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version 1.69, 2002/09/07 13:59:48

version 1.80, 2002/10/11 17:46:47

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<cvs_file>bianc_locam_01_la_1615</cvs_file>

<cvs_version/>

</info> <text> <front> </front> <body> <chap> <pb/>

</info> <text> <front> <pb/><section>

<s>ARISTOTELIS</s>

<s>ARISTOTELIS<lb/>LOCA MATHEMATICA<lb/>Ex vniuer&longs;is ip&longs;ius Operibus collecta, <lb/>& explicata.</s>

<s>LOCA MATHEMATICA</s>

<s>Ex vniuer&longs;is ip&longs;ius Operibus collecta, <lb/>& explicata.</s>

<s><emph type="italics"/>Aristotelicæ videlicet expo&longs;itionis complementum <lb/>hactenus de&longs;ideratum.<emph.end type="italics"/></s>

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<s><emph type="italics"/>Authore IOSEPHO BLANCANO Bononien&longs;i è Societate Ie&longs;u, <lb/>Mathematicarum in Gymna&longs;io Parmen&longs;i Profe&longs;&longs;ore.<emph.end type="italics"/></s>

<s>Ad Illu&longs;tri&longs;&longs;imum, ac Nobili&longs;&longs;imum</s>

<s>Ad Illu&longs;tri&longs;&longs;imum, ac Nobili&longs;&longs;imum<lb/>PETRVMFRANCISCVM MALASPINAM<lb/>Aedificiorum Marchionem, apud Cæ&longs;. Maie&longs;tatem <lb/>pro Sereni&longs;s. Parmen&longs;ium Duce Legatum.</s>

<s>PETRVMFRANCISCVM MALASPINAM</s>

<s>Aedificiorum Marchionem, apud Cæ&longs;. Maie&longs;tatem <lb/>pro Sereni&longs;s. </s>

<s>Parmen&longs;ium Duce Legatum.</s><figure/>

<s>BONONIÆ M. D C. </s>

<s>BONONIÆ M. D C. X V.</s>

<s>Apud Bartholomæum Cochium. </s>

<s>Superiorum permi&longs;&longs;u.</s>

<s>Sumptibus Hieronymi Tamburini.</s><pb pagenum="3"/><figure/>

<s>Sumptibus Hieronymi Tamburini.</s><pb/><pb pagenum="3"/>

</section>

<s>ILLVSTRISSIMO <lb/>AC NOBILISSIMO</s>

<section><s>ILLVSTRISSIMO <lb/>AC NOBILISSIMO<lb/>PETROFRANCISCO <lb/>MALASPINAE<lb/>ÆDIFICIORVM MARCHIONI.</s>

<s>PETROFRANCISCO <lb/>MALASPINAE</s>

<s>ÆDIFICIORVM MARCHIONI.</s><figure/>

<s><emph type="italics"/>En tandem Illustriß. Marchio opus no<lb/>strum de Locis Mathematicis apud Ari<lb/>stotelem, vnà cum Tractatione de natura <lb/>&longs;cientiarum Mathematicarum, necnon <lb/>Clarorum <expan abbr="Mathematicor&utilde;">Mathematicorum</expan> Chronologia; <lb/>quod tibi Mec&oelig;nati meo munificenti&longs;simo iure meritò <lb/>dicare, ac &longs;ub clarißimi tui nominis patrocinio in lucem <lb/>dare con&longs;titui. </s>

<s><emph type="italics"/>En tandem Illustriß. Marchio opus no<lb/>strum de Locis Mathematicis apud Ari<lb/>stotelem, vnà cum Tractatione de natura <lb/>&longs;cientiarum Mathematicarum, necnon <lb/>Clarorum Mathematicor&ubreve; Chronologia; <lb/>quod tibi Mec&oelig;nati meo munificenti&longs;simo iure meritò <lb/>dicare, ac &longs;ub clarißimi tui nominis patrocinio in lucem <lb/>dare con&longs;titui. </s>

<s>primùm quidem, vt mei perpetui erga te <lb/>amoris, & ob&longs;eruantiæ hoc vnum &longs;altem specimen exta<lb/>ret: tùm vt idoneum, æquumque propo&longs;itæ rei iudicem <lb/>nanci&longs;cerer. </s>

<s>cùm enim adiu&longs;tum arbitr&ubreve; duo potißimùm <lb/>requirantur, rerum &longs;cilicet cognitio, atque prudentia, quem <lb/>te rei, de qua agitur peritiorem, quemuè prudentiorem <lb/>inuenire potuerim? </s>

<s>cùm enim adiu&longs;tum <expan abbr="arbitr&utilde;">arbitrum</expan> duo potißimùm <lb/>requirantur, rerum &longs;cilicet cognitio, atque prudentia, quem <lb/>te rei, de qua agitur peritiorem, quemuè prudentiorem <lb/>inuenire potuerim? </s>

<s>tu enim cùm Phy&longs;iologiæ, ac Mathe<lb/>maticarum omnium Encyclopædiam mirum in modum<emph.end type="italics"/><pb pagenum="4"/><emph type="italics"/>excolueris, adintima Mathematicarum penetralia ita <lb/>per&longs;ua&longs;i&longs;ti, vt Archimedis, & Apollonij admirandis, ac <lb/>&longs;ubtilißimis Demon&longs;trationibus detinearis. </s>

<s>Quanta por<lb/>rò in rebus agendis prudentia valeas, toti penè Europæ <lb/>innotuit, cùm pro no&longs;tris Sereniß. Ducibus, non &longs;olùm ad <lb/>omnes ferè Italiæ, atque Germaniæ Principes, verùm etiam <lb/>ad Cæ&longs;aream Maie&longs;tatem rebus f&oelig;liciter ge&longs;tis Legatus <lb/>decimùm extiteris; ac demùm à Sereniß. </s>

<s>Duce Ranutio <lb/>inter primarios de Rep. </s>

<s>Con&longs;iliorum Authores ad&longs;citus <lb/>fueris. </s>

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<s>quæ &longs;i tibi accepta e&longs;&longs;e intellexero, iam tandem ma<lb/>ximorum munerum loco habenda e&longs;&longs;e cen&longs;e<lb/>bo. </s>

<s>incolumem tibi, ac f&oelig;licem D. Opt. <lb/>Max. </s>

<s>incolumem tibi, ac f&oelig;licem D. Opt. <lb/>Max. longæuitatem tueatur. <lb/></s>

<s>longæuitatem tueatur. <lb/></s>

<s><emph type="italics"/>Parmæ Idibus Maij M. DC. XIIII.<emph.end type="italics"/><lb/><figure id="fig1"/></s><pb pagenum="5"/><figure/>

<s><emph type="italics"/>Parmæ Idibus Maij M. DC. XIIII.<emph.end type="italics"/><lb/></s></section><pb pagenum="5"/><section>

<s>Liber de &longs;e ip&longs;o.</s>

<s><emph type="italics"/>Nec di&longs;cet Lector me &longs;olo interprete totum, <lb/>Nec &longs;ine me totum di&longs;cet Aristotelem.<emph.end type="italics"/></s><figure/>

<s><emph type="italics"/>Nec di&longs;cet Lector me &longs;olo interprete totum, <lb/>Nec &longs;ine me totum di&longs;cet Aristotelem.<emph.end type="italics"/></s>

<s>Ego Iordanus Ca&longs;&longs;ini Præpo&longs;itus Prouincialis Prouinciæ Venetæ Societatis <lb/>Ie&longs;u, ex auctoritate Adm. Reuer. P. nc&longs;tro Præpo&longs;iti Generalis P. Claudij <lb/>Aquæuiuæ, facultatem concedo, vt hoc opus P. Io&longs;ephi Blancani eiu&longs;dem <lb/>Societatis, quod in&longs;cribitur, Ari&longs;t. Loca Mathematica ex vniuer&longs;is ip&longs;ius <lb/>operibus collecta, & explicata, à deputatis Patribus recognitum, & ap<lb/>probatum typis mandari po&longs;&longs;it. </s>

<s>Ego Iordanus Ca&longs;&longs;ini Præpo&longs;itus Prouincialis Prouinciæ Venetæ Societatis <lb/>Ie&longs;u, ex auctoritate Adm. Reuer. </s>

<s>P. nc&longs;tro Præpo&longs;iti Generalis P. Claudij <lb/>Aquæuiuæ, facultatem concedo, vt hoc opus P. </s>

<s>Io&longs;ephi Blancani eiu&longs;dem <lb/>Societatis, quod in&longs;cribitur, Ari&longs;t. Loca Mathematica ex vniuer&longs;is ip&longs;ius <lb/>operibus collecta, & explicata, à deputatis Patribus recognitum, & ap<lb/>probatum typis mandari po&longs;&longs;it. </s>

<s>Parmæ die 15. Ianuarij 1615.</s>

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<s>& Reuerendi&longs;s. Archiepi&longs;c. </s>

<s>& Reuerendi&longs;s. Archiepi&longs;c. Bonon.</s>

<s>Bonon</s>

<s>Imprimatur</s>

<s>Fr. Hieronymus Onuphrius pro Reuerendi&longs;s. </s>

<s>Fr. Hieronymus Onuphrius pro Reuerendi&longs;s. P. Inqui&longs;itore Bonon.</s>

<s>P. Inqui&longs;itore Bonon<gap/></s>

</section><pb pagenum="6"/><section>

<s>LECTORI.</s>

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<s>Secunda, quòd plurima huiu&longs;modi loca à nemine, quod &longs;ciam, adhuc <lb/>declarata in tenebris magno no&longs;trorum malo delite&longs;cunt: cuiu&longs;modi <lb/>&longs;unt ad &longs;exaginta problemata, libellus de lineis in &longs;ecabilibus, libellus <lb/>de mundo, &longs;i tamen Ari&longs;totelis e&longs;t, & Mechanicæ quæ&longs;tiones, quamuis <lb/>enim Picolomineus in eas paraphra&longs;im ediderit, loca tamen earum dif<lb/>ficiliora non &longs;atis illu&longs;trauit. </s>

<s>Vt autem dixi 408. in vniuer&longs;um loca mi<lb/>nimùm numerantur, quibus illud Platonis in&longs;criptum e&longs;t <foreign lang="greek">ag aiome/trhtos <lb/>ud<gap/>i/s <gap/>i/to</foreign>; & in quibus Mathematicæ di&longs;ciplinæ rudes, & imperiti, quem <lb/>&longs;equuntur ducem Ari&longs;t.

<s>Vt autem dixi 408. in vniuer&longs;um loca mi<lb/>nimùm numerantur, quibus illud Platonis in&longs;criptum e&longs;t <foreign lang="greek">agaiome/trhtos <lb/>udei/s eisi/to</foreign>; & in quibus Mathematicæ di&longs;ciplinæ rudes, & imperiti, quem <lb/>&longs;equuntur ducem Ari&longs;t.

eum &longs;æpe de&longs;erere non &longs;ine turpi dedecoris no<lb/>ta coguntur; quo fit vt exempla illa Mathematica luc<gap/>m rebus aliquan<lb/>do allatura, t<gap/>n<gap/>bras cimmerijs, vt aiunt vmbris cra&longs;&longs;iores ij&longs;dem <lb/>obducant.</s>

eum &longs;æpe de&longs;erere non &longs;ine turpi dedecoris no<lb/>ta coguntur; quo fit vt exempla illa Mathematica lucem rebus aliquan<lb/>do allatura, tenebras cimmerijs, vt aiunt vmbris cra&longs;&longs;iores ij&longs;dem <lb/>obducant.</s>

<s>Tertia, quia Græcieorumdem locorum commentatores breuiter, & <lb/>ob&longs;curè admodum ea, quæ ad Mathematicum &longs;pectant, attingunt, hoc <lb/>enim ab ip&longs;is <expan abbr="cert&utilde;">certum</expan> ponitur, I ectorem e&longs;&longs;e, vt moris tunc erat, omnium <lb/>Philo &longs;ophorum, Mathematicis imbutum; at verò no&longs;tra ætate magna <lb/>cum Philo&longs;ophiæ iactura, quamplurimi earumdem di&longs;ciplinarum de&longs;ti<lb/>tuti præ&longs;idijs, ne Græcorum quidem Interpretum explanationes, ne<lb/>dum Ari&longs;t.

<s>Tertia, quia Græcieorumdem locorum commentatores breuiter, & <lb/>ob&longs;curè admodum ea, quæ ad Mathematicum &longs;pectant, attingunt, hoc <lb/>enim ab ip&longs;is <expan abbr="cert&utilde;">certum</expan> ponitur, Lectorem e&longs;&longs;e, vt moris tunc erat, omnium <lb/>Philo &longs;ophorum, Mathematicis imbutum; at verò no&longs;tra ætate magna <lb/>cum Philo&longs;ophiæ iactura, quamplurimi earumdem di&longs;ciplinarum de&longs;ti<lb/>tuti præ&longs;idijs, ne Græcorum quidem Interpretum explanationes, ne<lb/>dum Ari&longs;t.

ob&longs;curè dicta intelligunt.</s><pb pagenum="7"/>

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<s>quod &longs;i dicat, omnis triangulus habet tres æquales duo<lb/>busrectis: hic hærent, hinc anguntur: <expan abbr="cumq;">cumque</expan> ex his angu&longs;tijs, ac tricis <lb/>&longs;e minimè expedire valeant, aurea verba illa, quibus ingentes &longs;apientiæ <lb/>the&longs;auri continentur, alto &longs;ilentij velo contegere Mathematicarum eos <lb/>cogit in&longs;citia: vnde illud, quod Græcæ linguæ imperitis mutata oratio<lb/>ne acclamandum illis foret, Mathematicum e&longs;t, non legitur. </s>

<s>Nec mi<lb/>nus elegans illa altera expo&longs;itio; Diametrum e&longs;le incommen&longs;urabilem <lb/>co&longs;tæ; quod &longs;æpe apud Ari&longs;t.

<s>Nec mi<lb/>nus elegans illa altera expo&longs;itio; Diametrum e&longs;&longs;e incommen&longs;urabilem <lb/>co&longs;tæ; quod &longs;æpe apud Ari&longs;t.

legentibus occurrit, nihil aliud &longs;ibi velle, <lb/>quam Diametrum e&longs;&longs;e longiorem co&longs;ta, quam quidem a&longs;ymetriæ huius <lb/>ignorantiam Plato de legibus dial. </s>

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<s><emph type="italics"/>Vt &longs;e habet voluntas noua ad effectum nouum, <lb/>It a voluntas antiqua ad effectum antiquum. <lb/></s>

<s>Ergo permutatim, vt &longs;e habebit voluntas noua ad effectum <lb/>antiquum, <gap/>ta voluntas antiqua ad effectum nouum.<emph.end type="italics"/></s>

<s>Ergo permutatim, vt &longs;e habebit voluntas noua ad effectum <lb/>antiquum, ita voluntas antiqua ad effectum nouum.<emph.end type="italics"/></s>

<s><expan abbr="Spectat&utilde;">Spectatum</expan> admi&longs;&longs;i rilum teneatis amici? </s>

<s><expan abbr="Spectat&utilde;">Spectatum</expan> admi&longs;&longs;i ri&longs;um teneatis amici? </s>

<s>nego, ait; qui&longs;piam con&longs;equen<pb pagenum="8"/>tiam, non enim hoc e&longs;t argumentari à permutata ratione, deberet enim <lb/>inferre, &longs;ic, ergo ita &longs;e habebit voluntas noua ad antiquam, quemad<lb/>modum effectus nouus ad antiquum. </s>

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<s>Quid tandem <expan abbr="dic&etilde;dum">dicendum</expan> de quodam magni nominis Philo&longs;opho, om<lb/>nium tamen <expan abbr="Mathematicar&utilde;">Mathematicarum</expan> experte, qui in publica di&longs;putatione axio<lb/>ma illud Mathematicum, omne totum e&longs;t maius &longs;ua parte, in &longs;en&longs;u in <lb/>quo à Mathematicis effertur negare non erubuit, eò, quod in infinito, <lb/>vt aiebat non concederetur ab omnibus. </s>

<s>&longs;cilicet non intellig<gap/>bat ma<lb/>thematicum tantummodo tractare de Quantitate finita, ac terminata, <lb/>in qua axioma prædictum ab omnibus conceditur. </s>

<s>&longs;cilicet non intelligebat ma<lb/>thematicum tantummodo tractare de Quantitate finita, ac terminata, <lb/>in qua axioma prædictum ab omnibus conceditur. </s>

<s>Neque vero hic <lb/>nonnullorum infen&longs;us in Mathematicas animus quieuit, verum etiam <lb/>cò progre&longs;&longs;us e&longs;t, vt eas omnes omnino conuellere, atque ex albo &longs;cien<pb pagenum="9"/>tiarum, quamuis non Ari&longs;totele tantum, &longs;ed ip&longs;a etiam veritate repu<lb/>gnante, expungere conati &longs;int; <expan abbr="idq;">idque</expan> ne&longs;cio an vlla alia de cau&longs;a egerint, <lb/>quàm quod eas non &longs;atis calerent; non &longs;ecus <expan abbr="atq;">atque</expan> Ae&longs;opica illa Vulpes, <lb/>quæ cum cauda mutilata e&longs;&longs;et, caudarum mutilationem reliquis vulpi<lb/>bus vafrè per&longs;uadere conabatur. </s>

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<s>In horum igitur gratiam operam diligenter dedi, vt quantum in me <lb/>e&longs;&longs;et damna à me &longs;upra enarrata aliqua ex parte re&longs;arcirem. </s>

<s>Quapr<gap/>p<lb/><gap/>er loca hæc mathematica num rectè e&longs;&longs;ent è græco in latinum tran&longs;lata <lb/>diligenter prius expendi. </s>

<s>Quaprop<lb/>ter loca hæc mathematica num rectè e&longs;&longs;ent è græco in latinum tran&longs;lata

<lb/>diligenter prius expendi. </s>

<s>Deinde claritate, quàm potui max ma eadem <lb/>loca interpretatus &longs;um, & in horum, de quibus dixi gratiam, quædam <lb/>fanè tenuia pro&longs;equutus &longs;um, quæ alioquin libenter omi&longs;i&longs;&longs;em. </s>

<s>Tum fi<lb/>guras omnes, aut correxi, aut re&longs;titui, aut nouas appo&longs;ui. </s>

<s>Deinde claritate, quàm potui maxima eadem

<lb/>loca interpretatus &longs;um, & in horum, de quibus dixi gratiam, quædam

<s>Hocigitur <lb/>mo&longs;tro qualicunque labore poterit qui&longs;que omnia illa facile intelligere, <lb/><expan abbr="atq;">atque</expan> enumerata incommoda euitare, vnum tantummodo à L<gap/>ctore ma<lb/>thematicarum experte requiram, vt principia &longs;altem illa, &longs;cilicet defini<lb/>tiones, po&longs;tulata, & axiomata, quæ primò Euclideis libro præponuntur, <lb/>diligenter prius perlegat cum illa &longs;ua per&longs;picuitate ommbus &longs;int obuia; <lb/>cætera ego explicanda recipio. </s>

<lb/>fanè tenuia pro&longs;equutus &longs;um, quæ alioquin libenter omi&longs;i&longs;&longs;em. </s>

<s>Obiter etiam auctaria nonnulla partim <lb/>mathematica, partim naturalia in&longs;erui, quæ ob nouitatem, ac pulchri<lb/>cudinem grata Lectori, atque iucunda fore exi&longs;timaui.</s>

<s>Tum fi

<lb/>guras omnes, aut correxi, aut re&longs;titui, aut nouas appo&longs;ui. </s>

<s>Sciat præterea Lector no&longs;trum in&longs;titutum e&longs;&longs;e loca hæc mathemati<lb/>ca, quatenus mathematica &longs;unt declarare, &longs;iue ea &longs;upplere, quæ ex ma<lb/>thematicis petenda e&longs;&longs;ent: reliqua autem me tantum attingere, quan<lb/>tum harum rerum cum illis connexio po&longs;tulat.</s><pb pagenum="10"/>

<s>Hocigitur

<s>His omnibus placuit appendices opportune nonnullas addere, qua<lb/>rum prima de natura mathematicarum &longs;cientiarum: altera, qua omnes <lb/>demon&longs;trationes primi libri Euclidis breuiter ad Logicam normam ex<lb/>penduntur, vt pateat, quonam demon&longs;trationis genere <expan abbr="c&etilde;&longs;eri">cen&longs;eri</expan> <expan abbr="vnaqu&ecedil;q;">vnaqu&ecedil;que</expan> <lb/>debeat, & ex illis de cæteris iudicium fiat. </s>

<lb/>no&longs;tro qualicunque labore poterit qui&longs;que omnia illa facile intelligere,

<lb/><expan abbr="atq;">atque</expan> enumerata incommoda euitare, vnum tantummodo à Lectore ma

<s>Tandem in gratiam etiam <lb/>Mathematicorum tertiam appendicem appendi, qua omnia loca Ari&longs;t. <lb/>Geometrica ad Euclidis ordinem referuntur; vnde & ip&longs;i ad &longs;uas pr&ecedil;le<lb/>ctiones exornandas aliquid &longs;ubinde depromere queant.</s>

<lb/>thematicarum experte requiram, vt principia &longs;altem illa, &longs;cilicet defini

<lb/>tiones, po&longs;tulata, & axiomata, quæ primò Euclideis libro præponuntur,

<lb/>diligenter prius perlegat cum illa &longs;ua per&longs;picuitate omnibus &longs;int obuia;

<lb/>cætera ego explicanda recipio. </s>

<s>Obiter etiam auctaria nonnulla partim

<lb/>mathematica, partim naturalia in&longs;erui, quæ ob nouitatem, ac pulchri

<lb/>tudinem grata Lectori, atque iucunda fore exi&longs;timaui.</s>

<s>Sciat præterea Lector no&longs;trum in&longs;titutum e&longs;&longs;e loca hæc mathemati

<lb/>ca, quatenus mathematica &longs;unt declarare, &longs;iue ea &longs;upplere, quæ ex ma

<lb/>thematicis petenda e&longs;&longs;ent: reliqua autem me tantum attingere, quan

<lb/>tum harum rerum cum illis connexio po&longs;tulat.</s><pb pagenum="10"/>

<s>His omnibus placuit appendices opportune nonnullas addere, qua

<lb/>rum prima de natura mathematicarum &longs;cientiarum: altera, qua omnes

<lb/>demon&longs;trationes primi libri Euclidis breuiter ad Logicam normam ex

<lb/>penduntur, vt pateat, quonam demon&longs;trationis genere <expan abbr="c&etilde;&longs;eri">cen&longs;eri</expan> <expan abbr="vnaqu&ecedil;q;">vnaqu&ecedil;que</expan>

<lb/>debeat, & ex illis de cæteris iudicium fiat. </s>

<s>Tandem in gratiam etiam

<lb/>Mathematicorum tertiam appendicem appendi, qua omnia loca Ari&longs;t.

<lb/>Geometrica ad Euclidis ordinem referuntur; vnde & ip&longs;i ad &longs;uas pr&ecedil;le

<lb/>ctiones exornandas aliquid &longs;ubinde depromere queant.</s>

<s>Fruere igitur amice Lector hoc no&longs;tro qualiquali labore, quo ad ple<lb/>nam totius Ari&longs;t.

<s>Fruere igitur amice Lector hoc no&longs;tro qualiquali labore, quo ad ple

<lb/>nam totius Ari&longs;t.

intelligentiam, cui adhuc mathematicarum ignoratio <lb/>ob&longs;titit peruenire tandem po&longs;&longs;is: <expan abbr="illud&qacute;">illudque</expan>; experiaris, quod optimus qui<lb/>dam Philo&longs;ophus, cum totum hunc librum perlegi&longs;&longs;et, effatus e&longs;t, vide<lb/>licet, opus hoc <emph type="italics"/>Aristot elicæ expo&longs;itionis complementum ad hanc v&longs;que <lb/>diem de&longs;ideratum<emph.end type="italics"/> iure ac meritò nuncupari po&longs;&longs;e.</s>

intelligentiam, cui adhuc mathematicarum ignoratio

<s>Illud demum tanquam parergon addam, quod ego his elucubran<lb/>dis experientia didici, ad veram &longs;cilicet, ac perfectam to<lb/>tius Ari&longs;totelis intelligentiam linguæ in primis <lb/>græcæ, necnon mathematicarum om<lb/>nium di&longs;ciplinarum haud medio<lb/>crem cognitionem ne<lb/>ce&longs;&longs;ariam e&longs;&longs;e. <lb/></s>

<lb/>ob&longs;titit peruenire tandem po&longs;&longs;is: <expan abbr="illud&qacute;">illudque</expan>; experiaris, quod optimus qui

<lb/>dam Philo&longs;ophus, cum totum hunc librum perlegi&longs;&longs;et, effatus e&longs;t, vide

<lb/>licet, opus hoc <emph type="italics"/>Aristot elicæ expo&longs;itionis complementum ad hanc v&longs;que

<lb/>diem de&longs;ideratum<emph.end type="italics"/> iure ac meritò nuncupari po&longs;&longs;e.</s>

<s>Pr&ecedil;cipua qu&ecedil;dam, aut noua, autre&longs;taurata, <lb/>quæ obiter pertractantur.<lb/><arrow.to.target n="table1"/></s><table><table.target id="table1"/><row><cell><emph type="italics"/>1<emph.end type="italics"/></cell><cell><emph type="italics"/>De re&longs;olutione. numero marginali.<emph.end type="italics"/></cell><cell><emph type="italics"/>4<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>2<emph.end type="italics"/></cell><cell><emph type="italics"/>De figuris vacuum replentibus, vbi Aristotelis, & expo-&longs;itorum erratum aperitur. num.<emph.end type="italics"/></cell><cell><emph type="italics"/>121<emph.end type="italics"/></cell></row><row><cell/><cell><emph type="italics"/>Inibi, Apum mirabilis quædam in cellis &longs;uis hexagonis constituendis indu&longs;tria detegitur. num.<emph.end type="italics"/></cell><cell><emph type="italics"/>120<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>3<emph.end type="italics"/></cell><cell><emph type="italics"/>De ijs, quæ aquæ in&longs;ident, vnà cum noua demonstratione problema-tis illius Archimedis, quo metallorum mixtionem indi&longs;&longs;oluta Co-rona, explorauit. in additione. ante num.<emph.end type="italics"/></cell><cell><emph type="italics"/>124<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>4<emph.end type="italics"/></cell><cell><emph type="italics"/>De Cometa, recentiorum &longs;ententia. num.<emph.end type="italics"/></cell><cell><emph type="italics"/>136<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>5<emph.end type="italics"/></cell><cell><emph type="italics"/>De altitudine montis Cauca&longs;i.<emph.end type="italics"/></cell><cell><emph type="italics"/>148<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>6<emph.end type="italics"/></cell><cell><emph type="italics"/>De Terræ rotunditate, ac mundi duratione.<emph.end type="italics"/></cell><cell><emph type="italics"/>151<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>7<emph.end type="italics"/></cell><cell><emph type="italics"/>De Iride. in additione.<emph.end type="italics"/></cell><cell><emph type="italics"/>181<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>8<emph.end type="italics"/></cell><cell><emph type="italics"/>S<gap/>ytala quid.<emph.end type="italics"/></cell><cell><emph type="italics"/>250<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>9<emph.end type="italics"/></cell><cell><emph type="italics"/>Securis antiqua quæ, & qua ratione fieret.<emph.end type="italics"/></cell><cell><emph type="italics"/>258<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>10<emph.end type="italics"/></cell><cell><emph type="italics"/>Statera antiqua quæ,<emph.end type="italics"/></cell><cell><emph type="italics"/>259<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>11<emph.end type="italics"/></cell><cell><emph type="italics"/>De Ae&longs;tu Maris.<emph.end type="italics"/></cell><cell><emph type="italics"/>272<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>12<emph.end type="italics"/></cell><cell><emph type="italics"/>Araneorum indu&longs;tria nuper patefacta: vbi Democritus contra Ari&longs;t. defenditur.<emph.end type="italics"/></cell><cell><emph type="italics"/>293<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>13<emph.end type="italics"/></cell><cell><emph type="italics"/>De Lucis figuratione, & rerum &longs;imulacris in ob&longs;curo loco.<emph.end type="italics"/></cell><cell><emph type="italics"/>345<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>14<emph.end type="italics"/></cell><cell><emph type="italics"/>De Pupilla oculi.<emph.end type="italics"/></cell><cell><emph type="italics"/>408<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>15<emph.end type="italics"/></cell><cell><emph type="italics"/>De Mathematicarum natura. propè finem operis.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>16<emph.end type="italics"/></cell><cell><emph type="italics"/>Clarorum Mathematicorum Chronologia, in fine operis.<emph.end type="italics"/></cell><cell/></row></table><figure/><pb pagenum="12"/>

<s>Illud demum tanquam parergon addam, quod ego his elucubran

<lb/>dis experientia didici, ad veram &longs;cilicet, ac perfectam to

<lb/>tius Ari&longs;totelis intelligentiam linguæ in primis

<lb/>græcæ, necnon mathematicarum om

<lb/>nium di&longs;ciplinarum haud medio

<lb/>crem cognitionem ne

<lb/>ce&longs;&longs;ariam e&longs;&longs;e.

<s>Pr&ecedil;cipua qu&ecedil;dam, aut noua, autre&longs;taurata,

<lb/>quæ obiter pertractantur.

<lb/><arrow.to.target n="table1"/></s><table><table.target id="table1"/><row><cell><emph type="italics"/>1<emph.end type="italics"/></cell><cell><emph type="italics"/>De re&longs;olutione. numero marginali.<emph.end type="italics"/></cell><cell><emph type="italics"/>4<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>2<emph.end type="italics"/></cell><cell><emph type="italics"/>De figuris vacuum replentibus, vbi Aristotelis, & expo-&longs;itorum erratum aperitur. num.<emph.end type="italics"/></cell><cell><emph type="italics"/>121<emph.end type="italics"/></cell></row><row><cell/><cell><emph type="italics"/>Inibi, Apum mirabilis quædam in cellis &longs;uis hexagonis constituendis indu&longs;tria detegitur. num.<emph.end type="italics"/></cell><cell><emph type="italics"/>120<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>3<emph.end type="italics"/></cell><cell><emph type="italics"/>De ijs, quæ aquæ in&longs;ident, vnà cum noua demonstratione problema-tis illius Archimedis, quo metallorum mixtionem indi&longs;&longs;oluta Co-rona, explorauit. in additione. ante num.<emph.end type="italics"/></cell><cell><emph type="italics"/>124<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>4<emph.end type="italics"/></cell><cell><emph type="italics"/>De Cometa, recentiorum &longs;ententia. num.<emph.end type="italics"/></cell><cell><emph type="italics"/>136<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>5<emph.end type="italics"/></cell><cell><emph type="italics"/>De altitudine montis Cauca&longs;i.<emph.end type="italics"/></cell><cell><emph type="italics"/>148<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>6<emph.end type="italics"/></cell><cell><emph type="italics"/>De Terræ rotunditate, ac mundi duratione.<emph.end type="italics"/></cell><cell><emph type="italics"/>151<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>7<emph.end type="italics"/></cell><cell><emph type="italics"/>De Iride. in additione.<emph.end type="italics"/></cell><cell><emph type="italics"/>181<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>8<emph.end type="italics"/></cell><cell><emph type="italics"/>Scytala quid.<emph.end type="italics"/></cell><cell><emph type="italics"/>250<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>9<emph.end type="italics"/></cell><cell><emph type="italics"/>Securis antiqua quæ, & qua ratione fieret.<emph.end type="italics"/></cell><cell><emph type="italics"/>258<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>10<emph.end type="italics"/></cell><cell><emph type="italics"/>Statera antiqua quæ,<emph.end type="italics"/></cell><cell><emph type="italics"/>259<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>11<emph.end type="italics"/></cell><cell><emph type="italics"/>De Ae&longs;tu Maris.<emph.end type="italics"/></cell><cell><emph type="italics"/>272<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>12<emph.end type="italics"/></cell><cell><emph type="italics"/>Araneorum indu&longs;tria nuper patefacta: vbi Democritus contra Ari&longs;t. defenditur.<emph.end type="italics"/></cell><cell><emph type="italics"/>293<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>13<emph.end type="italics"/></cell><cell><emph type="italics"/>De Lucis figuratione, & rerum &longs;imulacris in ob&longs;curo loco.<emph.end type="italics"/></cell><cell><emph type="italics"/>345<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>14<emph.end type="italics"/></cell><cell><emph type="italics"/>De Pupilla oculi.<emph.end type="italics"/></cell><cell><emph type="italics"/>408<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>15<emph.end type="italics"/></cell><cell><emph type="italics"/>De Mathematicarum natura. propè finem operis.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>16<emph.end type="italics"/></cell><cell><emph type="italics"/>Clarorum Mathematicorum Chronologia, in fine operis.<emph.end type="italics"/></cell><cell/></row></table></section><pb pagenum="12"/><section>

<s><emph type="italics"/>PRIMVS INDEX LOCORVM ARIST.<emph.end type="italics"/></s>

<s><emph type="italics"/>Quæ in hoc opere explicantur, iuxta ordine librorum <lb/>ip&longs;ius ex vulgata editione Lugdunen&longs;i.<emph.end type="italics"/></s>

<s><emph type="italics"/>Quæ in hoc opere explicantur, iuxta ordine librorum

<lb/>ip&longs;ius ex vulgata editione Lugdunen&longs;i.<emph.end type="italics"/></s>

<s>In Prædicamentis.</s>

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1. &longs;ecti 3. lib.

1. de eo, quod est, omnis triangulus habet tres angulos æquales <lb/>æquales duobus rectis: Aequalitas Geometrica, quæ.<emph.end type="italics"/></s>

1. de eo, quod est, omnis triangulus habet tres angulos æquales

<lb/>æquales duobus rectis: Aequalitas Geometrica, quæ.<emph.end type="italics"/></s>

<s><emph type="italics"/>Cap.

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<s>Item de recto, & circulari. </s>

<s>Item de numero pari; impari<gap/><lb/>primo, & compo&longs;ito; æquilatero, & altera parte longiore.<emph.end type="italics"/></s>

<s>Item de numero pari, impari;

<lb/>primo, & compo&longs;ito; æquilatero, & altera parte longiore.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 11. Lineæ punctum inest per &longs;e, & c.<emph.end type="italics"/></s>

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<s>De I&longs;o&longs;cele. </s>

<s>De Alterna Proportione, <lb/>Item quod omnis triangulus habet tres, & c.<emph.end type="italics"/></s>

<s>De Alterna Proportione,

<lb/>Item quod omnis triangulus habet tres, & c.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 14. De ij&longs;aem cum præcedentibus.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 14. De ij&longs;dem cum præcedentibus.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 20. Magnitudines euadun<gap/> numeri. </s>

<s><emph type="italics"/>T. 20. Magnitudines euadunt numeri. </s>

<s>Item, quod non duo cubi cubus. </s>

<s>Item de <lb/>Mathematicis &longs;ubalternatis.<emph.end type="italics"/></s>

<s>Item de

<lb/>Mathematicis &longs;ubalternatis.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 23. Quadratura circuli &longs;ecundum Bry&longs;onem. </s>

<s>Item per&longs;ectam illam e&longs;&longs;e Demon<lb/>&longs;trationem, qua Geometræ o&longs;tendunt, quod Omnis triangulus habet tres, & c. <lb/></s>

<s>Item per&longs;ectam illam e&longs;&longs;e Demon

<lb/>&longs;trationem, qua Geometræ o&longs;tendunt, quod Omnis triangulus habet tres, & c.

<s>Per&longs;pectinam, & Mechanicam &longs;ubalternari Geometriæ, Mu&longs;icam, Arithmeticæ.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 24. De numero pari, impari, quodrangulo, cubo. </s>

<s>In Geometria quid irrationale, <lb/>refrangi, concurrere. </s>

<s>In Geometria quid irrationale,

<lb/>refrangi, concurrere. </s>

<s>Quid Astronomia con&longs;ideret.<emph.end type="italics"/></s>

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<s><emph type="italics"/>T. 29. Cur in Mathematicis non &longs;it Paralogi&longs;mus. </s>

<s>Item quid multiplicata propor<lb/>tio. </s>

<s>Item quid multiplicata propor

<s>Quid Cæneus dixerit. </s>

<s>Cur Affectiones Mathematicor&ubreve; maximè <expan abbr="conuertãtur">conuertantur</expan>.<emph.end type="italics"/></s>

<s>Cur Affectiones <expand abbr="Mathematicor&utilde;">Mathematicor&utilde;<expan><maximè <expan abbr="conuertãtur">conuertantur</expan>.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 30. De Lunæ &longs;phæricitate. </s>

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<s>& De &longs;ubalternatione, &c. </s>

<s>& Ma<lb/>thematicorum e&longs;t &longs;cire Propter quid: &longs;en&longs;itiuorum verò &longs;cire Quod.<emph.end type="italics"/></s>

<s>& Ma

<lb/>thematicorum e&longs;t &longs;cire Propter quid: &longs;en&longs;itiuorum verò &longs;cire Quod.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 37. I&longs;o&longs;celes, & Scalenum habere tres æquales, &c.<emph.end type="italics"/></s>

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<s><emph type="italics"/>T. 39. Habere tres angulos æquales, &c. </s>

<s>Item, quod omnis figura habet &longs;uos angu<lb/>los externos æquales quatuor tantum rectis.<emph.end type="italics"/></s>

<s>Item, quod omnis figura habet &longs;uos angu

<lb/>los externos æquales quatuor tantum rectis.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 43. Triangulum tres æquales, &c. </s>

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<s><emph type="italics"/>T. 1. Aequalitas, & inæqualitas. </s>

<s>Terram e&longs;&longs;e in medio mundi ab A&longs;tronomis per<lb/>fectè demon&longs;tratur. </s>

<s>Terram e&longs;&longs;e in medio mundi ab A&longs;tronomis per

<lb/>fectè demon&longs;tratur. </s>

<s>Item Quid con&longs;onantia.<emph.end type="italics"/></s>

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<s><emph type="italics"/>T. 7. Geometra, quædam accipit, quædam demon&longs;trat.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 11. Angulum in &longs;emicirculo rectum e&longs;&longs;e probari à Geometra per cau&longs;am materia<lb/>lem. </s>

<s><emph type="italics"/>T. 11. Angulum in &longs;emicirculo rectum e&longs;&longs;e probari à Geometra per cau&longs;am materia

<s>Zabarella correctus.<emph.end type="italics"/></s>

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<s>Colores in <lb/>Mu&longs;ica, qui. </s>

<s>Colores in

<lb/>Mu&longs;ica, qui. </s>

<s>tria genera veteris Mu&longs;icæ.<emph.end type="italics"/></s>

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Line 518

<s>Quadraturarur&longs;us Hippocratis, & Bry&longs;enis. </s>

<s>Mathe<lb/>maticæ non contentio&longs;æ. </s>

<s>Mathe

<lb/>maticæ non contentio&longs;æ. </s>

<s>Quadratio Antiphontis.<emph.end type="italics"/></s>

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<s><emph type="italics"/>T. 28. Quid con&longs;onantia Diapa&longs;on.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 68. Mathematicas Demon&longs;trationes babere cau&longs;am, quæ reducitur ad defini<lb/>tionem.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 68. Mathematicas Demon&longs;trationes babere cau&longs;am, quæ reducitur ad defini

<lb/>tionem.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 8. Denece&longs;&longs;ari<gap/>, quod e&longs;t in Mathematicis. </s>

<s><emph type="italics"/>T. 8. De nece&longs;&longs;ario, quod e&longs;t in Mathematicis. </s>

<s>& omnis triangulus habet tres an<lb/>gulos, &c.<emph.end type="italics"/></s>

<s>& omnis triangulus habet tres an

<lb/>gulos, &c.<emph.end type="italics"/></s>

<s>Ex 3. Phy&longs;ic.</s>

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<s><emph type="italics"/>T. 107. Centrum duplex grauit: & molis. </s>

<s>Qua ratione grauia ad mundi centrum <lb/>aptarentur.<emph.end type="italics"/></s>

<s>Qua ratione grauia ad mundi centrum

<lb/>aptarentur.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 109. Terram e&longs;&longs;e rotundam. </s>

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<s><emph type="italics"/>T. 66. Quænam planarum figurarum totum &longs;patium repleant. </s>

<s>Hinc de admirabili <lb/>Apum mgenio.<emph.end type="italics"/></s>

<s>Hinc de admirabili

<lb/>Apum mgenio.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. eodem. </s>

<s>Num plures Pyramides locum replere valeant, vbi Ari&longs;totiles, & omnes <lb/>expo&longs;itores erra&longs;&longs;e o&longs;tenduntur.<emph.end type="italics"/></s>

<s>Num plures Pyramides locum replere valeant, vbi Ari&longs;totiles, & omnes

<lb/>expo&longs;itores erra&longs;&longs;e o&longs;tenduntur.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 71. Terram e&longs;&longs;e cubum, cur dictum &longs;it.<emph.end type="italics"/></s>

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<s><emph type="italics"/>T. 33. Extare centrum Mundi, quò grauia tendunt, à quo leuia abeant.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 44. & <expan abbr="&longs;eq.">&longs;eque</expan> Cur quædam grautora quàm aqua, &longs;upernatent.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 44. & <expan abbr="&longs;eq.">&longs;eque</expan> Cur quædam grauiora quàm aqua, &longs;upernatent.<emph.end type="italics"/></s>

<s>Ex 2. de Generatione, & Corruptione.</s>

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<s><emph type="italics"/>Cap.

6. Sententia Ari&longs;totelis de Glaxia, partim defenditur: vera, deinde <lb/>aperitur.<emph.end type="italics"/></s><pb pagenum="15"/>

6. Sententia Ari&longs;totelis de Glaxia, partim defenditur: vera, deinde

<lb/>aperitur.<emph.end type="italics"/></s><pb pagenum="15"/>

<s><emph type="italics"/>Summa 4. cap 1. De Monte Parna&longs;&longs;o, dubia. </s>

<s>Mare extraneum, quod. </s>

<s>Errata quæ<lb/>dam veterum Geographorum, & Ari&longs;t.

<s>Errata quæ

<lb/>dam veterum Geographorum, & Ari&longs;t.

corriguntur. </s>

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2. De permutatione Aquarum, & continentis. </s>

<s>Noua ob&longs;eruatio de rotundi<lb/>tate Terræ, <expan abbr="atq;">atque</expan> Mundi duratione.<emph.end type="italics"/></s>

<s>Noua ob&longs;eruatio de rotundi

<lb/>tate Terræ, <expan abbr="atq;">atque</expan> Mundi duratione.<emph.end type="italics"/></s>

<s>Ex 2. Meteororum.</s>

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<s>Corona Ariadnæ. </s>

<s>Zonam torrid<gap/>m <lb/>falsò putabant inho&longs;pitalem. </s>

<s>Zonam torridam

<lb/>falsò putabant inho&longs;pitalem. </s>

<s>cur habitabilis.<emph.end type="italics"/></s>

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<s><emph type="italics"/>Summa 2. cap.

2. De Halone, &longs;eu Area, &longs;eu Corona, Mathematica demon&longs;iratio.<emph.end type="italics"/></s>

2. De Halone, &longs;eu Area, &longs;eu Corona, Mathematica demon&longs;tratio.<emph.end type="italics"/></s>

<s><emph type="italics"/>Cap.

4. De Iridis figura Mathematica demon&longs;tratio, &longs;ed deficiens. </s>

<s>Noua de eadem <lb/>tractatio.<emph.end type="italics"/></s>

<s>Noua de eadem

<lb/>tractatio.<emph.end type="italics"/></s>

<s><emph type="italics"/>Cap.

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<s>Ex 2. De Anima.</s>

<s><emph type="italics"/>T. 12. D finitioncm formalem, & cau&longs;alem explicat exemplo Quadrationis Geo<lb/>metricæ.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 12. Definitioncm formalem, & cau&longs;alem explicat exemplo Quadrationis Geo

<lb/>metricæ.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 86. Acutum, & Graue, vt differant.<emph.end type="italics"/></s>

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<s><emph type="italics"/>Cap.

1. Omnis triangulus babet tres, &c.<emph.end type="italics"/></s>

1. Omnis triangulus habet tres, &c.<emph.end type="italics"/></s>

<s><emph type="italics"/>Cap.

Line 756

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1. Initium Mathematicarum ab Aegyptiorum Saterdotibus. </s>

<s>Item, Automata, <lb/>quæ &longs;olstitia. </s>

<s>Item, Automata,

<lb/>quæ &longs;olstitia. </s>

<s>Diameter incommen&longs;.<emph.end type="italics"/></s>

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<s>Mathematicas puras carere cau&longs;is efficiente, & finali. </s>

<s>Ariftippus, vt Mathe<lb/>maticas &longs;ugillaret. </s>

<s>Ari&longs;tippus, vt Mathe

<lb/>maticas &longs;ugillaret. </s>

<s>Tetragoni&longs;mus est inuentio mediæ.<emph.end type="italics"/></s><pb pagenum="16"/>

Line 818

Line 870

<s>Omnis triangulus habet tres, &c. </s>

<s>Cur Angulus in &longs;emicir<lb/>culo rectus.<emph.end type="italics"/></s>

<s>Cur Angulus in &longs;emicir

<lb/>culo rectus.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 22. Omnis triangulus habet tres, &c.<emph.end type="italics"/></s>

Line 840

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<s>Ex 12. Methaphy&longs;.</s>

<s><emph type="italics"/>T. 44. Peculiari&longs;&longs;imam Philo&longs;ophiam, Mathematicorum videlicet, A&longs;tronomiam <lb/>pluralitatem C&oelig;lorum docere.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 44. Peculiari&longs;&longs;imam Philo&longs;ophiam, Mathematicorum videlicet, A&longs;tronomiam

<lb/>pluralitatem C&oelig;lorum docere.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 45. Numerus orbium c&oelig;le&longs;t ium &longs;ecundum Eudoxum.<emph.end type="italics"/></s>

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<s><emph type="italics"/>Cap.

3. Quaratione Mathematici tractant de Bo<gap/>o.<emph.end type="italics"/></s>

3. Qua ratione Mathematici tractant de Bono.<emph.end type="italics"/></s>

<s>In Mechanicas Quæ&longs;tiones.</s>

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<s><emph type="italics"/>Quæ&longs;t. 19. De Securi. </s>

<s>Securis veteris figura, & con&longs;tructio; vnà cum affectione <lb/>eius mirabili.<emph.end type="italics"/></s>

<s>Securis veteris figura, & con&longs;tructio; vnà cum affectione

<lb/>eius mirabili.<emph.end type="italics"/></s>

<s><emph type="italics"/>Quæ&longs;t. 20. De Statera. </s>

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<s><emph type="italics"/>Quæ&longs;t. 28. De Tollenone.<emph.end type="italics"/></s>

<s><emph type="italics"/>Quæ&longs;t. 29. De onere à duobus phalanga ge&longs;iato.<emph.end type="italics"/></s>

<s><emph type="italics"/>Quæ&longs;t. 29. De onere à duobus phalanga ge&longs;tato.<emph.end type="italics"/></s>

<s><emph type="italics"/>Quæ&longs;t. 30. De &longs;urgente à &longs;e&longs;&longs;ione.<emph.end type="italics"/></s>

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<s>100. De <gap/>&longs;tro, error Ari&longs;t. & veterum Geographorum.<emph.end type="italics"/></s>

<s>100. De I&longs;tro, error Ari&longs;t. & veterum Geographorum.<emph.end type="italics"/></s>

<s>In libello De lineis in&longs;ecabilibus.</s>

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<s><emph type="italics"/>Cap.

7. qua ratione in gre&longs;&longs;u &longs;iat bypotenu&longs;a. </s>

7. qua ratione in gre&longs;&longs;u fiat bypotenu&longs;a. </s>

<s>& ea quid &longs;it.<emph.end type="italics"/></s>

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4. Omnis triangulus habet tres, &c. </s>

<s>Ibidem Diametrum e&longs;&longs;e incommen<lb/>&longs;urabilem co&longs;tæ, habet cau&longs;am, & demon&longs;trationem.<emph.end type="italics"/></s>

<s>Ibidem Diametrum e&longs;&longs;e incommen

<lb/>&longs;urabilem co&longs;tæ, habet cau&longs;am, & demon&longs;trationem.<emph.end type="italics"/></s>

<s>In Ethicis ad Nicom.</s>

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9. Centrum circuli reperire.<emph.end type="italics"/></s>

<s><emph type="italics"/>Lib. 3. cap. 3. Diameter, & latus incommen&longs;urabilis: Item quid re&longs;olutio Geome<lb/>trica: Quid de&longs;ignatio.<emph.end type="italics"/></s>

<s><emph type="italics"/>Lib. 3. cap. 3. Diameter, & latus incommen&longs;urabilis: Item quid re&longs;olutio Geome

<lb/>trica: Quid de&longs;ignatio.<emph.end type="italics"/></s>

<s><emph type="italics"/>Lib. 5. cap.

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<s>Quid Proportionalitas. </s>

<s>Eam in 4. terminis con<lb/>&longs;i&longs;tere. </s>

<s>Eam in 4. terminis con

<lb/>&longs;i&longs;tere. </s>

<s>Item quid Permutata proportio. </s>

<s>Item quid Geometrica proportio. </s>

<s>Propor<lb/>tio continuata, & di&longs;iuncta quid.<emph.end type="italics"/></s>

<s>Propor

<lb/>tio continuata, & di&longs;iuncta quid.<emph.end type="italics"/></s>

<s><emph type="italics"/>cap.

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5. Duplum inter multiplices rationes primum tenet locum.<emph.end type="italics"/></s>

<s>Ex 1. lib. Mor. Eudemiorum.</s>

<s>Eudemiorum.</s>

<s><emph type="italics"/>Cap.

Omnis triangulus habet tres, &c.<emph.end type="italics"/></s>

7<gap/> Omnis triangulus habet tres, &c.<emph.end type="italics"/></s>

<s><emph type="italics"/>Cap.

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12. Triangulus habet tres, &c.<emph.end type="italics"/></s>

<s>Ex 7. lib. Mor. Eudemiorum.</s>

<s>Eudemiorum.</s>

<s><emph type="italics"/>Cap.

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<s><emph type="italics"/>Sectione 1. num. </s>

<s>3. De ortu &longs;yderum innerrantium: Succulæ, Hypades, Atlantides, <lb/>Virgiliæ, Pleiades. </s>

<s>3. De ortu &longs;yderum innerrantium: Succulæ, Hypades, Atlantides,

<lb/>Virgiliæ, Pleiades. </s>

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<s><emph type="italics"/>4. De inæquali &longs;olis vmbrarum incremento.<emph.end type="italics"/></s>

<s><emph type="italics"/>5. Cur Solis illuminationes &longs;emper rotundæ, quamuis per angulo&longs;a foramina ingre<lb/>diantur.<emph.end type="italics"/></s>

<s><emph type="italics"/>5. Cur Solis illuminationes &longs;emper rotundæ, quamuis per angulo&longs;a foramina ingre

<lb/>diantur.<emph.end type="italics"/></s>

<s><emph type="italics"/>6. Cur Luna &longs;emiplena videtur linea recta terminari? </s>

<s>vbi de illuminatione Lunæ, <lb/>quæ experientia docetur.<emph.end type="italics"/></s>

<s>vbi de illuminatione Lunæ,

<lb/>quæ experientia docetur.<emph.end type="italics"/></s>

<s><emph type="italics"/>7. Cur Sol, & Luna videantur plana?<emph.end type="italics"/></s>

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<s><emph type="italics"/>10. Cur in defectu Solis etiam illuminationes ip&longs;ius defectiuæ &longs;unt? </s>

<s>modus commodè <lb/>videndi eclyp&longs;im Solis.<emph.end type="italics"/></s>

<s>modus commodè

<lb/>videndi eclyp&longs;im Solis.<emph.end type="italics"/></s>

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<s>Hoc loco &longs;equentium probl. </s>

<s>cau&longs;a, <lb/>præmittitur totius Mu&longs;icæ ortus breuis tractatio.<emph.end type="italics"/></s>

<s>cau&longs;a,

<lb/>præmittitur totius Mu&longs;icæ ortus breuis tractatio.<emph.end type="italics"/></s>

<s><emph type="italics"/>3. Vox tam in hypate, quam in nete cantando rumpitur.<emph.end type="italics"/></s>

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<s><emph type="italics"/>15. Leges Mu&longs;icæ, quæ. </s>

<s>Genera, Diatonicum, Chromaticum, Encharmonium. <lb/></s>

<s>Genera, Diatonicum, Chromaticum, Encharmonium.

<s>Tetrachorda quæ.<emph.end type="italics"/></s>

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<s><emph type="italics"/>43. Idem cum 24. Iugum in lyra veteri quodnam fuerit, vna cum figura vete<lb/>ris lyræ.<emph.end type="italics"/></s>

<s><emph type="italics"/>43. Idem cum 24. Iugum in lyra veteri quodnam fuerit, vna cum figura vete

<s><emph type="italics"/>44. Cur &longs;uauius ad tibiam, quam ad lyram cantatur?<emph.end type="italics"/></s>

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<s><emph type="italics"/>48. Idem cum 7. quid Grauiden&longs;um.<emph.end type="italics"/></s>

<s><emph type="italics"/>49. Idem cum 30. In choris trag&oelig;diarum, nec &longs;ubdorius, nec &longs;ubphrygius modus <lb/>erat in v&longs;u.<emph.end type="italics"/></s>

<s><emph type="italics"/>49. Idem cum 30. In choris trag&oelig;diarum, nec &longs;ubdorius, nec &longs;ubphrygius modus

<lb/>erat in v&longs;u.<emph.end type="italics"/></s>

<s><emph type="italics"/>50. Cur grauior Melodia e&longs;t etiam mollior?<emph.end type="italics"/></s>

<s><emph type="italics"/>51. Dolia duo æqualia, quorum alterum plenum &longs;it, alterum dimidium, Diapa&longs;on <lb/>re&longs;onant.<emph.end type="italics"/></s>

<s><emph type="italics"/>51. Dolia duo æqualia, quorum alterum plenum &longs;it, alterum dimidium, Diapa&longs;on

<lb/>re&longs;onant.<emph.end type="italics"/></s>

<s>Ex &longs;ectione 23.</s>

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<s><emph type="italics"/>Cur duobus oculis res vna tantum videatur. </s>

<s>Cur aliquando rei vi&longs;æ gemina<lb/>tio accidat.<emph.end type="italics"/></s>

<s>Cur aliquando rei vi&longs;æ gemina

<lb/>tio accidat.<emph.end type="italics"/></s>

<s><emph type="italics"/>11. Cur di&longs;tractis oculis res vna duæ apparent?<emph.end type="italics"/></s>

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<s><emph type="italics"/>Primo. </s>

<s>De &longs;ubiecto Mathem. &longs;eu de materia intelligibili: vbi o&longs;tenduntur definitio<lb/>nes Mathematicæ e&longs;&longs;e perfecti&longs;&longs;imæ.<emph.end type="italics"/></s>

<s>De &longs;ubiecto Mathem. &longs;eu de materia intelligibili: vbi o&longs;tenduntur definitio

<lb/>nes Mathematicæ e&longs;&longs;e perfecti&longs;&longs;imæ.<emph.end type="italics"/></s>

<s>2. <emph type="italics"/>Demon&longs;trationes Mathematicas e&longs;&longs;e perfecti&longs;&longs;imas.<emph.end type="italics"/></s>

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<s>7. <emph type="italics"/>Clarorum Mathematicorum Chronicon.<emph.end type="italics"/></s>

<s><emph type="bold"/>Finis Primi Indicis.<emph.end type="bold"/></s><figure/><pb pagenum="22"/>

<s><emph type="bold"/>Finis Primi Indicis.<emph.end type="bold"/></s></section><pb pagenum="22"/><section>

<s><emph type="italics"/>ALTER INDEX<emph.end type="italics"/></s>

<s><emph type="italics"/>Quo loca Aristotelis Geometrica, in hoc Opere explicata, <lb/>ad Euclidem, &longs;ecundum propo&longs;itionum ordinem refe<lb/>runtur; vt Mathematicarum Profe&longs;&longs;ores habeant, <lb/>vnde &longs;uas prælectiones aliquando valeant locupletare.<emph.end type="italics"/></s>

<s><emph type="italics"/>Quo loca Aristotelis Geometrica, in hoc Opere explicata,

<lb/>ad Euclidem, &longs;ecundum propo&longs;itionum ordinem refe

<lb/>runtur; vt Mathematicarum Profe&longs;&longs;ores habeant,

<lb/>vnde &longs;uas prælectiones aliquando valeant locupletare.<emph.end type="italics"/></s>

<s><emph type="italics"/>In Primo Elem.

Euclidis.<emph.end type="italics"/></s>

<s>Ad verbum ip&longs;um <emph type="italics"/>(Elementum Euclidis)<emph.end type="italics"/> vide infra tex. 4. quinti <lb/>Methaph.</s>

<s>Ad verbum ip&longs;um <emph type="italics"/>(Elementum Euclidis)<emph.end type="italics"/> vide infra tex. 4. quinti

<lb/>Methaph.</s>

<s>Ad principia primi elementorum, vide infra tex. 5. pri. </s>

<s>Ad principia primi elementorum, vide infra tex. 5. pri. Po&longs;ter.</s>

<s>Po&longs;ter.</s>

<s>Ad definitionem 10. pri. pro angulo recto, vide 30. quæ&longs;t. </s>

<s>Mecha<lb/>nic. & cap.

<s>Mecha

<lb/>nic. & cap.

7. lib.

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<s>Ad axioma 10. quamuis Ari&longs;toteles nihil hac de re dicat; &longs;cias tamen velim <lb/>hoc vno axiomate qu&ecedil;&longs;tionem <expan abbr="quãdam">quandam</expan> inter Philo&longs;ophos valdè difficilem, <lb/>facile di&longs;&longs;olui. </s>

<s>Ad axioma 10. quamuis Ari&longs;toteles nihil hac de re dicat; &longs;cias tamen velim

<lb/>hoc vno axiomate qu&ecedil;&longs;tionem <expan abbr="quãdam">quandam</expan> inter Philo&longs;ophos valdè difficilem,

<s>ea e&longs;t, vtrum marmor, aut adamas, aliudue quidpiam infle<lb/>xibile &longs;ucce&longs;&longs;iuè findi, & aperiri po&longs;&longs;it. </s>

<lb/>facile di&longs;&longs;olui. </s>

<s>qui enim aiunt, &longs;ic refelluntur, quia <lb/>nimirum &longs;equeretur, duas rectas lineas habere &longs;egmentum commune: in<lb/>telligantur enim duæ lineæ, vna in vna &longs;uperficie, altera vero in altera, quæ <lb/>antequam incipiat apertio, congruant; Quoniam igitur tota illa apertio <lb/>non fit in in&longs;tanti, &longs;ed &longs;ucce&longs;&longs;iuè, facta iam aliqua apertionis parte con&longs;i<lb/>derentur prædictæ lineæ, erit igitur earum pars aliqua ab inuicem &longs;epara<lb/>ta, altera verò adhuc alteri congruens, ergo &longs;equetur, duas lineas habere <lb/>&longs;egmentum commune, quod e&longs;t impo&longs;&longs;ibile, quia contra 10. axioma.</s>

<s>ea e&longs;t, vtrum marmor, aut adamas, aliudue quidpiam infle

<lb/>xibile &longs;ucce&longs;&longs;iuè findi, & aperiri po&longs;&longs;it. </s>

<s>qui enim aiunt, &longs;ic refelluntur, quia

<lb/>nimirum &longs;equeretur, duas rectas lineas habere &longs;egmentum commune: in

<lb/>telligantur enim duæ lineæ, vna in vna &longs;uperficie, altera vero in altera, quæ

<lb/>antequam incipiat apertio, congruant; Quoniam igitur tota illa apertio

<lb/>non fit in in&longs;tanti, &longs;ed &longs;ucce&longs;&longs;iuè, facta iam aliqua apertionis parte con&longs;i

<lb/>derentur prædictæ lineæ, erit igitur earum pars aliqua ab inuicem &longs;epara

<lb/>ta, altera verò adhuc alteri congruens, ergo &longs;equetur, duas lineas habere

<lb/>&longs;egmentum commune, quod e&longs;t impo&longs;&longs;ibile, quia contra 10. axioma.</s>

<s>Ad Calcem axiomatum primi accommodetur tex. 1. primi Po&longs;ter.</s>

<s>Ad primam primi, po&longs;t ip&longs;ius explicationem, commodè declarari pote&longs;t, cur <lb/>Ari&longs;t. Demon&longs;trationes Geometricas appellet De&longs;criptiones, & De&longs;igna<lb/>tiones, vide cap.

<s>Ad primam primi, po&longs;t ip&longs;ius explicationem, commodè declarari pote&longs;t, cur

<lb/>Ari&longs;t. Demon&longs;trationes Geometricas appellet De&longs;criptiones, & De&longs;igna

<lb/>tiones, vide cap.

de Priori, & cap.

24. &longs;ecti primi, libri primi Priorum, & <lb/>tex. 4. quinti Methaph. </s>

24. &longs;ecti primi, libri primi Priorum, &

<lb/>tex. 4. quinti Methaph. & tex. 20. &longs;exti Methaph. & cap.

3. lib.

3. Ethic.

<s>& tex. 20. &longs;exti Methaph. </s>

<s>& cap.

3. lib.

3. Ethic. <lb/></s>

<s>Item ad primam primi, vide tex. 7. &longs;ecundi Po&longs;ter. loco 2.</s>

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1. Priorum.</s>

<s>Ad 21. primi, vide tex. 20. primi Po&longs;ter. </s>

<s>Ad 21. primi, vide tex. 20. primi Po&longs;ter. loco 2.</s>

<s>Ad 22. primi, vide locum 10. de lineis in&longs;ecabilibus.</s>

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1. Prior. & cap.

26. &longs;ecundi <expan abbr="Prior&utilde;">Priorum</expan>, & tex. </s>

26. &longs;ecundi <expan abbr="Prior&utilde;">Priorum</expan>, & tex. 2.

<lb/>primi Po&longs;ter. loco 4. & tex. 23. primi Po&longs;ter. vbi ait hanc e&longs;&longs;e poti&longs;&longs;imam

<lb/><expan abbr="demõ&longs;trationem">demon&longs;trationem</expan>. </s>

<s>2. <lb/>primi Po&longs;ter. loco 4. & tex. 23. primi Po&longs;ter. </s>

<s>vbi ait hanc e&longs;&longs;e poti&longs;&longs;imam <lb/><expan abbr="demõ&longs;trationem">demon&longs;trationem</expan>. </s>

<s>& tex. 37. primi Po&longs;ter. & tex. 39. primi Po&longs;ter. </s>

<s>& tex. 37. primi Po&longs;ter. & tex. </s>

<s>39. primi Po&longs;ter. </s>

<s>Ibidem <lb/>loco 4. & tex. 43. primi Po&longs;ter. </s>

<s>Ibidem

<lb/>loco 4. & tex. 43. primi Po&longs;ter. & tex. 2. &longs;ecundi Po&longs;ter. bis. </s>

<s>& tex. 2. &longs;ecundi Po&longs;ter. </s>

<s>& tex. 89. &longs;e<lb/>cundi Phy&longs;. & tex. </s>

<s>& tex. 89. &longs;e

<lb/>cundi Phy&longs;. & tex. 15. octaui Phy&longs;. & tex. 119. primi de C&oelig;lo. </s>

<s>15. octaui Phy&longs;. & tex. </s>

<s>119. primi de C&oelig;lo. </s>

<s>& tex. 25. <lb/>&longs;ecundi de C&oelig;lo. </s>

<s>& tex. 25.

<lb/>&longs;ecundi de C&oelig;lo. </s>

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<s>& remini&longs;c. <lb/></s>

<s>& remini&longs;c.

<s>& tex. 35. quinti Methaphy&longs;. & tex. 20. &longs;exti Methaphy&longs;. & tex. 22. &longs;exti

<lb/>Methaphy&longs;. & cap.

4. lib.

2. de Generat. animal. </s>

<s>& tex. 35. quinti Methaphy&longs;. & tex. </s>

<s>20. &longs;exti Methaphy&longs;. & tex. </s>

<s>22. &longs;exti <lb/>Methaphy&longs;. & cap.

4. lib.

2. de Generat. </s>

<s>animal. </s>

<s>& cap.

5. lib.

6. Ethic. & <lb/>cap.

6. Ethic. &

<lb/>cap.

2. Magnorum Moral. & cap.

10. Mag. Moral. & cap. 16. Mag. Moral. <lb/></s>

10. Mag. Moral. & cap. 16. Mag. Moral.

<lb/>& cap. 7. &longs;ecundi Eudem. & cap. 12. &longs;ecundi Eudem. & problema 6. &longs;ectio<pb pagenum="23"/>nis 30. tot Ari&longs;totelis loca illu&longs;trat vnica hæc Euclidis Demon&longs;tratio.</s>

<s>& cap. 7. &longs;ecundi Eudem. & cap. </s>

<s>12. &longs;ecundi Eudem. & problema 6. &longs;ectio<pb pagenum="23"/>nis 30. tot Ari&longs;totelis loca illu&longs;trat vnica hæc Euclidis Demon&longs;tratio.</s>

<s>Ad &longs;cholion præcedentis 32. primi, vide tex. 39. primi Po&longs;ter. loco 3. Item <lb/>tex. </s>

<s>Ad &longs;cholion præcedentis 32. primi, vide tex. 39. primi Po&longs;ter. loco 3. Item

<lb/>tex. 25. &longs;ecundi Po&longs;ter. loco vlt.</s>

<s>25. &longs;ecundi Po&longs;ter. loco vlt.</s>

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de Motu in Po&longs;tprædicam. </s>

<s>Qua<lb/>dratum augetur Gnomone circumpo&longs;ito.</s>

<s>Qua

<lb/>dratum augetur Gnomone circumpo&longs;ito.</s>

<s>Ad 14. propo&longs;. </s>

<s>2. opportunum e&longs;t Auditores de Quadratura circuli erudire, <lb/>vide igitur cap.

<s>2. opportunum e&longs;t Auditores de Quadratura circuli erudire,

<lb/>vide igitur cap.

de relatione in prædicam. </s>

<s>& cap.

31. &longs;ecundi Priorum, & <lb/>tex. 23. primi Po&longs;ter. & finem 1. cap. primi Elenchorum. </s>

31. &longs;ecundi Priorum, &

<lb/>tex. 23. primi Po&longs;ter. & finem 1. cap. primi Elenchorum. </s>

<s>lege primam Ar<lb/>chimedis de dimen&longs;ione circuli.</s>

<s>lege primam Ar

<lb/>chimedis de dimen&longs;ione circuli.</s>

<s><emph type="italics"/>In tertio Elem.<emph.end type="italics"/></s>

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<s>Ad 2. tertij, vide tex. 13. lib.

1. de Anima. </s>

1. de Anima. & locum 16. de lineis in&longs;ecab.</s>

<s>& locum 16. de lineis in&longs;ecab.</s>

<s>Ad 31. tertij, vide tex. 11. &longs;ecundi Po&longs;ter. & tex. 20. &longs;exti Methaph. loco 2.</s>

<s>Ad 31. tertij, vide tex. 11. &longs;ecundi Po&longs;ter. & tex. </s>

<s>20. &longs;exti Methaph. loco 2.</s>

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<s>lege tex. 66. <lb/>tertij de C&oelig;lo.</s>

<s>lege tex. 66.

<lb/>tertij de C&oelig;lo.</s>

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3. lib.

5. Ethyc. loco 4. & cap. 31. primi Ma<lb/>gnorum Moralium.</s>

5. Ethyc. loco 4. & cap. 31. primi Ma

<lb/>gnorum Moralium.</s>

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<s>Ad 12. definitionem 5. vide tex. 13. primi Po&longs;ter. </s>

<s>Ad 12. definitionem 5. vide tex. 13. primi Po&longs;ter. loco 3. & tex. 25. &longs;ecundi

<lb/>Po&longs;ter. & tex. 32. tertij de Anima. & cap.

3. lib.

5. Ethyc. loco 4.</s>

<s>loco 3. & tex. 25. &longs;ecundi <lb/>Po&longs;ter. </s>

<s>& tex. 32. tertij de Anima. </s>

<s>& cap.

3. lib.

5. Ethyc. loco 4.</s>

<s>Ad 16. propo&longs;. </s>

<s>5. vide tex. 25. &longs;ecundi Po&longs;ter. loco 2. ex hac Euclidis demon<lb/>&longs;tratione patet, vitio&longs;am e&longs;&longs;e illam Auerrois argumentationem, 8. Phy&longs;. <lb/></s>

<s>5. vide tex. 25. &longs;ecundi Po&longs;ter. loco 2. ex hac Euclidis demon

<lb/>&longs;tratione patet, vitio&longs;am e&longs;&longs;e illam Auerrois argumentationem, 8. Phy&longs;.

<lb/>comm. 15. &longs;cilicet.</s>

<s>comm. 15. &longs;cilicet.</s>

<s>Vt &longs;e habet voluntas antiqua ad antiquum effectum, <lb/>Ita &longs;e habet etiam voluntas noua ad effectum nouum: <lb/>Ergo permutando, ita &longs;e habebit voluntas antiqua ad effectum nouum. <lb/></s>

<s>Vt &longs;e habet voluntas antiqua ad antiquum effectum,

<lb/>Ita &longs;e habet etiam voluntas noua ad effectum nouum:

<lb/>Ergo permutando, ita &longs;e habebit voluntas antiqua ad effectum nouum.

<s>Quemadmodum voluntas noua ad effectum antiquum.</s>

<s>Non enim in permutando confert antecedentem ad antecedentem, & con<lb/>&longs;equentem ad con&longs;equentem, vt par erat, &longs;ed confert antecedentem ad <lb/>con&longs;equentem, quod non licet.</s>

<s>Non enim in permutando confert antecedentem ad antecedentem, & con

<lb/>&longs;equentem ad con&longs;equentem, vt par erat, &longs;ed confert antecedentem ad

<lb/>con&longs;equentem, quod non licet.</s>

<s><emph type="italics"/>In &longs;exto.<emph.end type="italics"/></s>

Line 1760

Line 1872

8. Topicorum loco 41.</s>

<s>Ad 13. &longs;exti, vide tex. 12. &longs;ecundi de Anima, & tex. </s>

<s>Ad 13. &longs;exti, vide tex. 12. &longs;ecundi de Anima, & tex. 3. tertij Methaphy&longs;.<emph type="italics"/>In &longs;eptimo.<emph.end type="italics"/></s>

<s>3. tertij Methaphy&longs;.<emph type="italics"/>In &longs;eptimo.<emph.end type="italics"/></s>

<s>Ad primam definitionem 7. vide tex. 5. primi Po&longs;ter.</s>

Line 1796

Line 1908

23. &longs;ecti 1. primi Priorum. </s>

<s>& tex. 48. <lb/>primi de C&oelig;lo.</s>

<s>& tex. 48.

<lb/>primi de C&oelig;lo.</s>

Line 1806

Line 1919

<s>& &longs;ecto 2. cap.

23. li<lb/>bri 1. Priorum. </s>

23. li

<lb/>bri 1. Priorum. </s>

<s>& cap. 22. lib.

2. Priorum. </s>

<s>& tex. 5. primi Po&longs;ter. & tex. </s>

<s>& tex. 5. primi Po&longs;ter. & tex. 44.

<lb/>primi Po&longs;ter. & cap.

15. primi Po&longs;ter. & tex. 119. primi de C&oelig;lo. </s>

<s>44. <lb/>primi Po&longs;ter. & cap.

15. primi Po&longs;ter. </s>

<s>& tex. 119. primi de C&oelig;lo. </s>

<s>& tex. <lb/>120. quarti Phy&longs;. & tex. </s>

<s>& tex.

<lb/>120. quarti Phy&longs;. & tex. 21. tertij de Anima. & cap.

1. primi Methaphy&longs;.

<lb/>& tex. 28. quarti Met. & tex. 34. quinti Met. & tex. 8. &longs;exti Met. & cap.

<lb/>lib.

2. de Generat. animal. </s>

<s>21. tertij de Anima. & cap.

1. primi Methaphy&longs;. <lb/></s>

<s>& tex. 28. quarti Met. </s>

<s>& tex. 34. quinti Met. </s>

<s>& tex. 8. &longs;exti Met. </s>

<s>& cap.

4. <lb/>lib.

2. de Generat. animal. </s>

Line 1860

Line 1978

3. cap.

3. Ethyc. & cap. 10. &longs;ecundi Eu<lb/>dem. </s>

3. Ethyc. & cap. 10. &longs;ecundi Eu

Line 1872

Line 1991

<s>Ad primam propo&longs;. </s>

<s>13. &longs;ecundum editionem Commandini, aut Zamberti. <lb/></s>

<s>13. &longs;ecundum editionem Commandini, aut Zamberti.

<s>vide initio Priorum, in verbum (Re&longs;olutio)</s>

<s>Atqne hæc &longs;unt, quæ ex Elementorum opere Ari&longs;toteles pa&longs;&longs;im v&longs;urpauit, <lb/>quæque nos infra explicabimus.</s>

<s>Atqne hæc &longs;unt, quæ ex Elementorum opere Ari&longs;toteles pa&longs;&longs;im v&longs;urpauit,

<lb/>quæque nos infra explicabimus.</s>

<s><emph type="bold"/>Finis Secundi Indicis.<emph.end type="bold"/></s>

<s>Quæ verò ad alias Mathematicas, Mu&longs;icam &longs;cilicet, Per&longs;pe<lb/>ctiuam, Mechanicam, & A&longs;tronomiam pertinens, facilè <lb/>poterunt ex primo Indice ad vnamquamque earum &longs;eor<lb/>&longs;um cum libuerit, &longs;ecerni.</s><figure/><pb pagenum="25"/>

<s>Quæ verò ad alias Mathematicas, Mu&longs;icam &longs;cilicet, Per&longs;pe

<lb/>ctiuam, Mechanicam, & A&longs;tronomiam pertinens, facilè

<lb/>poterunt ex primo Indice ad vnamquamque earum &longs;eor

<lb/>&longs;um cum libuerit, &longs;ecerni.</s></section><pb pagenum="25"/><section>

<s><emph type="italics"/>TERTIVS INDEX ALPHABETICVS,<emph.end type="italics"/></s>

<s><emph type="italics"/>cuius numeri re&longs;pondent numeris marginalibus Operis.<emph.end type="italics"/><lb/><arrow.to.target n="table2"/></s><table><table.target id="table2"/><row><cell><emph type="italics"/>A<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Abductio quid. eius inuentor. numero 16. marginali.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Acuti den&longs;um quid.<emph.end type="italics"/></cell><cell>399</cell></row><row><cell><emph type="italics"/>Aequalitas mathematica, quæ.<emph.end type="italics"/></cell><cell>10</cell></row><row><cell><emph type="italics"/>Ae&longs;tus maris natura.<emph.end type="italics"/></cell><cell>272</cell></row><row><cell><emph type="italics"/>Aequitas arithmetica, & æquitas &longs;ecundum dignitatem.<emph.end type="italics"/></cell><cell>330</cell></row><row><cell><emph type="italics"/>Agathir&longs;i populi.<emph.end type="italics"/></cell><cell>382</cell></row><row><cell><emph type="italics"/>Angulus quid. vt nominari debeat 10. angulum in &longs;emicirculo e&longs;&longs;e rectum o&longs;tendi per cau&longs;am materialem.<emph.end type="italics"/></cell><cell>71</cell></row><row><cell><emph type="italics"/>Angulus rectus variè con&longs;ideratur à Geometra, & à Fabro.<emph.end type="italics"/></cell><cell>301</cell></row><row><cell><emph type="italics"/>Antiphontis quadratura circuli.<emph.end type="italics"/></cell><cell>13</cell></row><row><cell><emph type="italics"/>Antennæ nauis problema.<emph.end type="italics"/></cell><cell>248</cell></row><row><cell><emph type="italics"/>Antipbonæ voces.<emph.end type="italics"/> 358. 363. 370. 371.</cell><cell>373</cell></row><row><cell><emph type="italics"/>Apum mirabilis indu&longs;tria.<emph.end type="italics"/></cell><cell>120</cell></row><row><cell><emph type="italics"/>Apotome linea, quæ.<emph.end type="italics"/></cell><cell>279</cell></row><row><cell><emph type="italics"/>Aquæ &longs;uperficiem e&longs;&longs;e &longs;phæricam ratione mathematica.<emph.end type="italics"/></cell><cell>107</cell></row><row><cell><emph type="italics"/>Arithmetica proportio.<emph.end type="italics"/></cell><cell>302</cell></row><row><cell><emph type="italics"/>Aranei industria patefacta, qua ad res inaca&longs;&longs; as tran&longs;eat.<emph.end type="italics"/></cell><cell>293</cell></row><row><cell><emph type="italics"/>filum emittit ex &longs;ece&longs;&longs;u contra Ari&longs;totelem pro Democrito.<emph.end type="italics"/></cell><cell><emph type="italics"/>ibidem.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>Astronomiæ principia duo, Apparentia, & Ob&longs;eruatio.<emph.end type="italics"/></cell><cell>8</cell></row><row><cell><emph type="italics"/>Automata, quæ.<emph.end type="italics"/> 199. 298.</cell><cell><emph type="italics"/>a. b.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>B<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Ba&longs;is ballarum in aqua, cur &longs;it alba, & cur non faciat vmbram.<emph.end type="italics"/></cell><cell>351</cell></row><row><cell><emph type="italics"/>Binomium linea, quæ.<emph.end type="italics"/></cell><cell>279</cell></row><row><cell><emph type="italics"/>Bra&longs;ilien&longs;es, qua ratione numer are &longs;oliti.<emph.end type="italics"/></cell><cell>340</cell></row><row><cell><emph type="italics"/>Bry&longs;onis quadratura circuli.<emph.end type="italics"/></cell><cell>35</cell></row><row><cell><emph type="italics"/>C<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Calippi opinio de numero C&oelig;lorum.<emph.end type="italics"/></cell><cell>236</cell></row><row><cell><emph type="italics"/>Cantilenam notam &longs;uauius, quam ignotam audimus.<emph.end type="italics"/></cell><cell>362</cell></row><row><cell><emph type="italics"/>Centrum circuli reperire. 303. Centrum mundi mathematicè o&longs;tenditur. 123. Cen-trum grauitatis, & molis.<emph.end type="italics"/> 38.</cell><cell>112</cell></row><row><cell><emph type="italics"/>Chordarum veterum nomina.<emph.end type="italics"/></cell><cell>359</cell></row><row><cell><emph type="italics"/>Circuli quadratura quid. an po&longs;&longs;ibilis. num. 1. Circulorum concentricorum maiores velocius moueri. 108. Circuli admiranda. 239. De circulorum concentricorum volutatione.<emph.end type="italics"/></cell><cell>263</cell></row><row><cell><emph type="italics"/>Coalternæ lineæ, quæ.<emph.end type="italics"/> 12. 14.</cell><cell>44</cell></row><row><cell><emph type="italics"/>C&oelig;lorum ordinem petendum ex A&longs;tronomis 109. item numerum<emph.end type="italics"/></cell><cell>233</cell></row><row><cell><emph type="italics"/>Colores in mu&longs;ica 78. Colores oculorum vnde.<emph.end type="italics"/></cell><cell>408</cell></row><row><cell><emph type="italics"/>Cometarum tractatiuncula ex recentioribus, qui eas in C&oelig;lo collocant. 136. e&longs;&longs;e &longs;u-pra aerem longi&longs;&longs;imo &longs;altem interuallo o&longs;tenditur mathematicè. 129. in additione.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Con&longs;onantia, quid. 64. quibus numeris con&longs;onantiæ con&longs;tent.<emph.end type="italics"/></cell><cell>210</cell></row><row><cell><emph type="italics"/>Conus, & cylindrus, cur variè mouentur.<emph.end type="italics"/></cell><cell>355</cell></row><pb pagenum="26"/><row><cell><emph type="italics"/>Cubus numerus. duo cubi cubus, quid &longs;ignificet.<emph.end type="italics"/></cell><cell>33</cell></row><row><cell><emph type="italics"/>Curru problema.<emph.end type="italics"/></cell><cell>252</cell></row><row><cell><emph type="italics"/>Cunei problema.<emph.end type="italics"/></cell><cell>256</cell></row><row><cell><emph type="italics"/>Cylindri, & coni motus comparatio problematica.<emph.end type="italics"/></cell><cell>335</cell></row><row><cell><emph type="italics"/>D<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Definitiones mathematicæ e&longs;&longs;e e&longs;&longs;entiales, & perfe&longs;&longs;imas. cap. 1. de nat. Math. definitionum v&longs;us in Mathematicis.<emph.end type="italics"/></cell><cell>81</cell></row><row><cell><emph type="italics"/>De&longs;criptio, & de&longs;cribere, quid.<emph.end type="italics"/> 2. 6. 7.</cell><cell>205</cell></row><row><cell><emph type="italics"/>De&longs;ignatio pro demon&longs;tratione mathematica.<emph.end type="italics"/></cell><cell>305</cell></row><row><cell><emph type="italics"/>Demon&longs;trationis perfectæ exemplum. 36. demon&longs;trationum mathematicarum præ-&longs;tantia. cap. 4. de nat. Mathem.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Dentiforcipis problema.<emph.end type="italics"/></cell><cell>260</cell></row><row><cell><emph type="italics"/>Denarij numeri perfectio. 339. cur <expan abbr="v&longs;q;">v&longs;que</expan> ad denari&ubreve; omnes <expan abbr="g&etilde;tes">gentes</expan> <expan abbr="numer&etilde;t">numerent</expan>.<emph.end type="italics"/></cell><cell>339. 8. <emph type="italics"/>&c.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>Diameter incommen&longs;urabilis costæ. 5. diametri etymon.<emph.end type="italics"/></cell><cell>337</cell></row><row><cell><emph type="italics"/>Diapa&longs;on quid. 90. 350. omnium con&longs;onantiarum pulcberrima.<emph.end type="italics"/></cell><cell>388</cell></row><row><cell><emph type="italics"/>Diapa&longs;on diapente.<emph.end type="italics"/></cell><cell>359</cell></row><row><cell><emph type="italics"/>Diapente con&longs;onantia, quæ.<emph.end type="italics"/></cell><cell>359</cell></row><row><cell><emph type="italics"/>Diate&longs;&longs;aron con&longs;onantia, quæ.<emph.end type="italics"/></cell><cell>359</cell></row><row><cell><emph type="italics"/>Di&longs;d apa&longs;on con&longs;onantia, quæ.<emph.end type="italics"/></cell><cell>359</cell></row><row><cell><emph type="italics"/>Die&longs;is, quid.<emph.end type="italics"/> 53.</cell><cell>226</cell></row><row><cell><emph type="italics"/>Dolia duo, quomodo aliquando Diapa&longs;on re&longs;onent.<emph.end type="italics"/></cell><cell>402</cell></row><row><cell><emph type="italics"/>Duplum inter multiplicia primum e&longs;t.<emph.end type="italics"/></cell><cell>322</cell></row><row><cell><emph type="italics"/>E<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Elementa mundi non componi ex figuris geometricis.<emph.end type="italics"/></cell><cell>120</cell></row><row><cell><emph type="italics"/>Elementa geometrica, quæ.<emph.end type="italics"/> 82.</cell><cell>213</cell></row><row><cell><emph type="italics"/>Eudoxi opinio de numero C&oelig;lorum.<emph.end type="italics"/></cell><cell>234</cell></row><row><cell><emph type="italics"/>Exempla mathematicorum, qualia. 11. non e&longs;&longs;e fal&longs;a.<emph.end type="italics"/></cell><cell>43</cell></row><row><cell><emph type="italics"/>Exemplorum veritas, & conformitas, quatenus requirantur.<emph.end type="italics"/></cell><cell>36</cell></row><row><cell><emph type="italics"/>F<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Figuram omnem planam habere &longs;uos angulos externos <expan abbr="quotc&ubreve;q;">quotc&ubreve;que</expan> æquales quatuor rectis angulis, quæ e&longs;t mira proprietas.<emph.end type="italics"/></cell><cell>59</cell></row><row><cell><emph type="italics"/>Figuræ &longs;imiles, quæ.<emph.end type="italics"/></cell><cell>70</cell></row><row><cell><emph type="italics"/>Figurarum planarum ordo 88. quæ nam totum locum repleant.<emph.end type="italics"/></cell><cell>96</cell></row><row><cell><emph type="italics"/>Figurarum &longs;olidarum, quænam totum locum repleant: vbi Ari&longs;t. & omnium expo&longs;i-torum rratum aperitur.<emph.end type="italics"/></cell><cell>121</cell></row><row><cell><emph type="italics"/>Figuratio lucis.<emph.end type="italics"/></cell><cell>345</cell></row><row><cell><emph type="italics"/>Figurationes pro demonftrationibus Mathem.<emph.end type="italics"/></cell><cell>194</cell></row><row><cell><emph type="italics"/>Filum Araneorum, ex qua parte corporis exeat, & ex qua materia.<emph.end type="italics"/></cell><cell>293</cell></row><row><cell><emph type="italics"/>Fluxus, ac refluxus maris.<emph.end type="italics"/></cell><cell>272</cell></row><row><cell><emph type="italics"/>Fu<gap/>ium l<gap/>ctorum problema.<emph.end type="italics"/></cell><cell>264</cell></row><row><cell><emph type="italics"/>G<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Galaxia quid. 131. Ari&longs;toteles defen&longs;us.<emph.end type="italics"/> 132.</cell><cell>140</cell></row><row><cell><emph type="italics"/>Galibei recens ob&longs;eruatio.<emph.end type="italics"/></cell><cell>141</cell></row><row><cell><emph type="italics"/>Generatria Mu&longs;icæ veteris. 78. Fusè explicantur.<emph.end type="italics"/></cell><cell>371</cell></row><row><cell><emph type="italics"/>Geodæ&longs;ia.<emph.end type="italics"/></cell><cell>207</cell></row><row><cell><emph type="italics"/>Geographiæ veteris plura errata,<emph.end type="italics"/> 145. 146. 147. 148.</cell><cell>149</cell></row><pb pagenum="27"/><row><cell><emph type="italics"/>Gnomon, quid. 3. &<emph.end type="italics"/></cell><cell>331</cell></row><row><cell><emph type="italics"/>Gnomones numeri.<emph.end type="italics"/></cell><cell>93</cell></row><row><cell><emph type="italics"/>Graue qua ratione ad centrum mundi de&longs;cenderet, ei&queacute; aptaretur.<emph.end type="italics"/></cell><cell>112</cell></row><row><cell><emph type="italics"/>Grauiden&longs;um, quid.<emph.end type="italics"/></cell><cell>399</cell></row><row><cell><emph type="italics"/>H<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Halonis demonfiratio.<emph.end type="italics"/></cell><cell>161</cell></row><row><cell><emph type="italics"/>Hippocratis chij quadratura circuli. 17. <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> quadratura lunulæ optima.<emph.end type="italics"/></cell><cell>17</cell></row><row><cell><emph type="italics"/>Hyades, Atlantides, & Succulæ.<emph.end type="italics"/></cell><cell>335</cell></row><row><cell><emph type="italics"/>Hypate, quid.<emph.end type="italics"/></cell><cell>360</cell></row><row><cell><emph type="italics"/>Hypotenu&longs;a in ince&longs;&longs;u animalium.<emph.end type="italics"/></cell><cell>294</cell></row><row><cell><emph type="italics"/>I<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Illuminationes Solis deficientis per foramina tran&longs;euntes, eur &longs;int defectiuæ.<emph.end type="italics"/></cell><cell>350</cell></row><row><cell><emph type="italics"/>modus videndi eclyp&longs;im facilis, ac iucundus.<emph.end type="italics"/></cell><cell><emph type="italics"/>ibidem.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>Ince&longs;&longs;us animalium lineis explicatur.<emph.end type="italics"/></cell><cell>294. <emph type="italics"/>& <expan abbr="&longs;eqq.">&longs;eqque</expan><emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>Incommen&longs;urabilia, quæ, & eorum inuentores.<emph.end type="italics"/></cell><cell>5</cell></row><row><cell><emph type="italics"/>Indiui&longs;ibilia mathematica e&longs;&longs;e priuationes. 189. oriri ex diui&longs;ione. 231. eorum duo genera.<emph.end type="italics"/></cell><cell>276</cell></row><row><cell><emph type="italics"/>Infinito, qua ratione vtantur Mathematici.<emph.end type="italics"/> 94.</cell><cell>96</cell></row><row><cell><emph type="italics"/>Iridis demon&longs;tratio &longs;ecundum Ari&longs;t. 163. & &longs;equentibus. Item noua de Iride tra-ctatio. ibidem in additione.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Iugum in lyra quid, & eius figura.<emph.end type="italics"/></cell><cell>396</cell></row><row><cell><emph type="italics"/>L<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Leges mu&longs;icales.<emph.end type="italics"/></cell><cell>204</cell></row><row><cell><emph type="italics"/>Libra maior, cur exactior. initio Mechanicarum quæ&longs;t.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Li<gap/>ea, quæ terminus est illuminationis in Luna, cur modo recta, modo curua vi-deatur.<emph.end type="italics"/></cell><cell>346</cell></row><row><cell><emph type="italics"/>Lineæ rationales, & irrationales, &c.<emph.end type="italics"/></cell><cell>279</cell></row><row><cell><emph type="italics"/>Lumen oculorum noctu videntium, in qua oculi parte manet. in addit. de Pupilla.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Lumen Solis, cur &longs;it circulare, quamuis per foramina angulo&longs;a ingrediatur.<emph.end type="italics"/></cell><cell>345</cell></row><row><cell><emph type="italics"/>Luna plana, cur appareat, cum &longs;it &longs;phærica. 347. cur in eadem altitudine cum Sole &longs;upra horizontem, maiorem vmbram efficiat.<emph.end type="italics"/></cell><cell>349</cell></row><row><cell><emph type="italics"/>Lunam e&longs;&longs;e &longs;phæricam. 48. illuminari &longs;phæricè quid: ibidem & de illuminatione Lu-næ. iterum e&longs;&longs;e &longs;phæricam ab eclyp&longs;ibus.<emph.end type="italics"/></cell><cell>111</cell></row><row><cell><emph type="italics"/>Luminarium Solis, & Lunæ ordo.<emph.end type="italics"/></cell><cell>133</cell></row><row><cell><emph type="italics"/>Lychanos, quid.<emph.end type="italics"/></cell><cell>360</cell></row><row><cell><emph type="italics"/>Lyræ veteris figura.<emph.end type="italics"/></cell><cell>396</cell></row><row><cell><emph type="italics"/>M<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Magalis, &longs;eu magas, & magadi&longs;&longs;are.<emph.end type="italics"/> 373.</cell><cell>393</cell></row><row><cell><emph type="italics"/>Mater<gap/>a intelligibilis. fusè verò. explicatur in tract. de nat. Mathem.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Mathema<gap/>icæ mediæ, &longs;eu &longs;ubalternatæ habent propter quid &longs;uarum demon&longs;iratio-nem.<emph.end type="italics"/></cell><cell>50</cell></row><row><cell><emph type="italics"/>Mathematici negant reperiri quantitatem indiui&longs;ibilem, &longs;eu minimam.<emph.end type="italics"/></cell><cell>100</cell></row><row><cell><emph type="italics"/>Mathematicæ non &longs;unt contentio&longs;æ. 83. ostendunt per cau&longs;am formalcm.<emph.end type="italics"/></cell><cell>91</cell></row><row><cell><emph type="italics"/>Mathematic<gap/>s inuenerunt Aegyptij Sacerdotes.<emph.end type="italics"/></cell><cell>198</cell></row><row><cell><emph type="italics"/>Mathematicæ o&longs;tendunt per cau&longs;am materialem, & formalem. 205. earum vtilitas. ibidem. De earum natura. in proprio tractatu.<emph.end type="italics"/></cell><cell/></row><pb pagenum="28"/><row><cell><emph type="italics"/>Mathematicæ maximè tractant de Bono, & Pulcro, qua ratione.<emph.end type="italics"/></cell><cell>237</cell></row><row><cell><emph type="italics"/>Mechanica facultas, quæ.<emph.end type="italics"/></cell><cell>238</cell></row><row><cell><emph type="italics"/>Melodia.<emph.end type="italics"/></cell><cell>331</cell></row><row><cell><emph type="italics"/>Melopeia quid.<emph.end type="italics"/></cell><cell>384</cell></row><row><cell><emph type="italics"/>Medium Demonstrationum Mathem. in earum tractatu.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Me&longs;e quid.<emph.end type="italics"/></cell><cell>360</cell></row><row><cell><emph type="italics"/>Mina in men&longs;uris quid.<emph.end type="italics"/></cell><cell>53</cell></row><row><cell><emph type="italics"/>Monochordiux<gap/>.<emph.end type="italics"/></cell><cell>359</cell></row><row><cell><emph type="italics"/>Motus nauigij, & remi comparatio. 247. Pulchra P. Nonij in id annotatio con-tra Ari&longs;t.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Modi mu&longs;ici.<emph.end type="italics"/></cell><cell>383</cell></row><row><cell><emph type="italics"/>Modorum antiquorum ordo, numerus, &c.<emph.end type="italics"/></cell><cell>383</cell></row><row><cell><emph type="italics"/>Motus primi mobilis, &longs;eu diurnus e&longs;t men&longs;ura c&oelig;lestium motuum.<emph.end type="italics"/></cell><cell>225</cell></row><row><cell><emph type="italics"/>Mu&longs;ici recentiores reprehen&longs;i. 331. & in fine Chronologiæ.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Mu&longs;icæ totius elementa.<emph.end type="italics"/></cell><cell>359</cell></row><row><cell><emph type="italics"/>Mu&longs;ica nuda, & cum melodia.<emph.end type="italics"/></cell><cell>331</cell></row><row><cell><emph type="italics"/>N<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Nete quid.<emph.end type="italics"/></cell><cell>360</cell></row><row><cell><emph type="italics"/>Nucifragi in&longs;trumenti problema.<emph.end type="italics"/></cell><cell>261</cell></row><row><cell><emph type="italics"/>Numerus, par, impar, primus, & compo&longs;itus, quadratus, &longs;eu æquilaterus, altera parte longior. 24. Cubus num.<emph.end type="italics"/></cell><cell>33</cell></row><row><cell><emph type="italics"/>Numeri capitales, qui.<emph.end type="italics"/></cell><cell>82</cell></row><row><cell><emph type="italics"/>Numerum parem e&longs;&longs;e cau&longs;am infiniti: imparem verò finiti.<emph.end type="italics"/></cell><cell>93</cell></row><row><cell><emph type="italics"/>Numerorum parium alij &longs;unt primi, alij non.<emph.end type="italics"/></cell><cell>224</cell></row><row><cell><emph type="italics"/>Numerus vnitarius.<emph.end type="italics"/></cell><cell>307</cell></row><row><cell><emph type="italics"/>O<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Oculi cur moueantur con&longs;imiliter.<emph.end type="italics"/></cell><cell>405</cell></row><row><cell><emph type="italics"/>Oculi anathome.<emph.end type="italics"/></cell><cell>408. <emph type="italics"/>&c.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>Omophonæ voces.<emph.end type="italics"/> 372.</cell><cell>392</cell></row><row><cell><emph type="italics"/>Oppo&longs;itio diametralis e&longs;t omnium maxima.<emph.end type="italics"/></cell><cell>327</cell></row><row><cell><emph type="italics"/>Ortus, & occa&longs;us &longs;yderum, quid, & quotuplex: vbide Orione, & Canicula.<emph.end type="italics"/></cell><cell>153</cell></row><row><cell><emph type="italics"/>P<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Paranete quæ voces, aut chordæ.<emph.end type="italics"/></cell><cell>360</cell></row><row><cell><emph type="italics"/>Parame&longs;e<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Parhypate<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Parelia, cur appareant nondum &longs;atis explicari.<emph.end type="italics"/></cell><cell>182</cell></row><row><cell><emph type="italics"/>Parna&longs;&longs;us mons, vbinam &longs;it. Item paropame&longs;&longs;us.<emph.end type="italics"/></cell><cell>145</cell></row><row><cell><emph type="italics"/>Partes quantitatis &longs;unt materia illius.<emph.end type="italics"/></cell><cell>211</cell></row><row><cell><emph type="italics"/>Per&longs;pectiuus, quatenus con&longs;ideret lineam.<emph.end type="italics"/></cell><cell>89</cell></row><row><cell><emph type="italics"/>Pa&longs;&longs;iones Mathematicorum cum &longs;ubiecto conuertuntur.<emph.end type="italics"/></cell><cell>47</cell></row><row><cell><emph type="italics"/>Pila chri&longs;tallina, vel vitrea, qua ratione comburat.<emph.end type="italics"/></cell><cell>60</cell></row><row><cell><emph type="italics"/>Planetæ, qua ratione moueantur duplici motu.<emph.end type="italics"/></cell><cell>130</cell></row><row><cell><emph type="italics"/>Plato &longs;olida ex planis componebat. 105. cur Elementis figuras Geometricas attri-bueret.<emph.end type="italics"/></cell><cell>122</cell></row><row><cell><emph type="italics"/>Planetarum ordo.<emph.end type="italics"/></cell><cell>271</cell></row><row><cell><emph type="italics"/>Principia Mathematicorum.<emph.end type="italics"/> 2.</cell><cell>118</cell></row><pb pagenum="29"/><row><cell><emph type="italics"/>Principia &longs;cientiarum duplicia. Ex quibus, & circa quod.<emph.end type="italics"/></cell><cell>61</cell></row><row><cell><emph type="italics"/>Principia Mathematica non pendere ab experientia.<emph.end type="italics"/></cell><cell>315</cell></row><row><cell><emph type="italics"/>Proportio alterna. 28. multiplicata, &longs;eu multiplex &longs;ecundum Cæneum.<emph.end type="italics"/></cell><cell>46</cell></row><row><cell><emph type="italics"/>P&longs;eudographia quid.<emph.end type="italics"/></cell><cell>83</cell></row><row><cell><emph type="italics"/>Proportionalitas quid.<emph.end type="italics"/></cell><cell>308</cell></row><row><cell><emph type="italics"/>Proportio continuata, & di&longs;iuncta quid. 310. alterna, &longs;eu permutata quid.<emph.end type="italics"/></cell><cell><emph type="italics"/>inibi.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>Proportio Geometrica. 311. Arithmetica.<emph.end type="italics"/></cell><cell>302</cell></row><row><cell><emph type="italics"/>Proportio &longs;ecundum dignita<gap/>em, e&longs;t Geometrica.<emph.end type="italics"/></cell><cell>330</cell></row><row><cell><emph type="italics"/>Problemata mu&longs;icalia varia à 360. <expan abbr="v&longs;q;">v&longs;que</expan> ad finem &longs;ectionis 19. problematum.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Punicum, mu&longs;icum in&longs;trumentum.<emph.end type="italics"/></cell><cell>370</cell></row><row><cell><emph type="italics"/>Pupillæ oculi etymon., & natura. 408. cur in oculo no&longs;tro imago pupillæ appareat. problem.<emph.end type="italics"/> 2.</cell><cell><emph type="italics"/>ibidem.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>Cur nigra in omnibus bominibus. probl.<emph.end type="italics"/> 5.</cell><cell/></row><row><cell><emph type="italics"/>Cur in Sole euane&longs;<gap/>at. probl.<emph.end type="italics"/> 6.</cell><cell/></row><row><cell><emph type="italics"/>Cur modo maior, modo minor appareat.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Pythagorici primi Mathematicis <expan abbr="operã">operam</expan> dedere, eas&queacute; ceteris &longs;cientijs præponeb&abreve;t.<emph.end type="italics"/></cell><cell>202</cell></row><row><cell><emph type="italics"/>Q<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Qv adra<gap/>ura circuli. vide circulus. de quadratura figurarum rectangularum, & quadraturæ duplex definitio cau&longs;alis, & formalis.<emph.end type="italics"/></cell><cell>185</cell></row><row><cell><emph type="italics"/>Quantitas an conctet ex indiui&longs;ibili. toto libello de lineis in&longs;ecabilibus argumentis mathematicis.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>R<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Remi problema.<emph.end type="italics"/></cell><cell>245</cell></row><row><cell><emph type="italics"/>Re&longs;olutio logica, & mathematica, vt conueniant. 4. &<emph.end type="italics"/></cell><cell>305</cell></row><row><cell><emph type="italics"/>Re&longs;ultus cadentium in terram, quibus angulis fiat.<emph.end type="italics"/></cell><cell>354</cell></row><row><cell><emph type="italics"/>Rythmus fusè explicatur.<emph.end type="italics"/></cell><cell>381</cell></row><row><cell><emph type="italics"/>Rubrum mare duplex.<emph.end type="italics"/></cell><cell>152</cell></row><row><cell><emph type="italics"/>S<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Scythala quid, & eius figura 250. &<emph.end type="italics"/></cell><cell>252</cell></row><row><cell><emph type="italics"/>Securis problema, vbi de<gap/>antiquæ &longs;<gap/>uris figura, & angulo pulchra demon&longs;tran-tur.<emph.end type="italics"/></cell><cell>258</cell></row><row><cell><emph type="italics"/>Semitonium, quid.<emph.end type="italics"/></cell><cell>360</cell></row><row><cell><emph type="italics"/>Solem e&longs;&longs;e terra multo maiorem: probatur.<emph.end type="italics"/></cell><cell>131</cell></row><row><cell><emph type="italics"/>Sphæram planum tangit in puncto. demon&longs;tratur.<emph.end type="italics"/></cell><cell>184</cell></row><row><cell><emph type="italics"/>Statera antiqua, quæ: eius figura, & problema.<emph.end type="italics"/></cell><cell>259</cell></row><row><cell><emph type="italics"/>Stereomatria, vt differat à Geometria.<emph.end type="italics"/></cell><cell>49</cell></row><row><cell><emph type="italics"/>Succula.<emph.end type="italics"/></cell><cell>253</cell></row><row><cell><emph type="italics"/>Symphonæ voces.<emph.end type="italics"/> 372.</cell><cell>392</cell></row><row><cell><emph type="italics"/>Symphonia.<emph.end type="italics"/></cell><cell>391</cell></row><row><cell><emph type="italics"/>T<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Temonis nauis problema.<emph.end type="italics"/></cell><cell>246</cell></row><row><cell><emph type="italics"/>Terram e&longs;&longs;e rotundam ex eclyp&longs;i. 114. Item aliter. 115. e&longs;&longs;e re&longs;pectu C&oelig;li paruam valde. 115. e&longs;&longs;e cubum cur Plato voluerit.<emph.end type="italics"/></cell><cell>122</cell></row><row><cell><emph type="italics"/>Terræ quantitas.<emph.end type="italics"/></cell><cell>115</cell></row><row><cell><emph type="italics"/>Terram paulatim reduci ad pefféctam rotunditatem.<emph.end type="italics"/></cell><cell>151</cell></row><row><cell><emph type="italics"/>Tetragoni&longs;mus. vide Quadratura.<emph.end type="italics"/></cell><cell/></row><pb pagenum="30"/><row><cell><emph type="italics"/>Teretizare, quid.<emph.end type="italics"/></cell><cell>366</cell></row><row><cell><emph type="italics"/>Tetrachordon, quid.<emph.end type="italics"/></cell><cell>386</cell></row><row><cell><emph type="italics"/>Tollenonis problema.<emph.end type="italics"/></cell><cell>267</cell></row><row><cell><emph type="italics"/>Tonus mu&longs;icus, qui; vnde oriatur.<emph.end type="italics"/></cell><cell>360</cell></row><row><cell><emph type="italics"/>Trochleæ problemata.<emph.end type="italics"/> 249. 250.</cell><cell>251</cell></row><row><cell><emph type="italics"/>Tunic<gap/> oculi. 408. in tractatu de Pupi<gap/>la.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>V<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Ventorum nomina, & &longs;itus.<emph.end type="italics"/></cell><cell>160. <emph type="italics"/>a.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>Vectis quotuplex, & c.<emph.end type="italics"/></cell><cell>244</cell></row><row><cell><emph type="italics"/>Veteres canere &longs;olitos non &longs;olum in choris, &longs;ed etiam in &longs;cenis.<emph.end type="italics"/> 371. 384.</cell><cell>400</cell></row><row><cell><emph type="italics"/>Virgiliæ, Pleiades.<emph.end type="italics"/></cell><cell>335</cell></row><row><cell><emph type="italics"/>Vi&longs;æ res gemmantur di&longs;tractis oculis.<emph.end type="italics"/></cell><cell>406</cell></row><row><cell><emph type="italics"/>Vi&longs;æ rei geminatio non fit altero oculo in latera torto, cur.<emph.end type="italics"/></cell><cell>407</cell></row><row><cell><emph type="italics"/>Vi&longs;us res vi&longs;as, cur non duplicet, etiam &longs;i duos oculos habeamus.<emph.end type="italics"/></cell><cell>405</cell></row><row><cell><emph type="italics"/>Vmbelici litoralis problema.<emph.end type="italics"/></cell><cell>254</cell></row><row><cell><emph type="italics"/>Vmbram terræ parum &longs;upra Lunam tran&longs;cendere.<emph.end type="italics"/></cell><cell>137</cell></row><row><cell><emph type="italics"/>Vmbrarum incrementa, & decrementa, cur inæqualia.<emph.end type="italics"/> 344.</cell><cell>348</cell></row><row><cell><emph type="italics"/>Vi&longs;us res geminat, &longs;i alter oculorum digito pellatur, cur.<emph.end type="italics"/></cell><cell>197</cell></row><row><cell><emph type="italics"/>Vocum mu&longs;icalium antiquæ appellationes.<emph.end type="italics"/></cell><cell>360</cell></row><row><cell><emph type="italics"/>Vox acuta velocior, grauis verò tarda, cur.<emph.end type="italics"/></cell><cell>77</cell></row><row><cell><emph type="italics"/>Voluminum &longs;ectio modo rectam lineam, modo curuam refert, cur.<emph.end type="italics"/></cell><cell>356</cell></row><row><cell><emph type="italics"/>Vnitas, cur indiui&longs;ibilis.<emph.end type="italics"/></cell><cell>22</cell></row><row><cell><emph type="italics"/>Z<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Zonas terræ, vt Arist. de&longs;ignet: & quæ &longs;ecundum ip&longs;um &longs;int habitabiles.<emph.end type="italics"/></cell><cell>156</cell></row><row><cell><emph type="italics"/>Zonam torridam quatuor reddunt habitabilem.<emph.end type="italics"/></cell><cell>159</cell></row></table>

<s><emph type="italics"/>cuius numeri re&longs;pondent numeris marginalibus Operis.<emph.end type="italics"/>

<lb/><arrow.to.target n="table2"/></s><table><table.target id="table2"/><row><cell><emph type="italics"/>A<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Abductio quid. eius inuentor. numero 16. marginali.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Acuti den&longs;um quid.<emph.end type="italics"/></cell><cell>399</cell></row><row><cell><emph type="italics"/>Aequalitas mathematica, quæ.<emph.end type="italics"/></cell><cell>10</cell></row><row><cell><emph type="italics"/>Ae&longs;tus maris natura.<emph.end type="italics"/></cell><cell>272</cell></row><row><cell><emph type="italics"/>Aequitas arithmetica, & æquitas &longs;ecundum dignitatem.<emph.end type="italics"/></cell><cell>330</cell></row><row><cell><emph type="italics"/>Agathir&longs;i populi.<emph.end type="italics"/></cell><cell>382</cell></row><row><cell><emph type="italics"/>Angulus quid. vt nominari debeat 10. angulum in &longs;emicirculo e&longs;&longs;e rectum o&longs;tendi per cau&longs;am materialem.<emph.end type="italics"/></cell><cell>71</cell></row><row><cell><emph type="italics"/>Angulus rectus variè con&longs;ideratur à Geometra, & à Fabro.<emph.end type="italics"/></cell><cell>301</cell></row><row><cell><emph type="italics"/>Antiphontis quadratura circuli.<emph.end type="italics"/></cell><cell>13</cell></row><row><cell><emph type="italics"/>Antennæ nauis problema.<emph.end type="italics"/></cell><cell>248</cell></row><row><cell><emph type="italics"/>Antipbonæ voces.<emph.end type="italics"/> 358. 363. 370. 371.</cell><cell>373</cell></row><row><cell><emph type="italics"/>Apum mirabilis indu&longs;tria.<emph.end type="italics"/></cell><cell>120</cell></row><row><cell><emph type="italics"/>Apotome linea, quæ.<emph.end type="italics"/></cell><cell>279</cell></row><row><cell><emph type="italics"/>Aquæ &longs;uperficiem e&longs;&longs;e &longs;phæricam ratione mathematica.<emph.end type="italics"/></cell><cell>107</cell></row><row><cell><emph type="italics"/>Arithmetica proportio.<emph.end type="italics"/></cell><cell>302</cell></row><row><cell><emph type="italics"/>Aranei industria patefacta, qua ad res inaca&longs;&longs; as tran&longs;eat.<emph.end type="italics"/></cell><cell>293</cell></row><row><cell><emph type="italics"/>filum emittit ex &longs;ece&longs;&longs;u contra Ari&longs;totelem pro Democrito.<emph.end type="italics"/></cell><cell><emph type="italics"/>ibidem.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>Astronomiæ principia duo, Apparentia, & Ob&longs;eruatio.<emph.end type="italics"/></cell><cell>8</cell></row><row><cell><emph type="italics"/>Automata, quæ.<emph.end type="italics"/> 199. 298.</cell><cell><emph type="italics"/>a. b.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>B<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Ba&longs;is ballarum in aqua, cur &longs;it alba, & cur non faciat vmbram.<emph.end type="italics"/></cell><cell>351</cell></row><row><cell><emph type="italics"/>Binomium linea, quæ.<emph.end type="italics"/></cell><cell>279</cell></row><row><cell><emph type="italics"/>Bra&longs;ilien&longs;es, qua ratione numer are &longs;oliti.<emph.end type="italics"/></cell><cell>340</cell></row><row><cell><emph type="italics"/>Bry&longs;onis quadratura circuli.<emph.end type="italics"/></cell><cell>35</cell></row><row><cell><emph type="italics"/>C<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Calippi opinio de numero C&oelig;lorum.<emph.end type="italics"/></cell><cell>236</cell></row><row><cell><emph type="italics"/>Cantilenam notam &longs;uauius, quam ignotam audimus.<emph.end type="italics"/></cell><cell>362</cell></row><row><cell><emph type="italics"/>Centrum circuli reperire. 303. Centrum mundi mathematicè o&longs;tenditur. 123. Cen-trum grauitatis, & molis.<emph.end type="italics"/> 38.</cell><cell>112</cell></row><row><cell><emph type="italics"/>Chordarum veterum nomina.<emph.end type="italics"/></cell><cell>359</cell></row><row><cell><emph type="italics"/>Circuli quadratura quid. an po&longs;&longs;ibilis. num. 1. Circulorum concentricorum maiores velocius moueri. 108. Circuli admiranda. 239. De circulorum concentricorum volutatione.<emph.end type="italics"/></cell><cell>263</cell></row><row><cell><emph type="italics"/>Coalternæ lineæ, quæ.<emph.end type="italics"/> 12. 14.</cell><cell>44</cell></row><row><cell><emph type="italics"/>C&oelig;lorum ordinem petendum ex A&longs;tronomis 109. item numerum<emph.end type="italics"/></cell><cell>233</cell></row><row><cell><emph type="italics"/>Colores in mu&longs;ica 78. Colores oculorum vnde.<emph.end type="italics"/></cell><cell>408</cell></row><row><cell><emph type="italics"/>Cometarum tractatiuncula ex recentioribus, qui eas in C&oelig;lo collocant. 136. e&longs;&longs;e &longs;u-pra aerem longi&longs;&longs;imo &longs;altem interuallo o&longs;tenditur mathematicè. 129. in additione.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Con&longs;onantia, quid. 64. quibus numeris con&longs;onantiæ con&longs;tent.<emph.end type="italics"/></cell><cell>210</cell></row><row><cell><emph type="italics"/>Conus, & cylindrus, cur variè mouentur.<emph.end type="italics"/></cell><cell>355</cell></row><pb pagenum="26"/><row><cell><emph type="italics"/>Cubus numerus. duo cubi cubus, quid &longs;ignificet.<emph.end type="italics"/></cell><cell>33</cell></row><row><cell><emph type="italics"/>Curru problema.<emph.end type="italics"/></cell><cell>252</cell></row><row><cell><emph type="italics"/>Cunei problema.<emph.end type="italics"/></cell><cell>256</cell></row><row><cell><emph type="italics"/>Cylindri, & coni motus comparatio problematica.<emph.end type="italics"/></cell><cell>335</cell></row><row><cell><emph type="italics"/>D<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Definitiones mathematicæ e&longs;&longs;e e&longs;&longs;entiales, & perfe&longs;&longs;imas. cap. 1. de nat. Math. definitionum v&longs;us in Mathematicis.<emph.end type="italics"/></cell><cell>81</cell></row><row><cell><emph type="italics"/>De&longs;criptio, & de&longs;cribere, quid.<emph.end type="italics"/> 2. 6. 7.</cell><cell>205</cell></row><row><cell><emph type="italics"/>De&longs;ignatio pro demon&longs;tratione mathematica.<emph.end type="italics"/></cell><cell>305</cell></row><row><cell><emph type="italics"/>Demon&longs;trationis perfectæ exemplum. 36. demon&longs;trationum mathematicarum præ-&longs;tantia. cap. 4. de nat. Mathem.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Dentiforcipis problema.<emph.end type="italics"/></cell><cell>260</cell></row><row><cell><emph type="italics"/>Denarij numeri perfectio. 339. cur <expan abbr="v&longs;q;">v&longs;que</expan> ad denari&ubreve; omnes <expan abbr="g&etilde;tes">gentes</expan> <expan abbr="numer&etilde;t">numerent</expan>.<emph.end type="italics"/></cell><cell>339. 8. <emph type="italics"/>&c.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>Diameter incommen&longs;urabilis costæ. 5. diametri etymon.<emph.end type="italics"/></cell><cell>337</cell></row><row><cell><emph type="italics"/>Diapa&longs;on quid. 90. 350. omnium con&longs;onantiarum pulcberrima.<emph.end type="italics"/></cell><cell>388</cell></row><row><cell><emph type="italics"/>Diapa&longs;on diapente.<emph.end type="italics"/></cell><cell>359</cell></row><row><cell><emph type="italics"/>Diapente con&longs;onantia, quæ.<emph.end type="italics"/></cell><cell>359</cell></row><row><cell><emph type="italics"/>Diate&longs;&longs;aron con&longs;onantia, quæ.<emph.end type="italics"/></cell><cell>359</cell></row><row><cell><emph type="italics"/>Di&longs;d apa&longs;on con&longs;onantia, quæ.<emph.end type="italics"/></cell><cell>359</cell></row><row><cell><emph type="italics"/>Die&longs;is, quid.<emph.end type="italics"/> 53.</cell><cell>226</cell></row><row><cell><emph type="italics"/>Dolia duo, quomodo aliquando Diapa&longs;on re&longs;onent.<emph.end type="italics"/></cell><cell>402</cell></row><row><cell><emph type="italics"/>Duplum inter multiplicia primum e&longs;t.<emph.end type="italics"/></cell><cell>322</cell></row><row><cell><emph type="italics"/>E<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Elementa mundi non componi ex figuris geometricis.<emph.end type="italics"/></cell><cell>120</cell></row><row><cell><emph type="italics"/>Elementa geometrica, quæ.<emph.end type="italics"/> 82.</cell><cell>213</cell></row><row><cell><emph type="italics"/>Eudoxi opinio de numero C&oelig;lorum.<emph.end type="italics"/></cell><cell>234</cell></row><row><cell><emph type="italics"/>Exempla mathematicorum, qualia. 11. non e&longs;&longs;e fal&longs;a.<emph.end type="italics"/></cell><cell>43</cell></row><row><cell><emph type="italics"/>Exemplorum veritas, & conformitas, quatenus requirantur.<emph.end type="italics"/></cell><cell>36</cell></row><row><cell><emph type="italics"/>F<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Figuram omnem planam habere &longs;uos angulos externos <expan abbr="quotc&ubreve;q;">quotc&ubreve;que</expan> æquales quatuor rectis angulis, quæ e&longs;t mira proprietas.<emph.end type="italics"/></cell><cell>59</cell></row><row><cell><emph type="italics"/>Figuræ &longs;imiles, quæ.<emph.end type="italics"/></cell><cell>70</cell></row><row><cell><emph type="italics"/>Figurarum planarum ordo 88. quæ nam totum locum repleant.<emph.end type="italics"/></cell><cell>96</cell></row><row><cell><emph type="italics"/>Figurarum &longs;olidarum, quænam totum locum repleant: vbi Ari&longs;t. & omnium expo&longs;i-torum rratum aperitur.<emph.end type="italics"/></cell><cell>121</cell></row><row><cell><emph type="italics"/>Figuratio lucis.<emph.end type="italics"/></cell><cell>345</cell></row><row><cell><emph type="italics"/>Figurationes pro demonftrationibus Mathem.<emph.end type="italics"/></cell><cell>194</cell></row><row><cell><emph type="italics"/>Filum Araneorum, ex qua parte corporis exeat, & ex qua materia.<emph.end type="italics"/></cell><cell>293</cell></row><row><cell><emph type="italics"/>Fluxus, ac refluxus maris.<emph.end type="italics"/></cell><cell>272</cell></row><row><cell><emph type="italics"/>Funium lectorum problema.<emph.end type="italics"/></cell><cell>264</cell></row><row><cell><emph type="italics"/>G<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Galaxia quid. 131. Ari&longs;toteles defen&longs;us.<emph.end type="italics"/> 132.</cell><cell>140</cell></row><row><cell><emph type="italics"/>Galibei recens ob&longs;eruatio.<emph.end type="italics"/></cell><cell>141</cell></row><row><cell><emph type="italics"/>Generatria Mu&longs;icæ veteris. 78. Fusè explicantur.<emph.end type="italics"/></cell><cell>371</cell></row><row><cell><emph type="italics"/>Geodæ&longs;ia.<emph.end type="italics"/></cell><cell>207</cell></row><row><cell><emph type="italics"/>Geographiæ veteris plura errata,<emph.end type="italics"/> 145. 146. 147. 148.</cell><cell>149</cell></row><pb pagenum="27"/><row><cell><emph type="italics"/>Gnomon, quid. 3. &<emph.end type="italics"/></cell><cell>331</cell></row><row><cell><emph type="italics"/>Gnomones numeri.<emph.end type="italics"/></cell><cell>93</cell></row><row><cell><emph type="italics"/>Graue qua ratione ad centrum mundi de&longs;cenderet, eiqué aptaretur.<emph.end type="italics"/></cell><cell>112</cell></row><row><cell><emph type="italics"/>Grauiden&longs;um, quid.<emph.end type="italics"/></cell><cell>399</cell></row><row><cell><emph type="italics"/>H<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Halonis demonfiratio.<emph.end type="italics"/></cell><cell>161</cell></row><row><cell><emph type="italics"/>Hippocratis chij quadratura circuli. 17. <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> quadratura lunulæ optima.<emph.end type="italics"/></cell><cell>17</cell></row><row><cell><emph type="italics"/>Hyades, Atlantides, & Succulæ.<emph.end type="italics"/></cell><cell>335</cell></row><row><cell><emph type="italics"/>Hypate, quid.<emph.end type="italics"/></cell><cell>360</cell></row><row><cell><emph type="italics"/>Hypotenu&longs;a in ince&longs;&longs;u animalium.<emph.end type="italics"/></cell><cell>294</cell></row><row><cell><emph type="italics"/>I<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Illuminationes Solis deficientis per foramina tran&longs;euntes, eur &longs;int defectiuæ.<emph.end type="italics"/></cell><cell>350</cell></row><row><cell><emph type="italics"/>modus videndi eclyp&longs;im facilis, ac iucundus.<emph.end type="italics"/></cell><cell><emph type="italics"/>ibidem.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>Ince&longs;&longs;us animalium lineis explicatur.<emph.end type="italics"/></cell><cell>294. <emph type="italics"/>& <expan abbr="&longs;eqq.">&longs;eqque</expan><emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>Incommen&longs;urabilia, quæ, & eorum inuentores.<emph.end type="italics"/></cell><cell>5</cell></row><row><cell><emph type="italics"/>Indiui&longs;ibilia mathematica e&longs;&longs;e priuationes. 189. oriri ex diui&longs;ione. 231. eorum duo genera.<emph.end type="italics"/></cell><cell>276</cell></row><row><cell><emph type="italics"/>Infinito, qua ratione vtantur Mathematici.<emph.end type="italics"/> 94.</cell><cell>96</cell></row><row><cell><emph type="italics"/>Iridis demon&longs;tratio &longs;ecundum Ari&longs;t. 163. & &longs;equentibus. Item noua de Iride tra-ctatio. ibidem in additione.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Iugum in lyra quid, & eius figura.<emph.end type="italics"/></cell><cell>396</cell></row><row><cell><emph type="italics"/>L<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Leges mu&longs;icales.<emph.end type="italics"/></cell><cell>204</cell></row><row><cell><emph type="italics"/>Libra maior, cur exactior. initio Mechanicarum quæ&longs;t.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Linea, quæ terminus est illuminationis in Luna, cur modo recta, modo curua vi-deatur.<emph.end type="italics"/></cell><cell>346</cell></row><row><cell><emph type="italics"/>Lineæ rationales, & irrationales, &c.<emph.end type="italics"/></cell><cell>279</cell></row><row><cell><emph type="italics"/>Lumen oculorum noctu videntium, in qua oculi parte manet. in addit. de Pupilla.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Lumen Solis, cur &longs;it circulare, quamuis per foramina angulo&longs;a ingrediatur.<emph.end type="italics"/></cell><cell>345</cell></row><row><cell><emph type="italics"/>Luna plana, cur appareat, cum &longs;it &longs;phærica. 347. cur in eadem altitudine cum Sole &longs;upra horizontem, maiorem vmbram efficiat.<emph.end type="italics"/></cell><cell>349</cell></row><row><cell><emph type="italics"/>Lunam e&longs;&longs;e &longs;phæricam. 48. illuminari &longs;phæricè quid: ibidem & de illuminatione Lu-næ. iterum e&longs;&longs;e &longs;phæricam ab eclyp&longs;ibus.<emph.end type="italics"/></cell><cell>111</cell></row><row><cell><emph type="italics"/>Luminarium Solis, & Lunæ ordo.<emph.end type="italics"/></cell><cell>133</cell></row><row><cell><emph type="italics"/>Lychanos, quid.<emph.end type="italics"/></cell><cell>360</cell></row><row><cell><emph type="italics"/>Lyræ veteris figura.<emph.end type="italics"/></cell><cell>396</cell></row><row><cell><emph type="italics"/>M<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Magalis, &longs;eu magas, & magadi&longs;&longs;are.<emph.end type="italics"/> 373.</cell><cell>393</cell></row><row><cell><emph type="italics"/>Materia intelligibilis. fusè verò. explicatur in tract. de nat. Mathem.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Mathematicæ mediæ, &longs;eu &longs;ubalternatæ habent propter quid &longs;uarum demon&longs;iratio-nem.<emph.end type="italics"/></cell><cell>50</cell></row><row><cell><emph type="italics"/>Mathematici negant reperiri quantitatem indiui&longs;ibilem, &longs;eu minimam.<emph.end type="italics"/></cell><cell>100</cell></row><row><cell><emph type="italics"/>Mathematicæ non &longs;unt contentio&longs;æ. 83. ostendunt per cau&longs;am formalcm.<emph.end type="italics"/></cell><cell>91</cell></row><row><cell><emph type="italics"/>Mathematicas inuenerunt Aegyptij Sacerdotes.<emph.end type="italics"/></cell><cell>198</cell></row><row><cell><emph type="italics"/>Mathematicæ o&longs;tendunt per cau&longs;am materialem, & formalem. 205. earum vtilitas. ibidem. De earum natura. in proprio tractatu.<emph.end type="italics"/></cell><cell/></row>

<s>Finis Tertij Indicis.</s><figure/><pb pagenum="31"/>

<pb pagenum="28"/><row><cell><emph type="italics"/>Mathematicæ maximè tractant de Bono, & Pulcro, qua ratione.<emph.end type="italics"/></cell><cell>237</cell></row><row><cell><emph type="italics"/>Mechanica facultas, quæ.<emph.end type="italics"/></cell><cell>238</cell></row><row><cell><emph type="italics"/>Melodia.<emph.end type="italics"/></cell><cell>331</cell></row><row><cell><emph type="italics"/>Melopeia quid.<emph.end type="italics"/></cell><cell>384</cell></row><row><cell><emph type="italics"/>Medium Demonstrationum Mathem. in earum tractatu.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Me&longs;e quid.<emph.end type="italics"/></cell><cell>360</cell></row><row><cell><emph type="italics"/>Mina in men&longs;uris quid.<emph.end type="italics"/></cell><cell>53</cell></row><row><cell><emph type="italics"/>Monochordium.<emph.end type="italics"/></cell><cell>359</cell></row><row><cell><emph type="italics"/>Motus nauigij, & remi comparatio. 247. Pulchra P. Nonij in id annotatio con-tra Ari&longs;t.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Modi mu&longs;ici.<emph.end type="italics"/></cell><cell>383</cell></row><row><cell><emph type="italics"/>Modorum antiquorum ordo, numerus, &c.<emph.end type="italics"/></cell><cell>383</cell></row><row><cell><emph type="italics"/>Motus primi mobilis, &longs;eu diurnus e&longs;t men&longs;ura c&oelig;lestium motuum.<emph.end type="italics"/></cell><cell>225</cell></row><row><cell><emph type="italics"/>Mu&longs;ici recentiores reprehen&longs;i. 331. & in fine Chronologiæ.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Mu&longs;icæ totius elementa.<emph.end type="italics"/></cell><cell>359</cell></row><row><cell><emph type="italics"/>Mu&longs;ica nuda, & cum melodia.<emph.end type="italics"/></cell><cell>331</cell></row><row><cell><emph type="italics"/>N<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Nete quid.<emph.end type="italics"/></cell><cell>360</cell></row><row><cell><emph type="italics"/>Nucifragi in&longs;trumenti problema.<emph.end type="italics"/></cell><cell>261</cell></row><row><cell><emph type="italics"/>Numerus, par, impar, primus, & compo&longs;itus, quadratus, &longs;eu æquilaterus, altera parte longior. 24. Cubus num.<emph.end type="italics"/></cell><cell>33</cell></row><row><cell><emph type="italics"/>Numeri capitales, qui.<emph.end type="italics"/></cell><cell>82</cell></row><row><cell><emph type="italics"/>Numerum parem e&longs;&longs;e cau&longs;am infiniti: imparem verò finiti.<emph.end type="italics"/></cell><cell>93</cell></row><row><cell><emph type="italics"/>Numerorum parium alij &longs;unt primi, alij non.<emph.end type="italics"/></cell><cell>224</cell></row><row><cell><emph type="italics"/>Numerus vnitarius.<emph.end type="italics"/></cell><cell>307</cell></row><row><cell><emph type="italics"/>O<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Oculi cur moueantur con&longs;imiliter.<emph.end type="italics"/></cell><cell>405</cell></row><row><cell><emph type="italics"/>Oculi anathome.<emph.end type="italics"/></cell><cell>408. <emph type="italics"/>&c.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>Omophonæ voces.<emph.end type="italics"/> 372.</cell><cell>392</cell></row><row><cell><emph type="italics"/>Oppo&longs;itio diametralis e&longs;t omnium maxima.<emph.end type="italics"/></cell><cell>327</cell></row><row><cell><emph type="italics"/>Ortus, & occa&longs;us &longs;yderum, quid, & quotuplex: vbide Orione, & Canicula.<emph.end type="italics"/></cell><cell>153</cell></row><row><cell><emph type="italics"/>P<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Paranete quæ voces, aut chordæ.<emph.end type="italics"/></cell><cell>360</cell></row><row><cell><emph type="italics"/>Parame&longs;e<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Parhypate<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Parelia, cur appareant nondum &longs;atis explicari.<emph.end type="italics"/></cell><cell>182</cell></row><row><cell><emph type="italics"/>Parna&longs;&longs;us mons, vbinam &longs;it. Item paropame&longs;&longs;us.<emph.end type="italics"/></cell><cell>145</cell></row><row><cell><emph type="italics"/>Partes quantitatis &longs;unt materia illius.<emph.end type="italics"/></cell><cell>211</cell></row><row><cell><emph type="italics"/>Per&longs;pectiuus, quatenus con&longs;ideret lineam.<emph.end type="italics"/></cell><cell>89</cell></row><row><cell><emph type="italics"/>Pa&longs;&longs;iones Mathematicorum cum &longs;ubiecto conuertuntur.<emph.end type="italics"/></cell><cell>47</cell></row><row><cell><emph type="italics"/>Pila chri&longs;tallina, vel vitrea, qua ratione comburat.<emph.end type="italics"/></cell><cell>60</cell></row><row><cell><emph type="italics"/>Planetæ, qua ratione moueantur duplici motu.<emph.end type="italics"/></cell><cell>130</cell></row><row><cell><emph type="italics"/>Plato &longs;olida ex planis componebat. 105. cur Elementis figuras Geometricas attri-bueret.<emph.end type="italics"/></cell><cell>122</cell></row><row><cell><emph type="italics"/>Planetarum ordo.<emph.end type="italics"/></cell><cell>271</cell></row><row><cell><emph type="italics"/>Principia Mathematicorum.<emph.end type="italics"/> 2.</cell><cell>118</cell></row><pb pagenum="29"/><row><cell><emph type="italics"/>Principia &longs;cientiarum duplicia. Ex quibus, & circa quod.<emph.end type="italics"/></cell><cell>61</cell></row><row><cell><emph type="italics"/>Principia Mathematica non pendere ab experientia.<emph.end type="italics"/></cell><cell>315</cell></row><row><cell><emph type="italics"/>Proportio alterna. 28. multiplicata, &longs;eu multiplex &longs;ecundum Cæneum.<emph.end type="italics"/></cell><cell>46</cell></row><row><cell><emph type="italics"/>P&longs;eudographia quid.<emph.end type="italics"/></cell><cell>83</cell></row><row><cell><emph type="italics"/>Proportionalitas quid.<emph.end type="italics"/></cell><cell>308</cell></row><row><cell><emph type="italics"/>Proportio continuata, & di&longs;iuncta quid. 310. alterna, &longs;eu permutata quid.<emph.end type="italics"/></cell><cell><emph type="italics"/>inibi.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>Proportio Geometrica. 311. Arithmetica.<emph.end type="italics"/></cell><cell>302</cell></row><row><cell><emph type="italics"/>Proportio &longs;ecundum dignitatem, e&longs;t Geometrica.<emph.end type="italics"/></cell><cell>330</cell></row><row><cell><emph type="italics"/>Problemata mu&longs;icalia varia à 360. <expan abbr="v&longs;q;">v&longs;que</expan> ad finem &longs;ectionis 19. problematum.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Punicum, mu&longs;icum in&longs;trumentum.<emph.end type="italics"/></cell><cell>370</cell></row><row><cell><emph type="italics"/>Pupillæ oculi etymon., & natura. 408. cur in oculo no&longs;tro imago pupillæ appareat. problem.<emph.end type="italics"/> 2.</cell><cell><emph type="italics"/>ibidem.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>Cur nigra in omnibus hominibus. probl.<emph.end type="italics"/> 5.</cell><cell/></row><row><cell><emph type="italics"/>Cur in Sole euane&longs;cat. probl.<emph.end type="italics"/> 6.</cell><cell/></row><row><cell><emph type="italics"/>Cur modo maior, modo minor appareat.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Pythagorici primi Mathematicis <expan abbr="operã">operam</expan> dedere, eas&queacute; ceteris &longs;cientijs præponeb&abreve;t.<emph.end type="italics"/></cell><cell>202</cell></row><row><cell><emph type="italics"/>Q<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Qvadratura circuli. vide circulus. de quadratura figurarum rectangularum, & quadraturæ duplex definitio cau&longs;alis, & formalis.<emph.end type="italics"/></cell><cell>185</cell></row><row><cell><emph type="italics"/>Quantitas an conctet ex indiui&longs;ibili. toto libello de lineis in&longs;ecabilibus argumentis mathematicis.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>R<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Remi problema.<emph.end type="italics"/></cell><cell>245</cell></row><row><cell><emph type="italics"/>Re&longs;olutio logica, & mathematica, vt conueniant. 4. &<emph.end type="italics"/></cell><cell>305</cell></row><row><cell><emph type="italics"/>Re&longs;ultus cadentium in terram, quibus angulis fiat.<emph.end type="italics"/></cell><cell>354</cell></row><row><cell><emph type="italics"/>Rythmus fusè explicatur.<emph.end type="italics"/></cell><cell>381</cell></row><row><cell><emph type="italics"/>Rubrum mare duplex.<emph.end type="italics"/></cell><cell>152</cell></row><row><cell><emph type="italics"/>S<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Scythala quid, & eius figura 250. &<emph.end type="italics"/></cell><cell>252</cell></row><row><cell><emph type="italics"/>Securis problema, vbi de antiquæ &longs;euris figura, & angulo pulchra demon&longs;tran-tur.<emph.end type="italics"/></cell><cell>258</cell></row><row><cell><emph type="italics"/>Semitonium, quid.<emph.end type="italics"/></cell><cell>360</cell></row><row><cell><emph type="italics"/>Solem e&longs;&longs;e terra multo maiorem: probatur.<emph.end type="italics"/></cell><cell>131</cell></row><row><cell><emph type="italics"/>Sphæram planum tangit in puncto. demon&longs;tratur.<emph.end type="italics"/></cell><cell>184</cell></row><row><cell><emph type="italics"/>Statera antiqua, quæ: eius figura, & problema.<emph.end type="italics"/></cell><cell>259</cell></row><row><cell><emph type="italics"/>Stereomatria, vt differat à Geometria.<emph.end type="italics"/></cell><cell>49</cell></row><row><cell><emph type="italics"/>Succula.<emph.end type="italics"/></cell><cell>253</cell></row><row><cell><emph type="italics"/>Symphonæ voces.<emph.end type="italics"/> 372.</cell><cell>392</cell></row><row><cell><emph type="italics"/>Symphonia.<emph.end type="italics"/></cell><cell>391</cell></row><row><cell><emph type="italics"/>T<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Temonis nauis problema.<emph.end type="italics"/></cell><cell>246</cell></row><row><cell><emph type="italics"/>Terram e&longs;&longs;e rotundam ex eclyp&longs;i. 114. Item aliter. 115. e&longs;&longs;e re&longs;pectu C&oelig;li paruam valde. 115. e&longs;&longs;e cubum cur Plato voluerit.<emph.end type="italics"/></cell><cell>122</cell></row><row><cell><emph type="italics"/>Terræ quantitas.<emph.end type="italics"/></cell><cell>115</cell></row><row><cell><emph type="italics"/>Terram paulatim reduci ad pefféctam rotunditatem.<emph.end type="italics"/></cell><cell>151</cell></row><row><cell><emph type="italics"/>Tetragoni&longs;mus. vide Quadratura.<emph.end type="italics"/></cell><cell/></row><pb pagenum="30"/><row><cell><emph type="italics"/>Teretizare, quid.<emph.end type="italics"/></cell><cell>366</cell></row><row><cell><emph type="italics"/>Tetrachordon, quid.<emph.end type="italics"/></cell><cell>386</cell></row><row><cell><emph type="italics"/>Tollenonis problema.<emph.end type="italics"/></cell><cell>267</cell></row><row><cell><emph type="italics"/>Tonus mu&longs;icus, qui; vnde oriatur.<emph.end type="italics"/></cell><cell>360</cell></row><row><cell><emph type="italics"/>Trochleæ problemata.<emph.end type="italics"/> 249. 250.</cell><cell>251</cell></row><row><cell><emph type="italics"/>Tunicæ oculi. 408. in tractatu de Pupilla.<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>V<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Ventorum nomina, & &longs;itus.<emph.end type="italics"/></cell><cell>160. <emph type="italics"/>a.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>Vectis quotuplex, & c.<emph.end type="italics"/></cell><cell>244</cell></row><row><cell><emph type="italics"/>Veteres canere &longs;olitos non &longs;olum in choris, &longs;ed etiam in &longs;cenis.<emph.end type="italics"/> 371. 384.</cell><cell>400</cell></row><row><cell><emph type="italics"/>Virgiliæ, Pleiades.<emph.end type="italics"/></cell><cell>335</cell></row><row><cell><emph type="italics"/>Vi&longs;æ res gemmantur di&longs;tractis oculis.<emph.end type="italics"/></cell><cell>406</cell></row><row><cell><emph type="italics"/>Vi&longs;æ rei geminatio non fit altero oculo in latera torto, cur.<emph.end type="italics"/></cell><cell>407</cell></row><row><cell><emph type="italics"/>Vi&longs;us res vi&longs;as, cur non duplicet, etiam &longs;i duos oculos habeamus.<emph.end type="italics"/></cell><cell>405</cell></row><row><cell><emph type="italics"/>Vmbelici litoralis problema.<emph.end type="italics"/></cell><cell>254</cell></row><row><cell><emph type="italics"/>Vmbram terræ parum &longs;upra Lunam tran&longs;cendere.<emph.end type="italics"/></cell><cell>137</cell></row><row><cell><emph type="italics"/>Vmbrarum incrementa, & decrementa, cur inæqualia.<emph.end type="italics"/> 344.</cell><cell>348</cell></row><row><cell><emph type="italics"/>Vi&longs;us res geminat, &longs;i alter oculorum digito pellatur, cur.<emph.end type="italics"/></cell><cell>197</cell></row><row><cell><emph type="italics"/>Vocum mu&longs;icalium antiquæ appellationes.<emph.end type="italics"/></cell><cell>360</cell></row><row><cell><emph type="italics"/>Vox acuta velocior, grauis verò tarda, cur.<emph.end type="italics"/></cell><cell>77</cell></row><row><cell><emph type="italics"/>Voluminum &longs;ectio modo rectam lineam, modo curuam refert, cur.<emph.end type="italics"/></cell><cell>356</cell></row><row><cell><emph type="italics"/>Vnitas, cur indiui&longs;ibilis.<emph.end type="italics"/></cell><cell>22</cell></row><row><cell><emph type="italics"/>Z<emph.end type="italics"/></cell><cell/></row><row><cell><emph type="italics"/>Zonas terræ, vt Arist. de&longs;ignet: & quæ &longs;ecundum ip&longs;um &longs;int habitabiles.<emph.end type="italics"/></cell><cell>156</cell></row><row><cell><emph type="italics"/>Zonam torridam quatuor reddunt habitabilem.<emph.end type="italics"/></cell><cell>159</cell></row></table><s>Finis Tertij Indicis.</s><pb pagenum="31"/>

</section><section>

<s>Vi&longs;um e&longs;t etiam opportunum Lectori fore, ea &longs;imul in vnum <lb/>loca colligere, in quibus Ari&longs;toteles mihi vi&longs;us e&longs;t in Ma<lb/>thematicis &longs;copum non attigi&longs;&longs;e, vt alij pr&ecedil;&longs;ertim Peripa<lb/>tetici facilius ea inuenire, <expan abbr="atq;">atque</expan> de ij&longs;dem iudicium ferre <lb/>po&longs;&longs;int.<lb/><arrow.to.target n="table3"/></s><pb pagenum="33"/><table><table.target id="table3"/><row><cell><emph type="italics"/>121<emph.end type="italics"/></cell><cell><emph type="italics"/>Nvmero marginali: vbi ait plura Octaedra, &longs;eu Pyramides re-plere locum: in quo omnes pariter expo&longs;itores lap&longs;i &longs;unt.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>124<emph.end type="italics"/></cell><cell><emph type="italics"/>Latitudinem figura, ait, cau&longs;am e&longs;&longs;e &longs;upernatationis. & aquam re&longs;i&longs;tere &longs;impliciter diui&longs;ioni.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>136<emph.end type="italics"/></cell><cell><emph type="italics"/>Cometas in &longs;uprema aeris regione collocat; cuius contrarium ibi line ari demonctratione ostenditur.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>147<emph.end type="italics"/></cell><cell><emph type="italics"/>Ait Tanaim, & Indum oriri ex monte Paropami&longs;&longs;o. & c.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>148<emph.end type="italics"/></cell><cell><emph type="italics"/>Ait, tertia parte noctis Cauca&longs;i verticem illuminari à Sole.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>149<emph.end type="italics"/></cell><cell><emph type="italics"/>Ait Danubium ex Pyreneo monte defluere.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>150<emph.end type="italics"/></cell><cell><emph type="italics"/>Ait fluuium quendam non minorem Rhodano in Liguria ab&longs;orberi, & iterum egredi.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>152<emph.end type="italics"/></cell><cell><emph type="italics"/>Ait Rubrum mare parum Atlantico Oceano commi&longs;ceri.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>159<emph.end type="italics"/></cell><cell><emph type="italics"/>Zonam torridam inhabit abilem exi&longs;timat.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>164<emph.end type="italics"/></cell><cell><emph type="italics"/>Putat Iridis angulos non po&longs;&longs;e vnum &longs;upra alterum collocari, &longs;ed tantummodo in orbem.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>182<emph.end type="italics"/></cell><cell><emph type="italics"/>Rationes, quas in Parelij dubitationibus affert, videntur inanes.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>236<emph.end type="italics"/></cell><cell><emph type="italics"/>In &longs;ubducendo c&oelig;le&longs;tium orbium numero, memoria labitur.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>243<emph.end type="italics"/></cell><cell><emph type="italics"/>Ait lineam O L, &longs;uperare lineam L R, quantitate P L, vt in figura.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>245<emph.end type="italics"/></cell><cell><emph type="italics"/>Remum ad vectem primi generis reducit.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>246<emph.end type="italics"/></cell><cell><emph type="italics"/>Temonem nauis reducit ad vectem primi generis.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>247<emph.end type="italics"/></cell><cell><emph type="italics"/>In motu Remi collato cum motu Nauigij, à Nonio erroris manifestè arguitur. eodem modo in motu Temonis.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>250<emph.end type="italics"/></cell><cell><emph type="italics"/>Ait maioribus trochleis, aut rotulis facilius onera &longs;ubleuari.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>256<emph.end type="italics"/></cell><cell><emph type="italics"/>Reducit cuneum ad vectem primi generis.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>270<emph.end type="italics"/></cell><cell><emph type="italics"/>Cur res in vorticibus ad medium ferantur, veram cau&longs;am a&longs;signa-re non videtur.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>275<emph.end type="italics"/></cell><cell><emph type="italics"/>Ait Danubium altero ramo in Mediterraneum, altero in Pontum effluere.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>293<emph.end type="italics"/></cell><cell><emph type="italics"/>Negat Araneum filum ab intrin&longs;eco emittere, Democritum iniuria refellens. quamuis hoc vltimum ad Phy&longs;icum pertineat.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>403<emph.end type="italics"/></cell><cell><emph type="italics"/>Problem. 2. &longs;ect. 2 3. non benè videtur a&longs;signare cau&longs;am variæ im-mer&longs;ionis nauigij.<emph.end type="italics"/></cell></row></table><figure/>

<s>Vi&longs;um e&longs;t etiam opportunum Lectori fore, ea &longs;imul in vnum

<lb/>loca colligere, in quibus Ari&longs;toteles mihi vi&longs;us e&longs;t in Ma

<lb/>thematicis &longs;copum non attigi&longs;&longs;e, vt alij pr&ecedil;&longs;ertim Peripa

<s>MATHEMATICA</s>

<lb/>tetici facilius ea inuenire, <expan abbr="atq;">atque</expan> de ij&longs;dem iudicium ferre

<lb/>po&longs;&longs;int.

<s>EX LIBRO</s>

<lb/><arrow.to.target n="table3"/></s><table><table.target id="table3"/><row><cell><emph type="italics"/>121<emph.end type="italics"/></cell><cell><emph type="italics"/>Nvmero marginali: vbi ait plura Octaedra, &longs;eu Pyramides re-plere locum: in quo omnes pariter expo&longs;itores lap&longs;i &longs;unt.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>124<emph.end type="italics"/></cell><cell><emph type="italics"/>Latitudinem figura, ait, cau&longs;am e&longs;&longs;e &longs;upernatationis. & aquam re&longs;i&longs;tere &longs;impliciter diui&longs;ioni.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>136<emph.end type="italics"/></cell><cell><emph type="italics"/>Cometas in &longs;uprema aeris regione collocat; cuius contrarium ibi line ari demonctratione ostenditur.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>147<emph.end type="italics"/></cell><cell><emph type="italics"/>Ait Tanaim, & Indum oriri ex monte Paropami&longs;&longs;o. & c.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>148<emph.end type="italics"/></cell><cell><emph type="italics"/>Ait, tertia parte noctis Cauca&longs;i verticem illuminari à Sole.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>149<emph.end type="italics"/></cell><cell><emph type="italics"/>Ait Danubium ex Pyreneo monte defluere.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>150<emph.end type="italics"/></cell><cell><emph type="italics"/>Ait fluuium quendam non minorem Rhodano in Liguria ab&longs;orberi, & iterum egredi.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>152<emph.end type="italics"/></cell><cell><emph type="italics"/>Ait Rubrum mare parum Atlantico Oceano commi&longs;ceri.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>159<emph.end type="italics"/></cell><cell><emph type="italics"/>Zonam torridam inhabit abilem exi&longs;timat.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>164<emph.end type="italics"/></cell><cell><emph type="italics"/>Putat Iridis angulos non po&longs;&longs;e vnum &longs;upra alterum collocari, &longs;ed tantummodo in orbem.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>182<emph.end type="italics"/></cell><cell><emph type="italics"/>Rationes, quas in Parelij dubitationibus affert, videntur inanes.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>236<emph.end type="italics"/></cell><cell><emph type="italics"/>In &longs;ubducendo c&oelig;le&longs;tium orbium numero, memoria labitur.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>243<emph.end type="italics"/></cell><cell><emph type="italics"/>Ait lineam O L, &longs;uperare lineam L R, quantitate P L, vt in figura.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>245<emph.end type="italics"/></cell><cell><emph type="italics"/>Remum ad vectem primi generis reducit.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>246<emph.end type="italics"/></cell><cell><emph type="italics"/>Temonem nauis reducit ad vectem primi generis.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>247<emph.end type="italics"/></cell><cell><emph type="italics"/>In motu Remi collato cum motu Nauigij, à Nonio erroris manifestè arguitur. eodem modo in motu Temonis.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>250<emph.end type="italics"/></cell><cell><emph type="italics"/>Ait maioribus trochleis, aut rotulis facilius onera &longs;ubleuari.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>256<emph.end type="italics"/></cell><cell><emph type="italics"/>Reducit cuneum ad vectem primi generis.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>270<emph.end type="italics"/></cell><cell><emph type="italics"/>Cur res in vorticibus ad medium ferantur, veram cau&longs;am a&longs;signa-re non videtur.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>275<emph.end type="italics"/></cell><cell><emph type="italics"/>Ait Danubium altero ramo in Mediterraneum, altero in Pontum effluere.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>293<emph.end type="italics"/></cell><cell><emph type="italics"/>Negat Araneum filum ab intrin&longs;eco emittere, Democritum iniuria refellens. quamuis hoc vltimum ad Phy&longs;icum pertineat.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>403<emph.end type="italics"/></cell><cell><emph type="italics"/>Problem. 2. &longs;ect. 2 3. non benè videtur a&longs;signare cau&longs;am variæ im-mer&longs;ionis nauigij.<emph.end type="italics"/></cell></row></table></section><pb/><pb pagenum="33"/>

<s>PRÆDICAMENTORVM</s>

<section><s>LOCA

<lb/>MATHEMATICA

<s>Per ordinem declarata.</s><figure/>

<lb/>EX LIBRO

<lb/>PRÆDICAMENTORVM

<lb/>Per ordinem declarata.</s>

</front><body><chap>

<s>Ex c. 3. De his, quæ ad aliquid. </s>

<s>Porrò quemadmodum vnus angulus vni angulo æqualis e&longs;t, ita

<lb/><expan abbr="aliquãdo">aliquando</expan> duo anguli &longs;unt vni angulo æquales, vt patet, &longs;i vnus angulus, v.g.

<lb/>angulus B A C, vbi ait <emph type="italics"/>(Scientia verò &longs;i non &longs;it,

<lb/>nihil probibet e&longs;&longs;e &longs;cibile, vt circuli quadratura, &longs;i e&longs;t &longs;cibilis,

<lb/>&longs;cientia quidem eius nondum e&longs;t)<emph.end type="italics"/> Cum velit Ari&longs;t.

o&longs;tendere,

<lb/>nó omnia correlatiua &longs;imul e&longs;&longs;e natura, id de &longs;cibili, & &longs;cien

<lb/>tia variè probat, præ&longs;ertim verò, quia multa &longs;int &longs;cibilia,

<lb/>quæ tamen nondum &longs;ciantur, vt patet, inquit, in Quadratu

<lb/>ra circuli, & &longs;cientia ip&longs;ius, quia quamuis ip&longs;a circuli quadratura &longs;it &longs;cibi

<lb/>lis, nondum tamen &longs;imul cum ip&longs;a, &longs;cientia illius extat. </s>

<s>Quæ vt perfectè

<lb/>intelligantur, &longs;ciendum e&longs;t, quadraturam circuli, quæ à Græcis tetrago

<lb/>ni&longs;mus dicitur, nihil aliud e&longs;&longs;e, quàm propo&longs;ito cuilibet circulo exhibere

<lb/>quadratum æquale. </s>

<s>Quæ æqualitas debet intelligi de areis, &longs;eu &longs;patijs, ita

<lb/>vt area circuli, &longs;eu &longs;patium illud, &longs;iue &longs;uperficies illa circularis, &longs;it æqualis

<lb/>areæ, &longs;eu &longs;uperficiei quadratæ. </s>

<s>Qua in re plurimi decipiuntur exi&longs;timantes

<lb/>per quadraturam cir culi inquiri æqualitatem linearum, ita vt circumferen

<lb/>tia circuli debeat e&longs;&longs;e æqualis ambitui, &longs;eu quatuor lateribus quadrati:

<lb/>quod omnino fal&longs;um e&longs;t.</s>

<s>Quadratio porrò circuli dupliciter proponi pote&longs;t, vel tanquam Theo

<lb/>rema, vel tanquam Problema <emph type="italics"/>(theorema autem e&longs;t propo&longs;itio, in qua nihil fa

<lb/>ciendum proponitur; problema verò aliquid fieri expo&longs;cit)<emph.end type="italics"/> neutrum autem tem

<lb/>pore Ari&longs;t.

erat adinuentum nam theorema inuentum e&longs;t po&longs;t ip&longs;um ducen

<lb/>tis circiter annis ab Archimede: problema verò nondum à quoquam per

<lb/>fectè potuit reperiri. </s>

<s>qua di&longs;tinctione &longs;aluari po&longs;&longs;unt nonnulli, vt Boetius

<lb/>hocloco, qui aiunt, &longs;e vidi&longs;&longs;e Demon&longs;trationem quadraturæ huius, &longs;i nimi

<lb/>rum intelligant theorema. </s>

<s>& alij etiam verum a&longs;&longs;erunt, dum negant hacte

<lb/>nus repertam e&longs;&longs;e, &longs;i nimirum de problemate loquantur, theorema Archi<pb pagenum="34"/>medis e&longs;t propo&longs;itio prima acuti&longs;&longs;imi libelli de Dimen&longs;ione circuli; e&longs;t au

<lb/>tem huiu&longs;modi. </s>

<s>Quilibet circulus æqualis e&longs;t triangulo rectangulo, cuius

<lb/>quidem &longs;emidiameter vni laterum, quæ circa rectum angulum &longs;unt, ambi

<lb/>tus verò ba&longs;i eius e&longs;t æqualis.</s><figure place="text" xlink:href="figures-la/009.01.034.1.tif"/>

<s>Sit, v.g. datus circulus, cuius &longs;emidiameter A B; & fit trian gulum rectangu

<lb/>lum A B C, cuius angulus B, &longs;it rectus, & latus B A, <expan abbr="con&longs;titu&etilde;s">con&longs;tituens</expan> angulum re

<lb/>ctum B, cum ba&longs;i B C, &longs;it æquale &longs;emidiametro A B; ba&longs;is verò B C, &longs;it æqua

<lb/>lis peripheriæ eiu&longs;dem circuli dati. </s>

<s>demon&longs;trat iam ibi Archimedes acuta

<lb/>æquè, ac euidenti demon&longs;tratione triangulum i&longs;tud æquale e&longs;&longs;e circulo illi.

<s>quod perinde e&longs;t, ac &longs;i o&longs;tendi&longs;&longs;et cuinam quadrato &longs;it æqualis, cum per vl

<lb/>timam 2. Eucl. po&longs;&longs;imus triangulo huic quadratum æquale con&longs;truere, quod

<lb/>con&longs;equenter dato circulo æquale erit. </s>

<s>Quod &longs;i in modum Problematis ita

<lb/>proponatur: Dato circulo æquale quadratum con&longs;truere, nondum inuenta

<lb/>e&longs;t ratio, quæ demon&longs;tratione confirmetur, qua id geometricè penitus, hoc

<lb/>e&longs;t ad æqualitatem mathematicam, &longs;eu exacti&longs;&longs;imam effici po&longs;&longs;it, <expan abbr="tota&qacute;">totaque</expan>; dif

<lb/>ficultas po&longs;ita e&longs;&longs;e videtur in inue&longs;tigando, quonam modo exhibeamus li

<lb/>neam rectam B C, æqualem peripheriæ circuli dati. </s>

<s>quam nullus hactenus

<lb/>geometricè illi æqualem potuit exhibere, <expan abbr="atq;">atque</expan> exhibita <expan abbr="euid&etilde;ti">euidenti</expan> demon&longs;tra

<lb/>tione comprobare; Quamuis Archimedes acumine &longs;anè mirabili in lib.

<lb/>lineis &longs;piralibus, eam <expan abbr="quoq;">quoque</expan> theorematicè, non tamen problematicè inue

<lb/>&longs;tigauit. </s>

<s>nam propo&longs;itione 18. illius <expan abbr="admirãdi">admirandi</expan> operis inuenit lineam rectam

<lb/>æqualem circumferentiæ primi circuli &longs;piralis lineæ; propo&longs; verò 19. repe

<lb/>rit aliam rectam æqualem circumferentiæ &longs;ecundi circuli. </s>

<s>vbi ait <emph type="italics"/>(Scientia verò &longs;i non &longs;it, <lb/>nihil probibet e&longs;&longs;e &longs;cibile, vt circuli quadratura, &longs;i e&longs;t &longs;cibilis, <lb/>&longs;cientia quidem eius nondum e&longs;t)<emph.end type="italics"/> Cum velit Ari&longs;t.

<s>tu ip&longs;um con&longs;ule,

<lb/>&longs;i admirandarum rerum contemplatione delectaris. </s>

o&longs;tendere, <lb/>nó omnia correlatiua &longs;imul e&longs;&longs;e natura, id de &longs;cibili, & &longs;cien<lb/>tia variè probat, præ&longs;ertim verò, quia multa &longs;int &longs;cibilia, <lb/>quæ tamen nondum &longs;ciantur, vt patet, inquit, in Quadratu<lb/>ra circuli, & &longs;cientia ip&longs;ius, quia quamuis ip&longs;a circuli quadratura &longs;it &longs;cibi<lb/>lis, nondum tamen &longs;imul cum ip&longs;a, &longs;cientia illius extat. </s>

<s>Quæ vt perfectè <lb/>intelligantur, &longs;ciendum e&longs;t, quadraturam circuli, quæ à Græcis tetrago<lb/>ni&longs;mus dicitur, nihil aliud e&longs;&longs;e, quàm propo&longs;ito cuilibet circulo exhibere <lb/>quadratum æquale. </s>

<s>Quæ æqualitas debet intelligi de areis, &longs;eu &longs;patijs, ita <lb/>vt area circuli, &longs;eu &longs;patium illud, &longs;iue &longs;uperficies illa circularis, &longs;it æqualis <lb/>areæ, &longs;eu &longs;uperficiei quadratæ. </s>

<s>Qua in re plurimi decipiuntur exi&longs;timantes <lb/>per quadraturam cir culi inquiri æqualitatem linearum, ita vt circumferen<lb/>tia circuli debeat e&longs;&longs;e æqualis ambitui, &longs;eu quatuor lateribus quadrati: <lb/>quod omnino fal&longs;um e&longs;t.</s>

<s>Quadratio porrò circuli dupliciter proponi pote&longs;t, vel tanquam Theo<lb/>rema, vel tanquam Problema <emph type="italics"/>(theorema autem e&longs;t propo&longs;itio, in qua nihil fa<lb/>ciendum proponitur; problema verò aliquid fseri expo&longs;cit)<emph.end type="italics"/> neutrum a<gap/> t<gap/>m tem<lb/>pore Ari&longs;t.

erat adinuentum nam theorema inuentum e&longs;t po&longs;t ip<gap/> m ducen<lb/>tis circiter annis ab Archimede: problema verò nondum à quoquam per<lb/>fectè potuit reperiri. </s>

<s>qua di&longs;tinctione &longs;aluari po&longs;&longs;unt nonnulli, vt Boetius <lb/>hocloco, qui aiunt, &longs;e vidi&longs;&longs;e Demon&longs;trationem quadraturæ huius, &longs;i nimi<lb/>rum intelligant theorema. </s>

<s>& alij etiam verum a&longs;&longs;erunt, dum negant hacte<lb/>nus repertam e&longs;&longs;e, &longs;i nimirum de problemate loquantur, theorema Archi<pb pagenum="34"/>medis e&longs;t propo&longs;itio prima acuti&longs;&longs;imi libelli de Dimen&longs;ione circuli; e&longs;t au<lb/>tem huiu&longs;modi. </s>

<s>Quilibet circulus æqualis e&longs;t triangulo rectangulo, cuius <lb/>quidem &longs;emidiameter vni laterum, quæ circa rectum angulum &longs;unt, ambi<lb/>tus verò ba&longs;i eius e&longs;t æqualis.</s><figure/>

<s>Sit, v.g. datus circulus, cuius &longs;emidiameter A B; & fit trian gulum rectangu<lb/>lum A B C, cuius angulus B, &longs;it rectus, & latus B A, <expan abbr="con&longs;titu&etilde;s">con&longs;tituens</expan> angulum re<lb/>ctum B, cum ba&longs;i B C, &longs;it æquale &longs;emidiametro A B; ba&longs;is verò B C, &longs;it æqua<lb/>lis peripheriæ eiu&longs;dem circuli dati. </s>

<s>demon&longs;trat iam ibi Archimedes acuta <lb/>æquè, ac euidenti demon&longs;tratione triangulum i&longs;tud æquale e&longs;&longs;e circulo illi. <lb/></s>

<s>quod perinde e&longs;t, ac &longs;i o&longs;tendi&longs;&longs;et cuinam quadrato &longs;it æqualis, cum per vl<lb/>timam 2. Eucl. po&longs;&longs;imus triangulo huic quadratum æquale con&longs;truere, quod <lb/>con&longs;equenter dato circulo æquale erit. </s>

<s>Multa hac de re Pap

<lb/>pus Alexandrinus lib.

<s>Quod &longs;i in modum Problematis ita <lb/>proponatur: Dato circulo æquale quadratum con&longs;truere, nondum inuenta <lb/>e&longs;t ratio, quæ demon&longs;tratione confirmetur, qua id geometricè penitus, hoc <lb/>e&longs;t ad æqualitatem mathematicam, &longs;eu exacti&longs;&longs;imam effici po&longs;&longs;it, <expan abbr="tota&qacute;">totaque</expan>; dif<lb/>ficultas po&longs;ita e&longs;&longs;e videtur in inue&longs;tigando, quonam modo exhibeamus li<lb/>neam rectam B C, æqualem peripheriæ circuli dati. </s>

<s>quam nullus hactenus <lb/>geometricè illi æqualem potuit exhibere, <expan abbr="atq;">atque</expan> exhibita <expan abbr="euid&etilde;ti">euidenti</expan> demon&longs;tra<lb/>tione comprobare; Quamuis Archimedes acumine &longs;anè mirabili in lib.

de <lb/>lineis &longs;piralibus, eam <expan abbr="quoq;">quoque</expan> theorematicè, non tamen problematicè inue<lb/>&longs;tigauit. </s>

<s>nam propo&longs;itione 18. illius <expan abbr="admirãdi">admirandi</expan> operis inuenit lineam rectam <lb/>æqualem circumferentiæ primi circuli &longs;piralis lineæ; propo&longs; verò 19. repe<lb/>rit aliam rectam æqualem circumferentiæ &longs;ecundi circuli. </s>

<s>tu ip&longs;um con&longs;ule, <lb/>&longs;i admirandarum rerum contemplatione delectaris. </s>

<s>Multa hac de re Pap<lb/>pus Alexandrinus lib.

4. Math. coll. </s>

<s>& Ioannes Buteo vnico volumine om<lb/>nes quadraturas tain pri&longs;corum, quam recentiorum <expan abbr="cõprehen&longs;us">comprehen&longs;us</expan> e&longs;t. </s>

<s>& Ioannes Buteo vnico volumine om

<lb/>nes quadraturas tain pri&longs;corum, quam recentiorum <expan abbr="cõprehen&longs;us">comprehen&longs;us</expan> e&longs;t. </s>

<s>Qua<lb/>re qui plura cupit, eos adeat; nos tamen infra &longs;uis locis explicabimus tres <lb/>illas celebres antiquorum Antiphontis, Bri&longs;&longs;onis, & Hippocratis quadra<lb/>turas, quamuis fal&longs;as, <expan abbr="quar&utilde;">quarum</expan> &longs;æpe meminit Ari&longs;t. & alij. </s>

<s>Qua

<lb/>re qui plura cupit, eos adeat; nos tamen infra &longs;uis locis explicabimus tres

<s>&longs;olet autem à non<lb/>nullis di&longs;putari, vtrum quadratura i&longs;ta problematica &longs;it po&longs;&longs;ibilis, nec ne, <lb/>cum videant eam à nemine, quamuis diu magno labore perqui&longs;itam, hacte<lb/>nus adinuentam e&longs;&longs;e. </s>

<lb/>illas celebres antiquorum Antiphontis, Bri&longs;&longs;onis, & Hippocratis quadra

<lb/>turas, quamuis fal&longs;as, <expan abbr="quar&utilde;">quarum</expan> &longs;æpe meminit Ari&longs;t. & alij. </s>

<s>ego quidem e&longs;&longs;e po&longs;&longs;ibilem exi&longs;timo, quis enim dubi<lb/>tare pote&longs;t, po&longs;&longs;e exi&longs;tere quadratum æquale circulo propo&longs;ito? </s>

<s>&longs;olet autem à non

<s>Quod &longs;i po<lb/>te&longs;t fieri, quare non etiam demon&longs;trari? </s>

<lb/>nullis di&longs;putari, vtrum quadratura i&longs;ta problematica &longs;it po&longs;&longs;ibilis, nec ne,

<lb/>cum videant eam à nemine, quamuis diu magno labore perqui&longs;itam, hacte

<s>pr&ecedil;fertim cum videamus ab Archi<lb/>mede iam inuentam e&longs;&longs;e, quatenus Theorema e&longs;t. </s>

<lb/>nus adinuentam e&longs;&longs;e. </s>

<s>& præterea con&longs;tet<gap/> Hip<lb/>pocratem quadra&longs;&longs;e lunulam, vt &longs;uo loco dicemus, & Archimedem in libel<pb pagenum="35"/>lo de quadratura Paraboles, quadra&longs;&longs;e ip&longs;am Parabolem, quæ tamen duæ fi<lb/>guræ, lunula &longs;cilicet, & parabola &longs;unt curuilineæ.</s>

<s>ego quidem e&longs;&longs;e po&longs;&longs;ibilem exi&longs;timo, quis enim dubi

<lb/>tare pote&longs;t, po&longs;&longs;e exi&longs;tere quadratum æquale circulo propo&longs;ito? </s>

<s>Quod &longs;i po

<lb/>te&longs;t fieri, quare non etiam demon&longs;trari? </s>

<s>pr&ecedil;fertim cum videamus ab Archi

<lb/>mede iam inuentam e&longs;&longs;e, quatenus Theorema e&longs;t. </s>

<s>& præterea con&longs;tet, Hip

<lb/>pocratem quadra&longs;&longs;e lunulam, vt &longs;uo loco dicemus, & Archimedem in libel<pb pagenum="35"/>lo de quadratura Paraboles, quadra&longs;&longs;e ip&longs;am Parabolem, quæ tamen duæ fi

<lb/>guræ, lunula &longs;cilicet, & parabola &longs;unt curuilineæ.</s>

Line 1972

Line 2158

<s>Ex cap.

de Priori <emph type="italics"/>(in &longs;cientijs demon&longs;tratiuis e&longs;t prius, & po&longs;terius ordine, <lb/>elementa enim priora &longs;unt ijs, quæ de&longs;cribuntur, nam principia prior a &longs;unt theore<lb/>matibus ordine)<emph.end type="italics"/> verba illa, nam principia, &c. </s>

de Priori <emph type="italics"/>(in &longs;cientijs demon&longs;tratiuis e&longs;t prius, & po&longs;terius ordine,

<lb/>elementa enim priora &longs;unt ijs, quæ de&longs;cribuntur, nam principia prior a &longs;unt theore

<s>quæ non &longs;unt in antiqua tran<lb/>&longs;latione de&longs;ump&longs;imus ex ca&longs;tigati&longs;&longs;imo græco codice editionis Francfor<lb/>dien&longs;is, propterea quod totum hunc locum declarant; &longs;unt autem i&longs;ta, <lb/><foreign lang="greek">ai/gar arxai/ pro/terai tw_n qewrhma/twn th ta/xei.</foreign> per &longs;cientias autem demon&longs;tra<lb/>tiuas intelligendas e&longs;&longs;e hoc loco ip&longs;as Mathematicas ex eo patet, quod illis <lb/>a&longs;&longs;ignet Ari&longs;t. De&longs;criptiones; nam hoc verbo, De&longs;criptiones, &longs;eu figuratio<lb/>nes, &longs;olet ip&longs;e Mathematicas Demon&longs;trationes innuere, quod in ip&longs;is figu<lb/>rationes, & De&longs;criptiones adhibeantur, vt alijs locis patebit: idcirco ver<lb/>ba illa à nobis addita ex græco, optim è <expan abbr="præced&etilde;tia">præcedentia</expan> exponunt, cum per ele<lb/>menta intelligantur principia, qualia &longs;unt initio Euclidis, & per de&longs;criptio<lb/>nes exponant theoremata. </s>

<lb/>matibus ordine)<emph.end type="italics"/> verba illa, nam principia, &c. </s>

<s>quod autem principia illa ordine priora &longs;int de<lb/>mon&longs;trationibus, &longs;iue ip&longs;as præcedant, ex ip&longs;a primi Euclidis in&longs;pectione <lb/>patere pote&longs;t.</s>

<s>quæ non &longs;unt in antiqua tran

<lb/>&longs;latione de&longs;ump&longs;imus ex ca&longs;tigati&longs;&longs;imo græco codice editionis Francfor

<lb/>dien&longs;is, propterea quod totum hunc locum declarant; &longs;unt autem i&longs;ta,

<lb/><foreign lang="greek">ai/gar arxai/ pro/terai tw_n qewrhma/twn th ta/xei.</foreign> per &longs;cientias autem demon&longs;tra

<lb/>tiuas intelligendas e&longs;&longs;e hoc loco ip&longs;as Mathematicas ex eo patet, quod illis

<lb/>a&longs;&longs;ignet Ari&longs;t. De&longs;criptiones; nam hoc verbo, De&longs;criptiones, &longs;eu figuratio

<lb/>nes, &longs;olet ip&longs;e Mathematicas Demon&longs;trationes innuere, quod in ip&longs;is figu

<lb/>rationes, & De&longs;criptiones adhibeantur, vt alijs locis patebit: idcirco ver

<lb/>ba illa à nobis addita ex græco, optim è <expan abbr="præced&etilde;tia">præcedentia</expan> exponunt, cum per ele

<lb/>menta intelligantur principia, qualia &longs;unt initio Euclidis, & per de&longs;criptio

<lb/>nes exponant theoremata. </s>

<s>quod autem principia illa ordine priora &longs;int de

<lb/>mon&longs;trationibus, &longs;iue ip&longs;as præcedant, ex ip&longs;a primi Euclidis in&longs;pectione

<lb/>patere pote&longs;t.</s>

Line 1984

Line 2184

<s>Ex cap.

de motu <emph type="italics"/>(Quadratum augetur Gnomone circumpo&longs;ito)<emph.end type="italics"/> Gnomon vox <lb/>græca inter alia &longs;ignificat in&longs;trumentum illud, quod Latini tum amu&longs;&longs;im, <lb/><figure id="fig2"/><lb/>tum normam appellant, Itali verò, Squadra, ad <lb/>cuius &longs;imilitudinem Geometræ denominarunt fi<lb/>guram quandam, &longs;eu portionem cuiu&longs;uis paralle<lb/>logrammi, vt videre e&longs;t in definitione &longs;ecunda <lb/>2. elem. </s>

de motu <emph type="italics"/>(Quadratum augetur Gnomone circumpo&longs;ito)<emph.end type="italics"/> Gnomon vox

<lb/>græca inter alia &longs;ignificat in&longs;trumentum illud, quod Latini tum amu&longs;&longs;im,

<s>& in præ&longs;enti figura, in qua quadratum <lb/>A B C D, circumpo&longs;ito gnomone E F G, augetur, <lb/>& fit maius quadratum H B I L.</s>

<lb/>tum normam appellant, Itali verò, Squadra, ad

<s>Idem etiam verum e&longs;t in quadrato arithmeti<lb/>co, &longs;iue in numero quadrato: is enim pariter ad<lb/>dito Gnomone augetur. </s>

<lb/>cuius &longs;imilitudinem Geometræ denominarunt fi

<lb/>guram quandam, &longs;eu portionem cuiu&longs;uis paralle

<lb/>logrammi, vt videre e&longs;t in definitione &longs;ecunda

<s>& in præ&longs;enti figura, in qua quadratum

<lb/>A B C D, circumpo&longs;ito gnomone E F G, augetur,

<lb/>& fit maius quadratum H B I L.</s>

<s>Idem etiam verum e&longs;t in quadrato arithmeti

<lb/>co, &longs;iue in numero quadrato: is enim pariter ad

<lb/>dito Gnomone augetur. </s>

<s>addito numero impari. <lb/></s>

<s>addito numero impari.

<s>quemadmodum infra 3. Phy&longs;. tex. 26. fusè explicabimus.</s>

<s><emph type="italics"/>Ex Primo Priorum re&longs;olutoriorum.<emph.end type="italics"/></s>

<s><emph type="italics"/>Ex Primo Priorum re&longs;olutoriorum.<emph.end type="italics"/></s>

</chap><chap>

<s>Aliquorum opinio e&longs;t, Ari&longs;totelem ho&longs;ce libros appella&longs;&longs;e re&longs;olu<lb/>torios, quod per illos doceat &longs;yllogi&longs;mum, ac demon&longs;trationem <lb/>iam factam in &longs;ua immediata principia re&longs;oluere, quam opinio<lb/>nem meum non e&longs;t, nunc refellere. </s>

<s>Aliquorum opinio e&longs;t, Ari&longs;totelem ho&longs;ce libros appella&longs;&longs;e re&longs;olu

<lb/>torios, quod per illos doceat &longs;yllogi&longs;mum, ac demon&longs;trationem

<s>per&longs;ua&longs;um tamen mihi e&longs;t, rem <lb/>multo aliter &longs;e habere, veram rationem huius tituli petendam e&longs;&longs;e ex peni<lb/>tiori Mathematicorum eruditione. </s>

<lb/>iam factam in &longs;ua immediata principia re&longs;oluere, quam opinio

<lb/>nem meum non e&longs;t, nunc refellere. </s>

<s>Sciendum <expan abbr="itaq;">itaque</expan> id, quod tradit Pappus <lb/>Alex. initio &longs;eptimi Mathem. collect. </s>

<s>per&longs;ua&longs;um tamen mihi e&longs;t, rem

<s>antiqui&longs;&longs;imos videlicet Geometras, <lb/>Euclidem, Apollonium Pergæum, & Ari&longs;t&ecedil;um &longs;crip&longs;i&longs;&longs;e libros de re&longs;olutio<lb/>ne, in quibus ars tradebatur, qua propo&longs;ito quouis theoremate, aut proble<lb/>mate po&longs;&longs;ent facile ex eo, tanquam vero accepto inue&longs;tigare aliquam veri<lb/>tatem, per quam deinde componerent illius, quod quærebatur, Demon&longs;tra<lb/>tionem; inue&longs;tigationem illam appellabant re&longs;olutionem: compo&longs;itionem <lb/>verò nominabant di&longs;cur&longs;um <expan abbr="ill&utilde;">illum</expan>, quo ex vero illo per re&longs;olutionem inuento, <pb pagenum="36"/>o&longs;tendebant conclu&longs;ionem. </s>

<lb/>multo aliter &longs;e habere, veram rationem huius tituli petendam e&longs;&longs;e ex peni

<lb/>tiori Mathematicorum eruditione. </s>

<s>Porrò Diogenes Laert. huius re&longs;olutionis in<lb/>uentorem facit Platonem: à quo eam Leodamas Tha&longs;ius didicit, cuius be<lb/>neficio, pluries deinde Geometricas demon&longs;trationes adinuenit. </s>

<s>Sciendum <expan abbr="itaq;">itaque</expan> id, quod tradit Pappus

<lb/>Alex. initio &longs;eptimi Mathem. collect. </s>

<s>antiqui&longs;&longs;imos videlicet Geometras,

<lb/>Euclidem, Apollonium Pergæum, & Ari&longs;t&ecedil;um &longs;crip&longs;i&longs;&longs;e libros de re&longs;olutio

<lb/>ne, in quibus ars tradebatur, qua propo&longs;ito quouis theoremate, aut proble

<lb/>mate po&longs;&longs;ent facile ex eo, tanquam vero accepto inue&longs;tigare aliquam veri

<lb/>tatem, per quam deinde componerent illius, quod quærebatur, Demon&longs;tra

<lb/>tionem; inue&longs;tigationem illam appellabant re&longs;olutionem: compo&longs;itionem

<lb/>verò nominabant di&longs;cur&longs;um <expan abbr="ill&utilde;">illum</expan>, quo ex vero illo per re&longs;olutionem inuento, <pb pagenum="36"/>o&longs;tendebant conclu&longs;ionem. </s>

<s>Porrò Diogenes Laert. huius re&longs;olutionis in

<lb/>uentorem facit Platonem: à quo eam Leodamas Tha&longs;ius didicit, cuius be

<lb/>neficio, pluries deinde Geometricas demon&longs;trationes adinuenit. </s>

<s>definitio <lb/><expan abbr="vtriu&longs;q;">vtriu&longs;que</expan> e&longs;t a pud Euclidem ad primam propo&longs;. </s>

<s>13. Elem. iuxta tran&longs;latio<lb/>nem Zamb<gap/>rti, & Commandini; vbi etiam <expan abbr="quinq;">quinque</expan> priora theoremata, pri<lb/>mò per re&longs;olutionem, deinde per compo&longs;itionem demon&longs;trantur, quæ tan<lb/>quam per&longs;picua exempla rei propo&longs;itæ in&longs;eruire po&longs;&longs;unt. </s>

<s>&longs;unt præterea fre<lb/>quentes huiu&longs;modi re&longs;olutiones in operibus Archimedis, Apollonij, & Pap<lb/>pi. </s>

<s>definitio

<lb/><expan abbr="vtriu&longs;q;">vtriu&longs;que</expan> e&longs;t a pud Euclidem ad primam propo&longs;. </s>

<s>13. Elem. iuxta tran&longs;latio

<lb/>nem Zamberti, & Commandini; vbi etiam <expan abbr="quinq;">quinque</expan> priora theoremata, pri

<lb/>mò per re&longs;olutionem, deinde per compo&longs;itionem demon&longs;trantur, quæ tan

<lb/>quam per&longs;picua exempla rei propo&longs;itæ in&longs;eruire po&longs;&longs;unt. </s>

<s>&longs;unt præterea fre

<lb/>quentes huiu&longs;modi re&longs;olutiones in operibus Archimedis, Apollonij, & Pap

<s>extat adhuc liber Datorum Euclidis, qui geometricis re&longs;olutionibus in

<lb/>&longs;eruiebat. </s>

<s>vtinam extarent etiam alij de re&longs;olutione, quorum auxilio non

<lb/>tantopere recentiores Mathematici in inueniendis Demo&longs;trationibus la

<lb/>borarent; hanc re&longs;olutionem, &longs;ic Pappus fu&longs;ius, quam Euclides explicat;

<lb/>re&longs;olutio e&longs;t via à quæ&longs;ito tanquam conce&longs;&longs;o per ea, quæ exip&longs;o con&longs;equun

<lb/>tur ad aliquod certum, & conce&longs;&longs;um: in re&longs;olutione enim id, quod quæritur

<lb/>tanquam factum, & verum &longs;upponentes, quid ex hoc &longs;equatur, con&longs;idera

<lb/>mus, quou&longs;que incidamus in aliquod iam cognitum, vel quod &longs;it è numero

<lb/>principiorum. </s>

<s>extat adhuc liber Datorum Euclidis, qui geometricis re&longs;olutionibus in<lb/>&longs;eruiebat. </s>

<s>Quod quidem erat fignum euidens, quæ&longs;itum quoque verum

<lb/>e&longs;&longs;e. </s>

<s>vtinam extarent etiam alij de re&longs;olutione, quorum auxilio non <lb/>tantopere recentiores Mathematici in inueniendis Demo&longs;trationibus la<lb/>borarent; hanc re&longs;olutionem, &longs;ic Pappus fu&longs;ius, quam Euclides explicat; <lb/>re&longs;olutio e&longs;t via à quæ&longs;ito tanquam conce&longs;&longs;o per ea, quæ exip&longs;o con&longs;equun<lb/>tur ad aliquod certum, & conce&longs;&longs;um: in re&longs;olutione enim id, quod quæritur <lb/>tanquam factum, & verum &longs;upponentes, quid ex hoc &longs;equatur, con&longs;idera<lb/>mus, quou&longs;que incidamus in aliquod iam cognitum, vel quod &longs;it è numero <lb/>principiorum. </s>

<s>Quod quidem erat fignum euidens, quæ&longs;itum quoque verum <lb/>e&longs;&longs;e. </s>

<s>eadem omnino habet Proclus in comm. ad &longs;extam primi elem. </s>

<s>Quod <lb/>porrò Ari&longs;t.

<s>Quod

<lb/>porrò Ari&longs;t. ip&longs;e hanc re&longs;olutionem Mathematicam cognouerit e&longs;&longs;e medij

ip&longs;e hanc re&longs;olutionem Mathematicam cognouerit e&longs;&longs;e medij <lb/>inqui&longs;itionem manife&longs;tum e&longs;t ex cap.

<lb/>inqui&longs;itionem manife&longs;tum e&longs;t ex cap. 3. lib. 3. Ethyc. vbi &longs;ic ait <emph type="italics"/>(Qui enim

<lb/>con&longs;ultat, quærere videtur, & re&longs;oluere prædicto modo, quemadmodum de&longs;igna

3. lib.

<lb/>tiones)<emph.end type="italics"/> vbi per de&longs;ignationes intelligit Geometricas demon&longs;trationes, vt

<lb/>&longs;upra innuimus, & infra probabimus; cum ergo con&longs;ultatio nihil aliud &longs;it,

3. Ethyc. vbi &longs;ic ait <emph type="italics"/>(Qui enim <lb/>con&longs;ultat, quærere videtur, & re&longs;oluere prædicto modo, quemadmodum de&longs;igna<lb/>tiones)<emph.end type="italics"/> vbi per de&longs;ignationes intelligit Geometricas demon&longs;trationes, vt <lb/>&longs;upra innuimus, & infra probabimus; cum ergo con&longs;ultatio nihil aliud &longs;it, <lb/>quam medij idonei ad finem in rebus agendis inqui&longs;itio, eamque dicat e&longs;&longs;e <lb/>&longs;imilem re&longs;olutioni Geometricæ, manife&longs;tum e&longs;t, ip&longs;am <expan abbr="quoq;">quoque</expan> re&longs;olutionem <lb/>e&longs;&longs;e medij in rebus &longs;peculatiuis idonei perue&longs;tigationem. </s>

<lb/>quam medij idonei ad finem in rebus agendis inqui&longs;itio, eamque dicat e&longs;&longs;e

<lb/>&longs;imilem re&longs;olutioni Geometricæ, manife&longs;tum e&longs;t, ip&longs;am <expan abbr="quoq;">quoque</expan> re&longs;olutionem

<lb/>e&longs;&longs;e medij in rebus &longs;peculatiuis idonei perue&longs;tigationem. </s>

<s>Exi&longs;timo igitur <lb/>cum docti&longs;&longs;imis Zabarella, Burana, Toleto, & alijs, Ari&longs;totilem non &longs;olum <lb/>hanc &longs;uam logicam ad mathematicarum &longs;cientiarum typum compegi&longs;&longs;e, <lb/>verum potius imitatum e&longs;&longs;e opus illud Euclidis de re&longs;olutione, atque ex eo <lb/>non &longs;olum plurima exempla Geometrica, verum etiam titulum de&longs;ump&longs;i&longs;&longs;e, <lb/>præ&longs;ertim cum argumentum e&longs;&longs;et ferè idem vtrobique, &longs;ed Ari&longs;t.

intentio <lb/>fuerit accommodare re&longs;olutionem omnibus <expan abbr="&longs;ci&etilde;tijs">&longs;cientijs</expan>; Euclidis verò, & alio<lb/>rum Geometriæ &longs;oli. </s>

<s>hinc patere pote&longs;t, cur hi libri re&longs;olutorij in&longs;cribantur, <lb/>quod &longs;cilicet tradunt methodum, qua valeamus quæ&longs;itum quoduis re&longs;olue<lb/>re, ide&longs;t, ex quæ&longs;ito tanquam vero inue&longs;tare aliquam veritatem, per quam <lb/>deinde propo&longs;itæ quæ&longs;tionis rationem methodo compo&longs;itiua reddamus. </s>

<s>Exi&longs;timo igitur

<lb/>cum docti&longs;&longs;imis Zabarella, Burana, Toleto, & alijs, Ari&longs;totilem non &longs;olum

<s>Et <lb/>verò cum reliquas appellationes Problematis, Theorematis, Propo&longs;itionis, <lb/>definitionum, po&longs;tulatorum, axiomatum, & alia huiu&longs;modi ex Geometri<lb/>cis ad omnes &longs;cientias tran&longs;tulerit, quid ni etiam re&longs;olutionem? </s>

<lb/>hanc &longs;uam logicam ad mathematicarum &longs;cientiarum typum compegi&longs;&longs;e,

<lb/>verum potius imitatum e&longs;&longs;e opus illud Euclidis de re&longs;olutione, atque ex eo

<s>maximè <lb/>verò, quia &longs;i horum lib.

<lb/>non &longs;olum plurima exempla Geometrica, verum etiam titulum de&longs;ump&longs;i&longs;&longs;e,

<lb/>præ&longs;ertim cum argumentum e&longs;&longs;et ferè idem vtrobique, &longs;ed Ari&longs;t.

intentio e&longs;&longs;et docere iam factum &longs;yllogi&longs;mum in &longs;ua <lb/>principia re&longs;oluere, parum e&longs;&longs;et vtilis; imò nec vtilis, &longs;ed &longs;uperfluum quid. <lb/></s>

intentio

<lb/>fuerit accommodare re&longs;olutionem omnibus <expan abbr="&longs;ci&etilde;tijs">&longs;cientijs</expan>; Euclidis verò, & alio

<lb/>rum Geometriæ &longs;oli. </s>

<s>hinc patere pote&longs;t, cur hi libri re&longs;olutorij in&longs;cribantur,

<lb/>quod &longs;cilicet tradunt methodum, qua valeamus quæ&longs;itum quoduis re&longs;olue

<lb/>re, ide&longs;t, ex quæ&longs;ito tanquam vero inue&longs;tare aliquam veritatem, per quam

<lb/>deinde propo&longs;itæ quæ&longs;tionis rationem methodo compo&longs;itiua reddamus. </s>

<s>Et

<lb/>verò cum reliquas appellationes Problematis, Theorematis, Propo&longs;itionis,

<lb/>definitionum, po&longs;tulatorum, axiomatum, & alia huiu&longs;modi ex Geometri

<lb/>cis ad omnes &longs;cientias tran&longs;tulerit, quid ni etiam re&longs;olutionem? </s>

<s>maximè

<lb/>verò, quia &longs;i horum lib.

intentio e&longs;&longs;et docere iam factum &longs;yllogi&longs;mum in &longs;ua

<lb/>principia re&longs;oluere, parum e&longs;&longs;et vtilis; imò nec vtilis, &longs;ed &longs;uperfluum quid.

<s>at verò vbinam docuit hanc re&longs;olutionem? </s>

<s>profecto nullibi. </s>

<s>quid opus e&longs;t <lb/>iam factum &longs;yllogi&longs;mum re&longs;oluere? </s>

<s>quid opus e&longs;t

<lb/>iam factum &longs;yllogi&longs;mum re&longs;oluere? </s>

<s>at verò propo&longs;itam quæ&longs;tionem re&longs;ol<lb/>uere veterum mathematicorum more, hoc opus, hic labor e&longs;t.</s>

<s>at verò propo&longs;itam quæ&longs;tionem re&longs;ol

<lb/>uere veterum mathematicorum more, hoc opus, hic labor e&longs;t.</s>

<s>Hanc porrò re&longs;olutionem attendendam e&longs;&longs;e primò penes formam, quam <lb/>docet primis duobus analyticis; &longs;ecundò penes materiam, quam tradit duo<pb pagenum="37"/>bus vltimis, non prætereundum. </s>

<s>Hanc porrò re&longs;olutionem attendendam e&longs;&longs;e primò penes formam, quam

<lb/>docet primis duobus analyticis; &longs;ecundò penes materiam, quam tradit duo<pb pagenum="37"/>bus vltimis, non prætereundum. </s>

<s>reliquas duas logicæ partes, Topicam &longs;ci<lb/>licet, & Elenchos, quæ &longs;yllogi&longs;mos probabilem, & apparentem docent, no<lb/>luit appellare re&longs;olutorios, quamuis inuentionem mediorum doceant, quia <lb/>iam mos i&longs;te inoleuerat apud Philo&longs;ophos, & Mathematicos, vt illa &longs;ola <lb/>pars, quæ ex materia nece&longs;&longs;aria doceret &longs;yllogi&longs;mum demon&longs;tratiuum con<lb/>&longs;truere, diceretur re&longs;olutio: cum Mathematici, qui primi de re&longs;olutione <lb/>&longs;crip&longs;erunt, talem materiam &longs;olum con&longs;iderent.</s>

<s>reliquas duas logicæ partes, Topicam &longs;ci

<lb/>licet, & Elenchos, quæ &longs;yllogi&longs;mos probabilem, & apparentem docent, no

<lb/>luit appellare re&longs;olutorios, quamuis inuentionem mediorum doceant, quia

<lb/>iam mos i&longs;te inoleuerat apud Philo&longs;ophos, & Mathematicos, vt illa &longs;ola

<lb/>pars, quæ ex materia nece&longs;&longs;aria doceret &longs;yllogi&longs;mum demon&longs;tratiuum con

<lb/>&longs;truere, diceretur re&longs;olutio: cum Mathematici, qui primi de re&longs;olutione

<lb/>&longs;crip&longs;erunt, talem materiam &longs;olum con&longs;iderent.</s>

Line 2074

Line 2343

23. &longs;ecti primi lib.

1. <emph type="italics"/>(Vt quod diameter incommen&longs;urabilis eo, quod <lb/>imparia æqualia paribus fiant, &longs;i fuerit po&longs;ita commen&longs;ur abilis. </s>

1. <emph type="italics"/>(Vt quod diameter incommen&longs;urabilis eo, quod

<lb/>imparia æqualia paribus fiant, &longs;i fuerit po&longs;ita commen&longs;ur abilis. </s>

<s>æqualia igitur fieri <lb/>imparia paribus ratiocinantur, diametrum vtrò incommen&longs;urabilem e&longs;&longs;e ex &longs;uppo<lb/>&longs;itione <expan abbr="mon&longs;trãt">mon&longs;trant</expan>, quoniam fal&longs;um accidit propter contradictionem)<emph.end type="italics"/> Euclides pri<lb/>mis duabus definitionibus 10. elem. </s>

<s>æqualia igitur fieri

<lb/>imparia paribus ratiocinantur, diametrum vtrò incommen&longs;urabilem e&longs;&longs;e ex &longs;uppo

<lb/>&longs;itione <expan abbr="mon&longs;trãt">mon&longs;trant</expan>, quoniam fal&longs;um accidit propter contradictionem)<emph.end type="italics"/> Euclides pri

<lb/>mis duabus definitionibus 10. elem. </s>

<s>definit, quæ nam &longs;int magnitudines <lb/>commen&longs;. </s>

<s>definit, quæ nam &longs;int magnitudines

<lb/>commen&longs;. </s>

<s>& quæ incommen&longs;. </s>

<s>&longs;ic; commen&longs;. </s>

<s>magnitudines dicuntur, quas <lb/><figure id="fig3"/><lb/>eadem men&longs;ura metitur, vt &longs;i fuerint duæ magnitu<lb/>dines, A, & B, quas eadem men&longs;ura C, ide&longs;t quan<lb/>titas C, metiatur, ide&longs;t <expan abbr="quãtitas">quantitas</expan> C, applicata quan<lb/>titati A, & per ip&longs;am aliquoties replicata ip&longs;am ad<lb/>æquatè ab&longs;umat, vt &longs;i linea C, quinquies &longs;uper li<lb/>neam A, replicata eam præcisè, & perfectè omninò <lb/>adæquaret: & eadem linea C, applicata lineæ B, & &longs;uper illam ter, v.g. re<lb/>petita ip&longs;am con&longs;umeret, diceretur <expan abbr="vtranq;">vtranque</expan> metiri, & proinde duas lineas <lb/>A, & B, e&longs;&longs;e comm. </s>

<s>magnitudines dicuntur, quas

<lb/>eadem men&longs;ura metitur, vt &longs;i fuerint duæ magnitu

<lb/>dines, A, & B, quas eadem men&longs;ura C, ide&longs;t quan

<lb/>titas C, metiatur, ide&longs;t <expan abbr="quãtitas">quantitas</expan> C, applicata quan

<lb/>titati A, & per ip&longs;am aliquoties replicata ip&longs;am ad

<lb/>æquatè ab&longs;umat, vt &longs;i linea C, quinquies &longs;uper li

<lb/>neam A, replicata eam præcisè, & perfectè omninò

<lb/>adæquaret: & eadem linea C, applicata lineæ B, & &longs;uper illam ter, v.g. re

<lb/>petita ip&longs;am con&longs;umeret, diceretur <expan abbr="vtranq;">vtranque</expan> metiri, & proinde duas lineas

<lb/>A, & B, e&longs;&longs;e comm. definit po&longs;tea <expan abbr="incomm&etilde;&longs;">incommen&longs;</expan> hoc modo, incomm, autem, qua

<lb/>rum nullam contingit communem men&longs;uram reperiri; vt &longs;i duarum linea

<lb/>rum, A, B, nunquam po&longs;&longs;et reperiri aliqua men&longs;u

<lb/>ra, quæ <expan abbr="vtranq;">vtranque</expan> adæquatè metiretur, v. g. &longs;i linea

<lb/>C, men&longs;uraret A, quater &longs;umpta, ter autem &longs;umpta

<lb/>non adæquaret omnino <expan abbr="lineã">lineam</expan> B, &longs;ed deficeret, vel ex

<lb/>cederet aliquantulum, <expan abbr="atq;">atque</expan> hoc fieret in quauis alia

<lb/>men&longs;ura, loco ip&longs;ius C, a&longs;&longs;umpta, &longs;iue maior, &longs;iue

<lb/>minor ip&longs;a C, vt <expan abbr="vtranq;">vtranque</expan> nunquam perfectè metiretur, e&longs;&longs;ent duæ illæ lineæ

<lb/>incommen&longs;. </s>

<s>definit po&longs;tea <expan abbr="incomm&etilde;&longs;">incommen&longs;</expan> hoc modo, incomm, autem, qua<lb/>rum nullam contingit communem men&longs;uram reperiri; vt &longs;i duarum linea<lb/><figure id="fig4"/><lb/>rum, A, B, nunquam po&longs;&longs;et reperiri aliqua men&longs;u<lb/>ra, quæ <expan abbr="vtranq;">vtranque</expan> adæquatè metiretur, v. g. </s>

<s>&longs;i linea <lb/>C, men&longs;uraret A, quater &longs;umpta, ter autem &longs;umpta <lb/>non adæquaret omnino <expan abbr="lineã">lineam</expan> B, &longs;ed deficeret, vel ex<lb/>cederet aliquantulum, <expan abbr="atq;">atque</expan> hoc fieret in quauis alia <lb/>men&longs;ura, loco ip&longs;ius C, a&longs;&longs;umpta, &longs;iue maior, &longs;iue <lb/>minor ip&longs;a C, vt <expan abbr="vtranq;">vtranque</expan> nunquam perfectè metiretur, e&longs;&longs;ent duæ illæ lineæ <lb/>incommen&longs;. </s>

<s>Extare porrò tales lineas, & &longs;uperficies, & corpora, <expan abbr="ea&qacute;">eaque</expan>; quam<lb/>plurima, ac penè infinita ex 10. Elem. manife&longs;tum e&longs;t. </s>

<s>inuentum autem hu<lb/>ius a&longs;ymmetriæ, quod Pythagoricis veteres attribuunt, mihi &longs;emper vi&longs;um <lb/>e&longs;t omni maius admiratione, cum nulla experientia, <expan abbr="nullus&qacute;">nullusque</expan>; effectus in ip<lb/>&longs;ius cognitionem potuerit pri&longs;cos illos Geometras inducere. </s>

<s>Extare porrò tales lineas, & &longs;uperficies, & corpora, <expan abbr="ea&qacute;">eaque</expan>; quam

<lb/>plurima, ac penè infinita ex 10. Elem. manife&longs;tum e&longs;t. </s>

<s>Quapropter <lb/>non immeritò diuinus ille Plato lib.

<s>inuentum autem hu

<lb/>ius a&longs;ymmetriæ, quod Pythagoricis veteres attribuunt, mihi &longs;emper vi&longs;um

<lb/>e&longs;t omni maius admiratione, cum nulla experientia, <expan abbr="nullus&qacute;">nullusque</expan>; effectus in ip

<lb/>&longs;ius cognitionem potuerit pri&longs;cos illos Geometras inducere. </s>

<s>Quapropter

<lb/>non immeritò diuinus ille Plato lib.

7. de legib. </s>

<s>huius a&longs;ymmetriæ ignora<lb/>tionem, adeo dete&longs;tatus e&longs;t, vt eam non hominum, &longs;ed &longs;uum, pecorumque <lb/>ignorantiam cen&longs;uerit. </s>

<s>huius a&longs;ymmetriæ ignora

<lb/>tionem, adeo dete&longs;tatus e&longs;t, vt eam non hominum, &longs;ed &longs;uum, pecorumque

<lb/>ignorantiam cen&longs;uerit. </s>

<s>inter lineas incommen&longs;. </s>

<s>&longs;unt diameter, & latus eiu&longs;<lb/>dem quadrati, quia nulla pote&longs;t reperiri men&longs;ura quantumuis exigua, vti <lb/><figure id="fig5"/><lb/>e&longs;t lineola E, in præ&longs;enti quadrato, etiam&longs;i illam in <lb/>infinitum &longs;ubdiuidas, quæ <expan abbr="vtranq;">vtranque</expan> lineam, diame<lb/>trum &longs;cilicet A C, & latus quoduis ex quatuor, v. g. <lb/></s>

<s>&longs;unt diameter, & latus eiu&longs;

<lb/>dem quadrati, quia nulla pote&longs;t reperiri men&longs;ura quantumuis exigua, vti

<lb/>e&longs;t lineola E, in præ&longs;enti quadrato, etiam&longs;i illam in

<lb/>infinitum &longs;ubdiuidas, quæ <expan abbr="vtranq;">vtranque</expan> lineam, diame

<lb/>trum &longs;cilicet A C, & latus quoduis ex quatuor, v. g.

<lb/>latus B C, præcisè omnino metiatur. </s>

<s>latus B C, præcisè omnino metiatur. </s>

<s>theorema <lb/>i&longs;tud demon&longs;tratur in vltima 10. Elem. eodem me<lb/>dio, quod ab Ari&longs;totele hic innuitur; Euclides ex <lb/>&longs;uppo&longs;itione alterius partis contradictionis ip&longs;ius <pb pagenum="38"/>propo&longs;itionis, quæfal&longs;a e&longs;t, nimirum &longs;uppo&longs;ito prædictas lineas e&longs;&longs;e comm. <lb/>deducit ad impo&longs;&longs;ibile, &longs;iue, vt ait hic Ari&longs;t.

<s>theorema

<lb/>i&longs;tud demon&longs;tratur in vltima 10. Elem. eodem me

<lb/>dio, quod ab Ari&longs;totele hic innuitur; Euclides ex

<lb/>&longs;uppo&longs;itione alterius partis contradictionis ip&longs;ius <pb pagenum="38"/>propo&longs;itionis, quæfal&longs;a e&longs;t, nimirum &longs;uppo&longs;ito prædictas lineas e&longs;&longs;e comm.

<lb/>deducit ad impo&longs;&longs;ibile, &longs;iue, vt ait hic Ari&longs;t.

fal&longs;um ratiocinatur, quod &longs;ci<lb/>licet idem numerus e&longs;&longs;et par, & impar, quod Ari&longs;t. &longs;ignificat, quando ait, <lb/>imparia æqualia paribus fiunt. </s>

fal&longs;um ratiocinatur, quod &longs;ci

<lb/>licet idem numerus e&longs;&longs;et par, & impar, quod Ari&longs;t. &longs;ignificat, quando ait,

<lb/>imparia æqualia paribus fiunt. </s>

<s>ex quo ab&longs;urdo deducitur fal&longs;am e&longs;&longs;e prædi<lb/>ctam &longs;uppo&longs;itionem, quæ a&longs;truebat e&longs;&longs;e comm. & proinde altera pars con<lb/>tradictionis, quæ e&longs;t, e&longs;&longs;e incomm. </s>

<s>ex quo ab&longs;urdo deducitur fal&longs;am e&longs;&longs;e prædi

<lb/>ctam &longs;uppo&longs;itionem, quæ a&longs;truebat e&longs;&longs;e comm. & proinde altera pars con

<lb/>tradictionis, quæ e&longs;t, e&longs;&longs;e incomm. vera a&longs;truitur. </s>

<s>vera a&longs;truitur. </s>

<s>ex quibus &longs;atis videtur ex<lb/>plicari hic locus. </s>

<s>videas igitur, quàm leuiter nonnulli no&longs;træ tempe&longs;tatis <lb/>ageometreti i&longs;tud exponant, dicentes diametrum e&longs;&longs;e incomm. co&longs;tæ, nihil <lb/>aliud &longs;igni&longs;icare, quam diametrum e&longs;&longs;e longiorem co&longs;ta, qua expo&longs;itione <lb/>nihil ineptius. </s>

<s>ex quibus &longs;atis videtur ex

<lb/>plicari hic locus. </s>

<s>videas igitur, quàm leuiter nonnulli no&longs;træ tempe&longs;tatis

<lb/>ageometreti i&longs;tud exponant, dicentes diametrum e&longs;&longs;e incomm. co&longs;tæ, nihil

<lb/>aliud &longs;igni&longs;icare, quam diametrum e&longs;&longs;e longiorem co&longs;ta, qua expo&longs;itione

<lb/>nihil ineptius. </s>

<s>Aduerte tandem figuram vulgatæ editionis e&longs;&longs;e ineptam, <lb/>cum habeat duo quadrata alterum &longs;uper diametro alterius, quorum maius <lb/>&longs;uperuacaneum e&longs;t.</s>

<s>Aduerte tandem figuram vulgatæ editionis e&longs;&longs;e ineptam,

<lb/>cum habeat duo quadrata alterum &longs;uper diametro alterius, quorum maius

<lb/>&longs;uperuacaneum e&longs;t.</s>

Line 2138

Line 2459

<s>Et cap.

24. &longs;ecti primi libri primi <emph type="italics"/>(Sed magis efficitur manife&longs;tum in de&longs;cri<lb/>ptionibus, vt quod æquicruris, qui ad ba&longs;im æquales &longs;int, ad centrum ductæ A B, <lb/>A C, &longs;i igitur æqualem accipiat A G, angulum ip&longs;i A B D, non omnino ex. </s>

24. &longs;ecti primi libri primi <emph type="italics"/>(Sed magis efficitur manife&longs;tum in de&longs;cri

<lb/>ptionibus, vt quod æquicruris, qui ad ba&longs;im æquales &longs;int, ad centrum ductæ A B,

<s>&longs;timans <lb/>æquales, qui &longs;emicirculorum, & rur&longs;us G, ip&longs;i D, non omnem a&longs;&longs;umens eum, qui &longs;e<lb/>cti. </s>

<lb/>A C, &longs;i igitur æqualem accipiat A G, angulum ip&longs;i A B D, non omnino ex. </s>

<s>amplius ab æqualibus existentibus totis angulis, & ablatorum æquales e&longs;&longs;e re<lb/>tiquos E, F, quod ex principio petet, ni&longs;i acceperit ab æqualibus demptis æqualia <lb/>derelinqui.)<emph.end type="italics"/> Primum &longs;cias characteres vulgatæ editionis, vna cum figura ip<lb/>&longs;is re&longs;pondente, e&longs;&longs;e mendo&longs;os; propterea ex textu græco vtrunque corri<lb/>gendum putaui in hunc, quem vidi&longs;ti modum. </s>

<s>&longs;timans

<lb/>æquales, qui &longs;emicirculorum, & rur&longs;us G, ip&longs;i D, non omnem a&longs;&longs;umens eum, qui &longs;e

<s>Secundo, per de&longs;criptiones <lb/>Ari&longs;t.

intelligere <expan abbr="demõ&longs;trationes">demon&longs;trationes</expan> Geometricas &longs;upra diximus, quod ex hoc <lb/>loco euidenter confirmatur, vbi manife&longs;tè loco de&longs;criptionis &longs;upponit li<lb/>nearem demon&longs;trationem. </s>

<s>amplius ab æqualibus existentibus totis angulis, & ablatorum æquales e&longs;&longs;e re

<lb/>tiquos E, F, quod ex principio petet, ni&longs;i acceperit ab æqualibus demptis æqualia

<lb/>derelinqui.)<emph.end type="italics"/> Primum &longs;cias characteres vulgatæ editionis, vna cum figura ip

<lb/>&longs;is re&longs;pondente, e&longs;&longs;e mendo&longs;os; propterea ex textu græco vtrunque corri

<lb/>gendum putaui in hunc, quem vidi&longs;ti modum. </s>

<s>Secundo, per de&longs;criptiones

<lb/>Ari&longs;t.

intelligere <expan abbr="demõ&longs;trationes">demon&longs;trationes</expan> Geometricas &longs;upra diximus, quod ex hoc

<lb/>loco euidenter confirmatur, vbi manife&longs;tè loco de&longs;criptionis &longs;upponit li

<lb/>nearem demon&longs;trationem. </s>

<s>In hoc <expan abbr="itaq;">itaque</expan> exemplo vult Ari&longs;t.

illud demon<lb/>&longs;trare, quod Euclides in 5. primi o&longs;tendit, alio tamen modo, &longs;cilicet I&longs;o&longs;ce<lb/>lium triangulorum, qui ad ba&longs;im &longs;unt anguli, inter &longs;e &longs;unt æquales. </s>

illud demon

<lb/>&longs;trare, quod Euclides in 5. primi o&longs;tendit, alio tamen modo, &longs;cilicet I&longs;o&longs;ce

<s>e&longs;t au<lb/>tem figura in omnibus textibus deprauata, quam &longs;ic puto <expan abbr="rè&longs;t&imacr;tuendam">rè&longs;tintuendam</expan> e&longs;&longs;e <lb/>ex quodam græco codice, qui characteres hoc modo appo&longs;uerat. </s>

<lb/>lium triangulorum, qui ad ba&longs;im &longs;unt anguli, inter &longs;e &longs;unt æquales. </s>

<s>&longs;it I&longs;o&longs;ce<lb/><figure id="fig6"/><lb/>les C A B, cuius ba&longs;is C B, Dico angulos &longs;upra ba&longs;im, <lb/>in quibus literæ E F, e&longs;&longs;e inuicem æquales. </s>

<s>e&longs;t au

<lb/>tem figura in omnibus textibus deprauata, quam &longs;ic puto <expan abbr="rè&longs;t&imacr;tuendam">rè&longs;tintuendam</expan> e&longs;&longs;e

<s>facto centro <lb/>in A, de&longs;cribatur circulus A B C, tran&longs;iens per puncta <lb/>C B, iam &longs;ic. </s>

<lb/>ex quodam græco codice, qui characteres hoc modo appo&longs;uerat. </s>

<s>omnes anguli &longs;emicirculi &longs;unt æquales in<lb/>ter &longs;e, ergo anguli A C G, A B D, &longs;unt æquales. </s>

<s>&longs;it I&longs;o&longs;ce

<s>Præte<lb/>rea cùm anguli ciu&longs;dem &longs;ectionis &longs;int æquales ad inui<lb/>cem, erunt anguli &longs;ectionis C B D G, nimirum anguli, <lb/>in quibus &longs;unt G, & D, inter &longs;e æquales: <expan abbr="cum&qacute;">cumque</expan>; hi duo <lb/>anguli &longs;ectionis &longs;int partes <expan abbr="angulor&utilde;">angulorum</expan> &longs;emicirculi A C G, <lb/>A B D, &longs;i illi ab his auferantur, auferuntur æquales anguli ab æqualibus an<lb/>gulis, ergo anguli, qui remanent, &longs;cilicet E, & F, erunt æquales, quod erat <lb/>demon&longs;trandum. </s>

<lb/>les C A B, cuius ba&longs;is C B, Dico angulos &longs;upra ba&longs;im,

<lb/>in quibus literæ E F, e&longs;&longs;e inuicem æquales. </s>

<s>facto centro

<lb/>in A, de&longs;cribatur circulus A B C, tran&longs;iens per puncta

<lb/>C B, iam &longs;ic. </s>

<s>omnes anguli &longs;emicirculi &longs;unt æquales in

<lb/>ter &longs;e, ergo anguli A C G, A B D, &longs;unt æquales. </s>

<s>Præte

<lb/>rea cùm anguli ciu&longs;dem &longs;ectionis &longs;int æquales ad inui

<lb/>cem, erunt anguli &longs;ectionis C B D G, nimirum anguli,

<lb/>in quibus &longs;unt G, & D, inter &longs;e æquales: <expan abbr="cum&qacute;">cumque</expan>; hi duo

<lb/>anguli &longs;ectionis &longs;int partes <expan abbr="angulor&utilde;">angulorum</expan> &longs;emicirculi A C G,

<lb/>A B D, &longs;i illi ab his auferantur, auferuntur æquales anguli ab æqualibus an

<lb/>gulis, ergo anguli, qui remanent, &longs;cilicet E, & F, erunt æquales, quod erat

<lb/>demon&longs;trandum. </s>

<s>hinc Ari&longs;t.

infert manife&longs;tum e&longs;&longs;e oportere in omni &longs;yllo<lb/>gi&longs;mo, reperiri vniuer&longs;ales, & affirmatiuas propo&longs;itiones, vt Factum e&longs;t in <lb/>præcedenti aliter e&longs;&longs;et petitio principij. </s>

infert manife&longs;tum e&longs;&longs;e oportere in omni &longs;yllo

<lb/>gi&longs;mo, reperiri vniuer&longs;ales, & affirmatiuas propo&longs;itiones, vt Factum e&longs;t in

<lb/>præcedenti aliter e&longs;&longs;et petitio principij. </s>

<s>Quænam vero &longs;it æqualitas, quam <lb/>Geometræ con&longs;iderant, infra cap.

<s>Quænam vero &longs;it æqualitas, quam

<lb/>Geometræ con&longs;iderant, infra cap. 1. &longs;ecti 3. explicabicur.</s>

1. &longs;ecti 3. explicabicur.</s>

Line 2178

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2. &longs;ecti 2. lib.

1. <emph type="italics"/>(Secundum veritatem quidem ex ijs, quæ &longs;ecundum <lb/>veritatem de&longs;cribuntur ine&longs;&longs;e, ad dialecticos autem &longs;yllogi&longs;mos ex propo&longs;itionibus <lb/>&longs;ecundum opinionem)<emph.end type="italics"/> verba illa; ex ijs, quæ <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> veritatem de&longs;cribuntur <pb pagenum="39"/>ine&longs;&longs;e; &longs;ic græcè, <foreign lang="greek">e/a tw_n xata\ aleiq/ei/an diagegramme/non,</foreign> vbi manife&longs;tè vtitur <lb/>verbo, De&longs;cribere, per quod &longs;uperius annotauimus apud Ari&longs;t. &longs;ignificari <lb/>Geometricas demon&longs;trationes, nam eas opponit dialecticis &longs;yllogi&longs;mis, &longs;e<lb/>quentibus verbis, cum dixit (ad diale cticos autem &longs;yllogi&longs;mos ex propo&longs;i<lb/>tionibus &longs;ecundum opinionem) hac adhibita con&longs;ideratione, quam inter<lb/>pres non videtur adhibui&longs;&longs;e, &longs;en&longs;us huius loci non erit ob&longs;curus.</s>

1. <emph type="italics"/>(Secundum veritatem quidem ex ijs, quæ &longs;ecundum

<lb/>veritatem de&longs;cribuntur ine&longs;&longs;e, ad dialecticos autem &longs;yllogi&longs;mos ex propo&longs;itionibus

<lb/>&longs;ecundum opinionem)<emph.end type="italics"/> verba illa; ex ijs, quæ <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> veritatem de&longs;cribuntur <pb pagenum="39"/>ine&longs;&longs;e; &longs;ic græcè, <foreign lang="greek">e/a tw_n xata\ aleiq/ei/an diagegramme/non,</foreign> vbi manife&longs;tè vtitur

<lb/>verbo, De&longs;cribere, per quod &longs;uperius annotauimus apud Ari&longs;t. &longs;ignificari

<lb/>Geometricas demon&longs;trationes, nam eas opponit dialecticis &longs;yllogi&longs;mis, &longs;e

<lb/>quentibus verbis, cum dixit (ad diale cticos autem &longs;yllogi&longs;mos ex propo&longs;i

<lb/>tionibus &longs;ecundum opinionem) hac adhibita con&longs;ideratione, quam inter

<lb/>pres non videtur adhibui&longs;&longs;e, &longs;en&longs;us huius loci non erit ob&longs;curus.</s>

<s>Ex eodem loco paulo po&longs;t <emph type="italics"/>(Quare principia quidem, quæ &longs;ecundum <expan abbr="vnum-quodq;">vnum<lb/>quodque</expan> &longs;unt experimenti est tradere: dico autem, vt a&longs;trologicam experientiam <lb/>a&longs;trologicæ &longs;cientiæ: acceptis enim apparentibus <expan abbr="&longs;uffici&etilde;ter">&longs;ufficienter</expan>, ita inuentæ &longs;unt a&longs;tro<lb/>logicæ demonstrationes)<emph.end type="italics"/> Cum rationem tradat inueniendorum mediorum ad <lb/>quodlibet problema demon&longs;trandum; nunc docet, non omnia in &longs;eientijs <lb/>po&longs;&longs;e probari, aut demou&longs;trari: principia enim &longs;cientiarum <expan abbr="nõ">non</expan> demon&longs;tran<lb/>tur, &longs;ed &longs;ola experientia manife&longs;ta &longs;unt; vt patet in A&longs;tronomia, quæ ab ex<lb/>perientia &longs;ua &longs;olet &longs;tabilire principia: principijs autem <expan abbr="experim&etilde;to">experimento</expan> con&longs;ti<lb/>tutis ex ip&longs;is reliqua problemata <expan abbr="demon&longs;trãtur">demon&longs;trantur</expan>. </s>

<s>Ex eodem loco paulo po&longs;t <emph type="italics"/>(Quare principia quidem, quæ &longs;ecundum <expan abbr="vnum-quodq;">vnum

<lb/>quodque</expan> &longs;unt experimenti est tradere: dico autem, vt a&longs;trologicam experientiam

<s>duo autem &longs;unt apud a&longs;tro<lb/>nomos genera experimenti, primum dicitur Phænomena, ide&longs;t, <expan abbr="appar&etilde;tiæ">apparentiæ</expan>; <lb/>& &longs;unt ea, quæ vulgo omnibus patent, vt Solem oriri, & occidere; a&longs;tra fer<lb/>ri circulariter, diem augeri modo, modo minui: & his &longs;imilia. </s>

<lb/>a&longs;trologicæ &longs;cientiæ: acceptis enim apparentibus <expan abbr="&longs;uffici&etilde;ter">&longs;ufficienter</expan>, ita inuentæ &longs;unt a&longs;tro

<lb/>logicæ demonstrationes)<emph.end type="italics"/> Cum rationem tradat inueniendorum mediorum ad

<s>alterum ge<lb/>nus dicitur ob&longs;eruationes, quæ tantummodo a&longs;tronomiæ peritis per ob&longs;er<lb/>uationem innote&longs;cunt, vt Solem inæqualiter ferri proprio motu per Zodia<lb/>cum; aliquando maiorem, aliquando minorem videri; plures dies immo<lb/>rari citra æquatiorem in parte Zodiaci boreali, quam in altera vltra æqua<lb/>torem au&longs;trali. </s>

<lb/>quodlibet problema demon&longs;trandum; nunc docet, non omnia in &longs;eientijs

<lb/>po&longs;&longs;e probari, aut demou&longs;trari: principia enim &longs;cientiarum <expan abbr="nõ">non</expan> demon&longs;tran

<lb/>tur, &longs;ed &longs;ola experientia manife&longs;ta &longs;unt; vt patet in A&longs;tronomia, quæ ab ex

<lb/>perientia &longs;ua &longs;olet &longs;tabilire principia: principijs autem <expan abbr="experim&etilde;to">experimento</expan> con&longs;ti

<lb/>tutis ex ip&longs;is reliqua problemata <expan abbr="demon&longs;trãtur">demon&longs;trantur</expan>. </s>

<s>duo autem &longs;unt apud a&longs;tro

<lb/>nomos genera experimenti, primum dicitur Phænomena, ide&longs;t, <expan abbr="appar&etilde;tiæ">apparentiæ</expan>;

<lb/>& &longs;unt ea, quæ vulgo omnibus patent, vt Solem oriri, & occidere; a&longs;tra fer

<lb/>ri circulariter, diem augeri modo, modo minui: & his &longs;imilia. </s>

<s>alterum ge

<lb/>nus dicitur ob&longs;eruationes, quæ tantummodo a&longs;tronomiæ peritis per ob&longs;er

<lb/>uationem innote&longs;cunt, vt Solem inæqualiter ferri proprio motu per Zodia

<lb/>cum; aliquando maiorem, aliquando minorem videri; plures dies immo

<lb/>rari citra æquatiorem in parte Zodiaci boreali, quam in altera vltra æqua

<lb/>torem au&longs;trali. </s>

<s>dies naturales e&longs;&longs;e inuicem inæquales, &c. </s>

<s>ex quibus deinde <lb/>ponunt eccentricos, & augem, ad &longs;aluandas tum apparentias, tum ob&longs;erua<lb/>tiones; & hac ratione a&longs;trologica &longs;cientia paulatim reperta e&longs;t, ac in dies <lb/>reperitur.</s>

<s>ex quibus deinde

<lb/>ponunt eccentricos, & augem, ad &longs;aluandas tum apparentias, tum ob&longs;erua

<lb/>tiones; & hac ratione a&longs;trologica &longs;cientia paulatim reperta e&longs;t, ac in dies

<lb/>reperitur.</s>

Line 2202

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3. &longs;ecti 2. lib.

1. <emph type="italics"/>(Vt an ne diameter incomm.)<emph.end type="italics"/> loquitur de a&longs;ymme<lb/>tria diametri, & co&longs;tæ eiu&longs;dem quadrati, de qua fusè egimus &longs;uperius in <lb/>cap.

1. <emph type="italics"/>(Vt an ne diameter incomm.)<emph.end type="italics"/> loquitur de a&longs;ymme

<lb/>tria diametri, & co&longs;tæ eiu&longs;dem quadrati, de qua fusè egimus &longs;uperius in

<lb/>cap.

23. &longs;ecti 1. huius libri; quæ &longs;i repetantur, optimè hunc <expan abbr="loc&utilde;">locum</expan> declarant.</s>

Line 2214

Line 2592

1. &longs;ecti 3. lib.

1. <emph type="italics"/>(Sit A, duo recti, in quo B, triangulus, in quo C, <lb/>æquicrus, ip&longs;i <expan abbr="itaq;">itaque</expan> C, ine&longs;t A. per B; ip&longs;i vero B, non amplius per aliud, per &longs;e <lb/>namque triangulus habet duos rectos)<emph.end type="italics"/> nullum aliud exemplum tam frequenter <lb/>v&longs;urpat Philo&longs;ophus, quam i&longs;tud ex Mathematicis de&longs;umptum de triangu<lb/>lo, &longs;cilicet, omnis triangulus habet tres angulos æquales duobus rectis an<lb/>gulis, cuius Demon&longs;tratio e&longs;t in 32. primi Elem.

1. <emph type="italics"/>(Sit A, duo recti, in quo B, triangulus, in quo C,

<lb/>æquicrus, ip&longs;i <expan abbr="itaq;">itaque</expan> C, ine&longs;t A. per B; ip&longs;i vero B, non amplius per aliud, per &longs;e

quod, vt probè intelliga<lb/>tur, explicandum e&longs;t penes quid attendenda &longs;it æqualitas inter angulum, & <lb/>angulum, quod facile a&longs;&longs;equemur, &longs;i meminerimus angulum e&longs;&longs;e in clinatio<lb/>nem illam, quam duæ lineæ non in directum po&longs;itæ faciunt: &longs;iue etiam (vt <lb/>melius percipiamus) angulum e&longs;&longs;e acumen illud, &longs;iue mucronem <expan abbr="ill&utilde;">illum</expan>, quem <lb/>duæ lineæ non in directum con&longs;titutæ faciunt, vt duarum linearum A B, A C, <lb/><figure id="fig7"/><lb/>inclinatio in puncto A, &longs;iue acumen illud, &longs;iue mucro, <lb/>e&longs;t ratio anguli. </s>

<lb/>namque triangulus habet duos rectos)<emph.end type="italics"/> nullum aliud exemplum tam frequenter

<lb/>v&longs;urpat Philo&longs;ophus, quam i&longs;tud ex Mathematicis de&longs;umptum de triangu

<s>&longs;olum igitur duo anguli erunt æqua<lb/>les, <expan abbr="quãdo">quando</expan> vnius acumen æquale erit acumini alterius; <lb/>etiam &longs;i lineæ con&longs;tituentes vnum angulum &longs;int lon<lb/>giores lineis alterum angulum con&longs;tituentibus, quia <lb/>quantitas anguli non attenditur penes longitudinem <pb pagenum="40"/><expan abbr="linear&utilde;">linearum</expan>, &longs;ed penes inclinationem, & mucronem, quem faciunt: vnde etiam&longs;i <lb/>duæ lineæ prædictæ A B, A C, productæ, &longs;iue etiam decurtatæ fuerint, dum<lb/>modo &longs;itus, &longs;iue po&longs;itio ip&longs;arum, quam ad inuicem habent, non varietur, <lb/>erit &longs;emper eadem quantitas anguli A. </s>

<lb/>lo, &longs;cilicet, omnis triangulus habet tres angulos æquales duobus rectis an

<lb/>gulis, cuius Demon&longs;tratio e&longs;t in 32. primi Elem.

<s>Aduertendum præterea rationem <lb/>anguli non po&longs;&longs;e &longs;aluari in &longs;olo puncto A, in quo lineæ concurrunt, &longs;ed ne<lb/>ce&longs;&longs;ariam e&longs;&longs;e aliquam quantitatem, quamuis exiguam, linearum A B, A C. <lb/></s>

quod, vt probè intelliga

<s>Notandum etiam, quod in nominatione angulorum, quæ fit per tres lite<lb/>ras, &longs;emper literam illam e&longs;&longs;e medio loco proferendam, quæ ad acumen ip<lb/>&longs;um po&longs;ita e&longs;t, vt in &longs;uperiori, litera A, debet &longs;emper media proferri, dicen<lb/>do angulum B A C, &longs;iue C A B, <expan abbr="n&utilde;quam">nunquam</expan> tamen licet dicere angulum A C B, <lb/>vel C B A. </s>

<lb/>tur, explicandum e&longs;t penes quid attendenda &longs;it æqualitas inter angulum, &

<lb/>angulum, quod facile a&longs;&longs;equemur, &longs;i meminerimus angulum e&longs;&longs;e in clinatio

<s>Porrò quemadmodum vnus angulus vni angulo æqualis e&longs;t, ita <lb/><expan abbr="aliquãdo">aliquando</expan> duo anguli &longs;unt vni angulo æquales, vt patet, &longs;i vnus angulus, v.g. <lb/>angulus B A C, diuidatur in duos angulos à linea A D. tunc enim duo angu<lb/><figure id="fig8"/><lb/>li partiales B A D, D A C, erunt æquales totali angulo <lb/>B A C, cum partes omnes &longs;imul &longs;umptæ &longs;int &longs;uo toti æqua<lb/>les. </s>

<lb/>nem illam, quam duæ lineæ non in directum po&longs;itæ faciunt: &longs;iue etiam (vt

<lb/>melius percipiamus) angulum e&longs;&longs;e acumen illud, &longs;iue mucronem <expan abbr="ill&utilde;">illum</expan>, quem

<lb/>duæ lineæ non in directum con&longs;titutæ faciunt, vt duarum linearum A B, A C,

<s>pariter tres anguli po&longs;&longs;unt æquari & vni, & duobus <lb/>alijs angulis, quando nimirum a cumina, &longs;iue mucrones il<lb/>li &longs;imul ad vnum punctum con&longs;tituti <expan abbr="adæquar&etilde;tur">adæquarentur</expan> mucro<lb/>niilli, quem con&longs;tituerent alij duo anguli, quibus illi tres <lb/>&longs;unt pares, v.g. &longs;int tres anguli trianguli A B C, <expan abbr="&longs;int&qacute;">&longs;intque</expan>; alij duo anguli recti, <lb/><figure id="fig9"/><lb/>quos linea perpendicularis D E, facit cum li<lb/>nea F G; &longs;it <expan abbr="inquã">inquam</expan> anguli recti D E F, D E G, <lb/>tunc tres anguli illius <expan abbr="triãguli">trianguli</expan> <expan abbr="dic&etilde;tur">dicentur</expan> æqua<lb/>les duobus hi&longs;ce rectis, &longs;i tres illi mucrones <lb/>trianguli fimul &longs;umpti, & vniti ad punctum <lb/>E, ad quod duo <expan abbr="quoq;">quoque</expan> mucrones angulorum <lb/><figure id="fig10"/><lb/>rectorum coeunt, congruent omnino duobus <lb/>prædictis angulis rectis, &longs;iue duobus illis mu<lb/>cronibus angulorum rectorum, &longs;iue con&longs;ti<lb/>tuent lineam rectam F E G, &longs;icuti faciunt <lb/>etiam duo illi anguli recti; &longs;iue etiam dica<lb/>mus, occupabunt idem &longs;patium omninò, & <lb/>præcisè, quod occupant duo recti: v.g. </s>

<lb/>inclinatio in puncto A, &longs;iue acumen illud, &longs;iue mucro,

<lb/>e&longs;t ratio anguli. </s>

<s>&longs;olum igitur duo anguli erunt æqua

<s>&longs;i mucro B, ibi poneretur, faceret <lb/>angulum F E H, & &longs;i ibi iuxta ip&longs;um apponeretur mucro A, faceret angulum <lb/>H E I. quem &longs;i deinceps &longs;ub&longs;equetur reliquus angulus C, con&longs;titueret <expan abbr="reli-qu&utilde;">reli<lb/>quum</expan> angulum I E G. iam, vt vides, illi tres anguli ad E, tran&longs;lati, &longs;unt æqua<lb/>les duobus rectis ad E, pariter con&longs;titutis, cum illi tres fiant partes duorum <lb/>rectorú, vel quia occupant idem &longs;patium, vel eandem lineam rectam F E G, <lb/>con&longs;tituant. </s>

<lb/>les, <expan abbr="quãdo">quando</expan> vnius acumen æquale erit acumini alterius;

<lb/>etiam &longs;i lineæ con&longs;tituentes vnum angulum &longs;int lon

<s>habet igitur omne triangulum &longs;iue &ecedil;quilaterum, &longs;iue &longs;calenum, <lb/>&longs;iue I&longs;o&longs;celes mirabilem hanc proprietatem, vt tres anguli, cuiu&longs;uis trian<lb/>guli &longs;int æquales duobus rectis angulis. </s>

<lb/>giores lineis alterum angulum con&longs;tituentibus, quia

<lb/>quantitas anguli non attenditur penes longitudinem <pb pagenum="40"/><expan abbr="linear&utilde;">linearum</expan>, &longs;ed penes inclinationem, & mucronem, quem faciunt: vnde etiam&longs;i

<s>Quam demon&longs;trationem primi om<lb/>nium Pythagorici perfecerunt, vt refert Proclus ad 32. primi Elem. Eucli<lb/>des deinde ibidem aliter, quam Pythagorici idem demon&longs;trauit. </s>

<lb/>duæ lineæ prædictæ A B, A C, productæ, &longs;iue etiam decurtatæ fuerint, dum

<lb/>modo &longs;itus, &longs;iue po&longs;itio ip&longs;arum, quam ad inuicem habent, non varietur,

<lb/>erit &longs;emper eadem quantitas anguli A. </s>

<s>Aduertendum præterea rationem

<lb/>anguli non po&longs;&longs;e &longs;aluari in &longs;olo puncto A, in quo lineæ concurrunt, &longs;ed ne

<lb/>ce&longs;&longs;ariam e&longs;&longs;e aliquam quantitatem, quamuis exiguam, linearum A B, A C.

<s>Notandum etiam, quod in nominatione angulorum, quæ fit per tres lite

<lb/>ras, &longs;emper literam illam e&longs;&longs;e medio loco proferendam, quæ ad acumen ip

<lb/>&longs;um po&longs;ita e&longs;t, vt in &longs;uperiori, litera A, debet &longs;emper media proferri, dicen

<lb/>do angulum B A C, &longs;iue C A B, <expan abbr="n&utilde;quam">nunquam</expan> tamen licet dicere angulum A C B,

<s>Porrò quemadmodum vnus angulus vni angulo æqualis e&longs;t, ita

<lb/><expan abbr="aliquãdo">aliquando</expan> duo anguli &longs;unt vni angulo æquales, vt patet, &longs;i vnus angulus, v.g.

<lb/>angulus B A C, diuidatur in duos angulos à linea A D. tunc enim duo angu

<lb/>li partiales B A D, D A C, erunt æquales totali angulo

<lb/>B A C, cum partes omnes &longs;imul &longs;umptæ &longs;int &longs;uo toti æqua

<s>pariter tres anguli po&longs;&longs;unt æquari & vni, & duobus

<lb/>alijs angulis, quando nimirum a cumina, &longs;iue mucrones il

<lb/>li &longs;imul ad vnum punctum con&longs;tituti <expan abbr="adæquar&etilde;tur">adæquarentur</expan> mucro

<lb/>niilli, quem con&longs;tituerent alij duo anguli, quibus illi tres

<lb/>&longs;unt pares, v.g. &longs;int tres anguli trianguli A B C, <expan abbr="&longs;int&qacute;">&longs;intque</expan>; alij duo anguli recti,

<lb/>quos linea perpendicularis D E, facit cum li

<lb/>nea F G; &longs;it <expan abbr="inquã">inquam</expan> anguli recti D E F, D E G,

<lb/>tunc tres anguli illius <expan abbr="triãguli">trianguli</expan> <expan abbr="dic&etilde;tur">dicentur</expan> æqua

<lb/>les duobus hi&longs;ce rectis, &longs;i tres illi mucrones

<lb/>trianguli fimul &longs;umpti, & vniti ad punctum

<lb/>E, ad quod duo <expan abbr="quoq;">quoque</expan> mucrones angulorum

<lb/>rectorum coeunt, congruent omnino duobus

<lb/>prædictis angulis rectis, &longs;iue duobus illis mu

<lb/>cronibus angulorum rectorum, &longs;iue con&longs;ti

<lb/>tuent lineam rectam F E G, &longs;icuti faciunt

<lb/>etiam duo illi anguli recti; &longs;iue etiam dica

<lb/>mus, occupabunt idem &longs;patium omninò, &

<lb/>præcisè, quod occupant duo recti: v.g. &longs;i mucro B, ibi poneretur, faceret

<lb/>angulum F E H, & &longs;i ibi iuxta ip&longs;um apponeretur mucro A, faceret angulum

<lb/>H E I. quem &longs;i deinceps &longs;ub&longs;equetur reliquus angulus C, con&longs;titueret <expan abbr="reli-qu&utilde;">reli

<lb/>quum</expan> angulum I E G. iam, vt vides, illi tres anguli ad E, tran&longs;lati, &longs;unt æqua

<lb/>les duobus rectis ad E, pariter con&longs;titutis, cum illi tres fiant partes duorum

<lb/>rectorú, vel quia occupant idem &longs;patium, vel eandem lineam rectam F E G,

<lb/>con&longs;tituant. </s>

<s>Quod &longs;i <lb/>quis huius rei <expan abbr="experi&etilde;tiam">experientiam</expan> aliquam velit; etiam&longs;i non exactam (cum æqua<lb/>litas mathematica non cadat &longs;ub &longs;en&longs;um, &longs;ed &longs;ola intelligentia percipiatur, <lb/>quippe quæ in materia intelligibili, non autem &longs;en&longs;ibili ver&longs;atur, & cuius <pb pagenum="41"/>æqualitas nullum di&longs;crimen, quantumuis minimum admittat, quod &longs;en&longs;ui <lb/>vitare ob &longs;ui imperfectione<gap/>on licet: vnde inter eæ, quæ mathematicè <lb/>&longs;unt æqualia, nullus intellectus aliquam valeat reper<gap/>re differentiam) &longs;umat <lb/>inquam triangulum quodpiam materiale, vt ex charta, quantum fieri po<lb/>ce&longs;t perfectum, deinde ducat lineam vnam perpendicularem &longs;uper aliam, <lb/>quæ &longs;cilicet faciat, cum illa duos angulos rectos. </s>

<s>po&longs;tea ab&longs;cindat tres an<lb/>gulos trianguli materialis, <expan abbr="eos&qacute;">eosque</expan>; ita &longs;imul componat, vt mucrones illorum <lb/>&longs;int vniti, & contigui ad punctum lineæ perpendicularis cum altera, vti e&longs;t <lb/>in &longs;uperiori figura punctnm E; & illicò apparebit tres illos angulos mate<lb/>riales obtegere adæquatè totum illud &longs;patium duorum rectorum, quos per<lb/>pendicularis con&longs;tituit. </s>

<s>Hoc autem experiri poteris in diuer&longs;is admodum <lb/>triangulis Scalenis, Rectangulis, I&longs;o&longs;celibus, Aequilateris, &c. </s>

<s>non &longs;ine de<lb/>lectatione, atque hic e&longs;t &longs;en&longs;us illorum verborum, omnis triangulus habet <lb/>tres &ecedil;quales duobus rectis. </s>

<s>Ab&longs;tineo à demon&longs;trationibus geometricis, quo<lb/>niam ij, qui Mathematicis &longs;unt imbuti, no&longs;tra hac opera parum indigent. <lb/></s>

<s>habet igitur omne triangulum &longs;iue &ecedil;quilaterum, &longs;iue &longs;calenum,

<lb/>&longs;iue I&longs;o&longs;celes mirabilem hanc proprietatem, vt tres anguli, cuiu&longs;uis trian

<lb/>guli &longs;int æquales duobus rectis angulis. </s>

<s>Quam demon&longs;trationem primi om

<lb/>nium Pythagorici perfecerunt, vt refert Proclus ad 32. primi Elem. Eucli

<lb/>des deinde ibidem aliter, quam Pythagorici idem demon&longs;trauit. </s>

<s>Quod &longs;i

<lb/>quis huius rei <expan abbr="experi&etilde;tiam">experientiam</expan> aliquam velit; etiam&longs;i non exactam (cum æqua

<lb/>litas mathematica non cadat &longs;ub &longs;en&longs;um, &longs;ed &longs;ola intelligentia percipiatur,

<lb/>quippe quæ in materia intelligibili, non autem &longs;en&longs;ibili ver&longs;atur, & cuius <pb pagenum="41"/>æqualitas nullum di&longs;crimen, quantumuis minimum admittat, quod &longs;en&longs;ui

<lb/>vitare ob &longs;ui imperfectionem non licet: vnde inter eæ, quæ mathematicè

<lb/>&longs;unt æqualia, nullus intellectus aliquam valeat reperire differentiam) &longs;umat

<lb/>inquam triangulum quodpiam materiale, vt ex charta, quantum fieri po

<lb/>ce&longs;t perfectum, deinde ducat lineam vnam perpendicularem &longs;uper aliam,

<lb/>quæ &longs;cilicet faciat, cum illa duos angulos rectos. </s>

<s>po&longs;tea ab&longs;cindat tres an

<lb/>gulos trianguli materialis, <expan abbr="eos&qacute;">eosque</expan>; ita &longs;imul componat, vt mucrones illorum

<lb/>&longs;int vniti, & contigui ad punctum lineæ perpendicularis cum altera, vti e&longs;t

<lb/>in &longs;uperiori figura punctnm E; & illicò apparebit tres illos angulos mate

<lb/>riales obtegere adæquatè totum illud &longs;patium duorum rectorum, quos per

<lb/>pendicularis con&longs;tituit. </s>

<s>Hoc autem experiri poteris in diuer&longs;is admodum

<lb/>triangulis Scalenis, Rectangulis, I&longs;o&longs;celibus, Aequilateris, &c. </s>

<s>non &longs;ine de

<lb/>lectatione, atque hic e&longs;t &longs;en&longs;us illorum verborum, omnis triangulus habet

<lb/>tres &ecedil;quales duobus rectis. </s>

<s>Ab&longs;tineo à demon&longs;trationibus geometricis, quo

<lb/>niam ij, qui Mathematicis &longs;unt imbuti, no&longs;tra hac opera parum indigent.

<s>&longs;i quis tamen volet, con&longs;ulat 32. primi Elem. </s>

<s>Ex hac igitur declaratione <lb/>licet cogno&longs;cere nonnullos ageometretos locum hunc, & &longs;imiles &longs;ub&longs;equen<lb/>tes non &longs;atis intelligere, dicentes, nihil aliud verba illa Ari&longs;t.

<s>Ex hac igitur declaratione

<lb/>licet cogno&longs;cere nonnullos ageometretos locum hunc, & &longs;imiles &longs;ub&longs;equen

velle &longs;ignifi<lb/>care, quàm omnem triangulum habere tres angulos, quod inquiunt, noti&longs;<lb/>&longs;imum e&longs;t. </s>

<lb/>tes non &longs;atis intelligere, dicentes, nihil aliud verba illa Ari&longs;t.

<s>Sed &longs;i incidant in &longs;equentia; æquales duobus rectis, tunc, cum <lb/>hæc non intelligant, ab&longs;tinent etiam à priorum declaratione, quibus præ<lb/>mi&longs;&longs;is facile e&longs;t Ari&longs;t.

velle &longs;ignifi

<lb/>care, quàm omnem triangulum habere tres angulos, quod inquiunt, noti&longs;

<lb/>&longs;imum e&longs;t. </s>

<s>Sed &longs;i incidant in &longs;equentia; æquales duobus rectis, tunc, cum

<lb/>hæc non intelligant, ab&longs;tinent etiam à priorum declaratione, quibus præ

<lb/>mi&longs;&longs;is facile e&longs;t Ari&longs;t.

textum percipere. </s>

<s>&longs;it A, duo recti, ide&longs;t, duo anguli <lb/>recti &longs;int pa&longs;&longs;io demon&longs;tranda, in quo B, triangulus, in quo C, æquicrus. </s>

<s>&longs;it A, duo recti, ide&longs;t, duo anguli

<lb/>recti &longs;int pa&longs;&longs;io demon&longs;tranda, in quo B, triangulus, in quo C, æquicrus. </s>

<s>ip&longs;i <lb/>itaque C, ide&longs;t triangulo æquicru&longs;i, ine&longs;t A, &longs;cilicet duo recti, hoc e&longs;t, ine&longs;t <lb/>æquicru&longs;i hæc, pa&longs;&longs;io habere tres angulos æquales duobus rectis per B, ide&longs;t <lb/>per <expan abbr="triangul&utilde;">triangulum</expan> vniuer&longs;ale, quia hæc proprietas e&longs;t trianguli propria, & <expan abbr="cõpe-tit">compe<lb/>tit</expan> æquicru&longs;i, non vt æquicrus e&longs;t, &longs;ed, vt triangulum e&longs;t; quare B, non crit <lb/>medium ip&longs;ius A, quia prædicta pa&longs;&longs;io. </s>

<s>ip&longs;i

<lb/>itaque C, ide&longs;t triangulo æquicru&longs;i, ine&longs;t A, &longs;cilicet duo recti, hoc e&longs;t, ine&longs;t

<s>A, non competit triangulo B, per <lb/>aliud, &longs;ed per &longs;e, de eo enim primo, & per &longs;e demon&longs;tratur in 32. primi Elem. <lb/>optimè Aegydius, & Niphus in hunc locum.</s>

<lb/>æquicru&longs;i hæc, pa&longs;&longs;io habere tres angulos æquales duobus rectis per B, ide&longs;t

<lb/>per <expan abbr="triangul&utilde;">triangulum</expan> vniuer&longs;ale, quia hæc proprietas e&longs;t trianguli propria, & <expan abbr="cõpe-tit">compe

<lb/>tit</expan> æquicru&longs;i, non vt æquicrus e&longs;t, &longs;ed, vt triangulum e&longs;t; quare B, non crit

<lb/>medium ip&longs;ius A, quia prædicta pa&longs;&longs;io. </s>

<s>A, non competit triangulo B, per

<lb/>aliud, &longs;ed per &longs;e, de eo enim primo, & per &longs;e demon&longs;tratur in 32. primi Elem.

<lb/>optimè Aegydius, & Niphus in hunc locum.</s>

<s>Ex eodem cap. <emph type="italics"/>(Non oportet autem exi&longs;timare penes id, quod exponimus, ali<lb/>quid accidere ab&longs;urdum, nihil enim vtimur eo, quod e&longs;t hoc aliquid e&longs;&longs;e. </s>

<s>Ex eodem cap. <emph type="italics"/>(Non oportet autem exi&longs;timare penes id, quod exponimus, ali

<lb/>quid accidere ab&longs;urdum, nihil enim vtimur eo, quod e&longs;t hoc aliquid e&longs;&longs;e. </s>

<s>&longs;ed &longs;icut <lb/>Geometra pedalem, & rectam hanc, & &longs;ine latitudine dicit, quæ non &longs;unt. </s>

<s>verum <lb/>non &longs;ic vtitur, tanquam ex his ratiocinans)<emph.end type="italics"/> Quoniam Ari&longs;t.

<s>&longs;ed &longs;icut

<lb/>Geometra pedalem, & rectam hanc, & &longs;ine latitudine dicit, quæ non &longs;unt. </s>

in exemplis affert <lb/>pro rebus characteres, A, B, C, po&longs;&longs;et qui&longs;piam &longs;u&longs;picari aliquod propterea <lb/>ab&longs;urdum accidere: cui &longs;u&longs;picioni Ari&longs;t.

<s>verum

<lb/>non &longs;ic vtitur, tanquam ex his ratiocinans)<emph.end type="italics"/> Quoniam Ari&longs;t.

re&longs;pondet, dicens, nihil inde ab&longs;ur<lb/>di accidere po&longs;&longs;e, quoniam ip&longs;e vtitur hi&longs;ce literis, <expan abbr="nõ">non</expan> quatenus literæ &longs;unt, <lb/>&longs;ed quatenus rerum vicem, pro quibus exponuntur, gerunt: quemadmodum <lb/>etiam Geometræ faciunt, qui lineam, quæ pedalis non e&longs;t, pedalem, & quæ <lb/>non e&longs;t recta, rectam; & quæ lata e&longs;t, non latam, &longs;upponunt, & tamen nihil <lb/>inde ab&longs;urdi contingit. </s>

in exemplis affert

<lb/>pro rebus characteres, A, B, C, po&longs;&longs;et qui&longs;piam &longs;u&longs;picari aliquod propterea

<lb/>ab&longs;urdum accidere: cui &longs;u&longs;picioni Ari&longs;t.

re&longs;pondet, dicens, nihil inde ab&longs;ur

<lb/>di accidere po&longs;&longs;e, quoniam ip&longs;e vtitur hi&longs;ce literis, <expan abbr="nõ">non</expan> quatenus literæ &longs;unt,

<lb/>&longs;ed quatenus rerum vicem, pro quibus exponuntur, gerunt: quemadmodum

<lb/>etiam Geometræ faciunt, qui lineam, quæ pedalis non e&longs;t, pedalem, & quæ

<lb/>non e&longs;t recta, rectam; & quæ lata e&longs;t, non latam, &longs;upponunt, & tamen nihil

<lb/>inde ab&longs;urdi contingit. </s>

<s>Ex quibus intelligimus per lineas illas &longs;en&longs;ibiles, &

<lb/>phy&longs;icas, quas Geometræ in &longs;uis figuris ducunt, intelligendas e&longs;&longs;e lineas ve

<lb/>rè Mathematicas omni latitudine carentes; vtitur enim inquit Ari&longs;t. Geo

<lb/>metra lineis phy&longs;icis, non tanquam phy&longs;icis, nec de eis tanquam de phy&longs;icis

<lb/>lineis ratiocinatur, &longs;ed ijs vtitur tanquam verè mathematicis. </s>

<s>Ex quibus intelligimus per lineas illas &longs;en&longs;ibiles, & <lb/>phy&longs;icas, quas Geometræ in &longs;uis figuris ducunt, intelligendas e&longs;&longs;e lineas ve<lb/>rè Mathematicas omni latitudine carentes; vtitur enim inquit Ari&longs;t. Geo<lb/>metra lineis phy&longs;icis, non tanquam phy&longs;icis, nec de eis tanquam de phy&longs;icis <lb/>lineis ratiocinatur, &longs;ed ijs vtitur tanquam verè mathematicis. </s>

<s>idem dicen

<s>idem dicen<lb/>dum e&longs;t de &longs;uperficiebus, necnon de corporibus, quæ ijdem Goometræ de<lb/>&longs;cribunt, vt per ea, de verè mathematicis di&longs;currant.</s><pb pagenum="42"/>

<lb/>dum e&longs;t de &longs;uperficiebus, necnon de corporibus, quæ ijdem Goometræ de

<lb/>&longs;cribunt, vt per ea, de verè mathematicis di&longs;currant.</s></chap><pb pagenum="42"/>

<chap>

<s><emph type="italics"/>Ex Libro &longs;ecundo Priorum.<emph.end type="italics"/></s>

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Line 2780

<s>Ex cap.

21. <emph type="italics"/>(Quod faciunt, qui coalternas putant &longs;cribere, latent enim ip&longs;<gap/><lb/>&longs;e ip&longs;os talia accip entes, quæ non est po&longs;&longs;ibile monstrare uon exiftentibus <lb/><gap/>o lternis)<emph.end type="italics"/> Vult Ari&longs;t.

21. <emph type="italics"/>(Quod faciunt, qui coalternas putant &longs;cribere, latent enim ip&longs;i

<lb/>&longs;e ip&longs;os talia accip entes, quæ non est po&longs;&longs;ibile monstrare uon exi&longs;tentibus

exemplo mathematico explicare, quid &longs;it pe<lb/>titio principij. </s>

<s>vbi per coalternas intelligit parallelas lineas, vox <lb/>enim græca <foreign lang="greek">parallhlos,</foreign> idem &longs;ignificat, ac mutuus, & coalternus. </s>

<s>quoad <lb/>exempli explicationem vtor figura textibus apponi &longs;olita, quæ e&longs;t præ&longs;ens. <lb/><figure id="fig11"/><lb/>probat Euclides in 28. primi Elem.

<lb/>coalternis)<emph.end type="italics"/> Vult Ari&longs;t.

quod &longs;i <lb/>linea recta quædam, vti E F, cadens &longs;uper <lb/>duas rectas, vti &longs;unt A B, C D, fe cerit angu<lb/>los alternos &ecedil;quales, angulos <expan abbr="nimir&utilde;">nimirum</expan> A G H, <lb/>G H D, ij enim dicuntur alterni; &longs;iue alios <lb/>dnos, nimirum B G H, G H C, hi enim &longs;unt <lb/><expan abbr="quoq;">quoque</expan> alterni; probat inquam has duas li<lb/>neas A B, C D, e&longs;&longs;e inuicem parallelas. </s>

exemplo mathematico explicare, quid &longs;it pe

<lb/>titio principij. </s>

<s>Iam &longs;i quis vellet probare, &longs;e duas <lb/>parallelas duxi&longs;&longs;e, hac ratione, quia &longs;cilicet fa ciunt prædictos angulos al<lb/>ternos æquales; & probaret facere angulos alternos æquales, quia &longs;unt pa<lb/>rallelæ, hic peteret principium, ide&longs;t, illud, quod principio <expan abbr="proband&utilde;">probandum</expan> erat, <lb/>afferret pro ratione, & cau&longs;a, quod dicitur peti principium, quia tunc pe<lb/>timus, vt concedatur nobis, id, quod principio, & primo omnium demon<lb/>&longs;trare propo&longs;ueramus. </s>

<s>vbi per coalternas intelligit parallelas lineas, vox

<lb/>enim græca <foreign lang="greek">parallhlos,</foreign> idem &longs;ignificat, ac mutuus, & coalternus. </s>

<s>aduerte, quod characteres, qui &longs;unt in &longs;equentibus <lb/>verbis huius loci, non appellant characteres figuræ appo&longs;itæ; in quo quidam <lb/>decepti, nullo pacto poterant locum hunc intelligere.</s>

<s>quoad

<lb/>exempli explicationem vtor figura textibus apponi &longs;olita, quæ e&longs;t præ&longs;ens.

<lb/>probat Euclides in 28. primi Elem.

quod &longs;i

<lb/>linea recta quædam, vti E F, cadens &longs;uper

<lb/>duas rectas, vti &longs;unt A B, C D, fe cerit angu

<lb/>los alternos &ecedil;quales, angulos <expan abbr="nimir&utilde;">nimirum</expan> A G H,

<lb/>G H D, ij enim dicuntur alterni; &longs;iue alios

<lb/>dnos, nimirum B G H, G H C, hi enim &longs;unt

<lb/><expan abbr="quoq;">quoque</expan> alterni; probat inquam has duas li

<lb/>neas A B, C D, e&longs;&longs;e inuicem parallelas. </s>

<s>Iam &longs;i quis vellet probare, &longs;e duas

<lb/>parallelas duxi&longs;&longs;e, hac ratione, quia &longs;cilicet fa ciunt prædictos angulos al

<lb/>ternos æquales; & probaret facere angulos alternos æquales, quia &longs;unt pa

<lb/>rallelæ, hic peteret principium, ide&longs;t, illud, quod principio <expan abbr="proband&utilde;">probandum</expan> erat,

<lb/>afferret pro ratione, & cau&longs;a, quod dicitur peti principium, quia tunc pe

<lb/>timus, vt concedatur nobis, id, quod principio, & primo omnium demon

<lb/>&longs;trare propo&longs;ueramus. </s>

<s>aduerte, quod characteres, qui &longs;unt in &longs;equentibus

<lb/>verbis huius loci, non appellant characteres figuræ appo&longs;itæ; in quo quidam

<lb/>decepti, nullo pacto poterant locum hunc intelligere.</s>

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22. lib.

2. Priorum <emph type="italics"/>(Vt &longs;i volens mon&longs;trare, quod diameter e&longs;t incom<lb/>men&longs;. </s>

2. Priorum <emph type="italics"/>(Vt &longs;i volens mon&longs;trare, quod diameter e&longs;t incom

<lb/>men&longs;. argueret Zenonis rationem, quod non e&longs;t moueri)<emph.end type="italics"/> &longs;uperius &longs;ecto 3. lib.

<s>argueret Zenonis rationem, quod non e&longs;t moueri)<emph.end type="italics"/> &longs;uperius &longs;ecto 3. lib.

1. <lb/>fusè explicauimus hanc a&longs;ymmetriam, quam &longs;i quis vellet demon&longs;trare ea<lb/>dem illa ratione, qua Zeno motum impugnabat, quia &longs;cilicet <expan abbr="m&etilde;&longs;ura">men&longs;ura</expan> com<lb/>munis, quæ debet vtramq, quantitatem men&longs;urare, debet in men&longs;urando <lb/>infinitas partes pertran&longs;ire, uimirum medietates medietatum in in&longs;initum, <lb/>e&longs;t autem impo&longs;&longs;ibile pertran&longs;ire infinitas huiu&longs;modi partes, & propterea <lb/>non poterit metiri, <expan abbr="neq;">neque</expan> vnam, <expan abbr="neq;">neque</expan> alteram ex <expan abbr="quãtitatibus">quantitatibus</expan>, quæ putaban<lb/>tur commen&longs;urabiles, afferret hic, inquit Ari&longs;t.

<lb/>fusè explicauimus hanc a&longs;ymmetriam, quam &longs;i quis vellet demon&longs;trare ea

<lb/>dem illa ratione, qua Zeno motum impugnabat, quia &longs;cilicet <expan abbr="m&etilde;&longs;ura">men&longs;ura</expan> com

<lb/>munis, quæ debet vtramq, quantitatem men&longs;urare, debet in men&longs;urando

<lb/>infinitas partes pertran&longs;ire, uimirum medietates medietatum in in&longs;initum,

<lb/>e&longs;t autem impo&longs;&longs;ibile pertran&longs;ire infinitas huiu&longs;modi partes, & propterea

<lb/>non poterit metiri, <expan abbr="neq;">neque</expan> vnam, <expan abbr="neq;">neque</expan> alteram ex <expan abbr="quãtitatibus">quantitatibus</expan>, quæ putaban

<lb/>tur commen&longs;urabiles, afferret hic, inquit Ari&longs;t.

non cau&longs;am pro cau&longs;a.</s>

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<s>Ex eodem cap. <emph type="italics"/>(Quoniam idem <expan abbr="vtiq;">vtique</expan> fal&longs;um per plures petitiones accidere <lb/>nihil forta&longs;&longs;e inconueniens, veluti coalternas coincidere; & &longs;i maior e&longs;t extrin&longs;ecus <lb/>angulus intrin&longs;eco; & &longs;i triangulus habet plures rectos duobus)<emph.end type="italics"/> per plures po&longs;i<lb/>tiones &longs;ubaudi fal&longs;as. </s>

<s>Ex eodem cap. <emph type="italics"/>(Quoniam idem <expan abbr="vtiq;">vtique</expan> fal&longs;um per plures petitiones accidere

<lb/>nihil forta&longs;&longs;e inconueniens, veluti coalternas coincidere; & &longs;i maior e&longs;t extrin&longs;ecus

<lb/>angulus intrin&longs;eco; & &longs;i triangulus habet plures rectos duobus)<emph.end type="italics"/> per plures po&longs;i

<lb/>tiones &longs;ubaudi fal&longs;as. </s>

<s>per coalternas intellige lineas æquidi&longs;tantes, &longs;eu pa<lb/>rallelas, vt in &longs;uperiori cap.

<s>per coalternas intellige lineas æquidi&longs;tantes, &longs;eu pa

<lb/>rallelas, vt in &longs;uperiori cap.

monuimus. </s>

<s>Cæterum Euclides propo&longs;. </s>

<s>28. pri<lb/>mi Elem.

<s>28. pri

<lb/>mi Elem.

o&longs;tendit, quod &longs;i fuerint duæ parallelæ veluti in præcedenti figura, <lb/>A B, C D, &longs;uper quas alia recta E F, incidat, nece&longs;&longs;ario faciet angulum ex<lb/>trin&longs;ecum E G B, v. g. </s>

<s>æqualem interno, & oppo&longs;ito, & ad ea&longs;dem partes, <lb/>angulo videlicet G H D. &longs;i ergo inquit Ari&longs;t &longs;upponamus i&longs;tud fal&longs;um, an<lb/>gulum &longs;cilicet E G B, externum e&longs;&longs;e maiorem angulo interno G H D, &longs;equi<lb/>tur etiam fal&longs;um, videlicet lineas <expan abbr="æquidi&longs;tãtes">æquidi&longs;tantes</expan> A B, C D, concurrere. </s>

<s>& pro<lb/>batur con&longs;equentia hoc modo, quia &longs;i angulus E G B, maior e&longs;t angulo <pb pagenum="43"/>G H D, appo&longs;ito vtiq<gap/> communi angulo B G H, erant primum, duo anguli <lb/>E G B, B G H, maiores, quam &longs;int duo B G H, G H D, quia &longs;i inæqualibus <lb/>æqualia addantur, tota erunt inæqualia, vt prius per 4, axioma: hoc loco <lb/>communis angulus additur &longs;emel maiori angulo, & &longs;emel minori; & ideo <lb/>totum illud, in quo e&longs;t maior angulus, adhuc maius e&longs;t altero toto, in quo <lb/>minor angulus continetur. </s>

<s>at illi duo E G B, B G H, per 13. primi, &longs;unt <lb/>æquales duobus rectis angulis, ergo duo <expan abbr="quoq;">quoque</expan> recti erunt maiores duobus <lb/>internis B G H, D H G, &longs;iue hi duo interni erunt minores duobus rectis. <lb/></s>

<s>At quando hi duo interni &longs;unt minores duobus rectis, tunc lineæ A B, C D, <lb/>&longs;unt concurrentes, &longs;i protrahantur ad partes prædictorum <expan abbr="angulor&utilde;">angulorum</expan>. </s>

o&longs;tendit, quod &longs;i fuerint duæ parallelæ veluti in præcedenti figura,

<lb/>A B, C D, &longs;uper quas alia recta E F, incidat, nece&longs;&longs;ario faciet angulum ex

<lb/>trin&longs;ecum E G B, v. g. æqualem interno, & oppo&longs;ito, & ad ea&longs;dem partes,

<lb/>angulo videlicet G H D. &longs;i ergo inquit Ari&longs;t &longs;upponamus i&longs;tud fal&longs;um, an

<lb/>gulum &longs;cilicet E G B, externum e&longs;&longs;e maiorem angulo interno G H D, &longs;equi

<lb/>tur etiam fal&longs;um, videlicet lineas <expan abbr="æquidi&longs;tãtes">æquidi&longs;tantes</expan> A B, C D, concurrere. </s>

<s>quod <lb/>P. Clauius luculenti, & hactenus de&longs;iderata demon&longs;tratione ad 28. primi <lb/>demon&longs;trauit. </s>

<s><expan abbr="Atq;">Atque</expan> hoc pacto ex prima fal&longs;a &longs;uppo&longs;itione, nimirum angu<lb/>lum illum externum e&longs;&longs;e maiorem interno, & oppo&longs;ito; &longs;equitur fal&longs;um, ni<lb/>mirum lineas parallelas concurrere.</s>

<s>Præterea &longs;i &longs;upponamus aliam fal&longs;itatem, &longs;cilicet triangulum habere tres <lb/>angulos maiores duobus rectis, &longs;equetur eadem iterum fal&longs;itas, &longs;cilicet pa<lb/><figure id="fig12"/><lb/>rallelas coincidere, & probatur &longs;ic; &longs;int enim <lb/><expan abbr="triãguli">trianguli</expan> A B C, tres anguli maiores, quam duo <lb/>recti anguli, & per punctum C, ducta &longs;it recta <lb/>C D, parallela lateri B A. quia ergo angulus <lb/>A, æqualis e&longs;t angulo &longs;ibi alterno A C D, per <lb/>29. primi, & quia totalis angulus B C D, æqua<lb/>lis e&longs;t duobus angulis B C A, A C D, quos tanquam &longs;uas partes adæquatas <lb/>continet, quorum alter, &longs;cilicet A C D, e&longs;t æqualis angulo A. erit idem to<lb/>talis angulus B C D, æqualis duobus angulis A, & A C B, trianguli propo&longs;i<lb/>ti. </s>

<s>& pro

<lb/>batur con&longs;equentia hoc modo, quia &longs;i angulus E G B, maior e&longs;t angulo <pb pagenum="43"/>G H D, appo&longs;ito <expan abbr="vtiq;">vtique</expan> communi angulo B G H, erant primum, duo anguli

<lb/>E G B, B G H, maiores, quam &longs;int duo B G H, G H D, quia &longs;i inæqualibus

<lb/>æqualia addantur, tota erunt inæqualia, vt prius per 4, axioma: hoc loco

<lb/>communis angulus additur &longs;emel maiori angulo, & &longs;emel minori; & ideo

<lb/>totum illud, in quo e&longs;t maior angulus, adhuc maius e&longs;t altero toto, in quo

<lb/>minor angulus continetur. </s>

<s>at illi duo E G B, B G H, per 13. primi, &longs;unt

<lb/>æquales duobus rectis angulis, ergo duo <expan abbr="quoq;">quoque</expan> recti erunt maiores duobus

<lb/>internis B G H, D H G, &longs;iue hi duo interni erunt minores duobus rectis.

<s>At quando hi duo interni &longs;unt minores duobus rectis, tunc lineæ A B, C D,

<lb/>&longs;unt concurrentes, &longs;i protrahantur ad partes prædictorum <expan abbr="angulor&utilde;">angulorum</expan>. </s>

<s>quod

<lb/>P. Clauius luculenti, & hactenus de&longs;iderata demon&longs;tratione ad 28. primi

<lb/>demon&longs;trauit. </s>

<s><expan abbr="Atq;">Atque</expan> hoc pacto ex prima fal&longs;a &longs;uppo&longs;itione, nimirum angu

<lb/>lum illum externum e&longs;&longs;e maiorem interno, & oppo&longs;ito; &longs;equitur fal&longs;um, ni

<lb/>mirum lineas parallelas concurrere.</s>

<s>Præterea &longs;i &longs;upponamus aliam fal&longs;itatem, &longs;cilicet triangulum habere tres

<lb/>angulos maiores duobus rectis, &longs;equetur eadem iterum fal&longs;itas, &longs;cilicet pa

<lb/>rallelas coincidere, & probatur &longs;ic; &longs;int enim

<lb/><expan abbr="triãguli">trianguli</expan> A B C, tres anguli maiores, quam duo

<lb/>recti anguli, & per punctum C, ducta &longs;it recta

<lb/>C D, parallela lateri B A. quia ergo angulus

<lb/>A, æqualis e&longs;t angulo &longs;ibi alterno A C D, per

<lb/>29. primi, & quia totalis angulus B C D, æqua

<lb/>lis e&longs;t duobus angulis B C A, A C D, quos tanquam &longs;uas partes adæquatas

<lb/>continet, quorum alter, &longs;cilicet A C D, e&longs;t æqualis angulo A. erit idem to

<lb/>talis angulus B C D, æqualis duobus angulis A, & A C B, trianguli propo&longs;i

<s>ergo totus i&longs;te angulus B C D, &longs;imul cum reliquo <expan abbr="triãguli">trianguli</expan> angulo B. con

<lb/>flabit compo&longs;itionem ex tribus angulis trianguli dati: & con&longs;equenter ta

<lb/>lis compo&longs;itio trium angulorum erit maior, quam &longs;int duo anguli recti. </s>

<s>ex

<lb/>quo &longs;equitur duas rectas B A, C D, &longs;uper quas cadit linea B C, faciens duos

<lb/>angulos internos, & ad ea&longs;dem partes, &longs;cilicet A B D, maiores duobus re

<lb/>ctis non e&longs;&longs;e parallelas, &longs;ed concurrentes (vt patet ex nuper citata demon

<lb/>&longs;tratione P. Clauij) quod fal&longs;um e&longs;t. </s>

<s>ergo totus i&longs;te angulus B C D, &longs;imul cum reliquo <expan abbr="triãguli">trianguli</expan> angulo B. con<lb/>flabit compo&longs;itionem ex tribus angulis trianguli dati: & con&longs;equenter ta<lb/>lis compo&longs;itio trium angulorum erit maior, quam &longs;int duo anguli recti. </s>

<s>& &longs;equitur ex &longs;ecunda fal&longs;a &longs;uppo&longs;itio

<s>ex <lb/>quo &longs;equitur duas rectas B A, C D, &longs;uper quas cadit linea B C, faciens duos <lb/>angulos internos, & ad ea&longs;dem partes, &longs;cilicet A B D, maiores duobus re<lb/>ctis non e&longs;&longs;e parallelas, &longs;ed concurrentes (vt patet ex nuper citata demon<lb/>&longs;tratione P. Clauij) quod fal&longs;um e&longs;t. </s>

<s>& &longs;equitur ex &longs;ecunda fal&longs;a &longs;uppo&longs;itio<lb/>ne. </s>

<s>ex quibus textus Ari&longs;t.

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<s>Ex cap.

26. <emph type="italics"/>(Vt &longs;i A, duo recti, in quo autem P., triangulus, in quo vero C, <lb/>&longs;en&longs;ibuis triangulus, &longs;u&longs;picari <expan abbr="namq;">namque</expan> po&longs;&longs;et aliquis non e&longs;&longs;e C, &longs;ciens, quod omnis <lb/>triangulus habet duos rectos: quare &longs;imul no&longs;cet, & ignorabit idem. </s>

26. <emph type="italics"/>(Vt &longs;i A, duo recti, in quo autem P., triangulus, in quo vero C,

<lb/>&longs;en&longs;ibuis triangulus, &longs;u&longs;picari <expan abbr="namq;">namque</expan> po&longs;&longs;et aliquis non e&longs;&longs;e C, &longs;ciens, quod omnis

<s>no&longs;ce enim <lb/>omnem triangulum, quod duobus rectis, non &longs;implex e&longs;t: &longs;ed hoc quidem eo, quod <lb/>vniuer&longs;alem habet &longs;cientiam: illud autem eo, quod &longs;ingularem. </s>

<lb/>triangulus habet duos rectos: quare &longs;imul no&longs;cet, & ignorabit idem. </s>

<s>&longs;ic igitur, vt vni<lb/>uer&longs;ale nouit C, quod duo recti; vt autem &longs;ingulare non nouit, quare non habebit <lb/>contrarias)<emph.end type="italics"/> vide, quæ diximus lib.

<s>no&longs;ce enim

<lb/>omnem triangulum, quod duobus rectis, non &longs;implex e&longs;t: &longs;ed hoc quidem eo, quod

<lb/>vniuer&longs;alem habet &longs;cientiam: illud autem eo, quod &longs;ingularem. </s>

<s>&longs;ic igitur, vt vni

<lb/>uer&longs;ale nouit C, quod duo recti; vt autem &longs;ingulare non nouit, quare non habebit

<lb/>contrarias)<emph.end type="italics"/> vide, quæ diximus lib.

1. &longs;ecto 3. cap.

1. ex quibus quidquid Ma<lb/>thematicum e&longs;t hic, clarum redditur. </s>

1. ex quibus quidquid Ma

<lb/>thematicum e&longs;t hic, clarum redditur. </s>

<s>reliqua verò, quæ ad Logicum &longs;pe<lb/>ctant, huius loci commentatores pro&longs;equuntur.</s>

<s>reliqua verò, quæ ad Logicum &longs;pe

<lb/>ctant, huius loci commentatores pro&longs;equuntur.</s>

<s>In cap.

Line 2392

Line 2960

<s>Notandum hic cum eruditi&longs;&longs;imo Burana, Abductionem hanc, de qua in hoc <lb/>cap.

<s>Notandum hic cum eruditi&longs;&longs;imo Burana, Abductionem hanc, de qua in hoc

<lb/>cap.

agitur e&longs;&longs;e vocem mathematicam, <expan abbr="cam&qacute;">camque</expan>; Ari&longs;t.

quemadmodum multa <lb/>alia à Mathematicis mutuatum ad omnes alias &longs;cientias tran&longs;tuli&longs;&longs;e. </s>

quemadmodum multa

<lb/>alia à Mathematicis mutuatum ad omnes alias &longs;cientias tran&longs;tuli&longs;&longs;e. </s>

<s>e&longs;&longs;e <pb pagenum="44"/>autem terminum mathematicum colligitur manife&longs;tè ex Proelo, qui lib.

3. <lb/>in comm. Elem.

<lb/>in comm. Elem.

Euclidis ad primam propo&longs;itionem primi Elementi, pag. <lb/></s>

Euclidis ad primam propo&longs;itionem primi Elementi, pag.

<s>121. &longs;ic ait, Abductio verò e&longs;t tran&longs;itus à propo&longs;ito problemate, vel theo<lb/>remate ad aliud, quo cognito, aut comparato Propo&longs;itum quoque per&longs;pi<lb/>cuum e&longs;t. </s>

<s>121. &longs;ic ait, Abductio verò e&longs;t tran&longs;itus à propo&longs;ito problemate, vel theo

<lb/>remate ad aliud, quo cognito, aut comparato Propo&longs;itum quoque per&longs;pi

<lb/>cuum e&longs;t. </s>

<s>Exempli cau&longs;a, cum cubi duplicatio propo&longs;ita e&longs;&longs;et ad inue&longs;ti<lb/>gandam quæ&longs;tionem in aliud tran&longs;tulere, quod illud propo&longs;itum con&longs;equi<lb/>tur, ad duarum nempe mediarum linearum inuentionem tran&longs;lata e&longs;t quæ<lb/>&longs;tio, & &longs;ic quærebant deinceps, quonam modo datis duabus rectis lineis, <lb/>duæ mediæ proportionales reperirentur. </s>

<s>Exempli cau&longs;a, cum cubi duplicatio propo&longs;ita e&longs;&longs;et ad inue&longs;ti

<lb/>gandam quæ&longs;tionem in aliud tran&longs;tulere, quod illud propo&longs;itum con&longs;equi

<lb/>tur, ad duarum nempe mediarum linearum inuentionem tran&longs;lata e&longs;t quæ

<lb/>&longs;tio, & &longs;ic quærebant deinceps, quonam modo datis duabus rectis lineis,

<lb/>duæ mediæ proportionales reperirentur. </s>

<s>Primum autem dicunt Hippocra<lb/>tem Chium pr&ecedil;dictorum titulorum, Abductionem feci&longs;&longs;e, qui & lunulæ qua<lb/>dratum fecit æquale, & alia multa in Geometria inuenit. </s>

<s>Primum autem dicunt Hippocra

<lb/>tem Chium pr&ecedil;dictorum titulorum, Abductionem feci&longs;&longs;e, qui & lunulæ qua

<lb/>dratum fecit æquale, & alia multa in Geometria inuenit. </s>

<s>hæc Proclus. </s>

<s>vbi <lb/>non di&longs;&longs;imulandum nos re&longs;titui&longs;&longs;e verbum, Abductionem, cuius loco inter<lb/>pres Procli vtitur inductionis voce, &longs;equuti & rationem, & græcum textum, <lb/>qui no&longs;tram hanc expo&longs;itionem euidenter po&longs;tulat, <foreign lang="greek">apagwgh\</foreign> enim valet & <lb/>inductionem, & abductionem, &longs;ed abductio omnino rei propo&longs;itæ quadrat.</s>

<s>vbi

<lb/>non di&longs;&longs;imulandum nos re&longs;titui&longs;&longs;e verbum, Abductionem, cuius loco inter

<lb/>pres Procli vtitur inductionis voce, &longs;equuti & rationem, & græcum textum,

<lb/>qui no&longs;tram hanc expo&longs;itionem euidenter po&longs;tulat, <foreign lang="greek">apagwgh\</foreign> enim valet &

<lb/>inductionem, & abductionem, &longs;ed abductio omnino rei propo&longs;itæ quadrat.</s>

<s>Notandum præterea Hippoetatem Chium fui&longs;&longs;e auctorem huius Abdu<lb/>ctionis, <expan abbr="eum&qacute;">eumque</expan>; feci&longs;&longs;e Abductionem à propo&longs;ito Problemate quadrandi cir<lb/>culi, vnde manife&longs;tè apparet, Ari&longs;totelem ex Mathematicis hunc terminum <lb/>mutuò accepi&longs;&longs;e, quandoquidem ex ij&longs;dem accepit etiam exemplum Abdu<lb/>ctionis Mathematicæ, imò etiam exemplum ip&longs;ius authoris Abductioni<gap/><lb/>Mathematicæ. </s>

<s>Notandum præterea Hippoetatem Chium fui&longs;&longs;e auctorem huius Abdu

<lb/>ctionis, <expan abbr="eum&qacute;">eumque</expan>; feci&longs;&longs;e Abductionem à propo&longs;ito Problemate quadrandi cir

<lb/>culi, vnde manife&longs;tè apparet, Ari&longs;totelem ex Mathematicis hunc terminum

<lb/>mutuò accepi&longs;&longs;e, quandoquidem ex ij&longs;dem accepit etiam exemplum Abdu

<lb/>ctionis Mathematicæ, imò etiam exemplum ip&longs;ius authoris Abductionis

<lb/>Mathematicæ. </s>

<s>&longs;yllogi&longs;mus autem Hippocratis, quo o&longs;tendebat circuli qua<lb/>draturam reducebatur ad has propo&longs;itiones, omnis rectilinea figura qua<lb/>dratur, &longs;ed circulus reducitur ad figuram rectilineam, ergo circulus qua<lb/>dratur. </s>

<s>&longs;yllogi&longs;mus autem Hippocratis, quo o&longs;tendebat circuli qua

<lb/>draturam reducebatur ad has propo&longs;itiones, omnis rectilinea figura qua

<lb/>dratur, &longs;ed circulus reducitur ad figuram rectilineam, ergo circulus qua

<lb/>dratur. </s>

<s>in probatione minoris facta e&longs;t Abductio, cum enim ip&longs;e vellet re<lb/>ctificare circumferentiam circuli per lunulas, nec valeret, alij per lineam, <lb/>quandam quadratricem, vt e&longs;t apud Pappum <expan abbr="Alexandrin&utilde;">Alexandrinum</expan>, & apud P. Cla<lb/>uium in fine &longs;exti Elem.

<s>in probatione minoris facta e&longs;t Abductio, cum enim ip&longs;e vellet re

<lb/>ctificare circumferentiam circuli per lunulas, nec valeret, alij per lineam,

<lb/>quandam quadratricem, vt e&longs;t apud Pappum <expan abbr="Alexandrin&utilde;">Alexandrinum</expan>, & apud P. Cla

<lb/>uium in fine &longs;exti Elem.

& alij aliter fru&longs;tra conarentur, facta e&longs;t Abductio <lb/>circa probationem minoris, in qua adhuc Mathematici verfantur; quæ pro<lb/>batio, &longs;i tandem inueniri po&longs;&longs;et, mox &longs;equeretur principale <expan abbr="propo&longs;it&utilde;">propo&longs;itum</expan> pro<lb/>blema, nimirum circulus quadraretur; vide quæ &longs;crip&longs;<gap/>mus in cap.

& alij aliter fru&longs;tra conarentur, facta e&longs;t Abductio

<lb/>circa probationem minoris, in qua adhuc Mathematici verfantur; quæ pro

<lb/>batio, &longs;i tandem inueniri po&longs;&longs;et, mox &longs;equeretur principale <expan abbr="propo&longs;it&utilde;">propo&longs;itum</expan> pro

<lb/>blema, nimirum circulus quadraretur; vide quæ &longs;crip&longs;imus in cap.

3. Præ<lb/>dicam. </s>

3. Præ

<lb/>dicam. de hac re, quia plurimum hunc conferunt. </s>

<s>de hac re, quia plurimum hunc conferunt. </s>

<s>&longs;ed iam ad textus expli<lb/>cationem veniamus.</s>

<s>&longs;ed iam ad textus expli

<lb/>cationem veniamus.</s>

<s>Ex eodem cap. <emph type="italics"/>(Veluti &longs;i K, e&longs;&longs;et quadrari, in quo autem E, rectilineum, in <lb/>quo verò F, circulus, &longs;i ip&longs;ius E F, vnum &longs;olum e&longs;&longs;et medium, hoc, quod e&longs;t, cum <lb/>lunulis æqualem fieri circulum rectilineo, e&longs;&longs;e po&longs;&longs;et propè ip&longs;um cogno&longs;cere, cum <lb/>vero B C, neque credibilius &longs;it, quam A C, <expan abbr="neq;">neque</expan> pauca media, non dico Abductio<lb/>nem: <expan abbr="neq;">neque</expan> quando B C, &longs;it immediatum, tale enim &longs;cientia est)<emph.end type="italics"/> Aduerte figuram <lb/>vulgatæ editionis e&longs;&longs;e mendo&longs;am, & propterea re&longs;tituendam e&longs;&longs;e, qualis pri<lb/>ma &longs;equens ex Simplicio ad tex. 11. primi Phy&longs;ic. </s>

<s>Ex eodem cap. <emph type="italics"/>(Veluti &longs;i K, e&longs;&longs;et quadrari, in quo autem E, rectilineum, in

<lb/>quo verò F, circulus, &longs;i ip&longs;ius E F, vnum &longs;olum e&longs;&longs;et medium, hoc, quod e&longs;t, cum

<lb/>lunulis æqualem fieri circulum rectilineo, e&longs;&longs;e po&longs;&longs;et propè ip&longs;um cogno&longs;cere, cum

<lb/>vero B C, neque credibilius &longs;it, quam A C, <expan abbr="neq;">neque</expan> pauca media, non dico Abductio

<s>hoc modo Hippocrates <lb/>Chius conabatur circulum ad quadrum redigere; fit circulus A B G C, qua<lb/>drandus; con&longs;tituatur <expan abbr="itaq;">itaque</expan> &longs;uper diametro cius B C, quadratum B C D F, <lb/>cuius diameter B D, &longs;ecatur bifariam in G, à circumferentia circuli dati, <lb/>quod patet ducta &longs;emidiametro H G, perpendiculari ex B C, quæ &longs;uo extre<lb/>mo puncto G, &longs;ecat bifariam, & <expan abbr="diametr&utilde;">diametrum</expan> B D, & circumferentiam B G C. <lb/>facto ergo centro G, de&longs;cribatur alter circulus per puncta B C D F, <expan abbr="conne-ctatur&qacute;">conne<lb/>ctaturque</expan>; recta G C. in triangulo orthogonio B C D, latus B D, &longs;ubtenditur <pb pagenum="45"/><figure id="fig13"/><lb/>angulo recto C, ergo quadratum eius ex eorol<lb/>lario 47. primi, duplum erit quadrati B C, quare <lb/>etiam circulus B C D F, duplus erit circuli A B<lb/>G C, per 2. duodecimi, & &longs;emicirculus B C D, <lb/>duplus erit &longs;emicirculi B A C: & quadrans B E<lb/>C G, æqualis erit &longs;emicirculo B A C: ablato igi<lb/>tur communi &longs;egmento B E C H, remanet lunu<lb/>la B A C E, æqualis triangulo B C G, quod trian<lb/>gulum &longs;i per vltimam &longs;ecundi quadretur, erit lu<lb/>nula B A C, con&longs;equenter quadrata. </s>

<lb/>nem: <expan abbr="neq;">neque</expan> quando B C, &longs;it immediatum, tale enim &longs;cientia est)<emph.end type="italics"/> Aduerte figuram

<lb/>vulgatæ editionis e&longs;&longs;e mendo&longs;am, & propterea re&longs;tituendam e&longs;&longs;e, qualis pri

<lb/>ma &longs;equens ex Simplicio ad tex. 11. primi Phy&longs;ic. hoc modo Hippocrates

<lb/>Chius conabatur circulum ad quadrum redigere; fit circulus A B G C, qua

<lb/>drandus; con&longs;tituatur <expan abbr="itaq;">itaque</expan> &longs;uper diametro cius B C, quadratum B C D F,

<lb/>cuius diameter B D, &longs;ecatur bifariam in G, à circumferentia circuli dati,

<lb/>quod patet ducta &longs;emidiametro H G, perpendiculari ex B C, quæ &longs;uo extre

<lb/>mo puncto G, &longs;ecat bifariam, & <expan abbr="diametr&utilde;">diametrum</expan> B D, & circumferentiam B G C.

<lb/>facto ergo centro G, de&longs;cribatur alter circulus per puncta B C D F, <expan abbr="conne-ctatur&qacute;">conne

<lb/>ctaturque</expan>; recta G C. in triangulo orthogonio B C D, latus B D, &longs;ubtenditur <pb pagenum="45"/><figure place="text" xlink:href="figures-la/009.01.045.1.tif"/>

<lb/>angulo recto C, ergo quadratum eius ex eorol

<lb/>lario 47. primi, duplum erit quadrati B C, quare

<lb/>etiam circulus B C D F, duplus erit circuli A B

<lb/>G C, per 2. duodecimi, & &longs;emicirculus B C D,

<lb/>duplus erit &longs;emicirculi B A C: & quadrans B E

<lb/>C G, æqualis erit &longs;emicirculo B A C: ablato igi

<lb/>tur communi &longs;egmento B E C H, remanet lunu

<lb/>la B A C E, æqualis triangulo B C G, quod trian

<lb/>gulum &longs;i per vltimam &longs;ecundi quadretur, erit lu

<lb/>nula B A C, con&longs;equenter quadrata. </s>

<s><expan abbr="hucu&longs;q;">hucu&longs;que</expan> be<lb/>nè procedit Hippocrates. </s>

<s>&longs;ed vt reliquum circu<lb/>li quadret, &longs;ic pergit, ponatur recta L M, dupla <lb/>ip&longs;ius B C, &longs;upra quam &longs;emicirculus de&longs;cribatur <lb/><figure id="fig14"/><lb/>L O M, cui in&longs;cribatur hexagoni <lb/>æquilateri dimidium L Q S M, & &longs;u<lb/>per tribus hexagoni lateribus, &longs;int <lb/>tres &longs;emicirculi, vt in figura. </s>

<s>& <expan abbr="quo-niã">quo<lb/>niam</expan> diameter L M, dupla e&longs;t <expan abbr="vniu&longs;-cuiu&longs;q;">vniu&longs;<lb/>cuiu&longs;que</expan> <expan abbr="diametror&utilde;">diametrorum</expan> B C, L Q, Q S, <lb/>S M, erit &longs;emicirculus L O M, &ecedil;qua<lb/>lis quatuor &longs;emicirculis prædictis <lb/>per 2. duodecimi, & per 4. &longs;ecundi <lb/>ablatis igitur tribus <expan abbr="&longs;egm&etilde;tis">&longs;egmentis</expan> com<lb/>munibus L N Q, Q O S, S P M, relinquetur trapezium L Q S M, æquale &longs;e<lb/>micirculo B A C, & tribus lunulis L H Q N, Q R S O, S X M P, ab&longs;cindan<lb/>tur <expan abbr="itaq;">itaque</expan> detrapezio tria triangula æqualia tribus lunulis, eo modo, quo &longs;u<lb/>pra in prima figura factum e&longs;t, & quod relinquetur æquale erit &longs;emicirculo <lb/>B A C. quod deinde quadretur per vlt. </s>

<s>&longs;ecundi, &longs;ed aduerte, quod quando <lb/>ait, ab&longs;cindantur de trapezio tria triangula æqualia lunulis, eo modo, quo <lb/>&longs;upra, committit deceptionem, quia eodem modo, quo &longs;upra minimè id fa<lb/>cere po&longs;&longs;umus, quia in &longs;uperiori figura triangula erant con&longs;tituta &longs;uper la<lb/>tus B C, quadrati B C D F, intra circulum de&longs;cripti, qui circulus facit cum <lb/>B C, maius <expan abbr="&longs;egment&utilde;">&longs;egmentum</expan>, quam faciat &longs;emicirculus L O M, cum lateribus L Q, <lb/>Q S, S M. & propterea &longs;emicirculus i&longs;te non habet eandem proportionem <lb/>ad vnamquamque lunularum &longs;uarum, quam habet &longs;emicirculus &longs;uperior <lb/>B C D, ad lunulam B A C E. <expan abbr="atq;">atque</expan> hæc e&longs;t fallacia, quam authorem &longs;uum mi<lb/>nimè latui&longs;&longs;e putandum, cuius Ari&longs;t. &longs;æpius mentionem in &longs;equentibus fa<lb/>ciet : quì enim fieri pote&longs;t, vt tam acutus inuentor, adeo manife&longs;tum erro<lb/>rem non vidi&longs;&longs;et, verum propter adinuenti excellentiam, authori &longs;uo pla<lb/>cuit paralogy&longs;mus. </s>

<s>mirabilis tamen &longs;emper habita e&longs;t illa &longs;uperior lunulæ <lb/>quadratio. </s>

<s><expan abbr="hucu&longs;q;">hucu&longs;que</expan> be

<lb/>nè procedit Hippocrates. </s>

<s>&longs;ed vt reliquum circu

<lb/>li quadret, &longs;ic pergit, ponatur recta L M, dupla

<lb/>ip&longs;ius B C, &longs;upra quam &longs;emicirculus de&longs;cribatur

<lb/>L O M, cui in&longs;cribatur hexagoni

<lb/>æquilateri dimidium L Q S M, & &longs;u

<lb/>per tribus hexagoni lateribus, &longs;int

<lb/>tres &longs;emicirculi, vt in figura. </s>

<s>& <expan abbr="quo-niã">quo

<lb/>niam</expan> diameter L M, dupla e&longs;t <expan abbr="vniu&longs;-cuiu&longs;q;">vniu&longs;

<lb/>cuiu&longs;que</expan> <expan abbr="diametror&utilde;">diametrorum</expan> B C, L Q, Q S,

<lb/>S M, erit &longs;emicirculus L O M, &ecedil;qua

<lb/>lis quatuor &longs;emicirculis prædictis

<lb/>per 2. duodecimi, & per 4. &longs;ecundi

<lb/>ablatis igitur tribus <expan abbr="&longs;egm&etilde;tis">&longs;egmentis</expan> com

<lb/>munibus L N Q, Q O S, S P M, relinquetur trapezium L Q S M, æquale &longs;e

<lb/>micirculo B A C, & tribus lunulis L H Q N, Q R S O, S X M P, ab&longs;cindan

<lb/>tur <expan abbr="itaq;">itaque</expan> detrapezio tria triangula æqualia tribus lunulis, eo modo, quo &longs;u

<lb/>pra in prima figura factum e&longs;t, & quod relinquetur æquale erit &longs;emicirculo

<lb/>B A C. quod deinde quadretur per vlt. </s>

<s>&longs;ecundi, &longs;ed aduerte, quod quando

<lb/>ait, ab&longs;cindantur de trapezio tria triangula æqualia lunulis, eo modo, quo

<lb/>&longs;upra, committit deceptionem, quia eodem modo, quo &longs;upra minimè id fa

<lb/>cere po&longs;&longs;umus, quia in &longs;uperiori figura triangula erant con&longs;tituta &longs;uper la

<lb/>tus B C, quadrati B C D F, intra circulum de&longs;cripti, qui circulus facit cum

<lb/>B C, maius <expan abbr="&longs;egment&utilde;">&longs;egmentum</expan>, quam faciat &longs;emicirculus L O M, cum lateribus L Q,

<lb/>Q S, S M. & propterea &longs;emicirculus i&longs;te non habet eandem proportionem

<lb/>ad vnamquamque lunularum &longs;uarum, quam habet &longs;emicirculus &longs;uperior

<lb/>B C D, ad lunulam B A C E. <expan abbr="atq;">atque</expan> hæc e&longs;t fallacia, quam authorem &longs;uum mi

<lb/>nimè latui&longs;&longs;e putandum, cuius Ari&longs;t. &longs;æpius mentionem in &longs;equentibus fa

<lb/>ciet : quì enim fieri pote&longs;t, vt tam acutus inuentor, adeo manife&longs;tum erro

<lb/>rem non vidi&longs;&longs;et, verum propter adinuenti excellentiam, authori &longs;uo pla

<lb/>cuit paralogy&longs;mus. </s>

<s>mirabilis tamen &longs;emper habita e&longs;t illa &longs;uperior lunulæ

<lb/>quadratio. </s>

<s>Ex quibus &longs;atis clara e&longs;&longs;e po&longs;&longs;unt ea, quæ ad <expan abbr="Mathematic&utilde;">Mathematicum</expan> per

<lb/>tinent, ad locum hunc de Abductione declarandum. </s>

<s>facta e&longs;t igitur abdu

<lb/>ctio ab Hippocrate in quadratione trium po&longs;teriorum lunularum, in qua

<lb/>rum quadratione diu immoratus, nunquam ni&longs;i cum paralogy&longs;mo quadra

<lb/>re valuit. </s>

<s>Ex quibus &longs;atis clara e&longs;&longs;e po&longs;&longs;unt ea, quæ ad <expan abbr="Mathematic&utilde;">Mathematicum</expan> per<lb/>tinent, ad locum hunc de Abductione declarandum. </s>

<s>Hæc pluribus, vt &longs;equentibus etiam textibus, in quibus huius te

<lb/>tragoni&longs;mi fit mentio &longs;atisfacere po&longs;&longs;imus. </s>

<s>facta e&longs;t igitur abdu<lb/>ctio ab Hippocrate in quadratione trium po&longs;teriorum lunularum, in qua<lb/>rum quadratione diu immoratus, nunquam ni&longs;i cum paralogy&longs;mo quadra<lb/>re valuit. </s>

<s>Hippocrates i&longs;te Chius e&longs;t alter <pb pagenum="46"/>ab illo Hippocrate Coo medicorum Magi&longs;tro, vt colligitur ex Alexandre

<lb/>Aphrod. in Primum Meteororum de Cometis.</s>

<s>Hæc pluribus, vt &longs;equentibus etiam textibus, in quibus huius te<lb/>tragoni&longs;mi fit mentio &longs;atisfacere po&longs;&longs;imus. </s>

</chap><chap>

<s>Hippocrates i&longs;te Chius e&longs;t alter <pb pagenum="46"/>ab illo Hippocrate Coo medicorum Magi&longs;tro, vt colligitur ex Alexandre <lb/>Aphrod. in Primum Meteororum de Cometis.</s>

<s><emph type="italics"/>Ex Primo Posteriorum re&longs;olutoriorum.<emph.end type="italics"/></s>

Line 2468

Line 3129

<s>Textu primo <emph type="italics"/>(Omnis doctrina, & omnis di&longs;ciplina di&longs;cur&longs;iua ex præexi<lb/>&longs;tenti fit cognitione. </s>

<s>Textu primo <emph type="italics"/>(Omnis doctrina, & omnis di&longs;ciplina di&longs;cur&longs;iua ex præexi

<lb/>&longs;tenti fit cognitione. </s>

<s>manife&longs;tum autem hoc &longs;peculantibus in omnibus, <lb/>Mathematicæ <expan abbr="namq;">namque</expan> &longs;cientiarum per hunc modum accedunt)<emph.end type="italics"/> quo mo<lb/>do Mathematicæ fiant ex præcedenti cognitione, &longs;cilicet Princi<lb/>piorum per&longs;picuè quilibet videbit, qui &longs;altem primum <expan abbr="Elem&etilde;torum">Elementorum</expan> Eucli<lb/>dis, vel è ianuis in&longs;pexerit; pr&ecedil;cedunt enim primo principiorum tria gene<lb/>ra, quorum primum continet definitiones &longs;ubiecti Geometriæ, vt definitio<lb/>nes lineæ, &longs;uperficiei, trianguli, &c: Secundum continet Po&longs;tulata. </s>

<s>Tertium <lb/>Axiomata, &longs;eu communes omnium conceptiones, & &longs;ententias, ex quibus <lb/>tanquam ex vberrimis, & chri&longs;taltinis fontibus Demon&longs;trationes Geome<lb/>tricæ deriuantur. </s>

<s>manife&longs;tum autem hoc &longs;peculantibus in omnibus,

<lb/>Mathematicæ <expan abbr="namq;">namque</expan> &longs;cientiarum per hunc modum accedunt)<emph.end type="italics"/> quo mo

<s>Idem vìdere licet in operibus aliorum Geometrarum, <lb/>Archimedis, Apollonij, Pappi, & cæterorum. </s>

<lb/>do Mathematicæ fiant ex præcedenti cognitione, &longs;cilicet Princi

<lb/>piorum per&longs;picuè quilibet videbit, qui &longs;altem primum <expan abbr="Elem&etilde;torum">Elementorum</expan> Eucli

<s>Aliæ &longs;iniliter mathematicæ, <lb/>vt Arithmetica, Per&longs;pectiua, Mu&longs;ica, Mechanica, A&longs;tronomia, non ni&longs;t ex <lb/>præmi&longs;&longs;is, ac manife&longs;ti&longs;simis principijs &longs;uas demon&longs;trationes deducunt. <lb/></s>

<lb/>dis, vel è ianuis in&longs;pexerit; pr&ecedil;cedunt enim primo principiorum tria gene

<lb/>ra, quorum primum continet definitiones &longs;ubiecti Geometriæ, vt definitio

<s>Nulla porrò alia &longs;cientia tam di&longs;tinctè &longs;ua præmittit principia, <expan abbr="tam&qacute;">tamque</expan>; per<lb/>&longs;picua, &longs;icuti Mathematicæ, vt non immeritò Philo&longs;ophus eas, tamquam <lb/>veræ &longs;cientiæ <expan abbr="typ&utilde;">typum</expan>, <expan abbr="eum&qacute;">eumque</expan>; omnibus numeris ab&longs;olutum &longs;ibi ob oculos pro<lb/>po&longs;uerit, ex quo veræ &longs;cientiæ de&longs;criptionem hi&longs;ce libris complecteretur.</s>

<lb/>nes lineæ, &longs;uperficiei, trianguli, &c: Secundum continet Po&longs;tulata. </s>

<s>Tertium

<lb/>Axiomata, &longs;eu communes omnium conceptiones, & &longs;ententias, ex quibus

<lb/>tanquam ex vberrimis, & chri&longs;taltinis fontibus Demon&longs;trationes Geome

<lb/>tricæ deriuantur. </s>

<s>Idem vìdere licet in operibus aliorum Geometrarum,

<lb/>Archimedis, Apollonij, Pappi, & cæterorum. </s>

<s>Aliæ &longs;iniliter mathematicæ,

<lb/>vt Arithmetica, Per&longs;pectiua, Mu&longs;ica, Mechanica, A&longs;tronomia, non ni&longs;t ex

<lb/>præmi&longs;&longs;is, ac manife&longs;ti&longs;simis principijs &longs;uas demon&longs;trationes deducunt.

<s>Nulla porrò alia &longs;cientia tam di&longs;tinctè &longs;ua præmittit principia, <expan abbr="tam&qacute;">tamque</expan>; per

<lb/>&longs;picua, &longs;icuti Mathematicæ, vt non immeritò Philo&longs;ophus eas, tamquam

<lb/>veræ &longs;cientiæ <expan abbr="typ&utilde;">typum</expan>, <expan abbr="eum&qacute;">eumque</expan>; omnibus numeris ab&longs;olutum &longs;ibi ob oculos pro

<lb/>po&longs;uerit, ex quo veræ &longs;cientiæ de&longs;criptionem hi&longs;ce libris complecteretur.</s>

<s>Tex. 2. <emph type="italics"/>(Quod enim omne triangulum habet duobus rectis æquales, præ&longs;ciuit: <lb/>quod autem hoc, quod e&longs;t in &longs;emicirculo triangulum e&longs;t, &longs;imul inducens cognouit)<emph.end type="italics"/><lb/>vide primo, quæ &longs;upra libro 1. Prior. &longs;ecto 3. cap.

<s>Tex. 2. <emph type="italics"/>(Quod enim omne triangulum habet duobus rectis æquales, præ&longs;ciuit:

<lb/>quod autem hoc, quod e&longs;t in &longs;emicirculo triangulum e&longs;t, &longs;imul inducens cognouit)<emph.end type="italics"/>

<lb/>vide primo, quæ &longs;upra libro 1. Prior. &longs;ecto 3. cap.

1. explicaui de angulis <lb/>trianguli. </s>

1. explicaui de angulis

<lb/>trianguli. </s>

<s>deinde &longs;cias, quod quando Ari&longs;t.

ait, hoc, quod e&longs;t in &longs;emicir cu<lb/>lo triangulum, &c. </s>

ait, hoc, quod e&longs;t in &longs;emicir cu

<lb/>lo triangulum, &c. </s>

<s>alludit ad demon&longs;trationem quandam, quam ip&longs;e infe<lb/>rius in exemplum adducet, & quæ e&longs;t in 3. Elem.

Euclidis 31. in qua talis fi<lb/>gura proponitur qualis e&longs;t præ&longs;ens, in qua vides triangulum A B C. in &longs;e<lb/><figure id="fig15"/><lb/>micirculo. </s>

<s>alludit ad demon&longs;trationem quandam, quam ip&longs;e infe

<lb/>rius in exemplum adducet, & quæ e&longs;t in 3. Elem.

<s>tunc autem dicitur triangulum in <lb/>&longs;emicirculo, quando ba&longs;is ip&longs;ius e&longs;t diameter <lb/>&longs;emicirculi, & reliqua duo latera ita concur<lb/>runt &longs;imul in angulum B, vt ip&longs;um paricer in <lb/>circumferentia con&longs;tituant, quibus pr&ecedil;mi&longs;sis <lb/>&longs;ic textum explicaueris: quod enim omne <lb/>triangulum habet tres angulos æquales duo<lb/>bus rectis angulis præ&longs;ciuit vniuer&longs;aliter per <lb/>32. primi; quod autem hoc particulare triangulum A B C, quod e&longs;t in &longs;e<lb/>micirculo habeat eandem proprietatem, &longs;imul, ac qui&longs;piam animaduertit <lb/>illud e&longs;&longs;e triangulum cogno&longs;cit, <expan abbr="ab&longs;q;">ab&longs;que</expan> vlla demon&longs;tratione, &longs;ed &longs;olum virtu<lb/>te illius maioris propo&longs;itionis; omne triangulum habet tres, &c.</s>

Euclidis 31. in qua talis fi

<lb/>gura proponitur qualis e&longs;t præ&longs;ens, in qua vides triangulum A B C. in &longs;e

<lb/>micirculo. </s>

<s>tunc autem dicitur triangulum in

<lb/>&longs;emicirculo, quando ba&longs;is ip&longs;ius e&longs;t diameter

<lb/>&longs;emicirculi, & reliqua duo latera ita concur

<lb/>runt &longs;imul in angulum B, vt ip&longs;um paricer in

<lb/>circumferentia con&longs;tituant, quibus pr&ecedil;mi&longs;sis

<lb/>&longs;ic textum explicaueris: quod enim omne

<lb/>triangulum habet tres angulos æquales duo

<lb/>bus rectis angulis præ&longs;ciuit vniuer&longs;aliter per

<lb/>32. primi; quod autem hoc particulare triangulum A B C, quod e&longs;t in &longs;e

<lb/>micirculo habeat eandem proprietatem, &longs;imul, ac qui&longs;piam animaduertit

<lb/>illud e&longs;&longs;e triangulum cogno&longs;cit, <expan abbr="ab&longs;q;">ab&longs;que</expan> vlla demon&longs;tratione, &longs;ed &longs;olum virtu

<lb/>te illius maioris propo&longs;itionis; omne triangulum habet tres, &c.</s>

<s>Tex. 5. <emph type="italics"/>(Vera quidem igitur oporter e&longs;&longs;e, quoniam non e&longs;t non ens &longs;cire, vt quod <lb/>diameter &longs;it commen&longs;urabi is)<emph.end type="italics"/> con&longs;ule ea, quæ &longs;crip&longs;imus ad cap.

<s>Tex. 5. <emph type="italics"/>(Vera quidem igitur oporter e&longs;&longs;e, quoniam non e&longs;t non ens &longs;cire, vt quod

<lb/>diameter &longs;it commen&longs;urabi is)<emph.end type="italics"/> con&longs;ule ea, quæ &longs;crip&longs;imus ad cap.

23. primi <lb/>Priorum, &longs;ecto 1. &longs;ine quibus locus hic &longs;atis intelligi nequit; ijs autem per<lb/>ceptis &longs;ic <expan abbr="loc&utilde;">locum</expan> hunc explicare po&longs;&longs;umus, cum diameter quadrati &longs;it incom<pb pagenum="47"/>men&longs;urabilis lateri &longs;ui quadrati, fal&longs;um erit dicere diametrum e&longs;&longs;e com<lb/>men&longs;urabilem prædicto lateri, quod autem fal&longs;um e&longs;t, illud non e&longs;t; igitur <lb/>impo&longs;sibile e&longs;t &longs;cire diametrum e&longs;&longs;e commen&longs;urabile.</s>

23. primi

<lb/>Priorum, &longs;ecto 1. &longs;ine quibus locus hic &longs;atis intelligi nequit; ijs autem per

<lb/>ceptis &longs;ic <expan abbr="loc&utilde;">locum</expan> hunc explicare po&longs;&longs;umus, cum diameter quadrati &longs;it incom<pb pagenum="47"/>men&longs;urabilis lateri &longs;ui quadrati, fal&longs;um erit dicere diametrum e&longs;&longs;e com

<lb/>men&longs;urabilem prædicto lateri, quod autem fal&longs;um e&longs;t, illud non e&longs;t; igitur

<lb/>impo&longs;sibile e&longs;t &longs;cire diametrum e&longs;&longs;e commen&longs;urabile.</s>

Line 2514

Line 3216

<s>Hoc eodem cap.

plura dicuntur de Principijs Demon&longs;trationis, &longs;iue &longs;cien<lb/>tiæ, vt &longs;unt Dignitates, Po&longs;itiones, Definitiones, & &longs;imilia, quæ quo modo <lb/>&longs;e habeant, & quo modo illis Demon&longs;trationes innitantur, optimè ex con<lb/>templatione primi libri Elem.

plura dicuntur de Principijs Demon&longs;trationis, &longs;iue &longs;cien

<lb/>tiæ, vt &longs;unt Dignitates, Po&longs;itiones, Definitiones, & &longs;imilia, quæ quo modo

<lb/>&longs;e habeant, & quo modo illis Demon&longs;trationes innitantur, optimè ex con

<lb/>templatione primi libri Elem.

Euclidis percipi pote&longs;t. </s>

<s>vt propterea benè ij <lb/>&longs;entiant, inter quos præcipui &longs;unt Toletus, & Zabarella, qui a&longs;&longs;erunt, Ari&longs;t. <lb/>Mathematicas &longs;cientias tamquam typum perfecti&longs;simarum &longs;cientiarum <lb/>&longs;ibi ob oculos propo&longs;ui&longs;&longs;e; ex quo typo veræ &longs;cientiæ de&longs;criptionem his li<lb/>bris complectaretur.</s>

<s>vt propterea benè ij

<lb/>&longs;entiant, inter quos præcipui &longs;unt Toletus, & Zabarella, qui a&longs;&longs;erunt, Ari&longs;t.

<lb/>Mathematicas &longs;cientias tamquam typum perfecti&longs;simarum &longs;cientiarum

<lb/>&longs;ibi ob oculos propo&longs;ui&longs;&longs;e; ex quo typo veræ &longs;cientiæ de&longs;criptionem his li

<lb/>bris complectaretur.</s>

<s>Eodem tex. 5. <emph type="italics"/>(Ponit enim Arithmeticus vnitatem indiui&longs;ibilem e&longs;&longs;e &longs;ecun<lb/>dum quantum)<emph.end type="italics"/> hoc quamquam non ponatur ab Arithmeticis expre&longs;sè, præ<lb/>&longs;upponitur tamen ab eis: nu&longs;quam enim Euclides in totis tribus Arithme<lb/>ticis libris, infra vnitatem de&longs;cendit, vt propterea appareat, ip&longs;am in quan<lb/>titate di&longs;creta e&longs;&longs;e minimum, & indiui&longs;ibile. </s>

<s>Eodem tex. 5. <emph type="italics"/>(Ponit enim Arithmeticus vnitatem indiui&longs;ibilem e&longs;&longs;e &longs;ecun

<lb/>dum quantum)<emph.end type="italics"/> hoc quamquam non ponatur ab Arithmeticis expre&longs;sè, præ

<lb/>&longs;upponitur tamen ab eis: nu&longs;quam enim Euclides in totis tribus Arithme

<lb/>ticis libris, infra vnitatem de&longs;cendit, vt propterea appareat, ip&longs;am in quan

<lb/>titate di&longs;creta e&longs;&longs;e minimum, & indiui&longs;ibile. </s>

<s>Verum dubitabit forrè qui&longs;<lb/>piam hoc modo, &longs;i vnitas minimum, <expan abbr="atq;">atque</expan> indiui&longs;ibile e&longs;t in quanto di&longs;creto, <lb/>qua igitur ratione Arithmetici practici eam diuidunt in dimidium, in trien<lb/>tem, in quadrantem, & alijs &longs;imiliter modis, vnde numeri illi, qui fractio<lb/>nes appellantur, exurgunt? </s>

<s>Verum dubitabit forrè qui&longs;

<lb/>piam hoc modo, &longs;i vnitas minimum, <expan abbr="atq;">atque</expan> indiui&longs;ibile e&longs;t in quanto di&longs;creto,

<s>Re&longs;pondemus, <expan abbr="quotie&longs;eunq;">quotie&longs;eunque</expan> vnitas diuiditur ab <lb/>Arithmeticis, tunc ip&longs;i eam accipiunt tanquam totum quoddam <expan abbr="cõtinuum">continuum</expan> <lb/>in plures partes diui&longs;ibile: &longs;iue tanquam aggregatum quoddam vnitatum, <lb/>quæ vnitates &longs;unt partes illius, vt quando dicunt, vnum horæ quadrantem, <lb/>vel duos horæ quadrantes, vel tres horæ quadrantes, accipiunt horam tan<lb/>quam aggregatum quatuor quadrantum, & propterea numeri illi 1/4. 2/4. 3/4. <lb/>& &longs;imiles fractiones, nihil aliud &longs;unt, quam numeri partium vnius horæ: ex <lb/>quo patet huiu&longs;modi fractiones omnes reduci ad numeros integros, qui <lb/>enim dicit tres quadrantes <gap/>/4. dicit tres partes alicuius totius, quod intel<lb/>ligitur diui&longs;um e&longs;&longs;e in 4. æquales partes, ex quibus illæ tres tantummodo <lb/>numerat.</s>

<lb/>qua igitur ratione Arithmetici practici eam diuidunt in dimidium, in trien

<lb/>tem, in quadrantem, & alijs &longs;imiliter modis, vnde numeri illi, qui fractio

<lb/>nes appellantur, exurgunt? </s>

<s>Re&longs;pondemus, <expan abbr="quotie&longs;eunq;">quotie&longs;eunque</expan> vnitas diuiditur ab

<lb/>Arithmeticis, tunc ip&longs;i eam accipiunt tanquam totum quoddam <expan abbr="cõtinuum">continuum</expan>

<lb/>in plures partes diui&longs;ibile: &longs;iue tanquam aggregatum quoddam vnitatum,

<lb/>quæ vnitates &longs;unt partes illius, vt quando dicunt, vnum horæ quadrantem,

<lb/>vel duos horæ quadrantes, vel tres horæ quadrantes, accipiunt horam tan

<lb/>quam aggregatum quatuor quadrantum, & propterea numeri illi 1/4. 2/4. 3/4.

<lb/>& &longs;imiles fractiones, nihil aliud &longs;unt, quam numeri partium vnius horæ: ex

<lb/>quo patet huiu&longs;modi fractiones omnes reduci ad numeros integros, qui

<lb/>enim dicit tres quadrantes 3/4. dicit tres partes alicuius totius, quod intel

<lb/>ligitur diui&longs;um e&longs;&longs;e in 4. æquales partes, ex quibus illæ tres tantummodo

<lb/>numerat.</s>

<s>Tex. 9. <emph type="italics"/>(Per &longs;e autem, <expan abbr="quæcunq;">quæcunque</expan> & in&longs;unt in eo, quod quid e&longs;t, vt triangulo li<lb/>nea, & lineæ punctum; &longs;ub&longs;tantia <expan abbr="namq;">namque</expan> ip&longs;orum ex his e&longs;t, & in oratione dicen<lb/>te, quid e&longs;t, in&longs;unt)<emph.end type="italics"/> aggreditur explicare quænam &longs;int ea, quæ per &longs;e dicun<lb/>tur: <expan abbr="quot&qacute;">quotque</expan>; modis dicatur aliquid per &longs;e. </s>

<s>Tex. 9. <emph type="italics"/>(Per &longs;e autem, <expan abbr="quæcunq;">quæcunque</expan> & in&longs;unt in eo, quod quid e&longs;t, vt triangulo li

<lb/>nea, & lineæ punctum; &longs;ub&longs;tantia <expan abbr="namq;">namque</expan> ip&longs;orum ex his e&longs;t, & in oratione dicen

<s>quorum primus e&longs;t, ea &longs;cilicet, <lb/>per &longs;e de aliquo &longs;ubiecto dici, <expan abbr="quæcunq;">quæcunque</expan> in definitione illius ponuntur, cu<lb/>iu&longs;modi &longs;unt linea, & punctum, quæ per &longs;e prædicantur, illa de triangulo, <lb/>i&longs;tud de linea; in de&longs;initione enim trianguli ponitur linea recta, quia linea <lb/>recta dum terminat illam &longs;uperficiem, quæ dicitur triangulus illi trianguli <lb/>naturam impertitur, & ideo triangulus definitur &longs;ic, triangulus e&longs;t figura <lb/>tribus lineis rectis terminata. </s>

<lb/>te, quid e&longs;t, in&longs;unt)<emph.end type="italics"/> aggreditur explicare quænam &longs;int ea, quæ per &longs;e dicun

<lb/>tur: <expan abbr="quot&qacute;">quotque</expan>; modis dicatur aliquid per &longs;e. </s>

<s>&longs;imiliter in definitione lineæ, non in&longs;initæ, <lb/>&longs;ed finitæ, & terminatæ ponitur punctum, quia duo puncta, quæ &longs;unt extre<lb/>ma illius, faciunt, vt ea &longs;it line a finita, & definitur &longs;ic, linea finita e&longs;t lon<lb/>gitudo, caius extrema &longs;unt puncta. </s>

<s>quorum primus e&longs;t, ea &longs;cilicet,

<s>quamuis autem hæc definitio apud Eu<lb/>clidem expre&longs;&longs;a non habeatur, tamen ex definitionibus ip&longs;ius præ&longs;ertim &longs;e<lb/>cunda, tertia, & quarta elici pote&longs;t.</s>

<lb/>per &longs;e de aliquo &longs;ubiecto dici, <expan abbr="quæcunq;">quæcunque</expan> in definitione illius ponuntur, cu

<lb/>iu&longs;modi &longs;unt linea, & punctum, quæ per &longs;e prædicantur, illa de triangulo,

<lb/>i&longs;tud de linea; in de&longs;initione enim trianguli ponitur linea recta, quia linea

<lb/>recta dum terminat illam &longs;uperficiem, quæ dicitur triangulus illi trianguli

<lb/>naturam impertitur, & ideo triangulus definitur &longs;ic, triangulus e&longs;t figura

<lb/>tribus lineis rectis terminata. </s>

<s>&longs;imiliter in definitione lineæ, non in&longs;initæ,

<lb/>&longs;ed finitæ, & terminatæ ponitur punctum, quia duo puncta, quæ &longs;unt extre

<lb/>ma illius, faciunt, vt ea &longs;it line a finita, & definitur &longs;ic, linea finita e&longs;t lon

<lb/>gitudo, caius extrema &longs;unt puncta. </s>

<s>quamuis autem hæc definitio apud Eu

<lb/>clidem expre&longs;&longs;a non habeatur, tamen ex definitionibus ip&longs;ius præ&longs;ertim &longs;e

<lb/>cunda, tertia, & quarta elici pote&longs;t.</s>

<s>Eodem tex. 9. <emph type="italics"/>(Et <expan abbr="quibu&longs;cunq;">quibu&longs;cunque</expan> iuexi&longs;tentium ip&longs;is, ip&longs;æ &longs;unt in oratione, quid<emph.end type="italics"/><pb pagenum="48"/><emph type="italics"/>est declarante, quemadmodum rectum ine&longs;t lineæ, & circulare: & impar, & p<gap/><lb/>numero, & primum, & compo&longs;itum, & æquilaterum, & altera parte longius. </s>

<lb/>numero, & primum, & compo&longs;itum, & æquilaterum, & altera parte longius. </s>

<s>& <lb/><expan abbr="o&itilde;bus">oimbus</expan> bis in&longs;unt in oratione, quid e&longs;t <expan abbr="declarãte">declarante</expan>, ibi quidem linea, hic vero numerus)<emph.end type="italics"/><lb/>quia locus hic benè exponitur à Toleto, & melius etiam à Conymbr. </s>

<s>addam <lb/>tantummodo quædam, quæ ad perfectam eius intelligentiam de&longs;iderantur. <lb/></s>

<s>Sciendum igitur primò, nu&longs;quam ab Euclide definiri rectum, circulare, <lb/>impar, par, primum, compo&longs;itum, æquilaterum, nec altera parte longius: <lb/><expan abbr="ver&utilde;">verum</expan> ab ip&longs;o in definitionibus primi definiri lineam rectam, non tamen cir<lb/>cularem expre&longs;sè. </s>

<s>in definitionibus deinde &longs;eptimi definiri <expan abbr="numer&utilde;">numerum</expan> parem, <lb/>& imparem, item numerum primum, & compofitum, & æquilaterum, & al<lb/>tera parte longiorem. </s>

<s>ex quibus definitionibus po&longs;&longs;unt erui definitiones re<lb/>cti, circularis, imparis, & cæterorum, quorum hic Ari&longs;toteles meminit. <lb/></s>

<s>Cæterum Euclides definitione 11. &longs;eptimi, &longs;ic definit numerum primum: <lb/>primus numerus e&longs;t, quem vnitas &longs;ola metitur. </s>

<s>numerus autem, vel vnitas <lb/>metiri dicitur alium numerum, quando &longs;æpius repetita ip&longs;um omnino ad<lb/>æquat, vt ternarius metitur nouenarium, quia ter repetitus ip&longs;um ad vn<lb/>guem explet. </s>

<s>illi igitur numeri dicuntur ab Arithmeticis primi, qui à nullo <lb/>alio, præterquam ab vnitate men&longs;urantur, quales &longs;unt, 2. 3. 5. 7. &c. </s>

<s>Defi<lb/>nitione verò 13. definit numerum compo&longs;itum &longs;ic; compo&longs;itus numerus e&longs;t, <lb/>quem numerus qui&longs;piam metitur, vt &longs;enarius erit compo&longs;itus, quia ip&longs;um <lb/>binarius metitur, nam ter repetitus, ip&longs;i perfectè adæquatur.</s>

<s>Per æquilaterum, intelligit quadratum, quadratus autem numerus defi<lb/>nitione 18. &longs;eptimi &longs;ic explicatur: Quadratus numerus e&longs;t, qui &longs;ub duobus <lb/>æqualibus numeris continetur, ide&longs;t, qui fit ex ductu vnius numeri in &longs;e ip<lb/><figure id="fig16"/><lb/>&longs;um, vt &longs;i ducantur 3. in 3. fient 9. qui continetur &longs;ub duobus <lb/>ternarijs; omnes autem ternarij &longs;unt æquales. </s>

<s>is autem nu<lb/>merus dicetur quadratus, quia, vt apparet in figura, nouem <lb/>ip&longs;ius vnitates po&longs;&longs;unt in plano ita ad inuicem collocari, vt <lb/>referant quadratum; & &longs;icuti quadratum geometricum ha<lb/>bet latera æqualia, ita etiam quadratum arithmeticum: &longs;i<lb/>ue numerus quadratus, habet &longs;ua latera æqualia, quot enim vnitates &longs;unt <lb/>in vno, tot etiam &longs;unt in reliquis, vt in præ&longs;enti &longs;unt tres vnitates in &longs;ingulis <lb/>lateribus. </s>

<s>pr&ecedil;terea quemadmodum quadratum geometricum re&longs;olni pote&longs;t <lb/>in plura quadrata, ita etiam arithmeticum, vt præ&longs;ens, qui re&longs;oluitur in <lb/>quatuor quadrata arithmetica. </s>

<s><expan abbr="Neq;">Neque</expan> enim pote&longs;t quilibet numerus, vt opi<lb/>nantur ageometreti, in hunc modum di&longs;poni, &longs;ed &longs;olum ij, qui producuntur <lb/>ex multiplicatione numeri alicuius in &longs;e ip&longs;um.</s>

<s>&

<lb/><expan abbr="o&itilde;bus">oimbus</expan> bis in&longs;unt in oratione, quid e&longs;t <expan abbr="declarãte">declarante</expan>, ibi quidem linea, hic vero numerus)<emph.end type="italics"/>

<s>Per altera parte longius, intelligit numerum, qui producitur à duobus <lb/><figure id="fig17"/><lb/>numeris inæqualibus inuicem multiplicatis, qualis e&longs;t <lb/>duodenarius, qui ex ductu trium in quatuor produci<lb/>tur, & refert figuram altera parte longiorem, &longs;iue, vt <lb/>ait Boetius longilateram, cuius vnum latus e&longs;t maius <lb/>altero, vt in appo&longs;ita figura videre licet. </s>

<lb/>quia locus hic benè exponitur à Toleto, & melius etiam à Conymbr. </s>

<s>atque hæc <lb/>&longs;unt, quæ ex Mathematicis petenda erant, ad huius <lb/>loci intelligentiam.</s>

<s>addam

<lb/>tantummodo quædam, quæ ad perfectam eius intelligentiam de&longs;iderantur.

<s>Sciendum igitur primò, nu&longs;quam ab Euclide definiri rectum, circulare,

<lb/>impar, par, primum, compo&longs;itum, æquilaterum, nec altera parte longius:

<lb/><expan abbr="ver&utilde;">verum</expan> ab ip&longs;o in definitionibus primi definiri lineam rectam, non tamen cir

<lb/>cularem expre&longs;sè. </s>

<s>in definitionibus deinde &longs;eptimi definiri <expan abbr="numer&utilde;">numerum</expan> parem,

<lb/>& imparem, item numerum primum, & compofitum, & æquilaterum, & al

<lb/>tera parte longiorem. </s>

<s>ex quibus definitionibus po&longs;&longs;unt erui definitiones re

<lb/>cti, circularis, imparis, & cæterorum, quorum hic Ari&longs;toteles meminit.

<s>Cæterum Euclides definitione 11. &longs;eptimi, &longs;ic definit numerum primum:

<lb/>primus numerus e&longs;t, quem vnitas &longs;ola metitur. </s>

<s>numerus autem, vel vnitas

<lb/>metiri dicitur alium numerum, quando &longs;æpius repetita ip&longs;um omnino ad

<lb/>æquat, vt ternarius metitur nouenarium, quia ter repetitus ip&longs;um ad vn

<lb/>guem explet. </s>

<s>illi igitur numeri dicuntur ab Arithmeticis primi, qui à nullo

<lb/>alio, præterquam ab vnitate men&longs;urantur, quales &longs;unt, 2. 3. 5. 7. &c. </s>

<s>Defi

<lb/>nitione verò 13. definit numerum compo&longs;itum &longs;ic; compo&longs;itus numerus e&longs;t,

<lb/>quem numerus qui&longs;piam metitur, vt &longs;enarius erit compo&longs;itus, quia ip&longs;um

<lb/>binarius metitur, nam ter repetitus, ip&longs;i perfectè adæquatur.</s>

<s>Per æquilaterum, intelligit quadratum, quadratus autem numerus defi

<lb/>nitione 18. &longs;eptimi &longs;ic explicatur: Quadratus numerus e&longs;t, qui &longs;ub duobus

<lb/>æqualibus numeris continetur, ide&longs;t, qui fit ex ductu vnius numeri in &longs;e ip

<lb/>&longs;um, vt &longs;i ducantur 3. in 3. fient 9. qui continetur &longs;ub duobus

<lb/>ternarijs; omnes autem ternarij &longs;unt æquales. </s>

<s>is autem nu

<lb/>merus dicetur quadratus, quia, vt apparet in figura, nouem

<lb/>ip&longs;ius vnitates po&longs;&longs;unt in plano ita ad inuicem collocari, vt

<lb/>referant quadratum; & &longs;icuti quadratum geometricum ha

<lb/>bet latera æqualia, ita etiam quadratum arithmeticum: &longs;i

<lb/>ue numerus quadratus, habet &longs;ua latera æqualia, quot enim vnitates &longs;unt

<lb/>in vno, tot etiam &longs;unt in reliquis, vt in præ&longs;enti &longs;unt tres vnitates in &longs;ingulis

<lb/>lateribus. </s>

<s>pr&ecedil;terea quemadmodum quadratum geometricum re&longs;olni pote&longs;t

<lb/>in plura quadrata, ita etiam arithmeticum, vt præ&longs;ens, qui re&longs;oluitur in

<lb/>quatuor quadrata arithmetica. </s>

<s><expan abbr="Neq;">Neque</expan> enim pote&longs;t quilibet numerus, vt opi

<lb/>nantur ageometreti, in hunc modum di&longs;poni, &longs;ed &longs;olum ij, qui producuntur

<lb/>ex multiplicatione numeri alicuius in &longs;e ip&longs;um.</s>

<s>Per altera parte longius, intelligit numerum, qui producitur à duobus

<lb/>numeris inæqualibus inuicem multiplicatis, qualis e&longs;t

<lb/>duodenarius, qui ex ductu trium in quatuor produci

<lb/>tur, & refert figuram altera parte longiorem, &longs;iue, vt

<lb/>ait Boetius longilateram, cuius vnum latus e&longs;t maius

<lb/>altero, vt in appo&longs;ita figura videre licet. </s>

<s>atque hæc

<lb/>&longs;unt, quæ ex Mathematicis petenda erant, ad huius

<lb/>loci intelligentiam.</s>

<s>Tex. 11. <emph type="italics"/>(Per &longs;e autem, & &longs;ecundum quod ip&longs;um, idem, vt per &longs;e lineæ inest<emph.end type="italics"/><pb pagenum="49"/><emph type="italics"/>punctum, & rectum; etenim &longs;ecundum quod linea, & triangulo, &longs;ecundum quod <lb/>triangulum duo recti: etenim per &longs;e triangulum duobus rectis æquale. </s>

<lb/>triangulum duo recti: etenim per &longs;e triangulum duobus rectis æquale. </s>

<s>Vniuer&longs;ale <lb/>autem e&longs;t tunc, quando in quolibet, & primo mon&longs;tratur, vt duos rectos habere, <lb/><expan abbr="neq;">neque</expan> figuræ e&longs;t vniuer&longs;ale, quamuis e&longs;t mon&longs;irare de figura, quod duos rectos habet, <lb/>&longs;ed non de qualibet figura, <expan abbr="neq;">neque</expan> vtitur qualibet figura monstrans, quadrangulum <lb/>enim figura a quidem est, non habet autem duobus rectis æquales. </s>

<s>Aequicrus verò <lb/>babet quidem <expan abbr="quodcunq;">quodcunque</expan> duobus rectis æquales, &longs;ed non primò, &longs;ed triangulum <lb/>prius. </s>

<s>quod igitur quoduis primum mon&longs;tratur duos rectos habens, aut <expan abbr="quodcunq;">quodcunque</expan> <lb/>aliud, huic primo ine&longs;t vniuer&longs;ale, & demonstratio de hoc vniuer&longs;aliter e&longs;t, de alijs <lb/>verò quodammodo, non per &longs;e, <expan abbr="neq;">neque</expan> de æquicrure e&longs;t vniuer&longs;aliter, &longs;ed in plus)<emph.end type="italics"/> pro <lb/>quorum intelligentia nece&longs;&longs;aria &longs;unt ea, quæ primo Priorum &longs;ecto 3. cap.

<s>Vniuer&longs;ale

<lb/>autem e&longs;t tunc, quando in quolibet, & primo mon&longs;tratur, vt duos rectos habere,

1. <lb/>&longs;crip&longs;imus. </s>

<lb/><expan abbr="neq;">neque</expan> figuræ e&longs;t vniuer&longs;ale, quamuis e&longs;t mon&longs;irare de figura, quod duos rectos habet,

<lb/>&longs;ed non de qualibet figura, <expan abbr="neq;">neque</expan> vtitur qualibet figura monstrans, quadrangulum

<s>deinde memineris figuram vniuer&longs;aliorem e&longs;&longs;e triangulo, & tri<lb/>angulum vniuer&longs;alius æquicrure. </s>

<lb/>enim figura a quidem est, non habet autem duobus rectis æquales. </s>

<s>quando ait (vt duos rectos habere) vult <lb/>dicere, habere duos angulos rectos non actu, &longs;ed potentia; quæ affectio e&longs;t <lb/>trianguli, quia, vt &longs;uperius diximus, habet tres angulos æquales duobus <lb/>rectis angulis: quæ proprietas vniuer&longs;aliter, & primò competit triangulo. <lb/></s>

<s>Aequicrus verò

<lb/>babet quidem <expan abbr="quodcunq;">quodcunque</expan> duobus rectis æquales, &longs;ed non primò, &longs;ed triangulum

<lb/>prius. </s>

<s>quod igitur quoduis primum mon&longs;tratur duos rectos habens, aut <expan abbr="quodcunq;">quodcunque</expan>

<lb/>aliud, huic primo ine&longs;t vniuer&longs;ale, & demonstratio de hoc vniuer&longs;aliter e&longs;t, de alijs

<lb/>verò quodammodo, non per &longs;e, <expan abbr="neq;">neque</expan> de æquicrure e&longs;t vniuer&longs;aliter, &longs;ed in plus)<emph.end type="italics"/> pro

<lb/>quorum intelligentia nece&longs;&longs;aria &longs;unt ea, quæ primo Priorum &longs;ecto 3. cap.

<lb/>&longs;crip&longs;imus. </s>

<s>deinde memineris figuram vniuer&longs;aliorem e&longs;&longs;e triangulo, & tri

<lb/>angulum vniuer&longs;alius æquicrure. </s>

<s>quando ait (vt duos rectos habere) vult

<lb/>dicere, habere duos angulos rectos non actu, &longs;ed potentia; quæ affectio e&longs;t

<lb/>trianguli, quia, vt &longs;uperius diximus, habet tres angulos æquales duobus

<lb/>rectis angulis: quæ proprietas vniuer&longs;aliter, & primò competit triangulo.

<s>non autem figuræ, quia figura e&longs;t vniuer&longs;alior. </s>

<s><expan abbr="neq;">neque</expan> i&longs;o&longs;celi, quia i&longs;o&longs;celes e&longs;t <lb/>re&longs;trictius triangulo. </s>

<s><expan abbr="neq;">neque</expan> i&longs;o&longs;celi, quia i&longs;o&longs;celes e&longs;t

<lb/>re&longs;trictius triangulo. </s>

<s>omittimus reliqua &longs;ingillatim exponere, tum quia &longs;a<lb/>tis clara &longs;unt, tum quia ab interpretibus benè explicantur.</s>

<s>omittimus reliqua &longs;ingillatim exponere, tum quia &longs;a

<lb/>tis clara &longs;unt, tum quia ab interpretibus benè explicantur.</s>

<s>Tex. 13. <emph type="italics"/>(Si quis igitur mon&longs;trauerit, quod rectæ <expan abbr="nõ">non</expan> coincidunt, videbitur <expan abbr="vtiq;">vtique</expan> <lb/>buius e&longs;&longs;e demonstratio, eo quod in omnibus e&longs;t rectis; non e&longs;t autem: &longs;i quidem <lb/>non quoniam &longs;ic æquales, fit hoc, &longs;ed &longs;ecundum quod <expan abbr="quomodocunq;">quomodocunque</expan> æquales)<emph.end type="italics"/> pro<lb/>ponit tres errores, qui circa demon&longs;trationem de vniuer&longs;ali contingunt, <lb/>quos omnes Geometricis exemplis illu&longs;trat; affert autem primo pro tertio <lb/>errore duo exempla, quorum primum in præmi&longs;&longs;is verbis continetur, <expan abbr="atq;">atque</expan> <lb/>ex 28. primi Elem.

<s>Tex. 13. <emph type="italics"/>(Si quis igitur mon&longs;trauerit, quod rectæ <expan abbr="nõ">non</expan> coincidunt, videbitur <expan abbr="vtiq;">vtique</expan>

<lb/>buius e&longs;&longs;e demonstratio, eo quod in omnibus e&longs;t rectis; non e&longs;t autem: &longs;i quidem

de&longs;umitur, quam propterea primo loco exponendam <lb/><figure id="fig18"/><lb/>cen&longs;ui. </s>

<lb/>non quoniam &longs;ic æquales, fit hoc, &longs;ed &longs;ecundum quod <expan abbr="quomodocunq;">quomodocunque</expan> æquales)<emph.end type="italics"/> pro

<lb/>ponit tres errores, qui circa demon&longs;trationem de vniuer&longs;ali contingunt,

<s>Quando igitur duæ rectæ con&longs;titu<lb/>tæ fuerint, vt A B, C D, in quas alia recta, <lb/>vt G F, incidens, faciat duos angulos in<lb/>ternos, re&longs;pectu rectarum A B, C D, & ad <lb/>ea&longs;dem partes rectæ E F, vt &longs;unt ex parte <lb/>&longs;ini&longs;tra anguli A G H, C H G; exparte ve<lb/>rò dextra B G H, D H G; &longs;i <expan abbr="inquã">inquam</expan> linea E F, <lb/>fecerit duos illos angulos ex parte &longs;ini&longs;tra &longs;imul &longs;umptos, æquales duobus <lb/>rectis angulis, vel duos ex parte dextra pariter æquales duobus rectis, pro<lb/>bat Euclides rectas A B, C D, non concurrere, &longs;iue parallelas e&longs;&longs;e. </s>

<lb/>quos omnes Geometricis exemplis illu&longs;trat; affert autem primo pro tertio

<lb/>errore duo exempla, quorum primum in præmi&longs;&longs;is verbis continetur, <expan abbr="atq;">atque</expan>

<s>Verum, <lb/>quia linea E F, pote&longs;t facere aliquando prædictos angulos non <expan abbr="tant&utilde;">tantum</expan> æqua<lb/>les duobus rectis, verum etiam rectos, quo etiam modo <expan abbr="probar&etilde;tur">probarentur</expan> cædem <lb/>lineæ e&longs;&longs;e parallelæ, vt in &longs;equenti figura, cum &longs;int anguli A G I, C I G, re<lb/><figure id="fig19"/><lb/>cti, probabitur de rectis A B, C D, æquidi&longs;tan<lb/>tia. </s>

<lb/>ex 28. primi Elem.

<s>Ex his facile textum in hunc modum expo<lb/>nemus; &longs;i quis igitur mon&longs;trauerit, quod rectæ <lb/>A B, C D, nunquam coincidunt, etiam&longs;i in in&longs;i<lb/>nitum producantur, &longs;eu quod &longs;unt æquidi&longs;tantes, <lb/>quando anguli prædicti interni &longs;unt duo recti, <lb/>videbitur <expan abbr="vtiq;">vtique</expan> huius e&longs;&longs;e demon&longs;tratio de vniuer&longs;ali per &longs;e, & de primo &longs;u<pb pagenum="50"/>biecto, vel &longs;ecundum quod ip&longs;um, eò quod probatur vniuer&longs;aliter de lineis <lb/>omnibus habentibus prædictos angulos rectos. </s>

de&longs;umitur, quam propterea primo loco exponendam

<s>non autem de omni, &longs;ecun<lb/>dum quod ip&longs;um, &longs;i quidem non competit affectio hæc, e&longs;&longs;e parallelas, li<lb/>neis habentibus illos angulos rectos actu; &longs;ed primò, & vniuer&longs;aliter, & &longs;e<lb/>cundum quod ip&longs;um competit lineis habentibus illos angulos æquales duo<lb/>bus rectis, <expan abbr="quomodocunq;">quomodocunque</expan> æquales &longs;int duobus rectis, &longs;iue ambo &longs;int recti, <lb/>&longs;iue vnus acutus, alter obtu&longs;us, &longs;ed tamen ambo &longs;imul æquentur duobus re<lb/>ctis, quales &longs;unt lineæ primæ figuræ. </s>

<lb/>cen&longs;ui. </s>

<s>In tertio igitur errore, vniuer&longs;ale exi<lb/>&longs;tit quidem, & habet nomen, &longs;ed tamen prætermittetur, &longs;eu &longs;trictius &longs;ume<lb/>tur, quam oportet. </s>

<s>Quando igitur duæ rectæ con&longs;titu

<lb/>tæ fuerint, vt A B, C D, in quas alia recta,

<s>alij latini, quos quidem viderim, præter Zabarellana <lb/>perperam omnino ob mathematicarum ignorantiam, exemplum i&longs;tud in<lb/>terpretantur.</s>

<lb/>vt G F, incidens, faciat duos angulos in

<lb/>ternos, re&longs;pectu rectarum A B, C D, & ad

<lb/>ea&longs;dem partes rectæ E F, vt &longs;unt ex parte

<lb/>&longs;ini&longs;tra anguli A G H, C H G; exparte ve

<lb/>rò dextra B G H, D H G; &longs;i <expan abbr="inquã">inquam</expan> linea E F,

<lb/>fecerit duos illos angulos ex parte &longs;ini&longs;tra &longs;imul &longs;umptos, æquales duobus

<lb/>rectis angulis, vel duos ex parte dextra pariter æquales duobus rectis, pro

<lb/>bat Euclides rectas A B, C D, non concurrere, &longs;iue parallelas e&longs;&longs;e. </s>

<s>Verum,

<lb/>quia linea E F, pote&longs;t facere aliquando prædictos angulos non <expan abbr="tant&utilde;">tantum</expan> æqua

<lb/>les duobus rectis, verum etiam rectos, quo etiam modo <expan abbr="probar&etilde;tur">probarentur</expan> cædem

<lb/>lineæ e&longs;&longs;e parallelæ, vt in &longs;equenti figura, cum &longs;int anguli A G I, C I G, re

<lb/>cti, probabitur de rectis A B, C D, æquidi&longs;tan

<s>Ex his facile textum in hunc modum expo

<lb/>nemus; &longs;i quis igitur mon&longs;trauerit, quod rectæ

<lb/>A B, C D, nunquam coincidunt, etiam&longs;i in in&longs;i

<lb/>nitum producantur, &longs;eu quod &longs;unt æquidi&longs;tantes,

<lb/>quando anguli prædicti interni &longs;unt duo recti,

<lb/>videbitur <expan abbr="vtiq;">vtique</expan> huius e&longs;&longs;e demon&longs;tratio de vniuer&longs;ali per &longs;e, & de primo &longs;u<pb pagenum="50"/>biecto, vel &longs;ecundum quod ip&longs;um, eò quod probatur vniuer&longs;aliter de lineis

<lb/>omnibus habentibus prædictos angulos rectos. </s>

<s>non autem de omni, &longs;ecun

<lb/>dum quod ip&longs;um, &longs;i quidem non competit affectio hæc, e&longs;&longs;e parallelas, li

<lb/>neis habentibus illos angulos rectos actu; &longs;ed primò, & vniuer&longs;aliter, & &longs;e

<lb/>cundum quod ip&longs;um competit lineis habentibus illos angulos æquales duo

<lb/>bus rectis, <expan abbr="quomodocunq;">quomodocunque</expan> æquales &longs;int duobus rectis, &longs;iue ambo &longs;int recti,

<lb/>&longs;iue vnus acutus, alter obtu&longs;us, &longs;ed tamen ambo &longs;imul æquentur duobus re

<lb/>ctis, quales &longs;unt lineæ primæ figuræ. </s>

<s>In tertio igitur errore, vniuer&longs;ale exi

<lb/>&longs;tit quidem, & habet nomen, &longs;ed tamen prætermittetur, &longs;eu &longs;trictius &longs;ume

<lb/>tur, quam oportet. </s>

<s>alij latini, quos quidem viderim, præter Zabarellana

<lb/>perperam omnino ob mathematicarum ignorantiam, exemplum i&longs;tud in

<lb/>terpretantur.</s>

<s>Ibidem <emph type="italics"/>(Et &longs;i triangulum non e&longs;&longs;et aliud, quam I&longs;o&longs;celes, &longs;ecundum quod I&longs;o<lb/>&longs;celes videretur <expan abbr="vtiq;">vtique</expan> ine&longs;&longs;e)<emph.end type="italics"/> i&longs;tud e&longs;t &longs;ecundum exemplum tertij erroris. </s>

<s>Ibidem <emph type="italics"/>(Et &longs;i triangulum non e&longs;&longs;et aliud, quam I&longs;o&longs;celes, &longs;ecundum quod I&longs;o

<lb/>&longs;celes videretur <expan abbr="vtiq;">vtique</expan> ine&longs;&longs;e)<emph.end type="italics"/> i&longs;tud e&longs;t &longs;ecundum exemplum tertij erroris. </s>

<s>Por<lb/>rò cum tres &longs;int &longs;pecies triangulorum, æquilaterum, I&longs;o&longs;celes, Scalenum, &longs;i <lb/>accideret, vt ex illis tribus vna tantum &longs;pecies, v. g. </s>

<s>Por

<lb/>rò cum tres &longs;int &longs;pecies triangulorum, æquilaterum, I&longs;o&longs;celes, Scalenum, &longs;i

<lb/>accideret, vt ex illis tribus vna tantum &longs;pecies, v. g. I&longs;o&longs;celes in mundo re

<lb/>periretur; <expan abbr="tunc&qacute;">tuncque</expan>; qui&longs;piam de I&longs;o&longs;cele o&longs;tenderet affectionem quampiam,

<lb/>putans &longs;e <expan abbr="o&longs;t&etilde;di&longs;&longs;e">o&longs;tendi&longs;&longs;e</expan> pa&longs;&longs;ionem de proprio &longs;ubiecto, & primo, falleretur, quia

<lb/>aifectio illa competeret I&longs;o&longs;celi, non vt huic &longs;peciei I&longs;o&longs;celis, &longs;ed quatenus

<lb/>e&longs;t triangulum, cui primo, & per &longs;e, & &longs;ecundum quod ip&longs;um conuenit. </s>

<s>I&longs;o&longs;celes in mundo re<lb/>periretur; <expan abbr="tunc&qacute;">tuncque</expan>; qui&longs;piam de I&longs;o&longs;cele o&longs;tenderet affectionem quampiam, <lb/>putans &longs;e <expan abbr="o&longs;t&etilde;di&longs;&longs;e">o&longs;tendi&longs;&longs;e</expan> pa&longs;&longs;ionem de proprio &longs;ubiecto, & primo, falleretur, quia <lb/>aifectio illa competeret I&longs;o&longs;celi, non vt huic &longs;peciei I&longs;o&longs;celis, &longs;ed quatenus <lb/>e&longs;t triangulum, cui primo, & per &longs;e, & &longs;ecundum quod ip&longs;um conuenit. </s>

<s>hoc <lb/>loco di&longs;ce&longs;&longs;imus à Zabarella, qui putat i&longs;tud e&longs;&longs;e exemplum primi erroris, <lb/>cum verba textus adeo clara &longs;int, vt expo&longs;itionem illius nullo modo admit<lb/>tant. </s>

<s>&longs;unt autem hæc textus verba <emph type="italics"/>(Et &longs;i triangulum non e&longs;&longs;et aliud, quam I&longs;o<lb/>&longs;celes, &longs;ecundum quod I&longs;o&longs;celes videretur <expan abbr="vtiq;">vtique</expan> ine&longs;&longs;e)<emph.end type="italics"/> quibus verbis manife&longs;tè <lb/>apparet Ari&longs;t.

accipere pro &longs;ubiecto vniuer&longs;ali non indiuiduum vnum, vt in <lb/>primo errore contingit, &longs;ed &longs;peciem loco generis, &longs;cilicet I&longs;o&longs;celes, quod <lb/>e&longs;t &longs;pecies trianguli pro genere ip&longs;o, nimirum pro Triangulo. </s>

<s>ait enim, &longs;i <lb/>non e&longs;&longs;et aliud, quam I&longs;o&longs;celes, <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> quod I&longs;o&longs;celes: quibus verbis cla<lb/>rè &longs;peciem, non indiuiduum, &longs;igni&longs;icat, ex his duobus exemplis manife&longs;tus <lb/>e&longs;t tertius error, qui erat, quando erat <emph type="italics"/>(vt in parte totum)<emph.end type="italics"/> <expan abbr="quod&qacute;">quodque</expan>; illis verbis <lb/>expo&longs;uerat <emph type="italics"/>(vei contingit etiam, vt in parte totum, in quo mon&longs;tratur: ijs emm, <lb/>quæ &longs;unt in parte inerit quidem demon&longs;tratio, & erit de omni, &longs;ed tamen non erit <lb/>buius primi vniuer&longs;aliter demon&longs;tratio. </s>

<s>hoc

<lb/>loco di&longs;ce&longs;&longs;imus à Zabarella, qui putat i&longs;tud e&longs;&longs;e exemplum primi erroris,

<lb/>cum verba textus adeo clara &longs;int, vt expo&longs;itionem illius nullo modo admit

<s>&longs;unt autem hæc textus verba <emph type="italics"/>(Et &longs;i triangulum non e&longs;&longs;et aliud, quam I&longs;o

<lb/>&longs;celes, &longs;ecundum quod I&longs;o&longs;celes videretur <expan abbr="vtiq;">vtique</expan> ine&longs;&longs;e)<emph.end type="italics"/> quibus verbis manife&longs;tè

<lb/>apparet Ari&longs;t.

accipere pro &longs;ubiecto vniuer&longs;ali non indiuiduum vnum, vt in

<lb/>primo errore contingit, &longs;ed &longs;peciem loco generis, &longs;cilicet I&longs;o&longs;celes, quod

<lb/>e&longs;t &longs;pecies trianguli pro genere ip&longs;o, nimirum pro Triangulo. </s>

<s>ait enim, &longs;i

<lb/>non e&longs;&longs;et aliud, quam I&longs;o&longs;celes, <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> quod I&longs;o&longs;celes: quibus verbis cla

<lb/>rè &longs;peciem, non indiuiduum, &longs;igni&longs;icat, ex his duobus exemplis manife&longs;tus

<lb/>e&longs;t tertius error, qui erat, quando erat <emph type="italics"/>(vt in parte totum)<emph.end type="italics"/> <expan abbr="quod&qacute;">quodque</expan>; illis verbis

<lb/>expo&longs;uerat <emph type="italics"/>(vei contingit etiam, vt in parte totum, in quo mon&longs;tratur: ijs emm,

<lb/>quæ &longs;unt in parte inerit quidem demon&longs;tratio, & erit de omni, &longs;ed tamen non erit

<lb/>buius primi vniuer&longs;aliter demon&longs;tratio. </s>

<s>dico auttm huius primi, &longs;ecundum quod

<lb/>buius demonstrationem, quando &longs;it primi vniuer&longs;aliter)<emph.end type="italics"/> ide&longs;t, quando vniuer&longs;ale

<lb/>&longs;ubiectum exi&longs;tit quidem, &longs;ed tamen non de ip&longs;o &longs;it demon&longs;tratio, &longs;ed de ali

<lb/>qua parte ip&longs;ius, v. g. de &longs;pecie aliqua demon&longs;tratur aliquid, quod deberet

<lb/>o&longs;tendi primò de ip&longs;o vniuer&longs;ali, cum illi primò competat.</s>

<s>dico auttm huius primi, &longs;ecundum quod <lb/>buius demonstrationem, quando &longs;it primi vniuer&longs;aliter)<emph.end type="italics"/> ide&longs;t, quando vniuer&longs;ale <lb/>&longs;ubiectum exi&longs;tit quidem, &longs;ed tamen non de ip&longs;o &longs;it demon&longs;tratio, &longs;ed de ali<lb/>qua parte ip&longs;ius, v. g. </s>

<s>de &longs;pecie aliqua demon&longs;tratur aliquid, quod deberet <lb/>o&longs;tendi primò de ip&longs;o vniuer&longs;ali, cum illi primò competat.</s>

<s>Ibidem <emph type="italics"/>(Et proportionale, quod alternatim, &longs;ecundum quod numeri, & &longs;ecun<lb/>dum quod lineæ, & &longs;ecundum quod &longs;olida, & &longs;ecundum quod tempora: quemad<lb/>modum & mon&longs;trabatur aliquando &longs;eor&longs;um, contingens <expan abbr="vtiq;">vtique</expan> de omnibus vnica <lb/>demon&longs;tratione mon&longs;irari; &longs;ed quia non &longs;unt nominatum quidam omnia hæc vnum, <lb/>numeri, longitudines, tempora &longs;olida, & &longs;pecie differunt à &longs;einuicem &longs;cor&longs;um <expan abbr="ac-cipiebãtur">ac<lb/>cipiebantur</expan>. </s>

<s>Ibidem <emph type="italics"/>(Et proportionale, quod alternatim, &longs;ecundum quod numeri, & &longs;ecun

<lb/>dum quod lineæ, & &longs;ecundum quod &longs;olida, & &longs;ecundum quod tempora: quemad

<s>nunc autem vniuer &longs;aliter mon&longs;tratur, <expan abbr="neq;">neque</expan> enim &longs;ecundum quod lineæ, <lb/>aut &longs;ecundum quod numeri, inerat; &longs;ed &longs;ecundum quod boc, quod vniuer &longs;ale &longs;up<lb/>ponunt e&longs;&longs;e)<emph.end type="italics"/> affert exemplum &longs;ecundi erroris, quiaccidit, quando vniuer&longs;a<lb/>le exi&longs;tit quidem, &longs;ed tamen e&longs;t innominatum, pro cuius explicatione &longs;cien<pb pagenum="51"/>dum quid &longs;it alterna proportio. </s>

<lb/>modum & mon&longs;trabatur aliquando &longs;eor&longs;um, contingens <expan abbr="vtiq;">vtique</expan> de omnibus vnica

<lb/>demon&longs;tratione mon&longs;irari; &longs;ed quia non &longs;unt nominatum quidam omnia hæc vnum,

<s>Alternam igitur proportionem definit Eu<lb/>clides definitione 12. quinti, &longs;ic, e&longs;t &longs;umptio antecedentis ad <expan abbr="anteced&etilde;tem">antecedentem</expan>, <lb/><figure id="fig20"/><lb/>& con&longs;equentis ad con&longs;equentem. </s>

<lb/>numeri, longitudines, tempora &longs;olida, & &longs;pecie differunt à &longs;einuicem &longs;cor&longs;um <expan abbr="ac-cipiebãtur">ac

<lb/>cipiebantur</expan>. </s>

<s>nunc autem vniuer &longs;aliter mon&longs;tratur, <expan abbr="neq;">neque</expan> enim &longs;ecundum quod lineæ,

<lb/>aut &longs;ecundum quod numeri, inerat; &longs;ed &longs;ecundum quod boc, quod vniuer &longs;ale &longs;up

<lb/>ponunt e&longs;&longs;e)<emph.end type="italics"/> affert exemplum &longs;ecundi erroris, quiaccidit, quando vniuer&longs;a

<lb/>le exi&longs;tit quidem, &longs;ed tamen e&longs;t innominatum, pro cuius explicatione &longs;cien<pb pagenum="51"/>dum quid &longs;it alterna proportio. </s>

<s>Alternam igitur proportionem definit Eu

<lb/>clides definitione 12. quinti, &longs;ic, e&longs;t &longs;umptio antecedentis ad <expan abbr="anteced&etilde;tem">antecedentem</expan>,

<lb/>& con&longs;equentis ad con&longs;equentem. </s>

<s>Explico, exponantur qua

<lb/>tuor quantitates proportionales, v.g. vt 6. ad 3. ita &longs;int 4. ad

<lb/>2. &longs;i igitur argumentemur &longs;ic, vt 6. ad 3. ita 4. ad 2. ergo al

<lb/>ternatim erit, vt 6. ad 4. ita 3. ad 2. &longs;iue dixerimus, vt pri

<lb/>mum ad &longs;ecundum, ita tertium ad quartum, igitur alterna

<lb/>tim erit, vt primum ad tertium, ita &longs;ecundum ad quartum: valebit con&longs;e

<lb/>quentia; quæ quidem probatur deinde propo&longs;itione 16. quinti de magnitu

<lb/>dinibus, hoc e&longs;t in vniuer&longs;um de lineis, &longs;uperficiebus, & &longs;olidis. </s>

<s>Explico, exponantur qua<lb/>tuor quantitates proportionales, v.g. vt 6. ad 3. ita &longs;int 4. ad <lb/>2. &longs;i igitur argumentemur &longs;ic, vt 6. ad 3. ita 4. ad 2. ergo al<lb/>ternatim erit, vt 6. ad 4. ita 3. ad 2. &longs;iue dixerimus, vt pri<lb/>mum ad &longs;ecundum, ita tertium ad quartum, igitur alterna<lb/>tim erit, vt primum ad tertium, ita &longs;ecundum ad quartum: valebit con&longs;e<lb/>quentia; quæ quidem probatur deinde propo&longs;itione 16. quinti de magnitu<lb/>dinibus, hoc e&longs;t in vniuer&longs;um de lineis, &longs;uperficiebus, & &longs;olidis. </s>

<s>quando igi

<s>quando igi<lb/>tur Ari&longs;t.

<lb/>tur Ari&longs;t.

ait, mon&longs;tramus proportionale, ide&longs;t, qua&longs;uis quatuor quantita<lb/>tes proportionales, habere hanc proprietatem, vt &longs;int etiam alternatim <lb/>proportionales, & non mon&longs;tramus vnica demon&longs;tratione de omni quouis <lb/>proportionali, &longs;ed &longs;eparatim de magnitudinibus in 16. quinti, de numeris <lb/>in 13. &longs;eptimi, & &longs;eor&longs;um de temporibus in a&longs;tronomia, vel phy&longs;ica; hoc <lb/>modo non o&longs;tendimus vniuer&longs;aliter de primo &longs;ubiecto, quia talis affectio <lb/>conuenit &longs;ingulis, non vt numeri, aut ma gnitudines, aut tempora &longs;unt, &longs;ed <lb/>&longs;ecundum quandam naturam illis omnibus communem, cui primò illa pa&longs;<lb/>&longs;io debetur; quæ quidem natura communis nomine caret, & propterea e&longs;t <lb/>cau&longs;a erroris.</s>

ait, mon&longs;tramus proportionale, ide&longs;t, qua&longs;uis quatuor quantita

<lb/>tes proportionales, habere hanc proprietatem, vt &longs;int etiam alternatim

<lb/>proportionales, & non mon&longs;tramus vnica demon&longs;tratione de omni quouis

<lb/>proportionali, &longs;ed &longs;eparatim de magnitudinibus in 16. quinti, de numeris

<lb/>in 13. &longs;eptimi, & &longs;eor&longs;um de temporibus in a&longs;tronomia, vel phy&longs;ica; hoc

<lb/>modo non o&longs;tendimus vniuer&longs;aliter de primo &longs;ubiecto, quia talis affectio

<lb/>conuenit &longs;ingulis, non vt numeri, aut ma gnitudines, aut tempora &longs;unt, &longs;ed

<lb/>&longs;ecundum quandam naturam illis omnibus communem, cui primò illa pa&longs;

<lb/>&longs;io debetur; quæ quidem natura communis nomine caret, & propterea e&longs;t

<lb/>cau&longs;a erroris.</s>

<s><emph type="italics"/>Nunc autem vniuer&longs;aliter demon&longs;tratur)<emph.end type="italics"/> nu&longs;quam apud Mathematicos in<lb/>uenio hanc demon&longs;trationem vniuer&longs;alem de illo communi omnibus præ<lb/>dictis, quare dicendum cum Zabarella, illud, nunc, e&longs;&longs;e intelligendum &longs;ic, <lb/>nunc autem, ide&longs;t, in præ&longs;entia autem deberet vniuer&longs;aliter demon&longs;trari, <lb/>quod tamen cum non &longs;iat, contingit nos decipi putantes vniuer&longs;aliter de<lb/>mon&longs;tra&longs;&longs;e. </s>

<s><emph type="italics"/>Nunc autem vniuer&longs;aliter demon&longs;tratur)<emph.end type="italics"/> nu&longs;quam apud Mathematicos in

<lb/>uenio hanc demon&longs;trationem vniuer&longs;alem de illo communi omnibus præ

<s>vel dicendum i&longs;tud verificari tantum de lineis, &longs;uperficiebus, & <lb/>&longs;olidis, de quibus &longs;imul in vnica natura communi, quæ e&longs;t magnitudo, de<lb/>mon&longs;tratur in 16. quinti vniuer&longs;aliter. </s>

<lb/>dictis, quare dicendum cum Zabarella, illud, nunc, e&longs;&longs;e intelligendum &longs;ic,

<lb/>nunc autem, ide&longs;t, in præ&longs;entia autem deberet vniuer&longs;aliter demon&longs;trari,

<s><expan abbr="atq;">atque</expan> hoc modo explicatum e&longs;t exem<lb/>plum &longs;ecundi erroris, qui verbis illis <emph type="italics"/>(Vel &longs;it quidem, &longs;ed innominatum &longs;it in <lb/>rebus &longs;pecie differentibus)<emph.end type="italics"/> continebatur.</s>

<lb/>quod tamen cum non &longs;iat, contingit nos decipi putantes vniuer&longs;aliter de

<lb/>mon&longs;tra&longs;&longs;e. </s>

<s>vel dicendum i&longs;tud verificari tantum de lineis, &longs;uperficiebus, &

<lb/>&longs;olidis, de quibus &longs;imul in vnica natura communi, quæ e&longs;t magnitudo, de

<lb/>mon&longs;tratur in 16. quinti vniuer&longs;aliter. </s>

<s><expan abbr="atq;">atque</expan> hoc modo explicatum e&longs;t exem

<lb/>plum &longs;ecundi erroris, qui verbis illis <emph type="italics"/>(Vel &longs;it quidem, &longs;ed innominatum &longs;it in

<lb/>rebus &longs;pecie differentibus)<emph.end type="italics"/> continebatur.</s>

Line 2686

Line 3589

<s>Ibidem <emph type="italics"/>(Propter hoc &longs;i quis mon&longs;trauerit &longs;ingulum triangulum. </s>

<s>demon&longs;tratio<lb/>ne aut vna, aut altera, quod duos rectos habet vnumquodque, <expan abbr="æquilatei&utilde;">æquilateium</expan> &longs;eor&longs;um, <lb/>& &longs;calenum, & æquicrus: nondum nouit triangulum, quod duobus rectis, <gap/>&longs;i &longs;o<lb/>phi&longs;tico modo, <expan abbr="neq;">neque</expan> vniuer&longs; aliter triangulum, <expan abbr="neq;">neque</expan> &longs;i vllum e&longs;t præter prædicta <lb/>triangulum alterum. </s>

<s>demon&longs;tratio

<lb/>ne aut vna, aut altera, quod duos rectos habet vnumquodque, <expan abbr="æquilatei&utilde;">æquilateium</expan> &longs;eor&longs;um,

<lb/>& &longs;calenum, & æquicrus: nondum nouit triangulum, quod duobus rectis, ni&longs;i &longs;o

<lb/>phi&longs;tico modo, <expan abbr="neq;">neque</expan> vniuer&longs; aliter triangulum, <expan abbr="neq;">neque</expan> &longs;i vllum e&longs;t præter prædicta

<lb/>triangulum alterum. </s>

<s>non enim &longs;ecundum quod triangulum, <expan abbr="neq;">neque</expan> omnetriangulum,

<lb/>ni&longs;i &longs;ecundum numerum, &longs;ecundum &longs;peciem autem non omne; & &longs;i nullum e&longs;t, quod

<lb/>non nouit)<emph.end type="italics"/> vltimo loco ponit exemplum primi erroris, quem &longs;upra verbis il

<lb/>lis <emph type="italics"/>(Quando vel nibil &longs;it accipere &longs;uperius, præter &longs;ingulare)<emph.end type="italics"/> expre&longs;&longs;erat, quod,

<lb/>vt benè intelligamus, opus e&longs;t ea, legere, quæ libro primo Priorum &longs;ecto 3.

<lb/>cap.

1. &longs;crip&longs;imus de proprietate illa trianguli, quod &longs;cilicet habet tres an

<lb/>gulos æquales duobus rectis angulis, quibus præmi&longs;&longs;is, &longs;ic deinde locum

<lb/>hunc interpretaberis; Propter hoc, quod præcedenti textu dictum e&longs;t; no

<lb/>tandum in primo errore vniuer&longs;ale, tanquam &longs;i non e&longs;&longs;et vniuer&longs;ale o&longs;ten

<lb/>ditur de &longs;ingulari, &longs;i quis igitur mon&longs;trauerit &longs;ingillatim de <expan abbr="vnoquoq;">vnoquoque</expan> trian

<lb/>gulo in &longs;ingulari, &longs;cilicet de vno æquilatero, tantum, & de vno Scaleno, &

<lb/>de vno I&longs;o&longs;cele, &longs;eparatim, vtens auteadem demon&longs;trationc dum de <expan abbr="vno&qacute;">vnoque</expan>; <pb pagenum="52"/>&longs;epatatim o&longs;tendit, aut vtens diuerfis demon&longs;trationibus, vna pro æquila

<lb/>tero, altera pro I&longs;o&longs;cele, tertia pro Scaleno, o&longs;tendens, quod <expan abbr="vnumquodq;">vnumquodque</expan>

<lb/>illorum habet tres angules æquales duobus rectis angulis; i&longs;te nondum no

<lb/>uit triangulum omne habere talem affectionem, ni&longs;i modo &longs;ophi&longs;tico, quia

<lb/>non cogno&longs;cit hanc affectionem illis <expan abbr="cõpetere">competere</expan> propter naturam illam com

<lb/>munem trianguli, cui primo, & per &longs;e competit; & neque vniuer&longs;aliter co

<lb/>gno&longs;cit triangulum omne e&longs;&longs;e tale, etiam &longs;i nullum aliud reperiatur trian

<lb/>gulum, præter illud æquilaterum, vel illud I&longs;o&longs;celes, vel illud Scalenum, de

<lb/>quibus &longs;eparatim <expan abbr="demõ&longs;trauit">demon&longs;trauit</expan>, & &longs;ecundum numernm, ide&longs;t de vnoquoque,

<lb/>quatenus e&longs;t vnum numero. </s>

<s>non nouit autem &longs;ecundum &longs;peciem, ideft fecun

<lb/>dum naturam, & formam communem illis tribus indiuiduis, quæ e&longs;t natu

<lb/>ra trianguli. </s>

<s>hoc autem e&longs;&longs;e exemplum primi erroris manife&longs;tè conuincitnr,

<lb/>tum ex verbis illis, quando nihil &longs;it &longs;uperius, præter &longs;ingulare, tum ex hu

<lb/>ius textus verbis illis <emph type="italics"/>(Singulum triangulum)<emph.end type="italics"/> & ex illis <emph type="italics"/>(Ni&longs;i &longs;ecundum nume

<lb/>rum)<emph.end type="italics"/> ide&longs;t, ni&longs;i de vno, quod &longs;it vnum numero. </s>

<s>propterea nos de &longs;in gulari

<lb/>triangulo omi&longs;&longs;a Zabarellæ &longs;ententia explicauimus tandem in confirma

<lb/>tionem no&longs;træ expo&longs;itionis in hæc tria errata illud non omittendum, &longs;atius

<lb/>e&longs;&longs;e dicere, Ari&longs;t.

attuli&longs;&longs;e pro tribus erratis tria exempla ordine retrogra

<lb/>do, quàm, quod facit Zabarella, primum e&longs;&longs;e pro tertio, &longs;ecundum pro pri

<lb/>mo, tertium verò pro &longs;ecundo; eo enim modo, Ari&longs;t.

<s>non enim &longs;ecundum quod triangulum, <expan abbr="neq;">neque</expan> omnetriangulum, <lb/>ni&longs;i &longs;ecundum numerum, &longs;ecundum &longs;peciem autem non omne; & &longs;i nullum e&longs;t, quod <lb/>non nouit)<emph.end type="italics"/> vltimo loco ponit exemplum primi erroris, quem &longs;upra verbis il<lb/>lis <emph type="italics"/>(Quando vel nibil &longs;it accipere &longs;uperius, præter &longs;ingulare)<emph.end type="italics"/> expre&longs;&longs;erat, quod, <lb/>vt benè intelligamus, opus e&longs;t ea, legere, quæ libro primo Priorum &longs;ecto 3. <lb/>cap.

confu&longs;ionem nulla ra

<lb/>tione, imò contra omnem rationem imponimus.</s>

1. &longs;crip&longs;imus de propriet<gap/>te illa trianguli, quod &longs;cilicet habet tres an<lb/>gulos æquales duobus rectis angulis, quibus præmi&longs;&longs;is, &longs;ic deinde locum <lb/>hunc interpretaberis; Propter hoc, quod præcedenti textu dictum e&longs;t; no<lb/>tandum in primo errore vniuer&longs;ale, tanquam &longs;i non e&longs;&longs;et vniuer&longs;ale o&longs;ten<lb/>ditur de &longs;ingulari, &longs;i quis igitur mon&longs;trauerit &longs;ingillatim de <expan abbr="vnoquoq;">vnoquoque</expan> trian<lb/>gulo in &longs;ingulari, &longs;cilicet de vno æquilatero, tantum, & de vno Scaleno, & <lb/>de vno I&longs;o&longs;cele, &longs;eparatim, vtens auteadem demon&longs;trationc dum de <expan abbr="vno&qacute;">vnoque</expan>; <pb pagenum="52"/>&longs;epatatim o&longs;tendit, aut vtens diuerfis demon&longs;trationibus, vna pro æquila<lb/>tero, altera pro I&longs;o&longs;cele, tertia pro Scaleno, o&longs;tendens, quod <expan abbr="vnumquodq;">vnumquodque</expan> <lb/>illorum habet tres angules æquales duobus rectis angulis; i&longs;te nondum no<lb/>uit triangulum omne habere talem affectionem, ni&longs;i modo &longs;ophi&longs;tico, quia <lb/>non cogno&longs;cit hanc affectionem illis <expan abbr="cõpetere">competere</expan> propter naturam illam com<lb/>munem trianguli, cui primo, & per &longs;e competit; & neque vniuer&longs;aliter co<lb/>gno&longs;cit triangulum omne e&longs;&longs;e tale, etiam &longs;i nullum aliud reperiatur trian<lb/>gulum, præter illud æquilaterum, vel illud I&longs;o&longs;celes, vel illud Scalenum, de <lb/>quibus &longs;eparatim <expan abbr="demõ&longs;trauit">demon&longs;trauit</expan>, & &longs;ecundum numernm, ide&longs;t de vnoquoque, <lb/>quatenus e&longs;t vnum numero. </s>

<s>non nouit autem &longs;ecundum &longs;peciem, ideft fecun<lb/>dum naturam, & formam communem illis tribus indiuiduis, quæ e&longs;t natu<lb/>ra trianguli. </s>

<s>hoc autem e&longs;&longs;e exemplum primi erroris manife&longs;tè conuincitnr, <lb/>tum ex verbis illis, quando nihil &longs;it &longs;uperius, præter &longs;ingulare, tum ex hu<lb/>ius textus verbis illis <emph type="italics"/>(Singulum triangulum)<emph.end type="italics"/> & ex illis <emph type="italics"/>(Ni&longs;i &longs;ecundum nume<lb/>rum)<emph.end type="italics"/> ide&longs;t, ni&longs;i de vno, quod &longs;it vnum numero. </s>

<s>propterea nos de &longs;in gulari <lb/>triangulo omi&longs;&longs;a Zabarellæ &longs;ententia explicauimus tandem in confirma<lb/>tionem no&longs;træ expo&longs;itionis in hæc tria errata illud non omittendum, &longs;atiu<gap/><lb/>e&longs;&longs;e dicere, Ari&longs;t.

attuli&longs;&longs;e pro tribus erratis tria exempla ordine retrogra<lb/>do, quàm, quod facit Zabarella, primum e&longs;&longs;e pro tertio, &longs;ecundum pro pri<lb/>mo, tertium verò pro &longs;ecundo; eo enim modo, Ari&longs;t.

confu&longs;ionem nulla ra<lb/>tione, imò contra omnem rationem imponimus.</s>

<s>Textu 14. continet quidem quædam mathematica, &longs;ed ferè eadem cum <lb/>&longs;uperioribus, quæ quia tum ex prædictis facile intelligi po&longs;&longs;unt, tum quia <lb/>benè ab expo&longs;itoribus explicantur, ne actum agamus, prætermittimus.</s>

<s>Textu 14. continet quidem quædam mathematica, &longs;ed ferè eadem cum

<lb/>&longs;uperioribus, quæ quia tum ex prædictis facile intelligi po&longs;&longs;unt, tum quia

<lb/>benè ab expo&longs;itoribus explicantur, ne actum agamus, prætermittimus.</s>

<s>Tex. 20. <emph type="italics"/>(Ni&longs;i magnitudines numeri &longs;int)<emph.end type="italics"/> hoc e&longs;t, ni&longs;i magnitudines &longs;int di<lb/>fcretæ, ita vt cadant &longs;ub numernm, vt &longs;i linea quæpiam diuidatur in partes <lb/>decem, vel duodecim, tunc euadit quantitas di&longs;creta, &longs;iue numerus. </s>

<s>Tex. 20. <emph type="italics"/>(Ni&longs;i magnitudines numeri &longs;int)<emph.end type="italics"/> hoc e&longs;t, ni&longs;i magnitudines &longs;int di

<lb/>fcretæ, ita vt cadant &longs;ub numernm, vt &longs;i linea quæpiam diuidatur in partes

<lb/>decem, vel duodecim, tunc euadit quantitas di&longs;creta, &longs;iue numerus. </s>

<s>& tunc <lb/>linca numerus e&longs;t. </s>

<s>& tunc

<lb/>linca numerus e&longs;t. </s>

<s>idem de &longs;uperficie, ac &longs;olido intelligendum.</s>

Line 2722

Line 3665

<s>Ibidem <emph type="italics"/>(Propter hoc Geometriæ non licet mon&longs;trare, quod contrariorum vna <lb/>e&longs;e &longs;cientia, &longs;ed neque quod duo cubi cubus)<emph.end type="italics"/> quo ad verba illa, duo cubi cubus<gap/><lb/>quæ ad nos pertinent, vult Ari&longs;t.

<s>Ibidem <emph type="italics"/>(Propter hoc Geometriæ non licet mon&longs;trare, quod contrariorum vna

<lb/>e&longs;e &longs;cientia, &longs;ed neque quod duo cubi cubus)<emph.end type="italics"/> quo ad verba illa, duo cubi cubus,

<lb/>quæ ad nos pertinent, vult Ari&longs;t.

docere, quod non debet Geometra o&longs;ten

<lb/>dere numerorum affectiones (per enbos enim intelligit numeros quo&longs;dam

<lb/>&longs;ic dictos, vt paulo po&longs;t o&longs;tendam) vt &longs;i quis vellet geometricè o&longs;tendere id,

<lb/>quod o&longs;tenditur in 4. noni Elem.

&longs;cilicet, &longs;i cubus numerus cubum numerum

<lb/>multiplicauerit, productus numerus erit pariter cubus. </s>

<s>nonnulli latinorum

<lb/>perperam textum hunc expo&longs;uerunt putantes reperiri &longs;olummodo cubos

<lb/>geometricos, at Euclides definit. </s>

<s>19. &longs;eptimi, &longs;ic arithmeticum cubum de

<lb/>finit, cubus numerus e&longs;t, qui &longs;ub tribus numeris æqualibus continetur, qua

<lb/>lis e&longs;t. </s>

<s>8. qui e&longs;t ad in&longs;tar cubi geometrici, & continetur&longs;ub tribus binarijs

<lb/>multiplicatis inuicem, quæ multiplicatio &longs;ic in&longs;tituitur, exponuntur tres bi

<lb/>narij, 2, 2, 2, primus ducitur in &longs;ecundum, & producitur.

<s>4. qui e&longs;t numerus quadratus huius figuræ, <figure place="text" xlink:href="figures-la/009.01.052.2.tif"/>, deinde

<lb/>tertius binarius ducitur in prædictum quadratum 4. & pro

<lb/>ducitur 8. qui dicitur cubus, quia &longs;i intelligantur duo qua

<lb/>ternarij, vnus &longs;upra alterum, vt in præ&longs;enti figura refe

<lb/>runt cubicam figuram, cuius tam longitudo, quam latitudo, <pb pagenum="53"/>& altitudo, e&longs;t 2. Similiter cubus numerus e&longs;t 27. quia &longs;it ex tribus terna

<lb/>rijs inuicem modo prædicto multiplicatis, 3. 3. 3. nam 3. in 3. ductis &longs;it 9.

<lb/>qui e&longs;t quadratus. </s>

<s>quo deinde ducto in tertium ter

<lb/>narium, producitur 27. qui e&longs;t cubus, & refert &longs;igu

<lb/>ram cubicam hanc. </s>

<s>Iam verò &longs;i cubus 8. multipli

<lb/>cet cubum 27. procreabitur 216. qui pariter cubus

<lb/>e&longs;t. </s>

<s><expan abbr="atq;">atque</expan> hoc &longs;ibi volunt verba illa, &longs;i duo cubi cubus,

<lb/>ide&longs;t, &longs;i duo numeri cubi multiplicentur mutuò, cu

<lb/>bus alter producetur; ex quibus videas, quam in

<lb/>eptè illi <expan abbr="interpret&etilde;tur">interpretentur</expan> hunc locum, qui dicunt, Ari

<lb/>&longs;totilem velle dicere non pertinere ad Geometram

<lb/>probare duos cubos geometricos &longs;ibi additos face

<lb/>re alium cubum, quod erat problema Delphicum de

<lb/>duplatione cubi, nondum inuentum; bis enim i&longs;ti peccant, primo in Logi

<lb/>cam, quia &longs;ic non tran&longs;iret Geometra de genere in genus, ip&longs;ius enim e&longs;t

<lb/>agere de duplatione cubi; &longs;ecundò in Mathematicas, cum nondum noue

<lb/>rint arithmeticos cubos; & præterca ignorent duos cubos &longs;ibi additos, non

<lb/>facere alium cubum. </s>

<s>Quod præterea hoc loco intelligendi &longs;int cubi arith

<lb/>metici certò certius con&longs;tat, ex &longs;equenti 24. textu, vbi &longs;ic dicitur <emph type="italics"/>(Veluti

<lb/>Arithmetica quidem, quid impar, aut par, aut quadrangulum, aut cubus.)<emph.end type="italics"/></s>

docere<gap/>, quod non debet Geometra o&longs;ten<lb/>dere numerorum affectiones (per enbos enim intelligit numeros quo&longs;dam <lb/>&longs;ic dictos, vt paulo po&longs;t o&longs;tendam) vt &longs;i quis vellet geometricè o&longs;tendere id, <lb/>quod o&longs;tenditur in 4. noni Elem.

&longs;cilicet, &longs;i cubus numerus cubum numerum <lb/>multiplicauerit, productus numerus erit pariter cubus. </s>

<s>nonnulli latinorum <lb/>perperam textum hunc expo&longs;uerunt putantes reperiri &longs;olummodo cubos <lb/>geometricos, at Euclides definit. </s>

<s>Ibidem <emph type="italics"/>(<expan abbr="Neq;">Neque</expan> alij &longs;cientiæ quod alterius, ni&longs;i <expan abbr="quæcunq;">quæcunque</expan> ita &longs;e habent inter &longs;e,

<lb/>vt &longs;it alterum &longs;ub altero, vt per&longs;pectiua ad Geometriam, & harmonica ad Arith

<lb/>meticam)<emph.end type="italics"/> excipit ab illa regula (qua prohibetur, quamuis &longs;cientiam in alie

<lb/>nam falcem immittere) &longs;cientias &longs;ubalternatas, quæ propriè in Mathemati

<lb/>cis reperiuntur, Per&longs;pectiua enim propriè &longs;ubalternatur Geometriæ, quia

<lb/>vtitur Demon&longs;trationibus linearibus, quas applicat lineis vi&longs;ualibus, & Mu

<lb/>&longs;ica &longs;ubalternatur Arithmeticæ, quia ab ip&longs;a mutuatur <expan abbr="demõ&longs;trationes">demon&longs;trationes</expan> nu

<lb/>merorum, quas applicat numeris &longs;onoris. </s>

<s>v.g. Per&longs;pectiua dicit, ea, quæ vi

<lb/>dentur eminus videri minora, quam quæ videntur cominus, quia illa viden

<lb/>tur &longs;ub angulo minori, hæc verò &longs;ub angulo maiori, quod verò remotiora

<lb/>videantur &longs;ub angulo minori, quam propinquiora cæteris paribus probat

<lb/>per 21. primi Elem.

&longs;it enim ma

<lb/>gnitudo vi&longs;a A B, remotior ab o

<lb/>culo in C, po&longs;ito, & vi&longs;a propin

<lb/>quior ab oculo in D. ductis lineis

<lb/>vi&longs;ualibus C A, C B: D A, D B; ab

<lb/>oculis C, & D, ad extremitates

<lb/>&longs;pectatæ magnitudinis, erit remo

<lb/>tioris vi&longs;ionis angulus C, minor

<lb/>angulo D, propinquioris, vt ex præallegata Demon&longs;tratione pater. </s>

<s>19. &longs;eptimi, &longs;ic arithmeticum cubum de<lb/>finit, cubus numerus e&longs;t, qui &longs;ub tribus numeris æqualibus continetur, qua<lb/>lis e&longs;t. </s>

<s>8. qui e&longs;t ad in&longs;tar cubi geometrici, & continetur&longs;ub tribus binarijs <lb/>multiplicatis inuicem, quæ multiplicatio &longs;ic in&longs;tituitur, exponuntur tres bi<lb/><figure id="fig21"/><lb/>narij, 2, 2, 2, primus ducitur in &longs;ecundum, & producitur. <lb/></s>

<s>4. qui e&longs;t numerus quadratus huius figuræ, <figure id="fig22"/>, deinde <lb/>tertius binarius ducitur in prædictum quadratum 4. & pro<lb/>ducitur 8. qui dicitur cubus, quia &longs;i intelligantur duo qua<lb/>zerna<gap/>ij, vnus &longs;upra alterum, vt in præ&longs;enti figura refe<lb/>runt cubicam figuram, cuius tam longitudo, quam latitudo, <pb pagenum="53"/>& altitudo, e&longs;t 2. Similiter cubus numerus e&longs;t 27. quia &longs;it ex tribus terna<lb/>rijs inuicem modo prædicto multiplicatis, 3. 3. 3. nam 3. in 3. ductis &longs;it 9. <lb/><figure id="fig23"/><lb/>qui e&longs;t quadratus. </s>

<s>Hine

<lb/>per&longs;picuè vides, qua ratione Per&longs;pectiua Geometriæ &longs;ubalternetur, &longs;iue

<lb/>quid &longs;it ip&longs;a &longs;ubalternatio, vbi medium e&longs;t Geometricum, conclu&longs;io autem

<lb/>optica. </s>

<s>Exemplum &longs;ubalternationis Muficæ &longs;it, <expan abbr="con&longs;onãtia">con&longs;onantia</expan> Diapa&longs;on, quam

<lb/>vulgò octauam appellant in data chorda collocare, hoc e&longs;t, vocem grauio

<lb/>rem facere duplam vocis acutioris &longs;umatur chorda A B, & diuidatur bifa

<lb/>riam, &longs;ine in æqualia in C; tota igitur chorda A B, ad dimidium A C, haber <pb pagenum="54"/><figure place="text" xlink:href="figures-la/009.01.054.1.tif"/>

<lb/>proportionem, quam 2. ad 1.

<lb/>&longs;iue duplam, ergo etiam &longs;o

<lb/>nus totius chordæ A B, ad <expan abbr="&longs;o-n&utilde;">&longs;o

<lb/>num</expan> chordæ dimidiæ A C, ha

<lb/>bebit eandem rationem, <expan abbr="nimir&utilde;">nimirum</expan> quam 2. ad 1. &longs;iue duplam. </s>

<s>&longs;ed &longs;onus chor

<lb/>dæ A B, ad &longs;onum chordæ A C, con&longs;onat diapa&longs;on, &longs;eu octauam, ergo in

<lb/>data chorda collocata e&longs;t con&longs;onantia diapa&longs;on, quod oportebat. </s>

<s>vides me

<lb/>dium e&longs;&longs;e arithmeticam, conclu&longs;ionem verò harmonicam. </s>

<s>Aliud exemplum

<lb/>Tonus, quod e&longs;t <expan abbr="interuall&utilde;">interuallum</expan> primæ vocis, Vt, ad &longs;ecundam, Rè, in duo æqua

<lb/>lia &longs;emitonia diuidi nequit, ratio e&longs;t Arithmetica, quia proportio &longs;uper

<lb/>particularis in duo æqualia arithmeticè &longs;ecari nequit; at Tonus con&longs;i&longs;tit in

<lb/>ratione &longs;uperparticulari, nempè in &longs;e&longs;quioctaua, ergo Tonus bifariam diui

<lb/>di nequit. </s>

<s>quo deinde ducto in tertium ter<lb/>narium, producitur 27. qui e&longs;t cubus, & refert &longs;igu<lb/>ram cubicam hanc. </s>

<s>de&longs;umptum e&longs;t ex Boetio.</s>

<s>Iam verò &longs;i cubus 8. multipli<lb/>cet cubum 27. procreabitur 216. qui pariter cubus <lb/>e&longs;t. </s>

<s><expan abbr="atq;">atque</expan> hoc &longs;ibi volunt verba illa, &longs;i duo cubi cubus, <lb/>ide&longs;t, &longs;i duo numeri cubi multiplicentur mutuò, cu<lb/>bus alter producetur; ex quibus videas, quam in<lb/>eptè illi <expan abbr="interpret&etilde;tur">interpretentur</expan> hunc locum, qui dicunt, Ari<lb/>&longs;totilem velle dicere non pertinere ad Geometram <lb/>probare duos cubos geometricos &longs;ibi additos face<lb/>re alium cubum, quod erat problema Delphicum de <lb/>duplatione cubi, nondum inuentum; bis enim i&longs;ti peccant, primo in Logi<lb/>cam, quia &longs;ic non tran&longs;iret Geometra de genere in genus, ip&longs;ius enim e&longs;t <lb/>agere de duplatione cubi; &longs;ecundò in Mathematicas, cum nondum noue<lb/>rint arithmeticos cubos; & præterca ignorent duos cubos &longs;ibi additos, non <lb/>facere alium cubum. </s>

<s>Quod præterea hoc loco intelligendi &longs;int cubi arith<lb/>metici certò certius con&longs;tat, ex &longs;equenti 24. textu, vbi &longs;ic dicitur <emph type="italics"/>(Veluti <lb/>Arithmetica quidem, quid impar, aut par, aut quadrangulum, aut cubus.)<emph.end type="italics"/></s>

<s>Tex. 23. <emph type="italics"/>(Est autem &longs;ic mon&longs;trare, quemadmodum Bry&longs;o quadraturam, &longs;ecun

<lb/>dum enim commune mon&longs;trant tales rationes)<emph.end type="italics"/> cum velit e&longs;tendere veram de

<lb/>mon&longs;trationem con&longs;tare debere ex proprijs, non autem ex communibus;

<lb/>primum affert exemplum demon&longs;trationis cuiu&longs;dam Bry&longs;onis, quæ ex com

<lb/>munibus procedat, vt autem benè intelligamus, quale&longs;nam &longs;int huin&longs;modi

<lb/>demon&longs;trationes, quæ per communia o&longs;tendunt, legenda prius ea &longs;unt, quæ

<lb/>&longs;crip&longs;imus de quadratura circuli in pr&ecedil;dicamento relationis. </s>

<s>Bry&longs;o itaque,

<lb/>vt tradit Alexander, in hunc modum conabatur quadrare <expan abbr="circul&utilde;">circulum</expan>. </s>

<s>&longs;it qua

<lb/>drandus circulus A B C D, cui circum&longs;eribatur quadratum E F G H. per

<lb/>7 quarti, & alterum quadratum I L M N, eidem in&longs;cribatur per 6. quarti,

<lb/>quid autem &longs;it circum&longs;cribere, & in&longs;cribere figuram circulo, ex definitione

<lb/>3. & 4. eiu&longs;dem libri petatur, quamuis

<lb/>ex in&longs;pectione figuræ <expan abbr="pr&ecedil;s&etilde;tis">pr&ecedil;sentis</expan> &longs;atis per

<lb/>cipi po&longs;&longs;it; deinde aliud <expan abbr="quadrat&utilde;">quadratum</expan> me

<lb/>dium inter prædicta duo con&longs;tituatur,

<lb/><expan abbr="&longs;it&qacute;">&longs;itque</expan>; O P Q R. </s>

<s>Iam &longs;ic o&longs;tendebat i&longs;tud

<lb/>medium quadratum e&longs;&longs;e æquale circu

<lb/>lo propo&longs;ito. </s>

<s><expan abbr="Quæcunq;">Quæcunque</expan> &longs;unt, &longs;imul ma

<lb/>iora eodem, & minora eodem, &longs;unt in

<lb/>uicem æqualia, &longs;ed circulus, & quadra

<lb/>tum medium, &longs;unt ambo maiora qua

<lb/>drato in&longs;cripto, & ambo minora qua

<lb/>drato circum&longs;cripto, ergo circulus, &

<lb/>quadratum medium, &longs;unt æqualia. </s>

<s>vte

<lb/>batur, inquit Ari&longs;t pr&ecedil;dicto principio,

<lb/>etiam numeris, lineis, temporibus, &

<lb/>qualitatibus communi, <expan abbr="neq;">neque</expan> deducto ex natura circuli, aut quadrati, de qui

<lb/>bus erat demon&longs;tratio. </s>

<s>præterea aduertendum e&longs;t, illud e&longs;&longs;e fal&longs;um, nam &longs;ex,

<lb/>& quinque, ambo &longs;unt maiores, quam quatuor, & minores, quam &longs;eptem,

<lb/>& tamen non &longs;unt æquales.</s>

<s>Ibidem <emph type="italics"/>(<expan abbr="Neq;">Neque</expan> alij &longs;cientiæ quod alterius, ni&longs;i <expan abbr="quæcunq;">quæcunque</expan> ita &longs;e habent inter &longs;e, <lb/>vt &longs;it alterum &longs;ub altero, vt per&longs;pectiua ad Geometriam, & harmonica ad Arith<lb/>meticam)<emph.end type="italics"/> excipit ab illa regula (qua prohibetur, quamuis &longs;cientiam in alie<lb/>nam falcem immittere) &longs;cientias &longs;ubalternatas, quæ propriè in Mathemati<lb/>cis reperiuntur, Per&longs;pectiua enim propriè &longs;ubalternatur Geometriæ, quia <lb/>vtitur Demon&longs;trationibus linearibus, quas applicat lineis vi&longs;ualibus, & Mu<lb/>&longs;ica &longs;ubalternatur Arithmeticæ, quia ab ip&longs;a mutuatur <expan abbr="demõ&longs;trationes">demon&longs;trationes</expan> nu<lb/>merorum, quas applicat numeris &longs;onoris. </s>

<s>In codem textu <emph type="italics"/>(<expan abbr="Vnumquodq;">Vnumquodque</expan> autem &longs;cimus, non &longs;ecundum accidens, quando

<lb/>&longs;ecundum illud cogno&longs;camus, &longs;ecundum quod ine&longs;t ex principijs illius, &longs;ecundam

<lb/>quod illud; vt duobus rectis æquales, habere, cui ine&longs;t per &longs;e, quod dictum e&longs;t ex<emph.end type="italics"/><pb pagenum="55"/><emph type="italics"/>principijs huius)<emph.end type="italics"/> affert nunc exemplum alterius demon&longs;trationis, quæ non

<lb/>ex communibus, vt præcedens Bry&longs;onis, &longs;ed ex proprijs principijs o&longs;tendit

<lb/>affectionem de &longs;ubiecto proprio. </s>

<s>E&longs;t autem illud exemplum toties decan

<lb/>tatum de triangulo habente tres angulos æquales duobus rectis angulis; id

<lb/>circo operæpretium e&longs;&longs;e puto explicare demon&longs;trationem, 32. primi Eucli

<lb/>dis, quæ i&longs;tud ex proprijs principijs demon&longs;trat, & quam hoc loco Ari&longs;to

<lb/>teles innuit, hoc enim modo ip&longs;ius Ari&longs;t.

mentem probè penetrare poteri

<s>v.g. Per&longs;pectiua dicit, ea, quæ vi<lb/>dentur eminus videri minora, quam quæ videntur cominus, quia illa viden<lb/>tur &longs;ub angulo minori, hæc verò &longs;ub angulo maiori, quod verò remotiora <lb/>videantur &longs;ub angulo minori, quam propinquiora cæteris paribus probat <lb/><figure id="fig24"/><lb/>per 21. primi Elem.

<s>&longs;it ergo <expan abbr="triãgulum">triangulum</expan> A B C. </s>

&longs;it enim ma<lb/>gnitudo vi&longs;a A B, remotior ab o<lb/>culo in C, po&longs;ito, & vi&longs;a propin<lb/>quior ab oculo in D. ductis lineis <lb/>vi&longs;ualibus C A, C B: D A, D B; ab <lb/>oculis C, & D, ad extremitates <lb/>&longs;pectatæ magnitudinis, erit remo<lb/>tioris vi&longs;ionis angulus C, minor <lb/>angulo D, propinquioris, vt ex præallegata Demon&longs;tratione pater. </s>

<s>Dico ag

<lb/>gregatum <expan abbr="tri&utilde;">trium</expan> ip&longs;ius angulorum A, B, C,

<lb/>e&longs;&longs;e æquale aggregato ex duobus angu

<lb/>lis rectis (vt autem melius intelligas, quæ

<lb/>&longs;equuntur, lege prius ea, quæ dicta &longs;unt

<lb/>in lib.

1. Priorum &longs;ecto 3. cap.

1.) produ

<lb/>catur latus B C, <expan abbr="v&longs;q;">v&longs;que</expan> in D, vt fiat angulus

<lb/>externus A C D; Iam &longs;ic, quoniam <expan abbr="pro-bat&utilde;">pro

<lb/>batum</expan> e&longs;t in 13. primi, duos angulos, quos

<lb/>facit linea A C, cum linea B D, &longs;cilicet angulos A C B, A C D, e&longs;&longs;e pares

<lb/>duobus rectis: & quia pariter in prima parte huins propo&longs;. </s>

<s>32. probatum

<lb/>e&longs;t ab Euclide duos angulos A B, e&longs;&longs;e æquales externo angulo A C D: &longs;i ter

<lb/>tius angulus reliquus A C B, &longs;umatur bis, &longs;emel cum duobus angulis A, B,

<lb/>& &longs;emel cum externo A C D, <expan abbr="add&etilde;tur">addentur</expan> æqualia æqualibus, & propterea tres

<lb/>anguli A, B, A C B, &longs;imul &longs;umpti, erunt æquales duobus A C D, A C B, &longs;imul

<lb/>&longs;umptis; &longs;ed his duobus &longs;unt æquales duo recti, ergo cum quæ &longs;unt æqualia

<lb/>vni tertio, &longs;int etiam æqualia inuicem, erit aggregatum trium angulorum

<lb/>A, B, A C B, æquale aggregato duorum rectorum; quod erat demon&longs;tran

<s>Medium <expan abbr="itaq;">itaque</expan> huius demon&longs;trationis, &longs;i res ad trutinam Logicam ex

<lb/>pendatur, e&longs;t, quod partes aggregati <expan abbr="tri&utilde;">trium</expan> <expan abbr="angulor&utilde;">angulorum</expan> A, B, A C B, &longs;unt æqua

<lb/>les partibus aggregati <expan abbr="duor&utilde;">duorum</expan>, & ideo <expan abbr="aggregat&utilde;">aggregatum</expan>, aggrega to æqua

<lb/>le e&longs;t. </s>

<s>Hine <lb/>per&longs;picuè vides, qua ratione Per&longs;pectiua Geometriæ &longs;ubalternetur, &longs;iue <lb/>quid &longs;it ip&longs;a &longs;ubalternatio, vbi medium e&longs;t Geometricum, conclu&longs;io autem <lb/>optica. </s>

<s>quod medium e&longs;t in genere cau&longs;æ materialis. </s>

<s>Exemplum &longs;ubalternationis Muficæ &longs;it, <expan abbr="con&longs;onãtia">con&longs;onantia</expan> Diapa&longs;on, quam <lb/>vulgò octauam appellant in data chorda collocare, hoc e&longs;t, vocem grauio<lb/>rem facere duplam vocis acutioris &longs;umatur chorda A B, & diuidatur bifa<lb/>riam, &longs;ine in æqualia in C; tota igitur chorda A B, ad dimidium A C, haber <pb pagenum="54"/><figure id="fig25"/><lb/>proportionem, quam 2. ad 1. <lb/>&longs;iue duplam, ergo etiam &longs;o<lb/>nus totius chordæ A B, ad <expan abbr="&longs;o-n&utilde;">&longs;o<lb/>num</expan> chordæ dimidiæ A C, ha<lb/>bebit eandem rationem, <expan abbr="nimir&utilde;">nimirum</expan> quam 2. ad 1. &longs;iue duplam. </s>

<s>quod verò partes illius

<lb/>&longs;int æquales partibus huius, probatur, per dignitatem <expan abbr="illã">illam</expan>, quæ &longs;unt æqualia

<lb/>vni tertio, &longs;unt etiam inter &longs;e. </s>

<s>partes porrò aggregati trium angulorum

<lb/>erant hæ, anguli A, B, vna; altera verò angulus A C B; partes verò aggre

<lb/>gati duorum rectorum erant A C B, A C D, quibus partibus, illæ &longs;unt æqua

<lb/>les, & ideo totum toti æquale. </s>

<s>quod medium e&longs;t omnino intrin&longs;ecum, & ex

<lb/>proprijs ip&longs;ius trianguli, &longs;iue ex proprijs angulorum ip&longs;ius, cum &longs;int ip&longs;ius

<lb/>partes. </s>

<s>quod pariter medium ex parte pa&longs;&longs;ionis, quæ demon&longs;tratur, e&longs;t ex

<lb/>proprijs, cum &longs;int partes illius materiales. </s>

<s>per materiam autem oportet

<lb/>hoc loco intelligere materiam intelligibilem, ide&longs;t quantitatem à qualita

<lb/>tibas ab&longs;tractam, & terminatam, de qua pluribus agemus infra in tractatu

<lb/>de natura mathematicarum. </s>

<s>Hinc videas eos magnopere decipi, qui pu

<lb/>tant, hanc demon&longs;trationem e&longs;&longs;e per extrin&longs;eca, eò quod ad demon&longs;tran

<lb/>dum producatur linea B C, in D, putantes lineam illam productam C D,

<lb/>e&longs;&longs;e demon&longs;trationis medium; lineæ <expan abbr="namq;">namque</expan> huiu&longs;modi, quæ in demon&longs;tra

<lb/>tionibus geometricis con&longs;truuntur, nunquam &longs;unt media propria demon

<lb/>&longs;trationum, &longs;ed tantummodo a&longs;&longs;umuntur ad probandum medium iam ex

<lb/>cogitatum e&longs;&longs;e veram cau&longs;am conclu&longs;ionis. </s>

<s>Hinc etiam manife&longs;tè colligas <pb pagenum="56"/>Mathematicas facultates habere demon&longs;trationes perfecti&longs;&longs;imas, quod

<lb/>ageometreti negare &longs;olent, &longs;ed audacter aiunt exempla Ari&longs;t.

non e&longs;&longs;e vera:

<lb/><expan abbr="neq;">neque</expan> requiri veritatem exemplorum; in <expan abbr="quor&utilde;">quorum</expan> <expan abbr="vtroq;">vtroque</expan> peccant, nam dictum

<lb/>illud v&longs;urpari &longs;olet, & debet de exemplis moralibus. </s>

<s>at vero requiri confor

<lb/>mitatem exemplorum cum regulis traditis, nemo &longs;anæ mentis dubitabit.

<s>&longs;ed &longs;onus chor<lb/>dæ A B, ad &longs;onum chordæ A C, con&longs;onat diapa&longs;on, &longs;eu octauam, ergo in <lb/>data chorda collocata e&longs;t con&longs;onantia diapa&longs;on, quod oportebat. </s>

<s>Vernm i&longs;ti confundunt conformitatem cum veritate. </s>

<s>vides me<lb/>dium e&longs;&longs;e arithmeticam, conclu&longs;ionem verò harmonicam. </s>

<s>Veritas exemplo tunc

<lb/>ine&longs;t, quando illud, quod in exemplo narratur, verè extitit, vt &longs;i quis in

<lb/>exemplum pudicitiæ afferret hi&longs;toriam Io&longs;ephi, <expan abbr="ver&utilde;">verum</expan> i&longs;tuà e&longs;&longs;et exemplum.

<s>Aliud exemplum <lb/>Tonus, quod e&longs;t <expan abbr="interuall&utilde;">interuallum</expan> primæ vocis, Vt, ad &longs;ecundam, Rè, in duo æqua<lb/>lia &longs;emitonia diuidi nequit, ratio e&longs;t Arithmetica, quia proportio &longs;uper<lb/>particularis in duo æqualia arithmeticè &longs;ecari nequit; at Tonus con&longs;i&longs;tit in <lb/>ratione &longs;uperparticulari, nempè in &longs;e&longs;quioctaua, ergo Tonus bifariam diui<lb/>di nequit. </s>

<s>quæ veritas in exemplis moralibus non &longs;emper e&longs;t nece&longs;&longs;aria, talia exempla

<lb/>&longs;unt &longs;æpè parabolæ, & fabulæ, quæ nunquam extiterunt, v. g. narratur ab

<lb/>Ari&longs;t.

<s>de&longs;umptum e&longs;t ex Boetio.</s>

de quodam filio, qui patrem crudeliter traxerat, qui po&longs;tea grandior

<lb/>factus, cum filium procrea&longs;&longs;et, ab eodem pariter raptatus e&longs;t ip&longs;e, v&longs;que ad

<lb/>eundem locum, quo ip&longs;e patrem &longs;uum impiè raptauerat. </s>

<s>Tex. 23. <emph type="italics"/>(Est autem &longs;ic mon&longs;trare, quemadmodum Bry&longs;o quadraturam, &longs;ecun<lb/>dum enim commune mon&longs;trant tales rationes)<emph.end type="italics"/> cum velit e&longs;tendere veram de<lb/>mon&longs;trationem con&longs;tare debere ex proprijs, non autem ex communibus; <lb/>primum affert exemplum demon&longs;trationis cuiu&longs;dam Bry&longs;onis, quæ ex com<lb/>munibus procedat, vt autem benè intelligamus, quale&longs;nam &longs;int huin&longs;modi <lb/>demon&longs;trationes, quæ per communia o&longs;tendunt, legenda prius ea &longs;unt, quæ <lb/>&longs;crip&longs;imus de quadratura circuli in pr&ecedil;dicamento relationis. </s>

<s>Bry&longs;o itaque, <lb/>vt tradit Alexander, in hunc modum conabatur quadrare <expan abbr="circul&utilde;">circulum</expan>. </s>

<s>&longs;it qua<lb/>drandus circulus A B C D, cui circum&longs;eribatur quadratum E F G H. per <lb/>7 quarti, & alterum quadratum I L M N, eidem in&longs;cribatur per 6. quarti, <lb/>quid autem &longs;it circum&longs;cribere, & in&longs;cribere figuram circulo, ex definitione <lb/><figure id="fig26"/><lb/>3. & 4. eiu&longs;dem libri petatur, quamuis <lb/>ex in&longs;pectione figuræ <expan abbr="pr&ecedil;s&etilde;tis">pr&ecedil;sentis</expan> &longs;atis per<lb/>cipi po&longs;&longs;it; deinde aliud <expan abbr="quadrat&utilde;">quadratum</expan> me<lb/>dium inter prædicta duo con&longs;tituatur, <lb/><expan abbr="&longs;it&qacute;">&longs;itque</expan>; O P Q R. </s>

<s>non e&longs;t nece&longs;&longs;e, ta

<lb/>lem extiti&longs;&longs;e filium, <expan abbr="neq;">neque</expan> patrem. </s>

<s>Iam &longs;ic o&longs;tendebat i&longs;tud <lb/>medium quadratum e&longs;&longs;e æquale circu<lb/>lo propo&longs;ito. </s>

<s>Verumtamen &longs;emper conformitas exem

<lb/>pli cum regulis, & præceptis, quæ traduntur nece&longs;&longs;aria e&longs;t, alioquin exem

<lb/>pla de&longs;truerent id, quod præceptio con&longs;truit, <expan abbr="illi&qacute;">illique</expan> contraria e&longs;&longs;et, quod om

<lb/>nino ab&longs;urdum foret. </s>

<s><expan abbr="Quæcunq;">Quæcunque</expan> &longs;unt, &longs;imul ma<lb/>iora eodem, & minora eodem, &longs;unt in<lb/>uicem æqualia, &longs;ed circulus, & quadra<lb/>tum medium, &longs;unt ambo maiora qua<lb/>drato in&longs;cripto, & ambo minora qua<lb/>drato circum&longs;cripto, ergo circulus, & <lb/>quadratum medium, &longs;unt æqualia. </s>

<s>non &longs;ecus, ac &longs;i quis vellet alium docere characteres

<lb/>latinos, <expan abbr="illi&qacute;">illique</expan>; barbaros, quos Gothicos vocant in exemplum proponeret. </s>

<s>vte<lb/>batur, inquit Ari&longs;t pr&ecedil;dicto principio, <lb/>etiam numeris, lineis, temporibus, & <lb/>qualitatibus communi, <expan abbr="neq;">neque</expan> deducto ex natura circuli, aut quadrati, de qui<lb/>bus erat demon&longs;tratio. </s>

<s>re

<lb/>quiritur igitur &longs;emper in omni exemplo conformitas cum eo, quod doce

<lb/>tur; in moralibus tamen non &longs;emper requiritur veritas, vti diximus; Alij

<lb/>verò dicunt non requiri in exemplis determinatam veritatem, &longs;ed &longs;atis e&longs;&longs;e,

<lb/>&longs;i exemplum verum &longs;it &longs;ecundum opinionem aliquorum: <expan abbr="quor&utilde;">quorum</expan> &longs;ententiam

<lb/>non improbamus. </s>

<s>præterea aduertendum e&longs;t, illud e&longs;&longs;e fal&longs;um, nam &longs;ex, <lb/>& quinque, ambo &longs;unt maiores, quam quatuor, & minores, quam &longs;eptem, <lb/>& tamen non &longs;unt æquales.</s>

<s>Exempla igitur ab Ari&longs;t.

pa&longs;&longs;im ex mathem aticis allata,

<lb/>congrua, <expan abbr="conformia&qacute;">conformiaque</expan>; omninò &longs;unt ip&longs;ius doctrinæ, aliter ip&longs;um perpetuò

<lb/>mentientem facimus. </s>

<s>In codem textu <emph type="italics"/>(<expan abbr="Vnumquodq;">Vnumquodque</expan> autem &longs;cimus, non &longs;ecundum accidens, quando <lb/>&longs;ecundum illud cogno&longs;camus, &longs;ecundum quod ine&longs;t ex principijs illius, &longs;ecundam <lb/>quod illud; vt duobus rectis æquales, habere, cui ine&longs;t per &longs;e, quod dictum e&longs;t ex<emph.end type="italics"/><pb pagenum="55"/><emph type="italics"/>principijs huius)<emph.end type="italics"/> affert nunc exemplum alterius demon&longs;trationis, quæ non <lb/>ex communibus, vt præcedens Bry&longs;onis, &longs;ed ex proprijs principijs o&longs;tendit <lb/>affectionem de &longs;ubiecto proprio. </s>

<s>E&longs;t autem illud exemplum toties decan<lb/>tatum de triangulo habente tres angulos æquales duobus rectis angulis; id<lb/>circo operæpretium e&longs;&longs;e puto explicare demon&longs;trationem, 32. primi Eucli<lb/>dis, quæ i&longs;tud ex proprijs principijs demon&longs;trat, & quam hoc loco Ari&longs;to<lb/>teles innuit, hoc enim modo ip&longs;ius Ari&longs;t.

mentem probè penetrare poteri<lb/><figure id="fig27"/><lb/>mus. </s>

<s>&longs;it ergo <expan abbr="triãgulum">triangulum</expan> A B C. </s>

<s>Dico ag<lb/>gregatum <expan abbr="tri&utilde;">trium</expan> ip&longs;ius angulorum A, B, C, <lb/>e&longs;&longs;e æquale aggregato ex duobus angu<lb/>lis rectis (vt autem melius intelligas, quæ <lb/>&longs;equuntur, lege prius ea, quæ dicta &longs;unt <lb/>in lib.

1. Priorum &longs;ecto 3. cap.

1.) produ<lb/>catur latus B C, <expan abbr="v&longs;q;">v&longs;que</expan> in D, vt fiat angulus <lb/>externus A C D; Iam &longs;ic, quoniam <expan abbr="pro-bat&utilde;">pro<lb/>batum</expan> e&longs;t in 13. primi, duos angulos, quos <lb/>facit linea A C, cum linea B D, &longs;cilicet angulos A C B, A C D, e&longs;&longs;e pares <lb/>duobus rectis: & quia pariter in prima parte huins propo&longs;. </s>

<s>32. probatum <lb/>e&longs;t ab Euclide duos angulos A B, e&longs;&longs;e æquales externo angulo A C D: &longs;i ter<lb/>tius angulus reliquus A C B, &longs;umatur bis, &longs;emel cum duobus angulis A, B, <lb/>& &longs;emel cum externo A C D, <expan abbr="add&etilde;tur">addentur</expan> æqualia æqualibus, & propterea tres <lb/>anguli A, B, A C B, &longs;imul &longs;umpti, erunt æquales duobus A C D, A C B, &longs;imul <lb/>&longs;umptis; &longs;ed his duobus &longs;unt æquales duo recti, ergo cum quæ &longs;unt æqualia <lb/>vni tertio, &longs;int etiam æqualia inuicem, erit aggregatum trium angulorum <lb/>A, B, A C B, æquale aggregato duorum rectorum; quod erat demon&longs;tran<lb/>dum. </s>

<s>Medium <expan abbr="itaq;">itaque</expan> huius demon&longs;trationis, &longs;i res ad trutinam Logicam ex<lb/>pendatur, e&longs;t, quod partes aggregati <expan abbr="tri&utilde;">trium</expan> <expan abbr="angulor&utilde;">angulorum</expan> A, B, A C B, &longs;unt æqua<lb/>les partibus aggregati <expan abbr="duor&utilde;">duorum</expan>, & ideo <expan abbr="aggregat&utilde;">aggregatum</expan>, aggrega to æqua<lb/>le e&longs;t. </s>

<s>quod medium e&longs;t in genere cau&longs;æ materialis. </s>

<s>quod verò partes illius <lb/>&longs;int æquales partibus huius, probatur, per dignitatem <expan abbr="illã">illam</expan>, quæ &longs;unt æqualia <lb/>vni tertio, &longs;unt etiam inter &longs;e. </s>

<s>partes porrò aggregati trium angulorum <lb/>erant hæ, anguli A, B, vna; altera verò angulus A C B; partes verò aggre<lb/>gati duorum rectorum erant A C B, A C D, quibus partibus, illæ &longs;unt æqua<lb/>les, & ideo totum toti æquale. </s>

<s>quod medium e&longs;t omnino intrin&longs;ecum, & ex <lb/>proprijs ip&longs;ius trianguli, &longs;iue ex proprijs angulorum ip&longs;ius, cum &longs;int ip&longs;ius <lb/>partes. </s>

<s>quod pariter medium ex parte pa&longs;&longs;ionis, quæ demon&longs;tratur, e&longs;t ex <lb/>proprijs, cum &longs;int partes illius materiales. </s>

<s>per materiam autem oportet <lb/>hoc loco intelligere materiam intelligibilem, ide&longs;t quantitatem à qualita<lb/>tibas ab&longs;tractam, & terminatam, de qua pluribus agemus infra in tractatu <lb/>de natura mathematicarum. </s>

<s>Hinc videas eos magnopere decipi, qui pu<lb/>tant, hanc demon&longs;trationem e&longs;&longs;e per extrin&longs;eca, eò quod ad demon&longs;tran<lb/>dum producatur linea B C, in D, putantes lineam illam productam C D, <lb/>e&longs;&longs;e demon&longs;trationis medium; lineæ <expan abbr="namq;">namque</expan> huiu&longs;modi, quæ in demon&longs;tra<lb/>tionibus geometricis con&longs;truuntur, nunquam &longs;unt media propria demon<lb/>&longs;trationum, &longs;ed tantummodo a&longs;&longs;umuntur ad probandum medium iam ex<lb/>cogitatum e&longs;&longs;e veram cau&longs;am conclu&longs;ionis. </s>

<s>Hinc etiam manife&longs;tè colligas <pb pagenum="56"/>Mathematicas facultates habere demon&longs;trationes perfecti&longs;&longs;imas, quod <lb/>ageometreti negare &longs;olent, &longs;ed audacter aiunt exempla Ari&longs;t.

non e&longs;&longs;e vera: <lb/><expan abbr="neq;">neque</expan> requiri veritatem exemplorum; in <expan abbr="quor&utilde;">quorum</expan> <expan abbr="vtroq;">vtroque</expan> peccant, nam dictum <lb/>illud v&longs;urpari &longs;olet, & debet de exemplis moralibus. </s>

<s>at vero requiri confor<lb/>mitatem exemplorum cum regulis traditis, nemo &longs;anæ mentis dubitabit. <lb/></s>

<s>Vernm i&longs;ti confundunt conformitatem cum veritate. </s>

<s>Veritas exemplo tunc <lb/>ine&longs;t, quando illud, quod in exemplo narratur, verè extitit, vt &longs;i quis in <lb/>exemplum pudicitiæ afferret hi&longs;toriam Io&longs;ephi, <expan abbr="ver&utilde;">verum</expan> i&longs;tuà e&longs;&longs;et exemplum. <lb/></s>

<s>quæ veritas in exemplis moralibus non &longs;emper e&longs;t nece&longs;&longs;aria, talia exempla <lb/>&longs;unt &longs;æpè parabolæ, & fabulæ, quæ nunquam extiterunt, v. g. </s>

<s>narratur ab <lb/>Ari&longs;t.

de quodam filio, qui patrem crudeliter traxerat, qui po&longs;tea grandior <lb/>factus, cum filium procrea&longs;&longs;et, ab eodem pariter raptatus e&longs;t ip&longs;e, v&longs;que ad <lb/>eundem locum, quo ip&longs;e patrem &longs;uum impiè raptauerat. </s>

<s>non e&longs;t nece&longs;&longs;e, ta<lb/>lem extiti&longs;&longs;e filium, <expan abbr="neq;">neque</expan> patrem. </s>

<s>Verumtamen &longs;emper conformitas exem<lb/>pli cum regulis, & præceptis, quæ traduntur nece&longs;&longs;aria e&longs;t, alioquin exem<lb/>pla de&longs;truerent id, quod præceptio con&longs;truit, <expan abbr="illi&qacute;">illique</expan> contraria e&longs;&longs;et, quod om<lb/>nino ab&longs;urdum foret. </s>

<s>non &longs;ecus, ac &longs;i quis vellet alium docere characteres <lb/>latinos, <expan abbr="illi&qacute;">illique</expan>; barbaros, quos Gothicos vocant in exemplum proponeret. </s>

<s>re<lb/>quiritur igitur &longs;emper in omni exemplo conformitas cum eo, quod doce<lb/>tur; in moralibus tamen non &longs;emper requiritur veritas, vti diximus; Alij <lb/>verò dicunt non requiri in exemplis determinatam veritatem, &longs;ed &longs;atis e&longs;&longs;e, <lb/>&longs;i exemplum verum &longs;it &longs;ecundum opinionem aliquorum: <expan abbr="quor&utilde;">quorum</expan> &longs;ententiam <lb/>non improbamus. </s>

<s>Exempla igitur ab Ari&longs;t.

pa&longs;&longs;im ex mathem aticis allata, <lb/>congrua, <expan abbr="conformia&qacute;">conformiaque</expan>; omninò &longs;unt ip&longs;ius doctrinæ, aliter ip&longs;um perpetuò <lb/>mentientem facimus. </s>

<s>Po&longs;tremò illud etiam e&longs;t aduertendum, fortè Ari&longs;t.

in <lb/>præ&longs;enti textu &longs;pecta&longs;&longs;e <expan abbr="nõ">non</expan> ad hanc Euclidianam demon&longs;trationem, &longs;ed po<lb/>tius ad Pithagoricam. </s>

<lb/>præ&longs;enti textu &longs;pecta&longs;&longs;e <expan abbr="nõ">non</expan> ad hanc Euclidianam demon&longs;trationem, &longs;ed po

<s>Pithagorei enim eam aliter, quamuis per idem me<lb/>dium, &longs;cilicet à cau&longs;a materiali, demon&longs;trabant; con&longs;truebant enim aliter, <lb/><expan abbr="neq;">neque</expan> vlla vtebantur diui&longs;ione. </s>

<lb/>tius ad Pithagoricam. </s>

<s>quod dictum velim propter nonnullos, qui ab <lb/>huiu&longs;modi diui&longs;ionibus abhorrent, <expan abbr="timent&qacute;">timentque</expan>; ne demon&longs;trationis perfectio<lb/>ni per eas plurimum derogetur. </s>

<s>Pithagorei enim eam aliter, quamuis per idem me

<lb/>dium, &longs;cilicet à cau&longs;a materiali, demon&longs;trabant; con&longs;truebant enim aliter,

<s>Pithagoreorum demon&longs;trationem vide <lb/>apud Clauium in &longs;cholio 32. primi Euclidis, quam ex Eudemo etiam Pro<lb/>clus in comm. eiu&longs;dem recitat.</s>

<lb/><expan abbr="neq;">neque</expan> vlla vtebantur diui&longs;ione. </s>

<s>quod dictum velim propter nonnullos, qui ab

<lb/>huiu&longs;modi diui&longs;ionibus abhorrent, <expan abbr="timent&qacute;">timentque</expan>; ne demon&longs;trationis perfectio

<lb/>ni per eas plurimum derogetur. </s>

<s>Pithagoreorum demon&longs;trationem vide

<lb/>apud Clauium in &longs;cholio 32. primi Euclidis, quam ex Eudemo etiam Pro

<lb/>clus in comm. eiu&longs;dem recitat.</s>

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<s>Ibidem <emph type="italics"/>(Demon&longs;tratio autem non computatur in aliud genus; m&longs;i, vt dictum <lb/>e&longs;t geometricæ demon&longs;trationes in Per&longs;pectiuas, aut Mcchamcas, & arithmeticæ in <lb/>harmonicas)<emph.end type="italics"/> exempla &longs;ubalternationis Per&longs;pectiuæ, & Mu&longs;icæ in tex. 20. at<lb/>tulimus; nunc Mechanicæ &longs;ubalternationis, quam hic Ari&longs;t.

<s>Ibidem <emph type="italics"/>(Demon&longs;tratio autem non computatur in aliud genus; m&longs;i, vt dictum

<lb/>e&longs;t geometricæ demon&longs;trationes in Per&longs;pectiuas, aut Mcchamcas, & arithmeticæ in

<lb/>harmonicas)<emph.end type="italics"/> exempla &longs;ubalternationis Per&longs;pectiuæ, & Mu&longs;icæ in tex. 20. at

<lb/>tulimus; nunc Mechanicæ &longs;ubalternationis, quam hic Ari&longs;t.

in&longs;inuat, exem<lb/>plum &longs;it illud, quod Archimedes prop.

in&longs;inuat, exem

<lb/>plum &longs;it illud, quod Archimedes prop.

14. primi Aequep. </s>

14. primi Aequep. demon&longs;trat, ni

<lb/>mirum centrum grauitatis omnis trianguli e&longs;&longs;e punctum illud, in quo rectæ

<lb/>lineæ ab angulis trianguli ad dimidia latera oppo&longs;ita ductæ concurrunt. </s>

<s>demon&longs;trat, ni<lb/>mirum centrum grauitatis omnis trianguli e&longs;&longs;e punctum illud, in quo rectæ <lb/>lineæ ab angulis trianguli ad dimidia latera oppo&longs;ita ductæ concurrunt. </s>

<s>&longs;it <lb/>triangulum A B C, à cuius angulis A, & B, ducantur duæ rectæ A D, B E, ita <lb/>vt bifariam &longs;ecent latera A C, B C, in punctis D, & E, & concurrant in F. <lb/></s>

<s>Dico F, e&longs;&longs;e centrum grauitatis propo&longs;iti trianguli. </s>

<s>&longs;it

<lb/>triangulum A B C, à cuius angulis A, & B, ducantur duæ rectæ A D, B E, ita

<lb/>vt bifariam &longs;ecent latera A C, B C, in punctis D, & E, & concurrant in F.

<s>Quoniam enim in 13. <lb/>Aequep. probauit centrum grauitatis e&longs;&longs;e in ea linea, quæ ducta ab angulo <lb/>quouis &longs;ecat oppo&longs;itum latus bifariam, crit in linea A D, <expan abbr="centr&utilde;">centrum</expan> grauitatis. <pb pagenum="57"/><figure id="fig28"/><lb/>&longs;ed eadem ratione erit etiam in linea B E, er<lb/>go non ni&longs;i in puncto F, quod <expan abbr="&longs;ol&utilde;">&longs;olum</expan> e&longs;t in vtra<lb/>que, quod erat demon&longs;trandum. </s>

<s>Dico F, e&longs;&longs;e centrum grauitatis propo&longs;iti trianguli. </s>

<s>Quoniam enim in 13.

<lb/>Aequep. probauit centrum grauitatis e&longs;&longs;e in ea linea, quæ ducta ab angulo

<lb/>quouis &longs;ecat oppo&longs;itum latus bifariam, crit in linea A D, <expan abbr="centr&utilde;">centrum</expan> grauitatis. <pb pagenum="57"/><figure place="text" xlink:href="figures-la/009.01.057.1.tif"/>

<lb/>&longs;ed eadem ratione erit etiam in linea B E, er

<lb/>go non ni&longs;i in puncto F, quod <expan abbr="&longs;ol&utilde;">&longs;olum</expan> e&longs;t in vtra

<lb/>que, quod erat demon&longs;trandum. </s>

<s>ex quibus ap<lb/>paret, qua ratione mechanica conclu&longs;io Geo<lb/>metriæ &longs;ubiaceat, dum lineari di&longs;cur&longs;u ip&longs;a <lb/>demon&longs;tratio perficitur. </s>

<s>Scias præterea cen<lb/>trum grauitatis e&longs;&longs;e tale punctum, ex quo &longs;i &longs;u<lb/>&longs;pendatur corpus triangulare vniformis cra&longs;<lb/>&longs;itici, manet &longs;emper horizonti parallelum, &longs;i <lb/>tamen antequam &longs;u&longs;penderetur, iacebat plano horizontis, æquidi&longs;tans; <lb/><expan abbr="neq;">neque</expan> &longs;i &longs;u&longs;pen&longs;um feratur huc illud nutat, &longs;ed &longs;emper in <expan abbr="cod&etilde;">codem</expan> &longs;itu per&longs;euerat.</s>

<s>ex quibus ap

<lb/>paret, qua ratione mechanica conclu&longs;io Geo

<lb/>metriæ &longs;ubiaceat, dum lineari di&longs;cur&longs;u ip&longs;a

<lb/>demon&longs;tratio perficitur. </s>

<s>Scias præterea cen

<lb/>trum grauitatis e&longs;&longs;e tale punctum, ex quo &longs;i &longs;u

<lb/>&longs;pendatur corpus triangulare vniformis cra&longs;

<lb/>&longs;itici, manet &longs;emper horizonti parallelum, &longs;i

<lb/>tamen antequam &longs;u&longs;penderetur, iacebat plano horizontis, æquidi&longs;tans;

<lb/><expan abbr="neq;">neque</expan> &longs;i &longs;u&longs;pen&longs;um feratur huc illud nutat, &longs;ed &longs;emper in <expan abbr="cod&etilde;">codem</expan> &longs;itu per&longs;euerat.</s>

<s>Tex. 24. <emph type="italics"/>(Veluti Arithmetica quidem quid impar, aut par; aut quadrangu<lb/>lum, aut cubus)<emph.end type="italics"/> cogno&longs;cas hinc certò certius quadrangulum, & cubum e&longs;&longs;e <lb/>&longs;pecies numerorum, &longs;icuti &longs;upra tex. 9. & 20. explicauimus, quò nunc te vi<lb/>ci&longs;&longs;im, vt præ&longs;entem locum intelligas, remittimus.</s>

<s>Tex. 24. <emph type="italics"/>(Veluti Arithmetica quidem quid impar, aut par; aut quadrangu

<lb/>lum, aut cubus)<emph.end type="italics"/> cogno&longs;cas hinc certò certius quadrangulum, & cubum e&longs;&longs;e

<lb/>&longs;pecies numerorum, &longs;icuti &longs;upra tex. 9. & 20. explicauimus, quò nunc te vi

<lb/>ci&longs;&longs;im, vt præ&longs;entem locum intelligas, remittimus.</s>

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<s>Ibidem <emph type="italics"/>(Geometrica verò quid irrationale, aut refrangi, aut concurrere)<emph.end type="italics"/> per <lb/>verbum, irrationale, non videtur Ari&longs;t.

<s>Ibidem <emph type="italics"/>(Geometrica verò quid irrationale, aut refrangi, aut concurrere)<emph.end type="italics"/> per

<lb/>verbum, irrationale, non videtur Ari&longs;t.

intellexi&longs;&longs;e proprietatem illam duo<lb/>rum linearum incommen&longs;urabilium longitudine, & potentia, quia v&longs;us fui&longs;<lb/>&longs;et verbo, <foreign lang="greek">a/rrpton.</foreign> quod apud Geometras v&longs;urpari &longs;olet in illa &longs;ignificatio<lb/>ne, &longs;ed v&longs;us e&longs;t verbo, <foreign lang="greek">a\logon,</foreign> quod latinè redditur improportionale.</s>

intellexi&longs;&longs;e proprietatem illam duo

<lb/>rum linearum incommen&longs;urabilium longitudine, & potentia, quia v&longs;us fui&longs;

<lb/>&longs;et verbo, <foreign lang="greek">a/rrpton.</foreign> quod apud Geometras v&longs;urpari &longs;olet in illa &longs;ignificatio

<lb/>ne, &longs;ed v&longs;us e&longs;t verbo, <foreign lang="greek">a\logon,</foreign> quod latinè redditur improportionale.</s>

<s>Per verbum <emph type="italics"/>(Refrangi)<emph.end type="italics"/> &longs;eu frangi, intelligit lineam aliquam rectam, non <lb/>in directum rendere, &longs;ed in aliquo puncto frangi, &longs;eu declinari à rectitudine, <lb/>ita vt con&longs;tituat angulum.</s>

<s>Per verbum <emph type="italics"/>(Refrangi)<emph.end type="italics"/> &longs;eu frangi, intelligit lineam aliquam rectam, non

<lb/>in directum rendere, &longs;ed in aliquo puncto frangi, &longs;eu declinari à rectitudine,

<lb/>ita vt con&longs;tituat angulum.</s>

<s>Per verbum <emph type="italics"/>(Concurrere)<emph.end type="italics"/> intelligit, non e&longs;&longs;e parallelas, &longs;ed ad idem ali<lb/>quod punctum coire, &longs;i protrahantur.</s>

<s>Per verbum <emph type="italics"/>(Concurrere)<emph.end type="italics"/> intelligit, non e&longs;&longs;e parallelas, &longs;ed ad idem ali

<lb/>quod punctum coire, &longs;i protrahantur.</s>

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<s>Ibidem <emph type="italics"/>(Et Astrologia &longs;imiliter)<emph.end type="italics"/> per A&longs;trologiam intelligit Ari&longs;t.

non iu<lb/>diciariam, quamuis à recentioribus hoc nomine vocetur, &longs;ed quam hodie <lb/>dicunt A&longs;tronomiam, <expan abbr="ait&qacute;">aitque</expan>; ip&longs;am con&longs;iderare quantitatem, figuram, mo<lb/>tum, & locum totius Mundi, ac partium ip&longs;ius integrantium, vt &longs;unt C&oelig;li, <lb/>& Elementa.</s>

non iu

<lb/>diciariam, quamuis à recentioribus hoc nomine vocetur, &longs;ed quam hodie

<lb/>dicunt A&longs;tronomiam, <expan abbr="ait&qacute;">aitque</expan>; ip&longs;am con&longs;iderare quantitatem, figuram, mo

<lb/>tum, & locum totius Mundi, ac partium ip&longs;ius integrantium, vt &longs;unt C&oelig;li,

<lb/>& Elementa.</s>

<s>Tex. 25. <emph type="italics"/>(<expan abbr="Neq;">Neque</expan> Geometra fal&longs;a &longs;upponit, quemadmodum quidam a&longs;&longs;eruere di<lb/>centes, quod non oportet fal&longs;o vti: Geometram verò mentiri dicentem pedalem, non <lb/>pedalem, aut rectam de&longs;criptam, non rectam <expan abbr="exist&etilde;tem">existentem</expan>: Geometra verò nihil con<lb/>cludit eò, quod bæc e&longs;t linea, &longs;ed quæ per hæc e&longs;tenduntur)<emph.end type="italics"/> innuit his verbis eam <lb/>materiam intelligibilem, quæ e&longs;t &longs;ubiectum Geometriæ: eam &longs;cilicet, quæ <lb/>&longs;ub figuris Geometricis &longs;en&longs;ibilibus, & <expan abbr="plerunq;">plerunque</expan> fal&longs;is latet; nam &longs;æpè Geo<lb/>metra vtitur linea quadam &longs;en&longs;ibili pro recta, quæ verè nec e&longs;t linea mathe<lb/>matica, nec recta; &longs;upponit aliquando talem lineam e&longs;&longs;e pedalem, quæ ve<lb/>rè non e&longs;t pedalis: Verumtamen non mentitur, quia re&longs;picit ad veram li<lb/>neam mathematicam, quæ &longs;ub illa intelligitur, & quæ recta concipitur; & <lb/>quidem hæc omnia verè concipiuntur, quoniam ita e&longs;&longs;e re vera po&longs;&longs;unt.</s>

<s>Tex. 25. <emph type="italics"/>(<expan abbr="Neq;">Neque</expan> Geometra fal&longs;a &longs;upponit, quemadmodum quidam a&longs;&longs;eruere di

<lb/>centes, quod non oportet fal&longs;o vti: Geometram verò mentiri dicentem pedalem, non

<lb/>pedalem, aut rectam de&longs;criptam, non rectam <expan abbr="exist&etilde;tem">existentem</expan>: Geometra verò nihil con

<lb/>cludit eò, quod bæc e&longs;t linea, &longs;ed quæ per hæc e&longs;tenduntur)<emph.end type="italics"/> innuit his verbis eam

<lb/>materiam intelligibilem, quæ e&longs;t &longs;ubiectum Geometriæ: eam &longs;cilicet, quæ

<lb/>&longs;ub figuris Geometricis &longs;en&longs;ibilibus, & <expan abbr="plerunq;">plerunque</expan> fal&longs;is latet; nam &longs;æpè Geo

<lb/>metra vtitur linea quadam &longs;en&longs;ibili pro recta, quæ verè nec e&longs;t linea mathe

<lb/>matica, nec recta; &longs;upponit aliquando talem lineam e&longs;&longs;e pedalem, quæ ve

<lb/>rè non e&longs;t pedalis: Verumtamen non mentitur, quia re&longs;picit ad veram li

<lb/>neam mathematicam, quæ &longs;ub illa intelligitur, & quæ recta concipitur; &

<lb/>quidem hæc omnia verè concipiuntur, quoniam ita e&longs;&longs;e re vera po&longs;&longs;unt.</s>

<s>Tex. 28. <emph type="italics"/>(Coaltern as verò coincidere)<emph.end type="italics"/> per coalternas intelligendas e&longs;&longs;e pa<lb/>rallelas lineas, alias, & nunc <expan abbr="quoq;">quoque</expan> monemus.</s>

<s>Tex. 28. <emph type="italics"/>(Coaltern as verò coincidere)<emph.end type="italics"/> per coalternas intelligendas e&longs;&longs;e pa

<lb/>rallelas lineas, alias, & nunc <expan abbr="quoq;">quoque</expan> monemus.</s>

<s>Tex. 29. <emph type="italics"/>(In Matbematicis verò non est &longs;imiliter paralogi&longs;mus, quoniam me<lb/>di&ubreve; e&longs;t &longs;emper, quod duplex, de hoc enim omni, & hoc rur&longs;us de alio dicitur omni)<emph.end type="italics"/><lb/>aduerte, quod quamuis nonnulli codices habeant pro, in mathematicis, in <pb pagenum="58"/>di&longs;cipli<gap/>is, idem tamen apud græcos <foreign lang="greek">maqhmata</foreign> &longs;unt, ac apud latinos di&longs;ci<lb/>plmæ; verbum autem <foreign lang="greek">maqhmata</foreign> v&longs;urpat hoc loco Ari&longs;toteles. </s>

<s>Tex. 29. <emph type="italics"/>(In Matbematicis verò non est &longs;imiliter paralogi&longs;mus, quoniam me

<lb/>di&ubreve; e&longs;t &longs;emper, quod duplex, de hoc enim omni, & hoc rur&longs;us de alio dicitur omni)<emph.end type="italics"/>

<s>Porrò non <lb/>e&longs;t in mathematicis, &longs;icut in alijs paralogi&longs;mus, quia in omni demon&longs;tra<lb/>tione maius extremum dicitur de omni medio, & rur&longs;us medium dicitur de <lb/>omni minori extremo, ac &longs;i diceret mathematicæ demon&longs;trationes &longs;unt in <lb/>primo modo, qui barbarè à latinis recentioribus Barbara appellatur. </s>

<lb/>aduerte, quod quamuis nonnulli codices habeant pro, in mathematicis, in <pb pagenum="58"/>di&longs;ciplinis, idem tamen apud græcos <foreign lang="greek">maqhmata</foreign> &longs;unt, ac apud latinos di&longs;ci

<lb/>plmæ; verbum autem <foreign lang="greek">maqhmata</foreign> v&longs;urpat hoc loco Ari&longs;toteles. </s>

<s>Hæc <lb/>e&longs;t autem pulcherrima mathematicarum commendatio, quippe præclarum <lb/>e&longs;t à laudato laudari. </s>

<s>Porrò non

<s>In mathematicis, inquit, non accidit &longs;imiliter para<lb/>logi&longs;mus, ide&longs;t, tam frequenter, quemadmodum in &longs;yllogi&longs;mis dialecticis, <lb/>quia modus argumentandi mathematicarum e&longs;t perfecti&longs;&longs;imus, quippe in <lb/>primo modo primæ figuræ.</s>

<lb/>e&longs;t in mathematicis, &longs;icut in alijs paralogi&longs;mus, quia in omni demon&longs;tra

<lb/>tione maius extremum dicitur de omni medio, & rur&longs;us medium dicitur de

<lb/>omni minori extremo, ac &longs;i diceret mathematicæ demon&longs;trationes &longs;unt in

<lb/>primo modo, qui barbarè à latinis recentioribus Barbara appellatur. </s>

<s>Hæc

<lb/>e&longs;t autem pulcherrima mathematicarum commendatio, quippe præclarum

<lb/>e&longs;t à laudato laudari. </s>

<s>In mathematicis, inquit, non accidit &longs;imiliter para

<lb/>logi&longs;mus, ide&longs;t, tam frequenter, quemadmodum in &longs;yllogi&longs;mis dialecticis,

<lb/>quia modus argumentandi mathematicarum e&longs;t perfecti&longs;&longs;imus, quippe in

<lb/>primo modo primæ figuræ.</s>

<s>Eodem tex. <emph type="italics"/>(Contingit autem quo&longs;dam non &longs;yllogi&longs;ticè dicere, & quod ex vtri&longs;<lb/>que con&longs;equentia accipiunt, quemadmodum & Cæneus facit, quod ignis in multi<lb/>plici proportione: etenim ignis celeriter gignitur, vt ait: & hæc est proportio. </s>

<s>Eodem tex. <emph type="italics"/>(Contingit autem quo&longs;dam non &longs;yllogi&longs;ticè dicere, & quod ex vtri&longs;

<lb/>que con&longs;equentia accipiunt, quemadmodum & Cæneus facit, quod ignis in multi

<s>&longs;ic <lb/>autem non e&longs;t &longs;yllog &longs;mus, ni&longs;i celerrimam proportio &longs;equatur multiplex: & ignem <lb/>celerrima in motu proportio)<emph.end type="italics"/> verba illa (in multiplici proportione) græcè &longs;ic <lb/>&longs;e habent, <foreign lang="greek">en th pollaplasioni analogia,</foreign> quod melius redditur latinè in mul<lb/>tiplici proportione, quemadmodum fecimus, quam in multiplicata, quem<lb/>admodum in vulgata editione. </s>

<lb/>plici proportione: etenim ignis celeriter gignitur, vt ait: & hæc est proportio. </s>

<s>&longs;ic

<lb/>autem non e&longs;t &longs;yllog &longs;mus, ni&longs;i celerrimam proportio &longs;equatur multiplex: & ignem

<lb/>celerrima in motu proportio)<emph.end type=