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version 1.14, 2002/08/06 17:19:52

version 1.30, 2002/08/11 00:50:23

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<?xml version="1.0"?>

<!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd" ><archimedes> <info> <author>Bianchini, Giuseppe</author> <title>Aristotelis loca mathematica</title> <date>1615</date>

<!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd" ><archimedes> <info>

<author>Biancani, Giuseppe</author>

<title>Aristotelis loca mathematica</title>

<place>Bologna</place> <translator></translator> <lang>la</lang> <cvs_file>bianc_locam_01_la_1615.xml</cvs_file><cvs_version>1.13</cvs_version><locator>0000000009</locator> </info> <text> <front> </front> <body> <chap> <pb/>

<place>Bologna</place>

<cvs_file>bianc_locam_01_la_1615</cvs_file>

<cvs_version></cvs_version>

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

<s>ARISTOTELIS</s>

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

24. &longs;ecti 1. lib. </s>

24. &longs;ecti 1. lib.

<s>1. de De&longs;criptionibus.<emph.end type="italics"/></s>

1. de De&longs;criptionibus.<emph.end type="italics"/></s>

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

2. &longs;ect 2. lib. </s>

2. &longs;ect 2. lib.

<s>1. de De&longs;criptionibus.<emph.end type="italics"/></s>

1. de De&longs;criptionibus.<emph.end type="italics"/></s>

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

3. &longs;ecti 2. lib. </s>

3. &longs;ecti 2. lib.

<s>1. de Incommen&longs;urabili.<emph.end type="italics"/></s>

1. de Incommen&longs;urabili.<emph.end type="italics"/></s>

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

1. &longs;ecti 3. lib. </s>

1. &longs;ecti 3. lib.

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

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

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

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4. loco 86. Elementa geometrica: Numeri capitales.<emph.end type="italics"/></s>

<s>In Elenchorum lib. </s>

<s>In Elenchorum lib.

1.</s>

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

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<s>Ex 1. Meteororum.</s>

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

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

<s>3. De magnitudine Terræ ad a&longs;tra, & &longs;olem collata.<emph.end type="italics"/></s>

3. De magnitudine Terræ ad a&longs;tra, & &longs;olem collata.<emph.end type="italics"/></s>

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

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4. De ordine Luminarium Solis, & Lunæ.<emph.end type="italics"/></s>

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

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

<s>3. de Mercurij stella. </s>

3. de Mercurij stella. </s>

<s>Item de Cometa: e&longs;&longs;e in C&oelig;lo.<emph.end type="italics"/></s>

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

<s><emph type="italics"/>Summa 1. cap. </s>

<s><emph type="italics"/>Summa 1. cap.

<s>1. De Marirubro.<emph.end type="italics"/></s>

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

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

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

<s>2. De ortu stellarum fixarum: Item de occa&longs;u earumdem.<emph.end type="italics"/></s>

2. De ortu stellarum fixarum: Item de occa&longs;u earumdem.<emph.end type="italics"/></s>

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

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<s>Ex 3. Meteor.</s>

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

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

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

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

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

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

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

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

<s>3. Pythagorei Mathematicas cæteris præferebant.<emph.end type="italics"/></s>

3. Pythagorei Mathematicas cæteris præferebant.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 47. Geometria habet &longs;uas præcognitiones.<emph.end type="italics"/></s>

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<s><emph type="italics"/>16. Idem ex contactu circuli cum linea recta.<emph.end type="italics"/></s>

<s>Ex lib.

<s>9. Hi&longs;toriæ Animalium.</s>

9. Hi&longs;toriæ Animalium.</s>

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

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<s>2. cap.

<s>1. Automata.<emph.end type="italics"/></s>

1. Automata.<emph.end type="italics"/></s>

<s>2. cap.

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

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

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

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<s>1. cap.

<s>7. Faber, & Geometra diuersè con&longs;iderant angulum rectum.<emph.end type="italics"/></s>

7. Faber, & Geometra diuersè con&longs;iderant angulum rectum.<emph.end type="italics"/></s>

<s>2. cap.

<s>6. De Arithmetica proportione.<emph.end type="italics"/></s>

6. De Arithmetica proportione.<emph.end type="italics"/></s>

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

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

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

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<s>5. cap.

<s>3. Vnitarius numerus. </s>

3. Vnitarius numerus. </s>

<s>Quid Proportionalitas. </s>

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

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

<s>6. Proportio Geometrica, & Arithmetica.<emph.end type="italics"/></s>

6. Proportio Geometrica, & Arithmetica.<emph.end type="italics"/></s>

<s>6. cap.

<s>5. Omnis triangulus, &c.<emph.end type="italics"/></s>

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

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

<s>8. Principia Mathematica non pendere ab experientia.<emph.end type="italics"/></s>

8. Principia Mathematica non pendere ab experientia.<emph.end type="italics"/></s>

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

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

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

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

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

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

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

<s>& cap.

7. lib.

1. Eth.</s>

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

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

<s>Demon&longs;trationes Geometricas appellet De&longs;criptiones, & De&longs;igna<lb/>tiones, vide cap. </s>

<s>Demon&longs;trationes Geometricas appellet De&longs;criptiones, & De&longs;igna<lb/>tiones, vide cap.

<s>de Priori, & cap. </s>

de Priori, & cap.

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

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

<s>4. quinti Methaph. </s>

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

<s>& cap.

3. lib.

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

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

<s>Item ad primam primi, vide tex. </s>

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<s>Ad 5. primi, vide cap. </s>

<s>Ad 5. primi, vide cap.

<s>24. &longs;ecti 1 lib. </s>

24. &longs;ecti 1 lib.

<s>1. Priorum.</s>

1. Priorum.</s>

<s>Ad 21. primi, vide tex. </s>

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

<s>Ad 28. primi, vide cap. </s>

<s>Ad 28. primi, vide cap.

21. & cap.

<s>22. &longs;ecundi <expan abbr="Prior&utilde;">Priorum</expan>, & tex. </s>

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

<s>13. primi Po&longs;ter.</s>

<s>Ad 32. primi, vide cap. </s>

<s>Ad 32. primi, vide cap.

<s>1. &longs;ecti 3. lib. </s>

1. &longs;ecti 3. lib.

<s>1. Prior. </s>

1. Prior. </s>

<s>& cap.

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

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

<s>2. <lb/>primi Po&longs;ter. </s>

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<s>tex 11. primi de Anima. </s>

<s>& cap.

1. de mem. </s>

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

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

<s>& cap.

4. lib.

<s>2. de Generat. </s>

2. de Generat. </s>

<s>animal. </s>

<s>& cap.

5. lib.

<s>6. Ethic. </s>

6. Ethic. </s>

<s>& <lb/>cap.

<s>2. Magnorum Moral. </s>

2. Magnorum Moral. </s>

<s>& cap.

10. Mag. </s>

<s>Moral. </s>

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

<s>Ad 2. definitionem 2. Gnomonis, vide cap. </s>

<s>Ad 2. definitionem 2. Gnomonis, vide cap.

<s>de Motu in Po&longs;tprædicam. </s>

de Motu in Po&longs;tprædicam. </s>

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

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

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

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

<s>de relatione in prædicam. </s>

de relatione in prædicam. </s>

<s>& cap.

<s>31. &longs;ecundi Priorum, & <lb/>tex. </s>

31. &longs;ecundi Priorum, & <lb/>tex. </s>

<s>23. primi Po&longs;ter. </s>

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

<s>Ad primam 3. vide cap. </s>

<s>Ad primam 3. vide cap.

9. lib.

<s>2. Ethycorum.</s>

2. Ethycorum.</s>

<s>Ad 2. tertij, vide tex. </s>

<s>13. lib.

<s>1. de Anima. </s>

1. de Anima. </s>

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

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<s>Ad commentarium P. </s>

<s>Clauij extremum lib. </s>

<s>Clauij extremum lib.

<s>4. elementorum. </s>

4. elementorum. </s>

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

<s>Ad 4. definitionem 5. vide cap. </s>

<s>Ad 4. definitionem 5. vide cap.

3. lib.

<s>2. Ethyc.</s>

2. Ethyc.</s>

<s>Ad 9. definitionem 5. vide cap. </s>

<s>Ad 9. definitionem 5. vide cap.

3. lib.

<s>5. Ethyc. </s>

5. Ethyc. </s>

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

<s>& cap.

3. lib.

<s>5. Ethyc. </s>

5. Ethyc. </s>

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<s>Ad 2. propo&longs;it. </s>

<s>6. vide cap.

2. lib.

<s>8. Topicorum loco 41.</s>

8. Topicorum loco 41.</s>

<s>Ad 13. &longs;exti, vide tex. </s>

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

<s>Ad 8. definitionem 7. vide cap. </s>

<s>Ad 8. definitionem 7. vide cap.

1. lib.

<s>1. Magnorum Moral.</s>

1. Magnorum Moral.</s>

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

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

<s>Ad primam definitionem 10. vide cap. </s>

<s>Ad primam definitionem 10. vide cap.

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

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

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

<s>Ad 118. decimi, vide cap. </s>

<s>Ad 118. decimi, vide cap.

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

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

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

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

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

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

<s>22. lib.

<s>2. Priorum. </s>

2. Priorum. </s>

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

<s>& cap.

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

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

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<s>21. tertij de Anima. </s>

<s>& cap.

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

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

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

<s>& cap.

4. <lb/>lib.

<s>2. de Generat. </s>

2. de Generat. </s>

<s>animal. </s>

<s>& lib.

3. cap.

<s>3. Ethyc. </s>

3. Ethyc. </s>

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

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

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

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

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

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

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

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

4. Math. </s>

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<s>Ex cap.

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

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

<s>quæ non &longs;unt in antiqua tran<lb/>&longs;latione de&longs;ump&longs;imus ex ca&longs;tigati&longs;&longs;imo græco codice editionis Francfor<lb/>dien&longs;is, propterea quod totum hunc locum declarant; &longs;unt autem i&longs;ta, <lb/><foreign lang="greek">ai/gar arxai/ pro/terai tw_n qewrhma/twn th ta/xei.</foreign> per &longs;cientias autem demon&longs;tra<lb/>tiuas intelligendas e&longs;&longs;e hoc loco ip&longs;as Mathematicas ex eo patet, quod illis <lb/>a&longs;&longs;ignet Ari&longs;t. </s>

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<s>Ex cap.

<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="fig2"></figure><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="fig2"></figure><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>

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

<s>13. Elem.

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

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

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

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<s>Quod <lb/>porrò Ari&longs;t.

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

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

3. lib.

<s>3. Ethyc. </s>

3. Ethyc. </s>

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

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

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

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

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

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

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

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<s>Ex cap.

<s>23. &longs;ecti primi lib. </s>

23. &longs;ecti primi lib.

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

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

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

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

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

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

<s>manife&longs;tum e&longs;t. </s>

manife&longs;tum e&longs;t. </s>

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

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

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

<s>7. de legib. </s>

7. de legib. </s>

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

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<s>latus B C, præcisè omnino metiatur. </s>

<s>theorema <lb/>i&longs;tud demon&longs;tratur in vltima 10. Elem. </s>

<s>theorema <lb/>i&longs;tud demon&longs;tratur in vltima 10. Elem.

<s>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/></s>

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/></s>

<s>deducit ad impo&longs;&longs;ibile, &longs;iue, vt ait hic Ari&longs;t.

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<s>Et cap.

<s>24. &longs;ecti primi libri primi <emph type="italics"/>(Sed magis efficitur manife&longs;tum in de&longs;cri<lb/>ptionibus, vt quod æquicruris, qui ad ba&longs;im æquales &longs;int, ad centrum ductæ A B, <lb/>A C, &longs;i igitur æqualem accipiat A G, angulum ip&longs;i A B D, non omnino ex. </s>

24. &longs;ecti primi libri primi <emph type="italics"/>(Sed magis efficitur manife&longs;tum in de&longs;cri<lb/>ptionibus, vt quod æquicruris, qui ad ba&longs;im æquales &longs;int, ad centrum ductæ A B, <lb/>A C, &longs;i igitur æqualem accipiat A G, angulum ip&longs;i A B D, non omnino ex. </s>

<s>&longs;timans <lb/>æquales, qui &longs;emicirculorum, & rur&longs;us G, ip&longs;i D, non omnem a&longs;&longs;umens eum, qui &longs;e<lb/>cti. </s>

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infert manife&longs;tum e&longs;&longs;e oportere in omni &longs;yllo<lb/>gi&longs;mo, reperiri vniuer&longs;ales, & affirmatiuas propo&longs;itiones, vt Factum e&longs;t in <lb/>præcedenti aliter e&longs;&longs;et petitio principij. </s>

<s>Quænam vero &longs;it æqualitas, quam <lb/>Geometræ con&longs;iderant, infra cap. </s>

<s>Quænam vero &longs;it æqualitas, quam <lb/>Geometræ con&longs;iderant, infra cap.

<s>1. &longs;ecti 3. explicabicur.</s>

1. &longs;ecti 3. explicabicur.</s>

<s>Ex cap.

<s>2. &longs;ecti 2. lib. </s>

2. &longs;ecti 2. lib.

<s>1. <emph type="italics"/>(Secundum veritatem quidem ex ijs, quæ &longs;ecundum <lb/>veritatem de&longs;cribuntur ine&longs;&longs;e, ad dialecticos autem &longs;yllogi&longs;mos ex propo&longs;itionibus <lb/>&longs;ecundum opinionem)<emph.end type="italics"/> verba illa; ex ijs, quæ <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> veritatem de&longs;cribuntur <pb pagenum="39"/>ine&longs;&longs;e; &longs;ic græcè, <foreign lang="greek">e/a tw_n xata\ aleiq/ei/an diagegramme/non,</foreign> vbi manife&longs;tè vtitur <lb/>verbo, De&longs;cribere, per quod &longs;uperius annotauimus apud Ari&longs;t. </s>

1. <emph type="italics"/>(Secundum veritatem quidem ex ijs, quæ &longs;ecundum <lb/>veritatem de&longs;cribuntur ine&longs;&longs;e, ad dialecticos autem &longs;yllogi&longs;mos ex propo&longs;itionibus <lb/>&longs;ecundum opinionem)<emph.end type="italics"/> verba illa; ex ijs, quæ <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> veritatem de&longs;cribuntur <pb pagenum="39"/>ine&longs;&longs;e; &longs;ic græcè, <foreign lang="greek">e/a tw_n xata\ aleiq/ei/an diagegramme/non,</foreign> vbi manife&longs;tè vtitur <lb/>verbo, De&longs;cribere, per quod &longs;uperius annotauimus apud Ari&longs;t. </s>

<s>&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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<s>Ex cap.

<s>3. &longs;ecti 2. lib. </s>

3. &longs;ecti 2. lib.

<s>1. <emph type="italics"/>(Vt an ne diameter incomm.)<emph.end type="italics"/> loquitur de a&longs;ymme<lb/>tria diametri, & co&longs;tæ eiu&longs;dem quadrati, de qua fusè egimus &longs;uperius in <lb/>cap. </s>

1. <emph type="italics"/>(Vt an ne diameter incomm.)<emph.end type="italics"/> loquitur de a&longs;ymme<lb/>tria diametri, & co&longs;tæ eiu&longs;dem quadrati, de qua fusè egimus &longs;uperius in <lb/>cap.

<s>23. &longs;ecti 1. huius libri; quæ &longs;i repetantur, optimè hunc <expan abbr="loc&utilde;">locum</expan> declarant.</s>

23. &longs;ecti 1. huius libri; quæ &longs;i repetantur, optimè hunc <expan abbr="loc&utilde;">locum</expan> declarant.</s>

<s>Ex cap.

<s>1. &longs;ecti 3. lib. </s>

1. &longs;ecti 3. lib.

<s>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. </s>

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.

<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="fig7"></figure><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="fig7"></figure><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>

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<s>Ex cap.

<s>21. <emph type="italics"/>(Quod faciunt, qui coalternas putant &longs;cribere, latent enim ip&longs;<gap/><lb/>&longs;e ip&longs;os talia accip entes, quæ non est po&longs;&longs;ibile monstrare uon exiftentibus <lb/><gap/>o lternis)<emph.end type="italics"/> Vult Ari&longs;t.

21. <emph type="italics"/>(Quod faciunt, qui coalternas putant &longs;cribere, latent enim ip&longs;<gap/><lb/>&longs;e ip&longs;os talia accip entes, quæ non est po&longs;&longs;ibile monstrare uon exiftentibus <lb/><gap/>o lternis)<emph.end type="italics"/> Vult Ari&longs;t.

exemplo mathematico explicare, quid &longs;it pe<lb/>titio principij. </s>

<s>vbi per coalternas intelligit parallelas lineas, vox <lb/>enim græca <foreign lang="greek">parallhlos,</foreign> idem &longs;ignificat, ac mutuus, & coalternus. </s>

<s>quoad <lb/>exempli explicationem vtor figura textibus apponi &longs;olita, quæ e&longs;t præ&longs;ens. <lb/><figure id="fig11"></figure><lb/>probat Euclides in 28. primi Elem. </s>

<s>quoad <lb/>exempli explicationem vtor figura textibus apponi &longs;olita, quæ e&longs;t præ&longs;ens. <lb/><figure id="fig11"></figure><lb/>probat Euclides in 28. primi Elem.

<s>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>

quod &longs;i <lb/>linea recta quædam, vti E F, cadens &longs;uper <lb/>duas rectas, vti &longs;unt A B, C D, fe cerit angu<lb/>los alternos &ecedil;quales, angulos <expan abbr="nimir&utilde;">nimirum</expan> A G H, <lb/>G H D, ij enim dicuntur alterni; &longs;iue alios <lb/>dnos, nimirum B G H, G H C, hi enim &longs;unt <lb/><expan abbr="quoq;">quoque</expan> alterni; probat inquam has duas li<lb/>neas A B, C D, e&longs;&longs;e inuicem parallelas. </s>

<s>Iam &longs;i quis vellet probare, &longs;e duas <lb/>parallelas duxi&longs;&longs;e, hac ratione, quia &longs;cilicet fa ciunt prædictos angulos al<lb/>ternos æquales; & probaret facere angulos alternos æquales, quia &longs;unt pa<lb/>rallelæ, hic peteret principium, ide&longs;t, illud, quod principio <expan abbr="proband&utilde;">probandum</expan> erat, <lb/>afferret pro ratione, & cau&longs;a, quod dicitur peti principium, quia tunc pe<lb/>timus, vt concedatur nobis, id, quod principio, & primo omnium demon<lb/>&longs;trare propo&longs;ueramus. </s>

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<s>Ex cap.

22. lib.

<s>2. Priorum <emph type="italics"/>(Vt &longs;i volens mon&longs;trare, quod diameter e&longs;t incom<lb/>men&longs;. </s>

2. Priorum <emph type="italics"/>(Vt &longs;i volens mon&longs;trare, quod diameter e&longs;t incom<lb/>men&longs;. </s>

<s>argueret Zenonis rationem, quod non e&longs;t moueri)<emph.end type="italics"/> &longs;uperius &longs;ecto 3. lib. </s>

<s>argueret Zenonis rationem, quod non e&longs;t moueri)<emph.end type="italics"/> &longs;uperius &longs;ecto 3. lib.

<s>1. <lb/>fusè explicauimus hanc a&longs;ymmetriam, quam &longs;i quis vellet demon&longs;trare ea<lb/>dem illa ratione, qua Zeno motum impugnabat, quia &longs;cilicet <expan abbr="m&etilde;&longs;ura">men&longs;ura</expan> com<lb/>munis, quæ debet vtramq, quantitatem men&longs;urare, debet in men&longs;urando <lb/>infinitas partes pertran&longs;ire, uimirum medietates medietatum in in&longs;initum, <lb/>e&longs;t autem impo&longs;&longs;ibile pertran&longs;ire infinitas huiu&longs;modi partes, & propterea <lb/>non poterit metiri, <expan abbr="neq;">neque</expan> vnam, <expan abbr="neq;">neque</expan> alteram ex <expan abbr="quãtitatibus">quantitatibus</expan>, quæ putaban<lb/>tur commen&longs;urabiles, afferret hic, inquit Ari&longs;t.

1. <lb/>fusè explicauimus hanc a&longs;ymmetriam, quam &longs;i quis vellet demon&longs;trare ea<lb/>dem illa ratione, qua Zeno motum impugnabat, quia &longs;cilicet <expan abbr="m&etilde;&longs;ura">men&longs;ura</expan> com<lb/>munis, quæ debet vtramq, quantitatem men&longs;urare, debet in men&longs;urando <lb/>infinitas partes pertran&longs;ire, uimirum medietates medietatum in in&longs;initum, <lb/>e&longs;t autem impo&longs;&longs;ibile pertran&longs;ire infinitas huiu&longs;modi partes, & propterea <lb/>non poterit metiri, <expan abbr="neq;">neque</expan> vnam, <expan abbr="neq;">neque</expan> alteram ex <expan abbr="quãtitatibus">quantitatibus</expan>, quæ putaban<lb/>tur commen&longs;urabiles, afferret hic, inquit Ari&longs;t.

non cau&longs;am pro cau&longs;a.</s>

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<s>Ex eodem cap. <emph type="italics"/>(Quoniam idem <expan abbr="vtiq;">vtique</expan> fal&longs;um per plures petitiones accidere <lb/>nihil forta&longs;&longs;e inconueniens, veluti coalternas coincidere; & &longs;i maior e&longs;t extrin&longs;ecus <lb/>angulus intrin&longs;eco; & &longs;i triangulus habet plures rectos duobus)<emph.end type="italics"/> per plures po&longs;i<lb/>tiones &longs;ubaudi fal&longs;as. </s>

<s>per coalternas intellige lineas æquidi&longs;tantes, &longs;eu pa<lb/>rallelas, vt in &longs;uperiori cap. </s>

<s>per coalternas intellige lineas æquidi&longs;tantes, &longs;eu pa<lb/>rallelas, vt in &longs;uperiori cap.

<s>monuimus. </s>

monuimus. </s>

<s>Cæterum Euclides propo&longs;. </s>

<s>28. pri<lb/>mi Elem.

<s>o&longs;tendit, quod &longs;i fuerint duæ parallelæ veluti in præcedenti figura, <lb/>A B, C D, &longs;uper quas alia recta E F, incidat, nece&longs;&longs;ario faciet angulum ex<lb/>trin&longs;ecum E G B, v. </s>

o&longs;tendit, quod &longs;i fuerint duæ parallelæ veluti in præcedenti figura, <lb/>A B, C D, &longs;uper quas alia recta E F, incidat, nece&longs;&longs;ario faciet angulum ex<lb/>trin&longs;ecum E G B, v. </s>

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<s>Ex cap.

<s>26. <emph type="italics"/>(Vt &longs;i A, duo recti, in quo autem P., triangulus, in quo vero C, <lb/>&longs;en&longs;ibuis triangulus, &longs;u&longs;picari <expan abbr="namq;">namque</expan> po&longs;&longs;et aliquis non e&longs;&longs;e C, &longs;ciens, quod omnis <lb/>triangulus habet duos rectos: quare &longs;imul no&longs;cet, & ignorabit idem. </s>

26. <emph type="italics"/>(Vt &longs;i A, duo recti, in quo autem P., triangulus, in quo vero C, <lb/>&longs;en&longs;ibuis triangulus, &longs;u&longs;picari <expan abbr="namq;">namque</expan> po&longs;&longs;et aliquis non e&longs;&longs;e C, &longs;ciens, quod omnis <lb/>triangulus habet duos rectos: quare &longs;imul no&longs;cet, & ignorabit idem. </s>

<s>no&longs;ce enim <lb/>omnem triangulum, quod duobus rectis, non &longs;implex e&longs;t: &longs;ed hoc quidem eo, quod <lb/>vniuer&longs;alem habet &longs;cientiam: illud autem eo, quod &longs;ingularem. </s>

<s>&longs;ic igitur, vt vni<lb/>uer&longs;ale nouit C, quod duo recti; vt autem &longs;ingulare non nouit, quare non habebit <lb/>contrarias)<emph.end type="italics"/> vide, quæ diximus lib. </s>

<s>&longs;ic igitur, vt vni<lb/>uer&longs;ale nouit C, quod duo recti; vt autem &longs;ingulare non nouit, quare non habebit <lb/>contrarias)<emph.end type="italics"/> vide, quæ diximus lib.

<s>1. &longs;ecto 3. cap. </s>

1. &longs;ecto 3. cap.

<s>1. ex quibus quidquid Ma<lb/>thematicum e&longs;t hic, clarum redditur. </s>

1. ex quibus quidquid Ma<lb/>thematicum e&longs;t hic, clarum redditur. </s>

<s>reliqua verò, quæ ad Logicum &longs;pe<lb/>ctant, huius loci commentatores pro&longs;equuntur.</s>

<s>In cap.

<s>31. de Abductione.</s>

31. de Abductione.</s>

<s>Notandum hic cum eruditi&longs;&longs;imo Burana, Abductionem hanc, de qua in hoc <lb/>cap. </s>

<s>Notandum hic cum eruditi&longs;&longs;imo Burana, Abductionem hanc, de qua in hoc <lb/>cap.

<s>agitur e&longs;&longs;e vocem mathematicam, <expan abbr="cam&qacute;">camque</expan>; Ari&longs;t.

agitur e&longs;&longs;e vocem mathematicam, <expan abbr="cam&qacute;">camque</expan>; Ari&longs;t.

quemadmodum multa <lb/>alia à Mathematicis mutuatum ad omnes alias &longs;cientias tran&longs;tuli&longs;&longs;e. </s>

<s>e&longs;&longs;e <pb pagenum="44"/>autem terminum mathematicum colligitur manife&longs;tè ex Proelo, qui lib. </s>

<s>e&longs;&longs;e <pb pagenum="44"/>autem terminum mathematicum colligitur manife&longs;tè ex Proelo, qui lib.

3. <lb/>in comm. </s>

<s>Elem.

<s>Euclidis ad primam propo&longs;itionem primi Elementi, pag. <lb/></s>

Euclidis ad primam propo&longs;itionem primi Elementi, pag. <lb/></s>

<s>121. &longs;ic ait, Abductio verò e&longs;t tran&longs;itus à propo&longs;ito problemate, vel theo<lb/>remate ad aliud, quo cognito, aut comparato Propo&longs;itum quoque per&longs;pi<lb/>cuum e&longs;t. </s>

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<s>in probatione minoris facta e&longs;t Abductio, cum enim ip&longs;e vellet re<lb/>ctificare circumferentiam circuli per lunulas, nec valeret, alij per lineam, <lb/>quandam quadratricem, vt e&longs;t apud Pappum <expan abbr="Alexandrin&utilde;">Alexandrinum</expan>, & apud P. </s>

<s>Cla<lb/>uium in fine &longs;exti Elem. </s>

<s>Cla<lb/>uium in fine &longs;exti Elem.

<s>& alij aliter fru&longs;tra conarentur, facta e&longs;t Abductio <lb/>circa probationem minoris, in qua adhuc Mathematici verfantur; quæ pro<lb/>batio, &longs;i tandem inueniri po&longs;&longs;et, mox &longs;equeretur principale <expan abbr="propo&longs;it&utilde;">propo&longs;itum</expan> pro<lb/>blema, nimirum circulus quadraretur; vide quæ &longs;crip&longs;<gap/>mus in cap. </s>

& alij aliter fru&longs;tra conarentur, facta e&longs;t Abductio <lb/>circa probationem minoris, in qua adhuc Mathematici verfantur; quæ pro<lb/>batio, &longs;i tandem inueniri po&longs;&longs;et, mox &longs;equeretur principale <expan abbr="propo&longs;it&utilde;">propo&longs;itum</expan> pro<lb/>blema, nimirum circulus quadraretur; vide quæ &longs;crip&longs;<gap/>mus in cap.

<s>3. Præ<lb/>dicam. </s>

3. Præ<lb/>dicam. </s>

<s>de hac re, quia plurimum hunc conferunt. </s>

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<s>2. <emph type="italics"/>(Quod enim omne triangulum habet duobus rectis æquales, præ&longs;ciuit: <lb/>quod autem hoc, quod e&longs;t in &longs;emicirculo triangulum e&longs;t, &longs;imul inducens cognouit)<emph.end type="italics"/><lb/>vide primo, quæ &longs;upra libro 1. Prior. </s>

<s>&longs;ecto 3. cap. </s>

<s>&longs;ecto 3. cap.

<s>1. explicaui de angulis <lb/>trianguli. </s>

1. explicaui de angulis <lb/>trianguli. </s>

<s>deinde &longs;cias, quod quando Ari&longs;t.

ait, hoc, quod e&longs;t in &longs;emicir cu<lb/>lo triangulum, &c. </s>

<s>alludit ad demon&longs;trationem quandam, quam ip&longs;e infe<lb/>rius in exemplum adducet, & quæ e&longs;t in 3. Elem. </s>

<s>alludit ad demon&longs;trationem quandam, quam ip&longs;e infe<lb/>rius in exemplum adducet, & quæ e&longs;t in 3. Elem.

<s>Euclidis 31. in qua talis fi<lb/>gura proponitur qualis e&longs;t præ&longs;ens, in qua vides triangulum A B C. in &longs;e<lb/><figure id="fig15"></figure><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="fig15"></figure><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>5. <emph type="italics"/>(Vera quidem igitur oporter e&longs;&longs;e, quoniam non e&longs;t non ens &longs;cire, vt quod <lb/>diameter &longs;it commen&longs;urabi is)<emph.end type="italics"/> con&longs;ule ea, quæ &longs;crip&longs;imus ad cap. </s>

<s>23. primi <lb/>Priorum, &longs;ecto 1. &longs;ine quibus locus hic &longs;atis intelligi nequit; ijs autem per<lb/>ceptis &longs;ic <expan abbr="loc&utilde;">locum</expan> hunc explicare po&longs;&longs;umus, cum diameter quadrati &longs;it incom<pb pagenum="47"/>men&longs;urabilis lateri &longs;ui quadrati, fal&longs;um erit dicere diametrum e&longs;&longs;e com<lb/>men&longs;urabilem prædicto lateri, quod autem fal&longs;um e&longs;t, illud non e&longs;t; igitur <lb/>impo&longs;sibile e&longs;t &longs;cire diametrum e&longs;&longs;e commen&longs;urabile.</s>

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>

<s>Hoc eodem cap. </s>

<s>Hoc eodem cap.

<s>plura dicuntur de Principijs Demon&longs;trationis, &longs;iue &longs;cien<lb/>tiæ, vt &longs;unt Dignitates, Po&longs;itiones, Definitiones, & &longs;imilia, quæ quo modo <lb/>&longs;e habeant, & quo modo illis Demon&longs;trationes innitantur, optimè ex con<lb/>templatione primi libri Elem. </s>

plura dicuntur de Principijs Demon&longs;trationis, &longs;iue &longs;cien<lb/>tiæ, vt &longs;unt Dignitates, Po&longs;itiones, Definitiones, & &longs;imilia, quæ quo modo <lb/>&longs;e habeant, & quo modo illis Demon&longs;trationes innitantur, optimè ex con<lb/>templatione primi libri Elem.

<s>Euclidis percipi pote&longs;t. </s>

Euclidis percipi pote&longs;t. </s>

<s>vt propterea benè ij <lb/>&longs;entiant, inter quos præcipui &longs;unt Toletus, & Zabarella, qui a&longs;&longs;erunt, Ari&longs;t. <lb/></s>

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<s>Aequicrus verò <lb/>babet quidem <expan abbr="quodcunq;">quodcunque</expan> duobus rectis æquales, &longs;ed non primò, &longs;ed triangulum <lb/>prius. </s>

<s>quod igitur quoduis primum mon&longs;tratur duos rectos habens, aut <expan abbr="quodcunq;">quodcunque</expan> <lb/>aliud, huic primo ine&longs;t vniuer&longs;ale, & demonstratio de hoc vniuer&longs;aliter e&longs;t, de alijs <lb/>verò quodammodo, non per &longs;e, <expan abbr="neq;">neque</expan> de æquicrure e&longs;t vniuer&longs;aliter, &longs;ed in plus)<emph.end type="italics"/> pro <lb/>quorum intelligentia nece&longs;&longs;aria &longs;unt ea, quæ primo Priorum &longs;ecto 3. cap. </s>

<s>1. <lb/>&longs;crip&longs;imus. </s>

1. <lb/>&longs;crip&longs;imus. </s>

<s>deinde memineris figuram vniuer&longs;aliorem e&longs;&longs;e triangulo, & tri<lb/>angulum vniuer&longs;alius æquicrure. </s>

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<s>13. <emph type="italics"/>(Si quis igitur mon&longs;trauerit, quod rectæ <expan abbr="nõ">non</expan> coincidunt, videbitur <expan abbr="vtiq;">vtique</expan> <lb/>buius e&longs;&longs;e demonstratio, eo quod in omnibus e&longs;t rectis; non e&longs;t autem: &longs;i quidem <lb/>non quoniam &longs;ic æquales, fit hoc, &longs;ed &longs;ecundum quod <expan abbr="quomodocunq;">quomodocunque</expan> æquales)<emph.end type="italics"/> pro<lb/>ponit tres errores, qui circa demon&longs;trationem de vniuer&longs;ali contingunt, <lb/>quos omnes Geometricis exemplis illu&longs;trat; affert autem primo pro tertio <lb/>errore duo exempla, quorum primum in præmi&longs;&longs;is verbis continetur, <expan abbr="atq;">atque</expan> <lb/>ex 28. primi Elem. </s>

<s>de&longs;umitur, quam propterea primo loco exponendam <lb/><figure id="fig18"></figure><lb/>cen&longs;ui. </s>

de&longs;umitur, quam propterea primo loco exponendam <lb/><figure id="fig18"></figure><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>

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<s>demon&longs;tratio<lb/>ne aut vna, aut altera, quod duos rectos habet vnumquodque, <expan abbr="æquilatei&utilde;">æquilateium</expan> &longs;eor&longs;um, <lb/>& &longs;calenum, & æquicrus: nondum nouit triangulum, quod duobus rectis, <gap/>&longs;i &longs;o<lb/>phi&longs;tico modo, <expan abbr="neq;">neque</expan> vniuer&longs; aliter triangulum, <expan abbr="neq;">neque</expan> &longs;i vllum e&longs;t præter prædicta <lb/>triangulum alterum. </s>

<s>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. </s>

<s>1. &longs;crip&longs;imus de propriet<gap/>te illa trianguli, quod &longs;cilicet habet tres an<lb/>gulos æquales duobus rectis angulis, quibus præmi&longs;&longs;is, &longs;ic deinde locum <lb/>hunc interpretaberis; Propter hoc, quod præcedenti textu dictum e&longs;t; no<lb/>tandum in primo errore vniuer&longs;ale, tanquam &longs;i non e&longs;&longs;et vniuer&longs;ale o&longs;ten<lb/>ditur de &longs;ingulari, &longs;i quis igitur mon&longs;trauerit &longs;ingillatim de <expan abbr="vnoquoq;">vnoquoque</expan> trian<lb/>gulo in &longs;ingulari, &longs;cilicet de vno æquilatero, tantum, & de vno Scaleno, & <lb/>de vno I&longs;o&longs;cele, &longs;eparatim, vtens auteadem demon&longs;trationc dum de <expan abbr="vno&qacute;">vnoque</expan>; <pb pagenum="52"/>&longs;epatatim o&longs;tendit, aut vtens diuerfis demon&longs;trationibus, vna pro æquila<lb/>tero, altera pro I&longs;o&longs;cele, tertia pro Scaleno, o&longs;tendens, quod <expan abbr="vnumquodq;">vnumquodque</expan> <lb/>illorum habet tres angules æquales duobus rectis angulis; i&longs;te nondum no<lb/>uit triangulum omne habere talem affectionem, ni&longs;i modo &longs;ophi&longs;tico, quia <lb/>non cogno&longs;cit hanc affectionem illis <expan abbr="cõpetere">competere</expan> propter naturam illam com<lb/>munem trianguli, cui primo, & per &longs;e competit; & neque vniuer&longs;aliter co<lb/>gno&longs;cit triangulum omne e&longs;&longs;e tale, etiam &longs;i nullum aliud reperiatur trian<lb/>gulum, præter illud æquilaterum, vel illud I&longs;o&longs;celes, vel illud Scalenum, de <lb/>quibus &longs;eparatim <expan abbr="demõ&longs;trauit">demon&longs;trauit</expan>, & &longs;ecundum numernm, ide&longs;t de vnoquoque, <lb/>quatenus e&longs;t vnum numero. </s>

1. &longs;crip&longs;imus de propriet<gap/>te illa trianguli, quod &longs;cilicet habet tres an<lb/>gulos æquales duobus rectis angulis, quibus præmi&longs;&longs;is, &longs;ic deinde locum <lb/>hunc interpretaberis; Propter hoc, quod præcedenti textu dictum e&longs;t; no<lb/>tandum in primo errore vniuer&longs;ale, tanquam &longs;i non e&longs;&longs;et vniuer&longs;ale o&longs;ten<lb/>ditur de &longs;ingulari, &longs;i quis igitur mon&longs;trauerit &longs;ingillatim de <expan abbr="vnoquoq;">vnoquoque</expan> trian<lb/>gulo in &longs;ingulari, &longs;cilicet de vno æquilatero, tantum, & de vno Scaleno, & <lb/>de vno I&longs;o&longs;cele, &longs;eparatim, vtens auteadem demon&longs;trationc dum de <expan abbr="vno&qacute;">vnoque</expan>; <pb pagenum="52"/>&longs;epatatim o&longs;tendit, aut vtens diuerfis demon&longs;trationibus, vna pro æquila<lb/>tero, altera pro I&longs;o&longs;cele, tertia pro Scaleno, o&longs;tendens, quod <expan abbr="vnumquodq;">vnumquodque</expan> <lb/>illorum habet tres angules æquales duobus rectis angulis; i&longs;te nondum no<lb/>uit triangulum omne habere talem affectionem, ni&longs;i modo &longs;ophi&longs;tico, quia <lb/>non cogno&longs;cit hanc affectionem illis <expan abbr="cõpetere">competere</expan> propter naturam illam com<lb/>munem trianguli, cui primo, & per &longs;e competit; & neque vniuer&longs;aliter co<lb/>gno&longs;cit triangulum omne e&longs;&longs;e tale, etiam &longs;i nullum aliud reperiatur trian<lb/>gulum, præter illud æquilaterum, vel illud I&longs;o&longs;celes, vel illud Scalenum, de <lb/>quibus &longs;eparatim <expan abbr="demõ&longs;trauit">demon&longs;trauit</expan>, & &longs;ecundum numernm, ide&longs;t de vnoquoque, <lb/>quatenus e&longs;t vnum numero. </s>

<s>non nouit autem &longs;ecundum &longs;peciem, ideft fecun<lb/>dum naturam, & formam communem illis tribus indiuiduis, quæ e&longs;t natu<lb/>ra trianguli. </s>

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<s>Ibidem <emph type="italics"/>(Propter hoc Geometriæ non licet mon&longs;trare, quod contrariorum vna <lb/>e&longs;e &longs;cientia, &longs;ed neque quod duo cubi cubus)<emph.end type="italics"/> quo ad verba illa, duo cubi cubus<gap/><lb/>quæ ad nos pertinent, vult Ari&longs;t.

docere<gap/>, quod non debet Geometra o&longs;ten<lb/>dere numerorum affectiones (per enbos enim intelligit numeros quo&longs;dam <lb/>&longs;ic dictos, vt paulo po&longs;t o&longs;tendam) vt &longs;i quis vellet geometricè o&longs;tendere id, <lb/>quod o&longs;tenditur in 4. noni Elem. </s>

<s>&longs;cilicet, &longs;i cubus numerus cubum numerum <lb/>multiplicauerit, productus numerus erit pariter cubus. </s>

&longs;cilicet, &longs;i cubus numerus cubum numerum <lb/>multiplicauerit, productus numerus erit pariter cubus. </s>

<s>nonnulli latinorum <lb/>perperam textum hunc expo&longs;uerunt putantes reperiri &longs;olummodo cubos <lb/>geometricos, at Euclides definit. </s>

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<s>Per&longs;pectiua dicit, ea, quæ vi<lb/>dentur eminus videri minora, quam quæ videntur cominus, quia illa viden<lb/>tur &longs;ub angulo minori, hæc verò &longs;ub angulo maiori, quod verò remotiora <lb/>videantur &longs;ub angulo minori, quam propinquiora cæteris paribus probat <lb/><figure id="fig24"></figure><lb/>per 21. primi Elem. </s>

<s>&longs;it enim ma<lb/>gnitudo vi&longs;a A B, remotior ab o<lb/>culo in C, po&longs;ito, & vi&longs;a propin<lb/>quior ab oculo in D. ductis lineis <lb/>vi&longs;ualibus C A, C B: D A, D B; ab <lb/>oculis C, & D, ad extremitates <lb/>&longs;pectatæ magnitudinis, erit remo<lb/>tioris vi&longs;ionis angulus C, minor <lb/>angulo D, propinquioris, vt ex præallegata Demon&longs;tratione pater. </s>

&longs;it enim ma<lb/>gnitudo vi&longs;a A B, remotior ab o<lb/>culo in C, po&longs;ito, & vi&longs;a propin<lb/>quior ab oculo in D. ductis lineis <lb/>vi&longs;ualibus C A, C B: D A, D B; ab <lb/>oculis C, & D, ad extremitates <lb/>&longs;pectatæ magnitudinis, erit remo<lb/>tioris vi&longs;ionis angulus C, minor <lb/>angulo D, propinquioris, vt ex præallegata Demon&longs;tratione pater. </s>

<s>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>

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

<s>Dico ag<lb/>gregatum <expan abbr="tri&utilde;">trium</expan> ip&longs;ius angulorum A, B, C, <lb/>e&longs;&longs;e æquale aggregato ex duobus angu<lb/>lis rectis (vt autem melius intelligas, quæ <lb/>&longs;equuntur, lege prius ea, quæ dicta &longs;unt <lb/>in lib. </s>

<s>1. Priorum &longs;ecto 3. cap. </s>

1. Priorum &longs;ecto 3. cap.

<s>1.) produ<lb/>catur latus B C, <expan abbr="v&longs;q;">v&longs;que</expan> in D, vt fiat angulus <lb/>externus A C D; Iam &longs;ic, quoniam <expan abbr="pro-bat&utilde;">pro<lb/>batum</expan> e&longs;t in 13. primi, duos angulos, quos <lb/>facit linea A C, cum linea B D, &longs;cilicet angulos A C B, A C D, e&longs;&longs;e pares <lb/>duobus rectis: & quia pariter in prima parte huins propo&longs;. </s>

1.) produ<lb/>catur latus B C, <expan abbr="v&longs;q;">v&longs;que</expan> in D, vt fiat angulus <lb/>externus A C D; Iam &longs;ic, quoniam <expan abbr="pro-bat&utilde;">pro<lb/>batum</expan> e&longs;t in 13. primi, duos angulos, quos <lb/>facit linea A C, cum linea B D, &longs;cilicet angulos A C B, A C D, e&longs;&longs;e pares <lb/>duobus rectis: & quia pariter in prima parte huins propo&longs;. </s>

<s>32. probatum <lb/>e&longs;t ab Euclide duos angulos A B, e&longs;&longs;e æquales externo angulo A C D: &longs;i ter<lb/>tius angulus reliquus A C B, &longs;umatur bis, &longs;emel cum duobus angulis A, B, <lb/>& &longs;emel cum externo A C D, <expan abbr="add&etilde;tur">addentur</expan> æqualia æqualibus, & propterea tres <lb/>anguli A, B, A C B, &longs;imul &longs;umpti, erunt æquales duobus A C D, A C B, &longs;imul <lb/>&longs;umptis; &longs;ed his duobus &longs;unt æquales duo recti, ergo cum quæ &longs;unt æqualia <lb/>vni tertio, &longs;int etiam æqualia inuicem, erit aggregatum trium angulorum <lb/>A, B, A C B, æquale aggregato duorum rectorum; quod erat demon&longs;tran<lb/>dum. </s>

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<s>20. at<lb/>tulimus; nunc Mechanicæ &longs;ubalternationis, quam hic Ari&longs;t.

in&longs;inuat, exem<lb/>plum &longs;it illud, quod Archimedes prop. </s>

in&longs;inuat, exem<lb/>plum &longs;it illud, quod Archimedes prop.

<s>14. primi Aequep. </s>

14. primi Aequep. </s>

<s>demon&longs;trat, ni<lb/>mirum centrum grauitatis omnis trianguli e&longs;&longs;e punctum illud, in quo rectæ <lb/>lineæ ab angulis trianguli ad dimidia latera oppo&longs;ita ductæ concurrunt. </s>

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<s>porrò quid inter multiplicem, & multipli<lb/>catam rationem inter&longs;it, optimè declarat no&longs;ter Clauius ad 4. definit. </s>

<s>lib.

5. <lb/>Elem.

<s>ex quo etiam loco pauca decerpam, quæ huic loco declarando con<lb/>ducunt. </s>

ex quo etiam loco pauca decerpam, quæ huic loco declarando con<lb/>ducunt. </s>

<s>Proportio igitur multiplex e&longs;t habitudo inter duas quantitates in<lb/>æquales, quarum maior continet minorem, bis, vel ter, vel quater, &c. </s>

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<s>37. (<emph type="italics"/>Vt æquicruri, & Scaleno hoc, quod e&longs;t duobus rectis æquales habere <lb/>&longs;ecandum commune aliquod ine&longs;t<emph.end type="italics"/>) quid &longs;it habcre tres æquales duobus rect<gap/><lb/>&longs;atis explicatum e&longs;t lib. </s>

r. </s>

<s>Priorum &longs;ecto 3. cap. </s>

<s>Priorum &longs;ecto 3. cap.

r. </s>

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

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<s>quid &longs;it habere tres æqua<lb/>les duobus rectis, &c. </s>

<s>fusè explicatum e&longs;t in lib. </s>

<s>fusè explicatum e&longs;t in lib.

<s>1. <lb/>Priorum &longs;ecto 3. cap. </s>

1. <lb/>Priorum &longs;ecto 3. cap.

<s>1. quò te nunc mitto.</s>

1. quò te nunc mitto.</s>

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<s>Eodem tex <emph type="italics"/>(Quando igitur cognofcimes, quod<lb/>quatuor exteriores &longs;unt æquales, quoniam I&longs;o&longs;celes, <lb/>adhuc defseit, propier quid I&longs;o&longs;celes? </s>

<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. </s>

<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 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 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>

<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>Eodem tex. <emph type="italics"/>(Vt &longs;i quis nouit, quod omnis triangulus habet tres duobus rectis <lb/>æquales)<emph.end type="italics"/> nihil &longs;requentius. </s>

<s>vide &longs;upra lib. </s>

<s>vide &longs;upra lib.

<s>1. Priorum &longs;ecto 3. cap. </s>

1. Priorum &longs;ecto 3. cap.

1.</s>

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<s>43. <emph type="italics"/>(Sed planum, quod et&longs;i e&longs;&longs;et &longs;entire triangulum, quod duobus rectis <lb/>æquales habet angulos)<emph.end type="italics"/> vide &longs;upra lib. </s>

<s>43. <emph type="italics"/>(Sed planum, quod et&longs;i e&longs;&longs;et &longs;entire triangulum, quod duobus rectis <lb/>æquales habet angulos)<emph.end type="italics"/> vide &longs;upra lib.

<s>1. Priorum &longs;ecto 3. cap. </s>

1. Priorum &longs;ecto 3. cap.

1.</s>

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<s>Cæterum per prin<lb/>cipia, ex quibus intelligit Dignitates, quia ex illis di&longs;currimus. </s>

<s>per princi<lb/>pia verò circa quod, intelligit Definitiones, quibus, vt apparet apud Eucli<lb/>dem, explicatur &longs;ubiectum, circa quod &longs;cientia ver&longs;atur; vt in definitioni<lb/>bus primi Elem. </s>

<s>docemur, quid &longs;it linea, quid triangulum, quid circulus, <lb/>quid magnitudines reliquæ, quæ &longs;unt materia, circa quam Geometria &longs;pe<lb/>culatur. </s>

docemur, quid &longs;it linea, quid triangulum, quid circulus, <lb/>quid magnitudines reliquæ, quæ &longs;unt materia, circa quam Geometria &longs;pe<lb/>culatur. </s>

<s>In &longs;eptimo verò traduntur definitiones numerorum, quid &longs;it nu<lb/>merus, quid impar, quid compo&longs;itus, quadratus, cubus, & reliquæ nume<lb/>rorum &longs;pecies, quæ &longs;unt materia &longs;eptimi, octaui, & noni, in quibus de Arith<lb/>metica tractatur.</s>

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<s>44. <emph type="italics"/>(Commen&longs;urabilem namq e&longs;&longs;e diametrum verè opinari, ab&longs;urdum e&longs;t)<emph.end type="italics"/><lb/>vide, quæ de <expan abbr="comm&etilde;&longs;urabilitate">commen&longs;urabilitate</expan> diametri quadrati cum latere expo&longs;uimus <lb/>lib. </s>

<s>1. Priorum &longs;ecto 1. cap. </s>

1. Priorum &longs;ecto 1. cap.

<s>23. ait igitur Ari&longs;t.

23. ait igitur Ari&longs;t.

ab&longs;urdum e&longs;&longs;e opinari dia<lb/>metrum e&longs;&longs;e commen&longs;urabilem co&longs;tæ, &longs;eu lateri eiu&longs;dem quadrati, reli<lb/>qua &longs;unt Logica.</s>

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<s>in quo Zabarella non probatur, qui &longs;olum <lb/>ait, terram e&longs;&longs;e in medio mundi, à Phy&longs;icis demon&longs;trari. </s>

<s><expan abbr="triãgulum">triangulum</expan> autem, <lb/>&longs;eu angulorum ip&longs;ius <expan abbr="prædicat&utilde;">prædicatum</expan> e&longs;t æqualitas, & inæqualitas: vt cum in 32. <lb/>primi Elem. </s>

<s>demon&longs;trat Euclides, omne triangulum habcre tres angulos <lb/>æquales duobus rectis.</s>

demon&longs;trat Euclides, omne triangulum habcre tres angulos <lb/>æquales duobus rectis.</s>

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<s>2. <emph type="italics"/>(Vt quod omnis triangulus duobus rectis æquales babet)<emph.end type="italics"/> vide anno<lb/>tata lib. </s>

<s>2. <emph type="italics"/>(Vt quod omnis triangulus duobus rectis æquales babet)<emph.end type="italics"/> vide anno<lb/>tata lib.

<s>1. Priorum &longs;ecto 3. cap. </s>

1. Priorum &longs;ecto 3. cap.

1.</s>

<s>Eodem tex. <emph type="italics"/>(Definitiones verò apparent omnes &longs;upponentes, & accipientes <lb/>ip&longs;um quid e&longs;t, vt Mathematicæ, quid vnitas, quid par, & impar)<emph.end type="italics"/> alludit ad de<lb/>finitiones 7. Elem. </s>

<s>vbi agitur de numeris. </s>

vbi agitur de numeris. </s>

<s>Quæ verò hoc loco de principijs <lb/>dicuntur, luculenti&longs;&longs;imè patent con&longs;ideranti definitiones, & axiomata, quæ <lb/>Mathematicis demon&longs;trationibus in omnibus ferè libris præmittuntur; ex <lb/>quibus &longs;tatim demon&longs;trationes deriuantur.</s>

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<s>Et paulo po&longs;t <emph type="italics"/>(<expan abbr="Neq;">Neque</expan> <expan abbr="vtiq;">vtique</expan> de plano figura, non enim e&longs;t planum figura, <expan abbr="neq;">neque</expan> fi<lb/>gura planum)<emph.end type="italics"/> alludit ad definitiones planarum figurarum, qualis e&longs;t circu<lb/>lus, cuius definitio e&longs;t inter definitiones primi Elem. </s>

<s>15. & e&longs;t huiu&longs;modi: <lb/>circulus e&longs;t figura plana, &longs;ub vnica linea comprehen&longs;a, quæ periphæria ap<lb/>pellatur, ad quam ab vno puncto eorum, quæ intra figuram &longs;unt po&longs;ita, ca<lb/>dentes omnes rectæ lineæ inter &longs;e &longs;unt æquales: in qua quidem definitione <lb/>non prædicatur planum de figura, nec figura de plano: <expan abbr="neq;">neque</expan> enim planum, <lb/>&longs;au plana &longs;uperficies e&longs;t figura &longs;ecundum &longs;e, ni&longs;i terminetur; <expan abbr="neq;">neque</expan> figura e&longs;t <lb/>plana &longs;uperficies, cum plurimæ &longs;int figuræ curuæ, & præterea &longs;olidæ quam<lb/>plurimæ.</s>

15. & e&longs;t huiu&longs;modi: <lb/>circulus e&longs;t figura plana, &longs;ub vnica linea comprehen&longs;a, quæ periphæria ap<lb/>pellatur, ad quam ab vno puncto eorum, quæ intra figuram &longs;unt po&longs;ita, ca<lb/>dentes omnes rectæ lineæ inter &longs;e &longs;unt æquales: in qua quidem definitione <lb/>non prædicatur planum de figura, nec figura de plano: <expan abbr="neq;">neque</expan> enim planum, <lb/>&longs;au plana &longs;uperficies e&longs;t figura &longs;ecundum &longs;e, ni&longs;i terminetur; <expan abbr="neq;">neque</expan> figura e&longs;t <lb/>plana &longs;uperficies, cum plurimæ &longs;int figuræ curuæ, & præterea &longs;olidæ quam<lb/>plurimæ.</s>

<s>Ibidem <emph type="italics"/>(Quoniam mon&longs;tratum e&longs;t I&longs;o&longs;celes habere tres angulos æquales duo<lb/>bus rectis, &longs;i id de omni triangulo mon&longs;tratum &longs;it)<emph.end type="italics"/> ex dictis lib. </s>

<s>1. Priorum &longs;ecto <lb/>3. cap. </s>

1. Priorum &longs;ecto <lb/>3. cap.

<s>1. petatur huius loci declaratio.</s>

1. petatur huius loci declaratio.</s>

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<s>Ibidem <emph type="italics"/>(Quod autem &longs;it, monstrat)<emph.end type="italics"/> vt per&longs;picuum e&longs;t in prima <expan abbr="demõ&longs;tra-tione">demon&longs;tra<lb/>tione</expan> primi Elem. </s>

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

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>

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<s>Cap. 13. (<emph type="italics"/>Con&longs;iderare, quod diameter est co&longs;tæ incommen&longs;urabilis<emph.end type="italics"/>) vide <lb/>quæ de hac re &longs;crip&longs;i lib. </s>

<s>Cap. 13. (<emph type="italics"/>Con&longs;iderare, quod diameter est co&longs;tæ incommen&longs;urabilis<emph.end type="italics"/>) vide <lb/>quæ de hac re &longs;crip&longs;i lib.

<s>1. Priorum &longs;ecto 1. cap. </s>

1. Priorum &longs;ecto 1. cap.

23.</s>

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<s>mu&longs;ica<lb/>lium, ait, &longs;i virga celerius feriat aerem, gigni motum celeriorem in aere, <lb/><expan abbr="atq;">atque</expan> hinc &longs;onum acutiorem reddi, experientia con&longs;tat: &longs;i autem eadem vir<lb/>ga tardius aerem feriat, gigni motum in aere tardiorem, ex quo etiam &longs;o<lb/>num grauem, vt experientia docet. </s>

<s>Ptolæmeus deinde lib. </s>

<s>Ptolæmeus deinde lib.

1. cap.

3. Harm. <lb/></s>

<s>cum ex alijs, tum ex celeritate oriri &longs;onum acutum, grauem verò ex tardi<lb/>tate a&longs;&longs;erit; vt &longs;i chorda eadem parum inten&longs;a pul&longs;etur, tardius aerem ver<lb/>berat, & ideo grauiorem &longs;onum efficit: &longs;i autem magis intendatur, validius <lb/>aerem pul&longs;abit, & proinde citiorem motum illi imprimet, & propterea <lb/>acutiorem &longs;onum reddet. </s>

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<s>ab hac &longs;ententia po&longs;teriores Mu&longs;ici non recel<lb/>&longs;erunt, vt videre e&longs;t apud Zarlinum.</s>

<s>In quo po&longs;tea con&longs;i&longs;tat ratio acnti anguli, explicat <expan abbr="induc&etilde;s">inducens</expan> definitionem <lb/>ip&longs;ius, quæ e&longs;t inter de&longs;initiones primi Elem. </s>

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

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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innuit. </s>

<s>quod genus for&longs;itan à calore denomina<lb/>batur, quòd ip&longs;ius notæ mu&longs;icales e&longs;&longs;ent coloratæ, vt hoc modo ab alijs ge<lb/>neribus digno&longs;eeretur<gap/> quam con&longs;uetudinem exi&longs;timat Zarlinus cap. </s>

<s>46. &longs;e<lb/>cundæ partis, etiam no&longs;tra tempe&longs;tate aliquo modo per&longs;euerare, cum vi<lb/>deamus in organis, & alijs huiu&longs;modi in&longs;trumentis, quæ pinnas, vulgò ta<lb/>&longs;tos, habent; illas inquam pinnas, quæ chromaticis interuallis deputatæ <lb/>&longs;unt, colore nigro tinctas e&longs;&longs;e.</s>

46. &longs;e<lb/>cundæ partis, etiam no&longs;tra tempe&longs;tate aliquo modo per&longs;euerare, cum vi<lb/>deamus in organis, & alijs huiu&longs;modi in&longs;trumentis, quæ pinnas, vulgò ta<lb/>&longs;tos, habent; illas inquam pinnas, quæ chromaticis interuallis deputatæ <lb/>&longs;unt, colore nigro tinctas e&longs;&longs;e.</s>

<s><emph type="italics"/>Libro Quarto.<emph.end type="italics"/></s>

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<s>Cap. 2. loco 32. (<emph type="italics"/>Vt qui lineam definiunt longitudmem &longs;ine latitudine e&longs;&longs;e<emph.end type="italics"/>) <lb/>&longs;uppenimus lectorem inteil exi&longs;&longs;e definitiones &longs;altem primi Elem. </s>

<s>in<lb/>ter quas d<gap/>f<gap/>nitio lineæ e&longs;t &longs;ecunda, <expan abbr="cadem&qacute;">cademque</expan>; cum hac Ari&longs;totelis.</s>

in<lb/>ter quas d<gap/>f<gap/>nitio lineæ e&longs;t &longs;ecunda, <expan abbr="cadem&qacute;">cademque</expan>; cum hac Ari&longs;totelis.</s>

<s><emph type="italics"/>Libro Octauo.<emph.end type="italics"/></s>

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<s>Cap. 2. loco 41. (<emph type="italics"/>Videntur dutem in di&longs;ciplinis, &longs;eu Mathematicis quædam <lb/>ob de&longs;initionis defectum non facile de&longs;cribi; vt & quoniam, quæ ad latus &longs;e<lb/>cat planum linea, &longs;imiliter diuidit & lineam, & locum: definitione autem dicta, <lb/>fi <gap/>m man fe&longs;tu<gap/>t&longs;t, quod dicitur, nam eandem ablationem habent lota, & <gap/>inea, <lb/><gap/>se latus planæ figuræ, est autem definitio eiu&longs;dem proportionis hæc<emph.end type="italics"/>) mendosè <lb/>lugitur à nonnullis (<emph type="italics"/>E&longs;t distem de&longs;initio eiu&longs;dem orationis hæc<emph.end type="italics"/>) quos puto de<lb/>ceptos ab æquiuoco <foreign lang="greek">lsgous</foreign> quod & orationem, & rationem, &longs;iue proportio<lb/>nem &longs;ignificat: hic autem &longs;ignificare proportionem res &longs;ubrecta &longs;atis mani<lb/>fe&longs;tat. </s>

<s>Notandum po&longs;tea cum Alexandro (quod & &longs;uperius alias commo<lb/>nui in cap. </s>

<s>Notandum po&longs;tea cum Alexandro (quod & &longs;uperius alias commo<lb/>nui in cap.

<s>de Priori, & alibi) per verbum (De&longs;cribi) fignificari hoc loco <lb/>geometricè demon&longs;trare, quoniam Geometræ <expan abbr="nõ">non</expan> ni&longs;i adhibit is de&longs;criptio<lb/>nibus, &longs;<gap/>u figutis demon&longs;trant. </s>

de Priori, & alibi) per verbum (De&longs;cribi) fignificari hoc loco <lb/>geometricè demon&longs;trare, quoniam Geometræ <expan abbr="nõ">non</expan> ni&longs;i adhibit is de&longs;criptio<lb/>nibus, &longs;<gap/>u figutis demon&longs;trant. </s>

<s>Vult autem Ari&longs;t.

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<s>Cap. 10. <emph type="italics"/>(Nam p&longs;eudograpbiæ non contentio&longs;æ (&longs;ecundum enim ea, quæ <lb/>&longs;ub arte &longs;unt, captio&longs;æ &longs;unt ratiocinationes) <expan abbr="neq;">neque</expan> &longs;i aliqua e&longs;t p&longs;eudogra<lb/>phia circa verum, vt Hippocratis quadratura, quæ per lunulas, &longs;ed, vt <lb/>Bry&longs;&longs;o quadrauit circulum; & tamet&longs;i quadretur circulus, quia tamen <lb/>non &longs;ecundum rem, ideo &longs;ophi&longs;ticus)<emph.end type="italics"/> qua ratione Hippocrates orbi quadrum <lb/>exhibere æquale tentauerit, explicatum e&longs;t abundè in 2. Priorum cap. </s>

<s>31. <lb/>& quo itidem modo Bry&longs;&longs;o lib. </s>

31. <lb/>& quo itidem modo Bry&longs;&longs;o lib.

<s>1. Po&longs;ter. </s>

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

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<s>Ibidem <emph type="italics"/>(Bry&longs;&longs;onis autem contentio&longs;a: & illam quidem non e&longs;t transferre, ni&longs;i <lb/>ad Geometriam &longs;olum; eo quod ex proprijs &longs;it principijs)<emph.end type="italics"/> <expan abbr="quãdo">quando</expan> ait <emph type="italics"/>(& illam qui<lb/>dem)<emph.end type="italics"/> intelligit quadrationem Hippocratis. </s>

<s>vide 2. Prior cap. </s>

<s>vide 2. Prior cap.

<s>31. & quæ pau<lb/>lo ante in præcedentibus locis diximus.</s>

31. & quæ pau<lb/>lo ante in præcedentibus locis diximus.</s>

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<s>&longs;e<lb/>ctiones appellat, orbem quadrare tentabat. </s>

<s>Eius den on&longs;trationem expli<lb/>caui ad cap. </s>

<s>Eius den on&longs;trationem expli<lb/>caui ad cap.

<s>31. de Abductione in 2. Priorum, quam inibi videas. </s>

31. de Abductione in 2. Priorum, quam inibi videas. </s>

<s>hoc &longs;olum <lb/>hic notandum pertinere ad Geometram, ip&longs;am refellere, quia ex fal&longs;a qua<lb/>dam præmi&longs;&longs;a ex Geometria de&longs;umpta, ratiocinabatur, idcirco debet (in<lb/>quit Ari&longs;t.) Geometra illius deceptionem inuenire. </s>

<s>Tetragoni&longs;mum autem <lb/>Antiphontis non e&longs;t Geometræ <expan abbr="cõfutare">confutare</expan>, quia aduer&longs;abatur principijs Geo<lb/>metriæ, &longs;upponebat enim circuli circumferentiam ex indiu<gap/>s, <expan abbr="minimis&qacute;">minimisque</expan>; <lb/>lineis rectis componi: cuius fal&longs;am demon&longs;trationem exp<gap/>ram i<gap/>uenies <lb/>ad cap. </s>

<s>10. primi Elench. </s>

10. primi Elench. </s>

<s>po&longs;&longs;umus addere tertiam rat<gap/> <gap/>cet <lb/>Hippocrates non procedebat per communia alijs &longs;ci<gap/>ad <lb/>tex. </s>

<s>23. primi Po&longs;ter. </s>

<s>cap.

<s>8. vbi ip&longs;ius p&longs;endographi<gap/>d<lb/>modum igitur Geometra di&longs;&longs;oluit fal&longs;as tantum<gap/>r<lb/>uatis Geometricis principijs procedunt; non au<gap/> <gap/>riæ <lb/>principia conuellunt: ita Phy&longs;ico non incumbit <expan abbr="cõtra">contra</expan> P<gap/> Me<lb/>li&longs;&longs;um naturæ principia de&longs;truentes di&longs;ceptare, aut falla<gap/> rationes <lb/>coarguere. </s>

8. vbi ip&longs;ius p&longs;endographi<gap/>d<lb/>modum igitur Geometra di&longs;&longs;oluit fal&longs;as tantum<gap/>r<lb/>uatis Geometricis principijs procedunt; non au<gap/> <gap/>riæ <lb/>principia conuellunt: ita Phy&longs;ico non incumbit <expan abbr="cõtra">contra</expan> P<gap/> Me<lb/>li&longs;&longs;um naturæ principia de&longs;truentes di&longs;ceptare, aut falla<gap/> rationes <lb/>coarguere. </s>

<s>Hoc volebat Ari&longs;toteles inferre.</s>

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<s>28. (<emph type="italics"/>Alio autem modo, &longs;orma, & exemplum: bæc autem e&longs;t ratio ip&longs;ius, <lb/>quod quid erat e&longs;&longs;e, & huius genera, vt ip&longs;ius diapa&longs;on duo ad vnum, & omnino <lb/>numerus, & partes, quæ in ratione &longs;unt<emph.end type="italics"/>) vt benè intelligas, quod in præ&longs;enti <lb/>textu <expan abbr="mathematic&utilde;">mathematicum</expan> e&longs;t, con&longs;ule prius, quæ &longs;crip&longs;i ad tex. </s>

<s>1. cap.

<s>primi 2. Po<lb/>&longs;ter. </s>

primi 2. Po<lb/>&longs;ter. </s>

<s>&longs;uper verba illa (<emph type="italics"/>Quid e&longs;t con&longs;onan<gap/>ia?<emph.end type="italics"/>) vbi per&longs;picuè videbis, cur <expan abbr="con-&longs;onãtiæ">con<lb/>&longs;onantiæ</expan>, quæ dicitur Diapa&longs;on, e&longs;&longs;entia, & definitio &longs;it ip&longs;a proportio dupla, <lb/>quæ &longs;ub his num. </s>

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<s>cum velit Ari&longs;t.

o&longs;tendere nece&longs;&longs;ita<lb/>tem, quæ in &longs;cientijs inter præmi&longs;&longs;as, &longs;cu medium, & conclu&longs;ionem reperi<lb/>tur, affert exemplum illud mathematicum &longs;ibi familiare, demon&longs;trationem <lb/>&longs;cilicet illam, qua o&longs;tenditur, omne triangulum habere tres angulos æqua<lb/>les duobus rectis angulis, cuius fu&longs;i&longs;&longs;imam explicationem inuenies &longs;upra in <lb/>primo Priorum, &longs;ecto 3. cap. </s>

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

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 id="fig33"></figure><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.

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<s>26. <emph type="italics"/>(Et hi quidem infinitum e&longs;&longs;e par; hoc enim compræhen&longs;um, & <lb/>ab impari terminatum tribuit ijs, quæ &longs;unt, infinitatem. </s>

<s>&longs;ignum autem <lb/>huius id e&longs;&longs;e, quod contingit in numcris, circumpo&longs;it is enim Gnomoni<lb/>bus circa vnum, & &longs;eor&longs;um, aliquando quidcm &longs;emper aliam fieri &longs;pe<lb/>ciem, aliquando autem vnam)<emph.end type="italics"/> vt melius percipiantur ea, quæ &longs;equuntur, lege <lb/>prius, quæ in cap. </s>

<s>de Motu in po&longs;t prædicamentis &longs;crip&longs;i de Gnomone, ad <lb/>&longs;imilitudinem enim Gnomonis illius Geometrici, inueniuntur etiam in nu<lb/>meris Gnomones Arithmetici. </s>

de Motu in po&longs;t prædicamentis &longs;crip&longs;i de Gnomone, ad <lb/>&longs;imilitudinem enim Gnomonis illius Geometrici, inueniuntur etiam in nu<lb/>meris Gnomones Arithmetici. </s>

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

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<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="fig35"></figure> non refert Gnomonem, quia lateribus in&ecedil;qualibus con<lb/>&longs;tat; <expan abbr="neq;">neque</expan> &longs;i hoc modo <figure id="fig36"></figure> 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. </s>

<s>quamuis theorematicè &longs;it axioma. <lb/></s>

quamuis theorematicè &longs;it axioma. <lb/></s>

<s>hunc porrò numerum parem dicebant terminatum e&longs;&longs;e ab impari, quia ori<lb/>tur ex diui&longs;ione cuiu&longs;uis rei, quæ vna &longs;it, &longs;umentes vnitatem pro impari. <lb/></s>

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<s>exponit. </s>

<s>alio præterea modo vtuntur infinito, vt quando &longs;upponunt data <lb/>quauis quantitate po&longs;&longs;e &longs;umi maiorem, vel etiam minorem in infinitum, vt <lb/>patet ex 6. po&longs;tulato primi Elem. </s>

<s>editionis Clauianæ. </s>

editionis Clauianæ. </s>

<s>numerum <expan abbr="quoq;">quoque</expan> au<lb/>geri po&longs;&longs;e in infinitum, e&longs;t &longs;ecundum po&longs;tulatum libri 7. Elem. </s>

<s>numerum <expan abbr="quoq;">quoque</expan> au<lb/>geri po&longs;&longs;e in infinitum, e&longs;t &longs;ecundum po&longs;tulatum libri 7. Elem.

<s>vel demum <lb/>quando probant quamlibet lineam po&longs;&longs;e diuidi bifariam, quia hinc &longs;equitur <lb/>po&longs;&longs;e &longs;ub diuidi in <expan abbr="infinit&utilde;">infinitum</expan>; his igitur modis Mathematicis <expan abbr="infinit&utilde;">infinitum</expan> in v&longs;u e&longs;t.</s>

vel demum <lb/>quando probant quamlibet lineam po&longs;&longs;e diuidi bifariam, quia hinc &longs;equitur <lb/>po&longs;&longs;e &longs;ub diuidi in <expan abbr="infinit&utilde;">infinitum</expan>; his igitur modis Mathematicis <expan abbr="infinit&utilde;">infinitum</expan> in v&longs;u e&longs;t.</s>

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<s>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. </s>

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

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

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<s>15. <emph type="italics"/>(Etenim triangulus habet tres angulos æquales duobus rectis angulis)<emph.end type="italics"/><lb/>lib. </s>

<s>15. <emph type="italics"/>(Etenim triangulus habet tres angulos æquales duobus rectis angulis)<emph.end type="italics"/><lb/>lib.

<s>1. Priorum, &longs;ecto 3. cap. </s>

1. Priorum, &longs;ecto 3. cap.

<s>1. huius rei explicationem reperies.</s>

1. huius rei explicationem reperies.</s>

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

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<s>10. primi Elem. </s>

<s>10. primi Elem.

<s>quæ docet <lb/>quamlibet lineam po&longs;&longs;e diuidi bifariam nulla e&longs;&longs;et, quia linea illa, quæ con<lb/>&longs;taret ex tribus Democriti atomis, nulla ratione bifariam &longs;ecari po&longs;&longs;et. </s>

quæ docet <lb/>quamlibet lineam po&longs;&longs;e diuidi bifariam nulla e&longs;&longs;et, quia linea illa, quæ con<lb/>&longs;taret ex tribus Democriti atomis, nulla ratione bifariam &longs;ecari po&longs;&longs;et. </s>

<s>pa<lb/>riter totus ferè decimus liber Elem. </s>

<s>pa<lb/>riter totus ferè decimus liber Elem.

<s>deceptiuus, & nullus e&longs;&longs;et, &longs;i enim da<lb/>rentur illæ atomi, ex quibus <expan abbr="quãtitas">quantitas</expan> conflaretur, nullæ e&longs;&longs;ent lineæ incom<lb/>men&longs;urabiles, quandoquidem omnes communi illa, ac indiuidua, commen<lb/>&longs;urarentur. </s>

deceptiuus, & nullus e&longs;&longs;et, &longs;i enim da<lb/>rentur illæ atomi, ex quibus <expan abbr="quãtitas">quantitas</expan> conflaretur, nullæ e&longs;&longs;ent lineæ incom<lb/>men&longs;urabiles, quandoquidem omnes communi illa, ac indiuidua, commen<lb/>&longs;urarentur. </s>

<s>po&longs;tulatum <expan abbr="quoq;">quoque</expan> illud, qualibet data magnitudine &longs;umi po&longs;&longs;e <lb/>minorem pror&longs;us irritum redderetur, quia data atomo, illa minor accipi <lb/>non po&longs;&longs;et.</s>

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<s>48. <emph type="italics"/>(Nihil autem refert grauitates, commen&longs;urabiles &longs;int, an incommen<lb/>&longs;ur abiles)<emph.end type="italics"/> quidnam &longs;it commen&longs;urabilitas, & incommen&longs;urabilitas, expli<lb/>catum e&longs;t lib. </s>

<s>1. Priorum, &longs;ecto 1. cap. </s>

1. Priorum, &longs;ecto 1. cap.

23.</s>

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<s>119. <emph type="italics"/>(Est autem impo&longs;&longs;ibile, & po&longs;&longs;ibile; fal&longs;um, & verum, ex &longs;uppo&longs;itio<lb/>ne quidem, dico autem, vt triangulum impo&longs;&longs;ibile e&longs;t duos rectos habere, &longs;i hæc)<emph.end type="italics"/><lb/>ide&longs;t, &longs;i &longs;upponantur fal&longs;a quædam, quæ &longs;upponi po&longs;&longs;unt, &longs;equetur impo&longs;&longs;i<lb/>bile e&longs;&longs;e triangulum habere tres angulos æquales duobus rectis angulis, vi<lb/>de, quæ &longs;crip&longs;i lib. </s>

<s>1. Priorum, &longs;ecto 3. cap. </s>

1. Priorum, &longs;ecto 3. cap.

<s>1. de hoc, quod e&longs;t, habere tres <lb/>angulos æquales duobus rectis. </s>

1. de hoc, quod e&longs;t, habere tres <lb/>angulos æquales duobus rectis. </s>

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<s>Ibidem <emph type="italics"/>(Et diameter commen&longs;urabilis est co&longs;tæ, &longs;i bæc)<emph.end type="italics"/> vide primo Priorum, <lb/>&longs;ecto 3. cap. </s>

<s>Ibidem <emph type="italics"/>(Et diameter commen&longs;urabilis est co&longs;tæ, &longs;i bæc)<emph.end type="italics"/> vide primo Priorum, <lb/>&longs;ecto 3. cap.

<s>23. hoc &longs;olum nunc addendum <emph type="italics"/>(Si hæc)<emph.end type="italics"/> v. </s>

23. hoc &longs;olum nunc addendum <emph type="italics"/>(Si hæc)<emph.end type="italics"/> v. </s>

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<s>&longs;i autem &longs;ecundum triangulum, vnum. <lb/></s>

<s>circulus non erit figura)<emph.end type="italics"/> In ordine figurarum conueniens e&longs;t, inquit, primam <lb/>facere circulum propter &longs;implici&longs;simam ip&longs;ius naturam, cum vnica, ac per<lb/>fecta circulari linea comprehendatur: <expan abbr="Triangul&utilde;">Triangulum</expan> verò &longs;ecundam, quoniam <lb/>duo anguli recti, ide&longs;t, quia triangulum habet tres angulos æquales duobus <lb/>rectis angulis; quod fusè explicatnm e&longs;t lib. </s>

<s>1. Priorum, &longs;ecto 3. cap. </s>

1. Priorum, &longs;ecto 3. cap.

<s>1. De<lb/>mum &longs;i primum locum dederimus triangulo, nullus alius remanet pro cir<lb/>culo, quod e&longs;t inconueniens, ergo circulus prima figura erit.</s>

1. De<lb/>mum &longs;i primum locum dederimus triangulo, nullus alius remanet pro cir<lb/>culo, quod e&longs;t inconueniens, ergo circulus prima figura erit.</s>

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

<s>Clauij &longs;uper 4. Elem. </s>

<s>Clauij &longs;uper 4. Elem.

<s>nos <lb/>ea tantum attingimus, quæ percipi po&longs;&longs;int ab homi<lb/>ne vix mathematicis tincto: &longs;ed tamen, quæ &longs;en&longs;um <lb/>Ari&longs;totelis patefaciunt. </s>

nos <lb/>ea tantum attingimus, quæ percipi po&longs;&longs;int ab homi<lb/>ne vix mathematicis tincto: &longs;ed tamen, quæ &longs;en&longs;um <lb/>Ari&longs;totelis patefaciunt. </s>

<s>Aliæ porrò figuræ replen<lb/>tes locum planum, quibus aliquando Architectores <lb/>vtuntur, vel &longs;unt irregulares, vel ad prædictas redu<lb/>ci po&longs;&longs;unt. </s>

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<s>Verum enim verò plures pyramides <lb/>regulares, &longs;iue plura Tetraedra non po&longs;&longs;e replere vacuum, <expan abbr="&longs;olidum&qacute;">&longs;olidumque</expan>; con<lb/>&longs;tituere, ex eo patet, quia &longs;i id præ&longs;tarent, conflarent nece&longs;&longs;ariò, vel vnum <lb/>ex <expan abbr="quinq;">quinque</expan> corporibus regularibus, de quibus in 13. Elemen. </s>

<s>vel aliud quod<lb/>piam; non aliud, nam, vt patet ex &longs;cholio 13. Elem. </s>

<s>vel aliud quod<lb/>piam; non aliud, nam, vt patet ex &longs;cholio 13. Elem.

<s>non dantur, ni&longs;i illa. <lb/></s>

non dantur, ni&longs;i illa. <lb/></s>

<s>quinque; <expan abbr="neq;">neque</expan> vllum ex illis, quia diameter huiu&longs;modi corporis, quod com<lb/>poneretur ex illis pyramidibus, e&longs;&longs;et dupla lateris eiu&longs;dem, vt patet, quia <lb/>pyramides illæ omnes concurrerent ad centrum &longs;phæræ illas omnes com<lb/>plectentis, quare latus vnius pyramidis à &longs;uperficie &longs;phæræ incipiens de&longs;i<lb/>neret in centrum, ergo latus i&longs;tud e&longs;&longs;et &longs;emidiameter, quapropter tota dia<lb/>meter illius &longs;ph&ecedil;ræ, & con&longs;equenter huius corporis in illa in&longs;cripti, e&longs;&longs;et du<lb/>pla lateris eiu&longs;dem figuræ &longs;olidæ in&longs;criptæ, &longs;ed nullo talis proportio diame<lb/>tri alicuius ex illis <expan abbr="quinq;">quinque</expan> &longs;olidis regularibus ad latus eiu&longs;dem reperitur, <lb/>quæ &longs;it nimirum dupla, vt patet ex vltimis demon&longs;trationibus 13. Elem. </s>

<s>ini<lb/>tio facto à 13. demon&longs;tratione, in quibus nulla reperitur proportio dupla <lb/>inter diametrum, & latus eiu&longs;dem alicuius ex illis &longs;olidis; ex quibus mani<lb/>fe&longs;tum e&longs;t, plures regulares pyramides quouis pacto &longs;imul vnitas nullo mo<lb/>do replere locum &longs;olidum. </s>

ini<lb/>tio facto à 13. demon&longs;tratione, in quibus nulla reperitur proportio dupla <lb/>inter diametrum, & latus eiu&longs;dem alicuius ex illis &longs;olidis; ex quibus mani<lb/>fe&longs;tum e&longs;t, plures regulares pyramides quouis pacto &longs;imul vnitas nullo mo<lb/>do replere locum &longs;olidum. </s>

<s>cum igitur animaduerterem, &longs;en&longs;um Ari&longs;t.

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<s>tu verò, &longs;i <lb/>adeo es mathematicis imbutus, con&longs;ule po&longs;tremas demon&longs;tra. </s>

<s>13. Elem.

<s>& <lb/>præcipuè &longs;cholium vltimum, vbi plura de his corporibus &longs;citu digni&longs;&longs;ima, <lb/><expan abbr="atq;">atque</expan> huc &longs;pectantia reperies ex his omnibus Mathematica, quæ no&longs;træ &longs;unt <lb/>partes, per&longs;picuè &longs;atis expo&longs;uimus.</s>

& <lb/>præcipuè &longs;cholium vltimum, vbi plura de his corporibus &longs;citu digni&longs;&longs;ima, <lb/><expan abbr="atq;">atque</expan> huc &longs;pectantia reperies ex his omnibus Mathematica, quæ no&longs;træ &longs;unt <lb/>partes, per&longs;picuè &longs;atis expo&longs;uimus.</s>

<s>Multo po&longs;t tempore, quàm hæc &longs;crip&longs;eram incidi fortè in cap. </s>

<s>Multo po&longs;t tempore, quàm hæc &longs;crip&longs;eram incidi fortè in cap.

<s>38. &longs;pecu<lb/>lationem 10. Benedicti de placitis Ari&longs;t. </s>

38. &longs;pecu<lb/>lationem 10. Benedicti de placitis Ari&longs;t. </s>

<s><expan abbr="reperi&qacute;">reperique</expan>; ab eo vno Ari&longs;t.

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<s>71 <emph type="italics"/>(Deinde &longs;i terra e&longs;t cubus &c.)<emph.end type="italics"/> lege definitiones 11. Elem. </s>

<s>71 <emph type="italics"/>(Deinde &longs;i terra e&longs;t cubus &c.)<emph.end type="italics"/> lege definitiones 11. Elem.

<s>quæ &longs;unt <lb/>admodum faciles, ibi reperies de&longs;initiones quinque corporum regularium, <lb/>quorum figuras Plato elementis tribuebat: qua verò id ratione faceret, ha<lb/>bes in &longs;phæra Clau. </s>

quæ &longs;unt <lb/>admodum faciles, ibi reperies de&longs;initiones quinque corporum regularium, <lb/>quorum figuras Plato elementis tribuebat: qua verò id ratione faceret, ha<lb/>bes in &longs;phæra Clau. </s>

<s>Simpl. </s>

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<s>&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>linea autem, quæ <lb/>facit tales angulos tendit ad centrum &longs;phæræ A, vt patet ad &longs;en&longs;um in figu<lb/>ra, & probari pote&longs;t geometricè ex primis tertij Elem. </s>

<s>linea autem, quæ <lb/>facit tales angulos tendit ad centrum &longs;phæræ A, vt patet ad &longs;en&longs;um in figu<lb/>ra, & probari pote&longs;t geometricè ex primis tertij Elem.

<s>ex quibus patet tam <lb/>læuia, quam grauia, quæ per talem lineam ferantur, re&longs;picere centrum A, <lb/>&longs;phæræ. </s>

ex quibus patet tam <lb/>læuia, quam grauia, quæ per talem lineam ferantur, re&longs;picere centrum A, <lb/>&longs;phæræ. </s>

<s>Vtrum autem i&longs;tud centrum &longs;it idem cum <expan abbr="c&etilde;tro">centro</expan> totius mundi, alius, <lb/>inquit, e&longs;t &longs;ermo, hoc e&longs;t, ad a&longs;tronomum pertinet. </s>

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<s><emph type="italics"/>Hoc loco de&longs;ideratur commentarius in cap. </s>

<s><emph type="italics"/>Hoc loco de&longs;ideratur commentarius in cap.

vlt. </s>

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

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<s>Svmma 1. cap. </s>

<s>Svmma 1. cap.

<s>3. <emph type="italics"/>(Moles autem terræ quanta &longs;it ad ambientes magnitudi<lb/>nes, non immamfestum, iam enim vi&longs;um est per a&longs;trologica theoremata, <lb/>quod multò etiam quibu&longs;dam a&longs;tris est minor)<emph.end type="italics"/> Quantitas terræ non &longs;o<lb/>lum ab&longs;olutè con&longs;iderata, ab A&longs;tronomis explorata habetur, vt vi<lb/>dere e&longs;t in &longs;phæra Clauij; &longs;ed etiam re&longs;pectiuè con&longs;iderata, ide&longs;t re&longs;pectu <lb/>aliorum elementorum, & ip&longs;orum etiam a&longs;trorum; cuius demon&longs;trationes <lb/>&longs;unt partim in libello Ari&longs;tarchi Samij, de magnitudine, & di&longs;tantia Solis, <lb/>& Lunæ, partim apud Ptolæmeum in magna Syntaxi, &longs;iue Almage&longs;to: par<lb/>tim apud Albategnium de &longs;cientia &longs;tellarum: partim demum apud Ticho<lb/>nem Brahe. </s>

3. <emph type="italics"/>(Moles autem terræ quanta &longs;it ad ambientes magnitudi<lb/>nes, non immamfestum, iam enim vi&longs;um est per a&longs;trologica theoremata, <lb/>quod multò etiam quibu&longs;dam a&longs;tris est minor)<emph.end type="italics"/> Quantitas terræ non &longs;o<lb/>lum ab&longs;olutè con&longs;iderata, ab A&longs;tronomis explorata habetur, vt vi<lb/>dere e&longs;t in &longs;phæra Clauij; &longs;ed etiam re&longs;pectiuè con&longs;iderata, ide&longs;t re&longs;pectu <lb/>aliorum elementorum, & ip&longs;orum etiam a&longs;trorum; cuius demon&longs;trationes <lb/>&longs;unt partim in libello Ari&longs;tarchi Samij, de magnitudine, & di&longs;tantia Solis, <lb/>& Lunæ, partim apud Ptolæmeum in magna Syntaxi, &longs;iue Almage&longs;to: par<lb/>tim apud Albategnium de &longs;cientia &longs;tellarum: partim demum apud Ticho<lb/>nem Brahe. </s>

<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>

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<s>Summa 1. cap. </s>

<s>Summa 1. cap.

<s>4. <emph type="italics"/>(Quæ igitur astrorum e&longs;t, velox quidem; longè autem: quæ <lb/>verò Lunæ deor&longs;um quidem, tarda autem: quæautem Solis ambo hæc babet &longs;uffi<lb/>cienter)<emph.end type="italics"/> quæ igitur a&longs;trorum, ide&longs;t latio a&longs;trorum e&longs;t velox, &longs;ed procul à ter<lb/>ra; Lunæ verò latio terræ quidem proxima, tarda tamen: at verò Solis la<lb/>tio medio modo &longs;e habet inter vtrumque, ide&longs;t, quia <expan abbr="neq;">neque</expan> nimis vt a&longs;tra di<lb/>&longs;tat, <expan abbr="neq;">neque</expan> tardè &longs;icut Luna circunfertur. </s>

4. <emph type="italics"/>(Quæ igitur astrorum e&longs;t, velox quidem; longè autem: quæ <lb/>verò Lunæ deor&longs;um quidem, tarda autem: quæautem Solis ambo hæc babet &longs;uffi<lb/>cienter)<emph.end type="italics"/> quæ igitur a&longs;trorum, ide&longs;t latio a&longs;trorum e&longs;t velox, &longs;ed procul à ter<lb/>ra; Lunæ verò latio terræ quidem proxima, tarda tamen: at verò Solis la<lb/>tio medio modo &longs;e habet inter vtrumque, ide&longs;t, quia <expan abbr="neq;">neque</expan> nimis vt a&longs;tra di<lb/>&longs;tat, <expan abbr="neq;">neque</expan> tardè &longs;icut Luna circunfertur. </s>

<s>exi&longs;timo Ari&longs;t.

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<s>Summa 2. cap. </s>

<s>Summa 2. cap.

<s>3. <emph type="italics"/>(Quod accidit circa Mercurij stellam, quia enim modicum <lb/><expan abbr="&longs;upera&longs;c&etilde;dis">&longs;upera&longs;cendis</expan>, &longs;æpè non apparet, it a vt po&longs;t tempus multum appareat)<emph.end type="italics"/> quod Mer<lb/>curius non ni&longs;i rarò con&longs;pici po&longs;&longs;it, cau&longs;a e&longs;t, quia parum à Sole elongatur, <lb/>&longs;iue ip&longs;um antecedat, &longs;iue &longs;ub&longs;equatur. </s>

3. <emph type="italics"/>(Quod accidit circa Mercurij stellam, quia enim modicum <lb/><expan abbr="&longs;upera&longs;c&etilde;dis">&longs;upera&longs;cendis</expan>, &longs;æpè non apparet, it a vt po&longs;t tempus multum appareat)<emph.end type="italics"/> quod Mer<lb/>curius non ni&longs;i rarò con&longs;pici po&longs;&longs;it, cau&longs;a e&longs;t, quia parum à Sole elongatur, <lb/>&longs;iue ip&longs;um antecedat, &longs;iue &longs;ub&longs;equatur. </s>

<s>ex quo fit, vt diu ferè &longs;imul cum So<lb/>le cit cumferatur, & propterea &longs;iue oriatur, &longs;iue occidat, parum &longs;upra ho<lb/>rizontem eleuatus apparere pote&longs;t, quod Ari&longs;t.

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<s>optimè expli<lb/>cant, quos tu con&longs;ule, ne actum agatur.</s>

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

<s>4. &longs;ummæ 2. lib. </s>

4. &longs;ummæ 2. lib.

<s>1. Meteor. </s>

1. Meteor. </s>

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

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<s>Summæ 2. cap. </s>

<s>Summæ 2. cap.

<s>5. <emph type="italics"/>(Ad hæc autem &longs;i quemadmodum o&longs;tenditur in ijs, quæ cir<lb/>ca Astrologiam &longs;peculationibus, Solis magnitudo maior e&longs;t quàm terræ; & diftax<lb/>tia multò maior a&longs;trorum ad terram quàm So is; &longs;icut Solis ad terram quàm Lu<lb/>næ; non <expan abbr="vtiq;">vtique</expan> longè alicubi à terra conus, qui à Sole, corijciet radios, <expan abbr="neq;">neque</expan> vtique <lb/>vmbraterræ, quæ vocatur nox, erit apud astra; &longs;ed ne<gap/>&longs;&longs;e Solem omnia a&longs;tra cir<lb/>cun&longs;picere, & nulli <gap/>p&longs;orum terram ob&longs;istere)<emph.end type="italics"/> ex dictis &longs;umma 1. cap. </s>

5. <emph type="italics"/>(Ad hæc autem &longs;i quemadmodum o&longs;tenditur in ijs, quæ cir<lb/>ca Astrologiam &longs;peculationibus, Solis magnitudo maior e&longs;t quàm terræ; & diftax<lb/>tia multò maior a&longs;trorum ad terram quàm So is; &longs;icut Solis ad terram quàm Lu<lb/>næ; non <expan abbr="vtiq;">vtique</expan> longè alicubi à terra conus, qui à Sole, corijciet radios, <expan abbr="neq;">neque</expan> vtique <lb/>vmbraterræ, quæ vocatur nox, erit apud astra; &longs;ed ne<gap/>&longs;&longs;e Solem omnia a&longs;tra cir<lb/>cun&longs;picere, & nulli <gap/>p&longs;orum terram ob&longs;istere)<emph.end type="italics"/> ex dictis &longs;umma 1. cap.

<s>3. huius, <lb/>& ex figura ibi de&longs;cripta, facilè e&longs;t intelligere præ&longs;entem locum; nam cum <lb/>Sol &longs;it multò maior terra, vt ibi probatur, ac minus di&longs;ter à terra quàm fixæ <lb/>&longs;tellæ, magis tamen quàm Luna, vt patet ex &longs;olari eclyp&longs;i, &longs;equitur nece&longs;&longs;a<lb/>riò vmbram terræ, quæ nox e&longs;t ip&longs;a, effici turbinatam, & valdè procul à ter<lb/>ra acumen coni vmbræ a&longs;cendet, &longs;ed paulò &longs;upra Lunam conus hic vmbræ <lb/>permittet radios Solis &longs;e ip&longs;um ambientes iterum &longs;imul committi, quod il<lb/>lis verbis <emph type="italics"/>(Conijciet radios)<emph.end type="italics"/> ide&longs;t committet radios expre&longs;&longs;it Ari&longs;t.

3. huius, <lb/>& ex figura ibi de&longs;cripta, facilè e&longs;t intelligere præ&longs;entem locum; nam cum <lb/>Sol &longs;it multò maior terra, vt ibi probatur, ac minus di&longs;ter à terra quàm fixæ <lb/>&longs;tellæ, magis tamen quàm Luna, vt patet ex &longs;olari eclyp&longs;i, &longs;equitur nece&longs;&longs;a<lb/>riò vmbram terræ, quæ nox e&longs;t ip&longs;a, effici turbinatam, & valdè procul à ter<lb/>ra acumen coni vmbræ a&longs;cendet, &longs;ed paulò &longs;upra Lunam conus hic vmbræ <lb/>permittet radios Solis &longs;e ip&longs;um ambientes iterum &longs;imul committi, quod il<lb/>lis verbis <emph type="italics"/>(Conijciet radios)<emph.end type="italics"/> ide&longs;t committet radios expre&longs;&longs;it Ari&longs;t.

cum igi<lb/>tur vmbra apud Lunam &longs;it &longs;atis gracilis, breui &longs;upra Lunam de&longs;inet, neque <lb/>vllo pacto ad affixa &longs;ydera protendetur, <expan abbr="neq;">neque</expan> illis renebras offundet. </s>

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<s>Summæ 2. cap. </s>

<s>Summæ 2. cap.

<s>6. Sunt qui velint Ari&longs;t. </s>

6. Sunt qui velint Ari&longs;t. </s>

<s>Galaxiam nihil aliud e&longs;&longs;e, quàm <lb/>quandam refractionem lucis &longs;tellarum illarum, quæ &longs;unt in ætherea Gala<lb/>xia, quæ inquam refractio fiat circa &longs;upremam aeris regionem ex occur&longs;u <lb/>exhalationum, quæ ibi perpetuò con&longs;eruantur, & vi earumdem &longs;tellarum <lb/>&longs;ur&longs;um &longs;emper attrahuntur, quæ refractio fiat ad eum modum, quo halo cir<lb/>ca Solem, & Lunam. </s>

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<s>Summa 4. cap. </s>

<s>Summa 4. cap.

<s>1. <emph type="italics"/>(In A&longs;ia igitur plurimi ex Parna&longs;&longs;o vocato monte videntur <lb/>&longs;tuentes)<emph.end type="italics"/> rectè dubitat Alexander, qua ratione mons Parna&longs;&longs;us ab Ari&longs;t.

1. <emph type="italics"/>(In A&longs;ia igitur plurimi ex Parna&longs;&longs;o vocato monte videntur <lb/>&longs;tuentes)<emph.end type="italics"/> rectè dubitat Alexander, qua ratione mons Parna&longs;&longs;us ab Ari&longs;t.

po<lb/>natur in A&longs;ia, cum certò certius con&longs;tet, ip&longs;um in Græcia Europæ regione <lb/>&longs;itum e&longs;&longs;e. </s>

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<s>vide The&longs;aurum geographicum Abrahami Ortelij. </s>

<s>Strabo lib. </s>

<s>Strabo lib.

<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 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 pagenum="98"/>&longs;um, Emodum, Imauum, & alijs nominibus: Macedones verò Cauca&longs;um <lb/>vocant.</s>

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

<s>3. cap.

<s>5. de hoc Ari&longs;t.

5. de hoc Ari&longs;t.

loco &longs;ie loquitur: <lb/>hic locus diligenter expendendus videtur tum quia difficillimus e&longs;t, <expan abbr="t&utilde;">tum</expan> quia <lb/>multis an&longs;am dedit reprehendendi Ari&longs;t.

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<s>vbi animaduertendum expo<lb/>&longs;itionem <expan abbr="hãc">hanc</expan> parùm differre à no&longs;tra modò allata, cùm <expan abbr="vtraq;">vtraque</expan> in idem tem<lb/>pus recidat; nam &longs;i dixerimus initio Crepu&longs;culi matutini illuminari ter<lb/>tiam partem Cauca&longs;i, tempus hoc coincidit cum initio tertiæ partis noctis, <lb/>quantitas enim Crepu&longs;culi in poli eleuatione 47. grad. </s>

<s>qualem habet Cau<lb/>ca&longs;us, per totam æ&longs;tatem tres horas plus minus continet, vt patet ex tabu<lb/>la quantitatis Crepu&longs;culi, quæ e&longs;t apud Nonium, & apud Clauium in &longs;phæ<lb/>ra vltimæ editionis; quæ quantitas reperiri geometrico calculo pote&longs;t, vt <lb/>docent Nonius, Clauius, & Maginus lib. </s>

<s>10. primi mob. </s>

10. primi mob. </s>

<s>quod quidem trium <lb/>circiter horarum tempus e&longs;t tertia ferè noctis pars in ijs regionibus, quibus <lb/>polus eleuatur 47. grad. </s>

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<s>Svmma 4. cap. </s>

<s>Svmma 4. cap.

<s>2. quod e&longs;t de permutatione, & vici&longs;&longs;itudine aquarum, <lb/>& continentis. </s>

2. quod e&longs;t de permutatione, & vici&longs;&longs;itudine aquarum, <lb/>& continentis. </s>

<s>Pergratum Lectori fore exi&longs;timaui, nec alienum ab <lb/>in&longs;tituto, &longs;i occa&longs;ione huius permutationis maris, ac terræ, rem ex<lb/>po&longs;uero &longs;citu digniffimam, quam pridem ob&longs;eruare c&oelig;pi, ac in dies <lb/>ob&longs;eruo, præ&longs;ertim cum nullus præteritorum &longs;criptorum, quod &longs;ciam, eam <lb/>literis mandauerit: Terræ &longs;cilicet totius molem paulatim reduci ad perfe<lb/>ctam &longs;phæricitatem, ita vt aliquando nece&longs;&longs;e &longs;it futurum ip&longs;am à mari inun<lb/>dari, <expan abbr="atq;">atque</expan> omninò inhabitabilem reddi. </s>

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<s>cau&longs;am au<lb/>tem re&longs;tauratri<gap/> huius terrenæ <expan abbr="rot&utilde;ditatis">rotunditatis</expan> e&longs;&longs;e aquas tum pluuiales, tum <lb/>fluuiales iamdiù ob&longs;eruauimus, vt ex &longs;equentibus ob&longs;eruationibus patebit.</s>

<s>Primò, videmus flumina quotidie montium radices corrodere, ac qua&longs;i <lb/>&longs;uffodere, ita vt pa&longs;&longs;im ex hoc, vel illo monte magnas faciant ruinas, ac pr&ecedil;<lb/>cipitia, <expan abbr="atq;">atque</expan> hiac inde prærupti appareant montes, vt meritò legamus apud <lb/>Iob cap. </s>

<s>14. allunione paulatim terra con&longs;umitur. </s>

14. allunione paulatim terra con&longs;umitur. </s>

<s>humum porrò illam ex <lb/>montibus delap&longs;am &longs;emper ad loca humiliora fluuij &longs;ecum detrahunt. </s>

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<s>Ad hæc; antiqua in montium verticibus con<lb/>&longs;tituta ædeficia, propterea intercidunt, quia terra hinc, & inde ab aquis <lb/>paulatim confumpta, <expan abbr="deor&longs;um&qacute;">deor&longs;umque</expan>; delap&longs;a, fundamenta ip&longs;orum nuda primò <lb/>relinquit; deindé terra etiam ip&longs;a, qua fundamenta innitebatur &longs;en&longs;im de<lb/>lap&longs;a, ip&longs;a <expan abbr="quoq;">quoque</expan> fundamenta vnà cum toto ædeficio nece&longs;&longs;e e&longs;t collabi, hu<lb/>ius &longs;igna infinita propemodum videri po&longs;&longs;unt; vnum tamen, quod toti orbi <lb/>confpicuum e&longs;t, non ommittam; Capitolium videlicet Romanum, cuius <lb/>modo fundamenta tota extant, quæ olim altè &longs;ub terram de&longs;cendebant. </s>

<s>vi<lb/>de pulcherrimam hac de re tractationem apud Georgium Agricolam lib. </s>

<s>vi<lb/>de pulcherrimam hac de re tractationem apud Georgium Agricolam lib.

3. <lb/>cap.

<s>1. qui amplius addit illud, quod & mihi maximè probatur; flumina ni<lb/>mirum producere montes, <expan abbr="colles&qacute;">collesque</expan>; hoc modo; vult enim initio mundi non <lb/>extiti&longs;&longs;e tot particulares montes ab inuicem di&longs;cretos, &longs;ed fui&longs;&longs;e perpetua <lb/>quædam terræ iuga eminentia quidem, &longs;ed non tot vallibus di&longs;&longs;ecta: v. </s>

1. qui amplius addit illud, quod & mihi maximè probatur; flumina ni<lb/>mirum producere montes, <expan abbr="colles&qacute;">collesque</expan>; hoc modo; vult enim initio mundi non <lb/>extiti&longs;&longs;e tot particulares montes ab inuicem di&longs;cretos, &longs;ed fui&longs;&longs;e perpetua <lb/>quædam terræ iuga eminentia quidem, &longs;ed non tot vallibus di&longs;&longs;ecta: v. </s>

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<s>quod primò Ari&longs;t.

te&longs;timonio in hoc cap. </s>

te&longs;timonio in hoc cap.

<s>comprobatur, <lb/>cum quo pariter &longs;entiunt veteres Geographi, & Hi&longs;torici omnes. </s>

comprobatur, <lb/>cum quo pariter &longs;entiunt veteres Geographi, & Hi&longs;torici omnes. </s>

<s>Ari&longs;t.

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<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>Huc facit locus quidam Strabonis ex lib. </s>

<s>Huc facit locus quidam Strabonis ex lib.

<s>12. de Pyramo Ciliciæ fluuio: <lb/>&longs;ic; montes verò egre&longs;&longs;us tantum limum in mare deducit, partim ex Ca<lb/>taonia, partim ex Ciliciæ campis, vt huiu&longs;modi de co oraculum feratur;</s>

12. de Pyramo Ciliciæ fluuio: <lb/>&longs;ic; montes verò egre&longs;&longs;us tantum limum in mare deducit, partim ex Ca<lb/>taonia, partim ex Ciliciæ campis, vt huiu&longs;modi de co oraculum feratur;</s>

<s><emph type="italics"/>Tempus erit rapidis olim cum Pyramus vndis <lb/>In &longs;acram veniet conge&longs;io litore, Cyprum:<emph.end type="italics"/></s>

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<s>Summæ 2. cap. </s>

<s>Summæ 2. cap.

<s>2. <emph type="italics"/>(Quapropter & circa Orionis orturm maximè fit tranquilli<lb/>tas)<emph.end type="italics"/> quando Medici, Philo&longs;ophi, Poetæ, ac reliqui auctores loquuntur de <lb/>ortu a&longs;trorum fixorum, aut con&longs;tellationum, quæ &longs;unt in firmamento, vti <lb/>e&longs;t Orion (& Canis, de quo po&longs;tea) intelligunt &longs;emper de ortu ip&longs;orum, qui <lb/>fit matutino tempore, quando &longs;cilicet vel &longs;imul cum Sole, vel paulò ante <lb/>Solem emergunt, ita vt videantur à nobis; qui ortus dicitur Co&longs;micus, tunc <lb/>propriè, quando &longs;imul a&longs;trum cum Sole oritur; quando autem incipit appa<lb/>rere manc ante Solem, dicitur ortus Heliacus. </s>

2. <emph type="italics"/>(Quapropter & circa Orionis orturm maximè fit tranquilli<lb/>tas)<emph.end type="italics"/> quando Medici, Philo&longs;ophi, Poetæ, ac reliqui auctores loquuntur de <lb/>ortu a&longs;trorum fixorum, aut con&longs;tellationum, quæ &longs;unt in firmamento, vti <lb/>e&longs;t Orion (& Canis, de quo po&longs;tea) intelligunt &longs;emper de ortu ip&longs;orum, qui <lb/>fit matutino tempore, quando &longs;cilicet vel &longs;imul cum Sole, vel paulò ante <lb/>Solem emergunt, ita vt videantur à nobis; qui ortus dicitur Co&longs;micus, tunc <lb/>propriè, quando &longs;imul a&longs;trum cum Sole oritur; quando autem incipit appa<lb/>rere manc ante Solem, dicitur ortus Heliacus. </s>

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<s>Summæ 2. cap. </s>

<s>Summæ 2. cap.

<s>3. de <expan abbr="v&etilde;tis">ventis</expan> <emph type="italics"/>(Oportet autem de &longs;itu &longs;imul rationes ex de&longs;criptio <lb/>ne con&longs;iderare)<emph.end type="italics"/> ide&longs;t rationes ventorum ex de&longs;criptione, ide&longs;t in figura ali<lb/>qua, vt in &longs;equenti con&longs;iderare; &longs;olet enim Ari&longs;t.

3. de <expan abbr="v&etilde;tis">ventis</expan> <emph type="italics"/>(Oportet autem de &longs;itu &longs;imul rationes ex de&longs;criptio <lb/>ne con&longs;iderare)<emph.end type="italics"/> ide&longs;t rationes ventorum ex de&longs;criptione, ide&longs;t in figura ali<lb/>qua, vt in &longs;equenti con&longs;iderare; &longs;olet enim Ari&longs;t.

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

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<s>Summæ 2. cap. </s>

<s>Summæ 2. cap.

<s>2. De Areæ figura <emph type="italics"/>(Refrangitur autem à con&longs;i&longs;tente caligine <lb/>circa Solem, aut Lunam vi&longs;us; quapropter non ex oppo&longs;ito &longs;icut iris, apparet. </s>

2. De Areæ figura <emph type="italics"/>(Refrangitur autem à con&longs;i&longs;tente caligine <lb/>circa Solem, aut Lunam vi&longs;us; quapropter non ex oppo&longs;ito &longs;icut iris, apparet. </s>

<s><expan abbr="Vn-diq;">Vn<lb/>dique</expan> autem &longs;imiliter refracto, nece&longs;&longs;e e&longs;t circulum e&longs;&longs;e, aut circuli partem. </s>

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<s><expan abbr="Atq;">Atque</expan> hæc cur Halonis figura orbicularis videatur, rationem <lb/>reddunt, <expan abbr="vna&qacute;">vnaque</expan>; textui lucem afferunt.</s>

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

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

<s>4. De Iridis figura.<emph.end type="italics"/></s>

4. De Iridis figura.<emph.end type="italics"/></s>

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<s>Hactenus de Iri<lb/>dis figura &longs;it &longs;atis.</s>

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

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

<s>5. De Parelio.<emph.end type="italics"/></s>

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

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<s>11. <emph type="italics"/>(Videtur autem non &longs;olum ip&longs;um quid e&longs;t cogno&longs;cere vtile e&longs;&longs;e <lb/>ad cogno&longs;cendas cau&longs;as accidentium &longs;ub&longs;tantijs: &longs;icut in Mathemati<lb/>cis quid rectum, & quid obliqaum, aut quid linea, & planum, ad co<lb/>gno&longs; cendum quot rectis, trianguli anguli &longs;unt æquales)<emph.end type="italics"/> quid &longs;it <expan abbr="vnum-quodq;">vnum<lb/>quodque</expan> ex prædictis patet tum ex definitionibus primi Elem. </s>

<s>tum ex com<lb/>mentarijs ip&longs;arum; quamuis autem ibi non definiatur <expan abbr="rect&utilde;">rectum</expan>, nec obliquum <lb/>in genere, definitur tamen linea recta, & obliqua, & plana &longs;uperficies, &longs;iue <lb/>planum, ex quibus facilè definitio recti, & obliqui colligi pote&longs;t: quæ defi<lb/>nitiones nece&longs;&longs;ariæ &longs;unt ad cogno&longs;cendum quot rectis angulis æquales &longs;int <lb/>tres anguli cuiufuis trianguli. </s>

tum ex com<lb/>mentarijs ip&longs;arum; quamuis autem ibi non definiatur <expan abbr="rect&utilde;">rectum</expan>, nec obliquum <lb/>in genere, definitur tamen linea recta, & obliqua, & plana &longs;uperficies, &longs;iue <lb/>planum, ex quibus facilè definitio recti, & obliqui colligi pote&longs;t: quæ defi<lb/>nitiones nece&longs;&longs;ariæ &longs;unt ad cogno&longs;cendum quot rectis angulis æquales &longs;int <lb/>tres anguli cuiufuis trianguli. </s>

<s>vide quæ de hac æqualitate &longs;crip&longs;i lib, primo <lb/>Priorum, &longs;ecto 3. cap. </s>

<s>vide quæ de hac æqualitate &longs;crip&longs;i lib, primo <lb/>Priorum, &longs;ecto 3. cap.

1.</s>

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<s>13. <emph type="italics"/>(Si igitur e&longs;t aliqua animæ operatio, aut pa&longs;&longs;io propria, continget vti<lb/>que ip&longs;am &longs;eparari: &longs;i verò nulla e&longs;t propria ip&longs;ius non vtique erit &longs;eparabilis. </s>

<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. </s>

<s>&pacute;robat li<lb/><figure id="fig62"></figure><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="fig62"></figure><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>

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animæ definitionem præ<lb/>mittit duplicem e&longs;&longs;e definitionem, alteram &longs;cilicet, quæ explicat &longs;olum rei <lb/>e&longs;&longs;entiam, quam dicunt formalem definitionem; alteram verò, quæ præte<lb/>rea explicat etiam rei cau&longs;am, quam dicunt cau&longs;alem definitionem: vtram<lb/>que autem exemplo Geometrico explicat.</s>

<s>In cap.

<s>igitur de relatione plura &longs;crip&longs;i de tetragoni&longs;mo, &longs;eu qua dratio<lb/>ne circuli, quæ huc &longs;pectant. </s>

igitur de relatione plura &longs;crip&longs;i de tetragoni&longs;mo, &longs;eu qua dratio<lb/>ne circuli, quæ huc &longs;pectant. </s>

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

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<s>&longs;i quis verò dicat tetragoni&longs;mum hunc quadri<lb/>lateri dati e&longs;&longs;e mediæ prædictæ inuentionem cau&longs;alem afferet definitionem, <lb/>cum rei cau&longs;am dicat. </s>

<s>Aduerte 10. Grammaticum immeritò accu&longs;are Ale<lb/>xandrum, quod dicat quadrationem hanc per inuentionem mediæ propor<lb/>tionalis tradi in 2. Elem. </s>

<s>Aduerte 10. Grammaticum immeritò accu&longs;are Ale<lb/>xandrum, quod dicat quadrationem hanc per inuentionem mediæ propor<lb/>tionalis tradi in 2. Elem.

<s>nam verè in 14. 2. traditur talis inuentio, quam<lb/>uis enim ibi nulla fiat expre&longs;&longs;a mentio huiu&longs;modi mediæ, in ip&longs;a tamen e<gap/><lb/>reperitur, ac per eam figuræ rectilineæ quadrantur: quod patet ex figura <lb/>14. prædictæ, quæ eadem e&longs;t cum figura 13. 6. qua docemur prædictam in<lb/>uentionem.</s>

nam verè in 14. 2. traditur talis inuentio, quam<lb/>uis enim ibi nulla fiat expre&longs;&longs;a mentio huiu&longs;modi mediæ, in ip&longs;a tamen e<gap/><lb/>reperitur, ac per eam figuræ rectilineæ quadrantur: quod patet ex figura <lb/>14. prædictæ, quæ eadem e&longs;t cum figura 13. 6. qua docemur prædictam in<lb/>uentionem.</s>

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<s>86. <emph type="italics"/>(Acutum mouet &longs;en&longs;um in tempore pauco multùm: graue autem in <lb/>multo parùm; non igitur velox e&longs;t acutum, graue autem tardum, fed fit illius qui<lb/>dem propter velocitatem motus huiu&longs;modi, huius autem propter tarditatem)<emph.end type="italics"/> vide <lb/>quæ de hac re primo topic. </s>

<s>cap.

<s>13. dicta &longs;unt, illa enim omnia in hunc lo<lb/>cum quadrant. </s>

13. dicta &longs;unt, illa enim omnia in hunc lo<lb/>cum quadrant. </s>

<s>Verum occurrit illa dubitatio; quod cum Ari&longs;t.

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<s>159. <emph type="italics"/>(Apparent autem, & fal&longs;a, de quibus &longs;imul exi&longs;timationem veram <lb/>habet, vt apparet &longs;ol vnius pedis, per&longs;ua&longs;um autem e&longs;t, eum maiorem e&longs;&longs;e babitata)<emph.end type="italics"/><lb/>habitata, ide&longs;t terra habitata. </s>

<s>Vide, quæ cap.

<s>3. &longs;ummæ 1. primi Meteor. <lb/></s>

3. &longs;ummæ 1. primi Meteor. <lb/></s>

<s>Item capite 5. &longs;ummæ 2. de Solis magnitudine &longs;crip&longs;i, ea enim huic loco <lb/>abundè &longs;atisfaciunt.</s>

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<s>21. <emph type="italics"/>(Vt incommen&longs;urabile, & diameter)<emph.end type="italics"/> vide, quæ de incom<lb/>men&longs;uratione diametri, & co&longs;tæ &longs;cripta &longs;unt lib. </s>

<s>21. <emph type="italics"/>(Vt incommen&longs;urabile, & diameter)<emph.end type="italics"/> vide, quæ de incom<lb/>men&longs;uratione diametri, & co&longs;tæ &longs;cripta &longs;unt lib.

<s>1. Priorum, cap. <lb/></s>

1. Priorum, cap. <lb/></s>

<s>23. vnde &longs;atis huic loco fieri pote&longs;t.</s>

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<s>32. <emph type="italics"/>(Sit igitur vt A, quidem album, ad B, quod nigrum; &longs;ic C, ad D; qua<lb/>re & p<gap/>rmutatim)<emph.end type="italics"/> ide&longs;t, quare & permutando (vt aiunt Geometræ) erit vt <lb/>A, ad C, ita B, ad D, hunc argumentandi modum à permutata proportio<lb/>ne explicaui in primo Po&longs;ter. </s>

<s>cap.

5. tex. </s>

<s>13. dicitur etiam alterna ratio; <lb/>& definitur ab Euclide definitione 12, 5.</s>

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<s>lege annotata primo <lb/>Po&longs;ter. </s>

<s>&longs;ecto 3. cap. </s>

<s>&longs;ecto 3. cap.

1.</s>

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<s>atque hæc <lb/>ip&longs;a cau&longs;a e&longs;t, cur illud, quod vnum tantum e&longs;t, duo tamen <lb/>emoto oculorum altero, videatur. </s>

<s>Vide Alhaze<lb/>num lib. </s>

<s>Vide Alhaze<lb/>num lib.

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

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

<s>11. & 12. & infra <lb/>Problem. </s>

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<s>Ibidem (<emph type="italics"/>Aut de diametri incommen&longs;ur abilitate, admirabile enim omnibus vi<lb/>detur, &longs;i quid, cum non &longs;it minimum non men&longs;uretur, decet autem in contrarium, <lb/>& in melius &longs;ecundum prouerbium con&longs;umare, quemadmod&ubreve; in his fit, cum di&longs;cant, <lb/>nihil enim magis vir Geometricus admiraretur, quàm &longs;i diamcter commen&longs;urabi<lb/>lis &longs;ieret<emph.end type="italics"/>) vide quæ de hac commen&longs;urabilitate &longs;crip&longs;i lib. </s>

<s>1. Priorum, &longs;ect. </s>

1. Priorum, &longs;ect. </s>

<s>1. <lb/>cap.

<s>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>

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>Summa 2. cap. </s>

<s>Summa 2. cap.

<s>3. (<emph type="italics"/>Pythagorici primi Mathematicis operam dedere, hæc præpo<lb/>nebant, & in cis enutriti, eorum principia, entium <expan abbr="quoq;">quoque</expan> cunctorum putant e&longs;&longs;e <lb/>principia<emph.end type="italics"/>) vtinam no&longs;trates Philo&longs;ophi Pythagoricos imitarentur; enimue<lb/>rò multò melius & &longs;ibi, & Philo&longs;ophiæ con&longs;ulerent. </s>

3. (<emph type="italics"/>Pythagorici primi Mathematicis operam dedere, hæc præpo<lb/>nebant, & in cis enutriti, eorum principia, entium <expan abbr="quoq;">quoque</expan> cunctorum putant e&longs;&longs;e <lb/>principia<emph.end type="italics"/>) vtinam no&longs;trates Philo&longs;ophi Pythagoricos imitarentur; enimue<lb/>rò multò melius & &longs;ibi, & Philo&longs;ophiæ con&longs;ulerent. </s>

<s>At verò non &longs;ine ma<lb/>gno artium, <expan abbr="atq;">atque</expan> di&longs;ciplinarum omnium di&longs;pendio à plurimis hac tempe<lb/>&longs;tate de&longs;pectui habentur; &longs;ed quid mirum cum quas &longs;cientiarum omnium <lb/>alumni Pythagorei omnibus &longs;cientijs anteferebant; eas no&longs;tri &longs;eculi quam<lb/>plures omnibus alijs facultatibus po&longs;thabeant.</s>

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<s>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. </s>

<s>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.

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

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

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

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<s>1. <lb/>&longs;cripta &longs;unt: necnon quæ in libro de Sen&longs;u in cap. </s>

<s>1. <lb/>&longs;cripta &longs;unt: necnon quæ in libro de Sen&longs;u in cap.

<s>8. Amplius inquit can&longs;am <lb/>formalem genericam eiu&longs;dem Diapa&longs;on e&longs;&longs;e numerum, & partes numeri, <lb/>&longs;ub numero enim continentur & duo, & vnum. </s>

8. Amplius inquit can&longs;am <lb/>formalem genericam eiu&longs;dem Diapa&longs;on e&longs;&longs;e numerum, & partes numeri, <lb/>&longs;ub numero enim continentur & duo, & vnum. </s>

<s>Occurrit hoc loco vnum <lb/>magnopere notandum, videlicet tam con&longs;onantias, quam di&longs;&longs;onantias ha<lb/>bere proportiones numerorum, hoc tamen di&longs;crimine, quod con&longs;onantiæ <lb/>habent &longs;olùm proportiones numerorum eorum, qui quaternario continen<lb/>tur, ex veterum præ&longs;ertim Pythagoreorum &longs;ententia, qui propterea vltra <lb/>quaternarium progredi vetabant. </s>

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<s>3. (<emph type="italics"/>Et ip&longs;ius Diapa&longs;on duplum, & numerus<emph.end type="italics"/>) &longs;cilicet cau&longs;æ formales <lb/>&longs;unt, quemadmodum &longs;upra tex. </s>

<s>2. huius cap. </s>

<s>2. huius cap.

<s>explicatum e&longs;t.</s><pb pagenum="141"/>

explicatum e&longs;t.</s><pb pagenum="141"/>

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<s>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. </s>

23.</s><figure></figure>

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<s>34. (<emph type="italics"/>Quemadmodum dicitur diametrum e&longs;&longs;e commen&longs;urabilem<emph.end type="italics"/>) vide an<lb/>notata 1. Priorum, fecto 1. cap. </s>

<s>34. (<emph type="italics"/>Quemadmodum dicitur diametrum e&longs;&longs;e commen&longs;urabilem<emph.end type="italics"/>) vide an<lb/>notata 1. Priorum, fecto 1. cap.

23.</s>

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<s>35. (<emph type="italics"/>Vt triangulo duos rectos habere<emph.end type="italics"/>) ide&longs;t affectio trianguli e&longs;t habe<lb/>re tres angulos æquales duobus rectis angulis. </s>

<s>Vide declarationem huius <lb/>lib. </s>

<s>Vide declarationem huius <lb/>lib.

<s>primo Priornm, &longs;ecto 3. cap. </s>

primo Priornm, &longs;ecto 3. cap.

1.</s>

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

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<s><emph type="italics"/>Vt &longs;i quis dicat diametrum po&longs;&longs;e commen&longs;arari, non tamen commen&longs;u<lb/>rabitur<emph.end type="italics"/>) & paulò infra (<emph type="italics"/>Commen&longs;urari enim impo&longs;&longs;ibile e&longs;t<emph.end type="italics"/>) expo&longs;i<lb/>tionem horum reperies 1. Priorum, &longs;ecto 1. cap. </s>

23.</s>

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<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>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. </s>

<s>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="fig68"></figure><lb/>fal&longs;um e&longs;t, nam omnes anguli, qui circa vnum <lb/>punctum, v. </s>

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="fig68"></figure><lb/>fal&longs;um e&longs;t, nam omnes anguli, qui circa vnum <lb/>punctum, v. </s>

<s>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. </s>

<s>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.

<s>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. </s>

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. </s>

<s>&longs;en<lb/>&longs;us e&longs;t omnes angulos ad ea&longs;dem partes con&longs;ti<lb/>tutos, v. </s>

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<s>22. (<emph type="italics"/>Vt puta &longs;i triangulum non putet mutari, non opinabitur modo duos <lb/>rectos habere, modo non, mutaretur enim<emph.end type="italics"/>) quia nimirum huius habemus &longs;cien<lb/>tiam per demon&longs;trationem 32. primi Elementorum. </s>

<s>quomodo autem tri<lb/>angulus habeat duos rectos, ide&longs;t tres angulos æquales duobus rectis angu<lb/>lis, explicatum e&longs;t primo Priorum, &longs;ecto 3. cap. </s>

<s>quomodo autem tri<lb/>angulus habeat duos rectos, ide&longs;t tres angulos æquales duobus rectis angu<lb/>lis, explicatum e&longs;t primo Priorum, &longs;ecto 3. cap.

1.</s>

<s>Ibidem (<emph type="italics"/>Verum aliquid quidem, aliquid verò non, vt puta parem numerum <lb/>primum nullum e&longs;&longs;e; aut quo&longs;dam quidem, quo&longs;dam verò non<emph.end type="italics"/>) definitione 11. <lb/>7. Elem. </s>

<s>&longs;ic numerus ille, qui à Mathematicis dicitur primus, definitur, pri<lb/>mus numerus e&longs;t, quem vnitas &longs;ola metitur, vnde patet inter numeros pa<lb/>res &longs;olum binarium e&longs;&longs;e primum, cum ip&longs;um &longs;ola vnitas bis replicata men<lb/>&longs;uraret. </s>

&longs;ic numerus ille, qui à Mathematicis dicitur primus, definitur, pri<lb/>mus numerus e&longs;t, quem vnitas &longs;ola metitur, vnde patet inter numeros pa<lb/>res &longs;olum binarium e&longs;&longs;e primum, cum ip&longs;um &longs;ola vnitas bis replicata men<lb/>&longs;uraret. </s>

<s>quaternarium autem, &longs;enarium, &c. </s>

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<s>Eodem tex. </s>

<s>&longs;ed cap. </s>

<s>&longs;ed cap.

<s>3. <emph type="italics"/>(Nox &longs;emper autem men&longs;ura numero vnum e&longs;t, verum <lb/>aliquando plura, vt puta die&longs;es duæ, non quidem &longs;ecundum cuditum, &longs;ed in ratio<lb/>nibus, & voces plures, quibus men&longs;uramus, & diameter duobus men&longs;uratur, & la<lb/>tus, & omnes magnitudines)<emph.end type="italics"/> ita corrigenda e&longs;t antiqua tran&longs;latio. </s>

3. <emph type="italics"/>(Nox &longs;emper autem men&longs;ura numero vnum e&longs;t, verum <lb/>aliquando plura, vt puta die&longs;es duæ, non quidem &longs;ecundum cuditum, &longs;ed in ratio<lb/>nibus, & voces plures, quibus men&longs;uramus, & diameter duobus men&longs;uratur, & la<lb/>tus, & omnes magnitudines)<emph.end type="italics"/> ita corrigenda e&longs;t antiqua tran&longs;latio. </s>

<s>quid die&longs;is <lb/>dictum &longs;it ad tex. </s>

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<s>Svmma r. </s>

<s>cap.

<s>2. <emph type="italics"/>(Si quis verò lineas, aut quæ has &longs;equuntur, dico autem <lb/>primas &longs;uperficies principia e&longs;&longs;e ponat. </s>

2. <emph type="italics"/>(Si quis verò lineas, aut quæ has &longs;equuntur, dico autem <lb/>primas &longs;uperficies principia e&longs;&longs;e ponat. </s>

<s>bæc non &longs;unt &longs;ub&longs;tantiæ &longs;eparabiles, <lb/>verùm &longs;ectiones, & diui&longs;iones, illæ quidem in &longs;uperficierum, hæc verò cor<lb/>porum, puncta verò linearum &longs;unt, & etiam ip&longs;arum earumdem termini; <lb/>hæc autem omnia in alijs &longs;unt, & nihil &longs;eparabile e&longs;t)<emph.end type="italics"/> ait puncta oriri ex &longs;ectio<lb/>ne lineæ, quamuis &longs;int etiam termini illius; lineas verò oriri ex diui&longs;ione <lb/>&longs;uperficierum, quamuis &longs;int etiam termini illarum. </s>

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<s>Summa 3. cap. </s>

<s>Summa 3. cap.

<s>2. <emph type="italics"/>(Vt puta &longs;ub Cane fiat frigus)<emph.end type="italics"/> ideft &longs;ub ortum Canis c&oelig;<lb/>læ&longs;tis, &longs;eu Caniculæ. </s>

2. <emph type="italics"/>(Vt puta &longs;ub Cane fiat frigus)<emph.end type="italics"/> ideft &longs;ub ortum Canis c&oelig;<lb/>læ&longs;tis, &longs;eu Caniculæ. </s>

<s>Vide quæ libro &longs;ecundo Meteororum, &longs;umma 2. cap. </s>

<s>Vide quæ libro &longs;ecundo Meteororum, &longs;umma 2. cap.

<s>2. <lb/>de hac &longs;tella &longs;crip&longs;imus.</s>

2. <lb/>de hac &longs;tella &longs;crip&longs;imus.</s>

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

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<s>Svmma 1. cap. </s>

<s>Svmma 1. cap.

<s>3. <emph type="italics"/>(Qui dicunt Mathematicas &longs;cientias nihil de bono, vel <lb/>pulchro dicere, fal&longs;um dicunt. </s>

3. <emph type="italics"/>(Qui dicunt Mathematicas &longs;cientias nihil de bono, vel <lb/>pulchro dicere, fal&longs;um dicunt. </s>

<s>dicunt. </s>

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<s>quam etiam <lb/>Pater Clauius demon&longs;trat propo&longs;. </s>

<s>2. lib.

<s>8. & propo&longs;. </s>

8. & propo&longs;. </s>

<s>1. lib.

4. Geom. </s>

<s>pract. <lb/></s>

<s>&longs;i autem de ip&longs;is circulis intelligerentur fal&longs;a e&longs;&longs;ent, non enim e&longs;t circulus <lb/>ad circulum, vt diameter ad diametrum; &longs;ed circuli &longs;unt inter &longs;e, quemad<lb/>modum à diametris ip&longs;orum quadrata per &longs;ecundam 12. Elem. </s>

<s>quadrata <lb/>autem &longs;unt inter &longs;e in duplicata ratione laterum per 20. 6. <expan abbr="eiusq;">eiusque</expan> corolla<lb/>rium; hoc e&longs;t &longs;i fiat, vt latus maioris quadrati ad latus minoris, ita latus mi<lb/>noris ad aliam tertiam lineam, erit quadratum maius ad minus, vt latus <lb/>ip&longs;ius ad tertiam illam lineam; non autem vt ad latus minoris. </s>

quadrata <lb/>autem &longs;unt inter &longs;e in duplicata ratione laterum per 20. 6. <expan abbr="eiusq;">eiusque</expan> corolla<lb/>rium; hoc e&longs;t &longs;i fiat, vt latus maioris quadrati ad latus minoris, ita latus mi<lb/>noris ad aliam tertiam lineam, erit quadratum maius ad minus, vt latus <lb/>ip&longs;ius ad tertiam illam lineam; non autem vt ad latus minoris. </s>

<s>cum ergo <lb/>circulus &longs;it ad circulum, vt quadratum diametri ad quadratum diametri, <lb/>& quadrata non <expan abbr="habeãt">habeant</expan> rationem laterum, &longs;eu diametrorum prædictorum, <lb/>&longs;ed illorum duplicatam, <expan abbr="neq;">neque</expan> circuli inuicem illam habere poterunt.</s>

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<s>Rhombus ex definitione 23. primi Elem. </s>

<s>Rhombus ex definitione 23. primi Elem.

<s>e&longs;t figura æquilatera qui<lb/><figure id="fig103"></figure><lb/>dem, &longs;ed non æquiangula, habet enim <lb/>binos oppo&longs;itos angulos acutos, & alies <lb/>binos oppo&longs;itos obtu&longs;os, talis e&longs;t præ<lb/>&longs;ens figura A B D C. </s>

e&longs;t figura æquilatera qui<lb/><figure id="fig103"></figure><lb/>dem, &longs;ed non æquiangula, habet enim <lb/>binos oppo&longs;itos angulos acutos, & alies <lb/>binos oppo&longs;itos obtu&longs;os, talis e&longs;t præ<lb/>&longs;ens figura A B D C. </s>

<s>In præ&longs;enti porrò quæ&longs;tione <lb/>&longs;upponitur punctum A, quod e&longs;t vnum extremum <lb/>in rhombo moueri &longs;uper latus A B, ver&longs;us B, & &longs;i<lb/>militer interim æqua velocitate moueri alterum <lb/>extremum B, &longs;uper idem latus A B_{2} ver&longs;us A, & in<lb/>terim dum hæc duo puncta hoc modo &longs;ibi obuiam <lb/>procedunt, moueri latus totum A B, eadem ve<lb/>locitate, ver&longs;us latus C D, ita vt &longs;emper ip&longs;i C D, <lb/>æquidi&longs;ter, <expan abbr="de&longs;cendat&qacute;">de&longs;cendatque</expan>; per latera A C, B D, quo<lb/>u&longs;que ip&longs;i C D, congruat.</s>

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<s>In 2. cap.

<s>recen&longs;et Planetarum ordinem, iuxta antiqui&longs;&longs;imorum <lb/>A&longs;tronomorum traditiones, qui ob paucas, <expan abbr="eas&qacute;">easque</expan>; imperfectas ob<lb/>&longs;eruationes muita ignorarunt, <expan abbr="atq;">atque</expan> in multis, & præcipuè in ordine Plane<lb/>tarum &longs;tatuendo, fal&longs;i &longs;unt: A&longs;tronomi enim po&longs;teriores, & maximè Ptolæ<lb/>meus, vnà cum recentioribus no&longs;tri &longs;eculi alium ordinem exactioribus ob<lb/>&longs;eruationibus, <expan abbr="atq;">atque</expan> demon&longs;trationibus a&longs;trucntes vetu&longs;ti&longs;&longs;imorum illorum <lb/>errores patefecerunt. </s>

recen&longs;et Planetarum ordinem, iuxta antiqui&longs;&longs;imorum <lb/>A&longs;tronomorum traditiones, qui ob paucas, <expan abbr="eas&qacute;">easque</expan>; imperfectas ob<lb/>&longs;eruationes muita ignorarunt, <expan abbr="atq;">atque</expan> in multis, & præcipuè in ordine Plane<lb/>tarum &longs;tatuendo, fal&longs;i &longs;unt: A&longs;tronomi enim po&longs;teriores, & maximè Ptolæ<lb/>meus, vnà cum recentioribus no&longs;tri &longs;eculi alium ordinem exactioribus ob<lb/>&longs;eruationibus, <expan abbr="atq;">atque</expan> demon&longs;trationibus a&longs;trucntes vetu&longs;ti&longs;&longs;imorum illorum <lb/>errores patefecerunt. </s>

<s>E&longs;t autem verus ordo, vt Luna &longs;it omnium terris pro<lb/>xima, deinde Mercurius, tùm Venus, po&longs;tea Sol, Mars, Iupiter, <expan abbr="Saturnus&qacute;">Saturnusque</expan>; <lb/>à terris alti&longs;&longs;imus, quos omnes &longs;tellarum affixarum &longs;phæra, quæ etiam fir<lb/>mamentum dicitur, complectitur. </s>

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<s>quod adhuc ma<lb/>nife&longs;tius erit, &longs;i con&longs;ideremus, quod quanto tardius quotidie Luna oritur, <lb/>tanto etiam maris æ&longs;tus tardius incipit. </s>

<s>Porrò vt appareat hanc e&longs;&longs;e vete<lb/>rum &longs;ententiam libet hic attexere quædam ex lib. </s>

<s>Porrò vt appareat hanc e&longs;&longs;e vete<lb/>rum &longs;ententiam libet hic attexere quædam ex lib.

<s>3. Strabonis, quæ ip&longs;e ex <lb/>Po&longs;&longs;idonio acceperat. </s>

3. Strabonis, quæ ip&longs;e ex <lb/>Po&longs;&longs;idonio acceperat. </s>

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

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<s>de creatione mundi tantummodo &longs;ine vlla expo&longs;itione, <expan abbr="atq;">atque</expan> con<lb/>firmatione proponit. </s>

<s>in eadem pror&longs;us &longs;ententia e&longs;t Rogerius Bachon inter <lb/>Opticos probati&longs;&longs;imus, cap. </s>

<s>in eadem pror&longs;us &longs;ententia e&longs;t Rogerius Bachon inter <lb/>Opticos probati&longs;&longs;imus, cap.

<s>5. de Speculis Mathematicis.</s>

5. de Speculis Mathematicis.</s>

<s>Aliorum demum opinationes, &longs;iue Angelo cuidam, &longs;iue virtuti totam <lb/>terram peruadenti hunc æ&longs;tum a&longs;cribentium, non e&longs;t meum refellere, cum <lb/>non phy&longs;icum, &longs;ed mathematicum agere in&longs;tituerim.</s>

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<s>A lint<gap/>igentiam igitur huius operis nece&longs;&longs;arium e&longs;t noui&longs;&longs;e, quæ nam <lb/>&longs;int quantitates commen&longs;urabiles, & quæ in commen&longs;urabiles. </s>

<s>quæ prima, <lb/>& &longs;ecunda definitione 10. Elem. </s>

<s>quæ prima, <lb/>& &longs;ecunda definitione 10. Elem.

<s>explicantur; <expan abbr="ego&qacute;">egoque</expan>; eas primo Priorum oc<lb/>ca&longs;ione a&longs;ymetriæ diametri cum co&longs;ta &longs;atis expo&longs;ui: vtrumuis locum vide<lb/>ris præ&longs;enti nece&longs;&longs;itati con&longs;ultum erit.</s>

explicantur; <expan abbr="ego&qacute;">egoque</expan>; eas primo Priorum oc<lb/>ca&longs;ione a&longs;ymetriæ diametri cum co&longs;ta &longs;atis expo&longs;ui: vtrumuis locum vide<lb/>ris præ&longs;enti nece&longs;&longs;itati con&longs;ultum erit.</s>

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<s>Ve<lb/>rùm neque &longs;ecundum &longs;e aliquam definitam naturam habebunt, &longs;ed collatæ &longs;ibi ip&longs;is <lb/>tam rationales, quàm irrationales erunt omnes.<emph.end type="italics"/></s>

<s>Hæc e&longs;t alia eorumdem ratio ad idem comprobandum: quam, vt benè <lb/>percipiamus, nonnulla prius ex definitionibus 10. Elem. </s>

<s>Hæc e&longs;t alia eorumdem ratio ad idem comprobandum: quam, vt benè <lb/>percipiamus, nonnulla prius ex definitionibus 10. Elem.

<s>&longs;unt explicanda: <lb/>vt quæ nam &longs;int lineæ rationales, quæ irrationales, quæ ex binis nomini<lb/>bus, quæ Apotomæ.</s>

&longs;unt explicanda: <lb/>vt quæ nam &longs;int lineæ rationales, quæ irrationales, quæ ex binis nomini<lb/>bus, quæ Apotomæ.</s>

<s>Propo&longs;ita igitur linea quapiam, v. </s>

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<s>indiu duæ autem lineæ &longs;ibi ip&longs;is co<gap/>men&longs;urabiles &longs;unt longitudine, cuminter &longs;e <lb/>&longs;iat æquales; quare po<gap/>emia quoque, quod &longs;i hoc e&longs;t, diuiduum erit quadratum.<emph.end type="italics"/></s><pb pagenum="207"/>

<s>Pergit adhuc nouis rationibus aduer&longs;arios refellere, dicens, &longs;i extarent <lb/>huiu&longs;modi indiuiduæ lineæ, &longs;equeretur omnes omninò lineas e&longs;&longs;e commen<lb/>&longs;urabiles, quod e&longs;t contra demon&longs;trata in 10. Elem. </s>

<s>quia cum omnes lineæ <lb/><expan abbr="con&longs;t&etilde;t">con&longs;tent</expan> per ip&longs;os ex lineis atomis, i&longs;tæ atomæ e&longs;&longs;ent omnium linearum com<lb/>munes men&longs;uræ, vnde & illæ, quæ dicuntur potentia tantum commen&longs;ura<lb/>biles, vt &longs;upra explicaui, erunt etiam commen&longs;urabiles longitudine. </s>

quia cum omnes lineæ <lb/><expan abbr="con&longs;t&etilde;t">con&longs;tent</expan> per ip&longs;os ex lineis atomis, i&longs;tæ atomæ e&longs;&longs;ent omnium linearum com<lb/>munes men&longs;uræ, vnde & illæ, quæ dicuntur potentia tantum commen&longs;ura<lb/>biles, vt &longs;upra explicaui, erunt etiam commen&longs;urabiles longitudine. </s>

<s>indiui<lb/>duæ verò ip&longs;æ, cum &longs;int inuicem æquales, erunt ip&longs;æ <expan abbr="quoq;">quoque</expan> commen&longs;urabi<lb/>les longitudine, quare & potentia, omnes enim longitudine commen&longs;ura<lb/>biles, &longs;unt etiam potentia commen&longs;urabiles, ex 9. 10. vnde &longs;equitur qua<lb/>drata earum omnia e&longs;&longs;e <expan abbr="quoq;">quoque</expan> commen&longs;urabilia: <expan abbr="atq;">atque</expan> hinc con&longs;equitur, in<lb/>quit, ea e&longs;&longs;e <expan abbr="quoq;">quoque</expan> diuidua (quam con&longs;ecutionem probat infra num. </s>

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<s>in omni autem æquilatero perpendicularis <lb/>in mediam ba&longs;im mcidit, quare, & in medium indiuiduæ.<emph.end type="italics"/></s>

<s>Ex 22. primi Elem. </s>

<s>Ex 22. primi Elem.

<s>ex tribus datis lineis, quarum quælibet duæ &longs;int, re<lb/>liqua maiores pote&longs;t con&longs;titui triangulum: poterit igitur ex tribus indiui<lb/><figure id="fig122"></figure><lb/>duis con&longs;titui <expan abbr="triãgulum">triangulum</expan>, <expan abbr="illud&qacute;">illudque</expan>; æquilaterum, cum omnes in<lb/>diuiduæ lineæ &longs;int æquales. </s>

ex tribus datis lineis, quarum quælibet duæ &longs;int, re<lb/>liqua maiores pote&longs;t con&longs;titui triangulum: poterit igitur ex tribus indiui<lb/><figure id="fig122"></figure><lb/>duis con&longs;titui <expan abbr="triãgulum">triangulum</expan>, <expan abbr="illud&qacute;">illudque</expan>; æquilaterum, cum omnes in<lb/>diuiduæ lineæ &longs;int æquales. </s>

<s>&longs;it igitur ex eis triangulum A B C, <lb/>&longs;i igitur ab angulo A, ducatur perpendicularis A D, ad ba&longs;im <lb/>B C, eam bif<gap/>riam &longs;ecabit ex &longs;cholio 26. primi, erit igitur li<lb/>nea B C, &longs;ecabilis, contra quam aduer&longs;arij opinantur.</s>

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<s>2. cap.

<s>1. <emph type="italics"/>(Sit&queacute; perinde ac admirabilia illa &longs;pontanea)<emph.end type="italics"/> Intelligit ma<lb/>chinas illas miro artificio confictas, quæ à &longs;e ip&longs;is intrin&longs;eco prin<lb/>cipio mouebantur, quas Græci veteres Automata, ide&longs;t &longs;pontanea, <lb/>vel &longs;pontina, vt vertit Interpres vocabant, cuiu&longs;modi &longs;unt Auto<lb/>mata Heronis Alexandrini, quæ adhuc extant græca, <expan abbr="quæ&qacute;">quæque</expan>; ab Abbate Gua<lb/>&longs;tallen&longs;i in Italicum &longs;unt conuer&longs;a. </s>

1. <emph type="italics"/>(Sit&queacute; perinde ac admirabilia illa &longs;pontanea)<emph.end type="italics"/> Intelligit ma<lb/>chinas illas miro artificio confictas, quæ à &longs;e ip&longs;is intrin&longs;eco prin<lb/>cipio mouebantur, quas Græci veteres Automata, ide&longs;t &longs;pontanea, <lb/>vel &longs;pontina, vt vertit Interpres vocabant, cuiu&longs;modi &longs;unt Auto<lb/>mata Heronis Alexandrini, quæ adhuc extant græca, <expan abbr="quæ&qacute;">quæque</expan>; ab Abbate Gua<lb/>&longs;tallen&longs;i in Italicum &longs;unt conuer&longs;a. </s>

<s>Automata hodie &longs;unt Horologia, quæ ex <lb/>multis dentatis rotis Germani con&longs;truunt.</s>

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<s>2. cap.

<s>4. <emph type="italics"/>(Nam & triangula figura duobus rectis æquale &longs;emper habet)<emph.end type="italics"/><lb/>vide quæ de hac re &longs;crip&longs;i lib. </s>

4. <emph type="italics"/>(Nam & triangula figura duobus rectis æquale &longs;emper habet)<emph.end type="italics"/><lb/>vide quæ de hac re &longs;crip&longs;i lib.

<s>1. Priorum, &longs;ecto 3. cap. </s>

1. Priorum, &longs;ecto 3. cap.

1.</s>

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<s>Ibidem <emph type="italics"/>(Et diametrum incommen&longs;urabilem e&longs;&longs;e cum co&longs;t a &longs;empiternum e&longs;t: at<lb/>tamen cau&longs;a eorum aliqua & demon&longs;tratio e&longs;t.<emph.end type="italics"/></s>

<s>Quælibro 1. Priorum, &longs;ecto 1. cap. </s>

<s>Quælibro 1. Priorum, &longs;ecto 1. cap.

<s>23. de hac re annotata &longs;unt, abundè <lb/>huic etiam loco &longs;atisfaciunt.</s>

23. de hac re annotata &longs;unt, abundè <lb/>huic etiam loco &longs;atisfaciunt.</s>

<s><emph type="italics"/>In Ethica, &longs;eu Moralia ad Nicomachum.<emph.end type="italics"/></s>

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<s>1. cap.

<s>7. <emph type="italics"/>(Faber enim, & Geometra diuer&longs;o modo rectum angulum <lb/><expan abbr="vtriq;">vtrique</expan> con&longs;iderant: ille quatenus <expan abbr="&longs;ol&utilde;">&longs;olum</expan> ad opus vtile e&longs;t, hic verò cum ve<lb/>ritatis &longs;peculator &longs;it, quid, & qualis &longs;it, indagat)<emph.end type="italics"/> Id quod dicit Ari&longs;t. <lb/></s>

7. <emph type="italics"/>(Faber enim, & Geometra diuer&longs;o modo rectum angulum <lb/><expan abbr="vtriq;">vtrique</expan> con&longs;iderant: ille quatenus <expan abbr="&longs;ol&utilde;">&longs;olum</expan> ad opus vtile e&longs;t, hic verò cum ve<lb/>ritatis &longs;peculator &longs;it, quid, & qualis &longs;it, indagat)<emph.end type="italics"/> Id quod dicit Ari&longs;t. <lb/></s>

<s>confirmatur ex eo, quod Fabri omnes vtuntur amu&longs;&longs;i, &longs;eu norma, <lb/><figure id="fig130"></figure><lb/>quæ nihil aliud e&longs;t quàm angulus rectus, quæ vulgò <lb/>&longs;quadra dicitur, vt eius auxilio angulum ip&longs;um re<lb/>ctum in opus conferant, <expan abbr="&longs;ic&qacute;">&longs;icque</expan>; normæ, aut amu&longs;&longs;is du<lb/>ctu &longs;ua ip&longs;i opera ad angulos rectos, ide&longs;t quadrata, <lb/>conficiunt. </s>

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

<s>contemplatur præterea Geometra omnes <lb/>angulos rectos e&longs;&longs;e inter &longs;e æquales, vt in 12. axiomate primi Elem. </s>

<s>contemplatur præterea Geometra omnes <lb/>angulos rectos e&longs;&longs;e inter &longs;e æquales, vt in 12. axiomate primi Elem.

<s>ponitur, <lb/>& &longs;imilia plura alia, quorum con&longs;iderationem Faber omninò negligit.</s>

ponitur, <lb/>& &longs;imilia plura alia, quorum con&longs;iderationem Faber omninò negligit.</s>

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<s>2. cap.

<s>9. <emph type="italics"/>(Vt circuli medium deprehendere non <lb/>cuiu&longs;libet, &longs;ed <expan abbr="&longs;ci&etilde;tis">&longs;cientis</expan> &longs;olummodo e&longs;t)<emph.end type="italics"/> Reperire medium, <lb/>&longs;iue centrum dati circuli docet Euclides propo&longs;itio<lb/>ne prima 3. hoc modo. </s>

9. <emph type="italics"/>(Vt circuli medium deprehendere non <lb/>cuiu&longs;libet, &longs;ed <expan abbr="&longs;ci&etilde;tis">&longs;cientis</expan> &longs;olummodo e&longs;t)<emph.end type="italics"/> Reperire medium, <lb/>&longs;iue centrum dati circuli docet Euclides propo&longs;itio<lb/>ne prima 3. hoc modo. </s>

<s>in datocirculo ducatur vt<lb/>cunque recta B C, quæ per 10. primi diuidatur bifa<lb/>riam in F, & per F, ducatur <expan abbr="perp&etilde;dicularis">perpendicularis</expan> A E F D, <lb/>quæ &longs;ecetur bifariam in E, <expan abbr="erit&qacute;">eritque</expan>; punctum E, non &longs;o<lb/>lum ip&longs;ius lineæ medium; &longs;ed etiam totius circuli <lb/>centrum, quemadm odum ibi demon&longs;trat Euclides.</s>

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<s>3. cap.

<s>3. <emph type="italics"/>(De æ<gap/>ernis autem nemo con&longs;ultat, vt <lb/>de mundo, aut diametro, & latere, quod nulla inter &longs;e <lb/>æquabilitate conueniant)<emph.end type="italics"/> Qua ratione diameter, & latus eiu&longs;dem quadrati <lb/>nulia æquabilitate, ide&longs;t nulla communi men&longs;ura inter &longs;e conueniant, fusè <lb/>explicatum e&longs;t libro Priorum, &longs;ecto 1. cap. </s>

3. <emph type="italics"/>(De æ<gap/>ernis autem nemo con&longs;ultat, vt <lb/>de mundo, aut diametro, & latere, quod nulla inter &longs;e <lb/>æquabilitate conueniant)<emph.end type="italics"/> Qua ratione diameter, & latus eiu&longs;dem quadrati <lb/>nulia æquabilitate, ide&longs;t nulla communi men&longs;ura inter &longs;e conueniant, fusè <lb/>explicatum e&longs;t libro Priorum, &longs;ecto 1. cap.

23.</s>

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<s>5. cap.

<s>3. <emph type="italics"/>(Quod en m proportione con&longs;tat, id non tam vnitario numero, <lb/>quàm numero in vniuer&longs;um proprium e&longs;t)<emph.end type="italics"/> Per vnitarium numerum intelligitur <lb/>numerus ex vnitatibus ab&longs;tractis con&longs;litus, ide&longs;t, cuius vnitates non &longs;int res <lb/>phy&longs;icæ, &longs;ed à naturalibus ab&longs;tractæ, qualis con&longs;iderat Arithmeticus: omni <lb/>tamen numero &longs;iue ab&longs;tracto, &longs;iue non, connenit proportiones &longs;u&longs;cipere, <lb/>id e&longs;t & numero, & rebus numeratis.</s><pb pagenum="218"/>

3. <emph type="italics"/>(Quod en m proportione con&longs;tat, id non tam vnitario numero, <lb/>quàm numero in vniuer&longs;um proprium e&longs;t)<emph.end type="italics"/> Per vnitarium numerum intelligitur <lb/>numerus ex vnitatibus ab&longs;tractis con&longs;litus, ide&longs;t, cuius vnitates non &longs;int res <lb/>phy&longs;icæ, &longs;ed à naturalibus ab&longs;tractæ, qualis con&longs;iderat Arithmeticus: omni <lb/>tamen numero &longs;iue ab&longs;tracto, &longs;iue non, connenit proportiones &longs;u&longs;cipere, <lb/>id e&longs;t & numero, & rebus numeratis.</s><pb pagenum="218"/>

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<s>Ibidem <emph type="italics"/>(Nim proportio æqualitas e&longs;t rationum)<emph.end type="italics"/> Per proportionem hoc lo<lb/>co intelligenda e&longs;t illa, quam nunc appellant proportionalitatem, quæ e&longs;t <lb/>duarum rationum, &longs;eu proportionum &longs;imilitudo, &longs;iue æqualitas, vt manife<lb/>&longs;tum e&longs;t ex 4. definit. </s>

<s>5. Elem.

v. </s>

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<s>5. cap.

<s>6. <emph type="italics"/>(Atque id vet propo<gap/>tione vel numero)<emph.end type="italics"/> ide&longs;t, vel proportio<lb/>nalitate Geometrica, vel Arithmetica; quæ autem &longs;it proportionalitas <lb/>Geometrica, dictum e&longs;t paulò ante in prioribus locis Mathematicis huius <lb/>quinti libri; quæ verò &longs;it proportionalitas Arithmetica dictum e&longs;t &longs;uperius <lb/>lib. </s>

6. <emph type="italics"/>(Atque id vet propo<gap/>tione vel numero)<emph.end type="italics"/> ide&longs;t, vel proportio<lb/>nalitate Geometrica, vel Arithmetica; quæ autem &longs;it proportionalitas <lb/>Geometrica, dictum e&longs;t paulò ante in prioribus locis Mathematicis huius <lb/>quinti libri; quæ verò &longs;it proportionalitas Arithmetica dictum e&longs;t &longs;uperius <lb/>lib.

2. cap.

<s>6. Verum hæc Arithmetica proportionalitas, meritò ab Ari&longs;tot. <lb/></s>

6. Verum hæc Arithmetica proportionalitas, meritò ab Ari&longs;tot. <lb/></s>

<s>hic contradi&longs;tincta e&longs;t à proportionalitate Geometrica: quia Arithmetica <lb/>hæc analogia attenditur &longs;olum, iuxta eundem exce&longs;&longs;um numerorum, non, <lb/>autem iuxta proportionem, &longs;eu habitudinem terminorum ad inuicem, quod <lb/>maximè in Geometrica &longs;pectatur. </s>

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<s>6. cap.

<s>5. <emph type="italics"/>(Verbi cau&longs;a triangulum tres angulos duobus rectis æquales ha<lb/>bere, velnon habere)<emph.end type="italics"/> lib. </s>

5. <emph type="italics"/>(Verbi cau&longs;a triangulum tres angulos duobus rectis æquales ha<lb/>bere, velnon habere)<emph.end type="italics"/> lib.

<s>1. Priorum, &longs;ecto 3. cap. </s>

1. Priorum, &longs;ecto 3. cap.

<s>1. fusè hanc trianguli affe<lb/>ctionem expo&longs;ui.</s>

1. fusè hanc trianguli affe<lb/>ctionem expo&longs;ui.</s>

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<s>6. cap.

<s>8. <emph type="italics"/>(Nam illud etiam con&longs;ideratione dignum videtur. </s>

8. <emph type="italics"/>(Nam illud etiam con&longs;ideratione dignum videtur. </s>

<s>quid &longs;it, quod <lb/>puer fieri Mathematicus pote&longs;t, &longs;apiens autem naturalis non pote&longs;t. </s>

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<s>7. cap.

<s>8. <emph type="italics"/>(In actionibus autem principium illud e&longs;t, cuius cau&longs;a res fit, <lb/>&longs;icut in Mathematicis &longs;uppo&longs;itiones; nam neque illic ratio e&longs;t, quæ doctrinam tra<lb/>dat principiorum, neque hic<emph.end type="italics"/>) Suppo&longs;itionum, &longs;iue principiorum Mathemati<lb/>corum tria &longs;unt genera, definitiones, po&longs;tulata, axiomata, quæ in ip&longs;o primi <lb/>Elementorum ve&longs;tibulo proponuntur: &longs;olaque terminorum explicatione <lb/><expan abbr="ab&longs;q;">ab&longs;que</expan> vllo di&longs;cur&longs;u, addi&longs;cuntur.</s>

8. <emph type="italics"/>(In actionibus autem principium illud e&longs;t, cuius cau&longs;a res fit, <lb/>&longs;icut in Mathematicis &longs;uppo&longs;itiones; nam neque illic ratio e&longs;t, quæ doctrinam tra<lb/>dat principiorum, neque hic<emph.end type="italics"/>) Suppo&longs;itionum, &longs;iue principiorum Mathemati<lb/>corum tria &longs;unt genera, definitiones, po&longs;tulata, axiomata, quæ in ip&longs;o primi <lb/>Elementorum ve&longs;tibulo proponuntur: &longs;olaque terminorum explicatione <lb/><expan abbr="ab&longs;q;">ab&longs;que</expan> vllo di&longs;cur&longs;u, addi&longs;cuntur.</s>

<s><emph type="italics"/>Ex primo Libro Magnorum Moralium.<emph.end type="italics"/></s>

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<s>Cap. 2. (<emph type="italics"/>Ab&longs;urdum enim &longs;it, volenti o&longs;tendere triangulum duobus rectis æqua<lb/>les habere angulos, &longs;umere principinm huiu&longs;modi, anima immortalis est<emph.end type="italics"/>) Repete, <lb/>quæ de hac trianguli proprietate fusè &longs;crip&longs;i lib. </s>

<s>1. Priorum, &longs;ect. </s>

1. Priorum, &longs;ect. </s>

<s>3. cap.

<s>1. <lb/>quam affectionem debet Geometra demon&longs;trare ex Geometriæ principijs, <lb/>quemadmodum facit Euclides in 32. primi, non autem ex principijs extrin<lb/>&longs;ecis, vt quod anima &longs;it immortalis.</s>

1. <lb/>quam affectionem debet Geometra demon&longs;trare ex Geometriæ principijs, <lb/>quemadmodum facit Euclides in 32. primi, non autem ex principijs extrin<lb/>&longs;ecis, vt quod anima &longs;it immortalis.</s>

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<s>& &longs;i tria gulum &longs;ecus, ita etiam, & quadratum commutabitur, <lb/>ex altera parte enim ei re&longs;pondet. </s>

<s>& &longs;i quadratum quatuor angulis rectis æquales, <lb/>non habuerit angulos ne quidem triangulum duobus rectis habebit æquales)<emph.end type="italics"/> Hanc <lb/>trianguli affectionem, habere &longs;cilicet, &longs;uos tres angulos æquales duobus re<lb/>ctis angulis abundè explicaui libro 1. Priorum, &longs;ecto 3. cap. </s>

<s>1. quam Eucli<lb/>des propo&longs;it. </s>

1. quam Eucli<lb/>des propo&longs;it. </s>

<s>32. primi demon&longs;trauit, ex qua demon&longs;tratione, tanquam ex <lb/>Geometrico principio &longs;equitur omne <expan abbr="quoq;">quoque</expan> quadrangulum habere quatuor <lb/>angulos æquales quatuor rectis angulis; omne <expan abbr="namq;">namque</expan> quadrangulum e&longs;t po<lb/>tentia duo triangula, cum diuidatur ducta ip&longs;ius diametro in duo <expan abbr="trìãgula">trìangula</expan>. <lb/></s>

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<s>Cap. 31. (<emph type="italics"/>A qui proportionale in quatuor nihilominus perficitur: nam quem<lb/>admodum A, ad B, ita C, ad D.<emph.end type="italics"/>) ide&longs;t proportionalitas in quatuor terminis <lb/>con&longs;i&longs;tit, quemadmodum pluribus &longs;upra lib. </s>

5. cap.

<s>3. Ethycorum explica<lb/>tum e&longs;t: quò nunc Lectorem ablego.</s>

3. Ethycorum explica<lb/>tum e&longs;t: quò nunc Lectorem ablego.</s>

<s><emph type="italics"/>Ex primo Libro Moralium Eudemiorum.<emph.end type="italics"/></s>

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<s>Cap. 7. <emph type="italics"/>(Nam&longs;i habenti trigono duos rectos, nece&longs;&longs;e e&longs;t tetragonum qua<lb/>tuor rectis con&longs;tare, manife&longs;tum e&longs;t, quod trigonus duos rectos habens <lb/>cau&longs;a eius exi&longs;tat. </s>

<s>Verùm &longs;i quid in trigono mutaris, nece&longs;&longs;arium e&longs;t, & <lb/>in tetragono mutes, vt &longs;i tres habuerunt, &longs;ex; & &longs;i quatuor, octo; &longs;in <lb/>verò non mutes, vt illud, ita hoc <expan abbr="quoq;">quoque</expan> habeat nece&longs;&longs;e e&longs;t)<emph.end type="italics"/> Lege prius, quæ &longs;upra <lb/>lib. </s>

<s>1. Magnor. </s>

1. Magnor. </s>

<s>moral. </s>

<s>cap.

<s>10. &longs;crip&longs;i, ex quibus po&longs;tea &longs;ic locum hunc in<lb/>terpretaberis, &longs;i triangulum habet tres angulos æquales duobus rectis an<lb/>gulis, nece&longs;&longs;e e&longs;t quodcunque quadrilaterum habere &longs;uos quatuor angulos <lb/>æquales quatuor rectis, quia omne quadrangulum continet duo triangula; <lb/>& &longs;i natura trianguli fuerit immutata ita, vt habeat tres angulos æquales <lb/>non duobus, &longs;ed tribus rectis, tunc nece&longs;&longs;e erit tetragonum <expan abbr="quoq;">quoque</expan> mutatum <lb/>e&longs;&longs;e, quia nece&longs;&longs;ariò habebit &longs;uos angulos æquales non quatuor tantum re<lb/>ctis, &longs;ed &longs;ex: pariter &longs;i triangulum habeat tres angulos quatuor rectis pa<lb/>res, quadrangulum &longs;uos habebit angulos, octo rectis æquiualentes. </s>

10. &longs;crip&longs;i, ex quibus po&longs;tea &longs;ic locum hunc in<lb/>terpretaberis, &longs;i triangulum habet tres angulos æquales duobus rectis an<lb/>gulis, nece&longs;&longs;e e&longs;t quodcunque quadrilaterum habere &longs;uos quatuor angulos <lb/>æquales quatuor rectis, quia omne quadrangulum continet duo triangula; <lb/>& &longs;i natura trianguli fuerit immutata ita, vt habeat tres angulos æquales <lb/>non duobus, &longs;ed tribus rectis, tunc nece&longs;&longs;e erit tetragonum <expan abbr="quoq;">quoque</expan> mutatum <lb/>e&longs;&longs;e, quia nece&longs;&longs;ariò habebit &longs;uos angulos æquales non quatuor tantum re<lb/>ctis, &longs;ed &longs;ex: pariter &longs;i triangulum habeat tres angulos quatuor rectis pa<lb/>res, quadrangulum &longs;uos habebit angulos, octo rectis æquiualentes. </s>

<s>His igi<lb/>tur ex Geometria &longs;atisfactum &longs;it.</s>

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<s>Cap. 10. <emph type="italics"/>(Multa verò opinione concipiunt, quæ penes nos non &longs;unt, vt diame<lb/>trum commen&longs;urabilem e&longs;&longs;e)<emph.end type="italics"/> Quæ lib. </s>

<s>Cap. 10. <emph type="italics"/>(Multa verò opinione concipiunt, quæ penes nos non &longs;unt, vt diame<lb/>trum commen&longs;urabilem e&longs;&longs;e)<emph.end type="italics"/> Quæ lib.

<s>1. Priorum, &longs;ecto 3. cap. </s>

1. Priorum, &longs;ecto 3. cap.

<s>23. de a&longs;yme<lb/>tria diametri, & co&longs;tæ eiu&longs;dem quadrati allata &longs;unt, &longs;atis huic etiam loco <lb/>facere po&longs;&longs;unt.</s>

23. de a&longs;yme<lb/>tria diametri, & co&longs;tæ eiu&longs;dem quadrati allata &longs;unt, &longs;atis huic etiam loco <lb/>facere po&longs;&longs;unt.</s>

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<s>Cap. 12. (<emph type="italics"/>Si triangulo duo recti, nece&longs;&longs;um e&longs;t hoc con&longs;equi<emph.end type="italics"/>) ide&longs;t, &longs;i triangu<lb/>lum habet tres angulos æquales duobus rectis, nece&longs;&longs;e e&longs;t con&longs;equi, vt &longs;upe<lb/>rius &longs;epius dixit, quod quadrilaterum habeat quatuor angulos æquales qua<lb/>tuor rectis, lib. </s>

1. Magn. </s>

<s>moral. </s>

<s>cap.

<s>10. &longs;atis de hac re dictum e&longs;t.</s><pb pagenum="223"/>

10. &longs;atis de hac re dictum e&longs;t.</s><pb pagenum="223"/>

<s><emph type="italics"/>Ex Septimo Moralium Eudem.<emph.end type="italics"/></s>

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<s>per mu&longs;icum au<lb/>tem modum intelligebant quandam vocum con&longs;titutionem, &longs;eu rithmum, <lb/>quem nos hodie vulgò ariam vocamus, vt doctè explicat Io&longs;ephus Zarlinus <lb/>in 4. parte In&longs;titut. </s>

<s>Mu&longs;icalium, necnon in lib. </s>

<s>Mu&longs;icalium, necnon in lib.

<s>6. &longs;upplem. </s>

6. &longs;upplem. </s>

<s>Denominati au<lb/>tem fuerunt prædicti, <expan abbr="alij&qacute;">alijque</expan>; plures modi à nationibus illis, apud quas ma<lb/>ximè in v&longs;u erant, vt Dorius à Dorien&longs;ibus; Phrygius à Phrygijs; Lydius à <lb/>Lydijs. </s>

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<s>Verùm hac de re legantur eruditi&longs;&longs;imi Dialogi de Mu&longs;ica Vincentij Ga<lb/>lilæi, cuius præcipuas rationes in fine huius operis, & chronologiæ videre <lb/>poteris. </s>

<s>Cæterum, qui plura de modis tam antiquis, quàm nouis de&longs;iderat, <pb pagenum="224"/>con&longs;ulat Io&longs;ephum Zarlinum in 4. parte In&longs;titutionum Mu&longs;icalium, necnon <lb/>lib. </s>

<s>Cæterum, qui plura de modis tam antiquis, quàm nouis de&longs;iderat, <pb pagenum="224"/>con&longs;ulat Io&longs;ephum Zarlinum in 4. parte In&longs;titutionum Mu&longs;icalium, necnon <lb/>lib.

<s>6. &longs;upplemen. </s>

6. &longs;upplemen. </s>

<s>virum vatia eruditione refertum, <expan abbr="de&qacute;">deque</expan>; Mu&longs;ica in primis <lb/>optimè meritum.</s>

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<s>Cap. 1. (<emph type="italics"/>Quare opus e&longs;t partim arithmetica æquitate vti, partim ea, quæ <lb/>e&longs;t &longs;ecundum dignitatem.<emph.end type="italics"/></s>

<s>Arithmeticam medietatem &longs;upra explicaui lib. </s>

<s>Arithmeticam medietatem &longs;upra explicaui lib.

2. cap.

<s>6. Ethi<lb/>corum. </s>

6. Ethi<lb/>corum. </s>

<s>per eam deinde, quæ e&longs;t &longs;ecundum dignitatem, intelligit <lb/>Geometricam, quam &longs;upra lib. </s>

<s>per eam deinde, quæ e&longs;t &longs;ecundum dignitatem, intelligit <lb/>Geometricam, quam &longs;upra lib.

5. cap.

<s>3. Ethicorum expo&longs;ui. </s>

3. Ethicorum expo&longs;ui. </s>

<s>vtimur enim <lb/>ea, quando opus e&longs;t di&longs;tribuere præmia non omnibus æqualiter, &longs;ed habita <lb/>ratione meritorum vniu&longs;cuiu&longs;que. </s>

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<s>Porrò &longs;ciendum <lb/>Ari&longs;t.

locum hunc ex Platonis lib. </s>

locum hunc ex Platonis lib.

8. de Rep. </s>

<s>accepi&longs;&longs;e, loco Mathematico <lb/>ob&longs;curi&longs;&longs;imo, vbi ille de Reip. </s>

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<s>1421. cap.

<s>12. <pb pagenum="225"/>illius commentarij propè finem præ&longs;ens Ari&longs;t.

12. <pb pagenum="225"/>illius commentarij propè finem præ&longs;ens Ari&longs;t.

locus ex præmi&longs;&longs;is ab eo bre<lb/>uiter, ac dilucidè declaratur. </s>

<s>quæ explanatio, quoniam mihi præ cæteris ar<lb/>ridet, eam hoc loco, explicatiorem tamen, referam. </s>

<s>Illud autem præ&longs;cien<lb/>dum e&longs;t, hæc quæ a Socrate lib. </s>

<s>Illud autem præ&longs;cien<lb/>dum e&longs;t, hæc quæ a Socrate lib.

<s>8. de Repub. </s>

8. de Repub. </s>

<s>recen&longs;entur, confingi à Mu&longs;is, <lb/>tanquam oraculum quoddam ob&longs;curi&longs;&longs;imum effata; quo arcana quædam <lb/>my&longs;teria de Rerump. </s>

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<s>Polit. </s>

<s>cap.

<s>2. tractaui, <expan abbr="earum&qacute;">earumque</expan>; proprieta<lb/>tes, quas hic Ari&longs;t.

2. tractaui, <expan abbr="earum&qacute;">earumque</expan>; proprieta<lb/>tes, quas hic Ari&longs;t.

recen&longs;et ibi connumeraui.</s>

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<s>Ibidem meminit etiam Rithmorum, & Harmonie. </s>

<s>Quid Rithmus dictum <lb/>e&longs;t &longs;uperius lib. </s>

<s>Quid Rithmus dictum <lb/>e&longs;t &longs;uperius lib.

<s>3. Politic. </s>

3. Politic. </s>

<s>e&longs;&longs;e quem nunc vulgò ariam cantores, ac tibici<lb/>nes appellant.</s>

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<s>Cap. 7 (<emph type="italics"/>Con&longs;iderandum vtrum omnibus vtendum &longs;it harmonijs, & rithmis<emph.end type="italics"/>) <lb/>Vide quæ &longs;upra lib. </s>

<s>Cap. 7 (<emph type="italics"/>Con&longs;iderandum vtrum omnibus vtendum &longs;it harmonijs, & rithmis<emph.end type="italics"/>) <lb/>Vide quæ &longs;upra lib.

<s>3. Politic. </s>

3. Politic. </s>

<s>cap.

<s>2. annotaui.</s>

2. annotaui.</s>

<s><emph type="italics"/>In Oeconomicis nihil Mathematicum reperi.<emph.end type="italics"/></s><figure></figure><pb pagenum="227"/>

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<s>3. <emph type="italics"/>(Quemadmodum tempora, ita &longs;yderum ortus, Orionis <lb/>Arcturi, Virgiliarum, Caniculæ, qui flatus, imbres&queacute; excitant, qui &longs;ereni<lb/>tates, frigora, tepore&longs;uè &longs;olent afferre)<emph.end type="italics"/> Intelligit de ortu co&longs;mico, qui <lb/>fit, quando a&longs;trum &longs;imul cum Sole oritur: quem ortum abundè in 2. <lb/>Meter. </s>

<s>&longs;umma 2. cap. </s>

<s>&longs;umma 2. cap.

<s>2. explicatum inuenies. </s>

2. explicatum inuenies. </s>

<s>Vt autem intelligas, quænam <lb/>&longs;int Orionis, Arcturi, Virgiliarum, & Caniculæ con&longs;tellationes, & in qua <lb/>c&oelig;li parte &longs;int collocatæ, &longs;atius e&longs;t globum aliquem a&longs;tronomicum, in quo <lb/>a&longs;teri&longs;mi omnes clarè depicti &longs;int, intueri, quàm hoc loco pluribus verbis <lb/>rem per &longs;e claram, ob&longs;curare. </s>

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<s>De Canicula &longs;atis dixi in 2. Meteor. </s>

<s>&longs;umma 2. cap. </s>

<s>&longs;umma 2. cap.

2.</s>

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<s>17. <emph type="italics"/>(Cur à Virgiliarum occa&longs;u ad Fauonij v&longs;que flatus, bi <lb/>poti&longs;&longs;imum pereant, qui morbo longo laborant, & &longs;enes, quam iuuenes potius?)<emph.end type="italics"/><lb/>Lege, quæ &longs;crip&longs;i de occa&longs;u &longs;yderum lib. </s>

<s>2. Meteor. </s>

2. Meteor. </s>

<s>&longs;umma 2. cap. </s>

<s>&longs;umma 2. cap.

<s>2. dein<lb/>de, quæ in &longs;uperiore proximè loco de Virgilijs: quibus hæc pauca <expan abbr="addãtur">addantur</expan>. <lb/></s>

2. dein<lb/>de, quæ in &longs;uperiore proximè loco de Virgilijs: quibus hæc pauca <expan abbr="addãtur">addantur</expan>. <lb/></s>

<s>cum intelligat de co&longs;mico Virgiliarum occa&longs;u, qui noctu apparet, <expan abbr="incipit&qacute;">incipitque</expan>; <lb/>tunc primum, quando oriente Sole, ip&longs;e occumbunt, nece&longs;&longs;e e&longs;t occa&longs;um <lb/>hunc incipere po&longs;t autumnale <expan abbr="æquinocti&utilde;">æquinoctium</expan> ferè &longs;e&longs;quimen&longs;e in no&longs;tris regio<lb/>nibus; cum enim Virgiliæ &longs;int in Tauro, nece&longs;&longs;e e&longs;t occidente Tauro initio <lb/>dixi, vt Sol &longs;it in oppo&longs;ito &longs;igno, videlicet in Scorpione; in quo a&longs;teri&longs;mo <lb/>Sol reperitur po&longs;t prædictum æquinoctium vno ferè men&longs;e cum dimidio. </s>

<s>de <lb/>hac re plura Plinius lib. </s>

<s>de <lb/>hac re plura Plinius lib.

18. cap.

31.</s>

<s><emph type="italics"/>Ex Sectione 15.<emph.end type="italics"/></s>

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<s>vt par, impar, quadratum, quadrantale, lon<lb/>gum, planum, primum, compo&longs;itum)<emph.end type="italics"/> Cur omnes nationes miro quodam con<lb/>&longs;en&longs;u &longs;uos numeros in denas, veluti in gradus quo&longs;dam diuidant, Ari&longs;toteles <lb/>cau&longs;am indagaturus, re&longs;pondet primò id fortè accidi&longs;&longs;e ob denarij numeri <lb/>perfectionem: cuius perfectionis hoe e&longs;t indicium, quod denarius continea<gap/><lb/>omnes numerorum &longs;pecies. </s>

<s>quæ quidem omnes numerorum &longs;pecies in defi<lb/>nitionibus 7. Elem. </s>

<s>quæ quidem omnes numerorum &longs;pecies in defi<lb/>nitionibus 7. Elem.

<s>exponuntur, quas con&longs;ulere debes. </s>

exponuntur, quas con&longs;ulere debes. </s>

<s>in denario numero <lb/>contineri numeros pares, ac impares, per &longs;e patet. </s>

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<s>in denario autem <expan abbr="cõtinetur">continetur</expan> etiam <lb/>hic numerus, e&longs;t enim octonarius numerus cubus, fit enim ex binario ter in <lb/>&longs;e ip&longs;um multiplicato, hoc modo; duo bis faciunt quatuor: rur&longs;us duo qua<lb/>ter faciunt octo; quem ex definitione numeri cubi, con&longs;tat e&longs;&longs;e cubum. </s>

<s>qua <lb/>ratione deinde reliqui numeri, longus, planus, primus, compo&longs;itus, in de<lb/>nario exi&longs;tant, facilè e&longs;t cogno&longs;cere, dummodo eorum definitiones tenean<lb/>tur, quæ initio 7. Elem. </s>

<s>traduntur.</s>

traduntur.</s>

<s>Ibidem <emph type="italics"/>(An quod denarius fons, <expan abbr="atq;">atque</expan> principium e&longs;t, quippe qui ex vno, duo<lb/>bus, tribus, & quatuor con&longs;tet)<emph.end type="italics"/> Aliam denarij dignitatem a&longs;&longs;ignat, quam ex <lb/>Plutarco lib. </s>

1. cap.

<s>3. de Placitis Philo&longs;ophorum, optimè po&longs;&longs;umus intel<lb/>ligere: vbi &longs;ic ait: Pythagorei aiebant denarium e&longs;&longs;e Naturam, quoniam <lb/>omnes gentes <expan abbr="v&longs;q;">v&longs;que</expan> ad decem natura duce numerabant. </s>

3. de Placitis Philo&longs;ophorum, optimè po&longs;&longs;umus intel<lb/>ligere: vbi &longs;ic ait: Pythagorei aiebant denarium e&longs;&longs;e Naturam, quoniam <lb/>omnes gentes <expan abbr="v&longs;q;">v&longs;que</expan> ad decem natura duce numerabant. </s>

<s>tum etiam, quia ex <lb/>quaternario con&longs;tabat, ide&longs;t, ex his quatuor numeris 1. 2. 3. 4. qui &longs;imul ad<lb/>diti faciunt decem: quaternarium enim Pythagorei multis de cau&longs;is, quas <lb/>apud Petrum Bungum de my&longs;terijs numerorum videre poteris, adeò extol<lb/>lebant, vt dicerent ex quaternario naturalia, omnia con&longs;tare, vt quaterni<lb/>tati omnia accepta referrent. </s>

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<s><emph type="italics"/>Iuro per omnipotentem animæ, qui Tetrada nofiræ <lb/>Perpetuos fontes naturæ infudit habentem.<emph.end type="italics"/></s>

<s>Pr&ecedil;cipua verò cau&longs;a, cur tantopere Pythagorei quaternitatem celebrarint, <lb/>refertur à Mar&longs;ilio Ficino cap. </s>

<s>Pr&ecedil;cipua verò cau&longs;a, cur tantopere Pythagorei quaternitatem celebrarint, <lb/>refertur à Mar&longs;ilio Ficino cap.

<s>24. compendij in Tymeum, his verbis: ex <lb/>quatuor elementis, geometrica, & harmonica ratione coniunctis inuicem, <lb/>vniuer&longs;um Mundum compo&longs;itum Pythagorei omnes exi&longs;timant: con&longs;onan<lb/>tiam horum in c&oelig;lo &longs;emper e&longs;&longs;e perfectam, &longs;ub c&oelig;lo autem aliquando dif<lb/>&longs;onantem. </s>

24. compendij in Tymeum, his verbis: ex <lb/>quatuor elementis, geometrica, & harmonica ratione coniunctis inuicem, <lb/>vniuer&longs;um Mundum compo&longs;itum Pythagorei omnes exi&longs;timant: con&longs;onan<lb/>tiam horum in c&oelig;lo &longs;emper e&longs;&longs;e perfectam, &longs;ub c&oelig;lo autem aliquando dif<lb/>&longs;onantem. </s>

<s>volebant ergò totum mundum, tam æthereum, quàm elementa<lb/>rem, con&longs;tare ex quatuor elementis; & ideò ex quaternario omnia con&longs;ta<lb/>re dicebant. </s>

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<s>comprobare, <expan abbr="atq;">atque</expan> intelligere po&longs;<lb/>&longs;umus. </s>

<s>e&longs;t autem 8. 9. Elem. </s>

<s>e&longs;t autem 8. 9. Elem.

<s>propo&longs;itio hæc: &longs;i decem numeri in eadem pro<pb pagenum="230"/>portione progrediantur ab vnitate incipientes, erunt ex illis quatuor cubi, <lb/>v.g. </s>

propo&longs;itio hæc: &longs;i decem numeri in eadem pro<pb pagenum="230"/>portione progrediantur ab vnitate incipientes, erunt ex illis quatuor cubi, <lb/>v.g. </s>

<s>in &longs;erie duplæ proportionis progrediantur hi decem termini: 1. 2. 4. 8. <lb/>16. 32. 64. 128. 256. 512. ex his decem numeris &longs;unt quatuor cubi, nimi<lb/>rum hi 1. 8. 64. 512. numerus cubus e&longs;t, qui fit ex tribus æqualibus numeris <lb/>in &longs;e multiplicatis. </s>

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<s>Difficile admodum e&longs;t problema i&longs;tud, & cuius &longs;olutio<lb/>nem nullus veterum, quod &longs;ciam, perfectè attigit: quamuis Vitellio nu. </s>

<s>39. <lb/>lib.

<s>2. hoc idem proponat, ac &longs;oluere contendat: Verùm nec Ari&longs;t.

2. hoc idem proponat, ac &longs;oluere contendat: Verùm nec Ari&longs;t.

nec Vi<lb/>tellio intellectui &longs;atisfaciunt mathematico: probabilia tamen afferunt. </s>

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<s><expan abbr="manife&longs;t&utilde;">manife&longs;tum</expan> eft enim, quò lon<lb/>gius radij C F, C H, producti fuerint, eò <lb/>maiorem fore <expan abbr="diametr&utilde;">diametrum</expan> illuminationis F H. <lb/>euadet enim L O, & &longs;imiliter ex productio<lb/>ne radiorum D K, D E, diameter alterius <lb/>illuminationis K E, augebitur, & fiet N M. <lb/>& con&longs;equenter duæ ip&longs;arum peripheriæ &longs;i<lb/>mul maiores fient, ac proinde ad vnius cir<lb/>culi &longs;imilitudinem ex <expan abbr="&longs;ec&utilde;da">&longs;ecunda</expan> notatione per<lb/>uenient. </s>

<s>& quamuis ex radiorum produ<lb/>ctione augeantur non &longs;olum prædictæ dia<lb/>metri illuminationum, &longs;ed etiam earum <lb/>differentiæ F K, & H E; hæ tamen differen<lb/>tiæ re&longs;pectu illorum nihil, quod &longs;en&longs;ibile &longs;it <lb/>augentur; quod inde oritur, quia angulus F C H, maior e&longs;t angulo F B K, <lb/>per 16. primi Elem. </s>

<s>& ideò crura F C, H C, magis dilatata &longs;unt quàm cru<lb/>ra F B, K B, & ideò &longs;i producantur, multò magis cre&longs;cit F H, dum euadit <lb/>M N, quàm F K, dum euadit M O. eodem modo magis cre&longs;cit K E, dum fit <lb/>O L, quàm H E, dum fit K L. quare ex &longs;ecunda notatione earum periphe<lb/>riæ ad vnius orbis figuram tandem concurrere videbuntur. </s>

& ideò crura F C, H C, magis dilatata &longs;unt quàm cru<lb/>ra F B, K B, & ideò &longs;i producantur, multò magis cre&longs;cit F H, dum euadit <lb/>M N, quàm F K, dum euadit M O. eodem modo magis cre&longs;cit K E, dum fit <lb/>O L, quàm H E, dum fit K L. quare ex &longs;ecunda notatione earum periphe<lb/>riæ ad vnius orbis figuram tandem concurrere videbuntur. </s>

<s>multò autem <lb/>euidentius ad rotunditatem accederent, &longs;i tertia illuminatio per tertium <lb/>aliquod punctum tran&longs;iens, &longs;ic perueniret; & quo plures, eò etiam perfectius, <lb/>omnes enim rotundæ e&longs;&longs;ent, & ex radiorum proce&longs;&longs;u augerentur, atque ad <lb/>vnius orbis formam &longs;e &longs;e reciperent.</s>

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<s>eius tantum experientiam non <lb/>iniucundam, qua i&longs;tud probat, non grauabor referre. </s>

<s>cap.

<s>igitur &longs;ecundo <lb/>de Figuratione lucis hæc habet: librum in &longs;ublimi locaui, qui e&longs;&longs;et loco lu<lb/>centis corporis, hunc inter & pauimentum figebatur tabella foramine mul<lb/>cangulo. </s>

igitur &longs;ecundo <lb/>de Figuratione lucis hæc habet: librum in &longs;ublimi locaui, qui e&longs;&longs;et loco lu<lb/>centis corporis, hunc inter & pauimentum figebatur tabella foramine mul<lb/>cangulo. </s>

<s>filum deinde ex vno libri angulo per foramen in pauimentum de<lb/>mi&longs;&longs;um, ita incidebat in pauimento, vt terminos foraminis raderet, cuius <lb/>ve&longs;tigia creta imitabar; qua ratione creabatur figura in pauimento &longs;imilis <lb/>foramini. </s>

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<s>per<lb/>moto enim Sole, & circulus, &longs;ecundum quem con&longs;picitur, reuertitur ad eadem <lb/>puncta. </s>

<s>&longs;ecundum enim infinitas inclinationes accidit inclinari: &longs;i quidem maxi<lb/>mi circuli per eadem puncta duci po&longs;&longs;unt infiniti)<emph.end type="italics"/> Vt rectè textum hunc intel<lb/>ligas, lege prius, quæ de Lunæ illuminatione lib. </s>

<s>1. Po&longs;t. </s>

1. Po&longs;t. </s>

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<s>quæ linea &longs;emper in Luna e&longs;t circula<lb/>ris, cum &longs;it in &longs;ph&ecedil;rico corpore: quan<lb/>do tamen Luna videtur &longs;emiplena, vt <lb/><expan abbr="quãdo">quando</expan> e&longs;t in D, vel in K. hæc linea K L, <lb/>videtur recta. </s>

<s>ratio huius e&longs;t, quia exi<lb/>&longs;tente Luna &longs;emiplena, circulus K D L, <lb/>qui e&longs;t ba&longs;is illuminationis &longs;olaris, e&longs;t in eodem plano cum oculo A, &longs;iue in <lb/>cadem rectitudine, vt apparet in figura, vbi, <gap/> linea K D L, &longs;umatur loco <pb pagenum="238"/>diametri prædicti circuli, & produci intelligatur ver&longs;us oculum A, per ip<lb/>ium tran&longs;ilit; quo in &longs;itu, &longs;i circulus oculo &longs;ubijciatur, non planam ip&longs;ius <lb/>&longs;uperficiem, &longs;ed circunferentiam <expan abbr="tant&utilde;">tantum</expan> a&longs;picit, <expan abbr="fit&qacute;">fitque</expan>; vt non lineam curuam, <lb/>&longs;ed rectam videre videatur, vt in præcedenti problemate diximus, & Per<lb/>&longs;pectiui demon&longs;trant, & Vitellio lib. </s>

<s>4. propo&longs;it. </s>

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

<s>5. & propo&longs;it. </s>

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<s>An, vt ea omnia, quorum quodnam plus, minu&longs;uè, di&longs;tet, incertum &longs;it, æquè <lb/>po&longs;ita e&longs;&longs;e videntur? </s>

<s>&longs;ic etiam res, quamuis vna, cùm plures tamen habeat partes, <lb/>ni&longs;i varius color ad&longs;it, partes illæ omnes, ex æquo collocatas videri nece&longs;&longs;e e&longs;t: quod <lb/>autem ex æquo videtur, nece&longs;&longs;arium etiam e&longs;t æquabile, ac planum apparere)<emph.end type="italics"/><lb/>Quæ&longs;tionem hanc demon&longs;tratiuè pertractat Vitellio lib. </s>

<s>4. propo&longs;it. </s>

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

<s>65. Eu<lb/>clides etiam theor. </s>

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<s><expan abbr="itaq;">itaque</expan> nece&longs;&longs;e e&longs;t radium à &longs;upe<lb/>riore procedentem intus cadere. </s>

<s>&longs;it Gnomon A D, Luna B Sol C, Lunæ radius B F.<emph.end type="italics"/><pb pagenum="240"/><figure id="fig138"></figure><lb/><emph type="italics"/>ergò vmbra Lunæ D F, Solis radius <lb/>C E; vmbra igitur nece&longs;&longs;ariò minor <lb/>e&longs;t, e&longs;t enim D E.)<emph.end type="italics"/> de hac re vide <lb/>Spheram P. Clauij, cap. </s>

<s>de Ordine <lb/>c&oelig;lorum, vnde huius textus expo<lb/>&longs;itionem in hunc modum licebit af<lb/>ferre. </s>

de Ordine <lb/>c&oelig;lorum, vnde huius textus expo<lb/>&longs;itionem in hunc modum licebit af<lb/>ferre. </s>

<s>Quando igitur Ari&longs;t.

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<s>&longs;ed in medio poti&longs;&longs;imum <expan abbr="&longs;pl&etilde;det">&longs;plendet</expan>. </s>

<s>lunula autem falcata exiguam habet latitudinem)<emph.end type="italics"/><lb/>Vt rectè, atque non &longs;ine delectatione problema præ&longs;ens intelligas, lege ea, <lb/>quæ in additione ad problema 5. huius &longs;ectionis &longs;crip&longs;imus de Figuratione <lb/>lucis: deinde operæpretium erit audire, quid dicat Gemma Fri&longs;ius in tra<lb/>ctatu de Radij aftronomici &longs;tructura, cap. </s>

<s>18. vbi loquitur de Solis deliquij <pb pagenum="241"/>dimen&longs;ione, his verbis, extat, inquit, alius modus dimetiédæ &longs;olaris eclyp<lb/>&longs;is, omnium facillimus, ac certi&longs;&longs;imus, cuius nos admonuit Era&longs;mus Rei<lb/>noldus in comm. </s>

18. vbi loquitur de Solis deliquij <pb pagenum="241"/>dimen&longs;ione, his verbis, extat, inquit, alius modus dimetiédæ &longs;olaris eclyp<lb/>&longs;is, omnium facillimus, ac certi&longs;&longs;imus, cuius nos admonuit Era&longs;mus Rei<lb/>noldus in comm. </s>

<s>in Theoricas Peurbarhij. </s>

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<s>deinde ait: &longs;i igitur nihil aliud e&longs;t <lb/>vmbra, quam id, quod non a&longs;picitur à Sole, & à Sole tamen videamus illu<lb/>minari totam bullæ ba&longs;im circulariter, nece&longs;&longs;e e&longs;t totam etiam bullam <expan abbr="vn-diq;">vn<lb/>dique</expan> à Sole illuminari, & con&longs;pici, quod tamen impo&longs;&longs;ibile e&longs;&longs;e demon&longs;tra<lb/>tur ab opticis: ip&longs;i enim demon&longs;trant, nullum corpus, quantumuis mini<lb/>mum, totum po&longs;&longs;e circum&longs;pici à quamuis maximo illuminante. </s>

<s>quod qui<lb/>dem antiquitus demon&longs;trauit Ari&longs;tarchus Samius in libello de di&longs;tantijs So<lb/><figure id="fig140"></figure><lb/>lis, & Lunæ: & <lb/>po&longs;tea Vitellio <lb/>lib. </s>

<s>quod qui<lb/>dem antiquitus demon&longs;trauit Ari&longs;tarchus Samius in libello de di&longs;tantijs So<lb/><figure id="fig140"></figure><lb/>lis, & Lunæ: & <lb/>po&longs;tea Vitellio <lb/>lib.

<s>2. propo&longs;. </s>

2. propo&longs;. </s>

<s>27. <lb/>& ex figura præ<lb/>&longs;enti facilè e&longs;t id <lb/>intelligere: <expan abbr="&itilde;">im</expan> qua <lb/>&longs;it Sol &longs;phæra A, <lb/>illuminans &longs;phæ<lb/>rulam B, extre<lb/>mi radij DF, EF. <pb pagenum="243"/>vmbra erit igitur G F H, ad partes C, Soli auer&longs;as. </s>

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<s>quapropter optimè intelliges textum hunc, &longs;i huiu&longs;modi <lb/>volumen bis &longs;ecueris, primo quidem &longs;ectione ba&longs;i voluminis parallela; &longs;e<lb/>cundo verò &longs;ectione tran&longs;uer&longs;ali, &longs;eu obliqua ad ba&longs;im: nam explicata pri<lb/>ma &longs;ectione apparebit eam e&longs;&longs;e lineam rectam: euoluta verò <expan abbr="&longs;ec&utilde;da">&longs;ecunda</expan> &longs;ectio<lb/>ne apparebit eam e&longs;&longs;e tortuo&longs;am, & flexuo&longs;am; Arift. </s>

<s>reddens rationem, <lb/>cur hæc &longs;it tortuo&longs;a, ait id e&longs;&longs;e, quia &longs;ectione obliqua exi&longs;tente, ide&longs;t ex vna <lb/>parte depre&longs;&longs;iori, & ex altera altiori, &longs;equitur, quod circuli, qui ex tali &longs;e<lb/>ctione oriuntur non remanent in eodem plano, dum euoluuntur; quare <expan abbr="neq;">neque</expan> <lb/>linea, ex qua illi circuli con&longs;tant, poterit e&longs;&longs;e in eodem plano, & ideo <expan abbr="neq;">neque</expan> <lb/>recta e&longs;&longs;e poterit, quia fieri nequit, vt eiu&longs;dem lineæ pars &longs;it in plano vno, <lb/>pars verò in altero; quod o&longs;tenditur in prima 11. Elem. </s>

<s>quæ e&longs;t hæc; rectæ <lb/>lineæ pars quædam non e&longs;t in &longs;ubiecto plano, pars verò in &longs;ublimi.</s>

quæ e&longs;t hæc; rectæ <lb/>lineæ pars quædam non e&longs;t in &longs;ubiecto plano, pars verò in &longs;ublimi.</s>

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<s>In 2. problema. </s>

<s>In verba illa <emph type="italics"/>(Sed quemadmodum linea bipedalis non<emph.end type="italics"/><lb/><figure id="fig143"></figure><lb/><emph type="italics"/>duplum, &longs;ed quadruplum quoddam de&longs;cribit, <lb/>&longs;ic, &c.)<emph.end type="italics"/> ide&longs;t, quemadmodum linea bi<lb/>pedalis, quæ quamuis &longs;it dupla lineæ pedalis non <lb/>tamen de&longs;cribit quadratum duplum quadrati il<lb/>lius, &longs;ed quadruplum: vt probatur in &longs;cholio 4. <lb/>2. Elem. </s>

<s>& videre e&longs;t in hac figura, vbi linea A B, <lb/>e&longs;t dupla lineæ A C. quadratum verò lineæ A B, <lb/>&longs;cilicet quadratum A B D E, e&longs;t <expan abbr="quadrupl&utilde;">quadruplum</expan> qua<lb/>drati lineæ A C, quadrati nimirum A C F G. re<lb/>liqua huius textus manife&longs;ta &longs;unt</s>

& videre e&longs;t in hac figura, vbi linea A B, <lb/>e&longs;t dupla lineæ A C. quadratum verò lineæ A B, <lb/>&longs;cilicet quadratum A B D E, e&longs;t <expan abbr="quadrupl&utilde;">quadruplum</expan> qua<lb/>drati lineæ A C, quadrati nimirum A C F G. re<lb/>liqua huius textus manife&longs;ta &longs;unt</s>

<s>Scias Lector, me nullum, horum de Mu&longs;ica Problematum (quemadmo<lb/>dum & in pluribus alijs mathematicis locis accidit) vidi&longs;&longs;e expo&longs;itorem, <lb/>præter vnum Petrum Aponentem, quem tamen tanquam omninò his rebus <lb/>elucidandis ineptum, reieci.</s><pb pagenum="247"/>

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<s>vbi per cantandi principium puto ip&longs;um intellige<lb/>re &longs;emitonium, &longs;iue die&longs;im, nam, vtip&longs;e ait primo <lb/><figure id="fig147"></figure><lb/>re &longs;emitonium, &longs;iue die&longs;im, nam, vt ip&longs;e ait primo <lb/>Po&longs;ter. </s>

<s>cap.

<s>38. die&longs;is h&ecedil;c, &longs;iue &longs;emitonium dicitur <lb/>principium cantus, quia minimum e&longs;t omnium in<lb/>teruallorum, quæ voce po&longs;&longs;int exprimi: <expan abbr="atq;">atque</expan> ex eo <lb/>alia interualla con&longs;tant, <expan abbr="e&longs;t&qacute;">e&longs;tque</expan>; veluti illorum ele<lb/>mentum. </s>

38. die&longs;is h&ecedil;c, &longs;iue &longs;emitonium dicitur <lb/>principium cantus, quia minimum e&longs;t omnium in<lb/>teruallorum, quæ voce po&longs;&longs;int exprimi: <expan abbr="atq;">atque</expan> ex eo <lb/>alia interualla con&longs;tant, <expan abbr="e&longs;t&qacute;">e&longs;tque</expan>; veluti illorum ele<lb/>mentum. </s>

<s>vide illius loci explanationem. </s>

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<s>Ex quibus patet quinam e&longs;&longs;ent enharmonij moduli, &longs;iue interualla, qui<lb/>bus enharmonium genus decantaretur. </s>

<s>Sciendum præterea ex lib. </s>

<s>Sciendum præterea ex lib.

<s>3. Mu&longs;i<lb/>corum Ptol. </s>

3. Mu&longs;i<lb/>corum Ptol. </s>

<s>modulos enharmonios fui&longs;&longs;e graues, & &longs;eueros, vt idcirco Do<lb/>rien&longs;es, quorum modi grauitate, ac &longs;eueritate præditi erant, ip&longs;is maximè <lb/>delectarentur. </s>

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<s>vide Prolæm. </s>

<s>lib.

1. <lb/>cap.

7. harm. </s>

<s>&longs;ed hoc loco ita accipiendas e&longs;&longs;e, vti dixi, manife&longs;tum e&longs;t ex <lb/>Ari&longs;t.

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<s>Velox autem, & acut&ubreve; facilè ob &longs;uam ve<lb/>locitatem latitat)<emph.end type="italics"/> Quid e&longs;&longs;et rithmus explicabitur in problemate 27. <expan abbr="&longs;equ&etilde;-ti">&longs;equen<lb/>ti</expan> ait: <emph type="italics"/>(Velox autem, & acutum)<emph.end type="italics"/> Cur vox acuta &longs;it velox, <expan abbr="dict&utilde;">dictum</expan> e&longs;t in 1. Top. <lb/></s>

<s>cap.

<s>13. reliqua &longs;unt &longs;atis clara.</s>

13. reliqua &longs;unt &longs;atis clara.</s>

<s>Probl. </s>

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<s>Iam explicandum e&longs;t breuiter, quid &longs;it rithmus, quem Latini numerum <lb/>dicunt partim ex Platone, partim ex Ari&longs;t. </s>

<s>Plato lib. </s>

<s>Plato lib.

2. de leg. </s>

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

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<s>30. <emph type="italics"/>(Cur <expan abbr="neq;">neque</expan> hypodorium, <expan abbr="neq;">neque</expan> hypop hrygium est in trag&oelig;diarum obo<lb/>ro? </s>

<s>An quia non habet anti&longs;trophon, vtpotè quæ &longs;cenica &longs;unt, imitationi&queacute;, accom<lb/>modata)<emph.end type="italics"/> huc pertinent ea, quæ ad cap. </s>

<s>An quia non habet anti&longs;trophon, vtpotè quæ &longs;cenica &longs;unt, imitationi&queacute;, accom<lb/>modata)<emph.end type="italics"/> huc pertinent ea, quæ ad cap.

2. lib.

<s>3. Polit. </s>

3. Polit. </s>

<s>&longs;crip&longs;i, de tonis, Do<lb/>rio, Phrygio, Lydio. </s>

<s>quibus nunc hæc addo, ex Boetio lib. </s>

<s>quibus nunc hæc addo, ex Boetio lib.

<s>4. tonus, &longs;en mo<lb/>dus erat quædam cantus con&longs;titutio, ab hypate <expan abbr="v&longs;q;">v&longs;que</expan> ad netem, proprio rith<lb/>mo modificata: ita vt modos Dorius alium rithmum, à Phrygio, & reliquis <lb/>di&longs;crepantem haberet. </s>

4. tonus, &longs;en mo<lb/>dus erat quædam cantus con&longs;titutio, ab hypate <expan abbr="v&longs;q;">v&longs;que</expan> ad netem, proprio rith<lb/>mo modificata: ita vt modos Dorius alium rithmum, à Phrygio, & reliquis <lb/>di&longs;crepantem haberet. </s>

<s>quilibet preterea modus &longs;uam certam &longs;edem in Mo<lb/>nochordio obtinebat, vnde &longs;equebatur vnum e&longs;&longs;e reliquis omnibus <lb/>grauiorem, alium e&longs;&longs;e omnium acuti&longs;&longs;imum, reliquos verò in<lb/>termedios, alijs grauiores fui&longs;&longs;e, vt in &longs;equenti figura, <lb/>in qua, tanquam in tabella, omnia, quæ de hi&longs;ce <lb/>modis dici &longs;olent, per&longs;picuè licet <lb/>intucri.</s><pb pagenum="262"/>

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<s>tres illi Hypodorius, Hy<lb/>pophrygius, Hypolydius, ita &longs;unt appellati, quod collocati e&longs;&longs;ent infra Do<lb/>rium, Phrygium, Lydium per vnum Tetrachordum, vt patet in figura. </s>

<s>&longs;ed <lb/>vt adhuc melius hanc rem intelligamus, dicendum e&longs;t cum Zarlino lib. </s>

<s>&longs;ed <lb/>vt adhuc melius hanc rem intelligamus, dicendum e&longs;t cum Zarlino lib.

<s>4. <lb/>In&longs;tit. </s>

4. <lb/>In&longs;tit. </s>

<s>Modos fui&longs;&longs;e varias &longs;pecies cantilenarum proprios rithmos haben<lb/>tes, certo ordine, ac certo carmine, <expan abbr="certo&qacute;">certoque</expan>; etiam in&longs;trumento decantari <lb/>&longs;olitas: denominabantur autem Doriæ, Phrygiæ, &c. </s>

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<s>&longs;i plura de modis, aut tonis <lb/>de&longs;ideras, con&longs;ule Ptolæm. </s>

<s>lib.

2. harm. </s>

<s>Boetium lib. </s>

<s>Boetium lib.

<s>4. Io&longs;ephum Zarli<lb/>num lib. </s>

4. Io&longs;ephum Zarli<lb/>num lib.

<s>4. In&longs;tit. </s>

4. In&longs;tit. </s>

<s>& lib.

<s>6. Supplem. </s>

6. Supplem. </s>

<s>mu&longs;icorum.</s>

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<s>Tempore igitur Terpandri cum &longs;eptem e&longs;&longs;ent <expan abbr="tant&utilde;">tantum</expan> chor<lb/>dæ in p&longs;alterijs, etiam&longs;i prima cum vltima con&longs;onantiam Diapa&longs;on re&longs;ona<lb/>ret, non tamen potuit hæc con&longs;onantia appellari Diaocto. </s>

<s>Boetius lib. </s>

<s>Boetius lib.

1. <lb/>cap.

<s>20. Mu&longs;icæ, prædicta a&longs;&longs;erit de Terpandro. </s>

20. Mu&longs;icæ, prædicta a&longs;&longs;erit de Terpandro. </s>

<s>Suidas ait <expan abbr="Terpandr&utilde;">Terpandrum</expan> fui&longs;&longs;e <lb/><expan abbr="Lesbi&utilde;">Lesbium</expan>, & Poetam Lyricum, qui primus lyram ex &longs;eptem chordis fecit, cùm <lb/>prius à Mercurio ex quatuor tantum con&longs;tructa fui&longs;&longs;et. </s>

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ex &longs;equentibus percipie<lb/>mus. </s>

<s>Pro intelligentia igitur huius problematis placet hic de&longs;cribere <expan abbr="demõ-&longs;trationem">demon<lb/>&longs;trationem</expan> 16. lib. </s>

<s>Pro intelligentia igitur huius problematis placet hic de&longs;cribere <expan abbr="demõ-&longs;trationem">demon<lb/>&longs;trationem</expan> 16. lib.

<s>3. docti&longs;&longs;imi Fabri &longs;tapulentis, qua ip&longs;e ve&longs;tigijs <expan abbr="antiquo-r&utilde;">antiquo<lb/>rum</expan> inhærens optimè præ&longs;enti quæ&longs;tioni &longs;atisfacit. </s>

3. docti&longs;&longs;imi Fabri &longs;tapulentis, qua ip&longs;e ve&longs;tigijs <expan abbr="antiquo-r&utilde;">antiquo<lb/>rum</expan> inhærens optimè præ&longs;enti quæ&longs;tioni &longs;atisfacit. </s>

<s>e&longs;t <expan abbr="a&utilde;t">aunt</expan> huiu&longs;modi: Bi&longs;dia<lb/>te&longs;&longs;aron, aut bis <expan abbr="Diap&etilde;te">Diapente</expan> <expan abbr="nullã">nullam</expan> con&longs;onantiam <expan abbr="cõponere">componere</expan> pote&longs;t, omnis <expan abbr="namq;">namque</expan> <lb/>con&longs;onantia, aut in proportione multiplici, aut in &longs;uperparticulari collo<lb/>canda e&longs;t, ex Pythagoreorum, <expan abbr="aliorum&qacute;">aliorumque</expan>; Mu&longs;icorum traditione; &longs;ed &longs;i duæ <lb/>Diate&longs;&longs;aron, aut duæ Diapente componantur, <expan abbr="neq;">neque</expan> multiplicem, neque &longs;u<lb/>perarticularem creant rationem, ergò additæ nullam efficere valent <expan abbr="cõ&longs;o-nantiam">con&longs;o<lb/>nantiam</expan>. </s>

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<s>Quamquam facilius &longs;it mouere exiguum, quam <expan abbr="magn&utilde;">magnum</expan>: <lb/>idem <expan abbr="itaq;">itaque</expan> in aere deberet accidere. </s>

<s>An quia non idem e&longs;t e&longs;&longs;e acutæ vocis à natu<lb/>ræ, <expan abbr="atq;">atque</expan> <expan abbr="acut&utilde;">acutum</expan> canere: verùm naturaliter imbecilla omnia acutæ &longs;unt vocis; prop<lb/>terea ectici &longs;unt acutæ vocis, quia parum aeris non multum ciere po&longs;&longs;unt: paucus <lb/>verò velociter fertur; in cantu verò acutum canert &longs;ignum e&longs;t roboris, quod enim <lb/>valdè <expan abbr="f&emacr;rtur">ferrtur</expan>, velociter fertur: & difficilè e&longs;t alta canere, at grauia &longs;unt humilia)<emph.end type="italics"/><lb/>Vt intelligas pr&ecedil;&longs;ens Problema, lege, quæ lib. </s>

1. Top. </s>

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<s>I. & Agell. </s>

<s>lib.

<s>16. Plutarchus <expan abbr="quoq;">quoque</expan> de mu&longs;ica &longs;æpè meminit Or<lb/>thiæ legis.</s>

16. Plutarchus <expan abbr="quoq;">quoque</expan> de mu&longs;ica &longs;æpè meminit Or<lb/>thiæ legis.</s>

<s>Difficilius deberet e&longs;&longs;e canere graue, quàm <expan abbr="acut&utilde;">acutum</expan>, quia graue e&longs;t in mul<lb/>to, & acutum in paruo, vt patet in cannis. </s>

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<s>primo harm. </s>

<s>cap.

<s>7. & alijs: &longs;ed illa Ari&longs;tot. </s>

7. & alijs: &longs;ed illa Ari&longs;tot. </s>

<s>&longs;ententiæ <lb/>minimè quadrant. </s>

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<s>7. vide igitur, quæ ibi expo&longs;ui; hoc loco quærendum re&longs;tat, quid &longs;it <lb/>illud Acutiden&longs;um. </s>

<s>pro qua re vide Ari&longs;toxenum lib. </s>

<s>pro qua re vide Ari&longs;toxenum lib.

<s>3. & Euclidem in I&longs;a<lb/>goge ad Mu&longs;icam: Zarlinum tandem lib. </s>

3. & Euclidem in I&longs;a<lb/>goge ad Mu&longs;icam: Zarlinum tandem lib.

<s>2. &longs;upplem. </s>

2. &longs;upplem. </s>

<s>& 5. quæ res, quamuis <lb/>plura dicant, adhuc ob antiquitatem non &longs;atis intelliguntur. </s>

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<s>excur&longs;us verò, & ar<lb/>matio erant partes, quibus con&longs;tabat Trag&oelig;dia, quemadmodum no&longs;træ in <lb/><expan abbr="quinq;">quinque</expan> actus diuiduntur. </s>

<s>vide Zarlinum cap. </s>

<s>vide Zarlinum cap.

<s>5. primæ partis In&longs;tit. </s>

5. primæ partis In&longs;tit. </s>

<s>vbi tra<lb/>dit fabulam quandam Delonam dictam, quæ in modum Trag&oelig;diæ habeba<lb/>tur, fui&longs;&longs;e diui&longs;am in 5. partes, Explorationem, Prouocationem, Iambicum, <lb/>Spondeum, & Ouationem, aut Saltationem.</s>

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<s>Archimedes enim demon&longs;t. </s>

<s>5. <lb/>lib.

<s>1. de ijs, quæ vehuntur in aqua acuti&longs;&longs;imè demon&longs;trat; &longs;olidarum ma<lb/>gnitudinum humido læuiorum, in humidum eò <expan abbr="v&longs;q;">v&longs;que</expan> demergi, vt tanta moles <lb/>humidi, quanta e&longs;t partis demer&longs;æ, eandem quam tota magnitudo, graui<lb/>tatem habeat. </s>

1. de ijs, quæ vehuntur in aqua acuti&longs;&longs;imè demon&longs;trat; &longs;olidarum ma<lb/>gnitudinum humido læuiorum, in humidum eò <expan abbr="v&longs;q;">v&longs;que</expan> demergi, vt tanta moles <lb/>humidi, quanta e&longs;t partis demer&longs;æ, eandem quam tota magnitudo, graui<lb/>tatem habeat. </s>

<s>quod idem Galilæus Galilæus, in Italico Di&longs;cur&longs;u de rebus, <lb/>quæ aquæ innatunt, &longs;ubtiliter <expan abbr="cõprobauit">comprobauit</expan>, vt videre e&longs;t apud ip&longs;um pag. </s>

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<s>Ad 6 Probl. <emph type="italics"/>(Cur nihil in eo delectamur, quod triangulum duobus rectis pa<lb/>res angulos internos habere &longs;pectamus)<emph.end type="italics"/> vide quæ lib. </s>

<s>Ad 6 Probl. <emph type="italics"/>(Cur nihil in eo delectamur, quod triangulum duobus rectis pa<lb/>res angulos internos habere &longs;pectamus)<emph.end type="italics"/> vide quæ lib.

<s>1. Priorum, &longs;ecto <lb/>3. cap. </s>

1. Priorum, &longs;ecto <lb/>3. cap.

<s>3. de hac trianguli proprietate annotaui, cuius etiam &longs;æpius Ari&longs;t. <lb/></s>

3. de hac trianguli proprietate annotaui, cuius etiam &longs;æpius Ari&longs;t. <lb/></s>

<s>meminit, nunquam tamen verbum illud, internos, præterquam hic, addi<lb/>dit; vt autem benè intelligas quinam &longs;int hi anguli interni, & qui externi, <lb/>& quod etiam rectis externi æquiualeat, lege quæ ad tex. </s>

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<s>vide propo&longs;. </s>

<s>26. lib.

<s>3. Vitell. </s>

3. Vitell. </s>

<s>quæ e&longs;thæc. </s>

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

<s>28. & 46. lib.

<s>3. pertractatur: quamuis <lb/>commentum illud Vitell. </s>

3. pertractatur: quamuis <lb/>commentum illud Vitell. </s>

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<s>vide Vitell. </s>

<s>prop.

103. & 104. li. </s>

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<s>3. 5. Metaph. </s>

<s>3. 5. Metaph.

<s>& omnibus Philo&longs;ophis. </s>

& omnibus Philo&longs;ophis. </s>

<s>quæ quidem materiæ accep<lb/>tio, &longs;imilis e&longs;t acceptioni materiæ phy&longs;icæ, ex qua tanquam ex parte <expan abbr="cõpo-&longs;itum">compo<lb/>&longs;itum</expan> con&longs;latur: conflatur enim ex materia, & forma, tanquam partibus. <lb/></s>

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<s>vide huius loci explicationem &longs;upra à <lb/>nobis allatam. </s>

<s>6. Metaph. </s>

<s>6. Metaph.

tex. </s>

<s>1. <emph type="italics"/>(Mathematicorum <expan abbr="quoq;">quoque</expan> principia, elements, <lb/>& cau&longs;æ &longs;unt)<emph.end type="italics"/> 11. Metaph. </s>

<s>1. <emph type="italics"/>(Mathematicorum <expan abbr="quoq;">quoque</expan> principia, elements, <lb/>& cau&longs;æ &longs;unt)<emph.end type="italics"/> 11. Metaph.

cap.

<s>1. &longs;ummæ 3. <emph type="italics"/>(Patet igitur tria e&longs;&longs;e genera &longs;pe<lb/>culatiuarum &longs;cientiarum, Naturalem, Mathematicam, Theologiam)<emph.end type="italics"/> ecce tibi, <lb/>quam clara &longs;it Ari&longs;t. </s>

1. &longs;ummæ 3. <emph type="italics"/>(Patet igitur tria e&longs;&longs;e genera &longs;pe<lb/>culatiuarum &longs;cientiarum, Naturalem, Mathematicam, Theologiam)<emph.end type="italics"/> ecce tibi, <lb/>quam clara &longs;it Ari&longs;t. </s>

<s>&longs;ententia.</s>

<s>Quod ad Platonis auctoritatem attinet, certum e&longs;t, eum in Mathematicis <lb/>cau&longs;am materialem, & formalem agnoui&longs;&longs;e, nam te&longs;te Ari&longs;t.

primo Metaph. <pb pagenum="12"/>cap. </s>

primo Metaph. <pb pagenum="12"/>cap.

<s>7. ip&longs;e non credebat aliarum cau&longs;arum, quàm formalis, & materialis, <lb/>quas Mathematici tractant, &longs;peculationem <expan abbr="philo&longs;ophicã">philo&longs;ophicam</expan> e&longs;&longs;e magnifa cien<lb/>dam; efficientem enim, & finalem <expan abbr="n&utilde;quam">nunquam</expan> explicuit, proptereaquod à Ma<lb/>thematicis, nunquam tractarentur. </s>

7. ip&longs;e non credebat aliarum cau&longs;arum, quàm formalis, & materialis, <lb/>quas Mathematici tractant, &longs;peculationem <expan abbr="philo&longs;ophicã">philo&longs;ophicam</expan> e&longs;&longs;e magnifa cien<lb/>dam; efficientem enim, & finalem <expan abbr="n&utilde;quam">nunquam</expan> explicuit, proptereaquod à Ma<lb/>thematicis, nunquam tractarentur. </s>

<s>Proclus præterea cap. </s>

<s>Proclus præterea cap.

10. lib.

<s>7. in Eu<lb/>clidem, ait; Mathematicam verò omninò rerum &longs;empit<gap/>rnarum vim ha<lb/>bentem, &longs;cientiam appellat Plato. </s>

7. in Eu<lb/>clidem, ait; Mathematicam verò omninò rerum &longs;empit<gap/>rnarum vim ha<lb/>bentem, &longs;cientiam appellat Plato. </s>

<s>& paulo po&longs;t, ne dicamus igitur, quod <lb/>Mathematicam à &longs;cientiarum numero Plato exp<gap/>llit. </s>

<s>& in fine cap.

<s>ait. <lb/><emph type="italics"/>(Mathematica tamen e&longs;t &longs;<gap/>entia, non vt à &longs;uppo&longs;itione immunis, &longs;ed vt propria<lb/>rum in anima rationum cognitrix, & vt cau&longs;as conclu&longs;ionum afferins)<emph.end type="italics"/> nota illud, <lb/>cau&longs;as conclu&longs;ionum afferens. </s>

ait. <lb/><emph type="italics"/>(Mathematica tamen e&longs;t &longs;<gap/>entia, non vt à &longs;uppo&longs;itione immunis, &longs;ed vt propria<lb/>rum in anima rationum cognitrix, & vt cau&longs;as conclu&longs;ionum afferins)<emph.end type="italics"/> nota illud, <lb/>cau&longs;as conclu&longs;ionum afferens. </s>

<s>concludit po&longs;tea &longs;ic, hæc omnia de Platonis <lb/>&longs;ententia pro Mathematicis dicta &longs;int.</s>

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<s>in Euclidem, totus e&longs;t in Mathematicis, præc puè verò in Geo<lb/>metria &longs;ummis laudibus cumulandis, <expan abbr="eas&qacute;">easque</expan>; e&longs;&longs;e perfecti&longs;&longs;imas &longs;cientias &longs;æ<lb/>pius non &longs;olum a&longs;&longs;erit, &longs;ed etiam demon&longs;trat. </s>

<s>ld igitur primum cap. </s>

<s>ld igitur primum cap.

10. lib. <lb/>

<s>primi aggreditur, vbi fusè o&longs;tendit ex Platone e&longs;&longs;e &longs;cientias, quæ <expan abbr="cõclu&longs;io-num">conclu&longs;io<lb/>num</expan> cau&longs;as afferant. </s>

primi aggreditur, vbi fusè o&longs;tendit ex Platone e&longs;&longs;e &longs;cientias, quæ <expan abbr="cõclu&longs;io-num">conclu&longs;io<lb/>num</expan> cau&longs;as afferant. </s>

<s>perfecti&longs;&longs;imas habere <expan abbr="demõ&longs;trationes">demon&longs;trationes</expan>. </s>

<s>& cap.

5. lib.

<s>2. <lb/>de Euclide loquens, ait <emph type="italics"/>(Præcipuè verò circa Geometricam elementorum in&longs;ti<lb/>tutionem eum qui&longs;piam admirabitur propter ordinem, & electionem eorum, quæ <lb/>per elementa di&longs;tribuit, etenim non ca a&longs;&longs;ump&longs;it omnia, quæ poterat dicere, &longs;ed ea <lb/>duntaxat, quæ elementari tradere potuit ordine. </s>

2. <lb/>de Euclide loquens, ait <emph type="italics"/>(Præcipuè verò circa Geometricam elementorum in&longs;ti<lb/>tutionem eum qui&longs;piam admirabitur propter ordinem, & electionem eorum, quæ <lb/>per elementa di&longs;tribuit, etenim non ca a&longs;&longs;ump&longs;it omnia, quæ poterat dicere, &longs;ed ea <lb/>duntaxat, quæ elementari tradere potuit ordine. </s>

<s>Adhuc autem omnis generis &longs;yl<lb/>logi&longs;morum modus, alios quidem à cau&longs;is fidem &longs;u&longs;cipientes, alios verò à certis no<lb/>tis perfectos, omnes autem inuincibiles, & certos, ad &longs;cientiam&queacute;, accommodatos)<emph.end type="italics"/><lb/>notanda &longs;nnt illa; à caufis fidem &longs;u&longs;cipientes, quibus præcipuè indicat, &longs;e in <lb/>Demon&longs;trationibus Euclidianis cau&longs;as agno&longs;cere.</s>

<s>Lib.

<s>deinde 3. in comm. </s>

deinde 3. in comm. </s>

<s>ad primam Euclidis propo&longs;it. </s>

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<s>ip&longs;e tex. </s>

<s>3. 5. Metaph. </s>

<s>3. 5. Metaph.

<s>id a&longs;&longs;erit. </s>

id a&longs;&longs;erit. </s>

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<s>Eudem. </s>

<s>cap.

<s>7. &longs;ic, in immobi<lb/>libus autem, vt in Mathematicis non per &longs;e, &longs;ed &longs;imilitudine quadam prin<lb/>cipia appellantur. </s>

7. &longs;ic, in immobi<lb/>libus autem, vt in Mathematicis non per &longs;e, &longs;ed &longs;imilitudine quadam prin<lb/>cipia appellantur. </s>

<s>cuius germana interpretatio non contra, &longs;ed pro Mathe<lb/>maticis e&longs;t: loquitur enim ibi Ari&longs;t.

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<s>ob eandem comparationem a&longs;&longs;e<lb/>rit etiam Mathematicas minus e&longs;&longs;e certas, quam Theologiam, quoniam, <lb/>&longs;cilicet hæc nihil &longs;upponit, <expan abbr="nihil&qacute;">nihilque</expan>; di&longs;currit, &longs;ed intuetur: illæ verò iactis <lb/>quibu&longs;dam principijs, quæ probari nequeunt, di&longs;currunt, <expan abbr="propterea&qacute;">proptereaque</expan>; in <lb/>ip&longs;o di&longs;cur&longs;u pote&longs;t error aliquis contingere. </s>

<s>eandem huius loci explica<lb/>tionem habes apud Proclum cap. </s>

<s>eandem huius loci explica<lb/>tionem habes apud Proclum cap.

10. lib.

<s>1. in Euclid. </s>

1. in Euclid. </s>

<s>&longs;ic; Verum quid &longs;ibi <lb/>velit Plato, quando in libris de Rep. </s>

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<s>concludamus tandem hæc pulcherrimo eiu&longs;dem 7. de Rep. </s>

<s>loco, quem etiam <lb/>Proclus lib. </s>

<s>loco, quem etiam <lb/>Proclus lib.

1. cap.

<s>8. ad verbum ferè &longs;ic recitat: Ideo & in Repub. </s>

8. ad verbum ferè &longs;ic recitat: Ideo & in Repub. </s>

<s>Socrates <lb/>rectè dixit, oculus, nimirum animæ, qui ab alijs &longs;tudijs exc&oelig;catur, <expan abbr="defodi-tur&qacute;">defodi<lb/>turque</expan>; à Mathematicis tantum di&longs;ciplinis recreari, <expan abbr="excitari&qacute;">excitarique</expan>; rur&longs;us ad eius, <lb/>qui e&longs;t, contemplationem, & à &longs;imulacris ad ea, quæ vera &longs;unt: nam pulchri<lb/>tudo, & ordo Mathematicarum rationum, firmitudoque, ac &longs;tabilitas con<lb/>templationis, nos ip&longs;is coniungit intellectibus, <expan abbr="perfectè&qacute;">perfectèque</expan>; in ip&longs;is obfirmat, <lb/>perpetuò quidem manentibus, & diuina pulchritudine collucentibus, ac mu<lb/>tuum ordinem &longs;eruantibus. </s>

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qui 6. Eth. </s>

<s>cap.

<s>8, <lb/>ait, quid e&longs;t, quòd puer fieri Mathem. </s>

8, <lb/>ait, quid e&longs;t, quòd puer fieri Mathem. </s>

<s>pote&longs;t, &longs;apiens, aut naturalis non po<lb/>te&longs;t? </s>

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<s>Con&longs;tat autem <lb/>vltimas duas ab ip&longs;o collocari in naturali peritia. </s>

<s>ade&longs;t etiam Procli autho <lb/>ritas, qui cap. </s>

<s>ade&longs;t etiam Procli autho <lb/>ritas, qui cap.

5. lib.

<s>1. &longs;ic habet; In&longs;trumentum <expan abbr="itaq;">itaque</expan> aptum ad iudicandum <lb/>cunctas res Mathematicas cogitationem ex Platonis &longs;ententia &longs;tatuimus, <lb/>quippequæ opinione quidem &longs;uperior e&longs;t, ab intelligentia verò &longs;uperatur. <lb/></s>

1. &longs;ic habet; In&longs;trumentum <expan abbr="itaq;">itaque</expan> aptum ad iudicandum <lb/>cunctas res Mathematicas cogitationem ex Platonis &longs;ententia &longs;tatuimus, <lb/>quippequæ opinione quidem &longs;uperior e&longs;t, ab intelligentia verò &longs;uperatur. <lb/></s>

<s>Per cogitationem verò intelligendum e&longs;&longs;e quendam mentis motum, ide&longs;t <lb/>di&longs;cur&longs;um, tum ex vi græcè vocis <foreign lang="greek">Nohmatos,</foreign> tum ex vi Latinæ vocis manife<lb/>&longs;tum e&longs;t; cogitatio enim dicitur qua&longs;i coagitatio. </s>

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<s>Septima e&longs;t, qua dicunt Geometras non habere materiam veram, & pro<lb/>priam, ea enim Phy&longs;ica e&longs;t, & proinde <expan abbr="neq;">neque</expan> cau&longs;am materialem. </s>

<s>Sed dicé<lb/>dum e&longs;t Geometras <expan abbr="quid&etilde;">quidem</expan> carere propria materia phy&longs;ica, non carere ta<lb/>men propria materia Mathematica, quæ e&longs;t illa intelligibilis, de qua cap. </s>

<s>1. <lb/>dictum e&longs;t.</s>

1. <lb/>dictum e&longs;t.</s>

<s>Octaua, e&longs;t cuiu&longs;dam dicentis, opinionem <expan abbr="commun&etilde;">communem</expan> e&longs;&longs;e, Mathematicam <lb/>non e&longs;&longs;e propriè &longs;cientiam; &longs;ed hoc mani&longs;e&longs;tè fal&longs;um e&longs;t, cum inter tot Phi<lb/>&longs;o&longs;ophos Græcos, Arabes, Latinos, &longs;olùm ip&longs;e duos, vel tres huius &longs;enten<lb/>tiæ in medium po&longs;&longs;it afferre, Piccolom. .&longs;. </s>

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<s>Duodecima e&longs;t, Mathematicas ab&longs;trahere à Bono: verùm eas ab ea libe<lb/>rat Ari&longs;t.

dum lib. </s>

dum lib.

<s>13. Metaphy&longs;. </s>

13. Metaphy&longs;. </s>

<s>ait: qui dicunt Mathematicas &longs;cientias ni<lb/>hil de bono, vel pulchro dicere, fal&longs;um dicunt: dicunt enim, & maximè <expan abbr="o&longs;t&etilde;-dunt">o&longs;ten<lb/>dunt</expan>, nam etiam&longs;i non nominant, quia tamen opera, & rationes o&longs;tendunt, <lb/>non ne dicunt de eis? </s>

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<s>quod maximè ij &longs;olent oppo<lb/>nere, qui &longs;cientias, vt ille cecinit cauponantur, &longs;eu qui eas quæ&longs;tuo&longs;as fa<lb/>ciunt. </s>

<s>Verùm hos animo mercatores potius, quàm Philo&longs;ophos amande<lb/>mus ad cap. </s>

<s>Verùm hos animo mercatores potius, quàm Philo&longs;ophos amande<lb/>mus ad cap.

<s>8. 9. & 10. libri primi Procli, vbi fusè de vtilitate <expan abbr="ear&utilde;">earum</expan> omnium <lb/>di&longs;&longs;erit. </s>

8. 9. & 10. libri primi Procli, vbi fusè de vtilitate <expan abbr="ear&utilde;">earum</expan> omnium <lb/>di&longs;&longs;erit. </s>

<s>Quod &longs;i Philo&longs;ophus &longs;it, qui hæc opponat; huic illa &longs;ufficiat vtili<lb/>tas, qua loca omnia Ari&longs;t. </s>

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<s>vt propterea mirandum &longs;it, cur non p&ecedil;nitus huiu&longs;<lb/>modi artes de medio tollantur, &longs;ed quod ait Cor. </s>

<s>Tacitus lib. </s>

<s>Tacitus lib.

<s>1. hi&longs;tor. </s>

1. hi&longs;tor. </s>

<s>hoc <lb/>genus hominum potentibus infidum, &longs;perantibus fallax, in Ciuitate no<lb/>&longs;tra, & vetabitur &longs;emper, & retinebitur. </s>

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<s>1. Ethic. </s>

<s>cap.

<s>7. <lb/>Geometram appellat veritatis &longs;peculatorem. </s>

7. <lb/>Geometram appellat veritatis &longs;peculatorem. </s>

<s>cùm præterea reliquæ &longs;cien<lb/>tiæ (Mathematicis medijs exceptis) rarò euidentiam pariant, rarò etiam <lb/>veritatem a&longs;&longs;equentur, &longs;ed ferè &longs;emper opiniones gignent, vnde eas meritò <lb/>non &longs;cientias, &longs;ed opiniones Plato voluit appellari: &longs;equitur ab&longs;olutè di<lb/>cendum e&longs;&longs;e, Mathematicas reliquarum &longs;cientiarum præ&longs;tanti&longs;&longs;imas e&longs;&longs;e; <lb/>quemadmodum inter opiniones præ&longs;tanti&longs;&longs;imum quid e&longs;t veritas.</s>

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<s>quàm admira<lb/>bilis e&longs;t 47. primi, pro cuius inuentione Pythagoras Mu&longs;is Hecatombas <lb/>immolauit? </s>

<s>In &longs;ecundò deinde Elem. </s>

<s>In &longs;ecundò deinde Elem.

lib.

<s>quàm &longs;ubtilis e&longs;t 14. quæ rectili<lb/>neo cuiuis quadratum exhibet æquale. </s>

quàm &longs;ubtilis e&longs;t 14. quæ rectili<lb/>neo cuiuis quadratum exhibet æquale. </s>

<s>In tertiò po&longs;tea, quot <expan abbr="quanta&qacute;">quantaque</expan>; an<pb pagenum="28"/>gulus ille contingentiæ continet miracula, qui quamuis quantus &longs;it, nequi <lb/>tamen à linea recta partiri. </s>

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<s>& cum intelligimus <expan abbr="quinq;">quinque</expan> tantum in tota rerum natura repe<lb/>riri po&longs;&longs;e &longs;olida regularia? </s>

<s>meritò igitur Cardanus lib. </s>

<s>meritò igitur Cardanus lib.

<s>16. de &longs;ubtilit. </s>

16. de &longs;ubtilit. </s>

<s>ait; <lb/>Euclidis &longs;unt duæ præcipuæ laudes, inconcu&longs;&longs;a dogmatum firmitas libri <lb/>elementorum, <expan abbr="perfectio&qacute;">perfectioque</expan>; adeò ab&longs;oluta, vt nullum opus iurè huic compa<lb/>rare audeas, quibus fit, vt adeò veritatis lux in eo refulgeat, vt ij &longs;oli in ar<lb/>duis quæ&longs;tionibus videantur po&longs;&longs;e verum à falsò di&longs;cernere, quì Euclidem <lb/>habeant familiarem.</s>

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<s>De &longs;phæra po&longs;tea, & cylindro varia de<lb/>mon&longs;trans, quanto & alios, ac &longs;o ip&longs;um &longs;patio &longs;uperat, vt dum inter cætera <lb/>diuino planè acumine o&longs;tendit cuiu&longs;libet &longs;phæræ &longs;uperficiem quadruplam <lb/>effe circuli eiu&longs;dem maximi. </s>

<s>Item &longs;i tria hæc, cylindrus, &longs;phæra, & conus, <lb/>&longs;int in eadem altitudine, <expan abbr="eorum&qacute;">eorumque</expan>; ba&longs;es &longs;int circuli maximi illius &longs;phæræ, <lb/>habere inuicem proportionem, quam <expan abbr="hab&etilde;t">habent</expan> hi numeri 3. 2. 1. quare ob tam <lb/>præclarum ingenij monumentum, &longs;epulcro ip&longs;ius marmoreo &longs;phæra, & cy<lb/>lindrus marmorei &longs;unt impo&longs;iti, quemadmodum Cicero lib. </s>

<s>5. Tu&longs;c narrat, <lb/>vbi etiam magnopere gloriatur, &longs;e cum in Sicilia Quæ&longs;tor e&longs;&longs;et, illud igno<lb/>ratum ab Syracufanis &longs;eptum vndique, ac ve&longs;titum vepribus, & dumetis in<lb/>daga&longs;&longs;e &longs;epulcrum. </s>

5. Tu&longs;c narrat, <lb/>vbi etiam magnopere gloriatur, &longs;e cum in Sicilia Quæ&longs;tor e&longs;&longs;et, illud igno<lb/>ratum ab Syracufanis &longs;eptum vndique, ac ve&longs;titum vepribus, & dumetis in<lb/>daga&longs;&longs;e &longs;epulcrum. </s>

<s>meritò igitur Cardanus lib. </s>

<s>meritò igitur Cardanus lib.

<s>16. de &longs;ubtil. </s>

16. de &longs;ubtil. </s>

<s>eum tanquam <lb/>ingeniorum Phænicem &longs;upra omnes &longs;ubtilitate præ&longs;tantes viros, <expan abbr="atq;">atque</expan> adeo <lb/>&longs;upra Ari&longs;tot. </s>

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<s>mi&longs;&longs;os <lb/>facio Hip&longs;iclem, Theodo&longs;ium tripolitam, Menelaum, Serenum, Pappum, & <lb/>alios, quorum opera omnem &longs;uperant admirationem, <expan abbr="ea&qacute;">eaque</expan>; mirabili adeo <lb/>connexione, ac certitudine tradita, vt nullus &longs;it, qui pri&longs;cis illis Geometris <lb/>ingenio cedere libenter nolit. </s>

<s>Quapropter cum Cardano lib. </s>

<s>Quapropter cum Cardano lib.

<s>16. de &longs;ubtil. <lb/></s>

16. de &longs;ubtil. <lb/></s>

<s>hanc partem concludamus; nihil mirum igitur, inquit, Geometriam e&longs;&longs;e <lb/>omnium &longs;cientiarum &longs;ubtili&longs;&longs;imam, quæ cùm tamen à manife &longs;ti&longs;&longs;imis ini<lb/>tium ducat, meritò an&longs;am præbuit, vt prima omnium etiam pueros doce<pb pagenum="29"/>retur. </s>

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tex. </s>

<s>3. lib.

<s>5. Metaph. </s>

5. Metaph.

<s>quem &longs;upra <expan abbr="c&utilde;">cum</expan> alijs <expan abbr="ex-plicat&utilde;">ex<lb/>plicatum</expan> habes. </s>

quem &longs;upra <expan abbr="c&utilde;">cum</expan> alijs <expan abbr="ex-plicat&utilde;">ex<lb/>plicatum</expan> habes. </s>

<s>&longs;ecundò, demon&longs;trat de eodem I&longs;o&longs;cele, angulos infra ba&longs;im <lb/>e&longs;&longs;e æquales, ratio, quia opponuntur &ecedil;qualibus lateribus <expan abbr="triangulor&utilde;">triangulorum</expan> quar<lb/>tæ præcedentis, quæ ratio videtur &longs;ignum quoddam æqualitatis eorum e&longs;&longs;e.</s>

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<s>non tamen qui&longs;piam exi&longs;timet idem iu<lb/>dicium de reliquis Mathematicis e&longs;&longs;e faciendum, A&longs;tronomia enim, Opti<lb/>ca, & ali&ecedil; vtuntur etiam alijs cau&longs;arum generibus in demon&longs;trando, vt &longs;up. <lb/></s>

<s>cap.

<s>5. de natura Mathem. </s>

5. de natura Mathem. </s>

<s>patuit. </s>

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<s>primus eclyp&longs;es Solis prædixit: quarum prima te&longs;te Piin. </s>

<s>lib 2. cap.

<s>12. <lb/>contigit ann. </s>

12. <lb/>contigit ann. </s>

<s>V. C. 170. men&longs;us e&longs;t Aegypti Pyramides ex vmbra. </s>

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<s>omne triangulum habere tres, &c. </s>

<s>quæ e&longs;t 32. primi Elem. </s>

<s>quæ e&longs;t 32. primi Elem.

<s>primi <lb/>Pythag. </s>

primi <lb/>Pythag. </s>

<s>demon&longs;trarunt. </s>

<s>47. primi Elem. </s>

<s>47. primi Elem.

<s>reperit, pro qua Mu&longs;is Hecatombas <lb/>immolauit. </s>

reperit, pro qua Mu&longs;is Hecatombas <lb/>immolauit. </s>

<s>primus Mathematicæ ludum aperuit. </s>

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<s>Vitr.

lib. </s>

lib.

7. Val. <lb/></s>

<s>OENIPEDES CHIVS Democriti &longs;yncronus, inuenit 12. & 23. <lb/>primi Elem. </s>

<s>OENIPEDES CHIVS Democriti &longs;yncronus, inuenit 12. & 23. <lb/>primi Elem.

<s>huius di&longs;cipulus fuit Zenodorus.</s>

huius di&longs;cipulus fuit Zenodorus.</s>

<s>ZENODORVS auctor tractatus de figuris I&longs;operimetris, qui e&longs;t <lb/>apud Clauium in &longs;phæra, & in Geometric. </s>

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<s>mu<lb/>&longs;icalium. </s>

<s>Pappus lib. </s>

<s>Pappus lib.

<s>3. eum Pythagoricum appellat; I&longs;idorus lib. </s>

3. eum Pythagoricum appellat; I&longs;idorus lib.

<s>3. ethym. <lb/></s>

3. ethym. <lb/></s>

<s>videtur eum paulo po&longs;t Pyth<gap/>goram collocare. </s>

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<s>Item de re&longs;olutione. </s>

<s>Item de locis &longs;olidis, <lb/>lib. </s>

<s>Item de locis &longs;olidis, <lb/>lib.

<s>5. Pappus.</s><pb pagenum="45"/>

5. Pappus.</s><pb pagenum="45"/>

<s>GEMINVS demon&longs;trauit linearum tres tantùm e&longs;&longs;e &longs;imilares, <expan abbr="rectã">rectam</expan>, <lb/>circularem, & &longs;piralem cylindricam. </s>

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<s>Propo&longs;itionem quintam primi ele<lb/>mentorum vniuer&longs;alius, quam Thales, demon&longs;trauit; o&longs;tendit enim æqua<lb/>les line as rectas ab vno puncto ad vnam &longs;imilium partium lineam, ide&longs;t, vel <lb/>ad rectam, vel ad circularem, vel ad cylindricam, <expan abbr="incid&etilde;tes">incidentes</expan>, facere angulos <lb/>ad b<gap/>&longs;im æquales. </s>

<s>Scripfit præterea lib. </s>

<s>Scripfit præterea lib.

<s>6. <expan abbr="geometricar&utilde;">geometricarum</expan> <expan abbr="enarration&utilde;">enarrationum</expan>. </s>

6. <expan abbr="geometricar&utilde;">geometricarum</expan> <expan abbr="enarration&utilde;">enarrationum</expan>. </s>

<s>Procl.</s>

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<s>XENOCRATES CHALCEDONIVS Platonis audit. </s>

<s>de geo<lb/>metria primum duos, deinde <expan abbr="quinq;">quinque</expan> lib. </s>

<s>de geo<lb/>metria primum duos, deinde <expan abbr="quinq;">quinque</expan> lib.

<s>compo&longs;uit. </s>

compo&longs;uit. </s>

<s>Item de numeris lib. </s>

<s>Item de numeris lib.

<s>1. <lb/>De a&longs;trologia lib. </s>

1. <lb/>De a&longs;trologia lib.

6. Diog. </s>

<s>Laert.</s>

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<s>lib.

<s>7. <lb/>vixit autem, & claruít, v&longs;que ad Ptolæmeum primum Aegypti Regem, vt <lb/>vult Procl &longs;it ne idem cum Euclide Megaren&longs;i anctore &longs;ectæ Megaricæ, du<lb/>bitatur. </s>

7. <lb/>vixit autem, & claruít, v&longs;que ad Ptolæmeum primum Aegypti Regem, vt <lb/>vult Procl &longs;it ne idem cum Euclide Megaren&longs;i anctore &longs;ectæ Megaricæ, du<lb/>bitatur. </s>

<s>Quintus geometrica clementa mira methodo contexuit. </s>

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<s>Item de re&longs;olu<lb/>tione, de fallacijs, de locis ad &longs;uperficiem lib. </s>

<s>Item de re&longs;olu<lb/>tione, de fallacijs, de locis ad &longs;uperficiem lib.

<s>2. Conicorum lib. </s>

2. Conicorum lib.

<s>4. Item <lb/>Pori&longs;matum lib. </s>

4. Item <lb/>Pori&longs;matum lib.

<s>3. quæ perierunt. </s>

3. quæ perierunt. </s>

<s>Pappus, & Procl.</s>

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auditor. </s>

<s>de Mu&longs;ica <lb/>lib. </s>

<s>de Mu&longs;ica <lb/>lib.

duos. </s>

<s>de Geometria etiam &longs;crip&longs;it. </s>

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<s>DICEARCHVS Siculus, Ari&longs;t, auditor, primus montium altitudi<lb/>nem perpendicularem dimen&longs;us e&longs;t; alti&longs;&longs;imum prodidit Pelion, nimirum <lb/>1250. pa&longs;&longs;uum. </s>

<s>Plinius lib. </s>

<s>Plinius lib.

2. c. </s>

<s>ARISTOXENVS Tarentinus Mu&longs;icus, auditor Ari&longs;t.

eius extant <lb/>harmonicorum lib. </s>

eius extant <lb/>harmonicorum lib.

3. Suid.</s>

<s>CONON Geometra, & A&longs;tronomus in&longs;ignis. </s>

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<s>ille verò alter Pontano no&longs;tro in Virgilium e&longs;t Archimedes Cononi &longs;ynero<lb/>nus, & familiaris. </s>

<s>Hunc plurimi faciebat Archimedes, <expan abbr="eius&qacute;">eiusque</expan>; propterea <lb/>mortem in lib. </s>

<s>Hunc plurimi faciebat Archimedes, <expan abbr="eius&qacute;">eiusque</expan>; propterea <lb/>mortem in lib.

<s>de Quadrat. </s>

de Quadrat. </s>

<s>Parab. </s>

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<s>cætera apud Vitr. <lb/></s>

<s>lib.

<s>9. ei Athenien&longs;es ob diuinas prædictiones publicè in gymna&longs;io &longs;tatuam <lb/>inaurata lingua &longs;tatuere. </s>

9. ei Athenien&longs;es ob diuinas prædictiones publicè in gymna&longs;io &longs;tatuam <lb/>inaurata lingua &longs;tatuere. </s>

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<s>extat epi&longs;tola ip&longs;ius ad Regem Ptolæmeum apud Eutocium, de ra<lb/>tione cubi duplicandi.</s>

<s>ARCHIMEDES Syracu&longs;anus ingeniorum Ph&oelig;nix: quadraginta <lb/>ip&longs;ius mira adinuenta Mechanica fui&longs;&longs;e, tradit Pappus lib. </s>

<s>ARCHIMEDES Syracu&longs;anus ingeniorum Ph&oelig;nix: quadraginta <lb/>ip&longs;ius mira adinuenta Mechanica fui&longs;&longs;e, tradit Pappus lib.

<s>8. quorum vnum <lb/>fuit; datum pondus data potentia mouere, in quo fertur dixi&longs;&longs;e.</s>

8. quorum vnum <lb/>fuit; datum pondus data potentia mouere, in quo fertur dixi&longs;&longs;e.</s>

<s><emph type="italics"/>Dic vbi contestam, & c&oelig;lum terram&queacute; mouebo.<emph.end type="italics"/></s>

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<s>quorum <lb/>memoria extat, hæc &longs;unt. </s>

<s>De æquaponderantibus lib. </s>

<s>De æquaponderantibus lib.

<s>2. Circuli dimen&longs;io. <lb/></s>

2. Circuli dimen&longs;io. <lb/></s>

<s>de lineis &longs;piralibus. </s>

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<s>&longs;cri<lb/>p&longs;it præterea de &longs;ectione proportionis, & &longs;patij. </s>

<s>de locis planis <lb/>lib. </s>

<s>de locis planis <lb/>lib.

<s>2. de perturbatis rationibus. </s>

2. de perturbatis rationibus. </s>

<s>de tactionibus. </s>

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<s>ISIDORVS Philo&longs;ophus, Hyp&longs;iclis Alexandrini præceptor; nam <lb/>Hyp&longs;icles in 15. Elem. </s>

<s>ISIDORVS Philo&longs;ophus, Hyp&longs;iclis Alexandrini præceptor; nam <lb/>Hyp&longs;icles in 15. Elem.

<s>vbi ponit inclinationes <expan abbr="quinq;">quinque</expan> <expan abbr="corpor&utilde;">corporum</expan> regularium, <lb/>ait &longs;e eas ab I&longs;idoro Magno præceptore &longs;uo accepi&longs;&longs;e. </s>

vbi ponit inclinationes <expan abbr="quinq;">quinque</expan> <expan abbr="corpor&utilde;">corporum</expan> regularium, <lb/>ait &longs;e eas ab I&longs;idoro Magno præceptore &longs;uo accepi&longs;&longs;e. </s>

<s>Plinius eum citat in <lb/>Geographicis. </s>

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<s>huius ianuæ cùm <lb/>ad eum audiendum Pomponius Magnus adiret, Imperij fa&longs;ces &longs;ubmi&longs;it.</s>

<s>SERENVS Antin&longs;en&longs;is: cuius &longs;unt Cylindricorum lib. </s>

<s>SERENVS Antin&longs;en&longs;is: cuius &longs;unt Cylindricorum lib.

<s>2. &longs;ubtili&longs;&longs;imi. <lb/></s>

2. &longs;ubtili&longs;&longs;imi. <lb/></s>

<s>videtur in hæc tempora po&longs;t Apollonium incidi&longs;&longs;e.</s>

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<s>condidit. <lb/></s>

<s>Adhuc extant eius lib. </s>

<s>Adhuc extant eius lib.

<s>3. in Arati Phænomena: & vnus A&longs;teri&longs;morum, <expan abbr="&longs;unt&qacute;">&longs;untque</expan>; <lb/>Græcè nuper editi.</s>

3. in Arati Phænomena: & vnus A&longs;teri&longs;morum, <expan abbr="&longs;unt&qacute;">&longs;untque</expan>; <lb/>Græcè nuper editi.</s>

<s>CLEOMEDES his &longs;eculis gr&ecedil;cè &longs;cribit Meteora, quibus tract at ea, <lb/>quæ in &longs;ph&ecedil;ra &longs;olent doceri. </s>

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<s>de diebus, & no<lb/>ctibus. </s>

<s>&longs;phæricorum lib. </s>

<s>&longs;phæricorum lib.

<s>3. de lineatione ædium. </s>

3. de lineatione ædium. </s>

<s>Commentaria <lb/>in Theud&ecedil; capita. </s>

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<s>lib.

3. cap.

2.</s>

<s>C. IVLIVS CÆSAR Monarcha, primus &longs;crip&longs;it Metaphra&longs;im in <lb/>Arati Ph&oelig;mena. </s>

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<s>SOLINVS, & P. MELA De &longs;itu orbis pariter con&longs;crip&longs;erat.</s>

<s>STRATON AMASENVS Philo&longs;ophus, lib. </s>

<s>STRATON AMASENVS Philo&longs;ophus, lib.

<s>7. Geographicos <lb/>edidit Suid.</s>

7. Geographicos <lb/>edidit Suid.</s>

<s>PLINIVS, omnis generis Mathemata &longs;uo operi <expan abbr="cõmi&longs;cuit">commi&longs;cuit</expan>. </s>

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<s>6. 12. deprehendit. </s>

<s>lib.

<s>6. de &longs;ubten&longs;is, &longs;eu chordis. </s>

6. de &longs;ubten&longs;is, &longs;eu chordis. </s>

<s>Item <lb/>lib.

<s>3. de &longs;ph&ecedil;ricis triangulis, qui extant.</s>

3. de &longs;ph&ecedil;ricis triangulis, qui extant.</s>

<s>PLVTARCHVS libellum de mu&longs;ica optima eruditione, ac doctrina <lb/>refertum reliquit; quem &longs;uperius &longs;æpè citauimus.</s>

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<s><emph type="italics"/>DVODECIMVM SECVLVM<emph.end type="italics"/><lb/><arrow.to.target n="table17"></arrow.to.target></s><table><table.target id="table17"></table.target><row><cell>Tertium autem à Chri&longs;to incipiens</cell><cell></cell></row><row><cell>Ab ann. Chri&longs;ti</cell><cell>201</cell></row><row><cell>Victore &longs;ummo Pont.</cell><cell></cell></row><row><cell>Imperante Septimio Seuero.</cell><cell></cell></row></table>

<s>PORPHIRIVS Philo&longs;ophus Platonicus &longs;crip&longs;it I&longs;agogem a&longs;tro<lb/>nomicarum rerum lib. </s>

<s>PORPHIRIVS Philo&longs;ophus Platonicus &longs;crip&longs;it I&longs;agogem a&longs;tro<lb/>nomicarum rerum lib.

3. Suid. </s>

<s>is e&longs;t, cuius e&longs;t I&longs;agoge, de quinque <lb/>vniuer&longs;alibus. </s>

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<s>S. AVGVSTINVS Epi&longs;copus lib. </s>

<s>S. AVGVSTINVS Epi&longs;copus lib.

<s>6. de Mu&longs;ica, Item de principijs <lb/>Geometriæ, & Arithmeticæ &longs;cribit.</s>

6. de Mu&longs;ica, Item de principijs <lb/>Geometriæ, & Arithmeticæ &longs;cribit.</s>

<s>MARINVS Philo&longs;ophus Neapolitanus Procli di&longs;cipulus. </s>

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<s>PELLES Aegien&longs;is. </s>

<s>Arithmeticorum lib. </s>

<s>Arithmeticorum lib.

2.</s>

<s>GEBER Arabs, cuius extat Opus a&longs;tronomicum 9. libris di&longs;tinctum, <lb/>quo Ptolemæi Almage&longs;tum expoint, ac corrigit. </s>

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<s>5. <expan abbr="elem&etilde;">elemen</expan>. </s>

<s>citat Iordanum, qui <lb/>&longs;crip&longs;it de Arithmetica lib. </s>

<s>citat Iordanum, qui <lb/>&longs;crip&longs;it de Arithmetica lib.

<s>12. & data Arithmetica. </s>

12. & data Arithmetica. </s>

<s>& de A&longs;tro<lb/>labio. </s>

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<s>in almag.</s><pb pagenum="61"/>

<s>IOANNES BVTEO Logi&longs;tices, lib. </s>

<s>IOANNES BVTEO Logi&longs;tices, lib.

5. de arca Noe. </s>

<s>de quadraturis cir<lb/>culorum. </s>

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<s>FRANCISCVS MAVROLY VS Abbas Siculus, <expan abbr="Co&longs;mographiã">Co&longs;mographiam</expan>. </s>

<s>Ari<lb/>thmeticorum lib. </s>

<s>Ari<lb/>thmeticorum lib.

<s>3. de lineis horarijs Photi&longs;mos. </s>

3. de lineis horarijs Photi&longs;mos. </s>

<s>& alia nonnulla, partim <lb/>nondum edita, quorum index habetur in &longs;ua Co&longs;mographia. </s>

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<s>Geometric. </s>

<s>lib.

5.</s>

<s><emph type="italics"/>XXVI. SECVLVM<emph.end type="italics"/><lb/><arrow.to.target n="table30"></arrow.to.target></s><table><table.target id="table30"></table.target><row><cell>Decimum verò &longs;eptimum Chr. ab ann. Domini</cell><cell>1601</cell></row><row><cell>Clemente 8. &longs;um. Pont.</cell><cell></cell></row><row><cell>Imp. Rodulpho 2. occid.</cell><cell></cell></row></table>

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<s>IO. BAPT. PORTA, eruditi&longs;&longs;imus æquè, ac nobili&longs;&longs;imus. </s>

<s>editi &longs;unt eius <lb/>lib. </s>

<s>editi &longs;unt eius <lb/>lib.

<s>9. de Refractione optices. </s>

9. de Refractione optices. </s>

<s>elementorum curuilineorum lib. </s>

<s>elementorum curuilineorum lib.

<s>3. Interpre<lb/>tatio primi Almage&longs;ti, cum comm. </s>

3. Interpre<lb/>tatio primi Almage&longs;ti, cum comm. </s>

<s>Theonis. </s>

<s>de Munitione lib. </s>

<s>de Munitione lib.

<s>3. Pneuma<lb/>ticorum lib. </s>

3. Pneuma<lb/>ticorum lib.

<s>3. Catoptrica nondum edita.</s>

3. Catoptrica nondum edita.</s>

<s>P. BERNARDINVS SALINVS de Soc. </s>

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<s>libri 11. in quibus &longs;uppo<lb/>&longs;ita recta æquali circumferentiæ plurima veluti corollaria demon&longs;trantur. <lb/></s>

<s>de horologijs lib. </s>

<s>de horologijs lib.

<s>2. varia problemata a&longs;tronomica lib. </s>

2. varia problemata a&longs;tronomica lib.

<s>1. de men&longs;uris geo<lb/>metricis lib. </s>

1. de men&longs;uris geo<lb/>metricis lib.

<s>1. quæ nondum edita a&longs;&longs;eruantur Genuæ in Colleg. </s>

1. quæ nondum edita a&longs;&longs;eruantur Genuæ in Colleg. </s>

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