F1 — ct) - f'(C ) = 0.
Hinc facile deducuntur ( 135 ) sequentes aequationes
f (21 + cix) = F ' x - ct), v = P ( x + ct) + P (21 + c1 - ),
c : = f ( 21 + (1 - x ) — f (x + c ! ), f (ce + 21 = F (-1)= f (c ),
f'(21 — * ) = F ( x ).
Quia vero ab x = o ad x = 21 rursus dependet p
ab initiali atque arbitrario statu 'aereae columnae , ic
circo poterit etiam asseri sequens aequatio.
f (ce + *-) = f ( c )
in praesenti est i = 1. 2, 3, 4 ......
æs. lu.—'r')
1
,
quia tamen ullam patiatur densitatis variationem. Aper-
tis itaque foraminibus in hisce postremis locis, nullo pa-
cto sonus mutari debet; quod experientiae consonumre-
peritur: imo non mutabitur sonus, licet tubo abscindatur
pars l—æ, quae ultra ↙
locum a: ad fundum usque pro-
tenditur. Atqui pars reliqua nihil aliud est nisi tubas
in utraque patens extremitate: ergo si de huiusmodi tubis
sermo sit, posita :

: o apud/uuum oriücium erit quo-

que apud alterum::o.

137.a In tubis itaque cylindricis, quorum ambo ori-
licia libera omnino sunt, habetur (129. 70.)
F'U—ct) -
f(l −↿− ct) ∙−−−−∙∙ o,F'( -
ct) --f(ct): 0.
Hinc facile deducuntur (135) sequentes aequationes
f(21 -l-ct -æ):F'(æ—-ct), v ::f'(æ ⊣− et)-t— f(ZI-i-ct—x).
es:/(21 -t-ct —x)-— f(æ-t—ct), f(ct-t-Zl):F'(—cc):f(c1),
f(2l — a: ): F' (æ)-
Quia vero ab a: 0 ad ..r:21 rursus dependet f
ab initiali atque arbitraria statu 'aereae columnae
,
ic-
circo poterit etiam asseri sequens aequatio.
f(ct—i- -—2'-£-) :f'(ct )

:

in praesenti est i: 1. 2, 3, 4 ......
≡⊲∙⋅⇀≣∎ lJ-r

: 301

22
Iterat ergo aerea columna per aequalia intervalla ic
oscillationes suas , quarum proinde numerus intra 1 &quot; erit
ic n =
21
Haud immoror inquisitioni distantiarum , ubi a
er vel quiescit, vel nativam retinet densitatem : hujusmo.
di namque investigatio similiter perficitur ac in Lubis,
quorum unum orificium apertum est . Satius forsan e
rit adnotare quod, facto i = 1 , exhibet ( 137 ) aequatio n '
relationem inter principalem tonum n ', redditum ab
elastico fluido intra tubum oscillante, et velocitatem c qua
sonus incederet si per ipsum fluidum propagaretur. Hinc
patet quomodo experimentis indagari possit velocitas c in
aliis elasticis flaidis ab aere atmosphaerico diversis : ex
tentaminibus Van - Rees, Frammeyer, et Moll prodiit so
ni velocitas sub temperie 10.° C
21
io gas oxigenio 3,7m, 9 :
bydrogenio 1233 , 3 ,
nitrogenio .
.
339 .
oxido nitrico 317 , 4 ,
acido salphuroso 229 , 2 ,
acido carbonico 370 , 7 ,
. .
suboxido carbonico . . 341,1
etc. etc.
301
. . 2! [terat ergo aerea columna per sequsl1a1ntervalla ∙∙∙∙⋮∙⋅−
oscillationes suas ∙ quarum ⋅proinde numerus intra 1&quot; crit
'
IC
Haud immoror inquisitioni distantiarum , ubi a-
er vel quiescit, vel nativam retinet densitatem: huiusmo-
di namque investigatio similiter perficitur ac in tubis,
quorum unum orificium apertum est. Satius forsan e-
rit adnotare quod, facto 1':1,exhibet (157) aequatio n'
0
∙−−∶ -2-l- relationem inter principalem tonum n', redditum ab
clastico flaido intra tubam oscillante, et velocitatem e qua
sonus incederet si per ipsum fluidum prcpagaretar. Hinc
patet quomodo experimentis indagari possit velocitasc in
aliis elasticis fluidis ab aere atmosphaerico diversis: ex
tentaminibas Van— Bees, Frammeyer, et Moll prodiit so-
ni velocitas sub temperie 10.0C
in gas oxigenio . . . . . . 317',g,
hydrogenio . . . . . 1233,3,
nitrogenio. . . . . . 339 . ∙
oxido uitrico . . . . 317 ,4
acido sulphuroso . . -
229 , 2
,
acido carbonico . . . . 370 , 7 ,
suboxido carbonico . . . 341 ,
1
,
etc. etc.302

138. Si tubus proponilur utrinque obseratus , quis
que videt fore v = o apud ambas extremitates; unde (129.7°)
f (c ) + F ( -ct)= 0,flfct) + F (l ct) = 0,
quarum ope determinatur motus inclusi aeris,
De propagatione soni per liquida , et per solida corpora.
139.* Quod spectat ad liquida corpora , in comperlo
est aquam v. g. contrahi perparum posse atque restitui
in suis partibus : itaque qua ratione turbatum posuimus
( 129..1 . ° ) aequilibrium , eadem in praesenti imaginemur
turbari . Propagato motu , densitas ré aquae libratae ver
tetur in je = pili + :) apud (2. , y , z ) ; et pressio o'
in w= '+Ae ; exprimit A numerum experimentis deter
minandum. Sumptis hic quoque X=0, Y=0, Z=o, et ra
tiocinando ut in citato n. ° assequemur
d
dQ
1 do dt (
dQ
dt
dr
A dL {1+ :)
dr seu
р. dr pi
A
tum facto c ” , perveniemus ad formulas (i' '.;&quot; . .
ji
į &quot; : 129, 1.0 ) . Non pluribus opus est ut intelligamus
( 129. 2.° 3,0 ) sonum per aquam diffundi aequabiliter ve
locitate.
VA
Numerus A potest determinari ex parvula contractione ,
quam juxta longitudinem à ( haud variata diametro ) pa
302
1381: Si tnbus praponitnr utrinqne obseratus , quis-
que videt fore v:o apud ambas extremitates; unde (129.?)
f(ctH-FX --ct):o,f7(l—i-ct)-i-F'(l—ct):a,
quarum Ope determinatur motus inclusi aeris.
De propagatione soni per liquida, et per solida corpora.
139:- Quod spectat ad liquida corpora, in comperto
est aquam v. g. contrahi perparum posse atque restitui
in suis partibus :itaqne qua ratione turbatum posuimus
( 129. ⋅↿∙∘ ) aequilibrium, eadem in praesenti imaginemur
turbari. Pr0pagato motn, densitas pf aquae libratae ver-
tetur in þ.:yJU—I—s) apud (.x.-,,] ,
z) :
et pressio a'
in ≔≖⇌≖⋝∣↰∟⋀⋮⋅⋮ eXprimit A numerum experimentis deter-
minandum. Sumptis hic quoque X:0, ↧↗−−−−⋅∘∙ Z:o, et ru-
tiocinando ut in citato 11.0 assequemur
d
dQ) ↼ 1 de ∙− dQ) (Et? A; JLu-Jr-s) ∙− &quot;( dt ∙
a d?
.
dr ' se.. a' d.- 4.-
A -
∙∣∣ -1 tum facto ?

: cz, perveniamus ad formulas (:

'.
t
'.
i'
i&quot;: 129, ↿∙∘ )
. Non pluribus opus est ut intelligamus
( 129. 2.0 3.&quot;) sonum per aquam diffundi aequabiliter ve-
locitate. ⋅
.: ⇂∕⋅−⋮∶⇡∣−⋅
Numerus A potest determinari'ex parvula contractione f:,
quam iuxta longitudinem l (haud variata diametro) pa-303
tilur columna aquea ob incrementum 5. superadditum
pressioni o '. Nam 1 : 1 - B = + ): , ideoque &lt; =
B \beta

: ' sed o=u'two=a' +As, igitur スー B 2

6.
A :
σολ
E \beta
In hypothesi pressionis . = 0 &quot; , 76) g, ac temperiei n=
10.• C, experimenta Dni Canton suppeditant B = 0,000046 ),
inter quem valorem et quos invenerunt DD . Parkins et
Oersted , nimirum
B=0,0000452 , B=0,0000482 ,
parvula est differentia. Ponatur hydrargyri densitas 1 ;
erit proxime u'= . : assumpta igitur g=9m, 8088,
13,5819
1
emerget
c=1483&quot; , 59.
Sonus videlicet propagatur per aquam plus quadruplo ce
lerius quam per aerem. D. Beudant dicit in hac se fuis
se sententia , ut e suis experimentis in mari institutis ta
lem deduceret soni velocitatem , quae 1500m saltem aequaret.
140.* Quisque videt soni velocitatem eadem ratione
posse in caeteris corporibus , sive liquidis , sive solidis ,
determinari , modo eorum partes contrahi perparum queant
atque restitui . Sic , manente 5 . = (0,76 ) 8 , obtinuit idem
ipse Canton hydrargyri contractionem B = 0,0000032 : as
sumpla igitur u = 1 , erit
c = 1576m , 35
1
303
titur columna aquea' ob incrementum m superadditum
pressioni w'- Nam 71: l—þ:p.'(1-l-s): p!
,
ideoque ::
P −⇀ 13 ⇤
⋅
m−⊸T;'sed ∏∙−−∶∏∎∙⊦∏∘∶−−∸⋅∄≖⋅−⊦∆⋮∙ igitur
'
A ∙− a'., ∙∙∙ wo).
8
5
In hypothesi pressionis uro :( o'&quot;, 76) g, ac temperiei :::,
10.(, C. experimenta Dni Cauton suppeditant þ:0,0000461,
inter quem valorem et quos invenerunt DD. Parltins et
Oersted , nimirum
,ezo,oooo45) ∙ þ:0,000048).
.
parvula est differentia. Ponatur hydrargyri densitas :1;
↿
15.5819
erit proxime pf: : assumpta igitur g:9&quot;?,,8088,
emerget ⋅
c:1483&quot;'
. 59.
Sonus videlicet prcpagatur per aquam plus quadruplo cc-
lerius quam per aerem. D. Bendant dicit in hac se fuis-
se seateutia , ut e suis exPerimentis in mari institutis ta-
lem deduceret soni velocitatem, quae 1500&quot; saltem aequaret.
1403 Quisque videt soni velocitatem eadem ratione
posse in caeteris corporibus
.
sive liquidis , sive soli-dis ,
determinari , modo eorum partes contrahi perparum queant-
atque restitui. Sic, manente wo:(0,76) g , obtinuit idem
ipse Canton hydrargyri contractionem [5:0,0000037t : as-
sumpta igitur pf:1 , erit
0:1576,&quot; ∙ 35304
velocitas , qua per hydrargyrom diffunditur sonus. Ante
quam usum contractionis \beta animadverteret Laplace ad de finiendam soni velocitatem per liquida et solida corpora ,
exhibuerat Chladni in sua Acustica aliam methodum
sane ingeniosam , ejusdem velocitatis investigandae in cor
poribus solidis.
141. # Innititur ista methodus analogiae, quam norunt
Physici inter oscillationes aeris in tubo cylindrico apud
ambas extremitales aperto et longitudinales oscillationes
virgae rigidae , cujus ambo extrema omnino libera sint.
Exprimat enimvero l oscillantis virgae longitudinem ; n'
principalem tonum , quem edit resonans virga ; c' quae
sitam velocitatem . Erit ( 137 )
n &quot; ; unde n ' : n &quot; = 0 : c' , ' =
21
Iam si velocitas soni per aerem repraesenterar per &quot; , ex
perimenta D.ni Chladoi praebent soni velocitatem c
per stannum . 717 를
per argentum
per cuprum .
12
per ferrum et vitrum ... 17
per varia lignorum genera 11 ad 17 , . .
Ad explorandam soni velocitatem per ferruin fusionis , in
promptu habebat D. Biot 376 tubos ex hoc metallo com .
pactos ; quibus singulis mediocris erat longitudo duorum
304
velocitas , qua per hydrargyrnm diffunditur sonus. Ante-
quam usum contractionis þ animadverteret Laplace ad de-
finiendum soni velocitatem per liquida et solida corpora ,
cxbibuerat Chladni in sua Acustica aliam methodum,
sane ingeniosam .
ejusdem velocitatis investigandae in cor-
poribus solidis.
1414: lnnititur ista methodus analogiae, quam norunt
Physici inter oscillationes aeris in tabo cylindrico apud
ambas extremitates aperto et lougitudiuales' oscillationes
virgae rigidae, cuius ambo extrema omnino libera sint.
Exprimat enimvero! oscillantis virgae longitudinem ; n&quot;
principalem tonum
, quem edit resonans virga ; c' quae-
sitam velocitatem. Erit (137) '
c, '
nn 11&quot;:-2-i-;nuden:n&quot;:c:c', c':c-—J.
»
Iam si velocitas soni per aerem repraesentetur per 1, ex-
perimenta D.!d Chladni praebent soni velocitatem c'
per stannum . ∙ ∙ ∙ ∙
, 7
vet--
per argentnm . . . . . . 9 ,
per cuprum . . . . .
.. . 12 ,
per ferrum et vitrum . . .
17
.
per varia lignorum genera . .
11 ad 17,
Ad explorandam soni velocitatem per ferrum' fusionis ,
in
promptu habebat. D. Biot 376 tubos ex hoc metallo com'
pactos ; quibus singulis mediocris erat longitudo duorum

* a305

metr. cum partibus millesimis 515. Sumptis experimentis,
prodiit soni velocitas 104 ; nisi quod jungebantur ii tu
bi ope plumbi, quod aliquanto sonum retardare videtur.

=== De vocis humanae origine. ===
142. Vocis humanae organum etsi considerari maxi
me debet tamquam instrumentum pneumaticum ftexili
et elastica materia ex parte compactum , non tamen ita
est ut cum instrumentis etiam fidicularibus aliquam non
babeat analogiam. Quod ut melius intelligatur , nonnulla
ex anatomicis sunt hic afferenda.
Palmo est viscus respirationi inserviens: in duas par
tes distinguitur , dexteram et sinistram , et duo magni lo
bi dicuntur , etsi quivis ex his duobus dividitur mino
ribus aliis. Substantia constat molli , spongiosa , rara et
vessiculosa ita ut ad aerem excipiendum aptissimus sit :
motu ergo dilatationis aere impletur , et constrictionis
motu eundem expellit ; atque aer ita expulsus primo per
multiplices canaliculos lobis interspereos , qui bronchia
dicuntur ; tum per duos ex utroque lobo emergentes ; de.
mum per ampliorem canalem emergit , qui ex praefa
tis duobus in unum conjunctis coalescit. Hic canalis seu tubus
ad oris usque radices ascendens trachea seu aspera arteria
nuncupatur ; in summitate asperae arteriae brevis canaliculus
habetur , qui larynx dicitur , cujus summitatem facit rima
ovalis a duabus membranis horizontaliter jacentibus relicta ;
quae rima glottis dicitur : atque huic superposita est epi
glottis ; tenuis scilicet et mobilis cartilago glottidem te
gens , quae ad hoc praecipue statuta esse videtur, ut dum
aliquid deglutimus , quidquam cibi aut potus in asperam
arteriam minime irruat, sed per contiguum canalem, qui
exophagus dicitur, et cujus orificium pharynx vocatur , de
more in stomachum demittatur. Itaque lobi pulmonis instar
}
305
metr. cum partibus millesimis 515. Sumptis experimentis.
prodiit soni velocitas 10;- ; nisi quod iungebantur ii ftu-
bi »ope plumbi, quod aliquanto sonum retardare videtur.
De vocis humanae origine.


142.1Vocis humanae organum etsi considerari maxi-
me debet tamquam instrumentum pneumaticum fiexili
et elastica materia ex parte compactam, non tamen ita
est ut cum instrumentis etiam fidicularibus aliquam non
habeat analogiam. Quod ut melius intelligatur, nonnulla
ex anatomicis sunt hic aderenda.
⋅ Palmo est viscus respirationi inserviens: in duas par-
tcc distinguitur , dexteram et sinistram ,
et duo magni lo-
bi dicuntur , etsi quivis ex his duobus dividitur mino-
ribus aliis. Substantia constat molli , spongiosa , rara et
vesaiculosa ita ut ad aerem excipiendum aptissimus sit:
motu ergo dilatationis aere impletur , et constrictioais
motu eumdem expellit; atque aer ita expulsus primo per
multiplices canaliculos' lobis interspereos , qui bronchia
dicuntur; tum per duos ex utroque lobo emergentes :dc-
mum per ampliorem canalem emergit , qui ex praefa-
tis duobus in unum coniunctis coalescit. Hic canalis seu tubas
ad oris usqne radices ascendens tracbea seu aspera arteria
nuncupatur; in summitate asperae arteriae brevis canaliculus
habetur , qui laryux dicitur, cuius summitatem facit rima
ovalis a duabus membranis horizontaliter jacentibus relicta;
quae rima glottis dicitur : atque huic superposita est epi-
glottis; tenuis scilicet et mobilis cartilago glottidem te-
gens, quae ad hoc praecipue statuta esse videtur, ut dum
aliquid deglutimus, quidquam cibi aut potus in asperam
arteriam minime irruat, sed per contiguum canalem, qui
cxophagus dicitur, et cuius orificium pharynx vocatur
, de
more in stomachum demittatur. Itaque lobi pulmonis instar306
re cur com
follium aerem excipiunt, cum compressi illum emittunt per
asperam arteriam : aer ita expulsus per asperam arteriam
in arctiorem laryngis canalem irroit , atque ita ex am
pliori in angustius spatium redactus compressionem pati
debet , oscillatoriumque motum concipere. Sed quia la
rynx flexili et elastica materia compingitur, iccirco ( 133)
ad motum tremulum ab aere vibrante excitabitur. Deinde
vero in eumdem aerem diversimode reagendo, prout magis
vel minus erit tensa , ejus Oscillationes diversimode quoque
modificabitur.
Obiter notamus antiquos et cum iis Galenum male
organum vocis humanae in trachea constituisse ; quam
arbitrabantur vices gerere tubi, per quem aer ad sonum
jam excitatus excurrit. Refelles hanc opinionem consi
derans aerem qui tracheam ascendit , libere ascende
et liberius habere spatium ; unde non est
primi debeat et oscillatorium motum habere : cum autem
glottis sit multo arctior quam trachea , per glottidem
transiens habet quidem unde comprimi possit. Haec de
voce humana : ad vocem enim quod spectat quorumdam
animalium , uti sunt multae aves ; hae cum etiam exse
cto collo , sola ventris compressione sonum edant , in his
utique trachea concurrit ad sonum ipsum modificandum .
Sed nil hinc eruitur contra jam dicta: in istis namque avi
bus trachea habetur supra glottidem , seu gloutis esse obser
vatur non ad summitatem , sed infra tracheam ; contra
ac est in homine , et plerisque aliis animalibus.
In monumentis Academiae Parisiensis ad an. 1741
observat Ferreinius intra laryngem duas haberi fibras ad
labrum glottidis ; quae fibrae ex impetu aeris per angu
stiorem laryngis canaliculum irrumpentis ad tremitum con
citantur , atque hoc tremitu resonant , quemadmodum in
fidibus contingit ; unde dictum est vocis humanae organum
analogiam habere aliquam ad instrumenta fidicularia . Sum
psit ille plures laryages cum sua glottide ; dunque insuf
306
folliam aerem excipiunt. tam compressi illam emittunt per
asperam arteriam: aer ita expulsus per asperam arteriam
in arctiorem laryngis canalem irruit ,. atque ita 'ex am-
pliori in angustias spatium redactus compressionem pati
debet, oscillatoriamque motum concipere. Sed quia la-
rynx flexili et elastica materia compingitur. iccirco (133)
ad motum tremulum ab aere vibrante excitabitur. Deinde
vero in eamdem aerem diversimode reagendo, prout magis
vel minus erit tensa, eius oscillationes diversimode quoque
modificabitur. ∙
Obiter notamus antiquos ,et cum iis Galenum male
organum vocis humanae in trachea constituisse; quam
arbitrabantur vices gerere tubi, per quem aer ad sonum
iam excitatus excurrit. Refelles hanc opinionem consi-
derans aerem. qui tracheam ascendit
,
libere ascende-
re,'et liberius habere spatium ; unde non est cur ,com-
primi debeat et oscillatorium motum habere: cum autem
glottis sit multo arctior quam trachea , per glottidem
transiens habet quidem unde comprimi possit. Haec de
voce humana : ad vocem enim quod spectat quorumdam
animalium , uti, sunt multae aves; hac cum etiam exse-
cto collo, sola ventris compressione sonum edant, in his
utique-trachea concurrit ad sonum ipsum modificandam.
Sed nil hinc eruitur contra iam dicta: in istis namque avi-
bus tracbea habetur supra glottidem , sen glottis esse obser-
vatur non ad summitatem, (sed infra tracheam ; contra
ac est in homine , et plerisque aliis animalibus. .
In. monumentis Academiae Parisiensis ad an. 1741
observat .Ferreinius intra laryngem duas haberi, fibras ad
labrum glottidis ; quae fibrae ex impetu aeris per angu-
stiorem laryugis canaliculata irrumpentis ad tremitum.con-
citantur, atque hoc tremitu resonant
, quemadmodum in
fidibus contingit; unde dictum est .vocis humanae organum
analogiam habere aliquam ad instrumenta fidicularia. Sum-
psit illa plures' larynges cum sua glottidc; dumque insuf-307
Aando sonus vocis animalis excitabatur, microscopio Gibras
praedictas inspiciendo tremor et vibratio in iisdem cerne
batur , prout in chordis musicis habetur dum resonant.
Atqui eo ipso ex earum tremitu ab irruente aere tamquam
a vi percutiente excitato sonus gigni debet, vel jam geni
tus modificationem quamdam recipere.
Rursus , sicut in chordis musicis contingit ' ut chorda
brevior det sonum acutiorem , graviorem longior : ita ani
madvertendum hic fuit an fibrarum illarum major minor
ve longitudo toni mutationem induceret. Compertum au
tem est quod , impedita illarum fibrarum parte ne tre
meret , tonus prodibat acutior.
Sumpsit etiam larynges bovis , canis, aliorumque ani
malium , deinde insufflando excitabatur mugitus bovis , et
conformis aliis animalibus sonus. Movendo autem musculos
ita , ut traherentur et distenderentur fibrae, excitabantur
mutationes soni , quae haberi solent in varia horum ani
malium voce.
Notetur illud : cum tensio vel remissio fibrarum glot
tidis et cartilagineae substantiue , qua larynx constat , ab
eodem musculo dependeat , ut notat Savart , consequitur
una cum fibris illis etiam laryngem tendi vel remitti.
Laxatis fibris, orificium glottidis ampliatur , et sonus pro
dit gravior ; tensis vero , orificium restringitur, et sonus
evadit acutior , ut in canibus observavit D. Magendie.

143. Si vox in larynge et glottide tanquam in proprio
organo fit ; quid ergo, inquies, os atque ejus partes con
ferunt ad formationem vocis ?
Respondeo oris cavitatem , linguam, dentes, labia con
currere ad modificationem perfectionemque vocis ; quae
larynge et glottide incipit quidem , sed non omnimode
ibi perficitur : nam quod in illis partibus sufficiens habea.
tur organum quin prorsus necessaria sint oris et linguae or
gana ad exhibendum aliquo modo sonum animalis pro
prium , apparet ex eo quod grues abscisso
in
et anseres 1
307
flando sonus vocis animalis excitabatur, microscupio fibras
praedictas inspiciendo tremor et vibratio in iisdem cerne-
batur, prout in chordis musicis habetur dum resonant.
Atqui eo ipso ex earum trcmitu ab irruente aere tamquam
a vi percutiente excitato sonus gigni debet. vel iam geni-
tus modificationem quamdam recipere.
Rursus , sicut in chordis musicis contingit 'ut chorda
brevior det sonum acutiorem, graviorem longior :
ita ani-
madvertendum hic fuit an librarum illarum maior minor-
ve longitudo toni mutationem induceret. Compertum au-
tem est quod ,
impedita illarum fibrarum parte ac tre-
meret. tonus prodibat acutior.
Sumpsit etiam larynges bovis , canis. aliorumque sni-
malium, deinde insufflaudo excitabatur mugitus bovis ,
et
conformis aliis animalibus sonus. Movendo autem musculos
ita, ut traherentur et distenderentur fibrae, excitabantur
'mutationes soni, quae haberi solent in varia horum ani-
malium voce.
Notetur illud: cum tensio vel remissio librarum glot—
tidis et cartilagineae substantiae, qua larynx constat , ab
eodem musculo dependeat , ut notat Savart, consequitur
una cum fibris illis etiam laryngem tendi vel remitti.
Laxatis' fibris, ,orificiu-m glottidis ampliatur, et sonus pro-
dit gravior; tensis vero , orificium restringitur. et sonus
evadit acutior, ut in canibus observavit D. Magendie.

143. Si vox in larynge et glottide tanquam in proprio
organo fit; quid ergo, inquies, os atque eius partes cou-
ferunt ad formationem vocis?
Respondeo oris cavitatem. linguam, dentes. labia con-
currere ad modificationem perfectionemque'vocis; (quae in
⇁
larynge et glottide incipit ⋅ quidem , sed non omnimode
⋅⋅
ibi perficitur: nam quod in illis partibus sufficiens habea-
tur organum qain prorsus necessaria sint oris et linguae or-
,gana ad exhibendum aliquo modo sonum animalis pro-
prium ,,apparet ex eo quod grues et anseres , abscisso308
capite , ex ventris compressione sonos edere possint iis si
miles, quos viventes edebant. Ad modificationem igitur per
fectionemque vocis in larynge et glottide inchoatae caete
ra concurrunt : neque haec modificatio in mera reflexione
consistit, sed in resonantia proportionata tono soni a
glottide emissi .
Ad articulatarum vocum formationem quod attinet ,
ea praecipue a mota linguae et labiorum repeti solet :
inter caeteros P. Fabri diligenter expendit quo pacto lin
gua et labia componantur ad cujusque syllabae efforma
tionem .

144. Dices: potest sonus excitari aerem expellendo
per angustius spatium ; atque ita sibilus per labiorum com
pressionem excitatur. Ergo dicendum videtur quod ex 90
la emissione aeris per angustius glottidis spatium vox effor
inari possit quin confugiamus ad tremitum laryngis et
fibrarum glottidis ; qui tremitus effectus erit soni quin in
sonum ipsum influat.
Respondeo : etsi sonus aliquis obtineri praecise pos
sit per hoc quod ex ampliore in angustius spatium aer
cogatar transire ; attamen quae hactenus diximus suadent
tremitum laryngis et fibrarum ad vocis formationem con .
currere; attenta praecipue varietate maxima , quae in vo
cis modificatione habetur. Novimus enim et singulos ho
mines modificari quam maxime vocem , et in diversis ho
minibus quam maxime diversum esse vocis sonum . Iam ve
ro cum habeatur sibilus per solam labiorum compressio
nem , inde expulso violenter aere , exigua est hujusmodi
soni diversitas; et omnes fere homines eumdem sonum ef
ficiunt , licet in diversa intensione : ergo cum contra in
voce diversitas sit maxima et proprius cuique sit homini so
nus , ad diversam fibrarum et laryngis materiam ac ten
sionem recurrendum potius videtur. Scio equidem ab in
strumentis pneumaticis etiam materia resistente compactis
et lingula instructis magnam edi posse varietatem sono
308
capite , ex ventris compressione sonos edere possint iis si-
miles, quos viventes edebant. Ad modificationem igitur per-
fectionemque vocis in laryuge et glottide inchoatae caete-
ra concurrunt: neque haec modificatio in mera reflexione
consistit, sed in resonantia prOportionata tono soni a
glottide emissi.
Ad articulatarum vocum formationem quod attinet ,
ea praecipue a motu linguae et labiorum repeti solet:
inter caeteros P. Fabri diligenter expendit quo pacto lin-
gua et labia componantur ad cuiusque syllabae efforma-
tionem.

144. Dices: potest sonus excitari aerem eXpellendo
per angustius spatium : atque ita sibilus per labiorum com-
pressionem excitatur. Ergo dicendum videtur quod ex so-
la emissione aeris per angustius glottidis spatium vox effor-
mari possit' quin confugiamus ad tremitum laryngis et
fibrarnm glottidis; qui tremitus effectus erit soni quin in'
sonum ipsum influat.
Respondeo: etsi sonus aliquis obtineri praecise pos-
sit per hoc quod ex ampliore in angustius spatium aer
cogatur transire; attamen quae hactenus diximus suadent
tremitum laryngis et librarum ad vocis formationem cou-
1:11rrere; attenta praecipue varietate maxima, quae in vo-
cis modificatione habetur. Novimus enim et singulos bo-
mines modificari quam maxime vocem, et in diversis ho-
minibus quam maxime diversum esse vocis sonum. Iam ve-
ro cum habeatur sibilus per solam labiorum compressio-
nem ,
inde expulso violenter aere , exigua est huiusmodi
soni diversitas; et omnes fere homines eumdem sonum ef-
ficiunt
,
licet in diversa intensione : ergo cum contra in
voce diversitas sit maxima et proprius cuique sit homini so-
nus , ad diversam librarum et laryngis materiam ac ten-
sionem recurrendum potius videtur. Scio equidem ab in-
strumentis pneumaticis etiam materia resistente compactis
et lingula instructis magnam edi posse varietatem sono-309
recessum
rum , atque ad instrumenta ista referri organum vocis ab
auctoribus non paucis. Verum non video quomodo glotti
dis fibrae se habeant ad vocis organum perinde ac lin
gula : si non ita haec movetur , ut epistomium alterne
aperiatur claudaturque ; licet ea citissime oscillet , nullus
inde prodibit sensibilis sonus . Iam vero glottidis fibrae
non sic oscillant , ut per mutuum accessum et
alterne claudatur aperiaturque ipsius glottidis foramen .
In glottidis fibris aeris irrumpentis impetu ad tremitum
concitalis auctores aliqui cum Ferreinio organum vocis ma
xime constituunt , illudque ad instrumenta fidicularia po
tissime revocant , minime considerantes quod hujusmodi fi
brae careant ea longitudine et crassitie , quae necessaria
esset ad graves atque ingentes humanae vocis tonos effi
ciendos,

145. Quaeres 

1.º qui sit defectus, ob quem mutis loquela deest. Cum plerique muti sint, quia sunt a nativitate surdi, quique proinde cum non possint alios loquentes audire ne loqui quidem discunt unquam; attamen sunt qui auditu pollent, at loquendi facultate destituuntur. Vitium in his multiplex esse potest; aut ex humorum nimietate et crassitie; aut ex fibrarum inelasticitate, qua etiam fit ut, timore insolito obrigescentibus fibris, vox impediatur in iis qui caeterum muti non sunt; vel ex nimia linguae turgescentia; vel alio vitio: adeoque non desunt exempla mutorum arte medica, aut etiam solius naturae auxilio loquelam adipiscentium. 

2.º Cur aves aliquae humanam vocem aemulentur, pleraeque non item. In psitlacis diligenter rem inspexit Kircherus, atque animadvertit pro corporis quantitate os satis magnum et excavatum, maxillas turgentes, linguam maxime flexilem, et rostrum superius contra indolem aliarum avium mobile; unde bruta pro majore vel minore aptitudine ad oris dilatationem, flexilitatem linguae, labiorum, vel rostri modificationem apta erunt ad sovum humanae vocis imitandum. Picae io
309
rum , atque ad instrumenta ista referri organum, vocis. ab
auctoribus non paucis. Verum non video quomodo glottidis fibrae se habeant ad vocis organum perinde ac lingula: si non ita haec movetur, ut epistomium alterne aperiatur claudaturque; licet ea citissime oscillet, nullus inde prodibit sensibilis sonus. Iam vero glottidis fibrae non sic oscillant, ut per mutuum accessum et recessum alterne claudatur aperiaturque ipsius glottidis foramen. In glottidis fibris aeris irrumpentis impetu ad tremitum concitatis auctores aliqui cum Ferreinio organum vocis maxime constituunt, illudque ad instrumenta fidicularia potissime revocant, minime considerantes quod huiusmodi fibrae careant ea longitudine et crassitie, quae necessaria esset ad graves atque ingentes humanae vocis tonos efiiciendos. 

145. Quaeres ↿∙∘ qui sit defectus , ob quem mutis
loquela deest. Cum plerique muti sint, quia sunt a nati-
vitate surdi
, quique proinde cum non possint alios loquen-
tes audire ne loqui quidem discunt unquam; attamen sunt
qui auditu pollent, at loquendi facultate destituuntur. Vitium
in his multiplex esse potest: aut ex humOrum nimietate
et crassitie; aut ex fibrarum inelasticitate , qua etiam fit
ut
,
timore insolito obrigescentibus fibris
, vox impedia-
'tur in iis qui caeterum muti non sunt; vel ex nimia lin-
guae turgescentia; vel alio vitio: adeoque non desunt exem-
pla mutorum arte medica
,
aut etiam solius naturae auxi-
lio loquelam adipiscentium. 2.(' Cur aves aliquae humanam
vocem aemulentur , pleraeque non item. In psittacis dili-
genter rem inspexit Kircherus , atque animadvertit pro
corporis quantitate os satis magnum et excavatum, maxil-
las turgentes, linguam maxime flexilem , et rostrum su-
perius contra indolem aliarum avium mobile; unde bru-
te pro majore vel minore aptitudine ad oris dilatationem
,
⋅ Hexilitaïem linguae ,
labiorum , vel rostri modificationem
apta'erunt ad sonum humanae vocis imitandum. Picae- iu-310
a
ter caeteras aves , et corvi antiquitus etiam ad voces hu
manas formandas instituebantur. 

3. ° An verum sit quod vox ita procreari possit ut infra laryngem genita videatur ,
ideoque sonus audiatur non tanquam ex ore , sed tanquam
ex alvo veniens. Ita de facto Pythonissae , quas Ethnicorum
historiae , et sacra scriptura etiam memorat, loquebantur:
aliquos tamen sine ulla suspicione daemoniaci operis ven
triloquos esse exemplis pluribus compertum est. Sicut enim
loquitur aerem expellendo , qui in larynge et glottide vo
cem excitat ; ita fieri potest ut aerem ore ac naribus at
lrahendo in gloutide item parem molum excitemus , sicque
non ex ore sed infra laryngem vox orta videatur,
प
be
AL

=== De auditus organo. ===
146. Externa auris pars palula est; et ex cartilagine intus concava atque elastica constat; quae in concham
sea cavitatem referentem conchae figuram desinit. Inser
vit ad colligendas uudas soni : hinc quasi natura duce qui
minus acuto pollet auditu , aut ad vocein nimis e lon
ginquo attendit , manum ad aures applicat , ut eo pacto
plures colligat aeris undas. Externa haec auris pars , quae
auricula simpliciter dicitur , musculis adornatur , quorum
ope sunt aliqui homines qui auriculam ad libitum mo
vent ; oves autem , equi et bruta alia multo facilius :
adnotant nonnulli Analomici ila necessariam esse exter
banc partem ut sonorus lenius allabatur
in internas cavitates, ut nonnisi confusa et quasi cum
inurmure fluentis aquae audiant ii, quibus auriculae abscis
sau sint. Animadvertendum tamen reptilia et aves hoc ex
lerno adminiculo carere.
Ad fundum conchae incipit meatus auditorius , qui est
canaliculus aliquanto tortuosus ; et ex majori latitudine in
minorem paullatim coarctator. Ita factum notat Val
9
nam aer 1
1
310.
ter caeteras aves , 'et corvi antiquitus etiam ad voces hn-
manas formandasinstituebantur. 3.0 An verum sit quod
'vox ita proci-cari possit ut infra laryngem genita videatur,
ideoque sonus audiatur non tanquam ex ore , sed tanquam
ex alvo veniens. Ita de facto Pythonissae , quas Ethnicorum
historiae ,
et sacra scriptura etiam memorat, loquebantur:
aliquos tamen sine ulla suspicione daemoniaci operis ven-
triloquos esse exemplis pluribus compertum est. Sicut enim
loquitur aerem expellendo , qui in larynge et glottide vo-
cem excitat; ita fieri potest ut aerem ore ac naribus at-
trahendo in glottide item parem motum excitemus, sicque
non ex ore sed infra laryngem vox orta videatur,
De auditu: organo.
146. Externa auris pars patula est, et ex cartilagi-
ne iutus concava atque elastica constat; quae in concham
sen cavitatem referentem conchae figuram desinit. Inser-
vit,ad colligendas undas soni: hinc qnasi natura duce qui
minus acuto pollet auditu
,
aut ad vocem nimis e lon-
giuquo attendit, manum ad aures applicat , ut eo pacto
plures colligat aeris undas. Externa haec auris pars, quae
auricula simpliciter dicitur
, musculis adornatur , quorum
0pe sunt aliqui homines qui auriculam ad Hibitum mo-
vent; oves autem
, equi et bruta alia multo facilius :
adnotaut nonnulli Anatomici itaqnecessariam esSe exter-
nam lianc partem ut aer sonorus lenius allahatur
in internas cavitates, ut nonnisi confusa et quasi- cum
murmure fluentis aquae audiant ii, quibus auriculae abscis-
sae sint. Animadverteudum tamen reptilia et aves hoc ex-
terno adminiculo carere.
Ad fundum conchae incipit meatus auditorins , qui est
canaliculus aliquanto tortuosus; et ex maiori latitudine in
minorem paullatim coarctatur. Ita factum notat Val-311
sa
salva at sonus intendatur magis , sicuti in recurvis lubis
a surdastris adhiberi solitis intenditur ; alii potius ad im
minuendum aeris impetum , ne in auris interiora fortius
impellat , has tortuositates in organo auditus a natura in
stilutas putant. In auditorio meatu humor quidam amarus
ac viscosus habetur , seu cerumen ; exsudat e glandulis
quas sebaceas vocant , et institutum est ut minima ani
malcula ab ingressu ad interiora auris arceantur .
Ad finem hujus canalis habetur membrana tympani:
haec membrana tenuissima , sicca , pellucida et valde ela
stica , obtensa est annullo , qui tamen totum circuitum
non complet ; et fere ad similitudinem pellis tympani mi
litaris cavitatem interiorem superambit : non est recte exten
sed curva nonnihil ; coacava scilicet respectu auris
externae , convexa ad partes internas . Fuit acerrima quae
stio , an membrana tympani omnem communicationem in
ter externam internamve aurem excludat , an contra per
via sit aeri externo. Argumentum pro communicatione va
lidum est , quod aliqui fumum ore exceptum per aurem
emittunt ; neque id semper imposturae vertendam est ,
ut compertum fuisse Nolletus ait a viro , cni Academia
regia jussum fecerat facti veritatem explorare. Argumen
tum contra communicationem est , quod Valsava , immis
so in aurem internam hydrargyro , quantumvis excute .
retur , nihil unquam per externam aurem defluxit ; quam
quam reponere soleat qui communicationem tuetur , quod
in cadaveris organo non est necesse partium structuram sal
vari.
Post pellem tympani habetur cavitas aere plena , quae
capsula dicitur , quaeque cum membrana praedicta tym
panum constituit. In hac sunt quatuor ossicula quae ap
pellantur malleus , incus , os orbiculare , stapia. Alii tria
tantum numerant , omisso osse orbiculari , vel quia, cum
ompium humani corporis ossiam minimum sit , adeo ut
non superet dimidium grani millii , animadversionem fu
31↿⋮ salva ut sonus intendatur magis , sicuti in recurvis tubis
'a snrdastris adhiberi solitis intenditur; alii potius ad im-
miuuendum aeris impetum , ne in auris interiora fortius
impellat, has tortuositates in organo auditus a natura in-
stitutas putant. In auditorio meatu humor quidam amarus
ac viscosus habetur , seu cerumen; exsudat e glandulis,
quas sebaceas vocant , et institutum est ut minima ani-
malcula ab ingressu ad interiora auris arceantur.
Ad finem hujus canalis habetur membrana tympani:
haec membrana tenuissima , sicca , pellucida et valde ela-
stica
, obtensa est annnllo , qui tamen totnm circuitum
non complet; et fere ad similitudinem pellis tympani «mi—
litaris cavitatem interiorem superambit: non est recte exten-
sa , sed 'curva nonnihil : concava scilicet respectu auris.
externae
, convexa ad partes internas. Fuit acerrima qnae-
stio , an membrana tympani omnem communicationem in-
ter externam internamve aurem excludat , an contra -per-
via sit aeri externo. Argumentum pro-communicatione va-
lidum est
, quod aliqui fumum ore exceptum per aurem
emittunt; neque id semper imposturae vertendam est,
ut compertum fuisse 'Nolletus ait a- viro, cni Academia
regia iussum fecerat facti veritatem explorare. Argumen-
tum eontra communicationem est
, quod Valsava ,
immis-
so in aurem internam hydrargyro , quantumvis excute-
retur, nihil unquam per externam aurem defluxit; quam-
quam reponere soleat qui communicationem tuetur
, quod
in cadaveris organo non est necesse partinm structuram sal-
var]. '
Post pellem tympani habetur- cavitas aere plena , quae
capsula dicitur
, quaeque eum membrana praedicta tym-
panum constituit. In hae sunt quatuor ossicula quae ap-
pellantur mallens
.
incus , os orbiculare
, stapia. Alii tria
tantum numerant
, omisso osse orbiculari, vel quia, cum
omnium humani uerporis ossium minimum sit, adeo ut
non superet dimidium grani millii
, animadversionem fu-312
1
1
gerit : vel quia ita adhaeret slapiae et incudi , at cum al
tero ex his confundi potuerit, Circa haec ossicula nolan
dum , quod ejusdem magnitudinis sint tam in infante quam
in adulto viro contra aliarum omnium corporis partium in- .
dolem , quae aetatis progressu augentur. Hoc ideo factum
esse docet Valsalva , ne augmento partium auditui inservien
tium alia sit sonorum ratio adulla aetate ac fuit ab ini
tio ; et ideas gravis atque acuti quas pueri imbibimus, ma
tare aetate proficiente cogamur.
In tympani cavitate habetur canalis quidam seu lu
ba Eustachiana dicta ab ipsius inventore : per hanc tu
bam ab interna auris cavitate obtinet communicatio cum
cavitate oris; desinit enim ista tuba ad radices uvae; quem
prope locum etiam interiora nasi communicant : hujus tu
bae ope fit , ut sonus ex oris cavitate auri communicetur,
ideoque qui dentibus stringit corpus resonans sobum au.
dit etiam auribus impeditis ; et surdastri hiante ore so
nos excipere solent , ut tali pacto juvelur melius auditio.
Praeter foramen ex quo tuba Eustachiana procedit ,
duo alia babentur in tympani cavitate foramina , quorum
unum dicitur fenestra ovalis , allerum fenestra rotunda.
Feuestra ovalis basi slapiae occluditur, rotunda solo mem
branulae tegumento obtegitur. .
Ex tympani cavitate per duo praedicta foramina , ovale
scilicet ac rotundum , itur in labyriothum , qui - est inte
rior alia cavitas in osse petroso ulterius excavata , et quo
dam liquido plena : in hac tres partes distingui solent ;
prima est vestibulum labyrinthi ; secunda constat tribus
ossiculis semicircularibus , quibus nomen labyrinthi pecu
liarins aliqui tribuunt ; tertia est cochlea seu limax, quae
ex osse constat in cochleae modum conlorto duos gyros
cum dimidio faciente. Elsi cochlea unus canalis videri
possit , est lameu revera duplex : dividitur enim secun
dum longitudinem medio segmento , parim osseo , partim
membranaceo , quod dicitur lamina spiralis. Cochlea in
1
1
1
i
1
1
1
•
2
0
1
1
1
.
312
gerit: vel quia ita adhaeret stapiae et incudi , at cum al-
tero ex his confundi potuerit. Circa haec ossicula notan-
dum , quod eiusdem magnitudinis sint tam in infante quam
in adulto viro contra aliarum omnium corporis partium in--
dolem, quae aetatis progressu augentur. Hoc ideo factum
esse docet Valsalva, ne augmento partium auditui inservien-
tium alia sit sonorum ratio adulta aetate ac fuit ab iui-
tio; et ideas gravis atque acuti quas pueri imbibitüus, mu-
tare aetate proficiente cogamur.
ln tympani cavitate habetur canalis quidam seu tu-
ba Eustachiaua dicta ab ipsius inventore: per hanc tu-
bam ab interna auris cavitate obtinet communicatio cum
cavitate oris; desinit enim ista tuba ad radices uvae; quem
prope locum etiam interiora nasi communicant :-huius tu-
bae upe fit
, ut sonus ex oris cavitate auri .communicetur,
ideoque qui dentibus stringit corpus resonans sonum su-
dit etiam auribus impeditis ; et surdastri hiante ore so-
nos excipere solent , ut tali pacto iuvetur melius auditio-
Praeter foramen ex quo tuba Eustachiana procedit,
duo alia habentur in tympani cavitate foramina , quorum
unum dicitur fenestra ovalis, alterum fenestra rotunda.
Feuestra ovalis basi stapiae occluditur, rotunda solo mem-
branulae tegumento obtegitur. .
Ex tympani cavitate per duo praedicta foramina
, ovsle
scilicet ac rotundum ,
itur in labyrinthum
, qui-est inte-
- rior alia cavitas in esse petroso ulterius excavata
,
et quo-
dam liquido plena: in hac,tres partes distingui solent;
prima est vestibulum labyrinthi ; secunda constat tribus
ossiculis semicircularibus , quibus nomen labyrinthi pecu-
liarius aliqui tribuunt; tertia est cochlea seu limax, quae
ex osse constat in cochleae modum contorto duos gyros
cum dimidio. faciente. Etsi cochlea unus canalis videri
possit ,
est tamen revera duplex: dividitur enim secun-
dum longitudinem medio segmento, partim osseo , partim
membranacea , quod dicitur lamina spiralis. Cochlea in313
avibus deest , si vera refert Boyle ; at ipsemet notat de
fectum hunc suppleri per cavitatem oblongam instar sacci.
Plures rami nervei per foramina perexigua ex eo ,
qui dicitur uervus auditorius , propagati per totam fere
aurem distribuuntur : in labyrinthum per quinque fora
mina ingrediuntur Gibrae nerveae , et ejus cavitatem inves
tiunt ; sed de nervis totum auris organum permeantibus
non vacat disserere. Id unum adnoto ex Mairano , spira
lem laminam fibrillis ita instructam esse ut quemadmo
dum ipsa ascendens ad cochleae apicem . semper angustior
fit , ita fibrae ipsae breviores semper evadunt.
147. Quaenam vero ex hactenus descriptis auris par
tibus pro praecipuo atque immediato auditionis organo sta
tueuda est ?. Aliqui membranam tympani assignarunt : at
experientia constat quod hac membrana lacerata vel erepta
adhuc manet auditio aliqua. Alii inepte in ossiculis intra
capsulam contentis organom auditus statuerunt , et sonum
ab anima immediate perceptibilem ex horum collisione
nasci affirmarunt. Praeter quam quod solus malleus in a
nimantibus quibusdam habeatur, falsum omnino est collidi
invicem haec ossicula atque inde sonum creari : adnexum
enim est caput mallei firmiter corpori incudis , et hujus
processus alter stapiae; adeoque cum aer exterous tympa
ni membranam impellit, omnia per modum unius intromit
tuntur et conjuncta simul sese restituunt ad locum pristi
num. Magis autem absona est illorum sententia , qui in
aere per capsam et labyrinthum existente auditus organum
collocabant: aerem hunc, quem innatum, insitum, vernacu
lum, implantatum dicebant, animatum statuere non vere
bantur. Communiter nunc auditus organum collocatur in
fibrillis nerveis per aurem internam, ac praesertim intra co
chleam disseminatis. Tremores itaque a corpore
excitati communicantur membranae tympani; tum per aea
rem in tympano existentem , nec non per ossiculorum se
riem, ad parietes asque labyrinthi et praecipue ad dupli
Sonoro
21
313
avibus deest , si vera refert Boyle ;
'at ipsemet notat de-
fectum hunc suppleri per cavitatem oblongam instar sacci.
Plures rami nervei per foramina perexigua ex eo ,- qui dicitur nervus auditorius, prcpagati per totam fere
aurem distribuuntur: in labyrinthum per quinque fora-
mina ingrediuntur fibrae nerveae, et eius cavitatem inves-
tiunt; sed de nervis totum auris organum permeantibus
non vacat disserere. Id unum adnoto ex Mairano , spira-
lem laminam fibrillis ita instructam esse ut quemadmo-
dum ipse ascendens ad cochleae apicem- semper angustior
fit
,
ita fibrae ipsae breviores semper evadunt.

147. Quaenam vero ex hactenus descriptis auris par-
tibus pro praecipuo atque immediatoauditionis organo sta-
tuenda est ?. Aliqui membranam tympani assignarent : at
experientia constat quod hac membrana lacerata vel erepta
adhuc manet auditio aliqua. Alii inepte in ossiculis intra
capsulam contentis organum auditus statuerunt
,
et sonum
ab anima- immediate perceptibilem ex horum collisione
nasci affirmarunt. Praeter quam quod solus malleus in a-
nimantibus quibusdam habeatur, falsum omnino est collidi
invicem haec ossicula atque inde sonum creari: adnexum
enim est caput mallei firmiter corpori incudis ,
et huius
processus alter stapiae; adeoque cum aer externus tympa-
ni membranam impellit, omnia per modum uniua intromit-
tuntnr et coniuncta simul sese restituunt ad, locum pristi-
num. Magis autem absona est illorum sententia , qui in..
aere per capsam et labyrinthum existente auditus organum
collocabant: aerem hunc, quem innatum, insitum, vernacu-
lum, implantatum dicebant, animatum statuere non vere-
bantur. Communiter nunc auditus organum collocatur in
fibrillis nerveis per aurem internam, ac praesertim intra co-
chleam disseminatis. Tremores itaque a sonoro corpore
excitati commnnicantur membranae tympani; tum per aes-.
rem in tympano existentem, nec non per ossiculorum se-
riem, ad parietes usque labyrinthi et praecipue ad dupli-
21
is314
cem fenestram , ovalem ac rolundam , transmissi deducuntur
ad liquidum cavitate labyrinthi contenlum ; inde vero ad fi
brillas nerveas praedictas, atque ad nervum ipsum audito
rium: unde fit, ut ex lege commercii anima ad sensationem
soni determinetur. Animadvertit Mairanus, quod sicut in
instrumentis quibusdam musicis binae et binae chordae pro
tonis singulis disponuntur, ita in lamina spirali nerveae fi
brillae dispositae sint : ex quo infert huc potius spectare
organum quam ad alias auris internas partes.
148. Quaeres 1º. Cur quibusdam grata , aliis pene ni
hil, aut etiam molesta sit harmonia. Alibi ( 121 ) dictumn
est chordam upisonam facile ad tremitum concitari: aliam
item , sed difficilius prout majorem minoremve cum chor
da percussa harmonicam proportionem habet. Alert Kir
cherus aliud experimentum , quod ad rem aptum est etiam
magis : experimentum ita se habet. Quinque sumantur
scyphi vitrei.ejusdem magnitudinis et capacitatis , et unus
quidem liquore impleatur, qui acquavite dicitur; alter vi
no ; tertius aqua puriori; quartus aqua communi ; quintus
crassiori aqua: tum ora cujusque poculi confricando sonus
quam fieri potest acntissimus excitetur. In primo quidem
• scypho spiritus ille maxime subsultabit; vinum moderatam su
bibit concitationem ; adhuc moderatior erit molus purio
ris aquae, et ita porro . Ex his intelligitur quomodo variae
in variis hominibus partium corporis commotiones orian
tur e sono. Cum autem animi molus, in quibus voluptas
consistit vel molestia , pendeant ex partium corporis affe
ctionibus; iis gratissima accidere poterit harmonia, quibus
ea solidorum ac fluidorum constitutio est , ut in iisdem com
motio consequatur impressionem factam in organo auditus
satis . vivida et animi moribus cum voluptate conjunctis ex
citandis apta: ii erunt ad harmoniam indifferentes, in qui
bus impressionem factam in organo auditus vix ulla con
sequitur alteratio solidarum fuidarumve corporis partium
quae pariat animi motus vel consonos, vel incongruos: iis
314
cem fenestram, ovalem ac rotundam, transmissi deducuntur
ad liquidum cavitate labyrinthi contentum; inde vero ad fi-
brillas nerveas praedictas, atque ad nervum ipsum audito-
rium: nnde fit, ut ex lege commerciianima ad sensationem
aoni determinetur. Animadvertit Mairanus, quod sicut in
instrumentis quibusdam musicis binae et binae chordae pro
tonis singulis disponuntur, ita ip lamina spirali nerveae fi-
brillae dispositae sint: ex quo infert huc potius spectare
organum quam ad alias auris internas partes.
148. Quaeres 1&quot;. Cur quibusdam grata, aliis pene ni-
hil, aut etiam molesta sit harmonia. Alibi (121 ) dictum
est chordam unisonam facile ad tremitum concitari: aliam
item, sed difficilius prout majorem minoremve cum chor-
da percussa harmonicum proportionem habet. Affert Kir-
cherus aliud experimentum, quod .ad rem aptum est etiam
magis : experimentum ita se habet. Quinque sumantur
scyphi vitrei-ejusdem magnitudinis et capacitatis, et unus
quidem liqum'e impleatur, qui acquavite dicitur; alter vi-
no; tertius aqua PUI'lOl'i; quartus aqua communi; quintus
crassiori aqua: tum ora cujusque poculi confricando sonus
quam fieri potest acutissimus excitetur. In primo quidem
scypho spiritus ille maxime subsultahit; vinum moderatam su-
bibit concitationem; adhuc moderatior erit motus purio-
ris aquae, et ita porro. Ex his intelligitur quomodo variae
in variis hominibus partium corporis commotiones orian-
tur e sono. Cum autem animi motus, in quibus voluptas
consistit vel molestia, pendeant ex partium corporis affe-
ctionibus; iis gratissima accidere poterit harmonia, quibus
easolidorum ac fluidorum constitutio est, ut in iisdem com-
motio consequatur impressionem factam in organo auditus
satis.vivida et animi motibus cum voluptate conjunctis ex-
citandis apta: ii erunt ad harmoniam indifferentes. tu qui-
bus impressionem factam in Organo auditus vix ulla con-
sequitur alteratio solidarum fluidarumve corporis partium ∙
quae pariat animi motus vel consouos, vel incongruos: iis1
1
315
denique molestia etiam accidet, quibus ex impressione ner
vorum acusticorum contingat incongrua motuum alteratio
in partibus corporis ad pracfatos animi molus inservienti
bus: quo fit etiam mechanice ut alii aliis sonorum gene
ribus vel delectentur magis, vel contra. Hanc tamen me
chanicam causam non arbitror esse sufficientem atque adae
quatam: admittenda est causa ex rationali natura hominis
pendens. Consistit harmonia in proportione illa , quam so
ni habent inter se ; unde fit ut in organo auditus vibra
tiones diversi generis, aliae frequentiores, aliaė tardio
res efficiantur: dum vibrationes istae organum anditus af
ficiunt, mens easdein comparat inter se, earumque propor
tionem animadvertit : si haec proportio ejusmodi sit ut fa
cile possit a mente percipi, et vibrationes facile comparari
queant, ex hoc gaudet mens; si confusa et inordinata vi
brationum sit comparatio , neque has mens facile con
ferre inter se potest, obruelur taedio: et quia imperi
tas in musica facilius comparat simpliciores consonantias quam
magis compositas, ideo musica planissima vulgo arridet ; qui
vero periti sunt et copiosioribus compositiouibus assueti, vix
patiuntur musicam nimis simplicem: item cum ex consue
tadine pendeat ut aliquas harmonicas proportiones faci
lius mens assequatur quam alias ; inde oritur at volu
ptas ex eo musices genere major sit, cui quis sit assue
tus; adeo ut ex hoc capite diversis nationibus diversae
placeant musicae species. Porro ab exercitatione etiam
profluit ut gratior accidat musica , etiam qua ex parte
mechanice voluptatem parit; ex assuetudine enim in fi
brillas nerveas docilitas inducitur ad recipiendas facilius
impressiones harmonicas.
2º. Cur duabus auribus unus idemque sonus au
diatur. Communis responsio est hujusmodi : cum in utra.
que aure creetur simillima impressio; non duplicem , sed
voam sensationem ab anima haber¡ necesse est. Qua in
re scite animadvertit Valsalva , summa industria provisum
315
denique molestia etiam accidet, quibus ex impressione ner-
vorum acnsticorum contingat incongrua motuum alteratio
in partibus corporis ad praefatos animi motus inservienti-
bus: quo fit etiam mechanica ut alii aliis sonorum gene-
ribus vel delectentur magis, 'vel contra. Hanc tamen me-
chanicam causam non arbitror esse sufficientem atque adae-i
quatam: admittenda est causa ex rationali natura hominis
pendens. Consistit harmonia in praportione illa, quam so-
ni habent inter se; unde fit ut in organo auditusvibraP-
tiones diversi generis, aliae frequentiores, aliae tardio-
res efficiantur: dum vibrationes istae organum auditus af-
Hciunt, mens easdem comparat inter se, earumque propor-
tionem animadvertit: si haec proportio ejusmodi sit ut fa-
cile possit a mente percipi, et vibrationes facile comparari
queant, ex hoc gaudet mens; si confusa et inordinata vi-
bratiouum sit comparatio , neque has mens facile con-
ferre inter se potest, obruetur taedio: et quia imperi-
tus in musica facilius comparat simpliciores consonantias quam
magis compositas, ideo musica planissima vulgo arridet ; qui
vero periti sunt et c0piosioribus compositionibus assueti, vix
patiuntur musicam nimis simplicem: item cum ex consue-,
tudine pendeat ut aliquas harmonicas preportiones faci-
lius mens assequetur quam alias; inde oritur ut volu-
ptas ex eo mus1ces genere major sit, cui quis sit assue-
tus; adeo ut ex hoc capite diversis nationibus diversae
placeant musicae species. Porro ab exercitatione etiam
profluit ut gratior accidat musica, etiam qua ex parte
mechanica voluptatem parit; ex assuetudine enim in fi-
brillas nerveas docilitas inducitur ad recipiendas facilius
impressiones harmonicas. ⋅
20. Cur duabus auribus unus idemque sonus au-
diatur. Communis responsio est huiusmodi: cum in utra-
que aure creetur simillima impressio; non duplicem, sed,
unam sensationem ab anima haberi necesse est. Qua in
re scite animadvertit Valsalva, summa industria provisum316
fuisse a natura ut in utraque aure quam simillima es
sent organa omnia ; adeo ut, dum in diversis hominibus
structura partium nonnihil variat, in eodem tamen bomi
mine nulla prorsus sit utriusque auris vel minima variatio .
Notetur illud : quemadmodum eadem chorda varios
tonos varia tensione edere potest, ita membrana tympani
lenditur diversimode ut variis tonis aple accomodetur ;
eapropter manubrium mallei eidem adnexum est, et ba
sis stapiae eodem modo membranae fenestrae ovalis: ten
sio autem et relaxatio membranae, nobis insciis , potest na
turaliter fieri. Itaque, ut primae vibrationes membranam
feriunt, admonita natura organum opportuna tensione aptat
ut respondens tonus in successivis vibrationibus fiat ma
gis vel minus sensibilis.
316
fuisse a natura ut in utraque aure quam simillima es-
sent organa omnia; adeo ut, dum in diversis hominibus
structura partium nonnihil variat, in eodem tamen homi-
miue nulla prorsus sit utriusque auris vel minima variatio.
Notetur illud: quemadmodum eadem chorda varios
tonos varia tensione edere potest, ita membrana tympani
tenditur diversimode ut variis tonis apte accomodetur;
eapropter manubrium mallei eidem adnexum est, et ba-
sis stapiae eodem modo membranae fenestrae ovalis: ten-
sio autem et relaxatio membranae, nobis insciis,potest na-
turaliter fieri. Itaque, ut primae vibrationes membranam
feriunt, admonita natura organum opportuna tensione aptat
ut respondens tonus in successivis vibrationibus fiat ma-
gis vel minus sensibilis.INDEX
RERUM QUAE IN PRIMO VOLUMINE CONTINENTUR.
MECHANICA E PRINCIPIA
Notiones praeambulae. pag. 1 .
Molus uniformis et varius : velocitas et quantitas mo
tas in motu uniformi. num . 1 .
Corporum indifferentia ad motum et ad quietem: quid
vires : quid earum aequilibrium ; et quomodo repraesen
tentur sive per lineas rectas, sive per numeros . n. 2, 3, 4.
Principiom motus • relativi : vires sunt ut quantitates
motus , n. 5, 6 .
Principium actionis et reactionis : mutatio status in
corpore haud repente gignitur a viribus extrinsecis , sed
per gradus indefinitae attenuationis capaces: quid vires in
stanlaneae et continuae. n. 7 .
De virium compositione et resolutione , deque earum
momentis et aequilibrio : aliquid quoque notatur de
vecte, axe in peritrochio, trochlca etc. pag. 6.
Compositio virium materiali puncto applicatarum: ae
quilibrium: varia circa virium resolutionem .. n. 8. 9. 10.
⋅ N D EX
RERUM QUAE IN ramo VOLUMINE CONTINENTUR.
' MECHANICAE PRINCIPIA
∙
W
⋅∙
Nott'ones praeambulae. pag. 1.
Motus uniformis et varius: velocitas et quantitas mo-
tus in motu uniformi. . . . . . . . . . num-1.
Corporum indifferentia ad motum et ad quietem: quid
vires: quid earum aequilibrium; et quomodo repraesen-
tentur sive per lineas rectas, sive per numeros. n.2, 3, 4.
. Principium motus 'relativi: vires sunt ut quantitates.
motus. ∙ ∙ ∙ ∙↴ ∙ ∙
.' ∙ ∙ ∙ ∙ ∙ ∙ n. 5, 6.
Principium actionis et reactionis: mutatio status in
corpore haud repente gignitur a viribus extrinsecis ,
sed
per gradus indefinitae attenuationis capaces: quid vires in-
stantaneae et-continuae.
. . . . . . . . . n. 7.»
De virium compositione et resolutione , deque earum
momentis et aequilibrio : aliquid quoque notatur de
vecte, axe in peritrochio, trochlea etc. pag. 6.
Compositio virium materiali puncto applicatarum: ae-
quilibrium: varia circa virium resolutionem.. n. 8. 9. 10.318
Compositio duarum virium extremis rectae rigidae punctis
applicatarum, et in eodem plano jacentium: aequilibrium circa
immobile punctum: principiam velocitatum virtualium in ordi
ne ad istiusmodi vires: momenta virium quoad punctum ( M) :
momentum resultantis aequatur summae ex momentis com
ponentium si hae in eamdem plagam circa ( M ) nituntur
movere puncta , quibus applicitae sunt; aequatur differentiae
si nituntur movere in plagas contrarias. n. 10. 10. 20.30,
Haec ipsa extenduntur ad quemvis numerum virium in
dato plano jacentium.
n. 10. 4º , 5º , 6º.
Binae vires haud jacentes in eodem plano nequeunt
ad unicam vim aequipollentem traduci : vires in aequili
brio constitutae; sollicitantesque vel solidum liberumque cor
pus, vel solidam corpus mobile duntaxat circa punctum fi
xum, vel solidum corpus mobile tantummodo circa asem
fixum : momenta quoad axem . n. 10: 70. ... 10 °.
Vires parallelae: vis inde resaltans: earum centrum :
momenta quoad planum: respondens theorema n . 11 , 12 ,
13. 10. 2º. 3º.
Parallelarum virium systema consistit in aequilibrio sub
duabus conditionibus simul explendis; altera est ut evane
scat earum summa; altera ut evanescat summa ex earum
momentis in ordine ad duo plana ipsis viribus paral
lela . n. 13. 4º. 5º.
.
Etsi vires non sunt parallelae, possunt tamen rednci
ad terna ejusmodi systemata, quorum primum coalescat ex
viribus parallelis axi OZ, secundum ex viribus jacentibus
in plano XOY simulque parallelis axi Qy, tertium ex vi
ribus agentibus juxta axem OX: aequilibrii conditiones quoad
systemata rigida viribus minime parallelis sollicitata; 1º , in
318
Compositio duarum virium extremis rectae rigidae punctis
applicatarum,etin eodem plano jacentium: aequilibrium circa
immobile punctum: princi pium velocitatum virtualium in ordi-
ne ad istiusmodi vires: momenta virium quoad punctum (M):
momentum resultantis aequatur summae ex momentis com-
ponentium si hae in eamdem plagam circa (M ) nituhtur
movere puncta, quibus applicitae sunt; aequatur differentiae
si nituntur movere in plagas contrarias. n. 10. 10. 2230.
∙ Haec ipsa extenduntur ad quemvis numerum virium in
dato plano jacentium.
. . . . . . n. 10. 40. 50. 6&quot;.
Binae vires haud jacentes in eodem plano nequeunt
ad unicam vim aequipollentem traduci : vires in aequili-
brio constitutae; sollicitantesque vel solidum liberumque cor-
pus, vel solidum corpus mobile duntaxat circa punctum fi-
xum, vel solidum corpus mobile tantummodo circa axem
fixum: momenta quoad axem. .'
. . n. 10: 70. 10&quot;-
Vires parallelae: vis inde resultans: earum centrum:
momenta quoad planum: respondens theorema &quot;. 11, 121
13. 10. 20. 30.
Parallelarum virium systema consistit in aequilibrio sub
duabus conditionibus simul explendis; altera est ut evane-
scat earum summa; altera ut evanescat summa ex earum
momentis in ordine ad duo plana ipsis viribus paral-
lela.
. . . . . . . . . . . . . n.13.4&quot;.5'-
Etsi vires non sunt parallelae, possunt tamen -reduci
ad terna eiusmodi systemata. quorum primum coalescat ex
viribus parallelis axi OZ, secundum ex viribus jacentibus
in plano XOV simulque parallelis axi QT, tertium ex vi-
ribus agentibus juxta axem OX: aequilibrii conditiones quoad
systemata rigida viribus minime parallelis sollicitata;1&quot;.in319
hypothesi systematis liberi; 2 °. in hypothesi systematis de
tenti puncto fixo; 3º . in hypothesi systematis detenti axe
fixo: conditio explenda ut plures vires minime parallelae
traduci possint ad unicam aequipollentem . n. 13. 6 ... 11º.
Duo solida corpora , datis viribus sollicitata , sese in
vicem aeque premendo apud datum mutui contaclus pan
ctum manent in aequilibrio : determinatur istiusmodi pres
sionis magnitudo. n. 13. 12 .
Solidum corpus , datis viribus sollicitatum, detinetur
duobus punctis fixis, sumptis in axe v. gr. OZ: determi
nantur pressiones exercitae in puncta illa juxta coordi
nalos axes OX, OY, OZ. n. 13. 13 .
Exempla aequilibrii in quibusdam machinis, praeci
so attritu : aequilibrium punctoruni materialium juncto
rum flis determinatae quidem longitudinis sed mobili
bas circa data puncta. n. 14. 15. 16 .
De centro gravitatis. pag. 33.
Varia de vi gravitatis, de corporum densitate , deque
specifica eorum gravitate: linea directionis. n . 17 , 18 , 19.
Generales formulae determinantes centrum gravita
tis: inveniri potest ratione mechanica: peculiari metho
do determinalur in triangulo et pyramide triangulari, n. 20.
De corporum collisione. pag . 37 .
Normalis collisio : 1º. corporum non elasticorum : 2 ° .
corporum perfecte elasticorum : 3º . corporum imperfe
cte elasticorum . n. 21 , 22, ... 25 .
319
hypothesi systematis liberi: &quot;. in hypothesi systematis de-
⋅ teuti puncto fixo: 30. in hypothesi systematis detenti axe
fixo: conditio explenda ut plures vires minime parallelae
traduci possint ad unicam aequipollentem. n. 13. 60... 1'l0.
Duo solida corpora, datis viribus sollicitata, sese in-
vicem aeque premendo apud datum mutui contactus pun-
ctum manent in aequilibrio: determinatur istiusmodi pres-
sionis magnitudo. . . . . . . . . . n. 13. 120.
Solidum corpus . datis viribus sollicitatum. detinetur
duobus punctis fixis, sumptis in axe v- gr. OZ: determi-
nantur pressiones exercitae in puncta illa iuxta coordi-'
natos axes OX, 0ï,OZ. . . . ∎∙ ∙ ∙ n.13.130.
Exempla aequilibrii. in quibusdam machinis, praeci-
so attritu : aequilibrium punctorum materialium iuncto-
rum filis determinatae quidem longitudinis sed mobili-
bus circa data puncta.
. . . . . . n.14.15.'16.
De centro gravitatis. pag. 33.
Varia de vi gravitatis, de corporum densitate.deque
specifica eorum gravitate: linea directionis. n. 17, 18, 19.
Geuerales formulae determinantes centrum gravita-
tis: inveniri potest ratione mechanica: peculiari metho-
do determinatur in triangulo et pyramide triangulari. n. 20-
Dä corporum collisione. pag. 37-
Normalis collisio: 10. corporum non elasticorum: 2&quot;.
corporum perfecte elasticorum : 3&quot;. corporum imperfe-
cte elasticorum. . .
-
. . . . . n.21,22,...25.320
Obliqua eorumdem corporum collisio. n . 26.
De motu rectilineo utcumque vario. pag. 42
Praemittantur nonnulla ex analysi infinitesimali, e
jusque ad res geometricas applicatione. n. 27. 10.2 ... 300.
Formulae spectantes ad motum rectilineum utcumque
varium : formulae quoad motum rectilineum uniformiter
varium: vis acceleratrix : vis motrix. n. 28.
Formulae pertinentes ad motum rectilineum utcum
que varium applicantur ad materiale punctum sollicita
tum vi acceleratrice, quae sit distantiae a dato centro pro
portionalis. n. 29.

De verticali gravium descensu atque ascensu . pag.
65.
Formulae, legesque huc spectantes: motus gravium in
machina Atwoodi. n . 30, 31 , 32 .
Quid si gravium descensus vel ascensus fiat in me
dio resistente sub ea conditione, ut resistentia medii sit pro .
portionalis quadrato velocitatis. n. 33.
De gravium descensu per plana inclinala ; de attritu ;
deque cochlea, et cuneo. pag. 71.
Formulae determinantes descensum gravium per plana
inclinata: gravium descensus per plana inclinata compara
tur cum verticali eorum descensu. n . 34, 35.
320
⋅∡ Obliqua eorumdem corporum collisio. . ∙⋅ n. 26.
De motu rectilineo utcumque uario. pag. 42
Praemittuntur nonnulla ex analysi infiuitesimali, e-
iusque ad res geometricas applicatione. n. 27. 10. 2&quot;....300.
Formulae spectantes ad motum rectilineum utcumque
varium: formulae quoad mo'tum rectilineum uniformiter
varium: vis acceleratrix: vis motrix. . . . . . n. 28.
Formulae pertinentes ad motum rectilineum utcum-
que varium applicantur ad materiale punctum sollicita-
tnm vi acceleratrice. quae sit distantiae a dato centro pro-
portionalis. ............n.ag.
!
De verticali gravium descensu atque ascensu. pag. 65.
Formulae, legesque huc spectantes: motus gravium in
machina Atwoodi. . . . . . . . . n. 30,31,32-
Quid si gravium descensus vel ascensus liat in me-
dio resistente sub ea conditione, ut resistentia medii sit pro-
portionalis quadrato velocitatis. . . . . . . n. 33-
De gravium descensu per plana inclinata; de attritu; ⇥
deque cochlea, et cuneo. ∙ pag. 71.
Formulae determinantes descensum gravium per plana
inclinata: gravium descensus per plana inclinata compara-
tur cum verticali eorum descensu. . . . n. 34, 35-321
Gravium descensus per plura plana inclinata sibi con
rigua . n. 36.
non.
Unde orialur attritus , caeteris paribus , est proportio
nalis pressioni : quomodo habeatur ratio attritus in motu
gravium per plana inclinata : grave in plano inclinato li
brandum potentia aliqua, sive habeatur ratio attritus , sive
n. 37. 10. 20 30
Aequilibrii leges in cochlea, et cuneo. n. 37. 4º. 5º.
Spectatur attritus in aequilibrio cochleae: itemque in
aequilibrio corporis ad rotatilem motum circa fixum cy
lindrum sollicitati : in machinis praeter resistentiam ex at
tritu spectanda etiam est resistentia ex funibus n. 37.6º.70.8° .
De motu gravium oblique projectorum . pag . 81 ,
Aequatio ad curvam, quam describunt gravia oblique
projecta; istiusmodi curva dicitur parabola. n. 38, 39.
Amplitudo jactus: maxima jactus amplitudo habetur sub
angulo projectionis semirecto: sub quo angulo projiciendum sit
grave ut offendat in datum scopum : altitudo jactus : ali
quid subjungitur de proprietatibus praefatae curvae. n. 40. 1º.
2 ° .... 70
Quid si gravia oblique projiciantur in medio resi
n. 41 .
stente.
De generalibus quibusdam proprietatibus motus curvili
nei, orti a viribus quarum una determinat materiale
punctum ad motum uniformem , altera ipsi materia
li puncto est continue applicata .
.
pag. 85.
Ubi in aliquo curvae puncto vis acceleratrix desinat
agere, excurret mobile per tangentem în punclo illo : ubi
321
Gnavium descensus-per plura plana inclinata sibi con-
ligua...............n.36.
Unde oriatur attritus. caeteris paribus, est proportio-
nalis pressioni: quomodo habeatur ratio attritus in motu
gravium per plana inclinata: grave in plano inclinato li-
brandum potentia aliqua, sive habeatur ratio attritus, sive
non. . , . . . . . . . . . . n. 37.10.2030.
Aequilibrii leges in cochlea, et cuneo. n. 37. 40. 50.
Spectatur attritus in aequilibrio cochleae: itemque in
aequilibrio corporis ad rotatilem motum circa fixum cy-
lindrum sollicitati: in machinis praeter resistentiam ex et-
tritu spectanda etiam est resistentia ex funibus n. 37.60.70.80.
De motu gravium oblique projectorum: pag. 81,
∙ Aequatio ad curvam, quam describunt gravia oblique
proiecta; istiusmodi curva dicitur parabola. . n. 38. 39.
Amplitudo iactus: maxima jactus amplitudo habetur sub
angulo projectiouis semirecto: sub quo angulo proiiciendum sit
grave ut offendat in datum scopum : altitudo jactus: ali-
quid subiungitur de proprietatibus praefatae curvae. n. 40. 10.
20 .... 70.
Quid si gravia oblique projiciantnr in medio resi-
stente. ↖∙∙∙∙∙∙∙∙∙∙⋅∙∙∙∥∙∡∎∙
De generalibus quibusdam praprietatibus motus curvili-
'nei, orti a viribus quarum una determinat materiale
punctum ad motum uniformem. altera ipsi materia-
li puncto est continue applicata.
. . . . pag. 85-
Ubi in aliquo curvae puncto vis acceleratrix desinat
agere, excurret mobile per tangentem in puncto illo: ubi322
tempore finito angulus, quem efformat vis acceleratrix cum
directione tangentis , fuerit semper acutus, acquiret mo
bile incrementum velocitatis finitum ; si semper obtusus,
patietur decrementum finitum ; si semper rectus , veloci
tas manebit constans: quadratum velocitatis adaequat vim
acceleratricem ductam in dimidium chordae, quae ex ejus
directione abscinditur ab osculatore circulo. n . 42.... 45.
Haec vera sunt de omnium virium genere; ponantur
vires acceleratrices ad centrum datum directae : jacebit cur
va in plano transeunte per rectam projectionis et per cen
trum virium: radius vector describet areas circa virium cen
trum temporibus proportionales: viceversa si radius ve
ctor describit areas circa punctum aliquod temporibus pro
portionales, vis acceleratrix erit constanter directa ad pun
ctum illud: velocitas, qua pollet mobile in eadem curva ,
exsistit reciproce proportionalis perpendiculo ducto a centro
virium in tangentem: vis acceleratrix in quovis curvae pun:
cto est directe ul radius vector, et reciproce ut factum
ex osculi radio in cnbum praefati perpendiculi : si ultra
punctum contactus sumitur arcus infinitesimus, a materiali
puncto describendus subsequente tempusculo, radiusque ve
ctor pertingens ad hujus arcus extremitatem producitur
donec occurrat tangenti, vis acceleralrix in contactus pun
cto erit directe ut pars radii vectoris producti intercepta
ac tangente , et reciproce ut quadratum tempuscu
li .
arcu
n. 46,
49.
Sive vires tendant ad centrum datum, sive non ; coor
dinatae puncti materialis in fine temporis e spectandae sunt
tanquam functiones ipsius t : formulae respicientes et veloci
tatem in quolibet curyae puncto, et binas componentes, al
teram juxta tangentem , alteram juxta normalem , in quas
resolvitur yis acceleratrix. n, 50. 10. 2º . 3º.
.
322
tempore linito angulus, quem etl'ormat- vis acceleratrix cum
directione tangentis ,
fuerit semper acutus, acquirat mo-
bile incrementum velocitatis Gnitum; si semper obtusus,
patietur decrementum (initum: si semper rectus, veloci-
tas mauebit constans: quadratum velocitatis adaequat vim.
acceleratricem ductam in dimidium chordae, quae ex eius
directione abscinditur ab osculatore circulo. n. 42.... 45.
Haec vera sunt de omnium virium genere; ponantur
vires acceleratrices ad centrum datum directae: iacebit cur-
va in plano transeunte per rectam projectiouis et per cen-
trum virium: radius vector describet areas circa virium cen-
trum temporibus proportionales: viceversa si radius ve-
ctor describit areas circa punctum aliquod temporibus pro-
portionales, vis acceleratrix erit constanter directa ad pun-
ctum illud: velocitas, qua pollet mobile in eadem curva
.
exsistit reciproce proportionalis perpendiculo ducto a centro
virium in tangentem: vis acceleratrix in quovis curvae pung-
cto est directe ut radius vector, et reciproce ut factum
ex osculi radio iu cubum praefati perpendiculi: si ultra
punctum contactus sumitur arcusiufiuitesimus, a materiali
puncto describendus subsequente tempusculo, radiosque ve-
ctor pertingens-ad huius arcus extremitatem producitur
donec occurrat tangenti, vis acceleratrix in contactus pun-
cto erit directe ut pars radii vectoris producti intercepta
arcu ac tangente ,
et reciproce ut quadratum tempuscu-
li. ∙∙∙∙∙⋅∙∙∙ ∙ ∙∙ ..n.46,...49-
Sive vires tendant ad centrum datum, sive non; coor-
dinatae puncti materialis in fine temporis t spectandae sunt
tanquam functiones ipsius :: formulae respicientes et veloci-
tatem in quolibet curvae puncto, et binas componentes, al-
teram juxta tangentem, alteram juxta normalcm. in. qu&quot;.
resolvitur vis acceleratrix. . . . . . n. 50.1'-2&quot;- 30,323
Resolata vi acceleratrice in ternas componentes axi
bus coordinatis parallelas, stabiliuntur formulae huc per
tinentes: applicantur formulae ad duas quaestiones, quarum al
tera respicit gravia oblique projecta in vacuo, altera respicit
gravia oblique projecta in medio resistente. n. 50. 4º. 5º . 6º.
Quomodo vis acceleratrix directa ad centrum expri
matur generatim per coordinatas polares : quomodo, data vi
acceleratrice directa ad centrum , inveniri possit aequatio
inter coordinatas polares ad lineam per quam movetur ma
teriale punclum: exemplum desumptum a vi acceleratrice ,
quae sit reciproce ut quadratum radii vectoris: sub hac le
ge poterit materiale punctum describere parabolam haben
tem suum focum in centro virium:
quaenam velocitas pro
jectionis ad id sit necessaria. n. 50. 7º. 8º... 15 °
Motus curvilineus impeditus : vis centrifuga. n. 51 .
De vi acceleratrice in motu circulari, existente
centro virium in centro circuli. pag . 109,
Istiusmodi motus ' est uniformis: vis acceleratrix obti
netur dividendo quadratum velocitatis per curvae circularis
radium: varia inde inferuntur et quoad projectionis velo citatem necessariam ad describendam cicularem curvam , et
quoad vires acceleratrices in diversis peripheriis circula
ribus. n. 52 , 53.
Vis centrifuga orta ex circulari telluris rotatione cir
ca suum axem : qua ratione decrescat ab aequatore ad po
los: qua ratione vis centrifuga imminuat gravitatem a po
lis ad aequatoren . n. 54.
323
Resoluta vi acceleratrice in ternas componentes axi-
bus coordiuatis parallelas, stabiliuntur formulae huc per-
tinentes: applicantur formulae ad duas quaestiones, quarum al-
tera respicitgravia oblique projecta in vacuo, altera respicit
gravia oblique proiecta in medio resistente. n. 50. 40. 50. 60.
Quomodo vis acceleratrix directa ad centrum lexpri- matur generatim per coordinatas polares: quomodo, data vi
acceleratrice directa ad centrum
.
inveniri possit aequatio
inter coordinatas polares ad lineam per quam movetur ma-
teriale punctum: exemplum desumptum a vi acceleratrice,
quae sit reciproce'ut quadratum radii vectoris: sub hac le-
ge poterit materiale punctum describere parabolam haben-
tem suum focum in centro virium: quaenam velocitas pro-
fectionis ad id sit necessaria. '
. . n. 50. 7&quot;. 80...150.
Motus curvilineus impeditus: vis centrifuga. n. St.
De vi acceleratrice in motu circulari, existente
centro m'rium' in centro circuli . pag. 109.
Istiusmodi motus 'est uniformis: vis acceleratrix obti-
netur dividendo quadratum velocitatis per curvae circularis
radium: varia iude inferuntur et quoad proiectionis velo-
citatem necessariam ad&quot; describendam cicularem curvam, et
quoad vires acceleratrices in diversis peripheriis circula-
ribus-.............n.52,53.
.
Vis centrifuga orta ex circulari telluris rotatione cir-
ca suum axem: qua ratione decrescat ab aequatore ad po-
los: qua ratione vis centrifuga imminuat gravitatem a po-
lis ad aequatorem.
. . .. . . . . ∙∙ ∙ n. 54-324
De vi acceleratrice in motu elliptico, existente
centro virium in foco ellipsis pag. 111,
Varia praemittuntur et circa rectas ita ductas ex puncto
quovis, ut tangant datam sphaeram ; et circa plaua tangentia
ducta per ejusmodi rectas ; et circa rectarum , arearumque
planarum projectiones in plano quolibet ; sed praecipue
circa ellipsim. n. 55. 1º, 2º ...14 °. . .
Quibus praemissis, demonstratur illud : existente cen
tro virium in foco ellipseos , vis acceleratrix in motu el
liptico est reciproce ut quadratum radii vectoris : quid in
duabus ellipsibus si quadrata temporum periodicorum sint ut cubi semiaxiun transversorum . n. 56.
Paucis subjunctis de ellipsi , parabola , et hyperbo
la, demonstratur quod, agentibus viribus in ratione reci
proca duplicata distantiarum a dato centro, praeter para
bolam poterit quoque mobile describere vel ellipsim
vel hyperbolam, existente focorum altero in centro virium:
quaenam projectionis velocitas requiratur ad ellipsim de
scribendam , quaenam ad hyperbolam. n, 67. 1.2.7 .
Obiter de lege virium in motu elliptico, ubi eae ten
dant ad ellipseos centrum . n. 57 , 8 .
De motu relativo punctorum materialium , tendentium
in se mutuo viribus acceleratricibus quae sint di
recte ut massae in quas tenditur, et reciproce ut qua
drata respondentium distantiarum . pag. 125.
Generales ad istiusmodi motum aequationes differen
tiales. n, 58,
324
- De ui acceleratrice in motu elliptica. existente
centro virium in foco ellipsis pag. 111.
Varia praemittuntur et circa rectas ita ductas ex puncto
quovis, ut tangant datam sphaeram; et circa plana tangentia
ducta per ejusmodi rectas; et circa rectarum, arearumque
planarum proiectiones in plano quolibet ; sed praecipue
circa ellipsim. . . . . . . . . n.55.10. 20 ...140.
Quibus praemissis, demonstratur illud: existente cen-
tro virium in foco ellipseos , vis acceleratrix in motu el-
liptico est reciproce ut quadratum radii vectoris: quid in
duabus ellipsibus si quadrata temporum periodicorum sint
ut cubi semiaxium transversorum . . . . . . n- 56.
Paucis subjunctis de ellipsi, parabola ,
et hyperbo-
la, demonstratur quod, agentibus viribus in ratione reci-
proca duplicata distantiarum a dato centro, praeter para-
bolam poterit quoque mobile describere vel ellipsim ,
vel hyperbolam, existente focorum altero in centro virium:
quaenam proiectiouis velocitas requiratur ad ellipsim de-
scribendam, quaenam ad hyperbolam. . n. 57. ↿∘∙ ⋍∘∙∙∙ 70.
Obiter de lege virium in motu elliptica, ubi eae ten- dant ad ellipseos centrum.
. . . . . . n. 57. 8&quot;.
De motu relativo punctorum &quot;materialium
,
tendentium
in se mutuo viribus acceleratricibus quae sint di-
recte ut massae in quas tenditur, et reciproce ut quab
drata respondentium distantiarum. pag. 125.
Gener-ales ad istiusmodi motum aequationes dideren-
tiüles- ∙∎∎ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ &quot;a 58.325
Spectantur duo tantum materialia puncta: vires per
turbantes ex reliquis punctis. n. 59, ... 62.
De pendulis ; deque gravium descensu per
arcus cycloidales. pag . 134.
Quid pendulum simplex ; quid compositum : vires
gignentes motum penduli simplicis n . 63.
Velocitates in puncto infimo acquistae a gravibus per
inaequales ejusdem circuli arcus descendentibus sunt ut
ipsorum arcuum chordae .
. n. 64.
Oscillationes penduli simplicis per arcus satis exi
guos , ulcumque ceteroquin inaequales , sunt ad sensum
isochronae seu aequidiuturnae : quid ex doctrina penduli
simplicis circa terrestrem gravitatem n. 65 , 66.
Centrum oscillationis in pendulo composito : etiam
oscillationes penduli compositi suņt isochronae, modo ta
men existant satis exiguae .
n. 67.
Oscillationes penduli simplicis in medio resistente :
primo in hypothesi resistentiae proportionalis simplici ve
locitati; deinde in hypothesi resistentiae proportionalis qua
drato velocitatis .
n. 68.
n . 69 .
Paucis praemissis de cycloide, demonstratur illud : ex
quocumque cycloidis puncto demittatur grave , eodem sem
per tempore perveniet ad punctum infimum
De attractione corporum in hypothesi attractionis agentis
in ratione directa massarum , et in reciproca
duplicata distantiarum .
Attractio corporum quorumcumque in materiale pun
clum situm sive extra corpus attrahens, sive intra. n . 70,71,72.
pag . 151 .
325
Spectautur duo tantum materialia puncta: vires per-
turbantes ex reliquis punctis. . . . . . n. 59....62.
De pendulis; deque gravium descensu per
arcus cycloidales. pag. 134.
Quid pendulum simplex; quid compositum : vires
gignentes motum penduli simplicis
. . . . . n. 63.
Velocitates in puncto intimo acquistae a gravibus per
inaequales ejusdem circuli arcus descendentibus sunt ut
ipsorum arcuum chordae
. . . . . .- . . n. 64.
Oscillationes penduli simplicis per arcus satis exi-
guos, utcumque ceteroquin iuaequales ,
sunt ad sensum
isochrouae seu aequidiuturuae: quid ex doctrina penduli
simplicis circa terrestrem gravitatem . . . n. 65
, 66.
Centrum oscillationis in pendulo composito: etiam
oscillationes penduli compositi sunt isochrouae, modo ta-
men existant satis exiguae . . . . . . . . n. 67.
Oscillationes penduli simplicis in medio resistente:
primo in hypothesi resistentiae proportionalis simplici ve-
locitati; deinde in hypothesi resistentiae proportionalis qua-
drato velocitatis
. . . .. . . . . . . n. 68.
Paucis praemissis de cycloide, demonstratur illud : ex
quocumque cycloidis puncto demittatur grave , eodem sem-
per tempore perveniet ad punctum infimum
. . n.. 69.
De attractione corporum in hypothesi attractionis agentis
in ratione directa massarum, et in reciproca
duplicata distantiarum. pag. 151 .
Attractio corporum quorumcumque in materiale pun-
ctum situm sive extra corpus attrahens, sireintrafn. 70,71,72.326
Expediuntur quae pertinent ad attractionem corpo
rum sphaericorum in punctum materiale n. 73,74,75.
Materiale punctum valde distans a corpore attrahente,
utcumque se habeat forma corporis, ea proxime ratione tendit
in ipsum corpus, qua tenderet si corporis partes in centro
gravitatis compenetrarentur: ubi dimensiones corporum quo
rumcuinque se mutuo attrahentium sint admodum exiguae
prae distantiis , quibus ipsa corpora disjunguntur , eorum
alterum tendet in alterum perinde ac si essent ambo in
suis gravitatis centris compenetrata ; haec assertio quoad
sphaerica corpora valet utcumque se habeat intercedens
distantia n. 76.
De gravitatione universali. pag. 159.
Ex mutua coelestium corporum gravitatione collata
cum terrestrii gravitate inferimus illud : gravitas ita ma
teriam afficit , ut singulae ejus particulae in alias omnes
et singulas gravitent in ratione directa massarum ad quas
tenditur , et reciproca duplicata distantiarum alterius ab
altera n . 77 , ...82.
.
Aliquid circa solarem et planeticas massas... n.83.10... 4.
Media telluris densitas determinata ex penduli aber
ratione ; itemque experimentis institutis in libra
siouis n. 83. 5. 6.°
tor
Quomodo ex marini aestus phoenomeno deduci pos
sit ratio inter lunarem ac terrestrem massam . n. 83.7 . °
326
Expediuntur quae pertinent ad attractionem corpo-
rum sphaericorum iu punctum materiale
. n. 73,74,75.
Materiale punctum- valde distans a corpore attraheute,
utcumque se habeat forma corporis, ea proxime ratione tendit
in ipsum corpus, qua tenderet si corporis partes in centro
gravitatis compenetrarentur: ubi dimensiones corporum quo-
rumcumque se mutuo attrahentium sint admodum exiguae
prae distantiis, quibus ipsa corpora disiunguntur , eorum
alterum tendet in alterum perinde ac si essent ambo in
suis gravitatis centris compenetrata ; haec assertio quoad
sphaerica corpora valet utcumque se habeat intercedens
distantia ...............n76
De gravitatione universali. pag. 159.
Ex mutua coelestium corporum gravitatione collata
cum terrestrii gravitate.iuferimus illud: gravitas ita'ma-
teriam allicit, ut singulae eius particulae in alias omnes
et singulas graviteut in ratione directa massarum ad quas
tenditur, et reciproca duplicata distantiarum alterius ab
altera . . . . . . . . . ∎∙ ∙ ∙ n. 77,...82.
Aliquid circa solarem et plaueticas massas...n.83.10...4.'
Media telluris densitas determinata ex penduli aber-
ratione : itemque experimentis institutis in libra tor-
Sioni. ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙∙ ∙ ∙ n. 830 5-0 S..
Quomodo ex marini aestus phoenomeuo deduci pos-
sit ratio inter lunarem ac terrestrem massam. n. 83. 7!327
Aliquid notatur de motu punctorum materialium
utcumque inter se connexorum pag. 169.
Formulae spectantes et ad translativum punctorum
motum juxta coordinatos axes, et ad rotatilem eorum mo
tum circum axes ipsos n. 84.
Moto punctorum systemate, perinde movebitur com
mune gravitatis centrum ac si , coeuntibus punctis in
ipsum centrum , applicarentur centro eaedem vires cum
iisdem directionibus , quibus pancta sollicitantur. n. 84.1.6
Principium de conservatione centri gravitatis : item
de conservatione arearum : necnon de conservatione vi
rium vivarum n. 84. 2.0 ... 5 ..
Relativus rigidi liberique systematis motus quoad
gravitatis centrum n. 84. 6. ° 7.0
Motus rigidi systematis circa axem fixum ; quibus
cuinque caeteroquin viribus acceleratricibus sollicitetur sy
stema : quid si vires acceleratrices consistant in sola gra
vitate ; huc spectat theoria penduli compositi : longitudo
penduli simplicis , quod suas perficit oscillationes eodem
tempore ac pendulum compositum : quid si nullae sint
vires acceleratrices : inertiae momenta quoad axem
principales systematis axes : principalia inertiae momen
n. 85. 1.° 2.° ... 7.0
.
ta .
De fluidorum corporum aequilibrio pag. 182.
Ex perfecta mobilitate , qua ponuntur gaudere flui
dorum corporum particulae , ostenditur principium de
aequalitate pressionis , atque inde eruuntur conditiones
requisitae ad aequilibrium cujusvis massae fluidae. n. 86.
327
Aliquid notatur de motu punctorum materialium
utcumque inter se connexorum pag. 169.
Formulae spectantes et ad translativum punctorum
motum iuxta coordinatos axes, et ad rotatilem eorum mo-
tum circum axes ipsos . . . . . . . . . n. 84.
Moto punctorum, systemate, perinde movebitur com-
mune gravitatis centrum ac si
, coeuntibus punctis in
ipsum centrum, applicarentur centro eaedem vires cum
iisdem directionibus , quibus puncta sollicitantur. n. 84.1.'
Principium de conservatione centri gravitatis: item
de conservatione arearum : necnon de conservatione vi-
rium vivarum . . . .
'. ⋅∙ ∙ ∙ n. 84. Z.&quot;...Sæ
Belativus rigidi liberique systematis motus quoad
gravitatis centrum
. . . . . . . . n.84. 6.&quot; 79
Motus rigidi systematis circa axem fixum .: quibus-
cumque caeteroquin viribus acceleratricibus sollicitetur sy-
stema :quid si vires acceleratrices consistant in sola gra-
vitate; huc spectat theoria penduli compositi: longitudo
penduli simplicis
, quod suas perficit oscillationes eodem
tempore ac pendulum compositum .: quid si nullae sint
vires acceleratrices : inertiae momenta quoad axem :
principales systematis axes : principalia inertiae momen-
ta.
. . . . . . . . . . n.85.1.0 Z.&quot;... 7.&quot;
De fluidorum corporum aequilibrio pag. 182.
Ex perfecta 'mobilitate
. qua ponuntur gaudere Hui-
dorum corporum particulae ,, ostenditur principium de
aequalitate pressionis , atque inde eruuntur conditiones
requisitae ad aequilibrium cujusvis massae Huidae. n. 86.328
Quid notandum circa superficiem massae fluidae li
bratae n. 87, 1. ° 2.° ... 5 .
Quid circa fluidum elasticitate pollens, ni 87. 6.0 7 .
De gravium homogeneorumque liquidorum
aequilibrio. pag. 187.
Liquida constituta in aequilibrio intra vasa : pres.
siones in areas sive horizontaliter , sive oblique demer
sas : centrum pressionis .
n. 98. 1. ° . , . 4.0
Solida liquidis immersa : aequilibrii positiones quoad
solidum liquido insidens n. 88.5 . , 89. 1.° 2. ° 3.°
Utrum aequilibrium sit stabile , nec ne. n. 90.
.
De gravium liquidorum aequilibrio in vasis
communicantibus.
pag. 195.
Quid si vasis communicantibus idem contineatur li
quidum : explicatio variorum effectuum ; antliae adspi
ranles , etc n. 91 , 92. 1.° 2.°
Quid si diversa contineantur liquida.
. n. 92. 3.0
De gravium elasticorumque fluidorum aequilibrio nec
non de altitudinibus dimetiendis ope barometri, et de
pondere ac densitate vaporum . pag. 199.
Conditio aequilibrii expressa per aequationem dif
ferentialem : perficitur integratio in hypothesi temperiei
constantis n. 93.
Inde eruitur formula inserviens ad altitudines di
328
Quid notandum circa superüciemi massae liuidae li-
bratae
. . . . . . . . . n. 87.1.02.&quot;...5.0
Quid circa fluidum elasticitate pollens. n: 87. 6.&quot; 7.&quot;
-De gravium homogeneorumque liquidarum
aequilibrio. pag. 187.
Liquida constituta in aequilibrio intra vasa: pres- ⋅
tiones in areas sive horizontaliter , sive oblique demer-
sas: centrum pressionis . . . . . n. 88. 1.&quot;
..,.
4.&quot;
∙
!'
Solida liquidis immersa : aequilibrii positiones quoad
solidum liquido insidens . . n. 88.5.&quot;, 89. 1.&quot; 2.0 3.&quot;
Utrum aequilibrium sit stabile, nec ne. . . n. 90.
De gravium liquidarum aequilibrio in 'vasis
communicantibus. pag. 195.
Quid si vasis communicantibus idem continaptur li-
quidum: explicatio variorum effectuum : antliae adspi-
TODIBBQ etc ∙ ∙ ∙ ∙ ∙
↼
∙ ∙ a ∙ &quot;.
5 91. 92. 1.02.0
Quid si diversa cbntiueantur liquida. . . n. 92. 3.&quot;
De gravium elasticorumque fluidorum aequilibrio nec
non de altitudinibus dimetiendis ope barometri, et de
pondere ac densitate. naporum. pag. 199.
Conditio aequilibrii expressa per aequationem dif-
ferentialem : perficitur integratio in hypothesi temperiei
constantis . . . .
.'&quot;.
. . ⋅ n. 93.
Inde eruitur formula inservieus ad altitudines di-329
metiendas ope barometri : varia observantur pro commo
diori formulae usu n. 94. 1. ° 2.° ... 6.•
Verticalis ascensus globi aereostatici : maxima glo
bi elatio . n. 95.
Maxima quantitas vaporis sese evolventis in vase un
dique clauso : vis elastica sicci aeris aucta ob evolu
tum vaporem : ratio inter densitatem aquei vaporis ac
densitatem sicci aeris sub eadem temperie eademque pres
sione : ratio inter eorum densitates ac pondera sub ea
dem temperie et diversis pressionibus : densitas aeris va
porosi librantis datam pressionem sub temperie data. n. 96.1 . °
2 .
Usus aquei vaporis in movendis machinis. n . 99.6. •
De aqua egrediente per angustum foramen
e vasis verticalibus sive cylindricis,
sive prismaticis. pag. 206 .
Nonnulla praemittuntur ex pluries iteratis experimen
tis .
n . 97.
Quaenam velocitas aquae egredientis: tempus impen
sum in descensu usque ad quamlibet altitudinem datam . n.98.
Quantitas aquae dato tempore egredientis : tempus
quo vas totum evacualur n. 99, 100,
Ratio inter tempora , quibus deplentur duo vasa ha
bentia et altitudines et orificia aequalia : quantitales aqua
rum successivis ' et aequalibus temporibus ex vasis ori
ficio efluentium : divisio vasorum in partes successivis dati
temporis unitatibus vacuandas n. 101 , 102.
22
' 329
metiendus ope barometri : varia observantur pro commo-
diori formulae usu . . . . . n. 94. 1..) .&quot;
... 6.&quot;
Verticalis ascensus globi aereostatici : maxima glo-
bi elatio. ∙ ∙ ∙ ∙ ∙ ' ∙ ∙ ∙ ∙ ∙ ∙
∙∎∎ ∙ n- 950
-Maxima quantitas vaporis sese evolventis in vase uu-
dique clauso : vis elastica sicci aeris aucta&quot; ob evolu-
tum vaporem : ratio inter densitatem aquei vaporis ac
densitatem sicci aeris sub eadem temperie eademque pres-
sione: ratio inter eorum densitates ac pondera sub ea-
dem temperie' et diva-sis- pressionibus: densitas aeris va-
porosi librantis datam pressionem sub temperie data. n. 961.&quot;
2.0
...
5.&quot;
Usus aquei vaporis in movendis machinis. n. 99. 6.&quot;
De aqua egrediente per angustum foramen
e vasis «verticalibus sive cylindricis,
sive prismaticis. pag. 206.
Nonnulla praemittuntur ex pluries iteratis experimen-
tis ∙ ∙ ∙ ∙ ∙ ⋅∙⋅ ∙ ∙ . ∙ ∙ ∙ ∙∎∎ ∙ ∙ ∙ &quot;o 970
Quaenam velocitas aquae egredientis: tempus impen-
sum in descensu usque ad quamlibet altitudinem datam. n.98.
,. Quantitas aquae dato tempore egredientis: tempus
quo vas tatum evacuatur . . . . . . n. 99,100.
Ratio inter tempora, quibus deplentur duo vasa ha-
bentia et altitudines 'et oriiicia aequalia : quantitates aqua-
rum successivis' et aequalibus temporibus ex vasis ori-
iicio efluentium: divisio vasorum in partes successivis dati
temporis unitatibus vacuandas . . . ∙∙ n. 101, 102.
22330
Contractio venae aqueae n. 103.
Ubinam perficiatur acceleratio , per quam velocitas
aquae descendentis admodum exigua mutatur in finalem
satisque grandem effluxus velocitatem . n . 104. 1.0
Quomodo motus aquae defluentis in regularibus al
veis traduci possit ad motum aquae prosilientis ex an
gustis vasorum orificiis n. 104. 2.• , ..5.
Illud cum Auctoribus non paucis assumitur tanquam
principium , quod nempe unumquodque liquidi in vase
quolibet descendentis tenuissimum et horizontale stra
tum coalescat iisdem constanter particulis communi , ea
que tantum verticali , velocitale donatis ; inde vero eruun
tur , quae pertinent ad ipsius liquidi motum n . 105.
Aliquid subjungiur circa generalem theoriam
motus corporum fluidorum . pag. 216.
Aequationes ad istiusmodi motum et quum massa
fluida homogenea vel heterogenea est incapax compressio
nis, et quum massa fluida pollet elasticitate. n. 106,107,108.
De tubis capillaribus. pag. 225.
sol Vires ex materia tubi , et ex materia liquidi ,
licitantes datam ipsius liquidi particulam : attentis viri
bus istis , suprema liquidi superficies vel induet curvam
concavamque figuram , vel curvam convexamque , vel ma
nebit, plana atque horizontalis, n. 109,1.9
330
Contractio venae aqueae . . . . -. . n. 103.
Ubinam perficiatur acceleratio, per quam velocitas
aquae descendentis admodum exigua mutatur in finalem
satisque grandem effluxus velocitatem. .
-
.n. 104. 1.&quot;
Quomodo motus aquae defluentis in regularibus al-
veis traduci possit ad motum aquae prosilientis ex au-
gustis vasorum orificiis
. . . . . n. 104. Z.&quot;. ..5.&quot;
Illud cum Auctoribus non paucis assumitur tanquam
principium, quod nempe unumquodque liquidi in vase
quolibet descendentis tenuissimum et horizontale stra-
tum coalescat iisdem constanter particulis communi , ea-
que tantum verticali, velocitate donatis :
inde vero eruun-
tur, qnae pertinent ad ipsius liquidi motum
. ∙⋅ n. 105-
Aliquid subjungiur circa generalem theoriam
motus corporum fluidorum. pag. 215-
Aequationes ad istiusmodi motum et quum massa
fluida homogenea vel heterogenea est incapax compressio-
nis, et quum massa fluida pollet elasticitate. n. 106,107,108.
De tubis capillaribus. pag. 225.
Vires ex materia tubi
,
et ex materia liquidi .
sol-
licitantes datam ipsius liquidi particulam: attentis viri-
bus istis , suprema liquidi superficies vel induet curvam
concavamque figuram , vel curvam couvexamque , vel ma-
nebit, plana atque horizontalis, ,. . . .. . n. 109.1.&quot;331
Quam attractionem exerceat massa liquida , cujus su
prema superficies est plana , in columellam liquidam per
pendiculariter illi superficiei planae insistentem . n. 109.2 .
Quam attractionem exerceat massa liquida , cujus su.
prema superficies est vel concavo -sphaerica vel convexo
sphaerica , in columellam liquidam perpendiculariter in
sistentem plano tangenti , dactó vel per punctum infimum
superficiei concavo-sphaericae vel per supremum superfi
ciei convexo -sphaericae. n. 109. 3.° 4.0 ... 70
Quid si massa liquida terminetur superficie concaya
vel convexa , quae non sit sphaerica. n. 109. 8.° ... 11.º
His declaratis , explicamus ascensum descensumque
liquorum in lubis capillaribus n. 110,
.
Nonnalla subjunguntur , quorum ratio desumitur ab
actione capillari . n. 111. 1.° 2.° ... 5 ° , 112
)
ACUSTICAE PRINCIPIA
Notiones praeambulae.
1
pag . 245.
Corpora, quae sonora dicuntur tunc sonum exci
tant quando ita agitantur , ut illorum partes tremulo ac
vibratorio satisque rapido concutiantur motu ; qui motus
communicatus aeri ambienti , et late diffusus afficit orga
nym auditus: vis acceleratrix in vibrante particula resonan
tis corporis.
. n . 113. 10. 20.
331
Quam attractiduem exerceat massa liquida , cuius su-
prema superficies est plana ,
in columellam liquidam per-
pendiculariter illi superficiei planae insistentem. n. 1092.&quot;
Quam attractionem exerceat massa liquida , cuius su-
prema- superficies est vel concavo-sphaerica vel convexo-
sphaerica, in columellam liquidam peu-pendiculariter in-
sistentem plano tangenti , dnctö vel per punctum infimum
superficiei concavo-sphaericae vel per supremum superfi-
ciei convexo-sphaericae. . . . n. 109. 3.&quot; 4.&quot;
.. . 7.&quot;
Quid si massa liquida terminetur superficie concava
vel convexa, quae non sit sphaerica. n. 109. 8.&quot;. ..
11.&quot;
His declaratis , explicamus ascensum descensumque
liquorum iu .tubis capillaribus . . . . . . n. 110.
Nonnulla subjunguntur ∙ quorum -ratio desumitur ab
actione capillari. . . . . n. 111. 1.&quot; 2.&quot;
. . . 5&quot;,112
AOUSTIGAE
W
PRINCIPIA
Notiones praeambulae. ∣
pag. 245.
Corpora, quae sonora dicuntur
,
tunc sonum exci-
tant quando ita agitantur, ut illorum partes tremulo ac
vibratorio satisque rapido concutiuntur motu; qui motus
communicatus aeri ambienti, et late diffusus afficit orga-
num auditus: vis acceleratrix in vibrante particula resonan-
tis corporis. . . . . . . '. . . . n. 113.1&quot;. 2&quot;.332
Progignitur quoque sonus ab aere vehementer compres
so , seseque statim restituente , n. 114.
.
Soni reflexio; inde echo. n . 115 .
Non solus aer est medium ideoneum transmissioni
sonorum. n. 116.
De intensitate soni; deque ejus gravitate,
et acutie .
pag. 248.
Sonus intensior ex eo gignitur quod in sonoro cor
pore plures ejusdem partes simul oscillant, et majus spa
tium singulis oscillationibus dato tempusculo percurrunt;
atque ita in aere ex numero item et majori oscillatione
partium aeris intensitas soni dependet ; remissior autem
sonus ex opposito.
n. 117.
Nonnulla explicantur circa soni intensitatem. n . 118.
ex Soni gravis et acuti discrimen repetendum est
numero vibrationum in partibus sonori corporis, ita ut
sonus gravior oriatur ex minus frequentibus vibrationibus so
nori corporis, ex crebrioribus contra sonus acutus ; idem
que de oscillationibus aeris in sono derivato. n. 119.
Quid consonantia , et quid dissonantia: varii conso
nantiae gradus: theoria chordaram vibrantium in hypothe
si vibrationum admodum exiguarum. n. 120. 1 ” 2 ”... 7 .
Varia proponuntur explicanda circa chordas vibran
tes . n. 121 .
332
Progignitur quoque sonus ab aere vdhemeuter compres-
so, seseque statim restituente. . . . . . . n. 114.
Soni reflexio; inde echo. . . . . . . n. 115.
Non solus aer est medium ideoneum transmissioni
SODOmm. ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ &quot;. 116.
De intensitate soni; deque eius gravitate,
et acutie. pag. 248.
Sonu's intensior ex eo gignitur quod in sonoro cor-
pore plures eiusdem partes simul oscillaut, et maius spa-
tium singulis oscillationibus dato tempusculo percurrunt:
atque ita in aere ex numero item et maiori oscillatione
partium aeris intensitas soni dependet; remissior autem
sonus ex opposito. . . . . . . . . . ,n. 117.
Nonnulla explicantur circa soni intensitatem.
. .n. 118.
Soni' gravis et acuti discrimen repetendum est ex
numero vibratiouum in partibus sonori corporis, ita ut
sonus gravior oriatur ex minus frequentibus vibrationibus so-
nori corporis, ex crebrioribus contra sonus acutus,- idem-
que de oscillationibus aeris in sono derivato. . n. 119.
Quid consonantia, et quid dissonantia: varii conso-
nantiae gradus: theoria chordarum vibrantium in hypothe-
si vibrationum admodum exiguarum. n- 120. 1&quot; 2&quot;... 7&quot;.
Varia proponuntur explicanda circa chordas vibran-
tes. ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙
Q &quot;. 1210333
Quomodo sonus trans obicem possit communicari ita,
ut tonus proprius sonori corporis permaneat. n. 122.
Unde asperitas aut lenitas soni proficiscatur. n. 123.
Transversae et longitudinales chordarum vibratio
nes: nodi in chordis vibrantibus: lineae nodales in super
ficiebus corporum resonantium : vibrationes laminarum ri
gidarum .
n. 124 .
De directa soni propagatione per aerem . pag. 265.
In iisdem circumstantiis sonus aequabili velocitate
in toto decursu devehitur; omnesque soni sive intensi ,
sive remissi, sive graves, sive acuti eadem velocitate dif
funduntur: qua ratione intensitas soni minuatur in pro
gressu . n. 125,
Undae sonorae constitutio. n. 126,
Soni et velocitas, et intensitas augetur a vento se
cundo, minuitur ab adverso . n . 127.
Experimenta instituta ad soni velocitatem determi
nandam; quae tamen experimenta non satis conveniunt :
hujus diversitatis rationes : quaenam utilitas ex determi
natione velocitatis qua propagatur sonus. .
n. 128
Generalis de fluidorum motu theoria applicatur ad
soni propagationem : soni velocitas eruta ex applicatione
theoriae comparatur cum velocitate quam praebent ex
perimenta .
n. 129. 10. 2º. 3º.
Crassities aerei strati, in quo particulae cientur una :
si impulsio in obicem facta quadrato velocitatis sumitur
-
22&quot;
333
Quqmodo sonus trans obicem possit communicari ita,
ut tonus proprius sonori corporis permaneat.
. n. 122.
Unde asperitas aut leuitas soni proficiscatur. n. 123.
Trausversae et longitudinales chordarum vibratio,-
nes: nodi in chordis vibrantibns: lineae nodales in super-'-
iiciebus corporum resonantium: vibrationes laminarum ri-
gidarum...........;..n.124.
De directa soni propagatione per aerem. pag. 265.
In iisdem circumstantiis sonus aequabili velocitate
in toto decursu devehitur; omnesque soni sive intensi ,
sive remissi, sive graves, sive acuti eadem velocitate dif-
fuuduntur: qua ratione intensitas soni minuatur in pro-
gressu..............-n.125.
Undae sonorae constitutio. . . . . . . n.126,
Soni et velocitas, et intensitas augetur a vento se-
eundi), mall!!! EI) adverw. ∙ ∙ ∙ ∙ ∙ ∙ n- 127.
Experimenta instituta ad soni velocitatem determi-
nandam; quae tamen experimenta non satis conveniunt:
hujus diversitatis rationes: quaenam utilitas ex determi-
natione velocitatis qua prOpagatur sonus. . . n. 128.
Generalis de fluidorum motu theoria applicatur ad
soni propagationem: soni velocitas eruta ex applicatione
theoriae comparatur cum velocitate quam praebent ex-
perimenta . . . . . . . . . . n.129.1o.2&quot;.3&quot;.
Crassities aerei strati, in quo particulae cientur uua:
si impulsio iu obicem facta quadrato velocitatis sumitur
22'334
proportionalis, rationem duplicatam distantiarum .sequetur
soni debilitatio.
n. 129. 4. 5 .
Cur pluribus corporibus simul resonantibus , inter
oscillationes in aere excitatas non habeatur confusio , omnes
que diversi soni inde orti ad aures distincte perveniant:
huc spectat principium de superpositione exiguorum mo
tuum. n. 129. 6.
Propagatio soni in cubis cylindricis indefinitae lon
gitudinis. n. 129.7 .
J
De reflexa soni propagatione per aerem pag. 289.
Cum in directa propagatione sonorus aer . offendit o
bicem aptum , reflectitur : varia ad echo spectantia ex
plicantur. n. 130.
Reflexio soni fit ad angulos incidentiae et reflexionis
aequales; regrediturque sonus eadem velocitate qua incedebat
antequam in obicem impingeret. n . 131 , 132, 1º. 2º
De instrumentis pneumaticis. pag. 294.
In instrumentis pneumaticis soni genesis repetenda non
est saltem praecipue ex oscillatione partium solidarum i
psius instrumenti : quo pacto sit explicanda : aer secun
dum fistulae longitudinem se habet instar chordae peragen
tis longitudinales vibrationes: etsi ex materia instrumenti
non habetur varietas quoad soni qualitatem, aut valde uota
bilem intensitatem ; varietas tamen habetur quoad meliorem
334
proportionalis, rationem duplicatam distantiarum .sequetur
soni dehilitatio. . . . . . . . ∙ ∙ n.129.40.50.
Cur pluribus corporibus simul resonantibns ,
inter
oscillationes in aere excitatas non habeatur confusio,omues-
que diversi soni inde orti ad aures distincte perveniant:
huc spectat principium de superpositione exiguorum mo-
tuum. ∙⋅∙ '.
. . . . . . . . . . n.129.6&quot;.
Propagatioi soni in. tubis cylindricis indefinitae lou-
gitudinisa, ∙ ∙ ∙ . . . . . .. . . n.129.7&quot;.
]
De refleæa soni propagatione per aerem pag. 289.
!. ∙ . .
Cum indirecta prOpagatione sonorus aer .oii'eudit o-
bicem aptum, reflectitur : varia ad echo spectantia ex-
Plimnturo. ∙∙ ∙∙ ∙∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ n. 1300
. Reflexio soni iit ad angulos incidentiae et reflexionis
aequales: regrediturque sonus eadem velocitate qua incedebat
antequam in obicem impingeret. . . .&quot;' 131, 132.1&quot;.2&quot;.
'pul '
De instrumentis pneumaticis. pag. 294.
In instrumentis pneumaticis soni genesis repetenda non
est saltem praecipue ex oscillatione partium solidarum i-
psius instrumenti: quo pacto sit explicanda: aer secun-
dum fistulae longitudinem se habet instar chordae peragen-
tis lougitudinales vibrationes: etsi ex materia instrumenti
non habetur varietas quoad soni qualitatem, aut valde uota-
bilem intensitatem; varietas tamen habetur quoad meliorem335
aliquam resonantiam : quid si intrumentum pneumaticum sit
compactum ex materia non resistente , quale v. g. esset in
strumentum membranaceum .. . n. 133.
Nonnulla proponuntur explicanda circa instrumenta
pneumatica. n . 134.
Tremulus aeris motus in tubis cylindricis determinatae
longitudinis :
1º. Quum tubus est firmiter obseratus apud alterum
orificium simulque apertus apud alterum n. 135, 136.
2°. Quum tubus est patens in utraque extremitate: in
de eruitur ratio investigandi velocitatem , qua propagatur
sonus in aliis fluidis elasticis diversis ab aere. n. 137.
3 °. Quum tubus est utrinque obseratus. n. 138.
De propagatione soni per liquida, et per solida
corpora . pag. 302.
Formulae huc spectantes: parvula contractio aquae et
hydrargiri ob auctam pressionem: usus istius contractionis in
determinanda velocitate soni per haec duo liquida. n . 139,140 .
Analogia inter oscillationes aeris in tubo cylindrico a
pud ambas extremitates aperto et longitudinales oscillationes
virgae rigidae suppeditat peculiarem methodum investigandi
velocitatem propagationis per solida corpora. n. 141 .
De vocis humanae origine. pag. 305.
Nonnulla ex anatomicis praemittuntur; quibus praemis
sis , stabilitur illud : vocis humanae organum etsi conside
rari maxime debeat tanquam instrumentum pneumaticum
335
aliquam resonantiam: quid si intrumentnm pneumaticum sit
compactum ex materia nou'reaistente, quale v.. g. esset in-
strumentum membranaceum. .. .. .. .. .. .. . n. 133.
Nonnulla proponuntur explicanda circa instrumenta
pneumatica. .. . .. .. .. .. .. .. .. .. .. ..
.n. 134.
Tremulus aeris motus'iu tubis cylindricis determinatae
longitudinis :
⇝
↿∘∙ ⊄⊇⇂⋯⋯∙⋯∣⋯∘⋅⊖⊱⇂ firmiter ohseratus apud alternm
orificium simulque apertus apud alterum . n. 135,136.
20. Quum tuhus est patens in utraque extremitate: in-
de eruitur ratio investigandi velocitatem
, qua propagatur
sonus in aliis fluidis elasticis diversis ab aere. n. 137.
(
3&quot;. Quum tubus est utrinque ohseratus. . n. 138.
i '
⋅
⋅ ↼
De prapagau'one soni per liquida, ettper &quot;solida ⊳∣ corpora. pag. 302.
.Fornrnlae huc spectantes: parvula contractio ailuae et
hydrargiri ob auctam pressionem: ususistius contractionis in
determinanda velocitate soni per haec duo liquida.ia.139, 140.
Analogia inter oscillationes aeris in tuho cylindrico a-
pud ambas extremitates aperto et lougitudinales oscillationes
virgae rigidae suppeditat peculiarem methodum investigandi
velocitatem propagatiouis per solida corpora. . n. 141.
De 'vocis humanae origine. pag. 305.
Nonuulla ex anatomicis praemittuntur; quibus praemis-
sis, stahilitur illud: vocis humanae organum etsi conside-
rari maxime debeat tanquam instrumentum pneumaticum
∩336
flexili et elastica materia ex parte compactum , non tamen ita
est ut cum instrumentis fidicularibus aliquam non habeat
analogiam . n. 142.
Quid, os atque ejus partes conferant ad formationem
vocis.
n. 143.
Variae refellantur sententiae de humanae vocis ori
gine; variaeque circa vocem humanam proponuntur quae
stiones. n . 144 , 145 .

De auditus organo . pag. 310.
Auris descriptio.
n. 146.
Quaenam ex auris partibus pro praecipuo atque im
mediato auditionis organo statuenda sit. n. 147.
Cur quibusdam grata, aliis pene nihil, aut etiam mo lesta sit harmonia ,
n. 148. 19.
Cur daabus auribus unus idemque sonus audiatur n.148.2 °.
1
336 '
Bexiliot'elgstica materia ex parte compactum, non tamen ita
eat ut cum, instrumentis iidicularibus aliquam non habeat
malogihmoo-o-0 ∙⋅∙∙∙⋅∙ ∙ ∙ ∙ ∙ ∙ ∙
&quot;0142.
Quid, os atque eius partes conferant ad formationem
'owa ∙∙∙ ∙ ∙ ∙ ∙ ∙ ∙∙∙∙∙ ∙ ∙
&quot;.1430
Variae refelluntur sententiae de humanae vocis ori-
gine, variaeque circa vocem humanam proponuntur quae-
'none'- ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ &quot;- 14401450
De auditus organo. pag. 310.
Auriadeacriptio. . . . . . . . . . n.146.
Quaenam ex auris partibus pro praecipuo atque im-
mediato auditionia organo statuenda- sit.
. . . n. 147.
Cur quibusdam grata, aliis pene nihil, aut etiam mo-
lestasit harmonia. ∙ ∙ ∙
. . . . . . n. 148. ∎∘∙
Cur duabus auribus unus idemque sonus audiatur n.148.2'.ERRATA CORRIGE
pag. lin .
1. 4. saepae
1. 5. decresit
4. 28. istanti
5. 13. rive
7. 29. poductis
8. 14. sin a
14 29. AH'.BC
15. 3. AF
saepe
decrescit
instanti .
siye .
prodactis
sin a .
AH'.BC
BF'
BF .
Y.
4. AF
24, 7. Sy
50. 6 et 7.
S * S

*

17. ſsfla)dx Sfaxdx.
52. 14.
f (x )dx f '( x )d.x2
2
2
56. 18.- ( tdx ,z + dz,u,...) -f(xtdx , z + de, u, ...).
dull .
Sfx )dx 22. ( x) dx
eck
58. 1.-C
+0 .
57. 4. del
1
Sfaydar
62. 3. W
v'dz'
11. dzi
63. 8. sint va
69. 12. quod
... 17.
v'du'
da
sintVC .
quoad
ngt 2gt
70. 7.- 7 . kalog(k2—12)
.
.log(k ?-- ).
1
-
1
!
ERBATA CORRIGE
pag. lin.
&quot;. 4. saepae saepe .
1. 5. decresit decrescit
4. 28. istanti instanti
.
5. 13. rive sive
.
7. 29. poductis productis .
8. 14. aina sin at
.
14 29. AH'.BC AH'.BC'
.
15. 3. AF' BF' .
⋅
∙ ∙ ∙ 4. AF BF
. 24, 7. &lt;nowiki&gt;:] z?&lt;/nowiki&gt;
.
50. 6et 7. f:&quot; f:&quot;
... 17. JfftæMx [f(xkiæ .
52. 14. figit&quot; f'(æ2)dæ* .
56. 18.—(æ-]-dæ,z-l—dz,u,...) --f(a—-[-dx , z—l-dz, u, -
.
)- - 57. 4. d,,p. dup.
.
.. .
22. 111-2635 f(ældæ
.
803 803
58. 1.-:.-C —]—G .
62. 3. 9) p
↿↿∙∙∙ v'dz' til—tf
.
d:, dz'
.
63. 8. siun/C sint;/C .
69. 12. quod ' quoad .
⋅ n : 2 c
...
17. ∘−⋚∟ ∉−≓∙∙
k: 70. 7. −∙∙ 2 2
∣⊂≄∣∘⊰≼∣∁≖−⋁≖⋟ −−⋅⊋−⋅ lOg(k —P )
--
. ∙∙∽∙∙⋅ −∙− ↼∙ -
∙−⊣ERRATA CORRIGE
pag. lin .
kdv Ka dy
70. 12 .
katus kype &quot;
71. 13 et 14. KC Kc
72. 23. u = ułgosinc u = a + g9 sinc .
75. 23. pressioni r.gMcosc' pressioni gMcosc' .
87. 2. Denotet enim a Denotet enim x .
IG &quot; IG &quot;
27. = IC &quot; : IC =
2 2
110. 9. R = Rcosa R = R , cosa
111. 5. 1880 to 288q'to .
da dala 146. 8 .
idt
148. 12. 69.º* 69. *
149. 6. x = A'B' - B'r - A'B ' - A'M x=A''B'-B'r=A'B'-A'M .''
x' ? c
x
151. 2 .
ic
(de)
Centre Ide
i
152. 78 et 20. r2
153. 22. (69)
154. 17.
72.°*
157. 8. SD
161. 26. 16931100
193. 23. u : M ' : fle ..
205. 7. aequeus
208. 14. aia
r2
( 70 ) .
72.*
GD .
19631100 .
Me : No :
aqueus ,
Q:.
i
3
ERRATA CORRIGE
∙∙∙ ∙∙∙∄≾≖∠≀⇂↗ *
kæ-I—uz ⋅
Kc
uza-l—gg sinc .
23. pressioni ngMcosc' pressioni gMcosc' .
pag. lin
70. 12:
liti—v- kZ.-v2
71. 13 et 14. KC
72. 23. uzu-l—gasinc '
75.
87. 2. Denotet enim a
∙⋅≆↴ IG&quot; ∙ IG&quot;
. 27.:IC :::—2— . 10: -—2-— .
110. 9- R::Rcosa: BzB, cos a .
111. 5. 1889'—]—-p' ∙ 28897'—-q)' .
' doc 146. 8. ; daz :
(2? (22?
148. 12. 6994! 6931:
149. 6. a::A&quot;B'-B'r-A&quot;B'-A'M sz&quot;B'-B'r:-A&quot;B'-A'M .
.... .. ⋅↕⋅≟≣∁ ⋅
...-7...
50 ac
152. 78 et20. fi ∙∘−⋮⋅⋅∙ ra .rz 153. 22. (69) (70)
.
154. 17. 7291: 724
157. 8. SD .GD .
161. 26. 16931100 - 19631100 .
193.23. p.':p.':p.., php.
205. 7. aequeus aqueus ,
208.1.£. a:a' «:a' .
Denotet enim æ .</text>
      <sha1>fjp62i4cgw4adm7piacq4u2zd516ssi</sha1>
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  </page>
  <page>
    <title>Navis glacifraga</title>
    <ns>0</ns>
    <id>300040</id>
    <revision>
      <id>3697745</id>
      <parentid>3654553</parentid>
      <timestamp>2022-08-17T10:14:22Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Naves]]; +[[Categoria:Genera navium]]</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2023" xml:space="preserve">[[Fasciculus:Polarstern.gif|thumb|''Polarstern'' (Latine ''Stella Polaris''), navis glacifraga Germanica per mare it]]
'''Navis glacifraga'''&lt;ref&gt;{{DAEL|Rompehielos|954}}; [[Reijo Pitkäranta]], ''Lexicon Finnico-Latino-Finnicum''. WSOY, 2001, s.v. ''jäänmurtaja''&lt;br&gt;Vel brevius '''glacifraga'''.&lt;/ref&gt; est [[navis]], quae per flumina vel maria [[glacies|glacie]] stricta [[navigatio|navigare]] potest, glaciem rumpens. Sive ipsa homines vel onera vehit, sive alias naves post se ducit, aquam liberam eis praebens. 

[[Fasciculus:Walking icebreaker on the Moscow river.jpg|thumb|Navicula glacifraga ambulatoria per [[Moscus|Moscum]] flumen it]]
Ab [[Medio aevo|Aevo medio]] notae sunt naves, quas homines vel equi per flumen vel canalim trahebant, ut iis glaciem frangerent; harum navium puppes gravide onerabantur, ut prora super glaciem efferretur et navis tracta suo pondere glaciem perrumperent. [[Machina vaporaria]] inventa [[navis vaporaria|naves vaporariae]] glacifragae construebantur, sed illae haud per omnem glaciem navigare poterant. Anno 1959 ''[[Lenin (navis glacifraga)|Lenin]]'', prima navis glacifraga cum [[Reactorium nucleare|reactorio nucleari]] ad glaciem [[Oceanus Glacialis|Oceani Glacialis]] in [[cursus marinus Borealis|cursu marino Boreali]] frangendam constructa est. Secunda navis glacifraga nuclearis, ''[[Arctica (navis glacifraga)|Arctica]]'', anno 1974 per [[Polus septentrionalis|Polum septentrionalem]] navigavit. Nunc diversae naves glacifragae construuntur, a maximis oceanicis usque ad parvas ambulatorias.

==Notae==
&lt;references /&gt;

==Nexus externi==

* [https://www.britannica.com/technology/icebreaker ''Navis glacifraga''] in ''Britannica'' {{ling|Anglice}}
* [https://snl.no/isbryter ''Navis glacifraga''] in ''Lexico Norvegico magno'' {{ling|Norvegice}}
* [https://bigenc.ru/technology_and_technique/text/2137149 ''Navis glacifraga''] in ''Encyclopaedia Russica magna'' {{ling|Russice}}

[[Categoria:Genera navium|glacifraga]]
[[Categoria:Glacies]]
[[Categoria:Regio Arctica]]</text>
      <sha1>6etvngvb0pmoykvagy8u2552v07q3yl</sha1>
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  </page>
  <page>
    <title>Usor:Demetrius Talpa/Harenarium</title>
    <ns>2</ns>
    <id>300124</id>
    <revision>
      <id>3697680</id>
      <parentid>3697585</parentid>
      <timestamp>2022-08-16T16:27:59Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>/*  */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9777" xml:space="preserve">&lt;small&gt;[[Fasciculus:Versiculi mnemonici.jpg|right|200px]]

''[[Magna charta libertatum|Magnam chartam libertatum]] talparum'' compositurus sum. Haec sunt libertates maximae includendae:
*[[Libertas fodiendi]]
*[[Libertas rodendi]]
*[[Libertas caecuttiendi]]
*[[Libertas dormiendi]]

===Miscellanea===
[[Fasciculus:Anagramma Quid est veritas = Est vir qui adest.gif|right|80px]]
{{coord|55|29|24|N|28|45|35|E|display=title}}
'''Panope''' vel '''Panopeus''' ([[Graece]] Πανόπη, Πανοπεύς) urbs est Graeca&lt;br&gt;

[https://books.google.fr/books?id=5Z1gAAAAcAAJ&amp;pg=PA205&amp;lpg=PA205&amp;dq=gatschinensis&amp;source=bl&amp;ots=gfK_drbDsn&amp;sig=ACfU3U0y8h5-f5zPeMe__iV70L4heT2OJg&amp;hl=la&amp;sa=X&amp;ved=2ahUKEwjMrfuEiZrxAhVFvIsKHay9AQwQ6AEwA3oECAUQAw#v=onepage&amp;q=gatschinensis&amp;f=false hortus imperialis Gatchinensis] [https://books.google.fr/books?id=lxw3glGfmwQC&amp;pg=PA207&amp;dq=gatschinensis&amp;hl=la&amp;sa=X&amp;ved=2ahUKEwiw5JWCiprxAhViposKHcXzCtgQ6AEwBXoECAcQAg#v=onepage&amp;q=gatschinensis&amp;f=false ] - [https://books.google.fr/books?id=7U5fAAAAcAAJ&amp;pg=PA70&amp;lpg=PA70&amp;dq=gatschinensis&amp;source=bl&amp;ots=gL8Ew-PKnf&amp;sig=ACfU3U3Qt-WgX8rADgVTZFKcn5aotF4oKw&amp;hl=la&amp;sa=X&amp;ved=2ahUKEwjMrfuEiZrxAhVFvIsKHay9AQwQ6AEwAXoECAMQAw#v=onepage&amp;q=gatschinensis&amp;f=false hortus Caesareus Gatchinensis]

[https://plant.depo.msu.ru/public/scan.jpg?pcode=MW0109044 Herbarium Universitatis Mosquensis] [https://books.google.ru/books?id=6YENywEACAAJ&amp;dq=herbarium+universitatis+mosquensis&amp;hl=la&amp;sa=X&amp;ved=2ahUKEwjgm6KB8ZfwAhVOtIsKHV3SDlsQ6AEwAHoECAAQAQ ] - [https://search.rsl.ru/ru/record/01003358157 Herbarium vivum sive Collectio plantarum siccarum Caesareae universitatis Mosquensis]  
[https://books.google.fr/books?id=LQA6AAAAcAAJ&amp;pg=RA2-PA168&amp;dq=silviculturam&amp;hl=la&amp;sa=X&amp;ved=2ahUKEwi45_WXuYz0AhVHEncKHeZNAFYQ6AF6BAgFEAI#v=onepage&amp;q=silviculturam&amp;f=false Silvicultura]

[https://old-smolensk.ru/?p=12670 Smolenscensis obsidionis liberationis obsessorum ac deditorum castrorum...] [https://rusneb.ru/catalog/000200_000018_RU_NLR_DIGIT_75071/ ]

[https://books.google.fr/books?id=uGV3SHOFe8YC ''Relationes status dioecesium in magno ducatu Lituaniae'']

[https://books.google.fr/books?id=fQFfAAAAcAAJ&amp;pg=PA97&amp;dq=zea+et+munichia+sunt&amp;hl=la&amp;sa=X&amp;ved=2ahUKEwjbjqiRv6L0AhWqtIsKHdIHByUQ6AF6BAgLEAI#v=onepage&amp;q=zea%20et%20munichia%20sunt&amp;f=false portus et statio]

[https://www.zvab.com/servlet/SearchResults?an=gotthold%20merten&amp;cm_sp=det-_-bdp-_-author Gotthold Adalbert (sic) Merten] [http://www.richardwolf.de/latein/capellan.htm geboren 1866, war Pfarrer im thüringischen Sonneberg und später Altphilologe am Realgymnasium in Lippstadt. Er starb 1946]


[https://books.google.ru/books?id=1iowAAAAYAAJ&amp;pg=PA388&amp;dq=o+superbe+quid+superbis+tua+superbia+te+superabit&amp;hl=la&amp;sa=X&amp;ved=2ahUKEwjBt8-Fj_v1AhVlkosKHajjCa4Q6AF6BAgFEAI#v=onepage&amp;q=o%20superbe%20quid%20superbis%20tua%20superbia%20te%20superabit&amp;f=false o/be...] — [[Lamspringe]]

[[Usor:,jlzifhsr/Manevichi|Manevichi]]

[https://la.wikipedia.org/w/index.php?title=Specialis:Quaerere&amp;limit=100&amp;offset=0&amp;profile=default&amp;search=insource%3Admgh.de%2Fde&amp;ns0=1 ad MGH nexus corrigendi]

{{Creanda|sv|Erik Väderhatt|Ericus Ventosi Pilei}}&lt;ref&gt;[https://books.google.ru/books?id=uEFSAQAAIAAJ&amp;q=Ventosi+pilei#v=snippet&amp;q=Ventosi%20pilei&amp;f=false ''Saxonis Grammatici Historia Danica'', vol. I, Hauniae 1839].&lt;/ref&gt;
==== ====
&lt;hr&gt;
&lt;small&gt;Insula Robusta&lt;ref&gt;{{ZK|Мошный, о-в}}&lt;/ref&gt; [[Serenscum]]&lt;ref&gt;[[Alexander Guagnini|Alexandri Guagnini]] [https://books.google.fr/books?id=xmG9pSVHMLMC&amp;q=Vorotiniam#v=snippet&amp;q=Vorotiniam&amp;f=false ''Sarmatiae Europeae descriptio'', 1578, p. 6].&lt;/ref&gt; [https://books.google.ru/books?id=bD9fAAAAcAAJ&amp;pg=PA9&amp;dq=portus+victoriae+australia&amp;hl=la&amp;sa=X&amp;ved=2ahUKEwiCvqeMmvr1AhXwo4sKHapqDXMQ6AF6BAgEEAI#v=onepage&amp;q=portus%20victoriae%20australia&amp;f=false Portus Victoriae] 
[https://books.google.ru/books?id=d6pDAQAAIAAJ&amp;pg=RA3-PA64&amp;dq=serrae+nivatae&amp;hl=la&amp;sa=X&amp;ved=2ahUKEwi7se7Hhav2AhUFrYsKHdGgCWwQ6AF6BAgDEAI#v=onepage&amp;q&amp;f=false Serra Nivata] [https://books.google.ru/books?id=o708AAAAYAAJ&amp;q=Cuchullin#v=snippet&amp;q=Cuchullin&amp;f=false Cuchullin] (et in Phingaleide) [https://books.google.ru/books?id=qdE_AAAAcAAJ&amp;q=Cualgnia#v=snippet&amp;q=Cualgnia&amp;f=false Cualgnia]

[https://books.google.ru/books?id=PcFOAAAAcAAJ&amp;pg=PA227&amp;dq=shudrae+sunt&amp;hl=de&amp;sa=X&amp;ved=2ahUKEwjo6_r1-aP5AhVsoosKHWfWCDkQ6AF6BAgHEAI#v=onepage&amp;q=shudrae%20sunt&amp;f=false de tribubus]—[https://books.google.ru/books?id=3ltgAAAAcAAJ&amp;pg=PA172&amp;dq=shudrae+sunt&amp;hl=de&amp;sa=X&amp;ved=2ahUKEwju0-OZ-aP5AhVk_SoKHR0hBlIQ6AF6BAgCEAI#v=onepage&amp;q=shudrae%20sunt&amp;f=false et haec] '''Arena mobilis'''&lt;ref&gt;{{Kirp|бархан}}&lt;/ref&gt; '''Selectio artficialis'''&lt;ref&gt;[[Ieremias Bonomelli|Ieremiae Bonomelli]] [https://books.google.ru/books?id=aL69veFB4usC&amp;q=selectio+artificialis#v=snippet&amp;q=selectio%20artificialis&amp;f=false ''Summa totius theologiae dogmaticae'', vol. 2, Mediolani 1876, p. 282]; {{Creanda|de|Tilman Pesch|Tilman Pesch}} [[Societas Iesu|SI]] [https://books.google.co.jp/books?hl=ru&amp;id=4Ifzp0MVoKMC&amp;q=selectio+artificialis#v=snippet&amp;q=selectio%20artificialis&amp;f=false ''Institutiones philosophiae naturalis secundum principia S. Thomae Aquinatis'', Friburgi Brisgoviae 1880, p. 748].&lt;/ref&gt; '''Pristavus'''&lt;ref&gt;[https://books.google.ru/books?id=Mz1GAAAAcAAJ&amp;q=pristavorum#v=snippet&amp;q=pristavorum&amp;f=false ]&lt;/ref&gt; &lt;br&gt;'''Navis bialveata'''&lt;ref&gt;{{DAEL|Catamarán|199}}&lt;/ref&gt; '''Celox'''&lt;ref&gt;V. ''yacht'' in: {{Morgan}}&lt;/ref&gt; '''Fregatta'''&lt;ref&gt;[[Matthaeus Gotardus Artus Dantiscanus]] ([https://data.cerl.org/thesaurus/cnp01919647 CERL]) [https://books.google.ru/books?id=xVdEAAAAcAAJ&amp;q=fregatta#v=snippet&amp;q=fregatta&amp;f=false ''Americæ pars VIII'', 1599, p. 70].&lt;/ref&gt; &lt;/small&gt;
&lt;hr&gt;&lt;references /&gt;
&lt;hr&gt;

===flumina incerta ===
&lt;div style=&quot;width:100%&quot;&gt;
&lt;div style=&quot;float:left; width:90%&quot;&gt;
&lt;div style=&quot;text-align:center; font-family:Cambria, Georgia, Times, 'Times New Roman', serif; font-size:130%; padding:0.2em 0.6em 0.2em 0.6em; border-right:1px solid #BEBEBE; background:#EDEDED&quot;&gt;&lt;/div&gt;
&lt;/div&gt;
&lt;div style=&quot;float:left; width:10%&quot;&gt;
&lt;div class=&quot;mw-customtoggle-1&quot; style=&quot;cursor:pointer; text-align:center; font-family: Cambria, Georgia, Times, 'Times New Roman', serif; padding:0.2em 0.6em 0.2em 0.6em; border-right:1px solid #BEBEBE; background:white; color:black&quot;&gt;MONSTRATVR&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-1&quot;&gt;
&lt;div class=&quot;toccolours mw-collapsible-content&quot; style=&quot;float:left; width:100%; border:0; background:#FFFFFF&quot;&gt;
&lt;small&gt;{{Creanda|ru|Мейерберг, Августин|Augustinus Meyrberg|Augustini Meyerberg}} [https://books.google.fr/books?hl=ru&amp;id=d7tMAAAAcAAJ&amp;q=Danecam#v=snippet&amp;q=Danecam&amp;f=false ''Iter in Moschoviam'',  p. 385].
{{div col|8}}
*Daneca 
*Zerdikium 
*[[:ru:Бетька|Beitma]]? &lt;ref&gt;''Бетьма/Ведьма?''&lt;/ref&gt;
*Atroba 
*Samara altera
*[https://wikimapia.org/#lang=ru&amp;lat=53.263981&amp;lon=49.465256&amp;z=11&amp;m=h&amp;show=/27545731/ru/Овраг-Аскульский Askula]&lt;ref&gt;[[:ru:Аскулы|Askulae vicus]]&lt;/ref&gt; 
*Komousium 
*Verovium
&lt;!--- Danecam?,  
Чердыка (Zerdikii) (“А ниже града Лаишева (Каз. Губ.) из Камы потекла протока Червык и пала в Волгу”. Книга Больш. Чертежу стр. 148.),  
Бейтмы (Beitmae)?, 
обеих Атроб (Atrobae), 
обеих Самар (Samarae), 
Сызрани (Syrani), 
Камышины (Komousii), 
Увары (Verovii) (Увара — название одного из главных и очень рыболовных устьев Волги в Астраханской Губернии. Эта река, при которой селение Уваринский учуг, впадает в море, к 35 верстах от Астрахани.), ---&gt;
{{div col end}}
&lt;/div&gt;
&lt;/div&gt;
&lt;/small&gt;
&lt;hr&gt;

'''Interpretes Homerici ceteri''': {{Creanda|cs|Antonín Liška (1791)|Antoninus Liška}} [[:s:cs:Soubor:Homer, Antonín Liška - Homérowa Odyssea - 1848.djvu|Bohemice]] — [https://web.archive.org/web/20070302034320/http://www.yunanmitolojisi.net/ilyada-1bolum/ Turcice] — [[:s:eo:Iliado/Kanto_Unua|Abraham Kofman]] [https://anno.onb.ac.at/cgi-content/anno-buch?apm=0&amp;aid=100032&amp;teil=0203&amp;seite=00000003&amp;zoom=1 Esperantice] — {{Creanda|sv|Johan Fredrik Johansson|Ioannes Fredericus Johansson}} Suecice — [[:d:Q52154093|Conradus Droste]] [http://www.biografischportaal.nl/persoon/51370290 Nederlandice] — {{creanda|nl|Jan Hendrik Glazemaker|Ioannes Henricus Glazemaker}} [https://lib.ugent.be/europeana/900000050989?pg=PP7 idem] 
&lt;hr&gt;
====incolis abundant, nomine carent====
{{div col|4}}&lt;small&gt;
*''Nižnevartovsk'' Ripa Vartovensis Inferior
*Balaschicha
*''Šachty'' Fodinae
*''Nižnekamsk'' Utbs Camae Inferioris
*''Dzeržinsk'' Nigrum (de priore ''Tschernoje'')
* Chimkae
* ''Angarsk'' Angaropolis
*''[[Podolsk]]''
* ''Prokop'evsk''
* ''Balakovo''
* ''Ljubercy''
* ''Severodvinsk'' Urbs Duinae Borealis (''in Urbe Duinae Boreali'') 
* ''Novočersassk	'' Circassium Novum
* ''Kamensk-Ural'skij'' Petra Uralensis
*''Èlektrostal' '' Electrichalybs
*''Salavat''
*''Železnodorožnyj'' Ferriviarium
*''Al'met'evsk''
*''Berezniki''
*''Rubcovsk''
*''Kopejsk''
*''Kovrov''
*''Krasnogorsk'' 
*''Chasavjurt''
*''Novomoskovsk'' Mosqua Nova
*''Pervoural'sk''
*''Neftejugansk''
*''Neftekamsk'' Naphticamium ut Solicamium
*''Novočeboksarsk'' Tscheboxari Novi
*''Čerkessk''
*''Orechovo-Zuevo''
*''Batajsk''
*''Ščëlkovo''
*''Nevinnomyssk''
*''Dimitrovgrad'' 
*''Novyj Urengoj''
*''Oktjabr'skij''
*''Domodedovo''
*''Obninsk''
*''Novošachtinsk'' Fodinae Novae
*''Seversk''
*''Puškino''
*''Žukovskij''
*''Kaspijsk'' Caspia
*''Nojabr'sk''
*''Ramenskoe''
*''Ačinsk''
*''Novokujbyševsk''
*Essentuki
*''Artëm''
*''Berdsk''
&lt;/small&gt;{{div col end}}
&lt;hr color=&quot;#CCCCCC&quot;&gt;
&lt;references /&gt;</text>
      <sha1>m6qfgt1kmax0v5k2glmih8e9lsu9uv8</sha1>
    </revision>
    <revision>
      <id>3697710</id>
      <parentid>3697680</parentid>
      <timestamp>2022-08-16T22:44:12Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>/*  */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10360" xml:space="preserve">&lt;small&gt;[[Fasciculus:Versiculi mnemonici.jpg|right|200px]]

''[[Magna charta libertatum|Magnam chartam libertatum]] talparum'' compositurus sum. Haec sunt libertates maximae includendae:
*[[Libertas fodiendi]]
*[[Libertas rodendi]]
*[[Libertas caecuttiendi]]
*[[Libertas dormiendi]]

===Miscellanea===
[[Fasciculus:Anagramma Quid est veritas = Est vir qui adest.gif|right|80px]]
{{coord|55|29|24|N|28|45|35|E|display=title}}
'''Panope''' vel '''Panopeus''' ([[Graece]] Πανόπη, Πανοπεύς) urbs est Graeca&lt;br&gt;

[https://books.google.fr/books?id=5Z1gAAAAcAAJ&amp;pg=PA205&amp;lpg=PA205&amp;dq=gatschinensis&amp;source=bl&amp;ots=gfK_drbDsn&amp;sig=ACfU3U0y8h5-f5zPeMe__iV70L4heT2OJg&amp;hl=la&amp;sa=X&amp;ved=2ahUKEwjMrfuEiZrxAhVFvIsKHay9AQwQ6AEwA3oECAUQAw#v=onepage&amp;q=gatschinensis&amp;f=false hortus imperialis Gatchinensis] [https://books.google.fr/books?id=lxw3glGfmwQC&amp;pg=PA207&amp;dq=gatschinensis&amp;hl=la&amp;sa=X&amp;ved=2ahUKEwiw5JWCiprxAhViposKHcXzCtgQ6AEwBXoECAcQAg#v=onepage&amp;q=gatschinensis&amp;f=false ] - [https://books.google.fr/books?id=7U5fAAAAcAAJ&amp;pg=PA70&amp;lpg=PA70&amp;dq=gatschinensis&amp;source=bl&amp;ots=gL8Ew-PKnf&amp;sig=ACfU3U3Qt-WgX8rADgVTZFKcn5aotF4oKw&amp;hl=la&amp;sa=X&amp;ved=2ahUKEwjMrfuEiZrxAhVFvIsKHay9AQwQ6AEwAXoECAMQAw#v=onepage&amp;q=gatschinensis&amp;f=false hortus Caesareus Gatchinensis]

[https://plant.depo.msu.ru/public/scan.jpg?pcode=MW0109044 Herbarium Universitatis Mosquensis] [https://books.google.ru/books?id=6YENywEACAAJ&amp;dq=herbarium+universitatis+mosquensis&amp;hl=la&amp;sa=X&amp;ved=2ahUKEwjgm6KB8ZfwAhVOtIsKHV3SDlsQ6AEwAHoECAAQAQ ] - [https://search.rsl.ru/ru/record/01003358157 Herbarium vivum sive Collectio plantarum siccarum Caesareae universitatis Mosquensis]  
[https://books.google.fr/books?id=LQA6AAAAcAAJ&amp;pg=RA2-PA168&amp;dq=silviculturam&amp;hl=la&amp;sa=X&amp;ved=2ahUKEwi45_WXuYz0AhVHEncKHeZNAFYQ6AF6BAgFEAI#v=onepage&amp;q=silviculturam&amp;f=false Silvicultura]

[https://old-smolensk.ru/?p=12670 Smolenscensis obsidionis liberationis obsessorum ac deditorum castrorum...] [https://rusneb.ru/catalog/000200_000018_RU_NLR_DIGIT_75071/ ]

[https://books.google.fr/books?id=uGV3SHOFe8YC ''Relationes status dioecesium in magno ducatu Lituaniae'']

[https://books.google.fr/books?id=fQFfAAAAcAAJ&amp;pg=PA97&amp;dq=zea+et+munichia+sunt&amp;hl=la&amp;sa=X&amp;ved=2ahUKEwjbjqiRv6L0AhWqtIsKHdIHByUQ6AF6BAgLEAI#v=onepage&amp;q=zea%20et%20munichia%20sunt&amp;f=false portus et statio]

[https://www.zvab.com/servlet/SearchResults?an=gotthold%20merten&amp;cm_sp=det-_-bdp-_-author Gotthold Adalbert (sic) Merten] [http://www.richardwolf.de/latein/capellan.htm geboren 1866, war Pfarrer im thüringischen Sonneberg und später Altphilologe am Realgymnasium in Lippstadt. Er starb 1946]


[https://books.google.ru/books?id=1iowAAAAYAAJ&amp;pg=PA388&amp;dq=o+superbe+quid+superbis+tua+superbia+te+superabit&amp;hl=la&amp;sa=X&amp;ved=2ahUKEwjBt8-Fj_v1AhVlkosKHajjCa4Q6AF6BAgFEAI#v=onepage&amp;q=o%20superbe%20quid%20superbis%20tua%20superbia%20te%20superabit&amp;f=false o/be...] — [[Lamspringe]]

[[Usor:,jlzifhsr/Manevichi|Manevichi]]

[https://la.wikipedia.org/w/index.php?title=Specialis:Quaerere&amp;limit=100&amp;offset=0&amp;profile=default&amp;search=insource%3Admgh.de%2Fde&amp;ns0=1 ad MGH nexus corrigendi]

{{Creanda|sv|Erik Väderhatt|Ericus Ventosi Pilei}}&lt;ref&gt;[https://books.google.ru/books?id=uEFSAQAAIAAJ&amp;q=Ventosi+pilei#v=snippet&amp;q=Ventosi%20pilei&amp;f=false ''Saxonis Grammatici Historia Danica'', vol. I, Hauniae 1839].&lt;/ref&gt;
==== ====
&lt;hr&gt;
&lt;small&gt;Insula Robusta&lt;ref&gt;{{ZK|Мошный, о-в}}&lt;/ref&gt; [[Serenscum]]&lt;ref&gt;[[Alexander Guagnini|Alexandri Guagnini]] [https://books.google.fr/books?id=xmG9pSVHMLMC&amp;q=Vorotiniam#v=snippet&amp;q=Vorotiniam&amp;f=false ''Sarmatiae Europeae descriptio'', 1578, p. 6].&lt;/ref&gt; [https://books.google.ru/books?id=bD9fAAAAcAAJ&amp;pg=PA9&amp;dq=portus+victoriae+australia&amp;hl=la&amp;sa=X&amp;ved=2ahUKEwiCvqeMmvr1AhXwo4sKHapqDXMQ6AF6BAgEEAI#v=onepage&amp;q=portus%20victoriae%20australia&amp;f=false Portus Victoriae] 
[https://books.google.ru/books?id=d6pDAQAAIAAJ&amp;pg=RA3-PA64&amp;dq=serrae+nivatae&amp;hl=la&amp;sa=X&amp;ved=2ahUKEwi7se7Hhav2AhUFrYsKHdGgCWwQ6AF6BAgDEAI#v=onepage&amp;q&amp;f=false Serra Nivata] [https://books.google.ru/books?id=o708AAAAYAAJ&amp;q=Cuchullin#v=snippet&amp;q=Cuchullin&amp;f=false Cuchullin] (et in Phingaleide) [https://books.google.ru/books?id=qdE_AAAAcAAJ&amp;q=Cualgnia#v=snippet&amp;q=Cualgnia&amp;f=false Cualgnia]

[https://books.google.ru/books?id=PcFOAAAAcAAJ&amp;pg=PA227&amp;dq=shudrae+sunt&amp;hl=de&amp;sa=X&amp;ved=2ahUKEwjo6_r1-aP5AhVsoosKHWfWCDkQ6AF6BAgHEAI#v=onepage&amp;q=shudrae%20sunt&amp;f=false de tribubus]—[https://books.google.ru/books?id=3ltgAAAAcAAJ&amp;pg=PA172&amp;dq=shudrae+sunt&amp;hl=de&amp;sa=X&amp;ved=2ahUKEwju0-OZ-aP5AhVk_SoKHR0hBlIQ6AF6BAgCEAI#v=onepage&amp;q=shudrae%20sunt&amp;f=false et haec] '''Arena mobilis'''&lt;ref&gt;{{Kirp|бархан}}&lt;/ref&gt; '''Selectio artficialis'''&lt;ref&gt;[[Ieremias Bonomelli|Ieremiae Bonomelli]] [https://books.google.ru/books?id=aL69veFB4usC&amp;q=selectio+artificialis#v=snippet&amp;q=selectio%20artificialis&amp;f=false ''Summa totius theologiae dogmaticae'', vol. 2, Mediolani 1876, p. 282]; {{Creanda|de|Tilman Pesch|Tilman Pesch}} [[Societas Iesu|SI]] [https://books.google.co.jp/books?hl=ru&amp;id=4Ifzp0MVoKMC&amp;q=selectio+artificialis#v=snippet&amp;q=selectio%20artificialis&amp;f=false ''Institutiones philosophiae naturalis secundum principia S. Thomae Aquinatis'', Friburgi Brisgoviae 1880, p. 748].&lt;/ref&gt; '''Pristavus'''&lt;ref&gt;[https://books.google.ru/books?id=Mz1GAAAAcAAJ&amp;q=pristavorum#v=snippet&amp;q=pristavorum&amp;f=false ]&lt;/ref&gt; &lt;br&gt;'''Navis bialveata'''&lt;ref&gt;{{DAEL|Catamarán|199}}&lt;/ref&gt; '''Celox'''&lt;ref&gt;V. ''yacht'' in: {{Morgan}}&lt;/ref&gt; '''Fregatta'''&lt;ref&gt;[[Matthaeus Gotardus Artus Dantiscanus]] ([https://data.cerl.org/thesaurus/cnp01919647 CERL]) [https://books.google.ru/books?id=xVdEAAAAcAAJ&amp;q=fregatta#v=snippet&amp;q=fregatta&amp;f=false ''Americæ pars VIII'', 1599, p. 70].&lt;/ref&gt; '''galera'''&lt;ref&gt;[[:d:Q93433786|Iacobi Coreni]] [https://books.google.ru/books?id=FSnQcGffYcYC&amp;q=galera#v=snippet&amp;q=galera&amp;f=false ''Clupeus patientiae'', Lugduni 1624, cap. XXVII]; [[:d:Q27572052|Adami Eberti]] [https://books.google.ru/books?id=m9E051lU_ZgC&amp;q=galerarum#v=snippet&amp;q=galerarum&amp;f=false ''Quinquaginta relationes ex Parnasso'', Hamburgi 1683, p. 256]; {{Creanda|cs|Amand Hermann|Amandus Hermann|Amandi Hermann}} [https://books.google.ru/books?id=8A0Cp0_WNdMC&amp;q=galerarum#v=snippet&amp;q=galerarum&amp;f=false ''Capistranus triumphans'', Coloniae 1700, pp. 533, 542, 543].&lt;/ref&gt;&lt;/small&gt;
&lt;hr&gt;&lt;references /&gt;
&lt;hr&gt;

===flumina incerta ===
&lt;div style=&quot;width:100%&quot;&gt;
&lt;div style=&quot;float:left; width:90%&quot;&gt;
&lt;div style=&quot;text-align:center; font-family:Cambria, Georgia, Times, 'Times New Roman', serif; font-size:130%; padding:0.2em 0.6em 0.2em 0.6em; border-right:1px solid #BEBEBE; background:#EDEDED&quot;&gt;&lt;/div&gt;
&lt;/div&gt;
&lt;div style=&quot;float:left; width:10%&quot;&gt;
&lt;div class=&quot;mw-customtoggle-1&quot; style=&quot;cursor:pointer; text-align:center; font-family: Cambria, Georgia, Times, 'Times New Roman', serif; padding:0.2em 0.6em 0.2em 0.6em; border-right:1px solid #BEBEBE; background:white; color:black&quot;&gt;MONSTRATVR&lt;/div&gt;
&lt;/div&gt;
&lt;/div&gt;
&lt;div class=&quot;mw-collapsible mw-collapsed&quot; id=&quot;mw-customcollapsible-1&quot;&gt;
&lt;div class=&quot;toccolours mw-collapsible-content&quot; style=&quot;float:left; width:100%; border:0; background:#FFFFFF&quot;&gt;
&lt;small&gt;{{Creanda|ru|Мейерберг, Августин|Augustinus Meyrberg|Augustini Meyerberg}} [https://books.google.fr/books?hl=ru&amp;id=d7tMAAAAcAAJ&amp;q=Danecam#v=snippet&amp;q=Danecam&amp;f=false ''Iter in Moschoviam'',  p. 385].
{{div col|8}}
*Daneca 
*Zerdikium 
*[[:ru:Бетька|Beitma]]? &lt;ref&gt;''Бетьма/Ведьма?''&lt;/ref&gt;
*Atroba 
*Samara altera
*[https://wikimapia.org/#lang=ru&amp;lat=53.263981&amp;lon=49.465256&amp;z=11&amp;m=h&amp;show=/27545731/ru/Овраг-Аскульский Askula]&lt;ref&gt;[[:ru:Аскулы|Askulae vicus]]&lt;/ref&gt; 
*Komousium 
*Verovium
&lt;!--- Danecam?,  
Чердыка (Zerdikii) (“А ниже града Лаишева (Каз. Губ.) из Камы потекла протока Червык и пала в Волгу”. Книга Больш. Чертежу стр. 148.),  
Бейтмы (Beitmae)?, 
обеих Атроб (Atrobae), 
обеих Самар (Samarae), 
Сызрани (Syrani), 
Камышины (Komousii), 
Увары (Verovii) (Увара — название одного из главных и очень рыболовных устьев Волги в Астраханской Губернии. Эта река, при которой селение Уваринский учуг, впадает в море, к 35 верстах от Астрахани.), ---&gt;
{{div col end}}
&lt;/div&gt;
&lt;/div&gt;
&lt;/small&gt;
&lt;hr&gt;

'''Interpretes Homerici ceteri''': {{Creanda|cs|Antonín Liška (1791)|Antoninus Liška}} [[:s:cs:Soubor:Homer, Antonín Liška - Homérowa Odyssea - 1848.djvu|Bohemice]] — [https://web.archive.org/web/20070302034320/http://www.yunanmitolojisi.net/ilyada-1bolum/ Turcice] — [[:s:eo:Iliado/Kanto_Unua|Abraham Kofman]] [https://anno.onb.ac.at/cgi-content/anno-buch?apm=0&amp;aid=100032&amp;teil=0203&amp;seite=00000003&amp;zoom=1 Esperantice] — {{Creanda|sv|Johan Fredrik Johansson|Ioannes Fredericus Johansson}} Suecice — [[:d:Q52154093|Conradus Droste]] [http://www.biografischportaal.nl/persoon/51370290 Nederlandice] — {{creanda|nl|Jan Hendrik Glazemaker|Ioannes Henricus Glazemaker}} [https://lib.ugent.be/europeana/900000050989?pg=PP7 idem] 
&lt;hr&gt;
====incolis abundant, nomine carent====
{{div col|4}}&lt;small&gt;
*''Nižnevartovsk'' Ripa Vartovensis Inferior
*Balaschicha
*''Šachty'' Fodinae
*''Nižnekamsk'' Utbs Camae Inferioris
*''Dzeržinsk'' Nigrum (de priore ''Tschernoje'')
* Chimkae
* ''Angarsk'' Angaropolis
*''[[Podolsk]]''
* ''Prokop'evsk''
* ''Balakovo''
* ''Ljubercy''
* ''Severodvinsk'' Urbs Duinae Borealis (''in Urbe Duinae Boreali'') 
* ''Novočersassk	'' Circassium Novum
* ''Kamensk-Ural'skij'' Petra Uralensis
*''Èlektrostal' '' Electrichalybs
*''Salavat''
*''Železnodorožnyj'' Ferriviarium
*''Al'met'evsk''
*''Berezniki''
*''Rubcovsk''
*''Kopejsk''
*''Kovrov''
*''Krasnogorsk'' 
*''Chasavjurt''
*''Novomoskovsk'' Mosqua Nova
*''Pervoural'sk''
*''Neftejugansk''
*''Neftekamsk'' Naphticamium ut Solicamium
*''Novočeboksarsk'' Tscheboxari Novi
*''Čerkessk''
*''Orechovo-Zuevo''
*''Batajsk''
*''Ščëlkovo''
*''Nevinnomyssk''
*''Dimitrovgrad'' 
*''Novyj Urengoj''
*''Oktjabr'skij''
*''Domodedovo''
*''Obninsk''
*''Novošachtinsk'' Fodinae Novae
*''Seversk''
*''Puškino''
*''Žukovskij''
*''Kaspijsk'' Caspia
*''Nojabr'sk''
*''Ramenskoe''
*''Ačinsk''
*''Novokujbyševsk''
*Essentuki
*''Artëm''
*''Berdsk''
&lt;/small&gt;{{div col end}}
&lt;hr color=&quot;#CCCCCC&quot;&gt;
&lt;references /&gt;</text>
      <sha1>dirvxxk5rhwbnwrtt3ayhachrqgwqkb</sha1>
    </revision>
  </page>
  <page>
    <title>Vicipaedia:Index paginarum cottidianarum/Anno 2022 promotae</title>
    <ns>4</ns>
    <id>300718</id>
    <revision>
      <id>3697672</id>
      <parentid>3697555</parentid>
      <timestamp>2022-08-16T15:39:42Z</timestamp>
      <contributor>
        <username>Giorno2</username>
        <id>30162</id>
      </contributor>
      <comment>/* Mense Augusto 2022 promotae */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6192" xml:space="preserve">== Mense Ianuario 2022 promotae ==
# [[Festum anni novi Sinici]]
# [[Iosephus Cooper]]
# [[Astronautica]]
# [[Altenburg (Vimaria)]]
# [[Cyclops (Euripides)]]
# [[Magi (Biblia)]]
# [[Associatio diurnariorum Esperanticorum internationalis]]
# [[Martinus Schäffer]]
# [[Corpus humanum]]
# [[Fenelo]]
# [[Nucifrangibulum]]
# [[Trippstein]]
# [[Domus Francisci Liszt (Vimaria)]]
# [[Udalricus Brandenburg]]
# [[Antigone (Sophocles)]]
# [[Antonius Blair]]
# [[Nucifrangibulum (ballatio)]]
# [[Marius Petipa]]
# [[Cellula Schwannia]]
# [[Hugenoti]]
# [[Sepulcretum Sancti Ioannis (Iena)]]
# [[Iphigenia Aulidensis]]
# [[Ioannes Vsevoložskij]]
# [[Delphi (urbs Graeciae)]]
# [[Peleus]]
# [[Museum de Re publica Vimariana]]
# [[Li Yu (scriptor)]]
# [[Magna Zimbabua]]
# [[Ioannes Gottfriedus Herder]]
# [[Subcontinens Indica]]
# [[Ossianus]]
== Mense Februario 2022 promotae ==
# [[Myelinum]]
# [[Tres Hierarchae]]
# [[Museum de Re publica democratica Germanica (Apolda)]]
# [[Elisabeth Streller]]
# [[Colles Nigri]]
# [[Matthias Castrén]]
# [[Sodalitas Rennsteig 1896]]
# [[Demosthenes (strategus)]]
# [[Monumentum Nationale Marinum Papahānaumokuākea]]
# [[Robertus Müller]]
# [[Ostreae craticula tostae]]
# [[Yuan Mei]]
# [[Nosocomium Sanctorum Georgii et Iacobi]]
# [[Amavangus]]
# [[Vesper (potio)]]
# [[Ioannes Racine]]
# [[Ioannes Racine]]
# [[Clipeus Arvernus]]
# [[Franciscus Georgius Rössler]]
# [[Simplicia montani victualia]]
# [[Guillelmus Wallaceus]]
# [[Wannsee Conventus]]
# [[Phèdre (Racine)]]
# [[Lex maiestatis]]
# [[Ucraina]]
# [[Equites (Aristophanes)]]
# [[Vladimirus Zelens'kyj]]
# [[Stela Mesae]]
== Mense Martio 2022 promotae ==
# [[Napoleo III (imperator Franciae)]]
# [[Latinitas culinaria]]
# [[Neurogenesis]]
# [[Genovefa Parisiensis]]
# [[Fugger]]
# [[Vespae (Aristophanes)]]
# [[Radix Iesse]]
# [[Psalmus 77]]
# [[Musica Renascentiae]]
# [[Acharnenses]]
# [[Galena Gorecka]]
# [[Ordinatio Regni Mineralium secundum Linnaeum]]
# [[Paryadres]]
# [[Orlando furioso]]
# [[Lingua Lazica]]
# [[Capella Bethleem (Praga)]]
# [[Pimpinella anisum]]
# [[Bracchium caecum]]
# [[Keith Floyd]]
# [[Paulus Jegorovas]]
# [[Conformatio terrestris]]
# [[Cultura interretialis]]
# [[Ioannes Biereye]]
# [[Mundus Russicus]]
# [[Ioanna Agalakova]]
# [[Pax (Aristophanes)]]
# [[Pax (Aristophanes)]]
# [[Ioannes Kotljarevs'kyj]]
# [[Evangelicum centrum Pauli Schneider Vimariense]]
# [[Paulus Schneider]]
# [[Hochepot]]

== Mense Aprili 2022 promotae ==
# [[Caliphatus]]
# [[Johannes Aurifaber (Vimariensis)]]
# [[McDonald's]]
# [[Conradus Ferdinandus Meyer]]
# [[Parlamentum Erfordiense]] 
# [[Ricardus Evans Schultes]]
# [[Associatio Esperantica Hiberniae]]
# [[Historia Anastasii de Honestis (Botticelli)]]
# [[Iulius Jüthner]]
# [[Dominica in Palmis de passione Domini]]
# [[Hippolytus (Euripides)]]
# [[Aula Thuringiae]]
# [[Chebeb]]
# [[Ioannes Stadius]]
# [[Pater noster]]
# [[Psalmus 22]]
# [[Dominica Resurrectionis Domini]]
# [[Oleum (generale)]]
# [[Hilarius Putnam]]
# [[Ars coquinaria mediaevalis]]
# [[Sirenes]]
# [[Orestes (Euripides)]] 
# [[Aloysius Schmaus]]
# [[Oedipus rex]]
# [[Loca monticulosa Kazachstanica]]
# [[Jean Paul]]
# [[Polymestor]]
# [[Insula Sancti Ludovici]]
# [[Dionysius Keefe]]
# [[Hecuba (Euripides)]]

== Mense Maio 2022 promotae ==
# [[Oksana Zabuzhko]]
# [[Andromaque (Racine)]]
# [[Fredericus Ludovicus Schröder]]
# [[Mistelbach]]
# [[Nessus]]
# [[Adversus Haereses (Irenaeus)]] 
# [[Ferdinandus Raimund]]
# [[Helena (Euripides)]]
# [[Hymnus Ucrainae]]
# [[De Christiana expeditione apud Sinas]]
# [[Bibliotheca Nationalis Ucrainae Vernadskiana]]
# [[Basidiomycota]]
# [[Lingua Coreana]]
# [[Aves (Aristophanes)]]
# [[Realismus scientificus]]
# [[Kep Enderby]]
# [[Omasa modo Cadomensi]]
# [[Nicolaus Iohannis]]
# [[Temperatura thermodynamica]]
# [[Candidus Fuldensis]]
# [[Candidus Fuldensis]]
# [[Propria ad mensam Imperatoris principia]]
# [[Aeacus]]
# [[Altiplanum]]
# [[Ferdinandus Maia]]
# [[Ranae (Aristophanes)]]
# [[Titus Brandsma]]
# [[Trachiniae]]
# [[Certamen poëticum Hoeufftianum]] 
# [[Ennominae]]
# [[Museum Transsilvanicum]]

== Mense Iunio 2022 promotae ==
# [[Gallus vinolentus]]
# [[Habilitas physica]]
# [[Petrus Frankopan]] 
# [[Sadko Hospes Dives]]
# [[Apenninus]]
# [[Spiritus Sanctus]]
# [[Danapris]]
# [[Hallgrimus Petraeus]]
# [[Topiramatum]]
# [[Franciscus Ruano]]
# [[Electra (Sophocles)]]
# [[Trinitas]]
# [[Anthropophagia]]
# [[Opus fundatum ecclesiarum munitarum]]
# [[Tubulus renalis colligens]]
# [[Clemens Brentano]]
# [[Capaneus]]
# [[Putin-mentula!]]
# [[Phoenissae]]
# [[Carolus Linnaeus]]
# [[Domus Boerorum]]
# [[Albertus Szent-Györgyi]]
# [[Tabasco (liquamen)]]
# [[A Moscua ad Galliculos]]
# [[Vladimirus Filatov]]
# [[Musica Romana antiqua]]
# [[Septem contra Thebas (Aeschylus)]]
# [[Elote]]
# [[Cultura Ucrainae]]
# [[Mauricius de Soliaco]]

== Mense Iulio 2022 promotae ==
# [[Dies sideralis]]
# [[Huldericus de Hutten]]
# [[Lalibela]]
# [[Ioannes Bellaius (cardinalis)]]
# [[Observatorium Zimmerwald]]
# [[Oedipus Coloneus (Sophocles)]]
# [[Gudrida Simonis filia]]
# [[Praesidium Franciscopolis]]
# [[Croesus]]
# [[Bloody Mary]]
# [[Jawaharlal Nehru]]
# [[Vestmannorum Insulae]]
# [[Antonius Guilielmus Amo]]
# [[Thesmophoriazusae]]
# [[Prunus spinosa]]
# [[Maunsel White]]
# [[Navis longa Russica, abi in malam rem!]]
# [[Kennethus Norton]]
# [[Salzkammergut]]
# [[Martinus Garrix]]
# [[Theodicaea]]
# [[Cleomenes I]]
# [[Ioannes Pedius Tethingerus]]
# [[Regio cervicalis anterior]]
# [[Lysistrata]]
# [[Isaacus Casaubonus]]
# [[Abbatia Sancti Victoris]]
# [[Genocidium Buchae factum]]
# [[Abductio Turcica]]
# [[Otto Grotewohl]]
# [[Aiax (Sophocles)]]

== Mense Augusto 2022 promotae ==
# [[Oberonia titania]]
# [[Colossi Memnonis]]
# [[Ioachimus Bellaius]]
# [[Fabella nautae naufragi]]
# [[Morbus Whippleianus]]
# [[Bethsabea]]
# [[De principe]]
# [[Meridianus primarius]]
# [[Eucharistia]]
# [[Franciscus de Neve]]
# [[Nucleus Telluris exterior]]  
# [[Philoctetes (Sophocles)]]
# [[Franciscus Huber]]
# [[Leviathan (Thomas Hobbius)]]
# [[Demophoon]]
# [[Ministerium Defensionis Foederationis Russicae]]
# [[Ordines Franciscani]]
[[Categoria:Pagina prima]]
[[Categoria:Paginae honoratae]]</text>
      <sha1>ixavm61xnkh5tpsqbqua1uua92u2aig</sha1>
    </revision>
  </page>
  <page>
    <title>Incursio Russica in Ucrainam (2022)</title>
    <ns>0</ns>
    <id>301723</id>
    <revision>
      <id>3697675</id>
      <parentid>3697057</parentid>
      <timestamp>2022-08-16T16:11:58Z</timestamp>
      <contributor>
        <username>LilyKitty</username>
        <id>18316</id>
      </contributor>
      <comment>eventa addo</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="80317" xml:space="preserve">{{eventuscurrens}}
{{Latinitas|-2}}
[[Fasciculus:War in Ukraine (2022) en.png|thumb|Rerum status primo certaminis die.]]
[[Fasciculus:2022 Russian invasion of Ukraine.svg|thumb|[[Tabula geographica]] incursionis.]]
'''Incursio Russica in Ucrainam''', sive '''praecipuum negotium militare''' (publica appellatio [[Russice|Russica]]&lt;ref&gt;[[Anglice]] 'special military operation'. Confer haec verba: &quot;In nexu ut in consensu cum articulo 51 partis{{dubsig}} 7 chartae Consociationis Nationum, cum sancione consilio Foederationis Russicae et consensu cum confirmatis Concilio Foederali die 22 Februarii istius anni foederibus de amicitia et auxilio mutuo cum Republica Populari Donetskensi et Republica Populari Luganskensi accipio ego&lt;!--quis sum?--&gt; solutionem{{dubsig}} praecipuum negotium militare habentem&quot; ([http://kremlin.ru/events/president/news/67843 ''В связи с этим в соответствии со статьёй 51 части 7 Устава ООН, с санкции Совета Федерации России и во исполнение ратифицированных Федеральным Собранием 22 февраля сего года договоров о дружбе и взаимопомощи с Донецкой Народной Республикой и Луганской Народной Республикой мною принято решение о проведении специальной военной операции'')]&lt;/ref&gt;), commotum est, cum [[arma]]tae [[miles|copiae]] [[Russia|Russicae]] in [[Ucraina]]m die [[24 Februarii]] [[2022]] invaderent, [[Bellum Russo-Ucrainum]] magnopere amplificantes, certamen quod anno [[2014]] post [[Res Novae Dignitatis|Res Novas Dignitatis]] ([[Ucrainice]] Революція гідності) coeperat. Quae [[incursio magna|incursio]] maximum [[exsul]]um discrimen [[Europa]]eanum post [[bellum mundanum secundum]] effecit,&lt;ref name=&quot;Blake-2022-03-15&quot;&gt;{{cite web |last=Blake |first=Daniel Keane, Elly |date=15 Martii [[2022]] |title=What is the Homes for Ukraine refugees scheme and how do you apply? |url=https://www.standard.co.uk/news/uk/host-ukraine-refugee-scheme-uk-london-russia-war-apply-b987910.html |access-date=15 Martii 2022 |website=Evening Standard}}.&lt;/ref&gt;&lt;ref&gt;{{cite news |title=Ukrainian exodus could be Europe's biggest refugee crisis since World War II |newspaper=El Pais |date=3 Martii [[2022]] |url=https://english.elpais.com/international/2022-03-03/ukrainian-exodus-could-be-europes-biggest-refugee-crisis-since-world-war-ii.html}}.&lt;/ref&gt; plus quam quinque et dimidio [[millio]]nibus Ucrainorum [[patria]]m relinquentibus&lt;ref name=&quot;UNHCR-Ukraine&quot;&gt;{{cite web |title=Situation Ukraine Refugee Situation |date=&lt;!--kept up-to-date--&gt; |website=United Nations High Commissioner for Refugees |url=https://data2.unhcr.org/en/situations/ukraine |access-date=15 Martii 2022}}.&lt;/ref&gt; et quarta totius numeri [[incola]]rum parte loco suo mota.&lt;ref&gt;{{cite news |first1=Rebecca |last1=Ratcliffe |first2=Abené |last2=Clayton |first3=Adam |last3=Gabbatt |first4=Léonie |last4=Chao-Fong |first5=Samantha |last5=Lock |first6=Tom |last6=Ambrose |date=19 Martii [[2022]] |title=Biden outlines 'consequences' if China aids Russia – as it happened |work=[[The Guardian]] |url=https://www.theguardian.com/world/live/2022/mar/18/russia-ukraine-war-latest-news-biden-to-warn-xi-against-backing-putin-as-russian-military-offensives-falter-live |access-date=28 Martii 2022 |archive-date=18 Martii 2022 |archive-url=https://web.archive.org/web/20220329020321/https://www.theguardian.com/world/live/2022/mar/18/russia-ukraine-war-latest-news-biden-to-warn-xi-against-backing-putin-as-russian-military-offensives-falter-live |url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite news |date=31 Martii [[2022]] |title=Ukraine war: Putin being misled by fearful advisers, US says |work=BBC News |url=https://www.bbc.com/news/world-europe-60936117 |access-date=31 Martii 2022}}.&lt;/ref&gt; [[Ordo mundi sanitarius]] rem necessitatem medicam appellavit.&lt;ref&gt;[https://www.who.int/emergencies/situations/ukraine-emergency OMS ad necessitatem medicam{{dubsig}} in Ucraina refert] {{Ling|Anglice}}.&lt;/ref&gt;&lt;ref&gt;Praeterea, incursio [[Indutiae Olympicae|indutias Olympicas]] violare &lt;!--a quibus?--&gt;dicebatur.&lt;/ref&gt;

Bello Russo-Ucraino anno [[2014]] conflato, Russia [[Crimaea a Foederatione Russica annexa|Crimaeam in suam dicionem redegit]] et [[secessionisticae vires Russicae in Donbas|secessionistae a Russia adiuvati]] partem [[Donbas]] [[regio]]nis in meridiana et orientali Ucraina corripuerunt, [[bellum in Donbas|bellum regionale]] ibi moventes.&lt;ref&gt;{{cite web |last=Kirby |first=Jen |date=28 Februari [[2022]] |title=Putin's invasion of Ukraine, explained |url=https://www.vox.com/2022/2/23/22948534/russia-ukraine-war-putin-explosions-invasion-explained |access-date=28 Februarii 2022 |website=[[Vox (website)|Vox]]}}.&lt;/ref&gt;&lt;ref&gt;{{cite web |date=28 Februarii [[2022]] |title=Conflict in Ukraine |url=https://cfr.org/global-conflict-tracker/conflict/conflict-ukraine |access-date=28 Februarii 2022 |website=Global Conflict Tracker |publisher=Council on Foreign Relations}}.&lt;/ref&gt; Russia anno [[2021]] exeunte [[praeludium incursionis Russicae in Ucrainam anno 2022|magnas vires militares]] ad fines suos cum Ucraina vocavit, usque ad 190&amp;thinsp;000 copias et eorum [[arma]]menta praemittens. [[Vladimirus Putin|Vladimirus Putinus]], praeses Russicus, in emissione [[radio]]phonica sub incursione opiniones [[irredentismus Russicus|irredentisticas]] nuntiavit,&lt;ref&gt;{{cite news |date=26 Februarii [[2022]] |title=Russia's invasion of Ukraine |newspaper=The Economist]|url=https://www.economist.com/briefing/2022/02/26/russias-invasion-of-ukraine |url-status=live |url-access=subscription |access-date=26 Februarii 2022 |archive-url=https://ghostarchive.org/archive/20220226/https://www.economist.com/briefing/2022/02/26/russias-invasion-of-ukraine |archive-date=26 Februarii 2022 |quote=Though the target of Mr. Putin's tirade on February 21st was Ukraine, the former Soviet republics now in NATO, Estonia, Latvia and Lithuania, have cause for alarm over his irredentism.}}.&lt;/ref&gt; Ucrainae [[ius civitatis]] denegavit,&lt;ref name=&quot;Putin Ukraine statehood&quot;&gt;{{cite news |last=Perrigo |first=Billy |date=22 Februarii [[2022]] |title=How Putin's Denial of Ukraine's Statehood Rewrites History |url=https://time.com/6150046/ukraine-statehood-russia-history-putin/ |magazine=Time |access-date=28 Februarii 2022}}.&lt;/ref&gt;&lt;ref&gt;{{cite web |date=22 Februarii 2022 |title=Putin Says He Does Not Plan to 'Restore Empire' |url=https://www.themoscowtimes.com/2022/02/22/putin-says-he-does-not-plan-to-restore-empire-a76519 |access-date=2 Martii 2022 |website=The Moscow Times}}.&lt;/ref&gt; ac perperam&lt;ref name=&quot;Tabarovsky-2022-02-27&quot;&gt;{{cite web |last1=Tabarovsky |first1=Izabella |last2=Finkel |first2=Evgeny |author-link2= |date=27 Februarii [[2022]] |title=Statement on the War in Ukraine by Scholars of Genocide, Nazism and World War II |work=The Jewish Journal of Greater Los Angeles |url=https://jewishjournal.com/news/worldwide/345515/statement-on-the-war-in-ukraine-by-scholars-of-genocide-nazism-and-world-war-ii/ |access-date=6 Aprilis 2022}}.&lt;/ref&gt; dixit Ucrainam a [[neonazimus in Ucraina|neonazistis]] regi, qui [[Russi in Ucraina|ethnicam minoritatem Russicam]] insectarentur.&lt;ref&gt;Abbruzzese 2022.&lt;/ref&gt; Praeterea dixit [[OTAN|Consociationem ex pacto Atlantico Septentrionali]] ([[OTAN]]) incolumitatem [[respublica|reipublicae]] Russicae per [[amplificatio OTAN|amplificationem ad orientem]] ex annis [[2000]] ineuntibus minitari, quam affirmationem OTAN ipsa in dubium vocavit.&lt;ref&gt;{{cite web |date=27 Ianuarii 2022 |title=NATO-Russia relations: the facts |url=https://www.nato.int/cps/en/natohq/topics_111767.htm |access-date=1 Martii 2022 |publisher=NATO |quote=NATO is a defensive alliance. Our purpose is to protect our member states. Every country that joins NATO undertakes to uphold its principles and policies. This includes the commitment that 'NATO does not seek confrontation and poses no threat to Russia,' as reaffirmed at the Brussels Summit this year. NATO enlargement is not directed against Russia. Every sovereign nation has the right to choose its own security arrangements. This is a fundamental principle of European security, one that Russia has also subscribed to and should respect. In fact, after the end of the Cold War, Russia committed to building an inclusive European security architecture, including through the Charter of Paris, the establishment of the OSCE, the creation of the Euro-Atlantic Partnership Council, and the NATO-Russia Founding Act.}}.&lt;/ref&gt; Russia postulavit, ne OTAN se diutius extenderet nec umquam cum Ucraina consociaret.&lt;ref&gt;{{cite news |last=Wiegrefe |first=Klaus |date=15 Februarii [[2022]] |title=NATO's Eastward Expansion: Is Vladimir Putin Right? |work=Der Spiegel |url=https://www.spiegel.de/international/world/nato-s-eastward-expansion-is-vladimir-putin-right-a-bf318d2c-7aeb-4b59-8d5f-1d8c94e1964d |access-date=28 Februarii 2022}}.&lt;/ref&gt;&lt;ref&gt;[https://www.bbc.com/news/world-europe-56720589 &quot;Why did Russia invade Ukraine and what does Putin want?&quot;] ([[BBC]]).&lt;/ref&gt; [[Civitates Foederatae]] aliaeque [[civitas|civitates]] dixerunt Russiam Ucrainam oppugnare aut invadere velle, quod autem legati Russici identidem usque ad 23 Februarii 2022 negabant,&lt;ref&gt;[https://www.nato.int/cps/en/natohq/topics_111767.htm &quot;NATO-Russia relations: the facts,&quot;] NATO. 27 Ianuarii 2022.&lt;/ref&gt; false, ut eventus manifesto (vide &quot;Cursum&quot; infra) mox docebant.

[[Vladimirus Zelens'kyj]], praeses Ucrainae, [[diplomatia|rationes diplomaticas]] abrogavit et [[lex militaris|legem militarem]] declaravit. [[Consociatio Nationum]] et magnae [[civitas sui iuris|civitates]], inter quas [[Britanniarum Regnum]], [[Francia]], [[Germania]], [[Iaponia]], [[Italia]], [[Polonia]], et [[Hispania]],&lt;ref&gt;[https://www.conchovalleyhomepage.com/news/national-news/the-latest-europe-warns-of-airspace-risks-around-ukraine/ &quot;Live updates: Hungary's Orban condemns Russian attack.&quot;]&lt;/ref&gt;&lt;ref&gt;[https://www.gazetaprawna.pl/wiadomosci/kraj/artykuly/8365614,sejm-potepil-stanowczo-rosyjska-agresje-i-wezwal-rosje-do-wycofania-sil-z-ukrainy.html Textus interretialis.]&lt;/ref&gt; atque [[Unio Europaea]] rem gestam condemnaverunt.&lt;ref&gt;[https://euobserver.com/world/154423 &quot;Massive EU sanctions to target Putin's war chest.&quot;]&lt;/ref&gt;

== Contextus ==
Ad causam incursionis intellegendam complura necessario consideranda sunt, quae Ucrainam et Russiam quodammodo inter se conectunt, scilicet historia pristina, historia Unionis Sovieticae, historia recentior [[cultura|culturae]] et [[civilitas|civilitatis]].

=== Historia pristina ===
:''Paginae principales: [[Historia Ucrainae]], [[Historia Russiae]], [[Russia Kioviensis]], [[Rhos]], [[Ruricidae]]''
[[Saeculum 9|Saeculo nono]] post Christum natum civitas [[Russia Kioviensis|Russiae Kioviensis]] orta est. Illa civitas in Europa Orientali sita nomen suum ex urbe [[Kiovia]] cepit, quae urbs magnum habet momentum non solum Ucrainis, sed etiam Russis (qui eam saepius tamquam incunabula culturae Russicae intuentur), propterea quod [[Ecclesia Orthodoxa Russica]] a [[Vladimirus I Magnus|Vladimiro I Magno]], qui anno [[988]] Kioviae [[Baptismus|sacro fonte ablutus erat]], originem suam trahit. 

[[Saeculum 14|Saeculo quarto decimo]] ineunte, postquam Russia Kioviensis invasione [[Imperium Mongolicum|Mongolo-Tartarorum]] (annis [[1237]]–[[1240]]) pervastata et infracta est, duo maiora territoria in eadem regione exstiterunt: [[Ducatus Volodimiriensis]], anno iam [[1157]] conditus et in septentriones et orientem solem spectans, et [[Ducatus Kioviensis]], inter occasum solis et meridiem vergens, qui tandem anno [[1362]] [[Magnus Ducatus Lituaniae|Magno Ducatui Lituaniae]] adiunctus est. Ducatus Volodimiriensis pedetemptim in [[Magnus Ducatus Moscoviae|Magnum Ducatum Moscoviae]] commutatus est; pars meridionalis, tunc &quot;confinium&quot; inter [[Polonia|Poloniam]] et Russiam appellata ([[Lingua Slavica orientalis antiqua|lingua Slavica orientali antiqua]] ''ѹкраина/ukraina,'' h. e. &quot;ad fines sita&quot;), pro sua parte in [[Res Publica Utriusque Nationis|Rem Publicam Utriusque Nationis]] transiit. 

=== Unio Sovietica dissoluta ===
:''Pagina principalis: [[Unio Rerum Publicarum Sovieticarum Socialisticarum]] (URSS)''
Ucraina, tempore [[Unio Rerum Publicarum Sovieticarum Socialisticarum|Unionis Sovieticae]] (1922–1991) pars eiusdem societatis, anno [[1991]] exeunte Unionem Sovieticam nuper dissolutam reliquerat. Quam [[dissolutio Unionis Sovieticae|dissolutionem]] praeses Vladimirus Putin in oratione, quam mense Aprili anni [[2005]] habuit, maximam eversionem geopoliticam et tragoediam populo Russico vocavit, cum multitudo civium Russicorum nunc extra fines Russicos viverent.&lt;ref&gt;[http://www.kremlin.ru/events/president/transcripts/22931 Oratio ad contionem sociorum Foederationis Russicae, die 25 Aprilis 2005 habita: ''&quot;Прежде всего следует признать, что крушение Советского Союза было крупнейшей геополитической катастрофой века. Для российского же народа оно стало настоящей драмой. Десятки миллионов наших сограждан и соотечественников оказались за пределами российской территории.&quot;'']&lt;/ref&gt; Sed etiam aliis in orationibus indicavit eosdem homines dolore dissolutae Unionis Sovieticae tantopere affectos esse, ut iis hucusque non contigisset differentias inter desideria et vivendi condiciones, quae nunc essent, diluere. Haec desideria praecipue Oleg Gazmanov, modorum factor Russicus, carmine suo ''Сделан в СССР (Factus sum in URSS)'' celebravit.&lt;ref&gt;[https://lyricstranslate.com/en/sdelan-v-sssr-сделан-в-ссср-made-ussr.html Textus carminis ''Сделан в СССР'']&lt;/ref&gt;

=== Historia recentior culturae et civilitatis ===
Hodie et Ucrainis et Russis sunt:
* duae linguae ([[lingua Ucrainica]] et [[lingua Russica]]);
* religiones propriae (variae Ecclesiae Orthodoxae Ucrainicae, Ecclesia Orthodoxa Russica);
* culturae propriae;
* conscientiae propriae (e. g. carmen patrium Ucrainicum, carmen patrium Russicum).

Vere 2014 organizatio paramilitaris nomine [[regiminis Asov]] (nomen Ucrainicum ad verbum expressum: &quot;grex seiunctus praecipuae rei gerendae militaris&quot;&lt;ref&gt;Conspicua est similis dictio inter Russicum ''специальной (praecipuae) военной (militaris) операции (operationis)'' et Ucrainicum ''спеціального (praecipuae) призначення (missionis)''; pariter conspicua sunt insignia non-cyrillica et &quot;Z&quot; vehiculorum militarium Russicorum et &quot;Ƶ&quot; ([[hamus lupi]]) regiminis Asov&lt;/ref&gt;), quae partim ultranationalistica, partim nazistica esse aestimatur, condita est. Potest fieri, ut scelera, quae a regimine Asov commissa esse dicuntur, una ex pluribus causis incursionis sit.

== Cursus ==
=== Phasis praeparatoria ===
'''Die 21 Februarii 2022'''&lt;br/&gt;
[[Vladimirus Putin]], praeses [[Foederatio Russica|Foederationis Russicae]] [[pactum Minscum secundum]] (diei [[12 Februarii]] [[2015]]) basem non esse declaravit. Eodem die internationaliter non approbatas [[Res publica popularis Luganskensis|Rempublicam popularem Prataliensem]] et [[Res publica popularis Donetskensis|Rempublicam popularem Donetskensem]] approbavit.

=== Phasis prima (diebus 24 Februarii - 7 Aprilis) ===
'''Die 24 Februarii 2022'''&lt;br/&gt;
Hora sexta [[Moscua|Moscuae]] ([[UTC+03:00]]) Vladimirus Putin, praeses [[Foederatio Russica|Foederationis Russicae]], cuiusdam operationis militaris initium annuntiavit, inter alia has praecipuas [[casus belli|causas]] afferens: se de amplificatione [[Consociatio ex pacto Atlantico Septentrionali|Consociationis ex pacto Atlantico Septentrionali (OTAN)]] in orientem versus deque accessu eius infrastructurae militaris ad fines [[Russia|Russiae]] sollicitum esse, quod in territoriis finitimis, quae Russiae antiquitus propriae essent, quaedam &quot;Anti-Russia&quot; fieret Russis infesta; assumptionem Ucrainae a Consociatione ex pacto Atlantico Septentrionali non ferendam esse; civitates OTAN &quot;nimios nationalistas et neonazistas in Ucraina&quot; adiuvare, qui incolis [[Crimaea|Crimaeae]] ac [[Sebastopolis]] liberam voluntatem iteratae coniunctionis cum Russia numquam ignoturi essent; res publicas populares in regione Donbas sitas opem a Russia expetiisse; illos neonazistas etiam in Crimaeam ituros esse [[bellum]] gestum et, sicuti olim oboedientes adiutores [[Adolphus Hitler|Adolphi Hitler]], homines inermes necatum; res igitur a se postulare, ut prompte ac statim ageret; Russiam ipsam tutam esse non posse Ucraina semper minitante; se autem non omnem Ucrainam occupare, sed eam dearmare et pacare velle; [[Civilitas|civilitatem]] Russicam in [[Libertas|libertate]] positam esse, quod ius omnibus, etiam incolis Ucrainae, patere deberet; se [[Maiestas (ius constitutionale)|summam rerum potestatem]] illarum civitatum, quae post dissolutionem Unionis Sovieticae factae essent, agnoscere.&lt;ref&gt;http://en.kremlin.ru/events/president/news/67843 {{Ling|Anglice}}&lt;/ref&gt;

[[Fasciculus:Житловий будинок у Києві (вул. Кошиця) після обстрілу.jpg|thumb|Kiovia, 25 Februarii]] 
'''Die 25 Februarii 2022'''&lt;br/&gt;
[[Kiovia|Kioviae]] hora quarta ([[UTC+02:00]]) pugnam circum urbem exardescere et rochetas militares [[Bomba|bombasque]] iaci relatum est. Resolutio apud [[Consilium securitatis|Consilium Securitatis]] [[Consociatio Nationum|Nationum Unitarum]] contra Russiam repudiata est, [[Res publica popularis Sinarum|Sinis]] suffragio abstinentibus, cum Sinarum legatus dixisset Sinas de progressu in Ucraina sollicitas esse. [[Consilium Europae]] decrevit, ut [[Russia|Foederationi Russicae]] ius repraesentationis sine mora denegaretur.&lt;ref&gt;[https://www.coe.int/en/web/portal/full-news/-/asset_publisher/y5xQt7QdunzT/content/council-of-europe-suspends-russia-s-rights-of-representation? Council of Europe suspends Russia’s rights of representation].&lt;/ref&gt;

'''Die 26 Februarii 2022'''&lt;br/&gt;
[[Fasciculus:Apartment block in Kharkiv damaged during Russian invasion.jpg|thumb|Aedes [[Charcovia]]e effectu [[missile|missilis]] partim disruptae.]]
''[[MV Namura Queen]],'' navis [[Iaponia|Iaponica]] generis ''[[bulk carrier]]'', Russica [[Navis longa|navi longa]] oppugnata est.

'''Die 27 Februarii 2022'''&lt;br/&gt;
Praeses Vladimirus Putin rettulit se vires nucleares expedivisse&lt;ref&gt;[https://www.reuters.com/world/india/war-with-ukraine-putin-puts-nuclear-deterrence-forces-alert-2022-02-27/ Putin puts nuclear deterrent on alert].&lt;/ref&gt;, quod summi magistratus principum civitatum OTAN verba provocantia adversum Russiam permitterent; qua de causa sese imperavisse, ut vires Russici exercitus ad deterrendum inservientes a ministro defensionis et a praeside consilii praetorii in praecipuum modum pugnandi redigerentur.&lt;ref&gt;swissinfo.ch, cum versione Russica: [https://www.swissinfo.ch/rus/в-белом-доме-заверяют--что-нато-не-представляет-угрозы-для-рф/47385836 ''высшие должностные лица ведущих стран НАТО допускают и агрессивные высказывания в адрес нашей страны, поэтому приказываю министру обороны и начальнику Генерального штаба перевести силы сдерживания российской армии в особый режим несения боевого дежурства.'']&lt;/ref&gt; Observatores [[Organizatio securitati cooperationique in Europa favendis|Organizationis securitati cooperationique in Europa favendis (OSCE)]] Ucrainam reliquerunt.

'''Die 28 Februarii 2022'''&lt;br/&gt;
In [[Ruthenia Alba]], prope fines Ucrainae, disceptationes coeptae sunt. Tamen pugnae continuantur.

'''Die 1 Martii 2022'''&lt;br/&gt;
[[Mariupolis]], urbs maior in Ucraina meridionali sita cum magno portu, prorsus fere cincta esse videtur, qua de causa [[Eduardus Basurin]], dux non agnitae [[Res publica popularis Donetskensis|rei publicae popularis Donetskensis]], Mariupolitanis duos transitus humanitarios ex urbe secundum [[Via Europaea E58|Viam Europaeam E58]] versus [[Foederatio Russica|Foederationem Russicam]] obtulit usque ad diem 2 Martii.&lt;ref&gt;[https://edition.cnn.com/europe/live-news/ukraine-russia-putin-news-03-01-22/h_0e3d20b474aa007bb1e4acc0d0fba984 Russian-backed separatist leader expects his forces to surround Mariupol on Tuesday] (CNN).&lt;/ref&gt; Kioviae locus [[Babij Jar]] impetui militari Russico expositus erat.&lt;ref&gt;[https://web.archive.org/web/20220301173043/https://www.jpost.com/international/article-699034 Russians attack Babyn Yar Holocaust massacre site in Kyiv] (Jerusalem Post).&lt;/ref&gt;

'''Die 2 Martii 2022'''&lt;br/&gt;
Vespere [[Suecia]] suum spatium aerium brevi tempore [[aeroplanum insectatorium|aeroplanis insectatoriis]] Russicis generis [[SU-24]] et [[SU-27]] violatum esse nuntiavit.&lt;ref&gt;''Dagens nyheter'': [https://www.dn.se/sverige/ryska-stridsplan-krankte-svenskt-luftrum-helt-oacceptabelt/ Ryska stridsplan kränkte svenskt luftrum: &quot;Helt oacceptabelt&quot;] (Russica aeroplana insectatoria Suecicum spatium aeriale violabant: &quot;Prorsus inacceptabile&quot;).&lt;/ref&gt;

'''Die 4 Martii 2022'''&lt;br/&gt;
Postquam impetus factus est in [[Ergasterium atomicum Zaporizhia|officinam atomicam Zaporizhiensem]], officina eius generis maxima Ucrainae, milites Russici hoc loco potiti sunt.&lt;ref&gt;[https://globalnews.ca/news/8658032/ukraine-russua-nuclear-plant-fire Russia troops capture Europe's largest power plant in Ukraine after intense battle].&lt;/ref&gt;

'''Die 5 Martii 2022'''&lt;br/&gt;
[[Mariupolis|Mariupolitanis]] non contigisse videtur ex urbe per transitus humanitarios profugere. Societas medicorum nomine [[Medici sine finibus|Medici Sine Finibus]] calamitatem humanitariam sic descripsit: &quot;Condicio humanitaria [[Mariupolis|Mariupoli]] est calamitosissima. Scimus ab administris nostris homines illic spe carentes saluti suae prospicere conari et, dum graves impetus continuantur, commeatus mox periculose defecturos esse. (…) Necesse est homines vitam cum dignitate vivere. Hoc significat: cum aditu ad [[Victus|victum]], aquam, [[Energia|vim electricam]], calefactorium.&quot;&lt;ref&gt;[https://https://www.msf.org/human-dignity-and-life-must-be-respected-besieged-mariupol-ukraine ''Médecins Sans Frontières,'' Human dignity and life must be respected in besieged Mariupol, Ukraine.]&lt;/ref&gt;

'''Die 6 Martii 2022'''&lt;br/&gt;
Impetus rochetarum factus est in Institutum [[Physica nuclearis|Physicae]] [[Charcovia|Charcoviensem]], in quo etiam [[reactorium nucleare]] inest.&lt;ref&gt;[https://www.dailymail.co.uk/news/article-10583203/Russian-forces-fire-rockets-SECOND-nuclear-facility-Kharkiv.html Russian forces 'fire rockets at SECOND nuclear facility'.]&lt;/ref&gt; Eodem die 4366 homines [[Contra bellum|bello]] [[Concursus populi|reclamantes]] in 56 urbibus Russicis custodiae mandati sunt, ita ut inde aestimari liceat numerum eorum, qui ab incursionis initio tota Russia comprehensi sunt, esse plus quam 10&amp;thinsp;000 hominum.&lt;ref&gt;[https://www.aljazeera.com/news/2022/3/6/detentions-across-russia-anti-war-protests-monitor Thousands arrested across Russia at anti-war protests.]&lt;/ref&gt; [[Ordo mundi sanitarius]] per [[Twitter]] sedem suam in Ucraina oppugnari nuntiavit.&lt;ref&gt;Ordo mundi sanitarius de [https://twitter.com/DrTedros/status/1500372569425387522?ref_src=twsrc%5Etfw%7Ctwcamp%5Etweetembed%7Ctwterm%5E1500372569425387522%7Ctwgr%5E%7Ctwcon%5Es1_&amp;ref_url=https%3A%2F%2Fwww.tagesschau.de%2Fnewsticker%2Fliveblog-ukraine-sonntag-101.html oppugnatione sedis suae] {{Ling|Anglice}}.&lt;/ref&gt;

'''Die 7 Martii 2022'''&lt;br/&gt;
Tertius congressus de [[Indutiae|indutiis]] fuit, sed nequiquam.

'''Die 8 Martii 2022'''&lt;br/&gt;
Consensus ad tutum transitum [[Fuga|fugituris]] dandum initus est. Eodem die, qui [[Dies internationalis feminarum]] fuit, [[Manifestatio|manifestationes]] contra bellum exstiterunt [[Petropolis|Petropoli]], floribus ad monumentum ''Magni Patrii Belli'' (Russice ''пaмятник Великой Отечественной Bойне'') depositis.&lt;ref&gt;[https://zona.media/chronicle/soldrty-ukr#44899  Протесты против вторжения  в Украину.]&lt;/ref&gt;

'''Die 9 Martii 2022'''&lt;br/&gt;
[[Mariupolis|Mariupoli]] impetus bombarum in [[valetudinarium]] [[Obstetricia|obstetricium]] factus est, quo multi homines necati sunt. Quam rem [[civitas Vaticana]] &quot;inacceptabilem&quot;,&lt;ref&gt;[https://www.reuters.com/world/europe/top-vatican-official-says-reported-bombing-childrens-hospital-ukraine-2022-03-09/ Top Vatican official says reported bombing of Ukraine children's hospital 'unacceptable']&lt;/ref&gt; Vladimirus Zelens'kyj &quot;atrocitatem&quot; vocavit.&lt;ref&gt;[https://edition.cnn.com/2022/03/09/europe/russia-invasion-ukraine-evacuations-03-09-intl/index.html Russia's bombing of maternity and children's hospital an 'atrocity', Zelensky says] ([[CNN]]).&lt;/ref&gt;

'''Die 10 Martii 2022'''&lt;br/&gt;
[[Attalea|Attaleae]], in urbe [[Turcia|Turciae]] meridionalis, [[Demetrius Kuleba]] et [[Sergius Lavrov]] egerunt de condicionibus [[Indutiae|indutiarum]], sed frustra.&lt;ref&gt;[https://www.bbc.com/news/world-europe-60687203 Ukraine war: No progress on ceasefire after Kyiv-Moscow talks] (BBC).&lt;/ref&gt;

'''Die 11 Martii 2022'''&lt;br/&gt;
Impetus Russicos in urbes Kioviam et Mariupolim auctos esse nuntiatum est. Kioviae impetus ex aere plerumque ex orientali et septentrionali parte fieri videntur. Mariupolis etiamnunc circumcincta est pontibus ac viis in urbem destructis.&lt;ref&gt;[https://interfax.com/newsroom/top-stories/76463/ Mariupol blocked. All bridges, approaches to town destroyed]&lt;/ref&gt; Coniectiones telorum illic videntur continuari, senatu urbis nuntiante hucusque ad minimum 1582 imbelles necatos esse.&lt;ref&gt;[https://www.reuters.com/world/ukraines-mariupol-says-1582-civilians-killed-by-russian-shelling-blockade-2022-03-11/ Ukraine's Mariupol says 1,582 civilians killed by Russian shelling and blockade]&lt;/ref&gt;

'''Die 12 Martii 2022'''&lt;br/&gt;
Praeses Ucrainae iam 1300 milites Ucrainicos mortuos esse nuntiavit. [[Melitopolis|Melitopoli]], in urbe [[Regio Zaporizhiensis|regionis Zaporizhiensis]] Ucrainae, [[concursus populi]] factus est, incolis postulantibus, ut [[demarchus]] urbis, Ivan Fedorov, qui a copiis Russicis [[Decessio coacta|abductus]] esse dicebatur, in libertatem vindicaretur.&lt;ref&gt;[https://www.bbc.com/news/world-europe-60719123 Ukraine war: Protests after Russians 'abduct' Melitopol mayor] (BBC).&lt;/ref&gt;

'''Die 13 Martii 2022'''&lt;br/&gt;
Prope [[Leopolis|Leopolim]], circiter 30 chiliometra a [[Polonia|Poloniae]] finibus, rochetas missas esse ferunt, quarum crepitus etiam in Polonia audiebatur.&lt;ref&gt;[https://wiadomosci.onet.pl/swiat/inwazja-rosji-na-ukraine-jaworow-atak-niemal-30-km-od-granicy-z-polska/nh404rr?utm_source=tw_wiadomosci&amp;utm_medium=social&amp;utm_campaign=onetsg_fb ''Nocny atak Rosjan. Niemal 30 km od granicy z Polską'' (Impetus nocturnus Russicus, fere 30 chiliometra a Poloniae finibus) {{Ling|Polonice}}.]&lt;/ref&gt;

'''Die 14 Martii 2022'''&lt;br/&gt;
In emissione nuntiorum Russici [[Trames televisionis|tramitis televisionis]] nomine [[Canalis primus (trames televisionis)|''Primi canalis'']] (Russice ''Первый канал'') [[Marina Ovsiannikova]] in cameram ostendit tabulam, in qua scripta erat bellum finiendum nec propagandae credendum esse.&lt;ref&gt;[https://www.bbc.com/news/av/world-europe-60745212 Ukraine war: Demonstrator disrupts Russia's flagship evening news broadcast] (BBC).&lt;/ref&gt;

'''Die 15 Martii 2022'''&lt;br/&gt;
Congressus [[interrete|interretialis]] de indutiis habitus est. Dum [[Kiovia|Kioviae]] [[periculum]] impetus maximum fit, [[ignitegium]] per 35 [[hora]]s declaratum est.&lt;ref&gt;[https://www.aljazeera.com/news/2022/3/15/kyiv-to-impose-35-hour-curfew-amid-fresh-russian-attacks Kyiv to impose 35-hour curfew amid fresh Russian attacks]&lt;/ref&gt;

'''Die 16 Martii 2022'''&lt;br/&gt;
[[Mariupolis|Mariupoli]] [[Valetudinarium|nosocomium]] invasum est quadringentis fere civibus captis.&lt;ref&gt;[https://www.bbc.com/news/world-europe-60757133 Ukraine war: Hostages as Russian forces occupy hospital, official says]&lt;/ref&gt; Impetus bombarum in [[theatrum]], quo cives fugerant, factus est.&lt;ref&gt;[https://www.bbc.com/news/world-europe-60772331 Russia attacks theatre sheltering civilians, Ukraine says]&lt;/ref&gt; [[Russia]] omnis exclusa est a [[Consilium Europae|Consilio Europae]], ex articulo 8 constituti eiusdem Consilii.&lt;ref&gt;[https://www.coe.int/en/web/portal/-/the-russian-federation-is-excluded-from-the-council-of-europe The Russian Federation is excluded from the Council of Europe]&lt;/ref&gt; [[Iudicium inter Civitates|Tribunal internationale Hagense]] rectionem Russiae ab invasione in Ucrainam desistere iussit.&lt;ref&gt;[https://www.theguardian.com/world/2022/mar/16/un-international-court-of-justice-orders-russia-to-halt-invasion-of-ukraine UN international court of justice orders Russia to halt invasion of Ukraine]&lt;/ref&gt;

'''Die 18 Martii 2022'''&lt;br/&gt;
Impetus bombarum in quaedam aedificia prope [[Aëroportus|aeroportum]] urbis [[Leopolis]] factus est.&lt;ref&gt;[https://www.bbc.com/news/world-europe-60506682 Ukraine war in maps: Tracking the Russian invasion]&lt;/ref&gt; [[Vladimirus Putin]] in stadio [[Moscua|Moscovitensi]] festum dedit ad octo annos [[Annexatio|annexionis]] [[Crimaea (res publica)|Crimaeae]] celebrandos.&lt;ref&gt;[https://www.bbc.com/news/world-europe-60793319 Putin hails Crimae annexation war with lessions on heroism]&lt;/ref&gt; [[Telum hypersonicum|Rochetam hypersonicam]] nomine [[Kh 47M2 Kinsal|''Kinsal'']] (Russice ''Кинжал'', Latine ''Pugio'') in [[Regio Ivano-Frankivskensis|regione Ivano-Frankivskensi]] in Ucraina occidentali adhibitam esse nuntiatum est.&lt;ref&gt;[https://ria.ru/20220319/sklad-1778982374.html Российские  военные  уничтожили скрад боеприпасов ВСУ ракетами &quot;Кинжал&quot;.]&lt;/ref&gt;
[[Fasciculus:Garden of the Righteous, Warsaw, March 19, 2022 Protest.jpg|thumb|''Hortus Iustorum'' Varsoviae. Arbor memorialis [[Anna Politkovskaya|Annae Politkovskayae]], die 19 Martii 2022.]]

'''Die 20 Martii 2022'''&lt;br/&gt;
[[Mariupolis|Mariupoli]] impetus bombarum in [[Schola artis|scholam artis]], quo cives fugerant, factus est quadringentis fere hominibus occisis. Quam rem [[Vladimirus Zelens'kyj]] [[scelus contra humanitatem]] vocavit.&lt;ref&gt;[https://www.abc.net.au/news/2022-03-20/ukraine-zelenskyy-russia-war-crimes-mariupol/100925240 Ukrainian art school shelting 400 residents in Mariupoli hit by Russian bombs, city council says]&lt;/ref&gt;&lt;ref&gt;[https://www.bbc.com/news/world-europe-60814913 In Mariupoli, children bear the brunt of Vladimir Putin's war]&lt;/ref&gt;

'''Die 21 Martii 2022'''&lt;br/&gt;
Mariupoli copiae Russicae, ut incolis facultatem urbis relinquendae facerent pariterque auxiliis humanitariis aditum in urbem ipsam darent, a Mariupolitanis [[Deditio|deditionem]] poposcerunt, id quod rectio Ucrainae reiecit.&lt;ref&gt;[https://www.bbc.com/news/world-europe-60816885 Ukraine conflict: Russia trying to starve Mariupol into surrender - MP]&lt;/ref&gt;

'''Die 23 Martii 2022'''&lt;br/&gt;
Ad [[Kiovia|Kioviam]] copiae Russicae feruntur bombis [[Phosphorus|phosphoricis]] usae esse,&lt;ref&gt;[https://dip.org.ua/en/news-of-the-mfa-of-ukraine/russians-used-phosphorus-bombs-near-kiev/ Russians used phosphorus bombs near Kiev]&lt;/ref&gt; id quod ex [[Conventus Genavenses|Conventibus Genavensibus]] nullo pacto fieri licet. Oxana Baulina, [[Diurnariorum ars|diurnaria]] Russa, Kioviae necata est.&lt;ref&gt;[https://www.bbc.com/news/world-europe-60855732 Ukraine war: Russian journalist Oksana Baulina killed in Kyiv shelling]&lt;/ref&gt;

'''Die 24 Martii 2022'''&lt;br/&gt;
Duces civitatum occidentalium, h. e. [[Consociatio ex pacto Atlantico Septentrionali|Consociationis ex pacto Atlantico Septentrionali]] (OTAN), civitatum G7, [[Unio Europaea|Unionis Europaeae]], [[Bruxellae (regio)|Bruxellis]] convenerunt, ut consulerent, quomodo Ucraina adiuvari posset.&lt;ref&gt;[https://www.bbc.com/news/world-europe-60849917 Western leaders meet to discuss Ukraine support]&lt;/ref&gt;

'''Die 25 Martii 2022'''&lt;br/&gt;
[[Iosephus Biden]], praeses [[Civitates Foederatae Americae|Civitatum Foederatarum]], in [[Polonia|Poloniam]] venit ad concordiam [[Solidaritas|solidaritatem]]&lt;nowiki/&gt;que cum populo Ucrainico affirmandam. Papa [[Franciscus (papa)|Franciscus]] pro populo Ucrainico oravit memor [[Vaticinatio|vaticinationis]] de pace, quae [[Fatima|Fatimae]], in oppido [[Portugallia|Portugalliae]], anno 1917 praedicta erat.&lt;ref&gt;[https://abcnews.go.com/International/wireStory/popes-peace-prayer-ukraine-recalls-ancient-prophesy-83667038 Pope's peace prayer for Ukraine recalls Fatima prophecy]&lt;/ref&gt;

'''Die 26 Martii 2022'''&lt;br/&gt;
Sergius Rudskoy, princeps consilii praetorii [[Copiae militares|virium]] Russicarum, annuntiavit copias Russiae animum attenturas esse ad orientalem Ucrainae regionem &quot;liberandam&quot;, significans fieri posse, ut [[strategia]] mutaretur.&lt;ref&gt;[https://www.bbc.com/news/world-europe-60872358 Russia targets east Ukraine, says first phrase over]&lt;/ref&gt;

'''Die 27 Martii 2022'''&lt;br/&gt;
[[Leopolis|Leopoli]] impetus bombarum factus est.

'''Die 28 Martii 2022'''&lt;br/&gt;
Urbs [[Mariupolis]] maxima ex parte destructa est et ad quadraginta milia hominum ab initio incursionis ad Russicum territorium [[Decessio coacta|abducti]] sunt.&lt;ref&gt;[https://www.bbc.com/news/world-europe-60894142 Russia transfers thousands of Mariupol civilians to its territory]&lt;/ref&gt; [[Diurnaria|Diarium]] Russicum liberale, cui nomen est ''[[Novaya Gazeta]]'', exemplaria divulgare vetitum est iussu [[Censura|censurae]] Russicae.&lt;ref&gt;[https://abcnews.go.com/International/live-updates/russia-ukraine/?id=83390885#83721085 Russia-Ucraine live updates: Russia's Novaya Gazeta newspaper suspends publication]&lt;/ref&gt;

'''Die 29 Martii 2022'''&lt;br/&gt;
[[Constantinopolis|Constantinopoli]] conventus de [[Indutiae|indutiis]] ineundis habitus est, in quo legatio Ucrainica proposuit, ut Ucraina [[Neutralitas|neutri parti se adiungeret]] et, ut solebat, [[Arma nuclearia|armis nuclearibus]] careret. Legatio Russica non mediocrem remissionem pugnarum annuntiavit, praesertim in regionibus Kioviensi et [[Czernihovia|Czernihoviensi]], id quod a civitatibus occidentalibus in dubium vocabatur.&lt;ref&gt;[https://www.reuters.com/world/europe/ukraine-sets-ceasefire-goal-new-russia-talks-breakthrough-looks-distant-2022-03-29/ Russia promises to scale back Ukraine war but West sceptical]&lt;/ref&gt;

'''Die 30 Martii 2022'''&lt;br/&gt;
Contra consensum indutiarum pridie deliberatum impetus bombarum [[Czernihovia|Czernihoviae]] et [[Kiovia|Kioviae]] factus est.&lt;ref&gt;[https://www.bbc.com/news/world-europe-60925713 War in Ukraine; Russia lauches news attacks after peace promise]&lt;/ref&gt; Secundum [[UNICEF]], plus quam [[Decies centena milia|duodecies centena milia]] impuberum ex patria fugati atque in [[Exsilium|exsilio]] sunt.&lt;ref&gt;[https://www.unicef.org/press-releases/two-million-refugee-children-flee-war-ukraine-search-safety-across-borders Two million refugee children flee war in Ukrainein search safety across borders]&lt;/ref&gt;

'''Die 31 Martii 2022'''&lt;br/&gt;
Rectio Russiae poenis oeconomicis repugnans minata est se subvectionem gasii naturalis esse omissuram, si negotia cum civitatibus occidentalibus non per [[Rubelus Russicus|Rubelos]] expedirentur.&lt;ref&gt;[https://www.bbc.com/news/business-60945248 Russia threatens to stop supplying gas if not paid in roubles]&lt;/ref&gt;

'''Die 1 Aprilis 2022'''&lt;br/&gt;
[[Bielogroda|Bielogrodae]], in urbe Russiae, tela ignifera ex duobus [[Helicopterum|helicopteris]] coniecta sunt in quoddam depositorium petrolei. Rectio Russiae culpam illius facti Ucrainae attribuit.&lt;ref&gt;[https://www.bbc.com/news/world-europe-60952125 War in Ucraine: Russia accuses Ukraine of attacking oildepot]&lt;/ref&gt; Milites Russici [[ergasterium atomicum]] [[Czernobela|Czernobelense]] reliquerunt.&lt;ref&gt;[https://www.bbc.com/news/world-europe-60945666 Ukraine war: Russian troops leave Chernobyl, Ukraine says]&lt;/ref&gt;

'''Die 2 Aprilis 2022'''&lt;br/&gt;
Copiae Russicae a militibus Ucrainicis ex urbe Irpin, in regione [[Kiovia|Kioviensi]] sita, pulsae sunt.&lt;ref&gt;[https://www.bbc.com/news/world-europe-60959667 Battle for Irpin: Russian forces pushed out of Kyiv suburb]&lt;/ref&gt;

'''Die 3 Aprilis 2022'''&lt;br/&gt;
[[Kiovia]] contentione militum Ucrainicorum liberata est. At [[Genocidium in Bucha|Buchae]], in oppido [[Regio Kioviensis|regionis Kioviensis]], multi homines more abominabili [[Mina terrestris|minis terrestribus]] necati sunt, id quod praeses Vladimirus Zelens'kyj [[Scelus bellicum|scelus belli]] et scelus contra humanitatem condemnavit.&lt;ref&gt;[https://www.bbc.com/news/world-europe-60970818 Ukraine war: Bucha street littered with burned-out tanks and corpes]&lt;/ref&gt; [[Odessa|Odessae]] impetus bombarum in aedificia petrolei factus est.

'''Die 4 Aprilis 2022'''&lt;br/&gt;
Vladimirus Zelens'kyj ipse Bucham iit ea loca visum, ubi [[homicidium]] factum erat, et illud scelus [[genocidium]] atrox iudicavit. Rectione Russiae negante tamen exstant [[satelles artificialis|satellitum]] imagines, quae [[oratio]]ni Russicae repugnant.&lt;ref&gt;[https://www.bbc.com/news/60981238 &quot;Bucha killings: Satellite image of bodies site contradicts Russian claims.&quot;]&lt;/ref&gt; [[Unio Europaea]] et [[Civitates Foederatae]] rem condemnaverunt, novam poenam contra Russiae rectionem postulantes.&lt;ref&gt;[https://www.cnbc.com/2022/04/04/ukraine-russia-facing-new-sanctions-after-alleged-atrocities-in-bucha.html Russia facing new sanctions after accusation of atrocities in Bucha, Ukraine]&lt;/ref&gt;

'''Die 5 Aprilis 2022'''&lt;br/&gt;
Vladimirus Zelens'kyj, praeses Ucrainae, [[Concilium Salutis]] de bello et genocidio in Ucraina nec non de pace allocutus est ope [[Telecommunicatio|telecommunicationis]].

'''Die 6 Aprilis 2022'''&lt;br/&gt;
[[Unio Europaea]] commercium [[Carbo|carbonis]] interdixit ut poena [[Genocidium in Bucha|genocidii Buchae]] facti. Papa [[Franciscus (papa)|Franciscus]] pro genocidii victimis oravit cum [[Vexillum Ucrainae|vexillo Ucrainae]] ex Bucha allato, et rem atrocem condemnavit.&lt;ref&gt;[https://www.reuters.com/world/europe/pope-holding-ukrainian-flag-condemns-atrocities-such-massacre-bucha-2022-04-06/ &quot;Pope kisses Ukrainian flag, condemns 'the massacre of Bucha.'&quot;]&lt;/ref&gt;

'''Die 7 Aprilis 2022'''&lt;br/&gt;
[[Conventus generalis (Nationes Unitae)|Conventus generalis Consociationis Nationum]] sententiam tulit, ut Russia e [[Consilium iurum humanorum Nationum Unitarum|Consilio iurum humanorum]] excluderetur.&lt;ref&gt;[https://thehill.com/policy/international/3261462-un-voting-to-suspend-russia-from-human-rights-council/ &quot;UN votes to suspend Russia from human rights council over killing in Bucha.&quot;]&lt;/ref&gt;

=== Phasis secunda (diebus 8 Aprilis - currentibus) ===
'''Die 8 Aprilis 2022'''&lt;br/&gt;
In [[statio ferriviaria|stationem ferriviariam]] urbis Kramatorsk, in [[Regio Donetskensis|regione Donetskensi]] in Ucraina orientali sitae, impetus [[missile|rochetarum]] factus est, quo plus quam 52 homines ex ea urbe fugituri necati sunt.&lt;ref&gt;[https://www.cbsnews.com/news/ukraine-russia-kramatorsk-train-station-donetsk-strike/ &quot;Ukraine says at least 52 people killed in Russian rocket attack on Kramatorsk train station.&quot;]&lt;/ref&gt; Vladimirus Zelens'kyj [[Boris Johnson]], [[primus minister|primum ministrum]] [[Britanniarum Regnum|Britannarum Regni]], Kioviae convenit. Deinde cum [[Ursula de Leyen]], [[Commissio Europaea|Commissionis Europaeae]] praeside, collocutus est.&lt;ref&gt;[https://www.dw.com/en/ukraine-eu-chief-offers-kyiv-first-track-to-membership/a-61409635 Ukraine: EU chief offers Kyiv fast track to membership].&lt;/ref&gt;

'''Die 9 Aprilis 2022'''&lt;br/&gt;
Corpora 132 civium reperta esse Makariviae, in oppido regionis Kioviensis, palam factum est.&lt;ref&gt;[https://www3.nhk.or.jp/nhkworld/en/news/20220409_19/ Bodies of 132 civilians reportedly found in Makariv, near Kyiv].&lt;/ref&gt;

'''Die 11 Aprilis 2022'''&lt;br/&gt;
Carolus Nehammer, primus minister [[Austria]]cus, Moscuam iit, ubi cum Vladimiro Putin collocutus finem belli impetrare conatus est.&lt;ref&gt;[https://globalnews.ca/news/8751491/austria-vladimir-putin-talks-russia-ukraine-war/ Austria's leader urges Vladimir Putin to stop Ukraine war durning in-person meeting].&lt;/ref&gt;

'''Die 12 Aprilis 2022'''&lt;br/&gt;
Vladimirus Putin se incursionem continuare velle dixit, donec &quot;nobile&quot; propositum patriae assequeretur.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61077648 Ukraine war: Putin vows continue invation until noble aims met]&lt;/ref&gt; [[Mariupolis|Mariupoli]] copiae Russicae portum urbis aggredientes [[Arma chemica|armis chemicis]] usae esse dicuntur.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61077641 Ukraine War: US &quot;deeply concerned at report of Mariupol chemical attack].&lt;/ref&gt;

'''Die 13 Aprilis 2022'''&lt;br/&gt;
Vladimirus Zelens'kyj invitationem [[Franciscus Gualterius Steinmeier|Francisci Gualterii Steinmeier]], praesidis Germaniae, rationibus politicis recusavit, quia ille tum, cum rerum externarum Germaniae minister erat, in gasio naturali ex Russia acquirendo iuverat.&lt;ref&gt;[https://www.teletrader.com/zelensky-refuses-to-host-germany-s-steinmeier-report/news/details/57672007 Zelensky refuses to host Germany-s Steinmeiner - report.]&lt;/ref&gt;

[[Fasciculus:Stamp of Ukraine s1985.jpg|thumb|[[Pittacium cursuale]]
'''Die 14 Aprilis 2022'''&lt;br/&gt;
Ucrainicum, duo dies ante [[navis|navem]] mersam publicatum, [[miles|militem]] Ucrainicum et navem pingit.&lt;ref&gt;{{cite news |date=14 Aprilis [[2022]] |title=Ukrainian soldiers who told Russian warship 'go f*** yourself' honoured on postal stamps |website=[[itv.com]] |url=https://www.itv.com/news/2022-04-14/ukrainian-soldiers-who-told-russian-warship-go-f-yourself-honoured-on-stamp |url-status=live |archive-url=https://web.archive.org/web/20220414202402/https://www.itv.com/news/2022-04-14/ukrainian-soldiers-who-told-russian-warship-go-f-yourself-honoured-on-stamp |archive-date=14 Aprilis 2022 |accessdate=14 Aprilis 2022}}.&lt;/ref&gt;&lt;ref&gt;{{cite news |last=Michael |first=Chris |date=12 Martii [[2022]] |title=Ukraine reveals 'Russian warship, go fuck yourself!' postage stamp |newspaper=[[The Guardian]] |url=https://www.theguardian.com/world/2022/mar/12/ukraine-reveals-russian-warship-go-fuck-yourself-postage-stamp |url-status=live |archive-url=https://web.archive.org/web/20220312120555/https://www.theguardian.com/world/2022/mar/12/ukraine-reveals-russian-warship-go-fuck-yourself-postage-stamp |archive-date=12 Martii 2022 |accessdate=14 Aprilis 2022}}.&lt;/ref&gt;]]
[[Moskva (navis bellica)|''Moskva'']], [[navis bellica]] et [[navis praetoria]] Classis Russicae, in [[Pontus Euxinus|Mari Nigro]] mersa est. Incertum erat, quomodo navis illud detrimentum passa esset, videlicet, an affecta esset [[missile|missilibus]] generis [[R 360 Neptunus (rocheta)|Neptuni]] (secundum rectionem Ucrainae), an [[Incendium|calamitate ignis]] (secundum rectionem Russiae).&lt;ref&gt;[https://www.bbc.com/news/world-europe-61114843 Russian warship Moskva has sunk - defence ministry].&lt;/ref&gt;

'''Die 15 Aprilis 2022'''&lt;br/&gt;
Copiae Russicae tela ignifera in Kioviam coniecerunt, postquam navis bellica ''Moskva'' mersa est.&lt;ref&gt;[https://www.theguardian.com/world/live/2022/apr/15/russia-ukraine-war-latest-russia-says-moskva-warship-has-sunk-after-ukraine-claims-missile-strike-live Russian Ukraine war latest: explosions in Kyiv after sinking of Russian warship; fighting reported at Mariupol steel plant - live].&lt;/ref&gt;

'''Die 16 Aprilis 2022'''&lt;br/&gt;
Rectio Russiae epistulam diplomaticam ad Civitates Foederatas misit admonens, ut, si de praebitione [[Arma|armorum]] non destitissent, &quot;consequentia non praedicenda&quot; fierent.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61122074 Ukraine round-up: Russia sends warning to US]&lt;/ref&gt; Insuper rectio Russiae tredecim hominibus politicis Britannarum Regni, inter eos primus minister [[Boris Johnson]], omni Russia interdixit.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61126391 Ukraine war: Russia bans Boris Johson from country over Ukraine war].&lt;/ref&gt;

'''Die 17 Aprilis 2022'''&lt;br/&gt;
Rectio Ucrainae condicionem a copiis Russicis positam, scilicet ut defensores urbis [[Mariupolis]] [[Deditio|arma deponerent]], reiecit. Papa Franciscus oratione sua, quam Romae per sollemnia [[Pascha (festum Christianum)|Paschatis]] habuerat, violentiam destructionemque a Russia actam condemnavit.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61135894 Ukraine round-up: Ukraine defies Mariupol deadline as Kharkiv shelled].&lt;/ref&gt;

'''Die 18 Aprilis 2022'''&lt;br/&gt;
[[Leopolis|Leopoli]], in urbe Ucrainae occidentalis, quae incursione Russica hucusque intacta erat, primus impetus telorum igniferorum factus est, quo homines ex populo necatos esse ferunt.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61141817 Ukraine war: First civilian deaths in Lviv shatter sense of safety]&lt;/ref&gt;

'''Die 19 Aprilis 2022'''&lt;br/&gt;
Vladimirus Zelens'kyj copias militares Russiae proelium magnum in regionibus [[regio Donetskensis|Donetskensi]] et [[regio Luganskensis|Luganskensi]] (Donbas coniunctim vocatis) inicitare affirmavit.&lt;ref&gt;[https://www.reuters.com/business/aerospace-defense/ukraine-says-battle-donbas-has-begun-russia-pushing-east-2022-04-18/ Ukraine says 'Battle of Donbas' has begun, Russia pushing in east] (Reuters).&lt;/ref&gt; [[Dionysius Šmyhal]] defensores civiles et milites Mariupolis usque ad ultimum sine deditione pugnaturos esse dixit.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61135901 Ukraine war: Mariupol defenders will fight to the end, says PM] (BBC).&lt;/ref&gt;

'''Die 20 Aprilis 2022'''&lt;br/&gt;
Dux militaris [[Mariupolis]] ultimam petitionem subsidii postulavit, dum [[Fabrica (officina)|fabrica]] metallurgica nomine Asovstal (Ucrainice Азовсталь), in qua [[ferrum]] et [[chalybs]] efficiebatur quaeque illi et mille fere hominibus adhuc refugio est, magno impetui telorum igniferorum subicitur.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61159812 Ukraine war: Mariupol commander makes 'last' plea for help] (BBC).&lt;/ref&gt;  

'''Die 21 Aprilis 2022'''&lt;br/&gt;
Vladimirus Putin impetui in fabricam metallurgicam Asovstal finem fieri et ipsum locum occludi iussit, ut, dum multi homines in Asovstal includuntur, &quot;ne musca quidem&quot; evolare posset. Mariupoli magnum [[sepulcretum]] hominum necatorum ex caelo inventum est.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61179734 Ukraine round-up: Putin backs off storming of Asovstal as Mariupol 'mass graves' found].&lt;/ref&gt;

'''Die 22 Aprilis 2022'''&lt;br/&gt;
Proelia in meridionali Ucraina et in orientali parte regionis Donbas aggravata sunt. Vladimirus Zelens'kyj dixit Russiam id agere, ut populus in iis regionibus occupatis habitans ad simulata suffragia vocaretur et, quibus factis, ita civitates fierent Russiae faventes.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61188943 Ukraine war: Russia aiming for full control of south].&lt;/ref&gt; Ucraina [[Moskva (navis bellica)|''Moskvam'']], navem bellicam die 14 Aprilis 2022 in [[Pontus Euxinus|Ponto]] mersam, &quot;hereditatem culturalem sub aqua&quot; appellavit.&lt;ref&gt;[https://mezha.media/en/2022/04/22/moskva-cruiser-has-become-an-object-of-underwater-cultural-heritage-of-ukraine/ Moskva cruiser has become an object of underwater cultural heritage of Ukraine].&lt;/ref&gt;

'''Die 23 Aprilis 2022'''&lt;br/&gt;
Duo duces Russici prope [[Chersonium]]&lt;ref&gt;[https://kyivindependent.com/uncategorized/military-intelligence-2-russian-generals-killed-near-kherson/ Military intelligence: 2 Russian generals killed near Kherson].&lt;/ref&gt; octoque imbelles, in quibus infans puellula trimestris, [[Odessa]]e telis igniferis necati sunt.&lt;ref&gt;[https://www.theguardian.com/world/live/2022/apr/23/russia-ukraine-war-latest-zelenskiy-warns-moscow-wants-to-capture-other-countries-as-moldova-expresses-concern-over-russian-plans-live Russia-Ukraine war: Baby killed in missile attack on Odesa; Zelenskiy says].&lt;/ref&gt;

'''Die 25 Aprilis 2022'''&lt;br/&gt;
Antonius Blinken, [[Civitatum Foederatarum Secretarius Civitatis]], Vladimirum Zelens'kyj convenit [[Kiovia|Kioviae]], una cum Lloyd Austin, Civitatum Foederatarum Secretario Defensionis, qui se sperare dixit Russicas amissiones in Ucraina factas administrationem Russiae dehortaturas esse, ne talia facinora alibi repeterentur.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61214176 Ukraine news: US wants to see a weakened Russia]&lt;/ref&gt;

'''Die 26 Aprilis 2022'''&lt;br/&gt;
[[Antonius Guterres]], [[Nationes Unitae|Nationum Unitarum]] secretarius generalis, Vladimirum Putin [[Moscua]]e convenit, quocum de auxilio humanitario Mariupolim ferendo deque hominibus ex urbe oppugnata servandis locutus est.&lt;ref&gt;[https://www.theguardian.com/world/2022/apr/26/un-guterres-putin-meeting-moscow-ukraine-war UN ready to evacuate people from Mariupol, says chief in Moscow visit]&lt;/ref&gt;

'''Die 27 Aprilis 2022'''&lt;br/&gt;
Vladimirus Putin exteras civitates monuit, ut caverent, ne rebus, quae in Ucraina gererentur, se interponerent; si vero civitas aliquae sese interponere conata esset, eandem rapidissimum responsum &quot;fulminis more&quot; laturam esse. [[Gazprom]] subvectionem gasii naturalis in [[Polonia]]m et [[Bulgaria]]m omisit.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61252320 Ukraine war: Putin warns against foreign intervention]&lt;/ref&gt;

'''Die 28 Aprilis 2022'''&lt;br/&gt;
[[Antonius Guterres]] Vladimirum Zelens'kyj Kioviae convenit profitens [[Consilium securitatis]] efficere non potuisse, ut bellum evitaretur aut conficeretur, eamque impotentiam esse causam magnae frustrationis et irae. [[Vladimirus Zelens'kyj]] affirmavit Antonio Guterres copiam fuisse, ut omnia [[scelus bellicum|scelera bellica]], quae a Russia in Ucraina essent commissa, videret ac testaretur.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61265635 Ukraine war: UN chief admits Security Council has failed]&lt;/ref&gt;

'''Die 29 Aprilis 2022'''&lt;br/&gt;
Rectio Ucrainae persecutionem decem militum Russicorum inchoavit, qui arguuntur [[genocidium in Bucha|scelera bellica Buchae]] commisisse, in quibus cives imbelles cruciatos, violatos, necatos esse constat.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61269480 Ukraine launches hunt for Russian soldiers accused of Bucha war crimes]&lt;/ref&gt;

'''Die 30 Aprilis 2022'''&lt;br/&gt;
Urbs [[Charcovia]] telis missilibus identidem quassata est, quamvis Vladimirus Zelens'kyj affirmavisset vires Ucrainicas in ea regione &quot;belligerando aliquantum succedere&quot;.&lt;ref&gt;[https://www.france24.com/en/live-news/20220430-kharkiv-shelled-as-russia-maintains-its-offensive Kharkiv shelled as Russia maintains its offensive]&lt;/ref&gt;

'''Die 1 Maii 2022'''&lt;br/&gt;
Aeroplana militaria Russica paulisper per spatium aerium et [[Suecia|Sueciae]]&lt;ref&gt;''Dagens nyheter'': [https://www.dn.se/sverige/ryskt-militarplan-har-krankt-svenskt-luftrum/ Aeroplanum militare Russicum spatium aerium Suecicum violavit.] {{Ling|Suecice}}&lt;/ref&gt; et [[Dania|Daniae]]&lt;ref&gt;''Politiken'': [https://politiken.dk/indland/art8747811/»Russerne-sender-os-en-hilsen-og-siger-at-de-er-trætte-af-sanktioner« Russi nobis salutem mittunt dicuntque se sanctionibus satiatos esse.] {{Ling|Danice}}&lt;/ref&gt; volaverunt. [[Mariupolis|Mariupoli]] viginti fere cives fabricam metallurgicam Asovstal reliquerunt.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61285178 Mariupol civilians leave besieged Asovstal steelworks]&lt;/ref&gt;

'''Die 2 Maii 2022'''&lt;br/&gt;
[[Sergius Lavrov]], rerum externarum Russiae minister, in percontatione televisifica interrogatus, quemadmodum Russia asseverare posset se pro &quot;de-nazificanda&quot; Ucraina pugnare, praesertim cum praeses [[Vladimirus Zelens'kyj]] ipse esset [[Iudaeus]], aperte respondit &quot;etiam [[Adolfus Hitler|Adolfum Hitler]] originem suam a Iudaeis traxisse; id omnino nihil significare&quot;.&lt;ref&gt;[https://www.bbc.com/news/world-middle-east-61296682 Israel outrage at Sergei Lavrov's claim that Hitler was part Jewish]&lt;/ref&gt; Quibus dictis Lavrov praesidem Ucrainae cum molitore [[Soa|Holocausti]] comparavit, id quod rectio [[Israel|Israelis]] indignissime tulit, renuntians ea dicta esse cum inexcusabilem flagitiosissimamque pronuntiationem tum terribilem errorem historicum.&lt;ref&gt;[https://www.latimes.com/world-nation/story/2022-05-02/israel-slams-russian-comments-nazism-antisemitism-ukraine Israel says Russian comments on Nazism and Ukraine are 'unforgivable']&lt;/ref&gt;

'''Die 3 Maii 2022'''&lt;br/&gt;
Vladimirus Putin cum [[Emmanuel Macron|Emmanuele Macron]] diutissime per [[telephonum]] collocutus est, monens [[Consociatio ex pacto Atlantico Septentrionali|civitatibus occidentalibus]], si pacem voluissent, desinendum esse arma in Ucrainam mittere.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61313732 Ukaine war: Putin tells Macron West should stop sending arms]&lt;/ref&gt; Russi impetum [[Missile|telorum missilium]] generis ''Onyx'' in aeroportum militarem [[Odessa|Odessensem]] fecerunt, ubi etiam [[aeronavium receptacula]], in quibus arma peregre illata deposita esse dicebantur, deleta sunt.&lt;ref&gt;[https://apa.az/en/cis-countries/russian-forces-hit-hangars-with-foreign-arms-in-odessa-with-onyx-missiles-375066 Russian forces hit hangars with foreign arms in Odessa with Onyx missiles]&lt;/ref&gt; Ministerium rerum externarum [[Russia|Russicae Foederationis]] ea, quae [[Sergius Lavrov|Sergio Lavrov]] antea dicere placuerant, iteravit et defendit, nempe originem praesidis Zelens'kyj sententiae [[Cremlinus (Moscua)|Cremlinicae]] non repugnare, quominus Ucraina a &quot;neonazistis&quot; regeretur, insuper afferens rectionem [[Israël (civitas)|Israelis]] &quot;neonazistas&quot; in Ucraina adiuvare.&lt;ref&gt;[https://www.cbsnews.com/news/russia-sergey-lavrov-hitler-remarks-accuses-israel-supporting-ukraine-neo-nazis/ Russia doubles down on foreign minister's Hitler remarks, accuses Israel of supporting &quot;neo-Nazis&quot; in Ukraine]&lt;/ref&gt;

'''Die 4 Maii 2022'''&lt;br/&gt;
[[Unio Europaea]] complures novorum consiliorum rationes, fortasse acerrimas omnium, quas hucusque imponebat, proposuit, inter alia plenam interdictionem [[Petroleum|petrolei]] ex Russia importandi et sanctiones in eos, qui [[Scelus bellicum|scelera belli]] commisisse putantur.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61318689 EU plans Russian oil ban and war crimes sanctions]&lt;/ref&gt; Ministerium agri culturae Ucrainae nuntiavit Russiam 400.000 tonnarum frumenti e regionibus occupatis asportanda curavisse, tertiam igitur partem copiae frumenti, quae in regionibus [[Chersonium|Chersoniensi]], [[Zaporizhia|Zaporizhiensi]], [[Donetsk|Donetskensi]] et [[Lugansk|Luganskensi]] seposita erat; si copia rei frumentariae deinceps esset imminuta, [[Fames|famem]] instaturam esse;&lt;ref&gt;[https://www.republicworld.com/world-news/russia-ukraine-crisis/ukraine-claims-400000-tonnes-of-grain-stolen-by-russian-forces-from-occupied-areas-articleshow.html Ukraine Claims 400,000 Tonnes Of Grain 'stolen' By Russian Forces From Occupied Areas]&lt;/ref&gt; urbem [[Mariupolis|Mariupolim]] a mense Martio iam fame urgeri.&lt;ref&gt;[https://www.ft.com/content/af7996a9-8c16-4421-a5b3-390315d3c7dc ‘Hell on earth’: survivors recount the assault on Mariupol]&lt;/ref&gt; Ministerium defensionis Russiae rettulit circiter 100 milites Russicos in quodam periculo [[Kaliningradum|Kaliningradi]] facto &quot;simulasse&quot; [[Rocheta|rochetas]] generis ''Iskander'' ad [[arma nuclearia]] portanda accommodatas &quot;electronice mittere&quot; necnon impetus, qui talem missionem sequi possent, prohibere.&lt;ref&gt;[https://www.themoscowtimes.com/2022/05/05/russia-simulates-nuclear-capable-strikes-near-eu-a77586 Russia Simulates Nuclear Strikes Near EU]&lt;/ref&gt;

'''Die 5 Maii 2022'''&lt;br/&gt;
Antonius Guterres, secretarius generalis Nationum Unitarum, dixit omnia esse experienda, ut homines, qui sese in fabricam metallurgicam Asovstal confugissent, &quot;ex iis locis infernis&quot; eriperentur. Vladimirus Putin [[Copiae militares|vires suas]] tutum transitum hominibus daturas esse affirmavit, si propugnatores Ucrainici sese dedituri essent. Rectio [[Israel (civitas)|Israelis]] nuntiavit praesidem Putin se telephonice de dictis [[Antisemitismus|antisemiticis]] a ministro Lavrov elocutis excusavisse.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61342436 Mariupol fighting: More evacuations from besieged city on Friday, UN says]&lt;/ref&gt;

'''Die 6 Maii 2022'''&lt;br/&gt;
Urgentibus causis [[Programma mundiale alimenti]] Nationum Unitarum expoposcit, ut [[portus]] in regione [[Odessa|Odessensi]] rursus patefierent, ita ut [[Cibus|cibaria]], quae in Ucraina bello vexata proveniunt, libere in ceteras orbis terrarum partes evehi atque exportari possent ad instantem crisim [[Fames|famis]] coercendam.&lt;ref&gt;[https://www.wfp.org/news/wfp-calls-urgent-opening-ukrainian-ports-help-rein-global-hunger-crisis WFP calls for urgent opening of Ukrainian ports to help rein in global hunger crisis]&lt;/ref&gt;

'''Die 7 Maii 2022'''&lt;br/&gt;
Vladimirus Zelens'kyj dixit Russiam, si de pacis condicionibus agere voluisset, in ea loca se recipere debere, quae ante incursionem tenuisset.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61359228 Ukraine war: Russia must withdraw to pre-invation position for a deal - Zelensky]&lt;/ref&gt; Aliquot diebus ante iam negaverat se ullas indutias, quae copiis Russicis permitterent in regionibus occupatis manere, esse accepturum.&lt;ref&gt;[https://www.marketwatch.com/story/zelensky-says-ukraine-cant-accept-any-peace-deal-that-leaves-russian-troops-on-his-countrys-territory-01651699666 Zelensky says Ukraine can’t accept any peace deal that leaves Russian troops on his country’s territory]&lt;/ref&gt;

'''Die 9 Maii 2022'''&lt;br/&gt;
Dum Moscuae [[Dies Victoriae]] in [[Forum Pulchrum|Foro Pulchro]] cum sollemni [[Pompa|pompa militari]] celebratur, Vladimirus Putin orationem ad populum habuit, in qua vetera cum recentioribus coniunxit.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61377886 Putin says Russia fighting for motherland in Ukraine in Victory Day speech]&lt;/ref&gt; Dixit enim patriae defensionem semper sanctam fuisse, cum fatum eius in discrimen venisset; nunc autem novam &quot;rem poenae&quot; in regione Donbas geri et &quot;[[Incursio magna|invasionem]]&quot; in terras Russiae antiquitus proprias, addita [[Crimaea]], palam praeparatam atque instructam esse; [[Kiovia|Kioviae]] comparationem [[Arma nuclearia|armorum nuclearium]] significatam esse; [[Consociatio ex pacto Atlantico Septentrionali|consociationem OTAN]] in territoriis Russiae adiacentibus omnia ad infrastructuram militarem pertinentia sedulo providere coepisse; periculum crevisse in dies; hanc aggressionem a Russia ad agendum coacta praeveniendo repulsam esse; illos, qui ''Magno Patrio Bello'' [[Nazismus|nazismum]] profligavissent, singulare exemplum [[Heros|virtutis fortitudinis]]&lt;nowiki/&gt;que in omne tempus Russis praebuisse; eos, qui nunc pugnarent, ea defendere, pro quibus eorum patres, avi, proavi pugnavissent.&lt;ref&gt;[http://kremlin.ru/events/president/news/68366 Выступление Президента России на военном параде (Oratio praesidis Russiae in pompa militari habita): ''&quot;Защита Родины, когда решалась её судьба, всегда была священной. (…) В открытую шла подготовка к очередной карательной операции на Донбассе, к вторжению на наши исторические земли, включая Крым. В Киеве заявляли о возможном приобретении ядерного оружия. Блок НАТО начал активное военное освоение прилегающих к нам территорий. (…) Опасность росла с каждым днём. (…) Россия дала упреждающий отпор агрессии. Это было вынужденное, своевременное и единственно правильное решение. (…) Те, кто сокрушил нацизм в годы Великой Отечественной войны, показали нам пример героизма на все времена. (…) Сегодня (они защищают) то, за что сражались (их) отцы и деды, прадеды.&quot;'']&lt;/ref&gt;
&lt;br/&gt;
Dixit praeses [[Francia|Franciae]], [[Emmanuel Macron]], Russiae ad dignitatem suam offendere non debere, &quot;non ad tentationem, non ad humiliationem, non ad ultitionis spiritum nos dedamus, quia pacis viae iam paucae et depopulatae.&quot;&lt;ref&gt;[https://lecourrier-du-soir.com/guerre-en-ukraine-macron-crache-le-morceau-la-paix-ne-se-construira-pas-dans-lhumiliation-de-la-russie/ Referentia secundum ''le courrier du soir''] {{Ling|Francogallice}}&lt;/ref&gt;

'''Die 10 Maii 2022'''&lt;br/&gt;
[[Emmanuel Macron]], iterum praeses [[Francia|Franciae]] creatus, [[Olaus Scholz|Olaum Scholz]], [[Cancellarius foederalis|cancellarium foederalem]] Germaniae, pridie [[Berolinum|Berolini]] convenerat, ubi in [[Conventus diurnariis edocendis|conventu diunariis edocendis]] de Ucraina in [[Unio Europaea|Unionem Europaeam]] accipienda locutus est, verens, ne essent complures anni, forsitan [[Decennium|decennia]], quoad Ucrainae liciturum esset sese ad Unionem Europaeam applicare.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61383632 Ukraine bid to join EU will take decades says Macron]&lt;/ref&gt; Dum [[Odessa|Odessae]] fiunt impetus rochetarum generis ''[[Kinsal]]'', relatum est milites Russicos tela in fabricam metallurgicam Asovstal conicere; adhuc centum fere imbelles cum aliquot defensoribus ibi perseverare.&lt;ref&gt;[https://www.bbc.com/news/live/world-europe-61353374 Ukraine latest news: Mariupol official says civilians still in steelworks]&lt;/ref&gt;&lt;ref&gt;[https://www.reuters.com/world/europe/least-100-civilians-remain-ukrainian-city-mariupols-steel-works-mayors-aide-2022-05-10/ Ukraine says Russia pounding Mariupol steel works, mayor's aide says 100 civilians remain]&lt;/ref&gt;

'''Die 11 Maii 2022'''&lt;br/&gt;
Ferunt copias Ucrainicas nonnullos [[Vicus rusticus|vicos rusticos]] ab urbe [[Charcovia]] ad septentriones et orientem solem vergentes a viribus Russicis recuperavisse easque retro ad fines compulisse.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61378196 Ukraine war: Russia pushed back from Kharkiv - refort from front line]&lt;/ref&gt;

'''Die 12 Maii 2022'''&lt;br/&gt;
Milites Russici instrumentis monitoriis deprehensi et excepti sunt, cum Ucrainos inermes peterent eosque a tergo necarent.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61425025 Ukraine conflict: Russian soldiers seen killing unarmed civilians]&lt;/ref&gt; Postquam [[Saulus Niinistö]], praeses [[Finnia|Finniae]], et [[Sanna Marin]], prima ministra Finniae, sententiam suam aperuerunt, ut Finnia [[Consociatio ex pacto Atlantico Septentrionali|Consociationi ex pacto Atlantico Septentrionali]] quam celerrime sese adiungeret, ministerium rerum externarum Russiae nuntiavit Foederationem Russicam coactum iri consilia voluntati Finnicae respondentia inire, ut crescens comminatio securitatis suae reprimeretur.&lt;ref&gt;[https://www.washingtonpost.com/world/2022/05/12/finland-nato-membership-russia-ukraine/ Finland moves to join NATO ‘without delay’ after Ukraine invasion]&lt;/ref&gt;

'''Die 14 Maii 2022'''&lt;br/&gt;
Copiae militares Ukrainae [[Charcovia]]m liberaverunt post proelium severum. Sed bellum summum in Ukraine lognum esse potest.&lt;ref&gt;[https://www.theguardian.com/world/2022/may/14/ukraine-has-won-the-battle-of-kharkiv-analysts-say-as-kyiv-warns-of-long-phase-of-war Ukraine has won the battle of Kharkiv, analysts says, as Kyiv warns of 'long phase of war']&lt;/ref&gt;

'''Die 16 Maii 2022'''&lt;br/&gt;
[[Factio socialis democratica (Suecia)|Factio socialis democratica]] rectionis [[Suecia]]e voluntatem ad [[OTAN]] cum Finlandia participandum confirmat.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61456726 Ukraine war: Sweden and Finland confirm Nato plans in historic shift].&lt;/ref&gt;

'''Die 17 Maii 2022'''&lt;br/&gt;
Omnes homines [[Mariupolis]] sub dedictionem iit ad ultimum post proelium octoginta dies.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61480988 Mariupol: The 80 days that lef a flourishing city in ruins]&lt;/ref&gt;

'''Die 18 Maii 2022'''&lt;br/&gt;
[[Antonius Guterres]], secreatrius generalis [[Nationes Unitae|Nationium Unitarum]] [[fames|discrimen alimentorum]] mundiale parere dixit, si [[portus]] [[Odessa]]e non liberatur et reconsturctus erit.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61503049 Ukraine invation could cause global food crisis, UN warns]&lt;/ref&gt;

'''Die 19 Maii 2022'''&lt;br/&gt;
Miles Russicus in custodia datur ratione sceli bellici civiem necandi eius [[culpa]]m confitetur apud primium [[iudicium]] de scelis bellcis in Ucraina.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61496428 Russian soldier pleads guilty in first war crims trial of Ukraine conflict]&lt;/ref&gt;

'''Die 20 Maii 2022'''&lt;br/&gt;
Rectio Ucrainae [[Ergasterium atomicum Zaporizhiense]] electricitatem ad Russiam non offere posse dixit, quia tempus longum necesse est ad circuitum electricitatis mutandum, sicut tempus [[pons|pontis]] inter [[Crimaea]]m et [[Paeninsula Tamanica]] fuit.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61524376 Ukraine says giant Zaporizhzhia nuclear plant can't supply Russia]&lt;/ref&gt;

'''Die 21 Maii 2022'''&lt;br/&gt;
Collegium Russicum [[Gazprom]] subvectionem gazii naturalis in [[Finlandia]]m omisit, ratione stipendii sed etiam voluntatis Finlandiae OTAN participandi putatur.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61524933 Russia halts gas supplies to Finland]&lt;/ref&gt;

'''Die 23 Maii 2022'''&lt;br/&gt;
Miles Russicus, Vadim Shishimarin, 21 annos natus, sententiam poenae perpetuae datur ratione [[scelus bellicum|scelii bellici]] qui civem 62 annos natum necavit.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61549569 Ukraine war: Russian soldier Vadim Shishimarin jailed for life over war crime]&lt;/ref&gt;

'''Die 24 Maii 2022'''&lt;br/&gt;
Voris Bondarev, [[legatus (civilitas)|legatus diplomatiae]] Russicus suum officium abrogavit, modo bellum sanguinarium et absurdum pro populi Ucrainis Russicisque a Vladimiro Putin [[motus contra bellum|condamnandi]].&lt;ref&gt;[https://www.bbc.com/news/world-europe-61555390 Russian diplomat quits over war in Ukraine]&lt;/ref&gt;

'''Die 25 Maii 2022'''&lt;br/&gt;
Proelium ad Severodonetsk, urbs [[regio Luganskensis|regionis Luganskensis]] magnum fit, dum [[via]] a Bakmut ad Severodonetsk fons progressus Ucrainae facta est.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61578156 Severodonetsk: Battle for key road as fighting reaches Ukraine city]&lt;/ref&gt;

'''Die 26 Maii 2022'''&lt;br/&gt;
Impetus bombarum in [[Charcovia]] fit, quod octo homines inter eos [[infans|infantem]] necavit.&lt;ref&gt;[https://www.bbc.com/news/live/world-europe-61593803 Ukraine war latest news: Shelling in Kharkiv kills eight including bady]&lt;/ref&gt; 

'''Die 28 Maii 2022'''&lt;br/&gt;
Copiae militares Ucrainae ex Severodonetsk regressae esse potest, dum res peior pro Ucraina fit vis Russicae causa.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61613479 Ukirane war: Troops could quit Severodonetsk amid Russian advance - official]&lt;/ref&gt;

'''Die 29 Maii 2022'''&lt;br/&gt;
[[Vladimirus Zelens'kyj]], praeces Ucrainae [[Charcovia]]m visit primum post Incursiom Russicam die 24 Februarii anno 2022, ad copias Ucrainae observandum.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61625512 Ucraine war: President Zelensky visit Kharkiv in first trip outside Kiyv relion]&lt;/ref&gt;

'''Die 30 Maii 2022'''&lt;br/&gt;
[[Unio Europaea]] unianimis consensum de interdiction [[petroleum]] importandi decreavit.&lt;ref&gt;[https://www.usatoday.com/story/news/nation/2022/05/30/ukraine-war-live-updates/9992994002/ EU may have deal on oil embargo]&lt;/ref&gt;.

'''Die 31 Maii 2022'''&lt;br/&gt;
Irina Venedictova, prosecutorix Ucrainae circa 15,00 actores scelorum bellicorum Russicos et actores [[propaganda]]e esse dixit per investigendum.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61652467 Ukraine reports 15,000 suspected wars crimes]&lt;/ref&gt;

'''Die 1 Iunii 2022'''&lt;br/&gt;
Rectio Russiae provisionem [[missile|missilium]] eminus res proelii aggravandae causa facere dixit.&lt;ref&gt;[https://www.bbc.com/news/world-us-canada-61655577 Ukraine war: Russia says US 'adding fuel to fire' be sendung longer-range rochets]&lt;/ref&gt;

'''Die 3 Iunii 2022'''&lt;br/&gt;
Post centum dies ex initio Incursionis Russicae, [[Vladimirus Zelens'kyj]] copias militates Russiae unum quinta [[territorium (pars administrativa)|territoriorum]] Ucrainae sub regno mittere dixist.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61675915 Ukraine war:Zelensky says Russia controls a fifth of Ukrainian territory]&lt;/ref&gt;,

'''Die 4 Iunii 2022'''&lt;br/&gt;
[[Monasterium]] arboris historicum in Mykolaiv, oriente Severidonetsk [[incendium]] subiectum est, impetus bombarum copiarum Russiae causa.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61688829 Ukraine round-up: Defiance in Mykolaiv and a wooden monastery ablaze]&lt;/ref&gt;
 
'''Die 5 Iunii 2022'''&lt;br/&gt;
Impetus a [[missile|missili]] a militaria Russicae fit in [[Kiovia]]m, caput Ukrauknaine war/inae.

'''Die 7 Iunii 2022'''&lt;br/&gt;
Uniones Europaeae legatus invasionem et politicam Russiae discrimen alimenti mundiali, causa [[paupertas|paupertatem]] et violationem [[securitas cibaria|securitatis cibariae]] facere aapud [[Conventus generalis (Nationes Unitae)|Conventus generals Nationum Unitarum]] accusavit.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61714234 Ukraine news: EU blames Russia for dood crisis prompting walkout]&lt;/ref&gt;

'''Die 8 Iunii 2022'''&lt;br/&gt;
[[Iuris consultus]] et activista [[iurum humanorum defensor|pro iuribus humanis]] Russici quales miles Russicos in Ucrainam irendum et pugnandum refugere dixient.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61607184 The Russian soldiers refusing to fight in Ukraine]&lt;/ref&gt;

'''Die 9 Iunii 2022'''&lt;br/&gt;
Caput [[Ordo mercaturae mundanae|Ordinis mercaturae mundanae]] crisem alimemti ratione Incursionis Russicae aggravare dixit et monet.&lt;ref&gt;[https://www.bbc.com/news/business-61727651 Ukraine war: WTO boss warns of global food crisis]&lt;/ref&gt; Ac in [[regio Donetskensis|regione Donetskensi]], rectio sub regimine Russico milites peregrinos&lt;ref&gt;Apud rectionem Russicam, [[miles mercennarius]] sunt.&lt;/ref&gt; Britannicos et [[Marocum|Marocum]] [[poena capitalis|poenam capitalem]] iudicat.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61751435 Ukraine round-up; Death sentences for foreign fighters and Putin upbeast on energy]&lt;/ref&gt;

'''Die 11 Iunii 2022'''&lt;br/&gt;
Auxilator Vladimurii Zelens'kyj circa 200 milites Uper diem nacati ex bello publicat. Sed rectio Ucrainae negotium de [[indutiae|indutiis]] cum Russia non facere dixit&lt;ref&gt;[https://www.bbc.com/news/world-europe-61742736 Ukraine causalites: Kyiv losing up to 200 troops a day - Zelensky aide]&lt;/ref&gt;

'''Die 13 Iunii 2022'''&lt;br/&gt;
Apud CREA&lt;ref&gt;[[Abbreviatio]] Angrica pro ''Centre for Rearch on Energy and Clean Air'' (Latine: Institutum pro [[energia]] et aëro puro)&lt;/ref&gt; organizationem re publica vacante [[Finnia]]e, [[Russia]] post incursionem anno 2022 ad hodie [[lucrum]] plus quam 90 [[billio]]nes [[$]] per negotium [[fomes fossilis|fomitum fossilium]] accquisit.&lt;ref&gt;[https://www.bbc.com/news/business-61785111 Ukraine war: Russia earns $97bn on energy export since invasion]&lt;/ref&gt;

'''Die 14 Iunii 2022'''&lt;br/&gt;
Papa [[Franciscus (papa)|Franciscus I]] incursionem Russicam in Ucrainam provocata a [[OTAN]] esse posse dixit. quamquam impetum inhumanum a copis Russiae condemmavit.&lt;ref&gt;[https://www.theguardian.com/world/2022/jun/14/pope-francis-ukraine-war-provoked-russian-troops Pope Francis says Ukraine was perhaps somehow provoked']&lt;/ref&gt;

'''Die 16 Iunii 2022'''&lt;br/&gt;
[[Emmanuel Macron]] praeses [[Francia]]e, [[Olaus Scholz]] primus minister [[Germania]]e et [[Marius Draghi]] primus minister [[Italia]]e in [[Kiovia]]m visiterunt et cum Vladimiro Zelens'kyj colliquium ad solidaritatem et integrationem Europaeam confirmandum facerunt.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61828229 Ukraine war: EU leaders back immediate candidate status for Kyiv]&lt;/ref&gt; 

'''Die 18 Iunii 2022'''&lt;br/&gt;
[[Vladimirus Putin]], praeses Russiae apud Conferentiam Internationalem Oeconomiae in [[Petropolis|Petropoli]] poenas oeconomicas a civitatibus occidentalibus et [[Unio Europaea|Unione Europaeae]] &quot;[[dementia]]m et [[stupiditas|stupiditatem]]&quot; dixit et condemnat.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61847300 Russia's Putin condemns 'mad and thoughtless' Western sanctions]&lt;/ref&gt;

'''Die 20 Iunii 2022'''&lt;br/&gt;
Rectio Ucrainae [[musica]]m Russicam creatam vel performatam a hominibus Russicis post anno 1991 perfomandum publice ac [[litterae Russicae|libros]] e Russia importandum interdit.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61859593 Uknaine to ban music by some Russians in media and public spaces]&lt;/ref&gt; Eodem die [[Ferrivia metropolitana Charcoviae|ferriviae metropolitanae]] atrium vehiculorum detrimentum maius passum est.

'''Die 21 Iunii 2022'''&lt;br/&gt;
Rectio Russiae rem &quot;severem&quot; parere causa interdictionis [[vectura]]e [[ferrivia]]e ad [[Kaliningradum]] a [[Lituania]] dixit.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61878929 Kalingrad: Russia warms Lithuania of consequences over rail transit blockade]&lt;/ref&gt;

'''Die 24 Iunii 2022'''&lt;br/&gt;
Rectio Ucrainae [[miles|militae]] e Ucrainae regredere ex Severodonetsk, urbe [[regio Luganskensis|regionis Luganskensis]] ordinat.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61920708 Ukraine war: Kyiv orders forces to withdraw from Severodonetsk]&lt;/ref&gt;

'''Die 27 Iunii 2022'''&lt;br/&gt;
Conventus [[G8|Grecis Septem]] in Germania fuit et civitates occidentales solidaritatem cum Ucraina ad finem belli confirmaverunt.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61954445 Ukraine war: G7 pledges to stay with Ukraine untill the end]&lt;/ref&gt;

'''Die 28 Iunii 2022'''&lt;br/&gt;
Impetus [[missile|missilorum]] fit in regionem Lysychansk et plus quam 21 homines cives neacti sunt.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61954068 Ukraine war: Russian forces 'storming' besieged city of Lusychansk]&lt;/ref&gt;

'''Die 29 Iunii 2022'''&lt;br/&gt;
Societas televisionis Ucrainae imaginem impetus missilorum ad [[emporium (aedificium)|emporium]] in Kremenchuk [[regio Poltavensis|regionis Poltavensis]] a copiis Russicae publicavit, ut testimonium sceli bellici.&lt;ref&gt;[https://www.bbc.com/news/av/world-europe-61978727 Ukraine war: CCTV shows missile striking shopping mall in Kremenchuk]&lt;/ref&gt; 

'''Die 30 Iunii 2022'''&lt;br/&gt;
Copiae Russicae ex insula serpantis vel ''Zmiinyi'' in [[Pontus Euxinus|Pontu Euxinu]] regessae sunt.&lt;ref&gt;[https://www.bbc.com/news/world-europe-61992491 Snake Island: Why Russia couldn't hold on  to stategic  Black Sea outcrop]&lt;/ref&gt;

'''Die 2 Iulii 2022'''&lt;br/&gt;
Officialis Ucrainae copiam Russicae impetum [[missile|missilorum]] in [[Odessa]] dare dixit, a quo 21 homines cives impubesque necati sunt.&lt;ref&gt;[https://www.bbc.com/news/world-europe-62006743 Ukraine war: Russian missile strikes kill 21 in Odesa region - emergency service]&lt;/ref&gt;

'''Die 3 Iulii 2022'''&lt;br/&gt;
Auctoritas [[Turcia]]e [[navis|navem]] [[frumentum]] e Berdynask Ucrainae transferendi cum vexillo Russiae in custodia davit.&lt;ref&gt;[https://www.bbc.com/news/world-europe-62010113 Turkey detains Russian-glagged  grain ship from Ukraine]&lt;/ref&gt;

'''Die 4 Iulii 2022'''&lt;br/&gt;
Copiae Ucrainae ubrem Lysychansk [[regio Luganskensis|regionis Lugenskensis]] sub regnum copiarum Russicarum ire admisit.&lt;ref&gt;[https://www.bbc.com/news/world-europe-62030051 Ukraine confirms Russia captured earstern city Lysychansk]&lt;/ref&gt;

'''Die 7 Iulii 2022'''&lt;br/&gt;
Iryna Venediktova, prosectorix generalis Ucrainae plus quam 21.000 actores [[scelus bellicum|sceli bellici]] investigendi esse puvlicat.&lt;ref&gt;[https://www.bbc.com/news/world-europe-62073669 Ukraine war: 21,000 alleged war crimes being investigated, prosector says]&lt;/ref&gt;

'''Die 11 Iulii 2022'''&lt;br/&gt;
[[Diamesolabeta]] Ucrainae Oleh Kotenko circa 7,200 [[miles|milites]] morti esse ratione belli dixit.&lt;ref&gt;[https://www.bbc.com/news/world-europe-62125343 Ukraine war: 7,200 Ukraine service personnel missing - ombudsmann]&lt;/ref&gt;

'''Die 12 Iulii 2022'''&lt;br/&gt;
Rectio Ucrainae [[matrimonium hominum eiusdem sexus]] legale admittere in bello potest.&lt;ref&gt;[https://www.bbc.com/news/world-europe-62134804 Ukraine to consider legalising same-sex marriage amid war]&lt;/ref&gt;

'''Die 14 Iulii 2022'''&lt;br/&gt;
Impetus [[missile|missilis]] a copis Russicis in [[Vinnicia]] fit, a quo p;us{{dubsig}} quam 23 homines necatu sunt.&lt;ref&gt;[https://www.bbc.com/news/world-europe-62163071 Ukraine war: 23 killed in Russian rocket attack in Vinnytsia]&lt;/ref&gt;

'''Die 16 Iulii 2022'''&lt;br/&gt;
Post inspectionem copiarum Russicarum [[minister defensionis (Russia)|ministro defensionis Russicae]], [[Sergius Šoigu]], [[Ministerium Defensionis Foederationis Russicae]] impetus in Ucrainam augendos declaravit.

'''Die 19 Iulii 2022'''&lt;br/&gt;
Vladimirus Putin in [[Teheranum]] caput Iraniae iit et [[Ali Khamenei]] praeses Iraniae et [[Recep Tayyip Erdoğan]] praeses Turciae visitat, ad conferentiam diplomaticam participandum.&lt;ref&gt;[https://www.bbc.com/news/world-europe-62218696 Ukraine war: Putin visits Iran in rare international trip]&lt;/ref&gt;

'''Die 21 Iulii 2022'''&lt;br/&gt;
[[Sergius Lavrov]], monister rerum externarum Russiae, obiectum proelii enpandere, non solum in territoria orientali dixit.&lt;ref&gt;[https://www.bbc.com/news/world-europe-62231936 Ukraine war: Russia's Lavrov says ready to expand war aims]&lt;/ref&gt;

'''Die 23 Iulii 2022'''&lt;br/&gt;
Impetus explosionis bombarum in [[portus|portu]] [[Odessa]] fit, quales horas post tempus ubi consensus de [[commercium|commercio]] [[frumentum|frumenti]] in pace decreatur.&lt;ref&gt;[https://www.bbc.com/news/world-europe-62276392 Ukraine war: Explosions rock Ukrainian post hours after grain deal]&lt;/ref&gt;

'''Die 26 Iulii 2022'''&lt;br/&gt;
[[Gazprom]] Russiae oblatum [[gasium naturale|gasi naturali]] pro [[Unio Europaea|Unione Europaea]] per via ''Nord Stream'' plus limitare decernat.&lt;ref&gt;[https://www.bbc.com/news/business-62291458 Gazprom: Norm Stream 1 to supply to EU to be cut further]&lt;/ref&gt;

'''Die 28 Iulii 2022'''&lt;br/&gt;
Copiae Russicae urbem in [[regio Kioviensis|regione Kioviensis]] impetum missili dat, a quo quinque homines necati sunt.&lt;ref&gt;[https://www.bbc.com/news/world-europe-62339767 Ukraine war: Russia hits Kyiv area as Ukraine seeks to retake south]&lt;/ref&gt;

'''Die 30 Iulii 2022'''&lt;br/&gt;
Rectio Ucrainae investigationem [[Nationes Unitae|Nationum Unitae]] et [[Motus internationalis Crucis Rubrae et Lunae Rubrae]] de [[bello captus|bello captis]] in carcero sub copias militarias Russicas postulat.&lt;ref&gt;[https://www.bbc.com/news/world-europe-62356211 Ukraine war: UN and Red Cross shoud investigate prinsion death, says Ukraine]&lt;/ref&gt;

'''Die 1 Augusti 2022'''&lt;br/&gt;
[[Navis]] Ucrainae commercii [[frumentum|frumenti]] ex [[Odessa]] iit sub consensu cum Russia, primum post inclusionem Russica anno 2022.&lt;ref&gt;[https://www.bbc.com/news/world-europe-62375580 Ukraine war: First grain ship leaves Under Russian deal]&lt;/ref&gt;

'''Die 3 Augusti 2022'''&lt;br/&gt;
Raphael Grossi, praeses [[Procuratio energiae atomicae internationalis|IAEA]] [[Ergasterium atomicum Zaporizhiense]] extra regnum ab impetu Russiae dixit.&lt;ref&gt;[https://www.bbc.com/news/world-europe-62412429 Ukraine war: IAEA says Zaporizhzhia nuclear plants out of control]&lt;/ref&gt;

'''Die 6 Augusti 2022'''&lt;br/&gt;
Raphael Grossi, praeses IAEA impetus militaris ad Ergasterium atomicum Zaporizhiense [[calamitas|calamitatem]] realiem et seriam facare posse dixit.&lt;ref&gt;[https://www.bbc.com/news/world-europe-62449982 Zaporizhzhia: Real risk of nuclear disaster in Ukraine - watchdog]&lt;/ref&gt;

'''Die 11 Augusti 2022'''&lt;br/&gt;
[[Nationes Unitae]] imptum militaris ad [[Ergasterium atomicum Zaporizhiense]] condemnant et civitates [[G8|G7]] hunc impetum condemnant et ergasterium ad rectionem Ucrainae cedere postulat.&lt;ref&gt;[https://www.bbc.com/news/world-europe-62505815 UN alarm as Ukraine nuckear power plant shelled again]&lt;/ref&gt;

'''Die 16 Augusti 2022'''&lt;br/&gt;
Copiae militares Ucrainae basum militarem Russicum in [[Crimaea]] destruunt, quod rectio Russiae hoc ut destructio collectiva condemnat.&lt;ref&gt;[https://www.bbc.com/news/world-europe-62560041 Ukraine war: Russia blames sabotage for news Crimea blast]&lt;/ref&gt;
== Notae ==
&lt;references/&gt;

==Bibliographia== 
*Abbruzzese, Jason. [[2022]]. &quot;Putin says he is fighting a resurgence of Nazism. That's not true&quot;. NBC News, 24 Februarii 2022. [https://www.nbcnews.com/politics/politics-news/live-blog/russia-ukraine-conflict-live-updates-n1289655/ncrd1289673 Textus interretialis.]

{{NexInt}}
* [[Res publica popularis Donetskensis]]

[[Categoria:Res militares]]
[[Categoria:Russia]]
[[Categoria:Ucraina]]
[[Categoria:2022]]</text>
      <sha1>qx63opjkpd1x7w6dvx5mpye00lpjvkh</sha1>
    </revision>
  </page>
  <page>
    <title>Nazlı Tolga</title>
    <ns>0</ns>
    <id>302995</id>
    <revision>
      <id>3697724</id>
      <parentid>3697442</parentid>
      <timestamp>2022-08-17T07:46:24Z</timestamp>
      <contributor>
        <username>Aranciomassimo6194</username>
        <id>157698</id>
      </contributor>
      <comment>/* Privata vita */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1453" xml:space="preserve">'''Nazlı Tolga Brenninkmeyer''', notior quam Nazlı Tolga (natus [[Ancyra|Ancyrae]] die [[8 Novembris]] [[1979]]), diurnarius Batavus-Turcicus est.

== Biographia ==
Post initium vitae suae in Kanal D diurnarius die [[31 Augusti]] anno [[1998]], in ephemeride universitate Marmara anno MMIII deducta est et nuntium programmatum Kanal D (1998-2002) spectavit, TV (2002-2003), Skytürk (2003-2007) et FOX (a die 3 Septembris 2007 ad diem 14 Iunii 2013). Nominata est Hostia Turcica Best in MMXI et MMXII et Felicissima Hostia Turcica in MMXIII.&lt;ref&gt;https://it.wikipedia.org/wiki/Nazlı_Tolga&lt;/ref&gt;

== Privata vita ==
2013 uxorem duxit in Negotiatorem [[Nederlandia|Batavicam]] Laurentium Brenninkmeyerum in [[Ecclesia Sancti Spiritus]] .

Postquam in [[Brasilia]] moratus est (2013), nunc (2017) [[Londinium|Londinii]] habitat. 

Loquitur [[Lingua Turcica|Turcico]], [[Lingua Lusitana|Lusitano]], [[Lingua Batava|Batavico]] et [[Lingua Anglica|Anglico]] . Is sororem et filiam habet. 

== Acta TV ==

* ''Kanal D Gece Haberleri'' ( Kanal D in 1998-2002);
* ''Nazlı Tolga ile Haber Masası'' et  (Skyturk in 2004-Septembris 2007);
* SHOW ANA HABER (anno 2002-2003);
* ''FOX  ANA HABER'' (anno 2008-2010);
* ''Nazlı Tolga ile Fox Ana Haber'' (in mense Septembri 2007 - Die 14 Iunii, 2013)

== Nota ==
== Nexus externi ==
*[https://www.instagram.com/nazlitolga]
[[Categoria:Nati 1979]]
[[Categoria:Diurnarii Turciae]]
[[Categoria:Diurnarii Nederlandiae]]</text>
      <sha1>e3ifbu57hd6gg58ad0m3upiee1hy0gu</sha1>
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  </page>
  <page>
    <title>Sergius Šoigu</title>
    <ns>0</ns>
    <id>303361</id>
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      <timestamp>2022-08-16T12:40:48Z</timestamp>
      <contributor>
        <username>LilyKitty</username>
        <id>18316</id>
      </contributor>
      <comment>de ministri defensionis Russiae</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1804" xml:space="preserve">{{L}}{{Capsa hominis Vicidata}}
'''Sergius Kužugeti filius Šoigu''' ([[Tuvinice]] ''Сергей Күжүгет оглу Шойгу'', natus die [[21 Martii]] [[1955]], [[Chadan]]i, [[Tuva]]e) est [[ingeniarius civilis]], [[politicus]] [[Russia|Russicus]] et [[dux exercitus (Russia)|dux exercitus]] (Russice: ''генерал армии''), qui ab anno 2012 munere [[Ministerium Defensionis Foederationis Russicae|ministri defensionis]] officio fungitur. Ex anno 2012 est praeses [[concilium ministrorum defensionis Foederationis Independentium Civitatum|concilium ministrorum defensionis ICF]].

De anno 1991 usque ad annum 2012 munere [[minister condicionum necessitatis (Russia)|minister condicionum necessitatis]] functus est. Ex eo munere anno 1999 praemium Foederationis Russicae nobilissimum [[Heros Foederationis Russicae|Herois Foederationis Russicae]] accepit. Anno 2012, brevi tempore, gubernator [[regio Moscuensis|regionis Moscuensis]] fuit.

Sergius Šoigu circulo ''[[silovicus|silovicorum]]'' (Russice: &quot;virium praefecti&quot;) attribuitur, hominum, qui [[intelligentiae ministeria Russiae|intelligentia ministerio]]{{dubsig}} [[copiae Russiae|copiisve Russicis]] laborant. Aestimatur Šoigu praesidi [[Foederatio Russica|Foederationis Russicae]], [[Vladimirus Putin|Vladimiro Putin]] persona familiaris.

== Pinacotheca ==
&lt;gallery&gt;
Fasciculus:Military exercises Center-2019-02.jpg| S. Šoigu (sinistre) cum [[Vladimirus Putin|Vladimiro Putin]]
Fasciculus:RAF A F9GenArmy 1974-1991.png|Sigillum [[dux exercitus (Russia)|ducis exercitus]]
Fasciculus:RIAN archive 470774 Gold Star medal (cropped).jpg|Praemium [[Heros Foederationis Russicae|Herois Foederationis Russicae]]
&lt;/gallery&gt;

==Nexus externi==
{{fontes biographici}}

{{Lifetime|1834||Soigu, Sergius}}
[[Categoria:Politici Russiae]]</text>
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      <timestamp>2022-08-16T18:52:30Z</timestamp>
      <contributor>
        <ip>213.87.89.5</ip>
      </contributor>
      <comment>/* Nexus externi */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1804" xml:space="preserve">{{L}}{{Capsa hominis Vicidata}}
'''Sergius Kužugeti filius Šoigu''' ([[Tuvinice]] ''Сергей Күжүгет оглу Шойгу'', natus die [[21 Martii]] [[1955]], [[Chadan]]i, [[Tuva]]e) est [[ingeniarius civilis]], [[politicus]] [[Russia|Russicus]] et [[dux exercitus (Russia)|dux exercitus]] (Russice: ''генерал армии''), qui ab anno 2012 munere [[Ministerium Defensionis Foederationis Russicae|ministri defensionis]] officio fungitur. Ex anno 2012 est praeses [[concilium ministrorum defensionis Foederationis Independentium Civitatum|concilium ministrorum defensionis ICF]].

De anno 1991 usque ad annum 2012 munere [[minister condicionum necessitatis (Russia)|minister condicionum necessitatis]] functus est. Ex eo munere anno 1999 praemium Foederationis Russicae nobilissimum [[Heros Foederationis Russicae|Herois Foederationis Russicae]] accepit. Anno 2012, brevi tempore, gubernator [[regio Moscuensis|regionis Moscuensis]] fuit.

Sergius Šoigu circulo ''[[silovicus|silovicorum]]'' (Russice: &quot;virium praefecti&quot;) attribuitur, hominum, qui [[intelligentiae ministeria Russiae|intelligentia ministerio]]{{dubsig}} [[copiae Russiae|copiisve Russicis]] laborant. Aestimatur Šoigu praesidi [[Foederatio Russica|Foederationis Russicae]], [[Vladimirus Putin|Vladimiro Putin]] persona familiaris.

== Pinacotheca ==
&lt;gallery&gt;
Fasciculus:Military exercises Center-2019-02.jpg| S. Šoigu (sinistre) cum [[Vladimirus Putin|Vladimiro Putin]]
Fasciculus:RAF A F9GenArmy 1974-1991.png|Sigillum [[dux exercitus (Russia)|ducis exercitus]]
Fasciculus:RIAN archive 470774 Gold Star medal (cropped).jpg|Praemium [[Heros Foederationis Russicae|Herois Foederationis Russicae]]
&lt;/gallery&gt;

==Nexus externi==
{{fontes biographici}}

{{Lifetime|1955||Soigu, Sergius}}
[[Categoria:Politici Russiae]]</text>
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    </revision>
  </page>
  <page>
    <title>Ratis</title>
    <ns>0</ns>
    <id>303562</id>
    <revision>
      <id>3697758</id>
      <parentid>3697238</parentid>
      <timestamp>2022-08-17T10:17:08Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Naves]]; +[[Categoria:Genera navium]]</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="706" xml:space="preserve">[[Fasciculus:Flooded haor.jpg|thumb|Ratis simplex ([[Bangladesa]]e anno 2010)]]
'''Ratis''' est connexarum trabium vel lignorum compages, quae per aquam agitur; [[navis]] genus simplicissimus est. [[Caulis|Сaules]] [[scirpus|scirpi]] vel [[phragmites|phragmitis]] pro lignis usurpari possunt. Rates magnae et arte factae et [[velum|vela]] ferre possunt. Cum trabes per flumina ipso fluxu transportantur, saepe in rates coniunguntur.

[[Ulixes]] carmine quinto ''[[Odyssea]]e'' ratem construit, quam [[Neptunus]] tempestate missa disrumpit.

== Nexus externi ==
{{forcellini}}[http://www.lexica.linguax.com/forc.php?searchedLG=ratis sub verbo ''ratis'']

[[Categoria:Genera navium]]

{{Myrias|Technologia}}</text>
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  </page>
  <page>
    <title>Iolaus</title>
    <ns>0</ns>
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      <timestamp>2022-08-16T14:08:54Z</timestamp>
      <contributor>
        <username>Marcus Terentius Bibliophilus</username>
        <id>2059</id>
      </contributor>
      <comment>/* De cultu heroico */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3134" xml:space="preserve">[[Fasciculus:Hercules and Iolaus mosaic - Anzio Nymphaeum.jpg|thumb|Hercules et Iolaus in [[Opus tessellatum|opere tessellato]] [[Antium (Latium)|Antii]] reperto in [[Nymphaeum|Nymphaeo]].]]
In mythologia Graeca '''Iolaus'''  fuit filius [[Iphiclus|Iphicli]], Herculis gemini fratris mortalis, et Automedusae. Ita per patrem [[Amphitryon|Amphitruonis]] et [[Alcmene|Alcmenae]] nepos, per matrem vero [[Pelops|Pelopis]] pronepos erat. [[Megara (mythologia)|Megaram]] primam Herculis uxorem in matrimonium accepit&lt;ref&gt;[[Apollodori bibliotheca]] II.4.11. [[Pausanias (scriptor)|Pausanias]] X.29.7.&lt;/ref&gt; postquam patruus furore correptus liberos quos ex ea genuerat interfecit.
== De rebus gestis ==
[[Patruus|Patrui]] [[Auriga|auriga]] erat et [[Olympia (certamina)|Olympicis ludis]] [[Quadrigae|quadrigarum]] certamen vicit&lt;ref&gt;[[Pausanias (scriptor)|Pausanias]] V.8.3-4. [[Hyginus mythographus|Hyginus]], ''Fabulae'' 273.11&lt;/ref&gt;. [[Argonautae|Argonautarum]] expeditioni et [[Aper Calydonius|apri Calydonii]] venationi interfuit. In [[Labores Herculis|duodecim laboribus]] patruum adsidue comitatus est et magno usui ei fuit inprimis adversus [[Hydra Lernaea|hydram]]&lt;ref&gt;[[Diodorus Siculus]] IV.11.20. [[Apollodori bibliotheca]] II.5.2&lt;/ref&gt;. [[Colonia|Coloniam]] [[Thebae (Boeotia)|Thebanorum]] quoque in [[Sardinia|Sardiniam]] duxit&lt;ref&gt;[[Diodorus Siculus]] IV.29.&lt;/ref&gt;. Postremo Herculis mortui filios, [[Heracleidae|Heraclidas]] dictos, tuetur et proelio sive ad Thebas, ut voluit poeta [[Pindarus]]&lt;ref&gt;''Pythia'' IX&lt;/ref&gt;, sive prope [[Marathon (Graecia)|Marathonem]], ut Athenienses malebant, [[Eurystheus (mythologia)|Eurystheum]] veterem Herculis inimicum vicit et occidit&lt;ref&gt;[[Pausanias (scriptor)|Pausanias]] I.44.10.&lt;/ref&gt;, quod argumentum tragoediae ''[[Heraclidae (Euripides)|Heraclidarum]]'' praebuit. Fabula addit deos viro senio confecto ad unam horam iuventutem reddidisse ita ut inimicum profligare posset&lt;ref&gt;[[Ovidius]], ''[[Metamorphoses (Ovidius)|Metamorphoses]]'' IX.396-401 qui donum [[Hebe|Hebae]] tribuit mariti precibus victae. [[Euripides]], ''[[Heraclidae (Euripides)|Heraclidae]]'' 843sqq.&lt;/ref&gt;.

== De cultu heroico ==
Multis in locis arae Iolao erectae erant, imprimis Thebis ubi paria [[Homosexualitas|homosexualia]] ad Iolai aram  fidem sibi invicem iurabant&lt;ref&gt;[[Aristoteles]] apud [[Plutarchus|Plutarchum]], ''Vita Pelopidae'' XVIII.5. Cf [[Pausanias (scriptor)|Pausanias]] IX.23.1.&lt;/ref&gt; quia multi aestimabant hoc quoque modo Iolaum ab Hercule amatum esse. Non minus enim per Iolaum quam per Herculem in [[Boeotia]] iurabatur&lt;ref&gt;[[Aristophanes]], ''[[Acharnenses]]'' 867.&lt;/ref&gt;Alibi quoque honorabatur ut in Sicilia et Athenis in Herculis templo&lt;ref&gt;[[Pausanias (scriptor)|Pausanias]] I.19.3.&lt;/ref&gt;. [[Agyrium|Agyrii]] certamina quoque in eius honorem fiebant et incolae capillos ei vovebant&lt;ref&gt;[[Diodorus Siculus]] IV.24.40.&lt;/ref&gt;.

== Notae ==
&lt;references/&gt;
== Nexus externi ==
*[https://journals.openedition.org/encyclopedieberbere//1583 Iolaos in ''Encyclopédie berbère'']
[[Categoria:Argonautae]]
[[Categoria:Hercules]]
[[Categoria:Religio Graeca]]
[[Categoria:Mythologia Thebana]]</text>
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    <revision>
      <id>3697665</id>
      <parentid>3697664</parentid>
      <timestamp>2022-08-16T14:09:10Z</timestamp>
      <contributor>
        <username>Marcus Terentius Bibliophilus</username>
        <id>2059</id>
      </contributor>
      <comment>/* De cultu heroico */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3136" xml:space="preserve">[[Fasciculus:Hercules and Iolaus mosaic - Anzio Nymphaeum.jpg|thumb|Hercules et Iolaus in [[Opus tessellatum|opere tessellato]] [[Antium (Latium)|Antii]] reperto in [[Nymphaeum|Nymphaeo]].]]
In mythologia Graeca '''Iolaus'''  fuit filius [[Iphiclus|Iphicli]], Herculis gemini fratris mortalis, et Automedusae. Ita per patrem [[Amphitryon|Amphitruonis]] et [[Alcmene|Alcmenae]] nepos, per matrem vero [[Pelops|Pelopis]] pronepos erat. [[Megara (mythologia)|Megaram]] primam Herculis uxorem in matrimonium accepit&lt;ref&gt;[[Apollodori bibliotheca]] II.4.11. [[Pausanias (scriptor)|Pausanias]] X.29.7.&lt;/ref&gt; postquam patruus furore correptus liberos quos ex ea genuerat interfecit.
== De rebus gestis ==
[[Patruus|Patrui]] [[Auriga|auriga]] erat et [[Olympia (certamina)|Olympicis ludis]] [[Quadrigae|quadrigarum]] certamen vicit&lt;ref&gt;[[Pausanias (scriptor)|Pausanias]] V.8.3-4. [[Hyginus mythographus|Hyginus]], ''Fabulae'' 273.11&lt;/ref&gt;. [[Argonautae|Argonautarum]] expeditioni et [[Aper Calydonius|apri Calydonii]] venationi interfuit. In [[Labores Herculis|duodecim laboribus]] patruum adsidue comitatus est et magno usui ei fuit inprimis adversus [[Hydra Lernaea|hydram]]&lt;ref&gt;[[Diodorus Siculus]] IV.11.20. [[Apollodori bibliotheca]] II.5.2&lt;/ref&gt;. [[Colonia|Coloniam]] [[Thebae (Boeotia)|Thebanorum]] quoque in [[Sardinia|Sardiniam]] duxit&lt;ref&gt;[[Diodorus Siculus]] IV.29.&lt;/ref&gt;. Postremo Herculis mortui filios, [[Heracleidae|Heraclidas]] dictos, tuetur et proelio sive ad Thebas, ut voluit poeta [[Pindarus]]&lt;ref&gt;''Pythia'' IX&lt;/ref&gt;, sive prope [[Marathon (Graecia)|Marathonem]], ut Athenienses malebant, [[Eurystheus (mythologia)|Eurystheum]] veterem Herculis inimicum vicit et occidit&lt;ref&gt;[[Pausanias (scriptor)|Pausanias]] I.44.10.&lt;/ref&gt;, quod argumentum tragoediae ''[[Heraclidae (Euripides)|Heraclidarum]]'' praebuit. Fabula addit deos viro senio confecto ad unam horam iuventutem reddidisse ita ut inimicum profligare posset&lt;ref&gt;[[Ovidius]], ''[[Metamorphoses (Ovidius)|Metamorphoses]]'' IX.396-401 qui donum [[Hebe|Hebae]] tribuit mariti precibus victae. [[Euripides]], ''[[Heraclidae (Euripides)|Heraclidae]]'' 843sqq.&lt;/ref&gt;.

== De cultu heroico ==
Multis in locis arae Iolao erectae erant, imprimis Thebis ubi paria [[Homosexualitas|homosexualia]] ad Iolai aram  fidem sibi invicem iurabant&lt;ref&gt;[[Aristoteles]] apud [[Plutarchus|Plutarchum]], ''Vita Pelopidae'' XVIII.5. Cf [[Pausanias (scriptor)|Pausanias]] IX.23.1.&lt;/ref&gt; quia multi aestimabant hoc quoque modo Iolaum ab Hercule amatum esse. Non minus enim per Iolaum quam per Herculem in [[Boeotia]] iurabatur&lt;ref&gt;[[Aristophanes]], ''[[Acharnenses]]'' 867.&lt;/ref&gt;. Alibi quoque honorabatur ut in Sicilia et Athenis in Herculis templo&lt;ref&gt;[[Pausanias (scriptor)|Pausanias]] I.19.3.&lt;/ref&gt;. [[Agyrium|Agyrii]] certamina quoque in eius honorem fiebant et incolae capillos ei vovebant&lt;ref&gt;[[Diodorus Siculus]] IV.24.40.&lt;/ref&gt;.

== Notae ==
&lt;references/&gt;
== Nexus externi ==
*[https://journals.openedition.org/encyclopedieberbere//1583 Iolaos in ''Encyclopédie berbère'']
[[Categoria:Argonautae]]
[[Categoria:Hercules]]
[[Categoria:Religio Graeca]]
[[Categoria:Mythologia Thebana]]</text>
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    <revision>
      <id>3697666</id>
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      <timestamp>2022-08-16T14:11:33Z</timestamp>
      <contributor>
        <username>Marcus Terentius Bibliophilus</username>
        <id>2059</id>
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      <comment>/* De rebus gestis */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3277" xml:space="preserve">[[Fasciculus:Hercules and Iolaus mosaic - Anzio Nymphaeum.jpg|thumb|Hercules et Iolaus in [[Opus tessellatum|opere tessellato]] [[Antium (Latium)|Antii]] reperto in [[Nymphaeum|Nymphaeo]].]]
In mythologia Graeca '''Iolaus'''  fuit filius [[Iphiclus|Iphicli]], Herculis gemini fratris mortalis, et Automedusae. Ita per patrem [[Amphitryon|Amphitruonis]] et [[Alcmene|Alcmenae]] nepos, per matrem vero [[Pelops|Pelopis]] pronepos erat. [[Megara (mythologia)|Megaram]] primam Herculis uxorem in matrimonium accepit&lt;ref&gt;[[Apollodori bibliotheca]] II.4.11. [[Pausanias (scriptor)|Pausanias]] X.29.7.&lt;/ref&gt; postquam patruus furore correptus liberos quos ex ea genuerat interfecit.
== De rebus gestis ==
[[Fasciculus:Heracles, Iolaus and Eros - Cista Ficoroni foot.jpg|thumb|[[Cista Ficoroni|Cista Ficoronia]]ː Eros inter Iolaum et Herculem.]]
[[Patruus|Patrui]] [[Auriga|auriga]] erat et [[Olympia (certamina)|Olympicis ludis]] [[Quadrigae|quadrigarum]] certamen vicit&lt;ref&gt;[[Pausanias (scriptor)|Pausanias]] V.8.3-4. [[Hyginus mythographus|Hyginus]], ''Fabulae'' 273.11&lt;/ref&gt;. [[Argonautae|Argonautarum]] expeditioni et [[Aper Calydonius|apri Calydonii]] venationi interfuit. In [[Labores Herculis|duodecim laboribus]] patruum adsidue comitatus est et magno usui ei fuit inprimis adversus [[Hydra Lernaea|hydram]]&lt;ref&gt;[[Diodorus Siculus]] IV.11.20. [[Apollodori bibliotheca]] II.5.2&lt;/ref&gt;. [[Colonia|Coloniam]] [[Thebae (Boeotia)|Thebanorum]] quoque in [[Sardinia|Sardiniam]] duxit&lt;ref&gt;[[Diodorus Siculus]] IV.29.&lt;/ref&gt;. Postremo Herculis mortui filios, [[Heracleidae|Heraclidas]] dictos, tuetur et proelio sive ad Thebas, ut voluit poeta [[Pindarus]]&lt;ref&gt;''Pythia'' IX&lt;/ref&gt;, sive prope [[Marathon (Graecia)|Marathonem]], ut Athenienses malebant, [[Eurystheus (mythologia)|Eurystheum]] veterem Herculis inimicum vicit et occidit&lt;ref&gt;[[Pausanias (scriptor)|Pausanias]] I.44.10.&lt;/ref&gt;, quod argumentum tragoediae ''[[Heraclidae (Euripides)|Heraclidarum]]'' praebuit. Fabula addit deos viro senio confecto ad unam horam iuventutem reddidisse ita ut inimicum profligare posset&lt;ref&gt;[[Ovidius]], ''[[Metamorphoses (Ovidius)|Metamorphoses]]'' IX.396-401 qui donum [[Hebe|Hebae]] tribuit mariti precibus victae. [[Euripides]], ''[[Heraclidae (Euripides)|Heraclidae]]'' 843sqq.&lt;/ref&gt;.

== De cultu heroico ==
Multis in locis arae Iolao erectae erant, imprimis Thebis ubi paria [[Homosexualitas|homosexualia]] ad Iolai aram  fidem sibi invicem iurabant&lt;ref&gt;[[Aristoteles]] apud [[Plutarchus|Plutarchum]], ''Vita Pelopidae'' XVIII.5. Cf [[Pausanias (scriptor)|Pausanias]] IX.23.1.&lt;/ref&gt; quia multi aestimabant hoc quoque modo Iolaum ab Hercule amatum esse. Non minus enim per Iolaum quam per Herculem in [[Boeotia]] iurabatur&lt;ref&gt;[[Aristophanes]], ''[[Acharnenses]]'' 867.&lt;/ref&gt;. Alibi quoque honorabatur ut in Sicilia et Athenis in Herculis templo&lt;ref&gt;[[Pausanias (scriptor)|Pausanias]] I.19.3.&lt;/ref&gt;. [[Agyrium|Agyrii]] certamina quoque in eius honorem fiebant et incolae capillos ei vovebant&lt;ref&gt;[[Diodorus Siculus]] IV.24.40.&lt;/ref&gt;.

== Notae ==
&lt;references/&gt;
== Nexus externi ==
*[https://journals.openedition.org/encyclopedieberbere//1583 Iolaos in ''Encyclopédie berbère'']
[[Categoria:Argonautae]]
[[Categoria:Hercules]]
[[Categoria:Religio Graeca]]
[[Categoria:Mythologia Thebana]]</text>
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    <revision>
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      <contributor>
        <username>Marcus Terentius Bibliophilus</username>
        <id>2059</id>
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      <comment>/* De rebus gestis */</comment>
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      <text bytes="3281" xml:space="preserve">[[Fasciculus:Hercules and Iolaus mosaic - Anzio Nymphaeum.jpg|thumb|Hercules et Iolaus in [[Opus tessellatum|opere tessellato]] [[Antium (Latium)|Antii]] reperto in [[Nymphaeum|Nymphaeo]].]]
In mythologia Graeca '''Iolaus'''  fuit filius [[Iphiclus|Iphicli]], Herculis gemini fratris mortalis, et Automedusae. Ita per patrem [[Amphitryon|Amphitruonis]] et [[Alcmene|Alcmenae]] nepos, per matrem vero [[Pelops|Pelopis]] pronepos erat. [[Megara (mythologia)|Megaram]] primam Herculis uxorem in matrimonium accepit&lt;ref&gt;[[Apollodori bibliotheca]] II.4.11. [[Pausanias (scriptor)|Pausanias]] X.29.7.&lt;/ref&gt; postquam patruus furore correptus liberos quos ex ea genuerat interfecit.
== De rebus gestis ==
[[Fasciculus:Heracles, Iolaus and Eros - Cista Ficoroni foot.jpg|thumb|[[Cista Ficoroni|Cista Ficoronia]]ː Eros inter Iolaum et Herculem.]]
[[Patruus|Patrui]] [[Auriga|auriga]] erat et [[Olympia (certamina)|Olympicis ludis]] [[Quadrigae|quadrigarum]] certamen vicit&lt;ref&gt;[[Pausanias (scriptor)|Pausanias]] V.8.3-4. [[Hyginus mythographus|Hyginus]], ''Fabulae'' 273.11&lt;/ref&gt;. [[Argonautae|Argonautarum]] expeditioni et [[Aper Calydonius|apri Calydonii]] venationi interfuit. In [[Labores Herculis|duodecim laboribus]] patruum adsidue comitatus est et magno usui ei fuit inprimis adversus [[Hydra Lernaea|hydram]]&lt;ref&gt;[[Diodorus Siculus]] IV.11.20. [[Apollodori bibliotheca]] II.5.2&lt;/ref&gt;. [[Colonia|Coloniam]] [[Thebae (Boeotia)|Thebanorum]] quoque in [[Sardinia|Sardiniam]] duxit&lt;ref&gt;[[Diodorus Siculus]] IV.29.&lt;/ref&gt;. Postremo Herculis iam mortui filios, [[Heracleidae|Heraclidas]] dictos, tuetur et proelio sive ad Thebas, ut voluit poeta [[Pindarus]]&lt;ref&gt;''Pythia'' IX&lt;/ref&gt;, sive prope [[Marathon (Graecia)|Marathonem]], ut Athenienses malebant, [[Eurystheus (mythologia)|Eurystheum]] veterem Herculis inimicum vicit et occidit&lt;ref&gt;[[Pausanias (scriptor)|Pausanias]] I.44.10.&lt;/ref&gt;, quod argumentum tragoediae ''[[Heraclidae (Euripides)|Heraclidarum]]'' praebuit. Fabula addit deos viro senio confecto ad unam horam iuventutem reddidisse ita ut inimicum profligare posset&lt;ref&gt;[[Ovidius]], ''[[Metamorphoses (Ovidius)|Metamorphoses]]'' IX.396-401 qui donum [[Hebe|Hebae]] tribuit mariti precibus victae. [[Euripides]], ''[[Heraclidae (Euripides)|Heraclidae]]'' 843sqq.&lt;/ref&gt;.

== De cultu heroico ==
Multis in locis arae Iolao erectae erant, imprimis Thebis ubi paria [[Homosexualitas|homosexualia]] ad Iolai aram  fidem sibi invicem iurabant&lt;ref&gt;[[Aristoteles]] apud [[Plutarchus|Plutarchum]], ''Vita Pelopidae'' XVIII.5. Cf [[Pausanias (scriptor)|Pausanias]] IX.23.1.&lt;/ref&gt; quia multi aestimabant hoc quoque modo Iolaum ab Hercule amatum esse. Non minus enim per Iolaum quam per Herculem in [[Boeotia]] iurabatur&lt;ref&gt;[[Aristophanes]], ''[[Acharnenses]]'' 867.&lt;/ref&gt;. Alibi quoque honorabatur ut in Sicilia et Athenis in Herculis templo&lt;ref&gt;[[Pausanias (scriptor)|Pausanias]] I.19.3.&lt;/ref&gt;. [[Agyrium|Agyrii]] certamina quoque in eius honorem fiebant et incolae capillos ei vovebant&lt;ref&gt;[[Diodorus Siculus]] IV.24.40.&lt;/ref&gt;.

== Notae ==
&lt;references/&gt;
== Nexus externi ==
*[https://journals.openedition.org/encyclopedieberbere//1583 Iolaos in ''Encyclopédie berbère'']
[[Categoria:Argonautae]]
[[Categoria:Hercules]]
[[Categoria:Religio Graeca]]
[[Categoria:Mythologia Thebana]]</text>
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    <revision>
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      <contributor>
        <username>Marcus Terentius Bibliophilus</username>
        <id>2059</id>
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      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3283" xml:space="preserve">[[Fasciculus:Hercules and Iolaus mosaic - Anzio Nymphaeum.jpg|thumb|Hercules et Iolaus in [[Opus tessellatum|opere tessellato]] [[Antium (Latium)|Antii]] reperto in [[Nymphaeum|Nymphaeo]].]]
In mythologia Graeca '''Iolaus'''  fuit filius [[Iphiclus|Iphicli]], Herculis gemini fratris mortalis, et Automedusae. Ita per patrem [[Amphitryon|Amphitruonis]] et [[Alcmene|Alcmenae]] nepos, per matrem vero [[Pelops|Pelopis]] pronepos erat. [[Megara (mythologia)|Megaram]] primam Herculis uxorem in matrimonium accepit&lt;ref&gt;[[Apollodori bibliotheca]] II.4.11. [[Pausanias (scriptor)|Pausanias]] X.29.7.&lt;/ref&gt; postquam patruus furore correptus liberos quos ex ea genuerat interfecit.

== De rebus gestis ==
[[Fasciculus:Heracles, Iolaus and Eros - Cista Ficoroni foot.jpg|thumb|[[Cista Ficoroni|Cista Ficoronia]]ː Eros inter Iolaum et Herculem.]]
[[Patruus|Patrui]] [[Auriga|auriga]] erat et [[Olympia (certamina)|Olympicis ludis]] [[Quadrigae|quadrigarum]] certamen vicit&lt;ref&gt;[[Pausanias (scriptor)|Pausanias]] V.8.3-4. [[Hyginus mythographus|Hyginus]], ''Fabulae'' 273.11&lt;/ref&gt;. [[Argonautae|Argonautarum]] expeditioni et [[Aper Calydonius|apri Calydonii]] venationi interfuit. In [[Labores Herculis|duodecim laboribus]] patruum adsidue comitatus est et magno usui ei fuit inprimis adversus [[Hydra Lernaea|hydram]]&lt;ref&gt;[[Diodorus Siculus]] IV.11.20. [[Apollodori bibliotheca]] II.5.2&lt;/ref&gt;. [[Colonia|Coloniam]] [[Thebae (Boeotia)|Thebanorum]] quoque in [[Sardinia|Sardiniam]] duxit&lt;ref&gt;[[Diodorus Siculus]] IV.29.&lt;/ref&gt;. Postremo Herculis iam mortui filios, [[Heracleidae|Heraclidas]] dictos, tuetur et proelio sive ad Thebas, ut voluit poeta [[Pindarus]]&lt;ref&gt;''Pythia'' IX&lt;/ref&gt;, sive prope [[Marathon (Graecia)|Marathonem]], ut Athenienses malebant, [[Eurystheus (mythologia)|Eurystheum]] veterem Herculis inimicum vicit et occidit&lt;ref&gt;[[Pausanias (scriptor)|Pausanias]] I.44.10.&lt;/ref&gt;, quod argumentum tragoediae ''[[Heraclidae (Euripides)|Heraclidarum]]'' praebuit. Fabula addit deos viro senio confecto ad unam horam iuventutem reddidisse ita ut inimicum profligare posset&lt;ref&gt;[[Ovidius]], ''[[Metamorphoses (Ovidius)|Metamorphoses]]'' IX.396-401 qui donum [[Hebe|Hebae]] tribuit mariti precibus victae. [[Euripides]], ''[[Heraclidae (Euripides)|Heraclidae]]'' 843sqq.&lt;/ref&gt;.

== De cultu heroico ==
Multis in locis arae Iolao erectae erant, imprimis Thebis ubi paria [[Homosexualitas|homosexualia]] ad Iolai aram  fidem sibi invicem iurabant&lt;ref&gt;[[Aristoteles]] apud [[Plutarchus|Plutarchum]], ''Vita Pelopidae'' XVIII.5. Cf [[Pausanias (scriptor)|Pausanias]] IX.23.1.&lt;/ref&gt; quia multi aestimabant hoc quoque modo Iolaum ab Hercule amatum esse. Non minus enim per Iolaum quam per Herculem in [[Boeotia]] iurabatur&lt;ref&gt;[[Aristophanes]], ''[[Acharnenses]]'' 867.&lt;/ref&gt;. Alibi quoque honorabatur ut in Sicilia et Athenis in Herculis templo&lt;ref&gt;[[Pausanias (scriptor)|Pausanias]] I.19.3.&lt;/ref&gt;. [[Agyrium|Agyrii]] certamina quoque in eius honorem fiebant et incolae capillos ei vovebant&lt;ref&gt;[[Diodorus Siculus]] IV.24.40.&lt;/ref&gt;.


== Notae ==
&lt;references/&gt;
== Nexus externi ==
*[https://journals.openedition.org/encyclopedieberbere//1583 Iolaos in ''Encyclopédie berbère'']
[[Categoria:Argonautae]]
[[Categoria:Hercules]]
[[Categoria:Religio Graeca]]
[[Categoria:Mythologia Thebana]]</text>
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  </page>
  <page>
    <title>Carabella</title>
    <ns>0</ns>
    <id>303567</id>
    <revision>
      <id>3697649</id>
      <parentid>3697619</parentid>
      <timestamp>2022-08-16T12:58:39Z</timestamp>
      <contributor>
        <username>IacobusAmor</username>
        <id>1163</id>
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      <minor />
      <comment>+Nexus externi ex en (10K)</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3249" xml:space="preserve">[[Fasciculus:Caravel Boa Esperanca Portugal.jpg|thumb|Carabella duobus malis cum velis triangularibus (Latinis dictis).]]
'''Carabella'''&lt;ref&gt;Vox [[lingua Italiana|Italiana]] originis [[Lusitane|Lusitanae]], interdum in textu Latino invenitur ([[Lilius Gyraldus|Lilii Gyraldi]] [https://books.google.ru/books?id=ZmZlam-Ns3EC&amp;q=carabella#v=snippet&amp;q=carabella&amp;f=false ''Opera quae extant omnia'', vol. I, Lugdini Batavorum 1696, p. 624]; [[Petrus Peckius|Petri Peckii]] [https://books.google.ru/books?id=yxS1DkgRKtcC&amp;q=carabella#v=snippet&amp;q=carabella&amp;f=false ''Ad rem nauticam pertinentes commentarii'', Amstelodami 1668, p.6]; [[Ludovicus Antonius Muratorius|Ludovici Antonii Muratorii]] [https://books.google.ru/books?id=Poxw1abcGxEC&amp;q=carabella#v=snippet&amp;q=carabella&amp;f=false ''Antiquitates Italicae medii aevi'', vol. V,  Arretii 1774, p. 275]). De ''carabo'' (i.e. nave) voce Latina mediaevali derivatur.&lt;/ref&gt; vel '''caravella'''&lt;ref&gt;{{Creanda|it|Giovanni Riccioli|Ioannes Baptista Ricciolius|Ioannis Baptistae Ricciolii}} [[Societas Iesu|SI]] [https://books.google.ru/books?id=_SsxAQAAMAAJ&amp;q=pontoni#v=snippet&amp;q=pontoni&amp;f=false ''Geographia et hydrographia reformata'', Venetiis 1672, p. 506]; Muratorius ibidem.&lt;/ref&gt; est genus [[navis velifera]]e duobus vel tribus [[malus (navigatio)|malis]], quod plerumque in [[Lusitania]], [[Hispania]], et [[Italia]] [[saeculum|saeculis]] a [[saeculum 12|duodecimo]] ad [[saeculum 15|quintum decimum]] in usu erat. Saeculo quinto decimo exeunte ''[[caracca]]e'', ad [[navigatio]]nem naves meliores, carabellas expulerunt.

[[Fasciculus:Title- Pinta (14868431197).jpg|thumb|''Pinta'', [[Christophorus Columbus|Columbi]] carabella.]]
[[Bartholomaeus Dias]] duabus carabellis [[promontorium Bonae Spei]] attendit. In [[exploratio|expeditione]] prima [[Christophorus Columbus|Columbi]] trium navium duae carabellae fuerunt, tertia autem et principalis ''caracca''; [[Vascus Gama]] duabus carabellis et duabus ''caraccis'' in [[India]]m navigavit; [[Expeditio Magellanica|expeditionis Magellanicae primae]] quattuor ''caraccae'' et una carabella participes fuerunt. 

==Notae==
&lt;references /&gt;

==Bibliographia==
* Schwarz, George Robert. [[2008]]. &quot;The History and Development of Caravels.&quot; Thesis, University of Cincinnati, Maius 2008. [http://nautarch.tamu.edu/Theses/pdf-files/Schwarz-MA2008.pdf PDF.] {{ling|Anglice}}

==Nexus externi==
{{Victionarium|Caravel|carabellam}}
* [https://www.britannica.com/technology/caravel de carabellis] (''Britannica'') {{ling|Anglice}}
* ''[https://web.archive.org/web/20071207154919/http://museu.marinha.pt/Museu/Site/PT Museu da Marinha]'' {{ling|Portugallice}}
* ''[https://web.archive.org/web/20071025015529/http://museu.marinha.pt/museu/site/pt/loja/reproducoes/ Museu da Marinha, fac-similes]'' {{ling|Portugallice}}
* ''[https://web.archive.org/web/20080215014748/http://www.instituto-camoes.pt/cvc/navegaport/c06.html Instituto Camões: Caravela]''
* Engstfeld, Axel. [[2002]]. ''[https://web.archive.org/web/20071208022611/http://www.arte.tv/de/wissen-entdeckung/abenteuer-arte/Durchbruch-am-Kap-des-Schreckens/1765302.html Durchbruch am Kap des Schreckens. ]'' {{ling|Theodisce}}

[[Categoria:Naves]]
[[Categoria:Verba Italiana mutuata]]
{{Myrias|Technologia}}</text>
      <sha1>98ttt5lf6wwsnyxbpnv354y2eihyhnk</sha1>
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    <revision>
      <id>3697735</id>
      <parentid>3697649</parentid>
      <timestamp>2022-08-17T10:12:22Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
      </contributor>
      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Naves]]; +[[Categoria:Genera navium]]</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3267" xml:space="preserve">[[Fasciculus:Caravel Boa Esperanca Portugal.jpg|thumb|Carabella duobus malis cum velis triangularibus (Latinis dictis).]]
'''Carabella'''&lt;ref&gt;Vox [[lingua Italiana|Italiana]] originis [[Lusitane|Lusitanae]], interdum in textu Latino invenitur ([[Lilius Gyraldus|Lilii Gyraldi]] [https://books.google.ru/books?id=ZmZlam-Ns3EC&amp;q=carabella#v=snippet&amp;q=carabella&amp;f=false ''Opera quae extant omnia'', vol. I, Lugdini Batavorum 1696, p. 624]; [[Petrus Peckius|Petri Peckii]] [https://books.google.ru/books?id=yxS1DkgRKtcC&amp;q=carabella#v=snippet&amp;q=carabella&amp;f=false ''Ad rem nauticam pertinentes commentarii'', Amstelodami 1668, p.6]; [[Ludovicus Antonius Muratorius|Ludovici Antonii Muratorii]] [https://books.google.ru/books?id=Poxw1abcGxEC&amp;q=carabella#v=snippet&amp;q=carabella&amp;f=false ''Antiquitates Italicae medii aevi'', vol. V,  Arretii 1774, p. 275]). De ''carabo'' (i.e. nave) voce Latina mediaevali derivatur.&lt;/ref&gt; vel '''caravella'''&lt;ref&gt;{{Creanda|it|Giovanni Riccioli|Ioannes Baptista Ricciolius|Ioannis Baptistae Ricciolii}} [[Societas Iesu|SI]] [https://books.google.ru/books?id=_SsxAQAAMAAJ&amp;q=pontoni#v=snippet&amp;q=pontoni&amp;f=false ''Geographia et hydrographia reformata'', Venetiis 1672, p. 506]; Muratorius ibidem.&lt;/ref&gt; est genus [[navis velifera]]e duobus vel tribus [[malus (navigatio)|malis]], quod plerumque in [[Lusitania]], [[Hispania]], et [[Italia]] [[saeculum|saeculis]] a [[saeculum 12|duodecimo]] ad [[saeculum 15|quintum decimum]] in usu erat. Saeculo quinto decimo exeunte ''[[caracca]]e'', ad [[navigatio]]nem naves meliores, carabellas expulerunt.

[[Fasciculus:Title- Pinta (14868431197).jpg|thumb|''Pinta'', [[Christophorus Columbus|Columbi]] carabella.]]
[[Bartholomaeus Dias]] duabus carabellis [[promontorium Bonae Spei]] attendit. In [[exploratio|expeditione]] prima [[Christophorus Columbus|Columbi]] trium navium duae carabellae fuerunt, tertia autem et principalis ''caracca''; [[Vascus Gama]] duabus carabellis et duabus ''caraccis'' in [[India]]m navigavit; [[Expeditio Magellanica|expeditionis Magellanicae primae]] quattuor ''caraccae'' et una carabella participes fuerunt. 

==Notae==
&lt;references /&gt;

==Bibliographia==
* Schwarz, George Robert. [[2008]]. &quot;The History and Development of Caravels.&quot; Thesis, University of Cincinnati, Maius 2008. [http://nautarch.tamu.edu/Theses/pdf-files/Schwarz-MA2008.pdf PDF.] {{ling|Anglice}}

==Nexus externi==
{{Victionarium|Caravel|carabellam}}
* [https://www.britannica.com/technology/caravel de carabellis] (''Britannica'') {{ling|Anglice}}
* ''[https://web.archive.org/web/20071207154919/http://museu.marinha.pt/Museu/Site/PT Museu da Marinha]'' {{ling|Portugallice}}
* ''[https://web.archive.org/web/20071025015529/http://museu.marinha.pt/museu/site/pt/loja/reproducoes/ Museu da Marinha, fac-similes]'' {{ling|Portugallice}}
* ''[https://web.archive.org/web/20080215014748/http://www.instituto-camoes.pt/cvc/navegaport/c06.html Instituto Camões: Caravela]''
* Engstfeld, Axel. [[2002]]. ''[https://web.archive.org/web/20071208022611/http://www.arte.tv/de/wissen-entdeckung/abenteuer-arte/Durchbruch-am-Kap-des-Schreckens/1765302.html Durchbruch am Kap des Schreckens. ]'' {{ling|Theodisce}}

[[Categoria:Genera navium|carabella]]
[[Categoria:Verba Italiana mutuata]]
{{Myrias|Technologia}}</text>
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  <page>
    <title>Alexander Halensis</title>
    <ns>0</ns>
    <id>303575</id>
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      <id>3697641</id>
      <timestamp>2022-08-16T12:09:16Z</timestamp>
      <contributor>
        <username>Utilo</username>
        <id>18337</id>
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      <comment>Paginam instituit, scribens '[[Fasciculus:Alexanderofhales.png|thumb|''Doctor Alexander Halensis'' a Georgio Glover [[Chalcographia|in aes incisus]], saec. 17 medio.]] Alexander Halensis&lt;ref&gt;etiam Alexander of Hales, Alensis, Halesius, Alesius&lt;/ref&gt; (natus circa annum [[1185]] in oppido [[Hales]] in [[Salopiensis comitatus|Salopiensi comitatu]] aut [[Glevum|Glevi]] in [[Glocestriensis comitatus|Glocestriensi comitatu]], mortuus die [[21 Augusti]] [[1245]] [[Lutetia]]e), dictus ''Doctor I...'</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2995" xml:space="preserve">[[Fasciculus:Alexanderofhales.png|thumb|''Doctor Alexander Halensis'' a Georgio Glover [[Chalcographia|in aes incisus]], saec. 17 medio.]]
Alexander Halensis&lt;ref&gt;etiam Alexander of Hales, Alensis, Halesius, Alesius&lt;/ref&gt; (natus circa annum [[1185]] in oppido [[Hales]] in [[Salopiensis comitatus|Salopiensi comitatu]] aut [[Glevum|Glevi]] in [[Glocestriensis comitatus|Glocestriensi comitatu]], mortuus die [[21 Augusti]] [[1245]] [[Lutetia]]e), dictus ''Doctor Irrefragibilis'' (ab [[Alexander IV]] [[papa]] in [[Bulla apostolica]] ''De Fontibus Paradisi'') et ''Theologorum Monarcha'', fuit [[Ordo Fratrum Minorum|Franciscanus]] [[Anglia|Anglicus]], [[theologia|theologus]] and [[philosophia|philosophus]] maximi momenti [[Scholastica (philosophia)|scholasticae]] methodi. Conditor sic dictae scholae antiquioris Franciscanorum habetur; imprimis ob commentarium sentiariarum [[Petrus Lombardus|Petri Lombardi]] notus est.

== Notae ==
&lt;references/&gt;

== Opera ==
;editiones antiquiores
*Summa universae theologiae. P. 3, Ihoannes de Colonia et Ioannes Manthen, Venetiis 1475. [http://digital.ub.uni-duesseldorf.de/ink/content/titleinfo/7970752 in interrete]
*Summa universae theologiae. Antonius Koberger, Norimbergae 1481.
:Vol. 1 (24.I.1482). Generalis discursus ī sūmā. [http://digital.ub.uni-duesseldorf.de/urn/urn:nbn:de:hbz:061:1-17982 in interrete]
:Vol. 2 (29.XI.1481). Tabula tractatuus hui[us] secūde ptis sūme alexādri. [http://digital.ub.uni-duesseldorf.de/urn/urn:nbn:de:hbz:061:1-331278 in interrete]

;editiones recentiores
*Glossa in quatuor libros sententiarum Petri Lombardi. Edita a Quaracchi Fathers. Bibliotheca Franciscana scholastica medii aevi, t. 12–15. Romae: Collegii S. Bonaventurae, 1951–1957.
*Quaestiones disputatae antequam esset frater. Editae a Quaracchi Fathers. Bibliotheca Franciscana scholastica medii aevi, t. 19–21. Quaracchi: Collegii S. Bonaventurae,1960.
*Summa universis theologia, (Summa fratris Alexandri), edita a Bernardini Klumper et Quaracchi Fathers, 4 vol. Romae: Collegii S. Bonaventurae, 1924–1948.

== Bibliographia ==
*Osborne, Kenan B. “Alexander of Hales,” in The History of Franciscan Theology edita ab eodem. St. Bonaventure, NY: Franciscan Institute Publications, 1994.
*Dettloff, Werner: Alexander Halesius. In: TRE 2 (1978), p. 245–248
*Gössmann, Elisabeth: Metaphysik und Heilsgeschichte. Eine theologische Untersuchung der Summa Halensis (= Mitteilungen des Grabmann-Instituts der Universität München, Sonderband), Grabmann-Institut zur Erforschung der Mittelalterlichen Theologie und Philosophie, Universitas Monacensis, Monanci 1964

== Nexus externi ==
{{CommuniaCat|Alexander of Hales|Alexandrum Halensem}}

{{bio-stipula}}

[[Categoria:Viri]]
[[Categoria:Nati saeculo 12]]
[[Categoria:Mortui 1245]]
[[Categoria:Auctores Latini mediaevales]]
[[Categoria:Doctores Ecclesiae Catholicae]]
[[Categoria:Franciscani]]
[[Categoria:Philosophi Angliae]]
[[Categoria:Scriptores Angliae]]
[[Categoria:Theologi]]</text>
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  <page>
    <title>Macaria (Herculis filia)</title>
    <ns>0</ns>
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    <revision>
      <id>3697682</id>
      <timestamp>2022-08-16T16:35:55Z</timestamp>
      <contributor>
        <username>Marcus Terentius Bibliophilus</username>
        <id>2059</id>
      </contributor>
      <comment>Paginam instituit, scribens 'In mythologia Graeca '''Macaria''' est Herculis filia quam [[Euripides]] in tragoediam ''[[Heraclidae (Euripides)|Heraclidarum]]'' induxit. [[Iolaus]] cum parvis Herculis liberis Eurysthei odium in [[Attica|Atticam]] fugit ubi patrocinium regis [[Demophoon|Demophontis]] impetravit. Rex armis profugos tueri paratus est sed vatum oracula immolationem virginis nobilis ad victoriam necessariam esse cecinerunt. Tum Macaria sese ad mortem voluntariam offert ut Deme...'</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="840" xml:space="preserve">In mythologia Graeca '''Macaria''' est Herculis filia quam [[Euripides]] in tragoediam ''[[Heraclidae (Euripides)|Heraclidarum]]'' induxit. [[Iolaus]] cum parvis Herculis liberis Eurysthei odium in [[Attica|Atticam]] fugit ubi patrocinium regis [[Demophoon|Demophontis]] impetravit. Rex armis profugos tueri paratus est sed vatum oracula immolationem virginis nobilis ad victoriam necessariam esse cecinerunt. Tum Macaria sese ad mortem voluntariam offert ut [[Demeter|Demetris]] filia placanda fratribus  gentique saluti foret. Quia nomen in argumento tantum et in personarum catalogo legitur, non in ipso fabulae textu philologi nonnulli coniecerunt [[Mythologia|mythographum]] posteriorem ex fonte proximo nomen heroidi imposuisse potius quam heroidem fonti nomen suum indidisse. 


[[Categoria:Hercules]]
[[Categoria:Mythologia Graeca]]</text>
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    <revision>
      <id>3697683</id>
      <parentid>3697682</parentid>
      <timestamp>2022-08-16T16:37:56Z</timestamp>
      <contributor>
        <username>Marcus Terentius Bibliophilus</username>
        <id>2059</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1066" xml:space="preserve">In mythologia Graeca '''Macaria''' est Herculis filia quam [[Euripides]] in tragoediam ''[[Heraclidae (Euripides)|Heraclidarum]]'' induxit. [[Iolaus]] cum parvis Herculis liberis Eurysthei odium in [[Attica|Atticam]] fugit ubi patrocinium regis [[Demophoon|Demophontis]] impetravit. Rex armis profugos tueri paratus est sed vatum oracula immolationem virginis nobilis ad victoriam necessariam esse cecinerunt. Tum Macaria sese ad mortem voluntariam offert ut [[Demeter|Demetris]] filia placanda fratribus  gentique saluti foret. Quia nomen in argumento tantum et in personarum catalogo legitur, non in ipso fabulae textu philologi nonnulli coniecerunt [[Mythologia|mythographum]] posteriorem ex fonte proximo nomen heroidi imposuisse potius quam heroidem fonti nomen suum indidisse. 

== Plura legere si cupis ==
P. Roussel, &quot;[https://www.persee.fr/doc/rbph_0035-0818_1922_num_1_2_6168 Le thème du sacrifice volontaire dans la tragédie d'Euripide]&quot;, ''Revue belge de Philologie et d'Histoire'', 1922ː 225-240
[[Categoria:Hercules]]
[[Categoria:Mythologia Graeca]]</text>
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    </revision>
    <revision>
      <id>3697684</id>
      <parentid>3697683</parentid>
      <timestamp>2022-08-16T16:38:32Z</timestamp>
      <contributor>
        <username>Marcus Terentius Bibliophilus</username>
        <id>2059</id>
      </contributor>
      <comment>/* Plura legere si cupis */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1067" xml:space="preserve">In mythologia Graeca '''Macaria''' est Herculis filia quam [[Euripides]] in tragoediam ''[[Heraclidae (Euripides)|Heraclidarum]]'' induxit. [[Iolaus]] cum parvis Herculis liberis Eurysthei odium in [[Attica|Atticam]] fugit ubi patrocinium regis [[Demophoon|Demophontis]] impetravit. Rex armis profugos tueri paratus est sed vatum oracula immolationem virginis nobilis ad victoriam necessariam esse cecinerunt. Tum Macaria sese ad mortem voluntariam offert ut [[Demeter|Demetris]] filia placanda fratribus  gentique saluti foret. Quia nomen in argumento tantum et in personarum catalogo legitur, non in ipso fabulae textu philologi nonnulli coniecerunt [[Mythologia|mythographum]] posteriorem ex fonte proximo nomen heroidi imposuisse potius quam heroidem fonti nomen suum indidisse. 

== Plura legere si cupis ==
*P. Roussel, &quot;[https://www.persee.fr/doc/rbph_0035-0818_1922_num_1_2_6168 Le thème du sacrifice volontaire dans la tragédie d'Euripide]&quot;, ''Revue belge de Philologie et d'Histoire'', 1922ː 225-240
[[Categoria:Hercules]]
[[Categoria:Mythologia Graeca]]</text>
      <sha1>3yb6z1bqy9fgpqcj1d2obhk5tq8clrs</sha1>
    </revision>
    <revision>
      <id>3697685</id>
      <parentid>3697684</parentid>
      <timestamp>2022-08-16T16:42:44Z</timestamp>
      <contributor>
        <username>Marcus Terentius Bibliophilus</username>
        <id>2059</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1282" xml:space="preserve">In mythologia Graeca '''Macaria''' est Herculis filia quam [[Euripides]] in tragoediam ''[[Heraclidae (Euripides)|Heraclidarum]]'' induxit. [[Iolaus]] cum parvis Herculis liberis Eurysthei odium in [[Attica|Atticam]] fugit ubi patrocinium regis [[Demophoon|Demophontis]] impetravit. Rex armis profugos tueri paratus est sed vatum oracula immolationem virginis nobilis ad victoriam necessariam esse cecinerunt. Tum Macaria sese ad mortem voluntariam offert ut [[Demeter|Demetris]] filia placanda fratribus  gentique saluti foret. Quia nomen in argumento tantum et in personarum catalogo legitur, non in ipso fabulae textu philologi nonnulli coniecerunt [[Mythologia|mythographum]] posteriorem ex fonte proximo nomen heroidi imposuisse potius quam heroidem fonti nomen suum indidisse. 

== Fontes ==
*[[Euripides]], ''[[Heraclidae (Euripides)|Heraclidae]]'' 474-630
*[[Pausanias (scriptor)|Pausanias]], ''[[Descriptio Graeciae]]'' I.32.6
*[[Strabo]], ''[[Geographica (Strabo)|Geographica]]'' VIII.6.19
== Plura legere si cupis ==
*P. Roussel, &quot;[https://www.persee.fr/doc/rbph_0035-0818_1922_num_1_2_6168 Le thème du sacrifice volontaire dans la tragédie d'Euripide]&quot;, ''Revue belge de Philologie et d'Histoire'', 1922ː 225-240
[[Categoria:Hercules]]
[[Categoria:Mythologia Graeca]]</text>
      <sha1>8c6xc1e06hyngv99frgjupvwslbvgsc</sha1>
    </revision>
    <revision>
      <id>3697686</id>
      <parentid>3697685</parentid>
      <timestamp>2022-08-16T16:43:51Z</timestamp>
      <contributor>
        <username>Marcus Terentius Bibliophilus</username>
        <id>2059</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1308" xml:space="preserve">In mythologia Graeca '''Macaria''' est Herculis filia quam [[Euripides]] in tragoediam ''[[Heraclidae (Euripides)|Heraclidarum]]'' induxit. [[Iolaus]] cum parvis Herculis liberis [[Eurystheus (mythologia)|Eurysthei]] odium in [[Attica|Atticam]] fugit ubi patrocinium regis [[Demophoon|Demophontis]] impetrat. Rex armis profugos tueri paratus est sed vatum oracula immolationem virginis nobilis ad victoriam necessariam esse cecinerunt. Tum Macaria sese ad mortem voluntariam offert ut [[Demeter|Demetris]] filia placanda fratribus  gentique saluti foret. Quia nomen in argumento tantum et in personarum catalogo legitur, non in ipso fabulae textu philologi nonnulli coniecerunt [[Mythologia|mythographum]] posteriorem ex fonte proximo nomen heroidi imposuisse potius quam heroidem fonti nomen suum indidisse. 

== Fontes ==
*[[Euripides]], ''[[Heraclidae (Euripides)|Heraclidae]]'' 474-630
*[[Pausanias (scriptor)|Pausanias]], ''[[Descriptio Graeciae]]'' I.32.6
*[[Strabo]], ''[[Geographica (Strabo)|Geographica]]'' VIII.6.19
== Plura legere si cupis ==
*P. Roussel, &quot;[https://www.persee.fr/doc/rbph_0035-0818_1922_num_1_2_6168 Le thème du sacrifice volontaire dans la tragédie d'Euripide]&quot;, ''Revue belge de Philologie et d'Histoire'', 1922ː 225-240
[[Categoria:Hercules]]
[[Categoria:Mythologia Graeca]]</text>
      <sha1>8so9arj87kijh7fwc6muvmbb1d6u9l9</sha1>
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    <revision>
      <id>3697688</id>
      <parentid>3697686</parentid>
      <timestamp>2022-08-16T16:44:51Z</timestamp>
      <contributor>
        <username>Marcus Terentius Bibliophilus</username>
        <id>2059</id>
      </contributor>
      <comment>/* Plura legere si cupis */</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1333" xml:space="preserve">In mythologia Graeca '''Macaria''' est Herculis filia quam [[Euripides]] in tragoediam ''[[Heraclidae (Euripides)|Heraclidarum]]'' induxit. [[Iolaus]] cum parvis Herculis liberis [[Eurystheus (mythologia)|Eurysthei]] odium in [[Attica|Atticam]] fugit ubi patrocinium regis [[Demophoon|Demophontis]] impetrat. Rex armis profugos tueri paratus est sed vatum oracula immolationem virginis nobilis ad victoriam necessariam esse cecinerunt. Tum Macaria sese ad mortem voluntariam offert ut [[Demeter|Demetris]] filia placanda fratribus  gentique saluti foret. Quia nomen in argumento tantum et in personarum catalogo legitur, non in ipso fabulae textu philologi nonnulli coniecerunt [[Mythologia|mythographum]] posteriorem ex fonte proximo nomen heroidi imposuisse potius quam heroidem fonti nomen suum indidisse. 

== Fontes ==
*[[Euripides]], ''[[Heraclidae (Euripides)|Heraclidae]]'' 474-630
*[[Pausanias (scriptor)|Pausanias]], ''[[Descriptio Graeciae]]'' I.32.6
*[[Strabo]], ''[[Geographica (Strabo)|Geographica]]'' VIII.6.19
== Plura legere si cupis ==
*P. Roussel, &quot;[https://www.persee.fr/doc/rbph_0035-0818_1922_num_1_2_6168 Le thème du sacrifice volontaire dans la tragédie d'Euripide]&quot;, ''Revue belge de Philologie et d'Histoire'', 1922ː 225-240
[[Categoria:Hercules]]
[[Categoria:Mythologia Graeca]]
[[Categoria:Heraclidae]]</text>
      <sha1>2deq60348i3r61sjfe6ry12mv4i3x29</sha1>
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    <revision>
      <id>3697690</id>
      <parentid>3697688</parentid>
      <timestamp>2022-08-16T16:48:18Z</timestamp>
      <contributor>
        <username>Marcus Terentius Bibliophilus</username>
        <id>2059</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1334" xml:space="preserve">In mythologia Graeca '''Macaria''' est Herculis filia quam [[Euripides]] in tragoediam ''[[Heraclidae (Euripides)|Heraclidarum]]'' induxit. [[Iolaus]] cum parvis Herculis liberis [[Eurystheus (mythologia)|Eurysthei]] odium in [[Attica|Atticam]] fugit ubi patrocinium regis [[Demophoon|Demophontis]] impetrat. Rex armis profugos tueri paratus est sed vatum oracula immolationem virginis nobilis ad victoriam necessariam esse cecinerunt. Tum Macaria sese ad mortem voluntariam offert ut [[Demeter|Demetris]] filia placanda fratribus  gentique saluti foret. Quia nomen in argumento tantum et in personarum catalogo legitur, non in ipso fabulae textu, philologi nonnulli coniecerunt [[Mythologia|mythographum]] posteriorem ex fonte proximo nomen heroidi imposuisse potius quam heroidem fonti nomen suum indidisse. 

== Fontes ==
*[[Euripides]], ''[[Heraclidae (Euripides)|Heraclidae]]'' 474-630
*[[Pausanias (scriptor)|Pausanias]], ''[[Descriptio Graeciae]]'' I.32.6
*[[Strabo]], ''[[Geographica (Strabo)|Geographica]]'' VIII.6.19
== Plura legere si cupis ==
*P. Roussel, &quot;[https://www.persee.fr/doc/rbph_0035-0818_1922_num_1_2_6168 Le thème du sacrifice volontaire dans la tragédie d'Euripide]&quot;, ''Revue belge de Philologie et d'Histoire'', 1922ː 225-240
[[Categoria:Hercules]]
[[Categoria:Mythologia Graeca]]
[[Categoria:Heraclidae]]</text>
      <sha1>iy4je6wh1crow7tipq6qe8bjanbs6cq</sha1>
    </revision>
    <revision>
      <id>3697691</id>
      <parentid>3697690</parentid>
      <timestamp>2022-08-16T16:53:23Z</timestamp>
      <contributor>
        <username>Marcus Terentius Bibliophilus</username>
        <id>2059</id>
      </contributor>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1361" xml:space="preserve">In mythologia Graeca '''Macaria''' est Herculis filia quam [[Euripides]] in tragoediam ''[[Heraclidae (Euripides)|Heraclidarum]]'' induxit. [[Iolaus]] cum parvis Herculis liberis [[Eurystheus (mythologia)|Eurysthei]] odium in [[Attica|Atticam]] fugit ubi patrocinium regis [[Demophoon|Demophontis]] impetrat. Rex armis profugos tueri paratus est sed vatum oracula immolationem virginis nobilis ad victoriam necessariam esse cecinerunt. Tum Macaria sese ad mortem voluntariam offert ut [[Demeter|Demetris]] filia placanda fratribus  gentique saluti foret. Quia nomen in argumento tantum et in personarum catalogo legitur, non in ipso fabulae textu, philologi nonnulli coniecerunt [[Mythologia|mythographum]] posteriorem ex fonte proximo nomen heroidi imposuisse potius quam heroidem fonti nomen suum in generosi facti memoriam indidisse. 

== Fontes ==
*[[Euripides]], ''[[Heraclidae (Euripides)|Heraclidae]]'' 474-630
*[[Pausanias (scriptor)|Pausanias]], ''[[Descriptio Graeciae]]'' I.32.6
*[[Strabo]], ''[[Geographica (Strabo)|Geographica]]'' VIII.6.19
== Plura legere si cupis ==
*P. Roussel, &quot;[https://www.persee.fr/doc/rbph_0035-0818_1922_num_1_2_6168 Le thème du sacrifice volontaire dans la tragédie d'Euripide]&quot;, ''Revue belge de Philologie et d'Histoire'', 1922ː 225-240
[[Categoria:Hercules]]
[[Categoria:Mythologia Graeca]]
[[Categoria:Heraclidae]]</text>
      <sha1>d9fet3km2zt349nmfsyfkkurkdfa8v1</sha1>
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    <title>Urbs Bissagensis</title>
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      <contributor>
        <ip>84.78.253.101</ip>
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      <comment>Redirigens ad [[Bissau]]</comment>
      <model>wikitext</model>
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      <text bytes="20" xml:space="preserve">#REDIRECT [[Bissau]]</text>
      <sha1>a1vewqp3mom678b1hmutjucxr0x49xs</sha1>
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    <revision>
      <id>3697717</id>
      <parentid>3697702</parentid>
      <timestamp>2022-08-17T04:09:57Z</timestamp>
      <contributor>
        <username>Xqbot</username>
        <id>9619</id>
      </contributor>
      <minor />
      <comment>automaton: rectificatio redirectionis duplicis → [[Bissavia]]</comment>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22" xml:space="preserve">#REDIRECT [[Bissavia]]</text>
      <sha1>pbyuhil2yadrkok3etu2u7hbdofl78e</sha1>
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    <title>Bissau</title>
    <ns>0</ns>
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    <redirect title="Bissavia" />
    <revision>
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      <contributor>
        <username>IacobusAmor</username>
        <id>1163</id>
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      <comment>IacobusAmor movit paginam [[Bissau]] ad [[Bissavia]] praeter redirectionem: Lemma</comment>
      <model>wikitext</model>
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      <text bytes="22" xml:space="preserve">#REDIRECT [[Bissavia]]</text>
      <sha1>pbyuhil2yadrkok3etu2u7hbdofl78e</sha1>
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  <page>
    <title>Usor:Friothaire</title>
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    <revision>
      <id>3697713</id>
      <timestamp>2022-08-17T02:02:16Z</timestamp>
      <contributor>
        <username>Friothaire</username>
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      <comment>Paginam instituit, scribens 'Advocatus sum in Pennsylvania.'</comment>
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      <text bytes="30" xml:space="preserve">Advocatus sum in Pennsylvania.</text>
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    <title>Usor:Friothaire/Harenarium</title>
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      <id>3697714</id>
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      <contributor>
        <username>Friothaire</username>
        <id>150629</id>
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      <comment>Paginam instituit, scribens 'Advocatus sum in Pennsylvania.'</comment>
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      <text bytes="30" xml:space="preserve">Advocatus sum in Pennsylvania.</text>
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    <title>Disputatio Usoris:Friothaire</title>
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      <id>3697715</id>
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      <contributor>
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        <id>150629</id>
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      <comment>Paginam instituit, scribens 'Advocatus sum in Pennsylvania.'</comment>
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      <text bytes="30" xml:space="preserve">Advocatus sum in Pennsylvania.</text>
      <sha1>88249yznhl0tisy4864rg9ez18neofl</sha1>
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  <page>
    <title>Navis remivaga</title>
    <ns>0</ns>
    <id>303582</id>
    <revision>
      <id>3697726</id>
      <timestamp>2022-08-17T09:40:05Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
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      <comment>nova</comment>
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      <text bytes="1648" xml:space="preserve">[[Fasciculus:Gizeh Sonnenbarke BW 2.jpg|thumb|[[Cheops|Cheopis]] navis, cum eo sepulta, quae quinquies binis remis movebatur]]
'''Navis remivaga'''{{Convertimus}}&lt;ref&gt;Cf. in [[Saturae Menippeae (Varro)|''Menippeis'']] [[Marcus Terentius Varro|Varro]]nis: &quot;nautae remivagam movent celocem&quot;.&lt;/ref&gt; est [[navigium]], quod [[remus (navigatio)|remis]] movetur. 

Antiquissimum genus navium est. [[Linter|Lintri]] parvi binis remis movebantur; magnae naves antiquitatis per numerum ordinum remorum dividebantur in [[uniremis|uniremes]], [[biremis|biremes]], [[triremis|triremes]], [[quadriremis|quadriremes]]. Saepe naves remivagae magnae et [[velum (navigatio)|vela]] habuerunt, ut nautae remis tantum vento adverso uterentur. Haec naves saepe ''[[galera]]e'' nuncupabantur&lt;ref&gt;[[:d:Q93433786|Iacobi Coreni]] [https://books.google.ru/books?id=FSnQcGffYcYC&amp;q=galera#v=snippet&amp;q=galera&amp;f=false ''Clupeus patientiae'', Lugduni 1624, cap. XXVII]; [[:d:Q27572052|Adami Eberti]] [https://books.google.ru/books?id=m9E051lU_ZgC&amp;q=galerarum#v=snippet&amp;q=galerarum&amp;f=false ''Quinquaginta relationes ex Parnasso'', Hamburgi 1683, p. 256]; {{Creanda|cs|Amand Hermann|Amandus Hermann|Amandi Hermann}} [https://books.google.ru/books?id=8A0Cp0_WNdMC&amp;q=galerarum#v=snippet&amp;q=galerarum&amp;f=false ''Capistranus triumphans'', Coloniae 1700, pp. 533, 542, 543].&lt;/ref&gt;.

==Notae==
&lt;references /&gt;

==Nexus externi==
{{forcellini}}[http://linguax.com/lexica/forc.php?searchedLG=remivagus s.v. ''remivagus'']
* [https://bigenc.ru/technology_and_technique/text/2377244 Naves remivagae] in ''Encyclopaedia Russica magna'' {{ling|Russice}}

[[Categoria:Naves]]
{{Myrias|Technologia}}</text>
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      <contributor>
        <username>Demetrius Talpa</username>
        <id>81729</id>
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      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Naves]]; +[[Categoria:Genera navium]]</comment>
      <model>wikitext</model>
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      <text bytes="1665" xml:space="preserve">[[Fasciculus:Gizeh Sonnenbarke BW 2.jpg|thumb|[[Cheops|Cheopis]] navis, cum eo sepulta, quae quinquies binis remis movebatur]]
'''Navis remivaga'''{{Convertimus}}&lt;ref&gt;Cf. in [[Saturae Menippeae (Varro)|''Menippeis'']] [[Marcus Terentius Varro|Varro]]nis: &quot;nautae remivagam movent celocem&quot;.&lt;/ref&gt; est [[navigium]], quod [[remus (navigatio)|remis]] movetur. 

Antiquissimum genus navium est. [[Linter|Lintri]] parvi binis remis movebantur; magnae naves antiquitatis per numerum ordinum remorum dividebantur in [[uniremis|uniremes]], [[biremis|biremes]], [[triremis|triremes]], [[quadriremis|quadriremes]]. Saepe naves remivagae magnae et [[velum (navigatio)|vela]] habuerunt, ut nautae remis tantum vento adverso uterentur. Haec naves saepe ''[[galera]]e'' nuncupabantur&lt;ref&gt;[[:d:Q93433786|Iacobi Coreni]] [https://books.google.ru/books?id=FSnQcGffYcYC&amp;q=galera#v=snippet&amp;q=galera&amp;f=false ''Clupeus patientiae'', Lugduni 1624, cap. XXVII]; [[:d:Q27572052|Adami Eberti]] [https://books.google.ru/books?id=m9E051lU_ZgC&amp;q=galerarum#v=snippet&amp;q=galerarum&amp;f=false ''Quinquaginta relationes ex Parnasso'', Hamburgi 1683, p. 256]; {{Creanda|cs|Amand Hermann|Amandus Hermann|Amandi Hermann}} [https://books.google.ru/books?id=8A0Cp0_WNdMC&amp;q=galerarum#v=snippet&amp;q=galerarum&amp;f=false ''Capistranus triumphans'', Coloniae 1700, pp. 533, 542, 543].&lt;/ref&gt;.

==Notae==
&lt;references /&gt;

==Nexus externi==
{{forcellini}}[http://linguax.com/lexica/forc.php?searchedLG=remivagus s.v. ''remivagus'']
* [https://bigenc.ru/technology_and_technique/text/2377244 Naves remivagae] in ''Encyclopaedia Russica magna'' {{ling|Russice}}

[[Categoria:Genera navium|remivaga]]
{{Myrias|Technologia}}</text>
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        <id>81729</id>
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      <comment>[[Project:HotCat|HotCat]]: -[[Categoria:Navis]]; +[[Categoria:Naves]]</comment>
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