Vita S. Wilfridi (Fridegodus)

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    <title>Vita S. Wilfridi (Fridegodus)</title>
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|OperaeTitulus= Vita S. Wilfridi 
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|Annus=saeculo IX
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'''[[Patrologia Latina/133|Migne ''Patrologia Latina Tomus 133'']]'''

133.1011B
==Observationes praeviae.==
133.0979A| Plures Wilfridum, eo nomine primum Eboracensem episcopum, scriptis suis celebravere auctores. In primis Beda Venerabilis res ab eo gestas describit in variis Historiae suae locis, maxime in lib. III, cap. 25, et lib. V, cap. 20. Bedae aequalis accedit Eddi seu Heddius, cognomento Stephanus, qui librum de Vita sancti Wilfridi composuisse fertur apud Joannem Pitseum aliosque recentiores. « Is fuit invitatus de Cantia a reverendissimo viro Wilfrido, ut primus post Jacobum cantandi magister Nortanhumbrorum Ecclesiis esset, » teste Beda in Historiae lib. IV, cap. 2; qua ex consuetudine Wilfridi gesta certo rescire potuit. Post Heddium occurrit Fridegodus, cui Odo Cantuariensis episcopus Vitam ejusdem sancti heroico metro texere praecepit. Legerat Willelmus Malmesburiae monachus eam Fridegodi scriptionem, de qua hoc profert judicium in lib. I de Pontificibus Cantuariensibus: « Exsecutus est id munus Fridegodus quidam versibus non improbandis: 133.0979B| nisi quod latinitatem perosus, graecitatem amat, Graecula verba frequentat, ut merito dictis ejus aptetur illud Plautinum:
Haec quidem praeter Sybillam leget nemo. Huic lucubrationi manum admovit Fridegodus statim post Wilfridi reliquiarum translationem in Ecclesiam Cantuariensem. Ipse etiam Odo, qui Ecclesiae Cantuariensi praefuit ab anno 834 in annos quatuor et viginti, Wilfridi gesta descripsisse perhibetur ab Eadmero. Eadmerus vero seu Edmerus monachus, Anselmo Cantuariensi pontifici a secretis, id postea praestitit anno 1121. Sic enim praefixa habet inscriptio codicis manu exarati bibliothecae Cottonianae: Vita sancti Wilfridi archiepiscopi per Eadmerum Cantuariensem anno MCXXI. Res ab se descriptas in exordio et in fine approbat « auctoritate venerabilium virorum Odonis archiepiscopi et Bedae 133.0980A| presbyteri » , nulla Heddii aut Fridegodi mentione facta, tametsi Fridegodi ipsissima verba non raro usurpat. Unde suspicio injicitur num Fridegodi fetum Odoni, qui hortator operis fuit, supponat. Sed tamen ex prologo ipsius Odonis, quem Eadmerus in fine Vitae ab se compositae citat, aliam Odonis a Fridegodi lucubratione fuisse conjectura est. Eadmeri fragmentum de translatione corporis sancti Wilfridi recitat Gervasius Dorobernensis monachus in libro De combustione et reparatione Dorobernensis ecclesiae; quod fragmentum ex alio Eadmeri opere desumptum est. Eadmeri aequalis fuit Willelmus Malmesburiae monachus, qui itidem facta sancti Wilfridi recantavit in libro secundo de Pontificibus Anglorum, ubi de episcopis Eboracensibus. Post eos omnes Petrus Blesensis, archidiaconus Bathoniensis, librum de Vita sancti Wilfridi composuit, eumque Galfrido Eboracensi archiepiscopo dedicavit, uti legitur in Monastico anglicano, pag. 172, ubi fragmentum 133.0980B| ex eodem Petro adducitur. His omnibus annumerandi sunt Anglorum scriptores, in his Thomas Stubbesius in libro de Episcopis Eboracensibus, quorum catalogus vetustus exstat apud Labbeum in tomo I Bibliothecae novae, pag. 322. Ex praedictis auctoribus obvii quibusque sunt Beda et Willelmus.
===2.===
Wilfridi diem natalem exprimunt tum Martyrologia recentiora, tum vulgata Bedae historia IV Idus Octobris; tametsi haec verba intrusa sunt in textum Bedae, quae apud Abrahamum Whelocum intra parenthesim conclusa, desunt in exemplari nostro Germanensi, uti et in Aluredi versione Saxonica. Dies certe praedictus est translationis, quo teste Eadmero corpus sancti Wilfridi decentiori loco repositus a Lanfranco est. At quo tandem die vir sanctus ad superos migravit, silent Vitae scriptores omnes, unus inter vetustos fastorum sacrorum conditores designat diem VIII Kal. Maii, is nempe qui 133.0981A| nomine Bedae vulgatus est metrici Martyrologii auctor in tomo X Spicilegii, his versibus:
&lt;poem&gt;
      Egbertus, digna virtutum laude coruscus,
      Astrigerum octavis veneranter scandit olympum,
      Quoque die praesul penetravit Wilfridus alma,
      Angelico vectus coetu trans sidera coeli. 
&lt;/poem&gt;
Celebrior tamen videtur fuisse solemnitas translationis IV Idus Octob., ut patet ex titulo quem Eadmerus post Prologum Vitae praeposuit.
===3.===
Monachi vitam ab anno aetatis quarto decimo egit Wilfridus, et regulam sancti Benedicti mirifice propagavit. Quo vero in loco monasticum susceperit habitum, non ita liquet. Et primo quidem annos natus quatuordecim apud Lindisfarnenses monachos diligenter « ea quae monasticae castitatis ac pietatis erant, et discere curabat et agere » inquit Beda in lib. V cap. 19: « Necdum quidem attonsus, verum eis qui tonsura majores sunt virtutibus, humilitatis et obedientiae, non mediocriter insignitus. In quo 133.0981B| monasterio cum aliquot annos Deo serviret animadvertit minime perfectam esse virtutis viam, quae tradebatur a Scotis: proposuitque animo venire Romam, et qui ad sedem apostolicam ritus ecclesiastici sive monasteriales servarentur, videre. » Illic perfectam vitae monasticae, hoc est Benedictinae, adeptus normam, reversus Lugdunum a Dalfino seu Annemundo loci episcopo tonsus est. Postea Alchfridus, Oswii Nortanhumbrorum regis filius. Wilfridum in Britanniam regressum « virum, ait Beda, doctissimum, magistrum habuit eruditionis Christianae. Hujus doctrinam omnibus Scotorum traditionibus jure praeferendam sciebat: unde ei etiam donaverat monasterium quadraginta familiarum in loco qui dicitur in HRipum. » Haec Beda in lib. III cap. 25, ubi Wilfridum abbatem consequenter appellat, ut proinde monastica insignia eo in monasterio suscepisse dicendus sit; paulo post ab Agilberto Gallo, occidentalium Saxonum episcopo, in praefato suo monasterio 133.0981C| presbyter factus rogatu Alchfridi. Scotis in synodo Streaneshalensi confutatis, in monasterium Lindisfarnense, ubi Scoti monachi degebant, monachos nostros induxit. Unde ipse apud Willelmum gloriatur, quod primus in Nortanhumbria « verum Pascha Scotis ejectis docuerit, et sanctissimi Benedicti regulam a monachis observari jusserit. » Complura aedificavit coenobia, « HRipense, Hagustaldense » in suo ipsius episcopatu; alia in Cantuariensi; Seleseiense apud australes Anglos, et alia in regno Merciorum. Huc spectant epitaphii ejus hi versus:
&lt;poem&gt;
      Inque locis istis monachorum examina crebra
      Colligit, ac monitis cavit quae Regula Patrum
&lt;/poem&gt;
Sedulus instituit. Quo in loco regula sancti Benedicti, quam Wilfridus tum in Britannia, tum Romae didicit et excoluit, Regula Patrum appellatur, uti et diserte apud Ambrosium Autpertum in libro De Vita sancti Paldonis abbatis et successorum. 133.0981D|
===4.===
Ad seriem gestorum Wilfridi quod attinet, natus est anno 634 aut insequenti siquidem obiit anno 709, annos natus septuaginta quinque. Adolescens annorum quatuordecim Lindisfarnense monasterium petiit, ibi aliquot annos moraturus. Inde Romam petiit anno circa 653, paulo ante aut certe paulo post mortem Honorii Cantuariensis episcopi, quam eo anno reponit Beda in lib. III cap. 20. Annis fere 133.0982A| quatuor tum in Lugduni mora, tum Romae exactis, reversus in Britanniam HRipense monasterium aedificat et abbatis nomine administrat. Anno prope 660 Colmannum, Scotorum antesignanum, refutat in synodo Streaneshalensi, tametsi Beda in lib. III cap. 26 signat annum 664. Verum Beda non satis sibi constat, dum eo loci scribit Agilbertum episcopum Gallum huic synodo interfuisse, ejusque jussu Wilfridum, adhuc presbyterum Colmanno restitisse; et tamen Wilfridum episcopum ordinatum esse anno ipso 664 Compendii in Gallia, ex cap. 28, ab Agilberto jam Parisiorum episcopo, qui finita ac soluta concione domum, id est in proprium episcopatum, quem apud occidentales Saxones obtinebat, rediisse dicitur cap. 26. Certe in Epitome Saxonica, Bedae Historiae subjecta, Agilbertus « anno 660 abiisse a Kenwalchio, et Parisiorum episcopatum in Galliis ad Sequanam flumen suscepisse » perhibetur. Wilfridus anno 664 ordinatus 133.0982B| episcopus Eboracensis, cum ex Gallia reversus didicisset Ceaddam Ecclesiae suae consecratum episcopum, secessit in HRipense monasterium. Dein a Wlfario Merciorum rege accitus, postea venit in Cantiam, ibique « presbyteros et diaconos ordinavit, usquedum Theodorus adveniret, » ex Bedae lib. III cap. 28, et lib. IV cap. 2, a quo Ceadda Eboraco pulsus, et Wilfridus in sede sua institutus. Anno 670 Oswius rex Nortanhumbrorum cum Romam ire vellet, Wilfridum itineris ducem sibi fieri rogaverat; sed re infecta moritur, apud Bedam in lib. IV cap. 5. « Anno 673 » Wilfridus concilio Heorutfordensi, Theodoro praesidente, « per proprios legatarios adfuisse » membratur ibidem. « Anno 677 orta inter ipsum et Egfridum regem dissensione, pulsus idem antistes a sede sui episcopatus, et duo in locum ejus substituti episcopi, Bosa, qui Deirorum, et Eata, qui Berniciorum provinciam gubernaret, » cap. 12. Interim Wilfridus post biennium Romam pergit causam dicturus. Confecta re provinciam australium 133.0982C| Saxonum convertit ad Christum, ac Seleseiense monasterium aedificavit, favente Edilwacho rege: ac Ceadwala itidem conspirante Vectae insulae populum Christianis institutis imbuit cap. 16. Anno 687, post mortem Cuthberti Lindisfarnensis episcopi, episcopatum Ecclesiae « illius anno uno servabat venerabilis antistes Wilfridus, donec eligeretur qui pro Cuthberto antistes ordinari deberet, » cap. 29. « Post longum exsilium, » annorum videlicet amplius decem, « in episcopatum Hagustaldensis Ecclesiae receptus » est ex lib. V cap. 3. « Anno secundo ( malim duodecimo) Aldfridi, qui post Egfridum regnavit, » alius ab Alchfrido, id est anno 697, « sedem suam et episcopatum, ipso rege invitante, recepit. Sed post quinque annos denuo accusatus ab eodem ipso rege et plurimis episcopis, praesulatu pulsus est, veniensque Romam » anno circa 703, a Joanne papa absolutus repedavit in Galliam. Meldis ad extrema vitae redactus, inducias annorum quinque 133.0982D| a Deo meruit. Postea in concilio Niddensi restitutus principio regni Ofredi Aldfridi filii, id est anno 705, mortuus est anno ejusdem Ofredi regis quarto, « post quadraginta et quinque annos episcopatus. » teste Beda in lib. V cap. 20, et quidem VIII Kal. Maii anni 709 ineuntis. Certe « quindecies ternos episcopus annos egisse » perhibetur in epitaphio suo apud Bedam relato.


INCIPIT PRAEFATIO.

==Praefatio.==
133.0981D|
&lt;poem&gt;
Inscius aegregios aegris conatibus actus
Ordior en, mutilum quo fert fiducia sensum.
Ausibus infidis aderit sed calculus ignis,
Forcipe vatidicum solitus purgare labellum.
133.0982D| Cunctipotens opifex, arcani pandulus index,
Spiritus internis animans dulcoribus antra,
Cordis amara mei, ne quid displodat inepti
Lingua tenax justi, quo non fervente serenas
Efficit haud Phoebus taedas aut Cynthia parcas.
133.0983A| Saecla priora quidem, lubrico phantasmate lusa,
Excoluere suos vano terrore tyrannos.
At postquam summi voluit regnator Olympi,
Virginis intactae talamos invisere gratis;
Dissepsit varios eliso principe lurcos,
Perque crucis lignum legis solvit maledictum.
Astriferas ubi glorificus recucurrit ad arces,
Praesens ipse suis jugiter cultoribus almum
Nubibus e summis direxit splendide Flatum.
Isque quater ternos ignita luce clientes
Imbuit et dignis numerum successibus auxit.
133.0984A| Hujus inexhaustae praedulcia pocula venae
Sensit Wilfridus fama super ardua notus:
Est votum nobis, si suppetat auctor herilis,
Nomen et elatas cujus depromere palmas.
Huc ades, o formose, Dei per climata testis.
Namque, fatebor enim, tua me pellexit honestas,
Scilicet indulcanda modis odisque canoris:
Coelo vita micans, terris oratio praestans.
Ergo age deciduas precibus suppleto loquelas,
Affer praesidium, ne sit sub falce ruendum.

INCIPIT VITA.
&lt;/poem&gt;
==I. Wilfridus in Britannia ortus luce indice.==
&lt;poem&gt;
133.0983B|
Tellus aequoreis circum gyratur ab undis,
Multipetax olim scelerum cultrixque deorum,
Anglicus extruso colit hanc anhleta [ an athleta] Britanno.
Hujus in occiduis convexi partibus orbis
Naturae solitum Wilfridus pertulit ortum.
Anxia dum genitrix parit hunc connixa sub umbris,
Flammivomis lampas radiis praesaga futuri
Coelitus enituit. Vicinia protinus omnis
Praepete concurrit nutanti pectore gressu,
Ignis edax socii subeat ne culmina tecti.
Haerent immoti, nondum miracula Christi
Amplexare cati [ i. e. docti], cum non stupuere cremari.
Est, aiunt, papae! divini numinis omen.
133.0983C| Interea taciti muliebri murmure discunt
Fortunata viri, quoniam nascentis imago
Tali promeruit donari forte sacello.
Hac face laetiferas pepulit puer iste tenebras
Cordibus e multis, gliscens virtutibus almis.
&lt;/poem&gt;
==II. Pie institutus, noverca relicta, palatium petit, dein monasterium sub Cudda.==
&lt;poem&gt;
Jam quia mellito dulcescit nectare guttur,
Prosequar arguta juvenem modulando cicuta.
Hic membris etenim comto glaucomate firmis,
Aequaevos habitu studuit superare venusto,
Nil puerile gerens: facundae munia linguae
Aspirante Deo gestis explebat apertis.
At perfecta virum postquam perfecerat aetas,
Natales statuit functa genitrice penates
133.0983D| Linquere et assiduis Domino famuliarier horis.
Quartum post decimum jamjam transegerat annum,
Immitis furias malens vitare novercae,
Cum sociis cessit sueta comitate decoris
Armatus properans juvenis, ceu nobile germen
Nobilium: sed operta fides umbone latebat.
Inde patris dextra pie commendatus avita,
Aulica perspicuis penetravit pergama plantis,
Insignis populi qua tempestate regebat
Dux praeclara nimis Aeonfled cognomine habenas.
133.0984B| Hujus magnificis actutum vultibus almus
Offertur multoque dehinc lepore beatur,
Acceptus cunctis fidei pollebat in actis.
Ac tum fortuitu memorandae Cudda senectae,
Vir probus ingenio, regum meritoque satelles,
Lubrica coenosi postponens gaudia saecli,
Maluit angustae certamen inire palaestrae,
Normalis sese dedans sub regimine vitae.
Is juvenem sacro suscepit amore docendum.
Exin tiro Dei fultus fervore fidei,
Ut verus Samuhel servabat jura magistri,
Fratribus aequali cunctis subjectus amore.
&lt;/poem&gt;
==III. Romam peregrinatur; ab Ercomberto detinetur quadriennio.==
&lt;poem&gt;
Jam gravis electam purgarat sarcina mentem,
133.0984C| Excoctumque diu solito flagramine pectus
Concipit internas Jesu spirante salebras.
A virtute volens virtutis in alta migrare,
Romuleum statuit concurrere gratis ad altar,
Scilicet ut veteris lustraret busta piacli,
Augmentumque suis quiret conquirere votis
Haec pia praefato narravit sensa magistro.
Archades hinc ambo voto statuere sub uno,
Culmen apostolicum peteret, creperumque piaret.
Et regina quidem faustum conferre levamen
Praetaxata volens, regi mandata propinquo
Dirigit ac Domini placidum committit alumnum.
Erchumbertus erat, plebis qui sceptra tenebat.
Hic sine fraude virum coluit, sine peste refecit.
Tandem triquadro postquam servaverat anno,
133.0984D| Pluribus emisit sociis fretum duce quodam,
Barbaries Biscop Baducing quem inculta vocabat.
Prospera flabra virum litus jecere sub aptum.
Gallica rura petit, Lugduni culmina scandit.
Mox juga corripuit, comitum deliquia sensit,
A duce deseritur, Christi tamen ore locatur.
&lt;/poem&gt;
==IV. Lugduni a Dalwino.==
&lt;poem&gt;
Tibia veriloquo cecinit miracula plectro:
Vis tamen innatae reticet haud mystica musae
133.0985A| Pangere magnificas athletae nobilis odas.
Moenia praefatae praesul memorabilis urbis
Christicolamque gregem septis servabat amoenis,
Francorum Dalwinus adeps: qui sicuti furvas
Pallenti lychino depellit rite tenebras.
Postquam fama volans pulsavit praesulis aures,
Accit continuo, praebens solamina justo,
Excipit hospitio, dapibus fovet oreque mulcet.
Praesensit quippe praecordia casta juventae,
Illustrisque viri faciem perfecte micantem.
Heus, inquit, hospes, mecum sis perpete sospes.
Dum juvat, uberibus pro votis utere glebis,
Farris et excreti communis copia vini.
Si vero consueta placent connubia forte,
Ne differre pares: generosi sanguinis altam
133.0985B| Immensis opibus cognatam trado puellam;
Meque patrem tibimet, temet mihi dedico prolem.
Haec placidus heros exprompserat ore jocundo.
Tiro dehinc agilis voti non immemor infit:
Ardent excoctis fibrarum omenta laternis,
Abdita conceptis dum detrahor, optime, votis.
Tethys cur tumidas Zephyro minitante salivas
Expetii? Frustra me saeva salacia pressit,
Cosmica si spreto repetam contagia Christo.
Sedis apostolicae viso primum cliothetro,
Aeria merear si longule vescier aura,
Te, pater, ex animo soboles devota videbo.
His ita compactis, opibus pro velle paratis,
Praesulis auxilio perrexit calle secundo.
&lt;/poem&gt;
==V. Romam tamen peraccedit, ubi a Bonifacio edoctus, papae donariis honoratus, Lugdunum revertitur.==
&lt;poem&gt;
133.0985C|
Quamvis informi Terpsicore paucula versu
Luserit, ardet amor quo pluscula cara superstes
Expediat, famulumque Dei super aethera tollat.
Namque iteris longos explevit honeste meatus,
Optatamque diu Petri properavit in aulam,
Tripudians pandis callido sub pectore venis.
Praeterea Andreae germana diaulia sancti
Ingressus, dictu mirabile! poplite cosso ( sic ),
Ingentes animos labiis patefecit apertis.
En ego commissis veniam deposco piaclis,
Inquit et excelsi testor per sceptra Tonantis,
Solve meae precibus molimina tarda loquelae,
Si quid habes fidei, pie doctor, apostole Christi.
133.0985D| Haec mox explicuit, non serius edita sumpsit,
Ingenium fandi capiens pariterque legendi.
Anglicus indicio superest adhuc quoque clerus,
Divinas pangens Latiis sermonibus odas.
Ille vaporatis afflatus haustibus erga
Auctorata volans altaria, liba piarum
Fundere jure precum studio satagebat inormi.
Unde sibi praeceptorem lucratus ad unguem,
133.0986A| Bis didicit bini pia dogmata docte fluenti,
Ritus paschales, varium lunaeque recursum,
Ignarasque suis latiatim sane rubricas.
Nomen erat tanto Bonifacius huic pedagogo,
Profuit iste viro nimium, quem glorificandum
Sub sacris papae venerabilis abdidit alis,
Exponens longos animi ferventis agones.
Quique fidem cernens, dextram praetendit, et ipsum
Liberius lata perfudit blande charita ( sic ).
Fultus sic pretiis sanctorum ex artubus almis
Relliquiis, pernix reduces supra memorati
Convertit gressus cara ad habitacula patris.
Commode cuncta meat, Lugduni celsa subintrat.
&lt;/poem&gt;
==VI. Post annos tres tondetur. Dalphino caeso, Britanniam repetit.==
&lt;poem&gt;
133.0986B| Quid narrem lacrymas alterna fronte profusas?
Gaudebant ambo, Wilfridus amore paterno,
Nec minus ipse pater dilecto pignore viso:
Quemque prius animo moerens laxarat amaro,
Permulcet, blandis dans basia casta lacertis.
Nunc singillatim modo miscent laeta vicissim.
Sero quidem rediit, verum tria lustra peregit
Hospite cum prisco, discens examine multo,
Abdita multimodae scrutarier antra sophiae.
Urit amor juvenem, nec tardat grata cupido,
Occuluit quod tecta pio synthemate virtus
Prodere. Nam crines summo de vertice passos
Pontificis dextra primum titulante recidit,
Optans vivificae sustollere stigma coronae.
Archipater gaudet heredem nactus opimum,
133.0986C| Cui queat eudoxas moriens committere caulas.
Arbiter ast rerum praesciverat hunc aliorsum
Vomere salvifico sulcare novalia Christo.
Francorum tum forte scelus vastissima pestis
Augebat, Balthild reginae nomine pistrix,
Olla velut fervens stygia fuligine tosta
Instabat laceris agios laniare creagris.
Non nostrum facinus metro discludere tantum.
Ecclesiae novem violaverat ense patronos [episcopos],
His licet exceptis, quos non capit ars numeralis.
Horum praefatus sine casu criminis unus,
Dalwinus fuerat verbo, re, speque beatus,
Quem fera multatrix lictoribus impietatis
Addixit saevas mucronis pendere poenas.
133.0986D| Hi mox arcessunt crudelia jura furentis,
Multivolae Domini beluae, quam jura tueri.
Ergo petens diri praefixum scamma duelli,
Accepit socium, quem fecerat ante propinquum.
Rennuit ipse tamen, huic ne foret exitus idem.
Sic Pater aeternam sumsit moriendo coronam,
Filius et fortes vixit tolerando labores,
Cum moerore gravi condivit funera caesi.
Protinus ipse parans reddi natalibus oris,
133.0987A| Caeruleum rapidis sulcabat Nerea [mare] remis.
Littora felici patuere salubria cursu.
&lt;/poem&gt;
==VII. Regiam adit. A rege praedia accipit et II Ripense monasterium.==
&lt;poem&gt;
Fasces imperii reges tum forte gemelli,
Alfridus imperitans, una genitor simul Oswiu,
Foedere consorti servabant scitaque vulgi,
Ardentes animis subdi coelestibus orgiis [ i. e. mysteriis].
Nec potuit latebris abdi fax missa nefandis.
Mox proceres inter rumusculus exiit aulae,
Scilicet adventasse virum virtute vigetum,
Optima clavigeris qui fatur dogmata coeli.
Regalis sane subiit proscenia fundi,
Pax domui tunc, infit, huic, captimque loquelas
Tractat evangelicas, quas indice discere Christo
133.0987B| Promeruit, latae penetrans gymnasia Romae,
Quaeque sub argutis congesserat optima Gallis.
Attonitus stupidis princeps sermonibus ultro,
Non puduit glauco telluri sternier ostro,
Ore petens agii confestim se benedici.
Explevit. Cum rege manens, sat multa benigno
Obtinuit merito pecuaria, praedia, census,
Atque monasterium vulgari nomine HRipis.
Qui non aeratas instabat condere gazas,
Dapsilis ast cunctis, dedans hermenia vitae,
Pauperibus largas praebebat sedulus escas,
Exemplo pendens cogitet quid spiritus ardens.
&lt;/poem&gt;
==VIII. Ab Aegilberto Gallo episcopo presbyter ordinatur.==
&lt;poem&gt;
Materiem fandi dabit amodo gratia tanti.
Interea Boreae praesul defertur ab oris.
133.0987C| Aegilbertus erat, doctrinae robore mactus,
Astabat ducibus, suadens quae mandat Jesus.
Is mox Wilfridi vitam meritumque notavit
Praecipuum. Sermone ducis sententia fertur,
Utile personam gradibus donando beari
Divinis, quam non hebetet censura palustris.
Allegat praesul facilis: verum ille juventae
Excursus trepidat, ne forsan mobilis aetas
Praecipitem pravae retrahat per competa sectae.
Tandem colla jugo subdens delecta petito,
Ordinis aptatur caelebs in honore secundi [ i. e. sacerdotii].
Tum magis atque magis fructu meliore gravescens,
Sufficiebat ovans cunctis documenta salutis.
&lt;/poem&gt;
==IX. Schismate exhorto de Paschate, Wilfridus Scotos revincit.==
&lt;poem&gt;
133.0987D|
Orta dehinc pestis geminum discludit in orbem
Unicolam verae fidei sat morbide plebem.
Dumque tremunt bifido corruptae schismate partes,
Eoo coiere simul, gelida sed et Arcto,
133.0988A| Scottica plebs, et Picta cohors, vulgusque Britannum.
Hinc regale decus, ephoeborumque manipli,
Turba senum, praesul violento deditus Euro.
Stant acies dubio certatim fine retentae,
Publica declivae fiunt divortia lunae.
Inde loquax positae Colmanus episcopus umbrae,
Dat praecepta prior subito lapsura fragore.
Nos seriem patriam, non frivola scripta tenemus,
Discipulo Eusebii Polycarpo dante Joannis.
Ille etenim bis septenae sub tempore phoebae,
Sanctum praefixit nobis fore Pascha colendum:
Atque nefas dixit, si quis contraria sentit.
Docte quidem morem narrans depromsit avitum,
Utpote praeteritae recolens choragia zoae [vitae].
Sed non aequatis appendit lancibus aequi
133.0988B| Testamenta Dei, veris aequata chylindris.
Condoluere Patres, caeco quia numine fratres
Nituntur validis tabularia menda tueri,
Obicibus faciles nec cedunt posse refelli.
Heroum jussu clivosi tramitis atqu
Anfractus lubricae prudenter devia sectae
Wilfridus legitur spermologus arte sagaci
Pellere, nota peregrinae cui famina glossae.
Hinc dubios inter cuneos orator adorsus:
Ter centum memini ter senos atque gerontas,
Limpida collectim Nicaeae exempla tulisse.
Quae si perspiciant, mox eluctabitur omnis
Desidiae scabies, torvis illata chelydris.
Metarunt etenim ter seni themata cycli
In sese refluos juncto monade recursus.
133.0988C| Hic tenet: ignorat sed hunc gens indiga veri,
Caecutiens lippis quid differt discere ocellis.
Dixerat, effertur legionum valde favore.
Addit praeterea plano sermone profatus:
Haec immota manent cunctos oracula soles.
Stirps quaecumque volet fixos obvertere fines,
Tartara flammiferis sudibus sub acerba rotetur.
Quo dicto siluit Princeps, canique sequaces
Frigida convulsis vexabant pectora pugnis.
Solvitur in tenerum confestim concio luctum.
Dulcia post modicum rex mitis protulit ora:
Condecet eniti poenis Acherontis adusti,
Exemptas justis mentes miscere suadelis,
Dicite, quaeso, Patres [ f. fidei] fervore calentes,
133.0988D| Numquid olympiaca Petro quis major in aula?
Quem nos et prisci dudum legere tyranni.
An Columchilla quo plebs infausta superbit?
Synodus elata respondit voce beata:
Petrus ab altithrono donatus desmate Jesu,
Difficit ignavae torquentia lora senectae,
133.0989A| Suscipit elisiis justum spiramina zetis [mansionibus],
Utpote coelestes retinens pro munere claves.
Hunc, inquit, plebes, hunc amplexemur et omnes,
Ne non vitalis pandatur regia templi.
Rex ait, et victi cesserunt improbe Picti,
Colmanusque suas inglorius abjicit arces,
Malens Ausonias victus dissolvere leges.
&lt;/poem&gt;
==X. Eboraci designatur episcopus. In Gallia vult ordinari.==
&lt;poem&gt;
Auxit rorigeri per quem micat astrea coeli
Victoris specimen, quem jam transvexerat Alpem.
Nam postquam dicto sedes viduata patrono;
Consulto Regis, necnon hortamine plebis,
Lactea suspensis qui praebeat ubera mandris,
133.0989B| Wilfridum patria creverunt voce legendum,
Lilia suppressis et norit comere lappis.
Territus ipse quidem viridis pro tempore succi,
Mille movens animo balearica tela nefandi
Hostis, et ignotas aegrotis sensibus auras.
Sed ne blasphemae ruat in praerupta catervae,
Aut vitet sceleris madidas lustrare lacunas;
Suscepit humilis quid fors jubet imperialis.
Ergo manet solitus succincto sudor alumno,
Emicuit quamvis genialis flamine vertex.
Convenere dies graduum, quibus ardua moles
Excipienda viro. Tunc fatur regibus ultro:
Inconcussa, duces, vestri sententia, quaeso,
Si fortunatis agitat mea dogmata fatis,
Primum praeteriti fas est meminisse laboris.
133.0989C| Nam superest recidiva lues dum lisque veterna,
Monstris haud patiar maculatis colla beari.
Gallica non paucis pavit me glarea aristis [annis],
Hesperiumque decus laribus servatur in illis.
Illac commissi properans sandalia juris
Accipiam, pontus ni forte repagula nostris
Classibus opponat, valido turbone coactus.
Indulgent proceres, magnates, necne calones.
&lt;/poem&gt;
==XI. Fretum trajicit. Consecratur.==
&lt;poem&gt;
Mox rebus gestis ornatur regia classis,
Coelicolae socias laeta exceptura phalanges.
Navita felicis defixit stigmata clavi.
Multiplices nectens malo trutinante rudentes.
Solvitur, et laxis fastigia spumea verrunt
133.0989D| Ocius antemnis pelagi, nec inepta capessunt
Littora, gratanti subeunt et pectore portum.
Interea placidi patuit legatio nautae.
Pontifices coeunt alacres, ut nota voluntas
In conventiculis adsumitur immaculatis,
Pertonuitque tubis aedes pretiosa beatis.
Stat duodena cohors multis stipata columnis,
Exsultant jubilo miscentes organa sistro,
Mystica veriloquo traduntur et arma ministro.
Occiput effuso gratis pinguescit olivo,
133.0990A| Candiduli primum renitent ornacula lini,
Aurifluae rutilant serrano murice zonae,
Comuntur roseo candentia podere colla,
Gemmata vehitur archontum more curuli,
Induit ormiscum subiit hoc schemate templum.
Postquam signa tulit multo radiata metallo,
Astitit, eucharis [eucharistia] libamina consecrat aris,
Omnibus expletis remeat, navemque subintrat.
&lt;/poem&gt;
==XII. Rediens tempestate jactatur. A praedonibus impetitur. Victor portum attingit.==
&lt;poem&gt;
Sed quia felices tentatio comprobat omnes
Indivisa comes; dum tensis alta pererrant
Marmora Psalmigraphi recolentes orgia velis
Ecce repentinis pontus fundotenus undis
Infremuit, casus agitans artemo minaces
133.0990B| Longe sub ignotis minitat labentia ripis
Agmina dira pati vicinae impendia mortis.
Incumbunt remis, vexatur languida puppis.
Cruribus ipse Pater navalia procubus arma
Confirmat positis, intentans lumina coelis.
Ecce sub infausti contraria littora portus
Gens effera suis casum spectabat in armis,
Praedari fragilem captiva plebe carinam.
Quos Pater intuitus, humili sermone precatus:
Naufragium servatis, ait, sed sponte mearum
Grandia contribuam, resipiscite, munera rerum.
Nos equidem gravis afflictat violentia ponti:
Sed velox aderit, fateor, miseratio Christi.
Stans volucres augur solitus servare sinistras,
133.0990C| Despumat tumidis oracula saeva labellis:
Eumenides furias vocitat sub murmure Parcas,
Manibus infernis ut fortis mandat Erinis [Erinnys]
Expedit admisso plebs cara cruore litari.
Hoc ait, et properans bellum ferale perurget.
Praesul item sanctas tendens ad sidera palmas,
Invocat auxilium divinitus adfore magnum.
Haec inter tereti juvenis scutale lapillo
Ornat, et infandi torquens per inania vatis,
Figens sincipiti trajecit in antra cerebri.
Corruit utque vafer, clamoribus aestuat aether
Cara dehinc recolit pastor miracula David:
Incubat ipse magis, avertitur impia pestis.
Conclamant acies truculento marte ruentes,
Vocibus infestis concursant agmina regis.
133.0990D| Quid plura? quoties crudelia bella lacessunt,
Excipiunt ictus Jesu vigilante cruentos.
Unde quater victi remeant, victorque sereno
Evehitur pelago, frustratus remige quino,
Optatumque tenent repetito limite portum.
&lt;/poem&gt;
==XII. Ceadda Eboraci substituto, in HRipense monasterium secedit. A Volfario rege Merciorum expetitur. Cantiae ministros ordinat, monasteria construit.==
&lt;poem&gt;
Hi quoque quos pridem regalis concio victos
133.0991A| Expulerat, fraude tantum potuere mereri.
Gallica detentus ponto dum rura vagatus,
Temperiem lente speraret adesse jocundam,
Mutavit malefida locum defectio regis.
Constituunt etenim perverso canone Coeddam,
Moribus acclinem, doctrinae robore fortem,
Praesulis eximii servare cubilia: sicque
Audacter vivo sponsam rapuere marito.
Obvia turba Patri luctu quatiente revelat,
Divertitque humilis sub nota mapalia HRipis.
Usque adeon, potuit quisquam nescire moveri?
Non hunc subreptae tanto discrimine sedes,
Non illum sumti multo luctamine fasces,
Aut labor, aut tumidi conterruit ira profundi:
Spe meliore manet latebris contectus in illis.
133.0991B| Saepe licet proprias videtur pascere ovillas,
Nulla datur requies, vicino urgente colono.
Quaeritur a ducibus, quorum Wlfarius unus,
Nobilibus pollens per Mercica regna triumphis;
Vix potuit fato pastorem efferre locello.
Qui postquam meruit tanto doctore potiri,
Affectu coluit, terris et honore ditavit.
Ekbertus vero, quem Cantia rura timebant,
Dum pastore caret, flexo rogitamine poscit,
Ordinet ut sacros decreto more ministros.
His ita digestis, natales prospere fines
Asseculis fultus repetit fastu sine multis.
Plures instituit normali tramite cellas.
Vitam privato servare monebat asylo.
Se quoque continuis ter ducti solibus anni
133.0991C| Accinxit solitis fidei pugnacibus armis:
Nec cessit diri spolians capitolia furis.
&lt;/poem&gt;
==XIV. A Theodoro sedi restituitur.==
&lt;poem&gt;
Emensis igitur praedicti temporis annis,
Mittitur Ausoniis pastorum praesul ab oris
Cultor justitiae Theodorus ac pietatis.
Comperit ut justum collata sede revulsum,
In commune dolet violatae foedera legis.
Mox foribus raptis Coeddam dimovit, et apte
Praefecit sedi longum pastore vacanti,
Wilfridumque suis redimivit protinus aulis.
Sic patiens meruit collato jure pacisci.
&lt;/poem&gt;
==XV. Templum resarcit. H Ripensem basilicam sacrat.==
&lt;poem&gt;
133.0991D| Redditus ergo suis, instabat promtius illis.
Ecclesiae vero fundamina cassa vetustae,
Culmina dissuto violabant trabe palumbes.
Humida contrito stillabant assere tecta:
Livida nudato suggrundia pariete passa
Imbricibus nullis: pluviae quacunque vagantur.
Pendula discissis fluitant laquearia tignis,
133.0992A| Fornice marcebant cataractae dilapidato.
His ita contuitis, exhorruit ilicet alti
Viribus ingenii reparare peribula templi.
Incumbunt fessi vasto sudore latomi,
Nec minus approperant opicizi ( sic ) emblemata proni
Arcus incultos hyalino [vitreo] claudere velo.
Pondus et informes atlantes [columnas] ferre priores
Jussit et expletum: lymphis perfunditur absis,
Albanturque suis lustrata altaria peplis.
Exin glorifico persolvit munia Christo.
Crescit summa viri, crescit dilectio sancti,
Virtutum genitrix almorum florida nutrix.
Non igitur coeptum dissolvit dexia [ i. e. prosperitas] votum,
Picria neve tulit. Seu dissologia [discordia] rupit,
133.0992B| Theosopho spirans animo cognata synergus
Oppida diffuso HRipis amplificare colurno.
Illic planata decusatim vomere terra,
Symmetriis perpendiculo perfecte libratis,
Ecclesiam statuit, talamum Christoque dicavit,
Tandem post paucas ornatis rebus imeras.
Convenere duces, necnon basileia pubes,
Tristatae comites, vulgi promiscua strages,
Ecclesiae proceres aedem pro more beantes,
Ornarunt altare Petri sub honore canori.
Stans in cancellis dat cunctis sperma salutis,
Regibus edixit, repetitaque rura recepit,
Verbo necne cibo triduo turbam satiavit.
Optima quaeque dedit libens exenia miris
133.0992C| Deflorata modis, capsacibus [atque gemellis (gemmulis)
Codex aurato conseptus grammate scriptus,
Auctus ( sic ) evangelicum servans in corpore textum.
Cumque benignivolo persolveret omnia corde,
Inflatur nullo Jesu moderamine typho.
Quapropter merita donatus digne chorea
Floruit, ut quondam Moysi sub tempore lampas,
Pandens aeternae callem patriamque quietis,
Atque domos sibimet superae religionis amicas.
&lt;/poem&gt;
==XVI. Baptizatos confirmat. Puerum mortuum suscitat, eumque baptizat et Deo sacrat.==
&lt;poem&gt;
Quid? loquar an sileam? verbi da, Christe, lucernam.
Exiguum dixi, liceat jam rumpere longas
133.0992D| Inscitia cogente moras, ne lingua pudorem
Praesumtrix tulerit. Neque enim modo sufficit ulla
Virtutum pelagus percurrere rite tuarum,
Quas meruere tui per te complere coloni.
E numero quorum Wilfridus marchio summus,
Nobiliter superae ducens per pascua vitae
Pastor alebat oves: ut propulit eminus hostes,
133.0993A| Monstrabat liquidis salvandos tinguere lymphis
In Patris et Nati necnon et Flaminis almi
Nomine, chrismales adhibens Pater ipse fluores.
Quem populi pulcra medium cinxere corona.
Ecce per offusas denso tamen agmine turmas
Plena metu mulier, defunctae funera prolis,
Sandapilam fugiens, fessis ostentat in ulnis.
Stat turbis immista fides, terrorque gementem
Afflictant tacitis circum praecordia flagris.
Tum Pater accessit, mox pallida conspicit ora.
Obstipuit, haesitque loco temeraria latrix,
Pronior incubuit, luctum singultibus auxit.
O Pater, elatam noli deponere dextram.
Orbatam nato matrem cur despicis uno?
En qui confirmas Christum fore [esse] cunctipotentem,
133.0993B| Imple fidem gestis, miserae da pignus alumnae.
Ingeminat querulas instanti murmure voces.
Concio concurrit gemitu collecta misellae.
Fit plangor grandis. Tandem compassus herilis
Praesul, agit tacitas sancto sub pectore grates,
Perfida quod novit Jesum rogitare caterva.
Moxque solo fixus claros in corde meatus
Tunc reserat, faciem lacrymis et corcula palmis
Afficit, Adonaique pio clamore citavit.
Consurgens autem praesensit adesse salutem.
Timpora cumque manu palpasset frigida; functus
Caesariem movit, vitali redditus aura.
Vulgus inexpletas confusis vocibus odas,
Haec ubi sunt patrata, dedit: puer illico sacris
133.0993C| Baptizatur aquis, miserae dans gaudia matri.
Protinus acceptum secum portavit alendum,
Jussa patri post septenos ut redderet annos.
Ast cogente fero contemsit dicta marito,
Exsul ad externos fugiens cum prole Britannos.
Nec multum latuit. Posthaec non sponte reduxit
Praefectus puerum quidam Patris in dominatum,
Exin divinis illum famulatibus addit.
Pluribus exemplo puer isdem profuit almo.
Tunc fax emicuit, quae inter crepitacula fulsit.
&lt;/poem&gt;
==XVII. Angliae felix status. Egfridus rex Nortanhumbrorum Pictos rebelles perdomat Wilfridi meritis.==
&lt;poem&gt;
Anglum tum gemina fulgebat lampade regnum.
Pontifices regis consulto digna gerebant,
Rex quoque praesulibus latria servibat honora.
133.0993D| Summa quies plebi legalia jura tenenti.
Frugiferis gravida gaudebant horrea glebis,
Epibatae [nautae] faciles ausu sprevere latrones.
Classica dira silent, nec formidatur egestas:
133.0994A| Promicat Ecclesia pacato foedere nexa.
Laetifer invidit, conamina mille reclusit,
Rumperet ut sanctam perverso schismate pacem.
Aggeribus crassis nisa est gens Picta rebellis,
Anglorum solitas jam tunc non ferre catenas.
Egfridus imperii largas retinebat habenas.
Conjunx alma viro Edildrid, famosa virago,
Addita conjugio vixit castissima virgo.
Princeps ergo leves animo flammante cohortes
Admovet, indomitoque reos mucrone trucidat.
Exclusere duos caesis ex hostibus amnes,
Manubiis fratri regis venere manipli,
Dant stragem late, redeunt in colla catenae:
Perfruitur magno rex victor longe triumpho.
Non armis etenim raro cum milite vicit:
133.0994B| Sed virtute Dei, meritis et praesulis almi,
Obtinuit gentes vacuo turgore minaces.
&lt;/poem&gt;
==XVIII. Mercii quoque perdomantur.==
&lt;poem&gt;
Nec minus innumeras, dum Mercica regna tumerent,
Cuspide confossas stravit rex iste catervas.
Quorum dux cessit fugiens, aevoque sequenti
Addidit Arctois borealia sceptra triumphis,
Fultus magnifico Wilfridi plane rogatu.
&lt;/poem&gt;
==XIX. Wilfridus pie vivit.==
&lt;poem&gt;
Virtus ergo viri nullo sudore remitti,
Neve supercilio subrectis nisibus extra
Justitiae verum diduci quibat oestrum [stimulum].
Sane per ignotas instanter mente pererrans,
Divinae gentes sationis tempore passim
Dilatans, exercitium non segne gerebat.
133.0994C| Excubiis pernox gaudens instare secretis,
Corpus perspicuis castum purgat in undis.
Ingeniti fervore laris ne torreat ustum.
Non ardor Phoebi, non algida tempora brumae
Dulcia conceptae rupere volumina curae:
Donec apostolicis jamjam defecta senectus
Praeceptis tanto meruit squalore resolvi.
Nec solus poti scyphum consumere rivi
Fas fore decrevit: jejunia laeta luebat,
Subjectosque viam praecurrens duxit in artam.
Conflua turba ducum soboles conferre sub ipsum
Gaudebat, mirata piam morum probitatem,
Seu servire Deo, seu nosse palatia justo.
Haec quoque falsiloquis praebebant jura linguis,
Ast athleta Dei contectus vellere pacis,
133.0994D| Obsequium mitis odiis suffibat acerbis.
&lt;/poem&gt;
==XX. S. Andreae basilicam condit. Elisum sospitati redintegrat.==
&lt;poem&gt;
Praeterea templum caesis instaurat arenis,
133.0995A| Circumdans caecis sinuato fornice cryptis,
Lenibus obductas pingens flavoribus aulas.
Mox recolit tardae collata juvamina linguae,
Andreaeque pii vocitamen contulit arae.
Invidia mortiferi non defuit aura chelydri.
Trullissant etenim muri dum futilis alta,
Corruit in praeceps colliso corpore frater.
Cruscula gressuros non norunt solvere tractus,
Elisisque tori [colli] marcescunt horride fibris.
Ultima cum timidae traheret suspiria vitae,
Concurrunt tristes ad haec funebria plebes.
Praescia mens Patris statim cognovit in antris.
Extimplo lacrymis puro de pectore fusis,
Hortatur reliquos Dominum obsecrare dolentes,
Ut reparet laesam pastor supremus ovillam.
133.0995B| Adfuit, et properam sumserunt membra salutem,
Erubuit laetum vita mediante repulsum.
Quapropter magni laudarunt munera Christi.
&lt;/poem&gt;
==XXI. Ermemburgae reginae impulsu Egifridus rex. Theodori consensu, Wilfridis dioecesim in tres partitur. Romam appellat Wilfridus. Insectantium poenas praedicit.==
&lt;poem&gt;
Daemonis ira fremens tantam vult subdole gnosiu
Fallere, delecto quo primum milite caeso
Invalidi fragiles perterreat agminis hastas.
Plurima maturans antiqua recurrit ad arma,
Corrupitque levem regis morbose jugalem.
Haec virosa malae jaciens tum tela pharetrae,
Corcula strophoso violavit vulnere regis,
Rancidulis infesta modis, ceu garrula perdix,
133.0995C| Culpabat justum collatis rebus abuti.
Gloria namque viro totis a partibus orbis,
Agricolae plures, fecundae gramina terrae,
Procurvi proceres, juvenum fetura locuples,
Insuper aethereae monachorum concio vitae.
Postquam pestiferum penetravit corda venenum,
Adscivere suis deceptum prone loquelis
Veri doctorem, justi quoque pene sequacem
Theodorum, supra cecinit quem rustica musa.
Annuit ipse quidem regalibus improbe coeptis,
Substituens ternos sub nomine praesulis intra
Contiguos Patri fines illo usque monarchas (episcopos).
Audiit Eutyches quid rex sibi contrahat, imas
Percontare volens naevo sine criminis iras,
133.0995D| Alta petit lento triclinia opipare gressu.
133.0996A| Ecce faces tanti stimulatoresque pericli
Obriguere metu, commentaturque maligna.
Cur laedor, Pater, inquit, ab his, quos nulla fatigat
Nostrarum, fateor, impune molestia legum?
Nos, aiunt, facinus quamvis cogente senatu
Ponimus, haud quodquam nefas decreta refelli.
Sic testes scelerum firi [fieri] potuere suorum.
Antistes vero nollens desistere coepto,
Quidquid ei pravum pseudo-theosebia nectit,
Romuleo sanxit voto fore discutiendum.
Intuitus quosdam foedo insultare cachinno:
O utinam laetos pueri feliciter annos
Ducatis! sed gaudia sors infesta resolvet.
Nec prius in refluos venient haec tempora soles,
Immatura nimis quam tristia fata luatis.
133.0996B| Haec ait, et sobrio compressit verba palato.
Alfwine post jugulum populi luxere profatum,
Inferias caeso dantes moerendo Clitoni
Cujus post tumulum fratris victoria fluxit.
&lt;/poem&gt;
==XXII. Mare transmisso ad Fresones deportatus verbum Dei praedicat.==
&lt;poem&gt;
His ita compertis rediit, laetoque negatas
Corde revisit oves, arentia pectora blandis
Paulatim reparat solamina dando loquelis.
Interea pernix obliquo littore celox [scapha]
Solvitur, aequoreos pertentatura meatus.
Quam circum monachi, dextra laevaque relicti,
Fausta patri pariter lacrymosis questibus orant.
Tunc inimica seges igni tradenda perenni,
Non contenta suae virus fudisse kakiae [malitiae,]
133.0996C| Legatos regi Francorum dirigit, atque
Oeconomum Domini praedati impune petivit.
Sed superi pietate ducis conamina falsa
Macario nullas quibant inferre ruinas.
Laetiferas vero scrobibus versute latentes
Decipulas sensit Winfridus praesul, eisdem
Perditus insidiis, tantum monogrammate lusus.
Praeterea flabris aliorsum prospere versis,
Freisonum laetis portum percepit in oris.
Gentiles stupuere nimis, sed summa mekotes ( sic )
Regem cum ducibus placatos reddidit omnes.
Categorans [praedicans] ergo divinum sperma popello,
Aperuit dulces in falsis rupibus amnes.
Tum pecudum fetura habilis, tum semen in arvis
133.0996D| Crevit, et exsultant sollemniter opiliones.
133.0997A| Post modicum multis tribuit baptisma salutis,
Et fundavit eas Wilbrordus quas habet arces,
Aequiparans tantum fidei fervore magistrum.
&lt;/poem&gt;
==XXIII. Frisonum regem in Wilfridum accendere nititur Ebroinus.==
&lt;poem&gt;
Nec tutela ducem, quem multae copia gazae
Francorum rupit, furiis accendit acerbis.
Syngraphidem sculpsit, munus venale notavit,
Quiret Adalgisum collatis subdole regem
Implanare opibus Efruinus [Ebroinus] philocompos
Legati properant, paraphrastica jussa revelant.
Rex vero phronymus scindens ferale volumen,
Aspernatus eos, flammis projecit in altis.
Imprecor aeterno pereat sic caumate, quisquis
Compactae quaerit fidei dissolvere nodos,
133.0997B| Infit: et infestos jussit remeare sodales.
Haud dubium quin lustratus fulgore superno,
Ad cunas agii quondam qui luxit ab alto.
&lt;/poem&gt;
==XXIV. Apostrophe ad reginam.==
&lt;poem&gt;
Imprudens mulier, justum furialibus armis.
Cur stimulas? sat erat patria expunxisse beatum.
Quid mare sollicitas? quid casses abde charybdi?
Num tibi quadripatens obedit machina ruris?
Tene jubente poli rutilabit dextimus axis?
Siste, capesse fidem, dynami capieris eadem,
Ocius amisso mutaberis ipsa marito,
Grassatrix scelerum fies inimica luporum,
Christi candidulos pasces modo mitior agnos.
Quem nunc persequeris, posthac devota sequeris,
133.0997C| Multa dehinc populis adhibens monimenta salutis.
Ista per ecstasin scalpello lusimus unco.
Coepta libet solito nunc insanire cothurno.
&lt;/poem&gt;
==XXV. Wilfridus ad Dagobertum se recipit.==
&lt;poem&gt;
Ver gelidos tepidis jam solverat aestibus Alpes,
Viribus adsumtis cum praesul laetus ab illis
Gallica rura petens exivit finibus, atque
Nota subit fidens Dagoberti moenia regis.
Is equidem merito, venturus ut adseret ordo,
Publica suscepto praebebat laeta patrono.
Perditus, ut nostis, inquit, popularibus alis,
Excepi moestas tenero olim somate erumnas.
Adscendi tremulum non multo remige lembum,
Pulsus ad ignavos adulescens exsul Hibernos.
133.0997D| En vobis, cujus merui tutamine reddi;
Per me, perque tuam, miles delecte, salutem,
Si qua fides animo, tantum servemus amicum.
133.0998A| Exsultant proceres: neque enim leve principis extra
Praecipuum fas est subjectos velle meare.
Obtulit egregias immenso oromate sedes,
Ire catillatum nec vult rex ultra beatum
Friguttire, petit vetitas alacrior arces.
Magnanimus vero votis regalibus heros.
Non cessit: verum rebus sociisque levatus,
Quin etiam sumto Deodato praesule pergit.
Rursus ab objecto properans sibi rege paratum
Vadit ad hospitium. Bertherus namque patentes
Imperitans terras multo terrore regebat.
Ipse pio solans justum suscepit amore.
Post epulas et post grati carchesia bacchi,
Rex coepit narrare Patri, qui livida civis
Viderit orsa mali sese violare volentis.
133.0998B| Ast ego praeteriti memor, inquit, sancte, pericli,
Ut quondam tellure mea privatus, amaras
Externo sub rege degens sensi cicatrices.
Foedere sed multo quamvis servatus ab humo,
Evasi saevos discedens liber amicos.
Me quoque muneribus voluerunt fallere latis,
Quo te deciperent glauci lacerando Britanni.
Quod si me fidei gentilis amore tenacis
Vendere despexit, te nec minus ipse fovebo.
Claxendix tonuit, comites rex promtulus auxit.
Sic Romana petunt pernicibus atria plantis.
&lt;/poem&gt;
==XXVI. Ipso praesente Romae synodus convocatur, in qua restituitur Wilfridus.==
&lt;poem&gt;
Cursores sed jam praemiserat archipolites
Theodorus. Tum Wilfridi pia fama beati
133.0998C| Ausonias notis penetravit nutibus aulas.
Euge ciunt cives certatim protinus omnes.
Exterit abiegnas Petri dignissime valvas.
Agatho apostolicas decorabat praesul habenas,
Justitia simplex, aequi non marcidus auspex.
Edictum posuit, patrium syntagma gregavit,
Atque quater duodena Patrum, nectente diade
Agmina conduxit, communi famine dixit:
Cultus et Ecclesiae fucatur sanctio priscae,
Ni coquat emersas calor ecclesiasticus herbas.
Anglica comperimus vexari pessime regna,
Dum praeter fixos fiunt discidia fines.
Arguto fratres fratrum discrimine clades.
Pendite ne rudibus pereant antiqua bilibris.
133.0998D| Primores Itali trutina pietatis adulti,
Talia doctiloquo retulerunt fata magistro:
Incipe, pastor, habes veterum decreta virorum.
133.0999A| Nunc illos admitte, placet, quos multa fatigant
Instigante fide spatiosae incommoda terrae.
Decernes sermone pio, quid subditus ordo,
Et quid quaeque colant, seu quid vitare jubentur
Ecclesiae, scelus hoc pacto vacuabitur omne.
Allusit: statimque sacras Wilfridus in aedes
Cogitur ipse suas jussus narrare querelas.
Obtulit expressum Patribus diploma legendum,
Ut patriae curas ovium ceu lucra piarum
Sedulus obsequio tractaverit, utque dicatas
Invidia cogente fores suspenderit, atque
Eheu pontificum fraudes qui more procorum
Ornatam proprio sponsam invasere patrono.
Non ego criminibus quemquam depravo peractis.
Praestolor Hesperia qui judicer, inquit, ab aula,
133.0999B| Omnia versifico longum exanthlare relatu.
Alternae perhibent sincera oracula partes
Expromisisse virum: prorsus hunc esse legendum,
Cujus apostolico solidandae robore vires,
Hoc virus ne serpat, et incautos malesano
Inficiat succo, dum Justum pessimat atrox.
Annuit, et scriptis legalibus archierarchus [papa]
Theodoro regique jubet sancita notari.
Ni pietate pari conservent jussa magistri,
Judicio Domini cunctos anathemate plecti,
Quaerentes animam sine re damnare beati.
&lt;/poem&gt;
==XXVII. Dagoberto bello impetito et caeso Wilfridus ob ipsum servatum convicio appetitur.==
&lt;poem&gt;
His ita fundatis, ne sicut apostata vilis,
Aut legum profugus sola quaerat perdere nota;
133.0999C| Limitibus planis excessit finibus illis.
Uritur infestus humanae prolis amicus,
Commaculare bonum perquirens livide votum.
Nam properans patuli dum verreret aequora campi,
Conglomerata suis furunt collegia signis
Noxia, regali, ceu fertur, sanguine fuso.
Civiles etenim Dagobertus Marchio dextras
Senserat, et rigidum capulotenus inguine ferrum.
Praelia vibratis intentant comminus hastis.
Arduus excusso quorum convicia telo
Unus pontifici foedo sic ingerit ore:
Delator patriae petulans, moribunde viator,
Gallica tu pridem violasti sceptra, relato,
Qui periit caesus, quem jure sequere, tyranno.
An pravum gessi, regem dum forte remisi?
133.0999D| Inquit, et opto pati gaudens pro nomine Christi.
Mox truculenta ruunt hastilia: deinde remissis
Acclines rogitant ancilibus ut benedicat.
Crediderim rutilo perculsi fulminis igne,
Quod patuit mundo, dum matris fluxit ab alvo.
&lt;/poem&gt;
==XXVIII. In Angliam reversus litteras exhibet pontificias.==
&lt;poem&gt;
Nil metuens ergo pertransit gramina tuto,
Securusque cavas intrat velociter alnos [naves].
133.1000A| Multivido fratrum post haec oramine fultus,
Auctorata ferens Petri vexilla beati,
Regales adiit, dans rectis gaudia, postes.
Bullatas offert signato cortice chartas.
Confestim doctae pariter coiere catervae,
Quid narret Romanus apex, audire paratae.
Archistrateios solio prospectat ab alto,
Jus crudele sacris ausus dictare loquelis.
&lt;/poem&gt;
==XXIX. Quibus spretis, ipse in carcerem conjicitur.==
&lt;poem&gt;
Post evangelicum catorthoma, post recitatas
Sedis apostolicae causas, indigne ferebant
Oppositi quidam raptam persolvere praedam.
Urgebantur acri furioso pectore bile:
Permulcentque, nefas: perverso fasmate [famine] regem
133.1000B| Spernere syngraphas furtivis rebus adeptas:
Symmystenque Dei gaudent sychophanta notari.
Tandem semoto Patrem trusere locello,
Blanda retractantem regis promissa superni,
Olim qui dixit, qui vos hic, me quoque spernit.
Tum regina gravis hujusce schismatis auctrix,
Praesumsit pretias [pretiosas] sancto divellere capsas,
Non impune sibi tantam sociata rapinam.
&lt;/poem&gt;
==XXX. Oratio ipsius ad suos.==
&lt;poem&gt;
Ergo opibus sociis pessumdatus inclytus heros,
Famina mirifico fratribus depromsit ab ore.
Gloria multa, loquens, multo captanda labore.
Cana Patrum series, longos experta dolores,
133.1000C| Vincere multiplices meruit divinitus hostes.
Heroum duodenus apex patiendo beatas
Sidereumque decus Jesu tribuente coronas
Excepit, multum modico lucratur ab ipso.
Nostra licet duram patiantur corpora sortem,
Mentes triticeo coelis indentur acervo,
Quo fruges animae capientur mox locupletes.
Praemia non quaerat, ni qui tolerando triumphat.
Dixit, et actutum lictorum torva vorago
Obstitit, et raptum tenebroso vinxit in antro.
&lt;/poem&gt;
==XXXI. In carcere illustratur luce superna.==
&lt;poem&gt;
Qualem te, pastor, laetum tristemve fatebor?
Laetabare quidem te Christi post fore risten ( sic );
Flebas damna gregis pastorum verbere laesi.
Actu coenoso dum stringeris undique vallo,
133.1000D| Num solito non quivisti fulgore potiri?
Exacuere leves inimici frustra ligones,
Nequiquam ancipites inspicant acrius enses.
Tempestiva tibi funduntur lumina coelis,
Inviso properat consueta porismate [meatu] lampas,
Pervigil excubias Jesu dum solvis amatas.
Testis adest custos, ergastula saeva serenis
Inspiciens radiare focis in tempore noctis.
133.1001A| Quid minus a Petro, cui lumen fulserat arto?
Digne quidem parili posthac donabere serto,
Quem voto sequeris, quem tormentis imitaris.
Et ne desperes, aequo splendore nitescis.
&lt;/poem&gt;
==XXXII. Pontificias litteras violari non patitur.==
&lt;poem&gt;
Interea multis vexatur fraudibus almos
Exsecrare apices, actusque probare feroces.
Vertex si capulo, dixit, caedatur acerbo,
Aut terebrent costas instantia pila tenellas;
Unum iota meo nunquam frustrabitur astu,
Quae sanxere Patres Petri documenta sequentes.
Sic legit fuscos primatum offendere vultus
Non et apostolicos consensit fallere ritus.
&lt;/poem&gt;
==XXXIII. Praesidis uxorem pene exanimem sanat.==
&lt;poem&gt;
133.1001B| Carpitur inde gravi conjunx dilecta veterno
Praesidis, infaustas ha ha! procurantis habenas.
Corpore contrito, compagibus atque solutis,
Mortua contractis fluitabant viscera nervis.
Sensu pene carens, extis jam condolomatis
Grande viro scrupulum congesserat, extima vitae
Spectanti, dum praeteritae spiramina reddat.
Protrahit illa moras: tandem velociter artas
Vir supplex satagit sancto reserare catenas.
Fletibus haud modicis deposcit opem medicantis.
Ergo Pater, solito pietatis munere fisus,
Passibus approperat conferre juvamen anhelis.
Adfuit, et turbas removeri jussit adactas.
Hinc precibus fusis, perfudit viscera lymphis:
Mox rediit priscos vitalis fervor in artus.
133.1001C| Nil nocuere virum crudelia lora beatum:
Quin alacri vultu lucum remeavit in ipsum,
Nec puduit tetras iterum curare cavernas.
Praefecto cognomen erat quem diximus Osferd ( al. Osfred,)
Conjugis ereptae vocitamen dicitur Ebbae,
Post modicum sacro gaudebat condita velo.
&lt;/poem&gt;
==XXXIV. Artius religatur, sed frustra. Solvitur.==
&lt;poem&gt;
At Praeses timidus, rerumque stupore coactus,
Malle pati propriae dixit dispendia vitae
Intimat, et saevo quam justum prodere regi
Isque tumens nimium praecipit acerbius illum
Compedibus rigidi damnatum carcere ferri
Stringere, mutata custodis sorte prioris.
Urgentur chalybes furcos aptare bicornes,
133.1001D| Et sterili funes nexare labore trilices,
Quis terrere queant athletae corda ligati.
Sed quo suppliciis animo mordacius instant,
Prorsus eo minimum se profecisse stupebant.
Nam quaecunque sacris nectebant vincula membris;
Aut diffusa cadunt, aut forte incongrua fiunt:
Artubus innocuis libertas cessit herilis.
O crudele genus! frustra contunditur incus.
O scelerata manus! quid prodest laedere corpus?
Non metuit jugulum, quem contremit indere lorum.
Digne satis gressus tremulae fugere catenae,
133.1002A| Pacifice qui currebant per competa vitae.
Ergo alacres pastor curans mitissimus agnos,
Lingua perdocuit, manibus baptismate tinxit.
&lt;/poem&gt;
==XXXV. Regina ob sublatum reliquiarium punitur. Wilfrido liberato convalescit.==
&lt;poem&gt;
Oppida commissi verum dum marchio regni
Permeat, illecebras explens pompose caducas,
Ecce repentino turbantur gaudia casu.
Dum raptis etenim conjunx regalis abuti
Audet relliquiis, spirans typho impietatis;
Vindex ira subit, satan per viscera gannit:
Quem coluit petulans, hujus sit hospita bacchans.
Talibus attonitis accessit regia nutrix
Anne patent, inquit, veteris commissa piacli?
Extera Wilfridum violant ergastula justum:
133.1002B| At tua terrifico quatiuntur daemone membra.
Sprevisti emeritas Petri de culmine chartas:
Excipis en dignas irato judice poenas.
Capsaces [phylacteria] collo demsisti immunda verendo:
Nunc moribunda Baal vinciris et ipsa veruto.
Naribus immanes ceu fundens torridus ignes,
Rex aderat, quem sic genitrix affatur honesta:
Jam memor esto tui, loetargum subrue, fili,
Ultio dilectam profligat summa jugalem.
Sed tu sceptrigeram si vis augescere dextram,
Vincula solve viri, quoquam sine crimine nexi:
Et si fers animo, jubeas discedere regno.
Annuit invitus, patuit mox carcer apertus,
Reddidit et claram nubes densissima stellam.
133.1002C| Protinus immeritam sensit regina medelam.
&lt;/poem&gt;
==XXXVI. Wilfridus liber deserit patriam. Palabundus adversariorum factione.==
&lt;poem&gt;
Excedens igitur patriam cognataque rura
Deserit, australesque petit velut advena terras.
Nobilis expulsum per devia quaeque viator
Obvius excepit, mundi quia summa potestas
Indefinito praefixerat omnia verbo,
Defore nil justis spondens pro se tribulatis.
Regia progenies Berthwaldus munere dives
Nec latuit: statimque furor hostilis inarsit.
Sed Pater a solito nunquam sudore fatiscens,
Instruxit gregibus monastica jura fugatis.
Jam fati tellure viri tunc invida pestis
Corda ducis zabuli praestrinxit hospitis apti,
133.1002D| Fratris Edilredi per Mercica regna decori,
Cujus in obsequio conjunx inimica manebat,
Ekfridi germana trucis tumidique tyranni.
Hi pulsare virum multo terrore serenum
Haud cessant, coguntque datam molimine pravo
Degenerare fidem. Sic compulit ire carentem
Hospitis auxilio, comitatum praeduce Christo.
Profecto alterius susceptus regis in aulis,
Femineas patitur instinctu daemonis iras.
Ermenburgis enim totius criminis altrix,
Cladibus, ut cecinit praemisso musa tenore,
133.1003A| Multimodis torsit tantum furibunda patronum:
Saeva soror cujus Centwine regia conjunx
Expulit indemnem nullum servata [servando] pudorem.
&lt;/poem&gt;
==XXXVII. Ad Australes Saxones se recipit. Regem Edilwalk cum suis Christo lucratur. Coenobia construit.==
&lt;poem&gt;
Quid strides, peritura seges, aut sulfure averni
Perpete torreris? en aspernata probabis.
Ipse sibi patriam praesul non longe jacentem
Adquiret, nondum divino vomere scissam.
Non licuit cultis committere semina sulcis.
Anne vacat saltem steriles invertere glebas?
Non timet exsilium qui gestat pectore Christum.
Gens igitur quaedam, scopulosis indita terris,
133.1003B| Saltibus incultis, et densis consita dumis,
Non facilem propriis aditum praebebat in arvis:
Gens ignara Dei, simulacris dedita vanis.
Hanc adit intrepidus, sincero dogmate fretus,
Adsistensque duci, pertractat munia Christi.
Foedere complacito statim captatur ab ipso,
Non auri pretio, ferro non hostis avaro:
Juravit pactos nunquam dissolvere nodos.
Hinc passim sacris permulcet corcula verbis,
Lucratusque ducem feliciter, atque jugalem,
Emicuit mediis fax indefessa tenebris.
Delubra contritis pereunt velocius aris.
Nec doluit madidas subvertens augur acerras.
Dulcia divino funduntur mella canistro,
Tinguntur plebes sanctum baptisma petentes.
133.1003C| Rex quoque vivifica perfusus nobilis unda,
Praedia delegat, coenobia praesul adornat.
Edilwalch princeps vocitatus dicitur iste,
Wilfridi meritis qui novit coelica tempe.
&lt;/poem&gt;
==XXXVIII. A Cedwala in regnum restituto excipitur in Westsaxoniam.==
&lt;poem&gt;
Teque canam, fugitive puer Cedvalla, negatis
Quem fudit sceptris nemoroso Ciltena [Westsaxonia] luco.
Audacem quippe Wilfrido subdere mentem,
Ut quires redhibere tui diademata regni,
Magnorum quaerens evadere fata malorum;
Crevisti patiens, rex fortis, ab exsule pasci.
Tempus post modicum fidei capis incrementum.
Nam vertex tremula quem dudum foverat ulva,
133.1003D| Praesulis auxilio rutilat cingente corona.
Tu quoque pontificem multa tellure ditasti.
Felix exsilium, quod poscit purpura regum!
Immo Pater felix, exsul pro foedere pacis!
&lt;/poem&gt;
==XXXIX. Egfrido mortuo, redit in gratiam cum Theodoro.==
&lt;poem&gt;
Rumor tunc prodit, quia bello plura cruento
Agmina corruerant alterna caede vorata:
Ekfridusque feri percussus cuspide Picti,
Exsilio sanctos desivit trudere pravo.
Praeterea pastor Dorobernica claustra gubernans
Theodorus, damnare nefas in praesule quondam
133.1004A| Commissum cupiens, jamjam vergente senecta
Expetiit revocare virum sat inepte repulsum.
Ergo redit humilis, implens exacta prioris.
Hos inter praesul Erchenwaldus venerandus.
Accedunt terni verae funabula lucis,
Exorsusque dehinc senior Theodorus infit:
Poenitet en, Fili, quia tot tormenta subisti.
Ultima porro meae properant confinia vitae:
Tu quocunque modo deliqui, quaeso, remitte.
Terruerant animum Wilfridi gesta beatum.
Tum coelis oculos attollens, brachia terris,
Indulsit facinus, sicut praecepit Iesus.
Ergo solo strati, poscunt miseramina Christi.
Inque vicem culpis perfecto corde remissis,
Gaudebant lacrymis sobria de fronte profusis.
&lt;/poem&gt;
==XL. Aldfrido et aliis commendatur.==
&lt;poem&gt;
133.1004B|
Jam fasces regni, jam publica sceptra peremti
Sumserat Aldfridus. Mox pastor scripsit eous
Sedibus abreptis reddi debere beatum
Christi mathiten post tanta pericula sennem:
Ne contingat huic moriens quod pertulit Ekfrid,
Qui voluit dictum primus spoliare patronum.
Hoc et idem cunctis statim designat amicis
Aelflede, quae virgineo succincta pudore
Pascebat nitidas Jesu prudenter ovillas.
Tuque clientelae, princeps, memor esto cupitae.
Scripsit Edilredo cognato morte sopito,
Qui studuit toto Wilfridum pellere mundo.
Nunc age, communi si vivas optime voto,
133.1004C| Saepe meis, fateor, parebas phronyme [prudenter] dictis.
Jam quoque Pastoris extremos crede rogatus,
Ut tua praecluibus gliscens cuprepia [excellentia] sceptris,
Ardua Romulei servet praecepta vigoris,
Quae statuere Patres Petri vectigal alentes,
Agatho praecipuus, Benedictus, Sergius atque.
Dux etiam monitis arrisit protinus istis,
Summo ceu decuit servans in honore magistrum,
Usque sub extremam vitae frustrabilis horam.
&lt;/poem&gt;
==XLI. Ab Aldfrido revocatur.==
&lt;poem&gt;
His ita consertis, Aldfridus stemmata regni
Non obscura gerens, Christi revocavit alumnum.
Certatim priscas possedit deinde casellas.
133.1004D| Pelluntur moechi, sedes vacuata patrono
Redditur: applaudunt cives felicia fantes.
Lucidus Ewroica [Eboraca] resplenduit axis in aula.
&lt;/poem&gt;
==XLII. Aldfrido iterum vexante, rursus confugit ad Edilredum.==
&lt;poem&gt;
Felices populi tanto pastore potiti!
Vix tamen exiguo manserunt gaudia lustro.
Nam modicum trita viguit perversius hydra.
Ardescunt veteres crebris pulsatibus ignes,
Ecclesiae puppis vexatur dire procellis,
Cogitur et rursum nauclerus mittere clavum,
Anchora ni firma primum solidetur arena.
133.1005A| Rex etenim dictus, ligurum [ligurionum] ferrugine tinctus,
Distendit variis veracia stamina pullis,
Alternando vices; nunc Patris subditus alis,
Nunc mendosorum credens inventa virorum.
Ni fallor, fida titubans statione carebat.
Ast pater immotam quaerens servare carinam,
Maluit elatum regis differe tumorem.
Idcirco cessit, divisaque rura reliquit,
Impatiens sanctam scindi se praesule normam,
Arvaque commissis monachis collata revelli,
Nec metuens magni praeferre oracula Petri.
Mox Aedilredi repetens solium memorati,
Acceptus colitur placida sub rege quiete,
Explens officium multa pietate supernum.
&lt;/poem&gt;
==XLIII. In concilium vocatus, paratas technas eludit.==
&lt;poem&gt;
133.1005B| Tantos post strepitus, quem non quivere molesta
Exsuperare fuga, nituntur fallere techna.
Denique collectim censura urgente ducali [regia]
Synodus astruitur, legatio menda jubetur
Caelibis accessum festino cogere gressu.
Concurrit facilis. Fratum tum saeva simultas
Undique stipatum fictis agitare querelis.
Machina non solido, sed mox interlita limo
Corruit, haud patiens vatem prognostica fantem
Lincea namque viro fuerant oculamino tanto:
Immo prophetales ad hoc potaverat haustus,
Internos etiam quo posset cernere nisus.
Tandem vafrinos blandis suggestibus astus
Increpitans, Italas suadebat credere chartas.
Ethnica sed nullis molliri corda lagonis [ sic ]
133.1005C| Cedebant, virusque tegunt sub mente nocivum.
&lt;/poem&gt;
==XLIV. Non vult sponte pontificatum abdicare. Romam iterum appellat.==
&lt;poem&gt;
Praeterea juvenis divini fautor honesti,
Expertus causas in sanctum aptare sinistras,
Accessit, regis dirum patefecit amentum,
Toxica letiferae volitant quo tela pharetrae.
Nec etenim puduit dicacem audire fidelem,
Qui mox exposuit, celer intra palatia repsit.
Summi stans igitur ducis inter castra perennis
Excipit horrisonas illaeso parmate fundas,
Nec potuit validis penetrari pelta sagittis.
Instabant etenim violento famine agentes,
Se Pater ut proprio vellet dispungere voto,
133.1005D| Indignumque sacris se judicet esse gerendis.
O malesuada cohors! o fraus! o sanctio discors!
Non eris immunis poenae, non criminis exsors.
Ipse quidem solitos gaudet perferre labores:
Tu Phlegeton subies, Erebum manesque pavesces.
Insignita piis tunc aperit ora loquelis.
Ecce quaterdenos, dicens, vexilla per annos
Intemerata tuli verae per cuncta Fidei,
Barrida [procacia] pravicolae quis stravi examina turbae,
Inclyta censurae servans invictus eoae
Famina, non falso ceu garrit pollice sculpta,
Mens inimica suo mox defectura veneno.
Nunc quia compellor, Petri discrimine purgor.
133.1006A| Non formido minas, exenia nulla capesso:
Meque meo nunquam vobis mucrone litabo.
Haec Pater oravit, plebeium jusque refellit.
&lt;/poem&gt;
==XLV. Ad Edilredum redit.==
&lt;poem&gt;
Solvitur infandum Jesu vincente coagmen,
Enituitque bonum praestantis praesulis omen.
Sic Aedilredi valvas repetivit hiantes,
Explanatque duci sceleris commenta maligni.
Isque benignivolam firmans in praesule curam,
Edixit priscum nullo frustramine foedus
Vellere, apostolica donec remearet ab aula.
&lt;/poem&gt;
==XLVI. Romam revisit.==
&lt;poem&gt;
Verum discordes flagrant ardentius irae.
Et quas Christicolis pastor fundaverat aedes,
Nisibus efferis averruncare parabant.
133.1006B| Extorres stavunt [ an statuunt] agio incassum palinodo,
Quos sacer electis gradibus formarat adelphos,
Dulsos coenosi lychinos audititi destos.
At pia ventilabro Christi sententia justo
Efflavit paleas, cocyti ardore cremandas,
Condivitque bonam coeli per horrea frugem.
Nam pater ipse greges et ovilia cara revisens
Ingemit, et precibus Domino commendat anhelis.
Aggressusque dehinc Romani pervia callis,
Ingreditur partas opulento remige lintres.
Carpebant placida libratis aequora velis,
Figitur et notis vehemens lautomia arenis.
Oblitas igitur peragrantes commode terras,
Divino comites potabat fonte fideles.
133.1006C| Non errare licet, non manticulare necesse est,
Gratia quos tanti servat ductoris opima.
Is etenim largas, quamvis devotus egeret,
Impensas eleis [liberalibus] palmis fundebat apertis.
Quapropter, ceu praecinuit Davitica psaltes,
Justa manebit ei vitae farrago perennis.
Transmissis tandem Jovialibus aeque cavernis,
Macarii penetrant rutilantia dindyma Petri,
Qua renitebat ovans summa virtute Joannes.
Affines moti festinant Euge per urbem.
Astipulant modulis comto sistemate, cerus [sacer]
Aedituusque graves oppandit agillimus aedes.
Introiit praesul testudine fultus opaca,
Qui tellure cubans unca suffragine suda [flexo poplite]
Mille piis instillat aquis, vi denique sancta
133.1006D| Coelica summorum reseravit claustra polorum.
Indidit aequisonas divinis auribus odas.
Oratu surgens, famulantum rite catervis
Ambitur, siccatque genas, suspiria pressat.
Ast ubi complacuit, sacra archisteria [templum] scandit.
Pontificum praesul, necnon et concio gaudet
Pistica [fidelis], Tarpeioque ruens de colle senectus.
Intestina Pater postquam dyrrachia [discidia] planxit,
Ad vicina redit lentis habitacula plantis.
&lt;/poem&gt;
==XLVII. Causam agit apud Papam, prolatis litteris apostolicis.==
&lt;poem&gt;
Interea scriptis legatio cassa modernis.
133.1007A| Berthwaldi de sorte ruit, puerile volumen
Opponens tanto nequiquam certe Patrono.
Ausoniae coeunt Graiae, mox atque soritae
Vestibulo cursim priscorum scita revolvunt.
Protinus admisso micuit syntagma berillo.
Ut jubar introiit, jubaris ceu nectar opimi
Protulit effevum ( sic ) liquido flavore libellum
Hisce characteribus. Christi tunc inquit alumnus:
Nulla damus patriae, fateor, discidia nostrae.
Non reges, non tristatas, regumve clientes,
Matris ad eximiae gremium concurrere gratis.
Elegi collata prius mihi jura probare.
Non ignota cano, vobis praecognita narro,
Jura dedere Patres Romanum stemma regentes.
Testis adest hujus sanctae periergia sedis,
133.1007B| Haec quia non sprevi, nam nec spernenda putavi.
Collatis carui crebris populatibus agris.
En qui me sensu graviter lymphante sequuntur,
Norunt si qua meo vitiamine rupta vicissim
Objectare queunt, vestro lustrabor ab ore.
Non ego pompisonis adii pretiosa lupatis [frenis]
Moenia, nec gilvis phalerato stromate mannis (equis tolutaribus:
Prosequor at nudus nudi vestigia Petri.
Posteritas sancite quibus subsistat amica
Legibus, et priscas nolite revellere chartas.
Obstipuere ( sic ) Patres haec sic fore [esse] comminiscentes.
Hac dixere pharo mundi rudis arsit origo,
Rabula nec deerit quo mens veneranda nitebit
133.1007C| Talia labentis patiuntur spargana vitae
Aemula, nec tardat spiris agitare malignis
Actio pravorum vitam moresque proborum.
Conquirenda tamen hujusce refragia sectae.
Et quamvis odiis proscriptus physiculatis [naturalibus]
Protinus arbitrio, Wilfride, piabere nostro.
Nec Romana prius falsi pilotria ( sic ) testes
Excedent, justas donec pro crimine poenas
Jure luant, discantque viro deferre probato.
Iste quidem Latias olim lallando papillas,
Pavit Christicolas multo dulcore catervas.
Oravere Patres: testatur et ipse Joannes (papa),
Haec dicens: Nullo morietur lampada fungo.
Insidiae veteres dormiscant ocius omnes.
Desipiunt hebetes tetrico moerore furentes.
133.1007D| Wilfridum damnare nefas: qui, sicuti torris,
Ereptus magno manet incombustus ab igne.
Ut petrus Christi, sic hic vestigia Petri
Percolit, aequoreis nunquam timefactus in undis.
Degeneres populi quid prodest velle beari?
Virtus si qua placet, patriis pepulistis ab oris.
En rutilum liquido summae virtutis agalma.
Sint rata quae sanxit, non haec obliteret usquam
Aut regalis apex, aut ecclesiasticus ordo.
Nobiscum sapiat, qui Christi stigmata gestat,
Ne ferale chaos evitandamque mephitim
133.1008A| Sentiscat rigidi retrusus in ima barathri.
Quidquid neutericis confinxit apocrypha biblis,
Cassetur priscis legatio censeo dictis.
Quis duo-quindecies, denos pariter quoque quinos,
Elogio archontes queat infamare maligno?
His coram fidei vir hic confessus honorem,
Extunc vitali merito craxatus [inscriptus] in albo,
Certis exspectat coelestia dona diebus,
Ut re percipiat, quod spe patienter et optat.
Huic qui dissentit, nimis, inquam, prorsus ineptit.

'' ( Sequentia, quae in ms. codice Corbeiensi desunt ex codice Cottonianae bibliothecae supplevit Mabillonius. )''

Haec eadem sacris jusserunt indere libris,
More cleanteo solitis bullata sigillis,
Regibus et clero mittentes, atque popello,
133.1008B| Ut dirimant odii crudelia jurgia prisci.
Et dixere Patri post oscula frontis aquosae:
Pacificus redeas; subjectis gaudia reddas;
Terris atque polis Jesu cleronomus [haeres] esto.
&lt;/poem&gt;

==XLVIII.==
&lt;poem&gt;
Devictis igitur falsi latoribus astus,
Pignere sanctorum multa quoque glore venustus,
Deserit Ausonias multis comitantibus oras.
Alpinosque petit, quo Celtica permeet arva.
Praeteriens notos pedetentim transit Alanos:
Tandem pacatos devenit ad usque Sicambros.
Hinc vexatur acri per singula membra veterno:
Nec quibat metas pedibus tranare cupitas.
Fertur equo primum; post haec gestamine fratrum,
Vocibus et toto pastorum corde dolentum;
133.1008C| Muratamque vehunt dilectum pondus ad urbem
Semianimem insueto componunt debite stratu:
Perculsique metu, quidnam contingat in actu
Exspectare timent, mortis formidine pressi.
Octonis etenim bis forte quaterque vigenis
Decubuit horis, pastus fraglamine mentis
Corporis aut escis, aut qualibet arte liquoris.
Demum cum quinti spirarent lumina solis,
Mittitur e summo (dictu mirabile!) coelo
Archanus Michael, nitido lampabilis ore;
Quem Pater intendens, non ut fantasma repellens,
Ceu stilbonta [splendentem], novum lumen, veneratur amicum.
Surgito concivis, nostri non portio vilis:
133.1008D| Et licei astricolas inter numerere catervas,
Inquit, ab orbatis jam nunc revocaberis agnis.
Postquam bis binis fratres solidaveris annis;
Cum mercede bona temet revocabo, sed insta
Christotochae dignas aedes fabricare Mariae:
Cujus amore tibi cumulantur certius anni.
Novit quid Petro, Petri quid solvis adelpho.
Continuo sanctos subiit vigor integer artus:
Angelicum superam numen recucurrit ad aethram.
Surgitur, et tristes renovantur digne sodales.
O defesse, sagax curat quem coelica pinax:
133.1009A| Et quia pellacis vitasti gaudia saecli,
Cum numero vitae meruisti praemia summae.
&lt;/poem&gt;
==XLIX.==
&lt;poem&gt;
Oceani tumidas exhinc remeavit ad undas,
Gurgitis et vasti circumflua terga repressit.
Cantia complacito patuerunt litora rastro.
Jussit apostolicum Berthwaldo ferre libellum,
Suscipit, ut decuit; Patris praecepta peregit.
Hinc divi [ f., divus] petiit, supra quem musa revolvit,
Laetus Edilredi petulas fidentius aulas.
Cui retulit cessim: laetantur et ambo vicissim.
&lt;/poem&gt;
==L.==
&lt;poem&gt;
Approperans ergo, laesus cunctamine nullo,
Aldfrido regi direxit epistile [epistolam] Petri.
133.1009B| Illicitum sapuit, non hoc sat juste recepit
Protulit ambages, sicut fecere priores:
Nolens directis veneramen ferre loquelis.
Quocirca meritum tulit immature periclum.
Ultrices etenim patiens in corpore poenas,
Cernere collibuit, quem dudum languide sprevit
Multaque conjiciens obiit, male gesta recensens
Testibus adhibitis se reddere velle negatis,
Si Domini auxilium vitae daret accumulatum.
Suscepit vacuas Aedwilf virosus habenas,
In sanctum furias exercens immoderatas:
Inque brevi periit, violento cive repulsus.
&lt;/poem&gt;
==LI.==
&lt;poem&gt;
Ut patriae curam susceptet rite tuendam,
Filius ascitur Alhdfridi saepe relati.
133.1009C| Qui gravitate proba, Wilfridi subditus ala,
Dilexit nimium, quia noverat esse beatum;
Is jussit proprii quantum tenet orbita regni;
Et coiere simul, Berthwaldo jure calante [vocante]
Tunc Patris alloquio rex censuit, omnis et ordo,
Servent ut pariter Wilfridi dogma verenter:
Pacis et Angligenae firmantur foedere gentes.
&lt;/poem&gt;
==LII.==
&lt;poem&gt;
Ex hinc convaluere nimis diorismata [definitiones] fortis
Athletae, pereuntque mali sophismata civis;
Germinis hydrei penitus tenuata cicatrix;
Additur atque bonis virtus apodictica (demonstrativa) victrix.
133.1009D| Ipse per obliquas pastor gradiens regiones,
Ut mater dociles doctrino lacte refovit;
Ut pater ignaros sobria succensuit [increpavit] ira
Ad lucis patriam cunctos metaliter hortans.
Praescivit porro Michaelis adesse profatum:
Optabatque gravem Silicernius adfore mortem.
Corruit, et subitus perclusit limmata [consilia] morbus
Agmina pullatis coeunt vernacula birris;
Commeat et toto vulgus miserabile ciclo.
Fit formido gravis, ne forte recedat ab illis.
133.1010A| Fratribus at plaudas Jesu recinentibus odas,
Substitit, et laeta sospes perfunditur aura.
&lt;/poem&gt;
==LIII.==
&lt;poem&gt;
Hactenus excolicum quanto sudore sacellum
Emerit exposui; numnam miracula quibo?
Ennea si melico chelim mihi carmine tangant,
Septenasque jugem [geminem] ciclico dulcore sorores
Aut sim magniloqui multus sectator Homeri;
Non plane absolvam: dictis jam cedo quibusdam.
Quos etenim daemon rabida possederat ira,
Effusis precibus fidei virtute resolvit:
Et queis debilitas artus devinxerat atra;
Aut oculo frustrans, aut linguae murmure damnans;
Aut gressus prohibens, aut reumate pectora stringens;
133.1010B| Et sensu demtos, et morbi tabe peraesos.
Ut frater congrex, scelerum, sicut patet, offex (impeditor),
Circumfusa means docet agmina Christicolarum,
Vivifici tetra plastica dat theoremata verbi,
Noviter increpitans pigrum decus Archiliorum [tribunos].
Haec ita sollerti perfecerat undique cura.
Sensit olympiadam dudum finirier auctam,
Et metamorphosin praesagit adesse cupitam.
Nec minus ipse tamen refovens normaliter omnes.
Aethralem vitam testatur jure sequendam.
Quid tricor dictis? dabitur nunc his quoque finis
Credo quidem scriptis: ut mors, dilectio fortis.
&lt;/poem&gt;
==LIV.==
&lt;poem&gt;
Crebis ergo virum febribus pulsatibus almum,
133.1010C| Concurrunt monachi virtutum munere macti:
Agmine pastorem tantum cinxere globoso.
His ita consaeptus, caput extulit, atque ita dixit:
Pullulet in vobis divini gratia roris.
Vos equidem, fratres, pridem revocatis euntem.
Parcite jam, quaeso, superum jam scandere regnum
Me juvat, et fragili praejudico carne resolvi.
Non vobis ultra forma conjungar in ista,
Donec mundanas turbent elementa tenebras.
Visibus et Domini praesens sit machina mundi.
Muscipulos nostis quos obicit aemulus hostis,
Illicibus quaerens animos vitiare figuris.
Ingerit incestos lascivis spebus amores.
Tunc oculos pulcris vanos illudit idaeis,
133.1010D| Aures et blandis demulget saepius arpis.
Ut fraudem doceat, largissima lucra ministrat.
Haec monui semper, haec vobis ipse frequenter
Opposui, nolite moras inferre migranti.
Urgeor, o defungar, et, o fratres, vigilate,
Pyr Flegeton calida ne vos absorbeat unda.
Haec ait, et durae constravit membra quieti.
Spiritus ergo viri Michael repetente beati,
Vimineo condens corpus Kyriale canistro
Exhausit vitro vitalem digne cruorem,
133.1011A| Inter moerentes, orantes, opperientes
Evasit volucer, carnis ceromate [visco] liber.
&lt;/poem&gt;
==LV.==
&lt;poem&gt;
Postquam membra sacro tumularunt sacra sepulcro,
Indusias sumsere sibi fratres venerandi:
Interulamque puer sancti sudore madentem
Corripuit, normatrici [abbatissae] tulit atque beatae,
Quam sibi flamineo [velo] sociaverat apte verendo
Egregius heros, redimitam castificando.
Hanc adit organico membris deserta vigore
Clinica, suppliciter poscens se tangere posse.
Pavit secretum depromere virgo fidele.
Tandem compatiens miserae subnixe petenti,
Vinxit aquis vestem, meruit quibus illa salutem.
133.1011B| Post haec bachantes dementi pectore quidam,
Vulcanum [ignem] tecto, quo transiit ille, adhibebant.
Sed vis defuncti viventem terruit ignem:
Saevos mitificans, virtutis et omina monstrans.
Annua bis senos confecerat orbita menses,
133.1012A| Subsidibus patris ex omni regione coactis,
Cum miranda nimis fulserunt signa sub astris.
Quaerebant etenim quis tot pastore vacantes
Excipiat caulas, et ovilia tanta reservet.
Hesperus aeriis sed jam densaverat umbras,
Pernox cura fratrum cum coelum respicit altum.
Emicuit cyclus tenebrosa per astra coruscus
Ceu matutinos igniret Phosphorus ortus.
Canduit hic radius monachum coenobia circum
Officiose, vias visus liquisse polinas.
Hinc patuit, Domini quia constat vinea firmis
Oratu Wilfridi aevum stabilita gigantis.
Mox apopempaeo grates persolvere summo
Accelerant, nomenque Patris constanter honorant.
Fistula rurestris cecinit prothemata cavis,
133.1012B| Callosas cani Meligor velut excola sannas,
Enervi trutinans exotica docmina [dogmata] plectro.
Nunc oleate mihi faveas industrius Odo.
Ut natum retuli stemma tum maxime clusi.
Non epistasin capit, ecstasinque veretur.
Pagina, Lector, ave: tibi caetera carpe tuapte.
&lt;/poem&gt;
[[Categoria:Patrologiae Cursus Completus]]</text>
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    <title>Pagina:Prisciani Institutionum grammaticarum libri II (v. 1, ed Hertz).djvu/345</title>
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      <text bytes="1925" sha1="a98ontulnyh703muipuhpcjrn7kuaeb" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;3&quot; user=&quot;Saumache&quot; /&gt;{{rh||LIBER VII 29—32|313}}&lt;/noinclude&gt;&lt;p&gt;In ‘ul’ brevem masculini vel communis Latina: ‘hic consŭl huius consulis’, ‘hic’ et ‘haec exŭl huius exulis’.

&lt;p&gt;In ‘an’ productam masculina Graeca: ‘hic Paeān huius Paeanis’.

&lt;p&gt;In ‘en’ correptam Latina masculina vel neutra: ‘hic flamĕn huius flaminis’, ‘hoc numĕn huius numinis’.

&lt;p&gt;In ‘en’ productam Latina vel Graeca masculini vel feminini generis: ‘hic liēn huius lienis’, ‘haec Sirēn huius Sirenis’.

&lt;p&gt;In ‘in’ productam Graeca masculina vel feminina: ‘hic delphīn huius 
delphinis’, ‘haec Trachīn huius Trachinis’.

&lt;p&gt;In ‘on’ productam Graeca masculina vel feminina: ‘hic Memnōn huius Memnonis’, ‘haec Sidōn huius Sidonis’.

&lt;p&gt;In ‘ar’ correptam Latina et Graeca et barbara masculini vel neutri generis: ‘hic Caesăr huius Caesaris’, ‘hic Bostăr huius Bostaris’, ‘hoc nectăr huius nectaris’, ‘hoc calcăr huius calcaris’.

{{r|32}}&lt;p&gt;In ‘ar’ productam Latina masculini vel communis vel neutri generis, et sunt vel monosyllaba vel ex monosyllabis composita: ‘hic Nār huius Naris’, ‘hic’ et ‘haec’ ‘et ‘hoc pār huius paris’, ‘dispār disparis’, ‘hoc fār huius farris’.

&lt;p&gt;In ‘er’ correptam masculina, feminina et neulra et communia, ut ‘hic patĕr huius patris’, ‘haec matĕr huius matris’, ‘hoc tubĕr huius tuberis’, ‘hic’ et ‘haec paupĕr huius pauperis’.

&lt;p&gt;In ‘er’ productam Graeca masculini et unum Latinum vel magis nothum neutri generis: ‘hic aēr huius aeris’, ‘hoc vēr huius veris’.

&lt;p&gt;In ‘ir’ unum femininum, ‘haec Gaddir huius Gaddiris’. quidam addunt ‘hic abaddir’, {{graeca|''ὁ βαίτυλος''}}. ‘huius abaddiris’, lapis, quem pro Iove devoravit Saturnus, sed in usu hoc non inveni.

&lt;p&gt;In ‘or’ correptam masculina et feminina et neutra et communia tam&lt;noinclude&gt;&lt;references/&gt;&lt;/noinclude&gt;</text>
      <sha1>a98ontulnyh703muipuhpcjrn7kuaeb</sha1>
    </revision>
    <revision>
      <id>281408</id>
      <parentid>281407</parentid>
      <timestamp>2026-09-20T16:02:41Z</timestamp>
      <contributor>
        <username>Saumache</username>
        <id>27923</id>
      </contributor>
      <origin>281408</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="1925" sha1="bnqns41y5jsm343hly3ikjzgof3mmhj" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;3&quot; user=&quot;Saumache&quot; /&gt;{{rh||LIBER VII 29—32|313}}&lt;/noinclude&gt;&lt;p&gt;In ‘ul’ brevem masculini vel communis Latina: ‘hic consŭl huius consulis’, ‘hic’ et ‘haec exŭl huius exulis’.

&lt;p&gt;In ‘an’ productam masculina Graeca: ‘hic Paeān huius Paeanis’.

&lt;p&gt;In ‘en’ correptam Latina masculina vel neutra: ‘hic flamĕn huius flaminis’, ‘hoc numĕn huius numinis’.

&lt;p&gt;In ‘en’ productam Latina vel Graeca masculini vel feminini generis: ‘hic liēn huius lienis’, ‘haec Sirēn huius Sirenis’.

&lt;p&gt;In ‘in’ productam Graeca masculina vel feminina: ‘hic delphīn huius 
delphinis’, ‘haec Trachīn huius Trachinis’.

&lt;p&gt;In ‘on’ productam Graeca masculina vel feminina: ‘hic Memnōn huius Memnonis’, ‘haec Sidōn huius Sidonis’.

&lt;p&gt;In ‘ar’ correptam Latina et Graeca et barbara masculini vel neutri generis: ‘hic Caesăr huius Caesaris’, ‘hic Bostăr huius Bostaris’, ‘hoc nectăr huius nectaris’, ‘hoc calcăr huius calcaris’.

&lt;p&gt;{{r|32}}In ‘ar’ productam Latina masculini vel communis vel neutri generis, et sunt vel monosyllaba vel ex monosyllabis composita: ‘hic Nār huius Naris’, ‘hic’ et ‘haec’ ‘et ‘hoc pār huius paris’, ‘dispār disparis’, ‘hoc fār huius farris’.

&lt;p&gt;In ‘er’ correptam masculina, feminina et neulra et communia, ut ‘hic patĕr huius patris’, ‘haec matĕr huius matris’, ‘hoc tubĕr huius tuberis’, ‘hic’ et ‘haec paupĕr huius pauperis’.

&lt;p&gt;In ‘er’ productam Graeca masculini et unum Latinum vel magis nothum neutri generis: ‘hic aēr huius aeris’, ‘hoc vēr huius veris’.

&lt;p&gt;In ‘ir’ unum femininum, ‘haec Gaddir huius Gaddiris’. quidam addunt ‘hic abaddir’, {{graeca|''ὁ βαίτυλος''}}. ‘huius abaddiris’, lapis, quem pro Iove devoravit Saturnus, sed in usu hoc non inveni.

&lt;p&gt;In ‘or’ correptam masculina et feminina et neutra et communia tam&lt;noinclude&gt;&lt;references/&gt;&lt;/noinclude&gt;</text>
      <sha1>bnqns41y5jsm343hly3ikjzgof3mmhj</sha1>
    </revision>
  </page>
  <page>
    <title>Pagina:Prisciani Institutionum grammaticarum libri II (v. 1, ed Hertz).djvu/347</title>
    <ns>104</ns>
    <id>82549</id>
    <revision>
      <id>281399</id>
      <parentid>271457</parentid>
      <timestamp>2026-09-20T15:43:42Z</timestamp>
      <contributor>
        <username>Saumache</username>
        <id>27923</id>
      </contributor>
      <comment>/* Emendata */</comment>
      <origin>281399</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="2350" sha1="0451jk2nd64yejgvjbza4rrlcu4kkzw" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;3&quot; user=&quot;Saumache&quot; /&gt;{{rh||LIBER VII 32—34|315}}&lt;/noinclude&gt;raro autem per nommativum coniungi etiam neutris, ut {{spatium||{{wl|Q1398|Virgilius}}}} in {{spatium||VIIII}}:&lt;poem&gt;{{gap|8em}}{{spatium||Nulla dies umquam memori vos eximet aev}}o,&lt;ref&gt;[[Aeneis/Liber IX#445|Aeneis 9.447]]&lt;/ref&gt;&lt;/poem&gt;et {{spatium||{{wl|Q193800|Iuvenalis}}}} in {{spatium||IV {{wl|Q2707099|saturarum}}}}:&lt;poem&gt;{{gap|17.5em}}{{spatium||tilulique cupido
{{gap|4em}}Haesuri saxis cinerum custodibu}}s,&lt;ref&gt;[[Saturae (Iuvenalis, Bucheler)/Liber IV/Satura X#140|Saturae (Iuvenalis) 4.10.143]]&lt;/ref&gt;&lt;/poem&gt;quamvis ‘hoc custos’ raro nisi figurate invenias et ‘haec custoda’ pluraliter nemo dicat. sic ergo neque ‘haec memora’ vel ‘auctora’ vel ‘paupera’ neutraliter puto facile in usu posse inveniri, sed, sicut supra dictum est, figurate sunt prolata et haec et alia plurima huiuscemodi apud auctores. {{r|34}}inveniuntur tarmen quaedam, quae obliquos casus communes possident, quamvis nominativus non sit communis, ut ‘plus’ cum sit neutri nominativus dumtaxat, genetivus eius communis est trium generum nec non et dativus et ablativus, unde pluralia quoque tam in ‘es’ quam in a invenimus: ‘hi’ et ‘hae plures’ et ‘haec plura’ vel ‘pluria’, unde compositum ‘compluria’. {{spatium||{{wl|Q172048|Terentius}}}} in {{spatium||{{wl|Q1217045|Phormione}}}}:&lt;poem&gt;{{gap|4em}}{{spatium||Multa advenienti †nova, ut fit, compluri}}a.&lt;ref&gt;[[Phormio/Actus IV#610|Phomrio 611]]&lt;/ref&gt;&lt;/poem&gt;ablativus quoque tam in e quam in i terminans invenitur, utpote communis trium generum, et accusativus pluralis tam in ‘es’ quam in ‘is’ {{spatium||{{wl|Q1398|Virgilius}}}} in {{spatium||I {{wl|Q674439|georgicon}}}}:&lt;poem&gt;{{gap|4em}}{{spatium||Seu pluris calor ille vias et caeca relaxat
{{gap|4em}}Spirament}}a.&lt;ref&gt;[[Georgicon (Greenough)/Liber I#85|Georgicon 1.89-90]]&lt;/ref&gt;&lt;/poem&gt;{{spatium||{{wl|Q6197|Horatius}}}} in {{spatium||I {{wl|Q1226213|epistularum}}}}:&lt;poem&gt;{{gap|4em}}{{spatium||Tractus uter pluris lepores, uter educet apro}}s.&lt;ref&gt;[[Epistulae (Horatius)/Liber I/Epistula XV#20|Epistulae (Horatius) 1.15.22]]&lt;/ref&gt;)&lt;/poem&gt;

&lt;p&gt;In ‘ur’ correptam Latina masculina et communia et neutra: ‘hic turtūr huius turturis’, ‘hic’ et ‘haec angūr auguris’ – {{spatium||{{wl|Q6197|Horatius}}}} in {{spatium||III {{wl|Q943884|carminum}}}}:&lt;noinclude&gt;&lt;references/&gt;&lt;/noinclude&gt;</text>
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    </revision>
  </page>
  <page>
    <title>Pagina:Prisciani Institutionum grammaticarum libri II (v. 1, ed Hertz).djvu/348</title>
    <ns>104</ns>
    <id>82550</id>
    <revision>
      <id>281406</id>
      <parentid>271446</parentid>
      <timestamp>2026-09-20T16:01:28Z</timestamp>
      <contributor>
        <username>Saumache</username>
        <id>27923</id>
      </contributor>
      <comment>/* Emendata */</comment>
      <origin>281406</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="2155" sha1="0izjfq8af72c2r1zpmwlhg9t9xm8s75" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;3&quot; user=&quot;Saumache&quot; /&gt;{{rh|316|PRISCIANI INST.}}&lt;/noinclude&gt;&lt;poem&gt;{{gap|11em}}{{spatium||nisi fallit augur
{{gap|4em}}Annosa corni}}x&lt;ref&gt;[[Carmina (Horatius)/Liber III/Carmen XVII#10|Carmina (Horatius) 3.17.12-13]]&lt;/ref&gt; –,&lt;/poem&gt;‘hoc Tibŭr huius Tiburis’.

&lt;p&gt;In ‘ur’ productam unum invenitur commune duum generum: ‘hic’ et ‘haec fūr huius furis’.

&lt;p&gt;{{r|35|VIII 35}}In ‘as’ correptam Graeca masculini vel feminini vel neutri vel communis generis: ‘hic Arcăs’ [proprium] ‘huius Arcadis’, ‘haec Pallăs huius Palladis’, ‘hoc Cerăs’ [proprium loci] ‘huius Ceratis’, ‘hic’ et ‘haec Arcăs’ [gentile] ‘huius Arcadis’.

&lt;p&gt;In ‘as’ productam Graeca et Latina masculini vel feminini vel neutri vel communis generis, ut ‘hic Atlās huius Atlantis’, ‘hic Maecenās huius Maecenatis’, ‘haec civitās huius civitatis’, ‘hoc vās huius vasis’, ‘hic’ et ‘haec Capenās huius Capenatis’.

&lt;p&gt;In ‘es’ correptam Latina masculina vel feminina vel communia duum vel trium generum, ut ‘hic Coclĕs huius Coclitis’, ‘haec segĕs huius segetis’, ‘hic’ et ‘haec milĕs huius militis’ – {{spatium||{{wl|Q7198|Ovidius}}}} in {{spatium||II {{wl|Q184742|metamorphoseon}}}}:&lt;poem&gt;{{gap|4em}}{{spatium||Miles erat Phoebes}}&lt;ref&gt;[[Metamorphoses (Ovidius)/Liber II#415|Metamorphoses 2.415]]&lt;/ref&gt;&lt;/poem&gt;de Callistone dicit –, ‘hic’ et ‘haec’ et ‘hoc terĕs’ – {{spatium||{{wl|Q1398|Virgilius}}}} in {{spatium||VII}}:&lt;poem&gt;{{gap|9em}}{{spatium||sed haec tereti mos est aptare flagello}}&lt;ref&gt;[[Aeneis/Liber VII#730|Aeneis 7.731]]&lt;/ref&gt; –,&lt;/poem&gt;‘hebĕs hebetis’, ‘sospĕs sospitis’ – {{spatium||{{wl|Q193800|Iuvenalis}}}} in {{spatium||V}}:&lt;poem&gt;{{gap|23em}}{{spatium||nec umquam
{{gap|4em}}Depositum tibi sospes erit}}&lt;ref&gt;[[Saturae (Iuvenalis, Bucheler)/Liber V/Satura XIII#175|Saturae (Iuvenalis)5.13.177]]&lt;/ref&gt; –,&lt;/poem&gt;‘hic’ et ‘haec’ et ‘hoc hospĕs huius hospitis’. {{spatium||{{wl|Q188646|Lucanus}}}} in {{spatium||V}}:&lt;poem&gt;{{gap|4em}}{{spatium||Hospes in externis audivit curia tecti}}s,&lt;ref&gt;[[Pharsalia/Liber V#10|Pharsalia 5.11]]&lt;/ref&gt;&lt;/poem&gt;&lt;noinclude&gt;&lt;references/&gt;&lt;/noinclude&gt;</text>
      <sha1>0izjfq8af72c2r1zpmwlhg9t9xm8s75</sha1>
    </revision>
  </page>
  <page>
    <title>Quaestio</title>
    <ns>0</ns>
    <id>83915</id>
    <revision>
      <id>281463</id>
      <parentid>275845</parentid>
      <timestamp>2026-09-20T17:29:12Z</timestamp>
      <contributor>
        <username>Coronus Tenebrosius</username>
        <id>31709</id>
      </contributor>
      <origin>281463</origin>
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      <format>text/x-wiki</format>
      <text bytes="463" sha1="66itl2sa1r2jagftcznle8gex6me4fd" xml:space="preserve">[[Carmina non adlecta]]


== QUAESTIO ==

&lt;poem&gt;
𝐌ultus in corde fuit mihi motus,
In historiam et mythos perspexi.
Causam universi quaerebam,
Et quaesivi rerum intimam mentem.

Sacras perlegi sapientum chartas,
Multa didici, sed cura crescebat.
Maior scientia maiores fert curas,
Veritatem tamen in corde non repperi.

Cum omnes dubitationes solvissem,
Menti lux veritatis affulsit.
Sed cor dubitationis semen spargit,
Et mens in incerto iterum perit.
&lt;/poem&gt;</text>
      <sha1>66itl2sa1r2jagftcznle8gex6me4fd</sha1>
    </revision>
  </page>
  <page>
    <title>Silentium</title>
    <ns>0</ns>
    <id>83916</id>
    <revision>
      <id>281464</id>
      <parentid>273060</parentid>
      <timestamp>2026-09-20T17:30:19Z</timestamp>
      <contributor>
        <username>Coronus Tenebrosius</username>
        <id>31709</id>
      </contributor>
      <origin>281464</origin>
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      <text bytes="202" sha1="ifj1omw5m8frgbc98cx5nxnvrwt97i5" xml:space="preserve">[[Carmina non adlecta]]

== SILENTIUM ==

&lt;poem&gt;
Arbores dormiunt
et leniter spirant.

Stellae sunt prope.
(Manu eas tangere possum,
sed timeo ne exstinguantur.)

Silentium in me est
altissimum.
&lt;/poem&gt;</text>
      <sha1>ifj1omw5m8frgbc98cx5nxnvrwt97i5</sha1>
    </revision>
  </page>
  <page>
    <title>Desiderium</title>
    <ns>0</ns>
    <id>83917</id>
    <revision>
      <id>281465</id>
      <parentid>273061</parentid>
      <timestamp>2026-09-20T17:31:21Z</timestamp>
      <contributor>
        <username>Coronus Tenebrosius</username>
        <id>31709</id>
      </contributor>
      <origin>281465</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="169" sha1="iuuohzvpvmn3pbx88uckzm8hw45deq9" xml:space="preserve">[[Carmina non adlecta]]

==DESIDERIUM==

&lt;poem&gt;
Esse volo cactus 
in deserto.

Esse volo ego 
cum deserto.

Spinas volo, spinas 
et flores,

pulcherrimos flores.
&lt;/poem&gt;</text>
      <sha1>iuuohzvpvmn3pbx88uckzm8hw45deq9</sha1>
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  </page>
  <page>
    <title>Tabulae neonicae</title>
    <ns>0</ns>
    <id>83918</id>
    <revision>
      <id>281466</id>
      <parentid>273112</parentid>
      <timestamp>2026-09-20T17:32:25Z</timestamp>
      <contributor>
        <username>Coronus Tenebrosius</username>
        <id>31709</id>
      </contributor>
      <origin>281466</origin>
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      <format>text/x-wiki</format>
      <text bytes="220" sha1="0tz1vpj7aoxnsp4jouxu1wryb777fa0" xml:space="preserve">[[Carmina non adlecta]]

== TABULAE NEONICAE ==

&lt;poem&gt;
Tantum lucis et caloris habent  
tabulae neonicae,

et me ad se trahunt;  
et nox ob eas fit  
magica.

Possident lumen quod homines  
iam pridem amiserunt.
&lt;/poem&gt;</text>
      <sha1>0tz1vpj7aoxnsp4jouxu1wryb777fa0</sha1>
    </revision>
  </page>
  <page>
    <title>Dolor</title>
    <ns>0</ns>
    <id>83919</id>
    <revision>
      <id>281467</id>
      <parentid>273063</parentid>
      <timestamp>2026-09-20T17:33:36Z</timestamp>
      <contributor>
        <username>Coronus Tenebrosius</username>
        <id>31709</id>
      </contributor>
      <origin>281467</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="225" sha1="5fal7nqcargtrq37gchwsk1d7r9bc1q" xml:space="preserve">[[Carmina non adlecta]]

DOLOR

&lt;poem&gt;
Si in eo est mea essentia  
ut cadam, cadam.

Fiat dolor!  
Nullus dolor est.

Labra vestra magna et atra sunt;  
dolorem meum devoraverunt.

Dolor meus evanuit  
in risu vestro.
&lt;/poem&gt;</text>
      <sha1>5fal7nqcargtrq37gchwsk1d7r9bc1q</sha1>
    </revision>
    <revision>
      <id>281468</id>
      <parentid>281467</parentid>
      <timestamp>2026-09-20T17:34:04Z</timestamp>
      <contributor>
        <username>Coronus Tenebrosius</username>
        <id>31709</id>
      </contributor>
      <origin>281468</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="231" sha1="m5c6nisqt6zrkebdaoriyevnywoob8h" xml:space="preserve">[[Carmina non adlecta]]

== DOLOR ==

&lt;poem&gt;
Si in eo est mea essentia  
ut cadam, cadam.

Fiat dolor!  
Nullus dolor est.

Labra vestra magna et atra sunt;  
dolorem meum devoraverunt.

Dolor meus evanuit  
in risu vestro.
&lt;/poem&gt;</text>
      <sha1>m5c6nisqt6zrkebdaoriyevnywoob8h</sha1>
    </revision>
  </page>
  <page>
    <title>Per membranam</title>
    <ns>0</ns>
    <id>83920</id>
    <revision>
      <id>281469</id>
      <parentid>273064</parentid>
      <timestamp>2026-09-20T17:35:03Z</timestamp>
      <contributor>
        <username>Coronus Tenebrosius</username>
        <id>31709</id>
      </contributor>
      <origin>281469</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="268" sha1="3jjejbozvrk9lqonud3xmxz3g2mvw6w" xml:space="preserve">[[Carmina non adlecta]]

== PER MEMBRANAM ==

&lt;poem&gt;
In umbra corporum florentium  
anima invisibilis iacet.  
Ea cupit, ea tendit,  
in altum se attollere.
  
Sed per membranam  
transire non potest.

Vincula crescunt invisibilia,  
ut corpori humano  
fiant.
&lt;/poem&gt;</text>
      <sha1>3jjejbozvrk9lqonud3xmxz3g2mvw6w</sha1>
    </revision>
  </page>
  <page>
    <title>Aves cantent</title>
    <ns>0</ns>
    <id>83938</id>
    <revision>
      <id>281470</id>
      <parentid>273087</parentid>
      <timestamp>2026-09-20T17:35:50Z</timestamp>
      <contributor>
        <username>Coronus Tenebrosius</username>
        <id>31709</id>
      </contributor>
      <origin>281470</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="208" sha1="gh41fs8fuup7xvkgzhrqdq8m8pnkko9" xml:space="preserve">[[Carmina non adlecta]]

== AVES CANTENT ==

&lt;poem&gt;
Aves circa me cantent,  
et omnia pulchra sint,
viridia.

O Deus, aves cantent  
circa me; ille flos  
qui in corde meo crescit  
numquam marcescat.
&lt;/poem&gt;</text>
      <sha1>gh41fs8fuup7xvkgzhrqdq8m8pnkko9</sha1>
    </revision>
  </page>
  <page>
    <title>Pictoris carmen</title>
    <ns>0</ns>
    <id>83939</id>
    <revision>
      <id>281471</id>
      <parentid>273071</parentid>
      <timestamp>2026-09-20T17:36:44Z</timestamp>
      <contributor>
        <username>Coronus Tenebrosius</username>
        <id>31709</id>
      </contributor>
      <origin>281471</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="194" sha1="hz5lp457iraiqeylepgeh7d8byhusr2" xml:space="preserve">[[Carmina non adlecta]]

== PICTORIS CARMEN ==

&lt;poem&gt;
O, si te attingere possem, pulchritudo,  
in forma hominis perfecta,  
dominus essem totius mundi,  
universi atque omnium aspecta.
&lt;/poem&gt;</text>
      <sha1>hz5lp457iraiqeylepgeh7d8byhusr2</sha1>
    </revision>
  </page>
  <page>
    <title>Cognitio</title>
    <ns>0</ns>
    <id>83940</id>
    <revision>
      <id>281472</id>
      <parentid>273088</parentid>
      <timestamp>2026-09-20T17:37:47Z</timestamp>
      <contributor>
        <username>Coronus Tenebrosius</username>
        <id>31709</id>
      </contributor>
      <origin>281472</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="233" sha1="rbbpqebjv4jx3v3lhc0ovxblkjl9tgr" xml:space="preserve">[[Carmina non adlecta]]

== COGNITIO ==

&lt;poem&gt;
Extra me ipsum  
dilatior fio.  
E tenebris in lucem  
dilatior fio.
  
Et semper me aliquid excitat  
in dilatione, 
 
et iterum a luce  
in cotidianitatem  
ignorantiae
redeo.
&lt;/poem&gt;</text>
      <sha1>rbbpqebjv4jx3v3lhc0ovxblkjl9tgr</sha1>
    </revision>
  </page>
  <page>
    <title>Apparitio</title>
    <ns>0</ns>
    <id>83941</id>
    <revision>
      <id>281473</id>
      <parentid>273078</parentid>
      <timestamp>2026-09-20T17:38:38Z</timestamp>
      <contributor>
        <username>Coronus Tenebrosius</username>
        <id>31709</id>
      </contributor>
      <origin>281473</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="228" sha1="modugeotdc04s9t1owx1fdbsl8ys4tx" xml:space="preserve">[[Carmina non adlecta]]


== APPARITIO ==

&lt;poem&gt;
Entia solitudinis. Duae planetae  
spatio obscuro cosmico seiunctae.

Non sunt planetae.  
Stellae sunt.

Lumen suum includunt  
in se. Corruunt  
in se:

foramen nigrum.
&lt;/poem&gt;</text>
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    </revision>
  </page>
  <page>
    <title>Puella</title>
    <ns>0</ns>
    <id>83942</id>
    <revision>
      <id>281474</id>
      <parentid>273080</parentid>
      <timestamp>2026-09-20T17:39:34Z</timestamp>
      <contributor>
        <username>Coronus Tenebrosius</username>
        <id>31709</id>
      </contributor>
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      <text bytes="191" sha1="kxu0xlkqppb6te4xsbxmu4o9koot2s9" xml:space="preserve">[[Carmina non adlecta]]

== PUELLA ==

&lt;poem&gt;
Ut serpens sinuatur  
albo ventre  
ad sonitus  
attonitos.

Amorem explere ea cupit.  
Iuvenis cor puellae abiciet,  
et ubera capesset.
&lt;/poem&gt;</text>
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    </revision>
  </page>
  <page>
    <title>Ad margines</title>
    <ns>0</ns>
    <id>83943</id>
    <revision>
      <id>281475</id>
      <parentid>274553</parentid>
      <timestamp>2026-09-20T17:41:23Z</timestamp>
      <contributor>
        <username>Coronus Tenebrosius</username>
        <id>31709</id>
      </contributor>
      <origin>281475</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="411" sha1="6364bf5v1o7k1avtaxvytn72lm2qv00" xml:space="preserve">[[Carmina non adlecta]]

== AD MARGINES ==

&lt;poem&gt;
Per dies per hortos
desertos ambulo,
aves in ramis sedent.

Folia tunc aspicio:
viridia sunt, fortasse non;
mihi ita tantum videtur.

Folia, folium viride est, sed iam
in marginibus flavescere
et lente siccari incipit.

Folium in uno arbore
senescit. Pira flavescunt. 
In gramine viridi
arida folia sparsa iacent.

Tempus se in marginibus
conglobatur.

&lt;/poem&gt;</text>
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    </revision>
  </page>
  <page>
    <title>Ego hic fui</title>
    <ns>0</ns>
    <id>83944</id>
    <revision>
      <id>281476</id>
      <parentid>276178</parentid>
      <timestamp>2026-09-20T17:43:01Z</timestamp>
      <contributor>
        <username>Coronus Tenebrosius</username>
        <id>31709</id>
      </contributor>
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== EGO HIC FUI ==

&lt;poem&gt;
Vos invito ut congregemur
et flammam ignis animi spectemus,
etiamsi ea calefieri non possumus.

Omnia in mentibus vestris sunt:
apis aliter quam nos
speciem lucis videt.
Res tantum exsistit
oculis nostris visa;
pro nobis.

Et unusquisque nostrum
ex angulo caeli sui
in terram prospicit.
Absoluto non caremus, quamquam
in certa domo collocati sumus,
in certo tempore;
aetate baroca, in domo baroco
proscenio et horto cultiore.

De absoluto cogitare possumus
quia non sumus absolutum.
Ut speculum,
totum frangi debet
ut sui conscius fiat.

Ego hic fui:
seditio particulae contra totum.
Frustra conatus
sui proprietatem probandi.

Quamvis separatus sim
et in Universo velut fragmentum avulsum fluiter,
omni gyro te desidero —
Fontem esse mei.

Et quaero: num umquam revertar
ad Plenitudinem meam,
a qua disiunctus fui,
ut Eam melius videre possem?

Tempore fatigati sumus
post longum iter.
Equi nostri capita verterunt
ad aeternitatem.

Obmutuimus
coram existentia.
&lt;/poem&gt;</text>
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  </page>
  <page>
    <title>Flos in tela araneae</title>
    <ns>0</ns>
    <id>83945</id>
    <revision>
      <id>281477</id>
      <parentid>273104</parentid>
      <timestamp>2026-09-20T17:44:48Z</timestamp>
      <contributor>
        <username>Coronus Tenebrosius</username>
        <id>31709</id>
      </contributor>
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      <text bytes="433" sha1="c6t77ubrizedzb9q7rasnb9uzhoq2se" xml:space="preserve">[[Carmina non adlecta]]

== FLOS IN TELA ARANEAE ==
&lt;poem&gt;
Quid est pulchritudo sine oculo
spectatoris?
Quid est corpus feminae sine manu
quae illud mulcet?
Frigidus marmor monumenti
flumine temporis perfusum.

Pulchritudo tua rosae similis est;
et antequam aestas transeat,
ventus validus eius petalos avellit.

Et omnes pili cadent, et omnia
inane et desertum manebit.
Et in postes veterum ianuarum
iam tela araneae haeret.
&lt;/poem&gt;</text>
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  </page>
  <page>
    <title>In mutatione</title>
    <ns>0</ns>
    <id>84234</id>
    <revision>
      <id>281478</id>
      <parentid>273909</parentid>
      <timestamp>2026-09-20T17:45:49Z</timestamp>
      <contributor>
        <username>Coronus Tenebrosius</username>
        <id>31709</id>
      </contributor>
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== IN MUTATIONE ==
&lt;poem&gt;
In flumine sto.
Flumen stat,
ego fluo.

Plenitudo Iesu,
vacuitas
Buddhae est.

Freedom is just another name
for nothing to lose.

Fortes profecti sunt,
quamvis haberent
quod amitterent.

Certum est:
si nihil agis,
omnia perdas.

Vita
in mutatione est.
&lt;/poem&gt;</text>
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    </revision>
  </page>
  <page>
    <title>De propagatione fructuum pulchritudinis</title>
    <ns>0</ns>
    <id>84235</id>
    <revision>
      <id>281479</id>
      <parentid>273910</parentid>
      <timestamp>2026-09-20T17:47:21Z</timestamp>
      <contributor>
        <username>Coronus Tenebrosius</username>
        <id>31709</id>
      </contributor>
      <origin>281479</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
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== DE PROPAGATIONE FRUCTUUM PULCHRITUDINIS ==

&lt;poem&gt;
Hominem musicam agnoscere,
quia eam in se gerit.
Hominem pulchritudinem agnoscere,
quia eam in se gerit.
Hominem bonitatem
et alterum hominem agnoscere,
quia eos in se gerit.

Agnoscimus ea
quae pars nostri sunt, quae amamus,
quae desideramus; agnoscimus ea
quia ea in nobis gerimus.

Ea quae nos non tangunt,
procul sunt a nobis, quamvis
fortasse prope; spectamus ea
tamquam mundum alienum et externum,
quia pars nostri non sunt.

Odimus, abominamur,
et perturbamur iis rebus
quas in nobis gerimus.

Metus noster et malum quod in mundo videmus
nos perturbant, quia id agnoscimus
in nobis, in nostra cupiditate et agressione,
etiam ut possibilitatem
quam nolumus.

Incertitudinem mundi circa nos, quae non sinit
nos perfici, agnoscimus in nobis
ut incertitudinem de nobis ipsis.

Eventus et nuntios externos in nos
tamquam sensus et affectus proicimus; post tempus
attenuatur limes inter mundum et nos:
et mundus nos occupat.

At nos non venimus huc
ut mundus nos occupet; venimus
ut nos ipsum transformemus:

in nostram pulchritudinem internam.
Quomodo in nobis pulchrum et bonum est — sic
extra nos fiat.

Sed facilius est alios culpare
quam pondus in nos recipere;
nam provocationes sunt — solum fortibus,
quibus nos non aequamus.

An fortasse, tamen…?

Pugnare est vivere,
et quisque suo modo pugnat,
relinquens post se
vestigia sui
— fructus suae pulchritudinis.
&lt;/poem&gt;</text>
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  </page>
  <page>
    <title>Victoria</title>
    <ns>0</ns>
    <id>84236</id>
    <revision>
      <id>281480</id>
      <parentid>273911</parentid>
      <timestamp>2026-09-20T17:48:26Z</timestamp>
      <contributor>
        <username>Coronus Tenebrosius</username>
        <id>31709</id>
      </contributor>
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      <text bytes="259" sha1="ac9691xrjjs1gjw8xcm1bkap8v5ghfo" xml:space="preserve">[[Carmina non adlecta]]

== VICTORIA ==

&lt;poem&gt;
Quotidie vita mea brevior fit.
Ideo hodiernus dies
plus mihi valet.

Quisque dies,
solis aut nubium,
nova victoria est.

Ita lente, victoria post victoriam,
et semper propior fio meo ultimo
— triumpho.
&lt;/poem&gt;</text>
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    </revision>
  </page>
  <page>
    <title>Odor amoris</title>
    <ns>0</ns>
    <id>84237</id>
    <revision>
      <id>281483</id>
      <parentid>275840</parentid>
      <timestamp>2026-09-20T17:52:21Z</timestamp>
      <contributor>
        <username>Coronus Tenebrosius</username>
        <id>31709</id>
      </contributor>
      <origin>281483</origin>
      <model>wikitext</model>
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      <text bytes="340" sha1="qly18pnycn8ibjl4p9zluflwsmc1rea" xml:space="preserve">[[Carmina non adlecta]]

== ODOR AMORIS ==

&lt;poem&gt;
Odorabat amor. Nox e stellis fulgentibus
leniter descendit, velut ros.
Digitis percurrit per crines 
meos hirsutos. Molliter 
me complexa est, velut 
aurora aestiva.

Profundius immergebam: infinitum est mare!
Deinde: evigilavi.

Dies purus et fulgens erat,
recentia sole perfusa.

&lt;/poem&gt;</text>
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    </revision>
    <revision>
      <id>281484</id>
      <parentid>281483</parentid>
      <timestamp>2026-09-20T17:53:25Z</timestamp>
      <contributor>
        <username>Coronus Tenebrosius</username>
        <id>31709</id>
      </contributor>
      <comment>/* ODOR AMORIS */</comment>
      <origin>281484</origin>
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== ODOR AMORIS ==

&lt;poem&gt;
Odorabat amor. Nox e stellis fulgentibus
leniter descendit, velut ros.
Digitis percurrit per crines 
meos hirsutos. Molliter 
me complexa est, velut 
aurora aestiva.

Profundius immergebam: 
infinitum est mare!
Deinde: evigilavi.

Dies purus et fulgens erat,
recentia sole perfusa.

&lt;/poem&gt;</text>
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    </revision>
  </page>
  <page>
    <title>Praesagia fine Martii</title>
    <ns>0</ns>
    <id>84238</id>
    <revision>
      <id>281485</id>
      <parentid>273913</parentid>
      <timestamp>2026-09-20T17:54:22Z</timestamp>
      <contributor>
        <username>Coronus Tenebrosius</username>
        <id>31709</id>
      </contributor>
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== PRAESAGIA FINE MARTII ==

&lt;poem&gt;
Aprilis non est mensis crudelissimus.
Multo peior est finis Martii.
Tempus est quo regnant
phantasmata.

Ratio et logica
sine vi remanent.
Imaginatio autem
nullam petit licentiam.

Et tunc vibrent memoriae,
parvi reges e tenebris.
In homine omnia excitantur;
et praesagit ver, et novum initium.
&lt;/poem&gt;</text>
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    </revision>
  </page>
  <page>
    <title>Inter verba</title>
    <ns>0</ns>
    <id>84242</id>
    <revision>
      <id>281486</id>
      <parentid>273984</parentid>
      <timestamp>2026-09-20T17:55:58Z</timestamp>
      <contributor>
        <username>Coronus Tenebrosius</username>
        <id>31709</id>
      </contributor>
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      <format>text/x-wiki</format>
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== INTER VERBA ==

&lt;poem&gt;
Verba dicta sunt,
colores inducti,
dies confectus est.

Poesis est quod manet
inter verba
non dictum.

Amator verborum esse debes
et vere ea diligere,
si vis esse poeta.

Tu verba non eligis,
illa te eligunt.

Tu tantum debes esse
satis liber
ut ea recipias.

Te vincent, coercebunt,
impedient; eris
in centum angustiis.

Sed si satis pertinax eris
in amore libertatis,
et cum verbis in libertate,
aliquando liber fies.
Satis liber ut sis
poeta.

Et invenies modum tuum
ad verba solvenda,
nam poesis verba non est.

Poesis est spatium libertatis et vitae
verbis notatum
quasi signis.

Poesis est quod manet
inter verba
non dictum.
&lt;/poem&gt;

 
== Commentarius ==

Cum poeta stilum suum invenerit, libertatem invenit atque modum verborum liberandorum.
Verba tum iam non sunt quae poetam premant, sed ipsa fluunt et id praebent quod scriptor voluit.
Id significat clavem suam invenisse, qua verba et linguam solvat atque reseret.Stilum invenit qui ei permittit se exprimere sine nimia cohibitione.
Hoc est spatium libertatis, cum poeta cum verbis in libertate est.
Aut, ut epigrammatice compendiose dicatur:


&lt;poem&gt;
'''ARS POETICA''' 
Minus est plus.
Non dictum regnat.
Verba indicant.
&lt;/poem&gt;</text>
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    </revision>
  </page>
  <page>
    <title>Nebulosae longinquae</title>
    <ns>0</ns>
    <id>84506</id>
    <revision>
      <id>281487</id>
      <parentid>274527</parentid>
      <timestamp>2026-09-20T17:56:57Z</timestamp>
      <contributor>
        <username>Coronus Tenebrosius</username>
        <id>31709</id>
      </contributor>
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== NEBULOSAE LONGINQUAE ==

&lt;poem&gt;
Longinquas vidi regiones, nebulosas et turbidas,
et in eis colles surgentes
velut pyramides regum veterum.

Inter colles nebula se volvit,
quasi serpens caelestis, nebula
quae pallio suo omnia velat.

Colles sunt velut corpus Terrae Matris.
Audacter in caelum surgunt et eminent,
cacumina acuta, silvis vestita, quasi
capita brassicae, quasi patellae inversae.

Caelum hos colles osculatur, eosque molliter
premit; inter eos nebula se volvit
velut fumus ex templo sacro.

Longinquas vidi regiones, nebulosas et turbidas,
et in eis colles velut templa gloriosa.
Sub eis autem vidi
agros iacentes, virentes,
aristas maizii aureas plenas,
oculo in longinquum fugientes.
&lt;/poem&gt;</text>
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    </revision>
  </page>
  <page>
    <title>Emergentia bacterii ...</title>
    <ns>0</ns>
    <id>84582</id>
    <revision>
      <id>281481</id>
      <parentid>274697</parentid>
      <timestamp>2026-09-20T17:50:10Z</timestamp>
      <contributor>
        <username>Coronus Tenebrosius</username>
        <id>31709</id>
      </contributor>
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      <format>text/x-wiki</format>
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== EMERGENTIA BACTERII SINE NOMINE QUAE IN UNIVERSUM CLAMAVIT ==

&lt;poem&gt;
Apprehensus in nebula somniorum et puerilium promissorum,
aspicis in fine cuniculi plenus tenebrarum
quomodo ianuae clauduntur, ianuae
post quas lux amplius non est.

Olim ibi habitabat Nada.¹
- “Quae Nada?” interrogas.
Illam, sororem Ulixis, quae, 
cum in Argentinam migravit,
nomen mutare debuit,
quia nomen eius Hispanice
- Nihil significat.²

Et num quis umquam se interrogavit 
quid hoc totum significet:
omnes illi gemitus, suspiria, clamores;
velut copulatio echini maris
in litore humido.

- O anima vulnerata, lacerata ungulis Fati,
superba esto! In dolore te leva, in clade
ne plora.³

Et num quis umquam se interrogavit 
quid hoc totum significet?
Quis est sensus omnium?
Hodie ploro, cras non ero,
post me puerum relinquo. Quis sum ego?
Velut quaestio Sphingis advenienti.⁴

Quis est sensus novi initii et novi finis,
in hac terra obscura, terra obscura
ad quam redeo?

Post me puerum relinquo
qui olim ego ipse fui,
in ungulis immitis Spei,
ut ploret, vulneratus gravibus auroris
novorum ortuum.

Et me ipsum interrogo, quis est sensus omnium, 
cum aliquando non aliud futurus sim
quam bacterium sine nomine
in immensitate universi
quod clamavit frustra, 
clamavit ad 
Caelum.
&lt;/poem&gt;
= =
London, Ontario, Canada, 

Septembris 2013

== Notae ==

¹ “Nada”: in lingua Croatica saepe est nomen femineum,
significans &quot;spem&quot;. In lingua Hispanica autem “nada” idem est ac &quot;nihil&quot;.

² Alusio ad Ulixem, qui Cyclopi Polyphemo dixit se vocari ''Οὖτις'', id est “Nemo”.

³ Reminiscentia fragmenti Archilochi:
&lt;poem&gt;
''Θῦμε, θῦμ᾽ ἀμήχανοισι κήδεσιν κυκώμενε,  
ἀνάστηθι καὶ ἀμύνου· ἔμπεδον δὲ θῦμον ἔχων… ''
&lt;/poem&gt;

⁴ Allusio ad tragoediam Sophocleam Oedipus Rex atque ad mythum de Sphinge.</text>
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      <timestamp>2026-09-20T17:51:21Z</timestamp>
      <contributor>
        <username>Coronus Tenebrosius</username>
        <id>31709</id>
      </contributor>
      <comment>/* EMERGENTIA BACTERII SINE NOMINE QUAE IN UNIVERSUM CLAMAVIT */</comment>
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== &lt;poem&gt;EMERGENTIA BACTERII SINE NOMINE 
QUAE IN UNIVERSUM CLAMAVIT &lt;/poem&gt;==

&lt;poem&gt;
Apprehensus in nebula somniorum et puerilium promissorum,
aspicis in fine cuniculi plenus tenebrarum
quomodo ianuae clauduntur, ianuae
post quas lux amplius non est.

Olim ibi habitabat Nada.¹
- “Quae Nada?” interrogas.
Illam, sororem Ulixis, quae, 
cum in Argentinam migravit,
nomen mutare debuit,
quia nomen eius Hispanice
- Nihil significat.²

Et num quis umquam se interrogavit 
quid hoc totum significet:
omnes illi gemitus, suspiria, clamores;
velut copulatio echini maris
in litore humido.

- O anima vulnerata, lacerata ungulis Fati,
superba esto! In dolore te leva, in clade
ne plora.³

Et num quis umquam se interrogavit 
quid hoc totum significet?
Quis est sensus omnium?
Hodie ploro, cras non ero,
post me puerum relinquo. Quis sum ego?
Velut quaestio Sphingis advenienti.⁴

Quis est sensus novi initii et novi finis,
in hac terra obscura, terra obscura
ad quam redeo?

Post me puerum relinquo
qui olim ego ipse fui,
in ungulis immitis Spei,
ut ploret, vulneratus gravibus auroris
novorum ortuum.

Et me ipsum interrogo, quis est sensus omnium, 
cum aliquando non aliud futurus sim
quam bacterium sine nomine
in immensitate universi
quod clamavit frustra, 
clamavit ad 
Caelum.
&lt;/poem&gt;

= =
London, Ontario, Canada, 

Septembris 2013

== Notae ==

¹ “Nada”: in lingua Croatica saepe est nomen femineum,
significans &quot;spem&quot;. In lingua Hispanica autem “nada” idem est ac &quot;nihil&quot;.

² Alusio ad Ulixem, qui Cyclopi Polyphemo dixit se vocari ''Οὖτις'', id est “Nemo”.

³ Reminiscentia fragmenti Archilochi:
&lt;poem&gt;
''Θῦμε, θῦμ᾽ ἀμήχανοισι κήδεσιν κυκώμενε,  
ἀνάστηθι καὶ ἀμύνου· ἔμπεδον δὲ θῦμον ἔχων… ''
&lt;/poem&gt;

⁴ Allusio ad tragoediam Sophocleam Oedipus Rex atque ad mythum de Sphinge.</text>
      <sha1>ezwqodt4xsbxqjbornv2jd4hsh3b3e3</sha1>
    </revision>
  </page>
  <page>
    <title>Pistolerus verus</title>
    <ns>0</ns>
    <id>84590</id>
    <revision>
      <id>281488</id>
      <parentid>274706</parentid>
      <timestamp>2026-09-20T17:59:13Z</timestamp>
      <contributor>
        <username>Coronus Tenebrosius</username>
        <id>31709</id>
      </contributor>
      <origin>281488</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="600" sha1="iabc9tdnadgyqgxujcpgtg69bahdrr2" xml:space="preserve">[[Carmina non adlecta]]

== PISTOLERUS VERUS ==

&lt;poem&gt;
1.
Cigaro fumo,
dynamitem iacio.
Ego sum banditus
Mexicanus.

2.
Sombrerus magnus,
umbo est latus,
parum nerecte inclinatus,
talis est eius status.

3.
Tequilam bibo, equum calco,
homo sum bonus,
asinum quendam percutio,
sine pardonus.

4.
Una femina in omni vico.
Pueri multi, quilibet - filius meus.
O gratias tibi ago,
omnipotens Deus!

5.
Poncho est latus,
sombrerus magnus,
Mexicanus sum natus,
- Pistolerus verus.

6.
Ubi cactus est,
umbra eius vera. 
Ibi requiescet
mens fessa mea. 
&lt;/poem&gt;


== Nota ==
In stilo goliardico scripta est.</text>
      <sha1>iabc9tdnadgyqgxujcpgtg69bahdrr2</sha1>
    </revision>
  </page>
  <page>
    <title>Hac nocte</title>
    <ns>0</ns>
    <id>84591</id>
    <revision>
      <id>281489</id>
      <parentid>274708</parentid>
      <timestamp>2026-09-20T18:00:11Z</timestamp>
      <contributor>
        <username>Coronus Tenebrosius</username>
        <id>31709</id>
      </contributor>
      <origin>281489</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="517" sha1="c9mjonqdc134rr33z12tm3cdi3plava" xml:space="preserve">[[Carmina non adlecta]]

== HAC NOCTE ==

&lt;poem&gt;
Nocte hac ego sum
canis qui in Lunam latrat.
Nocte hac ego sum
felis flava quae se insinuat
per desideria noctium tuarum
tenebrarum, inquietarum.

Nocte hac ego sum merula,
alauda et turtur
qui canit
sub fenestra tua.

Nocte hac ego sum
desiderium tuum inquietum,
vultus occultus.

Nocte hac ego sum desiderium
non effectum. Saxum vivum,
mutum, plenum quietis,
quod venit cum umbra piscium,
Lunam dorsis
equitans.

Oculi piscium, hac nocte,
implentur
lacrimis.
&lt;/poem&gt;</text>
      <sha1>c9mjonqdc134rr33z12tm3cdi3plava</sha1>
    </revision>
  </page>
  <page>
    <title>Pagina:Werkecarlf03gausrich.djvu/324</title>
    <ns>104</ns>
    <id>86109</id>
    <revision>
      <id>281394</id>
      <parentid>281085</parentid>
      <timestamp>2026-09-20T15:23:52Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <origin>281394</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="2469" sha1="o2qsoflm09bete1pm2p89afq1jn7aha" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;&lt;math display=&quot;block&quot;&gt;
\begin{align}
 &amp; +780,6 \\
 &amp; -411,0\cos x - 720,2\sin x \\
&amp;+\ 43,4\cos 2x-\ \ 2,2\sin 2x \\
&amp;-\ \ 4,3\cos 3x+\ \ 5,5 \sin 3x \\
&amp;-\ \ 1,1\cos 4x-\ \ 1,0\sin 4x \\
&amp;+\ \ 0,3\cos 5x-\ \ 0,3 \sin 5x \\
&amp;+\ \ 0,1\cos 6x
\end{align}
&lt;/math&gt; Distributamus secundo eandem periodum in quatuor periodos ternorum terminorum, quibus singulis itaque per formula primi ordinis &lt;math display=&quot;block&quot;&gt;
X'=\gamma+\gamma'\cos x+\delta'\sin x
&lt;/math&gt; satisfaciendum erit. Hic invenitur

{| 
|-
! Pro periodo
! ubi &lt;math display=&quot;inline&quot;&gt;y=3x&lt;/math&gt;
! &lt;math display=&quot;inline&quot;&gt;\gamma&lt;/math&gt;
! &lt;math display=&quot;inline&quot;&gt;\gamma'&lt;/math&gt;
! &lt;math display=&quot;inline&quot;&gt;\delta'&lt;/math&gt;
|-
| prima
| &lt;math display=&quot;inline&quot;&gt;0^{\circ}&lt;/math&gt;
| +776,3
| -368,3
| -718,8
|-
| secunda
| &lt;math display=&quot;inline&quot;&gt;90^{\circ}&lt;/math&gt;
| +786,0
| -414,5
| -676,0
|-
| tertia
| &lt;math display=&quot;inline&quot;&gt;180^{\circ}&lt;/math&gt;
| +785,0
| -453,0
| -721,1
|-
| quarta
| &lt;math display=&quot;inline&quot;&gt;270^{\circ}&lt;/math&gt;
| +775,0
| -408,2
| -765,0
|}

unde deducuntur formulae, sub quibus &lt;math display=&quot;inline&quot;&gt;\gamma,\gamma',\delta'&lt;/math&gt; exhibentur sequentes: &lt;math display=&quot;block&quot;&gt;
\begin{align}
&amp;\gamma\dots. +780,6-\ \ 4,3 \cos 3x+\ 5,5\sin 3x+0,1 \cos 6x \\
&amp;\gamma'\dots.-411,0+42,3\cos 3x- 3,2\sin 3x+ 0,3\cos 6x \\
&amp;\delta'\dots.-720,2+\ \ 1.2\cos 3x+44,5\sin 3x + 0,3\cos 6x
\end{align}
&lt;/math&gt; His valoribus in &lt;math display=&quot;inline&quot;&gt;\gamma+\gamma'\cos x + \delta'\sin x&lt;/math&gt; substitutis, partibus ultimis in &lt;math display=&quot;inline&quot;&gt;\gamma,\gamma',\delta'&lt;/math&gt;, quae &lt;math display=&quot;inline&quot;&gt;\cos 6x&lt;/math&gt; continent, secundum praecepta art. 26 tractatis, prorsus eadem formula eruitur, ad quam supra pervenimus.

&lt;br /&gt;
{{center|29.}}

Supersunt casus, ubi in functione &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt; vel cosinus vel sinus soli adsunt, quos in artt. 16, 17 generaliter tractavimus. Supponemus, ut in praecce., datam esse periodum &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; terminorum, puta &lt;math display=&quot;inline&quot;&gt;X = A, B, C, D&lt;/math&gt; etc. pro &lt;math display=&quot;inline&quot;&gt;x = a, b, c, d&lt;/math&gt; etc. Duo vero casus hic probe distinguendi sunt. Aliter enim tractari debet

&lt;math display=&quot;inline&quot;&gt;\qquad1^{\text{mo}}&lt;/math&gt; casus, ubi &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt; est complementum alicuius valorum sequentium &lt;math display=&quot;inline&quot;&gt;b, c, d&lt;/math&gt; etc. ad &lt;math display=&quot;inline&quot;&gt;360^{\circ}&lt;/math&gt; vel ad 0 vel ad multiplum peripheriae. Sit hic terminus &lt;math display=&quot;inline&quot;&gt;a + \frac{k}{\mu} 360^\circ&lt;/math&gt;,&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>o2qsoflm09bete1pm2p89afq1jn7aha</sha1>
    </revision>
    <revision>
      <id>281395</id>
      <parentid>281394</parentid>
      <timestamp>2026-09-20T15:36:46Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Emendata */</comment>
      <origin>281395</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="4753" sha1="qtkotdxmmpvq0tq00dy2pq405gncow0" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;3&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;&lt;math display=&quot;block&quot;&gt;
\begin{align}
 &amp; +780,6 \\
 &amp; -411,0\cos x - 720,2\sin x \\
&amp;+\ 43,4\cos 2x-\ \ 2,2\sin 2x \\
&amp;-\ \ 4,3\cos 3x+\ \ 5,5 \sin 3x \\
&amp;-\ \ 1,1\cos 4x-\ \ 1,0\sin 4x \\
&amp;+\ \ 0,3\cos 5x-\ \ 0,3 \sin 5x \\
&amp;+\ \ 0,1\cos 6x
\end{align}
&lt;/math&gt; Distributamus secundo eandem periodum in quatuor periodos ternorum terminorum, quibus singulis itaque per formula primi ordinis &lt;math display=&quot;block&quot;&gt;
X'=\gamma+\gamma'\cos x+\delta'\sin x
&lt;/math&gt; satisfaciendum erit. Hic invenitur

{| style=&quot;border-spacing:0px;&quot; align=&quot;center&quot;
|-
| style=&quot;border-right:1px solid black; border-bottom:1px solid black; text-align:center;&quot;|&lt;math display=&quot;inline&quot;&gt;\text{Pro periodo }&lt;/math&gt;
| style=&quot;border-right:1px solid black; border-bottom:1px solid black; text-align:center;&quot;|&lt;math display=&quot;inline&quot;&gt;\text{ ubi } y=3x\ &lt;/math&gt;
| style=&quot;border-right:1px solid black; border-bottom:1px solid black; text-align:center;&quot;|&lt;math display=&quot;inline&quot;&gt;\gamma&lt;/math&gt;
| style=&quot;border-right:1px solid black; border-bottom:1px solid black; text-align:center;&quot;|&lt;math display=&quot;inline&quot;&gt;\gamma'&lt;/math&gt;
| style=&quot;border-bottom:1px solid black; text-align:center;&quot;|&lt;math display=&quot;inline&quot;&gt;\delta'&lt;/math&gt;
|- style=&quot;text-align:center&quot;
| style=&quot;padding:0 10px 0 10px; text-align:left;border-right:1px solid black;&quot;|&lt;math display=&quot;inline&quot;&gt;\text{prima}&lt;/math&gt;
| style=&quot;padding:0 10px 0 10px;border-right:1px solid black;&quot;|&lt;math display=&quot;inline&quot;&gt;0^{\circ}&lt;/math&gt;
| style=&quot;padding:0 10px 0 10px;border-right:1px solid black;&quot;|&lt;math display=&quot;inline&quot;&gt;+776,3&lt;/math&gt;
| style=&quot;padding:0 10px 0 10px;border-right:1px solid black;&quot;|&lt;math display=&quot;inline&quot;&gt;-368,3&lt;/math&gt;
| style=&quot;padding:0 10px 0 10px;&quot;|&lt;math display=&quot;inline&quot;&gt;-718,8&lt;/math&gt;
|-style=&quot;text-align:center&quot;
| style=&quot;padding:0 10px 0 10px; text-align:left;border-right:1px solid black;&quot;|&lt;math display=&quot;inline&quot;&gt;\text{secunda}&lt;/math&gt;
| style=&quot;padding:0 10px 0 10px;border-right:1px solid black;&quot;|&lt;math display=&quot;inline&quot;&gt;90^{\circ}&lt;/math&gt;
| style=&quot;padding:0 10px 0 10px;border-right:1px solid black;&quot;|&lt;math display=&quot;inline&quot;&gt;+786,0&lt;/math&gt;
| style=&quot;padding:0 10px 0 10px;border-right:1px solid black;&quot;|&lt;math display=&quot;inline&quot;&gt;-414,5&lt;/math&gt;
| style=&quot;padding:0 10px 0 10px;&quot;|&lt;math display=&quot;inline&quot;&gt;-676,0&lt;/math&gt;
|-style=&quot;text-align:center&quot;
| style=&quot;padding:0 10px 0 10px; text-align:left;border-right:1px solid black;&quot;|&lt;math display=&quot;inline&quot;&gt;\text{tertia}&lt;/math&gt;
| style=&quot;padding:0 10px 0 10px;border-right:1px solid black;&quot;|&lt;math display=&quot;inline&quot;&gt;180^{\circ}&lt;/math&gt;
| style=&quot;padding:0 10px 0 10px;border-right:1px solid black;&quot;|&lt;math display=&quot;inline&quot;&gt;+785,0&lt;/math&gt;
| style=&quot;padding:0 10px 0 10px;border-right:1px solid black;&quot;|&lt;math display=&quot;inline&quot;&gt;-453,0&lt;/math&gt;
| style=&quot;padding:0 10px 0 10px;&quot;|&lt;math display=&quot;inline&quot;&gt;-721,1&lt;/math&gt;
|-style=&quot;text-align:center&quot;
| style=&quot;padding:0 10px 0 10px; text-align:left;border-right:1px solid black;&quot;|&lt;math display=&quot;inline&quot;&gt;\text{quarta}&lt;/math&gt;
| style=&quot;padding:0 10px 0 10px;border-right:1px solid black;&quot;|&lt;math display=&quot;inline&quot;&gt;270^{\circ}&lt;/math&gt;
| style=&quot;padding:0 10px 0 10px;border-right:1px solid black;&quot;|&lt;math display=&quot;inline&quot;&gt;+775,0&lt;/math&gt;
| style=&quot;padding:0 10px 0 10px;border-right:1px solid black;&quot;|&lt;math display=&quot;inline&quot;&gt;-408,2&lt;/math&gt;
| style=&quot;padding:0 10px 0 10px;&quot;|&lt;math display=&quot;inline&quot;&gt;-765,0&lt;/math&gt;
|}

unde deducuntur formulae, sub quibus &lt;math display=&quot;inline&quot;&gt;\gamma,\gamma',\delta'&lt;/math&gt; exhibentur sequentes: &lt;math display=&quot;block&quot;&gt;
\begin{align}
&amp;\gamma\ .\ .\ .\ .\ +780,6-\ \ 4,3 \cos 3x+\ 5,5\sin 3x+0,1 \cos 6x \\
&amp;\gamma'.\ .\ .\ .\ -411,0+42,3\cos 3x- 3,2\sin 3x+ 0,3\cos 6x \\
&amp;\delta'.\ .\ .\ .\ -720,2+\ \ 1,2\cos 3x+44,5\sin 3x + 0,3\cos 6x
\end{align}
&lt;/math&gt; His valoribus in &lt;math display=&quot;inline&quot;&gt;\gamma+\gamma'\cos x + \delta'\sin x&lt;/math&gt; substitutis, partibusque ultimis in &lt;math display=&quot;inline&quot;&gt;\gamma,\gamma',\delta'&lt;/math&gt;, quae &lt;math display=&quot;inline&quot;&gt;\cos 6x&lt;/math&gt; continent, secundum praecepta art. 26 tractatis, prorsus eadem formula eruitur, ad quam supra pervenimus.

&lt;br /&gt;
{{center|29.}}

Supersunt casus, ubi in functione &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt; vel cosinus vel sinus soli adsunt, quos in artt. 16, 17 generaliter tractavimus. Supponemus, ut in praecce., datam esse periodum &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; terminorum, puta &lt;math display=&quot;inline&quot;&gt;X = A, B, C, D&lt;/math&gt; etc. pro &lt;math display=&quot;inline&quot;&gt;x = a, b, c, d&lt;/math&gt; etc. Duo vero casus hic probe distinguendi sunt. Aliter enim tractari debet

&lt;math display=&quot;inline&quot;&gt;\qquad1^{\text{mo}}&lt;/math&gt; casus, ubi &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt; est complementum alicuius valorum sequentium &lt;math display=&quot;inline&quot;&gt;b, c, d&lt;/math&gt; etc. ad &lt;math display=&quot;inline&quot;&gt;360^{\circ}&lt;/math&gt; vel ad 0 vel ad multiplum peripheriae. Sit hic terminus &lt;math display=&quot;inline&quot;&gt;a + \frac{k}{\mu} 360^\circ&lt;/math&gt;,&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>qtkotdxmmpvq0tq00dy2pq405gncow0</sha1>
    </revision>
  </page>
  <page>
    <title>Pagina:Werkecarlf03gausrich.djvu/325</title>
    <ns>104</ns>
    <id>86110</id>
    <revision>
      <id>281396</id>
      <parentid>281086</parentid>
      <timestamp>2026-09-20T15:38:46Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <origin>281396</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="3873" sha1="42i7zi1ktfhrp8q0aha5kssj6pze3e1" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;atque &lt;math display=&quot;inline&quot;&gt;a + \frac{k}{\mu} 360^\circ + a = l \times 360^\circ&lt;/math&gt;, sive &lt;math display=&quot;inline&quot;&gt;a = (\mu l - k) \frac{180^\circ}{\mu}&lt;/math&gt;. Quare in hoc casu &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt; per &lt;math display=&quot;inline&quot;&gt;\frac{180^\circ}{\mu}&lt;/math&gt; divisibilis erit, adeoque &lt;math display=&quot;inline&quot;&gt;\sin \mu a = 0&lt;/math&gt;; porro habetur &lt;math display=&quot;inline&quot;&gt;b = (\mu l - k + 2) \frac{180^\circ}{\mu}&lt;/math&gt;, atque &lt;math display=&quot;inline&quot;&gt;\sin \mu b = 0&lt;/math&gt;, et perinde de sequentibus &lt;math display=&quot;inline&quot;&gt;c, d&lt;/math&gt; etc. Supponendo itaque, &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt; esse minorem quam &lt;math display=&quot;inline&quot;&gt;\frac{360^\circ}{\mu}&lt;/math&gt;, sed non negativum (quod permissum est, quum ab omnibus &lt;math display=&quot;inline&quot;&gt;a, b, c, d&lt;/math&gt; etc. multiplum proxime minus totius peripheriae subtrahere et ex residuis minimum, quod proprietate illa praeditum erit pro periodi initio accipere liceat), erit vel &lt;math display=&quot;inline&quot;&gt;a = 0&lt;/math&gt;, atque &lt;math display=&quot;inline&quot;&gt;b = \frac{1}{\mu} 360^\circ, c = \frac{2}{\mu} 360^\circ, d = \frac{3}{\mu} 360^\circ&lt;/math&gt; etc.; vel &lt;math display=&quot;inline&quot;&gt;a = \frac{1}{\mu} 180^\circ, b = \frac{3}{\mu} 180^\circ, c = \frac{5}{\mu} 180^\circ, d = \frac{7}{\mu} 180^\circ&lt;/math&gt; etc.

&lt;math display=&quot;inline&quot;&gt;\qquad\text{aliter }2^{\text{do}}&lt;/math&gt; casus, ubi non est complementum ullius valorum sequentium &lt;math display=&quot;inline&quot;&gt;b, c, d&lt;/math&gt; etc. ad &lt;math display=&quot;inline&quot;&gt;360^\circ&lt;/math&gt; vel 0 vel multiplum peripheriae. Hic &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt; non erit divisibilis per &lt;math display=&quot;inline&quot;&gt;\frac{180^\circ}{\mu}&lt;/math&gt;, adeoque &lt;math display=&quot;inline&quot;&gt;\sin \mu a = \sin \mu b = \sin \mu c = \sin \mu d&lt;/math&gt; etc. non &lt;math display=&quot;inline&quot;&gt;= 0&lt;/math&gt;.

In casu posteriore methodus generalis art. 16 statim applicari potest. In priore autem ex valoribus &lt;math display=&quot;inline&quot;&gt;a, b, c, d&lt;/math&gt; etc. (quos infra &lt;math display=&quot;inline&quot;&gt;360^\circ&lt;/math&gt; reductos esse supponimus), omnes antea reiicere oportet, qui sunt majores quam &lt;math display=&quot;inline&quot;&gt;180^\circ&lt;/math&gt;, quippe quorum complementa ad &lt;math display=&quot;inline&quot;&gt;360^\circ&lt;/math&gt; inter reliquos reperiuntur; insuperque, quando adsunt, valores 0 et &lt;math display=&quot;inline&quot;&gt;180^\circ&lt;/math&gt;, si in functione &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt; sinus soli occurruunt: sed quoniam applicatio methodi generalis hac ratione minus concinna evaderet, hunc casum alio modo infra absolvemus. Initium iam ab illo casu faciemus, ubi &lt;math display=&quot;inline&quot;&gt;\sin \mu a&lt;/math&gt; non est &lt;math display=&quot;inline&quot;&gt;= 0&lt;/math&gt;.

&lt;br /&gt;
{{center|30.}}

Sit primo &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt; functio formae &lt;math display=&quot;block&quot;&gt;
\alpha + \alpha' \cos x + \alpha'' \cos 2x + \alpha''' \cos 3x + \ldots + \alpha^n \cos nx
&lt;/math&gt; unde esse debet &lt;math display=&quot;inline&quot;&gt;\mu = n + 1&lt;/math&gt;. Hic habetur per lemma secundum art. 19 &lt;math display=&quot;block&quot;&gt;
\begin{align}
&amp;(\cos t - \cos b)(\cos t - \cos c)(\cos t - \cos d) \text{ etc.} = \frac{1}{2^{\mu-1}} \times \frac{\cos \mu t - \cos \mu a}{\cos t - \cos a}\\
&amp;= \frac{1}{2^{\mu-1} \sin a} \{ \sin \mu a + 2 \sin (\mu - 1)a \cos t + 2 \sin (\mu - 2)a \cos 2t + \text{etc.} + 2 \sin a \cos (\mu - 1)t \}
\end{align}
&lt;/math&gt; Facile scilicet confirmatur, productum ex aggregato secundae partis huius aequationis per &lt;math display=&quot;inline&quot;&gt;\cos t - \cos a&lt;/math&gt; fieri &lt;math display=&quot;inline&quot;&gt;= \sin a (\cos \mu t - \cos \mu a)&lt;/math&gt;. Quare fit quoque, scribendo &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt; pro &lt;math display=&quot;inline&quot;&gt;t&lt;/math&gt; &lt;math display=&quot;block&quot;&gt;
\begin{align}
(\cos a - \cos b)(\cos a - \cos c)(\cos a - \cos d) \text{ etc.} = \frac{1}{2^{\mu-1} \sin a} \{ \sin \mu a + 2 \sin (\mu - 1)a \cos a \\
+ 2 \sin (\mu - 2)a \cos 2a + \text{etc.} + 2 \sin a \cos (\mu - 1)a \}
\end{align}
&lt;/math&gt; Aggregatum in hac expressione fit &lt;math display=&quot;inline&quot;&gt;= \mu \sin \mu a&lt;/math&gt;, ut inde manifestum est, quod&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>42i7zi1ktfhrp8q0aha5kssj6pze3e1</sha1>
    </revision>
    <revision>
      <id>281397</id>
      <parentid>281396</parentid>
      <timestamp>2026-09-20T15:41:10Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <origin>281397</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="3872" sha1="667v3ajq1nry36r79w5g2o4zptldf4m" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;atque &lt;math display=&quot;inline&quot;&gt;a + \frac{k}{\mu} 360^\circ + a = l \times 360^\circ&lt;/math&gt;, sive &lt;math display=&quot;inline&quot;&gt;a = (\mu l - k) \frac{180^\circ}{\mu}&lt;/math&gt;. Quare in hoc casu &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt; per &lt;math display=&quot;inline&quot;&gt;\frac{180^\circ}{\mu}&lt;/math&gt; divisibilis erit, adeoque &lt;math display=&quot;inline&quot;&gt;\sin \mu a = 0&lt;/math&gt;; porro habetur &lt;math display=&quot;inline&quot;&gt;b = (\mu l - k + 2) \frac{180^\circ}{\mu}&lt;/math&gt;, atque &lt;math display=&quot;inline&quot;&gt;\sin \mu b = 0&lt;/math&gt;, et perinde de sequentibus &lt;math display=&quot;inline&quot;&gt;c, d&lt;/math&gt; etc. Supponendo itaque, &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt; esse minorem quam &lt;math display=&quot;inline&quot;&gt;\frac{360^\circ}{\mu}&lt;/math&gt;, sed non negativum (quod permissum est, quum ab omnibus &lt;math display=&quot;inline&quot;&gt;a, b, c, d&lt;/math&gt; etc. multiplum proxime minus totius peripheriae subtrahere et ex residuis minimum, quod proprietate illa praeditum erit pro periodi initio accipere liceat), erit vel &lt;math display=&quot;inline&quot;&gt;a = 0&lt;/math&gt;, atque &lt;math display=&quot;inline&quot;&gt;b = \frac{1}{\mu} 360^\circ, c = \frac{2}{\mu} 360^\circ, d = \frac{3}{\mu} 360^\circ&lt;/math&gt; etc.; vel &lt;math display=&quot;inline&quot;&gt;a = \frac{1}{\mu} 180^\circ, b = \frac{3}{\mu} 180^\circ, c = \frac{5}{\mu} 180^\circ, d = \frac{7}{\mu} 180^\circ&lt;/math&gt; etc.

&lt;math display=&quot;inline&quot;&gt;\qquad\text{aliter }2^{\text{do}}&lt;/math&gt; casus, ubi non est complementum ullius valorum sequentium &lt;math display=&quot;inline&quot;&gt;b, c, d&lt;/math&gt; etc. ad &lt;math display=&quot;inline&quot;&gt;360^\circ&lt;/math&gt; vel 0 vel multiplum peripheriae. Hic &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt; non erit divisibilis per &lt;math display=&quot;inline&quot;&gt;\frac{180^\circ}{\mu}&lt;/math&gt;, adeoque &lt;math display=&quot;inline&quot;&gt;\sin \mu a = \sin \mu b = \sin \mu c = \sin \mu d&lt;/math&gt; etc. non &lt;math display=&quot;inline&quot;&gt;= 0&lt;/math&gt;.

In casu posteriore methodus generalis art. 16 statim applicari potest. In priore autem ex valoribus &lt;math display=&quot;inline&quot;&gt;a, b, c, d&lt;/math&gt; etc. (quos infra &lt;math display=&quot;inline&quot;&gt;360^\circ&lt;/math&gt; reductos esse supponimus), omnes antea reiicere oportet, qui sunt majores quam &lt;math display=&quot;inline&quot;&gt;180^\circ&lt;/math&gt;, quippe quorum complementa ad &lt;math display=&quot;inline&quot;&gt;360^\circ&lt;/math&gt; inter reliquos reperiuntur; insuperque, quando adsunt, valores 0 et &lt;math display=&quot;inline&quot;&gt;180^\circ&lt;/math&gt;, si in functione &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt; sinus soli occurrunt: sed quoniam applicatio methodi generalis hac ratione minus concinna evaderet, hunc casum alio modo infra absolvemus. Initium iam ab illo casu faciemus, ubi &lt;math display=&quot;inline&quot;&gt;\sin \mu a&lt;/math&gt; non est &lt;math display=&quot;inline&quot;&gt;= 0&lt;/math&gt;.

&lt;br /&gt;
{{center|30.}}

Sit primo &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt; functio formae &lt;math display=&quot;block&quot;&gt;
\alpha + \alpha' \cos x + \alpha'' \cos 2x + \alpha''' \cos 3x + \ldots + \alpha^n \cos nx
&lt;/math&gt; unde esse debet &lt;math display=&quot;inline&quot;&gt;\mu = n + 1&lt;/math&gt;. Hic habetur per lemma secundum art. 19 &lt;math display=&quot;block&quot;&gt;
\begin{align}
&amp;(\cos t - \cos b)(\cos t - \cos c)(\cos t - \cos d) \text{ etc.} = \frac{1}{2^{\mu-1}} \times \frac{\cos \mu t - \cos \mu a}{\cos t - \cos a}\\
&amp;= \frac{1}{2^{\mu-1} \sin a} \{ \sin \mu a + 2 \sin (\mu - 1)a \cos t + 2 \sin (\mu - 2)a \cos 2t + \text{etc.} + 2 \sin a \cos (\mu - 1)t \}
\end{align}
&lt;/math&gt; Facile scilicet confirmatur, productum ex aggregato secundae partis huius aequationis per &lt;math display=&quot;inline&quot;&gt;\cos t - \cos a&lt;/math&gt; fieri &lt;math display=&quot;inline&quot;&gt;= \sin a (\cos \mu t - \cos \mu a)&lt;/math&gt;. Quare fit quoque, scribendo &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt; pro &lt;math display=&quot;inline&quot;&gt;t&lt;/math&gt; &lt;math display=&quot;block&quot;&gt;
\begin{align}
(\cos a - \cos b)(\cos a - \cos c)(\cos a - \cos d) \text{ etc.} = \frac{1}{2^{\mu-1} \sin a} \{ \sin \mu a + 2 \sin (\mu - 1)a \cos a \\
+ 2 \sin (\mu - 2)a \cos 2a + \text{etc.} + 2 \sin a \cos (\mu - 1)a \}
\end{align}
&lt;/math&gt; Aggregatum in hac expressione fit &lt;math display=&quot;inline&quot;&gt;= \mu \sin \mu a&lt;/math&gt;, ut inde manifestum est, quod&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>667v3ajq1nry36r79w5g2o4zptldf4m</sha1>
    </revision>
    <revision>
      <id>281398</id>
      <parentid>281397</parentid>
      <timestamp>2026-09-20T15:43:27Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Emendata */</comment>
      <origin>281398</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="3874" sha1="sj8uft3l5qo49br684yl85vgmmmicmr" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;3&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;atque &lt;math display=&quot;inline&quot;&gt;a + \frac{k}{\mu} 360^\circ + a = l \times 360^\circ&lt;/math&gt;, sive &lt;math display=&quot;inline&quot;&gt;a = (\mu l - k) \frac{180^\circ}{\mu}&lt;/math&gt;. Quare in hoc casu &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt; per &lt;math display=&quot;inline&quot;&gt;\frac{180^\circ}{\mu}&lt;/math&gt; divisibilis erit, adeoque &lt;math display=&quot;inline&quot;&gt;\sin \mu a = 0&lt;/math&gt;; porro habetur &lt;math display=&quot;inline&quot;&gt;b = (\mu l - k + 2) \frac{180^\circ}{\mu}&lt;/math&gt;, atque &lt;math display=&quot;inline&quot;&gt;\sin \mu b = 0&lt;/math&gt;, et perinde de sequentibus &lt;math display=&quot;inline&quot;&gt;c, d&lt;/math&gt; etc. Supponendo itaque, &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt; esse minorem quam &lt;math display=&quot;inline&quot;&gt;\frac{360^\circ}{\mu}&lt;/math&gt;, sed non negativum (quod permissum est, quum ab omnibus &lt;math display=&quot;inline&quot;&gt;a, b, c, d&lt;/math&gt; etc. multiplum proxime minus totius peripheriae subtrahere et ex residuis minimum, quod proprietate illa praeditum erit pro periodi initio accipere liceat), erit vel &lt;math display=&quot;inline&quot;&gt;a = 0&lt;/math&gt;, atque &lt;math display=&quot;inline&quot;&gt;b = \frac{1}{\mu} 360^\circ, c = \frac{2}{\mu} 360^\circ, d = \frac{3}{\mu} 360^\circ&lt;/math&gt; etc.; vel &lt;math display=&quot;inline&quot;&gt;a = \frac{1}{\mu} 180^\circ, b = \frac{3}{\mu} 180^\circ, c = \frac{5}{\mu} 180^\circ, d = \frac{7}{\mu} 180^\circ&lt;/math&gt; etc.

&lt;math display=&quot;inline&quot;&gt;\qquad\text{aliter }2^{\text{do}}&lt;/math&gt; casus, ubi non est complementum ullius valorum sequentium &lt;math display=&quot;inline&quot;&gt;b, c, d&lt;/math&gt; etc. ad &lt;math display=&quot;inline&quot;&gt;360^\circ&lt;/math&gt; vel 0 vel multiplum peripheriae. Hic &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt; non erit divisibilis per &lt;math display=&quot;inline&quot;&gt;\frac{180^\circ}{\mu}&lt;/math&gt;, adeoque &lt;math display=&quot;inline&quot;&gt;\sin \mu a = \sin \mu b = \sin \mu c = \sin \mu d&lt;/math&gt; etc. non &lt;math display=&quot;inline&quot;&gt;= 0&lt;/math&gt;.

In casu posteriore methodus generalis art. 16 statim applicari potest. In priore autem ex valoribus &lt;math display=&quot;inline&quot;&gt;a, b, c, d&lt;/math&gt; etc. (quos infra &lt;math display=&quot;inline&quot;&gt;360^\circ&lt;/math&gt; reductos esse supponimus), omnes antea reiicere oportet, qui sunt majores quam &lt;math display=&quot;inline&quot;&gt;180^\circ&lt;/math&gt;, quippe quorum complementa ad &lt;math display=&quot;inline&quot;&gt;360^\circ&lt;/math&gt; inter reliquos reperiuntur; insuperque, quando adsunt, valores 0 et &lt;math display=&quot;inline&quot;&gt;180^\circ&lt;/math&gt;, si in functione &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt; sinus soli occurrunt: sed quoniam applicatio methodi generalis hac ratione minus concinna evaderet, hunc casum alio modo infra absolvemus. Initium iam ab illo casu faciemus, ubi &lt;math display=&quot;inline&quot;&gt;\sin \mu a&lt;/math&gt; non est &lt;math display=&quot;inline&quot;&gt;= 0&lt;/math&gt;.

&lt;br /&gt;
{{center|30.}}

Sit primo &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt; functio formae &lt;math display=&quot;block&quot;&gt;
\alpha + \alpha' \cos x + \alpha'' \cos 2x + \alpha''' \cos 3x + \ldots + \alpha^n \cos nx
&lt;/math&gt; unde esse debebit &lt;math display=&quot;inline&quot;&gt;\mu = n + 1&lt;/math&gt;. Hic habetur per lemma secundum art. 19 &lt;math display=&quot;block&quot;&gt;
\begin{align}
&amp;(\cos t - \cos b)(\cos t - \cos c)(\cos t - \cos d) \text{ etc.} = \frac{1}{2^{\mu-1}} \times \frac{\cos \mu t - \cos \mu a}{\cos t - \cos a}\\
&amp;= \frac{1}{2^{\mu-1} \sin a} \{ \sin \mu a + 2 \sin (\mu - 1)a \cos t + 2 \sin (\mu - 2)a \cos 2t + \text{etc.} + 2 \sin a \cos (\mu - 1)t \}
\end{align}
&lt;/math&gt; Facile scilicet confirmatur, productum ex aggregato secundae partis huius aequationis per &lt;math display=&quot;inline&quot;&gt;\cos t - \cos a&lt;/math&gt; fieri &lt;math display=&quot;inline&quot;&gt;= \sin a (\cos \mu t - \cos \mu a)&lt;/math&gt;. Quare fit quoque, scribendo &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt; pro &lt;math display=&quot;inline&quot;&gt;t&lt;/math&gt; &lt;math display=&quot;block&quot;&gt;
\begin{align}
(\cos a - \cos b)(\cos a - \cos c)(\cos a - \cos d) \text{ etc.} = \frac{1}{2^{\mu-1} \sin a} \{ \sin \mu a + 2 \sin (\mu - 1)a \cos a \\
+ 2 \sin (\mu - 2)a \cos 2a + \text{etc.} + 2 \sin a \cos (\mu - 1)a \}
\end{align}
&lt;/math&gt; Aggregatum in hac expressione fit &lt;math display=&quot;inline&quot;&gt;= \mu \sin \mu a&lt;/math&gt;, ut inde manifestum est, quod&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>sj8uft3l5qo49br684yl85vgmmmicmr</sha1>
    </revision>
  </page>
  <page>
    <title>Pagina:Werkecarlf03gausrich.djvu/326</title>
    <ns>104</ns>
    <id>86111</id>
    <revision>
      <id>281400</id>
      <parentid>281350</parentid>
      <timestamp>2026-09-20T15:47:41Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Emendata */</comment>
      <origin>281400</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="2907" sha1="qbe6wpfpdz0kn21lmafg8s8dtpuc4qr" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;3&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;&lt;math display=&quot;block&quot;&gt;
\begin{align}
\text{terminus secundus} = \sin \mu a + \sin (\mu - 2)a\\
\text{ultimus} = \sin \mu a - \sin (\mu - 2)a\\
\text{porro terminus tertius} = \sin \mu a + \sin (\mu - 4)a\\
\text{terminus penultimus} = \sin \mu a - \sin (\mu - 4)a\\
\text{etc.}\qquad\qquad\qquad\qquad\qquad\qquad
\end{align}
&lt;/math&gt; Hinc colligitur, coëfficientem ipsius &lt;math display=&quot;inline&quot;&gt;A&lt;/math&gt; in formula art. 16 pro &lt;math display=&quot;inline&quot;&gt;T&lt;/math&gt; fieri &lt;math display=&quot;block&quot;&gt;
\frac{1}{\mu \sin \mu a} \{ \sin \mu a + 2 \sin (\mu - 1)a \cos t + 2 \sin (\mu - 2)a \cos 2t + \text{etc.} + 2 \sin a \sin (\mu - 1)t \}
&lt;/math&gt; Prorsus similes expressiones proveniunt pro coëfficientibus ipsorum &lt;math display=&quot;inline&quot;&gt;B, C, D&lt;/math&gt; etc., mutando tantummodo &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt; in &lt;math display=&quot;inline&quot;&gt;b, c, d&lt;/math&gt; etc. Quamobrem statuendo &lt;math display=&quot;block&quot;&gt;
\begin{align}
\varepsilon &amp;= \frac{1}{\mu} (A + B + C + D + \text{etc.}) \\
\varepsilon' &amp;= \frac{2}{\mu \sin \mu a} \{ A \sin (\mu - 1)a + B \sin (\mu - 1)b + C \sin (\mu - 1)c + D \sin (\mu - 1)d + \text{etc.} \} \\
\varepsilon'' &amp;= \frac{2}{\mu \sin \mu a} \{ A \sin (\mu - 2)a + B \sin (\mu - 2)b + C \sin (\mu - 2)c + D \sin (\mu - 2)d + \text{etc.} \} \\
\varepsilon''' &amp;= \frac{2}{\mu \sin \mu a} \{ A \sin (\mu - 3)a + B \sin (\mu - 3)b + C \sin (\mu - 3)c + D \sin (\mu - 3)d + \text{etc.} \} \\
&amp;\text{etc.} \\
\varepsilon^n &amp;= \frac{2}{\mu \sin \mu a} (A \sin a + B \sin b + C \sin c + D \sin d + \text{etc.})
\end{align}
&lt;/math&gt; erit &lt;math display=&quot;block&quot;&gt;
T = \varepsilon + \varepsilon' \cos t + \varepsilon'' \cos 2t + \varepsilon''' \cos 3t + \text{etc.} + \varepsilon^n \cos n t
&lt;/math&gt; Quum haec formula generaliter pro valore quocunque ipsius &lt;math display=&quot;inline&quot;&gt;t&lt;/math&gt; valere debeat, necessario cum &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt; identica fiet, mutata &lt;math display=&quot;inline&quot;&gt;t&lt;/math&gt; in &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;, adeoque coëfficientes &lt;math display=&quot;inline&quot;&gt;\varepsilon, \varepsilon', \varepsilon''&lt;/math&gt; etc. ipsis &lt;math display=&quot;inline&quot;&gt;\alpha, \alpha', \alpha''&lt;/math&gt; etc. resp. aequales.

&lt;br /&gt;
{{center|31.}}

Quum per praecepta supra explicata functio &lt;math display=&quot;inline&quot;&gt;X'&lt;/math&gt; formae &lt;math display=&quot;block&quot;&gt;
\begin{align}
\gamma + \gamma' \cos x + \gamma'' \cos 2x + \ldots + \gamma^m \cos mx \\
+ \delta' \sin x + \delta'' \sin 2x + \ldots + \delta^m \sin mx
\end{align}
&lt;/math&gt; quae omnibus &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; valoribus propositis satisfaciat, et in qua sit &lt;math display=&quot;inline&quot;&gt;m = \frac{1}{2}\mu - \frac{1}{2}&lt;/math&gt; vel &lt;math display=&quot;inline&quot;&gt;\frac{1}{2}\mu&lt;/math&gt;, prout &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; impar est vel par: operae pretium est, hanc functionem cum functione modo inventa &lt;math display=&quot;block&quot;&gt;
\varepsilon + \varepsilon' \cos x + \varepsilon'' \cos 2x + \varepsilon''' \cos 3x + \ldots + \varepsilon^{\mu-1} \cos (\mu - 1)x
&lt;/math&gt;&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>qbe6wpfpdz0kn21lmafg8s8dtpuc4qr</sha1>
    </revision>
  </page>
  <page>
    <title>Pagina:Werkecarlf03gausrich.djvu/327</title>
    <ns>104</ns>
    <id>86112</id>
    <revision>
      <id>281401</id>
      <parentid>281351</parentid>
      <timestamp>2026-09-20T15:50:46Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <origin>281401</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="3517" sha1="cp05j137b9qh983arcajeifuf0iscfi" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;comparare, quam per &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt; exprimemus. Comparando valores coëfficientium &lt;math display=&quot;inline&quot;&gt;\gamma, \gamma', \delta', \gamma'', \delta''&lt;/math&gt; etc. in artt. 20, 22 traditos cum valoribus coëfficientium &lt;math display=&quot;inline&quot;&gt;\varepsilon, \varepsilon', \varepsilon''&lt;/math&gt; etc. modo inventis, invenitur

&lt;math display=&quot;block&quot;&gt;
\begin{align}
&amp;\varepsilon = \gamma \\
&amp;\varepsilon' = \gamma' - \delta' \text{cotang } \mu a, \quad \ \ \varepsilon^{\mu-1} = \delta' \text{cosec } \mu a \\
&amp;\varepsilon'' = \gamma'' - \delta'' \text{cotang } \mu a, \ \quad \varepsilon^{\mu-2} = \delta'' \text{cosec } \mu a \\
&amp;\varepsilon''' = \gamma''' - \delta''' \text{cotang } \mu a, \quad \varepsilon^{\mu-3} = \delta''' \text{cosec } \mu a \\
&amp;\text{etc. usque ad} \\
&amp;(\tfrac{1}{2}) \varepsilon^m = \gamma^m - \delta^m \text{cotang } \mu a, \quad (\tfrac{1}{2}) \varepsilon^{\mu-m} = \delta^m \text{cosec } \mu a
\end{align}
&lt;/math&gt;

Factor &lt;math display=&quot;inline&quot;&gt;\frac{1}{2}&lt;/math&gt; ipsius &lt;math display=&quot;inline&quot;&gt;\varepsilon^m&lt;/math&gt; et &lt;math display=&quot;inline&quot;&gt;\varepsilon^{\mu-m}&lt;/math&gt; praepositus tunc tantummodo valet, quando &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; par est, omittique debet, quando &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; est impar. Quo pacto, in casu posteriore, ubi &lt;math display=&quot;inline&quot;&gt;\mu - m = m + 1&lt;/math&gt; manifesto omnes &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; coëfficientes &lt;math display=&quot;inline&quot;&gt;\varepsilon, \varepsilon', \varepsilon''&lt;/math&gt; etc. per &lt;math display=&quot;inline&quot;&gt;\gamma, \gamma', \delta'&lt;/math&gt; etc. determinati sunt; in priore vero, ubi &lt;math display=&quot;inline&quot;&gt;m = \mu - m&lt;/math&gt;, pro coëfficiente &lt;math display=&quot;inline&quot;&gt;\varepsilon^{\frac{1}{2}\mu}&lt;/math&gt; valores duos &lt;math display=&quot;inline&quot;&gt;2 \gamma^{\frac{1}{2}\mu} - 2 \delta^{\frac{1}{2}\mu} \cot \mu a, \ 2 \delta^{\frac{1}{2}\mu} \text{cosec } \mu a&lt;/math&gt; habemus, quorum aequalitatem ex aequatione &lt;math display=&quot;inline&quot;&gt;\gamma^{\frac{1}{2}\mu} \sin \frac{1}{2} \mu a = \delta^{\frac{1}{2}\mu} \cos \frac{1}{2} \mu a&lt;/math&gt; facile perspicere, et pro quibus igitur etiam hunc &lt;math display=&quot;inline&quot;&gt;\gamma^{\frac{1}{2}\mu} - \delta^{\frac{1}{2}\mu} \text{cotg } \mu a + \delta^{\frac{1}{2}\mu} \text{cosec } \mu a&lt;/math&gt; adoptare possumus.

Hinc facile deducitur, esse in utroque casu &lt;math display=&quot;block&quot;&gt;
\begin{align}
X'' - (X' - \delta' \sin x - \delta'' \sin 2x - \delta^m \sin 3x - \ldots - \delta^m \sin mx) \\
= - \text{cotg } \mu a (\delta' \cos x + \delta'' \cos 2x + \delta^m \cos 3x + \ldots + \delta^m \cos mx) \\
+ \text{cosec } \mu a (\delta' \cos (\mu - 1)x + \delta'' \cos (\mu - 2)x + \ldots + \delta^m \cos (\mu - m)x)
\end{align}
&lt;/math&gt; Hinc patet, &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt; ex &lt;math display=&quot;inline&quot;&gt;X'&lt;/math&gt; deduci, si pro quovis termino &lt;math display=&quot;inline&quot;&gt;\delta^\lambda \sin \lambda x&lt;/math&gt; substituatur &lt;math display=&quot;block&quot;&gt;
-\text{cotg } \mu a \ \delta^\lambda \cos \lambda x + \text{cosec } \mu a \ \delta^\lambda \cos (\mu - \lambda)x
&lt;/math&gt; sive &lt;math display=&quot;block&quot;&gt;
\frac{-\delta^\lambda \cos \mu a \cos \lambda x + \delta^\lambda \cos (\mu - \lambda)x}{\sin \mu a}
&lt;/math&gt; cuius praecepti comparationem cum iis, quae in art. 24 explicata sunt, lectoribus linquimus.

&lt;br /&gt;
{{center|32.}}

Si functio &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt; cum &lt;math display=&quot;inline&quot;&gt;\cos n x&lt;/math&gt; non abrumpitur, sed ulterius excurrit, terminis sequentibus per &lt;math display=&quot;inline&quot;&gt;\alpha^\mu \cos \mu x + \alpha^{\mu+1} \cos (\mu + 1)x + \ldots&lt;/math&gt; etc. expressis, habebimus&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>cp05j137b9qh983arcajeifuf0iscfi</sha1>
    </revision>
    <revision>
      <id>281402</id>
      <parentid>281401</parentid>
      <timestamp>2026-09-20T15:53:54Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <origin>281402</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="3518" sha1="62wls6zddk8ix9ckbvs51hdvtuvtnel" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;comparare, quam per &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt; exprimemus. Comparando valores coëfficientium &lt;math display=&quot;inline&quot;&gt;\gamma, \gamma', \delta', \gamma'', \delta''&lt;/math&gt; etc. in artt. 20, 22 traditos cum valoribus coëfficientium &lt;math display=&quot;inline&quot;&gt;\varepsilon, \varepsilon', \varepsilon''&lt;/math&gt; etc. modo inventis, invenitur

&lt;math display=&quot;block&quot;&gt;
\begin{align}
&amp;\varepsilon = \gamma \\
&amp;\varepsilon' = \gamma' - \delta' \text{cotang } \mu a, \quad \ \ \varepsilon^{\mu-1} = \delta' \text{cosec } \mu a \\
&amp;\varepsilon'' = \gamma'' - \delta'' \text{cotang } \mu a, \ \quad \varepsilon^{\mu-2} = \delta'' \text{cosec } \mu a \\
&amp;\varepsilon''' = \gamma''' - \delta''' \text{cotang } \mu a, \quad \varepsilon^{\mu-3} = \delta''' \text{cosec } \mu a \\
&amp;\text{etc. usque ad} \\
&amp;(\tfrac{1}{2}) \varepsilon^m = \gamma^m - \delta^m \text{cotang } \mu a, \quad (\tfrac{1}{2}) \varepsilon^{\mu-m} = \delta^m \text{cosec } \mu a
\end{align}
&lt;/math&gt;

Factor &lt;math display=&quot;inline&quot;&gt;\frac{1}{2}&lt;/math&gt; ipsius &lt;math display=&quot;inline&quot;&gt;\varepsilon^m&lt;/math&gt; et &lt;math display=&quot;inline&quot;&gt;\varepsilon^{\mu-m}&lt;/math&gt; praepositus tunc tantummodo valet, quando &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; par est, omittique debet, quando &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; est impar. Quo pacto, in casu posteriore, ubi &lt;math display=&quot;inline&quot;&gt;\mu - m = m + 1&lt;/math&gt; manifesto omnes &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; coëfficientes &lt;math display=&quot;inline&quot;&gt;\varepsilon, \varepsilon', \varepsilon''&lt;/math&gt; etc. per &lt;math display=&quot;inline&quot;&gt;\gamma, \gamma', \delta'&lt;/math&gt; etc. determinati sunt; in priore vero, ubi &lt;math display=&quot;inline&quot;&gt;m = \mu - m&lt;/math&gt;, pro coëfficiente &lt;math display=&quot;inline&quot;&gt;\varepsilon^{\frac{1}{2}\mu}&lt;/math&gt; valores duos &lt;math display=&quot;inline&quot;&gt;2 \gamma^{\frac{1}{2}\mu} - 2 \delta^{\frac{1}{2}\mu} \cot \mu a, \ 2 \delta^{\frac{1}{2}\mu} \text{cosec } \mu a&lt;/math&gt; habemus, quorum aequalitatem ex aequatione &lt;math display=&quot;inline&quot;&gt;\gamma^{\frac{1}{2}\mu} \sin \frac{1}{2} \mu a = \delta^{\frac{1}{2}\mu} \cos \frac{1}{2} \mu a&lt;/math&gt; facile perspicere, et pro quibus igitur etiam hunc &lt;math display=&quot;inline&quot;&gt;\gamma^{\frac{1}{2}\mu} - \delta^{\frac{1}{2}\mu} \text{cotg } \mu a + \delta^{\frac{1}{2}\mu} \text{cosec } \mu a&lt;/math&gt; adoptare possumus.

Hinc facile deducitur, esse in utroque casu &lt;math display=&quot;block&quot;&gt;
\begin{align}
X'' - (X' - \delta' \sin x - \delta'' \sin 2x - \delta''' \sin 3x - \ldots - \delta^m \sin mx) \\
= - \text{cotg } \mu a (\delta' \cos x + \delta'' \cos 2x + \delta^m \cos 3x + \ldots + \delta^m \cos mx) \\
+ \text{cosec } \mu a (\delta' \cos (\mu - 1)x + \delta'' \cos (\mu - 2)x + \ldots + \delta^m \cos (\mu - m)x)
\end{align}
&lt;/math&gt; Hinc patet, &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt; ex &lt;math display=&quot;inline&quot;&gt;X'&lt;/math&gt; deduci, si pro quovis termino &lt;math display=&quot;inline&quot;&gt;\delta^\lambda \sin \lambda x&lt;/math&gt; substituatur &lt;math display=&quot;block&quot;&gt;
-\text{cotg } \mu a \ \delta^\lambda \cos \lambda x + \text{cosec } \mu a \ \delta^\lambda \cos (\mu - \lambda)x
&lt;/math&gt; sive &lt;math display=&quot;block&quot;&gt;
\frac{-\delta^\lambda \cos \mu a \cos \lambda x + \delta^\lambda \cos (\mu - \lambda)x}{\sin \mu a}
&lt;/math&gt; cuius praecepti comparationem cum iis, quae in art. 24 explicata sunt, lectoribus linquimus.

&lt;br /&gt;
{{center|32.}}

Si functio &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt; cum &lt;math display=&quot;inline&quot;&gt;\cos n x&lt;/math&gt; non abrumpitur, sed ulterius excurrit, terminis sequentibus per &lt;math display=&quot;inline&quot;&gt;\alpha^\mu \cos \mu x + \alpha^{\mu+1} \cos (\mu + 1)x + \ldots&lt;/math&gt; etc. expressis, habebimus&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>62wls6zddk8ix9ckbvs51hdvtuvtnel</sha1>
    </revision>
    <revision>
      <id>281403</id>
      <parentid>281402</parentid>
      <timestamp>2026-09-20T15:55:22Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <origin>281403</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="3519" sha1="qwt7ogjleq1wkvf0bne9lwhm8uwgvnt" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;comparare, quam per &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt; exprimemus. Comparando valores coëfficientium &lt;math display=&quot;inline&quot;&gt;\gamma, \gamma', \delta', \gamma'', \delta''&lt;/math&gt; etc. in artt. 20, 22 traditos cum valoribus coëfficientium &lt;math display=&quot;inline&quot;&gt;\varepsilon, \varepsilon', \varepsilon''&lt;/math&gt; etc. modo inventis, invenitur

&lt;math display=&quot;block&quot;&gt;
\begin{align}
&amp;\varepsilon = \gamma \\
&amp;\varepsilon' = \gamma' - \delta' \text{cotang } \mu a, \quad \ \ \varepsilon^{\mu-1} = \delta' \text{cosec } \mu a \\
&amp;\varepsilon'' = \gamma'' - \delta'' \text{cotang } \mu a, \ \quad \varepsilon^{\mu-2} = \delta'' \text{cosec } \mu a \\
&amp;\varepsilon''' = \gamma''' - \delta''' \text{cotang } \mu a, \quad \varepsilon^{\mu-3} = \delta''' \text{cosec } \mu a \\
&amp;\text{etc. usque ad} \\
&amp;(\tfrac{1}{2}) \varepsilon^m = \gamma^m - \delta^m \text{cotang } \mu a, \quad (\tfrac{1}{2}) \varepsilon^{\mu-m} = \delta^m \text{cosec } \mu a
\end{align}
&lt;/math&gt;

Factor &lt;math display=&quot;inline&quot;&gt;\frac{1}{2}&lt;/math&gt; ipsius &lt;math display=&quot;inline&quot;&gt;\varepsilon^m&lt;/math&gt; et &lt;math display=&quot;inline&quot;&gt;\varepsilon^{\mu-m}&lt;/math&gt; praepositus tunc tantummodo valet, quando &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; par est, omittique debet, quando &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; est impar. Quo pacto, in casu posteriore, ubi &lt;math display=&quot;inline&quot;&gt;\mu - m = m + 1&lt;/math&gt; manifesto omnes &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; coëfficientes &lt;math display=&quot;inline&quot;&gt;\varepsilon, \varepsilon', \varepsilon''&lt;/math&gt; etc. per &lt;math display=&quot;inline&quot;&gt;\gamma, \gamma', \delta'&lt;/math&gt; etc. determinati sunt; in priore vero, ubi &lt;math display=&quot;inline&quot;&gt;m = \mu - m&lt;/math&gt;, pro coëfficiente &lt;math display=&quot;inline&quot;&gt;\varepsilon^{\frac{1}{2}\mu}&lt;/math&gt; valores duos &lt;math display=&quot;inline&quot;&gt;2 \gamma^{\frac{1}{2}\mu} - 2 \delta^{\frac{1}{2}\mu} \cot \mu a, \ 2 \delta^{\frac{1}{2}\mu} \text{cosec } \mu a&lt;/math&gt; habemus, quorum aequalitatem ex aequatione &lt;math display=&quot;inline&quot;&gt;\gamma^{\frac{1}{2}\mu} \sin \frac{1}{2} \mu a = \delta^{\frac{1}{2}\mu} \cos \frac{1}{2} \mu a&lt;/math&gt; facile perspicere, et pro quibus igitur etiam hunc &lt;math display=&quot;inline&quot;&gt;\gamma^{\frac{1}{2}\mu} - \delta^{\frac{1}{2}\mu} \text{cotg } \mu a + \delta^{\frac{1}{2}\mu} \text{cosec } \mu a&lt;/math&gt; adoptare possumus.

Hinc facile deducitur, esse in utroque casu &lt;math display=&quot;block&quot;&gt;
\begin{align}
X'' - (X' - \delta' \sin x - \delta'' \sin 2x - \delta''' \sin 3x - \ldots - \delta^m \sin mx) \\
= - \text{cotg } \mu a (\delta' \cos x + \delta'' \cos 2x + \delta''' \cos 3x + \ldots + \delta^m \cos mx) \\
+ \text{cosec } \mu a (\delta' \cos (\mu - 1)x + \delta'' \cos (\mu - 2)x + \ldots + \delta^m \cos (\mu - m)x)
\end{align}
&lt;/math&gt; Hinc patet, &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt; ex &lt;math display=&quot;inline&quot;&gt;X'&lt;/math&gt; deduci, si pro quovis termino &lt;math display=&quot;inline&quot;&gt;\delta^\lambda \sin \lambda x&lt;/math&gt; substituatur &lt;math display=&quot;block&quot;&gt;
-\text{cotg } \mu a \ \delta^\lambda \cos \lambda x + \text{cosec } \mu a \ \delta^\lambda \cos (\mu - \lambda)x
&lt;/math&gt; sive &lt;math display=&quot;block&quot;&gt;
\frac{-\delta^\lambda \cos \mu a \cos \lambda x + \delta^\lambda \cos (\mu - \lambda)x}{\sin \mu a}
&lt;/math&gt; cuius praecepti comparationem cum iis, quae in art. 24 explicata sunt, lectoribus linquimus.

&lt;br /&gt;
{{center|32.}}

Si functio &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt; cum &lt;math display=&quot;inline&quot;&gt;\cos n x&lt;/math&gt; non abrumpitur, sed ulterius excurrit, terminis sequentibus per &lt;math display=&quot;inline&quot;&gt;\alpha^\mu \cos \mu x + \alpha^{\mu+1} \cos (\mu + 1)x + \ldots&lt;/math&gt; etc. expressis, habebimus&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>qwt7ogjleq1wkvf0bne9lwhm8uwgvnt</sha1>
    </revision>
    <revision>
      <id>281404</id>
      <parentid>281403</parentid>
      <timestamp>2026-09-20T15:57:11Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Emendata */</comment>
      <origin>281404</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="3512" sha1="sckfs2p0ofikhipblkbg1nbfzuw5fkg" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;3&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;comparare, quam per &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt; exprimemus. Comparando valores coëfficientium &lt;math display=&quot;inline&quot;&gt;\gamma, \gamma', \delta', \gamma'', \delta''&lt;/math&gt; etc. in artt. 20, 22 traditos cum valoribus coëfficientium &lt;math display=&quot;inline&quot;&gt;\varepsilon, \varepsilon', \varepsilon''&lt;/math&gt; etc. modo inventis, invenitur

&lt;math display=&quot;block&quot;&gt;
\begin{align}
&amp;\varepsilon = \gamma \\
&amp;\varepsilon' = \gamma' - \delta' \text{cotang } \mu a, \quad \ \ \varepsilon^{\mu-1} = \delta' \text{cosec } \mu a \\
&amp;\varepsilon'' = \gamma'' - \delta'' \text{cotang } \mu a, \ \quad \varepsilon^{\mu-2} = \delta'' \text{cosec } \mu a \\
&amp;\varepsilon''' = \gamma''' - \delta''' \text{cotang } \mu a, \quad \varepsilon^{\mu-3} = \delta''' \text{cosec } \mu a \\
&amp;\text{etc. usque ad} \\
&amp;(\tfrac{1}{2}) \varepsilon^m = \gamma^m - \delta^m \text{cotang } \mu a, \quad (\tfrac{1}{2}) \varepsilon^{\mu-m} = \delta^m \text{cosec } \mu a
\end{align}
&lt;/math&gt;

Factor &lt;math display=&quot;inline&quot;&gt;\frac{1}{2}&lt;/math&gt; ipsius &lt;math display=&quot;inline&quot;&gt;\varepsilon^m&lt;/math&gt; et &lt;math display=&quot;inline&quot;&gt;\varepsilon^{\mu-m}&lt;/math&gt; praepositus tunc tantummodo valet, quando &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; par est, omittique debet, quando &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; est impar. Quo pacto, in casu posteriore, ubi &lt;math display=&quot;inline&quot;&gt;\mu - m = m + 1&lt;/math&gt; manifesto omnes &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; coëfficientes &lt;math display=&quot;inline&quot;&gt;\varepsilon, \varepsilon', \varepsilon''&lt;/math&gt; etc. per &lt;math display=&quot;inline&quot;&gt;\gamma, \gamma', \delta'&lt;/math&gt; etc. determinati sunt; in priore vero, ubi &lt;math display=&quot;inline&quot;&gt;m = \mu - m&lt;/math&gt;, pro coëfficiente &lt;math display=&quot;inline&quot;&gt;\varepsilon^{\frac{1}{2}\mu}&lt;/math&gt; valores duos &lt;math display=&quot;inline&quot;&gt;2 \gamma^{\frac{1}{2}\mu} - 2 \delta^{\frac{1}{2}\mu} \cot \mu a, \ 2 \delta^{\frac{1}{2}\mu} \text{cosec } \mu a&lt;/math&gt; habemus, quorum aequalitatem ex aequatione &lt;math display=&quot;inline&quot;&gt;\gamma^{\frac{1}{2}\mu} \sin \frac{1}{2} \mu a = \delta^{\frac{1}{2}\mu} \cos \frac{1}{2} \mu a&lt;/math&gt; facile perspicere, et pro quibus igitur etiam hunc &lt;math display=&quot;inline&quot;&gt;\gamma^{\frac{1}{2}\mu} - \delta^{\frac{1}{2}\mu} \text{cotg } \mu a + \delta^{\frac{1}{2}\mu} \text{cosec } \mu a&lt;/math&gt; adoptare possumus.

Hinc facile deducitur, esse in utroque casu &lt;math display=&quot;block&quot;&gt;
\begin{align}
X'' - (X' - \delta' \sin x - \delta'' \sin 2x - \delta''' \sin 3x - \ldots - \delta^m \sin mx) \\
= - \text{cotg } \mu a (\delta' \cos x + \delta'' \cos 2x + \delta''' \cos 3x + \ldots + \delta^m \cos mx) \\
+ \text{cosec } \mu a (\delta' \cos (\mu - 1)x + \delta'' \cos (\mu - 2)x + \ldots + \delta^m \cos (\mu - m)x)
\end{align}
&lt;/math&gt; Hinc patet, &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt; ex &lt;math display=&quot;inline&quot;&gt;X'&lt;/math&gt; deduci, si pro quovis termino &lt;math display=&quot;inline&quot;&gt;\delta^\lambda \sin \lambda x&lt;/math&gt; substituatur &lt;math display=&quot;block&quot;&gt;
-\text{cotg } \mu a \ \delta^\lambda \cos \lambda x + \text{cosec } \mu a \ \delta^\lambda \cos (\mu - \lambda)x
&lt;/math&gt; sive &lt;math display=&quot;block&quot;&gt;
\frac{-\delta^\lambda \cos \mu a \cos \lambda x + \delta^\lambda \cos (\mu - \lambda)x}{\sin \mu a}
&lt;/math&gt; cuius praecepti comparationem cum iis, quae in art. 24 explicata sunt, lectoribus linquimus.

&lt;br /&gt;
{{center|32.}}

Si functio &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt; cum &lt;math display=&quot;inline&quot;&gt;\cos n x&lt;/math&gt; non abrumpitur, sed ulterius excurrit, terminis sequentibus per &lt;math display=&quot;inline&quot;&gt;\alpha^\mu \cos \mu x + \alpha^{\mu+1} \cos (\mu + 1)x +&lt;/math&gt; etc. expressis, habebimus&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>sckfs2p0ofikhipblkbg1nbfzuw5fkg</sha1>
    </revision>
  </page>
  <page>
    <title>Pagina:Werkecarlf03gausrich.djvu/328</title>
    <ns>104</ns>
    <id>86113</id>
    <revision>
      <id>281405</id>
      <parentid>281352</parentid>
      <timestamp>2026-09-20T15:59:31Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <origin>281405</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="3948" sha1="4ysjcq7x9172l4vxrmloxjlygqtdjy8" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;&lt;math display=&quot;block&quot;&gt;
\begin{align}
\varepsilon &amp;= \alpha + \alpha^{\mu} \cos \mu a + \alpha^{2\mu} \cos 2\mu a + \text{etc.}\\[.5em]
\varepsilon' &amp;= \frac{1}{\sin \mu a} \{ \alpha' \sin \mu a + \alpha^{\mu+1} \sin 2\mu a + \alpha^{2\mu+1} \sin 3\mu a + \text{etc.}\\
&amp;\qquad\qquad- \alpha^{2\mu-1} \sin \mu a - \alpha^{3\mu-1} \sin 2\mu a - \alpha^{4\mu-1} \sin 3\mu a - \text{etc.}\}\\[.5em]
\varepsilon'' &amp;= \frac{1}{\sin \mu a} \{ \alpha'' \sin \mu a + \alpha^{\mu+2} \sin 2\mu a + \alpha^{2\mu+2} \sin 3\mu a + \text{etc.}\\
&amp;\qquad\qquad -\alpha^{2\mu-2} \sin \mu a - \alpha^{3\mu-2} \sin 2\mu a - \alpha^{4\mu-2} \sin 3\mu a - \text{etc.}\} \\[.5em]
&amp;\text{etc.} \\[.5em]
\varepsilon^{\mu-1} &amp;= \frac{1}{\sin \mu a} \{ \alpha^{\mu-1} \sin \mu a + \alpha^{2\mu-1} \sin 2\mu a + \alpha^{3\mu-1} \sin 3\mu a + \text{etc.} \\
&amp;\qquad\qquad- \alpha^{\mu+1} \sin \mu a - \alpha^{2\mu+1} \sin 2\mu a - \alpha^{3\mu+1} \sin 3\mu a - \text{etc.}\}
\end{align}
&lt;/math&gt; ex quibus formulis iudicari potest, quatenus differentiam coëfficientium &lt;math display=&quot;inline&quot;&gt;\varepsilon, \varepsilon', \varepsilon''&lt;/math&gt; etc. a veris &lt;math display=&quot;inline&quot;&gt;\alpha, \alpha', \alpha''&lt;/math&gt; etc. negligere liceat. In hoc itaque casu functio &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt; non erit &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt; ipsa, cum qua tamen in eo convenit, quod omnibus &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; valoribus propositis satisfacit. Omnes autem similes functiones his valoribus satisfacientes sub forma &lt;math display=&quot;inline&quot;&gt;X'' + Y(\cos \mu x - \cos \mu a)&lt;/math&gt; contenti erunt, designante &lt;math display=&quot;inline&quot;&gt;Y&lt;/math&gt; functionem indefinitam arcus &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; eiusdem formae ut &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt;, scilicet a sinubus liberam. Functio &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt; erit unica ordinis &lt;math display=&quot;inline&quot;&gt;\mu - 1&lt;/math&gt;: quaevis alia similis valoribus datis satisfaciens ad ordinem altiorem ascendet. Quodsi talis functio &lt;math display=&quot;inline&quot;&gt;X'''&lt;/math&gt; aliunde constaret, omnes similes functiones, per quos valores dati repraesentantur, in hac quoque forma &lt;math display=&quot;inline&quot;&gt;X''' + Y(\cos \mu x - \cos \mu a)&lt;/math&gt; contenti erunt, poteritque &lt;math display=&quot;inline&quot;&gt;Y&lt;/math&gt; ita determinari, ut functio ad ordinem &lt;math display=&quot;inline&quot;&gt;\mu - 1&lt;/math&gt; tum depressa prodeat, quae erit ipsa &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt;. Methodus vero facillima ad hunc finem videtur esse, si primo ex &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt; derivetur &lt;math display=&quot;inline&quot;&gt;X'&lt;/math&gt; per art. 24 atque hinc &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt; per art. 31. Sit itaque terminus quicunque in &lt;math display=&quot;inline&quot;&gt;X''' L \cos \lambda x&lt;/math&gt;, ponaturque &lt;math display=&quot;inline&quot;&gt;\lambda = k\mu + \lambda'&lt;/math&gt;, ita ut sit &lt;math display=&quot;inline&quot;&gt;\lambda' &lt; \mu&lt;/math&gt; atque &lt;math display=&quot;inline&quot;&gt;k&lt;/math&gt; integer. Tunc per art. 24 pro illo termino scribi debet in &lt;math display=&quot;inline&quot;&gt;X'&lt;/math&gt; &lt;math display=&quot;block&quot;&gt;
\begin{align}
&amp;L \cos k\mu a \cos \lambda' x - L \sin k\mu a \sin \lambda' x, \quad \text{si } \lambda' &lt; \frac{1}{2}\mu, \text{ neque vero } \lambda' = 0 \\
&amp;L \cos (k+1)\mu a \cos (\mu - \lambda') x + L \sin (k+1)\mu a \sin (\mu - \lambda') x, \quad \text{si } \lambda' &gt; \frac{1}{2}\mu \\
&amp;L \cos \frac{1}{2}\mu a \cos (k+\frac{1}{2})\mu a \cos \frac{1}{2}\mu x + L \sin \frac{1}{2}\mu a \cos (k+\frac{1}{2})\mu a \sin \frac{1}{2}\mu x, \quad \text{si } \lambda' = \frac{1}{2}\mu
\end{align}
&lt;/math&gt; Hinc autem prodit in &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt;, per art. praec. in casu primo &lt;math display=&quot;block&quot;&gt;
L \cos k\mu a \cos \lambda' x + L \sin k\mu a \text{ cotang } \mu a \cos \lambda' x - L \sin k\mu a \text{ cosec } \mu a \cos (\mu - \lambda') x
&lt;/math&gt; in casu secundo &lt;math display=&quot;block&quot;&gt;
\begin{align}
L \cos (k+1)\mu a \cos (\mu - \lambda') x - L \sin (k+1)\mu a \text{cotg } \mu a \cos (\mu - \lambda') x \\
\qquad + L \sin (k+1)\mu a \text{cosec } \mu a \cos \lambda' x
\end{align}
&lt;/math&gt;&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>4ysjcq7x9172l4vxrmloxjlygqtdjy8</sha1>
    </revision>
    <revision>
      <id>281409</id>
      <parentid>281405</parentid>
      <timestamp>2026-09-20T16:04:49Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <origin>281409</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="3971" sha1="ki33viycpjgph9b94xgddwmsv74983c" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;&lt;math display=&quot;block&quot;&gt;
\begin{align}
\varepsilon &amp;= \alpha + \alpha^{\mu} \cos \mu a + \alpha^{2\mu} \cos 2\mu a + \text{etc.}\\[.5em]
\varepsilon' &amp;= \frac{1}{\sin \mu a} \{ \alpha' \sin \mu a + \alpha^{\mu+1} \sin 2\mu a + \alpha^{2\mu+1} \sin 3\mu a + \text{etc.}\\
&amp;\qquad\qquad- \alpha^{2\mu-1} \sin \mu a - \alpha^{3\mu-1} \sin 2\mu a - \alpha^{4\mu-1} \sin 3\mu a - \text{etc.}\}\\[.5em]
\varepsilon'' &amp;= \frac{1}{\sin \mu a} \{ \alpha'' \sin \mu a + \alpha^{\mu+2} \sin 2\mu a + \alpha^{2\mu+2} \sin 3\mu a + \text{etc.}\\
&amp;\qquad\qquad -\alpha^{2\mu-2} \sin \mu a - \alpha^{3\mu-2} \sin 2\mu a - \alpha^{4\mu-2} \sin 3\mu a - \text{etc.}\} \\[.5em]
&amp;\text{etc.} \\[.5em]
\varepsilon^{\mu-1} &amp;= \frac{1}{\sin \mu a} \{ \alpha^{\mu-1} \sin \mu a + \alpha^{2\mu-1} \sin 2\mu a + \alpha^{3\mu-1} \sin 3\mu a + \text{etc.} \\
&amp;\qquad\qquad- \alpha^{\mu+1} \sin \mu a - \alpha^{2\mu+1} \sin 2\mu a - \alpha^{3\mu+1} \sin 3\mu a - \text{etc.}\}
\end{align}
&lt;/math&gt; ex quibus formulis iudicari potest, quatenus differentiam coëfficientium &lt;math display=&quot;inline&quot;&gt;\varepsilon, \varepsilon', \varepsilon''&lt;/math&gt; etc. a veris &lt;math display=&quot;inline&quot;&gt;\alpha, \alpha', \alpha''&lt;/math&gt; etc. negligere liceat. In hoc itaque casu functio &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt; non erit &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt; ipsa, cum qua tamen in eo convenit, quod omnibus &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; valoribus propositis satisfacit. Omnes autem similes functiones his valoribus satisfacientes sub forma &lt;math display=&quot;inline&quot;&gt;X'' + Y(\cos \mu x - \cos \mu a)&lt;/math&gt; contenti erunt, designante &lt;math display=&quot;inline&quot;&gt;Y&lt;/math&gt; functionem indefinitam arcus &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; eiusdem formae ut &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt;, scilicet a sinubus liberam. Functio &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt; erit unica ordinis &lt;math display=&quot;inline&quot;&gt;\mu - 1^{\text{ti}}&lt;/math&gt;: quaevis alia similis valoribus datis satisfaciens ad ordinem altiorem ascendet. Quodsi talis functio &lt;math display=&quot;inline&quot;&gt;X'''&lt;/math&gt; aliunde constaret, omnes similes functiones, per quos valores dati repraesentantur, in hac quoque forma &lt;math display=&quot;inline&quot;&gt;X''' + Y(\cos \mu x - \cos \mu a)&lt;/math&gt; contenti erunt, poteritque &lt;math display=&quot;inline&quot;&gt;Y&lt;/math&gt; ita determinari, ut functio ad ordinem &lt;math display=&quot;inline&quot;&gt;\mu - 1^{\text{tum}}&lt;/math&gt; depressa prodeat, quae erit ipsa &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt;. Methodus vero facillima ad hunc finem videtur esse, si primo ex &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt; derivetur &lt;math display=&quot;inline&quot;&gt;X'&lt;/math&gt; per art. 24 atque hinc &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt; per art. 31. Sit itaque terminus quicunque in &lt;math display=&quot;inline&quot;&gt;X''' L \cos \lambda x&lt;/math&gt;, ponaturque &lt;math display=&quot;inline&quot;&gt;\lambda = k\mu + \lambda'&lt;/math&gt;, ita ut sit &lt;math display=&quot;inline&quot;&gt;\lambda' &lt; \mu&lt;/math&gt; atque &lt;math display=&quot;inline&quot;&gt;k&lt;/math&gt; integer. Tunc per art. 24 pro illo termino scribi debebit in &lt;math display=&quot;inline&quot;&gt;X'&lt;/math&gt; &lt;math display=&quot;block&quot;&gt;
\begin{align}
&amp;L \cos k\mu a \cos \lambda' x - L \sin k\mu a \sin \lambda' x, \quad \text{si } \lambda' &lt; \frac{1}{2}\mu, \text{ neque vero } \lambda' = 0 \\
&amp;L \cos (k+1)\mu a \cos (\mu - \lambda') x + L \sin (k+1)\mu a \sin (\mu - \lambda') x, \quad \text{si } \lambda' &gt; \frac{1}{2}\mu \\
&amp;L \cos \frac{1}{2}\mu a \cos (k+\frac{1}{2})\mu a \cos \frac{1}{2}\mu x + L \sin \frac{1}{2}\mu a \cos (k+\frac{1}{2})\mu a \sin \frac{1}{2}\mu x, \quad \text{si } \lambda' = \frac{1}{2}\mu
\end{align}
&lt;/math&gt; Hinc autem prodit in &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt;, per art. praec. in casu primo &lt;math display=&quot;block&quot;&gt;
L \cos k\mu a \cos \lambda' x + L \sin k\mu a \text{ cotang } \mu a \cos \lambda' x - L \sin k\mu a \text{ cosec } \mu a \cos (\mu - \lambda') x
&lt;/math&gt; in casu secundo &lt;math display=&quot;block&quot;&gt;
\begin{align}
L \cos (k+1)\mu a \cos (\mu - \lambda') x - L \sin (k+1)\mu a \text{cotg } \mu a \cos (\mu - \lambda') x \\
\qquad + L \sin (k+1)\mu a \text{cosec } \mu a \cos \lambda' x
\end{align}
&lt;/math&gt;&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>ki33viycpjgph9b94xgddwmsv74983c</sha1>
    </revision>
    <revision>
      <id>281410</id>
      <parentid>281409</parentid>
      <timestamp>2026-09-20T16:08:55Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Emendata */</comment>
      <origin>281410</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="3986" sha1="5ad38pft5hey4xs1vjb74pdx0d6aft3" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;3&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;&lt;math display=&quot;block&quot;&gt;
\begin{align}
\varepsilon &amp;= \alpha + \alpha^{\mu} \cos \mu a + \alpha^{2\mu} \cos 2\mu a + \text{etc.}\\[.5em]
\varepsilon' &amp;= \frac{1}{\sin \mu a} \{ \alpha' \sin \mu a + \alpha^{\mu+1} \sin 2\mu a + \alpha^{2\mu+1} \sin 3\mu a + \text{etc.}\\
&amp;\qquad\qquad- \alpha^{2\mu-1} \sin \mu a - \alpha^{3\mu-1} \sin 2\mu a - \alpha^{4\mu-1} \sin 3\mu a - \text{etc.}\}\\[.5em]
\varepsilon'' &amp;= \frac{1}{\sin \mu a} \{ \alpha'' \sin \mu a + \alpha^{\mu+2} \sin 2\mu a + \alpha^{2\mu+2} \sin 3\mu a + \text{etc.}\\
&amp;\qquad\qquad -\alpha^{2\mu-2} \sin \mu a - \alpha^{3\mu-2} \sin 2\mu a - \alpha^{4\mu-2} \sin 3\mu a - \text{etc.}\} \\[.5em]
&amp;\text{etc.} \\[.5em]
\varepsilon^{\mu-1} &amp;= \frac{1}{\sin \mu a} \{ \alpha^{\mu-1} \sin \mu a + \alpha^{2\mu-1} \sin 2\mu a + \alpha^{3\mu-1} \sin 3\mu a + \text{etc.} \\
&amp;\qquad\qquad- \alpha^{\mu+1} \sin \mu a - \alpha^{2\mu+1} \sin 2\mu a - \alpha^{3\mu+1} \sin 3\mu a - \text{etc.}\}
\end{align}
&lt;/math&gt; ex quibus formulis iudicari potest, quatenus differentiam coëfficientium &lt;math display=&quot;inline&quot;&gt;\varepsilon, \varepsilon', \varepsilon''&lt;/math&gt; etc. a veris &lt;math display=&quot;inline&quot;&gt;\alpha, \alpha', \alpha''&lt;/math&gt; etc. negligere liceat. In hoc itaque casu functio &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt; non erit &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt; ipsa, cum qua tamen in eo convenit, quod omnibus &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; valoribus propositis satisfacit. Omnes autem similes functiones his valoribus satisfacientes sub forma &lt;math display=&quot;inline&quot;&gt;X'' + Y(\cos \mu x - \cos \mu a)&lt;/math&gt; contenti erunt, designante &lt;math display=&quot;inline&quot;&gt;Y&lt;/math&gt; functionem indefinitam arcus &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt; eiusdem formae ut &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt;, scilicet a sinubus liberam. Functio &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt; erit unica ordinis &lt;math display=&quot;inline&quot;&gt;\mu - 1^{\text{ti}}&lt;/math&gt;: quaevis alia similis valoribus datis satisfaciens ad ordinem altiorem ascendet. Quodsi talis functio &lt;math display=&quot;inline&quot;&gt;X'''&lt;/math&gt; aliunde constaret, omnes similes functiones, per quos valores dati repraesentantur, in hac quoque forma &lt;math display=&quot;inline&quot;&gt;X''' + Y(\cos \mu x - \cos \mu a)&lt;/math&gt; contenti erunt, poteritque &lt;math display=&quot;inline&quot;&gt;Y&lt;/math&gt; ita determinari, ut functio ad ordinem &lt;math display=&quot;inline&quot;&gt;\mu - 1^{\text{tum}}&lt;/math&gt; depressa prodeat, quae erit ipsa &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt;. Methodus vero facillima ad hunc finem videtur esse, si primo ex &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt; derivetur &lt;math display=&quot;inline&quot;&gt;X'&lt;/math&gt; per art. 24 atque hinc &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt; per art. 31. Sit itaque terminus quicunque in &lt;math display=&quot;inline&quot;&gt;X''' L \cos \lambda x&lt;/math&gt;, ponaturque &lt;math display=&quot;inline&quot;&gt;\lambda = k\mu + \lambda'&lt;/math&gt;, ita ut sit &lt;math display=&quot;inline&quot;&gt;\lambda' &lt; \mu&lt;/math&gt; atque &lt;math display=&quot;inline&quot;&gt;k&lt;/math&gt; integer. Tunc per art. 24 pro illo termino scribi debebit in &lt;math display=&quot;inline&quot;&gt;X'&lt;/math&gt; &lt;math display=&quot;block&quot;&gt;
\begin{align}
&amp;L \cos k\mu a \cos \lambda' x - L \sin k\mu a \sin \lambda' x, \quad \text{si } \lambda' &lt; \tfrac{1}{2}\mu, \text{ neque vero } \lambda' = 0 \\
&amp;L \cos (k+1)\mu a \cos (\mu - \lambda') x + L \sin (k+1)\mu a \sin (\mu - \lambda') x, \quad \text{si } \lambda' &gt; \tfrac{1}{2}\mu \\
&amp;L \cos \tfrac{1}{2}\mu a \cos (k+\tfrac{1}{2})\mu a \cos \tfrac{1}{2}\mu x + L \sin \tfrac{1}{2}\mu a \cos (k+\tfrac{1}{2})\mu a \sin \tfrac{1}{2}\mu x, \quad \text{si } \lambda' = \tfrac{1}{2}\mu
\end{align}
&lt;/math&gt; Hinc autem prodit in &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt;, per art. praec. in casu primo &lt;math display=&quot;block&quot;&gt;
L \cos k\mu a \cos \lambda' x + L \sin k\mu a \text{ cotang } \mu a \cos \lambda' x - L \sin k\mu a \text{ cosec } \mu a \cos (\mu - \lambda') x
&lt;/math&gt; in casu secundo &lt;math display=&quot;block&quot;&gt;
\begin{align}
L \cos (k+1)\mu a \cos (\mu - \lambda') x - L \sin (k+1)\mu a\, \text{cotg}\, \mu a \cos (\mu - \lambda') x \\
\qquad + L \sin (k+1)\mu a\, \text{cosec}\, \mu a \cos \lambda' x
\end{align}
&lt;/math&gt;&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>5ad38pft5hey4xs1vjb74pdx0d6aft3</sha1>
    </revision>
  </page>
  <page>
    <title>Pagina:Werkecarlf03gausrich.djvu/329</title>
    <ns>104</ns>
    <id>86114</id>
    <revision>
      <id>281411</id>
      <parentid>281353</parentid>
      <timestamp>2026-09-20T16:11:40Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <origin>281411</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="2735" sha1="9va7u6x9redzs6ltpzck7gihfx7duif" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;in casu tertio &lt;math display=&quot;block&quot;&gt;
\begin{align}
L \cos \tfrac{1}{2} \mu a \cos (k + \tfrac{1}{2}) \mu a \cos \tfrac{1}{2} \mu x - L \sin \tfrac{1}{2} \mu a \cos (k + \tfrac{1}{2}) \mu a \text{ cotg } \mu a \cos \tfrac{1}{2} \mu x \\
+ L \sin \tfrac{1}{2} \mu a \cos (k + \tfrac{1}{2}) \mu a \text{ cosec } \mu a \cos \tfrac{1}{2} \mu x
\end{align}
&lt;/math&gt; quae expressio in omnibus tribus casibus reducitur ad &lt;math display=&quot;block&quot;&gt;
\frac{L \sin (k + 1) \mu a \cos \lambda' x - L \sin k \mu a \cos (\mu - \lambda') x}{\sin \mu a}
&lt;/math&gt; Quoties autem &lt;math display=&quot;inline&quot;&gt;\lambda' = 0&lt;/math&gt;, habemus in &lt;math display=&quot;inline&quot;&gt;X'&lt;/math&gt; simpliciter &lt;math display=&quot;inline&quot;&gt;L \cos k \mu a&lt;/math&gt;, qui terminus sine variatione in &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt; retinetur.

&lt;br /&gt;
{{center|33.}}

Sit secundo &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt; functio formae &lt;math display=&quot;block&quot;&gt;
\delta' \sin x + \delta'' \sin 2x + \delta''' \sin 3x + \text{etc.} + \delta^n \sin nx
&lt;/math&gt; unde esse debetit &lt;math display=&quot;inline&quot;&gt;\mu = n&lt;/math&gt;. Hic fit, ex art. 30, coëfficiens ipsius &lt;math display=&quot;inline&quot;&gt;A&lt;/math&gt; in formula secunda art. 16 pro &lt;math display=&quot;inline&quot;&gt;T&lt;/math&gt; &lt;math display=&quot;block&quot;&gt;
\begin{align}
&amp;= \frac{1}{\mu \sin \mu a} \times \frac{\sin t}{\sin a} (\sin \mu a + 2 \sin (\mu - 
1)a \cos t + 2 \sin (\mu - 2)a \cos 2t + \text{etc.} \\
&amp;\qquad\qquad\qquad\qquad+ 2 \sin a \cos (\mu - 1)t)\\
&amp;= \frac{1}{\mu \sin \mu a} (2 \cos (\mu - 1)a \sin t + 2 \cos (\mu - 2)a \sin 2t + 2 \cos (\mu - 3)a \sin 3t + \text{etc.} \\
&amp;\qquad\qquad\qquad\qquad+ 2 \cos a \sin (\mu - 1)t + \sin \mu t)
\end{align}
&lt;/math&gt; Prorsus similes expressiones pro coëfficientibus ipsorum &lt;math display=&quot;inline&quot;&gt;B, C, D&lt;/math&gt; etc. prodeunt, mutato tantummodo arcu &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt; in &lt;math display=&quot;inline&quot;&gt;b, c, d&lt;/math&gt; etc. Quamobrem faciendo &lt;math display=&quot;block&quot;&gt;
\begin{align}
\zeta' &amp;= \frac{2}{\mu \sin \mu a} \{A \cos (\mu - 1)a + B \cos (\mu - 1)b + C \cos (\mu - 1)c + D \cos (\mu - 1)d + \text{etc.}\}\\
\zeta'' &amp;= \frac{2}{\mu \sin \mu a} \{A \cos (\mu - 2)a + B \cos (\mu - 2)b + C \cos (\mu - 2)c + D \cos (\mu - 2)d + \text{etc.}\}\\
\zeta''' &amp;= \frac{2}{\mu \sin \mu a} \{A \cos (\mu - 3)a + B \cos (\mu - 3)b + C \cos (\mu - 3)c + D \cos (\mu - 3)d + \text{etc.}\}\\
&amp;\text{etc.}\\
\zeta^{\mu-1} &amp;= \frac{2}{\mu \sin \mu a} \{A \cos a + B \cos b + C \cos c + D \cos d + \text{etc.}\}\\
\zeta^\mu &amp;= \frac{1}{\mu \sin \mu a} \{A + B + C + D + \text{etc.}\}
\end{align}
&lt;/math&gt; erit &lt;math display=&quot;block&quot;&gt;
T = \zeta' \sin t + \zeta'' \sin 2t + \zeta''' \sin 3t + \text{etc.} + \zeta^\mu \sin \mu t
&lt;/math&gt; Quum haec formula generaliter pro valore quocunque ipsius &lt;math display=&quot;inline&quot;&gt;t&lt;/math&gt; valeat, necessario&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>9va7u6x9redzs6ltpzck7gihfx7duif</sha1>
    </revision>
    <revision>
      <id>281412</id>
      <parentid>281411</parentid>
      <timestamp>2026-09-20T16:12:47Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <origin>281412</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="2737" sha1="6oi98vrnvf26drcjkh6s24pw6macnos" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;in casu tertio &lt;math display=&quot;block&quot;&gt;
\begin{align}
L \cos \tfrac{1}{2} \mu a \cos (k + \tfrac{1}{2}) \mu a \cos \tfrac{1}{2} \mu x - L \sin \tfrac{1}{2} \mu a \cos (k + \tfrac{1}{2}) \mu a \text{ cotg } \mu a \cos \tfrac{1}{2} \mu x \\
+ L \sin \tfrac{1}{2} \mu a \cos (k + \tfrac{1}{2}) \mu a \text{ cosec } \mu a \cos \tfrac{1}{2} \mu x
\end{align}
&lt;/math&gt; quae expressio in omnibus tribus casibus reducitur ad &lt;math display=&quot;block&quot;&gt;
\frac{L \sin (k + 1) \mu a \cos \lambda' x - L \sin k \mu a \cos (\mu - \lambda') x}{\sin \mu a}
&lt;/math&gt; 

Quoties autem &lt;math display=&quot;inline&quot;&gt;\lambda' = 0&lt;/math&gt;, habemus in &lt;math display=&quot;inline&quot;&gt;X'&lt;/math&gt; simpliciter &lt;math display=&quot;inline&quot;&gt;L \cos k \mu a&lt;/math&gt;, qui terminus sine variatione in &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt; retinetur.

&lt;br /&gt;
{{center|33.}}

Sit secundo &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt; functio formae &lt;math display=&quot;block&quot;&gt;
\delta' \sin x + \delta'' \sin 2x + \delta''' \sin 3x + \text{etc.} + \delta^n \sin nx
&lt;/math&gt; unde esse debetit &lt;math display=&quot;inline&quot;&gt;\mu = n&lt;/math&gt;. Hic fit, ex art. 30, coëfficiens ipsius &lt;math display=&quot;inline&quot;&gt;A&lt;/math&gt; in formula secunda art. 16 pro &lt;math display=&quot;inline&quot;&gt;T&lt;/math&gt; &lt;math display=&quot;block&quot;&gt;
\begin{align}
&amp;= \frac{1}{\mu \sin \mu a} \times \frac{\sin t}{\sin a} (\sin \mu a + 2 \sin (\mu - 
1)a \cos t + 2 \sin (\mu - 2)a \cos 2t + \text{etc.} \\
&amp;\qquad\qquad\qquad\qquad+ 2 \sin a \cos (\mu - 1)t)\\
&amp;= \frac{1}{\mu \sin \mu a} (2 \cos (\mu - 1)a \sin t + 2 \cos (\mu - 2)a \sin 2t + 2 \cos (\mu - 3)a \sin 3t + \text{etc.} \\
&amp;\qquad\qquad\qquad\qquad+ 2 \cos a \sin (\mu - 1)t + \sin \mu t)
\end{align}
&lt;/math&gt; Prorsus similes expressiones pro coëfficientibus ipsorum &lt;math display=&quot;inline&quot;&gt;B, C, D&lt;/math&gt; etc. prodeunt, mutato tantummodo arcu &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt; in &lt;math display=&quot;inline&quot;&gt;b, c, d&lt;/math&gt; etc. Quamobrem faciendo &lt;math display=&quot;block&quot;&gt;
\begin{align}
\zeta' &amp;= \frac{2}{\mu \sin \mu a} \{A \cos (\mu - 1)a + B \cos (\mu - 1)b + C \cos (\mu - 1)c + D \cos (\mu - 1)d + \text{etc.}\}\\
\zeta'' &amp;= \frac{2}{\mu \sin \mu a} \{A \cos (\mu - 2)a + B \cos (\mu - 2)b + C \cos (\mu - 2)c + D \cos (\mu - 2)d + \text{etc.}\}\\
\zeta''' &amp;= \frac{2}{\mu \sin \mu a} \{A \cos (\mu - 3)a + B \cos (\mu - 3)b + C \cos (\mu - 3)c + D \cos (\mu - 3)d + \text{etc.}\}\\
&amp;\text{etc.}\\
\zeta^{\mu-1} &amp;= \frac{2}{\mu \sin \mu a} \{A \cos a + B \cos b + C \cos c + D \cos d + \text{etc.}\}\\
\zeta^\mu &amp;= \frac{1}{\mu \sin \mu a} \{A + B + C + D + \text{etc.}\}
\end{align}
&lt;/math&gt; erit &lt;math display=&quot;block&quot;&gt;
T = \zeta' \sin t + \zeta'' \sin 2t + \zeta''' \sin 3t + \text{etc.} + \zeta^\mu \sin \mu t
&lt;/math&gt; Quum haec formula generaliter pro valore quocunque ipsius &lt;math display=&quot;inline&quot;&gt;t&lt;/math&gt; valeat, necessario&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>6oi98vrnvf26drcjkh6s24pw6macnos</sha1>
    </revision>
    <revision>
      <id>281413</id>
      <parentid>281412</parentid>
      <timestamp>2026-09-20T16:13:45Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <origin>281413</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="2733" sha1="ad0wessdyvrd2bswe7rl56246zr6mnb" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;in casu tertio &lt;math display=&quot;block&quot;&gt;
\begin{align}
L \cos \tfrac{1}{2} \mu a \cos (k + \tfrac{1}{2}) \mu a \cos \tfrac{1}{2} \mu x - L \sin \tfrac{1}{2} \mu a \cos (k + \tfrac{1}{2}) \mu a \text{ cotg } \mu a \cos \tfrac{1}{2} \mu x \\
+ L \sin \tfrac{1}{2} \mu a \cos (k + \tfrac{1}{2}) \mu a \text{ cosec } \mu a \cos \tfrac{1}{2} \mu x
\end{align}
&lt;/math&gt; quae expressio in omnibus tribus casibus reducitur ad &lt;math display=&quot;block&quot;&gt;
\frac{L \sin (k + 1) \mu a \cos \lambda' x - L \sin k \mu a \cos (\mu - \lambda') x}{\sin \mu a}
&lt;/math&gt; 

Quoties autem &lt;math display=&quot;inline&quot;&gt;\lambda' = 0&lt;/math&gt;, habemus in &lt;math display=&quot;inline&quot;&gt;X'&lt;/math&gt; simpliciter &lt;math display=&quot;inline&quot;&gt;L \cos k \mu a&lt;/math&gt;, qui terminus sine variatione in &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt; retinetur.

&lt;br /&gt;
{{center|33.}}

Sit secundo &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt; functio formae &lt;math display=&quot;block&quot;&gt;
\beta' \sin x + \beta'' \sin 2x + \beta''' \sin 3x + \text{etc.} + \beta^n \sin nx
&lt;/math&gt; unde esse debetit &lt;math display=&quot;inline&quot;&gt;\mu = n&lt;/math&gt;. Hic fit, ex art. 30, coëfficiens ipsius &lt;math display=&quot;inline&quot;&gt;A&lt;/math&gt; in formula secunda art. 16 pro &lt;math display=&quot;inline&quot;&gt;T&lt;/math&gt; &lt;math display=&quot;block&quot;&gt;
\begin{align}
&amp;= \frac{1}{\mu \sin \mu a} \times \frac{\sin t}{\sin a} (\sin \mu a + 2 \sin (\mu - 
1)a \cos t + 2 \sin (\mu - 2)a \cos 2t + \text{etc.} \\
&amp;\qquad\qquad\qquad\qquad+ 2 \sin a \cos (\mu - 1)t)\\
&amp;= \frac{1}{\mu \sin \mu a} (2 \cos (\mu - 1)a \sin t + 2 \cos (\mu - 2)a \sin 2t + 2 \cos (\mu - 3)a \sin 3t + \text{etc.} \\
&amp;\qquad\qquad\qquad\qquad+ 2 \cos a \sin (\mu - 1)t + \sin \mu t)
\end{align}
&lt;/math&gt; Prorsus similes expressiones pro coëfficientibus ipsorum &lt;math display=&quot;inline&quot;&gt;B, C, D&lt;/math&gt; etc. prodeunt, mutato tantummodo arcu &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt; in &lt;math display=&quot;inline&quot;&gt;b, c, d&lt;/math&gt; etc. Quamobrem faciendo &lt;math display=&quot;block&quot;&gt;
\begin{align}
\zeta' &amp;= \frac{2}{\mu \sin \mu a} \{A \cos (\mu - 1)a + B \cos (\mu - 1)b + C \cos (\mu - 1)c + D \cos (\mu - 1)d + \text{etc.}\}\\
\zeta'' &amp;= \frac{2}{\mu \sin \mu a} \{A \cos (\mu - 2)a + B \cos (\mu - 2)b + C \cos (\mu - 2)c + D \cos (\mu - 2)d + \text{etc.}\}\\
\zeta''' &amp;= \frac{2}{\mu \sin \mu a} \{A \cos (\mu - 3)a + B \cos (\mu - 3)b + C \cos (\mu - 3)c + D \cos (\mu - 3)d + \text{etc.}\}\\
&amp;\text{etc.}\\
\zeta^{\mu-1} &amp;= \frac{2}{\mu \sin \mu a} \{A \cos a + B \cos b + C \cos c + D \cos d + \text{etc.}\}\\
\zeta^\mu &amp;= \frac{1}{\mu \sin \mu a} \{A + B + C + D + \text{etc.}\}
\end{align}
&lt;/math&gt; erit &lt;math display=&quot;block&quot;&gt;
T = \zeta' \sin t + \zeta'' \sin 2t + \zeta''' \sin 3t + \text{etc.} + \zeta^\mu \sin \mu t
&lt;/math&gt; Quum haec formula generaliter pro valore quocunque ipsius &lt;math display=&quot;inline&quot;&gt;t&lt;/math&gt; valeat, necessario&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>ad0wessdyvrd2bswe7rl56246zr6mnb</sha1>
    </revision>
    <revision>
      <id>281414</id>
      <parentid>281413</parentid>
      <timestamp>2026-09-20T16:14:16Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <origin>281414</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="2733" sha1="n6uk5ulcg07ztrb6f1wtu2yzipvi4hi" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;in casu tertio &lt;math display=&quot;block&quot;&gt;
\begin{align}
L \cos \tfrac{1}{2} \mu a \cos (k + \tfrac{1}{2}) \mu a \cos \tfrac{1}{2} \mu x - L \sin \tfrac{1}{2} \mu a \cos (k + \tfrac{1}{2}) \mu a \text{ cotg } \mu a \cos \tfrac{1}{2} \mu x \\
+ L \sin \tfrac{1}{2} \mu a \cos (k + \tfrac{1}{2}) \mu a \text{ cosec } \mu a \cos \tfrac{1}{2} \mu x
\end{align}
&lt;/math&gt; quae expressio in omnibus tribus casibus reducitur ad &lt;math display=&quot;block&quot;&gt;
\frac{L \sin (k + 1) \mu a \cos \lambda' x - L \sin k \mu a \cos (\mu - \lambda') x}{\sin \mu a}
&lt;/math&gt; 

Quoties autem &lt;math display=&quot;inline&quot;&gt;\lambda' = 0&lt;/math&gt;, habemus in &lt;math display=&quot;inline&quot;&gt;X'&lt;/math&gt; simpliciter &lt;math display=&quot;inline&quot;&gt;L \cos k \mu a&lt;/math&gt;, qui terminus sine variatione in &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt; retinetur.

&lt;br /&gt;
{{center|33.}}

Sit secundo &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt; functio formae &lt;math display=&quot;block&quot;&gt;
\beta' \sin x + \beta'' \sin 2x + \beta''' \sin 3x + \text{etc.} + \beta^n \sin nx
&lt;/math&gt; unde esse debebit &lt;math display=&quot;inline&quot;&gt;\mu = n&lt;/math&gt;. Hic fit, ex art. 30, coëfficiens ipsius &lt;math display=&quot;inline&quot;&gt;A&lt;/math&gt; in formula secunda art. 16 pro &lt;math display=&quot;inline&quot;&gt;T&lt;/math&gt; &lt;math display=&quot;block&quot;&gt;
\begin{align}
&amp;= \frac{1}{\mu \sin \mu a} \times \frac{\sin t}{\sin a} (\sin \mu a + 2 \sin (\mu - 
1)a \cos t + 2 \sin (\mu - 2)a \cos 2t + \text{etc.} \\
&amp;\qquad\qquad\qquad\qquad+ 2 \sin a \cos (\mu - 1)t)\\
&amp;= \frac{1}{\mu \sin \mu a} (2 \cos (\mu - 1)a \sin t + 2 \cos (\mu - 2)a \sin 2t + 2 \cos (\mu - 3)a \sin 3t + \text{etc.} \\
&amp;\qquad\qquad\qquad\qquad+ 2 \cos a \sin (\mu - 1)t + \sin \mu t)
\end{align}
&lt;/math&gt; Prorsus similes expressiones pro coëfficientibus ipsorum &lt;math display=&quot;inline&quot;&gt;B, C, D&lt;/math&gt; etc. prodeunt, mutato tantummodo arcu &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt; in &lt;math display=&quot;inline&quot;&gt;b, c, d&lt;/math&gt; etc. Quamobrem faciendo &lt;math display=&quot;block&quot;&gt;
\begin{align}
\zeta' &amp;= \frac{2}{\mu \sin \mu a} \{A \cos (\mu - 1)a + B \cos (\mu - 1)b + C \cos (\mu - 1)c + D \cos (\mu - 1)d + \text{etc.}\}\\
\zeta'' &amp;= \frac{2}{\mu \sin \mu a} \{A \cos (\mu - 2)a + B \cos (\mu - 2)b + C \cos (\mu - 2)c + D \cos (\mu - 2)d + \text{etc.}\}\\
\zeta''' &amp;= \frac{2}{\mu \sin \mu a} \{A \cos (\mu - 3)a + B \cos (\mu - 3)b + C \cos (\mu - 3)c + D \cos (\mu - 3)d + \text{etc.}\}\\
&amp;\text{etc.}\\
\zeta^{\mu-1} &amp;= \frac{2}{\mu \sin \mu a} \{A \cos a + B \cos b + C \cos c + D \cos d + \text{etc.}\}\\
\zeta^\mu &amp;= \frac{1}{\mu \sin \mu a} \{A + B + C + D + \text{etc.}\}
\end{align}
&lt;/math&gt; erit &lt;math display=&quot;block&quot;&gt;
T = \zeta' \sin t + \zeta'' \sin 2t + \zeta''' \sin 3t + \text{etc.} + \zeta^\mu \sin \mu t
&lt;/math&gt; Quum haec formula generaliter pro valore quocunque ipsius &lt;math display=&quot;inline&quot;&gt;t&lt;/math&gt; valeat, necessario&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>n6uk5ulcg07ztrb6f1wtu2yzipvi4hi</sha1>
    </revision>
    <revision>
      <id>281415</id>
      <parentid>281414</parentid>
      <timestamp>2026-09-20T16:16:40Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Emendata */</comment>
      <origin>281415</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="2733" sha1="2uluyaduyufsyt5t5cud3kvm8e25csm" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;3&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;in casu tertio &lt;math display=&quot;block&quot;&gt;
\begin{align}
L \cos \tfrac{1}{2} \mu a \cos (k + \tfrac{1}{2}) \mu a \cos \tfrac{1}{2} \mu x - L \sin \tfrac{1}{2} \mu a \cos (k + \tfrac{1}{2}) \mu a \text{ cotg } \mu a \cos \tfrac{1}{2} \mu x \\
+ L \sin \tfrac{1}{2} \mu a \cos (k + \tfrac{1}{2}) \mu a \text{ cosec } \mu a \cos \tfrac{1}{2} \mu x
\end{align}
&lt;/math&gt; quae expressio in omnibus tribus casibus reducitur ad &lt;math display=&quot;block&quot;&gt;
\frac{L \sin (k + 1) \mu a \cos \lambda' x - L \sin k \mu a \cos (\mu - \lambda') x}{\sin \mu a}
&lt;/math&gt; 

Quoties autem &lt;math display=&quot;inline&quot;&gt;\lambda' = 0&lt;/math&gt;, habemus in &lt;math display=&quot;inline&quot;&gt;X'&lt;/math&gt; simpliciter &lt;math display=&quot;inline&quot;&gt;L \cos k \mu a&lt;/math&gt;, qui terminus sine variatione in &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt; retinetur.

&lt;br /&gt;
{{center|33.}}

Sit secundo &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt; functio formae &lt;math display=&quot;block&quot;&gt;
\beta' \sin x + \beta'' \sin 2x + \beta''' \sin 3x + \text{etc.} + \beta^n \sin nx
&lt;/math&gt; unde esse debebit &lt;math display=&quot;inline&quot;&gt;\mu = n&lt;/math&gt;. Hic fit, ex art. 30, coëfficiens ipsius &lt;math display=&quot;inline&quot;&gt;A&lt;/math&gt; in formula secunda art. 16 pro &lt;math display=&quot;inline&quot;&gt;T&lt;/math&gt; &lt;math display=&quot;block&quot;&gt;
\begin{align}
&amp;= \frac{1}{\mu \sin \mu a} \times \frac{\sin t}{\sin a} (\sin \mu a + 2 \sin (\mu - 
1)a \cos t + 2 \sin (\mu - 2)a \cos 2t + \text{etc.} \\
&amp;\qquad\qquad\qquad\qquad+ 2 \sin a \cos (\mu - 1)t)\\
&amp;= \frac{1}{\mu \sin \mu a} (2 \cos (\mu - 1)a \sin t + 2 \cos (\mu - 2)a \sin 2t + 2 \cos (\mu - 3)a \sin 3t + \text{etc.} \\
&amp;\qquad\qquad\qquad\qquad+ 2 \cos a \sin (\mu - 1)t + \sin \mu t)
\end{align}
&lt;/math&gt; Prorsus similes expressiones pro coëfficientibus ipsorum &lt;math display=&quot;inline&quot;&gt;B, C, D&lt;/math&gt; etc. prodeunt, mutato tantummodo arcu &lt;math display=&quot;inline&quot;&gt;a&lt;/math&gt; in &lt;math display=&quot;inline&quot;&gt;b, c, d&lt;/math&gt; etc. Quamobrem faciendo &lt;math display=&quot;block&quot;&gt;
\begin{align}
\zeta' &amp;= \frac{2}{\mu \sin \mu a} \{A \cos (\mu - 1)a + B \cos (\mu - 1)b + C \cos (\mu - 1)c + D \cos (\mu - 1)d + \text{etc.}\}\\
\zeta'' &amp;= \frac{2}{\mu \sin \mu a} \{A \cos (\mu - 2)a + B \cos (\mu - 2)b + C \cos (\mu - 2)c + D \cos (\mu - 2)d + \text{etc.}\}\\
\zeta''' &amp;= \frac{2}{\mu \sin \mu a} \{A \cos (\mu - 3)a + B \cos (\mu - 3)b + C \cos (\mu - 3)c + D \cos (\mu - 3)d + \text{etc.}\}\\
&amp;\text{etc.}\\
\zeta^{\mu-1} &amp;= \frac{2}{\mu \sin \mu a} \{A \cos a + B \cos b + C \cos c + D \cos d + \text{etc.}\}\\
\zeta^\mu &amp;= \frac{1}{\mu \sin \mu a} \{A + B + C + D + \text{etc.}\}
\end{align}
&lt;/math&gt; erit &lt;math display=&quot;block&quot;&gt;
T = \zeta' \sin t + \zeta'' \sin 2t + \zeta''' \sin 3t + \text{etc.} + \zeta^\mu \sin \mu t
&lt;/math&gt; Quum haec formula generaliter pro valore quocunque ipsius &lt;math display=&quot;inline&quot;&gt;t&lt;/math&gt; valeat, necessario&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>2uluyaduyufsyt5t5cud3kvm8e25csm</sha1>
    </revision>
  </page>
  <page>
    <title>Pagina:Werkecarlf03gausrich.djvu/330</title>
    <ns>104</ns>
    <id>86115</id>
    <revision>
      <id>281416</id>
      <parentid>281355</parentid>
      <timestamp>2026-09-20T16:17:42Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <origin>281416</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="3183" sha1="179g3n145hvw1h5uebrgt59ktq7p3mj" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;cum &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt; identica erit, mutando &lt;math display=&quot;inline&quot;&gt;t&lt;/math&gt; in &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;, unde coëfficientes &lt;math display=&quot;inline&quot;&gt;\zeta', \zeta'', \zeta'''&lt;/math&gt; etc. ipsis &lt;math display=&quot;inline&quot;&gt;\beta',\beta'',\beta'''&lt;/math&gt; etc. resp. aequales erunt.

&lt;br /&gt;
{{center|34.}}

Si ut in art. 31 omnibus valoribus propositis per functionem talem &lt;math display=&quot;block&quot;&gt;
\begin{align}
X' = \gamma + \gamma'\cos x + \gamma''\cos 2x + \text{etc.} + \gamma^m \cos mx \\
+ \delta'\sin x + \delta''\sin 2x + \text{etc.} + \delta^m \sin mx
\end{align}
&lt;/math&gt; satisfactum est, existente &lt;math display=&quot;inline&quot;&gt;m = \frac{1}{2}\mu - \frac{1}{2}&lt;/math&gt; vel &lt;math display=&quot;inline&quot;&gt;= \frac{1}{2}\mu&lt;/math&gt;, prout &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; impar vel par est, comparatio huius functionis cum hac &lt;math display=&quot;block&quot;&gt;
X'' = \zeta'\sin x + \zeta''\sin 2x + \zeta'''\sin 3x + \text{etc.} + \zeta^\mu \sin \mu x
&lt;/math&gt; quam in art. praec. eruimus, hasce aequationes suppeditant: &lt;math display=&quot;block&quot;&gt;
\begin{align}
\zeta' &amp;= \delta' + \gamma' \text{cotg } \mu a, &amp; \zeta'^{\mu-1} &amp;= \gamma' \text{cosec } \mu a \\
\zeta'' &amp;= \delta'' + \gamma'' \text{cotg } \mu a, &amp; \zeta'^{\mu-2} &amp;= \gamma'' \text{cosec } \mu a \\
\zeta''' &amp;= \delta''' + \gamma''' \text{cotg } \mu a, &amp; \zeta'^{\mu-3} &amp;= \gamma''' \text{cosec } \mu a \\
&amp;\text{etc. usque ad} \\
(\tfrac{1}{2})\zeta^m &amp;= \delta^m + \gamma^m \text{cotg } \mu a, &amp; (\tfrac{1}{2})\zeta^{\mu-m} &amp;= \gamma^m \text{cosec } \mu a
\end{align}
&lt;/math&gt;

ubi factor &lt;math display=&quot;inline&quot;&gt;\frac{1}{2}&lt;/math&gt; coëfficientibus &lt;math display=&quot;inline&quot;&gt;\zeta^m, \zeta^{\mu-m}&lt;/math&gt; praepositus pro eo tantum casu valet, ubi &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; par est, pro altero vero, ubi &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; impar est. omitti debet. In casu itaque posteriore ubi &lt;math display=&quot;inline&quot;&gt;\mu - m = m + 1&lt;/math&gt;, pro quovis coëfficiente &lt;math display=&quot;inline&quot;&gt;\zeta', \zeta''&lt;/math&gt; etc. valorem unum per &lt;math display=&quot;inline&quot;&gt;\gamma, \gamma', \gamma''&lt;/math&gt; etc. et &lt;math display=&quot;inline&quot;&gt;\delta', \delta''&lt;/math&gt; etc. habemus; in priore vero, ubi &lt;math display=&quot;inline&quot;&gt;\mu - m = m&lt;/math&gt;, pro coëfficiente &lt;math display=&quot;inline&quot;&gt;\zeta^{\frac{1}{2}\mu}&lt;/math&gt; duos aequales, pro quibus etiam valor &lt;math display=&quot;inline&quot;&gt;\delta^{\frac{1}{2}\mu} + \gamma^{\frac{1}{2}\mu} \cot \mu a + \gamma^{\frac{1}{2}\mu} \csc \mu a&lt;/math&gt; adoptari potest. Hinc colligitur

&lt;math display=&quot;block&quot;&gt;
\begin{align}
X'' &amp;= X' - \gamma - \gamma'\cos x - \gamma''\cos 2x - \text{etc.} - \gamma^m \cos mx \\
&amp;+ \text{cotg } \mu a (\gamma'\sin x + \gamma''\sin 2x + \gamma'''\sin 3x + \text{etc.} + \gamma^m \sin mx) \\
&amp;+ \text{cosec } \mu a (\gamma \sin \mu x + \gamma' \sin (\mu - 1)x + \gamma'' \sin (\mu - 2)x + \text{etc.} + \gamma^{\mu-m} \sin (\mu - m)x)
\end{align}
&lt;/math&gt;

Quamobrem ex &lt;math display=&quot;inline&quot;&gt;X'&lt;/math&gt; producitur &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt;, scribendo pro &lt;math display=&quot;inline&quot;&gt;\gamma&lt;/math&gt;, &lt;math display=&quot;inline&quot;&gt;\gamma \text{ cosec } \mu a \sin \mu x = \frac{\gamma \sin \mu x}{\sin \mu a}&lt;/math&gt; et, pro quovis termino &lt;math display=&quot;inline&quot;&gt;\gamma^\lambda \cos \lambda x&lt;/math&gt;&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>179g3n145hvw1h5uebrgt59ktq7p3mj</sha1>
    </revision>
    <revision>
      <id>281417</id>
      <parentid>281416</parentid>
      <timestamp>2026-09-20T16:20:20Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <origin>281417</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="3232" sha1="3v70s6cj32k7iwtiwb2ecdlocg42r7j" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;cum &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt; identica erit, mutando &lt;math display=&quot;inline&quot;&gt;t&lt;/math&gt; in &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;, unde coëfficientes &lt;math display=&quot;inline&quot;&gt;\zeta', \zeta'', \zeta'''&lt;/math&gt; etc. ipsis &lt;math display=&quot;inline&quot;&gt;\beta',\beta'',\beta'''&lt;/math&gt; etc. resp. aequales erunt.

&lt;br /&gt;
{{center|34.}}

Si ut in art. 31 omnibus valoribus propositis per functionem talem &lt;math display=&quot;block&quot;&gt;
\begin{align}
X' = \gamma + \gamma'\cos x + \gamma''\cos 2x + \text{etc.} + \gamma^m \cos mx \\
+ \delta'\sin x + \delta''\sin 2x + \text{etc.} + \delta^m \sin mx
\end{align}
&lt;/math&gt; satisfactum est, existente &lt;math display=&quot;inline&quot;&gt;m = \frac{1}{2}\mu - \frac{1}{2}&lt;/math&gt; vel &lt;math display=&quot;inline&quot;&gt;= \frac{1}{2}\mu&lt;/math&gt;, prout &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; impar vel par est, comparatio huius functionis cum hac &lt;math display=&quot;block&quot;&gt;
X'' = \zeta'\sin x + \zeta''\sin 2x + \zeta'''\sin 3x + \text{etc.} + \zeta^\mu \sin \mu x
&lt;/math&gt; quam in art. praec. eruimus, hasce aequationes suppeditant: &lt;math display=&quot;block&quot;&gt;
\begin{align}
 &amp;  &amp;  &amp; \zeta^{\mu}=\gamma\,\text{cosec}\,\mu a \\
\zeta' &amp;= \delta' + \gamma' \text{cotg } \mu a, &amp; \zeta'^{\mu-1} &amp;= \gamma' \text{cosec } \mu a \\
\zeta'' &amp;= \delta'' + \gamma'' \text{cotg } \mu a, &amp; \zeta'^{\mu-2} &amp;= \gamma'' \text{cosec } \mu a \\
\zeta''' &amp;= \delta''' + \gamma''' \text{cotg } \mu a, &amp; \zeta'^{\mu-3} &amp;= \gamma''' \text{cosec } \mu a \\
&amp;\text{etc. up to} \\
(\tfrac{1}{2})\zeta^m &amp;= \delta^m + \gamma^m \text{cotg } \mu a, &amp; (\tfrac{1}{2})\zeta^{\mu-m} &amp;= \gamma^m \text{cosec } \mu a
\end{align}
&lt;/math&gt;

ubi factor &lt;math display=&quot;inline&quot;&gt;\frac{1}{2}&lt;/math&gt; coëfficientibus &lt;math display=&quot;inline&quot;&gt;\zeta^m, \zeta^{\mu-m}&lt;/math&gt; praepositus pro eo tantum casu valet, ubi &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; par est, pro altero vero, ubi &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; impar est. omitti debet. In casu itaque posteriore ubi &lt;math display=&quot;inline&quot;&gt;\mu - m = m + 1&lt;/math&gt;, pro quovis coëfficiente &lt;math display=&quot;inline&quot;&gt;\zeta', \zeta''&lt;/math&gt; etc. valorem unum per &lt;math display=&quot;inline&quot;&gt;\gamma, \gamma', \gamma''&lt;/math&gt; etc. et &lt;math display=&quot;inline&quot;&gt;\delta', \delta''&lt;/math&gt; etc. habemus; in priore vero, ubi &lt;math display=&quot;inline&quot;&gt;\mu - m = m&lt;/math&gt;, pro coëfficiente &lt;math display=&quot;inline&quot;&gt;\zeta^{\frac{1}{2}\mu}&lt;/math&gt; duos aequales, pro quibus etiam valor &lt;math display=&quot;inline&quot;&gt;\delta^{\frac{1}{2}\mu} + \gamma^{\frac{1}{2}\mu} \cot \mu a + \gamma^{\frac{1}{2}\mu} \csc \mu a&lt;/math&gt; adoptari potest. Hinc colligitur

&lt;math display=&quot;block&quot;&gt;
\begin{align}
X'' &amp;= X' - \gamma - \gamma'\cos x - \gamma''\cos 2x - \text{etc.} - \gamma^m \cos mx \\
&amp;+ \text{cotg } \mu a (\gamma'\sin x + \gamma''\sin 2x + \gamma'''\sin 3x + \text{etc.} + \gamma^m \sin mx) \\
&amp;+ \text{cosec } \mu a (\gamma \sin \mu x + \gamma' \sin (\mu - 1)x + \gamma'' \sin (\mu - 2)x + \text{etc.} + \gamma^{\mu-m} \sin (\mu - m)x)
\end{align}
&lt;/math&gt;

Quamobrem ex &lt;math display=&quot;inline&quot;&gt;X'&lt;/math&gt; producitur &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt;, scribendo pro &lt;math display=&quot;inline&quot;&gt;\gamma&lt;/math&gt;, &lt;math display=&quot;inline&quot;&gt;\gamma \text{ cosec } \mu a \sin \mu x = \frac{\gamma \sin \mu x}{\sin \mu a}&lt;/math&gt; et, pro quovis termino &lt;math display=&quot;inline&quot;&gt;\gamma^\lambda \cos \lambda x&lt;/math&gt;&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>3v70s6cj32k7iwtiwb2ecdlocg42r7j</sha1>
    </revision>
    <revision>
      <id>281418</id>
      <parentid>281417</parentid>
      <timestamp>2026-09-20T16:21:03Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <origin>281418</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="3235" sha1="4wydchljs07p6irq7ju1w8wusa8idzw" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;cum &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt; identica erit, mutando &lt;math display=&quot;inline&quot;&gt;t&lt;/math&gt; in &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;, unde coëfficientes &lt;math display=&quot;inline&quot;&gt;\zeta', \zeta'', \zeta'''&lt;/math&gt; etc. ipsis &lt;math display=&quot;inline&quot;&gt;\beta',\beta'',\beta'''&lt;/math&gt; etc. resp. aequales erunt.

&lt;br /&gt;
{{center|34.}}

Si ut in art. 31 omnibus valoribus propositis per functionem talem &lt;math display=&quot;block&quot;&gt;
\begin{align}
X' = \gamma + \gamma'\cos x + \gamma''\cos 2x + \text{etc.} + \gamma^m \cos mx \\
+ \delta'\sin x + \delta''\sin 2x + \text{etc.} + \delta^m \sin mx
\end{align}
&lt;/math&gt; satisfactum est, existente &lt;math display=&quot;inline&quot;&gt;m = \frac{1}{2}\mu - \frac{1}{2}&lt;/math&gt; vel &lt;math display=&quot;inline&quot;&gt;= \frac{1}{2}\mu&lt;/math&gt;, prout &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; impar vel par est, comparatio huius functionis cum hac &lt;math display=&quot;block&quot;&gt;
X'' = \zeta'\sin x + \zeta''\sin 2x + \zeta'''\sin 3x + \text{etc.} + \zeta^\mu \sin \mu x
&lt;/math&gt; quam in art. praec. eruimus, hasce aequationes suppeditant: &lt;math display=&quot;block&quot;&gt;
\begin{align}
 &amp;  &amp;  &amp; \zeta^{\mu}=\gamma\,\text{cosec}\,\mu a \\
\zeta' &amp;= \delta' + \gamma' \text{cotg } \mu a, &amp; \zeta'^{\mu-1} &amp;= \gamma' \text{cosec } \mu a \\
\zeta'' &amp;= \delta'' + \gamma'' \text{cotg } \mu a, &amp; \zeta'^{\mu-2} &amp;= \gamma'' \text{cosec } \mu a \\
\zeta''' &amp;= \delta''' + \gamma''' \text{cotg } \mu a, &amp; \zeta'^{\mu-3} &amp;= \gamma''' \text{cosec } \mu a \\
&amp;\text{etc. usque ad} \\
(\tfrac{1}{2})\zeta^m &amp;= \delta^m + \gamma^m \text{cotg } \mu a, &amp; (\tfrac{1}{2})\zeta^{\mu-m} &amp;= \gamma^m \text{cosec } \mu a
\end{align}
&lt;/math&gt;

ubi factor &lt;math display=&quot;inline&quot;&gt;\frac{1}{2}&lt;/math&gt; coëfficientibus &lt;math display=&quot;inline&quot;&gt;\zeta^m, \zeta^{\mu-m}&lt;/math&gt; praepositus pro eo tantum casu valet, ubi &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; par est, pro altero vero, ubi &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; impar est. omitti debet. In casu itaque posteriore ubi &lt;math display=&quot;inline&quot;&gt;\mu - m = m + 1&lt;/math&gt;, pro quovis coëfficiente &lt;math display=&quot;inline&quot;&gt;\zeta', \zeta''&lt;/math&gt; etc. valorem unum per &lt;math display=&quot;inline&quot;&gt;\gamma, \gamma', \gamma''&lt;/math&gt; etc. et &lt;math display=&quot;inline&quot;&gt;\delta', \delta''&lt;/math&gt; etc. habemus; in priore vero, ubi &lt;math display=&quot;inline&quot;&gt;\mu - m = m&lt;/math&gt;, pro coëfficiente &lt;math display=&quot;inline&quot;&gt;\zeta^{\frac{1}{2}\mu}&lt;/math&gt; duos aequales, pro quibus etiam valor &lt;math display=&quot;inline&quot;&gt;\delta^{\frac{1}{2}\mu} + \gamma^{\frac{1}{2}\mu} \cot \mu a + \gamma^{\frac{1}{2}\mu} \csc \mu a&lt;/math&gt; adoptari potest. Hinc colligitur

&lt;math display=&quot;block&quot;&gt;
\begin{align}
X'' &amp;= X' - \gamma - \gamma'\cos x - \gamma''\cos 2x - \text{etc.} - \gamma^m \cos mx \\
&amp;+ \text{cotg } \mu a (\gamma'\sin x + \gamma''\sin 2x + \gamma'''\sin 3x + \text{etc.} + \gamma^m \sin mx) \\
&amp;+ \text{cosec } \mu a (\gamma \sin \mu x + \gamma' \sin (\mu - 1)x + \gamma'' \sin (\mu - 2)x + \text{etc.} + \gamma^{\mu-m} \sin (\mu - m)x)
\end{align}
&lt;/math&gt;

Quamobrem ex &lt;math display=&quot;inline&quot;&gt;X'&lt;/math&gt; producitur &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt;, scribendo pro &lt;math display=&quot;inline&quot;&gt;\gamma&lt;/math&gt;, &lt;math display=&quot;inline&quot;&gt;\gamma \text{ cosec } \mu a \sin \mu x = \frac{\gamma \sin \mu x}{\sin \mu a}&lt;/math&gt; et, pro quovis termino &lt;math display=&quot;inline&quot;&gt;\gamma^\lambda \cos \lambda x&lt;/math&gt;&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>4wydchljs07p6irq7ju1w8wusa8idzw</sha1>
    </revision>
    <revision>
      <id>281419</id>
      <parentid>281418</parentid>
      <timestamp>2026-09-20T16:22:30Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <origin>281419</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="3232" sha1="0wf1l03dfx84lz6x7i5a0dkzuirx4ax" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;cum &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt; identica erit, mutando &lt;math display=&quot;inline&quot;&gt;t&lt;/math&gt; in &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;, unde coëfficientes &lt;math display=&quot;inline&quot;&gt;\zeta', \zeta'', \zeta'''&lt;/math&gt; etc. ipsis &lt;math display=&quot;inline&quot;&gt;\beta',\beta'',\beta'''&lt;/math&gt; etc. resp. aequales erunt.

&lt;br /&gt;
{{center|34.}}

Si ut in art. 31 omnibus valoribus propositis per functionem talem &lt;math display=&quot;block&quot;&gt;
\begin{align}
X' = \gamma + \gamma'\cos x + \gamma''\cos 2x + \text{etc.} + \gamma^m \cos mx \\
+ \delta'\sin x + \delta''\sin 2x + \text{etc.} + \delta^m \sin mx
\end{align}
&lt;/math&gt; satisfactum est, existente &lt;math display=&quot;inline&quot;&gt;m = \frac{1}{2}\mu - \frac{1}{2}&lt;/math&gt; vel &lt;math display=&quot;inline&quot;&gt;= \frac{1}{2}\mu&lt;/math&gt;, prout &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; impar vel par est, comparatio huius functionis cum hac &lt;math display=&quot;block&quot;&gt;
X'' = \zeta'\sin x + \zeta''\sin 2x + \zeta'''\sin 3x + \text{etc.} + \zeta^\mu \sin \mu x
&lt;/math&gt; quam in art. praec. eruimus, hasce aequationes suppeditant: &lt;math display=&quot;block&quot;&gt;
\begin{align}
 &amp;  &amp;  &amp; \zeta^{\mu}=\gamma\,\text{cosec}\,\mu a \\
\zeta' &amp;= \delta' + \gamma' \text{cotg } \mu a, &amp; \zeta^{\mu-1} &amp;= \gamma' \text{cosec } \mu a \\
\zeta'' &amp;= \delta'' + \gamma'' \text{cotg } \mu a, &amp; \zeta^{\mu-2} &amp;= \gamma'' \text{cosec } \mu a \\
\zeta''' &amp;= \delta''' + \gamma''' \text{cotg } \mu a, &amp; \zeta^{\mu-3} &amp;= \gamma''' \text{cosec } \mu a \\
&amp;\text{etc. usque ad} \\
(\tfrac{1}{2})\zeta^m &amp;= \delta^m + \gamma^m \text{cotg } \mu a, &amp; (\tfrac{1}{2})\zeta^{\mu-m} &amp;= \gamma^m \text{cosec } \mu a
\end{align}
&lt;/math&gt;

ubi factor &lt;math display=&quot;inline&quot;&gt;\frac{1}{2}&lt;/math&gt; coëfficientibus &lt;math display=&quot;inline&quot;&gt;\zeta^m, \zeta^{\mu-m}&lt;/math&gt; praepositus pro eo tantum casu valet, ubi &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; par est, pro altero vero, ubi &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; impar est. omitti debet. In casu itaque posteriore ubi &lt;math display=&quot;inline&quot;&gt;\mu - m = m + 1&lt;/math&gt;, pro quovis coëfficiente &lt;math display=&quot;inline&quot;&gt;\zeta', \zeta''&lt;/math&gt; etc. valorem unum per &lt;math display=&quot;inline&quot;&gt;\gamma, \gamma', \gamma''&lt;/math&gt; etc. et &lt;math display=&quot;inline&quot;&gt;\delta', \delta''&lt;/math&gt; etc. habemus; in priore vero, ubi &lt;math display=&quot;inline&quot;&gt;\mu - m = m&lt;/math&gt;, pro coëfficiente &lt;math display=&quot;inline&quot;&gt;\zeta^{\frac{1}{2}\mu}&lt;/math&gt; duos aequales, pro quibus etiam valor &lt;math display=&quot;inline&quot;&gt;\delta^{\frac{1}{2}\mu} + \gamma^{\frac{1}{2}\mu} \cot \mu a + \gamma^{\frac{1}{2}\mu} \csc \mu a&lt;/math&gt; adoptari potest. Hinc colligitur

&lt;math display=&quot;block&quot;&gt;
\begin{align}
X'' &amp;= X' - \gamma - \gamma'\cos x - \gamma''\cos 2x - \text{etc.} - \gamma^m \cos mx \\
&amp;+ \text{cotg } \mu a (\gamma'\sin x + \gamma''\sin 2x + \gamma'''\sin 3x + \text{etc.} + \gamma^m \sin mx) \\
&amp;+ \text{cosec } \mu a (\gamma \sin \mu x + \gamma' \sin (\mu - 1)x + \gamma'' \sin (\mu - 2)x + \text{etc.} + \gamma^{\mu-m} \sin (\mu - m)x)
\end{align}
&lt;/math&gt;

Quamobrem ex &lt;math display=&quot;inline&quot;&gt;X'&lt;/math&gt; producitur &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt;, scribendo pro &lt;math display=&quot;inline&quot;&gt;\gamma&lt;/math&gt;, &lt;math display=&quot;inline&quot;&gt;\gamma \text{ cosec } \mu a \sin \mu x = \frac{\gamma \sin \mu x}{\sin \mu a}&lt;/math&gt; et, pro quovis termino &lt;math display=&quot;inline&quot;&gt;\gamma^\lambda \cos \lambda x&lt;/math&gt;&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>0wf1l03dfx84lz6x7i5a0dkzuirx4ax</sha1>
    </revision>
    <revision>
      <id>281420</id>
      <parentid>281419</parentid>
      <timestamp>2026-09-20T16:24:22Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <origin>281420</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="3242" sha1="o7yjrgfi2lci3k7hkdvzbq6g9v8gylo" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;cum &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt; identica erit, mutando &lt;math display=&quot;inline&quot;&gt;t&lt;/math&gt; in &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;, unde coëfficientes &lt;math display=&quot;inline&quot;&gt;\zeta', \zeta'', \zeta'''&lt;/math&gt; etc. ipsis &lt;math display=&quot;inline&quot;&gt;\beta',\beta'',\beta'''&lt;/math&gt; etc. resp. aequales erunt.

&lt;br /&gt;
{{center|34.}}

Si ut in art. 31 omnibus valoribus propositis per functionem talem &lt;math display=&quot;block&quot;&gt;
\begin{align}
X' = \gamma + \gamma'\cos x + \gamma''\cos 2x + \text{etc.} + \gamma^m \cos mx \\
+ \delta'\sin x + \delta''\sin 2x + \text{etc.} + \delta^m \sin mx
\end{align}
&lt;/math&gt; satisfactum est, existente &lt;math display=&quot;inline&quot;&gt;m = \frac{1}{2}\mu - \frac{1}{2}&lt;/math&gt; vel &lt;math display=&quot;inline&quot;&gt;= \frac{1}{2}\mu&lt;/math&gt;, prout &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; impar vel par est, comparatio huius functionis cum hac &lt;math display=&quot;block&quot;&gt;
X'' = \zeta'\sin x + \zeta''\sin 2x + \zeta'''\sin 3x + \text{etc.} + \zeta^\mu \sin \mu x
&lt;/math&gt; quam in art. praec. eruimus, hasce aequationes suppeditant: &lt;math display=&quot;block&quot;&gt;
\begin{align}
 &amp;  &amp;  &amp; \zeta^{\mu}=\gamma\,\text{cosec}\,\mu a \\
\zeta' &amp;= \delta' + \gamma' \text{cotg } \mu a, &amp; \zeta^{\mu-1} &amp;= \gamma' \text{cosec } \mu a \\
\zeta'' &amp;= \delta'' + \gamma'' \text{cotg } \mu a, &amp; \zeta^{\mu-2} &amp;= \gamma'' \text{cosec } \mu a \\
\zeta''' &amp;= \delta''' + \gamma''' \text{cotg } \mu a, &amp; \zeta^{\mu-3} &amp;= \gamma''' \text{cosec } \mu a \\
&amp;\text{etc. usque ad} \\
(\tfrac{1}{2})\zeta^m &amp;= \delta^m + \gamma^m \text{cotg } \mu a, &amp; (\tfrac{1}{2})\zeta^{\mu-m} &amp;= \gamma^m \text{cosec } \mu a
\end{align}
&lt;/math&gt;

ubi factor &lt;math display=&quot;inline&quot;&gt;\frac{1}{2}&lt;/math&gt; coëfficientibus &lt;math display=&quot;inline&quot;&gt;\zeta^m, \zeta^{\mu-m}&lt;/math&gt; praepositus pro eo tantum casu valet, ubi &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; par est, pro altero vero, ubi &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; impar est. omitti debet. In casu itaque posteriore ubi &lt;math display=&quot;inline&quot;&gt;\mu - m = m + 1&lt;/math&gt;, pro quovis coëfficiente &lt;math display=&quot;inline&quot;&gt;\zeta', \zeta''&lt;/math&gt; etc. valorem unum per &lt;math display=&quot;inline&quot;&gt;\gamma, \gamma', \gamma''&lt;/math&gt; etc. et &lt;math display=&quot;inline&quot;&gt;\delta', \delta''&lt;/math&gt; etc. habemus; in priore vero, ubi &lt;math display=&quot;inline&quot;&gt;\mu - m = m&lt;/math&gt;, pro coëfficiente &lt;math display=&quot;inline&quot;&gt;\zeta^{\frac{1}{2}\mu}&lt;/math&gt; duos aequales, pro quibus etiam valor &lt;math display=&quot;inline&quot;&gt;\delta^{\frac{1}{2}\mu} + \gamma^{\frac{1}{2}\mu}\,\text{cotg}\, \mu a + \gamma^{\frac{1}{2}\mu} \csc \mu a&lt;/math&gt; adoptari potest. Hinc colligitur

&lt;math display=&quot;block&quot;&gt;
\begin{align}
X'' &amp;= X' - \gamma - \gamma'\cos x - \gamma''\cos 2x - \text{etc.} - \gamma^m \cos mx \\
&amp;+ \text{cotg } \mu a (\gamma'\sin x + \gamma''\sin 2x + \gamma'''\sin 3x + \text{etc.} + \gamma^m \sin mx) \\
&amp;+ \text{cosec } \mu a (\gamma \sin \mu x + \gamma' \sin (\mu - 1)x + \gamma'' \sin (\mu - 2)x + \text{etc.} + \gamma^{\mu-m} \sin (\mu - m)x)
\end{align}
&lt;/math&gt;

Quamobrem ex &lt;math display=&quot;inline&quot;&gt;X'&lt;/math&gt; producitur &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt;, scribendo pro &lt;math display=&quot;inline&quot;&gt;\gamma&lt;/math&gt;, &lt;math display=&quot;inline&quot;&gt;\gamma \text{ cosec } \mu a \sin \mu x = \frac{\gamma \sin \mu x}{\sin \mu a}&lt;/math&gt; et, pro quovis termino &lt;math display=&quot;inline&quot;&gt;\gamma^\lambda \cos \lambda x&lt;/math&gt;&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>o7yjrgfi2lci3k7hkdvzbq6g9v8gylo</sha1>
    </revision>
    <revision>
      <id>281421</id>
      <parentid>281420</parentid>
      <timestamp>2026-09-20T16:24:50Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <origin>281421</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="3254" sha1="dqe7ernlnx8kznq88pdbw1odpi6nssw" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;cum &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt; identica erit, mutando &lt;math display=&quot;inline&quot;&gt;t&lt;/math&gt; in &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;, unde coëfficientes &lt;math display=&quot;inline&quot;&gt;\zeta', \zeta'', \zeta'''&lt;/math&gt; etc. ipsis &lt;math display=&quot;inline&quot;&gt;\beta',\beta'',\beta'''&lt;/math&gt; etc. resp. aequales erunt.

&lt;br /&gt;
{{center|34.}}

Si ut in art. 31 omnibus valoribus propositis per functionem talem &lt;math display=&quot;block&quot;&gt;
\begin{align}
X' = \gamma + \gamma'\cos x + \gamma''\cos 2x + \text{etc.} + \gamma^m \cos mx \\
+ \delta'\sin x + \delta''\sin 2x + \text{etc.} + \delta^m \sin mx
\end{align}
&lt;/math&gt; satisfactum est, existente &lt;math display=&quot;inline&quot;&gt;m = \frac{1}{2}\mu - \frac{1}{2}&lt;/math&gt; vel &lt;math display=&quot;inline&quot;&gt;= \frac{1}{2}\mu&lt;/math&gt;, prout &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; impar vel par est, comparatio huius functionis cum hac &lt;math display=&quot;block&quot;&gt;
X'' = \zeta'\sin x + \zeta''\sin 2x + \zeta'''\sin 3x + \text{etc.} + \zeta^\mu \sin \mu x
&lt;/math&gt; quam in art. praec. eruimus, hasce aequationes suppeditant: &lt;math display=&quot;block&quot;&gt;
\begin{align}
 &amp;  &amp;  &amp; \zeta^{\mu}=\gamma\,\text{cosec}\,\mu a \\
\zeta' &amp;= \delta' + \gamma' \text{cotg } \mu a, &amp; \zeta^{\mu-1} &amp;= \gamma' \text{cosec } \mu a \\
\zeta'' &amp;= \delta'' + \gamma'' \text{cotg } \mu a, &amp; \zeta^{\mu-2} &amp;= \gamma'' \text{cosec } \mu a \\
\zeta''' &amp;= \delta''' + \gamma''' \text{cotg } \mu a, &amp; \zeta^{\mu-3} &amp;= \gamma''' \text{cosec } \mu a \\
&amp;\text{etc. usque ad} \\
(\tfrac{1}{2})\zeta^m &amp;= \delta^m + \gamma^m \text{cotg } \mu a, &amp; (\tfrac{1}{2})\zeta^{\mu-m} &amp;= \gamma^m \text{cosec } \mu a
\end{align}
&lt;/math&gt;

ubi factor &lt;math display=&quot;inline&quot;&gt;\frac{1}{2}&lt;/math&gt; coëfficientibus &lt;math display=&quot;inline&quot;&gt;\zeta^m, \zeta^{\mu-m}&lt;/math&gt; praepositus pro eo tantum casu valet, ubi &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; par est, pro altero vero, ubi &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; impar est. omitti debet. In casu itaque posteriore ubi &lt;math display=&quot;inline&quot;&gt;\mu - m = m + 1&lt;/math&gt;, pro quovis coëfficiente &lt;math display=&quot;inline&quot;&gt;\zeta', \zeta''&lt;/math&gt; etc. valorem unum per &lt;math display=&quot;inline&quot;&gt;\gamma, \gamma', \gamma''&lt;/math&gt; etc. et &lt;math display=&quot;inline&quot;&gt;\delta', \delta''&lt;/math&gt; etc. habemus; in priore vero, ubi &lt;math display=&quot;inline&quot;&gt;\mu - m = m&lt;/math&gt;, pro coëfficiente &lt;math display=&quot;inline&quot;&gt;\zeta^{\frac{1}{2}\mu}&lt;/math&gt; duos aequales, pro quibus etiam valor &lt;math display=&quot;inline&quot;&gt;\delta^{\frac{1}{2}\mu} + \gamma^{\frac{1}{2}\mu}\,\text{cotg}\, \mu a + \gamma^{\frac{1}{2}\mu} \,\text{cosec}\, \mu a&lt;/math&gt; adoptari potest. Hinc colligitur

&lt;math display=&quot;block&quot;&gt;
\begin{align}
X'' &amp;= X' - \gamma - \gamma'\cos x - \gamma''\cos 2x - \text{etc.} - \gamma^m \cos mx \\
&amp;+ \text{cotg } \mu a (\gamma'\sin x + \gamma''\sin 2x + \gamma'''\sin 3x + \text{etc.} + \gamma^m \sin mx) \\
&amp;+ \text{cosec } \mu a (\gamma \sin \mu x + \gamma' \sin (\mu - 1)x + \gamma'' \sin (\mu - 2)x + \text{etc.} + \gamma^{\mu-m} \sin (\mu - m)x)
\end{align}
&lt;/math&gt;

Quamobrem ex &lt;math display=&quot;inline&quot;&gt;X'&lt;/math&gt; producitur &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt;, scribendo pro &lt;math display=&quot;inline&quot;&gt;\gamma&lt;/math&gt;, &lt;math display=&quot;inline&quot;&gt;\gamma \text{ cosec } \mu a \sin \mu x = \frac{\gamma \sin \mu x}{\sin \mu a}&lt;/math&gt; et, pro quovis termino &lt;math display=&quot;inline&quot;&gt;\gamma^\lambda \cos \lambda x&lt;/math&gt;&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>dqe7ernlnx8kznq88pdbw1odpi6nssw</sha1>
    </revision>
    <revision>
      <id>281422</id>
      <parentid>281421</parentid>
      <timestamp>2026-09-20T16:25:54Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Emendata */</comment>
      <origin>281422</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="3254" sha1="g4iflanh8asoc5p20hvyagi0eyealsr" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;3&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;cum &lt;math display=&quot;inline&quot;&gt;X&lt;/math&gt; identica erit, mutando &lt;math display=&quot;inline&quot;&gt;t&lt;/math&gt; in &lt;math display=&quot;inline&quot;&gt;x&lt;/math&gt;, unde coëfficientes &lt;math display=&quot;inline&quot;&gt;\zeta', \zeta'', \zeta'''&lt;/math&gt; etc. ipsis &lt;math display=&quot;inline&quot;&gt;\beta',\beta'',\beta'''&lt;/math&gt; etc. resp. aequales erunt.

&lt;br /&gt;
{{center|34.}}

Si ut in art. 31 omnibus valoribus propositis per functionem talem &lt;math display=&quot;block&quot;&gt;
\begin{align}
X' = \gamma + \gamma'\cos x + \gamma''\cos 2x + \text{etc.} + \gamma^m \cos mx \\
+ \delta'\sin x + \delta''\sin 2x + \text{etc.} + \delta^m \sin mx
\end{align}
&lt;/math&gt; satisfactum est, existente &lt;math display=&quot;inline&quot;&gt;m = \frac{1}{2}\mu - \frac{1}{2}&lt;/math&gt; vel &lt;math display=&quot;inline&quot;&gt;= \frac{1}{2}\mu&lt;/math&gt;, prout &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; impar vel par est, comparatio huius functionis cum hac &lt;math display=&quot;block&quot;&gt;
X'' = \zeta'\sin x + \zeta''\sin 2x + \zeta'''\sin 3x + \text{etc.} + \zeta^\mu \sin \mu x
&lt;/math&gt; quam in art. praec. eruimus, hasce aequationes suppeditant: &lt;math display=&quot;block&quot;&gt;
\begin{align}
 &amp;  &amp;  &amp; \zeta^{\mu}=\gamma\,\text{cosec}\,\mu a \\
\zeta' &amp;= \delta' + \gamma' \text{cotg } \mu a, &amp; \zeta^{\mu-1} &amp;= \gamma' \text{cosec } \mu a \\
\zeta'' &amp;= \delta'' + \gamma'' \text{cotg } \mu a, &amp; \zeta^{\mu-2} &amp;= \gamma'' \text{cosec } \mu a \\
\zeta''' &amp;= \delta''' + \gamma''' \text{cotg } \mu a, &amp; \zeta^{\mu-3} &amp;= \gamma''' \text{cosec } \mu a \\
&amp;\text{etc. usque ad} \\
(\tfrac{1}{2})\zeta^m &amp;= \delta^m + \gamma^m \text{cotg } \mu a, &amp; (\tfrac{1}{2})\zeta^{\mu-m} &amp;= \gamma^m \text{cosec } \mu a
\end{align}
&lt;/math&gt;

ubi factor &lt;math display=&quot;inline&quot;&gt;\frac{1}{2}&lt;/math&gt; coëfficientibus &lt;math display=&quot;inline&quot;&gt;\zeta^m, \zeta^{\mu-m}&lt;/math&gt; praepositus pro eo tantum casu valet, ubi &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; par est, pro altero vero, ubi &lt;math display=&quot;inline&quot;&gt;\mu&lt;/math&gt; impar est. omitti debet. In casu itaque posteriore ubi &lt;math display=&quot;inline&quot;&gt;\mu - m = m + 1&lt;/math&gt;, pro quovis coëfficiente &lt;math display=&quot;inline&quot;&gt;\zeta', \zeta''&lt;/math&gt; etc. valorem unum per &lt;math display=&quot;inline&quot;&gt;\gamma, \gamma', \gamma''&lt;/math&gt; etc. et &lt;math display=&quot;inline&quot;&gt;\delta', \delta''&lt;/math&gt; etc. habemus; in priore vero, ubi &lt;math display=&quot;inline&quot;&gt;\mu - m = m&lt;/math&gt;, pro coëfficiente &lt;math display=&quot;inline&quot;&gt;\zeta^{\frac{1}{2}\mu}&lt;/math&gt; duos aequales, pro quibus etiam valor &lt;math display=&quot;inline&quot;&gt;\delta^{\frac{1}{2}\mu} + \gamma^{\frac{1}{2}\mu}\,\text{cotg}\, \mu a + \gamma^{\frac{1}{2}\mu} \,\text{cosec}\, \mu a&lt;/math&gt; adoptari potest. Hinc colligitur

&lt;math display=&quot;block&quot;&gt;
\begin{align}
X'' &amp;= X' - \gamma - \gamma'\cos x - \gamma''\cos 2x - \text{etc.} - \gamma^m \cos mx \\
&amp;+ \text{cotg } \mu a (\gamma'\sin x + \gamma''\sin 2x + \gamma'''\sin 3x + \text{etc.} + \gamma^m \sin mx) \\
&amp;+ \text{cosec } \mu a (\gamma \sin \mu x + \gamma' \sin (\mu - 1)x + \gamma'' \sin (\mu - 2)x + \text{etc.} + \gamma^{\mu-m} \sin (\mu - m)x)
\end{align}
&lt;/math&gt;

Quamobrem ex &lt;math display=&quot;inline&quot;&gt;X'&lt;/math&gt; producitur &lt;math display=&quot;inline&quot;&gt;X''&lt;/math&gt;, scribendo pro &lt;math display=&quot;inline&quot;&gt;\gamma&lt;/math&gt;, &lt;math display=&quot;inline&quot;&gt;\gamma \text{ cosec } \mu a \sin \mu x = \frac{\gamma \sin \mu x}{\sin \mu a}&lt;/math&gt; et, pro quovis termino &lt;math display=&quot;inline&quot;&gt;\gamma^\lambda \cos \lambda x&lt;/math&gt;&lt;noinclude&gt;&lt;/noinclude&gt;</text>
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  <page>
    <title>Liber:RogerCotes HarmoniaMensurarum.djvu</title>
    <ns>106</ns>
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    <revision>
      <id>281491</id>
      <parentid>281283</parentid>
      <timestamp>2026-09-20T20:31:15Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>13304</id>
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|Title=Harmonia Mensurarum
|Language=la
|Volume=
|Author=[[Scriptor:Rogerius Cotesius|Rogerius Cotesius]]
|Translator=
|Editor=
|Illustrator=
|Publisher=Robertus Smith
|Address=Cantabrigia
|Year=1722
|Source=
|Image=9
|Progress=X
|Pages=&lt;pagelist /&gt;
|Remarks=
|Header=
|Footer=
}}
[[Categoria:Mathematica]]</text>
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  </page>
  <page>
    <title>Ordo diurnus</title>
    <ns>0</ns>
    <id>86251</id>
    <revision>
      <id>281392</id>
      <parentid>281389</parentid>
      <timestamp>2026-09-20T12:01:25Z</timestamp>
      <contributor>
        <username>Coronus Tenebrosius</username>
        <id>31709</id>
      </contributor>
      <comment>/* ORDO DIURNUS */</comment>
      <origin>281392</origin>
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      <text bytes="673" sha1="8lr0ho4jomwk4gxvc79njzqgs9jhjjl" xml:space="preserve">[[Carmina non adlecta]]

== ORDO DIURNUS ==

&lt;poem&gt;
Ad alia opera, quae mihi agenda sunt,
hodie me confero.
Exspectat me opus cloacarum
(in sensu verborum simplicissimo).
Et verba, quid valent, quid possunt,
hae verba, princeps ... ? 

Corpus bene erit motum,
et hauriet solem ac lucem 
post nocturnam pluviam.
Humida est terra. Non sentit
reditum meum neque propinquitatem.

Et vere, et metaphore vere,
omnibus in gradibus,
tempus est
proficisci.

In luto stans, radiis Solis perfusus,
sentio et scio:
tota Gea est somnium vivum.

Quam aliam rem veram,
praeter eam quae ex Terra oritur,
nos duo omnino scimus?

Aliquid audivimus, aliquid sentimus,
sed non vidimus.
&lt;/poem&gt;</text>
      <sha1>8lr0ho4jomwk4gxvc79njzqgs9jhjjl</sha1>
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    <revision>
      <id>281393</id>
      <parentid>281392</parentid>
      <timestamp>2026-09-20T12:08:29Z</timestamp>
      <contributor>
        <username>Coronus Tenebrosius</username>
        <id>31709</id>
      </contributor>
      <comment>/* ORDO DIURNUS */</comment>
      <origin>281393</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="672" sha1="97z7cgvyxsdq2slqioj9uk9mwean9n8" xml:space="preserve">[[Carmina non adlecta]]

== ORDO DIURNUS ==

&lt;poem&gt;
Ad alia opera, quae mihi agenda sunt,
hodie me confero.
Exspectat me opus cloacarum
(in sensu verborum simplicissimo).
Et verba, quid valent, quid possunt,
hae verba, princeps ...? 

Corpus bene erit motum,
et hauriet solem ac lucem 
post nocturnam pluviam.
Humida est terra. Non sentit
reditum meum neque propinquitatem.

Et vere, et metaphore vere,
omnibus in gradibus,
tempus est
proficisci.

In luto stans, radiis Solis perfusus,
sentio et scio:
tota Gea est somnium vivum.

Quam aliam rem veram,
praeter eam quae ex Terra oritur,
nos duo omnino scimus?

Aliquid audivimus, aliquid sentimus,
sed non vidimus.
&lt;/poem&gt;</text>
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  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/16</title>
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    <id>86252</id>
    <revision>
      <id>281423</id>
      <timestamp>2026-09-20T16:28:33Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Nondum emendata */ Paginam instituit, scribens 'Punicam, &amp;amp; agnum adeò macrum, ut eius exta in sole etiam viui inspicere liceret: pelluceve dixit Plautus in Rudente &amp;amp; Aulularia. Cur autem in aquas simulachra maiora veris cernantur, disputat Macrobius libro septimo Saturnalium.  Quod etiam terra sit rotûda, patet sic. Signa &amp;amp; stel læ non equaliter oriuntur, &amp;amp; occidunt omnibus hominibus vbique existentibus: sed prius oriuntur. &amp;amp; occidunt illis, qui sunt versus orientem. Et...</comment>
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      <text bytes="518" sha1="frkerhgbh7udhk9fnnnqyxi51q4ze69" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;Punicam, &amp;amp; agnum adeò macrum, ut eius exta in sole etiam viui inspicere liceret: pelluceve dixit Plautus in Rudente &amp;amp; Aulularia. Cur autem in aquas simulachra maiora veris cernantur, disputat Macrobius libro septimo Saturnalium.

Quod etiam terra sit rotûda, patet sic. Signa &amp;amp; stel læ non equaliter oriuntur, &amp;amp; occidunt omnibus hominibus vbique existentibus: sed prius oriuntur. &amp;amp; occidunt illis, qui sunt versus orientem. Et quòd ci tius &amp;amp; tardius oriûtur, &amp;amp; occidunt quibusdam, cau-&lt;noinclude&gt;&lt;/noinclude&gt;</text>
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  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/17</title>
    <ns>104</ns>
    <id>86253</id>
    <revision>
      <id>281424</id>
      <timestamp>2026-09-20T16:29:10Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Nondum emendata */ Paginam instituit, scribens 'sa est tumor terre, quod bene patet per ea quæ fiunc in sublimi, una enim &amp;amp; eadem eclipsis Lunæ numero, quæ apparent nobis in prima hora noctis, apparet orientalibus circa horam noctis tertiam. Vnde constat, quod illis prius fuit nox, &amp;amp; sol prius occidit quam nobis. Cuus rei causa est tum tumor terre.  Quod etiam terra habeat tumorem à septetrio: ne in austrum, &amp;amp; contrà, sic patet. Hominibus'</comment>
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      <text bytes="411" sha1="1wbktb8tg53zwwhjvaoc630nd85nhjx" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;sa est tumor terre, quod bene patet per ea quæ fiunc in sublimi, una enim &amp;amp; eadem eclipsis Lunæ numero, quæ apparent nobis in prima hora noctis, apparet orientalibus circa horam noctis tertiam. Vnde constat, quod illis prius fuit nox, &amp;amp; sol prius occidit quam nobis. Cuus rei causa est tum tumor terre.

Quod etiam terra habeat tumorem à septetrio: ne in austrum, &amp;amp; contrà, sic patet. Hominibus&lt;noinclude&gt;&lt;/noinclude&gt;</text>
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  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/18</title>
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    <id>86254</id>
    <revision>
      <id>281425</id>
      <timestamp>2026-09-20T16:29:32Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
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      <comment>/* Nondum emendata */ Paginam instituit, scribens 'existentibus versus septentrionem, quædam stellæ sunt sempiternæ apparitionis, scilicet quæ propinquæ accedunt ad polum arcticum, aliæ vero sunt sempiternæ occultationis, sicut illæ quæ sunt propinquæ polo antarctico. Si igitur aliquis procederet a septentrione versus austrum, in tantum posset procedere, quod stellæ, quæ prius erant ei sempiternæ apparitionis, ci iam tenderent in occasum, &amp;amp; quato magis accederet ad austrum, tanto plus mov...</comment>
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      <text bytes="1090" sha1="6w8fa5b5vrqvk8poknec0jxmfu1hwez" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;existentibus versus septentrionem, quædam stellæ sunt sempiternæ apparitionis, scilicet quæ propinquæ accedunt ad polum arcticum, aliæ vero sunt sempiternæ occultationis, sicut illæ quæ sunt propinquæ polo antarctico. Si igitur aliquis procederet a septentrione versus austrum, in tantum posset procedere, quod stellæ, quæ prius erant ei sempiternæ apparitionis, ci iam tenderent in occasum, &amp;amp; quato magis accederet ad austrum, tanto plus moveretur in occasum. Ille iterum idem homo posset videre stellas, quæ prius fuerant ei sempiternæ occultationis. Et e converso contingeret alicui procedenti ab austro versus septentrionem. Huius autem rei causa est tantum tumor terræ.

Item si terra esset plana ab oriente in occidentem, tam cito oriretur stellæ occidentalibus, quam orientalibus, quod patet esse falsum.

Item si terra esset plana a septentrione in austrum, &amp;amp; contrà, stellæ, quæ essent alicui sempiternæ apparitionis, semper apparerent eidem, quo cunque procederet, quod falsum est, sed quod plana sit, pra nimia eius quantitate hominum visui appareat.&lt;noinclude&gt;&lt;/noinclude&gt;</text>
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  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/19</title>
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    <revision>
      <id>281426</id>
      <timestamp>2026-09-20T16:30:05Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
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      <comment>/* Nondum emendata */ Paginam instituit, scribens ' QVOD AQVA SIT  ROTUNDA.  Qvod autem aqua habeat tumorem, &amp;amp; accedat ad rotunditatem, sic patet. Ponatur signum in littore maris, &amp;amp; exeat naus à portu, &amp;amp; in tantum elongetur, quod oculus existentis iuxta pedem mali non possit videri signum, stante vero naui oculus eiusdem existentis in summitate mali, bene videbit signum illud. Sed oculus existentis iuxta pede mali, melius deberet videre signum,'</comment>
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      <text bytes="410" sha1="ehc0mapj3t6t9bn6au01w0q8qyqxucz" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;
QVOD AQVA SIT

ROTUNDA.

Qvod autem aqua habeat tumorem, &amp;amp; accedat ad rotunditatem, sic patet. Ponatur signum in littore maris, &amp;amp; exeat naus à portu, &amp;amp; in tantum elongetur, quod oculus existentis iuxta pedem mali non possit videri signum, stante vero naui oculus eiusdem existentis in summitate mali, bene videbit signum illud. Sed oculus existentis iuxta pede mali, melius deberet videre signum,&lt;noinclude&gt;&lt;/noinclude&gt;</text>
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  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/20</title>
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    <id>86256</id>
    <revision>
      <id>281427</id>
      <timestamp>2026-09-20T16:30:45Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Nondum emendata */ Paginam instituit, scribens 'quam qui est in summitate mali, sicut patet per lineas ductas ab utroque ad signum. Et nulla alia huius rei causa est, quam tumor aquæ. Excludantur enim omnia alia impedimenta, sicut nebulæ &amp;amp; vapores ascendentes.  Item, cum aqua sit corpus homogeneum, totum cum partibus eiusdem crit rationis, sed partes aquæ (sicut in guttulis &amp;amp; roribus herbarum accidit) rotundam naturaliter appetut formam, ergo &amp;amp; totum, cuius sunt partes.  SCHOLI...</comment>
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      <text bytes="1035" sha1="7bohyumumugjl9ovjbyr4t7rroum1tj" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;quam qui est in summitate mali, sicut patet per lineas ductas ab utroque ad signum. Et nulla alia huius rei causa est, quam tumor aquæ. Excludantur enim omnia alia impedimenta, sicut nebulæ &amp;amp; vapores ascendentes.

Item, cum aqua sit corpus homogeneum, totum cum partibus eiusdem crit rationis, sed partes aquæ (sicut in guttulis &amp;amp; roribus herbarum accidit) rotundam naturaliter appetut formam, ergo &amp;amp; totum, cuius sunt partes.

SCHOLION VINET I. Homogeneum, eiusdem generis &amp;amp; naturæ, ex Græco ὁμογενής. Terra autem &amp;amp; aqua globum unum confluunt, quarnm dubitatum est aliquando utra maior esset. Quidam aquam putauerunt, sed si non fallunt, qui nuper orbem lustrauerunt descripseruntque, terra quam aquæ facies maior est, quod &amp;amp; Nonius noster monuit.

TERRAM ESSE CENTRUM 

mundi, immobilemque consistere. 

Qvod autem terra sit in medio firmamenti sita, sic patet. Existentibus in superficie terræ, stellæ apparent eiusmod quantitatis, siue sint in medio coeli, siue iuxta ortu, siue iuxta occa-&lt;noinclude&gt;&lt;/noinclude&gt;</text>
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  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/21</title>
    <ns>104</ns>
    <id>86257</id>
    <revision>
      <id>281428</id>
      <timestamp>2026-09-20T16:31:11Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Nondum emendata */ Paginam instituit, scribens 'sum: sum: &amp;amp; hoc ideo, quia æqualiter terra distat ab eis. Si n. terra magis accederet ad firmamentum in una parte quàm in alia, sequeretur qd’ aliquis existens in illa parte superficii terræ, quæ magis accederet ad firmamentum, non videret cœli medietaté. Sed hoc est cótra Ptolemæum &amp;amp; omnes Philosophos, dicentes, qv bicunq́; existat hó, lex signa ei oriuntur, &amp;amp; sex occidút, &amp;amp; medietas celi semper appareit ei, me dietas verò oc...</comment>
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      <text bytes="1103" sha1="p7a7s3kxoq8jpyf44wmxpsejfwyr4k4" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;sum: sum: &amp;amp; hoc ideo, quia æqualiter terra distat ab eis. Si n. terra magis accederet ad firmamentum in una parte quàm in alia, sequeretur qd’ aliquis existens in illa parte superficii terræ, quæ magis accederet ad firmamentum, non videret cœli medietaté. Sed hoc est cótra Ptolemæum &amp;amp; omnes Philosophos, dicentes, qv bicunq́; existat hó, lex signa ei oriuntur, &amp;amp; sex occidút, &amp;amp; medietas celi semper appareit ei, me dietas verò occultatur.

Illud item est signum, quòd terra sit tanquam centrum &amp;amp; punctus respectu firmamenti: quia si terra esset alicuius quantitatis respectu firmamenti: non contingeret medietatem cœli videri.

Item si intelligatur superficies plana super centrum terræ, diuidens eam in duo æqualia, &amp;amp; ipsum per consequens firma mérum. Occultus igitur existens in terræ centro videret medietaté celi: &amp;amp; idem existens in superficie terræ videret candem medietatem. Ex his colligit, qv insensibilis est quâtitas terræ, que est à superficie ad centrum. &amp;amp; p consequens quâti tas totius terræ insensibilis est respectu firmamenti.&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>p7a7s3kxoq8jpyf44wmxpsejfwyr4k4</sha1>
    </revision>
  </page>
  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/22</title>
    <ns>104</ns>
    <id>86258</id>
    <revision>
      <id>281429</id>
      <timestamp>2026-09-20T16:31:37Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Nondum emendata */ Paginam instituit, scribens 'Dicit etiam Alfraganus, &amp;amp; minima stellarum fixarum visu mutabilium, maior est tota terra: sed ipsa stella, respectu totius firmamenti, est sicut punctus &amp;amp; centrum: multò igitur fortius terra est punctus respectu firmamenti, cum sit minor ea.  Quòd autem terra in medio omnium teneatur immobilitèr, cùm sit summè grauis, sic persuaderè videtur eius grauitas. Omne graue naturaliter tendit ad centrum. Centrum quidem punctus in medio firmam...</comment>
      <origin>281429</origin>
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      <text bytes="1163" sha1="dje4l79gplg6ovsr5sywdmappz49wl8" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;Dicit etiam Alfraganus, &amp;amp; minima stellarum fixarum visu mutabilium, maior est tota terra: sed ipsa stella, respectu totius firmamenti, est sicut punctus &amp;amp; centrum: multò igitur fortius terra est punctus respectu firmamenti, cum sit minor ea.

Quòd autem terra in medio omnium teneatur immobilitèr, cùm sit summè grauis, sic persuaderè videtur eius grauitas. Omne graue naturaliter tendit ad centrum. Centrum quidem punctus in medio firmamenti. Terra igitur cum sit summè grauis, ad punctum illum naturaliter tendit.

Item, quicquid à medio mouetur, versus circumferentiam coeli ascendit: terra à medio mouetur: igitur ascendit quod pro impossibili relinquitur.

DE AMBITU TERRE ET DIA METRO.

T otius autem orbis terræ ambitus, authoritate Ambrosij, Theodosij, Macrobij, &amp;amp; Eratostenis Philosophorum, 252000. stadia continere diffinitur, vnicuiq́; quidem 360. partium zodiaci 700. stadia deputando. Sumpto n. astrolabio, vel quadrante in stellatæ noctis claritate, per vtrunque mediclinij foramen, polo perspecto notet graduum multitudo, in qua steterit mediclinum. Deinde procedat cosmimetra directe versus septentrionem à meridie, donec&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>dje4l79gplg6ovsr5sywdmappz49wl8</sha1>
    </revision>
  </page>
  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/23</title>
    <ns>104</ns>
    <id>86259</id>
    <revision>
      <id>281430</id>
      <timestamp>2026-09-20T16:32:11Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Nondum emendata */ Paginam instituit, scribens 'in alterius noctis claritate, viso ut prius polo, statuerit altius uno gradu mediclinium. Post hoc menfuretur huius itineris spatium, &amp;amp; inuenientur 790. stadia. Deinde datis vnicuique 360. graduum tot stadis, terreni orbis ambitus inuentus erit.  Ex his autem, iuxta circuli &amp;amp; diametri regulam diameter terræ sic inueniri poterit. Aufer vigesimam secundam partem de circuitu terræ &amp;amp; remanentis tertia pars, hoc est 80181. stadia &amp;amp;...</comment>
      <origin>281430</origin>
      <model>proofread-page</model>
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      <text bytes="1084" sha1="j3nuymhqvxj70mg4hven84ow2q7ou5v" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;in alterius noctis claritate, viso ut prius polo, statuerit altius uno gradu mediclinium. Post hoc menfuretur huius itineris spatium, &amp;amp; inuenientur 790. stadia. Deinde datis vnicuique 360. graduum tot stadis, terreni orbis ambitus inuentus erit.

Ex his autem, iuxta circuli &amp;amp; diametri regulam diameter terræ sic inueniri poterit. Aufer vigesimam secundam partem de circuitu terræ &amp;amp; remanentis tertia pars, hoc est 80181. stadia &amp;amp; semis, &amp;amp; tertia pars vnius stadij erit terreni orbis diameter siue spissitudo.

SCHOLION VINETI.

Ambrosius Theodosius Macrobius hic vnicus est autor, Macrobius scilicet ille, qui scripsit Saturnalium libros &amp;amp; commentaria in Somnium Scipionis, in quorum libro primo illa inuenies de terræ ambitu ex Eratosthene. Eratosthenis porro huius scripta Sacroboscus an viderit nescio: qua se ad nos pervenissent integra, sciremus, ambitumne terræ ducentorum &amp;amp; quinquaginta duorum milium stadiorum esse scripserit, quamodo traduit Plinius extremo libro secundo, Macrobius ille, Copella, &amp;amp; Sacroboscus: au verò ducentorum &amp;amp;&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>j3nuymhqvxj70mg4hven84ow2q7ou5v</sha1>
    </revision>
  </page>
  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/24</title>
    <ns>104</ns>
    <id>86260</id>
    <revision>
      <id>281431</id>
      <timestamp>2026-09-20T16:32:30Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Nondum emendata */ Paginam instituit, scribens 'quinquaginta solum, id est, μεριάδων ἐκουσι πέντε, habet Cleomedes libro primo. Huius certè vel solius auctoritas magni esse momenti debet, quod Eratosthenis rationes &amp;amp; demonstrationem adducat. Ptolomaus post Eratosthenem fuit qui pro septingentis quingenta tantum stadia terrestris meridiani tribuit vni cælestis meridiani gradui. Ita incomperta est terræ magnitudo propter metiendi difficultatem, seu illius mensuram terra, vt Eratosthe...'</comment>
      <origin>281431</origin>
      <model>proofread-page</model>
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      <text bytes="1573" sha1="29fz6g2hh3yvcnipzpqwsab89x123ej" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;quinquaginta solum, id est, μεριάδων ἐκουσι πέντε, habet Cleomedes libro primo. Huius certè vel solius auctoritas magni esse momenti debet, quod Eratosthenis rationes &amp;amp; demonstrationem adducat. Ptolomaus post Eratosthenem fuit qui pro septingentis quingenta tantum stadia terrestris meridiani tribuit vni cælestis meridiani gradui. Ita incomperta est terræ magnitudo propter metiendi difficultatem, seu illius mensuram terra, vt Eratosthenes, &amp;amp; alij, seu mari, vt Posidonius, eodem Cleomede autore, inire velis. Qui autem hic ambitus terræ vulgo dicitur, non solius est terra, sed terra &amp;amp; aqua simul qua vnam ambo dicta sunt sphæram constitueri: Cæterum de mensuris hos secerunt geometræ versiculos, memoria gratia.

Quattuor ex granis digitus componitur unus. Et quater in palmo digitus, quater in pede palmus. Quinque pedes passum faciunt, passus quoque centum. Viginti quinque stadium dant, sed miliare Octo dabunt stadia. duplicatum dat tibi leucam.

Leuca tamen Gallicum vocabulum, quo &amp;amp; Hispani vtuntur, non dua miliaria, sed vnum tantum &amp;amp; dimidium, id est, mille &amp;amp; quingentos passus continere definitur Ammiano Marcellino libro quinto decimo et sex todecimo. &amp;amp; Iornandi in Gothicis. Porrò rationem cer tam cuiusuis ambitus circuli ad suum diametrum geometræ haud dum potuerunt demonstrare. Accipitur tamen interim pro vera vulgo, tripla sesquiseptima, quomodo se habent, 22. ad 7. atque ita versiculi hi ex diametro persphæriam &amp;amp; contra, ex illa diametrum cuiusuis circuli minimo negotio invenire docent.&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>29fz6g2hh3yvcnipzpqwsab89x123ej</sha1>
    </revision>
    <revision>
      <id>281432</id>
      <parentid>281431</parentid>
      <timestamp>2026-09-20T16:32:54Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <origin>281432</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="1616" sha1="2qm7cmy3wplstdtk4bosobz35rwdcpl" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;quinquaginta solum, id est, μεριάδων ἐκουσι πέντε, habet Cleomedes libro primo. Huius certè vel solius auctoritas magni esse momenti debet, quod Eratosthenis rationes &amp;amp; demonstrationem adducat. Ptolomaus post Eratosthenem fuit qui pro septingentis quingenta tantum stadia terrestris meridiani tribuit vni cælestis meridiani gradui. Ita incomperta est terræ magnitudo propter metiendi difficultatem, seu illius mensuram terra, vt Eratosthenes, &amp;amp; alij, seu mari, vt Posidonius, eodem Cleomede autore, inire velis. Qui autem hic ambitus terræ vulgo dicitur, non solius est terra, sed terra &amp;amp; aqua simul qua vnam ambo dicta sunt sphæram constitueri: Cæterum de mensuris hos secerunt geometræ versiculos, memoria gratia.

Quattuor ex granis digitus componitur unus. Et quater in palmo digitus, quater in pede palmus. Quinque pedes passum faciunt, passus quoque centum. Viginti quinque stadium dant, sed miliare Octo dabunt stadia. duplicatum dat tibi leucam.

Leuca tamen Gallicum vocabulum, quo &amp;amp; Hispani vtuntur, non dua miliaria, sed vnum tantum &amp;amp; dimidium, id est, mille &amp;amp; quingentos passus continere definitur Ammiano Marcellino libro quinto decimo et sex todecimo. &amp;amp; Iornandi in Gothicis. Porrò rationem cer tam cuiusuis ambitus circuli ad suum diametrum geometræ haud dum potuerunt demonstrare. Accipitur tamen interim pro vera vulgo, tripla sesquiseptima, quomodo se habent, 22. ad 7. atque ita versiculi hi ex diametro persphæriam &amp;amp; contra, ex illa diametrum cuiusuis circuli minimo negotio invenire docent.

Circuitus circi per septem multiplicetur.&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>2qm7cmy3wplstdtk4bosobz35rwdcpl</sha1>
    </revision>
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  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/25</title>
    <ns>104</ns>
    <id>86261</id>
    <revision>
      <id>281433</id>
      <timestamp>2026-09-20T16:33:24Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Nondum emendata */ Paginam instituit, scribens 'Vigintique duo productum deinde secuto: Hinc numerus, quotiens qui dicitur, est diametrus; Si per vigintiduo multiplices diametrum. Per septemque seces numerum, qui prodit inde, Circuitum circi quotiens numerus tibi redit.  CAPITVLUM SECUNDVM De circulis, ex quibus sphæra materialis componitur: &amp;amp; illa super coelestis, quæ per istam imaginatur, componi intelligitur.  Orum autem circulorum quidam sunt maiores, quidam minores: ut sensui pate...</comment>
      <origin>281433</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="759" sha1="nqkeeh1in5kmclvev2cmb8mi5t2d1ba" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;Vigintique duo productum deinde secuto: Hinc numerus, quotiens qui dicitur, est diametrus; Si per vigintiduo multiplices diametrum. Per septemque seces numerum, qui prodit inde, Circuitum circi quotiens numerus tibi redit.

CAPITVLUM SECUNDVM De circulis, ex quibus sphæra materialis componitur: &amp;amp; illa super coelestis, quæ per istam imaginatur, componi intelligitur.

Orum autem circulorum quidam sunt maiores, quidam minores: ut sensui patet. Major autem circulus in sphæra dicitur, qui descriptus in superficie sphæræ super eius centrum diuidit sphæram in duo æqualia, Minor vero, qui descriptus i superficie sphæræ eam non diuidit in duo æqualia, sed in portiones inæquales. Inter circulos vero maiores, primo dicendum est de æquinoctiali.&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>nqkeeh1in5kmclvev2cmb8mi5t2d1ba</sha1>
    </revision>
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  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/26</title>
    <ns>104</ns>
    <id>86262</id>
    <revision>
      <id>281434</id>
      <timestamp>2026-09-20T16:34:28Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Nondum emendata */ Paginam instituit, scribens 'DE AEQVINOCTIALI  CIRCVLO.  Est igitur æquinoctialis circulus quidam, dividentens sphæram in duo æqualia, secundum quamlibet sui partem æquè distans ab utroque polo. Et dicitur æquinoctialis, quoniam quando Sol transit per illum (quod est bis in anno, in principio Arietis, scilicet, &amp;amp; in principio Libræ) est æquinoctiuû in universa terra: unde etiam appellatur æquator diei &amp;amp; noctis, quia adæquat diem artificialem nocti. Et dicitur cin...</comment>
      <origin>281434</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="1179" sha1="cvs0x5uv9rsmbgv9q34d8rsudyo3wad" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;DE AEQVINOCTIALI

CIRCVLO.

Est igitur æquinoctialis circulus quidam, dividentens sphæram in duo æqualia, secundum quamlibet sui partem æquè distans ab utroque polo. Et dicitur æquinoctialis, quoniam quando Sol transit per illum (quod est bis in anno, in principio Arietis, scilicet, &amp;amp; in principio Libræ) est æquinoctiuû in universa terra: unde etiam appellatur æquator diei &amp;amp; noctis, quia adæquat diem artificialem nocti. Et dicitur cingulus primi motus. Unde sciendum quod primus motus dicit motus primi mobilis, hoc est, nonæ sphæræ sive coeli vltimi, qui est ab oriente per occidentem, rediens iterum in orientem: qui etiam dicitur motus rationalis, ad similitudinem motus rationis, qui est in microcosmo, id est, in homine, scilicet quando sit consideratio à creatore per creaturas in creatorem, ibi sedendo. Secundus motus est firmamenti &amp;amp; planatarum, contrarius huic, ab occidente per oriente iterum rediens in occidente. Qui motus dicitur irrationalis sive sensualis, ad similitudinem motus microcosmi, qui est à corruptibilibus ad creatorem, iterum redies ad corruptibilia. Dicitur ergo cingulus primi motus, quia cingit, sive dividit primum&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>cvs0x5uv9rsmbgv9q34d8rsudyo3wad</sha1>
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  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/27</title>
    <ns>104</ns>
    <id>86263</id>
    <revision>
      <id>281435</id>
      <timestamp>2026-09-20T16:34:58Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Nondum emendata */ Paginam instituit, scribens 'mobile, scilicet, sphæram nonam in duo æqualia; æquè distans à polis mundi. Vnde notandum, quod polus mundi, qui nobis semper apparet, dicitur polus septentrionalis arcticus vel borealis. Septentrionalis dicitur à septentrione, hoc est, à maiori vrsa, qui dicitur à septem &amp;amp; trion, quod est bos: quia septem stellæ, quæ sunt in vrsa, tardè mouentur ad modum bouis, cum sint propinquæ polo. Vel dicuntur illæ septem stellæ septentriones, quasi...</comment>
      <origin>281435</origin>
      <model>proofread-page</model>
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      <text bytes="1400" sha1="r0f65xryc5z7s6nnf9axbjni3h4dypt" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;mobile, scilicet, sphæram nonam in duo æqualia; æquè distans à polis mundi. Vnde notandum, quod polus mundi, qui nobis semper apparet, dicitur polus septentrionalis arcticus vel borealis. Septentrionalis dicitur à septentrione, hoc est, à maiori vrsa, qui dicitur à septem &amp;amp; trion, quod est bos: quia septem stellæ, quæ sunt in vrsa, tardè mouentur ad modum bouis, cum sint propinquæ polo. Vel dicuntur illæ septem stellæ septentriones, quasi septem teriones, eò quod terunt partes circa polum. Arcticus quidem dicitur ab æxtros, quod est vrsa. Est enim iuxta maiorem vrsam.

SCHOLION VINETI.

Iuxta maiorem vrsam. Quidam libri habent, Iuxta minorem vrsam. Nam videtur hic auctor vrsam, qua vulgo maior dicitur, minorem vocare. Consule de hoc Vitruvium, Hyginium, Gellium, si quid habent certi.

Borealis vero dicitur quia est in illa parte à qua venit, Boreas. Polus vero oppositus: dicitur antarcticus, quasi contra arcticum, positus, dicitur &amp;amp; meridionalis, quia exparte meridiei est, dicitur etiam australis, quia est in illa parte à qua venit auster. Ista igitur duo puncta in firmamento stabilia, dicuntur poli mundi, quia sphæræ axem terminant, &amp;amp; ad illos voluitur mundus, quorum unus semper nobis apparet, reliquus vero semper occultatur. Vn de Virgilius in primo Georg. Hic vertex nobis semper sublimis, at illum Sub pedibus Styx atra videt manesque profundi.&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>r0f65xryc5z7s6nnf9axbjni3h4dypt</sha1>
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  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/28</title>
    <ns>104</ns>
    <id>86264</id>
    <revision>
      <id>281436</id>
      <timestamp>2026-09-20T16:36:23Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Nondum emendata */ Paginam instituit, scribens 'DE ZODIACO CIRCULO.  Est alius circulus in sphæra, qui interfecat æquinoctialem, &amp;amp; interfecatur ab eodem in duas partes æquales: &amp;amp; una eius medietas declinat versus septentrionem, alia versus austrum. Et dicitur iste circulus zodiacus à ἡσι quod est vita, quia secundum motum planetarum sub illo est omnis vita in rebus inferioribus. Vel dicitur à ἡσι quod est animal, quia cum diuidatur in 12. partes æquales, quælibet pars appellatur si...</comment>
      <origin>281436</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="1365" sha1="3t9n8cusshmg0bknojhfrjrt4avxgur" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;DE ZODIACO CIRCULO.

Est alius circulus in sphæra, qui interfecat æquinoctialem, &amp;amp; interfecatur ab eodem in duas partes æquales: &amp;amp; una eius medietas declinat versus septentrionem, alia versus austrum. Et dicitur iste circulus zodiacus à ἡσι quod est vita, quia secundum motum planetarum sub illo est omnis vita in rebus inferioribus. Vel dicitur à ἡσι quod est animal, quia cum diuidatur in 12. partes æquales, quælibet pars appellatur signum, &amp;amp; nomen habet speciale à nomine alicuius animalis, propter proprietatem aliquam conuenientem tam ipsi, quam animali. Vel propter dispositionem stellarum fixarum in illis partibus ad modum huiusmodi animantium. Iste vero circulus latinè dicitur signifer, quia fert signa, vel quia diuiditur in ea. Ab Aristotele vero in lib. 2. de generatione &amp;amp; corruptione, dicitur circulus obliquus, vbi dicit, quod secundum accessum &amp;amp; recessum Solis in circulo obliquo fiunt generationes &amp;amp; corruptiones in rebus inferioribus. Nomina autem signorum, ordinatio, &amp;amp; numerus in his patet versibus: 

Sunt Aries, Taurus Gemini, Cancer, Leo, Virgo. Libraque, 

Scorpius. Arcitcnens, Caper, Amphora, Pisces. 

Quodlibet autem signum diuiditur in 30. gradus. Vnde patet, quòd in toto zodiato sunt 560. gradus. Secundum autem Astronomos iterum quilibet gradus diuiditur in 60. minuta, quodlibet minu-&lt;noinclude&gt;&lt;/noinclude&gt;</text>
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  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/29</title>
    <ns>104</ns>
    <id>86265</id>
    <revision>
      <id>281437</id>
      <timestamp>2026-09-20T16:37:38Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Nondum emendata */ Paginam instituit, scribens 'tum in 60. secunda, quodlibet secundum in 60. ter tia, &amp;amp; sic deinceps vique ad decem. Et sicut diuiditur zodiacus ab astronomo, ita &amp;amp; quilibet circulus in sphæra, siue maior siue minor, in partes consimiles.  Cum omnis etiam circulus in sphæra, præter zodiacum, intelligatur sicut linea vel circumferentia, solus zodiacus intelligitur vt superficies, habens in latitudine sua duodecim gradus, de cuiusmodi gradibus iam locuti sumus. Vnde...</comment>
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      <text bytes="1297" sha1="799jy90sj7xapy7bypaewp29kjd2vov" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;tum in 60. secunda, quodlibet secundum in 60. ter tia, &amp;amp; sic deinceps vique ad decem. Et sicut diuiditur zodiacus ab astronomo, ita &amp;amp; quilibet circulus in sphæra, siue maior siue minor, in partes consimiles.

Cum omnis etiam circulus in sphæra, præter zodiacum, intelligatur sicut linea vel circumferentia, solus zodiacus intelligitur vt superficies, habens in latitudine sua duodecim gradus, de cuiusmodi gradibus iam locuti sumus. Vnde patet quòd quidam mentiuntur in Astrologia, dicentes, signa esse quadrata, nisi abutentes nomine, idem appellant quadratum &amp;amp; quadrangulũ. Signum enim habet gradus 39. in longitudine, 12. verò in latitudine.

Linea autem diuidens zodiacum in circuitu, ita quòd ex vna parte sui relinquat sex gradus, &amp;amp; ex alia parte alios sex, dicitur linea ecliptica, quoniam quando Sol &amp;amp; Luna sunt linealiter sub illa, contin git eclipsis Solis aut Lunæ. Solis, vt si fiat nouilunium, &amp;amp; Luna interponatur rectè in ter aspectum nostrũ, &amp;amp; corpus solare: Lunæ, vt in plenilunio, quando Sol Lunæ opponitur diametraliter. Vnde eclipsis Lunæ nihil aliud est, quà interpositio terræ, inter corpus Solis &amp;amp; Lunæ. Sol quidem semper decurrit sub ecliptica; omnes verò alij planetæ declinant vel versus septentrionem, vel versus austrum:&lt;noinclude&gt;&lt;/noinclude&gt;</text>
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  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/30</title>
    <ns>104</ns>
    <id>86266</id>
    <revision>
      <id>281438</id>
      <timestamp>2026-09-20T16:38:18Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Nondum emendata */ Paginam instituit, scribens 'quandoque autem sunt sub ecliptica. Pars vero zodiaci, quæ declinat ab æquinoctiali versus septentrionem, dicitur septentrionalis, vel borealis, vel arctica. Et illa sex signa, quæ sunt à principio Arietis vsque ad finem Virginis, dicuntur signa septentrionalia vel borealia. Alia vero pars zodiaci, quæ declinat ab æquinoctiali versus meridiem, dicitur meridionalis, vel australis, vel antarctica. Et sex signa quæ sunt à principio Libræ vsque i...</comment>
      <origin>281438</origin>
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      <text bytes="1601" sha1="5vj1pdpyifjwx246z5xuxpxysi8s6k8" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;quandoque autem sunt sub ecliptica. Pars vero zodiaci, quæ declinat ab æquinoctiali versus septentrionem, dicitur septentrionalis, vel borealis, vel arctica. Et illa sex signa, quæ sunt à principio Arietis vsque ad finem Virginis, dicuntur signa septentrionalia vel borealia. Alia vero pars zodiaci, quæ declinat ab æquinoctiali versus meridiem, dicitur meridionalis, vel australis, vel antarctica. Et sex signa quæ sunt à principio Libræ vsque in finem piscium, dicuntur meridionalia vel australia.

Cum autem dicitur, quod in Ariete est sol, vel in alio signo, sciendum, quod hæc præpolitio, in sumitur pro sub, secundum quod nunc accipimus signum. In alia autem significazione dicitur signum pyramis quadrilatera, cuius basis illa superficies, quam appellamus signum: vertex vero eius est in centro terræ. Et secundum hoc propriè loquendo possimus dicere planetas esse in signis. Tertio modo dicitur signum, ut intelligatur sex circuli transeuntes per polos zodiaci, &amp;amp; per principia 12 signorum. Illi sex circuli diuidunt totam superficiem sphæræ in duodecim partes, latas in medio, arctiores vero iuxta polos zodiaci: quolibet pars talis dicitur signum, &amp;amp; nomine habet speciale à nomine illius signi, quod intercipitur inter suas duas lineas. Et secundum hanc acceptio nem, stellæ, quæ sunt iuxta polos extra zodiacum, dicuntur esse in signis.

Iam intelligatur corpus quoddam, cuius basis sit signum, secundum quod nunc vltimo accipimus signum: acumen vero eius sit super axem zodiaci.

Tale igitur corpus in quarta significazione dicitur signum, secundum quam acceptio-&lt;noinclude&gt;&lt;/noinclude&gt;</text>
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    <revision>
      <id>281439</id>
      <parentid>281438</parentid>
      <timestamp>2026-09-20T16:39:06Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <origin>281439</origin>
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      <text bytes="1609" sha1="l0t4lh78y3xe3cl4688viyw1nawfplq" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;quandoque autem sunt sub ecliptica. Pars vero zodiaci, quæ declinat ab æquinoctiali versus septentrionem, dicitur septentrionalis, vel borealis, vel arctica. Et illa sex signa, quæ sunt à principio Arietis vsque ad finem Virginis, dicuntur signa septentrionalia vel borealia. Alia vero pars zodiaci, quæ declinat ab æquinoctiali versus meridiem, dicitur meridionalis, vel australis, vel antarctica. Et sex signa quæ sunt à principio Libræ vsque in finem piscium, dicuntur meridionalia vel australia.

Cum autem dicitur, quod in Ariete est sol, vel in alio signo, sciendum, quod hæc præpolitio, in sumitur pro sub, secundum quod nunc accipimus signum. In alia autem significazione dicitur signum pyramis quadrilatera, cuius basis illa superficies, quam appellamus signum: vertex vero eius est in centro terræ. Et secundum hoc propriè loquendo possimus dicere planetas esse in signis. Tertio modo dicitur signum, ut intelligatur sex circuli transeuntes per polos zodiaci, &amp;amp; per principia 12 signorum. Illi sex circuli diuidunt totam superficiem sphæræ in duodecim partes, latas in medio, arctiores vero iuxta polos zodiaci: quolibet pars talis dicitur signum, &amp;amp; nomine habet speciale à nomine illius signi, quod intercipitur inter suas duas lineas. Et secundum hanc acceptio nem, stellæ, quæ sunt iuxta polos extra zodiacum, dicuntur esse in signis.

Iam intelligatur corpus quoddam, cuius basis sit signum, secundum quod nunc vltimo accipimus signum: acumen vero eius sit super axem zodiaci.

Tale igitur corpus in quarta significazione dicitur signum, secundum quam acceptionem totus&lt;noinclude&gt;&lt;/noinclude&gt;</text>
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  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/31</title>
    <ns>104</ns>
    <id>86267</id>
    <revision>
      <id>281440</id>
      <timestamp>2026-09-20T16:39:38Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Nondum emendata */ Paginam instituit, scribens 'mundus. mundus diuiditur in duodecim partes æquales, quæ dicuntur signa: &amp;amp; sic, quicquid est in mundo: est in aliquo signo.  DE DVOBVS COLVRIS. Sunt autem alij duo circuli maiores in sphæra, qui dicuntur coluri, quorum officium est distinguere solstitia &amp;amp; æquinoctia. Dicitur auté colurus ἀκῶλον Græcè, quod est membrum, &amp;amp; ἐκώδια quod est bos syluester: quia quemadmodum.  SCHOLION VINETI. Αἱ ἀκῶλον Græcè) Imd αἱ...'</comment>
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      <text bytes="662" sha1="e84dv398l3smc7t8x2480232e4lnuub" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;mundus. mundus diuiditur in duodecim partes æquales, quæ dicuntur signa: &amp;amp; sic, quicquid est in mundo: est in aliquo signo.

DE DVOBVS COLVRIS. Sunt autem alij duo circuli maiores in sphæra, qui dicuntur coluri, quorum officium est distinguere solstitia &amp;amp; æquinoctia. Dicitur auté colurus ἀκῶλον Græcè, quod est membrum, &amp;amp; ἐκώδια quod est bos syluester: quia quemadmodum.

SCHOLION VINETI. Αἱ ἀκῶλον Græcè) Imd αἱ ἀκῶλον, quod est mutilum, mancum: ἐκ ὑψὶς cauda. Quoniam cum spatio 24. horarum zodiacus &amp;amp; equinoctialis toti orientur, &amp;amp; videri possint, colurorum nunquam supra terram vide&lt;noinclude&gt;&lt;/noinclude&gt;</text>
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  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/32</title>
    <ns>104</ns>
    <id>86268</id>
    <revision>
      <id>281441</id>
      <timestamp>2026-09-20T16:40:14Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
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      <comment>/* Nondum emendata */ Paginam instituit, scribens 'tur ea Sphæra Joannis, tur ea pars, quæ in antarcticico est circulo, quales arcticos Græci describunt. Quare hi, quasi amissa cauda, imperfecti &amp;amp; mutili nobis dicuntur.  cauda bouis sylvestris erecta, quæ est eius membrum, facit femicirculum, &amp;amp; non perfectum: ita colurus semper apparent nobis imperfectus: quoniam solum una eius medietas apparent, alia vero nobis occultatur.  Colurus igitur distinguens solstitia, transit per polos mund...</comment>
      <origin>281441</origin>
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      <text bytes="1206" sha1="8a8sxgudzhggeygs1iu5v8awstubqxo" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;tur ea Sphæra Joannis, tur ea pars, quæ in antarcticico est circulo, quales arcticos Græci describunt. Quare hi, quasi amissa cauda, imperfecti &amp;amp; mutili nobis dicuntur.

cauda bouis sylvestris erecta, quæ est eius membrum, facit femicirculum, &amp;amp; non perfectum: ita colurus semper apparent nobis imperfectus: quoniam solum una eius medietas apparent, alia vero nobis occultatur.

Colurus igitur distinguens solstitia, transit per polos mundi &amp;amp; per polos zodiaci, &amp;amp; maximas Solis declinationes, hoc est, per primos gradus Cancri &amp;amp; Capricorni. Unde primus punctus Cancri vbi colu rus iste interfecat zodaicum, dicitur punctus solstitij æstivialis. quia quando Sol est in eo, est solstitium, æstivale, &amp;amp; no potest Sol magis accedere ad zenith capitis nostri. Est autem zenith, punctus in firmamento directè suprapositus capitibus nostris. Arcus vero coluri, qui intercipitur inter punctum solstitij æstivialis &amp;amp; æquinoctialem, appellatur maxima Solis declinatio. Et est, secundum Ptolemæum, viginti trium graduum, &amp;amp; vnius &amp;amp; quinquaginta minutorum: secundum Almeone verò, viginti trium

''SCHOLION VINETI.''

''Hodie est 23. grad. &amp;amp; 30.minut. cuins varietatis''&lt;noinclude&gt;&lt;/noinclude&gt;</text>
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  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/33</title>
    <ns>104</ns>
    <id>86269</id>
    <revision>
      <id>281442</id>
      <timestamp>2026-09-20T16:40:41Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Nondum emendata */ Paginam instituit, scribens 'causam esse volunt motum illum vertium polos versus in octava sphæra observatum.  Similiter primus punctus Capricorni, vbi idem colurus ex alia parte intersecat zodiacum, dicitur punctus solstitij hyemalis: &amp;amp; arcus coluri interceptus inter puctum illum &amp;amp; æquinoctialem, dicitur alia maxima Solis declinatio, &amp;amp; æqualis priori.  Alter quidem colurus transit per polos mundi, &amp;amp; per prima puncta Arietis &amp;amp; Libræ, vbi sunt duo æqui...</comment>
      <origin>281442</origin>
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      <text bytes="1471" sha1="hnly9m99dwu7r4n2dwr7iw0gytozit4" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;causam esse volunt motum illum vertium polos versus in octava sphæra observatum.

Similiter primus punctus Capricorni, vbi idem colurus ex alia parte intersecat zodiacum, dicitur punctus solstitij hyemalis: &amp;amp; arcus coluri interceptus inter puctum illum &amp;amp; æquinoctialem, dicitur alia maxima Solis declinatio, &amp;amp; æqualis priori.

Alter quidem colurus transit per polos mundi, &amp;amp; per prima puncta Arietis &amp;amp; Libræ, vbi sunt duo æquinoctia: unde appellatur colurus distinguens æquinoctia. Isti autem duo coluri intersecant se se super polos mundi ad angulos rectos sphærales. Signa quidem solstitiorum &amp;amp; æquinoctiorum patet hic versibus.

Hec duo solstitium faciút, Cancer, Capricornus. Sed noctes æquant Aries, &amp;amp; Libra diebus.

DE MERIDIANO ET HORIZONTE.

Vnt iterum duo alij circuli maiores in sphæra, scilicet meridianus &amp;amp; horizon.

Est autem meridianus, circulus quidam trá-siens per polos mundi, &amp;amp; per zenith capitis nostri. Et dicitur meridianus, quia vbicunque sit homo &amp;amp; in quocunque tempore anni, quando Sol motu firmamenti peruenit ad suum meridianum, est illi meridies. Cólimi ratione dicitur circulus medij diei. Et notadum quod ciuitates, quarum una magis accedit ad orientem, quam alia, habent diversos meridianos. Arcus vero æquinoctialis interceptus inter duos meridianos dicitur longitudo ciuitatum. Si autem duæ ciuitates eundem habeant meridianum tunc æqualiter distat ab oriente &amp;amp; occidente.{{nop}}&lt;noinclude&gt;&lt;/noinclude&gt;</text>
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  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/34</title>
    <ns>104</ns>
    <id>86270</id>
    <revision>
      <id>281443</id>
      <timestamp>2026-09-20T16:41:07Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Nondum emendata */ Paginam instituit, scribens 'Horizon vero est circulus dividens inferius hemisphærium à superiori. unde appellatur horizon, id est, terminator visus. Dicitur etiam horizon circulus hemisphærij, eadem de causa. Est autem duplex horizon, rectus, &amp;amp; obliquus, siue decluius. Rectum horizontem &amp;amp; sphæram rectam habent illi, quorum zenith est in æquinoctiali, quia illorum horizon est circulus transiens per polos mundi dividens æquinoctiale ad angulos rectos sphærales, un...</comment>
      <origin>281443</origin>
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      <text bytes="1417" sha1="nfldeo0cn3w6nbcqqn0z11q44a1q72g" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;Horizon vero est circulus dividens inferius hemisphærium à superiori. unde appellatur horizon, id est, terminator visus. Dicitur etiam horizon circulus hemisphærij, eadem de causa. Est autem duplex horizon, rectus, &amp;amp; obliquus, siue decluius. Rectum horizontem &amp;amp; sphæram rectam habent illi, quorum zenith est in æquinoctiali, quia illorum horizon est circulus transiens per polos mundi dividens æquinoctiale ad angulos rectos sphærales, unde dicitur horizon rectus &amp;amp; sphæra recta. Obliuum horizontem siue decluem habent illi, quibus polus mundi eleuatur supra horizontem, &amp;amp; quoniam illorum horizon intersecat æquinoctiale ad angulos impares &amp;amp; obliquos, dicitur horizon obliquus, &amp;amp; sphæra obliqua siue decluius. Zenith autem capitis nostri semper est polus horizontis. Unde ex his patet, quod quanta est eleuatio poli mundi super horizontem, tanta est distantia zenithi ab æquinoctiali, quod sic patet. Cum in qualibet die naturali uterq́; colurus bis iugatur meridiano, siue idem sit quod meridianus, quicquid de uno probatur, &amp;amp; de reliquo. Sumatur igitur quarta pars coluri distinguentis solsticia, quæ est ab æquinoctiali usque ad polum mundi, sumatur iterum quarta pars eiusdem coluri, quæ est à zenith usque ad horizontem, cum zenith sit polus horizontis. Istæ duæ quartæ, cum sint quartæ eiusdem circuli, inter se sunt æquales: sed si ab æqualibus æqualia deman-&lt;noinclude&gt;&lt;/noinclude&gt;</text>
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  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/35</title>
    <ns>104</ns>
    <id>86271</id>
    <revision>
      <id>281444</id>
      <timestamp>2026-09-20T16:41:44Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Nondum emendata */ Paginam instituit, scribens 'tur, vel idem commune, residua erunt æqualia. dempto communi igitur arcu, scilicet qui est inter zenith &amp;amp; polum mundi, residua erunt æqualia, scilicet elevatio poli mundi supra horizontem, &amp;amp; distia zenith ab æquinoctiali.  DE QVATVOR CIRCULIS MINORIBUS.  Dicto de sex circulis maioribus, dicendum est de quattuor minoribus. Notandum igitur, quod Sol existens in primo puncto Cancri, siue in puncto solstitij æstivalis, raptu firmamenti de...</comment>
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      <text bytes="1199" sha1="n8g78l4wocw8a1m2n0xzl3gkc60nqmj" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;tur, vel idem commune, residua erunt æqualia. dempto communi igitur arcu, scilicet qui est inter zenith &amp;amp; polum mundi, residua erunt æqualia, scilicet elevatio poli mundi supra horizontem, &amp;amp; distia zenith ab æquinoctiali.

DE QVATVOR CIRCULIS MINORIBUS.

Dicto de sex circulis maioribus, dicendum est de quattuor minoribus. Notandum igitur, quod Sol existens in primo puncto Cancri, siue in puncto solstitij æstivalis, raptu firmamenti describit quædam circulum, qui ultimo descriptus est à Sole ex parte poli arcticici; unde appellatur cir-

SCHOLION VINETI.

''Firmamentum auctor primo capite vocabat octasam spharam. Sed hic firmamentum supremam spharam significet necesse est, aliquotque alius locis.''

culus solstitij æstivalis, ratione superius dicta, vel tropicus æstivalis, à ῥεοτι, quod est conuersio, quia tunc Sol incipit se convertere ad inferius hemisphærium, &amp;amp; recedere à nobis. Sol iterum existens in primo puncto Capricorni siue solstitij hyemalis, raptu firmamenti describit quendam circulum, qui ultimo describitur à Sole ex parte poli antarctici: unde appellatur circulus solstitij hyemalis, siue tropicus hyemalis, quia tunc Sol convertitur ad nos.&lt;noinclude&gt;&lt;/noinclude&gt;</text>
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  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/36</title>
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      <comment>/* Nondum emendata */ Paginam instituit, scribens 'SCHOLION VINETI.  Α τροπὴ q.) Hanc τροπὴν Cicero libro secundo de natura Deorum appellat reversionem, quod Sol reverti incipiat, ἐδὶ unde est profectus. Vocat autem auctor inferius hemisphærium partem mundi australem, ut Macrobius libro primo Saturnalium.  Cum autem zodiacus declinat ab æquinoctiali &amp;amp; polus zodiaci declinabit a polo mundi. Cum igitur moueatur octaua sphæra, &amp;amp; zodiacus, qui est pars octauæ sphæræ, mouebitur circa...'</comment>
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      <text bytes="1201" sha1="3h4e9j05g7l9cahnrwi0bwi5cvpbn0q" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;SCHOLION VINETI.

Α τροπὴ q.) Hanc τροπὴν Cicero libro secundo de natura Deorum appellat reversionem, quod Sol reverti incipiat, ἐδὶ unde est profectus. Vocat autem auctor inferius hemisphærium partem mundi australem, ut Macrobius libro primo Saturnalium.

Cum autem zodiacus declinat ab æquinoctiali &amp;amp; polus zodiaci declinabit a polo mundi. Cum igitur moueatur octaua sphæra, &amp;amp; zodiacus, qui est pars octauæ sphæræ, mouebitur circa axem mundi, &amp;amp; polus zodiaci mouebitur circa polum mūdi. Iste igitur circulus, quem describit polus Zodiaci circa polum mundi arcticum, dicitur circulus arcticus: ille vero circulus, quem describit alter polus zodiaci, circa polum mundi antarcticum, dicitur circulus antarcticus.

SCHOLION VINETI.

Arcticorum orbium descriptio hac multum diversa est ab ea; quam Cleomedes Proclus, &amp;amp; alij tradunt.

Quanta est etiam maxima Solis declinatio, scilicet ab æquinoctiali, tanta est distantia poli mundi ad polum zodiaci: quod sic patet. Sumatur colurus distinguens solstitia, qui transit per polos mundi &amp;amp; per polos zodiaci. Cum igitur omnes quartæ vnius &amp;amp; eiusdem circuli inter se sint æquales, quarta huius coluri, quæ&lt;noinclude&gt;&lt;/noinclude&gt;</text>
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    <revision>
      <id>281448</id>
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      <timestamp>2026-09-20T16:44:37Z</timestamp>
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        <username>Sjgallagher2</username>
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      <text bytes="1287" sha1="09yk87h8fnbtu45prjcergc3wg5l1cf" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;SCHOLION VINETI.

Α τροπὴ q.) Hanc τροπὴν Cicero libro secundo de natura Deorum appellat reversionem, quod Sol reverti incipiat, ἐδὶ unde est profectus. Vocat autem auctor inferius hemisphærium partem mundi australem, ut Macrobius libro primo Saturnalium.

Cum autem zodiacus declinat ab æquinoctiali &amp;amp; polus zodiaci declinabit a polo mundi. Cum igitur moueatur octaua sphæra, &amp;amp; zodiacus, qui est pars octauæ sphæræ, mouebitur circa axem mundi, &amp;amp; polus zodiaci mouebitur circa polum mūdi. Iste igitur circulus, quem describit polus Zodiaci circa polum mundi arcticum, dicitur circulus arcticus: ille vero circulus, quem describit alter polus zodiaci, circa polum mundi antarcticum, dicitur circulus antarcticus.

SCHOLION VINETI.

Arcticorum orbium descriptio hac multum diversa est ab ea; quam Cleomedes Proclus, &amp;amp; alij tradunt.

Quanta est etiam maxima Solis declinatio, scilicet ab æquinoctiali, tanta est distantia poli mundi ad polum zodiaci: quod sic patet. Sumatur colurus distinguens solstitia, qui transit per polos mundi &amp;amp; per polos zodiaci. Cum igitur omnes quartæ vnius &amp;amp; eiusdem circuli inter se sint æquales, quarta huius coluri, quæ est ab æquinoctiali vsque ad polum mundi, erit æqualis quartæ eiusdem coluri, quæ&lt;noinclude&gt;&lt;/noinclude&gt;</text>
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    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/37</title>
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      <comment>/* Nondum emendata */ Paginam instituit, scribens 'est ab æquinoctiali vsque ad polum mundi, erit æqualis quartæ eiusdem coluri, quæ est à De sacro Bosco. Cap. 1.  est à primo puncto Cancri visq́; ad polum zodiaci, igitur ab illis equalibus dēmpto có muni arcu; qui est à primo púcto Cancri visque ad polum mundi, residua erút aqualia, scilicet maxima Solis declinatio, &amp;amp; di stantia poli mundi ad polum zodiaci. Cum autem circulus arcticus secundum quamlibet sui partem æquè distet à polo mund...</comment>
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      <text bytes="1321" sha1="dt8j3tacufr9e10ka2avsq44f8p3h4i" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;est ab æquinoctiali vsque ad polum mundi, erit æqualis quartæ eiusdem coluri, quæ est à De sacro Bosco. Cap. 1.

est à primo puncto Cancri visq́; ad polum zodiaci, igitur ab illis equalibus dēmpto có muni arcu; qui est à primo púcto Cancri visque ad polum mundi, residua erút aqualia, scilicet maxima Solis declinatio, &amp;amp; di stantia poli mundi ad polum zodiaci. Cum autem circulus arcticus secundum quamlibet sui partem æquè distet à polo mundi, patet quod illa pars coluri, quæ est inter primum punctum Cancri, &amp;amp; circulum arcticum, serè est dupla ad maximam Solis declinationem, sive ad arcum eiusdem coluri, qui intercipitur inter circulum arcticum, &amp;amp; polum mundi arcticum, qui etiam arcus æqualis est maxima Solis declinationi. Cum enim colurus iste, sicut alij circuli, in sphæra sit 360. graduum, quarta eius erit 90. graduum. Cum igitur maxima Solis declinatio secundum Ptolomæum sit 23. graduum, &amp;amp; 51. minutorum: &amp;amp; totidem graduum sit arcus, qui est inter circulum arcticum &amp;amp; polum mundi arcticum, si ista duo simul iuncta, quæ serè faciunt 47. gradus, subtrahantur à 90. residuum crunt 42. gradus, quantus est arcus coluri, qui est inter primum punctum Cancri &amp;amp; circulum arcticum, &amp;amp; sic patet, quòd ille arcus serè duplus est ad maximam Solis declinationem.&lt;noinclude&gt;&lt;/noinclude&gt;</text>
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    <revision>
      <id>281447</id>
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      <timestamp>2026-09-20T16:43:16Z</timestamp>
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      <text bytes="1203" sha1="7dcgfkdxjrt00mmucux0n3o7mtc6f4k" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;est à primo puncto Cancri visq́; ad polum zodiaci, igitur ab illis equalibus dēmpto có muni arcu; qui est à primo púcto Cancri visque ad polum mundi, residua erút aqualia, scilicet maxima Solis declinatio, &amp;amp; di stantia poli mundi ad polum zodiaci. Cum autem circulus arcticus secundum quamlibet sui partem æquè distet à polo mundi, patet quod illa pars coluri, quæ est inter primum punctum Cancri, &amp;amp; circulum arcticum, serè est dupla ad maximam Solis declinationem, sive ad arcum eiusdem coluri, qui intercipitur inter circulum arcticum, &amp;amp; polum mundi arcticum, qui etiam arcus æqualis est maxima Solis declinationi. Cum enim colurus iste, sicut alij circuli, in sphæra sit 360. graduum, quarta eius erit 90. graduum. Cum igitur maxima Solis declinatio secundum Ptolomæum sit 23. graduum, &amp;amp; 51. minutorum: &amp;amp; totidem graduum sit arcus, qui est inter circulum arcticum &amp;amp; polum mundi arcticum, si ista duo simul iuncta, quæ serè faciunt 47. gradus, subtrahantur à 90. residuum crunt 42. gradus, quantus est arcus coluri, qui est inter primum punctum Cancri &amp;amp; circulum arcticum, &amp;amp; sic patet, quòd ille arcus serè duplus est ad maximam Solis declinationem.&lt;noinclude&gt;&lt;/noinclude&gt;</text>
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  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/38</title>
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    <revision>
      <id>281449</id>
      <timestamp>2026-09-20T16:45:50Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
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      </contributor>
      <comment>/* Nondum emendata */ Paginam instituit, scribens 'DE QVINQVE ZONIS.  Equinoctialis cum quatuor circulis minorebus dicuntur quinq́; paralleli, quasi æquidistantes, non quia quantum primus distat à secundo, tantum secūdus distet à tertio, quia hoc falsum est, sicut iam patuit, sed quia quilibet duo circuli simul iuncti secundum quamlibet sui partem æquè distant ab inuicem, &amp;amp; dicuntur parallelus æquinoctialis, parallelus solstitij æstivalis, parallelus solstitij hyēmalis, parallelus arcticu...</comment>
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      <text bytes="1482" sha1="roqtgmflmvg9ap0kwyiqfupcv7e337q" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;DE QVINQVE ZONIS.

Equinoctialis cum quatuor circulis minorebus dicuntur quinq́; paralleli, quasi æquidistantes, non quia quantum primus distat à secundo, tantum secūdus distet à tertio, quia hoc falsum est, sicut iam patuit, sed quia quilibet duo circuli simul iuncti secundum quamlibet sui partem æquè distant ab inuicem, &amp;amp; dicuntur parallelus æquinoctialis, parallelus solstitij æstivalis, parallelus solstitij hyēmalis, parallelus arcticus, &amp;amp; parallelus antarcticus.

Non tantum etiam quòd quatuor paralleli minores, scilicet duo tropici, &amp;amp; parallelus arcticus, &amp;amp; parallelus antarcticus distinguunt in cælo quinq́; zonas siue regiones; unde Virgilius in Georgicis. Quinque tenent cælum zone, quarum una corusco Semper sole rubens, &amp;amp; torrida semper ab igne, &amp;amp;c.

Distinguuntur etiam totidem plagæ in terra, directe prædictis zonis suppositæ, unde Ouidius primo Metamorphoseon.

Totidemque plagæ tellure premuntur.

Quarum quæ media est non est habitabilis æstu. 

Nix tegit alta duas; totidem inter vtranque locauit, Temperiemque dedit, mixta cum frigore flamma.

Illa igitur zona, quæ est inter duos tropicos, dicitur inhabitabilis propter calorem Solis discurrētis semper inter tropicos. Similiter plaga terræ, illi directè supposita dicitur inhabitabilis, proprer cadorem Solis discurrentis super illam. illæ verò duæ zonæ, quæ circumscribuntur à circulo arctico, &amp;amp; circulo antarcticco circa polos mūdi, inhabitabiles&lt;noinclude&gt;&lt;/noinclude&gt;</text>
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  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/39</title>
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    <revision>
      <id>281450</id>
      <timestamp>2026-09-20T16:46:33Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Nondum emendata */ Paginam instituit, scribens 'sunt propter nimiam frigiditatem, quia Sol ab eis maxime remouetur. Similiter intelligendum est de plagis terræ illis directe suppositis. Illæ autem duæ zonæ quaru una est inter tropicu æstivalem &amp;amp; circulum arcticum, &amp;amp; reliqua, q est inter tropicu hye malem &amp;amp; circulu antarcticu, habitabiles sunt, &amp;amp; temperatæ, caliditate torridæ zonæ existentis inter tropicos, &amp;amp; frigiditate zonarum, quæ sunt circa polos mundi extremarum. Id...</comment>
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      <text bytes="1219" sha1="quwnc59y14sfosgvllriu7r4udu46pf" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;sunt propter nimiam frigiditatem, quia Sol ab eis maxime remouetur. Similiter intelligendum est de plagis terræ illis directe suppositis. Illæ autem duæ zonæ quaru una est inter tropicu æstivalem &amp;amp; circulum arcticum, &amp;amp; reliqua, q est inter tropicu hye malem &amp;amp; circulu antarcticu, habitabiles sunt, &amp;amp; temperatæ, caliditate torridæ zonæ existentis inter tropicos, &amp;amp; frigiditate zonarum, quæ sunt circa polos mundi extremarum. Idem intellige de plagis terræ illis directe suppositis.

SCHOLION VINETI.

Recte apud Trogum Scythæ, gens septentionalis, de generis vetustate cu Aegyptius contendentes dicebat naturam, que calore &amp;amp; frigore regiones distinxisset, ad locorum patientiam homines quoq́; &amp;amp; alia animalia generasse. Nam quas zonas inhabitabiles a flu &amp;amp; fri gore permulti existimarunt, compertum multis navigationibus hoc nostro sæculo habitatores habere, (quod ante annos mille ac quingentos Iubam Numidia regem adversus plurimorum veterum opinionem probasse Solinus auctor est, &amp;amp; Polybium ante Iuba Strabo lib. secundo) earumque intemperiem nostros etiam homines temperato cælo natos &amp;amp; educatos, ferre tandem assueuisse. Quot enim Lusitanorum in Orientali, quot&lt;noinclude&gt;&lt;/noinclude&gt;</text>
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  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/40</title>
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    <revision>
      <id>281451</id>
      <timestamp>2026-09-20T16:46:58Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
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      <comment>/* Nondum emendata */ Paginam instituit, scribens 'reliquorum Hispanorum Colonias in Occidentali Zona torrida hodie inueniæs? Zonæ autem frigidæ non æquè nobis nota sunt, neque familiares.  ALIA FIGVRA DE DISTINctione Zonarum, qua simul causè distributionis earum &amp;amp; in cælo &amp;amp; in terra cernuntur.'</comment>
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      <text bytes="259" sha1="qs1uoh5zrobdyuu5783927vfxi018yd" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;reliquorum Hispanorum Colonias in Occidentali Zona torrida hodie inueniæs? Zonæ autem frigidæ non æquè nobis nota sunt, neque familiares.

ALIA FIGVRA DE DISTINctione Zonarum, qua simul causè distributionis earum &amp;amp; in cælo &amp;amp; in terra cernuntur.&lt;noinclude&gt;&lt;/noinclude&gt;</text>
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  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/41</title>
    <ns>104</ns>
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    <revision>
      <id>281452</id>
      <timestamp>2026-09-20T16:49:30Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
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      <comment>/* Nondum emendata */ Paginam instituit, scribens 'CAPITULUM TERTIVM De ortu &amp;amp; occasu signorum, &amp;amp; de diversitate dierum &amp;amp; noctium, &amp;amp; de divisione climatum.  Signorum autem ortus &amp;amp; occasus dupliciter ac scipitur, quoniam quantum ad poetas, &amp;amp; quantum ad astronomos. Est igitur ortus &amp;amp; occasus signorum, quò ad poetas, tripex, scilicet cosmicus, chronicus, &amp;amp; heliacus.  Cosmicus enim ortus siue mundanus est, quando signum vel stella super horizontem ex parte orientis...</comment>
      <origin>281452</origin>
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      <text bytes="1482" sha1="06s1xqthug0koiddzmp3dyzqdv4az6i" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;CAPITULUM TERTIVM De ortu &amp;amp; occasu signorum, &amp;amp; de diversitate dierum &amp;amp; noctium, &amp;amp; de divisione climatum.

Signorum autem ortus &amp;amp; occasus dupliciter ac scipitur, quoniam quantum ad poetas, &amp;amp; quantum ad astronomos. Est igitur ortus &amp;amp; occasus signorum, quò ad poetas, tripex, scilicet cosmicus, chronicus, &amp;amp; heliacus.

Cosmicus enim ortus siue mundanus est, quando signum vel stella super horizontem ex parte orientis de die ascendit. Et licet in qualibet die artificiali, sex signa sic orientur, tamen antonomasticè signum illud de cosmicè oriri, cum quo &amp;amp; in quo Sol manè oritur. Et hic ortus proprius &amp;amp; principalis &amp;amp; quotidianus dicitur. De hoc ortu exemplum in primo Georgicān habetur, vbi docetur satio fabarum &amp;amp; milij in vere, Sole existente in Tauro: sic,

Candidus auratis aperit cum cornibus annum Taurus, &amp;amp; * aduerso cedens Canis occidit astro,

Occasus vero cosmicus est, respectu oppositionis, scilicet quando Sol oritur cum aliquo signo, cu ius signi, signum oppositum occidit cosmicè. De hoc occasu dicitur in Georgicis, vbi docetur satio frumenti in sine autumni, Sole existente in Scorpione, qui cum oriatur cum Sole, Taurus signi eius oppositum, vbi sunt Pleiades, occidit: sic. 

Ante tibi Eoæ Athlantides abscondantur, 

Debita quàm sulcis committas semina.

Chronicus ortus siue temporalis est, quando signum vel stella post Solis occasum supra horizonetem ex parte orientis emergit chronicè, scilicet de&lt;noinclude&gt;&lt;/noinclude&gt;</text>
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  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/42</title>
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    <revision>
      <id>281453</id>
      <timestamp>2026-09-20T16:49:57Z</timestamp>
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        <username>Sjgallagher2</username>
        <id>29663</id>
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      <comment>/* Nondum emendata */ Paginam instituit, scribens 'nocte: &amp;amp; dicitur temporalis, quia tempus mathematicorum nascitur cum Solis occasu. De hoc or. Libr.1. Eleg.9. tu habemus in Ouidio lib.1.de Ponto, ubi conqueritur moram exilij sui, dicens: Quatuor autumnos Pleias orta facit. significans per quatuor autumnos, quatuor annos transijisse, postquam missus erat in exilium. Sed Virgilius voluit in autumno Pleiades occidere: ergo contrarij videntur. Sed ratio huius est, quod secundum Virgilium oc...</comment>
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      <text bytes="1382" sha1="0e4bdgqish5nxcqrktizszhmf5yujoi" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;nocte: &amp;amp; dicitur temporalis, quia tempus mathematicorum nascitur cum Solis occasu. De hoc or. Libr.1. Eleg.9. tu habemus in Ouidio lib.1.de Ponto, ubi conqueritur moram exilij sui, dicens: Quatuor autumnos Pleias orta facit. significans per quatuor autumnos, quatuor annos transijisse, postquam missus erat in exilium. Sed Virgilius voluit in autumno Pleiades occidere: ergo contrarij videntur. Sed ratio huius est, quod secundum Virgilium occidunt cosmicè, secundum Ouidium oriuntur chronicè: quod bene potest contingere codem die, sed differenter tamen, quia cosmicus occasus est, respectu temporis matutini: chronicus vero ortus, respectu vespertini est. Chronicus occasus est, respectu oppositionis, unde Lucanus lib.4. inquit, Tunc nox Thessalicas vrgebait parua sagittas. Heliacus ortus siue Solarius est, quando signum vel stella videri potest per clongationem Solis ab illo, quod prius videri non poterat Solis propinquitate. Exemplum huius ponit Ouid.lib.2.de Fastis, sic, Iam leuis obliqua tubsedit Aquarius vrna. Et Virgilius in Georg.lib.1. Genosiaque ardentis decedat stella Coronæ: Quæ iuxta Scorpionem existens non videtur, dum Sol erat in Scorpione. Occasus Heliacus est, quando Sol ad signum accedit, &amp;amp; illud sua præsentia &amp;amp; luminositate videri non permittit. Huius exemplum est in versu pramisso. * anverso Et * aduerso cedens canis occidit astro.&lt;noinclude&gt;&lt;/noinclude&gt;</text>
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  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/43</title>
    <ns>104</ns>
    <id>86279</id>
    <revision>
      <id>281454</id>
      <timestamp>2026-09-20T16:50:23Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Nondum emendata */ Paginam instituit, scribens 'SCHOLION VINETI.  Ioannes de Sacro Bosco multis locis se imperitum Graci sermonis ostendit, quemadmodum hic quoque, ubi chronicum pro acronicho ut diceret, non est dubium quin corrupti &amp;amp; mendosi libri in causa fuerint. Sunt autem ἀκρον sumimum, ὥς nox, ex quibus sit ἀκρονχος, &amp;amp; illud ἀκρωνχαι quod legitur apud Iulium Firmicu lib. 2. 8. quod vide an sit purum, &amp;amp; à quo recto singulari destellatur,  Porrò hic tibi quærendu...'</comment>
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      <text bytes="1558" sha1="nghh8r9di6k4sw162xlb8x4fgsxs00t" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;SCHOLION VINETI.

Ioannes de Sacro Bosco multis locis se imperitum Graci sermonis ostendit, quemadmodum hic quoque, ubi chronicum pro acronicho ut diceret, non est dubium quin corrupti &amp;amp; mendosi libri in causa fuerint. Sunt autem ἀκρον sumimum, ὥς nox, ex quibus sit ἀκρονχος, &amp;amp; illud ἀκρωνχαι quod legitur apud Iulium Firmicu lib. 2. 8. quod vide an sit purum, &amp;amp; à quo recto singulari destellatur,

Porrò hic tibi quærendum primùm, vtrum dicant Graci frequëtius αὐστολην, idest ortum ακρωνχον an Arcturum, Pleiadas, Arietem, &amp;amp; reliqua sydera acronicha, Ortū illum ακρωνχιον memini ego legere qui appellarent. Deinde sydera quæ per totam nocte oriuntur et occidunt, an acronychè id facere dici possint ut hic vult cum acronychè significatio noctis principio tantum conuenire videatur. Postremò si Sol erat in Geminis; quomodo non potius Sagittary occasus ille cosmicus appelletur, ut supra, Pleiadum. Mihi quidem certe vix videtur Lucani illud ad ullum occasus genus pertinere, sed tantum anni tempus &amp;amp; breues noctes designare, cui Saggitarius obliquè ingrediē, ei adversus signo in quo erat Sol, totam noctem supra horizontem versari celebriter videtur. Eius sunt versus libro quarto. Idem cum fortis animos præcepta subissent, Optauere diem nec segnis mergere ponto Tunc erat astra polus. nam Sol Iedaa tenebat Sydera, vicino cum lux altissima Cancro est. Nox tum Thessalias urgebait parua sagittas. Detegit orta dies stantes in rupibus Ietros, Pugnacesque mari, &amp;amp;c.&lt;noinclude&gt;&lt;/noinclude&gt;</text>
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  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/44</title>
    <ns>104</ns>
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    <revision>
      <id>281455</id>
      <timestamp>2026-09-20T16:50:57Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
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      <comment>/* Nondum emendata */ Paginam instituit, scribens 'INSTRUMENTUM QVO facillimè omnes diuersitates ortus poetici oculis subjiciuntur.  DE ORTV ET OCCASV signorum secundum astrologos seu de ascensionibus &amp;amp; descensionibus signorum, rectis &amp;amp; obliquis.'</comment>
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      <text bytes="204" sha1="cw4406c64eafesu33hegc0pcbrpl3mw" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;INSTRUMENTUM QVO facillimè omnes diuersitates ortus poetici oculis subjiciuntur.

DE ORTV ET OCCASV signorum secundum astrologos seu de ascensionibus &amp;amp; descensionibus signorum, rectis &amp;amp; obliquis.&lt;noinclude&gt;&lt;/noinclude&gt;</text>
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  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/45</title>
    <ns>104</ns>
    <id>86281</id>
    <revision>
      <id>281456</id>
      <timestamp>2026-09-20T16:51:32Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Nondum emendata */ Paginam instituit, scribens 'Sequitur Equitur de ortu &amp;amp; occasu signorum, prout sumunt astronomi, &amp;amp; prius in sphæra recta. Siendum est, quod tam in sphæra recta, quam obliqua ascendit æquinoctialis circulus semper uniformiter, scilicet in temporibus æqualibus æquales arcus ascendunt. Motus enim coeli uniformis.  SCHOLION VINETI.  Ascendit æquinoctialis.) Aequinoctialis oriuntur gradus 15. singulis horis æquinoctialibus, quibus 24. diem constitue vulgo dicuntur, qu...</comment>
      <origin>281456</origin>
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      <text bytes="875" sha1="gvd2u2q9ixg3gv4ygwyo6bic8xb95e2" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;Sequitur Equitur de ortu &amp;amp; occasu signorum, prout sumunt astronomi, &amp;amp; prius in sphæra recta. Siendum est, quod tam in sphæra recta, quam obliqua ascendit æquinoctialis circulus semper uniformiter, scilicet in temporibus æqualibus æquales arcus ascendunt. Motus enim coeli uniformis.

SCHOLION VINETI.

Ascendit æquinoctialis.) Aequinoctialis oriuntur gradus 15. singulis horis æquinoctialibus, quibus 24. diem constitue vulgo dicuntur, qui naturalis postea definitur. est: &amp;amp; angulus, quem facit æquinoctialis cum horizonte obliquo, non diversificatur in aliquibus horis. Partes vero zodiaci non de necessitate habent.

SCHOLION VINETI.

Et angulus, quem fa:) In sphæra recta æquinoctialis &amp;amp; horizon ad angulos se intersecant æquales, qui sunt recti in sphæra vero obliqua ad angulos inæqualles, quorum alter obiusus, ille, qui recto maior præter&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>gvd2u2q9ixg3gv4ygwyo6bic8xb95e2</sha1>
    </revision>
  </page>
  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/46</title>
    <ns>104</ns>
    <id>86282</id>
    <revision>
      <id>281457</id>
      <timestamp>2026-09-20T16:52:10Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Nondum emendata */ Paginam instituit, scribens 'nonaginta nonaginta gradus (tot comprehendit rectus in sphæra angulus) quos in meridiano circulo numerus ab æquinoctiali ad polum mundi, gradus complectitur sublimi ratis eiusdem poli supra horizontem. alter acutus, qui recto minor, eos gradus continet, qui reliqui sunt ex 90. pist quam ex ijs 90. subduxisti eandem poli altitudinem. Quantum enim supra 90. accedit vni ex illie angulis tantum decedit alteri. Non variantur autem si equatoris &amp;am...</comment>
      <origin>281457</origin>
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      <text bytes="1619" sha1="aewhk0l5m98xfeucgv4talgtbdnp5ka" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;nonaginta nonaginta gradus (tot comprehendit rectus in sphæra angulus) quos in meridiano circulo numerus ab æquinoctiali ad polum mundi, gradus complectitur sublimi ratis eiusdem poli supra horizontem. alter acutus, qui recto minor, eos gradus continet, qui reliqui sunt ex 90. pist quam ex ijs 90. subduxisti eandem poli altitudinem. Quantum enim supra 90. accedit vni ex illie angulis tantum decedit alteri. Non variantur autem si equatoris &amp;amp; orizontis anguli, quæadmodum auctor ait, sed zodiacus &amp;amp; horizon varüs inter se secant angulis, propter zodiaci ipsius obliquitatem, unde inæquales illæ ascensiones partium zodiaci. Annotavit &amp;amp; Petrus Nonius.

æquales ascensiones in utraque sphæra. quia quanto aliqua zodiaci pars rectius oritur tanto plus temporis ponitur in suo ortu. Huius signum est, quia sex signa oriuntur in longa vel in breui die artificia li, similiter &amp;amp; in nocte. Notandum igitur, quod orus vel occasus alicui signi, nihil aliud est quam illum partem æquinoctialis oriri, quæ oritur cum illo signo oriente, id est, ascendente supra horizontem: vel illum partem æquinoctialis occidere, quæ occidit cum illo signo occidente, id est, tendente ad occasum sub horizontem. Signum autem rectè oriri dicitur, cum quo maior pars æquinoctialis oritur; oblique vero, cum quo minor. Similiter etiam intelligendum est de occasu.

Et est sciendum, quòd in sphæra recta quartæ zodiaci inchoatæ à quattuor punctis, duobus scilicet solstitialibus, &amp;amp; duobus æquinoctiali-bus, adæquaturn suis ascensionibus, id est, quantum temporis colsumit quarta zodiaci in suo ortu, in tam&lt;noinclude&gt;&lt;/noinclude&gt;</text>
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  </page>
  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/47</title>
    <ns>104</ns>
    <id>86283</id>
    <revision>
      <id>281458</id>
      <timestamp>2026-09-20T16:53:02Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Nondum emendata */ Paginam instituit, scribens 'to tempore quarta æquinoctialis illi conterminalis peroritur: sed tamen partes illarum quartarum variantur. neque habent æquales ascensus, sicut iam patebit. Est enim regula. Quilibet duo arcus zodiaci æquales, &amp;amp; equaliter distantes ab aliquo quatuor punctorum iam dictorum, æquales habent ascensiones: &amp;amp; ex hoc sequitur, &amp;amp; signa opposita æquales habent ascensiones. &amp;amp; hoc est quod dicit Lucanus lib. 9. loquens de processu Catoni...</comment>
      <origin>281458</origin>
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      <text bytes="1441" sha1="ixlssd8hl5ukkcuruad8copxwffbb7w" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;to tempore quarta æquinoctialis illi conterminalis peroritur: sed tamen partes illarum quartarum variantur. neque habent æquales ascensus, sicut iam patebit. Est enim regula. Quilibet duo arcus zodiaci æquales, &amp;amp; equaliter distantes ab aliquo quatuor punctorum iam dictorum, æquales habent ascensiones: &amp;amp; ex hoc sequitur, &amp;amp; signa opposita æquales habent ascensiones. &amp;amp; hoc est quod dicit Lucanus lib. 9. loquens de processu Catonis in Libyam versus æquinoctialem,

Non obliqua meant, nec Tauro rectior exit Scorpius, aut Aries donat sua tempora Libræ, Aut astræa iubet lentos descendere pisces. Par Geminis Chiron, &amp;amp; idé quod Carcinus ardens. Humidus ἀγριός, nec plus Leo tollitur Vrna.

Hic dicit Lucanus: quod existentibus sub æquinoctiali, signa opposita æquales habent ascensiones &amp;amp; occasus. Oppositio autem signorum habetur per versum. Est lib.ari.scor.tau.sag.gemi.cap.ca.ar.le.pilvir.

SCHOLION VINETI.

Ex hoc sequitur.) Signa opposita, tametsi non aqua liter distant ab aliquo illorum quattuor punctorum, tamen cum aliquo alio signo vtrunq́; convienit: exempli gratia, Aries &amp;amp; virgo æquales habent ascensiones, quod equaliter distent à puncto solstitiali. Virgo eadem &amp;amp; Libra æquales: quæ equaliter distant ab æquinoctio: unde colligas Arietem cum Libra eundem bare ortum, quod axioma sit, inter se convenire, qua eidem conveniunt: &amp;amp; inter se esse paria, quæ eidem parta:&lt;noinclude&gt;&lt;/noinclude&gt;</text>
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  </page>
  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/48</title>
    <ns>104</ns>
    <id>86284</id>
    <revision>
      <id>281459</id>
      <timestamp>2026-09-20T16:53:47Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Nondum emendata */ Paginam instituit, scribens 'Lucani autem illud ex lib. nonò. Pharsalia, non obliqua meant non satis video quid sit: vereor enim ne velit Lucanus hoc significare, in sphæra recta signa omnia zodiaci rectè oriiri, &amp;amp; ascendere. Quæ autem signa in sphæra recta rectè, &amp;amp; quæ obliquè orientur, hic habes.  G. M.  Obl. Aries, Virgo, Libra, Pisces, 27 56 Obl. Taurus, Leo, Scorp. Aquar. 29 55 Rec. Gemi. Cancer. Sagit. Capri. 33 11  Porrò quia hi gradus sunt æquinoctialis,...</comment>
      <origin>281459</origin>
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      <text bytes="1399" sha1="49og394b7drsqxz86vm0k6ozebkyhtu" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;Lucani autem illud ex lib. nonò. Pharsalia, non obliqua meant non satis video quid sit: vereor enim ne velit Lucanus hoc significare, in sphæra recta signa omnia zodiaci rectè oriiri, &amp;amp; ascendere. Quæ autem signa in sphæra recta rectè, &amp;amp; quæ obliquè orientur, hic habes.

G. M.

Obl. Aries, Virgo, Libra, Pisces, 27 56 Obl. Taurus, Leo, Scorp. Aquar. 29 55 Rec. Gemi. Cancer. Sagit. Capri. 33 11

Porrò quia hi gradus sunt æquinoctialis, &amp;amp; æquinoctialis gradus 15. hora una oriuntur. facile hinc colleges, quanto tempore singula signa orientur.

Et est notandum, quod non valet talis argumentatio: Isti duo arcus sunt æquales, &amp;amp; simul incipiunt oriiri, &amp;amp; semper maior pars oritur de uno, quam de reliquo: ergo ille arcus citius peroritur, cujus maior pars semper oriebatur.

SCHOLION VINETI.

Maior pars oritur.) Oritur, pro orta videtur, aut orza est sicut dixit cap. primo cum terram esse centrum ostenderet, sex signa oriuntur &amp;amp; occidunt. Arcus autem zodiaci pars prior si celerius oritur, eius posterior pars tardius ascendit, &amp;amp; contrà. inde compensatio.

Instantia huius argumentationis manifesta est in partibus prædictarum quartarum. Si enim sumatur quarta pars zodiaci, quæ est à principio Arietis vsque ad finem Geminorum, semper maior pars oritur de quarta zodiaci, quam de quarta æquinoctialis sibi conterminali, &amp;amp; tamen illæ duæ quartæ&lt;noinclude&gt;&lt;/noinclude&gt;</text>
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  </page>
  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/49</title>
    <ns>104</ns>
    <id>86285</id>
    <revision>
      <id>281460</id>
      <timestamp>2026-09-20T16:55:02Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Nondum emendata */ Paginam instituit, scribens 'simul Simul peroriuntur. Idem intellige de quarta zodiaci, quæ est a principio Libræ usque ad finem Sagittarii.  Item si sumatur quarta zodiaci, quæ est a principio Cancri usque ad finem Virginis, semper maior pars oritur de quarta æquinoctialis, quam de quarta zodiaci, illi conterminali, &amp;amp; tamen illæ duæ quartæ simul peroriuntur. Idem intellige de quarta zodiaci, quæ est a primo puncto Capricorni usq́; ad finem piscium.  In sphæra autem...</comment>
      <origin>281460</origin>
      <model>proofread-page</model>
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      <text bytes="1641" sha1="kfud9od8gsevlr6nay6jtxrfjezlids" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;simul Simul peroriuntur. Idem intellige de quarta zodiaci, quæ est a principio Libræ usque ad finem Sagittarii.

Item si sumatur quarta zodiaci, quæ est a principio Cancri usque ad finem Virginis, semper maior pars oritur de quarta æquinoctialis, quam de quarta zodiaci, illi conterminali, &amp;amp; tamen illæ duæ quartæ simul peroriuntur. Idem intellige de quarta zodiaci, quæ est a primo puncto Capricorni usq́; ad finem piscium.

In sphæra autem obliqua siue declivi, duæ medietates zodiaci adæquantur suis ascensionibus: medietates dico, quæ sumuntur a duobus punctis æquinoctialibus: quia medietas zodiaci, quæ est a principio Arietis usque in finem Virginis, oritur cum medietate æquinoctialis sibi conterminali. Si militer alia medietas zodiaci oritur cum aliqua medietate æquinoctialis.

SCHOLION VINETI.

Innumeris in Iacobi Fabri commentariis tabellam hanc ad latitudinem 48. graduum factam, de ascensionibus signorum zodiaci.

&lt;table&gt;
&lt;tr&gt;
&lt;th&gt;
&lt;/th&gt;
&lt;th&gt;
G.
&lt;/th&gt;
&lt;th&gt;
M.
&lt;/th&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;
Aries,
&lt;/td&gt;
&lt;td&gt;
Pisces,
&lt;/td&gt;
&lt;td&gt;
14.
&lt;/td&gt;
&lt;td&gt;
50.
&lt;/td&gt;
&lt;td&gt;
obl.
&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;
Taurus,
&lt;/td&gt;
&lt;td&gt;
Aquarius,
&lt;/td&gt;
&lt;td&gt;
18.
&lt;/td&gt;
&lt;td&gt;
51.
&lt;/td&gt;
&lt;td&gt;
obl.
&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;
Gemini,
&lt;/td&gt;
&lt;td&gt;
Capricornus
&lt;/td&gt;
&lt;td&gt;
27.
&lt;/td&gt;
&lt;td&gt;
26.
&lt;/td&gt;
&lt;td&gt;
obl.
&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;
Cancet,
&lt;/td&gt;
&lt;td&gt;
Sagittarius
&lt;/td&gt;
&lt;td&gt;
36.
&lt;/td&gt;
&lt;td&gt;
58.
&lt;/td&gt;
&lt;td&gt;
rect.
&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;
Leo,
&lt;/td&gt;
&lt;td&gt;
Scorpius
&lt;/td&gt;
&lt;td&gt;
40.
&lt;/td&gt;
&lt;td&gt;
57.
&lt;/td&gt;
&lt;td&gt;
rect.
&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;
Virgo,
&lt;/td&gt;
&lt;td&gt;
Libra,
&lt;/td&gt;
&lt;td&gt;
40.
&lt;/td&gt;
&lt;td&gt;
58.
&lt;/td&gt;
&lt;td&gt;
rect.
&lt;/td&gt;
&lt;/tr&gt;
&lt;/table&gt;
Partes autem illarum medietatum variantur se&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>kfud9od8gsevlr6nay6jtxrfjezlids</sha1>
    </revision>
    <revision>
      <id>281461</id>
      <parentid>281460</parentid>
      <timestamp>2026-09-20T16:55:31Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <origin>281461</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="1642" sha1="syqp1u8ukdwu1a1qotcfzmc08uav331" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;simul Simul peroriuntur. Idem intellige de quarta zodiaci, quæ est a principio Libræ usque ad finem Sagittarii.

Item si sumatur quarta zodiaci, quæ est a principio Cancri usque ad finem Virginis, semper maior pars oritur de quarta æquinoctialis, quam de quarta zodiaci, illi conterminali, &amp;amp; tamen illæ duæ quartæ simul peroriuntur. Idem intellige de quarta zodiaci, quæ est a primo puncto Capricorni usq́; ad finem piscium.

In sphæra autem obliqua siue declivi, duæ medietates zodiaci adæquantur suis ascensionibus: medietates dico, quæ sumuntur a duobus punctis æquinoctialibus: quia medietas zodiaci, quæ est a principio Arietis usque in finem Virginis, oritur cum medietate æquinoctialis sibi conterminali. Si militer alia medietas zodiaci oritur cum aliqua medietate æquinoctialis.

SCHOLION VINETI.

Innumeris in Iacobi Fabri commentariis tabellam hanc ad latitudinem 48. graduum factam, de ascensionibus signorum zodiaci.

&lt;table&gt;
&lt;tr&gt;
&lt;th&gt;
&lt;/th&gt;
&lt;th&gt;
G.
&lt;/th&gt;
&lt;th&gt;
M.
&lt;/th&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;
Aries,
&lt;/td&gt;
&lt;td&gt;
Pisces,
&lt;/td&gt;
&lt;td&gt;
14.
&lt;/td&gt;
&lt;td&gt;
50.
&lt;/td&gt;
&lt;td&gt;
obl.
&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;
Taurus,
&lt;/td&gt;
&lt;td&gt;
Aquarius,
&lt;/td&gt;
&lt;td&gt;
18.
&lt;/td&gt;
&lt;td&gt;
51.
&lt;/td&gt;
&lt;td&gt;
obl.
&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;
Gemini,
&lt;/td&gt;
&lt;td&gt;
Capricornus
&lt;/td&gt;
&lt;td&gt;
27.
&lt;/td&gt;
&lt;td&gt;
26.
&lt;/td&gt;
&lt;td&gt;
obl.
&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;
Cancet,
&lt;/td&gt;
&lt;td&gt;
Sagittarius
&lt;/td&gt;
&lt;td&gt;
36.
&lt;/td&gt;
&lt;td&gt;
58.
&lt;/td&gt;
&lt;td&gt;
rect.
&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;
Leo,
&lt;/td&gt;
&lt;td&gt;
Scorpius
&lt;/td&gt;
&lt;td&gt;
40.
&lt;/td&gt;
&lt;td&gt;
57.
&lt;/td&gt;
&lt;td&gt;
rect.
&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;
Virgo,
&lt;/td&gt;
&lt;td&gt;
Libra,
&lt;/td&gt;
&lt;td&gt;
40.
&lt;/td&gt;
&lt;td&gt;
58.
&lt;/td&gt;
&lt;td&gt;
rect.
&lt;/td&gt;
&lt;/tr&gt;
&lt;/table&gt;
Partes autem illarum medietatum variantur se-&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>syqp1u8ukdwu1a1qotcfzmc08uav331</sha1>
    </revision>
  </page>
  <page>
    <title>Pagina:Sphaera Ioannis de Sacrobosco emendata. Elia Vineti Santonis scholia in eandem Sphaeram, ab ipso authore restituta. Adiunximus huic libro compendium in Sphaeram (IA bub gb hZh9BYVGr7wC 3).pdf/50</title>
    <ns>104</ns>
    <id>86286</id>
    <revision>
      <id>281462</id>
      <timestamp>2026-09-20T16:55:51Z</timestamp>
      <contributor>
        <username>Sjgallagher2</username>
        <id>29663</id>
      </contributor>
      <comment>/* Nondum emendata */ Paginam instituit, scribens 'cundum suas ascensiones, quoniam in illa medietate zodiaci, quæ est à principio Arietis vique ad finem Virginis, semper maior pars oritur de zodiaco, quam de æquinoctiali, &amp;amp; tamen illæ medietates simul peroruntur,  Econuerso contingit in reliqua medietate zodiaci, quæ est à principio Libræ vique ad finem Piscium. semper enim maior pars oritur de æquinoctiali, quando zodiaco: &amp;amp; tamen illæ medietates simul peroruntur. Vnde hic patet ins...</comment>
      <origin>281462</origin>
      <model>proofread-page</model>
      <format>text/x-wiki</format>
      <text bytes="921" sha1="32331m82cb6s64j51ixeai16p66rw2u" xml:space="preserve">&lt;noinclude&gt;&lt;pagequality level=&quot;1&quot; user=&quot;Sjgallagher2&quot; /&gt;&lt;/noinclude&gt;cundum suas ascensiones, quoniam in illa medietate zodiaci, quæ est à principio Arietis vique ad finem Virginis, semper maior pars oritur de zodiaco, quam de æquinoctiali, &amp;amp; tamen illæ medietates simul peroruntur,

Econuerso contingit in reliqua medietate zodiaci, quæ est à principio Libræ vique ad finem Piscium. semper enim maior pars oritur de æquinoctiali, quando zodiaco: &amp;amp; tamen illæ medietates simul peroruntur. Vnde hic patet instantia facta manifestior contra argumentationem superius dictam.

Arcus autem, qui succedunt Arieti vique ad finem Virginis in sphæra obliqua, minuunt ascensiones suas supra ascensiones eorundem arcuum in sphæra recta, quia minus oritur de æquinociali,

SCHOLION VINETI.

Arcus autem qui.) Si arcus, hi cœperint ab ipso primo Arietis puncto. Alias poterit hoc esse falsum, ut deprehendes ex collatione ascensionis singulorum signorum in tabula veriusque sphæra.&lt;noinclude&gt;&lt;/noinclude&gt;</text>
      <sha1>32331m82cb6s64j51ixeai16p66rw2u</sha1>
    </revision>
  </page>
  <page>
    <title>Scriptor:Philippus Nicodemus Frischlinus</title>
    <ns>102</ns>
    <id>86287</id>
    <revision>
      <id>281490</id>
      <timestamp>2026-09-20T19:33:08Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>13304</id>
      </contributor>
      <comment>nova</comment>
      <origin>281490</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="668" sha1="b97m76y84mz3uiguj1t9sh6tv9cmevf" xml:space="preserve">{{Scriptor
|IndicisNomen=Frischlinus, Philippus Nicodemus 
}}

==Opera==
* [[De studiis linguarum et liberalium artium]] 
* [[Rebecca]]
* [[Oratio de vita rustica]]
* [[Priscianus vapulans]]
* [[Hildegardis Magna]]
* [[Dido]]
* [[Venus]]
* [[Julius Caesar redivivus]] 
* [[Graecanica proverbia selectiora cum symbolis veterum quorundam philosophorum, regum et imperatorum]]
* [[Helvetiogermani]]
* [[Dialogus logicus contra Ramum]]
* [[Poppysmus tertius]] 
* [[LXII facetiae]] 
* [[Phasma]]
* [[Hebraeis]]
* [[Carmina (Frischlinus)|Carmina]]
* [[Horologiographia]]
* [[Nomenclator trilinguis Graecolatinogermanicus]]
* [[Scriptor:Aristophanes|Aristophanis]] versiones:</text>
      <sha1>b97m76y84mz3uiguj1t9sh6tv9cmevf</sha1>
    </revision>
  </page>
  <page>
    <title>Scriptor:Rogerius Cotesius</title>
    <ns>102</ns>
    <id>86288</id>
    <revision>
      <id>281492</id>
      <timestamp>2026-09-20T20:32:16Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>13304</id>
      </contributor>
      <comment>nova</comment>
      <origin>281492</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="46" sha1="j7ba68aubp7kjrpj69b42c6n6zf7f84" xml:space="preserve">{{Scriptor
|IndicisNomen=Cotesius, Rogerius
}}</text>
      <sha1>j7ba68aubp7kjrpj69b42c6n6zf7f84</sha1>
    </revision>
    <revision>
      <id>281493</id>
      <parentid>281492</parentid>
      <timestamp>2026-09-20T20:35:52Z</timestamp>
      <contributor>
        <username>Demetrius Talpa</username>
        <id>13304</id>
      </contributor>
      <origin>281493</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="211" sha1="0ljwf0of3kgusn516hkblm3b1es1fpa" xml:space="preserve">{{Scriptor
|IndicisNomen=Cotesius, Rogerius
}}
==Opera==
*[[Logometria]], 1714
*[[Harmonia mensurarum]], 1722
*[[Aestimatio errorum in mixta mathesis]]
*[[De methodo differentiali Newtoniana]]
*[[Canonotechnia]]</text>
      <sha1>0ljwf0of3kgusn516hkblm3b1es1fpa</sha1>
    </revision>
  </page>
</mediawiki>