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Colored diff for /texts/archimedes/xml/Attic/monte_mecha_01_la_1577.xml between version 1.31 and 1.32

version 1.31, 2003/03/08 22:39:28

version 1.32, 2003/03/19 20:40:59

Line 84

<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>

Line 260

<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>

Line 309

<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>

Line 454

<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>

Line 463

<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.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>

Line 474

<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 D li<lb/>berum, grauiu&longs;q, e&longs;&longs;e. </s>

Line 489

<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.2.1.17.5.1.30.0"> 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 507

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>

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<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>

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<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>

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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>magisqué centro innitetur in L, quàm <lb/>in D. </s>

<s>&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/>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.2.1.21.2.1.13.0"> 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>& <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>

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<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>

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<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 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<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 610

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>&longs;i verò moueatur ex A <lb/>in M per circum&longs;erentiam AM, &longs;ecundùm rectam eius motus <lb/>erit XM. </s>

<s>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 618

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>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>ideo pondus grauius erit in A, quàm in L. <lb/></s>

<s>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/>cumferentræ 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 684

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 721

Line 760

<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.2.1.33.3.1.16.0"> 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 &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, 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 736

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 753

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 815

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.2.1.39.3.1.3.0"> <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 838

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 851

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 915

Line 961

<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>

Line 933

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 949

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 967

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>

Line 992

Line 1054

<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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Line 1067

<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>

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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>quod <lb/>demon&longs;tre 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>accipiantur in libra vtcunque puncta AHkLB, ex quibus <lb/>data pondera &longs;pu&longs;pendantur. </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.2.1.91.3.1.1.0"> 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 gaui<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.2.1.105.1.1.1.0"> 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.2.1.105.1.1.2.0"> 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.2.1.125.3.1.1.0"> 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.2.1.139.4.1.1.0.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>

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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.2.1.165.13.1.1.0"> 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.2.1.177.15.1.1.0"> 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 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>

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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.2.1.201.6.1.1.0"> 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.2.1.219.1.1.1.0"> <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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