texts/archimedes/xml/carda_propo_015_la_1570.xml - Colored diff

Main History Search Repository tree

Return to carda_propo_015_la_1570.xml CVS log

Up to [CVSROOT] / texts / archimedes / xml

Colored diff for /texts/archimedes/xml/carda_propo_015_la_1570.xml between version 1.24 and 1.25

version 1.24, 2004/04/06 17:23:28

version 1.25, 2004/04/06 17:28:10

Line 3991

<s id="id000959">Quintum, corpora dura magis læduntur à latis, quia &longs;cindun</s><s id="id000960"><arrow.to.target n="marg173"/><lb/>tur, mollia autem à tenuibus, quia diuiduntur: nam mollitie excipi<lb/>unt aërem, & ita à latis non adeò patiuntur, & etiam, quoniam nec <lb/>franguntur, nec &longs;ponte &longs;cinduntur.</s>

<s id="id000959">Quintum, corpora dura magis læduntur à latis, quia &longs;cindun</s>

<s id="id000960"><arrow.to.target n="marg173"/><lb/>tur, mollia autem à tenuibus, quia diuiduntur: nam mollitie excipi<lb/>unt aërem, & ita à latis non adeò patiuntur, & etiam, quoniam nec <lb/>franguntur, nec &longs;ponte &longs;cinduntur.</s>

Line 4036

Line 4039

<s id="id000978">Sit graue a b c alligatum funibus in d ef, dico, <lb/><figure id="id.015.01.073.1.jpg" xlink:href="015/01/073/1.jpg"/><lb/>quòd facilius trahetur per fe quàm c b & e b, quàm <lb/>d a, quia &longs;i debet trahi ex a uel b, aut cadet, aut uis ex <lb/>a & b communicabitur c, igitur erit minor quàm in <lb/>c, & hoc naturaliter. </s>

<s id="id000979">Mathematica autem ratione quoniam ex a tra<lb/>hetur c, qua&longs;i per lineam d c: at attractio recta e&longs;t ualidior obliqua <lb/>igitur attractio c per d e&longs;t debilior, quàm per f. </s>

<s id="id000980">Rur&longs;us &longs;i e trahitur <lb/>per d cùm a peruenerit in d, erit perinde ac, &longs;i attractum e&longs;&longs;et per li<lb/>neam c d, &longs;ed linea c d mouet duobus motibus, uno ad &longs;uperiora, al</s><s id="id000981"><arrow.to.target n="marg177"/><lb/>tero ad latus, ergo lentius ad f per d c quàm f c, quod erat demon<lb/>&longs;trandum.</s>

<s id="id000981"><arrow.to.target n="marg177"/><lb/>tero ad latus, ergo lentius ad f per d c quàm f c, quod erat demon<lb/>&longs;trandum.</s>

<s id="id000982"><margin.target id="marg177"/>P<emph type="italics"/>er<emph.end type="italics"/> 59. <emph type="italics"/>buius.<emph.end type="italics"/></s>

Line 4051

Line 4057

<s id="id000986">Demon&longs;tratum e&longs;t &longs;uperius quòd &longs;i mobile &longs;it &longs;ph&etail;ricum, & tan</s><s id="id000987"><arrow.to.target n="marg179"/><lb/>gat planum in puncto, quòd mouetur per quancunque uim aptam <lb/>diuidere medium. </s>

<s id="id000986">Demon&longs;tratum e&longs;t &longs;uperius quòd &longs;i mobile &longs;it &longs;ph&etail;ricum, & tan</s>

<s id="id000987"><arrow.to.target n="marg179"/><lb/>gat planum in puncto, quòd mouetur per quancunque uim aptam <lb/>diuidere medium. </s>

<s id="id000988">Quia ergo &longs;i tangat in puncto facillime moue<lb/>tur, &longs;i in linea paulò difficilius, &longs;i per &longs;uperficiem adhuc difficilius, <lb/>igitur cum fiat attritio in motu quanto latius e&longs;t mobile eo diffici<lb/>lius mouetur. </s>

<s id="id000989">Sit ergo mobile a b, quod moueatur uer&longs;us c, & quia <lb/>pars b &longs;eu dimidium mouetur iuxta rationem me<lb/><figure id="id.015.01.073.2.jpg" xlink:href="015/01/073/2.jpg"/><lb/>dietatis, & pars a eodem modo, ergo conduplicata <lb/>difficultate, quia medietas b impedit medietatem, a <lb/>quanto latius e&longs;t, & longius a b, tanto difficilius <lb/><arrow.to.target n="marg180"/><lb/>mouetur. </s>

<s id="id000990">Et hoc intelligitur de corporibus ualde <lb/>latis propter dicta &longs;uperius.</s>

Line 4074

Line 4083

<s id="id000996">Graue de&longs;cendit naturaliter quatuor cau&longs;is: prima e&longs;t ponderis <lb/>magnitudo, unde quod grauius e&longs;t celerius de&longs;cendit. </s>

<s id="id000997">Secundò ob <lb/>paruam medij repugnantiam, ideo quanto medium e&longs;t rarius & <lb/>mobile tenuius, tanto celerius de&longs;cendit: contrà uerò tardius. </s>

<s id="id000998">Ter<lb/>tiò ob impetum aëris &longs;ub &longs;equentis: & ideo mobile quòd ex eadem </s><s id="id000999"><arrow.to.target n="marg182"/><lb/>materia con&longs;tat, &longs;emper de&longs;cendit parte acutiore &longs;uprapo&longs;ita, ne aër <lb/>cogatur celerius ferri: & quanto diutius de&longs;cendit, tanto magis in<lb/>tenditur motus, at que augetur, ut &longs;uprà de claratum e&longs;t. </s>

<s id="id000998">Ter<lb/>tiò ob impetum aëris &longs;ub &longs;equentis: & ideo mobile quòd ex eadem </s>

<s id="id000999"><arrow.to.target n="marg182"/><lb/>materia con&longs;tat, &longs;emper de&longs;cendit parte acutiore &longs;uprapo&longs;ita, ne aër <lb/>cogatur celerius ferri: & quanto diutius de&longs;cendit, tanto magis in<lb/>tenditur motus, at que augetur, ut &longs;uprà de claratum e&longs;t. </s>

<s id="id001000">Quarta cau&longs;a <lb/>e&longs;t, quod non impediatur ab aëre tran&longs;uer&longs;im moto, et à latere: ideo <lb/>leuia mobilia & magna non &longs;olum lentius de&longs;cendunt, quoniam <lb/><arrow.to.target n="marg183"/><lb/>paruam uim habeant, & magnam repugnantiam, &longs;ed quia tran&longs;uer<lb/><arrow.to.target n="marg184"/><lb/>&longs;im impul&longs;a minus mouentur motu recto, ut &longs;upra ui&longs;um e&longs;t. </s>

<s id="id001001">Por<pb pagenum="55" xlink:href="015/01/074.jpg"/>rò proportio ratione de&longs;cen&longs;us aucta, declarata e&longs;t paulo antè, <lb/>quare cum medium &longs;upponatur eiu&longs;dem generis, & figura non <lb/>eiu&longs;modi, nec leuitas, ut pror&longs;us non impellat, nedum ut moueat la<lb/>tus: figura quo que eadem ambobus relinquetur proportio motus <lb/>ad motum producta ex proportionibus incrementi in proportio<lb/><arrow.to.target n="marg185"/><lb/>nem ponderum, & iam habuimus proportionem incrementi ex <lb/><arrow.to.target n="marg186"/><lb/>motu aëris, ergo proportio unius motus producti ad alteram no<lb/>ta erit.</s>

<s id="id001001">Por

<pb pagenum="55" xlink:href="015/01/074.jpg"/>rò proportio ratione de&longs;cen&longs;us aucta, declarata e&longs;t paulo antè, <lb/>quare cum medium &longs;upponatur eiu&longs;dem generis, & figura non <lb/>eiu&longs;modi, nec leuitas, ut pror&longs;us non impellat, nedum ut moueat la<lb/>tus: figura quo que eadem ambobus relinquetur proportio motus <lb/>ad motum producta ex proportionibus incrementi in proportio<lb/><arrow.to.target n="marg185"/><lb/>nem ponderum, & iam habuimus proportionem incrementi ex <lb/><arrow.to.target n="marg186"/><lb/>motu aëris, ergo proportio unius motus producti ad alteram no<lb/>ta erit.</s>

<s id="id001002"><margin.target id="marg182"/>P<emph type="italics"/>ropo&longs;.<emph.end type="italics"/> 30.</s>

Line 4141

Line 4156

<s id="id001036">Aliter &longs;tante &longs;uppo&longs;itione ut Ludouicus Ferrarius ex demon<lb/>&longs;tratis à Ptolemæo quadratum b c, & e&longs;t 1 quad. e&longs;t æquale produ<lb/>cto ex b d in c e, quod e&longs;t 1, & a b in d c, igitur detracto 1, produ<lb/>cto b d in c e ex 1 quad. </s>

<s id="id001036">Aliter &longs;tante &longs;uppo&longs;itione ut Ludouicus Ferrarius ex demon<lb/>&longs;tratis à Ptolemæo quadratum b c, & e&longs;t 1 quad e&longs;t æquale produ<lb/>cto ex b d in c e, quod e&longs;t 1, & a b in d c, igitur detracto 1, produ<lb/>cto b d in c e ex 1 quad. </s>

<s id="id001037">quadrato c b, relinquitur productum ex <lb/>a b in c d 1 quad. </s>

<s id="id001038">m: 1, ergo diui&longs;o co per a b, quæ e&longs;t 1, relinquitur <lb/>c d 1 quad. </s>

<s id="id001039">m: 1 huius uerò quadratum per <expan abbr="ead&etilde;">eadem</expan> demon&longs;trata à Pto

<pb pagenum="56" xlink:href="015/01/075.jpg"/>lemæo, &etail;quale e&longs;t rectangulis ex b c in de, & b d in c e, igitur 1 quad. <lb/></s>

Line 4160

Line 4177

<s id="id001051">m: 2 æquantur 1 <lb/>quad. </s>

<s id="id001052">m: 1, igitur 1 cu. </s>

<s id="id001053">p: 1 æquantur 1 quad. </s>

<s id="id001055">Aliter ut Pacciolus, concurrant latera eptagoni b d, c e in a, & du<lb/>cantur perpendiculares a f, d g & c d, & &longs;it c e i ca 1 pos, & quia ut <lb/><arrow.to.target n="marg192"/><lb/>a e ad a c, ita d e ad b c, erit ergo b c (1 posp: 1)/(1 pos) quare b f (1/2 pos 1/2,)/(2 pos) & <lb/>quia d h e&longs;t dimidium d e, erit d h, & g f <lb/><figure id="id.015.01.075.1.jpg" xlink:href="015/01/075/1.jpg"/><lb/>1/2, cum ergo b f &longs;it (1/2 pos p: 1/2)/pos erit ergo di<lb/>ui&longs;a 1/2 pos per 1 pos, & exit 1/2, b f 1/2p: 1/2/pos <lb/>igitur detracta g f relinquetur g b 1/2/(1 pos). <lb/>& eius quadratum 1/4/(1 quad). igitur cum qua<lb/>dratum b d &longs;it 1, erit quadratum g d 1 m: <lb/>2/4/(2 quad)g c autem e&longs;t compo&longs;ita ex e f, quæ <lb/>e&longs;t 1/2p: 1/2/(1 pos) & f g quæ e&longs;t 1/2, erit igitur c <lb/>g 1 p: 1/2/(1 pos), & <expan abbr="quadrat&utilde;">quadratum</expan> eius 1 p: 1/pos e&longs;t 1/4/(1 quad.) quare <expan abbr="&qtilde;drat&utilde;">quadratum</expan> e d &qring;d e&longs;t <lb/><arrow.to.target n="marg193"/><lb/>compo&longs;itum ex quadratis c g & g d erit 2 p: 1/pos c a uerò e&longs;t æqua<lb/>lis c d, quia, ut demon&longs;tratum e&longs;t angulus d c e e&longs;t &longs;eptima pars <lb/>duorum rectorum, & angulus b c e ei duplus, quare cum c f a &longs;it re<lb/>ctus erit ex trige&longs;ima &longs;ecunda primi Elementorum f a c tres &longs;epti<lb/>mæ unius recti, ergo d a c 6/7 unius recti, d c a uerò 2/7 unius recti, quia <lb/><arrow.to.target n="marg194"/><lb/>e&longs;t &longs;eptima pars duorum rectorum, ígitur a d c e&longs;t 6/7 unius recti: igi<lb/>tur c d e&longs;t æqualis c a, ergo quadratum quadrato: igitur 1 quad. </s>

<s id="id001056">p: 2 <lb/>pos p: 1, æquatur 2 p: 1/(1 pos) igitur 1 quad. </s>

<s id="id001057">p: 2 pos, æquantur 1 p: 1/(1 pos). <lb/>Quare 1 cub. </s>

Line 4314

Line 4333

<s id="id001085">Omnis cylindrus &longs;phæræ habenti eundem magnum <lb/><arrow.to.target n="marg205"/><lb/>circulum, & altitudinem &longs;exquialter e&longs;t. </s>

<s id="id001086">Omnis &longs;phæra dupla e&longs;t <lb/><arrow.to.target n="marg206"/><lb/>cono, cuius ba&longs;is e&longs;t eius circulus magnus, & altitudo eadem, quæ <lb/>&longs;phæræ ip&longs;ius. </s>

<s id="id001087">Omnis &longs;uperficies &longs;phæræ quadrupla e&longs;t maiori <lb/><arrow.to.target n="marg207"/><lb/>&longs;uo circulo. </s>

<s id="id001088">Superficies portionis &longs;phæræ e&longs;t æqualis circulo, cu<lb/><arrow.to.target n="marg208"/><pb pagenum="58" xlink:href="015/01/077.jpg"/>ius &longs;emidiameter e&longs;t linea ducta à uertice portionis ad finem illius.</s>

<s id="id001088">Superficies portionis &longs;phæræ e&longs;t æqualis circulo, cu <lb/><arrow.to.target n="marg208"/>

<pb pagenum="58" xlink:href="015/01/077.jpg"/>ius &longs;emidiameter e&longs;t linea ducta à uertice portionis ad finem illius.</s>

Line 4561

Line 4583

<s id="id001200">Primum, tres æquales quantitates non po&longs;&longs;unt diuidi in tres, & <lb/>tres quantitates in continua proportione ordinatè, ut dixi, ni&longs;i u<lb/>triu&longs;que ordinis tres, ac tres inuicem &longs;int æquales.</s>

<s id="id001201">Secundum, tres quantitates in æquali exce&longs;&longs;u ordinate, ut dixi, <lb/>non po&longs;&longs;unt diuidi in tres, & tres quantitates, quæ &longs;int in eadem <lb/>proportione quantumcunque proportiones illæ duorum ordinum <lb/>sint diuer&longs;æ.</s>

<s id="id001202">Tertium, tres quantitates, quæ &longs;int in eadem proportione non <lb/>po&longs;&longs;unt diuidi ordinate in tres ac tres, quæ &longs;int in continua propor<lb/>tione ni&longs;i &longs;int ambæ proportiones eædem cum proportione ip&longs;a<lb/>rum quantitatum.</s>

Line 4854

Line 4876

<s id="id001320">tardius ad <lb/>a per b a, quam per c a, at per c a tardius, quam ex b in a per æqua<lb/>lem uim, ergo multo tardius ex b in a per c a uentum, quam per uen<lb/>tum ex b in a. </s>

<s id="id001321">Quærimus ergo compo&longs;itionem horum, ut &longs;it c <lb/>nauis, quæ debeat transferri ad a per uentum ex b, & &longs;equitur, <lb/>quod tardius, quam ex c per uentum ex c in a, & tardius ex b per <lb/>uentum ex cin a. </s>

<s id="id001322">Ergo malus, qui in prora e&longs;t conuoluto eo, qui <lb/>e&longs;t in puppi, ut etiam Ari&longs;toteles docet tantundem nititur ad re<lb/><arrow.to.target n="marg281"/><lb/>ctum ex cin æquidi&longs;tantem locum ab a quantum c di&longs;tat ab con<lb/>tra temo, qui in puppi e&longs;t dirigitur ad h, & &longs;i ualidius &longs;it uentus e<lb/>tiam adiuuante temonem, &longs;eu contra nitente, quantum licet mo<lb/>bili pondere nauis ad id latus, premitur enim nauis, qua&longs;i &longs;ubmer<lb/>gi debeat, uento in aduer&longs;um premente, ut &longs;i uentus repente huic <lb/>contrarius exoriatur, <expan abbr="pericul&utilde;">periculum</expan> &longs;ubeat, ne obruatur. </s>

<s id="id001323">Cum ergo uen<lb/>tus ex b feratur, æquidi&longs;tans c h, & c feratur per temonem in k, & ab <lb/>oppo&longs;itis æqualis actio &longs;equatur, imò tota impeditur, ex c in h fere<lb/>tur iuxta proportionem anguli, quem con&longs;tituit h c cum a c ad tot<lb/>um rectum, Si igitur ex c in a debuit ferri in duodecim horis ob

<s id="id001323">Cum ergo uen<lb/>tus ex b feratur, æquidi&longs;tans c h, & c feratur per temonem in k, & ab <lb/>oppo&longs;itis æqualis actio &longs;equatur, imò tota impeditur, ex c in h fere<lb/>tur iuxta proportionem anguli, quem con&longs;tituit h c cum a c ad to<lb/>um rectum, Si igitur ex c in a debuit ferri in duodecim horis ob

<pb pagenum="70" xlink:href="015/01/089.jpg"/>uim uenti, & uiæ longitudinem, angulus uerò h c a &longs;it &longs;exta re<lb/>cti pars, feretur ex c uer&longs;us a ad quantitatem b a in quatuorde<lb/>cim horis: igitur rur&longs;us quanta e&longs;t proportio c a ad b a tan<lb/>tum e&longs;t temporis, in quo fertur ex c ad a ad quatuor decim horas <lb/>per uentum b a.</s>

Line 4896

Line 4918

<s id="id001338">Vela tribus in locis di&longs;poni &longs;olent dolo b, quod in prora con<lb/>&longs;tituitur, & in malo, qui ponitur in medio ratione, quæ inferius <lb/>o&longs;tendetur, &longs;ed non ad unguem, quia cum malus in anteriorem <lb/>partem à uento impellatur, &longs;i e&longs;&longs;et in medio, &longs;emper præmeretur <lb/>nauis in anteriorem partem, ex quo duo magna incommoda &longs;eque <lb/>rentur: primùm ut periculum &longs;ubiret, ne inuer&longs;a in anteriorem par<pb pagenum="71" xlink:href="015/01/090.jpg"/>tem &longs;ubmergeretur. </s>

<pb pagenum="71" xlink:href="015/01/090.jpg"/>tem &longs;ubmergeretur. </s>

<s id="id001339">Secundum ne pre&longs;&longs;a in parte anteriore dif<lb/>ficilius aquas di&longs;&longs;ecaret, & ob id longe tardius, moueretur. </s>

<s id="id001340">Pro<lb/>pter hæc duo incommoda igitur malus etiam &longs;i unicus e&longs;&longs;et <lb/>(quod uulgati&longs;simum maioribus no&longs;tris | fuit) in parte magis <lb/>proræ proxima locabatur à gubernatoribus, ut e&longs;&longs;et qua&longs;i in trien<lb/>te à ro&longs;tro in be&longs;&longs;e à puppi: Rarum fuit, & memorabile, quod nunc <lb/>pa&longs;sim habet olim Antigoni <foreign lang="greek">triame/ou &</foreign> 1, uelorum trium: quorum <lb/>po&longs;tremum Epidromus ut ip&longs;a uoce intelligamus non fui&longs;&longs;e ue<lb/>lum in malo ip&longs;o medio, &longs;ed in puppi con&longs;titutum. </s>

<s id="id001341">Cau&longs;a Dolonis <lb/>inferius exponetur: quod autem e&longs;&longs;et paruum, & omnium mini<lb/>mum, ut nauis facile ab eo inuerteretur. </s>

Line 4913

Line 4938

<s id="id001352">Sed nos de uelorum uarieta<lb/>te loquimur, non ea', quæ ad malos pertinet. </s>

<s id="id001353">Con&longs;tat enim me<lb/>dio loco plus mouere, quam in extremis, ut infrà docebi<lb/>mus. </s>

<s id="id001354">Antiquo enim tempore opus non fuit malorum mul<lb/>titudine, quoniam &longs;ijderibus uias dirigebant ob id non ad <lb/>amu&longs;sim, quoniam linea dirigi non poterat maximè ob mo<lb/>tus obliquitatem in circulo ui&longs;us: ideò mali multi confu<lb/>&longs;ionem in cur&longs;u, & impedimentum in naui, maiu&longs;qúe pericu<lb/>lum attuli&longs;&longs;ent. </s>

<s id="id001355">At nunc inuenta pyxide, & lapidis Her<pb pagenum="72" xlink:href="015/01/091.jpg"/>culei auxilio pluribus locis uela di&longs;po&longs;ita melius dirigunt iter, ut <lb/>qua&longs;i cra&longs;&longs;a minerua depictum, & pote&longs;tate deformatum, ad amu&longs;<lb/>&longs;im contrahant. </s>

<s id="id001355">At nunc inuenta pyxide, & lapidis Her

<pb pagenum="72" xlink:href="015/01/091.jpg"/>culei auxilio pluribus locis uela di&longs;po&longs;ita melius dirigunt iter, ut <lb/>qua&longs;i cra&longs;&longs;a minerua depictum, & pote&longs;tate deformatum, ad amu&longs;<lb/>&longs;im contrahant. </s>

<s id="id001356">Motus ergo magnitudo non &longs;impliciter con&longs;tat, <lb/>&longs;ed comparatione &longs;uperficiei ueli ad uelum longitudine quidem, </s>

Line 5583

Line 5611

<s id="id001673">Voco motum inuer&longs;ionis, qui &longs;imilis e&longs;t motui &longs;phæræ, &longs;cili<lb/>cet circumuertendo graue à uertice, & manife&longs;tum e&longs;t, quòd in <lb/>quacunque figura, qua graue in&longs;idet plano per punctum ue</s><s id="id001674"><arrow.to.target n="marg339"/><lb/>lut ouata ip&longs;um mouetur à quauis ui, &longs;ed &longs;i in&longs;ideat per &longs;uperfi<lb/>ciem, quanto maior e&longs;t, & humilior, tanto difficilius mouetur, <lb/>ideò in corpore uiginti ba&longs;ium, quòd inter regularia uocata, plu<lb/>res habet, &longs;uperficies pro ratione æqualis ponderis, motus erit <lb/>longe facilior. </s>

<s id="id001674"><arrow.to.target n="marg339"/><lb/>lut ouata ip&longs;um mouetur à quauis ui, &longs;ed &longs;i in&longs;ideat per &longs;uperfi<lb/>ciem, quanto maior e&longs;t, & humilior, tanto difficilius mouetur, <lb/>ideò in corpore uiginti ba&longs;ium, quòd inter regularia uocata, plu<lb/>res habet, &longs;uperficies pro ratione æqualis ponderis, motus erit <lb/>longe facilior. </s>

<s id="id001675">Alia cau&longs;a e&longs;t inæqualitas partium, unde quæ ro<lb/>tunda &longs;unt, quia prominent, facile mouentur, & cum partes me<lb/>diæ in&longs;i&longs;tant plano, quanto minores erunt tanto facilius moue<lb/>buntur ratione ponderis. </s>

<s id="id001676">Vnde patet, quòd corpora ouata faci<lb/>lius mouentur, etiam quàm &longs;phærica, habent enim partem me<lb/>diam minorem, & paria &longs;unt ratione ince&longs;&longs;us plani, &longs;ed aëris mul<lb/>titudine tardius, quoniam enim &longs;phæra &longs;ub æquali ambitu plus <lb/>continet corporis, ergo ouatum æquale &longs;phæræ habet maio<lb/>rem ambitum ip&longs;a &longs;phæra. </s>

<s id="id001677">Hæc autem à Theone partim de<lb/>mon&longs;trata &longs;unt, partim ab Archimede, & partim à nobis, ergo <lb/>motus ouati e&longs;t fermè æqualis motui &longs;phæræ, & tardior e&longs;t con<lb/><figure id="id.015.01.110.1.jpg" xlink:href="015/01/110/1.jpg"/><lb/>citatus, quàm &longs;phæræ, quia à ma<lb/>iore excipitur aëre, & partes exte<lb/>riores non ita incumbunt in me<lb/>dium &longs;ecundum longitudinem. </s>

<s id="id001678">Cu<lb/>bus uero tardior e&longs;t propter æqua<lb/>litatem, & latitudinem &longs;uperficiei in<lb/>ferioris, omnium <expan abbr="aut&etilde;">autem</expan> minime pro<lb/>pter has cau&longs;as conus ambligonius, <lb/>& quanto magis fuerit, ratio uero <lb/>eleuationis e&longs;t, ut &longs;it cubus b c, cuius <lb/>medium grauitatis &longs;it b &longs;uper pla<pb pagenum="92" xlink:href="015/01/111.jpg"/>no de, & eleuetur ex a, & manife&longs;tum e&longs;t, quod in&longs;idebit per totam <lb/>lineam c f ip&longs;i plano, & proportio grauitatis totius &longs;u&longs;pen&longs;i in com<lb/>paratione ad grauitatem eius, qui inuertit, e&longs;t, uelut proportio par<lb/>tis terminatæ ad lineam c f uer&longs;us eum, qui eleuat ad partem, quæ <lb/>ultra e&longs;t, cum uerò hæ partes notæ &longs;int iuxta perpendiculum ex <lb/>centro grauitatis, manife&longs;tum e&longs;t, quod &longs;ciemus pondus corporis <lb/>a b cf, dum inuertitur in quo cunque &longs;itu ad pondus eius, dum &longs;u<lb/>&longs;penditur, & clarum e&longs;t, quòd cùm centrum, & medium grauitatis <lb/>fuerint in una linea per c f, tunc nulla erit grauitas.</s>

<pb pagenum="92" xlink:href="015/01/111.jpg"/>no de, & eleuetur ex a, & manife&longs;tum e&longs;t, quod in&longs;idebit per totam <lb/>lineam c f ip&longs;i plano, & proportio grauitatis totius &longs;u&longs;pen&longs;i in com<lb/>paratione ad grauitatem eius, qui inuertit, e&longs;t, uelut proportio par<lb/>tis terminatæ ad lineam c f uer&longs;us eum, qui eleuat ad partem, quæ <lb/>ultra e&longs;t, cum uerò hæ partes notæ &longs;int iuxta perpendiculum ex <lb/>centro grauitatis, manife&longs;tum e&longs;t, quod &longs;ciemus pondus corporis <lb/>a b cf, dum inuertitur in quo cunque &longs;itu ad pondus eius, dum &longs;u<lb/>&longs;penditur, & clarum e&longs;t, quòd cùm centrum, & medium grauitatis <lb/>fuerint in una linea per c f, tunc nulla erit grauitas.</s>

Line 5602

Line 5636

<s id="id001683">Sit a b, quæ &longs;i appen&longs;a e&longs;&longs;et ad æquidi<lb/><figure id="id.015.01.111.1.jpg" xlink:href="015/01/111/1.jpg"/><lb/>&longs;tantem terræ &longs;uperficiei, nulla ui po&longs;&longs;et ele</s><s id="id001684"><arrow.to.target n="marg341"/><lb/>uari, inflectatur ergo ad c punctum, omi&longs;&longs;a <lb/>c g, & manife&longs;tum e&longs;t, quod &longs;i b c in&longs;i&longs;teret <lb/><arrow.to.target n="marg342"/><lb/>ad perpendiculum, ponderaret a c &longs;i e&longs;&longs;et in <lb/>æquilibrio, ponatur ergo accliuis in c d per <lb/>notum angulum. </s>

<s id="id001684"><arrow.to.target n="marg341"/><lb/>uari, inflectatur ergo ad c punctum, omi&longs;&longs;a <lb/>c g, & manife&longs;tum e&longs;t, quod &longs;i b c in&longs;i&longs;teret <lb/><arrow.to.target n="marg342"/><lb/>ad perpendiculum, ponderaret a c &longs;i e&longs;&longs;et in <lb/>æquilibrio, ponatur ergo accliuis in c d per <lb/>notum angulum. </s>

<s id="id001685">Quia igitur b c ad c a no<lb/>ta e&longs;t, erit dicta &longs;uperiùs notum pondus <lb/>b h, po&longs;ita h c æquali c a, quare totius a b, <lb/>& iam fuit e k notum, & punctus d notus: <lb/>hoc enim infrà demon&longs;trabitur, qualis igitur proportio lineæ <lb/><arrow.to.target n="marg343"/><lb/>tran&longs;uer&longs;æ dl ad lineam de&longs;cendentem d m, talis differentiæ pon<lb/>derum c m, & c e, id e&longs;t partis ad partem. </s>

<s id="id001686">hæc autem inferiùs de<lb/>mon&longs;trabuntur. </s>

<s id="id001687">Neque enim ab&longs;urdum e&longs;t in materijs mi&longs;tis, ali<lb/><arrow.to.target n="marg344"/><lb/>quando uti nondum demon&longs;tratis cum fuerint mathematica, quia <lb/>obtinent principij rationem, quod etiam facit Archimedes. </s>

Line 5664

Line 5701

<s id="id001709"><arrow.to.target n="marg349"/><lb/>e&longs;&longs;e in pondere eleuando, quòd pondera auxilio uectium facilius <lb/>mouentur, quàm manibus. </s>

<s id="id001710">Rotulæ uerò in trochleis uectes &longs;unt, <lb/>& axis mi&longs;ta hypomochlij, ergo facilius pondus trahitur per u<lb/>nam rotulam, quàm &longs;i manu traheretur, at uerò per duas tres, <lb/>unde tris pa&longs;&longs;us longe facilius, & etiam facilius per quinque, unde <lb/>pentas pa&longs;&longs;us, nam quinque orbiculis, qua&longs;i totidem uectibus <lb/>diui&longs;um pondus manife&longs;tè fit leuius, & ut dictum e&longs;t, tanquam <lb/>totidem uectibus pondus eleuatur, e&longs;tqúe proportio produ

<pb pagenum="94" xlink:href="015/01/113.jpg"/>cta, &longs;emperque prior hypomochlij locum habet, ueruntamen minus <lb/>a&longs;&longs;umit laboris, po&longs;terior uerò uectis maiorem partem &longs;ibi ponde<lb/>ris &longs;eruat, uelut in &longs;uccula etiam iugum traiectum per plures colo<lb/>pes facilius uertitur. </s>

<s id="id001711">Et &longs;i quis dicat nónne totum pondus in&longs;idet <lb/>prim&etail; trochleæ per trochleam, intelligo nunc &longs;olùm rotulam cum <lb/>ip&longs;o axe, &longs;eu axiculo (ut dicunt) non autem in proprio &longs;ignificato, <lb/>in quo etiam funis traiectus, & in&longs;idens rotulæ, &longs;eu rotulis, nam <lb/>una trochlea plures continere' pote&longs;t orbiculos, & axes. </s>

<s id="id001712">Licet ergo <lb/>pondus in&longs;ideat primæ trochleæ, &longs;eu rotulæ, in eo tamen, quod tra<lb/>hitur, diuiditur', licet non æqualiter dico, præter id funis motum <lb/>intendi. </s>

Line 5683

Line 5722

<s id="id001718">Solent gemmarij uendere adamantem ponderis unius grani <lb/>uno coronato, duorum autem granorum tribus coronatis, qua<lb/>tuor autem, gratia exempli, quadraginta coronatis, qu&etail;ritur quan<lb/>tum ualebit adamas octo granorum, quoniam ergo proportio <lb/>non &longs;eruatur. </s>

<s id="id001719">E&longs;t enim in pondere utraque dupla, in pretio autem <lb/>ex prima habetur tripla, ex &longs;ecunda habetur proportio maior, <lb/>quàm tredecim ad unum, propterea utendum e&longs;t proportione <lb/>propinquiori, &longs;i &longs;atis faceret. </s>

<s id="id001719">E&longs;t enim in pondere utraque dupla, in precio autem <lb/>ex prima habetur tripla, ex &longs;ecunda habetur proportio maior, <lb/>quàm tredecim ad unum, propterea utendum e&longs;t proportione <lb/>propinquiori, &longs;i &longs;atis faceret. </s>

<s id="id001720">gratia exempli, in prima ad ditione fuit <lb/>unum granum, & acqui&longs;iuit proportionem triplam, in &longs;ecunda fue<lb/>runt duo grana, &longs;i ergo acqui&longs;i&longs;&longs;et &longs;olum &longs;excuplam proportio<lb/>nem, haberemus intentum. </s>

<s id="id001721">Propterea in i&longs;to ca&longs;u oportet demon<lb/>&longs;trare forma Geometrica, &longs;uppo&longs;ito, quòd &longs;it figura recta ex uno la <lb/><figure id="id.015.01.113.1.jpg" xlink:href="015/01/113/1.jpg"/><lb/>tere a b, ita ut angulus, uel minimus capiat b c æqualem a b, & ex <lb/>æquali b a c addito fiat b d tripla b c, & ex angulo b a e duplo b a d, <lb/>fiat b c d e quadragintupla a b, & iuxta rationem erit in infinitum. <lb/></s>

<s id="id001722">Siue &longs;it parabole, &longs;iue hyperbole, &longs;eu &longs;it alia coincidentium.</s>

Line 6078

Line 6117

<s id="id001918">igitur per <expan abbr="commun&etilde;">communem</expan> animi <expan abbr="&longs;ententiã">&longs;ententiam</expan> addito communi pondere a e, <lb/>erit pondus a b minus &longs;emper cum agina e&longs;t in d, <08> in ullo alio lo<lb/>co a b. </s>

<s id="id001919">Ergo pondus a b apprehen&longs;um in d <expan abbr="mouebi&ttilde;">mouebitur</expan> a b æquali ui <lb/><arrow.to.target n="marg395"/><lb/>maiore proportione, <08> in ullo alio loco. </s>

<s id="id001920">Ha&longs;tile ergo in medio ap<lb/>prehen&longs;um maiore ui mouebitur, quàm in ulla alia parte. </s>

<s id="id001921">Et &longs;i gra<pb pagenum="106" xlink:href="015/01/125.jpg"/>cilius &longs;it in anteriore parte propinquius comprehen&longs;um calci, & &longs;i <lb/>cra&longs;sius, uel grauius propius cu&longs;pidi. </s>

<s id="id001921">Et &longs;i gra

<pb pagenum="106" xlink:href="015/01/125.jpg"/>cilius &longs;it in anteriore parte propinquius comprehen&longs;um calci, & &longs;i <lb/>cra&longs;sius, uel grauius propius cu&longs;pidi. </s>

<s id="id001922">Semper igitur ob hanc cau<lb/>&longs;am mota ex medio grauitatis, &longs;eu uelo, &longs;eu ramo, &longs;eu manu uelo<lb/>cius mouentur, quàm ex alijs partibus. </s>

<s id="id001923">In remo etiam pote&longs;t acce<lb/>dere illud commodum, cuius meminit Ari&longs;toteles. </s>

<s id="id001924">Propter hoc igi<lb/>tur, qui malum in naui collo carunt tantùm unum, in medio fermè <lb/>eum collocarunt, ut antiqui: & qui duos aut tres, maiorem cra&longs;sio<lb/><arrow.to.target n="marg396"/><lb/>rem &longs;cilicet, & altiorem in medio con&longs;tituerunt.</s>

Line 6163

Line 6205

<s id="id001965">Multa ergo in huiu&longs;modi anomalis motibus con<lb/>&longs;ideranda &longs;unt, ut illorum impetum robur, ac locum definiamus.</s>

<s id="id001966">Ex hoc liquet, cur plumbeæ &longs;phærulæ longius ferantur à tor</s><s id="id001967"><arrow.to.target n="marg402"/><lb/>mento emi&longs;&longs;æ, quàm ligneæ, etiam &longs;i non frangantur.</s>

<s id="id001966">Ex hoc liquet, cur plumbeæ &longs;phærulæ longius ferantur à tor</s>

<s id="id001967"><arrow.to.target n="marg402"/><lb/>mento emi&longs;&longs;æ, quàm ligneæ, etiam &longs;i non frangantur.</s>

Line 6243

Line 6288

<s id="id002012">Sed, ut dixi, &longs;it baculus applicatus alicui la<lb/>pidi ea parte, qua facilius pote&longs;t impelli & trahi, & quæritur, quæ <lb/>maior &longs;it uis, an attrahendi? </s>

<s id="id002013">& dico quòd homo, uel conatur trahe<lb/>re toto corpore, & impellere, at que hoc modo magis trahit, quàm <lb/>impellet, quoniam corporis pondus melius adhibetur in tractione <lb/>quàm impul&longs;u: uel citra corporis pondus, &longs;ed &longs;ola ui membrorum: <lb/>& tunc magis impellit, quoniam impul&longs;us fit corpore prono in <expan abbr="anterior&etilde;">an<lb/>teriorem</expan> partem, quæ inclinatio, & motus e&longs;t naturalis magis, quàm <lb/>in attractione in partem po&longs;teriorem. </s>

<s id="id002014">Sed ubi nulla &longs;it diuer&longs;itas <lb/>neque horum, neque figurarum æqualis uis æqualem efficit motum: <lb/>quia impul&longs;us impellentis comparatione e&longs;t attractio re&longs;pectu al<lb/>terius. </s>

<s id="id002015">Verùm non e&longs;t eadem uis nec propè par impellendi, at que <lb/>attrahendi hominibus, cum attractio fiat per mu&longs;culos ad origi<pb pagenum="111" xlink:href="015/01/130.jpg"/>nem &longs;uam naturaliter &longs;e retrahentibus impul&longs;ui nullum in&longs;trumen<lb/>tum à natura delegatum inuenio, nam ad exten&longs;ionem mu&longs;culi &longs;a<lb/>nè ex aduer&longs;o &longs;unt fabricati: cum ergo duo &longs;int tantum motus mu<lb/>&longs;culorum ten&longs;io, dum <expan abbr="retrah&utilde;tur">retrahuntur</expan> ad principium &longs;uum, & remi&longs;sio, <lb/>dum membrum quie&longs;cit in naturali nullus erit locus impul&longs;ioni, <lb/>ni&longs;i ex con&longs;equentia non per &longs;e, quamobrem multo infirmiorem il<lb/>lum attractione in brachijs e&longs;&longs;e, nece&longs;&longs;e e&longs;t.</s>

<s id="id002015">Verùm non e&longs;t eadem uis nec propè par impellendi, at que <lb/>attrahendi hominibus, cum attractio fiat per mu&longs;culos ad origi

<pb pagenum="111" xlink:href="015/01/130.jpg"/>nem &longs;uam naturaliter &longs;e retrahentibus impul&longs;ui nullum in&longs;trumen<lb/>tum à natura delegatum inuenio, nam ad exten&longs;ionem mu&longs;culi &longs;a<lb/>nè ex aduer&longs;o &longs;unt fabricati: cum ergo duo &longs;int tantum motus mu<lb/>&longs;culorum ten&longs;io, dum <expan abbr="retrah&utilde;tur">retrahuntur</expan> ad principium &longs;uum, & remi&longs;sio, <lb/>dum membrum quie&longs;cit in naturali nullus erit locus impul&longs;ioni, <lb/>ni&longs;i ex con&longs;equentia non per &longs;e, quamobrem multo infirmiorem il<lb/>lum attractione in brachijs e&longs;&longs;e, nece&longs;&longs;e e&longs;t.</s>

Line 6329

Line 6377

<s id="id002062">Si ergo <expan abbr="etiã">etiam</expan> in <expan abbr="id&etilde;">idem</expan> tenderent, <expan abbr="nõ">non</expan> efficerent mille <lb/>ictus &qring;d tres, cuius demon&longs;tratio h&etail;c e&longs;t. </s>

<s id="id002063">Supponamus <expan abbr="proportion&etilde;">proportionem</expan> <lb/>b d c ad <expan abbr="&qtilde;rtam">quartam</expan> <expan abbr="part&etilde;">partem</expan> a d b ad dito re&longs;iduo ad b d c e&longs;&longs;e <expan abbr="&longs;ol&utilde;">&longs;olum</expan> <expan abbr="decuplã">decuplam</expan>: <lb/><expan abbr="t&utilde;c">tunc</expan> ex duob. </s>

<s id="id002064">ictibus centupla erit in d c ad <expan abbr="eã">eam</expan>, qu&etail; in b e, <expan abbr="etiã">etiam</expan> tribus <lb/>millecupla: nam <expan abbr="cõqua&longs;&longs;ata">conqua&longs;&longs;ata</expan> turri in primo ictu, id d decuplo magis <lb/>ad perpendiculum <08> in b d e <expan abbr="&longs;uma&ttilde;">&longs;umatur</expan> decima pars in ambitu d, & illa <lb/>erit ergo <expan abbr="tã">tam</expan> di&longs;&longs;oluta, & infirma ex &longs;uppo&longs;ito, <08> e&longs;t tota b e: &longs;ed ex &longs;e<lb/>cundo ictu decuplo magis <expan abbr="cõqua&longs;&longs;abi&ttilde;">conqua&longs;&longs;abitur</expan> illa pars, <08> b e ergo tota d c <lb/>centuplo magis <expan abbr="qua&longs;&longs;abi&ttilde;">qua&longs;&longs;abitur</expan> ex duob. </s>

<s id="id002065">ictibus c d turris, <08> b e, & ita in <lb/>tribus: ex <expan abbr="dec&etilde;">decem</expan> millibus ergo ictibus <expan abbr="etiã">etiam</expan> ad amu&longs;sim directis, <expan abbr="c&utilde;">cum</expan> ta<lb/><expan abbr="m&etilde;id">men id</expan> uix fieri po&longs;sit in <expan abbr="tãta">tanta</expan> multitudine <expan abbr="nõ">non</expan> plus <expan abbr="cõminue&ttilde;">comminuetur</expan> b d e, <08><lb/>ex decë c d <expan abbr="&ptilde;ter">propter</expan> <expan abbr="quã">quam</expan> <expan abbr="exigu&utilde;">exiguum</expan> <expan abbr="quippiã">quippiam</expan> in &longs;uperficie. </s>

<s id="id002065">ictibus c d turris, <08> b e, & ita in <lb/>tribus: ex <expan abbr="dec&etilde;">decem</expan> millibus ergo ictibus <expan abbr="etiã">etiam</expan> ad amu&longs;sim directis, <expan abbr="c&utilde;">cum</expan> ta<lb/><expan abbr="m&etilde;id">men id</expan> uix fieri po&longs;sit in <expan abbr="tãta">tanta</expan> multitudine <expan abbr="nõ">non</expan> plus <expan abbr="cõminue&ttilde;">comminuetur</expan> b d e, <08><lb/>ex decë c d <expan abbr="&ptilde;ter">pnter</expan> <expan abbr="quã">quam</expan> <expan abbr="exigu&utilde;">exiguum</expan> <expan abbr="quippiã">quippiam</expan> in &longs;uperficie. </s>

<s id="id002066">Imo ut <expan abbr="declarat&utilde;">declaratum</expan> <lb/>e&longs;t multo minus repetita ratione multiplicis. </s>

<s id="id002067">Ob id in arce <expan abbr="Mediolan&etilde;&longs;i">Medio<lb/>lanen&longs;i</expan> exterius lapidibus uiuis in <expan abbr="rotund&utilde;">rotundum</expan> diducta &longs;uperficie inter<lb/><figure id="id.015.01.133.2.jpg" xlink:href="015/01/133/2.jpg"/><lb/>uallo que <expan abbr="&qtilde;">quae</expan>drato hunc in <expan abbr="mod&utilde;">modum</expan> munit&etail; &longs;unt altiores tur<lb/>res. </s>

<s id="id002068">Fiat ergo murus cuius proportio a d c ad b d a &longs;it &longs;ex<lb/>quitertia, erit que angulus b d c <expan abbr="dodrãs">dodrans</expan> recti, & <expan abbr="par&utilde;">parum</expan> incli<lb/>natis, <expan abbr="&longs;iquid&etilde;">&longs;iquidem</expan> b d c erit quarta pars recti, & &longs;it tant&etail; ma<lb/>gnitudinis, at que duritiei, ac adeò benè coniunctus fer<lb/><arrow.to.target n="table16"/><lb/>reis cathenis, ac &longs;tolonibus, ut po&longs;sit re&longs;i&longs;tere <expan abbr="machinar&utilde;">machinarum</expan> <expan abbr="ferenti&utilde;">fe<lb/>rentium</expan> <expan abbr="&longs;ph&etail;rã">&longs;ph&etail;ram</expan> <expan abbr="librar&utilde;">librarum</expan> ducentarum (quæ &longs;anè maximæ &longs;unt) <lb/><figure id="id.015.01.133.3.jpg" xlink:href="015/01/133/3.jpg"/>quinquaginta: <expan abbr="t&utilde;c">tunc</expan> cum proportio &longs;exquitertia nouies repeti<lb/>ta, ut in numeris uides, efficiat quinquies replicatis nouem <lb/>ictibus, fiet proportio decupla quinquies producta, qu&etail; e&longs;t cen <lb/><expan abbr="t&utilde;">tum</expan> millium ad <expan abbr="un&utilde;">unum</expan> in quadraginta quin que ictibus. </s>

Line 6412

Line 6460

<s id="id002101">per&longs;&etail;pe <expan abbr="etiã">etiam</expan> omnino &longs;upra <expan abbr="fid&etilde;">fidem</expan>. </s>

<s id="id002102">Ita <expan abbr="nõ">non</expan> conquie&longs;cere oportet donec, <lb/>uel omnino à capto de&longs;inat ho&longs;tis, aut <expan abbr="loc&utilde;">locum</expan> occupet &longs;ibi <expan abbr="relict&utilde;">relictum</expan> po<lb/>tius <08> <expan abbr="qu&etilde;">quem</expan> elegerit. </s>

<s id="id002103">nam <expan abbr="experiment&utilde;">experimentum</expan> frequens docuit, ubi illæ ma<lb/>gn&etail; uires &longs;uo arbitrio <expan abbr="loc&utilde;">locum</expan>, <expan abbr="qu&etilde;">quem</expan> <expan abbr="eleger&utilde;t">elegerunt</expan> obtinere potuerint, <expan abbr="tand&etilde;">tandem</expan> <lb/>potiri locis <expan abbr="quãtumuis">quantumuis</expan> munitis in hoc &qring;d diximus <expan abbr="cõtra">contra</expan> <expan abbr="oppona&ttilde;">opponatur</expan>. <lb/></s>

<s id="id002104">Etenim <expan abbr="&longs;ept&etilde;">&longs;eptem</expan> modis <expan abbr="c&utilde;">cum</expan> urbes, at que arces <expan abbr="capian&ttilde;">capiantur</expan>, <expan abbr="quor&utilde;">quorum</expan> duo &longs;unt ex<lb/>tra <expan abbr="&ptilde;&longs;ent&etilde;">pre&longs;entem</expan> <expan abbr="con&longs;ideration&etilde;">con&longs;iderationem</expan> ob&longs;idio, <expan abbr="&qtilde;">quae</expan> magnitudine ambitus loci <expan abbr="tolli&ttilde;">tol<lb/>litur</expan>, & proditio, <expan abbr="&qtilde;">quae</expan> cu&longs;to<expan abbr="d&utilde;">dum</expan> <expan abbr="uigilãtia">uigilantia</expan>, cuniculi, euer&longs;io &longs;uperioris muri, <lb/>euer&longs;io ab imo per machinas, cuniculi, &longs;eu &longs;uffo&longs;sio, urbis euer&longs;io, &longs;eu <lb/><expan abbr="&etail;dificior&utilde;">&etail;dificiorum</expan>: & <expan abbr="&qtilde;uo">qua uo</expan>cant aggre&longs;sio, &longs;eu oppugnatio per &longs;calas, & crates <lb/><expan abbr="c&utilde;">cum</expan> &longs;agittarijs: his omnib. </s>

<s id="id002104">Etenim <expan abbr="&longs;ept&etilde;">&longs;eptem</expan> modis <expan abbr="c&utilde;">cum</expan> urbes, at que arces <expan abbr="capian&ttilde;">capiantur</expan>, <expan abbr="quor&utilde;">quorum</expan> duo &longs;unt ex<lb/>tra <expan abbr="&ptilde;&longs;ent&etilde;">pn&longs;entem</expan> <expan abbr="con&longs;ideration&etilde;">con&longs;iderationem</expan> ob&longs;idio, <expan abbr="&qtilde;">quae</expan> magnitudine ambitus loci <expan abbr="tolli&ttilde;">tol<lb/>litur</expan>, & proditio, <expan abbr="&qtilde;">quae</expan> cu&longs;to<expan abbr="d&utilde;">dum</expan> <expan abbr="uigilãtia">uigilantia</expan>, cuniculi, euer&longs;io &longs;uperioris muri, <lb/>euer&longs;io ab imo per machinas, cuniculi, &longs;eu &longs;uffo&longs;sio, urbis euer&longs;io, &longs;eu <lb/><expan abbr="&etail;dificior&utilde;">&etail;dificiorum</expan>: & <expan abbr="&qtilde;uo">qua uo</expan>cant aggre&longs;sio, &longs;eu oppugnatio per &longs;calas, & crates <lb/><expan abbr="c&utilde;">cum</expan> &longs;agittarijs: his omnib. </s>

<s id="id002105"><expan abbr="&longs;atisfact&utilde;">&longs;atisfactum</expan> puto, pr&etail;ter <08> oppugnationi pro<lb/>pter <expan abbr="humilitat&etilde;">humilitatem</expan> <expan abbr="muror&utilde;">murorum</expan>: <expan abbr="nã">nam</expan> lignis <expan abbr="opplen&ttilde;">opplentur</expan>, at que fa&longs;ciculis, terra que fo&longs;<lb/>&longs;&etail;: nihil. </s>

<s id="id002107">re&longs;i&longs;tit immen&longs;&etail; illi pote&longs;tati, & crudelitati <expan abbr="&longs;&etail;ui&longs;simor&utilde;">&longs;&etail;ui&longs;simorum</expan> ty<lb/><expan abbr="rãnor&utilde;">rannorum</expan>. </s>

Line 6432

Line 6480

<s id="id002121">Sed puerulo indito luxus, ignauiæ, crudelitatis atque <lb/>&longs;tultiti&etail; fontibus, mirabile dictu &longs;anè, & ad proportionem diuino<lb/>rum <expan abbr="in&longs;trumentor&utilde;">in&longs;trumentorum</expan> pertinens. </s>

<s id="id002122">Sed redeamus ad in&longs;titutum: Video <lb/>enim, quid po&longs;sit obijci, &longs;cilicet muros cra&longs;&longs;os, et altiores tueri <expan abbr="urb&etilde;">urbem</expan> <lb/>& ædificia illius po&longs;&longs;e ab&longs;que aggeris erectione, & &longs;i <expan abbr="diruan&ttilde;">diruantur</expan> manere <lb/>etiam nihilominus imo magis, quod e&longs;t terram, u&longs;que <expan abbr="quoniã">quoniam</expan> eadem <lb/>ratione manet, quia concuti non po&longs;sit à machinis: nec ho&longs;tes id cu<lb/>raturos, &longs;perantes hoc <expan abbr="&longs;ol&utilde;">&longs;olum</expan> &longs;ufficere, quod m&oelig;nia &longs;olo <expan abbr="æquen&ttilde;">æquentur</expan>, at que id <lb/><expan abbr="fact&utilde;">factum</expan> e&longs;t Mediolani, & in arce eius, <expan abbr="t&utilde;">tum</expan> Papi&etail; & in Cremonen&longs;i arce. <lb/></s>

<s id="id002123">Verùm ni fallor, ut paruis arcibus à tanta ui tormentorum nullum <lb/>e&longs;t <expan abbr="præ&longs;idi&utilde;">præ&longs;idium</expan>, aut &longs;alutis &longs;pes, ita neque <expan abbr="cõuenit">conuenit</expan>, ut muris humilibus ag<lb/>geri confidant, nam & pauci homines tanto labori non &longs;ufficerent, <lb/>& agger cum fo&longs;&longs;a effo&longs;&longs;a &longs;cilicet terra defen&longs;ores nimis in <expan abbr="angu&longs;t&utilde;">angu&longs;tum</expan> <lb/>cogeret. </s>

<s id="id002124">At in urbibus contra eueniet: muris enim erectis altius ma<lb/>chinæ lapidum fru&longs;tis hominem <expan abbr="occid&etilde;t">occident</expan>: an percu&longs;&longs;a &longs;uperiore par <lb/>te ob coniunctionem inferior concutitur, & in de <expan abbr="tot&utilde;">totum</expan> &longs;imul cadit, <lb/>ut uidimus Papi&etail;, quo <expan abbr="cad&etilde;te">cadente</expan>, & fo&longs;&longs;a impletur, & <foreign lang="greek">tEIkole/tois</foreign> facilior <lb/>aditus ad &longs;ubruendum reliquas partes <expan abbr="pr&etail;be&ttilde;">pr&etail;betur</expan>: imò percul&longs;i defen<pb pagenum="118" xlink:href="015/01/137.jpg"/>&longs;ores &longs;æpe muneris &longs;ui obliui&longs;cuntur, de&longs;ertaque ea parte liberum <lb/>ingre&longs;&longs;um ho&longs;tibus exhibent. </s>

<pb pagenum="118" xlink:href="015/01/137.jpg"/>&longs;ores &longs;æpe muneris &longs;ui obliui&longs;cuntur, de&longs;ertaque ea parte liberum <lb/>ingre&longs;&longs;um ho&longs;tibus exhibent. </s>

<s id="id002125">Tum uerò magis, quod non confi<lb/>dunt animo <expan abbr="nõ">non</expan> ad id parato, po&longs;&longs;e aggerem &longs;ufficientem, & in tam <lb/>breui tempore ex&longs;truere, & etiam intelligunt, antequam erigatur, <lb/>patere à lateribus introitum ho&longs;tibus.</s>

Line 6486

Line 6537

<s id="id002147">Quibus con&longs;titutis, qui &longs;tabunt iuxta l, & m longitudines aëris <lb/>moti, & loci, per quem tran&longs;it flabellum, &longs;entient magnum uentum, <lb/>quoniam cum corpus m x l ab extremis partibus &longs;it elatius a b ex<lb/>tremis, &longs;tantes, & alti tangentur à uento agitato. </s>

<s id="id002148">Si uero &longs;edeant, aer <lb/>primum non attinget illos, ut etiam quia &longs;ur&longs;um pellitur non per<lb/>ueniet ad illos, imò diffugiet, ergo non refrigerabuntur. </s>

<s id="id002149">Qui uerò <lb/>à lateribus l x m <expan abbr="&longs;tab&utilde;t">&longs;tabunt</expan> hiccinde, uelut in f g, &longs;i &longs;teterint, <expan abbr="nõ">non</expan> refrigera<lb/><expan abbr="b&utilde;tur">buntur</expan>, quia <expan abbr="quãdo">quando</expan> flabellum erit in l, uel m aer de&longs;cendet, ergo fugi<lb/>et ab illis, cum autem fuerit in x, erit in loco humiliori, & mouebi<pb pagenum="120" xlink:href="015/01/139.jpg"/>tur diuer&longs;a ratione, quippe ab f in h, & non ad latera, ergo ne que <lb/><figure id="id.015.01.139.1.jpg" xlink:href="015/01/139/1.jpg"/><lb/>contactu, neque motu, qui <lb/>fiet per æquidi&longs;tantem f, <lb/>& g non poterunt refrige<lb/>rari. </s>

<pb pagenum="120" xlink:href="015/01/139.jpg"/>tur diuer&longs;a ratione, quippe ab f in h, & non ad latera, ergo ne que <lb/><figure id="id.015.01.139.1.jpg" xlink:href="015/01/139/1.jpg"/><lb/>contactu, neque motu, qui <lb/>fiet per æquidi&longs;tantem f, <lb/>& g non poterunt refrige<lb/>rari. </s>

<s id="id002150">Sed &longs;i humili loco &longs;e<lb/>deant, quoniam aër de&longs;cen <lb/>dit, ex l & m uer&longs;us x, & <lb/>etiam, quia erunt proximi <lb/>h k, <expan abbr="quãdo">quando</expan> fuerit in x, <expan abbr="refrigerabun&ttilde;">refri<lb/>gerabuntur</expan> ualde. </s>

<s id="id002151">Qui <expan abbr="aut&etilde;">autem</expan> <lb/><expan abbr="er&utilde;t">erunt</expan> iuxta h & k minus <expan abbr="refrigerabun&ttilde;">re<lb/>frigerabuntur</expan> utri&longs;que, &longs;ed pau<lb/>lulum in reditibus propin <lb/>quis, & neque &longs;tantes, <expan abbr="neque&longs;ed&etilde;tes">neque <lb/>&longs;edentes</expan>, &longs;ed &longs;i altius attolla<lb/>tur h k. </s>

<s id="id002152">Rur&longs;us &longs;i b h k fue<lb/>rit grauior eodem, ut de<lb/>&longs;cendat tanto impetu, <expan abbr="quãto">quan<lb/>to</expan> a&longs;cendit attractum, ut <lb/>pote ex ligno tenui nucis, <lb/>tunc multo magis refrige<lb/>rabit, & procul, <expan abbr="nõ">non</expan> ob uim <lb/>ualidiorem, &longs;ed quoniam <lb/>celerius occur&longs;antes &longs;ibi <lb/>contrarijs motibus, ac <expan abbr="uehem&etilde;tibus">ue<lb/>hementibus</expan> fiet colli&longs;io par<lb/>tium aëris, & ideo in ambitum impelletur, & undique cubiculum <lb/>refrigerabit, quod non faciet maius longè flabellum lento motu <lb/>agitatum, aut ex materia leui. </s>

Line 6919

Line 6973

<s id="id002326">Denominationes in infinitum extendere.</s>

<s id="id002327">Inquit Euclides, &longs;i fuerint quotlibet quantitates ab uno in conti</s><s id="id002328"><arrow.to.target n="marg459"/><lb/><arrow.to.target n="marg460"/><lb/>nua proportione, erit tertius numerus quadratus, & omnes alij &longs;e<lb/>quentes uno intermi&longs;&longs;o. </s>

<s id="id002327">Inquit Euclides, &longs;i fuerint quotlibet quantitates ab uno in conti</s>

<s id="id002328"><arrow.to.target n="marg459"/><lb/><arrow.to.target n="marg460"/><lb/>nua proportione, erit tertius numerus quadratus, & omnes alij &longs;e<lb/>quentes uno intermi&longs;&longs;o. </s>

<s id="id002329">Tertia igitur in comparatione ad &longs;ecun<lb/>dam etiam, quod non &longs;it numerus, e&longs;t quadratum: e&longs;t enim tertia <lb/>ab uno quadratum &longs;ecundæ, quæ e&longs;t proportio. </s>

<s id="id002330">Detracto igitur <lb/>uno omnes quantitates loco pari &longs;unt quadratæ: ut &longs;cias ergo cu<lb/>ius &longs;unt quadratæ diuide per medium, & erit quadratum illius, er<lb/>go quadrage&longs;ima erit quadratum uige&longs;imæ, & uige&longs;ima decimæ, <lb/>& decima quintæ, & uige&longs;ima &longs;exta tertiæ decimæ, & ita de alijs. <lb/></s>

<s id="id002331">Iuxta hoc dicemus, quod &longs;ecunda erit <expan abbr="quadrat&utilde;">quadratum</expan>, & quarta quadra<lb/>tum quadrati, & octaua <expan abbr="quadrat&utilde;">quadratum</expan> quadrati quadrati. </s>

Line 6971

Line 7028

<s id="id002360">Ergo ultra propor<lb/>tionem &longs;unt tantum quatuor termini.</s>

<s id="id002361">Septima regula ad effugiendum omnes errores tu &longs;cis, quòd <lb/>4096 quadratum 64 e&longs;t &longs;extus a 64, ad quem habet proportionem <lb/>quadrati, & 64 e&longs;t &longs;imiliter &longs;extus ab uno illo &longs;cilicet non compu<pb pagenum="131" xlink:href="015/01/150.jpg"/>tato, & ita 64 habet rationem unius, & licet comparetur ad 2 rem, <lb/>& &longs;it &longs;extus ab eo, eo computato 4096 autem à 64 &longs;it &longs;eptimus, ta<lb/>men non e&longs;t eadem ratio, quia 64 non e&longs;t quadratum 2.</s>

<pb pagenum="131" xlink:href="015/01/150.jpg"/>tato, & ita 64 habet rationem unius, & licet comparetur ad 2 rem, <lb/>& &longs;it &longs;extus ab eo, eo computato 4096 autem à 64 &longs;it &longs;eptimus, ta<lb/>men non e&longs;t eadem ratio, quia 64 non e&longs;t quadratum 2.</s>

<s id="id002362">Propo&longs;itio cente&longs;ima trige&longs;ima &longs;eptima.</s>

Line 7188

Line 7248

<s id="id002376">Modos u&longs;us horum numerorum declarare.</s>

<s id="id002377">In quouis numero denominationis oportet tot addere o, quo<lb/><arrow.to.target n="marg464"/><pb pagenum="132" xlink:href="015/01/151.jpg"/>tus e&longs;t ordo, & facere tot numeros &longs;equentes; quotus e&longs;t ordo, & <lb/>&longs;emper minuere unam o, uelut quia quadrata <02> e&longs;t prima ad 2 ad<lb/>demus o, & fiet 20, nec alium qu&etail;remus numerum. </s>

<s id="id002377">In quouis numero denominationis oportet tot addere o, quo<lb/><arrow.to.target n="marg464"/>

<pb pagenum="132" xlink:href="015/01/151.jpg"/>tus e&longs;t ordo, & facere tot numeros &longs;equentes; quotus e&longs;t ordo, & <lb/>&longs;emper minuere unam o, uelut quia quadrata <02> e&longs;t prima ad 2 ad<lb/>demus o, & fiet 20, nec alium qu&etail;remus numerum. </s>

<s id="id002378">Sed quia cubi<lb/>ca e&longs;t &longs;ecundo loco, habebit prima nota 00, & fiet 300, & &longs;ecundum <lb/>3 unam 0, & fiet 30, & in quadrato quadrati addemus 000 primo, <lb/>& 00 &longs;ecundo, & o tertio, & ita habebimus 4000. 600. 40. &longs;ed quia <lb/>in tabula non e&longs;t 4 ultimum, addemus &longs;imilem primo &longs;emper. </s>

<s id="id002379">In <lb/>relato primo, ergo habebimus 50000. 1000. 1000. 50. & in cubo <lb/>quadrati 600000. 150000. 20000. 1500. 60. Manife&longs;tum e&longs;t, quòd <lb/>his uice uer&longs;a a&longs;&longs;ump&longs;imus 15 & 6 &longs;imiles prioribus addendo &longs;em<lb/>per ut dixi o minus, donec ad unam peruenerit. </s>

<s id="id002380">Et ita in relato &longs;e<lb/>cundo 7000000. 2100000. 350000. 35000. 2100. 70. & ita deinceps.</s>

Line 7307

Line 7370

<s id="id002431"><expan abbr="Quidã">Quidam</expan> per u&longs;uram <expan abbr="rediuiuã">rediuiuam</expan> fecit 40000 coronatos ex 40 in 40 <lb/><arrow.to.target n="marg471"/><lb/>annis. </s>

<s id="id002432">Qu&etail;ro <expan abbr="qutãa">qutana</expan> fuerit u&longs;ura, & <expan abbr="quãdo">quando</expan> habuit 1000 coronatos, <lb/><expan abbr="quidã">quidam</expan> uellent &longs;oluere per regulam trium quantitatum, in qua com<lb/>mitterentur maximi errores. </s>

<s id="id002433">Et in ea multi &longs;unt modi, & omnes fal<lb/>&longs;i præter hanc uiam nulla e&longs;t uera, adde quòd uellent multi per &longs;or<lb/>tem inuentam &longs;oluere augendo per &longs;ingulos annos, quod adeò <lb/>difficile e&longs;&longs;et, & penè foret impo&longs;sibile. </s>

<s id="id002434">Ideò diuides 40000 per 40 <lb/>numerum &longs;ortis exit 1000, igitur in 40 annis unum fit mille, &longs;unt <lb/>ergo 40 denominationes ab uno, quarum quadrage&longs;ima e&longs;t 1000, <lb/>igitur uige&longs;ima e&longs;t <02> 1000 | &longs;cilicet |31 3913/6283, igitur decima e&longs;t <02> eius <lb/><arrow.to.target n="marg472"/><lb/>5 3917/12566 huius radix, erit quinta quantitas 2 7/23, cuius <02> relata prima, <lb/><arrow.to.target n="table18"/><lb/>erit proportio 1 13/70, cuius quadratum e&longs;t 1 1889/4900 &longs;eu <lb/>1 67/165 pro &longs;ecunda quantitate, duces ergo primam, <lb/><figure id="id.015.01.156.1.jpg" xlink:href="015/01/156/1.jpg"/>quæ e&longs;t 83/70 in quintam, quæ e&longs;t reducta ad mino<lb/>res fractiones facilitatis cau&longs;a 53/23, & habebis &longs;ex<lb/>tam quantitatem 2 118/161, duco etiam quintam quan<lb/>titatem &longs;cilicet 53/23 in &longs;ecundam quæ e&longs;t 232/165, & fit &longs;e<lb/>ptimi anni quantitas, duco igitur &longs;eptem anno<lb/>rum numerum, qui e&longs;t 3 14/61 in 31 38/61 fit 102 992/6283. At in <lb/>&longs;ex annis additis ad uiginti, fit tanto minus, quan<lb/>to 31 38/61 ductum in differentiam &longs;eptem, & &longs;ex an<lb/>norum quæ e&longs;t 60/121, fit ergo 15 35/492. Quia ergo an<pb pagenum="138" xlink:href="015/01/157.jpg"/>nuatim &longs;olum u&longs;ura adijcitur &longs;orti, &longs;ufficiet diuidere 2 992/6283 per 15 35/492 <lb/>&longs;cilicet multiplicando per 12 numerum men&longs;ium 2 992/6283 fit 25 5621/6283 di<lb/>uide 25 5621/6283 per 15 35/492, exit men&longs;is unus, & dies 21, detrahe ex 27 an<lb/>nis, remanent anni 26, men&longs;es 10, dies 9, in quo tempore habuit <lb/>4000 aureos coronatos. </s>

<s id="id002434">Ideò diuides 40000 per 40 <lb/>numerum &longs;ortis exit 1000, igitur in 40 annis unum fit mille, &longs;unt <lb/>ergo 40 denominationes ab uno, quarum quadrage&longs;ima e&longs;t 1000, <lb/>igitur uige&longs;ima e&longs;t <02> 1000 |&longs;cilicet |31 3913/6283, igitur decima e&longs;t <02> eius <lb/><arrow.to.target n="marg472"/><lb/>5 3917/12566 huius radix, erit quinta quantitas 2 7/23, cuius <02> relata prima, <lb/><arrow.to.target n="table18"/><lb/>erit proportio 1 13/70, cuius quadratum e&longs;t 1 1889/4900 &longs;eu <lb/>1 67/165 pro &longs;ecunda quantitate, duces ergo primam, <lb/><figure id="id.015.01.156.1.jpg" xlink:href="015/01/156/1.jpg"/>quæ e&longs;t 83/70 in quintam, quæ e&longs;t reducta ad mino<lb/>res fractiones facilitatis cau&longs;a 53/23, & habebis &longs;ex<lb/>tam quantitatem 2 118/161, duco etiam quintam quan<lb/>titatem &longs;cilicet 53/23 in &longs;ecundam quæ e&longs;t 232/165, & fit &longs;e<lb/>ptimi anni quantitas, duco igitur &longs;eptem anno<lb/>rum numerum, qui e&longs;t 3 14/61 in 31 38/61 fit 102 992/6283. At in <lb/>&longs;ex annis additis ad uiginti, fit tanto minus, quan<lb/>to 31 38/61 ductum in differentiam &longs;eptem, & &longs;ex an<lb/>norum quæ e&longs;t 60/121, fit ergo 15 35/492. Quia ergo an

<pb pagenum="138" xlink:href="015/01/157.jpg"/>nuatim &longs;olum u&longs;ura adijcitur &longs;orti, &longs;ufficiet diuidere 2 992/6283 per 15 35/492 <lb/>&longs;cilicet multiplicando per 12 numerum men&longs;ium 2 992/6283 fit 25 5621/6283 di<lb/>uide 25 5621/6283 per 15 35/492, exit men&longs;is unus, & dies 21, detrahe ex 27 an<lb/>nis, remanent anni 26, men&longs;es 10, dies 9, in quo tempore habuit <lb/>4000 aureos coronatos. </s>

<s id="id002435">V&longs;ura autem fuit ut ui&longs;um 13/70, igitur per re<lb/>gulam trium duc 13 in 100 fit 1300, diuide 1300 per 70 exit 18 4/7 & <lb/>tanta fuit pro centum. </s>

<s id="id002436">Et cum computaueris in tribus annis, acqui<lb/>rit modico plus be&longs;&longs;e eius, quod habet. </s>

<s id="id002437">Et ita in 13 annis, & parua <lb/>illa parte perueniet ad decuplum eius, quod habet, &longs;cilicet 4000 au<lb/>reorum, & habebit aureos 40000, ut propo&longs;itum e&longs;t.</s>

Line 7462

Line 7528

<s id="id002479">Sit linea a b diui&longs;a in c uolo eius <lb/><figure id="id.015.01.159.1.jpg" xlink:href="015/01/159/1.jpg"/><lb/>partibus addere lineas, ut propo&longs;i</s><s id="id002480"><arrow.to.target n="marg482"/><lb/>tum e&longs;t, &longs;tatuo mediam c d inter a e & <lb/><arrow.to.target n="marg483"/><lb/>c b quæ &longs;it c d, & facio ut c d ad c a ita <lb/>c a ad a e, & ut d c ad c b ita c b ad b f, quia ergo d e media e&longs;t inter <lb/><arrow.to.target n="marg484"/><lb/>a c & c b, & ut ea ad a cita d c a c b ad c f erunt omnes in continua <lb/><arrow.to.target n="marg485"/><lb/>proportione, quare proportio e c ad c a ut c f ad b f & e c ad ea ut <lb/>c f ad c b quod e&longs;t propo&longs;itum.</s>

<s id="id002479">Sit linea a b diui&longs;a in c uolo eius <lb/><figure id="id.015.01.159.1.jpg" xlink:href="015/01/159/1.jpg"/><lb/>partibus addere lineas, ut propo&longs;i</s>

<s id="id002480"><arrow.to.target n="marg482"/><lb/>tum e&longs;t, &longs;tatuo mediam c d inter a e & <lb/><arrow.to.target n="marg483"/><lb/>c b quæ &longs;it c d, & facio ut c d ad c a ita <lb/>c a ad a e, & ut d c ad c b ita c b ad b f, quia ergo d e media e&longs;t inter <lb/><arrow.to.target n="marg484"/><lb/>a c & c b, & ut ea ad a cita d c a c b ad c f erunt omnes in continua <lb/><arrow.to.target n="marg485"/><lb/>proportione, quare proportio e c ad c a ut c f ad b f & e c ad ea ut <lb/>c f ad c b quod e&longs;t propo&longs;itum.</s>

<s id="id002481"><margin.target id="marg482"/>P<emph type="italics"/>er<emph.end type="italics"/> 13. <emph type="italics"/>&longs;ex <lb/>ti<emph.end type="italics"/> E<emph type="italics"/>lem.<emph.end type="italics"/></s>

Line 7487

Line 7556

<s id="id002488">Sit a, b, c, d, propo&longs;itæ line&etail;, <lb/><figure id="id.015.01.159.2.jpg" xlink:href="015/01/159/2.jpg"/><lb/>uolo diuidere a b ita in e ut <lb/>&longs;umpta &longs;ecundum proportio<lb/>nem alicuius quantitatis, puta <lb/>g ad a e &longs;ic b f ad e b & ut g ad <lb/>e b &longs;ic g a ad a e ut &longs;it propor<lb/>tio g e ad e f ut c ad d. </s>

<s id="id002489">Sint ergo <lb/>omnia <expan abbr="cõ&longs;tituta">con&longs;tituta</expan> & &longs;it g rectan<lb/>gulum ex a e in e b, cum ergo <lb/>g a contineat a e ut g continet e b, g autem continet e b &longs;ecundum <lb/>a e, igitur g a continet a e &longs;ecundum a c, ergo ex diffinitione qua</s><s id="id002490"><arrow.to.target n="marg487"/><lb/>drati a g e&longs;t quadratum a e. </s>

<s id="id002490"><arrow.to.target n="marg487"/><lb/>drati a g e&longs;t quadratum a e. </s>

<s id="id002491">Pari ratione b f e&longs;t quadratum b e. </s>

<s id="id002492">pro<lb/>portio igitur g e ad e f cum &longs;it ut c ad e ex &longs;uppo&longs;ito erit ut ip&longs;i pro<lb/>portioni addamus, & detrahamus ex duplo a b & dimidium re&longs;i<lb/>dui ducamus in &longs;e, & addamus aggregato quadrati a b cum ip&longs;a <pb pagenum="141" xlink:href="015/01/160.jpg"/>a b, & latus eius detracto dimidio re&longs;idui erit b c linea, quare diui<lb/>&longs;io nota, & e&longs;t ut dicamus : uolo diuidere datam lineam, ut quantita<lb/>tes adiectæ &longs;ub mutua proportione ad unam tertiam cum parti<lb/>bus obtineant inter &longs;e proportionem datam.</s>

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

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

Line 7807

Line 7882

<s id="id002632">qui bifa<lb/><expan abbr="riã">riam</expan> <expan abbr="compona&ttilde;">componatur</expan> ex duob. </s>

<s id="id002634">uelut 10 ex 25, & 25 & 49 & 1, <lb/><figure id="id.015.01.168.1.jpg" xlink:href="015/01/168/1.jpg"/><lb/>& <expan abbr="&longs;uma&ttilde;">&longs;umatur</expan> a b numerus quad. </s>

<s id="id002635">diui&longs;us in <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan>, ita quae c <lb/>d &longs;it portio minor eiu&longs;modi, ut adiecta illi <expan abbr="æ&qtilde;li">æquali</expan> c d gnomo <lb/>cir<expan abbr="c&utilde;&longs;criptus">cun&longs;criptus</expan> c k l <expan abbr="c&utilde;">cum</expan> <expan abbr="f&qtilde;drato">f quadrato</expan>, &longs;it <expan abbr="&etail;&qtilde;lis">&etail;qualis</expan> a b <expan abbr="&qtilde;drato">quadrato</expan>, detractis <lb/><expan abbr="igi&ttilde;">igitur</expan> c e & e d, <expan abbr="æ&qtilde;libus">æqualibus</expan> erunt duo <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> c k l <expan abbr="c&utilde;f">cum</expan> f qua<lb/>drato &etail;qualia duob. </s>

<s id="id002635">diui&longs;us in <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan>, ita quae c <lb/>d &longs;it portio minor eiu&longs;modi, ut adiecta illi <expan abbr="æ&qtilde;li">æquali</expan> c d gnomo <lb/>cir<expan abbr="c&utilde;&longs;criptus">cun&longs;criptus</expan> c k l <expan abbr="c&utilde;">cum</expan> <expan abbr="f&qtilde;drato">fquadrato</expan>, &longs;it <expan abbr="&etail;&qtilde;lis">&etail;qualis</expan> a b <expan abbr="&qtilde;drato">quadrato</expan>, detractis <lb/><expan abbr="igi&ttilde;">igitur</expan> c e & e d, <expan abbr="æ&qtilde;libus">æqualibus</expan> erunt duo <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> c k l <expan abbr="c&utilde;f">cunf</expan> qua<lb/>drato &etail;qualia duob. </s>

<s id="id002636"><expan abbr="&longs;upplem&etilde;tis">&longs;upplementis</expan> a b <expan abbr="c&utilde;">cum</expan> <expan abbr="&qtilde;drato">quadrato</expan> h g. </s>

<s id="id002637">Maio<lb/>ra <expan abbr="a&utilde;t">aut</expan> <expan abbr="&longs;upplem&etilde;ta">&longs;upplementa</expan> <expan abbr="exced&utilde;t">excedunt</expan> minora in duplo quad. </s>

<s id="id002638">c d <expan abbr="igi&ttilde;">igitur</expan> detractis <lb/>minoribus &longs;upplementis <expan abbr="cõmunibus">communibus</expan>, erit <expan abbr="dupl&utilde;">duplum</expan> quad. </s>

Line 7848

Line 7923

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

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

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

<s id="id002674">Vt uerò uideamus <expan abbr="re&longs;idu&utilde;">re&longs;iduum</expan>, proponamus quae uelim diui<lb/>dere 6 in duos numeros <expan abbr="&qtilde;d">quad</expan>, <expan abbr="prim&utilde;">primum</expan> &longs;cire debes &qring;d non po&longs;&longs;unt e&longs;&longs;e <pb pagenum="150" xlink:href="015/01/169.jpg"/>integri ex ratione dicta, quia oporteret ut e&longs;&longs;ent ambo impares aut <lb/>pares, & &longs;ic <expan abbr="differr&etilde;t">differrent</expan> numero pari, ergo oporteret ut e&longs;&longs;et unus me<lb/>dius numerus <expan abbr="&qtilde;d">quad</expan>. </s>

<pb pagenum="150" xlink:href="015/01/169.jpg"/>integri ex ratione dicta, quia oporteret ut e&longs;&longs;ent ambo impares aut <lb/>pares, & &longs;ic <expan abbr="differr&etilde;t">differrent</expan> numero pari, ergo oporteret ut e&longs;&longs;et unus me<lb/>dius numerus <expan abbr="&qtilde;d">quad</expan>. </s>

<s id="id002675">&longs;unt & ali&etail; rationes, &longs;ed neque unus po&longs;&longs;et e&longs;&longs;e inte<lb/>ger, & alius fractus, <expan abbr="nõ">non</expan> e&longs;&longs;et. </s>

<s id="id002677">6 numerus integer: <expan abbr="relinqui&ttilde;">relinquitur</expan> ergo ut <lb/>&longs;int duo fracti: &longs;ed in numeris fractis <expan abbr="&qtilde;d">quad</expan>. </s>

Line 7868

Line 7946

<s id="id002691">ille erit <expan abbr="qu&etilde;">quem</expan> qu&etail;rimus, <expan abbr="nã">nam</expan> diui&longs;o 6 per to<lb/>tum <expan abbr="ill&utilde;">illum</expan> numerum, inde &qring;d prouenit multiplicato per numeros <expan abbr="&qtilde;d">quad</expan>, <lb/><expan abbr="cõponentes">componentes</expan> illum <expan abbr="numer&utilde;">numerum</expan> productum, <expan abbr="producun&ttilde;">producuntur</expan> partes 6, quæ <expan abbr="er&utilde;t">erunt</expan> <lb/>numeri <expan abbr="&qtilde;d">quad</expan>. </s>

<s id="id002692">quia denominator utriu&longs;que partis ex &longs;uppo&longs;ito e&longs;t nume <lb/>rus <expan abbr="&qtilde;dratus">quadratus</expan>, qui multiplicatus e&longs;t per 6, & numeratores &longs;unt nume <lb/>ri <expan abbr="&qtilde;drati">quadrati</expan>, qui <expan abbr="cõponebant">componebant</expan> <expan abbr="numer&utilde;">numerum</expan> <expan abbr="product&utilde;">productum</expan>, et tales partes <expan abbr="&etail;quan&ttilde;">&etail;quantur</expan> <lb/>6, quia numerus productus <expan abbr="componi&ttilde;">componitur</expan> ex numeratoribus, & <expan abbr="produci&ttilde;">produ<lb/>citur</expan> tale <expan abbr="cõpo&longs;itum">compo&longs;itum</expan> ex 6 in <expan abbr="denominator&etilde;">denominatorem</expan>, & hic e&longs;t diui&longs;us per deno <lb/><expan abbr="minator&etilde;">minatorem</expan>, ergo prouenit 6, &longs;i <expan abbr="e&mtilde;">eum</expan> multiplicato 3 in 4 fit 12, diui&longs;o 12 per <lb/>4, exit nece&longs;&longs;ario idem 3. Pro colligendo ergo numeros omnes, qui <lb/><expan abbr="cõponuntur">componuntur</expan> ex <expan abbr="&qtilde;dratis">quadratis</expan>, propones tibi &longs;eriem <expan abbr="&qtilde;d">quad</expan>. </s>

<s id="id002693"><expan abbr="omni&utilde;">omnium</expan>, & inde iun<lb/>ges, & diuides per 6, & <expan abbr="c&utilde;">cum</expan> prodierit <expan abbr="&qtilde;dratus">quadratus</expan>, <expan abbr="inueni&ttilde;">inuenitur</expan> denominator, <lb/>& numeri <expan abbr="cõponentes">componentes</expan> ip&longs;um erunt numeratores, et &longs;uppo&longs;iti deno <lb/>minatoribus <expan abbr="cõ&longs;tituent">con&longs;tituent</expan> partes. </s>

<s id="id002694">Vt uerò cogno&longs;cas, ex quibus po&longs;<lb/>&longs;it componi primum ex imparibus, non oportet a&longs;&longs;umere ni&longs;i 135, <lb/>quia 7 diui&longs;um per 6 relinquit 1, & 9 diui&longs;um per 6, relinquit 3, & 35 <lb/>diui&longs;um per 6 relinquit 5. ergo non pote&longs;t componi numerus im<lb/>par, qui diuidatur per 6, ut &longs;uper&longs;it impar alius quàm 1. 3. 5. &longs;ed 1 & 3 <lb/>& 5, & 5 componunt 4 & 1, & 1 & 3 & 5 componunt 2, &longs;cilicet abie<lb/>cto 6, ergo tales numeri <expan abbr="&qtilde;drati">quadrati</expan> &longs;i &longs;int impares, uel ambo terminan<lb/>tur in 3, ut 9 & 81, qui faciunt 90, uel in 1 & 5, &longs;ed nullus numerus <lb/>quadratus diui&longs;us per 6 terminatur in 5, quia 1 ductum in &longs;e produ<lb/>cit 1, & 3 pro ducit 3, & 5 pro ducit 1, ut 5 in 5 facit 25, & 11 in 11 produ<pb pagenum="151" xlink:href="015/01/170.jpg"/>cit 121, quibus diui&longs;is per 6 &longs;upere&longs;t 1. Quod etiam &longs;ic demon&longs;tratur <lb/>de 5, & compo&longs;itis à 5, nam diui&longs;o 5 in 3 & 2, quadratum eius <expan abbr="cõponitur">compo<lb/>nitur</expan> ex duplo 3 in 2, in quo nihil &longs;upere&longs;t, &longs;i diuidatur per 6, & ex <lb/>quadrato 3, quòd e&longs;t 9, in quo &longs;upere&longs;t 3, & ex quadrato 2 quod e&longs;t </s>

<pb pagenum="151" xlink:href="015/01/170.jpg"/>cit 121, quibus diui&longs;is per 6 &longs;upere&longs;t 1. Quod etiam &longs;ic demon&longs;tratur <lb/>de 5, & compo&longs;itis à 5, nam diui&longs;o 5 in 3 & 2, quadratum eius <expan abbr="cõponitur">compo<lb/>nitur</expan> ex duplo 3 in 2, in quo nihil &longs;upere&longs;t, &longs;i diuidatur per 6, & ex <lb/>quadrato 3, quòd e&longs;t 9, in quo &longs;upere&longs;t 3, & ex quadrato 2 quod e&longs;t </s>

<s id="id002695"><arrow.to.target n="marg520"/><lb/>4, &longs;ed iunctis 4 & 3, & abiecto 6 &longs;upere&longs;t 1, ergo 5 in 5 <expan abbr="duct&utilde;">ductum</expan>, & diui<lb/>&longs;o producto relinquitur 1. Et &longs;imiliter capio 17, et <expan abbr="componi&ttilde;">componitur</expan> ex 12 & <lb/>5 quadratum, ergo 17 componitur ex quadrato 12, in quo nihil &longs;u<lb/>pere&longs;t, & duplo 5 in 12, in quo <expan abbr="etiã">etiam</expan> nihil &longs;upere&longs;t, &longs;i diuidatur per 6: <lb/>& ex quadrato 5, in quo &longs;upere&longs;t 1, ergo in nullo numero <expan abbr="cõpo&longs;ito">compo&longs;ito</expan> <lb/>ex 5 & 6, uel compo&longs;itis ex 6, poterit produci numerus, qui diui&longs;us <lb/>per 6 relinquat 5, igitur neque talis numerus potérit <expan abbr="cõponi">componi</expan> ex duo<lb/>bus quadratis, in quib. </s>

<s id="id002696">&longs;uper&longs;it 5 & 1, quia nullus e&longs;t, in quo &longs;uper<lb/>&longs;it 5 facta diui&longs;ione per 6. Ex quo colligitur una regula: quod &longs;i quis <lb/>dicat multiplicaui 27 in &longs;e, et diui&longs;i per 13, uellem &longs;cire quid &longs;upere&longs;t, <lb/>dico quod &longs;ine multiplicatione et diui&longs;ione poteris hoc &longs;cire ex de<lb/>mon&longs;tratione dicta, diuide ergo 27 per 13, & relinquitur 1, duc in &longs;e <lb/>fit 1: dices ergo, quod &longs;upererit 1, & ita &longs;i ducerem 28 in &longs;e, & diuide<lb/>rem per 11, dico quod &longs;upererit 3, nam diui&longs;o 28 per 11, relinquitur <lb/>6, duc in 6 fit 36, diuide per 11, relinquitur 3, ut dictum e&longs;t, & tantum <lb/><expan abbr="relinqui&ttilde;">relinquitur</expan> ducto 28 in &longs;e & fit 784, & diui&longs;o per 11. Reuertendo ergo <lb/>ad propo&longs;itum, pater quod ex duobus tantum numeris imparibus <lb/>quadratis pote&longs;t conflari ille numerus, <expan abbr="quor&utilde;">quorum</expan> radices diui&longs;æ per 6 <lb/>relinquunt 3. Sed de paribus uel &longs;upere&longs;t 2 uel 4 uel nihil, &longs;ed <expan abbr="&qtilde;dratum">quadra<lb/>tum</expan> 2 e&longs;t 4, & <expan abbr="&qtilde;dratum">quadratum</expan> 4 diui&longs;um per 6 etiam relinquit 4, ergo neque <lb/>ex duobus numeris, in quibus &longs;uper&longs;int 2, neque in quibus &longs;uper&longs;int <lb/>4, neque in quibus &longs;uper&longs;int in uno 2, in altero 4 <expan abbr="poter&utilde;t">poterunt</expan> quadrata, in <lb/>quibus &longs;emper &longs;upererit 4, & iuncta faciunt 8, in quod &longs;upere&longs;t 2, <expan abbr="cõ">con</expan>fla<lb/>re <expan abbr="numer&utilde;">numerum</expan> <expan abbr="dict&utilde;">dictum</expan> &longs;eu <expan abbr="quæ&longs;it&utilde;">quæ&longs;itum</expan>, qui po&longs;sit diuidi per 6: neque ex <expan abbr="&qtilde;d">quad</expan>. </s>

<s id="id002696">&longs;uper&longs;it 5 & 1, quia nullus e&longs;t, in quo &longs;uper<lb/>&longs;it 5 facta diui&longs;ione per 6. Ex quo colligitur una regula: quod &longs;i quis <lb/>dicat multiplicaui 27 in &longs;e, et diui&longs;i per 13, uellem &longs;cire quid &longs;upere&longs;t, <lb/>dico quod &longs;ine multiplicatione et diui&longs;ione poteris hoc &longs;cire ex de<lb/>mon&longs;tratione dicta, diuide ergo 27 per 13, & relinquitur 1, duc in &longs;e <lb/>fit 1: dices ergo, quod &longs;upererit 1, & ita &longs;i ducerem 28 in &longs;e, & diuide<lb/>rem per 11, dico quod &longs;upererit 3, nam diui&longs;o 28 per 11, relinquitur <lb/>6, duc in 6 fit 36, diuide per 11, relinquitur 3, ut dictum e&longs;t, & tantum <lb/><expan abbr="relinqui&ttilde;">relinquitur</expan> ducto 28 in &longs;e & fit 784, & diui&longs;o per 11. Reuertendo ergo <lb/>ad propo&longs;itum, pater quod ex duobus tantum numeris imparibus <lb/>quadratis pote&longs;t conflari ille numerus, <expan abbr="quor&utilde;">quorum</expan> radices diui&longs;æ per 6 <lb/>relinquunt 3. Sed de paribus uel &longs;upere&longs;t 2 uel 4 uel nihil, &longs;ed <expan abbr="&qtilde;dratum">quadra<lb/>tum</expan> 2 e&longs;t 4, & <expan abbr="&qtilde;dratum">quadratum</expan> 4 diui&longs;um per 6 etiam relinquit 4, ergo neque <lb/>ex duobus numeris, in quibus &longs;uper&longs;int 2, neque in quibus &longs;uper&longs;int <lb/>4, neque in quibus &longs;uper&longs;int in uno 2, in altero 4 <expan abbr="poter&utilde;t">poterunt</expan> quadrata, in <lb/>quibus &longs;emper &longs;upererit 4, & iuncta faciunt 8, in quod ̊&longs;upere&longs;t 2, <expan abbr="cõ">con</expan>fla<lb/>re <expan abbr="numer&utilde;">numerum</expan> <expan abbr="dict&utilde;">dictum</expan> &longs;eu <expan abbr="quæ&longs;it&utilde;">quæ&longs;itum</expan>, qui po&longs;sit diuidi per 6: neque ex <expan abbr="&qtilde;d">quad</expan>. </s>

<s id="id002697"><expan abbr="duor&utilde;">duo<lb/>rum</expan> <expan abbr="num&etilde;ror&utilde;">numerorum</expan>, in <expan abbr="quor&utilde;">quorum</expan> altero nihil &longs;uper&longs;it in reliquo &longs;uper&longs;it 2 uel <lb/>4, quia in aggregato <expan abbr="&qtilde;drator&utilde;">quadratorum</expan> &longs;emper &longs;upererit 4. Ergo relinqui<lb/>tur quod ille numerus componetur ex duobus quadratis, uel impa<lb/>ribus, quorum latera diui&longs;a per 6 relinquunt 3, uel ex duobus pari<lb/>bus, quorum latera diui&longs;a per 6 nihil relinquant. </s>

<s id="id002698">Oportet igitur <lb/>inuenire duos tales numeros quadratos numerorum imparium, in <lb/>quibus &longs;uper&longs;it 3, &longs;i diuidantur per 6, aut parium in quibus nihil &longs;u<lb/>per&longs;it, quorum aggregato diui&longs;o per 6 prodeat numerus <expan abbr="&qtilde;dratus'">quadratus'</expan>.</s>

Line 7912

Line 7993

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

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

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

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

Line 7955

Line 8036

<s id="id002735">Aliam ergo oportuit excogitare qu&etail; huiu&longs;modi e&longs;t. </s>

<s id="id002736">Sub axe a b, <lb/>qui circumuertitur ad molam contrahendam rotam, collocant par <lb/>uam quæ e&longs;t, ut ita dicam, pars axis ima cui in&longs;eruntur dentes in am<lb/>bitu ea ratione, ut dum mola tenditur, premant denticulos interio<lb/>res, atque ita elabitur, totiesque circumducitur manente g f, donec <lb/>colligatur mola, quæ non ut in priore reliquo extremo ulli rotæ <lb/>affixa e&longs;t, &longs;ed columnæ in continenti <lb/>opercula horologij. </s>

<s id="id002737">Cum ergo mola <lb/>tenta retrahat axem a b contrario mo<lb/><figure id="id.015.01.174.2.jpg" xlink:href="015/01/174/2.jpg"/><lb/>tu, & ille rotam mobilem, quæ cum <lb/>non po&longs;sit regredi propter auer&longs;os <lb/>dentes, mouet rotam f g contrario mo<lb/>tu, quæ circumacta per denticulos &longs;u<lb/>os curriculum agit, & reliqua omnia <lb/>nece&longs;&longs;aria. </s>

<s id="id002738">Cur autem cum laxatur mo <lb/>la, & uertit lentius c e rotam coniun<lb/>ctam, ideoque g f, & reliqua omnia <expan abbr="nõ">non</expan> tardetur tempus, & circumuo<pb pagenum="156" xlink:href="015/01/175.jpg"/>lutio indicis cau&longs;a e&longs;t alia longè quàm in priore, nam mola longior <lb/>fit cra&longs;sior, & durior adeoque robu&longs;ta, & rotæ leues, ac tempus dum <lb/>laxata fuerit munus &longs;uum iu&longs;to in tempore obeant: quare nece&longs;&longs;e <lb/>e&longs;t, ut ab initio uehementius agat, & celerius rotam cum axe qui tra<lb/>hitur à mola. </s>

<s id="id002738">Cur autem cum laxatur mo <lb/>la, & uertit lentius c e rotam coniun<lb/>ctam, ideoque g f, & reliqua omnia <expan abbr="nõ">non</expan> tardetur tempus, & circumuo

<pb pagenum="156" xlink:href="015/01/175.jpg"/>lutio indicis cau&longs;a e&longs;t alia longè quàm in priore, nam mola longior <lb/>fit cra&longs;sior, & durior adeoque robu&longs;ta, & rotæ leues, ac tempus dum <lb/>laxata fuerit munus &longs;uum iu&longs;to in tempore obeant: quare nece&longs;&longs;e <lb/>e&longs;t, ut ab initio uehementius agat, & celerius rotam cum axe qui tra<lb/>hitur à mola. </s>

<s id="id002739">Ergo excogitarunt aliud genus retinaculi forma co<lb/>chleæ quod ab initio moratur <expan abbr="uehem&etilde;ter">uehementer</expan> axem ne circumagatur, et <lb/>quanto magis mola explicatur eo minus retinet <expan abbr="impet&utilde;">impetum</expan> illius, adeo <lb/>ut uehementer retineat uehementem concitationem mediocriter <lb/>moderatam, &longs;egniter lentam, nullo modo iu&longs;tam: ita fit, ut &longs;emper <lb/>fermè æqualiter moueatur. </s>

<s id="id002740">Difficile e&longs;t tamen ad unguem &longs;eruare <lb/>moderationem, & æqualitatem, & magis etiam in his horologijs, <lb/>quæ uno circuitu molæ tempus <expan abbr="lõgius">longius</expan> exigunt: at difficilius etiam <lb/>efficere molam, quæ longo tempore duret, cum intenta ualde cele<lb/>rius moueat rotas, & ob id breui ab&longs;oluat circuitum, mollior au<lb/>tem citò remittatur. </s>

<s id="id002741">Et ob id longior & non adeò <lb/>dura melior e&longs;t. </s>

Line 8003

Line 8087

<s id="id002761">Secunda con&longs;tructio hanc habet rationem: Cum n rota indicis <lb/>coniuncta fuerit rotæ, quæ transfert malleum, nece&longs;&longs;e e&longs;t ut unà fe<pb pagenum="158" xlink:href="015/01/177.jpg"/>rantur: qui nimò illud magis mirum de quo illi non mirantur quia <lb/>frequens e&longs;t, &longs;cilicet cur aut quomodo &longs;i diui&longs;æ &longs;unt ut cir<expan abbr="ç&utilde;ducto">çunducto</expan> <lb/>indice non transferatur rota mallei, <expan abbr="põdere">pondere</expan> tamen uer&longs;ata rota in<lb/>dicis in idem incidat, ut horæ quæ pul&longs;u declarantur ad unguem <lb/>& in ei&longs;dem &longs;ectionibus <expan abbr="cõueniant">conueniant</expan> cum horis quas index o&longs;tendit.</s>

<s id="id002761">Secunda con&longs;tructio hanc habet rationem: Cum n rota indicis <lb/>coniuncta fuerit rotæ, quæ transfert malleum, nece&longs;&longs;e e&longs;t ut unà fe

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

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

Line 8654

Line 8741

<s id="id003022">uel quia con<lb/>&longs;onæ aut di&longs;&longs;onæ, uel quia concitat&etail; aut tardæ, uel quod maius e&longs;t quae<lb/> tendant in acutum ad alacritatem, uel in grauem de&longs;inant & remi&longs;&longs;um <lb/>&longs;onum ad <expan abbr="cõmi&longs;erationem">commi&longs;erationem</expan>, & lachrymas, aut etiam ex modo tetrachor<lb/>dorum. </s>

<s id="id003023">Illud &longs;anè non ob&longs;curum e&longs;t, <expan abbr="animã">animam</expan> cum &longs;ono maximè e&longs;&longs;e con<lb/><expan abbr="iunctã">iunctam</expan>, nam neque odoribus ut odores &longs;unt, neque &longs;aporibus, aut his quæ <lb/>tanguntur licet plurimum delectent, aut etiam lædant, anima mouetur <lb/>ad affectus, licet, ut dixi, magis homo delectetur, aut tri&longs;titia afficiatur <lb/>quemadmodum ex &longs;onorum uaria natura, quod etiam in mor&longs;is à Ta<lb/>rantula (arane&etail; genus e&longs;t) deprehenditur. </s>

<s id="id003024">Quinimò nec à luce nec à co<lb/>loribus aut pictura, ni&longs;i ut hæc ad memoriam <expan abbr="reuocãt">reuocant</expan> ea, propter quæ <lb/>ad hilaritatem aut tri&longs;titiam uel iram, uel commi&longs;erationem mouemur. <lb/></s>

<s id="id003025">Vnde <expan abbr="quo&longs;dã">quo&longs;dam</expan> reges ferunt iniurias acceptas iu&longs;si&longs;&longs;e depingi in aula ne <lb/>po&longs;&longs;ent obliui&longs;ci, at longè plures <expan abbr="curar&utilde;t">curarunt</expan>, ut potius <expan abbr="eor&utilde;">eorum</expan> facta egregia <pb pagenum="173" xlink:href="015/01/192.jpg"/>pingerentur continuata per memoriam uoluptate, quam dum illa àge <lb/>rent, <expan abbr="cõceperant">conceperant</expan>: nihilominus, neque color ip&longs;e, nec lux aut &longs;pectaculum <lb/>uel imagines po&longs;&longs;unt adeò mouere animi affectus, uel &longs;onus. </s>

<pb pagenum="173" xlink:href="015/01/192.jpg"/>pingerentur continuata per memoriam uoluptate, quam dum illa àge <lb/>rent, <expan abbr="cõceperant">conceperant</expan>: nihilominus, neque color ip&longs;e, nec lux aut &longs;pectaculum <lb/>uel imagines po&longs;&longs;unt adeò mouere animi affectus, uel &longs;onus. </s>

<s id="id003026">Nam <lb/>duo in uniuer&longs;um ex ui&longs;u ad animi affectus mouendos habentur, tene<lb/>bræ ad tri&longs;titiam & metum, pictura regionum <expan abbr="am&oelig;nar&utilde;">am&oelig;narum</expan> ad iucundita<lb/>tem, &longs;ed <expan abbr="irã">iram</expan> quæ moueant picturæ alacritatemúe aut <expan abbr="cõmi&longs;erationem">commi&longs;erationem</expan>, <lb/>non habemus. </s>

<s id="id003027">Videtur ergo ob hæc &longs;onus ip&longs;e magis animæ intimus <lb/><08> ullum aliud &longs;en&longs;ile. </s>

<s id="id003028">Quod &longs;i odoratus e&longs;t in <expan abbr="app&etilde;dicibus">appendicibus</expan> cerebri, ui <lb/>&longs;us in pupilla oculi, gu&longs;tus in linguæ neruis, ueri &longs;imile e&longs;t magis inti<lb/>mum e&longs;&longs;e auditum, &longs;cilicet in cerebro ip&longs;o, atque ob id magis ab illo mo<lb/>ueri animam. </s>

Line 8668

Line 8758

<s id="id003034">Vt enim corpus de&longs;erit aut impeditur à corporis commercio <lb/>corpus immoritur: hoc præ&longs;entiens animus, fiunt illa duo præuia ad <lb/>mortem timor & tri&longs;titia. </s>

<s id="id003035">Vt contrà, lætitia non e&longs;t ni&longs;i communicatio <lb/>animæ corpori, & quatenus communicatur &longs;olum de uita cogitat, atque<lb/> ob id qua&longs;i immortalis, qui lætatur obliui&longs;citur mortis. </s>

<s id="id003036">Ergo anim&etail; ra<lb/>tio illa erit, quæ ut cogno&longs;cit perfectè exhilaratur dulcedine uo cum, & <lb/>hoc fit in diapa&longs;on. </s>

<s id="id003037">Vt uerò imperfectè diapente, ut imperfectius dia<lb/>te&longs;&longs;aron, at cum ex diate&longs;&longs;aro & diapente perficitur diapa&longs;on, accidit ei <lb/><expan abbr="id&etilde;">idem</expan>, quod <expan abbr="quær&etilde;ti">quærenti</expan> gemmas in matrice dum inuenit, & ei qui ex tabulis <lb/>arcam <expan abbr="cõficit">conficit</expan>, & puero <expan abbr="c&utilde;">cum</expan> adole&longs;cit, & generaliter ei qui ex imperfectis <lb/>perfecta colligit: ex quintæ enim & quartæ &longs;en&longs;u <expan abbr="imperfectar&utilde;">imperfectarum</expan> con&longs;o<pb pagenum="174" xlink:href="015/01/193.jpg"/>nantiarum percipit perfectam diapa&longs;on. </s>

<pb pagenum="174" xlink:href="015/01/193.jpg"/>nantiarum percipit perfectam diapa&longs;on. </s>

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

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

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

Line 8682

Line 8775

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

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

<s id="id003050">Sed metus <expan abbr="c&utilde;">cum</expan> cau&longs;a, affectus propriè <expan abbr="nõ">non</expan> e&longs;t, &longs;ed potius &longs;cientia <expan abbr="quædã">quædam</expan>. <lb/></s>

<s id="id003051">Proprium enim perturbationum e&longs;t excedere rationem: at metus mor<lb/>tis, propri&etail; aut de filio, non e&longs;t à ratione alienús, nec excedit metas, modò <lb/>inanis non &longs;it aut fal&longs;us, ob hoc metum excludemus ab hoc negotio: <lb/>tum maximè ob id quod nulla mu&longs;ica e&longs;t quæ <expan abbr="met&utilde;">metum</expan> excitet cùm ea, <expan abbr="nõ">non</expan> <lb/>opus &longs;it in eo, qui &longs;it cum ratione coniunctus. </s>

<s id="id003051">Proprium enim perturbationum e&longs;t excedere rationem: at metus mor<lb/>tis, propri&etail; aut de filio, non e&longs;t à ratione alienús, nec excedit metas, modò <lb/>inanis non &longs;it aut fal&longs;us, ob hoc metum excludemus ab hoc negocio: <lb/>tum maximè ob id quod nulla mu&longs;ica e&longs;t quæ <expan abbr="met&utilde;">metum</expan> excitet cùm ea, <expan abbr="nõ">non</expan> <lb/>opus &longs;it in eo, qui &longs;it cum ratione coniunctus. </s>

<s id="id003052">Indicio e&longs;t quae potius <expan abbr="ill&utilde;">illum</expan> <lb/>excudit abrupta mu&longs;ica, &longs;icut & omnia alia quæ perturbant rationem, <lb/>ueluti <expan abbr="&longs;olan&utilde;">&longs;olanum</expan> & madrangora atque cicuta. </s>

<s id="id003053">Amorem igitur & odium <expan abbr="nõ">non</expan> <lb/>excitat mu&longs;ica, quia amor & odium alicuius &longs;unt amor & odium, mu&longs;i <lb/>ca <expan abbr="a&utilde;t">aut</expan> generales &longs;olum mouet animi affectus. </s>

<s id="id003054">Et commi&longs;eratio, licet &longs;it <lb/>Didonis aut Phillidis, tamen e&longs;t generaliter mi&longs;erentis. </s>

Line 8802

Line 8895

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

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

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

<s id="id003108">Ergo <lb/>cum idem fiat coniuncto amaro cum in&longs;ipido, ac deterius <expan abbr="c&utilde;">cum</expan> a&longs;trin<pb pagenum="172 [=178]" xlink:href="015/01/197.jpg"/>gente, uelut in acerbis glandibus, quibus nihil tri&longs;tius gu&longs;tari po<lb/>te&longs;t. </s>

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

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

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

Line 8863

Line 8959

<s id="id003144">Opor<lb/>tet autem imum e&longs;&longs;e ad unguem &longs;imile in colore colori anguli loci <lb/>& &longs;ummum parti quæ &longs;e oculis maximè &longs;ubiectam præbet & cla<lb/>ram: media uerò qualia ex umbris ob&longs;curari &longs;olent. </s>

<s id="id003145">Tertia ratio e&longs;t <lb/>pro modo partium iuxta <expan abbr="obliquitat&etilde;">obliquitatem</expan> a&longs;pectus: nam in&longs;picienti a b <lb/>in c d ex e oculo: depingemus in c d iuxta obli<lb/><figure id="id.015.01.199.1.jpg" xlink:href="015/01/199/1.jpg"/><lb/>quitatem &longs;uam, quia cum c d uideatur per line<lb/>as e a c & e b d, & eleuatum in &longs;itu a b, nece&longs;&longs;e e&longs;t <lb/>ut uideatur in &longs;itu a b, ergo eleuatum à c d. </s>

<s id="id003146">E&longs;t <lb/>& alia con&longs;ideratio proportionis ad proxima <lb/>remotaque, grati a exempli, &longs;i homo e&longs;&longs;et po&longs;t co<lb/>lumnam a b, lateret eius pars, quæ e&longs;t propinquior parieti c d, ergo <lb/>&longs;i depinxerimus hominis partes tantum dextram, reliquum &longs;ub um<lb/>bra, cogitur oculus iudicare columnam eleuatam a pariete. </s>

<s id="id003147">De<lb/>mum omnia hæc ita &longs;unt &longs;ubijcienda oculis, & per minimas diffe<pb pagenum="175 [=181]" xlink:href="015/01/200.jpg"/>rentias & animaduer&longs;iones ita dijudicanda, atque experimento &longs;ub<lb/>ijcienda, tum proprio, tum aliorum non artis in expertium, ut res <lb/>pror&longs;us ab&longs;oluta uideatur, atque in hoc multum refert multiplices <lb/>partes &longs;ecundum longitudinem coloribus di&longs;tinguere ad hoc a<lb/>ptis, qui &longs;unt ob&longs;curus, &longs;ub ob&longs;curus, cinereus, qualis &longs;ilicis candi<lb/>dus &longs;ine luce, demum etiam aliquid nigri adijciendum, nam diui&longs;io <lb/>&longs;ecundum longitudinem multum impedit, hanc repræ&longs;entationem <lb/>iuuant, & extrema benè coaptata, uelut &longs;capi imi, & capitula & &longs;u<lb/>premi, <expan abbr="t&utilde;">tum</expan> trabeationes ex materia coronæ, zofoni, t&oelig;nia, epi&longs;tylia, <lb/>plinthi, echini, hypotrachelia, a&longs;tagali, apophyges. </s>

<s id="id003147">De<lb/>mum omnia hæc ita &longs;unt &longs;ubijcienda oculis, & per minimas diffe

<pb pagenum="175 [=181]" xlink:href="015/01/200.jpg"/>rentias & animaduer&longs;iones ita dijudicanda, atque experimento &longs;ub<lb/>ijcienda, tum proprio, tum aliorum non artis in expertium, ut res <lb/>pror&longs;us ab&longs;oluta uideatur, atque in hoc multum refert multiplices <lb/>partes &longs;ecundum longitudinem coloribus di&longs;tinguere ad hoc a<lb/>ptis, qui &longs;unt ob&longs;curus, &longs;ub ob&longs;curus, cinereus, qualis &longs;ilicis candi<lb/>dus &longs;ine luce, demum etiam aliquid nigri adijciendum, nam diui&longs;io <lb/>&longs;ecundum longitudinem multum impedit, hanc repræ&longs;entationem <lb/>iuuant, & extrema benè coaptata, uelut &longs;capi imi, & capitula & &longs;u<lb/>premi, <expan abbr="t&utilde;">tum</expan> trabeationes ex materia coronæ, zofoni, t&oelig;nia, epi&longs;tylia, <lb/>plinthi, echini, hypotrachelia, a&longs;tagali, apophyges. </s>

<s id="id003148">Quæ etiam in <lb/>parte inferiore <expan abbr="c&utilde;">cum</expan> &longs;pira &longs;eu ba&longs;i & limbo & toro & plintho inferio<lb/>re, & &longs;tylobata, et alia t&oelig;nia &longs;umma diligentia, & cum eleuatione ac <lb/>magnitudine ultra columnæ limites extendantur. </s>

<s id="id003149">Sicin &longs;tylobata <lb/>ratio diapente con&longs;tat, cui &longs;olet addi utrinque &longs;exta pars pro coro<lb/>nice, manife&longs;tum e&longs;t autem, quod in ea con&longs;tat mu&longs;ica ratio diapa<lb/>&longs;on ex diapente & diate&longs;&longs;aro, compo&longs;iti nam duæ &longs;extæ partes, alte<lb/>ra utrinque adiecta tertiam conficiunt ut &longs;it diate&longs;&longs;aron &longs;uprà diapen<lb/>te. </s>

<s id="id003150">In regionibus autem & &longs;patijs depingendis eadem fermè &longs;eruan <lb/>da &longs;unt duobus tamen adiectis, <expan abbr="quor&utilde;">quorum</expan> unum e&longs;t ut longinqui&longs;sima <lb/>pars, <expan abbr="nõ">non</expan> per nigrum aut ob&longs;curum, &longs;ed c&oelig;ruleum <expan abbr="color&etilde;">colorem</expan>, qualis in <lb/>c&oelig;lo determinanda e&longs;t (ni&longs;i nox fingatur) nam c&oelig;lum longi&longs;simè <lb/>à nobis di&longs;tat, ita nubes coloribus proprijs, & montes cum niui<lb/>bus, & &longs;patia uelut fluminis alueus, mare, lacus, atque hæc omnia <lb/>per colores di&longs;tantiæ finguntur, uelut fluminis pars propior clara <lb/>& lympida, & colore aqueo cernitur remota ob&longs;cura, quæ maxi<lb/>mè procul abe&longs;t nigra. </s>

Line 8908

Line 9007

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

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

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

<s id="id003177">tertia ob illius con<pb pagenum="179 [=183]" xlink:href="015/01/202.jpg"/>&longs;umptionem. </s>

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

<pb pagenum="179 [=183]" xlink:href="015/01/202.jpg"/>&longs;umptionem. </s>

<s id="id003178">Propterea nil mirum e&longs;t &longs;i Nexo de his &longs;ubtiliter di<lb/>&longs;putauit, mirum fuit quod in tanta animi perturbatione ni&longs;i ad <lb/>amentia, ut illi putant, referatur. </s>

<s id="id003179">Sed quid iam amplius uagor, extat <lb/><arrow.to.target n="marg588"/><lb/>compendio&longs;a ratio con&longs;tructionis illius apud eundem Vitruuium <lb/>ubi Philander ex Atheneo &longs;onus hydradis &longs;uauis admodum atque <lb/><arrow.to.target n="marg589"/><lb/>iucundus auditu e&longs;t: ita ut omnes concinnitate capti conuerterent, <lb/>fuitque Alexendrin&etail; urbis inuentum authore Cte&longs;ibio ton&longs;ore, e&longs;t <lb/>autem magnæ Clep&longs;ydræ in&longs;trumentum non ab&longs;imile, &longs;unt enim <lb/>fi&longs;tulæ in aquam contortæ, quæ, cùm aqua à iuuene quopiam per<lb/>cutitur, axinis per organum tran&longs;euntibus inflantur, <expan abbr="periuc&utilde;dumqúe">periucundum<lb/>qúe</expan> &longs;onum emittunt. </s>

<s id="id003180">E&longs;t autem aræ rotundæ hoc in&longs;trumentum <lb/>per&longs;imile inuentumque Ptolemæi &longs;ecundi Euergit&etail; temporibus, de <lb/>quo eundem Cte&longs;ibium &longs;crip&longs;i&longs;&longs;e ferunt. </s>

Line 8936

Line 9038

<s id="id003190">Et &longs;i non erit ampla magnitudine <lb/>theatrum, media altitudinis tran&longs;uer&longs;aregio de&longs;ignetur, & in ea tre<lb/>decim cellæ duodecim æqualibus.

interuallis di&longs;tantes <expan abbr="confornicen&ttilde;">confornicentur</expan> <lb/>uti ea echea quæ &longs;upra &longs;cripta &longs;unt, ad neten hyperboleon &longs;onan<lb/>tia in cellis quæ &longs;unt in cornibus extremis utraque parte prima col<lb/>locentur, &longs;ecunda ab extremis diate&longs;&longs;aron ad <expan abbr="net&etilde;">neten</expan> diezeugmenon, <lb/>tertia diate&longs;&longs;aron ad neten parame&longs;on, quarta ad neten &longs;ynemme<lb/>non, quinta diate&longs;&longs;aron ad me&longs;en, &longs;exta diate&longs;&longs;aron ad hypaten me<lb/>&longs;en in medio unum diate&longs;&longs;aron ad hypaten hypaton. </s>

<s id="id003191">Quæ sequun<lb/>tur & ad intelligentiam prædictorum melius ex Gulielmo Philan<lb/>dro emendata &longs;ic tran&longs;cribemus: Eas regiones in tredecim cellas <lb/>diuidit æqualibus interuallis: id e&longs;t, cellas paribus uici&longs;sim inter<pb pagenum="178 [=184]" xlink:href="015/01/203.jpg"/>&longs;ticijs di&longs;po&longs;itas di&longs;tribuit &longs;ex hinc atque hinc & unam mediam, quæ <lb/>tamen non u&longs;us, &longs;ed partitionis & re&longs;pon&longs;us cau&longs;a fit in media pr&etail;<lb/>cinctione. </s>

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

<s id="id003192">In ima præcinctione ponuntur ua&longs;a qu&etail; habent harmo<lb/>ni&etail; <expan abbr="ration&etilde;">rationem</expan>, hoc modo. </s>

<s id="id003193">In <expan abbr="cornu&utilde;">cornuum</expan> cellis collocantur quæ <expan abbr="&longs;onit&utilde;">&longs;onitum</expan> ha<lb/>bent netes hyperboleon. </s>

<s id="id003194">Sub&longs;equuntur utrinque quæ &longs;unt ad neten <lb/>diezeugmenon interuallo con&longs;onantia diate&longs;&longs;aron. </s>

Line 8984

Line 9089

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

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

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

<s id="id003226">Hic uides quia <lb/>numerus 10 e&longs;t par, et quod adempta monade, relinquitur 9, qui e&longs;t <lb/>impar quòd medius qui pertinet ad quintum ordinem e&longs;t maxi<pb pagenum="186" xlink:href="015/01/205.jpg"/>mus, & e&longs;t 252, & e&longs;t coniugatio quinarij: hoc uolui dixi&longs;&longs;e, <lb/><figure id="id.015.01.205.1.jpg" xlink:href="015/01/205/1.jpg"/><arrow.to.target n="table24"/><lb/>ut intelligeres rationes colligendi &longs;ingulos ordines &longs;eor<lb/>&longs;um. </s>

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

<pb pagenum="186" xlink:href="015/01/205.jpg"/>mus, & e&longs;t 252, & e&longs;t coniugatio quinarij: hoc uolui dixi&longs;&longs;e, <lb/><figure id="id.015.01.205.1.jpg" xlink:href="015/01/205/1.jpg"/><arrow.to.target n="table24"/><lb/>ut intelligeres rationes colligendi &longs;ingulos ordines &longs;eor<lb/>&longs;um. </s>

<s id="id003227">Quod ergo attinet ad collectionem maximi numeri, <lb/>primus ordo &longs;eruit &longs;emper ultimo <expan abbr="relinqu&etilde;do">relinquendo</expan> monadem, <lb/>& &longs;ecundus penultimo, & tertius antepenultimo, & ita de <lb/><figure id="id.015.01.205.2.jpg" xlink:href="015/01/205/2.jpg"/>alijs, nam &longs;i &longs;ecundus uariatur 55 modis, & 'pen<lb/>ultimus uariabitur 55 modis. </s>

<s id="id003228">Et &longs;i tertius uaria<lb/>tur 165 modis, antepenultimus uariatur 165 mo <lb/>dis. </s>

<s id="id003229">Et ita de alijs.<lb/><arrow.to.target n="table25"/><lb/><arrow.to.target n="marg593"/></s>

Line 10444

Line 10552

<s id="id003555">De Ari&longs;totele <lb/>is multa inuenit &longs;uo Marte, & Theophra&longs;tus longè plura. </s>

<s id="id003556">De alijs, <lb/>dico tam medicis quàm philo&longs;ophis, hoc e&longs;t, quod queror, quod <lb/>in &longs;patio pene duorum millium annorum, non hoc quod ualde re<lb/>conditum erat, &longs;ed nec leue ullum experimentum, uel naturæ arca<lb/>num, uel uitæ &longs;alutare auxilium inuenerit. </s>

<s id="id003557">Sed litigant de nugis & <lb/>rebus inutilibus, & etiam qu&etail; &longs;ciri <expan abbr="nõ">non</expan> po&longs;&longs;unt, ac plerunque non &longs;ine <lb/>magna impietate. </s>

<s id="id003558">Quod uerò nece&longs;&longs;e &longs;it amittere uoluptatem, & <lb/>negotia prætermittere uolenti hanc uitam longam adipi&longs;ci, quæ <lb/>po&longs;tmodum etiam ualde in certa e&longs;t: dico quod quantum ad uolu<lb/>ptates & negotia, non e&longs;&longs;e nece&longs;&longs;e, &longs;ed &longs;olum &longs;uperfluas res, & dam<lb/>no&longs;as & irritas, quas etiam philo&longs;ophi & ciuitatum in&longs;titutores, & <lb/>morum cen&longs;ores docent debere uitari, etiam nullo propo&longs;ito emo<lb/>lumento, at reliqua <expan abbr="cõ&longs;uetudo">con&longs;uetudo</expan> efficit <expan abbr="nõ">non</expan> &longs;olum grata & tolerabilia, <lb/>&longs;ed etiam iucunda. </s>

<s id="id003558">Quod uerò nece&longs;&longs;e &longs;it amittere uoluptatem, & <lb/>negocia prætermittere uolenti hanc uitam longam adipi&longs;ci, quæ <lb/>po&longs;tmodum etiam ualde in certa e&longs;t: dico quod quantum ad uolu<lb/>ptates & negocia, non e&longs;&longs;e nece&longs;&longs;e, &longs;ed &longs;olum &longs;uperfluas res, & dam<lb/>no&longs;as & irritas, quas etiam philo&longs;ophi & ciuitatum in&longs;titutores, & <lb/>morum cen&longs;ores docent debere uitari, etiam nullo propo&longs;ito emo<lb/>lumento, at reliqua <expan abbr="cõ&longs;uetudo">con&longs;uetudo</expan> efficit <expan abbr="nõ">non</expan> &longs;olum grata & tolerabilia, <lb/>&longs;ed etiam iucunda. </s>

<s id="id003559">De incerto fine, quid e&longs;t certum apud homines, <lb/>ni&longs;i hoc nihil certum e&longs;&longs;e? </s>

<s id="id003560">Verum tamen &longs;i quis re&longs;piciat ad præ<lb/>mium tam &longs;ingulare e&longs;t, & nobile atque utile, ut non lu&longs;erit operam <lb/>immeritò, quicunque cum &longs;pe tam illu&longs;tris commodi, & tam exigua <lb/>iactura rerum, ac minore periculo &longs;e huic aleæ experiundæ commi<lb/>&longs;erit. </s>

<s id="id003561">Cum, &longs;i quis hoc ip&longs;um adipi&longs;catur, uerè dici po&longs;sit &longs;ummum <lb/>bonum adeptum e&longs;&longs;e: Non &longs;olum compos factus diuturnitatis ui<lb/>tæ, &longs;ed cum illa tot uoluptatum, quæ in longo tempore percipiun<lb/>tur &longs;cientiæ tot rerum, quas non ni&longs;i temporis longitudo o&longs;tende<lb/>re pote&longs;t, tot denique ca&longs;us uidere tum opum in crementum, quod <lb/>qua&longs;i certi&longs;simum e&longs;t in longa ætate & u&longs;u &longs;apientia & authoritate <lb/>plena, adeò ut fermè nece&longs;&longs;e &longs;it ad principatus &longs;peciem deuenire, <lb/>qui tamdiu uixerit, tum gloria ip&longs;a in comparabili. </s>

Line 10493

Line 10601

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

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

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

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

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

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

Line 11575

Line 11683

<s id="id004039">Moueatur l m &longs;uper a c, æquidi&longs;tans &longs;emper a b, ut &longs;eruet &longs;itum <lb/>quem habebat ita, quod <expan abbr="extrem&utilde;">extremum</expan> lineæ l m &longs;it &longs;emper in linea a c, & <lb/>l punctum quod gerit uicem a, de&longs;cendat tantum in linea l m, quan<lb/>tum l extremum in linea a c: dicit Philo&longs;ophus, quod a &longs;eu l &longs;emper <lb/>de&longs;cendet in linea a d, & erit in e a. </s>

<s id="id004040">Supponatur quae latus l m fit f g, & <lb/>erit l n, f t, ducatur <expan abbr="a&utilde;t">aut</expan> ex r puncto &longs;ectionis diametri, & lateris l m li <lb/><arrow.to.target n="marg790"/><lb/>near q, æquidi&longs;tans a f, <expan abbr="igi&ttilde;">igitur</expan> rhombus a q r f e&longs;t &longs;imilis rhombo toti <lb/>a b d c, & proportio a f ad fr, ut a c ad c d, &longs;ed a c e&longs;t &etail;qualis c d, <expan abbr="igi&ttilde;">igitur</expan> a f <lb/>e&longs;t æqualis f r, &longs;ed l de&longs;cendit in l m, <expan abbr="quant&utilde;">quantum</expan> e&longs;t a f ex &longs;uppo&longs;ito, <expan abbr="igi&ttilde;">igitur</expan> <lb/><expan abbr="punct&utilde;">punctum</expan> l &longs;emper erit in linea a d. </s>

<s id="id004041">Po&longs;t deficiunt quædam uerba: ob <lb/>quæ nemo intellexit &longs;ententiam Philo&longs;ophi, & <expan abbr="tam&etilde;">tamen</expan> au&longs;i &longs;unt impo<lb/>nere lectoribus, tan<08> intellexi&longs;&longs;ent, tres &longs;imul errores admittendo, <lb/>&longs;cilicet Ari&longs;totelem ob propriam ignorantiam, ut &longs;tultum accu&longs;an<lb/>do, qui fal&longs;a dicat, & demon&longs;trare nitatur: produnt &longs;e ip&longs;os cum <lb/>&longs;ua impudentia. </s>

<s id="id004041">Po&longs;t deficiunt quædam uerba: ob <lb/>quæ nemo intellexit &longs;ententiam Philo&longs;ophi, & <expan abbr="tam&etilde;">tamen</expan> au&longs;i &longs;unt impo<lb/>nere lectoribus, tanquam intellexi&longs;&longs;ent, tres &longs;imul errores admittendo, <lb/>&longs;cilicet Ari&longs;totelem ob propriam ignorantiam, ut &longs;tultum accu&longs;an<lb/>do, qui fal&longs;a dicat, & demon&longs;trare nitatur: produnt &longs;e ip&longs;os cum <lb/>&longs;ua impudentia. </s>

<s id="id004042">Et lectoribus imponere conantur, debet ergo &longs;ic <lb/>legi (“b in ip&longs;a b c diametro latum, ubi latus b d moueatur in late<lb/>re b a, & b æqualiter uer&longs;us d in b d, æqualis enim e&longs;t ip&longs;a b e”) <lb/>Tunc enim con&longs;tat ut hic dixi, m moueri per b c rectam ut l per a d: <lb/>Dicit ergo <expan abbr="c&utilde;">cum</expan> b d <expan abbr="mouea&ttilde;">moueatur</expan> in b a, tran&longs;it unico motu <expan abbr="totã">totam</expan> b a, & pun<pb pagenum="237" xlink:href="015/01/256.jpg"/><expan abbr="ct&utilde;">ctum</expan> tamen b, quod <expan abbr="moue&ttilde;">mouetur</expan> duobus motibus, non pertran&longs;it ni&longs;i b c, <lb/>quæ pote&longs;t e&longs;&longs;e minor b a: nam <expan abbr="cõ&longs;tat">con&longs;tat</expan> quod <expan abbr="quãdo">quando</expan> m erit in a, o erit <lb/>in e, & quia m de&longs;cendit in o, in eodem tempore, ergo o erit in c, & <lb/><expan abbr="trã&longs;iuit">tran&longs;iuit</expan> &longs;emper per rectam b c: igitur m e&longs;t minus <expan abbr="mot&utilde;">motum</expan> duobus mo<lb/>tibus quàm m l unico <expan abbr="tant&utilde;">tantum</expan>. </s>

<s id="id004043">Et quia aliquis dicere potui&longs;&longs;et non e&longs;t <lb/>mirum, quod m &longs;it minus motum duobus motibus quàm l m latus <lb/>unico tantum: quia m mouetur motu contrario motui lateris: nam <lb/>latus m o mouetur in latere b a a&longs;cendendo, et punctum m uer&longs;us o <lb/>in ip&longs;o m o de&longs;cendendo. </s>

<s id="id004044">Dicit Philo&longs;ophus, hoc e&longs;t mirum, quia <lb/>cum idem contingat in motu l, cuius latus mouetur per a c, & l per l <lb/>m recedendo in partem contrariam, nihilominus uelocius motum <lb/>e&longs;t l, quàm latus l m, quia a d e&longs;t longior a c. </s>

Line 12611

Line 12719

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

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

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

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

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

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

Line 12688

Line 12796

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

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

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

Line 12768

Line 12876

<s id="id004534">Et hoc e&longs;t quia in &etail;ternis motus e&longs;t quies, & ideo habent quandam <lb/>&longs;imilitudinem iuxta <expan abbr="perfection&etilde;">perfectionem</expan> &longs;uam, &longs;icut &longs;i e&longs;&longs;ent in circulo hoc <lb/><figure id="id.015.01.287.1.jpg" xlink:href="015/01/287/1.jpg"/><lb/>modo. </s>

<s id="id004535">Mutatio ergo <expan abbr="cõuenit">conue<lb/>nit</expan> in corporeis qu&etail; <expan abbr="pend&etilde;t">pendent</expan> <lb/>à corpore, &longs;icut lumini: qua<lb/>tenus enim &longs;unt ex corpo<lb/>reo, <expan abbr="occupãt">occupant</expan> diuer&longs;um <expan abbr="loc&utilde;">locum</expan>, <lb/>quatenus e&longs;t in corporei id, <lb/>agit &longs;ine tran&longs;itu per <expan abbr="medi&utilde;">medium</expan> <lb/>& in in&longs;tanti, ergo in corpo<lb/>rea &longs;impliciter mutationem <lb/>recipiunt, non in tempore <lb/>neque in loco. </s>

<s id="id004536">Videtur <expan abbr="aut&etilde;">autem</expan> <lb/>uelo<expan abbr="ci&longs;sim&utilde;">ci&longs;simum</expan> dupliciter <expan abbr="etiã">etiam</expan> <lb/>nobis iuxta &longs;en&longs;um, idque e&longs;t <lb/>in quo &longs;en&longs;us medij tran&longs;itum non percipit, & natura quod e&longs;t pri<lb/>mi mobilis. </s>

<s id="id004537">At dubitare quis pote&longs;t circa hoc, nam proprium mo<lb/>tus e&longs;t tangentia concutere, quietis autem minime: concutit autem <lb/>maximè quod ueloci&longs;simè mouetur, ob hoc arbitrati &longs;unt homi<lb/>nes quod ueloci&longs;simus motus multò plus di&longs;taret à natura quietis <lb/>quam tardus, &longs;ed hoc e&longs;t quia non eadem e&longs;t ratio uiolenti & natu<lb/>ralis: uiolenta enim non redeunt in &longs;e ip&longs;a, nec habent rationem cir<lb/>cularis, &longs;ed potius recti & infiniti, & ideò in his quæ mouentur mo<lb/>tu recto naturali cadit uiolentia, non autem in his quæ mouentur <lb/>motu circulari naturali: <expan abbr="cõ">con</expan> cu&longs;sio ergo e&longs;t in motu uiolento, & qua<lb/>li&longs;cunque motus uiolentus, quanto magis augetur tantò magis re<lb/>cedit à contrario, tantò magis remouetur à natura contrarij, & ha<lb/>bet actiones contrarias ualidiores.</s>

<s id="id004537">At dubitare quis pote&longs;t circa hoc, nam proprium mo<lb/>tus e&longs;t tangentia concutere, quietis autem minime: concutit autem <lb/>maximè quod ueloci&longs;simè mouetur, ob hoc arbitrati &longs;unt homi<lb/>nes quod ueloci&longs;simus motus multò plus di&longs;taret à natura quietis <lb/>quam tardus, &longs;ed hoc e&longs;t quia non eadem e&longs;t ratio uiolenti & natu<lb/>ralis: uiolenta enim non redeunt in &longs;e ip&longs;a, nec habent rationem cir<lb/>cularis, &longs;ed potius recti & infiniti, & ideò in his quæ mouentur mo<lb/>tu recto naturali cadit uiolentia, non autem in his quæ mouentur <lb/>motu circulari naturali: <expan abbr="cõ">com</expan> cu&longs;sio ergo e&longs;t in motu uiolento, & qua<lb/>li&longs;cunque motus uiolentus, quanto magis augetur tantò magis re<lb/>cedit à contrario, tantò magis remouetur à natura contrarij, & ha<lb/>bet actiones contrarias ualidiores.</s>

<s id="id004538">E&longs;t etiam aliud penè &longs;imile argumentum in figuris ip&longs;is, circulus <lb/>enim unica linea continetur, nulla tamen figura ab ea magis natura

Legend:

Removed from v.1.24
changed lines
	Added in v.1.25