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annotate texts/archimedesOldCVSRepository/archimedes/xml/galil_gal72_X01_it_1600.xml @ 12:f9a6b8344c3a

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DESpecs 2.0 Autumn 2009
author Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date Thu, 02 May 2013 11:14:40 +0200
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1 <?xml version="1.0"?>
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2 <archimedes xmlns:xlink="http:/www.w3.org/1999/xlink">
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3 <info>
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4 <author>Galileo Galilei</author>
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5 <title>notes on motion (Part of MS Gal. 72)</title>
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6 <date>ca. 1600 - ca. 1640</date>
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7 <place>Padua, Florence</place>
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8 <translator/>
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9 <lang>la</lang>
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10 <cvs_file>galil_gal72_X01_it_1600.xml</cvs_file>
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11 <cvs_version/>
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12 <locator>X01.xml</locator>
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13 </info>
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14 <text>
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15 <body>
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16 <pb xlink:href="X01/01/001.jpg" page="033_R"/>
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17 <p><s>Si in horizonte sumantur duo puncta, et ab altero ipsorum quaelibet linea inclinetur ad quam ex altero puncto horizontis altera recta ducatur, ex ea secans partem aequalem ei quae inter puncta horizontis intercipitur, casus per hanc ductam citius absolvitur, quam per quascumque alias rectas ex eodem puncto ad eamdem lineam protractas. In aliis autem, quae per angulos aequales supra et infra ab hac distiterint, casus fiunt temporibus aequalibus.</s></p>
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18 <p><s>Sint in linea horizontali <DELETION-1/> 'ab' <END-OF-DELETION-1/> duo puncta 'a', 'b' et a 'b' inclinetur recta 'bc', in qua ex termino 'b' sumatur 'bd' ipsi 'ba' aequalis, et iungatur 'ad'. Dico casum per 'ad' velocius fieri quam per quamlibet ex 'a' ad inclinatam 'bc' productam. Ex punctis enim 'a', 'd' ad ipsas 'ba', 'bd' perpendiculares, ducantur 'ae', 'de', sese in 'e' secantes, et quia in triangulo aequicruri 'abd' anguli 'bad', 'bda' sunt aequales, erunt reliqui ad rectos 'dae', 'ead' aequales, ergo centro 'e', intervallo autem 'ea', descriptus circulus per 'd' quoque transibit, et lineas 'ba', 'bd' tanget in punctis 'a', 'd'. Et cum 'a' sit terminus perpendiculi 'ae', casus per 'ad' citius absolvetur quam per quamcumque aliam ex eodem termino 'a' usque ad lineam 'bc' ultra circumferentiam circuli extensam: quod est primo demonstrandum.</s></p>
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19 <p><s>Quod si in perpendiculo <DELETION-1> 'ac' </DELETION-1> 'ae', quodcumque centrum 'f' et secundum intervallum 'fa' circulus 'agc' describatur, lineam 'bc' in punctis 'gc' secans, iunctae 'ag' 'ac' per angulos aequales, ex ante demonstratis a media 'ad' dirimentur; et <SUBSTITUTION-1> fit <SUBSTITUTED-BY-1> patet </SUBSTITUTED-BY-1></SUBSTITUTION-1> per ipsas eodem tempore fieri motus, cum ex apice perpendiculi ad circumferentiam 'agc' sint ductae.</s></p>
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20 <p><s>Lemma pro antecedenti.</s></p>
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21 <p><s>Si duo circuli se intus tangant, et linea recta interiorem circulum contingat, et alterum secet, tres lineae a contactu circulorum ad tria puncta tangentis et secantis lineae productae angulos duos aequales continebunt. Assumpta praesenti figura, protrahatur 'ad' usque ad 'h', et iungatur 'hf', secans 'gc' in 'i', et quia anguli in centris 'e', 'f' sunt aequales, cum similibus circumferentiis sectis a linea 'adh' insistant, erit linea 'fih' ipsi 'ed' parallela. Cumque 'ed' sit perpendicularis ad 'gc', ipsa quoque 'fih' ex centro 'f' ad lineam 'cg' perpendicularis erit, et, quod consequens est, arcum 'ghc' bifariam dividet, et angulus 'gah' angulo 'hac' erit aequalis. Quod erat probandum.</s></p>
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22 <p><s>p. 209</s></p>
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23 <pb xlink:href="X01/01/002.jpg" page="033_V"/>
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24 <pb xlink:href="X01/01/003.jpg" page="034_R"/>
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25 <p><s>Sit planum horizontis secundum lineam 'abc', ad quam sint duo plana inclinata secundum lineas 'db', 'da': dico idem mobile tardius moveri per 'da', quam per 'db' secundum rationem longitudinis 'da' ad longitudinem 'db'. Erigatur enim ex 'b' perpendicularis ad horizontem, quae sit 'be', ex 'd' vero ipsi 'bd' perpendicularis 'de', occurrens 'be' in 'e', et circa 'bde' triangulum circulus describatur, qui tanget 'ac' in puncto 'b', ex quo ipsi 'ad' parallela ducatur 'bf', et connectatur 'fd'. Patet, tarditatem per 'fb' esse consimilem tarditati per 'da'; quia vero tempore eodem movetur mobile per 'db' et 'fb', patet velocitates per 'bd' ad velocitates per 'fb' esse ut 'db' ad 'fb', ita ut semper iisdem temporibus duo mobilia, ex punctis 'd', 'f' venientia, linearum 'db', 'fb' partes integris lineis 'db', 'fb' proportione respondentes peregerint. Cum vero angulus 'bfd' in portione angulo 'dba' ad tangentem sit aequalis, angulus vero 'dbf' alterno 'bda', aequiangula erunt triangula 'bfd', 'abd', et ut 'bd' ad 'bf', ita 'ad' ad 'db': ergo, ut 'ad' ad <DELETION-1> 'df' </DELETION-1> 'db', ita velocitas per 'db' ad velocitatem per 'da', et, ex opposito, tarditas per 'da' ad tarditatem per 'bd'.</s></p>
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26 <p><s>Si hoc sumatur, reliqua demonstrari possent. Ponatur igitur, augeri vel imminui motus velocitatem secundum proportionem qua augentur vel minuuntur gravitatis momenta; et cum constet, eiusdem mobilis momenta gravitatis super plano 'db' ad momenta super plano 'da' esse ut longitudo 'da' ad longitudinem 'db', idcirco velocitatem per 'db' ad velocitatem per 'da' esse ut <SUBSTITUTION-1> 'ab' <SUBSTITUTED-BY-1> 'ad' </SUBSTITUTED-BY-1></SUBSTITUTION-1> ad 'db'.</s></p>
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27 <pb xlink:href="X01/01/004.jpg" page="034_V"/>
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28 <pb xlink:href="X01/01/005.jpg" page="035_R"/>
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29 <p><s>'ab' est media inter <DELETION-1> <ILLEGIBLE/> </DELETION-1> 'ca', 'as': nam rettangulum 'cas' aequatur rettangulo 'fad'. Si enim ducatur 'cf', erit triangulum 'caf' simile triangulo 'sad'.</s></p>
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30 <p><s><n-SUPERSCRIBED-BY-o/>n[umer]o 39</s></p>
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31 <p><s> ratio temporis 'ac' ad tempus 'ab' componitur ex ratione <DELETION-1> quadrati </DELETION-1> 'ac' ad <DELETION-1> quadratum </DELETION-1> 'ab' et altitudinis 'ad' <SUBSTITUTION-3> ad altitudinem 'ae', quae est ratio solidi ex 'ad' in quadratum 'ac' ad solidum ex <INSERTION-2> <DELETION-1> 'ab' in quadratum </DELETION-1> </INSERTION-2> 'ae' in quadratum 'ab'. Haec autem solida sunt aequalia, quia quadratum 'ca' ad quadratum 'ab' est ut linea 'ea' ad 'ad', nempe ut rettangulum 'fae' ad rettangulum 'fad'. <SUBSTITUTED-BY-3> ad mediam inter altitudines 'ad', 'ae', quae ratio est eadem cum ratione 'ba' ad 'ac'. Quadratum enim 'ab' ad <SYMBOL-QUADRATUM/>_[quadrat]um 'ac' est ut 'ad' ad 'ae', nempe ut <SYMBOL-RECTANGULUM/>_[rectangulum] 'fad' ad <SYMBOL-RECTANGULUM/>_[rectangulum] 'fae': sed ratio composita ex 'ca' ad 'ab' et ex 'ab' ad 'ca' est ratio aequalitatis. Ergo patet propositum. </SUBSTITUTED-BY-3></SUBSTITUTION-3> </s></p>
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32 <p><s>In numeris ab unitate consequentibus, summa cuius libet multitudinis ad aliam summam alterius multitudinis, si ab utraque dimidium maximi numeri auferatur, est ut quadratum multitudinis unius ad quadratum alterius multitudinis. Summa enim 'ab' est 36, ablato dimidio 8, remanet 32; summa 'ac' est 21, ablato dimidio 6, remanet 18: et 32 ad 18 est ut quadratum multitudinis 'ab', nempe 64, ad quadratum multitudinis 'ac', quod est 36.</s></p>
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33 <pb xlink:href="X01/01/006.jpg" page="035_V"/>
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34 <p><s><foreign lang="it">tempo per 'ac' al tempo per 'ab' è come 10 a 8.</foreign></s></p>
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35 <p><s>Ratio temporum <INSERTION-1> lationum </INSERTION-1> super planis <DELETION-1> inaequalibus </DELETION-1> quorum diversae sint inclinationes, et longitudines, nec non elevationes inaequales, componitur ex ratione longitudinum ipsorum et ex subdupla ratione elevationum eorumdem permutatim accepta.</s></p>
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36 <pb xlink:href="X01/01/007.jpg" page="036_R"/>
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37 <p><s>Dato perpendiculo et plano ad ipsum inclinato, quorum eadem sit altitudo idemque terminus sublimis, punctum in perpendiculo supra terminum communem reperire, ex quo si demittatur grave, quod postea convertatur per planum inclinatum, ipsum planum inclinatum conficiat eodem tempore, quo ipsum <DELETION-1> pl </DELETION-1> perpendiculum ex quiete conficeret.</s></p>
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38 <p><s>Sint perpendiculus, et planum inclinatum, quorum eadem sit altitudo, 'ab', 'ac': oportet in perpendiculo 'ba' producto ex 'a', <DELETION-1> in 'h' </DELETION-1> punctum invenire, ex quo demissum mobile conficiat spacium 'ac' eodem tempore, quo conficit perpendiculum 'ab' ex quiete in 'a'. Ponatur 'dce' ad angulos rectos ad 'ac', et secetur 'cd' aequalis 'ab', et iungatur 'ad', quae maior erit ipsa 'dc', et angulus 'adc' maior angulo 'cad' <INSERTION-2> (est enim 'ca' maior quam 'cd', seu 'ab'): </INSERTION-2> fiat angulus 'dae' aequalis angulo 'ade', et ad ipsam 'ae' perpendicularis sit 'ef', plano inclinato et extenso occurrens in 'f', et utraque 'ai', 'ag' ponatur ipsi 'cf' aequalis, et per 'g' horizonti aequidistans 'gh': dico 'h' esse punctum quod quaeritur.</s></p>
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39 <p><s>Intelligatur enim tempus casus per perpendiculum 'ab' esse 'ab', erit tempus per 'ac' ex 'a' <DELETION-1> in </DELETION-1> ipsa 'ac'. Cumque in <SYMBOL-3/>_[tri]ang[ul]o rectangulo 'aef' ab angulo recto 'e' perpendicularis ad basim 'af' sit acta 'ec', erit 'ae' media inter 'fa', 'ac', et 'ce' media inter 'ac', 'cf', hoc est inter 'ca', 'ai'; et cum ipsius 'ac' tempus ex 'a' sit 'ac', erit 'ae' tempus totius 'af', et 'ec' tempus ipsius 'ai'. Quia vero in <SYMBOL-3/>_[tri]angulo aequicruri 'aed', latus 'ae' est aequale lateri 'ed', erit 'ed' tempus per 'af': et est 'ec' tempus per 'ai': ergo 'cd', hoc est 'ab', erit tempus per 'if' ex quiete in 'a'; quod idem est ac si dicamus 'ab' esse tempus per 'ac' ex 'g', seu ex 'h': quod erat demonstrandum.</s></p>
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40 <pb xlink:href="X01/01/008.jpg" page="036_V"/>
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41 <pb xlink:href="X01/01/009.jpg" page="037_R"/>
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42 <p><s>Lemma. Sint tres lineae utcumque 'a', 'd', 'e', et inter 'a', 'd' media proportionalis sit 'b'; inter 'a', 'e' media proportionalis sit 'c'; inter 'e', 'd' tandem media sit 'g'. Dico ut 'c' ad 'b', ita esse 'g' ad 'd'. Quia enim 'b' est media inter 'a', 'd', erit quadratum 'b' aequale rettangulo 'ad'; similiter quadratum 'c' aequale rettangulo 'ae'; igitur, ut rectangulum 'ae' ad rectangulum 'ad', ita quadratum 'c' ad quadratum 'b'. Ut autem rectangulum 'ae' ad rectangulum 'ad', ita linea 'e' ad 'd'; ut vero linea 'e' ad lineam 'd', ita quadratum 'g' ad quadratum 'd'; ergo, ut quadratum 'c' ad quadratum 'b', ita quadratum 'g' ad quadratum 'd', et ut 'c' ad 'b', ita 'g' ad 'd'.</s></p>
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43 <p><s>Sint plana aequalia 'ab', <SUBSTITUTION-1> 'cd' <SUBSTITUTED-BY-1> 'cb' </SUBSTITUTED-BY-1></SUBSTITUTION-1> inaequaliter inclinata, et altitudo inclinationis plani 'ab' sit 'be', ipsius vero 'bc' sit 'bd'. Dico tempus casus super 'ba' ad tempus casus per 'bc', esse ut media proportionalis inter 'db', 'be' ad ipsam 'be'.</s></p>
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44 <p><s>Accipiatur 'fb' ipsis 'cb', 'ab' aequalis, et ipsarum 'fb', 'bd' media sit 'bs'; ipsarum vero 'fb', 'be' media sit 'br': et quia tempus casus 'bf' ad tempus casus 'bd' est ut 'sb' ad 'bd', tempus vero casus 'bd' ad tempus casus 'bc' ut 'bd' ad 'bc', ergo, ex aequali, tempus casus 'bf' ad tempus casus 'bc' ut 'sb' ad 'bc', et, convertendo, tempus casus 'bc' ad tempus casus 'bf' ut 'bc' ad 'bs'. Similiter autem demonstrabitur, ut tempus casus 'bf' ad tempus casus 'ba', ita linea 'rb' ad 'ba' vel 'bc'; ergo, ex aequali in analogia perturbata, ut tempus casus 'bc' ad tempus casus 'ba', ita 'rb' ad 'sb', et, conversim, ut tempus casus 'ba' ad tempus casus 'bc' ita 'sb' ad 'br'. Ex lemmate vero antecedenti, ut 'sb' ad 'br', ita media inter 'db', 'be' ad ipsam 'be': quare patet propositum. <DELETION-1> alit </DELETION-1> <lb/> Aliter, absque lemmate:</s></p>
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45 <pb xlink:href="X01/01/010.jpg" page="037_V"/>
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46 <p><s>Sit 'bi' media inter 'db', 'be', et 'is' parallela ad 'dc'; et quia ut tempus per 'ba' ad tempus per 'be', ita linea 'ba' ad lineam 'be'; ut autem <DELETION-1> 'be' ad </DELETION-1> tempus 'be' ad tempus 'bd', ita linea 'be' ad 'bi'; ut autem tempus 'bd' ad tempus 'bc', ita linea 'bd' ad 'bc', hoc est 'bi' ad 'bs'; ergo, ex aequali, ut tempus per 'ba' ad tempus per 'bc', ita linea 'ab', <DELETION-1> ad </DELETION-1> seu 'cb', ad 'bs', hoc est 'db' ad 'bi', seu 'ib' ad 'be': quod erat probandum.</s></p>
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47 <p><s>Tempora lationum in planis aequalibus, sed inaequaliter inclinatis, sunt inter se in subdupla ratione elevationum ipsorum planorum permutatim assumpta.</s></p>
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48 <pb xlink:href="X01/01/011.jpg" page="038_R"/>
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49 <p><s>Tempora casuum in planis quorum eadem sit altitudo, eamdem inter se servant rationem, sive illis idem impetus praecedat, sive ex quiete incipiant.</s></p>
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50 <p><s>Sint plana 'ab', 'ac', quorum eadem altitudo; extenso autem 'ba' <DELETION-1> q </DELETION-1> utcumque in 'd', fiat casus ex 'd' per ambo 'ac', 'ab'. Dico, tempus per 'ac' ad tempus per 'ab' esse in eadem ratione ac si principium casus foret in 'a'. Sit enim ipsarum 'bd', 'da' media <DELETION-1> d </DELETION-1> 'df', et ducta parallela ex 'f' ipsi 'bc', quae sit 'fg', erit 'ge' media inter 'ce', 'ea'. Facto igitur principio lationis ex 'd', tempora casuum per 'ac', 'ab' erunt inter se ut 'ag', 'af': quod si casus incipiat ex 'a', erunt tempora per 'ac', 'ab' inter se ut 'ac', 'ab' lineae: ergo patet propositum.</s></p>
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51 <pb xlink:href="X01/01/012.jpg" page="038_V"/>
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52 <pb xlink:href="X01/01/013.jpg" page="039_R"/>
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53 <p><s>Dato perpendiculo, ad ipsum lineam inflectere, in qua, cum ipsa habeat cum dato perpendiculo eandem altitudinem, fiat motus post casuum perpendiculo eodem tempore ac in eodem perpendiculo ex quiete.</s></p>
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54 <p><s>Sit datus perpendiculus 'ab' qui protrahatur, et ponatur 'bc' aequalis ipsi 'ab', et ducantur horizontes 'ce', 'ag': oportet, ex 'b' planum usque ad horizontem 'ce' inflectere, in quo fiat motus post casum ex 'a' eodem tempore ac in 'ab' ex quiete in 'a'. Sumatur 'cd' aequalis 'cb', et ponatur 'be' utrique 'bd', 'dc' aequalis: dico iam, per 'be' post casum 'ab' fieri motum eodem tempore ac in 'ab'. Producatur 'eb', occurrens horizonti superiori in 'g', et ipsarum 'eg', 'gb' media sit 'gf'; erit 'ef' ad 'fb' ut 'eg' ad 'gf', <SUBSTITUTION-1> sed <SUBSTITUTED-BY-1> et </SUBSTITUTED-BY-1></SUBSTITUTION-1> quadratum 'ef' ad quadratum 'fb' ut quadratum 'eg' ad quadratum 'gf'; sed quadratum 'eg' ad quadratum 'gf' est ut 'eg' ad 'gb'; est autem 'eg' dupla ad 'gb', quia aequales sunt 'ab', 'bc': ergo quadratum 'ef' duplum est quadrati 'fb', et linea 'bf' aequalis 'bc', cum tota 'be' duabus 'bd', 'bc' aequalis posita sit. Si igitur intelligatur 'ab' esse tempus casus per 'ab', erit 'gb' tempus casus per 'gb', et 'gf' tempus per totam 'ge'; ergo <DELETION-1> f </DELETION-1> 'bf' erit tempus per reliquam 'be' post casum ex 'g', seu ex 'a': quod est propositum.</s></p>
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55 <pb xlink:href="X01/01/014.jpg" page="039_V"/>
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56 <pb xlink:href="X01/01/015.jpg" page="040AR"/>
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57 <p><s>Dato perpendiculo et plano ad ipsum inflexo, in dato plano partem signare, in qua post casum in perpendiculo fiat motus eodem tempore, quo mobile perpendiculum confecit.</s></p>
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58 <p><s><DELETION-1> sit p </DELETION-1> Sit perpendiculum 'ab', et ad ipsum inflexa 'be': oportet in 'be' spacium signare, per quod mobile moveatur post casum 'ab' eodem tempore quo confecit ipsum 'ab'.</s></p>
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59 <p><s>Extenso plano 'eb', occurrat horizonti in 'd', et accipiatur 'bf' aequalis 'ab', et fiat, ut 'bd' ad 'df', ita 'fd' ad 'de': dico, tempus per 'be' post 'ab' aequari tempori per 'ab'. Si enim intelligatur 'ab' tempus per 'ab', erit 'db' tempus per 'db'; cumque sit ut 'bd' ad 'df', ita 'fd' ad 'de', erit 'df' tempus per totam 'de', et 'bf' per partem 'be' ex 'd'; sed tempus per 'be' post 'db' est idem ac post 'ab': ergo tempus per 'be' post 'ab' erit 'bf', aequale scilicet tempori 'ab': quod erat propositum.</s></p>
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60 <pb xlink:href="X01/01/016.jpg" page="040AV"/>
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61 <pb xlink:href="X01/01/017.jpg" page="040BR"/>
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62 <p><s>Sit modo cylindrus 'egt', respondens librae 'abcd', utcumque sectum in 'sg'. Dico, momentum totius cylindri pendentis ex 'c' ad momentum partis 'eg' pendentis ex 'b' esse ut <SYMBOL-RECTANGULUM/>_[rectangulum] 'dca' ad <SYMBOL-RECTANGULUM/>_[rectangulum] 'dba'. Ex demonstratis enim, momentum ponderis 'egt' ad momentum ponderis 'eg' habet rationem compositam ex ex [sic] pondere 'egt' ad pondus 'eg' et distantiae 'cd' ad distantiam 'db': pondus autem 'egt' ad pondus 'eg' est <DELETION-1> <ILLEGIBLE/> </DELETION-1> ut linea 'ac' ad 'ab': ergo momentum ponderis 'egt' ad momentum ponderis 'eg' habet rationem compositam ex 'cd' ad 'db' et ex 'ca' ad 'ab', quae est <SYMBOL-RECTANGULUM/>_[rectanguli] 'dca' ad <SYMBOL-RECTANGULUM/>_[rectangulum] 'dba'.</s></p>
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63 <pb xlink:href="X01/01/018.jpg" page="040BV"/>
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64 <pb xlink:href="X01/01/019.jpg" page="040CR"/>
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65 <p><s>Ponderum in libra suspensorum momenta habent rationem compositam ex ratione ipsorum ponderum et ex ratione distantiarum.</s></p>
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66 <p><s>Pendeant pondera 'de', 'f' ex distantiis 'ab', 'bc': dico, momentum ponderis 'de' ad momentum ponderis 'f' habere rationem compositam ex rationibus ponderis 'de' ad pondus 'f' et distantiae 'ab' ad distantiam 'bc'. Ut enim 'ab' ad 'bc', ita fiat pondus 'f' ad pondus 'do': cum ergo pondera 'f' et 'do' habeant rationem distantiarum 'ab', 'bc' permutatam, erit momentum ponderis 'f' aequale momento ponderis 'do'. Cum igitur sint 3 pondera utcumque 'ed', 'f' et 'do', erit ratio ponderis 'ed' ad 'do' composita ex rationibus 'ed' ad 'f' et 'f' ad 'do': ut autem pondus 'ed' ad pondus 'do', ita momentum 'ed' ad momentum 'do'; pendent enim ex eodem puncto: igitur, cum momentum 'do' sit aequale momento 'f', ratio momenti 'ed' ad momentum 'f' erit composita ex rationibus ponderis 'ed' ad pondus 'f' et ponderis 'f' ad pondus 'do'. Factum est autem pondus 'f' ad pondus 'do' ut distantia 'ab' ad distantiam 'bc': ergo patet, momentum ponderis 'ed' ad momentum ponderis 'f' habere rationem compositam ex rationibus ponderum 'ed', 'f' et distantiarum 'ab', 'bc'.</s></p>
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67 <p><s>Quod si suspendatur ex puncto 's', facta distantia 'bs' aequali distantiae 'bc', pondus 't' aequale ponderi 'f', erit eius momentum momento 'f' aequale; et similiter ponderum 'ed' et 't' momenta habebunt rationem compositam ex ponderibus 'ed', 't' et ex distantiis 'ab', 'bs'.</s></p>
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68 <p><s>Sit modo</s></p>
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69 <pb xlink:href="X01/01/020.jpg" page="040CV"/>
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70 <pb xlink:href="X01/01/021.jpg" page="042_R"/>
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71 <p><s>Amplitudo tota 465.</s></p>
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72 <p><s>Amplitudo tota 465.</s></p>
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73 <pb xlink:href="X01/01/022.jpg" page="042_V"/>
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74 <pb xlink:href="X01/01/023.jpg" page="043_R"/>
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75 <p><s><foreign lang="it">Passi la catenella per i punti 'fc', e dato lo scopo 'z', tira tanto la catena, che passi per 'z', e troverai la distanza 'sc' e l'angolo della elevazione etc.</foreign></s></p>
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76 <p><s><foreign lang="it">Dimostrasi che si come è impossibile tirar la catena in retto, così essere impossibile che [']l proietto vadia mai per dritto se non nella perpendicolare in su, come anco la catena a piombo si stende in retto.</foreign></s></p>
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77 <p><s><foreign lang="it">Si come la parabola del proietto è descritta da 2 moti, orizontale e perpendicolare, così la catenella risulta da 2 forze: oriz[onta]la, da chi la tira nell'estremità, e perpendicolare deorsum, da proprio peso.</foreign></s></p>
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78 <p><s><foreign lang="it">il grave in 'g' preme con manco forza che in 's', secondo la proporzione del <SYMBOL-RECTANGULUM/>_[rettangolo] 'fgc' al <SYMBOL-RECTANGULUM/>_[rettangolo] 'fsc' <DELETION-1> <ILLEGIBLE/> </DELETION-1>.</foreign></s></p>
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79 <pb xlink:href="X01/01/024.jpg" page="043_V"/>
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80 <pb xlink:href="X01/01/025.jpg" page="044AR"/>
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81 <p><s><foreign lang="it">Se il grave cadente arrivasse nell'istesso tempo al centro della terra, tanto cadendo da luogo lontanissimo da esso centro, quanto da uno vicinissimo, adunque quel grado di velocità che haverà acquistato in un tal tempo, nel passare una tal parte della minor lontananza, quel grado medesimo, dico, nell'istesso tempo acquisterà in maggior parte della maggior lontananza. Sia dunque la parte della minor lontananza passata nel tempo 'ab'; la linea 'ab' et il grado di velocità in 'b' sia l'istesso che quello in 'd' della parte maggiore 'cd', passata dal medesimo grave nel medesimo tempo 'ab'.</foreign></s></p>
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82 <pb xlink:href="X01/01/026.jpg" page="044AV"/>
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83 <pb xlink:href="X01/01/027.jpg" page="044BR"/>
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84 <p><s><foreign lang="it">Se il grave cadente arrivasse nel istesso tempo al centro della terra, tanto cadendo da luogo lontanissimo da esso centro quanto da uno vicinissimo, adunque quel grado di velocità che haverà aquistato <INSERTION-1> in un tal tempo, </INSERTION-1> nel passare una tal parte della minor lontananza, quel grado medesimo, dico, <SUBSTITUTION-1> <ILLEGIBLE/> <SUBSTITUTED-BY-1> nel istesso tempo </SUBSTITUTED-BY-1></SUBSTITUTION-1> acquisterà in maggior parte della maggior lontananza. Sia dunque la parte della minor lontananza passata nel tempo 'ab'; la linea 'ab' et il grado di velocità in 'b' sia l'istesso che quello in 'd' della parte maggiore 'cd', passata dal medesimo grave nel medesimo tempo 'ab'.</foreign></s></p>
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85 <pb xlink:href="X01/01/028.jpg" page="044BV"/>
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86 <pb xlink:href="X01/01/029.jpg" page="045_R"/>
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87 <p><s>Momenta gravitatis eiusdem mobilis super planum inclinatum et in perpendiculo permutatim respondent longitudini et <SUBSTITUTION-1> elevationis <SUBSTITUTED-BY-1> elevationi </SUBSTITUTED-BY-1></SUBSTITUTION-1> eiusdem plani.</s></p>
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88 <p><s>Sit ad orizontem 'ab' planum inclinatum 'ca', in quo sumatur quodcunque punctum 'c', et demissa perpendicularis ad orizontem 'cb' sit plani 'ca' altitudo seu elevatio: dico, momentum gravitatis mobilis 'd' super plano 'ca' ad totale suum momentum in perpendiculo 'cb' esse ut altitudo 'cb' ad eiusdem plani longitudinem 'ca'; id autem in mechanicis probatum est.</s></p>
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89 <pb xlink:href="X01/01/030.jpg" page="045_V"/>
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90 <pb xlink:href="X01/01/031.jpg" page="046_R"/>
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91 <p><s>Momenta gravitatis eiusdem mobilis super diversas planorum inclinationes habent inter se permutatim eandem rationem, quam eorum planorum longitudines, dum eidem elevationi respondeant.</s></p>
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92 <p><s>Sint diversae planorum inclinationes 'ab', 'ac', quae eidem elevationi 'ad' respondeant: dico, momentum gravitatis eiusdem mobilis super 'ab' ad momentum gravitatis super 'ac' eandem habere rationem, quam longitudo 'ac' habet ad longitudinem 'ab'. Ex praecedenti, enim, momentum gravitatis super 'ab' ad totale momentum in perpendiculo <SUBSTITUTION-1> 'ab' <SUBSTITUTED-BY-1> 'ad' </SUBSTITUTED-BY-1></SUBSTITUTION-1> est ut 'ad' ad 'ab': totale vero momentum per 'ad' ad momentum per 'ac' est ut 'ca' ad 'ad': ergo, ex aequali in <DELETION-1> <ILLEGIBLE/> </DELETION-1> analogia perturbata, momentum per 'ab' ad momentum per 'ac' erit ut longitudo 'ca' ad longitudinem 'ab'. Quod erat demonstrandum.</s></p>
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93 <pb xlink:href="X01/01/032.jpg" page="046_V"/>
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94 <pb xlink:href="X01/01/033.jpg" page="047_R"/>
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95 <p><s>Aliter ostendemus, mobile, temporibus aequalibus, pertransire 'ca', 'da'.</s></p>
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96 <p><s>Sit enim 'ba' aequalis ipsi 'da', et ducantur perpendiculares 'be', 'df': constat ex elementis mechanicis, momentum ponderis super plano secundum lineam 'abc' elevato ad momentum suum totale esse ut 'be' ad 'ba', eiusdem vero ponderis momentum super elevatione 'ad' <DELETION-1> <ILLEGIBLE/> </DELETION-1> ad totale suum momentum, eamdem ob causam, esse ut 'df' ad 'da' vel 'ba'; ergo eiusdem ponderis momentum super plano secundum 'da' inclinato ad momentum super inclinatione secundum 'abc' est ut linea 'df' ad lineam 'be'; quare spatia, quae pertransibit idem pondus temporibus aequalibus super inclinationibus 'ca', 'da', erunt inter se ut lineae 'be', 'df'. At ut 'be' ad 'df', ita demonstratur se habere 'ac' ad 'da'; ergo idem mobile temporibus aequalibus pertransibit lineas 'ca', 'da'. <DELETION-1> esse </DELETION-1> </s></p>
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97 <p><s>Esse autem ut 'be' ad 'df', ita 'ca' ad 'da', ita probatur.</s></p>
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98 <p><s>Iungatur 'cd', et per 'd' et 'b', ipsi 'af' parallelae, agantur <DELETION-1> 'dl' </DELETION-1> 'dgl', secans 'ca' in 'i', et 'bh': eritque angulus 'adi' aequalis angulo 'dca', cum circumferentiis 'la', 'ad' aequalibus insistant, estque angulus 'dac' communis. Ergo triangulorum aequiangulorum 'cad', 'dai' latera circa aequales angulos proportionalia erunt, et ut 'ca' ad 'ad', ita erit 'da' ad 'ai', idest <DELETION-1> <ILLEGIBLE/> </DELETION-1> 'ba' ad 'ai', seu 'ha' ad 'ag', hoc est 'be' ad 'df': quod erat probandum.</s></p>
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99 <p><s>Collige, existentibus planis inaequaliter inclinatis 'ad', 'ac', atque data longitudine 'ad', inveniri posse in plano 'ac' <DELETION-1> pro </DELETION-1> portionem, quae eodem tempore cum 'da' peragatur. Ducto enim perpendiculo 'df', et posita 'ab' aequali 'ad', ductoque perpendiculo 'be', fiat ut 'df' ad 'eb', ita <SUBSTITUTION-1> 'dc' <SUBSTITUTED-BY-1> 'da' </SUBSTITUTED-BY-1></SUBSTITUTION-1> ad 'ac'; eritque tempus per 'ca' aequale tempori per 'da'.</s></p>
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101 <pb xlink:href="X01/01/035.jpg" page="048_R"/>
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102 <p><s>Si in circulo ad horizontem erecto a puncto sublimi quotcumque ducantur linea rectae usque ad circumferentiam, per quas cadant gravia quotcumque, omnia temporibus aequalibus ad terminos suos pervenient.</s></p>
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103 <p><s>Sit enim circumferentia ad horizontem erecta 'abc', punctum sublime 'a', a quo lineae quotcumque ad circumferentiam usque protrahantur 'ac', 'ab', et per ipsas cadant mobilia: dico, temporibus aequalibus <SUBSTITUTION-1> ea <SUBSTITUTED-BY-1> illa </SUBSTITUTED-BY-1></SUBSTITUTION-1> perventura esse ad terminos 'c', 'b'. Sit enim 'ac' per centrum ducta, cui ex 'b' perpendicularis sit 'bd': patet 'ab' mediam esse proportionalem inter 'ca', 'ad'; quare, ex demonstratis, tempus quo mobile ex 'a' cadit in 'c', ad tempus casus ex 'a' in 'd' <DELETION-1> esse </DELETION-1> est ut linea 'ba' ad lineam 'ad'. Verum, similiter, ex demonstratis, tempus casus ex 'a' in 'b' ad tempus casus ex 'a' in 'd' est ut 'ba' ad 'ad': ergo tempora casuum 'ab', 'ac' erunt aequalia, cum eandem ad idem tempus casus 'ad' habeant rationem. Et similiter de reliquis omnibus motibus demonstrabitur: ergo patet propositum.</s></p>
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104 <p><s>Ex his colligitur, gravia eodem tempore pertransire plana inaequalia, et inaequaliter inclinata, dum, quam proportionem habet longitudo maioris plani ad longitudinem alterius, eamdem duplicatam habeat perpendiculariis maioris plani ad perpendicularem minoris. Cum enim <SYMBOL-QUADRATUM/>_[quadratum] 'ae' sit aequale <SYMBOL-RECTANGULUM/>_[rectangulo] 'caf', <SYMBOL-QUADRATUM/>_[quadratum] vero 'ba' <SYMBOL-RECTANGULUM/>_[rectangulo] 'cad'; <SYMBOL-RECTANGULUM/>_[rectangulum] vero 'caf' ad <SYMBOL-RECTANGULUM/>_[rectangulum] 'cad' est ut 'fa' ad 'ad'; ergo 'fa' ad 'ad' est ut <SYMBOL-QUADRATUM/>_[quadratum] 'ea' ad <SYMBOL-QUADRATUM/>_[quadratum] 'ba': ratio igitur perpendicularis 'fa' ad perpendicularem 'da' dupla est rationis 'ea' ad 'ab', etc.</s></p>
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105 <pb xlink:href="X01/01/036.jpg" page="048_V"/>
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106 <pb xlink:href="X01/01/037.jpg" page="049_R"/>
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107 <p><s>Fiat latio per plana inflexa 'ab', 'bc', et invenienda sit ratio temporis casus per 'ab' ad tempus casus per 'bc' post casum 'ab'.</s></p>
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108 <p><s>Ducatur horizon 'ae', cui <SUBSTITUTION-1> 'cp' <SUBSTITUTED-BY-1> 'cb' </SUBSTITUTED-BY-1></SUBSTITUTION-1> producta occurrat in 'e', et ipsarum 'ce', 'eb' media sit 'ed'. Dico, tempus per 'ab' ad tempus per 'bc' esse ut 'ab' ad 'bd'. Tempus enim per 'ab' ad tempus per 'eb' est ut 'ab' ad 'eb': tempus vero per 'eb' ad tempus per 'bc' est ut 'eb' ad 'bd': ergo tempus per 'ab' ad tempus per 'bc' est ut 'ab' ad 'bd': quod etc.</s></p>
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109 <p><s>Hic pr[a]emittenda videtur sequens propositio.</s></p>
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110 <p><s>Si linea, in qua fiat latio ex quiete, dividatur utcumque, tempus lationis prioris partis ad tempus lationis <SYMBOL-2/>_[secund][a]e partis est ut ipsamet prima pars ad excessum quo eadem pars superatur a media inter totam et <DELETION-1> et </DELETION-1> ipsam primam partem.</s></p>
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111 <p><s>Fiat latio per totam 'ab' ex quiete in 'a', quae utcumque divisa sit in 'c'; totius autem 'ba' et partis 'ac' media sit 'af', 'cf' vero excessus eiusdem mediae super primam 'ac'. Dico, te[m]pus lationis per 'ac' ad tempus per reliquam 'cb' esse ut 'ac' ad 'cf'. Quod patet: nam tempus per 'ac' ad tempus per totam 'ab' est ut 'ac' ad mediam 'af'; ergo, dividendo, tempus per 'ac' ad tempus per reliquam 'cb' erit ut 'ac' ad 'cf'. Si itaque intelligatur, tempus per 'ac' esse 'ac', tempus per 'cb' erit 'cf'.</s></p>
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112 <pb xlink:href="X01/01/038.jpg" page="049_V"/>
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113 <pb xlink:href="X01/01/039.jpg" page="050_R"/>
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114 <p><s>postea quam ostensum fuerit, tempora per 'ab', 'ac' esse aequalia, demonstrabitur tempus per 'ad' ad tempus per 'ae' esse ut 'da' ad mediam inter 'da', 'ae'. Nam tempus per 'da' ad tempus per 'ac' est ut 'da' ad 'ac' lineam: tempus autem per 'ac', id est per 'ab', ad tempus 'ae' est ut linea 'ba' ad 'ae', hoc est ut 'sa' ad 'ad': ergo, ex aequali in analogia perturbata, tempus per 'ad' ad tempus per 'ae' est ut linea 'sa' ad lineam 'ac'. Cumque 'ac', ex demonstratis, sit media inter 'sa', 'ab', et ut 'sa' ad 'ab' ita 'da' ad 'ae', ergo tempus per 'ad' ad tempus per 'ae' est ut 'da' ad mediam inter 'da', 'ae': quod erat probandum.</s></p>
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115 <pb xlink:href="X01/01/040.jpg" page="050_V"/>
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116 <pb xlink:href="X01/01/041.jpg" page="051_R"/>
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117 <p><s>Data inflexa ad datum perpendiculum, partem in inflexa accipere, in qua sola fiat motus eodem tempore atque in eadem cum perpendiculo.</s></p>
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118 <p><s>Sit perpendiculus 'ab', et ad ipsum inflexa 'bc'. Oportet in 'bc' partem accipere, in qua sola fiat motus eodem tempore ac in eadem cum perpendiculo 'ab'.</s></p>
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119 <p><s>Ducatur horizon 'ad', cui inclinata 'cb' <INSERTION-1> extensa </INSERTION-1> occurrat in 'e', ponaturque 'bf' aequalis 'ba', et, centro 'e', intervallo 'ef', circulus describatur 'fig', et 'fe' ad circumferentiam usque protahatur in 'g', et ut 'gb' ad 'bf', ita fiat 'bh' ad 'hf', et 'hi' circulum tangat in 'i'. Deinde, ex 'b' perpendicularis ad 'fc' erigatur 'bk', cui occurrat in 'l' linea 'eil'. Tandem ipsi 'el' perpendicularis ducatur 'lm', occurrens 'bc' in 'm'. Dico, <UNDERLINED> in linea 'bm' ex 'b' fieri motum eodem tempore ac ex 'a' per ambas 'ab', <DELETION-1> <ILLEGIBLE/> </DELETION-1> 'bm'. </UNDERLINED> </s></p>
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120 <p><s>Ponatur 'en' aequalis 'el'. Cumque ut <DELETION-1> b </DELETION-1> 'gb' ad 'bf', ita sit 'bh' ad 'hf', erit, permutando, ut 'gb' ad 'bh', ita 'bf' ad 'fh', et, dividendo, <DELETION-1> <ILLEGIBLE/> </DELETION-1> 'gh' ad 'hb', <DELETION-1> ita </DELETION-1> ut 'bh' ad 'hf'; quare rectangulum 'ghf' quadrato 'hb' erit aequale: sed idem rectangulum aequatur quoque quadrato 'hi': ergo 'bh' ipsi 'hi' est aequalis. Cumque in quadrilatero 'ilbh' latera 'hb', 'hi' sint aequalia, et anguli 'b', 'i' recti, erit latus quoque 'bl' ipsi 'li' aequale: est autem 'ei' aequalis 'ef': ergo tota 'le', seu 'ne', duabus 'lb', 'ef' est aequalis. Auferatur communis 'ef': ergo reliqua 'fn' ipsi 'lb' erit aequalis. At posita est 'fb' ipsi 'ba' aequalis: ergo 'lb' duabus 'ab', 'bn' aequatur. Rursus, si intelligatur tempus per 'ab' esse ipsam 'ab', erit tempus per 'eb' ipsi 'eb' aequale. Tempus autem per totam 'em' erit 'en', media scilicet inter 'me', 'eb', quare <DELETION-1> b </DELETION-1> reliquae 'bm' tempus <DELETION-1> ca <ILLEGIBLE/> </DELETION-1> casus post 'eb', seu post 'ab', erit ipsa 'bn'. Positum autem est, tempus per 'ab' esse 'ab': ergo tempus casus per ambas 'abm' est 'abn'. Cum autem tempus per 'eb' ex quiete in 'e' sit 'eb', tempus per 'bm' ex quiete in 'b' erit media proportionalis inter 'be', 'bm'; haec autem est 'bl'; tempus igitur per ambas 'abm' ex quiete in 'a' est 'abn': tempus vero per 'bm' solam ex quiete in 'b' est 'bl'; ostensum autem est 'bl' esse aequalem duabus 'ab', <DELETION-1> 'bm' </DELETION-1> 'bn'; ergo patet propositum.</s></p>
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121 <pb xlink:href="X01/01/042.jpg" page="051_V"/>
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122 <pb xlink:href="X01/01/043.jpg" page="052_R"/>
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123 <p><s>Sit ad horizontem 'ab' linea 'cd' utcumque inclinata, et in ipso horizonte quodlibet punctum notatum 'a'. Oportet in linea 'cd' punctum invenire, a quo in linea recta, usque ad 'a' protracta brevissimo tempore fiat motus.</s></p>
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124 <p><s>Erigatur ex 'a' perpendicularis 'ac' ad horizontem, et ex eodem demittatur perpendicularis ad 'cd', quae sit 'ae', et angulus 'cae' bifariam secetur per 'fa'. Dico, ex omnibus lineis quae a puncto 'a' ad lineam 'cd' protrahantur, 'fa' esse illam <DELETION-1> quae </DELETION-1> per quam motus brevissimo tempore absolvitur.</s></p>
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125 <p><s>Ducatur enim 'fg' ipsi 'ea' parallela; erit angulus 'gfa' angulo alterno 'fae' aequalis: sed angulus 'fae' ipsi 'fag' aequatur (cum totus 'cae' sit bifariam sectus): ergo 'gaf', 'gfa' aequales erunt, quare et latera 'gf', 'ga'. Si itaque, centro 'g' intervallo 'gf', circulus describatur, tanget ambas lineas 'cd', 'ab' in punctis 'f', 'a', eritque casus per 'fa' brevioris temporis, quam per rectas quascumque alias ex 'a' ad quaecumque puncta lineae 'cd' productas.</s></p>
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126 <p><s>Si linea recta supra orizontalem fuerit utcumque inclinata, planum <DELETION-1> per quod </DELETION-1> a dato puncto in orizontali usque ad inclinatam extensum, in quo descensus fit tempore omnium brevissimo, est illud quod bifariam dividit angulum contenctum a duabus perpendicularibus a dato puncto extensis, una ad orizontalem lineam, altera ad inclinatam.</s></p>
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127 <pb xlink:href="X01/01/044.jpg" page="052_V"/>
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128 <p><s>siquidem <lb/> siquidem</s></p>
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129 <pb xlink:href="X01/01/045.jpg" page="053_R"/>
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130 <p><s>Sit linea horizontis 'ac', perpendiculum vero 'bd', et in 'ac' accipiatur quodcumque punctum 'c': dico, quod si mobile debet ex 'c' ad lineam perpendiculi naturaliter per unicam lineam rectam moveri, ad eam perveniet tempore brevissimo si veniat per 'ce', quae lineam 'be', ipsi 'bc' aequalem, assumit.</s></p>
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131 <p><s>Centro enim 'b', intervallo 'be', circulus describatur, ductisque 'cf' et 'cg' utcumque, patebit motum per 'ce' citius absolvi quam per 'cf' <DELETION-1> <ILLEGIBLE/> </DELETION-1> aut 'cg'. Si enim ducatur tangens circulum 'ick', et ipsi 'cf' parallela <DELETION-1> duca </DELETION-1> 'elk', erit 'le' brevior quam 'cf': sed tempus per 'ce' aequatur tempori per 'le'. Similiter, ducta 'ehi' ipsi 'cg' parallela et aequali, constat 'cg' longiorem esse 'he': at tempus per 'ce' aequatur tempori per 'he'. Ergo patet propositum.</s></p>
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132 <p><s>Si ex aliquo puncto in orizontali sumpto descendat perpendiculum, ex alio vero puncto eiusdem orizontis ducendum sit usque ad perpendiculum planum per quod brevissimo tempore mobile descendat, tale planum erit illud quod de perpendiculo abscindit partem [a]equalem distantiae pu[n]cti accepti in orizonte a primo puncto perpendiculi.</s></p>
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133 <pb xlink:href="X01/01/046.jpg" page="053_V"/>
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134 <pb xlink:href="X01/01/047.jpg" page="054_R"/>
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135 <p><s><SYMBOL-RECTANGULUM/>_[rectangul]a [a]equalia sunt, unum quod continetur ab 'fb' et ab excessu 'ac' super 'ah', qui sit 'lc'; alterum est quod continetur ab 'ah', vel 'ai', vel 'al', et ab excessu 'fb' super 'bi', vel super 'cg': ergo ut 'al' ad 'lc', ita 'bf' ad 'fs', <UNDERLINED> et, per conversionem rationis, ut 'fb' ad 'bs', idest ut 'fb' ad 'cg', idest ut 'ab' ad 'ac', </UNDERLINED> nempe media 'bf' ad excessum sui super mediam 'cg'. Sed 'fb' ad 'cg' est ut 'ab' ad 'ac': posita igitur 'an' aequali 'ac', erit ut <DELETION-1> 'an' </DELETION-1> 'ab' ad 'bn', ita 'al' ad 'lc'.</s></p>
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136 <p><s>Totum opus tale videtur esse. Secetur 'an' aequalis 'ac', et ut 'ab' ad 'bn', ita fiat 'al' ad <DELETION-1> 'lg' </DELETION-1> 'lc', et ponatur 'ai' aequalis 'al', et ut 'ac' ad 'ib', ita fiat 'ib' ad 'ce': erit 'ce' linea qu[a]esita, nempe pars superior perpendiculi, ex qua mobile conficiet ipsam cum 'ab' tempore eodem ac solam 'ab'.</s></p>
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137 <pb xlink:href="X01/01/048.jpg" page="054_V"/>
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138 <p><s><SYMBOL-QUADRATUM/>_[quadratum] 'bg' aequatur <SYMBOL-QUADRATUM/>_[quadrat]is 'bf', 'fg' et 2 <SYMBOL-RECTANGULUM/>_[rectanguli] 'bfg': pro <SYMBOL-QUADRATUM/>_[quadrat]o 'bf' sume <SYMBOL-RECTANGULUM/>_[rectangulum] 'hfg'; erit <SYMBOL-QUADRATUM/>_[quadratum] 'bg' aequale <SYMBOL-2/>_[duo]bus <SYMBOL-RECTANGULUM/>_[rectangulis] <DELETION-1> <ILLEGIBLE/> </DELETION-1> 'bfg', <SYMBOL-QUADRATUM/>_[quadrat]o 'fg' et <SYMBOL-RECTANGULUM/>_[rectangul]o 'hfg', idest <SYMBOL-RECTANGULUM/>_[rectangulum] 'hgf' cum <SYMBOL-QUADRATUM/>_[quadrat]o 'fg': id autem idem est ac si dicas <SYMBOL-QUADRATUM/>_[quadratum] 'bg' esse aequale 2 <SYMBOL-RECTANGULUM/>_[rectangula] 'bfg', 2 <SYMBOL-RECTANGULUM/>_[rectangula] 'egf' et 2 <SYMBOL-QUADRATUM/>_[quadrata] 'fg'. Ex <SYMBOL-QUADRATUM/>_[quadrat]o 'bg' deme 2 <SYMBOL-RECTANGULUM/>_[rectangula] 'bfg' et 1 <SYMBOL-QUADRATUM/>_[quadratum] 'gf'; remanet <SYMBOL-QUADRATUM/>_[quadratum] 'bf' aequale 2 <SYMBOL-RECTANGULUM/>_[rectangulis] 'egf' minus 1 <SYMBOL-QUADRATUM/>_[quadratum] 'gf', quod est <SYMBOL-RECTANGULUM/>_[rectangulum] 'hfg' aequale <SYMBOL-QUADRATUM/>_[quadrat]o 'bf'.</s></p>
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139 <p><s>'ab' 36, 'bc' 24, 'bd' 16.</s></p>
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140 <p><s>'ad' 20, 'ds' 18.</s></p>
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141 <p><s>'ad' 20</s></p>
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142 <p><s>'ab' ad 'bc' ut 'cb' ad 'bd';</s></p>
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143 <p><s>'ad' ad 'ds' ut 'sd' ad 'db'.</s></p>
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144 <pb xlink:href="X01/01/049.jpg" page="055_R"/>
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145 <p><s>Quaeritur in 'ac' pars aequalis 'ab', quae conficiatur tempore aequali tempori per 'ab'. Ponatur 'ad' aequalis 'ab', et circa 'ac' semicirculus describatur, et ponatur 'af' aequalis dimidiae 'dc', et ab 'f' demittatur perpendicularis 'fe', et 'eg' ponatur aequalis 'ab': dico, 'eg' ex quiete in 'a' confici eodem tempore ac 'ab', med[i]a proportionalis inter 'ca', 'ag'.</s></p>
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146 <pb xlink:href="X01/01/050.jpg" page="055_V"/>
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147 <p><s>Productis lateribus 'ab', 'ac' versus 'd', 'e', et erectis perpendicularibus 'cg', 'bf', ponatur 'an' aequalis 'ac', et ut 'ab' ad 'bn', ita fiat 'al' ad 'lc', et ipsi 'al' secetur aequalis 'ai', ipsarumque 'ac', 'ib' tertia proportionalis sit <DELETION-1> 'cg' </DELETION-1> 'ce'; et diametro 'ae' semicirculus ducatur, secans 'cg' in 'g', ductaque per 'e' parallela 'ed', occurrenti 'ab' protractae in 'd', alter semicirculus describatur, secans perpendicularem 'bf' in 'f', et iungatur 'fa'. Constat iam, ut 'ab' ad 'bd', ita esse 'ac' ad 'ce', et mediam 'bf' ad mediam 'cg' <DELETION-1> hanc autem aequalem esse 'bi' </DELETION-1> ut 'ab' ad 'ac', et insuper 'ib' esse aequalem 'cg'. Cumque 'fb' maior sit 'cg', ponatur 'bs' ipsi 'cg' aequalis. E[t] quia ut 'ba' ad 'ac', seu 'an', ita 'fb' ad 'cg', seu 'bs', erit ut 'ab' ad 'bn', hoc est 'al' ad 'lc', ita 'bf' ad 'fs', et <SYMBOL-RECTANGULUM/>_[rectangulum] sub 'fb', 'lc' erit aequale <SYMBOL-RECTANGULUM/>_[rectangul]o sub 'al', 'fs', seu sub 'ai', 'fs'.</s></p>
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148 <p><s>Quaeritur versus 'c' pars quae conficiatur eodem tempore ac 'ad'.</s></p>
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149 <p><s>Sit tempus per 'ac', 'ac'; tempus per 'ad' erit 'ae'; ponatur 'cf' aequalis 'ae', et ipsarum 'ca', 'af' <SYMBOL-3/>_[terti]a proportionalis sit 'ag'. Dico 'gc' esse quod quaeritur. Cum enim tempus per totam 'ac' sit 'ac', tempus per 'ag' erit 'af', media inter 'ca', 'ag', et reliqua 'fc' erit tempus per 'gc': est autem 'fc' posita aequalis 'ae': ergo patet.</s></p>
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150 <p><s>In qualibet latione spacium quod conficitur versus finem eodem <DELETION-1> p </DELETION-1> tempore ac spatium versus principium, est medium proportionale inter totum lationis spatium et ipsum spatium versus principium.</s></p>
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151 <pb xlink:href="X01/01/051.jpg" page="056_R"/>
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152 <p><s>Fiat motus per 'abc' et per duas 'ab', 'bd', sitque 'ra' media inter 'ca', 'ab', et ipsi 'dc' parallela ducatur 'rs'. Dico iam, tempus per 'abc' ad tempus per 'abd' esse ut linea 'ac' ad lineam 'ar' cum 'sd'. Si enim protrahatur 'bd' usque ad occursum cum 'af', horizonti 'dc' parallela, erit 'fs' media inter 'df', 'fb'; et ut 'ca' ad 'ar', ita tempus per 'ca' ad tempus per 'ab'; ita ut, si ponatur 'ac' tempus per 'ac', erit 'ar' tempus per 'ab', et 'rc' tempus per 'bc'; et similiter 'sd' demonstrabitur esse tempus per 'bd' post casum ex 'f', vel ex 'a': ex quo patet tempus per totam 'ac' ad tempus per duas 'abd' esse, ut <DELETION-1> a <ILLEGIBLE/> </DELETION-1> 'ar' cum 'rc' ad 'ar' cum 'sd'.</s></p>
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153 <pb xlink:href="X01/01/052.jpg" page="056_V"/>
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154 <pb xlink:href="X01/01/053.jpg" page="057_R"/>
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155 <p><s><foreign lang="it">Da notarsi</foreign></s></p>
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156 <p><s>Sit ad horizontem 'ab' perpendicularis 'bc' et inclinata 'bd', in qua sumatur 'be', et ex 'e' ad 'bd' perpendicularis agatur 'ef', ipsi 'bc' occurrens in 'f': demonstrandum est tempus per 'be' aequari tempori per 'bf'. Ducatur ex 'e' perpendicularis ad 'ab', et sit 'eg': et quia impetus per 'bd' ad impetum per 'bc' est ut 'eg' ad 'be' <DELETION-1> <ILLEGIBLE/> </DELETION-1> (ut infra demonstratur); ut autem 'eg' ad 'be', ita 'be' ad 'bf', ob similitudinem triangulorum 'geb', 'bef'; ergo ut 'bf' spacium ad spacium 'be', ita impetus per 'bf' ad impetum per 'be': ergo eodem tempore fiet motus per 'bf' et per 'be'.</s></p>
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157 <p><s>haec <INSERTION-1> prima </INSERTION-1> propositio est iam demonstrata, et ideo, ut dupla, demictatur.</s></p>
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158 <p><s><foreign lang="it">Da notarsi</foreign></s></p>
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159 <p><s>Infra horizontem 'ab' ex eodem puncto 'c' duo rectae aequales utcumque inclinentur 'cd', 'ce', et ex terminis 'd', 'e' ad horizontem perpendiculares agantur 'da', 'eb', et lineae 'cd' constituatur angulus 'cdf', angulo 'bce' aequalis: dico, ut 'da' ad 'be' ita esse <SUBSTITUTION-1> <ILLEGIBLE/> <SUBSTITUTED-BY-1> 'dc' </SUBSTITUTED-BY-1></SUBSTITUTION-1> ad 'cf'. Ducatur perpendicularis 'cg': et quia angulus 'cdf' aequatur angulo 'bce', et rectus 'g' recto 'b', erit ut 'dc' ad 'cg', ita 'ce' ad 'eb': est autem 'cd' ipsi 'ce' aequalis: ergo 'cg' aequatur 'be'. Et cum angulus 'cdf' angulo 'bce' sit aequalis, et angulus 'fcd' co[m]munis, reliquus ad duos rectos 'dfc' reliquo 'dca' aequabitur, et anguli ad 'a' et 'g' sunt recti; ergo triangulum 'adc' <SYMBOL-3/>_[tri]ang[ul]o 'cgf' est simile: quare ut 'ad' ad 'dc', ita 'cg' ad 'cf', et, permutando, ut 'ad' ad 'cg', hoc est ad 'be', ita 'dc' ad 'cf': <INSERTION-1> quod </INSERTION-1> est probandum. Cum autem impetus per 'cd' ad impetum per 'cf' sit ut perpendiculus 'ad' ad perpendiculum 'be', constat, motus per 'cd' et 'cf' eodem tempore absolvi. Itaque distantiae quae in diversis inclinationibus eodem tempore conficiuntur, determinantur per lineam quae (ut facit 'df') lineis inclinatis occurrit secundum angulos aequales illis quos inclinatae ad horizontem constituunt, permutatim sumptos.</s></p>
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160 <pb xlink:href="X01/01/054.jpg" page="057_V"/>
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161 <p><s>Advertas cur cadentia ex 'a' sint semper una in locis sibi respondentibus, ut 'o', 's', ita ut angulus 'aos' sit aequalis angulo <DELETION-1> 'aos' </DELETION-1> 'bas'.</s></p>
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162 <pb xlink:href="X01/01/055.jpg" page="058_R"/>
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163 <p><s>Haec prima propositio non est motus materia, et haec pagina est hic posita propter secundam.</s></p>
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164 <p><s>Sit 'ic' perpendicularis ad diametrum circuli 'ab'; ductaque a puncto 'a' quaecumque linea circunferentiae et perpendiculari 'ci' occurrens, ut 'aid'. Dico, rectangulum 'dai' rectangulo 'bac' esse aequale.</s></p>
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165 <p><s>Si enim iungatur recta 'db', erit angulus in semicirculo ad punctum 'd' rectus; estque angulus 'c' quoque rectus, communis autem angulus ad 'a': ergo <SYMBOL-3/>_[tri]angulorum aequiangulorum 'dab', 'cai' latera erunt proportionalia, utque 'ba' ad 'ad', ita 'ia' ad 'ac'. Ergo patet propositum.</s></p>
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166 <p><s>Sint plana quaecumque inclinata 'ab', 'ac', et perpendiculus 'ae', cui ad rectos angulos 'bg', et sit inter 'ca', 'ad' media 'af'.</s></p>
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167 <p><s>Dico, tempus per 'ab' ad tempus per 'ac' esse ut 'ba' ad 'af'.</s></p>
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168 <p><s> Nam tempus per 'ab' ad tempus per 'ad' est ut 'ab' ad 'ad': tempus vero per 'ad' ad tempus per 'ac' est ut 'ad' ad 'af': ergo, ex aequali, tempus per 'ab' ad tempus per 'ac' est ut 'ab' ad 'af': quod erat ostendendum.</s></p>
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169 <pb xlink:href="X01/01/056.jpg" page="058_V"/>
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170 <pb xlink:href="X01/01/057.jpg" page="059_R"/>
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171 <p><s>Sit circuli circunferentia 'cbd', et diameter 'mc' ad horizontem erecta, et ducatur 'dc', non maior subtendente quadrantem, et a terminis 'd', 'c' aliae duae ad quodcumque punctum 'b'. Dico, mobile ex termino 'd' ferri per duas 'db', 'bc' lineas tempore breviori quam per 'dc' ex eodem termino 'd', vel per solam 'bc' ex termino 'b'.</s></p>
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172 <p><s>Ducta sit per 'd', ipsi 'cm' perpendicularis, 'mda', cui 'cb' extensa occurat in 'a'; sitque 'dn' ipsi 'mc' parallela, et 'bn' ad 'bd' perpendicularis, et circa triangulum rectangulum 'dbn' semicirculus describatur 'dfbn' <SUBSTITUTION-1> separans <SUBSTITUTED-BY-1> secans </SUBSTITUTED-BY-1></SUBSTITUTION-1> 'dc' in 'f'; et ipsarum 'cd', 'df' media sit proportionalis 'do', ipsarum autem 'ca', 'ab' sit 'av'. Sit autem 'ps' tempus quo peragitur tota 'dc', vel 'cb' (constat enim, eodem tempore peragi utramque 'dc', 'bc'), et quam rationem habet 'cd' ad 'do', hanc habeat tempus 'sp' ad tempus 'pr': erit tempus 'pr' id in quo mobile ex 'd' peragit 'df'; 'rs' vero id, in quo reliquum 'fc'.</s></p>
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173 <p><s>Cum vero 'ps' sit quoque tempus quo mobile ex 'b' peragit 'bc', si fiat ut 'bc' ad 'cd', ita 'sp' ad 'pt', erit 'pt' tempus casus ex 'a' in 'c', cum 'dc' media sit inter 'ac', 'cb', ex ante demonstratis. Fiat tandem ut 'ca' ad 'av', ita 'tp' ad 'pg': erit 'pg' tempus quo mobile ex 'a' venit in <DELETION-1> 'bg' </DELETION-1> 'b', 'gt' vero tempus residuum motus 'bc' consequentis post motum ex 'a' in 'b'. Cum vero 'dn', circuli 'dfbn' diameter, ad horizontem sit erecta, temporibus aequalibus peragentur 'df' et 'db' lineae: quare si demonstratum fuerit, mobile citius conficere 'bc' post casum 'db', quam 'fc' post peractam 'df', habebimus intentum. At eadem temporis celeritate conficiet mobile 'bc' veniens ex <DELETION-1> 'bd' </DELETION-1> 'db', ac si veniret ex 'ab', cum ex utroque casu 'db', 'ab' aequalia accipiat velocitatis momenta; ergo demonstrandum erit, breviori tempore peragi 'bc' post 'ab', quam 'fc' post 'df'. Explicatum est autem, tempus quo peragitur 'bc' post 'ab' esse 'gt'; tempus vero ipsius 'fc' post 'df' esse 'rs': ostendendum <DELETION-1> ergo </DELETION-1> </s></p>
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174 <pb xlink:href="X01/01/058.jpg" page="059_V"/>
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175 <p><s>itaque est, 'rs' maius esse quam 'gt'. Quod sic demonstratur: quia enim ut 'sp' ad 'pr', ita 'cd' ad 'do', per conversionem rationis et convertendo, ut 'rs' ad 'sp', in[ita] 'oc' ad 'cd', ut autem 'sp' ad 'pt', ita 'cd' ad 'ca'; et quia est ut 'tp' ad 'pg', ita 'ca' ad 'av', per conversionem rationis erit quoque ut 'pt' ad 'tg', ita 'ac' ad 'cv'; ergo, ex aequali, ut <SUBSTITUTION-1> <ILLEGIBLE/> <SUBSTITUTED-BY-1> 'rs' </SUBSTITUTED-BY-1></SUBSTITUTION-1> ad 'gt', ita 'oc' ad 'cv': est autem 'oc' maior quam 'cv', ut mox demonstrabitur: ergo tempus 'rs' maius est tempore 'gt': quod demonstrare oportebat. Cum vero 'cf' maior sit quam 'cb', 'fd' vero minor 'ba', habebit 'cd' ad 'df' maiorem rationem quam 'ca' ad 'ab'; ut autem 'cd' ad 'df', ita quadratum 'co' ad quadratum 'of', cum sint 'cd', 'do' et 'df' proportionales; ut vero 'ca' ad 'ab', ita quadratum 'cv' ad quadratum 'vb'; ergo 'co' ad 'of' maiorem rationem habet quam 'cv' ad 'vb': igitur, ex lemmate praedemonstrato, 'co' maior est quam 'cv'. Constat igitur, tempus per 'dc' ad tempus per 'dbc' esse ut 'doc' ad 'do' cum 'cv'.</s></p>
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176 <pb xlink:href="X01/01/059.jpg" page="060_R"/>
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177 <p><s>Tempus casus per planum inclinatum ad tempus casus per lineam suae altitudinis est ut eiusdem plani longitudo ad longitudinem suae altitudinis.</s></p>
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178 <p><s>Sit planum inclinatum 'ba' ad lineam horizontis 'ac', sitque linea altitudinis perpendicularis 'bc': dico, tempus casus quo mobile movetur per 'ba' ad tempus in quo cadit per 'bc' esse ut 'ba' ad 'bc'. Erigatur perpendicularis ad horizontem ex 'a', quae sit 'ad', cui occurrat in 'd' perpendicularis ad 'ab' ducta ex 'b', quae sit 'bd', et circa triangulum 'abd' circulus describatur: et quia 'da', 'bc' ambae sunt ad horizontem perpendiculares, constat, tempus casus per 'da' ad tempus casus per 'bc' esse ut media inter 'da', 'bc' ad ipsam 'bc'. Tempus autem casus per 'da' aequatur tempori casus per 'ba': media vero inter 'da' et 'bc' est ipsa 'ba': ergo patet propositum.</s></p>
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179 <p><s>Corollarium</s></p>
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180 <p><s>Ex hoc sequitur, casuum tempora per plana inclinata quorum eadem sit altitudo, esse inter se ut eorundem planorum longitudines. Si enim fuerit aliud planum inclinatum 'be', tempus casus per 'ba' ad tempus casus per 'bc' est ut 'ba' linea ad 'bc': tempus vero per 'bc' ad tempus casus per 'be' ut 'bc' ad 'be': ergo, ex aequali, patet propositum.</s></p>
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181 <pb xlink:href="X01/01/060.jpg" page="060_V"/>
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182 <pb xlink:href="X01/01/061.jpg" page="061_R"/>
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183 <p><s>attinentia ad motum</s></p>
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184 <p><s>'ea' 30 <lb/> 'ed' 42 <lb/> 'dc' 92 <lb/> media 'df' 62 1/2 <lb/> 'ei', 'ef' 20 1/2 <lb/> 'al' 85 <lb/> 'el' 55 <lb/> 'eb' 70 <lb/> 'ab' 100</s></p>
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185 <p><s>ut 'ea' ad 'ai', <DELETION-1> ita 'li' ad 'ie' </DELETION-1> ita 'ei' ad 'il'; et ut 'ed' ad 'df', ita 'cf' ad 'fe'.</s></p>
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186 <p><s>Probandum est, 'li' ad 'ie' maiorem habere rationem quam 'cf' ad 'fe'. Est autem 'li' ad 'ie' ut 'ia' ad 'ae'; 'cf' autem ad 'fe' ut 'fd' ad 'de': probare igitur debes, 'ia' ad 'ae' maiorem habere rationem quam 'fd' ad 'de', et, dividendo, 'ie' ad 'ea' maiorem habere rationem quam 'fe' ad 'ed'. Hoc autem manifestum est: nam eadem <DELETION-1> maior ad quam eadem habet minorem ratione </DELETION-1> maiorem habet rationem ad minorem. Componitur ergo demonstratio sic. Quia 'ea' minor est 'ed', 'ie' ad eam maiorem rationem habet quam 'fe' ad 'ed', et, componendo, 'ia' ad 'ae' maiorem rationem habet quam 'fd' ad 'de': verum ut 'ia' ad 'ae', ita est 'li' ad 'ie'; ut autem 'fd' ad 'de', ita 'cf' ad 'fe': ergo 'li' ad 'ie' maiorem rationem habet quam 'cf' ad 'fe', et, componendo, 'le' ad 'ei' maiorem habet rationem quam 'ce' ad 'ef': sunt autem 'ef', 'ei' aequales: ergo 'le' maior est quam 'ce'.</s></p>
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187 <p><s><BEGIN-OF-LINE-DESTROYED/> [In pla]no incli[nato] <BEGIN-OF-LINE-DESTROYED/> [assum]pta in 'eb' par[te] <BEGIN-OF-LINE-DESTROYED/> maiori quam 'ec' et mi[n]ori quam 'eb', punctum sublime reperire, ex quo cadens tempore <BEGIN-OF-LINE-DESTROYED/> [aeq]uali conficiat 'ec' et 'el'. <BEGIN-OF-LINE-DESTROYED/> [Qu]od autem oporteat, assumptam in 'eb' [ma]iorem esse quam 'ec', declaratur sic. <BEGIN-OF-LINE-DESTROYED/> [Duc]atur 'es' aequalis 'ec', et sumptis <BEGIN-OF-LINE-DESTROYED/> [med]iis 'sae', 'cde', 'ai', 'df', non esset <BEGIN-OF-LINE-DESTROYED/> aequalis 'ef', ut est necessarium: nam</s></p>
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188 <p><s>si id esset, foret quoque 'si' aequalis 'cf'; et <DELETION-1> <ILLEGIBLE/> </DELETION-1> cum sit ut 'cf' ad 'fe', ita 'fd' ad 'de' et <SUBSTITUTION-1> 'sa' <SUBSTITUTED-BY-1> 'ia' </SUBSTITUTED-BY-1></SUBSTITUTION-1> ad 'ae', esset, dividendo, 'fe' ad 'ed' [ut 'ie'] 'fc' ad 'ea', et esset 'ea' aequalis 'ed', quod est falsum. Quod autem oporteat, assumptam minorem esse quam 'be', sic ostenditur. Nam si ['fe'] [a]equatur 'ei', anguli 'efi', 'eif' erunt aequales, et <SYMBOL-ANGULUS/>_[angulus] 'fid' maior <SYMBOL-ANGULUS/>_[angul]o 'f', et latus <INSERTION-1> 'fd' </INSERTION-1> maius latere 'di', et <SYMBOL-QUADRATUM/>_[quadrat]um 'fd' maius <SYMBOL-QUADRATUM/>_[quadrat]is 'iad', et <SYMBOL-RECTANGULUM/>_[rectangulum] 'cde' maius <SYMBOL-RECTANGULUM/>_[rectangul]o 'bae' cum <SYMBOL-QUADRATUM/>_[quadrat]o 'ad', et <SYMBOL-RECTANGULUM/>_[rectangulum] 'ced' cum <SYMBOL-QUADRATUM/>_[quadrat]o 'ed' maius <SYMBOL-RECTANGULUM/>_[rectangul]o 'bea' cum <SYMBOL-QUADRATUM/>_[quadrat]is 'ead', et demptis <SYMBOL-3/>_[tri]bus <SYMBOL-QUADRATUM/>_[quadrat]is 'ead', <SYMBOL-RECTANGULUM/>_[rectangulum] 'ced' maius <SYMBOL-RECTANGULUM/>_[rectangul]o 'bea', quod est falsum, cum angulus 'c' sit rectus.</s></p>
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189 <p><s>Data igitur 'el' maiori 'es' et minori 'eb', quaeratur 'ea', ex qua cadens temporibus aequalibus conficiat 'aec' et <SUBSTITUTION-1> 'aeb' <SUBSTITUTED-BY-1> 'ael', </SUBSTITUTED-BY-1></SUBSTITUTION-1> sive 'ec' et <SUBSTITUTION-1> 'eb' <SUBSTITUTED-BY-1> 'el' </SUBSTITUTED-BY-1></SUBSTITUTION-1> post 'ae': <INSERTION-1> quod erit </INSERTION-1> dum 'ef', 'ei' sint aequales, <DELETION-1> <ILLEGIBLE/> </DELETION-1> positis 'ai', 'df' mediis 'lae', 'cde'.</s></p>
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190 <p><s>Attende. Quo vicinius fuerit 'l', 's', eo punctum 'a' remotius esse oportet, et quo vicinius fuerit 'l', 'b', eo a propius <BEGIN-OF-LINE-DESTROYED/> [con]tingit puncto 'e'; adeo ut, posito 'l' in 's', distantia 'ae' est infinita, et posito 'l' in 'b', 'a' recidit in 'e'. Insuper, dum 'l' sit in 's', puncta 'f', 'i' sunt in medio linearum 'ec', 'es'; dum vero 'l' sit in 'b', puncta 'f', 'i' sunt in 'e'.</s></p>
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191 <pb xlink:href="X01/01/062.jpg" page="061_V"/>
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192 <p><s><ILLEGIBLE/> Dupli[ci] aggressione possum progredi.</s></p>
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193 <p><s><ILLEGIBLE/> Prima: <lb/> Si 'ac' sit tempus per 'ac', erit 'bi' tempus per 'xa', et 'ab' tempus per 'ab', et 'bs' tempus per 'rb', et tempus <BEGIN-OF-LINE-DESTROYED/> [per 'ab'] post 'ra' erit excessus 'bs' super 'sa', <BEGIN-OF-LINE-DESTROYED/> [cui] oportet ostendere aequari 'ai'; ut verum est.</s></p>
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194 <p><s>Secunda: <lb/> Si 'xa' sit tempus per 'xa', erit 'ar' tempus per <DELETION-1> <ILLEGIBLE/> </DELETION-1> 'ra', et 'as' tempus per 'ab' ex quiete in 'a', et 'rs' tempus per totam 'rb', et excessus 'rs' super 'ra' (puta 'ao') erit tempus per 'ab' post 'ra'. Ostendendum ergo est, 'xa' cum 'ao' aequari 'as', ut verum est: quia ut 'ba' ad 'ar', ita 'ca' ad 'ax', idest 'na' ad 'as'; erit etiam 'bn' ad 'sr', ut 'ba' ad 'ar'.</s></p>
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195 <p><s>Restat ostendendum, 'bn' ad 'rt' esse quoque ut 'ba' ad 'ar'. Ponatur 'bv' aequalis 'ba': iam erit ut <DELETION-1> 'bn' a </DELETION-1> 'bv' ad 'vs', ita 'st' ad 'tr'.</s></p>
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196 <p><s><DELETION-1> Si 4 lineae, quarum prima et <SYMBOL-2/>_[secund]a sint aequales <SYMBOL-3/>_[tertia]e et <SYMBOL-4/>_[quart]ae simul sumptis, prima tamen </DELETION-1> </s></p>
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197 <p><s>Si fuerint <SYMBOL-4/>_[quattuor] lineae, quarum prima et <SYMBOL-2/>_[secund]a simul sumptae sint aequales <SYMBOL-3/>_[tertia]e et <SYMBOL-4/>_[quarta]e simul sumptis, sint autem prima et <SYMBOL-2/>_[secund]a minus inter se differentes quam <SYMBOL-3/>_[terti]a et <SYMBOL-4/>_[quart]a, <SYMBOL-RECTANGULUM/>_[rectangulum] primae et <SYMBOL-2/>_[secunda]e superat <SYMBOL-RECTANGULUM/>_[rectangulum] <SYMBOL-3/>_[tertia]e et <SYMBOL-4/>_[quart]ae <SYMBOL-RECTANGULUM/>_[rectangul]o contento ab excessu <SYMBOL-3/>_[tertia]e super primam in excessu primae super <SYMBOL-4/>_[quart]am.</s></p>
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198 <pb xlink:href="X01/01/063.jpg" page="062_R"/>
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199 <p><s>cum 'ab' sit dimidia 'bc', et sit eadem 'ab' tempus per 'ab', erit tota 'bc' tempus per ambas 'abc'.</s></p>
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200 <p><s>Accipiatur 'be', et sit 'bo' media inter 'eb', 'ba', et erit 'ob' tempus per 'eb' et per <SYMBOL-2/>_[du]plam ipsius 'bd'. Quod si 'db' conficitur tempore 'bo', 'cb' quo tempore conficietur fiat igitur ut 'db' ad 'bo', ita 'cb' ad aliam, quae sit, verbigratia, <INSERTION-1> 'bn', </INSERTION-1> et ostendatur <SYMBOL-2/>_[du]as <DELETION-1> <ILLEGIBLE/> </DELETION-1> 'ob', 'bn' maiores esse ipsa 'bc', et habebitur intentum.</s></p>
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201 <p><s>Redacta est res ad hoc lemma: sit 'eb' utcumque secta in 'a', et inter 'eb', 'ba' media sit 'bo', et ut 'eb' ad 'ba', ita sit 'ob' ad 'bn'. Dico <DELETION-1> non esse plus quam <SYMBOL-2/>_[du]pla ad 'ba'. Ostende 'oa' maiorem esse quam 'an'; 'ob', 'bn' esse plus quam <SYMBOL-2/>_[du]pla ad 'ab' </DELETION-1> </s></p>
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202 <p><s>'eb', 'bo', 'ba', 'bn' esse continue proportionales. Quia enim ut 'eb' ad 'bo', ita 'bo' ad 'ba', ratio 'eb' ad 'ba' erit <SYMBOL-2/>_[du]pla rationis 'ob' ad 'ba': et quia ut 'eb' ad 'ba', ita 'ob' ad 'bn', est autem ratio 'be' ad 'ba' <SYMBOL-2/>_[du]pla rationis 'ob' ad 'ba', erit quoque ratio 'ob' ad 'bn' <SYMBOL-2/>_[du]pla rationis 'bo' ad 'ba'. Verum ipsa ratio 'ob' ad 'bn' componitur ex rationibus 'ob' ad 'ba' et 'ab' ad 'bn': ergo ratio 'ab' ad 'bn' est eadem cum ratione 'ob' ad 'ba': ergo patet propositum.</s></p>
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203 <p><s>'ab' 20 <lb/> 'ib' 12 <lb/> 'bs' 15 3/10 tempus per 'bi' <lb/> 'br' 30 <lb/> 'bn' 38 7/17 tempus per <DELETION-1> <ILLEGIBLE/> </DELETION-1> 'rb' <lb/> media inter 'cr', 'rb' 46 <lb/> tempus per 'rc' 59 <lb/> tempus per 'bc' 20 2/5 <lb/> tempus per 'ibc' 35 1/2 <lb/> Si tempus per 'rb' esset 'rb', tempus per 'rc' esset 46 <lb/> mod[o] est 59.</s></p>
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204 <p><s>'ab' 20 <lb/> 'bc' 40 <lb/> 'bd' 50 <lb/> 'cd' 90 <lb/> 'de' 67 <lb/> 'be' 17 <lb/> 'ec' 23 <lb/> 'abe' 37 <lb/> tempus per 'bc' ex quiete in 'b' est 44 1/3 proxime; tempus per 'abc' 37.</s></p>
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205 <pb xlink:href="X01/01/064.jpg" page="062_V"/>
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206 <p><s>Attinentia ad motum</s></p>
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207 <pb xlink:href="X01/01/065.jpg" page="063_R"/>
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208 <p><s>Distantia, per quam mobile motu recto naturaliter movetur, perpendiculus, seu perpendicularis linea dicatur.</s></p>
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209 <p><s>Horizontalis linea, seu planum horizontale, appello lineam rectam, seu planum, cui perpendiculus ad rectos angulos incidit.</s></p>
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210 <p><s>Quae vero cum linea horizontali inaequales angulos constituunt, lineae seu plana inclinata dicantur.</s></p>
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211 <p><s>3</s></p>
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212 <pb xlink:href="X01/01/066.jpg" page="063_V"/>
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213 <pb xlink:href="X01/01/067.jpg" page="064_R"/>
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214 <p><s><DELETION-2> Quia 'ca' ad 'ab' est ut 'ba' ad 'bn', et ut 'na' cum 'ab' ab 'ab', ita 'ca' ad 'ae', erit, dividendo, 'na' ad 'ab' ut <DELETION-1> <ILLEGIBLE/> </DELETION-1> 'ce' ad 'ea'; fiat modo ut 'nb' ad 'ba', ita 'ea' ad 'ar': erit, componendo, ut 'na' ad 'ab', idest 'ce' ad <DELETION-1> 'ea', ita 'er' ad 'ra', et omnes antecedentes ad omnes consequentes, nempe 'cr' ad 're'. </DELETION-1> </DELETION-2> </s></p>
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215 <p><s>Dato perpendiculo et plano ad eum inclinatum [inclinato], partem in perpendiculo reperire, quae conficiatur tempore eodem ac planum post ipsam.</s></p>
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216 <p><s>Fiat ut 'ca' ad 'ab', ita 'ab' ad 'bn', et ut <DELETION-1> 'nb' </DELETION-1> 'na' cum 'ab' ad 'ab', ita ita [sic] 'ca' ad 'ae', et ut 'nb' ad 'ba', <INSERTION-1> seu 'ba' ad 'ac', </INSERTION-1> ita 'ea' ad 'ar', et ab 'r' ducatur ad 'ba' productam perpendicularis 'rx': dico, 'ax' esse quod quaeritur.</s></p>
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217 <p><s>Quia enim ut 'na' cum 'ab' ad 'ab', ita est 'ca' ad 'ae', erit, dividendo, ut 'na' ad 'ab', ita 'ce' ad 'ea'; et quia ut 'nb' ad 'ba', ita 'ea' ad 'ar', componendo erit ut 'na' ad 'ab', idest 'ce' ad 'ea', ita 'er' ad 'ra', et omnia antecedentia ad omnia consequentia, 'cr' ad 're'. Sunt igitur 'cr', 'er', 'ar' continue proportionales. Rursus, quia ut <SUBSTITUTION-1> 'nb' ad 'ba' <SUBSTITUTED-BY-1> 'ba' ad 'ac', </SUBSTITUTED-BY-1></SUBSTITUTION-1> ita 'ea' ad 'ar'; <DELETION-1> ut autem 'nb' ad 'ba', ita 'ba' ad 'ac', erit ut 'ba' ad 'ac', ita 'ca' ad 'ar'; verum </DELETION-1> ut <INSERTION-1> autem </INSERTION-1> 'ba' ad 'ac', ita 'xa' ad 'ar'; ergo utraque 'ea', 'ax' ad 'ar' eandem habent rationem: sunt ergo aequales.</s></p>
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218 <p><s>Modo si intelligamus, tempus per 'ra' esse ut 'ra', tempus per 'rc' erit 're', media inter 'cr', 'ra', et 'ae' erit tempus per 'ac' post 'ra': verum tempus per 'xa' est 'xa', dum 'ra' est tempus per 'ra'; ostensum autem est 'xa' esse aequo 'ae': ergo factum est quod facere oportebat.</s></p>
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219 <p><s>Quia factum est ut 'ca' ad 'ab', ita 'ab' ad 'bn', et ut 'na', 'ab' ad 'ba', ita 'ca' ad 'ax' seu 'ao', dividendo erit ut 'na' ad 'ab', ita 'co' ad 'oa'.</s></p>
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220 <p><s><foreign lang="it">Al molto Ill[ust]re et Ecc[ellen]te Sig[nor]e et P[ad]rone Oss[ervatissi]mo Galileo Galilei.</foreign></s></p>
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221 <pb xlink:href="X01/01/068.jpg" page="064_V"/>
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222 <pb xlink:href="X01/01/069.jpg" page="065_R"/>
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223 <p><s>Si fiat casus in perpendiculo, in quo a principio lationis sumatur pars, quovis temporis peracta, post quam sequatur motus inflexus per aliquod planum inclinatum, spacium quod in plano inclinato conficitur in tempore aequali tempori <INSERTION-1> casus </INSERTION-1> iam peracti in perpendiculo, ad spacium iam peractum in perpendiculo, maius erit quam duplum, minus vero quam triplum.</s></p>
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224 <p><s>Infra horizontem 'ae' sit perpendiculus 'ab', in quo ex principio 'a' fiat casus, cuius sumatur quaelibet pars 'ac'. Inde ex 'c' inclinetur planum 'cg', super quo post casum 'ac' inflectatur motus. Dico, quod spacium tali motu peractum per 'cg' in tempore aequali tempori casus per 'ac', est plus quam duplum, minus vero quam triplum <INSERTION-1> eiusdem spacii 'ac'. </INSERTION-1> </s></p>
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225 <p><s>Ponatur enim <UNDERLINED> unaquaeque ipsarum </UNDERLINED> 'cf', <UNDERLINED> 'cd' </UNDERLINED> ipsi 'ac' aequalis, et ut <UNDERLINED> 'ca' ad 'ad', ita fiat 'da' ad 'ab', ut vero </UNDERLINED> 'ce' ad 'ef', ita 'fe' ad 'eg': <UNDERLINED> erit iam ipsa 'cb' tripla 'ca', et tempus casus per 'ac' aequabitur tempori casus per 'cb' <INSERTION-2> post 'ac'. </INSERTION-2> </UNDERLINED> Si itaque ponatur, tempus casus per 'ac' esse ut linea 'ac', erit <UNDERLINED> 'cd' tempus casus per 'cb', et </UNDERLINED> 'ce' tempus per 'ec', et 'cf' tempus motus per 'cg'. Ostendendum itaque est, spacium 'cg' ipso 'ca' maius esse quam duplum, minus vero quam triplum. <DELETION-1> At </DELETION-1> <INSERTION-2> Cum enim sit </INSERTION-2> ut 'ce' ad 'ef', ita <DELETION-1> e </DELETION-1> 'fe' ad 'eg', erit et ita 'cf' ad 'fg'; <DELETION-1> <ILLEGIBLE/> </DELETION-1> minor autem est 'ec' ipsa 'ef'; quare et 'cf' minor erit 'fg', et 'gc' maior quam dupla ad 'fc' seu 'ac'. <DELETION-1> cu </DELETION-1> Cumque rursus 'fe' minor sit quam dupla ad 'ec' (est enim 'ec' maior 'ca', seu 'cf'), erit quoque 'gf' minor quam dupla ad 'fc', et 'gc' minor quam tripla ad 'cf', seu 'ca': quod erat ostendendum.</s></p>
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226 <p><s>Ex his constat, quod si inflexio post casum 'ac' fieret in horizontali 'icx', in tempore aequali tempori 'ac' conficeret spacium 'ci', duplum ad 'ca'. Positis enim 'ch', 'hi' inter se et ipsi 'ca' aequalibus, et extensis 'icx' in infinitum, erit ut 'ix' ad 'xh', ita 'hx' ad 'xc' et 'ih' ad 'hc'; quare tempus motus per <DELETION-1> <ILLEGIBLE/> </DELETION-1> 'ci' erit 'ch', seu 'ca'.</s></p>
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227 <p><s>Potest haec propositio universalius proferri: idem enim accidit si 'ab' non sit perpendicularis, sed utcumque inclinata.</s></p>
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228 <p><s>Attende quod si in inclinata 'cg' motus acceleratur in infinitum, videtur posse demonstrari, in orizontali extendi debere [a]equabiliter etiam in infinitum; quod etiam constat, si est [a]equabilis, esse etiam infinitum.</s></p>
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229 <pb xlink:href="X01/01/070.jpg" page="065_V"/>
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230 <p><s>Datis <SYMBOL-2/>_[duo]bus temporibus inaequalibus, et spacio quod in perpendiculo ex quiete conficitur in tempore breviori ex datis, a puncto supremo perpendiculi usque ad <DELETION-1> <ILLEGIBLE/> </DELETION-1> orizontem planum inflectere, super quo mobile descendat tempore <SUBSTITUTION-1> longiori ex datis aequale[aequali] <SUBSTITUTED-BY-1> aequale[aequali] longiori ex datis. </SUBSTITUTED-BY-1></SUBSTITUTION-1> </s></p>
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231 <pb xlink:href="X01/01/071.jpg" page="066_R"/>
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232 <p><s>Cum in planis inclinatis decrescat impetus mobilis, prout inclinatio minus participat de directione, et, quod consequens est, plus detrahat de gravitate mobili, consideretur num in mediis gravioribus, detrahentibus de gravitate mobilis, decrescat pariter impetus pro ratione gravitatis; ita ut in medio detrahente, v[erbi] g[ratia], dimidium gravitatis, impetus sit idem atque in inclinatione similiter detrahente gravitatis dimidium.</s></p>
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233 <pb xlink:href="X01/01/072.jpg" page="066_V"/>
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234 <pb xlink:href="X01/01/073.jpg" page="067_R"/>
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235 <p><s>Dato quolibet spacio in perpendiculo a principio lationis signato, quod in dato tempore conficiatur, datoque quocunque tempore minori, aliud aequale spacium, priori accepto in eodem perpendiculo reperire, quod in dato tempore conficiatur.</s></p>
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236 <p><s>Sit perpendiculum 'ab', in quo detur spacium 'ab', cuius tempus ex principio 'a' sit 'ab', sitque horizon 'cbe', et detur tempus ipso 'ab' minus, quod sit 'bc': oportet in eodem perpendiculo spacium eidem 'ab' aequale invenire, quod tempore 'bc' conficiatur. Iungatur linea 'ac', cunque 'bc' minor sit 'ba', erit angulus 'bac' minor angulo 'bca'; constituatur ipsi aequalis 'cae', et linea 'ae' horizonti in 'e' signo occurrat, ad quam perpendicularis ponatur 'ed', secans perpendiculum in 'd', et linea 'df' ipsi 'ba' secetur aequalis: dico, ipsam 'fd' ex 'a' confici in tempore 'bc'. Cum enim in triangulo rectangulo 'aed' <SUBSTITUTION-2> ad <SUBSTITUTED-BY-2> ab </SUBSTITUTED-BY-2></SUBSTITUTION-2> angulo recto 'e' perpendicularis ducta sit ad 'ad', erit 'ae' media inter 'da', 'ab', et 'be' media inter 'db', 'ba', hoc est inter 'fa', 'ab', est enim 'fa' aequalis 'db', cunque 'ab' positum sit esse tempus per 'ab', erit 'ae' tempus per totam 'ad', et 'eb' tempus per 'af'; cunque 'ae' sit aequalis ipsi 'ec', ob aequalitatem angulorum <DELETION-1> et </DELETION-1> 'eac' <DELETION-1> <ILLEGIBLE/> </DELETION-1> 'eca', relinquitur ut 'bc' tempus sit ipsius 'fd': quod erat ostendendum.</s></p>
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237 <p><s>12</s></p>
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238 <pb xlink:href="X01/01/074.jpg" page="067_V"/>
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239 <pb xlink:href="X01/01/075.jpg" page="068_R"/>
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240 <p><s>Sit circulus cuius diameter 'ab', et ipsi parallela tangens 'ce', et ex termino 'b' quaelibet linea 'bo' in circulo applicetur: dico, <DELETION-1> lineas </DELETION-1> perpendiculares, <DELETION-1> ad 'bo' </DELETION-1> quae a terminis 'b', 'o' ipsi 'bo' accomodantur, protractas, de linea 'ce' partem diametro circuli aequalem semper intercipere. Iungatur enim 'ao', et extendatur ad tangentem in 'f', quae ad 'bo' erit perpendicularis, cui ex 'b' parallela sit 'be': demonstrandum 'fe' diametro circuli esse aequalem.</s></p>
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241 <p><s>Id autem constat, quia in parallelogrammo 'abef' latera 'ab', 'fe' opposita aequalia sunt, ex Elementis. Vel dicas, quod ducta ex 'o', 'og' parallela ipsi 'ab', et 'bg' et 'bg' [sic] perpendiculari ad 'bo', abscindetur semper 'og' aequalis diametro circuli; quod patet ex triangulis 'aob', 'obg' similibus et aequalibus.</s></p>
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242 <p><s>Sit 'ac' perpendicularis ad horizontem 'cde', ponaturque inclinata 'bd', fiatque motus ex 'a' per 'abc' et per 'abd'. Dico, tempus per 'bc' post casum 'ab', ad tempus per 'bd' post eumdem casum 'ab', esse ut linea 'bc' ad 'bd'. Ducatur 'af' parallela 'dc',, et protrahatur <DELETION-1> b </DELETION-1> 'db' ad 'f'; erit iam tempus casus per 'fbd' ad tempus casus per 'abc' ut 'fd' linea ad lineam 'ac'. Est autem tempus casus per 'fb' ad tempus casus per 'ab' ut linea 'fb' ad lineam 'ab': ergo tempus casus reliquae 'bc' post 'ab' ad tempus casus reliquae 'bd' post 'fb' erit ut reliqua 'bc' ad reliquam 'bd'. Sed tempus casus per 'bd' post 'fb' est idem cum tempore per 'bd' post 'ab', cum 'af' <DELETION-1> est </DELETION-1> sit horizonti aequidistans: ergo patet propositum.</s></p>
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243 <p><s>Colligitur autem ex hoc, quod tempora casuum per 'bc' et 'bd', sive fiat principium motus ex termino 'b', sive praecedat motus, ex eadem tamen altitudine, eamdem inter se servant rationem, nempe eam quae est lineae 'bc' ad 'bd'.</s></p>
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244 <pb xlink:href="X01/01/076.jpg" page="068_V"/>
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245 <pb xlink:href="X01/01/077.jpg" page="069_R"/>
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246 <p><s>Fiat ut 'ba' cum <SYMBOL-2/>_[du]pla 'ac' ad 'ac', ita 'ca' ad 'ae', et ut 'ba' cum 'ac' ad 'ac', <SUBSTITUTION-1> ut <SUBSTITUTED-BY-1> ita </SUBSTITUTED-BY-1></SUBSTITUTION-1> 'ea' ad 'ar', et ab 'r' ducatur perpendicularis 'rx'. Et quia ut 'ba' cum <SYMBOL-2/>_[du]pla 'ac' ad 'ac', ita 'ca' ad 'ae', dividendo erit ut 'ba' cum 'ac' ad 'ac', ita 'ce' ad 'ea'. Et quia ut 'ba' ad 'ac' ita 'ea' ad 'ar', erit, componendo, ut 'ba' cum 'ac' ad 'ac', ita 'er' ad 'ra': sed ut 'ba' cum 'ac' ad 'ac', ita est 'ce' ad 'ea': ergo ut 'ce' ad 'ea', ita 'er' ad 'ra', et ambo antecedentia ad ambo consequentia nempe 'cr' ad 're'. Sunt itaque 'cr', 're', 'ra' proportionales. <DELETION-1> Et </DELETION-1> Amplius, quia ut 'ba' ad 'ac', ita positum est 'ea' ad 'ar'; et propter similitudinem <SYMBOL-3/>_[tri]angulorum, ut 'ba' ad 'ac', ita 'xa' ad 'ar'; ergo ut 'ea' ad 'ar', ita 'xa' ad 'ar': sunt itaque 'ea', 'xa' aequales.</s></p>
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247 <pb xlink:href="X01/01/078.jpg" page="069_V"/>
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248 <p><s>'ac' 9 <lb/> 'ab' 6 <lb/> 'ae' 3 3/8 <lb/> 'ar' 5 1/16 <lb/> 'ce' 5 5/8 <lb/> 'xa' 3 3/8</s></p>
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249 <p><s>Fiat ut 'baca' ad 'ac', ita 'ca' ad 'ae', erit dividendo ut 'bac' ad 'ac', ita 'ce' ad 'ea'; fiat modo ut 'ba' ad 'ac', ita 'ea' ad 'ar', et quia ut 'bac' ad 'ac', 'ce' ad 'ea', ut autem 'ac' ad 'ab', ita 'ra' ad 'ae'</s></p>
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250 <p><s>'bac' ad 'ac', 'ab' <lb/> 'ce' ad 'ea', 'ra', 'ax'</s></p>
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251 <p><s><DELETION-1> 'ce', 'ca', 'ce', 'ar', <lb/> 'ac', 'bac' <ILLEGIBLE/> </DELETION-1> </s></p>
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252 <p><s>'ab' 100 <lb/> 'ac' 20 <lb/> 'ao' 44 4/9 <lb/> 'ax' 222 1/5 <lb/> 'xb' 322 1/5 <lb/> m[edia] 'xba' 179 1/2 <lb/> m[edia] 'xab' 149 1/12</s></p>
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253 <p><s>tempus per 'ab' 30 5/12 <lb/> tempus per 'ao' 30 <lb/> totum tempus 60 1/3</s></p>
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254 <p><s>sit modo 'ai' 40 <lb/> 'az' 200 <lb/> media 'zba' 173 1/5 <lb/> media 'zab' 141 1/2 <lb/> tempus per 'ab' 31 7/10 <lb/> tempus per 'ia' 28 4/7 <lb/> 1 <DELETION-1> <ILLEGIBLE/> ?/7 </DELETION-1> <lb/> totum tempus 60 3/7</s></p>
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255 <p><s>16</s></p>
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256 <pb xlink:href="X01/01/079.jpg" page="070_R"/>
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257 <pb xlink:href="X01/01/080.jpg" page="070_V"/>
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258 <pb xlink:href="X01/01/081.jpg" page="071_R"/>
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259 <p><s>si fiat ut 'eb' ad 'ebc', ita 'ebc' ad aliam 'bs', erit excessus huius super 'ebc' ad 'bc' ut 'ebc' ad 'eb'.</s></p>
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260 <p><s>Fiat igitur ut 'ebc' ad 'eb', ita alia ad 'bc' <INSERTION-1> hoc
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261 autem fit ducta perpendiculari 'cn'; erit enim 'nbc' ad 'bc' ut 'cbe' ad 'be' </INSERTION-1> quae alia 'cbn' cum 'bc' est ea quae conficitur eodem tempore post 'bc' ac ipsa 'bc': quare si fiat ut haec alia cum 'bc' ad 'bc', ita 'cb' ad 'ab', erit 'ab' quaesitum.</s></p>
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262 <pb xlink:href="X01/01/082.jpg" page="071_V"/>
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263 <p><s>fiat ut 'ca' ad 'cab', ita 'bf' ad 'fg', et ut 'gb' ad 'ba', ita <SUBSTITUTION-1> 'ba' <SUBSTITUTED-BY-1> 'ca' </SUBSTITUTED-BY-1></SUBSTITUTION-1> ad 'ae': habebimus 'bg', quae peragitur tempore 'ab' post 'ab'.</s></p>
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264 <p><s>[Fiat ut 'ca' ad 'ab'] ita 'ab' ad aliam 'gf', cui addatur 'ab'; et fiat postea ut 'gb' ad 'ba', ita <SUBSTITUTION-1> 'ba' <SUBSTITUTED-BY-1> 'ca' </SUBSTITUTED-BY-1></SUBSTITUTION-1> ad 'ae', et erit 'ae' quaesitum.</s></p>
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265 <p><s>Posita 'cs' aequali 'ab', fiat ut 'ca' ad 'as', ita 'ba' ad <DELETION-1> 'ar' </DELETION-1> 'an', et ut 'ba' cum 'an' ad 'ab', ita <SUBSTITUTION-1> 'ba' <SUBSTITUTED-BY-1> 'ca' </SUBSTITUTED-BY-1></SUBSTITUTION-1> ad 'ae', etc.</s></p>
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266 <p><s>Vel fiat ut 'ca' ad 'ab', ita 'ab' ad 'bn', et ut 'ab' cum <SUBSTITUTION-1> 'bn' <SUBSTITUTED-BY-1> 'an' </SUBSTITUTED-BY-1></SUBSTITUTION-1> ad 'ab', ita <SUBSTITUTION-1> 'ab' <SUBSTITUTED-BY-1> 'ca' </SUBSTITUTED-BY-1></SUBSTITUTION-1> ad 'ae'.</s></p>
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267 <p><s>Data 'ac' quaeritur 'ae'. Posita 'bf' [a]equali 'ab', fiat ut 'bd' ad 'df' (idest ut 'ca' ad 'cab') ita 'fd' ad 'dg' (idest ita 'cab' ad aliam 'dg'); erit etiam ut 'db' ad <DELETION-1> 'bf' </DELETION-1> 'df', ita 'bf' ad 'fg' (idest ut 'ca' ad 'cab', ita 'ba', <INSERTION-1> seu 'bf', </INSERTION-1> ad <DELETION-1> illam aliam </DELETION-1> 'fg'): haec autem 'gf' cum 'fb' conficitur eodem tempore ac 'ab': quare si fiat ut 'gb' ad 'ba', ita <SUBSTITUTION-1> 'ba' <SUBSTITUTED-BY-1> 'ca' </SUBSTITUTED-BY-1></SUBSTITUTION-1> ad 'ae', erit 'ae' quod quaeritur.</s></p>
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268 <pb xlink:href="X01/01/083.jpg" page="072_R"/>
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269 <p><s>Pro inveniendo tempore minimo in quo conficiantur 'qab', attende numquid, posita 'ad' aequali 'ab', faciendum forte sit ut <DELETION-1> d </DELETION-1> 'ad' cum 'dc' ad <SUBSTITUTION-1> 'da' <SUBSTITUTED-BY-1> 'cd', </SUBSTITUTED-BY-1></SUBSTITUTION-1> ita 'ba' ad 'aq'. <DELETION-1> <ILLEGIBLE/> </DELETION-1> </s></p>
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270 <p><s>fiat ut 'ca' ad 'ab', ita 'ab' ad 'bn', et ut 'na' cum 'ab' ad 'ab', ita 'ca' ad 'ax'; erit 'ax' quaesitum.</s></p>
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271 <p><s>Ponatur 'ao' aequalis 'ax'; oportet 'or' esse mediam inter 'cr', 'ra', et ut 'co' ad 'oa', ita esse 'cr' ad 'ro', et <SYMBOL-QUADRATUM/>_[quadratum] 'co' ad <SYMBOL-QUADRATUM/>_[quadratum] 'oa' ut 'cr' ad 'ra', seu 'bx' ad 'xa': sed ut 'bx' ad 'xa', ita <SYMBOL-RECTANGULUM/>_[rectangulum] 'bxa' ad <SYMBOL-QUADRATUM/>_[quadratum] 'xa', seu 'ao': ergo <SYMBOL-QUADRATUM/>_[quadratum] 'co' debet aequari <SYMBOL-RECTANGULUM/>_[rectangul]o 'bxa', et 'co' esse mediam inter 'bx', 'xa'.</s></p>
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272 <p><s>tempus per 'ia' 24</s></p>
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273 <pb xlink:href="X01/01/084.jpg" page="072_V"/>
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274 <pb xlink:href="X01/01/085.jpg" page="073_R"/>
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275 <p><s>Dato tempore per 'ab', 'de' quaeritur tempus per 'cb'. Si tempus per 'ab' esset 'ab', tempus per 'ac' esset 'af', media inter 'ba', 'ac', et 'fb' esset tempus per 'cb': sed tempus per 'ab' non est 'ab', sed 'de': fac igitur ut 'ab' ad 'fb', ita 'de' ad 'eo', et erit 'eo' tempus per 'cb'. <DELETION-1> Fac </DELETION-1> Igitur primum cape mediam inter 'ba', 'ac', nempe 'af', et ut tota 'ab' ad excessum 'bf', ita 'de' ad 'eo', et habebis 'oe' tempus per 'cb'.</s></p>
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276 <p><s>Dato in horizontali quolibet spatio, ex cuius termino erectum sit perpendiculum, in quo sumatur pars aequalis dimidio spatii in horizontali signati, mobile ex tali altitudine descendens et in horizontali conversum conficiet horizontale spacium una cum perpendiculo breviori tempore, quam <DELETION-1> <ILLEGIBLE/> </DELETION-1> quodcumque aliud spatium perpendiculi cum eodem spatio horizontali.</s></p>
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277 <pb xlink:href="X01/01/086.jpg" page="073_V"/>
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278 <p><s>si tempus per 'ab' est 'ab', posita 'cd' aequali 'ab', tempus per 'cd' erit 'cd', et per totam 'cb' erit 'ce'.</s></p>
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279 <p><s>si 'ab' est tempus per 'ab', tempus per 'ad' seu per 'xb' est 'af'; quod si tempus per 'xb' esset 'xb', tempus per 'xa' esset media inter 'bx', 'xa', seu 'ad', 'db', quae est 'fd', tempus vero per reliquam 'ab' esset excessus 'ad' super 'df': sed quia tempus per totam 'xb' non est 'xb' sed 'af', oportet facere ut 'xb' seu 'da' ad 'af', ita 'df' ad aliam quae est 'fb', <DELETION-1> et 'ha' erit tempus per 'ab' ex 'x' </DELETION-1> et excessus ipsius 'af' super hanc 'fb' erit tempus per 'ab'; est igitur 'ah'</s></p>
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280 <pb xlink:href="X01/01/087.jpg" page="074_R"/>
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281 <pb xlink:href="X01/01/088.jpg" page="074_V"/>
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282 <p><s>Propositio.</s></p>
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283 <p><s>Dato plano inclinato et perpendiculo, quorum eadem sit elevatio, punctum sublime in perpendiculo extenso reperire, ex quo mobile decidens, et per planum inclinatum conversum, utrumque conficiat tempore eodem ac solum planum inclinatum ex quiete in eius superiori termino.</s></p>
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284 <pb xlink:href="X01/01/089.jpg" page="075_R"/>
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285 <p><s>Dato quolibet plano non ascendente, <DELETION-1> post </DELETION-1> perpendiculum ei adiu[n]gere, quod conficiatur eodem tempore <DELETION-1> <ILLEGIBLE/> </DELETION-1> ac ipsum planum datum post casum in perpendiculo. Constat autem, quod si datum planum fuerit orizontale, perpendiculum additum erit plani dimidium; si vero datum fuerit perpendiculare, adiunctum perpendiculum erit pars <SYMBOL-3/>_[terti]a. De horizontali demonstratum iam est, tempus quo talia 2 spatia conficiuntur esse omnium brevissimum.</s></p>
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286 <p><s>Sit datum planum, primo, perpendiculare 'ab', cui oporteat addere partem, quae ex quiete conficiatur tempore eodem ac perpendiculum 'ab' post ipsum additum.</s></p>
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287 <p><s><DELETION-1> faciendum </DELETION-1> Si tempus 'ca' debet esse aequale tempori 'ab', ergo totum tempus 'cb' erit <SYMBOL-2/>_[du]plum temporis 'ca': sed tempus per 'bc' ad tempus per 'ca' est ut media inter 'bc', 'ca': ergo media inter 'bc', 'ca' debet esse <SYMBOL-2/>_[du]pla 'ac'. <DELETION-1> sed si </DELETION-1> Posito igitur quod 'cd' sit media <DELETION-1> et <SYMBOL-2/>_[du]pla </DELETION-1> inter 'bc', 'ca', cum sit ut 'bc' ad 'cd', ita 'dc' ad 'ca', erit etiam ita 'bd' ad 'da'; ergo 'bd' debet esse <DELETION-1> media </DELETION-1> <SYMBOL-2/>_[du]pla 'da'. Posita ergo 'bd' <SYMBOL-2/>_[du]pla 'da' et 'dc' <SYMBOL-2/>_[du]pla 'ca', idest 'ac' aequali 'ad', factum erit etc.</s></p>
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288 <p><s>Modo non <INSERTION-1> sit </INSERTION-1> 'ab' perpendicularis, sed inclinata: constat similiter, tempus per 'cab' futurum esse <SYMBOL-2/>_[du]plum temporis per 'ca', seu per 'ab' post 'ca'. <DELETION-1> Ponatur 'ad' pars <SYMBOL-3/>_[terti]a 'ab', et perpendicularis 'dc' occurrat 'ac': tempus 'da' aequatur tempori 'ab', sed tempus quoque per 'ca' aequatur tempori per 'da'. </DELETION-1> </s></p>
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289 <pb xlink:href="X01/01/090.jpg" page="075_V"/>
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290 <p><s>ut 'cb' cum 'be' ad 'be', ita fiat <ILLEGIBLE/> 'rg' ad 'cb', et ut 'rg' cum 'eb' ad 'cb', ita fiat 'cb' ad 'ba': erit 'ba' quaesitum.</s></p>
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291 <pb xlink:href="X01/01/091.jpg" page="076_R"/>
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292 <p><s>Dum tempus per 'ab' sit 25, per 'db' est 53, et per 'dc' erit media inter 'bd', 'dc', nempe 'de' quae est 70 1/5; 'be' vero, quae est 17 1/5, erit tempus per 'bc'. Cum autem 'ba', 25, sit tempus per 'ab', posita in perpendiculo 'bf' aequali 'be', cuius partis perpendiculi erit tota 'af' tempus ? Sume ipsarum 'ba', 'af' <SYMBOL-3/>_[terti]am proportionalem 'ag', quae est 71 1/13.</s></p>
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293 <p><s>Pro invenienda igitur 'bg', sumenda est 'de' media inter 'bd', 'dc'; postea faciendum est, ut 'ba' ad <SYMBOL-2/>_[du]plam 'ab' cum 'be', ita 'be' ad <DELETION-1> 'bf' </DELETION-1> 'bg', seu, permutando, ut <SUBSTITUTION-1> 'ba' <SUBSTITUTED-BY-1> 'ab' </SUBSTITUTED-BY-1></SUBSTITUTION-1> ad 'be', ita <SYMBOL-2/>_[du]pla 'ba' cum 'be' ad 'bg'; et erit 'bg' quod quaeritur.</s></p>
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294 <p><s>Facilius et clarius. Dato perpendiculo 'ab' et plano ad ipsum inflexo 'bc', oportet in perpendiculo partem infra reperire, quae cum 'ab' conficiatur tempore eodem ac 'bc' cum eadem 'ab'. Ducta horizontali 'ad', extendatur 'cb' in 'd', et sit 'de' media 'bdc', cui ponatur aequalis 'bf'; et 'ba', 'af' <SYMBOL-3/>_[terti]a sit 'ag': erit 'bg' quod quaeritur. Posito enim 'db' tempore per 'db', erit 'ab' tempus per 'ab': et 'de' sit media 'bdc'; erit 'be' tempus 'bc' post 'db', idest post 'ab': sed eadem 'be', idest 'bf', est tempus 'bg' post 'ab': ergo</s></p>
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295 <p><s>Si ex puncto 'b' sumantur 'bc', 'bl', quae conficiantur tempore eodem, dico, ex quolibet puncto sublimi, ut 'a', citius confici 'abc' quam <DELETION-1> 'abf' </DELETION-1> 'abl'. Sed si ponatur 'bs' aequalis 'bc', citius 'abs' quam 'abc'. Potest tamen sumi 'a' adeo altum, ut ex eo citius conficiatur <DELETION-1> <ILLEGIBLE/> </DELETION-1> 'bc' cum eo quam alia maior quam 'bs', <INSERTION-1> licet minimum quid. </INSERTION-1> </s></p>
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296 <p><s><DELETION-2> Esse autem 'bf' semper maiorem quam 'be', sic probatur. Quia <SYMBOL-RECTANGULUM/>_[rectangulum] <SUBSTITUTION-1> 'lbe' <SUBSTITUTED-BY-1> 'lba' </SUBSTITUTED-BY-1></SUBSTITUTION-1> aequatur <SYMBOL-RECTANGULUM/>_[rectangul]o 'cbd', est autem 'lb' maior 'cb', ergo 'bd' maior 'ba': media autem inter 'cbd' est aequalis mediae 'lba': ergo, dempta 'ba' a media 'lba', reliqua 'bf' erit maior reliqua 'be', residuum mediae 'cbd', dempta 'bd'. <INSERTION-1> Adverte melius quid sequatur si mediae non sint minores ipsis 'db'. Dubito de paralogismo. </INSERTION-1> </DELETION-2> </s></p>
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297 <pb xlink:href="X01/01/092.jpg" page="076_V"/>
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298 <p><s>Quod erit si <SYMBOL-RECTANGULUM/>_[rectangulum] 'abi' sit aequale <SYMBOL-RECTANGULUM/>_[rectangul]o 'cal', id autem erit <UNDERLINED> si <SYMBOL-RECTANGULUM/>_[rectangulum] 'acl' cum <SYMBOL-QUADRATUM/>_[quadrat]o 'cb' sint aequalia <DELETION-1> <SYMBOL-QUADRATUM/>_[quadrat]o </DELETION-1> <SYMBOL-RECTANGULUM/>_[rectangul]o 'bai'; </UNDERLINED> si <SYMBOL-RECTANGULUM/>_[rectangulum] 'abi' sit aequale <SYMBOL-RECTANGULUM/>_[rectangul]o 'nai'</s></p>
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299 <p><s>Quia ut 'ab' ad 'bn', <DELETION-1> ita 'ai' ad </DELETION-1> ita 'ai' ad 'in', erit, permutando, ut 'ba' ad 'ai', ita 'bn' ad 'ni'.</s></p>
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300 <p><s>Quaeritur 'z', ex quo tempora 'zab', 'ab' sint aequalia.</s></p>
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301 <p><s>Modo videndum numquid tempus per 'oab' sit brevissimum, et primo qu[a]erere quantum sit posita 'oa' 703 1/8.</s></p>
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302 <p><s>Ponamus modo esse 'oa' 680, erit 'ax'</s></p>
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303 <pb xlink:href="X01/01/093.jpg" page="077_R"/>
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304 <p><s>Quia 'bi' est media 'cax', si ponatur 'ca' tempus per 'ac', erit 'bi' tempus per 'xa', et tempus per 'ra' erit illud ad quod tempus 'bi' sit ut 'xa' ad 'ar', seu 'ca' ad 'ab'. Vide igitur num ut 'ca' ad 'ab', ita sit 'bi' ad 'ia', seu ut 'ca' ad 'bi', ita sit 'ba' ad 'ai', seu ut 'an' ad 'bi', ita 'ba' ad 'ai', seu ut 'na' ad 'ab', ita 'bi' ad 'ia', seu ut 'ba' ad 'an', ita sit 'ai' ad 'ib', seu ut 'ha' ad 'ac', ita sit 'la' ad 'ib'; quod erit si <SYMBOL-RECTANGULUM/>_[rectangulum] 'abi' sit aequale <SYMBOL-RECTANGULUM/>_[rectangul]o 'cal' <DELETION-1> <ILLEGIBLE/> </DELETION-1> </s></p>
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305 <p><s>seu si <SYMBOL-RECTANGULUM/>_[rectangulum] 'aib' cum <SYMBOL-QUADRATUM/>_[quadrat]o 'ib', sit [a]equale <SYMBOL-RECTANGULUM/>_[rectangulo] 'nia' cum <SYMBOL-QUADRATUM/>_[quadrat]o 'ia'</s></p>
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306 <p><s>seu si (dempto communi <SYMBOL-RECTANGULUM/>_[rectangul]o 'nia'), <SYMBOL-RECTANGULUM/>_[rectangulum] 'bnia' cum <SYMBOL-QUADRATUM/>_[quadrat]o 'bi' sit equale <SYMBOL-QUADRATUM/>_[quadrat]o 'ia';</s></p>
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307 <p><s><MARK-FOR-INSERTION/> <SYMBOL-RECTANGULUM/>_[rectangul]o 'cla' cum <SYMBOL-QUADRATUM/>_[quadrat]o 'la', seu si <SYMBOL-RECTANGULUM/>_[rectangulum] 'aib' cum <SYMBOL-RECTANGULUM/>_[rectangul]o 'cax', sit [a]equale <SYMBOL-RECTANGULUM/>_[rectangul]o 'cld' cum <SYMBOL-QUADRATUM/>_[quadrat]o 'la'</s></p>
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308 <pb xlink:href="X01/01/094.jpg" page="077_V"/>
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309 <p><s>reperiatur altitudo 'n' ex qua conficiantur 'nab' et 'ab' sola tempore eodem; manifestum est quod ex omnibus punctis inter 'n', 'a' tempus per ambas lineas est brevius: quaeratur num tempus brevissimum sit in medio lineae 'na'. Videtur respondere.</s></p>
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310 <pb xlink:href="X01/01/095.jpg" page="078_R"/>
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311 <p><s>Scritta</s></p>
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312 <p><s>Sit horizontale planum, <DELETION-1> 'bd' </DELETION-1> in quo datum sit quodlibet spatium 'bc', et ex termino 'b' sit perpendiculum 'be', in quo 'ba' sit dimidium 'bc'. Dico, tempus quo mobile ex 'a' demissum conficiet ambo spatia 'ab', 'bc', esse temporum omnium brevissimum, quibus idem spacium 'bc' cum parte perpendiculi, sive maiori, sive minori parte 'ba', conficeretur.</s></p>
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313 <p><s>Sit <DELETION-1> primum </DELETION-1> sumpta maior, <INSERTION-1> vel minor </INSERTION-1> 'eb'. Ostendendum est tempus quo conficiuntur spatia 'eb', 'bc', longius esse tempore quo conficiuntur 'ab', 'bc'. Intelligatur, tempus per 'ab' esse ut 'ab'; erit quoque tempus motus in horizontali 'bc', cum 'bc' <SYMBOL-2/>_[du]pla sit ad 'ab', et per ambo spatia 'abc' tempus erit ut 'bc'. Sit 'bo' media inter 'eb', 'ba'; erit 'bo' tempus casus per 'eb'. Sit praeterea <DELETION-1> <ILLEGIBLE/> </DELETION-1> horizontale spatium 'bd' duplum ipsius 'be'; constat, tempus ipsius post casum 'eb' esse idem 'bo'. Fiat ut <SUBSTITUTION-1> 'db' <SUBSTITUTED-BY-1> 'de' </SUBSTITUTED-BY-1></SUBSTITUTION-1> ad 'ba', ita 'ob' ad <SUBSTITUTION-1> 'x' <SUBSTITUTED-BY-1> 'bn', </SUBSTITUTED-BY-1></SUBSTITUTION-1> et cum motus in horizontali sit aequabilis, sitque 'ob' tempus per 'bd' post casum ex 'e': erit <SUBSTITUTION-1> 'x' <SUBSTITUTED-BY-1> 'nb' </SUBSTITUTED-BY-1></SUBSTITUTION-1> tempus per 'bc' post casum ex eadem altitudine 'e'. Ex quo constat, 'ob' cum <SUBSTITUTION-1> 'x' <SUBSTITUTED-BY-1> 'bn' </SUBSTITUTED-BY-1></SUBSTITUTION-1> esse tempus per 'ebc'; cumque 'bc', <INSERTION-1> nempe dupla 'ba', </INSERTION-1> sit tempus per 'abc'. Ostendendum relinquitur 'ob' cum <SUBSTITUTION-1> 'x' <SUBSTITUTED-BY-1> 'bn' </SUBSTITUTED-BY-1></SUBSTITUTION-1> maiora esse quam <DELETION-1> 'bx' </DELETION-1> dupla 'ba'. <DELETION-1> Cum autem 'db' <SYMBOL-2/>_[du]pla sit 'be', et 'cb' <SYMBOL-2/>_[du]pla 'ba', erit ut </DELETION-1> Cum autem 'ob' media sit inter 'eb', 'ba', ratio 'eb' ad 'ba' dupla est rationis 'ob' ad 'ba'; et cum 'eb' ad 'ba' sit ut 'ob' ad 'bn', ratio autem 'be' ad 'ba' <SYMBOL-2/>_[du]pla sit rationis 'ob' ad 'ba', erit quoque ratio 'ob' ad 'bn' <SYMBOL-2/>_[du]pla rationis 'ob' ad 'ba': verum ipsa ratio 'ob' ad 'bn' componitur ex rationibus 'ob' ad 'ba' et 'ab' ad 'bn': ergo ratio 'ab' ad 'bn' est eadem cum ratione 'ob' <ILLEGIBLE/> ad 'ba'. Sunt igitur 4 'eb', 'bo', 'ba', 'bn' continue proportionales, et 'ob' cum 'bn' maiores quam <SYMBOL-2/>_[du]pla 'ba'.</s></p>
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314 <p><s><foreign lang="it">S[igno]r Galileo, Il S[igno]r Cav[alie]r[e] Guidoni è qui, e domattina andremo alla Certosa a desinare. Si cita però V[ostra] S[ignoria] a comparire in detto luogo alle 17 hore in circa, sotto pena di star senza desinare e di non haver l'olio che ella desidera. E le bacio le mani. <lb/> Dalle Rose à X di Gennaio 1630 <lb/> Di V[ostra] S[ignoria] Dev[otissi]mo Ser[vito]r <lb/> Niccolò Cini.</foreign></s></p>
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315 <pb xlink:href="X01/01/096.jpg" page="078_V"/>
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316 <p><s>Invenienda est in 'ac' pars aequalis ipsi 'ab', quae conficiatur eodem tempore quo ipsa 'ab'.</s></p>
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317 <p><s>'ab' longa 4; 'bc' 8 tempus per 'ab' 4; ergo 'abc' longitudo 12 conficitur in tempore 8.</s></p>
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318 <p><s>Sit 'db' 9; erit eius tempus 6, et ex <INSERTION-1> 'd' </INSERTION-1> conficietur 'db' 9 cum 'bce' 18 tempore 12, nempe 27 in tempore 12. Sed si 'be' conficitur (cum sit 18) tempore 6, 'bc' conficietur tempore 2 2/3, et tota 'dbc' in tempore 8 2/3: conficiuntur ergo citius 'abc' quam 'dbc'.</s></p>
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319 <p><s>Sit 'bd' 8; conficietur tempore radicis 32, idest 5 1/2, et erit 'be' 16, et 'dbc' conficietur tempore <DELETION-1> 8 1/4, 7 3/4 </DELETION-1> 8 1/4.</s></p>
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320 <p><s><foreign lang="it"><ILLEGIBLE/> Galileo Galilei <lb/> mio <ILLEGIBLE/></foreign></s></p>
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321 <pb xlink:href="X01/01/097.jpg" page="079_R"/>
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322 <p><s>Scritta. Dato perpendiculo et plano inclinato, quorum eadem sit elevatio, partem in inclinato reperire, quae sit aequalis perpendiculo et conficiatur eodem tempore.</s></p>
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323 <p><s>Sint 'ac', 'ab', et ponatur 'ad' aequalis 'ab', et reliqua 'dc' bifariam secetur in 'i'; et ut 'ac' ad 'ci', ita sit 'ci' ad 'ae', cui ponatur aequalis 'dg': erit 'eg' aequalis 'ad' et 'ab'. <DELETION-1> <ILLEGIBLE/> </DELETION-1> <SUBSTITUTION-2> hanc dico <INSERTION-1> insuper </INSERTION-1> <SUBSTITUTED-BY-2> Dico <INSERTION-1> insuper, </INSERTION-1> hanc </SUBSTITUTED-BY-2></SUBSTITUTION-2> confici tempore aequali tempori 'ab'. Quia, enim, ut 'ac' ad 'ci', ita 'ci' ad 'ae', seu <DELETION-1> 'gd' </DELETION-1> 'id' ad 'dg', erit, per conversionem rationis, ut 'ca' ad 'ai', ita 'di' ad 'ig', seu 'ci' ad 'ig': cum itaque sit ut totum 'ca' ad totum <INSERTION-1> 'ai', </INSERTION-1> ita ablatum 'ci' ad ablatum 'ig', erit reliquum 'ia' ad reliquum 'ag' ut totum 'ca' ad totum 'ai'. Est itaque 'ai' media inter 'ca', 'ag', <DELETION-1> et </DELETION-1> et 'ci' media inter 'ca', 'ae'. Si itaque ponatur, tempus per 'ab' esse 'ab', erit 'ac' tempus per 'ac', et 'ci', <INSERTION-1> seu 'id', </INSERTION-1> tempus per 'ae'. Cumque 'ai' media sit 'cag', sitque 'ca' tempus per totam 'ac'; erit 'ai' tempus per 'ag', et reliquum 'ic' per reliquum 'gc': fuit autem 'di' tempus per 'ae': <INSERTION-1> sunt itaque 'di', 'ic' tempora per utrasque 'ae', 'cg': </INSERTION-1> ergo reliquum 'da' erit tempus 'eg', aequale nempe tempori per 'ab'.</s></p>
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324 <p><s>Corollarium.</s></p>
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325 <p><s>Ex his constat, spatium quaesitum mediare inter partes superam et <SUBSTITUTION-1> infam <SUBSTITUTED-BY-1> inferam, </SUBSTITUTED-BY-1></SUBSTITUTION-1> quae temporibus aequalibus conficiuntur.</s></p>
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326 <p><s>'ai' 'ic'</s></p>
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327 <pb xlink:href="X01/01/098.jpg" page="079_V"/>
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328 <p><s>Spatia motus accelerati ex quiete et spatia motuum aequabilium ad motus acceleratos consequentia, <INSERTION-1> et temporibus iisdem confecta, </INSERTION-1> eandem inter se retinent rationem: sunt enim haec spatia dupla illorum. Tempora vero et gradus velocitatum acquisitarum eandem inter se habent rationem: haec tamen ratio subdupla est rationis spatiorum dictorum.</s></p>
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329 <p><s>Spatia motus accelerati 'ab', 'ac' et motuum aequabilium consequentium 'be', 'cd' eandem cum illis habent rationem: sunt enim dupla illorum. Tempora per 'ab', 'ac' sunt inter se ut gradus velocitatis in 'b' et in 'c': ratio vero haec subdupla est rationis 'ba' ad 'ac' vel 'be' ad 'cd'.</s></p>
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330 <pb xlink:href="X01/01/099.jpg" page="080_R"/>
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331 <p><s>In <SYMBOL-3/>_[tri]ang[ul]o rectangulo 'bcd' fiat angulo 'd' aequalis angulus 'cbe', et iungatur 'eb': erunt ergo 2 <SYMBOL-3/>_[tri]ang[ul]a 'dcb', 'ebc' similia. Dividatur tota 'dc' bifariam in 'h' et parallela 'hi' sit ipsi 'cb'. Dividatur pariter 'ec' bifariam in 'f', et ducatur 'fg' parallela 'bc', et fiat ut 'dh' ad 'hi' ita 'hi' ad 'hl', et iungatur 'li': erit <SYMBOL-3/>_[tri]ang[ul]um 'lih' simile <SYMBOL-3/>_[tri]ang[ul]o 'dhi', et ob id simile quoque ipsi 'efg'; sed 'hi' est aequalis 'gf' <INSERTION-1> (utriusque enim dupla est 'bc'), </INSERTION-1> ergo reliqua latera 'hl', 'fe' aequalia erunt; quare ter[t]ia proportionalis ipsarum 'lh', 'hi', <INSERTION-1> nempe </INSERTION-1> 'hd', erit aequalis <SYMBOL-3/>_[tertia]e proportionali ipsarum 'ef' et 'fg'. Sed <DELETION-1> 'hd' </DELETION-1> <SYMBOL-3/>_[terti]a proportionalis ipsarum 'lh', 'hi' est 'hd', dimidia nempe totius 'dc': ergo <SYMBOL-3/>_[terti]a proportionalis ipsarum 'ef', 'fg' aequabitur dimidiae 'cd', nempe ipsi 'ch'.</s></p>
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332 <p><s>Sed 'ch' est aequalis 'fl', cum 'cf' sit aequalis 'hl', et 'fh' communis: ergo <SYMBOL-3/>_[terti]a proportionalis ipsarum <DELETION-1> ipsarum </DELETION-1> 'ef', 'fg' erit 'fl', terminata in puncto 'l', ubi terminatur <SYMBOL-3/>_[terti]a proportionalis ipsarum 'dh', 'hi'. Ex hoc demonstrabitur proiectorum secundum elevationes a semirecta per angulos [a]equales factorum amplitudines parabolarum esse aequales.</s></p>
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333 <pb xlink:href="X01/01/100.jpg" page="080_V"/>
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334 <p><s><foreign lang="it">Sia 'ab' (50) tempo et impeto di 'ab'; tempo et impeto di 'yb' (68) è la media di 68 e 50, cioè <DELETION-1> 62 <ILLEGIBLE/> </DELETION-1> 58 4/11.</foreign></s></p>
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335 <pb xlink:href="X01/01/101.jpg" page="081_R"/>
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336 <p><s>Altitudines a 'ab' mensurate sunt cum scala maiorum transversalium 'bc'.</s></p>
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337 <pb xlink:href="X01/01/102.jpg" page="081_V"/>
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338 <pb xlink:href="X01/01/103.jpg" page="082_R"/>
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339 <p><s>Parabolarum quarum anguli Elevationum aequaliter differunt a semirecto <DELETION-1> in </DELETION-1> </s></p>
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340 <p><s>Scritta <lb/> In <SYMBOL-3/>_[tri]ang[ul]o rectangulo 'bcm' sint latera 'mc', 'cb' aequalia, et supra et infra lineam 'mb' constituantur <INSERTION-1> utcumque </INSERTION-1> anguli aequales 'mbe', 'mbd'; et quia semirectus 'm' aequatur internis 'd', 'b' <SYMBOL-3/>_[tri]ang[ul]i 'mdb', iisdem aequabitur alter semirectus 'mbc'. Quod si loco 'dbm' sumatur 'mbe', erit idem semirectus 'mbc' aequalis duobus 'd', 'mbe', et dempto communi 'mbe', erit reliquus 'ebc' aequalis ipsi 'd'. Dividantur 'cd', 'ce' bifariam in 'h' et 'f', et ducantur 'hi', 'fg' parallelae ipsi 'cb', et ut 'dh' ad 'hi', ita fiat 'ih' ad 'hl': erit <SYMBOL-3/>_[tri]ang[ul]us 'ihl' similis <SYMBOL-3/>_[tri]ang[ul]o 'ihd', cui etiam similis est 'egf'; cumque 'ih', 'gf' sint aequales (dimidiae nempe ipsius 'bc'), erit 'fe', <INSERTION-1> id est 'fc', </INSERTION-1> aequalis 'hl'. Et addita communi 'fh', erit 'ch' ipsi 'fl' aequalis. Si itaque intelligamus, per 'h' et 'b' parabolam esse descriptam, cuius altitudo erit 'hc', sublimitas vero 'hl', erit eius amplitudo 'cb', quae dupla est ad 'hi', media scilicet inter 'dh', seu 'ch', et 'hl'; eamque tanget 'db', aequalibus existentibus 'ch', 'hd'. Quod si, rursus, parabolam per 'fb' descriptam concipiamus <ILLEGIBLE/> a sublimitate 'fl' cum altitudine 'fc', quarum media <INSERTION-1> proportionalis </INSERTION-1> est 'fg', cuius <SYMBOL-2/>_[du]pla est orizontalis 'cb', erit pariter 'cb' eius amplitudo, illamque tanget 'eb', cum 'ef', 'fc' aequales sint. Distant autem anguli 'dbc', 'ebc' (elevationes scilicet ipsarum) aequaliter a semirecto 'mbc': ergo patet propositum.</s></p>
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341 <pb xlink:href="X01/01/104.jpg" page="082_V"/>
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342 <p><s><foreign lang="it">'ab' 60 tempo <lb/> 'bc' 24; suo tempo 38; <lb/> tutto il tempo per 'ab' e per la parabola <DELETION-1> 113 </DELETION-1> 98.</foreign></s></p>
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343 <p><s><foreign lang="it">'ae' 113 1/3 <lb/> 'af' 81</foreign></s></p>
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344 <p><s><foreign lang="it"><BEGIN-OF-LINE-DESTROYED/> 60 tempo <lb/> <BEGIN-OF-LINE-DESTROYED/> 20 tempo 34 5/8 <lb/> <BEGIN-OF-LINE-DESTROYED/> <SUBSTITUTION-1> 89 1/4 <SUBSTITUTED-BY-1> 69 1/4 </SUBSTITUTED-BY-1></SUBSTITUTION-1> <lb/> 106 scarso <lb/> <SUBSTITUTION-1> 90 <SUBSTITUTED-BY-1> 79 1/2 </SUBSTITUTED-BY-1></SUBSTITUTION-1> <BEGIN-OF-LINE-DESTROYED/> tra 106 e <SUBSTITUTION-1> 90 <SUBSTITUTED-BY-1> 79 1/2 </SUBSTITUTED-BY-1></SUBSTITUTION-1> <SUBSTITUTION-1> 104 <SUBSTITUTED-BY-1> 91 7/9 </SUBSTITUTED-BY-1></SUBSTITUTION-1> tempo per 'ae' <lb/> <BEGIN-OF-LINE-DESTROYED/> tempi per 'ab' e la parabola mostrano 104 5/8 <lb/> <BEGIN-OF-LINE-DESTROYED/> <BEGIN-OF-WORD-DESTROYED/>_?ina di nuovo</foreign></s></p>
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345 <pb xlink:href="X01/01/105.jpg" page="083_R"/>
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346 <pb xlink:href="X01/01/106.jpg" page="083_V"/>
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347 <p><s>Parabola 'bd' describitur ab elevatione 'ab' cum altitudine 'bc'. Ponatur, 'ab' esse tempus et impetum casus 'ab', sitque 'de' tangens parabolam: erit 'eb' aequalis 'bc': cumque 'bf' sit subdupla amplitudinis 'cd', erit quoque media inter sublimitatem 'ab' et altitudinem 'bc', eritque tempus casus et impetus per 'bc' in 'c'. Iuncta igitur 'af', erit mensura impetus in 'd' cadentis per 'abd'.</s></p>
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348 <p><s>Attende <SUBSTITUTION-1> quod <SUBSTITUTED-BY-1> numquid </SUBSTITUTED-BY-1></SUBSTITUTION-1> tempus et impetus per 'ab' cum parabola 'bd' est idem cum tempore et impetu per inclinatam 'ad'.</s></p>
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349 <p><s>sit 'ab' 80 'bc' 60</s></p>
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350 <p><s><ILLEGIBLE/> 'af' <END-OF-LINE-DESTROYED/> m?_<END-OF-WORD-DESTROYED/> <END-OF-LINE-DESTROYED/> ele[vatio] <END-OF-LINE-DESTROYED/> Es?_<END-OF-WORD-DESTROYED/> <END-OF-LINE-DESTROYED/></s></p>
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351 <pb xlink:href="X01/01/107.jpg" page="084_R"/>
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352 <p><s>Scritta <lb/> Tangat horizontalis 'ag' circulum, et a contactu sit diameter 'ab', et 2 cordae utcunque 'aeb'. Determinanda sit ratio temporis casus per 'ab' ad tempus descensus per ambas 'aeb'. Extendatur 'be' usque ad tangentem in 'g', et angulus 'bae' bifariam secetur, ducta 'af': dico, tempus per 'ab' ad tempus per 'aeb' esse ut 'ae' ad 'aef'. Cum enim angulus 'fab' aequalis sit angulo 'fae', angulus vero 'eag' angulo 'abf', erit totus 'gaf' <SYMBOL-2/>_[du]obus 'fab', 'abf' aequalis; quibus aequatur quoque angulus 'gfa'; ergo linea 'gf' ipsi 'ga' est aequalis. Et quia <SYMBOL-RECTANGULUM/>_[rectangulum] 'bge' aequatur <SYMBOL-QUADRATUM/>_[quadrat]o 'ga', erit quoque aequale <SYMBOL-QUADRATUM/>_[quadrato] 'gf', et 3 lineae 'bg', 'gf', 'ge' proportionales. Quod si ponatur, 'ae' esse tempus per 'ae', erit 'ge' tempus per 'ge', et 'gf' tempus per totam 'gb', et 'ef' tempus per 'eb', post descensum ex 'g' seu ex 'a' per 'ae': tempus igitur per 'ae' (seu per 'ab') ad tempus per <DELETION-1> 'aef' </DELETION-1> 'aeb' est ut 'ae' ad 'aef'.</s></p>
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353 <p><s>Aliter brevius. Secetur 'gf' aequalis 'ga'; constat, 'gf' esse mediam inter 'bg', 'ge'. Reliqua ut supra.</s></p>
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354 <pb xlink:href="X01/01/108.jpg" page="084_V"/>
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355 <p><s>Secta 'ca' utcumque in 'd', <SUBSTITUTION-1> reliqua <SUBSTITUTED-BY-1> pars </SUBSTITUTED-BY-1></SUBSTITUTION-1> vero 'cd' bifariam in 'i', dico quod si fiat ut tota 'ac' ad 'ce', ita 'id' ad 'dg', erit ut 'ca' ad 'ai', ita 'ia' ad 'ag'.</s></p>
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356 <p><s>Si totum 'ca' ad totum 'ai' est ut ablatum 'ia' ad ablatum 'ag', erit reliquum 'ci' ad reliquum 'ig', idest reliquum 'di' ad reliquum 'ig', ut totum 'ca' ad 'ai', seu 'ia' ad 'ag', et, per conversionem rationis, ut 'ac' ad 'ci', ita 'id' ad 'dg', seu 'ci' ad 'dg', idest ad 'ae': sed ita factum est. Componitur itaque: quia 'ci' ad 'ae', idest 'id' ad 'dg', est ut 'ac' ad 'ci', erit, per conversionem rationis, ut 'ca' ad 'ai', ita 'di' ad 'ig', seu 'ci' ad 'ig'; cum itaque sit ut totum 'ca' ad totum 'ai', ita ablatum 'ci' ad ablatum 'ig', erit etiam reliquum 'ia' ad reliquum 'ag' ut totum 'ca' ad totum 'ai': quod erat ostendendum.</s></p>
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357 <p><s>Faciendum ut 'ai' ad 'ig', ita 'ig' ad 'gd'. Ponatur 'ic' aequalis 'id', et fiat ut 'ac' ad 'ci', ita 'id' ad 'dg': erit, per conversionem rationis, ut 'ca' ad 'ai', ita 'di' ad 'ig', seu 'ci' ad 'ig'; et cum ut totum 'ca' ad totum 'ai', ita ablatum 'ci' ad ablatum 'ig', erit reliquum 'ia' ad reliquum 'ag' ut ablatum 'ci', seu 'di' ad 'ig', <DELETION-1> et dividendo </DELETION-1> et, per conversionem rationis, ut 'ai' ad 'ig', ita 'id' ad 'dg'.</s></p>
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358 <p><s>quia, ut 'ac' ad 'ce', ita 'ce' ad 'ed', componendo 'ae' ad 'ec', 'cd' ad 'de' 'ea', 'ac' 'dc', 'ce'</s></p>
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359 <pb xlink:href="X01/01/109.jpg" page="085_R"/>
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360 <p><s>Dato spacio, in quovis tempore peracto in perpendiculo, <DELETION-1> dataque proportione quacunque alterius spacii ad spacium peractum in perpendiculo, quae tamen maior sit quam dupla, minor vero quam <SYMBOL-3/>_[tri]pla <ILLEGIBLE/> </DELETION-1> ex termino huius spacii planum inflectere, super quo, tempore eodem, conficiatur spacium cuilibet spacio dato [a]equale, quod tamen spacii peracti in perpendiculo maius sit quam duplum, minus vero quam <SYMBOL-3/>_[tri]plum.</s></p>
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361 <p><s>Sit in perpendiculo 'as' tempore 'ac' peractum spacium 'ac', cuius 'ir' maius sit quam duplum, minus vero quam <SYMBOL-3/>_[tri]plum: oportet, ex termino 'c' planum inflectere super quo mobile eodem tempore 'ac' conficiat spacium ipsi 'ir' aequale. Sint 'rn', 'nm' ipsi 'ac' aequalia, et quam rationem habet 'im' ad 'mn' eandem habeat <DELETION-1> <ILLEGIBLE/> </DELETION-1> <INSERTION-1> 'ac' </INSERTION-1> ad aliam, cui aequalis sit 'ce', <INSERTION-1> orizonti 'ae' occurrens in 'e', </INSERTION-1> quae extendatur versus 'g', et accipiantur 'cf', 'fg', <INSERTION-1> 'go' </INSERTION-1> aequales ipsis 'rn', 'nm', <INSERTION-1> 'mi': </INSERTION-1> dico, tempus super inflexa <SUBSTITUTION-1> 'go' <SUBSTITUTED-BY-1> 'co' </SUBSTITUTED-BY-1></SUBSTITUTION-1> esse aequale tempori 'ac'. Cum enim sit ut 'og' ad 'gf', ita 'fc' ad 'ce', erit, componendo, ut 'of' ad 'fg', seu 'fc', ita 'fe' ad <DELETION-1> 'og' </DELETION-1> 'ec', et ut <DELETION-1> <ILLEGIBLE/> </DELETION-1> unum antecedentium ad unum consequentium, ita omnia ad omnia, nempe ut tota 'oe' ad <DELETION-1> 'fe' </DELETION-1> 'ef', ita 'fe' ad 'ec'. Sunt itaque 'oe', 'ef', 'ec' continue proportionales. Quod <SUBSTITUTION-1> si potest <SUBSTITUTED-BY-1> cum positum sit, </SUBSTITUTED-BY-1></SUBSTITUTION-1> tempus per 'ac' esse ut 'ac', erit 'ce' tempus per 'ec', et 'cf' tempus per 'co'; est autem 'cf' aequale ipsi 'ca': ergo factum est quod facere oportebat.</s></p>
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362 <p><s>Hinc patet, quod quo magis 'oc' accedit ad <SYMBOL-3/>_[tri]plicitatem ipsius 'ca', eo planum 'co' vergit versus perpendiculum, in quo tandem spacium peractum tempore aequali tempori 'ac' est <SYMBOL-3/>_[tri]plum ipsius 'ac'; quo vero magis eadem 'oc' accedit ad <SYMBOL-2/>_[du]plicitatem eiusdem 'ca', eo planum 'oc' accedit ad aequidistantiam cum orizonte 'ae', in quo tandem cum desinet, spacium 'oc' peractum tempore 'ca' erit duplum spacii 'ac'.</s></p>
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363 <pb xlink:href="X01/01/110.jpg" page="085_V"/>
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364 <p><s><DELETION-1> Assumo, eam esse cadentis mobilis per lineam 'al' accelerationem, ut pro ratione spacii peracti crescat velocitas ita, ut velocitas in 'c' ad velocitatem in 'b' sit ut spacium 'ca' ad spacium 'ba', etc. </DELETION-1> </s></p>
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365 <p><s><DELETION-1> Cum autem haec ita se habeant, ponatur 'ax' cum 'al' angulum continens, sumptisque partibus 'ab', 'bc', 'cd', 'de' etc. aequalibus, protrahantur 'bm', 'cn', 'do', 'ep' etc. Si itaque cadentis per 'al' velocitates in 'b', 'c', 'd', 'e' locis se habent ut distantia[e] 'ab', 'ac', 'ad', 'ae' etc., ergo se quoque habebunt ut lineae 'bm', 'cn', 'do', 'ep'. </DELETION-1> </s></p>
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366 <p><s><DELETION-2> Quia vero velocitas augetur consequenter in omnibus punctis lineae 'ae', et non tantum in adnotatis 'b', 'c', 'd', ergo velocitates illae omnes sese respicient ut lineae quae ab omnibus dictis punctis lineae 'ae' ipsis 'bm', 'cn', 'do' aequidistanter producuntur. Istae autem infinitae sunt, et constituunt triangulum 'aep': ergo velocitates in omnibus punctis lineae 'ab' ad velocitates in omnibus punctis lineae 'ac' ita se habent ut triangulus 'abm' ad triangulum <SUBSTITUTION-1> 'ado' <SUBSTITUTED-BY-1> 'acn' </SUBSTITUTED-BY-1></SUBSTITUTION-1>, et sic de reliquis, hoc est in duplicata proportione linearum 'ab', 'ac'. </DELETION-2> </s></p>
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367 <p><s><DELETION-2> Quia vero pro ratione incrementi accelerationis tempora quibus motus ipsi fiunt debent imminui, ergo tempus quo <INSERTION-1> mobile </INSERTION-1> permeat 'ab' ad tempus quo permeat 'ac' erit ut 'ab' linea ad eam quae inter 'ab', 'ac' media proportionalis existit. </DELETION-2> </s></p>
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368 <p><s>10</s></p>
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369 <pb xlink:href="X01/01/111.jpg" page="086_R"/>
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370 <p><s>Sit parabola 'bd' cuius amplitudo 'dc' sit <SYMBOL-2/>_[du]pla ad altitudinem 'cb'. Examinandum ipsam describi a minori impetu quam reliquae omnes, quarum eandem [eadem] sit amplitudo, aliam [alia] vero altitudo. Esto enim parabola 'gd' quam tangat 'hd', et fiat ut 'hg' ad 'gk', ita <DELETION-1> g </DELETION-1> 'kg' ad 'gl': erit, ex antedemonstratis, altitudo 'lg', ex qua cadens describet parabolam 'gd'. Ponatur 'ab' esse mensuram temporis et impetus: erit igitur 'ae' momentum lati <DELETION-1> per </DELETION-1> ex 'a' per parabolam 'bd' in 'd'. </s></p>
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371 <p><s>Inter 'ab' et 'gl' media sit 'gm'; erit 'gm' tempus et momentum in 'g' cadentis ex 'l'. Sit rursus inter 'bc', 'cg' media 'gn', quae erit momenti velocitatis et temporis mensura cadentis ex 'g' in 'c': cum assumptum sit 'bc' <INSERTION-1> seu 'ba' </INSERTION-1> esse tempus et momentum per 'bc', si igitur iungatur 'mn' erit 'mn' impetus proiecti per parabolam 'gd' in 'd'. <DELETION-2> Reperietur autem 'mn' maiorem esse ipsa 'ae'. Sit rursus altera parabola ad quam tangat 'rd' secans orizontem in 'p', et ut ipsorum 'ro', 'op' <SYMBOL-3/>_[ter]tia proportionalis sit 'oq', constat cadens ex 'q' describeretur parabolam 'od'. Sit inter 'oq' et 'ba' <INSERTION-1> seu 'go', </INSERTION-1> media 'os': erit 'os' momentum cadentis ex 'q' in 'o'. Sit pariter inter 'ac', 'cb' media 'ox': erit 'ox' momentum ex 'o' in 'c'. Iungatur 'xs' quae erit impetus proiecti per parabolam 'od' in 'd', maior quoque <INSERTION-1> impetus </INSERTION-1> proiecti per 'bd', nempe 'qe' </DELETION-2> </s></p>
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372 <p><s>Esse autem 'mn' maiorem quam 'ae' sic probatur: quia enim, 'gn' posita est media <INSERTION-1> inter </INSERTION-1> 'bc', 'cg', est autem 'bc' aequalis 'be', hoc est 'kg', <INSERTION-1> est enim unaquaeque subdupla 'dc'; </INSERTION-1> erit ut 'cg' ad 'gn', ita 'ng' ad 'gk', et ut 'cg' seu 'hg' ad 'gk', ita <SYMBOL-QUADRATUM/>_[quadratum] 'ng' ad <SYMBOL-QUADRATUM/>_[quadratum] 'gk'; ut autem 'hg' ad 'gk', ita facta est 'kg' ad 'gl'; ergo ut <SYMBOL-QUADRATUM/>_[quadratum] 'ng' ad <SYMBOL-QUADRATUM/>_[quadratum] 'gk', ita 'kg' ad 'gl'. Sed ut 'kg' ad 'gl', ita <SYMBOL-QUADRATUM/>_[quadratum] 'kg' ad <SYMBOL-QUADRATUM/>_[quadratum] 'gm'; media enim est 'gm' inter 'kg', 'gl', ergo 3 <SYMBOL-QUADRATUM/>_[quadrata] 'ng', 'gk', 'gm' sunt proportionalia, et 2 extrema 'ng', 'gm', idest 'mn', maius quam duplum <SYMBOL-QUADRATUM/>_[quadrati] 'kg', cuius <SYMBOL-QUADRATUM/>_[quadratum] 'ae' duplum est: ergo <SYMBOL-QUADRATUM/>_[quadratum] 'mn' maius est <SYMBOL-QUADRATUM/>_[quadrato] 'ae' et linea 'mn' maior linea 'ea'.</s></p>
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373 <p><s>Hic determinantur parabolae eiusdem amplitudinis, sed <DELETION-1> la </DELETION-1> altitudinum diversarum.</s></p>
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374 <p><s>Scritta</s></p>
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375 <pb xlink:href="X01/01/112.jpg" page="086_V"/>
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376 <p><s><foreign lang="it">Tutta 'ac' 140, e tanto sia il tempo e l'impeto in 'c', il quale impeto è di passare 280 nel tempo 140.</foreign></s></p>
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377 <p><s><foreign lang="it">'ab' 80; sarà il suo tempo la media tra 'ac', 'ab', cioè tra 140 e 80, che è 105. E però nell'orizontale 'bg' la velocità sarà di passare, nel tempo 105 di 'ab', <DELETION-1> cioè </DELETION-1> 160, che è il doppio di 'ab'. Ma il tempo di 'bc' dalla quiete in 'b', e [è] la media tra 'ac' 140 e 'bc' 60, che è 91; adunque diremo: in <INSERTION-1> questo </INSERTION-1> tempo 91, quanto si passerà di 'bg', della quale nel tempo di 'ab' che è 105 se ne passa 160 per la regola se ne passerà 138: e torna bene, chè tanta è 'cd'.</foreign></s></p>
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378 <p><s><foreign lang="it">Sia 'ab' 80, tempo ed impeto in 'b', che nella 'bg' in tempo 80, passerà 160. Il tempo di 'bc' sarà la media tra 'bc' 60 e 'ab' 80, che sarà 69. In questo tempo 69, quanto si passerà in 'bg', dove in 80 di tempo si passa 160 ? <DELETION-1> <ILLEGIBLE/> </DELETION-1> Si passa 138 e torna bene.</foreign></s></p>
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379 <p><s><foreign lang="it">'ab' 60, tempo et impeto; 'bc' 30. Sarà suo tempo et impeto la media tra 60 e 30, che è 42 1/3; adunque tutto 'l tempo di 'abd' è 102 1/3. L'ampiezza 'cd' è doppia della media tra 'ab', 'bc': è dunque 84 2/3. Ma tutta 'ac' è 90, e 'cd' 84 2/3: adunque 'ad' sarà 123 et il tempo di tutta 'ad' sarà quanto la media tra 'da', 'ag', che torna 100 e più, e mostra star bene.</foreign></s></p>
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380 <p><s><foreign lang="it">se nel tempo 140 si passa 280</foreign></s></p>
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381 <pb xlink:href="X01/01/113.jpg" page="086AR"/>
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382 <pb xlink:href="X01/01/114.jpg" page="086AV"/>
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383 <pb xlink:href="X01/01/115.jpg" page="087_R"/>
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384 <p><s>Scritta <lb/> Si in planis inaequalibus, quorum eadem sit elevatio, descendat <DELETION-1> fiat motus </DELETION-1> mobile, spatium quod <DELETION-1> <ILLEGIBLE/> </DELETION-1> in ima parte longioris conficitur tempore aequali tempori quo conficitur totum planum brevius, <DELETION-1> ad planum brevius eandem habet rationem </DELETION-1> est aequale <SUBSTITUTION-1> plano <SUBSTITUTED-BY-1> spatio </SUBSTITUTED-BY-1></SUBSTITUTION-1> quod componitur ex ipso breviori plano et ex parte ad quam idem brevius planum eam habet rationem, quam habet planum longius ad excessum quo longius <DELETION-1> planum </DELETION-1> brevius superat.</s></p>
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385 <p><s>Sit planum 'ac' longius, 'ab' vero brevius, quorum eadem sit elevatio 'ad', et ex ima parte 'ac' sumatur <DELETION-1> <ILLEGIBLE/> </DELETION-1> 'ce' aequale ipsi 'ab', et quam rationem habet totum 'ca' ad <DELETION-1> 'ax' </DELETION-1> 'ae', nempe ad excessum plani 'ca' super 'ab', hanc habeat 'ce' ad 'ef'. Dico, spatium 'fc' esse illud quod conficitur ex discessu ex 'a' tempore aequali tempori descensus per 'ab'. Cum enim totum 'ca' ad totum 'ae' sit ut ablatum 'ce' ad ablatum 'ef', erit reliquum 'ea' ad reliquum 'af' ut totum 'ca' ad totum 'ae'; sunt itaque tres 'ca', 'ae', 'af' continue proportionales. Quod si ponatur, tempus per 'ab' esse ut 'ab', erit tempus per 'ac' ut 'ac'; tempus vero per 'af' erit ut 'ae', et per reliquum <SUBSTITUTION-1> <ILLEGIBLE/> <SUBSTITUTED-BY-1> 'fc' </SUBSTITUTED-BY-1></SUBSTITUTION-1> erit ut 'ec': est autem 'ec' ipsi 'ab' aequale: ergo patet propositum.</s></p>
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386 <p><s>Esset problema pulcrum, in 'ac' partem ipsi 'ab' aequalem signare, quae conficiatur ex quiete in 'a' tempore aequali tempori per 'ab' ex quiete in 'a'.</s></p>
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387 <p><s>Sint plana 'ab', 'ac', quorum eadem sit elevatio 'ad', longius tamen sit 'ac'. Dico, motum versus inferiores partes plani 'ac' velociorem esse quam per 'ab'. Accipiatur 'ec' aequale 'ab', et ut 'ca' ad 'ae', ita sit 'ea' ad 'af'. Intelligatur 'ab' esse tempus per 'ab': erit 'ac' tempus per 'ac', et 'ae' erit tempus per 'af', 'ec' vero tempus per reliquum 'fc'. Est autem 'fc' maius quam 'ab', et conficitur tempore aequali. <DELETION-1> Constat autem, spatium quod conficitur in 'ac', tempore descensus per 'ab', esse medium inter 'ca', 'ab'. </DELETION-1> </s></p>
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388 <pb xlink:href="X01/01/116.jpg" page="087_V"/>
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389 <p><s>Datae parabolae elevationem invenire, ex qua decidens mobile parabolam datam describat.</s></p>
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390 <p><s>Sit data parabola 'bf', cuius altitudo 'bi', amplitudo vero 'if'. Ducta orizontali 'bl', accipiatur in perpendiculo 'be' aequalis 'bi', et connectatur 'elf', quae parabolam tanget in 'f' et orizontalem secabit in 'l'; fiat ut 'eb' ad 'bl', ita 'lb' <DELETION-1> l </DELETION-1> ad 'ba'. Dico, 'ab' esse elevationem ex qua decidens mobile, in 'b' conversum, describet parabolam 'bf'.</s></p>
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391 <p><s>Si enim intelligatur, tempus casus per 'eb' esse ipsum 'eb', et idem 'eb' esse momentum celeritatis in 'b', erit 'bl' <INSERTION-1> tempus et </INSERTION-1> momentum <DELETION-1> <ILLEGIBLE/> </DELETION-1> in 'b' cadentis ex 'a': cadens igitur ex 'a' in 'b', conversum in orizonte, tempore 'bl' <DELETION-1> <ILLEGIBLE/> </DELETION-1> transibit duplam 'ba'; ergo in eodem motu, tempore 'eb', transibit duplam 'bl': est enim ut tempus 'eb' ad tempus 'bl', ita dupla 'bl' ad duplam 'ba'. Dupla vero 'bl' est ipsa 'fi'; ergo tempore 'be' a cadenti ex 'a' conficietur orizontalis 'if': sed eodem tempore 'eb' conficitur perpendicularis 'bi' ex quiete in in 'b': ergo cadens ex 'a', conversus in 'b', eodem tempore conficit orizontalem 'if' et perpendicularem 'bi' ex quiete in 'b': describet ergo parabolam 'bf'.</s></p>
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392 <p><s>Constat, dimidiam basim esse mediam proportionalem inter altitudinem parabolae et elevationem supra parabolam, ex qua cadens illam designat.</s></p>
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393 <p><s>Scritta</s></p>
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394 <pb xlink:href="X01/01/117.jpg" page="088_R"/>
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395 <p><s>secundum eadem rationem respondentia, alia signantur in 'ac' per lineas innumeras parallelas ex punctis lineae 'ab' ad lineam 'ac' extensas, intercepta enim spacia 'ad', 'df', 'fh', et cetera, ad spacia 'ae', 'eg', 'gi', respondent singula singulis secundum rationem 'ac' ad 'ab'; suntque in singulis binis 'cb' respondentibus iidem velocitatis gradus. Ergo, ex praecedenti, tempora omnia simul sumpta lationum omnium per 'ab', ad tempora omnia similiter accepta lationum omnium per 'ac', eandem habebunt rationem quam spacia omnia lineae 'ab' ad spacia omnia lineae 'ac'; hoc autem idem est, ac tempus casus per 'ab' ad tempus casus per 'ac' esse ut linea 'ab' ad 'ac': quod erat demonstrandum.</s></p>
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396 <p><s><DELETION-2> hanc <DELETION-1> <ILLEGIBLE/> </DELETION-1> propositionem non arbitror veram, et credo <DELETION-1> propo </DELETION-1> ita debere proponi <DELETION-1> e </DELETION-1> et demonstrari. </DELETION-2> </s></p>
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397 <p><s><DELETION-1> Si in perpendiculo et in plano inclinato, quorum eadem sit longitudo, feratur idem mobile, tempora lationum erunt inter se ut planum inclinatum et pars perpendiculi, altitudinis ipsius plani </DELETION-1> </s></p>
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398 <pb xlink:href="X01/01/118.jpg" page="088_V"/>
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399 <p><s>Si in perpendiculo et in plano inclinato, quorum eadem sit altitudo, feratur idem mobile, tempora lationum erunt inter se ut plani inclinati et perpendiculi longitudines.</s></p>
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400 <p><s>Sint ad planum orizontis 'cb' perpendiculus 'ab' et planum inclinatum 'ac', quorum eadem sit altitudo, nempe ipsa perpendicularis 'ab', et per ipsa descendat idem mobile: dico, tempus lationis per 'ab' ad tempus lationis per 'ac' esse ut longitudo 'ab' ad longitudinem 'ac'. Cum enim <SUBSTITUTION-1> absumptum <SUBSTITUTED-BY-1> assumptum </SUBSTITUTED-BY-1></SUBSTITUTION-1> sit, in naturali descensu velocitatis momenta eadem semper reperiri in punctis aequaliter ab orizonte distantibus iusta perpendiculares distantias, <UNDERLINED> continue augeri secundum rationem elongationis perpendicularis a liena [linea] orizontali, in qua fuit lationis initium, </UNDERLINED> constat quod, producta linea orizontali 'am', quae ipsi 'bc' erit parallela, sumptisque in perpendiculo 'ab' quotcumque punctis 'e', 'g', 'i', 'l', et per ipsa ductis parallelis orizonti 'ed', 'gf', 'ih', 'lk', erit mobilis per 'ab' momentum seu gradus velocitatis in puncto 'e' idem cum gradu velocitatis lati per 'ac' in puncto 'd', cum punctorum 'e', 'd' eadem sit distantia perpendicularis ab <SUBSTITUTION-1> orizontis <SUBSTITUTED-BY-1> orizonte </SUBSTITUTED-BY-1></SUBSTITUTION-1> 'am': et similiter concludetur, in punctis 'f', 'g' idem esse velocitatis momentum, et rursus in punctis 'h', 'i', et 'k', 'l' et 'c', 'b'.</s></p>
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401 <p><s><UNDERLINED> Et quia velocitas semper intenditur pro ratione elongationis a termino 'a', constat in latione 'ab' tot esse velocitatis gradus seu momenta diversa, quot sunt in eadem linea 'ab' puncta magis ac magis a termino 'a' distantia; quibus totidem in linea 'ac' respondent, et per parallelas lineas determinantur, in quibus iidem sunt gradus velocitatis. Sunt igitur in linea 'ag' quasi innumera quaedam spaciola, quibus multitudine quidem aequalia, et bina sumpta </UNDERLINED> </s></p>
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402 <p><s><DELETION-1> credo esse utile si non <BEGIN-OF-LINE-DESTROYED/> [neces]sarium, demonstrasse, mobile in 'd' esse eiusdem momenti quam in 'c' </DELETION-1> </s></p>
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403 <pb xlink:href="X01/01/119.jpg" page="089AR"/>
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404 <p><s>Sit in perpendiculo 'ab' accepta pars 'ac', cuius tempus 'ac'; accepta rursus ubicumque parte 'db', ipsi 'ac' aequali, quaeritur tempus quo eadem 'db' post casum ex 'a' conficietur. Circa totam 'ab' semicirculus describatur 'aeb', et horizon ducatur 'ce', et iungatur 'ae', quae maior erit quam 'ce'; sit differentia 'af': dico, 'af' esse tempus per 'db'. Quia enim 'ae' est media inter 'ba', 'ac', estque 'ac' tempus per 'ac', erit 'ae' tempus per totam 'ab'; cumque 'ce' media <INSERTION-1> sit </INSERTION-1> inter 'da', 'ac' (est enim 'da' aequalis ipsi 'bc') erit 'ce' tempus per totam 'ad'; est autem 'ce' aequalis 'ef'; ergo 'ae' est tempus per totam 'ab', 'ef' vero per 'ad'; ergo 'af' erit tempus per 'db'.</s></p>
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405 <p><s>Dato in perpendiculo spacio quocumque a principio lationis peracto, datoque tempore casus, tempus reperire, quo aliud aequale spacium, ubicunque in perpendiculo acceptum, ab eodem mobili consequenter conficitur.</s></p>
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406 <p><s>13</s></p>
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407 <pb xlink:href="X01/01/120.jpg" page="089AV"/>
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408 <p><s><SYMBOL-3/>_[tri]ang[ul]um 'csf' est simile <SYMBOL-3/>_[tri]ang[ul]o 'dgc', ergo ut 'sf' ad 'fc', ita 'gc' ad 'cd'; et quia <SYMBOL-3/>_[tri]ang[ul]um 'cfg' est simile <SYMBOL-TRIANGULUM/>_[triangul]o 'dca', ergo ut <DELETION-1> 'fg' ad </DELETION-1> 'fg' ad 'cg', ita ita [sic] 'cd' ad 'da'; ergo, ex aequali, ut 'sf' ad 'cg', ita 'cg' ad 'da': ergo 'cg' est media inter 'sf', 'da', et ut 'da' ad <DELETION-1> 'cg' </DELETION-1> 'sf', <DELETION-1> <ILLEGIBLE/> </DELETION-1> ita <SYMBOL-QUADRATUM/>_[quadratum] 'da' ad <SYMBOL-QUADRATUM/>_[quadratum] 'cg'. Et rursus, cum <SYMBOL-TRIANGULUM/>_[triangulum] 'acd' simile sit <SYMBOL-TRIANGULUM/>_[triangulo] 'cgf', erit ut 'ad' ad 'dc', ita 'gc' ad 'cf', et, permutando, ut 'ad' ad 'cg', ita 'dc' ad 'cf', et ut <SYMBOL-QUADRATUM/>_[quadratum] 'da' ad <SYMBOL-QUADRATUM/>_[quadratum] 'cg', ita <SYMBOL-QUADRATUM/>_[quadratum] 'dc' ad <SYMBOL-QUADRATUM/>_[quadratum] 'cf'; sed ostensum est, <DELETION-1> ut </DELETION-1> <SYMBOL-QUADRATUM/>_[quadratum] 'da' ad <SYMBOL-QUADRATUM/>_[quadratum] 'cg' esse ut linea 'da' ad 'fs'; ergo, ut <SYMBOL-QUADRATUM/>_[quadratum] 'dc' ad <SYMBOL-QUADRATUM/>_[quadratum] 'cf', ita linea 'da' ad <DELETION-1> 'fe' </DELETION-1> 'fs'; ergo, ex praecedenti, cum planorum 'cd', 'cf' elevationes 'da', 'fs' duplam habeant rationem eorumdem planorum, tempora lationum per ipsa erunt aequalia.</s></p>
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409 <pb xlink:href="X01/01/121.jpg" page="089BR"/>
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410 <p><s>tempus casus per 'ac' ad tempus per 'af' est ut linea 'ca' ad 'at', et, dividendo, ut tempus per 'cf' ad tempus per 'fa', ita linea 'ct' ad 'ta'.</s></p>
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411 <p><s>Sed tempus per 'fa' ad tempus per 'ag' est ut linea 'ta' ad 'ar'; tempus autem per 'ag' at [ad] tempus per 'ab' est ut linea 'ar' ad 'ab', et, dividendo, tempus per 'ag' ad tempus 'gb' ut linea 'ar' [ad] 'rb'.</s></p>
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412 <pb xlink:href="X01/01/122.jpg" page="089BV"/>
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413 <pb xlink:href="X01/01/123.jpg" page="090_R"/>
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414 <p><s>Cadens ex 'a' in 'c', conversus, describit parabolam 'cd'; si vero momentum velocitatis in 'c' duplum foret, describeret parabolam 'ce', cuius 'eg' dupla esset ad 'gd': impetus enim duplus in 'c' permeat in orizonte duplum spacium tempore eodem. Sed ut acquiratur in 'c' momentum duplum, necesse est casum fieri ex <SYMBOL-4/>_[quadru]pla altitudine, nempe ex 'cb'. Pariter, ex altitudine <SYMBOL-4/>_[quadru]pla ad 'cb' describetur parabola 'cf', cuius amplitudo 'gf' <SYMBOL-2/>_[du]pla est ad 'ge'.</s></p>
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415 <p><s>Verum mobile in 'd' videtur supra impetum in 'c' addere impetum acquisitum per parabolam 'cd', quod respondet altitudini 'cg'. Mobile vero in 'e' idem momentum addit supra impetum quem habuit in 'c', qui erat <SYMBOL-2/>_[du]plus ad impetum alterius mobilis; ergo impetus mobilis in 'e' videtur esse sexquialterus ad impetum mobilis in 'd'. Similiter invenietur impetum in 'f' ad impetum in 'e' esse ut 5 ad 3.</s></p>
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416 <p><s>In elevatione igitur 'ea' si proiectum habuerit impetum sexquialterum ad impetum in 'd', proiecti secundum elevationem 'da' proiicientur secundum parabolas 'ec', 'dc' intra easdem parallelas, sed dixtantia 'eg' dupla erit ad 'dg'.</s></p>
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417 <p><s>Impetus in 'c' cadentis ex 'a' sit 100 <lb/> cadentis ex 'b' <DELETION-1> sit </DELETION-1> erit 200 <lb/> impetus in 'd' erit 200 <lb/> impetus in 'e' erit 300.</s></p>
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418 <p><s>Cadentis in 'a' ex 'h' impetus in 'a' erit 141; conversi vero per parabolam 'ae' impetus in 'e' erit duplicatus, nempe 282. Constat igitur, maiorem esse impetum venientis per parabolam 'ce' in 'e', quam venientis per parabolam 'ae'. Et si proiectum ex 'e', secundum elevationem 'eh', habet impetum ut 282, conficiet parabolam 'ea'; secundum elevationem vero 'ea' conficiet proiectum parabolam 'ec', si habuerit impetum ut <DELETION-1> <ILLEGIBLE/> </DELETION-1> 300. Ergo in elevazione [elevatione] semirecti 'eh' ab eadem vi longius eiaculatur, quam in elevatione 'ea', quae minor est semirecto.</s></p>
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419 <p><s>Impetus in 'f' est 500, venientis per parabolam 'cf'. Venientis vero per parabolam 'hf', impetus in 'f' est 400. Ex quo patet etiam, longius eiaculari ab eadem vi per elevationem semirecti, quam per minorem.</s></p>
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420 <p><s>Impetus in 'c' ex 's' erit 50; in 'r' erit 150.</s></p>
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421 <p><s>Impetus vero in 't' ex 'c' est fere 70 1/2; conversi per parabolam 'tr' in 'r' erit 141, minor nempe quam venientis ex 's' per 'c' in 'r', qui fuit 150. Unde consta[t], quod in elevatione semirecti 'rt' ab eadem vi longior fit pr[o]iectio, quam per elevationem 'rc'. <DELETION-1> <ILLEGIBLE/> </DELETION-1> </s></p>
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422 <pb xlink:href="X01/01/124.jpg" page="090_V"/>
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423 <p><s><BEGIN-OF-LINE-DESTROYED/> 'ab' est 27834 diam[eter] <BEGIN-OF-LINE-DESTROYED/> <BEGIN-OF-WORD-DESTROYED/> _'?d' erit diameter </s></p>
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424 <p><s><BEGIN-OF-LINE-DESTROYED/> quo absolvitur perpendic[ulum] <BEGIN-OF-LINE-DESTROYED/> <BEGIN-OF-WORD-DESTROYED/>_?o. Quodnam ergo erit tempus <BEGIN-OF-WORD-DESTROYED/>_?0 2/3</s></p>
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425 <p><s><BEGIN-OF-LINE-DESTROYED/>'cf' ad 'fb' ita tempus</s></p>
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426 <pb xlink:href="X01/01/125.jpg" page="091_R"/>
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427 <pb xlink:href="X01/01/126.jpg" page="091_V"/>
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428 <p><s>Si aliquod mobile duplici motu aequabili moveatur, nempe orizontali et perpendiculari, impetus lationis ex utroque motu compositae erit potentia aequalis ambobus momentis priorum motuum. Moveatur enim aliquod mobile aequabiliter <DELETION-1> in perpendiculo ad uno motu, et eodem tempore feratur aequabiliter per orizontem 'cb' </DELETION-1> duplici latione, et mutationi perpendiculari respondeat spacium 'ab', lationi vero orizontali eodem tempore confectae respondeat 'bc'. Cum igitur per motus aequabiles conficiantur eodem tempore spacia 'ab', 'bc', erunt harum lationum momenta <DELETION-1> ut </DELETION-1> inter se <INSERTION-1> ut </INSERTION-1> ipsae linae 'ab', 'bc': mobile vero, quod secundum hasce duas mutationes movetur, describet diametrum 'ac' eodem tempore quo facit mutationem perpendicularem 'ab' et orizontalem 'bc', eritque momentum suae velocitatis ut 'ac'; 'ac' autem potentia aequatur ipsis 'ab', 'bc'; ergo momentum compositum ex utrisque momentis 'ab', 'bc' est potentia tantum illis simul sumptis aequale: quod erat demonstrandum.</s></p>
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429 <p><s>In motu ex quiete eadem ratione intenditur velocitatis momentum, et tempus ipsius motus. Fiat enim motus per 'ab' ex quiete in 'a', et accipiatur quodlibet punctum 'c'; et ponatur 'ac' esse tempus casus per 'ac', et <DELETION-1> insuper </DELETION-1> momentum celeritatis in 'c' <INSERTION-1> acquisitae </INSERTION-1> esse <SUBSTITUTION-1> rursus <SUBSTITUTED-BY-1> pariter </SUBSTITUTED-BY-1></SUBSTITUTION-1> ut 'ac', sumaturque rursus quodlibet punctum 'b'. Dico, <SUBSTITUTION-1> tempus casus per 'ab' ad tempus per 'ac' esse ut momentum velocitatis in 'b' ad momentum in 'c' <SUBSTITUTED-BY-1> momentum velocitatis in 'b' ad momentum in 'c' esse ut tempus casus per 'ab' ad tempus per 'ac'. </SUBSTITUTED-BY-1></SUBSTITUTION-1> Sumatur 'as' media inter 'ba', 'ac', et cum positum sit tempus casus per 'ac' esse 'ac', erit 'as' tempus per 'ab'. Demonstrandum igitur est, momentum celeritatis in 'c' ad momentum caeleritatis [celeritatis] in 'b' esse ut 'ac' ad 'as'.</s></p>
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430 <p><s>Sumantur orizontales 'cd', dupla ad 'ca', 'be' vero dupla ad 'ba'. Constat, ex demonstratis, cadens per 'ac', conversum <DELETION-1> <ILLEGIBLE/> </DELETION-1> in orizonte 'cd', conficere 'cd' <INSERTION-1> motu aequabili </INSERTION-1> aequali tempore atque ipsa <INSERTION-1> confecit motu accelerato naturaliter </INSERTION-1> 'ac'; et similiter, 'be' confici eodem tempore atque 'ab': sed tempus ipsius 'ab' est 'as': ergo orizontalis 'be' conficitur tempore 'as'. Fiat ut tempus 'sa' ad tempus 'ac', ita 'eb' ad 'bl'; cumque motus per 'be' sit aequabilis, erit spacium 'bl' peractum tempore 'ac' secundum momentum celeritatis in 'b': sed secundum momentum caeleritatis [celeritatis] in 'c' eodem tempore 'ac' conficitur spacium 'cd'; momenta autem celeritatis sunt inter se ut spacia, quae iuxta ipsa <INSERTION-1> momenta </INSERTION-1> eodem conficiuntur tempore: ergo momentum celeritatis in 'c' ad momentum celeritatis in 'b' est ut 'dc' ad 'bl'. Quia vero ut 'dc' ad 'be', ita ipsarum dimidia, <INSERTION-1> nempe </INSERTION-1> 'ca' ad 'ab'; ut autem 'eb' ad 'bl', ita 'ba' ad 'as'; ergo, ex aequali, ut 'dc' ad 'bl', ita 'ca' ad 'as': hoc est, ut momentum celeritatis in 'c' ad momentum celeritatis in 'b', ita 'ca' ad 'as', hoc est, tempus per 'ca' ad tempus per 'ab': quod erat demonstrandum.</s></p>
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431 <p><s>Determinetur ergo impetus in singulis punctis parabolae 'bec' ex potentia momenti acquisiti per descensum 'ab', quod semper servatur idem <DELETION-1> et </DELETION-1> et determinat impetum orizontalem, et ex potentia alterius momenti acquisiti in descensu perpendiculari. Ut, v[erbi] g[ratia], in 'e' erit impetus determinatus a linea potente 'ab' et media inter 'db', 'bf', quae sit 'bg'.</s></p>
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432 <pb xlink:href="X01/01/127.jpg" page="092_R"/>
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433 <pb xlink:href="X01/01/128.jpg" page="092_V"/>
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434 <p><s>Sit 'qp' tempus per 'ac', et ut 'ac' ad <SUBSTITUTION-1> 'cb' <SUBSTITUTED-BY-1> 'cd', </SUBSTITUTED-BY-1></SUBSTITUTION-1> ita 'pq' ad 'qr'; erit 'qr' tempus per 'dc', seu per 'bc'. Sit ut 'cd' ad 'do', ita 'rq' ad 'qs': erit 'qs' tempus per 'df', et 'sr' tempus per 'fc' post 'df'. Fiat rursus ut 'ca' ad 'av', ita tempus per 'pq' ad 'qt'; erit 'qt' tempus per 'ab', <DELETION-1> <ILLEGIBLE/> </DELETION-1> 'tp' vero tempus per 'bc', post 'ab'.</s></p>
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435 <pb xlink:href="X01/01/129.jpg" page="093_R"/>
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436 <p><s><foreign lang="it">Molto Ill[ust]re et Ecc[ellen]te Sig[no]r Zio <lb/> Havendo hauto occasione di scriverli questi dua versi non ho voluto mancare di avvisarli il nostro bene stare: et <INSERTION-1> stiamo </INSERTION-1> allegramente <INSERTION-1> il simile piaccia a Dio segua di lei. </INSERTION-1> Però essendo che la Lena di su la Costa si muore di fame, perchè la Sanità non li da cosa alcuna però sarebbe bene cercare di rimediarci. Le cose <DELETION-1> g </DELETION-1> della peste passono assai bene che piaccia al Sig[no]r Dio liberarci afatto non sarò più lungo solo me li ricordo obbedentissimo et obligatissimo nipote e servitore, et pregandoli dal Altissimo il colmo di ogni sua maggiore felicità li baccio le mani. Di casa il dì 29 Genn[ai]o 1630 <lb/> Di V[ostra] S[ignoria] molto et Ecc[ellen]te <lb/> Aff[ezionatissi]mo Nipote e Servitore <lb/> Ceseri Galletti</foreign></s></p>
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437 <pb xlink:href="X01/01/130.jpg" page="093_V"/>
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438 <p><s>Scritta <lb/> Dato plano inclinato et perpendiculo, quorum eadem sit elevatio, punctum sublime in perpendiculo <INSERTION-1> extenso </INSERTION-1> reperire, ex quo mobile decidens, et per planum inclinatum conversum, utrumque conficiat tempore eodem ac solum planum inclinatum ex quiete in eius superiori termino.</s></p>
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439 <p><s>Sint planum inclinatum et perpendiculum 'ab', 'ac', quorum eadem sit elevatio, nempe 'ac': oportet, in perpendiculo ad partes 'a' extenso punctum sublime reperire, ex quo mobile decidens <DELETION-1> ep </DELETION-1> et per planum 'ab' conversum, partem assumptam perpendiculi et planum 'ab' conficiat tempore eodem ac si ex quiete in 'a' per solum planum 'ab' descenderet. Sit orizontalis linea 'bc', et secetur 'an' aequalis 'ac'; et ut 'ab' ad 'bn', ita fiat 'al' ad 'lc'; et ipsi 'al' ponatur aequalis 'ai', et <DELETION-1> quam proportionem habet 'ac' ad 'bi' </DELETION-1> ipsarum 'ac', 'bi' tertia proportionalis sit 'ce', in perpendiculo 'ac' producto signata. Dico, 'ce' esse spatium quaesitum, adeo ut, extenso perpendiculo supra 'a' <DELETION-1> et in eo posita parte </DELETION-1> et assumpta parte 'ax' ipsi 'ce' aequali, mobile ex 'x' conficiet utrumque spatium 'xab' <INSERTION-1> aequali tempore </INSERTION-1> ac solum 'ab' ex 'a'.</s></p>
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440 <p><s><SUBSTITUTION-1> Ponatur 'bs' aequalis 'be', seu 'cg' (constat enim, eas esse inter se aequales, cum sint mediae inter 'ac', 'ce') <ILLEGIBLE/> <SUBSTITUTED-BY-1> Ponatur horizontalis 'xr' aequidistans 'bc', cui occurrat 'ba' extensa in 'r'; </SUBSTITUTED-BY-1></SUBSTITUTION-1> deinde, <INSERTION-1> producta 'ab' in 'd', </INSERTION-1> ducatur 'ed' aequidistans 'cb', <SUBSTITUTION-1> et supra 'ad', 'ae' semicirculi describantur <SUBSTITUTED-BY-1> et supra 'ad' semicirculus describatur, </SUBSTITUTED-BY-1></SUBSTITUTION-1> et ex 'b' ipsi 'da' perpendicularis erigatur 'bf' usque ad circumferentiam: <DELETION-1> extendaturque 'bc' usque ad alteram circumferentiam in 'g'. Constat, 'cg' esse mediam inter 'ac', 'ce', et ideo aequalem ipsi 'bi' </DELETION-1> patet, <DELETION-1> similiter </DELETION-1> 'fb' esse mediam inter 'ab', 'bd', et ductam 'fa' mediam inter 'da', 'ab'. Ponatur 'bs' aequalis 'bi', <DELETION-1> seu 'cg', </DELETION-1> et 'fh' <DELETION-1> aequalis 'fb' </DELETION-1> aequalis 'fb'. Et quia ut 'ab' ad 'bd', ita 'ac' ad 'ce', estque 'bf' media inter 'ab', 'bd', et <SUBSTITUTION-1> 'cg' <SUBSTITUTED-BY-1> 'bi' </SUBSTITUTED-BY-1></SUBSTITUTION-1> media inter 'ac', 'ce', erit ut 'ba' ad 'ac', ita 'fb' ad <DELETION-1> 'cg' et 'ad' </DELETION-1> 'bs'; <DELETION-1> posita </DELETION-1> </s></p>
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441 <p><s>et cum sit ut 'ba' ad 'ac', seu 'an', ita 'fb' ad 'bs', erit, per conversionem rationis, 'bf' ad 'fs' ut 'ab' ad 'bn', hoc est 'al' ad 'lc'. <SYMBOL-RECTANGULUM/>_[Rectangulum] igitur sub 'fb', 'cl' aequatur <SYMBOL-RECTANGULUM/>_[rectangul]o sub 'al', 'sf'; hoc autem <SYMBOL-RECTANGULUM/>_[rectangulum] <INSERTION-1> 'al', 'sf' </INSERTION-1> est excessus <SYMBOL-RECTANGULUM/>_[rectangul]i sub 'al', 'bf', seu 'ai', 'bf', super <SYMBOL-RECTANGULUM/>_[rectangul]o 'ai', 'bs', seu 'aib'. <SYMBOL-RECTANGULUM/>_[Rectangulum] vero 'fb', 'lc' est excessus <SYMBOL-RECTANGULUM/>_[rectangul]i 'ac', 'bf' super <SYMBOL-RECTANGULUM/>_[rectangul]o 'al', 'bf'; <SYMBOL-RECTANGULUM/>_[rectangulum] autem 'ac', 'bf' aequatur <SYMBOL-RECTANGULUM/>_[rectangul]o 'abi' (est enim ut 'ba' ad 'ac', ita 'fb' ad <DELETION-1> 'cg' idest 'ad' </DELETION-1> 'bi'). Excessus igitur <SYMBOL-RECTANGULUM/>_[rectangul]i 'abi' super <SYMBOL-RECTANGULUM/>_[rectangul]o 'ai', 'bf', seu 'ai', 'fh', aequatur excessui <SYMBOL-RECTANGULUM/>_[rectangul]i 'ai', 'fh' super <SYMBOL-RECTANGULUM/>_[rectangul]o 'aib': ergo <DELETION-1> <ILLEGIBLE/> </DELETION-1> bina <SYMBOL-RECTANGULUM/>_[rectangul]a 'ai', 'fh' aequantur <SYMBOL-2/>_[duo]bus 'abi', 'aib', nempe <SYMBOL-2/>_[duo]bus 'aib' cum <SYMBOL-QUADRATUM/>_[quadrat]o 'bi'.</s></p>
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442 <p><s>Comune sumatur <SYMBOL-QUADRATUM/>_[quadratum] 'ai': erunt bina <SYMBOL-RECTANGULUM/>_[rectangul]a 'aib' cum <SYMBOL-2/>_[duo]bus <SYMBOL-QUADRATUM/>_[quadrat]is 'aib', nempe <SYMBOL-QUADRATUM/>_[quadratum] ipsum 'ab', aequale binis <SYMBOL-RECTANGULUM/>_[rectangul]is 'ai', 'fh' cum quadrato 'ai'. Comuniter rursus assumpto <SYMBOL-QUADRATUM/>_[quadrat]o 'bf', erunt duo <SYMBOL-QUADRATUM/>_[quadrat]a 'ab', 'bf', nempe unicum <SYMBOL-QUADRATUM/>_[quadratum] 'af', aequale binis <SYMBOL-RECTANGULUM/>_[rectangul]is 'ai', 'fh' cum <SYMBOL-2/>_[duo]bus <SYMBOL-QUADRATUM/>_[quadrat]is 'ai', 'fb', id est 'ai', 'fh'. Verum idem <SYMBOL-QUADRATUM/>_[quadratum] 'af' aequale est binis <SYMBOL-RECTANGULUM/>_[rectangul]is 'ahf' cum <SYMBOL-2/>_[duo]bus <SYMBOL-QUADRATUM/>_[quadrat]is 'ah', 'hf'; ergo bina <SYMBOL-RECTANGULUM/>_[rectangul]a 'ai', 'fh' cum <SYMBOL-QUADRATUM/>_[quadrat]is 'ai', 'fh' aequalia sunt binis <SYMBOL-RECTANGULUM/>_[rectangul]is 'ahf' <DELETION-1> h </DELETION-1> cum <SYMBOL-QUADRATUM/>_[quadrat]is 'ah', 'hf'; et dempto communi <SYMBOL-QUADRATUM/>_[quadrat]o 'hf', bina <SYMBOL-RECTANGULUM/>_[rectangul]a 'ai', 'fh' cum <SYMBOL-QUADRATUM/>_[quadrat]o 'ai' erunt aequalia binis <SYMBOL-RECTANGULUM/>_[rectangul]is 'ahf' cum <SYMBOL-QUADRATUM/>_[quadrat]o 'ah'. Cumque rectangulorum omnium 'fh' sit latus comune, erit linea 'ah' aequalis ipsi 'ai'. Si enim maior vel minor esset, <SYMBOL-RECTANGULUM/>_[rectangul]a quoque 'fha' et <SYMBOL-QUADRATUM/>_[quadratum] 'ha' maiora vel minora essent <SYMBOL-RECTANGULUM/>_[rectangul]is 'fh', 'ia' et <SYMBOL-QUADRATUM/>_[quadrat]o 'ia', contra id quod demonstratum est.</s></p>
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443 <p><s>Modo si intelligamus, tempus casus per 'ab' esse <DELETION-1> 'if' </DELETION-1> ut 'ab', tempus per 'ac' erit 'ac', et ipsa <SUBSTITUTION-1> 'cg' <SUBSTITUTED-BY-1> 'ib' </SUBSTITUTED-BY-1></SUBSTITUTION-1> media inter 'ac', 'ce', erit tempus per 'ce', seu per 'xa' ex quiete in 'x': cumque inter 'da', 'ab', seu 'rb', 'ba', media sit 'af', inter vero 'ab', 'bd', id est 'ra', 'ab', media sit 'bf', cui aequatur 'fh', erit, ex praedemonstratis, excessus 'ah' tempus per 'ab' ex quiete in 'r', seu post casum ex 'x'; dum tempus eiusdem 'ab' ex quiete in 'a' fuerit 'ab'. Tempus igitur per 'xa' est 'ib', per 'ab' vero post 'ra', seu 'xa', est 'ai'. Ergo patet propositum.</s></p>
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444 <pb xlink:href="X01/01/131.jpg" page="094_R"/>
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445 <p><s>erit 'bl' media 'mb', 'be', et 'el' media 'meb'; <DELETION-1> ergo </DELETION-1> et quia 'ne' aequatur 'hl', erit 'hb' excessus 'ne' super 'bl': verum 'hb' est etiam excessus 'ne' super 'nba', cum sit excessus 'be' super 'ba': ergo 2 'nb', 'ba' aequantur 'bl'.</s></p>
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446 <pb xlink:href="X01/01/132.jpg" page="094_V"/>
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447 <p><s>ut tempus per 'eag' sit idem cum tempore per 'ag', posito quod tempus per 'ag' sit 'ag', tempus per 'al' erit 'al', et 'fg' erit tempus per 'dg', et excessus 'gf' super 'fa', <INSERTION-1> 'ng', </INSERTION-1> erit tempus per 'ag' post 'da', et 'ia' media inter 'la', 'ae', erit tempus per 'ea'. Oportet igitur facere ut 'ai' cum 'ng' sint aequales 'ag': hoc erit cum excessus mediae 'fa' super mediam 'ai' sit aequalis excessui 'gf' super 'ag', secta 'go' aequali <DELETION-1> 'ag' </DELETION-1> 'ga'.</s></p>
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448 <pb xlink:href="X01/01/133.jpg" page="095_R"/>
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449 <pb xlink:href="X01/01/134.jpg" page="095_V"/>
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450 <p><s><SYMBOL-RECTANGULUM/>_[rettangul]um 'bae' superat <SYMBOL-RECTANGULUM/>_[rettangul]um 'bea' <SYMBOL-QUADRATUM/>_[quadrat]o 'ea'; <SYMBOL-RECTANGULUM/>_[rettangul]um 'cde' superat <SYMBOL-RECTANGULUM/>_[rettangul]um 'ced' <SYMBOL-QUADRATUM/>_[quadrat]o 'ed': sed <SYMBOL-RECTANGULUM/>_[rettangulum] 'bea' aequatur <SYMBOL-RECTANGULUM/>_[rettangul]o 'ced': excessus ergo <SYMBOL-RECTANGULUM/>_[rettangul]i 'cde', idest <SYMBOL-QUADRATUM/>_[quadrat]i 'fd', super <SYMBOL-RECTANGULUM/>_[rettangul]o 'bae', seu <SYMBOL-QUADRATUM/>_[quadrat]o 'ag', est idem cum excessu <SYMBOL-QUADRATUM/>_[quadrat]i 'ed' super <SYMBOL-QUADRATUM/>_[quadrat]o 'ae'. <SYMBOL-QUADRATUM/>_[Quadratum] vero 'ed' superat <SYMBOL-QUADRATUM/>_[quadratum] 'ea' <SYMBOL-QUADRATUM/>_[quadrat]o 'ad': ergo <SYMBOL-QUADRATUM/>_[quadratum] 'fd' superat <SYMBOL-QUADRATUM/>_[quadratum] 'ga' <SYMBOL-QUADRATUM/>_[quadrat]o 'ad'. Sed quadratum quoque 'dg' superat <SYMBOL-QUADRATUM/>_[quadrat]um 'ga' <SYMBOL-QUADRATUM/>_[quadrat]o 'ad': ergo <SYMBOL-QUADRATUM/>_[quadratum] 'gd' aequatur <SYMBOL-QUADRATUM/>_[quadrat]o 'df', et linea 'dg' lineae 'df', et angulus 'dgf' angulo 'dfg': ergo angulus 'f' maior angulo 'fge', et latus 'eg' latere 'ef'.</s></p>
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451 <p><s>Probatum est tempus perpendiculi sub 'e' post 'a' per totam <DELETION-1> 'ab' </DELETION-1> 'eb' longius esse tempore <DELETION-1> tempore </DELETION-1> per 'ec', idest partem perpendiculi quae conficitur tempore eodem cum 'ec', minorem esse quam 'eb', maiorem tamen quam 'ec': probandum restat quanta vero sit. Determinatur, posita 'en' aequali 'ef', et sumpta <SYMBOL-3/>_[terti]a proportionali post 'ea', 'an', quae <SYMBOL-3/>_[terti]a probandum restat quanto sit (dempta 'ae') maior quam 'ec'.</s></p>
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452 <pb xlink:href="X01/01/135.jpg" page="096_R"/>
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453 <pb xlink:href="X01/01/136.jpg" page="096_V"/>
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454 <p><s>Sit factum: et sit tempus per 'eac' ex 'e' idem cum tempore per 'ac' ex quiete in 'a'. Sit 'ea' tempus per 'ea'; erit 'fa' tempus per 'fa', et per totam 'fc' erit 'fh', seu 'fi', et per reliquam 'ac' ex 'f' erit 'ai': per ambas igitur 'eac' erit tempus 'eai', quod debet esse aequale tempori per 'ac' ex quiete in 'a'. Et quia 'fa' est tempus per 'fa', tempus per 'ac' ex quiete in 'a' erit 'ah', media nempe inter 'fa', 'ac'. Faciendum itaque est ut 'ah' sit aequalis utrisque 'eai', nempe protrahendae sunt 'ba', 'ca', ita ut (ducta orizontali 'fe') 'ea' cum excessu mediae inter 'cf', 'fa' super 'fa' (quod sit 'ai') sint aequales mediae inter 'ac', 'af', nempe ipsi 'ah'. Quod si ponatur, 'ca' esse tempus per 'ca', erit 'ba' tempus per 'ab', et 'ag' per 'ea'. Et posita 'fi' aequali 'ac', erit 'fi' <INSERTION-2> tempus per 'fi', et 'fo' tempus per totam 'fc', et 'oi', media inter 'if', 'fa' (est enim 'fa' aequalis 'ic'), erit tempus per 'fa', et 'pf' (excessus mediae 'of' super mediam <DELETION-1> <ILLEGIBLE/> </DELETION-1> 'oi') erit tempus per 'ac' ex 'f'. Faciendum itaque est ut 'pf' cum 'ag' sint aequales ipsi 'ac'. </INSERTION-2> </s></p>
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455 <pb xlink:href="X01/01/137.jpg" page="097_R"/>
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456 <p><s>Dato <n-SUPERSCRIBED-BY-o/>n[umer]o 'ba', 'bs', oportet alium ei addere.</s></p>
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457 <pb xlink:href="X01/01/138.jpg" page="097_V"/>
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458 <p><s>Dato perpendiculo 'ab' et inflexa <DELETION-1> <ILLEGIBLE/> </DELETION-1> 'ebg', cui perpendicularis sit 'bc', oportet semicirculum per 'e' describere, ita ut excessus mediae inter 'eg', 'gb', quae est 'gc', seu 'gd', <SUBSTITUTION-1> excessus autem 'bd' <SUBSTITUTED-BY-1> super 'gb' </SUBSTITUTED-BY-1></SUBSTITUTION-1> una cum perpendiculo 'bf', secto a perpendiculari 'gf', sint aequales mediae inter 'eb', 'bg', nempe 'bc'. Sit factum.</s></p>
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459 <p><s>Si 'cb' aequatur 'dbf', posita communi 'bg', 2 'cb', 'bg' erunt aequales <SYMBOL-2/>_[du]abus 'dg', 'bf', idest 'cg', 'bf'.</s></p>
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460 <p><s>posito 'ab' esse tempus per 'ab', erit 'eb' tempus per 'eb', et tempus per futuram 'bx' ex quiete in 'b' erit media inter 'eb', 'bx'; et ideo erigitur perpendicularis 'bo', ut in ea <DELETION-1> <ILLEGIBLE/> </DELETION-1> notetur media. Tempus vero per totam 'ebx' futurum est media inter 'xe', 'eb', quae erit 'eo', cuius excessus super 'eb' erit tempus per 'bx' post 'eb', qui excessus cum 'ba' (tempora scilicet per 'abx') debent aequari mediae 'bo'. Cum autem hoc fuerit (nempe excessum mediae 'oe', seu 'en', super 'eb' una cum 'ab', dico 'nba', esse [a]equales ipsi 'bo') posita communi 'be', erunt 2 'ne', <DELETION-1> seu </DELETION-1> idest 'oe', 'ab' aequales <SYMBOL-2/>_[du]abus 'ob', 'be': auferantur 'os', 'ba', aequales <SYMBOL-2/>_[dua]bus 'ob', 'bf'; reliqua 'fe' (quae datur) erit aequalis reliquae 'es'. Redactum ergo est opus, ut, centro 'e', intervallo 'f', circulo descripto, a centro 'e' educatur linea occurrens 'bo', ita ut pars extra circulum, qualis est 'so', sit aequalis 'ob'. Dum autem hoc fuerit, perpendicularis ex 's' ad 'so' occurret 'bf', ut in 'h', et erit aequalis 'hb', et tanget circulum, eritque eius <SYMBOL-QUADRATUM/>_[quadratum] aequale <SYMBOL-RECTANGULUM/>_[rectangulo] 'phf': cumque datae sint 'pf', 'fb', oport[et] ita dividere 'fb', ut <SYMBOL-RECTANGULUM/>_[rectangulum] 'phf' sit aequale <SYMBOL-QUADRATUM/>_[quadrato] 'hb'; quod erit dum 'ph' ad 'hb' sit ut 'bh' ad 'hf', et, componendo, ut 'pb' ad 'bh', ita 'bf' ad 'fh', et, permutando, ut 'pb' ad 'bf', ita 'bh' ad 'hf'. Dantur autem 'pb', 'bf': ergo dabitur 'bh'. Duc igitur a puncto 'h' tangentem 'hs', et per 's', 'eso', etc.</s></p>
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461 <p><s>fac ut <SYMBOL-2/>_[du]pla 'eb' ad 'bf', sic 'bf' ad 'fh'. Seu ut 'eb' ad 'bf', sic 'bf' ad aliam, cuius dimidium erit 'fh'.</s></p>
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462 <p><s>fac ut 'bx' ad 'xf', ita 'fb' ad 'bh', et habebis punctum 'h'.</s></p>
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463 <pb xlink:href="X01/01/139.jpg" page="098_R"/>
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464 <pb xlink:href="X01/01/140.jpg" page="098_V"/>
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465 <pb xlink:href="X01/01/141.jpg" page="099_R"/>
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466 <p><s>Si excessus 'od' aequatur 'di', <SYMBOL-RECTANGULUM/>_[rectangulum] 'pdo', idest <SYMBOL-QUADRATUM/>_[quadratum] 'da', ad <SYMBOL-RECTANGULUM/>_[rectangulum] 'ndi', idest ad <SYMBOL-QUADRATUM/>_[quadratum] 'de', erit ut linea 'pd' ad 'dn'; <SYMBOL-QUADRATUM/>_[quadratum] autem 'da' ad <SYMBOL-QUADRATUM/>_[quadratum] 'de' est ut <SYMBOL-QUADRATUM/>_[quadratum] 'ab' ad <SYMBOL-QUADRATUM/>_[quadratum] 'bc'; ergo faciendum est ut 'pd' ad 'nd' sit ut <SYMBOL-QUADRATUM/>_[quadratum] 'ab' ad <SYMBOL-QUADRATUM/>_[quadratum] 'bc': 'pd' autem componitur ex <SYMBOL-2/>_[dua]bus mediis 'df', 'fa', et 'nd' constat ex <SYMBOL-2/>_[dua]bus 'ead', ut 2 'dfa' ad <SYMBOL-2/>_[du]as 'dae' sint ut <SYMBOL-QUADRATUM/>_[quadratum] 'ab' ad <SYMBOL-QUADRATUM/>_[quadratum] 'bc'.</s></p>
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467 <p><s>Si 'ea' cum 'ag' aequantur 'af', excessus 'da' super 'ae' est aequalis excessui 'df' super 'fa', et 'fa', 'ad' aequantur 'fd', 'ea'.</s></p>
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468 <p><s>ut tempus per 'eab' sit aequale tempori per 'ab', faciendum est ut 'ea' cum 'ag' sint aequales 'af'.</s></p>
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469 <pb xlink:href="X01/01/142.jpg" page="099_V"/>
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470 <p><s>Si 'ha' aequatur 'ai', et 'hf', 'fb', et <SYMBOL-QUADRATUM/>_[quadratum] 'fa' <SYMBOL-QUADRATUM/>_[quadrat]is 'fba', demptis <SYMBOL-QUADRATUM/>_[quadrat]is 'fh', 'ha', 'fb', 'ia', <SYMBOL-RECTANGULUM/>_[rectangulum] bis sub 'fh', 'ha' aequabitur <SYMBOL-RECTANGULUM/>_[rectangulo] bis sub 'ai', 'ib' et <SYMBOL-QUADRATUM/>_[quadrat]o 'ib'.</s></p>
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471 <p><s><DELETION-1> <SYMBOL-QUADRATUM/>_[quadratum] 'fa' aequatur <SYMBOL-QUADRATUM/>_[quadrat]is 'ab', 'bf'; ergo <SYMBOL-RECTANGULUM/>_[rectangulum] 'dab' aequatur <SYMBOL-QUADRATUM/>_[quadrat]o 'ba' et <SYMBOL-RECTANGULUM/>_[rectangul]o 'abd': sed <SYMBOL-QUADRATUM/>_[quadrat]o 'ba' aequantur <SYMBOL-QUADRATUM/>_[quadrat]a 'bc', 'ca': ergo <SYMBOL-RECTANGULUM/>_[rectangulum] 'dab' superat <SYMBOL-QUADRATUM/>_[quadrat]a 'ac', 'cb' <SYMBOL-RECTANGULUM/>_[rectangul]o 'abd'. Protrahenda est igitur 'ab', donec <SYMBOL-RECTANGULUM/>_[rectangulum] 'dab' superet <SYMBOL-QUADRATUM/>_[quadrat]a 'bca' <SYMBOL-RECTANGULUM/>_[rectangul]o 'abd'. </DELETION-1> </s></p>
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472 <p><s>2 <SYMBOL-RECTANGULUM/>_[rectangul]a 'ahf' aequantur <SYMBOL-2/>_[dua]bus 'aib', 'fb', 'ac'; <INSERTION-1> est enim <SYMBOL-RECTANGULUM/>_[rectangulum] 'abi' aequale <SYMBOL-RECTANGULUM/>_[rectangul]o 'fb', 'ac', cum sit ut 'ac' ad 'cg', seu 'bi', ita 'ab' ad 'bf': </INSERTION-1> oportet igitur ut excessus <SYMBOL-RECTANGULUM/>_[rectangul]i 'ahf' super <SYMBOL-RECTANGULUM/>_[rectangul]o 'aib', seu 'ah', 'ib', sit aequalis excessui <SYMBOL-RECTANGULUM/>_[rectangul]i 'fb', 'ac' super <SYMBOL-RECTANGULUM/>_[rectangul]o <SUBSTITUTION-1> 'fah' <SUBSTITUTED-BY-1> 'fha'. </SUBSTITUTED-BY-1></SUBSTITUTION-1> <DELETION-1> sed <SYMBOL-RECTANGULUM/>_[rectangul]um 'ahf' superat <SYMBOL-RECTANGULUM/>_[rectangul]um 'ahib' ut linea 'fh' superat 'bi'; <SYMBOL-RECTANGULUM/>_[rectangulum] vero 'fb', 'ac' superat <SYMBOL-RECTANGULUM/>_[rectangulum] 'fha' ut linea 'ac' superat lineam 'ah'; ergo oportet excessum lineae 'ac' super 'ah' esse aequalem excessui 'fh' super 'bi'; ergo oportet lineas 'ac', 'ib', seu 'ac', 'cg', esse aequales lineae 'af'. Protraendae itaque sunt lineae 'ab', 'ac' in 'd', 'e', adeo ut media inter 'dab' si[t] aequalis <SYMBOL-2/>_[dua]bus 'ac' et mediae inter 'ac', 'ce'. </DELETION-1> </s></p>
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473 <p><s>Excessus autem <SYMBOL-RECTANGULUM/>_[rectangul]i 'ahf' super <SYMBOL-RECTANGULUM/>_[rectangul]o 'aib', seu 'ah', 'cg', est <SYMBOL-RECTANGULUM/>_[rectangulum] contentum a 'gc' et ab excessu 'fh', seu 'fb', super 'cg' et ab ipsa 'gc'. Excessus vero <SYMBOL-RECTANGULUM/>_[rectangul]i 'acbf' super <SYMBOL-RECTANGULUM/>_[rectangul]o 'ahf' est [a]equalis <SYMBOL-RECTANGULUM/>_[rectangul]o contencto ab excessu 'ac' super 'ah', seu 'ai', et ab ipsa <SUBSTITUTION-1> 'af' <SUBSTITUTED-BY-1> 'fh' </SUBSTITUTED-BY-1></SUBSTITUTION-1>. Si igitur ponatur 'al' aequalis 'ai', iste excessus erit <SYMBOL-RECTANGULUM/>_[rectangulum] 'fh', 'lc', cui debet esse aequalis alter excessus <SYMBOL-RECTANGULUM/>_[rectangul]i 'ahf' super <SYMBOL-RECTANGULUM/>_[rectangul]o 'aib', nempe (posita 'bo' aequali 'fb') <SYMBOL-RECTANGULUM/>_[rectangulum] 'aio'.</s></p>
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474 <p><s><SYMBOL-QUADRATUM/>_[quadratum] lineae aequalis <SYMBOL-2/>_[dua]bus 'haf' superat <SYMBOL-QUADRATUM/>_[quadratum] 'hf' <SYMBOL-RECTANGULUM/>_[rectangu]lo ex linea aequali <SYMBOL-3/>_[tri]bus 'fhaf' et ex excessu <SYMBOL-2/>_[dua]rum 'haf' super 'hf', quod in nostro casu debet esse 'ae': faciendum itaque est, quod <SYMBOL-RECTANGULUM/>_[rectangulum] <SYMBOL-3/>_[tri]um 'fhaf' in 'ea' cum <SYMBOL-QUADRATUM/>_[quadrat]o 'hf' sint aequalia <SYMBOL-QUADRATUM/>_[quadrat]o ex linea aequali <SYMBOL-2/>_[dua]bus 'haf'. Seu dicas, faciendum esse ut <SYMBOL-3/>_[tri]a <SYMBOL-RECTANGULUM/>_[rectangul]a trium laterum <SYMBOL-3/>_[tri]ang[ul]i 'haf' in 'ae' <DELETION-1> <ILLEGIBLE/> </DELETION-1> cum <SYMBOL-2/>_[duo]bus <SYMBOL-QUADRATUM/>_[quadrat]is 'ha', 'af' sint aequalia <SYMBOL-QUADRATUM/>_[quadrat]o ex 'haf' tanquam e una linea.</s></p>
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475 <p><s>faciendum est ut <SYMBOL-QUADRATUM/>_[quadratum] 'fa' ad <SYMBOL-QUADRATUM/>_[quadratum] 'fe' sit ut duae 'fha' ad <SYMBOL-2/>_[du]as 'fae'.</s></p>
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476 <pb xlink:href="X01/01/143.jpg" page="100_R"/>
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477 <pb xlink:href="X01/01/144.jpg" page="100_V"/>
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478 <p><s><foreign lang="it">al quantunque</foreign></s></p>
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479 <pb xlink:href="X01/01/145.jpg" page="101_R"/>
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480 <pb xlink:href="X01/01/146.jpg" page="101_V"/>
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481 <pb xlink:href="X01/01/147.jpg" page="102_R"/>
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482 <p><s>ad elevationem gr[adus] 22 proiectio in plano absumit amplitudinem ut 4 ad 3, nempe sexquitertiam altitudinis.</s></p>
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483 <p><s>momentum 'eb'</s></p>
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484 <pb xlink:href="X01/01/148.jpg" page="102_V"/>
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485 <p><s>Sit parabola 'cba' parallelogrammo 'cp' inscripta: dico, parallelogrammum parabolae esse sexquialterum; <INSERTION-1> hoc est, esse <SYMBOL-3/>_[tri]plum reliqui spacii 'apb' extra parabolam. </INSERTION-1> </s></p>
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486 <p><s>Si enim non sit, aut erit maius aut minus. Sit, primo, <SUBSTITUTION-1> minus, defectus <SUBSTITUTED-BY-1> maius: excessus </SUBSTITUTED-BY-1></SUBSTITUTION-1> autem quo spacium 'pc' <SUBSTITUTION-1> minus <SUBSTITUTED-BY-1> maius </SUBSTITUTED-BY-1></SUBSTITUTION-1> est quam <SUBSTITUTION-1> sexquialterum parabolae 'apb' <SUBSTITUTED-BY-1> <SYMBOL-3/>_[tri]plum spacii 'apb', </SUBSTITUTED-BY-1></SUBSTITUTION-1> vocetur 'x'. Divisoque parallelogrammo continue in spacia aequalia per lineas ipsis 'ac', 'pb' parallelas, deveniemus ad spacia, quorum unum ipso 'x' erit minus, quale sit 'ob', et per puncta ubi reliquae parallelae lineam parabolae secant, ducantur aequidistantes ipsi 'ap', donec figura quaedam spacio relicto extra parabolam circumscribatur, constans ex parallelogrammis 'ag', <DELETION-1> 'ge' 'ef' </DELETION-1> 'ke', 'lf', 'mh', 'ni', 'ob', quae figura spacium 'apb' extra parabolam minori quantitate superabit quam sit 'x', cum superet idem quantitate adhuc minori parallelogrammo 'ob'. Ergo idem parallelogrammum <INSERTION-1> 'cp' </INSERTION-1> maius erit <INSERTION-1> quam <SYMBOL-3/>_[tri]plum </INSERTION-1> dictae figurae circumscriptae: quod est impossibile.</s></p>
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487 <p><s>Nam est illa minus quam triplum; nam cum 'da' ad <SUBSTITUTION-1> 'dz' <SUBSTITUTED-BY-1> 'az' </SUBSTITUTED-BY-1></SUBSTITUTION-1> sit ut <SYMBOL-QUADRATUM/>_[quadratum] 'de' ad <SYMBOL-QUADRATUM/>_[quadratum] 'zg'; ut autem 'da' ad 'az', ita parallelogrammum 'dk', seu 'ke', ad parallelogrammum 'kz'; ergo ut <DELETION-1> 'ed' ad 'gz', seu 'la', ad 'ak', ita </DELETION-1> <SYMBOL-QUADRATUM/>_[quadratum] 'zg' ad <SYMBOL-QUADRATUM/>_[quadratum] 'de', idest <SYMBOL-QUADRATUM/>_[quadratum] 'ak' ad <SYMBOL-QUADRATUM/>_[quadratum] 'al', ita parallelogrammum 'ag' ad parallelogrammum 'ke'. Similiter ostendemus, reliqua parallelogramma 'lf', 'mh', 'ni', 'ob' esse inter se ut <SYMBOL-QUADRATUM/>_[quadrat]a linearum 'ak', 'al', 'am', 'an', 'ao', 'ap', sese aequaliter excedentium et quarum excessus minimae 'ak' est aequalis. Cum itaque sint huiusmodi spacia ut <SYMBOL-QUADRATUM/>_[quadratum] linearum sese aequaliter excedentium, quarum excessus minimae est aequalis; sintque alia, totidem numero, magnitudine vero unumquodque maximo 'ob' aequalia, parallelogrammum nempe 'cp' componentia; <INSERTION-1> constat, haec </INSERTION-1> <DELETION-1> quae </DELETION-1> ad spacia sese aequaliter excedentium linearum minora <SUBSTITUTION-1> sunt <SUBSTITUTED-BY-1> esse </SUBSTITUTED-BY-1></SUBSTITUTION-1> quam <SYMBOL-3/>_[tri]pla.</s></p>
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488 <p><s>Dico praeterea, non esse minus parallelogrammum 'cp' quam <SUBSTITUTION-1> sexquialterum <SUBSTITUTED-BY-1> <SYMBOL-3/>_[tri]plum </SUBSTITUTED-BY-1></SUBSTITUTION-1> ad idem spacium 'apb'. Si enim quis dicat esse minus, sit defectus 'x', et figura similiter inscribatur, constans ex parallelogrammis 'kq', 'lr', 'ms', 'nt', 'ov', quae sunt ut <SYMBOL-QUADRATUM/>_[quadrat]a linearum sese aequaliter excedentium etc., quae deficiat a dicto spacio minori quantitate quam sit 'x', cum deficiat per minorem quam sit 'ob', quae erit adhuc maior quam <SYMBOL-3/>_[terti]a pars parallelogrammi 'cp'; quod pariter est falsum, cum sit minor.</s></p>
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489 <pb xlink:href="X01/01/149.jpg" page="103_R"/>
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490 <p><s>Sublimitates Parabolarum eiusdem Amplitudinis partium 10000.</s></p>
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491 <pb xlink:href="X01/01/150.jpg" page="103_V"/>
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492 <p><s>Tabula altitudinum semiparabolarum infra elevationem gr[adus] 45, quarum impetus est semper idem, nempe sublimitas cum altitudine 10000.</s></p>
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493 <p><s>Sit impetus datus semper idem, nempe 'bd' ex altitudine et sublimitate composita linea 'db' 10000; et quia, dimidia amplitudo, nempe 'bf', mediat inter altitudinem et sublimitatem, intelligatur 'db' divisa ita, ut <SYMBOL-RECTANGULUM/>_[rectangulum] partium sit aequale <SYMBOL-QUADRATUM/>_[quadrat]o 'fb'. Quod si 'db' divisa sit bifariam in 'e', erit <SYMBOL-QUADRATUM/>_[quadratum] 'be' aequale <SYMBOL-RECTANGULUM/>_[rectangulo] partium ipsius 'db' et <SYMBOL-QUADRATUM/>_[quadrato] 'ae'; si ergo a <SYMBOL-QUADRATUM/>_[quadrato] 'be' dematur <SYMBOL-QUADRATUM/>_[quadratum] 'fb' (seu dicas <SYMBOL-RECTANGULUM/>_[rectangulum] illi aequale, a partibus contentum), remanebit <SYMBOL-QUADRATUM/>_[quadratum] 'ea', cuius radix <DELETION-1> 'adc' </DELETION-1> dempta ex 'eb', relinquet 'ba' altitudinem quaesitam. Amplitudo autem 'bc' iam calculata est ad singulos gradus elevationis.</s></p>
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494 <pb xlink:href="X01/01/151.jpg" page="104_R"/>
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495 <pb xlink:href="X01/01/152.jpg" page="104_V"/>
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496 <pb xlink:href="X01/01/153.jpg" page="105_R"/>
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497 <pb xlink:href="X01/01/154.jpg" page="105_V"/>
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498 <pb xlink:href="X01/01/155.jpg" page="106_R"/>
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499 <pb xlink:href="X01/01/156.jpg" page="106_V"/>
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500 <p><s><foreign lang="it">Prima proposizione. Che 'l proietto descrive la parabola.</foreign></s></p>
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501 <p><s><foreign lang="it">2a. Prova il moto composto di 2 equabili, orizontale e perpendicolare, essere in potenza eguale ad amendue.</foreign></s></p>
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502 <p><s><foreign lang="it">3. Considera il moto composto di 2, orizontale equabile e perpendicolare accelerato.</foreign></s></p>
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503 <p><s><foreign lang="it">4. Mostro come si debba determinar l'impeto del proietto in tutti punti della parabola.</foreign></s></p>
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504 <p><s><foreign lang="it">5. Trovare nell'asse prolungato della data parabola, il punto sublime dal quale il cadente descrive la parabola. Segue il corollario, che la metà dell'ampiezza è media tra l'altezza e la sublimità della parabola. Si aggiugne l'altro corollario, che è, le amplitudini delle parabole essere eguali quando le loro altezze e sublimità alternatamente sono eguali.</foreign></s></p>
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505 <p><s><foreign lang="it">6. Data la sublimità e l'altezza, trovar l'ampiezza della parabola.</foreign></s></p>
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506 <p><s><foreign lang="it">7. Nel descriver parabole di ampiezze eguali, minor impeto si ricerca in quella la cui ampiezza è doppia dell'altezza, che in qual si voglia altra. Segue per corollario, nelle parabole descritte dal medesimo impeto l'amplitudine massima esser di quella che nasce dall'elevazione dell'angolo semiretto.</foreign></s></p>
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507 <p><s><foreign lang="it">8. Le ampiezze de i tiri cacciati con l'istesso impeto, e per angoli egualmente mancanti, o eccedenti l'angolo semiretto, sono eguali.</foreign></s></p>
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508 <p><s><foreign lang="it">9. Le ampiezze sono eguali delle parabole, le altezze e sublimità delle quali si rispondono contrariamente</foreign></s></p>
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509 <p><s><foreign lang="it">10. I momenti delle parabole d'eguali ampiezze, son fra loro come i momenti delle altezze perpendicolari dalle quali si generano esse parabole.</foreign></s></p>
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510 <p><s><foreign lang="it">11. Il momento di qualsivoglia semiparabola è eguale al momento del cadente per la perpendicolare composta dell'altezza e sublimità della semiparabola.</foreign></s></p>
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511 <p><s><foreign lang="it">12. Dato l'impeto e l'ampiezza, trovar l'altezza della parabola.</foreign></s></p>
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512 <p><s><foreign lang="it">Simp[licio] Che la palla ricacciata in su descriva la medesima 'sx', mi par duro. <lb/> Sagr[redo] Ma se non vi par duro che, descrivendo la parabola intera 'yxs', possa ridescriver la 'sxy', non vedete <DELETION-1> <ILLEGIBLE/> </DELETION-1> che di necessità fa la 'sx' ?</foreign></s></p>
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513 <pb xlink:href="X01/01/157.jpg" page="107_R"/>
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514 <pb xlink:href="X01/01/158.jpg" page="107_V"/>
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515 <pb xlink:href="X01/01/159.jpg" page="108_R"/>
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516 <pb xlink:href="X01/01/160.jpg" page="108_V"/>
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517 <pb xlink:href="X01/01/161.jpg" page="109_R"/>
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518 <pb xlink:href="X01/01/162.jpg" page="109_V"/>
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519 <pb xlink:href="X01/01/163.jpg" page="110_R"/>
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520 <p><s><foreign lang="it">Per trovar l'altezze delle parabole <lb/> Dal <SYMBOL-QUADRATUM/>_[quadrat]o della metà dell'impeto (che è l'altezza con la sublimità della parabola) cava il <SYMBOL-QUADRATUM/>_[quadrat]o della metà dell'ampiezza della semiparabola; e la radice del rimanente, aggiunta alla metà dell'impeto, darà l'altezza cercata, quando l'elevazione è più di gr[ado] 45. Per la presente tavola che si fabrica, la metà dell'impeto è sempre 5000; <DELETION-1> ed il suo <SYMBOL-QUADRATUM/>_[quadrat]o 25000000 </DELETION-1> ma se l'elevazione sarà meno di gr[ado] 45, la detta radice del rimanente si de' sottrar dalla metà dell'impeto, ed il restante è l'altezza cercata.</foreign></s></p>
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521 <pb xlink:href="X01/01/164.jpg" page="110_V"/>
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522 <p><s>impetus in 'b' ex 'a' sit 100. Sitque 'bc' ipsi 'ba' aequalis: erit impetus in 'd' per 'abd' 142 proxime, et distantia 'cd' 200. Impetus in 'f' erit 125, distantia vero 'fi' 150: deberet autem esse 176 fere, ut servaretur ratio impetus in 'd' ad suam distantiam 'dc'. Impetus in 'h' fere 160, distantia eius 'hk' 250.</s></p>
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523 <p><s>impetus in 'd' e[x] 'e' ut 56938, in 'f' ut 44, in 'h' ut 63</s></p>
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524 <pb xlink:href="X01/01/165.jpg" page="111_R"/>
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525 <p><s>Sit 'ce' <SYMBOL-2/>_[du]pla ad 'ea', et 'fc' tangat parabolam 'ac'. Sit adhuc 'hd' aequalis 'ce' et maior quam dupla ad 'dg', et 'kh' tangat parabolam 'gh', et ut 'kg' ad 'gi', ita sit 'ig' ad 'gl'; <INSERTION-1> erit 'l' initium casus per parabolam 'gh', </INSERTION-1> et sit 'gx' media inter 'ae', 'gd'; <INSERTION-1> 'gs' vero media inter 'ig', 'gl': </INSERTION-1> demonstrandum est <SUBSTITUTION-1> 'lx' <SUBSTITUTED-BY-1> 'sx' </SUBSTITUTED-BY-1></SUBSTITUTION-1> maiorem esse quam 'fb'.</s></p>
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526 <p><s><SYMBOL-QUADRATUM/>_[quadratum] 'fb' aequatur <SYMBOL-QUADRATUM/>_[quadrat]is 'fa', 'ab', idest est duplum <SYMBOL-QUADRATUM/>_[quadrat]i 'gi'; et <SYMBOL-QUADRATUM/>_[quadratum] 'lx' ['sx'] aequatur <SYMBOL-QUADRATUM/>_[quadrat]is 'lg' ['sg'], 'gx': ostende ergo <SYMBOL-QUADRATUM/>_[quadrat]a 'lg' ['sg'], 'gx', vel <SYMBOL-QUADRATUM/>_[quadrat]um 'lx' ['sx'], esse plus quam <SYMBOL-2/>_[du]pla <DELETION-1> <SYMBOL-QUADRATUM/>_[quadrat]um </DELETION-1> <SYMBOL-QUADRATUM/>_[quadrat]i 'ig'.</s></p>
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527 <p><s><SYMBOL-QUADRATUM/>_[quadrat]um 'gx' aequatur <SYMBOL-QUADRATUM/>_[quadrat]o 'igd': ut 'dg' ad 'gx', ita 'gx' ad 'gi': ergo ut 'dg' ad 'gi', it[a] <SYMBOL-QUADRATUM/>_[quadratum] 'dg' ad <SYMBOL-QUADRATUM/>_[quadratum] 'gx'; ut autem 'dg', seu 'kg', ad 'gi', ita 'ig' ad 'gl'. Quia ut <SYMBOL-QUADRATUM/>_[quadratum] 'xg' ad <SYMBOL-QUADRATUM/>_[quadratum] 'gi', ita 'ig' ad 'gl'; ut autem 'ig' ad 'gl', ita <SYMBOL-QUADRATUM/>_[quadratum] 'ig' ad <SYMBOL-QUADRATUM/>_[quadratum] mediae inter 'ig', 'gl', <INSERTION-1> quae sit 'gs'; </INSERTION-1> ergo ut <SYMBOL-QUADRATUM/>_[quadratum] 'xg' ad <SYMBOL-QUADRATUM/>_[quadratum] 'gi', ita <SYMBOL-QUADRATUM/>_[quadratum] 'gi' ad <SYMBOL-QUADRATUM/>_[quadratum] <SUBSTITUTION-1> mediae inter 'ig', 'gl' <SUBSTITUTED-BY-1> 'gs'. </SUBSTITUTED-BY-1></SUBSTITUTION-1> Est autem 'xg' minor quam 'gi' (quia et 'dg' minor est quam 'gi'): ergo <SYMBOL-QUADRATUM/>_[quadratum] 'ig' minor est <SYMBOL-QUADRATUM/>_[quadrat]o mediae. Sed cum 3 <SYMBOL-QUADRATUM/>_[quadrat]a 'xg', 'gi' et mediae sint proportionalia, erunt extrema plus quam dup[l]a <SYMBOL-QUADRATUM/>_[quadrat]i 'gi'. <DELETION-1> Ergo multo plus quam <SYMBOL-2/>_[du]pla erunt <SYMBOL-QUADRATUM/>_[quadrat]a 'xg', 'gl' </DELETION-1> </s></p>
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528 <pb xlink:href="X01/01/166.jpg" page="111_V"/>
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529 <p><s><DELETION-2> Sit <SYMBOL-TRIANGULUM/>_[triangulum] rectangulum 'abc', latera habens aequalia 'ac', 'cb'. Fiant <SYMBOL-ANGULUS/>_[anguli] aequales 'dba', 'abe', et divisa 'ec' bifariam in 'f', et ducta 'fg' parallela 'cb', fiat ut 'ef' ad 'fg', ita 'gf' ad 'fl'. Dico quod si tota 'dc' bifariam secetur in 'h', ducta 'hi' parallela 'bc', erit ut 'dh' ad 'hi', ita 'ih' ad 'hl'. </DELETION-2> </s></p>
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530 <p><s><DELETION-2> Quia enim <DELETION-1> <ILLEGIBLE/> </DELETION-1> angulus 'cab' aequatur angulo 'cba', et 'dba' angulo 'abe', et angulus 'ceb' <SYMBOL-2/>_[duo]bus 'eab', 'abe' est aequalis, ergo 'ceb' ipsi 'cbd' aequabitur, et <SYMBOL-3/>_[tri]angulus 'ecb' <SYMBOL-3/>_[tri]angulo 'dcb' erit similis, et illis quoque et inter se similes sunt 'egf', 'dih'. Sed quia est ut 'ef' ad 'fg', ita 'gf' ad 'fl', erit <SYMBOL-3/>_[tri]angulus 'agf' ipsi 'egf' similis, et ipsi quoque 'dih'. </DELETION-2> </s></p>
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531 <pb xlink:href="X01/01/167.jpg" page="112_R"/>
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532 <p><s>Tabula altitudinum semiparabolarum a proiectis eodem impetu explosis descriptarum.</s></p>
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533 <pb xlink:href="X01/01/168.jpg" page="112_V"/>
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534 <pb xlink:href="X01/01/169.jpg" page="113_R"/>
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535 <pb xlink:href="X01/01/170.jpg" page="113_V"/>
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536 <pb xlink:href="X01/01/171.jpg" page="114_R"/>
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537 <p><s><BEGIN-OF-LINE-DESTROYED/> [Aequ]ales sunt amplitudines parabolarum, quarum altitudines et sublimitates e contrario sibi respondent.</s></p>
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538 <p><s><BEGIN-OF-LINE-DESTROYED/> [Parabolae] 'fh' altitudo 'gf' ad altitudinem 'cb' parabolae 'bd' eandem habeat rationem, <BEGIN-OF-LINE-DESTROYED/> [quam] sublimitas 'ba' ad sublimitatem 'fe'. Dico, amplitudines 'hg', 'dc' esse aequales. Cum enim prima 'gf' ad <SYMBOL-2/>_[secund]am 'cb' eandem habeat rationem quam <SYMBOL-3/>_[terti]a 'ba' ad <SYMBOL-4/>_[quart]am 'fe', <SYMBOL-RECTANGULUM/>_[rectangulum] 'gfe', primae et <SYMBOL-4/>_[quart]ae, aequale erit <SYMBOL-RECTANGULUM/>_[rectangulo] 'cba', <SYMBOL-2/>_[secund]ae et <SYMBOL-3/>_[terti]ae; ergo quadrata quae ihisce <SYMBOL-RECTANGULUM/>_[rectangul]is aequalia sunt, aequalia erunt inter se: <SYMBOL-RECTANGULUM/>_[rectangul]o vero 'gfe' aequale est <SYMBOL-QUADRATUM/>_[quadratum] dimidiae 'gh'; <SYMBOL-RECTANGULUM/>_[rectangul]o autem 'cba' aequatur <SYMBOL-QUADRATUM/>_[quadratum] dimidiae 'cd': ergo quadrata haec, et eorum latera, et laterum <SYMBOL-2/>_[du]pla, aequalia erunt. Haec autem sunt amplitudines 'gh', 'cd': ergo patet propositum.</s></p>
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539 <p><s>Scritta</s></p>
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540 <pb xlink:href="X01/01/172.jpg" page="114_V"/>
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541 <pb xlink:href="X01/01/173.jpg" page="115_R"/>
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542 <p><s>Sit parabola 'abc', <SUBSTITUTION-1> sitque 'cd' dupla ipsius 'da' <SUBSTITUTED-BY-1> cuius amplitudo 'cd' dupla sit altitudinis 'da', </SUBSTITUTED-BY-1></SUBSTITUTION-1> et illa tangat 'ec' in puncto 'c'. Erit <SUBSTITUTION-1> 'ce' <SUBSTITUTED-BY-1> 'ae' </SUBSTITUTED-BY-1></SUBSTITUTION-1> aequalis 'ad'; et cadens ex <DELETION-1> <ILLEGIBLE/> </DELETION-1> 'e' conversum in 'a' describit parabolam 'abc'. Sumatur in parabola quodlibet punctum 'b': contemplandum est quomodo pro describenda parabola 'ab' <DELETION-1> requiratur idem impetus cadentis ex 'e' usque ad 'a' </DELETION-1> ex 'a' reperiatur punctum 'e' ex <INSERTION-1> quo </INSERTION-1> decidat proiectum. Tangat 'bgf' <DELETION-1> isp </DELETION-1> ipsam in 'b', et ducatur orizontalis 'bh': erit 'ah' aequalis 'af'. Dico modo punctum 'e' reperiri, <DELETION-1> si f </DELETION-1> quia ut 'af' ad 'ag', ita est 'ga' ad 'ae'. Quod sic probatur. Ut 'da' ad 'ag', ita <SYMBOL-2/>_[du]pla 'da' ad <SYMBOL-2/>_[du]plam 'ag', nempe 'dc' ad 'hb'; et ut <SYMBOL-QUADRATUM/>_[quadratum] 'da' ad <SYMBOL-QUADRATUM/>_[quadratum] 'ag', ita <SYMBOL-QUADRATUM/>_[quadratum] 'dc' ad <SYMBOL-QUADRATUM/>_[quadratum] 'hb', et ita est linea 'da' ad 'ah', seu 'ea' ad 'af'. Constat igitur quod, si datae parabolae 'ab' inveniendus sit punctus sublimis 'e', ex quo cadens conficiat parabolam 'ab', posita 'af' aequali 'ah', et ducta 'fgb' quae parabolam tanget in 'b', sumpta <SYMBOL-3/>_[terti]a proportionalis ipsarum 'fa', 'ag', dabit 'ae' ex qua cadens, etc.: quod erat faciendum.</s></p>
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543 <p><s>Melius. Sit parabola 'ab' cuius amplitudo 'bh', et axis perpendicularis 'he', in quo invenienda sit altitudo ex qua cadens parabolam describat. Ponatur 'af' aequalis 'ah', et connectatur 'fb' secans orizontalem 'ag' in 'g', et <SUBSTITUTION-1> tangentem <SUBSTITUTED-BY-1> tangens </SUBSTITUTED-BY-1></SUBSTITUTION-1> parabolam in 'b'. Sitque ipsarum 'fa', 'ag' <SYMBOL-3/>_[terti]a proportionalis 'ae'. Dico 'e' esse punctum quaesitum. Si enim intelligamus 'ea' esse mensuram temporis casus ex 'e' in 'a' et impetus aquisiti in 'a', erit 'ag' (media nempe inter 'ea', 'af') tempus et impetus venientis ex 'f' in 'a', seu ex 'a' in 'h'.</s></p>
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544 <p><s>Sed impetus in 'a' <DELETION-1> <ILLEGIBLE/> </DELETION-1> cadentis ex 'e', tempore <DELETION-1> a </DELETION-1> 'ea', cum impetu aquisito in 'a' conficit in orizontali motu aequabili duplam 'ea'; ergo etiam eodem impetu, in tempore 'ag', conficiet duplam 'ga', nempe 'bh', et in perpendiculari motu ex quiete, eodem tempore <INSERTION-1> 'ga', </INSERTION-1> conficit 'ah'. Ergo eodem tempore conficiuntur amplitudo 'hb' et altitudo 'ah': describitur ergo parabola 'ab' ex casu ex 'e': quod quaerebatur.</s></p>
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545 <p><s>Scritta</s></p>
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546 <p><s>Tangat parabolam 'os'. Demonstrandum ut 'ob' ad 'bi', ita esse 'ib' ad 'ba', ita ut media sit 'bi' inter 'ab' 'bo'.</s></p>
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547 <p><s>momentum in 'g' 82 <lb/> momentum in 'f' 63</s></p>
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548 <pb xlink:href="X01/01/174.jpg" page="115_V"/>
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549 <p><s>tempus totius diametri est 280, cum eius longitudo fuerit per 48143; ergo tempus diametri, cuius longitudo 4000, erit 80 2/3.</s></p>
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550 <pb xlink:href="X01/01/175.jpg" page="116_R"/>
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552 <pb xlink:href="X01/01/177.jpg" page="117_R"/>
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554 <pb xlink:href="X01/01/179.jpg" page="118_R"/>
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555 <pb xlink:href="X01/01/180.jpg" page="118_V"/>
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556 <p><s>Data amplitudine et altitudine semiparabolae, sublimitatem reperire.</s></p>
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557 <p><s>Id statim colligitur ex eo quod dimidia amplitudo m[e]diat inter altitudinem et sublimitatem; ergo, diviso <SYMBOL-QUADRATUM/>_[quadrat]o dimidiae amplitudinis per altitudinem, habebimus sublimitatem quaesitam.</s></p>
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558 <p><s>Altitudines semiparabolarum, quarum eadem est amplitudo, aequantur dimidiae tangenti arcum suarum elevationum.</s></p>
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559 <p><s><SUBSTITUTION-1>Amplitudines semiparabolae <SUBSTITUTED-BY-1> Amplitudines semiparabolarum ad eodem impetu descriptarum. </SUBSTITUTED-BY-1></SUBSTITUTION-1></s></p>
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560 <p><s><SUBSTITUTION-1> Altezze delle Parabole <SUBSTITUTED-BY-1>Amplitudines semiparabolarum ad eodem impetu descriptarum, amplitudinibus congruentes. </SUBSTITUTED-BY-1></SUBSTITUTION-1></s></p>
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561 <p><s>Tabula continens altitudines et sublim[itates] <END-OF-LINE-DESTROYED/> semiparabolarum quarum amplitudo sit eadem, partium scilicet 10000, ad singulos gradus elevationis calculata.</s></p>
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562 <pb xlink:href="X01/01/181.jpg" page="119_R"/>
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563 <p><s>Tabula altitudinum semiparabolarum infra elevationem gr[adus] 45, quarum impetus sit idem etc.</s></p>
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564 <pb xlink:href="X01/01/182.jpg" page="119_V"/>
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565 <pb xlink:href="X01/01/183.jpg" page="120_R"/>
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566 <pb xlink:href="X01/01/184.jpg" page="120_V"/>
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567 <p><s>Tabula amplitudinum semiparabolarum quae a proiectis eodem impetu describuntur.</s></p>
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568 <pb xlink:href="X01/01/185.jpg" page="121_R"/>
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569 <pb xlink:href="X01/01/186.jpg" page="121_V"/>
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570 <pb xlink:href="X01/01/187.jpg" page="122_R"/>
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571 <p><s><DELETION-1> Data amplitudine et </DELETION-1> Ex datis amplitudinibus semiparabolarum in praecedenti tabula digestis, retenctoque [retentoque] communi impetu quo unaquaeque describitur, singularum semiparabolarum altitudines elicere.</s></p>
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572 <p><s>Sit amplitudo data 'bc'; impetus vero qui semper idem 'ob', hoc est aggregatum altitudinis et sublimitatis: reperienda est ac distinguenda ipsamet altitudo. Id autem consequemur, cum 'bo' ita divisa fuerit, ut 'bi', dimidia amplitudinis, media fuerit inter partes ipsius 'ob'. Factum iam sit, et cadat punctus divisionis in 'f'. Sitque eadem 'bo' bifariam secta in 'd'. Est igitur <SYMBOL-QUADRATUM/>_[quadratum] 'ib' aequale <SYMBOL-RECTANGULUM/>_[rectangul]o 'bfo'; <SYMBOL-QUADRATUM/>_[quadratum] vero 'do' aequatur eidem <SYMBOL-RECTANGULUM/>_[rectangul]o cum <SYMBOL-QUADRATUM/>_[quadrato] 'fd'. Si igitur ex <SYMBOL-QUADRATUM/>_[quadrat]o 'do' auferatur <SYMBOL-QUADRATUM/>_[quadratum] 'bi', quod <SYMBOL-RECTANGULUM/>_[rectangul]o 'bfd' ['bfo'] est aequale, remanebit <SYMBOL-QUADRATUM/>_[quadratum] 'fd', cuius latus 'df' additum lineae 'bd' dabit altitudinem 'bf' quaesitam. Ex datis itaque intentum assequimur. Operatio itaque talis erit.</s></p>
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573 <p><s>Ex <SYMBOL-QUADRATUM/>_[quadrato] dimidiae 'bo' datae aufer <SYMBOL-QUADRATUM/>_[quadratum] 'bi' notae; residui accipe radicem, quae erit 'fd'; hanc adde notae 'db', et habebis altitudinem 'bf' quaesitam.</s></p>
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574 <p><s>Scritta</s></p>
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575 <p><s>Semiparabolarum <DELETION-1> eodem </DELETION-1> proiectorum ab eodem impetu iuxta quascumque elevationes, per singulos gradus supra et infra 45, amplitudines calculo colligere, et in tabulas exigere.</s></p>
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576 <p><s>Constat ex praedemonstratis, parabolarum impetus esse aequales, cum illarum sublimitates, cum altitudinibus iunctae, aequales conficiunt perpendiculares supra orizontem: inter easdem ergo parallelas orizontales hae perpendiculares compraehendi debent. Ponatur itaque orizontali 'cb' perpendicularis 'ba' aequalis, et connectatur 'ac'. Erit 'acb' angulus gr[adus] 45, nempe semirectus; divisaque perpendiculari 'ba' bifariam in 'd', semiparabola 'dc' erit, <INSERTION-1> quae </INSERTION-1> a sublimitate 'ad' cum altitudine 'db' designatur, <SUBSTITUTION-1> et <ILLEGIBLE/> <SUBSTITUTED-BY-1> et quantitas </SUBSTITUTED-BY-1></SUBSTITUTION-1> impetus <INSERTION-1> eius </INSERTION-1> in 'c' <DELETION-1> quantitas </DELETION-1> eadem erit atque impetus in 'b' venientis mobilis ex quiete in 'a' per lineam 'ab'. Et si ducatur 'ag' aequidistans 'bc', <DELETION-1> reliquae om </DELETION-1> reliquarum omnium semiparabolarum quarum impetus idem <INSERTION-1> futurus </INSERTION-1> sit cum modo explicato, altitudines cum sublimitatibus iunctae <DELETION-1> in </DELETION-1> spatium inter parallelas 'ag', 'bc' explere debent. Insuper, cum iam demonstratum sit, semiparabolarum quarum tangentes aequaliter, sive supra sive infra, ab elevatione semirecta distant, amplitudines aequales esse, calculus quem pro maioribus elevationibus compilabimus, pro minoribus quoque deserviet. Elegimus praeterea numerum 10000, decemmillia, partium pro maxima amplitudine proiectionis ad elevationem gr[adus] 45. Tanta itaque supponatur esse linea 'ba' et amplitudo semiparabolae 'bc'. <DELETION-1> cuius dupla esset, nempe 20000, integrae parabolae amplitudo </DELETION-1>. Iam, ad opus accedentes, ducatur 'ce', angulum 'ecb' angulo 'acb' maiorem compraehendens, sitque semiparabola designanda, quae a linea 'ec' tangatur, <DELETION-1> eius </DELETION-1> et cuius sublimitas cum altitudine ipsam 'ba' adaequet.</s></p>
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577 <p><s>Ex tabula tangentium, per angulum datum 'bce' tangens ipsa 'be' accipiatur, et bifariam dividatur in 'f'; deinde ipsarum 'bf' et 'bi' (dimidia[e] 'bc') tertia reperiatur proportionalis, <DELETION-1> reperiatur </DELETION-1> quae necessario maior erit quam 'fa', cum 'bf' maior sit quam <SUBSTITUTION-1> 'bd' eodem 'fa' <SUBSTITUTED-BY-1> 'db'. </SUBSTITUTED-BY-1></SUBSTITUTION-1> <INSERTION-2> Id constat: quia <SYMBOL-RECTANGULUM/>_[rectangulum] sub 'fb' et <SYMBOL-3/>_[tertia]a proportionalis ipsarum 'fb', 'bd' aequale est <SYMBOL-QUADRATUM/>_[quadrat]o mediae 'db'; at <SYMBOL-RECTANGULUM/>_[rectangulum] sub 'bf', 'fa' <SUBSTITUTION-2> minor <SUBSTITUTED-BY-2> minus </SUBSTITUTED-BY-2></SUBSTITUTION-2> est <SYMBOL-QUADRATUM/>_[quadrat]o 'bd': deficit enim per <SYMBOL-QUADRATUM/>_[quadratum] 'fd'. </INSERTION-2> Sit igitur illa 'fo'. Semiparabolae igitur in <SYMBOL-3/>_[tri]ang[ul]o 'ecb' inscriptae iuxta tangentem 'ec', <DELETION-1> sublimitas est </DELETION-1> cuius 'cb' est amplitudo, 'bf' altitudo, sublimitas est 'fo'. Verum tota 'bo' supra parallelas extenditur: nos autem operae pretium habemus, <INSERTION-1> talem </INSERTION-1> eandem ex altitudine et sublimitate compositam ipsi 'ba' aequari; sic enim tum ipsa, tum semiparabola 'dc' <DELETION-1> ab eodem impetu </DELETION-1> a proiectis ex 'c' impetu eodem describentur. Reperienda igitur est altera semiparabola huic similis (innumerae enim iuxta elevationem anguli 'bce', maiores et minores, <DELETION-1> atque </DELETION-1> designare possunt, quae omnes inter se similes erunt), <DELETION-1> cuius </DELETION-1> cuius composita sublimitas cum altitudine, omologa scilicet ipsi 'bc', aequetur 'ba'. Fiat igitur ut 'ob' ad 'ba', ita amplitudo 'bc' ad 'cr', <INSERTION-1> et </INSERTION-1> inventa erit 'cr', quaesita amplitudo scilicet semiparabolae iuxta elevazionem [elevationem] anguli 'bce' designatam, cuius sublimitas cum altitudine iuncta intra spazium [spatium] a parallelis 'ga', 'cb' contenctum [contentum] habetur: quod quaerebatur.</s></p>
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578 <p><s>Operatio igitur [tal]is erit: Anguli dati 'bce' tangens reperiatur 'be', cuius medietati apponatur <SYMBOL-3/>_[terti]a proportionalis ipsius et medietatis 'bc', quae sit 'fo'; dein fiat ut 'ob' ad 'ba', ita 'bc' ad aliam, quae sit 'cr', amplitudo nempe quaesita. Exemplum apponamus. Sit angulus 'ecb' gr[adus] 55; eius tangens 11918, cuius dimidium, nempe 'bf', 5959; dimidia 'bc' 2500; harum dimidiarum <SYMBOL-3/>_[terti]a proportionalis 4195, quae addita ipsi 'bf', nempe 5959, dat 10154 pro ipsa 'bo'. Fiat rursus ut 'ob' ad 'ba', nempe ut 10154 ad 10000, ita 'bc', <DELETION-1> ad aliam </DELETION-1> nempe 10000 (utraque enim gra[dus] 45 est tangens), <INSERTION-1> ad </INSERTION-1> aliam, et habebimus amplitudinem 'rc' quaesitam 9848, qualium 'bc' 10000. <DELETION-1> illius </DELETION-1> Harum autem <SYMBOL-2/>_[du]pl[a]e sunt integrae amplitudines integrarum parabolarum, scilicet 19696 et 20000. Tantaque est amplitudo parabolae proiecti iuxta elevationem gr[adus] 35, cum aequaliter distent a gr[adus] 45.</s></p>
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579 <pb xlink:href="X01/01/188.jpg" page="122_V"/>
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580 <p><s><foreign lang="it">Sia l'angolo 'adc' gr[ado] 45: è manifesto che dalla sublimità 'ab' nascerà la parabola, la cui altezza 'bc'. Posto l'angolo 'edc' gr[ado] 55, si cerca la parabola alla elevazione di gr[ado] <DELETION-1> <ILLEGIBLE/> </DELETION-1> 55, la cui sublimità e altezza siano eguali alla 'ac'.</foreign></s></p>
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581 <p><s><foreign lang="it">Con falsa posizione cerca se di tal parabola fusse l'asse nella 'ec', con la tangente 'ed'. E però dividendo la 'ec' in mezo in 'f', onde l'altezza di tal parabola sia 'fc' e la sublimità 'fa': il che allora sarebbe quando la metà dell'ampieza 'cd' si trovasse esser media proporzionale tra la 'cf' e la 'fa'. <DELETION-1> nel quale media </DELETION-1> Ma tra 'ef' (cioè 'fc') et 'fa' media <SUBSTITUTION-1> la 'gf' <SUBSTITUTED-BY-1> una </SUBSTITUTED-BY-1></SUBSTITUTION-1> minore della metà di 'cd', alla quale è eguale la 'fi', per lo che tirata la 'ib' perpendicolare alla 'ec', cioè parallela alla 'ga', si haverà <INSERTION-1> essendo che la metà di 'cd' è media tra 'cb' e 'ba'. <DELETION-2> Posta dunque la 'fe' eguale alla metà di 'ca' </DELETION-2> Trova dunque quale è la sublimità tra la quale e la 'fc' sia media la metà dell'ampiezza 'cd', cioè la 'cb', e trovata che sia, pongasegli eguale la 'fo', et harassi </INSERTION-1> la sublimità 'of' descrivere la parabola, la cui altezza sia 'fc' ed ampiezza 'cd'.</foreign></s></p>
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582 <p><s><foreign lang="it">E' dunque tal parabola maggiore della cercata, secondo che la 'oc' è maggiore della 'ac', ma ben gl'è simile, sendo toccata dalla 'ed'. Convien dunque descriverne altra simile, diminuendo la sua sublimità e ampiezza secondo la proporzione di 'ca' a 'co'. <DELETION-1> Tirisi dunque la retta 'od', la quale seghi la 'ga' in 's', e per 's' passi la parallela alla alla [sic] 'ce', che sia 'tn': haremo il <SYMBOL-3/>_[tri]angolo 'sdn' simile all' 'ode', e 'l 'dtn' simile all' 'edc' </DELETION-1> Facciasi dunque come 'oc' a 'ca', così 'cd' a 'cn'.</foreign></s></p>
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583 <p><s><foreign lang="it">Si cerca l'ampiezza 'nd'.</foreign></s></p>
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584 <p><s><foreign lang="it">Data la tangente <SUBSTITUTION-1> 'de' <SUBSTITUTED-BY-1> 'ce' </SUBSTITUTED-BY-1></SUBSTITUTION-1> mediante l'angolo dato 'edc', dividasi in mezo in 'f', e delle 'fc', 'cs' sia <SYMBOL-3/>_[terz]a proporzionale 'fo', che sarà la sublimità della parabola 'fd'. <INSERTION-1> Congiugni 'cf' con 'fo', facendo 'oc'; </INSERTION-1> facciasi poi come 'oc' a 'ca', così 'cd' a 'dn', et haremo l'ampiezza cercata, cioè della parabola la cui sublimità, e altezza sono eguali alla 'ac', e per conseguenza nascono da impeti eguali de' proietti cacciati dal punto 'd'.</foreign></s></p>
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585 <pb xlink:href="X01/01/189.jpg" page="123_R"/>
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586 <p><s><foreign lang="it">trova di 'fc', 'cd', la <SYMBOL-3/>_[terz]a proporzionale 'fi'; <lb/> come 'ic' a 'ca', così 'fa', 'cs' a 'cn'; <lb/> trova 'es' per i <SYMBOL-QUADRATUM/>_[quadrat]i 'ec', 'cs'; <lb/> trova 'rn'; come 'cs' a 'sn', così <SUBSTITUTION-1> 'es' a 'sr' <SUBSTITUTED-BY-1> 'ec' a 'rn'; </SUBSTITUTED-BY-1></SUBSTITUTION-1> <lb/> vedi se la metà di 'sn' è media tra 'nq' e 'qo'.</foreign></s></p>
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587 <pb xlink:href="X01/01/190.jpg" page="123_V"/>
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588 <p><s><foreign lang="it">Delle vendite e baratti di bestie <lb/> Et per dare qualche regola circa le comprevendite baratti o altri contratti, che si facessero di alcune bestie baccine, cavalline, muline o asinine, et li difetti di essi, et in che tempo si posson rendere o farne il protesto, dichiarando nondimeno di ciascuna delle respettive, soprascritte bestie, si tenga ragione, eccetto però che delle non dome, delle quali il venditore o altri non sia tenuto stare se non delle malattie, et perciò con tale dichiaratione statuirne et ordinarne che chi havrà comperato o in alcun modo ricevuto da altri alcun bue, vacca, bufoli o altra bestia vaccina, e dal venditore gli sarà stata vantata per sana e da huomo da bene, o non sarà detto cosa alcuna come di sotto si dirà, o si troverà havere uno delli infrascritti difetti o malattie, la possa rendere et far rimenare, o farli il protesto ne tempi infrascritti, e gli sieno restituiti i sua danari e tutte le spese che havessi perciò giustamente fatte, e tali difetti, esempi sono gli infrascritti, cioè lunatico, mal di pietra, e mal caduco, o vero maestro tempo di quarantacinque dal dì di tal vendita o altro contratto.</foreign></s></p>
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589 <pb xlink:href="X01/01/191.jpg" page="124_R"/>
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590 <pb xlink:href="X01/01/192.jpg" page="124_V"/>
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591 <pb xlink:href="X01/01/193.jpg" page="125_R"/>
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592 <p><s><foreign lang="it">Molto Ill[ust]re et Ecc[ellentissi]mo Sig[no]r mio P[ad]ron Col[tissi]mo <lb/> Ho fatto diligenza per trovare il capretto o agnello, ma in questi paesi e a' mercati circonvicini prima che a Pasqua non si trova cosa buona, al qual tempo, quando V[ostra] S[ignoria] ne vorrà, haverò occasione di poterla servire. Rendo infinite grazie a V[ostra] S[ignoria] delli amorevoli offizii offerti per Giulio, ma io per me non intendo d'affaticare V[ostra] S[ignoria] per procacciarmi causa di nuovi rossori. Mando sei tordi che costano nove crazie, mentre co 'l fine gli faccio debita reverenza. Da S[an]ta Maria a Campoli 3 Marzo 1636 <lb/> Di V[ostra] S[ignoria] molto Ill[ust]re et Ecc[ellentissi]ma <lb/> Devotiss[i]mo e Oblig[atissi]mo se[vito]re <lb/> Alessandro Ninci</foreign></s></p>
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593 <pb xlink:href="X01/01/194.jpg" page="125_V"/>
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594 <p><s>Altitudines semiparabolarum, quarum eadem sit amplitudo, reperire.</s></p>
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595 <p><s>Id autem absolvitur per dimidiam tangentem arcum elevationis datae semiparabolae.</s></p>
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596 <p><s>Inventa, ex dictis, altitudine, sublimitates singularum semiparabolarum, quarum eadem sit amplitudo, facile reperies. Nam, cum dimidia amplitudo mediet inter altitudinem et sublimitatem, diviso <SYMBOL-QUADRATUM/>_[quadrato] mediae amplitudinis per altitudinem, habebimus sublimitatem, quae postea, addita altitudini, exibet impetum.</s></p>
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597 <p><s>Fabricemus ergo tabulam sublimitatum, sitque semper dimidia a[m]plitudo semiparabolae 5000. Eius <SYMBOL-QUADRATUM/>_[quadratum] semper idem 25000000. Elevatio sit gr[adus] 1, tangens ipsius 174 1/2, qualium tangens gr[adus] 45 est 10000.</s></p>
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598 <p><s>tangens gr[adus] 1, 174 1/2. Eius dimidium 87 1/4: per hunc numerum divide <SYMBOL-QUADRATUM/>_[quadratum] 25000000.</s></p>
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599 <p><s>Tabula <lb/> Altitudinum semiparabolis ad singulos grados elevationis</s></p>
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600 <pb xlink:href="X01/01/195.jpg" page="126_R"/>
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601 <p><s>Sit 'fg' orizon, et ex sublimi 'a' fiat motus per 'abf', et protracta 'ab' usque ad 'd', sit media inter 'da', 'ab' ipsa 'ac', et orizonti aequidistans sit 'ce'. Dico, tempus per 'ab' ad tempus per 'bf' esse ut 'ab' ad 'be'. Nam tempus per 'ab' ad tempus per 'bd' est ut 'ab' ad 'bc': tempus vero per 'bd' post 'ab' ad tempus per 'bf' post 'ab' est ut 'bd' ad 'bf', idest 'bc' ad 'be': ergo, ex aequali, tempus per 'ab' ad tempus per 'bf' est ut 'ab' ad 'be'.</s></p>
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602 <pb xlink:href="X01/01/196.jpg" page="126_V"/>
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604 <pb xlink:href="X01/01/198.jpg" page="127_V"/>
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605 <p><s>Serva</s></p>
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606 <p><s>Motuum qui a dato puncto usque ad datam lineam per rectas lineas conficiuntur, <SUBSTITUTION-1> illae <SUBSTITUTED-BY-1> ille </SUBSTITUTED-BY-1></SUBSTITUTION-1> brevissimo tempore absolvitur, qui in recta fit abscindens de data linea partem aequalem ei parti <DELETION-1> de </DELETION-1> lineae orizontalis, quae <DELETION-1> per </DELETION-1> per datum punctum usque ad datam lineam producitur, quae inter datum punctum et occursum intercipitur.</s></p>
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607 <p><s>Sit datum punctum 'a' et linea quaecumque 'bdc', et per 'a' orizonti aequidistans 'ab', quae lineae 'db' in 'b' occurat, et interceptae 'ab' ponatur aequalis 'bd'. Dico, motum per 'ad' absolvi tempore breviori ori [sic], quam per quamcumque aliam lineam ex puncto 'a' ad quodcumque punctum lineae 'bdc' <SUBSTITUTION-1> productae <SUBSTITUTED-BY-1> productam. </SUBSTITUTED-BY-1></SUBSTITUTION-1> </s></p>
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608 <p><s>Ducatur ad 'ba' perpendicularis 'ac', et ex 'd' ad ipsam 'bc' perpendicularis 'de', occurrens 'ac' in 'e': et quia in <SYMBOL-TRIANGULUM/>_[triangulo] aequicruri 'abd' anguli 'bad', 'bda' sunt aequales, ergo reliqui ad rectos, nempe 'ead', 'eda', aequales pariter erunt, et linea 'ea' aequalis ipsi 'ed'. Si itaque, centro 'e', intervallo 'ea', circulus describatur, transibit per 'd', ubi lineam 'bdc' tanget: quare lineae omnes quae supra vel infra 'ad' usque ad lineam 'bc' producentur, ultra circumferentiam circuli extendentur. Ex quo patet propositum.</s></p>
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609 <p><s>Sit ad orizontem 'ab' linea 'cd' utcumque inclinata, et in ipso orizonte quodlibet punctum notatum 'a': oportet in linea 'cd' punctum invenire, a quo in linea recta usque ad 'a' <SUBSTITUTION-1> protractas <SUBSTITUTED-BY-1> protracta </SUBSTITUTED-BY-1></SUBSTITUTION-1> brevissimo tempore fiat motus. Erigatur ex 'a' perpendicularis ad orizontem 'ac', et ex eodem demittatur perpendicularis ad 'cd', quae sit 'ae', et angulus 'cae' bifariam secetur per 'fa'. Dico, ex omnibus lineis quae a puncto 'a' ad lineam 'cd' protrahuntur, <DELETION-1> <ILLEGIBLE/> </DELETION-1> 'fa' esse illam per quam motus brevissimo tempore absolvitur. Ducatur enim 'fg' ipsi 'ea' parallela; erit <SYMBOL-ANGULUS/>_[angulus] 'gfa' <SYMBOL-ANGULUS/>_[angul]o <DELETION-1> 'gaf' </DELETION-1> alterno 'fae' aequalis; sed <SYMBOL-ANGULUS/>_[angulus] 'fae' ipsi 'fag' aequatur, cum totus 'cae' sit bifariam sectus: ergo 'gaf', 'gfa' aequales erunt, quare et latera 'gf', 'ga'. Si itaque, centro 'g', intervallo 'gf', circulus describatur, tanget ambas lineas 'cd', 'ab' in punctis 'f', 'a', eritque casus per 'fa' brevioris temporis quam per rectas quascumque alias ex 'a' ad quaecumque punta lineae 'cd' productas.</s></p>
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610 <pb xlink:href="X01/01/199.jpg" page="128_R"/>
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611 <p><s><foreign lang="it">Io suppongo (et forse potrò dimostrarlo) che il grave cadente naturalmente vada continuamente accrescendo la sua velocità secondo che accresce la distanza dal termine onde si partì: come, v[erbi] g[ratia], partendosi il grave dal punto 'a' et cadendo per la linea 'ab', suppongo che il grado di velocità nel punto <DELETION-1> di </DELETION-1> 'd' sia tanto maggiore che il grado di velocità in 'c', quanto la <DELETION-1> linea </DELETION-1> distanza 'da' è maggiore della 'ca', et così il grado di velocità in 'e' esser al grado di velocità in 'd' come 'ea' a 'da', et così in ogni punto della linea 'ab' trovarsi con gradi di velocità proporzionali alle distanze de i medesimi punti dal termine 'a'. Questo principio mi par molto naturale, et che risponda a tutte le esperienze che veggiamo negli strumenti et machine che operano percotendo, dove il percuziente fa tanto maggiore effetto quanto da più grande altezza casca: et supposto questo principio dimostrerò il resto.</foreign></s></p>
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612 <p><s><foreign lang="it">Faccia la linea 'ak' qualunque angolo con la 'af', et per li punti 'c', 'd', 'e', 'f', siano tirate le parallele 'cg', 'dh', 'ei', 'fk': et perchè <DELETION-1> come f <ILLEGIBLE/> 'fa' </DELETION-1> le linee 'fk', 'ei', 'dh', 'cg' sono tra di loro come le 'fa', 'ea', 'da', 'ca', adunque le velocità ne i punti 'f', 'e', 'd', 'c' sono come le linee 'fk', 'ei', 'dh', 'cg'. Vanno dunque continuatamente crescendo i gradi di velocità in tutti i punti della linea 'af' secondo l'incremento del[le] parallele tirate da tutti i medesimi punti. In oltre, perchè la veloci[tà] con la quale il mobile è venuto da 'a' in 'd' è composta di tut[ti] i gradi di velocità hauti in tutti i punti della linea 'ad', et la velocità con che ha passata la linea 'ac' è composta di tutti i gradi di velocità che ha <SUBSTITUTION-1> hauto <SUBSTITUTED-BY-1> hauti </SUBSTITUTED-BY-1></SUBSTITUTION-1> in tutti i punti della linea 'ac', adunque la velocità con che ha passata la linea 'ad', alla velocità con che ha passata la linea 'ac', ha quella proportione che hanno tutte linee parallele tirate da tutti i punti della linea <SUBSTITUTION-1> 'af' <SUBSTITUTED-BY-1> 'ad' </SUBSTITUTED-BY-1></SUBSTITUTION-1> sino alla 'ah', a tutte le parallele tirate da tutti i punti della linea 'ac' sino alla 'ag'; et questa proportione è quella che ha il triangolo 'adh' al <SYMBOL-3/>_[tri]angolo 'acg', ciò è il <SYMBOL-QUADRATUM/>_[quadrat]o 'ad' al <SYMBOL-QUADRATUM/>_[quadrat]o 'ac'.</foreign></s></p>
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613 <pb xlink:href="X01/01/200.jpg" page="128_V"/>
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614 <p><s><foreign lang="it">Adunque la velocità con che si è passata la linea 'ad', alla velocità con che si è passata la linea 'ac', ha doppia proporzione di quella che ha 'da' a 'ca'. Et perchè la velocità alla velocità ha contraria proporzione di quella che ha il tempo al tempo (imperò che il medesimo è crescere la velocità che sciemare il tempo), adunque <DELETION-1> la linea </DELETION-1> il tempo del moto in 'ad' al tempo del moto in 'ac' ha subduplicata proporzione di quella che ha la distanza 'ad' alla distanza 'ac'. Le distanze dunque dal principio del moto sono come i quadrati de i tempi, et, dividendo, gli spazii passati in tempi eguali sono come i numeri impari ab unitate: che risponde a quello che ho sempre detto et con esperienze osservato; et così tutti i veri si rispondono.</foreign></s></p>
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615 <p><s><foreign lang="it">Et se queste cose son vere, io dimostro che la velocità nel moto violento va decrescendo con la medesima proporzione con la quale, nella medesima linea retta, cresce nel moto naturale. Imperò che sia il principio del moto violento il punto 'b', et il fine il termine 'a'. Et per che il proietto non passa il termine 'a', adunque l'impeto che ha hauto in 'b' fu tanto, quanto poteva cacciarlo sino al termine 'a'; et l'impeto che il medesimo proietto ha in 'f' è tanto quanto può cacciarlo <DELETION-1> in </DELETION-1> al medesimo termine 'a'; et sendo il medesimo proietto in 'e', 'd', 'c', si trova congiunto con impeti potenti a spingerlo al medesimo termine 'a', nè più nè meno: adunque l'impeto va giustamente calando secondo che sciema la distanza del mobile dal termine 'a'. Ma secondo la medesima delle distanze dal termine 'a' va crescendo la velocità quando il medesimo grave caderà dal <DELETION-1> per </DELETION-1> punto 'a', come di sopra si è supposto et confrontato con le altre prime nostre osservazioni et dimostrazioni: adunque è manifesto quello che volevamo provare.</foreign></s></p>
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616 <pb xlink:href="X01/01/201.jpg" page="129_R"/>
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617 <p><s>Sit 'bd' media inter 'sd', 'dc', et centro 'd', intervallo 'b', secetur 'de', et per 'b' ipsi 'st' parallela 'bia'.</s></p>
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618 <p><s>Quia 'ts' tangit, et 'tc' secat, et 'ba' est parallela 'ts', erit <SYMBOL-TRIANGULUM/>_[triangulum] 'ati' similis <SYMBOL-3/>_[tri]angulo 'tcd'.</s></p>
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619 <p><s>Credo angulum 'sec' bifariam esse sectum per 'eb'.</s></p>
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620 <p><s>Angulus 'tds' duabus circumferentiis 'oc', 'ct' insistit; ergo illae sunt similes, et circumferentia 'do' similis est 'dct'; ergo ut linea 'do' ad 'oc', ita 'dt' ad 'tc': et quia <SYMBOL-RECTANGULUM/>_[rectangulum] 'dsc' aequatur <SYMBOL-QUADRATUM/>_[quadrato] 'st', ergo ut 'ds' ad 'st', ita 'ts' ad 'sc': ergo <SYMBOL-TRIANGULUM/>_[triangul]a 'dst', 'tsc' similia sunt, quibus et <SYMBOL-TRIANGULUM/>_[triangula] 'odc', 'icb' similia sunt.</s></p>
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621 <p><s>Quia est ut 'sd' ad 'de', ita 'de' ad 'dc', ergo <SYMBOL-TRIANGULUM/>_[triangulum] 'sde' similis est <SYMBOL-TRIANGULUM/>_[triangul]o 'dec', et ut 'se' ad 'ec', ita 'sd' ad 'de', et ita est 'sb' ad 'bc': ergo angulus 'ces' bifariam secatur linea 'eb'.</s></p>
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622 <pb xlink:href="X01/01/202.jpg" page="129_V"/>
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623 <pb xlink:href="X01/01/203.jpg" page="130_R"/>
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624 <p><s>media inter 'ag' 'af' 165</s></p>
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625 <pb xlink:href="X01/01/204.jpg" page="130_V"/>
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626 <p><s><SYMBOL-RECTANGULUM/>_[rectangulum] 'iae' esse omnium minimum 'lab', 'oan', 'paf' etc.: cum angulus <SUBSTITUTION-1> 'cex' <SUBSTITUTED-BY-1> 'cax' </SUBSTITUTED-BY-1></SUBSTITUTION-1> bifariam sectus sit, pendet ex eo, quod angulus 'aem' <SYMBOL-3/>_[tri]ang[ul]i 'aem' est aequalis angulo 'aix' <SYMBOL-3/>_[tri]ang[ul]i 'aix' et, quod consequens, est minor omnium 'alx', 'aox', etc., et maior omnium 'api', 'aci' etc.: probabitur ergo sic, <SYMBOL-RECTANGULUM/>_[rectangulum] 'iae' esse minus <SYMBOL-RECTANGULUM/>_[rectangul]o 'lab'. Cum enim angulus <SUBSTITUTION-1> 'amx' <SUBSTITUTED-BY-1> 'ame' </SUBSTITUTED-BY-1></SUBSTITUTION-1> sit aequalis angulo 'axi', et angulus 'mae' aequalis angulo 'xai' (est enim angulus 'a' bifariam sectus), ergo reIiquus 'mea' reliquo 'xia' aequabitur: sed angulus 'aem' maior est angulo 'abe': ergo <SYMBOL-ANGULUS/>_[angulus] 'ail' est maior <SYMBOL-ANGULUS/>_[angulo] 'eba'. Si igitur fiat <SYMBOL-ANGULUS/>_[angulus] <DELETION-1> 'aet' </DELETION-1> 'ait' <SYMBOL-ANGULUS/>_[angul]o 'abe' aequalis, erit, ob triangulorum similitudinem, ut 'ia' ad 'at', ita 'ba' ad 'ae', et <SYMBOL-RECTANGULUM/>_[rectangulum] 'iae' <SYMBOL-RECTANGULUM/>_[rectangul]o 'tab' aequale: ergo <SYMBOL-RECTANGULUM/>_[rectangulum] 'iae' est minus <SYMBOL-RECTANGULUM/>_[rectangul]o 'lab'.</s></p>
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627 <p><s>Similiter ostendetur esse quoque minus <SYMBOL-RECTANGULUM/>_[rectangulum] 'paf'. Cum enim <SYMBOL-ANGULUS/>_[angulus] 'aef', idest 'ail', sit maius angulo 'api', erit reliquus <SUBSTITUTION-1> 'aef' <SUBSTITUTED-BY-1> 'afe' </SUBSTITUTED-BY-1></SUBSTITUTION-1> minor reliquo 'aip'. Si igitur constituatur 'aiv' <SYMBOL-ANGULUS/>_[angulus] ipsi 'afe' aequalis, erit <SYMBOL-RECTANGULUM/>_[rectangulum] 'vaf' <SYMBOL-RECTANGULUM/>_[rectangul]o 'iae' aequale, ex quo patet propositum.</s></p>
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628 <p><s>Demon[s]trabitur etiam, quod rectangula talia quae a lineis ex 'a' ad lineam 'cx' ductis et a linea 'xm' sectis, ea quae <DELETION-1> fiu <ILLEGIBLE/> </DELETION-1> fiunt a lineis vicinioribus ipsi 'aei' semper minora sunt illis quae a remotioribus describuntur lineis.</s></p>
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629 <p><s>Constat insuper, quod media inter 'iae' est omnium mediarum minima, quae cadunt inter 'paf', 'lab', etc.</s></p>
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630 <p><s>Aliter brevius. Posito angulo 'ae8' aequali <SYMBOL-ANGULUS/>_[angul]o 'eam', erit linea 'e8' parallela 'am'; ergo perpendicularis ad 'mx', eritque aequalis '8a': quare, centro '8', intervallo '8e', circulus tanget 'mx' in 'e': unde patet propositum.</s></p>
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631 <p><s>Vide num 'ya' ad 'a3' sit ut 'dc' ad 'cz'.</s></p>
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632 <p><s>'s' per 'abc'; <lb/> 'q' per 'aec'; <lb/> 'g' per 'afc'; <lb/> 'q' per 'anc'.</s></p>
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633 <pb xlink:href="X01/01/205.jpg" page="131_R"/>
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634 <p><s><DELETION-1> accipiatur media inter 'rc', 'bt', cui aequalis ponatur 'do': erit reliqua 'co' eadem quae invenitur per mediam inter 'cd', 'df', quae est eadem 'do'. </DELETION-1> </s></p>
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635 <p><s>Considera momentum in singulis circumferentiae quadrantis pu[n]ctis imminui pro ratione <DELETION-1> perpendicularis </DELETION-1> accessus puncti perpendicularis, <UNDERLINED> ut 't', </UNDERLINED> ad centrum.</s></p>
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636 <p><s>ut 'bc' ad 'cd', sic 'cd'</s></p>
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637 <p><s>momentum super plano 'dc' ad totale momentum est ut linea 'tr' ad 'rd', ducta 'lb' aequidistante 'cd'.</s></p>
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638 <pb xlink:href="X01/01/206.jpg" page="131_V"/>
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639 <pb xlink:href="X01/01/207.jpg" page="132_R"/>
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640 <pb xlink:href="X01/01/208.jpg" page="132_V"/>
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641 <p><s>Detur 'ih'. dabitur 'io' per ablationem <SYMBOL-QUADRATUM/>_[quadrati] 'ih' ex <SYMBOL-QUADRATUM/>_[quadrat]o 'ho'. Deinde, ablata 'ih' ex 'bc', datur <SUBSTITUTION-1> 'bl' <SUBSTITUTED-BY-1> 'lc', </SUBSTITUTED-BY-1></SUBSTITUTION-1> cuius <SYMBOL-QUADRATUM/>_[quadratum], ablatum ex <SYMBOL-QUADRATUM/>_[quadrat]o 'ch' dato, dat <SYMBOL-QUADRATUM/>_[quadratum] 'lh' et ipsam 'lh', idest 'bi': ergo dabitur tota 'bo'.</s></p>
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642 <pb xlink:href="X01/01/209.jpg" page="133_R"/>
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643 <p><s><foreign lang="it">Sian 'ca', 'ab', <DELETION-1> note </DELETION-1> 'ad' note. Sarà nota anco 'de' e 'bf': e perchè 'dh' è nota, sendo eguale a 'de' et essendo il <SYMBOL-TRIANGULUM/>_[triangolo] 'hid' simile al noto 'fbc', sarà noto 'di'; et è nota 'dl', che sono i sini degl'archi 'hn', 'mn', li quali però saranno noti, e la loro proporzione. Sia il globo solare, il cui semidiametro 'ab', e sia l'arco 'bl' gr[ado] 30: sarà la linea 'ld' 866 di quali 'ab' è 1000. Prima, è manifesto che 2 punti 'b', 'l', posti nella superficie, passeranno i sini 'ld', 'ba' nell'istesso tempo: è in oltre chiaro, che ponendogli nelle linee 'de', 'ac' prolungate in infinito, i punti 'e', 'c' traverserebbono le medesime linee 'ba', 'ld' in tempi proporzionali ad esse; sì che, non si dando tal distanza infinita, i transiti per 'ba', 'ld' si faranno in tempi che fra di loro haranno minor proporzione che non ha la linea 'ba' alla 'dl'. E perchè, sendo 'dl' 866, 'ab' e [é] 1000, et il tempo per 'ld' al tempo per 'ba' deve esser come 7 a 8, facciasi come 7 a 8, così 866 a un'altra, <INSERTION-1> che sia 'di'; </INSERTION-1> sarà 947, e la rimanente 'ig' sarà 53. <DELETION-1> Tirisi per 'i' la parallela a 'bf', e per 'd' la parallela ad 'af' </DELETION-1> Adattisi la 'io' eguale a 'gd', e per 'a' passi la parallela 'ae', che concorra con 'dg' in 'e', e, centro 'a', facciasi 'l cerchio 'cef'</foreign></s></p>
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644 <pb xlink:href="X01/01/210.jpg" page="133_V"/>
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645 <p><s>scribo <lb/> scribbo <lb/> aqui</s></p>
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646 <pb xlink:href="X01/01/211.jpg" page="134_R"/>
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647 <pb xlink:href="X01/01/212.jpg" page="134_V"/>
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648 <pb xlink:href="X01/01/213.jpg" page="135_R"/>
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649 <pb xlink:href="X01/01/214.jpg" page="135_V"/>
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650 <pb xlink:href="X01/01/215.jpg" page="136_R"/>
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651 <pb xlink:href="X01/01/216.jpg" page="136_V"/>
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652 <p><s><foreign lang="it">ut 'ba' ad 'ac', ita 'bd' ad 'de'; ut autem 'ac' ad 'ci', ita 'de' ad 'ef': ergo ut 'ba' ad 'ci', ita 'bd' ad ad [sic] 'ef'.</foreign></s></p>
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653 <pb xlink:href="X01/01/217.jpg" page="137_R"/>
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654 <p><s>'is' ad 'or' ut 'ab' ad 'bd'</s></p>
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655 <pb xlink:href="X01/01/218.jpg" page="137_V"/>
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656 <p><s><foreign lang="it">Silvio Grazii stà in casa il Clarissimo Trivisano.</foreign></s></p>
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657 <p><s><foreign lang="it">'fe' ad 'ed' est ut 'dc' ad 'ca'; ut autem 'cd' ad 'eh', ita 'ac' ad 'cb': ergo, ex aequali, ut 'fe' ad 'eh', ita 'dc' ad 'cb'.</foreign></s></p>
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658 <p><s><foreign lang="it">Parti <DELETION-1> 'fd' </DELETION-1> 'eh' per 'ef', et tante volte quant'è l'avvenimento, entra 'dc' in 'cb'.</foreign></s></p>
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659 <p><s>monons<lb/>ms<lb/>m</s></p>
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660 <pb xlink:href="X01/01/219.jpg" page="138_R"/>
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661 <p><s>Si mobile aequabiliter latum duo pertranseat spacia, erunt tempora lationum inter se ut spatia peracta.</s></p>
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662 <p><s>Pertranseat enim mobile aequabiliter latum duo spacia 'AB', 'BC', et sit tempus motus ex 'A' in 'B', 'DE'; tempus vero lationis 'BC' esto 'EF'. Dico ut spacium 'AB' ad spacium 'BC' ita esse tempus 'DE' ad tempus 'EF'. Protractis enim utrinque spaciis et temporibus, sumantur quotcunque spacia in 'AG' ipsi 'AB' aequalia, et totidem tempora in 'DI', tempori 'DE' similiter aequalia; et rursus in 'CH' sumantur secundum quamcunque multitudinem <DELETION-1> tempora ip </DELETION-1> spacia ipsi 'CB' aequalia, et totidem tempora in 'FK', tempori 'EF' aequalia: erunt itaque spacium 'BG' et tempus 'EI' aeque multiplicia spacii 'BA' et temporis 'ED' iuxta quamcunque multiplicationem accepta, et similiter 'HB' spacium et 'KE' tempus spacii 'CB' temporisque 'FE' aeque multiplicia in qualibet multiplicatione. Et quia 'DE' est tempus lationis per 'AB', erit totum <DELETION-1> tempus </DELETION-1> 'EI' tempus totius 'BG', cum motus ponatur aequabilis sintque in 'EI' tot tempora ipsi 'DE' aequalia, quot sunt in 'BG' spatia aequalia 'BA'; et similiter ostendetur 'KE' esse tempus lationis per 'HB'. Cum autem motus ponatur aequabilis, si spacium 'GB' esset aequale ipsi 'BH', et tempus 'IE' tempori 'EK' esset aequale. Et si 'GB' maius sit quam 'BH', et 'IE' quam 'EK' <DELETION-1> <ILLEGIBLE/> </DELETION-1> maius erit, et si minus, minus. Tempus igitur 'IE' et spacium 'GB' aequae [aeque] multiplicia sunt, <INSERTION-1> iuxta quamcumque multiplicationem accepta, </INSERTION-1> temporis 'DE' et spacii 'AB' et vel una aequatur vel una deficiunt vel una excedunt tempus 'EK' et spacium 'BH', aeque multiplicia temporis 'EF' et spacii 'BC' in qualibet multiplicatione: ergo ut spacium 'AB' ad spacium 'BC', ita tempus 'DE' ad tempus 'EF': quod erat demonstrandum.</s></p>
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664 <p><s>Si fuerint quotlibet spacia, et alia illis multitudine paria, quae bina sumpta eandem habeant rationem, et per ipsa duo moveantur mobilia, ita ut in binis quibusque spaciis sibi respondentibus lationes sint <INSERTION-2> aequales et </INSERTION-2> aequabiles, <DELETION-2> <INSERTION-1> et aeque celeres </INSERTION-1> </DELETION-2> erunt ut omnia antecedentia spatia ad omnia consequentia, ita tempora lationum omnium antecedentium ad tempora lationum omnium consequentium spaciorum.</s></p>
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665 <p><s>Sint 'ab', 'bc', 'cd' spacia quotcumque, et alia multitudine aequalia 'ef', 'fg', 'gh'; et sit ut 'ab' ad 'ef', ita 'bc' ad 'fg' et 'cd' ad 'gh'. Duo autem mobilia <SUBSTITUTION-1> eadem celeritate et motu aequabili <SUBSTITUTED-BY-1> eodem motu et aequabili </SUBSTITUTED-BY-1></SUBSTITUTION-1> ferantur per duo spacia 'ab', 'ef', et tempora lationum sint 'ik', 'no'; 'kl' vero et 'op' sint tempora lationum quarumcumque aliarum aequalium et aequabilium per 'bc', 'fg'; tempora vero 'lm', 'pq' sint aliarum lationum aequalium inter se et aequabilium per 'cd', 'gh'. Dico, ut totum spacium 'ad' ad totum spacium <SUBSTITUTION-1> 'nq' <SUBSTITUTED-BY-1> 'eh', </SUBSTITUTED-BY-1></SUBSTITUTION-1> ita esse tempus totum <DELETION-1> 'if' </DELETION-1> 'im' ad tempus 'nq'.</s></p>
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666 <p><s>Cum enim motus per duo spacia 'ab', 'ef' sint aequales et aequabiles, erit, ex praecedenti, ut spacium 'ab' ad 'ef', ita tempus 'ik' ad 'no'; et similiter demonstrabitur ut 'bc' ad 'fg', ita 'kl' ad 'op', et ut 'cd' ad 'gh', ita 'lm' ad 'pq': et quia est ut 'ab' ad 'ef', ita 'bc' ad 'fg' et 'cd' ad 'gh', erit ut 'ik' ad 'no', ita 'kl' ad 'op' et 'lm' ad 'pq'. Cumque ru[rsus] sit ut 'ab' ad 'ef', ita 'bc' ad 'fg' et 'cd' ad 'gh', erit ut unum 'ab' ad unum 'ef', ita omnia 'ad' ad omnia <DELETION-1> 'ef' </DELETION-1> 'eh'; et similiter concludetur, ut unum <SUBSTITUTION-1> 'gk' <SUBSTITUTED-BY-1> 'ik' </SUBSTITUTED-BY-1></SUBSTITUTION-1> ad unum 'no', ita esse omnia 'im' ad omnia 'nq': est autem ut unum 'ab' ad unum 'ef', ita 'ik' ad 'no': ergo ut totum spacium 'ad' ad totum spacium 'eh', ita tempus 'im' ad tempus 'nq': quod erat ostendendum.</s></p>
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667 <pb xlink:href="X01/01/221.jpg" page="138AR"/>
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668 <pb xlink:href="X01/01/222.jpg" page="138AV"/>
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669 <pb xlink:href="X01/01/223.jpg" page="139_R"/>
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670 <p><s>Scritte <lb/> Si 2 mobilia aequabili motu ferantur, sint tamen velocitates inaequales, et in[a]equalia quoque spacia peracta, ratio temporum composita erit ex <DELETION-1> velocitatibus et spaciis </DELETION-1> ratione spaciorum <DELETION-1> et temporum </DELETION-1> et ex ratione <SUBSTITUTION-1> temporum <SUBSTITUTED-BY-1> velocitatum </SUBSTITUTED-BY-1></SUBSTITUTION-1> contrarie sumptarum.</s></p>
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671 <p><s>Sint 2 mobilia 'A', 'B', sitque velocitas ipsius 'A' ad velocitatem ipsius 'B' ut 'v' ad 't'; spacia autem peracta sint ut 's' ad 'r'. Dico, rationem temporis quo motum est 'A', ad tempus quo motum est 'B', compositam esse ex ratione velocitatis 't' ad velocitatem 'v' (sic enim contrarie sumitur) et ex ratione spacii 's' ad spacium 'r'. Sit tempus motus 'A' 'cd', et ut velocitas <DELETION-1> <ILLEGIBLE/> 'ad' ad </DELETION-1> 't' ad velocitatem 'v', ita sit tempus 'dc' ad tempus 'ef'. Et cum <DELETION-1> tempus </DELETION-1> 'cd' sit tempus in quo 'A' cum velocitate 'v' conficit spacium 's', sitque ut velocitas 't' mobilis 'B' ad velocitatem 'v', ita <DELETION-1> <ILLEGIBLE/> </DELETION-1> tempus 'dc' ad tempus 'ef', erit tempus 'ef' illud in quo mobile 'B' conficeret idem spacium 'S'. Fiat modo ut <DELETION-1> tempus </DELETION-1> spacium 's' ad spacium 'r', ita tempus 'ef' ad tempus 'gh': constat, 'gh' esse tempus quo 'B' conficeret spacium 'r'. Et quia ratio 'cd' ad 'gh' componitur ex rationibus 'cd' ad 'ef' et 'ef' ad 'gh', est autem <DELETION-1> <ILLEGIBLE/> </DELETION-1> ratio 'cd' ad 'ef' eadem cum ratione <DELETION-1> 't' ad </DELETION-1> velocitatum mobilium 'A', 'B' contrarie sumptarum, hoc est cum ratione 't' ad 'v'; ratio vero 'ef' ad 'gh' est eadem cum ratione spaciorum 's', 'r'; ergo patet propositum.</s></p>
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672 <p><s>Scritte <lb/> Si 2 mobilia aequabili motu ferantur, <DELETION-1> quorum velocitates sint inaequales </DELETION-1> ratio ipsarum velocitatum composita erit ex <SUBSTITUTION-1> ratione temporum et <SUBSTITUTED-BY-1> ratione </SUBSTITUTED-BY-1></SUBSTITUTION-1> spaciorum peractorum <INSERTION-1> <BEGIN-OF-LINE-DESTROYED/> [et ex] ratione <BEGIN-OF-LINE-DESTROYED/> [temporum] con[trarie] <BEGIN-OF-LINE-DESTROYED/> sumptorum. </INSERTION-1> </s></p>
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673 <p><s>Sint 2 mobilia 'A', 'B', aequabili motu <DELETION-1> sed inaequali velocitate </DELETION-1> lata; <DELETION-1> <ILLEGIBLE/> </DELETION-1> sint autem spacia ab illis peracta in ratione 'v' ad 't', tempora vero sint ut 's' ad 'r'. Dico, velocitatem mobilis 'A' ad velocitatem ipsius 'B' habere rationem compositam ex ratione spacii 'v' ad spacium 't' et temporis 'r' ad tempus 's'. Sit velocitas <DELETION-1> mobilis 'A' ut 'cd', et quam rationem habet spacium 'v' ad spacium 's', hanc habeat velocitas 'cd' ad aliam 'ef' </DELETION-1> 'cd' ea cum qua mobile 'A' conficit spacium 'v' in tempore 's', et quam rationem habet spacium 'v' ad spacium 't', hanc habeat velocitas 'cd' ad aliam 'ef'; erit 'ef' velocitas cum qua mobile 'B' conficit spacium 't' in tempore eodem 's': quod si fiat, ut tempus 'r' ad tempus 's', ita velocitas 'ef' ad aliam 'gh', erit velocitas 'gh' illa secundum quam mobile 'B' conficit spacium 't' in tempore 'r'. Habemus itaque velocitatem 'cd', cum qua mobile 'A' conficit spacium 'v' in tempore 's', et velocitatem 'gh' cum qua mobile 'B' conficit spacium 't' in tempore 'r'. Et est ratio 'cd' ad 'gh' composita ex rationibus 'cd' ad 'ef' et 'ef' ad 'gh'; ratio autem 'cd' ad 'ef' posita est eadem cum ratione <DELETION-1> <ILLEGIBLE/> </DELETION-1> spacii 'v' ad spacium <SUBSTITUTION-1> 'B' <SUBSTITUTED-BY-1> 't'; </SUBSTITUTED-BY-1></SUBSTITUTION-1> ratio vero 'ef' ad 'gh' est eadem cum ratione 'r' ad 's': ergo patet propositum.</s></p>
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674 <pb xlink:href="X01/01/224.jpg" page="139_V"/>
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675 <pb xlink:href="X01/01/225.jpg" page="140_R"/>
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676 <p><s>Sit linea orizontis 'ac', perpendiculum vero 'bd', et in 'ac' accipiatur quodcumque punctum 'c'. Dico, quod si mobile debet ex 'c' ad lineam perpendiculi naturaliter <INSERTION-1> per unicam lineam rectam moveri </INSERTION-1> moveri [sic], ad eam perveniet tempore brevissimo si veniat per 'ce', quae lineam 'be', ipsi 'bc' aequalem, adsumit. Centro enim 'b', intervallo 'be', circulus describatur, ductisque 'cf' et 'cg' utcumque, patebit, motum per 'ce' citius absolvi quam per 'cf' autem 'cg'. Si enim ducatur tangens circulum 'ick', et ipsi 'cf' <DELETION-1> <ILLEGIBLE/> </DELETION-1> parallela 'elk', erit 'le' brevior quam 'cf': sed tempus per 'ce' aequatur tempori per 'le'. Similiter, ducta <DELETION-1> 'cg' </DELETION-1> 'ehi' ipsi 'cg' parallela et aequali, constat 'cg' longiorem esse 'he': at tempus per 'ce' aequatur tempori per 'he'. Ergo patet propositum.</s></p>
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677 <pb xlink:href="X01/01/226.jpg" page="140_V"/>
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678 <pb xlink:href="X01/01/227.jpg" page="141_R"/>
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679 <pb xlink:href="X01/01/228.jpg" page="141_V"/>
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681 <p><s>Data inflexa ad datum perpendiculum, partem in inflexa accipere, in qua sola fiat motus eodem tempore atque in eadem cum perpendiculo.</s></p>
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682 <p><s>Sit perpendiculus 'ab', et ad ipsum inflexa 'bc': oportet, in 'bc' partem accipere, in qua <INSERTION-1> sola </INSERTION-1> fiat motus codem tempore ac in eadem cum perpendiculo 'ab'. Ducatur orizon 'ad', cui inclinata 'cb' occurrat in 'e', ponaturque 'bf' aequalis 'ba', <DELETION-1> et extensa 'fc', fiat 'eg' aequalis 'ef'. Et ut 'gb' ad 'bf', ita fiat 'bc' ad 'hf' </DELETION-1> et, centro 'e', intervallo 'ef', circulus describatur 'fig', et 'fe' ad circumferentiam usque protahatur in 'g', et ut 'gb' ad 'bf', ita fiat 'bh' ad 'hf', et 'hi' <INSERTION-1> circulum tangat in 'i'. </INSERTION-1> Deinde, ex 'b' perpendicularis ad 'fc' erigatur 'bk', cui occurrat in 'l', 'eil'; tandem ipsi 'el' perpendicularis ducatur 'lm', occurrens 'bc' in 'm'. Dico, in linea 'bm' ex 'b' fieri motum eodem tempore ac ex 'a' per ambas 'abm'.</s></p>
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683 <p><s><INSERTION-1> Ponatur 'en' aequalis 'el'; </INSERTION-1> cumque ut 'gb' ad 'bf', ita sit 'bh' ad 'hf', erit, permutando, ut 'gb' ad 'bh', ita 'bf' ad 'fh', et, dividendo, 'gh' ad 'hb', ut 'bh' ad 'hf'; quare <SYMBOL-RECTANGULUM/>_[rectangulum] 'ghf' <SYMBOL-QUADRATUM/>_[quadrato] 'hb' erit aequale: sed idem <SYMBOL-RECTANGULUM/>_[rectangulum] aequatur quoque <SYMBOL-QUADRATUM/>_[quadrat]o 'hi': ergo 'bh' ipsi 'hi' est aequalis. Cumque in quadrilatero 'ilbh' latera 'hb', 'hi' sint aequalia, et anguli 'b', 'i' recti, erit latus quoque 'bl' ipsi 'li' aequale; est autem 'ei' aequalis 'ef': ergo <DELETION-1> <ILLEGIBLE/> </DELETION-1> tota 'le', seu 'ne', duabus 'lb', 'ef' est aequalis. Auferat[ur] communis 'ef'; ergo reliqua 'fn' ipsi 'lb' aerit[erit] aequalis: at posita est 'fb' ipsi 'ba' aequalis: ergo 'lb' duabus 'ab', 'bn' aequatur.</s></p>
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684 <p><s>Rursus, si intelligatur, tempus per 'ab' esse ipsam 'ab', erit tempus per 'eb' ipsi 'eb' aequale; tempus autem per totam 'em' erit 'en', media scilicet inter 'me', 'eb'; quare reliquae 'bm' tempus <DELETION-1> erit 'bn' </DELETION-1> casus post 'eb', <DELETION-1> erit ipsa 'bn' seu </DELETION-1> seu post 'ab', erit ipsa 'bn'. Positum est autem, tempus per 'ab' esse 'ab': ergo tempus casus per ambas 'abm' est 'abn'. Cum autem tempus per 'eb' ex quiete in 'e' sit 'eb', tempus per 'bm' ex quiete in 'b' erit media proportionalis inter 'be', 'bm'; haec autem est 'bl'; tempus igitur per ambas 'abm' ex quiete in 'a' est 'abn'; tempus vero per 'bm' solam ex quiete in 'b' est 'bl'; ostensum autem est, 'bl' esse aequalem duobus 'ab', 'bn'; ergo patet propositum.</s></p>
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685 <pb xlink:href="X01/01/230.jpg" page="142_V"/>
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686 <pb xlink:href="X01/01/231.jpg" page="143_R"/>
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687 <p><s>Dato perpendiculo et plano ad ipsum inflexo, in dato plano partem signare, in qua post casum in perpendiculo fiat motus eodem tempore, quo mobile perpendiculum confecit.</s></p>
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688 <p><s>Sit perpendiculum 'ab' et ad ispsum [ipsum] inf[l]exa 'be': oportet, in 'be' spacium signare, per quod mobile moveatur <SUBSTITUTION-1> per <SUBSTITUTED-BY-1> post </SUBSTITUTED-BY-1></SUBSTITUTION-1> casum 'ab' eodem tempore quo confecit ipsum 'ab'.</s></p>
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689 <p><s>Extenso plano 'eb', occurrat orizonti in 'd', et accipiatur 'bf' aequalis 'ab', et fiat ut 'bd' ad 'df', ita 'fd' ad 'de'. Dico, tempus per 'be' post 'ab' [a]equari tempori per 'ab'. Si enim intelligatur, 'ab' tempus per 'ab', erit 'db' tempus per 'db'; cumque sit ut 'bd' ad 'df', ita 'fd' ad 'de', erit 'df' tempus per totam 'de', et 'bf' per partem 'be' ex 'd'. Sed tempus <DELETION-1> b </DELETION-1> per 'be' post 'db' est idem ac post 'ab': ergo tempus per 'be' post 'ab' erit 'bf', aequale scilicet tempori 'ab': quod erat propositum.</s></p>
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690 <p><s>Sit in perpendiculo 'ab' accepta pars 'ac', cuius tempus 'ac'; accepta rursus ubicumque parte 'db', ipsi 'ac' aequali, quaeritur tempus quo eadem 'db' post casum ex 'a' conficietur. Circa totam 'ab' semicirculus describatur 'aeb', et orizon ducatur 'ce', et iungatur 'ae', quae maior erit quam 'ce'; sit differentia 'af'. Dico, 'af' esse tempus per 'db'. Quia enim 'ae' est media inter 'ba', 'ac', estque 'ac' tempus per 'ac', erit 'ae' tempus per totam 'ab'; cumque 'ce' media sit inter 'da', 'ac' (est enim 'da' aequalis ipsi 'bc'), erit 'ce' tempus per totam 'ad'; est autem 'ce' aequalis 'ef'; ergo 'ae' est tempus per totam 'ab', 'ef' vero per 'ad'; ergo 'af' erit tempus per 'db'.</s></p>
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691 <pb xlink:href="X01/01/232.jpg" page="143_V"/>
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692 <p><s>Dato quolibet spacio in perpendiculo a principio lationis signato, quod in dato tempore conficiatur, datoque quocunque tempore minori, aliud aequale spacium, priori accepto aequale, in eodem perpendiculo reperire, quod in dato tempore conficiatur.</s></p>
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693 <p><s>Sit perpendiculum 'ab', in quo detur spacium 'ab', cuius tempus ex principio 'a' sit 'ab', sitque orizon 'cbe', et detur tempus ipso 'ab' minus, quod sit 'bc': oportet, in eodem perpendiculo spacium eidem 'ab' aequale invenire, quod tempore 'bc' conficiatur. Iungatur linea 'ac', cumque 'bc' minor sit 'ba', erit angulus 'bac' minor angulo 'bca'. Constituatur ipsi aequalis 'cae', et linea 'ae' orizonti in 'e' signo occurrat, ad quam perpendicularis ponatur 'ed', secans perpendiculum in 'd', et linea 'df' ipsi 'ba' secetur aequalis. Dico, ipsam 'fd' ex 'a' confici in tempore 'bc'. Cum enim in <SYMBOL-3/>_[tri]ang[ul]o rectangulo 'aed' ab angulo recto 'e' perpendicularis ducta sit ad 'ad', erit 'ae' media inter 'da', 'ab', et 'be' media inter 'db', 'ba', hoc est inter 'fa', 'ab' erit (est enim 'fa' aequalis 'db'); cumque 'ab' positum sit esse tempus per 'ab', erit 'ae' tempus per totam 'ad', et 'eb' tempus per 'af'; cumque 'ae' sit aequalis ipsi 'ec' ob aequalitatem angulorum 'eac', <SUBSTITUTION-1> 'ecd' <SUBSTITUTED-BY-1> 'eca', </SUBSTITUTED-BY-1></SUBSTITUTION-1> relinquitur ut 'bc' tempus sit ipsius 'fd': quod erat ostendendum.</s></p>
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694 <pb xlink:href="X01/01/233.jpg" page="144_R"/>
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695 <p><s>Tempus per 'ab', 'ab'; tempus per 'be' ex 'a', 'bf', posita 'df' media inter 'ed', 'db'; ergo tempora per 'abe' erunt 'abf'.</s></p>
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696 <p><s>Ponatur media inter 'eb', 'bd' ipsa 'bh'; erit 'bh' tempus 'be' ex 'b': oportet igitur facere ut 'bh' sit aequalis <SYMBOL-2/>_[dua]bus 'abf', hoc est ut 'ab' sit aequalis ipsi 'fh'.</s></p>
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697 <p><s>Factum sit ut tempus per 2 'abe' sit aequale tempori per solam 'be'. Divisa 'de' bifariam, semicirculus describatur, et ducatur perpendicularis 'bn', et iungatur 'dn': erit 'din' ['dn'] media inter 'ed', 'db', et 'bn' inter 'ebd'. Et existente 'ab' tempore per 'ab', et 'db' per 'db', secetur 'df' aequalis 'dn'; erit 'bf' tempus per 'be' post 'db', seu 'ab': unde tempus per 2 'abe' erit 'abf'; tempus vero per 'be' ex 'b' erit 'bn', cum sit media inter 'db', 'be': ergo 'bn' aequatur <SYMBOL-2/>_[dua]bus 'abf'.</s></p>
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698 <p><s>Posita communi 'bd', erunt 3 'db', 'bf', 'ba', hoc est 2 'dn', 'ba', aequales <SYMBOL-2/>_[dua]bus 'nb', 'bd'. Et ablatis aequalibus 'db', 'dr' et 'bs', 'ba', reliqua 'rn' reliquae 'ns' erit aequalis.</s></p>
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699 <pb xlink:href="X01/01/234.jpg" page="144_V"/>
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700 <pb xlink:href="X01/01/235.jpg" page="145_R"/>
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701 <p><s>Dato perpendiculo, ad ipsum lineam inflectere, in qua, cum ipsa habeat eandeam cum dato perpendiculo altitudinem, fiat motus post casum in perpendiculo <INSERTION-1> eodem tempore, </INSERTION-1> ac in eodem perpendiculo ex quiete.</s></p>
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702 <p><s>Sit datus perpendiculus 'ab', qui protrahatur, et ponatur 'bc' aequalis ipsi 'ab', et ducantur orizontes 'ce', 'ag': oportet, ex 'b' planum usque ad orizontem 'ce' inflectere, in quo fiat motus post casum ex 'a' eodem tempore, ac in 'ab' ex quiete in 'a'. Sumatur 'cd' aequalis 'cb', et ponatur 'be' utrique 'bd', 'dc' aequalis. Dico iam, per 'be' post casum 'ab' fieri motum eodem tempore ac in 'ab'. Producatur 'eb', occurrens orizonti superiori in 'g', <DELETION-1> et secetur 'bf' aequalis 'bc'; erit reliqua 'fe' ipsi 'bd' aequalis, et <SYMBOL-QUADRATUM/>_[quadratum] 'fe' duplum <SYMBOL-QUADRATUM/>_[quadrat]i 'fb'; sed </DELETION-1> et ipsarum 'eg', 'gb' media sit 'gf'; erit 'ef' ad 'fb' ut <DELETION-1> 'fg' </DELETION-1> 'eg' ad 'gf', et <SYMBOL-QUADRATUM/>_[quadratum] 'ef' ad <SYMBOL-QUADRATUM/>_[quadratum] 'fb' ut <SYMBOL-QUADRATUM/>_[quadratum] 'eg' ad <SYMBOL-QUADRATUM/>_[quadratum] 'gf'. Sed <SYMBOL-QUADRATUM/>_[quadratum] 'eg' ad <SYMBOL-QUADRATUM/>_[quadratum] 'gf' est ut 'eg' ad 'gb': est autem 'eg' dupla ad 'gb', quia aequales sunt 'ab', 'bc': ergo <SYMBOL-QUADRATUM/>_[quadratum] 'ef' duplum est <SYMBOL-QUADRATUM/>_[quadrat]i 'fb', et linea 'bf' aequalis 'bc', cum tota 'be' <SYMBOL-2/>_[dua]bus 'bd', 'bc' aequalis posita sit. Si igitur intelligatur, 'ab' esse tempus casus per 'ab', erit 'gb' tempus casus per 'gb', et 'gf' tempus per totam 'ge'; ergo 'bf' erit tempus per reliquam 'be' post casum ex 'g', seu ex 'a': quod est propositum.</s></p>
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703 <pb xlink:href="X01/01/236.jpg" page="145_V"/>
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704 <pb xlink:href="X01/01/237.jpg" page="146_R"/>
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705 <pb xlink:href="X01/01/238.jpg" page="146_V"/>
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706 <pb xlink:href="X01/01/239.jpg" page="147_R"/>
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707 <p><s>Contempletur quod, quemadmodum gravia omnia super orizonte quiescunt, licet maxima vel minima, ita in lineis inclinatis eadem volicate [velocitate] moventur, quemadmodum et in ipso quoque perpendiculo; quod bonum erit demonstrare, <DELETION-1> q </DELETION-1> dicendo quod, si gravius velocius, sequeretur quod gravius tardius, iunctis gravibus inaequalibus, etc.</s></p>
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708 <p><s>Movebuntur autem eadem celeritate non solum gravia inaequalia et homogen[e]a, sed etiam eterogenea, ut lignum et plumbum. Cum enim antea ostensum fuerit, magna et parva homogenea aequaliter moveri, dicas: sit 'b' sp[ha]era lignea et 'a' plumbea, adeo magna ut, cum in medio habeat cavitatem pro 'b', sit tamen gravior quam sphaera solida lignea ipsi a [sic] 'a' aequalis, ita ut per adversarium velocius mov[e]atur quam 'b': ergo si in cavitate 'i' ponatur 'b', tardius movebitur 'a' quam cum erat levior: quod est absurdum.</s></p>
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709 <p><s>t[empus] 'ab', t[empus] 'ae'; <lb/> l[ongitudo] 'bc', l[ongitudo] <DELETION-1> 'aed' </DELETION-1> 'ae'.</s></p>
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710 <p><s><DELETION-1> te[mpus] 'ac' ad 'ae', 'ab' <lb/> li[nea] 'ea', 'ab' </DELETION-1> <lb/> temp[us] 'da' ad temp[us] 'ac' ut lin[ea] 'da' ad lin[ea] 'ac'.</s></p>
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711 <p><s>Concludendum: tempus 'da' ad tempus 'ea' esse ut linea 'ca' ad 'ab'.</s></p>
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712 <p><s> Postea quam ostensum fuerit, tempora per 'ab', 'ac' esse aequalia, demonstrabitur tempus per 'ad' ad tempus per 'ae' esse ut 'da' ad mediam inter 'da', 'ae'. Nam tempus per 'da' ad tempus per 'ac' est ut 'da' ad 'ac' lineam; tempus autem per 'ac', <DELETION-1> ad <ILLEGIBLE/> </DELETION-1> idest per 'ab', ad tempus 'ae' est ut linea 'ba' ad 'ae', hoc est ut 'sa' ad 'ad': ergo, ex aequali in analogia perturbata, tempus per <SUBSTITUTION-1> 'da' <SUBSTITUTED-BY-1> 'ad' </SUBSTITUTED-BY-1></SUBSTITUTION-1> ad tempus per 'ae' est ut linea 'sa' ad lineam 'ac'. Cumque 'ac', ex demonstratis, sit media inter 'sa', 'ab', et ut 'sa' ad 'ab', ita 'da' ad 'ae', ergo <DELETION-1> ut </DELETION-1> tempus per 'ad' ad tempus per 'ae' est ut 'da' ad mediam inter 'da', 'ae': quod erat probandum.</s></p>
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713 <p><s>Paralogismus.</s></p>
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714 <pb xlink:href="X01/01/240.jpg" page="147_V"/>
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715 <p><s>Sit ad orizontem 'ab' perpendicularis 'bc' et inclinata 'bd', in qua sumatur 'be', et ex 'e' ad 'bd' perpendicularis agatur 'ef', ipsi 'bc' occurrens in 'f'. Demonstrandum sit tempus per 'be' aequari tempori per 'bf'.</s></p>
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716 <p><s>Ducatur ex 'e' perpendicularis ad 'ab', quae sit 'eg': <DELETION-1> g </DELETION-1> et quia impetus per 'bd' ad impetum per 'bc' est ut 'eg' ad 'be' (ut infra demonstratur); ut autem 'eg' ad 'be', ita 'be' ad 'bf', ob similitudinem triangulorum 'geb', 'bef'; ergo ut 'bf' spacium ad spacium <SUBSTITUTION-1> 'bd' <SUBSTITUTED-BY-1> 'be', </SUBSTITUTED-BY-1></SUBSTITUTION-1> ita impetus per 'bf' ad impetum per 'be': ergo eodem tempore fiet motus per 'bf' et per 'be'.</s></p>
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717 <p><s>Advertas cur cadentia ex 'a' sint semper una in locis sibi respondentibus, ut 'o', 's', ita ut <SYMBOL-ANGULUS/>_[angulus] 'aos' sit aequalis angulo 'bas'.</s></p>
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718 <p><s>Infra orizontem 'ab' ex eodem puncto 'c' duae rectae aequales utcumque inclinentur 'cd', 'ce', et ex terminis 'd', 'e' ad orizontem perpendiculares agantur 'da', 'eb', et lineae 'cd' in puncto 'd' constituatur <SYMBOL-ANGULUS/>_[angulum] 'cdf' <SYMBOL-ANGULUS/>_[angulo] 'bce' aequalis. Dico, ut 'da' ad 'be' ita esse 'dc' ad 'cf'.</s></p>
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719 <p><s>Ducatur perpendicularis 'cg': et quia 'cdf' aequatur angulo 'bce', et rectus 'g' recto 'b', erit ut 'dc' ad 'cg', ita 'ce' ad 'eb': est autem 'cd' ipsi 'ce' aequalis: ergo 'cg' aequatur 'be'. Et cum angulus 'cdf' angulo 'bce' sit aequalis, et <SYMBOL-ANGULUS/>_[angulus] 'fcd' communis, reliquus ad duos rectos 'dfc' reliquo 'dca' aequabitur, et anguli ad 'a' et 'g' sunt recti; ergo <SYMBOL-TRIANGULUM/>_[triangulum] 'adc' <SYMBOL-TRIANGULUM/>_[triangul]o 'cgf' est simile: quare ut 'ad' ad 'dc', ita 'cg' ad 'cf', et, permutando, ut 'ad' ad 'cg', hoc est ad 'be', ita 'dc' ad 'cf': quod erat probandum.</s></p>
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720 <p><s>Cum autem impetus per 'cd' ad impetum per 'cf' sit ut perpendiculus 'ad' ad perpendiculum 'be', constat, motus per 'cd' et 'cf' eodem tempore absolvi. Itaque distantiae quae in diversis inclinationibus eodem tempore conficiuntur, determinantur per lineam quae (ut facit 'df') <DELETION-1> in </DELETION-1> lineis inclinatis occurrit secundum angulos aequales illis quos inclinatae ad orizontem constituunt, permutatim sumptos.</s></p>
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721 <pb xlink:href="X01/01/241.jpg" page="148_R"/>
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722 <p><s>Sit 'qp' tempus per 'ac' et ut 'ac' ad 'cd' ita fiat 'pq' ad 'qr' erit 'qr' tempus per 'dc' seu per 'bc'. Sit ut 'cd' ad 'do' ita 'rq' ad 'qs' erit 'qs' tempus per 'df' et 'sr' tempus per 'fc' post 'df'. Fiat rursus ut 'ca' ad 'av' ita tempus 'pq' ad 'qt' <DELETION-1> <ILLEGIBLE/> </DELETION-1> erit 'qt' tempus per 'ab' 'tp' vero tempus per 'bc' post 'ab'.</s></p>
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723 <pb xlink:href="X01/01/242.jpg" page="148_V"/>
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724 <pb xlink:href="X01/01/243.jpg" page="149_R"/>
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725 <pb xlink:href="X01/01/244.jpg" page="149_V"/>
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726 <p><s>quaeritur ratio 'co' ad 'cv'.</s></p>
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727 <p><s><SYMBOL-RECTANGULUM/>_[rectangulum] 'cdf' aequatur <SYMBOL-RECTANGULUM/>_[rectangulo] 'rc', 'ds'; <SYMBOL-RECTANGULUM/>_[rectangulum] 'acb' aequatur <SYMBOL-RECTANGULUM/>_[rectangulo] 'rcn'; ergo <SYMBOL-RECTANGULUM/>_[rectangul]um 'cdf' ad <SYMBOL-RECTANGULUM/>_[rectangulum] 'acb' est ut diameter 'ds' ad diametrum 'nc': ut autem 'cn' ad 'ds', ita 'cd' ad 'df', ob similitudinem portionum 'cbc' et 'df': ut autem 'cd' ad 'df', ita <SYMBOL-QUADRATUM/>_[quadratum] 'co' ad <SYMBOL-QUADRATUM/>_[quadratum] 'of'.</s></p>
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728 <pb xlink:href="X01/01/245.jpg" page="150_R"/>
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729 <p><s>Sit 'do' media inter 'cd', 'df'; 'av' media inter 'ca', 'ab'; 'ck' media inter 'ac', 'cb'; et accipiatur utcumque 'ds'.</s></p>
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730 <p><s><UNDERLINED> fiat ut 'od' ad 'df', ita 'sd' ad 'dr'; </UNDERLINED> ut 'cd' ad 'do', ita 'sd' ad 'dr', seu ut 'bc' ad 'cd', <DELETION-1> seut </DELETION-1> seu ut 'dc' ad 'ca'.</s></p>
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731 <p><s>Fiat ut 'bc' ad <DELETION-1> 'ca' </DELETION-1> 'ck', ita 'sd' ad 'dt'; ut autem 'va' ad 'ab', ita 'td' ad 'dg'.</s></p>
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732 <p><s>Probetur 'gt' minorem esse quam 'sr'.</s></p>
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733 <p><s>ostende 'co' maiorem esse 'cv'.</s></p>
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734 <p><s>ostende</s></p>
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735 <p><s>sit ut <DELETION-1> 'ob' </DELETION-1> 'od', 'dc', 'ca', 'av', ita 'rd', 'ds', <DELETION-1> d </DELETION-1> 'td', 'dg'.</s></p>
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736 <p><s>Quia enim ut <SUBSTITUTION-1> 'sp' <SUBSTITUTED-BY-1> 'sd' </SUBSTITUTED-BY-1></SUBSTITUTION-1> ad <SUBSTITUTION-1> 'pr' <SUBSTITUTED-BY-1> 'dr', </SUBSTITUTED-BY-1></SUBSTITUTION-1> ita 'cd' ad 'do', per conversionem rationis et convertendo, ut 'rs' ad <SUBSTITUTION-1> 'sp' <SUBSTITUTED-BY-1> 'sd', </SUBSTITUTED-BY-1></SUBSTITUTION-1> ita 'oc' ad 'cd'; ut autem <SUBSTITUTION-1> 'sp' <SUBSTITUTED-BY-1> 'sd' </SUBSTITUTED-BY-1></SUBSTITUTION-1> ad <SUBSTITUTION-1> 'pt' <SUBSTITUTED-BY-1> 'dt', </SUBSTITUTED-BY-1></SUBSTITUTION-1> ita 'cd' <DELETION-1> ad 'ca', </DELETION-1> <UNDERLINED> hoc est 'kc', </UNDERLINED> ad 'ca': et quia est <DELETION-1> 'ad' </DELETION-1> ut <DELETION-1> d </DELETION-1> <SUBSTITUTION-1> 'tp' ad 'pg' <SUBSTITUTED-BY-1> 'td' ad 'dg', </SUBSTITUTED-BY-1></SUBSTITUTION-1> ita 'ca' ad 'av', per conversionem rationis erit quoque ut <SUBSTITUTION-1> 'pt' <SUBSTITUTED-BY-1> 'dt' </SUBSTITUTED-BY-1></SUBSTITUTION-1> ad 'tg', ita <SUBSTITUTION-1> 'ca' <SUBSTITUTED-BY-1> 'ac' </SUBSTITUTED-BY-1></SUBSTITUTION-1> ad 'cv': ergo, ex aequali, ut 'rs' ad 'gt', ita 'oc' ad 'cv'. Ostenditur autem, per lemmata, 'co' maior quam 'cv'; ergo tempus 'rs' maius est tempore 'gt': est autem 'rs' tempus quo peragitur 'fc' post 'df', 'gt' vero tempus quo peragitur 'bc' post 'ab': ergo patet propositum.</s></p>
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737 <pb xlink:href="X01/01/246.jpg" page="150_V"/>
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738 <pb xlink:href="X01/01/247.jpg" page="151_R"/>
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739 <p><s>Sit 'gd' erecta ad orizontem, <DELETION-1> et ipsi </DELETION-1> 'df' vero <DELETION-1> <ILLEGIBLE/> </DELETION-1> inclinata: dico, eodem tempore fieri motum ex 'g' in 'd' et ex 'f' in 'd'.</s></p>
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740 <p><s>Momentum enim super 'fd' est idem ac super contingente in 'e', quae ipsi 'fd' esset parallela; ergo momentum super 'fd' ad totale momentum erit ut 'ca' ad 'ab', idest <DELETION-1> 'ad' </DELETION-1> 'ae'. Verum ut 'ca' ad 'ae', ita 'id' ad 'da', et dupla 'fd' ad duplam 'dg'; ergo momentum super 'fd' ad totale momentum, scilicet per 'gd', est ut 'fd' ad 'gd': ergo eodem tempore fiet motus per 'fd' et 'gd'.</s></p>
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741 <pb xlink:href="X01/01/248.jpg" page="151_V"/>
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742 <pb xlink:href="X01/01/249.jpg" page="152_R"/>
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743 <p><s>Sit ut 'ba' ad 'ad', ita 'da' ad 'ac', et sit 'be' gradus velocitatis in 'b', et ut 'ba' ad 'ad', ita sit 'be' ad 'cf'; erit 'cf' gradus velocitatis in 'c'. Cum itaque sit ut 'ca' ad 'ad', ita 'cf' ad 'be', erit etiam ut <SYMBOL-QUADRATUM/>_[quadratum] 'ca' ad <SYMBOL-QUADRATUM/>_[quadratum] 'ad', ita <SYMBOL-QUADRATUM/>_[quadratum] 'cf' ad <SYMBOL-QUADRATUM/>_[quadratum] 'be': ut autem <SYMBOL-QUADRATUM/>_[quadratum] 'ca' ad <SYMBOL-QUADRATUM/>_[quadratum] 'ad', ita 'ca' ad 'ab'; ut igitur <DELETION-1> <SYMBOL-QUADRATUM/>_[quadratum] </DELETION-1> 'ca' ad 'ab', ita <SYMBOL-QUADRATUM/>_[quadratum] 'cf' ad <SYMBOL-QUADRATUM/>_[quadratum] 'be': sunt ergo pun[c]ta 'e', 'f' in parabola.</s></p>
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744 <p><s><foreign lang="it">con un grado d'impeto fa 2 miglia all'ora; con 4 gradi d'impeto farà 8 miglia in un'ora, e 16 in 2 ore.</foreign></s></p>
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745 <p><s><foreign lang="it">4 miglia con 10 di velocità in 6<superscript>4</superscript> hore. 9 miglia con 15 di velocità in hore</foreign></s></p>
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746 <p><s><foreign lang="it">per 'ab' velocitas ut 4; per 'ac' velocitas ut 13 1/2.</foreign></s></p>
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747 <p><s><foreign lang="it">4 miglia con 15 di velocità in 4, 8 miglia in 8</foreign></s></p>
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748 <p><s><foreign lang="it">per 'ab' velocitas ut <UNDERLINED> 10 </UNDERLINED>; per 'ac' ut 15</foreign></s></p>
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749 <p><s>tempus per 'ab' 4; 'ac' 6</s></p>
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750 <p><s>'ab' 4 <lb/> 'ad' 6; 'be' 20 <lb/> 'ac' 9; 'cf' 30.</s></p>
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751 <pb xlink:href="X01/01/250.jpg" page="152_V"/>
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752 <pb xlink:href="X01/01/251.jpg" page="153_R"/>
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753 <pb xlink:href="X01/01/252.jpg" page="153_V"/>
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754 <pb xlink:href="X01/01/253.jpg" page="154_R"/>
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755 <pb xlink:href="X01/01/254.jpg" page="154_V"/>
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756 <pb xlink:href="X01/01/255.jpg" page="155_R"/>
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757 <pb xlink:href="X01/01/256.jpg" page="155_V"/>
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758 <pb xlink:href="X01/01/257.jpg" page="156_R"/>
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759 <pb xlink:href="X01/01/258.jpg" page="156_V"/>
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760 <p><s>sit 'ad' media inter 'fa' ad <lb/> ut <DELETION-1> 'fx' </DELETION-1> 'fa' ad 'a' <SYMBOL-ALPHA/>_'ALPHA' ita 'a' <SYMBOL-ALPHA/>_'ALPHA' ad 'ad' <lb/> <DELETION-1> 'f' <SYMBOL-ALPHA/>_'ALPHA' 'a' </DELETION-1> <lb/> <SYMBOL-ALPHA/>_'ALPHAa' ad <lb/> 'f' <SYMBOL-ALPHA/>_'ALPHA' <SYMBOL-ALPHA/>_'ALPHAd'</s></p>
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761 <pb xlink:href="X01/01/259.jpg" page="157_R"/>
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762 <pb xlink:href="X01/01/260.jpg" page="157_V"/>
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763 <p><s>'ce' in 'se' et divide per 'eb' et iisdem num veniat tempus 'aec' </s></p>
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764 <p><s>Inveniendum sit tempus quo conficiuntur <SYMBOL-2/>_[due] 'acb' in rationem ad tempus quo conficitur sola 'ab' fiat 'abg' <SYMBOL-ANGULUS/>_[angulus] rectus et semicirculus 'abg' describatur et protrahatur 'ac' ad 's' et connectantur 'gs' 'bs'</s></p>
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765 <pb xlink:href="X01/01/261.jpg" page="158_R"/>
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766 <p><s>Si ut 'eb' ad 'bd', ita est 'db' ad 'bc', erit ita 'ed' ad 'de'; et quia 'eb' est dupla 'bc', erit <SYMBOL-QUADRATUM/>_[quadratum] 'ed' duplum <SYMBOL-QUADRATUM/>_[quadrato] 'dc'.</s></p>
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767 <pb xlink:href="X01/01/262.jpg" page="158_V"/>
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768 <pb xlink:href="X01/01/263.jpg" page="159_R"/>
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769 <pb xlink:href="X01/01/264.jpg" page="159_V"/>
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770 <pb xlink:href="X01/01/265.jpg" page="160_R"/>
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771 <p><s>Sit 'ba' aequalis ipsi 'da', et ducantur perpendiculares 'be', 'df': constat ex elementis mec[h]anicis, momentum ponderis super plano secundum lineam 'abc' elevato ad momentum suum totale esse ut 'be' ad 'ba', eiusdem vero ponderis momentum super elevatione 'ad' ad totale suum momentum, eamdem ob causam esse <INSERTION-1> ut </INSERTION-1> 'df' ad 'da', vel 'ba'; ergo eiusdem ponderis momentum super plano secundum 'da' inclinato ad momentum super inclinatione secundum 'abc' est ut linea 'df' ad lineam 'be'; quare, spacia quae pertransibit idem pondus temporibus aequalibus super inclinationibus 'ca', 'da', erunt inter se ut lineae 'be', 'df'. At ut 'be' ad 'df', ita demon[s]tratur se habere 'ac' ad 'da': ergo idem mobile temporibus aequalibus pertransibit lineas 'ca', 'da'.</s></p>
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772 <p><s>Esse autem ut 'be' ad 'df', ita 'ca' ad 'da', ita demonstratur. Iungatur 'cd', et per 'd' et 'b', ipsi 'af' parallellae, agantur 'dgl', <DELETION-1> 'bh' </DELETION-1> secans 'ca' in 'i', et 'bh'; eritque angulus 'adi' aequalis angulo 'dca', <DELETION-1> extensam 'dg' usque ad circumferentiam </DELETION-1> cum circumferentiis 'la', 'ad' aequalibus insistant, estque angulus 'dac' communis. Ergo <SYMBOL-3/>_[tri]angulorum aequiangulorum 'cad', 'dai' latera circa aequales angulos proportionalia erunt, et ut 'ca' ad 'ad', ita erit 'da' ad 'ai', idest 'ba' ad 'ai', seu 'ha' ad 'ag', <DELETION-1> <ILLEGIBLE/> </DELETION-1> hoc est 'be' ad 'df': quod erat probandum.</s></p>
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773 <pb xlink:href="X01/01/266.jpg" page="160_V"/>
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774 <pb xlink:href="X01/01/267.jpg" page="161_R"/>
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775 <p><s>Sit 'ab' tempus per 'ab', et posita 'cd' aequali eidem 'ab', quaeratur tempus per 'cd'. Sit 'ae' media inter 'ca', 'ab', et 'af' media inter 'da', <SUBSTITUTION-1> 'ad' <SUBSTITUTED-BY-1> 'ab': </SUBSTITUTED-BY-1></SUBSTITUTION-1> et quia 'ab' est tempus per 'ab', erit 'ae' tempus per 'ac'; <DELETION-1> et 'be' tempus per 'be' </DELETION-1> et quia 'af' est media inter 'da', 'ab', erit 'af' tempus per totam <DELETION-1> 'ab' </DELETION-1> 'ad': fuit autem 'ae' tempus per 'ac': ergo 'ef' est tempus per 'cd'.</s></p>
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776 <p><s>sit 'ab' <SUBSTITUTION-1> 18 <SUBSTITUTED-BY-1> 8 </SUBSTITUTED-BY-1></SUBSTITUTION-1> <lb/> 'cd' <SUBSTITUTION-1> 18 <SUBSTITUTED-BY-1> 8 </SUBSTITUTED-BY-1></SUBSTITUTION-1> <lb/> 'bc' 10 <lb/> 'ac' 18 <lb/> 'ad' 26 <lb/> 'ae' 12 'af' 14 1/2 <lb/> 'ef' 1 1/2</s></p>
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777 <pb xlink:href="X01/01/268.jpg" page="161_V"/>
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778 <p><s>Dato perpendiculo et plano ad ipsum inclinato, quorum eadem sit altitudo idemque terminus sublimis, punctum in perpendiculo supra terminum communem reperire, ex quo si demittatur grave, quod postea convertatur per planum inclinatum, ipsum planum inclinatum conficiat eodem tempore, quo ipsum perpendiculum ex quiete conficeret.</s></p>
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779 <p><s>Sint perpendiculus et planum inclinatum, quorum eadem sit altitudo, 'ab', 'ac': oportet in perpendiculo 'ba', producto ex 'a' in 'h', punctum invenire, ex quo demissum mobile conficiat spacium 'ac' eodem tempore, quo conficit perpendiculum 'ab' ex quiete in 'a'. Ponatur 'dce' ad rectos angulos ad 'ac', et secetur 'cd' aequalis 'ab', et iungatur 'ad', quae maior erit ipsa 'dc', <DELETION-1> <ILLEGIBLE/> </DELETION-1> et angulus 'adc' maior angulo 'cad'. Fiat angulus 'dae' aequalis angulo 'ade', et ad ipsam 'ae' perpendicularis sit 'ef', plano inclinato et extenso occurrens in 'f', et <INSERTION-1> utraque 'ai', </INSERTION-1> 'ag' ponatur ipsi 'cf' aequalis, et per 'g' <DELETION-1> ad <ILLEGIBLE/> </DELETION-1> orizonti aequidistans 'gh'. Dico, 'h' esse punctum quod quaeritur.</s></p>
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780 <p><s>Intelligatur enim, tempus casus per perpendiculum 'ab' esse 'ab'; erit tempus per 'ac' ex 'a' ipsa 'ac': cumque in <SYMBOL-3/>_[tri]angulo rectangulo 'aef' ab angulo recto 'e' perpendicularis ad basim 'af' sit acta 'ec', erit 'ae' media inter 'fa', 'ac', et 'ce' media inter 'ac', 'cf', hoc est inter 'ca', 'ai': et cum ipsius 'ac' tempus ex 'a' sit 'ac', erit 'ae' tempus totius 'af', et 'ec' tempus ipsius <DELETION-1> 'ia' </DELETION-1> 'ai'. Quia vero in <SYMBOL-3/>_[tri]angulo aequicruri 'aed' latus 'ae' est aequale lateri 'ed', erit 'ed' tempus per 'af', et est 'ec' tempus per 'ai': ergo 'cd', hoc est 'ab', erit tempus per 'if' ex quiete in 'a'; quod idem est ac si dicamus 'ab' esse tempus per 'ac' ex 'g', seu ex 'h': quod erat determinandum.</s></p>
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781 <pb xlink:href="X01/01/269.jpg" page="162_R"/>
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782 <p><s>Datis <SYMBOL-2/>_[duo]bus planis orizontalibus a perpendiculo sectis, dataque qualibet proportione minoris ad maiorem, oportet in perpendiculo pun[c]tum sublime reperire, ex quo mobilia cadentia, et in horizontalibus reflexa, temporibus casuum suorum spatia in horizontalibus conficiant datam inter se habentia rationem.</s></p>
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783 <p><s>Secentur plana horizontalia 'cd', 'be' a perpendiculo 'acb', sitque data proportio <INSERTION-1> 'n' </INSERTION-1> minoris ad maiorem <DELETION-1> 'n' ad </DELETION-1> 'fg'. Oportet, in perpendiculo punctum sublime reperire, ex quo mobile cadens, in plano 'cd' reflexum, tempore aequali tempori sui casus spatium conficiat, quod ad <DELETION-1> <ILLEGIBLE/> </DELETION-1> spatium confectum ab altero mobili, ex eodem puncto subblimi [sublimi] venienti, tempore aequali tempori sui casus, motu reflexo per 'be' planum, <UNDERLINED> conficeret </UNDERLINED> habeat rationem <DELETION-1> <ILLEGIBLE/> </DELETION-1> eamdem cum data 'n' ad 'fg'. Ponatur 'gh' aequalis ipsi 'n'; et ut 'fh' ad 'hg', ita fiat 'bc' ad 'cl': dico, 'l' esse punctum quaesitum. Accepta enim 'cm' <DELETION-1> <ILLEGIBLE/> </DELETION-1> <SYMBOL-2/>_[du]pla 'cl', ducatur 'lm', plano 'be' occurrens in 'o'; erit 'bo' <SYMBOL-2/>_[du]pla 'bl': et quia ut 'fh' ad 'hg', ita 'bc' ad 'cl', erit, componendo, ut <BEGIN-OF-LINE-DESTROYED/> ['fg' ad 'gh', idest ad 'n',] <END-OF-LINE-DESTROYED/> ita 'bl' ad 'lc', et 'bo' ad 'cm'. Cum autem 'cm' <SYMBOL-2/>_[du]pla sit ad 'lc', patet, <DELETION-1> <ILLEGIBLE/> </DELETION-1> spatium 'cm' esse illud quod a mobili venienti ex 'l' post casum 'lo' ['lc'] conficitur in plano 'cd', et eadem ratione 'bo' esse illud quod conficitur post casum 'lb' in tempore aequali tempori casus per 'lb', cum 'bo' sit <SYMBOL-2/>_[du]pla ad 'bl'. Ergo patet propositum. Quod si intelligamus, 'cm' et 'bo' esse circulorum circumferentias circa centrum quo grave tendit descriptorum, habebimus distantiam 'l', unde deducitur ratio velocitatum in illis circulis latorum: ex qua et ex ratione conversionum distantia centri elici?_<END-OF-WORD-DESTROYED/> <BEGIN-OF-LINE-DESTROYED/></s></p>
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786 <p><s>Sit circuli quadrans 'acib', et ex 'b', ipsi 'ac' parallela, ducatur 'be'; et in ea assumpto centro, circulus 'boes' describat[ur], ita ut secet circumferentiam quadrantis, quod sit in 'i'; et connectatur 'cb' et 'ci', quae usque ad 's' extendatur: dico, lineam 'ci' ipsa 'co' semper esse <SUBSTITUTION-1> maiorem <SUBSTITUTED-BY-1> minorem. </SUBSTITUTED-BY-1></SUBSTITUTION-1> Iungatur 'ai', quae circulum 'boe' tanget; si enim iungatur 'di', erit aequalis ipsi 'db'; cum vero 'db' tangat quadrantem, tanget etiam 'di'; ergo ad diametrum 'ai' erit perpendicularis: quare et ipsa 'ai' circulum 'boe' tanget. Et quia angulus 'aic' maior est angulo 'abc', cum maiori insistat periferiae [peripheriae], ergo angulus quoque 'sin' ipso 'abc' maior erit: quare portio 'ies' maior est portione 'bo', et linea 'cs', centro vicinior, maior ipsa 'cb': quare et 'co' maior 'ci'. <SUBSTITUTION-2> cum <SYMBOL-RECTANGULUM/>_[rectangulum] 'bco' sit aequale <SYMBOL-RECTANGULUM/>_[rectangulo] 'sci'. <SUBSTITUTED-BY-2> <INSERTION-1> <ILLEGIBLE/> </INSERTION-1> </SUBSTITUTED-BY-2></SUBSTITUTION-2> </s></p>
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787 <p><s>Idem autem magis accidet, si, ut in altera figura, 'bic' quadrante fuerit minor. Nam perpendicularis 'db' circulum secabit 'cib'; quare 'di' quoque, cum ipsi 'db' sit aequalis; et angulus 'dia' erit obtusus, et ideo 'ain' circulum quoque 'bin' ['bie'] secabit. Cumque angulus 'abc' minor sit angulo 'aic', qui aequatur ipsi 'sin'; iste autem est adhuc minor eo qui ad contactum in 'i' fieret per lineam 'si'; ergo portio 'sei' est longe maior portione 'bo': unde etc.</s></p>
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788 <p><s>Sit circuli circumferentia 'cbd', et diameter <DELETION-1> 'cb' </DELETION-1> 'mc' ad orizontem erecta, et ducatur 'dc', non maior subtendente quadrantem, et a terminis 'd', 'c' aliae duae ad quodcumque punctum 'b'. Dico, mobile ex termino 'd' <DELETION-1> velocius </DELETION-1> ferri per duas 'db', 'bc' lineas tempore breviori quam per 'dc' ex eodem termino 'd', vel per solam 'bc' ex termino 'b'. Ducta sit per 'd', ipsi 'cm' perpendicularis, 'mda', cui 'cb' extensa occurat in 'a'; sitque 'dn' ipsi 'mc' parallela, et 'bn' ad 'bd' perpendicularis, et circa <SYMBOL-3/>_[tri]angulum rectangulum 'dbn' semicirculus describatur 'dfbn', <SUBSTITUTION-1> segans <SUBSTITUTED-BY-1> secans </SUBSTITUTED-BY-1></SUBSTITUTION-1> 'dc' in 'f'. Et ipsarum 'cd', 'df' media sit proportionalis 'do', ipsarum autem 'ca', 'ab', 'av'. Sit autem 'ps' tempus quo peragitur tota 'dc', vel 'cb' (constat enim eodem tempore peragi utramque 'dc', 'bc'), et quam rationem habet 'cd' ad 'do', hanc habeat tempus 'sp' ad tempus 'pr': erit tempus 'pr' id in quo mobile ex 'd' peragit 'df'; 'rs' vero id in quo reliquum 'fc'. Cum vero 'ps' sit quoque tempus quo mobile ex 'b' peragit 'bc', si fiat ut 'bc' ad 'cd', ita 'sp' ad 'pt', erit 'pt' tempus casus ex 'a' in 'c', cum 'dc' media sit inter 'ac', 'cb', ex ante demonstratis. Fiat tandem ut 'ca' ad 'av', ita 'tp' ad 'pg': erit 'pg' tempus quo mobile ex 'a' venit in <DELETION-1> <ILLEGIBLE/> </DELETION-1> 'b', 'gt' vero tempus residuum motus 'bc' <DELETION-1> post </DELETION-1> consequentis post motum <DELETION-1> 'ab' </DELETION-1> ex 'a' in 'b'.</s></p>
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789 <p><s>Cum vero 'dn', circuli 'dfbn' diameter, ad orizontem sit erecta, temporibus aequalibus peragentur 'df', 'db' lineae: quare si demonstratum fuerit, mobile citius conficere 'bc' post casum 'db', quam 'fc' post peractam 'df', habebimus intentum. At eadem temporis celeritate conficiet mobile 'bc' veniens ex 'db' ac si veniret ex 'ab', cum ex utroque casu 'db', 'ab' aequalia accipiat velocitatis momenta; ergo demonstrandum erit, breviori tempore peragi 'bc' post 'ab', quam 'fc' post 'df'. Explicatum est autem, tempus quo peragitur 'bc' post 'ab' esse 'gt'; tempus vero ipsius 'fc' post 'df' esse 'rs': ostendendum itaque est 'rs' maius esse quam 'gt'. Quod sic demonstratur. Quia enim ut 'sp' ad 'pr', ita 'cd' ad 'do', per conversionem rationis et convertendo, ut 'rs' ad 'sp', ita 'oc' ad 'cd', ut autem 'sp' ad 'pt', ita 'cd' ad 'ca'; et quia est ut 'tp' ad 'pg', ita 'ca' ad 'av', per conversionem rationis erit quoque ut 'pt' ad 'tg', ita 'ac' ad 'cv'; ergo, ex aequali, ut 'rs' ad 'gt', ita 'oc' ad 'cv': <DELETION-1> oste cum vero 'cf' sit maior 'cb' </DELETION-1> est autem 'oc' maior quam 'cv', ut mox demonstrabitur: ergo tempus 'rs' maius est tempore 'gt': quod demonstrare oportebat. Cum vero 'cf' maior sit quam 'cb', 'fd' vero minor 'ba', habebit 'cd' ad 'df' maiorem rationem quam 'ca' ad 'ab'; ut autem 'cd' ad 'df', ita <SYMBOL-QUADRATUM/>_[quadratum] 'co' ad <SYMBOL-QUADRATUM/>_[quadratum] 'of', cum sint 'cd', 'do', 'df' proportionales; ut vero 'ca' ad 'ab', ita <SYMBOL-QUADRATUM/>_[quadratum] 'cv' ad <SYMBOL-QUADRATUM/>_[quadratum] 'vb'; ergo 'co' ad 'of' maiorem rationem habet quam 'cv' ad 'vb': igitur, ex lemmate pr[a]edemonstrato, 'co' maior est quam 'cv'. Constat igitur, <DELETION-1> velo </DELETION-1> tempus per 'dc' ad tempus per 'dbc' esse ut <SUBSTITUTION-1> 'ce' ad 'cu' <SUBSTITUTED-BY-1> 'doc' ad 'do' cum 'cv'. </SUBSTITUTED-BY-1></SUBSTITUTION-1> </s></p>
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790 <pb xlink:href="X01/01/272.jpg" page="163_V"/>
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791 <p><s>Factus sit motus ex <SUBSTITUTION-1> 'ab' <SUBSTITUTED-BY-1> 'a' </SUBSTITUTED-BY-1></SUBSTITUTION-1> in 'b' naturaliter acceleratus: dico, quod si velocitas in omnibus punctis 'ab' fuisset eadem ac reperitur <DELETION-1> <ILLEGIBLE/> </DELETION-1> in puncto 'b', duplo citius fuisset peractum spacium 'ab'; quia velocitates omnes in singulis punctis 'ab' lineae, ad totidem velocitates quarum unaquaeque esset aequalis velocitati 'bc', eam habent rationem quam <SYMBOL-3/>_[tri]angulus 'abc' ad rectangulum 'abcd'. Sequitur ex hoc, quod si ad orizontem 'cd' fuerit planum 'ba' elevatum, sitque 'bc' dupla ad 'ba', mobile ex 'a' in 'b', et successive ex 'b' in 'c', temporibus aequalibus esse pervencturum: nam postquam est in 'b', per reliquam 'bc' uniformi velocitate et eadem movetur, qua in ipsomet termino 'b' post casum 'ab'. Patet rursus, totum tempus per 'abe' ad tempus per 'ab' esse sesquialterum.</s></p>
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792 <p><s>Si post casum per aliquod planum inclinatum sequatur motus per planum orizontis, erit tempus casus per planum <DELETION-1> inclila inchi </DELETION-1> inclinatum ad tempus motus per quamlibet lineam orizontis ut dupla longitudo plani inclinati ad <DELETION-1> a </DELETION-1> lineam acceptam orizontis. Sit linea orizontis 'cb', planum inclinatum 'ab', et post casum per 'ab' sequatur motus per orizontem, in quo sumatur quaelibet linea 'bd'. Dico, tempus casus per 'ab' ad tempus motus per 'bd' esse ut dupla 'ab' ad 'bd'. Sumpta enim 'bc' ipsius 'ab' dupla, constat ex praedemonstratis, tempus casus per 'ab' aequari tempori motus per 'bc': sed <INSERTION-1> tempus </INSERTION-1> motus per 'bc' ad <INSERTION-1> tempus </INSERTION-1> motus per 'bd' est ut linea 'cb' ad lineam 'bd': ergo tempus motus per 'ab' ad tempus motus per 'bd' est ut <SYMBOL-2/>_[du]pla 'ab' linea ad lineam 'bd'.</s></p>
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793 <p><s>Tempus casus per planum inclinatum ad tempus casus per lineam suae altitudinis est ut eiusdem plani longitudo ad longitudinem suae altitudinis.</s></p>
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794 <p><s>Sit planum inclinatum 'ba' ad lineam orizontis 'ac', sitque linea altitudinis perpendicularis 'bc'. Dico, tempus casus quo mobile movetur per 'ba' ad tempus in quo cadit per 'bc' esse ut 'ba' ad 'bc'. Erigatur perpendicularis <DELETION-1> per </DELETION-1> ad orizontem ex 'a', quae sit 'ad', cui occurrat in 'd' perpendicularis ad 'ab' ducta ex 'b', quae sit 'bd', et circa <SYMBOL-3/>_[tri]angulum 'abd' circulus describatur: et quia 'da', 'bc' ambae sunt ad orizontem perpendiculares, constat, tempus casus per 'da' ad tempus casus per 'bc' esse ut media inter 'da' <DELETION-1> ad </DELETION-1> et 'bc' ad ipsam 'bc'. Tempus autem casus per 'da' aequatur tempori casus per 'ba': media vero inter 'da' et 'bc' est ipsa 'ba': ergo patet propositum.</s></p>
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795 <p><s>Corollarium</s></p>
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796 <p><s>Ex hoc sequitur, casuum tempora per plana inclinata quorum eadem sit altitudo, esse inter se ut eorumdem planorum longitudines. Si enim fuerit aliud planum inclinatum 'be', tempus casus per 'ba' ad tempus casus per 'bc' est ut 'ba' linea ad 'bc': tempus vero per 'bc' ad tempus casus per 'be' ut 'bc' ad 'be': ergo, ex aequali, patet propositum.</s></p>
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797 <pb xlink:href="X01/01/273.jpg" page="164_R"/>
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798 <p><s>Sint ad orizontem 'db' quotcumque lineae ex eadem altitudine 'a' demissae 'ab', 'ac', 'ad', et sumpto quolibet puncto 'g', per ipsum orizonti parallela sit 'gfe', sitque media inter 'ba', 'ag' ipsa 'ar', et per 'r' altera parallela 'rtv': constat, lineas 'at', 'av' esse medias inter 'ca', 'af' et 'da', 'ae'. Dico, quod si assumatur 'ab' esse tempus quo mobile cadit ex 'a' in 'b', tempus 'rb' esse illud quo conficitur 'gb', 'tc' vero esse tempus ipsius 'cf', et 'vd' ipsius 'ed'. Id autem constat: nam, cum 'ar' sit media inter 'ba', 'ag', sitque 'ba' tempus casus totius 'ab', tempus 'ar' erit tempus casus per 'ag'; ergo reliquum temporis 'rb' erit tempus casus per 'gb' post 'ag'; et idem dicetur de aliis temporibus 'tc', 'vd' et lineis 'fc', 'ed'. Patet insuper, tempora casuum per 'gb', 'fc', 'ed' esse ut lineas 'gb', 'fc', 'ed'; non tamen a magnitudinibus ipsarum linearum 'gb', 'fc', 'ed' esse determinandas eorumdem temporum quantitates, si temporis mensura ponatur 'ab', in quo tempore conficiatur linea 'ab', sed desumendas esse a lineis 'rb', 'tc', 'vd'.</s></p>
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799 <p><s>Sit circulus, cuius diameter 'ab', et ipsi parallela tangens 'ce', et ex termino 'b' quaelibet linea 'bo' in circulo applicetur: <DELETION-1> a cuius </DELETION-1> dico, perpendiculares quae a terminis 'b', 'o' ipsi 'bo' accomodantur, protractas, de linea 'ce' partem diametro circuli aequalem semper intercipere. Iungatur enim 'ao', et extendatur ad tangentem in 'f', quae ad 'bo' erit perpendicularis, cui ex 'b' parallela sit 'be': demostrandum, 'fe' diametro circuli esse aequalem. Id autem constat, quia in parallelogrammo 'abef' latera 'ab', 'fe' opposita aequalia sunt ex Elementis.</s></p>
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800 <p><s><INSERTION-1> Vel dicas, quod ducta ex 'o', 'og' parallela ipsi 'ab', et 'bg' perpendiculari ad 'bo', abscindetur semper 'og' aequalis diametro circuli; quod patet ex <SYMBOL-TRIANGULUM/>_[triangul]is 'aob', 'obg' similibus et aequalibus. </INSERTION-1> </s></p>
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801 <p><s>Sit 'ac' perpendicularis ad orizontem 'cde', ponaturque inclinata 'bd', fiatque motus ex 'a' per 'abc' et per 'abd'. Dico, tempus per 'bc' post casum 'ab' ad tempus per 'bd' post eumdem casum 'ab' esse ut linea 'bc' ad 'bd'. Ducatur 'af' parallela 'dc' et protrahatur 'db' ad 'f'; erit iam tempus casus per 'fbd' ad tempus casus per 'abc' ut 'fd' linea ad lineam 'ac': est autem tempus casus per 'fb' ad tempus casus per 'ab' ut linea 'fb' ad lineam 'ab': ergo tempus casus reliquae 'bc' post 'ab' ad tempus casus reliquae 'bd' post 'fb' erit ut reliqua 'bc' ad reliquam 'bd'. Sed tempus casus per 'bd' post 'fb' est idem cum tempore per 'bd' post 'ab', cum 'af' orizonti aequidistans sit: ergo patet propositum.</s></p>
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802 <p><s>Colligitur autem ex hoc, quod tempora casuum per 'bc' et 'bd', sive fiat principium motus et termino 'b', sive praecedat motus, ex eadem tamen altitudine, eandem inter se servant rationem, nempe eam quae est lineae 'bc' ad 'bd'.</s></p>
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803 <pb xlink:href="X01/01/274.jpg" page="164_V"/>
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804 <p><s>Si in semicirculo <ILLEGIBLE/> quae cum perpendiculo non habeat terminum communem, motus per illam cicius [citius] absolvitur quam per diametrum perpendicularem.</s></p>
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805 <p><s>Si enim 'bb' fuerit perpendiculus, ducta quaelibet linea 'ca' in semicirculo non terminetur ad 'b': patet quod, si connectatur linea 'cb', erit 'ca' ipsa 'cb' brevior et minus inclinata; ex quo patet propositum.</s></p>
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806 <p><s>Si in circulo, cuius diameter sit ad perpendiculum, ducatur linea quae a diametro secetur, motus per ipsam tardius absolvetur quam per diametrum perpendicularem. In praecedenti enim figura sit linea quaelibet; et quia ipsa erit longior quam 'cb' et magis inclinata, propositum fit manifestum.</s></p>
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807 <p><s>Mirandum. Numquid motus per perpendiculum 'ad' velocior sit quam per inclinationem 'ab' ? Videtur esse; nam aequalia spacia citius conficiuntur per 'ad' quam per 'ab'. Attamen videtur etiam non esse; nam, ducta orizontali 'bc', tempus per 'ab' ad tempus per 'ac' est ut 'ab' ad 'ac': ergo eadem momenta velocitatis per 'ab' et per 'ac'. Est enim una eademque velocitas illa quae, temporibus in[a]equalibus, spacia transit inaequalia, eandem quam tempora rationem habentia.</s></p>
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808 <p><s>Momenta velocitatum cadentis ex subblimi [sublimi] sunt inter se ut radices distantiarum peractarum, nempe in subduplicata ratione illarum.</s></p>
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809 <pb xlink:href="X01/01/275.jpg" page="165_R"/>
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810 <pb xlink:href="X01/01/276.jpg" page="165_V"/>
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811 <pb xlink:href="X01/01/277.jpg" page="166_R"/>
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812 <p><s>Cum semidiameter sit 100000, quadrantis circumferentia est 157143 157042; <lb/> seu si semidiameter sit 1000, <SUBSTITUTION-1> quadrans erit <SUBSTITUTED-BY-1> circumferentia quadrantis </SUBSTITUTED-BY-1></SUBSTITUTION-1> plus 1570 minus 1571.</s></p>
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813 <p><s>si <SYMBOL-QUADRATUM/>_[quadratum] sit 1000000, <SUBSTITUTION-1> <SYMBOL-QUADRATUM/>_[quadrans] erit plus 660000 minus 660500 <SUBSTITUTED-BY-1> <SYMBOL-QUADRATUM/>_[quadrans] erit 785250. </SUBSTITUTED-BY-1></SUBSTITUTION-1> </s></p>
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814 <p><s>Tempus quo conficieretur circumferentia quadrantis, si esset recta et ad perpendiculum, 125331.</s></p>
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815 <p><s>Tempus per 'ac' ad tempus per 2 'aec' est ut 1000 ad 937 1/2 fere; tempus per 'ec' ad tempus per 2 'egc' ut 1000 ad 866 3/5; tempus per '8c' ad <SYMBOL-2/>_[du]as suas ut 1000 ad 733 2/3.</s></p>
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816 <p><s>'ad' longa puncta 180; sit tempus casus per ipsam <m-SUPERSCRIBED-BY-1/>m[inutum] 180, et per ambas 'adc' <m-SUPERSCRIBED-BY-1/>m[inutum] 270. <lb/> 'ac' --- 254 3/5 --- <m-SUPERSCRIBED-BY-1/>m[inuti] 254 3/5 <lb/> 'ae' --- 138 --- tempus casus per illam <m-SUPERSCRIBED-BY-1/>m[inutum] 164 <lb/> 'ec' --- 138 --- tempus casus per eam post 'ac' <m-SUPERSCRIBED-BY-1/>m[inutum] 75 et per ambas 'aec' <m-SUPERSCRIBED-BY-1/>m[inutum] <SUBSTITUTION-1> 196 <SUBSTITUTED-BY-1> 239 </SUBSTITUTED-BY-1></SUBSTITUTION-1> <lb/> 'af' recta --- 70 1/2 tempus --- 113 1/2 <lb/> 'fe' --- 70 1/2 tempus casus post 'af' --- 48 <DELETION-1> 1/2 </DELETION-1> et per ambas 'afe' <m-SUPERSCRIBED-BY-1/>m[inutum] <SUBSTITUTION-2> 154 <SUBSTITUTED-BY-2> 161 1/2 </SUBSTITUTED-BY-2></SUBSTITUTION-2> <lb/> 'eg' --- 70 1/2 tempus --- 39 <lb/> 'gc' --- 70 1/2 --- 36 et per ambas 'egc' 75 <lb/> et per 4 'afegc' 236 1/2.</s></p>
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817 <p><s>Considera num tempus per 'ac' ad tempus per <SYMBOL-2/>_[du]as 'aec' sit ut radix radicis lineae quae a centro 'b' super 'ac' cadit <DELETION-1> perdi </DELETION-1> perpendiculariter, ad radicem radicis perpendicularis ex eodem centro super 'ae'.</s></p>
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818 <p><s>Tempus per 2 'egc' ex quiete in 'e' est 66326; deberet autem esse 71757, si casus per 'egc' ad casum per 'ec' haberet eandem rationem <DELETION-1> quam </DELETION-1> quam casus per 'aec' ad casum per 'ac': movetur ergo citius per 'egc' quam per 'aec'. Et ex quiete in 8 tempus per <SYMBOL-2/>_[du]as '8c' ad tempus per solam '8c' est ut 14378 ad 19598: longe igitur adhuc citius movetur quam per 2 'egc'.</s></p>
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819 <p><s>ra?_<END-OF-WORD-DESTROYED/> <BEGIN-OF-WORD-DESTROYED/>_?ciuntur per <ILLEGIBLE/> <END-OF-LINE-DESTROYED/></s></p>
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820 <pb xlink:href="X01/01/278.jpg" page="166_V"/>
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824 <p><s>Si in orizonte sumantur duo puncta, et ab altero ipsorum quaelibet linea inclinetur, <DELETION-1> in qua sumatur a termino in orizonte pars aequalis ei quae inter puncta in orizonte signata intercipitur, et ad </DELETION-1> ad quam ex altero puncto orizontis altera recta ducatur, <DELETION-1> <ILLEGIBLE/> </DELETION-1> ex ea secans partem aequalem ei quae inter puncta orizontis intercipitur, casus per hanc ductam citius absolvitur <DELETION-1> per </DELETION-1> quam per <DELETION-1> <ILLEGIBLE/> </DELETION-1> quascunque alias rectas ex eodem puncto ad eandem lineam inclinatam protractas. In aliis autem, quae per angulos aequales supra et infra <DELETION-1> hanc </DELETION-1> ab hac distiterint, casus fiunt temporibus aequalibus.</s></p>
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825 <p><s>Sint in linea orizontali duo puncta 'a', 'b', et a 'b' inclinetur recta 'bc', in qua ex termino 'b' sumatur 'bd', ipsi 'ba' aequalis, et iungatur 'ad': <DELETION-1> a qua per angulos aequales dirimantur duae 'ag', 'ac' </DELETION-1> dico, casum per 'ad' velocius fieri quam per quamlibet ex 'a' ad inclinatam 'bc' productam. <DELETION-1> et insuper tempora casuum per 'ag', 'ac' esse aequalia </DELETION-1> Ex punctis enim 'a', 'd' ad ipsas 'ba', 'bd' perpendiculares ducantur 'ae', 'de', sese in 'e' secantes: et quia in <SYMBOL-3/>_[tri]angulo aequicruri 'abd' anguli 'bad', 'bda' sunt aequales, erunt reliqui ad rectos 'dae', 'eda' aequales; ergo, centro 'e', intervallo autem 'ea', descriptus circulus per 'd' quoque transibit, et lineas 'ba', 'bd' tanget in punctis 'a', 'd'. Et cum 'a' sit terminus perpendiculi 'ae', casus per 'ad' citius absolvetur quam per quamcunque aliam ex eodem termino 'a' usque ad lineam 'bc' ultra circumferentiam circuli extensam: quod est primo demonstrandum.</s></p>
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826 <p><s>Quod si in perpendiculo 'ae' sumatur infra 'e' quodcumque centrum 'f', et secundum intervallum 'fa' circulus 'agc' describatur, <DELETION-1> iunctis </DELETION-1> lineam 'bc' in punctis 'g', 'c' secans, iunctae 'ag', 'ac' per angulos aequales, ex antedemonstratis, a media 'ad' dirimentur; et patet, per ipsas eodem tempore fieri motum, cum ex apice perpendiculi ad circumferentiam 'agc' sint ductae.</s></p>
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827 <p><s>Si duo circuli se intus tangant, et <DELETION-1> a con </DELETION-1> linea recta interiorem circulum contingat et alterum secet, tres lineae a contactu circulorum ad tria puncta tangentis et secantis lineae productae angulos duos aequales continebunt.</s></p>
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828 <p><s>Assumpta praesenti figura, protrahatur 'ad' usque ad 'h', et iungatur 'hf', secans 'gc' in 'i': et quia anguli in centris 'e', 'f' sunt aequales, cum similibus circumferentiis sectis a linea 'adh' insistant, erit linea 'fih' ipsi 'ed' parallela. Cumque 'ed' sit perpendicularis ad 'gc', ipsa quoque 'fih' ex centro 'f' ad lineam 'cg' perpendicularis erit, et quod consequens est, arcum 'ghc' bifariam dividet, et angulus 'gah' angulo 'hac' erit aequalis, etc.</s></p>
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829 <pb xlink:href="X01/01/282.jpg" page="168_V"/>
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830 <pb xlink:href="X01/01/283.jpg" page="169_R"/>
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831 <pb xlink:href="X01/01/284.jpg" page="169_V"/>
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832 <p><s>Fiat motus per 'abc' et per duas 'abd', sitque 'ra' media inter 'ca', 'ab', et <DELETION-1> per </DELETION-1> ipsi 'dc' parallela ducatur 'rs'. Dico iam, tempus per 'abc' ad tempus per 'abd' esse ut linea 'ac' ad 'ar' cum 'sd'. Si enim protrahatur 'db' usque ad occursum cum 'af', orizonti 'dc' parallela, erit 'fs' media inter 'df', 'fb'; et ut 'ca' ad 'ar', ita tempus per 'ca' ad tempus per 'ab'; ita ut, si ponatur 'ac' tempus per 'ac', erit 'ar' tempus per 'ab', et 'rc' tempus per 'bc'; et similiter 'sd' demonstrabitur esse tempus per 'bd' post casum ex 'f', vel ex 'a': ex quo patet, tempus per totam 'ac' ad tempus per duas 'abd' esse ut 'ar' cum 'rc' ad 'ar' cum 'sd'. <DELETION-1> Demonstrabitur autem eandem esse rationem 'abr' ad 'abs' </DELETION-1> </s></p>
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833 <pb xlink:href="X01/01/285.jpg" page="170_R"/>
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834 <p><s>Sit data 'ef' maior 'ba', minor vero quam <SYMBOL-2/>_[du]pla eiusdem 'ba'; accipiatur 'ed' aequalis 'ba', et reliquae 'df' ponatur aequalis 'di', et ut 'ei' ad 'id', ita fiat 'df' ad aliam 'fx', atque ex 'b' reflectatur planum 'bo' aequale 'ex'. Dico, planum 'bo' esse illud super quo post descensum 'ab' mobile in tempore aequali tempori descensus per 'ab' pertransit ascendendo spatium [a]equale dato 'ef'. Cum enim sit ut 'ei' ad 'id', ita 'df' ad 'fx', erit, componendo, ut 'ed' ad 'di', ita 'dx' ad 'xf'; hoc est ut 'ed' ad 'df', ita 'dx' ad 'xf', et, permutando, ut 'ed' ad 'dx', ita 'df' ad <SUBSTITUTION-1> 'xf' <SUBSTITUTED-BY-1> 'fx', </SUBSTITUTED-BY-1></SUBSTITUTION-1> et, componendo, ut 'ex' ad 'xd', ita 'dx' ad 'xf'; hoc est ut 'bo' ad 'or', ita 'ro' ad 'os'.</s></p>
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835 <p><s>Quod si ponamus, tempus per 'ab' esse 'ab', erit tempus 'ob' ipsa 'ob', et 'ro' tempus per 'os', et reliquum 'br' tempus per reliquum 'sb', descendendo ex 'o' in 'b': sed tempus descensus per 'sb' ex quiete in 'o' est aequale tempori ascensus ex 'b' in 's' post descensum 'ab': ergo <DELETION-1> 'br', seu 'ba', est tempus 'bs' est sp </DELETION-1> 'bo' est planum ex 'b' elevatum, super quo post descensum per 'ab' conficitur in tempore 'br', seu 'ba', spatium 'bs', aequale spatio dato 'ef': quod facere oportebat.</s></p>
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836 <p><s>Si post casum in perpendiculo fiat reflexus motus in linea orizontali</s></p>
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837 <pb xlink:href="X01/01/286.jpg" page="170_V"/>
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838 <pb xlink:href="X01/01/287.jpg" page="171_R"/>
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839 <pb xlink:href="X01/01/288.jpg" page="171_V"/>
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840 <pb xlink:href="X01/01/289.jpg" page="172_R"/>
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841 <p><s>Si in circulo <INSERTION-1> ad orizontem erecto </INSERTION-1> a puncto sublimi quotcumque ducantur lineae rectae usque ad circumferentiam per quas cadant gravia quotcumque, omnia <DELETION-1> <ILLEGIBLE/> </DELETION-1> temporibus aequalibus ad terminos suos pervenient.</s></p>
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842 <p><s>Sit enim circumferentia ad orizontem erecta 'abc' punctum sublime 'a' a quo lineae quotcumque ad circumferentiam usque protrahantur 'ac' 'ab' et per ipsas cadant mobilia <DELETION-1> aequalium momentorum </DELETION-1>, dico temporibus aequalibus ea perventura esse ad terminos 'c' 'b'. Sit enim <DELETION-1> <ILLEGIBLE/> </DELETION-1> 'ac' per centrum ducta, cui ex 'b' perpendicularis sit 'bd'. Patet 'ab' mediam esse proportionalem inter 'ca' 'ad' quare ex demonstratis tempus quo mobile ex 'a' cadit in 'c' ad tempus casus ex 'a' in 'd' est ut linea 'ba' ad 'ad'. Verum similiter ex demonstratis tempus casus ex 'a' in <DELETION-1> 'bc' </DELETION-1> 'b' ad tempus casus ex 'a' in 'd' est ut 'ba' ad 'ad' ergo tempora casuum 'ab' 'ac' erunt aequalia cum eandem ad idem tempus casus 'ad' habeant rationem. Et similiter de reliquis omnibus motibus demonstrabitur, ergo patet propositum.</s></p>
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843 <p><s>Ex his colligitur gravia eodem tempore pertransire plana inaequalia et inaequaliter inclinata dum quam proportionem habet longitudo maioris plani ad longitudinem alterius, eandem <INSERTION-1> duplicatam </INSERTION-1> habeat perpendicularis maioris plani ad <SUBSTITUTION-1> eam quae inter hanc et perpendicularem <SUBSTITUTED-BY-1> perpendicularem </SUBSTITUTED-BY-1></SUBSTITUTION-1> minoris <DELETION-1> plani sit media proportionalis </DELETION-1> : cum enim <SYMBOL-QUADRATUM/>_[quadratum] 'ae' sit aequale <SYMBOL-RECTANGULUM/>_[rectangulo] 'caf' <SYMBOL-QUADRATUM/>_[quadratum] vero 'ba' <SYMBOL-RECTANGULUM/>_[rectangulo] 'cad' <SYMBOL-RECTANGULUM/>_[rectangulum] vero 'caf' ad <SYMBOL-RECTANGULUM/>_[rectangulum] 'cad' est ut 'fa' ad 'ad' ergo 'fa' ad 'ad' est ut <SYMBOL-QUADRATUM/>_[quadratum] 'ea' ad <SYMBOL-QUADRATUM/>_[quadratum] 'ba' ratio igitur perpendicularis 'fa' ad perpendicularem 'da' dupla est rationis 'ea' ad 'ab'.</s></p>
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844 <p><s>Aliter ostendemus mobile temporibus aequalibus pertransire 'ca' 'da'.</s></p>
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845 <p><s>Sit enim 'ba' aequalis ipsi 'da' et ducantur perpendiculares 'be' 'df' constat ex elementis mec[h]anicis momentum ponderis super plano secundum lineam 'abc' elevato ad momentum suum totale esse ut 'be' ad 'ba' eiusdem vero ponderis momentum super elevatione 'ad' ad totale suum momentum eamdem ob causam esse ut 'df' ad 'da' vel 'ba' ergo eiusdem ponderis momentum super plano secundum 'da' inclinato ad momentum super inclinatione secundum 'abc' <SUBSTITUTION-1> esse <SUBSTITUTED-BY-1> est </SUBSTITUTED-BY-1></SUBSTITUTION-1> ut linea 'df' ad lineam 'be' quare spacia quae pertransibit idem pondus temporibus aequalibus super inclinationibus 'ca' 'da' erunt inter se ut lineae 'be' 'df' at ut 'be' ad 'df' ita demonstratur se habere 'ac' ad 'da' ergo idem mobile temporibus aequalibus pertransibit lineas 'ca', 'da'.</s></p>
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846 <p><s>Esse autem ut 'be' ad 'df' ita 'ca' ad 'da' ita probatur. Iungatur 'cd' et per 'd' et 'b' ipsi 'af' parallelae agantur 'dgl' secans 'ca' in 'i' et 'bh' eritque angulus 'adi' aequalis angulo 'dca', cum circumferentiis 'la' 'ad' aequalibus insistant, estque angulus 'dac' communis ergo <SYMBOL-3/>_[tri]angulorum [a]equiangulorum 'cad' 'dai' latera circa aequales angulos proportionalia erunt, et ut 'ca' ad 'ad' ita erit 'da' ad 'ai' idest 'ba' ad 'ai' seu 'ha' ad 'ag' hoc est 'be' ad 'df' quod erat probandum.</s></p>
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847 <p><s>Collige existentibus planis inaequaliter inclinatis 'ad' 'ac' atque data longitudine 'ad' inveniri posse in plano 'ac' portionem quae eodem tempore cum 'da' peragatur ducto enim perpendiculo 'df' et posita 'ab' aequali 'ad' ductoque perpendiculo 'be' fiat ut 'df' ad 'eb' ita 'da' ad <DELETION-1> 'ab' </DELETION-1> 'ac' eritque tempus per 'ca' aequale tempori per 'da'.</s></p>
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848 <pb xlink:href="X01/01/290.jpg" page="172_V"/>
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849 <p><s>Sit planum orizzontis secundum lineam 'abc', ad quam sint duo plana inclinata secundum lineas 'db', 'da'. Dico, idem mobile tardius moveri per 'da' quam per 'db' secundum rationem longitudinis 'da' ad longitudinem 'db'.</s></p>
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850 <p><s>Erigatur enim ex 'b' perpendicularis ad orizontem, quae sit 'be', ex 'd' vero ipsi 'bd' perpendicularis 'de', occurrens 'be' in 'e', et circa 'bde' <SYMBOL-3/>_[tri]angulum circulus describatur, qui tanget 'ac' in puncto 'b', ex quo ipsi 'ad' parallela ducatur 'bf', et connettatur 'fd'. Patet, <DELETION-1> ve </DELETION-1> tarditatem per 'fb' esse consimilem tarditati per 'da'; quia vero tempore eodem movetur mobile per 'db' et 'fb', patet, velocitates per 'db' ad velocitates per 'fb' esse ut 'db' ad 'fb', ita ut semper iisdem temporibus duo mobilia, ex punctis 'd', 'f' venientia, linearum 'db', 'fb' partes integris lineis 'db', 'fb' proportione <DELETION-1> conficiantur </DELETION-1> respondentes peregerint. Cum vero angulus 'bfd' in portione angulo 'dba' ad tangentem sit aequalis, angulus vero 'dbf' alterno 'bda', aequiangula erunt <SYMBOL-3/>_[tri]angula 'bfd', 'abd', et ut 'bd' ad 'bf', ita 'ad' ad 'db': ergo ut 'ad' ad 'db', ita velocitas per 'db' ad velocitatem per 'da', et, ex opposito, tarditas per 'da' ad tarditatem per 'db'.</s></p>
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851 <p><s>Si hoc sumatur, reliqua demonstrari possent. Ponatur igitur, augeri vel imminui motus velocitatem secundum proportionem qua augentur vel minuuntur gravitatis momenta; et cum constet, eiusdem mobilis momenta gravitatis super plano 'db' ad momenta super plano 'da' esse ut longitudo 'da' ad longitudinem 'db', idcirco velocitatem per 'db' ad velocitatem per 'da' esse ut 'ad' ad 'db'.</s></p>
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852 <p><s>Lemma. Sit 'dc' ad diametrum 'ba' perpendicularis, et a termino 'b' educatur 'bed' utcunque, et connectatur 'fb': dico, 'fb' inter 'db', 'be' esse mediam. Connectatur 'ef', et per 'b' ducatur tangens 'bg', quae erit ipsi 'cd' parallela; quare angulus 'dbg' angulo 'fdb' erit aequalis: at eidem 'gbd' aequatur quoque angulus 'efb' in portione alterna: ergo similia sunt <SYMBOL-3/>_[tri]angula 'fbd', 'feb', et ut 'bd' ad 'bf', ita 'fb' ad 'be'.</s></p>
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853 <p><s>Lemma. Sit linea 'ac' maior ipsa 'df', et habeat 'ab' ad 'bc' maiorem rationem quam 'de' ad 'ef': dico, 'ab' ipsa 'de' maiorem esse. Quia enim 'ab' ad 'bc' maiorem rationem habet quam 'de' ad 'ef', quam rationem habet 'ab' ad 'bc', hanc habebit 'de' ad minorem quam 'ef'. Sit 'eg': et quia 'ab' ad 'bc' est ut 'de' ad 'eg', erit ut 'ca' ad 'ab', ita 'gd' ad 'de': est autem 'ca' maior 'gd': ergo et 'ba' ipsa 'de' maior erit.</s></p>
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854 <pb xlink:href="X01/01/291.jpg" page="173_R"/>
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855 <p><s>totale momentum ad momentum per 'bd' est ut 'ba' ad 'bc' <DELETION-1> 'bc' </DELETION-1> </s></p>
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856 <p><s>mo[mentum] 'ba' mom[entum] tot[ale] mo[mentum] 'bd'</s></p>
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857 <pb xlink:href="X01/01/292.jpg" page="173_V"/>
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858 <pb xlink:href="X01/01/293.jpg" page="174_R"/>
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859 <p><s>linea 'ab' ad 'ac'.</s></p>
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860 <p><s>tempus 'ba' ad 'ac'.</s></p>
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861 <p><s>ut tempus per 'ab' ad tempus per 'ag', ita linea 'ab' ad lineam 'ar'; ergo, dividendo, ut tempus per 'bc' ad tempus per 'ag', ita linea 'br' ad 'rg'.</s></p>
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862 <p><s>ut tempus 'da' {118} ad tempus 'ab' {60}, ita linea 'da' ad lineam 'ab'.</s></p>
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863 <p><s>tempus 'ad' 118; tempus per 'ac' 95; tempus 'ed' 23.</s></p>
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864 <pb xlink:href="X01/01/294.jpg" page="174_V"/>
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865 <pb xlink:href="X01/01/295.jpg" page="175_R"/>
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866 <pb xlink:href="X01/01/296.jpg" page="175_V"/>
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867 <pb xlink:href="X01/01/297.jpg" page="176_R"/>
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868 <p><s><DELETION-2> Sit <SYMBOL-3/>_[tri]angulum rectangulum 'abc', et 'ab' sit aequalis 'bc', et secetur bifariam 'ac' in 'd', et connectatur 'bd', sitque 'ai' ipsi 'cb' parallela; <DELETION-1> post </DELETION-1> positaque 'ae' ipsi 'ab' aequali, erunt 'ca', 'ae', 'ad' continue proportionales. Secetur 'cb' bifariam in 'f' et connectatur 'ef': dico, quod si protrahatur quaelibet linea ex puncto ad lineam 'ai', ut puta 'cghi', esse proportionales 'ci', 'ig', 'ih'. </DELETION-2> Falsa est.</s></p>
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870 <pb xlink:href="X01/01/299.jpg" page="177_R"/>
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871 <p><s><DELETION-2> <SUBSTITUTION-1> addere <SUBSTITUTED-BY-1> necessariam </SUBSTITUTED-BY-1></SUBSTITUTION-1> hanc propositionem ad praecedentem existimo </DELETION-2> </s></p>
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872 <p><s><DELETION-2> Velocitates mobilium quae inaequali momento incipiunt motum, sunt semper inter se in eadem proportione ac si aequabili motu progrederentur: ut, verbi gratia, mobile per 'ac' incipit motum cum momento ad momentum per 'ad' ut 'da' ad 'ac'; si aequabili motu progrederetur, tempus per 'ac' ad tempus per 'ad' esset <DELETION-1> esset </DELETION-1> ut 'ac' ad 'ad', quod in accelerato dubito quidem; et ideo demonstra?_<END-OF-WORD-DESTROYED/> </DELETION-2> </s></p>
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873 <p><s>Aliter sic. <lb/> Tempus per 'ac' ad tempus per 'ab', ex praecedentibus, est ut linea 'ac' ad lineam 'ab'; <DELETION-1> sed tempus </DELETION-1> sed etiam ad tempus 'ad' <DELETION-1> est </DELETION-1> habet eamdem rationem, cum 'ab' sit media inter 'ac', 'ad'; ergo tempora 'ad', 'ab' erunt aequalia.</s></p>
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874 <pb xlink:href="X01/01/300.jpg" page="177_V"/>
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875 <p><s>Si ex eodem puncto orizontis duca[n]tur perpendiculus et planum inclinatum, et in plano inclinato sumatur quodlibet punctum, a quo ipsi plano perpendicularis linea usque ad perpendiculum protrahatur, lationes in parte perpendiculi inter orizontem et occursum perpendicularis intercepta, et in parte plani inclinati inter eandem perpendicularem et orizontem intercepta, eodem tempore absolventur.</s></p>
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876 <p><s>Sint ex eodem puncto 'c' orizontis 'ab' perpendiculus 'cd' et planum inclinatum 'ce', et in 'ce' sumpto quolibet puncto 'f', ex eo ad 'ec' perpendicularis agatur 'fg', occurrens perpendiculo in puncto 'g': dico, lationes per 'cg' et per 'cf' eodem tempore confici. Demictatur ex eodem puncto 'f' perpendicularis ad orizontem 'fh', quae erit perpendiculo 'cd' parallela, et angulus 'hfc' coalterno 'fcg' aequalis, et rectus 'chf' recto 'cfg': quare aequiangula erunt triangula 'chf', 'cfg', et ut 'hf' ad 'fc', ita 'fc' ad 'cg'; ut autem 'hf' ad 'fc', ita momentum gravitatis mobilis in plano 'ce' ad totale suum momentum in perpendiculo 'cd'. Habet igitur distantia 'cf' ad distantiam 'cg' eandem rationem quam gravitatis momentum super plano 'ce' ad totale momentum per perpendiculum <SUBSTITUTION-1> 'cd' <SUBSTITUTED-BY-1> 'cg' </SUBSTITUTED-BY-1></SUBSTITUTION-1>, quare eodem tempore conficientur lationes per 'cf' et 'cg'.</s></p>
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877 <pb xlink:href="X01/01/301.jpg" page="178_R"/>
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878 <p><s>Sint ad horizontem 'db' quotcumque lineae ab eadem altitudine 'a' demiss[a]e 'ab', 'ac', 'ad', et sumpto quolibet puncto 'g', per ipsum horizonti parallela sit 'gfe'; sitque media inter 'ba', <SUBSTITUTION-1> 'ag' <SUBSTITUTED-BY-1> 'ab' </SUBSTITUTED-BY-1></SUBSTITUTION-1> ipsa 'ar', et per 'r' altera parallela 'rtu': constat, lineas 'at', 'au' esse medias inter 'ca', 'af' et 'da', 'ae'.</s></p>
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879 <p><s>Dico, quod si assumatur 'ab' esse tempus quo mobile cadit ex 'a' in 'b', tempus 'rb' esse illud quo conficitur 'gb', 'tc' vero esse tempus ipsius 'cf', et 'ud' ipsius 'ed'. Id autem constat: nam, cum 'ar' sit media inter <SUBSTITUTION-1> 'da' <SUBSTITUTED-BY-1> 'ba', </SUBSTITUTED-BY-1></SUBSTITUTION-1> 'ag', sitque 'ba' tempus casus totius 'ab', tempus 'ar' erit tempus casus per 'ag'; ergo reliquum temporis 'rb' erit tempus casus per 'gb' post 'ag'; et idem dicetur de aliis temporibus 'tc', 'vd', et lineis 'fc', 'ed'. Patet insuper, tempora casuum per 'gb', 'fc', 'ed' esse ut lineas 'gb', 'fc', 'ed'; non tamen a magnitudinibus ipsarum linearum 'gb', 'fc', 'ed' esse determinandas eorumdem temporum quantitates, si temporis mensura ponatur 'ab', in quo tempore conficiatur linea 'ab', sed desumendas esse a lineis 'rb', 'tc', 'ud'.</s></p>
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880 <pb xlink:href="X01/01/302.jpg" page="178_V"/>
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881 <pb xlink:href="X01/01/303.jpg" page="179_R"/>
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882 <p><s>Si in perpendiculo et in plano inclinato, <DELETION-1> feratur </DELETION-1> quorum eadem sit altitudo, feratur idem mobile, tempora lationum erunt inter se ut plani inclinati et perpendiculi longitudines.</s></p>
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883 <p><s>Sint ad planum orizontis 'cb' perpendiculus 'ab' et planum inclinatum 'ac', quorum eadem sit altitudo, nempe ipsa perpendicularis 'ab', et per ipsa descendat idem mobile: dico, tempus lationis per 'ab' ad tempus lationis per 'ac' esse ut longitudo 'ab' ad longitudinem 'ac'.</s></p>
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884 <p><s> Cum enim assumptum sit, in naturali descensu velocitatis momenta <INSERTION-1> eadem semper reperiri in punctis aequaliter ab orizonte distantibus iuxta perpendiculares distantias, </INSERTION-1> <UNDERLINED> continue augeri secundum rationem elongationis perpendicularis a linea orizontali, in qua <DELETION-2> pu </DELETION-2> fuit lationis initium, </UNDERLINED> constat quod, producta linea orizontali 'am', quae ipsi 'bc' erit parallela, sumptisque in perpendiculo 'ab' quotcumque punctis 'e', 'g', 'i', 'l', et per ipsa ductis parallelis orizonti 'ed', 'gf', 'ih', 'lk', erit mobilis per 'ab' momentum seu gradus velocitatis in puncto 'e' idem cum gradu velocitatis lati per 'ac' in puncto 'd', cum punctorum 'e', 'd' eadem sit distantia perpendicularis ab orizonte 'am': et similiter concludetur in pun[c]tis 'f', 'g' idem esse velocitatis momentum, et rursus in punctis 'h', 'i', et 'k', 'l', et 'c', 'b'.</s></p>
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885 <p><s><UNDERLINED> Et quia velocitas semper intenditur pro ratione elongationis a termino 'a', constat, in latione 'ab' tot esse velocitatis gradus, seu momenta, <INSERTION-1> diversa, </INSERTION-1> quot sunt in eadem linea 'ab' puncta magis ac magis a termino 'a' distantia; quibus totidem in linea 'ac' respendent [respondent], et per parallelas lineas determinantur, in quibus iidem sunt gradus velocitatis. </UNDERLINED> </s></p>
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886 <p><s>Sunt igitur in linea 'ab' quasi innumera quaedam spaciola, quibus multitudine quidem aequalia, et <INSERTION-1> bina sumpta </INSERTION-1> secundum eandem rationem respondentia, alia signantur in 'ac' per lineas innumeras parallelas ex punctis lineae 'ab' ad lin[e]am 'ac' extensas <INSERTION-1> (intercepta enim spacia 'ad', 'df', 'fh', etc. ad spacia 'ae', 'eg', 'gi', etc. respondent singula singulis secundum rationem 'ac' ad 'ab'); </INSERTION-1> suntque in singulis binis sibi respondentibus iidem velocitatis gradus.</s></p>
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887 <p><s> Ergo, ex praecedenti, tempora omnia simul sumpta lationum omnium per 'ab', ad tempora omnia similiter accepta lationum omnium per 'ac', eandem habebunt rationem quam spacia omnia lineae 'ab' ad spacia omnia lin[e]ae 'ac'; hoc autem idem est, ac tempus casus per 'ab' ad tempus casus per 'ac' esse ut linea 'ab' ad 'ac': quod erat demonstrandum.</s></p>
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888 <pb xlink:href="X01/01/304.jpg" page="179_V"/>
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889 <p><s>Tempora lationum per diversas lineas inclinatas, quarum eadem sit altitudo perpendicularis, sunt inter se ut earumdem linearum longitudines.</s></p>
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890 <p><s>Sint ad orizontem 'bd' diversa plana inclinata 'ab', 'ac', quorum eadem sit altitudo 'ad' perpendicularis: dico, tempus casus per 'ab' ad tempus casus per 'ac' esse ut 'ab' longitudo ad longitudinem 'ac'. Ex antecedenti enim, tempus casus per 'ab' ad tempus casus per perpendiculum 'ad' est ut 'ab' linea ad lineam 'ad', et, per eandem, ut 'ad' linea ad ipsam 'ac', ita tempus casus per 'ad' ad tempus casus per 'ac': ergo, ex aequali, ut longitudo 'ab' ad longitudinem 'ac', ita tempus casus per 'ab' ad tempus casus 'ac': quod erat probandum.</s></p>
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891 <p><s>Si in linea naturalis descensus a principio lationis sumantur duae distantiae inaequales, momenta velocitatis cum quibus mobile permeat illas distantias sunt inter se in duplicata proportione ipsarum distantiarum.</s></p>
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892 <p><s>Sit linea naturalis [des]census 'ab', in qua ex <DELETION-1> termino </DELETION-1> principio lationis 'a' sumantur duae distantiae 'ac', 'ad'. Dico, momenta velocitatis cum quibus mobile permeat 'ad', ad momenta velocitatis cum quibus permeavit 'ac', esse <DELETION-1> ut </DELETION-1> in dupli[cata] proportione distantiarum 'ad', <ILLEGIBLE/> Ponatur linea 'ae' ad 'ab' quemlibet angulum continens</s></p>
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893 <pb xlink:href="X01/01/305.jpg" page="179AR"/>
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894 <pb xlink:href="X01/01/306.jpg" page="179AV"/>
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895 <p><s>Momenta gravitatis eiusdem mobilis super plano inclinato et in perpendiculo permutatim respondent longitudini et elevationis [elevationi] eiusdem plani.</s></p>
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896 <p><s>Sit ad orizontem 'ab' planum inclinatum 'ca', <DELETION-1> cuius </DELETION-1> in quo sumatur quodcumque punctum 'c', et demissa perpendicularis ad orizontem 'cb' sit plani 'ca' altitudo seu elevatio: dico, momentum gravitatis mobilis 'd' super plano 'ca' ad totale suum momentum in perpendiculo 'cb' esse ut altitudo 'cb' ad eiusdem plani longitudinem 'ca'. Id autem in mecanicis probatum est.</s></p>
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897 <pb xlink:href="X01/01/307.jpg" page="179BR"/>
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898 <pb xlink:href="X01/01/308.jpg" page="179BV"/>
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899 <p><s>Momenta gravitatis eiusdem mobilis super diversas planorum inclinationes habent inter se permutatim eandem rationem, quam eorumdem planorum longitudines, dum eidem elevationi respondeant.</s></p>
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900 <p><s>Sint diversae planorum inclinationes 'ab', 'ac', quae eidem elevationi 'ad' respondeant. Dico, mementum [momentum] gravitatis eiusdem mobilis super 'ab' ad momentum gravitatis super 'ac' eandem habere rationem, quam longitudo 'ac' habet ad longitudinem 'ab'. Ex praecedenti, enim, momentum gravitatis super 'ab' ad totale momentum in perpendiculo 'ad' est ut 'ad' ad 'ab'; totale vero momentum per 'ad' ad momentum per 'ac' est ut 'ca' ad 'ad': ergo, ex aequali in analogia perturbata, momentum per 'ab' ad momentum per 'ac' erit ut longitudo 'ca' ad longitudinem <ILLEGIBLE/> [Quod erat demonstrandum.]</s></p>
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901 <pb xlink:href="X01/01/309.jpg" page="180_R"/>
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902 <p><s>Si ex eodem puncto <INSERTION-1> orizontis </INSERTION-1> ducantur perpendiculus et planum inclinatum, et in plano inclinato sumatur quodlibet punctum, a quo <INSERTION-1> ipsi plano </INSERTION-1> perpendicularis linea usque ad perpendiculum protrahatur, lationes in parte perpendiculi inter orizontem et occursum perpendicularis intercepta, et in parte plani inclinati inter eandem perpendicularem et orizontem intercepta, eodem tempore absolventur.</s></p>
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903 <p><s>Sint ex eodem puncto 'c' orizontis 'ab' perpendiculus 'cd' et planum inclinatum 'ce', et in 'ce' sumpto quolibet puncto 'f', ex eo ad 'ec' perpendicularis agatur 'fg', occurrens perpendiculo in puncto 'g': dico, lationes per 'cg' et per 'cf' eodem tempore confici.</s></p>
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904 <p><s>Demittatur ex eodem puncto 'f' perpendicularis ad orizontem 'fh', quae erit perpendiculo 'cd' parallela, et angulus 'hfc' coalterno 'fcg' aequalis, et rectus 'chf' recto 'cfg': quare aequiangula erunt triangula 'chf', 'cfg', et ut 'hf' ad 'fc', ita 'fc' ad 'cg'. Ut autem 'hf', 'fc', ita momentum gravitatis mobilis in plano 'ce' ad totale suum momentum in perpendiculo 'cd'. Habet igitur distantia 'cf' ad distantiam 'cg' eandem rationem quam gravitatis momentum super plano 'ce' ad totale momentum per perpendiculum 'cg': quare eodem tempore conficientur lationes per 'cf' et 'cg'.</s></p>
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905 <pb xlink:href="X01/01/310.jpg" page="180_V"/>
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906 <pb xlink:href="X01/01/311.jpg" page="181_R"/>
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907 <p><s>Factus sit motus ex 'a' in 'b' naturaliter acceleratus: dico, quod si velocitas in omnibus punctis 'ab' fuisset eadem ac reperitur in puncto 'b', duplo citius fuisset peractum spacium 'ab'; quia velocitates omnes in singulis punctis 'ab' lineae, ad totidem velocitates quarum unaquaeque esset aequalis velocitati 'bc', eam habent rationem quam triangulus 'abc' ad rectangulum 'abcd'.</s></p>
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908 <p><s>Sequitur ex hoc, quod si ad <DELETION-1> <ILLEGIBLE/> </DELETION-1> orizontem 'cd' fuerit planum 'ba' elevatum, sitque 'bc' dupla ad 'ba', mobile ex 'a' in 'b', et successive ex 'b' in 'c', temporibus aequalibus esse perven[c]turum: nam postquam est in 'b', per reliquam 'bc' uniformi velocitate et eadem movetur, qua in ipsomet termino 'b' post casum 'ab'. Patet rursus, totum tempus per 'abe' ad tempus per 'ab' esse sesquialterum.</s></p>
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909 <p><s>Huic demonstrationi necessarium mihi videtur ostendisse antea, motum orizontalem uniformiter progredi in infinitum.</s></p>
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910 <pb xlink:href="X01/01/312.jpg" page="181_V"/>
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911 <pb xlink:href="X01/01/313.jpg" page="182AR"/>
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912 <p><s><foreign lang="it">Notabile per i proietti nel determinare quanto detragga la propension naturale in giù al moto preterna[tura]le della proiezzione.</foreign></s></p>
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913 <p><s>Si impetus violentus disponatur secundum numeros pares, descensus naturalis demit dimidium, ut constat in exemplis 'D', 'F', 'E', 'B', 'C', 'A'; verum si dispositio sit secundum numeros impares, naturalis descensus demit minus quam dimidium iuxta numerum partium dispositarum, ut patet in exemplis 'G', 'H', 'I', 'L'. In 'G', enim, partes dispositae iuxta impetum <DELETION-1> non </DELETION-1> violentum non retardatum sunt 3, nempe 5, 1O, 15; ex quibus <DELETION-1> <p-SUPERSCRIBED-BY-a/>p[rim]a </DELETION-1> in prima demitur 1, <DELETION-1> ex <SYMBOL-2/>_[secund]a 3 </DELETION-1> et relinquitur 4; dempto ex <SYMBOL-2/>_[secund]a 4, relinquitur 6; dempto ex <SYMBOL-3/>_[terti]a, nempe ex 15, 9, relinquitur idem numerus 6, qui deficit a dimidio 15 per 3, qui est numerus parzium [partium] 5, 1O, 15. In exemplo 'H' numerus partium est 4, <DELETION-1> <ILLEGIBLE/> </DELETION-1> subtractiones motus naturalis sunt 6, 4, 2, quae conficiunt 12, <DELETION-1> qui deficit a dimidio 2 </DELETION-1> cuius duplum deficit a 28 per 4. In exemplo 'I' subtractiones 8, 6, 4, 2 exibent 20, cuius duplus [duplum] deficit a 45 per 5, qui similiter est numerus partium etc. In 'L' pariter apparet, subtractiones, nempe 156, duplicatae deficere per 13 (qui est numerum partium motus violenti) a 325, etc.</s></p>
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914 <pb xlink:href="X01/01/314.jpg" page="182AV"/>
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915 <p><s>Motuum qui a dato puncto usque ad datam lineam per rectas lineas conficiuntur, ille brevissimo tempore absolvitur, qui in recta fit abscindens de data linea partem aequalem ei parti lineae horizontalis, quae per datum punctum usque ad datam lineam producitur, quae inter datum punctum et occursum intercipitur. Sit datum punctum 'a' et linea quaecumque 'bdc', et per 'a' horizonti aequidistans 'ab', quae lineae 'db' in 'b' occurat, et interceptae 'ab' ponatur aequalis 'bd'. Dico, motum per 'ad' absolvi tempore breviori, quam per quamcumque aliam lineam ex puncto 'a' ad quodcumque punctum lineae 'bdc' productam. Ducatur ad 'ba' perpendicularis 'ac', et ex 'd' ad ipsam 'bc' perpendicularis 'de', occurrens 'ac' in 'e': et quia in <SYMBOL-3/>_[tri]angulo <BEGIN-OF-LINE-DESTROYED/> [aequi]cruri 'abd' anguli 'bad', 'bda' sunt aequales, ergo reliqui ad rectos 'ead', 'eda', <BEGIN-OF-LINE-DESTROYED/> [aequales] pariter erunt, et linea 'ea' aequalis ipsi 'ed'. Si itaque, centro 'e', intervallo <BEGIN-OF-LINE-DESTROYED/> ['ea', cir]culus describatur, transibit per 'd', ubi lineam 'bdc' tanget: quare lineae om[nes quae supr]a vel infra 'ad' usque ad lineam 'bc' producentur, ultra circumferentiam cir[culi] <BEGIN-OF-LINE-DESTROYED/> [ex]tendentur. Ex quo patet propositum.</s></p>
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916 <pb xlink:href="X01/01/315.jpg" page="182BR"/>
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917 <pb xlink:href="X01/01/316.jpg" page="182BV"/>
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918 <pb xlink:href="X01/01/317.jpg" page="183_R"/>
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919 <p><s>media inter 'ad' 'te' 84090 <lb/> tempus per 'te' 84090</s></p>
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920 <pb xlink:href="X01/01/318.jpg" page="183_V"/>
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921 <pb xlink:href="X01/01/319.jpg" page="184_R"/>
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922 <pb xlink:href="X01/01/320.jpg" page="184_V"/>
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923 <p><s>longitudo 'ao' 127 tempus per 'ao' 151</s></p>
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924 <pb xlink:href="X01/01/321.jpg" page="185_R"/>
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925 <p><s>Lemma. Sit linea 'ac' maior ipsa 'df', et habeat 'ab' ad 'bc' maiorem rationem quam 'de' ad 'ef': dico, 'ab' ipsa 'de' maiorem esse. Quia enim 'ab' ad 'bc' maiorem rationem habet quam 'de' ad 'ef'; <DELETION-1> quam rationem habebit </DELETION-1> quam rationem <DELETION-1> habebit </DELETION-1> habet 'ab' ad 'bc', hanc habebit 'de' ad minorem quam 'ef'. Sit <DELETION-1> <ILLEGIBLE/> </DELETION-1> 'eg', et quia 'ab' ad 'bc' est ut 'de' ad 'eg', erit ut 'ca' ad 'ab', ita 'gd' ad 'de': est autem 'ca' maior 'dg': ergo et 'ba' ipsa 'de' maior erit.</s></p>
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926 <pb xlink:href="X01/01/322.jpg" page="185_V"/>
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927 <pb xlink:href="X01/01/323.jpg" page="186_R"/>
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928 <pb xlink:href="X01/01/324.jpg" page="186_V"/>
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929 <p><s>Dicimus, tempus quo mobile permeat lineas 'db', 'bc' brevius esse tempore quo permeat solam 'bc'. Sit 'ae' aequalis 'bc': si itaque fuerint motus initia puncta 'a', 'b', eodem tempore peragentur lineae 'bc' et 'ae'. Sit tempus quo conficitur 'ae', <INSERTION-1> vel 'bc', </INSERTION-1> !ipsum 'mn', et quam rationem habet 'ae' ad mediam inter 'ae', 'ac', hanc habeat 'nm' ad 'nx'; erit 'nx' tempus totius 'ac': quam vero rationem habet 'ca' ad mediam inter 'ca', 'ab', hanc habeat tempus 'xn' ad 'nr'; <DELETION-1> hanc </DELETION-1> erit 'rn' tempus ipsius 'ab', 'rx' vero ipsius 'bc' post 'ab' (quam 'xr' oportet minorem esse ipsa 'mn').</s></p>
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930 <p><s>Ostendatur, citius transiri 'bc' post 'ab' quam 'fc' post 'df'. Sit 'ds' tempus quo peragitur tota 'dc', vel 'bc', et quam rationem habet media inter 'cd', 'df' ad 'df', hanc habeat tempus 'sd' ad 'dr'; constat, tempus ipsius 'fc' esse 'rs': quia vero tempus ipsius 'bc', seu 'ae', est idem 'ds', fiat ut 'ea' ad mediam inter 'ea' et totam 'ac', ita 'sd' ad 'dt', eritque 'dt' tempus totius 'ac'. Quod si rursus fiat ut tota 'ca' ad mediam inter 'ca', 'ab', ita 'td' ad 'dv', erit 'vt' tempus ipsius 'bc' post 'ab': hoc autem ostendendum est, esse minus ipso 'rs'.</s></p>
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931 <p><s>Nota. Sit in circumferentia utcumque ducta 'do', <DELETION-1> 'db' </DELETION-1> et iungatur <DELETION-1> <ILLEGIBLE/> </DELETION-1> 'co': dico dico [sic], citius moveri ex 'd' in 'o' quam ex 'o' in 'c'. Ostensum enim est aequali tempore moveri ex 'o' in 'c', atque ex ex [sic] ['d'] in 'c'; verum ex 'd' in 'o' patet celerius fieri motum quam ex 'd' in 'c'.</s></p>
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932 <pb xlink:href="X01/01/325.jpg" page="187_R"/>
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933 <p><s>tempus 'ae' sit 120 <lb/> tempus per 'ec' si esset orizontalis esset 60</s></p>
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934 <p><s>tempus 'ec' 120 per 'ae' 77 <lb/> tempus 'ae' 317 'ec' 120 </s></p>
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935 <p><s>ut <ILLEGIBLE/> '?y' ad 'yi' ita</s></p>
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936 <p><s>sit</s></p>
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937 <p><s><DELETION-1> <SYMBOL-SOL/>_'SOLk' 'kc' </DELETION-1> <lb/> <SYMBOL-SOL/>_'SOLd' 'dc' 'cz' </s></p>
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938 <p><s>'yg?' <ILLEGIBLE/></s></p>
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939 <p><s><ILLEGIBLE/></s></p>
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940 <p><s>'gm' 'g23' </s></p>
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941 <pb xlink:href="X01/01/326.jpg" page="187_V"/>
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942 <pb xlink:href="X01/01/327.jpg" page="188_R"/>
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943 <p><s><DELETION-2> ut 'ca' ad 'ab', ita 'ab' ad 'ae'; ergo ut <SYMBOL-QUADRATUM/>_[quadratum] 'ca' ad <SYMBOL-QUADRATUM/>_[quadratum] 'ba', vel <SYMBOL-QUADRATUM/>_[quadratum] 'ba' ad <SYMBOL-QUADRATUM/>_[quadratum] 'ae', ita 'ca' ad 'ae', vel 'ea' ad 'as'; ergo punctum 's' reperitur, si fiat ut <SYMBOL-QUADRATUM/>_[quadratum] 'ba' ad <SYMBOL-QUADRATUM/>_[quadratum] 'ae', ita 'ea' ad 'as': fiet autem hoc, si ipsarum 'ba', 'ae' accipiatur <SYMBOL-4/>_[quart]a <DELETION-1> pop </DELETION-1> proportionalis 'as'. At <SYMBOL-QUADRATUM/>_[quadratum] 'ba' ad <SYMBOL-QUADRATUM/>_[quadratum] 'ae' est ut <SYMBOL-RECTANGULUM/>_[rectangulum] ex diametro in 'an' ad <SYMBOL-RECTANGULUM/>_[rectangulum] ex diametro in 'ar', quibus sunt aequalia; ergo ut 'ea' ad 'as', ita 'na' ad 'ar', idest altitudo lineae 'ba' ad altitudinem lineae 'ae'. </DELETION-2> </s></p>
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944 <p><s><DELETION-2> Linea ergo 'ba' ad 'as' est ut cubus 'ba' ad cubum <DELETION-1> 'be' </DELETION-1> 'ae'. Fiat ut 'rn' ad 'be', ita 'be' ad 'no', et ut 'ba' ad 'an', ita 'no' ad 'bd'. </DELETION-2> </s></p>
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945 <p><s><DELETION-2> 'bv' ad 've' est ut 'ca' ad 'ae', hoc est 'ea' ad 'as'. </DELETION-2> </s></p>
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946 <p><s>Quia ut 'ac' ad 'cs', ita 'sc' ad 'ce' <DELETION-1> ergo </DELETION-1> et 'as' ad 'se', ergo ut 'ac' ad 'ce', idest 'vb' ad 'be', ita <SYMBOL-QUADRATUM/>_[quadratum] 'as' ad <SYMBOL-QUADRATUM/>_[quadratum] 'se'; est aut[em] ut 'vb' ad 'be', ita <SYMBOL-QUADRATUM/>_[quadratum] 'vb' ad <SYMBOL-QUADRATUM/>_[quadratum] 'bi': ergo ut <SYMBOL-QUADRATUM/>_[quadratum] 'as' ad <SYMBOL-QUADRATUM/>_[quadratum] 'se', ita <SYMBOL-QUADRATUM/>_[quadratum] 'vb' ad <SYMBOL-QUADRATUM/>_[quadratum] 'bi', et ut</s></p>
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947 <pb xlink:href="X01/01/328.jpg" page="188_V"/>
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948 <pb xlink:href="X01/01/329.jpg" page="189_R"/>
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949 <pb xlink:href="X01/01/330.jpg" page="189_V"/>
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950 <p><s><DELETION-1> In <SYMBOL-2/>_[duo]bus planis quomodocumque inclinatis tempora casuum habent ipsorum planorum proportionem subduplicatam. </DELETION-1> </s></p>
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951 <p><s>tempus per 'ac' ad tempus per 'ab' habet rationem compositam ex 'ca', 'ab' et 'da' ad 'as'.</s></p>
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952 <p><s>si <DELETION-1> <ILLEGIBLE/> </DELETION-1> diameter 'ab' sit 4000, arcus <DELETION-1> <ILLEGIBLE/> </DELETION-1> 'bd' conficitur temporibus <SUBSTITUTION-1> 124 <SUBSTITUTED-BY-1> 62 </SUBSTITUTED-BY-1></SUBSTITUTION-1> </s></p>
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953 <p><s>perpendicularis cuius longitudo <SUBSTITUTION-1> 48142 <SUBSTITUTED-BY-1> 48143, </SUBSTITUTED-BY-1></SUBSTITUTION-1> conficitur tempore 280.</s></p>
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954 <p><s>tempus 'ba' ad t[empus] 'ad' ut 'ba' ad 'ad'</s></p>
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955 <pb xlink:href="X01/01/331.jpg" page="190_R"/>
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956 <pb xlink:href="X01/01/332.jpg" page="190_V"/>
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957 <pb xlink:href="X01/01/333.jpg" page="191_R"/>
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958 <pb xlink:href="X01/01/334.jpg" page="191_V"/>
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959 <pb xlink:href="X01/01/335.jpg" page="192_R"/>
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960 <p><s>tempus per 'gx'</s></p>
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961 <p><s>tempus per 'ed' 54119 <lb/> tempus per 'cgx'</s></p>
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962 <p><s><DELETION-1> 'cgx' 'ldg' 511259 tempus 511259 <lb/> 'gx' 472242 </DELETION-1></s></p>
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963 <p><s>tempus per 'eg' 83801</s></p>
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964 <pb xlink:href="X01/01/336.jpg" page="192_V"/>
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965 <pb xlink:href="X01/01/337.jpg" page="193_R"/>
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966 <p><s><foreign lang="it"><DELETION-1> Particolari privilegii dell'artiglieria sopra gl'altri strumenti mecanici. </DELETION-1> </foreign></s></p>
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967 <p><s><foreign lang="it"><DELETION-1> Della sua forza, et onde proceda. </DELETION-1> </foreign></s></p>
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968 <p><s><foreign lang="it"><DELETION-1> Se operi con magior forza in una certa dist[anza] o da vicino. </DELETION-1> </foreign></s></p>
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969 <p><s><foreign lang="it">Se la palla vadia per linea retta, non sen[do tirata] a perpendicolo.</foreign></s></p>
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970 <p><s><foreign lang="it">Che linea descriva la palla nel suo [moto].</foreign></s></p>
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971 <p><s><foreign lang="it"><DELETION-1> La causa et il tempo dello stornare il pezzo. </DELETION-1> </foreign></s></p>
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972 <p><s><foreign lang="it">Impedimenti che rendono il pezo difettoso et il tiro incerto.</foreign></s></p>
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973 <p><s><foreign lang="it">Del metterle a cavallo e scavalcarIe.</foreign></s></p>
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974 <p><s><foreign lang="it">Della fabrica del colibro.</foreign></s></p>
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975 <p><s><foreign lang="it">Dell'esamine circa la bontà et giustezza del pezzo.</foreign></s></p>
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976 <p><s><foreign lang="it">Se quanto più e [è] lungo il pezzo più tira lontano, e perchè.</foreign></s></p>
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977 <p><s><foreign lang="it">A quale elevatione tiri più da lontano, et perchè.</foreign></s></p>
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978 <p><s><foreign lang="it">Che nel tornare la palla ingiù nel perpendicolo, torna con le medesime forze et velocità con che andò in su.</foreign></s></p>
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979 <p><s><foreign lang="it">Diverse palle artifitiate et lanterne, et lor uso.</foreign></s></p>
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980 <pb xlink:href="X01/01/338.jpg" page="193_V"/>
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981 <pb xlink:href="X01/01/339.jpg" page="194_R"/>
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982 <p><s>Ex puncto 'C' horizontalis lineae 'LCX' duo plana utcumque inflectantur 'CD', 'CE', quae secentur recta quadam 'DF', ita ut anguli 'CDF', 'DFC' angulis 'XCE', 'LCD' permutatim sumptis sint aequales. Dico, tempora descensuum per 'CD', 'CF' esse aequalia (fient autem anguli permutatim aequales, si unus angulorum, verbigratia 'CDF', aequalis fiat angulo 'XCE' ad</s></p>
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983 <p><s>aliam partem posito, <SUBSTITUTION-1> reliqu <ILLEGIBLE/> <SUBSTITUTED-BY-1> reliquus </SUBSTITUTED-BY-1></SUBSTITUTION-1> enim 'CFD', reliquo 'DCL' aequalis erit; nam cum tres anguli trianguli 'DCF' aequales sint tribus 'LCD', 'DCF', 'FCX', utpote duobus rectis aequales, si dematur comunis 'DCF', erunt duo 'CDF', 'DFC' duobus 'XCE', 'LCD' aequales, ac propterea, cum fecerimus angulum 'CDF' angulo 'XCE' aequalem, habebimus quoque angulum 'CFD' aequalem angulo 'LCD'). Ponatur</s></p>
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984 <pb xlink:href="X01/01/340.jpg" page="194_V"/>
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985 <pb xlink:href="X01/01/341.jpg" page="195_R"/>
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986 <pb xlink:href="X01/01/342.jpg" page="195_V"/>
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987 <pb xlink:href="X01/01/343.jpg" page="196_R"/>
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988 <pb xlink:href="X01/01/344.jpg" page="196_V"/>
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989 </body>
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990 </text>
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991 </archimedes>