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Colored diff for /texts/archimedes/xml/carda_propo_015_la_1570.xml between version 1.26 and 1.27

version 1.26, 2004/04/06 17:37:36

version 1.27, 2004/04/06 17:42:16

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<s id="id000001">HIERONYMI <lb/>CARDANI MEDIO<lb/>LANENSIS, CIVISQVE BONO<lb/>NIENSIS, PHILOSOPHI, MEDICI ET <lb/>Mathematici clari&longs;simi,</s>

<s id="id000002">OPVS NOVVM DE <lb/>PROPORTIONIBVS NVMERORVM, MO<lb/>TVVM, PONDERVM, SONORVM, ALIARVMQVE RERVM <lb/>men&longs;urandarum, non &longs;olùm Geometrico more &longs;tabilitum, &longs;ed etiam <lb/>uarijs experimentis & ob&longs;eruationibus rerum in natura, &longs;olerti <lb/>demon&longs;tratione illu&longs;tratum, ad multiplices u&longs;us ac<lb/>commodatum, & in V libros dige&longs;tum.</s>

<s id="id000003">PRAETEREA.</s>

Line 427

</row>

<row>

<cell>LXIIII.</cell>

<cell>O<emph type="italics"/>mne mobile quantò latius tanto tardius mouitur in plano.<emph.end type="italics"/></cell>

<cell>O<emph type="italics"/>mne mobile quantò latius tanto tardius moustur in plano.<emph.end type="italics"/></cell>

</row>

<row>

Line 542

</row>

<row>

<cell>LXXXVII.</cell>

<cell>P<emph type="italics"/>roportiones craßitiei aquæ ad <expan abbr="a&etilde;r&etilde;">aerem</expan> in <expan abbr="cõparatione">comparatione</expan> ad radios demon&longs;trare.<emph.end type="italics"/></cell>

<cell>P<emph type="italics"/>roportiones craßitiei aquæ ad <expan abbr="a&etilde;r&etilde;">aerrem</expan> in <expan abbr="cõparatione">comparatione</expan> ad radios demon&longs;trare.<emph.end type="italics"/></cell>

</row>

<row>

Line 572

</row>

<row>

<cell>XCIII.</cell>

<cell>P<emph type="italics"/>ropter qd in <expan abbr="cõcußione">concußione</expan> <expan abbr="etiã">etiam</expan> leui nauis loco moueatar <expan abbr="o&longs;t&etilde;dere">o&longs;tendere</expan>.<emph.end type="italics"/> V<emph type="italics"/>nde manife<expan abbr="&longs;i&utilde;">&longs;tum</expan> e&longs;t duas naues &longs;ibi <expan abbr="inuic&etilde;">inuicem</expan> occur&longs;antes retrocedere, & <expan abbr="quãt&utilde;">quantum</expan> <expan abbr="retrocedãt">retrocedant</expan> ambæ.<emph.end type="italics"/></cell>

<cell>P<emph type="italics"/>ropter qd in <expan abbr="cõcußione">concußione</expan> <expan abbr="etiã">etiam</expan> leui nauis loco moueatar <expan abbr="o&longs;t&etilde;dere">o&longs;tendere</expan>.<emph.end type="italics"/> V<emph type="italics"/>nde manifi <expan abbr="&longs;i&utilde;">&longs;ium</expan> e&longs;t duas naues &longs;ibi <expan abbr="inuic&etilde;">inuicem</expan> occur&longs;antes retrocedere, & <expan abbr="quãt&utilde;">quantum</expan> <expan abbr="retrocedãt">retrocedant</expan> ambæ.<emph.end type="italics"/></cell>

</row>

Line 583

</row>

<row>

<cell>C<emph type="italics"/>uiu&longs;uis trigoni rectanguli, aut cuius duo anguli &longs;int in dupla proportione, aut qui circulo in&longs;criptus &longs;it cognita quantitate unius lateris in comparatione ad dimetien<expan abbr="t&etilde;">tem</expan>, &longs;i proportio duorum laterum cognita fuerit, <expan abbr="er&utilde;t">erunt</expan> omnia eius latera cognita.<emph.end type="italics"/></cell>

<cell>C<emph type="italics"/>uiu&longs;uis trigoni rectanguli, aut cuius duo auguli &longs;int in dupla proportione, aut qui circulo in&longs;criptus &longs;it cognita quantitate unius lateris in comparatione ad dimetien <expan abbr="t&etilde;">tem</expan>, &longs;i proportio duorum laterum cognita fuerit, <expan abbr="er&utilde;t">erunt</expan> omnia eius latera cognita.<emph.end type="italics"/></cell>

</row>

<row>

Line 613

</row>

<row>

<cell>P<emph type="italics"/>roportionem pretij gemmarum ex tribus in eodem genere cognitis inuenire.<emph.end type="italics"/></cell>

<cell>P<emph type="italics"/>roportionem precij gemmarum ex tribus in eodem genere cognitis inuenire.<emph.end type="italics"/></cell>

</row>

<row>

Line 643

</row>

<row>

<cell>S<emph type="italics"/>i in circulo duæ diametri ad rectum angulum &longs;e &longs;ecauerint: aliæ uerò ad perpendiculum ex diametro exierint ad circum ferentiam, &longs;ingulæ &longs;upra diametrum erunt maiores portionibus reliquis diametri &longs;uperioribus, infra autem minores.<emph.end type="italics"/> D<emph type="italics"/>imidium autem portionis &longs;uperioris re&longs;iduum ad centrum maius &longs;agitta habebit.<emph.end type="italics"/> I<emph type="italics"/>n aliqua præterea portionis &longs;uperioris parte, quæ uer&longs;us diametrum tran&longs;uer&longs;um po&longs;ita e&longs;t, maior e&longs;t differentia partis diametri ei <expan abbr="corre&longs;põdentis">corre&longs;pondentis</expan>, <expan abbr="&qtilde;">quae</expan> line æ tran&longs;uer&longs;æ.<emph.end type="italics"/></cell>

<cell>S<emph type="italics"/>i in circulo duæ diametri ad rectum angulum &longs;e &longs;ecauerint: aliæ uerò ad perpendiculum ex diametro exicrint ad circum ferentiam, &longs;ingulæ &longs;upra diametrum erunt ma iores portionibus reliquis diametri &longs;uperioribus, infra autem minores.<emph.end type="italics"/> D<emph type="italics"/>imidium autem portionis &longs;uperioris re&longs;iduum ad centrum maius &longs;agitta habebit.<emph.end type="italics"/> I<emph type="italics"/>n aliqua præterea portionis &longs;uperioris parte, quæ uer&longs;us diametrum tran&longs;uer&longs;um po&longs;ita e&longs;t, maior e&longs;t differentia partis diametri ei <expan abbr="corre&longs;põdentis">corre&longs;pondentis</expan>, <expan abbr="&qtilde;">quae</expan> line æ tran&longs;uer&longs;æ.<emph.end type="italics"/></cell>

</row>

<row>

Line 673

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<row>

<cell>CXIII.</cell>

<cell>C<emph type="italics"/>ur uirga longius mittatur à puero quam à uiro inuestigare.<emph.end type="italics"/></cell>

<cell>C<emph type="italics"/>ur uirga longius mittatur à puero quam à uiro inueftigare.<emph.end type="italics"/></cell>

</row>

<row>

Line 744

</row>

<row>

<cell>CXXVII.</cell>

<cell>D<emph type="italics"/>ata poli altitudine ortus amplitudinem demonstrare.<emph.end type="italics"/></cell>

<cell>D<emph type="italics"/>ata poli altitudine ortus amplitudinem demonftrare.<emph.end type="italics"/></cell>

</row>

<row>

Line 860

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<row>

<cell>CXLIX.</cell>

<cell>D<emph type="italics"/>atam lineam &longs;ic diuidere, ut proportio quadratorum ad duplum unius partis in alteram &longs;it, ut lineæ datæ ad lineam datam.<emph.end type="italics"/></cell>

<cell>D<emph type="italics"/>atam lineam &longs;ic diuidere, ut proportio quadratorum ad dupium unius partis in alteram &longs;it, ut lineæ datæ ad lineam datam.<emph.end type="italics"/></cell>

</row>

<row>

Line 885

</row>

<row>

<cell>CLIIII.</cell>

<cell>S<emph type="italics"/>i lineæ datæ alia linea adiungatur, ab extremitatibus autem prioris lineæ duæ rectæ in unum punctum concurrant proportionem habentes, quam mediam inter tota m & adiectam, & adiectam erit punctus, concur&longs;us à puncto extremo lineæ adiectæ di&longs;tans per lineam mediam.<emph.end type="italics"/> Q<emph type="italics"/>uod &longs;i ab extremo alicuius lineæ æqualis mediæ, &longs;eu peripheria circuli, cuius &longs;emidiameter &longs;it media linea duæ lineæ ad prædicta puncta producantur, ip&longs;æ erunt in proportione mediæ ad adiectam.<emph.end type="italics"/></cell>

<cell>S<emph type="italics"/>i lineæ datæ alia linea adiungatur, ab extremitatibus autem prioris lineæ duæ rectæ in unum punctum concurrant proportionem habentes, quam mediam inter tota m & adiectam, & adiectam erit punctus, concur&longs;us à puncto extremo lineæ adiectæ di&longs;tans per lineam mediam.<emph.end type="italics"/> Q<emph type="italics"/>uod &longs;i ab extremo alicuius lineæ æqua'is mediæ, &longs;eu peripheria circuli, cuius &longs;emidiameter &longs;it media linea duæ lineæ ad prædicta puncta producantur, ip&longs;æ erunt in proportione mediæ ad adiectam.<emph.end type="italics"/></cell>

</row>

<row>

Line 1031

</row>

<row>

<cell>CLXXXIII.</cell>

<cell>S<emph type="italics"/>patium uitæ naturalis per &longs;pacium uitæ fortuitum declarare.<emph.end type="italics"/></cell>

<cell>S<emph type="italics"/>pacium uitæ naturalis per &longs;pacium uitæ fortuitum declarare.<emph.end type="italics"/></cell>

</row>

<row>

Line 1067

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<row>

<cell>CLXXXIX.</cell>

<cell>S<emph type="italics"/>i pondus minus ad longitudinem minorem &longs;ub æquali proportione coaptetur, facilius deor&longs;um trahetur quàm quod maius e&longs;t & propius.<emph.end type="italics"/></cell>

<cell>S<emph type="italics"/>i pondus minus ad longitudinem minorem &longs;ub æquali proportione coaptetar, facilius deor&longs;um trahetur quàm quod maius e&longs;t & propius.<emph.end type="italics"/></cell>

</row>

<row>

<cell>S<emph type="italics"/>i fuerit primum graue minus &longs;ecundo, & &longs;ecundum minus tertio, proportio autem primi ad &longs;ecundum multo maior quàm &longs;ecundi ad tertium, pos&longs;ibile erit propo&longs;itis uiribus ei&longs;dem addere pondus <expan abbr="&longs;ec&utilde;do">&longs;ecundo</expan>, ut ip&longs;um & tertium moueatur faciliùs ab ei&longs;dem uiribus, & primo uel &longs;ecundo quàm antea.<emph.end type="italics"/></cell>

<cell>S<emph type="italics"/>i fuerit primum graue minus &longs;ecundo, & &longs;ecundum minus tertio, proportio autem primi ad &longs;ecundum multo maior quàm &longs;ecundi ad tertium, po&longs;ibile erit propo&longs;itis uiribus ei&longs;dem addere pondus <expan abbr="&longs;ec&utilde;do">&longs;ecundo</expan>, ut ip&longs;um & tertium moueatur faciliùs ab ei&longs;dem uiribus, & primo uel &longs;ecundo quàm antea.<emph.end type="italics"/></cell>

</row>

<row>

Line 1284

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<row>

<cell>CCXXXI.</cell>

<cell>T<emph type="italics"/>res e&longs;&longs;e mundos atque inter ip&longs;os nullam e&longs;&longs;e proportionem: nec numero eos definiri.<emph.end type="italics"/></cell>

<cell>T<emph type="italics"/>res e&longs;&longs;e mundos atque inter ip&longs;os nullam e&longs;&longs;e proportionem: nec numero cos definiri.<emph.end type="italics"/></cell>

</row>

<row>

<cell>CCXXXII.</cell>

<cell>O<emph type="italics"/>mnis motus naturalis quanto uelocior e&longs;t tanto propior e&longs;t & magis &longs;imilflimus quieti.<emph.end type="italics"/></cell>

<cell>O<emph type="italics"/>mnis motus naturalis quanto uelocior e&longs;t tanto propior e&longs;t & magis &longs;imil limus quieti.<emph.end type="italics"/></cell>

</row>

<row>

Line 1337

<s id="id000051">Proportiones &longs;ecundum genus notæ dicuntur, cùm nouimus, <lb/>quòd &longs;int maiores, aut minores. </s>

<s id="id000052">Nam cùm æquales &longs;unt, &longs;imul ne<lb/>cesse e&longs;t, ut cogno&longs;camus genus, & &longs;peciem.</s>

<s id="id000052">Nam cùm æquales &longs;unt, &longs;imul ne<lb/>ceffe e&longs;t, ut cogno&longs;camus genus, & &longs;peciem.</s>

<s id="id000053">Quinta diffinitio.</s>

Line 2345

<pb pagenum="12" xlink:href="015/01/031.jpg"/>dem tamen generis, cum illis dico quòd proportio e ad d e&longs;t com<lb/>po&longs;ita ex proportionibus e ad a c, & e ad b c. </s>

<s id="id000262">Po&longs;ita ergo e tanquam &longs;u<lb/>periore numero, & a c & c b inferioribus, erit ex octaua propo&longs;itio<lb/>ne huius proportio productorum ex e in a c, & coniunctorum, & <lb/>ex con&longs;equenti per primam &longs;ecundi Elementorum producti ex e in <lb/>a b ad productum ex a c in c b compo&longs;ita ex proportionibus e ad <lb/>a c, & e ad c b: at quod fit ex a c in c b, e&longs;t æquale ei quod fit ex a b in <lb/>d, eo quòd a b, a c, c b & d &longs;unt omiologæ per decimam &longs;extam &longs;exti <lb/><expan abbr="Elem&etilde;torum">Elementorum</expan>: Proportio igitur producti ex e in a b ad productum <lb/>ex d in a b e&longs;t compo&longs;ita ex proportionibus e ad a c, & e ad e b: At <lb/>proportio producti ex e in a b ad productum ex d in a b, e&longs;t uelut e <lb/><arrow.to.target n="marg32"/><lb/>ad d. </s>

<s id="id000262">Po&longs;ita ergo e tan<08> &longs;u<lb/>periore numero, & a c & c b inferioribus, erit ex octaua propo&longs;itio<lb/>ne huius proportio productorum ex e in a c, & coniunctorum, & <lb/>ex con&longs;equenti per primam &longs;ecundi Elementorum producti ex e in <lb/>a b ad productum ex a c in c b compo&longs;ita ex proportionibus e ad <lb/>a c, & e ad c b: at quod fit ex a c in c b, e&longs;t æquale ei quod fit ex a b in <lb/>d, eo quòd a b, a c, c b & d &longs;unt omiologæ per decimam &longs;extam &longs;exti <lb/><expan abbr="Elem&etilde;torum">Elementorum</expan>: Proportio igitur producti ex e in a b ad productum <lb/>ex d in a b e&longs;t compo&longs;ita ex proportionibus e ad a c, & e ad e b: At <lb/>proportio producti ex e in a b ad productum ex d in a b, e&longs;t uelut e <lb/><arrow.to.target n="marg32"/><lb/>ad d. </s>

<s id="id000263">per &longs;uppo&longs;ita igitur proportio e ad d e&longs;t compo&longs;ita ex propor<lb/>tionibus e ad a c, & e ad b c, quod fuit demon&longs;trandum.</s>

Line 2698

<s id="id000385">Sint aliquot quantitates a b c d e f g h in <lb/>continua proportione. </s>

<s id="id000386">Arithmetica di&longs;po&longs;it&etail; <lb/>ita ut minima <expan abbr="ear&utilde;">earum</expan> qu&etail; &longs;it h, &longs;it &etail;qualis diffe<lb/>renti&etail; quantitatum <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> ordinem di&longs;po<lb/><expan abbr="&longs;itar&utilde;">&longs;itarum</expan>, uelut differentia a & b, & b & c, & c & <lb/>d, et ita de alijs, addantur <expan abbr="a&utilde;t">aut</expan> <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> &longs;in <lb/>gulis harum, quæ &longs;int i k l m n o p, ita ut <expan abbr="&etilde;os">eos</expan> <lb/>fiant &etail;quales <expan abbr="c&utilde;">cum</expan> &longs;uis &longs;upplementis ip&longs;i line&etail; <lb/>à maiori. </s>

<s id="id000386">Arithmetica di&longs;po&longs;it&etail; <lb/>ita ut minima <expan abbr="ear&utilde;">earum</expan> qu&etail; &longs;it h, &longs;it &etail;qualis diffe<lb/>renti&etail; quantitatum <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> ordinem di&longs;po<lb/><expan abbr="&longs;itar&utilde;">&longs;itarum</expan>, uelut differentia a & b, & b & c, & c & <lb/>d, et ita de alijs, addantur <expan abbr="a&utilde;t">aut</expan> <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> &longs;in <lb/>gulis harum, quæ &longs;int i k l m n o p, ita ut <expan abbr="o&etilde;s">oes</expan> <lb/>fiant &etail;quales <expan abbr="c&utilde;">cum</expan> &longs;uis &longs;upplementis ip&longs;i line&etail; <lb/>à maiori. </s>

<s id="id000387">E&longs;tque <expan abbr="id&etilde;">idem</expan> ac &longs;i e&longs;&longs;ent aliquot quanti<pb pagenum="18" xlink:href="015/01/037.jpg"/>tates, & <expan abbr="diuideren&ttilde;">diuiderentur</expan> &longs;ingul&etail; <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> numerum <expan abbr="illar&utilde;">illarum</expan>, &longs;i quatuor in <lb/>quatuor partes æquales, &longs;i quinque in quinque, &longs;i decem in decem, ea ra<lb/>tione ut ultima <expan abbr="diuidere&ttilde;">diuideretur</expan>, ubi e&longs;t finis primæ partis, penultima ubi <lb/>e&longs;t finis &longs;ecundæ partis, ante penultima ubi e&longs;t finis tertiæ, & &longs;ic de <lb/>alijs. </s>

<s id="id000388">Vocabo ergo primas <expan abbr="quãtitates">quantitates</expan> propo&longs;itas a b c d e f g h quan<lb/>titates primi ordinis, &longs;ed quantitates æquales quæ <expan abbr="con&longs;tãt">con&longs;tant</expan> ex quan<lb/>titatis. </s>

<s id="id000389">primi ordinis, & &longs;upplementis, appellabo quantitates &longs;ecun<lb/>di ordinis: ex quo patet quòd prima <expan abbr="quãtitas">quantitas</expan> erit ex utro que ordine, <lb/>quia non e&longs;t diui&longs;a, reliquæ omnes differunt, quantitates uerò quas <lb/>adiunxi nominabo <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan>, & &longs;unt una minus <expan abbr="quã">quam</expan> quantitates <lb/>ordinum: ut &longs;i <expan abbr="quãtitates">quantitates</expan> ordinum &longs;int octo, erunt &longs;upplementa &longs;e<lb/>ptem, & &longs;i quantitates <expan abbr="ordin&utilde;">ordinum</expan>, e&longs;&longs;ent &longs;eptem e&longs;&longs;ent <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> &longs;ex, <lb/>quia inter &longs;upplementa <expan abbr="nõ">non</expan> <expan abbr="adnumera&ttilde;">adnumeratur</expan> quantitas indiui&longs;a. </s>

Line 3015

<s id="id000532">Propo&longs;itio trige&longs;ima tertia.</s>

<s id="id000532">Propo&longs;itio trige&longs;ima ertia.</s>

<s id="id000533">Omnia duo mobilia æqualis undique magnitudinis, quæ æquali <lb/>in tempore æqualia &longs;patia pertran&longs;eunt in diuer&longs;is &longs;ub&longs;tantia me<lb/>dijs, nece&longs;&longs;e e&longs;t, ut &longs;it ponderis ad pondus, quemadmodum medij <lb/>ad medium, proportio duplicata.<lb/><arrow.to.target n="marg88"/></s>

Line 3093

<s id="id000564">Tertium &longs;ic &longs;it a b humi<lb/>lior uox, quæ excre&longs;cat &longs;e<lb/>mitonio minore &longs;olum in <lb/>c, & &longs;it d e dupla ad ab &longs;e<lb/>cundum naturam, ut in uo<lb/>cibus medijs fiet, ut &longs;i e debeat excre&longs;cere &longs;emitonio minore per de<lb/>cimam nonam quinti <expan abbr="Elem&etilde;torum">Elementorum</expan> f e dupla c b, & in acutis ubi ex<lb/>creuerit ad diapa&longs;on quadrupla: pueri autem uox, quæ iam diapa<lb/>&longs;on altior e&longs;t d e, erit bis diapa&longs;on, & ideò quadrupla b c, &longs;ed in acu<lb/>tioribus erit dupla, nullus enim puer e&longs;t adeo fractæ uocis, qui &longs;u<lb/>pra humillimam non a&longs;cendat per diapa&longs;on, igitur interuallum uo<lb/>cum erit octuplum a d, b c, &longs;ed communiter a&longs;cendunt ad bis diapa<lb/>&longs;on, igitur interuallum unius uocis etiam cum &longs;emitonio propor<lb/>tionem habentis e&longs;t æquale fermè toti a b, cum autem in diapa&longs;on <lb/>&longs;int duodecim &longs;emitonia, & duo comata, manife&longs;tum e&longs;t, quod ex<lb/>ten&longs;io illa erit maxima in <expan abbr="cõparatíone">comparatíone</expan> grauioris uo cis a b. </s>

<s id="id000565">Et ideò <lb/>minimum in crementum in humilioribus uocibus, ubi quis coga<pb pagenum="40 [=30]" xlink:href="015/01/049.jpg"/>tur a&longs;cendere, maximum e&longs;&longs;e uidetur, adeò ut ægrè à pluribus fera<lb/>tur, à quibu&longs;dam non omnino feratur.</s>

<s id="id000565">Et ideò <lb/>minimum in crementum in humilioribus uocibus, ubi quis coga

<pb pagenum="40 [=30]" xlink:href="015/01/049.jpg"/>tur a&longs;cendere, maximum e&longs;&longs;e uidetur, adeò ut ægrè à pluribus fera<lb/>tur, à quibu&longs;dam non omnino feratur.</s>

<s id="id000566">SCHOLIVM.</s>

Line 3156

Line 3159

<s id="id000590">Sit a corpus quie&longs;cens in pauimento b, & mouetur in eo occul</s><s id="id000591"><arrow.to.target n="marg96"/><lb/>to motu uer&longs;us centrum, ut &longs;uprà ui&longs;um e&longs;t, contra<lb/><figure id="id.015.01.050.1.jpg" xlink:href="015/01/050/1.jpg"/><lb/>rius illi &longs;it motus ad c, &longs;i ergo a quie&longs;ceret in c moue<lb/>retur ad b occulto motu certa ui, ergo eadem re&longs;titit, <lb/>ne traheretur ad c. </s>

<s id="id000590">Sit a corpus quie&longs;cens in pauimento b, & mouetur in eo occul</s>

<s id="id000591"><arrow.to.target n="marg96"/><lb/>to motu uer&longs;us centrum, ut &longs;uprà ui&longs;um e&longs;t, contra<lb/><figure id="id.015.01.050.1.jpg" xlink:href="015/01/050/1.jpg"/><lb/>rius illi &longs;it motus ad c, &longs;i ergo a quie&longs;ceret in c moue<lb/>retur ad b occulto motu certa ui, ergo eadem re&longs;titit, <lb/>ne traheretur ad c. </s>

<s id="id000592">Manife&longs;tum e&longs;t autem, quod hic <lb/><arrow.to.target n="marg97"/><lb/>motus occultus e&longs;t minor manife&longs;to.<lb/><arrow.to.target n="marg98"/></s>

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<s id="id000630">Hoc quomodo non po&longs;sit fieri &longs;uprà docuimus, nunc etiam ge</s><s id="id000631"><arrow.to.target n="marg109"/><lb/>neraliter dicam, cum con&longs;i&longs;tant hæc in duobus terminis, productio <lb/>uerò præ&longs;upponit quatuor terminos, ut in prima propo&longs;itione, aut <lb/>&longs;altem tres, atque in his medius habet rationem mouentis, & moti, <lb/>ergo cum in huiu&longs;modi <expan abbr="nõ">non</expan> &longs;int quatuor termini, nec tres, è quibus <lb/>unus &longs;it mouens, & motum proportio non poterit produci. </s>

<s id="id000630">Hoc quomodo non po&longs;sit fieri &longs;uprà docuimus, nunc etiam ge</s>

<s id="id000631"><arrow.to.target n="marg109"/><lb/>neraliter dicam, cum con&longs;i&longs;tant hæc in duobus terminis, productio <lb/>uerò præ&longs;upponit quatuor terminos, ut in prima propo&longs;itione, aut <lb/>&longs;altem tres, atque in his medius habet rationem mouentis, & moti, <lb/>ergo cum in huiu&longs;modi <expan abbr="nõ">non</expan> &longs;int quatuor termini, nec tres, è quibus <lb/>unus &longs;it mouens, & motum proportio non poterit produci. </s>

<s id="id000632">Illud <lb/>etiam patet exemplo, nam &longs;i e&longs;&longs;et lapis, aut nauis ob&longs;i&longs;tens ut &longs;ex, & <lb/>e&longs;&longs;ent homines uiribus &longs;inguli, ut quatuor cum dimidio, tres mo<lb/>uerent in proportione dupla &longs;exquiquarta perdicta &longs;uperius eo<lb/>dem loco, at &longs;i proportio duci po&longs;&longs;et aliquorum hominum nume<lb/>rus po&longs;&longs;et mouere in duplicata proportione ad unguem &longs;cilicet <lb/>5 1/16 ut e&longs;&longs;et uix hominum collectorum 30 3/8 at nullus e&longs;t numerus ho<lb/>minum qui collectus faciat hunc numerum, nam &longs;ex homines ex<lb/>plent numerum 27, & &longs;eptem 31 1/2, & ideò non pote&longs;t duci propor<lb/>tio. </s>

<s id="id000633">Et ideò maximus e&longs;t error dicendo decem homines mouent na <lb/>uim proportione tripla, ergo triginta alij additis illis &longs;imiles robo<lb/>re mouebunt à proportione uiginti &longs;eptupla &longs;cilicet ducta nonu<pb pagenum="33" xlink:href="015/01/052.jpg"/>pla in triplam. </s>

<pb pagenum="33" xlink:href="015/01/052.jpg"/>pla in triplam. </s>

<s id="id000634">Sed &longs;umpta proportione alio modo producitur. </s>

<s id="id000635">Ve<lb/>lut &longs;i dicam, homines decem mouent nauim, aut <expan abbr="fer&utilde;t">ferunt</expan> pondus pro<lb/>portione tripla, igitur quadraginta homines idem facient propor<lb/>tione duodecupla &longs;cilicet quadrupla in triplam ducta. </s>

<s id="id000636">Cum ergo <lb/>addo triginta homines, qui mouent in proportione nonupla, non <lb/>oportet ducere nonuplam in triplam, &longs;ed totum numerum accipe<lb/>re, & quam proportionem habet ad partem, tandem habet uis mo<lb/>uens ad uim <expan abbr="mou&etilde;tem">mouentem</expan>. </s>

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<s id="id000668">Et libella nihil retinet ni&longs;i quantum e&longs;t pondus eius quod cu<pb pagenum="35" xlink:href="015/01/054.jpg"/>pit ad centrum peruenire, & pondus ei appen&longs;um non prohi<lb/>bet motum, etiam &longs;i e&longs;&longs;et infinitum, ni&longs;i quatenus non uult recede<lb/>re ex directo centri mundi: & ut grauat hypomochlion faciens im<lb/>pre&longs;sionem.</s>

<s id="id000668">Et libella nihil retinet ni&longs;i quantum e&longs;t pondus eius quod cu

<pb pagenum="35" xlink:href="015/01/054.jpg"/>pit ad centrum peruenire, & pondus ei appen&longs;um non prohi<lb/>bet motum, etiam &longs;i e&longs;&longs;et infinitum, ni&longs;i quatenus non uult recede<lb/>re ex directo centri mundi: & ut grauat hypomochlion faciens im<lb/>pre&longs;sionem.</s>

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<s id="id000712">Sint duo mobilia a & b in eodem pun<lb/><figure id="id.015.01.055.1.jpg" xlink:href="015/01/055/1.jpg"/><lb/>cto, quæ æqualiter uer&longs;us eandem partem <lb/>moueantur æqualibus in temporibus, inui<lb/>cem tamen in æqualiter, ita quod a in f & b <lb/>in g temporibus ab&longs;oluant circulum, & ho<lb/>rum differentia &longs;it h. </s>

<s id="id000713">Dum itaque a perficit <lb/>circulum b perueniat in c, igitur c d b e&longs;t dif<lb/>ferentia, quæ &longs;uperanda e&longs;t, & proportio <lb/>circuli ad b c ut g ad f, quare reliqui ad reli<lb/>quum, ut re&longs;idui ad re&longs;iduum, &longs;cilicet circu<lb/>li ad c d b, ut g ad h, & b c ad c d b ut f ad h, coniungantur igitur in k <lb/>tempore, eruntque k f g h omiologa, ut productum ex circulo in b c <lb/>diui&longs;o per certam quantitatem & cum circulo & b c & c d b diffe<lb/>rentia, & &longs;it &longs; productum ex f in g, dico quod diui&longs;a &longs; per h exibit k <lb/>tempus coniunctionis primæ, &longs;it itaque d locus coniunctionis, dico <lb/>igitur quod differentia &longs;patij pertran&longs;iti a b, a & a, b in reditu ex con<lb/>iunctione prima ad d e&longs;t unus circulus completus, non enim po&longs;<lb/>&longs;unt e&longs;&longs;e plures, nam &longs;equeretur, quòd a aliquando pertran&longs;i&longs;&longs;et b, <lb/>et &longs;ic non e&longs;&longs;et prima coniunctio, nec pote&longs;t e&longs;&longs;e minus, nam &longs;ic cum <lb/>a & b &longs;int in d ultra perfectas circulationes uterque eorum pertran<lb/>&longs;iuit arcum b c, igitur nullo modo differentia pote&longs;t e&longs;&longs;e minor cir<lb/>culo, neque maior, ut declaratum e&longs;t, igitur e&longs;t unus circulus ad un<pb pagenum="37" xlink:href="015/01/056.jpg"/>guem. </s>

<s id="id000714">Hoch declarato ponatur m spatium compositum ex circulis <lb/>pertran&longs;itis a b a cum &longs;patio b d, etenim &longs;patium, quod pertran&longs;it <lb/>b a coniunctione in a, ad coniunctionem primam in d, & erit ex de<lb/>mon&longs;tratis horum differentia circulus qui uocetur o, & &longs;it p &longs;pa<lb/>tium, quod pertran&longs;it b in tempore eodem, in quo a pertran&longs;it o, & <lb/>&longs;it q differentia o, & p qu&etail; in circulo e&longs;t c d l b, quia igitur in eodem <lb/>tempore a pertran&longs;it m & b, n, erit m ad n, ut a ad b, & eadem ratio<lb/>ne a ad b, ut o ad p, igitur ex undecima quinti Euclidis m ad n, ut o <lb/>ad p, quare cum o &longs;it differentia m & n, & q, differentia o & p erit ex <lb/>decima nona quinti Euclidis, m ad o, ut o ad q, & ita circulus e&longs;t ana<lb/>logus inter &longs;patium pertran&longs;itum à motore uelociori, & inter diffe<lb/>rentiam &longs;patij quæ accidit, dum uelocior motor pertran&longs;it circu<lb/>lum, id e&longs;t quòd circulus a c d e&longs;t analogus inter c d l b, & circulos <lb/>pertran&longs;itos a b a cum portione b d. </s>

<s id="id000715">Reuertor igitur ad propo&longs;i<lb/>tum, cum &longs;it m ad o, ut o ad q, & m ad o, ut n ad p, ex &longs;exta decima <lb/>quinti Euclidis, erit ex undecima eiu&longs;dem n ad p, ut o ad q, quare ex <lb/>&longs;exta decima &longs;exti Elementorum ducto o, id e&longs;t circulo, &longs;eu maiore <lb/>numero in p &longs;patium pertran&longs;itum a b, &longs;eu ducto fin g, & diui&longs;o per <lb/>q differentiam &longs;patiorum, &longs;eu per h exibit n, &longs;eu &longs;patium quod <lb/>pertran&longs;it b ab una coniunctione ad aliam quod erat demon<lb/>&longs;trandum.</s>

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<pb pagenum="38" xlink:href="015/01/057.jpg"/>ergo iam &longs;upponatur numerari a b & c non numerabitur a b a, er<lb/>go a non perficiet circuitus, ergo non redibit ad primum <expan abbr="loc&utilde;">locum</expan>, ergo <lb/>non erit iunctus cum b & c. </s>

<s id="id000727">Quod &longs;i dicas a b c coniungi in decem <lb/>&longs;eptem annis numero non numerato ab ali <lb/><figure id="id.015.01.057.1.jpg" xlink:href="015/01/057/1.jpg"/><lb/>quo illorum temporum, auferantur perfe<lb/>ctæ circulationes, & <expan abbr="remaneb&utilde;t">remanebunt</expan> dimidium <lb/>ex a, duæ quintæ ex b, tres &longs;eptimæ ex c, igi<lb/>tur oportebit ut hæ portiones &longs;int æqua<lb/>les, ut po&longs;t perfectas circulationes in idem <lb/>punctum, <expan abbr="cõueniant">conueniant</expan>, ergo 1/2 & 2/5 & 3/7 æqui<lb/>ualebunt, quare proportio 7 ad 3 & 5 ad 2 <lb/>& 2 ad 1, e&longs;t una, quare permutando 3 ad 2 <lb/>ut 7 ad 5, &longs;ed 7 & 5 &longs;unt contra &longs;e primi, ergo in &longs;ua proportione mi <lb/>nimi per dicta in &longs;eptimo Elementorum: ergo tria, & duo non &longs;unt <lb/>in eadem proportione. </s>

<s id="id000728">Rur&longs;us dicantur conuenire in annis qua</s><s id="id000729"><arrow.to.target n="marg125"/><lb/>tuordecim cum dimidio, ergo in uiginti nouem conuenient ite<lb/>rum: ergo per &longs;ecundam partem erit &longs;eptem ad unum, ut duo ad <lb/>unum, igitur permutando unius ad unum, ut &longs;eptem ad duo, &longs;ed <lb/>unum e&longs;t æquale uni, ergo duo erunt æqualia &longs;eptem. </s>

<s id="id000728">Rur&longs;us dicantur conuenire in annis qua</s>

<s id="id000729"><arrow.to.target n="marg125"/><lb/>tuordecim cum dimidio, ergo in uiginti nouem conuenient ite<lb/>rum: ergo per &longs;ecundam partem erit &longs;eptem ad unum, ut duo ad <lb/>unum, igitur permutando unius ad unum, ut &longs;eptem ad duo, &longs;ed <lb/>unum e&longs;t æquale uni, ergo duo erunt æqualia &longs;eptem. </s>

<s id="id000730">Rur&longs;us dica<lb/>mus, quod in tempore annorum <02> quadrata decem &longs;imiliter aufe<lb/>ram integras reuolutiones, quas potero, & erunt <02> 2 1/2 m: 1, & <02> 2/5 & <lb/><02> 10/49 æqualia. </s>

<s id="id000731">Hic uides infinita &longs;equi in conuenientia, quæ longum <lb/>e&longs;&longs;et numerare, nam &longs;eptem e&longs;&longs;et æquale quinque, & proportio reci&longs;i <lb/>ad potentia rethe, ut numeri ad numerum. </s>

<s id="id000732">Igitur non conueniunt <lb/>ante &longs;eptuaginta annos.<lb/><arrow.to.target n="marg126"/></s>

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<s id="id000773">Et &longs;i totum <lb/>&longs;patium cum circuitibus erit unius generis, erunt duplicata, & tri<lb/>plicata, & quadruplicata eiu&longs;dem generis: quare cum &longs;patia ip&longs;a <lb/>detractis circuitibus uelut rhete habeant naturam reci&longs;i, & &longs;patia <lb/>ip&longs;a tota &longs;int eiu&longs;dem generis, erunt &longs;patia, quæ relinquuntur eiu&longs;<lb/>dem generis. </s>

<s id="id000774">Erunt tamen incommen&longs;a nece&longs;&longs;ariò, &longs;i partes fuerint <lb/>incommen&longs;æ toti. </s>

<s id="id000775">Ponatur a c incommen&longs;a toti circulo dico, quod <lb/>a k <expan abbr="etiã">etiam</expan> e&longs;t incommen&longs;a toti circulo: & <expan abbr="etiã">etiam</expan> a k, & k c. </s>

<s id="id000776">Quia enim a c <lb/>e&longs;t incommen&longs;a circulo, & k a cum toto circulo &longs;emel e&longs;t commen<pb pagenum="41" xlink:href="015/01/060.jpg"/>&longs;a a c, quia multiplex ei. </s>

<s id="id000776">Quia enim a c <lb/>e&longs;t incommen&longs;a circulo, & k a cum toto circulo &longs;emel e&longs;t commen

<s id="id000777">igitur cum circulus, & a k diuidantur in cir<lb/><arrow.to.target n="marg133"/><lb/>culum et a k, & circulus &longs;it incommen&longs;us circulo, cum a k erit aggre<lb/></s><s id="id000778">gatum ex circulo, & a k incommen&longs;um ip&longs;i a k, & a k pariter incom<lb/><arrow.to.target n="marg134"/><lb/>men&longs;a circulo. </s>

<pb pagenum="41" xlink:href="015/01/060.jpg"/>&longs;a a c, quia multiplex ei. </s>

<s id="id000777">igitur cum circulus, & a k diuidantur in cir<lb/><arrow.to.target n="marg133"/><lb/>culum et a k, & circulus &longs;it incommen&longs;us circulo, cum a k erit aggre<lb/></s>

<s id="id000778">gatum ex circulo, & a k incommen&longs;um ip&longs;i a k, & a k pariter incom<lb/><arrow.to.target n="marg134"/><lb/>men&longs;a circulo. </s>

<s id="id000779">Rur&longs;us quia a k e&longs;t incommen&longs;a circulo cum a k, & <lb/>circulus cum a k &longs;it multiplex ad a c, erit a k incommen&longs;a a c, quare <lb/><arrow.to.target n="marg135"/><lb/>erit c k incommen&longs;a a k & a c, & circulo ad dita a k. </s>

<s id="id000780">Si ergo a c &longs;it <lb/>commen&longs;a circulo, erunt omnes portiones e genere numeri, & &longs;i <lb/><arrow.to.target n="marg136"/><lb/>potentia rhete erunt omnes, uel potentia rhete, uel circulis detra<lb/>ctis, ut a k & a l reci&longs;a: & a c &longs;it potentia &longs;ecunda rhete, id e&longs;t radix cu<lb/>bica erunt omnes c d, d e, e f, potentia &longs;ecunda rhete, et radices cubi<lb/>cæ numeri, &longs;eu latera corporum rhete, a k uero & a l, & huiu&longs;modi <lb/>in infinitum reci&longs;a potentia rhete.<lb/><arrow.to.target n="marg137"/></s>

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<s id="id000822">Sit orbis a b cuius cen<lb/><figure id="id.015.01.062.2.jpg" xlink:href="015/01/062/2.jpg"/><lb/>centrum c, manubrium c <lb/>d f e, &longs;eu uero tangat circu<lb/>lum g, &longs;eu more gemmas <lb/>&longs;culpentium aligetur al<lb/>teri orbi funiculo a l b, & <lb/>&longs;it in uertice axis k m or<lb/>biculus &longs;olidus aut &longs;emi<lb/>circulari forma m, dico <lb/>quod proportio motus a <lb/>b ad motum m e&longs;t produ<lb/>cta ex duabus proportio<lb/>nibus c n <expan abbr="&longs;emidimeti&etilde;tis">&longs;emidimetientis</expan>, <lb/>& &longs;emidimetientis m ad k <lb/>o, quare ut rectanguli c n <lb/>in dimidium dimetientis <lb/>m ad quadratum o, ut enim a b ad ol orbem, id e&longs;t <expan abbr="peripheriar&utilde;">peripheriarum</expan> ita <lb/>c n ad o k, quoniam o l mouetur toties in una circuitione a b, quo<lb/>ties <expan abbr="peripheriã">peripheriam</expan> o l <expan abbr="contine&ttilde;">continetur</expan> in peripheria a b, ergo quoties o k con<lb/>tinetur in c n toties in una circuitione a b o l circumuertitur, &longs;ed <lb/>quoties circumuertitur ol, toties etiam m, quia uterque mouetur eo<lb/>dem circuitu k m axis, ergo quoties m circumducitur in circuitu a <lb/>b toties o k continetur in c n, ergo &longs;i fiat comparatio &longs;emidiametri <lb/>m ad c n, erit producta proportio circuitus a b ad circuitum m ex <lb/>proportione c n ad o k, et &longs;emidimetientis m ad <expan abbr="id&etilde;">idem</expan> o k, ergo per 26 <lb/>proportio numeri circuitus unius p <expan abbr="alter&utilde;">alterum</expan> e&longs;t, ut rectanguli &longs;ub c n, <lb/>& &longs;emidimetiente m ad quadratum k o, quod erat <expan abbr="demon&longs;trand&utilde;">demon&longs;trandum</expan>.</s>

<s id="id000823">Manife&longs;tum e&longs;t autem ex ip&longs;a &longs;ola con&longs;titutione, quod &longs;i a b mo</s><s id="id000824"><arrow.to.target n="marg142"/><pb pagenum="44" xlink:href="015/01/063.jpg"/>uetur &longs;ur&longs;um à dextro in &longs;ini&longs;trum in inferiore parte, mouebitur à <lb/>&longs;ini&longs;tro in dextrum, & uterque circulorum g & k in &longs;uperiore parte, <lb/>& in inferiore mouebitur contrario motu, &longs;cilicet in &longs;uperiore à &longs;ini<lb/>&longs;tro in dextrum, & inferiore à dextro in &longs;ini&longs;trum, illi uerò duo or<lb/>bes &longs;imili motu mouebuntur tam in parte &longs;uperiore, quàm inferio<lb/>re, & proportio motuum eorum inter &longs;e erit uelut dimetientium <lb/>eorundem.<lb/><arrow.to.target n="marg143"/></s>

<s id="id000823">Manife&longs;tum e&longs;t autem ex ip&longs;a &longs;ola con&longs;titutione, quod &longs;i a b mo</s>

<pb pagenum="44" xlink:href="015/01/063.jpg"/>uetur &longs;ur&longs;um à dextro in &longs;ini&longs;trum in inferiore parte, mouebitur à <lb/>&longs;ini&longs;tro in dextrum, & uterque circulorum g & k in &longs;uperiore parte, <lb/>& in inferiore mouebitur contrario motu, &longs;cilicet in &longs;uperiore à &longs;ini<lb/>&longs;tro in dextrum, & inferiore à dextro in &longs;ini&longs;trum, illi uerò duo or<lb/>bes &longs;imili motu mouebuntur tam in parte &longs;uperiore, quàm inferio<lb/>re, & proportio motuum eorum inter &longs;e erit uelut dimetientium <lb/>eorundem.<lb/><arrow.to.target n="marg143"/></s>

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<s id="id000841">Galenus libro quinto de Simplicibus medicamentis, quem &longs;e</s><s id="id000842"><arrow.to.target n="marg148"/><lb/>quuti &longs;unt alij medici, ponit quatuor ordines <expan abbr="medicamentor&utilde;">medicamentorum</expan> iux<lb/>ta qualitates calidi, frigidi, &longs;icci, & humidi, & primus e&longs;t cum <expan abbr="medicament&utilde;">medi<lb/>camentum</expan> non &longs;entitur quale &longs;it licet operetur, uelut cam&etail;melon, ab<lb/>&longs;ynthium, & oriza: &longs;ecundus e&longs;t, cum &longs;entitur, &longs;ed non lædit, ut nux <lb/>myri&longs;tica, &longs;aluia, ozimum: tertius e&longs;t cum &longs;entitur, & lædit, &longs;ed <lb/>non de&longs;truit, neque corrumpit corpus, uelut a&longs;&longs;arum apium &longs;ta<lb/>phi&longs;agria, cappares, myrrha, ruta: quartus e&longs;t, cum de&longs;truit ue<lb/>lut pyretrum, piper, euphorbium cæpe aggre&longs;te, & &longs;inapis, cina

<s id="id000841">Galenus libro quinto de Simplicibus medicamentis, quem &longs;e</s>

<s id="id000842"><arrow.to.target n="marg148"/><lb/>quuti &longs;unt alij medici, ponit quatuor ordines <expan abbr="medicamentor&utilde;">medicamentorum</expan> iux<lb/>ta qualitates calidi, frigidi, &longs;icci, & humidi, & primus e&longs;t cum <expan abbr="medicament&utilde;">medi<lb/>camentum</expan> non &longs;entitur quale &longs;it licet operetur, uelut cam&etail;melon, ab<lb/>&longs;ynthium, & oriza: &longs;ecundus e&longs;t, cum &longs;entitur, &longs;ed non lædit, ut nux <lb/>myri&longs;tica, &longs;aluia, ozimum: tertius e&longs;t cum &longs;entitur, & lædit, &longs;ed <lb/>non de&longs;truit, neque corrumpit corpus, uelut a&longs;&longs;arum apium &longs;ta<lb/>phi&longs;agria, cappares, myrrha, ruta: quartus e&longs;t, cum de&longs;truit ue<lb/>lut pyretrum, piper, euphorbium cæpe aggre&longs;te, & &longs;inapis, cina

<pb pagenum="45" xlink:href="015/01/064.jpg"/>momum autem, & gingiber numerantur inter medicinas calídas <lb/>tertij gradus, & hoc opus comparatur ad corpus &longs;icut dicit Gale<lb/>nus, & Serapio non ad linguam, ut medici no&longs;tri temporis interpre<lb/>tantur. </s>

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<s id="id000926">Sit a mobile, grauitatis centrum b, cuius pars ei pro<lb/><arrow.to.target n="marg161"/><lb/><figure id="id.015.01.070.3.jpg" xlink:href="015/01/070/3.jpg"/><lb/>ximior &longs;it c a, dico quod de&longs;cendat motu naturali c a, <lb/>parte tangendo terram, quia enim totum a non pote&longs;t <lb/>de&longs;cendere ad centrum de&longs;cendit b, quia eadem e&longs;t na<lb/>tura partis, & totius: totius autem terræ natura e&longs;t ut <lb/>centrum, totius &longs;it centrum grauitatis, quare b breuiore uia fertur <lb/><arrow.to.target n="marg162"/><lb/>ad centrum, ergo per c d proximiorem partem ip&longs;i b. </s>

<s id="id000927">Sed pars pro<lb/>ximior nece&longs;&longs;ariò e&longs;t grauior, quia centrum e&longs;t in medio grauita<pb pagenum="52" xlink:href="015/01/071.jpg"/>tis, ergo omne mobile de&longs;cendit motu naturali per &longs;ui grauio<lb/>rem partem.<lb/><arrow.to.target n="marg163"/></s>

<s id="id000927">Sed pars pro<lb/>ximior nece&longs;&longs;ariò e&longs;t grauior, quia centrum e&longs;t in medio grauita

<pb pagenum="52" xlink:href="015/01/071.jpg"/>tis, ergo omne mobile de&longs;cendit motu naturali per &longs;ui grauio<lb/>rem partem.<lb/><arrow.to.target n="marg163"/></s>

Line 5715

Line 5752

<s id="id001715">Propo&longs;itio cente&longs;ima prima.</s>

<s id="id001716">Proportionem pretij gemmarum ex tribus in eodem genere co<lb/>gnitis inuenire.<lb/><arrow.to.target n="marg350"/></s>

<s id="id001716">Proportionem precij gemmarum ex tribus in eodem genere co<lb/>gnitis inuenire.<lb/><arrow.to.target n="marg350"/></s>

Line 6544

Line 6581

<s id="id002150">Sed &longs;i humili loco &longs;e<lb/>deant, quoniam aër de&longs;cen <lb/>dit, ex l & m uer&longs;us x, & <lb/>etiam, quia erunt proximi <lb/>h k, <expan abbr="quãdo">quando</expan> fuerit in x, <expan abbr="refrigerabun&ttilde;">refri<lb/>gerabuntur</expan> ualde. </s>

<s id="id002151">Qui <expan abbr="aut&etilde;">autem</expan> <lb/><expan abbr="er&utilde;t">erunt</expan> iuxta h & k minus <expan abbr="refrigerabun&ttilde;">re<lb/>frigerabuntur</expan> utri&longs;que, &longs;ed pau<lb/>lulum in reditibus propin <lb/>quis, & neque &longs;tantes, <expan abbr="neque&longs;ed&etilde;tes">neque <lb/>&longs;edentes</expan>, &longs;ed &longs;i altius attolla<lb/>tur h k. </s>

<s id="id002152">Rur&longs;us &longs;i b h k fue<lb/>rit grauior eodem, ut de<lb/>&longs;cendat tanto impetu, <expan abbr="quãto">quan<lb/>to</expan> a&longs;cendit attractum, ut <lb/>pote ex ligno tenui nucis, <lb/>tunc multo magis refrige<lb/>rabit, & procul, <expan abbr="nõ">non</expan> ob uim <lb/>ualidiorem, &longs;ed quoniam <lb/>celerius occur&longs;antes &longs;ibi <lb/>contrarijs motibus, ac <expan abbr="uehem&etilde;tibus">ue<lb/>hementibus</expan> fiet colli&longs;io par<lb/>tium aëris, & ideo in ambitum impelletur, & undique cubiculum <lb/>refrigerabit, quod non faciet maius longè flabellum lento motu <lb/>agitatum, aut ex materia leui. </s>

<s id="id002153">Idem multo magis contingeret, ubi <lb/>duo e&longs;&longs;ent flabella laquearibus appen&longs;a, quæ ad perpendiculum <lb/><expan abbr="a&etilde;rem">aerem</expan> mouerent, &longs;eu quod &longs;uperficies eo modo &longs;e haberent: & &longs;i <lb/>flabella rotunda e&longs;&longs;ent, tunc maiorem ambitum aëris occuparent, <lb/>& uelocius deficientibus angulis mouebuntur.</s>

<s id="id002153">Idem multo magis contingeret, ubi <lb/>duo e&longs;&longs;ent flabella laquearibus appen&longs;a, quæ ad perpendiculum <lb/><expan abbr="a&etilde;rem">aerrem</expan> mouerent, &longs;eu quod &longs;uperficies eo modo &longs;e haberent: & &longs;i <lb/>flabella rotunda e&longs;&longs;ent, tunc maiorem ambitum aëris occuparent, <lb/>& uelocius deficientibus angulis mouebuntur.</s>

<s id="id002154">Propo&longs;itio cente&longs;ima uige&longs;ima &longs;ecunda.</s>

Line 7884

Line 7921

<s id="id002634">uelut 10 ex 25, & 25 & 49 & 1, <lb/><figure id="id.015.01.168.1.jpg" xlink:href="015/01/168/1.jpg"/><lb/>& <expan abbr="&longs;uma&ttilde;">&longs;umatur</expan> a b numerus quad. </s>

<s id="id002635">diui&longs;us in <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan>, ita quae c <lb/>d &longs;it portio minor eiu&longs;modi, ut adiecta illi <expan abbr="æ&qtilde;li">æquali</expan> c d gnomo <lb/>cir<expan abbr="c&utilde;&longs;criptus">cun&longs;criptus</expan> c k l <expan abbr="c&utilde;">cum</expan> <expan abbr="f&qtilde;drato">fquadrato</expan>, &longs;it <expan abbr="&etail;&qtilde;lis">&etail;qualis</expan> a b <expan abbr="&qtilde;drato">quadrato</expan>, detractis <lb/><expan abbr="igi&ttilde;">igitur</expan> c e & e d, <expan abbr="æ&qtilde;libus">æqualibus</expan> erunt duo <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> c k l <expan abbr="c&utilde;f">cunf</expan> qua<lb/>drato &etail;qualia duob. </s>

<s id="id002636"><expan abbr="&longs;upplem&etilde;tis">&longs;upplementis</expan> a b <expan abbr="c&utilde;">cum</expan> <expan abbr="&qtilde;drato">quadrato</expan> h g. </s>

<s id="id002637">Maio<lb/>ra <expan abbr="a&utilde;t">aut</expan> <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> <expan abbr="exced&utilde;t">excedunt</expan> minora in duplo quad. </s>

<s id="id002637">Maio<lb/>ra <expan abbr="a&utilde;t">aunt</expan> <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> <expan abbr="exced&utilde;t">excedunt</expan> minora in duplo quad. </s>

<s id="id002638">c d <expan abbr="igi&ttilde;">igitur</expan> detractis <lb/>minoribus &longs;upplementis <expan abbr="cõmunibus">communibus</expan>, erit <expan abbr="dupl&utilde;">duplum</expan> quad. </s>

<s id="id002639">c d <expan abbr="c&utilde;">cum</expan> f qua<lb/>drato &etail;qualia h g <expan abbr="&qtilde;drato">quadrato</expan>. </s>

<s id="id002640">Ergo propo&longs;ito numero, putà 3 ducam in &longs;e <lb/>fit 9, <expan abbr="ducã">ducam</expan> 2 <expan abbr="minor&etilde;">minorem</expan> in &longs;e fit 4, duplicabo fit 8, detraho ex 9, <expan abbr="relinqui&ttilde;">relinquitur</expan> <lb/>1 numerus <expan abbr="&qtilde;dratus">quadratus</expan>, <expan abbr="igi&ttilde;">igitur</expan> <expan abbr="dicã">dicam</expan> &qring;d 3 <expan abbr="c&utilde;">cum</expan> duplo 2, & erit <expan abbr="tot&utilde;">totum</expan> 7, e&longs;t unus <lb/>numerus, alter <02> 1. 1. 1, & <expan abbr="hor&utilde;">horum</expan> <expan abbr="&qtilde;d">quad</expan>. </s>

Line 7944

Line 7981

<s id="id002689">numeris <expan abbr="&qtilde;d">quad</expan>. </s>

<s id="id002690">nam &longs;i e&longs;&longs;et impar, <expan abbr="nõ">non</expan> po&longs;&longs;et prodire numerus integer, & <lb/><expan abbr="c&utilde;">cum</expan> prouenerit numerus <expan abbr="&qtilde;d">quad</expan>. </s>

<s id="id002691">ille erit <expan abbr="qu&etilde;">quem</expan> qu&etail;rimus, <expan abbr="nã">nam</expan> diui&longs;o 6 per to<lb/>tum <expan abbr="ill&utilde;">illum</expan> numerum, inde &qring;d prouenit multiplicato per numeros <expan abbr="&qtilde;d">quad</expan>, <lb/><expan abbr="cõponentes">componentes</expan> illum <expan abbr="numer&utilde;">numerum</expan> productum, <expan abbr="producun&ttilde;">producuntur</expan> partes 6, quæ <expan abbr="er&utilde;t">erunt</expan> <lb/>numeri <expan abbr="&qtilde;d">quad</expan>. </s>

<s id="id002692">quia denominator utriu&longs;que partis ex &longs;uppo&longs;ito e&longs;t nume <lb/>rus <expan abbr="&qtilde;dratus">quadratus</expan>, qui multiplicatus e&longs;t per 6, & numeratores &longs;unt nume <lb/>ri <expan abbr="&qtilde;drati">quadrati</expan>, qui <expan abbr="cõponebant">componebant</expan> <expan abbr="numer&utilde;">numerum</expan> <expan abbr="product&utilde;">productum</expan>, et tales partes <expan abbr="&etail;quan&ttilde;">&etail;quantur</expan> <lb/>6, quia numerus productus <expan abbr="componi&ttilde;">componitur</expan> ex numeratoribus, & <expan abbr="produci&ttilde;">produ<lb/>citur</expan> tale <expan abbr="cõpo&longs;itum">compo&longs;itum</expan> ex 6 in <expan abbr="denominator&etilde;">denominatorem</expan>, & hic e&longs;t diui&longs;us per deno <lb/><expan abbr="minator&etilde;">minatorem</expan>, ergo prouenit 6, &longs;i <expan abbr="e&mtilde;">eum</expan> multiplicato 3 in 4 fit 12, diui&longs;o 12 per <lb/>4, exit nece&longs;&longs;ario idem 3. Pro colligendo ergo numeros omnes, qui <lb/><expan abbr="cõponuntur">componuntur</expan> ex <expan abbr="&qtilde;dratis">quadratis</expan>, propones tibi &longs;eriem <expan abbr="&qtilde;d">quad</expan>. </s>

<s id="id002692">quia denominator utriu&longs;que partis ex &longs;uppo&longs;ito e&longs;t nume <lb/>rus <expan abbr="&qtilde;dratus">quadratus</expan>, qui multiplicatus e&longs;t per 6, & numeratores &longs;unt nume <lb/>ri <expan abbr="&qtilde;drati">quadrati</expan>, qui <expan abbr="cõponebant">componebant</expan> <expan abbr="numer&utilde;">numerum</expan> <expan abbr="product&utilde;">productum</expan>, et tales partes <expan abbr="&etail;quan&ttilde;">&etail;quantur</expan> <lb/>6, quia numerus productus <expan abbr="componi&ttilde;">componitur</expan> ex numeratoribus, & <expan abbr="produci&ttilde;">produ<lb/>citur</expan> tale <expan abbr="cõpo&longs;itum">compo&longs;itum</expan> ex 6 in <expan abbr="denominator&etilde;">denominatorem</expan>, & hic e&longs;t diui&longs;us per deno <lb/><expan abbr="minator&etilde;">minatorem</expan>, ergo prouenit 6, &longs;i <expan abbr="e&mtilde;">emm</expan> multiplicato 3 in 4 fit 12, diui&longs;o 12 per <lb/>4, exit nece&longs;&longs;ario idem 3. Pro colligendo ergo numeros omnes, qui <lb/><expan abbr="cõponuntur">componuntur</expan> ex <expan abbr="&qtilde;dratis">quadratis</expan>, propones tibi &longs;eriem <expan abbr="&qtilde;d">quad</expan>. </s>

<s id="id002693"><expan abbr="omni&utilde;">omnium</expan>, & inde iun<lb/>ges, & diuides per 6, & <expan abbr="c&utilde;">cum</expan> prodierit <expan abbr="&qtilde;dratus">quadratus</expan>, <expan abbr="inueni&ttilde;">inuenitur</expan> denominator, <lb/>& numeri <expan abbr="cõponentes">componentes</expan> ip&longs;um erunt numeratores, et &longs;uppo&longs;iti deno <lb/>minatoribus <expan abbr="cõ&longs;tituent">con&longs;tituent</expan> partes. </s>

<s id="id002694">Vt uerò cogno&longs;cas, ex quibus po&longs;<lb/>&longs;it componi primum ex imparibus, non oportet a&longs;&longs;umere ni&longs;i 135, <lb/>quia 7 diui&longs;um per 6 relinquit 1, & 9 diui&longs;um per 6, relinquit 3, & 35 <lb/>diui&longs;um per 6 relinquit 5. ergo non pote&longs;t componi numerus im<lb/>par, qui diuidatur per 6, ut &longs;uper&longs;it impar alius quàm 1. 3. 5. &longs;ed 1 & 3 <lb/>& 5, & 5 componunt 4 & 1, & 1 & 3 & 5 componunt 2, &longs;cilicet abie<lb/>cto 6, ergo tales numeri <expan abbr="&qtilde;drati">quadrati</expan> &longs;i &longs;int impares, uel ambo terminan<lb/>tur in 3, ut 9 & 81, qui faciunt 90, uel in 1 & 5, &longs;ed nullus numerus <lb/>quadratus diui&longs;us per 6 terminatur in 5, quia 1 ductum in &longs;e produ<lb/>cit 1, & 3 pro ducit 3, & 5 pro ducit 1, ut 5 in 5 facit 25, & 11 in 11 produ

Line 7953

Line 7990

<s id="id002695"><arrow.to.target n="marg520"/><lb/>4, &longs;ed iunctis 4 & 3, & abiecto 6 &longs;upere&longs;t 1, ergo 5 in 5 <expan abbr="duct&utilde;">ductum</expan>, & diui<lb/>&longs;o producto relinquitur 1. Et &longs;imiliter capio 17, et <expan abbr="componi&ttilde;">componitur</expan> ex 12 & <lb/>5 quadratum, ergo 17 componitur ex quadrato 12, in quo nihil &longs;u<lb/>pere&longs;t, & duplo 5 in 12, in quo <expan abbr="etiã">etiam</expan> nihil &longs;upere&longs;t, &longs;i diuidatur per 6: <lb/>& ex quadrato 5, in quo &longs;upere&longs;t 1, ergo in nullo numero <expan abbr="cõpo&longs;ito">compo&longs;ito</expan> <lb/>ex 5 & 6, uel compo&longs;itis ex 6, poterit produci numerus, qui diui&longs;us <lb/>per 6 relinquat 5, igitur neque talis numerus potérit <expan abbr="cõponi">componi</expan> ex duo<lb/>bus quadratis, in quib. </s>

<s id="id002696">&longs;uper&longs;it 5 & 1, quia nullus e&longs;t, in quo &longs;uper<lb/>&longs;it 5 facta diui&longs;ione per 6. Ex quo colligitur una regula: quod &longs;i quis <lb/>dicat multiplicaui 27 in &longs;e, et diui&longs;i per 13, uellem &longs;cire quid &longs;upere&longs;t, <lb/>dico quod &longs;ine multiplicatione et diui&longs;ione poteris hoc &longs;cire ex de<lb/>mon&longs;tratione dicta, diuide ergo 27 per 13, & relinquitur 1, duc in &longs;e <lb/>fit 1: dices ergo, quod &longs;upererit 1, & ita &longs;i ducerem 28 in &longs;e, & diuide<lb/>rem per 11, dico quod &longs;upererit 3, nam diui&longs;o 28 per 11, relinquitur <lb/>6, duc in 6 fit 36, diuide per 11, relinquitur 3, ut dictum e&longs;t, & tantum <lb/><expan abbr="relinqui&ttilde;">relinquitur</expan> ducto 28 in &longs;e & fit 784, & diui&longs;o per 11. Reuertendo ergo <lb/>ad propo&longs;itum, pater quod ex duobus tantum numeris imparibus <lb/>quadratis pote&longs;t conflari ille numerus, <expan abbr="quor&utilde;">quorum</expan> radices diui&longs;æ per 6 <lb/>relinquunt 3. Sed de paribus uel &longs;upere&longs;t 2 uel 4 uel nihil, &longs;ed <expan abbr="&qtilde;dratum">quadra<lb/>tum</expan> 2 e&longs;t 4, & <expan abbr="&qtilde;dratum">quadratum</expan> 4 diui&longs;um per 6 etiam relinquit 4, ergo neque <lb/>ex duobus numeris, in quibus &longs;uper&longs;int 2, neque in quibus &longs;uper&longs;int <lb/>4, neque in quibus &longs;uper&longs;int in uno 2, in altero 4 <expan abbr="poter&utilde;t">poterunt</expan> quadrata, in <lb/>quibus &longs;emper &longs;upererit 4, & iuncta faciunt 8, in quod &longs;upere&longs;t 2, <expan abbr="cõ">con</expan>fla<lb/>re <expan abbr="numer&utilde;">numerum</expan> <expan abbr="dict&utilde;">dictum</expan> &longs;eu <expan abbr="quæ&longs;it&utilde;">quæ&longs;itum</expan>, qui po&longs;sit diuidi per 6: neque ex <expan abbr="&qtilde;d">quad</expan>. </s>

<s id="id002696">&longs;uper&longs;it 5 & 1, quia nullus e&longs;t, in quo &longs;uper<lb/>&longs;it 5 facta diui&longs;ione per 6. Ex quo colligitur una regula: quod &longs;i quis <lb/>dicat multiplicaui 27 in &longs;e, et diui&longs;i per 13, uellem &longs;cire quid &longs;upere&longs;t, <lb/>dico quod &longs;ine multiplicatione et diui&longs;ione poteris hoc &longs;cire ex de<lb/>mon&longs;tratione dicta, diuide ergo 27 per 13, & relinquitur 1, duc in &longs;e <lb/>fit 1: dices ergo, quod &longs;upererit 1, & ita &longs;i ducerem 28 in &longs;e, & diuide<lb/>rem per 11, dico quod &longs;upererit 3, nam diui&longs;o 28 per 11, relinquitur <lb/>6, duc in 6 fit 36, diuide per 11, relinquitur 3, ut dictum e&longs;t, & tantum <lb/><expan abbr="relinqui&ttilde;">relinquitur</expan> ducto 28 in &longs;e & fit 784, & diui&longs;o per 11. Reuertendo ergo <lb/>ad propo&longs;itum, pater quod ex duobus tantum numeris imparibus <lb/>quadratis pote&longs;t conflari ille numerus, <expan abbr="quor&utilde;">quorum</expan> radices diui&longs;æ per 6 <lb/>relinquunt 3. Sed de paribus uel &longs;upere&longs;t 2 uel 4 uel nihil, &longs;ed <expan abbr="&qtilde;dratum">quadra<lb/>tum</expan> 2 e&longs;t 4, & <expan abbr="&qtilde;dratum">quadratum</expan> 4 diui&longs;um per 6 etiam relinquit 4, ergo neque <lb/>ex duobus numeris, in quibus &longs;uper&longs;int 2, neque in quibus &longs;uper&longs;int <lb/>4, neque in quibus &longs;uper&longs;int in uno 2, in altero 4 <expan abbr="poter&utilde;t">poterunt</expan> quadrata, in <lb/>quibus &longs;emper &longs;upererit 4, & iuncta faciunt 8, in quod ̊&longs;upere&longs;t 2, <expan abbr="cõ">con</expan>fla<lb/>re <expan abbr="numer&utilde;">numerum</expan> <expan abbr="dict&utilde;">dictum</expan> &longs;eu <expan abbr="quæ&longs;it&utilde;">quæ&longs;itum</expan>, qui po&longs;sit diuidi per 6: neque ex <expan abbr="&qtilde;d">quad</expan>. </s>

<s id="id002697"><expan abbr="duor&utilde;">duo<lb/>rum</expan> <expan abbr="num&etilde;ror&utilde;">numerorum</expan>, in <expan abbr="quor&utilde;">quorum</expan> altero nihil &longs;uper&longs;it in reliquo &longs;uper&longs;it 2 uel <lb/>4, quia in aggregato <expan abbr="&qtilde;drator&utilde;">quadratorum</expan> &longs;emper &longs;upererit 4. Ergo relinqui<lb/>tur quod ille numerus componetur ex duobus quadratis, uel impa<lb/>ribus, quorum latera diui&longs;a per 6 relinquunt 3, uel ex duobus pari<lb/>bus, quorum latera diui&longs;a per 6 nihil relinquant. </s>

<s id="id002698">Oportet igitur <lb/>inuenire duos tales numeros quadratos numerorum imparium, in <lb/>quibus &longs;uper&longs;it 3, &longs;i diuidantur per 6, aut parium in quibus nihil &longs;u<lb/>per&longs;it, quorum aggregato diui&longs;o per 6 prodeat numerus <expan abbr="&qtilde;dratus'">quadratus'</expan>.</s>

Line 8198

Line 8235

<s id="id002811">Ex hoc patet quod reliqua duo &longs;patia <lb/><arrow.to.target n="marg539"/><lb/>non po&longs;&longs;unt e&longs;&longs;e æqualia rectilineo. </s>

<s id="id002812">Nam <lb/>&longs;patium b a c demon&longs;tratum e&longs;t æquale e&longs;<lb/>&longs;e rectilineo, & b ad non e&longs;t æquale rectili<lb/>neo, <expan abbr="igi&ttilde;">igitur</expan> <expan abbr="&longs;pati&utilde;">&longs;patium</expan> c a d non pote&longs;t e&longs;&longs;e æquale <lb/>angulo rectilineo, nam &longs;i &longs;ic &longs;it b a c &etail;quale <lb/>f g h & c a d h g k, <expan abbr="igi&ttilde;">igitur</expan> <expan abbr="tot&utilde;">totum</expan>, b a d erit &etail;quale <lb/><arrow.to.target n="marg540"/><lb/>toti f g k quod e&longs;t contra <expan abbr="&longs;uppo&longs;it&utilde;">&longs;uppo&longs;itum</expan>, ideò neque<lb/>b a e quia b a c & d a e &longs;unt <expan abbr="æ&qtilde;lia">æqualia</expan> rectilineis <lb/>per &longs;e, & <expan abbr="etiã">etiam</expan> pariter accepta. </s>

<s id="id002813">Totum <expan abbr="a&utilde;t">aut</expan> <expan abbr="&longs;pati&utilde;">&longs;patium</expan> a e&longs;t <expan abbr="&etail;&qtilde;le">&etail;quale</expan> quatuor, re<lb/>ctis ergo <expan abbr="re&longs;idu&utilde;">re&longs;iduum</expan>, &longs;cilicet &longs;patia c a d & b a c pariter accepta &longs;unt <expan abbr="&etail;&qtilde;lia">&etail;qua<lb/>lia</expan> rectilineis &longs;patijs, &longs;ed <expan abbr="&longs;pati&utilde;">&longs;patium</expan> e a d non e&longs;t <expan abbr="æ&qtilde;le">æquale</expan> rectilineo, ergo per<lb/>demon&longs;trata hic, nec b a e, <expan abbr="nã">nam</expan> &longs;i &longs;it, &longs;it ergo b a e æquale h g k & quia <lb/>ambo &longs;patia b a e & c a d &longs;unt <expan abbr="æ&qtilde;lia">æqualia</expan> rectilineo ex demon&longs;tratis, &longs;it <lb/>ergo æqualia f g k, erit ergo ex communi animi &longs;ententia &longs;patium f <lb/>g h æquale &longs;patio c a d, quod e&longs;t contra primam partem corrolarij.</s>

<s id="id002813">Totum <expan abbr="a&utilde;t">aunt</expan> <expan abbr="&longs;pati&utilde;">&longs;patium</expan> a e&longs;t <expan abbr="&etail;&qtilde;le">&etail;quale</expan> quatuor, re<lb/>ctis ergo <expan abbr="re&longs;idu&utilde;">re&longs;iduum</expan>, &longs;cilicet &longs;patia c a d & b a c pariter accepta &longs;unt <expan abbr="&etail;&qtilde;lia">&etail;qua<lb/>lia</expan> rectilineis &longs;patijs, &longs;ed <expan abbr="&longs;pati&utilde;">&longs;patium</expan> e a d non e&longs;t <expan abbr="æ&qtilde;le">æquale</expan> rectilineo, ergo per<lb/>demon&longs;trata hic, nec b a e, <expan abbr="nã">nam</expan> &longs;i &longs;it, &longs;it ergo b a e æquale h g k & quia <lb/>ambo &longs;patia b a e & c a d &longs;unt <expan abbr="æ&qtilde;lia">æqualia</expan> rectilineo ex demon&longs;tratis, &longs;it <lb/>ergo æqualia f g k, erit ergo ex communi animi &longs;ententia &longs;patium f <lb/>g h æquale &longs;patio c a d, quod e&longs;t contra primam partem corrolarij.</s>

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<s id="id002970">Rur&longs;us proportio 128 ad 64 <expan abbr="cõponi&ttilde;">componitur</expan> ex proportionibus </s>

<s id="id002971">80 ad 64, <expan abbr="&qtilde;">quae</expan> <expan abbr="habe&ttilde;">habetur</expan> <lb/>pro ditono ut <expan abbr="dict&utilde;">dictum</expan> e&longs;t, & e&longs;t &longs;exquiquarta proportio. </s>

<s id="id002972">At 128 cum 80 e&longs;t in <lb/>proportione &longs;uperpartiente tres quintas, <expan abbr="&qtilde;">quae</expan> <expan abbr="iter&utilde;">iterum</expan> e&longs;t con&longs;ona. </s>

<s id="id002973">Regula <expan abbr="e&mtilde;">eum</expan> <lb/>e&longs;t quae ubi con&longs;onantia uo <expan abbr="c&utilde;">cum</expan> <expan abbr="diuida&ttilde;">diuidatur</expan> in duas partes, <expan abbr="quar&utilde;">quarum</expan> una &longs;it con&longs;o<lb/>nans, <expan abbr="reliquã">reliquam</expan> <expan abbr="etiã">etiam</expan> e&longs;&longs;e <expan abbr="con&longs;onant&etilde;">con&longs;onantem</expan>, at <expan abbr="nõ">non</expan> <expan abbr="cõuerti&ttilde;">conuertitur</expan>. </s>

<s id="id002973">Regula <expan abbr="e&mtilde;">emm</expan> <lb/>e&longs;t quae ubi con&longs;onantia uo <expan abbr="c&utilde;">cum</expan> <expan abbr="diuida&ttilde;">diuidatur</expan> in duas partes, <expan abbr="quar&utilde;">quarum</expan> una &longs;it con&longs;o<lb/>nans, <expan abbr="reliquã">reliquam</expan> <expan abbr="etiã">etiam</expan> e&longs;&longs;e <expan abbr="con&longs;onant&etilde;">con&longs;onantem</expan>, at <expan abbr="nõ">non</expan> <expan abbr="cõuerti&ttilde;">conuertitur</expan>. </s>

<s id="id002974">S&etail;pe. </s>

<s id="id002976">fit ut ex duabus <lb/></s>

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<s id="id002981">Cum ergo octo ad <expan abbr="&longs;ept&etilde;">&longs;eptem</expan> di&longs;&longs;ona &longs;it, <lb/>quippe nimis remota e&longs;t h&etail;c proportio à &longs;en<lb/>&longs;u humano: <expan abbr="quamobr&etilde;">quamobrem</expan> ex regula data, ne<lb/>que proportio <expan abbr="&longs;ept&etilde;">&longs;eptem</expan> ad &longs;ex. </s>

<s id="id002982">Sed dubitabis <lb/>meritò, quia <expan abbr="c&utilde;">cum</expan> diate&longs;&longs;aron diuidatur <expan abbr="bifariã">bifa<lb/>riam</expan>, in <expan abbr="diton&utilde;">ditonum</expan> & <expan abbr="&longs;emitoni&utilde;">&longs;emitonium</expan>, ac rur&longs;us in <expan abbr="&longs;emiditon&utilde;">&longs;e<lb/>miditonum</expan> & <expan abbr="ton&utilde;">tonum</expan>, quarum altera <expan abbr="cõ&longs;onans">con&longs;onans</expan> e&longs;t, reliqua <expan abbr="nõ">non</expan>. </s>

<s id="id002983"><expan abbr="Vide&ttilde;">Videtur</expan> ergo <lb/>infirmari regula illa, quae con&longs;onantia diui&longs;a &longs;i una pars <expan abbr="cõ&longs;onet">con&longs;onet</expan>, alia non <lb/>po&longs;sit e&longs;&longs;e di&longs;&longs;onans, <expan abbr="nã">nam</expan> con&longs;tat <expan abbr="coni&utilde;">conium</expan> & <expan abbr="&longs;emitoni&utilde;">&longs;emitonium</expan> tam per &longs;e quam in <lb/><expan abbr="cõpo&longs;itione">compo&longs;itione</expan> di&longs;&longs;onare: & <expan abbr="nõ">non</expan> <expan abbr="par&utilde;">parum</expan> &longs;ed acerbè. </s>

<s id="id002984"><expan abbr="Ver&utilde;">Verum</expan> re&longs;pondeo diate&longs;&longs;a<lb/>ron, ut dixi, numerari inter ambiguas coniugationes, quatenus <expan abbr="e&mtilde;">eum</expan> per <lb/>&longs;e e&longs;t, di&longs;&longs;onans e&longs;t: at que &longs;ic in <expan abbr="con&longs;onant&etilde;">con&longs;onantem</expan> & di&longs;&longs;onantem diuidi pote&longs;t: <lb/>quatenus <expan abbr="a&utilde;t">aut</expan> pars e&longs;t diapa&longs;on <expan abbr="cõ&longs;onans">con&longs;onans</expan> in acutis: quan <08> <expan abbr="etiã">etiam</expan> adiecta <lb/>ditono aut &longs;emiditono &longs;uprà efficiat <expan abbr="&longs;extã">&longs;extam</expan> maiorem aut <expan abbr="minor&etilde;">minorem</expan> parum <lb/>benè &longs;onantes. </s>

<s id="id002984"><expan abbr="Ver&utilde;">Verum</expan> re&longs;pondeo diate&longs;&longs;a<lb/>ron, ut dixi, numerari inter ambiguas coniugationes, quatenus <expan abbr="e&mtilde;">emm</expan> per <lb/>&longs;e e&longs;t, di&longs;&longs;onans e&longs;t: at que &longs;ic in <expan abbr="con&longs;onant&etilde;">con&longs;onantem</expan> & di&longs;&longs;onantem diuidi pote&longs;t: <lb/>quatenus <expan abbr="a&utilde;t">aut</expan> pars e&longs;t diapa&longs;on <expan abbr="cõ&longs;onans">con&longs;onans</expan> in acutis: quan <08> <expan abbr="etiã">etiam</expan> adiecta <lb/>ditono aut &longs;emiditono &longs;uprà efficiat <expan abbr="&longs;extã">&longs;extam</expan> maiorem aut <expan abbr="minor&etilde;">minorem</expan> parum <lb/>benè &longs;onantes. </s>

<s id="id002985">At quintupla proportio ut ab initio propo&longs;itum e&longs;t, <expan abbr="cõ&longs;tat">con&longs;tat</expan> <lb/>bis diapa&longs;on, & &longs;exquiquarta, ut planè <expan abbr="manife&longs;t&utilde;">manife&longs;tum</expan> e&longs;t: &longs;exquiquarta <expan abbr="a&utilde;t">aut</expan>

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</row>

</table>

<s id="id002997">E&longs;t & alia diui&longs;io toni in &longs;emitonia, <expan abbr="&qtilde;">quae</expan> e&longs;t uaria <expan abbr="pon&etilde;do">ponendo</expan> <expan abbr="ton&utilde;">tonum</expan> inter 18 <lb/>& 16, media uox e&longs;t 17 &longs;emitonium maius inter 17 & 16, &longs;ed minus inter <lb/>18 & 17, <expan abbr="quor&utilde;">quorum</expan> differentia e&longs;t 1/288. Hic &longs;ubit admiratio quomodo <expan abbr="&longs;emitoni&utilde;">&longs;emi<lb/>tonium</expan> minus <expan abbr="apte&ttilde;">aptetur</expan> tam gratè in &longs;ymphonijs, maius <expan abbr="a&utilde;t">aut</expan> <expan abbr="nequaquã">nequaquam</expan>. </s>

<s id="id002997">E&longs;t & alia diui&longs;io toni in &longs;emitonia, <expan abbr="&qtilde;">quae</expan> e&longs;t uaria <expan abbr="pon&etilde;do">ponendo</expan> <expan abbr="ton&utilde;">tonum</expan> inter 18 <lb/>& 16, media uox e&longs;t 17 &longs;emitonium maius inter 17 & 16, &longs;ed minus inter <lb/>18 & 17, <expan abbr="quor&utilde;">quorum</expan> differentia e&longs;t 1/288. Hic &longs;ubit admiratio quomodo <expan abbr="&longs;emitoni&utilde;">&longs;emi<lb/>tonium</expan> minus <expan abbr="apte&ttilde;">aptetur</expan> tam gratè in &longs;ymphonijs, maius <expan abbr="a&utilde;t">aunt</expan> <expan abbr="nequaquã">nequaquam</expan>. </s>

<s id="id002998">Ptole<lb/>m&etail;us hoc negaret, quia &longs;exquiquinta &longs;eu &longs;emiditonus <expan abbr="cõ&longs;tat">con&longs;tat</expan> tono inte<lb/>gro, qui e&longs;t inter 90 & 80, & &longs;emitonio <expan abbr="plu&longs;quã">plu&longs;quam</expan> maiore quod e&longs;t inter <lb/>96 & 90, & e&longs;t &longs;exquiquinta decima: <expan abbr="&qtilde;">quae</expan> maior e&longs;t tono maiore 1/255. Pro<lb/>pterea dicemus cau&longs;am e&longs;&longs;e quae po&longs;ito &longs;emiditono inter 81 & 96, id e&longs;t, <lb/>27 & 32 &longs;ublato tono, id e&longs;t, 234 & 216, remanebit 13 differentia 256 ad <lb/>243, &longs;eu qualis e&longs;t 96 ad 91 & 1/8 quæ e&longs;t ut 768 ad 729 et redit ad <expan abbr="id&etilde;">idem</expan>, &longs;cili<pb pagenum="171" xlink:href="015/01/190.jpg"/>cet, ut 256 ad 243, 13 autem e&longs;t paulo plus decimanona, ergo multo mi<lb/>nus &longs;emitonio minore. </s>

<s id="id002999">&longs;ecundum <expan abbr="m&etilde;tem">mentem</expan> ergo Ptolemæi, po&longs;ito tono <lb/>inter 135, & 120, & &longs;emitonio maiore inter 128 & 120 remanebit &longs;emito<lb/>nium minus fermè inter 19 & 18, id e&longs;t, 133 & 126, qu&etail; proportio differt <lb/>à 135 & 138. Si quis autem bene animaduertat, &longs;exquioctuage&longs;ima illa <lb/>adimitur, ex tono & additur &longs;emitonio minori, & hæc e&longs;t cau&longs;a quòd <lb/>&longs;emitonium maius Ptolemæi &longs;it concinnum, quia additur tonis imper<lb/>fectis. </s>

<s id="id003000">Dimidium autem &longs;emitonij minoris e&longs;t inter 36 & 35, & uocatur <lb/><expan abbr="cõma">comma</expan>: & e&longs;t minus & maius: maius e&longs;t inter 35 & 34, rur&longs;us <expan abbr="cõma">comma</expan> mi<lb/>nus diuiditur in duas die&longs;es, minorem, quæ e&longs;t inter 72 & 71, & maio<lb/>rem, qu&etail; e&longs;t inter 71 & 70, & ideò manet difficultas quomodo intenta <lb/>uoce per die&longs;im fiat melior con&longs;onantia? </s>

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<s id="id003026">Nam <lb/>duo in uniuer&longs;um ex ui&longs;u ad animi affectus mouendos habentur, tene<lb/>bræ ad tri&longs;titiam & metum, pictura regionum <expan abbr="am&oelig;nar&utilde;">am&oelig;narum</expan> ad iucundita<lb/>tem, &longs;ed <expan abbr="irã">iram</expan> quæ moueant picturæ alacritatemúe aut <expan abbr="cõmi&longs;erationem">commi&longs;erationem</expan>, <lb/>non habemus. </s>

<s id="id003027">Videtur ergo ob hæc &longs;onus ip&longs;e magis animæ intimus <lb/><08> ullum aliud &longs;en&longs;ile. </s>

<s id="id003028">Quod &longs;i odoratus e&longs;t in <expan abbr="app&etilde;dicibus">appendicibus</expan> cerebri, ui <lb/>&longs;us in pupilla oculi, gu&longs;tus in linguæ neruis, ueri &longs;imile e&longs;t magis inti<lb/>mum e&longs;&longs;e auditum, &longs;cilicet in cerebro ip&longs;o, atque ob id magis ab illo mo<lb/>ueri animam. </s>

<s id="id003029">Neque <expan abbr="e&mtilde;">eum</expan> in <expan abbr="a&etilde;re">aere</expan> concepto à concauitatibus auris, qui no<lb/>&longs;tri pars non e&longs;t: neque à tympano, cùm &longs;uperflua fui&longs;&longs;et cauitas interior <lb/>omnis: neque enim inter pupillam & cerebrum pars ulla cernitur ad ui<lb/>&longs;um adiuuandum idonea: &longs;ed &longs;olus &longs;ufficit con&longs;en&longs;us pupill&etail; cum cere<lb/>bro: nam ad nos per &longs;piritus differtur imago, non <expan abbr="e&mtilde;">eum</expan> ui&longs;us e&longs;&longs;et unus, <lb/>nec in uno tempore fieret, &longs;ed ueluti è <expan abbr="&longs;ec&utilde;do">&longs;ecundo</expan> &longs;peculo & decimo &longs;imul, <lb/>& eodem tempore reflectitur imago, ut à primo ita &longs;en&longs;us ui&longs;us ex pu<lb/>pilla in cerebro & in corde & anima &longs;imul relucet. </s>

<s id="id003029">Neque <expan abbr="e&mtilde;">emm</expan> in <expan abbr="a&etilde;re">aere</expan> concepto à concauitatibus auris, qui no<lb/>&longs;tri pars non e&longs;t: neque à tympano, cùm &longs;uperflua fui&longs;&longs;et cauitas interior <lb/>omnis: neque enim inter pupillam & cerebrum pars ulla cernitur ad ui<lb/>&longs;um adiuuandum idonea: &longs;ed &longs;olus &longs;ufficit con&longs;en&longs;us pupill&etail; cum cere<lb/>bro: nam ad nos per &longs;piritus differtur imago, non <expan abbr="e&mtilde;">emm</expan> ui&longs;us e&longs;&longs;et unus, <lb/>nec in uno tempore fieret, &longs;ed ueluti è <expan abbr="&longs;ec&utilde;do">&longs;ecundo</expan> &longs;peculo & decimo &longs;imul, <lb/>& eodem tempore reflectitur imago, ut à primo ita &longs;en&longs;us ui&longs;us ex pu<lb/>pilla in cerebro & in corde & anima &longs;imul relucet. </s>

<s id="id003030">At ergo non potuit <lb/>in tympano uel neruo den&longs;iore fieri auditus, &longs;ed in cerebro ip&longs;o, ob &qring;d <lb/>magis moueret affectus. </s>

<s id="id003031">Sed & magis incorporeus e&longs;t &longs;onus, ut qui <lb/>in&longs;trumentum proprium non afficiat, ni&longs;i cum immoderatus fuerit, at <lb/>omnis color, omnis lux oculum afficit, ac, ut ita dicam, tingit, neque &longs;uc<lb/>ce&longs;siones illas ob id adeò minutas oculus percipere pote&longs;t ut auris, <lb/>&longs;ed coinquinatur, ut ita dicam, priorum obiectorum reliquijs atque ima<lb/>ginibus. </s>

<s id="id003032">Vt in uniuer&longs;um con&longs;tet puriorem e&longs;&longs;e auditus &longs;en&longs;um etiam <lb/>animæ no&longs;træ propiorem quàm ui&longs;um.</s>

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<s id="id003050">Sed metus <expan abbr="c&utilde;">cum</expan> cau&longs;a, affectus propriè <expan abbr="nõ">non</expan> e&longs;t, &longs;ed potius &longs;cientia <expan abbr="quædã">quædam</expan>. <lb/></s>

<s id="id003051">Proprium enim perturbationum e&longs;t excedere rationem: at metus mor<lb/>tis, propri&etail; aut de filio, non e&longs;t à ratione alienús, nec excedit metas, modò <lb/>inanis non &longs;it aut fal&longs;us, ob hoc metum excludemus ab hoc negocio: <lb/>tum maximè ob id quod nulla mu&longs;ica e&longs;t quæ <expan abbr="met&utilde;">metum</expan> excitet cùm ea, <expan abbr="nõ">non</expan> <lb/>opus &longs;it in eo, qui &longs;it cum ratione coniunctus. </s>

<s id="id003052">Indicio e&longs;t quae potius <expan abbr="ill&utilde;">illum</expan> <lb/>excudit abrupta mu&longs;ica, &longs;icut & omnia alia quæ perturbant rationem, <lb/>ueluti <expan abbr="&longs;olan&utilde;">&longs;olanum</expan> & madrangora atque cicuta. </s>

<s id="id003053">Amorem igitur & odium <expan abbr="nõ">non</expan> <lb/>excitat mu&longs;ica, quia amor & odium alicuius &longs;unt amor & odium, mu&longs;i <lb/>ca <expan abbr="a&utilde;t">aut</expan> generales &longs;olum mouet animi affectus. </s>

<s id="id003053">Amorem igitur & odium <expan abbr="nõ">non</expan> <lb/>excitat mu&longs;ica, quia amor & odium alicuius &longs;unt amor & odium, mu&longs;i <lb/>ca <expan abbr="a&utilde;t">aunt</expan> generales &longs;olum mouet animi affectus. </s>

<s id="id003054">Et commi&longs;eratio, licet &longs;it <lb/>Didonis aut Phillidis, tamen e&longs;t generaliter mi&longs;erentis. </s>

<s id="id003055">Qu&etail;ramus er<lb/>go rur&longs;us qui &longs;int affectus generales animi. </s>

<s id="id003056">Et &longs;anè <expan abbr="uiden&ttilde;">uidentur</expan> e&longs;&longs;e lætitia <lb/>atque tri&longs;titia: impetus & remi&longs;sio: &longs;&etail;uitia ac mi&longs;ericordia & audacia. </s>

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<s id="id003084">nunc uo<expan abbr="can&ttilde;">cantur</expan> <expan abbr="in&longs;trum&etilde;ta">in&longs;trumenta</expan> quib. </s>

<s id="id003085">caruerunt anti<lb/>qui) <expan abbr="nõ">non</expan> alia e&longs;t ratio, quàm <expan abbr="&qtilde;">quae</expan> dicta e&longs;t <expan abbr="con&longs;tituendar&utilde;">con&longs;tituendarum</expan> con&longs;onantiarum <lb/>in ditonis & &longs;emiditonis &longs;extaque utraque. </s>

<s id="id003086">Vt <expan abbr="e&mtilde;">eum</expan> quatuor con&longs;onantiæ <lb/>&longs;uauiores <expan abbr="efficeren&ttilde;">efficerentur</expan>, nece&longs;&longs;e fuit <expan abbr="unã">unam</expan>, &longs;cilicet <expan abbr="diapent&etilde;">diapentem</expan> uariari. </s>

<s id="id003086">Vt <expan abbr="e&mtilde;">emm</expan> quatuor con&longs;onantiæ <lb/>&longs;uauiores <expan abbr="efficeren&ttilde;">efficerentur</expan>, nece&longs;&longs;e fuit <expan abbr="unã">unam</expan>, &longs;cilicet <expan abbr="diapent&etilde;">diapentem</expan> uariari. </s>

<s id="id003087">Exempli <lb/>gratia, &longs;int fides expo&longs;it&etail; octo, & ut <expan abbr="con&longs;titua&ttilde;">con&longs;tituatur</expan> proportio h ad c, ut 128 <lb/><figure id="id.015.01.195.1.jpg" xlink:href="015/01/195/1.jpg"/><arrow.to.target n="table22"/><lb/>ad 80, id e&longs;t ut 8 ad 5, c facta e&longs;t remi&longs;sior octoge&longs;ima, quare <expan abbr="c&utilde;">cum</expan> <lb/>81 diapente habeat ad 121 <expan abbr="c&utilde;">cum</expan> dimidio, erit ad 80 maior 1 1/2, id e&longs;t <lb/>octuage&longs;ima parte 120, quare intentior diapente. </s>

<s id="id003088">At in diapa&longs;o <lb/>omnia ad <expan abbr="id&etilde;">idem</expan> redeunt: <expan abbr="hor&utilde;">horum</expan> etiam cau&longs;a &longs;emitonia nigra illa ad<lb/>dita &longs;unt. </s>

<s id="id003089">Sed h&etail;c tractatio proprium <expan abbr="loc&utilde;">locum</expan> exigeret, &longs;ecus e&longs;&longs;et ni<lb/>mis curio&longs;i illa huc traducere. </s>

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<s id="id003095">Melius feci&longs;&longs;et Ptolem&etail;us, &longs;i <expan abbr="hãc">hanc</expan> proportionem ad &longs;apores & odores <lb/>et picturas, <expan abbr="quemadmod&utilde;">quemadmodum</expan> inuenimus nos, applica&longs;&longs;et, uel ut Vitruuius <lb/>ad machinas, poterat <expan abbr="e&mtilde;">eum</expan> hoc &longs;cire, cum Vitruuius plu&longs;<08> centum quin<lb/>quaginta annis <expan abbr="Ptolem&etail;&utilde;">Ptolem&etail;um</expan> antece&longs;&longs;erit. </s>

<s id="id003095">Melius feci&longs;&longs;et Ptolem&etail;us, &longs;i <expan abbr="hãc">hanc</expan> proportionem ad &longs;apores & odores <lb/>et picturas, <expan abbr="quemadmod&utilde;">quemadmodum</expan> inuenimus nos, applica&longs;&longs;et, uel ut Vitruuius <lb/>ad machinas, poterat <expan abbr="e&mtilde;">emm</expan> hoc &longs;cire, cum Vitruuius plu&longs;<08> centum quin<lb/>quaginta annis <expan abbr="Ptolem&etail;&utilde;">Ptolem&etail;um</expan> antece&longs;&longs;erit. </s>

<s id="id003096">Et quan<08> Latinè &longs;crip&longs;erit, non <lb/>tam turpè erat latina legi&longs;&longs;e, aut <expan abbr="cõuer&longs;a">conuer&longs;a</expan> ab alio quopiam intellexi&longs;&longs;e, <08><lb/>ne&longs;ciui&longs;&longs;e nece&longs;&longs;aria pulchraque inuenta aliorum clarorum uirorum, & <lb/>quod deterius erat, <expan abbr="rer&utilde;">rerum</expan> memorabilium loco fabulas &longs;ubtexui&longs;&longs;e. </s>

<s id="id003097">Ergo <lb/>ut ad rem ueniam: mu&longs;ica proportio bifariam <expan abbr="inueni&ttilde;">inuenitur</expan> in &longs;aporibus: &longs;im<lb/>pliciter, & ex comparatione, & &longs;impliciter quidem &longs;umma &longs;uauitas ad <lb/>diapa&longs;on refertur: e&longs;t enim &longs;uaui&longs;simus concen&longs;us in &longs;aporibus, ergo <lb/>dulce ei <expan abbr="re&longs;põdet">re&longs;pondet</expan>, ut &longs;implex, quid enim &longs;uauius e&longs;&longs;e pote&longs;t in utro que ge<lb/>nere. </s>

<s id="id003098">At pinguis, qualis in carnibus & ouis benè pr&etail;paratis ad <expan abbr="diap&etilde;te">diapente</expan> <lb/>refertur, e&longs;t enim & ip&longs;e &longs;uaui&longs;simus po&longs;t dulce, at que in &longs;uo genere perfe<lb/>ctus, diate&longs;&longs;aron uerò optimè &longs;al&longs;o <expan abbr="cõuenit">conuenit</expan>. </s>

<s id="id003099">Hic enim per &longs;e improbus <lb/>e&longs;t & in&longs;uauis, &longs;icut etiam &longs;apor &longs;al&longs;us e&longs;t, diate&longs;&longs;aron <expan abbr="a&utilde;t">aut</expan> cum diapente <lb/>perficit diapa&longs;on, & cum diapa&longs;o inutile e&longs;t, et di&longs;cordat, ita &longs;apor &longs;al&longs;us <lb/>cum pingui &longs;ummam delectationem affert: cum dulci adeò parum con<lb/>gruit, ut melius &longs;ocietur <expan abbr="c&utilde;">cum</expan> amaro, uelut in oliuis benè &longs;al&longs;is. </s>

<s id="id003099">Hic enim per &longs;e improbus <lb/>e&longs;t & in&longs;uauis, &longs;icut etiam &longs;apor &longs;al&longs;us e&longs;t, diate&longs;&longs;aron <expan abbr="a&utilde;t">aunt</expan> cum diapente <lb/>perficit diapa&longs;on, & cum diapa&longs;o inutile e&longs;t, et di&longs;cordat, ita &longs;apor &longs;al&longs;us <lb/>cum pingui &longs;ummam delectationem affert: cum dulci adeò parum con<lb/>gruit, ut melius &longs;ocietur <expan abbr="c&utilde;">cum</expan> amaro, uelut in oliuis benè &longs;al&longs;is. </s>

<s id="id003100">Ergo &longs;al<lb/>&longs;us &longs;apor cum diate&longs;&longs;aro ad <expan abbr="ungu&etilde;">unguem</expan> congruit rur&longs;us &longs;emiditonus <expan abbr="c&utilde;">cum</expan> in&longs;i<lb/>pido, & a&longs;tringens cum ditono conueniunt ad unguem, nam uterque <expan abbr="nõ">non</expan> <lb/>illepidus, & cum dulci conuenit, ita &longs;emiditonus & ditonus cum diapa<pb pagenum="171 [=177]" xlink:href="015/01/196.jpg"/>&longs;o conueniunt, uterque etiam horum &longs;aporum parum mouet &longs;en<lb/>&longs;um, & inter &longs;e &longs;unt qua&longs;i &longs;imiles quod ditono accidit & &longs;emidito<lb/>no, &longs;ed & neuter horum cum pingui conuenit, neque ditonus aut &longs;e<lb/>miditonus cum diapente congruit, di&longs;cordat enim h&etail;c compo&longs;itio <lb/>non parum. </s>

<s id="id003101">Rur&longs;us & in hoc &longs;imiles &longs;unt quod diate&longs;&longs;aron cum di<lb/>tono & &longs;emiditono plurimum conuenit, ita & in&longs;ipidum, & a&longs;trin<lb/>gens cum &longs;al&longs;o bellè <expan abbr="cõueniunt">conueniunt</expan>. </s>

<s id="id003102">Diate&longs;&longs;aron enim cum ditono &longs;ex<lb/>tam efficit maiorem, & cum &longs;emiditono minorem qu&etail; utrique con&longs;o<lb/>nant, non tamen plus &longs;uaues per &longs;e &longs;unt, quòd dulci & pingui care<lb/>ant, ut nec &longs;exta maior aut minor, &qring;d neque diapa&longs;on perficiant neque <lb/>diapente: Acris <expan abbr="aut&etilde;">autem</expan> &longs;apor &longs;exta maiori &longs;imilis e&longs;t, acidus minori: <lb/>mutuo conueniunt cum in&longs;ipido acris, & cum a&longs;tringente acidus, <lb/>quemadmodum & &longs;exta maior cum &longs;emiditono, & minor cum di<lb/>tono copulatur perficientes diapa&longs;on: &longs;ed minus &longs;uauem, quia ab<lb/>e&longs;t diapente ibi, quia abe&longs;t pingue: au&longs;terum uero cum acri mode<lb/>rato conuenit, propterea bene uterque cum in&longs;ipido iungitur, unde <lb/>illud Epigrammatici:</s>

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<s id="id003109">Manife&longs;tum e&longs;t igitur optimè conuenire hanc &longs;aporum diui<lb/>&longs;ionem cum mu&longs;ica proportione.</s>

<s id="id003110">Cumque &longs;apores ex &longs;eptem planetis pendent manife&longs;tè, Saturnus <lb/><expan abbr="e&mtilde;">eum</expan> habet a&longs;tringens, quoniam frigidus e&longs;t & &longs;iccus. </s>

<s id="id003110">Cumque &longs;apores ex &longs;eptem planetis pendent manife&longs;tè, Saturnus <lb/><expan abbr="e&mtilde;">emm</expan> habet a&longs;tringens, quoniam frigidus e&longs;t & &longs;iccus. </s>

<s id="id003111">Iupiter pingue <lb/><expan abbr="cõtraria">contraria</expan> ratione, & <expan abbr="quoniã">quoniam</expan> hic &longs;uauis e&longs;t, ille tri&longs;tis, acre & au&longs;terum <lb/><expan abbr="cõueniunt&longs;oli">conueniunt &longs;oli</expan>, apparetque in eis uis maxima ad <expan abbr="&longs;pirit&utilde;">&longs;piritum</expan> uitalem <expan abbr="cõfir">confir</expan><lb/>mandum, uiresque <expan abbr="o&etilde;s">eos</expan> adauget, uelut & Sol. </s>

<s id="id003111">Iupiter pingue <lb/><expan abbr="cõtraria">contraria</expan> ratione, & <expan abbr="quoniã">quoniam</expan> hic &longs;uauis e&longs;t, ille tri&longs;tis, acre & au&longs;terum <lb/><expan abbr="cõueniunt&longs;oli">conueniunt &longs;oli</expan>, apparetque in eis uis maxima ad <expan abbr="&longs;pirit&utilde;">&longs;piritum</expan> uitalem <expan abbr="cõfir">confir</expan> <lb/>mandum, uiresque <expan abbr="o&etilde;s">oens</expan> adauget, uelut & Sol. </s>

<s id="id003112">Venus habet dulce: de<lb/>mon&longs;tratione hoc non indiget. </s>

<s id="id003113">Mars &longs;al&longs;um & <expan abbr="c&utilde;">cum</expan> peruer&longs;è di&longs;po&longs;i<lb/>tus e&longs;t, <expan abbr="amar&utilde;">amarum</expan>. </s>

<s id="id003114">Luna in&longs;ipidum. </s>

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<s id="id003116">Huiu&longs;modi ergo ratione con&longs;iderata <lb/>Luna ad <expan abbr="&longs;emiditon&utilde;">&longs;emiditonum</expan> pertinebit Mercurius ad <expan abbr="&longs;extã">&longs;extam</expan> minorem, Sol <lb/>ad &longs;extam maiorem, Mars ad <expan abbr="tetrachord&utilde;">tetrachordum</expan>, Saturnus ad ditonum, <lb/>Iupiter ad diapente, Venus ad diapa&longs;on, unde plena illius dona uul<lb/>garis felicitatis opum honoris amoris & uoluptatis, po&longs;t quem e&longs;t <lb/>Iupiter, ut &longs;ine his duobus omnino nulla po&longs;sit e&longs;&longs;e felicitas.</s>

<s id="id003117">Sed & in circulo &longs;igniferi aliquam mu&longs;ica proportio habebit ra<lb/>tionem: diapa&longs;on <expan abbr="e&mtilde;">eum</expan> erit & totius ad dimidium, & be&longs;sis ad trien<lb/>tem, & dimidij ad quadrantem, & trientis ad <expan abbr="&longs;extant&etilde;">&longs;extantem</expan>, diapente <expan abbr="a&utilde;t">aut</expan> <lb/>totius circuli ad be&longs;&longs;em, & dodrantis ad <expan abbr="dimidi&utilde;">dimidium</expan>, & dimidij ad tri<lb/>entem, & <expan abbr="quadrãtis">quadrantis</expan> ad <expan abbr="&longs;extant&etilde;">&longs;extantem</expan>, diate&longs;&longs;aron <expan abbr="a&utilde;t">aut</expan> totius circuli ad do<lb/>drantem, & be&longs;sis ad <expan abbr="dimidi&utilde;">dimidium</expan>, & trientis ad <expan abbr="quadrãtem">quadrantem</expan>: itaque in hoc <lb/>&longs;olo <expan abbr="c&utilde;">cum</expan> Ptolem&etail;o concordamus, in reliquis duobus ne&longs;cio qua ra<lb/>tione Ptolem&etail;us omi&longs;erit unam <expan abbr="cõiugationem">coniugationem</expan>, nam <expan abbr="c&utilde;">cum</expan> e&longs;&longs;ent qua<lb/>tuor in diapa&longs;on & diapente, tres tantum numerauit. </s>

<s id="id003117">Sed & in circulo &longs;igniferi aliquam mu&longs;ica proportio habebit ra<lb/>tionem: diapa&longs;on <expan abbr="e&mtilde;">emm</expan> erit & totius ad dimidium, & be&longs;sis ad trien<lb/>tem, & dimidij ad quadrantem, & trientis ad <expan abbr="&longs;extant&etilde;">&longs;extantem</expan>, diapente <expan abbr="a&utilde;t">aut</expan> <lb/>totius circuli ad be&longs;&longs;em, & dodrantis ad <expan abbr="dimidi&utilde;">dimidium</expan>, & dimidij ad tri<lb/>entem, & <expan abbr="quadrãtis">quadrantis</expan> ad <expan abbr="&longs;extant&etilde;">&longs;extantem</expan>, diate&longs;&longs;aron <expan abbr="a&utilde;t">aunt</expan> totius circuli ad do<lb/>drantem, & be&longs;sis ad <expan abbr="dimidi&utilde;">dimidium</expan>, & trientis ad <expan abbr="quadrãtem">quadrantem</expan>: itaque in hoc <lb/>&longs;olo <expan abbr="c&utilde;">cum</expan> Ptolem&etail;o concordamus, in reliquis duobus ne&longs;cio qua ra<lb/>tione Ptolem&etail;us omi&longs;erit unam <expan abbr="cõiugationem">coniugationem</expan>, nam <expan abbr="c&utilde;">cum</expan> e&longs;&longs;ent qua<lb/>tuor in diapa&longs;on & diapente, tres tantum numerauit. </s>

<s id="id003118">Reliquas <expan abbr="a&utilde;t">aut</expan> <lb/>quatuor per integra &longs;igna numerare licebit, ad <expan abbr="ration&etilde;">rationem</expan>, tamen a&longs;pe<lb/>ctuum deducere non po&longs;&longs;umus, propterea efficaciam quandam ha<lb/>bent etiam &longs;ignorum mutationes, &longs;ed harmoniam non perficiunt, <lb/>nam & &longs;i &longs;umamus &longs;exquiquartam & &longs;exquiquintam, ut in his &longs;ex<lb/>quialteram, &longs;eu diapente con&longs;tituamus, aut tria aut &longs;ex &longs;igna acci<lb/>pere oportebit: utrunque fuerit, reliqua pars ad diate&longs;&longs;aron pertinere <lb/>minimè pote&longs;t: quamobrem conuenientius e&longs;&longs;et meo iudicio, ut to<lb/>tus circulus non ad diapa&longs;on, uelut Ptolemæus, referretur, &longs;ed po<lb/>tius ad diapa&longs;on diapente: ita enim con&longs;titutis quatuor, quinque, <lb/>&longs;ex, duodecimque numeris, con&longs;taret tota ratio harmonica, diui&longs;o e<lb/>tiam diapente in ditonum & &longs;emiditonum. </s>

<s id="id003118">Reliquas <expan abbr="a&utilde;t">aunt</expan> <lb/>quatuor per integra &longs;igna numerare licebit, ad <expan abbr="ration&etilde;">rationem</expan>, tamen a&longs;pe<lb/>ctuum deducere non po&longs;&longs;umus, propterea efficaciam quandam ha<lb/>bent etiam &longs;ignorum mutationes, &longs;ed harmoniam non perficiunt, <lb/>nam & &longs;i &longs;umamus &longs;exquiquartam & &longs;exquiquintam, ut in his &longs;ex<lb/>quialteram, &longs;eu diapente con&longs;tituamus, aut tria aut &longs;ex &longs;igna acci<lb/>pere oportebit: utrunque fuerit, reliqua pars ad diate&longs;&longs;aron pertinere <lb/>minimè pote&longs;t: quamobrem conuenientius e&longs;&longs;et meo iudicio, ut to<lb/>tus circulus non ad diapa&longs;on, uelut Ptolemæus, referretur, &longs;ed po<lb/>tius ad diapa&longs;on diapente: ita enim con&longs;titutis quatuor, quinque, <lb/>&longs;ex, duodecimque numeris, con&longs;taret tota ratio harmonica, diui&longs;o e<lb/>tiam diapente in ditonum & &longs;emiditonum. </s>

<s id="id003119">&longs;ed de hoc &longs;atis.</s>

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<pb pagenum="174 [=180]" xlink:href="015/01/199.jpg"/>to ruborem, ciliorum <expan abbr="cõtractionem">contractionem</expan>, tumorem faciei in ambulante <lb/>inclinationem quandam, flexionem cruris atque &longs;imilia. </s>

<s id="id003136">quintum e&longs;t <lb/>lux coloribus <expan abbr="exhib&etilde;da">exhibenda</expan>, &longs;ed de horum nullo propo&longs;itum e&longs;t hic lo<lb/>qui, quando quidem hæc u&longs;u magis & con&longs;ideratione, quàm ratio<lb/>ne con&longs;tent proportioneúe, nec &longs;int adeò admiranda ut neque &longs;im<lb/>plex magnitudo <expan abbr="quã&longs;exto">qua&longs;exto</expan> loco reponere po&longs;&longs;umus. </s>

<s id="id003136">quintum e&longs;t <lb/>lux coloribus <expan abbr="exhib&etilde;da">exhibenda</expan>, &longs;ed de horum nullo propo&longs;itum e&longs;t hic lo<lb/>qui, quando quidem hæc u&longs;u magis & con&longs;ideratione, quàm ratio<lb/>ne con&longs;tent proportioneúe, nec &longs;int adeò admiranda ut neque &longs;im<lb/>plex magnitudo <expan abbr="quã&longs;exto">quan&longs;exto</expan> loco reponere po&longs;&longs;umus. </s>

<s id="id003137">Tria ergo ui<lb/>dentur e&longs;&longs;e præcipua quorum nunc ratio habenda e&longs;&longs;et, ut &longs;int in <lb/>totum nouem, &longs;ed unum ex his relinquemus, tum quia alienum ab <lb/>hac con&longs;ideratione, tum quia alibi pertractatum atque etiam ab alijs, <lb/>neque adeò admiratione dignum &longs;cilicet magnitudo picturarum re<lb/>&longs;pondens magnitudini corporum iuxta &longs;itus differentiam, nam <lb/>qu&etail; altiores &longs;unt paulo latiores atque in &longs;uperiori magis parte quam <lb/>in inferiore, multò autem longiores e&longs;&longs;e oportet, &longs;ic & quæ à latere <lb/>erunt eadem ratione iuxta a&longs;pectus ingredientium rationem. </s>

<s id="id003138">Ve<lb/>rum hoc ut dixi omittamus, & de duplici miraculo in pictura lo<lb/>quamur, &longs;cilicet di&longs;tantia magna quam in parua tabella referimus, <lb/>et corporeitate quam in plano repr&etail;&longs;entamus. </s>

<s id="id003139">Horum autem duo<lb/>rum aliqua communia &longs;unt aliqua propria. </s>

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<s id="id003189">“De Va&longs;is uerò æneis theatri quod melius e&longs;t quàm ut eundem <lb/>authorem con&longs;ulamus, dicentem ua&longs;a &etail;rea pro ratione magnitudi<lb/>nis theatri ita fabricentur, ut cum <expan abbr="tang&utilde;tur">tanguntur</expan>, &longs;onitum facere po&longs;sint <lb/>inter &longs;e diate&longs;&longs;aron diapente, ex ordine addit diapa&longs;on, po&longs;tea inter <lb/>&longs;edes theatri con&longs;titutis cellis ratione mu&longs;ica ibi collocentur: ita uti <lb/>nullum parietem tangant circaque habeant locum <expan abbr="uacu&utilde;">uacuum</expan> et à &longs;ummo <lb/>capite &longs;patium, ponantque inuer&longs;a & habeant in parte qu&etail; &longs;pectat ad <lb/>&longs;cenam &longs;uppo&longs;itos cuneos ne minus alios &longs;emipede, contraque eas <lb/>cellas relinquantur apertur&etail; inferiorum graduum cubilibus lon<lb/>g&etail; pedes duos altæ &longs;emipedem. </s>

<s id="id003190">Et &longs;i non erit ampla magnitudine <lb/>theatrum, media altitudinis tran&longs;uer&longs;aregio de&longs;ignetur, & in ea tre<lb/>decim cellæ duodecim æqualibus.

interuallis di&longs;tantes <expan abbr="confornicen&ttilde;">confornicentur</expan> <lb/>uti ea echea quæ &longs;upra &longs;cripta &longs;unt, ad neten hyperboleon &longs;onan<lb/>tia in cellis quæ &longs;untin cornibus extremis utraque parte prima col<lb/>locentur, &longs;ecunda ab extremis diate&longs;&longs;aron ad <expan abbr="net&etilde;">neten</expan> diezeugmenon, <lb/>tertia diate&longs;&longs;aron ad neten parame&longs;on, quarta ad neten &longs;ynemme<lb/>non, quinta diate&longs;&longs;aron ad me&longs;en, &longs;exta diate&longs;&longs;aron ad hypaten me<lb/>&longs;en in medio unum diate&longs;&longs;aron ad hypaten hypaton. </s>

interuallis di&longs;tantes <expan abbr="confornicen&ttilde;">confornicentur</expan> <lb/>uti ea echea quæ &longs;upra &longs;cripta &longs;unt, ad neten hyperboleon &longs;onan<lb/>tia in cellis quæ &longs;untin cornibus extremis utraque parte prima col<lb/>locentur, &longs;ecunda ab extremis diate&longs;&longs;aron ad <expan abbr="net&etilde;">netem</expan> diezeugmenon, <lb/>tertia diate&longs;&longs;aron ad neten parame&longs;on, quarta ad neten &longs;ynemme<lb/>non, quinta diate&longs;&longs;aron ad me&longs;en, &longs;exta diate&longs;&longs;aron ad hypaten me<lb/>&longs;en in medio unum diate&longs;&longs;aron ad hypaten hypaton. </s>

<s id="id003191">Quæ sequun<lb/>tur & ad intelligentiam prædictorum melius ex Gulielmo Philan<lb/>dro emendata &longs;ic tran&longs;cribemus: Eas regiones in tredecim cellas <lb/>diuidit æqualibus interuallis: id e&longs;t, cellas paribus uici&longs;sim inter

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<s id="id003361"><arrow.to.target n="marg621"/><lb/>bis manife&longs;tum e&longs;t, quòd pars quæ re&longs;picit nos quantò remotior <lb/>fuerit à Sole, <expan abbr="tãto">tanto</expan> magis illuminatur. </s>

<s id="id003362">Manife&longs;tum e&longs;t etiam & expe<lb/>rimento & ratione, quòd illud quod magis lucet, & e&longs;t <expan abbr="illuminat&utilde;">illuminatum</expan> <lb/>à Sole in nocte, maius uidetur, &longs;icut etiam de facibus nocturnis. </s>

<s id="id003363">Et <lb/>rur&longs;us, quod &longs;ub &longs;tantia orbium circa loca quæ habentur pro polis <lb/>e&longs;t den&longs;ior, & quod res in medio den&longs;o apparent maiores, &longs;icut de <lb/>pi&longs;cibus in aqua, denarijs & baculis. </s>

<s id="id003364">Demon&longs;tratum <expan abbr="a&utilde;t">aut</expan> e&longs;t in præ<lb/>cedenti, quod quando &longs;tella fuerit in polo orbis &longs;uperioris, quòd <lb/>tunc maximè retrocedit, & ideò cum in tempore maximi retro ce&longs;<lb/>&longs;us &longs;it in oppo&longs;ito Solis <expan abbr="d&utilde;">dum</expan> tres &longs;uperiores &longs;unt in oppo&longs;itu Solis, <lb/>multo maiores duabus ex cau&longs;is e&longs;&longs;e uidentur, & iuxta proportio<lb/>nem propinquitatis ad Solem commutant quantitatem & tanto <lb/>minores apparent, quia non po&longs;&longs;unt, commutare <expan abbr="formã">formam</expan>, uelut Lu<lb/>na propter æqualitatem &longs;ub&longs;tanti&etail; & luminis proprij copiam, qu&etail; <lb/>non &longs;init di&longs;cerni uarietatem figur&etail;. </s>

<s id="id003364">Demon&longs;tratum <expan abbr="a&utilde;t">aunt</expan> e&longs;t in præ<lb/>cedenti, quod quando &longs;tella fuerit in polo orbis &longs;uperioris, quòd <lb/>tunc maximè retrocedit, & ideò cum in tempore maximi retro ce&longs;<lb/>&longs;us &longs;it in oppo&longs;ito Solis <expan abbr="d&utilde;">dum</expan> tres &longs;uperiores &longs;unt in oppo&longs;itu Solis, <lb/>multo maiores duabus ex cau&longs;is e&longs;&longs;e uidentur, & iuxta proportio<lb/>nem propinquitatis ad Solem commutant quantitatem & tanto <lb/>minores apparent, quia non po&longs;&longs;unt, commutare <expan abbr="formã">formam</expan>, uelut Lu<lb/>na propter æqualitatem &longs;ub&longs;tanti&etail; & luminis proprij copiam, qu&etail; <lb/>non &longs;init di&longs;cerni uarietatem figur&etail;. </s>

<s id="id003365">In Luna autem &longs;ecus e&longs;t, nam in

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<s id="id003708"><margin.target id="marg695"/>Q<emph type="italics"/>uæ&longs;t.<emph.end type="italics"/> 27.</s>

<s id="id003709">Hanc generaliter propo&longs;uit Ari&longs;toteles in Mechanicis, <expan abbr="o&longs;tendi&ttilde;">o&longs;tenditur</expan> <lb/><expan abbr="e&mtilde;">eum</expan> quod &longs;i a b in e, & d e in f æqualia <lb/>pondera in medio <expan abbr="&longs;u&longs;pendãtur">&longs;u&longs;pendantur</expan>, quod <lb/><figure id="id.015.01.238.2.jpg" xlink:href="015/01/238/2.jpg"/><lb/>grauius erit a b quam d e. </s>

<s id="id003709">Hanc generaliter propo&longs;uit Ari&longs;toteles in Mechanicis, <expan abbr="o&longs;tendi&ttilde;">o&longs;tenditur</expan> <lb/><expan abbr="e&mtilde;">emm</expan> quod &longs;i a b in e, & d e in f æqualia <lb/>pondera in medio <expan abbr="&longs;u&longs;pendãtur">&longs;u&longs;pendantur</expan>, quod <lb/><figure id="id.015.01.238.2.jpg" xlink:href="015/01/238/2.jpg"/><lb/>grauius erit a b quam d e. </s>

<s id="id003710">Et hoc e&longs;t <lb/>certum quia a & b extrema plus di<lb/>&longs;tant ab hypomochlio. </s>

<s id="id003711">Sit igitur g h re&longs;ecta æqualis hic cinde d e, <lb/>pondus e&longs;t æquale a b, erit g h minus pondere d e in k, igitur per <lb/>communem animi &longs;ententiam k e&longs;t æquale uerò ponderi a g & h b, <lb/>igitur cum a g & h b plus ponderent in &longs;itu &longs;uo quam in &longs;itu d e, <lb/>patet propo&longs;itum quoad Ari&longs;totelem attinet, &longs;cilicet quod a b e&longs;t <lb/>grauior d e.</s>

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<s id="id003948">Hanc propo&longs;itionem non &longs;crip&longs;i quòd e&longs;&longs;et magni momenti, &longs;ed <lb/>propter modum probandi, &longs;i enim re&longs;picis ex uno oppo&longs;ito &longs;cilicet <lb/>quod peripheria circuli &longs;it maior trianguli lateribus, o&longs;tendo de<lb/>mon&longs;tratione non ducente ad inconueniens, &longs;ed &longs;implici quod ip&longs;a <lb/>peripheria e&longs;t minor trianguli lateribus, & hoc nunquam fuit <expan abbr="fact&utilde;">factum</expan> <lb/>ab aliquo, imò uidetur plane impo&longs;sibile. </s>

<s id="id003949">Et e&longs;t res admirabilior <lb/>quæ inuenta &longs;it ab orbe condito, &longs;cilicet o&longs;tendere aliquid ex &longs;uo <lb/>oppo&longs;ito, demon&longs;tratione non ducente ad impo&longs;sibile & ita, ut <expan abbr="nõ">non</expan> <lb/>po&longs;sit demon&longs;trari ea <expan abbr="demõ">demon</expan>&longs;tratione ni&longs;i per illud <expan abbr="&longs;uppo&longs;it&utilde;">&longs;uppo&longs;itum</expan> quod <lb/>e&longs;t contrarium conclu&longs;ioni, uelut &longs;i quis demon&longs;traret quòd So<lb/>crates e&longs;t albus quia e&longs;t niger, & non po&longs;&longs;et demon&longs;trare aliter, & <lb/>ideo e&longs;t longè maius Chry&longs;ippeo Syllogi&longs;mo.<lb/><arrow.to.target n="marg772"/></s>

<s id="id003949">Et e&longs;t res admirabilior <lb/>quæ inuenta &longs;it ab orbe condito, &longs;cilicet o&longs;tendere aliquid ex &longs;uo <lb/>oppo&longs;ito, demon&longs;tratione non ducente ad impo&longs;sibile & ita, ut <expan abbr="nõ">non</expan> <lb/>po&longs;sit demon&longs;trari ea <expan abbr="demõ">demom</expan>&longs;tratione ni&longs;i per illud <expan abbr="&longs;uppo&longs;it&utilde;">&longs;uppo&longs;itum</expan> quod <lb/>e&longs;t contrarium conclu&longs;ioni, uelut &longs;i quis demon&longs;traret quòd So<lb/>crates e&longs;t albus quia e&longs;t niger, & non po&longs;&longs;et demon&longs;trare aliter, & <lb/>ideo e&longs;t longè maius Chry&longs;ippeo Syllogi&longs;mo.<lb/><arrow.to.target n="marg772"/></s>

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<s id="id003965">Fulmen edit tonitru in quo & caua nebula <lb/>excipit aërem, & multum impetuque maximo delatum, <expan abbr="ob&longs;trep&utilde;t">ob&longs;trepunt</expan> au<lb/>tem metalla magis quam ligna eo quòd magis ob <expan abbr="continuitat&etilde;">continuitatem</expan> par <lb/>tes moueantur. </s>

<s id="id003966">Indicio e&longs;t, quod intenta ut æs & tenuia <expan abbr="maior&etilde;">maiorem</expan> &longs;tre<lb/>pitum edunt: & dum &longs;onant tremunt, aurum autem parum &longs;onat, <lb/>quoniam den&longs;i&longs;simum e&longs;t, et minus intentum <expan abbr="arg&etilde;tum">argentum</expan>, minus den <lb/>&longs;um, & magis intentum, quod autem intentum e&longs;t totum &longs;imul mo<lb/>uetur, & ob id &longs;tridet: lignum <expan abbr="aut&etilde;">autem</expan> & tabula &longs;onat, non quia ut me<lb/>tallum percutiat aërem, &longs;ed quia in eo aër percutitur. </s>

<s id="id003967">Cra&longs;&longs;um <expan abbr="aut&etilde;">autem</expan> <lb/>metallum & lignum non adeò &longs;onant: metallum quoniam non mo<lb/>uet aërem, non enim mouetur: lignum quoniam non mouetur, nec <lb/>in eo qui e&longs;t inclu&longs;us aër, aër autem facilè mouetur, & ob id in ligno <lb/>cauo, etiam&longs;i cra&longs;&longs;um &longs;it, &longs;trepitus magnus editur. </s>

<s id="id003968">Ergo et&longs;i tenue <lb/>&longs;it metallum, quod infixum e&longs;t tabul&etail;, re&longs;onat multum: <expan abbr="nõ">non</expan> quia mo<lb/>ueatur, &longs;ed quoniam <expan abbr="a&etilde;rem">aerem</expan> in tabula <expan abbr="cõ">con</expan> cutit. </s>

<s id="id003968">Ergo et&longs;i tenue <lb/>&longs;it metallum, quod infixum e&longs;t tabul&etail;, re&longs;onat multum: <expan abbr="nõ">non</expan> quia mo<lb/>ueatur, &longs;ed quoniam <expan abbr="a&etilde;rem">aerrem</expan> in tabula <expan abbr="cõ">com</expan> cutit. </s>

<s id="id003969">Neque enim tabula per <lb/>&longs;e &longs;ola, quæ etiam nimis tunderetur &longs;onum edere magnum pote&longs;t <lb/>quoniam cedit: Oportet <expan abbr="aut&etilde;">autem</expan> non cedere quod re&longs;onat, neque metal<lb/>lum &longs;i cra&longs;&longs;um, &longs;ed hebetem <expan abbr="&longs;on&utilde;">&longs;onum</expan> etiam tabul&etail; infixum reddit, quo<lb/>niam neque moueri pote&longs;t infixum & cra&longs;&longs;um, nec cauerno&longs;um e&longs;t, & <lb/>tamen excipit ictum, ne lignum re&longs;onet. </s>

<s id="id003970">Velox autem ictus <expan abbr="nõ">non</expan> acu<lb/>tum <expan abbr="&longs;on&utilde;">&longs;onum</expan> reddit, & &longs;i cum impetu &longs;it: indicio e&longs;t tonitru & machin&etail; <lb/>bellicæ igne&etail;, contrà angu&longs;ta fi&longs;tula <expan abbr="acut&utilde;">acutum</expan> &longs;onum reddit, <expan abbr="etiã">etiam</expan> remi&longs;<lb/>&longs;è inflata. </s>

<s id="id003971">Igitur aër &longs;oni cau&longs;a e&longs;t &longs;ecundum <expan abbr="mot&utilde;">motum</expan>, ubi ergo multus <lb/>aër & magnus motus ibi &longs;onus magnus. </s>

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<s id="id003978"><arrow.to.target n="marg778"/><lb/>Ari&longs;totelem non <expan abbr="a&longs;&longs;ecut&utilde;">a&longs;&longs;ecutum</expan> fui&longs;&longs;e, aut potius ad <lb/><arrow.to.target n="marg779"/><lb/>nos <expan abbr="corruptã">corruptam</expan> &longs;cripturam perueni&longs;&longs;e: nam qui <lb/>expo <expan abbr="n&utilde;t">nunt</expan> multo minus <expan abbr="intellig&utilde;t">intelligunt</expan>. </s>

<s id="id003979">Sit ergo cur<lb/>rus humilis &longs;cytalis <expan abbr="iucumb&etilde;s">iucumbes</expan> a b c. </s>

<s id="id003979">Sit ergo cur <lb/>rus humilis &longs;cytalis <expan abbr="iucumb&etilde;s">iucumbens</expan> a b c. </s>

<s id="id003980">Diximus <lb/><expan abbr="aut&etilde;">autem</expan> &longs;uprà quid e&longs;&longs;et &longs;cytala & currus rotis, <expan abbr="&qtilde;">quae</expan> <lb/>&longs;untlonge maiores &longs;cytalis e f g h, <expan abbr="demõ&longs;tran">demon&longs;tran</expan> <lb/><expan abbr="d&utilde;">dum</expan> e&longs;t <expan abbr="&longs;cytalã">&longs;cytalam</expan>, quamuis minoris ambitus ma<lb/>gis mouere <08> rotam, <expan abbr="c&utilde;">cum</expan> ergo de una demon<lb/>&longs;trauerimus, de <expan abbr="o&itilde;bus">oibus</expan> erit <expan abbr="intelligend&utilde;">intelligendum</expan>. </s>

<s id="id003980">Diximus <lb/><expan abbr="aut&etilde;">autem</expan> &longs;uprà quid e&longs;&longs;et &longs;cytala & currus rotis, <expan abbr="&qtilde;">quae</expan> <lb/>&longs;untlonge maiores &longs;cytalis e f g h, <expan abbr="demõ&longs;tran">demon&longs;tran</expan> <lb/><expan abbr="d&utilde;">dum</expan> e&longs;t <expan abbr="&longs;cytalã">&longs;cytalam</expan>, quamuis minoris ambitus ma<lb/>gis mouere <08> rotam, <expan abbr="c&utilde;">cum</expan> ergo de una demon<lb/>&longs;trauerimus, de <expan abbr="o&itilde;bus">oimbus</expan> erit <expan abbr="intelligend&utilde;">intelligendum</expan>. </s>

<s id="id003981">Quia <lb/>ergo &longs;cytala k l m habet hypomochlion in k et <lb/>m, & <expan abbr="põdus">pondus</expan> premit in l, <expan abbr="igi&ttilde;">igitur</expan> rota uer&longs;atilis mo<lb/><arrow.to.target n="marg780"/><lb/><expan abbr="uebi&ttilde;">uebitur</expan> tanto facilius procedendo, quanta e&longs;t <expan abbr="lõ">lon</expan>gitudo l m & l k, &longs;ed & <lb/>rotul&etail; ill&etail; <expan abbr="uer&longs;ab&utilde;t">uer&longs;abunt</expan> hypomochlion, &qring;d e&longs;t l <expan abbr="cõparatione">comparatione</expan> k & m col<lb/>lopum, <expan abbr="igi&ttilde;">igitur</expan> facilius multo <expan abbr="uer&longs;abi&ttilde;">uer&longs;abitur</expan> currus à &longs;cytalis <08> rotis. </s>

<s id="id003981">Quia <lb/>ergo &longs;cytala k l m habet hypomochlion in k et <lb/>m, & <expan abbr="põdus">pondus</expan> premit in l, <expan abbr="igi&ttilde;">igitur</expan> rota uer&longs;atilis mo<lb/><arrow.to.target n="marg780"/><lb/><expan abbr="uebi&ttilde;">uebitur</expan> tanto facilius procedendo, quanta e&longs;t <expan abbr="lõ">lom</expan> gitudo l m & l k, &longs;ed & <lb/>rotul&etail; ill&etail; <expan abbr="uer&longs;ab&utilde;t">uer&longs;abunt</expan> hypomochlion, &qring;d e&longs;t l <expan abbr="cõparatione">comparatione</expan> k & m col<lb/>lopum, <expan abbr="igi&ttilde;">igitur</expan> facilius multo <expan abbr="uer&longs;abi&ttilde;">uer&longs;abitur</expan> currus à &longs;cytalis <08> rotis. </s>

<s id="id003982">Et hoc <lb/>e&longs;t quod dixit Philo&longs;ophus. </s>

<s id="id003983">In utri&longs;que. </s>

<s id="id003985">his <expan abbr="reuolui&ttilde;">reuoluitur</expan> circulus et mo<lb/>tus <expan abbr="impelli&ttilde;">impellitur</expan>, intelligit <expan abbr="mutuã">mutuam</expan> <expan abbr="commutation&etilde;">commutationem</expan> hypomochlij cum col<lb/>lopibus, nam ut <expan abbr="trahãtur">trahantur</expan> rotul&etail; <expan abbr="&qtilde;">quae</expan> &longs;unt hypomochlij loco, collopes <lb/><expan abbr="terminan&ttilde;">terminantur</expan> in medio: ut <expan abbr="a&utilde;t">aut</expan> <expan abbr="uerta&ttilde;">uertatur</expan> axis, qui & hypomochlion in me<lb/>dio <expan abbr="collop&utilde;">collopum</expan> initium &longs;int rotulæ. </s>

<s id="id003985">his <expan abbr="reuolui&ttilde;">reuoluitur</expan> circulus et mo<lb/>tus <expan abbr="impelli&ttilde;">impellitur</expan>, intelligit <expan abbr="mutuã">mutuam</expan> <expan abbr="commutation&etilde;">commutationem</expan> hypomochlij cum col<lb/>lopibus, nam ut <expan abbr="trahãtur">trahantur</expan> rotul&etail; <expan abbr="&qtilde;">quae</expan> &longs;unt hypomochlij loco, collopes <lb/><expan abbr="terminan&ttilde;">terminantur</expan> in medio: ut <expan abbr="a&utilde;t">aunt</expan> <expan abbr="uerta&ttilde;">uertatur</expan> axis, qui & hypomochlion in me<lb/>dio <expan abbr="collop&utilde;">collopum</expan> initium &longs;int rotulæ. </s>

<s id="id003986">Ex quo <expan abbr="&longs;equi&ttilde;">&longs;equitur</expan>, &qring;d quanto <expan abbr="lõgiores">longiores</expan> <lb/>erunt l k l t & l m, tanto facilius <expan abbr="mouebun&ttilde;">mouebuntur</expan> currus, at quanto humi<lb/>liores, modò non obruantur in terra, quoniam tardius mouentur, <lb/>quæ minorem habent circuitum, quæ autem tardius mouentur, fa<lb/>cilius mouentur, ut &longs;uprà &longs;æpius demon&longs;tratum e&longs;t: Ob has ergo <lb/>duas cau&longs;as pondera facilius feruntur curribus cum &longs;cytalis, quàm <lb/>cum rotis magnis modò terra non obruantur.</s>

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de Subtilitate, at nunc quod ad de<lb/><arrow.to.target n="marg781"/><lb/>mon&longs;trationem attinet <expan abbr="eor&utilde;">eorum</expan> &longs;ubijciam. </s>

<s id="id003994">&longs;ingul&etail; rotul&etail; diffi <lb/>culter <expan abbr="mouen&ttilde;">mouentur</expan>, igitur nece&longs;&longs;e e&longs;t &longs;ingulas participes e&longs;&longs;e grauitatis, <lb/>igitur & totam <expan abbr="grauitat&etilde;">grauitatem</expan> e&longs;&longs;e diui&longs;am: quare ut in <expan abbr="pr&etail;ced&etilde;ti">pr&etail;cedenti</expan> facilius <lb/>moueri. </s>

<s id="id003995">Habent & rotul&etail; ip&longs;&etail; centrum &longs;eu axem hypomochlij, &longs;eu <lb/><arrow.to.target n="marg782"/><lb/>fulcimenti loco, ambitum <expan abbr="a&utilde;t">aut</expan> iuxta &longs;emidiametrum, uelut collopes <pb pagenum="234" xlink:href="015/01/253.jpg"/>&longs;eu uectes, quare tanto facilius mouebuntur quanto maiores <expan abbr="er&utilde;t">erunt</expan>, <lb/><figure id="id.015.01.253.1.jpg" xlink:href="015/01/253/1.jpg"/><lb/>& ut plures. </s>

<s id="id003995">Habent & rotul&etail; ip&longs;&etail; centrum &longs;eu axem hypomochlij, &longs;eu <lb/><arrow.to.target n="marg782"/><lb/>fulcimenti loco, ambitum <expan abbr="a&utilde;t">aunt</expan> iuxta &longs;emidiametrum, uelut collopes <pb pagenum="234" xlink:href="015/01/253.jpg"/>&longs;eu uectes, quare tanto facilius mouebuntur quanto maiores <expan abbr="er&utilde;t">erunt</expan>, <lb/><figure id="id.015.01.253.1.jpg" xlink:href="015/01/253/1.jpg"/><lb/>& ut plures. </s>

<s id="id003996">Vna enim alterius loco fungitur uectis. </s>

<s id="id003997">Trochlea qui<lb/>dem e&longs;t, ut uides, in&longs;trumentum longum &longs;uprà angu&longs;tius, &longs;ed non, <lb/>cra&longs;&longs;um, in quo plures orbiculi &longs;olent collo cari, unde &longs;æpe numero <lb/>trochleæ nomine intelligimus orbiculos ei inclu&longs;os, circa quos fu<lb/>nis uocatur, ut in trochleis & orbiculi & funes includuntur. </s>

<s id="id003998">Succu<lb/>lis etiam &longs;olent capita funium trahi: ut uectis auxilio imò nonnum<lb/>quàm rotarum facilius pondera eleuantur.<lb/><arrow.to.target n="marg783"/></s>

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<s id="id004003">“E&longs;t autem ei quod diuinitus generandum e&longs;t circuitus, quem nu<lb/>merus <expan abbr="cõtinet">continet</expan> perfectus. </s>

<s id="id004004">Humanæ uerò, in quo primum argumen<lb/>tationes &longs;uperantes, ut &longs;uperatæ tres di&longs;tantiæ: quatuor autem ter<lb/>minos accipientes, &longs;imilium & di&longs;similium, ab <expan abbr="undanti&utilde;">undantium</expan> & deficien<lb/>tium cuncta corre&longs;pondentia, & rationem habentia inuicem effece<lb/>runt. </s>

<s id="id004005">Quorum &longs;exquitertium fundamentum quinario <expan abbr="iunct&utilde;">iunctum</expan> duas <lb/>efficit harmonias ter aucta quidem: æqualem æqualiter centum to<lb/>ties, quandam autem æqualem quidem, longitudine <expan abbr="a&utilde;t">aut</expan> &longs;ingulum <lb/>quidem numerorum à diametris <expan abbr="ration&etilde;">rationem</expan> habentibus quinarij indi <lb/>gentibus uno &longs;ingulis: non habentibus rationem <expan abbr="a&utilde;t">aut</expan> duobus, cen<lb/>tum autem cuborum ternarij. </s>

<s id="id004005">Quorum &longs;exquitertium fundamentum quinario <expan abbr="iunct&utilde;">iunctum</expan> duas <lb/>efficit harmonias ter aucta quidem: æqualem æqualiter centum to<lb/>ties, quandam autem æqualem quidem, longitudine <expan abbr="a&utilde;t">aunt</expan> &longs;ingulum <lb/>quidem numerorum à diametris <expan abbr="ration&etilde;">rationem</expan> habentibus quinarij indi <lb/>gentibus uno &longs;ingulis: non habentibus rationem <expan abbr="a&utilde;t">aunt</expan> duobus, cen<lb/>tum autem cuborum ternarij. </s>

<s id="id004006">Totus autem hic numerus geometri <lb/>cus talem authoritatem habet ad potiorem deterioremque <expan abbr="generation&etilde;">genera<lb/>tionem</expan>. </s>

<s id="id004007">Quem locum Ari&longs;toteles ita declarat. </s>

<s id="id004008">Quorum &longs;exquiter<lb/>tium fundamentum quinario coniunctum duas exhibet harmo<lb/>nias, <expan abbr="inqui&etilde;s">inquies</expan>, <expan abbr="quãdo">quando</expan> numerus diagrammatis huius <expan abbr="efficia&ttilde;">efficiatur</expan> &longs;olidus.”</s>

<s id="id004008">Quorum &longs;exquiter<lb/>tium fundamentum quinario coniunctum duas exhibet harmo<lb/>nias, <expan abbr="inqui&etilde;s">inquiens</expan>, <expan abbr="quãdo">quando</expan> numerus diagrammatis huius <expan abbr="efficia&ttilde;">efficiatur</expan> &longs;olidus.”</s>

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Line 11648

<s id="id004012">Par e&longs;t ut in diuina generatione numerus </s>

<s id="id004013"><arrow.to.target n="marg785"/><lb/><expan abbr="accipere&ttilde;">acciperetur</expan> perfectus: ut intelligat generationem confe&longs;tim &longs;equi cor <lb/>ruptionem: nam &longs;ermo e&longs;t de corruptione, corrumpitur <expan abbr="a&utilde;t">aut</expan> unum<lb/>quodque ut aliud generetur, malum enim e&longs;t ob bonum, non contrà. <lb/></s>

<s id="id004013"><arrow.to.target n="marg785"/><lb/><expan abbr="accipere&ttilde;">acciperetur</expan> perfectus: ut intelligat generationem confe&longs;tim &longs;equi cor <lb/>ruptionem: nam &longs;ermo e&longs;t de corruptione, corrumpitur <expan abbr="a&utilde;t">aunt</expan> unum<lb/>quodque ut aliud generetur, malum enim e&longs;t ob bonum, non contrà. <lb/></s>

<s id="id004014">Liquet autem ex Euclide talem numerum e&longs;&longs;e octies mille <expan abbr="cent&utilde;">centum</expan> ui<lb/>ginti octo. </s>

<s id="id004015">Et hic e&longs;t finis <expan abbr="omni&utilde;">omnium</expan> urbium diuinus, cuius <expan abbr="quadrupl&utilde;">quadruplum</expan> <lb/>uelut in c&oelig;li re&longs;titutionibus, ac continuato ordine &longs;olet ob&longs;eruari, <lb/>e&longs;t propè annus magnus: ueri&longs;imile e&longs;t enim <expan abbr="tãto">tanto</expan> tempore <expan abbr="cõfundi">confundi</expan> <lb/>decima, &longs;cilicet totius circuitus parte. </s>

<s id="id004016">Humanæ uerò intelligit qua<lb/><figure id="id.015.01.253.2.jpg" xlink:href="015/01/253/2.jpg"/><arrow.to.target n="table29"/><lb/>tuor à monade numeros, aut in quauis ratione principium li<lb/>neam &longs;uperficiem corpus, ut <expan abbr="un&utilde;">unum</expan>, duo, quatuor, octo pariter <lb/>octo: duodecim decem octo uiginti <expan abbr="&longs;ept&etilde;">&longs;eptem</expan>: inter hæc &longs;unt tria <lb/>&longs;patia, & octo cum uiginti &longs;eptem &longs;unt di&longs;similia & deficien<lb/>tia: maiora <expan abbr="e&mtilde;">eum</expan> &longs;unt &longs;uis partibus à quibus numerantur. </s>

<s id="id004016">Humanæ uerò intelligit qua<lb/><figure id="id.015.01.253.2.jpg" xlink:href="015/01/253/2.jpg"/><arrow.to.target n="table29"/><lb/>tuor à monade numeros, aut in quauis ratione principium li<lb/>neam &longs;uperficiem corpus, ut <expan abbr="un&utilde;">unum</expan>, duo, quatuor, octo pariter <lb/>octo: duodecim decem octo uiginti <expan abbr="&longs;ept&etilde;">&longs;eptem</expan>: inter hæc &longs;unt tria <lb/>&longs;patia, & octo cum uiginti &longs;eptem &longs;unt di&longs;similia & deficien<lb/>tia: maiora <expan abbr="e&mtilde;">emm</expan> &longs;unt &longs;uis partibus à quibus numerantur. </s>

<s id="id004017">Contrà de<lb/>cem octo & duodecim &longs;unt &longs;imilia atque ab <expan abbr="undãtia">undantia</expan>, & corre&longs;ponden

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Line 11713

<s id="id004034">Pro intellectu reliquorum ab eo dictorum, & quorundam mira<lb/>bilium, proponatur alius rhombus illi &etail;qualis, in tabula pictus deli<lb/>neatis lateribus & diametris, qui fit l m o n, & diametri l p o & m p <lb/>n, & ab&longs;cindatur hic ex &longs;uperficie, & &longs;uperponatur ita, ut puncta l m <lb/>o n ordinatim cadant, & aptentur <expan abbr="p&utilde;ctis">punctis</expan> a b d c, & p aptetur ip&longs;i k. <lb/></s>

<s id="id004035">Et tunc &longs;i rhombus l o totus moueretur, nece&longs;&longs;e e&longs;t, ut moueatur &longs;e<lb/>cundum latus aliquod, ut pote l m, & &etail;quidi&longs;tans a b, igitur dicetur <lb/><figure id="id.015.01.255.1.jpg" xlink:href="015/01/255/1.jpg"/><lb/>moueri &longs;uper latus aliquod, &longs;cilicet a c: atque hic e&longs;t mo<lb/>tus, quem Ari&longs;toteles uocat <expan abbr="mot&utilde;">motum</expan> a b &longs;uper latus a c. <lb/></s>

<s id="id004036">Si <expan abbr="a&utilde;t">aut</expan> fingamus quie&longs;cere latus aliquod l o, uel pars <lb/>lateris, non po&longs;&longs;et omnino moueri in &longs;uperficie a d <lb/>rhombi: et ita <expan abbr="nõ">non</expan> perinde e&longs;&longs;et ac &longs;i a d rhombus mo<lb/>ueretur, quod tamen &longs;upponit Ari&longs;toteles. </s>

<s id="id004036">Si <expan abbr="a&utilde;t">aunt</expan> fingamus quie&longs;cere latus aliquod l o, uel pars <lb/>lateris, non po&longs;&longs;et omnino moueri in &longs;uperficie a d <lb/>rhombi: et ita <expan abbr="nõ">non</expan> perinde e&longs;&longs;et ac &longs;i a d rhombus mo<lb/>ueretur, quod tamen &longs;upponit Ari&longs;toteles. </s>

<s id="id004037">Neque <expan abbr="etiã">etiam</expan> <lb/>&longs;i quie&longs;ceret punctum aliud quam p haberet ratio<lb/>nem motus regularis, quod ab illo &longs;upponitur: reli<lb/>quum e&longs;t igitur, ut rhombus l o moueatur uice rhombi a d &longs;eruan<lb/>do centrum, id e&longs;t punctum p in puncto k. </s>

<s id="id004038">Dicamus ergo primum <lb/>de motu compo&longs;ito Ari&longs;totelis, & pò&longs;t de no&longs;tro.</s>

<s id="id004039">Moueatur l m &longs;uper a c, æquidi&longs;tans &longs;emper a b, ut &longs;eruet &longs;itum <lb/>quem habebat ita, quod <expan abbr="extrem&utilde;">extremum</expan> lineæ l m &longs;it &longs;emper in linea a c, & <lb/>l punctum quod gerit uicem a, de&longs;cendat tantum in linea l m, quan<lb/>tum l extremum in linea a c: dicit Philo&longs;ophus, quod a &longs;eu l &longs;emper <lb/>de&longs;cendet in linea a d, & erit in e a. </s>

<s id="id004040">Supponatur quae latus l m fit f g, & <lb/>erit l n, f t, ducatur <expan abbr="a&utilde;t">aut</expan> ex r puncto &longs;ectionis diametri, & lateris l m li <lb/><arrow.to.target n="marg790"/><lb/>near q, æquidi&longs;tans a f, <expan abbr="igi&ttilde;">igitur</expan> rhombus a q r f e&longs;t &longs;imilis rhombo toti <lb/>a b d c, & proportio a f ad fr, ut a c ad c d, &longs;ed a c e&longs;t &etail;qualis c d, <expan abbr="igi&ttilde;">igitur</expan> a f <lb/>e&longs;t æqualis f r, &longs;ed l de&longs;cendit in l m, <expan abbr="quant&utilde;">quantum</expan> e&longs;t a f ex &longs;uppo&longs;ito, <expan abbr="igi&ttilde;">igitur</expan> <lb/><expan abbr="punct&utilde;">punctum</expan> l &longs;emper erit in linea a d. </s>

<s id="id004040">Supponatur quae latus l m fit f g, & <lb/>erit l n, f t, ducatur <expan abbr="a&utilde;t">aunt</expan> ex r puncto &longs;ectionis diametri, & lateris l m li <lb/><arrow.to.target n="marg790"/><lb/>near q, æquidi&longs;tans a f, <expan abbr="igi&ttilde;">igitur</expan> rhombus a q r f e&longs;t &longs;imilis rhombo toti <lb/>a b d c, & proportio a f ad fr, ut a c ad c d, &longs;ed a c e&longs;t &etail;qualis c d, <expan abbr="igi&ttilde;">igitur</expan> a f <lb/>e&longs;t æqualis f r, &longs;ed l de&longs;cendit in l m, <expan abbr="quant&utilde;">quantum</expan> e&longs;t a f ex &longs;uppo&longs;ito, <expan abbr="igi&ttilde;">igitur</expan> <lb/><expan abbr="punct&utilde;">punctum</expan> l &longs;emper erit in linea a d. </s>

<s id="id004041">Po&longs;t deficiunt quædam uerba: ob <lb/>quæ nemo intellexit &longs;ententiam Philo&longs;ophi, & <expan abbr="tam&etilde;">tamen</expan> au&longs;i &longs;unt impo<lb/>nere lectoribus, tanquam intellexi&longs;&longs;ent, tres &longs;imul errores admittendo, <lb/>&longs;cilicet Ari&longs;totelem ob propriam ignorantiam, ut &longs;tultum accu&longs;an<lb/>do, qui fal&longs;a dicat, & demon&longs;trare nitatur: produnt &longs;e ip&longs;os cum <lb/>&longs;ua impudentia. </s>

<s id="id004041">Po&longs;t deficiunt quædam uerba: ob <lb/>quæ nemo intellexit &longs;ententiam Philo&longs;ophi, & <expan abbr="tam&etilde;">tamen</expan> au&longs;i &longs;unt impo<lb/>nere lectoribus, tan<08> intellexi&longs;&longs;ent, tres &longs;imul errores admittendo, <lb/>&longs;cilicet Ari&longs;totelem ob propriam ignorantiam, ut &longs;tultum accu&longs;an<lb/>do, qui fal&longs;a dicat, & demon&longs;trare nitatur: produnt &longs;e ip&longs;os cum <lb/>&longs;ua impudentia. </s>

<s id="id004042">Et lectoribus imponere conantur, debet ergo &longs;ic <lb/>legi (“b in ip&longs;a b c diametro latum, ubi latus b d moueatur in late<lb/>re b a, & b æqualiter uer&longs;us d in b d, æqualis enim e&longs;t ip&longs;a b e”) <lb/>Tunc enim con&longs;tat ut hic dixi, m moueri per b c rectam ut l per a d: <lb/>Dicit ergo <expan abbr="c&utilde;">cum</expan> b d <expan abbr="mouea&ttilde;">moueatur</expan> in b a, tran&longs;it unico motu <expan abbr="totã">totam</expan> b a, & pun<pb pagenum="237" xlink:href="015/01/256.jpg"/><expan abbr="ct&utilde;">ctum</expan> tamen b, quod <expan abbr="moue&ttilde;">mouetur</expan> duobus motibus, non pertran&longs;it ni&longs;i b c, <lb/>quæ pote&longs;t e&longs;&longs;e minor b a: nam <expan abbr="cõ&longs;tat">con&longs;tat</expan> quod <expan abbr="quãdo">quando</expan> m erit in a, o erit <lb/>in e, & quia m de&longs;cendit in o, in eodem tempore, ergo o erit in c, & <lb/><expan abbr="trã&longs;iuit">tran&longs;iuit</expan> &longs;emper per rectam b c: igitur m e&longs;t minus <expan abbr="mot&utilde;">motum</expan> duobus mo<lb/>tibus quàm m l unico <expan abbr="tant&utilde;">tantum</expan>. </s>

<s id="id004043">Et quia aliquis dicere potui&longs;&longs;et non e&longs;t <lb/>mirum, quod m &longs;it minus motum duobus motibus quàm l m latus <lb/>unico tantum: quia m mouetur motu contrario motui lateris: nam <lb/>latus m o mouetur in latere b a a&longs;cendendo, et punctum m uer&longs;us o <lb/>in ip&longs;o m o de&longs;cendendo. </s>

<s id="id004044">Dicit Philo&longs;ophus, hoc e&longs;t mirum, quia <lb/>cum idem contingat in motu l, cuius latus mouetur per a c, & l per l <lb/>m recedendo in partem contrariam, nihilominus uelocius motum <lb/>e&longs;t l, quàm latus l m, quia a d e&longs;t longior a c. </s>

<s id="id004045">Ex quo patet, quae qu&etail;&longs;tio <lb/>Philo&longs;ophi e&longs;t una tantum, & non duæ. </s>

<s id="id004046">Et e&longs;t cur motum duobus <lb/>motibus in rhombo, in uno mouetur uelocius latere tantum moto <lb/>uno motu, in alio tardius? </s>

<s id="id004047">Et quia aliquis dicere po&longs;&longs;et, &qring;d b c po&longs;<lb/>&longs;et e&longs;&longs;e <expan abbr="lõgior">longior</expan> a c: Dicit Philo&longs;ophus, uerum e&longs;t, &longs;ed ego po&longs;&longs;um in<lb/>uenire talem rhombum, qui etiam habeat a c longiorem, & tunc ni<lb/>hilominus <expan abbr="&longs;equi&ttilde;">&longs;equitur</expan> quod dico. </s>

<s id="id004048">Aliud <expan abbr="a&utilde;t">aut</expan>, quod docet ex hac demon<lb/>&longs;tratione, e&longs;t quae ex duobus motibus rectis diuer&longs;is pote&longs;t fieri unus <lb/>motus rectus diuer&longs;us: igitur idem punctum, puta formica poterit <lb/>&longs;imul, & &longs;emel moueri duobus motibus rectis diuer&longs;is. </s>

<s id="id004048">Aliud <expan abbr="a&utilde;t">aunt</expan>, quod docet ex hac demon<lb/>&longs;tratione, e&longs;t quae ex duobus motibus rectis diuer&longs;is pote&longs;t fieri unus <lb/>motus rectus diuer&longs;us: igitur idem punctum, puta formica poterit <lb/>&longs;imul, & &longs;emel moueri duobus motibus rectis diuer&longs;is. </s>

<s id="id004049">Et hoc e&longs;t, <lb/>quia primus motus e&longs;t rectus &longs;olum &longs;ecundum formam, & non &longs;e<lb/>cundum materiam: & alter &longs;ecundus, &longs;cilicet mi&longs;tus e&longs;t &longs;ecundum <lb/>materiam & non &longs;ecundum formam per rectam.</s>

<s id="id004050"><margin.target id="marg790"/>P<emph type="italics"/>er<emph.end type="italics"/> 24. <emph type="italics"/>&longs;exti <emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<s id="id004051">Ex hoc <expan abbr="&longs;equi&ttilde;">&longs;equitur</expan> aliud magis <expan abbr="mir&utilde;">mirum</expan>, et e&longs;t iuxta <expan abbr="no&longs;tr&utilde;">no&longs;trum</expan> motum rhom<lb/>bi l o in rhombo a d, fixo centro p in centro k, & <expan abbr="mouea&ttilde;">moueatur</expan> quomodo <lb/>libet, l, dico quod l f &longs;emper æqualis erit a f, quia <expan abbr="e&mtilde;">eum</expan> k l & k a &longs;unt æ<lb/><figure id="id.015.01.256.1.jpg" xlink:href="015/01/256/1.jpg"/><lb/>quales, <expan abbr="c&utilde;">cum</expan> e&longs;&longs;ent una linea ante motum ducta, l a erit <lb/>angulus k l a, æqualis angulo k a l, &longs;ed angulus k a c <lb/><arrow.to.target n="marg791"/><lb/>e&longs;t æqualis angulo k l m, cum angulus k l m e&longs;&longs;et <expan abbr="id&etilde;">idem</expan> <lb/>angulo k a b, & angulus k a b e&longs;t <expan abbr="æ&qtilde;lis">æqualis</expan> angulo k a c, <lb/><arrow.to.target n="marg792"/><lb/>igitur angulus k l m e&longs;t æqualis angulo k a c, <expan abbr="igi&ttilde;">igitur</expan> re&longs;i<lb/>duus fl a e&longs;t æqualis re&longs;iduo f a l, quare f a æqualis <lb/><arrow.to.target n="marg793"/><lb/>fl. </s>

<s id="id004051">Ex hoc <expan abbr="&longs;equi&ttilde;">&longs;equitur</expan> aliud magis <expan abbr="mir&utilde;">mirum</expan>, et e&longs;t iuxta <expan abbr="no&longs;tr&utilde;">no&longs;trum</expan> motum rhom<lb/>bi l o in rhombo a d, fixo centro p in centro k, & <expan abbr="mouea&ttilde;">moueatur</expan> quomodo <lb/>libet, l, dico quod l f &longs;emper æqualis erit a f, quia <expan abbr="e&mtilde;">emm</expan> k l & k a &longs;unt æ<lb/><figure id="id.015.01.256.1.jpg" xlink:href="015/01/256/1.jpg"/><lb/>quales, <expan abbr="c&utilde;">cum</expan> e&longs;&longs;ent una linea ante motum ducta, l a erit <lb/>angulus k l a, æqualis angulo k a l, &longs;ed angulus k a c <lb/><arrow.to.target n="marg791"/><lb/>e&longs;t æqualis angulo k l m, cum angulus k l m e&longs;&longs;et <expan abbr="id&etilde;">idem</expan> <lb/>angulo k a b, & angulus k a b e&longs;t <expan abbr="æ&qtilde;lis">æqualis</expan> angulo k a c, <lb/><arrow.to.target n="marg792"/><lb/>igitur angulus k l m e&longs;t æqualis angulo k a c, <expan abbr="igi&ttilde;">igitur</expan> re&longs;i<lb/>duus fl a e&longs;t æqualis re&longs;iduo f a l, quare f a æqualis <lb/><arrow.to.target n="marg793"/><lb/>fl. </s>

<s id="id004052">Si igitur quantum procedit latus m l in a c, <expan abbr="tãtum">tantum</expan> <lb/>de&longs;cendat punctum in linea l m punctum perpetuo, erit in linea a c, <lb/>& per eam mouebitur. </s>

<s id="id004053">Vnde &longs;equitur quod</s>

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Line 12142

<s id="id004191">Propo&longs;itorum circulorum a & b centra iungam recta a b, &longs;uper </s>

<s id="id004192"><arrow.to.target n="marg837"/><lb/>quam ut &longs;emidiametrum de&longs;cribo circulum b c, & ex puncto a ad <lb/><arrow.to.target n="marg838"/><lb/>perpendiculum a d, ex quo ab&longs;cindo æqualem &longs;emidiametro b e li<lb/><arrow.to.target n="marg839"/><lb/><figure id="id.015.01.264.1.jpg" xlink:href="015/01/264/1.jpg"/><lb/>neam d f, ex f duco a d perpendi<lb/>culum f g, ex g in a duco a g, & æ<lb/>qualem angulo g a d, b a h ab&longs;cin <lb/>do h k <expan abbr="&etail;qual&etilde;">&etail;qualem</expan> d f &longs;eu b e, duco <expan abbr="a&utilde;t">aut</expan> <lb/><arrow.to.target n="marg840"/><lb/>b e, ut &longs;it <expan abbr="æquidi&longs;tãs">æquidi&longs;tans</expan> h k, duco h e, <lb/><arrow.to.target n="marg841"/><lb/><expan abbr="quã">quam</expan> dico contangere utrunque <expan abbr="circul&utilde;">cir<lb/>culum</expan> b k: produco b k, & quia duæ <lb/>lineæ b a & a k &longs;unt &etail;quales duo<lb/>bus lineis a g & a f, duæ enim <lb/>prodeunt ab eodem centro, reli<lb/>quæ &longs;unt re&longs;idua æqualium d f & h k, & angulus b a k æqualis <lb/><arrow.to.target n="marg842"/><lb/>g a f, ex &longs;uppo&longs;ito erit angulus g f a æqualis angulo b k a, g f a au<lb/>tem rectus fuit, quia g f ad perpendiculum erecta fuit, itaque b k a <lb/>rectus e&longs;t, & ideo b k h rectus, quare <expan abbr="c&utilde;">cum</expan> b e & k h &longs;int æquales, & æ<lb/><arrow.to.target n="marg843"/><lb/>quidi&longs;tantes, erit angulus e oppo&longs;itus b h k rectus, igitur duo angu<lb/>li e b k & e h k duobus rectis æquales, quare cum &longs;int æquales inui<lb/><arrow.to.target n="marg844"/><lb/>cem, quia oppo&longs;iti in parallelogrammo uterque eorum rectus erit. <lb/><arrow.to.target n="marg845"/><lb/>Recti ergo &longs;unt anguli e & h, & lineæ b e & a h ex centris circulo<lb/>rum, & angulos Illos con&longs;tituit lineæ e h, igitur e h contangit u<lb/><arrow.to.target n="marg846"/><lb/>trunque circulum.</s>

<s id="id004192"><arrow.to.target n="marg837"/><lb/>quam ut &longs;emidiametrum de&longs;cribo circulum b c, & ex puncto a ad <lb/><arrow.to.target n="marg838"/><lb/>perpendiculum a d, ex quo ab&longs;cindo æqualem &longs;emidiametro b e li<lb/><arrow.to.target n="marg839"/><lb/><figure id="id.015.01.264.1.jpg" xlink:href="015/01/264/1.jpg"/><lb/>neam d f, ex f duco a d perpendi<lb/>culum f g, ex g in a duco a g, & æ<lb/>qualem angulo g a d, b a h ab&longs;cin <lb/>do h k <expan abbr="&etail;qual&etilde;">&etail;qualem</expan> d f &longs;eu b e, duco <expan abbr="a&utilde;t">aunt</expan> <lb/><arrow.to.target n="marg840"/><lb/>b e, ut &longs;it <expan abbr="æquidi&longs;tãs">æquidi&longs;tans</expan> h k, duco h e, <lb/><arrow.to.target n="marg841"/><lb/><expan abbr="quã">quam</expan> dico contangere utrunque <expan abbr="circul&utilde;">cir<lb/>culum</expan> b k: produco b k, & quia duæ <lb/>lineæ b a & a k &longs;unt &etail;quales duo<lb/>bus lineis a g & a f, duæ enim <lb/>prodeunt ab eodem centro, reli<lb/>quæ &longs;unt re&longs;idua æqualium d f & h k, & angulus b a k æqualis <lb/><arrow.to.target n="marg842"/><lb/>g a f, ex &longs;uppo&longs;ito erit angulus g f a æqualis angulo b k a, g f a au<lb/>tem rectus fuit, quia g f ad perpendiculum erecta fuit, itaque b k a <lb/>rectus e&longs;t, & ideo b k h rectus, quare <expan abbr="c&utilde;">cum</expan> b e & k h &longs;int æquales, & æ<lb/><arrow.to.target n="marg843"/><lb/>quidi&longs;tantes, erit angulus e oppo&longs;itus b h k rectus, igitur duo angu<lb/>li e b k & e h k duobus rectis æquales, quare cum &longs;int æquales inui<lb/><arrow.to.target n="marg844"/><lb/>cem, quia oppo&longs;iti in parallelogrammo uterque eorum rectus erit. <lb/><arrow.to.target n="marg845"/><lb/>Recti ergo &longs;unt anguli e & h, & lineæ b e & a h ex centris circulo<lb/>rum, & angulos Illos con&longs;tituit lineæ e h, igitur e h contangit u<lb/><arrow.to.target n="marg846"/><lb/>trunque circulum.</s>

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Line 12731

<s id="id004433">Primus e&longs;t eorum quæ agimus <lb/>quibus prudentia dominatur, atque hic admodum certus e&longs;t, ut in <lb/>negotijs publicis priuatis que uidemus, e&longs;t <expan abbr="aut&etilde;">autem</expan> duplex, ciuilis & mili <lb/>taris. </s>

<s id="id004434">Secundus e&longs;t naturalium, e&longs;t autem maximè euidens in tribus <lb/>medicina, agricultura & nauigatione. </s>

<s id="id004435">Tertius e&longs;t eorum quæ &longs;unt <lb/>&longs;ecundum naturam, &longs;ed non per cau&longs;as, uelut a&longs;trologia & phy&longs;io<lb/>gnomia. </s>

<s id="id004436">Eius <expan abbr="a&utilde;t">aut</expan> tres &longs;unt partes phy&longs;iognomia, metopo&longs;copia & <lb/>chiromantia, namque a&longs;trologia et&longs;i per cau&longs;as &longs;it, magis tamen per <lb/>&longs;igna o&longs;tendere uidetur, nam quod Iuppiter in a&longs;cendente bonos <lb/>præbeat mores, cur magis hoc in loco uel illo, magna e&longs;t quæ&longs;tio. <lb/></s>

<s id="id004436">Eius <expan abbr="a&utilde;t">aunt</expan> tres &longs;unt partes phy&longs;iognomia, metopo&longs;copia & <lb/>chiromantia, namque a&longs;trologia et&longs;i per cau&longs;as &longs;it, magis tamen per <lb/>&longs;igna o&longs;tendere uidetur, nam quod Iuppiter in a&longs;cendente bonos <lb/>præbeat mores, cur magis hoc in loco uel illo, magna e&longs;t quæ&longs;tio. <lb/></s>

<s id="id004437">Quartus e&longs;t con&longs;en&longs;us omnium nobi&longs;cum atque fatale uin culum, in <lb/>quo genere ponuntur fulgrum ca&longs;us, exta, & augurium & hygro<lb/>mantia. </s>

<s id="id004438">In quinto modo ponuntur ea quæ cum anima no&longs;tra con<lb/>&longs;en&longs;um habent, eiu&longs;modi &longs;unt uitæ aut genij aut eroes. </s>

<s id="id004439">Sextus uerò <lb/>e&longs;t ex origine, uelut &longs;unt Prophetæ & uates Sybillæque, quorum uis <lb/>alia in &longs;e ip&longs;is, ut prophetarum, alia uaporis ut Delphici oraculi, alia <lb/>aqu&etail; uelut in Colophonio oraculo. </s>

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