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Colored diff for /texts/archimedes/xml/carda_propo_015_la_1570.xml between version 1.22 and 1.34

version 1.22, 2004/03/30 14:01:26

version 1.34, 2008/01/30 14:32:14

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<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>

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<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 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>

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<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

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<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>

Line 8663

<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>

Line 8785

<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>

Line 8814

<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>

Line 8863

<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>

Line 8915

<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>

Line 8939

<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>

Line 8948

<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>

Line 8985

<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>

Line 9074

<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

Line 9791

<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

Line 10943

<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>

Line 11507

<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>

Line 11559

<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>

Line 11582

<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>

Line 11612

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>

Line 11632

<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>

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

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, 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>

Line 11727

<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>

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>

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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