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Colored diff for /texts/archimedes/xml/carda_propo_015_la_1570.xml between version 1.15 and 1.33

version 1.15, 2004/02/19 17:23:10

version 1.33, 2008/01/25 15:51:41

Line 7601

<s id="id002492">pro<lb/>portio igitur g e ad e f cum &longs;it ut c ad e ex &longs;uppo&longs;ito erit ut ip&longs;i pro<lb/>portioni addamus, & detrahamus ex duplo a b & dimidium re&longs;i<lb/>dui ducamus in &longs;e, & addamus aggregato quadrati a b cum ip&longs;a

<pb pagenum="141" xlink:href="015/01/160.jpg"/>a b, & latus eius detracto dimidio re&longs;idui erit b clinea, quare diui<lb/>&longs;io nota, & e&longs;t ut dicamusu: olo diuidere datam lineam, ut quantita<lb/>tes adiectæ &longs;ub mutua proportione ad unam tertiam cum parti<lb/>bus obtineantinter &longs;e proportionem datam.</s>

<pb pagenum="141" xlink:href="015/01/160.jpg"/>a b, & latus eius detracto dimidio re&longs;idui erit b c linea, quare diui<lb/>&longs;io nota, & e&longs;t ut dicamus : uolo diuidere datam lineam, ut quantita<lb/>tes adiectæ &longs;ub mutua proportione ad unam tertiam cum parti<lb/>bus obtineant inter &longs;e proportionem datam.</s>

<s id="id002493"><margin.target id="marg487"/>P<emph type="italics"/>er<emph.end type="italics"/> 1. <emph type="italics"/>&longs;ecuu <lb/>di<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>

<s id="id002493"><margin.target id="marg487"/>P<emph type="italics"/>er<emph.end type="italics"/> 1. <emph type="italics"/>&longs;ecun<lb/>di<emph.end type="italics"/> E<emph type="italics"/>lement.<emph.end type="italics"/></s>

<s id="id002494">Propo&longs;itio cente&longs;imaquadrage&longs;imanona.</s>

Line 7680

<s id="id002525">Propo&longs;itio cente&longs;imaquinquage&longs;ima&longs;ecunda.</s>

<s id="id002526">Si linea in duas partes æquales duas que in æquales diuidatur, fue<lb/>ritque proportio aggregati ex maiore & dimidio ad ip&longs;am maiorem <lb/>uelut ex minore, & aliqua linea ad ip&longs;am minorem, & rur&longs;us aggre<lb/>gati ex minore dimidio ad ip&longs;am minorem, uelut aggregati ex ma<lb/>iore & alia addita ad ip&longs;am maiorem, erit proportio dimidij'ad par <lb/>tem unam inæqualem, uelut alterius partis inæqualis ad &longs;uam ad<lb/>ditam mutuò, & etiam proportio ad ditarum inuicem, uelut pro<lb/>portio partium inæqualium duplicata, & rur&longs;us ip&longs;um dimidium <lb/>lineæ a&longs;&longs;umptæ medium erit proportione inter additas. </s>

<s id="id002526">Si linea in duas partes æquales duas que in æquales diuidatur, fue<lb/>ritque proportio aggregati ex maiore & dimidio ad ip&longs;am maiorem <lb/>uelut ex minore, & aliqua linea ad ip&longs;am minorem, & rur&longs;us aggre<lb/>gati ex minore dimidio ad ip&longs;am minorem, uelut aggregati ex ma<lb/>iore & alia addita ad ip&longs;am maiorem, erit proportio dimidij ad par<lb/>tem unam inæqualem, uelut alterius partis inæqualis ad &longs;uam ad<lb/>ditam mutuò, & etiam proportio additarum inuicem, uelut pro<lb/>portio partium inæqualium duplicata, & rur&longs;us ip&longs;um dimidium <lb/>lineæ a&longs;&longs;umptæ medium erit proportione inter additas. </s>

<s id="id002527">Demum <lb/>proportio dimidij cum ad dita maiore ad dimidium cum addita mi<lb/>nore, uelut maioris partis ad minorem.<lb/><arrow.to.target n="marg496"/></s>

Line 7694

<pb pagenum="143" xlink:href="015/01/162.jpg"/>c b ad b d, uelut g a ad a d, & hoc e&longs;t primum. </s>

<s id="id002531">Quia ergo c a e&longs;t æ<lb/>qualis c b, erit c a ad b d, uelut g a ad a d, & iam fuit a d ad c a, ut b d <lb/>ad f b, per conuer&longs;am igitur a d ad b d, ut g a ad a d, & ut b d ad fb, <lb/>interpo&longs;itis ergo a d & d b inter a g & b f cum compo&longs;ita &longs;it pro<lb/>portio a g ad b f ex proportione a g ad a d, & ad d b, & d b <lb/>ad b f, & proportio a d ad d b, &longs;it æqualis proportioni <lb/><figure id="id.015.01.162.1.jpg" xlink:href="015/01/162/1.jpg"/><lb/>a g ad a d, & d b ad b f, igitur proportio a g ad b f. </s>

<s id="id002532">Per de<lb/>mon&longs;trata ab Alchindo e&longs;t duplicata proportioni a d ad <lb/>d b quod e&longs;t &longs;ecundum. </s>

<s id="id002533">Rur&longs;us quia ex primo demon<lb/>&longs;trato, uel eius conuer&longs;o proportio a d ad a c e&longs;t uelut b d <lb/>ad b f, & d b ad a c, ut a d ad a g, proportiones ergo <lb/><figure id="id.015.01.162.2.jpg" xlink:href="015/01/162/2.jpg"/><lb/>a d & d b ad a c componunt proportionem produ<lb/>ducti a d in d b, quod &longs;it h ad quadratum a c quod &longs;it <lb/>k, & &longs;imiliter proportio b d ad b f & a d ad a g com<lb/>ponunt proportionem producti ex b d in a d, quod <lb/>&longs;itl ad productum b f in a g, quod &longs;it m, per demon&longs;trata ab Eucli<lb/>de in &longs;exto Elementorum, igitur proportio h ad k ut l ad m, &longs;ed h & </s>

<s id="id002533">Rur&longs;us quia ex primo demon<lb/>&longs;trato, uel eius conuer&longs;o proportio a d ad a c e&longs;t uelut b d <lb/>ad b f, & d b ad a c, ut a d ad a g, proportiones ergo <lb/><figure id="id.015.01.162.2.jpg" xlink:href="015/01/162/2.jpg"/><lb/>a d & d b ad a c componunt proportionem produ<lb/>cti a d in d b, quod &longs;it h ad quadratum a c quod &longs;it <lb/>k, & &longs;imiliter proportio b d ad b f & a d ad a g com<lb/>ponunt proportionem producti ex b d in a d, quod <lb/>&longs;it l ad productum b f in a g, quod &longs;it m, per demon&longs;trata ab Eucli<lb/>de in &longs;exto Elementorum, igitur proportio h ad k ut l ad m, &longs;ed h & </s>

<s id="id002534"><arrow.to.target n="marg497"/><lb/>l &longs;unt æquales, quia producuntur ex ei&longs;dem, igitur per demon&longs;tra<lb/>ta in quinto Elementorum Euclidis, k e&longs;t æquale m, ergo a c e&longs;t me<lb/>dia pro portione inter b f & g a, quod e&longs;t tertium. </s>

Line 7716

<pb pagenum="144" xlink:href="015/01/163.jpg"/>una e&longs;t &longs;uperflua, quia ut dixi, una &longs;e quitur ad aliam. </s>

<s id="id002544">Ex hoc pa<lb/>tet cur Dio cles a&longs;&longs;ump&longs;erit lineam unam, quæ e&longs;t a c, quæ &longs;e ha<lb/>bet ad a d, & d b, ut uici&longs;sim a d, & d b ad additas, quod e&longs;t pri<lb/>mum demon&longs;tratum. </s>

<s id="id002545">Sic enim omittit primum quod proponit Ar <lb/>chimedes, & a&longs;&longs;umit quod proximum e&longs;t: & ideò Archimedes non <lb/>pro bat, nec præ&longs;upponit, quod à Diocle probatur, &longs;cilicet datum <lb/>e&longs;&longs;e punctum d in linea a b, &longs;ed &longs;olum in linea g f, ideò cogitur pro<lb/>bare &longs;ecundum quod demon&longs;tratur ab Eutocio, & à nobis demon <lb/>&longs;tratum e&longs;t &longs;uprà. </s>

<s id="id002546">Archimedes <expan abbr="a&utilde;t">aunt</expan> a&longs;&longs;umit <expan abbr="lineã">lineam</expan> extra circulum, <expan abbr="quã">quam</expan> <lb/>uo cat b f, quæ e&longs;t æqualis b c medietati: aliam a&longs;&longs;umit quam uocat <lb/>b h, cuius proportio ad b d e&longs;t &longs;icut quadrati ad a d quadratum a b. <lb/></s>

<s id="id002546">Archimedes <expan abbr="a&utilde;t">aut</expan> a&longs;&longs;umit <expan abbr="lineã">lineam</expan> extra circulum, <expan abbr="quã">quam</expan> <lb/>uocat b f, quæ e&longs;t æqualis b c medietati: aliam a&longs;&longs;umit quam uocat <lb/>b h, cuius proportio ad b d e&longs;t &longs;icut quadrati ad a d quadratum a b. <lb/></s>

<s id="id002547">Con&longs;tat ergo quod proportio g d ad d f e&longs;t data. </s>

<s id="id002548">Et &longs;imiliter f g ad <lb/>g d, & e&longs;t 1 præ proportione data. </s>

<s id="id002549">Vnde notandum quod datum <lb/>dicitur, &longs;impliciter cognitum alio modo, dicitur datum po&longs;itione, <lb/>quod e&longs;t certum & tale, uelut &longs;i quis dicat, diuide 10 in duos nume<lb/>ros quadratos: hoc non e&longs;t datum, pote&longs;t enim diuidi pluribus mo <lb/>dis. </s>

Line 7768

<s id="id002570">H&etail;c propo&longs;itio e&longs;t admirabilis: & etiam de&longs;crip&longs;i, ut multa &longs;ecre<lb/>ta Dialecticæ potius <expan abbr="aperiren&ttilde;">aperirentur</expan> quam quod huic propo&longs;ito <expan abbr="mult&utilde;">multum</expan> <lb/>congrueret. </s>

<s id="id002571">Ideò potius &longs;cholij cau&longs;a po&longs;ita e&longs;t quam ip&longs;ius tracta<lb/>tionis: ut <expan abbr="mod&utilde;">modum</expan> demon&longs;trandi magis quam id, &qring;d <expan abbr="demon&longs;tra&ttilde;">demon&longs;tratur</expan>, re<lb/>&longs;picere oporteat. </s>

<s id="id002572"><expan abbr="Con&longs;titua&ttilde;">Con&longs;tituatur</expan> ergo (per uiam problematis) linea a b <lb/>& proportio c ad d, & fiat d e ad c, ut c ad d, & a b ad e ut b f ad d, & <lb/>ut g ad c, eritque g media inter a f & f b, quod licet &longs;olum &longs;upponatur <lb/>ab Appollonio, <expan abbr="tam&etilde;">tamen</expan> facilè demon&longs;tratur & à Commandino adie<lb/>cta e&longs;t <expan abbr="demõ">demom</expan> &longs;tratio. </s>

<s id="id002573">Concurrant ergo ex a & b du&etail; line&etail; in aliquod </s>

Line 7805

<s id="id002583">Cum ergo h&etail;c demon&longs;tratio &longs;it ex &longs;en&longs;u in uno puncto h, ideò ad <lb/>quælibet puncta traduci pote&longs;t, quæ potero imaginari, & ita pri<lb/>ma uo cabitur &longs;en&longs;us, <expan abbr="&longs;ec&utilde;da">&longs;ecunda</expan> imaginandi: Et <expan abbr="quoniã">quoniam</expan> in demon&longs;tran<lb/>do non a&longs;&longs;umimus aliquid, quod &longs;it proprium alicui puncto, ni&longs;i <lb/>proportionem h a ad h b &longs;imilem e&longs;&longs;e c ad d, ideo hoc pertinet ad <lb/>intellectum, & e&longs;t tertium. </s>

<s id="id002584">Etidem dico &longs;i k e&longs;&longs;et ultra h quod po<lb/>te&longs;t contingere. </s>

<s id="id002585">modò k a ad k b &longs;it ut c ad d & k f &longs;it &etail;qualis g idem <lb/>&longs;equetur, & comprehenditur &longs;ub tertio & pertinet ad intellectum, <lb/>& quoniam demon&longs;tratur quod punctum k ubicun que &longs;umatur, e&longs;t <lb/>in &etail;quali <expan abbr="di&longs;tãtia">di&longs;tantia</expan> à puncto f&longs;cilicet per g lineam, erit &longs;emper in peri<lb/>pheria circuli, & hoc pote&longs;t e&longs;&longs;e in infinitis locis &longs;impliciter & extra <lb/>infinitum nihil e&longs;t, igitur &longs;ub hoc continetur conuer&longs;um &longs;cilicet, <lb/>quod a quolibet puncto circuli ductis lineis ad a & b ip&longs;&etail; erunt in <lb/>proportione c ad d. </s>

<s id="id002586">Et ita ab&longs;que principijs Geometricis concluditur <lb/>propo&longs;itio Geometrica & hoc e&longs;t <foreign lang="greek">w_erila/mpousin</foreign> & fermè &longs;ummum in<lb/>tellectus humani. </s>

<s id="id002586">Et ita ab&longs;que principijs Geometricis concluditur <lb/>propo&longs;itio Geometrica & hoc e&longs;t <foreign lang="greek">perila/mpousin</foreign> & fermè &longs;ummum in<lb/>tellectus humani. </s>

<s id="id002587">Et pote&longs;t demon&longs;trari Geometricè duobus uer<lb/>bis. </s>

<s id="id002589"><expan abbr="f&longs;upponi&ttilde;">f&longs;upponitur</expan> æqualis g eo quòd h e&longs;t in peripheria circu<lb/>li erit media inter a f & f b, quare cum angulus f &longs;it communis, erit <lb/>proportio a h ad h b, laterum re&longs;picientium angulum f in utroque </s>

Line 7813

<s id="id002590"><arrow.to.target n="marg510"/><lb/>triangulo, uelut h f lateris in maiori ad f b latus in minori, quare <lb/><arrow.to.target n="marg511"/><lb/>cum ex &longs;uppo&longs;ito h f ad fb &longs;it ut c ad d, erit a ad b, ut c ad d. </s>

<s id="id002591">Et uides <lb/>Apollonium, & Pappium quanta &longs;uperflua adij ciant in hac &longs;ecun<lb/><arrow.to.target n="marg512"/><lb/>da parte demon&longs;trationis, quæ e&longs;t prima apud illos, & ducunt <expan abbr="unã">unam</expan> <lb/>lineam non nece&longs;&longs;ariam ex puncto b ad latus fh. </s>

<s id="id002592">Vt <expan abbr="antiquor&utilde;">antiquorum</expan> ple <lb/>rique non tantum potuerint Geometria & ingenio, quæ ferunt excel <lb/>lenti&longs;sima in illis, quantum nos ex Dialectica <foreign lang="greek">w_e?ila/mpousin</foreign> inducen <lb/>tes. </s>

<s id="id002592">Vt <expan abbr="antiquor&utilde;">antiquorum</expan> ple <lb/>rique non tantum potuerint Geometria & ingenio, quæ ferunt excel<lb/>lenti&longs;sima in illis, quantum nos ex Dialectica <foreign lang="greek">pẹrila/mpousin</foreign> inducen <lb/>tes. </s>

<s id="id002593">e&longs;t enim &longs;ingulare hoc exemplum.<lb/><arrow.to.target n="marg513"/></s>

Line 7837

<s id="id002600">Ex hoc pater qualiter ex uera demon&longs;tratione &longs;en&longs;u o&longs;ten&longs;a per<lb/>uenimus ad quotquot imaginando, inde intellectu abiectis condi<lb/>tionibus non nece&longs;&longs;arijs facimus infinitum & uniuer&longs;ale. </s>

<s id="id002601">Demum <lb/>&longs;ine artis &longs;pe cialis auxilio o&longs;tendimus Iheorema uniuer&longs;ale (quod <lb/>etiam poterat o&longs;tendi Geometricè, &longs;ed longè pulchrius e&longs;t, ac &longs;ubli<lb/>mius per <foreign lang="greek">w_erilampousin</foreign>, qa hoc ip&longs;o infinita alia do cemus generaliter <lb/>per &longs;implicem <expan abbr="compreh&etilde;&longs;ionem">comprehen&longs;ionem</expan> o&longs;tendere) &longs;cilicet quod à quouis <lb/>puncto peripheri&etail; circuli, cuius &longs;emidiameter e&longs;t media proportio<lb/>ne inter totam exten&longs;am à centro u&longs;que exterius, & partem quæ' e&longs;t à <lb/>centro ad punctum de&longs;criptum &longs;ub proportione continua <expan abbr="datar&utilde;">datarum</expan> <lb/>linearum lineæ ductæ ex eo ad punctum exterius, & punctum de<lb/>&longs;criptum &longs;unt in proportione datarum linearum.</s>

<s id="id002601">Demum <lb/>&longs;ine artis &longs;pecialis auxilio o&longs;tendimus theorema uniuer&longs;ale (quod <lb/>etiam poterat o&longs;tendi Geometricè, &longs;ed longè pulchrius e&longs;t, ac &longs;ubli<lb/>mius per <foreign lang="greek">perilampousin</foreign>, qua hoc ip&longs;o infinita alia docemus generaliter <lb/>per &longs;implicem <expan abbr="compreh&etilde;&longs;ionem">comprehen&longs;ionem</expan> o&longs;tendere) &longs;cilicet quod à quouis <lb/>puncto peripheri&etail; circuli, cuius &longs;emidiameter e&longs;t media proportio<lb/>ne inter totam exten&longs;am à centro u&longs;que exterius, & partem quæ' e&longs;t à <lb/>centro ad punctum de&longs;criptum &longs;ub proportione continua <expan abbr="datar&utilde;">datarum</expan> <lb/>linearum lineæ ductæ ex eo ad punctum exterius, & punctum de<lb/>&longs;criptum &longs;unt in proportione datarum linearum.</s>

<s id="id002602">Propo&longs;itio cente&longs;imaquinquage&longs;imaquinta.</s>

Line 7900

<s id="id002625">Cum autem uolueris duo quadrata quæ differant in 100, tunc <lb/>per regulam datam &longs;i auferes 1, peruenires ad numeros magnos & <lb/>fractos, & ideo melius e&longs;t quia numerus e&longs;t par, ut detrahas nume<lb/>rum parem quadratum, ita quod re&longs;iduum po&longs;sit diuidi per <expan abbr="dupl&utilde;">duplum</expan> <lb/>radicis, ut in hoc non detraho neque quia remanet impar, nec 16 quia <lb/>84 <expan abbr="re&longs;idu&utilde;">re&longs;iduum</expan> non <expan abbr="põt">pont</expan> diuidi per 8 ita ut exeat integer numerus, ergo <lb/><expan abbr="detrahã">detraham</expan> 4 & <expan abbr="relinque&ttilde;">relinquetur</expan> 96, diuido per <expan abbr="dupl&utilde;">duplum</expan> radicis quod e&longs;t 4 exit <lb/>24, cuius quadratum &qring;d e&longs;t 576 addito 100 facit 676 <expan abbr="quadrat&utilde;">quadratum</expan> 26. <lb/>Et ita ex 433 non auferam &longs;ed 9, quia relinquetur 24 qui pote&longs;t diui<lb/>di per &longs;e, duplum <02> 9 & exit 4 cuius <expan abbr="quadrat&utilde;">quadratum</expan> e&longs;t 16, addito 33 fit 49.</s>

<s id="id002625">Cum autem uolueris duo quadrata quæ differant in 100, tunc <lb/>per regulam datam &longs;i auferes 1, peruenires ad numeros magnos & <lb/>fractos, & ideo melius e&longs;t quia numerus e&longs;t par, ut detrahas nume<lb/>rum parem quadratum, ita quod re&longs;iduum po&longs;sit diuidi per <expan abbr="dupl&utilde;">duplum</expan> <lb/>radicis, ut in hoc non detraho neque quia remanet impar, nec 16 quia <lb/>84 <expan abbr="re&longs;idu&utilde;">re&longs;iduum</expan> non <expan abbr="põt">pont</expan> diuidi per 8 ita ut exeat integer numerus, ergo <lb/><expan abbr="detrahã">detraham</expan> 4 & <expan abbr="relinque&ttilde;">relinquetur</expan> 96, diuido per <expan abbr="dupl&utilde;">duplum</expan> radicis quod e&longs;t 4 exit <lb/>24, cuius quadratum qua e&longs;t 576 addito 100 facit 676 <expan abbr="quadrat&utilde;">quadratum</expan> 26. <lb/>Et ita ex 433 non auferam &longs;ed 9, quia relinquetur 24 qui pote&longs;t diui<lb/>di per &longs;e, duplum <02> 9 & exit 4 cuius <expan abbr="quadrat&utilde;">quadratum</expan> e&longs;t 16, addito 33 fit 49.</s>

<s id="id002626">Secunda regula, cum uolueris propo&longs;ito uno numero quadra<lb/>to illum diuidere infinitis modis in duos numeros quadratos, cape <lb/>quemuis numerum quadratum per primum exemplum regul&etail; pri <lb/>mæ, & cum eo diuide numerum propo&longs;itum, & qui proueniet erit <lb/>quadratus, <expan abbr="h&utilde;c">hunc</expan> ergo duces in partes numeri quadrati qu&etail; &longs;unt nu<lb/>meri <expan abbr="&qtilde;drati">quadrati</expan>, & fient duo quadrati numeri, & illi <expan abbr="compon&etilde;t">component</expan> <expan abbr="numer&utilde;">numerum</expan> <lb/><expan abbr="quadrat&utilde;">quadratum</expan> <expan abbr="prior&etilde;">priorem</expan> quem diui&longs;i&longs;ti. </s>

Line 7949

<s id="id002662">Et ita poteris diuidere 32 in infinitos alios <expan abbr="&qtilde;d">quad</expan>.</s>

<s id="id002663">Sexta regula, ponamus modò &qring;d uelim diuidere 10, <expan abbr="cõpo&longs;it&utilde;">compo&longs;itum</expan> ex <lb/>duob. </s>

<s id="id002663">Sexta regula, ponamus modò quod uelim diuidere 10, <expan abbr="cõpo&longs;it&utilde;">compo&longs;itum</expan> ex <lb/>duob. </s>

<s id="id002665">9 & 1, & non <expan abbr="dupl&utilde;">duplum</expan> numero <expan abbr="&qtilde;d">quad</expan>. </s>

<s id="id002666">ita &qring;d &longs;it diui&longs;us in alios <lb/>duos: <expan abbr="ducã">ducam</expan> 10 in 25 <expan abbr="cõpo&longs;it&utilde;">compo&longs;itum</expan> ex duob. </s>

<s id="id002666">ita quod &longs;it diui&longs;us in alios <lb/>duos: <expan abbr="ducã">ducam</expan> 10 in 25 <expan abbr="cõpo&longs;it&utilde;">compo&longs;itum</expan> ex duob. </s>

<s id="id002668">fit 250/25, at 250 <expan abbr="cõponi&ttilde;">componitur</expan> aliter <lb/>ex duob. </s>

<s id="id002670"><08> 225/25 & 25/25, &longs;cilicet 169/25 & 81/25, id e&longs;t 6 19/25 & 3 6/25, qui &longs;unt <expan abbr="&qtilde;d">quad</expan>. <lb/></s>

<s id="id002671">2 3/5 & 1 4/5, & ita uolo diuidere 13 in duo alia <expan abbr="&qtilde;drata">quadrata</expan> <08> 9 & 4, duco 13 in <lb/>25 & fit 325/25, qui nece&longs;&longs;ario <expan abbr="cõponi&ttilde;">componitur</expan> ex 225/25 & 100/25, &longs;ed ego uolo &qring;d <expan abbr="cõpo">compo</expan> <lb/><expan abbr="na&ttilde;">natur</expan> aliter, uelut ex 289/25 & 63/25, & ita ex 11 14/25 & 1 11/25, qui &longs;unt numeri <expan abbr="&qtilde;d">quad</expan>. </s>

<s id="id002672">com <lb/>ponentes 13, & <02> &longs;unt 3 2/5 & 1 1/5, & in his opus e&longs;t in du&longs;tria, &longs;cilicet ut <lb/><expan abbr="multiplice&ttilde;">multiplicetur</expan> per numeros <expan abbr="&qtilde;d">quad</expan>. </s>

<s id="id002673">ut proueniant numeri illi <expan abbr="bifariã">bifariam</expan> compo <lb/>&longs;iti ex <expan abbr="&qtilde;dratis">quadratis</expan>. </s>

<s id="id002673">ut proueniant numeri illi <expan abbr="bifariã">bifariam</expan> comp <lb/>&longs;iti ex <expan abbr="&qtilde;dratis">quadratis</expan>. </s>

<s id="id002674">Vt uerò uideamus <expan abbr="re&longs;idu&utilde;">re&longs;iduum</expan>, proponamus quae uelim diui <lb/>dere 6 in duos numeros <expan abbr="&qtilde;d">quad</expan>, <expan abbr="prim&utilde;">primum</expan> &longs;cire debes &qring;d non po&longs;&longs;unt e&longs;&longs;e

Line 7977

<s id="id002685">fit <expan abbr="tot&utilde;">totum</expan>, <lb/>ergo ex medietate in <expan abbr="&qtilde;d">quad</expan>. </s>

<s id="id002686">idem fit medietas, &longs;ed medietas e&longs;t nume<lb/>rus <expan abbr="&qtilde;d">quad</expan>. </s>

<s id="id002687">ergo 3 e&longs;&longs;et numerus <expan abbr="&qtilde;d">quad</expan>. </s>

<s id="id002688">&qring;d e&longs;t fal&longs;um, oportet <expan abbr="igi&ttilde;">igitur</expan> ut nume <lb/>ri illi &longs;int inæ quales, & ut 6 diuidatur in duas partes in&etail;quales, hoc <lb/><expan abbr="a&utilde;t">aunt</expan> fit diuidendo quemlibet <expan abbr="numer&utilde;">numerum</expan> parem, qui <expan abbr="cõponi&ttilde;">componitur</expan> ex duob. <lb/></s>

<s id="id002688">quod e&longs;t fal&longs;um, oportet <expan abbr="igi&ttilde;">igitur</expan> ut nume <lb/>ri illi &longs;int inæ quales, & ut 6 diuidatur in duas partes in&etail;quales, hoc <lb/><expan abbr="a&utilde;t">aut</expan> fit diuidendo quemlibet <expan abbr="numer&utilde;">numerum</expan> parem, qui <expan abbr="cõponi&ttilde;">componitur</expan> ex duob. <lb/></s>

<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>

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 relin quitur 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 relin quat 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, & relin quitur 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, relin quitur <lb/>6, duc in 6 fit 36, diuide per 11, relin quitur 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/>relin quunt 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 &qring;&longs;upere&longs;t 2, <expan abbr="cõ">com</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;">numerrorum</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="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 8004

<pb pagenum="152" xlink:href="015/01/171.jpg"/>qui diui&longs;us per 6 &longs;upererit 3, & in paribus qui poterit diuidi per 6. <lb/>Quia <expan abbr="componun&ttilde;">componuntur</expan> ex huiu&longs;modi: uelut 3 in &longs;e facit 9, & 25 in &longs;e facit <lb/>225, qui <expan abbr="i&utilde;cti">iuncti</expan> <expan abbr="faci&utilde;t">faciunt</expan> 234, diui&longs;o 235 per 6 exit 39, qui <expan abbr="iter&utilde;">iterum</expan> diui&longs;us per 6 <lb/>&longs;upere&longs;t 3, & &longs;imiliter capio 6 & 12, <expan abbr="quor&utilde;">quorum</expan> <expan abbr="&qtilde;drata">quadrata</expan> &longs;unt 36 & 144, & <lb/><expan abbr="aggregat&utilde;">aggregatum</expan> 180, qui diui&longs;us per 6 exit 30, qui <expan abbr="iter&utilde;">iterum</expan> pote&longs;t diuidi per <lb/>6. Et hoc quia <expan abbr="quilibetillor&utilde;">quilibetillorum</expan> pote&longs;t diuidi per <expan abbr="&qtilde;drat&utilde;">quadratum</expan> 6 in paribus, <lb/>ergo aggregato diui&longs;o per 6 &qring;d prodit, <expan abbr="iter&utilde;">iterum</expan> poterit diuidi per 6. <lb/>Et in imparibus quo dlibet <expan abbr="&qtilde;drator&utilde;">quadratorum</expan> exuperat &longs;upra &longs;enarios in 3, <lb/><expan abbr="igi&ttilde;">igitur</expan> <expan abbr="aggregat&utilde;">aggregatum</expan> diui&longs;um in 2 pariet <expan abbr="numer&utilde;">numerum</expan> qui diui&longs;us per 3, exibit <lb/>numerus impar <expan abbr="cõpo&longs;itus">compo&longs;itus</expan> ex &longs;enarijs & 3. Illud ergo <expan abbr="quadrat&utilde;">quadratum</expan>, &qring;d <lb/>prodibit, uel erit <expan abbr="cõpo&longs;itum">compo&longs;itum</expan> ex &longs;enarijs, uel &longs;upererit 3. Sed <expan abbr="c&utilde;">cum</expan> 3 nume <lb/>ret 6, ergo tres <expan abbr="&qtilde;drati">quadrati</expan> numeri &longs;cilicet duo, qui <expan abbr="cõponunt">componunt</expan> <expan abbr="numer&utilde;">numerum</expan>, <lb/><arrow.to.target n="marg521"/><lb/>& qui prodit per <expan abbr="diui&longs;ion&etilde;">diui&longs;ionem</expan> 6, erunt <expan abbr="cõpo&longs;iti">compo&longs;iti</expan> inter &longs;e, ergo & radices il <lb/>lorum. </s>

<s id="id002701"><expan abbr="Igi&ttilde;">Igitur</expan> radix numeri <expan abbr="&qtilde;drati">quadrati</expan>, qui prouenit diui&longs;o aggregato <expan abbr="quadrator&utilde;">qua<lb/>dratorum</expan> per 6 e&longs;t ex <expan abbr="eod&etilde;">eodem</expan> ordine <expan abbr="impari&utilde;">imparium</expan>, &longs;i impares numeri <expan abbr="&qtilde;drati">quadrati</expan> <lb/><expan abbr="fuer&utilde;t">fuerunt</expan>, aut <expan abbr="pari&utilde;">parium</expan> &longs;i pares. </s>

<s id="id002702">At hoc e&longs;&longs;e <expan abbr="nõ">non</expan> pote&longs;t, <expan abbr="nã">nam</expan> fracti illi numeri, <lb/>qui <expan abbr="er&utilde;t">erunt</expan> radices, <expan abbr="nõ">non</expan> <expan abbr="er&utilde;t">erunt</expan> minimi, &longs;ed diui&longs;i per 3 o&longs;tendent minores, <lb/>quod e&longs;t contra &longs;uppo&longs;itum, quare nullo modo 6 pote&longs;t diuidi in <lb/>duos numeros quadratos, neque integros, neque fractos, quod erat <lb/>demon&longs;trandum. </s>

<s id="id002703">Habes igitur ex hoc demon&longs;trationem quando <lb/><expan abbr="nõ">non</expan> po&longs;sit diuidi, & quado po&longs;sit, quod po&longs;sit, & quomodo &longs;imul.</s>

<s id="id002703">Habes igitur ex hoc demon&longs;trationem quando <lb/><expan abbr="nõ">non</expan> po&longs;sit diuidi, & quando po&longs;sit, quod po&longs;sit, & quomodo &longs;imul.</s>

<s id="id002704"><margin.target id="marg521"/>P<emph type="italics"/>er<emph.end type="italics"/> 29. <emph type="italics"/>&longs;e<lb/>ptimi<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

Line 8019

<s id="id002708">Contingit quando que &qring;d <expan abbr="horologior&utilde;">horologiorum</expan> tem <lb/><figure id="id.015.01.171.1.jpg" xlink:href="015/01/171/1.jpg"/><lb/>pus breue e&longs;t, uolumus <expan abbr="a&utilde;t">aunt</expan> maius efficere: id <lb/>duob. </s>

<s id="id002708">Contingit quandoque &qring;d <expan abbr="horologior&utilde;">horologiorum</expan> tem<lb/><figure id="id.015.01.171.1.jpg" xlink:href="015/01/171/1.jpg"/><lb/>pus breue e&longs;t, uolumus <expan abbr="a&utilde;t">aut</expan> maius efficere: id <lb/>duob. </s>

<s id="id002709">modis po&longs;&longs;umus, <expan abbr="quor&utilde;">quorum</expan> unus diffici<lb/>lior e&longs;t &longs;ed perpetuus, & longè nobilior, nam <lb/>grauitas ponderis uer&longs;atilis efficit <expan abbr="quid&etilde;">quidem</expan> <expan abbr="tardior&etilde;">tar<lb/>diorem</expan>, &longs;ed di fficilius <expan abbr="mobil&etilde;">mobilem</expan>, & ob id grauio<lb/>re <expan abbr="põdere">pondere</expan> in <expan abbr="digent&etilde;">digentem</expan>. </s>

<s id="id002710">Sit ergo rota a b uer&longs;ati<lb/>lis, quæ certam men&longs;uram exigit pro quacunque funis parte corre&longs;peron <lb/>dentis uni denti ex centum, in quos di&longs;tincta &longs;it, curriculum <expan abbr="a&utilde;t">aunt</expan> c d <lb/>quinque <expan abbr="denti&utilde;">dentium</expan>, per &qring;drota &longs;exaginta dentes <expan abbr="hab&etilde;s">habens</expan> <expan abbr="circumuolua&ttilde;">circumuoluatur</expan> in <lb/><expan abbr="cõuer&longs;ione">conuer&longs;ione</expan>, <expan abbr="igi&ttilde;">igitur</expan> prim&etail; rot&etail; uities <expan abbr="circumfere&ttilde;">circumferetur</expan>, <expan abbr="&longs;ec&utilde;da">&longs;ecunda</expan> <expan abbr="d&etilde;tesque">dentesque</expan> M. CC. <lb/>rur&longs;us ad <expan abbr="hãc">hanc</expan> <expan abbr="&longs;ecundã">&longs;ecundam</expan> tertia <expan abbr="necta&ttilde;">nectatur</expan> cum curriculo &longs;ex <expan abbr="denti&utilde;">dentium</expan>, atque in <lb/>ea <expan abbr="d&etilde;tes">dentes</expan> &longs;eptuaginta duo, ut in una <expan abbr="cõuer&longs;ione">conuer&longs;ione</expan> &longs;int xiiij cccc, dentes <lb/><expan abbr="igi&ttilde;">igitur</expan> tot dentes in una <expan abbr="cõuer&longs;ione">conuer&longs;ione</expan> prim&etail; rot&etail; circumuoluentur. </s>

<s id="id002710">Sit ergo rota a b uer&longs;ati<lb/>lis, quæ certam men&longs;uram exigit pro quacunque funis parte corre&longs;peron<lb/>dentis uni denti ex centum, in quos di&longs;tincta &longs;it, curriculum <expan abbr="a&utilde;t">aut</expan> c d <lb/>quinque <expan abbr="denti&utilde;">dentium</expan>, per &qring;drota &longs;exaginta dentes <expan abbr="hab&etilde;s">habens</expan> <expan abbr="circumuolua&ttilde;">circumuoluatur</expan> in <lb/><expan abbr="cõuer&longs;ione">conuer&longs;ione</expan>, <expan abbr="igi&ttilde;">igitur</expan> prim&etail; rot&etail; uities <expan abbr="circumfere&ttilde;">circumferetur</expan>, <expan abbr="&longs;ec&utilde;da">&longs;ecunda</expan> <expan abbr="d&etilde;tesque">dentesque</expan> M. CC. <lb/>rur&longs;us ad <expan abbr="hãc">hanc</expan> <expan abbr="&longs;ecundã">&longs;ecundam</expan> tertia <expan abbr="necta&ttilde;">nectatur</expan> cum curriculo &longs;ex <expan abbr="denti&utilde;">dentium</expan>, atque in <lb/>ea <expan abbr="d&etilde;tes">dentes</expan> &longs;eptuaginta duo, ut in una <expan abbr="cõuer&longs;ione">conuer&longs;ione</expan> &longs;int xiiij cccc, dentes <lb/><expan abbr="igi&ttilde;">igitur</expan> tot dentes in una <expan abbr="cõuer&longs;ione">conuer&longs;ione</expan> prim&etail; rot&etail; circumuoluentur. </s>

<s id="id002711">Iam <lb/>uerò tempus illud poterit duplicari ac triplicari iuxta <expan abbr="tarditat&etilde;">tarditatem</expan> tem <lb/>poris uer&longs;atilis: <expan abbr="quãto">quanto</expan> <expan abbr="igi&ttilde;">igitur</expan> pondero&longs;ius fuerit illud <expan abbr="t&etilde;pus">tempus</expan>, tanto tar<lb/>dius <expan abbr="mouebi&ttilde;">mouebitur</expan>, pauciores que circumuolutiones nece&longs;&longs;ari&etail; <expan abbr="er&utilde;t">erunt</expan> ad <expan abbr="expl&etilde;dam">ex<lb/>plendam</expan> unam <expan abbr="di&etilde;">diem</expan>: id e&longs;t horas 24, &longs;ed hoc in <expan abbr="cõmodi">commodi</expan> accedet, quòd <lb/>reuolutio indicis tanto tardior erit, ut <expan abbr="nõ">non</expan> iu&longs;tè o&longs;ten dat horas: pro

<pb pagenum="153" xlink:href="015/01/172.jpg"/>po&longs;itum igitur e&longs;t, ut pondera tardius ferantur, index <expan abbr="a&utilde;t">aunt</expan>, & qu&etail; ad <lb/>indicem &longs;equuntur horarum demon&longs;trationes celerius aut eodem <lb/>modo ferantur. </s>

<pb pagenum="153" xlink:href="015/01/172.jpg"/>po&longs;itum igitur e&longs;t, ut pondera tardius ferantur, index <expan abbr="a&utilde;t">aut</expan>, & qu&etail; ad <lb/>indicem &longs;equuntur horarum demon&longs;trationes celerius aut eodem <lb/>modo ferantur. </s>

<s id="id002712">Ponamus ergo po&longs;t<08> eadem e&longs;t ratio celerioris & <lb/>æqué uelocis, ponderis <expan abbr="a&utilde;t">aunt</expan> tardius de&longs;cendentis, aut <expan abbr="cõtrà">contrà</expan> tardio<lb/>ris, aut æqualiter cir cumducti in dicis, celerioris <expan abbr="a&utilde;t">aunt</expan> de&longs;cen&longs;us pon<lb/>deris, quod ad nullam <expan abbr="utilitat&etilde;">utilitatem</expan> profuturum uideo. </s>

<s id="id002712">Ponamus ergo po&longs;t<08> eadem e&longs;t ratio celerioris & <lb/>æqué uelocis, ponderis <expan abbr="a&utilde;t">aut</expan> tardius de&longs;cendentis, aut <expan abbr="cõtrà">contrà</expan> tardio<lb/>ris, aut æqualiter circumducti in dicis, celerioris <expan abbr="a&utilde;t">aut</expan> de&longs;cen&longs;us pon<lb/>deris, quod ad nullam <expan abbr="utilitat&etilde;">utilitatem</expan> profuturum uideo. </s>

<s id="id002713">Sit ergo ut pon <lb/>dus uelim tardius de&longs;cendere, rotam <expan abbr="a&utilde;t">aunt</expan> &etail;qualiter circumferri, dico <lb/>quod ex tempore mobili &longs;eu uer&longs;atili (& e&longs;t ferrum, quod in &longs;um<lb/>mo horologij citra ultraque <expan abbr="fer&ttilde;">fertur</expan> tam in horologijs ponderum <08> mo <lb/>læ) id fieri non pote&longs;t: nam quantum tardabitur rota tertia &longs;ecunda <lb/>& prima, atque ob id de&longs;cen&longs;us ponderum, tantum remorabitur rota <lb/>prima quæ indicem o&longs;tendit, ergo tantum index tardabitur quan<lb/>rum <expan abbr="põdera">pondera</expan>, & ut uno uerbo dicam, cùm <expan abbr="ead&etilde;">eadem</expan> rota index circumfe<lb/>ratur, & <expan abbr="põdus">pondus</expan> de&longs;cendat, <expan abbr="quant&utilde;">quantum</expan> unum tardatur tantum & aliud.</s>

<s id="id002713">Sit ergo ut pon<lb/>dus uelim tardius de&longs;cendere, rotam <expan abbr="a&utilde;t">aut</expan> &etail;qualiter circumferri, dico <lb/>quod ex tempore mobili &longs;eu uer&longs;atili (& e&longs;t ferrum, quod in &longs;um<lb/>mo horologij citra ultraque <expan abbr="fer&ttilde;">fertur</expan> tam in horologijs ponderum <08> mo <lb/>læ) id fieri non pote&longs;t: nam quantum tardabitur rota tertia &longs;ecunda <lb/>& prima, atque ob id de&longs;cen&longs;us ponderum, tantum remorabitur rota <lb/>prima quæ indicem o&longs;tendit, ergo tantum index tardabitur quan<lb/>tum <expan abbr="põdera">pondera</expan>, & ut uno uerbo dicam, cùm <expan abbr="ead&etilde;">eadem</expan> rota index circumfe<lb/>ratur, & <expan abbr="põdus">pondus</expan> de&longs;cendat, <expan abbr="quant&utilde;">quantum</expan> unum tardatur tantum & aliud.</s>

<s id="id002714">Secundus modus e&longs;t, ut rota una totum tempus cum indice in ui <lb/>gintiquatuor horis circumuoluatur, & currulis in quo funis minor <lb/>fiat: nece&longs;&longs;e e&longs;t <expan abbr="igi&ttilde;">igitur</expan>, ut circumuoluta rota aut &longs;emel aut bis, <expan abbr="&ttilde;er">turer</expan>, qua<lb/>ter decies, & <expan abbr="circumuolua&ttilde;">circumuoluatur</expan> pleno cir cuitu index, et &longs;ine errore: quo<lb/>niam tempus & dentes men&longs;uræ re&longs;pondent: igitur &longs;ub ei&longs;dem cir<lb/>cuitibus numero eodemque tempore minus ex fune <expan abbr="de&longs;cend&etilde;t">de&longs;cendent</expan> in cur <lb/>ruli paruo <08> magno: quare mutatione indiget currulis, aut ut funis <lb/>circumuoluens rotam curriculum habeat <expan abbr="annex&utilde;">annexum</expan> rotæ o&longs;ten denti <lb/>horas, in qua pauciores &longs;int dentes: nam in eodem tempore, & cir<lb/>cuitu paucioribus uicibus circumuoluitur rota funis quæ grauita<lb/>te temporis, & multitudine <expan abbr="denti&utilde;">dentium</expan> certam <lb/><figure id="id.015.01.172.1.jpg" xlink:href="015/01/172/1.jpg"/><lb/>&longs;eruabit <expan abbr="men&longs;urã">men&longs;uram</expan>. </s>

<s id="id002714">Secundus modus e&longs;t, ut rota una totum tempus cum indice in ui<lb/>gintiquatuor horis circumuoluatur, & currulis in quo funis minor <lb/>fiat: nece&longs;&longs;e e&longs;t <expan abbr="igi&ttilde;">igitur</expan>, ut circumuoluta rota aut &longs;emel aut bis, <expan abbr="&ttilde;er">tur</expan>, qua<lb/>ter decies, & <expan abbr="circumuolua&ttilde;">circumuoluatur</expan> pleno circuitu index, et &longs;ine errore: quo<lb/>niam tempus & dentes men&longs;uræ re&longs;pondent: igitur &longs;ub ei&longs;dem cir<lb/>cuitibus numero eodemque tempore minus ex fune <expan abbr="de&longs;cend&etilde;t">de&longs;cendent</expan> in cur<lb/>ruli paruo <08> magno: quare mutatione indiget currulis, aut ut funis <lb/>circumuoluens rotam curriculum habeat <expan abbr="annex&utilde;">annexum</expan> rotæ o&longs;ten denti <lb/>horas, in qua pauciores &longs;int dentes: nam in eodem tempore, & cir<lb/>cuitu paucioribus uicibus circumuoluitur rota funis quæ grauita<lb/>te temporis, & multitudine <expan abbr="denti&utilde;">dentium</expan> certam <lb/><figure id="id.015.01.172.1.jpg" xlink:href="015/01/172/1.jpg"/><lb/>&longs;eruabit <expan abbr="men&longs;urã">men&longs;uram</expan>. </s>

<s id="id002715">Sed in hoc nece&longs;&longs;e e&longs;t gra<lb/>uius efficere pondus, aut leuius <expan abbr="t&etilde;pus">tempus</expan> <expan abbr="quoniã">quo<lb/>niam</expan> funis debilius circumuertit <expan abbr="rotã">rotam</expan>: minus <lb/><expan abbr="tñ">tnm</expan> tardè <08> &longs;it pro paruitatis circuitus ratione.</s>

<s id="id002715">Sed in hoc nece&longs;&longs;e e&longs;t gra<lb/>uius efficere pondus, aut leuius <expan abbr="t&etilde;pus">tempus</expan> <expan abbr="quoniã">quo<lb/>niam</expan> funis debilius circumuertit <expan abbr="rotã">rotam</expan>: minus <lb/><expan abbr="tñ">tn</expan> tardè quod &longs;it pro paruitatis circuitus ratione.</s>

<s id="id002716">Tertius modus facilior e&longs;t, & magis com <lb/><expan abbr="p&etilde;dio&longs;us">pendio&longs;us</expan>: Sit horologium a b c, in quo rota <lb/>d quæ funem <expan abbr="cõtinet">continet</expan> ba&longs;is horologij e f, cui <lb/>firmiter &longs;int <expan abbr="app&etilde;&longs;&etail;">appen&longs;&etail;</expan> du&etail; trochle&etail; g & h, & fu <lb/>nis una parte tro chle&etail; appen&longs;us in k, <expan abbr="duca&ttilde;">ducatur</expan> <lb/>ad inferiorem aliam tro chleam lin&longs;eraturque<lb/>ibi orbiculo &longs;uo, & redeat à dextra &longs;uperius <lb/><expan abbr="in&longs;era&ttilde;que">in&longs;eraturque</expan> orbiculo &longs;uperioris tro chle&etail;, dedu <lb/><expan abbr="ca&ttilde;que">caturque</expan> uer&longs;us <expan abbr="&longs;ini&longs;trã">&longs;ini&longs;tram</expan>: atque ibi <expan abbr="de&longs;cend&etilde;s">de&longs;cendens</expan> habe <lb/>at <expan abbr="põdus">pondus</expan> tractorium in m, <expan abbr="deduca&ttilde;que">deducaturque</expan> &longs;upra <lb/>ad <expan abbr="rotã">rotam</expan> horologij d, et cir cumuolutus exeat <lb/>ip&longs;um, & <expan abbr="de&longs;c&etilde;dat">de&longs;cendat</expan> ad tro <expan abbr="chleãn">chleann</expan>, &longs;ub que ea circumuolutus <expan abbr="iter&utilde;">iterum</expan> a&longs;cen

<s id="id002716">Tertius modus facilior e&longs;t, & magis com<lb/><expan abbr="p&etilde;dio&longs;us">pendio&longs;us</expan>: Sit horologium a b c, in quo rota <lb/>d quæ funem <expan abbr="cõtinet">continet</expan> ba&longs;is horologij e f, cui <lb/>firmiter &longs;int <expan abbr="app&etilde;&longs;&etail;">appen&longs;&etail;</expan> du&etail; trochle&etail; g & h, & fu <lb/>nis una parte trochle&etail; appen&longs;us in k, <expan abbr="duca&ttilde;">ducatur</expan> <lb/>ad inferiorem aliam trochleam l in&longs;eraturque <lb/>ibi orbiculo &longs;uo, & redeat à dextra &longs;uperius <lb/><expan abbr="in&longs;era&ttilde;que">in&longs;eraturque</expan> orbiculo &longs;uperioris trochle&etail;, dedu<lb/><expan abbr="ca&ttilde;que">caturque</expan> uer&longs;us <expan abbr="&longs;ini&longs;trã">&longs;ini&longs;tram</expan>: atque ibi <expan abbr="de&longs;cend&etilde;s">de&longs;cendens</expan> habe <lb/>at <expan abbr="põdus">pondus</expan> tractorium in m, <expan abbr="deduca&ttilde;que">deducaturque</expan> &longs;upra <lb/>ad <expan abbr="rotã">rotam</expan> horologij d, et circumuolutus exeat <lb/>ip&longs;um, & <expan abbr="de&longs;c&etilde;dat">de&longs;cendat</expan> ad tro<expan abbr="chleãn">chlean</expan>, &longs;ub que ea circumuolutus <expan abbr="iter&utilde;">iterum</expan> a&longs;cen<pb pagenum="154" xlink:href="015/01/173.jpg"/>dat à dextra parte, et circumuoluatur h cochle&etail; rediens ad &longs;ini&longs;tram <lb/>ibique de&longs;cendens connectatur trochleæ in inferiori in o, cuius imæ <lb/>parti annectatur pondus remorans in imo annexum parte troch<lb/>leæ p. </s>

<pb pagenum="154" xlink:href="015/01/173.jpg"/>dat à dextra parte, et circumuoluatur h co chle&etail; rediens ad &longs;ini&longs;tram <lb/>ibique de&longs;cendens connectatur tro chleæ in inferiori in o, cuius imæ <lb/>parti annectatur pondus remorans in imo annexum parte tro ch<lb/>leæp. </s>

<s id="id002717">Cum ergo trahitur n tro chlea, trahitur funis adeò ut pon<lb/>dus m, tandem a&longs;cendat cum tro chleal prope k: quia ergo in duo<lb/>decim horis pondus m de&longs;cenderet per k l funem reuolutionibus <lb/>circa d rotam dicamus uiginti, ergo &longs;i debet de&longs;cendere à k ad l, per <lb/>funem duplicatam k l cum ip&longs;am nece&longs;&longs;e &longs;it obequitantem d reuo<lb/>lutionibus quadraginta circumuolui d, nam tota o h n d m g l k lon <lb/>gè maior e&longs;t duplo k l, nece&longs;&longs;e e&longs;t m de&longs;cendere tardius quàm in du <lb/>plo temporis, quo de&longs;cenderet per rectum funem k l, quod erat de<lb/>mon&longs;trandum. </s>

<s id="id002718">Et hanc appendicem uidi apud Cæ&longs;arem Odonum <lb/>Apulum medicum, uirum elegantem lepidique ingenij. </s>

<s id="id002719">Memento <lb/>uerò quod ubi orbiculi non cederent funi, uel quia duriores in cir<lb/>cumuolutione, uel quia latius exciperent illum reduplicato fune <lb/>circa illos omnin o circumducuntur, &longs;ed difficilius ideò egent gra<lb/>uiori pondere.</s>

<s id="id002720">Propo&longs;itio cente&longs;imaquinquage&longs;ima&longs;eptima.</s>

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<s id="id002724">Sunt horum duo genera primum, & anti <lb/><figure id="id.015.01.173.1.jpg" xlink:href="015/01/173/1.jpg"/><lb/>quius licet multo po&longs;terius eo quod pon<lb/>deribus ducitur, quod funiculo ex inte&longs;ti<lb/>nis ouium &longs;eu fidibus liræ agitur. </s>

<s id="id002725">Sit igitur <lb/>axis f k erectus &longs;uper plano, cui per longum <lb/>coniuncta mola multiplicis &longs;piræ in fine, cu <lb/>ius cannectatur ferreo circulo, qui habeatur lo co cap&longs;ulæ b c, quæ <lb/>circumuolui po&longs;sit: huic <expan abbr="circ&utilde;ductus">circunductus</expan> funis d e multipliciter in pun <lb/>cto g, &longs;it autem e h in modum pyramidis &longs;en&longs;im in acutum, &longs;ed non <lb/>ualde per <expan abbr="&longs;pirã">&longs;piram</expan> exculptam de&longs;inentis, cui rota in uertice in&longs;erta den <lb/>&longs;iculo, & uertatur h e, colligens funiculum tractum in &longs;pira uer&longs;us <lb/>apicem: unde funiculus circumuoluet b g d, <expan abbr="cap&longs;ulã">cap&longs;ulam</expan> uer&longs;us c, traher <lb/>ergo molam, & con&longs;trin get uiolenter <expan abbr="quãtum">quantum</expan> fert longitudo funis <lb/>quæ circumuolui pote&longs;t a b e ad h: & cum trahitur in d eremittitur, <lb/>non pote&longs;t mola &longs;tatim retrahere reluctantibus denticulis h l rotæ, <lb/>& alijs quæ implicantur curriculo m, a igitur mola con&longs;tructa uio<lb/>lenter mouet b g d, cap&longs;ulam motu contrario à c in d & in g & in b, <lb/>quare funis d e trahitur, & trahit e h illum circumuoluendo contra<lb/>rio motu priori, is mouet denticulo rotam h l, illa per curriculum in <lb/>aliam <expan abbr="rotã">rotam</expan>, & &longs;ic deinceps donec tempus moueatur, & rota indicis. <lb/></s>

<s id="id002725">Sit igitur <lb/>axis f k erectus &longs;uper plano, cui per longum <lb/>coniuncta mola multiplicis &longs;piræ in fine, cu<lb/>ius c annectatur ferreo circulo, qui habeatur loco cap&longs;ulæ b c, quæ <lb/>circumuolui po&longs;sit: huic <expan abbr="circ&utilde;ductus">circunductus</expan> funis d e multipliciter in pun<lb/>cto g, &longs;it autem e h in modum pyramidis &longs;en&longs;im in acutum, &longs;ed non <lb/>ualde per <expan abbr="&longs;pirã">&longs;piram</expan> exculptam de&longs;inentis, cui rota in uertice in&longs;erta den<lb/>&longs;iculo, & uertatur h e, colligens funiculum tractum in &longs;pira uer&longs;us <lb/>apicem: unde funiculus circumuoluet b g d, <expan abbr="cap&longs;ulã">cap&longs;ulam</expan> uer&longs;us c, trahet <lb/>ergo molam, & con&longs;tringet uiolenter <expan abbr="quãtum">quantum</expan> fert longitudo funis <lb/>quæ circumuolui pote&longs;t a b e ad h: & cum trahitur in d eremittitur, <lb/>non pote&longs;t mola &longs;tatim retrahere reluctantibus denticulis h l rotæ, <lb/>& alijs quæ implicantur curriculo m, a igitur mola con&longs;tructa uio<lb/>lenter mouet b g d, cap&longs;ulam motu contrario à c in d & in g & in b, <lb/>quare funis d e trahitur, & trahit e h illum circumuoluendo contra<lb/>rio motu priori, is mouet denticulo rotam h l, illa per curriculum in <lb/>aliam <expan abbr="rotã">rotam</expan>, & &longs;ic deinceps donec tempus moueatur, & rota indicis. <lb/></s>

<s id="id002726">Hic ade&longs;t cap&longs;ula, & quod circumuertitur à claue non e&longs;t axis mol&etail; <lb/>&longs;ed extra molam, &longs;cilicet e h. </s>

<s id="id002727">Et quoniam hac ratione quanto mola a

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<pb pagenum="155" xlink:href="015/01/174.jpg"/>magis <expan abbr="explicabi&ttilde;">explicabitur</expan>, tanto lentius trahet, & uertet e h, ideò hoc ex &longs;tru <lb/>ctura auxilium præ&longs;tatur, ut funis in inferiore parte <expan abbr="cõplexus">complexus</expan> latio<lb/>res orbes, & è regione tanto uehementius uertat e h: & ita uis quæ <lb/>remittitur ob molæ laxitatem, augetur tantundem ob &longs;itum & ma<lb/>gnitudinem &longs;pirarum ut di&longs;tantiorum &longs;ua extremitate ab hypomo <lb/>chlio, quod e&longs;t axis coni e h, &longs;eu in&longs;tar axis.</s>

<s id="id002728">Alterum genus horologiorum cum mola &longs;ine fune loco cap&longs;ul&etail; <lb/>habet <expan abbr="rotã">rotam</expan> plano &longs;ub &longs;tratam, plenam denticulis axis, quo circum<lb/>agitur uiolenter, non e&longs;t extra molam, &longs;ed ei annexa e&longs;t mola intus, <lb/>exterius <expan abbr="a&utilde;t">aunt</expan> rot&etail;; ergo circumducto axe mol&etail; uim patitur circulus <lb/>exterior, &longs;ed non <expan abbr="moue&ttilde;">mouetur</expan>, quoniam clauo <expan abbr="impedi&ttilde;">impeditur</expan>. </s>

<s id="id002728">Alterum genus horologiorum cum mola &longs;ine fune loco cap&longs;ul&etail; <lb/>habet <expan abbr="rotã">rotam</expan> plano &longs;ub &longs;tratam, plenam denticulis axis, quo circum<lb/>agitur uiolenter, non e&longs;t extra molam, &longs;ed ei annexa e&longs;t mola intus, <lb/>exterius <expan abbr="a&utilde;t">aut</expan> rot&etail;; ergo circumducto axe mol&etail; uim patitur circulus <lb/>exterior, &longs;ed non <expan abbr="moue&ttilde;">mouetur</expan>, quoniam clauo <expan abbr="impedi&ttilde;">impeditur</expan>. </s>

<s id="id002729">Vbi mola quan<lb/>tum decet con&longs;tricta e&longs;t &longs;ublato clauo &longs;tatim &longs;ecum trahit rotam, & <lb/>illa <expan abbr="curricul&utilde;">curriculum</expan> rotas que alias, & tempus agitur, & index uertitur. </s>

<s id="id002730">Sed <lb/>in hoc idem e&longs;t in commodum &longs;ine remedio <lb/><figure id="id.015.01.174.1.jpg" xlink:href="015/01/174/1.jpg"/><lb/>quod fuit in priore. </s>

<s id="id002731">Vbi enim c&oelig;perit laxa<lb/>ri mola tanto tardius progrediuntur rotæ <lb/>atque index. </s>

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<s id="id002747">Hoc fieri pote&longs;t in &longs;ingulo genere horologij trium <expan abbr="de&longs;criptor&utilde;">de&longs;criptorum</expan>. <lb/></s>

<s id="id002748">Propterea &longs;ufficiat de uno o&longs;tendi&longs;&longs;e. </s>

<s id="id002749">Sed & in &longs;ingulo genere &longs;unt <lb/>multi modi, unius tamen reddidi&longs;&longs;e <expan abbr="ration&etilde;">rationem</expan> &longs;ufficiat. </s>

<s id="id002750">Hoc <expan abbr="a&utilde;t">aunt</expan> qua<lb/>tuor habet difficultates: prima ut horarum ictus conueniant cum <lb/>indice: &longs;ecunda ut conuer&longs;o indice conuertatur, & rota ictuum: ter <lb/>tia ut ictuum numerus cum numero indicis conueniat. </s>

<s id="id002750">Hoc <expan abbr="a&utilde;t">aut</expan> qua<lb/>tuor habet difficultates: prima ut horarum ictus conueniant cum <lb/>indice: &longs;ecunda ut conuer&longs;o indice conuertatur, & rota ictuum: ter<lb/>tia ut ictuum numerus cum numero indicis conueniat. </s>

<s id="id002751">Vnde mul<lb/>ta &longs;unt horologia, in quibus ictus unus &longs;olum auditur &longs;ingulis ho<lb/>ris, atque hic modus facilis e&longs;t: quarta cur in horum pleri&longs; que &longs;i non <lb/>pul&longs;ata &longs;tatim hora <expan abbr="transfera&ttilde;ur">transferaturur</expan> index, non ce&longs;&longs;at pul&longs;atio: imò nec <lb/>retineri pote&longs;t, donec pondus illud de&longs;cenderit. </s>

<s id="id002751">Vnde mul<lb/>ta &longs;unt horologia, in quibus ictus unus &longs;olum auditur &longs;ingulis ho<lb/>ris, atque hic modus facilis e&longs;t: quarta cur in horum pleri&longs; que &longs;i non <lb/>pul&longs;ata &longs;tatim hora <expan abbr="transfera&ttilde;ur">transferatur</expan> index, non ce&longs;&longs;at pul&longs;atio: imò nec <lb/>retineri pote&longs;t, donec pondus illud de&longs;cenderit. </s>

<s id="id002752">Ergo primi & ter<lb/>tij ratio hæc habeatur, cum rota qu&etail; indicis rotam circumagit, per<lb/>uenerit ad horæ finem, denticulo &longs;oluit aliam, eleuans obicem, illa <lb/>mouetur à pondere proprio alio, &longs;cilicet ab illo quod tempus agit: <lb/>aut &longs;i &longs;it horologium molæ à mola alia propria, quæ malleos cir<lb/>cumacta perpetuò mouet, atque motura e&longs;&longs;et &longs;emper, donec pondus <lb/>ad terram de&longs;cenderet: uerum dum mouetur de&longs;cendit ferrum pro <lb/>quouis ictu quod in rotæ limbum incidit, & donec inciderit in eam <lb/>partem quæ lenis e&longs;t dilabitur, nec retinetur, & ita eleuatur rur&longs;us,

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<pb pagenum="158" xlink:href="015/01/177.jpg"/>rantur: quinimò illud magis mirum de quo illi non mirantur quia <lb/>frequens e&longs;t, &longs;cilicet cur aut quomodo &longs;i diui&longs;æ &longs;unt ut cir <expan abbr="ç&utilde;ducto">çunducto</expan> <lb/>indice non transferatur rota mallei, <expan abbr="põdere">pondere</expan> tamen uer&longs;ata rota in<lb/>dicis in idem incidat, ut horæ quæ pul&longs;u declarantur ad unguem <lb/>& in ei&longs;dem &longs;ectionibus <expan abbr="cõueniant">conueniant</expan> cum horis quas index o&longs;ten dit.</s>

<s id="id002762">Ver&grave;m quia multis modis contingit ordinem horologiorum <lb/>peruerti: in &longs;imilibus quidem &longs;i hora indicis &longs;imul & pul&longs;us unà <lb/>circumferuntur, &longs;ed tardius ambo index traducitur ad locum debi<lb/>tum, inde ponderi aliquid additur. </s>

<s id="id002762">Verum quia multis modis contingit ordinem horologiorum <lb/>peruerti: in &longs;imilibus quidem &longs;i hora indicis &longs;imul & pul&longs;us unà <lb/>circumferuntur, &longs;ed tardius ambo index traducitur ad locum debi<lb/>tum, inde ponderi aliquid additur. </s>

<s id="id002763">Si uerò antè proce&longs;&longs;erit quam. <lb/></s>

<s id="id002764">Sol in dicet ablato pondere, &longs;ines tempus fluere u&longs;que ad indicis lo<lb/>cum &longs;ine motu horologij, pondus quoque ip&longs;um minues. </s>

<s id="id002765">At &longs;i pon<lb/>dus pul&longs;us in terram deuenerit uel propè, expecta donec &longs;uper li<lb/>nea index fuerit, inde trahe, neque. </s>

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<s id="id002792">Ex hoc etiam patet quod &longs;pacium con<lb/><arrow.to.target n="marg532"/><lb/><expan abbr="tent&utilde;">tentum</expan> à peripheria circuli nulli angulo rectilineo &etail;quale e&longs;&longs;e pote&longs;t. <lb/></s>

<s id="id002792">Ex hoc etiam patet quod &longs;patium con<lb/><arrow.to.target n="marg532"/><lb/><expan abbr="tent&utilde;">tentum</expan> à peripheria circuli nulli angulo rectilineo &etail;quale e&longs;&longs;e pote&longs;t. <lb/></s>

<s id="id002793">nam dimidium e&longs;&longs;et æquale dimidio, quod e&longs;t contra demon&longs;trata.</s>

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<s id="id002810"><margin.target id="marg538"/>P<emph type="italics"/>er modum<emph.end type="italics"/><lb/>8. <emph type="italics"/>primi<emph.end type="italics"/> E<emph type="italics"/>l.<emph.end type="italics"/></s>

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

<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 &qring;d 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="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">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;pacio 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="id002828">SCHOLIVM.</s>

<s id="id002829">Atque hæc omnia &longs;equuntur de mente Euclidis, quæ tamen ui<lb/>dentur difficillima creditu, quoniam anguli rectilinei, et ex periphe

<s id="id002829">Atque hæc omnia &longs;equuntur de mente Euclidis, quæ tamen ui<lb/>dentur difficillima creditu, quoniam anguli rectilinei, et ex periphe<pb pagenum="161" xlink:href="015/01/180.jpg"/>ria & recta &longs;unt ex genere quantitatis continuæ, & quòd detur ma<lb/>ius & minus & nunquam detur &etail;quale, uidetur ab&longs;urdum ne dum <lb/>admirabile. </s>

<pb pagenum="161" xlink:href="015/01/180.jpg"/>ria & recta &longs;unt ex genere quantitatis continuæ, & quòd detur ma<lb/>ius & minus & nunquam detur &etail;quale, uidetur ab&longs;urdum ne dum <lb/>admirabile. </s>

<s id="id002830">Et maximè quod etiam anguli ex peripheria & recta <lb/>&longs;unt diuer&longs;orum generum inter &longs;e & infinitorum. </s>

<s id="id002831">Pr&etail;terea i&longs;tud re<lb/>pugnare uidetur ip&longs;imet Euclidi, dicenti duabus magnitu dinibus <lb/><arrow.to.target n="marg547"/><lb/><arrow.to.target n="marg548"/><lb/>propo&longs;itis inæqualibus, &longs;i de maiore earum plus dimidio detraha<lb/>tur, atque iterum de re&longs;iduo maius dimidio, & rur&longs;us de eo quod re<lb/>linquitur plus dimidio, nece&longs;&longs;e erit ut tandem minor minore quan<lb/>titas relinquatur. </s>

<s id="id002832">Neque illud argumentum uidetur concludere an<lb/>gulus contactus, ex recta, & circuli circumferentia non pote&longs;t recta <lb/>diuidi, & rectilineus pote&longs;t diuidi, ergo rectilin eus &longs;emper e&longs;t ma<lb/>ior angulo contactus, quia hoc contingit in angulo contactus pro <lb/>pter modum anguli, non paruitatem: &longs;i cut etiam non ualet de figu<lb/><figure id="id.015.01.180.1.jpg" xlink:href="015/01/180/1.jpg"/><lb/>ra a lunari, & quadrangulo b. </s>

<s id="id002833">nam pote&longs;t b diuidi <lb/>ab angulo ad angulum recta & a non pote&longs;t, & <lb/>tamen a maius e&longs;t quam b, cum contineat ip&longs;am. <lb/></s>

<s id="id002834">Proponantur ergo duo circuli a d e & a f g qui &longs;e contingant in a, & <lb/>corum centra &longs;int b & c & ducantur rectæ a f d & a g e & con&longs;tat <lb/>&qring;d portiones a d & a f &longs;imiles &longs;unt, <lb/><figure id="id.015.01.180.2.jpg" xlink:href="015/01/180/2.jpg"/><lb/>itemque a e & a g, ducta enim a b c <lb/><arrow.to.target n="marg549"/><lb/>per centra circulorum ex contactu <lb/>tran&longs;ibit per illa: quare anguli h a g <lb/>& h a e &longs;untijdem & &longs;imiliter h a f <lb/>& h a d ijdem, portiones ergo af & <lb/>a d itemque a g & a e &longs;imiles &longs;unt: an<lb/>gulus igitur g a e ex peripherijs & <lb/><arrow.to.target n="marg550"/><lb/>e a d ex rectis &longs;unt ijdem in puncto <lb/>a: &longs;ed quod ad ba&longs;sim maior e&longs;t ba<lb/>&longs;is g e quam e d: hoc enim &longs;uppono <lb/>quod per &longs;e e&longs;t manife&longs;tum toties <lb/><expan abbr="diuid&etilde;do">diuidendo</expan> arcum d e ut fiat minor recta g e. </s>

<s id="id002834">Proponantur ergo duo circuli a d e & a f g qui &longs;e contingant in a, & <lb/>eorum centra &longs;int b & c & ducantur rectæ a f d & a g e & con&longs;tat <lb/>qui portiones a d & a f &longs;imiles &longs;unt, <lb/><figure id="id.015.01.180.2.jpg" xlink:href="015/01/180/2.jpg"/><lb/>itemque a e & a g, ducta enim a b c <lb/><arrow.to.target n="marg549"/><lb/>per centra circulorum ex contactu <lb/>tran&longs;ibit per illa: quare anguli h a g <lb/>& h a e &longs;unt ijdem & &longs;imiliter h a f <lb/>& h a d ijdem, portiones ergo af & <lb/>a d itemque a g & a e &longs;imiles &longs;unt: an<lb/>gulus igitur g a e ex peripherijs & <lb/><arrow.to.target n="marg550"/><lb/>e a d ex rectis &longs;unt ijdem in puncto <lb/>a: &longs;ed quod ad ba&longs;sim maior e&longs;t ba<lb/>&longs;is g e quam e d: hoc enim &longs;uppono <lb/>quod per &longs;e e&longs;t manife&longs;tum toties <lb/><expan abbr="diuid&etilde;do">diuidendo</expan> arcum d e ut fiat minor recta g e. </s>

<s id="id002835">Quia ergo &longs;unt du&etail; ma<lb/>gnitudines, quarum ter mini &longs;unt ijdem ex una parte, &longs;cilicet pun<lb/>ctum a, ex alia autem unus e&longs;t maior altero, &longs;cilicet g e quam e f & <lb/><arrow.to.target n="marg551"/><lb/>a d e peripheria e&longs;t maior recta a g e. </s>

<s id="id002836">Ergo per regulam dialecti<lb/>cam &longs;i &longs;ub eadem proportione procederent, maius e&longs;&longs;et &longs;patium <lb/>&longs;emper inter peripherias quàm rectas. </s>

<s id="id002837">igitur angulus peripheria<lb/>rum e&longs;t maior angulo à rectis contento. </s>

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<s id="id002882"><margin.target id="marg566"/>P<emph type="italics"/>er<emph.end type="italics"/> 4. <emph type="italics"/>&longs;exti<emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<s id="id002883"><margin.target id="marg567"/>P<emph type="italics"/>er<emph.end type="italics"/> 10. <emph type="italics"/>diff<lb/>tertij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<s id="id002883"><margin.target id="marg567"/>P<emph type="italics"/>er<emph.end type="italics"/> 10. <emph type="italics"/>diff <lb/>tertij<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<s id="id002884"><margin.target id="marg568"/>P<emph type="italics"/>er præce<lb/>dentem.<emph.end type="italics"/></s>

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<s id="id002929">Propo&longs;itionem motuum omnium <expan abbr="&longs;tellar&utilde;">&longs;tellarum</expan> ad &longs;olem con&longs;iderare.</s>

<s id="id002930">Videtur Sol qua&longs;i Rex in C&oelig;lo, nam omnes orbes cum illius <lb/><arrow.to.target n="marg577"/><lb/>motu conueniunt, & uideturres admiratione digna his, qui non <lb/>nouerunt, quanta &longs;it concordia omnium rerum, de qua infrà dice<lb/>mus. </s>

<s id="id002930">Videtur Sol qua&longs;i Rex in C&oelig;lo, nam omnes orbes cum illius <lb/><arrow.to.target n="marg577"/><lb/>motu conueniunt, & uidetur es admiratione digna his, qui non <lb/>nouerunt, quanta &longs;it concordia omnium rerum, de qua infrà dice<lb/>mus. </s>

<s id="id002931">Ergo Luna primum hoc habet, ut linea æqualis motu Solis <lb/>&longs;emper media &longs;it inter lineam æqualis motus Lun&etail; & loci maximè <lb/>inæqualitatis motus eius, ubi &longs;cilicet tardi&longs;simè mouetur, Veneris <lb/>autem & Mercurij ut motus æquales idem &longs;emper &longs;int cum motu <lb/>æquali, & locus cumloco ip&longs;ius Solis ad unguem præterid quod <lb/>infrà dicemus. </s>

<s id="id002932">Trium uerò <expan abbr="&longs;uperior&utilde;">&longs;uperiorum</expan> ratio &longs;ic <expan abbr="cõ&longs;tat">con&longs;tat</expan> ad Solem ut à <lb/>Prolem&etail;o <expan abbr="ob&longs;eruat&utilde;">ob&longs;eruatum</expan> e&longs;t ex Hipparcho. </s>

<s id="id002932">Trium uerò <expan abbr="&longs;uperior&utilde;">&longs;uperiorum</expan> ratio &longs;ic <expan abbr="cõ&longs;tat">con&longs;tat</expan> ad Solem ut à <lb/>Ptolem&etail;o <expan abbr="ob&longs;eruat&utilde;">ob&longs;eruatum</expan> e&longs;t ex Hipparcho. </s>

<s id="id002933">In omnire&longs;titutione cuiu&longs;<lb/>libet planet&etail; &longs;uperioris numerus <expan abbr="reuolution&utilde;">reuolutionum</expan> Solis &etail;qualis e&longs;t nu<lb/>mero <expan abbr="re&longs;titution&utilde;">re&longs;titutionum</expan> planet&etail; <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> <expan abbr="mot&utilde;">motum</expan> æqualitatis & in&etail;qualita <lb/>tis pariter acceptis. </s>

<s id="id002934">Velut Saturnus in annis quinquaginta nouem <lb/>die una & horis decem octo quinquage&longs;ies &longs;epties per motum in&etail;<lb/>qualem ad <expan abbr="ungu&etilde;">unguem</expan>, per æqualem autem duabus reuolutionibus par <lb/>te in&longs;uper una & quadraginta quin que minutijs, quæ re&longs;pondent di<lb/>ei uni, & horis decem octo ex motu Solis, & ita bis Saturnus reuol <lb/>uitur &longs;ecundum motum æqualitatis & quinquage&longs;ies &longs;epties per <lb/>motum inæqualem & &longs;imiliter. </s>

<s id="id002935">Iupiter in annis 70, diebus trecen<lb/>tis &longs;exaginta, horis quatuor, &longs;exaginta quinque reuolutiones in&etail;qua <lb/>les perficiet & &longs;ex &etail;quales, deficientibus ex &etail;qualibus quatuor par<lb/>tibus & dextante quod e&longs;t <expan abbr="quãtum">quantum</expan> peragraret Solin quatuor die<lb/>bus, & dextante diei ad perfectionem &longs;cilicet annorum &longs;eptuaginta <lb/>atque unius. </s>

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<s id="id002938">Mercurij quo que &longs;tella in quadraginta &longs;ex annis & una die <lb/>& hora una fermè quadraginta &longs;ex fermè perficit reuolutiones æ<lb/>qualis motus & in&longs;uper gradum unum cum portione re&longs;pondenti <lb/>portioni temporis, id e&longs;t, horæ fermè uni: in æqualitatis autem cen<lb/>&longs;um quadraginta quin que. </s>

<s id="id002939">Atque h&etail;c &longs;unt manife&longs;ti&longs;sima et ut dixi ad<lb/>miranda &longs;unt, præterea alia minus generalia, aut minus manife&longs;ta <lb/>aut non tanti momenti quæ con&longs;ultò prætermitto, non e&longs;t. </s>

<s id="id002941">locus <lb/>hic do cendi artes &longs;ingulas &longs;ed &longs;olum ea tra ctandi quæ ad argumen

<s id="id002941">locus <lb/>hic docendi artes &longs;ingulas &longs;ed &longs;olum ea tractandi quæ ad argumen<pb pagenum="168" xlink:href="015/01/187.jpg"/>tum pertinent. </s>

<pb pagenum="168" xlink:href="015/01/187.jpg"/>tum pertinent. </s>

<s id="id002942">Igitur ut ad rem redeam. </s>

<s id="id002943">Solis cum octauo Orbe ea <lb/>ratio e&longs;t, ut linea quam ille permeat eadem &longs;it quam qu&etail; fix&etail; &longs;tellæ, <lb/>non. </s>

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<s id="id002957">omnia <expan abbr="cõprehendi">comprehendi</expan> uidentur. </s>

<s id="id002958">1. omnes uo <expan abbr="c&utilde;">cum</expan> differentiæ, <lb/><expan abbr="quanquã">quanquam</expan> ex octo <expan abbr="tant&utilde;">tantum</expan> uo cibus con&longs;tet. </s>

<s id="id002959">Pò&longs;t &longs;unt quæ in <expan abbr="&qtilde;drupla">quadrupla</expan>, <lb/>unde bis diapa&longs;on, po&longs;t quæ in tripla, nam propior e&longs;t monadi &longs;eu &etail;<lb/>qualitati: &longs;ed non adeò &longs;implex ut bis diapa&longs;on. </s>

<s id="id002960">Vocant <expan abbr="a&utilde;t">aunt</expan> hanc <lb/>diapa&longs;on diapente: inde <expan abbr="&longs;ub&longs;equi&ttilde;">&longs;ub&longs;equitur</expan> octupla qu&etail; uix in uocib. </s>

<s id="id002960">Vocant <expan abbr="a&utilde;t">aut</expan> hanc <lb/>diapa&longs;on diapente: inde <expan abbr="&longs;ub&longs;equi&ttilde;">&longs;ub&longs;equitur</expan> octupla qu&etail; uix in uocibus </s>

<s id="id002961">huma<lb/>nis habetur: <expan abbr="frequ&etilde;s">frequens</expan> in in&longs;trumentis, uo <expan abbr="ca&ttilde;que">caturque</expan> tris diapa&longs;on inde &longs;ex<lb/>cupla, &longs;eu bis diapa&longs;on diapente. </s>

<s id="id002962">Quintupla <expan abbr="a&utilde;t">aunt</expan> minus <expan abbr="cõcors">concors</expan> e&longs;t: <lb/>&longs;ed de hac inferius dicemus, atque de multiplicib. </s>

<s id="id002962">Quintupla <expan abbr="a&utilde;t">aut</expan> minus <expan abbr="cõcors">concors</expan> e&longs;t: <lb/>&longs;ed de hac inferius dicemus, atque de multiplicibus </s>

<s id="id002963">dicta &longs;unto. </s>

<s id="id002964">Sed de <lb/><expan abbr="cõ">com</expan> centu ex particula &longs;uperaddita &longs;exquialtera &longs;exquitertia atque alijs <lb/>nunc agendum. </s>

<s id="id002965">Clarum e&longs;t. </s>

<s id="id002967">has e&longs;&longs;e &longs;implici&longs;simas. </s>

<s id="id002968">Cum ergo du <lb/>pla proportio non magis po&longs;sit diuidi æqualibus interuallis atque<lb/>&longs;implicibus proportionibus quàm in &longs;exquialteram & &longs;exquiter<lb/>tiam, uelutinter 4 & 2 interpo&longs;ito 3. nam proportio 3 ad 2 e&longs;t &longs;ex<lb/>quialtera, & 4 ad 3 &longs;exquitertia: nec melius pote&longs;t diuidi, at &longs;exqui<lb/>alteram & &longs;exquitertiam quantumuis magnis numeris diuidere <lb/>non licebat melius aut commodius quam per &longs;exquioctauas: uelu<lb/>ti &longs;umpto numero 64 cui duplus e&longs;t 128, inter medius 96 qui cum <lb/>64 &longs;exquialteram facit proportionem, quæ &longs;uaui&longs;sima e&longs;t omni<lb/>um deductis multiplicibus, uo caturque diapente. </s>

<s id="id002969">At quæ e&longs;t 128 ad <lb/>96 &longs;exquitertia e&longs;t minu&longs;que benè &longs;onat per &longs;e, &longs;ed in acutioribus uo<lb/>cibus &longs;olum cum alijs benè &longs;onat, uelut cum diapente, perficiens <lb/>diapa&longs;on, interuallum, ergo inter 96 & 64 diui&longs;um per &longs;exquiocta<pb pagenum="169" xlink:href="015/01/188.jpg"/>uas producit 72 et 81, <expan abbr="nã">nam</expan> 72 ad 64 e&longs;t &longs;exquio<expan abbr="ctau&utilde;">ctauum</expan>, &longs;icut 81 ad 72. uerùm <lb/>id accidebat in <expan abbr="cõmodi">commodi</expan> quae 81 ad 64 <expan abbr="nullã">nullam</expan> habet <expan abbr="proportion&etilde;">proportionem</expan> <expan abbr="commodã">commodam</expan>, <lb/>& multo minus 96 ad 81, quare ui&longs;um e&longs;t Ptolem&etail;o ut &longs;ubtracta mona<lb/>de <expan abbr="fier&etilde;t">fierent</expan> termini 64, 72, 80, & 96, proportio <expan abbr="a&utilde;t">aut</expan> 80 ad 64 <expan abbr="cõ&longs;tituit">con&longs;tituit</expan> &longs;exqui<lb/><expan abbr="quartã">quartam</expan> atque <expan abbr="diton&utilde;">ditonum</expan>, proportio quoque 96 ad 72 <expan abbr="&longs;exquitertiã">&longs;exquitertiam</expan> <expan abbr="&longs;emiditon&utilde;">&longs;emiditonum</expan> que. <lb/></s>

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

<pb pagenum="169" xlink:href="015/01/188.jpg"/>uas producit 72 et 81, <expan abbr="nã">nam</expan> 72 ad 64 e&longs;t &longs;exquio <expan abbr="ctau&utilde;">ctauum</expan>, &longs;icut 81 ad 72. uerùm <lb/>id accidebat in <expan abbr="cõmodi">commodi</expan> quae 81 ad 64 <expan abbr="nullã">nullam</expan> habet <expan abbr="proportion&etilde;">proportionem</expan> <expan abbr="commodã">commodam</expan>, <lb/>& multominus 96 ad 81, quare ui&longs;um e&longs;t Ptolem&etail;o ut &longs;ubtracta mona <lb/>de <expan abbr="fier&etilde;t">fierent</expan> termini 64, 72, 80, & 96, proportio <expan abbr="a&utilde;t">aunt</expan> 80 ad 64 <expan abbr="cõ&longs;tituit">con&longs;tituit</expan> &longs;exqui <lb/><expan abbr="quartã">quartam</expan> atque <expan abbr="diton&utilde;">ditonum</expan>, proportio quo que 96 ad 72 <expan abbr="&longs;exquitertiã">&longs;exquitertiam</expan> <expan abbr="&longs;emiditon&utilde;">&longs;emiditonum</expan> que. <lb/></s>

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

<s id="id002977">con&longs;onantibus di&longs;&longs;onans <expan abbr="cõpo&longs;itio">compo&longs;itio</expan> <expan abbr="oria&ttilde;">oriatur</expan>, uelut ex duplici <expan abbr="diap&etilde;te">diapente</expan>, aut <lb/><expan abbr="diap&etilde;te">diapente</expan> <expan abbr="c&utilde;">cum</expan> ditono, &longs;ed ut ad <expan abbr="propo&longs;it&utilde;">propo&longs;itum</expan> reuertar, alia diapa&longs;on e&longs;t inter 80 <lb/>& 40, at inter 48 & 40 e&longs;t &longs;emiditonus ut <expan abbr="o&longs;t&etilde;&longs;um">o&longs;ten&longs;um</expan> e&longs;t, uelut inter 96 & <lb/>80, nam inter 45 & 40 e&longs;t proportio &longs;exquioctaua, inter 48 <expan abbr="a&utilde;t">aunt</expan> & 45 &longs;ex<lb/>quiquinta decima, <expan abbr="igi&ttilde;">igitur</expan> ex regula data proportio 80 ad 48 <expan abbr="&qtilde;">quae</expan> e&longs;t &longs;uperbi<lb/>partiens tertias &longs;eu &longs;olida <expan abbr="c&utilde;">cum</expan> be&longs;&longs;e &longs;eu &longs;exta maior erit <expan abbr="cõ&longs;onans">con&longs;onans</expan>. </s>

<s id="id002977">con&longs;onantibus di&longs;&longs;onans <expan abbr="cõpo&longs;itio">compo&longs;itio</expan> <expan abbr="oria&ttilde;">oriatur</expan>, uelut ex duplici <expan abbr="diap&etilde;te">diapente</expan>, aut <lb/><expan abbr="diap&etilde;te">diapente</expan> <expan abbr="c&utilde;">cum</expan> ditono, &longs;ed ut ad <expan abbr="propo&longs;it&utilde;">propo&longs;itum</expan> reuertar, alia diapa&longs;on e&longs;t inter 80 <lb/>& 40, at inter 48 & 40 e&longs;t &longs;emiditonus ut <expan abbr="o&longs;t&etilde;&longs;um">o&longs;ten&longs;um</expan> e&longs;t, uelut inter 96 & <lb/>80, nam inter 45 & 40 e&longs;t proportio &longs;exquioctaua, inter 48 <expan abbr="a&utilde;t">aut</expan> & 45 &longs;ex<lb/>quiquinta decima, <expan abbr="igi&ttilde;">igitur</expan> ex regula data proportio 80 ad 48 <expan abbr="&qtilde;">quae</expan> e&longs;t &longs;uperbi<lb/>partiens tertias &longs;eu &longs;olida <expan abbr="c&utilde;">cum</expan> be&longs;&longs;e &longs;eu &longs;exta maior erit <expan abbr="cõ&longs;onans">con&longs;onans</expan>. </s>

<s id="id002978">Iam er <lb/>go uidemus detractione aut additione &longs;exquio ctuage&longs;imæ, concinnas <lb/>reddi uulgatiores armonias: <expan abbr="tertiã">tertiam</expan> utran que <expan abbr="maior&etilde;">maiorem</expan> &longs;cilicet & <expan abbr="minor&etilde;">minorem</expan>, ac <lb/>rur&longs;us <expan abbr="&longs;extã">&longs;extam</expan> <expan abbr="maior&etilde;">maiorem</expan> atque minore <expan abbr="&qtilde;">quae</expan> in minoribus numeris &longs;cilicet à mo<lb/>nade ad octo po&longs;itæ &longs;unt. </s>

<s id="id002979">Vides præterea <expan abbr="&longs;emiditon&utilde;">&longs;emiditonum</expan> in &longs;exquiquinta <lb/><arrow.to.target n="table21"/><lb/><expan abbr="cõ&longs;tare">con&longs;tare</expan>: adeò ut à &longs;enario infra nihil inutile <lb/><figure id="id.015.01.188.1.jpg" xlink:href="015/01/188/1.jpg"/>reddatur. </s>

<s id="id002980">Diate&longs;&longs;aron <expan abbr="a&utilde;t">aunt</expan> cum primum di <lb/>uidi pote&longs;t, &longs;i &longs;ecus diuidatur <08> in <expan abbr="diton&utilde;">ditonum</expan> <lb/>& <expan abbr="&longs;emitoni&utilde;">&longs;emitonium</expan>, aut in &longs;emiditonum & <expan abbr="ton&utilde;">tonum</expan>, <lb/>&longs;cilicet in duo <expan abbr="tant&utilde;">tantum</expan> interualla, non <expan abbr="cõmodius">commo<lb/>dius</expan> <expan abbr="quã">quam</expan> inter octo & &longs;eptem & &longs;ex diuidi <lb/>pote&longs;t. </s>

<s id="id002980">Diate&longs;&longs;aron <expan abbr="a&utilde;t">aut</expan> cum primum di <lb/>uidi pote&longs;t, &longs;i &longs;ecus diuidatur <08> in <expan abbr="diton&utilde;">ditonum</expan> <lb/>& <expan abbr="&longs;emitoni&utilde;">&longs;emitonium</expan>, aut in &longs;emiditonum & <expan abbr="ton&utilde;">tonum</expan>, <lb/>&longs;cilicet in duo <expan abbr="tant&utilde;">tantum</expan> interualla, non <expan abbr="cõmodius">commo<lb/>dius</expan> <expan abbr="quã">quam</expan> inter octo & &longs;eptem & &longs;ex diuidi <lb/>pote&longs;t. </s>

<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;">emm</expan> per <lb/>fe 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">aunt</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">aunt</expan>

<pb pagenum="170" xlink:href="015/01/189.jpg"/>ditonus: bis diapa&longs;on <expan abbr="a&utilde;t">aunt</expan> quindecim uo cibus. </s>

<pb pagenum="170" xlink:href="015/01/189.jpg"/>ditonus: bis diapa&longs;on <expan abbr="a&utilde;t">aut</expan> quindecim uo cibus. </s>

<s id="id002986">Omnes igitur decem, & <lb/><expan abbr="&longs;ept&etilde;">&longs;eptem</expan> uoces, <expan abbr="&qtilde;">quae</expan> &longs;exdecim interuallis <expan abbr="di&longs;tinguun&ttilde;">di&longs;tinguuntur</expan>, con&longs;onantes &longs;unt: & ex <lb/>genere ditoni, & &longs;exquiquartæ, &longs;ed paulo minus benè <expan abbr="&longs;onãt">&longs;onant</expan> <08> ditonus <lb/>ip&longs;e. </s>

<s id="id002987">Igitur <expan abbr="quintuplã">quintuplam</expan> multiplicem ad &longs;ex <expan abbr="quiquartã">quiquartam</expan> reduximus. </s>

<s id="id002988">Verum <lb/>ut o&longs;ten&longs;um e&longs;t & decima&longs;eptima, <expan abbr="&qtilde;">quae</expan> bis diapa&longs;on <expan abbr="cõ&longs;tat">con&longs;tat</expan>, & &longs;emiditono <lb/>benè &longs;onat, h&etail;c <expan abbr="a&utilde;t">aunt</expan> inter non aginta &longs;ex & uiginti: quadrupla <expan abbr="igi&ttilde;">igitur</expan> e&longs;t & <lb/>&longs;uperquadripartiens quintas. </s>

<s id="id002988">Verum <lb/>ut o&longs;ten&longs;um e&longs;t & decima&longs;eptima, <expan abbr="&qtilde;">quae</expan> bis diapa&longs;on <expan abbr="cõ&longs;tat">con&longs;tat</expan>, & &longs;emiditono <lb/>benè &longs;onat, h&etail;c <expan abbr="a&utilde;t">aut</expan> inter nonaginta &longs;ex & uiginti: quadrupla <expan abbr="igi&ttilde;">igitur</expan> e&longs;t & <lb/>&longs;uperquadripartiens quintas. </s>

<s id="id002989">Diapa&longs;on quo que cum &longs;exta maiore & mi<lb/>nore eandem habentrationem quam 16 ad 5, & 10 ad 3, triplam utranque, <lb/>&longs;ed altera &longs;exquiquinta, altera &longs;exquitertia: bis diapa&longs;on uerò <expan abbr="c&utilde;">cum</expan> ei&longs;dem <lb/>ut uiginti ad tria, & 32 ad quin que &longs;excupla utraque: &longs;ed altera &longs;uperbipar<lb/>tiens tertias, altera quintas. </s>

<s id="id002990"><expan abbr="Manife&longs;t&utilde;">Manife&longs;tum</expan> e&longs;t igitur hanc diui&longs;ionem <expan abbr="nõ">non</expan> &longs;o<lb/>lum concinnam magis e&longs;&longs;e & &longs;uauem &longs;ed omnem <expan abbr="tonor&utilde;">tonorum</expan> & &longs;emitonio<lb/>rum <expan abbr="nece&longs;sitat&etilde;">nece&longs;sitatem</expan> effugere. </s>

<s id="id002991">Quòd uerò in cau&longs;a fuit ut toni & &longs;emitonia <lb/>in u&longs;u e&longs;&longs;ent, id e&longs;t, quoniam in <expan abbr="di&longs;c&etilde;do">di&longs;cendo</expan> nece&longs;&longs;e e&longs;t eandem &longs;eruari ratio<lb/>nem in <expan abbr="crementor&utilde;">crementorum</expan>, ne que arithmeticam &longs;ed <expan abbr="geometricã">geometricam</expan>. </s>

<s id="id002992">Ideò <expan abbr="a&longs;c&etilde;&longs;us">a&longs;cen&longs;us</expan> per <lb/>tonos & &longs;emitonia <expan abbr="cõmodus">commodus</expan> fuit, nam duplicem <expan abbr="&longs;ol&utilde;">&longs;olum</expan> differentiam pue <lb/>ri u&longs;u a&longs;&longs;equi coguntur. </s>

<s id="id002993">At uerò poterat & per &longs;exqui&longs;extam diuidi dia <lb/>te&longs;&longs;aron, ut inter triginta &longs;ex & quadraginta nouem interpo&longs;itis 42, ue<lb/>rùm triplex <expan abbr="&longs;equeba&ttilde;">&longs;equebatur</expan> in <expan abbr="cõueniens">conueniens</expan>: primum ut diate&longs;&longs;aron ad amu&longs;sim <lb/>non &longs;eruaretur, &longs;ed incidebat in cacophoniam, addita quadrage&longs;ima o<lb/>ctaua parte: deficiente <expan abbr="a&utilde;t">aunt</expan> in duabus &longs;exqui&longs;eptimis numeris &longs;eu propor<lb/>tione &longs;exquitertia: ut inter 49 & 64 loco 48 & 64, uelut <expan abbr="etiã">etiam</expan> inter 48 ad <lb/>36, additaigitur monade in termino medio utrin que fit di&longs;&longs;onantia. </s>

<s id="id002993">At uerò poterat & per &longs;exqui&longs;extam diuidi dia<lb/>te&longs;&longs;aron, ut inter triginta &longs;ex & quadraginta nouem interpo&longs;itis 42, ue<lb/>rùm triplex <expan abbr="&longs;equeba&ttilde;">&longs;equebatur</expan> in <expan abbr="cõueniens">conueniens</expan>: primum ut diate&longs;&longs;aron ad amu&longs;sim <lb/>non &longs;eruaretur, &longs;ed incidebat in cacophoniam, addita quadrage&longs;ima o<lb/>ctaua parte: deficiente <expan abbr="a&utilde;t">aut</expan> in duabus &longs;exqui&longs;eptimis numeris &longs;eu propor<lb/>tione &longs;exquitertia: ut inter 49 & 64 loco 48 & 64, uelut <expan abbr="etiã">etiam</expan> inter 48 ad <lb/>36, addita igitur monade in termino medio utrin que fit di&longs;&longs;onantia. </s>

<s id="id002994">Se<lb/>cundum inconueniens, e&longs;t quae &longs;ic diuidente non &longs;eruabatur ratio &longs;exqui<lb/>quartæ & &longs;exquiquintæ &longs;eu ditoni & &longs;emiditoni, quæ uoces benè &longs;o<lb/>nant. </s>

<s id="id002995">Tertium inconueniens erat, quòd hæcratio diuidendi diapentes <lb/>minimè &longs;atisfaciebat, uelutinter 324 & 216. Interponere enim nece&longs;&longs;e <lb/>erat 252 & 294, unde incongrua rur&longs;us erat diui&longs;io. </s>

<s id="id002996">His tot cau&longs;is cum <lb/>proportiones maiores non fatisfacerent ut &longs;exqui quinta quæ diate&longs;&longs;a<lb/>ron nullo modo æqualiter diuidere pote&longs;t, & in diapente deficit &longs;exqui <lb/>uige&longs;imaquarta, ut inter 25 & 36, coacti &longs;unt cum nec &longs;exqui&longs;exta nec <lb/>&longs;exqui&longs;eptima idoneæ e&longs;&longs;ent ad &longs;exquio ctauam confugere.</s>

Line 8757

Line 8745

</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">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

<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>

<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>

<s id="id003001">nam de remi&longs;sione po&longs;&longs;emus <lb/>dicere quòd accipitur loco &longs;exquio ctuage&longs;imæ: &longs;ed in &longs;exquioctuage<lb/>&longs;ima remittitur de tono &longs;ecundum mentem Ptolemæi, in die&longs;i intendi<lb/>tur &longs;emitonium minus, &longs;icut o&longs;tendit experimentum, &longs;ed for&longs;an conue <lb/>niunt quia intentio &longs;emitonij minoris deducit &longs;emiditonum ad &longs;exqui <lb/>quintam: e&longs;t enim differentia &longs;emitonij minoris intenti hoc modo ad <lb/>&longs;emitonium minus, ut 136 ad 135: &longs;ed hoc e&longs;t longè minus &longs;exquioctua <lb/>ge&longs;ima, unum &longs;at e&longs;t, hanc e&longs;&longs;e ultimam diui&longs;ionem toni in octo par<lb/>tes, & ut in diatonico toni dominantur, ita in chromatico &longs;emitonia in <lb/>enarmonico die&longs;es, &longs;ed die&longs;es fugitando (utita dicam) ac aures uelli<lb/>cando, mirum in modum oblectant audientes: uelut toni &longs;tando, un<lb/>de etiam nomen, &longs;emitonia medium modum obtinent.</s>

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<s id="id003004">Virum enim hunc &longs;upra omnem humani ingenij <expan abbr="metã">metam</expan> fui&longs;&longs;e <expan abbr="nõ">non</expan> nega<lb/>mus: &longs;ed hanc partem quam hic agit, adeò infeliciter tractat, ut malim <lb/>credere <expan abbr="tot&utilde;">totum</expan> illum tertium <expan abbr="libr&utilde;">librum</expan> fui&longs;&longs;e ab aliquo alio <expan abbr="adiect&utilde;">adiectum</expan>. </s>

<s id="id003005">Etenim <lb/>quid turpius &longs;apienti homini <08> imitari uulgares illos? </s>

<s id="id003006"><expan abbr="&longs;ept&etilde;">&longs;eptem</expan> planetæ, <lb/>&longs;eptem mundi miracula, <expan abbr="&longs;ept&etilde;">&longs;eptem</expan> artes liberales: quid enim &longs;imilitudo nu<pb pagenum="172" xlink:href="015/01/191.jpg"/>meri iuuare pote&longs;t, aut quàm afferre utilitatem? </s>

<pb pagenum="172" xlink:href="015/01/191.jpg"/>meri iuuare pote&longs;t, aut quàm afferre utilitatem? </s>

<s id="id003007">nimis certè in <expan abbr="dign&utilde;">dignum</expan> e&longs;t <lb/>uti <expan abbr="argum&etilde;to">argumento</expan> à &longs;imilitudine &longs;umpto: tum maximè adeò leui. </s>

<s id="id003008">Sed quo<lb/>niam con&longs;tat omnia quæ in mundo &longs;unt ordine coniuncta e&longs;&longs;e, & ne<lb/>ce&longs;sitate uinciri, ideò cùm finis ip&longs;e uerus &longs;it, non tam debemus Ptole<lb/>mæum damnare, quae non probauerit, quàm laudare, quod <expan abbr="ueritat&etilde;">ueritatem</expan> &longs;ine <lb/>ratione &longs;it a&longs;&longs;ectus. </s>

<s id="id003009">Sæpe enim accidit huiu&longs;modi uiris adeò pr&etail;&longs;tan<lb/>tibus ut ueritas detegatur, quam cùm illi, ut mos e&longs;t <expan abbr="homin&utilde;">hominum</expan>, rationi<lb/>bus adornare nituntur, tran&longs;gredientes metam muneris, in ab&longs;urda & <lb/>ineptias <expan abbr="incid&utilde;t">incidunt</expan>. </s>

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<s id="id003011">Verùm dubium e&longs;t, an &longs;oni propter nume <lb/>ros iucundi &longs;int, an propter aliud? </s>

<s id="id003012">& &longs;i propter aliud, cur ergo numeri <lb/>ad hoc &longs;unt nece&longs;&longs;arij? </s>

<s id="id003013">& cur ob&longs;eruare eos oportet ne ab illorum ordi <lb/>ne di&longs;iungi po&longs;sint? </s>

<s id="id003014">Hoc <expan abbr="a&utilde;t">aunt</expan> perfacilè <expan abbr="intelligi&ttilde;">intelligitur</expan>, & à nobis aliâs decla<lb/>ratum e&longs;t, &longs;cilicet delectare nos, quæ percipiuntur quæque ratione facta <lb/>uidentur, <expan abbr="quoniã">quoniam</expan> in his naturæ uis relucet & imago uniuer&longs;i, ergo dele <lb/>ctant nos, quoniam natur&etail; ordine nos con&longs;tamus. </s>

<s id="id003014">Hoc <expan abbr="a&utilde;t">aut</expan> perfacilè <expan abbr="intelligi&ttilde;">intelligitur</expan>, & à nobis aliâs decla<lb/>ratum e&longs;t, &longs;cilicet delectare nos, quæ percipiuntur quæque ratione facta <lb/>uidentur, <expan abbr="quoniã">quoniam</expan> in his naturæ uis relucet & imago uniuer&longs;i, ergo dele<lb/>ctant nos, quoniam natur&etail; ordine nos con&longs;tamus. </s>

<s id="id003015">Illud difficilius lon <lb/>gè &qring;d <expan abbr="tam&etilde;">tamen</expan> diligenti ob&longs;eruatione <expan abbr="dign&utilde;">dignum</expan> uidetur, &longs;cilicet, quonam pa <lb/>cto harmonia cum rebus c&oelig;le&longs;tibus aut humanis <expan abbr="cõiuncta">coniuncta</expan> &longs;it. </s>

<s id="id003016">For&longs;an <lb/>& illud ab re non e&longs;&longs;et intelligere, cur nullum animal pr&etail;ter hominem <lb/>capax &longs;it harmoniæ? </s>

<s id="id003017">an for&longs;an <expan abbr="quoniã">quoniam</expan> &longs;olus homo ratione participet, <lb/>& ob id &longs;olus gaudet ratione? </s>

<s id="id003018">ordinata <expan abbr="a&utilde;t">aunt</expan> ratione <expan abbr="cõ&longs;tant">con&longs;tant</expan> aut &longs;ola aut <lb/>maximè, numerus autem quid aliud e&longs;t quàm ordinis <expan abbr="&longs;eparator&utilde;">&longs;eparatorum</expan> ima<lb/>go. </s>

<s id="id003018">ordinata <expan abbr="a&utilde;t">aut</expan> ratione <expan abbr="cõ&longs;tant">con&longs;tant</expan> aut &longs;ola aut <lb/>maximè, numerus autem quid aliud e&longs;t quàm ordinis <expan abbr="&longs;eparator&utilde;">&longs;eparatorum</expan> ima<lb/>go. </s>

<s id="id003019">Porrò hæc accipienda &longs;unt ex his quæ &longs;en&longs;ibus deprehenduntur, <lb/>qualia &longs;unt quae animus mouetur & uarios affectus in duit iuxta harmo<lb/>niæ diuer&longs;itatem lætiti&etail;, tri&longs;titi&etail;, impetus, remi&longs;sionis, timoris, &longs;pei, ira<lb/>cundiæ, & commi&longs;erationis. </s>

<s id="id003020">Nos enim maximè octo affectus mouent <lb/>mu&longs;icæ modulationes. </s>

<s id="id003021">Secundum quid autem mouent? </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;">emm</expan> in <expan abbr="a&etilde;re">aerre</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 deffertur 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="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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<pb pagenum="174" xlink:href="015/01/193.jpg"/>nantiarum percipit perfectam diapa&longs;on. </s>

<s id="id003038">Videamus ergo an aliquid &longs;it <lb/>&longs;imile in animæ facultatibus, nec <expan abbr="dubi&utilde;">dubium</expan> e&longs;t quin ex &longs;en&longs;ib. </s>

<s id="id003038">Videamus ergo an aliquid &longs;it <lb/>&longs;imile in animæ facultatibus, nec <expan abbr="dubi&utilde;">dubium</expan> e&longs;t quin ex &longs;en&longs;ibus. </s>

<s id="id003039">exterioribus <lb/>atque interioribus fiat intelligentia. </s>

<s id="id003040">Et &longs;en&longs;us <expan abbr="quid&etilde;">quidem</expan> exteriores &longs;exquiter <lb/>tia <expan abbr="cõ&longs;tant">con&longs;tant</expan>: e&longs;t enim <expan abbr="illor&utilde;">illorum</expan> imperfecta cognitio: maior longè memori&etail; <lb/>unius & rationis reliquarumque <expan abbr="facultat&utilde;">facultatum</expan>, ex quibus <expan abbr="intellig&etilde;tia">intelligentia</expan> oritur. <lb/></s>

<s id="id003041">Iam uerò habemus exactam <expan abbr="&longs;imilitudin&etilde;">&longs;imilitudinem</expan> facultatum anim&etail; human&etail;, <expan abbr="&qtilde;">quae</expan> <lb/>cogno&longs;cit. </s>

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<s id="id003048">Et <expan abbr="uide&ttilde;">uidetur</expan> mu&longs;ica nec hoc &etail;qualiter monere, &longs;ed <expan abbr="prim&utilde;">primum</expan> uideamus <lb/>an hi&longs;oli affectus &longs;int maximi, quippe dee&longs;&longs;e <expan abbr="uiden&ttilde;">uidentur</expan> amor atque odium. <lb/></s>

<s id="id003049">Et mihi dubium non e&longs;t quin hi potenti&longs;simi &longs;int <expan abbr="omni&utilde;">omnium</expan> præter <expan abbr="met&utilde;">metum</expan>. <lb/></s>

<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, ppri&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="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">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>

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<pb pagenum="175" xlink:href="015/01/194.jpg"/>nes habentibus atque a&longs;peris. </s>

<s id="id003061">Hæc autem, ut ita dicam, nulla e&longs;t mu&longs;ica. <lb/></s>

<s id="id003062">Sed neque mu&longs;ica ulla tri&longs;titiam gignit, cum ut dixi, tri&longs;titia nil aliud &longs;it <08><lb/>mortis imago, mu&longs;ica <expan abbr="a&utilde;t">aunt</expan> uitam fouet. </s>

<s id="id003063">Vnde <expan abbr="nõ">non</expan> immeritò fertur Xeno <lb/>philus mu&longs;icus <expan abbr="cent&utilde;">centum</expan> quinque annis &longs;ine aliquo <expan abbr="incõmodo">incommodo</expan> uixi&longs;&longs;e, quod <lb/>&longs;ingulare e&longs;&longs;e exemplum in humana uita refert Plinius. </s>

<s id="id003064">Relin quitur igi <lb/>tur tandem, ut mu&longs;ica maximè moueat tres affectus lætitiam, remi&longs;sio<lb/>nem & mi&longs;ericordiam. </s>

<s id="id003065">Et quod ex his po&longs;tmodum ad labores in&longs;urga<lb/>mus intentius, hoc non e&longs;t ex mu&longs;ic&etail; ui aut facultate, &longs;ed <expan abbr="cõ&longs;equentibus">con&longs;equentibus</expan> <lb/>ad illa alia cau&longs;is. </s>

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<s id="id003069"><arrow.to.target n="marg580"/><lb/>affectuum, <expan abbr="c&utilde;">cum</expan> remittuntur uoces aut intendantur ad <expan abbr="ear&utilde;">earum</expan> intentionem. <lb/></s>

<s id="id003070">Sed non e&longs;t æqualis ratio, quoniam natura no&longs;tra ad <expan abbr="remi&longs;sion&etilde;">remi&longs;sionem</expan> natu<lb/>raliter inclinata e&longs;t, ad intentionem non ita, &longs;ed per uim <expan abbr="quandã">quandam</expan> aut me<lb/>dio uoluptatis, aut cum anima purior e&longs;t à corporis impedimentis. </s>

<s id="id003071">Et <lb/>ob id ad &longs;tudia nil aptius e&longs;t pura &longs;obrietate: nihil ineptius crapula atque<lb/>temulentia. </s>

<s id="id003072">At l&etail;titi&etail; cau&longs;&etail; &longs;unt, & <expan abbr="cõ">com</expan> cordia uo <expan abbr="c&utilde;">cum</expan>, & mutatio ex a&longs;pera <lb/>in &longs;uauem, <expan abbr="nõ">non</expan> &longs;ecus ac eius qui euadit è paupertate uel è mole&longs;tia aliqua <lb/>aut dolore aut alio <expan abbr="incõmodo">incommodo</expan>, tum inten&longs;io uo <expan abbr="c&utilde;">cum</expan> ac liber &longs;onus. </s>

<s id="id003072">At l&etail;titi&etail; cau&longs;&etail; &longs;unt, & <expan abbr="cõ">con</expan>cordia uo <expan abbr="c&utilde;">cum</expan>, & mutatio ex a&longs;pera <lb/>in &longs;uauem, <expan abbr="nõ">non</expan> &longs;ecus ac eius qui euadit è paupertate uel è mole&longs;tia aliqua <lb/>aut dolore aut alio <expan abbr="incõmodo">incommodo</expan>, tum inten&longs;io uo <expan abbr="c&utilde;">cum</expan> ac liber &longs;onus. </s>

<s id="id003073">Vnde <lb/>in l&etail;titia &longs;olent homines exclamare. </s>

<s id="id003074">At ad <expan abbr="cõmi&longs;erationem">commi&longs;erationem</expan> mouendam <lb/>omnia remitti oportet ex magna in parua, adeoque deficientem ex a&longs;pera <lb/>in leuem, ex ueloci in tardam, ex di&longs;&longs;ona in con&longs;onantem. </s>

<s id="id003075">Antiqui ergo <lb/>(ut author e&longs;t Cælius Rhodiginius) Dorico ad temperantiam & mode <lb/><arrow.to.target n="marg581"/><lb/>rationem utebantur, &longs;cilicet quòd non haberet præcipites lap&longs;us, neque<lb/>arduas intentiones: Phrygio ad impetum & bellicum ardorem, &longs;cilicet <lb/>per a&longs;peras intentiones: Lydio ad fletus & lamentationes per ca&longs;us & <lb/>remi&longs;siones longas ac &longs;uaues: ideo funeribus peculiaris: Mixolydio ad <lb/>commi&longs;erationem, ut defectiones interponantur & breues abruptæque<lb/>remi&longs;siones, iuuantque in hoc plurimum & &longs;en&longs;us uerborum, familiaris <lb/>hic tragædijs: Aeolicus qui & Ionicus tranquillitatis animi author e&longs;t <lb/>&longs;o mnumque conciliat: Dorico non ab&longs;imilis &longs;ed &longs;uauior & mollior: ideò <lb/>chromatici generis. </s>

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<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">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>

<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="id003104">Piper enim acre e&longs;t, & uinum au&longs;terum e&longs;t. </s>

<s id="id003105">Et iu&longs;ta querela Cicero<lb/>nis in Epi&longs;tolis familiaribus, qui à maluis fatetur &longs;e uictum, ut deci<lb/>derit in lienteriam: conueniunt ambo hi &longs;apores <expan abbr="c&utilde;">cum</expan> dulci & pingui, <lb/>uelut & utraque &longs;exta maior & minor cum diapa&longs;on & diapente, at <lb/>neuter cum &longs;al&longs;o, nam neque diate&longs;&longs;aron cum &longs;extamaiore uel mino<lb/>re iungi pote&longs;t. </s>

<s id="id003106">Amarus autem &longs;apor tono per&longs;imilis e&longs;t, di&longs;&longs;onus <lb/>enim per &longs;e e&longs;t &longs;emper, & amarus per&longs;e odio&longs;us tonus origo e&longs;t o<lb/>mnium <expan abbr="con&longs;onantiar&utilde;">con&longs;onantiarum</expan>, ita omnes fructus, &longs;eu dulces &longs;eu a&longs;tringen<lb/>tes, &longs;eu acidi, &longs;eu acres prius amari &longs;unt: tonus præterea nulla cum <lb/>con&longs;onantia peius coit quàm cum diapa&longs;o, ita neque amarus &longs;apor <lb/>infelicius iungnur quàm cum dulci, amarus quo que &longs;apor cum nul<lb/>lo magis conuenit <expan abbr="quã">quam</expan> cum &longs;al&longs;o, ita tonus additus diate&longs;&longs;aro, perfi <lb/>cit diapente dulci&longs;simam con&longs;onantiam, ut multi oliuas benè&longs;al&longs;as <lb/>prætulerint fa&longs;ianis: tantum conuenit &longs;al&longs;o cum amaro, amarus, <lb/>quo que &longs;apor leuis non abhorret à pingui, deteriorem <expan abbr="tam&etilde;">tamen</expan> aliquan <lb/>to efficit, ut intortis ex ab&longs;ynthio ouis & ca&longs;eo, atque in uitibus in <lb/>quibus coma ab&longs;ynthij in cocta fuit parum, degenerat tamen &longs;apor <lb/>ille à pingui: ita tono addito ad diapente fit &longs;exta maior, non adeò <lb/>&longs;uauis ut diapente, attamen <expan abbr="nõ">non</expan> pror&longs;us in&longs;uauis. </s>

<s id="id003106">Amarus autem &longs;apor tono per&longs;imilis e&longs;t, di&longs;&longs;onus <lb/>enim per &longs;e e&longs;t &longs;emper, & amarus per&longs;e odio&longs;us tonus origo e&longs;t o<lb/>mnium <expan abbr="con&longs;onantiar&utilde;">con&longs;onantiarum</expan>, ita omnes fructus, &longs;eu dulces &longs;eu a&longs;tringen<lb/>tes, &longs;eu acidi, &longs;eu acres prius amari &longs;unt: tonus præterea nulla cum <lb/>con&longs;onantia peius coit quàm cum diapa&longs;o, ita neque amarus &longs;apor <lb/>infelicius iungitur quàm cum dulci, amarus quo que &longs;apor cum nul<lb/>lo magis conuenit <expan abbr="quã">quam</expan> cum &longs;al&longs;o, ita tonus additus diate&longs;&longs;aro, perfi<lb/>cit diapente dulci&longs;simam con&longs;onantiam, ut multi oliuas benè&longs;al&longs;as <lb/>prætulerint fa&longs;ianis: tantum conuenit &longs;al&longs;o cum amaro, amarus, <lb/>quo que &longs;apor leuis non abhorret à pingui, deteriorem <expan abbr="tam&etilde;">tamen</expan> aliquan<lb/>to efficit, ut intortis ex ab&longs;ynthio ouis & ca&longs;eo, atque in uitibus in <lb/>quibus coma ab&longs;ynthij in cocta fuit parum, degenerat tamen &longs;apor <lb/>ille à pingui: ita tono addito ad diapente fit &longs;exta maior, non adeò <lb/>&longs;uauis ut diapente, at tamen <expan abbr="nõ">non</expan> pror&longs;us in&longs;uauis. </s>

<s id="id003107">Similiter &longs;i tonus <lb/>addatur ad &longs;emiditonum aut ad ditonum ex altero fit diate&longs;&longs;aron, <lb/>qui non concordat ex reliquo tritonus omnium a&longs;perrimus. </s>

<s id="id003108">Ergo <lb/>cum idem fiat coniuncto amaro cum in&longs;ipido, ac deterius <expan abbr="c&utilde;">cum</expan> a&longs;trin

<pb pagenum="172 [=178]" xlink:href="015/01/197.jpg"/>gente, uelut in acerbis glandibus, quibus nihil tri&longs;tius gu&longs;tari po<lb/>te&longs;t. </s>

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

<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;">emm</expan> habet a&longs;tringens, quoniam frigidus e&longs;t & &longs;iccus. </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;">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">aunt</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="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">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, duo decimque 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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<s id="id003165">In capitulis enim dextra ac &longs;ini&longs;tra <lb/>&longs;unt foramina homotonorum, per qu&etail; tenduntur ergatis aut &longs;ucu<lb/>lis & uectibus è neruo torti funes, qui non præcluduntur, nec præ<lb/>ligantur ni&longs;i &longs;onitus ad artificis aures certos & &etail;quales fuerint. </s>

<s id="id003166">Bra<lb/>chia enim quæ in eas tentiones includuntur cum exten duntur æ<lb/>qualiter & parter utraque plagam emittere debent. </s>

<s id="id003167">Quod &longs;i non ho<lb/>motona fuerint, impedient directam telorum mi&longs;sionem. </s>

<s id="id003168">Item the<lb/>atris ua&longs;a ærea, qu&etail; in cellis &longs;ub gradib. </s>

<s id="id003168">Item the<lb/>atris ua&longs;a ærea, qu&etail; in cellis &longs;ub gradibus. </s>

<s id="id003169">mathematica ratione <expan abbr="collocan&ttilde;">collo<lb/>cantur</expan>, & <expan abbr="&longs;onit&utilde;">&longs;onitum</expan> di&longs;crimina, qu&etail; Gr&etail;ci <foreign lang="greek">)hx=eia</foreign> <expan abbr="uocãt">uocant</expan>, ad &longs;ymphonias mu <lb/>&longs;icas &longs;iue concentus <expan abbr="componun&ttilde;">componuntur</expan>, diui&longs;a in circinatione diate&longs;&longs;aron <lb/>& diapente & diapa&longs;on, uti uox &longs;c&oelig;nici &longs;onitus <expan abbr="cõueniens">conueniens</expan> in di&longs;po <lb/>&longs;itionibus, tactu <expan abbr="c&utilde;">cum</expan> o&longs;tenderit aucta <expan abbr="c&utilde;">cum</expan> <expan abbr="increm&etilde;to">incremento</expan> clarior et &longs;uauior <lb/>ad <expan abbr="&longs;pectator&utilde;">&longs;pectatorum</expan> perueniat aures. </s>

<s id="id003170">Hydraulicas quo que machinas & cæ<lb/>tera <expan abbr="&qtilde;">quae</expan> &longs;unt &longs;imilia his organis &longs;ine mu&longs;icis rationib. </s>

<s id="id003170">Hydraulicas quo que machinas & cæ<lb/>tera <expan abbr="&qtilde;">quae</expan> &longs;unt &longs;imilia his organis &longs;ine mu&longs;icis rationibus. </s>

<s id="id003171">efficere nemo <lb/>poterit. </s>

<s id="id003172">Capiamus ergo primum illud &qring;d e&longs;t manife&longs;tius, &longs;cilicet de <lb/>hydraulicis organis quorum meminit Suetonius in Nerone: Reli<lb/>quam diei partem per organa hydraulica noui & ignoti generis cir <lb/>cunduxit, o&longs;tenden&longs;que &longs;ingula de ratione ac difficultate cuiu&longs;que di&longs;<lb/>&longs;erens iam &longs;e prolaturum, ut con&longs;tet illa fui&longs;&longs;e magni opificij quæ <lb/>no&longs;tra &etail;tate de&longs;iere.” Re&longs;tat unicum & ualde leue <expan abbr="exempl&utilde;">exemplum</expan> auiculæ <lb/>æneæ uelligneæ re&longs;onantis. </s>

<s id="id003173">Certum e&longs;t <expan abbr="a&etilde;re">aerre</expan> effici &longs;onum, &longs;ed ita mi <lb/>&longs;ceri aquæ, ut dulcior & mollior non &longs;olum euadat, &longs;ed etiam acuti<lb/>or ac modulatior. </s>

<s id="id003173">Certum e&longs;t <expan abbr="a&etilde;re">aere</expan> effici &longs;onum, &longs;ed ita mi<lb/>&longs;ceri aquæ, ut dulcior & mollior non &longs;olum euadat, &longs;ed etiam acuti<lb/>or ac modulatior. </s>

<s id="id003174">Eadem autem ratio maris: &longs;ed cum aquæ corpus <lb/>moueatur, uidetur difficile &longs;eruare proportionem. </s>

<s id="id003175">ea prima diffi<lb/>cultas. </s>

<s id="id003176">&longs;ecunda e&longs;t, quod cùm aqua moueatur, uix ficri po&longs;&longs;e uide<lb/>tur ut totum &longs;eruet uocis integrum tenorem. </s>

<s id="id003176">&longs;ecunda e&longs;t, quod cùm aqua moueatur, uix fieri po&longs;&longs;e uide<lb/>tur ut totum &longs;eruet uocis integrum tenorem. </s>

<s id="id003177">tertia ob illius con

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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 diapent, ex ordine addit diapa&longs;on, po&longs;tea inter <lb/>&longs;edes theatri con&longs;titutis cellis ratione mu&longs;ica ibi collo centur: 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 & hab eant 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="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 duo decim æqualib.

<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;">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 fequun<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

<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

<pb pagenum="178 [=184]" xlink:href="015/01/203.jpg"/>&longs;ticijs di&longs;po&longs;itas di&longs;tribuit &longs;ex hinc atque hinc & unam mediam, quæ <lb/>tamen non u&longs;us, &longs;ed partitionis & re&longs;pon&longs;us cau&longs;a fit in media pr&etail;<lb/>cinctione. </s>

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<s id="id003213">In media quæ &longs;unt ad me&longs;en, quod ea ad proslambanomenon <lb/>habet con&longs;onantiam diapa&longs;on, & ad lychanon hypaton diapente.” </s>

<s id="id003214"><arrow.to.target n="marg591"/><lb/>Hæc autem ex &longs;igura patent in opere de Subtilitate de&longs;cripta.</s>

<s id="id003214"><arrow.to.target n="marg591"/><lb/>Hæc autem ex figura patent in opere de Subtilitate de&longs;cripta.</s>

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<s id="id003220">Coniugationes cuiu&longs;uis numeri breuiter inuenire.</s>

<s id="id003221">Sint gratia exempli <expan abbr="dec&etilde;">decem</expan> homines, & patet quod po&longs;&longs;ent e&longs;&longs;e &longs;in <lb/><arrow.to.target n="marg592"/><lb/>guli, & hoc <expan abbr="dec&etilde;">decem</expan> modis, quia &longs;unt <expan abbr="dec&etilde;">decem</expan>, ut Petrus & Ioannes: item, <lb/>po&longs;&longs;unt e&longs;&longs;e omnes &longs;imul, & hoc uno modo tantum, & po&longs;&longs;unt e&longs;&longs;e <lb/>duo, & hoc pote&longs;t uariari <expan abbr="&qtilde;">quae</expan> draginta quinque modis: & po&longs;&longs;unt e&longs;&longs;e <lb/>octo, & manife&longs;tum e&longs;t, quod <expan abbr="totid&etilde;">totidem</expan> modis uariantur, &longs;cilicet qua<lb/>draginta quinque, nam cum erunt octo, duo <expan abbr="quirelinqu&utilde;tur">quirelinquuntur</expan>, uariari <lb/>po&longs;&longs;unt 45 modis, ergo & illi octo ad <expan abbr="ungu&etilde;">unguem</expan> totidem modis. </s>

<s id="id003221">Sint gratia exempli <expan abbr="dec&etilde;">decem</expan> homines, & patet quod po&longs;&longs;ent e&longs;&longs;e &longs;in<lb/><arrow.to.target n="marg592"/><lb/>guli, & hoc <expan abbr="dec&etilde;">decem</expan> modis, quia &longs;unt <expan abbr="dec&etilde;">decem</expan>, ut Petrus & Ioannes: item, <lb/>po&longs;&longs;unt e&longs;&longs;e omnes &longs;imul, & hoc uno modo tantum, & po&longs;&longs;unt e&longs;&longs;e <lb/>duo, & hoc pote&longs;t uariari <expan abbr="&qtilde;">qua</expan>draginta quinque modis: & po&longs;&longs;unt e&longs;&longs;e <lb/>octo, & manife&longs;tum e&longs;t, quod <expan abbr="totid&etilde;">totidem</expan> modis uariantur, &longs;cilicet qua<lb/>draginta quinque, nam cum erunt octo, duo <expan abbr="quirelinqu&utilde;tur">qui relinquuntur</expan>, uariari <lb/>po&longs;&longs;unt 45 modis, ergo & illi octo ad <expan abbr="ungu&etilde;">unguem</expan> totidem modis. </s>

<s id="id003222">Et &longs;i<lb/>militer tres quot modis uariantur tot modis <expan abbr="&longs;ept&etilde;">&longs;eptem</expan>, & quot modis <lb/>quatuor tot&longs;ex: quinque autem quia &longs;unt dimidium decem, pluribus <lb/>modis uariantur. </s>

<s id="id003223">Etideò pro ordine huius detrahes <expan abbr="un&utilde;">unum</expan>, ut &longs;i &longs;int <lb/>undecim uiri pones decem, &longs;i decem pones <expan abbr="nou&etilde;">nouem</expan>, & colliges natu<lb/>ralem &longs;eriem numerorum, utinfrà uides uno &longs;emper termino defi<lb/>ciente: & expriore ordine, ubi uidebis &longs;emper <expan abbr="etiã">etiam</expan> duplicari nume<lb/>ros: ut 3. 6. in de &longs;ub 6. 10. & 20 àlatere, & &longs;ub 20 35. & à latere 70 du<lb/>plum 35, & &longs;ub <lb/><arrow.to.target n="table23"/><lb/><figure id="id.015.01.204.1.jpg" xlink:href="015/01/204/1.jpg"/>70 126, & à late<lb/>re 252, & hoc pro<lb/>cognitione &qring;d <lb/>rectè &longs;is opera<lb/>tus. </s>

<s id="id003223">Et ideò pro ordine huius detrahes <expan abbr="un&utilde;">unum</expan>, ut &longs;i &longs;int <lb/>undecim uiri pones decem, &longs;i decem pones <expan abbr="nou&etilde;">nouem</expan>, & colliges natu<lb/>ralem seriem numerorum, ut infrà uides uno &longs;emper termino defi<lb/>ciente: & ex priore ordine, ubi uidebis &longs;emper <expan abbr="etiã">etiam</expan> duplicari nume<lb/>ros: ut 3. 6. in de &longs;ub 6. 10. & 20 àlatere, & &longs;ub 20 35. & à latere 70 du<lb/>plum 35, & &longs;ub <lb/><arrow.to.target n="table23"/><lb/><figure id="id.015.01.204.1.jpg" xlink:href="015/01/204/1.jpg"/>70 126, & à late<lb/>re 252, & hoc pro <lb/>cognitione &qring;d <lb/>rectè &longs;is opera<lb/>tus. </s>

<s id="id003224">Secundò a<lb/>nimaduertes <expan abbr="&longs;equ&etilde;tes">&longs;e<lb/>quentes</expan> ordines <lb/>fieri ex recta li<lb/>nea priorum, ue <lb/>lut &longs;extus ordo e&longs;t 7. 28. 84. 210. 462. ita incipiendo in primo ordi<lb/>ne à 7, & tendendo ad dextram, inuenies illos eo&longs;dem numeros ad <lb/>unguem, & ita in &longs;eptimo ordine 8. 36. 120. 330. à &longs;ini&longs;tra inuento 8 <lb/>in primo ordine, & procedendo ad dextram, inuenies 36. 120. & <lb/>330. Tertium e&longs;t quod numeri ultimi à medio &longs;unt ijdem, ut 462 & <lb/>462. 330 & 330. 165 & 165. 55 & 55. 11 & 11. Et &longs;eor&longs;um, ut dixi, rema<lb/>net 1. Oportetigitur colligere numeros angulares, ut à latere ui<lb/>des, & fit 2047 numerus coniugationum, tot enim modis po&longs;&longs;unt <lb/>uariari. </s>

<s id="id003225">Et &longs;i e&longs;&longs;ent decem tantum, ut ab initio propo&longs;ui, primus or<lb/>do finitur ad 10, &longs;ecundus ad 45, tertius ad 120, quartus ad 210, quin <lb/>tus ad 252, &longs;extus redit ad 210, &longs;eptimus ad 120, octauus ad 45, no<lb/>nus ad 10, decimus ad 1. Etita colligeretur &longs;umma ex extremis nu<lb/>meris angularibus 1023. Et tot erunt coniugationes. </s>

<s id="id003226">Hic uides quia <lb/>numerus 10 e&longs;t par, et quod adempta monade, relinquitur 9, qui e&longs;t <lb/>impar quòd medius qui pertinet ad quintum ordinem e&longs;t maxi

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<s id="id003232">Hæc autem ratio &longs;atisfacit multum, & e&longs;t ne<lb/>ce&longs;&longs;aria in temperiebus corporis humani. </s>

<s id="id003233">Vt in <lb/>&longs;ecundo, De dentibus. </s>

<s id="id003234">Et etiam ut quælibet di<lb/>&longs;ciplina quàm breui&longs;simè tradi po&longs;sit, ut gratia <lb/>exempli, medicina tota in una pagina, dico me<lb/>dicina <expan abbr="nõ">non</expan> &longs;olum Græcorum, &longs;ed etiam Arabum <lb/>& Latinorum, & etiam longè plus: nam &longs;i tradatur uigintiquatuor <lb/>regulis fimplicibus, & ex illis fiant coniugationes 16777215, mani <lb/>fe&longs;tum e&longs;t quod erunt regulæ omnes hæ multo plures, quàm con<lb/>tineantur in omnibus libris Græcorum, & Arabum, & Latino<lb/>rum, qui extant. </s>

<s id="id003234">Et etiam ut quælibet di<lb/>&longs;ciplina quàm breui&longs;simè tradi po&longs;sit, ut gratia <lb/>exempli, medicina tota in una pagina, dico me<lb/>dicina <expan abbr="nõ">non</expan> &longs;olum Græcorum, &longs;ed etiam Arabum <lb/>& Latinorum, & etiam longè plus: nam &longs;i tradatur uiginti quatuor <lb/>regulis simplicibus, & ex illis fiant coniugationes 16777215, mani <lb/>fe&longs;tum e&longs;t quod erunt regulæ omnes hæ multo plures, quàm con<lb/>tineantur in omnibus libris Græcorum, & Arabum, & Latino<lb/>rum, qui extant. </s>

<s id="id003235">Et tamen per&longs;picuum e&longs;t, uigintiquatuor regulas <lb/>una pagina commodi&longs;simè contineri. </s>

<s id="id003236">Et hoc aliâs docui, quan<lb/>quàm credam me erra&longs;&longs;e in &longs;upputatione, nam locum inuenire non <lb/>potui. </s>

<s id="id003237">Vnum e&longs;t id certum, quòd hæc ratio quàm nunc explicabo, <lb/>e&longs;t uera & demon&longs;tratiua, & facillima.</s>

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<s id="id003239">Et ideò inueni hanc, quæ (ut <lb/>dixi) facillima e&longs;t: adde numero propo&longs;ito monadem, in de confla<lb/>ri inuenias numerum à monade in eodem ordine, & ab eo detra<lb/>cta monade habes numerum coniugationum. </s>

<s id="id003240">Exemplum, &longs;i &longs;int <lb/>10 adde 1 fit 11. Vndecimus ergo numerus in proportione dupla <lb/>e&longs;t 1024, detrahe 1 & relinquantur 1023 numerus coniugationum, <lb/>ut in priore &longs;upputatione. </s>

<s id="id003241">Item &longs;i &longs;int 11 numeri adde 1 fit 12, duo de<lb/>cimus ergo numerus in proportione dupla e&longs;t 2048, detrahe 1 re<lb/>lin quuntur 2047, coniugationes 11, ut prius in &longs;uprà &longs;cripto exem<lb/>plo. </s>

<s id="id003242">Et ita pro uiginti quatuor regulis adde 1 fit 25, uige&longs;imus quin<lb/>cus igitur numerus in ordine duplæ proportionis à monade e&longs;t <lb/>16777216, ergo detracta monade relin quitur numerus (ut dixi) re<lb/>gularum & coniugationum uigintiquatuor regularum, quæ ta<lb/>men non &longs;int contrariæ inuicem: nam tunc e&longs;&longs;ent pauciores. </s>

<s id="id003242">Et ita pro uiginti quatuor regulis adde 1 fit 25, uige&longs;imus quin<lb/>tus igitur numerus in ordine duplæ proportionis à monade e&longs;t <lb/>16777216, ergo detracta monade relinquitur numerus (ut dixi) re<lb/>gularum & coniugationum uiginti quatuor regularum, quæ ta<lb/>men non &longs;int contrariæ inuicem: nam tunc e&longs;&longs;ent pauciores. </s>

<s id="id003243">Et <lb/>quia in i&longs;tis numeris duplicandis po&longs;&longs;es facile incidere in errorem, <lb/>diuide ultimum per 16, & &longs;i nihil &longs;upere&longs;t, rectè proce&longs;sit opus: &longs;in

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<s id="id003414">Duc ergo, ut ex<lb/>plores ueritatem, 6 in 9 fit 54, duc 2 in 1 fit 2, iunge fit 56 <lb/>diuide per 7 exit 8 perfectio quæ&longs;ita.</s>

<s id="id003415">Per idem intelliges detractionem ex ma&longs;&longs;a argenti perfectionis <lb/>7, detraxi quartam partem perfectionis 10, uolo &longs;cire do drantem

<s id="id003415">Per idem intelliges detractionem ex ma&longs;&longs;a argenti perfectionis <lb/>7, detraxi quartam partem perfectionis 10, uolo &longs;cire dodrantem <pb pagenum="200" xlink:href="015/01/219.jpg"/>

qualis relinquatur perfectionis, duc quadrantem in 10 fit 30, duc 12 <lb/>in 7 fit 84, detrahe 30 ex 84, relinquitur 54, divide 54 per 9, re&longs;idu<lb/>um 12 & 3, exit 6 perfectio re&longs;idui.</s>

<lb/>

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<s id="id003425">Sint duæ quanitates a b & k l, & ab&longs;ci&longs;&longs;æ duæ partes &longs;imiles ex <lb/>utragque b c & l m, & &longs;it propo&longs;ita aliqua proportio, quæ &longs;it h, & <lb/>&longs;umatur portio b d ip&longs;ius b c, &longs;ecundum proportionem h, & &longs;i<lb/>

<s id="id003425">Sint duæ quanitates a b & k l, & ab&longs;ci&longs;&longs;æ duæ partes &longs;imiles ex <lb/>utraque b c & l m, & &longs;it propo&longs;ita aliqua proportio, quæ &longs;it h, & <lb/>&longs;umatur portio b d ip&longs;ius b c, &longs;ecundum proportionem h, & &longs;i<lb/>

<pb pagenum="201" xlink:href="015/01/220.jpg"/>militer l n ip&longs;ius l m, iuxta pro<lb/><figure id="id.015.01.220.1.jpg" xlink:href="015/01/220/1.jpg"/><lb/>portionem h, &longs;umatur rur&longs;us <lb/><arrow.to.target n="marg626"/><lb/>de ip&longs;ius a b pars &longs;ecundum h, <lb/><arrow.to.target n="marg627"/><lb/>& n o ip&longs;ius k l, &longs;ecundum ean <lb/>dem proportionem. </s>

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<pb pagenum="204" xlink:href="015/01/223.jpg"/>portionem b e ad c k data e&longs;t, & c f ad e d, ut b c ad b a, erit ut a c ad <lb/>l e contineat a b ad b k, ut f e ad e l, c b ad b k, &longs;ed a b ad a d, ut d c ad <lb/>d h, igitur a b ad b d, ut d e ad c h. </s>

<s id="id003485">Sunt ergo duæ quantitates a b & <lb/>d c, quæ eandem habent compo&longs;itam proportionem ad g k & k b, <lb/>& h l & l e, quare per præcedentem proportionis h l ad l e, ad pro<lb/>portinem g k ad k b, ut h l detracto prouentu d e, diui&longs;i per propor<lb/>tionem ad d e ad proportionem g k, detracto prouentu a b, diui&longs;i <lb/>per eandem proportionem ad ip&longs;um a b. </s>

<s id="id003485">Sunt ergo duæ quantitates a b & <lb/>d c, quæ eandem habent compo&longs;itam proportionem ad g k & k b, <lb/>& h l & l e, quare per præcedentem proportionis h l ad l e, ad pro<lb/>portionem g k ad k b, ut h l detracto prouentu d e, diui&longs;i per propor<lb/>tionem ad d e ad proportionem g k, detracto prouentu a b, diui&longs;i <lb/>per eandem proportionem ad ip&longs;um a b. </s>

<s id="id003486">Si igitur nota e&longs;t l e & h l, <lb/>erit nota proportio re&longs;idui h l detracto prouentu d e diui&longs;i per pro<lb/>portionem, quare nota detractio g k detracto prouentu a b diui&longs;i <lb/>per eandem proportionem ad a b. </s>

<s id="id003487">E&longs;t autem a b nota, & propor<lb/>tio nota, & ideo prouentus, & cum &longs;it proportio nota, erit ergo <lb/>re&longs;iduum notum, cui addito prouentu fit tota g k nota, quod fuit <lb/>demon&longs;trandum.</s>

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<s id="id003546">Vt con&longs;tet iam in natura animalium <lb/>& in genere caloris habentis magnum motum, & &longs;ub&longs;tantiam te<lb/>nuem hoc inueniri po&longs;&longs;e, ut excludamus plantas de <expan abbr="quar&utilde;">quarum</expan> uita lon<lb/>gi&longs;sima &longs;atis con&longs;tat, &longs;ed quia caret motu euidenti calor in illis, & <lb/>&longs;ub&longs;tantia e&longs;t cra&longs;&longs;a animalium comparatione, non laboro. </s>

<s id="id003547">At de <lb/>elephanto omnes confitentur quòd &longs;it omnium ingenio &longs;i&longs;simum, <lb/>adeò ut multi homines illo indu&longs;tria & cognitione inferiores e&longs;&longs;e <lb/>uideantur. </s>

<s id="id003548">Neque etiam ueri&longs;imile e&longs;t quod natura hominem fecerit <lb/>hac in parte illo inferiorem, præ&longs;ertim cum de nullo alio animali <lb/>apud Ari&longs;totelem dubium &longs;it, & ubi modo aliquod dubium e&longs;&longs;et <lb/>propter querelam Theophra&longs;ti, & illud quod &longs;olet prædicari de <lb/>ceruis, tanto magis ueri&longs;imile e&longs;t indignum fui&longs;&longs;e hominem conce<lb/>dere tot animalibus in diuturnitate uitæ. </s>

<s id="id003549">Quam obrem cum hæc <lb/>tractatio ad libros de tuenda Sanitate &longs;pectaret, homines ad eos re<lb/>leego, nam ob id illos con&longs;crip&longs;i quòd uiderem Galenum nec hoc

<s id="id003549">Quam ob rem cum hæc <lb/>tractatio ad libros de tuenda Sanitate &longs;pectaret, homines ad eos re<lb/>lego, nam ob id illos con&longs;crip&longs;i quòd uiderem Galenum nec hoc <pb pagenum="210" xlink:href="015/01/229.jpg"/>uidi&longs;&longs;e nec multa alia, &longs;ed eorum loco longas & inutiles di&longs;putatio<lb/>nes inter&longs;erui&longs;&longs;e. </s>

<pb pagenum="210" xlink:href="015/01/229.jpg"/>uidi&longs;&longs;e nec multa alia, &longs;ed eorum loco longas & inutiles di&longs;putatio<lb/>nes inter&longs;erui&longs;&longs;e. </s>

<s id="id003550">Verùm etiam, quoniam eam tractationem diuul<lb/>&longs;it, ut alia cogamus quærere in libris de Alimentis, alia, de cibis bo<lb/>ni & mali &longs;ucci: tum uerò & tractatio ip&longs;a eduliorum e&longs;t imperfe<lb/>cta, & multa etiam deficiunt circa genera: in quo e&longs;t ex cu&longs;andus ob <lb/>uarietatem regionis & ætatis. </s>

<s id="id003551">Dee&longs;t præterea maxima pars, quæ <lb/>nec ibi nec alibi habetur, &longs;cilieet, de ciborum præparatione. </s>

<s id="id003551">Dee&longs;t præterea maxima pars, quæ <lb/>nec ibi nec alibi habetur, &longs;cilicet, de ciborum præparatione. </s>

<s id="id003552">Quod <lb/>etiam hæc latuerint tot præclaros uiros, quid mirum? </s>

<s id="id003553">cum Hippo<lb/>crates uixerit &longs;eculo illo agre&longs;ti, in quo non e&longs;t mirandum, quod ali <lb/>quid, pauca quædam & ab&longs;tru&longs;a omi&longs;erit, &longs;ed quod tam multa tam <lb/>bene inuenerit, ut fuerit, &longs;icut de Pindaro dicitur, imò longè uerius <lb/>quam de Pindaro inimitabilis. </s>

<s id="id003554">De Galeno quid mirum, qui non <lb/>ni&longs;i ueterum &longs;cripta collegit, atque utinam <expan abbr="&longs;alt&etilde;">&longs;altem</expan> bene. </s>

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<s id="id003582"><expan abbr="Declarat&utilde;">Declaratum</expan> e&longs;t. </s>

<s id="id003584">&longs;uperius quod unus motus in <lb/><expan abbr="eod&etilde;">eodem</expan> mobili <expan abbr="ali&utilde;">alium</expan> impedit & retardat. </s>

<s id="id003585">Cum ergo h k &longs;it in &longs;pacio a b <lb/>l m & aqua <expan abbr="rapia&ttilde;">rapiatur</expan> motu, dico ad d e mouebit ad d e, & motu dico <lb/>qui uidetur circularis, nam mouetur motu eius à quo <expan abbr="&longs;u&longs;tine&ttilde;">&longs;u&longs;tinetur</expan>. </s>

<s id="id003585">Cum ergo h k &longs;it in &longs;patio a b <lb/>l m & aqua <expan abbr="rapia&ttilde;">rapiatur</expan> motu, dico ad d e mouebit ad d e, & motu dico <lb/>qui uidetur circularis, nam mouetur motu eius à quo <expan abbr="&longs;u&longs;tine&ttilde;">&longs;u&longs;tinetur</expan>. </s>

<s id="id003586">Mo<lb/>uetur etiam ad d e, quoniam pars illa e&longs;t humilior, nam &longs;emper de<lb/>&longs;cendit, omne <expan abbr="a&utilde;t">aunt</expan> quod mouetur partim e&longs;t in termino, à quo, par<lb/>tim ad quem, ideo partim iam aqua illa cum de&longs;cendat humilior e&longs;t <lb/>locus, igitur nauis ad <expan abbr="ill&utilde;">illum</expan> locum feretur. </s>

<s id="id003586">Mo<lb/>uetur etiam ad d e, quoniam pars illa e&longs;t humilior, nam &longs;emper de<lb/>&longs;cendit, omne <expan abbr="a&utilde;t">aut</expan> quod mouetur partim e&longs;t in termino, à quo, par<lb/>tim ad quem, ideo partim iam aqua illa cum de&longs;cendat humilior e&longs;t <lb/>locus, igitur nauis ad <expan abbr="ill&utilde;">illum</expan> locum feretur. </s>

<s id="id003587">Tertio, quia latus k impelli <lb/>tur, in maiore circulo, ideo maiore impetu <lb/><figure id="id.015.01.231.1.jpg" xlink:href="015/01/231/1.jpg"/><lb/>quàm h, quare <expan abbr="de&longs;c&etilde;det">de&longs;cendet</expan> & circulo mouebi<lb/>tur, <expan abbr="nã">nam</expan> &longs;i h quie&longs;ceret <expan abbr="palã">palam</expan> e&longs;t, &qring;d nauis circu <lb/>lariter <expan abbr="mouere&ttilde;">moueretur</expan>, &longs;ed h fungitur uice <expan abbr="quie&longs;c&etilde;tis">quie&longs;cen<lb/>tis</expan>, quia tardius <expan abbr="moue&ttilde;">mouetur</expan> <expan abbr="quã">quam</expan> k, <expan abbr="igi&ttilde;">igitur</expan> k moue<lb/>bitur ad d e & motu circulari aut participe <lb/>eius. </s>

<s id="id003588">Quarta cau&longs;a e&longs;t, quoniam h cupit <expan abbr="de&longs;c&etilde;dere">de<lb/>&longs;cendere</expan>, ut graue. </s>

<s id="id003589">ergo ferri, ubi minus impe <lb/>diatur à motu <expan abbr="uiol&etilde;to">uiolento</expan>, at minus <expan abbr="impedi&ttilde;">impeditur</expan> in <lb/>circulo, de qua a b, qa a b <expan abbr="c&utilde;">cum</expan> maioris &longs;it ambitus a qua in co ulterius <lb/><expan abbr="fer&ttilde;">fertur</expan> <expan abbr="quã">quam</expan> in d e, ob hæc <expan abbr="o&itilde;a">oina</expan> & in mari & fluminibus ac lacubus <expan abbr="c&utilde;">cum</expan> na <lb/>ues fuerint in ambitu uorticis <expan abbr="iã">iam</expan> <expan abbr="rapiun&ttilde;">rapiuntur</expan> ad <expan abbr="ill&utilde;">illum</expan>, & circulari motu: <lb/>isque motus e&longs;t <expan abbr="indici&utilde;">indicium</expan> &longs;ubmer&longs;ionis, <expan abbr="quoniã">quoniam</expan> indicat <expan abbr="aquã">aquam</expan>, ibi propè <lb/><expan abbr="de&longs;c&etilde;dere">de&longs;cendere</expan> rectà uer&longs;us <expan abbr="c&etilde;tr&utilde;">centrum</expan>, & ob id <expan abbr="prud&etilde;tes">prudentes</expan> naut&etail; magna ui uen <lb/>toru & <expan abbr="remor&utilde;">remorum</expan> &longs;&etail;pe <expan abbr="&longs;eruãt">&longs;eruant</expan> &longs;e, pr&etail;o <expan abbr="ccupãtes">ccupantes</expan> <expan abbr="mot&utilde;">motum</expan> <expan abbr="elic&utilde;">elicum</expan> recto motu. <lb/></s>

<s id="id003589">ergo ferri, ubi minus impe<lb/>diatur à motu <expan abbr="uiol&etilde;to">uiolento</expan>, at minus <expan abbr="impedi&ttilde;">impeditur</expan> in <lb/>circulo, de qua a b, qua a b <expan abbr="c&utilde;">cum</expan> maioris &longs;it ambitus a qua in co ulterius <lb/><expan abbr="fer&ttilde;">fertur</expan> <expan abbr="quã">quam</expan> in d e, ob hæc <expan abbr="o&itilde;a">oina</expan> & in mari & fluminibus ac lacubus <expan abbr="c&utilde;">cum</expan> na<lb/>ues fuerint in ambitu uorticis <expan abbr="iã">iam</expan> <expan abbr="rapiun&ttilde;">rapiuntur</expan> ad <expan abbr="ill&utilde;">illum</expan>, & circulari motu: <lb/>isque motus e&longs;t <expan abbr="indici&utilde;">indicium</expan> &longs;ubmer&longs;ionis, <expan abbr="quoniã">quoniam</expan> indicat <expan abbr="aquã">aquam</expan>, ibi propè <lb/><expan abbr="de&longs;c&etilde;dere">de&longs;cendere</expan> rectà uer&longs;us <expan abbr="c&etilde;tr&utilde;">centrum</expan>, & ob id <expan abbr="prud&etilde;tes">prudentes</expan> naut&etail; magna ui uen<lb/>toru & <expan abbr="remor&utilde;">remorum</expan> &longs;&etail;pe <expan abbr="&longs;eruãt">&longs;eruant</expan> &longs;e, pr&etail;o<expan abbr="ccupãtes">ccupantes</expan> <expan abbr="mot&utilde;">motum</expan> <expan abbr="elic&utilde;">elicum</expan> recto motu. <lb/></s>

<s id="id003590">Cur <expan abbr="a&utilde;t">aunt</expan> aqua <expan abbr="&qtilde;">quae</expan> e&longs;t in a, non potius <expan abbr="fera&ttilde;">feratur</expan> per obliquam lineam ad d <lb/>uel g, <08> ad e uel c inde ex illis ad d uel g, præ&longs;ertim <expan abbr="c&utilde;">cum</expan> ad&longs;it breuior

<s id="id003590">Cur <expan abbr="a&utilde;t">aut</expan> aqua <expan abbr="&qtilde;">quae</expan> e&longs;t in a, non potius <expan abbr="fera&ttilde;">feratur</expan> per obliquam lineam ad d <lb/>uel g, <08> ad e uel c inde ex illis ad d uel g, præ&longs;ertim <expan abbr="c&utilde;">cum</expan> ad&longs;it breuior

<pb pagenum="213" xlink:href="015/01/232.jpg"/>a e & e d et a g breuior a e et c (ut docet Euclides) cau&longs;a e&longs;t quia aqua <lb/>quæ de&longs;cendit per e d & c g maiore impetu de&longs;cendit quàm per ad <lb/>uel a g ut demon&longs;tratum e&longs;t, ergo non poterit quæ e&longs;t in e d uel e g <lb/>loco dimoueri, nec cedere aquæ per obliquam lineam de&longs;cendenti.</s>

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<s id="id003598">Sed cur rur&longs;us contractiora dum &longs;unt crura, homo facilius exur<lb/>git? </s>

<s id="id003599">Proponantur c f contracta ad perpendiculum, & in clinetur b a <lb/>in o ut fiant b o & f e equidi&longs;tantes, ita enim commodius &longs;urgimus: <lb/>nec aliter qui &longs;unt imbecilliores: quia ergo b e&longs;t in directo f, ideo <lb/>mu&longs;culi femoris inferiores ob crus, & &longs;uperiores ob pectus &longs;unt <lb/>magis ten&longs;i & anteriores cruris itidem, ideo maiore ui trahunt par <lb/>ticulam. </s>

<s id="id003599">Proponantur c f contracta ad perpendiculum, & inclinetur b a <lb/>in o ut fiant b o & f e aequidi&longs;tantes, ita enim commodius &longs;urgimus: <lb/>nec aliter qui &longs;unt imbecilliores: quia ergo b e&longs;t in directo f, ideo <lb/>mu&longs;culi femoris inferiores ob crus, & &longs;uperiores ob pectus &longs;unt <lb/>magis ten&longs;i & anteriores cruris itidem, ideo maiore ui trahunt par<lb/>ticulam. </s>

<s id="id003600">Vnde manente fixo f & capite etiam & pectore grauitate <lb/>&longs;ua adiuuantibus, facilius homo exurgit quam ad latos angulos <lb/>cum contractio, ut dixi, mu&longs;culorum et inclinatio partium &longs;uperio<lb/>rum fiat maior.</s>

<s id="id003601">Rur&longs;us pro prima parte problematis, dico quòd quanto altior <lb/>e&longs;t b f tanto facilius exurgit, nam &longs;upponatur angu<lb/><figure id="id.015.01.232.2.jpg" xlink:href="015/01/232/2.jpg"/><lb/>lus reflixionis a h e æqualis a h c, & b c k æqualis h k f, <lb/>igitur cum b f &longs;it breuior b f, erit h k breuior b c & f k, <lb/>f c. quare b c femur, & f c crus erunt uiolentius exten<lb/>&longs;a quàm in &longs;itu h k, k f ergo, mu&longs;culi fa cilius erigent <lb/>&longs;edentem altiore loco quàm humiliore, quod erat de<lb/>mon&longs;trandum.</s>

<s id="id003601">Rur&longs;us pro prima parte problematis, dico quòd quanto altior <lb/>e&longs;t b f tanto facilius exurgit, nam &longs;upponatur angu<lb/><figure id="id.015.01.232.2.jpg" xlink:href="015/01/232/2.jpg"/><lb/>lus reflexionis a h e æqualis a h c, & b c k æqualis h k f, <lb/>igitur cum b f &longs;it breuior b f, erit h k breuior b c & f k, <lb/>f c. quare b c femur, & f c crus erunt uiolentius exten<lb/>&longs;a quàm in &longs;itu h k, k f ergo, mu&longs;culi facilius erigent <lb/>&longs;edentem altiore loco quàm humiliore, quod erat de<lb/>mon&longs;trandum.</s>

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<s id="id003652">Sint pondera b minus, c maius, & ducatur aggre<lb/><figure id="id.015.01.235.2.jpg" xlink:href="015/01/235/2.jpg"/><lb/>gatum ex a uiribus & b minore pondere in e, & ei <lb/>addatur quadratum dimidij a, dico quod radix &longs;eu<lb/>latus huius detracto dimidio a e&longs;t pondus auxiliare <lb/>æquale, &longs;it productum a b in e &longs;uperficies & quadra<lb/>tum dimidij a &longs;it e, ita quod tota d e &longs;it &longs;uperficies <lb/>quadrata, cuius latus &longs;it f g: f h autem dimidium a di<lb/>co h g e&longs;&longs;e pondus auxiliare æquale. </s>

<s id="id003652">Sint pondera b minus, c maius, & ducatur aggre<lb/><figure id="id.015.01.235.2.jpg" xlink:href="015/01/235/2.jpg"/><lb/>gatum ex a uiribus & b minore pondere in e, & ei <lb/>addatur quadratum dimidij a, dico quod radix &longs;eu <lb/>latus huius detracto dimidio a e&longs;t pondus auxiliare <lb/>æquale, &longs;it productum a b in e &longs;uperficies & quadra<lb/>tum dimidij a &longs;it e, ita quod tota d e &longs;it &longs;uperficies <lb/>quadrata, cuius latus &longs;it f g: f h autem dimidium a di<lb/>co h g e&longs;&longs;e pondus auxiliare æquale. </s>

<s id="id003653">Quia enim f g

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<s id="id003693">Omitto modo de motu aucto per <lb/>uim humanam, de quo uidetur quærere Ari&longs;toteles, quili<lb/>bet enim aër addit &longs;uper motum iam acqui&longs;itum & fit hoc <lb/>argumentum centies ac millies maius, quoniam m e&longs;t qui <lb/>diuidit, pondus autem non ponetrat. </s>

<s id="id003694">Sicut ergo cuneus <lb/>magis diuidit lignum quam claua, ita quod mouetur &longs;ine <lb/>proportione (ut ita dicam) non &longs;olum ob <expan abbr="impet&utilde;">impetum</expan> nece&longs;&longs;e <lb/>e&longs;t ut uehementer diuidat lignum aut lapidem &longs;ubiectum, <lb/>& non in proportione di&longs;tanti&etail;. </s>

<s id="id003695">Sicut &longs;i pondus in forma <lb/>&longs;ecuris, & ip&longs;a &longs;ecuris diuidit longe magis ligna quam cla<lb/>uis maioris ponderis & maiore ui de&longs;cendens: ita pondus motum <lb/>quam immotum. </s>

<s id="id003696">Hoc adeò per&longs;picuam habet cau&longs;&longs;am, ut quanto <lb/>plura uerba addererentur, eo redderetur res difficilior. </s>

<s id="id003696">Hoc adeò per&longs;picuam habet cau&longs;&longs;am, ut quanto <lb/>plura uerba adderentur, eo redderetur res difficilior. </s>

<s id="id003697">Habet ergo <lb/>propriam &longs;olum grauitatem & motum occultum. </s>

<s id="id003698">C&etail;terum e&longs;t ter<lb/>tium, genus <expan abbr="medi&utilde;">medium</expan>, cum idem pondus appen&longs;um e&longs;t, ue<lb/><figure id="id.015.01.237.2.jpg" xlink:href="015/01/237/2.jpg"/><lb/>lut f quod dico e&longs;&longs;e maius & minus occultum quam &longs;i ia<lb/>ceret in plano, quoniam &longs;icut tuber & cauitas in qua iacet <lb/>&longs;imul tempore &longs;unt, natura tamen tuber e&longs;t prius cauitate, <lb/>ita pondus appen&longs;um prius e&longs;t, contrà nixum uinculi na<lb/>tura & quodammodo tempore, &longs;emper enim grauat, & illud &longs;em<lb/>per re&longs;i&longs;tit &longs;upra illius grauitatem: Sed pondus quod e&longs;t in plano <lb/>occultam omnino habet actionem bifariamque di&longs;ting uitur a pon<lb/>dere &longs;u&longs;pen&longs;o: Primum quòd pondus quod quie&longs;cit & contra in<lb/>tendi principium &longs;imul non &longs;olum &longs;unt tempore &longs;ed etiam natu<lb/>ra. </s>

<s id="id003699">Sed in appen&longs;o, ut dixi, pondus prius grauat quam uincu

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Line 10978

<s id="id003723"><arrow.to.target n="marg698"/><lb/>&longs;it dupla a c, & <foreign lang="greek">a b a g</foreign> dupla, & diuidantur a b a c & <foreign lang="greek">a b a g</foreign> in quotuis <lb/>partes &etail;quales inuicem, nam &longs;upponitur a b &etail;qualis <foreign lang="greek">a b</foreign> & a c æqua<lb/>lis <foreign lang="greek">a g</foreign>, & ducantur rectæ lineæ decu&longs;&longs;atim & ad rectos angulos, & <lb/><expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> id &longs;tatuantur re&longs;tes, quod decu&longs;&longs;a<lb/><figure id="id.015.01.239.1.jpg" xlink:href="015/01/239/1.jpg"/><lb/>tim po&longs;itæ utiliores <expan abbr="er&utilde;t">erunt</expan>, omitto quod de<lb/>centius ob &longs;patiorum minorem differenti<lb/>am. </s>

<s id="id003724">Adducam &longs;olùm tres Philo&longs;ophi ratio<lb/>nes: prima, quoniam ligna non adeò facilè <lb/>finduntur nec incuruantur tran&longs;uer&longs;im tra<lb/>cta, ut recta & &longs;ecundum longitudinem, Et <lb/>ideò longè plus durabit <foreign lang="greek">a b g d</foreign> <expan abbr="quã">quam</expan> a b c d, <lb/>& cum &longs;pondis rectoribus, & ideò etiam <lb/>cum re&longs;tibus magis intentis: & erit firmior <lb/>& pulchrior. </s>

<s id="id003725">Secunda ratio e&longs;t, quod cum <lb/>re&longs;tes in &longs;ecunda con&longs;titutione æquales inuicem &longs;int, in prima quæ <lb/>&longs;ecundum latitudinem dupl&etail;, qu&etail; longiores erunt magis laxabun<lb/>tur tran&longs;uer&longs;alibus, & ita turpiores & incommodæ breui redden<lb/>tur, & in &longs;ecunda con&longs;titutione &etail;qualiter &longs;u&longs;tinebunt pondus & re<lb/>uolutionem cubantis, tum ob æqualitatem longitudinis inter &longs;e, <lb/>tum ob &longs;itum &longs;imilem inter &longs;e, tum ad humanum decubitum <expan abbr="di&longs;simil&etilde;">di&longs;si<lb/>milem</expan>, nam (ut o&longs;ten&longs;um e&longs;t) in præcedenti magis grauat pondus in <lb/>extremis quam in medio, & magis laxantur ob id quæ &longs;unt &longs;ecun<lb/>dum eundem fitum. </s>

<s id="id003726">Et hanc cau&longs;&longs;am expo&longs;itores non intellexe<lb/>runt multi, multo minus tertiam, in qua faciunt demon&longs;trationem <lb/>Geometricam & computantrem numeris. </s>

<s id="id003726">Et hanc cau&longs;&longs;am expo&longs;itores non intellexe<lb/>runt multi, multo minus tertiam, in qua faciunt demon&longs;trationem <lb/>Geometricam & computantem numeris. </s>

<s id="id003727">Deinde non animaduer <lb/>tunt quod in &longs;ecunda figura a&longs;&longs;umunt quinque lineas, cum in prima <lb/>tantum a&longs;&longs;ump&longs;i&longs;&longs;ent quatuor. </s>

<s id="id003728">Peius omnibus e&longs;t quod demon<lb/>&longs;tratio hæc cum de tran&longs;uer&longs;is ad magis tran&longs;uer&longs;as lineas &longs;it non <lb/>e&longs;t ad propo&longs;itum Ari&longs;totelis, qui in duabus primis rationibus <lb/>tran&longs;uer&longs;as comparauit his, quæ à latere ad latus & à capite ad ca<lb/>put deducuntur, ita ubi trifariam decepti &longs;unt, ibi maximè glori<lb/>antur. </s>

<s id="id003729">Mi&longs;erum nunc philo&longs;ophandi genus: uoluntque &longs;upercilium <lb/>e&longs;&longs;e loco doctrinæ. </s>

<s id="id003730">Sint igitur lineæ ductæ ut uides, dico omnes <lb/>pariter acceptas in prima figura, e&longs;&longs;e longiores omnibus pariter ac<lb/><arrow.to.target n="marg699"/><lb/>ceptis in &longs;ecunda figura, quod intendit <expan abbr="demõ">demom</expan> &longs;trare Ari&longs;toteles. </s>

<s id="id003731">O<lb/>&longs;ten&longs;o ergo de duabus, idem &longs;uppo&longs;ito numero equali de omnibus

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<s id="id003741">Si duo circuli &longs;uper eodem centro eodem motu transferuntur, <lb/>æquale &longs;patiu m &longs;uperant.</s>

<s id="id003742">Sint duo circuli a b, c d &longs;uper eodem centro e qui transferantur <lb/><figure id="id.015.01.240.1.jpg" xlink:href="015/01/240/1.jpg"/><lb/><arrow.to.target n="marg705"/><lb/>&longs;uper axe per <expan abbr="&longs;pati&utilde;">&longs;patium</expan> c g dum re&longs;oluitur c d, <lb/>tum ergo a erit in f, quia c d contingit pla<lb/>num c g, igitur e c e&longs;t ad <expan abbr="perp&etilde;diculum">perpendiculum</expan> c g, <lb/><arrow.to.target n="marg706"/><lb/>ergo punctum a e&longs;t in f & a f æqualis c g, <lb/><arrow.to.target n="marg707"/><lb/>igitur a b circulus &longs;olum reuolutus e&longs;t &longs;e<lb/>mel, & tantum perambulauit &longs;pacij quan<lb/>tum e d & æquali uelo citate, cùm tamen &longs;eor&longs;um &longs;it proportio &longs;pa<lb/>tij ad <expan abbr="&longs;pati&utilde;">&longs;patium</expan> ut circuli ad circulum. </s>

<s id="id003742">Sint duo circuli a b, c d &longs;uper eodem centro e qui transferantur <lb/><figure id="id.015.01.240.1.jpg" xlink:href="015/01/240/1.jpg"/><lb/><arrow.to.target n="marg705"/><lb/>&longs;uper axe per <expan abbr="&longs;pati&utilde;">&longs;patium</expan> c g dum re&longs;oluitur c d, <lb/>tum ergo a erit in f, quia c d contingit pla<lb/>num c g, igitur e c e&longs;t ad <expan abbr="perp&etilde;diculum">perpendiculum</expan> c g, <lb/><arrow.to.target n="marg706"/><lb/>ergo punctum a e&longs;t in f & a f æqualis c g, <lb/><arrow.to.target n="marg707"/><lb/>igitur a b circulus &longs;olum reuolutus e&longs;t &longs;e<lb/>mel, & tantum perambulauit &longs;patij quan<lb/>tum e d & æquali uelo citate, cùm tamen &longs;eor&longs;um &longs;it proportio &longs;pa<lb/>tij ad <expan abbr="&longs;pati&utilde;">&longs;patium</expan> ut circuli ad circulum. </s>

<s id="id003743">Hæc e&longs;t &longs;ubtili&longs;sima <expan abbr="quæ&longs;tion&utilde;">quæ&longs;tionum</expan> <lb/><arrow.to.target n="marg708"/><lb/><expan abbr="propo&longs;itar&utilde;">propo&longs;itarum</expan> ab Ari&longs;to tele in mechanicis, quam &longs;ic quidam &longs;oluunt. <lb/></s>

<s id="id003744">Supponunt duo: <expan abbr="prim&utilde;">primum</expan> &longs;i quid ab aliquo mouetur nihil conferens <lb/><figure id="id.015.01.240.2.jpg" xlink:href="015/01/240/2.jpg"/><lb/>ad illum motum, <lb/>ex&longs;e ip&longs;o per tan <lb/>tum mouebitur <lb/><expan abbr="&longs;pati&utilde;">&longs;patium</expan>, per quan<lb/>tum ab illo mo<lb/>tore mouebitur: <lb/>Secundum, <expan abbr="ead&etilde;">eadem</expan> <lb/>potentia in <expan abbr="eod&etilde;">eodem</expan> <lb/>tempore diuer&longs;o <lb/>modo duo mobi <lb/>lia mouebit &etail;qua <lb/>lia, cum <expan abbr="un&utilde;">unum</expan> mo<lb/>tui a&longs;&longs;entietur aliud <expan abbr="nõ">non</expan>. </s>

<s id="id003745">quod &longs;i hæc mobilia &longs;eiuncta fui&longs;&longs;ent, quod <lb/>aptitudinem haberet <expan abbr="&longs;eiunct&utilde;">&longs;eiunctum</expan> uelo cius moueretur, quàm dum con <lb/>iunctum e&longs;t. </s>

<s id="id003746">Cum ergo inquiunt circulus c d moueatur ab a b cir<lb/>culo, nec conferat quic<08> ad motum, ideo tantum tran&longs;ibit &longs;pacium

<s id="id003746">Cum ergo inquiunt circulus c d moueatur ab a b cir<lb/>culo, nec conferat quic<08> ad motum, ideo tantum tran&longs;ibit &longs;patium <pb pagenum="222" xlink:href="015/01/241.jpg"/>c d quantum a b per primum &longs;uppo&longs;itum. </s>

<s id="id003747">Sed quoniam proposi<lb/>to circulo alio non circa idem centrum, utpote k l reuoluetur & <lb/>perueniet ad h ex demon&longs;tratis. </s>

<pb pagenum="222" xlink:href="015/01/241.jpg"/>c d quantum a b per primum &longs;uppo&longs;itum. </s>

<s id="id003747">Sed quoniam propofi<lb/>to circulo alio non circa idem centrum, utpote k l reuoluetur & <lb/>perueniet ad h ex demon&longs;tratis. </s>

<s id="id003748"><expan abbr="Re&longs;ponde&ttilde;">Re&longs;pondetur</expan> ad hoc, quod idem e&longs;t, <lb/>quia unus circulus tantum per &longs;e mouetur circa centrum, reliqui <lb/>omnes non per&longs;e circa centrum, &longs;ed ab alio circulo primo mouen<lb/>tur, ideò nihil refert &longs;eu &longs;int circa idem centrum &longs;eu circa aliud, hoc <lb/>enim fortuitum e&longs;t. </s>

<s id="id003749">Ideo ad argumentum re&longs;pondent cauillo&longs;am <lb/>e&longs;&longs;e <expan abbr="hãc">hanc</expan> di&longs;putationem, cum &longs;upponatidem ambobus circulis per <lb/>&longs;e centrum e&longs;&longs;e. </s>

<s id="id003750">Sed non e&longs;t per&longs;e, uerùm per <expan abbr="accid&etilde;s">accidens</expan>. </s>

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<s id="id003808">Sit a b c æquilaterus circulo in&longs;criptus, </s>

<s id="id003809"><arrow.to.target n="marg716"/><lb/>cuius centrum d, ducaturque ad e f rectà per <lb/>centrum, & ducantur d b & d c, eritque ex hoc <lb/><arrow.to.target n="marg717"/><lb/>triangulus a b d &etail;quilaterus triangulo a c d, <lb/><arrow.to.target n="marg718"/><lb/>quare angulus b a d æqualis c a d, igitur ar<lb/>cus b e æqualis c e, igitur arcus b e e&longs;t &longs;exta <lb/><arrow.to.target n="marg719"/><lb/>pars circuli, quare b e recta latus exagoni, <lb/>quare b e erit æqualis d e, igitur cum anguli <lb/><arrow.to.target n="marg720"/><lb/>a d f &longs;int utrin que recti, crit d f æqualis f e, itaque<lb/><arrow.to.target n="marg721"/><lb/>f d, tertia pars fa & fb dimidium a b quia b c. </s>

<s id="id003809"><arrow.to.target n="marg716"/><lb/>cuius centrum d, ducaturque ad e f rectà per <lb/>centrum, & ducantur d b & d c, eritque ex hoc <lb/><arrow.to.target n="marg717"/><lb/>triangulus a b d &etail;quilaterus triangulo a c d, <lb/><arrow.to.target n="marg718"/><lb/>quare angulus b a d æqualis c a d, igitur ar<lb/>cus b e æqualis c e, igitur arcus b e e&longs;t &longs;exta <lb/><arrow.to.target n="marg719"/><lb/>pars circuli, quare b e recta latus exagoni, <lb/>quare b e erit æqualis d e, igitur cum anguli <lb/><arrow.to.target n="marg720"/><lb/>a d f &longs;int utrin que recti, erit d f æqualis f e, itaque <lb/><arrow.to.target n="marg721"/><lb/>f d, tertia pars fa & fb dimidium a b quia b c. </s>

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<s id="id003937"><arrow.to.target n="marg766"/><lb/>h f k, &longs;i non faciant trian gulum æquicruri<lb/>um b c d &longs;uper b c, & cuius ambo latera pa <lb/>riter accepta &longs;int æqualia a b & a c. </s>

<s id="id003938">Et du<lb/>cam contingentem & habebo trapezium <lb/><arrow.to.target n="marg767"/><lb/>h b, c k. </s>

<s id="id003939">Quare &longs;i peripheria circuli b c e&longs;t <lb/><figure id="id.015.01.250.1.jpg" xlink:href="015/01/250/1.jpg"/><lb/>minor d b & d c pariter acceptis, habeo <expan abbr="intent&utilde;">intentum</expan>, &longs;i non toties <expan abbr="diuidã">diuidam</expan> <lb/>peripheriam per æqualia ut fiat figura polygonia &longs;uper b c æquila<lb/>tera & æquiangula, cuius differentia a peripheria &longs;it minor differen <lb/>tia d b & d c à trapezio b h, k c, id e&longs;t, tribus eius lateribus, nam cum <lb/>d h & d k &longs;int maiores h k, con&longs;tat quod d b & d e &longs;unt maiores h b, <lb/>& k c & h k igitur &longs;it differentia illa l, & <expan abbr="differ&etilde;tia">differentia</expan> peripheri&etail; à lineis <lb/>polyg oniæ minorl: igitur cum peripheria &longs;it æqualis aut maior <lb/>d b & d c, & differentia a lateribus polygoniæ minor quàm d b & <lb/>d c, a b, h b, h k, k c, erit minor proportio peripheriæ ad latera poly<lb/><arrow.to.target n="marg768"/><lb/>goniæ quàm d b & d c ad tria latera trapezij, quare minor propor<lb/><arrow.to.target n="marg769"/><lb/>tio peripheriæ ad d b & d c quàm laterum polygoniæ ad tria latera <lb/><arrow.to.target n="marg770"/><lb/>trapezij, &longs;ed latera polygoniæ &longs;unt minora tribus laterib. </s>

<s id="id003939">Quare &longs;i peripheria circuli b c e&longs;t <lb/><figure id="id.015.01.250.1.jpg" xlink:href="015/01/250/1.jpg"/><lb/>minor d b & d c pariter acceptis, habeo <expan abbr="intent&utilde;">intentum</expan>, &longs;i non toties <expan abbr="diuidã">diuidam</expan> <lb/>peripheriam per æqualia ut fiat figura polygonia &longs;uper b c æquila<lb/>tera & æquiangula, cuius differentia a peripheria &longs;it minor differen<lb/>tia d b & d c à trapezio b h, k c, id e&longs;t, tribus eius lateribus, nam cum <lb/>d h & d k &longs;int maiores h k, con&longs;tat quod d b & d e &longs;unt maiores h b, <lb/>& k c & h k igitur &longs;it differentia illa l, & <expan abbr="differ&etilde;tia">differentia</expan> peripheri&etail; à lineis <lb/>polygoniæ minori: igitur cum peripheria &longs;it æqualis aut maior <lb/>d b & d c, & differentia a lateribus polygoniæ minor quàm d b & <lb/>d c, a b, h b, h k, k c, erit minor proportio peripheriæ ad latera poly<lb/><arrow.to.target n="marg768"/><lb/>goniæ quàm d b & d c ad tria latera trapezij, quare minor propor<lb/><arrow.to.target n="marg769"/><lb/>tio peripheriæ ad d b & d c quàm laterum polygoniæ ad tria latera <lb/><arrow.to.target n="marg770"/><lb/>trapezij, &longs;ed latera polygoniæ &longs;unt minora tribus lateribus. </s>

<s id="id003940">trapezij, <lb/><arrow.to.target n="marg771"/><lb/>igitur peripheria b c e&longs;t minor d b & d e, quod erat <expan abbr="demon&longs;trand&utilde;">demon&longs;trandum</expan>.</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;ubij ciam. </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">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>

<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 in clu&longs;os, circa quos fu<lb/>nis uo catur, ut in tro chleis & orbiculi & funes in cluduntur. </s>

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

<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">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 poteric <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>

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<s id="id004056"><margin.target id="marg793"/>P<emph type="italics"/>er<emph.end type="italics"/> 6. <emph type="italics"/>primi<emph.end type="italics"/><lb/>E<emph type="italics"/>lem.<emph.end type="italics"/></s>

<s id="id004057">Quod <expan abbr="punct&utilde;">punctum</expan> l <expan abbr="mouebi&ttilde;">mouebitur</expan> duob. </s>

<s id="id004057">Quod <expan abbr="punct&utilde;">punctum</expan> l <expan abbr="mouebi&ttilde;">mouebitur</expan> duobus </s>

<s id="id004058">motib. </s>

<s id="id004058">motibus. </s>

<s id="id004059">uno recto in linea, &longs;cilicet <lb/><arrow.to.target n="marg794"/><lb/>l m, & altero circulari. </s>

<s id="id004060">&longs;. </s>

<s id="id004061">circa <expan abbr="centr&utilde;">centrum</expan> k, & <expan abbr="tñ">tnm</expan> <expan abbr="mouebi&ttilde;">mouebitur</expan> uerè motu re<lb/>cto <expan abbr="t&mtilde;">tmm</expan> in alia linea, &longs;cilicet a c, & hoc e&longs;t <expan abbr="prim&utilde;">primum</expan> admirabile. </s>

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<s id="id004153">Ex hac etiam patet, quòd con&longs;tituto angulo in cen<lb/><arrow.to.target n="marg822"/><lb/>tro a b c, & ducta linea a d à puncto a, &longs;ciemus quo re&longs;i<lb/>&longs;ilit in linea b c: ducta enim c d, faciemus angulum c d e <lb/><arrow.to.target n="marg823"/><lb/>æqualem a b c, & erit angulus a d g æqualis angulo e d <lb/>h, igitur d e re&longs;ilit ex a b a d linea.</s>

<s id="id004153">Ex hac etiam patet, quòd con&longs;tituto angulo in cen<lb/><arrow.to.target n="marg822"/><lb/>tro a b c, & ducta linea a d à puncto a, &longs;ciemus quo re&longs;i<lb/>lit in linea b c: ducta enim c d, faciemus angulum c d e <lb/><arrow.to.target n="marg823"/><lb/>æqualem a b c, & erit angulus a d g æqualis angulo e d <lb/>h, igitur d e re&longs;ilit ex a b a d linea.</s>

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<s id="id004297">Et ex illo loco apparet, quod Auerroes ne&longs;ciuit Geo

<pb pagenum="250" xlink:href="015/01/269.jpg"/>metriam, ficut &longs;emper fuit mos Philo&longs;ophorum <expan abbr="cõtentio&longs;orum">contentio&longs;orum</expan>, ut <lb/>nil &longs;ciant, &longs;ed &longs;olum garrire. </s>

<pb pagenum="250" xlink:href="015/01/269.jpg"/>metriam, sicut &longs;emper fuit mos Philo&longs;ophorum <expan abbr="cõtentio&longs;orum">contentio&longs;orum</expan>, ut <lb/>nil &longs;ciant, &longs;ed &longs;olum garrire. </s>

<s id="id004298">audierat hoc ab aliquo malo Geome<lb/>tra, & repo&longs;uitin &longs;uos libros: nam nos, ut &longs;uprà uidi&longs;ti, demon&longs;tra<lb/>uimus oppo&longs;itum. </s>

<s id="id004299">Quod uerò &longs;it macula illa ex umbra terræ, ue<lb/>rum non e&longs;t, quoniam una e&longs;&longs;et & non diui&longs;a, & occuparet totam il <lb/>lius faciem: nec e&longs;t uerum quod mutaret &longs;itum, quia &longs;uperficies ter<lb/>ræ e&longs;t nonupla &longs;uperficiei Lunæ. </s>

<s id="id004300">Sicut terræ &longs;uperficies e&longs;t minor <lb/>trige&longs;ima parte &longs;uperficiei Solis. </s>

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<s id="id004314">Sit peluis a b aqua plena: &longs;peculum in ea c d e f quadratum, aut <lb/>perfecte, aut oblongum &longs;u bmer&longs;um in ea: Sol primum &longs;olus in g

<s id="id004314">Sit peluis a b aqua plena: &longs;peculum in ea c d e f quadratum, aut <lb/>perfecte, aut oblongum &longs;ub mer&longs;um in ea: Sol primum &longs;olus in g <pb pagenum="251" xlink:href="015/01/270.jpg"/>oculus ex aduer&longs;o in <lb/>h, ita ut ad æquales <lb/>angulos po&longs;sit uide<lb/>re <expan abbr="Sol&etilde;">Solem</expan> in k, dico &qring;d <lb/>depre&longs;&longs;o oculo in m, <lb/>uidebit alium Solem <lb/>maiorem uer&longs;us mar<lb/>ginem aduer&longs;um in l, <lb/>& longè &longs;plendidio<lb/>rem: quia enim radij <lb/><expan abbr="reflect&utilde;tur">reflectuntur</expan> ex k, ut ro <lb/>bu&longs;ti & à medio den<lb/>&longs;iore ad rarius, qui <lb/>non <expan abbr="inflecten&ttilde;">inflectentur</expan>, erunt <lb/>pauci, & ideò Sol in <lb/>k minor apparebit, et <lb/>languidior: maior au<lb/><figure id="id.015.01.270.1.jpg" xlink:href="015/01/270/1.jpg"/><lb/>tem pars deflectetur à <expan abbr="perp&etilde;diculari">perpendiculari</expan> ad m, igitur Sol apparebit ma<lb/>ior & ualidior longè &longs;plendentibus radijs, adeò ut uix ferri po&longs;sit. <lb/></s>

<pb pagenum="251" xlink:href="015/01/270.jpg"/>oculus ex aduer&longs;o in <lb/>h, ita ut ad æquales <lb/>angulos po&longs;sit uide<lb/>re <expan abbr="Sol&etilde;">Solem</expan> in k, dico &qring;d <lb/>depre&longs;&longs;o oculo in m, <lb/>uidebit alium Solem <lb/>maiorem uer&longs;us mar <lb/>ginem aduer&longs;um in l, <lb/>& longè &longs;plendidio<lb/>rem: quia enim radij <lb/><expan abbr="reflect&utilde;tur">reflectuntur</expan> ex k, ut ro <lb/>bu&longs;ti & à medio den <lb/>&longs;iore ad rarius, qui <lb/>non <expan abbr="inflecten&ttilde;">inflectentur</expan>, erunt <lb/>pauci, & ideò Sol in <lb/>k minor apparebit, et <lb/>languidior: maior au <lb/><figure id="id.015.01.270.1.jpg" xlink:href="015/01/270/1.jpg"/><lb/>tem pars deflectetur à <expan abbr="perp&etilde;diculari">perpendiculari</expan> ad m, igitur Sol apparebit ma<lb/>ior & ualidior longè &longs;plendentibus radijs, adeò ut uix ferri po&longs;sit. <lb/></s>

<s id="id004315">Sed quoniam angulus ex &longs;uppo&longs;ito m l &longs; maior e&longs;t h k e, igitur cum <lb/>oculus iudicet &longs;e uidere a d æquales angulos, uidebitur g depre&longs;<lb/>&longs;ior & propior labro in t, &longs;icut n m e&longs;t infra h, ita t infra g, quare <expan abbr="etiã">etiam</expan> <lb/>ut angulus m l &longs; &longs;it æqualis angulo t l f, nece&longs;&longs;e e&longs;t ut l &longs;it ultra k: ali<lb/>ter t uideretur qua&longs;i tangere aquam. </s>

<s id="id004316">In hora autem deliquij Solis, <lb/>uelut hodie v. Idus Aprilis hora &longs;exta diei, <expan abbr="c&utilde;">cum</expan> diligenti&longs;simi &longs;tatue<lb/>rint medium eclip&longs;is in quinta, & &longs;uppo&longs;ita fuerit ob&longs;curatio à Io<lb/>anne Stadio partium nouem cum be&longs;&longs;e, & tempus horæ unius & <lb/>m: 26, fuit tamen maior & longior: quoniam luminaria <expan abbr="fuer&utilde;t">fuerunt</expan> pro<lb/>piora una parte caudæ Draconis, quam ip&longs;e po&longs;uerit in tabulis, & <lb/>hoc quia &longs;upponit &etail;quinoctium tardius diebus duobus <expan abbr="quã">quam</expan> apud <lb/>Alphon&longs;um: & for&longs;an &longs;ufficiebat una dies, &longs;cilicet ut e&longs;&longs;et die deci<lb/>ma Martij horis decemocto à meridie: nam tunc omnia re&longs;pon<lb/>dent ob&longs;eruationi: in qua apparuerunt quatuor Lunæ: & quidem <lb/>ab initio fuerunt duæ orientaliores è regione, &longs;cilicet o p, & una o c <lb/>cidentalior n, & tantum di&longs;tabat n a k quantum o: Et clarum erat <lb/>quòd p erat, &longs;icut &longs;ecunda iris parua & non candida, &longs;ed rubra pur<lb/>pureo mi&longs;ta, quoniam ex reflexu o oriebatur: apparebat autem a la <lb/>tere illo, quoniam Luna dextram partem obtegebat, ideo illa erat <lb/>minus lumino&longs;a, & uerus Sol erat in k, modò Lunæ, modò Solis <lb/>imaginem referens ubi tran&longs;i&longs;&longs;et eclip&longs;is medium, non amplius <lb/>tres illæ Lunæ apparuerunt à dextra & à &longs;ini&longs;tra, &longs;ed una ultra nos

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<s id="id004318">Cau&longs;a e&longs;t, quoniam Luna <expan abbr="c&utilde;">cum</expan> <lb/>permeet Solem non è regione <lb/>recta lineæ oppo&longs;itæ no&longs;tro ui <lb/>&longs;ui, & &longs;olum <expan abbr="mom&etilde;to">momento</expan>, & in lon <lb/>gis <expan abbr="tempor&utilde;">temporum</expan> interuallis po&longs;sit <lb/>obtegere illum. </s>

<s id="id004319">Sit ergo ut Sol <lb/>obtegatur à Luna medijs par<lb/>tibus, & &longs;int radij extremi in <lb/>&longs;peculo: a c & a d, igitur erunt <lb/>tanquam duo Soles, &longs;ed uterque<lb/>illorum geminatur, ideò fiunt <lb/>tres: medius enim ob Lunæ <lb/>per&longs;picuitatem integer, appa<lb/>ret, ideò modò &longs;ub forma So<lb/>lis, modò Lunæ laterones am<lb/>bo &longs;ub forma Lunæ: ideò <expan abbr="er&utilde;t">erunt</expan> <lb/>tres, quib. </s>

<s id="id004319">Sit ergo ut Sol <lb/>obtegatur à Luna medijs par<lb/>tibus, & &longs;int radij extremi in <lb/>&longs;peculo: a c & a d, igitur erunt <lb/>tanquam duo Soles, &longs;ed uterque<lb/> illorum geminatur, ideò fiunt <lb/>tres: medius enim ob Lunæ <lb/>per&longs;picuitatem integer, appa<lb/>ret, ideò modò &longs;ub forma So<lb/>lis, modò Lunæ laterones am<lb/>bo &longs;ub forma Lunæ: ideò <expan abbr="er&utilde;t">erunt</expan> <lb/>tres, quibus. </s>

<s id="id004320">ad dita Luna p, quæ <lb/>e&longs;t reflexa a &longs;ecunda, fient qua<lb/>tuor. </s>

<s id="id004321">At dices cur non fit refle<lb/>xus &longs;ecundum directum oculi, <lb/>ut Lunæ appareant ultra citra<lb/>que Solem? </s>

<s id="id004322">Dico quod Luna <lb/>diuidente orbem reflexus fit ad latera, quia radij tran&longs;uer&longs;im ferun<lb/>tur: cum autem non diuiditur fit pror&longs;um & retror&longs;um. </s>

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<s id="id004403">Proponatur a quantitas, qu&etail; debeat mutari ab uir<lb/><arrow.to.target n="marg889"/><lb/>tute quæ non fit in materia, & palam e&longs;t quod non po<lb/>terit permutari in in&longs;tanti, quia &longs;imul e&longs;&longs;et, & non e&longs;&longs;et <lb/>ergo repugnaret, neque etiam pote&longs;t non e&longs;&longs;e, ut demon&longs;tratum e&longs;t <lb/>in Hyperchen, quia repugnant nece&longs;&longs;ario & e&longs;&longs;entiæ Dei, neque mo<lb/>uetur à certa proportione, quia b caret omni quantitate, ergo ni<lb/><arrow.to.target n="marg890"/><lb/>hil o&longs;tendit uim ip&longs;ius b e&longs;&longs;e finitam, quod ergo moueatur tardè ce<pb pagenum="257" xlink:href="015/01/276.jpg"/>leriter paruum magnum, i&longs;tud contingit totum ex conditionibus <lb/>a, id e&longs;t, materiæ & quantitatis: uelut, gratia exempli, &longs;i a e&longs;&longs;et in ua<lb/>&longs;culo palmi, non po&longs;&longs;et implere iugerum, & hoc <expan abbr="nõ">non</expan> o&longs;tendit ullam <lb/>imperfectionem in b. </s>

<pb pagenum="257" xlink:href="015/01/276.jpg"/>leriter paruum magnum, i&longs;tud contingit totum ex conditionibus <lb/>a, id e&longs;t, materiæ & quantitatis: uelut, gratia exempli, &longs;i a e&longs;&longs;et in ua<lb/>&longs;culo palmi, non po&longs;&longs;et implere iugerum, & hoc <expan abbr="nõ">non</expan> o&longs;tendit ullam <lb/>imperfectionem in b. </s>

<s id="id004404">Et &longs;icut homines omnes &longs;unt in carcere huius <lb/>mundi, & tamen uidentur e&longs;&longs;e &longs;ibiliberi, & appellant <expan abbr="&longs;ol&utilde;">&longs;olum</expan> illos e&longs;&longs;e <lb/>in carcere qui &longs;unt in erga&longs;tulo, ita omnis materia, & omnis quan<lb/>titas habet conditiones, per quas (ut ita <expan abbr="dicã">dicam</expan>) con&longs;tringitur, & repu <lb/>gnat eas mutari, & ideò <expan abbr="uitã">uitam</expan> agunt &longs;ine ulla proportione. </s>

<s id="id004405">Quod ue <lb/>rò dictum e&longs;t, &longs;upra dictum fuit, per exemplum dictum e&longs;t, <expan abbr="nõ">non</expan> quia <lb/>ita &longs;it, finge ergo quod in aliquo pariete, non &longs;it albitudo, ni&longs;i unius <lb/>gradus, illa non operabitur ni&longs;i per unum <expan abbr="grad&utilde;">gradum</expan>, etiam &longs;i calx e&longs;&longs;et <lb/>infinitè alba, & &longs;imiliter de luce Solis, ergo omnes mentes mouent <lb/>&longs;ine proportione, & non po&longs;&longs;unt dici finitæ uel infinitæ, quia ip&longs;æ <lb/>&longs;unt expertes omnis quantitatis, imò omnis relationis ad quantita <lb/>tem, & hoc e&longs;t quod latuit multos, & maximè propter dictum Phi<lb/>lo&longs;ophi, e&longs;t ergo omnis operatio iuxta id quod e&longs;t in materia, & <lb/>non quod una mens maiores habeat uires, alia cum non &longs;it in eis, <lb/>neque maius neque minus.</s>

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<s id="id004410">Non enim pote&longs;t e&longs;&longs;e comparatio iuxta magnitudines motas, <lb/><arrow.to.target n="marg891"/><lb/>quoniam uel &longs;unt corpora c&oelig;le&longs;tia, uel elementaria, <expan abbr="elem&etilde;taria">elementaria</expan> e&longs;&longs;e <lb/>non po&longs;&longs;unt, quia illa cum &longs;int corruptioni obnoxia, id e&longs;t, tran&longs;mu <lb/>tationi, &longs;ecundum qualitatem <expan abbr="nõ">non</expan> po&longs;&longs;unt e&longs;&longs;e &longs;ubiecta in corporca<lb/>rum &longs;ub&longs;tantiarum, neque à primis &longs;ub&longs;tantijs moueri, neque etiam ex<lb/>cipere primò lumen &longs;uum, &longs;ed mouentur per uim influxam à c&oelig;le<lb/>&longs;tibus corporibus, neque etiam per motum corporum <expan abbr="c&oelig;le&longs;ti&utilde;">c&oelig;le&longs;tium</expan>, nam <lb/>illa non mouentur &longs;ecundum proportionem mentis ad corpus, &longs;ed <lb/>iuxta rationem finis, à qua circum&longs;cribuntur, & ideo quod Satur<lb/>nus moueatur uelo ciore motu, quàm Iuppiter ab Oriente in Occi<lb/>dentem, hoc non e&longs;t, quia uitæ quæ mouet Saturnum fit robu&longs;tior <lb/>uita qu&etail; mouet Iouem, cum &longs;int una & eadem: uel &longs;i dicas quod &longs;int <lb/>diuer&longs;æ uita Saturni, non tamen e&longs;t ualidior in comparatione ad <lb/>&longs;uum c&oelig;lum, uita Iouis non moueret celerius Saturnum ab Occi<lb/>dente in Orientem, quàm uita Iouis Iouem, quod e&longs;t fal&longs;um, &longs;ed ta<lb/>lis motus uelo citas e&longs;t ratione finis, quia oportet ut pariter mouea<lb/>tur eo motu, & quia c&oelig;lum Saturni e&longs;t maius, ideo celerius moue<lb/>tur quam Iouis, & hoc ratione corporis mobilis, & <expan abbr="nõ">non</expan> ratione pro<lb/>portionis ad corpus. </s>

<s id="id004411">Dico etiam, quod non habent poteftatem <lb/>aliam, per quam &longs;ubeant proportionem, nam qu&etail;ritur cuius com

<s id="id004411">Dico etiam, quod non habent potestatem <lb/>aliam, per quam &longs;ubeant proportionem, nam qu&etail;ritur cuius com

<pb pagenum="258" xlink:href="015/01/277.jpg"/>paratione illa proportio oriatur, nam non ad corpora, quia neque <lb/>ad c&oelig;le&longs;tia, neque mortalia, ut dictum e&longs;t, ni&longs;i fin gamus alia corpora, <lb/>quod e&longs;t ab&longs;urdum, neque etiam ratione incorporeorum, nam non <lb/>po&longs;&longs;unt de&longs;truere &longs;e inuicem, quia inferior non pote&longs;t tollere &longs;upe<lb/>riorem, neque multo minus pote&longs;t uelle. </s>

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<s id="id004449">Videbitur ab initio paradoxum, &longs;ed ubi & modum & demon<lb/><arrow.to.target n="marg895"/><lb/>&longs;trationem ip&longs;am deprehenderis, intelliges ita e&longs;&longs;e iuxta luminis na <lb/>turalis rationem, tum uerò maximè, cum id adiecero non prohibe<lb/>re me, quin ut partes in homine numerentur. </s>

<s id="id004450">Sed aliud e&longs;t partes in <lb/>homine dinumerare, quæ numero ip&longs;o non di&longs;tinguuntur, &longs;ed &longs;i <lb/>plures homines &longs;eor&longs;um de earum numero interroges &longs;inguli di<lb/>uer&longs;a, nec <expan abbr="exiguõ">exiguom</expan> interuallo differentia re&longs;pondebunt, &longs;ed unus de<lb/>cem puta, alius centum, alius innumerabiles pronunciabit. </s>

<s id="id004451">Quin <lb/>etiam qui&longs;que qua ratione uelis illas di&longs;tinguere interrogabit, at non <lb/>&longs;ic de numero gregis pauidum, aut de pecunijs, in quibus nemo ab <lb/>altero di&longs;&longs;entiet, ni&longs;i cum in numerando errorem admi&longs;erit. </s>

<s id="id004452">Igitur <lb/>dico non e&longs;&longs;e numerum in incorporeis, nam finitus erit uel infini<lb/>tus: &longs;i infinitus, numerus non erit, quoniam primum nullus Deus <lb/>erit nulla prima &longs;ub&longs;tantia: nam quomodo Deus erit aut Domi<lb/>nus infinitorum, aut primus ubi non e&longs;t ultimum? </s>

<s id="id004453">Sed neque nume<lb/>rus aliquis certus earum e&longs;&longs;e pote&longs;t, cum primum non magis hic <lb/>quàm ille: neque enim definiuntur ullo termino, &longs;eu centum, &longs;eu mil<lb/>le aut millies mille: nec cum &longs;ubij ciantur quantitati continuæ pote<lb/>runt &longs;ubijci numero, uel alteri cuipiam accidenti. </s>

<s id="id004454">Sed omnia &longs;unt <lb/>unum, ita tamen quod perfectius e&longs;t atque imperfectius diffu&longs;um ab <lb/>ip&longs;o infinito, cuius in extremo cohærent mentes no&longs;træ & animæ, <lb/>& c&oelig;lum, quæ communicatæ inferioribus atque corporibus illa <lb/>agunt, mutant & &longs;eruant. </s>

<s id="id004455">Ip&longs;um quàm ultimum e&longs;&longs;e, e&longs;t in mundo, <lb/>quod e&longs;t corpus, & eius pars præ cipua c&oelig;lum deinde reliqua. <lb/></s>

<s id="id004456">Omniaque mouentur & transferuntur immobili primo principio, <lb/>quod cum illis <expan abbr="coniunct&utilde;">coniunctum</expan> e&longs;t: nam reliqua incorporea ab ip&longs;o pro<lb/>&longs;luunt. </s>

<s id="id004457">E&longs;t & ratio Ari&longs;totelis in tertio decimo Theologicorum &longs;er <lb/><arrow.to.target n="marg896"/><lb/>monum, Deus non e&longs;t unus numeri ratione, &longs;ed ita ut non &longs;it plura,

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<s id="id004470">Sed non e&longs;t neque geometrica, neque arithmetica, nec mu&longs;ica, nec <lb/>per recen&longs;um, e&longs;&longs;ent enim quantitates quæ <expan abbr="compararen&ttilde;">compararentur</expan>: unaqu&etail;que<lb/>enim harum inter quantitates con&longs;tituta: at illa e&longs;t ut producentis <lb/>ad productum. </s>

<s id="id004471">Et non comparantur quoad æternitatem, quia ut <lb/>aliâs declaraui, omnis &longs;ub&longs;tantia e&longs;t æterna: quanto magis incor<lb/>porea. </s>

<s id="id004472">Quia ergo primum per <expan abbr="præced&etilde;tem">præcedentem</expan> habet rationem totius, <lb/>& e&longs;t infinitum, <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> ea parte qua recedit, quia primum non e&longs;t, <lb/>plus di&longs;tat a primo quam à tertio, igitur de&longs;cendendo u&longs;que ad pri<lb/>ma elementa. </s>

<s id="id004473">Sed obijcies de qualitatibus & accidentibus: dico <lb/>quod habent <expan abbr="medi&utilde;">medium</expan> e&longs;&longs;e, licet tempore infinito uin cantur à &longs;ub&longs;tan <lb/>tijs, ill&etail; tamen etiam uin cuntur & ab&longs;que participatione perfectionis <lb/>illius cum <expan abbr="accid&etilde;tia">accidentia</expan> participent e&longs;&longs;entia & tempore, & &longs;i quis dicat, <lb/>cur ergo Sol & lupiter <expan abbr="nõ">non</expan> &longs;unt locati in &longs;upremis orbibus, cum &longs;int <lb/>nobiliores & maiores & potentiores cæteris erraticis: dieo quòd <lb/>fuit ob mundum inferiorem, quoniam &longs;i fui&longs;&longs;ent altiores mundus <lb/>inferior frigore corrumperetur, quando quidem uel &longs;ic frigore pre<lb/>mantur, in hyeme etiam &longs;ub torrida plaga, & &longs;ub polis ac iuxta eos

<s id="id004473">Sed obijcies de qualitatibus & accidentibus: dico <lb/>quod habent <expan abbr="medi&utilde;">medium</expan> e&longs;&longs;e, licet tempore infinito uincantur à &longs;ub&longs;tan<lb/>tijs, ill&etail; tamen etiam uincuntur & ab&longs;que participatione perfectionis <lb/>illius cum <expan abbr="accid&etilde;tia">accidentia</expan> participent e&longs;&longs;entia & tempore, & &longs;i quis dicat, <lb/>cur ergo Sol & lupiter <expan abbr="nõ">non</expan> &longs;unt locati in &longs;upremis orbibus, cum &longs;int <lb/>nobiliores & maiores & potentiores cæteris erraticis: dico quòd <lb/>fuit ob mundum inferiorem, quoniam &longs;i fui&longs;&longs;ent altiores mundus <lb/>inferior frigore corrumperetur, quando quidem uel &longs;ic frigore pre<lb/>mantur, in hyeme etiam &longs;ub torrida plaga, & &longs;ub polis ac iuxta eos

<pb pagenum="263" xlink:href="015/01/282.jpg"/>&longs;emper. </s>

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<pb pagenum="264" xlink:href="015/01/283.jpg"/>uitæ moueant: & quod mouet nece&longs;&longs;ariò mouet, & quod non po<lb/>te&longs;t mouere, quoniam omnia æterna &longs;unt: & in &etail;ternis idem e&longs;t e&longs;&longs;e <lb/>ac po&longs;&longs;e: igitur inter mundum incorporeum & uitarum nulla e&longs;t <lb/>proportio uera, &longs;ed &longs;olum &longs;ublimis, nec numerus: ni&longs;i ut à nobis fin <lb/>gitur. </s>

<s id="id004491">Velut &longs;i dicamus in tabula, & in negocio e&longs;t principium me<lb/>dium finis, & hæc po&longs;&longs;unt dici tria quatenus di&longs;tinguuntur: &longs;ed <expan abbr="nõ">non</expan> <lb/>ob hoc dicendum e&longs;t tabulam, aut negocium habere tres partes, <lb/>multo minus e&longs;&longs;e tria negocia aut tres tabulas.</s>

<s id="id004491">Velut &longs;i dicamus in tabula, & in negocio e&longs;t principium me<lb/>dium finis, & hæc po&longs;&longs;unt dici tria quatenus di&longs;tinguuntur: &longs;ed <expan abbr="nõ">non</expan> <lb/>ob hoc dicendum e&longs;t tabulam, aut negotium habere tres partes, <lb/>multo minus e&longs;&longs;e tria negocia aut tres tabulas.</s>

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<s id="id004502">Igitur cum in pari tempore qu&etail; uelo citer mouen <lb/>tur plus &longs;patij &longs;uperent, maius etiam relinquunt medium inter lo<lb/>cum, & locum, & ob id magis remota &longs;unt à quiete, & magis illi <expan abbr="cõtraria">con<lb/>traria</expan>: hæc igitur e&longs;t ratio cur quæ uelo cius moueantur, minus quie <lb/>ti &longs;imilia aut proxima exi&longs;timentur. </s>

<s id="id004503">Dico ergo, quod illa quæ natu<lb/>raliter uelo ci&longs;simè mouentur, &longs;unt magis &longs;imilia & magis proxima <lb/>ip&longs;is quie&longs;centibus quàm quæ tardè: cum enim omnis motus natu <lb/>ralis nece&longs;&longs;ariò <expan abbr="etiã">etiam</expan> &longs;it regularis, ut qui à uirtute Dei fiat, erit uel per <lb/>lineam obliquam aut <expan abbr="rectã">rectam</expan>. </s>

<s id="id004504">Quoniam uerò <expan abbr="multar&utilde;">multarum</expan> recta e&longs;t per<lb/>fecti&longs;sima, & obliquarum circularis, erit omnis motus naturalis cir <lb/>cularis aut rectus: dico ergo quòd in utroque <expan abbr="uer&utilde;">uerum</expan> e&longs;t quod dicitur. <lb/></s>

<s id="id004505">Et <expan abbr="prim&utilde;">primum</expan> in circulari ille motus e&longs;t propinquior quieti, in quo par<lb/>tes &longs;unt propinquiores &longs;uo loco, &longs;ed &longs;i ueloci&longs;simus &longs;it motus, nun<lb/>quàm ita &longs;unt extra &longs;uum locum, qui enim in pote&longs;tate &longs;int proxi<lb/>mæ ei: ergo partes ill&etail; inde &longs;e habent ac &longs;i quiefcerent. </s>

<s id="id004506">Secunda ra<lb/>tio, quia quod uelo ci&longs;simè <expan abbr="moue&ttilde;">mouetur</expan>, ab&longs;que dubio tanto tempore quie <lb/>&longs;cit in &longs;uo loco quantò quod tardè: exemplum. </s>

<s id="id004507">Luna in triginta an <lb/>nis quie&longs;cit in principio arietis <expan abbr="quadring&etilde;teis">quadringenteis</expan> per &longs;ex horas, id e&longs;t, <lb/>centum diebus in quadringentis uicibus, Saturnus <expan abbr="c&etilde;tum">centum</expan> diebus <lb/>&longs;ed &longs;emel tantum: ergo tantum Luna quie&longs;cit, quantum Saturnus, <lb/><expan abbr="cõparatione">comparatione</expan> ad idem tempus addita pari ratione in alijs partibus, <lb/>&longs;ed cum uelo cius moueatur Luna quàm Saturnus minus quie&longs;ce<lb/>re uidebitur Luna in alijs partibus quàm Saturnus, & tantundem <lb/>in principio arietis Luna ut Saturnus, ergo cum Luna tantundem <lb/>in principio arietis quie&longs;cat, quantum Saturnus in triginta annis, & <lb/>in alijs partibus minus quàm Saturnus, igitur ab&longs;olutè Luna plus <lb/>quie&longs;cit in principio arietis, quàm Saturnus dato tempore æquali <lb/>triginta <expan abbr="annor&utilde;">annorum</expan>. </s>

<s id="id004508">Et formatur demon&longs;tratio hoc modo: Luna quan <lb/>do e&longs;t in loco ip&longs;o, puta in principio arietis, ibidem e&longs;t actu, & quie <lb/>&longs;cit per tantundem temporis <expan abbr="quant&utilde;">quantum</expan> Saturnus, & in omnibus alijs <lb/>locis data paritate, e&longs;t &longs;emper propior ip&longs;i principio arietis pote&longs;ta <lb/>te quam Saturnus, igitur Luna plus quie&longs;cit in principio arietis <lb/>quam Saturnus, quia dum ibidem &longs;unt æqualiter <expan abbr="quie&longs;c&utilde;t">quie&longs;cunt</expan>, & dum <lb/>&longs;unt extra, Luna &longs;emper e&longs;t propior & pote&longs;tate magis in illo loco, <lb/>igitur Luna magis quie&longs;cit in principio arietis quàm Saturnus. </s>

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