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version 1.73, 2002/09/13 18:22:04

version 1.74, 2002/09/14 07:28:57

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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> </front> <body> <chap> <pb xlink:href="009/01/001.jpg"/>

<s>ARISTOTELIS</s>

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

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

<s>Sumptibus Hieronymi Tamburini.</s><pb xlink:href="009/01/002.jpg"/><pb xlink:href="009/01/003.jpg"/>

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

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<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>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>tu enim cùm Phy&longs;iologiæ, ac Mathe<lb/>maticarum omnium Encyclopædiam mirum in modum<emph.end type="italics"/><pb xlink:href="009/01/004.jpg"/><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. Duce Ranutio <lb/>inter primarios de Rep. </s>

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

<s><emph type="italics"/>Parmæ Idibus Maij M. DC. XIIII.<emph.end type="italics"/><lb/></s><pb xlink:href="009/01/005.jpg"/>

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

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<s>LECTORI.</s>

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<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.

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

ob&longs;curè dicta intelligunt.</s><pb xlink:href="009/01/007.jpg"/>

<s>Quarta. </s>

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

<s>nego, ait; qui&longs;piam con&longs;equen<pb xlink:href="009/01/008.jpg"/>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>

<s>quæ vitio&longs;a argumentatio quan<lb/>tumuis læuis &longs;it, & manife&longs;ta, quo&longs;dam tamen magni nominis philo&longs;o<lb/>phantes adeò tor&longs;it, vt adhuc torqueat.</s>

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<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>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>

<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 xlink:href="009/01/009.jpg"/>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>

<s>Verum enim verò optimè &longs;cio, ea, <lb/>qu&ecedil; hactenus dicta &longs;unt non in omnes no&longs;tri temporis Philo&longs;ophos qua<lb/>drare, cum non pauci hodie quoque &longs;int, qui more antiquorum Mathe<lb/>maticis &longs;uffulti, optimè &longs;uis &longs;tudijs con&longs;ulentes, reliquam Philo&longs;ophiam <lb/>non &longs;ine magno compendio aggrediuntur. </s>

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<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>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>

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<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>

<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><pb pagenum="12"/>

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

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<s><emph type="italics"/>T. 25. Geometram non mentiri in &longs;uis exemplis.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 28. De Parallelis.<emph.end type="italics"/></s><pb pagenum="13"/>

<s><emph type="italics"/>T. 28. De Parallelis.<emph.end type="italics"/></s><pb xlink:href="009/01/013.jpg"/>

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

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<s><emph type="italics"/>T. 31. Quonam infinito vtantur Mathematici.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 71. De infinito Mathematica.<emph.end type="italics"/></s><pb pagenum="14"/>

<s><emph type="italics"/>T. 71. De infinito Mathematica.<emph.end type="italics"/></s><pb xlink:href="009/01/014.jpg"/>

<s>Ex 4. Phy&longs;ic.</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 xlink:href="009/01/015.jpg"/>

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

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

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

<s>Tetragoni&longs;mus est inuentio mediæ.<emph.end type="italics"/></s><pb xlink:href="009/01/016.jpg"/>

<s><emph type="italics"/>Tex. 8. Geodæ&longs;ia quid.<emph.end type="italics"/></s>

Line 904

<s><emph type="italics"/>Quæ&longs;t. 15. De Vmbelicis.<emph.end type="italics"/></s>

<s><emph type="italics"/>Quæ&longs;t. 16. De ligni oblongi, ac breuis flexura.<emph.end type="italics"/></s><pb pagenum="17"/>

<s><emph type="italics"/>Quæ&longs;t. 16. De ligni oblongi, ac breuis flexura.<emph.end type="italics"/></s><pb xlink:href="009/01/017.jpg"/>

<s><emph type="italics"/>Quæ&longs;t. 17. De Cuneo.<emph.end type="italics"/></s>

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

3. Automata.<emph.end type="italics"/></s><pb pagenum="18"/>

3. Automata.<emph.end type="italics"/></s><pb xlink:href="009/01/018.jpg"/>

<s>De Generatione Animal.</s>

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<s>2. Iterum Diametri ethymologia.<emph.end type="italics"/></s><pb pagenum="19"/>

<s>2. Iterum Diametri ethymologia.<emph.end type="italics"/></s><pb xlink:href="009/01/019.jpg"/>

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<s><emph type="italics"/>28. Cur cantilenæ quædam leges de cebantur?<emph.end type="italics"/></s>

<s><emph type="italics"/>30. De Harmonijs, &longs;eu Modis, &longs;eu Tonis pri&longs;corum.<emph.end type="italics"/></s><pb pagenum="20"/>

<s><emph type="italics"/>30. De Harmonijs, &longs;eu Modis, &longs;eu Tonis pri&longs;corum.<emph.end type="italics"/></s><pb xlink:href="009/01/020.jpg"/>

<s><emph type="italics"/>31. Vetustiores fui&longs;&longs;e magis Melopæos.<emph.end type="italics"/></s>

Line 1422

<s><emph type="italics"/>7. Quantitas ip&longs;ius num videatur?<emph.end type="italics"/></s>

<s><emph type="italics"/>8. Cur modo maior, modo minor videatur, & cui&longs;dam lepida deceptio.<emph.end type="italics"/></s><pb pagenum="21"/>

<s><emph type="italics"/>8. Cur modo maior, modo minor videatur, & cui&longs;dam lepida deceptio.<emph.end type="italics"/></s><pb xlink:href="009/01/021.jpg"/>

<s>Additamentum de natura Mathematicarum di&longs;ciplinarum.</s>

Line 1442

<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><pb pagenum="22"/>

<s><emph type="bold"/>Finis Primi Indicis.<emph.end type="bold"/></s><pb xlink:href="009/01/022.jpg"/>

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

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2. Magnorum Moral. & cap.

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>

10. Mag. Moral. & cap. 16. Mag. Moral. <lb/>& cap. 7. &longs;ecundi Eudem. & cap. 12. &longs;ecundi Eudem. & problema 6. &longs;ectio<pb xlink:href="009/01/023.jpg"/>nis 30. tot Ari&longs;totelis loca illu&longs;trat vnica hæc Euclidis Demon&longs;tratio.</s>

Line 1788

<s>9. vide problem. </s>

<s>3. &longs;ectionis 15. loco 4.</s><pb pagenum="24"/>

<s>3. &longs;ectionis 15. loco 4.</s><pb xlink:href="009/01/024.jpg"/>

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

Line 1880

<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><pb pagenum="25"/>

<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 xlink:href="009/01/026.jpg"/><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 xlink:href="009/01/027.jpg"/><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 xlink:href="009/01/028.jpg"/><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 xlink:href="009/01/029.jpg"/><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 xlink:href="009/01/030.jpg"/><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>Finis Tertij Indicis.</s><pb pagenum="31"/>

<s>Finis Tertij Indicis.</s><pb xlink:href="009/01/031.jpg"/>

<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><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>

<s>MATHEMATICA</s>

Line 1904

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

<s>Ex c. 3. De his, quæ ad aliquid. </s>File "Macintosh HD:Users:petermcl:Desktop:biancani_test_stuff:texts:bianc-test.2.xml"; Line 2219: <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,

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<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>& 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 xlink:href="009/01/034.jpg"/>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>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 id="id.009.01.034.1.jpg" xlink:href="009/01/034/1.jpg" place="text"/>

<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>

Line 1964

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<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<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>& præterea con&longs;tet<gap/> Hip<lb/>pocratem quadra&longs;&longs;e lunulam, vt &longs;uo loco dicemus, & Archimedem in libel<pb xlink:href="009/01/035.jpg"/>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 1984

Line 1985

<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 place="text" xlink:href="figures-la/009.01.035.1.tif"/><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, <lb/><figure id="id.009.01.035.1.jpg" xlink:href="009/01/035/1.jpg" place="text"/><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>

<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>

Line 2010

Line 2011

<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>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 xlink:href="009/01/036.jpg"/>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>

Line 2056

Line 2057

<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>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 2078

Line 2079

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

<s>magnitudines dicuntur, quas <lb/><figure place="text" xlink:href="figures-la/009.01.037.1.tif"/><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/><figure place="text" xlink:href="figures-la/009.01.037.2.tif"/><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>magnitudines dicuntur, quas <lb/><figure id="id.009.01.037.1.jpg" xlink:href="009/01/037/1.jpg" place="text"/><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/><figure id="id.009.01.037.2.jpg" xlink:href="009/01/037/2.jpg" place="text"/><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>

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Line 2101

<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 place="text" xlink:href="figures-la/009.01.037.3.tif"/><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>&longs;unt diameter, & latus eiu&longs;<lb/>dem quadrati, quia nulla pote&longs;t reperiri men&longs;ura quantumuis exigua, vti <lb/><figure id="id.009.01.037.3.jpg" xlink:href="009/01/037/3.jpg" place="text"/><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>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 xlink:href="009/01/038.jpg"/>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>

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Line 2149

<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>

<s>&longs;it I&longs;o&longs;ce<lb/><figure place="text" xlink:href="figures-la/009.01.038.1.tif"/><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>&longs;it I&longs;o&longs;ce<lb/><figure id="id.009.01.038.1.jpg" xlink:href="009/01/038/1.jpg" place="text"/><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>

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Line 2171

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 xlink:href="009/01/039.jpg"/>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>

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Line 2209

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.

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 place="text" xlink:href="figures-la/009.01.039.1.tif"/><lb/>inclinatio in puncto A, &longs;iue acumen illud, &longs;iue mucro, <lb/>e&longs;t ratio anguli. </s>

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="id.009.01.039.1.jpg" xlink:href="009/01/039/1.jpg" place="text"/><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<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>

<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 xlink:href="009/01/040.jpg"/><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>

<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>

<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>

<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 place="text" xlink:href="figures-la/009.01.040.1.tif"/><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>

<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="id.009.01.040.1.jpg" xlink:href="009/01/040/1.jpg" place="text"/><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>

<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 place="text" xlink:href="figures-la/009.01.040.2.tif"/><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 place="text" xlink:href="figures-la/009.01.040.3.tif"/><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>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="id.009.01.040.2.jpg" xlink:href="009/01/040/2.jpg" place="text"/><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="id.009.01.040.3.jpg" xlink:href="009/01/040/3.jpg" place="text"/><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>

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<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 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>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 xlink:href="009/01/041.jpg"/>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>

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<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"/>

<s><emph type="italics"/>Ex Libro &longs;ecundo Priorum.<emph.end type="italics"/></s>

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<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 place="text" xlink:href="figures-la/009.01.042.1.tif"/><lb/>probat Euclides in 28. primi Elem.

<s>quoad <lb/>exempli explicationem vtor figura textibus apponi &longs;olita, quæ e&longs;t præ&longs;ens. <lb/><figure id="id.009.01.042.1.jpg" xlink:href="009/01/042/1.jpg" place="text"/><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>

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<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>& pro<lb/>batur con&longs;equentia hoc modo, quia &longs;i angulus E G B, maior e&longs;t angulo <pb xlink:href="009/01/043.jpg"/>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>

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<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 place="text" xlink:href="figures-la/009.01.043.1.tif"/><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>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="id.009.01.043.1.jpg" xlink:href="009/01/043/1.jpg" place="text"/><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>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>

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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.

<s>e&longs;&longs;e <pb xlink:href="009/01/044.jpg"/>autem terminum mathematicum colligitur manife&longs;tè ex Proelo, qui lib.

3. <lb/>in comm. Elem.

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<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. 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>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. 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 xlink:href="009/01/045.jpg"/><figure id="id.009.01.045.1.jpg" xlink:href="009/01/045/1.jpg" place="text"/><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>

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<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 place="text" xlink:href="figures-la/009.01.045.2.tif"/><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>&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="id.009.01.045.2.jpg" xlink:href="009/01/045/2.jpg" place="text"/><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>

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<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>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>Hippocrates i&longs;te Chius e&longs;t alter <pb xlink:href="009/01/046.jpg"/>ab illo Hippocrate Coo medicorum Magi&longs;tro, vt colligitur ex Alexandre <lb/>Aphrod. in Primum Meteororum de Cometis.</s>

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<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 place="text" xlink:href="figures-la/009.01.046.1.tif"/><lb/>micirculo. </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/><figure id="id.009.01.046.1.jpg" xlink:href="009/01/046/1.jpg" place="text"/><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>

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<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 xlink:href="009/01/047.jpg"/>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>

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<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>

<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 xlink:href="009/01/048.jpg"/><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>

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<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 place="text" xlink:href="figures-la/009.01.048.1.tif"/><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>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="id.009.01.048.1.jpg" xlink:href="009/01/048/1.jpg" place="text"/><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>

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<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/><figure place="text" xlink:href="figures-la/009.01.048.2.tif"/><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>Per altera parte longius, intelligit numerum, qui producitur à duobus <lb/><figure id="id.009.01.048.2.jpg" xlink:href="009/01/048/2.jpg" place="text"/><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>

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<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>

<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 xlink:href="009/01/049.jpg"/><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>

<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>

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<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.

de&longs;umitur, quam propterea primo loco exponendam <lb/><figure place="text" xlink:href="figures-la/009.01.049.1.tif"/><lb/>cen&longs;ui. </s>

de&longs;umitur, quam propterea primo loco exponendam <lb/><figure id="id.009.01.049.1.jpg" xlink:href="009/01/049/1.jpg" place="text"/><lb/>cen&longs;ui. </s>

<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>

<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 place="text" xlink:href="figures-la/009.01.049.2.tif"/><lb/>cti, probabitur de rectis A B, C D, æquidi&longs;tan<lb/>tia. </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/><figure id="id.009.01.049.2.jpg" xlink:href="009/01/049/2.jpg" place="text"/><lb/>cti, probabitur de rectis A B, C D, æquidi&longs;tan<lb/>tia. </s>

<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>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 xlink:href="009/01/050.jpg"/>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>

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<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>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>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 xlink:href="009/01/051.jpg"/>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/><figure place="text" xlink:href="figures-la/009.01.051.1.tif"/><lb/>& con&longs;equentis ad con&longs;equentem. </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/><figure id="id.009.01.051.1.jpg" xlink:href="009/01/051/1.jpg" place="text"/><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>

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<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 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>

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 xlink:href="009/01/052.jpg"/>&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>

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<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 place="text" xlink:href="figures-la/009.01.052.1.tif"/><lb/>narij, 2, 2, 2, primus ducitur in &longs;ecundum, & producitur. <lb/></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="id.009.01.052.1.jpg" xlink:href="009/01/052/1.jpg" place="text"/><lb/>narij, 2, 2, 2, primus ducitur in &longs;ecundum, & producitur. <lb/></s>

<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/>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 place="text" xlink:href="figures-la/009.01.053.1.tif"/><lb/>qui e&longs;t quadratus. </s>

<s>4. qui e&longs;t numerus quadratus huius figuræ, <figure id="id.009.01.052.2.jpg" xlink:href="009/01/052/2.jpg" place="text"/>, 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 xlink:href="009/01/053.jpg"/>& 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="id.009.01.053.1.jpg" xlink:href="009/01/053/1.jpg" place="text"/><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>

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<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/><figure place="text" xlink:href="figures-la/009.01.053.2.tif"/><lb/>per 21. primi Elem.

<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="id.009.01.053.2.jpg" xlink:href="009/01/053/2.jpg" place="text"/><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>

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<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>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 xlink:href="009/01/054.jpg"/><figure id="id.009.01.054.1.jpg" xlink:href="009/01/054/1.jpg" place="text"/><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>

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<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 place="text" xlink:href="figures-la/009.01.054.2.tif"/><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>&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="id.009.01.054.2.jpg" xlink:href="009/01/054/2.jpg" place="text"/><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>

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<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>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 xlink:href="009/01/055.jpg"/><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 place="text" xlink:href="figures-la/009.01.055.1.tif"/><lb/>mus. </s>

mentem probè penetrare poteri<lb/><figure id="id.009.01.055.1.jpg" xlink:href="009/01/055/1.jpg" place="text"/><lb/>mus. </s>

<s>&longs;it ergo <expan abbr="triãgulum">triangulum</expan> A B C. </s>

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<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.

<s>Hinc etiam manife&longs;tè colligas <pb xlink:href="009/01/056.jpg"/>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>

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<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>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 xlink:href="009/01/057.jpg"/><figure id="id.009.01.057.1.jpg" xlink:href="009/01/057/1.jpg" place="text"/><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>

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<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"/><lb/>aduerte, quod quamuis nonnulli codices habeant pro, in mathematicis, in <pb xlink:href="009/01/058.jpg"/>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>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>

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<s>Ibidem (<emph type="italics"/>Conuertuntur autem magis, quæ&longs;unt in mathematicis, quoniam nul<lb/>lum accidens accipiunt (m quo quidem ijs præ&longs;t&abreve;t, quæ di&longs;putationibus traduntur) <lb/>&longs;ed definitiones<emph.end type="italics"/>) Hæc e&longs;t altera mathematicarum laus, vnde earum quoque <lb/>præ&longs;tantia elucet, quia &longs;cilicet mathematicæ pro medijs vtuntur definitio<pb pagenum="59"/>nibus &longs;ubiecti, aut pa&longs;&longs;ionis, quæ nuilo modo &longs;unt accidentalia conclu&longs;ioni, <lb/>v. g. in prima Euclidis demon&longs;tratione per definitionem &longs;ubiecti probantur <lb/>tres lineæ e&longs;&longs;e æquales, quia nimirum &longs;int &longs;emidiametri circulorum æqua<lb/>lium, quæ e&longs;t ip&longs;arum definitio. </s>

<s>Ibidem (<emph type="italics"/>Conuertuntur autem magis, quæ&longs;unt in mathematicis, quoniam nul<lb/>lum accidens accipiunt (m quo quidem ijs præ&longs;t&abreve;t, quæ di&longs;putationibus traduntur) <lb/>&longs;ed definitiones<emph.end type="italics"/>) Hæc e&longs;t altera mathematicarum laus, vnde earum quoque <lb/>præ&longs;tantia elucet, quia &longs;cilicet mathematicæ pro medijs vtuntur definitio<pb xlink:href="009/01/059.jpg"/>nibus &longs;ubiecti, aut pa&longs;&longs;ionis, quæ nuilo modo &longs;unt accidentalia conclu&longs;ioni, <lb/>v. g. in prima Euclidis demon&longs;tratione per definitionem &longs;ubiecti probantur <lb/>tres lineæ e&longs;&longs;e æquales, quia nimirum &longs;int &longs;emidiametri circulorum æqua<lb/>lium, quæ e&longs;t ip&longs;arum definitio. </s>

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<s>Deinde adhuc magis te conuerte, & cer<lb/>nes illuminationem dimidiatæ Lunæ &longs;imilem: verte adhuc te ip&longs;um donec <lb/>&longs;it &longs;phæra ita lumini oppo&longs;ita, vt inter ip&longs;am, & lumen oculus tuus &longs;it me<lb/>dius; apparebit tunc tota illuminatio, quæ erit in&longs;tar plenilunij. </s>

<s>perge ad<pb pagenum="60"/>huc te ip&longs;um conuertere, & videbis paulatim lumen oculo tuo decre&longs;cere <lb/>non aliter ac in Luna &longs;ene&longs;cente. </s>

<s>perge ad<pb xlink:href="009/01/060.jpg"/>huc te ip&longs;um conuertere, & videbis paulatim lumen oculo tuo decre&longs;cere <lb/>non aliter ac in Luna &longs;ene&longs;cente. </s>

<s><expan abbr="atq;">atque</expan> hoe e&longs;t &longs;phæricè illuminari, fierique <lb/>&longs;phærica illuminationis augmenta. </s>

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<s>nunc igitur paraphra<gap/><lb/>&longs;olum huius loci dabo. </s>

<s>Triangnlo I&longs;o&longs;celi, & Scaleno connenit pa&longs;&longs;io i<gap/><lb/>habere tres angulos æquales duobus rectis angulis &longs;ecundum aliquod<gap/><pb pagenum="61"/>mune, quia illis competit, quatenus ambo &longs;unt figura quædam, ide&longs;t, qua<lb/>tenus <expan abbr="vtrumq;">vtrumque</expan> illorum triangulum e&longs;t; triangulo <expan abbr="namq;">namque</expan> omni primo com<lb/>petit habere tres angulos æquales duobus rectis.</s>

<s>Triangnlo I&longs;o&longs;celi, & Scaleno connenit pa&longs;&longs;io i<gap/><lb/>habere tres angulos æquales duobus rectis angulis &longs;ecundum aliquod<gap/><pb xlink:href="009/01/061.jpg"/>mune, quia illis competit, quatenus ambo &longs;unt figura quædam, ide&longs;t, qua<lb/>tenus <expan abbr="vtrumq;">vtrumque</expan> illorum triangulum e&longs;t; triangulo <expan abbr="namq;">namque</expan> omni primo com<lb/>petit habere tres angulos æquales duobus rectis.</s>

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<s>&longs;ed fortè <expan abbr="dic&etilde;dum">dicendum</expan>, <lb/>Ari&longs;t.

con&longs;idera&longs;&longs;e, Minam re&longs;pectu Talenti, re&longs;pectu enim illius dici pote&longs;t <lb/>principium, cum &longs;ex millia minarum in Attico talento continerentur.</s><figure place="text" xlink:href="figures-la/009.01.061.1.tif"/>

con&longs;idera&longs;&longs;e, Minam re&longs;pectu Talenti, re&longs;pectu enim illius dici pote&longs;t <lb/>principium, cum &longs;ex millia minarum in Attico talento continerentur.</s><figure id="id.009.01.061.1.jpg" xlink:href="009/01/061/1.jpg" place="text"/>

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<s>quoniain trian<lb/>gulus: & hoc quoniam figura rectilinea, &c.)<emph.end type="italics"/> exem<lb/>plo geometrico vult o&longs;tendere demon&longs;trationem <lb/>vniuer&longs;alem e&longs;&longs;e particulari præ&longs;tantiorem: e&longs;t <lb/>autem exemplum de pulcherrima, <expan abbr="atq;">atque</expan> admira<lb/>bili proprietate, quæ omnibus figuris rectilineis <lb/>conuenit, e&longs;t <expan abbr="&qacute;">que</expan>; huiu&longs;modi: Omnis figuræ rectili<lb/>neæ anguli externi omnes &longs;imul &longs;umpti, &longs;unt æqua <lb/>les quatuor rectis angulis, quæ affectio demon<lb/>&longs;tratur in &longs;cholio 32. primi Elem.

dicuntur autern <lb/>anguli externi, qui productis lateribus fiunt, vt in <lb/>triangulo pra&longs;enti anguli externi &longs;unt, B D C, <pb pagenum="62"/>D F E, F B A, ita vt quælibet figura tot angulos externos &longs;ortiatur, quot <lb/>habet latera; cum exproductis lateribus oriantur. </s>

dicuntur autern <lb/>anguli externi, qui productis lateribus fiunt, vt in <lb/>triangulo pra&longs;enti anguli externi &longs;unt, B D C, <pb xlink:href="009/01/062.jpg"/>D F E, F B A, ita vt quælibet figura tot angulos externos &longs;ortiatur, quot <lb/>habet latera; cum exproductis lateribus oriantur. </s>

<s>Vt autem propo&longs;itio ve<lb/>rificetur, &longs;ingula latera ordinatim &longs;unt producenda, hoc e&longs;t, ver&longs;us eandem <lb/>partem, vt in figuris appo&longs;itis vides. </s>

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<s>Et paulo po&longs;t <emph type="italics"/>(Qutmadmod&ubreve; &longs;t vi<lb/>trum perforatum videremus, & lumen <lb/>permeans, planum vtique e&longs;&longs;et propter <lb/>quid comburit)<emph.end type="italics"/> Ioquitur de ea com<lb/>bu&longs;tione, cuæ fit per refractionem <lb/>media &longs;phæra vitrea. </s>

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<s>48. decimi libri; non au<lb/>tem de ea, quæ fit per reflexionem <lb/>ex &longs;peculo concauo quando combu<lb/>&longs;tio fit per refractionem, cau&longs;atur à <lb/>radijs Solis vitrum permeantibus, <lb/>in quo ita franguntur, vt egredien<lb/>tes è vitro &longs;imul vniantur, ex qua <lb/>vnione ita calor intenditur, vt ibi <lb/>comburat. </s>

<s>vt in appo&longs;ita figura cer<lb/>nere facile e&longs;t; in qua radij à Sole <lb/>manentes, &longs;phæram vitream perua<pb pagenum="63"/>dunt, <expan abbr="atq;">atque</expan> in exitu ita refraguntur, vt ad A, punctum coaceruati, ibi po&longs;<lb/>&longs;int, &longs;i quid combu&longs;tibile occurrat, comburere. </s>

<s>vt in appo&longs;ita figura cer<lb/>nere facile e&longs;t; in qua radij à Sole <lb/>manentes, &longs;phæram vitream perua<pb xlink:href="009/01/063.jpg"/>dunt, <expan abbr="atq;">atque</expan> in exitu ita refraguntur, vt ad A, punctum coaceruati, ibi po&longs;<lb/>&longs;int, &longs;i quid combu&longs;tibile occurrat, comburere. </s>

<s>Si igitur, inquit Ari&longs;t.

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cau&longs;am formalem con&longs;onantiæ, & con&longs;equenter defini<lb/>tionem ip&longs;ius. </s>

<s>definiunt igitur Mu&longs;ici con&longs;onantiam hoc modo; Con&longs;onan<lb/>tia e&longs;t compo&longs;itio &longs;oni grauis, & acuti, quæ &longs;uauiter auribus accidit; & quo<lb/>rum &longs;onorum proportio ad inuicem &longs;it &longs;icuti proportio numerorum, qui <lb/>quaternario includuntur: vt e&longs;t proportio 2. ad 1. vel 3. ad 1. vel 4. ad 1. <lb/>vel 3. ad 2. vel 4. ad 3. <expan abbr="Quotie&longs;eunq;">Quotie&longs;eunque</expan> igitur duo &longs;oni habuerin quampiam <pb pagenum="64"/>ex <expan abbr="quinq;">quinque</expan> prædictis proportionibus, &longs;i &longs;imul coaluerint, ita vt ex eis vnue <lb/>tantum &longs;onus efficiatur; &longs;onus ille erit concordans, & auribus gratus. </s>

<s>definiunt igitur Mu&longs;ici con&longs;onantiam hoc modo; Con&longs;onan<lb/>tia e&longs;t compo&longs;itio &longs;oni grauis, & acuti, quæ &longs;uauiter auribus accidit; & quo<lb/>rum &longs;onorum proportio ad inuicem &longs;it &longs;icuti proportio numerorum, qui <lb/>quaternario includuntur: vt e&longs;t proportio 2. ad 1. vel 3. ad 1. vel 4. ad 1. <lb/>vel 3. ad 2. vel 4. ad 3. <expan abbr="Quotie&longs;eunq;">Quotie&longs;eunque</expan> igitur duo &longs;oni habuerin quampiam <pb xlink:href="009/01/064.jpg"/>ex <expan abbr="quinq;">quinque</expan> prædictis proportionibus, &longs;i &longs;imul coaluerint, ita vt ex eis vnue <lb/>tantum &longs;onus efficiatur; &longs;onus ille erit concordans, & auribus gratus. </s>

<s><expan abbr="atq;">atque</expan> <lb/>hæc e&longs;t &longs;ententia pri&longs;corum præ&longs;ertim Pythagoreorum, qui propterea di<lb/>cebantnon licere Mu&longs;ico vltra quaternarium pertran&longs;ire, eò quod &longs;olæ pro<lb/>portiones, vt diximus, numerorum quaternario contentorum, concordem, <lb/>ac con&longs;onantem concentum efficere poterant: quod vt adhuc melius per<lb/><figure place="text" xlink:href="figures-la/009.01.064.1.tif"/><lb/>cipiamus, accipe exemplum. </s>

<s><expan abbr="atq;">atque</expan> <lb/>hæc e&longs;t &longs;ententia pri&longs;corum præ&longs;ertim Pythagoreorum, qui propterea di<lb/>cebantnon licere Mu&longs;ico vltra quaternarium pertran&longs;ire, eò quod &longs;olæ pro<lb/>portiones, vt diximus, numerorum quaternario contentorum, concordem, <lb/>ac con&longs;onantem concentum efficere poterant: quod vt adhuc melius per<lb/><figure id="id.009.01.064.1.jpg" xlink:href="009/01/064/1.jpg" place="text"/><lb/>cipiamus, accipe exemplum. </s>

<s>Sint duæ chordæ <lb/>A, & B, æqualis cra&longs;&longs;itici, & æquè ten&longs;æ. </s>

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vbi triangulum æquilaterum con&longs;truit, & po&longs;tea probat <lb/>illud e&longs;&longs;e triangulum æquilaterum. </s>

<s>Certum tamen e&longs;t, Geometram luppo<lb/>nere triangulum in communi, cum inter definitiones ip&longs;ius contineatur, <pb pagenum="65"/>quod tamen non ob&longs;tat, quominus probare po&longs;&longs;it, aliquando po&longs;&longs;e <expan abbr="cõ&longs;trni">con&longs;trni</expan>, <lb/>& e&longs;&longs;e aliquod particulare triangulum, vt fit in prædicta demon&longs;tratione, <lb/>Euclidis.</s>

<s>Certum tamen e&longs;t, Geometram luppo<lb/>nere triangulum in communi, cum inter definitiones ip&longs;ius contineatur, <pb xlink:href="009/01/065.jpg"/>quod tamen non ob&longs;tat, quominus probare po&longs;&longs;it, aliquando po&longs;&longs;e <expan abbr="cõ&longs;trni">con&longs;trni</expan>, <lb/>& e&longs;&longs;e aliquod particulare triangulum, vt fit in prædicta demon&longs;tratione, <lb/>Euclidis.</s>

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<s>quod fortè non benè aduertens Iacobus Zabarella, <lb/>alioquin in his &longs;atis oculatus incidit in errorem, dicens, &longs;e nullo pacto vi<lb/>dere medium Euclidianæ demon&longs;trationis e&longs;&longs;e cau&longs;am materialem; quod <lb/>tamen nos mox aperiemus. </s>

<s>per angulum in &longs;emicirculo intelligas eum, qui <lb/>fit à lineis ductis ab extremitatibus diametri, & &longs;imul in quoduis punctum <lb/><figure place="text" xlink:href="figures-la/009.01.065.1.tif"/><lb/>circumferentiæ coeuntibus, vt in figura <lb/>præ&longs;enti vides lineas A C, B C, ad C, pun<lb/>ctum conuenire, <expan abbr="ibi&qacute;">ibique</expan>; facere angulum, <lb/>A C B, qui dicitur angulus in &longs;emicircu<lb/>lo, quia de&longs;criptus e&longs;t in &longs;emicirculo A<lb/>C B. <expan abbr="e&longs;t&qacute;">e&longs;tque</expan>; &longs;anè mirabilis hæc &longs;emicirculi <lb/>proprietas, cum <expan abbr="vbicunq;">vbicunque</expan> punctum C, in <lb/>periphæria &longs;umptum fuerit, &longs;emper ta<lb/>men angulus A C B, fiat rectus. </s>

<s>per angulum in &longs;emicirculo intelligas eum, qui <lb/>fit à lineis ductis ab extremitatibus diametri, & &longs;imul in quoduis punctum <lb/><figure id="id.009.01.065.1.jpg" xlink:href="009/01/065/1.jpg" place="text"/><lb/>circumferentiæ coeuntibus, vt in figura <lb/>præ&longs;enti vides lineas A C, B C, ad C, pun<lb/>ctum conuenire, <expan abbr="ibi&qacute;">ibique</expan>; facere angulum, <lb/>A C B, qui dicitur angulus in &longs;emicircu<lb/>lo, quia de&longs;criptus e&longs;t in &longs;emicirculo A<lb/>C B. <expan abbr="e&longs;t&qacute;">e&longs;tque</expan>; &longs;anè mirabilis hæc &longs;emicirculi <lb/>proprietas, cum <expan abbr="vbicunq;">vbicunque</expan> punctum C, in <lb/>periphæria &longs;umptum fuerit, &longs;emper ta<lb/>men angulus A C B, fiat rectus. </s>

<s>quod Euclides eodem pror&longs;us medio, quod <lb/>Ari&longs;t.

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ex hac præterea materia conflatur <lb/>definitio minoris extremi, vel &longs;ubiecti; d<gap/>m dicitur, angulus in &longs;emicircu<lb/>lo e&longs;t dimidium duorum rectorum. </s>

<s>&longs;yllogi&longs;mus enim reducitur tandem ad <pb pagenum="66"/>hanc formam, dimidium duorum rectorum e&longs;t rectus, angulus in &longs;emicir<lb/>culo e&longs;t dimidium duorum <expan abbr="rector&utilde;">rectorum</expan>, ergo angulus in &longs;emicirculo e&longs;t rectus. <lb/></s>

<s>&longs;yllogi&longs;mus enim reducitur tandem ad <pb xlink:href="009/01/066.jpg"/>hanc formam, dimidium duorum rectorum e&longs;t rectus, angulus in &longs;emicir<lb/>culo e&longs;t dimidium duorum <expan abbr="rector&utilde;">rectorum</expan>, ergo angulus in &longs;emicirculo e&longs;t rectus. <lb/></s>

<s>vides in minori propo&longs;itione contineri definitionem &longs;ubiecti materialem? <lb/></s>

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<s>1. &longs;exti, &longs;ic explicat: &longs;imi<lb/>les figuræ rectilíneæ &longs;unt; quæ & angulos &longs;ingulos, &longs;ingulis angulis æquales <lb/>habent, <expan abbr="atq;">atque</expan> etiam latera, quæ circa angulos æquales &longs;unt proportionalia. <lb/></s>

<s>vt &longs;i duo triangula appo&longs;ita habeant angulos æquales, <expan abbr="angul&utilde;">angulum</expan> A, angulo D: <lb/>angulum B, angulo E. angulum C, angulo F. & præterea latera, quæ &longs;une <pb pagenum="67"/><figure place="text" xlink:href="figures-la/009.01.067.1.tif"/><lb/>circa angulos æquales, v. g. cirea an<lb/>gulos A, & D, habeant proportiona<lb/>lia, hoc e&longs;t, vt latus A B, ad latus A C; <lb/>ita &longs;it latus D E, ad latus D F; & &longs;ic de <lb/>lateribus alijs circa reliquos angulos <lb/>æquales; erunt tunc prædicta duo tri<lb/>angula fimilia.</s>

<s>vt &longs;i duo triangula appo&longs;ita habeant angulos æquales, <expan abbr="angul&utilde;">angulum</expan> A, angulo D: <lb/>angulum B, angulo E. angulum C, angulo F. & præterea latera, quæ &longs;une <pb xlink:href="009/01/067.jpg"/><figure id="id.009.01.067.1.jpg" xlink:href="009/01/067/1.jpg" place="text"/><lb/>circa angulos æquales, v. g. cirea an<lb/>gulos A, & D, habeant proportiona<lb/>lia, hoc e&longs;t, vt latus A B, ad latus A C; <lb/>ita &longs;it latus D E, ad latus D F; & &longs;ic de <lb/>lateribus alijs circa reliquos angulos <lb/>æquales; erunt tunc prædicta duo tri<lb/>angula fimilia.</s>

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huiu&longs;modi, Angulus acutus <lb/>e&longs;t, qui minor recto e&longs;t. </s>

<s>Demum explicat, cur nam gladius dicatur acutus, <lb/>quia nimirum habet angulum acutum &longs;uperficialem, ide&longs;t, quem duæ &longs;uper<lb/>ficies &longs;imul in acie gladij concurrentes efficiunt.</s><pb pagenum="68"/>

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<s>Vult autem Ari&longs;t.

exemplo mathematico <lb/>o&longs;tendere, difficile e&longs;&longs;e di&longs;putare, aut <gap/> <expan abbr="rgum&etilde;tari">rgumentari</expan>, ni&longs;i prius rectè a&longs;&longs;ignetur <pb pagenum="69"/>definitio illius rei, de qua di&longs;&longs;eritur. </s>

exemplo mathematico <lb/>o&longs;tendere, difficile e&longs;&longs;e di&longs;putare, aut <gap/> <expan abbr="rgum&etilde;tari">rgumentari</expan>, ni&longs;i prius rectè a&longs;&longs;ignetur <pb xlink:href="009/01/069.jpg"/>definitio illius rei, de qua di&longs;&longs;eritur. </s>

<s>Porrò exemplum mathematicum hic <lb/>allatum &longs;ic videtur explicandum: Conetur aliquis demon&longs;trare hanc pro<lb/>po&longs;itionem; &longs;i linea ducta fuerit æquidi&longs;tans lateri vnius plani trianguli, &longs;e<lb/>cabit & latera, & locum, ideft &longs;uperficiem illam triangularem &longs;imiliter, ide &longs;t <lb/><figure place="text" xlink:href="figures-la/009.01.069.1.tif"/><lb/>in eadem proportione, vt in triangulo A B C, <lb/>linea D E, parallela ba&longs;i B C, &longs;ecat latera A B, <lb/>& A C, in punctis D, & E, in eadem ratione, <lb/>in qua etiam fecat totum triangulum, ita vt <lb/>eadem &longs;it proportio lineæ A D, ad D B, & lineæ <lb/>A E, ad E C, quæ e&longs;t partium totalis trianguli <lb/>A B C, &longs;eilicet quæ e&longs;t partis A D E, ad partem <lb/>E D C, fiue ad partem D E B. quod con&longs;tat ex <lb/>&longs;ecunda 6. Elem. </s>

<s>Porrò exemplum mathematicum hic <lb/>allatum &longs;ic videtur explicandum: Conetur aliquis demon&longs;trare hanc pro<lb/>po&longs;itionem; &longs;i linea ducta fuerit æquidi&longs;tans lateri vnius plani trianguli, &longs;e<lb/>cabit & latera, & locum, ideft &longs;uperficiem illam triangularem &longs;imiliter, ide &longs;t <lb/><figure id="id.009.01.069.1.jpg" xlink:href="009/01/069/1.jpg" place="text"/><lb/>in eadem proportione, vt in triangulo A B C, <lb/>linea D E, parallela ba&longs;i B C, &longs;ecat latera A B, <lb/>& A C, in punctis D, & E, in eadem ratione, <lb/>in qua etiam fecat totum triangulum, ita vt <lb/>eadem &longs;it proportio lineæ A D, ad D B, & lineæ <lb/>A E, ad E C, quæ e&longs;t partium totalis trianguli <lb/>A B C, &longs;eilicet quæ e&longs;t partis A D E, ad partem <lb/>E D C, fiue ad partem D E B. quod con&longs;tat ex <lb/>&longs;ecunda 6. Elem. </s>

<s>Inquit ergo Ari&longs;t. </s>

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<s>Per capitales numeros intelligo &longs;implices <lb/>ab v<gap/>itate, <expan abbr="v&longs;q;">v&longs;que</expan> ad nouem inclu&longs;iuè. </s>

<s>& quando ait, alium numerum cogno<lb/>&longs;cere multiplicatum, &longs;ignificat vtile valdè e&longs;&longs;e ad quotidianum v&longs;um <lb/>cogno&longs;cere, quemnam numerum producant numeri capitales, <lb/>&longs;i ad inuicem multiplicentur, quamuis huiu&longs;modi co<lb/>gnitio facilis, ac leuis &longs;it: qua de cau&longs;a vide<lb/>mus v&longs;que in hanc diem pueros diu in <lb/>Abaco memoriter perdi&longs;cen<lb/>do detineri.</s><pb pagenum="70"/>

<s><emph type="italics"/>Ex Primo Elenchorum.<emph.end type="italics"/></s>

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<s>Ad &longs;inem cap. <emph type="italics"/>(Aut vt Antiphon quadra<lb/>uit)<emph.end type="italics"/> &longs;imile peccatum pecca&longs;&longs;e Antiphon<lb/>tem in orbe quadrando, ac Hippocratem, <lb/>Ari&longs;t.

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<s>Cæterum <lb/>Antiphontem in hunc modum orbem ad <lb/>quadrum redigere tenta&longs;&longs;e, tradit Simpli<lb/>cius. </s>

<s>circulo quadrando in&longs;cribebat pri<lb/>mò quadratum A B C D. deinde in &longs;ingu<lb/>lis quatuor &longs;egmentis in&longs;cribebat totidem <lb/>trigona æquilatera, vt patet in ad&longs;cripta <pb pagenum="71"/>figura. </s>

<s>po&longs;tea &longs;uper &longs;ingula latera horum triangulorum in reliquis &longs;egmen<lb/>tis in&longs;cribebat adhuc triangula &longs;imilia triangulo A I E. alia in&longs;uper trigona <lb/>&longs;uper latera i&longs;torum con&longs;tituebat, donec ambitus figuræ illius multilateræ <lb/>in circulo delinearæ, circumferentiæ circuli aptaretur. </s>

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<s>At Bry&longs;&longs;o, eo quod per communia alijs &longs;cientijs deduceret ratiocinatio<lb/>nem propterea p&longs;eudographia Antiphontis non litigio&longs;a quidem, &longs;ed <lb/>tamen fallax extitit, non enim per communia alijs &longs;cientijs <lb/>procedat; vnde nec transferri poterat ip&longs;ius fal&longs;a de<lb/>&longs;criptio, &longs;eu demon&longs;tratio extra Geometriæ li<lb/>mites, quod cau&longs;a e&longs;t contentionis.</s>

<s><emph type="italics"/>Log<gap/>corum locorum finis.<emph.end type="italics"/></s><pb pagenum="72"/>

<s><emph type="italics"/>Log<gap/>corum locorum finis.<emph.end type="italics"/></s><pb xlink:href="009/01/072.jpg"/>

<s>EX PRIMO LIBRO</s>

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<s>2.1. continetur: quibus per&longs;pectis facilis erit phy&longs;ico totius <lb/>loci intelligentia.</s><pb pagenum="73"/>

<s>2.1. continetur: quibus per&longs;pectis facilis erit phy&longs;ico totius <lb/>loci intelligentia.</s><pb xlink:href="009/01/073.jpg"/>

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1. quam nece&longs;&longs;e e&longs;t, con&longs;ulas. </s>

<s>pro medio autem <lb/>huius pa&longs;&longs;ionis accipit lineam perpendicularem, quam innuit verbis illis <lb/><emph type="italics"/>(quoniam enim hoc rectum e&longs;t<emph.end type="italics"/>) vt in figura &longs;it triangulum A B C, <expan abbr="&longs;it&qacute;">&longs;itque</expan>; vt latus <lb/><figure place="text" xlink:href="figures-la/009.01.073.1.tif"/><lb/>A C, &longs;it perpendiculare <expan abbr="c&utilde;">cum</expan> latere B C, & pro<lb/>ducatur B C, in D; tunc triangulum A B C, <lb/>habere tres angulos, A, B, & A C B, æquales <lb/>duobus rectis planum erit: nam <expan abbr="c&utilde;">cum</expan> latus A C, <lb/>&longs;it perpendiculare (quod Ari&longs;t.

<s>pro medio autem <lb/>huius pa&longs;&longs;ionis accipit lineam perpendicularem, quam innuit verbis illis <lb/><emph type="italics"/>(quoniam enim hoc rectum e&longs;t<emph.end type="italics"/>) vt in figura &longs;it triangulum A B C, <expan abbr="&longs;it&qacute;">&longs;itque</expan>; vt latus <lb/><figure id="id.009.01.073.1.jpg" xlink:href="009/01/073/1.jpg" place="text"/><lb/>A C, &longs;it perpendiculare <expan abbr="c&utilde;">cum</expan> latere B C, & pro<lb/>ducatur B C, in D; tunc triangulum A B C, <lb/>habere tres angulos, A, B, & A C B, æquales <lb/>duobus rectis planum erit: nam <expan abbr="c&utilde;">cum</expan> latus A C, <lb/>&longs;it perpendiculare (quod Ari&longs;t.

dicit, cum <expan abbr="re-ct&utilde;">re<lb/>ctum</expan> hoc &longs;it) erunt duo anguli deinceps A C B, <lb/>A C D, recti, ex definitione lineæ perpendicu<lb/>laris, cum ergo duo anguli A, & B, externo, <expan abbr="recto&qacute;">rectoque</expan>; A C D, &longs;int æquales per <lb/>32. primi, & reliquus angulus A C B, communis, ide&longs;t, &longs;it angulus triangu<lb/>li, & angulus vnus lineæ perpendicularis, & ideò rectus; manife&longs;tè apparet, <lb/>tres angulos A, B, A C B, e&longs;&longs;e æquales nece&longs;&longs;ariò duobus rectis, ex po&longs;itio<lb/>ne illius recti, &longs;iue lateris perpendicularis, quia ex verò, verum nece&longs;&longs;ariò <lb/>&longs;equitur; non tamen po&longs;ita hac pa&longs;&longs;ione, &longs;iue conclu&longs;ione, habere &longs;cilicet <lb/>tres angulos æquales duobus rectis, nece&longs;&longs;ariò &longs;equitur illud e&longs;&longs;e rectum, <lb/>idelt latus illud A C, e&longs;&longs;e perpendiculare ad latus B C, quia verum <lb/>&longs;equi pote&longs;t ex verò, & falsò. </s>

<s>valebit tamen hæc con&longs;e quen<lb/>tia, &longs;i triangulum non habet hanc proprietatem, ne<lb/>que illud rectum e&longs;t, ide&longs;t, <expan abbr="neq;">neque</expan> latus prædi<lb/>ctum crit <expan abbr="perp&etilde;diculare">perpendiculare</expan>, quia fal&longs;um <lb/>non, ni&longs;i exfal&longs;o &longs;equitur.</s><pb pagenum="74"/>

<s><emph type="italics"/>Ex Tertio Phy&longs;icorum.<emph.end type="italics"/></s>

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<s>Pythagorici enim (à quibus i&longs;ta mutuatus <lb/>e&longs;t Ari&longs;t.

numeros impares &longs;olos appellabant Gnomones, eò quod in for<lb/>mam normæ æquilateræ, &longs;iue Gnomonis con&longs;titui po&longs;&longs;int, vt patet in his <lb/><figure place="text" xlink:href="figures-la/009.01.074.1.tif"/><lb/>nimirum in ternario, quinario, &longs;eptenario, & &longs;ic de <lb/>reliquis imparibus. </s>

numeros impares &longs;olos appellabant Gnomones, eò quod in for<lb/>mam normæ æquilateræ, &longs;iue Gnomonis con&longs;titui po&longs;&longs;int, vt patet in his <lb/><figure id="id.009.01.074.1.jpg" xlink:href="009/01/074/1.jpg" place="text"/><lb/>nimirum in ternario, quinario, &longs;eptenario, & &longs;ic de <lb/>reliquis imparibus. </s>

<s>pares autem numeri, quia ne<lb/>queunt in figuram normæ æquilateræ di&longs;poni, cum <lb/>non habeant vnitatem pro angulo, & paria po&longs;tea la<lb/>tera, vt oportet, non merentur appellari Gnomones, vt quaternarius, &longs;i di<lb/>&longs;ponatur &longs;ic <figure place="text" xlink:href="figures-la/009.01.074.2.tif"/> non refert Gnomonem, quia lateribus in&ecedil;qualibus con<lb/>&longs;tat; <expan abbr="neq;">neque</expan> &longs;i hoc modo <figure place="text" xlink:href="figures-la/009.01.074.3.tif"/> quia dee&longs;t huic figuræ angularis vnitas, quæ <lb/>illi nece&longs;iaria e&longs;t. </s>

<s>pares autem numeri, quia ne<lb/>queunt in figuram normæ æquilateræ di&longs;poni, cum <lb/>non habeant vnitatem pro angulo, & paria po&longs;tea la<lb/>tera, vt oportet, non merentur appellari Gnomones, vt quaternarius, &longs;i di<lb/>&longs;ponatur &longs;ic <figure id="id.009.01.074.2.jpg" xlink:href="009/01/074/2.jpg" place="text"/> non refert Gnomonem, quia lateribus in&ecedil;qualibus con<lb/>&longs;tat; <expan abbr="neq;">neque</expan> &longs;i hoc modo <figure id="id.009.01.074.3.jpg" xlink:href="009/01/074/3.jpg" place="text"/> quia dee&longs;t huic figuræ angularis vnitas, quæ <lb/>illi nece&longs;iaria e&longs;t. </s>

<s>Pythagorici igitur dicebant, numerum parem ideò e&longs;&longs;e <lb/>infinitum ip&longs;um, quia videbant ip&longs;um e&longs;&longs;e cau&longs;am perpetuæ diui&longs;ionis, cum <lb/>quælibet res quanta &longs;it diui&longs;ibilis bifariam, ide&longs;t in duo &longs;ecundum numerum <lb/>parem, & &longs;ubdiui&longs;ibilis po&longs;tea bifariam, & &longs;ic in infinitum, vt de linea pro<lb/>blematicè probatur in 10. primi Elem.

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<s>7. Arith<lb/>metices lordani, vbi i&longs;tud idem demon&longs;trat, quæ e&longs;t hæc. </s>

<s>&longs;it vnitas, & &longs;uo or<lb/>dine &longs;equantur impares, vt in &longs;equenti hac &longs;erie apparet 1. 3. 5. 7. 9. & c. <lb/><figure place="text" xlink:href="figures-la/009.01.074.4.tif"/><lb/>&longs;i igitur vnitati addatur ternarius in Gnomo<lb/>nis modum, vt vides in prima figura, produ<lb/>cetur quaternarius numerus, qui e&longs;t numerus <lb/>quadratus (quid &longs;it quadratus numerus expli<lb/>caui in Logicis tex. 9. primi Po&longs;ter.) etfi huic <lb/>quaternario addatur &longs;equens impar, qui e&longs;t <lb/>quinarius in modum Gnomonis, vt in &longs;ecund<gap/><lb/>figura, &longs;it numerus nouenarius, qui pariter e&longs;t quadratus. </s>

<s>&longs;it vnitas, & &longs;uo or<lb/>dine &longs;equantur impares, vt in &longs;equenti hac &longs;erie apparet 1. 3. 5. 7. 9. & c. <lb/><figure id="id.009.01.074.4.jpg" xlink:href="009/01/074/4.jpg" place="text"/><lb/>&longs;i igitur vnitati addatur ternarius in Gnomo<lb/>nis modum, vt vides in prima figura, produ<lb/>cetur quaternarius numerus, qui e&longs;t numerus <lb/>quadratus (quid &longs;it quadratus numerus expli<lb/>caui in Logicis tex. 9. primi Po&longs;ter.) etfi huic <lb/>quaternario addatur &longs;equens impar, qui e&longs;t <lb/>quinarius in modum Gnomonis, vt in &longs;ecund<gap/><lb/>figura, &longs;it numerus nouenarius, qui pariter e&longs;t quadratus. </s>

<s>et&longs;i huic &longs;imiliter <lb/>addatur &longs;e quens impar, nimirum &longs;eptenarius, conflabitur &longs;edenarius, qui <lb/>numerus pariter quadratus e&longs;t, vt in tertia figura, & hoc modo, &longs;i in infini<pb pagenum="75"/>tum procedatur, numeri &longs;emper quadrati progignentur. </s>

<s>et&longs;i huic &longs;imiliter <lb/>addatur &longs;e quens impar, nimirum &longs;eptenarius, conflabitur &longs;edenarius, qui <lb/>numerus pariter quadratus e&longs;t, vt in tertia figura, & hoc modo, &longs;i in infini<pb xlink:href="009/01/075.jpg"/>tum procedatur, numeri &longs;emper quadrati progignentur. </s>

<s>Vides igitur, qui <lb/>ratione Gnomonum, &longs;iue imparium additione fiat &longs;emper eadem &longs;pecies, <lb/>&longs;cilicet quadratus numerus, quod &longs;ignum e&longs;t, inquiunt, imparem numerum <lb/>non infinitatis, &longs;ed finitatis e&longs;&longs;e auctorem. </s>

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<s>quæ &longs;ic explicantur: &longs;int ab vnitate pares di&longs;po&longs;iti ordinatim <lb/>hoc modo, 1. 2. 4. 6. &c. </s>

<s>&longs;i igitur vnitati binarius coaceruetur, fit numerus <lb/><figure place="text" xlink:href="figures-la/009.01.075.1.tif"/><lb/>triangularis, vt in prima figura. </s>

<s>&longs;i igitur vnitati binarius coaceruetur, fit numerus <lb/><figure id="id.009.01.075.1.jpg" xlink:href="009/01/075/1.jpg" place="text"/><lb/>triangularis, vt in prima figura. </s>

<s>&longs;i huic ternario <lb/>coaceruetur &longs;equens par, fiet altera &longs;pecies, ni<lb/>mirum <gap/>exagonus numerus, vt in &longs;ecunda figu<lb/>ra. </s>

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<s>Tex. 120. ter in hoc textu meminit commen&longs;urabilitatis, & incommen<lb/>&longs;urabilitatis, quæ e&longs;t diametri ad co&longs;tam: cuius explicationem vide <lb/>primo Priorum, &longs;ecto primo, cap.

23.</s><pb pagenum="76"/>

23.</s><pb xlink:href="009/01/076.jpg"/>

<s><emph type="italics"/>Ex Quinto Phy&longs;icorum.<emph.end type="italics"/></s>

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<s>Tex. 36. <emph type="italics"/>(Sit <expan abbr="itaq;">itaque</expan> linea, in qua A G E, infinita ad partes E; & alia vtrinque <lb/>infinita, in qua <foreign lang="greek">b</foreign> B; &longs;i <expan abbr="itaq;">itaque</expan> de&longs;cribat circulum linea A G E, cir ca centrum G, fe-<emph.end type="italics"/><lb/><figure place="text" xlink:href="figures-la/009.01.076.1.tif"/><lb/><emph type="italics"/>retur circulariter linea A G E, &longs;ecans ali<lb/>quando lineam <foreign lang="greek">b</foreign> B, tempore finito; totum <lb/>enim tempus, in quo circulariter latum <lb/>e&longs;t C&oelig;lum finitum e&longs;t, & ablatum igitur, <lb/>quo &longs;ecans ferebatur; erit igitur aliqued <lb/>prmcipium, quo primum linea A G E, li<lb/>neam <foreign lang="greek">b</foreign> B, &longs;ecuit. </s>

<s>Tex. 36. <emph type="italics"/>(Sit <expan abbr="itaq;">itaque</expan> linea, in qua A G E, infinita ad partes E; & alia vtrinque <lb/>infinita, in qua <foreign lang="greek">b</foreign> B; &longs;i <expan abbr="itaq;">itaque</expan> de&longs;cribat circulum linea A G E, cir ca centrum G, fe-<emph.end type="italics"/><lb/><figure id="id.009.01.076.1.jpg" xlink:href="009/01/076/1.jpg" place="text"/><lb/><emph type="italics"/>retur circulariter linea A G E, &longs;ecans ali<lb/>quando lineam <foreign lang="greek">b</foreign> B, tempore finito; totum <lb/>enim tempus, in quo circulariter latum <lb/>e&longs;t C&oelig;lum finitum e&longs;t, & ablatum igitur, <lb/>quo &longs;ecans ferebatur; erit igitur aliqued <lb/>prmcipium, quo primum linea A G E, li<lb/>neam <foreign lang="greek">b</foreign> B, &longs;ecuit. </s>

<s>&longs;ed impo&longs;&longs;ibile est; non <lb/>est igitur circulariter verti infinit&<gap/>, quare <lb/><expan abbr="neq;">neque</expan> mundum, &longs;i e&longs;&longs;et infinitus)<emph.end type="italics"/> quamuis <lb/>textus hic parum &longs;it mathematicus, <lb/>quia tamen &longs;upponit figuram mathe<lb/>maticam, quæ in codicibus pariter, ac <lb/>commentarijs de&longs;ideratur, illam pla<lb/>cuit apponere. </s>

<s>in qua quidem, quamuis duæ lineæ infinitæ &longs;upponantur, vna <lb/>ad alteram <expan abbr="tãtum">tantum</expan> partem in qua E: altera verò ad <expan abbr="vtramq;">vtramque</expan> partem <foreign lang="greek">b,</foreign> & B, <pb pagenum="77"/>non potuerunt tamen de&longs;cribi, ni&longs;i finitæ; appo&longs;itæ idcircò &longs;unt ad partes <lb/>illas, ad quas deberent e&longs;&longs;e infinitæ lineolæ quædam infinitatem indicantes. <lb/></s>

<s>in qua quidem, quamuis duæ lineæ infinitæ &longs;upponantur, vna <lb/>ad alteram <expan abbr="tãtum">tantum</expan> partem in qua E: altera verò ad <expan abbr="vtramq;">vtramque</expan> partem <foreign lang="greek">b,</foreign> & B, <pb xlink:href="009/01/077.jpg"/>non potuerunt tamen de&longs;cribi, ni&longs;i finitæ; appo&longs;itæ idcircò &longs;unt ad partes <lb/>illas, ad quas deberent e&longs;&longs;e infinitæ lineolæ quædam infinitatem indicantes. <lb/></s>

<s>debemus po&longs;tea, vt mentem Ari&longs;t.

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<s>quam vnam <expan abbr="hab&etilde;um">habenum</expan>. <lb/></s>

<s>diui&longs;io enim in plana non perinde e&longs;&longs;icitur, vt qui&longs;piam <expan abbr="diuid&etilde;s">diuidens</expan> in par<lb/>tes diuidat totum, &longs;ed vt in &longs;pecie diuer&longs;a: patct igitur &longs;phæram e&longs;&longs;e &longs;olidarum <lb/>primam)<emph.end type="italics"/> qui &longs;olida diuidunt in plana, ca diuidunt <expan abbr="&longs;ec&utilde;dum">&longs;ecundum</expan> numerum &longs;uper<lb/>&longs;icierum, quibus ambiuntur, v. g. diuidunt cubum in &longs;ex &longs;uperficies, quia <lb/>cubus &longs;ex quadratis planis &longs;uperficiebus continetur: qua ratione nequcunt <pb pagenum="78"/>&longs;ohæram in plana vlla re&longs;oluere, <expan abbr="neq;">neque</expan> in alias plures &longs;uperficies, quia &longs;phæ<lb/>ra ambitur vnica tantum &longs;uperficie &longs;phærica. </s>

<s>diui&longs;io enim in plana non perinde e&longs;&longs;icitur, vt qui&longs;piam <expan abbr="diuid&etilde;s">diuidens</expan> in par<lb/>tes diuidat totum, &longs;ed vt in &longs;pecie diuer&longs;a: patct igitur &longs;phæram e&longs;&longs;e &longs;olidarum <lb/>primam)<emph.end type="italics"/> qui &longs;olida diuidunt in plana, ca diuidunt <expan abbr="&longs;ec&utilde;dum">&longs;ecundum</expan> numerum &longs;uper<lb/>&longs;icierum, quibus ambiuntur, v. g. diuidunt cubum in &longs;ex &longs;uperficies, quia <lb/>cubus &longs;ex quadratis planis &longs;uperficiebus continetur: qua ratione nequcunt <pb xlink:href="009/01/078.jpg"/>&longs;ohæram in plana vlla re&longs;oluere, <expan abbr="neq;">neque</expan> in alias plures &longs;uperficies, quia &longs;phæ<lb/>ra ambitur vnica tantum &longs;uperficie &longs;phærica. </s>

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<s>Tex. 31. <emph type="italics"/>(At verò, quod aquæ &longs;uperficies talis &longs;it, manife&longs;tum e&longs;t hac &longs;uppo&longs;i<lb/>tione &longs;umpta, quod apta natura e&longs;t &longs;emper con<gap/>luere aqua ad magis concauum: ma<lb/>gis autem concauum e&longs;t, quod centro propinquius est. </s>

<s>ducantur ergo ex centro A,<emph.end type="italics"/><lb/><figure place="text" xlink:href="figures-la/009.01.078.1.tif"/><lb/><emph type="italics"/>linea A B, & linea A C, & producatur, in qua B C, <lb/>duct<gap/> igitur ad ba&longs;im linea, in qua A D, minor e&longs;t eis, <lb/>quæ ex centro. </s>

<s>ducantur ergo ex centro A,<emph.end type="italics"/><lb/><figure id="id.009.01.078.1.jpg" xlink:href="009/01/078/1.jpg" place="text"/><lb/><emph type="italics"/>linea A B, & linea A C, & producatur, in qua B C, <lb/>duct<gap/> igitur ad ba&longs;im linea, in qua A D, minor e&longs;t eis, <lb/>quæ ex centro. </s>

<s>magis igitur concauus locus e&longs;t, quare <lb/>influet aqua, donec <expan abbr="vtiq;">vtique</expan> æquetur. </s>

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<s>educatur <expan abbr="quoq;">quoque</expan> recta alia ex centro A, quæ pertingat <lb/>ad B C, quæ ba&longs;is e&longs;t trianguli B A C, & producatur vlterius quantumlibet <lb/>in E. intelligatur demum circumferentia tran&longs;ire per puncta B, & C, quia <lb/>illæ duæ lineæ A B, A C, &longs;unt æquales, quæ circumferentia alteram A D, quæ <lb/>fuit protracta, &longs;ecet in E. </s>

<s>Iam &longs;ic argumentatur: aqua natura &longs;ua &longs;emper <lb/>de&longs;luit ad locum magis concauum, ide&longs;t, ad loca centro A, terræ propin<lb/>quiora, quale e&longs;&longs;et in figura locus D, re&longs;pectu locorum B, & C, quia A D, <lb/>linea minor e&longs;t ijs, quæ ex centro eductæ &longs;unt A B, A C. quapropter aqua <lb/>debet de&longs;luere ex B, ad D, vel ex C, ad idem D, donec pertingat ad E. qui <lb/>locus non e&longs;t decliuior punctis B, & C. quare cum loca B, E, C, quæ &longs;unt ex<pb pagenum="79"/>trema linearum, &longs;int æquè decliuia, nece&longs;&longs;e e&longs;t aquæ &longs;uperficiem apud ip&longs;a <lb/>con&longs;i&longs;tere, tunc enim debet quie&longs;cere, aliter nunquam quie&longs;ceret; &longs;ed vide<lb/>mus aquam manentem, & quietam, ergo quie&longs;cit circa puncta B, E, C, à <lb/>centro terræ æquidi&longs;tantia, per quæ tran&longs;it linea circularis coniungens illa; <lb/>et&longs;i &longs;uperficies per eiu&longs;modi loca pertran&longs;iret, e&longs;&longs;et &longs;phærica: &longs;ed &longs;uper&longs;i<lb/>cies aquæ tran&longs;it per talia loca, ergo &longs;phærica e&longs;t. </s>

<s>Iam &longs;ic argumentatur: aqua natura &longs;ua &longs;emper <lb/>de&longs;luit ad locum magis concauum, ide&longs;t, ad loca centro A, terræ propin<lb/>quiora, quale e&longs;&longs;et in figura locus D, re&longs;pectu locorum B, & C, quia A D, <lb/>linea minor e&longs;t ijs, quæ ex centro eductæ &longs;unt A B, A C. quapropter aqua <lb/>debet de&longs;luere ex B, ad D, vel ex C, ad idem D, donec pertingat ad E. qui <lb/>locus non e&longs;t decliuior punctis B, & C. quare cum loca B, E, C, quæ &longs;unt ex<pb xlink:href="009/01/079.jpg"/>trema linearum, &longs;int æquè decliuia, nece&longs;&longs;e e&longs;t aquæ &longs;uperficiem apud ip&longs;a <lb/>con&longs;i&longs;tere, tunc enim debet quie&longs;cere, aliter nunquam quie&longs;ceret; &longs;ed vide<lb/>mus aquam manentem, & quietam, ergo quie&longs;cit circa puncta B, E, C, à <lb/>centro terræ æquidi&longs;tantia, per quæ tran&longs;it linea circularis coniungens illa; <lb/>et&longs;i &longs;uperficies per eiu&longs;modi loca pertran&longs;iret, e&longs;&longs;et &longs;phærica: &longs;ed &longs;uper&longs;i<lb/>cies aquæ tran&longs;it per talia loca, ergo &longs;phærica e&longs;t. </s>

<s>Huius etiam habes acu<lb/>ti&longs;&longs;imam Archimedis demon&longs;trationem initio libelli de ijs, quæ vehuntur <lb/>in aqua, quam in &longs;uam &longs;phæram retulit Clauius.</s>

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<s>Tex. 46. <emph type="italics"/>(Reliquum e&longs;t orbes quidem moueri, stellas verò quie&longs;cere, & infixas <lb/>ip&longs;is orbibus ferri; &longs;olum enim &longs;ic nullum ab&longs;urdum accidit. </s>

<s>celeriorem enim e&longs;&longs;e <lb/>maioris circuli velocitatem, rationabile e&longs;t circa idem centrum infixis: vt enim in <lb/>alijs maius corpus velocius fertur propria latione, &longs;ic, & in circularibus: maius <lb/>enim e&longs;t eorum, quæ auferuntur <gap/>b eis, quæ ex centro, maioris circuli &longs;egmentum)<emph.end type="italics"/><lb/>ex intellectione vltimæ periodi textus totius intelligentia pendet: &longs;it igitur <lb/><figure place="text" xlink:href="figures-la/009.01.079.1.tif"/><lb/>figura præ&longs;ens, in qua cum &longs;int duo circuli concen<lb/>trici, vnus altero maior, <expan abbr="eductæ&qacute;">eductæque</expan>; &longs;int ex <expan abbr="c&etilde;tro">centro</expan> duæ <lb/>&longs;emidiametri A D, A E, quæ <expan abbr="vtrunq;">vtrunque</expan> circulum &longs;e<lb/>cant, apparet maius e&longs;&longs;e <expan abbr="&longs;egment&utilde;">&longs;egmentum</expan> D E, quod è ma<lb/>iori circulo &longs;emidiametri ex <expan abbr="c&etilde;tro">centro</expan> eductæ auferunt, <lb/>quam &longs;egmentum B C, minoris circuli, quod ei&longs;dem <lb/>&longs;emidia metris intercipitur. </s>

<s>celeriorem enim e&longs;&longs;e <lb/>maioris circuli velocitatem, rationabile e&longs;t circa idem centrum infixis: vt enim in <lb/>alijs maius corpus velocius fertur propria latione, &longs;ic, & in circularibus: maius <lb/>enim e&longs;t eorum, quæ auferuntur <gap/>b eis, quæ ex centro, maioris circuli &longs;egmentum)<emph.end type="italics"/><lb/>ex intellectione vltimæ periodi textus totius intelligentia pendet: &longs;it igitur <lb/><figure id="id.009.01.079.1.jpg" xlink:href="009/01/079/1.jpg" place="text"/><lb/>figura præ&longs;ens, in qua cum &longs;int duo circuli concen<lb/>trici, vnus altero maior, <expan abbr="eductæ&qacute;">eductæque</expan>; &longs;int ex <expan abbr="c&etilde;tro">centro</expan> duæ <lb/>&longs;emidiametri A D, A E, quæ <expan abbr="vtrunq;">vtrunque</expan> circulum &longs;e<lb/>cant, apparet maius e&longs;&longs;e <expan abbr="&longs;egment&utilde;">&longs;egmentum</expan> D E, quod è ma<lb/>iori circulo &longs;emidiametri ex <expan abbr="c&etilde;tro">centro</expan> eductæ auferunt, <lb/>quam &longs;egmentum B C, minoris circuli, quod ei&longs;dem <lb/>&longs;emidia metris intercipitur. </s>

<s>Verumtamen &longs;i circuli <lb/>ambo &longs;imul moueantur, maior circulus æquali tem<lb/>pore maius illud &longs;patium D E, & minor minus B C, <lb/>pertran&longs;ibit: idem igitur de c&oelig;le&longs;tibus orbibus di<lb/>cendum, qui quamuis omnes diurnum &longs;imul motum <lb/>ab&longs;oluunt, maiores tamen celerius conuertuntur: quo fit, vt &longs;tellæ maiori<lb/>bus circulis infixæ, <expan abbr="atq;">atque</expan> delatæ, maiori celeritate &longs;uos cur&longs;us peragant, ne<lb/>que oportet eas, dum mouentur c&oelig;lum di&longs;&longs;ecare, quod accideret, &longs;i pro<lb/>prio motu veluti pri&longs;ces per aquam progrederentur.</s>

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<s>Tex. 57. <emph type="italics"/>(De ordine autem ip&longs;orum, quo quidem modo &longs;ingula di&longs;penantur<gap/>, vt <lb/>quædam &longs;int priora, quædam posteriora, & quomodo &longs;patijs &longs;e h<gap/>be&abreve;<gap/> ad inuicem,<emph.end type="italics"/><pb pagenum="80"/><emph type="italics"/>ex ijs circa A&longs;trologiam, con&longs;ideretur: dicitur enim &longs;ufficienter)<emph.end type="italics"/> &longs;umit hoc loco <lb/>A&longs;trologiam, pro A&longs;tronomia, &longs;i iuxta recentiores loqui velimus. </s>

<s>Tex. 57. <emph type="italics"/>(De ordine autem ip&longs;orum, quo quidem modo &longs;ingula di&longs;penantur<gap/>, vt <lb/>quædam &longs;int priora, quædam posteriora, & quomodo &longs;patijs &longs;e h<gap/>be&abreve;<gap/> ad inuicem,<emph.end type="italics"/><pb xlink:href="009/01/080.jpg"/><emph type="italics"/>ex ijs circa A&longs;trologiam, con&longs;ideretur: dicitur enim &longs;ufficienter)<emph.end type="italics"/> &longs;umit hoc loco <lb/>A&longs;trologiam, pro A&longs;tronomia, &longs;i iuxta recentiores loqui velimus. </s>

<s>Dicit igi<lb/>tur ordinem c&oelig;lorum, ac &longs;yderum, item &longs;itum, & proportiones magnitu<lb/>dinum corundem, cum per naturalis &longs;cientiæ princip ia &longs;ciri nequeant, ex <lb/>rationibus A&longs;tronomorum petenda e&longs;&longs;e, apud quos i&longs;ta &longs;ufficienter <expan abbr="demon-&longs;tr&etilde;tur">demon<lb/>&longs;trentur</expan>. </s>

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<s>Quare &longs;i vnum est tale, palam e&longs;t, quod & alia <expan abbr="vtiq;">vtique</expan> erunt <lb/>talia)<emph.end type="italics"/> &longs;icuti <expan abbr="præced&etilde;s">præcedens</expan> &longs;phæricitatis Lunæ ratio ex Per&longs;pectiua de&longs;umpta e&longs;t, <lb/>ita præ&longs;ens ex A&longs;tronomia, ex eò enim, quod eclyp&longs;is Solis habeat figuram <lb/>lunulæ, ide&longs;t, &longs;i in&longs;tar Lunæ falcatæ, probant A&longs;tronomi Lunam e&longs;&longs;e &longs;phæri<lb/>cam. </s>

<s>intellige tamen partem illam Solis, quæ non eclyp&longs;atur, habere figu<lb/>ram lunulæ, pars enim à Luna obumbrata non videtur, et&longs;i videretur oua<lb/>lem quandam &longs;peciem, præ&longs;eferret: pars igitur, illa e&longs;t corniculata, quia <lb/><figure place="text" xlink:href="figures-la/009.01.080.1.tif"/><lb/>cum Solis defectio ex interpo&longs;itione Lunæ inter nos, & <lb/>Solem contingat, & Luna &longs;it &longs;phærica, nece&longs;&longs;ariò &longs;phæ<lb/>ricè, & circulariter Solem obumbrabit; quare pars illa <lb/>non obumbrata remanet falcata, & corniculata, vt in <lb/>præ&longs;enti figura vidcre e&longs;t; vbi cernis, Lunam Solem or<lb/>biculariter offu&longs;care in linea A D C, partem Solis de<lb/>tectam <expan abbr="contentã">contentam</expan> lineis curuis A B C D, e&longs;&longs;e lunularem, <lb/>& falcatam; cum ergo in hunc modum fiat Solis deli<lb/>quium, &longs;ignum certum e&longs;t, Lunam e&longs;&longs;e &longs;phæricam.</s>

<s>intellige tamen partem illam Solis, quæ non eclyp&longs;atur, habere figu<lb/>ram lunulæ, pars enim à Luna obumbrata non videtur, et&longs;i videretur oua<lb/>lem quandam &longs;peciem, præ&longs;eferret: pars igitur, illa e&longs;t corniculata, quia <lb/><figure id="id.009.01.080.1.jpg" xlink:href="009/01/080/1.jpg" place="text"/><lb/>cum Solis defectio ex interpo&longs;itione Lunæ inter nos, & <lb/>Solem contingat, & Luna &longs;it &longs;phærica, nece&longs;&longs;ariò &longs;phæ<lb/>ricè, & circulariter Solem obumbrabit; quare pars illa <lb/>non obumbrata remanet falcata, & corniculata, vt in <lb/>præ&longs;enti figura vidcre e&longs;t; vbi cernis, Lunam Solem or<lb/>biculariter offu&longs;care in linea A D C, partem Solis de<lb/>tectam <expan abbr="contentã">contentam</expan> lineis curuis A B C D, e&longs;&longs;e lunularem, <lb/>& falcatam; cum ergo in hunc modum fiat Solis deli<lb/>quium, &longs;ignum certum e&longs;t, Lunam e&longs;&longs;e &longs;phæricam.</s>

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<s>Tex. 107. <emph type="italics"/>(Quod autem dubitatur, hoc e&longs;t; videre autem non e&longs;t difficile, &longs;i pa<lb/>rum con&longs;iderauerimus, & di&longs;tinxerimus, quonam modo cen&longs;eamus quantamuis ma<lb/>gnitudinem grauem ad medium ferri. </s>

<s>manife&longs;tum enim e&longs;t, quod non quou&longs;que ex<lb/>tremum tangat ip&longs;um centrum; &longs;ed maior pars vincat, oportet, <expan abbr="quou&longs;q;">quou&longs;que</expan> &longs;uo medio <lb/>ip&longs;um medium compræhendat; <expan abbr="hucn&longs;q;">hucn&longs;que</expan> enim habet propen&longs;ioncm)<emph.end type="italics"/> &longs;en&longs;us Ari&longs;to<lb/>telis e&longs;t, debere nos exi&longs;timare, quod &longs;i quæpiam grauis magnitudo de&longs;cen<lb/>dat ad centrum mundi, eam non perman&longs;uram, &longs;latim ac ip&longs;ius extremum <lb/>centrum mundi attigent; &longs;ed cò <expan abbr="v&longs;q;">v&longs;que</expan> de&longs;cen&longs;uram, <expan abbr="quou&longs;q;">quou&longs;que</expan> ip&longs;ius medium, <lb/>mundi medium, &longs;iue centrum a&longs;&longs;equutum &longs;it; maior enim ip&longs;ius pars, in qua <lb/>&longs;cilicet medium e&longs;t, minorem partem propellit, donec vtrinque à centro <lb/>mundi æquè emineat; omne enim graue <expan abbr="hucu&longs;q;">hucu&longs;que</expan> habet propen&longs;ionem, &longs;iue <pb pagenum="81"/><expan abbr="hucu&longs;q;">hucu&longs;que</expan> grauitat, v. g. &longs;i lapis illuc de&longs;cenderet, non quie&longs;ceret &longs;tatim ac <lb/>prima ip&longs;ius pars ad mundi centrum pertingeret, &longs;ed reliquæ ip&longs;ius partes <lb/>adhuc grauitarent, <expan abbr="&longs;ic&qacute;">&longs;icque</expan>; vlterius primam partem impellerent, donec lapi<lb/>áis medium, mundi medio congrueret: quo facto lapis quie&longs;ceret. </s>

<s>manife&longs;tum enim e&longs;t, quod non quou&longs;que ex<lb/>tremum tangat ip&longs;um centrum; &longs;ed maior pars vincat, oportet, <expan abbr="quou&longs;q;">quou&longs;que</expan> &longs;uo medio <lb/>ip&longs;um medium compræhendat; <expan abbr="hucn&longs;q;">hucn&longs;que</expan> enim habet propen&longs;ioncm)<emph.end type="italics"/> &longs;en&longs;us Ari&longs;to<lb/>telis e&longs;t, debere nos exi&longs;timare, quod &longs;i quæpiam grauis magnitudo de&longs;cen<lb/>dat ad centrum mundi, eam non perman&longs;uram, &longs;latim ac ip&longs;ius extremum <lb/>centrum mundi attigent; &longs;ed cò <expan abbr="v&longs;q;">v&longs;que</expan> de&longs;cen&longs;uram, <expan abbr="quou&longs;q;">quou&longs;que</expan> ip&longs;ius medium, <lb/>mundi medium, &longs;iue centrum a&longs;&longs;equutum &longs;it; maior enim ip&longs;ius pars, in qua <lb/>&longs;cilicet medium e&longs;t, minorem partem propellit, donec vtrinque à centro <lb/>mundi æquè emineat; omne enim graue <expan abbr="hucu&longs;q;">hucu&longs;que</expan> habet propen&longs;ionem, &longs;iue <pb xlink:href="009/01/081.jpg"/><expan abbr="hucu&longs;q;">hucu&longs;que</expan> grauitat, v. g. &longs;i lapis illuc de&longs;cenderet, non quie&longs;ceret &longs;tatim ac <lb/>prima ip&longs;ius pars ad mundi centrum pertingeret, &longs;ed reliquæ ip&longs;ius partes <lb/>adhuc grauitarent, <expan abbr="&longs;ic&qacute;">&longs;icque</expan>; vlterius primam partem impellerent, donec lapi<lb/>áis medium, mundi medio congrueret: quo facto lapis quie&longs;ceret. </s>

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<s>Ibidem <emph type="italics"/>(Circa autem eclyp&longs;es, &longs;emper curuam habet termin&abreve;tem lmeam: qua<lb/>re qaon'am eclyp&longs;im palitur propter terræ obiectionem, terræ <expan abbr="circumfer&etilde;tia">circumferentia</expan> &longs;phæ<lb/>rica exi&longs;tens, figuræ cau&longs;a erit)<emph.end type="italics"/> probat rotunditatem terræ ab eclyp&longs;i lunari, <lb/>ex eo, quod Luna &longs;phæricè eclyp&longs;etur, quod innuitur illis verbis, &longs;emper <pb pagenum="82"/>curuam hzbet terminantem lineam, linea &longs;cilicet, quæ terminat partem <lb/>eclyp&longs;atam à non eclyp&longs;ata, &longs;emper apparet circularis; cum autem hæc li<lb/>nea &longs;it terminus vmbræ terræ, quæ lumen obumbrat, &longs;ignum <expan abbr="manife&longs;t&utilde;">manife&longs;tum</expan> e&longs;t <lb/>vmbram ip&longs;am e&longs;&longs;e rotundam; nam cum Luna deficiat propter terræ obie<lb/>ctionem inter ip&longs;am, & Solem, ita, vt vmbra terræ protendatur <expan abbr="v&longs;q;">v&longs;que</expan> ad Lu<lb/>nam, <expan abbr="eam&qacute;">eamque</expan>; in omni eclyp&longs;atione, &longs;iue eclyp&longs;is &longs;it &longs;upra terram, &longs;iue infra, <lb/>ad quamlibet <expan abbr="deniq;">denique</expan> partem terræ fiat, orbiculariter eam contegit, &longs;ignum <lb/>per&longs;picuum e&longs;t terram proijcere quoquouer&longs;us vmbram rotundam, quæ vt <lb/>in &longs;phæra o&longs;tenditur, e&longs;t rotunda ad modum coni; cum ergo vmbra terræ <lb/>ex quauis parte proijciatur, &longs;it rotunda, certò certius colligitur, <expan abbr="terram&qacute;">terramque</expan>; <lb/><expan abbr="quoq;">quoque</expan> ip&longs;am rotunda &longs;igura præditam e&longs;&longs;e. </s>

<s>Ibidem <emph type="italics"/>(Circa autem eclyp&longs;es, &longs;emper curuam habet termin&abreve;tem lmeam: qua<lb/>re qaon'am eclyp&longs;im palitur propter terræ obiectionem, terræ <expan abbr="circumfer&etilde;tia">circumferentia</expan> &longs;phæ<lb/>rica exi&longs;tens, figuræ cau&longs;a erit)<emph.end type="italics"/> probat rotunditatem terræ ab eclyp&longs;i lunari, <lb/>ex eo, quod Luna &longs;phæricè eclyp&longs;etur, quod innuitur illis verbis, &longs;emper <pb xlink:href="009/01/082.jpg"/>curuam hzbet terminantem lineam, linea &longs;cilicet, quæ terminat partem <lb/>eclyp&longs;atam à non eclyp&longs;ata, &longs;emper apparet circularis; cum autem hæc li<lb/>nea &longs;it terminus vmbræ terræ, quæ lumen obumbrat, &longs;ignum <expan abbr="manife&longs;t&utilde;">manife&longs;tum</expan> e&longs;t <lb/>vmbram ip&longs;am e&longs;&longs;e rotundam; nam cum Luna deficiat propter terræ obie<lb/>ctionem inter ip&longs;am, & Solem, ita, vt vmbra terræ protendatur <expan abbr="v&longs;q;">v&longs;que</expan> ad Lu<lb/>nam, <expan abbr="eam&qacute;">eamque</expan>; in omni eclyp&longs;atione, &longs;iue eclyp&longs;is &longs;it &longs;upra terram, &longs;iue infra, <lb/>ad quamlibet <expan abbr="deniq;">denique</expan> partem terræ fiat, orbiculariter eam contegit, &longs;ignum <lb/>per&longs;picuum e&longs;t terram proijcere quoquouer&longs;us vmbram rotundam, quæ vt <lb/>in &longs;phæra o&longs;tenditur, e&longs;t rotunda ad modum coni; cum ergo vmbra terræ <lb/>ex quauis parte proijciatur, &longs;it rotunda, certò certius colligitur, <expan abbr="terram&qacute;">terramque</expan>; <lb/><expan abbr="quoq;">quoque</expan> ip&longs;am rotunda &longs;igura præditam e&longs;&longs;e. </s>

<s>hanc eandem rationem, &longs;i libue<lb/>rit, fu&longs;ius pertractatam videre poteris apud P. Clauium in &longs;phæra.</s>

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<s>Quare non &longs;olum ex his manife&longs;tum e&longs;t rotundam e&longs;&longs;e figuram terræ, &longs;ed & &longs;phæræ <lb/>non magnæ: non enim tam celeriter in&longs;igne quippiam faceret, tran&longs;latis nobis adeò <lb/>parum)<emph.end type="italics"/> hic textus ei, qui &longs;phæram mundi audiuerit perfacilis e&longs;t: propte<lb/>rea eum breuiter &longs;ic paraphra&longs;ticè exponam. </s>

<s>Terram e&longs;&longs;e rotundam, <expan abbr="atq;">atque</expan> <lb/>re&longs;pectu c&oelig;le&longs;tium corporum non magnam, &longs;ignum e&longs;t, quod facto à nobis <lb/>paruo itinere &longs;iue ad meridionalem plagam, &longs;iue ad <expan abbr="&longs;ept&etilde;trionalem">&longs;eptentrionalem</expan> (quam <lb/>Vr&longs;am dicit) magnopere mutatur horizon: quod apparet primo ex varia<lb/>tionc a&longs;trorum, nam quæ in primo loco &longs;upra no&longs;trum verticem <expan abbr="trã&longs;ibant">tran&longs;ibant</expan>, <lb/>in &longs;ecundo loco non amplius, &longs;ed alia, <expan abbr="atq;">atque</expan> alia valde ab inuicem &longs;eiuncta <lb/><figure place="text" xlink:href="figures-la/009.01.082.1.tif"/><lb/>exfacto quamuis paruo itinere tran&longs;eunt. </s>

<s>Terram e&longs;&longs;e rotundam, <expan abbr="atq;">atque</expan> <lb/>re&longs;pectu c&oelig;le&longs;tium corporum non magnam, &longs;ignum e&longs;t, quod facto à nobis <lb/>paruo itinere &longs;iue ad meridionalem plagam, &longs;iue ad <expan abbr="&longs;ept&etilde;trionalem">&longs;eptentrionalem</expan> (quam <lb/>Vr&longs;am dicit) magnopere mutatur horizon: quod apparet primo ex varia<lb/>tionc a&longs;trorum, nam quæ in primo loco &longs;upra no&longs;trum verticem <expan abbr="trã&longs;ibant">tran&longs;ibant</expan>, <lb/>in &longs;ecundo loco non amplius, &longs;ed alia, <expan abbr="atq;">atque</expan> alia valde ab inuicem &longs;eiuncta <lb/><figure id="id.009.01.082.1.jpg" xlink:href="009/01/082/1.jpg" place="text"/><lb/>exfacto quamuis paruo itinere tran&longs;eunt. </s>

<s>&longs;it in <lb/>præ&longs;enti figura terra, vbi A, in qua facta parua <lb/>mutatione ex loco F, in locum G, fieret magna <lb/>mutatio <expan abbr="a&longs;tror&utilde;">a&longs;trorum</expan> ver&longs;icalium B, in C, quæ mul<lb/>tum ab inuicem di&longs;tant. </s>

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<s>aliud præterea &longs;ignum hums horizontis permutationis e&longs;t, quod <lb/>&longs;tellæ, quæ in priori loco &longs;upra horizontem apparebant, mutato paululum <lb/>loco ad alterutram plagam, &longs;tatim ab&longs;conduntur; aliæ verò nouæ <expan abbr="appar&etilde;t">apparent</expan> <lb/>vt in Acgypto, & Cypro, &longs;tella, quæ dicitur Canobus &longs;upra horizontem <lb/>a&longs;cendit; quæ &longs;i paululum Vr&longs;am, &longs;eu &longs;eptentrionem ambulaueris, &longs;tatim <lb/>latitabit. </s>

<s>Demum ciu&longs;dem citæ mutationis &longs;initoris indicium etiam &longs;it, <pb pagenum="83"/>quod regiones &longs;eptentrionales incolentibus plurima &longs;unt a&longs;tra, quæ nun<lb/>quam occidunt, quamuis horizontem leuiter per&longs;iringant, quæ tamen Cy<lb/>prijs, <expan abbr="atq;">atque</expan> Aegyptijs oriuntur, <expan abbr="atq;">atque</expan> occidunt. </s>

<s>Demum ciu&longs;dem citæ mutationis &longs;initoris indicium etiam &longs;it, <pb xlink:href="009/01/083.jpg"/>quod regiones &longs;eptentrionales incolentibus plurima &longs;unt a&longs;tra, quæ nun<lb/>quam occidunt, quamuis horizontem leuiter per&longs;iringant, quæ tamen Cy<lb/>prijs, <expan abbr="atq;">atque</expan> Aegyptijs oriuntur, <expan abbr="atq;">atque</expan> occidunt. </s>

<s>ex quibus & rotunditas, & <lb/>paruitas terræ colligi pote&longs;t. </s>

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<s>Tex. <emph type="italics"/>(Ad hæc nece&longs;&longs;e e&longs;t non omne corpus e&longs;&longs;e diui&longs;ibile dicere, &longs;ed repugnare <lb/>certi&longs;&longs;i nis &longs;cientijs; nam Mathematicæ ip&longs;um quideæ intelligibile, accipiunt diui<lb/>&longs;ibile)<emph.end type="italics"/> ip&longs;um intelligibile, ide&longs;t, quantitatem ab&longs;tractam tam continuam, <lb/>quam di&longs;eretam, quam &longs;tatuunt Philo&longs;ophi e&longs;&longs;e &longs;ubiectam materiam ma<lb/>thematicarum. </s>

<s>quam ideo appellant intelligibilem, quia cum &longs;it ab&longs;tracta <lb/>per intellectum à &longs;en&longs;ibilibus affectionibus, re&longs;tat vt &longs;it tantummodo intel<pb pagenum="84"/>lectu perceptibilis. </s>

<s>Hanc eandem &longs;upponunt e&longs;&longs;e diui&longs;ibilem in infinitum, <lb/>vt &longs;upra 3. Phy&longs;. textu 31. dictum e&longs;t.</s>

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<s>tales &longs;unt, <lb/>quibus fieri po&longs;&longs;unt pauimenta, oportet enim, vt &longs;imul vnitæ nihil vacui in <lb/>pauimento relinquant. </s>

<s>huiu&longs;modi &longs;unt triangula æquilatera (de his enim <lb/>intelligendus e&longs;t textus) quadrata, & hexagona, &longs;iue &longs;exilatera regularia; <lb/><figure place="text" xlink:href="figures-la/009.01.084.1.tif"/><lb/>nam &longs;ex triangula æquilatera &longs;imul iuncta in plano paui<lb/>re po&longs;&longs;unt, vt patet in figura præ&longs;enti; ratio huius e&longs;t, <lb/>quia omnes anguli circa idem punctum (y. </s>

<s>huiu&longs;modi &longs;unt triangula æquilatera (de his enim <lb/>intelligendus e&longs;t textus) quadrata, & hexagona, &longs;iue &longs;exilatera regularia; <lb/><figure id="id.009.01.084.1.jpg" xlink:href="009/01/084/1.jpg" place="text"/><lb/>nam &longs;ex triangula æquilatera &longs;imul iuncta in plano paui<lb/>re po&longs;&longs;unt, vt patet in figura præ&longs;enti; ratio huius e&longs;t, <lb/>quia omnes anguli circa idem punctum (y. </s>

<s>g. A, in hac <lb/>figura) in plano, quotquot fuerint con&longs;tituti, &longs;unt æqua<lb/>les quatuor rectis, ex coroll. </s>

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<s>&longs;ecundo 15. primi Elemen<lb/>ti: cum igitur &longs;ex anguli, trianguli æquilateri <expan abbr="æquiualeãt">æquiualeant</expan> <lb/>quatuor rectis angulis, con&longs;tituti omnes circa punctum <lb/>A, totum locum circa illud implere po&longs;&longs;unt. </s>

<s>Quadratum etiam replere lo<lb/><figure place="text" xlink:href="figures-la/009.01.084.2.tif"/><lb/>cum manife&longs;tum e&longs;t, cum enim ip&longs;ius anguli &longs;intrecti, &longs;i <lb/>quatuor quadrata ad idem punctum A, copulentur, vt in <lb/>figura apparet, replebunt eadem de cau&longs;a vacuum.</s>

<s>Quadratum etiam replere lo<lb/><figure id="id.009.01.084.2.jpg" xlink:href="009/01/084/2.jpg" place="text"/><lb/>cum manife&longs;tum e&longs;t, cum enim ip&longs;ius anguli &longs;intrecti, &longs;i <lb/>quatuor quadrata ad idem punctum A, copulentur, vt in <lb/>figura apparet, replebunt eadem de cau&longs;a vacuum.</s>

<s>Hexagonum quoque regulare, ide&longs;t æquilaterum, & <lb/>æquiangulum idem præ&longs;tare pote&longs;t; cum enim tres angu<lb/>li ip&longs;ius æquiualeant quatuor rectis, &longs;i tria hexagona ad <lb/>idem punctum A, vt in &longs;igura adaptentur, nece&longs;&longs;ariò ni<lb/>hil vacui inter ip&longs;a relinquetur, vt in figura hac o&longs;tenditur. </s>

<s>præter has tres <lb/><figure place="text" xlink:href="figures-la/009.01.084.3.tif"/><lb/>figuras, nulla alia reperitur, quæ i&longs;tud efficere pol<lb/>&longs;it. </s>

<s>præter has tres <lb/><figure id="id.009.01.084.3.jpg" xlink:href="009/01/084/3.jpg" place="text"/><lb/>figuras, nulla alia reperitur, quæ i&longs;tud efficere pol<lb/>&longs;it. </s>

<s>cuius demon&longs;trationem perfectam videre pote<lb/>ris in fine commentarij P. Clauij &longs;uper 4. Elem.

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<s>cum igitur tres tantum ex figuris planis <lb/>totum repleant, hæ &longs;olæ poterunt elementis attri<lb/>bui, ac propterea non &longs;ufficient, ni&longs;i pro tribus elementis. </s>

<s>quare quartum <lb/><expan abbr="ab&longs;q;">ab&longs;que</expan> figura relinquetur; quod e&longs;t ab&longs;urdum.</s><pb pagenum="85"/>

<s>quare quartum <lb/><expan abbr="ab&longs;q;">ab&longs;que</expan> figura relinquetur; quod e&longs;t ab&longs;urdum.</s><pb xlink:href="009/01/085.jpg"/>

<s><emph type="italics"/>Admirabilis quædam A&pgrave;um industria.<emph.end type="italics"/></s>

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videretur &longs;ibi non con&longs;tare, <lb/>vel perperam exi&longs;tima&longs;&longs;e plura Tetraedra complere &longs;oliditatem. </s>

<s>deceptus <pb pagenum="86"/>fortè fuit Ari&longs;t.

<s>deceptus <pb xlink:href="009/01/086.jpg"/>fortè fuit Ari&longs;t.

cò quod videret Ico&longs;aedrum con&longs;tare ex viginti pyramidi<lb/>bus, verùm illæ non &longs;unt regulares, ide&longs;t <expan abbr="nõ">non</expan> &longs;unt Tetraedra, vt po&longs;tea o&longs;ten<lb/>dam. </s>

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ab om<lb/>ni crrore vindicabimus. </s>

<s>Hoc tamen certum e&longs;t, ex prædictis, Græcos om<lb/>nes pariter, ac Latinos, illos &longs;equentes, lapos e&longs;&longs;e, a&longs;&longs;erentes duodecim py<lb/>ramides complere &longs;olidum locum, <expan abbr="atq;">atque</expan> Dodecaedrum con&longs;tituere; nam py<lb/>ramides Dodecaedron con&longs;tituentes non &longs;unt regulares, ide&longs;t, non &longs;unt Te<pb pagenum="87"/>traedra (de quibus tamen Ari&longs;t.

<s>Hoc tamen certum e&longs;t, ex prædictis, Græcos om<lb/>nes pariter, ac Latinos, illos &longs;equentes, lapos e&longs;&longs;e, a&longs;&longs;erentes duodecim py<lb/>ramides complere &longs;olidum locum, <expan abbr="atq;">atque</expan> Dodecaedrum con&longs;tituere; nam py<lb/>ramides Dodecaedron con&longs;tituentes non &longs;unt regulares, ide&longs;t, non &longs;unt Te<pb xlink:href="009/01/087.jpg"/>traedra (de quibus tamen Ari&longs;t.

loquitur) vt patet ex &longs;upradictis. </s>

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<s>Tex. 33. <emph type="italics"/>(Deinde ad &longs;imiles videtur angulos ignis quidem &longs;ur&longs;um ferri, <lb/>terra autem deor&longs;um, & omninò quod grauitatem babet, quare nece&longs;&longs;e <lb/>est ferri ad medium. </s>

<s>boc autem vtrum accidit ad ip&longs;um tcrræ medium, <lb/>an ad vniuer&longs;i, quoniam idem ip&longs;orum &longs;it, alius &longs;ermo e&longs;t)<emph.end type="italics"/> cum vellet <lb/><figure place="text" xlink:href="figures-la/009.01.087.1.tif"/><lb/>probare Ari&longs;toteles dari <expan abbr="p&utilde;ctum">punctum</expan> quoddam in m<gap/>dio <lb/>mundi, ad quod grauia de&longs;cendant, & concurrent: <lb/>& à quo leuia a&longs;c<gap/>ndat; vtitur, præter alias, etism <lb/>ratione aliqua ex parte mathematica; quæ e&longs;t huiu&longs;<lb/>modi. </s>

<s>boc autem vtrum accidit ad ip&longs;um tcrræ medium, <lb/>an ad vniuer&longs;i, quoniam idem ip&longs;orum &longs;it, alius &longs;ermo e&longs;t)<emph.end type="italics"/> cum vellet <lb/><figure id="id.009.01.087.1.jpg" xlink:href="009/01/087/1.jpg" place="text"/><lb/>probare Ari&longs;toteles dari <expan abbr="p&utilde;ctum">punctum</expan> quoddam in m<gap/>dio <lb/>mundi, ad quod grauia de&longs;cendant, & concurrent: <lb/>& à quo leuia a&longs;c<gap/>ndat; vtitur, præter alias, etism <lb/>ratione aliqua ex parte mathematica; quæ e&longs;t huiu&longs;<lb/>modi. </s>

<s>videmus ignem, & cætera l&ecedil;uia a&longs;cendere à <lb/>terra &longs;ur&longs;um ad angulos æquales; &longs;imiliter videmus <lb/>terram, & c&ecedil;tera grauia de&longs;cendere ad terram dcor<lb/>&longs;um ad angulos æquales, quod &longs;ignum e&longs;t omnia i&longs;ta <lb/>idem mundi medium re&longs;picere: v.g. &longs;it terra in &longs;igu<lb/>ra præ&longs;enti circulus E C D, cuius medium, &longs;ine c<gap/><pb pagenum="88"/>trum A. via, qua a&longs;cendit ignis &longs;it in linea A C B, quæ facit angulos in &longs;u<lb/>perficie terræ æquales, nimirum angulos B C D, B C E. &longs;imiliter terra per <lb/>candem lineam <expan abbr="faci&etilde;s">faciens</expan> eo&longs;dem angulos æquales de&longs;cendit. </s>

<s>videmus ignem, & cætera l&ecedil;uia a&longs;cendere à <lb/>terra &longs;ur&longs;um ad angulos æquales; &longs;imiliter videmus <lb/>terram, & c&ecedil;tera grauia de&longs;cendere ad terram dcor<lb/>&longs;um ad angulos æquales, quod &longs;ignum e&longs;t omnia i&longs;ta <lb/>idem mundi medium re&longs;picere: v.g. &longs;it terra in &longs;igu<lb/>ra præ&longs;enti circulus E C D, cuius medium, &longs;ine c<gap/><pb xlink:href="009/01/088.jpg"/>trum A. via, qua a&longs;cendit ignis &longs;it in linea A C B, quæ facit angulos in &longs;u<lb/>perficie terræ æquales, nimirum angulos B C D, B C E. &longs;imiliter terra per <lb/>candem lineam <expan abbr="faci&etilde;s">faciens</expan> eo&longs;dem angulos æquales de&longs;cendit. </s>

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<s><emph type="italics"/>EX PRIMO METEORORVM.<emph.end type="italics"/></s>

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<s>Porrò facile e&longs;t demon&longs;trare Solem e&longs;&longs;e terra multò maiorem, <lb/>terram verò maiorem Luna, <expan abbr="id&qacute;">idque</expan>; ex eclyp&longs;i lunari, cuius imaginem habes <lb/>in figura &longs;equenti; vbi vmbra terræ e&longs;t D B E, in quam Luna nigricans im<lb/>mergitur, ac lumine deficit, reliqua cognitu &longs;unt facilia: quia igitur A&longs;tro<lb/>nomi ob&longs;eruarunt vmbram terræ paulò &longs;upra Lunam pertingere, cum &longs;upe<lb/>riora a&longs;tra non adeat, hinc collegerunt eam nece&longs;&longs;ariò e&longs;&longs;e acuminatam, &longs;eu <lb/>conicam, vt figura refert. </s>

<s>Cum ergo terra vmbram proijciat turbinatam, <lb/>nece&longs;&longs;ariò corpus Solis, quod ip&longs;am illuminat, eadem maior erit: quoti<lb/>diana enim experientia docemur, corpore illuminante exi&longs;tente maiore <lb/>quà &longs;it illuminatum, vmbram proijci fa&longs;tigiatam: cum deinde Solem val<lb/>de a terra di&longs;tare certum &longs;it, optimè infertur, eum re&longs;pectu terræ e&longs;&longs;e maxi<lb/>mum: quanto enim duæ lineæ, &longs;iue radij B A, B C. à terra ad partes Solis <lb/><figure place="text" xlink:href="figures-la/009.01.089.1.tif"/><lb/>magis elongantur, tan<lb/>to maius corpus <expan abbr="illu-minãs">illu<lb/>minans</expan> intercipiunt. </s>

<s>Cum ergo terra vmbram proijciat turbinatam, <lb/>nece&longs;&longs;ariò corpus Solis, quod ip&longs;am illuminat, eadem maior erit: quoti<lb/>diana enim experientia docemur, corpore illuminante exi&longs;tente maiore <lb/>quà &longs;it illuminatum, vmbram proijci fa&longs;tigiatam: cum deinde Solem val<lb/>de a terra di&longs;tare certum &longs;it, optimè infertur, eum re&longs;pectu terræ e&longs;&longs;e maxi<lb/>mum: quanto enim duæ lineæ, &longs;iue radij B A, B C. à terra ad partes Solis <lb/><figure id="id.009.01.089.1.jpg" xlink:href="009/01/089/1.jpg" place="text"/><lb/>magis elongantur, tan<lb/>to maius corpus <expan abbr="illu-minãs">illu<lb/>minans</expan> intercipiunt. </s>

<s>ha<lb/>ctenus de magnitudine <lb/>terræ ad Solem. </s>

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<s>Eodem cap. <emph type="italics"/>(Con&longs;iderautes vtique, quæ nunc c&longs;tenduntur per Mathematica<emph.end type="italics"/><pb pagenum="90"/><emph type="italics"/>&longs;u<gap/>ficienter, for: è vtique de&longs;isterent ab hac puerili opinione; valde enim &longs;implex <lb/>e&longs;t pu<gap/>re <expan abbr="vnumquodq;">vnumquodque</expan> eorum quæ feruntur e&longs;&longs;e paruum magnitudinibus, quia vi<lb/>detur <gap/>&longs;picientibus, binc nobis &longs;ic)<emph.end type="italics"/> vtinam i&longs;ta, necnon alia his &longs;imilia, quæ <lb/>pa&longs;&longs;im apud Ari&longs;t.

<s>Eodem cap. <emph type="italics"/>(Con&longs;iderautes vtique, quæ nunc c&longs;tenduntur per Mathematica<emph.end type="italics"/><pb xlink:href="009/01/090.jpg"/><emph type="italics"/>&longs;u<gap/>ficienter, for: è vtique de&longs;isterent ab hac puerili opinione; valde enim &longs;implex <lb/>e&longs;t pu<gap/>re <expan abbr="vnumquodq;">vnumquodque</expan> eorum quæ feruntur e&longs;&longs;e paruum magnitudinibus, quia vi<lb/>detur <gap/>&longs;picientibus, binc nobis &longs;ic)<emph.end type="italics"/> vtinam i&longs;ta, necnon alia his &longs;imilia, quæ <lb/>pa&longs;&longs;im apud Ari&longs;t.

occurrunt, <expan abbr="pleriq;">plerique</expan> no&longs;træ ætatis con&longs;iderarent, qui nulla <lb/>ratione probari po&longs;&longs;e exi&longs;timant, Solem, v. g. terra e&longs;&longs;e centies &longs;exagies &longs;e<lb/>xies maiorem; &longs;ed c<gap/> iam, quod peius e&longs;t, negant e&longs;&longs;e maiorem; ad demon<lb/>&longs;trationes autem a&longs;tronomicas dicunt &longs;e exi&longs;timare eas e&longs;&longs;e fallaces; at que <lb/>impo&longs;libile e&longs;&longs;e nos res adeo à nobis di&longs;taptes &longs;ufficienter perue&longs;tigare: <lb/>quanto &longs;apientius, ac prudentius eorum Magi&longs;ter Ari&longs;t.

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<s>quapropter in <lb/>lis quidem locis neque fieri cometem ip&longs;um. </s>

<s>quando verò ad Boream &longs;ubdefecerit, <lb/>accipere comam, quia magna e&longs;t circun&longs;erentia, quæ e&longs;t &longs;upra horizontem; quæ au-<emph.end type="italics"/><pb pagenum="91"/><emph type="italics"/>tem e&longs;t &longs;ubtus, pars circuli parua; facilè enim vi&longs;um hominum p<gap/>tingere tunc ad <lb/>Solem)<emph.end type="italics"/> cur cometa in regione au&longs;trali vltra Solis, <expan abbr="anni&qacute;">annique</expan>; vias con&longs;titutus <lb/>non appareret, cau&longs;am referebat Hippocrates paruitatem circuli, qucm <lb/>motu diurno cometa de&longs;cribebat, ob quam adeò parum &longs;upra horizontem <lb/>attolleretur, vt <expan abbr="nõ">non</expan> po&longs;&longs;et vi&longs;us no&longs;ter ab ip&longs;o ad Solem reflecti; quod &longs;ecun<lb/>dum ip&longs;um erat nece&longs;&longs;arium ad cometarum apparitionem. </s>

<s>quando verò ad Boream &longs;ubdefecerit, <lb/>accipere comam, quia magna e&longs;t circun&longs;erentia, quæ e&longs;t &longs;upra horizontem; quæ au-<emph.end type="italics"/><pb xlink:href="009/01/091.jpg"/><emph type="italics"/>tem e&longs;t &longs;ubtus, pars circuli parua; facilè enim vi&longs;um hominum p<gap/>tingere tunc ad <lb/>Solem)<emph.end type="italics"/> cur cometa in regione au&longs;trali vltra Solis, <expan abbr="anni&qacute;">annique</expan>; vias con&longs;titutus <lb/>non appareret, cau&longs;am referebat Hippocrates paruitatem circuli, qucm <lb/>motu diurno cometa de&longs;cribebat, ob quam adeò parum &longs;upra horizontem <lb/>attolleretur, vt <expan abbr="nõ">non</expan> po&longs;&longs;et vi&longs;us no&longs;ter ab ip&longs;o ad Solem reflecti; quod &longs;ecun<lb/>dum ip&longs;um erat nece&longs;&longs;arium ad cometarum apparitionem. </s>

<s>I oquitur igitur <lb/>Hippocrates de circulis, quos diurna conuer&longs;ione cometes circumducir, qui <lb/>omninò &longs;imiles &longs;unt ijs, quos etiam Sol, <expan abbr="reliqua&qacute;">reliquaque</expan>; a&longs;tra eodem motu de &longs;i<lb/>gnant. </s>

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<s>&longs;ed vt ab auctoritate, in quam obiter incidimus <expan abbr="initi&utilde;">initium</expan> faciamus, <lb/>non e&longs;t exi&longs;timandum nonnuilos &longs;olum ex recentioribus id con&longs;tanter a&longs;&longs;e<pb pagenum="92"/>uera&longs;&longs;e, &longs;ed &longs;uperiori etiam ætate id ip&longs;um Hieron. </s>

<s>&longs;ed vt ab auctoritate, in quam obiter incidimus <expan abbr="initi&utilde;">initium</expan> faciamus, <lb/>non e&longs;t exi&longs;timandum nonnuilos &longs;olum ex recentioribus id con&longs;tanter a&longs;&longs;e<pb xlink:href="009/01/092.jpg"/>uera&longs;&longs;e, &longs;ed &longs;uperiori etiam ætate id ip&longs;um Hieron. </s>

<s>Cardan. libro de &longs;ubtili<lb/>tate conatus e&longs;t, <expan abbr="neq;">neque</expan> irrito conatu, demon&longs;trare; qui præterea idem cum <lb/>&longs;e ip &longs;o &longs;en&longs;i&longs;&longs;e ait Albumazar. </s>

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<s>3. Ratio, qua etiam ante nouas ob&longs;eruationes vti &longs;olebant, de&longs;umitur <lb/>ex motu cometæ diurno, quo &longs;cilicet oritur, & occidit, quemadmodum cæ<lb/>tera &longs;ydera, hoc e&longs;t &longs;patio 24. horarum diurnam conuer&longs;ionem circa totam <lb/>terram ab&longs;oluit. </s>

<s>&longs;i igitur comete e&longs;&longs;et in &longs;ublimiori aeris regione, vbi cæte<lb/>ra ignita meteora collocantur, <expan abbr="moueretur&qacute;">mouereturque</expan>; diurno motu circa terram, &longs;e<lb/>queretur nece&longs;&longs;ariò eum tanta velocitate videri à nobis circumferri, vt po<lb/>tius fulgor quidam, &longs;eu radius pertran&longs;iens ab oriente in occidentem appa<lb/>reret, quam &longs;tella qu&ecedil;dam: <expan abbr="id&qacute;">idque</expan>; propter propinquitatem; a&longs;tra enim ob ni<lb/>miam di&longs;tantiam videntur tardè moueri, quamuis veloci&longs;&longs;imè moueantur.</s><figure place="text" xlink:href="figures-la/009.01.092.1.tif"/>

<s>&longs;i igitur comete e&longs;&longs;et in &longs;ublimiori aeris regione, vbi cæte<lb/>ra ignita meteora collocantur, <expan abbr="moueretur&qacute;">mouereturque</expan>; diurno motu circa terram, &longs;e<lb/>queretur nece&longs;&longs;ariò eum tanta velocitate videri à nobis circumferri, vt po<lb/>tius fulgor quidam, &longs;eu radius pertran&longs;iens ab oriente in occidentem appa<lb/>reret, quam &longs;tella qu&ecedil;dam: <expan abbr="id&qacute;">idque</expan>; propter propinquitatem; a&longs;tra enim ob ni<lb/>miam di&longs;tantiam videntur tardè moueri, quamuis veloci&longs;&longs;imè moueantur.</s><figure id="id.009.01.092.1.jpg" xlink:href="009/01/092/1.jpg" place="text"/>

<s>Quod melius ex &longs;equenti figura <lb/><expan abbr="cõuincitur">conuincitur</expan>, vbi circulus interior e&longs;t <lb/>terra, cuius &longs;emidiameter A B. cir<lb/>culus verò exterior e&longs;t cometæ gy<lb/>rus, quem ip&longs;e &longs;patio 24. horarum <lb/>percurrit, qui &longs;ecundum veram pro<lb/>portionem deberet adhuc ip&longs;i terræ <lb/>propinquior, ac proinde minor e&longs;&longs;e, <lb/>iuxta aeris &longs;upremam partem. </s>

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<s>experientia tamen con&longs;tat, <lb/>cometas videri &longs;upra horizontem tot horis, quot &longs;tellæ fixæ, &longs;ub quibus mo<lb/>uentur: non ergo e&longs;t in &longs;upremo aere. </s>

<s>Quod &longs;i &longs;iat figura, in qua exterior <lb/>cometæ ambitus adeò magnus &longs;it, vt ip&longs;ius portio D C, &longs;upra horizontem <lb/>exi&longs;tens, re&longs;pondeat tempori, quo cometa &longs;upra no&longs;trum pariter horizon<lb/>tem &longs;pectatur, ea figura terræ &longs;emidiametrum A B. toties multiplicabit, vt <lb/>ip&longs;i Lunæ circuitui proximè accedat.</s><pb pagenum="93"/>

<s>Præterea aiunt, quis &longs;anæ mentis dixerit, Meteoron vlium ex materia <lb/>vaga, ac fluxa con&longs;tans, po&longs;&longs;e tanta pernicitate moueri, vt diurnam con<lb/>uer&longs;ionem ab&longs;oluat? </s>

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<s>quì igitur fieri potuerit, vt in hac corruptibili <expan abbr="m&utilde;di">mundi</expan> parte ex ma<lb/>teria adeò &longs;luxa, & vaga, quam illis Ari&longs;teteles &longs;upponit, tandiu perdura<lb/>re potui&longs;&longs;ent.</s>

<s><expan abbr="Atq;">Atque</expan> hæ &longs;unt rationes, quibus plurimi a&longs;tronomorum recentiorum, co<pb pagenum="94"/>metas ho&longs;ce motum æthereæ regioni conformem, contrà quam Ari&longs;t.

<s><expan abbr="Atq;">Atque</expan> hæ &longs;unt rationes, quibus plurimi a&longs;tronomorum recentiorum, co<pb xlink:href="009/01/094.jpg"/>metas ho&longs;ce motum æthereæ regioni conformem, contrà quam Ari&longs;t.

opi<lb/>natus e&longs;t, obtinui&longs;&longs;e, munifeftum e&longs;&longs;e volunt; ac proinde eorum locum, & <lb/>cur&longs;um in c&oelig;le&longs;ti mundi parte extiti&longs;&longs;e, &longs;e comproba&longs;&longs;e exi&longs;timant: qua de <lb/>re prudentis Lectoris e&longs;to iudicium: <expan abbr="neq;">neque</expan> enim, vt ille cecinit, no&longs;trum e&longs;t, <lb/>tantas componere lites.</s>

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<s>Eodem cap. <emph type="italics"/>(Amplius autem e&longs;t tertia quæd<gap/>m opinio de ip&longs;o, dicunt enim<emph.end type="italics"/><pb pagenum="95"/><emph type="italics"/>quidam lac e&longs;&longs;e reflexionem no&longs;tri vi&longs;us ad Solem; &longs;icut & &longs;tellam comatam; im<lb/>po&longs;&longs;ibile autem e&longs;t & hoc, &longs;i enim videns quieuerit & &longs;peculum, & quod videtur <lb/>omne in eodem puncto &longs;peculi eadem apparebit <expan abbr="vtiq;">vtique</expan> pars imaginis, &longs;i autem mo<lb/>ueatur &longs;peculum, & quod videtur, in eadem quidem di&longs;tantia ad videns, & quie<lb/>&longs;cens; ad inuicem autem <expan abbr="neq;">neque</expan> æquè velociter, <expan abbr="neq;">neque</expan> in eadem &longs;emper di&longs;tantia im<lb/>po&longs;&longs;ibile eandem imaginem in eadem e&longs;&longs;e parte &longs;peculi. </s>

<s>Eodem cap. <emph type="italics"/>(Amplius autem e&longs;t tertia quæd<gap/>m opinio de ip&longs;o, dicunt enim<emph.end type="italics"/><pb xlink:href="009/01/095.jpg"/><emph type="italics"/>quidam lac e&longs;&longs;e reflexionem no&longs;tri vi&longs;us ad Solem; &longs;icut & &longs;tellam comatam; im<lb/>po&longs;&longs;ibile autem e&longs;t & hoc, &longs;i enim videns quieuerit & &longs;peculum, & quod videtur <lb/>omne in eodem puncto &longs;peculi eadem apparebit <expan abbr="vtiq;">vtique</expan> pars imaginis, &longs;i autem mo<lb/>ueatur &longs;peculum, & quod videtur, in eadem quidem di&longs;tantia ad videns, & quie<lb/>&longs;cens; ad inuicem autem <expan abbr="neq;">neque</expan> æquè velociter, <expan abbr="neq;">neque</expan> in eadem &longs;emper di&longs;tantia im<lb/>po&longs;&longs;ibile eandem imaginem in eadem e&longs;&longs;e parte &longs;peculi. </s>

<s>Quæ autem in lactis circu<lb/>lo feruntur a&longs;tra, & Sol, ad quem fit reflexio, mouentur manentibus nobis, & &longs;i<lb/>militer, & æqualiter ad nos di&longs;tantia; à &longs;e ip&longs;is autem non æqualiter: aliquando <lb/>enim medijs noctibus Delphin oritur, aliquando verò diluculo. </s>

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<s>Ibidem <emph type="italics"/>(Quæ autem in lactis cir culo feruntur astra, & Sol, ad quem fit refle<lb/>xio, mouentur mancntibus nobis, & &longs;imiliter, & æqualiter ad nos di&longs;t antia à &longs;e <lb/>ip&longs;is autem non æqualiter)<emph.end type="italics"/> quæ hic ab Ari&longs;totele dicuntur <expan abbr="nõ">non</expan> &longs;unt v&longs;quequae; <lb/>vera propter apogæum, ac porigæum Solis, quæ quidem duo ab omnibus <lb/>a&longs;tronomis a&longs;&longs;eruatur: quando igitur Sol e&longs;t in apogæo, maiori multo in<pb pagenum="96"/>teruallo di&longs;tat à nobis, quàm quando e&longs;t in perigæo, interuallum enim illud <lb/>con&longs;tat diametris terræ duobus, & quadraginta, hoc e&longs;t milliarijs 208000. <lb/>ferè, ide&longs;t octonis millibus &longs;upra ducenta millia. </s>

<s>Ibidem <emph type="italics"/>(Quæ autem in lactis cir culo feruntur astra, & Sol, ad quem fit refle<lb/>xio, mouentur mancntibus nobis, & &longs;imiliter, & æqualiter ad nos di&longs;t antia à &longs;e <lb/>ip&longs;is autem non æqualiter)<emph.end type="italics"/> quæ hic ab Ari&longs;totele dicuntur <expan abbr="nõ">non</expan> &longs;unt v&longs;quequae; <lb/>vera propter apogæum, ac porigæum Solis, quæ quidem duo ab omnibus <lb/>a&longs;tronomis a&longs;&longs;eruatur: quando igitur Sol e&longs;t in apogæo, maiori multo in<pb xlink:href="009/01/096.jpg"/>teruallo di&longs;tat à nobis, quàm quando e&longs;t in perigæo, interuallum enim illud <lb/>con&longs;tat diametris terræ duobus, & quadraginta, hoc e&longs;t milliarijs 208000. <lb/>ferè, ide&longs;t octonis millibus &longs;upra ducenta millia. </s>

<s>quæ differentia facit vt Sol <lb/>manife&longs;tè appareat nobis minor apogæus, quàm perigæus. </s>

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<s>rationem hanc di<lb/>uer&longs;itatis a&longs;pectus a&longs;tronomicè magis explicatam reperies apud Clauium <lb/>in &longs;phæra. </s>

<s>Porrò hæc ratio quamuis adeo certa, ac no&longs;tra tempe&longs;tate vul<lb/>gata, parum tamen à nonnullis de rebus Meteorologicis commentaria con<lb/>farcinantibus intellecta, minimè eos ab&longs;terrere potuit, quin prædictam opi<lb/>nionem, non &longs;olum Ari&longs;toteli imponerent, verum etiam ip&longs;i <expan abbr="tãquam">tanquam</expan> veram <pb pagenum="97"/>a&longs;truerent: huiu&longs;modi patiuntur incommoda, qui <expan abbr="ab&longs;q;">ab&longs;que</expan> Mathematicarum <lb/>auxilio Philo&longs;ophiam aggrediuntur.</s>

<s>Porrò hæc ratio quamuis adeo certa, ac no&longs;tra tempe&longs;tate vul<lb/>gata, parum tamen à nonnullis de rebus Meteorologicis commentaria con<lb/>farcinantibus intellecta, minimè eos ab&longs;terrere potuit, quin prædictam opi<lb/>nionem, non &longs;olum Ari&longs;toteli imponerent, verum etiam ip&longs;i <expan abbr="tãquam">tanquam</expan> veram <pb xlink:href="009/01/097.jpg"/>a&longs;truerent: huiu&longs;modi patiuntur incommoda, qui <expan abbr="ab&longs;q;">ab&longs;que</expan> Mathematicarum <lb/>auxilio Philo&longs;ophiam aggrediuntur.</s>

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<s>Strabo lib.

15. <lb/>&longs;ic: Indiam à &longs;eptentrione Tauri extrema terminant, ab Ariana v&longs;que in <lb/>orientale mare, quæ extrema indigenæ particulatim nominant Poropami&longs;<gap/><pb pagenum="98"/>&longs;um, Emodum, Imauum, & alijs nominibus: Macedones verò Cauca&longs;um <lb/>vocant.</s>

15. <lb/>&longs;ic: Indiam à &longs;eptentrione Tauri extrema terminant, ab Ariana v&longs;que in <lb/>orientale mare, quæ extrema indigenæ particulatim nominant Poropami&longs;<gap/><pb xlink:href="009/01/098.jpg"/>&longs;um, Emodum, Imauum, & alijs nominibus: Macedones verò Cauca&longs;um <lb/>vocant.</s>

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<s>Lo<lb/>cum hunc fu<gap/>è pertractat eruditi&longs;&longs;imus Iacobus Mazonius &longs;ectione 3. & 4. <lb/>de Comparatione Platonis, & Ari&longs;t.

quo in opere plurima habet ex Mathe<lb/>maticis de&longs;umpta, quibus naturalem Philo&longs;ophiam mirificè illu&longs;trat, <expan abbr="mani-fe&longs;tum&qacute;">mani<lb/>fe&longs;tumque</expan>; reddit, quàm nece&longs;&longs;ariæ &longs;int Mathematicæ ad philo&longs;ophicæ veri<pb pagenum="99"/>tatis in&longs;pectionem. </s>

<s>Is igitur &longs;ect. </s>

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<s>po&longs;tea &longs;ic di&longs;currit; His ergo ita &longs;e habentibus, dico nos in<lb/>uenire po&longs;&longs;e viam, qua &longs;altem rudi Minerua, montis altitudinem comper<lb/>tam habeamus. </s>

<s>&longs;i enim in principio Crepu&longs;culi v. g. matutini (ita enim, vt <lb/>&longs;upra annotaui intelligendus e&longs;t Ari&longs;t.) illuminatur tertia pars, nece&longs;&longs;arium <lb/>vidctur tertiam illam partem &longs;upra cam regionem collocari, ex qua Cre<lb/>pu&longs;culum in planitie apparere incipit, &longs;ed illa regio ex Alhazino, & Vitell. <lb/>de Crepu&longs;culis milliaribus 52. à terra recedit, ergo duæ tertiæ montis par<lb/>tes, quæ Solem initio auroræ non vident, &longs;unt 52. milliaria ad perpendicu<lb/>lum, & tertia alia pars illuminata e&longs;t ad perpendiculum 26. milliaria: ita <lb/>vt totius montis altitudo perpendicularis &longs;it 78. mill. &longs;ed papè in quos acu<pb pagenum="100"/>leos imprudens me conieci? </s>

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<s>videtur enim his verbis velle dicere, quod quan<lb/>do habitantibus planitiem, quæ e&longs;t ad pedem montis Cauca&longs;i, vel horizon<lb/>tem eiu&longs;dem, incipit Crepu&longs;culum, ij&longs;dem etiam tunc tertia montis pars <lb/>appareat illuminata; in quo &longs;en&longs;u errat po&longs;tea in colligenda montis altitu<lb/>dine, quamuis enim verum e&longs;&longs;et partem illuminatam eminere totam &longs;upra <lb/>52. milliaria, non tamen &longs;equitur ip&longs;am &longs;olam eminere, &longs;ed alia etiam pars <lb/>eminere pote&longs;t, quod &longs;ic geometricè demon&longs;trabo. </s>

<s>de&longs;cribatur enim figura <lb/><figure place="text" xlink:href="figures-la/009.01.100.1.tif"/><pb pagenum="101"/>illa, qua ad vaporum altitudines indagandas vtuntur Alhazenus, Vitellio, <lb/>& Clauius, in qua terræ globus e&longs;t F L G E, regiò vaporum, & exhalatio<lb/>num M X N T. horizon a&longs;tronomicus O P. phy&longs;icus Q R, tangens terram <lb/>in puncto F, vbi etiam ponendus e&longs;t huius horizontis habitator, vnà cum. <lb/></s>

<s>de&longs;cribatur enim figura <lb/><figure id="id.009.01.100.1.jpg" xlink:href="009/01/100/1.jpg" place="text"/><pb xlink:href="009/01/101.jpg"/>illa, qua ad vaporum altitudines indagandas vtuntur Alhazenus, Vitellio, <lb/>& Clauius, in qua terræ globus e&longs;t F L G E, regiò vaporum, & exhalatio<lb/>num M X N T. horizon a&longs;tronomicus O P. phy&longs;icus Q R, tangens terram <lb/>in puncto F, vbi etiam ponendus e&longs;t huius horizontis habitator, vnà cum. <lb/></s>

<s>Cauca&longs;o F V. </s>

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<s>Quod &longs;i &longs;equamur alteram expo&longs;itionem, vt nimirum Ari&longs;tor. lo<lb/>quatur non de tertia montis parte, &longs;ed noctis, ita vt dicat, circa initium <lb/>tertiæ partis noctis apicem montis illu&longs;trari, altitudo eius erit tantum<pb pagenum="102"/>modo 180. quot continet latus F V. vt vidimus, quæ quamuis illa minor &longs;it, <lb/>adhuc tamen ab&longs;urda e&longs;t.</s>

<s>Quod &longs;i &longs;equamur alteram expo&longs;itionem, vt nimirum Ari&longs;tor. lo<lb/>quatur non de tertia montis parte, &longs;ed noctis, ita vt dicat, circa initium <lb/>tertiæ partis noctis apicem montis illu&longs;trari, altitudo eius erit tantum<pb xlink:href="009/01/102.jpg"/>modo 180. quot continet latus F V. vt vidimus, quæ quamuis illa minor &longs;it, <lb/>adhuc tamen ab&longs;urda e&longs;t.</s>

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<s>Ad fin<gap/>m eiu&longs;dem cap. <emph type="italics"/>(Et circa Ligu&longs;ticam non minor Rhodano ab&longs;orbetur <lb/>qui<gap/>am &longs;l<gap/>us, & iterum egreditur &longs;ecundum alium locum)<emph.end type="italics"/> <expan abbr="incompert&utilde;">incompertum</expan> & hoc <pb pagenum="103"/>Ari&longs;t.

vt &longs;uperiora, ob Geographiæ illius &longs;eculi imperfectionem, nu&longs;quam <lb/>enim in tota Liguria quidpiam tale reperitur.</s>

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<s>Ex <lb/>continua etiam hac inter montes corro&longs;ione facta manife&longs;tè apparet, flumi<lb/>num alueos in montanis modò e&longs;&longs;e humiliores quàm olim, quamuis contra<lb/>rium accidat alueis &longs;luuiorum per plana decurrentium, qui modò altiores <lb/>&longs;unt <expan abbr="quã">quam</expan> exordio mundi, vt paulò po&longs;t <expan abbr="o&longs;t&etilde;dam">o&longs;tendam</expan>. </s>

<s>Illud autem liquidò apparet <lb/>ex &longs;ignis, &longs;eu &longs;ymbolis, &longs;eu ex &longs;imilitudine terræ, aut lapidis, quæ in alti&longs;&longs;imis <lb/>fluminum ripis hinc inde pa&longs;&longs;im <expan abbr="vid&etilde;tur">videntur</expan>, quæ indicio &longs;unt montes illos iam <pb pagenum="104"/>olim fui&longs;&longs;e continuos, <expan abbr="atq;">atque</expan> vnam, <expan abbr="eandem&qacute;">eandemque</expan>; terram <expan abbr="contin&etilde;tem">continentem</expan>, ante qua<gap/><lb/>flumen eos ab inuicem &longs;epararet; <expan abbr="flumen&qacute;">flumenque</expan>; ip&longs;um olim altius, vbi &longs;unt &longs;igna <lb/>illa ambula&longs;&longs;e; quemadmodum in Pyramo Ciliciæ amne ob&longs;eruauit Strabo, <lb/>dum libro 12. de illius ripis hæc tradit, mira præterea e&longs;t montis cæ&longs;ura, <lb/>per quam alueus ducitur; nam quemadmodum in petris per medium &longs;ci&longs;&longs;is <lb/>contingit, alterius partis depre&longs;&longs;ioribus ita conuenire alterius partis emi<lb/>nentias, vt coniungi po&longs;&longs;int: &longs;ic videre e&longs;t imminentes flumini petras vtrin<lb/>que ferè <expan abbr="v&longs;q;">v&longs;que</expan> ad montis &longs;umma pertendentes duorum, triumuè iugerum <lb/>&longs;patio concauitates qua&longs;dam eminentijs oppo&longs;itas habere. </s>

<s>Illud autem liquidò apparet <lb/>ex &longs;ignis, &longs;eu &longs;ymbolis, &longs;eu ex &longs;imilitudine terræ, aut lapidis, quæ in alti&longs;&longs;imis <lb/>fluminum ripis hinc inde pa&longs;&longs;im <expan abbr="vid&etilde;tur">videntur</expan>, quæ indicio &longs;unt montes illos iam <pb xlink:href="009/01/104.jpg"/>olim fui&longs;&longs;e continuos, <expan abbr="atq;">atque</expan> vnam, <expan abbr="eandem&qacute;">eandemque</expan>; terram <expan abbr="contin&etilde;tem">continentem</expan>, ante qua<gap/><lb/>flumen eos ab inuicem &longs;epararet; <expan abbr="flumen&qacute;">flumenque</expan>; ip&longs;um olim altius, vbi &longs;unt &longs;igna <lb/>illa ambula&longs;&longs;e; quemadmodum in Pyramo Ciliciæ amne ob&longs;eruauit Strabo, <lb/>dum libro 12. de illius ripis hæc tradit, mira præterea e&longs;t montis cæ&longs;ura, <lb/>per quam alueus ducitur; nam quemadmodum in petris per medium &longs;ci&longs;&longs;is <lb/>contingit, alterius partis depre&longs;&longs;ioribus ita conuenire alterius partis emi<lb/>nentias, vt coniungi po&longs;&longs;int: &longs;ic videre e&longs;t imminentes flumini petras vtrin<lb/>que ferè <expan abbr="v&longs;q;">v&longs;que</expan> ad montis &longs;umma pertendentes duorum, triumuè iugerum <lb/>&longs;patio concauitates qua&longs;dam eminentijs oppo&longs;itas habere. </s>

<s>hæc Strabo de <lb/>vno, quod nos in pluribus ob&longs;eruauimus. </s>

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<s>&longs;ic <lb/>etiam templa Epi&longs;copalia, quæ <expan abbr="plerunq;">plerunque</expan> &longs;atis peruetu&longs;ta &longs;unt, admodum <lb/>infra terram con&longs;piciuntur. </s>

<s>Idem affirmant c&oelig;mentarij, & architectores <pb pagenum="105"/>omnes, quibus <expan abbr="vbiq;">vbique</expan> terrarum<gap/>, dum in planis ædeficiorum fundamenta ex<lb/>canant, occurrit primò terra quædam, quam ip&longs;i motam appellant, quæ li<lb/>gnis, ruderibus, ferramentis, numi&longs;matis, &longs;epulturis, <expan abbr="varijs&qacute;">varijsque</expan>; rebus per<lb/>mixta e&longs;t; qua eruta, reperitur terra alia, quam nunquam fui&longs;&longs;e motam, ap<lb/>paret, ex eo quod &longs;olida, ac benè compacta &longs;it, neque vllis externis rebus, <lb/>præ&longs;ertim artificiatis admixta, terra illa, quam motam dicunt, variam va<lb/>rijs in locis &longs;ortita e&longs;t altitudinem, prout aquæ plurimum, vel minimum <lb/>montanæ terræ huc, vel illuc comportarunt: alicubi vt hic Parmæ erit &longs;ex <lb/>vlnarum, alibi viginti, vt Mutinæ; alibi triginta, vt Romæ, nonnullis in lo<lb/>cis. </s>

<s>Idem affirmant c&oelig;mentarij, & architectores <pb xlink:href="009/01/105.jpg"/>omnes, quibus <expan abbr="vbiq;">vbique</expan> terrarum<gap/>, dum in planis ædeficiorum fundamenta ex<lb/>canant, occurrit primò terra quædam, quam ip&longs;i motam appellant, quæ li<lb/>gnis, ruderibus, ferramentis, numi&longs;matis, &longs;epulturis, <expan abbr="varijs&qacute;">varijsque</expan>; rebus per<lb/>mixta e&longs;t; qua eruta, reperitur terra alia, quam nunquam fui&longs;&longs;e motam, ap<lb/>paret, ex eo quod &longs;olida, ac benè compacta &longs;it, neque vllis externis rebus, <lb/>præ&longs;ertim artificiatis admixta, terra illa, quam motam dicunt, variam va<lb/>rijs in locis &longs;ortita e&longs;t altitudinem, prout aquæ plurimum, vel minimum <lb/>montanæ terræ huc, vel illuc comportarunt: alicubi vt hic Parmæ erit &longs;ex <lb/>vlnarum, alibi viginti, vt Mutinæ; alibi triginta, vt Romæ, nonnullis in lo<lb/>cis. </s>

<s>Comprobatur tandem hæc no&longs;tra ob&longs;eruatio ex arte illa, qua per ea&longs;<lb/>dem fluuiales aquas &longs;olent, tam loca depre&longs;&longs;iora per aggerationem paula<lb/>tim replere, <expan abbr="atq;">atque</expan> eleuare: quàm etiam altiora per aquarum earumdem cor<lb/>ro&longs;ionem deprimere. </s>

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<s><expan abbr="quart&utilde;">quartum</expan> e&longs;t illi Bo&longs;phorus Tracius; quod <lb/>vnà cum præcedentibus &longs;atius e&longs;t apud ip&longs;um, vel potius apud eius expo&longs;i<lb/>torem Vicomercatum videre, vt breuitati con&longs;ulatur. </s>

<s>Accedit & Plinij te<lb/>&longs;timonium, qui tradit multas terras na&longs;ci, non &longs;olum fluminum inuectu, &longs;<gap/>d <pb pagenum="106"/>etiam marium rece&longs;&longs;u; &longs;ic mare ab Ambraciæ portu 10. millia pa&longs;&longs;uum; ab <lb/>Athenarum verò <expan abbr="quinq;">quinque</expan> millia, & alijs in locis plus minu&longs;uè rece&longs;&longs;i&longs;&longs;e &longs;cri<lb/>bit. </s>

<s>Accedit & Plinij te<lb/>&longs;timonium, qui tradit multas terras na&longs;ci, non &longs;olum fluminum inuectu, &longs;<gap/>d <pb xlink:href="009/01/106.jpg"/>etiam marium rece&longs;&longs;u; &longs;ic mare ab Ambraciæ portu 10. millia pa&longs;&longs;uum; ab <lb/>Athenarum verò <expan abbr="quinq;">quinque</expan> millia, & alijs in locis plus minu&longs;uè rece&longs;&longs;i&longs;&longs;e &longs;cri<lb/>bit. </s>

<s>Huc facit locus quidam Strabonis ex lib.

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<s>Quapropter ni&longs;i <lb/>igne illo, quem &longs;acræ literæ innuunt catacly&longs;mus ille præueniatur, aqua <lb/>mundus interiturus e&longs;&longs;et. </s>

<s>&longs;ed de his hactenus.</s><pb pagenum="107"/>

<s>&longs;ed de his hactenus.</s><pb xlink:href="009/01/107.jpg"/>

<s>Quoad magnum illud Diluuium, quod Ari&longs;t.

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<s>A&longs;tra verò incrrantia, & <lb/>planetæ Sole tardiores oriuntur <expan abbr="vtroq;">vtroque</expan> modo. </s>

<s>nam cùm ip&longs;a Sol, quippe il<lb/>lis velocior primum a&longs;&longs;equitur, ea &longs;uo lumine obtegit, <expan abbr="e&longs;t&qacute;">e&longs;tque</expan>; hic occa&longs;us eo<lb/>rum heliacus: cum verò eadem præterierit, ac po&longs;t &longs;e reliquerit fit, vt mo<lb/>tu diurno toto c&oelig;lo conuer&longs;o, mane ante Solem effulgeant, &longs;iue heliacè <lb/>oriantur: & cum quotidie magis Sol ab illis recedat, ip&longs;aque magis à Sole <pb pagenum="108"/>elongentur, fit, vt quotidie magis ortum Solis anticipent, & citius mane au<lb/>te Solem videantur. </s>

<s>nam cùm ip&longs;a Sol, quippe il<lb/>lis velocior primum a&longs;&longs;equitur, ea &longs;uo lumine obtegit, <expan abbr="e&longs;t&qacute;">e&longs;tque</expan>; hic occa&longs;us eo<lb/>rum heliacus: cum verò eadem præterierit, ac po&longs;t &longs;e reliquerit fit, vt mo<lb/>tu diurno toto c&oelig;lo conuer&longs;o, mane ante Solem effulgeant, &longs;iue heliacè <lb/>oriantur: & cum quotidie magis Sol ab illis recedat, ip&longs;aque magis à Sole <pb xlink:href="009/01/108.jpg"/>elongentur, fit, vt quotidie magis ortum Solis anticipent, & citius mane au<lb/>te Solem videantur. </s>

<s><expan abbr="&longs;ic&qacute;">&longs;icque</expan>; tanto in dies citius, vt deinde media etiam nocte <lb/>oriantur; tum ante mediam noctem po&longs;tea paulò ante occa&longs;um Solis. </s>

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<s>Eodem cap. <emph type="italics"/>(Ete&longs;iæ autem flant post ver&longs;iones, & Canis ortum)<emph.end type="italics"/> per ver&longs;io<lb/>nes intelligit tropicos, quod & tropici etymon confirmat, <expan abbr="c&utilde;">cum</expan> tropicus idem <lb/>valeat, ac conuer&longs;iuus. </s>

<s>circa Canis ortum eadem &longs;unt notanda, quæ &longs;upra <lb/>de ortu Orionis annotaui; intelligit enim eum Canis ortum, qui mane fiat <pb pagenum="109"/>primum paulò ante Solis ortum, cum &longs;cilicet incipit apparere.</s>

<s>circa Canis ortum eadem &longs;unt notanda, quæ &longs;upra <lb/>de ortu Orionis annotaui; intelligit enim eum Canis ortum, qui mane fiat <pb xlink:href="009/01/109.jpg"/>primum paulò ante Solis ortum, cum &longs;cilicet incipit apparere.</s>

<s>Cum porrò in c&ecedil;lo &longs;it Canis maior, & Canis minor, qui & Procyon, ide&longs;t <lb/>Anticanis dicitur, exi&longs;timo Canem maiorem e&longs;&longs;e eum, qui vulgò Canicula <lb/>nominatur, <expan abbr="&longs;olet&qacute;">&longs;oletque</expan>; vehementes, ac noxios calores excitare. </s>

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<s>Eodem cap. <emph type="italics"/>(Duobus enim exi&longs;tentibus &longs;egmentis habitabilis regionis: vno <lb/>quidem ad &longs;uperiorem polum, qui no&longs;ter e&longs;t; altero ad alterum, & ad meridiem: <lb/><expan abbr="ea&qacute;">eaque</expan>, tympani &longs;peciem habeant, talem enim figuram terræ excidunt ex centro ip&longs;ius <lb/>ductæ lineæ, & faciunt duos conos, bunc quidem habentem ba&longs;im tropicum, alte<lb/>rum autem habentem ba&longs;im circulum &longs;emper manifestum, verticem autem in me<lb/>dio terræ. </s>

<s>eodem autem modo ad inferiorem polum alij duo coni terræ &longs;egmenta fa<lb/>ciunt)<emph.end type="italics"/> vt benè duas ha&longs;ce terræ portiones, quas &longs;olas habitabiles putat Ari<lb/>&longs;tot. concipias, <expan abbr="reliquaq;">reliquaque</expan> huius loci intelligas, in&longs;pice &longs;equentem figuram. <lb/><figure place="text" xlink:href="figures-la/009.01.109.1.tif"/><lb/>Maior circulus &longs;it c&oelig;lum, in quo polus L, articus; M, antarticus, ille eleua<lb/>tus &longs;upra no&longs;trum horizontem S N, 45. gradibus, i&longs;te verò totidem infra <lb/>depre&longs;&longs;us. </s>

<s>eodem autem modo ad inferiorem polum alij duo coni terræ &longs;egmenta fa<lb/>ciunt)<emph.end type="italics"/> vt benè duas ha&longs;ce terræ portiones, quas &longs;olas habitabiles putat Ari<lb/>&longs;tot. concipias, <expan abbr="reliquaq;">reliquaque</expan> huius loci intelligas, in&longs;pice &longs;equentem figuram. <lb/><figure id="id.009.01.109.1.jpg" xlink:href="009/01/109/1.jpg" place="text"/><lb/>Maior circulus &longs;it c&oelig;lum, in quo polus L, articus; M, antarticus, ille eleua<lb/>tus &longs;upra no&longs;trum horizontem S N, 45. gradibus, i&longs;te verò totidem infra <lb/>depre&longs;&longs;us. </s>

<s><expan abbr="&longs;int&qacute;">&longs;intque</expan>; diametri circuli &longs;emper apparentium maximi S R, necnon <pb pagenum="110"/>diametri &longs;emper occultorum maximi Y N: tropicorum item T Q, Cancri, <lb/>X O, Capricorni, vt vides in figura. </s>

<s><expan abbr="&longs;int&qacute;">&longs;intque</expan>; diametri circuli &longs;emper apparentium maximi S R, necnon <pb xlink:href="009/01/110.jpg"/>diametri &longs;emper occultorum maximi Y N: tropicorum item T Q, Cancri, <lb/>X O, Capricorni, vt vides in figura. </s>

<s>Terra &longs;it A B C H G F E D Z K. à cu<lb/>ius centro Z, educantur primo duæ lineæ rectæ Z R, Z S. ad circulum &longs;em<lb/>per apparentium maximum, quæ in terra tran&longs;eant per puncta B, K. & iun<lb/>gatur linea B K: iam vides conum S R Z, cuius ba&longs;is e&longs;t circulus &longs;emper ap<lb/>parens S R, vertex autem Z, in centro terræ, vt ait Ari&longs;tot. </s>

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<s>Eodem cap. <emph type="italics"/>(Et quidem ad latitudinem <expan abbr="v&longs;q;">v&longs;que</expan> ad inhabitabilia &longs;cimus hahita-<emph.end type="italics"/><pb pagenum="111"/><emph type="italics"/>tam, hic enim propter frigus non amplius habitant, illic autem propter æ&longs;tum)<emph.end type="italics"/><lb/>illic autem, ide&longs;t &longs;ub Zona torrida, compertum autem e&longs;t nunc totam ferè <lb/>torridam Zonam, & quidem alicubi percommodè habitari, cuius cau&longs;æ &longs;unt <lb/>quatuor, quæ ip&longs;um latuerunt. </s>

<s>Eodem cap. <emph type="italics"/>(Et quidem ad latitudinem <expan abbr="v&longs;q;">v&longs;que</expan> ad inhabitabilia &longs;cimus hahita-<emph.end type="italics"/><pb xlink:href="009/01/111.jpg"/><emph type="italics"/>tam, hic enim propter frigus non amplius habitant, illic autem propter æ&longs;tum)<emph.end type="italics"/><lb/>illic autem, ide&longs;t &longs;ub Zona torrida, compertum autem e&longs;t nunc totam ferè <lb/>torridam Zonam, & quidem alicubi percommodè habitari, cuius cau&longs;æ &longs;unt <lb/>quatuor, quæ ip&longs;um latuerunt. </s>

<s>prima <expan abbr="ea&qacute;">eaque</expan>; toti Zonæ torridæ communis, <lb/>e&longs;t perpetuum æquinoctium, quo Sol tantum &longs;upra, quantum infra terram <lb/>immoratur. </s>

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figuras, imò demon&longs;tratio<lb/>nes ip&longs;as Mathematicorum, de&longs;criptiones appellare, vt &longs;æpius in Logicis <lb/>monuimus.</s>

<s><emph type="italics"/>De&longs;criptus &longs;it igitur, vt clarior res euadat horizontis circulus quapropter, & <lb/>rotundus)<emph.end type="italics"/> vt in &longs;equenti figura circulus A G B H, de&longs;criptus horizontem <lb/>referret,</s><figure place="text" xlink:href="figures-la/009.01.111.1.tif"/><pb pagenum="112"/>

<s><emph type="italics"/>De&longs;criptus &longs;it igitur, vt clarior res euadat horizontis circulus quapropter, & <lb/>rotundus)<emph.end type="italics"/> vt in &longs;equenti figura circulus A G B H, de&longs;criptus horizontem <lb/>referret,</s><figure id="id.009.01.111.1.jpg" xlink:href="009/01/111/1.jpg" place="text"/><pb xlink:href="009/01/112.jpg"/>

<s><emph type="italics"/>Oportet autem ip&longs;ius alteram portionem intelligere, quæ nobis habitatur; quæ <lb/>eodem modo diuidi poterit)<emph.end type="italics"/> ide&longs;t oportet intelligere ip&longs;ius horizontis, vel ter<lb/>ræ habitatæ partem, quæ quamuis rotunda non &longs;it, poterit tamen, ac &longs;i ro<lb/>tunda e&longs;&longs;et in figura circulari repre&longs;entari, <expan abbr="atq;">atque</expan> in plures partes eo modo, <lb/>quo circulus &longs;ecatur, &longs;ecari.</s>

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<s>Alij autem <lb/>&longs;unt, &longs;ecundum quos non &longs;unt contrarij venti, ab I, quem vocant Tra&longs;ciam, qui me<lb/>dius e&longs;t inter Argesten, & Apparitiam, à K, autem, quem vocant Me&longs;en, Mtdius <lb/>enim e&longs;t Cæciæ, & Aparetiæ. </s>

<s>Diameter autem K I, iuxta circulum &longs;emper con&longs;pi<lb/>cuum e&longs;&longs;e &longs;olet, non tamen exactè)<emph.end type="italics"/> ide&longs;t linea K I, &longs;olet in horizonte referre <lb/>diametrum circuli omnium &longs;emper apparentium maximi, eo quod &longs;it ferè <lb/>&longs;ub diametro illius, in qualibet enim &longs;phæra obliqua, ide&longs;t, in qua polus cle<pb pagenum="113"/>natur, intelligunt A&longs;tronomi circulum quendam &longs;emper apparentium ma<lb/>ximum, quem de&longs;cribunt ex ip&longs;o polo, tanquam centro, & interuailo v&longs;que <lb/>ad horizontem, circa ip&longs;um polum: hunc appellant &longs;emper apparentium, <lb/>maximum, quia intra hunc alios quamplurimos concipiunt circa eundem <lb/>polum, quorum minores &longs;emper &longs;unt polo propinquiores. </s>

<s>Diameter autem K I, iuxta circulum &longs;emper con&longs;pi<lb/>cuum e&longs;&longs;e &longs;olet, non tamen exactè)<emph.end type="italics"/> ide&longs;t linea K I, &longs;olet in horizonte referre <lb/>diametrum circuli omnium &longs;emper apparentium maximi, eo quod &longs;it ferè <lb/>&longs;ub diametro illius, in qualibet enim &longs;phæra obliqua, ide&longs;t, in qua polus cle<pb xlink:href="009/01/113.jpg"/>natur, intelligunt A&longs;tronomi circulum quendam &longs;emper apparentium ma<lb/>ximum, quem de&longs;cribunt ex ip&longs;o polo, tanquam centro, & interuailo v&longs;que <lb/>ad horizontem, circa ip&longs;um polum: hunc appellant &longs;emper apparentium, <lb/>maximum, quia intra hunc alios quamplurimos concipiunt circa eundem <lb/>polum, quorum minores &longs;emper &longs;unt polo propinquiores. </s>

<s>huius igitur dia<lb/>metrum vult Ari&longs;t.

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<s>Notandum etiam Aream, de qua mox dicam explicari po&longs;&longs;e tam per re<lb/>flexionem, quàm per refractionem: per reflexionem, quia &longs;upponunt Philo<lb/>&longs;ophi e&longs;&longs;e in acre rorido innumcra &longs;pecula parua inuicem valdè proxima, <lb/>ide&longs;t guttulas, per quas re&longs;lectatur ad oculum no&longs;trum &longs;pecies &longs;yderis. </s>

<s>per <lb/>re&longs;ractionem verò, vt vult Vitellio, quia &longs;umit totum illum aerem humi<lb/>dum magis den&longs;um e&longs;&longs;e aere paro, qui e&longs;t circa oculos no&longs;tros, & hoc modo <lb/>con&longs;tituit diucr&longs;a media in den&longs;itate, per quam fiat vi&longs;io; corpus inquam <pb pagenum="114"/>illud humidum den&longs;ius, & aerem deinde circa oculum rarius. </s>

<s>per <lb/>re&longs;ractionem verò, vt vult Vitellio, quia &longs;umit totum illum aerem humi<lb/>dum magis den&longs;um e&longs;&longs;e aere paro, qui e&longs;t circa oculos no&longs;tros, & hoc modo <lb/>con&longs;tituit diucr&longs;a media in den&longs;itate, per quam fiat vi&longs;io; corpus inquam <pb xlink:href="009/01/114.jpg"/>illud humidum den&longs;ius, & aerem deinde circa oculum rarius. </s>

<s>Vicomerca<lb/>tus igitur quamuis vtatur voce reflexionis in Halone, non tamen ex prædi<lb/>ctis videtur reprchenden dus.</s>

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<s>ab co<lb/>dem enim &longs;igno ad idem &longs;ignum æquales frangentur &longs;uper circuli lineam &longs;emper. </s>

<s>&longs;it<emph.end type="italics"/><lb/><figure place="text" xlink:href="figures-la/009.01.114.1.tif"/><lb/><emph type="italics"/>enim à puncto, in quo A, ad B, fracta, & ea, quæ est <lb/>A C B, & quæ A F B, & quæ A D B, æquales autem <lb/>& hæ A C, A F, A D, inuiccm. </s>

<s>&longs;it<emph.end type="italics"/><lb/><figure id="id.009.01.114.1.jpg" xlink:href="009/01/114/1.jpg" place="text"/><lb/><emph type="italics"/>enim à puncto, in quo A, ad B, fracta, & ea, quæ est <lb/>A C B, & quæ A F B, & quæ A D B, æquales autem <lb/>& hæ A C, A F, A D, inuiccm. </s>

<s>& quæ ad B, inui<lb/>cem &longs;cilicet C B, E B, D B. & protrahatur A E B, <lb/>quare trianguli æquales, etenim &longs;uper æqualem, quæ <lb/>e&longs;t A E B, ducantur autem <expan abbr="perp&etilde;diculares">perpendiculares</expan> ad A E B, <lb/>ex angulis; à C, quidem, quæ e&longs;t C E; ab F, autem, <lb/>quæ e&longs;t F E; à D, autem, quæ e&longs;t D E, æquales itaque <lb/>hæ, in æqualibus enim tri&abreve;gulis, & in vno plano om<lb/>nes, ad rectum emm omnes ei, quæ e&longs;t A E B. & ad <lb/>vnum punctum E, copulantur, circulus igitur erit <lb/>de&longs;cripta, centrum autem E. &longs;it autem B, quidem Sol, <lb/>A, autem vi&longs;us, quæ autem e&longs;t circa C D F, circun<lb/>ferentia nubes, à qua refrangitur vi&longs;us ad Solem)<emph.end type="italics"/><lb/>quia &longs;uppono Aream, &longs;iue Halonem fieri per re<lb/>fractionem, vt vult etiam Vitellio, propterea <lb/><expan abbr="præmitt&etilde;dum">præmittendum</expan> e&longs;t principium quoddam, quo tra<lb/>ctatio de refractione innititur; e&longs;t autem huiu&longs;<lb/>modi; ea, quæ <expan abbr="vid&etilde;tur">videntur</expan> per refractionem, &longs;iue &longs;ub <lb/>aliquo refractionis angulo, manentibus nobis & <lb/>a&longs;tro, & medio ij&longs;dem in locis, non po&longs;&longs;unt vide<lb/>ri &longs;ub diuer&longs;o angulo à priori, ncc per con&longs;<gap/><expan abbr="qu&etilde;s">quens</expan> <lb/>alibi apparere. </s>

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lineas vi&longs;uales à &longs;ydere B, ad oculos no&longs;tros A, per <lb/>nubem roridam C D F, procedentes, in nube con&longs;imiliter refrangi, ide&longs;t <expan abbr="vn-diq;">vn<lb/>dique</expan> circa Solem, Lunamuè facere angulos refractionis æquales. </s>

<s>quod etiam <pb pagenum="115"/>patet ex 48. 10. Vitellionis; vt in figura, in qua &longs;ydus B, oculus A, nubes <lb/>C D F, radij vi&longs;uales tres refracti in nube &longs;int B C A, B D A, B E A, facien<lb/>tes con&longs;imilem refractionem, ide&longs;t angulos refractos B C A, B D A, B E A, <lb/>æquales in punctis C, D, F: <expan abbr="atq;">atque</expan> hoc e&longs;t con&longs;imilem facere refractionem. <lb/></s>

<s>quod etiam <pb xlink:href="009/01/115.jpg"/>patet ex 48. 10. Vitellionis; vt in figura, in qua &longs;ydus B, oculus A, nubes <lb/>C D F, radij vi&longs;uales tres refracti in nube &longs;int B C A, B D A, B E A, facien<lb/>tes con&longs;imilem refractionem, ide&longs;t angulos refractos B C A, B D A, B E A, <lb/>æquales in punctis C, D, F: <expan abbr="atq;">atque</expan> hoc e&longs;t con&longs;imilem facere refractionem. <lb/></s>

<s>Supponit &longs;ecundò lineas à &longs;ydere ad nubem, v&longs;que exten&longs;as e&longs;&longs;e æquales, vt <lb/>&longs;unt B C, B D, B F: &longs;imiliter reliquas tres à nube ad vi&longs;um A. pares e&longs;&longs;e C A, <lb/>D A, F A. his &longs;uppo&longs;itis, &longs;i deinde protrahatur recta A B, ab oculo ad &longs;ydus, <lb/>exurgunt tria triangula omninò æqualia, & &longs;imilia, cuni duo latera vnius <lb/>&longs;int æqualia duobus alterius <expan abbr="vtrunq;">vtrunque</expan> vtrique, & angulus angulo, & præterea <lb/>ba&longs;is &longs;it communis; ideò per quartam primi &longs;unt omninò æqualia. </s>

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<s>Ibidem <emph type="italics"/>(Hemi&longs;pberio enim exi&longs;t exte &longs;uper horizontis circulum in quo A. cen<lb/>tro autem K, alio autem quodam oriente puncto, in quo G, &longs;i lineæ, quæ à K, &longs;ecun<lb/>dum conum excidentes faciant velut axem lineam in qua G K, & à K. ad M, co<lb/>pulatæ refrangantur ab hemi&longs;phærio ad G, &longs;uper maiorem angulum, circuli circun<lb/>ferentiam incident lineæ, quæ à K, & &longs;i quidem in ortu, aut in occa&longs;u a&longs;tri reflexio <lb/>fiat, &longs;emicirculus ab <expan abbr="horizõte">horizonte</expan> a&longs;&longs;umetur &longs;uper terram factus. </s>

<s>&longs;i autem &longs;upra, minor<emph.end type="italics"/><pb pagenum="116"/><figure place="text" xlink:href="figures-la/009.01.116.1.tif"/><lb/><emph type="italics"/>&longs;emper &longs;emicirculo, minus autem, <lb/>cum in meridie fuerit a&longs;trum)<emph.end type="italics"/> quod <lb/>&longs;upra monui, iterum moneo, <expan abbr="re-tin&etilde;dam">re<lb/>tinendam</expan> vocem reflexionis, <expan abbr="quã-uis">quan<lb/>uis</expan> in antiqua tran&longs;latione lega<lb/>tur refractio, e&longs;t enim apud om<lb/>nes in confe&longs;&longs;o Iridem fieri per <lb/>reflexionem. </s>

<s>&longs;i autem &longs;upra, minor<emph.end type="italics"/><pb xlink:href="009/01/116.jpg"/><figure id="id.009.01.116.1.jpg" xlink:href="009/01/116/1.jpg" place="text"/><lb/><emph type="italics"/>&longs;emper &longs;emicirculo, minus autem, <lb/>cum in meridie fuerit a&longs;trum)<emph.end type="italics"/> quod <lb/>&longs;upra monui, iterum moneo, <expan abbr="re-tin&etilde;dam">re<lb/>tinendam</expan> vocem reflexionis, <expan abbr="quã-uis">quan<lb/>uis</expan> in antiqua tran&longs;latione lega<lb/>tur refractio, e&longs;t enim apud om<lb/>nes in confe&longs;&longs;o Iridem fieri per <lb/>reflexionem. </s>

<s>E&longs;t igitur in &longs;upe<lb/>riori figura, quam textui, vt par <lb/>erat re&longs;titui, horizon G K O. cuius centrum K. in quo e&longs;t vi&longs;us no&longs;ter, <expan abbr="&longs;it&qacute;">&longs;itque</expan>; <lb/>hemi&longs;phærium no&longs;trum in arcu G A M O, repræ&longs;entatum, <expan abbr="&longs;it&qacute;">&longs;itque</expan>; nubes rori<lb/>da, in qua Iris appareat, vbi M, quod punctum M, nubem referens, in figu<lb/>ra ponitur in hemi&longs;phærij ambitu, quod c&oelig;lum repræ&longs;entat, cum tamen <lb/>nubes parum à terra &longs;ubuchatur; id enim ad demon&longs;trationem ferè perinde <lb/>e&longs;t. </s>

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<s>Ibidem <emph type="italics"/>(Sit enim in <expan abbr="ori&etilde;te">oriente</expan> pri<lb/>mum vbi G, & refracta &longs;it K M, <lb/>ad G, & planum erectum &longs;it in quo <lb/>A, à triangulo in quo G K M, cir<lb/>culus igitur erit &longs;ectio &longs;phæ<gap/>æ, qui <lb/>maximus &longs;it in quo A, differet enim <lb/>mbil &longs;i quod<gap/><expan abbr="&ubreve;q;">&ubreve;que</expan> eorum, quæ &longs;uper <lb/>G K, &longs;ecundum triangul&ubreve; K M G, <lb/>erectum fuerit planum. </s>

<s>lineæ igitur <lb/>ab ijs, quæ G, K, ductæ in bac ratio<lb/>ne non cen&longs;tituentur ad aliud, & <lb/>aliud punctum, quàm &longs;emicirculi <lb/>in quo A. </s>

<s>Quoniam enim puncta <lb/>G, K, data &longs;unt, & quæ K M, vtique data erit; & quæ M G, ad M K; datam igi<lb/><gap/>ur circunferentiam tanget M, fit <expan abbr="itaq;">itaque</expan> bæc in qua M N, quare &longs;ectio circunferen-<emph.end type="italics"/><pb pagenum="117"/><emph type="italics"/>tiarum data e&longs;t. </s>

<s>Quoniam enim puncta <lb/>G, K, data &longs;unt, & quæ K M, vtique data erit; & quæ M G, ad M K; datam igi<lb/><gap/>ur circunferentiam tanget M, fit <expan abbr="itaq;">itaque</expan> bæc in qua M N, quare &longs;ectio circunferen-<emph.end type="italics"/><pb xlink:href="009/01/117.jpg"/><emph type="italics"/>tiarum data e&longs;t. </s>

<s>apud autem aliud punctum, quam ip&longs;ius M N, circunferentiæ, ab <lb/>ij&longs;dem punctis, eadem ratio in eodem plano non con&longs;i&longs;tit)<emph.end type="italics"/> eorum omnium, quæ <lb/>demon&longs;tranda &longs;unt, præmittenda &longs;unt duo nece&longs;&longs;aria fundamenta. </s>

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<s>Alterum e&longs;t apud Opticos vulga<lb/>tum; ea &longs;cilicet, quæ per reflexionem (de quorum numero e&longs;t Iris) viden<lb/>tur, videri, tunc &longs;olum, quando angulus incidentiæ fuerit æqualis angulo <lb/>reflexionis, quia tunc breui&longs;&longs;imis lineis fit vi&longs;io; quibus &longs;oli, natura (&longs;i fieri <lb/><figure place="text" xlink:href="figures-la/009.01.117.1.tif"/><lb/>pote&longs;t) vtitur. </s>

<s>Alterum e&longs;t apud Opticos vulga<lb/>tum; ea &longs;cilicet, quæ per reflexionem (de quorum numero e&longs;t Iris) viden<lb/>tur, videri, tunc &longs;olum, quando angulus incidentiæ fuerit æqualis angulo <lb/>reflexionis, quia tunc breui&longs;&longs;imis lineis fit vi&longs;io; quibus &longs;oli, natura (&longs;i fieri <lb/><figure id="id.009.01.117.1.jpg" xlink:href="009/01/117/1.jpg" place="text"/><lb/>pote&longs;t) vtitur. </s>

<s>v. g. in figura præ&longs;enti &longs;it &longs;pe<lb/>culum C D E, obiectum A, oculus B, linea in<lb/>cidentiæ e&longs;t A D, & angulus pariter inciden<lb/>tiæ e&longs;t A D C. linea verò D B, e&longs;t linea refle<lb/>xionis, & angulus pariter reflexionis e&longs;t B D<lb/>E, qui duo anguli ni&longs;i fuerint æquales, nun<lb/>quam videbitur obiectum A, ab oculo B, hinc <lb/>e&longs;t, quod aliquando po&longs;ito &longs;peculo, obiectum <lb/>quamuis illi aduer&longs;um, à nobis pariter ante <lb/>&longs;peculum con&longs;titutis, videri nequit, quia &longs;ci<lb/>licet in tali po&longs;itione &longs;peculi, obiecti, & no&longs;tri, nulla linea incidentiæ, ide&longs;t, <lb/>quæ ab obiecto in &longs;peculum tendit, facere pote&longs;t angulum cum &longs;peculo, qui <lb/>dicitur angulus incidentiæ, æqualem angulo illi, quem facit linea eadem re<lb/>flexa à &longs;peculo ad oculum, quem dicunt angulum re&longs;lexionis. </s>

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<s>Inquit igitur Ari&longs;t. &longs;it enim in oriente, &c. </s>

<s>vbi aggreditur probare vnum <lb/>ex tribus illis, quæ &longs;upra propo&longs;uit, nimirum tunc Iridem e&longs;&longs;e &longs;emicircu<lb/>lum, quando a&longs;trum fuerit in oriente, &longs;iue in horizonte, vbi G. &longs;i igitur per <lb/>triangulum G M K, intelligamus <expan abbr="plan&utilde;">planum</expan> exten&longs;um, in quo A, in figura, adeo <lb/>magnum, vt totum &longs;ece<gap/> hemi&longs;phærium, faciet in &longs;uperficie hemi&longs;phærij &longs;e<pb pagenum="118"/>ctionem, quæ erit portio maximi circuli, per 6. Theodo&longs;ij, cum planum &longs;e<lb/>cans hemi&longs;phærium, tran&longs;eat per <expan abbr="centr&utilde;">centrum</expan> ip&longs;ius, quæ &longs;ectio, &longs;iue circuli por<lb/>tio repræ&longs;entatur in figura, per &longs;emicirculum in quo A, &longs;iue in quo G A M<lb/>R O. nihil autem refert quodcunque intelligas planum &longs;uper axem G K O, <lb/>tran&longs;iens &longs;iue per triangulum G K M, &longs;iue per aliud illi &longs;imile. </s>

<s>vbi aggreditur probare vnum <lb/>ex tribus illis, quæ &longs;upra propo&longs;uit, nimirum tunc Iridem e&longs;&longs;e &longs;emicircu<lb/>lum, quando a&longs;trum fuerit in oriente, &longs;iue in horizonte, vbi G. &longs;i igitur per <lb/>triangulum G M K, intelligamus <expan abbr="plan&utilde;">planum</expan> exten&longs;um, in quo A, in figura, adeo <lb/>magnum, vt totum &longs;ece<gap/> hemi&longs;phærium, faciet in &longs;uperficie hemi&longs;phærij &longs;e<pb xlink:href="009/01/118.jpg"/>ctionem, quæ erit portio maximi circuli, per 6. Theodo&longs;ij, cum planum &longs;e<lb/>cans hemi&longs;phærium, tran&longs;eat per <expan abbr="centr&utilde;">centrum</expan> ip&longs;ius, quæ &longs;ectio, &longs;iue circuli por<lb/>tio repræ&longs;entatur in figura, per &longs;emicirculum in quo A, &longs;iue in quo G A M<lb/>R O. nihil autem refert quodcunque intelligas planum &longs;uper axem G K O, <lb/>tran&longs;iens &longs;iue per triangulum G K M, &longs;iue per aliud illi &longs;imile. </s>

<s>Præmitten<lb/>dum præterea non po&longs;&longs;e in &longs;emicirculo &longs;uperiori, quod e&longs;t planum, & &longs;ectio <lb/>trianguli G K M, poni alias duas lineas. </s>

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<s>&longs;it <expan abbr="au-t&etilde;">au<lb/>tem</expan> ambitus ille in figura &longs;equenti notatus literis L M N. qui ambitus L M N, <lb/>non e&longs;t <expan abbr="concipi&etilde;dus">concipiendus</expan> in eodem plano &longs;emicirculi G A N O, quemadmodum <lb/>falsò pingitur in figura; &longs;ed debemus ip&longs;um concipere tanquam erectum ad <lb/>angulos rectos cum prædicto &longs;emicirculo, necnon cum horizonte G K O. <lb/></s>

<s>Iam &longs;i <expan abbr="triãgulum">triangulum</expan> G M K, prioris figuræ circumuoluatur circa axem G K O, <lb/>punctum ip&longs;ius M, de&longs;cribit prædictum ambitum L M N. hunc ambitum <lb/>inquit Ariltot. linea K M, attinger, <expan abbr="erit&qacute;">eritque</expan>; hic ambitus datus, vt dictum e&longs;t. <lb/><figure place="text" xlink:href="figures-la/009.01.118.1.tif"/><lb/>Erit præterea &longs;ectio circunferentiarum ho<lb/>rizontis, & huius amb tus data, cuius extre<lb/>ma puncta e&longs;&longs;ent L, & N. &longs;i enim <expan abbr="cõcipiamus">concipiamus</expan> <lb/>in figura non &longs;olum horizontis <gap/>iametrum <lb/>G K O, &longs;ed etiam circunferentiam (in qua <lb/>circunferentia e&longs;&longs;ent duo illa puncta L, & N, <lb/>vt in præ&longs;enti de&longs;criptione melius intelli ge<lb/>tur, in qua horizon G N O L, & ambitus <lb/>prædictus e&longs;t L M N, qui debet intelligi ele<lb/>uatus &longs;upra horizontem perpendiculariter) <lb/>tunc &longs;ectio ip&longs;ius mutua cum horizonte e&longs;&longs;et <pb pagenum="119"/>linea N P L, cuius extrema puncta &longs;unt L, N, quæ data erunt, cum &longs;int ex<lb/>trema lineæ K M, circumlatæ; & quemadmodum dabatur &longs;uperius punctum <lb/>M. eadem ratione ex Datis, dabitur punctum N, & L. quare etiam &longs;ectio <lb/>N P L, quæ inter data puncta continetur, data erit ex 26. Datorum.</s>

<s>Iam &longs;i <expan abbr="triãgulum">triangulum</expan> G M K, prioris figuræ circumuoluatur circa axem G K O, <lb/>punctum ip&longs;ius M, de&longs;cribit prædictum ambitum L M N. hunc ambitum <lb/>inquit Ariltot. linea K M, attinger, <expan abbr="erit&qacute;">eritque</expan>; hic ambitus datus, vt dictum e&longs;t. <lb/><figure id="id.009.01.118.1.jpg" xlink:href="009/01/118/1.jpg" place="text"/><lb/>Erit præterea &longs;ectio circunferentiarum ho<lb/>rizontis, & huius amb tus data, cuius extre<lb/>ma puncta e&longs;&longs;ent L, & N. &longs;i enim <expan abbr="cõcipiamus">concipiamus</expan> <lb/>in figura non &longs;olum horizontis <gap/>iametrum <lb/>G K O, &longs;ed etiam circunferentiam (in qua <lb/>circunferentia e&longs;&longs;ent duo illa puncta L, & N, <lb/>vt in præ&longs;enti de&longs;criptione melius intelli ge<lb/>tur, in qua horizon G N O L, & ambitus <lb/>prædictus e&longs;t L M N, qui debet intelligi ele<lb/>uatus &longs;upra horizontem perpendiculariter) <lb/>tunc &longs;ectio ip&longs;ius mutua cum horizonte e&longs;&longs;et <pb xlink:href="009/01/119.jpg"/>linea N P L, cuius extrema puncta &longs;unt L, N, quæ data erunt, cum &longs;int ex<lb/>trema lineæ K M, circumlatæ; & quemadmodum dabatur &longs;uperius punctum <lb/>M. eadem ratione ex Datis, dabitur punctum N, & L. quare etiam &longs;ectio <lb/>N P L, quæ inter data puncta continetur, data erit ex 26. Datorum.</s>

Line 5580

Line 5581

<s>Fortè etiam addi pote&longs;t, quod alibi exi&longs;ten<lb/>te puncto R, quàm in M, non po&longs;&longs;ent anguli incidentiæ, & reflexionis e&longs;&longs;e <lb/>æquales, quæ cau&longs;a e&longs;&longs;et cur &longs;ub alio angulo, quam prædicto G M K, Iris <lb/>non appareret.</s>

<s>Prædicta omnia &longs;unt &longs;ecundum Ari&longs;tot. di&longs;cur&longs;um, & figurationem dicta, <lb/>nam &longs;ecundum veritatem po&longs;&longs;<gap/>nt in eadem nube con&longs;titui plures anguli <lb/>æquales, nec tamen in eodem orbe, &longs;ed vnus &longs;upra <expan abbr="alter&utilde;">alterum</expan>; vt in figura præ<lb/><figure place="text" xlink:href="figures-la/009.01.119.1.tif"/><lb/>&longs;enti, &longs;i nubes e&longs;&longs;et vbi B D. <lb/>oculus in C, Sol in A. e&longs;&longs;ent <lb/>duo anguli A B C, A D C, æ<lb/>quales per 33. 3. qui tamen <lb/>non &longs;unt in gyrum con&longs;tituti, <lb/>po&longs;&longs;et igitur, per <expan abbr="illor&utilde;">illorum</expan> vtrun<lb/>que Sol Iridem efficere. </s>

<s>Prædicta omnia &longs;unt &longs;ecundum Ari&longs;tot. di&longs;cur&longs;um, & figurationem dicta, <lb/>nam &longs;ecundum veritatem po&longs;&longs;<gap/>nt in eadem nube con&longs;titui plures anguli <lb/>æquales, nec tamen in eodem orbe, &longs;ed vnus &longs;upra <expan abbr="alter&utilde;">alterum</expan>; vt in figura præ<lb/><figure id="id.009.01.119.1.jpg" xlink:href="009/01/119/1.jpg" place="text"/><lb/>&longs;enti, &longs;i nubes e&longs;&longs;et vbi B D. <lb/>oculus in C, Sol in A. e&longs;&longs;ent <lb/>duo anguli A B C, A D C, æ<lb/>quales per 33. 3. qui tamen <lb/>non &longs;unt in gyrum con&longs;tituti, <lb/>po&longs;&longs;et igitur, per <expan abbr="illor&utilde;">illorum</expan> vtrun<lb/>que Sol Iridem efficere. </s>

Line 5588

Line 5589

<s>Fortè confu giendum e&longs;t ad illud, quod Maurolycus, & <lb/>10. Bapti&longs;ta Porta ob&longs;eruarunt; debere <expan abbr="nimir&utilde;">nimirum</expan> di&longs;tantiam ab oculo ad cen<lb/>trum Iridis e&longs;&longs;e æqualem altitudini, &longs;iue &longs;emidiametro Iridis. </s>

<s>Ita vt non &longs;o<pb pagenum="120"/>lum requiratur idem angulus, &longs;ed etiam tanta Iridis altitudo, <expan abbr="quãta">quanta</expan> requi<lb/>ritur vt angulus in orbem con&longs;tituatur, ex quo Iris po&longs;&longs;it apparere. </s>

<s>Ita vt non &longs;o<pb xlink:href="009/01/120.jpg"/>lum requiratur idem angulus, &longs;ed etiam tanta Iridis altitudo, <expan abbr="quãta">quanta</expan> requi<lb/>ritur vt angulus in orbem con&longs;tituatur, ex quo Iris po&longs;&longs;it apparere. </s>

<s>hæc à <lb/>nemine hactenus animaduer&longs;a placuit addere, vt ex ijs demon&longs;tratio Iridis <lb/>omnibus numeris aliquando ab&longs;olui po&longs;&longs;it, quod infra (ni fallor, fauente <lb/>Deo) præ&longs;tabimus.<lb/><arrow.to.target n="marg166"/></s>

Line 5606

Line 5607

<s>Dicit <expan abbr="itaq;">itaque</expan> Ari&longs;t.

cum data &longs;it proportio linearum K M, & M G, in &longs;upe<lb/>riori &longs;ecunda figura numeri 164. quam nunc iterum in&longs;picere opertet; ex<lb/><figure place="text" xlink:href="figures-la/009.01.120.1.tif"/><lb/>ponatur alia linea recta B D. quæ diui<lb/>datur in partes B, & D. proportionales <lb/>cum lineis K M, G M, per 10. 6. cum <lb/>ergo K M, &longs;it minor quàm G M, per 19. <lb/>primi, quia in triangulo G M K, oppo<lb/>nitur minori angulo, erit <expan abbr="quoq;">quoque</expan> B, minor quàm D, addatur iam ip&longs;i B. linea <lb/>nea F, ita vt &longs;it tota F B, tertia proportionalis ad duas B, & D, per 11. 6. <lb/>hoc ordine, vt F B, ad D. ita D, ad B. </s>

cum data &longs;it proportio linearum K M, & M G, in &longs;upe<lb/>riori &longs;ecunda figura numeri 164. quam nunc iterum in&longs;picere opertet; ex<lb/><figure id="id.009.01.120.1.jpg" xlink:href="009/01/120/1.jpg" place="text"/><lb/>ponatur alia linea recta B D. quæ diui<lb/>datur in partes B, & D. proportionales <lb/>cum lineis K M, G M, per 10. 6. cum <lb/>ergo K M, &longs;it minor quàm G M, per 19. <lb/>primi, quia in triangulo G M K, oppo<lb/>nitur minori angulo, erit <expan abbr="quoq;">quoque</expan> B, minor quàm D, addatur iam ip&longs;i B. linea <lb/>nea F, ita vt &longs;it tota F B, tertia proportionalis ad duas B, & D, per 11. 6. <lb/>hoc ordine, vt F B, ad D. ita D, ad B. </s>

<s>Deinde vt &longs;e habet F, ad K G. ita &longs;it <lb/>B, ad aliam, quæ &longs;it K P, in eadem figura per 12. 6. & à puncto P, ad M, iun<lb/>gatur recta P M. </s>

Line 5618

Line 5619

<s>Ibidem <emph type="italics"/>(Erit etiam, quod quæ F, ad K G. & quæ B, ad K P. & quæ D, ad P M. <lb/>non enim &longs;it, &longs;ed aut ad minorem, aut ad maiorem ea, quæ P M, nibil emm differet. <lb/></s>

<s>&longs;it enim ad P R. eandem ergo rationem G K, & K P, & P R, inuicem habebunt, <lb/>quam quæ F, B, D: quæ autem F, B, D, proportionales crant, quod quidem D, ad <lb/>B. quæ F B, ad D: quare quod quæ P G, ad P R, quæ P R, ad eam, quæ P K. &longs;i igi<lb/>tur ab ijs, quæ K G, quæ G R, & K R, ad R, coniungantur, coniunctæ hæ eandem <lb/>habebunt rationem, quam quæ G P, ad eam, quæ P R, circa eundem enim angulum <lb/>P, proportion aliter, & quæ trianguli G P R, & eius, qui K R P. quare & quæ G R, <lb/>ad eam quæ K R, eandem rationem habebit, quam & quæ G P, ad eam quæ P R, <lb/>habet autem & quæ M G, ad M K, eam rationem, quam quæ D, ad eam quæ B, <lb/>quare ambæ à punctis G K, non &longs;olum ad circun&longs;erentiam M N, con&longs;tituentur ean<lb/>aem habentes rationem, &longs;ed & alibi, quod quidem impo&longs;&longs;ibile)<emph.end type="italics"/> incipit, vt dixi, <pb pagenum="121"/>probare P, e&longs;&longs;e polum prædicti ambitus, &longs;ic. </s>

<s>&longs;it enim ad P R. eandem ergo rationem G K, & K P, & P R, inuicem habebunt, <lb/>quam quæ F, B, D: quæ autem F, B, D, proportionales crant, quod quidem D, ad <lb/>B. quæ F B, ad D: quare quod quæ P G, ad P R, quæ P R, ad eam, quæ P K. &longs;i igi<lb/>tur ab ijs, quæ K G, quæ G R, & K R, ad R, coniungantur, coniunctæ hæ eandem <lb/>habebunt rationem, quam quæ G P, ad eam, quæ P R, circa eundem enim angulum <lb/>P, proportion aliter, & quæ trianguli G P R, & eius, qui K R P. quare & quæ G R, <lb/>ad eam quæ K R, eandem rationem habebit, quam & quæ G P, ad eam quæ P R, <lb/>habet autem & quæ M G, ad M K, eam rationem, quam quæ D, ad eam quæ B, <lb/>quare ambæ à punctis G K, non &longs;olum ad circun&longs;erentiam M N, con&longs;tituentur ean<lb/>aem habentes rationem, &longs;ed & alibi, quod quidem impo&longs;&longs;ibile)<emph.end type="italics"/> incipit, vt dixi, <pb xlink:href="009/01/121.jpg"/>probare P, e&longs;&longs;e polum prædicti ambitus, &longs;ic. </s>

<s>Primò enim &longs;ciendum in præ<lb/>mi&longs;&longs;a con&longs;tructione e&longs;&longs;e, vt F, ad G K, & B, ad K P, ita D, ad P M. nam &longs;i non <lb/>&longs;it eadem ratio D, ad P M, cum alijs prædictis, erit eadem ratio eiu&longs;dem D, <lb/>ad aliam maiorem, vel minorem ipfa P M. &longs;it ad minorem P R. nihil enim <lb/>refert &longs;iue dixeris habere eandem rationem ad minorem, &longs;iue ad maiorem, <lb/>ergo permutando erunt G K, K P, P R, proportionales cum F, B, D. &longs;ed li<lb/>neæ F, B, D, erant proportionales <expan abbr="compon&etilde;do">componendo</expan> hoc modo, vt F B, ad D, ita <lb/>D, ad B: quare &longs;imiliter erunt vt G P, ad P R, ita P R, ad P K. per 18. 5. &longs;i igi<lb/>tur à punctis G, & K, figuræ nu. </s>

Line 5638

Line 5639

<s>Porrò &longs;ub angulo G M K, linearum G M, M K, Iris <lb/>apparet: quare apparebit etiam &longs;ub alijs omnibus, quæ à punctis G K, duci <lb/>po&longs;&longs;unt ad extremum lineæ P M, quia erunt in eadem ratione cum illis; cum <lb/>non de&longs;inant in eundem <expan abbr="&longs;emicircul&utilde;">&longs;emicirculum</expan>, &longs;ed in ambitum Iridis M N, in quo M, <lb/>punctum imaginamur circumduci. </s>

<s>Ex quibus pater P, e&longs;&longs;e polum Iridis, ex <lb/>quo per puncta M, vbi &longs;it reflexio, de&longs;cribitur arcus attingens omnes Iridis <lb/>reflexiones.</s><pb pagenum="122"/>

Line 5658

Line 5659

<s>Atque hoc accidit Sole, vel Luna in horizonte exi&longs;tentibus; quod <lb/>erat primo loco demon&longs;trandum.</s>

<s>Porrò &longs;ciendum po&longs;&longs;e nos breuius polum prædictum inuenire, &longs;i nimirum <lb/><figure place="text" xlink:href="figures-la/009.01.122.1.tif"/><lb/>ad M, ducatur M P, faciens angulum K P M, æqua<lb/>lem angulo G M K, per 23. primi, erunt enim duo <lb/>triangula <expan abbr="æquiãgula">æquiangula</expan> G P M, K P M, angulus enim <lb/>P, e&longs;t communis, angulus verò M K P, e&longs;t æqualis <lb/>duobus G, & G M K, per 32. primi, ergo etiam <lb/>duobus ad M, &longs;iue toti G M P, & reliquus K M P, <lb/>reliquo, quare per 4.6. latera circa angulos æqua<lb/>les proportionalia erunt, & omologa G M, ad M K, ita G P, ad P M, quæ <lb/>æqualibus augulis &longs;ubtenduntur. </s>

<s>Porrò &longs;ciendum po&longs;&longs;e nos breuius polum prædictum inuenire, &longs;i nimirum <lb/><figure id="id.009.01.122.1.jpg" xlink:href="009/01/122/1.jpg" place="text"/><lb/>ad M, ducatur M P, faciens angulum K P M, æqua<lb/>lem angulo G M K, per 23. primi, erunt enim duo <lb/>triangula <expan abbr="æquiãgula">æquiangula</expan> G P M, K P M, angulus enim <lb/>P, e&longs;t communis, angulus verò M K P, e&longs;t æqualis <lb/>duobus G, & G M K, per 32. primi, ergo etiam <lb/>duobus ad M, &longs;iue toti G M P, & reliquus K M P, <lb/>reliquo, quare per 4.6. latera circa angulos æqua<lb/>les proportionalia erunt, & omologa G M, ad M K, ita G P, ad P M, quæ <lb/>æqualibus augulis &longs;ubtenduntur. </s>

<s><expan abbr="ea&longs;d&etilde;">ea&longs;dem</expan> autem proprietates habcbant etiam <lb/>triangula Ari&longs;t.

Line 5676

Line 5677

<s>Polus autem circuli, in quo P, erit &longs;ub horizonte eo, in quo A C, eleuato puncto, <lb/>in quo G. in eadem autem & polus, & centrum circuli, & terminantis nunc ortum, <lb/>e&longs;t enim i&longs;te, in quo G P. </s>

<s>Quoniam autem &longs;upra diametrum, quæ A C, quod K G, <lb/>centrum vtique erit &longs;ub horizonte priori eius, in quo A C, in linea K P, in quo <foreign lang="greek">w,</foreign><emph.end type="italics"/><pb pagenum="123"/><figure place="text" xlink:href="figures-la/009.01.123.1.tif"/><lb/><emph type="italics"/>Quare minor erit &longs;uperior &longs;ectio &longs;emicir<lb/>culo, in qua S T, (nam Q S T, &longs;emicir<lb/>culus est, nunc autem inter&longs;ectus e&longs;t ab <lb/>horizonte A C; <expan abbr="itaq;">itaque</expan> Q S, di&longs;parens erit) <lb/>eleuato ip&longs;o Sole)<emph.end type="italics"/> demon&longs;trat propo&longs;i<lb/>tionem &longs;ecundam nimirum Sole &longs;upra <lb/>horizontem elcuato, ambitum Iridis <lb/>e&longs;&longs;e minorem circuli portionem, &longs;iue <lb/>&longs;emicirculo minorem. </s>

<s>Quoniam autem &longs;upra diametrum, quæ A C, quod K G, <lb/>centrum vtique erit &longs;ub horizonte priori eius, in quo A C, in linea K P, in quo <foreign lang="greek">w,</foreign><emph.end type="italics"/><pb xlink:href="009/01/123.jpg"/><figure id="id.009.01.123.1.jpg" xlink:href="009/01/123/1.jpg" place="text"/><lb/><emph type="italics"/>Quare minor erit &longs;uperior &longs;ectio &longs;emicir<lb/>culo, in qua S T, (nam Q S T, &longs;emicir<lb/>culus est, nunc autem inter&longs;ectus e&longs;t ab <lb/>horizonte A C; <expan abbr="itaq;">itaque</expan> Q S, di&longs;parens erit) <lb/>eleuato ip&longs;o Sole)<emph.end type="italics"/> demon&longs;trat propo&longs;i<lb/>tionem &longs;ecundam nimirum Sole &longs;upra <lb/>horizontem elcuato, ambitum Iridis <lb/>e&longs;&longs;e minorem circuli portionem, &longs;iue <lb/>&longs;emicirculo minorem. </s>

<s>&longs;it igitur in fi<lb/>gura &longs;uperiori, quam textui <expan abbr="cõgruen-tem">congruen<lb/>tem</expan> re&longs;tituimus, linea A C, horizon<lb/>talis, &longs;upra quam Sol &longs;it eleuatus in <lb/>circulo altitudinis in loco G, axis au<lb/>rem coni, quem reflexè faciunt &longs;it <lb/>G K <foreign lang="greek">w</foreign> P. alia igitur omnia, quæ &longs;upra exi&longs;tente in ortu a&longs;tro o&longs;ten&longs;a &longs;unt, hic <lb/>pariter o&longs;tendi po&longs;&longs;unt, &longs;cilicet Iridem fieri tantum per lineas proportiona<lb/>les, & æquales lineis G M, M K, quia Iris videri nequit, ni&longs;i in tali, ac deter<lb/>minata reflexione, & angulo, vt initio &longs;uppo&longs;ui; & quia lineæ illis propor<lb/>tionales non po&longs;&longs;unt alibi con&longs;titui, quam in ambitu circulari, & in diuer&longs;is <lb/>planis, &longs;equitur, vt &longs;upra Iridem e&longs;&longs;e circularem M N L; <expan abbr="eius&qacute;">eiusque</expan>; polum P, & <lb/>centrum <foreign lang="greek">w,</foreign> inueniemus &longs;imiliter in axe G K <foreign lang="greek">w</foreign> P, & quia axis hic &longs;ecat hori<lb/>zontem in K, in hac vltima figura propter eleuationem Solis &longs;upra A C, in <lb/>G, &longs;equitur partem axis, in qua <foreign lang="greek">w,</foreign> & P, exi&longs;tunt, infra horizontem deprimi. <lb/></s>

Line 5702

Line 5703

<s>Aduertendum præterea Vicomercatum inordinatè citare librum Dato<lb/>rum Euclidis, & <expan abbr="quandoq;">quandoque</expan> etiam malè citare Euclidem ip&longs;um. </s>

<s>peius verò <pb pagenum="124"/>faciunt ij, qui has demon&longs;trrationes <expan abbr="ab&longs;q;">ab&longs;que</expan> vlla libri Datorum mentione ex<lb/>plicare conantur, cum manife&longs;tè illo innitantur.</s>

<s>peius verò <pb xlink:href="009/01/124.jpg"/>faciunt ij, qui has demon&longs;trrationes <expan abbr="ab&longs;q;">ab&longs;que</expan> vlla libri Datorum mentione ex<lb/>plicare conantur, cum manife&longs;tè illo innitantur.</s>

<s>Cæterum &longs;i quis breues, ac dilucidas harum rerum demon&longs;trationes re<lb/>quirat, is legat 74. 75. 76. propo&longs;itiones 10. Vitell. vel &longs;equentem no&longs;tram <lb/>de Iride additionem. </s>

Line 5726

Line 5727

<s>quæ omnia adhibita &longs;phæra materia<lb/>li, eaque a&longs;tronomicè ad &longs;uam eleuationem <lb/>accommodata, nullo negotio li<lb/>cebit intueri.</s><pb pagenum="125"/>

<s>quæ omnia adhibita &longs;phæra materia<lb/>li, eaque a&longs;tronomicè ad &longs;uam eleuationem <lb/>accommodata, nullo negotio li<lb/>cebit intueri.</s><pb xlink:href="009/01/125.jpg"/>

<s><emph type="italics"/>Additio de Iride.<emph.end type="italics"/></s>

Line 5744

Line 5745

<s>non in hoc &longs;itu, & ouali figura, &longs;ed <lb/><figure place="text" xlink:href="figures-la/009.01.125.1.tif"/><lb/>e&longs;&longs;e perfectè &longs;emicircularem, <expan abbr="habere&qacute;">habereque</expan>; talem po&longs;itionem, vt pars ip&longs;ius B F, <lb/>&longs;it citra chartam eleuata, <expan abbr="ip&longs;i&qacute;">ip&longs;ique</expan>; perpendicularis, pars verò E B, vltra pagi<pb pagenum="126"/>nam rectà recedat, <expan abbr="&longs;ic&qacute;">&longs;icque</expan>; diameter Iridis E F, faciat angulos rectos cum <lb/>linea horizontali A C L, in quo &longs;itu oculo C, totus ex oppo&longs;ito directè &longs;pe<lb/>ctaretur, non aliter ac Iridem ip&longs;am con&longs;picere &longs;olemus. </s>

<s>non in hoc &longs;itu, & ouali figura, &longs;ed <lb/><figure id="id.009.01.125.1.jpg" xlink:href="009/01/125/1.jpg" place="text"/><lb/>e&longs;&longs;e perfectè &longs;emicircularem, <expan abbr="habere&qacute;">habereque</expan>; talem po&longs;itionem, vt pars ip&longs;ius B F, <lb/>&longs;it citra chartam eleuata, <expan abbr="ip&longs;i&qacute;">ip&longs;ique</expan>; perpendicularis, pars verò E B, vltra pagi<pb xlink:href="009/01/126.jpg"/>nam rectà recedat, <expan abbr="&longs;ic&qacute;">&longs;icque</expan>; diameter Iridis E F, faciat angulos rectos cum <lb/>linea horizontali A C L, in quo &longs;itu oculo C, totus ex oppo&longs;ito directè &longs;pe<lb/>ctaretur, non aliter ac Iridem ip&longs;am con&longs;picere &longs;olemus. </s>

<s>Quod autem ne<lb/>ce&longs;&longs;aria &longs;it nubes ro&longs;cida, pulcherrima hac experientia <expan abbr="cõprobatur">comprobatur</expan>: &longs;i enim <lb/>in Sole po&longs;iti ore aquam efflantes leui a&longs;pergine aerem Soli, ac nobis ad<lb/>uer&longs;um irroremus, actutum Iridis arcum guttulis illis, quamuis volitanti<lb/>bus inhærcntem &longs;umma voluptate &longs;pectabimus. </s>

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<s>primò, ex angulo reflexionis determinato, qui videlicet &longs;it ferè <lb/>&longs;emirectus. </s>

<s>&longs;ecundò, ex paribus di&longs;tantijs C D, D B, huiu&longs;modi enim plures <lb/>anguli, qui ad Iridem &longs;unt nece&longs;&longs;arij (debent enim &longs;ingulæ Iridis partes &longs;ub <lb/>huiu&longs;modi angulo repre&longs;entari) non po&longs;&longs;unt aliter quàm in gyrum <expan abbr="cõ&longs;titui">con&longs;titui</expan>, <pb pagenum="127"/>quem gyrum optimè concipiemus, &longs;i imaginemur triangulum A B C, cir<lb/>cumuerti circa lineam horizontalem A C L, fixam, tanquam circa axem. </s>

<s>&longs;ecundò, ex paribus di&longs;tantijs C D, D B, huiu&longs;modi enim plures <lb/>anguli, qui ad Iridem &longs;unt nece&longs;&longs;arij (debent enim &longs;ingulæ Iridis partes &longs;ub <lb/>huiu&longs;modi angulo repre&longs;entari) non po&longs;&longs;unt aliter quàm in gyrum <expan abbr="cõ&longs;titui">con&longs;titui</expan>, <pb xlink:href="009/01/127.jpg"/>quem gyrum optimè concipiemus, &longs;i imaginemur triangulum A B C, cir<lb/>cumuerti circa lineam horizontalem A C L, fixam, tanquam circa axem. </s>

<s>in <lb/>hac enim conuer&longs;ione angulus Iridis B, de&longs;cribet circulum, qui erit Iris, & <lb/>pertr an&longs;ibit omnes angulos, qui in tali Solis, oculi, ac nubis &longs;itu, arcum ef<lb/>ficere &longs;unt idonei.</s>

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<s>Ex his etiam Iridis definitio in hunc modum concinnari pote&longs;t, Iris <lb/>e&longs;t arcus multicolor in nube rorida, ex radiorum Solis, aut Lunæ reflexio<lb/>ne &longs;ab &longs;tatuto angulo effulgens.</s>

<s>Octauo loco Problemata nonnulla re&longs;oluemus.</s><pb pagenum="128"/>

<s>Octauo loco Problemata nonnulla re&longs;oluemus.</s><pb xlink:href="009/01/128.jpg"/>

<s><emph type="italics"/>Problema Primum.<emph.end type="italics"/></s>

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<s><emph type="italics"/>Problema Secundum.<emph.end type="italics"/></s>

<s>Maior quidem, imò etiam integer circulus, &longs;ed ab oculo in &longs;ummitate <lb/>montis con&longs;tituto, <expan abbr="Sole&qacute;">Soleque</expan>; iam multum eleuato videri pote&longs;t, vt in <lb/>hac &longs;ecunda figura cernitur, vbi euecto Sole ad locum E, &longs;upra horizontem <lb/><figure place="text" xlink:href="figures-la/009.01.128.1.tif"/><lb/>A B, poterit oculus in vertice montis C, po&longs;itus Iridem F G H I, comple<lb/>tam videre, quia infra lineam E C D, in qua exi&longs;tunt Sol, oculus, & Iridis <lb/>centrum, nihil e&longs;t ad partes D, vbi nubes irrorat, quod Iridis apparitioni <lb/>&longs;it impedimento.</s><pb pagenum="129"/>

<s>Maior quidem, imò etiam integer circulus, &longs;ed ab oculo in &longs;ummitate <lb/>montis con&longs;tituto, <expan abbr="Sole&qacute;">Soleque</expan>; iam multum eleuato videri pote&longs;t, vt in <lb/>hac &longs;ecunda figura cernitur, vbi euecto Sole ad locum E, &longs;upra horizontem <lb/><figure id="id.009.01.128.1.jpg" xlink:href="009/01/128/1.jpg" place="text"/><lb/>A B, poterit oculus in vertice montis C, po&longs;itus Iridem F G H I, comple<lb/>tam videre, quia infra lineam E C D, in qua exi&longs;tunt Sol, oculus, & Iridis <lb/>centrum, nihil e&longs;t ad partes D, vbi nubes irrorat, quod Iridis apparitioni <lb/>&longs;it impedimento.</s><pb xlink:href="009/01/129.jpg"/>

<s>Cur quanto Sol altior e&longs;t, tanto inferior, <expan abbr="tanto&qacute;">tantoque</expan>; &longs;emicir<lb/>culo minor appareat Iris?</s>

<s><emph type="italics"/>Problema Tertium.<emph.end type="italics"/></s>

<s>Qvia eleuato Sole ad E, vt in hac tertia figura, nece&longs;&longs;ario centrum Iri<lb/>dis D, infra horizontem A B, deprimetur, cum in eadem recta E C D. <lb/><figure place="text" xlink:href="figures-la/009.01.129.1.tif"/><lb/>Sol E, oculus C, <expan abbr="centr&utilde;&qacute;">centrunque</expan>; Iridis D, exi&longs;tant: vnde nece&longs;&longs;ariò &longs;equitur Iridis <lb/>portionem F G H, &longs;upra horizontem extantem, &longs;emicirculo minorem e&longs;&longs;e.</s>

<s>Qvia eleuato Sole ad E, vt in hac tertia figura, nece&longs;&longs;ario centrum Iri<lb/>dis D, infra horizontem A B, deprimetur, cum in eadem recta E C D. <lb/><figure id="id.009.01.129.1.jpg" xlink:href="009/01/129/1.jpg" place="text"/><lb/>Sol E, oculus C, <expan abbr="centr&utilde;&qacute;">centrunque</expan>; Iridis D, exi&longs;tant: vnde nece&longs;&longs;ariò &longs;equitur Iridis <lb/>portionem F G H, &longs;upra horizontem extantem, &longs;emicirculo minorem e&longs;&longs;e.</s>

<s>Cur Iris in&longs;equentes fugit, fugientes verò in&longs;equitur?</s>

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<s>Pvlcherrimum i&longs;tud phænomenon primus omnium Philippus Mendæus <lb/>Platonis di&longs;cipulus, ob&longs;eruauit; Cuius ratio e&longs;t, quia arcus non ni&longs;i &longs;ub <lb/>determinato angulo, di&longs;tantijs etiam illis paribus, ac tandem idone a a&longs;per<lb/>gino&longs;æ nubis multiplicatione &longs;pectatur; quapropter &longs;i quis per aerem to<lb/>tum <expan abbr="vndiq;">vndique</expan> ro&longs;cidum inambulet, <expan abbr="vbicunq;">vbicunque</expan> illi anguli, <expan abbr="illæ&qacute;">illæque</expan>; conditiones af<lb/>fuerint Iris apparebit: quod &longs;i in aperta planitie obequitans arcu con&longs;pe<lb/>cto, additis equo calcaribus citatum cur&longs;um ad eum direxerit, fugientem <lb/>ante &longs;e Iridem &longs;umma cum iucunditate mirabitur.</s>

<s>Ex dictis pr&ecedil;tere a patet, &longs;impliciter nimis eos hallucinari, qui exi&longs;timant <lb/>in plana, aut concaua nubis &longs;uperficie Iridem tantummodo apparere po&longs;&longs;e.</s><pb pagenum="130"/>

<s>Curlunares Irides fiunt rariores?</s>

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<s>Ibi verò <emph type="italics"/>(Si autem procul ab&longs;it, &c.)<emph.end type="italics"/> reddit rationem, cur parelium non <lb/>appareat in nube à Sole valde remota &longs;ecundum vulgarem æ&longs;timationem, <lb/>vnde vulgarem etiam rationem affert, ait enim, nubem illam e&longs;&longs;e veluti &longs;pe<lb/>culum Solis repre&longs;entatiuum, &longs;peculum autem tàm longè à Sole po&longs;itum, <lb/>reddi debile, & proptereá non po&longs;&longs;e Solis imaginem referre: Verùm ratio <lb/>hæc nulla e&longs;&longs;e videtur, quis enim ignorat non propterea e&longs;&longs;e remotius à So<lb/>le, quamuis maiorem habere videatur à Sole lateralem di&longs;tantiam, vt pau<lb/>lò ante dixi? </s>

<s>Eandem rationem illi dubitationi accommodat, cur <expan abbr="neq;">neque</expan> vi<lb/>deatur &longs;upra Solem, quamuis non ei quadret, pote&longs;t enim aliqua nubes vi<pb pagenum="131"/>deri &longs;upra Solem, quæ tamen remotior &longs;it à Sole, quam illa, in qua Parclium <lb/>gignitur. </s>

<s>Eandem rationem illi dubitationi accommodat, cur <expan abbr="neq;">neque</expan> vi<lb/>deatur &longs;upra Solem, quamuis non ei quadret, pote&longs;t enim aliqua nubes vi<pb xlink:href="009/01/131.jpg"/>deri &longs;upra Solem, quæ tamen remotior &longs;it à Sole, quam illa, in qua Parclium <lb/>gignitur. </s>

<s>Ait po&longs;tea <emph type="italics"/>(A latere autem, &c.)<emph.end type="italics"/> cur appareat in nube fatis Soli <lb/>à latere vicina, in di&longs;tantiam à Sole refert: &longs;ed quæ dudum dicta &longs;unt, i&longs;tud <lb/><expan abbr="quoq;">quoque</expan> refellunt. </s>

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<s>&longs;ed <lb/>&longs;icut recto in quantum rectum multa accidunt, vt tangere æ<gap/>eam &longs;phæram &longs;ecun<lb/>dum punctum, non tamen tanget hoc, rectum ip&longs;um &longs;eparatum: in&longs;eparabile enim, <lb/>&longs;i quidem cum corpore quodam &longs;emper e&longs;t)<emph.end type="italics"/> Propo&longs;itio 2. tertij Elem.

&pacute;robat li<lb/><figure place="text" xlink:href="figures-la/009.01.131.1.tif"/><lb/>neam rectam, duo quælibet puncta <expan abbr="quãtumuis">quantumuis</expan> pro<lb/>pinqua in circuli ambitu a&longs;&longs;umpta coniungentem <lb/>cadere intra circulum. </s>

&pacute;robat li<lb/><figure id="id.009.01.131.1.jpg" xlink:href="009/01/131/1.jpg" place="text"/><lb/>neam rectam, duo quælibet puncta <expan abbr="quãtumuis">quantumuis</expan> pro<lb/>pinqua in circuli ambitu a&longs;&longs;umpta coniungentem <lb/>cadere intra circulum. </s>

<s>v. g. puncta A B, quantum<lb/>uis &longs;ibi inuicem propinqua fnerint, attamen &longs;i line a <lb/>A B, ea coniungat, ip&longs;a cadet intra circulum, & <lb/>veluti chorda &longs;ubtendet arcum A B, quantulum<lb/>cunque. </s>

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<s>ex qua demon&longs;tratione colligitur in corol<lb/>lario eius lineam rectam tangentem circulum ip<lb/>&longs;um in vnico puncto tangere. </s>

<s>v. g. rectam C D, tan<lb/>gere circulum in puncto E. &longs;i enim dixeris tangere <lb/>in duobus admodum propinquis, vt in E F, tunc non erit amplius tangens, <lb/>&longs;ed &longs;ecans, quia vt modo dixi, pars lineæ rectæ, quæ <expan abbr="cõiungeret">coniungeret</expan> puncta E F, <pb pagenum="132"/>intra circulum per &longs;ecundam præallegatam caderet, quod e&longs;t ab&longs;urdum, <lb/>quia contra hypothe&longs;im, cum &longs;upponamus illam &longs;olùm tangere, non autem <lb/>&longs;ecare circulum. </s>

<s>v. g. rectam C D, tan<lb/>gere circulum in puncto E. &longs;i enim dixeris tangere <lb/>in duobus admodum propinquis, vt in E F, tunc non erit amplius tangens, <lb/>&longs;ed &longs;ecans, quia vt modo dixi, pars lineæ rectæ, quæ <expan abbr="cõiungeret">coniungeret</expan> puncta E F, <pb xlink:href="009/01/132.jpg"/>intra circulum per &longs;ecundam præallegatam caderet, quod e&longs;t ab&longs;urdum, <lb/>quia contra hypothe&longs;im, cum &longs;upponamus illam &longs;olùm tangere, non autem <lb/>&longs;ecare circulum. </s>

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<s>Ex hac Euclidis doctrina Theodo&longs;ius primo &longs;phæricorum, <lb/>propo&longs;itione 3. probat planum, &longs;iue &longs;uperficiem planam tangere &longs;phæram <lb/>in vnico puncto, vt hoc loco innuit Philo&longs;ophus. </s>

<s>probat autem hac ferè ra<lb/><figure place="text" xlink:href="figures-la/009.01.132.1.tif"/><lb/>tione. </s>

<s>probat autem hac ferè ra<lb/><figure id="id.009.01.132.1.jpg" xlink:href="009/01/132/1.jpg" place="text"/><lb/>tione. </s>

<s>&longs;it &longs;phæra A B C, quæ tangat quodpiam planum <lb/>in duobus punctis A, B, &longs;i fieri pote&longs;t. </s>

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<s>propterea nunc tantum propria huius loci <expan abbr="de-clarãda">de<lb/>claranda</expan> re&longs;tant. </s>

<s>loquitur igitur hic Philo &longs;ophus non de quadratione circuli, <pb pagenum="133"/>&longs;ed figuræ rectilineæ illius, quæ dicitur Altera parte longior, qualis e&longs;t præ<lb/>&longs;ens figura A B C D, cuius quadrandæ ratio e&longs;t huiu&longs;modi. </s>

<s>loquitur igitur hic Philo &longs;ophus non de quadratione circuli, <pb xlink:href="009/01/133.jpg"/>&longs;ed figuræ rectilineæ illius, quæ dicitur Altera parte longior, qualis e&longs;t præ<lb/>&longs;ens figura A B C D, cuius quadrandæ ratio e&longs;t huiu&longs;modi. </s>

<s>per 13. 6. inue<lb/><figure place="text" xlink:href="figures-la/009.01.133.1.tif"/><lb/>niatur recta linea media proportionalis inter <lb/>duo latera figuræ A B, B C, <expan abbr="ea&qacute;">eaque</expan>; &longs;it B D, in &longs;e<lb/>quenti figura. </s>

<s>per 13. 6. inue<lb/><figure id="id.009.01.133.1.jpg" xlink:href="009/01/133/1.jpg" place="text"/><lb/>niatur recta linea media proportionalis inter <lb/>duo latera figuræ A B, B C, <expan abbr="ea&qacute;">eaque</expan>; &longs;it B D, in &longs;e<lb/>quenti figura. </s>

<s>e&longs;&longs;e autem mediam proportio<lb/>nalem nihil aliud e&longs;t quam ita e&longs;&longs;e A B, ad B D, <lb/>&longs;icut B D, ad B C. <expan abbr="dicitur&qacute;">diciturque</expan>; media proportio<lb/>nalis, quia in hac habitudine medium locum obtinet. </s>

<s>quadratum autem li<lb/>neæ B D, æquale e&longs;t rectangulo dato A B C D, per 17.6. Inuentio porrò hu<lb/>ius mediæ proportionalis, quia facilis e&longs;t, & &longs;citu iucunda, eam &longs;ic habeto. <lb/><figure place="text" xlink:href="figures-la/009.01.133.2.tif"/><lb/>accipe duo latera A B, & B C, <expan abbr="quadrãdi">quadrandi</expan> rectan<lb/>guli, <expan abbr="ea&qacute;">eaque</expan>; in directum con&longs;titue, vt vnicam re<lb/>ctam con&longs;tituant A C, vt apparet in figura; de<lb/>inde diui&longs;a tota A C, bifariam in E, facto cen<lb/>tro in E, de&longs;cribe &longs;emicirculum &longs;uper lineam <lb/>A C, demum à puncto B, in quo duo latera con<lb/>iunguntur, erigatur linea perpendicularis <expan abbr="v&longs;q;">v&longs;que</expan> <lb/>ad periphæriam, quæ &longs;it B D. hæc enim B D, e&longs;t media proportionalis inter <lb/>latera A B, B D, quam nimirum habitudinem habet A B, ad B D, eam quo<lb/>que obtinet B D, ad B C. </s>

<s>quadratum autem li<lb/>neæ B D, æquale e&longs;t rectangulo dato A B C D, per 17.6. Inuentio porrò hu<lb/>ius mediæ proportionalis, quia facilis e&longs;t, & &longs;citu iucunda, eam &longs;ic habeto. <lb/><figure id="id.009.01.133.2.jpg" xlink:href="009/01/133/2.jpg" place="text"/><lb/>accipe duo latera A B, & B C, <expan abbr="quadrãdi">quadrandi</expan> rectan<lb/>guli, <expan abbr="ea&qacute;">eaque</expan>; in directum con&longs;titue, vt vnicam re<lb/>ctam con&longs;tituant A C, vt apparet in figura; de<lb/>inde diui&longs;a tota A C, bifariam in E, facto cen<lb/>tro in E, de&longs;cribe &longs;emicirculum &longs;uper lineam <lb/>A C, demum à puncto B, in quo duo latera con<lb/>iunguntur, erigatur linea perpendicularis <expan abbr="v&longs;q;">v&longs;que</expan> <lb/>ad periphæriam, quæ &longs;it B D. hæc enim B D, e&longs;t media proportionalis inter <lb/>latera A B, B D, quam nimirum habitudinem habet A B, ad B D, eam quo<lb/>que obtinet B D, ad B C. </s>

<s>Quadratum igitur huius B D, hoc e&longs;t quadratum, <lb/>cuius quatuor latera &longs;iut æqualia lineæ B D, quale e&longs;t præ&longs;ens, æquale erit <lb/><figure place="text" xlink:href="figures-la/009.01.133.3.tif"/><lb/>dato &longs;uperiori rectangulo A B C D, <expan abbr="atq;">atque</expan> hoc modo per<lb/>acta erit quadratio, &longs;eu tetragoni&longs;mus dati quadrilateri <lb/>A B C D. </s>

<s>Quadratum igitur huius B D, hoc e&longs;t quadratum, <lb/>cuius quatuor latera &longs;iut æqualia lineæ B D, quale e&longs;t præ&longs;ens, æquale erit <lb/><figure id="id.009.01.133.3.jpg" xlink:href="009/01/133/3.jpg" place="text"/><lb/>dato &longs;uperiori rectangulo A B C D, <expan abbr="atq;">atque</expan> hoc modo per<lb/>acta erit quadratio, &longs;eu tetragoni&longs;mus dati quadrilateri <lb/>A B C D. </s>

<s>Vides igitur, qua ratione quadratum con&longs;ti<lb/>tuatur æquale dato quadrilatero; & qua rationem inuen<lb/>tio illius mediæ proportionalis &longs;it cau&longs;a quadraturæ re<lb/>ctanguli, & proinde &longs;i quis dicat quadrationem hanc e&longs;&longs;e <lb/>effectionem rectanguli æquilateri, ide&longs;t quadrati, æqualis dato quadrilate<lb/>ro, hic definitionem formalem &longs;olum afferet: quæ definitio, vt dixit in Lo<lb/>gicis, e&longs;t in&longs;tar conclu&longs;ionis. </s>

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ibi dicat <lb/><emph type="italics"/>(Vox acuta quidem velox)<emph.end type="italics"/> hic autem <emph type="italics"/>(Non igitur velox e&longs;t acutum<emph.end type="italics"/>) repugnan<lb/>tia dicere videtur. </s>

<s>cui dubitationi &longs;ic occurrendum; vt dicamus ibi Philo<lb/>&longs;ophum dicere vocem acutam e&longs;&longs;e velocem, quatenus acumen vocis oritur <pb pagenum="134"/>ex velocitate motus aerem impellentis. </s>

<s>cui dubitationi &longs;ic occurrendum; vt dicamus ibi Philo<lb/>&longs;ophum dicere vocem acutam e&longs;&longs;e velocem, quatenus acumen vocis oritur <pb xlink:href="009/01/134.jpg"/>ex velocitate motus aerem impellentis. </s>

<s>hic verò di&longs;tinguere acutum à ve<lb/>loci, tanquam effectum à cau&longs;a.</s>

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<s>ex quibus <lb/>&longs;en&longs;us verberum Ari&longs;t.

manife&longs;tus e&longs;t.</s><pb pagenum="135"/>

manife&longs;tus e&longs;t.</s><pb xlink:href="009/01/135.jpg"/>

<s>Eodem cap. <emph type="italics"/>(Velut Diapa&longs;on, & Diapente)<emph.end type="italics"/> quid &longs;it con&longs;onantia Diapa<lb/>&longs;on, explicaui in primo Po&longs;ter. tex. 1. Diapente verò e&longs;t con&longs;onantia ex duo<lb/><figure place="text" xlink:href="figures-la/009.01.135.1.tif"/><lb/>bus &longs;onis coale&longs;cens, quorum proportio &longs;it vt <lb/>3. ad 2. quæ dicitur &longs;e&longs;quialtera. </s>

<s>Eodem cap. <emph type="italics"/>(Velut Diapa&longs;on, & Diapente)<emph.end type="italics"/> quid &longs;it con&longs;onantia Diapa<lb/>&longs;on, explicaui in primo Po&longs;ter. tex. 1. Diapente verò e&longs;t con&longs;onantia ex duo<lb/><figure id="id.009.01.135.1.jpg" xlink:href="009/01/135/1.jpg" place="text"/><lb/>bus &longs;onis coale&longs;cens, quorum proportio &longs;it vt <lb/>3. ad 2. quæ dicitur &longs;e&longs;quialtera. </s>

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<s>Cap. 3. <emph type="italics"/>(Quemadmodum igitur, &longs;i quem lateat &longs;uppo&longs;itus oculo digitus, non <lb/>&longs;olum app trebit, &longs;ed etiam putabitur duo, quod e&longs;t vnum. </s>

<s>Si verò non lateat appa<lb/>rebit quidem, non putabitur tamen)<emph.end type="italics"/> e&longs;t hæc optica deceptio, quæ tunc accidit, <lb/>cum aliquod obiectum intuentes, interim digito alterum oculum &longs;ur&longs;um <lb/>pellimus, ita vt oculi propterea varient &longs;itum re&longs;pectu obiecti, &longs;iue non eo<pb pagenum="136"/>dem &longs;itu vterque obiectum intueatur, hoc e&longs;t, vt optici aiunt, axes vi&longs;uales <lb/>non amplius concurrunt &longs;imul in rem vi&longs;am. </s>

<s>Si verò non lateat appa<lb/>rebit quidem, non putabitur tamen)<emph.end type="italics"/> e&longs;t hæc optica deceptio, quæ tunc accidit, <lb/>cum aliquod obiectum intuentes, interim digito alterum oculum &longs;ur&longs;um <lb/>pellimus, ita vt oculi propterea varient &longs;itum re&longs;pectu obiecti, &longs;iue non eo<pb xlink:href="009/01/136.jpg"/>dem &longs;itu vterque obiectum intueatur, hoc e&longs;t, vt optici aiunt, axes vi&longs;uales <lb/>non amplius concurrunt &longs;imul in rem vi&longs;am. </s>

<s>Vnde &longs;equitur &longs;peciem rei in<lb/>tentionalem oculis vario &longs;itu affectis imprimi, ac proinde eam eundem &longs;i<lb/>tum in vtroque oculo minimè obtinere, &longs;ed ea, quæ oculo à &longs;uo naturali <lb/>&longs;latu dimoto accidit ab altera alterius oculi differt; quapropter vario <lb/>ctiam modo, duplici nimirum, obiectum repre&longs;entant. </s>

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3. propo&longs;it. </s>

<s>11. & 12. & infra <lb/>Problem. 7. &longs;ectionis 31.</s><pb pagenum="137"/>

<s>11. & 12. & infra <lb/>Problem. 7. &longs;ectionis 31.</s><pb xlink:href="009/01/137.jpg"/>

<s>EX PRIMO</s>

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<s>1. <lb/>cap.

1. Videtur inquit mirum à principio Geometriam aggredienti diame<lb/>trum, & latus eiu&longs;dem quadrati non commen&longs;urari, cum in neutro eorum <lb/>detur minimum, &longs;eu indiui&longs;ibile, videtur enim omne diui&longs;ibile po&longs;&longs;e men&longs;u<pb pagenum="138"/>rari. </s>

<s>po&longs;tea tamen cum in Geometria ver&longs;atus fuerit, maximè admirare<lb/>tur, &longs;i audiret diametrum e&longs;&longs;e lateri commen&longs;urabilem.</s>

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<s>Arithmeti<lb/>ca, cuius v&longs;us quàm latè patet? </s>

<s>Mu&longs;ica practica, qua quotidie ip&longs;i oblecta<pb pagenum="139"/>mur; <expan abbr="Deo&qacute;">Deoque</expan>; Optimo Maximo laudes debitas concinimus. </s>

<s>Mu&longs;ica practica, qua quotidie ip&longs;i oblecta<pb xlink:href="009/01/139.jpg"/>mur; <expan abbr="Deo&qacute;">Deoque</expan>; Optimo Maximo laudes debitas concinimus. </s>

<s>Mechanica pra<lb/>ctica, cuius op<gap/>gentia pondera, vel exigua vi, <expan abbr="inuita&qacute;">inuitaque</expan>; natura &longs;u&longs;g; <expan abbr="de&qacute;">deque</expan>; <lb/>commouentur. </s>

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<s>Tex. 28. (<emph type="italics"/>Vti diametrum commen&longs;urabilem e&longs;&longs;e<emph.end type="italics"/>) legenda &longs;unt ea, quæ libro <lb/>primo Priorum, &longs;ecto 1. cap.

23. de hac commen&longs;urabilitate, & incommen<lb/>&longs;urabilitate tractata &longs;unt.</s><pb pagenum="140"/>

23. de hac commen&longs;urabilitate, & incommen<lb/>&longs;urabilitate tractata &longs;unt.</s><pb xlink:href="009/01/140.jpg"/>

<s><emph type="italics"/>Ex Quinto Metaphy&longs;icæ.<emph.end type="italics"/></s>

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explicatum e&longs;t.</s>

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<s>Tex. 17. <emph type="italics"/>(Veluti diametrum commen&longs;urabilem e&longs;&longs;e impo&longs;&longs;ibile est)<emph.end type="italics"/> huius expo<lb/>&longs;itionem inuenies 1. Priorum, &longs;ecto 1. cap.

23.</s><figure place="text" xlink:href="figures-la/009.01.141.1.tif"/>

23.</s><figure id="id.009.01.141.1.jpg" xlink:href="009/01/141/1.jpg" place="text"/>

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<s>Tex. 20. (<emph type="italics"/>De&longs;eriptiones <expan abbr="quoq;">quoque</expan> actu inueniuntur, diuidentes nanque <lb/>inuenirent, quod &longs;i diui&longs;æ e&longs;&longs;ent, manife&longs;i è e&longs;&longs;ent, nunc autem in&longs;unt potentia, cur <lb/>triangulus duo recti? </s>

<s>quia qui circa vnum punctum anguli duobus rectis æquales<emph.end type="italics"/><pb pagenum="142"/><emph type="italics"/>&longs;unt, &longs;i igitur quæ ad latus educeretur, videnti mox e&longs;&longs;et manife&longs;tum<emph.end type="italics"/>) per de&longs;cri<lb/>ptiones, vel figurationes, vel de&longs;ignationes intelligendas e&longs;&longs;e demon&longs;tra<lb/>tiones Geometricas &longs;æpius &longs;upra dictum e&longs;t, & pariter ex hoc loco com<lb/>probatur. </s>

<s>quia qui circa vnum punctum anguli duobus rectis æquales<emph.end type="italics"/><pb xlink:href="009/01/142.jpg"/><emph type="italics"/>&longs;unt, &longs;i igitur quæ ad latus educeretur, videnti mox e&longs;&longs;et manife&longs;tum<emph.end type="italics"/>) per de&longs;cri<lb/>ptiones, vel figurationes, vel de&longs;ignationes intelligendas e&longs;&longs;e demon&longs;tra<lb/>tiones Geometricas &longs;æpius &longs;upra dictum e&longs;t, & pariter ex hoc loco com<lb/>probatur. </s>

<s>Dicit igitur, quod demon&longs;trationes &longs;uas Geometræ inueniunt, <lb/>reducendo ad actum ea, quæ erant in potentia, diuidentes enim educunt in <lb/>actum, figuras, angulos, lineas, & cætera huiu&longs;inodi, quæ prius &longs;olùm erat <lb/>in potentia, ex quibus po&longs;tea &longs;uas demon&longs;trationes perficiunt (<emph type="italics"/>Cur triangu<lb/>lus duo recti<emph.end type="italics"/>) affert exemplum eius, quod proximè dixerat, &longs;cilicet Geome<lb/>tras demon&longs;trare producendo ad actum entia quædam Mathematica, quod <lb/>exemplum, vt intelligas ijs opus habes, quæ primo Priorum, &longs;ecto 3. cap.

1. <lb/>con&longs;cripta &longs;unt (<emph type="italics"/>Cur triangulus duo recti?<emph.end type="italics"/>) ide&longs;t, cur triangulus habet tres <lb/>angulos æquales duobus rectis angulis (<emph type="italics"/>Quia qui circa vnum punctum anguli <lb/>duobus rectis angulis æquales &longs;unt<emph.end type="italics"/>) ni&longs;i hoc dictum ad bonum trahatur &longs;en&longs;um, <lb/><figure place="text" xlink:href="figures-la/009.01.142.1.tif"/><lb/>fal&longs;um e&longs;t, nam omnes anguli, qui circa vnum <lb/>punctum, v. g. A, &longs;unt con&longs;tituti, æquales &longs;unt <lb/>non duobus, vt e&longs;t in textu, &longs;ed quatuor rectis, <lb/>vt patet ex corollario 2. 15. primi Elem.

1. <lb/>con&longs;cripta &longs;unt (<emph type="italics"/>Cur triangulus duo recti?<emph.end type="italics"/>) ide&longs;t, cur triangulus habet tres <lb/>angulos æquales duobus rectis angulis (<emph type="italics"/>Quia qui circa vnum punctum anguli <lb/>duobus rectis angulis æquales &longs;unt<emph.end type="italics"/>) ni&longs;i hoc dictum ad bonum trahatur &longs;en&longs;um, <lb/><figure id="id.009.01.142.1.jpg" xlink:href="009/01/142/1.jpg" place="text"/><lb/>fal&longs;um e&longs;t, nam omnes anguli, qui circa vnum <lb/>punctum, v. g. A, &longs;unt con&longs;tituti, æquales &longs;unt <lb/>non duobus, vt e&longs;t in textu, &longs;ed quatuor rectis, <lb/>vt patet ex corollario 2. 15. primi Elem.

quot<lb/>quot enim anguli con&longs;tituantur ad punctum A, <lb/>omnes &longs;imul erunt æquales quatuor rectis, quos <lb/>faciunt præ&longs;entes lineæ B C, D E. vniuer&longs;i enim <lb/>illi congruent his quatuor rectis: &longs;ed Ari&longs;t. &longs;en<lb/>&longs;us e&longs;t omnes angulos ad ea&longs;dem partes con&longs;ti<lb/>tutos, v. g. ad partes &longs;uperiores lineæ B C, e&longs;&longs;e <lb/>æquales duobus rectis B A D, D A C, vt o&longs;tenditur in 13. primi, necnon <lb/>etiam patere pote&longs;t ex corollario 2. 15. eiu&longs;dem. </s>

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<s>tales &longs;unt quatuor anguli <lb/>ad &longs;uperiores partes lineæ B C, & ad punctum A, con&longs;tituti, qui, vt patet, <lb/><figure place="text" xlink:href="figures-la/009.01.142.2.tif"/><lb/>&longs;unt æquales duobus rectis B A D, D A C, <lb/>tales etiam &longs;unt in hac &longs;ecunda figura tres <lb/>anguli B C A, A C D, D C E, qui quidem <lb/>æquales &longs;unt duobus rectis angulis. </s>

<s>tales &longs;unt quatuor anguli <lb/>ad &longs;uperiores partes lineæ B C, & ad punctum A, con&longs;tituti, qui, vt patet, <lb/><figure id="id.009.01.142.2.jpg" xlink:href="009/01/142/2.jpg" place="text"/><lb/>&longs;unt æquales duobus rectis B A D, D A C, <lb/>tales etiam &longs;unt in hac &longs;ecunda figura tres <lb/>anguli B C A, A C D, D C E, qui quidem <lb/>æquales &longs;unt duobus rectis angulis. </s>

<s>hoc <lb/>&longs;en&longs;i&longs;&longs;e Ari&longs;t.

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<s>nunc &longs;olùm <pb pagenum="143"/><figure place="text" xlink:href="figures-la/009.01.143.1.tif"/><lb/>hæc addenda &longs;unt. </s>

<s>nunc &longs;olùm <pb xlink:href="009/01/143.jpg"/><figure id="id.009.01.143.1.jpg" xlink:href="009/01/143/1.jpg" place="text"/><lb/>hæc addenda &longs;unt. </s>

<s>Re&longs;pondet Ari&longs;t.

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<s>quapropter in A&longs;trologia tale vn&ubreve; prin<lb/>cipium, & men&longs;ura e&longs;t. </s>

<s>motum <gap/> æqualem, & veloci&longs;&longs;ini&ubreve; &oelig;li &longs;up<lb/>ponunt, ad quem cæteros tudicant<emph.end type="italics"/>) intelligit motum diureum, quam <lb/>primo c&oelig;lo, &longs;eu mobili a&longs;eribunt, hic enim veloci&longs;&longs;imus e&longs;t omnium reli<pb pagenum="144"/>quorum c&oelig;le&longs;tium motuum, ac &longs;impliciffimus, & valdè vniformis, ac regu<lb/>laris, & propterea minimum habet tempus, ide&longs;t tempus vnius diei natura<lb/>lis, quo tempore totum primum mobile circulationem integram perficit. <lb/></s>

<s>motum <gap/> æqualem, & veloci&longs;&longs;ini&ubreve; &oelig;li &longs;up<lb/>ponunt, ad quem cæteros tudicant<emph.end type="italics"/>) intelligit motum diureum, quam <lb/>primo c&oelig;lo, &longs;eu mobili a&longs;eribunt, hic enim veloci&longs;&longs;imus e&longs;t omnium reli<pb xlink:href="009/01/144.jpg"/>quorum c&oelig;le&longs;tium motuum, ac &longs;impliciffimus, & valdè vniformis, ac regu<lb/>laris, & propterea minimum habet tempus, ide&longs;t tempus vnius diei natura<lb/>lis, quo tempore totum primum mobile circulationem integram perficit. <lb/></s>

<s>per minimum tempus, po&longs;&longs;uut etiam intelligi partes diei, quæ &longs;unt horæ, & <lb/>horarum partes. </s>

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<s>cum ergò quadratum maius, & minus &longs;int æquiangula, <lb/>quia habent omnes angulos rectos; & præterea habeant latera circa æqua<lb/>les angulos proportionalia, &longs;icut enim latera maioris quadrati circa vnum <lb/>angulum rectum &longs;unt in proportione æqualitatis; ita <expan abbr="quoq;">quoque</expan> latera minoris <lb/>circa vnum angulum rectum &longs;unt illis proportionalia, cum &longs;int inuicem pa<lb/>riter in proportione æqualitatis, erunt nece&longs;&longs;ariò &longs;imilia hæc duo quadrata. <lb/></s>

<s>duæ ctiam, exempli gratia, lineæ rectæ &longs;unt inuicem &longs;imiles, quamuis vna <lb/>&longs;it maior altera.</s><pb pagenum="145"/>

<s>duæ ctiam, exempli gratia, lineæ rectæ &longs;unt inuicem &longs;imiles, quamuis vna <lb/>&longs;it maior altera.</s><pb xlink:href="009/01/145.jpg"/>

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<s>In Luna pariter plures nece&longs;&longs;arios illi orbes ad motus <lb/>ip&longs;ius &longs;aluandos prætermittit. </s>

<s>Ex &longs;ententia tamen Tychonis Brahe hos or<lb/>bes, ac circulos tanquam ab inuicem di&longs;tinctos abrogare debemus.</s><pb pagenum="146"/>

<s>Ex &longs;ententia tamen Tychonis Brahe hos or<lb/>bes, ac circulos tanquam ab inuicem di&longs;tinctos abrogare debemus.</s><pb xlink:href="009/01/146.jpg"/>

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<s>pluralitas <expan abbr="itaq;">itaque</expan> &longs;phærarum tanta &longs;it)<emph.end type="italics"/> textum hunc <lb/>per paraphra&longs;im &longs;ic explico; Calippus igitur eundem quidem ordinem, at<lb/>que di&longs;tantiam &longs;phærarum cum Eudoxo ponebat: <expan abbr="eandem&qacute;">eandemque</expan>; pluralitatem <lb/>orbium mouentium Saturnum, ac Jouem; quatuor <expan abbr="nimir&utilde;">nimirum</expan> <expan abbr="vnicuiq;">vnicuique</expan> eorum. <lb/></s>

<s>&longs;ed putabat &longs;oli duas addendas, ac Lunæ &longs;imiliter, &longs;i quis eorum <expan abbr="appar&etilde;tias">apparentias</expan> <pb pagenum="147"/>&longs;aluare vellet. </s>

<s>&longs;ed putabat &longs;oli duas addendas, ac Lunæ &longs;imiliter, &longs;i quis eorum <expan abbr="appar&etilde;tias">apparentias</expan> <pb xlink:href="009/01/147.jpg"/>&longs;aluare vellet. </s>

<s>cæteris verò errantium, Marti, Veneri, & Mercurio <expan abbr="vnicuiq;">vnicuique</expan> <lb/>vnam. </s>

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<s>nam & &longs;i non <lb/>nominant, quia tamen opera, & rationes ostendunt, non ne dicunt de eis? <lb/></s>

<s>pulchra <expan abbr="namq;">namque</expan> maximè &longs;pecies &longs;unt, ordo, commen&longs;uratio, & definu&ubreve;, quæ <lb/>maximè à Mathematicis &longs;cientijs o&longs;tenduntur, &c.)<emph.end type="italics"/> placuit hæc in Mathemati<lb/>carum commendationem, ac defen&longs;ionem apponere, cum non de&longs;int hac <lb/>no&longs;tra tempe&longs;tate ageometrcti complures, qui cas libenter &longs;ugillare &longs;olent.</s><pb pagenum="148"/>

<s><emph type="italics"/>IN MECHANICAS QVÆSTIONES.<emph.end type="italics"/></s>

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<s>Mathematicæ enim mediæ, de quorum nu<lb/>mero e&longs;t facultas Mechanica, con&longs;iderant quantitatem rei alicuius <lb/>determinatæ, &longs;ic A&longs;tronomia circa c&oelig;le&longs;tium corporum, <expan abbr="mo-tuum&qacute;">mo<lb/>tuumque</expan>; quantitates, Per&longs;pectiua circa linearum vi&longs;ua<lb/>lium; Mu&longs;ica circa &longs;onorum quantitates ver<lb/>&longs;antur. </s>

<s>quæ placuit annotare, vt &longs;cien<lb/>tiæ huius naturam per&longs;pectam <lb/>haberemus.</s><pb pagenum="149"/>

<s>quæ placuit annotare, vt &longs;cien<lb/>tiæ huius naturam per&longs;pectam <lb/>haberemus.</s><pb xlink:href="009/01/149.jpg"/>

<s>De dignitatibus, <expan abbr="admirandis&qacute;">admirandisque</expan>; Circuli proprietatibus.</s>

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<s>Admiranda igitur ea, quæ circa li<lb/>bram &longs;iunt, ad circulum <expan abbr="tãquam">tanquam</expan> cau&longs;am referuntur, quæ verò circa vectem <lb/>ad ip&longs;am libram: alia autem ferè omnia, quæ circa mechanicas contingunt <lb/>motiones, ad vectem reducuntur.</s>

<s>Præter prædicta aliud tandem mirum ip&longs;i ine&longs;t, quia nimirum cum innu<lb/>mera &longs;int puncta in vna <expan abbr="eadem&qacute;">eademque</expan>; linea, quæ &longs;emidiameter e&longs;t, omnia tamen <pb pagenum="150"/>quando &longs;emidiameter circa centrum mouetur, quamuis cum ip&longs;a mouean<lb/>tur, inæquali veiocitate mouentur; Nam punctum illud &longs;emper velocius <lb/>mouetur, quod remotius e&longs;t à centro circuli, &longs;eu à manente &longs;emidiametri <lb/>termino, & proinde illud tardius, quod centro proximius e&longs;t. </s>

<s>Præter prædicta aliud tandem mirum ip&longs;i ine&longs;t, quia nimirum cum innu<lb/>mera &longs;int puncta in vna <expan abbr="eadem&qacute;">eademque</expan>; linea, quæ &longs;emidiameter e&longs;t, omnia tamen <pb xlink:href="009/01/150.jpg"/>quando &longs;emidiameter circa centrum mouetur, quamuis cum ip&longs;a mouean<lb/>tur, inæquali veiocitate mouentur; Nam punctum illud &longs;emper velocius <lb/>mouetur, quod remotius e&longs;t à centro circuli, &longs;eu à manente &longs;emidiametri <lb/>termino, & proinde illud tardius, quod centro proximius e&longs;t. </s>

<s><expan abbr="Atq;">Atque</expan> ex hac <lb/>mira circuli proprietate, <expan abbr="pleraq;">pleraque</expan> miraculorum accidunt circuli motioni<lb/>bus, vt in &longs;equentibus quæ&longs;tionibus manife&longs;tum erit.</s>

<s>Quoniam autem &longs;ecundum contrarias &longs;imul motiones mouetur circulus, <lb/>& alterum quidem diametri extremum vbi A, in figura præ&longs;enti antror&longs;um <lb/><figure place="text" xlink:href="figures-la/009.01.150.1.tif"/><lb/>mouetur; alterum verò vbi B, retror<lb/>&longs;um, efficiunt nonnulli, vt ab vnica mo<lb/>tione multi contrariò &longs;imul mouean<lb/>tur denticulati circuli: vt &longs;unt ij, quos <lb/>in locis proponunt &longs;acris, quorum alij <lb/>&longs;unt ænei, alij ferrei. </s>

<s>Quoniam autem &longs;ecundum contrarias &longs;imul motiones mouetur circulus, <lb/>& alterum quidem diametri extremum vbi A, in figura præ&longs;enti antror&longs;um <lb/><figure id="id.009.01.150.1.jpg" xlink:href="009/01/150/1.jpg" place="text"/><lb/>mouetur; alterum verò vbi B, retror<lb/>&longs;um, efficiunt nonnulli, vt ab vnica mo<lb/>tione multi contrariò &longs;imul mouean<lb/>tur denticulati circuli: vt &longs;unt ij, quos <lb/>in locis proponunt &longs;acris, quorum alij <lb/>&longs;unt ænei, alij ferrei. </s>

<s>&longs;i enim circulus <lb/>A B, alterum circulum C D, contige<lb/>rit, mota diametro A B, ita vt A, an<lb/>tror&longs;um eat, commouebit alteram dia<lb/>metrum C D, ita vt C, retror&longs;um, hoc e&longs;t in contrarium ip&longs;i A, veniat, in <lb/>contrarium igitur mouebitur &longs;ecundus circulus C D, ad circulum A B, & <lb/>rur&longs;us circulus E F. in contrarium ip&longs;i C D, commouebitur ab ip&longs;o C D, ob <lb/>eandem rationem. </s>

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<s>horum autem cau<lb/>&longs;a, quoniam quæ circulum de&longs;cribit linea, duabus fertur lationibus, quæ nul<lb/>lam inuicem obtinent analogiam: quod antequam probemus, &longs;ciendum <lb/>e&longs;t, quod, quidquid duobus motibus inuicem proportionatis, mouetur, ne<lb/>ce&longs;&longs;e e&longs;t, quod motu exillis mixto progrediatur per lineam rectam, quæ dia<lb/>meter e&longs;t quadrilateri, cuius latera habeant illam proportionem, quam <lb/>duo illi motus. </s>

<s>&longs;it enim in figura proportio lateris A B, ad latus A C, quam <lb/>ctiam habent duo motus, &longs;ecundum quos latum quodpiam feratur, <expan abbr="&longs;it&qacute;">&longs;itque</expan>; la<lb/>tum illud A, & feratur motu vno ver&longs;us B, per lineam A B, altero verò mo<lb/>tu feratur ver&longs;us C. quod fiet &longs;i cogitemus latus A B, <expan abbr="de&longs;c&etilde;dere">de&longs;cendere</expan> ver&longs;us M C, <pb pagenum="151"/><figure place="text" xlink:href="figures-la/009.01.151.1.tif"/><lb/>ip&longs;i æquidi&longs;tanter, dum punctum A, mouetnr <lb/>ad B. his duabus lationibus A, latum. </s>

<s>&longs;it enim in figura proportio lateris A B, ad latus A C, quam <lb/>ctiam habent duo motus, &longs;ecundum quos latum quodpiam feratur, <expan abbr="&longs;it&qacute;">&longs;itque</expan>; la<lb/>tum illud A, & feratur motu vno ver&longs;us B, per lineam A B, altero verò mo<lb/>tu feratur ver&longs;us C. quod fiet &longs;i cogitemus latus A B, <expan abbr="de&longs;c&etilde;dere">de&longs;cendere</expan> ver&longs;us M C, <pb xlink:href="009/01/151.jpg"/><figure id="id.009.01.151.1.jpg" xlink:href="009/01/151/1.jpg" place="text"/><lb/>ip&longs;i æquidi&longs;tanter, dum punctum A, mouetnr <lb/>ad B. his duabus lationibus A, latum. </s>

<s>nece&longs;&longs;a<lb/>niò motu mixto progre dietur per diametrum <lb/>A M, quod &longs;ic probari pote&longs;t; &longs;it iam A, mo<lb/>tum primo motu <expan abbr="v&longs;q;">v&longs;que</expan> ad D, linea verò ex &longs;e<lb/>cundo motu &longs;it in G F E, quo motu punctum <lb/>A, quod erat in D, <expan abbr="tract&utilde;">tractum</expan> erit in F. quod pun<lb/>ctum e&longs;t in diametro A M, quoniam enim mo<lb/>uetur duobus motibus, cum lineis A B, A C, proportionalibus, motus au<lb/>tem <expan abbr="hucu&longs;q;">hucu&longs;que</expan> &longs;unt A D, A E, quæ debent e&longs;&longs;e proportionales, cum A B, A C <lb/>compleatur rectangulum A D F E, erunt &longs;imiliter proportionalia F E, D E, <lb/>cum &longs;int æqualia duobus D A, A E, quare per 26. 6. cum quadrilaterum <lb/>paruum A D F E, &longs;it &longs;imile toti A B M C, erit A M, <expan abbr="vtriu&longs;q;">vtriu&longs;que</expan> diameter, ergò <lb/>punctum F, in quo e&longs;t A, e&longs;t in diametro A M. eodem modo, de quouis pun<lb/>cto in linea A B, ad quod A, perueniat, probabitur ab altero motu de&longs;cen<lb/>di&longs;&longs;e v&longs;que ad diametrum. </s>

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<s>&longs;i enim dixeris illud po&longs;&longs;e de&longs;cribere rectam lineam, tunc circa rectam il<lb/>lam tanquam diametrum de&longs;cribam quadrilaterum, & po&longs;tea o&longs;tendam, vt <lb/>proximè o&longs;ten&longs;um e&longs;t, illud latum e&longs;&longs;e &longs;ecundum laterum illius proportio<lb/>nem, quare impo&longs;&longs;ibile e&longs;t id, quod mouetur duabus lationibus nullam in<lb/>uicem rationem habentibus, ferri per lineam rectam: quapropter <expan abbr="dic&etilde;dum">dicendum</expan> <lb/>e&longs;t hoc modo <expan abbr="lat&utilde;">latum</expan>, nece&longs;&longs;ariò ferri circulariter, &longs;iue per lineam circularem. <lb/></s>

<s>Quod autemea, quæ de&longs;cribit circulum linea, dum altero eins manente <lb/>extremo circumagitur, duabus &longs;imul feratur lationibus, ex quibus motus <lb/>orbicularis oriatur, manife&longs;tum e&longs;t ex &longs;uperioribus, quia & antror&longs;um, & <lb/>retror&longs;um impellitur; tùm etiam, quia &longs;i rectà tenderet recta <expan abbr="de&longs;crib&etilde;s">de&longs;cribens</expan> cir<lb/><figure place="text" xlink:href="figures-la/009.01.151.2.tif"/><lb/>culum, nunquam ad diametri perpendiculum <lb/>perueniret, &longs;ed tamen peruenit, ita vt &longs;it ip&longs;a <lb/>à centro perpendicularis diametro. </s>

<s>Quod autemea, quæ de&longs;cribit circulum linea, dum altero eins manente <lb/>extremo circumagitur, duabus &longs;imul feratur lationibus, ex quibus motus <lb/>orbicularis oriatur, manife&longs;tum e&longs;t ex &longs;uperioribus, quia & antror&longs;um, & <lb/>retror&longs;um impellitur; tùm etiam, quia &longs;i rectà tenderet recta <expan abbr="de&longs;crib&etilde;s">de&longs;cribens</expan> cir<lb/><figure id="id.009.01.151.2.jpg" xlink:href="009/01/151/2.jpg" place="text"/><lb/>culum, nunquam ad diametri perpendiculum <lb/>perueniret, &longs;ed tamen peruenit, ita vt &longs;it ip&longs;a <lb/>à centro perpendicularis diametro. </s>

<s>&longs;it circuli <lb/>figura A B C D, in qua extremum diametri <lb/>B, feratur ad alterum extremum vbi D, per <lb/>ip&longs;ius diametri B D, circumuolutionem circa <lb/>centrum F, nece&longs;&longs;e e&longs;t aliquando B, perueniat <lb/>ad C. &longs;i igitur B, feretur duabus lationibus <lb/>aliquo modo proportionatis, v. g. vt e&longs;t pro<lb/>portio lateris B E, ad E C, latus, &longs;equeretur <lb/>ex demon&longs;tratis ip&longs;um B, ferri per <expan abbr="rectã">rectam</expan> B C, <lb/>quæ diameter e&longs;&longs;et quadrilateri B E C G. &longs;ed <pb pagenum="152"/>quia in nu&longs;la proportione fertur, propterea per circularem lineam B E C, <lb/>progreditur ad C, ita vt ip&longs;a diameter B D, in po&longs;itione A C, fiat perpendi<lb/>cularis priori diametro B D. ex quibus &longs;equitur eam moueri duobus moti<lb/>bus nullam rationem habentibus; quod erat intentum.</s>

<s>&longs;it circuli <lb/>figura A B C D, in qua extremum diametri <lb/>B, feratur ad alterum extremum vbi D, per <lb/>ip&longs;ius diametri B D, circumuolutionem circa <lb/>centrum F, nece&longs;&longs;e e&longs;t aliquando B, perueniat <lb/>ad C. &longs;i igitur B, feretur duabus lationibus <lb/>aliquo modo proportionatis, v. g. vt e&longs;t pro<lb/>portio lateris B E, ad E C, latus, &longs;equeretur <lb/>ex demon&longs;tratis ip&longs;um B, ferri per <expan abbr="rectã">rectam</expan> B C, <lb/>quæ diameter e&longs;&longs;et quadrilateri B E C G. &longs;ed <pb xlink:href="009/01/152.jpg"/>quia in nu&longs;la proportione fertur, propterea per circularem lineam B E C, <lb/>progreditur ad C, ita vt ip&longs;a diameter B D, in po&longs;itione A C, fiat perpendi<lb/>cularis priori diametro B D. ex quibus &longs;equitur eam moueri duobus moti<lb/>bus nullam rationem habentibus; quod erat intentum.</s>

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<s>quoniam <lb/>enim propius e&longs;t manenti eius, quæ minor e&longs;t extremum, quàm extremum <lb/>maioris, propterea plus à centro, cui propius e&longs;t, retrahitur à priori mo<lb/>tu, <expan abbr="hinc&qacute;">hincque</expan>; motus eius tardior redditur, ide&longs;t, quia centro propius e&longs;t; hinc <lb/>fit, vt extremum illud de&longs;cribat lineam circularem quidem, &longs;ed tamen <lb/>curuiorem quam de&longs;cribat extremum longioris lineæ, quæ circulum minus <lb/>curuum, &longs;eu magis ad rectam lineam accedentem delineat. </s>

<s>omni quidem <lb/>igitur lineæ circulum de&longs;cribenti i&longs;tud accidit, vt duobus feratur motioni<lb/>bus; vna quidem, quæ illi naturalis, ac &longs;ecundum circunferentiam, qua re<lb/>ctà tenderet ni&longs;i impediretur: altera verò, quæ illi innaturalis, qua in tran&longs;<lb/>uer&longs;um agitur, &longs;eu &longs;ecus centrum, ob quam cogitur in gyrum duci, minor <lb/>autem linea &longs;ecundum hanc motionem innaturalem plus fertur, quàm ma<lb/>ior, ide&longs;t plus ip&longs;ius progre&longs;&longs;io inflectitur in orbem; quia enim e&longs;t centro <lb/><figure place="text" xlink:href="figures-la/009.01.152.1.tif"/><lb/>vicinior, quod quodammodo retra<lb/>hit à motu naturali, propterea ma<lb/>gis vincitur, quàm remotior. </s>

<s>omni quidem <lb/>igitur lineæ circulum de&longs;cribenti i&longs;tud accidit, vt duobus feratur motioni<lb/>bus; vna quidem, quæ illi naturalis, ac &longs;ecundum circunferentiam, qua re<lb/>ctà tenderet ni&longs;i impediretur: altera verò, quæ illi innaturalis, qua in tran&longs;<lb/>uer&longs;um agitur, &longs;eu &longs;ecus centrum, ob quam cogitur in gyrum duci, minor <lb/>autem linea &longs;ecundum hanc motionem innaturalem plus fertur, quàm ma<lb/>ior, ide&longs;t plus ip&longs;ius progre&longs;&longs;io inflectitur in orbem; quia enim e&longs;t centro <lb/><figure id="id.009.01.152.1.jpg" xlink:href="009/01/152/1.jpg" place="text"/><lb/>vicinior, quod quodammodo retra<lb/>hit à motu naturali, propterea ma<lb/>gis vincitur, quàm remotior. </s>

<s>Quod <lb/>ex his erit <expan abbr="manife&longs;t&utilde;">manife&longs;tum</expan>. </s>

<s>&longs;it circulus vbi <lb/>B C E D, & alter in eo minor, vbi <lb/>N M O P, circa idem centrum A. & <lb/>proijciantur diametri in magno qui<lb/>dem C D, B E, in minori verò M O, <lb/>N P. & altera parte longius quadri<lb/>laterum compleatur D K R C. &longs;i igi<lb/>tur &longs;emidiameter A B, circumacta <lb/>de&longs;cribit circulum maiorem, reuer<lb/>titur tandem ad locum B A, vnde di<lb/>gre&longs;&longs;a e&longs;t. </s>

<s>&longs;imiliter M A, circumuoluta <pb pagenum="153"/>redibit ad priorem po&longs;itionem in M A. </s>

<s>&longs;imiliter M A, circumuoluta <pb xlink:href="009/01/153.jpg"/>redibit ad priorem po&longs;itionem in M A. </s>

<s>Tardius autem fertur M A, quàm <lb/>B A, vt dictum e&longs;t, quia maior illi fit retractio à recta progre&longs;&longs;ione. </s>

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quo putat &longs;e cau&longs;am aperui&longs;&longs;e, cur lon<lb/>gior &longs;emidiameter velocius moueatur: quod num rectè attigerit, non puto <lb/>operæpretium e&longs;&longs;e hoc loco di&longs;cutere, præ&longs;ertim cum ad naturalem Philo<lb/>&longs;ophum &longs;pectet.</s>

<s>Mihi tamen maximè con&longs;iderandum videtur hoc ip&longs;um quod a&longs;&longs;eruit, & <pb pagenum="154"/>ex &longs;e patet, remotiores &longs;cilicet partes diametrorum à centro velocius mo<lb/>ueri, quàm viciniores; ex hac enim maiori velocitate &longs;equitur maiore etiam <lb/>vi moueri, vnde & potentiæ mouenti in extremo eius vis augebitur, & plus <lb/>poterit quam &longs;ola &longs;ine vecte, e&longs;t enim vectis duæ &longs;emidiametri altera alte<lb/>ram longior; ex quibus fortè apparet vnde vectis vires oriantur.</s>

<s>Mihi tamen maximè con&longs;iderandum videtur hoc ip&longs;um quod a&longs;&longs;eruit, & <pb xlink:href="009/01/154.jpg"/>ex &longs;e patet, remotiores &longs;cilicet partes diametrorum à centro velocius mo<lb/>ueri, quàm viciniores; ex hac enim maiori velocitate &longs;equitur maiore etiam <lb/>vi moueri, vnde & potentiæ mouenti in extremo eius vis augebitur, & plus <lb/>poterit quam &longs;ola &longs;ine vecte, e&longs;t enim vectis duæ &longs;emidiametri altera alte<lb/>ram longior; ex quibus fortè apparet vnde vectis vires oriantur.</s>

<s>His igitur tanquam huius Mechanicæ facultatis principijs po&longs;itis, ad va<lb/>rias Quæ&longs;tiones di&longs;cutiendas accedit.</s>

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<s>Cvr autem maiores libræ minoribus &longs;int exactiores, palàm e&longs;t ex <lb/>præmi&longs;&longs;is principijs. </s>

<s>con&longs;iderare enim oportet, quod in motu li<lb/>bræ de&longs;cribitur quidam circulus, cuius diameter &longs;unt ip&longs;a libræ <lb/>brachia, centrum verò e&longs;t fpartum, &longs;iue trutina; hoc enim pun<lb/>ctum in motu libræ manet: duo verò brachia &longs;unt veluti duæ &longs;emidiametri <lb/><figure place="text" xlink:href="figures-la/009.01.154.1.tif"/><lb/>à centro exeuntes, vt in figura cerne<lb/>re e&longs;t, in qua centrum, &longs;iue &longs;partum <lb/>e&longs;t vbi C, reliqua &longs;unt manife&longs;ta. </s>

<s>con&longs;iderare enim oportet, quod in motu li<lb/>bræ de&longs;cribitur quidam circulus, cuius diameter &longs;unt ip&longs;a libræ <lb/>brachia, centrum verò e&longs;t fpartum, &longs;iue trutina; hoc enim pun<lb/>ctum in motu libræ manet: duo verò brachia &longs;unt veluti duæ &longs;emidiametri <lb/><figure id="id.009.01.154.1.jpg" xlink:href="009/01/154/1.jpg" place="text"/><lb/>à centro exeuntes, vt in figura cerne<lb/>re e&longs;t, in qua centrum, &longs;iue &longs;partum <lb/>e&longs;t vbi C, reliqua &longs;unt manife&longs;ta. </s>

<s>In <lb/>eadem porrò figura libra maior &longs;it <lb/>A B. minor verò circa idem &longs;partum <lb/>C, &longs;it F G. </s>

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<s>Quamobrem machinantur ij, qui purpuram vendunt, vt pendendo <lb/>defraudent, tum in medio libræ non ponentes &longs;partum, vt hoc modo bra<lb/>chium ex vna parte longius factum facilius moueatur, & proinde à minori <lb/>purpuræ pondere; tum etiam <expan abbr="plumb&utilde;">plumbum</expan> in lancem illam infundentes inquam <lb/>merces imponitur, vel partem illam lancis, quam magis grauitare cu<lb/>piunt ex ligno radici proximo, vel ex nodo&longs;o facientes: lignum <lb/>enim, quod radici proximum e&longs;t, graue admodum e&longs;t, <lb/>quemadmodum etiam nodus; quia nodus e&longs;t, <lb/>quædam radix. </s>

<s><expan abbr="Atq;">Atque</expan> hæc e&longs;t huius pri<lb/>mæ quæ&longs;tionis paraphra&longs;is.</s><pb pagenum="155"/>

<s><expan abbr="Atq;">Atque</expan> hæc e&longs;t huius pri<lb/>mæ quæ&longs;tionis paraphra&longs;is.</s><pb xlink:href="009/01/155.jpg"/>

<s><emph type="italics"/>QVÆSTIO SECVNDA<emph.end type="italics"/></s>

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<s>Si autem deor&longs;um constitutum <lb/>fuerit, non a&longs;cendit, &longs;ed manet? </s>

<s>An quia &longs;ur&longs;um &longs;parto quidem exi&longs;ten<lb/>te plus libræ extra perpendiculum fit, &longs;partum enim e&longs;t perpendiculum, <lb/>quare nece&longs;&longs;e est deor&longs;um ferri id, quod plus est, donec a&longs;cendat, quæ bifariam li<lb/>bram diuidit ad ip&longs;um perpendicul&ubreve;, cum onus incumbat ad libræ partem tractam.<emph.end type="italics"/><lb/><figure place="text" xlink:href="figures-la/009.01.155.1.tif"/><lb/><emph type="italics"/>&longs;it libra vbi recta B C, &longs;partum autem <lb/>A D: hoc igitur &longs;ur&longs;um erecto, perpendi<lb/>culum erit vbi A D M. &longs;i igitur in ip&longs;o B, <lb/>ponatur onus, B, quidem de&longs;cendet vbi E; <lb/>C, autem a&longs;cendat vbi H, quamobrem ea, <lb/>quæ bifariam libram &longs;ecat, primò quidem <lb/>erit D M, ip&longs;ius perpendiculi; incumben<lb/>te autem onere erit D G, quare libræ ip<lb/>&longs;ius vbi E H, quod extra A M, perpendi<lb/>culum e&longs;t, vbi e&longs;t D D H, maius e&longs;t dimidio. </s>

<s>An quia &longs;ur&longs;um &longs;parto quidem exi&longs;ten<lb/>te plus libræ extra perpendiculum fit, &longs;partum enim e&longs;t perpendiculum, <lb/>quare nece&longs;&longs;e est deor&longs;um ferri id, quod plus est, donec a&longs;cendat, quæ bifariam li<lb/>bram diuidit ad ip&longs;um perpendicul&ubreve;, cum onus incumbat ad libræ partem tractam.<emph.end type="italics"/><lb/><figure id="id.009.01.155.1.jpg" xlink:href="009/01/155/1.jpg" place="text"/><lb/><emph type="italics"/>&longs;it libra vbi recta B C, &longs;partum autem <lb/>A D: hoc igitur &longs;ur&longs;um erecto, perpendi<lb/>culum erit vbi A D M. &longs;i igitur in ip&longs;o B, <lb/>ponatur onus, B, quidem de&longs;cendet vbi E; <lb/>C, autem a&longs;cendat vbi H, quamobrem ea, <lb/>quæ bifariam libram &longs;ecat, primò quidem <lb/>erit D M, ip&longs;ius perpendiculi; incumben<lb/>te autem onere erit D G, quare libræ ip<lb/>&longs;ius vbi E H, quod extra A M, perpendi<lb/>culum e&longs;t, vbi e&longs;t D D H, maius e&longs;t dimidio. </s>

<s>&longs;i igitur amoueatur onus ab ip&longs;o E, ne<lb/>ce&longs;&longs;e e&longs;t H, deor&longs;um ferri, minus enim e&longs;t ip&longs;um E D. &longs;i quidem igitur &longs;ur&longs;um ha<lb/>buerit &longs;partum, propter hoc a&longs;cendit libra. </s>

<s>&longs;i autem deor&longs;um fuerit, id quod &longs;ub<lb/>stat, contrarium facit; plus enim dimidio fit libræ, quæ deor&longs;um e&longs;t, pars, quàm <lb/>quod perpendiculum &longs;ecet; quapropter non a&longs;cendit, pars enim eleuata leuior e&longs;t.<emph.end type="italics"/><lb/><figure place="text" xlink:href="figures-la/009.01.155.2.tif"/><lb/><emph type="italics"/>&longs;it libra vbi N G, perpendiculum autem <lb/>K L M, bifariam igitur &longs;ecatur N G. im<lb/>po&longs;ito autem onere in ip&longs;o N, erit quidem <lb/>N, vbi O, ip&longs;um autem G, vbi R; K L, au<lb/>tem vbi K P, quare maius e&longs;t L P O, quàm <lb/>L R, ip&longs;o P L. </s>

<s>&longs;i autem deor&longs;um fuerit, id quod &longs;ub<lb/>stat, contrarium facit; plus enim dimidio fit libræ, quæ deor&longs;um e&longs;t, pars, quàm <lb/>quod perpendiculum &longs;ecet; quapropter non a&longs;cendit, pars enim eleuata leuior e&longs;t.<emph.end type="italics"/><lb/><figure id="id.009.01.155.2.jpg" xlink:href="009/01/155/2.jpg" place="text"/><lb/><emph type="italics"/>&longs;it libra vbi N G, perpendiculum autem <lb/>K L M, bifariam igitur &longs;ecatur N G. im<lb/>po&longs;ito autem onere in ip&longs;o N, erit quidem <lb/>N, vbi O, ip&longs;um autem G, vbi R; K L, au<lb/>tem vbi K P, quare maius e&longs;t L P O, quàm <lb/>L R, ip&longs;o P L. </s>

<s>Ablato igitur onere, ne<lb/>ce&longs;&longs;e e&longs;t manere; incumbit enim, ceu onus <lb/>exce&longs;&longs;us medietatis in quo P L.)<emph.end type="italics"/> Aduer<lb/>te textum græcum e&longs;&longs;e mendo&longs;um, la<lb/>tinum vero mendo&longs;i&longs;&longs;imum. </s>

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<s>Porrò quoniam Piccolomineus, <lb/>& &longs;i plurimum, vt ip&longs;e fatetur, in&longs;udauerit, non tamen &longs;olutionem huius <lb/>quæ&longs;tionis e&longs;t a&longs;&longs;ecutus, eam tibi ex Mechanicis Guidibaldi tradam. </s>

<s>Ari&longs;t<lb/>igitur ponit duas libræ &longs;pecies, &longs;iue potius duas eiu&longs;dem libræ po&longs;itiones, <lb/>vnam, quæ haber &longs;partum, &longs;iue perpendiculum &longs;upra; alteram, quæ infra. <lb/><figure place="text" xlink:href="figures-la/009.01.155.3.tif"/><lb/>vt in præ&longs;enti figura, &longs;it libra B C, cuius <lb/>&longs;partum, &longs;iue perpendiculum A D, &longs;it &longs;ur<lb/>&longs;um, ita vt in puncto A, &longs;it affixum perpen<lb/>diculum, & circa idem punctum A, tan<lb/>quam circa centrum tota libra circum<lb/>uertatur. </s>

<s>Ari&longs;t<lb/>igitur ponit duas libræ &longs;pecies, &longs;iue potius duas eiu&longs;dem libræ po&longs;itiones, <lb/>vnam, quæ haber &longs;partum, &longs;iue perpendiculum &longs;upra; alteram, quæ infra. <lb/><figure id="id.009.01.155.3.jpg" xlink:href="009/01/155/3.jpg" place="text"/><lb/>vt in præ&longs;enti figura, &longs;it libra B C, cuius <lb/>&longs;partum, &longs;iue perpendiculum A D, &longs;it &longs;ur<lb/>&longs;um, ita vt in puncto A, &longs;it affixum perpen<lb/>diculum, & circa idem punctum A, tan<lb/>quam circa centrum tota libra circum<lb/>uertatur. </s>

<s>hæc e&longs;t prima libræ collocatio. </s>

<s>&longs;it deinde libra B C, cuius &longs;partum, <lb/>&longs;iue perpendiculum A D, &longs;it deor&longs;um, vt in altera figura, <expan abbr="&longs;it&qacute;">&longs;itque</expan>; circa pun<pb pagenum="156"/><figure place="text" xlink:href="figures-la/009.01.156.1.tif"/><lb/>ctum A, tanquam circa <expan abbr="centr&utilde;">centrum</expan>, aut axem <lb/>ita fixum, vt ip&longs;i libræ conuer&longs;io innita<lb/>tur, quæ e&longs;t altera libræ po&longs;itio. </s>

<s>&longs;it deinde libra B C, cuius &longs;partum, <lb/>&longs;iue perpendiculum A D, &longs;it deor&longs;um, vt in altera figura, <expan abbr="&longs;it&qacute;">&longs;itque</expan>; circa pun<pb xlink:href="009/01/156.jpg"/><figure id="id.009.01.156.1.jpg" xlink:href="009/01/156/1.jpg" place="text"/><lb/>ctum A, tanquam circa <expan abbr="centr&utilde;">centrum</expan>, aut axem <lb/>ita fixum, vt ip&longs;i libræ conuer&longs;io innita<lb/>tur, quæ e&longs;t altera libræ po&longs;itio. </s>

<s>Quærit <lb/>igitur, cur &longs;i in libra &longs;ur&longs;um <expan abbr="hab&etilde;te">habente</expan> per<lb/>pendiculum, & centrum, ponatur ex vna <lb/>parte onus quodpiam, v. g. in parte B, vt in prima textus figura factum e&longs;t, <lb/>libra de primo &longs;itu B C, mouetur ad &longs;itum E H, &longs;ed tamen ablato pondere <lb/>reuertitur &longs;ua &longs;pontè ad pri&longs;tinum &longs;itum B C. &longs;i autem in libra, cuius per<lb/>pendiculum, ac centrum deor&longs;um &longs;it, vt in &longs;ecunda figura textus, pondus <lb/>imponatur, ip&longs;a quidem à &longs;itu B C, ad &longs;itum O R, transferretur; verumta<lb/>men ablato onere, <expan abbr="nõ">non</expan> amplius ad priorem po&longs;itionem, vti prior, reucrtitur.</s>

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<s>Ratio igitur, cur tunc li<lb/>bra amoto pondere ad horizontis æquilibrium reuertatur e&longs;t, quia pondus <lb/>libræ impo&longs;itum in altera tantum libræ parte, grauitando impellit libram <lb/>ad alium &longs;itum E H, ita vt maior pars libræ con&longs;tituatur ex altera parte li<lb/>neæ directionis prioris A D M, in qua etiam parte exi&longs;tit centrum granita<lb/>tis libræ ip&longs;ius, e&longs;t enim circa D, quod centrum vi ponderis incumbentis in <lb/>E, cogitur paulùm a&longs;cendere, <expan abbr="atq;">atque</expan> contra ip&longs;ius naturalem inclinationem à <lb/>mundi centro recedere, vt &longs;i in libra B C, appendatur onus in B, vt in pri<lb/>ma textus figura; B, de&longs;cendet ad E, & C, a&longs;cendet ad H, & centrum graui<lb/>tatis D, paulùm a&longs;cendet à centro mundi, & linca A D M, quæ libram bi<lb/>fariam &longs;ecabat modo tran&longs;lato perpendiculo in A D G, non amplius cam <lb/>bifariam &longs;ecabit; &longs;ed libræ E H, maior pars erit vltra perpendiculum A D<lb/>M, quæ maior pars e&longs;t D D H.</s>

<s>Si igitur nunc onus amoueatur libræ E H, centrum grauitatis, quod e&longs;t <pb pagenum="157"/>ad D, remanet vltra priorem Directionis lineam; & quia pondus non am<lb/>plius illi æ que ponderat, grauitabit, & quia libra cùm affixa &longs;it ad A, nequit <lb/>deor&longs;um recta tendere, circumferretur circa A, trahente ip&longs;am grauitatis <lb/>centro, cum nihil ob&longs;it, donec iterum perpendiculum A D G, priori &longs;itui <lb/>A D M, congruat: hac enim ratione centrum grauitatis, quantum pote&longs;t, <lb/>iuxta naturam &longs;uam de&longs;cendet, <expan abbr="libra&qacute;">libraque</expan>; ad pri&longs;tinum æquilibrij B C, &longs;itum <lb/>re&longs;tituetur. </s>

<s>Si igitur nunc onus amoueatur libræ E H, centrum grauitatis, quod e&longs;t <pb xlink:href="009/01/157.jpg"/>ad D, remanet vltra priorem Directionis lineam; & quia pondus non am<lb/>plius illi æ que ponderat, grauitabit, & quia libra cùm affixa &longs;it ad A, nequit <lb/>deor&longs;um recta tendere, circumferretur circa A, trahente ip&longs;am grauitatis <lb/>centro, cum nihil ob&longs;it, donec iterum perpendiculum A D G, priori &longs;itui <lb/>A D M, congruat: hac enim ratione centrum grauitatis, quantum pote&longs;t, <lb/>iuxta naturam &longs;uam de&longs;cendet, <expan abbr="libra&qacute;">libraque</expan>; ad pri&longs;tinum æquilibrij B C, &longs;itum <lb/>re&longs;tituetur. </s>

<s>Si autem deor&longs;um fuerit &longs;partum in &longs;ecunda figura textus, im<lb/>po&longs;ito pondere contrarium accidit, quia maior pars libræ, & in qua cen<lb/>trum grauitatis e&longs;t, in tali motu de&longs;cendit: altera autem pars minor, ac læ<lb/>uior &longs;ur&longs;um tollitur. </s>

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<s>contrarium enim videtur debere <lb/>fieri, nam mouenti additur grauitas vectis, & ideò pondus augetur, ergò <lb/>difficilius ip&longs;um cum vecte, quàm &longs;ine eo mouere deberet.</s>

<s>Vectis porrò e&longs;t in&longs;trumentum oblongum, quo ad &longs;ubleuandum graue <lb/>quodpiam vtuntur opifices, quod innititur cuidam fulcimento, quod græcè <lb/>hypomoclion dicitur: hypomoclion antem oncri leuando, quanrum &longs;ieri <pb pagenum="158"/>pote&longs;t proximum e&longs;&longs;e debet, vt vectis pars longior &longs;it ad partes potentiæ <lb/>mouentis. </s>

<s>Vectis porrò e&longs;t in&longs;trumentum oblongum, quo ad &longs;ubleuandum graue <lb/>quodpiam vtuntur opifices, quod innititur cuidam fulcimento, quod græcè <lb/>hypomoclion dicitur: hypomoclion antem oncri leuando, quanrum &longs;ieri <pb xlink:href="009/01/158.jpg"/>pote&longs;t proximum e&longs;&longs;e debet, vt vectis pars longior &longs;it ad partes potentiæ <lb/>mouentis. </s>

<s>vt plurimum verò fulcimentum e&longs;t inter pondus, & potentiam: <lb/>aliquando etiam e&longs;t ex altero vectis extremo, ita vt onus &longs;it inter fulturam, <lb/>& potentiam; aliquando potentia e&longs;t inter vtrunque, vnde tres vectis &longs;pe<lb/>cies exi&longs;tunt. </s>

<s>vt in &longs;ubiectis figuris apparet. </s>

<s>In prima, vectis e&longs;t A B, fultu<lb/><figure place="text" xlink:href="figures-la/009.01.158.1.tif"/><lb/>ra E, onus C. potentia autem &longs;eu vis, <lb/>&longs;eu aliud pondus <expan abbr="mou&etilde;s">mouens</expan> &longs;it vbi D. quæ <lb/>deor&longs;um in D, præmens eleuabit &longs;ur<lb/>&longs;um ex altera parte onus C. & vectis <lb/>circa fulturam E, tanquam centrum <lb/>conuertetur. </s>

<s>In prima, vectis e&longs;t A B, fultu<lb/><figure id="id.009.01.158.1.jpg" xlink:href="009/01/158/1.jpg" place="text"/><lb/>ra E, onus C. potentia autem &longs;eu vis, <lb/>&longs;eu aliud pondus <expan abbr="mou&etilde;s">mouens</expan> &longs;it vbi D. quæ <lb/>deor&longs;um in D, præmens eleuabit &longs;ur<lb/>&longs;um ex altera parte onus C. & vectis <lb/>circa fulturam E, tanquam centrum <lb/>conuertetur. </s>

<s>In altera figura pondus <lb/>e&longs;t inter fulturam, & potentiam, ful<lb/>tura autem in altera extremitate, vt <lb/>patet in figura, hic autem potentia <lb/>non præmit deor&longs;um in D: &longs;ed &longs;ur&longs;um <lb/>vectem eleuando pondus C, attollitur. <lb/></s>

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<s>In hac quæ&longs;tione re&longs;pexit Ari&longs;t. &longs;olùm ad primam vectis &longs;peciem. </s>

<s>Illud <lb/>demum, quod dixit eandem habere rationem potentiam ad pondus, quàm <lb/>partes vectis inuicem demon&longs;tratum e&longs;t po&longs;tea acuti&longs;&longs;imè ab Archimede <pb pagenum="159"/>propo&longs;itione 6. & 7. de æqueponderantibus: & no&longs;tra <expan abbr="t&etilde;pe&longs;tate">tempe&longs;tate</expan> alio quam<lb/>uis modo, & vnica demon&longs;tratione à Guido Vbaldo in &longs;uis Mechanicis pro<lb/>po&longs;itione 1. de Vecte, quæ e&longs;t huiu&longs;modi; Potentia &longs;u&longs;tinens pondus vecti <lb/>appen&longs;um, eandem ad ip&longs;um pondus proportionem habet, quam vectis di<lb/>&longs;tantia inter fulcimentum, ac ponderis &longs;u&longs;pen&longs;ionem, ad di&longs;tantiam, à fulci<lb/>mento ad potentiam interiectam. </s>

<s>Illud <lb/>demum, quod dixit eandem habere rationem potentiam ad pondus, quàm <lb/>partes vectis inuicem demon&longs;tratum e&longs;t po&longs;tea acuti&longs;&longs;imè ab Archimede <pb xlink:href="009/01/159.jpg"/>propo&longs;itione 6. & 7. de æqueponderantibus: & no&longs;tra <expan abbr="t&etilde;pe&longs;tate">tempe&longs;tate</expan> alio quam<lb/>uis modo, & vnica demon&longs;tratione à Guido Vbaldo in &longs;uis Mechanicis pro<lb/>po&longs;itione 1. de Vecte, quæ e&longs;t huiu&longs;modi; Potentia &longs;u&longs;tinens pondus vecti <lb/>appen&longs;um, eandem ad ip&longs;um pondus proportionem habet, quam vectis di<lb/>&longs;tantia inter fulcimentum, ac ponderis &longs;u&longs;pen&longs;ionem, ad di&longs;tantiam, à fulci<lb/>mento ad potentiam interiectam. </s>

<s>quod de omni vecte ab eo demon&longs;tratur, <lb/>cuius propo&longs;itionis &longs;en&longs;us e&longs;t hic; in &longs;uperiori prima figura &longs;i pars vectis <lb/>E B, fuerit, v.g. qua drupla partis A E; etiam pondus C, erit quadruplo ma<lb/>ius pondere, &longs;eu vi in D, quæ ip &longs;um C, ope vectis &longs;u&longs;tinet. </s>

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vectem &longs;ecundi ge<lb/>neris ad &longs;olutionem non adhibuit, vt par erat, & propterea ob&longs;cu<lb/>rior eua&longs;it, ita etiam in præ&longs;enti, qu&ecedil;&longs;tionem ad ve ctem primi ge<lb/>neris reducit, quæ ad alterum reducendà erat: <expan abbr="atq;">atque</expan> hinc ob&longs;curi<lb/>tas, atque prolixitas &longs;olutionis manauit. </s>

<s>E&longs;t enim propriè Temo, &longs;iue gu<lb/>bernaculum nauis, vectis &longs;ecundi generis, vt mox explicabo, e&longs;t enim temo <lb/><figure place="text" xlink:href="figures-la/009.01.159.1.tif"/><lb/>in&longs;trumentum in extrema nanis par<lb/>te, &longs;eu puppi affixum, vt in figura prç<lb/>&longs;enti vides tabellam, in qua B C D, <lb/>cuius manubrium A B, intra nauim <lb/>recipitur, quæ tabella, &longs;eu temo in <lb/>duobus cardinibus, vbi C, & D, cir<lb/>cumuertitur à Nauis gubernatore, <lb/>manubrium vbi A, tractante; ex qua <lb/>conuer&longs;ione nauigium, quò vult ip&longs;e <lb/>gubernator facilè dirigit, ip&longs;umque <lb/>nauigium huc illuc quamuis adeò magnum ip&longs;e &longs;olus impellit, & agitat. </s>

<s>E&longs;t enim propriè Temo, &longs;iue gu<lb/>bernaculum nauis, vectis &longs;ecundi generis, vt mox explicabo, e&longs;t enim temo <lb/><figure id="id.009.01.159.1.jpg" xlink:href="009/01/159/1.jpg" place="text"/><lb/>in&longs;trumentum in extrema nanis par<lb/>te, &longs;eu puppi affixum, vt in figura prç<lb/>&longs;enti vides tabellam, in qua B C D, <lb/>cuius manubrium A B, intra nauim <lb/>recipitur, quæ tabella, &longs;eu temo in <lb/>duobus cardinibus, vbi C, & D, cir<lb/>cumuertitur à Nauis gubernatore, <lb/>manubrium vbi A, tractante; ex qua <lb/>conuer&longs;ione nauigium, quò vult ip&longs;e <lb/>gubernator facilè dirigit, ip&longs;umque <lb/>nauigium huc illuc quamuis adeò magnum ip&longs;e &longs;olus impellit, & agitat. </s>

<s>e&longs;t <lb/>enim temo vectis, cuius auxilio vires mirum in modum augentur, nam to<lb/>ta A B, e&longs;t ip&longs;a Vectis longitudo, cuius hypomoclion e&longs;t mare, cui contra<pb pagenum="160"/>nititur tabella B E; onus autem e&longs;t puppis, quod onus præ&longs;ertim in cardini<lb/>bus C D, mouenti re&longs;i&longs;tit, & quod præcipuè mouere gubernator intendit. <lb/></s>

<s>e&longs;t <lb/>enim temo vectis, cuius auxilio vires mirum in modum augentur, nam to<lb/>ta A B, e&longs;t ip&longs;a Vectis longitudo, cuius hypomoclion e&longs;t mare, cui contra<pb xlink:href="009/01/160.jpg"/>nititur tabella B E; onus autem e&longs;t puppis, quod onus præ&longs;ertim in cardini<lb/>bus C D, mouenti re&longs;i&longs;tit, & quod præcipuè mouere gubernator intendit. <lb/></s>

<s>cum igitur motum onus &longs;it intra vectis extrema, hypomoclion in extremo <lb/>ad B E, vbi in motu temonis tabella mare vrget, quod minimè cedit, <expan abbr="ip&longs;a&qacute;">ip&longs;aque</expan>; <lb/>in hoc motu ferè maneat, & fiat qua&longs;i centrum, circa quod totus temo cir<lb/>cumducitur, patet temonem e&longs;&longs;e vectem &longs;ecundæ &longs;peciei, vt dicebam. </s>

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<s>dif<lb/>ficultas e&longs;t in verbis illis <emph type="italics"/>(Prima enim pars celerrimè fertur)<emph.end type="italics"/> & in illis <emph type="italics"/>(Sic ip<lb/>&longs;ius continui in fine imbecili&longs;&longs;ima eft latio)<emph.end type="italics"/> videtur velle dicere, quod quando <lb/>continuum aliquod proiectum fertur per aera, pars ip&longs;ius anterior ea e&longs;t, <lb/>quæ præ cæteris partibus principaliter mouetur, & ad cuius motum reliquæ <lb/>po&longs;teriores tanquam &longs;ub&longs;e quentes moueantur; qua&longs;i dicat tota vis lationis <lb/>e&longs;t in anteriori parte: &longs;iue ip&longs;i impetus maior ine&longs;t: videmus enim proiecta, <lb/>quorum vna pars e&longs;t cæteris grauior, quia ei parti melius imprimitur mo<lb/>tus, eam etiam fieri anteriorem in latione, quamuis initio fuerit po&longs;terior. <lb/></s>

<s>&longs;ic etiam quando graue fertur deor&longs;um, dicimus ip&longs;um ferri &longs;ecundum cen<lb/>trum grauitatis ip&longs;ius, <expan abbr="ibi&qacute;">ibique</expan>; maiorem vim grauitandi exi&longs;tere, &longs;ic in proie<lb/>ctis partem anteriorem dicere po&longs;&longs;umus e&longs;&longs;e, &longs;ecundum quam totum conti<lb/>nuum fertur: <expan abbr="ibi&qacute;">ibique</expan>; totum e&longs;&longs;e impetum lationis, & propterea etiam maio<lb/>ri impetu, <expan abbr="atq;">atque</expan> celerrimè ferri: & <expan abbr="con&longs;equ&etilde;ter">con&longs;equenter</expan> partem po&longs;teriorem, quam<lb/>uis priorem æqua velocitate con&longs;equatur, non tamen tanto impetu, cum ip<lb/>&longs;a ad alterius impetum moueatur, & propterca latio ip&longs;ius e&longs;t admodum <lb/>imbecillis.</s><pb pagenum="161"/>

<s>Si quis &longs;agittam per aerem latam à &longs;uo motu vellet deflectere, eam faci<lb/>lius in po&longs;teriore parte à &longs;uo cur&longs;u deuiaret, quàm in anteriore. </s>

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<s><expan abbr="atq;">atque</expan> hinc oritur nece&longs;&longs;itas <expan abbr="e&utilde;">eum</expan> po&longs;teriori parti affigendi. </s>

<s>&longs;ubdit po&longs;tea aliam <lb/>eiu&longs;dem rationem, quia nimirum parua motione facta in puppi multo ma<lb/>ius interuallum cogitur mutare prora; nam idem angulus, quo eius lineæ <lb/>&longs;unt longiores, eò maiorem &longs;ubten&longs;am &longs;ibi lineam re&longs;picit, quod facilè in <lb/><figure place="text" xlink:href="figures-la/009.01.161.1.tif"/><lb/>ad&longs;cripta figura intueri licet; in qua duæ <lb/>lineæ A B, A C, continent angulum A, cui <lb/>angulo &longs;ubtenduntur tres lineæ parallelæ <lb/>F G, D E, B C, quarum B C, maxima e&longs;t, <lb/>quia ibi maiores, &longs;iue remotiores &longs;unt ab <lb/>angulo A, duæ rectæ A B, A C, ip&longs;um con<lb/>tinentes, quod Geometricè per 4. 6. pro<lb/>bari pote&longs;t. </s>

<s>&longs;ubdit po&longs;tea aliam <lb/>eiu&longs;dem rationem, quia nimirum parua motione facta in puppi multo ma<lb/>ius interuallum cogitur mutare prora; nam idem angulus, quo eius lineæ <lb/>&longs;unt longiores, eò maiorem &longs;ubten&longs;am &longs;ibi lineam re&longs;picit, quod facilè in <lb/><figure id="id.009.01.161.1.jpg" xlink:href="009/01/161/1.jpg" place="text"/><lb/>ad&longs;cripta figura intueri licet; in qua duæ <lb/>lineæ A B, A C, continent angulum A, cui <lb/>angulo &longs;ubtenduntur tres lineæ parallelæ <lb/>F G, D E, B C, quarum B C, maxima e&longs;t, <lb/>quia ibi maiores, &longs;iue remotiores &longs;unt ab <lb/>angulo A, duæ rectæ A B, A C, ip&longs;um con<lb/>tinentes, quod Geometricè per 4. 6. pro<lb/>bari pote&longs;t. </s>

<s>&longs;ic etiam facta motione, vel <lb/>parua in puppi, tota nauis transfertur ad <lb/>alium &longs;itum, ita vt prora multum aliò transferatur, quod non accideret, &longs;i <lb/>eadem motio fieret ad medium nauigij. </s>

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<s>B E, enim ponitur <lb/>æqualis ip&longs;i A D, &longs;ed minus interuallum propter re&longs;i&longs;tentiam aquæ ex &longs;up<lb/>po&longs;itione percurret, quale e&longs;t B F, quod minus e&longs;t quàm A D, quare etiam li<lb/>nea B G, abbreuiabitur, <expan abbr="erit&qacute;">eritque</expan>; veluti F Y, quæ etiam erit minor ip&longs;a D G, <lb/>quæ facta e&longs;t D Y, propter duo &longs;indlia triangula D Y A, B Y F, &longs;imilia au<lb/>tem triangula &longs;unt ea, quorum anguli vnius &longs;unt æquales angulis alterius, <lb/>quo po&longs;ito &longs;unt etiam latera vnius proportionalia lateribus alterius, vt pa<lb/>tet ex prima definitione 6. necnon ex quarta eiu&longs;dem demon&longs;tratione. </s>

<s>hæc <pb pagenum="162"/><figure place="text" xlink:href="figures-la/009.01.162.1.tif"/><lb/>quidem duo triangula &longs;unt &longs;imi<lb/>lia, & rectè concluditur F Y, mi<lb/>nus e&longs;&longs;e quàm D Y, &longs;ed tamen <lb/>non videntur i&longs;ta propo&longs;itum <lb/>o&longs;tendere, quod erat, plus nauim <lb/>procedere, quàm palmulam re<lb/>trocedere. </s>

<s>hæc <pb xlink:href="009/01/162.jpg"/><figure id="id.009.01.162.1.jpg" xlink:href="009/01/162/1.jpg" place="text"/><lb/>quidem duo triangula &longs;unt &longs;imi<lb/>lia, & rectè concluditur F Y, mi<lb/>nus e&longs;&longs;e quàm D Y, &longs;ed tamen <lb/>non videntur i&longs;ta propo&longs;itum <lb/>o&longs;tendere, quod erat, plus nauim <lb/>procedere, quàm palmulam re<lb/>trocedere. </s>

<s>Fateor quidem tex<lb/>tum hunc e&longs;&longs;e ob&longs;curi&longs;&longs;imum, <lb/><expan abbr="id&qacute;">idque</expan>; propterea fortè quia e&longs;t admodum corruptus, præ&longs;ertim circa chara<lb/>cteres, qui corrigendi &longs;unt vti nos facimus. </s>

Line 7196

Line 7197

<s>Supponit autem ip&longs;e <lb/>auctor remi palmulam retrocedere, quoties nauigium in anteriora progre<pb pagenum="163"/>ditur, <expan abbr="locum&qacute;">locumque</expan>; &longs;calmi, &longs;uper quo circulari motu remus vertitur, in medio <lb/>ip&longs;ius remi po&longs;itum e&longs;&longs;e, vt &longs;cilicet tantum di&longs;tet à manubrio, quantum à <lb/>palmula. </s>

<s>Supponit autem ip&longs;e <lb/>auctor remi palmulam retrocedere, quoties nauigium in anteriora progre<pb xlink:href="009/01/163.jpg"/>ditur, <expan abbr="locum&qacute;">locumque</expan>; &longs;calmi, &longs;uper quo circulari motu remus vertitur, in medio <lb/>ip&longs;ius remi po&longs;itum e&longs;&longs;e, vt &longs;cilicet tantum di&longs;tet à manubrio, quantum à <lb/>palmula. </s>

<s>Duæ <expan abbr="itaq;">itaque</expan> rectæ lineæ ponantur æquales A B, & D E, quæ quidem <lb/>in C, puncto medio &longs;e inuicem &longs;ecent, & connectantur A B, & D E: remus <lb/>autem in initio vnius remigationis po&longs;itionem habeat rectam lineam A B, <lb/><expan abbr="&longs;it&qacute;">&longs;itque</expan>; A, manubrium; B, palmula; C, verò &longs;ealmus. </s>

<s>Cum igitur A, remi ca<lb/>put in fine ip&longs;ius remigationis eò tran&longs;latum fuerit D, non erit B, vbi E; &longs;i <lb/><figure place="text" xlink:href="figures-la/009.01.163.1.tif"/><lb/>enim ibi fuerit; remus igitur po&longs;itionem <lb/>habebit rectam lineam D E; & quoniam <lb/>contrapo&longs;iti anguli, qui ad C, æquales &longs;unt, <lb/>& duo latera A C, & D C, trianguli A D C, <lb/>duobus lateribus B C, & C E, trianguli B<lb/>E C, æqualia etiam &longs;unt: reliqui igitur an<lb/>guli, <expan abbr="atq;">atque</expan> ba&longs;es ip&longs;orum <expan abbr="triãgulorum">triangulorum</expan> æqua<lb/>les erunt per 4. propo&longs;itionem primi libri <lb/>Euclidis, & propterea tantum &longs;patium per<lb/>curret B, quantum A: &longs;calmus verò C, im<lb/>motus omninò erit: & nauigium idcircò, in <lb/>quo ip&longs;e &longs;calmus, immotum etiam erit con<lb/>tra hypothe&longs;im. </s>

<s>Cum igitur A, remi ca<lb/>put in fine ip&longs;ius remigationis eò tran&longs;latum fuerit D, non erit B, vbi E; &longs;i <lb/><figure id="id.009.01.163.1.jpg" xlink:href="009/01/163/1.jpg" place="text"/><lb/>enim ibi fuerit; remus igitur po&longs;itionem <lb/>habebit rectam lineam D E; & quoniam <lb/>contrapo&longs;iti anguli, qui ad C, æquales &longs;unt, <lb/>& duo latera A C, & D C, trianguli A D C, <lb/>duobus lateribus B C, & C E, trianguli B<lb/>E C, æqualia etiam &longs;unt: reliqui igitur an<lb/>guli, <expan abbr="atq;">atque</expan> ba&longs;es ip&longs;orum <expan abbr="triãgulorum">triangulorum</expan> æqua<lb/>les erunt per 4. propo&longs;itionem primi libri <lb/>Euclidis, & propterea tantum &longs;patium per<lb/>curret B, quantum A: &longs;calmus verò C, im<lb/>motus omninò erit: & nauigium idcircò, in <lb/>quo ip&longs;e &longs;calmus, immotum etiam erit con<lb/>tra hypothe&longs;im. </s>

<s>&longs;upponitur enim in que&longs;tio<lb/>ne, quod nauigium illa remigatione in anteriora moueatur, remi verò pal<lb/>mula retrocedat. </s>

Line 7224

Line 7225

<s>Aæqualis porrò o&longs;ten&longs;us e&longs;t an<lb/>guius B Z T, angulo A D T, præterea angulus D A T; angulo T B Z, æqua<lb/>lis: angulus igitur B Z T, angulo T B Z, maior erit, & propterea latus B T, <lb/>trianguli B T Z, latere T Z, maius erit: tota igitur recta linea A B, tota <lb/>D Z, maior erit: & idcircò cum remus po&longs;itionem habuerit rectam lineam <lb/>D Z palmula erit vltra Z. </s>

<s>E&longs;to igitur in K, & connectantur rectæ lineæ B D, <lb/>& B K: &longs;patium igitur decur&longs;um ab ip&longs;a palmula non erit B Z, &longs;ed B K: quod <pb pagenum="164"/>quidem minus etiam o&longs;tendemus e&longs;&longs;e ip&longs;o D A. </s>

<s>E&longs;to igitur in K, & connectantur rectæ lineæ B D, <lb/>& B K: &longs;patium igitur decur&longs;um ab ip&longs;a palmula non erit B Z, &longs;ed B K: quod <pb xlink:href="009/01/164.jpg"/>quidem minus etiam o&longs;tendemus e&longs;&longs;e ip&longs;o D A. </s>

<s>Nam quoniam duo latera <lb/>B D, & D K, trianguli B D K, duobus lateribus B D, & D E, <expan abbr="triãguli">trianguli</expan> B E D, <lb/>æqualia &longs;unt, &longs;ed minor e&longs;t angulus B D K, angulo B D E: minorigitur erit <lb/>ba&longs;is B K, ba&longs;e B E, per 24. primi, quod demon&longs;trandum erat</s>

Line 7246

Line 7247

<s>Si Remiges nauigium mouere po&longs;&longs;unt, maius &longs;emper &longs;pa<lb/>tium remi manubrium percurrit, quàm nauigium.</s>

<s>Sit enim remus A C, manubrium A, &longs;calmus B, qui propter nauigij <lb/>motum &longs;patium percurrat à B, in D, in quo loco ip&longs;eremus A C, &longs;i<lb/><figure place="text" xlink:href="figures-la/009.01.164.1.tif"/><lb/>tum rectitudinis habeat E F. </s>

<s>Sit enim remus A C, manubrium A, &longs;calmus B, qui propter nauigij <lb/>motum &longs;patium percurrat à B, in D, in quo loco ip&longs;eremus A C, &longs;i<lb/><figure id="id.009.01.164.1.jpg" xlink:href="009/01/164/1.jpg" place="text"/><lb/>tum rectitudinis habeat E F. </s>

<s>Spatium <lb/>itaque, quod A, conficit, curna linea <lb/>&longs;it A E, cui recta linea re&longs;pondeat A Z, in re<lb/>ctam E F, perpendieularis. </s>

Line 7256

Line 7257

<s>4. 6. libri Eucli<lb/>dis: maior e&longs;t autem A G, ipfa B G, & maior <lb/>igitur erit A Z, quam B D. & proinde maius <lb/>&longs;patium remi manubrium percurrit, quam <lb/>nauigium, quod demon&longs;trandum erat.</s>

<s>Quod &longs;i à puncto B, rectam lineam vtrinque <lb/>ducamus H K, ad remi men&longs;uram, rectos facientem angulos cum B D, <expan abbr="re-ctam&qacute;">re<lb/>ctamque</expan>; A Z, &longs;ecantem in I, manife&longs;tè intelligemus ip&longs;am rectam A Z, con<lb/>&longs;tare ex A I, & I Z, quarum prior re&longs;pondet curuæ A H, quæ motu proprio <lb/>manubrij de&longs;cripta e&longs;t; po&longs;terior verò æqualis e&longs;t rectæ B D, quæ motu na<lb/>uigij decur&longs;a e&longs;t.</s><pb pagenum="165"/>

<s><emph type="italics"/>PROPOSITIO SECVNDA.<emph.end type="italics"/></s>

Line 7266

Line 7267

<s>Nam &longs;i a loco &longs;uo dimota fuerit: <lb/>&longs;patium igitur permeet C D, ad po&longs;teriora: quo quidem decur&longs;o, <lb/>remus A C, po&longs;itionem rectitudmis habeat F D, &longs;calmus <expan abbr="itaq;">itaque</expan> B, tran&longs;latus <lb/>erit in G. </s>

<s>Excitetur autem à puncto B, in <expan abbr="vtramq;">vtramque</expan> partem linea E B R, ad <lb/><figure place="text" xlink:href="figures-la/009.01.165.1.tif"/><lb/>rectos angulos &longs;uper B G, & à <expan abbr="p&utilde;cto">puncto</expan> A, recta A H, <lb/>&longs;uper D F: itemque à puncto E, recta C E, &longs;uper <lb/>E R; ip&longs;arum verò rectarum linearum E R, & <lb/>A H, &longs;ectio &longs;it in K, &longs;ed C F., & D F, &longs;it in Z, & quo<lb/>niam A K, id &longs;patium e&longs;t, quod motu proprio re<lb/>mi manubrium permeauit, curuilineo enim re<lb/>&longs;pondeat A R, recta autem B G, id &longs;patium e&longs;t, <lb/>quod nauigium confecit: ip&longs;æ igitur rectæ lineæ <lb/>H K, & B G, æquales erunt. </s>

<s>Excitetur autem à puncto B, in <expan abbr="vtramq;">vtramque</expan> partem linea E B R, ad <lb/><figure id="id.009.01.165.1.jpg" xlink:href="009/01/165/1.jpg" place="text"/><lb/>rectos angulos &longs;uper B G, & à <expan abbr="p&utilde;cto">puncto</expan> A, recta A H, <lb/>&longs;uper D F: itemque à puncto E, recta C E, &longs;uper <lb/>E R; ip&longs;arum verò rectarum linearum E R, & <lb/>A H, &longs;ectio &longs;it in K, &longs;ed C F., & D F, &longs;it in Z, & quo<lb/>niam A K, id &longs;patium e&longs;t, quod motu proprio re<lb/>mi manubrium permeauit, curuilineo enim re<lb/>&longs;pondeat A R, recta autem B G, id &longs;patium e&longs;t, <lb/>quod nauigium confecit: ip&longs;æ igitur rectæ lineæ <lb/>H K, & B G, æquales erunt. </s>

<s>Atqui in duobus æqui<lb/>angulis triangulis E B C, & B A K, vel per 26. <lb/>propo&longs;itionem primi Euclidis, vel 4. 6. æquales <lb/>e&longs;&longs;e concludes A K, & E C, rectas lineas: quapro<lb/>pter æqualis erit E C, rectæ B G, per communem <lb/>&longs;ententiam: eidem autem B G, æqualis e&longs;t E Z, <lb/>in parallelogrammo, per 34. propo&longs;itionem ip<lb/>&longs;ius primi libri: æqualis igitur erit recta E Z, re<lb/>ctæ E C, pars toti, quod e&longs;t impo&longs;&longs;ibile. </s>

Line 7278

Line 7279

<s>Remus enim incipiente motu po&longs;itionem habeat A C, de&longs;inente <lb/>verò rectitudinis &longs;itum F G. &longs;calmus igitur B, propter nauigij <lb/>motum, &longs;patium con&longs;iciet B D. </s>

<s>Excitetur à puncto B, in <expan abbr="vtramq;">vtramque</expan> <lb/>partem perpendicularis E Z, in quam veniant a punctis A, & C, <lb/>ad rectos angulos rectæ lineæ A E, & C Z: &longs;patium autem A E, à manubrio <pb pagenum="166"/><figure place="text" xlink:href="figures-la/009.01.166.1.tif"/><lb/>decur&longs;um motu proprio &longs;patij B D, duplum <lb/>&longs;it: recta verò linea C H, curuæ re&longs;pondeat <lb/>C G, quæ à remi palmula de&longs;cripta e&longs;t. </s>

<s>Excitetur à puncto B, in <expan abbr="vtramq;">vtramque</expan> <lb/>partem perpendicularis E Z, in quam veniant a punctis A, & C, <lb/>ad rectos angulos rectæ lineæ A E, & C Z: &longs;patium autem A E, à manubrio <pb xlink:href="009/01/166.jpg"/><figure id="id.009.01.166.1.jpg" xlink:href="009/01/166/1.jpg" place="text"/><lb/>decur&longs;um motu proprio &longs;patij B D, duplum <lb/>&longs;it: recta verò linea C H, curuæ re&longs;pondeat <lb/>C G, quæ à remi palmula de&longs;cripta e&longs;t. </s>

<s>Di<lb/>co ip&longs;as rectas lineas B D, & C H, æquales <lb/>e&longs;&longs;e. </s>

Line 7308

Line 7309

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

<s>Ex hac, & præcedenti infertur, quod &longs;i remi manubrium motu proprio <lb/>maius &longs;patium decurrat, quàm nauigium, &longs;iue id &longs;it duplum, &longs;iue mi<pb pagenum="167"/>nus duplo, &longs;iue maius duplo, &longs;patium, quod nauigium interim decurrit ad <lb/>anteriora, & quod palmula remi in contrarium &longs;imul iuncta, ei quod ip&longs;um <lb/>remi manubrium motu proprio conficit, æqualia erunt. </s>

<s>Ex hac, & præcedenti infertur, quod &longs;i remi manubrium motu proprio <lb/>maius &longs;patium decurrat, quàm nauigium, &longs;iue id &longs;it duplum, &longs;iue mi<pb xlink:href="009/01/167.jpg"/>nus duplo, &longs;iue maius duplo, &longs;patium, quod nauigium interim decurrit ad <lb/>anteriora, & quod palmula remi in contrarium &longs;imul iuncta, ei quod ip&longs;um <lb/>remi manubrium motu proprio conficit, æqualia erunt. </s>

<s>&longs;emper enim B D, <lb/>æqualis e&longs;t H Z: tota verò C Z, quæ æqualis e&longs;t A E, ex &longs;uis partibus C H, <lb/>& H Z, con&longs;tabit.</s>

Line 7322

Line 7323

<s>Si celerius feratur nauigium, quàm remi manubrium, mo<lb/>uebitur palmula in vlteriora, <expan abbr="nil&qacute;">nilque</expan>; vnquam retroce<lb/>det, <expan abbr="id&qacute;">idque</expan>; &longs;patium decurret, quo nauigij motus <lb/>motum manubrij &longs;uperat.</s>

<s>Habeat enim remus incipiente motu po&longs;itionem A C: de&longs;inente <lb/><figure place="text" xlink:href="figures-la/009.01.167.1.tif"/><lb/>verò <expan abbr="&longs;it&utilde;">&longs;itum</expan> rectitudinis F G. &longs;cal<lb/>mus igitur B, propter nauigij <lb/>motum tran&longs;latus, erit in D, &longs;it <lb/><expan abbr="itaq;">itaque</expan> &longs;patium B D, maius quàm A H, à re<lb/>mi manubrio motu proprio decur&longs;um: &longs;ic <lb/>enim celerius dicetur ferri <expan abbr="nauigi&utilde;">nauigium</expan>, quàm <lb/>manubrium. </s>

<s>Habeat enim remus incipiente motu po&longs;itionem A C: de&longs;inente <lb/><figure id="id.009.01.167.1.jpg" xlink:href="009/01/167/1.jpg" place="text"/><lb/>verò <expan abbr="&longs;it&utilde;">&longs;itum</expan> rectitudinis F G. &longs;cal<lb/>mus igitur B, propter nauigij <lb/>motum tran&longs;latus, erit in D, &longs;it <lb/><expan abbr="itaq;">itaque</expan> &longs;patium B D, maius quàm A H, à re<lb/>mi manubrio motu proprio decur&longs;um: &longs;ic <lb/>enim celerius dicetur ferri <expan abbr="nauigi&utilde;">nauigium</expan>, quàm <lb/>manubrium. </s>

<s>Dico, quòd palmula C, in <lb/>vlteriora mouebitur. </s>

Line 7332

Line 7333

<s>eadem <lb/>enim celeritate mouentur A, in H, & C, <lb/>ver&longs;us I, circa &longs;calmum. </s>

<s>Atqui per hypo<lb/>the&longs;im celerius fertur nauigium, quam A. <lb/>in H, celerius igitur ip&longs;um nauigium fer<lb/>tur, quàm C, ver&longs;us I. &longs;ed mouetur idem <pb pagenum="168"/>C. ip&longs;a nauigij celeritate ver&longs;us K; celerius igitur ferretur C, ad K, quam <lb/>ad I, quapropter nihil vnquam retrocedet ip&longs;um C, imò verò in vlteriora <lb/>progredietur, <expan abbr="&longs;patium&qacute;">&longs;patiumque</expan>; decurret C K, quod quidem relinquitur detracto <lb/>I C, ex I K. &longs;i enim remi palmula tota ip&longs;a nauigij celeritate moueretur, vl<lb/>tra K, progrederetur, cum B, perueniret ad D: &longs;ed retrahitur interim, pro<lb/>pter eum motum, qui fit circa B. </s>

<s>Atqui per hypo<lb/>the&longs;im celerius fertur nauigium, quam A. <lb/>in H, celerius igitur ip&longs;um nauigium fer<lb/>tur, quàm C, ver&longs;us I. &longs;ed mouetur idem <pb xlink:href="009/01/168.jpg"/>C. ip&longs;a nauigij celeritate ver&longs;us K; celerius igitur ferretur C, ad K, quam <lb/>ad I, quapropter nihil vnquam retrocedet ip&longs;um C, imò verò in vlteriora <lb/>progredietur, <expan abbr="&longs;patium&qacute;">&longs;patiumque</expan>; decurret C K, quod quidem relinquitur detracto <lb/>I C, ex I K. &longs;i enim remi palmula tota ip&longs;a nauigij celeritate moueretur, vl<lb/>tra K, progrederetur, cum B, perueniret ad D: &longs;ed retrahitur interim, pro<lb/>pter eum motum, qui fit circa B. </s>

<s>Sic igitur palmulæ celeritate, quæ à mo<lb/>tu nauigij prouenit retardata, decur&longs;um &longs;patium erit C K. </s>

Line 7376

Line 7377

<s>mouens <lb/>potentia e&longs;t ventus, qui mouet in antenna F C G. quanto igitur &longs;ublimior <lb/>e&longs;t antenna, tanto longior euadit vectis E C, <expan abbr="tanto&qacute;">tantoque</expan>; maiores fiunt venti <lb/>vires. </s>

<s>dixi autem onus e&longs;&longs;e in D, quia &longs;i nauis vento ob&longs;i&longs;teret, ip&longs;a inuerte<lb/>retur hac ratione, vt puppis A, eleuata, prora B, demergeretur, manente <pb pagenum="169"/><figure place="text" xlink:href="figures-la/009.01.169.1.tif"/><lb/>veluti centro parte E. quia ve<lb/>rò ob maris liquiditatem na<lb/>uis minimè obfi&longs;tit, &longs;ed facilè <lb/>cedens à ventis vrgetur, hinc <lb/>fit, vt meritò dixerim pondus <lb/>nauis e&longs;&longs;e ad D, fulcimentum <lb/>verò ad E.</s>

<s>dixi autem onus e&longs;&longs;e in D, quia &longs;i nauis vento ob&longs;i&longs;teret, ip&longs;a inuerte<lb/>retur hac ratione, vt puppis A, eleuata, prora B, demergeretur, manente <pb xlink:href="009/01/169.jpg"/><figure id="id.009.01.169.1.jpg" xlink:href="009/01/169/1.jpg" place="text"/><lb/>veluti centro parte E. quia ve<lb/>rò ob maris liquiditatem na<lb/>uis minimè obfi&longs;tit, &longs;ed facilè <lb/>cedens à ventis vrgetur, hinc <lb/>fit, vt meritò dixerim pondus <lb/>nauis e&longs;&longs;e ad D, fulcimentum <lb/>verò ad E.</s>

<s>Quæ&longs;tio &longs;eptima, & &longs;atis pec <lb/>&longs;e clara e&longs;t; <expan abbr="neq;">neque</expan> Mathemati<lb/>ci e&longs;t eam exponere.</s>

Line 7406

Line 7407

<s>Præterea circulares etiam, &longs;i cui obulam fiunt corpori, illud &longs;imi<lb/>liter &longs;ecundum pu&longs;illum tan gunt: rectilineæ verò figuræ, rectitudine &longs;ua <lb/>plani multum contingerent. </s>

<s>Ad hæc motor mouens huiu&longs;inodi rotas, eas <lb/>mouet, quò nutant: nam quando rota erecta e&longs;t &longs;uper pauimentum, dia<lb/>meter ip&longs;ius, quæ à contactu pauimenti ad angulos rectos, ad &longs;upremum <pb pagenum="170"/>rotæ perducitur totum rotæ pondus in duas æquas partes diuidit, ita vt ta<lb/>le pondus in æquilibrio con&longs;tituatur, cum ex vna parte tantum &longs;it, quantum <lb/>ex altera; ex quo fit, vt vel exigua vis ip&longs;am impellere valeat: quando enim <lb/>duo æqualia pondera &longs;unt in æquilibrio, quelibet vis pote&longs;t ea ab æquilibrio <lb/>dimouere. </s>

<s>Ad hæc motor mouens huiu&longs;inodi rotas, eas <lb/>mouet, quò nutant: nam quando rota erecta e&longs;t &longs;uper pauimentum, dia<lb/>meter ip&longs;ius, quæ à contactu pauimenti ad angulos rectos, ad &longs;upremum <pb xlink:href="009/01/170.jpg"/>rotæ perducitur totum rotæ pondus in duas æquas partes diuidit, ita vt ta<lb/>le pondus in æquilibrio con&longs;tituatur, cum ex vna parte tantum &longs;it, quantum <lb/>ex altera; ex quo fit, vt vel exigua vis ip&longs;am impellere valeat: quando enim <lb/>duo æqualia pondera &longs;unt in æquilibrio, quelibet vis pote&longs;t ea ab æquilibrio <lb/>dimouere. </s>

<s>quando po&longs;tea rota e&longs;t in motu, vel cum primum ei motus fuerit <lb/>à motore inditus, &longs;emper nutat ad partes illas, ad quas primum fuit incita<lb/>ta per impre&longs;&longs;am motionem, quapropter nullo negotio ad ea&longs;dem partes, <lb/>&longs;eu antror&longs;um mouetur; quò enim <expan abbr="vnumquodq;">vnumquodque</expan> vergit, illuc facillimè fer<lb/>tur: quemadmodum è contrario difficillimum e&longs;t in contrariam nutus &longs;ui <lb/>partem vnumquodque pellere. </s>

Line 7416

Line 7417

<s>& quia &longs;icut diameter ad diametrum, ita maio<lb/>ris circuli periphæria ad minoris periphæriam (vt po&longs;tea o&longs;tendam) & quia <lb/>quo <expan abbr="lõgior">longior</expan> diameter e&longs;t, eò facilius, vt initio probaui, mouetur, fit vt etiam <lb/>periphæria maioris facilius, quàm minoris moueatur, &longs;iue dixeris, quod an<lb/>gulus maioris circuli ad angulum minoris nutum quendam habet; & quia <lb/>facilius mouetur angulus maioris, quàm minoris, fit, vt maior rota adhi<lb/>beatur ad minorem mouendam: & quia intra maiorem infinitæ circa idem <lb/>centrum concipi po&longs;&longs;unt, hinc fit, vt rotæ maiores facilius moueantur, & <lb/>motæ moueant cæteras intra &longs;e contentas. </s>

<s>quod dictum e&longs;t de nutu anguli <lb/>maioris circuli ad angulum minoris ex appo&longs;ita figura facilè patebit, vbi <lb/><figure place="text" xlink:href="figures-la/009.01.170.1.tif"/><lb/>pro minore angulo intelligendus e&longs;t arcus C B, <lb/>pro maiore autem arcus D E, quorum <expan abbr="vterq;">vterque</expan> vo<lb/>catur angulus, quoniam angulo A, qui e&longs;t in cen<lb/>tro opponuntur. </s>

<s>quod dictum e&longs;t de nutu anguli <lb/>maioris circuli ad angulum minoris ex appo&longs;ita figura facilè patebit, vbi <lb/><figure id="id.009.01.170.1.jpg" xlink:href="009/01/170/1.jpg" place="text"/><lb/>pro minore angulo intelligendus e&longs;t arcus C B, <lb/>pro maiore autem arcus D E, quorum <expan abbr="vterq;">vterque</expan> vo<lb/>catur angulus, quoniam angulo A, qui e&longs;t in cen<lb/>tro opponuntur. </s>

<s>Atque hæc &longs;ufficiant deijs, quæ <lb/>primo modo moueutur.</s>

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<s>Secundum igitur inclinatio<lb/>nem hanc, quæ in obliquum e&longs;t, ide&longs;t, quæ &longs;ecundum circunferentiam &longs;it, <lb/>ip&longs;am rotam mouens facillimè mouet. </s>

<s>altera latio e&longs;t, &longs;ecundum quam cir<pb pagenum="171"/>culus à &longs;eip&longs;o &longs;ecundum diametrum mouetur, ide&longs;t circa &longs;uum centrum re<lb/>trahit continuò extrema diametri; ne recta &longs;ecundum naturalem lationem <lb/>ferantur, &longs;ed in orbem circulariter circa centrum gyrentur. </s>

<s>altera latio e&longs;t, &longs;ecundum quam cir<pb xlink:href="009/01/171.jpg"/>culus à &longs;eip&longs;o &longs;ecundum diametrum mouetur, ide&longs;t circa &longs;uum centrum re<lb/>trahit continuò extrema diametri; ne recta &longs;ecundum naturalem lationem <lb/>ferantur, &longs;ed in orbem circulariter circa centrum gyrentur. </s>

<s>hæc Ari&longs;t. </s>

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<s>3. probatum e&longs;t ex Archimede, & Gui<lb/>do Vbaldo, di&longs;tantiæ autem &longs;int æquales, er<gap/>nt etiam pondus, & potentia <lb/>æqualia, ide&longs;t, &longs;i pondus e&longs;&longs;et vnius libræ, &longs;u&longs;tineretur à tanta vi, <expan abbr="quãta">quanta</expan> opus <lb/>e&longs;t ad libram vnam &longs;u&longs;tinendam, & non amplius. </s>

<s>vt autem clarè appareat <lb/>vectis in trochlea, & hypomoclion, & æquales di&longs;tantiæ, &longs;it figura, in qua <lb/><figure place="text" xlink:href="figures-la/009.01.171.1.tif"/><lb/>pondus D, ductario funi D C B E, alligatum. </s>

<s>vt autem clarè appareat <lb/>vectis in trochlea, & hypomoclion, & æquales di&longs;tantiæ, &longs;it figura, in qua <lb/><figure id="id.009.01.171.1.jpg" xlink:href="009/01/171/1.jpg" place="text"/><lb/>pondus D, ductario funi D C B E, alligatum. </s>

<s>poten<lb/>tia <expan abbr="&longs;u&longs;tin&etilde;s">&longs;u&longs;tinens</expan> E. axis autem erit diameter rotulæ B A C, <lb/>nam potentia premit rotulam in B, & pondus in C, & <lb/>cum rotula &longs;u&longs;tineatur in A, à &longs;u&longs;pen&longs;orio F A. erit <lb/>punctum A, hypomoclion, quia in motu vectis eua<lb/>dit centrum, <expan abbr="e&longs;t&qacute;">e&longs;tque</expan>; punctum manens. </s>

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manife&longs;tum etiam e&longs;t ma<lb/>iorem vim quamlibet, quam &longs;it ea, quæ &longs;u&longs;tinet, po&longs;&longs;e <lb/>idem pondus &longs;ur&longs;um mouere. </s>

<s>hæc & præ&longs;enti loco, & <lb/>&longs;equentibus lucem afferre po&longs;&longs;unt.</s><pb pagenum="172"/>

<s>hæc & præ&longs;enti loco, & <lb/>&longs;equentibus lucem afferre po&longs;&longs;unt.</s><pb xlink:href="009/01/172.jpg"/>

<s><emph type="italics"/>QVÆSTIO NONA<emph.end type="italics"/></s>

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<s>Vt autem con&longs;tat ex &longs;equenti <lb/>quæ&longs;tione 11. &longs;cyntala erat in&longs;trumentum quoddam vectorium, quod ro<lb/>tas, &longs;icut currus, aliter tamen factas, habebat, porrò <foreign lang="greek">sxutalh\,</foreign> ide&longs;t &longs;cytala <lb/>inter alia &longs;ignificat <expan abbr="bacul&utilde;">baculum</expan>, &longs;iue hgnum oblongum, ac teres, qualia ea &longs;unt, <lb/>quibus vtimur in &longs;ucculis, vulgò Na&longs;pe; & in axe in peritrochio, vt videre <lb/>e&longs;t apud <expan abbr="Guid&utilde;">Guidum</expan> Vbaldum. </s>

<s>hinc factum e&longs;t, vt apud Lacædemonios &longs;cytala <lb/>&longs;ignificaret quoddam genus epi&longs;tolæ, quam &longs;cytalem laconicam dicebant, <lb/>quia in charta in&longs;tar zonæ oblonga, & circa &longs;cytalam, hoc e&longs;t circa bacillum <lb/>quendam &longs;piratim circumuoluta exarabatur; ita vt yer&longs;us &longs;cripturæ &longs;ecun<lb/>dum &longs;urculi longitudinem ducerentur, ex quo &longs;iebat, vt per iuncturas mem<lb/>branæ, literæ, ac verba procederent, membranam hanc ex &longs;cytala reuolu<lb/>tam, & aliter complicatam Imperatori mittebant, re&longs;olutio autem mem<pb pagenum="173"/>branæ literas truncas, atque mutilas reddebat; cum partim continerentur <lb/>citra iuncturas, partim vltra: eæquè partes, quæ &longs;imul fuerant &longs;criptæ, & <lb/>continuatæ, po&longs;t re&longs;olutionem erant ab innicem valde di&longs;&longs;itæ. </s>

<s>hinc factum e&longs;t, vt apud Lacædemonios &longs;cytala <lb/>&longs;ignificaret quoddam genus epi&longs;tolæ, quam &longs;cytalem laconicam dicebant, <lb/>quia in charta in&longs;tar zonæ oblonga, & circa &longs;cytalam, hoc e&longs;t circa bacillum <lb/>quendam &longs;piratim circumuoluta exarabatur; ita vt yer&longs;us &longs;cripturæ &longs;ecun<lb/>dum &longs;urculi longitudinem ducerentur, ex quo &longs;iebat, vt per iuncturas mem<lb/>branæ, literæ, ac verba procederent, membranam hanc ex &longs;cytala reuolu<lb/>tam, & aliter complicatam Imperatori mittebant, re&longs;olutio autem mem<pb xlink:href="009/01/173.jpg"/>branæ literas truncas, atque mutilas reddebat; cum partim continerentur <lb/>citra iuncturas, partim vltra: eæquè partes, quæ &longs;imul fuerant &longs;criptæ, & <lb/>continuatæ, po&longs;t re&longs;olutionem erant ab innicem valde di&longs;&longs;itæ. </s>

<s>quapropter <lb/>Imperator commenti totius con&longs;cius, eandem membranam &longs;cytali alteri <lb/>priori omninò &longs;imili, <expan abbr="æquali&qacute;">æqualique</expan>; eodem modo, quo prius circumponebat, <expan abbr="&longs;ic&qacute;">&longs;icque</expan>; <lb/>iuncturæ priores redibant, quæ literas, ac verba mutila, & imperfecta in <lb/>integrum re&longs;tituebant, vt facilè legi po&longs;&longs;ent. </s>

<s>hoc illi vtebantur &longs;ecreto, cum <lb/>literas ad Imperatores &longs;uos mi&longs;&longs;as, ho&longs;tibus occultas e&longs;&longs;e volebant.</s>

<s>Ex quibus conijcere licet &longs;cytalam fui&longs;&longs;e lignum oblongum, & teres, &longs;iue <lb/>vt Geometræ dicunt, Cylindrum; in cuius tamen extremitatibus e&longs;&longs;ent <lb/>margines duo aliquantulum prominentes, ceu binæ rotæ, cum ip&longs;o tamen <lb/>continuæ, & connexæ, vt cum ip&longs;o &longs;imul conuoluerentur; non tamen tan<lb/><figure place="text" xlink:href="figures-la/009.01.173.1.tif"/><lb/>quam circa axem. </s>

<s>Ex quibus conijcere licet &longs;cytalam fui&longs;&longs;e lignum oblongum, & teres, &longs;iue <lb/>vt Geometræ dicunt, Cylindrum; in cuius tamen extremitatibus e&longs;&longs;ent <lb/>margines duo aliquantulum prominentes, ceu binæ rotæ, cum ip&longs;o tamen <lb/>continuæ, & connexæ, vt cum ip&longs;o &longs;imul conuoluerentur; non tamen tan<lb/><figure id="id.009.01.173.1.jpg" xlink:href="009/01/173/1.jpg" place="text"/><lb/>quam circa axem. </s>

<s>cuius hanc accipe fi<lb/>guram. </s>

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re&longs;pondet dicens, quod graue e&longs;t ægrè moneri non &longs;olum <lb/>contra nutum &longs;uum, idc&longs;t &longs;ur&longs;um, &longs;ed etiam in obliquum, &longs;eu ad latera, quia <lb/>grauia deor&longs;um <expan abbr="nutãt">nutant</expan>, non &longs;ur&longs;um, nec in tran&longs;uer&longs;um: ideo libræ cum one<lb/>re, quia &longs;unt grauiores, & rota ferrea quàm lignea, & ferrea ctiam maior, <lb/>quàm minor grauior e&longs;t, ideò difficilius agitatur.</s>

<s>Contra quam re&longs;pon&longs;ionem &longs;ic fortè obijcies; in præcedenti enim quæ<pb pagenum="174"/>&longs;tione dictum e&longs;t ab Ari&longs;t.

<s>Contra quam re&longs;pon&longs;ionem &longs;ic fortè obijcies; in præcedenti enim quæ<pb xlink:href="009/01/174.jpg"/>&longs;tione dictum e&longs;t ab Ari&longs;t.

maiores trochleas, & &longs;cytalas, minoribus facilius <lb/>commoueri, hic autem dicit maiorem rotam difficilius quàm minorem mo<lb/>ueri. </s>

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<s>Crediderim ego facilius portari magna onera per &longs;cytalas, propter ip&longs;a<lb/>rum firmitatem, currus enim <expan abbr="ip&longs;orumq;">ip&longs;orumque</expan> rotæ &longs;unt multò debiliores, neque <lb/>maioribus oneribus &longs;ufficiunt. </s>

<s>Concludit po&longs;tea quæ&longs;tionem dicens, quia <pb pagenum="175"/>igitur &longs;cytalæ ab ip&longs;o onere non ita premuntur quin moueri melius po&longs;&longs;int <lb/>quàm currus, imò ab ip&longs;o onere iam commoto, ip&longs;æ quoque incitentur, & <lb/>præterea à potentia per planum infernè, benè &longs;ub&longs;tratum, & complanatum <lb/>trahantur, fit, vt qua&longs;i in duobus locis ip&longs;arum rotæ impellantur ab onere <lb/>&longs;upra, & à potentia infra; &longs;icque facilius quam currus ingentia præ&longs;ertim <lb/>onera vehunt.</s>

<s>Concludit po&longs;tea quæ&longs;tionem dicens, quia <pb xlink:href="009/01/175.jpg"/>igitur &longs;cytalæ ab ip&longs;o onere non ita premuntur quin moueri melius po&longs;&longs;int <lb/>quàm currus, imò ab ip&longs;o onere iam commoto, ip&longs;æ quoque incitentur, & <lb/>præterea à potentia per planum infernè, benè &longs;ub&longs;tratum, & complanatum <lb/>trahantur, fit, vt qua&longs;i in duobus locis ip&longs;arum rotæ impellantur ab onere <lb/>&longs;upra, & à potentia infra; &longs;icque facilius quam currus ingentia præ&longs;ertim <lb/>onera vehunt.</s>

<s><emph type="italics"/>De Funda.<emph.end type="italics"/></s>

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<s>quærit igitur, <lb/>cur quanto gracilius fucrit corpus &longs;ucculæ A B, tanto facilius vertitur. <lb/></s>

<s>Ratio e&longs;t, quia collops, quemadmodum etiam iugum, reducitur ad vectem, <pb pagenum="176"/><figure place="text" xlink:href="figures-la/009.01.176.1.tif"/><lb/>cuius hypomoclion e&longs;t in medio <lb/>&longs;ucculæ, &longs;iue in axe ip&longs;ius &longs;ucculæ; <lb/>potentia verò e&longs;t in &longs;ummitatibus <lb/>collopum, vt in C, E, F, D, pon<lb/>dus verò e&longs;t vbi funis ductarius <lb/>cum onere pendet è &longs;uccula in &longs;u<lb/>perficie nimirum, vt vbi L, quare <lb/>pars vectis inter axim, & &longs;uperfi<lb/>ciem &longs;ucculæ eadem e&longs;t, quæ inter <lb/>hypomoclium, & pondus. </s>

<s>Ratio e&longs;t, quia collops, quemadmodum etiam iugum, reducitur ad vectem, <pb xlink:href="009/01/176.jpg"/><figure id="id.009.01.176.1.jpg" xlink:href="009/01/176/1.jpg" place="text"/><lb/>cuius hypomoclion e&longs;t in medio <lb/>&longs;ucculæ, &longs;iue in axe ip&longs;ius &longs;ucculæ; <lb/>potentia verò e&longs;t in &longs;ummitatibus <lb/>collopum, vt in C, E, F, D, pon<lb/>dus verò e&longs;t vbi funis ductarius <lb/>cum onere pendet è &longs;uccula in &longs;u<lb/>perficie nimirum, vt vbi L, quare <lb/>pars vectis inter axim, & &longs;uperfi<lb/>ciem &longs;ucculæ eadem e&longs;t, quæ inter <lb/>hypomoclium, & pondus. </s>

<s>quanto <lb/>igitur &longs;ucculæ corpus gracilius fuerit, tanto hæc pars minuetur; & con&longs;e<lb/>quenter altera inter hypomoclium, & potentiam productior euadet: eaque <lb/>propter facilius à motore ver&longs;abitur.</s>

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Line 7683

<s>in A, verò onus ip&longs;ius ligni, deor&longs;um tendens <lb/>premit, quod nunc e&longs;t in&longs;tar potentiæ motricis, quare A, & C, &longs;unt &longs;ibi in<lb/>uicem, & potentiæ, & pondera. </s>

<s>In minori autem ligno, onus ligni in D, <pb pagenum="177"/><figure place="text" xlink:href="figures-la/009.01.177.1.tif"/><lb/>fultura manus in E, potentia alterius ma<lb/>nus in F. iam inquir Ari&longs;t.

<s>In minori autem ligno, onus ligni in D, <pb xlink:href="009/01/177.jpg"/><figure id="id.009.01.177.1.jpg" xlink:href="009/01/177/1.jpg" place="text"/><lb/>fultura manus in E, potentia alterius ma<lb/>nus in F. iam inquir Ari&longs;t.

maius lignum <lb/>A B C, magis flectitur, quamuis cra&longs;&longs;ius <lb/>&longs;it, quàm lignum D E F, quod e&longs;t tenuuius, <lb/>&longs;ed multò breuius; quia in maiori onus <lb/>ip&longs;ius ligni, quod circa A, deor&longs;um pre<lb/>mit <expan abbr="lõgius">longius</expan> di&longs;tat ab hypomoclio B, quàm <lb/>in minori ligno. </s>

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<s>Verum &longs;atius e&longs;t cum Guido Vbaldo reducere cuneum ad duos &longs;ecundi <lb/>generis vectes, quorum fultura &longs;it in cunei apice extremo, pondus verò in<lb/>tra vectem, ea nimirum pars ligni, que à cuneo vrgetur, ac diuellitur. </s>

<s>cuneo <lb/>præterea vires adduntur ex valida mallei percu&longs;&longs;ione; malleus autem ip&longs;e <lb/>magna vi percutit, quia motus mouet, &longs;eu quia mouens malleum, mouet <lb/>ip&longs;um etiam dum e&longs;t in ip&longs;a latione, vnde ip&longs;a lationis celeritate malleus <lb/>fit valentior: <expan abbr="hoc&qacute;">hocque</expan>; modo paruos cunci vectes maiores con&longs;equuntur vires, <lb/><figure place="text" xlink:href="figures-la/009.01.177.2.tif"/><lb/>quàm ip&longs;a vectium magnitudo po&longs;tulet. <lb/></s>

<s>cuneo <lb/>præterea vires adduntur ex valida mallei percu&longs;&longs;ione; malleus autem ip&longs;e <lb/>magna vi percutit, quia motus mouet, &longs;eu quia mouens malleum, mouet <lb/>ip&longs;um etiam dum e&longs;t in ip&longs;a latione, vnde ip&longs;a lationis celeritate malleus <lb/>fit valentior: <expan abbr="hoc&qacute;">hocque</expan>; modo paruos cunci vectes maiores con&longs;equuntur vires, <lb/><figure id="id.009.01.177.2.jpg" xlink:href="009/01/177/2.jpg" place="text"/><lb/>quàm ip&longs;a vectium magnitudo po&longs;tulet. <lb/></s>

<s>&longs;it cuneus A B C. lignum autem &longs;cinden<lb/>dum D E F G, <expan abbr="vectes&qacute;">vectesque</expan> duo &longs;int A C, & <lb/>B C, quorum commune hypomoclion e&longs;t <lb/>in C, onus autem vectis B C, e&longs;t pars li<lb/>gni G, hæc enim ip&longs;i contranititur, <expan abbr="atq;">atque</expan> <lb/>ab eo expellitur. </s>

<s>potentia verò mouens <lb/>vectem e&longs;t in malleo, dum &longs;uperius latus <lb/>cunei A B, percutit. </s>

<s>alter huic auer&longs;us <lb/>vectis e&longs;t latus A C, cuius fultura e&longs;t C, <lb/>cadem cum priori, onus propul&longs;atum D, <pb pagenum="178"/>potentia cum altero communis e&longs;t in latere A B, à malleo validè percu&longs;&longs;o. <lb/></s>

<s>alter huic auer&longs;us <lb/>vectis e&longs;t latus A C, cuius fultura e&longs;t C, <lb/>cadem cum priori, onus propul&longs;atum D, <pb xlink:href="009/01/178.jpg"/>potentia cum altero communis e&longs;t in latere A B, à malleo validè percu&longs;&longs;o. <lb/></s>

<s>cunei igitur virtus partim ex vectibus, partim ex percu&longs;&longs;ione con&longs;tat.</s>

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<s>Auxilio huius in&longs;trumenti machinato<lb/>res parua vi attollunt ingentia pondera. </s>

<s>communiter autem con&longs;tat ex plu<lb/><figure place="text" xlink:href="figures-la/009.01.178.1.tif"/><lb/>ribus orbiculis, qui partim &longs;uperiori loculamento, <lb/>partim infetiori inditi &longs;unt, per quos orbiculos cer<lb/>ta lege circumductus e&longs;t ductarius funis, qui deinde <lb/>in &longs;ui po&longs;trema parte à potentia tractus omnes illos <lb/>orbiculos, per quos tran&longs;it circumuoluens inferius <lb/>loculamentum, cui appen&longs;um e&longs;t pondus, vnà cum <lb/>ponderc attollit. </s>

<s>communiter autem con&longs;tat ex plu<lb/><figure id="id.009.01.178.1.jpg" xlink:href="009/01/178/1.jpg" place="text"/><lb/>ribus orbiculis, qui partim &longs;uperiori loculamento, <lb/>partim infetiori inditi &longs;unt, per quos orbiculos cer<lb/>ta lege circumductus e&longs;t ductarius funis, qui deinde <lb/>in &longs;ui po&longs;trema parte à potentia tractus omnes illos <lb/>orbiculos, per quos tran&longs;it circumuoluens inferius <lb/>loculamentum, cui appen&longs;um e&longs;t pondus, vnà cum <lb/>ponderc attollit. </s>

<s>figuram &longs;implicis trochleæ, con<lb/>&longs;tantis &longs;cilicet ex duobus tantum orbiculis, facilita<lb/>tis cau&longs;a exhibebo, in hac enim melius apparebit, <lb/>qua ratione trochlea ad vectem reducatur. </s>

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Line 7753

<s>&longs;ed vbinam hic <lb/>vectis? </s>

<s>con&longs;idera diametrum M L, inferioris orbiculi, hæc enim ea e&longs;t, quæ <pb pagenum="179"/>vectem gerit. </s>

<s>con&longs;idera diametrum M L, inferioris orbiculi, hæc enim ea e&longs;t, quæ <pb xlink:href="009/01/179.jpg"/>vectem gerit. </s>

<s>huius enim extrema L M, à fune tanguntur, & ab eius medio <lb/>B, onus pender, & grauitat; & quia funis in M, &longs;ur&longs;um trahitur, <expan abbr="&longs;ecum&qacute;">&longs;ecumque</expan>; ex <lb/>parte illa &longs;ur&longs;um eieuat diametrum L M, erit potentia mouens, & eleuans <lb/>in M. pondus verò intra vectem ad B, medium vectis; quare fulcimentum <lb/>erit in reliquo extremo L, vbi funis &longs;u&longs;tinet loculamentum, & vbi diameter, <lb/>&longs;eu vectis innititur. </s>

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<s>placet au<lb/>tem his, quæ de cuneo, & &longs;ecuri dicta &longs;unt, nonnulla ex Guido Vbal<lb/>do loco corollarij adijcere, videlicet. </s>

<s>Ad huiu&longs;modi facultatis in<lb/>&longs;trumentum ca <expan abbr="quoq;">quoque</expan> omnia commodè referri po&longs;&longs;unt, quæ percu&longs;<lb/>&longs;ione, &longs;iue impul&longs;u incidunt, diuidunt, perforant, <expan abbr="huiu&longs;modi&qacute;">huiu&longs;modique</expan>; alia obeunt <lb/>munera; vt en&longs;es, gladij, mucrones, &longs;ecures, terebræ, & &longs;imilia: &longs;erra <expan abbr="quoq;">quoque</expan> <lb/>ad hoc reducitur, dentes enim percutiunt, <expan abbr="cunei&qacute;">cuneique</expan>; in&longs;tar exi&longs;tunt.</s><pb pagenum="180"/>