Wikisource lawikisource https://la.wikisource.org/wiki/Pagina_prima MediaWiki 1.47.0-wmf.20 first-letter Media Specialis Disputatio Usor Disputatio Usoris Vicifons Disputatio Vicifontis Fasciculus Disputatio Fasciculi MediaWiki Disputatio MediaWiki Formula Disputatio Formulae Auxilium Disputatio Auxilii Categoria Disputatio Categoriae Scriptor Disputatio Scriptoris Pagina Disputatio Paginae Liber Disputatio Libri TimedText TimedText talk Modulus Disputatio Moduli Event Event talk 0 34177 281552 138692 2026-09-22T09:44:56Z Cantons-de-l'Est 4777 Unused parameter. See [[:Categoria:Page transclusions with invalid attributes]] for more details. 281552 wikitext text/x-wiki {{Textquality|75%}} {{titulus |Scriptor=Titus Maccius Plautus |OperaeTitulus=Amphitruo |OperaeWikiPagina=Amphitruo (Lindsay) |Annus= ~206 a.C.n. |SubTitulus= Actus I |Editio= ex T. Macci Plauti ''Comoediae - vol. I'', Oxford, excudebat Joannes Johnson, 1904 |Liber= T. Macci Plauti Comoediae, I.djvu }} {{Liber |Ante=Prologus |AnteNomen=Amphitruo (Lindsay)/Prologus |Post= Actus II |PostNomen= Amphitruo (Lindsay)/Actus II }} <pages index="T. Macci Plauti Comoediae, I.djvu" from=34 to=51 fromsection=1 tosection=1 /> {{Liber |Ante=Prologus |AnteNomen=Amphitruo (Lindsay)/Prologus |Post= Actus II |PostNomen= Amphitruo (Lindsay)/Actus II }} lhpuewpfsf8e6fk5o8i1lmc7das985m 104 86116 281495 281356 2026-09-21T12:24:15Z Sjgallagher2 29663 281495 proofread-page text/x-wiki <noinclude><pagequality level="1" user="Sjgallagher2" /></noinclude><math display="block"> \gamma^{\lambda}\text{ cotg }\mu a \sin \lambda x + \gamma^{\lambda}\text{ cosec } \mu a\sin(\mu-\lambda)x\ \text{ sive }\ \frac{\gamma^{\lambda}\cos \mu a\sin \lambda x+\gamma^{\lambda}\sin(\mu-\lambda)x}{\sin \mu a} </math> quod praeceptum lectores cum iis, quae in art. 24 tradidimus, ipsi comparent. <br /> {{center|35.}} Si functio <math display="inline">X</math> cum <math display="inline">\sin n x</math> non abrumpitur, sed ulterius excurrit, terminis sequentibus per <math display="inline">\delta^{k+1} \sin (\mu + 1)x + \delta^{\mu+2} \sin (\mu + 2)x</math> etc. expressis, habebimus: <math display="block"> \begin{align} \zeta' = \frac{1}{\sin \mu a} \{ \delta' \sin \mu a + \delta^{\mu+1} \sin 2 \mu a + \delta^{2\mu+1} \sin 3 \mu a + \text{etc.} \\ + \delta^{2\mu-1} \sin \mu a + \delta^{3\mu-1} \sin 2 \mu a + \delta^{4\mu-1} \sin 3 \mu a + \text{etc.} \}\\ \zeta'' = \frac{1}{\sin \mu a} \{ \delta'' \sin \mu a + \delta^{\mu+2} \sin 2 \mu a + \delta^{2\mu+2} \sin 3 \mu a + \text{etc.} \\ + \delta^{2\mu-2} \sin \mu a + \delta^{3\mu-2} \sin 2 \mu a + \delta^{4\mu-2} \sin 3 \mu a + \text{etc.} \}\\ \zeta''' = \frac{1}{\sin \mu a} \{ \delta''' \sin \mu a + \delta^{\mu+3} \sin 2 \mu a + \delta^{2\mu+3} \sin 3 \mu a + \text{etc.} \\ + \delta^{2\mu-3} \sin \mu a + \delta^{3\mu-3} \sin 2 \mu a + \delta^{4\mu-3} \sin 3 \mu a + \text{etc.} \}\\ \end{align} </math> etc. Pro coëfficiente ultimo autem <math display="block"> \zeta^\mu = \frac{1}{\sin \mu a} (\delta^\mu \sin \mu a + \delta^{2\mu} \sin 2 \mu a + \delta^{3\mu} \sin 3 \mu a + \text{etc.}) </math> Hae formulae ostendunt, quatenus differentia inter <math display="inline">X</math> et <math display="inline">X''</math> negligi possit. Haec posterior formula simplicissima erit inter omnes similes, per quas <math display="inline">\mu</math> valoribus propositis satisfit; hae vero omnes in formula <math display="inline">X'' + Y \sin x (\cos \mu x - \cos \mu a)</math> contentae erunt, designante <math display="inline">Y</math>, ut in art. 32 functionem indefinitam arcus <math display="inline">x</math> a sinubus liberam. Et generaliter, si <math display="inline">X''</math> est functio quaecunque eiusdem formae ut <math display="inline">X</math>, i.e. solos sinus continens, per quam <math display="inline">\mu</math> valoribus datis satisfit, formula <math display="inline">X'' + Y \sin x (\cos \mu x - \cos \mu a)</math> omnes huiusmodi functiones continebit, quae si <math display="inline">Y</math> rite determinatur, ad ordinem <math display="inline">\mu^{\text{tum}}</math> deprimi potest, quo pacto necessario functio <math display="inline">X''</math> ipsa prodire debet. Prorsus simili modo ut in art. 32 regula generalis sequens ad hunc finem eruitur: Pro quovis termino in <math display="inline">X''</math> tali <math display="inline">L \sin \lambda x</math>, ubi <math display="inline">\lambda</math> est maior quam <math display="inline">\mu</math>, substituere oportet in <math display="inline">X''</math>, faciendo <math display="inline">\lambda = k \mu + \lambda'</math>, ita ut <math display="inline">k \mu</math> sit multiplum ipsius <math display="inline">\mu</math> proxime minus quam <math display="inline">\lambda</math> adeoque <math display="inline">\lambda'</math> inter limites 1 et <math display="inline">\mu</math> incl. situs, terminos <math display="block"> L \frac{\sin (k+1)\mu a}{\sin \mu a} \sin \lambda' x + L \frac{\sin k\mu a}{\sin \mu a} \sin (\mu - \lambda') x </math> qui, quoties fit <math display="inline">\lambda' = \mu</math>, ad unum <math display="inline">L \frac{\sin \lambda a}{\sin \mu a} \sin \mu x</math> reducuntur.{{nop}}<noinclude></noinclude> g6u8dv0n425ozy1a0oxna9ce4vv9f2g 281496 281495 2026-09-21T12:24:48Z Sjgallagher2 29663 281496 proofread-page text/x-wiki <noinclude><pagequality level="1" user="Sjgallagher2" /></noinclude><math display="block"> \gamma^{\lambda}\text{ cotg }\mu a \sin \lambda x + \gamma^{\lambda}\text{ cosec } \mu a\sin(\mu-\lambda)x\ \text{ sive }\ \frac{\gamma^{\lambda}\cos \mu a\sin \lambda x+\gamma^{\lambda}\sin(\mu-\lambda)x}{\sin \mu a} </math> quod praeceptum lectores cum iis, quae in art. 24 tradidimus, ipsi comparent. <br /> {{center|35.}} Si functio <math display="inline">X</math> cum <math display="inline">\sin n x</math> non abrumpitur, sed ulterius excurrit, terminis sequentibus per <math display="inline">\delta^{\mu+1} \sin (\mu + 1)x + \delta^{\mu+2} \sin (\mu + 2)x</math> etc. expressis, habebimus: <math display="block"> \begin{align} \zeta' = \frac{1}{\sin \mu a} \{ \delta' \sin \mu a + \delta^{\mu+1} \sin 2 \mu a + \delta^{2\mu+1} \sin 3 \mu a + \text{etc.} \\ + \delta^{2\mu-1} \sin \mu a + \delta^{3\mu-1} \sin 2 \mu a + \delta^{4\mu-1} \sin 3 \mu a + \text{etc.} \}\\ \zeta'' = \frac{1}{\sin \mu a} \{ \delta'' \sin \mu a + \delta^{\mu+2} \sin 2 \mu a + \delta^{2\mu+2} \sin 3 \mu a + \text{etc.} \\ + \delta^{2\mu-2} \sin \mu a + \delta^{3\mu-2} \sin 2 \mu a + \delta^{4\mu-2} \sin 3 \mu a + \text{etc.} \}\\ \zeta''' = \frac{1}{\sin \mu a} \{ \delta''' \sin \mu a + \delta^{\mu+3} \sin 2 \mu a + \delta^{2\mu+3} \sin 3 \mu a + \text{etc.} \\ + \delta^{2\mu-3} \sin \mu a + \delta^{3\mu-3} \sin 2 \mu a + \delta^{4\mu-3} \sin 3 \mu a + \text{etc.} \}\\ \end{align} </math> etc. Pro coëfficiente ultimo autem <math display="block"> \zeta^\mu = \frac{1}{\sin \mu a} (\delta^\mu \sin \mu a + \delta^{2\mu} \sin 2 \mu a + \delta^{3\mu} \sin 3 \mu a + \text{etc.}) </math> Hae formulae ostendunt, quatenus differentia inter <math display="inline">X</math> et <math display="inline">X''</math> negligi possit. Haec posterior formula simplicissima erit inter omnes similes, per quas <math display="inline">\mu</math> valoribus propositis satisfit; hae vero omnes in formula <math display="inline">X'' + Y \sin x (\cos \mu x - \cos \mu a)</math> contentae erunt, designante <math display="inline">Y</math>, ut in art. 32 functionem indefinitam arcus <math display="inline">x</math> a sinubus liberam. Et generaliter, si <math display="inline">X''</math> est functio quaecunque eiusdem formae ut <math display="inline">X</math>, i.e. solos sinus continens, per quam <math display="inline">\mu</math> valoribus datis satisfit, formula <math display="inline">X'' + Y \sin x (\cos \mu x - \cos \mu a)</math> omnes huiusmodi functiones continebit, quae si <math display="inline">Y</math> rite determinatur, ad ordinem <math display="inline">\mu^{\text{tum}}</math> deprimi potest, quo pacto necessario functio <math display="inline">X''</math> ipsa prodire debet. Prorsus simili modo ut in art. 32 regula generalis sequens ad hunc finem eruitur: Pro quovis termino in <math display="inline">X''</math> tali <math display="inline">L \sin \lambda x</math>, ubi <math display="inline">\lambda</math> est maior quam <math display="inline">\mu</math>, substituere oportet in <math display="inline">X''</math>, faciendo <math display="inline">\lambda = k \mu + \lambda'</math>, ita ut <math display="inline">k \mu</math> sit multiplum ipsius <math display="inline">\mu</math> proxime minus quam <math display="inline">\lambda</math> adeoque <math display="inline">\lambda'</math> inter limites 1 et <math display="inline">\mu</math> incl. situs, terminos <math display="block"> L \frac{\sin (k+1)\mu a}{\sin \mu a} \sin \lambda' x + L \frac{\sin k\mu a}{\sin \mu a} \sin (\mu - \lambda') x </math> qui, quoties fit <math display="inline">\lambda' = \mu</math>, ad unum <math display="inline">L \frac{\sin \lambda a}{\sin \mu a} \sin \mu x</math> reducuntur.{{nop}}<noinclude></noinclude> oop8l7s0092wmh1a3ek2ns5ev07l4yi 281497 281496 2026-09-21T12:31:39Z Sjgallagher2 29663 /* Emendata */ 281497 proofread-page text/x-wiki <noinclude><pagequality level="3" user="Sjgallagher2" /></noinclude><math display="block"> \gamma^{\lambda}\text{ cotg }\mu a \sin \lambda x + \gamma^{\lambda}\text{ cosec } \mu a\sin(\mu-\lambda)x\ \text{ sive }\ \frac{\gamma^{\lambda}\cos \mu a\sin \lambda x+\gamma^{\lambda}\sin(\mu-\lambda)x}{\sin \mu a} </math> quod praeceptum lectores cum iis, quae in art. 24 tradidimus, ipsi comparent. <br /> {{center|35.}} Si functio <math display="inline">X</math> cum <math display="inline">\sin n x</math> non abrumpitur, sed ulterius excurrit, terminis sequentibus per <math display="inline">\delta^{\mu+1} \sin (\mu + 1)x + \delta^{\mu+2} \sin (\mu + 2)x</math> etc. expressis, habebimus: <math display="block"> \begin{align} \zeta' = \frac{1}{\sin \mu a} \{ \delta' \sin \mu a + \delta^{\mu+1} \sin 2 \mu a + \delta^{2\mu+1} \sin 3 \mu a + \text{etc.} \\ + \delta^{2\mu-1} \sin \mu a + \delta^{3\mu-1} \sin 2 \mu a + \delta^{4\mu-1} \sin 3 \mu a + \text{etc.} \}\\ \zeta'' = \frac{1}{\sin \mu a} \{ \delta'' \sin \mu a + \delta^{\mu+2} \sin 2 \mu a + \delta^{2\mu+2} \sin 3 \mu a + \text{etc.} \\ + \delta^{2\mu-2} \sin \mu a + \delta^{3\mu-2} \sin 2 \mu a + \delta^{4\mu-2} \sin 3 \mu a + \text{etc.} \}\\ \zeta''' = \frac{1}{\sin \mu a} \{ \delta''' \sin \mu a + \delta^{\mu+3} \sin 2 \mu a + \delta^{2\mu+3} \sin 3 \mu a + \text{etc.} \\ + \delta^{2\mu-3} \sin \mu a + \delta^{3\mu-3} \sin 2 \mu a + \delta^{4\mu-3} \sin 3 \mu a + \text{etc.} \}\\ \end{align} </math> etc. Pro coëfficiente ultimo autem <math display="block"> \zeta^\mu = \frac{1}{\sin \mu a} (\delta^\mu \sin \mu a + \delta^{2\mu} \sin 2 \mu a + \delta^{3\mu} \sin 3 \mu a + \text{etc.}) </math> Hae formulae ostendunt, quatenus differentia inter <math display="inline">X</math> et <math display="inline">X''</math> negligi possit. Haec posterior formula simplicissima erit inter omnes similes, per quas <math display="inline">\mu</math> valoribus propositis satisfit; hae vero omnes in formula <math display="inline">X'' + Y \sin x (\cos \mu x - \cos \mu a)</math> contentae erunt, designante <math display="inline">Y</math>, ut in art. 32 functionem indefinitam arcus <math display="inline">x</math> a sinubus liberam. Et generaliter, si <math display="inline">X'''</math> est functio quaecunque eiusdem formae ut <math display="inline">X</math>, i.e. solos sinus continens, per quam <math display="inline">\mu</math> valoribus datis satisfit, formula <math display="inline">X''' + Y \sin x (\cos \mu x - \cos \mu a)</math> omnes huiusmodi functiones continebit, quae si <math display="inline">Y</math> rite determinatur, ad ordinem <math display="inline">\mu^{\text{tum}}</math> deprimi potest, quo pacto necessario functio <math display="inline">X''</math> ipsa prodire debet. Prorsus simili modo ut in art. 32 regula generalis sequens ad hunc finem eruitur: Pro quovis termino in <math display="inline">X'''</math> tali <math display="inline">L \sin \lambda x</math>, ubi <math display="inline">\lambda</math> est maior quam <math display="inline">\mu</math>, substituere oportet in <math display="inline">X''</math>, faciendo <math display="inline">\lambda = k \mu + \lambda'</math>, ita ut <math display="inline">k \mu</math> sit multiplum ipsius <math display="inline">\mu</math> proxime minus quam <math display="inline">\lambda</math> adeoque <math display="inline">\lambda'</math> inter limites 1 et <math display="inline">\mu</math> incl. situs, terminos <math display="block"> L \frac{\sin (k+1)\mu a}{\sin \mu a} \sin \lambda' x + L \frac{\sin k\mu a}{\sin \mu a} \sin (\mu - \lambda') x </math> qui, quoties fit <math display="inline">\lambda' = \mu</math>, ad unum <math display="inline">L \frac{\sin \lambda a}{\sin \mu a} \sin \mu x</math> reducuntur.{{nop}}<noinclude></noinclude> s3mg8bz2x0keqec8po6kop4e88l6ooe 104 86117 281498 281093 2026-09-21T12:36:23Z Sjgallagher2 29663 /* Emendata */ 281498 proofread-page text/x-wiki <noinclude><pagequality level="3" user="Sjgallagher2" /></noinclude><br /> {{center|36.}} Transformationes in art. praec. atque in art. 32 traditae concinnius ex theoremate quodam generali deduci possunt, quod quum per se quoque satis elegans sit, paucis hic adhuc attingemus. {{small-caps|Theorema.}} ''Designantibus <math display="inline">\lambda, \lambda', \lambda''</math> numeros integros quoscunque, <math display="inline">\mu</math> numerum integrum, qui differentias inter illos, <math display="inline">\lambda'-\lambda, \lambda''-\lambda', \lambda-\lambda''</math> metitur (e.g. unitatem), <math display="inline">x</math> arcum indefinitum, <math display="inline">a</math> arcum definitum: functiones'' <math display="block"> \begin{align} P = \sin(\lambda'-\lambda'')a \cos \lambda x + \sin(\lambda''-\lambda)a \cos \lambda' x + \sin(\lambda-\lambda')a \cos \lambda'' x \\ Q = \sin(\lambda'-\lambda'')a \sin \lambda x + \sin(\lambda''-\lambda)a \sin \lambda' x + \sin(\lambda-\lambda')a \cos \lambda'' x \end{align} </math> ''per <math display="inline">\cos \mu x - \cos \mu a</math> erunt divisibles.'' ''Demonstr''. Quando <math display="inline">\lambda</math> per <math display="inline">\mu</math> divisibilis est, ideoque etiam <math display="inline">\lambda', \lambda''</math> per <math display="inline">\mu</math> divisibiles erunt, facile confirmatur, valorem ipsarum <math display="inline">P, Q</math>, si substituatur <math display="inline">x = a</math>, esse identice <math display="inline">= 0</math>; quare <math display="inline">P</math> non mutabitur, si pro <math display="block"> \begin{align} \cos \lambda x &\text{ substituitur } \cos \lambda x - \cos \lambda a \\ \cos \lambda' x &\text{ substituitur } \cos \lambda' x - \cos \lambda' a \\ \cos \lambda'' x &\text{ substituitur } \cos \lambda'' x - \cos \lambda'' a \end{align} </math> neque <math display="inline">Q</math>, si pro <math display="block"> \begin{align} \sin \lambda x &\text{ substituitur } \sin \lambda x - \frac{\sin \lambda a \sin \mu x}{\sin \mu a} \\ \sin \lambda' x &\text{ substituitur } \sin \lambda' x - \frac{\sin \lambda' a \sin \mu x}{\sin \mu a} \\ \sin \lambda'' x &\text{ substituitur } \sin \lambda'' x - \frac{\sin \lambda'' a \sin \mu x}{\sin \mu a} \end{align} </math> Sed hae sex expressiones per <math display="inline">\cos \mu x - \cos \mu a</math> divisibles sunt, quod pro valore positivo ipsius <math display="inline">\lambda</math> de prima et quarta ostendisse sufficit. Scilicet facile per multiplicationem confirmatur, esse <math display="block"> \begin{align} &\sin \mu a (\cos \lambda x - \cos \lambda a) \\ &= (\cos \mu x - \cos \mu a) \{ 2 \sin \mu a \cos (\lambda - \mu)x + 2 \sin 2 \mu a \cos (\lambda - 2 \mu)x \\ &\qquad\qquad+ 2 \sin 3 \mu a \cos (\lambda - 3 \mu)x + \text{ etc.} + 2 \sin (\lambda - \mu)a \cos \mu x + \sin \lambda a \}\\ &\sin \mu a \sin \lambda x - \sin \lambda a \sin \mu x \\ &= (\cos \mu x - \cos \mu a) \{ 2 \sin \mu a \sin (\lambda - \mu)x + 2 \sin 2 \mu a \sin (\lambda - 2 \mu)x \\ &\qquad\qquad+ 2 \sin 3 \mu a \sin (\lambda - 3 \mu)x + \text{ etc.} + 2 \sin (\lambda - \mu)a \sin \mu x \} \end{align} </math> Casus, ubi <math display="inline">\lambda</math> est negativus, ad hunc sponte reducitur. Hinc patet, functiones<noinclude></noinclude> kqpfszv7eawhf4wnaxjawvqcfpmvkyy 104 86118 281499 281094 2026-09-21T12:40:42Z Sjgallagher2 29663 /* Emendata */ 281499 proofread-page text/x-wiki <noinclude><pagequality level="3" user="Sjgallagher2" /></noinclude><math display="inline">P, Q</math> ex partibus per <math display="inline">\cos \mu x - \cos \mu a</math> divisibiles compositas, ideoque ipsas quoque per hunc divisorem divisibiles esse. II. Quando <math display="inline">\lambda</math> per <math display="inline">\mu</math> non est divisibilis, sit <math display="inline">l</math> numerus integer arbitrarius per <math display="inline">\mu</math> divisibilis, ponaturque <math display="inline">\lambda = l + \theta</math>, <math display="inline">\lambda' = l' + \theta</math>, <math display="inline">\lambda'' = l'' + \theta</math>, unde etiam <math display="inline">l', l''</math> per <math display="inline">\mu</math> divisibles erunt. Iam patet, si ponatur <math display="block"> \begin{align} \sin (l' - l'') a \cos lx + \sin (l'' - l) a \cos l'x + \sin (l - l') a \cos l''x = P'\\ \sin (l' - l'') a \sin lx + \sin (l'' - l) a \sin l'x + \sin (l - l') a \sin l''x = Q' \end{align} </math> fieri <math display="block"> P = P' \cos \theta x - \theta' \sin \theta x, \quad Q = P' \sin \theta x + Q' \cos \theta x, </math> atque functiones <math display="inline">P', Q'</math>, quippe quae sub casum primum iam absolutum pertinent, per <math display="inline">\cos \mu x - \cos \mu a</math> divisibiles: hinc manifesto etiam <math display="inline">P</math> et <math display="inline">Q</math> per <math display="inline">\cos \mu x - \cos \mu a</math> divisibiles erunt. Q. E. D. Ceterum demonstratio casus primi ita perfecta est, ut non sine quibusdam explicationibus applicari possit, quoties <math display="inline">\sin \mu a = 0</math>; tunc vero fit <math display="inline">P = 0, Q = 0</math>, ita ut demonstratione omnino non opus sit.</li></ol> Quodsi itaque ponitur <math display="inline">\lambda - \lambda' = k \mu, \lambda - \lambda'' = (k+1) \mu</math>, patet, per <math display="inline">\cos \mu x - \cos \mu a</math> divisibiles esse <math display="block"> \begin{align} \sin \mu a \cos \lambda x - \sin (k+1) \mu a \cos \lambda' x + \sin k \mu a \cos (\lambda' - \mu) x\\ \sin \mu a \sin \lambda x - \sin (k+1) \mu a \sin \lambda' x + \sin k \mu a \sin (\lambda' - \mu) x \end{align} </math> adeoque etiam <math display="block"> \begin{align} \cos \lambda x - \frac{\sin (k+1) \mu a \cos \lambda' x - \sin k \mu a \cos (\mu - \lambda') x}{\sin \mu a}\\ \sin \lambda x - \frac{\sin (k+1) \mu a \sin \lambda' x + \sin k \mu a \sin (\mu - \lambda') x}{\sin \mu a} \end{align} </math> unde ratio substitutionum in artt. 32, 35 statim elucet: quotiens enim ex divisione posteriore e solis sinubus constabit, adeoque manifesto denuo per <math display="inline">\sin x</math> divisibilis erit. <br /> {{center|37.}} In artt. 30-36 supposuimus, <math display="inline">\sin \mu a</math> non esse <math display="inline">= 0</math>: superest itaque, ut easdem disquisitiones pro eo casu resumamus, ubi <math display="inline">\sin \mu a = 0</math>. Hic statim supponemus, esse <math display="inline">a = 0</math>, vel <math display="inline">a = \frac{180^\circ}{\mu}</math>.{{nop}}<noinclude></noinclude> l1kqcygspb6hz376sjc1x6coihsc2ys 104 86119 281500 281357 2026-09-21T12:44:19Z Sjgallagher2 29663 281500 proofread-page text/x-wiki <noinclude><pagequality level="1" user="Sjgallagher2" /></noinclude>Sit primo <math display="inline">X</math> functio formae <math display="block"> \gamma + \gamma' \cos x + \gamma'' \cos 2x + \text{etc.} + \gamma^n \cos nx </math> adeoque multitudo coëfficientium incognitorum <math display="inline">= 1 + n</math>. Iam scimus, ex omnibus valoribus propositis functionis <math display="inline">X</math> eos reiicii debere, qui respondent valori ipsius maior quam 180°: quatuor itaque casus hic sunt distinguendi: # quando <math display="inline">\mu</math> par, atque <math display="inline">a = 0</math>, erit 180° valor <math display="inline">\frac{1}{2} \mu + 1^{\text{tus}}</math> ipsius <math display="inline">x</math>; quare quum sequentes reiicii debeant, remanent valores <math display="inline">\frac{1}{2} \mu + 1</math>. Hinc esse debet <math display="inline">n = \frac{1}{2} \mu</math>. # quando <math display="inline">\mu</math> par, atque <math display="inline">a = \frac{180^\circ}{\mu}</math>, valor <math display="inline">\frac{1}{2} \mu^{\text{tus}}</math> ipsius <math display="inline">x</math> erit <math display="inline">180^\circ - \frac{180^\circ}{\mu}</math>; sequentes, qui fiunt maiores quam 180°, reiiciendi sunt. Hinc esse debet <math display="inline">n = \frac{1}{2} \mu - 1</math>. # quando <math display="inline">\mu</math> impar est, atque <math display="inline">a = 0</math>, fit valor <math display="inline">\frac{1}{2} \mu + \frac{1}{2}^{\text{tus}} = 180^\circ - \frac{180^\circ}{\mu}</math>, et # quando <math display="inline">\mu</math> impar est, atque <math display="inline">a = \frac{180^\circ}{\mu}</math>, fit valor <math display="inline">\frac{1}{2} \mu + \frac{1}{2}^{\text{tus}} = 180^\circ</math>: sequentes in utroque casu reiicii debent, adeoque erit <math display="inline">n = \frac{1}{2} \mu - \frac{1}{2}</math>. Iam quoniam methodus in praecc. adhibita ad casum praesentem, ubi pars valorum datorum a periodo completa antea rescindenda esset, non sine quibusdam ambagibus applicari posset, methodum sequentem praeferimus. Si per praecerta artt. 20, 22 functio formae <math display="block"> \begin{align} \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} </math> investigatur, per quam omnibus <math display="inline">\mu</math> valoribus datis satisfit, et in qua <math display="inline">m = \frac{1}{2} \mu - \frac{1}{2}</math> vel <math display="inline">= \frac{1}{2} \mu</math>, prout <math display="inline">\mu</math> impar est vel par, coëfficientes <math display="inline">\delta', \delta'', \delta'''</math> etc. sponte fient <math display="inline">= 0</math>. Nullo enim negotio patet, in expressione tali <math display="block"> A \sin \lambda a + B \sin \lambda b + C \sin \lambda c + D \sin \lambda d + \text{etc.} </math> fieri vel partem primam <math display="inline">= 0</math>, atque ultimam <math display="inline">= -B \sin \lambda b</math>, penultimam <math display="inline">= -C \sin \lambda c</math>, antepenultimam <math display="inline">= -D \sin \lambda d</math> etc. puta quando <math display="inline">a = 0</math>; vel ultimam <math display="inline">= -A \sin \lambda a</math>, penultimam <math display="inline">= -B \sin \lambda b</math>, antepenultimam <math display="inline">= -C \sin \lambda c</math> etc., quando <math display="inline">a = \frac{180^\circ}{\mu}</math>, quum pro talibus valoribus ipsius <math display="inline">x</math>, quorum alter alterius complementum ad 360° est, valores functionis <math display="inline">X</math> aequales sint. Quamobrem functio <math display="block"> X' = \gamma + \gamma' \cos x + \gamma'' \cos 2x + \text{etc.} + \gamma^m \cos mx </math><noinclude></noinclude> 73byykzetjl5prka888ognoirnhzcrm 281501 281500 2026-09-21T12:45:53Z Sjgallagher2 29663 /* Emendata */ 281501 proofread-page text/x-wiki <noinclude><pagequality level="3" user="Sjgallagher2" /></noinclude>Sit primo <math display="inline">X</math> functio formae <math display="block"> \gamma + \gamma' \cos x + \gamma'' \cos 2x + \text{etc.} + \gamma^n \cos nx </math> adeoque multitudo coëfficientium incognitorum <math display="inline">= 1 + n</math>. Iam scimus, ex omnibus valoribus propositis functionis <math display="inline">X</math> eos reiicii debere, qui respondent valori ipsius maior quam 180°: quatuor itaque casus hic sunt distinguendi: # quando <math display="inline">\mu</math> par, atque <math display="inline">a = 0</math>, erit 180° valor <math display="inline">\frac{1}{2} \mu + 1^{\text{tus}}</math> ipsius <math display="inline">x</math>; quare quum sequentes reiicii debeant, remanent valores <math display="inline">\frac{1}{2} \mu + 1</math>. Hinc esse debet <math display="inline">n = \frac{1}{2} \mu</math>. # quando <math display="inline">\mu</math> par, atque <math display="inline">a = \frac{180^\circ}{\mu}</math>, valor <math display="inline">\frac{1}{2} \mu^{\text{tus}}</math> ipsius <math display="inline">x</math> erit <math display="inline">180^\circ - \frac{180^\circ}{\mu}</math>; sequentes, qui fiunt maiores quam 180°, reiiciendi sunt. Hinc esse debet <math display="inline">n = \frac{1}{2} \mu - 1</math>. # quando <math display="inline">\mu</math> impar est, atque <math display="inline">a = 0</math>, fit valor <math display="inline">\frac{1}{2} \mu + \frac{1}{2}^{\text{tus}} = 180^\circ - \frac{180^\circ}{\mu}</math>, et # quando <math display="inline">\mu</math> impar est, atque <math display="inline">a = \frac{180^\circ}{\mu}</math>, fit valor <math display="inline">\frac{1}{2} \mu + \frac{1}{2}^{\text{tus}} = 180^\circ</math>: sequentes in utroque casu reiicii debent, adeoque erit <math display="inline">n = \frac{1}{2} \mu - \frac{1}{2}</math>. Iam quoniam methodus in praecc. adhibita ad casum praesentem, ubi pars valorum datorum a periodo completa antea rescindenda esset, non sine quibusdam ambagibus applicari posset, methodum sequentem praeferimus. Si per praecerta artt. 20, 22 functio formae <math display="block"> \begin{align} \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} </math> investigatur, per quam omnibus <math display="inline">\mu</math> valoribus datis satisfit, et in qua <math display="inline">m = \frac{1}{2} \mu - \frac{1}{2}</math> vel <math display="inline">= \frac{1}{2} \mu</math>, prout <math display="inline">\mu</math> impar est vel par, coëfficientes <math display="inline">\delta', \delta'', \delta'''</math> etc. sponte fient <math display="inline">= 0</math>. Nullo enim negotio patet, in expressione tali <math display="block"> A \sin \lambda a + B \sin \lambda b + C \sin \lambda c + D \sin \lambda d + \text{etc.} </math> fieri vel partem primam <math display="inline">= 0</math>, atque ultimam <math display="inline">= -B \sin \lambda b</math>, penultimam <math display="inline">= -C \sin \lambda c</math>, antepenultimam <math display="inline">= -D \sin \lambda d</math> etc. puta quando <math display="inline">a = 0</math>; vel ultimam <math display="inline">= -A \sin \lambda a</math>, penultimam <math display="inline">= -B \sin \lambda b</math>, antepenultimam <math display="inline">= -C \sin \lambda c</math> etc., quando <math display="inline">a = \frac{180^\circ}{\mu}</math>, quum pro talibus valoribus ipsius <math display="inline">x</math>, quorum alter alterius complementum ad 360° est, valores functionis <math display="inline">X</math> aequales sint. Quamobrem functio <math display="block"> X' = \gamma + \gamma' \cos x + \gamma'' \cos 2x + \text{etc.} + \gamma^m \cos mx </math><noinclude></noinclude> 99cl8etdeigiozd5up69s04fqzozhd1 104 86120 281502 281358 2026-09-21T12:51:36Z Sjgallagher2 29663 /* Emendata */ 281502 proofread-page text/x-wiki <noinclude><pagequality level="3" user="Sjgallagher2" /></noinclude>in qua coëfficientes <math display="inline">\gamma, \gamma', \gamma''</math> etc. determinantur per formulas <math display="block"> \begin{align} \gamma & = \frac{1}{\mu}(A + B + C + D + \text{etc.})\\ \gamma' & = \frac{2}{\mu}(A \cos a + B \cos b + C \cos c + D \cos d + \text{etc.})\\ \gamma'' & = \frac{2}{\mu}(A \cos 2a + B \cos 2b + C \cos 2c + D \cos 2d + \text{etc.})\\ \end{align} </math> etc., ultimus autem, quando <math display="inline">\mu</math> par est atque adeo <math display="inline">m = \frac{1}{2}\mu</math>, per hanc <math display="block"> \gamma^m = \frac{1}{\mu}(A \cos ma + B \cos mb + C \cos mc + D \cos md + \text{etc.}) </math> necessario cum functione <math display="inline">X</math> identica erit, siquidem haec non est gradus altioris quam supra definivimus. Namque in casibus 3 et 4 <math display="inline">X</math> est ordinis <math display="inline">\frac{1}{2}\mu - \frac{1}{2}^{\text{ti}}</math>, i.e. eiusdem ut <math display="inline">X'</math> et proin per art. 24 cum <math display="inline">X'</math> identica. In casu primo <math display="inline">X</math> est eiusdem ordinis ut <math display="inline">X'</math> et in casu secundo non maioris, quare tum hic tum illic omnes termini saltem usque ad ordinem <math display="inline">\frac{1}{2}\mu - 1^{tum}</math> in utraque functione convenient (art. 24). Terminos ordinis <math display="inline">\frac{1}{2}\mu^{\text{ti}}</math> in his functionibus quoque convenire debere, inde per eundem art.24 patet, quod in <math display="inline">X</math> aequatio conditionalis <math display="inline">K \sin ma = L \cos ma</math> locum habet; scilicet fit <math display="inline">L = 0</math>, atque in casu primo <math display="inline">\sin ma = 0</math>, in secundo, ubi <math display="inline">X</math> ad ordinem <math display="inline">\frac{1}{2}\mu - 1</math> tantummodo ascendit, <math display="inline">K = 0</math>. Ceterum in casu secundo <math display="inline">X'</math> ordinis altioris esse videtur quam <math display="inline">X</math>, sed in hoc casu terminus ordinis <math display="inline">\frac{1}{2}\mu^{\text{ti}}</math> in <math display="inline">X'</math> quoque evanescit, quum fiat <math display="block"> \gamma^{\frac{1}{2}\mu} = \frac{1}{\mu}(A \cos 90^\circ + B \cos 270^\circ + C \cos 450^\circ + D \cos 630^\circ + \text{etc.}) = 0 </math> ita ut in hoc quoque casu <math display="inline">X'</math> revera sit ordinis <math display="inline">m - 1^{\text{ti}}</math> sive <math display="inline">\frac{1}{2}\mu - 1^{\text{ti}}</math>. <br /> {{center|38.}} Si functio <math display="inline">X</math> cum termino <math display="inline">\cos nx</math> non abrumpitur, sed ulterius excurrit: denotatis terminis sequentibus per <math display="inline">\alpha^{n+1} \cos(n+1)x + \alpha^{n+2} \cos(n+2)x + \text{etc.}</math> erit per artt. 21, 23 <math display="block"> \begin{align} \gamma &= \alpha \pm \alpha^{\mu} \pm \alpha^{2\mu} \pm \text{etc.}\\ \gamma' &= \alpha' \pm \alpha^{\mu-1} \pm \alpha^{\mu+1} \pm \alpha^{2\mu-1} \pm \alpha^{2\mu+1} \pm \text{etc.}\\ \gamma'' &= \alpha'' \pm \alpha^{\mu-2} \pm \alpha^{\mu+2} \pm \alpha^{2\mu-2} \pm \alpha^{2\mu+2} \pm \text{etc.}\\ \gamma''' &= \alpha''' \pm \alpha^{\mu-3} \pm \alpha^{\mu+3} \pm \alpha^{2\mu-3} \pm \alpha^{2\mu+3} \pm \text{etc.} \end{align} </math> et sic porro usque ad ultimum <math display="inline">\gamma^m</math>, quando <math display="inline">\mu</math> impar est, vel ad penultimum <math display="inline">\gamma^{m-1}</math>, quando <math display="inline">\mu</math> par est; signum inferius hic valet, quoties <math display="inline">a = \frac{180^\circ}{\mu}</math>, adeoque<noinclude></noinclude> fmsnnnvye4qa0ktzqwukar8r9fcv6gf 281503 281502 2026-09-21T12:52:00Z Sjgallagher2 29663 281503 proofread-page text/x-wiki <noinclude><pagequality level="3" user="Sjgallagher2" /></noinclude>in qua coëfficientes <math display="inline">\gamma, \gamma', \gamma''</math> etc. determinantur per formulas <math display="block"> \begin{align} \gamma & = \frac{1}{\mu}(A + B + C + D + \text{etc.})\\ \gamma' & = \frac{2}{\mu}(A \cos a + B \cos b + C \cos c + D \cos d + \text{etc.})\\ \gamma'' & = \frac{2}{\mu}(A \cos 2a + B \cos 2b + C \cos 2c + D \cos 2d + \text{etc.})\\ \end{align} </math> etc., ultimus autem, quando <math display="inline">\mu</math> par est atque adeo <math display="inline">m = \frac{1}{2}\mu</math>, per hanc <math display="block"> \gamma^m = \frac{1}{\mu}(A \cos ma + B \cos mb + C \cos mc + D \cos md + \text{etc.}) </math> necessario cum functione <math display="inline">X</math> identica erit, siquidem haec non est gradus altioris quam supra definivimus. Namque in casibus 3 et 4 <math display="inline">X</math> est ordinis <math display="inline">\frac{1}{2}\mu - \frac{1}{2}^{\text{ti}}</math>, i.e. eiusdem ut <math display="inline">X'</math> et proin per art. 24 cum <math display="inline">X'</math> identica. In casu primo <math display="inline">X</math> est eiusdem ordinis ut <math display="inline">X'</math> et in casu secundo non maioris, quare tum hic tum illic omnes termini saltem usque ad ordinem <math display="inline">\frac{1}{2}\mu - 1^{\text{tum}}</math> in utraque functione convenient (art. 24). Terminos ordinis <math display="inline">\frac{1}{2}\mu^{\text{ti}}</math> in his functionibus quoque convenire debere, inde per eundem art.24 patet, quod in <math display="inline">X</math> aequatio conditionalis <math display="inline">K \sin ma = L \cos ma</math> locum habet; scilicet fit <math display="inline">L = 0</math>, atque in casu primo <math display="inline">\sin ma = 0</math>, in secundo, ubi <math display="inline">X</math> ad ordinem <math display="inline">\frac{1}{2}\mu - 1</math> tantummodo ascendit, <math display="inline">K = 0</math>. Ceterum in casu secundo <math display="inline">X'</math> ordinis altioris esse videtur quam <math display="inline">X</math>, sed in hoc casu terminus ordinis <math display="inline">\frac{1}{2}\mu^{\text{ti}}</math> in <math display="inline">X'</math> quoque evanescit, quum fiat <math display="block"> \gamma^{\frac{1}{2}\mu} = \frac{1}{\mu}(A \cos 90^\circ + B \cos 270^\circ + C \cos 450^\circ + D \cos 630^\circ + \text{etc.}) = 0 </math> ita ut in hoc quoque casu <math display="inline">X'</math> revera sit ordinis <math display="inline">m - 1^{\text{ti}}</math> sive <math display="inline">\frac{1}{2}\mu - 1^{\text{ti}}</math>. <br /> {{center|38.}} Si functio <math display="inline">X</math> cum termino <math display="inline">\cos nx</math> non abrumpitur, sed ulterius excurrit: denotatis terminis sequentibus per <math display="inline">\alpha^{n+1} \cos(n+1)x + \alpha^{n+2} \cos(n+2)x + \text{etc.}</math> erit per artt. 21, 23 <math display="block"> \begin{align} \gamma &= \alpha \pm \alpha^{\mu} \pm \alpha^{2\mu} \pm \text{etc.}\\ \gamma' &= \alpha' \pm \alpha^{\mu-1} \pm \alpha^{\mu+1} \pm \alpha^{2\mu-1} \pm \alpha^{2\mu+1} \pm \text{etc.}\\ \gamma'' &= \alpha'' \pm \alpha^{\mu-2} \pm \alpha^{\mu+2} \pm \alpha^{2\mu-2} \pm \alpha^{2\mu+2} \pm \text{etc.}\\ \gamma''' &= \alpha''' \pm \alpha^{\mu-3} \pm \alpha^{\mu+3} \pm \alpha^{2\mu-3} \pm \alpha^{2\mu+3} \pm \text{etc.} \end{align} </math> et sic porro usque ad ultimum <math display="inline">\gamma^m</math>, quando <math display="inline">\mu</math> impar est, vel ad penultimum <math display="inline">\gamma^{m-1}</math>, quando <math display="inline">\mu</math> par est; signum inferius hic valet, quoties <math display="inline">a = \frac{180^\circ}{\mu}</math>, adeoque<noinclude></noinclude> 0npzpyzn8pwbhwoyby2lugmbzvp11ia 104 86121 281504 281359 2026-09-21T12:58:23Z Sjgallagher2 29663 /* Emendata */ 281504 proofread-page text/x-wiki <noinclude><pagequality level="3" user="Sjgallagher2" /></noinclude>in casu 2 et 4, superius in casu 1 et 2; denique pro ultimo habebitur in casu primo <math display="block"> \gamma^m = \alpha^m + \alpha^{3m} + \alpha^{5m} + \text{etc.} </math> in casu secundo autem <math display="inline">\gamma^m = 0</math>. Hae aequationes ostendunt, quatenus differentiam inter functiones <math display="inline">X</math> et <math display="inline">X'</math> negligere permissum esse possit. Haec posterior functio inter omnes, quae <math display="inline">\mu</math> valoribus propositis satisfaciunt, simplicissima erit, quae omnes sub forma <math display="inline">X' + Y \sin (\frac{1}{2} \mu x - \frac{1}{2} \mu a)</math> contenti erunt, quae in casu 1 et 3 ad <math display="inline">X' + Y \sin \frac{1}{2} \mu x</math>, in casibus 2 et 4 vero ad <math display="inline">X' + Y \cos \frac{1}{2} \mu x</math> reducitur. Manifesto autem, si haec expressio eiusdem formae esse debet ut <math display="inline">X</math> i.e. a sinibus libera, <math display="inline">Y</math> esse debet <math display="block"> \begin{align} &\text{in casu primo formae } &g \sin x + g' \sin 2x + g'' \sin 3x + \text{etc.} \\ &\text{in casu secondo formae } &g + g' \cos x + g'' \cos 2x + \text{etc.} \\ &\text{in casu tertio formae } &g \sin \tfrac{1}{2} x + g' \sin \tfrac{3}{2} x + g'' \sin \tfrac{5}{2} x + \text{etc.}\\ &\text{in casu quarto formae } &g \cos \tfrac{1}{2} x + g' \cos \tfrac{3}{2} x + g'' \cos \tfrac{5}{2} x + \text{etc.}\\ \end{align} </math> Et generalius, designante <math display="inline">X''</math> functionem quamcunque ipsi <math display="inline">X</math> similem, quae <math display="inline">\mu</math> valoribus propositis satisfacit, omnes huiusmodi formae sub formula <math display="inline">X'' + Y \sin \frac{1}{2} \mu x</math> vel <math display="inline">X'' + Y \cos \frac{1}{2} \mu x</math> contentae erunt, ubi <math display="inline">Y</math> functionem indefinitam eius, quam modo docuimus formae designat. Hoc ita perficere licet, ut sic functio ad ordinem <math display="inline">\frac{1}{2} \mu, \frac{1}{2} \mu - 1, \frac{1}{2} \mu - \frac{1}{2}, \frac{1}{2} \mu - \frac{1}{2}</math> depressa prodeat, quae manifesto cum <math display="inline">X'</math> identica erit. Regula autem generalis pro reductione talis functionis <math display="inline">X''</math> ad <math display="inline">X'</math> ex art. 24 facile deducitur. Pro quovis termino <math display="inline">L \cos \lambda x</math> in <math display="inline">X''</math> substitui debet in <math display="inline">X'</math>, facto <math display="inline">\lambda = k \mu \pm \lambda'</math>, ita ut <math display="inline">\lambda'</math> non sit maior quam <math display="inline">\frac{1}{2} \mu</math>, terminus <math display="inline">\pm L \cos \lambda' x</math>, ubi signum inferius accipendum est, quoties simul <math display="inline">a = \frac{180^\circ}{\mu}</math> atque <math display="inline">k</math> par, superius in casibus reliquis; denique quoties in casu secundo, i.e. pro <math display="inline">a = \frac{180^\circ}{\mu}</math> et valore pari ipsius <math display="inline">\mu</math>, evadit <math display="inline">\lambda' = \frac{1}{2} \mu</math>, pro <math display="inline">L \cos \lambda x</math> statim poni debet 0 in <math display="inline">X'</math>, sive terminus ille omnino neglegi. <br /> {{center|39.}} Si secundo functio <math display="inline">X</math> est formae <math display="block"> \beta' \sin x + \beta'' \sin 2x + \beta''' \sin 3x + \text{etc.} + \beta^n \sin nx </math> adeoque multitudo coëfficientium incognitorum <math display="inline">= n</math>, etiam multitudo valorum<noinclude></noinclude> c5gz51g19vtf7h2qql192qmvdhaah2x 104 86122 281505 281360 2026-09-21T13:02:57Z Sjgallagher2 29663 /* Emendata */ 281505 proofread-page text/x-wiki <noinclude><pagequality level="3" user="Sjgallagher2" /></noinclude>datorum, subductis superfluis, esse debet <math display="inline">= n</math>, ut ad coëfficientium determinationem completam sufficiant. Iam quum ut superflui in hoc casu reiiciendi sint valores functionis <math display="inline">X</math> ii, qui respondent valori ipsius <math display="inline">x</math> maiori quam <math display="inline">180^\circ</math> nec non valori <math display="inline">0</math> et <math display="inline">180^\circ</math>, habebimus pro quatuor casibus supra distinctis: <math display="block"> \begin{align} 1. & \text{ quando } \mu\text{ par, }\qquad &&a=0, && \text{ erit } n=\tfrac{1}{2}\mu-1 \\ 2. & \text{ quando } \mu\text{ par, }\qquad &&a=\tfrac{180^{\circ}}{\mu}, & & \text{ erit } n=\tfrac{1}{2}\mu \\ 3. & \text{ quando } \mu\text{ impar, }\qquad &&a=0, && \text{ erit } n=\tfrac{1}{2}\mu- \tfrac{1}{2} \\ 4. & \text{ quando } \mu\text{ impar, }\qquad &&a=\tfrac{180^{\circ}}{\mu}, && \text{ erit } n=\tfrac{1}{2}\mu-\tfrac{1}{2} \end{align} </math> Iam prorsus simili modo ut in art. 37, in functione <math display="block"> \begin{align} \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} </math> ad normam artt. 20, 22 eruta, quae omnibus <math display="inline">\mu</math> valoribus satisfacit, et in qua <math display="inline">m</math> vel <math display="inline">= \frac{1}{2}\mu - \frac{1}{2}</math>, vel <math display="inline">= \frac{1}{2}\mu</math>, coëfficientes <math display="inline">\gamma, \gamma', \gamma''</math> etc. sponte evanescent. Quum enim in serie <math display="inline">A, B, C, D</math> etc. vel termini ultimi ordine retrogrado in casu praesenti vel fiant <math display="inline">= -B, -C, -D</math> etc. vel <math display="inline">= -A, -B, -C</math> etc., prout <math display="inline">a = 0</math>, vel <math display="inline">= \frac{180^\circ}{\mu}</math>, insuperque pro illo casu <math display="inline">A = 0</math>, manifesto <math display="block"> A \cos \lambda a + B \cos \lambda b + C \cos \lambda c + D \cos \lambda d + \text{etc.} </math> pro quovis valore ipsius <math display="inline">x</math> erit <math display="inline">= 0</math>. Quamobrem functio <math display="inline">X' =</math> <math display="block"> \delta' \sin x + \delta'' \sin 2x + \delta''' \sin 3x + \text{etc.} + \delta^m \sin mx </math> in qua coëfficientes <math display="inline">\delta', \delta'', \delta'''</math> etc. determinantur per aequationes <math display="block"> \begin{align} \delta' &= \tfrac{2}{\mu} (A \sin a + B \sin b + C \sin c + D \sin d + \text{etc.}) \\ \delta'' &= \tfrac{2}{\mu} (A \sin 2a + B \sin 2b + C \sin 2c + D \sin 2d + \text{etc.}) \\ \delta''' &= \tfrac{2}{\mu} (A \sin 3a + B \sin 3b + C \sin 3c + D \sin 3d + \text{etc.}) \end{align} </math> etc., ultimus autem, quando <math display="inline">\mu</math> par est, adeoque <math display="inline">m = \frac{1}{2}\mu</math>, per hanc <math display="block"> \delta^m = \frac{1}{\mu} (A \sin ma + B \sin mb + C \sin mc + D \sin md + \text{etc.}) </math> necessario cum <math display="inline">X</math> identica erit, quod eodem modo, ut in art. 37, facile demonstratur. Ceterum functio <math display="inline">X'</math> in casu primo, ubi <math display="inline">a = 0</math>, <math display="inline">\mu</math> par, revera ad or-<noinclude></noinclude> sijx9dopmgh53u1wa8z7pclf9w12ptq 104 86123 281506 281361 2026-09-21T13:07:04Z Sjgallagher2 29663 /* Emendata */ 281506 proofread-page text/x-wiki <noinclude><pagequality level="3" user="Sjgallagher2" /></noinclude>dinem <math display="inline">\frac{1}{2} \mu - 1</math> tantum ascendit, quum fiat <math display="block"> \begin{align} \delta^m &= \frac{1}{\mu} (A \sin 0 + B \sin 180^\circ + C \sin 360^\circ + D \sin 540^\circ + \text{etc.}) \\ &= 0 \end{align} </math> <br /> {{center|40.}} Si functio <math display="inline">X</math> non est ordinis <math display="inline">n^{\text{ti}}</math>, ut supposuimus, sed ulterius excurrit, aequationes sequentes docebunt, quomodo differentia inter <math display="inline">X'</math> et <math display="inline">X</math> a coëfficientibus sequentibus pendeat (v. artt. 21, 23) <math display="block"> \begin{align} \delta' &= \beta' \mp \beta^{\mu-1} \pm \beta^{\mu+1} - \beta^{2\mu-1} + \beta^{2\mu+1} \mp \text{etc.} \\ \delta'' &= \beta'' \mp \beta^{\mu-2} \pm \beta^{\mu+2} - \beta^{2\mu-2} + \beta^{2\mu+2} \mp \text{etc.} \\ \delta''' &= \beta''' \mp \beta^{\mu-3} \pm \beta^{\mu+3} - \beta^{2\mu-3} + \beta^{2\mu+3} \mp \text{etc.} \end{align} </math> et sic porro usque ad ultimum <math display="inline">\delta^m</math> vel penultimum <math display="inline">\delta^{m-1}</math>, prout <math display="inline">\mu</math> impar est vel par; signa superiore hic valent, quando <math display="inline">a = 0</math>, inferiora, quando <math display="inline">a = \frac{180^\circ}{\mu}</math>: denique pro ultimo habetur in casu (1) <math display="inline">\delta^m = 0</math>, in casu (2) vero <math display="inline">\delta^m = \beta^m - \beta^{3m} + \beta^{5m} - \beta^{7m} + \text{etc}</math>. Omnes functiones periodicae, per quas <math display="inline">\mu</math> valoribus propositis satisfit, et ex quibus <math display="inline">X'</math> est simplicissima, sub forma <math display="inline">X' + Y \sin (\frac{1}{2} \mu x - \frac{1}{2} \mu a)</math> sive generalius sub forma <math display="inline">X'' + Y \sin (\frac{1}{2} \mu x - \frac{1}{2} \mu a)</math> contentae erunt, designante <math display="inline">X''</math> functionem talem quamcunque, quae formula pro casu 1 et 3 ad <math display="inline">X'' + Y \sin \frac{1}{2} \mu x</math>, pro casu 2 et 4 autem ad <math display="inline">X'' + Y \cos \frac{1}{2} \mu x</math> reducitur; <math display="inline">Y</math> vero, siquidem alias functiones non consideramus, nisi quae ipsi <math display="inline">X</math> sunt similes, i.e. e solis sinibus compositae, necessario debet esse: <math display="block"> \begin{align} &\text{in casu 1 formae } &g + g' \cos x + g'' \cos 2x + \text{etc.} \\ &\text{in casu 2 formae } &g \sin x + g' \sin 2x + g'' \sin 3x + \text{etc.} \\ &\text{in casu 3 formae } &g \cos \tfrac{1}{2} x + g' \cos \tfrac{3}{2} x + g'' \cos \tfrac{5}{2} x + \text{etc.}\\ &\text{in casu 4 formae } &g \sin \tfrac{1}{2} x + g' \sin \tfrac{3}{2} x + g'' \sin \tfrac{5}{2} x + \text{etc.}\\ \end{align} </math> Functionem <math display="inline">Y</math> hic ita determinare licebit, ut prodeat functio ad ordinem <math display="inline">\frac{1}{2} \mu - 1, \frac{1}{2} \mu, \frac{1}{2} \mu - \frac{1}{2}, \frac{1}{2} \mu - \frac{3}{2}</math> depressa, quae cum <math display="inline">X'</math> necessario identica erit. Pro reductione functionis <math display="inline">X''</math> ad <math display="inline">X'</math> regula generalis sequens habetur: Quivis terminus in <math display="inline">X''</math> talis <math display="inline">L \sin \lambda x = L \sin (k \mu \pm \lambda') x</math>, transmutetur aut in <math display="inline">\pm L \sin \lambda' x</math> (quoties <math display="inline">a = 0</math>, vel <math display="inline">k</math> par), aut in <math display="inline">\mp L \sin \lambda' x</math> (quoties nec <math display="inline">a = 0</math>, nec <math display="inline">k</math> par, i.e. quoties simul <math display="inline">a = \frac{180^\circ}{\mu}</math> atque <math display="inline">k</math> impar): denique quoties in casu<noinclude></noinclude> rtupvtpznruwk4oyk4cb4hrv6nv8a42 104 86124 281512 281100 2026-09-21T14:00:44Z Sjgallagher2 29663 /* Emendata */ 281512 proofread-page text/x-wiki <noinclude><pagequality level="3" user="Sjgallagher2" /></noinclude>primo i.e. pro <math display="inline">a = 0</math>, et valore pari ipsius <math display="inline">\mu</math> evadit <math display="inline">\lambda' = \frac{1}{2} \mu</math>, terminus <math display="inline">L \sin \lambda x</math> omnino destruatur. <br /> {{center|41.}} Quum omnes casus speciales in artt. 29-40 considerati ad casum generalem in artt. 20-28 absolute reducti sint, omnia artificia, per quae in hoc casu calculus abbreviatur, qualia in artt. 25, 26, 27 explicavimus, etiam ad illos applicari poterunt. Quamobrem non opus erit, huic disquisitioni immorari, cui sequens exemplum ad artt. 39, 40 pertinens, finem imponet. Aequatio centri pro novo planeta ''Iunone'', exhibita excentricitate 0,254236, calculata est per methodum indirectam per singulos denos gradus, ut sequitur {| style="border-spacing:0px;" align="center" |- style="text-align:center" | colspan="2" style="border-right:1px solid black;"|&nbsp; | colspan="4"|<math display="inline">\text{Aequatio Centri }=X</math> |- style="text-align:center" | colspan="2" style="padding: 0 10px 0 10px;text-align:left;border-bottom:1px solid black; border-right:1px solid black;"|<math display="inline">\text{Anomalia media } =x</math> | style="padding: 0 10px 0 10px;border-bottom:1px solid black;"|<math display="inline">-</math> | style="padding: 0 10px 0 10px;border-bottom:1px solid black;"|&nbsp; | style="text-align:left;padding: 0 10px 0 10px;border-bottom:1px solid black;"|<math display="inline">+</math> | style="padding: 0 10px 0 10px;border-bottom:1px solid black;"|&nbsp; |- style="text-align:center" | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0^\circ</math> | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">360^\circ</math> | style="padding: 0 10px 0 10px;"|<math display="inline">0</math> | style="padding: 0 10px 0 10px;"|&nbsp; | style="padding: 0 10px 0 10px;"|&nbsp; | style="padding: 0 10px 0 10px;"|<math display="inline">0</math> |- style="text-align:center" | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">10 </math> | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">350 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">3^\circ</math> | style="padding: 0 10px 0 10px;"|<math display="inline">50'</math> | style="padding: 0 10px 0 10px;"|<math display="inline">38''30\ .\ .\ .\ .\ </math> | style="padding: 0 10px 0 10px;"|<math display="inline">13838''30</math> |- style="text-align:center" | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">20 </math> | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">340 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">7 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">38 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">21,47\ .\ .\ .\ .\ </math> | style="padding: 0 10px 0 10px;"|<math display="inline">27501,47</math> |- style="text-align:center" | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">30 </math> | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">330 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">11 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">20 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">8,79 \ .\ .\ .\ .\ </math> | style="padding: 0 10px 0 10px;"|<math display="inline">40808,79</math> |- style="text-align:center" | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">40 </math> | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">320 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">14 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">52 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">48,06\ .\ .\ .\ .\ </math> | style="padding: 0 10px 0 10px;"|<math display="inline">53568,06</math> |- style="text-align:center" | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">50 </math> | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">310 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">18 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">12 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">49,21\ .\ .\ .\ .\ </math> | style="padding: 0 10px 0 10px;"|<math display="inline">65569,21</math> |- style="text-align:center" | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">60 </math> | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">300 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">21 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">16 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">17,02\ .\ .\ .\ .\ </math> | style="padding: 0 10px 0 10px;"|<math display="inline">76577,02</math> |- style="text-align:center" | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">70 </math> | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">290 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">23 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">58 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">42,92\ .\ .\ .\ .\ </math> | style="padding: 0 10px 0 10px;"|<math display="inline">86322,92</math> |- style="text-align:center" | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">80 </math> | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">280 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">26 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">14 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">55,85\ .\ .\ .\ .\ </math> | style="padding: 0 10px 0 10px;"|<math display="inline">94495,85</math> |- style="text-align:center" | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">90 </math> | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">270 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">27 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">58 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">52,36\ .\ .\ .\ .\ </math> | style="padding: 0 10px 0 10px;"|<math display="inline">100732,36</math> |- style="text-align:center" | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">100 </math> | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">260 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">29 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">3 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">28,13\ .\ .\ .\ .\ </math> | style="padding: 0 10px 0 10px;"|<math display="inline">104608,13</math> |- style="text-align:center" | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">110 </math> | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">250 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">29 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">20 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">33,68\ .\ .\ .\ .\ </math> | style="padding: 0 10px 0 10px;"|<math display="inline">105633,68</math> |- style="text-align:center" | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">120 </math> | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">240 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">28 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">41 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">2,10 \ .\ .\ .\ .\ </math> | style="padding: 0 10px 0 10px;"|<math display="inline">103262,10</math> |- style="text-align:center" | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">130 </math> | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">230 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">26 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">55 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">22,77\ .\ .\ .\ .\ </math> | style="padding: 0 10px 0 10px;"|<math display="inline">96922,77</math> |- style="text-align:center" | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">140 </math> | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">220 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">23 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">55 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">2,70 \ .\ .\ .\ .\ </math> | style="padding: 0 10px 0 10px;"|<math display="inline">86102,70 </math> |- style="text-align:center" | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">150 </math> | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">210 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">19 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">35 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">0,79 \ .\ .\ .\ .\ </math> | style="padding: 0 10px 0 10px;"|<math display="inline">70500,79</math> |- style="text-align:center" | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">160 </math> | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">200 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">13 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">57 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">40,52\ .\ .\ .\ .\ </math> | style="padding: 0 10px 0 10px;"|<math display="inline">50260,52</math> |- style="text-align:center" | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">170 </math> | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">190 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">7 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">16 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">58,33\ .\ .\ .\ .\ </math> | style="padding: 0 10px 0 10px;"|<math display="inline">26218,33</math> |- style="text-align:center" | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">180 </math> | style="padding: 0 10px 0 10px;border-right:1px solid black;"|<math display="inline">180 </math> | style="padding: 0 10px 0 10px;"|<math display="inline">0 </math> | style="padding: 0 10px 0 10px;"|&nbsp; | style="padding: 0 10px 0 10px;"|&nbsp; | style="padding: 0 10px 0 10px;"|<math display="inline">0</math> |} Discerpimus hanc periodum 36 terminorum in sex minores senorum terminorum; valores seni functionis <math display="inline">X</math> in singulis periodis contenti exhibebuntur per formulam talem <math display="block"> \begin{align} \gamma + \gamma' \cos x + \gamma'' \cos 2x + \gamma''' \cos 3x \\ + \delta' \sin x + \delta'' \sin 2x + \delta''' \sin 3x \end{align} </math><noinclude></noinclude> oj4htozy65ctfx51oeeka6m09avu3cc 104 86125 281507 281362 2026-09-21T13:13:27Z Sjgallagher2 29663 281507 proofread-page text/x-wiki <noinclude><pagequality level="1" user="Sjgallagher2" /></noinclude>ubi pro coëfficientibus <math display="inline">\gamma, \gamma', \delta'</math> etc. valores sequentes invenimus: (table) Singuli coëfficientes <math display="inline">\gamma, \gamma', \gamma'', \gamma''', \delta', \delta'', \delta'''</math> rursus sub formam talem <math display="block"> \begin{align} \varepsilon + \varepsilon' \cos 6x + \varepsilon'' \cos 12x + \varepsilon''' \cos 18x \\ + \zeta' \sin 6x + \zeta'' \sin 12x + \zeta''' \sin 18x \end{align} </math> reducentur: nullo vero negotio perspicitur, pro quattuor prioribus evanescere debere <math display="inline">\varepsilon, \varepsilon', \varepsilon'', \varepsilon'''</math>; et pro tribus posterioribus, <math display="inline">\zeta', \zeta'', \zeta'''</math>. Hoc modo invenitur <math display="block"> \begin{align} \gamma &= +\quad 64'',848 \sin 6x + 0'',052 \sin 12x \\ \gamma' &= -\ \ 251'',277 \sin 6x - 0'',167 \sin 12x \\ \gamma'' &= +\ \ 879'',002 \sin 6x + 0'',500 \sin 12x \\ \gamma''' &= -1764'',511 \sin 6x - 0'',824 \sin 12x \\ \delta' &= -104044'',264 +\ 213'',968 \cos 6x + 0'',132 \cos 12x + 0'',000 \cos 18x \\ \delta'' &= +\ \ 16275'',150 -\ 868'',020 \cos 6x - 0'',489 \cos 12x - 0'',003 \cos 18x \\ \delta''' &= -\quad1763'',689 +1762'',868 \cos 6x + 0'',819 \cos 12x + 0'',002 \cos 18x \end{align} </math> His valoribus pro <math display="inline">\gamma, \gamma'</math> etc. substitutis, praeceptisque art. praec. observatis, prodit functio sequens pro aequatione centri, in qua singuli coëfficientes intra centesimam minuti secundi partem exacti sunt. <math display="block"> \begin{array}{l|l} -104044''264 \sin x & -1''643 \sin 9x \\ +\ 16275,150 \sin 2x & +0,494 \sin 10x \\ -\ \ \ 3527,378 \sin 3x & -0,149 \sin 11x \\ +\quad873,511 \sin 4x & +0,052 \sin 12x \\ -\quad232,622 \sin 5x & -0,017 \sin 13x \\ +\ \quad64,848 \sin 6x & +0,006 \sin 14x \\ -\ \quad18,655 \sin 7x & -0,004 \sin 15x \\ +\quad\ \ \ 5,491 \sin 8x & +0,003 \sin 16x \end{array} </math><noinclude></noinclude> osybsey6bknst6087im29gg90cpyb70 281508 281507 2026-09-21T13:27:04Z Sjgallagher2 29663 281508 proofread-page text/x-wiki <noinclude><pagequality level="1" user="Sjgallagher2" /></noinclude>ubi pro coëfficientibus <math display="inline">\gamma, \gamma', \delta'</math> etc. valores sequentes invenimus: {| style="border-spacing:0px;" align="center" |- style="text-align:center" | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">\text{Periodus}</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">y = 6x</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">\gamma</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">\gamma'</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">\gamma''</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">\gamma'''</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">\delta'</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">\delta''</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;"|<math display="inline">\delta'''</math> |- style="text-align:center" | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">\text{prima}</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0^\circ</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-103830,145</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+15406,638</math> | style="padding:0 10px 0 10px;"|<math display="inline">0</math> |- style="text-align:center" | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">\text{secunda}</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">60</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+56''205</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-217''757</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+761''671</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-1528,925</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-103937,346</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+15841,387</math> | style="padding:0 10px 0 10px;"|<math display="inline">-892,667</math> |- style="text-align:center" | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">\text{tertia}</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">120</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+56,115</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-217,467</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+760,805</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-1527,397</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-104151,314</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+16709,402</math> | style="padding:0 10px 0 10px;"|<math display="inline">-2645,530</math> |- style="text-align:center" | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">\text{quarta}</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">180</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-104258,100</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+17142,684</math> | style="padding:0 10px 0 10px;"|<math display="inline">-3525,740</math> |- style="text-align:center" | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">\text{quinta}</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">240</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-56,115</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-217,467</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-760,805</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+1527,397</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-104151,314</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+16709,402</math> | style="padding:0 10px 0 10px;"|<math display="inline">-2645,530</math> |- style="text-align:center" | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">\text{sexta}</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">300</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-56,205</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-217,757</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-781,671</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+1528,925</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-103937,346</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+15841,387</math> | style="padding:0 10px 0 10px;"|<math display="inline">-882,667</math> |} Singuli coëfficientes <math display="inline">\gamma, \gamma', \gamma'', \gamma''', \delta', \delta'', \delta'''</math> rursus sub formam talem <math display="block"> \begin{align} \varepsilon + \varepsilon' \cos 6x + \varepsilon'' \cos 12x + \varepsilon''' \cos 18x \\ + \zeta' \sin 6x + \zeta'' \sin 12x + \zeta''' \sin 18x \end{align} </math> reducentur: nullo vero negotio perspicitur, pro quattuor prioribus evanescere debere <math display="inline">\varepsilon, \varepsilon', \varepsilon'', \varepsilon'''</math>; et pro tribus posterioribus, <math display="inline">\zeta', \zeta'', \zeta'''</math>. Hoc modo invenitur <math display="block"> \begin{align} \gamma &= +\quad 64'',848 \sin 6x + 0'',052 \sin 12x \\ \gamma' &= -\ \ 251'',277 \sin 6x - 0'',167 \sin 12x \\ \gamma'' &= +\ \ 879'',002 \sin 6x + 0'',500 \sin 12x \\ \gamma''' &= -1764'',511 \sin 6x - 0'',824 \sin 12x \\ \delta' &= -104044'',264 +\ 213'',968 \cos 6x + 0'',132 \cos 12x + 0'',000 \cos 18x \\ \delta'' &= +\ \ 16275'',150 -\ 868'',020 \cos 6x - 0'',489 \cos 12x - 0'',003 \cos 18x \\ \delta''' &= -\quad1763'',689 +1762'',868 \cos 6x + 0'',819 \cos 12x + 0'',002 \cos 18x \end{align} </math> His valoribus pro <math display="inline">\gamma, \gamma'</math> etc. substitutis, praeceptisque art. praec. observatis, prodit functio sequens pro aequatione centri, in qua singuli coëfficientes intra centesimam minuti secundi partem exacti sunt. <math display="block"> \begin{array}{l|l} -104044''264 \sin x & -1''643 \sin 9x \\ +\ 16275,150 \sin 2x & +0,494 \sin 10x \\ -\ \ \ 3527,378 \sin 3x & -0,149 \sin 11x \\ +\quad873,511 \sin 4x & +0,052 \sin 12x \\ -\quad232,622 \sin 5x & -0,017 \sin 13x \\ +\ \quad64,848 \sin 6x & +0,006 \sin 14x \\ -\ \quad18,655 \sin 7x & -0,004 \sin 15x \\ +\quad\ \ \ 5,491 \sin 8x & +0,003 \sin 16x \end{array} </math><noinclude></noinclude> n7n0jnxubkmpn7su1bbv6u36ubloojv 281509 281508 2026-09-21T13:28:07Z Sjgallagher2 29663 281509 proofread-page text/x-wiki <noinclude><pagequality level="1" user="Sjgallagher2" /></noinclude>ubi pro coëfficientibus <math display="inline">\gamma, \gamma', \delta'</math> etc. valores sequentes invenimus: {| style="border-spacing:0px;" align="center" |- style="text-align:center" | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">\text{Periodus}</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">y = 6x</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">\gamma</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">\gamma'</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">\gamma''</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">\gamma'''</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">\delta'</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">\delta''</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;"|<math display="inline">\delta'''</math> |- style="text-align:center" | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">\text{prima}</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0^\circ</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-103830,145</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+15406,638</math> | style="padding:0 10px 0 10px;"|<math display="inline">0</math> |- style="text-align:center" | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">\text{secunda}</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">60</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+56''205</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-217''757</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+761''671</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-1528,825</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-103937,346</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+15841,387</math> | style="padding:0 10px 0 10px;"|<math display="inline">-892,667</math> |- style="text-align:center" | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">\text{tertia}</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">120</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+56,115</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-217,467</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+760,805</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-1527,397</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-104151,314</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+16709,402</math> | style="padding:0 10px 0 10px;"|<math display="inline">-2645,530</math> |- style="text-align:center" | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">\text{quarta}</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">180</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-104258,100</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+17142,684</math> | style="padding:0 10px 0 10px;"|<math display="inline">-3525,740</math> |- style="text-align:center" | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">\text{quinta}</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">240</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-56,115</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-217,467</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-760,805</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+1527,397</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-104151,314</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+16709,402</math> | style="padding:0 10px 0 10px;"|<math display="inline">-2645,530</math> |- style="text-align:center" | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">\text{sexta}</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">300</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-56,205</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-217,757</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-781,671</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+1528,825</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-103937,346</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+15841,387</math> | style="padding:0 10px 0 10px;"|<math display="inline">-882,667</math> |} Singuli coëfficientes <math display="inline">\gamma, \gamma', \gamma'', \gamma''', \delta', \delta'', \delta'''</math> rursus sub formam talem <math display="block"> \begin{align} \varepsilon + \varepsilon' \cos 6x + \varepsilon'' \cos 12x + \varepsilon''' \cos 18x \\ + \zeta' \sin 6x + \zeta'' \sin 12x + \zeta''' \sin 18x \end{align} </math> reducentur: nullo vero negotio perspicitur, pro quattuor prioribus evanescere debere <math display="inline">\varepsilon, \varepsilon', \varepsilon'', \varepsilon'''</math>; et pro tribus posterioribus, <math display="inline">\zeta', \zeta'', \zeta'''</math>. Hoc modo invenitur <math display="block"> \begin{align} \gamma &= +\quad 64'',848 \sin 6x + 0'',052 \sin 12x \\ \gamma' &= -\ \ 251'',277 \sin 6x - 0'',167 \sin 12x \\ \gamma'' &= +\ \ 879'',002 \sin 6x + 0'',500 \sin 12x \\ \gamma''' &= -1764'',511 \sin 6x - 0'',824 \sin 12x \\ \delta' &= -104044'',264 +\ 213'',968 \cos 6x + 0'',132 \cos 12x + 0'',000 \cos 18x \\ \delta'' &= +\ \ 16275'',150 -\ 868'',020 \cos 6x - 0'',489 \cos 12x - 0'',003 \cos 18x \\ \delta''' &= -\quad1763'',689 +1762'',868 \cos 6x + 0'',819 \cos 12x + 0'',002 \cos 18x \end{align} </math> His valoribus pro <math display="inline">\gamma, \gamma'</math> etc. substitutis, praeceptisque art. praec. observatis, prodit functio sequens pro aequatione centri, in qua singuli coëfficientes intra centesimam minuti secundi partem exacti sunt. <math display="block"> \begin{array}{l|l} -104044''264 \sin x & -1''643 \sin 9x \\ +\ 16275,150 \sin 2x & +0,494 \sin 10x \\ -\ \ \ 3527,378 \sin 3x & -0,149 \sin 11x \\ +\quad873,511 \sin 4x & +0,052 \sin 12x \\ -\quad232,622 \sin 5x & -0,017 \sin 13x \\ +\ \quad64,848 \sin 6x & +0,006 \sin 14x \\ -\ \quad18,655 \sin 7x & -0,004 \sin 15x \\ +\quad\ \ \ 5,491 \sin 8x & +0,003 \sin 16x \end{array} </math><noinclude></noinclude> pcceqsvy645oa7zmg7ov83wgx2i1kn3 281510 281509 2026-09-21T13:29:08Z Sjgallagher2 29663 281510 proofread-page text/x-wiki <noinclude><pagequality level="1" user="Sjgallagher2" /></noinclude>ubi pro coëfficientibus <math display="inline">\gamma, \gamma', \delta'</math> etc. valores sequentes invenimus: {| style="border-spacing:0px;" align="center" |- style="text-align:center" | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">\text{Periodus}</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">y = 6x</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">\gamma</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">\gamma'</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">\gamma''</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">\gamma'''</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">\delta'</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">\delta''</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;"|<math display="inline">\delta'''</math> |- style="text-align:center" | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">\text{prima}</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0^\circ</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-103830,105</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+15406,638</math> | style="padding:0 10px 0 10px;"|<math display="inline">0</math> |- style="text-align:center" | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">\text{secunda}</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">60</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+56''205</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-217''757</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+761''671</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-1528,825</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-103937,346</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+15841,387</math> | style="padding:0 10px 0 10px;"|<math display="inline">-892,667</math> |- style="text-align:center" | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">\text{tertia}</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">120</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+56,115</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-217,467</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+760,805</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-1527,397</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-104151,314</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+16709,402</math> | style="padding:0 10px 0 10px;"|<math display="inline">-2645,530</math> |- style="text-align:center" | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">\text{quarta}</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">180</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-104258,100</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+17142,684</math> | style="padding:0 10px 0 10px;"|<math display="inline">-3525,740</math> |- style="text-align:center" | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">\text{quinta}</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">240</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-56,115</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-217,467</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-760,805</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+1527,397</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-104151,314</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+16709,402</math> | style="padding:0 10px 0 10px;"|<math display="inline">-2645,530</math> |- style="text-align:center" | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">\text{sexta}</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">300</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-56,205</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-217,757</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-781,671</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+1528,825</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-103937,346</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+15841,387</math> | style="padding:0 10px 0 10px;"|<math display="inline">-882,667</math> |} Singuli coëfficientes <math display="inline">\gamma, \gamma', \gamma'', \gamma''', \delta', \delta'', \delta'''</math> rursus sub formam talem <math display="block"> \begin{align} \varepsilon + \varepsilon' \cos 6x + \varepsilon'' \cos 12x + \varepsilon''' \cos 18x \\ + \zeta' \sin 6x + \zeta'' \sin 12x + \zeta''' \sin 18x \end{align} </math> reducentur: nullo vero negotio perspicitur, pro quattuor prioribus evanescere debere <math display="inline">\varepsilon, \varepsilon', \varepsilon'', \varepsilon'''</math>; et pro tribus posterioribus, <math display="inline">\zeta', \zeta'', \zeta'''</math>. Hoc modo invenitur <math display="block"> \begin{align} \gamma &= +\quad 64'',848 \sin 6x + 0'',052 \sin 12x \\ \gamma' &= -\ \ 251'',277 \sin 6x - 0'',167 \sin 12x \\ \gamma'' &= +\ \ 879'',002 \sin 6x + 0'',500 \sin 12x \\ \gamma''' &= -1764'',511 \sin 6x - 0'',824 \sin 12x \\ \delta' &= -104044'',264 +\ 213'',968 \cos 6x + 0'',132 \cos 12x + 0'',000 \cos 18x \\ \delta'' &= +\ \ 16275'',150 -\ 868'',020 \cos 6x - 0'',489 \cos 12x - 0'',003 \cos 18x \\ \delta''' &= -\quad1763'',689 +1762'',868 \cos 6x + 0'',819 \cos 12x + 0'',002 \cos 18x \end{align} </math> His valoribus pro <math display="inline">\gamma, \gamma'</math> etc. substitutis, praeceptisque art. praec. observatis, prodit functio sequens pro aequatione centri, in qua singuli coëfficientes intra centesimam minuti secundi partem exacti sunt. <math display="block"> \begin{array}{l|l} -104044''264 \sin x & -1''643 \sin 9x \\ +\ 16275,150 \sin 2x & +0,494 \sin 10x \\ -\ \ \ 3527,378 \sin 3x & -0,149 \sin 11x \\ +\quad873,511 \sin 4x & +0,052 \sin 12x \\ -\quad232,622 \sin 5x & -0,017 \sin 13x \\ +\ \quad64,848 \sin 6x & +0,006 \sin 14x \\ -\ \quad18,655 \sin 7x & -0,004 \sin 15x \\ +\quad\ \ \ 5,491 \sin 8x & +0,003 \sin 16x \end{array} </math><noinclude></noinclude> qnnbh47zh2f37knj88rfqepaixl5gfk 281511 281510 2026-09-21T13:31:28Z Sjgallagher2 29663 /* Emendata */ 281511 proofread-page text/x-wiki <noinclude><pagequality level="3" user="Sjgallagher2" /></noinclude>ubi pro coëfficientibus <math display="inline">\gamma, \gamma', \delta'</math> etc. valores sequentes invenimus: {| style="border-spacing:0px;" align="center" |- style="text-align:center" | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">\text{Periodus}</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">y = 6x</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">\gamma</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">\gamma'</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">\gamma''</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">\gamma'''</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">\delta'</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;border-right:1px solid black;"|<math display="inline">\delta''</math> | style="padding:0 10px 0 10px;border-bottom:1px solid black;"|<math display="inline">\delta'''</math> |- style="text-align:center" | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">\text{prima}</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0^\circ</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-103830,105</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+15406,638</math> | style="padding:0 10px 0 10px;"|<math display="inline">0</math> |- style="text-align:center" | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">\text{secunda}</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">60</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+56''205</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-217''757</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+761''671</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-1528,825</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-103937,346</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+15841,387</math> | style="padding:0 10px 0 10px;"|<math display="inline">-882,667</math> |- style="text-align:center" | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">\text{tertia}</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">120</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+56,115</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-217,467</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+760,805</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-1527,397</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-104151,314</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+16709,402</math> | style="padding:0 10px 0 10px;"|<math display="inline">-2645,530</math> |- style="text-align:center" | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">\text{quarta}</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">180</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">0</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-104258,100</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+17142,684</math> | style="padding:0 10px 0 10px;"|<math display="inline">-3525,740</math> |- style="text-align:center" | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">\text{quinta}</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">240</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-56,115</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-217,467</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-760,805</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+1527,397</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-104151,314</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+16709,402</math> | style="padding:0 10px 0 10px;"|<math display="inline">-2645,530</math> |- style="text-align:center" | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">\text{sexta}</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">300</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-56,205</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-217,757</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-761,671</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+1528,825</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">-103937,346</math> | style="padding:0 10px 0 10px;border-right:1px solid black;"|<math display="inline">+15841,387</math> | style="padding:0 10px 0 10px;"|<math display="inline">-882,667</math> |} Singuli coëfficientes <math display="inline">\gamma, \gamma', \gamma'', \gamma''', \delta', \delta'', \delta'''</math> rursus sub formam talem <math display="block"> \begin{align} \varepsilon + \varepsilon' \cos 6x + \varepsilon'' \cos 12x + \varepsilon''' \cos 18x \\ + \zeta' \sin 6x + \zeta'' \sin 12x + \zeta''' \sin 18x \end{align} </math> reducentur: nullo vero negotio perspicitur, pro quattuor prioribus evanescere debere <math display="inline">\varepsilon, \varepsilon', \varepsilon'', \varepsilon'''</math>; et pro tribus posterioribus, <math display="inline">\zeta', \zeta'', \zeta'''</math>. Hoc modo invenitur <math display="block"> \begin{align} \gamma &= +\quad 64'',848 \sin 6x + 0'',052 \sin 12x \\ \gamma' &= -\ \ 251'',277 \sin 6x - 0'',167 \sin 12x \\ \gamma'' &= +\ \ 879'',002 \sin 6x + 0'',500 \sin 12x \\ \gamma''' &= -1764'',511 \sin 6x - 0'',824 \sin 12x \\ \delta' &= -104044'',264 +\ 213'',968 \cos 6x + 0'',132 \cos 12x + 0'',000 \cos 18x \\ \delta'' &= +\ \ 16275'',150 -\ 868'',020 \cos 6x - 0'',489 \cos 12x - 0'',003 \cos 18x \\ \delta''' &= -\quad1763'',689 +1762'',868 \cos 6x + 0'',819 \cos 12x + 0'',002 \cos 18x \end{align} </math> His valoribus pro <math display="inline">\gamma, \gamma'</math> etc. substitutis, praeceptisque art. praec. observatis, prodit functio sequens pro aequatione centri, in qua singuli coëfficientes intra centesimam minuti secundi partem exacti sunt. <math display="block"> \begin{array}{l|l} -104044''264 \sin x & -1''643 \sin 9x \\ +\ 16275,150 \sin 2x & +0,494 \sin 10x \\ -\ \ \ 3527,378 \sin 3x & -0,149 \sin 11x \\ +\quad873,511 \sin 4x & +0,052 \sin 12x \\ -\quad232,622 \sin 5x & -0,017 \sin 13x \\ +\ \quad64,848 \sin 6x & +0,006 \sin 14x \\ -\ \quad18,655 \sin 7x & -0,004 \sin 15x \\ +\quad\ \ \ 5,491 \sin 8x & +0,003 \sin 16x \end{array} </math><noinclude></noinclude> rc66guaan23w96yym4jmdzyjo3bttss 104 86195 281513 281372 2026-09-21T14:48:55Z Sjgallagher2 29663 281513 proofread-page text/x-wiki <noinclude><pagequality level="3" user="Sjgallagher2" /></noinclude>{{center| {{x-larger|ELIAS VINETVS}} {{larger|De Ioan. de Sacro Bosco.}} {{larger|AD IOANNEM TACITUM.}} φιλίτρον }} {{dropinitial|I}}OANNI ''de Sacro Bosco patria fuit, quæ nunc Anglia insula, olim Albion &amp; Britannia appellata. Lutetiæ literas &amp; philosophiam didicit, doctorque; Parisiensis fuit. Scripsit de Sphæra mundi, de astrolabo, de algorithmo (supputandi artem ita vocarunt barbari) &amp; de computo ecclesiastico, ad annum Christi MCCLVI, vt ex eo carmine liquet, quo is libellus de computo concluditur. Lutetiæ sepultus est, in sodalium Maturinalium claustris: cuius medio tumulo insculpta sphæra, ac circum illam hoc epitaphium,'' <poem> ''De Sacro Bosco qui compotista Ioannes'' ''Tempora qui sequeris, memor esto quod morieris'' ''Si miser es, plora. Miserans pro me precor ora.'' </poem> ''Ex quo quidem vel solo carmine, quale id fuerit sæ-''<noinclude></noinclude> mc4va6loabuerw3z8pnjx6ulnmpo28t 104 86198 281514 281375 2026-09-21T14:52:10Z Sjgallagher2 29663 281514 proofread-page text/x-wiki <noinclude><pagequality level="3" user="Sjgallagher2" /></noinclude>{{center|''Sphæræ Joannis'' {{x-larger|Capitulum Primum}} }} {{dropinitial|S}}Phæra igitur ab Euclide sic describitur: Sphæra est transitus circúferentiæ dimidij circuli quae (fixa diametro) eousque circunducitur, quousque; ad locum suum redeat, id est: Sphæra est tale rotúdum &amp; solidum, quod describitur ab arcu semicirculi circunducto. Sphæra etiam a Theodosio sic describitur: Sphæra est solidum quoddam una superficie contentum, in cuius medio punctus est, a quo omnes lineæ ductae ad circumferentiam sunt æquales: &amp; ille punctus dicitur centrum sphæræ. Linea vero recta, transiens per centrum sphæræ, applicans extremitates suas ad circunferentiam ex vtraque parte, circa quam sphæra voluitur, dicitur axis sphæræ. Duo verò puncta axem terminantia dicuntur poli sphæræ. {{center|{{larger|''SCHOLION VINETI.''}}}} ''Euclidis definitio sphæra est hæc in elemento undecimo. Sphæra est quando semicirculi manente diametro, circunductus semicirculus in idem rursus redit, vnde cœpit circunscribi figura illa. Huius definitionis sententia ex subjectis picturis patere potest. Quam autem ponit auctor, corruperunt Græcorum Euclidis interpretes. Si enim linea in latum fluens describit tantum superficiem, circumferentia semicirculi circumducta, describet superficiem globi solum, non globum solidum.''<noinclude></noinclude> k20uma18bfruvflhbhpgsrryvdbm6p8 104 86199 281515 281376 2026-09-21T15:53:58Z Sjgallagher2 29663 /* Emendata */ 281515 proofread-page text/x-wiki <noinclude><pagequality level="3" user="Sjgallagher2" /></noinclude>{{center| [[File:DeSacrobosco DeSphaera Fig1.png|inline|500px]] {{x-larger|D I V I S I O &nbsp; S P H Æ R Æ}} {{x-larger|M V N D I .}} }} {{dropinitial|S}}Phæra autem mundi dupliciter diuiditur, secundum substantiam, &amp; secundum accidens. Secundum substantiam in sphæras nouem. ''In sphæras nouem.) Recentiores {{sic|Astologi}} decimam sphæram addiderunt, propter tertium in octaua sphæra animaduersum motum, quem dixerunt motum trepidationis, &amp; motum accessus &amp; recessus; de quo Purbachius in Theoricis.'' Scilicet sphæram nouam, quæ primus motus siue primum mobile dicitur: &amp; in sphæram stellarum fixarum, quæ firmamentum nuncupatur: &amp; in septem sphæras septem planetarum, quarum quaedam sunt maiores, quædam minores secundum quod plus accedunt vel recedunt à firmamento. Vnde inter illas sphæras, sphæra Saturni<noinclude></noinclude> 807i6luxhlk8dbyib41cc6aiu8ihm31 104 86200 281516 281377 2026-09-21T15:56:37Z Sjgallagher2 29663 /* Emendata */ 281516 proofread-page text/x-wiki <noinclude><pagequality level="3" user="Sjgallagher2" /></noinclude>maxima, sphæra verò Lunæ minima, prout in sequenti figura continentur. ''Figura ostendens numerum ac ordinem sphærarum cœlestium, indicansque divisionem mundi secundum [s]ubstantiam.'' {{center| [[File:DeSacrobosco DeSphaera Fig2.png|inline|500px]] }} Secundum accidens autem dividitur in sphæram rectam &amp; sphæram obliquam. Illi autem dicuntur habere sphæram rectam, qui manent sub æquinoctia-<noinclude></noinclude> 4l9pq4t6qpyxwvag0ueixqtanp6b122 104 86201 281517 281378 2026-09-21T16:04:21Z Sjgallagher2 29663 /* Emendata */ 281517 proofread-page text/x-wiki <noinclude><pagequality level="3" user="Sjgallagher2" /></noinclude>li, si aliquis ibi manere possit. Et dicitur eis recta, quia neuter polorum magis altero illis eleuatur: vel quoniam eorum horizon intersecat æquinoctialem, &amp; intersecatur ab eodem ad angulos rectos sphærales. Illi verò dicuntur hæc sphæram obliquam, quicunque habitant citra æquinoctialem, vel ultra. Illis enim supra horizontem alter polorum semper eleuatur, alter verò semper deprimitur: vel quoniam illorum horizon artificialis intersecat æquinoctialem, &amp; intersecatur ab eodem ad angulos impares &amp; obliquos. {{center|{{larger|''SCHOLION VINETI.''}}}} ''Horizon artificialis, quis hic dicatur, dum quærant &amp; contendant interpretes, tu eum interim accipito, qui capite secundo obliquis &amp; decliuis appellabitur.'' {{center| [[File:DeSacrobosco DeSphaera Fig3.png|inline|500px]] {{larger|DE PARTIBVS MUNDI,}} Et quæ sunt partes eiusdem. }} {{dropinitial|V}}Niversalis autem mundi machina in duo di uiditur, in ætheream scilicet &amp; elementarem regionem. Elementaris quidem alterationi continuæ peruia existens, in quatuor diuiditur. Est enim terra tanquam mundi centrum in medio omnium<noinclude></noinclude> 4vc3frqkftqgxp00sh4cm9f90lewpdg 104 86202 281518 281379 2026-09-21T16:12:07Z Sjgallagher2 29663 /* Emendata */ 281518 proofread-page text/x-wiki <noinclude><pagequality level="3" user="Sjgallagher2" /></noinclude>sita, circa quam aqua, circa aquam æret, circa ærem ignis, illic purus &amp; non turbidus orbem Lunæ attinges, vt ait Aristoteles in libro Meteororum. Sic enim ea disposuit Deus gloriosus &amp; sublimis. Et hæc quatuor elementa dicuntur, quæ vicissim à semetipsis alterantur, corrumputur, &amp; generantur. Sunt autem elementa corpora simplicia, quæ in partes diversarum formarum minime diuidi possunt; ex quorum comixtione diversæ generatorum species fit. Quorum trium quodlibet terram orbiculariter vndique circundat nisi quantum siccitas terræ humori aquæ oblistit ad vitam animantium tuendam. Omnia etiam præter terram mobilia existút, quæ vt centrum mundi, ponderositate sui, magnum extremorum motum vndique equaliter fugiens, rotundæ sphæræ medium possidet. Circa elementarem quidem regionem ætherea regio lucida ab omni variatione, sua immutabili essentia, immunis existens, motu continuo circulariter incedit, &amp; hæc à Philosophis quinta nuncupatur essentia: cuius nouem sunt Sphæræ, si- {{center|{{larger|''SCHOLION VINETI.''}}}} ''Quinta nuncupatur essentia) Ab Aristotele [Greek] quidem, idest, essentia lib primo de Cælo, in libro vero de mundo (quem eiusdem esse Aristotelis quidam negant) [Greek], id est, elementum, alius a quattuor illis, igne, ære, aqua, terra.'' cut in proximo pertractatum est, scilicet, Lunæ, Mercurij; Veneris, Solis, Martis, Iouis Saturni, stellarum fixarum, &amp; cœli ultimi. Istarum<noinclude></noinclude> ccrbtntsadd7dxs45s83dhdo90qeej5 281519 281518 2026-09-21T16:16:33Z Sjgallagher2 29663 281519 proofread-page text/x-wiki <noinclude><pagequality level="3" user="Sjgallagher2" /></noinclude>sita, circa quam aqua, circa aquam æret, circa ærem ignis, illic purus &amp; non turbidus orbem Lunæ attinges, vt ait Aristoteles in libro Meteororum. Sic enim ea disposuit Deus gloriosus &amp; sublimis. Et hæc quatuor elementa dicuntur, quæ vicissim à semetipsis alterantur, corrumputur, &amp; generantur. Sunt autem elementa corpora simplicia, quæ in partes diversarum formarum minime diuidi possunt; ex quorum comixtione diversæ generatorum species fit. Quorum trium quodlibet terram orbiculariter vndique circundat nisi quantum siccitas terræ humori aquæ oblistit ad vitam animantium tuendam. Omnia etiam præter terram mobilia existút, quæ vt centrum mundi, ponderositate sui, magnum extremorum motum vndique equaliter fugiens, rotundæ sphæræ medium possidet. Circa elementarem quidem regionem ætherea regio lucida ab omni variatione, sua immutabili essentia, immunis existens, motu continuo circulariter incedit, &amp; hæc à Philosophis quinta nuncupatur essentia: cuius nouem sunt Sphæræ, si- {{center|{{larger|''SCHOLION VINETI.''}}}} ''Quinta nuncupatur essentia) Ab Aristotele οὐσίχ quidem, idest, essentia lib primo de Cælo, in libro vero de Mundo (quem eiusdem esse Aristotelis quidam negant) στοιχεῖον, id est, elementum, aliud a quattuor illis, igne, ære, aqua, terra.'' cut in proximo pertractatum est, scilicet, Lunæ, Mercurij; Veneris, Solis, Martis, Iouis Saturni, stellarum fixarum, &amp; cœli ultimi. Istarum<noinclude></noinclude> nrppbmklizmpcgbxd3fraivwtcbrks4 281520 281519 2026-09-21T16:18:18Z Sjgallagher2 29663 281520 proofread-page text/x-wiki <noinclude><pagequality level="3" user="Sjgallagher2" /></noinclude>sita, circa quam aqua, circa aquam æret, circa ærem ignis, illic purus &amp; non turbidus orbem Lunæ attinges, vt ait Aristoteles in libro Meteororum. Sic enim ea disposuit Deus gloriosus &amp; sublimis. Et hæc quatuor elementa dicuntur, quæ vicissim à semetipsis alterantur, corrumputur, &amp; generantur. Sunt autem elementa corpora simplicia, quæ in partes diversarum formarum minime diuidi possunt; ex quorum comixtione diversæ generatorum species fit. Quorum trium quodlibet terram orbiculariter vndique circundat nisi quantum siccitas terræ humori aquæ oblistit ad vitam animantium tuendam. Omnia etiam præter terram mobilia existút, quæ vt centrum mundi, ponderositate sui, magnum extremorum motum vndique equaliter fugiens, rotundæ sphæræ medium possidet. Circa elementarem quidem regionem ætherea regio lucida ab omni variatione, sua immutabili essentia, immunis existens, motu continuo circulariter incedit, &amp; hæc à Philosophis quinta nuncupatur essentia: cuius nouem sunt Sphæræ, si- {{center|{{larger|''SCHOLION VINETI.''}}}} ''Quinta nuncupatur essentia) Ab Aristotele οὐσία quidem, idest, essentia lib primo de Cælo, in libro vero de Mundo (quem eiusdem esse Aristotelis quidam negant) στοιχεῖον, id est, elementum, aliud a quattuor illis, igne, ære, aqua, terra.'' cut in proximo pertractatum est, scilicet, Lunæ, Mercurij; Veneris, Solis, Martis, Iouis Saturni, stellarum fixarum, &amp; cœli ultimi. Istarum<noinclude></noinclude> ksvtjfl7kns64ozt2nj89butsjwb49e 104 86203 281521 281380 2026-09-21T16:18:51Z Sjgallagher2 29663 281521 proofread-page text/x-wiki <noinclude><pagequality level="1" user="Sjgallagher2" /></noinclude>autem autem sphærarum qualibet superior inferioré sphæricè circundat: quarum quidem duo sunt motus. Vnus est coeli vltimi super duas axis extremitates, scilicet, polum arcticum &amp; antarcticum, ab oriëte per occidentem iterum rediens in orientem, quem æquinoctialis circulus per medium diuidit. Est etià alius inferiorum sphærarum motus obliquu huic oppositus super polos suos distantés à primis viginti tribus gradibus &amp; triginta tribus minutis. {{center|{{larger|''SCHOLION VINETI.''}}}} A primis scilicet polis mundi, id est, arcticó &amp; antarcticó. Pro viginti tribus minutis hoc loco solum Fabri Stavulensis exemplar habet vnum et quinquaginta, fortasse, quod is numerus sit cap. secundo, ubi de coluris et circulis minoribus.” Quoniam aut hec ex Ptolemæo ferè sunt, qua obseruarunt diversa recentiores Astrologi, ea lege in Purbachy Theoricis. Sed primus omnes alias sphæras secum impetu suo rapit intra dié &amp; noctem circa terram semel, illis tamen contra nitentibus. vt octaua sphæra in centum annis gradu uno. Húc siquidem motum secundú diuidit per mediu Zodiacus, sub quo quilibet septem plane tarum sphæram habet, propriam, in qua desertur motu proprio contra [[File:DeSacrobosco DeSphaera Fig4.png|inline]]<noinclude></noinclude> lns0l7asb6xdmlbdb9yzj3nil4z2qut 281537 281521 2026-09-21T18:09:46Z Sjgallagher2 29663 /* Emendata */ 281537 proofread-page text/x-wiki <noinclude><pagequality level="3" user="Sjgallagher2" /></noinclude>autem sphærarum quælibet superior inferiorem sphæricè circundat: quarum quidem duo sunt motus. Vnus est cœli vltimi super duas axis extremitates, scilicet, polum arcticum &amp; antarcticum, ab oriente per occidentem iterum rediens in orientem, quem æquinoctialis circulus per medium diuidit. Est etiam alius inferiorum sphærarum motus obliquum huic oppositus super polos suos distantes à primis viginti tribus gradibus &amp; triginta tribus minutis. {{center|{{larger|''SCHOLION VINETI.''}}}} ''A primis scilicet polis mundi, id est, arctico &amp; antarctico. Pro viginti tribus minutis hoc loco solum Fabri Stavulensis exemplar habet vnum et quinquaginta, fortasse, quod is numerus sit cap. secundo, vbi de Coluris et circulis minoribus. Quoniam autem hæc ex Ptolemæo ferè sunt, quæ obseruarunt diversa recentiores Astrologi, ea lege in Purbachi Theoricis.'' Sed primus omnes alias sphæras secum impetu suo rapit intra diem &amp; noctem circa terram semel, illis tamen contrà nitentibus, vt octaua sphæra in centum annis gradu uno. Hunc siquidem motum secundum diuidit per medium Zodiacus, sub quo quilibet septem planetarum sphæram habet propriam, in qua desertur motu proprio contra [[File:DeSacrobosco DeSphaera Fig4.png|inline]]<noinclude></noinclude> ofs9h3zb1o69rmuykntqhiazljysg0m 104 86204 281522 281381 2026-09-21T16:19:07Z Sjgallagher2 29663 281522 proofread-page text/x-wiki <noinclude><pagequality level="1" user="Sjgallagher2" /></noinclude>coeli intimi motum, &amp; in diversis spatijis temporis ipsum metitur: vt Saturnus in triginta annis: Iuppiter in duodecim: Mars in duobus: Sol in trecentis sexaginta quinque diebus &amp; sex horis ferè: Venus &amp; Mercurius similiter ferè cum Sole. Luna vero in viginti septem &amp; octo horis. COELUM MOVERI CIRCULARITER &amp; esse figuræ Sphæricæ. Qvod autem coelum euoluatur ab oriente in occidentem, signum est. Stellæ quæ oriuntur in oriente, semper elevantur paulatim &amp; successivè, quousque in medium coeli veniant; &amp; sunt semper in eadem propinquitate &amp; remotione adiuicem: &amp; ita semper se habentes, tendunt in occasum continuè, &amp; uniformiter. Est &amp; aliud signum. Stellæ quæ sunt iuxta polu arcticum, quæ nunquam nobis occidunt, mouétur continuè, &amp; uniformiter circa polum describendo circulos suos, &amp; semper sunt in æquali distantia ad inuicem &amp; propinquitate. Unde per istos duos motus continuos stellarum tam tendentium ad occasionum, quam non, patet, quòd firmamentum movetur ab oriente in occidentem. [[File:DeSacrobosco DeSphaera Fig5.png|inline]]<noinclude></noinclude> kk6y2jmtpn7e6qlxxmwsaon9nnz2kk4 281539 281522 2026-09-21T18:21:48Z Sjgallagher2 29663 /* Emendata */ 281539 proofread-page text/x-wiki <noinclude><pagequality level="3" user="Sjgallagher2" /></noinclude>cœli intimi motum, &amp; in diuersis spatijis temporum ipsum metitur: vt Saturnus in triginta annis: Iuppiter in duodecim: Mars in duobus: Sol in trecentis sexaginta quinque diebus &amp; sex horis ferè: Venus &amp; Mercurius similiter ferè cum Sole. Luna verò in viginti septem &amp; octo horis. {{center| COELVM MOVERI CIRCULARITER &amp; esse figuræ Sphæricæ. }} {{dropinitial|Q}}Vod autem cœlum euoluatur ab oriente in occidentem, signum est. Stellæ quæ oriuntur in oriente, semper elevantur paulatim &amp; successiuè, quousque in medium cœli veniant; &amp; sunt semper in eadem propinquitate &amp; remotione adiuicem: &amp; ita semper se habentes, tendunt in occasum continuè, &amp; uniformiter. Est &amp; aliud signum. Stellæ quæ sunt iuxta polum arcticum, quæ nunquam nobis occidunt, mouentur continuè, &amp; uniformiter circa polum describendo circulos suos, &amp; semper sunt in æquali distantia ad inuicem &amp; propinquitate. Unde per istos duos motus continuos stellarum tam tendentium ad occaum, quam non, patet, quòd firmamentum mouetur ab oriente in occidentem. {{center|[[File:DeSacrobosco DeSphaera Fig5.png|inline|400px]]}}<noinclude></noinclude> h3x3vghct88zuiv60obsqdvwfygolg7 104 86205 281523 281382 2026-09-21T16:19:31Z Sjgallagher2 29663 281523 proofread-page text/x-wiki <noinclude><pagequality level="1" user="Sjgallagher2" /></noinclude>Quod autem cælum sit rotundurn, tripex est ratio, similitudo, commoditas, &amp; necessitas. Similitudo, quoniam mundus sensibilis factus est ad similitudinem mundi archetypi, in quo nec est principium, nec finis. Vnde ad huius similitudinem factus mundus sensibilis, habet formam rotundam, in qua non est assignare principium, neque finem. {{center|{{larger|''SCHOLION VINETI.''}}}} Αρχέτυπος substant. prima forma, idea, &amp; exemplar, quod imitetur. Mundus Archetyprus (vt archetypas nugas, archetypos Cleantas dixerunt Iunenalis &amp; Martialis) hic dicitur ea mundi forma, quam mente concepit Deus mundum facturus, qua tui cogitatio æterna est, vt Deus ipse. Sic hunc locu interpretantur. Commoditas, quia omnium corporum isoperimetrorum sphæra maximum est: omnium etiam formarum rotunda capacissima est. quoniam igitur maximum &amp; rotundum, ideo capacissimum: unde cum mundus omnia contineat, talis forma fuit illi utilis &amp; commoda. {{center|{{larger|''SCHOLION VINETI.''}}}} ισόν equale, ωδῆ circum, μήτω mensura τοπίου τοα linea circundans, &amp; ambitus. Si itaque fuerint duæ insulæ, verbi gratia, ambitus vicenum stadiorum quatuor altera triquetra sit, altera rotunda sit circuli; speciem habeat, isoperimetrarum illarum maior erit rotunda. Sic babebit, si ex eodem luto vas rotundum<noinclude></noinclude> fkfs6jc2bvixq15bcsh3jud2wn3p4yy 281540 281523 2026-09-21T18:30:28Z Sjgallagher2 29663 /* Incerta */ 281540 proofread-page text/x-wiki <noinclude><pagequality level="2" user="Sjgallagher2" /></noinclude>Quod autem cœlum sit rotundum, triplex est ratio, similitudo, commoditas, &amp; necessitas. Similitudo, quoniam mundus sensibilis factus est ad similitudinem mundi archetypi, in quo nec est principium, nec finis. Vnde ad huius similitudinem factus mundus sensibilis, habet formam rotundam, in qua non est assignare principium, neque finem. {{center|{{larger|''SCHOLION VINETI.''}}}} ''Αρχέτυπος substant. prima forma, idea, &amp; exemplar, quod imitetur. Mundus Archetyprus (vt archetypas nugas, archetypos Cleantas dixerunt Iuuenalis &amp; Martialis) hic dicitur ea mundi forma, quam mente concepit Deus mundum facturus, qua Dei cogitatio æterna est, vt Deus ipse. Sic hunc locum interpretantur.'' Commoditas, quia omnium corporum isoperimetrorum sphæra maximum est: omnium etiam formarum rotunda capacissima est. Quoniam igitur maximum &amp; rotundum, ideo capacissimum: unde cum mundus omnia contineat, talis forma fuit illi utilis &amp; commoda. {{center|{{larger|''SCHOLION VINETI.''}}}} ''ῐ̓́σον equale, ωδῆ circum, μέτον mensura τοπίου τοα linea circundans, &amp; ambitus. Si itaque fuerint duæ insulæ, verbi gratia, ambitus vicenum stadiorum quarum altera triquetra sit, altera rotunda sit circulique speciem habeat, isoperimetrarum illarum maior erit rotunda. Sic habebit, si ex eodem luto vas rotundum''<noinclude></noinclude> it4xqtnrcuamsjv2agehdj0e7zzdsj8 281541 281540 2026-09-21T18:30:59Z Sjgallagher2 29663 281541 proofread-page text/x-wiki <noinclude><pagequality level="2" user="Sjgallagher2" /></noinclude>Quod autem cœlum sit rotundum, triplex est ratio, similitudo, commoditas, &amp; necessitas. Similitudo, quoniam mundus sensibilis factus est ad similitudinem mundi archetypi, in quo nec est principium, nec finis. Vnde ad huius similitudinem factus mundus sensibilis, habet formam rotundam, in qua non est assignare principium, neque finem. {{center|{{larger|''SCHOLION VINETI.''}}}} ''Αρχέτυπος substant. prima forma, idea, &amp; exemplar, quod imitetur. Mundus Archetypus (vt archetypas nugas, archetypos Cleantas dixerunt Iuuenalis &amp; Martialis) hic dicitur ea mundi forma, quam mente concepit Deus mundum facturus, qua Dei cogitatio æterna est, vt Deus ipse. Sic hunc locum interpretantur.'' Commoditas, quia omnium corporum isoperimetrorum sphæra maximum est: omnium etiam formarum rotunda capacissima est. Quoniam igitur maximum &amp; rotundum, ideo capacissimum: unde cum mundus omnia contineat, talis forma fuit illi utilis &amp; commoda. {{center|{{larger|''SCHOLION VINETI.''}}}} ''ῐ̓́σον equale, ωδῆ circum, μέτον mensura τοπίου τοα linea circundans, &amp; ambitus. Si itaque fuerint duæ insulæ, verbi gratia, ambitus vicenum stadiorum quarum altera triquetra sit, altera rotunda sit circulique speciem habeat, isoperimetrarum illarum maior erit rotunda. Sic habebit, si ex eodem luto vas rotundum''<noinclude></noinclude> nke8hbepa0iyo31lxv8rd016pl980cr 14 86245 281551 281327 2026-09-22T09:44:47Z Cantons-de-l'Est 4777 More infos. 281551 wikitext text/x-wiki Haec categoria continet paginas quae unum vel plures parametros includunt, quos additamentum "Proofread Page" – ad transclusions perficiendas adhibitum – ut incorrectos notat. Haec categorizatio a machina Vicifons peragitur cum parametro non adhibito aut ignoto est. [[Categoria:Corrigenda]] ii3zbqj64wgks2csz08dnv81fzuj1m5 104 86249 281528 281383 2026-09-21T16:21:45Z Sjgallagher2 29663 281528 proofread-page text/x-wiki <noinclude><pagequality level="1" user="Sjgallagher2" /></noinclude>&amp; quaSphæræ Joannis &amp; quadratum finxerit. Theon in lib. i magnæ Syntaxes Ptolemæi. [[File:DeSacrobosco DeSphaera Fig6.png|inline|right|300px]] Necessestas, quoniam si mundus esset alterius formæ, quam rotunda; scilicet trilateræ, vel quadrilateræ, vel multilateræ, sequentur duo impossibilia, scilicet quod aliquis loc’ esset vacuus &amp; corpus sine loco: quorum utriquæ est falsum, sicut patet in angulis eleucatis &amp; circunvolutis. [[File:DeSacrobosco DeSphaera Fig7.png|inline|300px]] Item sicut dicit Alfraganus, si celum esset planum, aliqua pars coeli esset nobis propinquior alia, illa scilicet, quæ esset supra caput nostrum.<noinclude></noinclude> ddhpary29t3rmjk7ydqnyustpk5h47j 281542 281528 2026-09-21T18:35:22Z Sjgallagher2 29663 /* Emendata */ 281542 proofread-page text/x-wiki <noinclude><pagequality level="3" user="Sjgallagher2" /></noinclude>''&amp; quadratum finxeri. Theon in lib. i magnæ Syntaxios Ptolemæi.'' [[File:DeSacrobosco DeSphaera Fig6.png|inline|right|300px]] Necessestas, quoniam si mundus esset alterius formæ, quàm rotunda; scilicet trilateræ, vel quadrilateræ, vel multilateræ, sequerentur duo impossibilia, scilicet quod aliquis locus esset vacuus &amp; corpus sine loco: quorum utrumquæ est falsum, sicut patet in angulis eleuatis &amp; circunuolutis. [[File:DeSacrobosco DeSphaera Fig7.png|inline|300px]] Item sicut dicit Alfraganus, si cœlum esset planum, aliqua pars cœli esset nobis propinquior alia, illa scilicet, quæ esset supra caput nostrum.<noinclude></noinclude> mp21rxi7trzhvavih2vhfmjb5okhxr6 104 86250 281524 281384 2026-09-21T16:19:44Z Sjgallagher2 29663 281524 proofread-page text/x-wiki <noinclude><pagequality level="1" user="Sjgallagher2" /></noinclude>{{center|{{larger|''SCHOLION VINETI.''}}}} Alfraganus tuus differentia secunda nugatur, quia cæli in orbem acti puncta singula à centro semper æqualiter distantia converterentur. Annotavit Petrus Nonius. Igitur stella ibi existens, esset nobis propinquior, quam stella in ortu vel occasu. Sed quæ nobis propinquiora sunt, maiora videntur. Ergo Sol, vel alia stella existens in medio cæli, maior deberet videri, quam in ortu existens vel in occasu. Cuuius contrarium videmus contingere: maior enim apparet Sol, vel alia stella existens in oriente, vel occidente, quam in medio cæli. Sed cum rei veritas ita non sit, huius apparentiæ causa eit, quod in tempore hyemali vel pluuiali vapores quidam ascendunt inter aspectum nostrum &amp; Solem, vel aliam stellam. &amp; cum illi vapores sint corpus diaphanum, disagregant radios nostros vituales, ita quod non comprehenditur rem in sua naturali &amp; vera quantitate, sicut patet in denario proiecto in fundo aquæ limpidæ, qui propter similem disagregationè radiorum apparet majoris, quam suæ veræ quantitatis. SCHOLION VINETI. Diaphanum necnon apud probatos autores legavit Rhodiginus; ego διαφανείο τανι μεμνήσκω. Sunt aut διαφαναι, aqua, vitrum, crystallas, vapores, &amp; alia huiusmodi, ita rara, utque ea videbepossis. Pellucida latinos dixisse purum, sicut &amp; cribrum, &amp; laternam<noinclude></noinclude> hdsarutyrs8etyyu2vbgma8q4ycy9g2 281525 281524 2026-09-21T16:19:50Z Sjgallagher2 29663 281525 proofread-page text/x-wiki <noinclude><pagequality level="1" user="Sjgallagher2" /></noinclude>{{center|{{larger|''SCHOLION VINETI.''}}}} Alfraganus tuus differentia secunda nugatur, quia cæli in orbem acti puncta singula à centro semper æqualiter distantia converterentur. Annotavit Petrus Nonius. Igitur stella ibi existens, esset nobis propinquior, quam stella in ortu vel occasu. Sed quæ nobis propinquiora sunt, maiora videntur. Ergo Sol, vel alia stella existens in medio cæli, maior deberet videri, quam in ortu existens vel in occasu. Cuuius contrarium videmus contingere: maior enim apparet Sol, vel alia stella existens in oriente, vel occidente, quam in medio cæli. Sed cum rei veritas ita non sit, huius apparentiæ causa eit, quod in tempore hyemali vel pluuiali vapores quidam ascendunt inter aspectum nostrum &amp; Solem, vel aliam stellam. &amp; cum illi vapores sint corpus diaphanum, disagregant radios nostros vituales, ita quod non comprehenditur rem in sua naturali &amp; vera quantitate, sicut patet in denario proiecto in fundo aquæ limpidæ, qui propter similem disagregationè radiorum apparet majoris, quam suæ veræ quantitatis. {{center|{{larger|''SCHOLION VINETI.''}}}} Diaphanum necnon apud probatos autores legavit Rhodiginus; ego διαφανείο τανι μεμνήσκω. Sunt aut διαφαναι, aqua, vitrum, crystallas, vapores, &amp; alia huiusmodi, ita rara, utque ea videbepossis. Pellucida latinos dixisse purum, sicut &amp; cribrum, &amp; laternam<noinclude></noinclude> 1hu1f4baf51xyut65omqia9a409f6uv 281543 281525 2026-09-21T18:43:26Z Sjgallagher2 29663 /* Incerta */ 281543 proofread-page text/x-wiki <noinclude><pagequality level="2" user="Sjgallagher2" /></noinclude>{{center|{{larger|''SCHOLION VINETI.''}}}} ''Alfraganus tuus differentia secunda nugatur, quia cœli in orbem acti puncta singula à centro semper æqualiter distantia converterentur Annotauit Petrus Nonius.'' Igitur stella ibi existens, esset nobis propinquior, quam stella in ortu vel occasu. Sed quæ nobis propinquiora sunt, maiora videntur. Ergo Sol, vel alia stella existens in medio cœli, maior deberet videri, quam in ortu existens vel in occasu. Cuius contrarium videmus contingere: maior enim apparet Sol, vel alia stella existens in oriente, vel occidente, quam in medio cœli. Sed cum rei veritas ita non sit, huius apparentiæ causa eit, quod in tempore hyemali vel pluuiali vapores quidam ascendunt inter aspectum nostrum &amp; Solem, vel aliam stellam. &amp; cum illi vapores sint corpus diaphanum, disgregant radios nostros vituales, ita quod non compræhendunt rem in sua naturali &amp; vera quantitate, sicut patet in denario proiecto in fundo aquæ limpidæ, qui propter similem disagregationem radiorum apparet maioris, quam suæ veræ quantitatis. {{center|{{larger|''SCHOLION VINETI.''}}}} ''Diaphanum necnon apud probatos autores legavit Rhodiginus; ego διαφανείο tantum memini. Sunt aut διαφαν aer, aqua, vitrum, crystallus, vapores, &amp; alia huiusmodi, ita rara, utque ea videre possis. Pellutida [L]atinos dixisse puto, sicut &amp; cribrum, &amp; laternam''<noinclude></noinclude> 5xrotkp479y01t2cwubbkbhyfqrzda9 104 86252 281529 281423 2026-09-21T16:22:25Z Sjgallagher2 29663 281529 proofread-page text/x-wiki <noinclude><pagequality level="1" user="Sjgallagher2" /></noinclude>''Punicam, &amp; agnum adeò macrum, ut eius exta in sole etiam viui inspicere liceret: pelluceve dixit Plautus in Rudente &amp; Aulularia. Cur autem in aquas simulachra maiora veris cernantur, disputat Macrobius libro septimo Saturnalium.'' {{center|[[File:DeSacrobosco DeSphaera Fig8.png|inline|600px]]}} Quod etiam terra sit rotûda, patet sic. Signa &amp; stel læ non equaliter oriuntur, &amp; occidunt omnibus hominibus vbique existentibus: sed prius oriuntur. &amp; occidunt illis, qui sunt versus orientem. Et quòd ci tius &amp; tardius oriûtur, &amp; occidunt quibusdam, cau-<noinclude></noinclude> jqsepbhkk75vsmmoesob5nhlj19dmb1 281544 281529 2026-09-21T18:45:25Z Sjgallagher2 29663 281544 proofread-page text/x-wiki <noinclude><pagequality level="1" user="Sjgallagher2" /></noinclude>''Punicam, &amp; agnum adeò macrum, ut eius exta in sole etiam viui inspicere liceret: pellucere dixit Plautus in Rudente &amp; Aulularia. Cur autem in aquas simulachra maiora veris cernantur, disputat Macrobius libro septimo Saturnalium.'' {{center|[[File:DeSacrobosco DeSphaera Fig8.png|inline|600px]]}} Quod etiam terra sit rotunda, patet sic. Signa &amp; stellæ non æqualiter oriuntur, &amp; occidunt omnibus hominibus vbique existentibus: sed prius oriuntur, &amp; occidunt illis, qui sunt versus orientem. Et quòd citius &amp; tardius oriuntur, &amp; occidunt quibusdam, cau-<noinclude></noinclude> akb4ofattynb2ja3scp61lmkad3hqv2 281545 281544 2026-09-21T18:45:29Z Sjgallagher2 29663 /* Emendata */ 281545 proofread-page text/x-wiki <noinclude><pagequality level="3" user="Sjgallagher2" /></noinclude>''Punicam, &amp; agnum adeò macrum, ut eius exta in sole etiam viui inspicere liceret: pellucere dixit Plautus in Rudente &amp; Aulularia. Cur autem in aquas simulachra maiora veris cernantur, disputat Macrobius libro septimo Saturnalium.'' {{center|[[File:DeSacrobosco DeSphaera Fig8.png|inline|600px]]}} Quod etiam terra sit rotunda, patet sic. Signa &amp; stellæ non æqualiter oriuntur, &amp; occidunt omnibus hominibus vbique existentibus: sed prius oriuntur, &amp; occidunt illis, qui sunt versus orientem. Et quòd citius &amp; tardius oriuntur, &amp; occidunt quibusdam, cau-<noinclude></noinclude> rttpjq0u29gg0ulxg13wgt4ivimxho9 104 86253 281530 281424 2026-09-21T16:23:04Z Sjgallagher2 29663 281530 proofread-page text/x-wiki <noinclude><pagequality level="1" user="Sjgallagher2" /></noinclude>sa est tumor terre, quod bene patet per ea quæ fiunc in sublimi, una enim &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; sol prius occidit quam nobis. Cuus rei causa est tum tumor terrae. {{center|[[File:DeSacrobosco DeSphaera Fig9.png|inline|500px]]}} Quod etiam terra habeat tumorem à septetrio: ne in austrum, &amp; contrà, sic patet. Hominibus<noinclude></noinclude> jr4o93ig52dv7v9uyklgqapjs3fylpk 281546 281530 2026-09-21T18:47:49Z Sjgallagher2 29663 /* Emendata */ 281546 proofread-page text/x-wiki <noinclude><pagequality level="3" user="Sjgallagher2" /></noinclude>sa est tumor terrae, quod bene patet per ea quæ fiunt in sublimi, una enim &amp; eadem eclipsis Lunæ numero, quæ apparet nobis in prima hora noctis, apparet orientalibus circa horam noctis tertiam. Vnde constat, quòd illis prius fuit nox, &amp; sol prius occidit quam nobis. Cuius rei causa est tamen tumor terrae. {{center|[[File:DeSacrobosco DeSphaera Fig9.png|inline|500px]]}} Quòd etiam terra habeat tumorem à septetrione in austrum, &amp; contrà, sic patet. Hominibus<noinclude></noinclude> dloh8jhegpqbblbnjvrzxwqscduezqd 104 86254 281547 281425 2026-09-21T18:52:13Z Sjgallagher2 29663 /* Emendata */ 281547 proofread-page text/x-wiki <noinclude><pagequality level="3" user="Sjgallagher2" /></noinclude>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, ei iam tenderent in occasum, &amp; quantò magis accederet ad austrum, tantò plus moverentur in occasum. Ille iterum idem homo posset videre stellas, quæ prius fuerant ei sempiternæ occultationis. Et è conuerso 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 orirentur stellæ occidentalibus, quàm orientalibus, quod patet esse falsum. Item si terra esset plana à septentrione in austrum, &amp; contrà, stellæ, quæ essent alicui sempiternæ apparitionis, semper apparerent eidem, quocunque procederet, quod falsum est, sed quòd plana sit, præ nimia eius quantitate hominum visui apparet.<noinclude></noinclude> toq01n1iyhc3v804f1iq9g64uhvv3lq 104 86255 281531 281426 2026-09-21T16:23:34Z Sjgallagher2 29663 281531 proofread-page text/x-wiki <noinclude><pagequality level="1" user="Sjgallagher2" /></noinclude> {{center|[[File:DeSacrobosco DeSphaera Fig10.png|inline|500px]]}} QVOD AQVA SIT ROTUNDA. Qvod autem aqua habeat tumorem, &amp; accedat ad rotunditatem, sic patet. Ponatur signum in littore maris, &amp; exeat naus à portu, &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,<noinclude></noinclude> 9y1mxf4yntsqnze55n8fzkua2216tnu 104 86256 281526 281427 2026-09-21T16:20:11Z Sjgallagher2 29663 281526 proofread-page text/x-wiki <noinclude><pagequality level="1" user="Sjgallagher2" /></noinclude>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; vapores ascendentes. Item, cum aqua sit corpus homogeneum, totum cum partibus eiusdem crit rationis, sed partes aquæ (sicut in guttulis &amp; roribus herbarum accidit) rotundam naturaliter appetut formam, ergo &amp; totum, cuius sunt partes. {{center|{{larger|''SCHOLION VINETI.''}}}} Homogeneum, eiusdem generis &amp; naturæ, ex Græco ὁμογενής. Terra autem &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; 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-<noinclude></noinclude> bbfafxzk85grw54jzqpmbkyqzbo3gct 281532 281526 2026-09-21T16:24:02Z Sjgallagher2 29663 281532 proofread-page text/x-wiki <noinclude><pagequality level="1" user="Sjgallagher2" /></noinclude>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; vapores ascendentes. [[File:DeSacrobosco DeSphaera Fig11.png|inline|left|300px]]Item, cum aqua sit corpus homogeneum, totum cum partibus eiusdem crit rationis, sed partes aquæ (sicut in guttulis &amp; roribus herbarum accidit) rotundam naturaliter appetut formam, ergo &amp; totum, cuius sunt partes. {{center|{{larger|''SCHOLION VINETI.''}}}} Homogeneum, eiusdem generis &amp; naturæ, ex Græco ὁμογενής. Terra autem &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; 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-<noinclude></noinclude> 7beqmvqj1r4mrit1x3hoe26ciqtdevw 104 86257 281533 281428 2026-09-21T17:54:17Z Sjgallagher2 29663 281533 proofread-page text/x-wiki <noinclude><pagequality level="1" user="Sjgallagher2" /></noinclude>[[File:DeSacrobosco_DeSphaera_Fig12.png|inline|right|200px]]sum: &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; omnes Philosophos, dicentes, qv bicunq́; existat hó, lex signa ei oriuntur, &amp; sex occidút, &amp; medietas celi semper appareit ei, me dietas verò occultatur. Illud item est signum, quòd terra sit tanquam centrum &amp; punctus respectu firmamenti: quia si terra esset alicuius quantitatis respectu firmamenti: non contingeret medietatem cœli videri. [[File:DeSacrobosco_DeSphaera_Fig13.png|inline|left|250px]]Item si intelligatur superficies plana super centrum terræ, diuidens eam in duo æqualia, &amp; ipsum per consequens firma mérum. Occultus igitur existens in terræ centro videret medietaté celi: &amp; idem existens in superficie terræ videret candem medietatem. Ex his colligit, qv insensibilis est quâtitas terræ, que est à superficie ad centrum. &amp; p consequens quâti tas totius terræ insensibilis est respectu firmamenti.<noinclude></noinclude> ig50kt1f7o4p8wuf48qy2qdaxja5h6j 104 86258 281534 281429 2026-09-21T17:55:35Z Sjgallagher2 29663 281534 proofread-page text/x-wiki <noinclude><pagequality level="1" user="Sjgallagher2" /></noinclude>Dicit etiam Alfraganus, &amp; minima stellarum fixarum visu mutabilium, maior est tota terra: sed ipsa stella, respectu totius firmamenti, est sicut punctus &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. [[File:DeSacrobosco_DeSphaera_Fig14.png|inline|right|250px]] {{dropinitial|T}}otius autem orbis terræ ambitus, authoritate Ambrosij, Theodosij, Macrobij, &amp; Eratostenis Philosophorum, 252000. stadia continere diffinitur, vnicuique quidem 360. partium zodiaci 700. stadia deputando. Sumpto enim 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<noinclude></noinclude> lz7z5p7jpp834bt43x8fwbx91ji2ilt 104 86259 281527 281430 2026-09-21T16:20:28Z Sjgallagher2 29663 281527 proofread-page text/x-wiki <noinclude><pagequality level="1" user="Sjgallagher2" /></noinclude>in alterius noctis claritate, viso ut prius polo, statuerit altius uno gradu mediclinium. Post hoc menfuretur huius itineris spatium, &amp; inuenientur 790. stadia. Deinde datis vnicuique 360. graduum tot stadis, terreni orbis ambitus inuentus erit. Ex his autem, iuxta circuli &amp; diametri regulam diameter terræ sic inueniri poterit. Aufer vigesimam secundam partem de circuitu terræ &amp; remanentis tertia pars, hoc est 80181. stadia &amp; semis, &amp; tertia pars vnius stadij erit terreni orbis diameter siue spissitudo. {{center|{{larger|''SCHOLION VINETI.''}}}} Ambrosius Theodosius Macrobius hic vnicus est autor, Macrobius scilicet ille, qui scripsit Saturnalium libros &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; quinquaginta duorum milium stadiorum esse scripserit, quamodo traduit Plinius extremo libro secundo, Macrobius ille, Copella, &amp; Sacroboscus: au verò ducentorum &amp;<noinclude></noinclude> pi0cihj3hxyhvvkjm5jstw75xe6pw52 281535 281527 2026-09-21T18:01:22Z Sjgallagher2 29663 281535 proofread-page text/x-wiki <noinclude><pagequality level="1" user="Sjgallagher2" /></noinclude>in alterius noctis claritate, viso ut prius polo, statuerit altius uno gradu mediclinium. Post hoc menfuretur huius itineris spatium, &amp; inuenientur 790. stadia. Deinde datis vnicuique 360. graduum tot stadis, terreni orbis ambitus inuentus erit. Ex his autem, iuxta circuli &amp; diametri regulam diameter terræ sic inueniri poterit. Aufer vigesimam secundam partem de circuitu terræ &amp; remanentis tertia pars, hoc est 80181. stadia &amp; semis, &amp; tertia pars vnius stadij erit terreni orbis diameter siue spissitudo. [[File:DeSacrobosco_DeSphaera_Fig15.png|inline|left|200px]] {{center|{{larger|''SCHOLION VINETI.''}}}} Ambrosius Theodosius Macrobius hic vnicus est autor, Macrobius scilicet ille, qui scripsit Saturnalium libros &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; quinquaginta duorum milium stadiorum esse scripserit, quamodo traduit Plinius extremo libro secundo, Macrobius ille, Copella, &amp; Sacroboscus: au verò ducentorum &amp;<noinclude></noinclude> 5h2cz2rja5478vim3s6h21prnzwjg48 281536 281535 2026-09-21T18:02:03Z Sjgallagher2 29663 281536 proofread-page text/x-wiki <noinclude><pagequality level="1" user="Sjgallagher2" /></noinclude>in alterius noctis claritate, viso ut prius polo, statuerit altius uno gradu mediclinium. Post hoc menfuretur huius itineris spatium, &amp; inuenientur 790. stadia. Deinde datis vnicuique 360. graduum tot stadis, terreni orbis ambitus inuentus erit. Ex his autem, iuxta circuli &amp; diametri regulam diameter terræ sic inueniri poterit. Aufer vigesimam secundam partem de circuitu terræ &amp; remanentis tertia pars, hoc est 80181. stadia &amp; semis, &amp; tertia pars vnius stadij erit terreni orbis diameter siue spissitudo. [[File:DeSacrobosco_DeSphaera_Fig15.png|inline|left|200px]] {{center|{{larger|''SCHOLION VINETI.''}}}} ''Ambrosius Theodosius Macrobius hic vnicus est autor, Macrobius scilicet ille, qui scripsit Saturnalium libros &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; quinquaginta duorum milium stadiorum esse scripserit, quamodo traduit Plinius extremo libro secundo, Macrobius ille, Copella, &amp; Sacroboscus: au verò ducentorum &amp;''<noinclude></noinclude> 03wqddw1sbk5zjev0cm5dvxt0nmysic 281538 281536 2026-09-21T18:19:32Z Sjgallagher2 29663 /* Nondum emendata */ 281538 proofread-page text/x-wiki <noinclude><pagequality level="1" user="Sjgallagher2" /></noinclude>in alterius noctis claritate, viso ut prius polo, statuerit altius uno gradu mediclinium. Post hoc menfuretur huius itineris spatium, &amp; inuenientur 790. stadia. Deinde datis vnicuique 360. graduum tot stadis, terreni orbis ambitus inuentus erit. Ex his autem, iuxta circuli &amp; diametri regulam diameter terræ sic inueniri poterit. Aufer vigesimam secundam partem de circuitu terræ &amp; remanentis tertia pars, hoc est 80181. stadia &amp; semis, &amp; tertia pars vnius stadij erit terreni orbis diameter siue spissitudo. {{center|[[File:DeSacrobosco_DeSphaera_Fig15.png|inline|500px]] }} {{center|{{larger|''SCHOLION VINETI.''}}}} ''Ambrosius Theodosius Macrobius hic vnicus est autor, Macrobius scilicet ille, qui scripsit Saturnalium libros &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; quinquaginta duorum milium stadiorum esse scripserit, quamodo traduit Plinius extremo libro secundo, Macrobius ille, Copella, &amp; Sacroboscus: au verò ducentorum &amp;''<noinclude></noinclude> kt3721lquyttub6gsfktzd4imcq153m 0 86289 281548 2026-09-21T19:08:19Z Sjgallagher2 29663 Paginam instituit, scribens '{{header | OperaeTitulus = De Sphaera | Scriptor = Iohannes de Sacrobosco | Annus = ca. 1230 | Genera = Physica | Liber = 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 | Editio = Sphaera Ioannis de Sacrobosco Emendata. Elias Vinetus, Venetiis 1586. }} {{incomplete}} <pa...' 281548 wikitext text/x-wiki {{header | OperaeTitulus = De Sphaera | Scriptor = Iohannes de Sacrobosco | Annus = ca. 1230 | Genera = Physica | Liber = 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 | Editio = Sphaera Ioannis de Sacrobosco Emendata. Elias Vinetus, Venetiis 1586. }} {{incomplete}} <pages index="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" from=1 to =1 /> {{rule}} <pages index="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" from=2 to =4 /> {{rule}} <pages index="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" from=5 to =5 /> <pages index="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" from=6 to =25 /> qoszxc5t0cwizh59pb9rbj38lb4q5a4 104 86290 281549 2026-09-21T23:30:51Z Trooper57 22870 /* Nondum emendata */ Paginam instituit, scribens 'C ε l lil 2 tum corpus pilis duriuſculis eleganter flayis veſtitum eſt, conſtantibus ſubſtantia inſtat ſcra¬ rum equinarum. Si termis hiio outem humanam attingat. urit inſtar ignis. taſilienſibus: infectum eſt parallelogramm figura. ſed in parte poſticy orbiculari: anterior ſectio ſemidigitum longa poſterior digitum. Crura habet ſe unum par in oriori ſectione. duo in puſteriore. ocellos nigros: barbam conſtantem ¶duobus pilis dentatis...' 281549 proofread-page text/x-wiki <noinclude><pagequality level="1" user="Trooper57" /></noinclude>C ε l lil 2 tum corpus pilis duriuſculis eleganter flayis veſtitum eſt, conſtantibus ſubſtantia inſtat ſcra¬ rum equinarum. Si termis hiio outem humanam attingat. urit inſtar ignis. taſilienſibus: infectum eſt parallelogramm figura. ſed in parte poſticy orbiculari: anterior ſectio ſemidigitum longa poſterior digitum. Crura habet ſe unum par in oriori ſectione. duo in puſteriore. ocellos nigros: barbam conſtantem ¶duobus pilis dentatis orilioribus: apte abſcondere e barbam & pedes poteſta in altum gſilire eſtitur teſta dura. in dorio obſoleti lignci golo¬ teriori vero parte paralle logtammi magulam nigtam habet flauo ſuperſparſam in poſteriore autem parte tora. nimirum alis, clogantiſſima linea nige ſplendentes tondum ſocundum longitudinem. Alæ arce com¬ pligata corpori: inferius corpus totum hepatigi coloris ctura ſplendide ni¬ gra ut & barba o Aliquo modo conyenit cum Iupreſti Scarabcocido, niſi quod caput citra primam inſectionem exſertum non habeat de quo vide Thomx kφνηεη Londinatis Thcatrum Infectorum lib i cap xix o Praſilienſibus. uſca apis magnitudine: crura habet ſex priora duo bre¬ iora ſequentia longior. Sultimum par longiſlimum. habenthæ poſteriora ſuras amplas latas apum modo cornigula duo in capite alas quatusr. Tota muſca coloris viridis cum tuteo minti. Inſignis & mite ſplendet, in fronte ſapphyrin coloris. oculi nigri corniculares lutgi goloris. ala vatiogata magulis brunnis. uſca alia ſignta & magnitudine vulgari ſimilis. ſed corpore breyiori. &in poſtica parit habente proceſſum tibie formem: ctura ſex duo anteriora ut in vulaatibus. ſed ultima du¬ pba longitudinis retrorſum inflexa ut in Locuſtis duo longa corniculs in fronte alas quatuor communes alijs: ꝯcellos ſapphytinos ſplendentes catera in totum goloris nigrigantis Hultas intoni muicas circa Camarigibi & Tapruſſū aliaque loca mediterranea commun bus quidem ſimiles. ſed flaxi coloris & paululum ligneo rariegatas. alde infeſtas hominibus moleſtiſſimas. Reperitur &alis. cius corporis longitdo nondum digitum æquat: ocelli prominentes rotundi cpatici coloris crura ſer obſcuro flauo & ſuſco rariegata ala quatuor. dug exte tiotes digitum long. dug interiores ſemidiuitur. ac tanſparentes omnes tenueſque a genteis venulis & punctulis fuſcis variegatx. Totum corpus ex argenteo ε iſco colore eſt mintum. VI I I. C A I Quici. Iacatinga. Vermia terreſtrū. Praſilienſibus. iniectum eſt corpore oyali inſtar Scatabei. Corporis longitudo Lunius & trientis digiti Crura habet ſex, quodlibet tribus intemnodiis conſtans e que ultimapas grurum ſerrata. quodlibet habet duos unguiculos. Caput & ocelli par duo σοιηηλ ριορεοσulos eorientis. longa poſtorius verſa & in argum porrecta, duos & ſemis di¬ gitos longa. longo ecedentia corporis longitudinum. & ſingula habent internodia leσom. undecimo inſcruntur in caput craſſitie aquant fi¬ lum craſſius. Caput & anterior ſectio obſcuri eſt brunni coloris cum¬ ſplendido nigio minti. poſterior & alæ brunni obſcure ſplendentis Ter medium autem alatum ſeu dorſum ganſerſim lines lata flaua paliciens S tendit Crura lurea & ad internodia obſcuro brunno maculata. Cornua diuerſicoloria lureo & nigro. nimitum ad genigula nigra catera alternatim lutea s Braſilionſibus. Infectum frequens plur ioſis menſibus in hortis., ſilys & campis. Huos digitos longum caput cum oculis piſi mediporis magnitudinem habet. totun¬ dum. oculi grandiſſimi ellyptieꝝ figura. Os quando aperit ſatis grando habet & inferius la¬ bium ſeu maxillam inferiorem bifariam in medio diuiſam ſuperiorem integram in oro qua¬ tuor dentes ſalcatos ſuperiores duos fortiores. quomlibet quatuor aculeis munitum. inie¬ οησ νηοaσulco in cntremitute. Super os in ſummiture capitis duas habet eminentias partu¬ quaſi biſidam frontem & ibidem duo cornicula manime exilia. Thorat & venter conti¬ mnum ſunt ſomidigitum longum. ſabꝝ medioeris craſſitie, qi adiungitur cauda unum quadtantem digit ſones. undocim intemodiis conſtans. figur tilateræ pyramidalis & in extremitate furgta Crura haberſen in thoragē & ventre. haud longa piloſa, in extremitate duobus eiliſſimis unguiculis corpus etiam piloſum. Alas habet quatuor dorio comuntas. utrimque duis. quas ſemper in diro dum a latere cxpanſas gerit. inter volandum & ſedendum ſiue rependum. Eſt autem quælibet ala pene duos digitos longa ſemidigitum lata. ubi latiſ. ſima.<noinclude></noinclude> 34kwjhn3z5yzebexzosb4zrikriqxw1 281550 281549 2026-09-21T23:31:07Z Trooper57 22870 /* Problematic */ 281550 proofread-page text/x-wiki <noinclude><pagequality level="2" user="Trooper57" /></noinclude>{{Imago deest}} C ε l lil 2 tum corpus pilis duriuſculis eleganter flayis veſtitum eſt, conſtantibus ſubſtantia inſtat ſcra¬ rum equinarum. Si termis hiio outem humanam attingat. urit inſtar ignis. taſilienſibus: infectum eſt parallelogramm figura. ſed in parte poſticy orbiculari: anterior ſectio ſemidigitum longa poſterior digitum. Crura habet ſe unum par in oriori ſectione. duo in puſteriore. ocellos nigros: barbam conſtantem ¶duobus pilis dentatis orilioribus: apte abſcondere e barbam & pedes poteſta in altum gſilire eſtitur teſta dura. in dorio obſoleti lignci golo¬ teriori vero parte paralle logtammi magulam nigtam habet flauo ſuperſparſam in poſteriore autem parte tora. nimirum alis, clogantiſſima linea nige ſplendentes tondum ſocundum longitudinem. Alæ arce com¬ pligata corpori: inferius corpus totum hepatigi coloris ctura ſplendide ni¬ gra ut & barba o Aliquo modo conyenit cum Iupreſti Scarabcocido, niſi quod caput citra primam inſectionem exſertum non habeat de quo vide Thomx kφνηεη Londinatis Thcatrum Infectorum lib i cap xix o Praſilienſibus. uſca apis magnitudine: crura habet ſex priora duo bre¬ iora ſequentia longior. Sultimum par longiſlimum. habenthæ poſteriora ſuras amplas latas apum modo cornigula duo in capite alas quatusr. Tota muſca coloris viridis cum tuteo minti. Inſignis & mite ſplendet, in fronte ſapphyrin coloris. oculi nigri corniculares lutgi goloris. ala vatiogata magulis brunnis. uſca alia ſignta & magnitudine vulgari ſimilis. ſed corpore breyiori. &in poſtica parit habente proceſſum tibie formem: ctura ſex duo anteriora ut in vulaatibus. ſed ultima du¬ pba longitudinis retrorſum inflexa ut in Locuſtis duo longa corniculs in fronte alas quatuor communes alijs: ꝯcellos ſapphytinos ſplendentes catera in totum goloris nigrigantis Hultas intoni muicas circa Camarigibi & Tapruſſū aliaque loca mediterranea commun bus quidem ſimiles. ſed flaxi coloris & paululum ligneo rariegatas. alde infeſtas hominibus moleſtiſſimas. Reperitur &alis. cius corporis longitdo nondum digitum æquat: ocelli prominentes rotundi cpatici coloris crura ſer obſcuro flauo & ſuſco rariegata ala quatuor. dug exte tiotes digitum long. dug interiores ſemidiuitur. ac tanſparentes omnes tenueſque a genteis venulis & punctulis fuſcis variegatx. Totum corpus ex argenteo ε iſco colore eſt mintum. VI I I. C A I Quici. Iacatinga. Vermia terreſtrū. Praſilienſibus. iniectum eſt corpore oyali inſtar Scatabei. Corporis longitudo Lunius & trientis digiti Crura habet ſex, quodlibet tribus intemnodiis conſtans e que ultimapas grurum ſerrata. quodlibet habet duos unguiculos. Caput & ocelli par duo σοιηηλ ριορεοσulos eorientis. longa poſtorius verſa & in argum porrecta, duos & ſemis di¬ gitos longa. longo ecedentia corporis longitudinum. & ſingula habent internodia leσom. undecimo inſcruntur in caput craſſitie aquant fi¬ lum craſſius. Caput & anterior ſectio obſcuri eſt brunni coloris cum¬ ſplendido nigio minti. poſterior & alæ brunni obſcure ſplendentis Ter medium autem alatum ſeu dorſum ganſerſim lines lata flaua paliciens S tendit Crura lurea & ad internodia obſcuro brunno maculata. Cornua diuerſicoloria lureo & nigro. nimitum ad genigula nigra catera alternatim lutea s Braſilionſibus. Infectum frequens plur ioſis menſibus in hortis., ſilys & campis. Huos digitos longum caput cum oculis piſi mediporis magnitudinem habet. totun¬ dum. oculi grandiſſimi ellyptieꝝ figura. Os quando aperit ſatis grando habet & inferius la¬ bium ſeu maxillam inferiorem bifariam in medio diuiſam ſuperiorem integram in oro qua¬ tuor dentes ſalcatos ſuperiores duos fortiores. quomlibet quatuor aculeis munitum. inie¬ οησ νηοaσulco in cntremitute. Super os in ſummiture capitis duas habet eminentias partu¬ quaſi biſidam frontem & ibidem duo cornicula manime exilia. Thorat & venter conti¬ mnum ſunt ſomidigitum longum. ſabꝝ medioeris craſſitie, qi adiungitur cauda unum quadtantem digit ſones. undocim intemodiis conſtans. figur tilateræ pyramidalis & in extremitate furgta Crura haberſen in thoragē & ventre. haud longa piloſa, in extremitate duobus eiliſſimis unguiculis corpus etiam piloſum. Alas habet quatuor dorio comuntas. utrimque duis. quas ſemper in diro dum a latere cxpanſas gerit. inter volandum & ſedendum ſiue rependum. Eſt autem quælibet ala pene duos digitos longa ſemidigitum lata. ubi latiſ. ſima.<noinclude></noinclude> 7jirfgk1ywxqly3jacihb3xrss24rbl