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version 1.28, 2003/02/28 10:27:30

version 1.34, 2003/03/28 20:55:54

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<info>

<author>Del Monte, Guidobaldo</author><title>Mechanicorum Liber</title> <date>1577</date><place>Pisauri</place><translator></translator><lang>LA</lang><cvs_file>monte_mecha_02_la_1577</cvs_file><cvs_version>2635.10</cvs_version><locator>036.xml</locator></info>

<author>Monte, Guidobaldo del</author><title>Mechanicorum Liber</title> <date>1577</date><place>Pisauri</place><translator></translator><lang>LA</lang><cvs_file>monte_mecha_02_la_1577</cvs_file><cvs_version>2635.10</cvs_version><locator>036.xml</locator></info>

<text>

<front>

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<s id="id.2.1.1.14.1.3.0"> ego enim in hac præ&longs;ertim <lb/>facultate Archimedis ve&longs;tigijs hærere &longs;emper vo <lb/>lui: & licet eius lucubrationes ad <expan abbr="mechanicã">mechanicam</expan>per<pb xlink:href="036/01/009.jpg"/>tinentes multis ab hinc annis pa&longs;&longs;im &longs;oleant do<lb/>ctis de&longs;iderari: eruditi&longs;&longs;imus tamen libellus de æ<lb/>queponderantibus præ manibus <expan abbr="homin&utilde;">hominum</expan>adhuc <lb/>ver&longs;atur, in quò tanquam in copio&longs;i&longs;&longs;ima p&oelig;nu <lb/>omnia ferè mechanica dogmata repo&longs;ita mihi vi<lb/>dentur; quem &longs;anè libellum, &longs;i ætatis no&longs;træ mathe<lb/>matici &longs;ibi magis familiarem adhibui&longs;&longs;ent; reperi&longs;<lb/>&longs;ent &longs;anè <expan abbr="&longs;ent&etilde;tias">&longs;ententias</expan>multas, quas modó ip&longs;i firmas, <lb/>& ratas e&longs;&longs;e docent; &longs;ubtili&longs;&longs;imè, atquè <expan abbr="veri&longs;&longs;imè">veri&longs;<lb/>&longs;ime</expan>conuul&longs;as, & labefactatas. </s>

<s id="id.2.1.1.14.1.4.0"> &longs;ed hoc vi<lb/>derint ip&longs;i. </s>

<s id="id.2.1.1.14.1.5.0"> ego enim ad Pappum redeo, qui <lb/>ad v&longs;um mathematicarum vberiorem, <expan abbr="emulumentorumquè">emulu<lb/>mentorumque</expan>acce&longs;&longs;iones amplificandas peni<lb/>tus conuer&longs;us, de quinque principibus machi<lb/>nis, Vecte nempè, Trochlea, Axe in peri<lb/>trochio, Cuneo, & Cochlea, multa <expan abbr="egregiè">egre<lb/>gie</expan>philo&longs;ophatus e&longs;t; demon&longs;trauit què quicquid <lb/>in machinis, aut cogitari peritè, aut acutè <lb/>definiri, aut certò &longs;tatui pote&longs;t, id omne <expan abbr="quinquè">quin<lb/>que</expan>illis infinita vi præditis machinis referen<lb/>dum e&longs;&longs;e. </s>

<s id="id.2.1.1.14.1.6.0"> atquè vtinam iniuria temporis ni<lb/>hil è tanti viri &longs;criptis abra&longs;i&longs;&longs;et: nec enim tam <lb/>den&longs;a in&longs;citiæ caligo vniuer&longs;um propè terra<lb/>rum orbem obtexi&longs;&longs;et, neque tanta mechani<lb/>cæfacultatis e&longs;&longs;et ignoratio con&longs;ecuta, vt ma<lb/>thematicarum proceres exi&longs;timarentur illi, qui <lb/>modò inepti&longs;&longs;ima quadam di&longs;tinctione, diffi|cultate<pb xlink:href="036/01/010.jpg"/>s nonnullas, nec illas tamen &longs;atis ar<lb/>duas, & ob&longs;curas è medio tollunt. </s>

<s id="id.2.1.1.14.1.6.0">atquè vtinam iniuria temporis ni<lb/>hil è tanti viri &longs;criptis abra&longs;i&longs;&longs;et: nec enim tam <lb/>den&longs;a in&longs;citiæ caligo vniuer&longs;um propè terra<lb/>rum orbem obtexi&longs;&longs;et, neque tanta mechani<lb/>cæ facultatis e&longs;&longs;et ignoratio con&longs;ecuta, vt ma<lb/>thematicarum proceres exi&longs;timarentur illi, qui <lb/>modò inepti&longs;&longs;ima quadam di&longs;tinctione, diffi<pb xlink:href="036/01/010.jpg"/>cultates nonnullas, nec illas tamen &longs;atis ar<lb/>duas, & ob&longs;curas è medio tollunt. </s>

<s id="id.2.1.1.14.1.7.0"> reperiun<lb/>tur enim aliqui, no&longs;traq; ætate emunctæ naris <lb/>mathematici, qui mechanicam, tùm <expan abbr="mathematicè">mathe<lb/>matice</expan>&longs;eor&longs;um, tùm phi&longs;icè con&longs;iderari po&longs;<lb/>&longs;e affirmant; ac &longs;i aliquando, vel &longs;ine demon<lb/>&longs;trationibus geometricis, vel &longs;ine vero motu <lb/>res mechanicæ con&longs;iderari po&longs;&longs;int: qua &longs;anè di<lb/>&longs;tinctione (vt leuius cum illis agam) nihil aliud mi<lb/>hi commini&longs;ci videntur, quàm vt dum &longs;e, tùm <lb/>phi&longs;icos, tùm mathematicos proferant, vtra<lb/>que (quod aiunt) &longs;ella excludantur. </s>

<s id="id.2.1.1.14.1.8.0"> nequè <lb/>enim amplius mechanica, &longs;i à machinis ab&longs;tra<lb/>hatur, & &longs;eiungatur, mechanica pote&longs;t appel<lb/>lari. </s>

<s id="id.2.1.1.14.1.9.0"> Emicuit tamen inter i&longs;tas tenebras (quam<lb/>uis alij quoquè nonnulli fuerint præclari&longs;&longs;imi) <lb/>Solis in&longs;tar Federicus Commandinus, qui multis <lb/>docti&longs;&longs;imis elucubrationibus ami&longs;&longs;um mathema<lb/>ticarum patrimonium non modò re&longs;taurauit, <lb/>verùm etiam auctiùs, & locupletiùs effecit. </s>

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<s id="id.2.1.5.4.1.8.0"> quod <lb/>demon&longs;trare oportebat. </s>

<s id="id.2.1.6.1.1.1.0"> <margin.target id="note3"></margin.target>4. <emph type="italics"/>primi Archimedis de æqueponderantibus.<emph.end type="italics"/> </s>

<s id="id.2.1.6.1.1.1.0"><margin.target id="note3"></margin.target>4. <emph type="italics"/>primi Archi<lb/>medis de <lb/>æqueponde<lb/>rantibus.<emph.end type="italics"/></s>

<s id="id.2.1.6.1.1.2.0"> <margin.target id="note4"></margin.target>1. <emph type="italics"/>Huius<emph.end type="italics"/> </s>

<s id="id.2.1.6.1.1.3.0"> <margin.target id="note5"></margin.target>1. <emph type="italics"/>Huius.<emph.end type="italics"/> </s>

</figure>

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<s id="id.2.1.9.2.1.1.0"> Libra horizonti æquidi&longs;tans æqualia in ex<lb/>tremitatibus, æqualiterq; à centro in ip&longs;a libra <lb/>collocato, di&longs;tantia habens pondera; &longs;iue inde <lb/>moueatur, &longs;iue minus; vbicunq; relicta, manebit. <figure id="id.036.01.023.1.jpg" xlink:href="036/01/023/1.jpg"></figure> </s>

<s id="id.2.1.9.3.1.1.0"> Sit libra recta linea A <lb/>B horizonti æquidi&longs;tans, <lb/>cuius centrum C in ea<lb/>dem &longs;it linea AB; di&longs;tan<lb/>tia verò CA &longs;it di&longs;tantiæ <lb/>CB æqualis: &longs;intq; pon<lb/>dera in AB æqualia, quo<lb/>rum centra grauitatis &longs;int <lb/>in puntis AB. </s>

<s id="id.2.1.9.3.1.1.0">Sit libra recta linea A <lb/>B horizonti æquidi&longs;tans, <lb/>cuius centrum C in ea<lb/>dem &longs;it linea AB; di&longs;tan<lb/>tia verò CA &longs;it di&longs;tantiæ <lb/>CB æqualis: &longs;intq; pon<lb/>dera in AB æqualia, quo<lb/>rum centra grauitatis &longs;int <lb/>in <expan abbr="puntis">punctis</expan> AB. </s>

<s id="id.2.1.9.3.1.1.0.a"> Moueatur <lb/>libra, vt in DE, ibiquè <lb/>relinquatur. </s>

<s id="id.2.1.9.3.1.2.0"> Dico primùm libram DE non moueri, in eoquè &longs;itu <lb/>manere. </s>

<s id="id.2.1.9.3.1.3.0"> Quoniam enim pondera AB &longs;unt æqualia; erit magni<lb/>tudinis ex vtroq; pondere, videlicet A, & B compo&longs;itæ centrum <lb/>grauitatis C. quare idem punctum C, & centrum libræ, & <expan abbr="centr&utilde;">centrum</expan><lb/>grauitatis totius ponderis erit. </s>

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<s id="id.2.1.9.4.1.1.0"> Cum verò in iis, quæ dicta &longs;unt, grauitatis tantùm magnitudi<lb/>num, quæ in extremitatibus libræ po&longs;itæ &longs;unt æquales, ab&longs;q; <expan abbr="líbræ">li<lb/>bræ</expan>grauitate con&longs;iderauerimus; quoniam tamen adhuc libræ bra<lb/>chia &longs;unt æqualia, idcirco idem libræ, eius grauitate con&longs;iderata, <lb/>vnà cum ponderibus, vel &longs;ine ponderibus eueniet. </s>

<s id="id.2.1.9.4.1.2.0"> idem enim cen<lb/>trum grauitatis fine ponderibus libræ tantùm grauitatis centrum <lb/>erit. </s>

<s id="id.2.1.9.4.1.3.0"> Similiter &longs;i pondera in libræ extremitatibus appendantur, vt <lb/>fieri &longs;olet, idem cueniet; dummodo ex &longs;u&longs;pen&longs;ionum punctis ad <lb/>centra grauitatum ponderum ductæ lineæ (quocunq; modo mo<lb/>ueatur libra) &longs;i protrahantur, in centrum mundi concurrant. </s>

<s id="id.2.1.9.4.1.3.0">Similiter &longs;i pondera in libræ extremitatibus appendantur, vt <lb/>fieri &longs;olet, idem eueniet; dummodo ex &longs;u&longs;pen&longs;ionum punctis ad <lb/>centra grauitatum ponderum ductæ lineæ (quocunq; modo mo<lb/>ueatur libra) &longs;i protrahantur, in centrum mundi concurrant. </s>

<s id="id.2.1.9.4.1.4.0"> vbi <lb/>enim pondera hoc modo &longs;unt appen&longs;a, ibi graue&longs;cunt, ac&longs;i in ii&longs;<lb/>dem punctis centra grauitatum haberent. </s>

<s id="id.2.1.9.4.1.5.0"> præterea, quæ &longs;equun<lb/>tur, eodem pror&longs;us modo con&longs;iderare poterimus. </s>

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<s id="id.2.1.13.3.1.3.0"> deinde quò propius e&longs;t ip&longs;i A, quouis alio remotiori <lb/>grauius e&longs;&longs;e. </s>

<s id="id.2.1.13.3.1.4.0"> Vt pondus in A grauius e&longs;&longs;e, quàm in D; & in D, <lb/>quàm in L. &longs;imiliter in A grauius, quam in N; & in N grauius, <lb/>quàm in M. </s>

<s id="id.2.1.13.3.1.4.0.a"> Vnum tantùm con&longs;iderando pondus in altero libræ <lb/><arrow.to.target n="note22"></arrow.to.target>brachio &longs;ur&longs;um deor&longs;umq; moto. </s>

<s id="id.2.1.13.3.1.5.0"> Quia (inquiunt) po&longs;itat rutina <lb/>in CF, pondus in A longius e&longs;t à trutina, quàm in D: & in D <lb/>longius, quàm in L. ductis enim DO LP ip&longs;i CF perpendicula<lb/><arrow.to.target n="note23"></arrow.to.target>ribus, li<*>ea AC maior e&longs;t, quàm DO, & DO ip&longs;a LP. quod <lb/><arrow.to.target n="note24"></arrow.to.target>idem euenit in punctis NM. </s>

<s id="id.2.1.13.3.1.5.0">Quia (inquiunt) po&longs;ita trutina <lb/>in CF, pondus in A longius e&longs;t à trutina, quàm in D: & in D <lb/>longius, quàm in L. ductis enim DO LP ip&longs;i CF perpendicula<lb/><arrow.to.target n="note23"></arrow.to.target>ribus, linea AC maior e&longs;t, quàm DO, & DO ip&longs;a LP. quod <lb/><arrow.to.target n="note24"></arrow.to.target>idem euenit in punctis NM. </s>

<s id="id.2.1.13.3.1.5.0.a"> deinde ex quo loco (aiunt) pon<lb/>dus velocius mouetur, ibi grauius e&longs;t; velocius autem ex A, quàm <lb/>ab alio &longs;itu mouetur; ergo in A grauius e&longs;t. </s>

<s id="id.2.1.13.3.1.6.0"> &longs;imili modo, quò <lb/>propius e&longs;t ip&longs;i A, velocius quoque mouetur; ergo in D gra<lb/><arrow.to.target n="note25"></arrow.to.target>uius erit, quàm in L. </s>

<s id="id.2.1.13.3.1.6.0.a"> Altera deinde cau&longs;a, quam ex rectiori, & obli <lb/><arrow.to.target n="note26"></arrow.to.target>quiori motu deducunt, e&longs;t; quò pondus in arcubus æqualibus re<lb/>ctius de&longs;cendit, grauius e&longs;&longs;e videtur; cum pondus liberum, atq; <lb/><arrow.to.target n="note27"></arrow.to.target>&longs;olutum &longs;uaptè natura rectè moueatur; &longs;ed in A rectius de&longs;cen<lb/>dit; ergo in A grauius erit. </s>

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<s id="id.2.1.17.5.1.1.0"> Producatur FG v&longs;q; ad mundi cen<lb/>trum, quod &longs;it S. & à puncto S circu<lb/>lum AFBG contingens ducatur. </s>

<s id="id.2.1.17.5.1.2.0"> neq; <lb/>enim linea à puncto S circulum con<lb/>tingere pote&longs;t in A; nam ducta AS <lb/>triangulum ACS duos haberet angu<lb/>los rectos, nempè SAC ACS, quod <arrow.to.target n="note33"></arrow.to.target><lb/>e&longs;t impo&longs;sibile. </s>

<s id="id.2.1.17.5.1.3.0"> neq; &longs;upra punctum A <lb/>in circumferentia AF continget; cir<lb/>culum enim &longs;ecatet. </s>

<s id="id.2.1.17.5.1.3.0">neq; &longs;upra punctum A <lb/>in circumferentia AF continget; cir<lb/>culum enim &longs;ecaret. </s>

<s id="id.2.1.17.5.1.4.0"> tanget igitur in<lb/>fra, &longs;itq; SO. connectantur deinde SD <lb/>SL, quæ circumferentiam AOG in <lb/>punctis KH &longs;ecent. </s>

<s id="id.2.1.17.5.1.5.0"> & Ck CH con<lb/>iungantur. </s>

<s id="id.2.1.17.5.1.6.0"> Et quoniam pondus, quanto <lb/>propius e&longs;t ip&longs;i F, magis quoque inni<lb/>titur centro; vt pondus in D magis ver<lb/>&longs;ionis puncto C innititur tanquam <lb/>centro; hoc e&longs;t in D magis &longs;upra li<lb/>neam CD grauitat, quàm &longs;i e&longs;&longs;et in A <lb/>&longs;upra lineam CA; & adhuc magis in <lb/>L &longs;upra lineam CL; Nam cùm tres <lb/>anguli cuiu&longs;cunq; trianguli duobus re<lb/><figure id="id.036.01.033.1.jpg" xlink:href="036/01/033/1.jpg"></figure><lb/>ctis &longs;int æquales, & trianguli DCk æquicruris angulus DCk <lb/>minor &longs;it angulo LCH æquicruris trianguli LCH: erunt reli<lb/>qui ad ba&longs;im &longs;cilicet CDk CkD &longs;imul &longs;umpti reliquis CLH <lb/>CHL maiores. </s>

<s id="id.2.1.17.5.1.7.0"> & horum dimidii; hoc e&longs;t angulus CDS angu<lb/>lo CLS maior erit. </s>

<s id="id.2.1.17.5.1.8.0"> cùm itaq; CLS &longs;it minor, linea CL ma<lb/>gis adhærebit motui naturali ponderis in L pror&longs;us &longs;oluti. </s>

<s id="id.2.1.17.5.1.9.0"> hoc <lb/>e&longs;t lineæ LS, quàm CD motui DS. </s>

<s id="id.2.1.17.5.1.9.0.a"> pondus enim in L libe<lb/>berum, atq; &longs;olutum in centrum mundi per LS moueretur, pon<lb/>dusq; in D per DS. </s>

<s id="id.2.1.17.5.1.9.0.a">pondus enim in L <expan abbr="libe">li</expan><lb/>berum, atq; &longs;olutum in centrum mundi per LS moueretur, pon<lb/>dusq; in D per DS. </s>

<s id="id.2.1.17.5.1.9.0.b"> quoniam verò pondus in L totum &longs;uper LS <lb/>grauitat, in D verò &longs;uper DS: pondus in L magis &longs;upra lineam <lb/>CL grauitabit, quàm exi&longs;tens in D &longs;upra lineam DC. ergo <lb/>linea CL pondus magis &longs;u&longs;tentabit, quàm linea CD. </s>

<s id="id.2.1.17.5.1.9.0.c"> <expan abbr="Eodemqué">Eodem<lb/>que</expan>modo, quò pondus propius fuerit ip&longs;i F, magis ob hanc cau<lb/>&longs;am à linea CL &longs;u&longs;tineri o&longs;tendetur-&longs;emper enim angulus CLS <pb xlink:href="036/01/034.jpg"/>minor e&longs;&longs;et. </s>

<s id="id.2.1.17.5.1.9.0.c"><expan abbr="Eodemqué">Eodem<lb/>que</expan> modo, quò pondus propius fuerit ip&longs;i F, magis ob hanc cau<lb/>&longs;am à linea CL &longs;u&longs;tineri o&longs;tendetur; &longs;emper enim angulus CLS <pb xlink:href="036/01/034.jpg"/>minor e&longs;&longs;et. </s>

<s id="id.2.1.17.5.1.10.0"> quod etiam patet; quia &longs;i <lb/>lineæ CL, & LS in vnam coinciderent <lb/>lineam, quod euenit in FCS; tunc linea <lb/>CF totum &longs;u&longs;tineret pondus in F, im<lb/>mobilemq; redderet: neq; vllam pror<lb/>&longs;us grauitatem in circumferentia circu<lb/>li haberet. </s>

<s id="id.2.1.17.5.1.11.0"> Idem ergo pondus propter <lb/>&longs;ituum diuer&longs;itatem grauius, leuiu&longs;q; erit. </s>

<s id="id.2.1.17.5.1.12.0"> <lb/>non autem quia ratione &longs;itus interdum <lb/>maiorem re vera acquirat grauitatem, <lb/>interdum verò amittat, cùm eiu&longs;dem &longs;it <lb/>&longs;emper grauitatis, vbicunque reperiatur; <lb/>&longs;ed quia magis, minu&longs;uè in circumferen<lb/>tia grauitat, vt in D magis &longs;upra circum<lb/>ferentiam DA grauitat, quàm in L &longs;upra <lb/>circumferentiam LD. </s>

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<s id="id.2.1.17.5.1.15.0"> e&longs;t enim angulus CAS <lb/>acutus; cùm angulus ACS &longs;it rectus. </s>

<s id="id.2.1.17.5.1.16.0"> lineæ igitur CA CD ali <lb/>qua ex parte, non tamen ex æquo ponderi renituntur. </s>

<s id="id.2.1.17.5.1.17.0"> & quotie&longs; <lb/>cunque angulus in circumferentia circuli à lineis à centro <lb/>mundi S, & centro C prodeuntibus, fuerit acutus; idem eue<lb/>nire &longs;imiliter o&longs;tendemus. </s>

<s id="id.2.1.17.5.1.18.0"> Quoniam autem mixtus angulus CLD <pb n="11" xlink:href="036/01/035.jpg"/>æqualis e&longs;t angulo CDA, cùm à &longs;emidiametris, eademq; circumfe<lb/>rentia contineantur; & angulus C<emph type="italics"/>L<emph.end type="italics"/>S angulo CDS e&longs;t minor; <lb/>erit reliquus <emph type="italics"/>s<emph.end type="italics"/>LD reliquo SDA maior. </s>

<s id="id.2.1.17.5.1.18.0">Quoniam autem mixtus angulus CLD <pb n="11" xlink:href="036/01/035.jpg"/>æqualis e&longs;t angulo CDA, cùm à &longs;emidiametris, eademq; circumfe<lb/>rentia contineantur; & angulus C<emph type="italics"/>L<emph.end type="italics"/>S angulo CDS e&longs;t minor; <lb/>erit reliquus <emph type="italics"/>S<emph.end type="italics"/>LD reliquo SDA maior. </s>

<s id="id.2.1.17.5.1.19.0"> quare circumferentia <lb/>DA, hoc e&longs;t de&longs;cen&longs;us ponderis in D propior erit motui natu<lb/>rali ponderis in D &longs;oluti, lineæ &longs;cilicet DS, quàm circumferen<lb/>tia LD lineæ LS. </s>

<s id="id.2.1.17.5.1.19.0.a"> minus igitur linea CD ponderi in D reniti<lb/>tur, quàm linea CL ponderi in L. </s>

<s id="id.2.1.17.5.1.19.0.b"> linea ideo CD minus &longs;u&longs;tinet, <lb/>quàm CL; pondu&longs;q; magis liberum erit in D, quàm in L: <lb/>cùm pondus naturaliter magis per DA moueatur, quàm per LD. <lb/>quare grauius erit in D, quàm in L. &longs;imiliter o&longs;tendemus CA <lb/>minus &longs;u&longs;tinere, quàm CD: pondu&longs;q; magis in A, quàm in Dli <lb/>berum, grauiu&longs;q, e&longs;&longs;e. </s>

<s id="id.2.1.17.5.1.19.0.b">linea ideo CD minus &longs;u&longs;tinet, <lb/>quàm CL; pondu&longs;q; magis liberum erit in D, quàm in L: <lb/>cùm pondus naturaliter magis per DA moueatur, quàm per LD. <lb/>quare grauius erit in D, quàm in L. &longs;imiliter o&longs;tendemus CA <lb/>minus &longs;u&longs;tinere, quàm CD: pondu&longs;q; magis in A, quàm in D li<lb/>berum, grauiu&longs;q, e&longs;&longs;e. </s>

<s id="id.2.1.17.5.1.20.0"> Ex parte deinde inferiori ob ea&longs;dem cau&longs;as, <lb/>quò pondus propius fuerit ip&longs;i G, magis detinebitur, vt in H ma<lb/>gis à linea CH, quàm in K à linea CK. nam cùm angulus CHS <lb/>maior &longs;it angulo CkS, ad rectitudinem magis appropinquabunt <arrow.to.target n="note34"></arrow.to.target><lb/>&longs;e &longs;e lineæ CHHS, quàm Ck kS; atq; ob id pondus magis deti<lb/>nebitur à CH, quàm à Ck &longs;i enim CH HS in vnam conuenirent <lb/>lineam vt euenit pondere exi&longs;tente in G; tunc linea CG totum &longs;u<lb/>&longs;tineret' pondus in G, ita vt immobilis per&longs;i&longs;teret. </s>

<s id="id.2.1.17.5.1.21.0"> quò igitur <lb/>minor erit angulus linea CH, & de&longs;cen&longs;u ponderis &longs;oluti, &longs;cilicet <lb/>HS contentus, eò minus quoq; eiu&longs;modi linea pondus detinebit. </s>

<s id="id.2.1.17.5.1.22.0"> <lb/>& vbiminus detinebitur, ibi magis liberum, grauiu&longs;q; exi&longs;tet. </s>

Line 489

Line 488

<s id="id.2.1.17.5.1.27.0"> <expan abbr="Quoniã">Quoniam</expan>autem angulus CHS ma<lb/>ior e&longs;t angulo CKS, <expan abbr="d&etilde;ptis">demptis</expan>æqualibus angulis CHG CkH; erit <lb/>reliquus SHG reliquo SKH maior. </s>

<s id="id.2.1.17.5.1.28.0"> circumferentia igitur k H, hoc <lb/>e&longs;t de&longs;cen&longs;us ponderis in k, propior erit motui naturali ponderis in <lb/>k &longs;oluti, hoc e&longs;t lineæ k S, quàm circumferentia HG lineæ HS. mi <lb/>nus idcirco detinet linea Ck, quàm CH: cùm pondus naturali<lb/>ter magis moueatur per k H, quàm per HG. </s>

<s id="id.2.1.17.5.1.28.0.a"> &longs;imili ratione o&longs;ten<lb/>detur, quò minor erit angulus SkH, lineam Ck minus &longs;u&longs;tinere. </s>

<s id="id.2.1.17.5.1.29.0"> <pb xlink:href="036/01/036.jpg"/>exi&longs;tente igitur pondere in O, quia angu<lb/>lus SOC non &longs;olum minor e&longs;t angulo <lb/>CKS, verùm etiam omnium angulorum <lb/>à punctis CS prodeuntium, verticemq; <lb/>in circumferuntia OkG habentium mi<lb/>nimus; erit anglus SOK, & angulo SkH, <lb/>& eiu&longs;modi omnium minimus. </s>

<s id="id.2.1.17.5.1.29.0"><pb xlink:href="036/01/036.jpg"/>exi&longs;tente igitur pondere in O, quia angu<lb/>lus SOC non &longs;olum minor e&longs;t angulo <lb/>CKS, verùm etiam omnium angulorum <lb/>à punctis CS prodeuntium, verticemq; <lb/>in circumferuntia OkG habentium mi<lb/>nimus; erit <expan abbr="anglus">angulus</expan> SOK, & angulo SkH, <lb/>& eiu&longs;modi omnium minimus. </s>

<s id="id.2.1.17.5.1.30.0"> ergo de<lb/>&longs;cen&longs;us ponderis in O propior erit motui <lb/>naturali ip&longs;ius in O &longs;oluti, quàm in alio <lb/>&longs;itu circumferentiæ OkG. lineaq; CO <lb/>minus pondus &longs;u&longs;tinebit, quàm &longs;i pon<lb/>dusin quouis alio fuerit &longs;itu eiu&longs;dem cir<lb/>cumferentiæ OG. </s>

<s id="id.2.1.17.5.1.30.0">ergo de<lb/>&longs;cen&longs;us ponderis in O propior erit motui <lb/>naturali ip&longs;ius in O &longs;oluti, quàm in alio <lb/>&longs;itu circumferentiæ OkG. </s>

<s>lineaq; CO <lb/>minus pondus &longs;u&longs;tinebit, quàm &longs;i pon<lb/>dus in quouis alio fuerit &longs;itu eiu&longs;dem cir<lb/>cumferentiæ OG. </s>

<s id="id.2.1.17.5.1.30.0.a"> &longs;imiliter quoniam con<lb/>tingentiæ angulus SOk, & angulo SDA, <lb/>& SAO, ac quibu&longs;cunq; &longs;imilibus e&longs;t mi <lb/>nor; erit de&longs;cen&longs;us ponderis in O motui <lb/>naturali ip&longs;ius ponderis in O &longs;oluti pro<lb/>pior, quàm in alio &longs;itu circumferentiæ <lb/>ODF. </s>

<s id="id.2.1.17.5.1.30.0.b"> Præte reaquoniam linea GO pon<lb/>dus in O dum deor&longs;um mouetur, impelle<lb/>re nonpote&longs;t, ita vt vltra lineam OS mo<lb/>ueatur; cùm linea OS circulum non &longs;ecet, <lb/><figure id="id.036.01.036.1.jpg" xlink:href="036/01/036/1.jpg"></figure><lb/>&longs;ed contingat; angulu&longs;q; SOC &longs;it rectus, & non acutus; pondus <lb/>in O nihil &longs;upra lineam CO grauitabit. </s>

<s id="id.2.1.17.5.1.31.0"> neq; centro innitetur. </s>

<s id="id.2.1.17.5.1.32.0"> quem <lb/>admodum in quouis alio puncto &longs;upra O accideret. </s>

<s id="id.2.1.17.5.1.33.0"> erit igitur pon<lb/>dus in O magis ob has cau&longs;as liberum, atq; &longs;olutum in hoc &longs;itu, <lb/>quàm in quouis alio circumferentiæ FOG. acidcirco in hoc <lb/>grauius erit, hoc e&longs;t magis grauitabit, quàm in alio &longs;itu. </s>

<s id="id.2.1.17.5.1.33.0">erit igitur pon<lb/>dus in O magis ob has cau&longs;as liberum, atq; &longs;olutum in hoc &longs;itu, <lb/>quàm in quouis alio circumferentiæ FOG. </s>

<s>ac idcirco in hoc <lb/>grauius erit, hoc e&longs;t magis grauitabit, quàm in alio &longs;itu. </s>

<s id="id.2.1.17.5.1.34.0"> & quò <lb/>propius fuerit ip&longs;i O remotiori grauius erit. </s>

<s id="id.2.1.17.5.1.35.0"> lineaq; CO horizonti <lb/>æquidi&longs;tans erit. </s>

<s id="id.2.1.17.5.1.36.0"> non tamen puncti C horizonti (vt ip&longs;i exi&longs;ti<lb/>mant) &longs;ed ponderis in O con&longs;tituti, cùm ex centro grauitatis <lb/>ponderis &longs;ummendus &longs;it horizon. </s>

Line 508

Line 509

<s id="id.2.1.19.1.2.1.0"> Si autem libræ brachium ip&longs;o CO <lb/>fuerit maius, putá quantitate CD; erit <lb/>quoq; pondus in O grauius. </s>

<s id="id.2.1.19.1.2.2.0"> circulus de<lb/>&longs;cribatur OH, cuius centrum &longs;it D, &longs;e <arrow.to.target n="note35"></arrow.to.target><lb/>midiameterq; DO. tanget circulus OH <lb/>circulum FOG in puncto O, lineamq; <arrow.to.target n="note36"></arrow.to.target><lb/>OS, quæ ponderis in O rectus, natura<lb/>li&longs;q; e&longs;t de&longs;cen&longs;us, in eodem puncto con <lb/>tinget. </s>

<s id="id.2.1.19.1.2.2.0">circulus de<lb/>&longs;cribatur OH, cuius centrum &longs;it D, &longs;e<arrow.to.target n="note35"></arrow.to.target><lb/>midiameterq; DO. </s>

<s>tanget circulus OH <lb/>circulum FOG in puncto O, lineamq; <arrow.to.target n="note36"></arrow.to.target><lb/>OS, quæ ponderis in O rectus, natura<lb/>li&longs;q; e&longs;t de&longs;cen&longs;us, in eodem puncto con<lb/>tinget. </s>

<s id="id.2.1.19.1.2.3.0"> & quoniam angulus SOH mi<lb/>nor e&longs;t angulo SOG, erit de&longs;cen&longs;us <lb/>ponderis in O per circumferentiam OH <lb/>motui naturali OS propior, quàm per <lb/>circumferentiam OG. </s>

<s id="id.2.1.19.1.2.3.0.a"> magis ergo li<lb/>berum, atq; &longs;olutum, ac per con&longs;equens <lb/>grauius erit in O, centro libræ exi&longs;ten<lb/>te in D, quàm in C. &longs;imiliter o&longs;ten<lb/>detur, quò maius fuerit brachium DO, <lb/>pondus in O adhuc grauius e&longs;&longs;e. <figure id="id.036.01.037.1.jpg" xlink:href="036/01/037/1.jpg"></figure> </s>

<s id="id.2.1.19.3.1.1.0"> Siverò idem circulus AFBG, <lb/>cuius centrum &longs;it R, propius fuerit <lb/>mundi centro S; circulumqué à pun<lb/>cto S ducatur contingens ST; punctum <lb/>T (vbi grauius e&longs;t pondus) magis <lb/>à puncto A di&longs;tabit, quàm punctum <lb/>O. ducantur enim à punctis OT ip&longs;i <lb/>CS perpendiculares OMTN; conne<lb/>ctanturq; RT; &longs;itq; centrum R in li<lb/>nea CS; lineaq; ARB ip&longs;i ACB æqui <lb/><arrow.to.target n="note37"></arrow.to.target>di&longs;tans. </s>

<s id="id.2.1.19.3.1.2.0"> Quoniam igitur triangula COS <lb/>RTS &longs;unt rectangula; erit SC ad CO, <lb/>vt CO ad CM. &longs;imiliter SR ad RT, <lb/>vt RT ad RN. cùm itaq; &longs;it RT ip<lb/><arrow.to.target n="note38"></arrow.to.target>&longs;i CO æqualis, & SC ip&longs;a SR maior: <lb/>maiorem habebit proportionem SC <lb/>ad CO, quàm SR ad RT. quare ma <lb/>iorem quoq; proportionem habebit <lb/>CO ad CM, quàm RT ad RN. </s>

<s id="id.2.1.19.3.1.2.0">Quoniam igitur triangula COS <lb/>RTS &longs;unt rectangula; erit SC ad CO, <lb/>vt CO ad CM. </s>

<s id="id.2.1.19.3.1.2.0.a"> mi <lb/><arrow.to.target n="note39"></arrow.to.target>nor ergo erit CM, quàm RN. &longs;ecetur <lb/>igitur RN in P, ita vt RP &longs;it ip&longs;i <lb/><figure id="id.036.01.038.1.jpg" xlink:href="036/01/038/1.jpg"></figure><lb/>CM æqualis; & à puncto P ip&longs;is MONT æquidi&longs;tans ducatur <lb/>PQ, quæ circumferentiam AT &longs;ecet in Q: deniq; connectatur <lb/><expan abbr="Rq.">Rque</expan>quoniam enim duæ CO CM duabus RQRP &longs;unt æqua <lb/><arrow.to.target n="note40"></arrow.to.target>les, & angulus CMO angulo RPQ e&longs;t æqualis; erit & angu<lb/>lus MCO angulo PRQ æqualis. </s>

<s>&longs;imiliter SR ad RT, <lb/>vt RT ad RN. </s>

<s>cùm itaq; &longs;it RT ip<lb/><arrow.to.target n="note38"></arrow.to.target>&longs;i CO æqualis, & SC ip&longs;a SR maior: <lb/>maiorem habebit proportionem SC <lb/>ad CO, quàm SR ad RT. </s>

<s>quare ma<lb/>iorem quoq; proportionem habebit <lb/>CO ad CM, quàm RT ad RN. </s>

<s>&longs;ecetur <lb/>igitur RN in P, ita vt RP &longs;it ip&longs;i <lb/><figure id="id.036.01.038.1.jpg" xlink:href="036/01/038/1.jpg"></figure><lb/>CM æqualis; & à puncto P ip&longs;is MONT æquidi&longs;tans ducatur <lb/>PQ, quæ circumferentiam AT &longs;ecet in Q: deniq; connectatur <lb/>RQ. </s>

<s>quoniam enim duæ CO CM duabus RQRP &longs;unt æqua<lb/><arrow.to.target n="note40"></arrow.to.target>les, & angulus CMO angulo RPQ e&longs;t æqualis; erit & angu<lb/>lus MCO angulo PRQ æqualis. </s>

<s id="id.2.1.19.3.1.3.0"> angulus autem MCA rectus <lb/><arrow.to.target n="note41"></arrow.to.target>recto PRA e&longs;t æqualis; ergo reliquus OCA reliquo QRA <lb/>æqualis, & circumferentia OA circumferentiæ QA æqualis quo<lb/>que erit. </s>

<s id="id.2.1.19.3.1.4.0"> punctum idcirco T, quia magis à puncto A di&longs;tat, <lb/>quàm Q; magis quoq; à puncto A di&longs;tabit, quàm punctum O. <lb/>&longs;imiliter o&longs;tendetur, quò propius fuerit circulus mundi centro, eun<lb/>dem magis di&longs;tare. </s>

<s id="id.2.1.19.3.1.4.0">punctum idcirco T, quia magis à puncto A di&longs;tat, <lb/>quàm Q; magis quoq; à puncto A di&longs;tabit, quàm punctum O. <lb/></s>

<s>&longs;imiliter o&longs;tendetur, quò propius fuerit circulus mundi centro, eun<lb/>dem magis di&longs;tare. </s>

<s id="id.2.1.19.3.1.5.0"> atq; ita vt prius demon&longs;trabitur pondus in cir<lb/>cumferentia TAF centro R inniti, in circumferentia verò TG <lb/>à linea detineri; atq; in puncto T grauius e&longs;&longs;e. </s>

Line 532

Line 540

<s id="id.2.1.21.1.2.1.0"> Si autem punctum G e&longs;&longs;et <lb/>in centro mundi; tunc quò <lb/>pondus propius fuerit ip&longs;i G, <lb/>grauius erit: & vbicunq; po<lb/>natur pondus præterquàm in <lb/>ip&longs;o G, &longs;emper centro C inni<lb/>tetur, vt in K. nam ducta <lb/>G k, efficiet hæc (&longs;ecun<lb/>dùm quam fit ponderis natu<lb/>ralis motus) vná cum libræ <lb/>brachio k C angulum acu<lb/>tum. </s>

<s>nam ducta <lb/>G k, efficiet hæc (&longs;ecun<lb/>dùm quam fit ponderis natu<lb/>ralis motus) vná cum libræ <lb/>brachio k C angulum acu<lb/>tum. </s>

<s id="id.2.1.21.1.2.2.0"> æquicruris enim trian<lb/>guli CkG ad ba&longs;im anguli <lb/>ad k, & G &longs;unt &longs;emper acuti. </s>

<s id="id.2.1.21.1.2.3.0"> <lb/><figure id="id.036.01.039.1.jpg" xlink:href="036/01/039/1.jpg"></figure><lb/>Conferantur autem inuicem hæc duo, pondus videlicet in k, & <lb/>pondus in D: erit pondus in k grauius, quàm in D. nam iuncta <lb/>DG, cùm tres anguli cuiu&longs;cunque trianguli duobus &longs;int rectis <lb/>æquales, & trianguli CDG æquicruris angulus DCG maior &longs;it <lb/>angulo kCG æquicruris trianguli CkG: erunt reliqui ad ba&longs;im an<lb/>guli DGC GDC &longs;imul &longs;umpti reliquis KGCGkC &longs;imul &longs;umptis <lb/>minores. </s>

<s id="id.2.1.21.1.2.4.0"> horumq; dimidii; angulus &longs;cilicet CDG angulo CKG <lb/>minor erit. </s>

Line 546

Line 555

<s id="id.2.1.21.1.2.9.0.b"> <lb/>Similiter o&longs;tendetur pondus, <lb/>quò fuerit ip&longs;i F propius, vt <lb/>in L, minus grauitare: pro<lb/>pius verò ip&longs;i G, vt in H, <lb/>grauius e&longs;&longs;e. <figure id="id.036.01.040.1.jpg" xlink:href="036/01/040/1.jpg"></figure> </s>

<s id="id.2.1.21.2.1.1.0"> Si verò centrum mundi <lb/>S e&longs;&longs;et inter puncta CG; <lb/>primùm quidem &longs;imili<lb/>ter o&longs;tendetur pondus vbi <lb/>cunq; po&longs;itum centro C <lb/>initi, vt in H. ductis enim <lb/>HG HS, angulus ad <lb/>ba&longs;im GHC æquicruris tri <lb/>anguli CHG e&longs;t &longs;emper <lb/>acutus: quare & SHC ip<lb/>&longs;o minor erit quoq; &longs;em<lb/>per acutus. </s>

<s>ductis enim <lb/>HG HS, angulus ad <lb/>ba&longs;im GHC æquicruris tri <lb/>anguli CHG e&longs;t &longs;emper <lb/>acutus: quare & SHC ip<lb/>&longs;o minor erit quoq; &longs;em<lb/>per acutus. </s>

<s id="id.2.1.21.2.1.2.0"> ducatur au<lb/>tem à puncto S ip&longs;i CS <lb/>perpendicularis Sk. </s>

<s id="id.2.1.21.2.1.3.0"> di<lb/><figure id="id.036.01.040.2.jpg" xlink:href="036/01/040/2.jpg"></figure><lb/>co pondus grauius e&longs;&longs;e in k, quàm in alio &longs;itu circumferentiæ FKG. <lb/>& quò propius fuerit ip&longs;i F, vel G, minus grauitare. </s>

<s id="id.2.1.21.2.1.4.0"> Accipiantur <lb/>ver&longs;us F puncta DL, connectanturq; LC LS DC DS, produ<lb/>canturq; LS DS k SHS v&longs;q; ad circuli circumferentiam in EM <lb/>NO; connectanturq; CE, CM, CN, CO. </s>

Line 555

Line 565

<s id="id.2.1.21.2.1.5.0.a"> maior autem e&longs;t LS, quàm DS; & SM ip&longs;a SE. </s>

<s id="id.2.1.21.2.1.5.0.b"> <pb n="14" xlink:href="036/01/041.jpg"/>ergo LS SE &longs;imul &longs;umptæ ip&longs;is DS SM maiores erunt. </s>

<s id="id.2.1.21.2.1.6.0"> eademq; <arrow.to.target n="note45"></arrow.to.target><lb/>ratione kN minorem e&longs;&longs;e DM o&longs;tendetur. </s>

<s id="id.2.1.21.2.1.7.0"> rur&longs;us quoniam re<lb/>ctangulum OSH æquale e&longs;t rectangulo kSN; ob eandem cau&longs;am <lb/>HO maior erit kN. eodemq; pror&longs;us modo kN omnibus a<lb/>liis per punctum S tran&longs;euntibus minorem e&longs;&longs;e demon&longs;trabitur. </s>

<s id="id.2.1.21.2.1.7.0">rur&longs;us quoniam re<lb/>ctangulum OSH æquale e&longs;t rectangulo kSN; ob eandem cau&longs;am <lb/>HO maior erit kN. </s>

<s>eodemq; pror&longs;us modo kN omnibus a<lb/>liis per punctum S tran&longs;euntibus minorem e&longs;&longs;e demon&longs;trabitur. </s>

<s id="id.2.1.21.2.1.8.0"> <lb/>& quoniam æquicrurium triangulorum CLE DCM latera LC <lb/>CE lateribus DC CM &longs;unt æqualia; ba&longs;is verò LE maior e&longs;t <lb/>DM: erit angulus LCE angulo DCM maior. </s>

<s id="id.2.1.21.2.1.9.0"> quare ad ba&longs;im <arrow.to.target n="note46"></arrow.to.target><lb/>anguli C<emph type="italics"/>L<emph.end type="italics"/>E CEL &longs;imul &longs;umpti angulis CDM CMD mi<lb/>nores erunt. </s>

<s id="id.2.1.21.2.1.10.0"> & horum dimidii, angulus &longs;cilicet CLS angulo CDS <lb/>minor erit. </s>

<s id="id.2.1.21.2.1.11.0"> ergo pondus in <emph type="italics"/>L<emph.end type="italics"/>magis &longs;upra lineam LC, quàm <lb/>in D &longs;upra DC grauitabit, magisqué centro innitetur in L, quàm <lb/>in D. &longs;imiliter o&longs;tendetur in D magis <expan abbr="c&etilde;tro">centro</expan>C inniti, quàm in k. </s>

<s id="id.2.1.21.2.1.11.0">ergo pondus in <emph type="italics"/>L<emph.end type="italics"/> magis &longs;upra lineam LC, quàm <lb/>in D &longs;upra DC grauitabit. </s>

<s id="id.2.1.21.2.1.12.0"> ergo <lb/>ponds in k grauius erit, quàm in D; & in D, quàm in L. eademq; pror <lb/>&longs;us ratione quoniam kN minor e&longs;t HO, erit angulus CKS an<lb/>gulo CHS maior. </s>

<s id="id.2.1.21.2.1.11.0.a">magisqué centro innitetur in L, quàm <lb/>in D. </s>

<s id="id.2.1.21.2.1.11.0.b">&longs;imiliter o&longs;tendetur in D magis <expan abbr="c&etilde;tro">centro</expan> C inniti, quàm in k. </s>

<s id="id.2.1.21.2.1.12.0">ergo <lb/><expan abbr="ponds">pondus</expan> in k grauius erit, quàm in D; & in D, quàm in L. eademq; pror<lb/>&longs;us ratione quoniam kN minor e&longs;t HO, erit angulus CKS an<lb/>gulo CHS maior. </s>

<s id="id.2.1.21.2.1.13.0"> quare pondus in H magis centro C innite<lb/>tur, quàm in k. </s>

<s id="id.2.1.21.2.1.14.0"> & hoc modo o&longs;tendetur, vbicunq; in circum<lb/>ferentia FDG fuerit pondus, minus in K centro C inniti, quàm <lb/>in alio &longs;itu: & quò propius fuerit ip&longs;i F, vel G, magis inniti. </s>

<s id="id.2.1.21.2.1.15.0"> dein<lb/>de quoniam angulus CkS maior e&longs;t CDS, & CDk æqualis <lb/>e&longs;t CkH: erit reliquus SkH reliquo SDk minor. </s>

<s id="id.2.1.21.2.1.16.0"> quare cir<lb/>cumferentia k H propior erit motui naturali recto ponderis in K <lb/>&longs;oluti, lineæ &longs;cilicet k S, quàm circumferentia D k motui DS. & <lb/>ideo linea CD magis ip&longs;i ponderi in D renititur, quàm CK <lb/>ponderi in k con&longs;tituto. </s>

<s id="id.2.1.21.2.1.16.0">quare cir<lb/>cumferentia k H propior erit motui naturali recto ponderis in K <lb/>&longs;oluti, lineæ &longs;cilicet k S, quàm circumferentia D k motui DS. </s>

<s id="id.2.1.21.2.1.17.0"> hacq; ratione o&longs;tendetur angulum <lb/>SHG maiorem e&longs;&longs;e SkH: & per con&longs;equens lineam CH magis <lb/>ponderi in H reniti, quàm CK ponderi in K. &longs;imiliter demon<lb/>&longs;trabitur lineam C<emph type="italics"/>L<emph.end type="italics"/>magis pondus &longs;u&longs;tinere, quàm CD: ob <lb/>ea&longs;demq; cau&longs;as o&longs;tendetur pondus in K minus &longs;upra lineam Ck <lb/>grauitare, quàm in quouis alio &longs;itu fuerit circumferentiæ FDG. <lb/>& quò propius fuerit ip&longs;i F, vel G, minus grauitare. </s>

<s id="id.2.1.21.2.1.16.0.a">& <lb/>ideo linea CD magis ip&longs;i ponderi in D renititur, quàm CK <lb/>ponderi in k con&longs;tituto. </s>

<s id="id.2.1.21.2.1.18.0"> grauius ergo <lb/>erit in k, quàm in alio &longs;itu: minu&longs;q; graue erit, quò propius fue<lb/>rit ip&longs;i F. vel G. <pb xlink:href="036/01/042.jpg"/> </s>

<s id="id.2.1.21.2.1.17.0">hacq; ratione o&longs;tendetur angulum <lb/>SHG maiorem e&longs;&longs;e SkH: & per con&longs;equens lineam CH magis <lb/>ponderi in H reniti, quàm CK ponderi in K. </s>

<s>&longs;imiliter demon<lb/>&longs;trabitur lineam C<emph type="italics"/>L<emph.end type="italics"/> magis pondus &longs;u&longs;tinere, quàm CD: ob <lb/>ea&longs;demq; cau&longs;as o&longs;tendetur pondus in K minus &longs;upra lineam Ck <lb/>grauitare, quàm in quouis alio &longs;itu fuerit circumferentiæ FDG. <lb/></s>

<s>& quò propius fuerit ip&longs;i F, vel G, minus grauitare. </s>

<s id="id.2.1.21.2.1.18.0">grauius ergo <lb/>erit in k, quàm in alio &longs;itu: minu&longs;q; graue erit, quò propius fue<lb/>rit ip&longs;i F, vel G. <pb xlink:href="036/01/042.jpg"/></s>

<s id="id.2.1.22.1.1.1.0"> <margin.target id="note42"></margin.target>35 <emph type="italics"/>Tertii.<emph.end type="italics"/> </s>

Line 581

Line 597

<s id="id.2.1.23.1.1.3.0"> pondus ergo ma <lb/>nebit. <figure id="id.036.01.042.1.jpg" xlink:href="036/01/042/1.jpg"></figure> </s>

<s id="id.2.1.23.2.1.1.0"> Quoniam autem in his hactenus demon&longs;tratis, nullam de gra<lb/>uitate brachii libræ mentionem fecimus, idcirco &longs;i brach&longs;i quoq; <lb/>grauitatem con&longs;iderare voluerimus, centrum grauitatis magnitu<lb/>dinis ex pondere, brachioq; compo&longs;itæ inueniri poterit, circulo<lb/>rumq; circumferentiæ &longs;ecundum di&longs;tantiam à centro libræ ad <lb/>hoc ip&longs;um grauitatis centrum de&longs;cribentur, ac &longs;i in ip&longs;o (vt re ue<lb/>ra e&longs;t) pondus con&longs;titutum fuerit; omnia, &longs;icuti ab&longs;q; libræ bra<lb/>chii grauitate con&longs;iderata inuenimus; hoc quoq; modo eius con&longs;i<lb/>derata grauitate reperiemus. </s>

<s id="id.2.1.23.2.1.1.0">Quoniam autem in his hactenus demon&longs;tratis, nullam de gra<lb/>uitate brachii libræ mentionem fecimus, idcirco &longs;i brachii quoq; <lb/>grauitatem con&longs;iderare voluerimus, centrum grauitatis magnitu<lb/>dinis ex pondere, brachioq; compo&longs;itæ inueniri poterit, circulo<lb/>rumq; circumferentiæ &longs;ecundum di&longs;tantiam à centro libræ ad <lb/>hoc ip&longs;um grauitatis centrum de&longs;cribentur, ac &longs;i in ip&longs;o (vt re ue<lb/>ra e&longs;t) pondus con&longs;titutum fuerit; omnia, &longs;icuti ab&longs;q; libræ bra<lb/>chii grauitate con&longs;iderata inuenimus; hoc quoq; modo eius con&longs;i<lb/>derata grauitate reperiemus. </s>

<s id="id.2.1.24.1.1.1.0"> <margin.target id="note47"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>

Line 589

Line 605

<s id="id.2.1.25.1.2.1.0"> Ex dictis igitur, con&longs;iderando li<lb/>bram, vt longè à mundi centro a<lb/>be&longs;t, quemadmodum ip&longs;i fecere, &longs;i<lb/>cuti etiam actu e&longs;t, apparet fal&longs;itas <lb/>dicentium pondus in A grauius e&longs;&longs;e, <lb/>quàm in alio &longs;itu. </s>

<s id="id.2.1.25.1.2.2.0"> &longs;imulq; fal&longs;um e&longs;&longs;e, <lb/>quò pondus à linea FG magis di&longs;tat <lb/>grauiuis e&longs;&longs;e. </s>

<s id="id.2.1.25.1.2.2.0">&longs;imulq; fal&longs;um e&longs;&longs;e, <lb/>quò pondus à linea FG magis di&longs;tat <lb/><expan abbr="grauiuis">grauius</expan> e&longs;&longs;e. </s>

<s id="id.2.1.25.1.2.3.0"> nam punctum O pro<lb/>pius e&longs;t ip&longs;i FG, quàm punctum A. <lb/>e&longs;t enim linea à puncto O ip&longs;i FG <arrow.to.target n="note48"></arrow.to.target><lb/>perpendicularis ip&longs;a CA minor. </s>

<s id="id.2.1.25.1.2.4.0"> de<lb/>inde ex puncto A pondus velocius mo <lb/>ueri, quàm ab alio &longs;itu, e&longs;t quoque <lb/>fal&longs;um. </s>

<s id="id.2.1.25.1.2.5.0"> ex puncto enim O pondus ve<lb/>locius mouebitur, quàm ex puncto <lb/>A; cùm in O &longs;it magis liberum, atq; <lb/>&longs;olutum, quàm in alio &longs;itu: de&longs;cen&longs;us <lb/>qué ex puncto O propior &longs;it motui na<lb/>turali recto, quàm quilibet alius de<lb/>&longs;cen&longs;us. <figure id="id.036.01.043.1.jpg" xlink:href="036/01/043/1.jpg"></figure> </s>

<s id="id.2.1.25.2.1.1.0"> Præterea cùm ex re<lb/>ctiori, & obliquiori <expan abbr="defc&etilde;">defcem</expan><lb/>&longs;u o&longs;tendunt, pondus in <lb/>A grauiur e&longs;&longs;e, quàm in <lb/>D; & in D, quàm in <lb/>L; primùm quidem fal<lb/>&longs;um exi&longs;timant, &longs;i pon<lb/>dus aliquod collocatum <lb/>fuerit in quocunq; &longs;itu <lb/>circunferentiæ, vt in D, <lb/>rectum eius de&longs;cen&longs;um <lb/>per rectam lineam DR <lb/>ip&longs;i FG parallelam, tam <lb/>quàm &longs;ecundùm mo|tum<figure id="id.036.01.043.2.jpg" xlink:href="036/01/043/2.jpg"></figure><pb xlink:href="036/01/044.jpg"/>naturalem fieri de<lb/>bere; &longs;icuti prius dictum <lb/>e&longs;t. </s>

<s id="id.2.1.25.2.1.1.0">Præterea cùm ex re<lb/>ctiori, & obliquiori <expan abbr="de&longs;c&etilde;&longs;u">de&longs;cen<lb/>&longs;u</expan> o&longs;tendunt, pondus in <lb/>A <expan abbr="grauiur">grauior</expan> e&longs;&longs;e, quàm in <lb/>D; & in D, quàm in <lb/>L; primùm quidem fal<lb/>&longs;um exi&longs;timant, &longs;i pon<lb/>dus aliquod collocatum <lb/>fuerit in quocunq; &longs;itu <lb/>circunferentiæ, vt in D, <lb/>rectum eius de&longs;cen&longs;um <lb/>per rectam lineam DR <lb/>ip&longs;i FG parallelam, tam <lb/>quàm &longs;ecundùm mo<figure id="id.036.01.043.2.jpg" xlink:href="036/01/043/2.jpg"></figure><pb xlink:href="036/01/044.jpg"/>tum naturalem fieri de<lb/>bere; &longs;icuti prius dictum <lb/>e&longs;t. </s>

<s id="id.2.1.25.2.1.2.0"> In quocunq; enim <lb/>&longs;itu pondus aliquod con<lb/>&longs;tituatur, &longs;i naturalem <lb/>eius ad propium locum <lb/>motionem &longs;pectemus, <lb/>cùm rectá ad eum <expan abbr="&longs;uaptè">&longs;ua<lb/>pte</expan>natura moueatur, &longs;up<lb/>po&longs;ita totius vniuer&longs;i figu<lb/>ra, eiu&longs;modi erit; vt <lb/>&longs;emper <expan abbr="&longs;pati&utilde;">&longs;patium</expan>, per quod <lb/>naturaliter mouetur, ra<lb/>tionem habere videatur <lb/><figure id="id.036.01.044.1.jpg" xlink:href="036/01/044/1.jpg"></figure><lb/>lineæ à circumferentia ad centrum productæ. </s>

<s id="id.2.1.25.2.1.2.0">In quocunq; enim <lb/>&longs;itu pondus aliquod con<lb/>&longs;tituatur, &longs;i naturalem <lb/>eius ad propium locum <lb/>motionem &longs;pectemus, <lb/>cùm rectá ad eum &longs;ua<lb/>ptè natura moueatur, &longs;up<lb/>po&longs;ita totius vniuer&longs;i figu<lb/>ra, eiu&longs;modi erit; vt <lb/>&longs;emper <expan abbr="&longs;pati&utilde;">&longs;patium</expan>, per quod <lb/>naturaliter mouetur, ra<lb/>tionem habere videatur <lb/><figure id="id.036.01.044.1.jpg" xlink:href="036/01/044/1.jpg"></figure><lb/>lineæ à circumferentia ad centrum productæ. </s>

<s id="id.2.1.25.2.1.3.0"> non igitur natura<lb/>les de&longs;cen&longs;us recti cuiuslibet &longs;oluti ponderis per lineas fieri po&longs;<lb/>&longs;unt inter &longs;e &longs;e parallelas; cùm omnes in centrum mundi conue<lb/>niant. </s>

<s id="id.2.1.25.2.1.4.0"> &longs;upponunt deinde ponderis ex D in A per rectam lineam <lb/>ver&longs;us centrum mundi motum eiu&longs;dem e&longs;&longs;e quantitatis, ac &longs;i fui&longs;<lb/>&longs;et ex O in C: ita vt punctum A æqualiter à centro mundi &longs;it <lb/>di&longs;tans, vt C. quod e&longs;t etiam fal&longs;um; nam punctum A magis <lb/>à centro mundi di&longs;tat, quàm C: maior enim e&longs;t linea à cen<lb/><arrow.to.target n="note49"></arrow.to.target>tro mundi v&longs;q; ad A, quàm à centro mundi v&longs;q; ad C: cùm li<lb/>nea à centro mundi v&longs;q; ad A rectum &longs;ubtendat angulum à li<lb/>neis AC, & à puncto C ad centrum mundi contentum. </s>

<s>quod e&longs;t etiam fal&longs;um; nam punctum A magis <lb/>à centro mundi di&longs;tat, quàm C: maior enim e&longs;t linea à cen<lb/><arrow.to.target n="note49"></arrow.to.target>tro mundi v&longs;q; ad A, quàm à centro mundi v&longs;q; ad C: cùm li<lb/>nea à centro mundi v&longs;q; ad A rectum &longs;ubtendat angulum à li<lb/>neis AC, & à puncto C ad centrum mundi contentum. </s>

<s id="id.2.1.25.2.1.5.0"> ex qui<lb/>bus non &longs;olum &longs;uppo&longs;itio illa, qua libram DE in AB redire demon<lb/>&longs;trant, verùm etiam omnes ferè ip&longs;orum demon&longs;trationes ruunt. </s>

<s id="id.2.1.25.2.1.6.0"> <lb/>ni&longs;i forta&longs;&longs;e dixerint, hæc omnia propter maximam à centro mun<lb/>di v&longs;q; ad nos di&longs;tantiam adeo in&longs;en&longs;ibilia e&longs;&longs;e, vt propter in&longs;en<lb/>&longs;ibilitatem tanquam vera &longs;upponi po&longs;sint: cùm omnes <expan abbr="quid&etilde;">quidem</expan>alii, qui <lb/>hæc tractauerunt, tanquam nota &longs;uppo&longs;uerint. </s>

<s id="id.2.1.25.2.1.7.0"> præ&longs;ertim quia <lb/>&longs;en&longs;ibilitas illa non efficit, quin de&longs;cen&longs;us ponderis ex L in D <lb/>(vt eorum verbis vtar) minus capiat de directo, quàm de&longs;cen<lb/>&longs;us DA. &longs;imiliter arcus DA magis de directo capiet, quàm cir<lb/>cumferentia EV. quocirca vera erit &longs;uppo&longs;itio; aliæq; demon<lb/>&longs;trationes in &longs;uo robore permanebunt. </s>

<s id="id.2.1.25.2.1.8.0"> Concedamus etiam pon <pb n="16" xlink:href="036/01/045.jpg"/>dus in A grauius e&longs;&longs;e, quàm in alio &longs;itu; rectumq; ponderis de<lb/>&longs;cen&longs;um per rectam lineam ip&longs;i FG parallelam fieri debere; & <lb/>quælibet puncta in lineis horizonti æquidi&longs;tantibus accepta æ<lb/>qualiter à centro mundi di&longs;tare: non tamen propterea &longs;equetur, <lb/>veram e&longs;&longs;e demon&longs;trationem, qua inferunt pondus in A grauius <lb/>e&longs;&longs;e, quàm in alio &longs;itu, vt in L. &longs;i enim verum e&longs;&longs;et, quò pon<lb/>dus hoc modo rectius de&longs;cendit, ibi grauius e&longs;&longs;e; &longs;equeretur etiam, <lb/>quò idem pondus in æqualibus arcubus æqualiter rectè de&longs;cende <lb/>ret, vt in ii&longs;dem locis æqualem haberet grauitatem, quod fal<lb/>&longs;um e&longs;&longs;e ita demon&longs;tratur. </s>

<s>&longs;imiliter arcus DA magis de directo capiet, quàm cir<lb/>cumferentia EV. </s>

<s>quocirca vera erit &longs;uppo&longs;itio; aliæq; demon<lb/>&longs;trationes in &longs;uo robore permanebunt. </s>

<s>&longs;i enim verum e&longs;&longs;et, quò pon<lb/>dus hoc modo rectius de&longs;cendit, ibi grauius e&longs;&longs;e; &longs;equeretur etiam, <lb/>quò idem pondus in æqualibus arcubus æqualiter rectè de&longs;cende<lb/>ret, vt in ii&longs;dem locis æqualem haberet grauitatem, quod fal<lb/>&longs;um e&longs;&longs;e ita demon&longs;tratur. </s>

<s id="id.2.1.26.1.1.1.0"> <margin.target id="note48"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/>15 <emph type="italics"/>Tertii.<emph.end type="italics"/> </s>

Line 611

Line 631

<s id="id.2.1.27.1.1.1.0"> Sint circumferentiæ AL AM inter &longs;e &longs;e æquales; & conne<lb/>ctatur LM, quæ AB &longs;ecet in X: erit LM ip&longs;i FG æquidi&longs;tans, <lb/>ip&longs;iq; AB perpendicularis. </s>

<s id="id.2.1.27.1.1.2.0"> & XM ip&longs;i XL æqualis erit. </s>

<s id="id.2.1.27.1.1.3.0"> &longs;i igi<arrow.to.target n="note50"></arrow.to.target><lb/>tur pondus ex L moueatur in A per circumferentiam LA, rectus <lb/>eius motus erit &longs;ecundùm lineam LX. &longs;i verò moueatur ex A <lb/>in M per circum&longs;erentiam AM, &longs;ecundùm rectam eius motus <lb/>erit XM. quare de&longs;cen&longs;us ex L in A æqualis erit de&longs;cen&longs;ui ex A <lb/>in M; tum ob circumferentias æquales, tum propter rectas li <lb/>neas ip&longs;i AB perpendiculares æquales. </s>

<s id="id.2.1.27.1.1.3.0.a">&longs;i verò moueatur ex A <lb/>in M per circumferentiam AM, &longs;ecundùm rectam eius motus <lb/>erit XM. </s>

<s id="id.2.1.27.1.1.3.0.b">quare de&longs;cen&longs;us ex L in A æqualis erit de&longs;cen&longs;ui ex A <lb/>in M; tum ob circumferentias æquales, tum propter rectas li <lb/>neas ip&longs;i AB perpendiculares æquales. </s>

<s id="id.2.1.27.1.1.4.0"> ergo idem pondus in L <lb/>æquè graue erit, vt in A, quod e&longs;t fal&longs;um. </s>

<s id="id.2.1.27.1.1.5.0"> cum longé grauius &longs;it <lb/>in A, quàm in L. </s>

Line 619

Line 641

<s id="id.2.1.28.1.1.1.0"> <margin.target id="note50"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/>3 <emph type="italics"/>Tertii.<emph.end type="italics"/> </s>

<s id="id.2.1.29.1.1.1.0"> Quamuis autem AMLA æqualiter &longs;ecundùm ip&longs;os de directo <lb/>capiant; dicent forta&longs;&longs;e, quia tamen principium de&longs;cen&longs;us ex L <lb/>&longs;cilicet LD minus de directo capit, quàm principium de&longs;cen&longs;us <lb/>ex A, &longs;cilicet AN; pondus in A grauius erit, quàm in L. nam <lb/>cùm circumferentia AN &longs;it ip&longs;i LD (vt &longs;upra po&longs;itum e&longs;t) <lb/>æqualis, quæ &longs;ecundùm ip&longs;os de directo capit CT; LD verò <lb/>de directo capit PO. ideo pondus grauius erit in A, quàm in L. <lb/>quod &longs;i verum e&longs;&longs;et, &longs;equeretur idem pondus in eodem &longs;itu diuer<lb/>&longs;o duntaxat modo con&longs;ideratum in habitudine ad eundem &longs;itum, <lb/>tum grauius, tum leuius e&longs;&longs;e. </s>

<s id="id.2.1.29.1.1.1.0.a">nam <lb/>cùm circumferentia AN &longs;it ip&longs;i LD (vt &longs;upra po&longs;itum e&longs;t) <lb/>æqualis, quæ &longs;ecundùm ip&longs;os de directo capit CT; LD verò <lb/>de directo capit PO. </s>

<s id="id.2.1.29.1.1.1.0.b">ideo pondus grauius erit in A, quàm in L. <lb/></s>

<s id="id.2.1.29.1.1.1.0.c">quod &longs;i verum e&longs;&longs;et, &longs;equeretur idem pondus in eodem &longs;itu diuer<lb/>&longs;o duntaxat modo con&longs;ideratum in habitudine ad eundem &longs;itum, <lb/>tum grauius, tum leuius e&longs;&longs;e. </s>

<s id="id.2.1.29.1.1.2.0"> quod e&longs;t impo&longs;sibile. </s>

<s id="id.2.1.29.1.1.3.0"> hoc e&longs;t, &longs;i <lb/>de&longs;cen&longs;um con&longs;ideremus ponderis in L, quatenus ex L in A de<lb/>&longs;cendit, grauius erit, quàm &longs;i eiu&longs;dem ponderis de&longs;cen&longs;um con<lb/>&longs;ideremus ex L in D tantùm. </s>

<s id="id.2.1.29.1.1.4.0"> neq; enim negare po&longs;&longs;unt ex ei&longs;<lb/>demmet dictis, quin de&longs;cen&longs;us ponderis ex L in A de directo ca <lb/>piat LX, &longs;iue PC. de&longs;cen&longs;us verò AM, quin &longs;imiliter de directo <pb xlink:href="036/01/046.jpg"/>capiat XM: cùm ip&longs;i <lb/>quoq; hoc modo acci<lb/>piant, atq; ita accipe<lb/>re &longs;it nece&longs;&longs;e. </s>

<s id="id.2.1.29.1.1.4.0">neq; enim negare po&longs;&longs;unt ex ei&longs;<lb/>demmet dictis, quin de&longs;cen&longs;us ponderis ex L in A de directo ca<lb/>piat LX, &longs;iue PC. </s>

<s>de&longs;cen&longs;us verò AM, quin &longs;imiliter de directo <pb xlink:href="036/01/046.jpg"/>capiat XM: cùm ip&longs;i <lb/>quoq; hoc modo acci<lb/>piant, atq; ita accipe<lb/>re &longs;it nece&longs;&longs;e. </s>

<s id="id.2.1.29.1.1.5.0"> &longs;i enim li<lb/>bram DE in AB redire <lb/>demon&longs;trare volunt, com<lb/>parando de&longs;cen&longs;us pon<lb/>deris in D cum de&longs;cen<lb/>&longs;u ponderis in E, nece&longs;&longs;e <lb/>e&longs;t, vt o&longs;tendant rectum <lb/>de&longs;cen&longs;um OC corre<lb/>&longs;pondentem circumferen<lb/>tiæ DA maiorem e&longs;&longs;e re<lb/>cto de&longs;cen&longs;u TH circum<lb/><figure id="id.036.01.046.1.jpg" xlink:href="036/01/046/1.jpg"></figure><lb/>ferentiæ EV corre&longs;pondente. </s>

<s id="id.2.1.29.1.1.6.0"> &longs;i enim partem tantùm totius de<lb/>&longs;cen&longs;us ex D in A acciperent, vt D k; o&longs;tenderentq; magis cape<lb/>re de directo de&longs;cen&longs;um Dk, quàm æqualis portio de&longs;cen&longs;us ex <lb/>puncto E. &longs;equetur pondus in D &longs;ecundùm ip&longs;os grauius e&longs;&longs;e pon<lb/>dere in E; & v&longs;q; ad k tantùm deor&longs;um moueri: ita vt libra mo<lb/>ta &longs;it in kI. &longs;imiliter &longs;i libram KI in AB redire demon&longs;trare vo<lb/>lunt accipiendo portionem de&longs;cen&longs;us ex k in A; hoc e&longs;t k S; <lb/>o&longs;tenderentq; k S magis de directo capere, quàm ex aduer&longs;o æ<lb/>qualis de&longs;cen&longs;us ex puncto I: &longs;imili modo &longs;equetur pondus in k <lb/>grauius e&longs;&longs;e, quàm in I; & v&longs;q; ad S tantùm moueri. </s>

<s>&longs;equetur pondus in D &longs;ecundùm ip&longs;os grauius e&longs;&longs;e pon<lb/>dere in E; & v&longs;q; ad k tantùm deor&longs;um moueri: ita vt libra mo<lb/>ta &longs;it in kI. </s>

<s>&longs;imiliter &longs;i libram KI in AB redire demon&longs;trare vo<lb/>lunt accipiendo portionem de&longs;cen&longs;us ex k in A; hoc e&longs;t k S; <lb/>o&longs;tenderentq; k S magis de directo capere, quàm ex aduer&longs;o æ<lb/>qualis de&longs;cen&longs;us ex puncto I: &longs;imili modo &longs;equetur pondus in k <lb/>grauius e&longs;&longs;e, quàm in I; & v&longs;q; ad S tantùm moueri. </s>

<s id="id.2.1.29.1.1.7.0"> & &longs;i rur&longs;us <lb/>o&longs;tenderent portionem de&longs;cen&longs;us ex S in A, atq; ita deinceps, re<lb/>ctiorem e&longs;&longs;e æquali de&longs;cen&longs;u ponderis oppo&longs;iti; &longs;emper &longs;equetur <lb/>libram SI ad AB propius accedere, nunquam tamen in AB per<lb/>uenire demon&longs;trabunt. </s>

<s id="id.2.1.29.1.1.8.0"> &longs;i igitur libram DE in AB redire demon<lb/>&longs;trare volunt, nece&longs;&longs;e e&longs;t, vt de&longs;cen&longs;um ponderis ex D in A de di <lb/>recro capere quantitatem lineæ ex puncto D ip&longs;i AB ad rectos <lb/>angulos ductæ accipiant. </s>

<s id="id.2.1.29.1.1.9.0"> atq; ita, &longs;i æquales de&longs;cen&longs;us DA AN <lb/>inuicem comparemus, qui æqualiter de directo capient OC CT, <lb/>cueniet idem pondus in D æquè graue e&longs;&longs;e, vt in A. &longs;i verò por<lb/>tiones tantum ex D A accipiamus; grauius erit in A, quàm <lb/>in D. ergo ex diuer&longs;itate tantùm modi con&longs;iderandi, idem pon<lb/>dus, & grauius, & leuius e&longs;&longs;e continget. </s>

<s id="id.2.1.29.1.1.9.0">atq; ita, &longs;i æquales de&longs;cen&longs;us DA AN <lb/>inuicem comparemus, qui æqualiter de directo capient OC CT, <lb/>eueniet idem pondus in D æquè graue e&longs;&longs;e, vt in A. </s>

<s>&longs;i verò por<lb/>tiones tantum ex D A accipiamus; grauius erit in A, quàm <lb/>in D. </s>

<s>ergo ex diuer&longs;itate tantùm modi con&longs;iderandi, idem pon<lb/>dus, & grauius, & leuius e&longs;&longs;e continget. </s>

<s id="id.2.1.29.1.1.10.0"> non autem exip&longs;a na<pb n="17" xlink:href="036/01/047.jpg"/>tura rei. </s>

<s id="id.2.1.29.1.1.11.0"> In&longs;uper ip&longs;orum &longs;uppo&longs;itio non a&longs;&longs;erit, pondus &longs;ecun<lb/>dùm &longs;itum grauius e&longs;&longs;e, quantò in eodem &longs;itu minus obliquum <lb/>e&longs;t principium ip&longs;ius de&longs;cen&longs;us. </s>

<s id="id.2.1.29.1.1.12.0"> Suppo&longs;itio igitur &longs;uperius alla<lb/>ta, hoc e&longs;t, &longs;ecundùm &longs;itum pondus grauius e&longs;&longs;e, quantò in eo <lb/>dem &longs;itu minus obliquus e&longs;t de&longs;cen&longs;us; non &longs;olum ex his, quæ <lb/>diximus, vllo modo concedi pote&longs;t; &longs;ed quoniam huius oppo&longs;i<lb/>tum o&longs;tendere quoq; non e&longs;t difficile: &longs;cilicet idem pondus in <lb/>æqualibus circumferentiis, quò minus obliquus e&longs;t de&longs;cen&longs;us, ibi <lb/>minus grauitare. </s>

<s id="id.2.1.29.2.1.1.0"> Sint enim vt prius cir <lb/>cumferentræ AL AM <lb/>inter &longs;e &longs;e æquales; &longs;itq; <lb/>punctum L propè F. & <lb/>connectatur LM, quæ <lb/>ip&longs;i AB perpendicularis <lb/>erit. </s>

<s id="id.2.1.29.2.1.1.0">Sint enim vt prius cir<lb/><expan abbr="cumferentræ">cumferentiae</expan> AL AM <lb/>inter &longs;e &longs;e æquales; &longs;itq; <lb/>punctum L propè F. </s>

<s>& <lb/>connectatur LM, quæ <lb/>ip&longs;i AB perpendicularis <lb/>erit. </s>

<s id="id.2.1.29.2.1.2.0"> & LX ip&longs;i XM <lb/>æqualis. </s>

<s id="id.2.1.29.2.1.3.0"> deinde propè <lb/>M inter MG quoduis <lb/>accipiatur punctum P. <lb/>fiatq; circumferentia PO <lb/>circumferentiæ AM æ<lb/>qualis. </s>

<s id="id.2.1.29.2.1.4.0"> erit punctum O <lb/><figure id="id.036.01.047.1.jpg" xlink:href="036/01/047/1.jpg"></figure><expan abbr="propè"><lb/>prope</expan>A. connectanturq; CL, CO, CM, CP, OP. & à <lb/>puncto P ip&longs;i OC perpendicularis ducatur PN. </s>

<s id="id.2.1.29.2.1.4.0">erit punctum O <lb/><figure id="id.036.01.047.1.jpg" xlink:href="036/01/047/1.jpg"></figure><expan abbr="propè"><lb/>prope</expan> A. </s>

<s>connectanturq; CL, CO, CM, CP, OP. </s>

<s>& à <lb/>puncto P ip&longs;i OC perpendicularis ducatur PN. </s>

<s id="id.2.1.29.2.1.4.0.a"> & quoniam cir<lb/>cumferentia AM circumferentiæ OP e&longs;t æqualis: erit angu<lb/>lus <arrow.to.target n="note51"></arrow.to.target>ACM æqualis angulo OCP; & angulus CXM rectus re<lb/>cto CNP e&longs;t æqualis: erit quoq; reliquus XMC trianguli MCX <arrow.to.target n="note52"></arrow.to.target><lb/>reliquo NPC trianguli PCN æqualis. </s>

<s id="id.2.1.29.2.1.5.0"> &longs;ed & latus CM lateri <arrow.to.target n="note53"></arrow.to.target><lb/>CP e&longs;t æquale: ergo triangulum MCX triangulo PCN æquale <lb/>erit. </s>

<s id="id.2.1.29.2.1.6.0"> latu&longs;q; MX lateri NP æquale. </s>

<s id="id.2.1.29.2.1.7.0"> quare linea PN ip&longs;i LX æqua <lb/>lis erit. </s>

<s id="id.2.1.29.2.1.8.0"> ducatur præterea à puncto O linea OT ip&longs;i AC æqui <lb/>di&longs;tans, quæ NP &longs;ecet in V. atq; ip&longs;i OT à puncto P perpendi<lb/>cularis ducatur, quæ quidem inter OV cadere non pote&longs;t; nam <lb/>cùm angulus ONV &longs;it rectus; erit OVN acutus. </s>

<s id="id.2.1.29.2.1.8.0">ducatur præterea à puncto O linea OT ip&longs;i AC æqui<lb/>di&longs;tans, quæ NP &longs;ecet in V. </s>

<s>atq; ip&longs;i OT à puncto P perpendi<lb/>cularis ducatur, quæ quidem inter OV cadere non pote&longs;t; nam <lb/>cùm angulus ONV &longs;it rectus; erit OVN acutus. </s>

<s id="id.2.1.29.2.1.9.0"> quare OVP <arrow.to.target n="note54"></arrow.to.target><lb/>obtu&longs;us erit. </s>

<s id="id.2.1.29.2.1.10.0"> non igitur linea à puncto P ip&longs;i OT intra OV <pb xlink:href="036/01/048.jpg"/>perpendicularis cadet. </s>

<s id="id.2.1.29.2.1.11.0"> <lb/>duo enim anguli vnius <lb/>trianguli, vnus quidem <lb/>rectus, alter verò ob<lb/>tu&longs;us e&longs;&longs;et. </s>

<s id="id.2.1.29.2.1.12.0"> quod e&longs;t im<lb/>po&longs;sibile. </s>

<s id="id.2.1.29.2.1.13.0"> cadet ergo in <lb/>linea OT in parte VT. <lb/>&longs;itq; PT. erit PT &longs;ecun<lb/>dùm ip&longs;os rectus circum<lb/>ferentiæ OP de&longs;cen&longs;us. </s>

<s id="id.2.1.29.2.1.13.0">cadet ergo in <lb/>linea OT in parte VT. <lb/></s>

<s>&longs;itq; PT. erit PT &longs;ecun<lb/>dùm ip&longs;os rectus circum<lb/>ferentiæ OP de&longs;cen&longs;us. </s>

<s id="id.2.1.29.2.1.14.0"> <lb/>Quoniam igitur angulus <lb/>ONV e&longs;t rectus; erit <lb/><arrow.to.target n="note55"></arrow.to.target>linea OV ip&longs;a ON ma<lb/>ior. </s>

<s id="id.2.1.29.2.1.15.0"> quare OT ip&longs;a <lb/><figure id="id.036.01.048.1.jpg" xlink:href="036/01/048/1.jpg"></figure><lb/>quoq; ON maior exi&longs;tet. </s>

<s id="id.2.1.29.2.1.16.0"> Cùm itaq; linèa OP angulos &longs;ubten<lb/>dat rectos ONP OTP; erit quadratum ex OP quadratis ex <lb/><arrow.to.target n="note56"></arrow.to.target>ON NP &longs;imul &longs;umptis æquale. </s>

<s id="id.2.1.29.2.1.17.0"> &longs;imiliter quadratis ex OT TP <lb/>&longs;imul æquale. </s>

<s id="id.2.1.29.2.1.18.0"> quare quadrata &longs;imul ex ON NP quadratis ex <lb/>OT TP &longs;imul æqualia erunt. </s>

<s id="id.2.1.29.2.1.19.0"> quadratum autem ex OT maius <lb/>e&longs;t quadrato ex ON; cum linea OT &longs;it ip&longs;a ON maior. </s>

<s id="id.2.1.29.2.1.20.0"> ergo qua<lb/>dratum ex NP maius erit quadrato ex TP. ac propterea linea <lb/>TP minor erit linea PN, & linea LX. minus obliquus igitur e&longs;t <lb/>de&longs;cen&longs;us arcus LA, quàm arcus OP. </s>

<s id="id.2.1.29.2.1.20.0">ergo qua<lb/>dratum ex NP maius erit quadrato ex TP. </s>

<s>ac propterea linea <lb/>TP minor erit linea PN, & linea LX. </s>

<s>minus obliquus igitur e&longs;t <lb/>de&longs;cen&longs;us arcus LA, quàm arcus OP. </s>

<s id="id.2.1.29.2.1.20.0.a"> ergo pondus in L, ex ip<lb/>&longs;orum dictis, grauius erit, quàm in O. quod ex iis, quæ &longs;upra di<lb/>ximus e&longs;t manife&longs;tè fal&longs;um, cùm pondus in O grauius &longs;it, quàm <lb/>in L. </s>

<s id="id.2.1.29.2.1.20.0.b"> non igitur ex rectiori, & obliquiori motu ita accepto col<lb/>ligi pote&longs;t, &longs;ecundùm &longs;itum pondus grauius e&longs;&longs;e, quantò in eo <lb/>dem &longs;itu minus obliquus e&longs;t de&longs;cen&longs;us. </s>

<s id="id.2.1.29.2.1.21.0"> Atq; hinc oritur omnis <lb/>fermé ip&longs;orum error in hacre, atq; deceptio: nam quamuis per <lb/>accidens interdum ex fal&longs;is &longs;equatur verum, per &longs;e tamen ex fal<lb/>&longs;is fal&longs;um &longs;equitur, quemadmodum ex veris &longs;emper verum, nil <lb/>idcirco mirum, &longs;i dum fal&longs;a accipiunt; illi&longs;q; tanquam veri&longs;si<lb/>mis innituntur; fal&longs;i&longs;sima omninò colligunt, atq; concludunt. </s>

Line 675

Line 712

<s id="id.2.1.31.1.1.5.0"> <lb/>po&longs;tremò tamen ob ponderum de&longs;cen&longs;uum comparationem colli<lb/>gentes inferunt, pondus in D deor&longs;um moueri, & pondus in E <lb/>&longs;ur&longs;um, vtraq; &longs;imul in libra inuicem connexa accipientes. </s>

<s id="id.2.1.31.1.1.6.0"> <expan abbr="verùm">ve<lb/>rum</expan>ex ii&longs;demmet, quibus vtuntur, principiis, ac demon&longs;tratio<lb/>nibus, oppo&longs;itum eius, quod defendere conantur, facillimè col<lb/>ligi pote&longs;t. </s>

<s id="id.2.1.31.1.1.7.0"> Nam &longs;i comparetur de&longs;cen&longs;us ponderis in D cum a<lb/>&longs;cen&longs;u ponderis in E, vt ductis EK DH ip&longs;i AB perpendicula<lb/>ribus; cùm angulus DCH &longs;it æqualis angulo ECk; & angulus <arrow.to.target n="note57"></arrow.to.target><lb/>DHC rectus æqualis e&longs;t recto E k C; & latus DC lateri CE æqua <lb/>le: erit triangulum CDH triangulo CEk æquale, & latus DH la-<arrow.to.target n="note58"></arrow.to.target><pb xlink:href="036/01/050.jpg"/>teri Ek æquale. </s>

<s id="id.2.1.31.1.1.8.0"> cùm <lb/>autem angulus DCA <lb/>&longs;it angulo ECB æqua<lb/>lis: erit quoq; circum<lb/>ferentia DA cirferen<lb/>tiæ BE æqualis. </s>

<s id="id.2.1.31.1.1.8.0">cùm <lb/>autem angulus DCA <lb/>&longs;it angulo ECB æqua<lb/>lis: erit quoq; circum<lb/>ferentia DA <expan abbr="cirferen">circumferen</expan><lb/>tiæ BE æqualis. </s>

<s id="id.2.1.31.1.1.9.0"> dum <lb/>itaq; pondus in D de<lb/>&longs;cendit per circumfe<lb/>rentiam DA, pondus <lb/>in E per circumferen<lb/>tiam EB ip&longs;i DA æ<lb/>qualem a&longs;cendit. </s>

<s id="id.2.1.31.1.1.10.0"> & de<lb/>&longs;cen&longs;us <expan abbr="põderis">ponderis</expan>in D de <lb/>directo (more <expan abbr="ip&longs;or&utilde;">ip&longs;orum</expan>) <lb/><figure id="id.036.01.050.1.jpg" xlink:href="036/01/050/1.jpg"></figure><lb/>capiet DH; a&longs;cen&longs;us verò ponderis in E de directo capiet Ek ip<lb/>&longs;i DH æqualem: erit itaq; de&longs;cen&longs;us ponderis in D a&longs;cen&longs;ui pon<lb/>deris in E æqualis, & qualis erit propen&longs;io vnius ad motum deor<lb/>sum, talis etiam erit re&longs;i&longs;tentia alterius ad motum &longs;ur&longs;um. </s>

<s id="id.2.1.31.1.1.11.0"> re<lb/>&longs;i&longs;tentia &longs;cilicet violentiæ ponderis in E in a&longs;cen&longs;u naturali po<lb/>tentiæ ponderis in D in de&longs;cen&longs;u contrà nitendo apponitur; cùm <lb/>&longs;it ip&longs;i æqualis. </s>

Line 685

Line 722

<s id="id.2.1.31.1.1.15.0"> &longs;i enim moueret; nece&longs;&longs;e e&longs;&longs;et, pondus in D maiorem <lb/>habere virtutem de&longs;cendendo, quàm pondus in E a&longs;cendendo; <lb/>&longs;ed hæc &longs;unt æqualia: ergo pondera manebunt. </s>

<s id="id.2.1.31.1.1.16.0"> & grauitas pon<lb/>deris in D grauitati ponderis in E æqualis erit. </s>

<s id="id.2.1.31.1.1.17.0"> Præterea quoniam <lb/>&longs;upponunt, quò pondus à linea directionis FG magis di&longs;tat, eò <lb/>grauius e&longs;&longs;e: Idcirco ductis quoq; à punctis DE ip&longs;i FG perpen<lb/>dicularibus DO EI; &longs;imili modo demon&longs;trabitur, triangulum <lb/>CDO triangulo CEI æqualem e&longs;&longs;e: & lineam DO ip&longs;i EI æqua<lb/>lem. </s>

<s id="id.2.1.31.1.1.18.0"> tam igitur di&longs;tat à linea FG pondus in D, quàm pondus in <lb/>E. ex ip&longs;orum igitur rationibus, atq; &longs;uppo&longs;itionibus, pondera <lb/>in DE æquè grauia erunt. </s>

<s id="id.2.1.31.1.1.18.0">tam igitur di&longs;tat à linea FG pondus in D, quàm pondus in <lb/>E. </s>

<s>ex ip&longs;orum igitur rationibus, atq; &longs;uppo&longs;itionibus, pondera <lb/>in DE æquè grauia erunt. </s>

<s id="id.2.1.31.1.1.19.0"> Amplius quid prohibet, quin libram <lb/>DE ex nece&longs;sitate in FG moueri &longs;imili ratione o&longs;tendatur? </s>

<s id="id.2.1.31.1.1.20.0"> Pri<pb n="19" xlink:href="036/01/051.jpg"/>mùm quidem ex eorummet demon&longs;trationibus colligi pote&longs;t, a<lb/>&longs;cen&longs;um ponderis in E ver&longs;us B rectiorem e&longs;&longs;e a&longs;cen&longs;u ponderis <lb/>in D ver&longs;us F; hoc e&longs;t minus capere de directo a&longs;cen&longs;um pon<lb/>deris in D in arcubus æqualibus a&longs;cen&longs;u ponderis in E. </s>

<s id="id.2.1.31.1.1.20.0.a"> &longs;uppona<lb/>tur ergo &longs;ecundùm &longs;itum pondus leuius e&longs;&longs;e, quantò in eodem &longs;i<lb/>tu minus rectus e&longs;t a&longs;cen&longs;us: quæ quidem &longs;uppo&longs;itio, adeò ma<lb/>nife&longs;ta e&longs;&longs;e videtur, veluti ip&longs;orum altera. </s>

Line 710

Line 748

<s id="id.2.1.33.3.1.2.0"> &longs;imiliter pondus in E &longs;olutum <lb/>per lineam ES mouebitur. </s>

<s id="id.2.1.33.3.1.3.0"> quare &longs;i <lb/>(vt rei veritas e&longs;t) ponderis de&longs;cen<lb/>&longs;us magis, minu&longs;uè obliquus dicetur <lb/>&longs;ecundùm rece&longs;&longs;um, & acce&longs;&longs;um ad <lb/>&longs;patia per lineas DSES de&longs;ignata, <lb/>iuxta naturales ip&longs;orum ad propria lo <lb/>ca lationes; con&longs;picuum e&longs;t, minus <lb/>obliquum e&longs;&longs;e de&longs;cen&longs;um ip&longs;ius E <lb/>per EG, quàm ip&longs;ius D per DA: <lb/>cùm angulum SEG angulo SDA <lb/>minorem e&longs;&longs;e &longs;upra o&longs;ten&longs;um &longs;it. </s>

<s id="id.2.1.33.3.1.4.0"> qua <lb/>re in E pondus magis grauitabit, <lb/>quàm in D. quod e&longs;t penitus oppo<lb/>&longs;itum eius, quod ip&longs;i o&longs;tendere cona<lb/>ti &longs;unt. </s>

<s id="id.2.1.33.3.1.5.0"> In&longs;urgent autem forta&longs;&longs;e <lb/>contranos, &longs;i igitur (dicent) pondus <lb/>in E grauius e&longs;t pondere in D, libra <lb/><figure id="id.036.01.052.1.jpg" xlink:href="036/01/052/1.jpg"></figure><lb/>DE in hoc &longs;itu minimè per&longs;i&longs;tet, quod <expan abbr="equid&etilde;">equidem</expan>tueri propo&longs;uimus: <lb/>&longs;ed in FG mouebitur. </s>

<s id="id.2.1.33.3.1.5.0">In&longs;urgent autem forta&longs;&longs;e <lb/>contrarios, &longs;i igitur (dicent) pondus <lb/>in E grauius e&longs;t pondere in D, libra <lb/><figure id="id.036.01.052.1.jpg" xlink:href="036/01/052/1.jpg"></figure><lb/>DE in hoc &longs;itu minimè per&longs;i&longs;tet, quod <expan abbr="equid&etilde;">equidem</expan> tueri propo&longs;uimus: <lb/>&longs;ed in FG mouebitur. </s>

<s id="id.2.1.33.3.1.6.0"> quibus re&longs;pondemus, plurimum referre, &longs;iue <lb/>con&longs;ideremus pondera, quatenus &longs;unt inuicem di&longs;iuncta, &longs;iue quate <lb/>nus &longs;unt &longs;ibi inuicem connexa. </s>

<s id="id.2.1.33.3.1.7.0"> alia e&longs;t enim ratio ponderis in E &longs;ine <lb/>connexione ponderis in D, alia verò eiu&longs;dem alteri ponderi con<lb/>nexi; ita vt alterum &longs;ine altero moueri non po&longs;sit. </s>

<s id="id.2.1.33.3.1.8.0"> nam ponde<lb/>ris in E, quatenus e&longs;t &longs;ine alterius ponderis connexione, rectus <lb/>naturalis de&longs;cen&longs;us e&longs;t per lineam ES; quatenus verò connexum <lb/>e&longs;t ponderi in D, eius naturalis de&longs;cen&longs;us non erit amplius per <lb/>lineam ES, &longs;ed per lineam ip&longs;i CS parallelam. </s>

Line 721

Line 759

<s id="id.2.1.33.3.1.13.0"> Quare pondera in <lb/>DE, quatenus &longs;unt &longs;ibi inuicem connexa, &longs;i ip&longs;orum naturalem mo <lb/>tum &longs;pectemus, non &longs;ecundùm lineas DS ES, &longs;ed &longs;ecundùm <lb/>LDH MEk ip&longs;i CS æquidi&longs;tantes mouebuntur. </s>

<s id="id.2.1.33.3.1.14.0"> ponderis <expan abbr="verò">ve<lb/>ro</expan>in E liberi, ac &longs;oluti, naturalis propen&longs;io erit per ES: ponderis <lb/>autem in D &longs;imiliter &longs;oluti erit per DS. ac propterea non e&longs;t incon<lb/>ueniens idem pondus modò in E, modò in D, grauius e&longs;&longs;e in E, <lb/>quàm in D. </s>

<s id="id.2.1.33.3.1.14.0.a"> &longs;i verò pondera in ED &longs;ibi inuicem connexa, quate<lb/>nusq; &longs;unt connexa con&longs;iderauerimus; erit ponderis in E natura<lb/>lis propen&longs;io per lineam MEK: grauitas enim alterius ponde<lb/>ris in D efficit, nè pondus in E per lineam ES grauitet, &longs;ed per <lb/>Ek. </s>

<s id="id.2.1.33.3.1.15.0"> quod ip&longs;um quoq; grauitas ponderis in E efficit, nè &longs;cilicet <lb/>pondus in D per rectam DS degrauet; &longs;ed &longs;ecundùm DH: vtra<lb/>que enim &longs;e impediunt, nè ad propria loca permeent. </s>

<s id="id.2.1.33.3.1.15.0">quod ip&longs;um quoq; grauitas ponderis in E efficit, nè &longs;cilicet <lb/>pondus in D per rectam DS degrauet; &longs;ed &longs;ecundùm DH: vtra<lb/>que enim &longs;e impediunt, nè ad propria loca <expan abbr="permeent">permeant</expan>. </s>

<s id="id.2.1.33.3.1.16.0"> Cùm igi<lb/>tur naturalis de&longs;cen&longs;us rectus ponderum in DE &longs;it &longs;ecundùm <lb/>LDH MEK: erit &longs;imliter rectus eorum a&longs;cen&longs;us &longs;ecundùm ea&longs; <lb/>dem lineas HDL KEM. atq; a&longs;cen&longs;us ponderis in E magis, mi<lb/>nu&longs;uè obliquus dicetur; quantò &longs;ecundùm &longs;patium magis, <expan abbr="minu&longs;uè">mi<lb/>nu&longs;ue</expan>iuxta lineam Mk mouebitur. </s>

<s id="id.2.1.33.3.1.16.0">Cùm igi<lb/>tur naturalis de&longs;cen&longs;us rectus ponderum in DE &longs;it &longs;ecundùm <lb/>LDH MEK: erit <expan abbr="&longs;imliter">similiter</expan> rectus eorum a&longs;cen&longs;us &longs;ecundùm ea&longs;<lb/>dem lineas HDL KEM. atq; a&longs;cen&longs;us ponderis in E magis, mi<lb/>nu&longs;uè obliquus dicetur; quantò &longs;ecundùm &longs;patium magis, mi<lb/>nu&longs;uè iuxta lineam Mk mouebitur. </s>

<s id="id.2.1.33.3.1.17.0"> hocq; pror&longs;us modo iuxta li<lb/>neam LH &longs;ummendus e&longs;t, tùm de&longs;cen&longs;us, tùm a&longs;cen&longs;us ponde<lb/>ris in D. &longs;i itaq; pondus in E deor&longs;um per EG moueretur; pon<lb/>dus in D &longs;ur&longs;um per DF moueret. </s>

<s id="id.2.1.33.3.1.17.0">hocq; pror&longs;us modo iuxta li<lb/>neam LH &longs;ummendus e&longs;t, tùm de&longs;cen&longs;us, tùm a&longs;cen&longs;us ponde<lb/>ris in D. </s>

<s>&longs;i itaq; pondus in E deor&longs;um per EG moueretur; pon<lb/>dus in D &longs;ur&longs;um per DF moueret. </s>

<s id="id.2.1.33.3.1.18.0"> & quoniam angulus CEK <arrow.to.target n="note60"></arrow.to.target><lb/>æqualis e&longs;t angulo CDL, & angulus CEG angulo CDF æqua<lb/>lis; erit reliquus GEK reliquo LDF æqualis. </s>

<s id="id.2.1.33.3.1.19.0"> cùm autem &longs;up<lb/>po&longs;itio illa, quæ ait, &longs;ecundúm &longs;itum pondus grauius e&longs;&longs;e, <expan abbr="quantò">quan<lb/>to</expan>in eodem &longs;itu minus obliquus e&longs;t de&longs;cen&longs;us; tanquam clara, <lb/>atq; con&longs;picua admittatur; proculdubio hæc quoq; accipienda <lb/>erit; nempè, &longs;ecundúm &longs;itum pondus grauius e&longs;&longs;e, quantò in eo<lb/>dem &longs;itu minus obliquus e&longs;t a&longs;cen&longs;us. </s>

<s id="id.2.1.33.3.1.20.0"> cùm non minus manife&longs;ta, <pb xlink:href="036/01/054.jpg"/>rationiq; &longs;it con&longs;entanea. </s>

<s id="id.2.1.33.3.1.21.0"> æqualis <lb/>igitur erit de&longs;cen&longs;us ponderis in E <lb/>a&longs;cen&longs;ui ponderis in D. eandem <lb/>enim obliquitatem habet de&longs;cen&longs;us <lb/>ponderis in E, quam habet a&longs;cen<lb/>&longs;us ponderis in D; & qualis erit <lb/>propen&longs;io vnius ad motum deor&longs;um, <lb/>talis quoq; erit re&longs;i&longs;tentia alterius ad <lb/>motum &longs;ur&longs;um. </s>

<s id="id.2.1.33.3.1.21.0">æqualis <lb/>igitur erit de&longs;cen&longs;us ponderis in E <lb/>a&longs;cen&longs;ui ponderis in D. </s>

<s>eandem <lb/>enim obliquitatem habet de&longs;cen&longs;us <lb/>ponderis in E, quam habet a&longs;cen<lb/>&longs;us ponderis in D; & qualis erit <lb/>propen&longs;io vnius ad motum deor&longs;um, <lb/>talis quoq; erit re&longs;i&longs;tentia alterius ad <lb/>motum &longs;ur&longs;um. </s>

<s id="id.2.1.33.3.1.22.0"> <expan abbr="nõ">non</expan>ergo pondus in E <lb/>pondus in D &longs;ur&longs;um mouebit. </s>

<s id="id.2.1.33.3.1.23.0"> neq; <lb/>pondus in D deor&longs;um mouebitur, ita <lb/>vt &longs;ur&longs;um moueat pondus in E. nam <lb/><expan abbr="c&utilde;">cum</expan>angulus CEB &longs;it ip&longs;i CDA æqua<lb/><arrow.to.target n="note61"></arrow.to.target>lis, & Angulus CEM &longs;it angulo <lb/>CDH æqualis; erit reliquus MEB <lb/>reliquo HDA æqualis. </s>

<s id="id.2.1.33.3.1.24.0"> de&longs;cen&longs;us <lb/>igitur ponderis in D a&longs;cen&longs;ui ponde<lb/>ris in E æqualis erit. </s>

Line 737

Line 777

<s id="id.2.1.33.4.1.1.0"> Alia deinde ratio, li<lb/>bram &longs;imiliter DE in AB <lb/>redire o&longs;tendens, cùm in<lb/>quiunt, exi&longs;tente trutina in <lb/>CF meta e&longs;t CG. </s>

<s id="id.2.1.33.4.1.1.0.a"> & quo<lb/>niam angulus DCG maior <lb/>e&longs;t angulo ECG; pondus <lb/>in D grauius erit pondere <lb/>in E; ergo libra DE in AB <lb/>redibit: nihil meo iudicio <lb/>concludit. </s>

<s id="id.2.1.33.4.1.2.0"> figmentumq; <lb/>hoc de trutina, & meta po<lb/>tius omittendum, ac &longs;ilen|tio<figure id="id.036.01.054.2.jpg" xlink:href="036/01/054/2.jpg"></figure><pb n="21" xlink:href="036/01/055.jpg"/><expan abbr="prætereund&utilde;">prætereundum</expan>e&longs;&longs;et, quàm <expan abbr="verb&utilde;">verbum</expan><expan abbr="vll&utilde;">vllum</expan>in eius confutatione &longs;umen<lb/>dum; cùm &longs;it pror&longs;us voluntarium. </s>

<s id="id.2.1.33.4.1.2.0">figmentumq; <lb/>hoc de trutina, & meta po<lb/>tius omittendum, ac &longs;ilen<figure id="id.036.01.054.2.jpg" xlink:href="036/01/054/2.jpg"></figure><pb n="21" xlink:href="036/01/055.jpg"/>tio <expan abbr="prætereund&utilde;">prætereundum</expan> e&longs;&longs;et, quàm <expan abbr="verb&utilde;">verbum</expan> <expan abbr="vll&utilde;">vllum</expan> in eius confutatione &longs;umen<lb/>dum; cùm &longs;it pror&longs;us voluntarium. </s>

<s id="id.2.1.33.4.1.3.0"> nece&longs;sitas enim cur pondus <lb/>in D ex maiore angulo &longs;it grauius; curq; maior angulus maioris <lb/>&longs;it cau&longs;a grauitatis; nu&longs;quam apparet. </s>

<s id="id.2.1.33.4.1.4.0"> &longs;i autem comparentur in<lb/>uicem anguli, cùm angulus GCD &longs;it æqualis angulo FCE; &longs;i angu<lb/>lus GCD e&longs;t cau&longs;a grauitatis; quare angulus FCE &longs;imiliter gra<lb/>uitatis non e&longs;t cau&longs;a? </s>

<s id="id.2.1.33.4.1.5.0"> Huius autem rei eam in medium rationem <lb/>afferre videntur, quoniam CG e&longs;t meta, & CF trutina. </s>

Line 754

Line 794

<s id="id.2.1.35.1.1.1.0"> Sit eadem libra AB, cu<lb/>ius medium C. &longs;it deinde <lb/>tota FG trutina. </s>

<s id="id.2.1.35.1.1.2.0"> eaq; im<lb/>mobilis exi&longs;tat; quæ libram <lb/>AB in puncto C &longs;u&longs;tineat. </s>

<s id="id.2.1.35.1.1.3.0"> <lb/>moueaturq; libra in DE. & <lb/>quoniam trutina e&longs;t, & &longs;u<lb/>pra, & infra libram, quis <lb/>nam angulus erit cau&longs;a gra<lb/>uitatis, cùm libra DE in <lb/><figure id="id.036.01.055.1.jpg" xlink:href="036/01/055/1.jpg"></figure><expan abbr="eod&etilde;"><lb/>eodem</expan>&longs;emper puncto &longs;u&longs;tineatur? </s>

<s id="id.2.1.35.1.1.3.0"><lb/>moueaturq; libra in DE. </s>

<s>& <lb/>quoniam trutina e&longs;t, & &longs;u<lb/>pra, & infra libram, quis <lb/>nam angulus erit cau&longs;a gra<lb/>uitatis, cùm libra DE in <lb/><figure id="id.036.01.055.1.jpg" xlink:href="036/01/055/1.jpg"></figure><expan abbr="eod&etilde;"><lb/>eodem</expan> &longs;emper puncto &longs;u&longs;tineatur? </s>

<s id="id.2.1.35.1.1.4.0"> dicent for&longs;an, &longs;i trutina à potentia <lb/>in F &longs;u&longs;titencatur, tunc CG erit tanquam meta, & angulus <lb/>DCG grauitatis erit cau&longs;a. </s>

<s id="id.2.1.35.1.1.5.0"> &longs;i verò &longs;u&longs;tineatur in G, tunc FCE <lb/>erit cau&longs;a grauitatis, CF verò tanquam meta erit. </s>

<s id="id.2.1.35.1.1.6.0"> cuius quidem <lb/>rei nulla videtur e&longs;&longs;e cau&longs;a, ni&longs;i immaginaria. </s>

Line 816

Line 857

<s id="id.2.1.39.3.1.1.0"> Sit libra AB horizonti <lb/>æquidi&longs;tans, cuius centrum <lb/>C &longs;it &longs;upra libram, perpen<lb/>diculumq; &longs;it CD. &longs;intq; in <lb/>AB ponderum æqualium <lb/>centra grauitatis po&longs;ita: mo<lb/>taq; &longs;it libra in EF. </s>

<s id="id.2.1.39.3.1.1.0.a"> Dico <lb/>pondus in E maiorem ha<lb/>bere grauitatem, quàm pon<lb/>dus in F. & ob id libram <lb/>EF in AB redire. </s>

<s id="id.2.1.39.3.1.1.0.a">Dico <lb/>pondus in E maiorem ha<lb/>bere grauitatem, quàm pon<lb/>dus in F. </s>

<s>& ob id libram <lb/>EF in AB redire. </s>

<s id="id.2.1.39.3.1.2.0"> Produ<lb/>catur primùm CD v&longs;q; ad <lb/>mundi <expan abbr="centr&utilde;">centrum</expan>, quod &longs;it S. de <lb/>inde AC CB EC CF HS <lb/><expan abbr="cõnectantur">connectantur</expan>, à puncti&longs;q; EF <lb/>ip&longs;i HS æquidi&longs;tantes du<lb/>cantur Ek GFL. </s>

<s id="id.2.1.39.3.1.2.0.a"> Quoniam <lb/>igitur naturalis de&longs;cen&longs;us re<lb/>ctus totius magnitudinis, <lb/>libræ &longs;cilicet EF &longs;ic con&longs;ti<lb/>tutæ vná cum ponderibus, <lb/>e&longs;t &longs;cundùm grauitatis cen<lb/>trum H per rectam HS; erit <lb/><figure id="id.036.01.059.1.jpg" xlink:href="036/01/059/1.jpg"></figure><lb/>quoq; ponderum in EF ita po&longs;sitorum de&longs;cen&longs;us &longs;ecundùm re<lb/>ctas Ek FL ip&longs;i HS parallelas; &longs;icuti &longs;upra demon&longs;trauimus. </s>

<s id="id.2.1.39.3.1.2.0.a">Quoniam <lb/>igitur naturalis de&longs;cen&longs;us re<lb/>ctus totius magnitudinis, <lb/>libræ &longs;cilicet EF &longs;ic con&longs;ti<lb/>tutæ vná cum ponderibus, <lb/>e&longs;t <expan abbr="&longs;cundùm">secundum</expan> grauitatis cen<lb/>trum H per rectam HS; erit <lb/><figure id="id.036.01.059.1.jpg" xlink:href="036/01/059/1.jpg"></figure><lb/>quoq; ponderum in EF ita po&longs;sitorum de&longs;cen&longs;us &longs;ecundùm re<lb/>ctas Ek FL ip&longs;i HS parallelas; &longs;icuti &longs;upra demon&longs;trauimus. </s>

<s id="id.2.1.39.3.1.3.0"> <pb xlink:href="036/01/060.jpg"/>De&longs;cen&longs;us igitur, & a&longs;cen<lb/>&longs;us ponderum in EF ma<lb/>gis, minu&longs;uè obliquus di<lb/>cetur &longs;ecundùm acce&longs;&longs;um, <lb/>& rece&longs;&longs;um iuxta lineas Ek <lb/>FL de&longs;ignatum. </s>

<s id="id.2.1.39.3.1.4.0"> <expan abbr="Quoniã">Quoniam</expan>au<lb/><expan abbr="t&etilde;">tem</expan>duo latera AD DC duo<lb/>bus lateribus BD DE &longs;unt <lb/>æqualia; anguliq; ad D &longs;unt <lb/><arrow.to.target n="note65"></arrow.to.target>recti; erit latus AC lateri <lb/>CB æquale. </s>

<s id="id.2.1.39.3.1.4.0"><expan abbr="Quoniã">Quoniam</expan> <expan abbr="aut&etilde;">au<lb/>tem</expan> duo latera AD DC duo<lb/>bus lateribus BD DE &longs;unt <lb/>æqualia; anguliq; ad D &longs;unt <lb/><arrow.to.target n="note65"></arrow.to.target>recti; erit latus AC lateri <lb/>CB æquale. </s>

<s id="id.2.1.39.3.1.5.0"> & cùm pun<lb/>ctum C &longs;it immobile; dum <lb/>puncta AB mouentur, cir<lb/>culi circumferentiam de&longs;cri<lb/>bent, cuius &longs;emidiameter <lb/>erit AC. quare centro C, <lb/>circulus de&longs;cribatur AEBF. <lb/>puncta AB EF in circuli <lb/>circumferentia erunt. </s>

<s id="id.2.1.39.3.1.6.0"> &longs;ed <lb/>cùm EF &longs;it ip&longs;i AB æqua <lb/><arrow.to.target n="note66"></arrow.to.target>lis; erit circumferentia <lb/>EAF circumferentiæ AFB <lb/>æqualis. </s>

<s id="id.2.1.39.3.1.7.0"> quare dempta <lb/><figure id="id.036.01.060.1.jpg" xlink:href="036/01/060/1.jpg"></figure><lb/>communi AF, erit circumferentia EA circumferentiæ FB æqua <lb/>lis. </s>

<s id="id.2.1.39.3.1.8.0"> Quoniam autem mixtus angulus CEA e&longs;t æqualis mixto <lb/>CFB; & HFB ip&longs;o CFB e&longs;t maior; angulus verò HEA ip&longs;o <lb/>CEA minor; erit angulus HFB angulo HEA maior. </s>

<s id="id.2.1.39.3.1.9.0"> à quibus <lb/><arrow.to.target n="note67"></arrow.to.target>&longs;i auferantur anguli HFG HEk æquales; erit angulus GFB an <lb/>gulo kEA maior. </s>

<s id="id.2.1.39.3.1.10.0"> ergo de&longs;cen&longs;us ponderis in E minus obliquus <lb/>erit a&longs;cen&longs;u ponderis in F. & quamquam pondus in E de&longs;cen<lb/>dendo, & pondus in F a&longs;cendendo per circumferentias mouean<lb/>tur æquales; quia tamen pondus in E ex hoc loco rectius de&longs;cen<lb/>dit, quàm pondus in F a&longs;cendit: idcirco naturalis potentia pon<lb/>deris in E re&longs;i&longs;tentiam violentiæ ponderis F &longs;uperabit. </s>

<s id="id.2.1.39.3.1.10.0">ergo de&longs;cen&longs;us ponderis in E minus obliquus <lb/>erit a&longs;cen&longs;u ponderis in F. </s>

<s>& quamquam pondus in E de&longs;cen<lb/>dendo, & pondus in F a&longs;cendendo per circumferentias mouean<lb/>tur æquales; quia tamen pondus in E ex hoc loco rectius de&longs;cen<lb/>dit, quàm pondus in F a&longs;cendit: idcirco naturalis potentia pon<lb/>deris in E re&longs;i&longs;tentiam violentiæ ponderis F &longs;uperabit. </s>

<s id="id.2.1.39.3.1.11.0"> quare <lb/>maiorem grauitatem habebit pondus in E, quàm pondus in F. </s>

<s id="id.2.1.39.3.1.11.0.a"> <lb/>ergo pondus in E deor&longs;um, pondus verò in F &longs;ur&longs;um mouebitur: <pb n="24" xlink:href="036/01/061.jpg"/>donec libra EF in AB redeat. </s>

<s id="id.2.1.39.3.1.12.0"> quod demon&longs;trare oportebat. </s>

Line 839

Line 882

<s id="id.2.1.41.1.1.1.0"> Huius autem effectus ratio ab Ari&longs;totele po&longs;ita, hic manife&longs;ta in <arrow.to.target n="note68"></arrow.to.target><lb/>tueri pote&longs;t. </s>

<s id="id.2.1.41.1.1.2.0"> &longs;it enim punctum N vbi CS EF &longs;e inuicem &longs;ecant. </s>

<s id="id.2.1.41.1.1.3.0"> <lb/>& quoniam HE e&longs;t ip&longs;i HF æqualis; erit NE maior NF. li<lb/>nea ergo CS, quam perpendiculum vocat, libram EF in partes di <lb/>uidet inæquales. </s>

<s id="id.2.1.41.1.1.3.0"><lb/>& quoniam HE e&longs;t ip&longs;i HF æqualis; erit NE maior NF. </s>

<s>li<lb/>nea ergo CS, quam perpendiculum vocat, libram EF in partes di<lb/>uidet inæquales. </s>

<s id="id.2.1.41.1.1.4.0"> cùm itaq; pars libræ NE &longs;it maior NF; atq; id, <lb/>quod plus e&longs;t, nece&longs;&longs;e e&longs;t, deor&longs;um ferri: libra ergo EF ex parte E <lb/>deor&longs;um mouebitur, donec in AB redeat. </s>

Line 852

Line 896

<s id="id.2.1.43.1.1.4.0"> &longs;i igitur centro C, &longs;pa<lb/>tioq; CD, circulus de&longs;cri<lb/>batur DHM; erunt pun<lb/>cta DH in circuli circum<lb/>ferentia. </s>

<s id="id.2.1.43.1.1.5.0"> Quoniam au<lb/>tem CH ip&longs;i EF e&longs;t per<lb/>pendicularis; continget li<lb/>nea EHS circulum DHM <lb/>in puncto H. </s>

<s id="id.2.1.43.1.1.5.0.a"> pondus igi<lb/>tur in H (&longs;icuti &longs;upra de<lb/>mon&longs;trauimus) grauius <lb/><figure id="id.036.01.061.1.jpg" xlink:href="036/01/061/1.jpg"></figure><lb/>erit, quàm in alio &longs;itu circuli DHM. </s>

<s id="id.2.1.43.1.1.5.0.b"> ergo magnitudo ex EF <lb/>ponderibus, & libra EF compo&longs;ita, cuius centrum grauitatis e&longs;t <lb/>in H, in hoc &longs;itu magis grauitabit, quàm in quocunq; alio &longs;itu <pb xlink:href="036/01/062.jpg"/>circuli fuerit punctum H. <lb/>ab hoc igitur &longs;itu velo<lb/>cius, quàm à quocunq; <lb/>alio mouebitur. </s>

<s id="id.2.1.43.1.1.5.0.b">ergo magnitudo ex EF <lb/>ponderibus, & libra EF compo&longs;ita, cuius centrum grauitatis e&longs;t <lb/>in H, in hoc &longs;itu magis grauitabit, quàm in quocunq; alio &longs;itu <pb xlink:href="036/01/062.jpg"/>circuli fuerit punctum H. <lb/></s>

<s>ab hoc igitur &longs;itu velo<lb/>cius, quàm à quocunq; <lb/>alio mouebitur. </s>

<s id="id.2.1.43.1.1.6.0"> & &longs;i H <lb/>propius fuerit ip&longs;i D mi <lb/>nus grauitabit, minu&longs;q; <lb/>ab eo &longs;itu mouebitur. </s>

<s id="id.2.1.43.1.1.7.0"> <lb/>&longs;emper enim de&longs;cen&longs;us <lb/>obliquior e&longs;t, & minus re<lb/>ctus. </s>

<s id="id.2.1.43.1.1.8.0"> libra ergo EF velo<lb/>cius ab hoc &longs;itu mouebi<lb/>tur, quàm ab alio &longs;itu. </s>

Line 898

Line 943

<s id="id.2.1.47.4.1.2.0"> Ducantur <lb/>à punctis EF ip&longs;i AB <lb/>perpendiculares EL FM, <lb/>quæ inter &longs;e æquidi&longs;tan<lb/>tes <arrow.to.target n="note71"></arrow.to.target><figure id="id.036.01.065.1.jpg" xlink:href="036/01/065/1.jpg"></figure>erunt; &longs;itq; punctum N, vbi AB EF &longs;e inuicem &longs;ecant. </s>

<s id="id.2.1.47.4.1.3.0"> <lb/>Quoniam igitur angulus FNM e&longs;t æqualis angulo ENL, & an<lb/>gulus <arrow.to.target n="note72"></arrow.to.target>F MN rectus recto ELN æqualis, ac reliquus NFM reli<lb/>quo <arrow.to.target n="note73"></arrow.to.target>NEL e&longs;t etiam æqualis; erit triangulum NLE triangu<lb/>lo NMF &longs;imile. </s>

<s id="id.2.1.47.4.1.4.0"> vt igitur NE ad EL, ita NF ad FM; & per <arrow.to.target n="note74"></arrow.to.target><lb/>mutando vt EN ad NF, ita EL ad FM. &longs;ed cùm &longs;it HE ip&longs;i <arrow.to.target n="note75"></arrow.to.target><lb/>HF æqualis, erit EN maior NF; quare & EL maior erit FM. </s>

<s id="id.2.1.47.4.1.4.0.a"> <lb/>& quoniam dum pondus in E per circumferentiiam EA de&longs;cendit, <lb/>pondus in F per circumferentiam FB ip&longs;i circumferentiæ EA <lb/>æqualem a&longs;cendit; de&longs;cen&longs;u&longs;q; ponderis in E de directo (vt ip<lb/>&longs;i dicunt) capit EL: a&longs;cen&longs;us verò ponderis in F de directo ca<lb/>pit FM; minus de directo capiet a&longs;cen&longs;us ponderis in F, quàm <lb/>de&longs;cen&longs;us ponderis in E. maiorem igitur grauitatem habebit pon<lb/>dus in E, quàm pondus in F. </s>

<s id="id.2.1.47.4.1.4.0.a"><lb/>& quoniam dum pondus in E per <expan abbr="circumferentiiam">circumferentiam</expan> EA de&longs;cendit, <lb/>pondus in F per circumferentiam FB ip&longs;i circumferentiæ EA <lb/>æqualem a&longs;cendit; de&longs;cen&longs;u&longs;q; ponderis in E de directo (vt ip<lb/>&longs;i dicunt) capit EL: a&longs;cen&longs;us verò ponderis in F de directo ca<lb/>pit FM; minus de directo capiet a&longs;cen&longs;us ponderis in F, quàm <lb/>de&longs;cen&longs;us ponderis in E. maiorem igitur grauitatem habebit pon<lb/>dus in E, quàm pondus in F. </s>

<s id="id.2.1.48.1.1.1.0"> <margin.target id="note71"></margin.target>28 <emph type="italics"/>Primi.<emph.end type="italics"/> </s>

Line 909

Line 954

<s id="id.2.1.49.1.1.1.0"> Producatur CD ex vtraq; parte in OP, quæ lineam EF in <lb/>puncto S &longs;ecet. </s>

<s id="id.2.1.49.1.1.2.0"> & quoniam (vt aiunt) quò magis pondus à li<lb/>nea directionis OP di&longs;tat, eò fit grauius; idcirco hoc quoq; me <lb/>dio pondus in E maiorem habere grauitauitatem pondere in F o<lb/>&longs;tendetur. </s>

<s id="id.2.1.49.1.1.2.0">& quoniam (vt aiunt) quò magis pondus à li<lb/>nea directionis OP di&longs;tat, eò fit grauius; idcirco hoc quoq; me <lb/>dio pondus in E maiorem habere <expan abbr="grauitauitatem">grauitatem</expan> pondere in F o<lb/>&longs;tendetur. </s>

<s id="id.2.1.49.1.1.3.0"> Ducantur à punctis EF ip&longs;i OP perpendiculares EQ <lb/>FR. &longs;imili ratione o&longs;tendetur, triangulum QES triangulo RFS <lb/>&longs;imile e&longs;&longs;e; lineamq; EQ ip&longs;a RF maiorem e&longs;&longs;e. </s>

<s id="id.2.1.49.1.1.4.0"> pondus itaq; <lb/>in E magis à linea OP di&longs;tabit, quàm pondus in F; ac propterea <lb/>pondus in E maiorem habebit grauitatem pondere in F. ex quibus <lb/>reditus libræ EF in AB manife&longs;tus apparet. </s>

<s id="id.2.1.49.3.1.1.0"> Si autem centrum libræ <lb/>&longs;it infra libram, tunc pon<lb/>dus depre&longs;&longs;um maiorem <lb/>habere grauitatem eleuato <lb/>ii&longs;dem mediis o&longs;tendetur. </s>

<s id="id.2.1.49.3.1.2.0"> <lb/>ducantur à punctis EF ip<lb/>&longs;i AB perpendiculares EL <lb/>FM. &longs;imiliter demon&longs;tra<lb/>bitur EL maiorem e&longs;&longs;e <lb/>FM; & ob id de&longs;cen&longs;us <lb/>ponderis in F minus de di <lb/>recto capiet, quàm a&longs;cen<lb/><figure id="id.036.01.066.1.jpg" xlink:href="036/01/066/1.jpg"></figure><lb/>&longs;us ponderis in E: quocirca re&longs;i&longs;tentia violentiæ ponderis in E &longs;u<lb/>perabit naturalem propen&longs;ionem ponderis in F. ergo pondus in E <lb/>pondere in F grauius erit. </s>

<s id="id.2.1.49.3.1.2.0"><lb/>ducantur à punctis EF ip<lb/>&longs;i AB perpendiculares EL <lb/>FM. </s>

<s>&longs;imiliter demon&longs;tra<lb/>bitur EL maiorem e&longs;&longs;e <lb/>FM; & ob id de&longs;cen&longs;us <lb/>ponderis in F minus de di <lb/>recto capiet, quàm a&longs;cen<lb/><figure id="id.036.01.066.1.jpg" xlink:href="036/01/066/1.jpg"></figure><lb/>&longs;us ponderis in E: quocirca re&longs;i&longs;tentia violentiæ ponderis in E &longs;u<lb/>perabit naturalem propen&longs;ionem ponderis in F. </s>

<s>ergo pondus in E <lb/>pondere in F grauius erit. </s>

<s id="id.2.1.49.4.1.1.0"> Producatur etiam CD ex vtraq; parte in OP; ip&longs;iq; à punctis <lb/>EF perpendiculares ducantur EQ FR. eodem pror&longs;us modo <lb/>o&longs;tendetur, lineam EQ maiorem e&longs;&longs;e FR. pondus ideò in E ma<lb/>gis à linea directionis OP di&longs;tabit, quàm pondus in F. maio<lb/>rem igitur grauitatem habebit pondus in E, quàm pondus in F. <lb/>ex quibus &longs;equitur, libram EF ex parte E deor&longs;um moueri. </s>

<s id="id.2.1.49.4.1.1.0">Producatur etiam CD ex vtraq; parte in OP; ip&longs;iq; à punctis <lb/>EF perpendiculares ducantur EQ FR. </s>

<s>eodem pror&longs;us modo <lb/>o&longs;tendetur, lineam EQ maiorem e&longs;&longs;e FR. </s>

<s>pondus ideò in E ma<lb/>gis à linea directionis OP di&longs;tabit, quàm pondus in F. </s>

<s>maio<lb/>rem igitur grauitatem habebit pondus in E, quàm pondus in F. <lb/></s>

<s>ex quibus &longs;equitur, libram EF ex parte E deor&longs;um moueri. </s>

<s id="id.2.1.49.5.1.1.0"> Ari&longs;toteles itaq; has duas tantùm quæ&longs;tiones propo&longs;uit, ter<lb/>tiamq; reliquit; &longs;cilicet cùm centrum libræ in ip&longs;a e&longs;t libra: hanc <lb/>autem ommi&longs;sit, vt notam, quemadmodum res valde notas præ<lb/>termittere &longs;olet. </s>

<s id="id.2.1.49.5.1.2.0"> nam cui dubium, &longs;i pondus in eius centro gra<lb/>uitatis &longs;u&longs;tineatur, quin maneat? </s>

<s id="id.2.1.49.5.1.3.0"> Ea verò, quæ ex ip&longs;ius &longs;enten<lb/>tia attulimus, aliquis reprehendere po&longs;&longs;et, nos integram eius &longs;enten<lb/>tiam minimè protuli&longs;&longs;e affimans. </s>

<s id="id.2.1.49.5.1.3.0">Ea verò, quæ ex ip&longs;ius &longs;enten<lb/>tia attulimus, aliquis reprehendere po&longs;&longs;et, nos integram eius &longs;enten<lb/>tiam minimè protuli&longs;&longs;e <expan abbr="affimans">affirmans</expan>. </s>

<s id="id.2.1.49.5.1.4.0"> nam cùm in &longs;ecunda parte &longs;e<lb/>cundæ quæ&longs;tionis proponit, cur libra, trutina deor&longs;um con&longs;tituta, <lb/>quando deor&longs;um lato pondere qui&longs;piam id amouet, non a&longs;cen<lb/>dit, &longs;ed manet? </s>

<s id="id.2.1.49.5.1.5.0"> non a&longs;&longs;erit adhuc libram deor&longs;um moueri; &longs;ed <lb/>manere. </s>

<s id="id.2.1.49.5.1.6.0"> quod in vltima quoq; conclu&longs;ione colligi&longs;&longs;e videtur. </s>

Line 934

Line 985

<s id="id.2.1.49.7.1.1.0"> Sit enim libra AB <lb/>horizonti æquidi&longs;tans, <lb/>cuius centrum E &longs;it <lb/>infra libram. </s>

<s id="id.2.1.49.7.1.2.0"> quia ve <lb/>rò Ari&longs;toteles libram, <lb/>&longs;icuti actu e&longs;t, con&longs;ide<lb/>rat; ideò nece&longs;&longs;e e&longs;t <lb/>trutinam, vel aliquid <lb/>aliud infra centrum E <lb/>collocare, vt EF <lb/>(quod quidem truti<lb/>na erit) ita vt centrum <lb/>E &longs;u&longs;tineat. </s>

<s id="id.2.1.49.7.1.3.0"> &longs;itq; per<lb/><figure id="id.036.01.067.1.jpg" xlink:href="036/01/067/1.jpg"></figure><lb/>pendiculum ECD. & vt libra AB ab hoc moueatur &longs;itu; dicit <lb/>Ari&longs;toteles, ponatur pondus in B, quod cùm &longs;it graue, libram ex <lb/>parte B deor&longs;um mouebit; putá in G. ita vt propter impedimen<lb/>tum deor&longs;um amplius moueri non poterit. </s>

<s id="id.2.1.49.7.1.3.0">&longs;itq; per<lb/><figure id="id.036.01.067.1.jpg" xlink:href="036/01/067/1.jpg"></figure><lb/>pendiculum ECD. </s>

<s>& vt libra AB ab hoc moueatur &longs;itu; dicit <lb/>Ari&longs;toteles, ponatur pondus in B, quod cùm &longs;it graue, libram ex <lb/>parte B deor&longs;um mouebit; putá in G. </s>

<s>ita vt propter impedimen<lb/>tum deor&longs;um amplius moueri non poterit. </s>

<s id="id.2.1.49.7.1.4.0"> non enim dicit Ari<lb/>&longs;toteles, moueatur libra ex parte B deor&longs;um, quou&longs;q; libuerit; dein <lb/>de relinquatur, vt nos diximus: &longs;ed præcipit, vt in ip&longs;o B po<lb/>natur pondus, quod ex ip&longs;ius natura deor&longs;um &longs;emper mouebi<lb/>tur; donec libra trutinæ, &longs;iue alicui alii adhæreat. </s>

<s id="id.2.1.49.7.1.5.0"> & quando B erit <lb/>in G, erit libra in GH; in quo &longs;itu, ablato pondere, manebit: <lb/>cùm maior pars libræ à perpendiculo &longs;it ver&longs;us G, quæ e&longs;t DG, <lb/>quàm DH. </s>

<s id="id.2.1.49.7.1.5.0.a"> nec deor&longs;um amplius mouebitur; nam libra, vel <lb/>trutinæ, vel alteri cuipiam, quod centrum libræ &longs;u&longs;tineat, incum<lb/>bet. </s>

<s id="id.2.1.49.7.1.6.0"> &longs;i enim huic non adhæreret, libra ex parte G deor&longs;um ex <lb/>ip&longs;ius &longs;ententia moueretur; cùm id, quod plus e&longs;t, &longs;cilicet DG, <lb/>deor&longs;um ferri &longs;it nece&longs;&longs;e. </s>

<s id="id.2.1.49.8.1.1.0"> Cæterum quis adhuc dicere poterit, &longs;i paruum imponatur pon<lb/>dus in B, mouebitur quidem libra deor&longs;um, non autem v&longs;q; ad <lb/>G. in quò &longs;itu &longs;ecundùm Ari&longs;totelem, ablato pondere, mane<lb/>re deberet. </s>

<s id="id.2.1.49.8.1.1.0">Cæterum quis adhuc dicere poterit, &longs;i paruum imponatur pon<lb/>dus in B, mouebitur quidem libra deor&longs;um, non autem v&longs;q; ad <lb/>G. </s>

<s>in quò &longs;itu &longs;ecundùm Ari&longs;totelem, ablato pondere, mane<lb/>re deberet. </s>

<s id="id.2.1.49.8.1.2.0"> quod experimento patet; cùm in vna tantùm libræ <lb/>extremitate, impo&longs;ito onere, hocq; vel maiore, vel minore, libra <lb/>plus, minu&longs;uè inclinetur. </s>

<s id="id.2.1.49.8.1.3.0"> Quod e&longs;t quidem veri&longs;&longs;imum, centro &longs;upra <lb/>libram, non autem infra, neq; in ip&longs;a libra collocato. </s>

<s id="id.2.1.49.8.1.4.0"> Vt exempli <lb/>gratia. </s>

Line 950

Line 1004

<s id="id.2.1.49.10.1.1.0"> Sit libra horizonti æ<lb/>quidi&longs;tans AB, cuius cen<lb/>trum C &longs;it &longs;upra libram, <lb/>perpendiculumq; CD ho<lb/>rizonti perpendiculare, <lb/>quod ex parte D produca<lb/>tur in H. </s>

<s id="id.2.1.49.10.1.1.0.a"> Quoniam enim <lb/>con&longs;iderata libræ grauita<lb/>te, erit punctum D libræ <lb/>centrum grauitatis. </s>

<s id="id.2.1.49.10.1.2.0"> &longs;i ergo <lb/>in B paruum imponatur <lb/>pondus, cuius centrum <lb/><figure id="id.036.01.068.1.jpg" xlink:href="036/01/068/1.jpg"></figure><lb/>grauitatis &longs;it in puncto B; magnitudinis ex libra AB, & pondere <lb/>in B compo&longs;itæ non erit amplius centrum grauitatis D; &longs;ed erit in <lb/><arrow.to.target n="note76"></arrow.to.target>linea DB, vt in E: ita vt DE ad EB &longs;it, vt pondus in B ad gra<lb/>uitatem libræ AB. Connectatur CE. </s>

<s id="id.2.1.49.10.1.2.0">&longs;i ergo <lb/>in B paruum imponatur <lb/>pondus, cuius centrum <lb/><figure id="id.036.01.068.1.jpg" xlink:href="036/01/068/1.jpg"></figure><lb/>grauitatis &longs;it in puncto B; magnitudinis ex libra AB, & pondere <lb/>in B compo&longs;itæ non erit amplius centrum grauitatis D; &longs;ed erit in <lb/><arrow.to.target n="note76"></arrow.to.target>linea DB, vt in E: ita vt DE ad EB &longs;it, vt pondus in B ad gra<lb/>uitatem libræ AB. </s>

<s id="id.2.1.49.10.1.2.0.a"> Quoniam autem pun<lb/>ctum Ce&longs;t immobile, dum libra mouetur, punctum E circuli cir<lb/>cumferentiam EFG de&longs;cribet, cuius &longs;emidiameter CE, & cen<lb/>trum C. quia verò CD horizonti e&longs;t perpendicularis, linea CE <lb/>horizonti perpendicularis nequaquam erit. </s>

<s>Connectatur CE. </s>

<s>quia verò CD horizonti e&longs;t perpendicularis, linea CE <lb/>horizonti perpendicularis nequaquam erit. </s>

<s id="id.2.1.49.10.1.3.0"> quare magnitudo ex <lb/>AB, & pondere in B compo&longs;ita minimè in hoc &longs;itu manebit; &longs;ed <lb/><arrow.to.target n="note77"></arrow.to.target>deor&longs;um &longs;ecundùm eius grauitatis centrum E per circumferen<lb/>tiam EFG mouebitur; donec CE horizonti perpendicularis eua<lb/>dat; hoc e&longs;t, donec CE in CDF perueniat. </s>

<s id="id.2.1.49.10.1.4.0"> atq; tunc libra AB <lb/>mota erit in kL, in quo &longs;itu libra vná cum pondere manebit. </s>

<s id="id.2.1.49.10.1.5.0"> nec <lb/>deor&longs;um amplius mouebitur. </s>

<s id="id.2.1.49.10.1.6.0"> Si verò in B ponatur pondus graui<lb/>us; centrum grauitatis totius magnitudinis erit ip&longs;i B propius, vt in <lb/>M. & tunc libra deor&longs;um, donec iuncta CM in linea CDH per <lb/>ueniat, mouebitur. </s>

<s id="id.2.1.49.10.1.6.0">Si verò in B ponatur pondus graui<lb/>us; centrum grauitatis totius magnitudinis erit ip&longs;i B propius, vt in <lb/>M. </s>

<s>& tunc libra deor&longs;um, donec iuncta CM in linea CDH per <lb/>ueniat, mouebitur. </s>

<s id="id.2.1.49.10.1.7.0"> Ex maiore igitur, & minore pondere in B po<lb/>&longs;ito, libra plus, minu&longs;uè inclinabitur. </s>

<s id="id.2.1.49.10.1.8.0"> ex quo &longs;equitur pondus B <lb/>quarta circuli parte minorem &longs;emper circumferentiam de&longs;cribe<lb/>re, cùm angulus FCE &longs;it &longs;emper acutus. </s>

<s id="id.2.1.49.10.1.9.0"> nunquam enim punctum <lb/>B v&longs;q; ad lineam CH perueniet, cùm centrum grauitatis ponde<lb/>ris, & libræ &longs;imul &longs;emper inter DB exi&longs;tat. </s>

Line 968

Line 1025

<s id="id.2.1.51.1.2.1.0"> Habeat autem libra AB <lb/>centrum C in ip&longs;a libra, atq; <lb/>in eius medio: erit C libræ <lb/>centrum quoq; grauitatis; <lb/>à quo ip&longs;i AB, horizontiq; <lb/>perpendicularis ducatur FC <lb/>G. ponatur deinde in B <lb/>quoduis pondus; erit totius <lb/>magnitudinis centrum gra<lb/>uitatis putá in E; ita vt CE <lb/><figure id="id.036.01.069.1.jpg" xlink:href="036/01/069/1.jpg"></figure><lb/>ad EB &longs;it, vt pondus in B ad libræ grauitatem. </s>

<s>ponatur deinde in B <lb/>quoduis pondus; erit totius <lb/>magnitudinis centrum gra<lb/>uitatis putá in E; ita vt CE <lb/><figure id="id.036.01.069.1.jpg" xlink:href="036/01/069/1.jpg"></figure><lb/>ad EB &longs;it, vt pondus in B ad libræ grauitatem. </s>

<s id="id.2.1.51.1.2.2.0"> & quoniam CE <lb/>non e&longs;t horizonti perpendicularis, libra AB, atq; pondus in B <lb/>in hoc &longs;itu nunquam manebunt; &longs;ed deor&longs;um ex parte B mouebun<lb/>tur, donec CE horizonti fiat perpendicularis. </s>

<s id="id.2.1.51.1.2.3.0"> hoc e&longs;t donec li<lb/>bra AB in FG perueniat. </s>

<s id="id.2.1.51.1.2.4.0"> ex quo patet, quolibet pondus in B <lb/>circuli quartam &longs;emper de&longs;cribere. </s>

<s id="id.2.1.51.2.1.1.0"> Sit autem centrum Cin<lb/>fra libram AB. &longs;itq; DCE <lb/>perpendiculum. </s>

<s id="id.2.1.51.2.1.1.0">Sit autem centrum C in<lb/>fra libram AB. </s>

<s>&longs;itq; DCE <lb/>perpendiculum. </s>

<s id="id.2.1.51.2.1.2.0"> &longs;imiliter <lb/>po&longs;ito in B pondere, cen<lb/>trum grauitatis magnitudi<lb/>nis ex AB libra, & ponde<lb/>re in B compo&longs;itæ in linea <lb/>DB erit; vt in F; ita vt DF <lb/>ad FB &longs;it, vt pondus in B <lb/><figure id="id.036.01.069.2.jpg" xlink:href="036/01/069/2.jpg"></figure><lb/>ad libræ pondus. </s>

<s id="id.2.1.51.2.1.3.0"> Iungatur CF. & quoniam CD horizonti e&longs;t <lb/>perpendicularis; linea CF horizonti nequaquam perpendicula<lb/>ris exi&longs;tet. </s>

<s id="id.2.1.51.2.1.3.0">Iungatur CF. </s>

<s>& quoniam CD horizonti e&longs;t <lb/>perpendicularis; linea CF horizonti nequaquam perpendicula<lb/>ris exi&longs;tet. </s>

<s id="id.2.1.51.2.1.4.0"> quare magnitudo ex AB libra, ac pondere in B com<lb/>po&longs;ita in hoc &longs;itu nunquam per&longs;i&longs;tet; &longs;ed deor&longs;um, ni&longs;i aliquid <lb/>impediat, mouebitur; donec CF in DCE perueniat: in quo &longs;itu <lb/>libra vná cum pondere manebit. </s>

<s id="id.2.1.51.2.1.5.0"> & punctum B erit vt in G, atq; <lb/>punctum A in H, libraq; GH non amplius centrum infra, &longs;ed &longs;u<lb/>pra ip&longs;am habebit. </s>

<s id="id.2.1.51.2.1.6.0"> quod idem &longs;emper eueniet; quamuis mini<lb/>mum imponatur pondus in B. ergo priu&longs;quam B perueniat ad <lb/>G; nece&longs;&longs;e e&longs;t libram, &longs;iue trutinæ deor&longs;um po&longs;itæ, vel alicui <pb xlink:href="036/01/070.jpg"/>alteri, quod centrum C &longs;u<lb/>&longs;tineat, occurrere; ibiq; ad<lb/>hærere. </s>

<s id="id.2.1.51.2.1.6.0">quod idem &longs;emper eueniet; quamuis mini<lb/>mum imponatur pondus in B. </s>

<s>ergo priu&longs;quam B perueniat ad <lb/>G; nece&longs;&longs;e e&longs;t libram, &longs;iue trutinæ deor&longs;um po&longs;itæ, vel alicui <pb xlink:href="036/01/070.jpg"/>alteri, quod centrum C &longs;u<lb/>&longs;tineat, occurrere; ibiq; ad<lb/>hærere. </s>

<s id="id.2.1.51.2.1.7.0"> ex hoc &longs;equitur, pon<lb/>dus in B vltra lineam Dk <lb/>&longs;emper moueri; ac circuli <lb/>quarta maiorem &longs;emper cir<lb/><expan abbr="cumfer&etilde;tiam">cumferentiam</expan>de&longs;cribere: e&longs;t <lb/>enim angulus FCE &longs;emper <lb/>obtu&longs;us, cùm angulus DCF <lb/>&longs;emper &longs;it acutus. </s>

<s id="id.2.1.51.2.1.8.0"> quò au<lb/><figure id="id.036.01.070.1.jpg" xlink:href="036/01/070/1.jpg"></figure><lb/>tem pondus in B fuerit leuius, maiorem tamen adhuc circumfe<lb/>rentiam de&longs;cribet. </s>

<s id="id.2.1.51.2.1.9.0"> nam quò pondus in G leuius fuerit, eò ma<lb/>gis pondus in G eleuabitur; libraq; GH ad &longs;itum horizonti æqui<lb/>di&longs;tantem propius accedet. </s>

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<s id="id.2.1.51.3.1.5.0"> & quamquam libra brachia habeat inæqualia, idem eue<lb/>niet; eodemq; pro&longs;us modo o&longs;tendetur, centrum libræ diuer&longs;imo <lb/>dè collocatum varios producere effectus. </s>

<s id="id.2.1.51.4.1.1.0"> Sit enim libra AB hori<lb/>zonti æquidi&longs;tans; & in AB <lb/>&longs;int pondera inæqualia, quo <lb/>rum grauitatis centrum &longs;it <lb/>C: &longs;u&longs;pendaturq; libra in <lb/>eodem puncto C. & mo<lb/>ueatur libra in DE. </s>

<s id="id.2.1.51.4.1.1.0">Sit enim libra AB hori<lb/>zonti æquidi&longs;tans; & in AB <lb/>&longs;int pondera inæqualia, quo <lb/>rum grauitatis centrum &longs;it <lb/>C: &longs;u&longs;pendaturq; libra in <lb/>eodem puncto C. </s>

<s>& mo<lb/>ueatur libra in DE. </s>

<s id="id.2.1.51.4.1.1.0.a"> mani <lb/><arrow.to.target n="note78"></arrow.to.target>fe&longs;tum e&longs;t libram non &longs;o<lb/>lum in DE, &longs;ed in quouis <lb/>alio &longs;itu manere. <figure id="id.036.01.070.2.jpg" xlink:href="036/01/070/2.jpg"></figure> </s>

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<s id="id.2.1.51.8.1.1.0"> Ex his determinatis &longs;i libra &longs;it <lb/>arcuata, vel libræ brachia angulum <lb/>con&longs;tituant; centrumq; diuer&longs;imo <lb/>dè collocetur (quamquam hæc pro<lb/>priè non &longs;it libra) varios tamen <lb/>huius quoq; effectus o&longs;tendere pote<lb/>rimus. </s>

<s id="id.2.1.51.8.1.2.0"> Vt &longs;it libra ACB, cuius <lb/>centrum, circa quod vertitur, &longs;it C. <lb/>ductaq; AB, &longs;it arcus &longs;iue angulus <lb/><figure id="id.036.01.071.3.jpg" xlink:href="036/01/071/3.jpg"></figure><lb/>ACB &longs;upra lineam AB; & in AB grauitatis centra ponderum <lb/>ponantur, quæ in hoc &longs;itu maneant. </s>

<s id="id.2.1.51.8.1.2.0">Vt &longs;it libra ACB, cuius <lb/>centrum, circa quod vertitur, &longs;it C. <lb/></s>

<s>ductaq; AB, &longs;it arcus &longs;iue angulus <lb/><figure id="id.036.01.071.3.jpg" xlink:href="036/01/071/3.jpg"></figure><lb/>ACB &longs;upra lineam AB; & in AB grauitatis centra ponderum <lb/>ponantur, quæ in hoc &longs;itu maneant. </s>

<s id="id.2.1.51.8.1.3.0"> moueatur deinde libra ab <pb xlink:href="036/01/072.jpg"/>hoc &longs;itu, putá in ECF. </s>

<s id="id.2.1.51.8.1.3.0.a"> Dico li<lb/>bram ECF in ACB redire. </s>

<s id="id.2.1.51.8.1.4.0"> to<lb/>tius magnitudinis centrum grauita<lb/>tis inueniatur D. & CD iunga<lb/>tur. </s>

<s id="id.2.1.51.8.1.4.0">to<lb/>tius magnitudinis centrum grauita<lb/>tis inueniatur D. </s>

<s>& CD iunga<lb/>tur. </s>

<s id="id.2.1.51.8.1.5.0"> Quoniam enim pondera AB <lb/><arrow.to.target n="note81"></arrow.to.target>manent, linea CD horizonti per<lb/>pendicularis erit. </s>

<s id="id.2.1.51.8.1.6.0"> quando igitur <lb/>libra erit in ECF, linea CD erit <lb/>putá in CG; quæ cùm non &longs;it ho<lb/><figure id="id.036.01.072.1.jpg" xlink:href="036/01/072/1.jpg"></figure><lb/>rizonti perpendicularis; libra ECF in ACB redibit. </s>

<s id="id.2.1.51.8.1.7.0"> quod idem <lb/>eueniet, &longs;i centrum C &longs;upra libram con&longs;tituatur, vt in H. </s>

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<s id="id.2.1.53.10.1.1.0"> Sit AB libra, cuius centrum C; &longs;intq; duo pondera EF ex pun<lb/>ctis BG &longs;u&longs;pen&longs;a: diuidaturq; BG in H, ita vt BH ad HG <lb/>eandem habeat proportionem, quam pondus E ad pondus F. </s>

<s id="id.2.1.53.10.1.1.0.a"> <lb/>Dico pondera EF tàm in BG ponderare, quàm &longs;i vtraq; ex pun<lb/>cto H &longs;u&longs;pendantur. </s>

<s id="id.2.1.53.10.1.2.0"> fiat AC ip&longs;i CH æqualis. </s>

<s id="id.2.1.53.10.1.3.0"> & vt AC ad <lb/>CG, ita fiat pondus E ad pondus L. &longs;imiliter vt AC ad CB, <lb/>ita fiat pondus F ad pondus M. ponderaq; LM ex puncto A &longs;u<lb/>&longs;pendantur. </s>

<s id="id.2.1.53.10.1.3.0">& vt AC ad <lb/>CG, ita fiat pondus E ad pondus L. </s>

<s>&longs;imiliter vt AC ad CB, <lb/>ita fiat pondus F ad pondus M. </s>

<s>ponderaq; LM ex puncto A &longs;u<lb/>&longs;pendantur. </s>

<s id="id.2.1.53.10.1.4.0"> Quoniam enim AC e&longs;t æqualis CH, erit BC ad <lb/>CH vt pondus M ad pondus F. </s>

<s id="id.2.1.53.10.1.4.0.a"> & quoniam maior e&longs;t BC, <lb/>quàm CH; erit & pondus M ip&longs;o F maius. </s>

<s id="id.2.1.53.10.1.5.0"> diuidatur igitur pon<lb/>dus M in duas partes QR, &longs;itq; pars Q ip&longs;i F æqualis; erit BC <lb/><arrow.to.target n="note82"></arrow.to.target>ad CH, vt RQ ad Q: & diuidendo, vt BH ad HC, ita R ad q. <lb/><arrow.to.target n="note83"></arrow.to.target>deinde conuertendo, vt CH ad HB, ita Q ad R. </s>

<s id="id.2.1.53.10.1.5.0.a"> Præterea quo<lb/>niam CH e&longs;t æqualis ip&longs;i CA, erit HC ad CG, vt pondus <lb/>E ad pondus L: maior autem e&longs;t HC, quàm CG; erit & pon<pb n="31" xlink:href="036/01/075.jpg"/>dus E pondere L maius. </s>

<s id="id.2.1.53.10.1.6.0"> diuidatur itaq; pondus E in duas partes <lb/>NO ita, vt pars O &longs;it ip&longs;i L æqualis, erit HC ad CG, vt to<lb/>tum NO ad O; & diuidendo, vt HG ad GC, ita N ad O: <arrow.to.target n="note84"></arrow.to.target><lb/>conuertendoq; vt CG ad GH, ita O ad N. & iterum com<lb/>ponendo, vt CH ad HG, ita ON ad N. vt autem GH <arrow.to.target n="note85"></arrow.to.target><lb/>ad HB, ita e&longs;t F ad ON. quare ex æquali, vt CH ad HB, ita F <arrow.to.target n="note86"></arrow.to.target><lb/>ad N. &longs;ed vt CH ad HB ita e&longs;t Q ad R: erit igitur Q ad R, vt <arrow.to.target n="note87"></arrow.to.target><lb/>F ad N; & permutando, vt Q ad F, ita R ad N. e&longs;t autem pars <arrow.to.target n="note88"></arrow.to.target><lb/>Q ip&longs;i F æqualis; quare & pars R ip&longs;i N æqualis erit. </s>

<s>& iterum com<lb/>ponendo, vt CH ad HG, ita ON ad N. </s>

<s>vt autem GH <arrow.to.target n="note85"></arrow.to.target><lb/>ad HB, ita e&longs;t F ad ON. </s>

<s>quare ex æquali, vt CH ad HB, ita F <arrow.to.target n="note86"></arrow.to.target><lb/>ad N. &longs;ed vt CH ad HB ita e&longs;t Q ad R: erit igitur Q ad R, vt <arrow.to.target n="note87"></arrow.to.target><lb/>F ad N; & permutando, vt Q ad F, ita R ad N. </s>

<s>e&longs;t autem pars <arrow.to.target n="note88"></arrow.to.target><lb/>Q ip&longs;i F æqualis; quare & pars R ip&longs;i N æqualis erit. </s>

<s id="id.2.1.53.10.1.7.0"> Itaq; cùm <lb/>pondus L &longs;it ip&longs;i O æquale, & pondus F ip&longs;i Q etiam æquale, atq; <lb/>pars R ip&longs;i N æqualis; erunt pondera LM ip&longs;is EF ponderibus <lb/>æqualia. </s>

<s id="id.2.1.53.10.1.8.0"> & quoniam e&longs;t, vt AC ad CG, ita pondus E ad pon<lb/>dus L; pondera EL æqueponderabunt. </s>

<s id="id.2.1.53.10.1.9.0"> &longs;imiliter quoniam e&longs;t, vt <arrow.to.target n="note89"></arrow.to.target><lb/>AC ad CB, ita pundus F ad pondus M; pondera quoq; FM <lb/>æqueponderabunt. </s>

<s id="id.2.1.53.10.1.9.0">&longs;imiliter quoniam e&longs;t, vt <arrow.to.target n="note89"></arrow.to.target><lb/>AC ad CB, ita <expan abbr="pundus">pondus</expan> F ad pondus M; pondera quoq; FM <lb/>æqueponderabunt. </s>

<s id="id.2.1.53.10.1.10.0"> Pondera igitur LM ponderibus EF in BG <arrow.to.target n="note90"></arrow.to.target><lb/>appen&longs;is æqueponderabunt. </s>

<s id="id.2.1.53.10.1.11.0"> cùm autem di&longs;tantia CA æqualis &longs;it <lb/>di&longs;tantiæ CH; &longs;i igitur vtraq; pondera EF in H appendantur, <lb/>pondera LM ip&longs;is EF ponderibus in H appen&longs;is æquepondera<lb/>bunt. </s>

<s id="id.2.1.53.10.1.12.0"> &longs;ed LM ip&longs;is EF in GB quoq; æqueponderant: æquè <arrow.to.target n="note91"></arrow.to.target><lb/>igitur grauia erunt pondera EF in GB, vt in H appen&longs;a. </s>

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<s id="id.2.1.53.11.1.1.0.b"> <lb/>fiat CA ip&longs;i CH æqualis, & vt CA ad CB, ita fiat pondus F ad <lb/>aliud D, quod appendatur in A. </s>

<s id="id.2.1.53.11.1.1.0.c"> Quoniam enim CH e&longs;t æqua<pb xlink:href="036/01/076.jpg"/><figure id="id.036.01.076.1.jpg" xlink:href="036/01/076/1.jpg"></figure><lb/>lis CA, erit CH ad CB, vt F ad D; & maior quidem e&longs;t CB, <lb/>quàm CH; idcirco D pondere F maius erit. </s>

<s id="id.2.1.53.11.1.2.0"> Diuidatur ergo D <lb/>in duas partes Gk, &longs;itq; G ip&longs;i F æqualis; erit vt BC ad CH, <lb/>vt Gk ad G; & diuidendo, vt BH ad HC, ita K ad G; & conuer <lb/><arrow.to.target n="note92"></arrow.to.target>tendo, vt CH ad HB, ita G ad k. </s>

<s id="id.2.1.53.11.1.3.0"> Vt autem CH ad HB, ita e&longs;t <lb/><arrow.to.target n="note93"></arrow.to.target>F ad E. vt igitur G ad k, ita e&longs;t F ad E; & permutando vt G <lb/><arrow.to.target n="note94"></arrow.to.target>ad F, ita k ad E. &longs;unt autem GF æqualia; erunt & kE inter &longs;e <lb/>&longs;e æqualia. </s>

<s id="id.2.1.53.11.1.3.0">Vt autem CH ad HB, ita e&longs;t <lb/><arrow.to.target n="note93"></arrow.to.target>F ad E. </s>

<s>vt igitur G ad k, ita e&longs;t F ad E; & permutando vt G <lb/><arrow.to.target n="note94"></arrow.to.target>ad F, ita k ad E. </s>

<s>&longs;unt autem GF æqualia; erunt & kE inter &longs;e <lb/>&longs;e æqualia. </s>

<s id="id.2.1.53.11.1.4.0"> cùm itaq; pars G &longs;it ip&longs;i F æqualis, & K ip&longs;i E; erit <lb/>totum C k ip&longs;is EF ponderibus æquale. </s>

<s id="id.2.1.53.11.1.5.0"> & quoniam AC e&longs;t ip<lb/>&longs;i CH æqualis; &longs;i igitur pondera EF ex puncto H &longs;u&longs;pendantur, <lb/>pondus D ip&longs;is EF in H appen&longs;is æqueponderabit. </s>

<s id="id.2.1.53.11.1.6.0"> &longs;ed & ip&longs;is <lb/>æqueponderat in CB, hoc e&longs;t F in B, & E in C; cùm &longs;it vt AC <lb/>ad CB, ita F ad. D. </s>

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<s id="id.2.1.53.12.1.1.0"> Sit deniq; libra AB, & ex punctis AB &longs;u&longs;pen&longs;a &longs;int pondera <lb/>EF; &longs;itq; centrum libræ C intra pondera; diuidaturq; AB in <lb/>D, ita vt AD ad DB &longs;it, vt pondus F ad pondus E. </s>

<s id="id.2.1.53.12.1.1.0.a"> Dico pon<lb/>dera EF tàm in AB ponderare, quám &longs;i vtraq; ex puncto D &longs;u&longs;pen<lb/>dantur. </s>

<s id="id.2.1.53.12.1.2.0"> fiat CG æqualis ip&longs;i CD; & vt DC ad CA, ita fiat <lb/>pondus E ad aliud H; quod appendatur in D. vt autem GC ad <lb/>CB, ita fiat pondus F ad aliud K; appendaturq; k in G. </s>

<s id="id.2.1.53.12.1.2.0.a"> <expan abbr="Quoniã">Quoniam</expan>enim <lb/>e&longs;t, vt BC ad CG, hoc e&longs;t ad CD, ita pondus k ad F; erit K ma <lb/>ius pondere F. quare diuidatur pondus k in L, & MN; fiatq; <lb/>pars L ip&longs;i F æqualis; erit vt BC ad CD, vt totum LMN ad <lb/>L; & diuidendo, vt BD ad DC, ita pars MN ad partem L. vt <arrow.to.target n="note95"></arrow.to.target><lb/>igitur BD ad DC, ita pars MN ad F. vt autem AD ad DB, <lb/>ita F ad E: quare ex æquali, vt AD ad DC, ita MN ad E. cùm <arrow.to.target n="note96"></arrow.to.target><expan abbr="verò"><lb/>vero</expan>AD &longs;it ip&longs;a CD maior; erit & pars MN pondere E <lb/>maior: diuidatur ergo MN in duas partes MN, &longs;itq; M æqua <lb/>lis ip&longs;i E. erit vt AD ad DC, vt NM ad M; & diuidendo, vt <arrow.to.target n="note97"></arrow.to.target><lb/>AC ad CD, ita N ad M: conuertendoq; vt DC ad CA, ita M <lb/>ad N. vt autem DC ad CA, ita e&longs;t E ad H; erit igitur M ad N <arrow.to.target n="note98"></arrow.to.target><lb/>vt E ad H; & permutando, vt M ad E, ita N ad H. &longs;ed ME <arrow.to.target n="note99"></arrow.to.target><lb/>&longs;unt inter &longs;e æqualia, erunt NH inter &longs;e&longs;e quoq; æqualia. </s>

<s id="id.2.1.53.12.1.2.0.a"><expan abbr="Quoniã">Quoniam</expan> enim <lb/>e&longs;t, vt BC ad CG, hoc e&longs;t ad CD, ita pondus k ad F; erit K ma <lb/>ius pondere F. </s>

<s>quare diuidatur pondus k in L, & MN; fiatq; <lb/>pars L ip&longs;i F æqualis; erit vt BC ad CD, vt totum LMN ad <lb/>L; & diuidendo, vt BD ad DC, ita pars MN ad partem L. </s>

<s>vt <arrow.to.target n="note95"></arrow.to.target><lb/>igitur BD ad DC, ita pars MN ad F. </s>

<s>vt autem AD ad DB, <lb/>ita F ad E: quare ex æquali, vt AD ad DC, ita MN ad E. </s>

<s>cùm <arrow.to.target n="note96"></arrow.to.target><expan abbr="verò"><lb/>vero</expan> AD &longs;it ip&longs;a CD maior; erit & pars MN pondere E <lb/>maior: diuidatur ergo MN in duas partes MN, &longs;itq; M æqua <lb/>lis ip&longs;i E. </s>

<s>erit vt AD ad DC, vt NM ad M; & diuidendo, vt <arrow.to.target n="note97"></arrow.to.target><lb/>AC ad CD, ita N ad M: conuertendoq; vt DC ad CA, ita M <lb/>ad N. </s>

<s>vt autem DC ad CA, ita e&longs;t E ad H; erit igitur M ad N <arrow.to.target n="note98"></arrow.to.target><lb/>vt E ad H; & permutando, vt M ad E, ita N ad H. </s>

<s>&longs;ed ME <arrow.to.target n="note99"></arrow.to.target><lb/>&longs;unt inter &longs;e æqualia, erunt NH inter &longs;e&longs;e quoq; æqualia. </s>

<s id="id.2.1.53.12.1.3.0"> & quo<lb/>niam ita e&longs;t AC ad CD, vt H ad E: pondera HE æqueponde<lb/>rabunt. <arrow.to.target n="note100"></arrow.to.target> </s>

<s id="id.2.1.53.12.1.4.0"> &longs;imiliter quoniam e&longs;t vt GC ad CB, ita F ad k, ponde<pb xlink:href="036/01/078.jpg"/><figure id="id.036.01.078.1.jpg" xlink:href="036/01/078/1.jpg"></figure><lb/><arrow.to.target n="note101"></arrow.to.target>ra etiam kF æqueponderabunt. </s>

<s id="id.2.1.53.12.1.5.0"> pondera igitur Ek HF in li<lb/>bra AB, cuius centrum C, æqueponderabunt. </s>

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<s id="id.2.1.53.12.1.9.0"> & cùm <lb/>&longs;it CG ip&longs;i CD æqualis, &longs;i igitur pondera EF ex puncto D &longs;u&longs;pen<lb/>dantur, pondera EF in D appen&longs;a ip&longs;i LM æqueponderabunt. </s>

<s id="id.2.1.53.12.1.10.0"> quare <lb/>LM tàm ip&longs;is EF in AB appen&longs;is æqueponderat, quàm in pun<lb/>cto D appen&longs;is. </s>

<s id="id.2.1.53.12.1.11.0"> libra enim &longs;emper eodem modo manet. </s>

<s id="id.2.1.53.12.1.12.0"> Ponde<lb/><arrow.to.target n="note103"></arrow.to.target>ra ergo EF tàm in AB ponderabunt, quàm in puncto D. quod <lb/>demon&longs;tre oportebat. </s>

<s id="id.2.1.53.12.1.12.0">Ponde<lb/><arrow.to.target n="note103"></arrow.to.target>ra ergo EF tàm in AB ponderabunt, quàm in puncto D. </s>

<s id="id.2.1.53.12.1.9.0.a">quod <lb/><expan abbr="demon&longs;tre">demonstrare</expan> oportebat. </s>

<s id="id.2.1.54.1.1.1.0"> <margin.target id="note82"></margin.target>17 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>

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<s id="id.2.1.55.2.1.2.0.a"> pondera LM ip&longs;is EF in GB appen&longs;is (vt &longs;upra dictum e&longs;t) <lb/>æqueponderabunt. </s>

<s id="id.2.1.55.2.1.3.0"> Sint deinde puncta NO centra grauitatis pon<lb/>derum EF; connectanturq; GN BO; iungaturq; NO, quæ tan<lb/>quam libra erit; quæ etiam efficiat lineas GN BO inter &longs;e &longs;e æqui<lb/>di&longs;tantes e&longs;&longs;e; à punctoq; H horizonti perpendicularis ducatur <lb/>HP, quæ NO &longs;ecet in P, atq; ip&longs;is GN BO &longs;it æquidi&longs;tans. <lb/> </s>

<s id="id.2.1.55.2.1.3.0.a"> deniq; connectatur GO, quæ HP &longs;ecet in R. </s>

<s id="id.2.1.55.2.1.4.0"> Quoniam igitur <lb/>HR e&longs;t lateri BO trianguli GBO æquidi&longs;tans; erit GH ad HB, <lb/>vt GR ad RO. &longs;imiliter quoniam RP e&longs;t lateri GN trianguli <arrow.to.target n="note104"></arrow.to.target><lb/>OGN æquidi&longs;tans; erit GR ad RO, vt NP ad PO. quare <lb/>vt GH ad HB, ita e&longs;t NP ad PO. vt autem GH ad HB, ita <arrow.to.target n="note105"></arrow.to.target><lb/>e&longs;t pondus F ad pondus E; vt igitur NP ad PO, ita e&longs;t pondus <lb/>F ad pondus E. </s>

<s id="id.2.1.55.2.1.4.0">Quoniam igitur <lb/>HR e&longs;t lateri BO trianguli GBO æquidi&longs;tans; erit GH ad HB, <lb/>vt GR ad RO. </s>

<s>&longs;imiliter quoniam RP e&longs;t lateri GN trianguli <arrow.to.target n="note104"></arrow.to.target><lb/>OGN æquidi&longs;tans; erit GR ad RO, vt NP ad PO. </s>

<s>quare <lb/>vt GH ad HB, ita e&longs;t NP ad PO. </s>

<s>vt autem GH ad HB, ita <arrow.to.target n="note105"></arrow.to.target><lb/>e&longs;t pondus F ad pondus E; vt igitur NP ad PO, ita e&longs;t pondus <lb/>F ad pondus E. </s>

<s id="id.2.1.55.2.1.4.0.a"> punctum ergo P centrum erit grauitatis magni<lb/>tudinis ex vtri&longs;q; EF ponderibus compo&longs;itæ. </s>

<s id="id.2.1.55.2.1.5.0"> Intelligantur itaq; <arrow.to.target n="note106"></arrow.to.target><lb/>pondera EF ita e&longs;&longs;e à libra NO connexa, ac &longs;i vna tantùm e&longs;&longs;et <lb/>magnitudo ex vtri&longs;q; EF compo&longs;ita, in puncti&longs;q; BG appen&longs;a. </s>

<s id="id.2.1.55.2.1.6.0"> &longs;i <lb/>igitur ponderum &longs;u&longs;pen&longs;iones BG &longs;oluantur, manebunt pondera <arrow.to.target n="note107"></arrow.to.target><lb/>EF ex HP &longs;u&longs;pen&longs;a; &longs;icuti in GB prius manebant. </s>

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<s id="id.2.1.57.3.1.1.0.a"> Dico pondus F ad pondus E eam in grauitate proportionem ha<lb/>bere, quam habet di&longs;tantia CA ad di&longs;tantiam AD. </s>

<s id="id.2.1.57.3.1.1.0.b"> fiat enim vt <lb/>CA ad AD, ita pondus F ad aliud pondus, quod &longs;it G. </s>

<s id="id.2.1.57.3.1.1.0.c"> Dico pri <lb/>múm pondera GF ex puncto C &longs;u&longs;pen&longs;a tantùm ponderare, quan<lb/>tùm pondera EF ex punctis DC. </s>

<s id="id.2.1.57.3.1.1.0.d"> Secetur DC bifariam in H, & <lb/>ex H appendantur vtraq; pondera EF. ponderabunt EF &longs;imul <lb/>&longs;umpta in eo &longs;itu, quantùm ponderant in DC. ponatur BA <arrow.to.target n="note108"></arrow.to.target><lb/>æqualis AH, &longs;eceturq; BA in K, ita vt &longs;it KA æqualis AD: <lb/>deinde ex puncto B appendatur pondus L duplum ponderis F, <lb/>hoc e&longs;t æquale duobus ponderibus EF, quod quidem æqueponde<lb/>rabit ponderibus EF in H appen&longs;is, hoc e&longs;t appen&longs;is in DC. </s>

<s id="id.2.1.57.3.1.1.0.d">Secetur DC bifariam in H, & <lb/>ex H appendantur vtraq; pondera EF. </s>

<s id="id.2.1.57.3.1.1.0.e"> <expan abbr="Quoniã">Quoniam</expan><lb/>igitur, vt CA ad AD, ita e&longs;t pondus F ad pondus G; erit compo<lb/>nendo vt CA AD ad AD, hoc e&longs;t vt Ck ad AD, ita ponde<lb/>ra <arrow.to.target n="note109"></arrow.to.target>FG ad pondus G. &longs;ed cùm &longs;it, vt CA ad AD, ita F pon<lb/>dus ad pondus G; erit conuertendo, vt DA ad AC, ita pondus <arrow.to.target n="note110"></arrow.to.target><lb/>G ad pondus F; & con&longs;equentium dupla, vt DA ad duplam ip&longs;ius <lb/>AC, ita pondus G ad duplum ponderis F, hoc e&longs;t ad pondus <lb/>L. </s>

<s>ponderabunt EF &longs;imul <lb/>&longs;umpta in eo &longs;itu, quantùm ponderant in DC. ponatur BA <arrow.to.target n="note108"></arrow.to.target><lb/>æqualis AH, &longs;eceturq; BA in K, ita vt &longs;it KA æqualis AD: <lb/>deinde ex puncto B appendatur pondus L duplum ponderis F, <lb/>hoc e&longs;t æquale duobus ponderibus EF, quod quidem æqueponde<lb/>rabit ponderibus EF in H appen&longs;is, hoc e&longs;t appen&longs;is in DC. </s>

<s id="id.2.1.57.3.1.1.0.f"> Quare vt Ck ad DA, ita pondera EF ad pondus G; & vt <pb xlink:href="036/01/082.jpg"/><figure id="id.036.01.082.1.jpg" xlink:href="036/01/082/1.jpg"></figure><lb/><arrow.to.target n="note111"></arrow.to.target>AD ad <expan abbr="duplã">duplam</expan>ip&longs;ius AC, ita pondus G ad pondus L; ergo ex æquali, <lb/>vt Ck ad <expan abbr="duplã">duplam</expan>ip&longs;ius AC, ita pondera FG ad pondus L. &longs;ed vt Ck <lb/>ad duplam AC, ita dimidia CK, videlicet AH, hoc e&longs;t BA, ad <lb/>AC. </s>

<s>&longs;ed cùm &longs;it, vt CA ad AD, ita F pon<lb/>dus ad pondus G; erit conuertendo, vt DA ad AC, ita pondus <arrow.to.target n="note110"></arrow.to.target><lb/>G ad pondus F; & con&longs;equentium dupla, vt DA ad duplam ip&longs;ius <lb/>AC, ita pondus G ad duplum ponderis F, hoc e&longs;t ad pondus <lb/>L. </s>

<s id="id.2.1.57.3.1.1.0.f">Quare vt Ck ad DA, ita pondera EF ad pondus G; & vt <pb xlink:href="036/01/082.jpg"/><figure id="id.036.01.082.1.jpg" xlink:href="036/01/082/1.jpg"></figure><lb/><arrow.to.target n="note111"></arrow.to.target>AD ad <expan abbr="duplã">duplam</expan> ip&longs;ius AC, ita pondus G ad pondus L; ergo ex æquali, <lb/>vt Ck ad <expan abbr="duplã">duplam</expan> ip&longs;ius AC, ita pondera FG ad pondus L. </s>

<s>&longs;ed vt Ck <lb/>ad duplam AC, ita dimidia CK, videlicet AH, hoc e&longs;t BA, ad <lb/>AC. </s>

<s id="id.2.1.57.3.1.1.0.g"> Vt igitur BA ad AC, ita FG pondera ad pondus L. </s>

<s id="id.2.1.57.3.1.1.0.h"> Qua <lb/>re ex &longs;exta eiu&longs;dem primi Archimedis, duo pondera FG ex pun<lb/>cto C &longs;u&longs;pen&longs;a tantùm ponderabunt, quantùm pondus L ex B; <lb/>hoc e&longs;t quantùm pondera EF ex punctis DC &longs;u&longs;pen&longs;a. </s>

<s id="id.2.1.57.3.1.2.0"> Itaq; quo<lb/>niam pondera FG tantùm ponderant, quantum pondera EF; &longs;u<lb/>blato communi pondere F, tàm ponderabit pondus G in C ap<lb/>pen&longs;um, quàm pondus E in D. </s>

<s id="id.2.1.57.3.1.2.0.a"> ac propterea pondus F ad pon<lb/><arrow.to.target n="note112"></arrow.to.target>dus E eam in grauitate proportionem habet, quam habet ad pon<lb/>dus G. &longs;ed pondus F ad G erat, vt CA ad AD:. ergo & F pon<lb/>dus ad pondus E eam in grauitate proportionem habebit, quam ha<lb/>bet CA ad AD. quod demon&longs;trare oportebat. </s>

<s id="id.2.1.57.3.1.2.0.a">ac propterea pondus F ad pon<lb/><arrow.to.target n="note112"></arrow.to.target>dus E eam in grauitate proportionem habet, quam habet ad pon<lb/>dus G. </s>

<s>&longs;ed pondus F ad G erat, vt CA ad AD: ergo & F pon<lb/>dus ad pondus E eam in grauitate proportionem habebit, quam ha<lb/>bet CA ad AD. </s>

<s>quod demon&longs;trare oportebat. </s>

<s id="id.2.1.58.1.1.1.0"> <margin.target id="note108"></margin.target>5 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>

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<s id="id.2.1.59.4.1.1.0"> Sit libra BAC, cu<lb/>ius centrum A; in pun<lb/>ctis verò BC pondera <lb/>appendantur æqualia G <lb/>F: &longs;itq; primùm cen<lb/>trum A vtcunque inter <lb/>BC. </s>

<s id="id.2.1.59.4.1.1.0.a"> Dico pondus F ad <lb/>pondus G eam in graui<lb/><figure id="id.036.01.083.1.jpg" xlink:href="036/01/083/1.jpg"></figure><lb/>tate proportionem habere, quam habet di&longs;tantia CA ad di&longs;tan<lb/>tiam AB. </s>

<s id="id.2.1.59.4.1.1.0.b"> fiat vt BA ad AC, ita pondus F ad aliud H, quod ap<lb/>pendatur in B: pondera HF ex A æqueponderabunt. </s>

<s id="id.2.1.59.4.1.2.0"> &longs;ed cùm <arrow.to.target n="note113"></arrow.to.target><lb/>pondera FG &longs;int æqualia, habebit pondus H ad pondus G ean<lb/>dem proportionem, quam habet ad F. vt igitur CA ad AB, ita <arrow.to.target n="note114"></arrow.to.target><lb/>e&longs;t H ad G. vt autem H ad G, ita e&longs;t grauitas ip&longs;ius H ad graui<lb/>tatem ip&longs;ius G; cùm in eodem puncto B &longs;int appen&longs;a. </s>

<s id="id.2.1.59.4.1.3.0"> quare vt CA <lb/>ad AB, ita grauitas ponderis H ad grauitatem ponderis G. cùm au<lb/>tem grauitas ponderis F in C appen&longs;i &longs;it æqualis grauitati ponderis <lb/>H in B; erit grauitas ponderis F ad grauitatem ponderis G, vt CA <lb/>ad AB, videlicet vt di&longs;tantia ad di&longs;tantiam. </s>

<s>vt igitur CA ad AB, ita <arrow.to.target n="note114"></arrow.to.target><lb/>e&longs;t H ad G. </s>

<s>vt autem H ad G, ita e&longs;t grauitas ip&longs;ius H ad graui<lb/>tatem ip&longs;ius G; cùm in eodem puncto B &longs;int appen&longs;a. </s>

<s id="id.2.1.59.4.1.3.0">quare vt CA <lb/>ad AB, ita grauitas ponderis H ad grauitatem ponderis G. </s>

<s>cùm au<lb/>tem grauitas ponderis F in C appen&longs;i &longs;it æqualis grauitati ponderis <lb/>H in B; erit grauitas ponderis F ad grauitatem ponderis G, vt CA <lb/>ad AB, videlicet vt di&longs;tantia ad di&longs;tantiam. </s>

<s id="id.2.1.59.4.1.4.0"> quod demon&longs;trare <lb/>oportebat. </s>

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<s id="id.2.1.61.1.1.1.0.a"> Dico <lb/>&longs;imiliter ita e&longs;&longs;e gra<lb/><figure id="id.036.01.083.2.jpg" xlink:href="036/01/083/2.jpg"></figure><lb/>uitatem ponderis F ad grauitatem ponderis E, vt di&longs;tantia CA ad <lb/>di&longs;tantiam AD. </s>

<s id="id.2.1.61.1.1.1.0.b"> fiat AB æqualis ip&longs;i AD, & in B appendatur <lb/>pondus G æquale ponderi E, & ponderi F. </s>

<s id="id.2.1.61.1.1.1.0.c"> Quoniam enim AB e&longs;t <lb/>æqualis AD; pondera GE æqueponderabunt. </s>

<s id="id.2.1.61.1.1.2.0"> &longs;ed cùm grauitas <lb/>ponderis F ad grauitatem ponderis G &longs;it, vt CA ad AB, & graui<lb/>tas ponderis E &longs;it æqualis grauitati ponderis G; erit grauitas pon-<lb/>deris F ad grauitatem ponderis E, vt CA ad AB, hoc e&longs;t vt CA <lb/>ad AD. quod demon&longs;trare oportebat. </s>

<s id="id.2.1.61.1.1.2.0">&longs;ed cùm grauitas <lb/>ponderis F ad grauitatem ponderis G &longs;it, vt CA ad AB, & graui<lb/>tas ponderis E &longs;it æqualis grauitati ponderis G; erit grauitas pon-<lb/>deris F ad grauitatem ponderis E, vt CA ad AB, hoc e&longs;t vt CA <lb/>ad AD. </s>

<s>quod demon&longs;trare oportebat. </s>

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<s id="id.2.1.62.1.1.1.0"> <margin.target id="note115"></margin.target><emph type="italics"/>Stateræ ratio.<emph.end type="italics"/> </s>

<s id="id.2.1.63.1.1.1.0"> Sit enim &longs;tate<lb/>ræ &longs;capus AB, cu<lb/>ius trutina &longs;it in <lb/>C; &longs;itq; &longs;tateræ <lb/>appendiculum E. <lb/>appendatur in A <lb/>pondus D, quod <lb/>æqueponderet ap<lb/>pendiculo E in F <lb/><figure id="id.036.01.084.1.jpg" xlink:href="036/01/084/1.jpg"></figure><lb/>appen&longs;o. </s>

<s id="id.2.1.63.1.1.1.0">Sit enim &longs;tate<lb/>ræ &longs;capus AB, cu<lb/>ius trutina &longs;it in <lb/>C; &longs;itq; &longs;tateræ <lb/>appendiculum E. <lb/></s>

<s>appendatur in A <lb/>pondus D, quod <lb/>æqueponderet ap<lb/>pendiculo E in F <lb/><figure id="id.036.01.084.1.jpg" xlink:href="036/01/084/1.jpg"></figure><lb/>appen&longs;o. </s>

<s id="id.2.1.63.1.1.2.0"> aliud quoq; appendatur pondus G in A, quod etiam <lb/>appendiculo E in B appen&longs;o æqueponderet. </s>

<s id="id.2.1.63.1.1.3.0"> Dico grauitatem <lb/>ponderis D ad grauitatem ponderis G ita e&longs;&longs;e, vt CF ad CB. </s>

<s id="id.2.1.63.1.1.3.0.a"> <lb/>Quoniam enim grauitas ponderis D e&longs;t æqualis grauitati ponde<lb/>ris E in F appen&longs;i, & grauitas ponderis G e&longs;t æqualis grauitati pon<lb/>deris E in B; erit grauitas ponderis D ad grauitatem ponderis E in <lb/>F, vt grauitas ponderis G ad grauitatem ponderis E in B: & permu<lb/><arrow.to.target n="note116"></arrow.to.target>tando, vt grauitas ponderis D ad grauitatem ponderis G, ita graui<lb/>tas ip&longs;ius E in F, ad grauitatem ip&longs;ius E in B; grauitas autem pon <lb/><arrow.to.target n="note117"></arrow.to.target>deris E in F ad grauitatem ponderis E in B e&longs;t, vt CF ad CB; vt <lb/>igitur grauitas ponderis D ad grauitatem ponderis G, ita e&longs;t CF <lb/>ad CB &longs;i ergo pars &longs;capi CB in partes diuidatur æquales, &longs;olo <lb/>pondere E, & propius, & longius à puncto C po&longs;ito; ponderum <lb/>grauitates, quæ ex puncto A &longs;u&longs;penduntur inter &longs;e &longs;e notæ erunt. </s>

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<s id="id.2.1.65.2.1.1.0"> Sit &longs;capus AB, cuius tru<lb/>tina &longs;it in C; &longs;itq; &longs;tateræ ap<lb/>pendiculum E, quod appen<lb/>datur in A; &longs;intqué pon<lb/>dera DG inæqualia, quorum <lb/>inter &longs;e &longs;e grauitatum propor<lb/>tiones quærimus: appenda<lb/>tur pondus D in B, ita vt ip&longs;i <lb/><figure id="id.036.01.085.1.jpg" xlink:href="036/01/085/1.jpg"></figure><lb/>E æqueponderet. </s>

<s id="id.2.1.65.2.1.2.0"> &longs;imiliter pondus G appendatur in F, quod ei<lb/>dem ponderi E æqueponderet. </s>

<s id="id.2.1.65.2.1.3.0"> dico D ad G ita e&longs;&longs;e, vt CF ad <lb/>CB. </s>

<s id="id.2.1.65.2.1.3.0.a"> Quoniam enim pondera DE æqueponderant, erit D ad E, <arrow.to.target n="note118"></arrow.to.target><lb/>vt CA ad CB. cùm autem pondera quoque GE æquepon<lb/>derent, erit pondus E ad pondus G, vt FC ad CA; quare ex æqua <lb/>li pondus D ad pondus G ita erit, vt CF ad CB. quod o&longs;tende <arrow.to.target n="note119"></arrow.to.target><lb/>re quoq; oportebat. </s>

<s id="id.2.1.65.2.1.3.0.a">Quoniam enim pondera DE æqueponderant, erit D ad E, <arrow.to.target n="note118"></arrow.to.target><lb/>vt CA ad CB. </s>

<s>cùm autem pondera quoque GE æquepon<lb/>derent, erit pondus E ad pondus G, vt FC ad CA; quare ex æqua <lb/>li pondus D ad pondus G ita erit, vt CF ad CB. </s>

<s>quod o&longs;tende<arrow.to.target n="note119"></arrow.to.target><lb/>re quoq; oportebat. </s>

<s id="id.2.1.66.1.1.1.0"> <margin.target id="note118"></margin.target>6 <emph type="italics"/>Primi Archim. de æquep.<emph.end type="italics"/> </s>

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<s id="id.2.1.67.2.1.1.0"> Quotcunque datis in libra ponderibus <lb/>vbicunque appen&longs;is, centrum libræ inuenire, <lb/>ex quo &longs;i &longs;u&longs;pendatur libra, data pondera ma<lb/>neant. <figure id="id.036.01.086.1.jpg" xlink:href="036/01/086/1.jpg"></figure> </s>

<s id="id.2.1.67.3.1.1.0"> Sit libra AB, &longs;intq; data quotcunque pondera CDEFG. <lb/>accipiantur in libra vtcunque puncta AHkLB, ex quibus <lb/>data pondera &longs;pu&longs;pendantur. </s>

<s id="id.2.1.67.3.1.1.0">Sit libra AB, &longs;intq; data quotcunque pondera CDEFG. <lb/></s>

<s id="id.2.1.67.3.1.1.0.a">accipiantur in libra vtcunque puncta AHkLB, ex quibus <lb/>data pondera <expan abbr="&longs;pu&longs;pendantur">suspendantur</expan>. </s>

<s id="id.2.1.67.3.1.2.0"> Centrum libræ inuenire oportet, <lb/>ex quo &longs;i fiat &longs;u&longs;pen&longs;io, data pondera maneant. </s>

<s id="id.2.1.67.3.1.3.0"> Diuidatur <pb n="37" xlink:href="036/01/087.jpg"/><figure id="id.036.01.087.1.jpg" xlink:href="036/01/087/1.jpg"></figure><lb/>AH in M, ita vt HM ad MA, &longs;it vt grauitas ponderis <lb/>C ad grauitatem ponderis D. </s>

<s id="id.2.1.67.3.1.3.0.a"> deinde diuidatur BL in N, ita <lb/>vt LN ad NB, &longs;it vt grauitas ponderis G ad grauitatem pon<lb/>deris F. diuidaturq; MN in O, ita vt MO ad ON &longs;it, vt <lb/>grauitas ponderum FG ad grauitatem ponderum CD. </s>

<s id="id.2.1.67.3.1.3.0.a">deinde diuidatur BL in N, ita <lb/>vt LN ad NB, &longs;it vt grauitas ponderis G ad grauitatem pon<lb/>deris F. </s>

<s id="id.2.1.67.3.1.3.0.b"> <expan abbr="tandemqué">tandem<lb/>que</expan>diuidatur kO in P, ita vt kP ad PO, &longs;it vt grauitas pon<lb/>derum CDFG ad grauitatem ponderis E. </s>

<s>diuidaturq; MN in O, ita vt MO ad ON &longs;it, vt <lb/>grauitas ponderum FG ad grauitatem ponderum CD. </s>

<s id="id.2.1.67.3.1.3.0.b">tandem<lb/>què diuidatur kO in P, ita vt kP ad PO, &longs;it vt grauitas pon<lb/>derum CDFG ad grauitatem ponderis E. </s>

<s id="id.2.1.67.3.1.3.0.c"> Quoniam igitur pon <lb/>dera CDFG tàm ponderant in O, quàm CD in M, & FG in N; <arrow.to.target n="note120"></arrow.to.target><lb/>æqueponderabunt pondera CD in M, & FG in N, & pondus E <lb/>in K, &longs;i ex puncto P &longs;u&longs;pendantur. </s>

<s id="id.2.1.67.3.1.4.0"> cùm verò pondera CD tan<lb/>tùm ponderent in M, quantùm in AH, & FG in N, quantùm <lb/>in LB; pondera CDFG ex AHLB punctis &longs;u&longs;pen&longs;a, & pon<lb/>dus E ex k, &longs;i ex P &longs;u&longs;pendantur, æqueponderabunt, atq; mane<lb/>bunt. </s>

<s id="id.2.1.67.3.1.5.0"> Inuentum e&longs;t ergo centrum libræ P, ex quo data pondera <lb/>manent. </s>

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<s id="id.2.1.70.1.1.1.0"> <margin.target id="note121"></margin.target>8 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>

</figure>

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<s id="id.2.1.89.3.1.1.0.a"> <lb/>&longs;i verò alii &longs;int quoq; vectes AF AG, quorum fulcimenta &longs;int <lb/>HK; pondu&longs;q; AC in vecte AG ex punctis AQ &longs;it appen&longs;um; <lb/>in vecte autem AF in punctis AP: lineaq; DE producta &longs;ecet <lb/>AF in L, & AG in M. </s>

<s id="id.2.1.89.3.1.1.0.b"> dico potentiam in F pondus AC &longs;u&longs;tinen<lb/>tem ad ip&longs;um pondus eam habere proportionem, quam habet kL <pb n="44" xlink:href="036/01/101.jpg"/>ad kF; & potentiam in B ad pondus eam habere, quam NE ad <lb/>NB; & potentiam in G ad pondus eam, quam HM ad HG. </s>

<s id="id.2.1.89.3.1.1.0.c"> <lb/>Quoniam enim DL horizonti e&longs;t perpendicularis, pondus AC <lb/>vbicunq; in linea DL fuerit appen&longs;um, eodem modo, quo reperi<lb/>tur, manebit. </s>

<s id="id.2.1.89.3.1.2.0"> quare in vecte AB &longs;i &longs;u&longs;pen&longs;iones, quæ &longs;unt ad AO <lb/>&longs;oluantur, pondus AC in E appen&longs;um eodem modo manebit, &longs;i<lb/>cutinunc manet; hoc e&longs;t &longs;ublato puncto A, & linea QO, codem <lb/>modo pondus in E appen&longs;um manebit, vt ab ip&longs;is AO pun<lb/>ctis &longs;u&longs;tinebatur; ex commentario Federici Commandini in &longs;extam <lb/>Archimedis <expan abbr="propo&longs;ion&etilde;">propo&longs;ionem</expan>de quadratura parabolæ, & ex prima huius <lb/>de libra. </s>

<s id="id.2.1.89.3.1.2.0">quare in vecte AB &longs;i &longs;u&longs;pen&longs;iones, quæ &longs;unt ad AO <lb/>&longs;oluantur, pondus AC in E appen&longs;um eodem modo manebit, &longs;i<lb/>cuti nunc manet; hoc e&longs;t &longs;ublato puncto A, & linea QO, codem <lb/>modo pondus in E appen&longs;um manebit, vt ab ip&longs;is AO pun<lb/>ctis &longs;u&longs;tinebatur; ex commentario Federici Commandini in &longs;extam <lb/>Archimedis <expan abbr="propo&longs;ion&etilde;">propo&longs;itionem</expan> de quadratura parabolæ, & ex prima huius <lb/>de libra. </s>

<s id="id.2.1.89.3.1.3.0"> Itaq; quoniam pondus AC eandem ad libram habet con&longs;ti<lb/>tutionem, &longs;iue in AO &longs;u&longs;tineatur, &longs;iue ex puncto E &longs;it appen&longs;um; <lb/>eadem potentia in B idem pondus AC, &longs;iue in E, &longs;iue in AO <lb/>&longs;u&longs;pen&longs;um &longs;u&longs;tinebit. </s>

<s id="id.2.1.89.3.1.4.0"> potentia verò in B &longs;u&longs;tinens pondus AC <lb/>in E appen&longs;um ad ip&longs;um pondus ita &longs;e habet, vt NE ad NB; po<lb/>tentia <arrow.to.target n="note145"></arrow.to.target>igitur in B &longs;u&longs;tinens pondus AC ex punctis AO &longs;u&longs;pen<lb/>&longs;um ad ip&longs;um pondus ita erit, vt NE ad NB. </s>

<s id="id.2.1.89.3.1.4.0.a"> Non aliter o&longs;ten <lb/>detur pondus AC ex puncto L &longs;u&longs;pen&longs;um manere, &longs;icuti à pun<lb/>ctis AP &longs;u&longs;tinetur; potentiamq; in F ad ip&longs;um pondus ita e&longs;&longs;e, vt kL <lb/>ad KF. </s>

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<s id="id.2.1.91.3.1.1.0"> Et &longs;i vectes AB <lb/>AF AG habeant <lb/>fulcimenta in A, <lb/>& pondus &longs;it NO; <lb/>deinde ab eius <lb/>centro grauitatis <lb/>D ducatur ip&longs;i A <lb/>B, & horizonti <lb/><expan abbr="perp&etilde;dicularis">perpendicularis</expan>D <lb/>MEL; &longs;intq; po<lb/>tentiæ in FBG: <lb/>&longs;imiliter o&longs;tende<lb/>tur ita e&longs;&longs;e poten<lb/><figure id="id.036.01.102.1.jpg" xlink:href="036/01/102/1.jpg"></figure><lb/>tiam in G pondus NO &longs;u&longs;tinentem ad ip&longs;um pondus, vt AM <lb/>ad AG; ac potentiam in B, vt AE ad AB; & potentiam in F, <lb/>vt AL ad AF. </s>

<s id="id.2.1.91.4.1.1.0"> Sit deinde <lb/>vectis AB ho<lb/>rizonti æqui<lb/>di&longs;tans, cuius <lb/>fulcimentum <lb/>D; & &longs;it BE <lb/>pondus, cuius <lb/>centrum ??? graui<lb/>tatis &longs;it F &longs;u<lb/>pra vectem: à <lb/>punctoq; F ho <lb/>rizonti, & ip&longs;i <lb/>AB ducatur <lb/><figure id="id.036.01.102.2.jpg" xlink:href="036/01/102/2.jpg"></figure><lb/>FH; pondu&longs;q; à puncto B, & PQ &longs;u&longs;tineatur. </s>

<s id="id.2.1.91.4.1.1.0">Sit deinde <lb/>vectis AB ho<lb/>rizonti æqui<lb/>di&longs;tans, cuius <lb/>fulcimentum <lb/>D; & &longs;it BE <lb/>pondus, cuius <lb/>centrum <expan abbr="gaui">graui</expan><lb/>tatis &longs;it F &longs;u<lb/>pra vectem: à <lb/>punctoq; F ho<lb/>rizonti, & ip&longs;i <lb/>AB ducatur <lb/><figure id="id.036.01.102.2.jpg" xlink:href="036/01/102/2.jpg"></figure><lb/>FH; pondu&longs;q; à puncto B, & PQ &longs;u&longs;tineatur. </s>

<s id="id.2.1.91.4.1.2.0"> Sint deinde alii ve<lb/>ctes BL BM, quorum fulcimenta &longs;int NO; lineaq; FH producta &longs;e<lb/>cet BM in k, & BL in G; pondus autem in vecte BL in pun<lb/>ctis BP &longs;u&longs;tineatur; in vecte autem BM à puncto B, & PR. </s>

<s id="id.2.1.91.4.1.2.0.a"> Di<lb/>co potentiam in L pondus BE vecte BL &longs;u&longs;tinentem ad ip&longs;um <lb/>pondus eam habere proportionem, quam NG ad NL; & po<pb n="45" xlink:href="036/01/103.jpg"/>tentiam in A ad pondus eam habere, quam DH ad DA; poten<lb/>tiamq; in M ad pondus eam, quam Ok ad OM. </s>

<s id="id.2.1.91.4.1.2.0.b"> Quoniam e<lb/>nim à centro grauitatis F ducta e&longs;t kF horizonti perpendicularis, <lb/>ex quocunq; puncto lineæ kF &longs;u&longs;tineatur pondus, manebit; vt <arrow.to.target n="note146"></arrow.to.target><lb/>nunc &longs;e habet. </s>

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<s id="id.2.1.92.1.1.1.0"> <margin.target id="note146"></margin.target>1 <emph type="italics"/>Huius de libra.<emph.end type="italics"/> </s>

<s id="id.2.1.92.1.1.2.0"> <margin.target id="note147"></margin.target>1 <emph type="italics"/>Huius<*><emph.end type="italics"/> </s>

<s id="id.2.1.92.1.1.2.0"><margin.target id="note147"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/></s>

<s id="id.2.1.93.1.1.1.0"> Si verò LAM e&longs;&longs;ent fulcimenta, & potentiæ in NDO; &longs;imi <lb/>liter o&longs;tendetur ita e&longs;&longs;e potentiam in N ad pondus, vt LG ad L <lb/>N; & potentiam in D, vt AH ad AD; & potentiam in O, vt <lb/>Mk ad MO. <pb xlink:href="036/01/104.jpg"/> </s>

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<s id="id.2.1.95.10.1.1.0"> Connectantur enim BO BP, <lb/>&longs;imiliter o&longs;tendetur angulum <lb/>PKB minorem e&longs;&longs;e OHB. </s>

<s id="id.2.1.95.10.1.1.0.a"> & <lb/>quoniam angulus FHB æqua<lb/>lis e&longs;t angulo GkB; erit angu<lb/>lus GkN angulo FHM ma<lb/>ior: quare & linea GN ma<lb/>ior erit ip&longs;a FM. ideoq; linea <lb/>nea BN minor erit linea BM. </s>

<s id="id.2.1.95.10.1.1.0.a">& <lb/>quoniam angulus FHB æqua<lb/>lis e&longs;t angulo GkB; erit angu<lb/>lus GkN angulo FHM ma<lb/>ior: quare & linea GN ma<lb/>ior erit ip&longs;a FM. ideoq; linea <lb/><expan abbr="nea"></expan> BN minor erit linea BM. </s>

<s id="id.2.1.95.10.1.1.0.b"> <lb/>Cùm autem maior &longs;it BF ip&longs;a <lb/>BM; erit BM ip&longs;a BA minor. </s>

<s id="id.2.1.95.10.1.2.0"> Si<lb/>miliq; modo o&longs;tendetur, quò <lb/>propius fuerit BG ip&longs;i BC, li<lb/>neam BN &longs;emper minorem <lb/>e&longs;&longs;e. </s>

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<s id="id.2.1.99.2.1.1.0"> Sit deinde vectis AB horizonti æquidi&longs;tans, cuius fulcimen<lb/>tum B; & centrum grauitatis H ponderis CD &longs;it &longs;upra vectem; <lb/>moueaturq; vectis in BE, pondu&longs;q; in FG. </s>

<s id="id.2.1.99.2.1.1.0.a"> dico minorem po<lb/>tentiam in E &longs;u&longs;tinere pondus FG vecte EB, quàm potentia in <lb/>A pondus CD vecte AB. </s>

<s id="id.2.1.99.2.1.1.0.b"> &longs;it k centrum grauitatis ponderis FG, <lb/>& à centris grauitatum Hk ip&longs;orum horizontibus perpendicu<lb/><arrow.to.target n="note162"></arrow.to.target>lares ducantur HL kM. </s>

<s id="id.2.1.99.2.1.1.0.c"> Quoniam enim (ex &longs;upra demon&longs;tratis) <lb/><arrow.to.target n="note163"></arrow.to.target>BM minor e&longs;t BL, & BE ip&longs;i BA æqualis; minorem habebit <lb/><arrow.to.target n="note164"></arrow.to.target>proportionem BM ad BE, quàm BL ad BA. &longs;ed vt BM ad <lb/>BE, ita potentia in E &longs;u&longs;tinens pondus FG ad ip&longs;um pondus; & <lb/>vt BL ad BA, ita potentia in A ad pondus CD; minorem <lb/>habebit proportionem potentia in E ad pdndus FG, quàm poten <lb/><arrow.to.target n="note165"></arrow.to.target>tia in A ad pondus CD. </s>

<s id="id.2.1.99.2.1.1.0.c">Quoniam enim (ex &longs;upra demon&longs;tratis) <lb/><arrow.to.target n="note163"></arrow.to.target>BM minor e&longs;t BL, & BE ip&longs;i BA æqualis; minorem habebit <lb/><arrow.to.target n="note164"></arrow.to.target>proportionem BM ad BE, quàm BL ad BA. &longs;ed vt BM ad <lb/>BE, ita potentia in E &longs;u&longs;tinens pondus FG ad ip&longs;um pondus; & <lb/>vt BL ad BA, ita potentia in A ad pondus CD; minorem <lb/>habebit proportionem potentia in E ad pondus FG, quàm poten<lb/><arrow.to.target n="note165"></arrow.to.target>tia in A ad pondus CD. </s>

<s id="id.2.1.99.2.1.1.0.d"> Ergo potentia in E minor erit poten<lb/>tia in A. &longs;imiliter o&longs;tendetur, quò magis pondus eleuabitur, &longs;em<lb/>per minorem potentiam pondus &longs;u&longs;tinere. </s>

<s id="id.2.1.99.2.1.2.0"> Sit autem vectis in <lb/>BO, & pondus in PQ, cuius cenrtum grauitatis &longs;it R. </s>

<s id="id.2.1.99.2.1.2.0">Sit autem vectis in <lb/>BO, & pondus in PQ, cuius centrum grauitatis &longs;it R. </s>

<s id="id.2.1.99.2.1.2.0.a"> dico maio<lb/>rem potentiam in O requiri ad &longs;u&longs;tinendum pondus PQ vecte BO, <lb/>quàm pondus CD vecte BA. </s>

<s id="id.2.1.99.2.1.2.0.b"> ducatur à puncto R horizonti per<lb/><arrow.to.target n="note166"></arrow.to.target>pendicularis RS. </s>

<s id="id.2.1.99.2.1.2.0.c"> & quoniam BS maior e&longs;t BL, habebit BS ad <lb/>BO maiorem proportionem, quàm BL ad BA; quare maior erit <lb/>potentia in O &longs;u&longs;tinens pondus PQ, quàm potentia in A &longs;u&longs;ti<lb/>nens pondus CD. & hoc modo o&longs;tendetur' quò vectis BO ma<lb/>gis à vecte AB deor&longs;um tendens di&longs;tabit, &longs;emper maiorem ponderi <pb n="51" xlink:href="036/01/115.jpg"/>&longs;u&longs;tinendo requiri potentiam. </s>

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<s id="id.2.1.101.1.1.1.0"> Hinc quoq; vt &longs;upra patet pontentiam in A ad potentiam in E e&longs; <lb/>&longs;e, vt BL ad BM; potentiamq; in A ad potentiam in O, vt BL <lb/>ad BS. atque potentiam in E ad potentiam in O, vt BM <lb/>ad BS. </s>

<s id="id.2.1.101.2.1.1.0"> Præterea &longs;i in B alia intelligatur potentia, ita vt duæ &longs;int poten<lb/>tiæ pondus &longs;u&longs;tinentes; minor erit potentia in B &longs;u&longs;tinens pon<lb/>dus PQ vecte BO, quàm pondus CD vecte B32x aduer&longs;o au<lb/>tem maior requiritur potentia in B ad &longs;u&longs;tinendum pondus FG ve <lb/>cte BE, quàm pondus CD vecte AB. ducta enim kN ip&longs;i EB <lb/>perpendicularis, erit EN ip&longs;i AL æqualis: quare EM ip&longs;a LA <lb/>maior erit. </s>

<s id="id.2.1.101.2.1.1.0">Præterea &longs;i in B alia intelligatur potentia, ita vt duæ &longs;int poten<lb/>tiæ pondus &longs;u&longs;tinentes; minor erit potentia in B &longs;u&longs;tinens pon<lb/>dus PQ vecte BO, quàm pondus CD vecte BA aduer&longs;o au<lb/>tem maior requiritur potentia in B ad &longs;u&longs;tinendum pondus FG ve<lb/>cte BE, quàm pondus CD vecte AB. ducta enim kN ip&longs;i EB <lb/>perpendicularis, erit EN ip&longs;i AL æqualis: quare EM ip&longs;a LA <lb/>maior erit. </s>

<s id="id.2.1.101.2.1.2.0"> ergo maiorem habebit proportionem EM ad E<emph type="italics"/>B<emph.end type="italics"/>, <arrow.to.target n="note167"></arrow.to.target><expan abbr="quàm"><lb/>quam</expan>LA ad A<emph type="italics"/>B<emph.end type="italics"/>; & LA ad A<emph type="italics"/>B<emph.end type="italics"/>maiorem, quàm SO ad O<emph type="italics"/>B<emph.end type="italics"/>; <arrow.to.target n="note168"></arrow.to.target><lb/>quæ &longs;unt proportiones potentiæ ad pondus. </s>

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<s id="id.2.1.104.1.1.2.0"> <margin.target id="note170"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>

<s id="id.2.1.105.1.1.1.0"> Sit autem vectis A<emph type="italics"/>B<emph.end type="italics"/><lb/>horizonti æquidi&longs;tans, <lb/>cuius fulcimentum <emph type="italics"/>B<emph.end type="italics"/>, <lb/>grauitati&longs;q; centrum H <lb/>ponderis AC &longs;it &longs;upra <lb/>vectem: moueaturq; ve<lb/>ctis in <emph type="italics"/>B<emph.end type="italics"/>E, ac pondus <lb/>in EF, potentiaq; in G. <lb/>&longs;imiliter vt &longs;upra o&longs;ten<lb/>detur potentiam in G <lb/>pondus EF &longs;ui&longs;tinen<lb/><figure id="id.036.01.115.1.jpg" xlink:href="036/01/115/1.jpg"></figure><lb/>tem minorem e&longs;&longs;e potentia in D pondus AC &longs;u&longs;tinente. </s>

<s id="id.2.1.105.1.1.1.0">Sit autem vectis A<emph type="italics"/>B<emph.end type="italics"/> <lb/>horizonti æquidi&longs;tans, <lb/>cuius fulcimentum <emph type="italics"/>B<emph.end type="italics"/>, <lb/>grauitati&longs;q; centrum H <lb/>ponderis AC &longs;it &longs;upra <lb/>vectem: moueaturq; ve<lb/>ctis in <emph type="italics"/>B<emph.end type="italics"/>E, ac pondus <lb/>in EF, potentiaq; in G. <lb/>&longs;imiliter vt &longs;upra o&longs;ten<lb/>detur potentiam in G <lb/>pondus EF <expan abbr="&longs;ui&longs;tinen">sustinen</expan><lb/><figure id="id.036.01.115.1.jpg" xlink:href="036/01/115/1.jpg"></figure><lb/>tem minorem e&longs;&longs;e potentia in D pondus AC &longs;u&longs;tinente. </s>

<s id="id.2.1.105.1.1.2.0"> cùm <pb xlink:href="036/01/116.jpg"/>enim minor &longs;it BM ip&longs;a <lb/>BL, minorem habebit <lb/>proportionem MB ad <lb/>BG, quàm LB ad BD. <lb/>atq; hoc modo o&longs;ten<lb/>detur, quò pondus ve<lb/>cte magis eleuabitur, mi<lb/>norem &longs;emper. ad pon<lb/>dus &longs;u&longs;tinendum requi<lb/>ri potentiam. </s>

<s id="id.2.1.105.1.1.4.0"> Simili<lb/>ter &longs;i moucatur vectis <lb/>in BO, potentiaq; &longs;u<lb/><figure id="id.036.01.116.1.jpg" xlink:href="036/01/116/1.jpg"></figure><lb/>&longs;tinens in N, o&longs;tendetur potentiam in N maiorem e&longs;&longs;e potentia in <lb/>D. maiorem enim habet proportionem SB ad BN, quàm LB <lb/>ad BD. o&longs;tendetur etiam, quò magis pondus deprimetur; ma<lb/>iorem &longs;emper (vt &longs;u&longs;tineatur) requiri potentiam. quod demon <lb/>&longs;trare oportebat. </s>

<s id="id.2.1.105.1.1.4.0">Simili<lb/>ter &longs;i moueatur vectis <lb/>in BO, potentiaq; &longs;u<lb/><figure id="id.036.01.116.1.jpg" xlink:href="036/01/116/1.jpg"></figure><lb/>&longs;tinens in N, o&longs;tendetur potentiam in N maiorem e&longs;&longs;e potentia in <lb/>D. maiorem enim habet proportionem SB ad BN, quàm LB <lb/>ad BD. o&longs;tendetur etiam, quò magis pondus deprimetur; ma<lb/>iorem &longs;emper (vt &longs;u&longs;tineatur) requiri potentiam. quod demon<lb/>&longs;trare oportebat. </s>

<s id="id.2.1.105.1.1.5.0"> quod demon<lb/>&longs;trare oportebat. </s>

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<s id="id.2.1.105.6.1.1.0"> Ex iis etiam demon&longs;trabitur, &longs;i centrum grauitatis eiu&longs;dem pon<lb/>deris, &longs;iue propinquius, &longs;iue remotius fuerit à vecte AB horizon<lb/>ti æquidi&longs;tante, eandem potentiam in A pondus nihilominus <lb/>&longs;u&longs;tinere: vt&longs;i centrum grauitatis H ponderis BD longius ab&longs;it <lb/>à vecte BA, quàm centrum grauitatis N ponderis PV, dum<lb/>modo ducta à puncto H perpendicularis HL horizonti, vectiq; <lb/>AB tran&longs;eat per N; &longs;itq; pondus PV ponderi BD æquale; <lb/>erit tùm pondus BD, tùm pondus PV, ac &longs;i ambo in L e&longs;<lb/>&longs;ent appen&longs;a; atque &longs;unt æqualia, cùm loco vnius ponderis ac<lb/>cipiantur, eadem igitur potentia in A &longs;u&longs;tinens pondus BD, <lb/>pondus quoq; PV &longs;u&longs;tinebit. </s>

<s id="id.2.1.105.6.1.2.0"> Vecte autem EF, quò centrum <lb/>grauitatis longius fuerit à vecte, eò facilius potentia idem pon<lb/>dus &longs;u&longs;tinebit: vt &longs;i centrum grauitatis k ponderis FG longius <lb/>&longs;it à vecte EF, quàm centrum grauitatis X ponderis YZ; ita ta<lb/>men vt ducta à puncto k vecti FE perpendicularis tran&longs;eat per <lb/>X; &longs;itq; pondus FG ponderi YZ æquale; & à punctis kX ip<lb/>&longs;o<*>um horizontibus perpendiculares ducantur KM X9; erit C9 <lb/>maior CM; ac propterea pondus FG in vecte erit, ac &longs;i in M e&longs; <lb/>&longs;et appen&longs;um, & pondus YZ, ac &longs;i in 9 e&longs;&longs;et appen&longs;um. </s>

<s id="id.2.1.105.6.1.2.0">Vecte autem EF, quò centrum <lb/>grauitatis longius fuerit à vecte, eò facilius potentia idem pon<lb/>dus &longs;u&longs;tinebit: vt &longs;i centrum grauitatis k ponderis FG longius <lb/>&longs;it à vecte EF, quàm centrum grauitatis X ponderis YZ; ita ta<lb/>men vt ducta à puncto k vecti FE perpendicularis tran&longs;eat per <lb/>X; &longs;itq; pondus FG ponderi YZ æquale; & à punctis kX ip<lb/>&longs;orum horizontibus perpendiculares ducantur KM X9; erit C9 <lb/>maior CM; ac propterea pondus FG in vecte erit, ac &longs;i in M e&longs; <lb/>&longs;et appen&longs;um, & pondus YZ, ac &longs;i in 9 e&longs;&longs;et appen&longs;um. </s>

<s id="id.2.1.105.6.1.3.0"> quo <pb xlink:href="036/01/118.jpg"/><figure id="id.036.01.118.1.jpg" xlink:href="036/01/118/1.jpg"></figure><lb/><arrow.to.target n="note171"></arrow.to.target>niam autem maiorem habet proportionem C9 ad CE, quàm <lb/>CM ad CE, maior potentia in E &longs;u&longs;tinebit pondus YZ, quàm <lb/>FG. </s>

<s id="id.2.1.105.6.1.3.0.a"> In vecte autem QR è conuer&longs;o demon&longs;trabitur, &longs;cilicet <lb/>quò centrum grauitatis eiu&longs;dem ponderis &longs;it longius à vecte, eò <lb/>maiorem e&longs;&longs;e potentiam pondus &longs;u&longs;tinentem. </s>

<s id="id.2.1.105.6.1.4.0"> maior enim e&longs;t <lb/>CT, quàm CI; & ob id maiorem habebit proportionem CT <lb/>ad CR, quàm CI ad CR. </s>

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<s id="id.2.1.106.1.1.1.0"> <margin.target id="note171"></margin.target>8 <emph type="italics"/>Quinti.<emph.end type="italics"/> </s>

<s id="id.2.1.107.1.1.1.0"> RROPOSITIO VIIII. </s>

<s id="id.2.1.107.1.1.1.0">PROPOSITIO VIIII. </s>

<s id="id.2.1.107.2.1.1.0"> Potentia pondus &longs;u&longs;tinens infra vectem ho<lb/>rizonti æquidi&longs;tantem ip&longs;ius centrum grauitatis <pb n="53" xlink:href="036/01/119.jpg"/>habens, quò magis ab hoc &longs;itu vecte pondus ele<lb/>uabitur maiori &longs;emper potentia, vt &longs;u&longs;tineatur, <lb/>egebit. </s>

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<s id="id.2.1.109.2.1.1.0"> Sit deinde vectis AB horizonti æquidi&longs;tans, cuius fulcimentum <lb/>B; pondu&longs;q; CD habeat centrum grauitatis O infra vectem; &longs;itq; <lb/>potentia in A &longs;u&longs;tinens pondus CD. </s>

<s id="id.2.1.109.2.1.1.0.a"> Moueatur deinde vectis in <pb n="54" xlink:href="036/01/121.jpg"/>BE BF, pondu&longs;q; transferatur in GH kL. </s>

<s id="id.2.1.109.2.1.1.0.b"> Dico maiorem re<lb/>quiri potentiam in E, vt pondus &longs;u&longs;tineatur, quàm in A; & ma<lb/>iorem in A, quàm in F. ducantur à centris grauitatum horizon<lb/>tibus perpendiculares NM OP QR, quæ ex parte NOQ <lb/>protractæ in centrum mundi conuenient. </s>

<s id="id.2.1.109.2.1.2.0"> &longs;imiliter vt &longs;upra o&longs;ten <lb/>detur BM <expan abbr="maior&etilde;">maiorem</expan>e&longs;&longs;e BP, & <emph type="italics"/>B<emph.end type="italics"/>P maiorem BR; & BM ad BE ma<lb/>iorem <arrow.to.target n="note177"></arrow.to.target>habere proportionem, qaàm BP ad BA; & BP ad BA ma<lb/>iorem, quàm BR ad BF: & propter hoc potentiam in E maio<lb/>rem e&longs;&longs;e potentia in A; & potentiam in A maiorem potentia in <lb/>F. & quò vectis magis à &longs;itu AB eleuabitur, &longs;emper o&longs;tendetur, <lb/>maiorem requiri potentiam ponderi &longs;u&longs;tinendo. &longs;i verò depri<lb/>metur, minorem. </s>

<s id="id.2.1.109.2.1.2.0">&longs;imiliter vt &longs;upra o&longs;ten <lb/>detur BM <expan abbr="maior&etilde;">maiorem</expan> e&longs;&longs;e BP, & <emph type="italics"/>B<emph.end type="italics"/>P maiorem BR; & BM ad BE ma<lb/>iorem <arrow.to.target n="note177"></arrow.to.target>habere proportionem, <expan abbr="qaàm">quam</expan> BP ad BA; & BP ad BA ma<lb/>iorem, quàm BR ad BF: & propter hoc potentiam in E maio<lb/>rem e&longs;&longs;e potentia in A; & potentiam in A maiorem potentia in <lb/>F. & quò vectis magis à &longs;itu AB eleuabitur, &longs;emper o&longs;tendetur, <lb/>maiorem requiri potentiam ponderi &longs;u&longs;tinendo. &longs;i verò depri<lb/>metur, minorem. </s>

<s id="id.2.1.110.1.1.1.0"> <margin.target id="note177"></margin.target>7 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>

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<s id="id.2.1.115.1.1.2.0"> quod idem <lb/>potentiæ mouenti eueniet. </s>

<s id="id.2.1.115.2.1.1.0"> RROPOSITIO XI. </s>

<s id="id.2.1.115.2.1.1.0">PROPOSITIO XI. </s>

<s id="id.2.1.115.3.1.1.0"> Si vectis di&longs;tantia inter fulcimentum, & poten<lb/>tiam ad di&longs;tantiam fulcimento, punctoq;, vbi <lb/>à centro grauitatis ponderis horizonti ducta <lb/>perpendicularis vectem &longs;ecat, interiectam ma<lb/>iorem habuerit proportionem, quàm pondus <lb/>ad potentiam; pondus vtiq; à potentia moue<lb/>bitur. </s>

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<s id="id.2.1.115.4.1.1.0.a"> Di<lb/>co pondus Cà potentia in B moueri. </s>

<s id="id.2.1.115.4.1.2.0"> fiat vt BD ad DA, ita <lb/>pondus E ad potentiam in B; atq; pondus E quoq; appendatur <lb/>in A: patet potentiam in B æqueponderare ip&longs;i E; hoc e&longs;t pon<lb/>dus <arrow.to.target n="note181"></arrow.to.target>E &longs;u&longs;tinere. </s>

<s id="id.2.1.115.4.1.3.0"> & quoniam BD ad DA maiorem habet pro<lb/>portionem, quàm Cad potentiam in B; & vt BD ad DA, ita <pb xlink:href="036/01/126.jpg"/>e&longs;t pondus E ad po<lb/>tentiam: igitur E ad <lb/>potentiam maiorem <lb/>habebit proportio<lb/>nem, quàm pondus <lb/>C ad eandem poten<lb/><arrow.to.target n="note182"></arrow.to.target>tiam. </s>

<s id="id.2.1.115.4.1.4.0"> quare pondus <lb/>E maius erit ponde<lb/><figure id="id.036.01.126.1.jpg" xlink:href="036/01/126/1.jpg"></figure><lb/>re C. & cùm potentia ip&longs;<*> E æqueponderet, potentia igitur ip&longs;i <lb/>C non æqueponderabit, &longs;ed &longs;ua ui deor&longs;um verget. </s>

<s id="id.2.1.115.4.1.4.0">quare pondus <lb/>E maius erit ponde<lb/><figure id="id.036.01.126.1.jpg" xlink:href="036/01/126/1.jpg"></figure><lb/>re C. & cùm potentia ip&longs;a E æqueponderet, potentia igitur ip&longs;i <lb/>C non æqueponderabit, &longs;ed &longs;ua ui deor&longs;um verget. </s>

<s id="id.2.1.115.4.1.5.0"> pondus igitur <lb/>C à potentia in B mouebitur vecte AB, cuius fulcimentum <lb/>e&longs;t D. </s>

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<s id="id.2.1.125.3.1.1.0"> Sit pondus A vt centum, potentia verò mouens &longs;it vt decem; <lb/>&longs;itq; datus vectis BC. </s>

<s id="id.2.1.125.3.1.1.0.a"> oportet potentiam, quæ e&longs;t decem pondus <lb/>A centum vecte BC mouere. </s>

<s id="id.2.1.125.3.1.2.0"> Diuidatur BC in D, ita vt CD <lb/>ad DB eandem habeat proportionem, quàm habet centum ad <lb/>decem, hoc e&longs;t decem ad vnum; etenim &longs;i D ficret fulcimentum, <lb/>con&longs;tat potentiam vt decem in C æqueponderare ponderi A in B <arrow.to.target n="note194"></arrow.to.target><lb/>appen&longs;o: hoc e&longs;t pondus A &longs;u&longs;tinere. </s>

<s id="id.2.1.125.3.1.2.0">Diuidatur BC in D, ita vt CD <lb/>ad DB eandem habeat proportionem, quàm habet centum ad <lb/>decem, hoc e&longs;t decem ad vnum; etenim &longs;i D fieret fulcimentum, <lb/>con&longs;tat potentiam vt decem in C æqueponderare ponderi A in B <arrow.to.target n="note194"></arrow.to.target><lb/>appen&longs;o: hoc e&longs;t pondus A &longs;u&longs;tinere. </s>

<s id="id.2.1.125.3.1.3.0"> accipiatur inter BD quod <lb/>uis punctum E, & fiat E fulcimentum. </s>

<s id="id.2.1.125.3.1.4.0"> Quoniam enim maior <arrow.to.target n="note195"></arrow.to.target><lb/>e&longs;t proportio CE ad EB, quàm CD ad DB; maiorem habebit <lb/>proportionem CE ad EB, quàm pondus A ad potentiam decem <lb/>in C: potentia igitur decem in C pondus A centum in B appen<lb/>&longs;um vecte BC, cuius fulcimentum &longs;it E, mouebit. <arrow.to.target n="note196"></arrow.to.target> </s>

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<s id="id.2.1.127.1.2.1.0"> Hoc autem fieri non po<lb/>te&longs;t exi&longs;tente vecte BC, cuius <lb/>fulcimentum &longs;it B, & pondus <lb/>A centum in C appen&longs;um: po<lb/>natur enim potentia &longs;u&longs;tinens <lb/>pondus A vtcunq; inter BC, <lb/><arrow.to.target n="note200"></arrow.to.target>vt in D, &longs;emper potentia ma<lb/><arrow.to.target n="note201"></arrow.to.target>ior erit pondere A. quare opor<lb/><figure id="id.036.01.130.1.jpg" xlink:href="036/01/130/1.jpg"></figure><lb/>tet datam potentiam maiorem e&longs;&longs;e pondere A. &longs;it igitur poten<lb/>tia data vt centum quinquaginta. </s>

<s id="id.2.1.127.1.2.2.0"> diuidatur BC in D, ita vt CB <lb/>ad BD &longs;it, vt centum quinquaginta ad centum; hoc e&longs;t tria ad duo: <lb/><arrow.to.target n="note202"></arrow.to.target>& &longs;i ponatur potentia in D, patet potentiam in D &longs;u&longs;tinere pon<lb/>dus A in C appep&longs;um. </s>

<s id="id.2.1.127.1.2.2.0">diuidatur BC in D, ita vt CB <lb/>ad BD &longs;it, vt centum quinquaginta ad centum; hoc e&longs;t tria ad duo: <lb/><arrow.to.target n="note202"></arrow.to.target>& &longs;i ponatur potentia in D, patet potentiam in D &longs;u&longs;tinere pon<lb/>dus A in C <expan abbr="appep&longs;um">appensum</expan>. </s>

<s id="id.2.1.127.1.2.3.0"> accipiatur itaq; inter DC quoduis pun<lb/><arrow.to.target n="note203"></arrow.to.target>ctum E, ponaturq; potentia mouens in E; & cùm maior &longs;it pro<lb/>portio EB ad BC, quàm DB ad BC; habebit EB ad BC maio<lb/>rem proportionem, quàm pondus A ad potentiam in E. </s>

<s id="id.2.1.127.1.2.3.0.a"> poten<lb/><arrow.to.target n="note204"></arrow.to.target>tia igitur vt centum quinquaginta in E pondus A centum in C <lb/>appen&longs;um vecte BC, cuius fulcimentum e&longs;t B, mouebit. </s>

<s id="id.2.1.127.1.2.4.0"> quod <lb/>facere oportebat. </s>

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<s id="id.2.1.135.3.1.1.0"> Diuidatur AM in Q, ita vt AQ ad QM &longs;it, ut grauitas ue<lb/>ctis AB ad grauitatem ponderis P; deinde ut CF ad CQ, ita fat <lb/>grauitas AB, & P &longs;imul ad potentiam, quæ ponatur in B: patet <lb/>potentiam in B uectem AB unà cum pondere P &longs;u&longs;tinere. </s>

<s id="id.2.1.135.3.1.2.0"> Si ue-<arrow.to.target n="note211"></arrow.to.target><expan abbr="rò"><lb/>ro</expan>e&longs;&longs;et CA ad CM, vt AB ad P; e&longs;&longs;et punctum C eorum centrum <arrow.to.target n="note212"></arrow.to.target><lb/>grauitatis, & ideo vectis AB vná cum pondere P ab&longs;q; potentia in <arrow.to.target n="note213"></arrow.to.target><lb/>B manebit. </s>

<s id="id.2.1.135.3.1.2.0">Si ue<arrow.to.target n="note211"></arrow.to.target><expan abbr="rò"><lb/>ro</expan> e&longs;&longs;et CA ad CM, vt AB ad P; e&longs;&longs;et punctum C eorum centrum <arrow.to.target n="note212"></arrow.to.target><lb/>grauitatis, & ideo vectis AB vná cum pondere P ab&longs;q; potentia in <arrow.to.target n="note213"></arrow.to.target><lb/>B manebit. </s>

<s id="id.2.1.135.3.1.3.0"> &longs;ed &longs;i ponderum grauitatis centrum e&longs;&longs;et inter CF, vt <lb/>in O; fiat vt CF ad CO, ita AB&P &longs;imul ad potentiam, quæ <lb/>in B, & vectem AB, & pondus P &longs;u&longs;tinebit. <pb xlink:href="036/01/136.jpg"/><figure id="id.036.01.136.1.jpg" xlink:href="036/01/136/1.jpg"></figure> </s>

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<s id="id.2.1.139.4.1.1.0.a"> dico poten<lb/>tiam in G ponderi A <lb/>æqualem e&longs;&longs;e. </s>

<s id="id.2.1.139.4.1.2.0"> Sit FG <lb/>æquidi&longs;tans CB. </s>

<s id="id.2.1.139.4.1.2.0.a"> <lb/>Quoniam igitur pon<lb/><arrow.to.target n="note217"></arrow.to.target>dus A manet; erit <lb/><figure id="id.036.01.140.1.jpg" xlink:href="036/01/140/1.jpg"></figure><lb/>CB horizonti plano perpendicularis <*> quare FG eidem plano per<lb/><arrow.to.target n="note218"></arrow.to.target>pendicularis erit. </s>

<s id="id.2.1.139.4.1.3.0"> Sint CF <expan abbr="p&utilde;cta">puncta</expan>in orbiculo, à quibus funes CB FG <lb/>in horizontis <expan abbr="plan&utilde;">planum</expan>ad rectos angulos de&longs;cendunt; tangent BC FG <lb/><expan abbr="orbicul&utilde;">orbiculum</expan>CEF in punctis CF. <expan abbr="orbicul&utilde;">orbiculum</expan>enim <expan abbr="&longs;ecarenõ">&longs;ecarenon</expan>po&longs;&longs;unt. </s>

<s id="id.2.1.139.4.1.3.0">Sint CF <expan abbr="p&utilde;cta">puncta</expan> in orbiculo, à quibus funes CB FG <lb/>in horizontis <expan abbr="plan&utilde;">planum</expan> ad rectos angulos de&longs;cendunt; tangent BC FG <lb/><expan abbr="orbicul&utilde;">orbiculum</expan> CEF in punctis CF. <expan abbr="orbicul&utilde;">orbiculum</expan> enim &longs;ecare <expan abbr="nõ">non</expan> po&longs;&longs;unt. </s>

<s id="id.2.1.139.4.1.4.0"> con<lb/>nectantur DC DF; erit CF recta linea, & anguli DCB DFG recti. </s>

<s id="id.2.1.139.4.1.5.0"> <lb/><arrow.to.target n="note219"></arrow.to.target><expan abbr="Quoniã">Quoniam</expan><expan abbr="aut&etilde;">autem</expan>BC tùm horizonti, tùm ip&longs;i CF e&longs;t perpendicularis; <lb/>erit linea CF horizonti æquidi&longs;tans. </s>

<s id="id.2.1.139.4.1.6.0"> cùm verò <expan abbr="põdus">pondus</expan>appen&longs;um &longs;it <lb/><arrow.to.target n="note220"></arrow.to.target>in BC, & potentia &longs;it in G; quod idem e&longs;t, ac &longs;i e&longs;&longs;et in F; erit <lb/>CF tanquam libra, &longs;iue vectis, cuius centrum, &longs;iue fulcimentum e&longs;t <lb/>D; nam in axiculo orbuculus &longs;u&longs;tinetur; atq; punctum D, cùm &longs;it <lb/>centrum axiculi, & orbiculi, etiam vtri&longs;que circumuolutis <lb/>immobile remanet. </s>

<s id="id.2.1.139.4.1.6.0">cùm verò <expan abbr="põdus">pondus</expan> appen&longs;um &longs;it <lb/><arrow.to.target n="note220"></arrow.to.target>in BC, & potentia &longs;it in G; quod idem e&longs;t, ac &longs;i e&longs;&longs;et in F; erit <lb/>CF tanquam libra, &longs;iue vectis, cuius centrum, &longs;iue fulcimentum e&longs;t <lb/>D; nam in axiculo <expan abbr="orbuculus">orbiculus</expan> &longs;u&longs;tinetur; atq; punctum D, cùm &longs;it <lb/>centrum axiculi, & orbiculi, etiam vtri&longs;que circumuolutis <lb/>immobile remanet. </s>

<s id="id.2.1.139.4.1.7.0"> Itaq; cùm di&longs;tantia DC &longs;it æqualis di&longs;tantiæ <lb/>DF, potentiaq; in F ponderi A in C appen&longs;o æqueponderet, cùm <lb/><arrow.to.target n="note221"></arrow.to.target>pondus &longs;u&longs;tineat, ne deor&longs;um vergat; erit potentia in F, &longs;iue in G <lb/>(nam idem e&longs;t) con&longs;tituta ponderi A æqualis. </s>

<s id="id.2.1.139.4.1.8.0"> Idem enim effi<lb/>cit potentia in G, ac &longs;i in G aliud e&longs;&longs;et appen&longs;um pondus æquale <lb/>ponderi A; quæ pondera in CF appen&longs;a æquæponderabunt. </s>

<s id="id.2.1.139.4.1.9.0"> Præ<lb/>terea, cùm in neutram fiat motus partem, idem erit vnico exi<pb n="64" xlink:href="036/01/141.jpg"/>&longs;tente fune BC EFG hoc modo orbiculo circumuoluto, ac &longs;i duo <lb/>e&longs;&longs;ent funes BC FG alligati in vecte, &longs;iue libra CF. </s>

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<s id="id.2.1.141.7.1.2.0"> & quoniam dum orbi<lb/>culus circumuertitur, circumferen<lb/>tia circuli CEF &longs;emper e&longs;t æquidi<lb/>&longs;tans circumferentiæ axiculi GHk; <lb/>circa enim axiculum circumuerti<lb/>tur; & circulorum æquidi&longs;tantes cir<lb/>cumferentiæ idem habent centrum; <lb/>erit punctum D &longs;emper & orbiculi, <lb/><figure id="id.036.01.142.1.jpg" xlink:href="036/01/142/1.jpg"></figure><lb/>& axiculi centrum. </s>

<s id="id.2.1.141.7.1.3.0"> Itaq; cùm DC &longs;it æqualis DF, & DG ip&longs;i <lb/>Dk; erit GC ip&longs;i kF æqualis. </s>

<s id="id.2.1.141.7.1.4.0"> &longs;i igitur in vecte, &longs;iue libra CF <lb/>pondera appendantur æqualia, æqueponderabunt. </s>

<s id="id.2.1.141.7.1.5.0"> di&longs;tantia enim <lb/>CG æqualis e&longs;t di&longs;tantiæ kF; axiculu&longs;<*>; GHK immobilis gerit <lb/>vicem centri, &longs;iue fulcimenti. </s>

<s id="id.2.1.141.7.1.5.0">di&longs;tantia enim <lb/>CG æqualis e&longs;t di&longs;tantiæ kF; axiculu&longs;q; GHK immobilis gerit <lb/>vicem centri, &longs;iue fulcimenti. </s>

<s id="id.2.1.141.7.1.6.0"> immobili igitur manente axicu<lb/>lo, &longs;i ponatur in F potentia &longs;u&longs;tinens pondus in C appen&longs;um; erit <lb/>potentia in F ip&longs;i ponderi æqualis. </s>

<s id="id.2.1.141.7.1.7.0"> quod erat o&longs;tendendum. </s>

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<s id="id.2.1.144.1.1.1.0"> <margin.target id="note225"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>

<s id="id.2.1.144.1.1.2.0"> <margin.target id="note226"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>

</figure>

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<s id="id.2.1.145.6.1.2.0"> Dico unamquamq; poten<lb/>tiam in BD ponderis C &longs;ubtriplam e&longs;&longs;e. </s>

<s id="id.2.1.145.7.1.1.0"> Quoniam enim altera <lb/>potentia e&longs;t in D colloca<lb/>ta, & pondus C in eodem <lb/>puncto D e&longs;t appen&longs;um; <lb/>potentia in D partem <lb/>ponderis C &longs;u&longs;t^{i}nebit ip<lb/>&longs;i potentiæ D æqualem. </s>

<s id="id.2.1.145.7.1.1.0">Quoniam enim altera <lb/>potentia e&longs;t in D colloca<lb/>ta, & pondus C in eodem <lb/>puncto D e&longs;t appen&longs;um; <lb/>potentia in D partem <lb/>ponderis C &longs;u&longs;tinebit ip<lb/>&longs;i potentiæ D æqualem. </s>

<s id="id.2.1.145.7.1.2.0"> <lb/><figure id="id.036.01.148.1.jpg" xlink:href="036/01/148/1.jpg"></figure><lb/>quare potentia in B partem &longs;u&longs;tinebit reliquam, quæ pars dupla erit <lb/>ip&longs;ius potentiæ in B; cùm pondus ad potentiam eandem habeat <lb/>proportionem, quam AB ad AD: & potentiæ in BD &longs;unt æqua<lb/>les; ergo potentia in B duplam &longs;u&longs;tinebit partem eius, quam &longs;u&longs;ti<lb/>net potentia in D. </s>

<s id="id.2.1.145.7.1.2.0.a"> diuidatur ergo pondus C in duas partes, qua <lb/>rum vna &longs;it reliquæ dupla; quod fiet, &longs;i in tres partes æquales EFG <lb/>diui&longs;erimus: tunc enim FG dupla erit ip&longs;ius E. </s>

<s id="id.2.1.145.7.1.2.0.b"> Itaq; potentia <lb/>in D partem E &longs;u&longs;tinebit, & potentiam in B reliquas FG. vtreq; <lb/>igitur inter &longs;e &longs;e æquales potentiæ in BD &longs;imul totum &longs;u&longs;tinebunt <lb/>pondus C. </s>

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<s id="id.2.1.154.1.1.2.0"> <margin.target id="note237"></margin.target>2 <emph type="italics"/>Huius. de vecte.<emph.end type="italics"/> </s>

<s id="id.2.1.155.1.1.1.0"> Si verò funis religetur in E, <lb/>& &longs;ecundùm quatuor adhuc <lb/>circumuoluatur orbiculos, per <lb/>ueniatq; ad P. &longs;imiliter o&longs;ten <lb/>detur potentiam in P &longs;ubqua<lb/>druplam e&longs;&longs;e ponderis A. <lb/>idem enim e&longs;t, ac &longs;i funis re<lb/>ligatus e&longs;&longs;et in L, potentiaq; <lb/>&longs;u&longs;tineret pondus fune tribus <lb/>tantùm orbiculis circumdu<lb/>cto, quorum centra e&longs;&longs;ent B <lb/><expan abbr="Cq.">Cque</expan>orbiculus enim cuius <lb/>centrum D e&longs;t p&oelig;nitus inu<lb/>tilis. <figure id="id.036.01.157.1.jpg" xlink:href="036/01/157/1.jpg"></figure> </s>

<s id="id.2.1.155.1.1.1.0">Si verò funis religetur in E, <lb/>& &longs;ecundùm quatuor adhuc <lb/>circumuoluatur orbiculos, per <lb/>ueniatq; ad P. &longs;imiliter o&longs;ten <lb/>detur potentiam in P &longs;ubqua<lb/>druplam e&longs;&longs;e ponderis A. <lb/>idem enim e&longs;t, ac &longs;i funis re<lb/>ligatus e&longs;&longs;et in L, potentiaq; <lb/>&longs;u&longs;tineret pondus fune tribus <lb/>tantùm orbiculis circumdu<lb/>cto, quorum centra e&longs;&longs;ent B <lb/>CQ. </s>

<s>orbiculus enim cuius <lb/>centrum D e&longs;t p&oelig;nitus inu<lb/>tilis. <figure id="id.036.01.157.1.jpg" xlink:href="036/01/157/1.jpg"></figure></s>

<s id="id.2.1.155.1.3.1.0"> PROPOSITIO VIII. </s>

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<s id="id.2.1.158.1.1.3.0"> <margin.target id="note242"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/>8 <emph type="italics"/>huius<emph.end type="italics"/> </s>

<s id="id.2.1.158.1.1.4.0"> <margin.target id="note243"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/>8 <emph type="italics"/>Huius<emph.end type="italics"/> </s>

</figure>

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<s id="id.2.1.161.8.1.2.0"> <lb/>CED trochleæ ponderi A alli<lb/>gatæ ex kH; &longs;itq; KH ad rectos <lb/>angulos horizonti, ita vt pon<lb/>dus &longs;emper trochleæ motum, &longs;i<lb/>ue &longs;ur&longs;um, &longs;iue deor&longs;um factum <lb/>&longs;equatur; &longs;itq; orbiculi centrum <lb/>K; & funis orbiculo circumuo<lb/>lutus &longs;it BCDEF, qui relige<lb/>tur in B, ita vt in B immobilis <lb/>maneat; & &longs;it potentia in F mo<lb/>uens pondus A. </s>

<s id="id.2.1.161.8.1.2.0.a"> dico potentia m <lb/>in F &longs;emper mouere <expan abbr="põdus">pondus</expan>A ve<lb/>cte horizonti æquidi&longs;tante. </s>

<s id="id.2.1.161.8.1.3.0"> &longs;int <lb/>BC EF inter &longs;e &longs;e, ip&longs;iq; kH æ<lb/>quidi&longs;tantes, & eiu&longs;dem kH ho<lb/>rizonti perpendiculares, tangen<lb/>te&longs;q; <expan abbr="circul&utilde;">circulum</expan>CED in EC <expan abbr="p&utilde;ctis">punctis</expan>; <lb/>et connectatur EC, quæ per cen<arrow.to.target n="note245"></arrow.to.target><lb/>trum k tran&longs;ibit, horizontiq; <lb/>æquidi&longs;tans erit; &longs;icuti prius di<lb/>ctum e&longs;t. </s>

<s id="id.2.1.161.8.1.4.0"> Quoniam enim or<lb/>biculus CED circa eius cen<lb/>trum K vertitur; ideo dum vis <lb/>in F trahit &longs;ur&longs;um punctum E, <lb/>deberet punctum C de&longs;cende <lb/>re, ac trahere deor&longs;um B; &longs;ed fu<lb/><figure id="id.036.01.167.1.jpg" xlink:href="036/01/167/1.jpg"></figure><lb/>nis in B e&longs;t immobilis, & BC de&longs;cedere non pote&longs;t; quare dum <lb/>potentia in F trahit &longs;ur&longs;um E, totus orbiculus &longs;ur&longs;um mouebitur; <lb/>ac per con&longs;equens tota trochlea, & pondus; & EkC erit tanquam <arrow.to.target n="note246"></arrow.to.target><lb/>vectis, cuius fulcimentum erit C; e&longs;t enim punctum C propter BC <lb/>ferè immobile, potentia verò mouens vectem e&longs;t in F fune EF, <pb xlink:href="036/01/168.jpg"/>& pondus in k appen&longs;um. </s>

<s id="id.2.1.161.8.1.4.0">Quoniam enim or<lb/>biculus CED circa eius cen<lb/>trum K vertitur; ideo dum vis <lb/>in F trahit &longs;ur&longs;um punctum E, <lb/>deberet punctum C de&longs;cende <lb/>re, ac trahere deor&longs;um B; &longs;ed fu<lb/><figure id="id.036.01.167.1.jpg" xlink:href="036/01/167/1.jpg"></figure><lb/>nis in B e&longs;t immobilis, & BC <expan abbr="de&longs;cedere">descendere</expan> non pote&longs;t; quare dum <lb/>potentia in F trahit &longs;ur&longs;um E, totus orbiculus &longs;ur&longs;um mouebitur; <lb/>ac per con&longs;equens tota trochlea, & pondus; & EkC erit tanquam <arrow.to.target n="note246"></arrow.to.target><lb/>vectis, cuius fulcimentum erit C; e&longs;t enim punctum C propter BC <lb/>ferè immobile, potentia verò mouens vectem e&longs;t in F fune EF, <pb xlink:href="036/01/168.jpg"/>& pondus in k appen&longs;um. </s>

<s id="id.2.1.161.8.1.5.0"> <lb/>quòd &longs;i punctum C omnino fue<lb/>rit immobile, moueaturq; ve<lb/>ctis EC in NC; & diuidatur <lb/>NC bifariam in L: erunt CL <lb/>LN ip&longs;is Ck KE æquales. </s>

<s id="id.2.1.161.8.1.6.0"> <lb/>quare &longs;i vectis EC e&longs;&longs;et in CN, <lb/>punctum k e&longs;&longs;et in L; & &longs;i du<lb/>catur LM horizonti perpendi<lb/>cularis, quæ &longs;it etiam æqualis <lb/>kH; e&longs;&longs;et pondus A, hoc e&longs;t <lb/>punctum H in M. </s>

<s id="id.2.1.161.8.1.6.0.a"> &longs;ed quoniam <lb/>potentia in F dum tendit &longs;ur<lb/>&longs;um mouendo orbiculum, &longs;em<lb/>per mouetur &longs;uper rectam EFG, <lb/>quæ &longs;emper e&longs;t quoq; æquidi<lb/>&longs;tans BC; nece&longs;&longs;e erit orbicu<lb/>lum trochleæ &longs;emper inter li<lb/>neas EG BC e&longs;&longs;e: & centrum <lb/>k, cum &longs;it in medio, &longs;uper <lb/>rectam lineam HkT &longs;emper <lb/>moueri. </s>

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<s id="id.2.1.165.13.1.1.0"> Sit pondus A, &longs;int duo orbiculi, quor&umacr; <expan abbr="c&etilde;">cem</expan><lb/>tra k I trochleæ ponderi alligatæ k <foreign lang="greek">a</foreign>; ita vt <lb/>pondus motum trochleæ &longs;ur&longs;um, & deor&longs;um <lb/>&longs;emper &longs;equatur: &longs;it deinde orbiculus, cuius cen<lb/>trum L, trochleæ &longs;ur&longs;um appen&longs;æ in <35>; &longs;itq; <lb/>funis circa omnes orbiculos circumuolutus BC<lb/>DEFGHZMNO, religatu&longs;q; in B; &longs;itq; po<lb/>tentia in O mouens pondus A. </s>

<s id="id.2.1.165.13.1.1.0">Sit pondus A, &longs;int duo orbiculi, <expan abbr="quor&utilde;">quorum</expan> <expan abbr="c&etilde;tra;">cen<lb/>tra</expan> k I trochleæ ponderi alligatæ k <foreign lang="greek">a</foreign>; ita vt <lb/>pondus motum trochleæ &longs;ur&longs;um, & deor&longs;um <lb/>&longs;emper &longs;equatur: &longs;it deinde orbiculus, cuius cen<lb/>trum L, trochleæ &longs;ur&longs;um appen&longs;æ in <35>; &longs;itq; <lb/>funis circa omnes orbiculos circumuolutus BC<lb/>DEFGHZMNO, religatu&longs;q; in B; &longs;itq; po<lb/>tentia in O mouens pondus A. </s>

<s id="id.2.1.165.13.1.1.0.a"> dico &longs;patium, <lb/>quod mouendo pertran&longs;it potentia in O, qua<lb/>druplum e&longs;&longs;e &longs;patii moti ponderis A. </s>

<s id="id.2.1.165.13.1.1.0.b"> mouean<lb/>tur orbiculi trochleæ ponderi alligatæ; & dum <lb/>centrum k e&longs;t in R, centrum I &longs;it in S, & pon<lb/>dus A, hoc e&longs;t punctum <foreign lang="greek">a</foreign>in <foreign lang="greek">b</foreign>: erunt IS kR <lb/><foreign lang="greek">ab</foreign>inter &longs;e &longs;e æquales, itemq; k I ip&longs;i RS e<lb/>rit æqualis. </s>

<s id="id.2.1.165.13.1.2.0"> orbiculi enim inter &longs;e &longs;e eandem <lb/>&longs;emper &longs;eruant di&longs;tantiam; & k <foreign lang="greek">a</foreign>ip&longs;i R <foreign lang="greek">b</foreign>æ<lb/>qualis erit. </s>

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<s id="id.2.1.167.11.1.1.0"> Sit trochlea habens orbiculum, cuius <lb/>centrum A; & &longs;it pondus B alligatum fu<lb/>ni CDEFG, qui circa orbiculum &longs;it re<lb/>uolutus, ac tandem religatus in G: &longs;itq; <lb/>potentia in H &longs;u&longs;tinens pondus. </s>

<s id="id.2.1.167.11.1.2.0"> dico po<lb/>tentiam in H duplam e&longs;&longs;e ponderis B. du<lb/>catur DF per <expan abbr="centr&utilde;">centrum</expan>A horizonti æquidi<lb/>&longs;tans. </s>

<s id="id.2.1.167.11.1.3.0"> <expan abbr="quoniã">quoniam</expan>igitur potentia in H &longs;u&longs;tinet <lb/><expan abbr="trochleã">trochleam</expan>, quæ &longs;u&longs;tinet <expan abbr="orbicul&utilde;in">orbiculunin</expan>eius <expan abbr="c&etilde;tro">centro</expan><lb/>A, qui pondus &longs;u&longs;tinet; erit potentia &longs;u&longs;ti<lb/>nens <expan abbr="orbicul&utilde;">orbiculum</expan>, ac &longs;i in A <expan abbr="cõ&longs;tituta">con&longs;tituta</expan>e&longs;&longs;et; ip&longs;a <lb/>ergo in A exi&longs;tente, pondere verò in D <lb/>appen&longs;o, funiq; CD religato; erit DF <lb/>tanquam vectis, cuius fulcimentum erit <lb/>F, pondus in D, & potentia in A. </s>

<s id="id.2.1.167.11.1.3.0"><expan abbr="quoniã">quoniam</expan> igitur potentia in H &longs;u&longs;tinet <lb/><expan abbr="trochleã">trochleam</expan>, quæ &longs;u&longs;tinet <expan abbr="orbicul&utilde;">orbiculum</expan> in eius <expan abbr="c&etilde;tro">centro</expan> <lb/>A, qui pondus &longs;u&longs;tinet; erit potentia &longs;u&longs;ti<lb/>nens <expan abbr="orbicul&utilde;">orbiculum</expan>, ac &longs;i in A <expan abbr="cõ&longs;tituta">con&longs;tituta</expan> e&longs;&longs;et; ip&longs;a <lb/>ergo in A exi&longs;tente, pondere verò in D <lb/>appen&longs;o, funiq; CD religato; erit DF <lb/>tanquam vectis, cuius fulcimentum erit <lb/>F, pondus in D, & potentia in A. </s>

<s id="id.2.1.167.11.1.3.0.a"> po<lb/><arrow.to.target n="note253"></arrow.to.target>tentia verò ad pondus e&longs;t, vt DF ad <lb/>ad FA, & DF dupla e&longs;t ip&longs;ius FA; Po<lb/><figure id="id.036.01.180.1.jpg" xlink:href="036/01/180/1.jpg"></figure><lb/>tentia igitur in A, &longs;iue in H, quod idem e&longs;t, ponderis B dupla erit. </s>

<lb/>

<s id="id.2.1.167.11.1.4.0"> quod demon&longs;trare oportebat. </s>

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<s id="id.2.1.168.1.1.1.0"> <margin.target id="note253"></margin.target>3 <emph type="italics"/>Huius. de vecte.<emph.end type="italics"/> </s>

<s id="id.2.1.169.1.1.1.0"> Præterea con&longs;iderandum occurrit, cùm hæc omnia maneant, <lb/>idem e&longs;&longs;e vnico exi&longs;tente fune CD EFG hoc modo orbiculo cicum <lb/>uoluto, ac &longs;i duo e&longs;&longs;ent funes CD FG in vecte &longs;iue libra DF al<lb/>ligati. </s>

<s id="id.2.1.169.1.1.1.0">Præterea con&longs;iderandum occurrit, cùm hæc omnia maneant, <lb/>idem e&longs;&longs;e vnico exi&longs;tente fune CD EFG hoc modo orbiculo <expan abbr="cicum">circum</expan> <lb/>uoluto, ac &longs;i duo e&longs;&longs;ent funes CD FG in vecte &longs;iue libra DF al<lb/>ligati. </s>

<s id="id.2.1.169.2.1.1.0"> ALITER. </s>

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<s id="id.2.1.171.7.1.1.0"> Sit orbiculus, cuius centrum A, tro<lb/>chleæ infernè affixæ; & &longs;it funis BCD <lb/>EFG non &longs;olum huic orbiculo circumuo<lb/>lutus, verùm etiam orbiculo trochleæ &longs;u<lb/>perioris, cuius centrum k; &longs;itq; funis in <lb/>B &longs;uperiori trochleæ religatus; & in G &longs;it ap<lb/>pen&longs;um pondus H; potentiaq; in L &longs;u&longs;ti<lb/>neat pondus H. </s>

<s id="id.2.1.171.7.1.1.0.a"> dico potentiam in L tri<lb/>plam e&longs;&longs;e ponderis H. </s>

<s id="id.2.1.171.7.1.1.0.b"> &longs;i enim duæ e&longs;&longs;ent <lb/>potentiæ pondus H &longs;u&longs;tidentes, vna in <lb/>K, altera in B, erunt vtræq; &longs;imul triplæ <lb/><arrow.to.target n="note255"></arrow.to.target>ponderis H potentia enim in k dupla e&longs;t <lb/>ponderis H, & potentia in B ip&longs;i ponderi <lb/>æqualis. </s>

<s id="id.2.1.171.7.1.1.0.b">&longs;i enim duæ e&longs;&longs;ent <lb/>potentiæ pondus H <expan abbr="&longs;u&longs;tidentes">sustinentes</expan>, vna in <lb/>K, altera in B, erunt vtræq; &longs;imul triplæ <lb/><arrow.to.target n="note255"></arrow.to.target>ponderis H potentia enim in k dupla e&longs;t <lb/>ponderis H, & potentia in B ip&longs;i ponderi <lb/>æqualis. </s>

<s id="id.2.1.171.7.1.2.0"> & quoniam &longs;ola potentia in L <lb/>vtri&longs;q; &longs;cilicet potentiæ in KB e&longs;t æqua<lb/>lis. </s>

<s id="id.2.1.171.7.1.3.0"> &longs;u&longs;tinet enim potentia in L; tùm po<lb/>tentiam in K, tùm potentiam in B; idem <lb/>qué efficit potentia in L, ac &longs;i duæ e&longs;&longs;ent <lb/>potentiæ, vna in k, altera in B: Tri<lb/>pla igitur erit potentia in L ponderis H. <lb/>quod der<*>on&longs;trare o<*>ortebat. <figure id="id.036.01.184.1.jpg" xlink:href="036/01/184/1.jpg"></figure> </s>

<s id="id.2.1.171.7.1.3.0">&longs;u&longs;tinet enim potentia in L; tùm po<lb/>tentiam in K, tùm potentiam in B; idem <lb/>qué efficit potentia in L, ac &longs;i duæ e&longs;&longs;ent <lb/>potentiæ, vna in k, altera in B: Tri<lb/>pla igitur erit potentia in L ponderis H. <lb/>quod dermon&longs;trare oportebat. <figure id="id.036.01.184.1.jpg" xlink:href="036/01/184/1.jpg"></figure></s>

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<s id="id.2.1.175.3.1.1.0"> Sit trochlea inferior, duos habens orbiculos, <lb/>quorum centra AB; &longs;it qué trochlea &longs;uperior <lb/>duos &longs;imiliter habens orbiculos, quorum cen<lb/>tra CD; funi&longs;q; EFGHKLMNOP &longs;it cir<lb/>ca omnes orbiculos reuolutus, qui &longs;it religatus <lb/>in E; & in P appendatur pondus Q; &longs;itq; po<lb/>tentia in R. </s>

<s id="id.2.1.175.3.1.1.0.a"> dico potentiam in R quadruplam <lb/>e&longs;&longs;e ponderis q. Cùm enim &longs;i duæ intelligan<lb/>tur potentiæ, vna in k, altera in D, potentia <lb/><arrow.to.target n="note257"></arrow.to.target>in k &longs;u&longs;tinens pondus Q fune k LMNOP æ<lb/>qualis erit ponderi; erunt duæ &longs;imul potentiæ, <lb/>vna in D, altera in k, pondus Q &longs;u&longs;tinentes, <lb/>triplæ eiu&longs;dem ponderis. </s>

<s id="id.2.1.175.3.1.2.0"> Potentia verò in C <lb/>dupla e&longs;t potentiæ in k, & per con&longs;equens pon<lb/>deris Q; idem enim e&longs;t, ac &longs;i in k appen&longs;um e&longs; <lb/><arrow.to.target n="note258"></arrow.to.target>&longs;et pondus æquale ponderi Q, cuius dupla e&longs;t <lb/>potentia in C; duæ igitur potentiæ in DC qua<lb/>druplæ &longs;unt ponderis q. & cùm potentia in R <lb/>orbiculis &longs;u&longs;tineat pondus Q, erit <expan abbr="pot&etilde;tia">potentia</expan>in R, <lb/>ac &longs;i duæ e&longs;&longs;ent potentiæ, vna in D, altera in C, <lb/>& vtræq; &longs;imul pondus Q &longs;u&longs;tinerent. </s>

<s id="id.2.1.175.3.1.3.0"> ergo po<lb/>tentia in R quadrupla e&longs;t ponderis q. quod <lb/>oport<*>bat demon&longs;trare. <figure id="id.036.01.186.1.jpg" xlink:href="036/01/186/1.jpg"></figure> </s>

<s id="id.2.1.175.3.1.3.0">ergo po<lb/>tentia in R quadrupla e&longs;t ponderis q. quod <lb/>oportebat demon&longs;trare. <figure id="id.036.01.186.1.jpg" xlink:href="036/01/186/1.jpg"></figure></s>

<s id="id.2.1.175.3.3.1.0"> COROLLARIVM </s>

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<s id="id.2.1.177.15.1.1.0"> PROPOSITIO XVIIII. </s>

<s id="id.2.1.177.16.1.1.0"> Si vtriu&longs;q; duarum trochlearum &longs;ingulis orbi <lb/>culis, quarum altera &longs;upernè appen&longs;a, altera <expan abbr="verò">ve<lb/>ro</expan>infernè à &longs;u&longs;tinente potentia rententa fuerit, <lb/>funis circumuoluatur; altero eius extremo alicu<lb/>bi religato, alteri autem pondere appen&longs;o; du<lb/>pla erit ponderis potentia. </s>

<s id="id.2.1.177.16.1.1.0">Si vtriu&longs;q; duarum trochlearum &longs;ingulis orbi <lb/>culis, quarum altera &longs;upernè appen&longs;a, altera ve<lb/>rò infernè à &longs;u&longs;tinente potentia <expan abbr="rententa">retenta</expan> fuerit, <lb/>funis circumuoluatur; altero eius extremo alicu<lb/>bi religato, alteri autem pondere appen&longs;o; du<lb/>pla erit ponderis potentia. </s>

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<s id="id.2.1.178.1.1.1.0"> <margin.target id="note259"></margin.target>3 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>

</figure>

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<s id="id.2.1.180.1.1.1.0"> <margin.target id="note260"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/> </s>

<s id="id.2.1.181.1.1.1.0"> Si autem in N &longs;it potentia mouens pondus M, erit &longs;patium <lb/>ponderis M duplum &longs;patii potentiæ in N. quod ex duodecima <lb/>huius manife&longs;tum e&longs;t; &longs;patium enim puncti L deor&longs;um ten<lb/>dentis duplum e&longs;t &longs;pat^{1}i N &longs;ur&longs;um; erit igitur è conuer&longs;o &longs;patium <lb/>potentiæ in N deor&longs;um tendentis dimidium &longs;aptii ponderis M &longs;ur<lb/>&longs;um moti. </s>

<s id="id.2.1.181.1.1.1.0">Si autem in N &longs;it potentia mouens pondus M, erit &longs;patium <lb/>ponderis M duplum &longs;patii potentiæ in N. quod ex duodecima <lb/>huius manife&longs;tum e&longs;t; &longs;patium enim puncti L deor&longs;um ten<lb/>dentis duplum e&longs;t &longs;patii N &longs;ur&longs;um; erit igitur è conuer&longs;o &longs;patium <lb/>potentiæ in N deor&longs;um tendentis dimidium <expan abbr="&longs;aptii">spatii</expan> ponderis M &longs;ur<lb/>&longs;um moti. </s>

<s id="id.2.1.181.2.1.1.0"> Sicut autem ex tertia, quinta, &longs;eptima huius, &c. </s>

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<s id="id.2.1.187.3.1.1.0.a"> dico pro<lb/>portionem potentiæ ad pondus &longs;e&longs;quiter<lb/>tiam e&longs;&longs;e. <figure id="id.036.01.200.1.jpg" xlink:href="036/01/200/1.jpg"></figure> </s>

<s id="id.2.1.187.4.1.1.0"> Quoniam enim potentia in E &longs;u&longs;tinens <lb/><arrow.to.target n="note268"></arrow.to.target>pondus D fune ECB AKPO &longs;ubtripla e&longs;t <lb/><arrow.to.target n="note269"></arrow.to.target>ip&longs;ius D, ip&longs;ius autem E dupla e&longs;t potentia <lb/>in H; erit potentia in H &longs;ub&longs;e&longs;quialtera pon<lb/>deris D. &longs;imili quoq; modo quoniam po<lb/>tentia in O_{3} quæ e&longs;t, ac &longs;i e&longs;&longs;et in centro or<lb/><arrow.to.target n="note270"></arrow.to.target>biculi ABC, &longs;ubtripla e&longs;t ponderis D; ip<lb/>&longs;ius autem O dupla e&longs;t potentia in N; erit <lb/>quoq; potentia in N &longs;ub&longs;e&longs;quialtera ponde<lb/>ris D. quare duæ &longs;imul potentiæ in HN pon <lb/>dus D &longs;uperant tertia parte, &longs;e &longs;e habentq; ad <lb/>D in ratione &longs;e&longs;quitertia: & cùm potentia <lb/>in M duabus &longs;it potentiis in HN &longs;imul &longs;um<lb/>ptis æqualis, &longs;uperabit itidem potentia in <lb/>M pondus D tertia parte. </s>

<s id="id.2.1.187.4.1.1.0">Quoniam enim potentia in E &longs;u&longs;tinens <lb/><arrow.to.target n="note268"></arrow.to.target>pondus D fune ECB AKPO &longs;ubtripla e&longs;t <lb/><arrow.to.target n="note269"></arrow.to.target>ip&longs;ius D, ip&longs;ius autem E dupla e&longs;t potentia <lb/>in H; erit potentia in H &longs;ub&longs;e&longs;quialtera pon<lb/>deris D. &longs;imili quoq; modo quoniam po<lb/>tentia in O, quæ e&longs;t, ac &longs;i e&longs;&longs;et in centro or<lb/><arrow.to.target n="note270"></arrow.to.target>biculi ABC, &longs;ubtripla e&longs;t ponderis D; ip<lb/>&longs;ius autem O dupla e&longs;t potentia in N; erit <lb/>quoq; potentia in N &longs;ub&longs;e&longs;quialtera ponde<lb/>ris D. quare duæ &longs;imul potentiæ in HN pon<lb/>dus D &longs;uperant tertia parte, &longs;e &longs;e habentq; ad <lb/>D in ratione &longs;e&longs;quitertia: & cùm potentia <lb/>in M duabus &longs;it potentiis in HN &longs;imul &longs;um<lb/>ptis æqualis, &longs;uperabit itidem potentia in <lb/>M pondus D tertia parte. </s>

<s id="id.2.1.187.4.1.2.0"> ergo proportio <lb/>potentiæ in M ad pondus D &longs;e&longs;quitertia <lb/>e&longs;t. </s>

<s id="id.2.1.187.4.1.3.0"> quod demon&longs;trare oportebat. </s>

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<s id="id.2.1.193.9.1.1.0"> PROPOSITIO XXIIII. </s>

<s id="id.2.1.193.10.1.1.0"> Si tribus duarum trochlearum orbiculis, qua <lb/>rum altera vnius dumtaxat orbiculi &longs;upernè à <lb/>potentia &longs;u&longs;tineatur, altera verò duorum <expan abbr="infernè">infer<lb/>ne</expan>, ponderiq, alligata fuerit con&longs;tituta, cir<lb/>cundetur funis; vtroq; eius extremo alicubi, &longs;ed <lb/>non &longs;uperiori trochleæ religato: duplum erit <lb/>pondus potentiæ. </s>

<s id="id.2.1.193.10.1.1.0">Si tribus duarum trochlearum orbiculis, qua <lb/>rum altera vnius dumtaxat orbiculi &longs;upernè à <lb/>potentia &longs;u&longs;tineatur, altera verò duorum <expan abbr="infernè">infer<lb/>ne</expan>, ponderiq; alligata fuerit con&longs;tituta, cir<lb/>cundetur funis; vtroq; eius extremo alicubi, &longs;ed <lb/>non &longs;uperiori trochleæ religato: duplum erit <lb/>pondus potentiæ. </s>

<s id="id.2.1.193.11.1.1.0"> Sint AB centra orbiculorum <lb/>trochleæ ponderi C alligatæ; D ve<lb/>rò &longs;it centrum orbiculi trochleæ &longs;u<lb/>perioris; &longs;it deinde funis per om<lb/>nes orbiculos circumuolutus, reli<lb/>gatu&longs;q; in EF; & &longs;it potentia in <lb/>G &longs;u&longs;tinens pondus C. </s>

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Line 3279

<s id="id.2.1.195.3.1.1.0"> Hinc autem con&longs;iderandum <lb/>e&longs;t quomodo fiat motus; quia, <lb/>cùm funis &longs;it religatur in F, vectis <lb/>NO in prima figura habebit ful<lb/>cimentum O, pondus in medio, <lb/>& potentia in N. &longs;imiliter quo<lb/>niam funis e&longs;t religatus in E, ve<lb/>ctis PQ habebit <expan abbr="fulciment&utilde;">fulcimentum</expan>P, & <lb/>pondus in medio, & potentia in <lb/>q. idcirco partes orbiculorum <lb/>in N, & Q &longs;ur&longs;um mouebuntur; <lb/>orbiculi ergo non in eandem, &longs;ed <lb/>in contrarias mouebuntur partes, <lb/>videlicet vnus dextro&longs;um, alter&longs;i<lb/>ni&longs;tror&longs;um. </s>

<s id="id.2.1.195.3.1.1.0">Hinc autem con&longs;iderandum <lb/>e&longs;t quomodo fiat motus; quia, <lb/>cùm funis &longs;it religatur in F, vectis <lb/>NO in prima figura habebit ful<lb/>cimentum O, pondus in medio, <lb/>& potentia in N. &longs;imiliter quo<lb/>niam funis e&longs;t religatus in E, ve<lb/>ctis PQ habebit <expan abbr="fulciment&utilde;">fulcimentum</expan> P, & <lb/>pondus in medio, & potentia in <lb/>q. idcirco partes orbiculorum <lb/>in N, & Q &longs;ur&longs;um mouebuntur; <lb/>orbiculi ergo non in eandem, &longs;ed <lb/>in contrarias mouebuntur partes, <lb/>videlicet vnus <expan abbr="dextro&longs;um">dextrorsum</expan>, alter &longs;i<lb/>ni&longs;tror&longs;um. </s>

<s id="id.2.1.195.3.1.2.0"> & quoniam potentiæ <lb/>in NQ eædem &longs;unt, quæ &longs;unt in <lb/>LM; potentiæ igitur in LM æ<lb/>quales &longs;ur&longs;um mouebuntur. </s>

<s id="id.2.1.195.3.1.3.0"> ve<lb/>ctis igitur LM in neutram moue<lb/>bitur partem. </s>

<s id="id.2.1.195.3.1.4.0"> quare neq; orbicu<lb/>lus circumuertetur. </s>

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<s id="id.2.1.201.6.1.1.0"> Hactenus proportiones ponderis ad potentiam multiplices, <lb/>& &longs;ubmultiplices; deinde &longs;uperparticulares, <expan abbr="&longs;ub&longs;uperparticulare&longs;qué">&longs;ub&longs;uperparticu<lb/>lare&longs;que</expan>declaratæ fuerunt: nunc autem reliquum e&longs;t, vt propor<lb/>tiones inter pondus, & potentiam &longs;uperpartientes, & multi<lb/>plices &longs;uperparticulares, multiplicesqué &longs;uperpartientes mani<lb/>fe&longs;tentur. </s>

<s id="id.2.1.201.6.1.1.0">Hactenus proportiones ponderis ad potentiam multiplices, <lb/>& &longs;ubmultiplices; deinde &longs;uperparticulares, &longs;ub&longs;uperparticu<lb/>lare&longs;qué declaratæ fuerunt: nunc autem reliquum e&longs;t, vt propor<lb/>tiones inter pondus, & potentiam &longs;uperpartientes, & multi<lb/>plices &longs;uperparticulares, multiplicesqué &longs;uperpartientes mani<lb/>fe&longs;tentur. </s>

<s id="id.2.1.201.7.1.1.0"> PROPOSITIO XXVI. </s>

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<s id="id.2.1.207.1.1.1.0"> Eodem modo, quo &longs;uperpartientes inuenimus, has quo<lb/>que omnes multiplices &longs;uperparticulares reperiemus. </s>

<s id="id.2.1.207.1.1.2.0"> vt fiat <arrow.to.target n="note289"></arrow.to.target><lb/>pondus B ad potentiam in A, vt quinq; ad vnum; potentia ve <arrow.to.target n="note290"></arrow.to.target><expan abbr="rò"><lb/>ro</expan>in C ad potentiam in A, vt duo ad vnum; quod fiet, &longs;i fu<lb/>nis &longs;it religatus in D, non autem trochleæ &longs;uperiori, vel in F: erit <lb/>pondus B ad potentiam in C, vt quinq; ad duo; hoc e&longs;t duplum <lb/>&longs;e&longs;quialterum. </s>

<s id="id.2.1.207.1.1.2.0">vt fiat <arrow.to.target n="note289"></arrow.to.target><lb/>pondus B ad potentiam in A, vt quinq; ad vnum; potentia ve<arrow.to.target n="note290"></arrow.to.target><lb/>ro in C ad potentiam in A, vt duo ad vnum; quod fiet, &longs;i fu<lb/>nis &longs;it religatus in D, non autem trochleæ &longs;uperiori, vel in F: erit <lb/>pondus B ad potentiam in C, vt quinq; ad duo; hoc e&longs;t duplum <lb/>&longs;e&longs;quialterum. </s>

<s id="id.2.1.208.1.1.1.0"> <margin.target id="note289"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/>9 <emph type="italics"/>huius.<emph.end type="italics"/> </s>

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<s id="id.2.1.209.3.1.1.0"> Fiat potentia in A pondus B &longs;u&longs;tinens &longs;uboctupla ponderis B; <arrow.to.target n="note291"></arrow.to.target><lb/>& potentia in C potentiæ in A &longs;it tripla; erit pondus B ad po<lb/>tentiam in C, vt octo ad tria. </s>

<s id="id.2.1.209.3.1.2.0"> & è conuer&longs;o omnem potentiæ ad <pb xlink:href="036/01/216.jpg"/>pondus proportionem multipticem &longs;uperpartientem in ueniemus. </s>

<s id="id.2.1.209.3.1.2.0">& è conuer&longs;o omnem potentiæ ad <pb xlink:href="036/01/216.jpg"/>pondus proportionem <expan abbr="multipticem">multiplicem</expan> &longs;uperpartientem in ueniemus. </s>

<s id="id.2.1.209.3.1.3.0"> <lb/>& vt in cæteris reperiemus ita e&longs;&longs;e pondus ad potentiam pondus <lb/>&longs;u&longs;tinentem, vt &longs;patium potentiæ mouentis ad &longs;patium pon<lb/>deris. </s>

<s id="id.2.1.210.1.1.1.0"> <margin.target id="note291"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/>9 <emph type="italics"/>huius Ex<emph.end type="italics"/>17 <emph type="italics"/>huius.<emph.end type="italics"/> </s>

<s id="id.2.1.211.1.1.1.0"> Notandum autem e&longs;t, quòd cùm in præcedentibus demo&longs;tratio <lb/>nibus &longs;æpius dictum fuerit, potentiam pondus &longs;u&longs;tinentem ip&longs;ius <lb/>ponderis duplam e&longs;&longs;e, vel triplam, & huiu&longs;modi; vt in decima<lb/>quinta huius o&longs;ten&longs;um e&longs;t; quia tamen potentia non &longs;olum pon<lb/>dus, verùm etiam trochleam &longs;u&longs;tinet; idcirco maioris longè vir<lb/>tutis, maiori&longs;q; ip&longs;i ponderi proportionis con&longs;tituenda videtur <lb/>ip&longs;a potentia. </s>

<s id="id.2.1.211.1.1.1.0">Notandum autem e&longs;t, quòd cùm in præcedentibus <expan abbr="demo&longs;tratio">demonstratio</expan><lb/>nibus &longs;æpius dictum fuerit, potentiam pondus &longs;u&longs;tinentem ip&longs;ius <lb/>ponderis duplam e&longs;&longs;e, vel triplam, & huiu&longs;modi; vt in decima<lb/>quinta huius o&longs;ten&longs;um e&longs;t; quia tamen potentia non &longs;olum pon<lb/>dus, verùm etiam trochleam &longs;u&longs;tinet; idcirco maioris longè vir<lb/>tutis, maiori&longs;q; ip&longs;i ponderi proportionis con&longs;tituenda videtur <lb/>ip&longs;a potentia. </s>

<s id="id.2.1.211.1.1.2.0"> quod quidem verum e&longs;t, &longs;i etiam trochleæ graui<lb/>tatem con&longs;iderare voluerimus. </s>

<s id="id.2.1.211.1.1.3.0"> &longs;ed quoniam inter potentiam, & <lb/>pondus proportionem quærimus: ideo hanc trochleæ grauitatem <lb/>ommi&longs;imus, quam &longs;iquis etiam con&longs;iderare voluerit, vim ip&longs;i po<lb/>tentiæ æqualem trochleæ addere poterit. </s>

<s id="id.2.1.211.1.1.4.0"> Quod ip&longs;um etiam in <lb/>fune ob&longs;eruari poterit. </s>

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<s id="id.2.1.219.1.1.1.0"> <expan abbr="Animaduertend&utilde;">Animaduertendum</expan>quoq; e&longs;t in mo <lb/>uendis ponderibus, potentiam ali<lb/>quando for&longs;itan melius mouere mo <lb/>uendo &longs;e deor&longs;um, quàm mouendo <lb/>&longs;e &longs;ur&longs;um. </s>

<s id="id.2.1.219.1.1.2.0"> vt circumuoluatur adhuc <lb/>funis per alium trochleæ &longs;uperioris <lb/>orbiculum, cuius centrum C, funi&longs;q; <lb/><arrow.to.target n="note303"></arrow.to.target>perueniat in D; erit <expan abbr="pot&etilde;tia">potentia</expan>in D &longs;u&longs;ti<lb/><expan abbr="n&etilde;s">nens</expan><expan abbr="põdus">pondus</expan>B &longs;imiliter duodecim, <expan abbr="qu&etilde;">quem</expan><lb/>admodum erat in A. </s>

<s id="id.2.1.219.1.1.2.0">vt circumuoluatur adhuc <lb/>funis per alium trochleæ &longs;uperioris <lb/>orbiculum, cuius centrum C, funi&longs;q; <lb/><arrow.to.target n="note303"></arrow.to.target>perueniat in D; erit <expan abbr="pot&etilde;tia">potentia</expan> in D <expan abbr="&longs;u&longs;tin&etilde;s">&longs;u&longs;tinens</expan> <expan abbr="põdus">pondus</expan> B &longs;imiliter duodecim, <expan abbr="qu&etilde;">quem</expan><lb/>admodum erat in A. </s>

<s id="id.2.1.219.1.1.2.0.a"> Ideo poten<lb/>tia vt tredecim in D pondus B mo<lb/>uebit. </s>

<s id="id.2.1.219.1.1.3.0"> & quia mouet &longs;e deor&longs;um, <lb/>forta&longs;&longs;e trahet facilius, quàm in A; <lb/>atq; tempus e&longs;t idem, &longs;icut etiam <lb/>erat in A. <figure id="id.036.01.222.1.jpg" xlink:href="036/01/222/1.jpg"></figure> </s>

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<s id="id.2.1.223.11.1.4.0"> Potentia igitur in F &longs;u&longs;tinens pondus k, ne deor&longs;um ver<lb/>gat, ponderi K æqueponderabit; ip&longs;iq; M æqualis erit. </s>

<s id="id.2.1.223.11.1.5.0"> idem enim <lb/>præ&longs;tat potentia, quod pondus M. </s>

<s id="id.2.1.223.11.1.5.0.a"> pondus igitur K ad poten<lb/><arrow.to.target n="note308"></arrow.to.target>tiam in F erit, vt CF ad CB; & conuertendo, potentia ad <lb/>pondus erit, vt CB ad CF, hoc e&longs;t, &longs;emidiameter axis ad &longs;emi<pb n="108" xlink:href="036/01/229.jpg"/>diametrum tympani vnà cum &longs;cytala DF. </s>

<s id="id.2.1.223.11.1.5.0.b"> Similiter etiam o&longs;ten<lb/>detur, &longs;i potentia pondus &longs;u&longs;tinens fuerit in q. tunc enim &longs;u&longs;ti<lb/>neret vecte CQ; & ad pondus eam haberet proportionem, quam <arrow.to.target n="note309"></arrow.to.target><lb/>habet CB ad <expan abbr="Cq.">Cque</expan>Videlicet &longs;emidiameter axis ad &longs;emidiame<lb/>trum tympani vná cum &longs;cytala <expan abbr="Eq.">Eque</expan>quod demon&longs;trare opor<lb/>tebat. </s>

<s id="id.2.1.223.11.1.5.0.b">Similiter etiam o&longs;ten<lb/>detur, &longs;i potentia pondus &longs;u&longs;tinens fuerit in q. tunc enim &longs;u&longs;ti<lb/>neret vecte CQ; & ad pondus eam haberet proportionem, quam <arrow.to.target n="note309"></arrow.to.target><lb/>habet CB ad Cq. Videlicet &longs;emidiameter axis ad &longs;emidiame<lb/>trum tympani vná cum &longs;cytala Eq. quod demon&longs;trare opor<lb/>tebat. </s>

<s id="id.2.1.224.1.1.1.0"> <margin.target id="note307"></margin.target>6. <emph type="italics"/>Primi Archim. de æquepon.<emph.end type="italics"/> </s>

<s id="id.2.1.224.1.1.3.0"> <margin.target id="note308"></margin.target><emph type="italics"/>Cor.<emph.end type="italics"/>4. <emph type="italics"/>quinti.<emph.end type="italics"/> </s>

<s id="id.2.1.224.1.1.4.0"> <margin.target id="note309"></margin.target>2 <emph type="italics"/>Huuius. de vecte.<emph.end type="italics"/> </s>

<s id="id.2.1.224.1.1.4.0"><margin.target id="note309"></margin.target>2 <emph type="italics"/><expan abbr="Huuius">Huius</expan>. de vecte.<emph.end type="italics"/></s>

<s id="id.2.1.225.1.1.1.0"> COROLLARIVM. </s>

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<s id="id.2.1.225.4.1.1.0"> Hoc autem loco con&longs;iderandum occurrit, quòd &longs;i in alia &longs;cyta<lb/>la appendatur pondus, vt in T, &longs;u&longs;tinens pondus k; it a nempè, vt <lb/>pondus in T appen&longs;um, pondusq; k circa axem con&longs;titutum <lb/>maneant; erit pondus in T grauius pondere M in F appen&longs;o. </s>

<s id="id.2.1.225.4.1.2.0"> <lb/>iungatur enim TB, & à puncto C horizonti perpendicularis du<lb/>catur CI, quæ lineam TB &longs;ecet in I; tandemq; connectatur <lb/>TC, quæ æqualis erit CF. </s>

<s id="id.2.1.225.4.1.2.0.a"> Quoniam autem pondera appen&longs;a <lb/>&longs;unt in TB, perindè &longs;e &longs;e habebunt, ac &longs;i in punctis TB ip&longs;orum <lb/>centra grauitatum haberent; vt antca dictum e&longs;t. </s>

<s id="id.2.1.225.4.1.2.0.a">Quoniam autem pondera appen&longs;a <lb/>&longs;unt in TB, perindè &longs;e &longs;e habebunt, ac &longs;i in punctis TB ip&longs;orum <lb/>centra grauitatum haberent; vt antea dictum e&longs;t. </s>

<s id="id.2.1.225.4.1.3.0"> & quia ma<lb/>nent, erit punctum I (ex prima huius de libra) amborum &longs;imul <lb/>grauitatis centrum; cùm &longs;it CI horizonti perpendicularis. </s>

<s id="id.2.1.225.4.1.4.0"> &longs;ed <lb/>quoniam angulus BCI e&longs;t rectus, erit BIC acutus, lineaq; BI <arrow.to.target n="note310"></arrow.to.target><lb/>ip&longs;a BC maior erit. </s>

<s id="id.2.1.225.4.1.5.0"> quare angulus CIT erit obtu&longs;us; atq; <arrow.to.target n="note311"></arrow.to.target><lb/>ideo line^{a} CT ip&longs;a T^{I} maior erit. </s>

<s id="id.2.1.225.4.1.5.0">quare angulus CIT erit obtu&longs;us; atq; <arrow.to.target n="note311"></arrow.to.target><lb/>ideo linea CT ip&longs;a TI maior erit. </s>

<s id="id.2.1.225.4.1.6.0"> Cùm autem CT maior &longs;it <lb/>TI, & IB maior BC; maiorem habebit proportionem TC ad <lb/>CB, quàm TI ad IB; & conuertendo, minorem habebit pro<pb xlink:href="036/01/230.jpg"/><figure id="id.036.01.230.1.jpg" xlink:href="036/01/230/1.jpg"></figure><lb/>portionem BC ad CT, hoc e&longs;t ad CF, quàm BI ad IT; vt ex <lb/>vige&longs;ima &longs;exta quinti elementorum (iuxta Commandini editio<lb/>nem) patet. </s>

<s id="id.2.1.225.4.1.7.0"> Quoniam verò punctum I e&longs;t ponderum in TB <lb/><arrow.to.target n="note312"></arrow.to.target>exi&longs;tentium centrum grauitatis; erit pondus in T ad pondus in B, <lb/>vt BI ad IT. </s>

<s id="id.2.1.225.4.1.7.0.a"> pondus verò in F ad idem pondus in B e&longs;t, vt BC <lb/>ad CF; maiorem igitur proportionem habebit pondus in T ad <lb/>pondus in B, quàm pondus in F ad idem pondus in B. </s>

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<s id="id.2.1.227.1.1.1.0"> Si verò loco ponderis in T animata potentia &longs;u&longs;tinens pon<lb/>dus k con&longs;tituatur; quæ ita degrauet &longs;e, ac &longs;i in centrum mundi <lb/>tendere vellet; quemadmodum &longs;uapte natura efficit pondus in T <lb/>appen&longs;um; erit hæc eadem ponderi in T appen&longs;o æqualis; alio<lb/>quin non &longs;u&longs;tineret; quæ quidem ip&longs;a potentia in F collocata ma<pb n="109" xlink:href="036/01/231.jpg"/>ior erit. </s>

<s id="id.2.1.227.1.1.2.0"> &longs;icuti enim &longs;e &longs;e habet pondus in T ad pondus in F, ita <lb/>& potentia in T ad potentiam in F; cùm potentiæ &longs;int ponderi<lb/>bus æquales. </s>

<s id="id.2.1.227.1.1.3.0"> verùm &longs;i vnaquæq; potentia &longs;eor&longs;um &longs;umpta, tàm <lb/>in T, quàm in F &longs;u&longs;tinens pondus <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan><expan abbr="circ&utilde;ferentiam">circunferentiam</expan>THFN <lb/>moueri &longs;e vellet, veluti apprehen&longs;a manu &longs;cytala; tunc eademmet <lb/>potentia, vel in F, vel in T con&longs;tituta idem pondus k &longs;u&longs;tinere po<lb/>terit; cùm &longs;emper in cuiu&longs;cunq; extremitate &longs;cytalæ ponatur, ab <lb/>eodem centro C æquidi&longs;tans fuerit, ac &longs;ecundum eandem circum<lb/>ferentiam ab eodem centro æqualiter &longs;emper di&longs;tantem perpen&longs;io<lb/>nem habeat. </s>

<s id="id.2.1.227.1.1.4.0"> neq; enim (&longs;icuti pondus) proprio nutu magis in <lb/>centrum ferri exoptat, qu<*>m circulariter moueri; cùm vtrunq;, &longs;eu <lb/>quemlibet alium motum nullo pror&longs;us re&longs;piciat di&longs;crimine. </s>

<s id="id.2.1.227.1.1.4.0">neq; enim (&longs;icuti pondus) proprio nutu magis in <lb/>centrum ferri exoptat, quam circulariter moueri; cùm vtrunq;, &longs;eu <lb/>quemlibet alium motum nullo pror&longs;us re&longs;piciat di&longs;crimine. </s>

<s id="id.2.1.227.1.1.5.0"> pro<lb/>pterea non eodem modo res &longs;e &longs;e habet, &longs;iue pondera, &longs;iue anímatæ <lb/>potentiæ ii&longs;dem locis eodem munere abeundo fuerint con&longs;titutæ. </s>

<s id="id.2.1.227.2.1.1.0"> Potentia autem mouet pondus vecte FB, videlicet dum po<lb/>tentia in F circumuertit tympanum, circumuertit etiam axem; & <lb/>FB fit tamquam vectis, cuius fulcimentum C, potentia mouens <lb/>in F, & podus in B appen&longs;um. </s>

<s id="id.2.1.227.2.1.1.0">Potentia autem mouet pondus vecte FB, videlicet dum po<lb/>tentia in F circumuertit tympanum, circumuertit etiam axem; & <lb/>FB fit tamquam vectis, cuius fulcimentum C, potentia mouens <lb/>in F, & <expan abbr="podus">pondus</expan> in B appen&longs;um. </s>

<s id="id.2.1.227.2.1.2.0"> & dum punctum F peruenit in N; <lb/>punctum H erit in F, & punctum B erit in O; ita vt ducta NO <lb/>tran&longs;eat per C; eodemq; tempore pondus k motum erit in P, ita <lb/>vt OBP &longs;it æqualis ip&longs;i BL, cùm &longs;it idem funis. </s>

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<s id="id.2.1.230.1.1.1.0"> <margin.target id="note315"></margin.target>23 <emph type="italics"/>Octaui libri Pappi.<emph.end type="italics"/> </s>

<s id="id.2.1.231.1.1.1.0"> COROLLAR VM. </s>

<s id="id.2.1.231.1.1.1.0">COROLLARIVM. </s>

<s id="id.2.1.231.2.1.1.0"> Ex his manife&longs;tum e&longs;t, quò facilius pondus mo<lb/>uetur, tempus quoq; eò maius e&longs;&longs;e; & quò dif<lb/>ficilius, eò tempus minuse&longs;&longs;e. </s>

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<s id="id.2.1.231.6.1.3.0"> & &longs;i CB axis <lb/>&longs;emidiameter e&longs;&longs;et, & CA &longs;emidiameter tympani cùm &longs;cytalis; <lb/><arrow.to.target n="note316"></arrow.to.target>patet potentiam vt decem in A ponderi &longs;exaginta in B æquepon<lb/>derare. </s>

<s id="id.2.1.231.6.1.4.0"> Accipiatur autem inter BC quoduis punctum D; fiatq; <lb/>BD &longs;emidiameter axis, & DA &longs;emidiameter tympani cùm &longs;cy<lb/>talis; ponaturq; pondus &longs;exaginta in B fune circa axem, & potentia <lb/><arrow.to.target n="note317"></arrow.to.target><emph type="italics"/>in A. </s>

<s id="id.2.1.231.6.1.4.0.a"> Quoniam enim AD ad DB maiorem habet proportio<lb/>nem, quam AC ad CB; maiorem habebit proportionem AD ad <lb/>DB, quam pondus &longs;exaginta in B appen&longs;um ad potentiam vt decem<emph.end type="italics"/><lb/><arrow.to.target n="note318"></arrow.to.target>in A. </s>

<s id="id.2.1.231.6.1.4.0.b"> Quare potentia in A pondus &longs;exaginta axe imperitro<lb/>chio mouebit, cuius axis &longs;emidiameter e&longs;t BD, & DA &longs;emidia<lb/>meter tympani cùm &longs;cytalis. </s>

<s id="id.2.1.231.6.1.4.0.b">Quare potentia in A pondus &longs;exaginta axe in peritro<lb/>chio mouebit, cuius axis &longs;emidiameter e&longs;t BD, & DA &longs;emidia<lb/>meter tympani cùm &longs;cytalis. </s>

<s id="id.2.1.231.6.1.5.0"> quod erat faciendum. </s>

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<s id="id.2.1.233.5.3.1.0"> DE CVNEO. </s>

<s id="id.2.1.233.6.1.1.0"> Aristoteles in quæ&longs;tioni<lb/>bus Mechanicis quæ&longs;tione deci<lb/>ma&longs;eptima a&longs;&longs;erit, cuneum &longs;cin<lb/>dendo ponderi duorum vicem <lb/>pror&longs;us gerere vectium &longs;ibi inui<lb/>cem contrariorum hoc niodo. </s>

<s id="id.2.1.233.6.1.1.0">Aristoteles in quæ&longs;tioni<lb/>bus Mechanicis quæ&longs;tione deci<lb/>ma&longs;eptima a&longs;&longs;erit, cuneum &longs;cin<lb/>dendo ponderi duorum vicem <lb/>pror&longs;us gerere vectium &longs;ibi inui<lb/>cem contrariorum hoc <expan abbr="niodo">modo</expan>. </s>

<s id="id.2.1.233.7.1.1.0"> Sit cuneus ABC, cu<lb/>ius vertex B, & &longs;it AB <lb/>æqualis BC; quod au<lb/>tem &longs;cindendum e&longs;t, <lb/>&longs;it DEFG; &longs;itq; pars <lb/>cunei HB k intra DE <lb/>FG, & HB æqualis <lb/>&longs;it ip&longs;i Bk. </s>

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<s id="id.2.1.233.7.1.2.0.a"> eodemq; modo CB <lb/>fit vectis, cuius fulci<lb/><figure id="id.036.01.237.1.jpg" xlink:href="036/01/237/1.jpg"></figure><lb/>mentum e&longs;t K, & pondus &longs;imiliter in B. </s>

<s id="id.2.1.233.7.1.2.0.b"> &longs;ed dum percutitur cu<lb/>neus, maiori adhuc ip&longs;ius portione ip&longs;um DEFG ingreditur, <lb/>quàm prius e&longs;&longs;et: &longs;it autem portio hæc MBL; &longs;itq; M B ip&longs;i BL <lb/>æqualis. </s>

<s id="id.2.1.233.7.1.3.0"> & cùm MB BI. &longs;int ip&longs;is HB BK maiores; erit ML maior <pb xlink:href="036/01/238.jpg"/>Hk. </s>

<s id="id.2.1.233.7.1.4.0"> dum igitur ML <lb/>erit in &longs;itu Hk; opor<lb/>ter, vt fiatmaior &longs;ci&longs;sio; <lb/>& D moueatur ver&longs;us <lb/>O, G autem ver&longs;us N: <lb/>& quò maior pars cu<lb/>nei intra DEFG ingre<lb/>dietur, eò maior fiet <lb/>&longs;ci&longs;sio; & DG ma<lb/>gis adhuc impellentur <lb/>ver&longs;us ON. </s>

<s id="id.2.1.233.7.1.4.0">dum igitur ML <lb/>erit in &longs;itu Hk; opor<lb/><expan abbr="ter">tet</expan>, vt fiat maior &longs;ci&longs;sio; <lb/>& D moueatur ver&longs;us <lb/>O, G autem ver&longs;us N: <lb/>& quò maior pars cu<lb/>nei intra DEFG ingre<lb/>dietur, eò maior fiet <lb/>&longs;ci&longs;sio; & DG ma<lb/>gis adhuc impellentur <lb/>ver&longs;us ON. </s>

<s id="id.2.1.233.7.1.4.0.a"> pars igi<lb/>tur KG eius, quod &longs;cin<lb/>ditur, mouebitur à ve<lb/>cte AB, cuius fulcimen<lb/>tum e&longs;t H, & pondus <lb/><figure id="id.036.01.238.1.jpg" xlink:href="036/01/238/1.jpg"></figure><lb/>in B; ita vt punctum B ip&longs;ius vectis AB impellat partem KG. <lb/>& pars HD mouebitur à vecte CB, cuius fulcimentum e&longs;t k; ita <lb/>vt B vecte CB partem HD impellat. </s>

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<s id="id.2.1.233.22.1.2.0.a"> &longs;it dein<lb/>de linea EF, per quam <lb/>tran&longs;eat planum hori<lb/>zonti æquidi&longs;tans; &longs;itq; <lb/>BD in eadem linea EF; <lb/>& dum cuneus percuti<lb/>tur, dumq; mouetur ver<lb/><figure id="id.036.01.243.1.jpg" xlink:href="036/01/243/1.jpg"></figure><lb/>&longs;us E, &longs;emper BD &longs;it in linea EF. quod verò &longs;cindendum e&longs;t <lb/>&longs;it GHLM, intra quod &longs;it pars cunei kBI. manife&longs;tum e&longs;t, <pb xlink:href="036/01/244.jpg"/>dum cuneus uer&longs;us E <lb/>mouetur, partem kG <lb/>ver&longs;us N moueri; & par<lb/>tem HI uer&longs;us O. per<lb/>cutiatur cuneus, ita vt <lb/>AC &longs;it in linea NO; <lb/>tunc k erit in A, & I in <lb/>C: & k ex &longs;uperius di<lb/>ctis motum erit &longs;uper <lb/>kA, & I &longs;uper IC. <lb/>quare dum cuneus mo<lb/><figure id="id.036.01.244.1.jpg" xlink:href="036/01/244/1.jpg"></figure><lb/>uetur, pars KG &longs;uper BA latus cunei mouebitur, & pars IH &longs;uper <lb/>latus BC. </s>

<s id="id.2.1.233.22.1.2.0.b"> pars igitur kG &longs;uper planum mouetur horizonti incli<lb/>natum, cuius inclinatio e&longs;t angulus FBA. &longs;imiliter IH moue<lb/>tur &longs;uper planum BC in angulo FBC. </s>

<s id="id.2.1.233.22.1.2.0.c"> Partes ergo eius, quod <lb/>&longs;cinditur &longs;uper plana horizonti inclinata mouebuntur. </s>

<s id="id.2.1.233.22.1.3.0"> & quam<lb/>quam planum BC &longs;it &longs;ub horizonte; pars tamen IH &longs;uper IC mo<lb/>uetur, tamquam &longs;i BC e&longs;&longs;et &longs;upra <expan abbr="horizont&etilde;">horizontem</expan>in angulo DBC. partes <lb/>enim eius quod &longs;inditur, eodem tempore, ab eadem potentia mo<lb/>uentur; eadem ergo erit ratio motus partis IH, ac partis KG. &longs;i<lb/>militer eadem e&longs;t ratio, &longs;iue EF &longs;it horizonti æquidi&longs;tans, &longs;iue <lb/>horizonti perpendicularis, vel alio modo. </s>

<s id="id.2.1.233.22.1.3.0">& quam<lb/>quam planum BC &longs;it &longs;ub horizonte; pars tamen IH &longs;uper IC mo<lb/>uetur, tamquam &longs;i BC e&longs;&longs;et &longs;upra <expan abbr="horizont&etilde;">horizontem</expan> in angulo DBC. partes <lb/>enim eius quod <expan abbr="&longs;inditur">scinditur</expan>, eodem tempore, ab eadem potentia mo<lb/>uentur; eadem ergo erit ratio motus partis IH, ac partis KG. &longs;i<lb/>militer eadem e&longs;t ratio, &longs;iue EF &longs;it horizonti æquidi&longs;tans, &longs;iue <lb/>horizonti perpendicularis, vel alio modo. </s>

<s id="id.2.1.233.22.1.4.0"> nece&longs;&longs;e e&longs;t enim poten<lb/>tiam cuneum mouentem eandem e&longs;&longs;e, cùm cætera eadem rema <lb/>neant. </s>

<s id="id.2.1.233.22.1.5.0"> eadem igitur erit ratio. </s>

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<s id="id.2.1.235.1.1.4.0"> &longs;imiliter demon<lb/>&longs;trabimus angulum VEF æqualem e&longs;&longs;e ENP, & VED æqualem <lb/>EQS. cùm autem angulus ABC minor &longs;it angulo DEF; erit <lb/>& angulus TBC minor VEN. quare & BGk minor ENP. <lb/>&longs;imili modo BIM minor EQS. </s>

<s id="id.2.1.235.1.1.4.0.a"> quoniam autem cuneus ABC <lb/>duobus mouet vectibus AB BC, quorum fulcimenta &longs;unt in B; <lb/>& pondera in GI: &longs;imiliter cuneus DEF duobus vectibus mouet <lb/>DE EF, quorum fulcimenta &longs;unt in E; & pondera in N Q: per <lb/>præcedentem pondera GH IL facilius vectibus AB BC mo<lb/>uebuntur, quàm pondera NO QR vectibus DE EF. </s>

<s id="id.2.1.235.1.1.4.0.b"> ponde<lb/>ra ergo GH IL facilius cuneo ABC mouebuntur, quàm ponde<lb/>ra NO QR cuneo DEF. </s>

<s id="id.2.1.235.1.1.4.0.c"> & quia eadem e&longs;t ratio in mouendo, <lb/>atq, in &longs;cindendo; facilius idcirco aliquod cuneo ABC &longs;cindetur <lb/>quàm cuneo DEF. &longs;imiliterq; o&longs;tendetur, quò minor e&longs;t angu<lb/>lus ad verticem cunei, eò facilius aliquod moueri, vel &longs;cindi. </s>

<s id="id.2.1.235.1.1.4.0.c">& quia eadem e&longs;t ratio in mouendo, <lb/>atq; in &longs;cindendo; facilius idcirco aliquod cuneo ABC &longs;cindetur <lb/>quàm cuneo DEF. &longs;imiliterq; o&longs;tendetur, quò minor e&longs;t angu<lb/>lus ad verticem cunei, eò facilius aliquod moueri, vel &longs;cindi. </s>

<s id="id.2.1.235.1.1.5.0"> quod <lb/>demon&longs;trare oportebat. </s>

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<s id="id.2.1.237.10.1.1.0"> DE COCHLEA. </s>

<s id="id.2.1.237.11.1.1.0"> Pappvs in eodem octauo libro <lb/>multa pertractans de cochlea, do <lb/>cet quomodo conficienda &longs;it; & <lb/>quomodo magna huiu&longs;modi in<lb/>&longs;trumento moueantnr pondera; <lb/>nec non alia theoremata ad eius <lb/>cognitionem valdè vtilia. </s>

<s id="id.2.1.237.11.1.1.0">Pappvs in eodem octauo libro <lb/>multa pertractans de cochlea, do<lb/>cet quomodo conficienda &longs;it; & <lb/>quomodo magna huiu&longs;modi in<lb/>&longs;trumento moueantur pondera; <lb/>nec non alia theoremata ad eius <lb/>cognitionem valdè vtilia. </s>

<s id="id.2.1.237.11.1.2.0"> Quoniam autem in<lb/>ter cætera pollicetur, &longs;e o&longs;tendere velle, co<lb/>chleam nihil aliud e&longs;&longs;e præter a&longs;&longs;umptum cu<lb/>neum percu&longs;sionis expertem vecte motionem <lb/>facientem; hoc autem in ip&longs;o de&longs;ideratur; pro<lb/>pterea idip&longs;um o&longs;tendere conabimur, nec non <lb/>eiu&longs;dem cochleæ ad vectem, libramq; reductio<lb/>nem; vt ip&longs;ius tandem completa habeatur co<lb/>gnitio. <pb xlink:href="036/01/254.jpg"/><figure id="id.036.01.254.1.jpg" xlink:href="036/01/254/1.jpg"></figure> </s>

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<s id="id.2.1.237.14.1.1.0"> PROPOSIO I. </s>

<s id="id.2.1.237.14.1.1.0">PROPOSITIO I. </s>

<s id="id.2.1.237.15.1.1.0"> Cuneus hoc modocirca cylindrum accommo<lb/>datus, nihil e&longs;t aliud; ni&longs;i cochlea duas habens he<lb/>lices in vnic o punctoinuicem coniunctas. <figure id="id.036.01.255.1.jpg" xlink:href="036/01/255/1.jpg"></figure> </s>

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<s id="id.2.1.237.20.1.4.0"> & &longs;iue co<lb/>chlea fuerit horizonti perpendicularis, <lb/>&longs;iue horizonti æquidi&longs;tans, vel alio mo<lb/>do collocata, nihil refert: &longs;emper enim <lb/>eadem erit ratio. <pb xlink:href="036/01/258.jpg"/><figure id="id.036.01.258.1.jpg" xlink:href="036/01/258/1.jpg"></figure> </s>

<s id="id.2.1.237.21.1.1.0"> Si verò (vt in tertia figura) &longs;upra cochleam imponatur aliquod, <lb/>vt B, quod quidem tylum vocant, ita accommodatum, vt inferio <lb/>ri parte helices habeat concauas ip&longs;i cochleæ appo&longs;itè admodum <lb/>congruentes; per&longs;picuum &longs;atis e&longs;&longs;e poterit, ip&longs;um B, dum coclhea <lb/>circumuertitur, &longs;uper helices cochleæ eo pror&longs;us modo moueri; <lb/>quo pondus iuxta primam <expan abbr="figurã">figuram</expan>mouebatur: dummodo tylum ap<lb/>tetur, vt docet Pappus in octauo libro; ita &longs;cilicet vt tantùm <expan abbr="antè">an<lb/>te</expan>, retrouè axi cylindri æquidi&longs;tans moueatur. <figure id="id.036.01.258.2.jpg" xlink:href="036/01/258/2.jpg"></figure> </s>

<s id="id.2.1.237.21.1.1.0">Si verò (vt in tertia figura) &longs;upra cochleam imponatur aliquod, <lb/>vt B, quod quidem tylum vocant, ita accommodatum, vt inferio <lb/>ri parte helices habeat concauas ip&longs;i cochleæ appo&longs;itè admodum <lb/>congruentes; per&longs;picuum &longs;atis e&longs;&longs;e poterit, ip&longs;um B, dum <expan abbr="coclhea">cochlea</expan> <lb/>circumuertitur, &longs;uper helices cochleæ eo pror&longs;us modo moueri; <lb/>quo pondus iuxta primam <expan abbr="figurã">figuram</expan> mouebatur: dummodo tylum ap<lb/>tetur, vt docet Pappus in octauo libro; ita &longs;cilicet vt tantùm <expan abbr="antè">an<lb/>te</expan>, retrouè axi cylindri æquidi&longs;tans moueatur. <figure id="id.036.01.258.2.jpg" xlink:href="036/01/258/2.jpg"></figure></s>

<s id="id.2.1.237.22.1.1.0"> Et &longs;i loco tyli, quod helices habet concauas in parte inferiori, con<lb/>&longs;tituatur, vt in quarta figura, cylindrus concauus vt D, & in eius <lb/>concaua &longs;uperficie de&longs;cribantur helices, in cidanturq; ita, vt aptè <pb n="123" xlink:href="036/01/259.jpg"/>cùm cochlea congruant (eodem enim modo de&longs;cribentur helices <lb/>in &longs;uperficie concauia cylindri, &longs;icuti fit in conuexa) &longs;i deinde co<lb/>chlea in &longs;uis polis firmetur, &longs;cilicet in &longs;uo axe, circumuertaturq;; <lb/>patet D ad motum circumuer&longs;ionis cochleæ quemmadmodum ty<lb/>lum moueri. </s>

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<s id="id.2.1.238.1.1.1.0"> <margin.target id="note323"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/>4. <emph type="italics"/>&longs;exti.<emph.end type="italics"/> </s>

<s id="id.2.1.239.1.1.1.0"> Quomodo autem hoc ad libram reducatur mnnife&longs;tum e&longs;t ex <lb/>nona octaui libri eiu&longs;dem Pappi. </s>

<s id="id.2.1.239.1.1.1.0">Quomodo autem hoc ad libram reducatur <expan abbr="mnnife&longs;tum">manifestum</expan> e&longs;t ex <lb/>nona octaui libri eiu&longs;dem Pappi. </s>

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<s id="id.2.1.240.1.1.3.0"> <margin.target id="note326"></margin.target>4 <emph type="italics"/>Primi.<emph.end type="italics"/> </s>

</figure>

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<s id="id.2.1.245.2.1.7.0"> <pb n="128" xlink:href="036/01/269.jpg"/>Intelligatur itaq; manente cochlea pondus R moueri à potentia <lb/>in L vecte Lk &longs;uper helicen Ck: vel quod idem e&longs;t, &longs;icut etiam <lb/>&longs;upra diximus, &longs;i pondus R aptetur ita, vt moueri non po&longs;sit, ni <lb/>&longs;i &longs;uper rectam PQ axi cylindri æquidi&longs;tantem; circumuertaturq; <lb/>cochlea, potentia exi&longs;tente in L; mouebitur pondus R &longs;uper he<lb/>licen CD eodem modo, ac &longs;i à vecte Lk moueretur. </s>

<s id="id.2.1.245.2.1.8.0"> idem enim <lb/>e&longs;t, &longs;iue pondus manente cochlea &longs;uper helicen moueatur; &longs;iue he<lb/>lix circumuertatur, ita vt pondus &longs;uper ip&longs;am moueatur. </s>

<s id="id.2.1.245.2.1.9.0"> cùm <lb/>ab eadem potentia in L moueatur. </s>

<s id="id.2.1.245.2.1.10.0"> &longs;imiliter o&longs;tendetur, quò lon. <lb/>gior &longs;it LI, adhuc pondus facilius &longs;emper moueri. </s>

<s id="id.2.1.245.2.1.10.0">&longs;imiliter o&longs;tendetur, quò lon<lb/>gior &longs;it LI, adhuc pondus facilius &longs;emper moueri. </s>

<s id="id.2.1.245.2.1.12.0"> à minori enim <arrow.to.target n="note334"></arrow.to.target><lb/>potentia moueretur. </s>

<s id="id.2.1.245.2.1.13.0"> quod erat propo&longs;itum. </s>

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