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Colored diff for /texts/archimedes/xml/Attic/bianc_locam_01_la_1615.xml between version 1.67 and 1.68

version 1.67, 2002/09/06 14:57:04

version 1.68, 2002/09/07 13:46:08

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<s>PETRVMFRANCISCVM MALASPINAM</s>

<s>Aedificiorum Marchionem, apud Cæ&longs;. </s>

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

<s>Maie&longs;tatem <lb/>pro Sereni&longs;s. </s>

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

<s>BONONIÆ M. </s>

<s>BONONIÆ M. D C. </s>

Line 52

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

<s><emph type="italics"/>En tandem Illustriß. </s>

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

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

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

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

<s>Quanta por<lb/>rò in rebus agendis prudentia valeas, toti penè Europæ <lb/>innotuit, cùm pro no&longs;tris Sereniß. </s>

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

<s>Ducibus, non &longs;olùm ad <lb/>omnes ferè Italiæ, atque Germaniæ Principes, verùm etiam <lb/>ad Cæ&longs;aream Maie&longs;tatem rebus f&oelig;liciter ge&longs;tis Legatus <lb/>decimùm extiteris; ac demùm à Sereniß. </s>

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

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

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

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

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

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

<s>longæuitatem tueatur. <lb/></s>

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

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

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

<s>Reuer. </s>

<s>P. nc&longs;tro Præpo&longs;iti Generalis P. </s>

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

<s>Claudij <lb/>Aquæuiuæ, facultatem concedo, vt hoc opus P. </s>

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

<s>Parmæ die 15. Ianuarij 1615.</s>

<s><emph type="italics"/>Iordanus Ca&longs;&longs;ini P.<emph.end type="italics"/></s>

<s><emph type="italics"/>Iordanus Ca&longs;&longs;ini P.<emph.end type="italics"/>Don Marcellus Balda&longs;&longs;inus pro Illu&longs;tri&longs;s. </s>

<s>Don Marcellus Balda&longs;&longs;inus pro Illu&longs;tri&longs;s. </s>

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

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

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

<s>Bonon</s>

<s>Imprimatur</s>

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

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

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

<s>Inqui&longs;itore Bonon<gap/></s><pb pagenum="6"/>

<s>LECTORI.</s>

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ait duo cubi, cu<lb/>bus, ip&longs;um loqui putant de duplatione Geometrici cubi, nondum in<lb/>uenta; non intelligentes, eum ibi de numeris cubis &longs;ermonem habere. <lb/></s>

<s>Auerroes ip&longs;e tantus vir 5. Phy&longs;. commen. </s>

<s>Auerroes ip&longs;e tantus vir 5. Phy&longs;. commen. 15. quàm &longs;e Mathematicis, <lb/>reliqui&longs;que Philo&longs;ophis irridendum præbet dum à permutata propor<lb/>tione putat &longs;erectè in hunc modum pluribus apud ip&longs;um verbis explica<lb/>tum, argumentari,</s>

<s>15. quàm &longs;e Mathematicis, <lb/>reliqui&longs;que Philo&longs;ophis irridendum præbet dum à permutata propor<lb/>tione putat &longs;erectè in hunc modum pluribus apud ip&longs;um verbis explica<lb/>tum, argumentari,</s>

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

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

<s>Quo fit, vt cæteros ageo<lb/>metretos ita antecellant, vt eorum Magi&longs;tri appellari po&longs;&longs;int, & <expan abbr="debeãt">debeant</expan>; <lb/>tales fuerunt in primis Vicomercatus, Cardanus, Zabarella, Toletus, <lb/>Bonamicus, & alij, quibus paulo antiquiores fuerunt D. Tho. & Scotus, <lb/>Hiomnes Mathematicarum auxilio, quantum inter reliquos philo&longs;o<lb/>phantes excelluerint, nemo e&longs;t qui non nouerit. </s>

<s>& Scotus, <lb/>Hiomnes Mathematicarum auxilio, quantum inter reliquos philo&longs;o<lb/>phantes excelluerint, nemo e&longs;t qui non nouerit. </s>

<s>Illud hoc loco minimè <lb/>tacendum, Iacobum Zabarellam in &longs;uis logicæ commentarijs te&longs;tari &longs;e <lb/>bis &longs;umma diligentia totum Euclidem perlegi&longs;&longs;e, vt perfectè Ari&longs;t.

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

<s>Tandem in gratiam etiam <lb/>Mathematicorum tertiam appendicem appendi, qua omnia loca Ari&longs;t. <lb/></s>

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

<s>Geometrica ad Euclidis ordinem referuntur; vnde & ip&longs;i ad &longs;uas pr&ecedil;le<lb/>ctiones exornandas aliquid &longs;ubinde depromere queant.</s>

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

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

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

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

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

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eodem, de exemplis, quibus vtuntar Geometræ.<emph.end type="italics"/></s>

<s>In &longs;ecundo Priorum Re&longs;ol.</s>

<s>In &longs;ecundo Priorum Re&longs;ol.<emph type="italics"/>Cap.

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

21. de lineis Paralellis, &longs;eu Coalternis.<emph.end type="italics"/></s>

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

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

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

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

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<s><emph type="italics"/>Textu primo, De Præcognitis Mathematicarum.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 2. Omnis triangulus habet tres, & c.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 2. Omnis triangulus habet tres, & c.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 5. De Diametro incommen&longs;urabili. </s>

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

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

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

<s><emph type="italics"/>T. 13. De Parallelis. </s>

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

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

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

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

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

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

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

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

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<s><emph type="italics"/>T. 25. Permutatim proportionale quid; exemplum in triangulis.<emph.end type="italics"/></s>

<s>In primo lib. </s>

<s>In primo lib. Topicorum.</s>

<s>Topicorum.</s>

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

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<s>In Elenchorum lib.

1.</s>

1.</s>

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

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

<s>Ex 1. Phy&longs;ic.</s>

<s>Ex 1. Phy&longs;ic.</s>

<s><emph type="italics"/>T. 11. De Quadraturis circuli Hippocratis, & Antiphontis.<emph.end type="italics"/></s>

<s>Ex 2. Phy&longs;ic.</s>

<s>Ex 2. Phy&longs;ic.</s>

<s><emph type="italics"/>T. 20. Quatenus Per&longs;pectmus con&longs;ideret lineam.<emph.end type="italics"/></s>

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

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

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

<s><emph type="italics"/>T. 76. Quinam numeri dicantur Gnomones.<emph.end type="italics"/></s>

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

<s>Ex 4. Phy&longs;ic.</s>

<s>Ex 4. Phy&longs;ic.</s>

<s><emph type="italics"/>T. 120. De commen&longs;urab. </s>

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

<s>Ex 5. Phy&longs;ic.</s>

<s>Ex 5. Phy&longs;ic.</s>

<s><emph type="italics"/>T. 6. De chordis, graui, acuta; media, & vltima.<emph.end type="italics"/></s>

<s>Ex 8. Phy&longs;ic.</s>

<s>Ex 8. Phy&longs;ic.</s>

<s><emph type="italics"/>T. 15. Omnis triangulus habet tres æquales, &c.<emph.end type="italics"/></s>

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<s><emph type="italics"/>T. 36. Ratione vtitur lineari; vt probet mundum e&longs;&longs;e finitum.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 48. Commen&longs;urab. </s>

<s><emph type="italics"/>T. 48. Commen&longs;urab. & incommen&longs;urab. </s>

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

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3. De Ventorum&longs;itu.<emph.end type="italics"/></s>

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

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

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

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<s>Rationes Ari&longs;totelis refelluntur.<emph.end type="italics"/></s>

<s>Ex 1. De Anima.</s>

<s>Ex 1. De Anima.</s>

<s><emph type="italics"/>Tex. 11. Quid rectum, quid obliquum. </s>

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<s><emph type="italics"/>T. 13. Sphæra planum tangit in puncto.<emph.end type="italics"/></s>

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

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

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

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<s><emph type="italics"/>T. 159. De Solis magnitudine ad terram.<emph.end type="italics"/></s>

<s>Ex 3. De Anima.</s>

<s>Ex 3. De Anima.</s>

<s><emph type="italics"/>T. 21. Incommen&longs;urabile.<emph.end type="italics"/></s>

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<s><emph type="italics"/>T. 32. Permutata proportio.<emph.end type="italics"/></s>

<s>Ex lib. De Sen&longs;u.</s>

<s>De Sen&longs;u.</s>

<s><emph type="italics"/>Capite 6. Die&longs;is.<emph.end type="italics"/></s>

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<s>Diapen&longs;e.<emph.end type="italics"/></s>

<s>Ex lib. De Memoria, & Rem.</s>

<s>De Memoria, & Rem.</s>

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

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3. Mathemata facile remini&longs;cibilia.<emph.end type="italics"/></s>

<s>Ex lib. De Somnijs.</s>

<s>De Somnijs.</s>

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

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3. Cur Oculus digito dimotus res geminatas videat.<emph.end type="italics"/></s>

<s>Ex 1. Methaphy&longs;.</s>

<s>Ex 1. Methaphy&longs;.</s>

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

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

<s>Ex 2. Methaphy&longs;.</s>

<s>Ex 2. Methaphy&longs;.</s>

<s><emph type="italics"/>T. 14. Leges apud Mu&longs;icos quid.<emph.end type="italics"/></s>

<s>Ex 3. Methaphy&longs;.</s>

<s>Ex 3. Methaphy&longs;.</s>

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

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

<s>Ex 4. Methaphy&longs;.</s>

<s>Ex 4. Methaphy&longs;.</s>

<s><emph type="italics"/>T. 4. Quæ &longs;int primæ, & quæ &longs;ecundæ inter Mathematicas.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 28. Diameter, commen&longs;urabilis.<emph.end type="italics"/></s>

<s>Ex 5. Methaphy&longs;.</s>

<s>Ex 5. Methaphy&longs;.</s>

<s><emph type="italics"/>T. 2. Exemplum cau&longs;æ formalis ex Diapa&longs;on. </s>

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<s><emph type="italics"/>T. 34. Diameter incommen&longs;urabilis.<emph.end type="italics"/></s>

<s>Ex 6. Methaphy&longs;.</s>

<s>Ex 6. Methaphy&longs;.</s>

<s><emph type="italics"/>T. 1. Principia, elementa, & cau&longs;æ in Mathem.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 1. Principia, elementa, & cau&longs;æ in Mathem.<emph.end type="italics"/></s>

<s><emph type="italics"/>T. 8. Diameter. </s>

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

<s>Ex 10. Methaphy&longs;.</s>

<s>Ex 10. Methaphy&longs;.</s>

<s><emph type="italics"/>T. 4. Motum diurnum men&longs;uram reliquorum. </s>

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<s>Diuer&longs;um in Math. quid.<emph.end type="italics"/></s>

<s>Ex 11. Methaphy&longs;.</s>

<s>Ex 11. Methaphy&longs;.</s>

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

2. Ortus punctorum, linearum, &longs;uperficierum.<emph.end type="italics"/></s>

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

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

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

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<s><emph type="italics"/>T. 47. Orbium c&oelig;le&longs;tium numerus, & fabrica ex Calippo.<emph.end type="italics"/></s>

<s>Ex 13. Methaphy&longs;.</s>

<s>Ex 13. Methaphy&longs;.</s>

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

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

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

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

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<s><emph type="italics"/>Quæ&longs;t. 10. De Libra vacua.<emph.end type="italics"/></s>

<s><emph type="italics"/>Qùæ&longs;t. </s>

<s><emph type="italics"/>Qùæ&longs;t. 11. De Curru, & &longs;cytala.<emph.end type="italics"/></s>

<s>11. De Curru, & &longs;cytala.<emph.end type="italics"/></s>

<s><emph type="italics"/>Quæ&longs;t. 13. De lugo. </s>

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

<s>In libello De Mundo ad Alex.</s>

<s>In libello De Mundo ad Alex.<emph type="italics"/>Cap.

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

2. Ordo Planetarum.<emph.end type="italics"/></s>

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<s>In libro De Admirandis audit.</s>

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

<s><emph type="italics"/>Lib. 7. cap. 8. De principijs Mathem.<emph.end type="italics"/></s>

<s><emph type="italics"/>Lib. 7. cap. 8. De principijs Mathem.<emph.end type="italics"/></s>

<s>Ex 1. Magnorum Moralium.</s>

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30. Proportionale in quatuor terminis con&longs;i&longs;tit.<emph.end type="italics"/></s>

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

<s>Moralium Eudemiorum.</s>

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

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

<s>Eudemiorum.</s>

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

<s>Eudemiorum.</s>

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12. Diametralis oppo&longs;itio.<emph.end type="italics"/></s>

<s>Ex 3. lib. Politicorum.</s>

<s>Politicorum.</s>

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

2. Modi Dorius, & Phrygius apud Mu&longs;icos, quì.<emph.end type="italics"/></s>

<s>Ex 4. lib. Polit.</s>

<s>Polit.</s>

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

3. Modus Doricus, & Phrygius.<emph.end type="italics"/></s>

<s>Ex 5. lib. Polit.</s>

<s>Polit.</s>

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

1. Aequitas Arithmetica, & quæ &longs;ecundum dignitatem.<emph.end type="italics"/></s>

<s>Ex 8. Polit.</s>

<s>Ex 8. Polit.</s>

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

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

<s>1. Cur ba&longs;es bullarum in aquis &longs;unt albæ?<emph.end type="italics"/></s>

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<s><emph type="italics"/>6. De voluminum &longs;ectione.<emph.end type="italics"/></s>

<s><emph type="italics"/>12. Idem cum præcedenti 3.<emph.end type="italics"/></s>

<s><emph type="italics"/>12. Idem cum præcedenti 3.<emph.end type="italics"/></s>

<s><emph type="italics"/>13. Idem cum 4 &longs;uperiori. </s>

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

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

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

<s><emph type="italics"/>32. De ip&longs;ius Diapa&longs;on ethymo.<emph.end type="italics"/></s>

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<s><emph type="italics"/>40. Cur &longs;olam Diapa&longs;on magadari &longs;olent?<emph.end type="italics"/></s>

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<s>5. <emph type="italics"/>De 4. Mathematicis medijs: A&longs;tronomia, Per&longs;pectiua, Mu&longs;ica, Mechanica.<emph.end type="italics"/></s>

<s>6. <emph type="italics"/>Appendix de re&longs;olutione omnium demonstrationum primi Euclidis.<emph.end type="italics"/></s>

<s>6. <emph type="italics"/>Appendix de re&longs;olutione omnium demonstrationum primi Euclidis.<emph.end type="italics"/></s>

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

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

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

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

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

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

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

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

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

<s>Ad definitionem 10. pri. </s>

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

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

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

<s>& cap.

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

7. lib.

1. Eth.</s>

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

<s>Ad Calcem axiomatum primi accommodetur tex. </s>

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

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

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

de Priori, & cap.

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

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

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

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

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

<s>& cap.

3. lib.

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

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

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

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

<s>7. &longs;ecundi Po&longs;ter. loco 2.</s>

<s>Ad 5. primi, vide cap.

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1. Priorum.</s>

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

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

<s>20. primi Po&longs;ter. </s>

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

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21. & cap.

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

22. &longs;ecundi <expan abbr="Prior&utilde;">Priorum</expan>, & tex. 13. primi Po&longs;ter.</s>

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

<s>Ad 32. primi, vide cap.

1. &longs;ecti 3. lib.

1. Prior. </s>

1. Prior. & cap.

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

<s>& cap.

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

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

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

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

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

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

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

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

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

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

<s>Ibidem <lb/>loco 4. & tex. </s>

<s>43. primi Po&longs;ter. </s>

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

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

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

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

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

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

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

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

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

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

<s>& cap.

<s>tex 11. primi de Anima. & cap.

1. de mem. </s>

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

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

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

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

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

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

<s>& cap.

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

4. lib.

2. de Generat. </s>

2. de Generat. </s>

<s>animal. </s>

<s>animal. </s>

<s>& cap.

5. lib.

6. Ethic. </s>

6. Ethic. & <lb/>cap.

<s>& <lb/>cap.

2. Magnorum Moral. & cap.

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

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

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

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

<s>12. &longs;ecundi Eudem. </s>

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

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

<s>Ad &longs;cholion præcedentis 32. primi, vide tex. </s>

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

<s>39. primi Po&longs;ter. loco 3. Item <lb/>tex. </s>

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

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

<s>Ad 45. primi, vide locum 9. de lineis in&longs;ecabilibus.</s>

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<s>Item locum 14. de ij&longs;dem.</s>

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

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

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

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<s>& cap.

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

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

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

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

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

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

<s>Ad primam 3. vide cap.

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2. Ethycorum.</s>

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

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

1. de Anima. </s>

<s>13. lib.

1. de Anima. </s>

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

<s>Ad 31. tertij, vide tex. </s>

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

<s>11. &longs;ecundi Po&longs;ter. & tex. </s>

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

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

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

<s>Ad commentarium P. </s>

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

<s>Clauij extremum lib.

<s>Ad commentarium P. Clauij extremum lib.

4. elementorum. </s>

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

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

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

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3. lib.

2. Ethyc.</s>

2. Ethyc.</s>

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

3. lib.

5. Ethyc. </s>

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

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

<s>Ad 10. definitionem 5. vide tex. 29. primi Po&longs;ter. loco 2.</s>

<s>Ad 10. definitionem 5. vide tex. </s>

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

<s>29. primi Po&longs;ter. loco 2.</s>

<s>Ad 12. definitionem 5. vide tex. </s>

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

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

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

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

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

<s>& cap.

3. lib.

5. Ethyc. </s>

5. Ethyc. loco 4.</s>

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

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

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

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

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

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

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8. Topicorum loco 41.</s>

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

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

<s>12. &longs;ecundi de Anima, & tex. </s>

<s>3. tertij Methaphy&longs;.</s>

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

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

<s>Ad primam definitionem 7. vide tex. </s>

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

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

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

1. lib.

1. Magnorum Moral.</s>

1. Magnorum Moral.</s>

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

<s>Ad 4. propo&longs;. </s>

<s>9. vide tex. 20. primi Po&longs;ter. loco 2.</s>

<s>20. primi Po&longs;ter. loco 2.</s>

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

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

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

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

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

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

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

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

<s>Ad 118. decimi, vide cap.

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2. Priorum. </s>

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

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

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

<s>& cap.

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

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

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

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

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

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

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

<s>& cap.

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

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

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

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

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

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

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

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

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

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

<s>& cap.

4. <lb/>lib.

2. de Generat. </s>

2. de Generat. animal. </s>

<s>animal. </s>

<s>& lib.

3. cap.

3. Ethyc. </s>

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

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

<s>tot Ari&longs;t.

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

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

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

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

Line 1910

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

<s>3. De his, quæ ad aliquid. </s>

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

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

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

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

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

<s>quod perinde e&longs;t, ac &longs;i o&longs;tendi&longs;&longs;et cuinam quadrato &longs;it æqualis, cum per vl<lb/>timam 2. Eucl. </s>

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

<s>po&longs;&longs;imus triangulo huic quadratum æquale con&longs;truere, quod <lb/>con&longs;equenter dato circulo æquale erit. </s>

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

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

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

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

<s>26. fusè explicabimus.</s>

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

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

<s>Porrò Diogenes Laert. </s>

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

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

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

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

<s>eadem omnino habet Proclus in comm. </s>

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

<s>ad &longs;extam primi elem. </s>

<s>Quod <lb/>porrò Ari&longs;t.

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3. lib.

3. Ethyc. </s>

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

<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>

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

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

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

<s>re<lb/>petita ip&longs;am con&longs;umeret, diceretur <expan abbr="vtranq;">vtranque</expan> metiri, & proinde duas lineas <lb/>A, & B, e&longs;&longs;e comm. </s>

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

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

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

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

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

<s>theorema <lb/>i&longs;tud demon&longs;tratur in vltima 10. Elem. eodem me<lb/>dio, quod ab Ari&longs;totele hic innuitur; Euclides ex <lb/>&longs;uppo&longs;itione alterius partis contradictionis ip&longs;ius <pb pagenum="38"/>propo&longs;itionis, quæfal&longs;a e&longs;t, nimirum &longs;uppo&longs;ito prædictas lineas e&longs;&longs;e comm. <lb/></s>

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

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

fal&longs;um ratiocinatur, quod &longs;ci<lb/>licet idem numerus e&longs;&longs;et par, & impar, quod Ari&longs;t. &longs;ignificat, quando ait, <lb/>imparia æqualia paribus fiunt. </s>

<s>ex quo ab&longs;urdo deducitur fal&longs;am e&longs;&longs;e prædi<lb/>ctam &longs;uppo&longs;itionem, quæ a&longs;truebat e&longs;&longs;e comm. </s>

<s>& proinde altera pars con<lb/>tradictionis, quæ e&longs;t, e&longs;&longs;e incomm. </s>

<s>ex quo ab&longs;urdo deducitur fal&longs;am e&longs;&longs;e prædi<lb/>ctam &longs;uppo&longs;itionem, quæ a&longs;truebat e&longs;&longs;e comm. & proinde altera pars con<lb/>tradictionis, quæ e&longs;t, e&longs;&longs;e incomm. </s>

<s>vera a&longs;truitur. </s>

<s>ex quibus &longs;atis videtur ex<lb/>plicari hic locus. </s>

<s>videas igitur, quàm leuiter nonnulli no&longs;træ tempe&longs;tatis <lb/>ageometreti i&longs;tud exponant, dicentes diametrum e&longs;&longs;e incomm. </s>

<s>videas igitur, quàm leuiter nonnulli no&longs;træ tempe&longs;tatis <lb/>ageometreti i&longs;tud exponant, dicentes diametrum e&longs;&longs;e incomm. co&longs;tæ, nihil <lb/>aliud &longs;igni&longs;icare, quam diametrum e&longs;&longs;e longiorem co&longs;ta, qua expo&longs;itione <lb/>nihil ineptius. </s>

<s>co&longs;tæ, nihil <lb/>aliud &longs;igni&longs;icare, quam diametrum e&longs;&longs;e longiorem co&longs;ta, qua expo&longs;itione <lb/>nihil ineptius. </s>

<s>Aduerte tandem figuram vulgatæ editionis e&longs;&longs;e ineptam, <lb/>cum habeat duo quadrata alterum &longs;uper diametro alterius, quorum maius <lb/>&longs;uperuacaneum e&longs;t.</s>

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quod, vt probè intelliga<lb/>tur, explicandum e&longs;t penes quid attendenda &longs;it æqualitas inter angulum, & <lb/>angulum, quod facile a&longs;&longs;equemur, &longs;i meminerimus angulum e&longs;&longs;e in clinatio<lb/>nem illam, quam duæ lineæ non in directum po&longs;itæ faciunt: &longs;iue etiam (vt <lb/>melius percipiamus) angulum e&longs;&longs;e acumen illud, &longs;iue mucronem <expan abbr="ill&utilde;">illum</expan>, quem <lb/>duæ lineæ non in directum con&longs;titutæ faciunt, vt duarum linearum A B, A C, <lb/><figure id="fig7"/><lb/>inclinatio in puncto A, &longs;iue acumen illud, &longs;iue mucro, <lb/>e&longs;t ratio anguli. </s>

<s>&longs;olum igitur duo anguli erunt æqua<lb/>les, <expan abbr="quãdo">quando</expan> vnius acumen æquale erit acumini alterius; <lb/>etiam &longs;i lineæ con&longs;tituentes vnum angulum &longs;int lon<lb/>giores lineis alterum angulum con&longs;tituentibus, quia <lb/>quantitas anguli non attenditur penes longitudinem <pb pagenum="40"/><expan abbr="linear&utilde;">linearum</expan>, &longs;ed penes inclinationem, & mucronem, quem faciunt: vnde etiam&longs;i <lb/>duæ lineæ prædictæ A B, A C, productæ, &longs;iue etiam decurtatæ fuerint, dum<lb/>modo &longs;itus, &longs;iue po&longs;itio ip&longs;arum, quam ad inuicem habent, non varietur, <lb/>erit &longs;emper eadem quantitas anguli A. </s>

<s>Aduertendum præterea rationem <lb/>anguli non po&longs;&longs;e &longs;aluari in &longs;olo puncto A, in quo lineæ concurrunt, &longs;ed ne<lb/>ce&longs;&longs;ariam e&longs;&longs;e aliquam quantitatem, quamuis exiguam, linearum A B, A C. <lb/></s>

<s>Notandum etiam, quod in nominatione angulorum, quæ fit per tres lite<lb/>ras, &longs;emper literam illam e&longs;&longs;e medio loco proferendam, quæ ad acumen ip<lb/>&longs;um po&longs;ita e&longs;t, vt in &longs;uperiori, litera A, debet &longs;emper media proferri, dicen<lb/>do angulum B A C, &longs;iue C A B, <expan abbr="n&utilde;quam">nunquam</expan> tamen licet dicere angulum A C B, <lb/>vel C B A. </s>

<s>Porrò quemadmodum vnus angulus vni angulo æqualis e&longs;t, ita <lb/><expan abbr="aliquãdo">aliquando</expan> duo anguli &longs;unt vni angulo æquales, vt patet, &longs;i vnus angulus, v.g. <lb/>angulus B A C, diuidatur in duos angulos à linea A D. tunc enim duo angu<lb/><figure id="fig8"/><lb/>li partiales B A D, D A C, erunt æquales totali angulo <lb/>B A C, cum partes omnes &longs;imul &longs;umptæ &longs;int &longs;uo toti æqua<lb/>les. </s>

<s>angulus B A C, diuidatur in duos angulos à linea A D. tunc enim duo angu<lb/><figure id="fig8"/><lb/>li partiales B A D, D A C, erunt æquales totali angulo <lb/>B A C, cum partes omnes &longs;imul &longs;umptæ &longs;int &longs;uo toti æqua<lb/>les. </s>

<s>pariter tres anguli po&longs;&longs;unt æquari & vni, & duobus <lb/>alijs angulis, quando nimirum a cumina, &longs;iue mucrones il<lb/>li &longs;imul ad vnum punctum con&longs;tituti <expan abbr="adæquar&etilde;tur">adæquarentur</expan> mucro<lb/>niilli, quem con&longs;tituerent alij duo anguli, quibus illi tres <lb/>&longs;unt pares, v.g. &longs;int tres anguli trianguli A B C, <expan abbr="&longs;int&qacute;">&longs;intque</expan>; alij duo anguli recti, <lb/><figure id="fig9"/><lb/>quos linea perpendicularis D E, facit cum li<lb/>nea F G; &longs;it <expan abbr="inquã">inquam</expan> anguli recti D E F, D E G, <lb/>tunc tres anguli illius <expan abbr="triãguli">trianguli</expan> <expan abbr="dic&etilde;tur">dicentur</expan> æqua<lb/>les duobus hi&longs;ce rectis, &longs;i tres illi mucrones <lb/>trianguli fimul &longs;umpti, & vniti ad punctum <lb/>E, ad quod duo <expan abbr="quoq;">quoque</expan> mucrones angulorum <lb/><figure id="fig10"/><lb/>rectorum coeunt, congruent omnino duobus <lb/>prædictis angulis rectis, &longs;iue duobus illis mu<lb/>cronibus angulorum rectorum, &longs;iue con&longs;ti<lb/>tuent lineam rectam F E G, &longs;icuti faciunt <lb/>etiam duo illi anguli recti; &longs;iue etiam dica<lb/>mus, occupabunt idem &longs;patium omninò, & <lb/>præcisè, quod occupant duo recti: v.g. </s>

<s>&longs;int tres anguli trianguli A B C, <expan abbr="&longs;int&qacute;">&longs;intque</expan>; alij duo anguli recti, <lb/><figure id="fig9"/><lb/>quos linea perpendicularis D E, facit cum li<lb/>nea F G; &longs;it <expan abbr="inquã">inquam</expan> anguli recti D E F, D E G, <lb/>tunc tres anguli illius <expan abbr="triãguli">trianguli</expan> <expan abbr="dic&etilde;tur">dicentur</expan> æqua<lb/>les duobus hi&longs;ce rectis, &longs;i tres illi mucrones <lb/>trianguli fimul &longs;umpti, & vniti ad punctum <lb/>E, ad quod duo <expan abbr="quoq;">quoque</expan> mucrones angulorum <lb/><figure id="fig10"/><lb/>rectorum coeunt, congruent omnino duobus <lb/>prædictis angulis rectis, &longs;iue duobus illis mu<lb/>cronibus angulorum rectorum, &longs;iue con&longs;ti<lb/>tuent lineam rectam F E G, &longs;icuti faciunt <lb/>etiam duo illi anguli recti; &longs;iue etiam dica<lb/>mus, occupabunt idem &longs;patium omninò, & <lb/>præcisè, quod occupant duo recti: v.g. </s>

<s>&longs;i mucro B, ibi poneretur, faceret <lb/>angulum F E H, & &longs;i ibi iuxta ip&longs;um apponeretur mucro A, faceret angulum <lb/>H E I. quem &longs;i deinceps &longs;ub&longs;equetur reliquus angulus C, con&longs;titueret <expan abbr="reli-qu&utilde;">reli<lb/>quum</expan> angulum I E G. iam, vt vides, illi tres anguli ad E, tran&longs;lati, &longs;unt æqua<lb/>les duobus rectis ad E, pariter con&longs;titutis, cum illi tres fiant partes duorum <lb/>rectorú, vel quia occupant idem &longs;patium, vel eandem lineam rectam F E G, <lb/>con&longs;tituant. </s>

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<s>Ab&longs;tineo à demon&longs;trationibus geometricis, quo<lb/>niam ij, qui Mathematicis &longs;unt imbuti, no&longs;tra hac opera parum indigent. <lb/></s>

<s>&longs;i quis tamen volet, con&longs;ulat 32. primi Elem. </s>

<s>&longs;i quis tamen volet, con&longs;ulat 32. primi Elem. </s>

<s>Ex hac igitur declaratione <lb/>licet cogno&longs;cere nonnullos ageometretos locum hunc, & &longs;imiles &longs;ub&longs;equen<lb/>tes non &longs;atis intelligere, dicentes, nihil aliud verba illa Ari&longs;t.

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re&longs;pondet, dicens, nihil inde ab&longs;ur<lb/>di accidere po&longs;&longs;e, quoniam ip&longs;e vtitur hi&longs;ce literis, <expan abbr="nõ">non</expan> quatenus literæ &longs;unt, <lb/>&longs;ed quatenus rerum vicem, pro quibus exponuntur, gerunt: quemadmodum <lb/>etiam Geometræ faciunt, qui lineam, quæ pedalis non e&longs;t, pedalem, & quæ <lb/>non e&longs;t recta, rectam; & quæ lata e&longs;t, non latam, &longs;upponunt, & tamen nihil <lb/>inde ab&longs;urdi contingit. </s>

<s>Ex quibus intelligimus per lineas illas &longs;en&longs;ibiles, & <lb/>phy&longs;icas, quas Geometræ in &longs;uis figuris ducunt, intelligendas e&longs;&longs;e lineas ve<lb/>rè Mathematicas omni latitudine carentes; vtitur enim inquit Ari&longs;t. Geo<lb/>metra lineis phy&longs;icis, non tanquam phy&longs;icis, nec de eis tanquam de phy&longs;icis <lb/>lineis ratiocinatur, &longs;ed ijs vtitur tanquam verè mathematicis. </s>

<s>Geo<lb/>metra lineis phy&longs;icis, non tanquam phy&longs;icis, nec de eis tanquam de phy&longs;icis <lb/>lineis ratiocinatur, &longs;ed ijs vtitur tanquam verè mathematicis. </s>

<s>idem dicen<lb/>dum e&longs;t de &longs;uperficiebus, necnon de corporibus, quæ ijdem Goometræ de<lb/>&longs;cribunt, vt per ea, de verè mathematicis di&longs;currant.</s><pb pagenum="42"/>

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

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

<s>æqualem interno, & oppo&longs;ito, & ad ea&longs;dem partes, <lb/>angulo videlicet G H D. &longs;i ergo inquit Ari&longs;t &longs;upponamus i&longs;tud fal&longs;um, an<lb/>gulum &longs;cilicet E G B, externum e&longs;&longs;e maiorem angulo interno G H D, &longs;equi<lb/>tur etiam fal&longs;um, videlicet lineas <expan abbr="æquidi&longs;tãtes">æquidi&longs;tantes</expan> A B, C D, concurrere. </s>

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<s>At quando hi duo interni &longs;unt minores duobus rectis, tunc lineæ A B, C D, <lb/>&longs;unt concurrentes, &longs;i protrahantur ad partes prædictorum <expan abbr="angulor&utilde;">angulorum</expan>. </s>

<s>quod <lb/>P. Clauius luculenti, & hactenus de&longs;iderata demon&longs;tratione ad 28. primi <lb/>demon&longs;trauit. </s>

<s>Clauius luculenti, & hactenus de&longs;iderata demon&longs;tratione ad 28. primi <lb/>demon&longs;trauit. </s>

<s><expan abbr="Atq;">Atque</expan> hoc pacto ex prima fal&longs;a &longs;uppo&longs;itione, nimirum angu<lb/>lum illum externum e&longs;&longs;e maiorem interno, & oppo&longs;ito; &longs;equitur fal&longs;um, ni<lb/>mirum lineas parallelas concurrere.</s>

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

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

3. <lb/>in comm. Elem.

<s>Elem.

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>

<s>Exempli cau&longs;a, cum cubi duplicatio propo&longs;ita e&longs;&longs;et ad inue&longs;ti<lb/>gandam quæ&longs;tionem in aliud tran&longs;tulere, quod illud propo&longs;itum con&longs;equi<lb/>tur, ad duarum nempe mediarum linearum inuentionem tran&longs;lata e&longs;t quæ<lb/>&longs;tio, & &longs;ic quærebant deinceps, quonam modo datis duabus rectis lineis, <lb/>duæ mediæ proportionales reperirentur. </s>

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<s>&longs;yllogi&longs;mus autem Hippocratis, quo o&longs;tendebat circuli qua<lb/>draturam reducebatur ad has propo&longs;itiones, omnis rectilinea figura qua<lb/>dratur, &longs;ed circulus reducitur ad figuram rectilineam, ergo circulus qua<lb/>dratur. </s>

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

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

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

3. Præ<lb/>dicam. </s>

3. Præ<lb/>dicam. </s>

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

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<s>Ex eodem cap. <emph type="italics"/>(Veluti &longs;i K, e&longs;&longs;et quadrari, in quo autem E, rectilineum, in <lb/>quo verò F, circulus, &longs;i ip&longs;ius E F, vnum &longs;olum e&longs;&longs;et medium, hoc, quod e&longs;t, cum <lb/>lunulis æqualem fieri circulum rectilineo, e&longs;&longs;e po&longs;&longs;et propè ip&longs;um cogno&longs;cere, cum <lb/>vero B C, neque credibilius &longs;it, quam A C, <expan abbr="neq;">neque</expan> pauca media, non dico Abductio<lb/>nem: <expan abbr="neq;">neque</expan> quando B C, &longs;it immediatum, tale enim &longs;cientia est)<emph.end type="italics"/> Aduerte figuram <lb/>vulgatæ editionis e&longs;&longs;e mendo&longs;am, & propterea re&longs;tituendam e&longs;&longs;e, qualis pri<lb/>ma &longs;equens ex Simplicio ad tex. </s>

<s>11. primi Phy&longs;ic. </s>

<s>hoc modo Hippocrates <lb/>Chius conabatur circulum ad quadrum redigere; fit circulus A B G C, qua<lb/>drandus; con&longs;tituatur <expan abbr="itaq;">itaque</expan> &longs;uper diametro cius B C, quadratum B C D F, <lb/>cuius diameter B D, &longs;ecatur bifariam in G, à circumferentia circuli dati, <lb/>quod patet ducta &longs;emidiametro H G, perpendiculari ex B C, quæ &longs;uo extre<lb/>mo puncto G, &longs;ecat bifariam, & <expan abbr="diametr&utilde;">diametrum</expan> B D, & circumferentiam B G C. <lb/>facto ergo centro G, de&longs;cribatur alter circulus per puncta B C D F, <expan abbr="conne-ctatur&qacute;">conne<lb/>ctaturque</expan>; recta G C. in triangulo orthogonio B C D, latus B D, &longs;ubtenditur <pb pagenum="45"/><figure id="fig13"/><lb/>angulo recto C, ergo quadratum eius ex eorol<lb/>lario 47. primi, duplum erit quadrati B C, quare <lb/>etiam circulus B C D F, duplus erit circuli A B<lb/>G C, per 2. duodecimi, & &longs;emicirculus B C D, <lb/>duplus erit &longs;emicirculi B A C: & quadrans B E<lb/>C G, æqualis erit &longs;emicirculo B A C: ablato igi<lb/>tur communi &longs;egmento B E C H, remanet lunu<lb/>la B A C E, æqualis triangulo B C G, quod trian<lb/>gulum &longs;i per vltimam &longs;ecundi quadretur, erit lu<lb/>nula B A C, con&longs;equenter quadrata. </s>

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<s>Hæc pluribus, vt &longs;equentibus etiam textibus, in quibus huius te<lb/>tragoni&longs;mi fit mentio &longs;atisfacere po&longs;&longs;imus. </s>

<s>Hippocrates i&longs;te Chius e&longs;t alter <pb pagenum="46"/>ab illo Hippocrate Coo medicorum Magi&longs;tro, vt colligitur ex Alexandre <lb/>Aphrod. </s>

<s>Hippocrates i&longs;te Chius e&longs;t alter <pb pagenum="46"/>ab illo Hippocrate Coo medicorum Magi&longs;tro, vt colligitur ex Alexandre <lb/>Aphrod. in Primum Meteororum de Cometis.</s>

<s>in Primum Meteororum de Cometis.</s>

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

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

<s>&longs;ecto 3. cap.

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>

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<s>Eodem tex. </s>

<s>Eodem tex. 5. <emph type="italics"/>(Ponit enim Arithmeticus vnitatem indiui&longs;ibilem e&longs;&longs;e &longs;ecun<lb/>dum quantum)<emph.end type="italics"/> hoc quamquam non ponatur ab Arithmeticis expre&longs;sè, præ<lb/>&longs;upponitur tamen ab eis: nu&longs;quam enim Euclides in totis tribus Arithme<lb/>ticis libris, infra vnitatem de&longs;cendit, vt propterea appareat, ip&longs;am in quan<lb/>titate di&longs;creta e&longs;&longs;e minimum, & indiui&longs;ibile. </s>

<s>5. <emph type="italics"/>(Ponit enim Arithmeticus vnitatem indiui&longs;ibilem e&longs;&longs;e &longs;ecun<lb/>dum quantum)<emph.end type="italics"/> hoc quamquam non ponatur ab Arithmeticis expre&longs;sè, præ<lb/>&longs;upponitur tamen ab eis: nu&longs;quam enim Euclides in totis tribus Arithme<lb/>ticis libris, infra vnitatem de&longs;cendit, vt propterea appareat, ip&longs;am in quan<lb/>titate di&longs;creta e&longs;&longs;e minimum, & indiui&longs;ibile. </s>

<s>Verum dubitabit forrè qui&longs;<lb/>piam hoc modo, &longs;i vnitas minimum, <expan abbr="atq;">atque</expan> indiui&longs;ibile e&longs;t in quanto di&longs;creto, <lb/>qua igitur ratione Arithmetici practici eam diuidunt in dimidium, in trien<lb/>tem, in quadrantem, & alijs &longs;imiliter modis, vnde numeri illi, qui fractio<lb/>nes appellantur, exurgunt? </s>

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<s>Eodem tex. </s>

<s>Eodem tex. 9. <emph type="italics"/>(Et <expan abbr="quibu&longs;cunq;">quibu&longs;cunque</expan> iuexi&longs;tentium ip&longs;is, ip&longs;æ &longs;unt in oratione, quid<emph.end type="italics"/><pb pagenum="48"/><emph type="italics"/>est declarante, quemadmodum rectum ine&longs;t lineæ, & circulare: & impar, & p<gap/><lb/>numero, & primum, & compo&longs;itum, & æquilaterum, & altera parte longius. </s>

<s>9. <emph type="italics"/>(Et <expan abbr="quibu&longs;cunq;">quibu&longs;cunque</expan> iuexi&longs;tentium ip&longs;is, ip&longs;æ &longs;unt in oratione, quid<emph.end type="italics"/><pb pagenum="48"/><emph type="italics"/>est declarante, quemadmodum rectum ine&longs;t lineæ, & circulare: & impar, & p<gap/><lb/>numero, & primum, & compo&longs;itum, & æquilaterum, & altera parte longius. </s>

<s>& <lb/><expan abbr="o&itilde;bus">oimbus</expan> bis in&longs;unt in oratione, quid e&longs;t <expan abbr="declarãte">declarante</expan>, ibi quidem linea, hic vero numerus)<emph.end type="italics"/><lb/>quia locus hic benè exponitur à Toleto, & melius etiam à Conymbr. </s>

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<s>Ibidem <emph type="italics"/>(Et &longs;i triangulum non e&longs;&longs;et aliud, quam I&longs;o&longs;celes, &longs;ecundum quod I&longs;o<lb/>&longs;celes videretur <expan abbr="vtiq;">vtique</expan> ine&longs;&longs;e)<emph.end type="italics"/> i&longs;tud e&longs;t &longs;ecundum exemplum tertij erroris. </s>

<s>Por<lb/>rò cum tres &longs;int &longs;pecies triangulorum, æquilaterum, I&longs;o&longs;celes, Scalenum, &longs;i <lb/>accideret, vt ex illis tribus vna tantum &longs;pecies, v. </s>

<s>Por<lb/>rò cum tres &longs;int &longs;pecies triangulorum, æquilaterum, I&longs;o&longs;celes, Scalenum, &longs;i <lb/>accideret, vt ex illis tribus vna tantum &longs;pecies, v. g. </s>

<s>I&longs;o&longs;celes in mundo re<lb/>periretur; <expan abbr="tunc&qacute;">tuncque</expan>; qui&longs;piam de I&longs;o&longs;cele o&longs;tenderet affectionem quampiam, <lb/>putans &longs;e <expan abbr="o&longs;t&etilde;di&longs;&longs;e">o&longs;tendi&longs;&longs;e</expan> pa&longs;&longs;ionem de proprio &longs;ubiecto, & primo, falleretur, quia <lb/>aifectio illa competeret I&longs;o&longs;celi, non vt huic &longs;peciei I&longs;o&longs;celis, &longs;ed quatenus <lb/>e&longs;t triangulum, cui primo, & per &longs;e, & &longs;ecundum quod ip&longs;um conuenit. </s>

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<s>ait enim, &longs;i <lb/>non e&longs;&longs;et aliud, quam I&longs;o&longs;celes, <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan> quod I&longs;o&longs;celes: quibus verbis cla<lb/>rè &longs;peciem, non indiuiduum, &longs;igni&longs;icat, ex his duobus exemplis manife&longs;tus <lb/>e&longs;t tertius error, qui erat, quando erat <emph type="italics"/>(vt in parte totum)<emph.end type="italics"/> <expan abbr="quod&qacute;">quodque</expan>; illis verbis <lb/>expo&longs;uerat <emph type="italics"/>(vei contingit etiam, vt in parte totum, in quo mon&longs;tratur: ijs emm, <lb/>quæ &longs;unt in parte inerit quidem demon&longs;tratio, & erit de omni, &longs;ed tamen non erit <lb/>buius primi vniuer&longs;aliter demon&longs;tratio. </s>

<s>dico auttm huius primi, &longs;ecundum quod <lb/>buius demonstrationem, quando &longs;it primi vniuer&longs;aliter)<emph.end type="italics"/> ide&longs;t, quando vniuer&longs;ale <lb/>&longs;ubiectum exi&longs;tit quidem, &longs;ed tamen non de ip&longs;o &longs;it demon&longs;tratio, &longs;ed de ali<lb/>qua parte ip&longs;ius, v. </s>

<s>de &longs;pecie aliqua demon&longs;tratur aliquid, quod deberet <lb/>o&longs;tendi primò de ip&longs;o vniuer&longs;ali, cum illi primò competat.</s>

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<s>Alternam igitur proportionem definit Eu<lb/>clides definitione 12. quinti, &longs;ic, e&longs;t &longs;umptio antecedentis ad <expan abbr="anteced&etilde;tem">antecedentem</expan>, <lb/><figure id="fig20"/><lb/>& con&longs;equentis ad con&longs;equentem. </s>

<s>Explico, exponantur qua<lb/>tuor quantitates proportionales, v.g. </s>

<s>Explico, exponantur qua<lb/>tuor quantitates proportionales, v.g. vt 6. ad 3. ita &longs;int 4. ad <lb/>2. &longs;i igitur argumentemur &longs;ic, vt 6. ad 3. ita 4. ad 2. ergo al<lb/>ternatim erit, vt 6. ad 4. ita 3. ad 2. &longs;iue dixerimus, vt pri<lb/>mum ad &longs;ecundum, ita tertium ad quartum, igitur alterna<lb/>tim erit, vt primum ad tertium, ita &longs;ecundum ad quartum: valebit con&longs;e<lb/>quentia; quæ quidem probatur deinde propo&longs;itione 16. quinti de magnitu<lb/>dinibus, hoc e&longs;t in vniuer&longs;um de lineis, &longs;uperficiebus, & &longs;olidis. </s>

<s>vt 6. ad 3. ita &longs;int 4. ad <lb/>2. &longs;i igitur argumentemur &longs;ic, vt 6. ad 3. ita 4. ad 2. ergo al<lb/>ternatim erit, vt 6. ad 4. ita 3. ad 2. &longs;iue dixerimus, vt pri<lb/>mum ad &longs;ecundum, ita tertium ad quartum, igitur alterna<lb/>tim erit, vt primum ad tertium, ita &longs;ecundum ad quartum: valebit con&longs;e<lb/>quentia; quæ quidem probatur deinde propo&longs;itione 16. quinti de magnitu<lb/>dinibus, hoc e&longs;t in vniuer&longs;um de lineis, &longs;uperficiebus, & &longs;olidis. </s>

<s>quando igi<lb/>tur Ari&longs;t.

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<s>Ibidem <emph type="italics"/>(<expan abbr="Neq;">Neque</expan> alij &longs;cientiæ quod alterius, ni&longs;i <expan abbr="quæcunq;">quæcunque</expan> ita &longs;e habent inter &longs;e, <lb/>vt &longs;it alterum &longs;ub altero, vt per&longs;pectiua ad Geometriam, & harmonica ad Arith<lb/>meticam)<emph.end type="italics"/> excipit ab illa regula (qua prohibetur, quamuis &longs;cientiam in alie<lb/>nam falcem immittere) &longs;cientias &longs;ubalternatas, quæ propriè in Mathemati<lb/>cis reperiuntur, Per&longs;pectiua enim propriè &longs;ubalternatur Geometriæ, quia <lb/>vtitur Demon&longs;trationibus linearibus, quas applicat lineis vi&longs;ualibus, & Mu<lb/>&longs;ica &longs;ubalternatur Arithmeticæ, quia ab ip&longs;a mutuatur <expan abbr="demõ&longs;trationes">demon&longs;trationes</expan> nu<lb/>merorum, quas applicat numeris &longs;onoris. </s>

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

<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"/><lb/>per 21. primi Elem.

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

<s>Hine <lb/>per&longs;picuè vides, qua ratione Per&longs;pectiua Geometriæ &longs;ubalternetur, &longs;iue <lb/>quid &longs;it ip&longs;a &longs;ubalternatio, vbi medium e&longs;t Geometricum, conclu&longs;io autem <lb/>optica. </s>

<s>Exemplum &longs;ubalternationis Muficæ &longs;it, <expan abbr="con&longs;onãtia">con&longs;onantia</expan> Diapa&longs;on, quam <lb/>vulgò octauam appellant in data chorda collocare, hoc e&longs;t, vocem grauio<lb/>rem facere duplam vocis acutioris &longs;umatur chorda A B, & diuidatur bifa<lb/>riam, &longs;ine in æqualia in C; tota igitur chorda A B, ad dimidium A C, haber <pb pagenum="54"/><figure id="fig25"/><lb/>proportionem, quam 2. ad 1. <lb/>&longs;iue duplam, ergo etiam &longs;o<lb/>nus totius chordæ A B, ad <expan abbr="&longs;o-n&utilde;">&longs;o<lb/>num</expan> chordæ dimidiæ A C, ha<lb/>bebit eandem rationem, <expan abbr="nimir&utilde;">nimirum</expan> quam 2. ad 1. &longs;iue duplam. </s>

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<s>Veritas exemplo tunc <lb/>ine&longs;t, quando illud, quod in exemplo narratur, verè extitit, vt &longs;i quis in <lb/>exemplum pudicitiæ afferret hi&longs;toriam Io&longs;ephi, <expan abbr="ver&utilde;">verum</expan> i&longs;tuà e&longs;&longs;et exemplum. <lb/></s>

<s>quæ veritas in exemplis moralibus non &longs;emper e&longs;t nece&longs;&longs;aria, talia exempla <lb/>&longs;unt &longs;æpè parabolæ, & fabulæ, quæ nunquam extiterunt, v. </s>

<s>quæ veritas in exemplis moralibus non &longs;emper e&longs;t nece&longs;&longs;aria, talia exempla <lb/>&longs;unt &longs;æpè parabolæ, & fabulæ, quæ nunquam extiterunt, v. g. </s>

<s>narratur ab <lb/>Ari&longs;t.

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<s>quod dictum velim propter nonnullos, qui ab <lb/>huiu&longs;modi diui&longs;ionibus abhorrent, <expan abbr="timent&qacute;">timentque</expan>; ne demon&longs;trationis perfectio<lb/>ni per eas plurimum derogetur. </s>

<s>Pithagoreorum demon&longs;trationem vide <lb/>apud Clauium in &longs;cholio 32. primi Euclidis, quam ex Eudemo etiam Pro<lb/>clus in comm. </s>

<s>Pithagoreorum demon&longs;trationem vide <lb/>apud Clauium in &longs;cholio 32. primi Euclidis, quam ex Eudemo etiam Pro<lb/>clus in comm. eiu&longs;dem recitat.</s>

<s>eiu&longs;dem recitat.</s>

<s>Ibidem <emph type="italics"/>(Sed quemadmod&ubreve; harmonica per Arithmeticam)<emph.end type="italics"/> vide &longs;upra tex. </s>

<s>Ibidem <emph type="italics"/>(Sed quemadmod&ubreve; harmonica per Arithmeticam)<emph.end type="italics"/> vide &longs;upra tex. 20.</s>

<s>Ibidem <emph type="italics"/>(Demon&longs;tratio autem non computatur in aliud genus; m&longs;i, vt dictum <lb/>e&longs;t geometricæ demon&longs;trationes in Per&longs;pectiuas, aut Mcchamcas, & arithmeticæ in <lb/>harmonicas)<emph.end type="italics"/> exempla &longs;ubalternationis Per&longs;pectiuæ, & Mu&longs;icæ in tex. </s>

<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.

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>

<s>&longs;it <lb/>triangulum A B C, à cuius angulis A, & B, ducantur duæ rectæ A D, B E, ita <lb/>vt bifariam &longs;ecent latera A C, B C, in punctis D, & E, & concurrant in F. <lb/></s>

<s>&longs;it <lb/>triangulum A B C, à cuius angulis A, & B, ducantur duæ rectæ A D, B E, ita <lb/>vt bifariam &longs;ecent latera A C, B C, in punctis D, & E, & concurrant in F. <lb/></s>

<s>Dico F, e&longs;&longs;e centrum grauitatis propo&longs;iti trianguli. </s>

<s>Quoniam enim in 13. <lb/>Aequep. </s>

<s>Quoniam enim in 13. <lb/>Aequep. probauit centrum grauitatis e&longs;&longs;e in ea linea, quæ ducta ab angulo <lb/>quouis &longs;ecat oppo&longs;itum latus bifariam, crit in linea A D, <expan abbr="centr&utilde;">centrum</expan> grauitatis. <pb pagenum="57"/><figure id="fig28"/><lb/>&longs;ed eadem ratione erit etiam in linea B E, er<lb/>go non ni&longs;i in puncto F, quod <expan abbr="&longs;ol&utilde;">&longs;olum</expan> e&longs;t in vtra<lb/>que, quod erat demon&longs;trandum. </s>

<s>probauit centrum grauitatis e&longs;&longs;e in ea linea, quæ ducta ab angulo <lb/>quouis &longs;ecat oppo&longs;itum latus bifariam, crit in linea A D, <expan abbr="centr&utilde;">centrum</expan> grauitatis. <pb pagenum="57"/><figure id="fig28"/><lb/>&longs;ed eadem ratione erit etiam in linea B E, er<lb/>go non ni&longs;i in puncto F, quod <expan abbr="&longs;ol&utilde;">&longs;olum</expan> e&longs;t in vtra<lb/>que, quod erat demon&longs;trandum. </s>

<s>ex quibus ap<lb/>paret, qua ratione mechanica conclu&longs;io Geo<lb/>metriæ &longs;ubiaceat, dum lineari di&longs;cur&longs;u ip&longs;a <lb/>demon&longs;tratio perficitur. </s>

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<s>Tex. 24. <emph type="italics"/>(Veluti Arithmetica quidem quid impar, aut par; aut quadrangu<lb/>lum, aut cubus)<emph.end type="italics"/> cogno&longs;cas hinc certò certius quadrangulum, & cubum e&longs;&longs;e <lb/>&longs;pecies numerorum, &longs;icuti &longs;upra tex. 9. & 20. explicauimus, quò nunc te vi<lb/>ci&longs;&longs;im, vt præ&longs;entem locum intelligas, remittimus.</s>

<s>9. & 20. explicauimus, quò nunc te vi<lb/>ci&longs;&longs;im, vt præ&longs;entem locum intelligas, remittimus.</s>

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

<s>vn<lb/>de proportio multiplex habet &longs;ub &longs;e genera infinita, quando enim maior <lb/>continet minorem bis, dicitur Dupla: quando ter, Tripla: quando quater, <lb/>Quadrupla: & &longs;ic in infinitum: v. </s>

<s>2. ad 1. e&longs;t proportio dupla; 3. ad 1. tri<lb/>pla; 4. ad 1. quadrupla, &c. </s>

<s>omnes tamen continentur &longs;ub genere multipli<lb/>cis rationis. </s>

<s>porrò &longs;i qu&ecedil;piam proportio ex genere multiplici progrediatur <lb/>per plures terminos, v. </s>

<s>porrò &longs;i qu&ecedil;piam proportio ex genere multiplici progrediatur <lb/>per plures terminos, v. g. </s>

<s>proportio quadrupla progrediatur hoc modo, <lb/>1. 4. 16. 64. 256. &c. </s>

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<s>Ibidem (<emph type="italics"/>Conuertuntur autem magis, quæ&longs;unt in mathematicis, quoniam nul<lb/>lum accidens accipiunt (m quo quidem ijs præ&longs;t&abreve;t, quæ di&longs;putationibus traduntur) <lb/>&longs;ed definitiones<emph.end type="italics"/>) Hæc e&longs;t altera mathematicarum laus, vnde earum quoque <lb/>præ&longs;tantia elucet, quia &longs;cilicet mathematicæ pro medijs vtuntur definitio<pb pagenum="59"/>nibus &longs;ubiecti, aut pa&longs;&longs;ionis, quæ nuilo modo &longs;unt accidentalia conclu&longs;ioni, <lb/>v. </s>

<s>in prima Euclidis demon&longs;tratione per definitionem &longs;ubiecti probantur <lb/>tres lineæ e&longs;&longs;e æquales, quia nimirum &longs;int &longs;emidiametri circulorum æqua<lb/>lium, quæ e&longs;t ip&longs;arum definitio. </s>

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<s>Po&longs;t nonnulla (<emph type="italics"/>Vt Per&longs;pectiua ad Geometriam, & Mechanica ad Stereome<lb/>tricam, & Harmonica ad Arithmeticam, vt Apparentia ad A&longs;trologicam<emph.end type="italics"/>) &longs;upra <lb/>tex. 20. exempla &longs;ubalternationum Per&longs;pectiuæ, & Mechanicæ cum Geo<lb/>metria &longs;unt allata. </s>

<s>20. exempla &longs;ubalternationum Per&longs;pectiuæ, & Mechanicæ cum Geo<lb/>metria &longs;unt allata. </s>

<s>hic primo notandum Stereometriam non ef&longs;e &longs;cientiam <lb/>di&longs;tinctam à Geometria, ni&longs;i &longs;icuti partem à toto: nam cum Geometria <lb/>con&longs;ideret quantitatem, &longs;ecundum tres dimen&longs;iones, longitudinem, latitu<lb/>dinem, & profunditatem, oritur triplex illius diui&longs;ro, de lineis, de &longs;uperfi<lb/>ciebus, de &longs;olidis. </s>

<s>pars igitur, quæ de &longs;olidis tractat, <expan abbr="pattim&qacute;">pattimque</expan>; continetur <lb/>11. 12. 13. 14. & 15. Euclidis, partim aliorum Geometrarum libris, vt li<lb/>bro Archim. </s>

<s>de Sphæra, & Cyl. </s>

<s>& &longs;imilibus, dicitur Stereometria à græco <lb/><foreign lang="greek">steoeov,</foreign> ide&longs;t &longs;olidum. </s>

<s>Porrò cur malit Ari&longs;t. Mechanicam &longs;ubalternari Ste<lb/>reometriæ, quam toti Geometriæ, qua tamen, vt videre e&longs;t apud Archime<lb/>dem, innititur, fortè ea ratio e&longs;t, quia Mechanica præcipuè con&longs;iderat ma<lb/>chinas, quæ corpora &longs;unt, & propterea præcipuè, & primò debet Stereome<lb/>triæ, quæ corpora pariter contemplatur, &longs;ubalternari. </s>

<s>Quod ait Apparen<lb/>tia ad A&longs;irol. </s>

<s>Quod ait Apparen<lb/>tia ad A&longs;irol. </s>

<s>inteiligit per Apparentia vulgarem quandam Nautarum, & <lb/>Agricolarum a&longs;tronomiam, quæ quodammodo &longs;ubaiternatur, & pendet ex <lb/>&longs;cientia A&longs;trologiæ; indiget enim cognitione ortus, & motus a&longs;trorum, <lb/>præ&longs;ertim Lunæ, Hyadum, Pleiadum, & Canis. </s>

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<s>Po&longs;t nonnulla (<emph type="italics"/>Hic enim ip&longs;um quidem quod &longs;en&longs;itiuorum e&longs;t &longs;cire, ip&longs;um ve<lb/>rò Propter quid Mathematicorum; hi <expan abbr="namq;">namque</expan> habent cau&longs;arum demon&longs;trationes, <lb/>&c.<emph.end type="italics"/>) &longs;en&longs;us e&longs;t in &longs;ubalternatis, & dependentibus di&longs;ciplinis, quas &longs;en&longs;itiuas <lb/>appellat, quia de rebus &longs;en&longs;ibilibus &longs;unt, vt in Per&longs;pectiua de obiectis vifibi<lb/>libus, & in Mu&longs;ica de &longs;onis cogno&longs;citur Quod, ide&longs;t effectus: cuius effectus <lb/>cau&longs;a, &longs;eu Propter quid &longs;citur auxilio Mathematicarum, ide&longs;t, traditur à <lb/>&longs;cientijs &longs;ubalternantibus. </s>

<s>alicuius effectus in Per&longs;pectiua cau&longs;a inqui<lb/>ritur, & inuenitur ope Geometriæ, cuiilla &longs;ubiacet. </s>

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1. Priorum &longs;ecto 3. cap.

1.</s>

1.</s>

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1. Priorum &longs;ecto 3. cap.

1.</s>

1.</s>

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<s>Ad finem tex. </s>

<s>Ad finem tex. 43. <emph type="italics"/>(Principia enim duplicia &longs;unt, ex quibus, & circa quod: <lb/>quæ quidem igitur, ex quibus, communia &longs;unt: quæ autem circa quod propria, vt <lb/>numerus, magnitudo)<emph.end type="italics"/> nonnulli codices corruptè legunt (vt numerus magni<lb/>tudine) &longs;ed ex græco tex. </s>

<s>43. <emph type="italics"/>(Principia enim duplicia &longs;unt, ex quibus, & circa quod: <lb/>quæ quidem igitur, ex quibus, communia &longs;unt: quæ autem circa quod propria, vt <lb/>numerus, magnitudo)<emph.end type="italics"/> nonnulli codices corruptè legunt (vt numerus magni<lb/>tudine) &longs;ed ex græco tex. </s>

<s>corrigendi &longs;unt, vti fecimus. </s>

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<s>&longs;i in medio, aut non)<emph.end type="italics"/> Zabarella locum hunc, etiam <lb/>quatenus ad Mathematicum attinet, optimè declarat. </s>

<s>In quæ<lb/>&longs;tionibus, & demon&longs;trationibus duo &longs;unt, &longs;ubiectum, & <expan abbr="prædicat&utilde;">prædicatum</expan>, <expan abbr="vtriu&longs;q;">vtriu&longs;que</expan> <lb/>cau&longs;æ exi&longs;tunt, & quæruntur: v. </s>

<s>Luna, terra, Sol, & triangulum &longs;unt &longs;u<lb/>biectum in demon&longs;tratione, quorum prædicata &longs;unt, Lunæ quidem, & So<lb/>lis, eclyp&longs;is. </s>

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1. Priorum &longs;ecto 3. cap.

1.</s>

1.</s>

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<s>Tex. 7. <emph type="italics"/>(Quid enim &longs;ignificat triangulum, accipit Geometra)<emph.end type="italics"/> vt manife&longs;tum <lb/>e&longs;t in 20. dednitione primi Elem.</s>

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

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

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

<s>adeò vt hæc &longs;it demon&longs;tratio omnibus numeris ab&longs;oluta per cau&longs;am mate<lb/>rialem, vt benè &longs;entit Ari&longs;t. </s>

<s>adeò vt hæc &longs;it demon&longs;tratio omnibus numeris ab&longs;oluta per cau&longs;am mate<lb/>rialem, vt benè &longs;entit Ari&longs;t. </s>

<s>Reliqua ad logicum pertinent, etiam&longs;i per cha<lb/>racteres more mathematicorum exponantur.</s>

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<s>Tex. 25. <emph type="italics"/>(Vt propter quid, & permutatim proportionale? </s>

<s>& c.<emph.end type="italics"/>) quod quan<lb/>titates, quæ &longs;unt proportionales, &longs;int etiam alternatim, &longs;eu permutatim <lb/>proportionales explicatum e&longs;t ad tex. </s>

<s>& c.<emph.end type="italics"/>) quod quan<lb/>titates, quæ &longs;unt proportionales, &longs;int etiam alternatim, &longs;eu permutatim <lb/>proportionales explicatum e&longs;t ad tex. 13. primi Po&longs;ter. quæ etiam nece&longs;&longs;a<lb/>ria &longs;unt ad hunc locum benè intelligendum. </s>

<s>13. primi Po&longs;ter. quæ etiam nece&longs;&longs;a<lb/>ria &longs;unt ad hunc locum benè intelligendum. </s>

<s>Illud autem commune propter <lb/>quod ea, quæ &longs;unt proportionalia, &longs;int etiam permutatim proportionalia, <lb/>e&longs;t quoddam innominatum, de quo ibi dictum e&longs;t, quod cum conueniat li<lb/>neis, & numeris, & tamen &longs;eparatim de vtri&longs;que illa pa&longs;&longs;io demon&longs;tretur, <lb/>quærit cuinam primò, & per &longs;e conueniat hæc pa&longs;&longs;io, e&longs;&longs;e permutatim pro<lb/>portionale; &longs;cilicet quidnam &longs;it illud innominatum; in quo deinde commu<lb/>nicent lineæ, & numeri, vt inde habeant e&longs;&longs;e etiam permutatim propor<lb/>tionalia.</s>

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

<s>vt &longs;i duo triangula appo&longs;ita habeant angulos æquales, <expan abbr="angul&utilde;">angulum</expan> A, angulo D: <lb/>angulum B, angulo E. angulum C, angulo F. & præterea latera, quæ &longs;une <pb pagenum="67"/><figure id="fig31"/><lb/>circa angulos æquales, v. </s>

<s>cirea an<lb/>gulos A, & D, habeant proportiona<lb/>lia, hoc e&longs;t, vt latus A B, ad latus A C; <lb/>ita &longs;it latus D E, ad latus D F; & &longs;ic de <lb/>lateribus alijs circa reliquos angulos <lb/>æquales; erunt tunc prædicta duo tri<lb/>angula fimilia.</s>

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<s>Ibidem (<emph type="italics"/>Vt extrin&longs;ecos æquales e&longs;&longs;e<emph.end type="italics"/>) ide&longs;t extrin&longs;ecos angulos cuiu&longs;uis fi<lb/>guræ rectilineæ æquales e&longs;&longs;e quatuor rectis angulis: vide quæ &longs;crip&longs;imus de <lb/>hac re ad tex. </s>

<s>39. &longs;ecundi Po&longs;ter. quæ huic pariter loco &longs;atisfaciunt.</s>

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

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1. cap.

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

3. Harm. <lb/>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>

<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>

<s>videmus etiam, quod cannæ organo<lb/>rum maiores cum plus aeris moucant, & idcirco tardius, &longs;onum grauiorem <lb/>emittunt, quàm cannæ graciliores, quæ quia parum aeris cient, & ideo ce<lb/>lerius, &longs;onum acutum edunt. </s>

<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>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.

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<s>Eodem cap. (<emph type="italics"/>Rur&longs;um &longs;i eorundem; quæ &longs;unt &longs;ub eodem nomine diuer&longs;æ d ffe<lb/>rentiæ &longs;unt; vt coloris, qui e&longs;t in corporibus, & in melodijs<emph.end type="italics"/>) veteres Mu&longs;ici can<lb/>tilenas omnes ad tria genera reuocarunt, videlicet Enharmonicum, Chro<lb/>maticum, & Diatonicum; quæ diftinguebantur inuicem ex varia diui&longs;ione <lb/>interuallorum, ex quibus ip&longs;orum Monochordia conftabant: &longs;iue ex varijs <lb/>vocum interuallis, v. </s>

<s>quia in vno continebantur plures toni, vt in Diato<lb/>nico; in alio plures die&longs;es, vt in Enharmonico; in tertio verò plura &longs;emito<lb/>nia, vt in Chromatico: quæ vox deducitur à chroma, græco, quod latinis <lb/>e&longs;t color; quare Chromaticum latinè redditur coloratum. </s>

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

<s>Porrò exemplum mathematicum hic <lb/>allatum &longs;ic videtur explicandum: Conetur aliquis demon&longs;trare hanc pro<lb/>po&longs;itionem; &longs;i linea ducta fuerit æquidi&longs;tans lateri vnius plani trianguli, &longs;e<lb/>cabit & latera, & locum, ideft &longs;uperficiem illam triangularem &longs;imiliter, ide &longs;t <lb/><figure id="fig32"/><lb/>in eadem proportione, vt in triangulo A B C, <lb/>linea D E, parallela ba&longs;i B C, &longs;ecat latera A B, <lb/>& A C, in punctis D, & E, in eadem ratione, <lb/>in qua etiam fecat totum triangulum, ita vt <lb/>eadem &longs;it proportio lineæ A D, ad D B, & lineæ <lb/>A E, ad E C, quæ e&longs;t partium totalis trianguli <lb/>A B C, &longs;eilicet quæ e&longs;t partis A D E, ad partem <lb/>E D C, fiue ad partem D E B. quod con&longs;tat ex <lb/>&longs;ecunda 6. Elem. </s>

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

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

<s>Si quis <lb/>vellet hoc demon&longs;trare nondum præmi&longs;&longs;a defi<lb/>nitione eorum, quæ habent eandem rationem, &longs;iue nondum definitione al<lb/>lata quantitatum proportionalium, hic difficile id valeret o&longs;tendere: at ve<lb/>rò allata prins definitione quantitatum proportionalium facile demon&longs;tra<lb/>bit. </s>

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<s>Porrò Euclides definit. </s>

<s>&longs;eptima 5. paulo ali<lb/>ter definit quantitates proportionales e&longs;&longs;e illas, quæ eandem habent ratio<lb/>nem, v. </s>

<s>&longs;eptima 5. paulo ali<lb/>ter definit quantitates proportionales e&longs;&longs;e illas, quæ eandem habent ratio<lb/>nem, v. g. </s>

<s>&longs;i &longs;it, vt prima ad &longs;ecundam, ita tertia ad quartam. </s>

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

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

1. Po&longs;ter. tex. 23. <expan abbr="&longs;ol&utilde;modo">&longs;olummodo</expan> id hoc loco no<lb/>tandum per p&longs;eudographiam intelligere, vt apertè etiam inferius explicat, <lb/>Geometricam demon&longs;trationem fallacem, eò quod demon&longs;trationes geo<lb/>metricæ fiant adhibitis de&longs;criptionibus, &longs;eu figurationibus: p&longs;endographia <lb/>autem latinè idem e&longs;t, ac fal&longs;a de&longs;criptio; quemadmodum è contrariò, &longs;i<lb/>cuti &longs;upra in Topicis, & alibi ob&longs;eruaui, per de&longs;cribere intelligit geometri<lb/>cè demon&longs;trare, & per de&longs;criptiones intelligit demon&longs;trationes geometri<lb/>cas. </s>

<s>23. <expan abbr="&longs;ol&utilde;modo">&longs;olummodo</expan> id hoc loco no<lb/>tandum per p&longs;eudographiam intelligere, vt apertè etiam inferius explicat, <lb/>Geometricam demon&longs;trationem fallacem, eò quod demon&longs;trationes geo<lb/>metricæ fiant adhibitis de&longs;criptionibus, &longs;eu figurationibus: p&longs;endographia <lb/>autem latinè idem e&longs;t, ac fal&longs;a de&longs;criptio; quemadmodum è contrariò, &longs;i<lb/>cuti &longs;upra in Topicis, & alibi ob&longs;eruaui, per de&longs;cribere intelligit geometri<lb/>cè demon&longs;trare, & per de&longs;criptiones intelligit demon&longs;trationes geometri<lb/>cas. </s>

<s>Qua ratione item Hippocrates ex ijs, quæ &longs;ub arte Geometriæ &longs;unt, <lb/>procederet ibi dictum e&longs;t, propter quod non e&longs;t contentio&longs;a, quamuis fallax <lb/>ip&longs;ius demon&longs;tratio: appellat enim Ari&longs;t illas demon&longs;trationes contentio<lb/>&longs;as, quæ non procedunt ex proprijs illius &longs;cientiæ, in qua fiunt, &longs;ed ex com<lb/>munibus alijs &longs;cientijs: captio&longs;as verò, & &longs;ophi&longs;ticas, quæ ex proprijs &longs;cien<lb/>tiæ, in qua fiunt, decipiunt. </s>

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<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.

10. primi Elench. </s>

10. primi Elench. 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>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. cap.

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<s>Tex. 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>

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

<s>1. cap.

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

<s>&longs;it vnitas, & &longs;uo or<lb/>dine &longs;equantur impares, vt in &longs;equenti hac &longs;erie apparet 1. 3. 5. 7. 9. & c. <lb/><figure id="fig37"/><lb/>&longs;i igitur vnitati addatur ternarius in Gnomo<lb/>nis modum, vt vides in prima figura, produ<lb/>cetur quaternarius numerus, qui e&longs;t numerus <lb/>quadratus (quid &longs;it quadratus numerus expli<lb/>caui in Logicis tex. 9. primi Po&longs;ter.) etfi huic <lb/>quaternario addatur &longs;equens impar, qui e&longs;t <lb/>quinarius in modum Gnomonis, vt in &longs;ecund<gap/><lb/>figura, &longs;it numerus nouenarius, qui pariter e&longs;t quadratus. </s>

<s>9. primi Po&longs;ter.) etfi huic <lb/>quaternario addatur &longs;equens impar, qui e&longs;t <lb/>quinarius in modum Gnomonis, vt in &longs;ecund<gap/><lb/>figura, &longs;it numerus nouenarius, qui pariter e&longs;t quadratus. </s>

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

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

<s>cui &longs;i &longs;equens addatur par, &longs;cilicet &longs;enarius, <lb/>fiet iterum noua numeri forma, v. </s>

<s>cui &longs;i &longs;equens addatur par, &longs;cilicet &longs;enarius, <lb/>fiet iterum noua numeri forma, v. g. </s>

<s>dodecago<lb/>nus, vt in tertia figura. </s>

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<s>Tex. 31. <emph type="italics"/>(Vtuntur etiam Mathematici infinito)<emph.end type="italics"/> <expan abbr="aliquãdo">aliquando</expan> Mathematici du<lb/>cunt lineas quantumuis longas, &longs;eu indefinitæ longitudinis, quas etiam in<lb/>finitas appellant: & hoc modo vtuntur infinito, vt infra tex. </s>

<s>71. ip&longs;e Ari&longs;t. <lb/>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.

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<s>Tex. 33. <emph type="italics"/>(Vt &longs;i quis minimam qu&abreve;dam e&longs;&longs;e dicat magnitudinem, hic enim <lb/>minimum introducens, maxima <expan abbr="vbiq;">vbique</expan> amoueret mathematicor&ubreve;)<emph.end type="italics"/> ide&longs;t, <lb/>&longs;i quis, vt Democritus po&longs;uerit in magnitudinibus e&longs;&longs;e minima, <lb/>&longs;eu indiui&longs;ibilia, ex quibus entia mathematica componerentur, <lb/>hic euerteret maxima mathematicorum, ide&longs;t maxime ip&longs;orum demon&longs;tra<lb/>tiones, atque etiam effata euerterentur: v. </s>

<s>10. primi Elem.

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

<s>&longs;i in triangulo pag. </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.

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. g. </s>

<s>&longs;i &longs;upponamus li<lb/>neas e&longs;&longs;e compo&longs;itas ex indiui&longs;ibilibus, con&longs;ectarium erit diametrum e&longs;&longs;e <lb/>commen&longs;urabilem co&longs;tæ, quia indiui&longs;ibile illud, ex quo vtraque linea con<lb/>&longs;tat, erit <expan abbr="vtriu&longs;q;">vtriu&longs;que</expan> men&longs;ura communis.</s>

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

<s>diui&longs;io enim in plana non perinde e&longs;&longs;icitur, vt qui&longs;piam <expan abbr="diuid&etilde;s">diuidens</expan> in par<lb/>tes diuidat totum, &longs;ed vt in &longs;pecie diuer&longs;a: patct igitur &longs;phæram e&longs;&longs;e &longs;olidarum <lb/>primam)<emph.end type="italics"/> qui &longs;olida diuidunt in plana, ca diuidunt <expan abbr="&longs;ec&utilde;dum">&longs;ecundum</expan> numerum &longs;uper<lb/>&longs;icierum, quibus ambiuntur, v. </s>

<s>diuidunt cubum in &longs;ex &longs;uperficies, quia <lb/>cubus &longs;ex quadratis planis &longs;uperficiebus continetur: qua ratione nequcunt <pb pagenum="78"/>&longs;ohæram in plana vlla re&longs;oluere, <expan abbr="neq;">neque</expan> in alias plures &longs;uperficies, quia &longs;phæ<lb/>ra ambitur vnica tantum &longs;uperficie &longs;phærica. </s>

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<s>&longs;it igitur in præcedenti figura A, centrum mundi, <lb/>ex quo educantur duæ rectæ lineæ æquales A B, A C, quæ deinde alia recta <lb/>B C, coniungantur. </s>

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

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

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<s>Tex. <emph type="italics"/>(Luna autem o&longs;tenditur per ea, quæ circa vi&longs;um, quod &longs;phærica &longs;it: non <lb/>enim <expan abbr="vtiq;">vtique</expan> fieret accre&longs;cens, & decre&longs;cens, plurim&ubreve; quidem alter a ex parte curua, <lb/>altera concaua, aut <expan abbr="vtrmq;">vtrmque</expan> curua, &longs;emel autem bipartita)<emph.end type="italics"/> ait per ea, quæ circa <lb/>vi&longs;um, ide&longs;t per opticem probari Lunam e&longs;&longs;e &longs;phæricam: &longs;ed con&longs;ule, quæ <lb/>primo Po&longs;ter. tex. </s>

<s>3. de hac re &longs;crip&longs;i, & plenam etiam huius loci intelligen<lb/>tiam a&longs;&longs;equeris, præ&longs;ertim &longs;i experimentum ibi traditum inieris.</s>

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

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

<s>&longs;i lapis illuc de&longs;cenderet, non quie&longs;ceret &longs;tatim ac <lb/>prima ip&longs;ius pars ad mundi centrum pertingeret, &longs;ed reliquæ ip&longs;ius partes <lb/>adhuc grauitarent, <expan abbr="&longs;ic&qacute;">&longs;icque</expan>; vlterius primam partem impellerent, donec lapi<lb/>áis medium, mundi medio congrueret: quo facto lapis quie&longs;ceret. </s>

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<s>quam<lb/>uis enim &longs;ingulis men&longs;ibus Luna terminetur modo linea concaua, vt quan<lb/>do noua e&longs;t; modo recta, vt quando diuidua e&longs;t: modo vtrinque curua, vt <lb/>cum à diuiduæad plenilunium tendit. </s>

<s>quod fu&longs;ius primo Po&longs;ter. tex. </s>

<s>quod fu&longs;ius primo Po&longs;ter. tex. 30. ex<lb/>po&longs;ui. </s>

<s>30. ex<lb/>po&longs;ui. </s>

<s>in eclyp&longs;ibus tamen &longs;emper curuam habet lineam illam, quæ partem <lb/>ec'yp&longs;atam de&longs;init; vt paulo po&longs;t explicabo. </s>

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

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

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

<s>Clauium in &longs;phæra.</s>

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

<s>&longs;i autem terra e&longs;&longs;et <lb/>maior, v. </s>

<s>&longs;i autem terra e&longs;&longs;et <lb/>maior, v. g. </s>

<s>circulus medius, tunc facta maio<lb/>ri mutatione ex D, in E, fieret eadem a&longs;trorum <lb/>variatio ex B, in C; &longs;ed cum nos experiamur <lb/>&longs;ieri magnam a&longs;trorum mutationem, ex parua <lb/>locorum intercapedine, &longs;ignum e&longs;t magnope<lb/>re mutari horizontem, ac proinde terram e&longs;&longs;e <lb/>rotundam, ac re&longs;pectu c&oelig;le&longs;tium corporum <lb/>paruam. </s>

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

<s>has ea&longs;dem rationes fu&longs;ius explicatas repe<lb/>ries apud P. </s>

<s>has ea&longs;dem rationes fu&longs;ius explicatas repe<lb/>ries apud P. Clauium in &longs;phæra.</s>

<s>Clauium in &longs;phæra.</s>

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<s>Tex. 112. <emph type="italics"/>(Matbematicorum etiam, qui circum ferentiæ magnitudinemratio<lb/>cinari tentant, ad 400. dicunt ftadiorum millia, &c.)<emph.end type="italics"/> quam &longs;ubtilibus rationi<lb/>bus inue&longs;tigauerint A&longs;tronomi quantitatem terræ, optimè, ac dilucidè ex<lb/>ponitur à P. Clauio in &longs;phæra: quem &longs;i libet, con&longs;ule, ne inani labore opu<lb/>&longs;culum i&longs;tud exere&longs;cat.</s>

<s>Clauio in &longs;phæra: quem &longs;i libet, con&longs;ule, ne inani labore opu<lb/>&longs;culum i&longs;tud exere&longs;cat.</s>

<s><emph type="italics"/>Ex Tertio de C&oelig;lo.<emph.end type="italics"/></s>

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<s>aduerte tamen hanc &longs;phæræ compo&longs;itionem nullo modo <lb/>habere partes actu, cum &longs;phæra &longs;it vnica &longs;implici &longs;uperficie terminata; &longs;ed <lb/>quæ tantum &longs;int à prædictis imaginatis circulis de&longs;ignatæ: at verò aliæ fi<lb/>guræ, quæ pluribus planis terminantur, vt cubus, octaedrum, & &longs;imilia, quæ <lb/>Ari&longs;t.

vocat rectiliheas, quia terminantur &longs;uperficiebus rectilineis actu di<lb/>&longs;tinctis ab inuicem ex natura &longs;ua, non per no&longs;tram de&longs;ignationem, ideò re<lb/>ctè dicuntur componi ex pyramidibus, v. </s>

<s>dicimus cubum componi ex &longs;ex <lb/>pyramidibus, quia cum habeat &longs;ex ba&longs;es, cogitamus &longs;upra <expan abbr="vnamquamq;">vnamquamque</expan> il<lb/>larum &longs;ingulas pyramides erigi, quarum omnium vertices ad idem punctum <lb/>medium intra cubum imaginatum coeant. </s>

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<s>huiu&longs;modi &longs;unt triangula æquilatera (de his enim <lb/>intelligendus e&longs;t textus) quadrata, & hexagona, &longs;iue &longs;exilatera regularia; <lb/><figure id="fig44"/><lb/>nam &longs;ex triangula æquilatera &longs;imul iuncta in plano paui<lb/>re po&longs;&longs;unt, vt patet in figura præ&longs;enti; ratio huius e&longs;t, <lb/>quia omnes anguli circa idem punctum (y. </s>

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

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

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

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<s>præter has tres <lb/><figure id="fig46"/><lb/>figuras, nulla alia reperitur, quæ i&longs;tud efficere pol<lb/>&longs;it. </s>

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

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

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

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>

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

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<s><expan abbr="Notã">Notam</expan> dum præterea, quod per pyramidem debemus intelligere pyramidem <lb/>regularem, quæ dicitur etiam Tetraedrum, <expan abbr="e&longs;t&qacute;">e&longs;tque</expan>; &longs;ecunda inter <expan abbr="quinq;">quinque</expan> cor<lb/>pora regularia rectilinea, quæ alias Platonica corpora dicuntur. </s>

<s><expan abbr="eorum&qacute;">eorumque</expan>; <lb/>defraitiones &longs;unt in 11. Elem. </s>

<s><expan abbr="eorum&qacute;">eorumque</expan>; <lb/>defraitiones &longs;unt in 11. Elem. </s>

<s>Tetraedrum autem &longs;ic definitur, e&longs;t figura &longs;o<lb/>lida &longs;ub quatuor triangulis æquilateris, <expan abbr="atq;">atque</expan> inuicem æqualious contenta: <lb/>de hac inquam e&longs;t &longs;ermo. </s>

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<s>Verum quidem e&longs;t octo cubos &longs;imul adactos &longs;oliditatem conficere, <lb/>quia ad id nece&longs;&longs;arij &longs;unt octo anguli &longs;olidi, quos octo cubi præbere po&longs;&longs;unt, <lb/>cum anguli ip&longs;orum &longs;int recti, & &longs;olidi. </s>

<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.

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.

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>

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nullo <lb/>modo po&longs;&longs;e verificari de repletione &longs;olidi per plura Tetraedra, & omnes <lb/>tamen commentatores auctoritate Ari&longs;t.

decepti pro ip&longs;o &longs;tarent, dubius, <lb/><expan abbr="anceps&qacute;">ancepsque</expan>; diu hæ&longs;i, neque quid quam mea Minerua a&longs;&longs;erere au&longs;us &longs;um, &longs;ed P. <lb/>Clauium præceptorem meum per literas con&longs;ului, qui in hunc modum hu<lb/>mani&longs;&longs;imè re&longs;pondit; cubus implet locum quater &longs;umptus, ad idem enim <lb/>punctum quatuor cubi coaptantur: &longs;ic etiam pyramis &longs;exies &longs;umpta, &longs;eu &longs;ex <lb/>pyramides ad idem punctum iunctæratione &longs;ub&longs;tantium <expan abbr="triangulor&utilde;">triangulorum</expan> æqui<lb/>laterorum. </s>

<s>Clauium præceptorem meum per literas con&longs;ului, qui in hunc modum hu<lb/>mani&longs;&longs;imè re&longs;pondit; cubus implet locum quater &longs;umptus, ad idem enim <lb/>punctum quatuor cubi coaptantur: &longs;ic etiam pyramis &longs;exies &longs;umpta, &longs;eu &longs;ex <lb/>pyramides ad idem punctum iunctæratione &longs;ub&longs;tantium <expan abbr="triangulor&utilde;">triangulorum</expan> æqui<lb/>laterorum. </s>

<s>Verum hac ratione non videntur implere locum lolidum, fa<lb/>teor; &longs;ed tamen Ari&longs;t.

in co tex. </s>

in co tex. non loquitur de repletione loci &longs;olidi. </s>

<s>non loquitur de repletione loci &longs;olidi. </s>

<s>hæc <lb/>ip&longs;e. </s>

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

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

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

hoc loco <lb/>erroris notari, dum a&longs;&longs;eruit duodecim pyramides replere <expan abbr="loc&utilde;">locum</expan> corporeum, <lb/>ide&longs;t, vt exponit ip&longs;e, &longs;ex pyramides &longs;uper hexagonam aliquam figuram <lb/>&longs;uperficialem, & &longs;ex &longs;ub eadem, id præ&longs;tarent, cum potius maius vacuum <lb/>remaneat ad quamlibet partium &longs;upra, & infra, quam plenum. </s>

<s>hæc ip&longs;e. </s>

<s>&longs;ed <lb/>expo&longs;itio i&longs;ta puerili, ne dum Ari&longs;t.

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

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>

<s>Simpl. etiam hoc loco &longs;atisfacit.</s>

<s>etiam hoc loco &longs;atisfacit.</s>

<s><emph type="italics"/>Ex Quarto de C&oelig;lo.<emph.end type="italics"/></s>

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<s>boc autem vtrum accidit ad ip&longs;um tcrræ medium, <lb/>an ad vniuer&longs;i, quoniam idem ip&longs;orum &longs;it, alius &longs;ermo e&longs;t)<emph.end type="italics"/> cum vellet <lb/><figure id="fig47"/><lb/>probare Ari&longs;toteles dari <expan abbr="p&utilde;ctum">punctum</expan> quoddam in m<gap/>dio <lb/>mundi, ad quod grauia de&longs;cendant, & concurrent: <lb/>& à quo leuia a&longs;c<gap/>ndat; vtitur, præter alias, etism <lb/>ratione aliqua ex parte mathematica; quæ e&longs;t huiu&longs;<lb/>modi. </s>

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

<s>&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.

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

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<s>Sol porrò, & pla<lb/>netæ, qui motibus proprijs hunc circulum peragunt, dicuntur moueri duo<lb/>bus motibus, & quidem contrarijs: quoniam dum Sol. </s>

<s>per Zodiacum <lb/>graditur motu proprio, interim etiam à primo mobili fertur ab ortu in oc<lb/>ca&longs;um: ex quibus duobus motibus fit vnus tantum Solis motus &longs;piralis, qui <lb/>mixtus e&longs;t, ide&longs;t, qui fit à duobus motoribus; vnde re vera Sol non mouetur <lb/>duobus motibus contrarijs re ip&longs;a di&longs;tinctis; hoc enim impo&longs;&longs;ibile e&longs;t: &longs;ed <lb/>motu mixto ex duobus, qui &longs;piralis e&longs;t, circa mundum de&longs;cribens &longs;piras ab <lb/>v<gap/>o tropico ad alterum: qui, vt dixi, cau&longs;atur à duobus motoribus, qui &longs;unt <lb/>Sol ip&longs;e, mouens &longs;e ip&longs;um per Zodiacum: & primum mobile mouens in&longs;u<lb/>per ip&longs;um Solem, & Zodiacum ab ortu in occa&longs;um circa mundum.</s><pb pagenum="89"/>

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

occurrunt, <expan abbr="pleriq;">plerique</expan> no&longs;træ ætatis con&longs;iderarent, qui nulla <lb/>ratione probari po&longs;&longs;e exi&longs;timant, Solem, v. </s>

occurrunt, <expan abbr="pleriq;">plerique</expan> no&longs;træ ætatis con&longs;iderarent, qui nulla <lb/>ratione probari po&longs;&longs;e exi&longs;timant, Solem, v. g. </s>

<s>terra e&longs;&longs;e centies &longs;exagies &longs;e<lb/>xies maiorem; &longs;ed c<gap/> iam, quod peius e&longs;t, negant e&longs;&longs;e maiorem; ad demon<lb/>&longs;trationes autem a&longs;tronomicas dicunt &longs;e exi&longs;timare eas e&longs;&longs;e fallaces; at que <lb/>impo&longs;libile e&longs;&longs;e nos res adeo à nobis di&longs;taptes &longs;ufficienter perue&longs;tigare: <lb/>quanto &longs;apientius, ac prudentius eorum Magi&longs;ter Ari&longs;t.

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<s>Idem proportionaliter imaginari debes de circulis, quos cometa tam vltra <lb/>Capricornum, quàm citra Cancrum delineat; nam eorum, qui &longs;unt vltra <lb/>Capricornum ad au&longs;trum minores adhuc &longs;ectiones &longs;upra horizontem exi<lb/>&longs;terent, quàm opus &longs;it ad cometen &longs;pectandum. </s>

<s><expan abbr="At&qacute;ue">Atque</expan> hæc cau&longs;a e&longs;t ex &longs;en<lb/>tentia Hippocr. </s>

<s><expan abbr="At&qacute;ue">Atque</expan> hæc cau&longs;a e&longs;t ex &longs;en<lb/>tentia Hippocr. cur in illa au&longs;trali plaga <expan abbr="n&utilde;quam">nunquam</expan> cometes effu<gap/>g<gap/>at. </s>

<s>cur in illa au&longs;trali plaga <expan abbr="n&utilde;quam">nunquam</expan> cometes effu<gap/>g<gap/>at. </s>

<s>è con<lb/>trario autem, quia ad boream &longs;ectiones illæ maximæ &longs;unt, <expan abbr="aptæ&qacute;">aptæque</expan>; ad refra<lb/>ctionem vi&longs;us no&longs;tri v&longs;que ad Solem, idcircò in hac mundi parte cometas <lb/>con&longs;picere &longs;olemus. </s>

<s>Reliqua Vicomercatus, <expan abbr="atq;">atque</expan> Alexand. </s>

<s>Reliqua Vicomercatus, <expan abbr="atq;">atque</expan> Alexand. optimè expli<lb/>cant, quos tu con&longs;ule, ne actum agatur.</s>

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

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

4. &longs;ummæ 2. lib.

1. Meteor. </s>

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

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

<s>In præ&longs;enti cap. </s>

<s>In præ&longs;enti cap. Ari&longs;t. &longs;uam de Cometis &longs;ententiam exponit: Come<lb/>tam nimirum infra Lunam in elementari mundo procreari, & ignitum <lb/>quoddam Meteoron, ex lenta, pingui, <expan abbr="&longs;icca&qacute;">&longs;iccaque</expan>; materia à terra in &longs;u<lb/>prem<gap/>m aeris regionem attracta, exi&longs;tere; <expan abbr="ibi&qacute;">ibique</expan>; rapti aeris calore, <lb/>vel elementi ignis (quod illic e&longs;&longs;e putat) vicinitate, vel etiam vi a&longs;trorum <lb/>incendi, <expan abbr="atq;">atque</expan> impelli. </s>

<s>Ari&longs;t. &longs;uam de Cometis &longs;ententiam exponit: Come<lb/>tam nimirum infra Lunam in elementari mundo procreari, & ignitum <lb/>quoddam Meteoron, ex lenta, pingui, <expan abbr="&longs;icca&qacute;">&longs;iccaque</expan>; materia à terra in &longs;u<lb/>prem<gap/>m aeris regionem attracta, exi&longs;tere; <expan abbr="ibi&qacute;">ibique</expan>; rapti aeris calore, <lb/>vel elementi ignis (quod illic e&longs;&longs;e putat) vicinitate, vel etiam vi a&longs;trorum <lb/>incendi, <expan abbr="atq;">atque</expan> impelli. </s>

<s>Han&cacute; porrò opinionem & &longs;i probabilibus tantum ra<lb/>tionibus confirmatam vulgò tamen <expan abbr="v&longs;q;">v&longs;que</expan> ad hanc diem receptam, cum fal<lb/>&longs;am e&longs;&longs;e a&longs;tronomi exi&longs;timent, non erit abs re rationes eas ex &longs;ecundo pro<lb/>gymn. </s>

<s>Tichonis volumine, de&longs;umptas hic breuiter referre, quibus a&longs;trono<lb/>mus ille eos &longs;upra Lunam in ætherea regione collocauit: quas quidem ra<lb/>tiones ille ex diuturnis ob&longs;eruationibus per exqui&longs;ita organa factis adinue<lb/>nit: ea&longs;que Mathematicis linearum, ac numerorum demon&longs;trationibus <lb/>explicauit.</s>

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

<s>Cardan. </s>

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

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

<s>quibus etiam ex antiquis Seneca annumeran<lb/>dus e&longs;t. </s>

<s>pr&ecedil;dicti autem recentiores omnes varijs demon&longs;trationibus ex ac<lb/>curata ob&longs;eruatione erutis illud certò certius con&longs;irmare contendunt: <expan abbr="id&qacute;">idque</expan>; <lb/>non in vno dumtaxat, &longs;ed in <expan abbr="quinq;">quinque</expan> cometis; quorum demon&longs;trationes apud <lb/>Tychonem partim in progymn. </s>

<s>partim in epi&longs;t. </s>

<s>partim in epi&longs;t. fu&longs;ius explicatas reperies.</s>

<s>fu&longs;ius explicatas reperies.</s>

<s>2. Quarum poti&longs;&longs;ima illa e&longs;t, quæ ex parallaxi, &longs;eu a&longs;pectus diuer&longs;itate <lb/>de&longs;umitur, certi&longs;&longs;imum enim e&longs;t lumen illud e&longs;&longs;e altero &longs;ublimius, quod mi<lb/>norem exhibet parallaxim: expertos autem &longs;e e&longs;&longs;e hi omnes, affirmant ho<lb/>&longs;ce quinque cometas multò minorem pati parallaxim, quam Lunam; imò <lb/>quempiam minorem, quàm Sol ip&longs;e patiatur, quo po&longs;ito manife&longs;tè conuin<lb/>ceretur eos omnes &longs;upra Lunam in ætherea regione efful&longs;i&longs;&longs;e.</s>

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<s>Secunda anno 1600. in Cygno, quæ nec dum <lb/>extinguitur. </s>

<s>Tertia anno 1604. inter Sagittarij &longs;tellas vi&longs;a e&longs;t, de quibus vi<lb/>de P. </s>

<s>Tertia anno 1604. inter Sagittarij &longs;tellas vi&longs;a e&longs;t, de quibus vi<lb/>de P. Clauium in &longs;phæra breuiter de illis tractantem: aut &longs;i mauis, & vacat, <lb/>vide quoad primam primum volumen progymna&longs;matum Tychonis Brahe, <lb/>vbi etiam aliorum a&longs;tronomorum de eadem certi&longs;&longs;imas commentationes <lb/>reperies. </s>

<s>Clauium in &longs;phæra breuiter de illis tractantem: aut &longs;i mauis, & vacat, <lb/>vide quoad primam primum volumen progymna&longs;matum Tychonis Brahe, <lb/>vbi etiam aliorum a&longs;tronomorum de eadem certi&longs;&longs;imas commentationes <lb/>reperies. </s>

<s>con&longs;ule etiam de reliquis duabus Ioannis Kepleri Cæ&longs;areæ Maie<lb/>&longs;tatis Mathematici commentaria; & coactus libenter fateberis noui ali<lb/>quid c&oelig;lo aduenire po&longs;&longs;e.</s>

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<s>Summæ 2. cap.

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

6. Sunt qui velint Ari&longs;t. 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>

<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>

<s>& quemadmodum halo, &longs;iue area omnibus <expan abbr="vndecunq;">vndecunque</expan> <lb/>a&longs;picientibus &longs;emper videntur in eodem c&oelig;li loco, hoc e&longs;t è regione Solis, <lb/>vel Lunæ; &longs;imiliter Galaxia in aere omnibus <expan abbr="vndecunq;">vndecunque</expan> intuentibus appa<lb/>reat in eadem c&oelig;li parte, ide&longs;t ex aduersò eorumdem &longs;yderum, quæ c&oelig;le<lb/>&longs;tem lacteam viam conficiunt. </s>

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<s>ab hoc autem ab&longs;cinditur Tanais pars exi&longs;tens in Meotidem paludem fluit au<lb/>tem, & Indus ex ip&longs;o, omnium fluuiorum fluxio maxima)<emph.end type="italics"/> hæc omnia &longs;unt fal&longs;a, <lb/>& impo&longs;&longs;ibilia; nam cum Bactrus Bactrianam regionem irriget, quæ e&longs;t vl<lb/>tra Per&longs;iam, Choa&longs;pes verò Per&longs;iam ip&longs;am, Indus <expan abbr="deniq;">denique</expan> in India oriatur: <lb/>quì fieri pote&longs;t, vt in Regionibus adeò inuicem di&longs;&longs;itis orti fluuij ab eodem <lb/><expan abbr="quoq;">quoque</expan> Paropame&longs;&longs;o monte ortum ducant. </s>

<s>nec minus fal&longs;um e&longs;t illud de Ta<lb/>nai, quod &longs;it qua&longs;i ip&longs;ius Araxis ramus quidam, Tanais enim ex Riphæis <lb/><expan abbr="mõtibus">montibus</expan> Scythiæ delabitur in Meotidem paludem longè longius ab Araxi. <lb/></s>

<s><expan abbr="eum&qacute;">eumque</expan>; terminum inter Europam, & A&longs;iam Geographi con&longs;tituunt, vnde <lb/>Diony&longs;ius Afer &longs;ic cecinit:</s>

<s><emph type="italics"/>Europam, <expan abbr="atq;">atque</expan> A&longs;iam Tanais di&longs;terminat amnis.<emph.end type="italics"/></s>

<s>verùm huiu&longs;modi errata Ari&longs;t. </s>

<s>verùm huiu&longs;modi errata Ari&longs;t. <expan abbr="atq;">atque</expan> adeò Geographis illius temporis con<lb/>donanda &longs;unt, cum nondum Geographia &longs;atis exculta e&longs;&longs;et.</s>

<s><expan abbr="atq;">atque</expan> adeò Geographis illius temporis con<lb/>donanda &longs;unt, cum nondum Geographia &longs;atis exculta e&longs;&longs;et.</s>

<s><emph type="italics"/>De altitudine montis Cauca&longs;i.<emph.end type="italics"/></s>

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<s>Eod. cap. <emph type="italics"/>(Cauca&longs;us autem maximus mons e&longs;t eorum qui ad or<gap/><expan abbr="&etilde;tem">entem</expan> æ&longs;tiua<lb/>lem, & multitudine, & altitudine &longs;igna autem altitudinis quidem, quia <lb/>videtur & à vocatis Profundis, & à nauigantibus in Stagnum in&longs;uper il<lb/>lu&longs;trantur à Sole ip&longs;ius &longs;ummitates, v&longs;que ad tertiam partem nocte, & ab <lb/>aurora, & iterum a ve&longs;pera)<emph.end type="italics"/> Cauca&longs;us mons &longs;itus e&longs;t inter mare Euxinum, & <lb/>Ca&longs;pium, &longs;upra Cholchidem, & Iberiam regiones, vbi polus eleuatur 47. <lb/>circiter grad. </s>

<s>cap. <emph type="italics"/>(Cauca&longs;us autem maximus mons e&longs;t eorum qui ad or<gap/><expan abbr="&etilde;tem">entem</expan> æ&longs;tiua<lb/>lem, & multitudine, & altitudine &longs;igna autem altitudinis quidem, quia <lb/>videtur & à vocatis Profundis, & à nauigantibus in Stagnum in&longs;uper il<lb/>lu&longs;trantur à Sole ip&longs;ius &longs;ummitates, v&longs;que ad tertiam partem nocte, & ab <lb/>aurora, & iterum a ve&longs;pera)<emph.end type="italics"/> Cauca&longs;us mons &longs;itus e&longs;t inter mare Euxinum, & <lb/>Ca&longs;pium, &longs;upra Cholchidem, & Iberiam regiones, vbi polus eleuatur 47. <lb/>circiter grad. </s>

<s>ac re&longs;pectu Græciæ, & maris Euxini vergit ad eam mundi pla<lb/>gam, vnde illis æ&longs;tiuo tempore Sol oritur. </s>

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<s>Alter huius <lb/>loci &longs;en&longs;us e&longs;t, ait Mazonius, vt non de tertia montis parte, &longs;ed de tertia <lb/>noctis portione loquatur, ita vt manè. </s>

<s>initio tertiæ, & vltimæ noctis <lb/>parte, cacumen Cauca&longs;i illuminetur. </s>

<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. 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>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.

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>

<s>&longs;iue ergo dicamus id contingere initio Crepu&longs;culi, <lb/>&longs;iue initio tertiæ partis noctis, erit idem tempus, trium &longs;cilicet horarum. <lb/></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. &longs;iue ergo dicamus id contingere initio Crepu&longs;culi, <lb/>&longs;iue initio tertiæ partis noctis, erit idem tempus, trium &longs;cilicet horarum. <lb/></s>

<s>&longs;i ergo, inquit Mazonius, &longs;equamur priorem declarationem, nece&longs;&longs;arium <lb/>e&longs;t dicere, quod ea tertia pars montis, quæ initio auroræ Solis lumine per<lb/>funditur, &longs;it ea montis altitudo, qua ip&longs;e exuperat illam aeris regionem, <lb/>vnde Crepu&longs;culum incipit apparere. </s>

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

<s>&longs;i enim in principio Crepu&longs;culi v. </s>

<s>&longs;i enim in principio Crepu&longs;culi v. g. </s>

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

<s>de Crepu&longs;culis milliaribus 52. à terra recedit, ergo duæ tertiæ montis par<lb/>tes, quæ Solem initio auroræ non vident, &longs;unt 52. milliaria ad perpendicu<lb/>lum, & tertia alia pars illuminata e&longs;t ad perpendiculum 26. milliaria: ita <lb/>vt totius montis altitudo perpendicularis &longs;it 78. mill. </s>

<s>&longs;ed papè in quos acu<pb pagenum="100"/>leos imprudens me conieci? </s>

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

<s>Sol A B C, qui initio Crepu&longs;culi infra horizontem O P, depri<lb/>mitur gr. </s>

<s>Sol A B C, qui initio Crepu&longs;culi infra horizontem O P, depri<lb/>mitur gr. 18. vti ab A&longs;tronomis compertum e&longs;t, hoc e&longs;t, arcum D P, e&longs;&longs;e <lb/>grad. </s>

<s>18. vti ab A&longs;tronomis compertum e&longs;t, hoc e&longs;t, arcum D P, e&longs;&longs;e <lb/>grad. </s>

<s>18. radius autem C I K, tangens terram, incipit illuminare halitus, <lb/>qui &longs;unt ad K, in extremo horizonte &longs;en&longs;ibili F K. quique po&longs;&longs;unt videri ab <lb/>oculo in F, ide&longs;t ab huius horizontis habitatore. </s>

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<s>quibus ab ip&longs;is demon&longs;tra<lb/>tis, &longs;i H F, terræ &longs;emidiameter, quæ continet milliar. </s>

<s>3579. ponatur &longs;inus <lb/>totus 100000. & latus F K, ponatur tangens anguli ad H, quem pr&ecedil;dicti au<lb/>thores probant e&longs;&longs;e grad. </s>

<s>3579. ponatur &longs;inus <lb/>totus 100000. & latus F K, ponatur tangens anguli ad H, quem pr&ecedil;dicti au<lb/>thores probant e&longs;&longs;e grad. 8. 54. erit F K, tangens partium 15659. fiat igi<lb/>tur per 2. pro. </s>

<s>8. 54. erit F K, tangens partium 15659. fiat igi<lb/>tur per 2. pro. </s>

<s>trjang. </s>

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<s>iam ip&longs;ius latus F K, inno<lb/>tuit, angulus verò ad F, e&longs;t rectus; at angulus ad K, &longs;ic manife&longs;tabitur; in <lb/>quadrilatero F K I H, quatuor anguli &longs;unt æquales 4. rectis ex 32. primi. </s>

<s>duo <lb/>autem F, & I, &longs;unt recti ex 18. 3. ergo reliqui duo H, & K, æquales erunt duo<lb/>bus rectis, quorum alter H, e&longs;t gr. </s>

<s>duo <lb/>autem F, & I, &longs;unt recti ex 18. 3. ergo reliqui duo H, & K, æquales erunt duo<lb/>bus rectis, quorum alter H, e&longs;t gr. 17. 48. vt præditi Mathematici <expan abbr="o&longs;t&etilde;dunt">o&longs;tendunt</expan>, <lb/>reliquus igitur ad K, erit gr. </s>

<s>17. 48. vt præditi Mathematici <expan abbr="o&longs;t&etilde;dunt">o&longs;tendunt</expan>, <lb/>reliquus igitur ad K, erit gr. </s>

<s>162. 12. vt compleat duos rectos. </s>

<s>qui &longs;i detra<lb/>hatur à duobus rectis, qui &longs;unt deinceps ad lineam F K, reliquus angulus <lb/>F K V, erit gr. </s>

<s>qui &longs;i detra<lb/>hatur à duobus rectis, qui &longs;unt deinceps ad lineam F K, reliquus angulus <lb/>F K V, erit gr. 17. 48. &longs;i ergo latus F K, notum ponatur &longs;inus totus 100000. <lb/>latus verò F V, tangens anguli noti, erit ip&longs;a 32100. fiat igitur,<lb/><arrow.to.target n="table5"/></s>

<s>17. 48. &longs;i ergo latus F K, notum ponatur &longs;inus totus 100000. <lb/>latus verò F V, tangens anguli noti, erit ip&longs;a 32100. fiat igitur,<lb/><arrow.to.target n="table5"/></s><table><table.target id="table5"/><row><cell>vt F K, &longs;inus totus,</cell><cell>ad milliar.</cell><cell>ita F V, tangens</cell><cell>ad milliar.</cell></row><row><cell>100000.</cell><cell>560.</cell><cell>32100.</cell><cell>180.</cell></row></table>

<table><table.target id="table5"/><row><cell>vt F K, &longs;inus totus,</cell><cell>ad milliar.</cell><cell>ita F V, tangens</cell><cell>ad milliar.</cell></row><row><cell>100000.</cell><cell>560.</cell><cell>32100.</cell><cell>180.</cell></row></table>

<s><expan abbr="inueniemus&qacute;">inueniemusque</expan>; latus F V, continere milliar. </s>

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<s>vnde patet Mazonium erra&longs;&longs;e in colligenda hoc modo Cauca&longs;i al<lb/>titudine, ex prima Crepu&longs;culi illuminatione in horizonte Cauca&longs;i facta, <lb/>cum ex præmi&longs;&longs;o calculo con&longs;tet partem montis F V, totam tunc temporis <lb/>e&longs;&longs;e tenebro&longs;am, quamuis &longs;uperet multò regionem vaporum, contrà quàm <lb/>ip&longs;e putabat, &longs;uperat enim eam milliar. </s>

<s>128. quare duæ tertiæ montis erunt <lb/>non 52. mill. </s>

<s>128. quare duæ tertiæ montis erunt <lb/>non 52. mill. vt ip&longs;e ait, &longs;ed mill. </s>

<s>vt ip&longs;e ait, &longs;ed mill. </s>

<s>180. & proinde tota altitudo erit mill. </s>

<s>180. & proinde tota altitudo erit mill. 270. <lb/>quod &longs;anè ridiculum e&longs;t, cum nullius montis altitudo &longs;e&longs;quimilliare tran<lb/>&longs;cendat. </s>

<s>270. <lb/>quod &longs;anè ridiculum e&longs;t, cum nullius montis altitudo &longs;e&longs;quimilliare tran<lb/>&longs;cendat. </s>

<s>Quod &longs;i &longs;equamur alteram expo&longs;itionem, vt nimirum Ari&longs;tor. </s>

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

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

<s>Si v<gap/>rò dixerimus Ari&longs;t.

intelligere hæc omnia, non re&longs;pectu horizontis <lb/>Cauca&longs;i, &longs;ed alterius, cuius habitator in principio &longs;ui Crepu&longs;culi tertiam <lb/>Cauca&longs;i partem iam illu&longs;tratam videat, vti accideret &longs;i Cauca&longs;us &longs;tatuere<lb/>tur in L K, vbi incipit Crepu&longs;culum habitanti in F. tunc e&longs;&longs;et altitudo tanta, <lb/>quanta colligit Mazonius, &longs;i tamen Ari&longs;t.

intelligatur de tertia montis par<lb/>te; e&longs;t enim L K, altitudo habituum 52. mill. </s>

intelligatur de tertia montis par<lb/>te; e&longs;t enim L K, altitudo habituum 52. mill. & duæ tertiæ montis, quare <lb/>totus mons erit 78. &longs;i autem intelligatur circa tertiam noctis partem, mon<lb/>tis apicem illuminatum videri ab habitatore F, &longs;ic altitudo eins erit tan<lb/>tummodo 52. mill. </s>

<s>& duæ tertiæ montis, quare <lb/>totus mons erit 78. &longs;i autem intelligatur circa tertiam noctis partem, mon<lb/>tis apicem illuminatum videri ab habitatore F, &longs;ic altitudo eins erit tan<lb/>tummodo 52. mill. </s>

<s>quæ tamen adhuc omnem veritatem nimium &longs;uperat. <lb/></s>

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<s>Tandem monendus mihi Lector e&longs;t, in demon&longs;tratione Magini, quæ e&longs;t <lb/>apud Mazonium &longs;ect. </s>

<s>4. citati operis; a&longs;&longs;umi radium Solis tangentem terræ <lb/>globum, qui cum horizonte faciat angulum gr. </s>

<s>4. citati operis; a&longs;&longs;umi radium Solis tangentem terræ <lb/>globum, qui cum horizonte faciat angulum gr. 18. quod fal&longs;um e&longs;t, &longs;olus <lb/>enim radius centralis, qui à centro Solis ad centrum terræ ducitur talem <lb/>facit angulum, <expan abbr="atq;">atque</expan> hac de cau&longs;a ip&longs;e colligit altitudinem no&longs;tra maiorem; <lb/>no&longs;tra e&longs;t 270. mill. </s>

<s>18. quod fal&longs;um e&longs;t, &longs;olus <lb/>enim radius centralis, qui à centro Solis ad centrum terræ ducitur talem <lb/>facit angulum, <expan abbr="atq;">atque</expan> hac de cau&longs;a ip&longs;e colligit altitudinem no&longs;tra maiorem; <lb/>no&longs;tra e&longs;t 270. mill. </s>

<s>&longs;ua verò 276. vbi etiam, &longs;icut & nos a&longs;&longs;umit horizon<lb/>tem Cauca&longs;i.</s>

<s>Aduertendum tandem Mazonium admodum aduer&longs;antia loquutum e&longs;&longs;e, <lb/>&longs;ect. </s>

<s>enim 3. demon&longs;tratinè concludit altitudinem 76. mill. </s>

<s>enim 3. demon&longs;tratinè concludit altitudinem 76. mill. &longs;ect. </s>

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

<s>verò 4. &longs;i<lb/>mul <expan abbr="c&utilde;">cum</expan> Magino demon&longs;tratiuè pariter colligit altitudinem eiu&longs;dem 276. m. <lb/></s>

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3. <lb/>cap.

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>

<s>mons no&longs;ter Apenninus erat iugum, &longs;iue dor&longs;um quoddam terræ eminens <lb/>quidem, &longs;ed nullis vallibus in tot particulares colles, aut montes di&longs;&longs;ectum; <lb/>&longs;ed po&longs;tquam flumina à &longs;ummitate ip&longs;ius deor&longs;um fluere c&oelig;perunt; paula<lb/>tim corrodentes humum in dies magis, ac magis effecerunt valles, <expan abbr="atq;">atque</expan> hac <lb/>ratione in colles, <expan abbr="montes&qacute;">montesque</expan>; plurimes totus Apenninus diui&longs;us e&longs;t. </s>

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

<s>qua in arte exercitati&longs;&longs;imum P. </s>

<s>qua in arte exercitati&longs;&longs;imum P. Augu&longs;tinum Spernac<lb/>ciatum no&longs;træ Societatis videmus modo de mandato Summi Pontificis Pa<lb/>dum, ac Renum Bononien&longs;em ob aggerationem &longs;tagnantes in mari emitte<lb/>re; cui totus hic no&longs;ter di&longs;cur&longs;us maximè probatur. </s>

<s>Augu&longs;tinum Spernac<lb/>ciatum no&longs;træ Societatis videmus modo de mandato Summi Pontificis Pa<lb/>dum, ac Renum Bononien&longs;em ob aggerationem &longs;tagnantes in mari emitte<lb/>re; cui totus hic no&longs;ter di&longs;cur&longs;us maximè probatur. </s>

<s>Ex quibus omnibus &longs;e<lb/>quitur &longs;uperficiem terræ tam montium, quam planorum quotidie variari. <lb/></s>

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hoc capite exi&longs;timat po&longs;t <lb/>multa &longs;ecula reuolui, hoc veritati e&longs;&longs;e con&longs;entaneum argumento &longs;unt, ac <lb/>pariter admirationi varia <expan abbr="cõchiliorum">conchiliorum</expan> genera, quæ tùm in Apennino mon<lb/>te, tùm in Alpibus ob&longs;eruaui; <expan abbr="Ìdem&qacute;">Ìdemque</expan>; in alijs mundi partibus inueniri pu<lb/>to; præ&longs;ertim in tam immen&longs;a copia, <expan abbr="atq;">atque</expan> intra vi&longs;cera montium colloca<lb/>ta, quæ nulla vis humana illuc contuli&longs;&longs;er, ni&longs;i temporibus catacly&longs;mi ebul<lb/>lientibus aquis maris &longs;uper terram facta fui&longs;&longs;et hæc varia rerum maritima<lb/>rum cum terre&longs;tribus commixtio: quæ quidem optimè ex Pomponio Mela <lb/>comprobantur, qui libro 1. de Numidia &longs;ic narrat: interius, & longè &longs;atis <lb/>à litore, &longs;i fides res capit, mirum admodum, &longs;pinæ pi&longs;cium, <expan abbr="Muric&utilde;">Muricum</expan>, <expan abbr="O&longs;treo-rum&qacute;">O&longs;treo<lb/>rumque</expan>; fragmenta, &longs;axi atritu, vti &longs;olent fluctibus, & non differentia mari<lb/>nis, infixæ cautibus anchoræ, <expan abbr="alia&qacute;">aliaque</expan>; huiu&longs;modi &longs;igna, & ve&longs;tigia effu&longs;i olim <lb/><expan abbr="v&longs;q;">v&longs;que</expan> ad ea loca pelagi, in campis nihil alentibus e&longs;&longs;e inuenirique narrantur. <lb/></s>

<s>neque locus ille Ouid. </s>

<s>neque locus ille Ouid. Met. </s>

<s>15. extra rem:</s>

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<s>fit au<lb/>tem ortus hic Orionis, heliacus, & matutinus, de quo Ari&longs;t.

hoc loco, & alij <lb/>auctores, no&longs;tra hac tempe&longs;tate paulò ante Solis ingre&longs;&longs;um in Cancrum, &longs;i<lb/>ue ante &longs;ol&longs;titium æ&longs;tiuum circa 22. Iunij.</s>

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<s>vocantur autem &longs;ecundum po<lb/>&longs;itionem locorum venti &longs;ic; Zephyrus quidem ab A, hoc enim e&longs;t occidens æquino<lb/>ctialis. </s>

<s>Boreas autem, & Aparetias à G. hic enim Vr&longs;a, contrarius autem huic <lb/>Notus ab H. </s>

<s>Boreas autem, & Aparetias à G. hic enim Vr&longs;a, contrarius autem huic <lb/>Notus ab H. </s>

<s>Meridies enim e&longs;t hic, à quo flat, & H, ip&longs;i G, contrarium e&longs;t; &longs;ecun<lb/>dum enim diametrum &longs;unt. </s>

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

<s>Sol (vt in præ&longs;enti figura) <lb/>videatur ab oculo A, media nube C D F, &longs;ub an<lb/>gulo refractionis B C A, vel B F A, & alijs &longs;imilibus angulis in eadem nube; <lb/>manente igitur oculo A, & a&longs;tro B, necnon nube C D E. eodem in loco, im<lb/>po&longs;&longs;ibile e&longs;t Solem videri ab eodem oculo &longs;ub diuer&longs;o angulo à priori, nec <lb/>con&longs;equenter alibi apparere, quam in B. </s>

<s>Nunc ad textus declarationem, in <lb/>quo continetur Geometrica demon&longs;tratio rotunditatis Areæ, quam &longs;ic bre<lb/>uiter prius veteres excogitarunt: Viderunt primò Solem in Area apparere <lb/>in orbem, & con&longs;imiliter: hinc intulerunt nece&longs;&longs;e e&longs;&longs;e apparere etiam per <lb/>con&longs;imiles, &longs;iue æquales refractionis angulos; quia diuer&longs;i anguli, diuer&longs;am <lb/>etiam <expan abbr="appar&etilde;tiam">apparentiam</expan> efficiunt: atqui con&longs;imiles, &longs;iue æquales refractionis an<lb/>gulos nece&longs;&longs;e e&longs;t in circulum <expan abbr="cõ&longs;titui">con&longs;titui</expan>, vt mox con&longs;tabit; cau&longs;a igitur rotun<lb/>ditatis huius, e&longs;t angulorum refractionis æqualitas. </s>

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

<s>ducan<lb/>tur nunc ex angulis C, D, F, tres perpendiculares ad rectam A B, quæ &longs;int <lb/>C E, D E, F E, in figura; quæ tres nece&longs;&longs;ariò erunt æquales, cum &longs;int ductæ <lb/>ab angulis æqualibus æqualium triangulorum ad communem ba&longs;im, & di<lb/>uident nece&longs;&longs;ariò ba&longs;im in eodem puncto E, cum diuidant triangula æqua<lb/>lia proportionaliter; <expan abbr="erunt&qacute;">eruntque</expan>; propterea hæ tres rectæ in eodem plano, quod <lb/>in nube concipitur ex 5. 11. Quare &longs;i concipiamus &longs;uperficiem, &longs;iue planum <lb/>delineari circa E, ad interuallum linearum æqualium C E, D E, F E, de<lb/>&longs;criptus erit circulus per 9. tertij, cuius circumferentia C D F. </s>

<s>Ex quibus <lb/>patet tria illa puncta C, D, E, per quæ Sol tran&longs;paret e&longs;&longs;e in orbem di&longs;po&longs;i<lb/>ta. </s>

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Line 5488

<s>in oriente G, &longs;it a&longs;trum. </s>

<s>&longs;i ergò lineæ vi&longs;uales à K, ad M, nubem tenden<lb/>tes reflectantur &longs;uper maiorem angulum M K G, ad G, erit reflexarum vna <lb/>veluti M G. </s>

<s>&longs;i ergò lineæ vi&longs;uales à K, ad M, nubem tenden<lb/>tes reflectantur &longs;uper maiorem angulum M K G, ad G, erit reflexarum vna <lb/>veluti M G. </s>

<s>Porro omnes lineæ viluales, quæ adnubem M, incidunt, nece&longs;<lb/>&longs;ariò, vt probabo, cadent in ambitum circularem. </s>

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

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

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

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

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<s>Primum <lb/>e&longs;t; ea, quæ videmus per reflexionem &longs;ub quopiam angulo, manentibus no<lb/>bis &longs;peculo, & obiecto ij&longs;dem in locis, non po&longs;&longs;unt videri &longs;ub alio diuer&longs;o <lb/>angulo, nec alibi con&longs;equenter apparere. </s>

<s>in &longs;uperiori figura, quam <lb/>textui re&longs;tituimus exi&longs;tente Sole in G, oculo in K, & nube in M. ex qua ra<lb/>dius Solis G M, re&longs;lectatur ad vi&longs;um in K, per <expan abbr="lineã">lineam</expan> M K, &longs;ub angulo G M K, <lb/>impo&longs;&longs;ibile e&longs;t manentibus illis, vt dixi, videri Solem in nube M, &longs;ub diuer<lb/>&longs;o angulo à priori, nec alibi apparere. </s>

<s>Alterum e&longs;t apud Opticos vulga<lb/>tum; ea &longs;cilicet, quæ per reflexionem (de quorum numero e&longs;t Iris) viden<lb/>tur, videri, tunc &longs;olum, quando angulus incidentiæ fuerit æqualis angulo <lb/>reflexionis, quia tunc breui&longs;&longs;imis lineis fit vi&longs;io; quibus &longs;oli, natura (&longs;i fieri <lb/><figure id="fig53"/><lb/>pote&longs;t) vtitur. </s>

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

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

<s>G R, K R, ad aliud punctum, <lb/>vti e&longs;t R, quæ habeant eandem inuicem proportionem, quam habent prio<lb/>res duæ G M, K M, quod probatur, quia &longs;i &longs;int vt G M, ad K M, ita G R, ad <lb/>K R, cum G R, &longs;it centro K, propinquior quam G M, erit etiam eadem G R, <lb/>longior ip&longs;a G M, per 15. 3. & tamen deberet e&longs;&longs;e æqualis illi; quemadmo<lb/>dum K M, e&longs;t æqualis alteri K R; nequeunt autem duæ lineæ inæquales inui<lb/>cem, habere eandem rationem ad duas inuicem æquales: ergo non habent <lb/>eandem rationem G M, & K M, quam habent G R, & K R. quod &longs;i punctum <lb/>R, &longs;umatur &longs;upra M, erit &longs;imilis <expan abbr="demõ&longs;tratio">demon&longs;tratio</expan>, &longs;i literæ M, & R, loca permu<lb/>tent. </s>

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<s>G, enim e&longs;t in ortu. </s>

<s>K, verò in centro <lb/>horizontis, &longs;equitur, quod etiam linea G K, cuins ip&longs;a &longs;unt extrema, data <lb/>&longs;it, & po&longs;itione, & magnitudine, per 26. Datorum Euclidis. </s>

<s>K, verò in centro <lb/>horizontis, &longs;equitur, quod etiam linea G K, cuins ip&longs;a &longs;unt extrema, data <lb/>&longs;it, & po&longs;itione, & magnitudine, per 26. Datorum Euclidis. </s>

<s>eadem quoque <lb/>ratione data erit K M, linea; &longs;iue quia e&longs;t æqualis ip&longs;i G K, &longs;iue quia per <lb/>a&longs;trolabium po&longs;&longs;umus ip&longs;ius longitudinem, & po&longs;itionem inue&longs;tigare; qua<lb/>re & punctum M, datum erit per 27. Datorum, quare & linea G M, data <lb/>erit quoad &longs;itum, & magnitudinem per 26. Datorum. </s>

<s>Quare per primam <lb/>Datorum erit data proportio linearum G M, M K, punctum <expan abbr="itaq;">itaque</expan> M, tange t <lb/>ambitum datum, qui ba&longs;is e&longs;t coni, quem linea K M, de&longs;cribit in reuolutio<lb/>ne axis G K O, &longs;uper polis G, O. cum enim data &longs;it K M, po&longs;itu, & magni<lb/>tudine, <expan abbr="ea&qacute;">eaque</expan>; &longs;it latus prædicti coni, &longs;equitur periphæriam, vel ambitum ba<lb/>&longs;is coni e&longs;&longs;e datum per &longs;imilem definitionem 5. definitioni Datorum. </s>

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

<s>Iam &longs;i <expan abbr="triãgulum">triangulum</expan> G M K, prioris figuræ circumuoluatur circa axem G K O, <lb/>punctum ip&longs;ius M, de&longs;cribit prædictum ambitum L M N. hunc ambitum <lb/>inquit Ariltot. linea K M, attinger, <expan abbr="erit&qacute;">eritque</expan>; hic ambitus datus, vt dictum e&longs;t. <lb/><figure id="fig54"/><lb/>Erit præterea &longs;ectio circunferentiarum ho<lb/>rizontis, & huius amb tus data, cuius extre<lb/>ma puncta e&longs;&longs;ent L, & N. &longs;i enim <expan abbr="cõcipiamus">concipiamus</expan> <lb/>in figura non &longs;olum horizontis <gap/>iametrum <lb/>G K O, &longs;ed etiam circunferentiam (in qua <lb/>circunferentia e&longs;&longs;ent duo illa puncta L, & N, <lb/>vt in præ&longs;enti de&longs;criptione melius intelli ge<lb/>tur, in qua horizon G N O L, & ambitus <lb/>prædictus e&longs;t L M N, qui debet intelligi ele<lb/>uatus &longs;upra horizontem perpendiculariter) <lb/>tunc &longs;ectio ip&longs;ius mutua cum horizonte e&longs;&longs;et <pb pagenum="119"/>linea N P L, cuius extrema puncta &longs;unt L, N, quæ data erunt, cum &longs;int ex<lb/>trema lineæ K M, circumlatæ; & quemadmodum dabatur &longs;uperius punctum <lb/>M. eadem ratione ex Datis, dabitur punctum N, & L. quare etiam &longs;ectio <lb/>N P L, quæ inter data puncta continetur, data erit ex 26. Datorum.</s>

<s>linea K M, attinger, <expan abbr="erit&qacute;">eritque</expan>; hic ambitus datus, vt dictum e&longs;t. <lb/><figure id="fig54"/><lb/>Erit præterea &longs;ectio circunferentiarum ho<lb/>rizontis, & huius amb tus data, cuius extre<lb/>ma puncta e&longs;&longs;ent L, & N. &longs;i enim <expan abbr="cõcipiamus">concipiamus</expan> <lb/>in figura non &longs;olum horizontis <gap/>iametrum <lb/>G K O, &longs;ed etiam circunferentiam (in qua <lb/>circunferentia e&longs;&longs;ent duo illa puncta L, & N, <lb/>vt in præ&longs;enti de&longs;criptione melius intelli ge<lb/>tur, in qua horizon G N O L, & ambitus <lb/>prædictus e&longs;t L M N, qui debet intelligi ele<lb/>uatus &longs;upra horizontem perpendiculariter) <lb/>tunc &longs;ectio ip&longs;ius mutua cum horizonte e&longs;&longs;et <pb pagenum="119"/>linea N P L, cuius extrema puncta &longs;unt L, N, quæ data erunt, cum &longs;int ex<lb/>trema lineæ K M, circumlatæ; & quemadmodum dabatur &longs;uperius punctum <lb/>M. eadem ratione ex Datis, dabitur punctum N, & L. quare etiam &longs;ectio <lb/>N P L, quæ inter data puncta continetur, data erit ex 26. Datorum.</s>

<s>Illud nunc in memoriam <expan abbr="reuocãdum">reuocandum</expan>, quod paulò ante probaui, nimirum <lb/>proportionem linearum G M, K M, non po&longs;&longs;e &longs;eruari in alijs lineis, quæ &longs;int <lb/>in eodem plano trianguli G M K, &longs;i ducantur ab ij&longs;dem punctis G, K. pote&longs;t <lb/>tamen &longs;eruari in alijs duabus, quæ cadant in prædictum ambitum, &longs;iue <expan abbr="cir-cunfer&etilde;tiam">cir<lb/>cunferentiam</expan> L M N, <expan abbr="quæ&qacute;">quæque</expan>; &longs;int in alio plano, <expan abbr="quã">quam</expan> in plano trianguli G M K, <lb/>quod tamen tran&longs;eat per axem G K O, <expan abbr="&longs;it&qacute;">&longs;itque</expan>; vnum ex planis illis, de quibus <lb/>&longs;upra dictum e&longs;t. </s>

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<s>Ibidem <emph type="italics"/>(Extraponatur igitur quædam linea, quæ D B, & &longs;eindatur vt M G, ad<lb/>M K, &longs;ic quæ D, ad B, maior autem quæ M G, ea quàm M K, quoniam &longs;uper ma<lb/>iorem angulum reflexio coni, maiori enim angulo &longs;ubtenditur trianguli M K G. <lb/></s>

<s>Maior igitur e&longs;t & ip&longs;a D, ip&longs;a B. addatur igitur ad eam, quæ B, ea in qua F, vt <lb/>&longs;it quod D, ad B, quæ B F, ad D. </s>

<s>Maior igitur e&longs;t & ip&longs;a D, ip&longs;a B. addatur igitur ad eam, quæ B, ea in qua F, vt <lb/>&longs;it quod D, ad B, quæ B F, ad D. </s>

<s>Deinde quod F, ad K G, quæ B, ad aliam fiat, <lb/>quæ K P. & à P, ad M, copuletur quæ P M, erit igitur P. polus circuli, ad quem <lb/>lineæ, quæ à K, incidunt)<emph.end type="italics"/> <expan abbr="hucu&longs;q;">hucu&longs;que</expan> o&longs;tendit lineas vi&longs;uales cadere ad M, pun<lb/>ctum in Iridis periphæriam, pergit deinceps inue&longs;tigare polum, & po&longs;tea <lb/>centrum eiu&longs;dem ambitus, vtraque autem exi&longs;tere in horizonte reperit, vt <lb/>hinc inferat Iridis portionem illam, quæ oriente Sole &longs;upra horizontem ap<lb/>paret, e&longs;&longs;e &longs;emicirculum, vt propo&longs;uerat. </s>

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

cum data &longs;it proportio linearum K M, & M G, in &longs;upe<lb/>riori &longs;ecunda figura numeri 164. quam nunc iterum in&longs;picere opertet; ex<lb/><figure id="fig56"/><lb/>ponatur alia linea recta B D. quæ diui<lb/>datur in partes B, & D. proportionales <lb/>cum lineis K M, G M, per 10. 6. cum <lb/>ergo K M, &longs;it minor quàm G M, per 19. <lb/>primi, quia in triangulo G M K, oppo<lb/>nitur minori angulo, erit <expan abbr="quoq;">quoque</expan> B, minor quàm D, addatur iam ip&longs;i B. linea <lb/>nea F, ita vt &longs;it tota F B, tertia proportionalis ad duas B, & D, per 11. 6. <lb/>hoc ordine, vt F B, ad D. ita D, ad B. </s>

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

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

<s>Dico P, e&longs;&longs;e polum circuli, quem dixi Iridis, & in quem li<lb/>neæ à K, procedentes turbinis formam effingunt, probatur autem ab Ari&longs;t. <lb/>in &longs;equentibus.</s>

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<s>164. <expan abbr="iungãtur">iungantur</expan> lineæ ad R, quæ &longs;int G R, K R, <lb/>erit vt G R, ad K R, ita G P, ad P R. quia orta <expan abbr="s&utilde;t">sunt</expan> duo <expan abbr="triãgula">triangula</expan> G P R, K P R, <lb/>quæ habent eundem angulum ad P. & latera proportionalia circa dictum <lb/>angulum. </s>

<s>e&longs;t etiam vt G P, ad P R, in maiori triangulo, ita P R, ad K P, in <lb/>minori, ex con&longs;tructione, quare per 6. 6. erunt illa duo triangula æquian<lb/>gula; ergò per 4. 6. erunt latera circum æquales angulos proportionalia; <lb/>quare erit vt G P, ad P R. ita G R, ad R K: erat autem vt K M, ad G M, ita <lb/>B, ad D. & ita etiam G P, ad P R; ergò per 11. 5. vt K M, ad M G. ita K R, <lb/>ad R G, intra eandem circunferentiam, & in eodem plano: quod e&longs;&longs;e im<lb/>po&longs;&longs;ibile &longs;upra o&longs;tendimus, hoc autem impo&longs;&longs;ibile, &longs;equitur &longs;i neges e&longs;&longs;e vt <lb/>F, ad G K; & B, ad K P, ita D, ad P M.</s>

<s>Ibidem <emph type="italics"/>(Quoni&abreve; igitur quæ D, <expan abbr="neq;">neque</expan> ad minorem ea, quæ P M, <expan abbr="neq;">neque</expan> ad maiorem <lb/>(&longs;imiliter enim demon&longs;ir abimus) palam e&longs;t, quod ad ip&longs;am <expan abbr="vtiq;">vtique</expan> erit, in qua P M, <lb/>quare erit, quod quæ M P, ad P K, quæ P G, ad M P. </s>

<s>Si igitur eo in quo P, polo <lb/>vtens, di&longs;tantia autem ea, in qua P M, circulns de&longs;cribatur, omnes angulos attin<lb/>get, quos reflexæ faciunt, quæ à K, G. &longs;i autem non, &longs;imiliter o&longs;tendentur eandem <lb/>babere rationem, quæ alibi, quam in &longs;emicirculo con&longs;tituuntur; quod quidem erat <lb/>impo&longs;&longs;ibile)<emph.end type="italics"/> quoniam igitur, inquit, linea D, <expan abbr="neq;">neque</expan> ad minorem, <expan abbr="neq;">neque</expan> ad ma<lb/>iorem quam P M, habet eam rationem, quæ e&longs;t ip&longs;ius F, ad G K, aut ip&longs;ius <lb/>B, ad K P. &longs;imiliter enim demon&longs;tratur ab&longs;urdum &longs;equi. </s>

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<s>&longs;imilia dicta &longs;unt in Halone. </s>

<s>Cum ergò ip&longs;ius centrum <lb/><foreign lang="greek">a</foreign>, &longs;it in diametro horizontis G K <foreign lang="greek">w</foreign> P O, manife&longs;tum fit portionem eius, quæ <lb/>&longs;upra horizontem eminet, e&longs;&longs;e &longs;emicirculum, qui in figura notatur lineis <lb/>L M N. </s>

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

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<s>Ibidem <emph type="italics"/>(Iterum &longs;it horizon quidem in quo A C. oriatur autem &longs;upra hunc G, <lb/>axis autem &longs;it nunc in quo G P. </s>

<s>Ibidem <emph type="italics"/>(Iterum &longs;it horizon quidem in quo A C. oriatur autem &longs;upra hunc G, <lb/>axis autem &longs;it nunc in quo G P. </s>

<s>Alia igitur omnia &longs;imiliter o&longs;tendentur vt & prius. <lb/></s>

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

<s>Quoniam autem &longs;upra diametrum, quæ A C, quod K G, <lb/>centrum vtique erit &longs;ub horizonte priori eius, in quo A C, in linea K P, in quo <foreign lang="greek">w,</foreign><emph.end type="italics"/><pb pagenum="123"/><figure id="fig58"/><lb/><emph type="italics"/>Quare minor erit &longs;uperior &longs;ectio &longs;emicir<lb/>culo, in qua S T, (nam Q S T, &longs;emicir<lb/>culus est, nunc autem inter&longs;ectus e&longs;t ab <lb/>horizonte A C; <expan abbr="itaq;">itaque</expan> Q S, di&longs;parens erit) <lb/>eleuato ip&longs;o Sole)<emph.end type="italics"/> demon&longs;trat propo&longs;i<lb/>tionem &longs;ecundam nimirum Sole &longs;upra <lb/>horizontem elcuato, ambitum Iridis <lb/>e&longs;&longs;e minorem circuli portionem, &longs;iue <lb/>&longs;emicirculo minorem. </s>

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

<s>Cæterum &longs;i quis breues, ac dilucidas harum rerum demon&longs;trationes re<lb/>quirat, is legat 74. 75. 76. propo&longs;itiones 10. Vitell. </s>

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

<s>vel &longs;equentem no&longs;tram <lb/>de Iride additionem. </s>

<s>ego enim longiorem hanc, <expan abbr="atq;">atque</expan> impeditam Ari&longs;t.

tra<lb/>ctationem in gratiam textus illius, vt in&longs;tituti mei ratio po&longs;tulabat, per&longs;e<lb/>quutus &longs;um.</s>

tra<lb/>ctationem in gratiam textus illius, vt in&longs;tituti mei ratio po&longs;tulabat, per&longs;e<lb/>quutus &longs;um.</s>

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<s>In ijs autem diebus, <lb/>qui ad hyemates ver&longs;iones, quia non mult&ubreve; &longs;unt &longs;upra terram &longs;ectiones cir culorum, <lb/>contrarium nece&longs;&longs;arium fieri, modicum enim eleuato in quo G, in meridie fit Sol)<emph.end type="italics"/><lb/>quærit cur po&longs;t æquinoctium autumnale v&longs;que ad vernum, hoc e&longs;t hyemali <lb/>tempore, Iris appareat etiam Sole meridiem occupante: reliquo autem <lb/>tempore æ&longs;tiuo, quod e&longs;t ab æquinoctio verno ad autumnale appareat tan<lb/>tum Sole vel in ortu, aut occa&longs;u exi&longs;tente, vel parum &longs;upra terram &longs;ublato. <lb/></s>

<s>cau&longs;a autem huius refert in &longs;ectiones parallelorum circulorum, quos Sol <lb/>diurno motu inter <expan abbr="vtrunq;">vtrunque</expan> <expan abbr="tropic&utilde;">tropicum</expan> de&longs;cribit: nam &longs;ectiones parallelorum, <lb/>qui &longs;unt ad Vr&longs;am, ide&longs;t in parte &longs;phæræ Boreali, qui omnes &longs;unt inter æqua<lb/>torem, & tropicum Cancri; &longs;ectiones inquam horum circulorum, quæ &longs;unt <lb/>&longs;upra horizontem, maiores &longs;unt &longs;ectionibus infra horizontem depre&longs;&longs;is, & <lb/>&longs;emper eò maiores, quò propiores &longs;unt Cancro, ita vt magna yaldè &longs;it ea <lb/>portio, quæ e&longs;t &longs;upra terram, exigua verò admodum, quæ infra (intelligan<lb/>tur hæc in &longs;phæra obliqua, cuius polus eleuetur grad. 45. circiter) quare <lb/>quando a&longs;trum G, con&longs;cenderit meridiem, adeò P, polus Iridis, & etiam <foreign lang="greek">w,</foreign><lb/>centrum eius infra terram deprimitur, vt aut nihil, aut in&longs;en&longs;ibile quid de <lb/>Iridis ambitu &longs;upra terram eleuari po&longs;&longs;it, contrarium accidit in parallelis <lb/>meridionalibus, quia eorum &longs;ectiones &longs;uperiores &longs;unt &longs;emper inferioribus <lb/>minores, quapropter etiam &longs;i a&longs;trum ad meridiem eleuetur, parum tamen <lb/>attollitur, & con&longs;equenter centrum <foreign lang="greek">w,</foreign> Iridis parum infra horizontem <lb/>de&longs;cendit, ac propterea etiam in meridie pars ip&longs;ius &longs;atis ma<lb/>gna con&longs;picitur. </s>

<s>45. circiter) quare <lb/>quando a&longs;trum G, con&longs;cenderit meridiem, adeò P, polus Iridis, & etiam <foreign lang="greek">w,</foreign><lb/>centrum eius infra terram deprimitur, vt aut nihil, aut in&longs;en&longs;ibile quid de <lb/>Iridis ambitu &longs;upra terram eleuari po&longs;&longs;it, contrarium accidit in parallelis <lb/>meridionalibus, quia eorum &longs;ectiones &longs;uperiores &longs;unt &longs;emper inferioribus <lb/>minores, quapropter etiam &longs;i a&longs;trum ad meridiem eleuetur, parum tamen <lb/>attollitur, & con&longs;equenter centrum <foreign lang="greek">w,</foreign> Iridis parum infra horizontem <lb/>de&longs;cendit, ac propterea etiam in meridie pars ip&longs;ius &longs;atis ma<lb/>gna con&longs;picitur. </s>

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

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

<s>Sed contra prædicta de angulo Iridis determinato eadem nobis obijcies, <lb/>quæ nos &longs;upra ad finem numeri 164. Ari&longs;t. & alijs obiecimus, plures <expan abbr="nimi-r&utilde;">nimi<lb/>rum</expan> po&longs;&longs;e con&longs;titui angulos æquales angulo Iridis B, in plano trianguli A B C, <lb/>qui non &longs;int in eodem orbe con&longs;tituti, in quo &longs;unt omnes anguli B. </s>

<s>Iridem <lb/>reflectentes, <expan abbr="quiq;">quique</expan> reflexionem faciant ad eundem oculum C, vnde &longs;equitur <lb/>prædictam Iridis altitudinem non e&longs;&longs;e, vti diximus, determinatam, cum <lb/>po&longs;&longs;it angulus B, alios &longs;ibi æquales tam &longs;upra, quàm infra habere, qua ra<lb/>tione deberet etiam Iris, & altius, & inferius apparere.</s>

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<s>Cur oriente, aut occumbente Sole, Iris &longs;emicirculus e&longs;t?</s>

<s>Cau&longs;a huius hæc e&longs;t; &longs;upra enim dictum e&longs;t, in omni Iridis appari<lb/>tione tria hæc, Solem, oculum, & Iridis centrum e&longs;&longs;e in eadem re<lb/>cta linea, v. </s>

<s>Cau&longs;a huius hæc e&longs;t; &longs;upra enim dictum e&longs;t, in omni Iridis appari<lb/>tione tria hæc, Solem, oculum, & Iridis centrum e&longs;&longs;e in eadem re<lb/>cta linea, v. g. </s>

<s>in linea A C D, præcedentis figuræ, cum igitur Sol <lb/>tam oriens, quam occidens &longs;it in horizonte, v. </s>

<s>in linea A C D, præcedentis figuræ, cum igitur Sol <lb/>tam oriens, quam occidens &longs;it in horizonte, v. g. </s>

<s>in A, horizontis <lb/>puncto, &longs;imiliter oculus &longs;it in C, horizontis centro, con&longs;ectarium e&longs;t, cen<lb/>trum etiam Iridis D, e&longs;&longs;e pariter in horizontis &longs;uperficie, quare &longs;ecabitur <lb/>ab horizonte per centrum, vnde etiam &longs;equitur ip&longs;ius Iridis portionem <lb/>E B F, quæ &longs;upra horizontem extat e&longs;&longs;e &longs;emicirculum. </s>

<s>Quod &longs;i horizon non <lb/>ob&longs;taret, <expan abbr="integr&utilde;">integrum</expan> Iris compleret orbem, <expan abbr="cerneretur&qacute;">cernereturque</expan>; toto ambitu B F M E.</s>

<s>An <expan abbr="quando&qacute;">quandoque</expan>; maior &longs;emicirculo appareat?</s>

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

1.</s>

1.</s>

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&pacute;robat li<lb/><figure id="fig62"/><lb/>neam rectam, duo quælibet puncta <expan abbr="quãtumuis">quantumuis</expan> pro<lb/>pinqua in circuli ambitu a&longs;&longs;umpta coniungentem <lb/>cadere intra circulum. </s>

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

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

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

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<s>ex quibus ea etiam, <lb/>quæ ad phy&longs;icum &longs;pectant manife&longs;ta fiunt, nimirum &longs;icut entia mathemati<lb/>ca à materia non exi&longs;tunt &longs;eparata, quia &longs;ic nullam haberent operationem; <lb/>ita etiam anima, &longs;i nullam habet propriam operationem non exi&longs;tet à cor<lb/>pore &longs;eparata.</s>

<s><emph type="italics"/>Ex Secundo de Anima.<emph.end type="italics"/></s>

<s><emph type="italics"/>Ex Secundo de Anima.<emph.end type="italics"/></s>

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<s>quadratum autem li<lb/>neæ B D, æquale e&longs;t rectangulo dato A B C D, per 17.6. Inuentio porrò hu<lb/>ius mediæ proportionalis, quia facilis e&longs;t, & &longs;citu iucunda, eam &longs;ic habeto. <lb/><figure id="fig65"/><lb/>accipe duo latera A B, & B C, <expan abbr="quadrãdi">quadrandi</expan> rectan<lb/>guli, <expan abbr="ea&qacute;">eaque</expan>; in directum con&longs;titue, vt vnicam re<lb/>ctam con&longs;tituant A C, vt apparet in figura; de<lb/>inde diui&longs;a tota A C, bifariam in E, facto cen<lb/>tro in E, de&longs;cribe &longs;emicirculum &longs;uper lineam <lb/>A C, demum à puncto B, in quo duo latera con<lb/>iunguntur, erigatur linea perpendicularis <expan abbr="v&longs;q;">v&longs;que</expan> <lb/>ad periphæriam, quæ &longs;it B D. hæc enim B D, e&longs;t media proportionalis inter <lb/>latera A B, B D, quam nimirum habitudinem habet A B, ad B D, eam quo<lb/>que obtinet B D, ad B C. </s>

<s>Quadratum igitur huius B D, hoc e&longs;t quadratum, <lb/>cuius quatuor latera &longs;iut æqualia lineæ B D, quale e&longs;t præ&longs;ens, æquale erit <lb/><figure id="fig66"/><lb/>dato &longs;uperiori rectangulo A B C D, <expan abbr="atq;">atque</expan> hoc modo per<lb/>acta erit quadratio, &longs;eu tetragoni&longs;mus dati quadrilateri <lb/>A B C D. </s>

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

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<s>Vide, quæ cap.

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>

<s><emph type="italics"/>Ex Tertio de Anima.<emph.end type="italics"/></s>

<s><emph type="italics"/>Ex Tertio de Anima.<emph.end type="italics"/></s>

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<s>Tex. 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. cap.

5. tex. </s>

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

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

<s><emph type="italics"/>Ex Libro de Senfit.<emph.end type="italics"/></s>

<s><emph type="italics"/>Ex Libro de Senfit.<emph.end type="italics"/></s>

<s>Cap, 6. <emph type="italics"/>(Et qui in Die&longs;i &longs;onus latet, quamuis contin<gap/>um exi&longs;tentem audit <lb/>omnem cantum, di&longs;t antia enim eius ad extremos &longs;onos latet)<emph.end type="italics"/> quid &longs;it <lb/>Die&longs;is apud Mu&longs;icos explicatum e&longs;t primo Po&longs;ter. tex. 38. cum <lb/>autem Die&longs;is &longs;it minima di&longs;tantia, &longs;eu vt loquuntur Mu&longs;ici, mini<lb/>mum <expan abbr="interuall&utilde;">interuallum</expan> inter duas voces, hinc fit vt hæc minima di&longs;tantia inter ex<lb/>tremos &longs;onos non exaudiatur, quemadmodum nec minima particula alicu<lb/>ius magni corporis à longè vi&longs;i <expan abbr="nõ">non</expan> percipitur, &longs;ed latetinter extrema illius.</s>

<s>38. cum <lb/>autem Die&longs;is &longs;it minima di&longs;tantia, &longs;eu vt loquuntur Mu&longs;ici, mini<lb/>mum <expan abbr="interuall&utilde;">interuallum</expan> inter duas voces, hinc fit vt hæc minima di&longs;tantia inter ex<lb/>tremos &longs;onos non exaudiatur, quemadmodum nec minima particula alicu<lb/>ius magni corporis à longè vi&longs;i <expan abbr="nõ">non</expan> percipitur, &longs;ed latetinter extrema illius.</s>

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<s>Eodem cap. <emph type="italics"/>(Velut Diapa&longs;on, & Diapente)<emph.end type="italics"/> quid &longs;it con&longs;onantia Diapa<lb/>&longs;on, explicaui in primo Po&longs;ter. tex. </s>

<s>Eodem cap. <emph type="italics"/>(Velut Diapa&longs;on, & Diapente)<emph.end type="italics"/> quid &longs;it con&longs;onantia Diapa<lb/>&longs;on, explicaui in primo Po&longs;ter. tex. 1. Diapente verò e&longs;t con&longs;onantia ex duo<lb/><figure id="fig67"/><lb/>bus &longs;onis coale&longs;cens, quorum proportio &longs;it vt <lb/>3. ad 2. quæ dicitur &longs;e&longs;quialtera. </s>

<s>1. Diapente verò e&longs;t con&longs;onantia ex duo<lb/><figure id="fig67"/><lb/>bus &longs;onis coale&longs;cens, quorum proportio &longs;it vt <lb/>3. ad 2. quæ dicitur &longs;e&longs;quialtera. </s>

<s>&longs;int duæ <lb/>chordæ æqualis cra&longs;fitiei, <expan abbr="atq;">atque</expan> æquè ten&longs;æ: vna <lb/>tamen habeat ad alteram proportionem &longs;e&longs;<lb/>quialteram, vt in figura apparet; &longs;i &longs;imul pul<lb/>&longs;entur, edent con&longs;onantiam Diapente. </s>

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<s>lege annotata primo <lb/>Po&longs;ter. &longs;ecto 3. cap.

1.</s>

1.</s>

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<s>Extat adhuc <lb/>de huiu&longs;modi machinis liber Heronis Alexandrini, quem nuper ex græco <lb/>latinum reddidit docti&longs;&longs;imus Abbas Gua&longs;tallenfis. </s>

<s>de huiu&longs;modi artificio&longs;is <lb/>operibus, quibus &longs;æpè pri&longs;ci ita admirationi fuere, vt præ&longs;tigia quædam ar<lb/>tificium ignorantibus, viderentur, intelligit hoc loco Ari&longs;t.</s>

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<s>Tex. 47. (<emph type="italics"/>Qui Geometriam di&longs;cit aliqua præ&longs;cire contingit<emph.end type="italics"/>) ide&longs;t definitio<lb/>nes, po&longs;tulata, axiomata, quæ &longs;unt tria principiorum genera, ex quibus to<lb/>ta Geometria deducitur.</s>

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

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

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<s>19. <expan abbr="problemat&utilde;">problematum</expan> vbi <expan abbr="tanquã">tanquam</expan> in proprio loco i&longs;ta fu&longs;ius pertractabuntur.</s>

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

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

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<s>Eodem tex. </s>

<s>Eodem tex. 3. (<emph type="italics"/>Item & in cæteris tunc &longs;cire vnumquodque arbitramur torum, <lb/>quorum &longs;unt demonstrationes, cum quid e&longs;t &longs;ciamus, vt puta quid tetragoni&longs;mus, <lb/>quòd inuentio mediæ<emph.end type="italics"/>) eadem reperies &longs;uperius in &longs;ecundo de Anima, tex. </s>

<s>3. (<emph type="italics"/>Item & in cæteris tunc &longs;cire vnumquodque arbitramur torum, <lb/>quorum &longs;unt demonstrationes, cum quid e&longs;t &longs;ciamus, vt puta quid tetragoni&longs;mus, <lb/>quòd inuentio mediæ<emph.end type="italics"/>) eadem reperies &longs;uperius in &longs;ecundo de Anima, tex. </s>

<s>12. <lb/>fu&longs;ius explicata.</s>

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<s>Vocabulum tamen i&longs;tud Geo<lb/>dæ&longs;iæ fuit po&longs;tea ad latiorem tran&longs;latum &longs;ignificationem: extat enim Geo<lb/>dæ&longs;ia Heronis Mechanici antiqui &longs;criptoris, quampridem Baroccius lati<lb/>nitate donauit, quæ quidem ars e&longs;t eadem cum Geometria practica, cum <lb/>non &longs;olum diui&longs;iones, &longs;ed men&longs;urationes omnes etiam per dioptricam f<lb/>cultatem, &longs;eu per lineas vi&longs;uales doceat inue&longs;tigare.</s>

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

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

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<s>illas <lb/>duas recentiores &longs;ubalternantes, has verò &longs;ecundas &longs;ubalternatas vocant. <lb/></s>

<s>Exempla &longs;ubalternationum varia attuli in Logicis tex. </s>

<s>Exempla &longs;ubalternationum varia attuli in Logicis tex. 20. & 23. primi Po<lb/>&longs;ter. </s>

<s>20. & 23. primi Po<lb/>&longs;ter. </s>

<s>vbi clarè licet intueri quid &longs;it &longs;ubalternatio, vnde etiam præ&longs;ens lo<lb/>cus illu&longs;tratur.</s>

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

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

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

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<s>Tex. 2. (<emph type="italics"/>Alia verò cau&longs;a e&longs;t forma, & exemplar: hæc autem e&longs;t ratio ip<lb/>&longs;ius quid erat e&longs;&longs;e, & borum genera, vt ip&longs;ius Diapa&longs;on duo ad vnum, <lb/>& &longs;impliciter numerus, & partes, quæ in ra ione &longs;unt<emph.end type="italics"/>) affert exem<lb/>plum cau&longs;æ formalis ex Mu&longs;ica petitum; <expan abbr="ait&qacute;">aitque</expan>; cau&longs;am formalem <lb/>illius con&longs;onantiæ, quæ Diapa&longs;on dicitur, <expan abbr="e&longs;t&qacute;">e&longs;tque</expan>; omnium perfecti&longs;&longs;ima, e&longs;&longs;e <lb/>duplam proportionem, ide&longs;t, quæ e&longs;t inter duo, & vnum, id, quod omnes <lb/>Mu&longs;ici <expan abbr="fat&etilde;tur">fatentur</expan>. </s>

<s>quod vtinelius intelligas, repete, quæ in 2. Po&longs;ter. ad tex. </s>

<s>quod vtinelius intelligas, repete, quæ in 2. Po&longs;ter. ad tex. 1. <lb/>&longs;cripta &longs;unt: necnon quæ in libro de Sen&longs;u in cap.

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

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>

<s>Recentiores tamen y&longs;que ad &longs;enarium <lb/>procedunt, quippe, qui omnes vocum con&longs;onantias admittunt, quæ pro<lb/>portionibus numerorum &longs;enario contentorum præditæ &longs;int. </s>

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<s>qua de re pluri<lb/>bus Zarlinus colloquio 2. definit. </s>

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<s>Tex. 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>Tex. 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. 2. huius cap.

<s>2. huius cap.

explicatum e&longs;t.</s>

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

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<s>Tex. 12. <emph type="italics"/>(Principium <expan abbr="itaq;">itaque</expan> &longs;cibilis, circa <expan abbr="vnumquodq;">vnumquodque</expan> ip&longs;um vnum, non e&longs;t au<lb/>tem idem in cunct is generibus vnum, &longs;ed hic quidem die&longs;is, hic verò vocalis, aut <lb/>muta)<emph.end type="italics"/> ide&longs;t, in Mu&longs;ica quidem principium omnium, & elementum e&longs;t die<lb/>&longs;is, quæ e&longs;t minima vox, aut &longs;onus, qui &longs;ub Mu&longs;ici con&longs;iderationem cadat. <lb/></s>

<s>Porrò ad tex. </s>

<s>Porrò ad tex. 38. primi Po&longs;ter. de die&longs;i plura &longs;unt dicta.</s>

<s>38. primi Po&longs;ter. de die&longs;i plura &longs;unt dicta.</s>

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<s>Tex. eodem <emph type="italics"/>(Metaphoricè autem, quæ in Geometria po<lb/>tentia dicitur)<emph.end type="italics"/> potentiam vnius lineæ appellant Geometræ <lb/>quadratum illius, ide&longs;t quadratum &longs;uper ip&longs;am con&longs;tru<lb/>ctum. </s>

<s>quadratum in quo C, dicitur potentia lineæ <lb/>D B, quia &longs;uper illam con&longs;tructum e&longs;t.</s>

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primo Priornm, &longs;ecto 3. cap.

1.</s>

1.</s>

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

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

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di&longs;cur&longs;u.</s>

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

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

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

1. <lb/>con&longs;cripta &longs;unt (<emph type="italics"/>Cur triangulus duo recti?<emph.end type="italics"/>) ide&longs;t, cur triangulus habet tres <lb/>angulos æquales duobus rectis angulis (<emph type="italics"/>Quia qui circa vnum punctum anguli <lb/>duobus rectis angulis æquales &longs;unt<emph.end type="italics"/>) ni&longs;i hoc dictum ad bonum trahatur &longs;en&longs;um, <lb/><figure id="fig68"/><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.

quot<lb/>quot enim anguli con&longs;tituantur ad punctum A, <lb/>omnes &longs;imul erunt æquales quatuor rectis, quos <lb/>faciunt præ&longs;entes lineæ B C, D E. vniuer&longs;i enim <lb/>illi congruent his quatuor rectis: &longs;ed Ari&longs;t. &longs;en<lb/>&longs;us e&longs;t omnes angulos ad ea&longs;dem partes con&longs;ti<lb/>tutos, v. </s>

<s>ad partes &longs;uperiores lineæ B C, e&longs;&longs;e <lb/>æquales duobus rectis B A D, D A C, vt o&longs;tenditur in 13. primi, necnon <lb/>etiam patere pote&longs;t ex corollario 2. 15. eiu&longs;dem. </s>

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<s>Ibidem (<emph type="italics"/>Cur in &longs;emicirculo vniuer&longs;aliter rectus? </s>

<s>quia &longs;i tres æquales, & quæ <lb/>ba&longs;is e&longs;t duo, & quæ ex medio &longs;upra stat recta, videnti manifestum erit ei, qui illud <lb/>&longs;ciat<emph.end type="italics"/>) In 2. Po&longs;ter. tex. </s>

<s>11. inuenies hu ius loci expo&longs;itionem. </s>

<s>nunc &longs;olùm <pb pagenum="143"/><figure id="fig70"/><lb/>hæc addenda &longs;unt. </s>

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<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>

1.</s>

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<s>pares, non e&longs;&longs;e primos, cum <lb/>eos non &longs;ola vnitas, &longs;ed alius numerus metiatur: quaternarium enim bina<lb/>rius bis replicatus men&longs;urat: &longs;enarium men&longs;urat & binarius, & ternarius: <lb/>quare verum erit exi&longs;timare inter pares numeros aliquos e&longs;&longs;e primos, ide&longs;t <lb/>binarium, aliquos verò non, ide&longs;t cæteros pares vltra binarium.</s>

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

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

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<s>Ibidem <emph type="italics"/>(Et in Mu&longs;ica Die&longs;is primus &longs;en&longs;ibilis &longs;onus, quia minimum)<emph.end type="italics"/> ide&longs;t mi<lb/>nimum interuallum, quod à Mu&longs;icis con&longs;ideretur, e&longs;t men&longs;ura maiorum in<lb/>teruallorum. </s>

<s>ad tex. 38. primi Po&longs;ter. &longs;atis dictum e&longs;t de Die&longs;i, quæ videas.</s>

<s>38. primi Po&longs;ter. &longs;atis dictum e&longs;t de Die&longs;i, quæ videas.</s>

<s>Eodem tex. </s>

<s>Eodem tex. &longs;ed cap.

<s>&longs;ed cap.

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>

<s>38. primi Po&longs;ter. quando autem ait <emph type="italics"/>(Vt puta duæ die&longs;es)<emph.end type="italics"/><lb/>ide&longs;t duæ die&longs;es &longs;unt men&longs;ura vnius interualli mu&longs;ici, qui tonus appellatur: <lb/>quæ quidem duæ die&longs;es non &longs;unt men&longs;ura &longs;en&longs;ibilis, quæ &longs;cilicet auribus per<lb/>cipiatur, &longs;ed tantummodò exi&longs;tunt in numerorum proportionibus, ibi per <lb/>intellectum excogitatis, quando ait <emph type="italics"/>(Et voces plures quibus men&longs;uramus)<emph.end type="italics"/><lb/>quando vtimur eodem interuallo, &longs;iue eadem voce ad cantus men&longs;uram, <lb/>tunc &longs;unt plures men&longs;uræ numero, quamuis vna tantam &longs;pe<gap/> Ait <emph type="italics"/>(Et dia<lb/>meter duobus men&longs;uratur)<emph.end type="italics"/> v. </s>

<s>quid die&longs;is <lb/>dictum &longs;it ad tex. 38. primi Po&longs;ter. quando autem ait <emph type="italics"/>(Vt puta duæ die&longs;es)<emph.end type="italics"/><lb/>ide&longs;t duæ die&longs;es &longs;unt men&longs;ura vnius interualli mu&longs;ici, qui tonus appellatur: <lb/>quæ quidem duæ die&longs;es non &longs;unt men&longs;ura &longs;en&longs;ibilis, quæ &longs;cilicet auribus per<lb/>cipiatur, &longs;ed tantummodò exi&longs;tunt in numerorum proportionibus, ibi per <lb/>intellectum excogitatis, quando ait <emph type="italics"/>(Et voces plures quibus men&longs;uramus)<emph.end type="italics"/><lb/>quando vtimur eodem interuallo, &longs;iue eadem voce ad cantus men&longs;uram, <lb/>tunc &longs;unt plures men&longs;uræ numero, quamuis vna tantam &longs;pe<gap/> Ait <emph type="italics"/>(Et dia<lb/>meter duobus men&longs;uratur)<emph.end type="italics"/> v. </s>

<s>g. duobus &longs;emidiametris: v<gap/> pedibus. <lb/></s>

<s>duobus &longs;emidiametris: v<gap/> pedibus. <lb/></s>

<s>& latus pariter quadrati, duobus. </s>

<s>pedibus me<gap/> <expan abbr="m&qacute;">mque</expan>; mo<lb/>do reliquæ omnes magnitudines po&longs;&longs;unt ab ea<gap/> me<gap/> replica<lb/>ta men&longs;urari.</s>

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<s>Tex. 11. <emph type="italics"/>(Similia verò &longs;i cum non &longs;int eadem &longs;impliciter, nec &longs;ecundum &longs;ab&longs;t an<lb/>tiam &longs;ubiectam in d. </s>

<s>Tex. 11. <emph type="italics"/>(Similia verò &longs;i cum non &longs;int eadem &longs;impliciter, nec &longs;ecundum &longs;ab&longs;t an<lb/>tiam &longs;ubiectam in d. </s>

<s>fferentia &longs;ecundum formam eadem &longs;it: quemadmodum quadra<lb/>tum maius minori &longs;imile e&longs;t, & lineæ inæquales, hæ enim &longs;imiles quidem, ver&ubreve; non <lb/>cædem &longs;impliciter &longs;unt)<emph.end type="italics"/> Prima definitio &longs;exti definit &longs;imiles figuras eas e&longs;&longs;e, <lb/>quæ &longs;unt æquiangulæ inuicem, & quæ habent latera proportionalia circa <lb/>æquales angulos. </s>

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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>

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

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<s>Hactenus de numero c&oelig;lorum.</s>

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

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

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

<s>Qvidquid Mathematicum in his quæ&longs;tionibus occurret, illud, vt <lb/>plurimum per paraphra&longs;im exponemus, ita tamen, vt tex. </s>

<s>Qvidquid Mathematicum in his quæ&longs;tionibus occurret, illud, vt <lb/>plurimum per paraphra&longs;im exponemus, ita tamen, vt tex. Ari&longs;t. <lb/>& figuræ textui re&longs;pondentes per eam, quantum fieri poterit re<lb/>&longs;tituantur, & &longs;i quæ &longs;e offerent difficilia, pro viribus &longs;oluantur. <lb/></s>

<s>Ari&longs;t. <lb/>& figuræ textui re&longs;pondentes per eam, quantum fieri poterit re<lb/>&longs;tituantur, & &longs;i quæ &longs;e offerent difficilia, pro viribus &longs;oluantur. <lb/></s>

<s>E&longs;t autem nonnullis in locis textus tam græcus, quàm latinus adeò corrup<lb/>tus, ac deprauatus, vt nullo modo emendari queat.</s>

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<s>Quod autemea, quæ de&longs;cribit circulum linea, dum altero eins manente <lb/>extremo circumagitur, duabus &longs;imul feratur lationibus, ex quibus motus <lb/>orbicularis oriatur, manife&longs;tum e&longs;t ex &longs;uperioribus, quia & antror&longs;um, & <lb/>retror&longs;um impellitur; tùm etiam, quia &longs;i rectà tenderet recta <expan abbr="de&longs;crib&etilde;s">de&longs;cribens</expan> cir<lb/><figure id="fig73"/><lb/>culum, nunquam ad diametri perpendiculum <lb/>perueniret, &longs;ed tamen peruenit, ita vt &longs;it ip&longs;a <lb/>à centro perpendicularis diametro. </s>

<s>&longs;it circuli <lb/>figura A B C D, in qua extremum diametri <lb/>B, feratur ad alterum extremum vbi D, per <lb/>ip&longs;ius diametri B D, circumuolutionem circa <lb/>centrum F, nece&longs;&longs;e e&longs;t aliquando B, perueniat <lb/>ad C. &longs;i igitur B, feretur duabus lationibus <lb/>aliquo modo proportionatis, v. </s>

<s>vt e&longs;t pro<lb/>portio lateris B E, ad E C, latus, &longs;equeretur <lb/>ex demon&longs;tratis ip&longs;um B, ferri per <expan abbr="rectã">rectam</expan> B C, <lb/>quæ diameter e&longs;&longs;et quadrilateri B E C G. &longs;ed <pb pagenum="152"/>quia in nu&longs;la proportione fertur, propterea per circularem lineam B E C, <lb/>progreditur ad C, ita vt ip&longs;a diameter B D, in po&longs;itione A C, fiat perpendi<lb/>cularis priori diametro B D. ex quibus &longs;equitur eam moueri duobus moti<lb/>bus nullam rationem habentibus; quod erat intentum.</s>

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

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

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

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

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<s>con&longs;iderare enim oportet, quod in motu li<lb/>bræ de&longs;cribitur quidam circulus, cuius diameter &longs;unt ip&longs;a libræ <lb/>brachia, centrum verò e&longs;t fpartum, &longs;iue trutina; hoc enim pun<lb/>ctum in motu libræ manet: duo verò brachia &longs;unt veluti duæ &longs;emidiametri <lb/><figure id="fig75"/><lb/>à centro exeuntes, vt in figura cerne<lb/>re e&longs;t, in qua centrum, &longs;iue &longs;partum <lb/>e&longs;t vbi C, reliqua &longs;unt manife&longs;ta. </s>

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

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

<s>Iam vt præmi&longs;&longs;um e&longs;t, ea<lb/>dem vi, vel eodem onere in lance B, <lb/>po&longs;ito, mouebitur velocius brachium <lb/>libræ maioris, quàm minoris &longs;it ma<lb/>ior tran&longs;lata ad <expan abbr="loc&utilde;">locum</expan> D E, ergò com<lb/>mota e&longs;t per arcum B E, vel A D. </s>

<s>Minor autem libra acta e&longs;&longs;et per mino<lb/>rem arcum G I, vel F H, melius autem apparet arcus B E, maior, quam mi<lb/>nor G I, <expan abbr="atq;">atque</expan> hoc e&longs;t, quòd maiores libras exactiores facit. </s>

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<s>&longs;it deinde libra B C, cuius &longs;partum, <lb/>&longs;iue perpendiculum A D, &longs;it deor&longs;um, vt in altera figura, <expan abbr="&longs;it&qacute;">&longs;itque</expan>; circa pun<pb pagenum="156"/><figure id="fig79"/><lb/>ctum A, tanquam circa <expan abbr="centr&utilde;">centrum</expan>, aut axem <lb/>ita fixum, vt ip&longs;i libræ conuer&longs;io innita<lb/>tur, quæ e&longs;t altera libræ po&longs;itio. </s>

<s>Quærit <lb/>igitur, cur &longs;i in libra &longs;ur&longs;um <expan abbr="hab&etilde;te">habente</expan> per<lb/>pendiculum, & centrum, ponatur ex vna <lb/>parte onus quodpiam, v. </s>

<s>Quærit <lb/>igitur, cur &longs;i in libra &longs;ur&longs;um <expan abbr="hab&etilde;te">habente</expan> per<lb/>pendiculum, & centrum, ponatur ex vna <lb/>parte onus quodpiam, v. g. </s>

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

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<s>his præmi&longs;&longs;is &longs;ic quæ&longs;tioni &longs;atisfacit, & primò primæ parti, <lb/>quando nimirum &longs;partum &longs;upernè collocatum e&longs;t. </s>

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

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

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

<s>quod de omni vecte ab eo demon&longs;tratur, <lb/>cuius propo&longs;itionis &longs;en&longs;us e&longs;t hic; in &longs;uperiori prima figura &longs;i pars vectis <lb/>E B, fuerit, v.g. </s>

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

<s>qua drupla partis A E; etiam pondus C, erit quadruplo ma<lb/>ius pondere, &longs;eu vi in D, quæ ip &longs;um C, ope vectis &longs;u&longs;tinet. </s>

<s>quod etiam trans<lb/>ferre debes ad &longs;ecundam figuram.</s>

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<s>B, verò in mari palmula. </s>

<s>&longs;i igitur A, vbi D, transferatur, per totum &longs;pa<lb/>tium A D, non permeabit tantumdem &longs;patij B, <expan abbr="v&longs;q;">v&longs;que</expan> ad E. </s>

<s>&longs;i igitur A, vbi D, transferatur, per totum &longs;pa<lb/>tium A D, non permeabit tantumdem &longs;patij B, <expan abbr="v&longs;q;">v&longs;que</expan> ad E. </s>

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

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<s>Verum hoc non demon&longs;trat; <expan abbr="neq;">neque</expan> ex præmi&longs;&longs;is deduci pote&longs;t. </s>

<s>po&longs;tea <lb/>&longs;ubdit <emph type="italics"/>(Stans autem erit medium vbi e&longs;t G, in contrarium enim ip&longs;i, qu<foreign lang="greek">q</foreign>d in mari <lb/>e&longs;t, extremo B, procedit, vbi extremum in nauigio e&longs;t A, non procederet autcm <lb/>vbi est D, ni&longs;i commoueretur nauigi&ubreve;, & eò transferretur vbi e&longs;t remi principium)<emph.end type="italics"/><lb/>vbi in textu mendosè legitur C, pro G.</s>

<s>Sen&longs;us porrò horum verborum e&longs;t hic; &longs;i remus cirea &longs;calmum G, verte<lb/>retur, & tamen nauis ab eo non propelleretur, &longs;ed &longs;taret, tunc medium na<lb/>uis maneret vbi G, per motum enim remi impellitur in contrarias partes <lb/>ip&longs;i palmulæ B, quæ e&longs;t in mari, quia &longs;equitur motum alterius extremi A, <lb/>manubrij &longs;cilicet remi, qui e&longs;t in naui: quod autem nauigium à remo mo<lb/>neatur, &longs;ignum e&longs;t, quia manubrium A, non procederet vbi e&longs;t D, ni&longs;i pari<lb/>ter cum remo nauigium illor&longs;um con&longs;equeretur. </s>

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<s>Scalmus porrò quamquam circularis remi motus expers <lb/>&longs;it; motu tamen nauigij commouetur. </s>

<s>Remus igitur po&longs;itionem habeat in <lb/>fine ip&longs;ius remigationis rectam lineam D Z, quæ quidem rectam A B, &longs;ecec <lb/>in T, inter B, & C; rectam verò B E, in Z. </s>

<s>Remus igitur po&longs;itionem habeat in <lb/>fine ip&longs;ius remigationis rectam lineam D Z, quæ quidem rectam A B, &longs;ecec <lb/>in T, inter B, & C; rectam verò B E, in Z. </s>

<s>Et quoniam duo coalterni anguli <lb/>C A D, & C B E, æquales <expan abbr="o&longs;t&etilde;&longs;i">o&longs;ten&longs;i</expan> &longs;unt, & angulus A T D, contrapo&longs;ito B T Z, <lb/>æqualis e&longs;t: duo igitur triangula A T D, & B Z T, æquiangula erunt per 32. <lb/>primi, & communem &longs;ententiam. </s>

<s>Similia <expan abbr="itaq;">itaque</expan> erunt ip&longs;a triangula, <expan abbr="late-ra&qacute;">late<lb/>raque</expan>; habebunt proportionalia per 4. 6. &longs;icut A T, ad B T, ita D A, ad B Z. <lb/></s>

<s>Maior e&longs;t autem A T, quàm B T: maior igitur D A, quàm B Z, quod etiam <lb/>per <expan abbr="cõmunem">communem</expan> <expan abbr="&longs;ent&etilde;tiam">&longs;ententiam</expan> neglecta <expan abbr="triangulor&utilde;">triangulorum</expan> &longs;imilitudine concludi pote&longs;t.</s>

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<s>Vti<lb/>mur aurem tralatione, <expan abbr="atq;">atque</expan> demon&longs;trationis figura Victoris Fau&longs;ti. </s>

<s>Aduer<lb/>tendum e&longs;t tamen, quod cum remus po&longs;itionem habuerit D Z, remi palmu<lb/>la erit infra Z. </s>

<s>Aduer<lb/>tendum e&longs;t tamen, quod cum remus po&longs;itionem habuerit D Z, remi palmu<lb/>la erit infra Z. </s>

<s>Nam quoniam <expan abbr="triãguli">trianguli</expan> A D C, duo latera A C, & D C, æqua<lb/>lia po&longs;ita &longs;unc: duo igitur anguli, qui ad D, & A, æquales erunt: angulus <lb/>igitur A D T, angulo D A T, maior erit: & idcircò latus A T, trianguli A<lb/>T D, latere D T, maius erit per 19. primi. </s>

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

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

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

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

<s>Sit enim remus A C, manubrium A, &longs;calmus B, qui propter nauigij <lb/>motum &longs;patium percurrat à B, in D, in quo loco ip&longs;eremus A C, &longs;i<lb/><figure id="fig85"/><lb/>tum rectitudinis habeat E F. </s>

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

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<s>Esto iterum remus A C, manubrium A, &longs;calmus B: tantum autem &longs;pa<lb/>tium conficiat nauigium; quantum motu proprio A. Dico, quod C, <lb/>remi palmula immota manebit. </s>

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

<s>Excitetur autem à puncto B, in <expan abbr="vtramq;">vtramque</expan> partem linea E B R, ad <lb/><figure id="fig86"/><lb/>rectos angulos &longs;uper B G, & à <expan abbr="p&utilde;cto">puncto</expan> A, recta A H, <lb/>&longs;uper D F: itemque à puncto E, recta C E, &longs;uper <lb/>E R; ip&longs;arum verò rectarum linearum E R, & <lb/>A H, &longs;ectio &longs;it in K, &longs;ed C F., & D F, &longs;it in Z, & quo<lb/>niam A K, id &longs;patium e&longs;t, quod motu proprio re<lb/>mi manubrium permeauit, curuilineo enim re<lb/>&longs;pondeat A R, recta autem B G, id &longs;patium e&longs;t, <lb/>quod nauigium confecit: ip&longs;æ igitur rectæ lineæ <lb/>H K, & B G, æquales erunt. </s>

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<s>Si remi manubrium motu proprio duplum confecerit &longs;pa<lb/>tium, quàm nauigium, tantum prouehetur ea remiga<lb/>tione nauigium, quantum palmula retroce&longs;&longs;erit.</s>

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

<s>Excitetur à puncto B, in <expan abbr="vtramq;">vtramque</expan> <lb/>partem perpendicularis E Z, in quam veniant a punctis A, & C, <lb/>ad rectos angulos rectæ lineæ A E, & C Z: &longs;patium autem A E, à manubrio <pb pagenum="166"/><figure id="fig87"/><lb/>decur&longs;um motu proprio &longs;patij B D, duplum <lb/>&longs;it: recta verò linea C H, curuæ re&longs;pondeat <lb/>C G, quæ à remi palmula de&longs;cripta e&longs;t. </s>

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<s>In de&longs;cripta enim figura ponatur B D, minor quam A E, &longs;ed eius dimi<lb/>dio maior. </s>

<s>Dico, quod ip&longs;a B D, maior e&longs;t quàm C H. </s>

<s>Dico, quod ip&longs;a B D, maior e&longs;t quàm C H. </s>

<s>Nam B D, & <lb/>H Z, æquales &longs;unt: Ad hæc A E, & C Z, æquales &longs;unt rectæ lineæ; ma<lb/>ior igitur erit H Z, dimidio ip&longs;ius A E: quapropter reliqua C H, mi<lb/>nor dimidio erit eiu&longs;dem A E, & minor igitur erit C H, quàm B D. </s>

<s>Spa<lb/>tium autem B D, id e&longs;t, quod nauigium conficit, &longs;patium verò C H, remi <lb/>palmula in contrarium decurrit; idcircò prior pars Theorematis vera e&longs;t. <lb/></s>

<s>Po&longs;terior autem &longs;imiliter o&longs;tendetur. </s>

<s>&longs;i enim B D, minor e&longs;t dimidio ip&longs;ius <lb/>A E: minor igitur erit, & H Z, dimidio eiu&longs;dem A E; & quoniam A E, & <lb/>C Z, æquales &longs;unt: reliqua igitur C H, dimidio eiu&longs;dem A E, maior erit: & <lb/>proinde minor erit B D, quàm C H. </s>

<s>Nauigium igitur minus &longs;patium de<lb/>curret in anteriora, quam remi palmula in contrarium, quod demon&longs;tran<lb/>dum &longs;u&longs;cepimus.</s>

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

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

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

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

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

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

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

<s>Videtur autem <lb/>&longs;olo remorum impul&longs;u hoc fieri non po&longs;&longs;e, &longs;ed alia in&longs;uper virtute impel<lb/>lente opus e&longs;&longs;e, vt venti, vel aquæ.</s>

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

<s>dixi autem onus e&longs;&longs;e in D, quia &longs;i nauis vento ob&longs;i&longs;teret, ip&longs;a inuerte<lb/>retur hac ratione, vt puppis A, eleuata, prora B, demergeretur, manente <pb pagenum="169"/><figure id="fig89"/><lb/>veluti centro parte E. quia ve<lb/>rò ob maris liquiditatem na<lb/>uis minimè obfi&longs;tit, &longs;ed facilè <lb/>cedens à ventis vrgetur, hinc <lb/>fit, vt meritò dixerim pondus <lb/>nauis e&longs;&longs;e ad D, fulcimentum <lb/>verò ad E.</s>

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

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<s>mouentur, quia circulares figuræ <lb/>parua &longs;ui parte, & qua&longs;i in puncto planum, &longs;eu pauimentum contingunt, vn<lb/>de fit, vt <expan abbr="neq;">neque</expan> offen&longs;ent, <expan abbr="neq;">neque</expan> impingant; cuius cau&longs;a e&longs;t, quia à terra &longs;emo<lb/>tus e&longs;t angulus, ide&longs;t tali angulo planum contingunt, vt ab eo &longs;tatim rotæ <lb/>curuatura à terra eleuari incipiat, & propterea parum terræ hæreat: in fi<lb/>guris verò rectilineis, in quadrata. </s>

<s>&longs;ecus accidit, quia ab angulo ad an<lb/>gulum linea recta tenditur, vnde in ip&longs;ius volutatione po&longs;t contactum vnius <lb/>anguli tota recta linea &longs;equens, plano adaptabitur, & non &longs;emouebitur &longs;ta<lb/>tim in altum, & ideò multum offen&longs;abit, & impinget, <expan abbr="tarde&qacute;">tardeque</expan>; idcircò mo<lb/>uebitur. </s>

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

<s>hæc Ari&longs;t. </s>

<s>hæc Ari&longs;t. </s>

<s>Re<lb/>&longs;tat vt &longs;atisfaciam promi&longs;&longs;is.</s>

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<s>1. lib.

4. Geom. </s>

4. Geom. pract. <lb/></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.

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<s>nec minus vera videtur re&longs;pon&longs;io, cum ait <emph type="italics"/>(An quia <lb/>quanto maior fuerit illa, quæ à centro e&longs;t, in æquali <expan abbr="t&etilde;pore">tempore</expan> maius mouetur (patium)<emph.end type="italics"/><lb/>quæ quidem vera &longs;unt, &longs;i intelligantur hoc modo, nimirum, quod quando <lb/>plures <expan abbr="circul&longs;conc&etilde;trici">circul&longs;concentrici</expan>, <expan abbr="atq;">atque</expan> inuicem connexi fuerint, ita vt vans &longs;ine alijs <lb/>moueri nequeat, tunc quanto maior fuerit diameter, & con&longs;equenter cir<lb/>cunferentia, tanto velocius mouebitur. </s>

<s>&longs;i autem intelligantur de duobus <lb/>circulis ab inuicem &longs;eparatis, quorum vnus <expan abbr="ab&longs;q;">ab&longs;que</expan> altero moueri pote&longs;t, vt &longs;ie <lb/>quando vtimur modo rotula magna, modo parua ad aquam hauriendam <lb/>non videntur vera, in quo &longs;en&longs;u manife&longs;tè loquitur Ari&longs;t. </s>

<s>Quapropter vt &longs;in<lb/>cerè loquar, nunc ne&longs;cio, qua ratione Ari&longs;t ab errore excu&longs;are valeam, alijs <lb/>fortè occurret.</s>

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<s>Cvr libræ, quæ omni incumbente pondere &longs;unt vacuæ ab impo&longs;ito <lb/>pondere facilius mouentur, quàm &longs;i quopiam inexi&longs;tente pondere <lb/>aliud rur&longs;us onus &longs;uperaddatur. </s>

<s>&longs;imiliter etiam rota, & huiu&longs;modi <lb/>quippiam, quod grauius quidem e&longs;t, difficilius commouetur quàm <lb/>læue, v. </s>

<s>&longs;imiliter etiam rota, & huiu&longs;modi <lb/>quippiam, quod grauius quidem e&longs;t, difficilius commouetur quàm <lb/>læue, v. g. </s>

<s>rota ferrea difficilius, quàm lignea. </s>

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<s>Notandum primò, quæ Græcis <foreign lang="greek">*krexai,</foreign> ide&longs;t Crocæ dicuntur, Latinis <lb/>Vmbilicos appellari; de his enim loquitur Cic. </s>

<s>Notandum primò, quæ Græcis <foreign lang="greek">*krexai,</foreign> ide&longs;t Crocæ dicuntur, Latinis <lb/>Vmbilicos appellari; de his enim loquitur Cic. 2. de Oratore, vbi <lb/>&longs;ic, non audeo dicere de talibus viris, &longs;ed tamen ita narrare &longs;ole<lb/>bat Sceuola, conchas, eos, & vmbilicos ad Caietam, & ad Lucri<lb/>num legere con&longs;ueui&longs;&longs;e. </s>

<s>2. de Oratore, vbi <lb/>&longs;ic, non audeo dicere de talibus viris, &longs;ed tamen ita narrare &longs;ole<lb/>bat Sceuola, conchas, eos, & vmbilicos ad Caietam, & ad Lucri<lb/>num legere con&longs;ueui&longs;&longs;e. </s>

<s>hos autem vmbilicos exponunt Grammatici e&longs;&longs;e <lb/>lapillos paruos, acrotundos, polito&longs;que, de quibus etiam Ari&longs;t.

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<s>quod &longs;i &longs;cire aueas quantum iuuet, re&longs;pondeo ip&longs;um vires potentiæ <lb/>duplicare; adeo vt &longs;i quatuor. </s>

<s>homines erant nece&longs;&longs;arij ad pondus tol<lb/>lendum, auxilio huius &longs;implicis trochleæ duo tantum &longs;ufficiant. </s>

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<s>quod &longs;i plures aliæ rotulæ tam <lb/>&longs;upernè, quàm infernè addantur, vt&longs;olet in maioribus trochleis, quas ve<lb/>teres Poly&longs;pa&longs;tos, ide&longs;t multum trahentes dixerunt, augebuntur vires in in<lb/>finitum. </s>

<s>quod dixi de virium duplicatione con&longs;tat ex 6. & 7. propo&longs;itione <lb/>Archimedis de Aequip. </s>

<s>quod dixi de virium duplicatione con&longs;tat ex 6. & 7. propo&longs;itione <lb/>Archimedis de Aequip. </s>

<s>quia enim in vecte no&longs;tro L M, dupla e&longs;t proportio <lb/>inter L M, & L B, eadem etiam proportio erit inter pondus, & potentiam, <lb/>quare pondus C, duplum erit potentiæ in M, hoc e&longs;t à minore potentia &longs;ibi <lb/>&longs;ubdupla &longs;u&longs;tinebitur: & à quauis adhuc <expan abbr="quantumcunq;">quantumcunque</expan> maiore eleuabitur.</s>

<s>Qui plura de trochlea de&longs;iderat, adeat Guidi Vbaldi, Mechanica, cuius <lb/>auxilio fateor me verum &longs;en&longs;um harum Mechanicarum Ari&longs;t. & præ&longs;ertim <lb/>huius loci enuclea&longs;&longs;e. </s>

<s>quæ &longs;i cum Piccolominei expo&longs;itione contuleris, vide<lb/>bis eum nequaquam cognoui&longs;&longs;e, vbi nam vectis in trochlea lateret, eumque <lb/>tam &longs;uperiorem, quàm inferiorem <expan abbr="rotulã">rotulam</expan> æquè vectem facere; in quo etiam <lb/>Io. Bapti&longs;ta Benedictus pariter erra&longs;&longs;e videtur in &longs;uis &longs;peculationibus, cum <lb/>inferiores tantummodo vice vectium fungantur, vt probatum e&longs;t.</s>

<s>Bapti&longs;ta Benedictus pariter erra&longs;&longs;e videtur in &longs;uis &longs;peculationibus, cum <lb/>inferiores tantummodo vice vectium fungantur, vt probatum e&longs;t.</s>

<s><expan abbr="Atq;">Atque</expan> ex his &longs;atis mihi videtur textus, ac &longs;en&longs;us Ari&longs;t.

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<s>Secundò, o&longs;tendam angulum &longs;ecuris, qui <lb/>curuilineus e&longs;t, æqualem e&longs;&longs;e angulo trianguli æquilateri, qui rectilineus e&longs;t. <lb/></s>

<s>Proclus igitur in comm. </s>

<s>Proclus igitur in comm. 23. primi Euclidis, &longs;ic ait: o&longs;ten&longs;um fuit ab anti<lb/>quis, &longs;cilicet Geometris, quod angulus figuræ illius, quæ &longs;ecuri &longs;imilis e&longs;t, <lb/>æqualis e&longs;t angulo rectilineo, quippe qui duabus tertijs anguli recti æqualis <lb/>e&longs;t. </s>

<s>23. primi Euclidis, &longs;ic ait: o&longs;ten&longs;um fuit ab anti<lb/>quis, &longs;cilicet Geometris, quod angulus figuræ illius, quæ &longs;ecuri &longs;imilis e&longs;t, <lb/>æqualis e&longs;t angulo rectilineo, quippe qui duabus tertijs anguli recti æqualis <lb/>e&longs;t. </s>

<s>hanc anguli &longs;ecuris affectionem, cum nec ille, nec alij, quod &longs;ciam de<lb/>mon&longs;trent, ego paulò po&longs;t demon&longs;trabo. </s>

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<s>de<lb/>inde ex centro C, interuallo. </s>

<s>v. g. <lb/>C B, de&longs;cribatur circulus B F; &longs;i<lb/>militer eodem interuallo B D, ex <lb/>centro D, de&longs;cribatur circulus <lb/>B E; tandem ex B, centro, atque <lb/>eodem interuallo ducatur alius <lb/>circulus D E F C, qui priores duos &longs;ecabit in punctis E F. <expan abbr="cõ&longs;ideremus">con&longs;ideremus</expan> iam, <lb/>reliquis circulorum partibus ommi&longs;&longs;is, curuilineam figuram B E F, quam <lb/>e&longs;&longs;e veteris &longs;ecuris formam ex <expan abbr="&longs;ent&etilde;tia">&longs;ententia</expan> Proclinon e&longs;t dubitandum, cum cir<lb/>c<gap/>is &longs;e mutuò per centra &longs;ecantibus con&longs;tituatur, vt vult ip&longs;e, & præterea <lb/>habeat angulos E F, tantos, quantos ip&longs;e tradit, vt mox patebit; linea au<lb/>tem A B C, &longs;ecuris manubrium refert.</s>

<s>g. <lb/>C B, de&longs;cribatur circulus B F; &longs;i<lb/>militer eodem interuallo B D, ex <lb/>centro D, de&longs;cribatur circulus <lb/>B E; tandem ex B, centro, atque <lb/>eodem interuallo ducatur alius <lb/>circulus D E F C, qui priores duos &longs;ecabit in punctis E F. <expan abbr="cõ&longs;ideremus">con&longs;ideremus</expan> iam, <lb/>reliquis circulorum partibus ommi&longs;&longs;is, curuilineam figuram B E F, quam <lb/>e&longs;&longs;e veteris &longs;ecuris formam ex <expan abbr="&longs;ent&etilde;tia">&longs;ententia</expan> Proclinon e&longs;t dubitandum, cum cir<lb/>c<gap/>is &longs;e mutuò per centra &longs;ecantibus con&longs;tituatur, vt vult ip&longs;e, & præterea <lb/>habeat angulos E F, tantos, quantos ip&longs;e tradit, vt mox patebit; linea au<lb/>tem A B C, &longs;ecuris manubrium refert.</s>

<s>Quod autem tam angulus E, quàm angulus F, &longs;int æquales duabus tertijs <lb/>vnius angulirecti, &longs;iue quod idem e&longs;t angulo trianguli æquilateri, manife<lb/>ftum erithoc modo. </s>

<s>De&longs;cribatur iterum &longs;ecuralis figura prædicto mode, <lb/><gap/><expan abbr="itq;">itque</expan> ea A B C. ducantur præterea ad &longs;ingulos angulos tres rectæ A B, B C, <lb/>C A, quæ con&longs;tituunt trianguium æquilaterum A B C, tria enim ip&longs;ius late<lb/><figure id="fig98"/><lb/>ra &longs;ubtendunt tres arcus æquales A B, B C, C A, <lb/>&longs;unt enim tres &longs;extantes æqualium circulorum, <lb/><gap/>t facilè colligi pote&longs;t ex 15. 4. ex quo etiam &longs;e<lb/>quitur tres ilias circulorum portiones, quas re<lb/>ctè cum &longs;uis arcubus con&longs;tituunt e&longs;&longs;e inuicem <lb/>æquales, & limiles portiones nimirum A B E, <lb/>B C D, C A F. hinc pr&ecedil;terea &longs;equitur angulos ip<lb/>&longs;arum e&longs;&longs;e inuicem æquales, angulos, v.g. </s>

<s>A B E, <lb/>C B D, mixtos e&longs;&longs;e æquales, quod facilè e&longs;t per imaginatiam &longs;uperpo&longs;itio<lb/>nem demon&longs;trare. </s>

<s>cum igitur prædicti duo anguli &longs;int æquales, &longs;itque intet <lb/>eos medius alius angulus E B C, qui pariter mixtus e&longs;t, &longs;i ip&longs;e addatur tanl <lb/>angulo C B D, quàm angulo A B E, inuicem æqualibus, erunt duo anguli <pb pagenum="181"/>A B C, rectilineus, & E B D, curuilineus æquales. </s>

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<s>quod <lb/>epigramma carminibus loco linearum con&longs;tat, quæ in &longs;ecuris formam con<lb/>&longs;tituta &longs;unt.</s>

<s>Sciendum namque e&longs;t Simmiam, poeticam hanc &longs;ecurim concinna&longs;&longs;e in <lb/>gratiam Epei illius, qui equum Troianum ligneum fuerat architectatus, vt <lb/>e&longs;t apud Virg. </s>

<s>Et ip&longs;e doli fabricator Epeus. </s>

<s>Et ip&longs;e doli fabricator Epeus. </s>

<s>qui cum &longs;oluendi voti cau&longs;a <lb/>vellet &longs;ecurim, &longs;iue bipennem, qua in equi Durij molitione v&longs;us fuerat, Mi<lb/>neruæ Deæ, quod &longs;ibi in eo opere faciendo auxilio fui&longs;&longs;et, dedicare, <expan abbr="eam&qacute;">eamque</expan>; <lb/>vt Ari&longs;t.

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<s>quæ dedicatio, &longs;iue epigramma, quod adhuc extat, deinceps &longs;e<lb/>curis Simmiæ vocitata e&longs;t; ex qua figura bipennis illius, equi Durij fabrica<lb/>tricis nobis adhuc magna cum voluptate innotuit. </s>

<s>Porrò gratum, <expan abbr="atq;">atque</expan> ad <lb/>ea, quæ diximus intelligenda vtile Lectori fore arbitrati &longs;umus, ip&longs;am Sim<lb/>miæ bipennem ex operibus Theocriti, quibus addi &longs;olet, huc referre; quam <lb/>P. Ricardus E&longs;ius de no&longs;tra Societate linguæ græcæ periti&longs;&longs;imus, in hunc <lb/>modum tran&longs;tulit. </s>

<s>Ricardus E&longs;ius de no&longs;tra Societate linguæ græcæ periti&longs;&longs;imus, in hunc <lb/>modum tran&longs;tulit. </s>

<s>hoc autem ordine legenda e&longs;t: lectio à manubrio <lb/>incipiat, deinde legatur carmen; forti&longs;&longs;imæ Deæ, quod &longs;ub&longs;e<lb/>quatur; dedit Epeus, & &longs;ic in orbem lectio, <expan abbr="v&longs;q;">v&longs;que</expan> ad me<lb/>dium circumducatur. </s>

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conijcio <lb/>primum ha&longs;ta oblonga, qualis e&longs;t in <lb/>præ&longs;enti figura A B, ex cuius altero <lb/>extremo B, pendebat appendicu<lb/>lum, quod propriè æquipondium <lb/>dicitur: ex altera verò extremitate <lb/>A, lanx vna pendebat; in qua carnes, aliæuè merces ponderabantur: in me<lb/>dia <expan abbr="deniq;">denique</expan> ha&longs;ta paribus interuallis plures trutinæ, ex quibus &longs;ingulis modo <lb/>hac, modo illa, prout pondus emptoris po&longs;tulabat &longs;u&longs;pendebatur, <expan abbr="atq;">atque</expan> in<lb/>terim tantum mercis lanci imponebatur, donec æquipondio præpondera<lb/>ret in æquilibrio. </s>

<s>&longs;ingulæ autem trutinæ ad aliquod determinatum pondus <lb/>trutinandum, erant con&longs;titutæ, v. </s>

<s>&longs;ingulæ autem trutinæ ad aliquod determinatum pondus <lb/>trutinandum, erant con&longs;titutæ, v. g. </s>

<s>vna ad &longs;ex libras, altera ad octo, &c. <lb/></s>

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<s>eadem &longs;it proportio inter pondus mer<lb/>cis, & pondus æquipondij, quæ e&longs;t permutatim inter di&longs;tantias vtrinque ab <lb/>a&longs;&longs;umpta trutina, quæ in trutinando hypomoclij vicem gerit: nam &longs;tatera <lb/>reducitur ad vectem; pondus erit æquipondium; & merces in lance erit po<lb/>tentia mouens: &longs;unt autem in tota &longs;tateræ ha&longs;ta trutinæ plures, hoc enim <lb/>modo tota fit vniformis quoad pondus. </s>

<s>æquipondium præterea debet ha<lb/>bere tantum pondus, quantum e&longs;t in nuda lance, vt &longs;ic tota &longs;tatera &longs;it per &longs;e <lb/>&longs;ola æquilibrabilis: & præterea debet habere pondus &longs;tatum, a c legitimum, <lb/>v. </s>

<s>vnius libræ, aut duarum, aut trium, prout magis <expan abbr="trutinãdæ">trutinandæ</expan> merci ido<lb/>neum erit, & hoc erit proprium æquipondij pondus. </s>

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<s>Quod autem punctus A, motu illo de&longs;eri<lb/>bat lineam A D, punctus verò B. lineam B C, manife&longs;tum erit hoc modo. </s>

<s>&longs;it <lb/>v. </s>

<s>&longs;it <lb/>v. g. </s>

<s>punctum A, motu proprio delatum, <expan abbr="v&longs;q;">v&longs;que</expan> ad punctum E, medium late<lb/>ris A B, erit interim totum latus A B, tran&longs;latum vbi e&longs;t F G, hoc e&longs;t, ad &longs;ui <lb/>itineris dimidium, quia horum motus ponuntur æquales: hoc autem motu <lb/>ip&longs;um punctum A, erit nece&longs;&longs;ariò in K, hoc e&longs;t in linea A D, vt dicebamus. <lb/></s>

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<s>Notandum autem, quod re&longs;tes æquales &longs;unt cum &longs;uis curuaturis. </s>

<s>re<lb/>&longs;tis A B, cum &longs;ua curuatura B C, æqualis e&longs;t re&longs;ti C D, vnà cum eius curua<lb/>tura D H, & aliæ eodem modo &longs;e habent, quia eadem demon&longs;tratio omni<lb/>bus accommodari pote&longs;t: quia enim figura A B G M, parallelogrammum <lb/>e&longs;t, æqualia enim &longs;unt latera B G, A M, & quot foramina &longs;unt in vno, tot <lb/>etiam &longs;unt in altero, <expan abbr="ea&qacute;">eaque</expan>; inuicem æquidi&longs;tant, &longs;equitur omnes re&longs;tes e&longs;&longs;e <lb/>parallelas, & æquales, per 33, primi. </s>

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<s><expan abbr="Atq;">Atque</expan> hic e&longs;t &longs;en&longs;us Ari&longs;t.

quamuis tex. </s>

quamuis tex. ip&longs;ius propter nimiam tam in græ<lb/>cis, quàm in latinis codicibus corruptionem, totus re&longs;titui nequiuerit.</s>

<s>ip&longs;ius propter nimiam tam in græ<lb/>cis, quàm in latinis codicibus corruptionem, totus re&longs;titui nequiuerit.</s><figure/><pb pagenum="192"/>

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

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<s>&longs;i verò addatur onus G, tunc quidem paulò difficilius intingemus, <lb/>&longs;ed tamen vas plenum po&longs;tea multò facilius, quod opus, & labor e&longs;t, &longs;ur&longs;um <lb/>educemus: operæpretium igitur e&longs;t, onus illud plumbi, aut lapidis adiun<lb/>gere in extremo A, quia &longs;ic pondus illud tanquam quædam potentia vecte <lb/>A B, vtens &longs;ur&longs;um hydriam plenam rapiet, <expan abbr="hac&qacute;">hacque</expan>; ratione nos labore leua<lb/>bit, <expan abbr="totum&qacute;">totumque</expan>; hauriendi opus demi&longs;&longs;ione, <expan abbr="atq;">atque</expan> eleuatione <expan abbr="con&longs;tãs">con&longs;tans</expan>, alleuabit,</s><pb pagenum="194"/>

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