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| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
|---|---|
| date | Wed, 29 Nov 2017 16:55:37 +0100 |
| parents | 22d6a63640c6 |
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<?xml version="1.0"?> <archimedes xmlns:xlink="http://www.w3.org/1999/xlink" > <info> <author>Monte, Guidobaldo del</author> <title>In Duos Archimedis Aequeponderatium libros paraphrasis</title> <date>1588</date> <place>Pesaro</place> <translator></translator> <lang>la</lang> <cvs_file>monte_aeque_077_la_1588.xml</cvs_file> <cvs_version></cvs_version> <locator>077.xml</locator> </info> <text> <front> </front> <body> <chap id="N10019"> <pb xlink:href="077/01/001.jpg" id="p.0001"></pb> <p id="N1001D" type="head"> <s id="N1001F">GVIDIVBALDI <lb></lb>E MARCHIONIBVS <lb></lb>MONTIS <lb></lb>IN DVOS ARCHIMEDIS <lb></lb>ÆQVEPONDERANTIVM <lb></lb>LIBROS</s> </p> <p id="N1002B" type="head"> <s id="N1002D">PARAPHRASIS <lb></lb>Scholijs illuſtrata.</s> </p> <figure id="id.077.01.001.1.jpg" xlink:href="077/01/001/1.jpg"></figure> <p id="N10034" type="head"> <s id="N10036">PISAVRI <lb></lb>Apud Hieronymum Concordiam; <lb></lb>M D LXXXVIII. <lb></lb><emph type="italics"></emph>Superiorum Conceſſu.<emph.end type="italics"></emph.end></s> </p> <pb xlink:href="077/01/002.jpg"></pb> <pb xlink:href="077/01/003.jpg"></pb> <p id="N10046" type="head"> <s id="N10048">SERENISSIMO <lb></lb>FRANC.^{CO} MARIAE <lb></lb>II. VRBINI DVCI.</s> </p> <p id="N1004E" type="head"> <s id="N10050">GVIDVSVBALDVS <lb></lb>E' MARCHIONIBVS MONTIS S.</s> </p> <p id="N10054" type="main"> <s id="N10056">Iam decemnium elapſum eſt, DVX Sere<lb></lb>niſſime, ex quo de rebus machanicis volu<lb></lb>men, veras (ni fallor) mirabilium mechani<lb></lb>corum effectuum cauſas manifeſtans, in lu<lb></lb>cem dedi; vbi non nulla antiquiora, <expan abbr="præci-puaq;">præci<lb></lb>pua〈que〉</expan> illuſtrium græcorum authorum pla<lb></lb>cita ad ſuſceptum negotium pertinentia, <lb></lb>tanquam rectę rationi magis conſentanea amplexatus ſum. </s> <s id="N1006A"><lb></lb>quibus ſanè, tanquam ſolidiſſimis innixa fundamentis, theo<lb></lb>remata multa, ac varia conſtruxi. </s> <s id="N1006F">quippe quæ, licet non inua<lb></lb>lidis quo〈que〉 demonſtrationum præſidijs à me ipſo munita <lb></lb>fuerint; pleriſquè tamen, qui non admodum fortaſſe in huiuſ<lb></lb>modi rerum cauſis inueſtigandis verſati exiſtunt, noua pror<lb></lb>ſus (vt accepi) ac ferme inaudita, nec ſatis (vt opinor) apud eos <lb></lb>firma, at〈que〉 ideo illis non omnino ſatisfeciſſe, viſa ſunt. </s> <s id="N1007B">Quo<lb></lb>circa cogitanti mihi, qua ratione fieri poſſet, vt opus illud à <lb></lb>me editum, quàm plurimorum ſibi gratiam in dies magis con<lb></lb>ciliaret, in mentem venit, non aliunde id mihi oportuniùs <expan abbr="cõtingere">con<lb></lb>tingere</expan> potuiſſe, quàm ſi priſcos ipſos, & grauiſſimos alioqui <lb></lb>authores de hac re elegantiſſimè diſſerentes illis offerrem. </s> <s id="N1008B">ra<lb></lb>tus, vt ſolidiſſimâ eorum doctrinâ, quæ à me propoſita, & ex<pb xlink:href="077/01/004.jpg"></pb>plicata fuere theoremata, firmiora redderentur. </s> <s id="N10092">ſimulquè alio<lb></lb>rum ambiguitati, ne dicam imbecillitam ſuccurreretur. </s> <s id="N10096">vel ſal<lb></lb>tem ipſi grauiſſima eorum authoritate non nullorum captiua<lb></lb>rent intellectum, in obſequium meliùs, rectiùſquè <expan abbr="ſentientiũ">ſentientium</expan>, <lb></lb>at〈que〉 intelligentium. </s> <s id="N100A2">Nihil enim tam, aut a conſuetudine, aut <lb></lb>ab opinione remotum eſſe ſolet, quod ſola authoritate proba<lb></lb>ri non poſſit. </s> <s id="N100A8">Verùm ne huiuſmodi negotium in recenſendis <lb></lb>multorum ad propoſitam veritatem confirmandam teſtimo<lb></lb>nijs latiùs, quàm par eſſet, protraheretur; mihi conſtitui, ex mul<lb></lb>tis vnicum tantùm, eumquè reliquorum omnium hac in par <lb></lb>te facilè principem deligere: qui, & meam cauſam tueretur: & <lb></lb>illis, ſi fieri poſſet, ſatisfaceret: vt〈que〉 grave; coràm illis ipſe ſe offerens, <lb></lb>tanquam meo quo〈que〉 nomine miſſus intelligeretur; quibuſ<lb></lb>dam meis notis non inſignitum certè, ſed aſſociatum eundem <lb></lb>prodire volui. </s> <s id="N100BA">Eſt autem grauiſſimus hic author Syracuſius ille <lb></lb>Archimedes de mechanicis elementis conſultiſſimè diſſerens. </s> <s id="N100BE"><lb></lb>cuius nimirum dignitati, at〈que〉 authoritati, vt omnes probè à <lb></lb>me conſultum intelligerent; decreui, vt 〈que〉madmodum inter <lb></lb>alios illius ordinis viros primatum obtinet, ita nulli alij, quàm <lb></lb>amplitudini tuę DVX Sereniſſime, hac noſtra ętate, doctrina, <lb></lb>rerumquè omnium cognitione ſingulari, citra controuerſiam <lb></lb>Principi ſupremo, ſuum in primis hoc tempore præſtaret obſe<lb></lb>quium. </s> <s id="N100CD">quod incredibili ſanè animi mei iucunditate conti<lb></lb>giſſe fateor; non ſolùm, vt rurſum aliquam ſingularis meæ er<lb></lb>ga amplitudinem tuam obſeruantiæ, ac venerationis, tot, tan<lb></lb>tiſquè nominibus iam pridem debitę teſtificationem ederem; <lb></lb>verùm etiam, vt munuſculo illi meo tanto Principi audentiùs <lb></lb>fortaſſe antea oblato, ne prorſus prę ſua tenuitate deſpiceretur, <lb></lb>opem ferret. </s> <s id="N100DB">quanquam ne〈que〉 id quidem, pro eximia animi <lb></lb>tam excelſi magnitudine, ſuſpicandum fuit. </s> <s id="N100DF">Per hunc ergo <expan abbr="tã">tam</expan> <lb></lb>celebrem authorem ad te Princeps optime, ac pręſtantiſſime <lb></lb>lætabundus accedo. </s> <s id="N100E9">Is enim mihi, 〈que〉madmodum & ego ipſi, <lb></lb>ad te aditum patefeciſſe videtur; & ſicut eundem tibi <expan abbr="lõge">longe</expan> gra<lb></lb>tiſſimum futurum confido; ita me tui amantiſſimum, & obſer<lb></lb>uantiſſimum, vt eâdem, qua conſueuiſti, benignitate proſe<lb></lb>quaris, oro ſuplex, & obſecro. </s> <s id="N100F3">Aueto dulce præſidium, ac ętatis <lb></lb>noſtræ ſplendidum decus; & eſto perpetuò fęlix. </s> </p> <pb xlink:href="077/01/005.jpg" pagenum="1"></pb> <p id="N100FA" type="head"> <s id="N100FC">GVIDIVBALDI <lb></lb>E MARCHIONIBVS <lb></lb>MONTIS.</s> </p> <p id="N10102" type="head"> <s id="N10104">PRAEFATIO:</s> </p> <p id="N10106" type="main"> <s id="N10108">Mechanica facultas <expan abbr="nõ">non</expan> ſolùm ab imperitis, <lb></lb>verùm etiam ab eruditis admirabilis ſem<lb></lb>per habita fuit; eorum enim, quę in admi<lb></lb>rationem homines trahunt, duo eſſe gene<lb></lb>ra Ariſtoteles in principio <expan abbr="ſuarũ">ſuarum</expan> <expan abbr="quęſtionũ">quęſtionum</expan> <lb></lb>Meehanicarum aſſeruit; quorum ſanè alte <lb></lb>rum ad ea pertinet, quæ natura quidem, <lb></lb>proximis tamen ipſorum cauſis latentibus in lucem <expan abbr="prodeũt">prodeunt</expan>; <lb></lb>alterum verò ſpectat ad ea, quę pręter naturam, & arte fiunt; <lb></lb>quibus natura ſuperari videtur (quamquam & ipſa plurimùm <lb></lb>momenti ad ſe ipſam euincendam tune quo〈que〉 afferat) & <lb></lb>quod naturę uiribus in lucem prodire nequit, id arte fieri con<lb></lb>tingat, ob idquè maiorem adhuc admirationem excitat, quòd <lb></lb>ars naturę ęmula, quaſi aduerſus naturam <expan abbr="ipugnãs">ipugnans</expan>, cam ſupe<lb></lb>ret, & <expan abbr="tanquã">tanquan</expan> vim ipſi in ferre videatur; cuius ſanè operationis <lb></lb>cauſa quo〈que〉 cognita admirationem parit; cùm exigua admo <lb></lb>dum ad tanti operis productionem appareat. </s> <s id="N10142">admirabile eſt ſa<lb></lb>nè ipſius artis magiſterium, cùm adeò potens ſit, vt effectus na<lb></lb>turę repugnantes producere tentet. </s> <s id="N10148">quippè quibus, niſi ita ſen<lb></lb>ſibus ſubijciàntur; vt tangi propemodum, & conſpici poſſint, <lb></lb>vix fides adhibeatur; idquè <expan abbr="nõ">non</expan> ſine admiratione adhuc cogni<lb></lb>tum, ac perſuaſum nobis eſſe poſſit. </s> <s id="N10156">huiuſmodi autem mira<lb></lb>bilium operum opifex eſt ipſa mechanica diſciplina, tam na<lb></lb>turę ęmula, quàm oppugnatrix valida. </s> <s id="N1015C">Hęc enim grauia pro<lb></lb>prio fermè nutu ſurſum attolli, magnaquè pondera ab exigua <pb xlink:href="077/01/006.jpg" pagenum="2"></pb>admodum virtute moueri, aliaquè id genus huiuſmodi ſpe<lb></lb>ctanda proponit. </s> <s id="N10166">vt tum imperitis ex ipſorummet effectuum <lb></lb>intuitu, tum eruditis in cauſarum varia contemplatione ad<lb></lb>mirationem pariat. </s> <s id="N1016C">veluti ſi ea ſpectemus, quę neruis, vel ali<lb></lb>quo mouétur inſtrumento; vel quę ſpiritibus <expan abbr="cõcinnuntur">concinnuntur</expan>, & <lb></lb>fiunt; de quibus Heron, & alij pertractarunt; vel deni〈que〉 alijs <lb></lb>modis. </s> <s id="N10178">quamquam nos in ijs, quæ dicenda ſunt, de ea mecha<lb></lb>nicæ facultatis parte, quæ ad <expan abbr="põdera">pondera</expan>, <expan abbr="diſtãtiaſ〈que〉">diſtantiaſ〈que〉</expan> inter ipſa <expan abbr="exiſtẽtes">exi<lb></lb>ſtentes</expan> pertinet, <expan abbr="quorũ">quorum</expan> ſtatus ad ęquilibrium reduci poteſt, ver<lb></lb>ba faciemus. </s> <s id="N10188">quæ <expan abbr="quidẽ">quidem</expan> pars totius mechanicę facultatis prin<lb></lb>ceps exiſtit. </s> <s id="N1018C">ea enim eſt, in qua artem ſuperare naturam aper<lb></lb>tiùs <expan abbr="cõſpicitur">conſpicitur</expan>: quod quidem, qua ratione contingat, hinc pla<lb></lb>num euadet. </s> </p> <p id="N10196" type="main"> <s id="N10198">Ars quippe ex Ariſtotele phiſicorum ſecundo, & ex proæ<lb></lb>mio quæſtionum mechanicarum triplici modo in ſuis opifi<lb></lb>cijs ſeſe habere videtur. </s> <s id="N1019E">Nam vel immitatur naturam; vel ea <lb></lb>perficit, quæ natura perficere non poteſt; vel deni〈que〉 ea, quæ <lb></lb>pręter naturam fiunt, operatur; in quibus tamen omnibus o<lb></lb>perandi rationibus, ſi diligenter eas conſideremus, artem ſem<lb></lb>per immitari naturam perſpiciemus. </s> <s id="N101A8">Primùm quidem multas <lb></lb>artes naturam immitari aperte videmus, vt ſculpturam, & hu<lb></lb>iuſmodi alias. </s> <s id="N101AE">Quando autem ars ea perficit, quæ ſola natu<lb></lb>ra perficere non poteſt, vt in arte medica euenire ſolet; <expan abbr="naturã">naturam</expan> <lb></lb>ipſam pariter emulatur, & naturæ aſſociata, velut inſtrumen<lb></lb>tum eius, naturalem effectum perficere dicitur: tuncquè <expan abbr="eodẽ">eodem</expan> <lb></lb>modo operatur, ac ſi natura rem ipſam abſ〈que〉 artis ope perfice <lb></lb>repoſſet, quod planè artis præſtantiam manifeſtat: quippè <lb></lb>cùm niſi ars ipſi naturæ <expan abbr="manũ">manum</expan> porrigat, natura ipſa proprios <lb></lb>effectus perficere ex ſeſe minimè poſſit. </s> <s id="N101C6">At verò ſi ars <expan abbr="naturã">naturam</expan> <lb></lb>immitando ipſam ſuperauerit; vt ea, quæ ab arte fiunt, præter <lb></lb>naturam eueniant, longè adhuc præſtantiùs artis ingenium <lb></lb>apparebit. </s> <s id="N101D2">ſiquidem immitando naturam (paradoxum id for <lb></lb>tè videbitur, cùm tamen veriſſimum ſit) præter naturæ ordi<lb></lb>nem operari dicatur. </s> <s id="N101D8">Ars. <expan abbr="n.">enim</expan> mirabili artificio naturam ipsa na<lb></lb>tura ſuperat; ita nimirum res diſponendo, vt ipſa efficeret na<lb></lb>tura, ſi eiuſmodi ſibi producendos ſtatueret effectus. </s> <s id="N101E2">quod qui <lb></lb>dem ſubiecto exemplo magis perſpicuum fiet. </s> </p> <pb xlink:href="077/01/007.jpg" pagenum="3"></pb> <p id="N101E9" type="main"> <s id="N101EB">Sint enim duo pondera <lb></lb> <arrow.to.target n="fig1"></arrow.to.target><lb></lb>AB in aliquo vecte, A ma<lb></lb>ius, B minus; quorum ſi<lb></lb>mul ita in vecte diſpoſito<lb></lb>rum ſit centrum grauitatis <lb></lb>C. ſit autem ſub vecte in<lb></lb>ter CA fulcimentum in D. <lb></lb>& quoniam pondera AB penes C grauitatis centrum inclinan<lb></lb>tur? </s> <s id="N10202">tunc C deorſum naturaliter mouebitur; ac per conſe〈qué〉s <lb></lb><expan abbr="pōdus">pondus</expan> quo〈que〉 B deorſum tendet. </s> <s id="N10209">Sed ſi B deorſum mouetur, <lb></lb>A certè ſurſum eleuabitur. </s> <s id="N1020D">quippe quod, <expan abbr="quãuis">quamuis</expan>, vt graue eſt, <lb></lb>at〈que〉 ſolutum abſ〈que〉 connexione ponderis B deorſum tende <lb></lb>ret; attamen vt adnexum ponderi B, intercedente vecte AB, <lb></lb>ſurſum mouebitur: & (vt ita dicam) pondus A contra pro<lb></lb>priam naturam naturaliter aſcendet. </s> <s id="N1021B">Vndè <expan abbr="perſpicuũ">perſpicuum</expan> eſt, hos <lb></lb>motus effectus eſſe naturales. </s> <s id="N10223">Quid igitur efficit ars ipſa? </s> <s id="N10225">nil <lb></lb>fanè aliud, quàm quòd resita diſponit, & accomodat; vt ſimi<lb></lb>les effectus inde prodeant at〈que〉 ſi naturales omnino exiſtant, <lb></lb>quare opus erit, ut Ars naturam immitetur, ſiquidem effectus <lb></lb>naturales prouenire debent. </s> <s id="N1022F">propterea vectem, fulcimentum<lb></lb>què eodem modo diſponit; & loco ponderis B aliquam con<lb></lb><gap></gap>ſtituit potentiam, quæ pręmendo parem vim habeat grauita<lb></lb>ti ipſius B; at〈que〉 tunc ipſa potentia mouens, quę minoreſt gra<lb></lb>uitate ponderis A, ipſum A grauius nihilominus attollet. <lb></lb>quod quamuis propriæ ipſius naturæ repugnet, naturaliter <expan abbr="tamẽ">ta<lb></lb>men</expan> ab ipſa potentia in B exiſtente <expan abbr="ſursũ">ſursum</expan> feretur: res enim ita di<lb></lb>ſpoſitæ talem habent naturam, vt A quidem ſurſum, B vero <lb></lb>deorſum moueri debeant. </s> <s id="N10246">quę ſanè ex noſtro Mechanicorum <lb></lb>libro, & ex ijs, quæ in hoc pertractantur; compertiſſimè red<lb></lb>dentur, & quod diximus devecte, de alijs quo〈que〉 in ſtrumen<lb></lb>tis mechanicis intelligendum eſt. </s> <s id="N1024E">quorum quidem apparatus <lb></lb>ſunt artis opera, effectus autem ipſius penè naturæ: cùm eius <lb></lb>momenta, inclinationesquè ſequantur, veluti præcipuas eiuſ<lb></lb>modi operum effectrices cauſas: quippè quæ ſunt omnino ad<lb></lb>mirabiles, ac pręſtantiſſime; 〈que〉madmodum ex ipſarum <expan abbr="contẽplatione">con<lb></lb>templatione</expan> patere poteſt. </s> <s id="N1025A">cuius rei <expan abbr="argumẽtũ">argumentum</expan> illud indicaſſe ſat <lb></lb>eſto, <expan abbr="nimirũ">nimirum</expan> eas à ſummis uiris, Ariſtotele, & Archimede fuiſſe <pb xlink:href="077/01/008.jpg" pagenum="4"></pb>pertractatas. </s> <s id="N1026A">Ariſtoteles. <expan abbr="n.">enim</expan> in principio <expan abbr="Quęſtionũ">Quęſtionum</expan> <expan abbr="mechanica-rũ">mechanica<lb></lb>rum</expan> multa, ea〈qué〉 pręcipua ad cauſas rei mechanicæ <expan abbr="dignoſcẽdas">dignoſcendas</expan> <lb></lb>aperuit; 〈qué〉 ſecutus Archimedes in his libris mechanica prin<lb></lb>cipia explicatiùs patefecit, eaquè planiora reddidit. </s> <s id="N10282">Nec propte<lb></lb>rea Ariſtoteles diminutus extitit: etenim <expan abbr="eorũ">eorum</expan>, quę ab ipſo pro<lb></lb>poſita, & explicata fuere, problematum cauſas egregiè patefe<lb></lb>cit. </s> <s id="N1028E">ſed quoniam Archimedi ſcopus fuit mechanicę diſciplinę <lb></lb>rudimenta explanare; propterea ad magis particularia <expan abbr="enucleã">enucleam</expan> <lb></lb>da deſcendere voluit. </s> <s id="N10298">Ariſtoteles. <expan abbr="n.">enim</expan> (gratia <expan abbr="exẽpli">exempli</expan>) <expan abbr="quęrẽs">quęrens</expan> cur <lb></lb>vecte magna mouemus pondera? </s> <s id="N102A8">cauſam eſſe ait <expan abbr="longitudinẽ">longitudinem</expan> <lb></lb>vectis maiorem ad partem potentiæ: & rectè quidem; cùm ex <lb></lb>principio ab ipſo conſtituto manifeſtum ſit, ea, quę ſunt in <lb></lb>longiori à centro <expan abbr="diſtãtia">diſtantia</expan>, <expan abbr="maiorẽ">maiorem</expan> quo〈que〉 habere virtuté. </s> <s id="N102BC">Ar<lb></lb>chimedes verò vlteriùs adhuc progredi voluit, hoc admiſſo, <expan abbr="nẽ">nem</expan> <lb></lb>pè quod eſt in longiori diſtantia maiorem uim habere, quàm <lb></lb>id, quod eſt in breuiori, inquirere etiam voluit, quanta ſit vis <lb></lb>eius, quod eſt in longiori diſtantia ad id, quod eſt in breuiori; <lb></lb>ita vt inter hęc nota reddatur qualis, & quę ſit eorum propor<lb></lb>tio determinata. </s> <s id="N102CE">at〈que〉 ideo <expan abbr="fundamẽtum">fundamentum</expan> illud mechanicum <lb></lb>pręſtantiſſimum manifeſtauit; videlicet ita ſeſe habere pon<lb></lb>dus ad pondus, vt diſtantia ad inſtantiam, vnde pondera ſu<lb></lb>ſpenduntur, ſeſe permutatim habet. </s> <s id="N102DA">quo ignoto, res mechani<lb></lb>cę nullo modo pertractari poſſe videntur. </s> <s id="N102DE">quandoquidem <lb></lb>huic tota mechanica facultas tanquam vnico, pręcipuo〈que〉 <lb></lb><expan abbr="fundamẽto">fundamento</expan> innititur. </s> <s id="N102E7">Quare Archimedes <expan abbr="Ariſtotelẽ">Ariſtotelem</expan> ſequi vide<lb></lb>tur; quod non ſolùm patet exijs, quæ dicta ſunt; verùm etiam <lb></lb>ſi Archimedis poſtulata <expan abbr="cõſiderauerimus">conſiderauerimus</expan>, quibus <expan abbr="cõſtituẽdis">conſtituendis</expan>, <lb></lb>ea, quæ de principijs mechanicis Ariſtoteles patefecit, Archi<lb></lb>medé ſupponere <expan abbr="cõperiemus">comperiemus</expan>. vt deinceps ſuo loco <expan abbr="perſpicuũ">perſpicuum</expan> <lb></lb>fiet. </s> <s id="N10307">In ratione pręterea, ac modo <expan abbr="cõſiderãdi">conſiderandi</expan> mechanica, maxi<lb></lb>ma ambo affinitate coniuncti in cedere vidétur. </s> <s id="N1030F">Ariſtoteles. <expan abbr="n.">enim</expan> <lb></lb> <arrow.to.target n="marg1"></arrow.to.target> res mechanicas tum Mathematica, tú naturalia ſapere, ac reſpi<lb></lb>cere aſſeruit: quod <expan abbr="quidẽ">quidem</expan> & Archimedes optimè nouit: <expan abbr="nã">nam</expan> quę <lb></lb>Mathematicè ſunt conſideranda, geometricè demonſtrauit, <lb></lb>vt ſunt diſtantiæ, proportiones, & alia huiuſmodi: quæ verò <lb></lb>ſunt naturalia, naturaliter <expan abbr="quoq;">quo〈que〉</expan> <expan abbr="cõſiderauit">conſiderauit</expan>; vt ea, quæ ad gra<lb></lb>uitatis centrum ſpectant, & quæ ſurſum, & quę deorſum moue <pb xlink:href="077/01/009.jpg" pagenum="5"></pb>ri debent; & cętera huiuſmodi. </s> <s id="N10337">Ex quibus patet <expan abbr="maximũ">maximum</expan> eſſe <lb></lb>inter tantos viros in his pertractandis conſenſum. </s> <s id="N1033F">Ambiget <lb></lb>fortaſſe quiſpiam, nunquid hęc principia rectè ab illis fuerint <lb></lb>pertractata? </s> <s id="N10345">ſed ſtatim omnis ceſſat dubitandi occaſio, ſi tan<lb></lb>torum virorum pręſtantia ad memoriam reuocetur; quibus, <lb></lb>citra controuerſiam in diſciplinis ab ipſis traditis, omnes eru<lb></lb>diti <expan abbr="palmã">palmam</expan> deferunt. </s> <s id="N10351">vt 〈que〉madmodum <expan abbr="abſq;">abſ〈que〉</expan> Ariſtotele duce, <lb></lb>at〈que〉 doctore, nemo ad rectè <expan abbr="philoſophãdum">philoſophandum</expan>, ita ne〈que〉 <expan abbr="etiã">etiam</expan> <lb></lb>ad Mathematicam, <expan abbr="pręcipueq́ue">pręcipue〈que〉</expan> Mechanicam diſciplinam <lb></lb><expan abbr="abſq;">abſ〈que〉</expan> Archimede ſeſe <expan abbr="quiſpiã">quiſpiam</expan> diſponere poſſit: quorum ſanè <lb></lb>apud peritiores authoritas meritò ob id ſuprema extat; quòd <lb></lb>ab ipſis res eo meliori, <expan abbr="pręſtantioriq́">pręſtantiori〈que〉</expan>; modo pertractatę <expan abbr="fuerũt">fuerunt</expan>, <lb></lb>quo ipſarum rerum natura, at〈que〉 doctrinę ratio poſtulabat. & <lb></lb>qui ſcientiarum cupidi ſunt, illos ſequi, eorum què ſcripta ſępè <lb></lb>ſępius attentè perlegere debent. </s> <s id="N1037E">Pręterea philoſophię, ac Ma<lb></lb>thematicę profeſſores in hoc conueniunt; quòd cùm aliqua ad <lb></lb>philoſophiam ſpectantia tractant; mirum in modum Ariſto<lb></lb>telem laudibus extollunt. </s> <s id="N10386">qui verò Mathematicas pertractare <lb></lb>ſtudét, ſtatim ad Archimedis laudes pariter ſe <expan abbr="cōferũt">conferunt</expan>. tametſi <lb></lb>circa ea, quę nó ſunt Archimedis verſentur; vt <expan abbr="quã">quam</expan> plurimi fece<lb></lb>re, quod <expan abbr="quidẽ">quidem</expan> optimo factum eſt conſilio. </s> <s id="N10396">etenim ſi ea, quæ <lb></lb>mathematica ope indigent, laudare volunt, ad Archimedem <lb></lb>confugiendum eſt; vt ſi inuentionem, ſubtiliſſimum Archi<lb></lb>medis inuentum afferant, quo modum adinuenit cognoſcen<lb></lb>dę quantitatis argenti, quod erat in corona Regis aurea, vt Vi<lb></lb>truuius teſtatur; & alia huiuſmodi; ſi admirabilia, ſtatim affe<lb></lb>rant Archimedis ſphęram in globo vitreo elaboratam, in qua <lb></lb>omnes cęleſtis ſphæræ motus relucebant; ita ut natura potiùs <lb></lb>Archimedem immitata, quàm Archimedes naturam illuſiſſe <arrow.to.target n="marg2"></arrow.to.target><lb></lb>videatur; nauim præterea graui pondere oneratam è mari in <lb></lb>littus ab Archimede eductam; aliaquè id genus plurima. </s> <s id="N103AF">De<lb></lb>ni〈que〉 ſi res Mathematicas ciuitatibus eſſe vtiles oſtendere vo<lb></lb>lunt, ea, quæ ab Archimede contra Marcellum in defenſio<lb></lb>ne patriæ facta fuere, in medium afferant, quo tempore bellica <lb></lb>opera adeo mirabilia effecit, vt ſolus Archimedes contra bel<lb></lb>licoſiſſimos Romanos pugnare ſufficiens videretur. </s> <s id="N103BB">quæ qui<lb></lb>dem omnia Mechanica diſciplina <expan abbr="cõfecta">confecta</expan> ſunt. </s> <s id="N103C3">Quid igitur <pb xlink:href="077/01/010.jpg" pagenum="6"></pb>Mechanica admirabilius, & vtilius? </s> <s id="N103C9">è qua tot, tantaquè ad <lb></lb>humani generis vtilitatem conferentia prodeunt? </s> <s id="N103CD">eximia cer<lb></lb>tè, & præclara admodum hæc Archimedis geſta fuere; quæ ta<lb></lb>men, ſi ad alia quamplurima, quæ de ipſo dici, ac afferri poſ<lb></lb>ſunt, conferantur; exigua ſanè mihi videntur. </s> <s id="N103D5">Nam quæ ha<lb></lb>ctenus commemorata ſunt, (quamquam fortaſſe <expan abbr="nõ">non</expan> omnia) <lb></lb>multa tamen, huiuſmodiquè ſimilia alij quo〈que〉 effecerunt, <lb></lb>& adhuc extant fortaſſe viri eo ingenij acumine pręditi, qui <lb></lb>talia aggredi non vererentur: ſed <expan abbr="nõnulla">nonnulla</expan> egregia <expan abbr="extãt">extant</expan> ipſius <lb></lb>Archimedis opera, quorum ſimilia, nec antea, nec poſt <expan abbr="ipsũ">ipsum</expan> <lb></lb>facta fuere, ne〈que〉 in futurum facienda fore à nemine ſint ex<lb></lb>pectanda. </s> <s id="N103F1">omnium enim admirabiliſſima, præſtantiſſima<lb></lb>què ſunt eius ſcripta, in quibus, & ingenij acumen, inuentio<lb></lb>nes ſubtiliſſimæ, perfectaquè doctrina planè conſpicitur. </s> <s id="N103F7">adeo <lb></lb>enim his omnibus Archimedis ſcripta aliorum ſcripta mathe<lb></lb>maticorum excellunt, ſuperantquè; vt quæ aliorum, facilè <lb></lb>quidem inter ſeſe comparari, cum ijs verò, quę ab Archimede <lb></lb>nobis relicta fuerunt; nullo modo poſſint. </s> <s id="N10401">ut apertiſsimè <lb></lb>(alijs interim omiſsis) conſpicuum redditur ex ijs, quæ de <lb></lb>ſphęra & cylindro, & ex ijs, quę de æ〈que〉ponderantibus ſcri<lb></lb>pta reliquit: quippè quę ob eorum <expan abbr="pręſtãtiam">pręſtantiam</expan>, ac dignitatem <lb></lb>meritò literis aureis eſſent imprimenda. </s> <s id="N1040F">liber enim de ſphęra, <lb></lb>& cylindro inter Archimedis ſcripta <expan abbr="excellẽs">excellens</expan> adeò <expan abbr="habit^{9}">habitus</expan> fuit; <lb></lb>vt ad eius <expan abbr="ſepulcrũ">sepulcrum</expan> appoſita fuerit ſphęra, & <expan abbr="cylindr^{9}">cylindrus</expan>: <expan abbr="quib^{9}">quibus</expan> a <lb></lb>Cicerone conſpectis; ſtatim illud Archimedis <expan abbr="ſepulcrũ">sepulcrum</expan> eſſe in<lb></lb>tellexit: de cuius inuentione ob uiri <expan abbr="excellentiã">excellentiam</expan> maximè glo<lb></lb>riatur: Deindè qua ratione ipſum à temerario vanę orationis <lb></lb>proferendæ auſu, (dum ſic loquitur, da mihi vbi ſiſtam, ter<lb></lb>ramquè mouebo) vindicare poſſemus; niſi hęc, quæ de æ〈que〉<lb></lb>ponderantibus extant, ſcripta reliquiſſet? ex his enim habita <lb></lb>notitia proportionis ponderum, & diſtantiarum, ſit manife<lb></lb>ſtum non eſſe à ratione, nequè à natura prorſus alienum, poſſe <lb></lb>terram moueri, ſi daretur conſiſtendi locus. </s> <s id="N10433">quod etiam ex <lb></lb>noſtro volumine Mechanico annis ab hinc aliquot elapſis e<lb></lb>dito varijs quoquè inſtrumentis parere poteſt. <expan abbr="quandoquidẽ">quandoquidem</expan> <lb></lb>multis modis, datum pondus à data potentia moueri, ibi <expan abbr="oſtẽ">oſtem</expan> <lb></lb>ſumeſt. </s> <s id="N10445">vbi demonſtrationes à nobis conſtitutę ijs, quæ apud <pb xlink:href="077/01/011.jpg" pagenum="7"></pb>Archimedem preſenti opere habentur, totam eorum vim fer<lb></lb>ri volunt acceptam. </s> <s id="N1044D">Etne quidpiam, quod ſtudioſis mecha<lb></lb>nicæ facultatis prodeſſe poſſit, prętermitteretur, ad horum <lb></lb>Archimedis librorum interprætationem aliquid operis con<lb></lb>tuliſſe placuit; ſatisquè nobis feciſſe videbimur; ſi ſaltem ſtu<lb></lb>dioſi nos Archimedis veſtigia ſecutos fuiſſe cognouerint. <lb></lb>Et quamuis opus hoc fuerit ab Eutocio Aſcalonita nonnullis <lb></lb>commentarijs illuſtratum, quia tamen propter Archimedis <lb></lb><expan abbr="ſcriptorũ">ſcriptorum</expan> obſcuritaté multa adhuc remanét abſtruſa, nec pror<lb></lb>ſus omnibus peruia; pręſertim gręcarum literarum experti<lb></lb>bus; cùm liber hic in latinum verſus multis in locis obſcurus, <lb></lb>alijsquè pleris〈que〉 quodammodo mancus meritò ſuſpicetur; <lb></lb>ita vt adhuc in tenebris iacere videatur; gręcusquè præterea <lb></lb>codex impreſſus, 〈que〉m ſecuti ſumus, multis in locis aliqua <lb></lb>correctione egere videatur; idcirco ab huiuſmodi munere <lb></lb>pręſtando deſiſtere noluimus: quin ſimul hos libros in <expan abbr="latinũ">latinum</expan> <lb></lb>ſermonem verteremus; commentarijsquè illuſtratos redde<lb></lb>remus. </s> <s id="N1046F">Cùm præſertim hinc tutus ad mechanicam <expan abbr="diſciplinã">diſciplinam</expan> <lb></lb>pateat aditus. </s> <s id="N10477">Quare vt mens huius pręclariſſimi Mathema<lb></lb>tici magis, at〈que〉 magis, quàm fieri poſsit, pro virili noſtra <lb></lb>perſpicua reddatur; & huius ſcientiæ cupidi in adipiſcendis <lb></lb>pulcherrimis hiſce theorematibus minùs laborent; à commu<lb></lb>ni genere interprętandi aliquantulum in præſentia diſcedere <lb></lb>nobis viſum eſt oportunum. </s> <s id="N10483">Nam qui res mathematicas in<lb></lb>terprætati ſunt, ſuos commentarios ſeorſum à demonſtratio<lb></lb>nibus collocauere: nos verò, quę noſtra ſunt, verbis ipſius <arrow.to.target n="marg3"></arrow.to.target><lb></lb>Archimedis inſeruimus, & hoc tantùm in ipſis demonſtra<lb></lb>tionibus, non in propoſitionibus, & huiuſmodi alijs, hac <lb></lb>planè habita diſtinctione, vt quæ ſunt Archimedis (his, vel <lb></lb><emph type="italics"></emph>his literarum notis<emph.end type="italics"></emph.end>) cognoſcantur, ipſiusquè tantùm Ar<lb></lb>chimedis eſſe intelligantur. </s> <s id="N1049B">Quę verò alterius ſunt cha<lb></lb>racteris, utquę huius exiſtent formæ, noſtra eſſe ſemper <lb></lb>ſint exiſtimanda. </s> <s id="N104A1">& quoad fieri potuit, verba omnia, quę <lb></lb>nobis declaratione aliqua, nec non correctione indigere viſa <lb></lb>ſunt (ijs tamen omiſſis, quę parui, imò nullius ſunt momenti, <lb></lb>vt eſt literarum immutatio, & huiuſmodi alia) dilucidè expli<lb></lb>care, at〈que〉 emendare ſtuduimus. </s> <s id="N104AB">quibus etiam hanc adhibui <pb xlink:href="077/01/012.jpg" pagenum="8"></pb>mus diligentiam, quod quamuis ea, quæ noſtra, ſunt, verbis <lb></lb>ſint Archimedis inſerta; ſiquis tamen verba tantùm Archi<lb></lb>medis legere maluerit, rectè id aſſequi poterit; ſiquidem ne <lb></lb>verbum quidem Archimedis omiſimus: quinnimo ea ita di<lb></lb>ſpoſuimus, vt ſuum prorſus retineant ſenſum, poſſintquè <expan abbr="cōtinuatè">con<lb></lb>tinuatè</expan> legi; ac ſi nihil inter ipſa inſertum fuerit. </s> <s id="N104BF">quod qui<lb></lb>dem ſtudioſis non inutile fore iudicauimus; qui abſ〈que〉 no<lb></lb>ſtris additionibus <expan abbr="Archimedē">Archimedem</expan> tantùm habebunt; <expan abbr="cũ">cum</expan> noſtris <lb></lb>verò additionibus Archimedis demonſtrationes continua<lb></lb>tas, & explicatas habebunt. </s> <s id="N104CD">Huberionis autem doctrinæ gra<lb></lb>tia permulta adiunximus ſcholia, in quibus paſſim ordinem, <lb></lb>Authoriſquè artificium patefecimus; nec non multa lemma<lb></lb>ta ad Archimedis demonſtrationes neceſſaria <expan abbr="demõſtraui-mus">demonſtraui<lb></lb>mus</expan>; aliaquè nonnulla ad explicationem, ſubiectamquè ma<lb></lb>teriam valde vtilia adiecimus. </s> <s id="N104DD">Vt etiam Archimedis dicta <lb></lb>magis eluceſcant, antequam ad explicationem verborum <lb></lb>ipſius accedamus, nonnulla prius declarare oportunum no<lb></lb>bis viſum eſt ad ea, quæ in his libris Archimedis ſupponit <lb></lb>tanquam cognita. </s> <s id="N104E7">Deinde conſiderandus proponitur ſcopus, <lb></lb>at〈que〉 intentio Archimedis; diuiſio item librorum; huiuſ<lb></lb>modiquè alia, quæ ſummam afferent facilitatem ad intel<lb></lb>ligendam: mentem Archimedis. </s> </p> <p id="N104EF" type="margin"> <s id="N104F1"><margin.target id="marg1"></margin.target><emph type="italics"></emph>in princip. <lb></lb>〈que〉ſt. </s> <s id="N104F9">Me<lb></lb>chan.<emph.end type="italics"></emph.end></s> </p> <p id="N104FF" type="margin"> <s id="N10501"><margin.target id="marg2"></margin.target><emph type="italics"></emph>Claudianus<emph.end type="italics"></emph.end></s> </p> <p id="N10509" type="margin"> <s id="N1050B"><margin.target id="marg3"></margin.target><emph type="italics"></emph>declaratio <lb></lb>huius para <lb></lb>phraſis.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.012.1.jpg" xlink:href="077/01/012/1.jpg"></figure> <p id="N1051B" type="main"> <s id="N1051D">Cùm itaquè ſupponat, nos exquiſitam habere notitiam <lb></lb>centri grauitatis; illius definitionem afferre libuit: pro cuius <lb></lb>tamen faciliori notitia illud quo〈que〉 in primis admonen<lb></lb> <arrow.to.target n="marg4"></arrow.to.target> dum duximus; nimirum quatuor reperiri centra. <expan abbr="Centrũ">Centrum</expan> ui<lb></lb>delicet vniuerſi, centrum magnitudinis, centrum figuræ, & <lb></lb>centrum grauitatis, quod quidem grauitatis centrum rectè <lb></lb>definitur à Pappo Alexandrino in octauo libro mathemati<lb></lb>carum collectio num hoc pacto. </s> </p> <p id="N10535" type="margin"> <s id="N10537"><margin.target id="marg4"></margin.target><gap></gap></s> </p> <p id="N1053B" type="head"> <s id="N1053D">DEFINITIO CENTRI GRAVITATIS</s> </p> <p id="N1053F" type="main"> <s id="N10541">Centrum grauitatis vniuſcuiuſ〈que〉 corporis eſt punctum <lb></lb>quoddam intra poſitum, à quo ſi graue appenſum mente <lb></lb>conçipiatur, dum fertur, quieſcit<gap></gap> & ſerua<gap></gap> eam, quam in <lb></lb>principio habebat poſitionem, neque in ipſa latione circum- <pb xlink:href="077/01/013.jpg" pagenum="9"></pb>uertitur. </s> </p> <p id="N10553" type="head"> <s id="N10555">EIVSDEM ALIA DEFINITIO.</s> </p> <p id="N10557" type="main"> <s id="N10559">Centrum grauitatis vniuſcuiuſ〈que〉 ſolidæ figuræ eſt <expan abbr="punctũ">punctum</expan> <lb></lb>illud intra poſitum, circa quod vndi〈que〉 partes ęqualium mo <lb></lb>mentorum conſiſtunt. </s> <s id="N10563">ſi. <expan abbr="n.">enim</expan> per tale centrum ducatur <expan abbr="planũ">planum</expan> fi<lb></lb>guram quomodo cun〈que〉 ſecans, ſemper in partes æ〈que〉ponde<lb></lb>rantes ipſam diuidet. </s> </p> <p id="N10571" type="main"> <s id="N10573">Hanc poſtremam definitionem, ſeu potiùs deſcriptionem <lb></lb>tradidit Federicus Commandinus in libro de centro grauita<lb></lb>tis ſolidorum. </s> <s id="N10579">ex quipus ſanè definitionibus eluceſcit natura, <lb></lb> <arrow.to.target n="fig2"></arrow.to.target><lb></lb>at〈que〉 facultas <expan abbr="cẽtri">centri</expan> grauitatis. <lb></lb>vt ſi punctum A fuerit <expan abbr="centrũ">centrum</expan> <lb></lb>grauitatis corporis BC, tunc <lb></lb>ex Pappi ſententia, ſi BC <expan abbr="ſuſpẽ">ſuſpem</expan> <lb></lb>datur ex A, magnitudo BC <lb></lb>eadem, qua reperitur, diſpo<lb></lb>ſitione locata manebit; ne〈que〉 <lb></lb>partes ullas ipſius corporis, vt quę ſunt ad <lb></lb> <arrow.to.target n="fig3"></arrow.to.target><lb></lb>BC, circumuerti, ne〈que〉 omnino ſuum <lb></lb>mutare ſitum depræhendetur. </s> <s id="N105A5">ſi verò vt <lb></lb><expan abbr="Cõmandino">Commandino</expan> placuit, A fuerit centrum <lb></lb>grauitatis magnitudinis BCD, eadem<lb></lb>què per punctum A vtcun〈que〉 <expan abbr="ſecũdùm">ſecundùm</expan> <lb></lb>rectitudinem diuidatur, veluti per EAF. <lb></lb>tunc pars EBF ipſi ECDF æ〈que〉ponde<lb></lb>rabit, quamuis EBF, & ED ſint magni<lb></lb>tudines inæquales. </s> <s id="N105B8">ſæpenumero enim e<lb></lb>uenire ſolet, vt in diuiſione figuræ per eius centrum graui<lb></lb>tatis ipſa aliquando in partes diuidatur æquales, ali<lb></lb>quando in partes inæquales: vt ſuo loco oſtendemus: <arrow.to.target n="marg5"></arrow.to.target><lb></lb>ſemper tamen in partes diuiditur hinc inde æ〈que〉pon<lb></lb>derantes; non tamen ſeorſum conſtitutas, ab inuicen<lb></lb>què ſeiunctas, & veluti ad æquilibrium examinatas; vt pu<lb></lb>ta ſi EBF decem pondo ponderet; ED quo〈que〉 totidem <lb></lb>pependiſſe oporteat. </s> <s id="N105CD">res quippe non ſic ſe habet, ſed cas eſſe <lb></lb>in eo ſitu æ〈que〉ponderantes, in quo reperiuntur; vt neutra <pb xlink:href="077/01/014.jpg" pagenum="10"></pb>alteri pręponderet. </s> <s id="N105D5">ex quibus colligi poteſt, ſi graue quidpiam <lb></lb>in centro mundi collo catum fuerit, oportere centrum graui<lb></lb>tatis illius in centro mundi conſtitutum eſſe: ſiquidem vt <lb></lb>graue illud tunc quieſcat, partes vndi〈que〉 ipſum ambientes ę<lb></lb>qualium momentorum exiſtere, at〈que〉 manere oporteat. <lb></lb>Quare dum aſſeritur, graue quod cum〈que〉 naturali propen<lb></lb>ſione ſedem in mundi centro appetere, nil aliud ſignifica<lb></lb>tur, quàm quòd eiuſmodi graue proprium centrum grauitatis <lb></lb>cum centro vniuerſi coaptare expetit, vt optimè quieſcere va<lb></lb>leat. </s> <s id="N105E9">Ex quo ſequitur motum deorſum alicuius grauis fieri <lb></lb>per rectam lineam, quæ centrum grauitatis ipſius grauis, cen<lb></lb>trumquè mundi connectit. </s> <s id="N105EF">quandoquidem grauia deorſum <lb></lb>rectà feruntur. </s> <s id="N105F3">Vnde manifeſtum eſt, Grauia ſecundum gra<lb></lb>uitatis centrum deorſum tendere. </s> <s id="N105F7">quod nos in noſtro Mecha<lb></lb>nicorum libro ſuppoſuimus. </s> </p> <p id="N105FB" type="margin"> <s id="N105FD"><margin.target id="marg5"></margin.target><emph type="italics"></emph>in fine pri<lb></lb>mi huius.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.014.1.jpg" xlink:href="077/01/014/1.jpg"></figure> <figure id="id.077.01.014.2.jpg" xlink:href="077/01/014/2.jpg"></figure> <p id="N1060F" type="main"> <s id="N10611">Ex ijs omnibus, quæ hactenus de centro grauitatis dicta <lb></lb>ſunt, perſpicuum eſt, vnumquod〈que〉 graue in eius centro <lb></lb>grauitatis propriè grauitare, veluti nomen ipſum centri gra<lb></lb>uitatis idipſum manifeſtè præſeferre videtur. </s> <s id="N10619">ita vt tota vis, <lb></lb>grauitaſquè ponderis in ipſo grauitatis centro coaceruata, col<lb></lb>lectaquè eſſe, ac tanquam in ipſum vndiquè fluere videatur. <lb></lb>Nam ob <expan abbr="grauitatẽ">grauitatem</expan> pondus in <expan abbr="cẽtrum">centrum</expan> vniuerſi naturaliter per <lb></lb>uenire cupit; centrum verò graui tatis (exdictis) eſt id, quod <lb></lb>propriè in centrum mundi tendit. </s> <s id="N1062D">in centro igitur grauitatis <lb></lb>pondus propriè grauitat. </s> <s id="N10631">Præterea quando aliquod pondus <lb></lb>ab aliqua potentia in centro grauitatis ſuſtinetur; tunc pon<lb></lb>dus ſtatim manet, totaquè ipſius ponderis grauitas ſenſu per<lb></lb>cipitur. </s> <s id="N10639">quod etiam contingit, ſi ſuſteneatur pondus in ali<lb></lb>quo puncto, à quo per centrum grauitatis ducta recta linea <lb></lb>in centrum mundi tendat. </s> <s id="N1063F">hoc nam〈que〉 modo idem eſt, ac <lb></lb> <arrow.to.target n="marg6"></arrow.to.target> ſi <expan abbr="põdus">pondus</expan> in eius centro grauitatis propriè ſuſtineretur. </s> <s id="N1064B">Quod <lb></lb>quidem non contingit, ſi ſuſtineatur pondus in alio pun<lb></lb>cto. </s> <s id="N10651">ne〈que〉 enim pondus manet, quin potiùs <expan abbr="antequã">antequam</expan> ipſius <lb></lb>grauitas percipi poſſit, vertitur vti〈que〉 pondus, donec ſimi <lb></lb>liter à ſuſpenſionis puncto ad centrum grauitatis ducta re<lb></lb>cta linea in vniuerſi centrum recto tramite feratur. <lb></lb>quæ quidem ex prima noſtrorum Mechanicorum pro- <pb xlink:href="077/01/015.jpg" pagenum="11"></pb>poſitione ſunt manifeſta, quando autem hæc linea eſt hori<lb></lb>zonti erecta, tunc idem prorſus eſt (vt mox diximus) perinde <lb></lb>ac ſi pondus in centro grauitatis ad vnguem ſuſtineretur. <lb></lb>Quocirca ſi pònderis grauitas minimè percipi poteſt, niſi in <lb></lb><expan abbr="cẽtro">centro</expan> grauitatis ipſius, <expan abbr="põdus">pondus</expan> certè in ipſo propriè grauitat. </s> </p> <p id="N10672" type="margin"> <s id="N10674"><margin.target id="marg6"></margin.target><gap></gap></s> </p> <p id="N10678" type="main"> <s id="N1067A">Centrum figuræ apud Mathematicos eſt punctum, à quo <lb></lb>ſemidiametri exeunt; vel per quod <expan abbr="trãſeunt">tranſeunt</expan> diametri, vt circu<lb></lb>li centrum, & ellipſis, necnon oppoſitarum ſectionum. </s> </p> <p id="N10684" type="main"> <s id="N10686">Centrum verò magnitudinis eſt id, quod medium figuræ <lb></lb>obtinet; vel quod ęqualiter ab exteriori ſuperficie diſtat. </s> <s id="N1068A">vt <lb></lb>ſphærę centrum. </s> </p> <p id="N1068E" type="main"> <s id="N10690">Centrum deni〈que〉 mundi eſt punctum in medio vniuerſi <lb></lb>ſitum, omniumquè rerum infimum. </s> </p> <p id="N10694" type="main"> <s id="N10696">Cæterùm ad meliorem horum notitiam obſeruandum eſt, <lb></lb>hęc centra aliquando ſimul omnia inter ſe conuenire, <expan abbr="aliquã">aliquam</expan> <lb></lb>do nonnulla; aliquando autem minimè. </s> <s id="N106A0">ſimul verò omnia <lb></lb>conueniunt. </s> <s id="N106A4">vt centrum vniuerſi, centrum magnitudinis ter<lb></lb>ræ (ſphęræ ſcilicet ex aqua, terraquè compoſitę, quam nos bre<lb></lb>uitatis ſtudio terram tantùm nuncupabimus) centrum figu<lb></lb>rę terrę; ac centrum grauitatis terrę. </s> <s id="N106AC">Cùm enim terra ſit ſphæ<lb></lb>rica (vt omnes fatentur.) eius medium erit centrum figurę, à <lb></lb>quo ſemidiametri exeunt. </s> <s id="N106B2">idipſum què erit centrum magnitu<lb></lb>dinis, ſiquidem ipſius figurę medium obtinet. </s> <s id="N106B6">Pręterea idem <lb></lb>punctum eſt centrum grauitatis terrę. </s> <s id="N106BA">& quoniam terra in me <lb></lb>dio <expan abbr="mūdi">mundi</expan> quieſcit, erit hoc <expan abbr="centrũ">centrum</expan> grauitatis in centro vniuerſi <lb></lb>collocatum. </s> <s id="N106C8">& hoc duntaxat modo centra omnia in <expan abbr="vnũ">vnum</expan> con<lb></lb>uenire poſſunt. </s> <s id="N106D0">quamquam verò ſphęra, quę continet <expan abbr="terrā">terram</expan> & <lb></lb>aquą, compoſita eſt ex corporibus diuerſę ſpeciei, <expan abbr="differẽtiſquè">differentiſquè</expan> <lb></lb>grauitatis, nimirum ex terra, & aqua; non <expan abbr="tamẽ">tamen</expan> efficitur, quin <lb></lb><expan abbr="mediũ">medium</expan> ipſius cum centro grauitatis conſpiret in vnum. <expan abbr="Nã">Nam</expan> ex <lb></lb>Ariſto telis ſententia terra circa mundi centrum vndi〈que〉 <expan abbr="cõſi">conſi</expan> <arrow.to.target n="marg7"></arrow.to.target><lb></lb>ſtit; & Archimedes affirmat, <expan abbr="etiã">etiam</expan> <expan abbr="humidũ">humidum</expan> manens eſſe <arrow.to.target n="marg8"></arrow.to.target> <expan abbr="ſphęri-cũ">ſphęri<lb></lb>cum</expan>, cuius <expan abbr="cẽtrum">centrum</expan> eſt <expan abbr="centrũ">centrum</expan> vniuerſi. </s> <s id="N10710">ſi ita 〈que〉 terra, & aqua ma<lb></lb><expan abbr="nẽt">nent</expan>, <expan abbr="quieſcũtquè">quieſcuntquè</expan> circa <expan abbr="centrũ">centrum</expan> vniuerſi, ergo <expan abbr="centrũ">centrum</expan> <expan abbr="mūdi">mundi</expan> <expan abbr="ipſo-rũ">ipſo<lb></lb>rum</expan> ſimul <expan abbr="cẽtrũ">centrum</expan> grauitatis exiſtit. </s> <s id="N10731">at〈que〉 adeo quatuor prędicta <lb></lb>centra in <expan abbr="vnũ">vnum</expan> ſimul conueniunt punctum. </s> <s id="N10739">Quod <expan abbr="autẽ">autem</expan> tria ſi<lb></lb>mul centra in vnum coeant, ſatis <expan abbr="conſpicuū">conſpicuum</expan> eſſe poterit cuiquè <pb xlink:href="077/01/016.jpg" pagenum="12"></pb>ſphæram aliquam, putà ligneam, vel alterius (ſimilaris <expan abbr="tamẽ">tamen</expan>) <lb></lb>naturæ intuenti; ſiquidem eius medium erit centrum magni<lb></lb>tudinis, & centrum figuræ; idemquè punctum erit ipſius cen<lb></lb><arrow.to.target n="marg9"></arrow.to.target>trum grauitatis; circa quod vndi〈que〉 partes æ〈que〉ponderant. <lb></lb>& quoniam hæc ſphæra non eſt in centro mundi; propterea <lb></lb>tria tantùm centra ſimul conuenient. </s> <s id="N1075D">ſi verò ſphęra non ſimi<lb></lb>laris, ſed diſſimilaris fuerit, veluti altera ipſius meditate plum<lb></lb>bea, altera verò medietate lignea exiſtente, tunc eius medium <lb></lb>erit quippe centrum magnitudinis, & figurę, grauitatis verò <lb></lb>centrum nequaquam. </s> <s id="N10767">Nam partes vndi〈que〉 circa medium æ<lb></lb>〈que〉ponderare non poſſent; ſed grauitatis centrum ad grauio<lb></lb>rem partem, nimirum plumbeam declinabit. </s> <s id="N1076D">& hoc modo <lb></lb>duo tantùm centra inter ſe conuenient. </s> <s id="N10771">vt etiam (modo ta<lb></lb>men diuerſo) accidit ellipſi; cuius centrum eſt centrum figu<lb></lb>rę, ſiquidem per ipſum tranſeunt diametri; idemquè <expan abbr="punctũ">punctum</expan> <lb></lb> <arrow.to.target n="marg10"></arrow.to.target> eſt ipſius centrum grauitatis. </s> <s id="N10781">quod cùm non ſit propriè me<lb></lb>dium figuræ, non erit quo〈que〉 centrum magnitudinis. <expan abbr="mediū">medium</expan> <lb></lb>enim figuræ propriè circulo, ac ſphæræ tantùm competit. <lb></lb>Quare duo centra hoc quo〈que〉 modo ſimul tantùm conue<lb></lb>nient. </s> <s id="N1078F">In figura paraboles recta linea terminatę centrum gra<lb></lb> <arrow.to.target n="marg11"></arrow.to.target>uitatis intra figuram reperitur, quippè quod ne〈que〉 centrum <lb></lb>figuræ, ne〈que〉 centrum magnitudinis eſſe poteſt. </s> <s id="N10799">etenim in <lb></lb>hac figura non poteſt dari medium, vnde ne〈que〉 centrum ma<lb></lb>gnitudinis dabitur, & quoniam in parabole diametri ſunt in<lb></lb>terſe ęquidiſtantes, vt ex primo libro conicorum Apollonij <lb></lb>Pergei conſtat; ne〈que〉 etiam centrum figuræ dabitur. </s> <s id="N107A3">ſic igi<lb></lb>tur centra nullo modo conuenient. </s> </p> <p id="N107A7" type="margin"> <s id="N107A9"><margin.target id="marg7"></margin.target><emph type="italics"></emph>lib. </s> <s id="N107AF">de cælo<emph.end type="italics"></emph.end></s> </p> <p id="N107B3" type="margin"> <s id="N107B5"><margin.target id="marg8"></margin.target><emph type="italics"></emph>lib. </s> <s id="N107BB">de iis <lb></lb>quę uehun<lb></lb>tur in aqua<emph.end type="italics"></emph.end></s> </p> <p id="N107C3" type="margin"> <s id="N107C5"><margin.target id="marg9"></margin.target>16 <emph type="italics"></emph>Federi<lb></lb>ci <expan abbr="cõm">comm</expan>. de <lb></lb>centro gra<lb></lb>uitatis ſoli <lb></lb>dorum.<emph.end type="italics"></emph.end></s> </p> <p id="N107DA" type="margin"> <s id="N107DC"><margin.target id="marg10"></margin.target>4. <emph type="italics"></emph>Fed. </s> <s id="N107E3">com<lb></lb>man. </s> <s id="N107E7">de cen<lb></lb>tro graui<lb></lb>tatis ſolido <lb></lb>rum.<emph.end type="italics"></emph.end></s> </p> <p id="N107F1" type="margin"> <s id="N107F3"><margin.target id="marg11"></margin.target><emph type="italics"></emph>in ſecundo <lb></lb>libro huius<emph.end type="italics"></emph.end></s> </p> <p id="N107FD" type="main"> <s id="N107FF">Nouiſſe quo〈que〉 oportet centrum grauitatis communius <lb></lb>eſſe, in pluribuſquè reperiri, quàm centra magnitudinis, & fi<lb></lb>guræ: centrum verò figuræ communius eſſe centro magnitu<lb></lb>dinis. <expan abbr="Nã">Nam</expan> quodlibet corpus, & quęlibet figura neceſſe eſt, vt habeat <lb></lb><expan abbr="cẽtrũ">centrum</expan> grauitatis intrinſecùs, vel extrinſecùs. </s> <s id="N10810">intrinſecùs vt <lb></lb><expan abbr="cẽtrũ">centrum</expan> grauitatis alicuius corporis regularis, quod eſt in medio <lb></lb>figuræ, vel alicuius figuræ vt A; cuius centrum grauitatis ſit <lb></lb>in ambitu figuræ, vt in puncto B; extrinſecùs verò vt figura <lb></lb>C, cuius centrum grauitatis extrinſecus ſit, vt in D; quod <lb></lb>eſt intelligendum, ſi graue C in centrum mundi tenderet, <pb xlink:href="077/01/017.jpg" pagenum="13"></pb>tunc centrum D cum centro mundi <expan abbr="cõ-">con<lb></lb></expan> <arrow.to.target n="fig4"></arrow.to.target><lb></lb>ueniret; figuraquè C quieſceret circa cen<lb></lb>trum vniuerſi, veluti ſe habet circa <expan abbr="cẽtrum">centrum</expan> <lb></lb>D. partes enim figuræ talem poſſunt ha<lb></lb>bere ſitum, vt inter ſe ę〈que〉ponderare poſ<lb></lb>ſint. </s> <s id="N10839">vt ex ſubiectis figuris perſpicuum eſt. <lb></lb>& ad huc clariùs, ſi intelligatur figura, vt <lb></lb>E circulo tum exteriori, tum interiori ter <lb></lb>minata, cuius centrum grauitatis extra fi<lb></lb>guram erit in F. quod quidem cum cir<lb></lb>culorum centro conueniet. </s> <s id="N10845">circa quod <lb></lb>(exiſtente centro F in centro mundi) <lb></lb>partes vndi〈que〉 ę〈que〉ponderabunt: cùm <lb></lb>omnes ęqualiter à centro grauitatis <expan abbr="diſtẽt">diſtent</expan>. <lb></lb>præterea in hac figura E centrum graui<lb></lb>tatis (quamuis ſit extra figuram) cum cen<lb></lb>tro figuræ, <expan abbr="cẽtroquè">centroquè</expan> magnitudinis ipſius <lb></lb>figuræ conuenire, fortaſſe non erit incon<lb></lb>ueniens aſſerere. </s> <s id="N1085F">At verò figuræ AC nul<lb></lb>lo pacto figuræ, magnitudinisquè <expan abbr="centrũ">centrum</expan> <lb></lb>habebunt. </s> <s id="N10869">& quamuis dictum ſit <expan abbr="centrũ">centrum</expan> <lb></lb>grauitatis corporum regularium eſſe me<lb></lb>dium ipſorum, non tamen propterea dicendum eſt, idem eſſe <lb></lb>centrum magnitudinis, at〈que〉 figuræ, niſi impropriè; <expan abbr="mediũ">medium</expan> <lb></lb>enim his impropriè attribuitur, ſicuti etiam centrum figuræ; <lb></lb>cùm lineæ ex ipſo prodeuntes non ſint ipſorum corporum <lb></lb>(quatenus regularia ſunt) ſemidiametri. </s> <s id="N1087F">quare centrum gra<lb></lb>uitatis reperiri poteſt abſ〈que〉 alijs centris; at non è conuerſo. <lb></lb>Rurſus commune magis eſt <expan abbr="cẽtrum">centrum</expan> figuræ centro magnitu<lb></lb>dinis; quia præter circulum, & ſphæram, quæ tam figuræ, <expan abbr="quã">quam</expan> <lb></lb>magnitudinis centrum habent, nonnullæ figuræ ſuum ha<lb></lb>bent figuræ centrum in ipſis, & extra ipſas; in ipſis, vt ellipſis, <lb></lb>cuius centrum intùs habetur; ſemicirculus etiam, dimidia què <lb></lb>ſphæra centrum habent in limbo. </s> <s id="N10897">extra figuram verò veluti <lb></lb>hyperbolæ centrum, quod extra figuram exiſtit; vbi nempè <lb></lb>diametri concurrunt. </s> <s id="N1089D">Quæ quidem omnia ſunt figuræ cen<lb></lb>tra; magnitudinis verò minimè. </s> <s id="N108A1">verùm obijciet hoc loco for<pb xlink:href="077/01/018.jpg" pagenum="14"></pb>taſſe quiſpiam, vel ambas, inquiens, centri grauitatis defini<lb></lb>tiones allatas, diminutas eſſe; vel ijs, quæ modò à nobis de <expan abbr="cẽ">cem</expan> <lb></lb>tro grauitatis dicta ſunt, repugnare; cùm oſtenderimus cen<lb></lb>trum grauitatis aliquando eſſe, vel in ambitu figuræ, vel extra <lb></lb>figuram; definitiones verò allatę ſemper ſupponunt illud eſſe <lb></lb>in ipſis intra <expan abbr="poſitũ">poſitum</expan>. <expan abbr="Cõfirmaturquè">Confirmaturquè</expan> difficultas, quandoqui<lb></lb>dem, ne〈que〉 huiuſmodi centrum extra figuram conſtitutum, <lb></lb>fuiſſe Archimedi prorſus ignotum, exiſtimare debemus; vt <lb></lb>colligere licet ex nono poſtulato huius libri; cùm inquit. <lb></lb><emph type="italics"></emph>Omnis figuræ, cuius perimeter ſit ad eandem partem concauus, centrum <lb></lb>grauitatis intra ipſam eſſe oportet.<emph.end type="italics"></emph.end> quaſi non repugnet figurę peri<lb></lb>metrum non ad eandem partem concauum habenti, extra <lb></lb>ipſam grauitatis centrum obtinere. </s> <s id="N108D0">Cui obiectioni in hunc <lb></lb>modum occurri poterit, ſi dixerimus, quòd quamuis exempli <lb></lb>gratia in figura C dictum ſit centrum grauitatis D extra fi<lb></lb>guram exiſtere, id ipſum etiam intra figuram eſſe affirmati <lb></lb>poterit. </s> <s id="N108DA">ſiquidem ambitus figurę C centrum D intra ſe <expan abbr="cõ">com</expan> <lb></lb>tinct; ita vt reſpectu tötius ſit intra. </s> <s id="N108E2">idemquè dicendum eſt de <lb></lb>altera figura A. hoc autem euidentiſſimum eſt in figura E. <lb></lb>& hic eſt ſenſus definitionum centri grauitatis. </s> <s id="N108E8">His ita〈que〉 pri<lb></lb>mùm cognitis conſideranda eſt intentio Archimedis in his li<lb></lb>bris, quę quidem vt plurimum à librorum inſcriptionibus e<lb></lb>luceſcere ſolet. </s> </p> <figure id="id.077.01.018.1.jpg" xlink:href="077/01/018/1.jpg"></figure> <p id="N108F4" type="head"> <s id="N108F6">DE SCOPO HORVM LIBRORVM</s> </p> <p id="N108F8" type="main"> <s id="N108FA">Si Archimedis propoſitum in his libris ex ipſa operis in<lb></lb>ſcriptione, vt in alijs quo〈que〉 aliorum authorum volumini<lb></lb>bus fieri vt plurimùm ſolet, inueſtigandum erit, partim ſanè <lb></lb>conſpicuum illud eſſe videbitur, partim verò ignotum adeò, <lb></lb>vt potiùs nullius fermè rei ſe habiturum eſſe ſermonem profi<lb></lb>teatur Archimedes. </s> <s id="N10906">quid enim (obſecro) verbis illis ſignificari <lb></lb>potuit, 〈que〉 primi libri initio ita ſe <expan abbr="habẽt">habent</expan>. <foreign lang="grc">Aρχιμήδους ἐπιπέδων ἰσορ<lb></lb>ροπιχω̄ν, ὴ κέντρα βάρων ἐπιπέδων.</foreign> hoc eſt. <emph type="italics"></emph>Archimedis planorum æ〈que〉pon<lb></lb>derantium, vel centra grauitatum planorum.<emph.end type="italics"></emph.end> quando quidem vide<lb></lb>tur Archimedes rem prorſus <expan abbr="inutilẽ">inutilem</expan>, quinnimò naturę repu<lb></lb>gnantem ſibi contemplandam proponere. </s> <s id="N10924">dùm enim polli- <pb xlink:href="077/01/019.jpg" pagenum="15"></pb>cetur ſe eſſe pertractaturum de planis æquæponderantibus, ſi<lb></lb>ue de centris grauitatum planorum; cùm ea, quæ æ〈que〉ponde <lb></lb>rare debent, ponderare quo〈que〉 oporteat; ſi plana æ〈que〉ponde<lb></lb>rare <expan abbr="debẽt">debent</expan>, grauitate quadam illa prædita eſſe neceſſe eſt. </s> <s id="N10934">quod <lb></lb>valdè à planorum natura abhorret, cùm grauitas, nonniſi cor<lb></lb>poribus, ne〈que〉 tamen omnibus competat. </s> <s id="N1093A">ipſe tamen, dum <lb></lb>plana æ〈que〉ponderantia, vel centra grauitatum planorum ſe <lb></lb>explicaturum pollicetur, apertè ſupponit plana, ac ſuperficies <lb></lb>graues exiſtere, rem ſanè immaginariam prorſus, ipſiusquè rei <lb></lb>naturæ nullatenus reſpondentem. </s> <s id="N10944">ita vt Archimedes circa ea, <lb></lb>quæ omnino rei naturæ aduerſantur, negotium ſumpſiſſe vi<lb></lb>deatur. </s> <s id="N1094A">Verùm enimuero ſi Authoris <expan abbr="mẽtem">mentem</expan> acuratiùs intuea<lb></lb>mur, rem planè egregiam, naturæquè rei apprimè conſenta<lb></lb>neam ipſum pertractandam ſumpſiſſe depræhendemus. </s> <s id="N10954">Nam <lb></lb>quamuis plana, quatenus plana ſunt, nullam habeant graui<lb></lb>tatem, non eſt tamen à rei natura, ne〈que〉 à ratione alienum, <lb></lb>quin poſſimus planorum, ſuperficierum què centra grauitatis <lb></lb>depræhendere, ex quibus ſi ſuſpendantur, planorum partes <lb></lb>vndiquè ęqualium momentorum conſiſtentes maneant. <expan abbr="quã-doquidem">quan<lb></lb>doquidem</expan> centrum grauitatis talis eſt naturæ, vt ſi mente <expan abbr="cõ-cipiamus">con<lb></lb>cipiamus</expan>, rem aliquam in eius centro grauitatis appenſam eſ<lb></lb>ſe, eo prorſus modo, quo reperitur, quieſcat, & maneat. </s> <s id="N1096E">vt <lb></lb>antea declarauimus. </s> <s id="N10972">& quamuis re ipſa, actù〈que〉 plana <expan abbr="ſeorsũ">ſeorsum</expan> <lb></lb>à corporibus reperiri ne〈que〉ant; in ipſis tamen hæc ipſorum <lb></lb>circa centra grauitatis æ〈que〉ponderatio ad actum facilè redigi <lb></lb>poterit. </s> <s id="N1097E">Vt ſit ſolidum AB priſ<lb></lb> <arrow.to.target n="fig5"></arrow.to.target><lb></lb>ma, <expan abbr="cui^{9}">cuius</expan> latera AE CF DB ſint <lb></lb>horizonti erecta, ſuperiorquè ba<lb></lb>ſis ACD, 〈que〉m ad modum & in<lb></lb>ferior EFB ſit horizonti æquidi<lb></lb>ſtans; ſit autem plani ACD cen<lb></lb>trum grauitatis G, ex quo G ſi <lb></lb>ſuſpendatur totum AB patet <lb></lb>planum ACD horizonti æqui<lb></lb>diſtans permanere, ac propterea <lb></lb>circa <expan abbr="cẽtrum">centrum</expan> grauitatis G æ〈que〉<lb></lb>ponderare. </s> <s id="N1099F">quod quidem, quamuis egeat demonſtratione, <pb xlink:href="077/01/020.jpg" pagenum="16"></pb> <arrow.to.target n="marg12"></arrow.to.target> in præſentia omittatur; infraquè ſuo loco oſtendendum. </s> <s id="N109A9">ſat <lb></lb>autem nobis nunc ſit oſtendiſſe, hæc ad praxim reduci, ma<lb></lb>nibuſquè (vt dicitur.) contrectari poſſe. </s> <s id="N109AF">Quòd ſi hæc ita ſe ha<lb></lb>bent, huiuſmodi conſideratio non erit vana, ne〈que〉 vt inuti<lb></lb>lis reijcienda. </s> <s id="N109B5">Sed vlteriùs adhuc progrediamur, dicamuſ<lb></lb>què, quoniam planum ACD, quatenus eſt corpori coniun<lb></lb>ctum, horizonti æquidiſtans permanere debet; ſi ſeorſum à <lb></lb>corpore illud intelligamus, vt ſi ADC ex eius centro graui<lb></lb>tatis G ſuſpendatur, tunc quocun〈que〉 modo reperiatur, hoc <lb></lb>eſt ſiue horizonti ęquidiſtans, ſiuè <lb></lb>minùs, idipſum permanſurum ni<lb></lb><arrow.to.target n="fig6"></arrow.to.target><lb></lb>hilominus intelligere poſſumus, <lb></lb>parteſquè vndi〈que〉 æqualium mo<lb></lb>mentorum conſiſtentes. </s> <s id="N109CE">Ne〈que〉 <lb></lb>enim Ariſto teles grauibus dunta<lb></lb>xat, ſed etiam leuibus momenta <lb></lb>tribuit, idipſum què (vt Eutocius <lb></lb>in horum librorum comentarijs <lb></lb>refert) Ptolæmeo quo〈que〉 placuit, vt habetur in líbro (à nobis <lb></lb>ramen deſiderato) 〈que〉m de momentis ſcripſit. </s> <s id="N109DC">Pręterea alij<lb></lb>quo〈que〉 Philoſophi id ipſum ſenſiſſe videntur. </s> <s id="N109E0">quod eſt qui<lb></lb>dem rationi conſentaneum, ſuperuolant enim, quæ leuia ſunt, <lb></lb>& ſi mente concipiatur <expan abbr="eadẽ">eadem</expan> figura leuis cuiuſpiam eſſe, tunc <lb></lb>ſi detineatur in G, partes vndi〈que〉 ęqualium <expan abbr="momentorũ">momentorum</expan> <lb></lb>conſiſtent, eſſetquè G (vt ita dicam) centrum leuitatis. </s> <s id="N109F2">Quo<lb></lb>niam autem circa centrum grauitatis ę〈que〉ponderationem <lb></lb>conſideramus, id circo plana, tanquam no bis apparentia gra<lb></lb>uitatem habere, mente concipimus. </s> <s id="N109FA">Non eſt igitur à ratio<lb></lb>ne alienum, æ〈que〉ponderantiam in planis, vt grauibus conſi<lb></lb>deratis intelligere, conciperequè. </s> <s id="N10A00">Nec quicquam nobis offi<lb></lb>cit, quòd definitiones centri grauitatis priùs allatæ non pla<lb></lb>norum, ſed corporum centra explicarunt, ita vt grauitatis <expan abbr="cẽ-trũ">cen<lb></lb>trum</expan> ad corpora, <expan abbr="nõ">non</expan> ad plana ſit <expan abbr="referendũ">referendum</expan>. Hoc enim ideo fa<lb></lb><expan abbr="ctũ">ctum</expan> eſt, quia propriè <expan abbr="centrũ">centrum</expan> grauitatis reſpicit corpora; non ta<lb></lb>men propterea impropriè reſpicit plana, ſed quia primò reſpi<lb></lb>cit corpora; in <expan abbr="quib^{9}">quibus</expan> actu ineſſe <expan abbr="depræhẽditur">depræhenditur</expan>. propterea <expan abbr="eędẽ-met">eędem<lb></lb>met</expan> definitiones planis quo〈que〉 in <expan abbr="hũc">hunc</expan> <expan abbr="modũ">modum</expan> aptari <expan abbr="poterũt">poterunt</expan>. </s> </p> <pb xlink:href="077/01/021.jpg" pagenum="17"></pb> <p id="N10A3B" type="margin"> <s id="N10A3D"><margin.target id="marg12"></margin.target><emph type="italics"></emph>in fine pri<lb></lb>mi libri.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.021.1.jpg" xlink:href="077/01/021/1.jpg"></figure> <figure id="id.077.01.021.2.jpg" xlink:href="077/01/021/2.jpg"></figure> <p id="N10A4F" type="head"> <s id="N10A51">DEFINITIO CENTRI GRAVITATIS PLANORVM.</s> </p> <p id="N10A53" type="main"> <s id="N10A55">Centrum grauitatis vniuſcuiuſ〈que〉 plani eſt punctum quod<lb></lb>dam intra poſitum, à quo ſi planum appenſum mente con<lb></lb>cipiatur, dum fertur, quieſcit; & ſeruat eam, quam in princi<lb></lb>pio habebat poſitionem, ne〈que〉 in ipſa latione <expan abbr="circũuertitur">circumuertitur</expan>. </s> </p> <p id="N10A61" type="head"> <s id="N10A63">EIVSDEM ALIA DEFINITIO.</s> </p> <p id="N10A65" type="main"> <s id="N10A67">Centrum grauitatis vniuſcuiuſ〈que〉 plani eſt punctum il<lb></lb>lud intra poſitum, circa quod vndi〈que〉 partes æqualium mo <lb></lb>mentorum conſiſtunt. </s> <s id="N10A6D">ſi enim per tale centrum recta du<lb></lb>catur linea figuram quomodocun〈que〉 ſecans, ſemper in par<lb></lb>tes æ〈que〉ponderantes ipſam diuidet. </s> </p> <p id="N10A75" type="main"> <s id="N10A77">Vt Ita〈que〉 in planis quo〈que〉 centrum grauitatis conſide<lb></lb>ratur, ita etiam plana grauitate prædita conſiderare, non e<lb></lb>rit abſurdum. </s> <s id="N10A7D">ſi enim impoſſibile eſſet conſiderare plana gra<lb></lb>uitate prædita, centrum quo〈que〉 grauitatis in ipſis nullo mo<lb></lb>do concipi poſſet; at〈que〉 perſpicuum eſt, centrum grauitatis in <lb></lb>ipſis admitti, ac deſignari poſſe, igitur & plana grauitate inſi<lb></lb>gnita. </s> <s id="N10A87">Et ſi mathematicus conſiderat corpora ſecluſa interim <lb></lb>ipſorum grauitate, & leuitate: & Aſtronomus corpora conſi<lb></lb>derans cæleſtia, quæ ne〈que〉 grauia, ne〈que〉 leuia ſunt, non pro<lb></lb>pterea <expan abbr="cõſiderat">conſiderat</expan> ea ex propria <expan abbr="ipſorũ">ipſorum</expan> natura, ne〈que〉 grauia, ne <lb></lb>〈que〉 leuia eſſe; etenim quamuis grauia, vel leuia eſſent, nihilo <lb></lb>minus ne〈que〉 grauia, ne〈que〉 leuia eſſe ea conſideraret. </s> <s id="N10A9B">quòd ſi <lb></lb>Mathematicus hoc pacto huiuſmodi corpora intelligere po<lb></lb>teſt; quid prohibet rurſum <expan abbr="eadẽ">eadem</expan>, <expan abbr="quãuis">quamuis</expan> vt talia, ne〈que〉 grauia, <lb></lb>ne〈que〉 leuia ſint; vel grauia, vel leuia eſſe concipere? <expan abbr="〈quẽ〉ad-modum">〈que〉mad<lb></lb>modum</expan> hoc quo〈que〉 <expan abbr="exẽ">exem</expan> <lb></lb> <arrow.to.target n="fig7"></arrow.to.target><lb></lb>plo res magis eluceſcet: <lb></lb>veluti ſi intelligamus ex <lb></lb>AC appenſa eſſe plana <lb></lb>DE, quæ ſint æqualia; ſu<lb></lb>ſpendaturquè AC in me <lb></lb>dio prorſus in B; cur mente intelligere non poſſumus, <lb></lb><expan abbr="quantitatẽ">quantitatem</expan>, <expan abbr="ſpaciũquè">ſpaciumquè</expan> D <expan abbr="æ〈que〉põderare">æ〈que〉ponderare</expan> ſpacio E; cùm ſint æqua<lb></lb>lia? <gap></gap> ſi planorum alterum, putà D, maius eſſet ipſo E; tunc <pb xlink:href="077/01/022.jpg" pagenum="18"></pb>ſtatim non ſolùm ę〈que〉ponderare non poſſe, verùm etiam pla<lb></lb>num D deorſum tendere concipiemus. </s> <s id="N10ADE">& hoc nulla alia de <lb></lb>cauſa, quàm quòd cùm D maius ſit, quàm E, ſtatim <expan abbr="ipsũ">ipsum</expan> <lb></lb>D, quàm E grauius quo〈que〉 eſſe concipimus. </s> <s id="N10AE8">Conſiderare <lb></lb>igitur plana cum grauitate non eſt omnino à ratione <expan abbr="alienũ">alienum</expan>. <lb></lb>Quare vtrum 〈que〉 titulum, nempe planorum æ〈que〉ponderan<lb></lb>tium, vel centra grauitatis <expan abbr="planorũ">planorum</expan>, admittendum duximus. <lb></lb>Verùm quoniam Archimedes ſecundum librum ſimplici vo<lb></lb>cabulo, nimirum (quaſi ſimul omnia complectens) <emph type="italics"></emph>æ〈que〉pon<lb></lb>derantium<emph.end type="italics"></emph.end> in ſcripſit; idcirco tam primum, quàm ſecundum li<lb></lb>brum (æ〈que〉ponderantium) inſcribendum exiſtimamus. </s> <s id="N10B06">eo<lb></lb>què libentiùs; quoniam ipſemet Eutocius horum quo〈que〉 li<lb></lb>brorum explanator hoſce libros hoc tantùm nomine æ〈que〉<lb></lb>ponderantium nuncupauit: alijquè omnes, qui hos Archime<lb></lb>dis libros nominant; hoc titulo de æ〈que〉ponderantibus nun<lb></lb>cupant. </s> <s id="N10B12">Præterea titulus hic magis operi congruere mihi vide<lb></lb>tur; quoniam nonnulla Archimedes in principio pertractat, <lb></lb>quæ tam ſolidis, quàm planis communia exiſtunt; quamuis <lb></lb>cætera ad plana ſint <expan abbr="tantũ">tantum</expan> <expan abbr="referẽda">referenda</expan>. in quibus omnibus de re <lb></lb>admodum vtili, & ad <expan abbr="quãplurima">quamplurima</expan> <expan abbr="cõduẽcti">conduencti</expan> pertractat. <expan abbr="quãdoqui">quandoqui</expan> <lb></lb><expan abbr="dẽ">dem</expan> ex ijs, quæ ab Archimede his libris docemur, in <expan abbr="multarũ">multarum</expan> <expan abbr="re-rũ">re<lb></lb>rum</expan> <expan abbr="cognitionẽ">cognitionem</expan> peruenire poſſumus. </s> <s id="N10B3F">quod facilè conſtat inpri<lb></lb>mis ipſiuſmet Archimedis <expan abbr="exẽplo">exemplo</expan>. <expan abbr="ſiquidẽ">ſiquidem</expan> hac methodo ipſe <lb></lb>in libro de quadratura paraboles <expan abbr="cõparãdo">comparando</expan> plana in libra <expan abbr="cõ">com</expan> <lb></lb>ſtituta, ipſius paraboles <expan abbr="quadraturã">quadraturam</expan> miro artificio adinuenit. <lb></lb>Deinceps ex cognitione <expan abbr="cẽtroiũ">centrorum</expan> grauitatis planorum, nos in <lb></lb>cognitionem centrorum grauitatum ſolidorum deducimur. <lb></lb>Deni〈que〉 adeo proficua eſt hæc doctrina, quam nobis in his <lb></lb>libris Archimedes præſtat; vt affirmare non verear, nullum <lb></lb>eſſe Theorema, nullum què problema ad rem mechanicam <lb></lb>pertinens, quod in ſui ſpeculatione peculiare <expan abbr="nõ">non</expan> aſſumat <expan abbr="fun-damẽtum">fun<lb></lb>damentum</expan> ex ijs, quæ Archimedes in his libris ediſſerit. </s> <s id="N10B74">〈que〉m<lb></lb>admodum (cæteris interim omiſſis) patet ex vulgata illa pro<lb></lb>poſitione enunciante, ita ſe habere pondus ad pondus, vt di<lb></lb>ſtantia ad diſtantiam permutatim ſe habet, ex quibus ſuſpen<lb></lb>duntur. </s> <s id="N10B7E">quæ præclariſſimè ab ipſo in primo libro demonſtra<lb></lb>tur. </s> <s id="N10B82">Et quamuis Iordanus Nemorarius (〈que〉m ſecutus eſt <pb xlink:href="077/01/023.jpg" pagenum="19"></pb>Nicolaus Tartalea, & alij) in libello de ponderibus hanc <expan abbr="eã-dem">ean<lb></lb>dem</expan> propoſitionem quo〈que〉 demonſtrare conatus ſit; & ad <lb></lb><expan abbr="cã">cam</expan> oſtendendam pluribus medijs fuerit vſus; nulli tamen pro<lb></lb>bationi demonſtrationis nomen conuenire poteſt. </s> <s id="N10B95">cùm vix <lb></lb>ex probabilibus, & ijs, quæ nullo modo neceſſitatem <expan abbr="afferũt">afferunt</expan>, <lb></lb>& fortaſſe ne〈que〉 ex probabilibus ſuas componat rationes. <lb></lb>Cùm in mathematicis demonſtrationes requirantur exquiſi<lb></lb>tiſſimæ. </s> <s id="N10BA3">ac propterea ne〈que〉 inter Mechanicos videtur mihi <lb></lb>Iordanus ille eſſe recenſendus. </s> <s id="N10BA7">Quapropter ad Archimedem <lb></lb>confugiendum eſt, ſi fundamenta mechanica, veraquè huius <lb></lb>ſcientiæ principia perdiſcere cupimus: qui (meo iudicio) ad <lb></lb>hoc potiſſimùm reſpexit; vt elementa mechanica traderet. </s> <s id="N10BAF">vt <lb></lb>etiam Pappus in octauo Mathematicarum collectionum li<lb></lb>bro ſentit; quod quidem ex diuiſione, ac progreſſu horum li<lb></lb>brorum facilè dignoſcetur. </s> </p> <figure id="id.077.01.023.1.jpg" xlink:href="077/01/023/1.jpg"></figure> <p id="N10BBB" type="head"> <s id="N10BBD">DE DIVISIONE HORVM LIBRORVM.</s> </p> <p id="N10BBF" type="main"> <s id="N10BC1">Diuiditur enim in primis hic tractatus in duos libros diui<lb></lb>ſus, in poſtulata, & theoremata: theoremata verò ſubdiui<lb></lb>duntur in duas ſectiones, quarum prima continet priora o<lb></lb>cto theoremata; ad alteram verò reliqua theoremata <expan abbr="ſpectãt">ſpectant</expan>. <lb></lb>quæ quidem adhuc in alias duas partes diuidi poteſt; nempè <lb></lb>in theoremata primo libro examinata, & in ea, quæ ſecun<lb></lb>dus liber contemplatur. </s> <s id="N10BD3">Hanc autem horum librorum con<lb></lb>ſtituimus diuiſionem, quoniam imprimis Archimedes, (o<lb></lb>miſſis poſtulatis, quæ primum locum obtinere debent) quæ<lb></lb>dam tractauit communia in prioribus octo theorematibus; <lb></lb>quorum ſcopus eſt inuenire fundamentum illud <expan abbr="præcipuũ">præcipuum</expan> <lb></lb>mechanicum, quòd ſcilicet ita ſe habet grauitas ad grauita<lb></lb>tem, vt diſtantia ad diſtantiam permutatim. </s> <s id="N10BE5">ad quod <expan abbr="demõſtrandum">demon<lb></lb>ſtrandum</expan> quin〈que〉 præmittit theoremata, quæ paulatim <lb></lb>deducunt nos in cognitionem demonſtrationis præfati fun<lb></lb>damenti. </s> <s id="N10BED">quo loco illud ſummoperè notandum eſt, nimi<lb></lb>rum fundamentum illud, nec non octo priora theorema<lb></lb>ta communia eſſe tam planis, quàm ſolidis; at〈que〉 promiſ<lb></lb>cuè de vtriſ〈que〉 <expan abbr="Archimedẽ">Archimedem</expan> demonſtrare. </s> <s id="N10BF9">quòd ſi quis aliter <pb xlink:href="077/01/024.jpg" pagenum="20"></pb>ſenſerit, demonſtrationeſquè tantùm de planis <expan abbr="cõcludere">concludere</expan> exi<lb></lb>ſtimauerit, vel de ſolidis, non autem <expan abbr="quibuſcũ〈que〉">quibuſcun〈que〉</expan>, ſed vel de <lb></lb>rectilineis, vel de homogeneis tantùm, & de ijs, quæ inter ſe <lb></lb>ſunt eiuſdem ſpeciei, longè aberrat à ſcopo, & mente Archi<lb></lb>medis. </s> <s id="N10C0F">etenim in his ſemper loquitur. </s> <s id="N10C11">vel de grauibus ſimpli<lb></lb>citer, veluti in primis tribus theorematibus; vel de magnitu<lb></lb>dinibus, vt in reliquis quin〈que〉 quod quidem nomen tam <lb></lb>planis, quàm ſolidis quibuſcun〈que〉 eſt <expan abbr="cõmune">commune</expan>, vt etiam ij, <lb></lb>qui parùm in Mathematicis verſati ſunt, ſatis norunt. </s> <s id="N10C1F">ſicu<lb></lb>ti etiam Euclides, dum quinti libri propoſitiones pertracta<lb></lb>uit, quantitatem continuam ſub nomine magnitudinis <expan abbr="cõ">com</expan> <lb></lb>prehendit. </s> <s id="N10C2B">quòd <expan abbr="autẽ">autem</expan> nomen grauis ſit <expan abbr="cõmune">commune</expan>, iam ſatis <lb></lb>per ſe conſtat. </s> <s id="N10C37">Perſpicuum eſt igitur priora hæc octo Theo<lb></lb>remata communia eſſe, tam planis, quàm ſolidis. </s> <s id="N10C3B">ac non ſo<lb></lb>lùm ſolidis eiuſdem ſpeciei, & homogeneis, verùm etiam ſoli <lb></lb>dis diuerſæ ſpeciei, & hęterogeneis, vt ſuo loco manifeſtum <lb></lb>fiet. </s> <s id="N10C43">Iactoquè hoc fundamento, quod Archimedes in <expan abbr="duob^{9}">duobus</expan> <lb></lb>propoſitionibus, ſexta nempè, & ſeptima demonſtrauit; in o<lb></lb>ctaua tanquam corrollarium colligit. </s> <s id="N10C49">Deinceps peculiariter <lb></lb>pertractat de centro grauitatis planorum, nec amplius plana <lb></lb>nominat magnitudinis nomine, ſed proprijs cuiuſcun〈que〉 <lb></lb>nominibus; vt parallelogrammi, trianguli, & aliorum huiuſ<lb></lb>modi. </s> <s id="N10C53">& in hac parte deſcendit ad particularia. </s> <s id="N10C55">quippè cùm <lb></lb>& ſi non actu fortaſſe, virtute tamen cuiuſlibet particularis <lb></lb>plani centrum grauitatis nos doceat. </s> <s id="N10C5B">in primo enim libro <lb></lb>ſat ſi bi viſum eſt oſtendiſſe centra grauitatum <expan abbr="triangulorũ">triangulorum</expan>, <lb></lb>ac parallelogrammorum, ex quibus cæterarum figurarum, <lb></lb>veluti pentagoni, hexagoni, & aliorum ſimilium centra gra<lb></lb>uitatis inueſtigare non admodum erit difficile. </s> <s id="N10C65">ſiquidem hu<lb></lb>iuſmodi plana in triangula diuiduntur. </s> <s id="N10C69">vt in ſine primi li<lb></lb>bri attingemus. </s> <s id="N10C6D">In ſecundo autem libro altiùs ſe extollit, & <lb></lb>moro ſuo circa ſubtiliſſima theoremata verſatur; nempè cir<lb></lb>ca centrum grauitatis conice ſectionis, quæ parabole nun<lb></lb>cupatur. </s> <s id="N10C75">nonnullaquè præmittit theoremata, quæ ſunt tan<lb></lb>quam præuie diſpoſitiones ad inueſtigandam demonſtra<lb></lb>tionem centri grauitatis in parabole. </s> <s id="N10C7B">Ita〈que〉 perſpicuum eſt, <lb></lb>Archimedem propriè elementa mechanica tradere. </s> <s id="N10C7F">quando- <pb xlink:href="077/01/025.jpg" pagenum="21"></pb>quidem duo pertractat, quæ ſunt tanquam elementa huius <lb></lb>ſcientiæ. </s> <s id="N10C87">fundamentum nempè illud præſtantiſſimum iam <lb></lb>toties præfatum, deinde centra grauitatis planorum oſtendit. <lb></lb>& quamuis hi duo Archimedis libelli pauca continere videan<lb></lb>tur, non tamen pauca docuiſſe Archimedem exiſtimandum <lb></lb>eſt. </s> <s id="N10C91">multa enim ſunt mole exigua, quæ tamen virtute maxima <lb></lb>habentur. </s> <s id="N10C95">quod planè Archimedis ſcriptis accidit; hiſquè prę<lb></lb>ſertim, ex quibus patet aditus ad multa, ac penè infinita theo<lb></lb>remata, problemataquè mechanica. </s> <s id="N10C9B">nihil enim in hoc gene<lb></lb>re demonſtrari poteſt, quod his non indigeat ſcriptis. </s> <s id="N10C9F">& <lb></lb>quod admirabilius eſt, nos non ſolùm pro fundamento ſu<lb></lb>ſcipere poſſe ad aliquod demonſtrandum theoremata in his <lb></lb>libris demonſtrata, verùm etiam ab his demonſtrationibus <lb></lb>perdiſcerere ipſum modum argumentandi, & demonſtrandi; <lb></lb>vt ſuis locis oſtendemus. </s> <s id="N10CAB">ita vt verè concludendum ſit, nemi<lb></lb>nem prorſus inter mechanicos connumerandum fore, qui <lb></lb>hæc Archimedis ſcripta ignorat. </s> <s id="N10CB1">ignoratis enim principijs <lb></lb>nulla eſt ſcientia, vt apud omnes ſapientes perſpicuum eſt. <lb></lb>Ipſum igitur Archimedem audiamus, eiuſquè ſcripta diligen<lb></lb>tiſſimè perpendamus. </s> </p> <pb xlink:href="077/01/026.jpg" pagenum="22"></pb> <pb xlink:href="077/01/027.jpg" pagenum="23"></pb> <p id="N10CBF" type="head"> <s id="N10CC1">GVIDIVBALDI <lb></lb>EMARCHIONIBVS <lb></lb>MONTIS. <lb></lb>IN PRIMVM ARCHIMEDIS <lb></lb>AEQVEPONDERANTIVM <lb></lb>LIBRVM <lb></lb>PARAPHRASIS <lb></lb>SCHOLIIS ILLVSTRATA.</s> </p> <p id="N10CD1" type="head"> <s id="N10CD3">Archimedis tamen huius primi libri <lb></lb>titulus ſic ſe habet.</s> </p> <p id="N10CD7" type="head"> <s id="N10CD9"><emph type="italics"></emph>ARCHIMEDIS PLANORVM AEQVEPONDERANTIVM, <lb></lb>VEL CENTRA GRAVITATVM PLANORVM.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.027.1.jpg" xlink:href="077/01/027/1.jpg"></figure> <p id="N10CE4" type="head"> <s id="N10CE6">ARCHIMEDIS POSTVLATA.</s> </p> <p id="N10CE8" type="head"> <s id="N10CEA">I.</s> </p> <p id="N10CEC" type="main"> <s id="N10CEE">Grauia æqualia ex æqualibus diſtantijs æ〈que〉<lb></lb>ponderare. </s> </p> <p id="N10CF2" type="head"> <s id="N10CF4">SCHOLIVM.</s> </p> <p id="N10CF6" type="main"> <s id="N10CF8">Dvobvs modis grauia in diſtantijs <lb></lb>collocata intelligi poſſunt. </s> <s id="N10CFC">quod & <lb></lb>in cæteris poſtulatis, & in propoſi<lb></lb>tionibus intelligendum eſt. </s> <s id="N10D02">etenim <lb></lb>vel grauia <expan abbr="sũt">sunt</expan> appenſa, vt in prima fi<lb></lb>gura æqualia grauia AB ſunt in CD <lb></lb>appenſa; ita vt diſtantia EC ſit <expan abbr="diſtãtiæ">di<lb></lb>ſtantiæ</expan> ED æqualis. </s> <s id="N10D10">intelligaturquè <lb></lb>CD tanquam libra, quæ ſuſpendatur <lb></lb>in E. vel vt in ſecunda figura grauia AB habent ipſorum <lb></lb>centra grauitatis, quæ ſint CD, in ipſa DC linea, in pun- <pb xlink:href="077/01/028.jpg" pagenum="24"></pb>ctis <expan abbr="nẽpè">nempè</expan> CD <lb></lb> <arrow.to.target n="fig8"></arrow.to.target><lb></lb>conſtituta. </s> <s id="N10D27">li<lb></lb>braquè ſimili<lb></lb>ter ex puncto <lb></lb>E ſuſpendatur; <lb></lb>ſitquè <expan abbr="diſtãtia">diſtantia</expan> <lb></lb>EC diſtantiæ <lb></lb>ED æqualis. <lb></lb><expan abbr="erũt">erunt</expan> vti〈que〉 in <lb></lb>vtra〈que〉 figura <lb></lb>pondera AB <lb></lb>in diſtantijs ę<lb></lb>qualibus con<lb></lb>ſtituta. </s> <s id="N10D44">ac pro<lb></lb>pterea æ〈que〉ponderabunt, at〈que〉 manebunt. </s> <s id="N10D48">nulla enim ratio <lb></lb>afferri poteſt, cur ex parte A, vel ex parte B deorſum, vel ſur<lb></lb>ſum fieri debeat motus; cùm omnia ſint paria. </s> <s id="N10D4E">ea verò æ〈que〉<lb></lb>ponderare debere, aliqua ratione manifeſtari poteſt ex eo, <lb></lb>quod oſtenſum eſt à nobis in noſtro mechanicorum libro, <lb></lb>tractatu de libra: quod quidem ab Ariſto tele quo〈que〉 in prin<lb></lb>cipio quæſtionum mechanicarum elici poteſt: idem ſcilicet <lb></lb>pondus longius a centro grauius eſſe eodem pondere ipſi cen<lb></lb>tro propinquiori. </s> <s id="N10D5C">Vnde ſi duo eſſent pondera æqualia alte<lb></lb>rum altero propinquius centro, quod remotius eſt, grauius al<lb></lb>tero appareret. </s> <s id="N10D62">ſi igitur grauia æqualia à centro æqualiter di<lb></lb>ſtabunt, æ〈que〉 grauia erunt. </s> <s id="N10D66">ac propterea æ〈que〉ponderabunt. <lb></lb>quod quidem ſupponit Archimedes. </s> <s id="N10D6A">Punctum autem illud, <lb></lb>quod Archimedes accipit, vnde ſumuntur diſtantiæ, ex qui<lb></lb>bus grauia ſuſpenduntur, veluti punctum E, Ariſtoteles cent<lb></lb>rum appellat. </s> <s id="N10D72">& hæc quidem æ〈que〉ponderatio tam ponderi<lb></lb>bus in libra appenſis, quàm in ipſa (vt dictum eſt) conſtitutis <lb></lb>competit: dummodo ea, quibus appenduntur pondera, libe<lb></lb>re ſemper in centrum mundi tendere poſſint. </s> <s id="N10D7A">vtro〈que〉 enim <lb></lb>modo in punctis CD grauitant, vt diximus etiam in eodem <lb></lb>tractatu de libra. </s> <s id="N10D80">Nouiſſe tamen oportet Archimedem in his <lb></lb>libris potiùs intellexiſſe pondera eſſe in diſtantijs collocata, vt <lb></lb>in ſecunda figura, quàm appenſa; vt ex quarta, & quinta <pb xlink:href="077/01/029.jpg" pagenum="25"></pb>primi libri propoſitione pater. </s> <s id="N10D8A">demonſtrationes enim cla<lb></lb>riores redduntur. </s> </p> <figure id="id.077.01.029.1.jpg" xlink:href="077/01/029/1.jpg"></figure> <figure id="id.077.01.029.2.jpg" xlink:href="077/01/029/2.jpg"></figure> <p id="N10D95" type="main"> <s id="N10D97">Porrò non ignoran<lb></lb>dum hoc Archimedis <lb></lb>poſtulatum verificari <lb></lb>de ponderibus quocun<lb></lb>〈que〉 ſitu diſpoſitis, ſiue <lb></lb>CED fuerit horizonti <lb></lb><expan abbr="æquidiſtãs">æquidiſtans</expan>, ſiuè minùs; <lb></lb>vt in hac prima figura, <lb></lb>codem modo ſemper <lb></lb>verum eſſe pondera æ<lb></lb>qualia CD ex ęquali<lb></lb>bus diſtantijs EC ED <lb></lb>æ〈que〉ponderare, vt in<lb></lb>fra (poſt ſcilicet <expan abbr="quartã">quartam</expan> <lb></lb>huius propoſitionem) <lb></lb>perſpicuum erit. </s> <s id="N10DBE">Qua<lb></lb>re cùm Archimedes <expan abbr="tã">tam</expan> <lb></lb>in hoc poſtulato, <expan abbr="quã">quam</expan> <lb></lb>in ſe〈que〉ntibus, ſuppo<lb></lb>nit pondera in diſtan<lb></lb>tijs eſſe collocata, intel<lb></lb>ligendum eſt <expan abbr="diſtãtias">diſtantias</expan> <lb></lb>ex vtra〈que〉 parte in ea<lb></lb>dem recta linea exiſte<lb></lb>re. </s> <s id="N10DDE">Nam ſi (vt in ſecun<lb></lb>da figura) <expan abbr="diſtãtia">diſtantia</expan> AB <lb></lb>fuerit ęqualis diſtantię BC, quæ non indirectum iaceant, <lb></lb>ſed angulum conſtituant; tunc pondera AB, quamuis ſint <lb></lb>ęqualia, non ę〈que〉ponderabunt. </s> <s id="N10DEC">niſi quando (vt in tertia fi<lb></lb>gura) iuncta AC, bifariamquè diuiſa in D, ductaquè BD, <lb></lb>fuerit hęc horizonti perpendicularis, vt in eodem tractatu <lb></lb>noſtro expoſuimus. </s> <s id="N10DF4">Diſtantias igitur in eadem recta linea <lb></lb>ſemper exiſtere intelligendum eſt. </s> <s id="N10DF8">vt ex demonſtrationibus <lb></lb>Archimedis perſpicuum eſt. </s> </p> <pb xlink:href="077/01/030.jpg" pagenum="26"></pb> <p id="N10DFF" type="head"> <s id="N10E01">II.</s> </p> <p id="N10E03" type="main"> <s id="N10E05">Aequalia verò grauia ex inæqualibus <expan abbr="diſtãtijs">diſtantijs</expan> <lb></lb>non æqueponderare, ſed præponderare ad gra<lb></lb>ue ex maiori diſtantia. </s> </p> <p id="N10E0B" type="head"> <s id="N10E0D">SCHOLIVM.</s> </p> <p id="N10E0F" type="main"> <s id="N10E11">Si enim <expan abbr="diſtã">diſtam</expan> <lb></lb> <arrow.to.target n="fig9"></arrow.to.target><lb></lb>tia EC maior <lb></lb>fuerit diſtantia <lb></lb>ED, grauibus <lb></lb>AB ſimiliter æ<lb></lb>qualibus <expan abbr="exiſtẽ">exiſtem</expan> <lb></lb>tibus, & in CD poſitis, tunc concedendum videtur graue A <lb></lb>præponderare ipſi B, quandoquidem EC longior eſt, quàm <lb></lb>ED. ſupponit autem Archimedes hoc poſtulatum reſpiciens <lb></lb>fortaſſe ad ea, quæ Ariſtoteles in principio quæſtionum me<lb></lb>chanicarum oſtendit, vbi colligit Ariſtoteles idem pondus ce<lb></lb>leriùs ferri, quò magis à centro diſtat, vel quod idem eſt, duo <lb></lb>pondera æqualia inæqualiter à centro diſtantia, quod magis <lb></lb>diſtat, celeriùs ferri. </s> <s id="N10E3A">quod autem æqualium ponderum cele<lb></lb>riùs fertur, grauius exiſtit; erit igitur A grauius, quàm B. <lb></lb>quia EC longior eſt, quàm ED. Nos quo〈que〉 (vt diximus) <lb></lb>in libro noſtrorum Mechanicorum tractatu de libra, alijs <lb></lb>quo〈que〉 rationibus oſtendimus, quo pondus eſt in longiori <lb></lb>diſtantia grauius eſſe. </s> <s id="N10E46">ex quibus ſequitur propter longiorem <lb></lb>diſtantiam EC pondus A præponderare ponderi B. ac pro<lb></lb>pterea deorſum ferri. </s> </p> <figure id="id.077.01.030.1.jpg" xlink:href="077/01/030/1.jpg"></figure> <p id="N10E50" type="head"> <s id="N10E52">III.</s> </p> <p id="N10E54" type="main"> <s id="N10E56">Grauibus ex aliquibus diſtantijs <expan abbr="æ〈que〉ponderãtibus">æ〈que〉ponderan<lb></lb>tibus</expan>, ſi alteri grauium aliquid adijciatur, non æ<lb></lb>〈que〉ponderare; ſed ad graue, cui adiectum fuit, <lb></lb>deorſum ferri. </s> </p> <pb xlink:href="077/01/031.jpg" pagenum="27"></pb> <p id="N10E61" type="head"> <s id="N10E63">SCHOLIVM</s> </p> <p id="N10E65" type="main"> <s id="N10E67">Grauia enim <lb></lb> <arrow.to.target n="fig10"></arrow.to.target><lb></lb>AB ſiuè æqua<lb></lb>lia, ſiue in ęqua<lb></lb>lia æ〈que〉ponde<lb></lb>rent ex diſtan<lb></lb>tijs AC CB, al<lb></lb>teri verò gra<lb></lb>uium, putà B, <lb></lb>adijciatur pon<lb></lb>dus D. perſpicuum eſt pondera BD ſimul magis ponderare, <lb></lb>quàm A. ſi enim B ę〈que〉ponderat ipſi A; erit pondus B in <lb></lb>hoc ſitu æ〈que〉graue, vt A: pondera igitur BD in hoc ſitu <expan abbr="nõ">non</expan> <lb></lb>erunt æ〈que〉grauia, vt pondus A. ſed grauiora exiſtent, quàm <lb></lb>A. quare BD deorſum tendent. </s> </p> <figure id="id.077.01.031.1.jpg" xlink:href="077/01/031/1.jpg"></figure> <p id="N10E90" type="head"> <s id="N10E92">IIII.</s> </p> <p id="N10E94" type="main"> <s id="N10E96">Similiter autem, ſi ab altero grauium auferatur <lb></lb>aliquid, non æ〈que〉ponderare; verùm ad graue, à <lb></lb>quo nil ablatum eſt, deorſum tendere. </s> </p> <p id="N10E9C" type="head"> <s id="N10E9E">SCHOLIVM.</s> </p> <p id="N10EA0" type="main"> <s id="N10EA2">Ae〈que〉ponderent grauia BD ſimul, & A ſecundùm <arrow.to.target n="marg13"></arrow.to.target> di<lb></lb>ſtantias CB CA; vt in eadem figura, & ab altero eorum, putà <lb></lb>BD, auferatur D, remanebunt grauia BA; eritquè A gra<lb></lb>uius ipſo B. Nam ſi BD ſimul æ〈que〉ponderant ipſi A, B <lb></lb>tantùm eidem A non æ〈que〉ponderabit, ſed leuius erit. </s> <s id="N10EB4">vnde <lb></lb>ſequitur ex parte A motum fieri deorſum. </s> </p> <pb xlink:href="077/01/032.jpg" pagenum="28"></pb> <p id="N10EBB" type="margin"> <s id="N10EBD"><margin.target id="marg13"></margin.target><emph type="italics"></emph>eadem figu<lb></lb>ra.<emph.end type="italics"></emph.end></s> </p> <p id="N10EC7" type="head"> <s id="N10EC9">V</s> </p> <p id="N10ECB" type="main"> <s id="N10ECD">Aequalibus, ſimilibuſquè figuris planis inter ſe <lb></lb>coaptatis, centra quo〈que〉 grauitatum inter ſe coa<lb></lb>ptati oportet. </s> </p> <p id="N10ED3" type="head"> <s id="N10ED5">SCHOLIVM.</s> </p> <p id="N10ED7" type="main"> <s id="N10ED9">Aequales, <expan abbr="ſimilesq́">ſimiles〈que〉</expan>; ſint <lb></lb> <arrow.to.target n="fig11"></arrow.to.target><lb></lb>figuræ ABC DEF, qua<lb></lb>rum centra grauitatis ſint <lb></lb>GH; ſi ABC ſuperpona<lb></lb>tur ipſi DEF, & hoc <expan abbr="ſecũ">ſecum</expan> <lb></lb>dùm laterum <expan abbr="æqualitatẽ">æqualitatem</expan>, <lb></lb>hoc eſt ſi latus AB fuerit <lb></lb>æquale lateri DE, tunc <lb></lb>ponatur AB ſuper DE; ſimiliter AC ſuper DF, & BC ſuper <lb></lb>EF; tunc manifeſtum eſt centrum grauitatis G ſuper centro <lb></lb>grauitatis H ad unguem conuenire; ita vt ſint vnum tan <expan abbr="tũ">tum</expan> <lb></lb>punctum. </s> <s id="N10F06">Plana enim quæ ſe inuicem contingunt, non ef<lb></lb>ficiunt, niſi vnum tantùm planum. </s> <s id="N10F0A">Solius autem figuræ ex <lb></lb>planis ABC DEF inuicen coaptatis, vnum tantùm erit cen<lb></lb>trum grauitatis, vt nos in noſtro mechanicorum libro ſup<lb></lb>poſuimus; centra igitur grauitatis inter ſeſe conuenire neceſ<lb></lb>ſe eſt. </s> <s id="N10F14">ſi enim centra grauitatis inter ſe non conuenirent, v<lb></lb>na tantùm figura duo poſſet centra grauitatis habere. </s> <s id="N10F18">quod <lb></lb>eſſet omnino <expan abbr="incõueniens">inconueniens</expan>. Dixit autem Archimedes oporte<lb></lb>re has figuras eſſe ſimiles, & æquales, nam figuræ æquales, <lb></lb>ſed non ſimiles, item ſimiles, & <expan abbr="nõ">non</expan> æquales eſſe poſſunt. </s> <s id="N10F28">qua<lb></lb>re, vt inter ſeſe coaptari poſſint, & ſimiles, & æquales eſſe ne<lb></lb>ceſſe eſt. </s> </p> <figure id="id.077.01.032.1.jpg" xlink:href="077/01/032/1.jpg"></figure> <p id="N10F32" type="head"> <s id="N10F34">VI</s> </p> <p id="N10F36" type="main"> <s id="N10F38">Inæ qualium autem, ſed ſimilium centra graui<lb></lb>tatum eſſe ſimiliter poſita. </s> </p> <pb xlink:href="077/01/033.jpg" pagenum="29"></pb> <p id="N10F3F" type="head"> <s id="N10F41">SCHOLIVM.</s> </p> <p id="N10F43" type="main"> <s id="N10F45">Inæquales ſint figuræ, ſi<lb></lb> <arrow.to.target n="fig12"></arrow.to.target><lb></lb>miles verò ABCD EFGH, <lb></lb>quarum cétra grauitatis ſint <lb></lb>KL. ſupponit Archimedes <lb></lb>hęc grauitatis centra KL eſ<lb></lb>ſe in figuris ABCD EFGH <lb></lb>ſimiliter poſita. cùm enim <lb></lb>ſimilium figurarum, & late<lb></lb>ra, & ſpacia ſint ſimilia, neceſſe eſt in ipſis ſimili quo 〈que〉 mo<lb></lb>do centra grauitatis eſſe poſita. </s> <s id="N10F62">vt in ſe〈que〉nti clariùs apparebit. <lb></lb>quomodo autem Archimedes intelligat hanc poſitionis ſimi<lb></lb>litudinem, hoc modo definit. </s> </p> <figure id="id.077.01.033.1.jpg" xlink:href="077/01/033/1.jpg"></figure> <p id="N10F6C" type="head"> <s id="N10F6E">VII.</s> </p> <p id="N10F70" type="main"> <s id="N10F72">Dicimus quidem puncta in ſimilibus figuris eſ<lb></lb>ſe ſimiliter poſita, à quibus ad æquales angulos <lb></lb>ductæ rectæ lineæ cum homologis lateribus angu<lb></lb>los æquales efficiunt. </s> </p> <p id="N10F7A" type="head"> <s id="N10F7C">SCHOLIVM.</s> </p> <p id="N10F7E" type="main"> <s id="N10F80">In ſimilibus figuris ABCD EFGH ſint homologa latera <lb></lb>AB EF, BCFG, CD GH, AD EH. anguli verò æquales, qui <lb></lb>ad AE, BF, CG, DH, primum quidem oſtendendum eſt fie<lb></lb>ri poſſe, ut à duobus punctis intra figuras conſtitutis, duci <lb></lb>poſſint rectę lineę ad angulos æquales, quę cum lateribus an<lb></lb>gulos ęquales efficiant. </s> <s id="N10F8C">Quaſi dicat Archimedes, quoniam <lb></lb>ſupponere poſſumus puncta in ſimilibus figuris eſſe ſimiliter <lb></lb>poſita, ideo ſupponere quo〈que〉 poſſumus centra grauitatis in <lb></lb>ipſis eſſe ſimiliter poſita. </s> <s id="N10F94">Ita〈que〉 ſint figuræ ABCD EFGH ſi<lb></lb>miles, vt dictum eſt, ſumaturquè in ABCD vtcum〈que〉 pun<lb></lb>ctum K à quo ducatur KA KB KC KD. deinde fiat an<pb xlink:href="077/01/034.jpg" pagenum="30"></pb><arrow.to.target n="fig13"></arrow.to.target><lb></lb>gulus FEL angulo BAK æqualis; & EFL ipſi ABK. Iun<lb></lb>ganturquè GL LH. Dico L eſſe ſimiliter poſitum, vt K. <lb></lb>Quoniam enim anguli BAK ABK ſunt angulis FEL EFL <lb></lb>æquales, erit reliquus BKA ipſi FLE æqualis, eritquè ob ſi<lb></lb> <arrow.to.target n="marg14"></arrow.to.target>militudinem triangulorum KA ad AB, vt LE ad EF. eſt <lb></lb>verò AB ad AD, vt EF ad EH propter ſimilitudinem fi<lb></lb><arrow.to.target n="marg15"></arrow.to.target>gurarum, erit igitur ex æquali AK ad AD, vt LE ad EH, <lb></lb>& quoniam angulus BAD angulo FEH eſt æqualis, & BAK <lb></lb>ipſi FEL æqualis; erit & reliquus angulus KAD angulo <lb></lb> <arrow.to.target n="marg16"></arrow.to.target> LEH æqualis. </s> <s id="N10FC1">Quare triangulum KAD triangulo LEH ſi<lb></lb>mile exiſtit, eodemquè modo oſtendetur BKG ſimile eſſe <lb></lb>FLG, & KCD ipſi LGH. ex quibus conſtat angulos KBC <lb></lb>LFG, KCB LGF, & huiuſmodi reliquos reliquis æquales eſſe. <lb></lb>& ob id puncta KL in figuris ABCD EFGH eſſe ſimili<lb></lb>ter poſita. </s> </p> <p id="N10FCD" type="margin"> <s id="N10FCF"><margin.target id="marg14"></margin.target>4 <emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N10FD8" type="margin"> <s id="N10FDA"><margin.target id="marg15"></margin.target>22 <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N10FE3" type="margin"> <s id="N10FE5"><margin.target id="marg16"></margin.target>6 <emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.034.1.jpg" xlink:href="077/01/034/1.jpg"></figure> <p id="N10FF2" type="main"> <s id="N10FF4">Ita〈que〉 demonſtrato dari poſſe puncta in figuris ſimiliter <lb></lb>poſita, potuit ſanè Archimedes antecedens poſtulatum ſup<lb></lb>ponere, nempè inæqualium, ſed ſimilium figurarum centra <lb></lb>grauitatis eſſe ſimiliter poſita. </s> <s id="N10FFC">quod quidem poſtulatum eſt <lb></lb>rationi valde conſentaneum. </s> <s id="N11000">ex dictis enim (ſuppoſitis KL <lb></lb>centris grauitatum) triangulum ABK triangulo EFL ſimi<lb></lb> <arrow.to.target n="marg17"></arrow.to.target> le exiſtit; veluti BKC ipſi FLG. & reliqua reliquis. </s> <s id="N1100A">Quare vt <lb></lb>AK ad KB, ſic EL ad LF, ac permutando vt AK ad EL, <lb></lb>ita BK ad FL. ſimiliter oſtendetur ita eſſe BK ad FL, vt <lb></lb>KC ad LG, & KD ad LH. quare centra grauitatis KL <pb xlink:href="077/01/035.jpg" pagenum="31"></pb>proportionaliter ab angulis diſtant. </s> </p> <p id="N11016" type="margin"> <s id="N11018"><margin.target id="marg17"></margin.target>4 <emph type="italics"></emph>ſexti<emph.end type="italics"></emph.end><lb></lb>16 <emph type="italics"></emph>quinti<emph.end type="italics"></emph.end></s> </p> <p id="N11028" type="main"> <s id="N1102A"><expan abbr="Ducãtur">Ducantur</expan> pręterea à punctis KL ad latera perpendiculares <lb></lb>KM KN KO KP, LQ LR LS LT. & quoniam anguli <lb></lb>KMA LQE ſunt recti, ac propterea æquales, & KAM LEQ <lb></lb>ſunt æquales, ut oſtenſum eſt; erit reliquus MKA reliquo <lb></lb>QLE ęqualis, triangulumquè AKM triangulo ELQ ſimile. <lb></lb>vt igitur AK ad KM; ſic EL ad <expan abbr="Lq.">L〈que〉</expan> & permutando AK <arrow.to.target n="marg18"></arrow.to.target><lb></lb>ad EL, vt KM ad <expan abbr="Lq.">L〈que〉</expan> pariquè ratione oſtendetur triangu<lb></lb>lum BKM triangulo FLQ ſimile exiſtere; eſſequè BK ad <lb></lb>FL, vt KM ad <expan abbr="Lq.">L〈que〉</expan> ſimiliterquè in alijs triangulis oſten<lb></lb>detur, ita eſſe Bk ad FL, vt KN ad LR; & Ck ad GL eſſe, vt <lb></lb>kO ad LS; at〈que〉 kD ad LH, vt kP ad LT. quia verò AK <lb></lb>EL, Bk FL, Ck GL, Dk HL in eadem ſunt proportione, vt <lb></lb>proximè demonſtratum fuit; in eadem quo〈que〉 proportione <lb></lb>erit kM ad LQ, & KN ad LR; & KO ad LS, at〈que〉 kP ad <lb></lb>LT. ex quibus ſequitur centra grauitatis KL, non ſolùm ab <lb></lb>angulis in eadem proportione diſtare; verùm etiam à late<lb></lb>ribus in eadem quo〈que〉 proportione diſtare. </s> <s id="N1105E">Ita〈que〉 cognito, <lb></lb>quomodo intelligar Archimedes centra grauitatis in ſimili<lb></lb>bus figuris eſſe ſimiliter poſita; nunc conſiderandum eſt præ <lb></lb>cedens poſtulatum, quatenus nimirum oporteat grauitatis <expan abbr="cẽ">cem</expan> <lb></lb>tra in ſimilibus figuris ſimiliter eſſe conſtituta. </s> <s id="N1106C">Nam inti<lb></lb>miùs conſiderando hanc ſimilem horum grauitatis <expan abbr="centrorũ">centrorum</expan> <lb></lb>poſitionem, congruum, & neceſſarium videtur, ſimiles figu<lb></lb>ras ſecundùm eandem proportionem eſſe æ〈que〉pon<expan abbr="derãtes">derantes</expan>; <lb></lb>eademquè ratione (ob earum ſimilitudinem) circa grauita<lb></lb>tis centra æ〈que〉ponderare, veluti ſi figuræ: AC EG (quarum <lb></lb>centra grauitatis ſint KL) à rectis lineis PN TR vtcumquè <lb></lb>diuidantur, quæ per centra KL tranſeant; dummodo in figu<lb></lb>ris ſint ſimiliter ductæ; hoc eſt, vel latera, vel angulos in <expan abbr="eadẽ">eadem</expan> <lb></lb>proportione diſpeſcant: vt ſit AP ad PD, vt ET ad TH. æ<lb></lb>〈que〉ponderabunt vti〈que〉 partes PABN PNCD, veluti partes <lb></lb>TEFR TRGH. & hæc non eſt ſimplex æ〈que〉ponderatio; ve<lb></lb>rùm etiam (vt ita dicam) ſimilis, & æqualis æ〈que〉ponderatio. <lb></lb>cùm ſit ſecundùm eandem proportionem, quandoquidem <lb></lb>eſt PB ipſi TF ſimilis, cùm triangula AKB ELF, AKP ELT, <lb></lb>BKN FLR, ſint inter ſe ſimilia, quæ quidem efficiunt, figuras <pb xlink:href="077/01/036.jpg" pagenum="32"></pb>PB TF inter ſe ſimiles eſſe. </s> <s id="N1109C">ob eademquè cauſam eſt PC ſi<lb></lb>milis TG. quod quidem ex demonſtratis etiam facilè con<lb></lb>ſtat. </s> <s id="N110A2">cùm anguli ſint ęquales, & latera proportionalia. </s> <s id="N110A4">Vt au<lb></lb>tem clariùs intelligatur hæc ſimilis, & æqualis æ〈que〉pondera<lb></lb>rio, adducere libuit nonnulla ex ijs, quæ poſteriùs tractanda <lb></lb>ſumentur. </s> <s id="N110AC">Ita〈que〉 intelligatur punctum V centrum eſſe gra<lb></lb> <arrow.to.target n="fig14"></arrow.to.target><lb></lb>uitatis figuræ PB, X verò centrum grauitatis figure TF. ſi<lb></lb>militer punctum Y centrum eſſe grauitatis figuræ PC, Z <lb></lb>verò figurę TG. Iunganturquè VY XZ. quæ quidem per <lb></lb>centra grauitatis KL tranſibunt. </s> <s id="N110BB">quòd ex ijs, quę dicenda <lb></lb>ſunt, manifeſtum erit, percipuè〈que〉 ex octaua proportione <lb></lb>primi huius. </s> <s id="N110C1">quod tamen interim ſupponatur. </s> <s id="N110C3">At verò quo<lb></lb>niam PB PC ę〈que〉ponderant ſecundùm proportionem, <lb></lb>quam habet YK ad KV; TF verò & TG ę〈que〉ponderant <lb></lb>ſecundùm proportionem, quam habet ZL ad LX. eſt. <expan abbr="n.">enim</expan> <lb></lb>ac ſi AN eſſet appenſa in V, & PC in Y; ER in X, & <lb></lb>TG in Z. vt in ſe〈que〉ntibus manifeſta erunt. </s> <s id="N110D3">Atverò quo<lb></lb> <arrow.to.target n="marg19"></arrow.to.target>niam AN ſimilis eſt ipſi ER, habebit AN ad ER <expan abbr="duplã">duplam</expan> <lb></lb>proportionem eius, quam habet latus PN ad TR. pariquè <lb></lb>ratione quoniam PC ſimilis eſt TG, habebit PC ad TG <lb></lb>duplam proportionem eius, quam habet idem latus PN ad <lb></lb> <arrow.to.target n="marg20"></arrow.to.target> TR. quare ita ſe habet AN ad ER, ut PC ad TG. & per<lb></lb> <arrow.to.target n="marg21"></arrow.to.target> mutando AN ad PC, vt ER ad TG. Sed vt AN ad PC, ita eſt <lb></lb>Y K ad KV, & vt ER ad TG. ſic ZL ad LX. eandem igitur <pb xlink:href="077/01/037.jpg" pagenum="33"></pb><expan abbr="proportionẽ">proportionem</expan> habebit YK ad KV, quam ZL ad LX. Quare <lb></lb>AN PC, & ER TG ſecundùm eandem proportionem æ<lb></lb>〈que〉ponderabunt. </s> <s id="N110FE">quod quidem contingit ex ſimilitudine fi<lb></lb>gurarum, & ex centris grauitatum KL ſimiliter poſitis, quę <lb></lb>quidem magnitudines, ſi non eſſent ſimiles, diuiſę <expan abbr="quidẽ">quidem</expan> per <lb></lb>centrum grauitatis, partes vti〈que〉 ę〈que〉ponderarent; non ta<lb></lb>men ſemper ſecundùm eandem proportionem. </s> <s id="N11108">quod tamen <lb></lb>ſemper figuris ſimilibus (cùm in ipſis grauitatis centra ſint ſi <lb></lb>militer poſita) contingit; dummodo (vt dictum eſt) diui<lb></lb>dantur. </s> <s id="N11110">Vnde conſtat, quam ſit conueniens grauitatis centra <lb></lb>in figuris hac ratione eſſe conſtituta. </s> <s id="N11114">ex quibus omnibus per<lb></lb>ſpicuum eſt, centra grauitatis debere in figuris ſimilibus eſſe ſi <lb></lb>militer poſita. </s> <s id="N1111A">vt Archimedes in <expan abbr="pręcedẽti">pręcedenti</expan> poſtulato pręmiſit. </s> </p> <p id="N1111C" type="margin"> <s id="N1111E"><margin.target id="marg18"></margin.target>4 <emph type="italics"></emph>ſexti<emph.end type="italics"></emph.end><lb></lb>16 <emph type="italics"></emph>quinti<emph.end type="italics"></emph.end></s> </p> <p id="N1112E" type="margin"> <s id="N11130"><margin.target id="marg19"></margin.target>20 <emph type="italics"></emph>ſexti<emph.end type="italics"></emph.end></s> </p> <p id="N11139" type="margin"> <s id="N1113B"><margin.target id="marg20"></margin.target>11 <emph type="italics"></emph>quinti<emph.end type="italics"></emph.end></s> </p> <p id="N11144" type="margin"> <s id="N11146"><margin.target id="marg21"></margin.target>16 <emph type="italics"></emph>quinti<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.037.1.jpg" xlink:href="077/01/037/1.jpg"></figure> <p id="N11153" type="head"> <s id="N11155">VIII.</s> </p> <p id="N11157" type="main"> <s id="N11159">Si magnitudines ex æqualibus diſtantijs æ〈que〉<lb></lb>ponderant, & ipſis æquales ex ijſdem diſtantijs æ<lb></lb>〈que〉ponderabunt. </s> </p> <p id="N1115F" type="head"> <s id="N11161">SCHOLIVM.</s> </p> <p id="N11163" type="main"> <s id="N11165">Hoc eſt perſpicuum, <expan abbr="nã">nam</expan> <lb></lb> <arrow.to.target n="fig15"></arrow.to.target><lb></lb>ſi magnitudines AB ex di<lb></lb>ſtantijs CA CB ę〈que〉pon<lb></lb>derant: ſit autem D ipſi A <lb></lb>ęqualis, & E ipſi B. <expan abbr="auferã">auferam</expan> <lb></lb>turquè magnitudines AB à <lb></lb>linea AB, ipſarumquè loco ponatur D in A, & E in B, ma<lb></lb>gnitudines DE ſimiliter <expan abbr="ę〈que〉pondęrabũt">ę〈que〉pondęrabunt</expan>. qua ratione enim <lb></lb>magnitudines AB inter ſeſe ę〈que〉ponderare dicuntur; eadem <lb></lb>prorſus, & magnitudines DE ex ijſdem diſtantijs ę〈que〉pon<lb></lb>derabunt. </s> <s id="N1118C">quandoquidem omnia data ſunt paria. </s> <s id="N1118E">illud ta<lb></lb>men non eſt pretereundum, nimirum non oportere DE ipſis <lb></lb>AB ęquales eſſe in magnitudine, ſed in grauitate. </s> <s id="N11194">poteſt enim <pb xlink:href="077/01/038.jpg" pagenum="34"></pb>magnitudinum inęqualium minor maiore grauior exiſtere, <lb></lb>ob naturæ diuerſitatem, ac propterea cùm inquit Archimedes <lb></lb><emph type="italics"></emph>& ipſis aquales<emph.end type="italics"></emph.end>, ſiue ſint magnitudine æquales, vel inæquales, in<lb></lb>telligendum eſt eſſe omnino æquales in grauitate. </s> <s id="N111A5">grauitas. <expan abbr="n.">enim</expan> <lb></lb>cauſa eſt, vt magnitudines æ〈que〉ponderare debeant. </s> </p> <figure id="id.077.01.038.1.jpg" xlink:href="077/01/038/1.jpg"></figure> <p id="N111B1" type="head"> <s id="N111B3">VIIII,</s> </p> <p id="N111B5" type="main"> <s id="N111B7">Omnis figuræ, cuius perimeter ſit ad <expan abbr="eandẽ">eandem</expan> par<lb></lb>tem concauus, centrum grauitatis intra figuram <lb></lb>eſſe oportet. </s> </p> <p id="N111C1" type="head"> <s id="N111C3">SCHOLIVM.</s> </p> <figure id="id.077.01.038.2.jpg" xlink:href="077/01/038/2.jpg"></figure> <p id="N111C8" type="main"> <s id="N111CA">Quid intelligat Ar<lb></lb>chimedes per has figu<lb></lb>ras ad eandem partem <lb></lb>concauas, apertiùs ſi<lb></lb>gnificauit initio libro<lb></lb>rum de ſphęra, & cylin<lb></lb>dro. </s> <s id="N111D8">vbi primùm vult <lb></lb>has figuras eſſe termina<lb></lb>tas; quod non ſolùm in<lb></lb>telligendum eſt decur<lb></lb>uilineis, verùm etiam <lb></lb>de rectilineis, & de mi<lb></lb>xtis. </s> <s id="N111E6">rectilineę quidem <lb></lb>erunt trium, quattuor, <lb></lb>quin〈que〉 & plurium la<lb></lb>terum; quamuis latera <lb></lb>non ſint æqualia, ne<lb></lb>〈que〉 anguli ęquales, vt <pb xlink:href="077/01/039.jpg" pagenum="35"></pb>ABCDE, cuius omnes anguli ſunt flexi ad interiorem figuræ <lb></lb>partem. </s> <s id="N111F8">& hoc modo perimeter huius figuræ erit ad eandem <lb></lb>partem concauus. </s> <s id="N111FC">vnde excluduntur figuræ, exempli gratia <lb></lb>FGHKL; cùm angulus K non ſit ſinuoſus, & concauus ad <lb></lb>eandem partem, vt reliqui anguli; qui ſunt ſinuoſi verſus inte<lb></lb>riorem partem figurę K vero ad exteriorem. </s> <s id="N11206">ſimili modo <lb></lb>intelligendum eſt de curuilineis, vt circuli, ellipſes, vel alterius <lb></lb>generis figuræ, vt ſunt MN, quæ ſuam habent concauitatem <lb></lb>ad eandem partem: ſed curuline˛ OP non ſunt ad eandem <lb></lb>partem concauę. </s> <s id="N11214">Mixtæ quo〈que〉 figuræ, ut ſunt portiones cir<lb></lb>culi, hyperbolę ac parabolę rectis linenis terminatę, vel alte<lb></lb>rius generis figurę, vt ſunt QR. hę quidem omnes ſunt ad <expan abbr="eãdem">ean<lb></lb>dem</expan> partem concauę. Mixtæ verò ST minimè Regulam au<lb></lb>tem quandam vniuerſalem ex verbis Archimedis loco citato <lb></lb>elicere poſſumus, vt cognoſcere valeamus, an figuræ ſint ad <lb></lb>eandem partem concauæ, vel minùs vt ſcilicet in oblata figu<lb></lb>ra vbicum〈que〉 duo ſumi poſſint puncta, quæ ſi recta linea <expan abbr="cõnectantur">con<lb></lb>nectantur</expan>, tota recta li<lb></lb><arrow.to.target n="fig16"></arrow.to.target><lb></lb>nea, vel ipſius pars ali<lb></lb>qua extra figuram non <lb></lb>cadat. </s> <s id="N11239">vt in figuris A, <lb></lb>quæ ſunt ad <expan abbr="eandẽ">eandem</expan> par<lb></lb>tem concauæ, vtcum<lb></lb>〈que〉 duo ſumantur <expan abbr="pũ-cta">pun<lb></lb>cta</expan> BC, quæ conne<lb></lb>ctantur, tota uti〈que〉 re<lb></lb>cta linea inter puncta <lb></lb>BC exiſtens, extra figu<lb></lb>ram non cadet. </s> <s id="N11253">Quòd <lb></lb>ſi hæc linea cum termino, hoc eſt eum latere figurę conueni<lb></lb>ret, vt ſi figuræ latus fuerit rectum, in quo duo ſumantur pun<lb></lb>cta, nihilominus recta linea inter hæc puncta extra figuram <lb></lb>non cadet: quandoquidem figuræ terminus extra figuram mi<lb></lb>nimè reperitur at〈que〉 hac ratione quomodocun〈que〉, & <expan abbr="vbicũ〈que〉">vbicum<lb></lb>〈que〉</expan> in his figuris duo ſumantur puncta, idem ſemper contin<lb></lb>get. </s> <s id="N11263">Quod tamen figuris D ſemper euenite non poteſt in qui<lb></lb>bus (cùm non ſint ad eandem partem concauę) duo ſumere <pb xlink:href="077/01/040.jpg" pagenum="36"></pb>poſſumus puncta EG, inter quę tota recta linea EG extra <lb></lb>figuram cadet. </s> <s id="N1126D">vel fumere poſſumus puncta FG, ita vt rectę <lb></lb>lineę FG pars EG extra figuram cadat. </s> <s id="N11271">figurę igitur, quæ <lb></lb>ad eandem partem ſunt concauæ, illę ſunt, quę ſinuoſitatem, <lb></lb>concauitatemquè ſuam habent ſemper interiorem ipſius fi<lb></lb>gurę partem reſpicientem. </s> <s id="N11279">Harum què rectè ſupponit Archi<lb></lb>medes centrum grauitatis ſemper eſſe intra ipſam figuram. <lb></lb>ita vt ne〈que〉 centrum eſſe poſſit in ambitu ipſius figurę ete<lb></lb>nim ſi extra figuram, ſiue in ambitu ipſius eſſe poſſet, num<lb></lb>quam circa centrum grauitatis partes figurę vndiquè <expan abbr="ę〈que〉põderarent">ę〈que〉pon<lb></lb> <arrow.to.target n="marg22"></arrow.to.target>derarent</expan>: ne〈que〉 facta ex grauitatis centro ſuſpenſione figura <lb></lb>vbicum〈que〉, & in omni ſitu maneret. </s> <s id="N1128F">quod ramen ex ratione <lb></lb>centri grauitatis efficere deberet. </s> <s id="N11293">tota nimirum figura ex vna <lb></lb>eſſet parte, & ex altera nihil eſſet, quod ipſi figurę ę〈que〉ponde<lb></lb>rare poſſet. </s> <s id="N11299">Neceſſe eſt igitur centrum grauitatis cuiuſlibet fi<lb></lb>gurę ad eandem partem concauę eſſe in ſpacio à figurę ambi<lb></lb>tu contento. </s> <s id="N1129F">vt figurę AB <lb></lb> <arrow.to.target n="fig17"></arrow.to.target><lb></lb>centrum grauitatis erit in<lb></lb>tra ipſam, putà in C. quod <lb></lb>quidem non euenit ſemper <lb></lb>in alijs figuris, quę ſuum <expan abbr="cõ">com</expan> <lb></lb>cauitatis ambitum interio<lb></lb>rem figurę partem <expan abbr="nõ">non</expan> reſpi<lb></lb>cientem habent. </s> <s id="N112BC">cùm varijs <lb></lb>modis poſſit centrum graui<lb></lb>tatis in figuris eſſe <expan abbr="collocatũ">collocatum</expan>. <lb></lb>vt ſuperius quo〈que〉 diximus. <lb></lb>Nam figurę D <expan abbr="centrũ">centrum</expan> gra<lb></lb>uitatis erit extra ambitum fi<lb></lb>gurę, vt in E. figura verò F <lb></lb>ita ſe habere poterit, vt cen<lb></lb>trum grauitatis ſit in perime<lb></lb>tro, vt in G. euenit <expan abbr="autẽ">autem</expan> aliquando vt in figura HK <expan abbr="centrũ">centrum</expan> <lb></lb>grauitatis L intra ipſam figuram reperiatur; quamuis conca<lb></lb>uitates la torum interiorem partem minimè <expan abbr="reſpiciãt">reſpiciant</expan>. Sed hęc <lb></lb>poſſunt eſſe, & non eſſe, vt in figura M, cuius centrum extra <lb></lb>eſſe poteſt in N. quamuis (vt antea diximus) centrum graui- <pb xlink:href="077/01/041.jpg" pagenum="37"></pb>tatis intra figuram ſemper exiſtere aliquo modo intelligi po<lb></lb>teſt. </s> </p> <p id="N112F2" type="margin"> <s id="N112F4"><margin.target id="marg22"></margin.target><emph type="italics"></emph>per def. <lb></lb><expan abbr="cẽt">cent</expan>. grau.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.041.1.jpg" xlink:href="077/01/041/1.jpg"></figure> <figure id="id.077.01.041.2.jpg" xlink:href="077/01/041/2.jpg"></figure> <p id="N11309" type="main"> <s id="N1130B">Refert Eutocius hoc loco, Geminum rectè dicere, dum aſſe<lb></lb>rit Archimedem dignitates petitiones apellare. </s> <s id="N1130F">æqualia enim <lb></lb>grauia ex diſtantijs æqualibus æ〈que〉ponderare, dignitas eft; & <lb></lb>quæ deinceps. <expan abbr="Verũ">Verum</expan> ſi hæc principia ab Archimede tradita re<lb></lb>ctè perpendamus, omnia dignitates eſſe minimè reperiemus. <lb></lb>nam ſeptimum poſtulatum eſt definitio, non dignitas. </s> <s id="N1131D">veluti <lb></lb>alia fortaſſe nonnulla non ſunt dignitates, vt ſecundum; quod <lb></lb>aliquo modo probari poteſt, vt diximus. </s> <s id="N11323">ſextum quo〈que〉 po<lb></lb>tiùs eſt ſuppoſito, quàm dignitas. </s> <s id="N11327">Quoniam autem vt clarè <lb></lb>conſpicitur Archimedes ſub vno tantùm titulo pauca hæc <lb></lb>principia complecti voluit; quippe quod inſtitutum quàm plu<lb></lb>rimis mathematicis ſolemne fuit, qui principia vnico tantum <lb></lb>nomine nuncuparunt, modò vno, modò altero; nimirum, <lb></lb>vel petitionis, vel dignitatis, vt refert Proclus ſecundo libro, & <lb></lb>tertio ſuorum commentariorum in primum elementorum. </s> <s id="N11335">Eu<lb></lb>clidis; qui de Archimede peculiariter mentionem faciens, in<lb></lb>quit illum in his libris principia vnico tantùm nomine (peti<lb></lb>tionis ſcilicet) nuncupaſſe. </s> <s id="N1133D">Hæc tamen potiùs petitionum, <lb></lb>quàm definitionum, vel dignitatum nomine nuncupare vo<lb></lb>luit; nam ſi dignitates appellaſſet; ea principia, quæ non ſunt <lb></lb>dignitates, inter dignitates malè collocaſſet. </s> <s id="N11345">nulla quippè defi<lb></lb>nitio dignitas dici debet; quandoquidem definitio terminos <lb></lb>declarat, at〈que〉 conſtituit. </s> <s id="N1134B">dignitas verò notos terminos copu<lb></lb>lat. </s> <s id="N1134F">Pariquè ratione ſi definitionis nomine hæc principia nun<lb></lb>cupaſſet. </s> <s id="N11353">dignitates malè ſub hoc nomine complexus fuiſſet, <lb></lb>quæ nullo modo rem definiunt, ſed cùm ſint communes no<lb></lb>tiones, ſtatim cùm eas intellectus apprehendit, quieſcit. </s> <s id="N11359">Qua<lb></lb>re omnia ſub petitionum nomine recte collocauit, non eſt. <expan abbr="n.">enim</expan> <lb></lb>abſurdum dignitates, definitioneſquè poſſe apellari petitio<lb></lb>nes. </s> <s id="N11365">etenim petimus, quæ ſunt concedenda, at〈que〉 dignitates <lb></lb>ſunt concedendę, ergo eas petere quo〈que〉 poſſumus. </s> <s id="N11369">Definitio<lb></lb>nibus verò rectè quo〈que〉 hoc nomen conuenire poteſt. </s> <s id="N1136D">Nam <lb></lb>dùm definitio terminos conſtituat, at〈que〉 declaret, cur non pe<lb></lb>tere poſſumus, terminos ſic ſe habere, vel ſiceſſe rectè definitos? <lb></lb>vt exempli gratia, petit Archimedes puncta in figuris fimiliter <pb xlink:href="077/01/042.jpg" pagenum="38"></pb>poſita, ita ſehabere, vt ſunt ab ipſo definita, vel rectè eſſe defi<lb></lb>nita puncta, quæ ſunt in figuris ſimilibus poſita. </s> <s id="N1137B">Quapropter <lb></lb>hæc principia, quoniam pauca ſunt, ſub petitionum nomine <lb></lb>Archimedes rectè collocauit. </s> <s id="N11381">quòd ſi multa extitiſſent, ea for<lb></lb>taſſe diſtinxiſſet. </s> </p> <p id="N11385" type="main"> <s id="N11387"><emph type="italics"></emph>His ſuppoſitis.<emph.end type="italics"></emph.end> <expan abbr="poſtquã">poſtquam</expan> Archimedes <expan abbr="prĩcipia">principia</expan> poſuit, ad theore<lb></lb>mata ſe conuertit, & inquit, <emph type="italics"></emph>his ſuppoſitis<emph.end type="italics"></emph.end>, quaſi dicat, ea, quæ <lb></lb>poſuimus, ſufficiunt ad oſtendenda theoremata, veluti. </s> </p> <p id="N113A0" type="head"> <s id="N113A2">PROPOSITIO. I.</s> </p> <p id="N113A4" type="main"> <s id="N113A6">Grauia, quæ ex æqualibus diſtantijs æ〈que〉pon<lb></lb>derant, æqualia ſunt. </s> </p> <p id="N113AA" type="main"> <s id="N113AC">Sint AD, & B grauia, <lb></lb> <arrow.to.target n="fig18"></arrow.to.target><lb></lb>quæ ex æqualibus diſtantijs <lb></lb>CA CB æ〈que〉ponderent. </s> <s id="N113B7">di<lb></lb>co grauia AD, & B inter<lb></lb>ſeſe æqualia eſſe. <emph type="italics"></emph>ſi enim<emph.end type="italics"></emph.end> (ſi fie<lb></lb>ri poteſt) <emph type="italics"></emph>eſſent inæqualia<emph.end type="italics"></emph.end>; vt ſi <lb></lb>AD eſſet grauius, quàm B, <lb></lb>ſit D exceſſus, quo AD grauius eſt, quàm B. <emph type="italics"></emph>ablato<emph.end type="italics"></emph.end> ita〈que〉 <lb></lb><emph type="italics"></emph>exceſſu<emph.end type="italics"></emph.end> D <emph type="italics"></emph>à maiori<emph.end type="italics"></emph.end> AD, <emph type="italics"></emph>reliqua<emph.end type="italics"></emph.end> grauia, quæ relinquuntur AB, <lb></lb> <arrow.to.target n="marg23"></arrow.to.target> erunt inter ſe ęqualia; quę ex ęqualibus diſtantijs CA CB æ<lb></lb>〈que〉ponderare deberent; tamen <emph type="italics"></emph>non æ〈que〉ponderabunt. </s> <s id="N113F3">cùm<emph.end type="italics"></emph.end> enim <lb></lb>poſitum ſit AD B ę〈que〉ponderare, & <emph type="italics"></emph>ab altero a〈que〉ponderan-<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg24"></arrow.to.target> <emph type="italics"></emph>tium<emph.end type="italics"></emph.end> AD <emph type="italics"></emph>aliquod ſit ablatum<emph.end type="italics"></emph.end> D; reliqua grauia AB ex ęqua<lb></lb>libus diſtantijs CA CB non ę〈que〉ponderabunt quod fieri <lb></lb>non poteſt; ſiquidem AB inter ſe ſunt ęqualia. <emph type="italics"></emph>Grauia igitur, <lb></lb>quæ ex æqualibus <expan abbr="distãtijs">distantijs</expan> æ〈que〉ponderant, æqualia ſunt.<emph.end type="italics"></emph.end> quod de<lb></lb>monſtrare oportebat. </s> </p> <p id="N11423" type="margin"> <s id="N11425"><margin.target id="marg23"></margin.target>4. <emph type="italics"></emph>poſtula<lb></lb>tum huius<emph.end type="italics"></emph.end></s> </p> <p id="N11430" type="margin"> <s id="N11432"><margin.target id="marg24"></margin.target><emph type="italics"></emph>contra pri<lb></lb>mum post <lb></lb>huius.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.042.1.jpg" xlink:href="077/01/042/1.jpg"></figure> <p id="N11442" type="head"> <s id="N11444">SCHOLIVM.</s> </p> <p id="N11446" type="main"> <s id="N11448">Cùm ſit ſcopus Archimedis (vt diximus) in primis octo <lb></lb>theorematibus, fundamentum tradere in hac ſcientia præci- <pb xlink:href="077/01/043.jpg" pagenum="39"></pb>puum, nempè magnitudinum grauitates inter ſe ita ſe habe<lb></lb>re, vt diſtantiæ permutatim ex quibus ſuſpenduntur ſe <expan abbr="habẽt">habent</expan>. <lb></lb>primùm incipit oſtendere, quomodo ſe habeant grauia in di<lb></lb>ſtantijs ęqualibus poſita; primùmquè in hac prima propoſitio <lb></lb>ne oſtendit, ſi grauia ę〈que〉ponderant ex diſtantijs ęqualibus, <lb></lb>ęqualia eſſe. </s> <s id="N1145E">in ſe〈que〉nti verò, ſi grauia ſunt inęqualia, ex di<lb></lb>ſtantijs ęqualibus nullo modo æ〈que〉ponderare oſtendet; ſed <lb></lb>præponderare ad maius. </s> </p> <p id="N11464" type="head"> <s id="N11466">PROPOSITIO. II.</s> </p> <p id="N11468" type="main"> <s id="N1146A">Inæqualia grauia ex æqualibus diſtantijs non <lb></lb>æ〈que〉ponderabunt, ſed præponderabit ad maius. </s> </p> <figure id="id.077.01.043.1.jpg" xlink:href="077/01/043/1.jpg"></figure> <p id="N11471" type="main"> <s id="N11473">Sint gra<lb></lb>uia inęqua<lb></lb>lia AB C in <lb></lb>diſtantijs <expan abbr="ęqualib^{9}">ę<lb></lb>qualibus</expan> DA <lb></lb>DC. ſitquè <lb></lb>grauius AB, <lb></lb>quàm C. di<lb></lb>co grauia AB C non ę〈que〉ponderare, ſed maius AB <expan abbr="deorsũ">deorsum</expan> <lb></lb>ferri. </s> <s id="N1148B">ſit B exceſſus, quo AB ſuperat C. <emph type="italics"></emph>ablato<emph.end type="italics"></emph.end> ita〈que〉 à ma<lb></lb>iori AB <emph type="italics"></emph>exceſſu<emph.end type="italics"></emph.end> B, reliqua grauia AC ęqualia ex diſtantijs <lb></lb>DA DC <emph type="italics"></emph>æ〈que〉ponderabunt. </s> <s id="N114A0">cùm æqualia grauia ex distantiis æquali-<emph.end type="italics"></emph.end> <arrow.to.target n="marg25"></arrow.to.target><lb></lb><emph type="italics"></emph>bus æ〈que〉ponderent.<emph.end type="italics"></emph.end> ſi ita〈que〉 grauia AC ę〈que〉ponderant, <emph type="italics"></emph>adiecto <lb></lb>igitur<emph.end type="italics"></emph.end> ipſi A <emph type="italics"></emph>ablato<emph.end type="italics"></emph.end> B, <emph type="italics"></emph>præponderabit ad maius<emph.end type="italics"></emph.end>, hoc eſt ab deor <arrow.to.target n="marg26"></arrow.to.target><lb></lb>ſum tendet. <emph type="italics"></emph>quoniam æ〈que〉ponderantium altero<emph.end type="italics"></emph.end> nempè A <emph type="italics"></emph>adiectum <lb></lb>fuit<emph.end type="italics"></emph.end> B. Grauius igitur præponderat leuiori, ambobus in <expan abbr="diſtãtijs">diſtan<lb></lb>tijs</expan> ęqualibus poſitis. </s> <s id="N114DC">quod demonſtrare oportebat. </s> </p> <p id="N114DE" type="margin"> <s id="N114E0"><margin.target id="marg25"></margin.target>1 <emph type="italics"></emph>poſt hu<lb></lb>ius.<emph.end type="italics"></emph.end></s> </p> <p id="N114EB" type="margin"> <s id="N114ED"><margin.target id="marg26"></margin.target>3 <emph type="italics"></emph>post hu<lb></lb>ius.<emph.end type="italics"></emph.end></s> </p> <p id="N114F8" type="head"> <s id="N114FA">SCHOLIVM.</s> </p> <p id="N114FC" type="main"> <s id="N114FE">Hæc duo theoremata in gręco exemplari impreſſo ſequun<lb></lb>tur <expan abbr="quidẽ">quidem</expan> poſtulata, & reliquis theorematibus ſunt prępoſita. <pb xlink:href="077/01/044.jpg" pagenum="40"></pb>quia verò inter principia collocari non poſſunt; cùm ſuas ha<lb></lb>beant propoſitiones, ſuaſquè ſeorſum habeant demonſtratio<lb></lb>nes, ideo inter propoſitiones ipſa collocare nobis viſum eſt. <lb></lb>cùm pręſertim nonnulla ex ſe〈que〉ntibus theorematibus, po<lb></lb>tiſſimùm verò proximum eiuſdem cum his duobus ordinis, <lb></lb>& naturæ ſint. </s> <s id="N11514">Ne〈que〉 enim propterea peruertitur ordo; non <lb></lb>enim hę propoſitiones in alium transferuntur locum. </s> <s id="N11518">ſed <expan abbr="tã-tùm">tan<lb></lb>tùm</expan> inter alias numeris adnotantur. </s> <s id="N11520">exiſtimandum enim eſt, <lb></lb>Archimedem propoſitiones in ſerie propoſitionum collocaſ<lb></lb>ſe. </s> <s id="N11526">hanc verò exiguam mutationem accidiſſe <expan abbr="oblongitudinẽ">oblongitudinem</expan> <lb></lb>temporis; cuius proprium eſt, res potiùs deſtruere, quàm ac<lb></lb>comodare. </s> <s id="N11530">Hoc autem nobis hanc præbebit commoditatem, <lb></lb>vt, quando libuerit, has propoſitiones numeris nominare <lb></lb>poſſimus. </s> <s id="N11536">id ipſumquè numeri poſtulata diſtinguentes præ<lb></lb>ſtant, quamuis in Gręco codice poſtulata (Gręcorum more) <lb></lb>numeris adnotata non ſint. </s> </p> <p id="N1153C" type="head"> <s id="N1153E">PROPOSITIO. III.</s> </p> <p id="N11540" type="main"> <s id="N11542">Inæqualia grauia ex diſtantijs inæqualibus æ<lb></lb> <arrow.to.target n="marg27"></arrow.to.target> 〈que〉ponderabunt, maius quidem ex minori. </s> </p> <p id="N1154A" type="margin"> <s id="N1154C"><margin.target id="marg27"></margin.target>A</s> </p> <figure id="id.077.01.044.1.jpg" xlink:href="077/01/044/1.jpg"></figure> <p id="N11553" type="main"> <s id="N11555"><emph type="italics"></emph>Sint in æqualia grauia AD, B<emph.end type="italics"></emph.end>; <lb></lb> <arrow.to.target n="marg28"></arrow.to.target> <emph type="italics"></emph>ſit què maius AD<emph.end type="italics"></emph.end>, exceſſus ve <lb></lb>rò, quo AD ſuperat B, ſit <lb></lb>D. <emph type="italics"></emph><expan abbr="æ〈que〉põderentquè">æ〈que〉ponderentquè</expan><emph.end type="italics"></emph.end> AD B <emph type="italics"></emph>ex <lb></lb>diſtantiis AC C B. oſtendendum <lb></lb>eſt, minorem eſſe<emph.end type="italics"></emph.end> <expan abbr="diftantiã">diftantiam</expan> <emph type="italics"></emph>AC <lb></lb>ipſa CB. Non ſit quidem, ſi fie<lb></lb>ri potest<emph.end type="italics"></emph.end>, AC minor, quàm CB; erit nimirum, vel ęqualis, <lb></lb>vel maior. </s> <s id="N1158E">Quòd ſi AC fuerit ęqualis ipſi CB, <emph type="italics"></emph>ablato enim <lb></lb>exceſſu<emph.end type="italics"></emph.end> D, <emph type="italics"></emph>quo AD ſuperat B. cùm ab a〈que〉ponderantium altero ab<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg29"></arrow.to.target> <emph type="italics"></emph>latum ſit aliquid<emph.end type="italics"></emph.end>, grauia AB non æ〈que〉ponderabunt; ſed <emph type="italics"></emph>præ-<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg30"></arrow.to.target> <emph type="italics"></emph>ponderabit ad B. non præponderabit autem; exiſtente enim AC aqua <lb></lb>li CB<emph.end type="italics"></emph.end>, cùm ab inęqualibus grauibus AD B ablatus ſit ex<lb></lb>ceſſus D, <emph type="italics"></emph>grauia<emph.end type="italics"></emph.end>, quæ relinquuntur AB, erunt inter ſe <emph type="italics"></emph>æqualia<emph.end type="italics"></emph.end>; <pb xlink:href="077/01/045.jpg" pagenum="41"></pb>quæ <emph type="italics"></emph>ex diſtantiis æqualibus<emph.end type="italics"></emph.end> AC CB <emph type="italics"></emph>æ〈que〉ponderarent.<emph.end type="italics"></emph.end> at non ę〈que〉 <lb></lb>ponderant, quod eſt abſurdum. </s> <s id="N115DE">diſtantia igitur AC ipſi CB <lb></lb>æqualis eſſe non poteſt. <emph type="italics"></emph>ſi uerò AC maior fuerit CB<emph.end type="italics"></emph.end>; ablato ſi<lb></lb>militer exceſſu D, nihilominus ęqualia grauia AB non ę〈que〉 <lb></lb>ponderabunt, ſed <emph type="italics"></emph>inclinabitur ad A. æqualia enim grauia<emph.end type="italics"></emph.end> AB <emph type="italics"></emph>ex <emph.end type="italics"></emph.end> <arrow.to.target n="marg31"></arrow.to.target><lb></lb><emph type="italics"></emph>distantiis inæqualibus non æ〈que〉ponderant, ſed inclinatur ad maiorem <lb></lb>distantiam<emph.end type="italics"></emph.end> AC. ergo totum AD multò magis præponderabit, <lb></lb>quàm B. quod fieri non poteſt. </s> <s id="N11609">poſita enim ſunt æ〈que〉ponde<lb></lb>rare. </s> <s id="N1160D">Quare AC maior eſſe non poteſt, quàm CB. ſed oſtenſa <lb></lb>eſt, ne〈que〉 ipſi CB æqualis eſſe: <emph type="italics"></emph>ac propterea minor eſt AC, quàm <lb></lb>CB. Manifestum eſt ita〈que〉 grauia ex distantiis inæqualibus æ〈que〉pon<lb></lb>derantia, inæqualia eſſe; maiuſquè in minori<emph.end type="italics"></emph.end> diſtantia <emph type="italics"></emph>existere.<emph.end type="italics"></emph.end> quod <lb></lb>oportebat demonſtrare. </s> </p> <p id="N11623" type="margin"> <s id="N11625"><margin.target id="marg28"></margin.target>B</s> </p> <p id="N11629" type="margin"> <s id="N1162B"><margin.target id="marg29"></margin.target>4 <emph type="italics"></emph>post hu<lb></lb>ius.<emph.end type="italics"></emph.end></s> </p> <p id="N11636" type="margin"> <s id="N11638"><margin.target id="marg30"></margin.target>1 <emph type="italics"></emph>poſt hu<lb></lb>ius.<emph.end type="italics"></emph.end></s> </p> <p id="N11643" type="margin"> <s id="N11645"><margin.target id="marg31"></margin.target>2 <emph type="italics"></emph>post hu<lb></lb>ius.<emph.end type="italics"></emph.end></s> </p> <p id="N11650" type="head"> <s id="N11652">SCHOLIVM.</s> </p> <p id="N11654" type="main"> <s id="N11656">In propoſitione verba illa, <emph type="italics"></emph>maius quidem ex minori<emph.end type="italics"></emph.end>, non <expan abbr="habẽtur">haben<arrow.to.target n="marg32"></arrow.to.target><lb></lb>tur</expan> integra in codice græco, qui ſic habet, <foreign lang="grc">καὶ τό ἀπὸ το̄ν ἐλάσσονος</foreign><lb></lb>vbi deſiderari videtur <foreign lang="grc">μέιζον</foreign>, vt integrè ita legatur, <foreign lang="grc">καὶ τὸ μείζον <lb></lb>ἀπὸ τοῡ ἐλάσσονος.</foreign></s> </p> <p id="N11676" type="margin"> <s id="N11678"><margin.target id="marg32"></margin.target>A</s> </p> <p id="N1167C" type="main"> <s id="N1167E"><emph type="italics"></emph>Sitquè maius A.<emph.end type="italics"></emph.end> Græcus codex, <foreign lang="grc">καὶ ἔσω τὸ α</foreign>, vbi ſimiliter <arrow.to.target n="marg33"></arrow.to.target> ſup<lb></lb>plendum eſt, <foreign lang="grc">καὶ ἔσω μείζον τὸ α</foreign> Hæc verò ita ſunt omnino reſti<lb></lb>tuenda, quia in vltima demonſtrationis concluſione inquit <lb></lb>Archimedes, <emph type="italics"></emph>Manifeſtum est ita〈que〉 grauia ex diſtantiis inæqualibus <lb></lb>æ〈que〉ponderantia inæqualia eſſe; maiuſquè in minori existere.<emph.end type="italics"></emph.end></s> </p> <p id="N1169E" type="margin"> <s id="N116A0"><margin.target id="marg33"></margin.target>B</s> </p> <p id="N116A4" type="main"> <s id="N116A6"><expan abbr="Poſtquã">Poſtquam</expan> Archimedes <expan abbr="duab^{9}">duabus</expan> primis <expan abbr="poſitionib^{9}">poſitionibus</expan> <expan abbr="oſtẽdit">oſtendit</expan>, <expan abbr="qũo">quno</expan> <lb></lb>ſe <expan abbr="hẽant">henant</expan> grauia ex <expan abbr="diſtãtijs">diſtantijs</expan> <expan abbr="ęqualib^{9};">ęqualibus</expan> in hac tertia <expan abbr="cõuertiſſe">conuertitſe</expan> ad <lb></lb><expan abbr="oſtẽdẽdũ">oſtendendum</expan>, <expan abbr="qũo">quno</expan> ſe <expan abbr="hẽnt">hennt</expan> ex <expan abbr="diſtãtijs">diſtantijs</expan> <expan abbr="inęqualib^{9}">inęqualibus</expan>. & <expan abbr="qm̃">quem</expan> in <expan abbr="ſecũdo">ſecundo</expan> <lb></lb>poſtulato <expan abbr="aſsũpſit">aſsumpſit</expan>, <expan abbr="qũo">quno</expan> ſe <expan abbr="hẽnt">hennt</expan> grauia ęqualia in <expan abbr="diſtãtijs">diſtantijs</expan> in ę<lb></lb>qualibus <expan abbr="cõſtituta">conſtituta</expan>; <expan abbr="nimirũ">nimirum</expan> <expan abbr="qd">quod</expan> eſt in <expan abbr="lõgiori">longiori</expan> <expan abbr="diſtãtia">diſtantia</expan>, <expan abbr="prępõde-rat">pręponde<lb></lb>rat</expan> ei, <expan abbr="qd">quod</expan> eſt in breuiori. <expan abbr="nũc">nunc</expan> <expan abbr="oſtẽdit">oſtendit</expan>, <expan abbr="qũo">quno</expan> inęqualia grauia ſe <lb></lb><expan abbr="hẽnt">hennt</expan>, ita vt <expan abbr="ę〈que〉põderẽt">ę〈que〉ponderent</expan>, in <expan abbr="diſtãtijs">diſtantijs</expan> in ęqualibus poſita. <expan abbr="demõ">demom</expan> <lb></lb>ſtratquè graue maius in breuiori <expan abbr="diſtãtia">diſtantia</expan> <expan abbr="eẽ">eem</expan> oportere, <expan abbr="min^{9}">minus</expan> ve<lb></lb>rò graue in <expan abbr="lõgiori">longiori</expan>. & ecce quomodo Archimedes <expan abbr="paulatĩ">paulatim</expan> de <lb></lb>ducit nos in <expan abbr="cognitionẽ">cognitionem</expan> principalis <expan abbr="fundamẽti">fundamenti</expan>, <expan abbr="qd">quod</expan> ſcilicet gra<lb></lb>ue ad graue eſt, vt <expan abbr="diſtãtia">diſtantia</expan> ad <expan abbr="diſtãtiã">diſtantiam</expan> <expan abbr="pmutatim">permutatim</expan>. </s> <s id="N11749">Ex hoc. <expan abbr="n.">enim</expan> pri<lb></lb>mùm cognoſcimus grauius in minori, leuius <expan abbr="autẽ">autem</expan> in maiori <lb></lb>diſtantia eſſe debere, ſi ę〈que〉ponderare debent. </s> </p> <pb xlink:href="077/01/046.jpg" pagenum="42"></pb> <p id="N1175A" type="head"> <s id="N1175C">PROPOSITIO. IIII.</s> </p> <p id="N1175E" type="main"> <s id="N11760">Si due magnitudines æquales non idem <expan abbr="centrũ">centrum</expan> <lb></lb>grauitatis habuerint, magnitudinis ex vtriſ〈que〉 <lb></lb>magnitudinibus compoſitæ centrum grauitatis <lb></lb>erit medium rectæ lineæ grauitatis centra magni<lb></lb>tudinum coniungentis. </s> </p> <p id="N1176E" type="main"> <s id="N11770"><emph type="italics"></emph>Sit <expan abbr="quidẽ">quidem</expan> A <emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="fig19"></arrow.to.target><lb></lb><emph type="italics"></emph><expan abbr="centrũ">centrum</expan> grauita<lb></lb>tis magnitudi<lb></lb>nis A. B uerò <emph.end type="italics"></emph.end><lb></lb>ſit <expan abbr="cẽtrũ">centrum</expan> gra<lb></lb>uitatis <emph type="italics"></emph>magni<lb></lb>tudinis B iun<lb></lb>staquè AB bifariam diuidatur in C. dico magnitudinis ex utriſquè ma<lb></lb>gnitudinibus compoſitæ centrum<emph.end type="italics"></emph.end> grauitatis <emph type="italics"></emph>eſſe punctum C. ſi. <expan abbr="n.">enim</expan> non; ſit <lb></lb>utrarumquè magnitudinum AB centrum grauitatis D, ſi fieri <expan abbr="põt">potest</expan>. Quòd <lb></lb>autem ſit in linea AB, præoſtenſum est. </s> <s id="N117AF">Quoniam igitur punstum D <expan abbr="cẽ">cem</expan><emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg34"></arrow.to.target> <emph type="italics"></emph><expan abbr="trũ">trum</expan> eſt grauitatis magnitudinis ex AB <expan abbr="cõpoſitæ">compoſitæ</expan>, <expan abbr="ſuſpẽſo">ſuſpenſo</expan> <expan abbr="pũcto">puncto</expan> D<emph.end type="italics"></emph.end>, magni<lb></lb>tudines AB <emph type="italics"></emph>æ〈que〉ponderabunt. </s> <s id="N117D6">magnitudines igitur AB<emph.end type="italics"></emph.end> ęquales <emph type="italics"></emph>æ〈que〉 <lb></lb>ponderant ex diſtantiis AD DB<emph.end type="italics"></emph.end> in ęqualibus exiſtentibus; <emph type="italics"></emph>quod fie<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg35"></arrow.to.target> <emph type="italics"></emph>ri non poteſt. </s> <s id="N117F1">æqualia. <expan abbr="n.">enim</expan><emph.end type="italics"></emph.end> grauia <emph type="italics"></emph>ex diſtantiis in a qualibus non <expan abbr="æ〈que〉põde-rãt">æ〈que〉ponde<lb></lb>rant</expan>.<emph.end type="italics"></emph.end> <expan abbr="Nõ">non</expan> eſt igitur D <expan abbr="ipſarũ">ipſarum</expan> <expan abbr="magnitudinũ">magnitudinum</expan> <expan abbr="cẽtrũ">centrum</expan> grauitatis.. <emph type="italics"></emph>Qua <lb></lb>re manifestum est punstum C <expan abbr="centrũ">centrum</expan> eſſe grauitatis magnitudinis ex AB <lb></lb>compoſitæ.<emph.end type="italics"></emph.end> quod demonſtrare oportebat. </s> </p> <p id="N11823" type="margin"> <s id="N11825"><margin.target id="marg34"></margin.target><emph type="italics"></emph>def. </s> <s id="N1182B">centri <lb></lb>grauit. <lb></lb>contra 2. <lb></lb>post huius<emph.end type="italics"></emph.end></s> </p> <p id="N11835" type="margin"> <s id="N11837"><margin.target id="marg35"></margin.target>2 <emph type="italics"></emph>post hu<lb></lb>ius.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.046.1.jpg" xlink:href="077/01/046/1.jpg"></figure> <p id="N11846" type="head"> <s id="N11848">SCHOLIVM.</s> </p> <figure id="id.077.01.046.2.jpg" xlink:href="077/01/046/2.jpg"></figure> <p id="N1184D" type="main"> <s id="N1184F">Poſſunt magnitudines ęquales <expan abbr="idẽ">idem</expan> <expan abbr="centrũ">centrum</expan> <lb></lb>grauitatis habere, vt duo <expan abbr="parallelogrãma">parallelogramma</expan> æ<lb></lb>qualia ad rectos ſibi <expan abbr="inuicẽ">inuicem</expan> angulos exiſten<lb></lb>tia: <expan abbr="triãgulũ">triangulum</expan> quo〈que〉 & <expan abbr="parallelogrãmũ">parallelogrammum</expan> in<lb></lb>terſe æqualia. <expan abbr="p̃terea">propterea</expan> cubos, piramides, cylin<lb></lb>dros, & huiuſmodi alias magnitudines ęqua<lb></lb>les <expan abbr="idẽ">idem</expan> grauitatis <expan abbr="cẽtrũ">centrum</expan> <expan abbr="hẽre">herre</expan> intelligere poſſu<lb></lb>mus. </s> <s id="N11887">propterea in propoſitione cùm inquit Archimedes <lb></lb><emph type="italics"></emph>ſi duæ magnitudines æquales non idem centrum grauitatis <emph.end type="italics"></emph.end> <pb xlink:href="077/01/047.jpg" pagenum="43"></pb><emph type="italics"></emph>habuerint.<emph.end type="italics"></emph.end> intelligendum eſt his verbis Archimedem ſuppo<lb></lb>nere magnitudines ita eſſe conſtitutas, vt à centro ad centrum <lb></lb>duci poſſit recta linea. </s> <s id="N1189D">quod idem obſeruandum eſt in prima <lb></lb>propoſitione ſecundi libri huius. </s> </p> <p id="N118A1" type="main"> <s id="N118A3">Súmoperè aút <expan abbr="animaduertẽda">animaduertenda</expan> ſunt <expan abbr="nõnulla">nonnulla</expan>, quibus vtitur <lb></lb>Archimedes in hac propoſitione, cùm ſint communiſſima, <lb></lb>& maximè vtilia in hac ſcientia. </s> <s id="N118AD">ac primùm quidem conſide<lb></lb>randum occurrit, quid ſibi vult Archimedes per magnitudi<lb></lb>nem ex vtriſ〈que〉 magnitudinibus AB compoſitam. </s> <s id="N118B3">Nam ma<lb></lb>gnitudines AB ſunt inuicem ſeparatę, & ſunt duę, ipſe autem <lb></lb>vtram〈que〉 vnam tantùm conſiderat. </s> <s id="N118B9">quod quidem ita <expan abbr="intelli-gendũ">intelli<lb></lb>gendum</expan> eſt. <expan abbr="quoniã">quoniam</expan> ſcilicet recta linea AB eas coniungit; ideo <lb></lb>Archimedes conſiderat vnam tantùm eſſe <expan abbr="magnitudinẽ">magnitudinem</expan>; quę <lb></lb>conſtat ex ipſis AB, & efficitur vna magnitudo à linea AB. <lb></lb>cuius munus eſt non ſolùm connectere magnitudines AB, <lb></lb>ita vtne〈que〉 ad ſe ampliùs accedere, ne〈que〉 recedere inuicem <lb></lb>poſſint; ſintquè ab hac linea quaſi compulſę eundem ſemper <lb></lb>interſe ſeruare ſi tum: verum etiam ſi ſuſpendantur ex C, in<lb></lb>telligendum eſt linea AB in rectitudinem iacere, inſuperquè <lb></lb>ſuſtinere magnitudines AB. Ne〈que〉 magis vna eſt magnitudo <lb></lb>quadrilaterum, <expan abbr="pẽtagonum">pentagonum</expan>, cubus, & huiuſmodi aliæ, quàm <lb></lb>ſit magnitudo, quæ componitur ex magnitudinibus AB v<lb></lb>nà cum linea AB. quòd ſi eſt vna tantùm magnitudo, ergo <lb></lb>vnum habet <expan abbr="cẽtrum">centrum</expan> grauitatis. </s> <s id="N118E9">Archimedes igitur quęrit cen<lb></lb>trum grauitatis huiuſce magnitudinis; demonſtratquè cen<lb></lb>trum eſſe in puncto C. quod eſt medium lineæ AB. notan<lb></lb>dum eſt autem Archimedem non conſiderare grauitatem li<lb></lb>neę AB. vt potè, quę longitudo tantùm exiſtat. </s> <s id="N118F3">Quòd ſi quis <lb></lb>etiam mente concipere vellet lineam AB grauitate <expan abbr="pręditã">pręditam</expan> <lb></lb>eſſe; nihilominus centrum grauitatis lineę AB ſimiliter eſſet <lb></lb>in eius medio C. nam longitudo AC longitudini CB eſt <lb></lb>æqualis; ac propterea hę quidem longitudines eſſent inter ſeſe <lb></lb>ę〈que〉ponderantes. </s> <s id="N11903">Quare, ſiue <expan abbr="cõſiderata">conſiderata</expan> grauitate lineę AB, <lb></lb>ſiue minùs, centrum grauitatis magnitudinis ex AB compo<lb></lb>ſitę eſt <expan abbr="mediũ">medium</expan> rectę lineę, quæ centra grauitatis <expan abbr="magnitudinũ">magnitudinum</expan> <lb></lb>coniungit. </s> <s id="N11913">Et hoc modo ſi plures etiam eſſent magnitudines <lb></lb>à recta linea coniunctę, eodem modo eas pro vna tantùm ma<pb xlink:href="077/01/048.jpg" pagenum="44"></pb>gnitudine ex <expan abbr="plurib^{9}">pluribus</expan> magnitudinibus compoſita accipere po<lb></lb>terimus, veluti Archimedes in ſe〈que〉ntibus accipiet. </s> </p> <p id="N1191D" type="main"> <s id="N1191F">Argumentandi modus in eſt in hac demonſtratione maxi<lb></lb>ma conſideratione dignus, & huius ſcientiæ maximè pro<lb></lb>prius. </s> <s id="N11925">cùm enim dixiſſet Archimedes poſito centro grauitatis <lb></lb>magnitudinis ex AB compoſitæ in puncto D, ſtatim infert. <lb></lb><emph type="italics"></emph>Quoniam igitur punctum D centrum eſt grauitatis magnitudinis ex <lb></lb>AB compoſita, ſuſpenſo puncto D, magnitudines AB æ〈que〉pondera<lb></lb>bunt.<emph.end type="italics"></emph.end> hoc eſt ſi magnitudo ex AB compoſita ſuſpendatur ex <lb></lb>D, manebit, vt reperitur; nec amplius in alteram partem in cli <lb></lb>nabit. </s> <s id="N11938">quod euenit ob naturam centri grauitatis, quod talis <lb></lb>eſt naturæ (ſicuti initio explicauimus) ut ſi graue in eius cen<lb></lb>tro grauitatis ſuſtineatur, eo modo manet, quo reperitur, <expan abbr="dũ">dum</expan> <lb></lb>ſuſpenditur; parteſquè undiquè æ〈que〉ponderant. </s> <s id="N11944">& ob id ſi <lb></lb>magnitudo ex AB compoſita ſuſpendatur in eius centro gra<lb></lb>uitatis, manet; parteſquè AB æ〈que〉ponderant. </s> <s id="N1194A">ac propterea <lb></lb>quando in ſe〈que〉ntibus quærit Archimedes, quoniam grauia <lb></lb>æ〈que〉ponderare debent, tunc tantùm quærit ipſorum <expan abbr="cẽtrum">centrum</expan> <lb></lb>grauitatis, ut in ſexta, ſeptimaquè propoſitione in quit Archi<lb></lb>medes magnitudines ę〈que〉ponderare ex diſtantijs, quę permu<lb></lb>tatim proportionem habent, ut ipſarum grauitates, in <expan abbr="demõ">demom</expan> <lb></lb>ſtratione tamen quærit, vbi nam eſt <expan abbr="cẽtrum">centrum</expan> grauitatis magni<lb></lb>tudinis ex vtrisquè compoſitę. </s> <s id="N11966">quo inuento, ſtatim neceſſariò <lb></lb>ſequitur, magnitudines, ſi ex ipſo centro ſuſpendantur, æ〈que〉 <lb></lb>ponderare. </s> </p> <p id="N1196C" type="main"> <s id="N1196E">Hinc colligere poſſumus alterum argumentandi modum, <lb></lb>conuerſo nempè modo, veluti in eadem figura, ſi dicamus <lb></lb>grauia AB ſuſpenſa ex C æ〈que〉ponderant, ſtatim inferre <lb></lb>poſſumus, punctum C ipſorum ſimul grauium, hoc eſt ma<lb></lb>gnitudinis ex ipſis AB compoſitę centrum eſſe grauitatis. <lb></lb>Quare ad ſe inuicem conuertuntur, hoc punctum eſt horum <lb></lb>grauium centrum grauitatis; ergo hęc grauia ex hoc puncto <lb></lb>æqùeponderant; & è conuerſo, nempè hæc grauia ex hoc pun<lb></lb>cto æ〈que〉ponderant, ergo idem punctum eſt ipſorum <expan abbr="cẽtrum">centrum</expan> <lb></lb>grauitatis. </s> <s id="N11986">ſed ad uertendum hanc ſequi <expan abbr="conuertibilitatẽ">conuertibilitatem</expan>, <expan abbr="quã-do">quan<lb></lb>do</expan> præfatum punctum eſt in recta linea, quæ centra grauita<lb></lb>tum ponderum coniungit; deinde quando hęc linea non eſt <pb xlink:href="077/01/049.jpg" pagenum="45"></pb>horizonti perpendicularis. </s> <s id="N11998">ſecus aurem minimè. </s> <s id="N1199A">Nam ſi pon<lb></lb>dera AB ſint in libra ADB, quę ſit arcuata, vel angulum <expan abbr="cō-ſtituat">con<lb></lb>ſtituat</expan>, ſiue intelligatur libra recta linea AB, cui affixa ſit <lb></lb>perpendicularis CD. vt in tractatu de libra noſtrorum Me<lb></lb>chanicorum diximus. </s> <s id="N119A8">ſuſpendantur autem pondera AB ex <lb></lb> <arrow.to.target n="fig20"></arrow.to.target><lb></lb>D, & æ〈que〉ponderent; <expan abbr="nõ">non</expan> <lb></lb>ſequitur tamen, ergo D <lb></lb><expan abbr="cẽtrum">centrum</expan> eſt grauitatis ma<lb></lb>gnitudinis ex AB com<lb></lb>poſitę. </s> <s id="N119C0">centrum enim gra<lb></lb>uitatis in linea exiſtit AB <lb></lb>quæ centra grauitatis ma<lb></lb>gnitudinum AB coniun<lb></lb>git, nempe in C. Verùm coniungat recta linea AB centra <lb></lb> <arrow.to.target n="fig21"></arrow.to.target><lb></lb>grauitatis æqualium ponderum AB, lineaquè <lb></lb>AB, cuius medium ſit C, in centrum mundi <expan abbr="tẽ-dat">ten<lb></lb>dat</expan>, magnitudoquè ex ipſis AB compoſita vbi<lb></lb>cun〈que〉 ſuſpendatur in linea AB, vt in E; ma<lb></lb>nebunt vti〈que〉 pondera AB ex E ſuſpenſa, vt in <lb></lb>prima propoſitione de libra noſtrorum Mecha<lb></lb>nicorum oſtendimus. </s> <s id="N119E1">cùm C ſit ipſorum <expan abbr="centrū">centrum</expan> <lb></lb>grauitatis, & EC ſit horizonti erecta. </s> <s id="N119E9">Et quam<lb></lb>uis magnitudo ex ipſis AB compoſita ex E ſu<lb></lb>ſpenſa maneat; non propterea ſequitur ergo E <lb></lb>centrum eſt grauitatis magnitudinis ex ipſis AB <lb></lb>compoſitę. </s> <s id="N119F3">niſi fortè accidat ſuſpenſio ex puncto <lb></lb>C. Præterea verò aduertendum eſt in hoc caſu <expan abbr="põdera">pon<lb></lb>dera</expan> AB, dici quidem poſſe, manere, non autem <lb></lb>æ〈que〉ponderare. </s> <s id="N119FF">omnia nimirum, quę æ〈que〉ponderant, ma<lb></lb>nent; ſed non è conuerſo, quæ manent, æ〈que〉ponderant. </s> <s id="N11A03">Nam <lb></lb>ſi pondus A maius fuerit pondere B; ſiue B maius, quàm <lb></lb>A, vbicun〈que〉 fiat ſuſpenſio in linea AB, ſemper ob <expan abbr="eãdem">eandem</expan> <lb></lb>cauſam, quomodocun〈que〉 ſint pondera, manebunt; non ta<lb></lb>men æ〈que〉ponderabunt. </s> <s id="N11A11">Vt enim pondera æ〈que〉ponderent, <lb></lb>requiritur, vt pars parti, virtuſquè vnius virtuti alterius hinc <lb></lb>inde reſiſtere, & æquipollere poſſit; vt propriè dici poſſint <expan abbr="põ">pom</expan> <lb></lb>dera æ〈que〉ponderare. </s> <s id="N11A1D">& vt hoc euenire poſſit, oportet, vt par <pb xlink:href="077/01/050.jpg" pagenum="46"></pb>tes ex determinatis diſtantijs determinatas quo〈que〉 habeant <lb></lb>grauitates; ſi ex dato puncto æ〈que〉ponderare debent. </s> <s id="N11A25">Quòd <lb></lb>ſi in hoc caſu datum fuerit punctum C, ex quo pondera AB <lb></lb>ex æqualibus diſtantijs CA CB ę〈que〉ponderare debeant: o<lb></lb>porteret, vt pondera AB (ex demonſtratis) ſemper eſſent æ<lb></lb>qualia. <expan abbr="Quoniã">Quoniam</expan> <expan abbr="autẽ">autem</expan> <expan abbr="quomodocũ〈que〉">quomodocun〈que〉</expan> ſint pondera, hoc eſt; ſi <lb></lb>ue pondus A maius, ſiue minus fuerit, quàm B, manent, ſi <lb></lb>igitur dixerimus, ergo pondus A ponderi B ę〈que〉ponderat; <lb></lb>eſſet omnino inconueniens. </s> <s id="N11A41">cùm ex ijsdem diſtantijs <expan abbr="eidẽ">eidem</expan> <expan abbr="põ">pom</expan> <lb></lb>deri pondus quandoquè maius, quandoquè minus ę〈que〉pon<lb></lb>derare non poſſit; vt in hoc caſu accidere poteſt. </s> <s id="N11A4F">Quocirca <lb></lb>nec propriè dici poſſunt pondera, ſiue in libra AB, ſiue ex <lb></lb>diſtantijs CA CB conſtituta eſſe. </s> <s id="N11A55">Vndè ne〈que〉 Archimedis <lb></lb>propoſitiones in hoc caſu ſunt intelligendę quandoquidem <lb></lb>in his propriè quærit ponderum, magnitudinumquè æ〈que〉<lb></lb>ponderationes. </s> <s id="N11A5D">ne〈que〉 enim in hac quarta demonſtratione in <lb></lb>hoc caſu potuiſſet Archimedes abſurdum oſtendere, ſi C <expan abbr="nõ">non</expan> <lb></lb>eſt grauitatis centrum magnitudinis ex AB compoſitæ, ſit <lb></lb>E. facta igitur ex E ſuſpenſione, magnitudines æquales AB <lb></lb>ex in æqualibus diſtantijs EA EB ę〈que〉ponderabunt. </s> <s id="N11A6B">quod <lb></lb>fieri non poteſt. </s> <s id="N11A6F">non enim hoc eſt abſurdum; cùm pondera <lb></lb>ex E ſuſpenſa <expan abbr="maneãt">maneant</expan> idcirco quando linea AB eſt <expan abbr="horizõ">horizom</expan> <lb></lb>ti erecta; propriè ad rem noſtram minimè pertinet. </s> <s id="N11A7D">Ex dictis <lb></lb>igitur ſemper valet conſe〈que〉ntia, hoc punctum horum pon<lb></lb>derum centrum eſt grauitatis, ergo ſi ex hoc ſuſpendantur, <expan abbr="põ">pom</expan> <lb></lb>dera ę〈que〉ponderant. </s> <s id="N11A89">non autem è conuerſo. </s> <s id="N11A8B">niſi quando ar<lb></lb>gumentatio ſumitur ſemper ex recta linea, quæ centra graui<lb></lb>tatis magnitudinum coniungit, & quando hęc linea non eſt <lb></lb> <arrow.to.target n="fig22"></arrow.to.target><lb></lb>horizonti erecta. </s> <s id="N11A98">hac enim <lb></lb>ratione quocun〈que〉 modo <lb></lb>recta linea ſe habeat, ſem<lb></lb>per ſequitur idem. </s> <s id="N11AA0">Vt ſi li<lb></lb>nea AB fuerit, ſiue <expan abbr="nõ">non</expan> fue<lb></lb>rit horizonti æquidiſtans, <lb></lb>ipſius medium C centrum <lb></lb>erit grauitatis magnitudi<lb></lb>nis ex magnitudinibus AB æqualibus compoſitę. </s> <s id="N11AB0">vnde ſequi<pb xlink:href="077/01/051.jpg" pagenum="47"></pb>tur, ſi appendantur pondera AB ex C, æ〈que〉ponderare. </s> <s id="N11AB6">& <lb></lb>è conuerſo, ſi AB pondera ex C æ〈que〉ponderant, ergo C <lb></lb>centrum grauitatis exiſtit. </s> <s id="N11ABC">ex quibus ſequitur lineam AB, <expan abbr="põ">pom</expan> <lb></lb>deraquè manere eo modo, quo reperiuntur. </s> <s id="N11AC4">vt in noſtro me<lb></lb>chanicorum libro in codem tractatu de libra demonſtraui<lb></lb>mus, & aduerſus illos, qui aliter ſentiunt, abundè ſatis <arrow.to.target n="marg36"></arrow.to.target> diſpu<lb></lb>tauimus. </s> </p> <p id="N11AD0" type="margin"> <s id="N11AD2"><margin.target id="marg36"></margin.target><emph type="italics"></emph>poſt quar<lb></lb>tam propo<lb></lb>ſitionem.<emph.end type="italics"></emph.end><lb></lb>*</s> </p> <figure id="id.077.01.051.1.jpg" xlink:href="077/01/051/1.jpg"></figure> <figure id="id.077.01.051.2.jpg" xlink:href="077/01/051/2.jpg"></figure> <figure id="id.077.01.051.3.jpg" xlink:href="077/01/051/3.jpg"></figure> <p id="N11AEC" type="main"> <s id="N11AEE">In demonſtratione autem huius quartæ propoſitionis in<lb></lb>quit Archimedes. <emph type="italics"></emph>Quòd autem ſit in linea AB, præostenſum eſt.<emph.end type="italics"></emph.end> qua <lb></lb>ſi dicat Archimedes, ſe priùs oſtendiſſe centrum grauitatis ma <lb></lb>gnitudinis ex AB compoſitæ eſſe in linea AB; quod tamen <lb></lb>in ijs, quæ dicta ſunt, non videtur expreſſum. </s> <s id="N11AFE">virtute tamen ſi <lb></lb>conſideremus ea, quę in prima, tertiaquè propoſitione dicta <lb></lb>ſunt, facilè ex his concludi poteſt, centrum grauitatis magni<lb></lb>tudinis ex duabus magnitudinibus compoſitæ eſſe in recta li<lb></lb>nea, quæ ipſarum centra grauitatis coniungit. </s> <s id="N11B08">Quare memi<lb></lb>niſſe oportet eorum, quę a nobis in expoſitione primi poſtu<lb></lb>lati huius dicta fuere, nempè Archimedem ſupponere, diſtan<lb></lb>tias eſſe in vna, eademquè recta linea conſtitutas. </s> <s id="N11B10">ideoquè in <lb></lb>prima propoſitio nec inquit, Grauia, quę ex <expan abbr="diſtãtijs">diſtantijs</expan> ęquali<lb></lb>bus <expan abbr="æ〈que〉põderãt">æ〈que〉ponderant</expan>, æqualia eſſe inter ſe; Archimedes què <expan abbr="demõ">demom</expan> <lb></lb>ſtrat, quòd quando æ〈que〉ponderant, ſunt æqualia: ex dictis <lb></lb>ſequitur, ſi æ〈que〉ponderant, ergo centrum grauitatis magni<lb></lb>tudinis ex ipſis compoſitę erit in eo puncto, vbi æ〈que〉ponde<lb></lb>rant; hoc eſt in medio diſtantiarum, lineę ſcilicet, quę <expan abbr="grauiũ">grauium</expan> <lb></lb>centra grauitatis coniungit. </s> <s id="N11B30">quod idem eſt, ac ſi Archimedes <lb></lb>dixiſſet. </s> <s id="N11B34">Grauia, quę habent centrum grauitatis in medio li<lb></lb>neę, quę magnitudinum centra grauitatis coniungit, ęqua<lb></lb>lia ſunt inter ſe. </s> <s id="N11B3A">cuius quidem hęc quarta propoſitio videtur <lb></lb>eſſe conuerſa. </s> <s id="N11B3E">quamuis Archimedes loco grauium nominet <lb></lb>magnitudines. </s> <s id="N11B42">Pręterea in tertia propoſitione, quoniam <expan abbr="oſtẽ-dit">oſten<lb></lb>dit</expan> Archimedes, inęqualia grauia ę〈que〉ponderare ex <expan abbr="diſtãtijs">diſtantijs</expan> <lb></lb>inęqualibus, ita vt grauius ſit in minori diſtantia, ſequitur er<lb></lb>go centrum grauitatis eſt in eo puncto, vbi æ〈que〉ponderant; <lb></lb>& idem eſt, ac ſi dixiſſet, in æqualium grauium centrum gra<lb></lb>uitatis eſt in recta linea, quæ ipſorum centra grauitatis con<lb></lb>iungit; ita vt ſit propinquius grauiori, remotius uerò leuiori. <pb xlink:href="077/01/052.jpg" pagenum="48"></pb>vnde ſequitur centrum grauitatis ipſorum grauium ubicum<lb></lb>〈que〉 eſſe poſſe in recta linea, quę ipſorum centra grauitatis <expan abbr="cõ">con<lb></lb>iungit</expan>. </s> <s id="N11B64">Ex quibus concludi poteſt, <expan abbr="cẽtrum">centrum</expan> grauitatis magni<lb></lb>tudinis ex duabus magnitudinibus compoſitę eſſe in recta li<lb></lb>nea, quæ ipſorum centra grauitatis connectit. </s> </p> <p id="N11B6E" type="main"> <s id="N11B70">Poſtremò notandum eſt, Archimedem ea, quæ in ſuperio<lb></lb>ribus propoſitionibus nuncupauit grauia, in hac quarta pro<lb></lb>poſitione, veluti etiam in ſe〈que〉ntibus, non ampliùs grauia, <lb></lb>ſed (vti diximus) magnitudines nominare. </s> <s id="N11B78">quod quidem his <lb></lb>de cauſis id ab ipſo factum exiſtimo. </s> <s id="N11B7C">primùm enim, quia in <lb></lb>his expreſse quærit centrum grauitatis; quod quidem <expan abbr="cẽtrum">centrum</expan>, <lb></lb>quamuis ſit centrum grauitatis, potiùs reſpicit <expan abbr="magnitudinẽ">magnitudinem</expan>, <lb></lb>quàm graue aliquod. </s> <s id="N11B8C">Nam cùm dicimus centrum grauitatis, <lb></lb>ſtatim innuimus ſitum, ſitum inquàm determinatum figu<lb></lb>ræ, in qua eſt; ſiquidem centrum grauitatis eſt punctum, & <lb></lb>(vt ita dicam) punctum grauitatis eius, in quo eſt. </s> <s id="N11B94">& ideo, <lb></lb>quoniam magnitudo formam habet dete mina tam, <expan abbr="centrũ">centrum</expan> <lb></lb>grauitatis rectè poteſt reſpicere ſitum reſpectu magnitudinis, <lb></lb>in qua eſt; quod tamen efficere non poteſt reſpectu grauis. <lb></lb>etenim graue, ut graue eſt, non habet formam determina <expan abbr="tã">tam</expan>; <lb></lb>cùm eadem grauitas eſſe poſſit in cubo, in piramide, aliiſquè <lb></lb>corporibus quibuſcun〈que〉, modò minoribus, modò maiori<lb></lb>bus, pro ut ſunt diuerſarum ſpecierum. </s> <s id="N11BAC">quare centrum grauita<lb></lb>tis non poteſt reſpicere ſitum in grauibus, quatenus grauia <expan abbr="cõ">con<lb></lb>ſiderantur</expan>; ſed quatenus magnitudines exiſtunt. </s> <s id="N11BB6">Præterea Ar<lb></lb>chimedes loco grauium magnitudines nominat, quia eas di<lb></lb>uiſibiles conſiderat, quod eſt proprium magnitudinis; vt in ſe<lb></lb>xta, ſeptima, & octaua propoſitione. </s> <s id="N11BBE">& quamuis, dum <expan abbr="diuidũ">diuidum</expan> <lb></lb>tur magnitudines, grauia quo〈que〉 diuiſa proueniant; non ta<lb></lb>men propterea grauia diuiduntur, ut grauia. <expan abbr="nõ">non</expan>.n. </s> <s id="N11BCC">hoc ipſis <lb></lb>competit, vt grauibus; ſed vt magnitudinibus, quæ ſunt per <lb></lb>ſe diuiſibiles. </s> <s id="N11BD2">Archimedes igitur his de cauſis nomen <expan abbr="grauiũ">grauium</expan> <lb></lb>in magnitudines mutauit. </s> <s id="N11BDA">in ſuperioribus enim theoremati<lb></lb>bus pertractauit, quomodo res æ〈que〉ponderant ex diſtantijs <lb></lb>modò æqualibus, modò in æqualibus. </s> <s id="N11BE0">& quoniam res <expan abbr="ę〈que〉põ-derant">ę〈que〉pon<lb></lb>derant</expan>, pro ut ſunt magis grauia, & minùs grauia; non ut <expan abbr="sũt">sunt</expan> <lb></lb>maiores, vel minores magnitudines, ſiquidem talis naturæ <pb xlink:href="077/01/053.jpg" pagenum="49"></pb>eſſe poteſt minor magnitudo, quę maiore magnitudine alte<lb></lb>rius nature grauior exiſtat; proindé Archimedes in ſuperiori<lb></lb>bus rectè grauia nuncupauit; optimèquè in his magnitudines <lb></lb>vocat. </s> <s id="N11BF8">At verò aduertendum eſt, quòd quamuis Archimedes <lb></lb>in his magnitudines nominet, non propterea exiſtimandum <lb></lb>eſt, eum intelligere magnitudines tantùm; ſed magnitudines <lb></lb>grauitate pręditas, ita ut in ipſis omnino grauitatem reſpiciat. <lb></lb>Etenim pluribus modis intelligere poſſumus magnitudines, <lb></lb>vel enim ut ſint inter ſe eiuſdem ſpeciei, vel diuerſæ; nec <expan abbr="nõ">non</expan> <lb></lb>inſuper homogeneæ, vel heterogeneæ. </s> <s id="N11C0A">vt in hac propoſitione <lb></lb><expan abbr="quãdo">quando</expan> Archimedes <expan abbr="pponit">proponit</expan> duas magnitudines ęquales, <expan abbr="tũc">tuc</expan> <lb></lb>intelligere poſſumus eas eſſe eiuſdem ſpeciei, & homogeneas; <lb></lb>quæ, cùm ſint æquales, erit & grauitas vnius grauitati alterius <lb></lb>æqualis. </s> <s id="N11C17">ſi verò conſideremus eas eſſe diuerſæ ſpeciei, & e<lb></lb>tiam heterogeneas; tunc quando Archimedes proponit has <lb></lb>magnitudines æquales; intelligendum eſt, eas eſſe æquales in <lb></lb>grauitate; quæ quidem efficit, vt demonſtratio, quod propo<lb></lb>ſitum eſt, concludat. </s> <s id="N11C21">vt ex eius demonſtratione patet. </s> <s id="N11C23">Et his <lb></lb>quo〈que〉 modis intelligere poſſumus magnitudines in ſe〈que〉n<lb></lb>tibus vſ〈que〉 ad nonam propoſitionem in quibus ſcilicet intel<lb></lb>ligere poſſumus magnitudines eſſe non ſolùm eiuſdem ſpe<lb></lb>ciei, vel diuerſæ, verùm etiam & homogeneas. </s> <s id="N11C2D">& heteroge<lb></lb>neas. </s> <s id="N11C31">ut poſt ſeptimam clariùs oſtendemus. </s> <s id="N11C33">Verùm de<lb></lb>monſtrationes clariores redduntur, ſi intelligamus magnitu<lb></lb>dines eſſe eiuſdem ſpeciei, & homogeneas, in quibus graui<lb></lb>tas magnitudini reſpondet, vt ſi ipſarum altera fuerit alte<lb></lb>rius dupla, & grauitas vnius grauitatis alterius dupla exiſtat. <lb></lb>Quòd ſi magnitudo fuerit alterius tripla, vel quadrupla, &c. <lb></lb>erit & grauitas grauitatis tripla, vel quadrupla, & ſic dein<lb></lb>ceps. </s> <s id="N11C43">deinde ſi magnitudo bifariam diuiſa fuerit, & ipſius gra<lb></lb>uitas in duas ęquas partes ſit quo〈que〉 diuiſa. </s> <s id="N11C47">quòd ſi magnitu<lb></lb>do in plures diuidatur partes, & grauitas quo〈que〉 in totidem <lb></lb>eiuſdem proportionis diuiſa proueniat. </s> </p> <pb xlink:href="077/01/054.jpg" pagenum="50"></pb> <p id="N11C50" type="head"> <s id="N11C52">PROPOSITIO. V.</s> </p> <p id="N11C54" type="main"> <s id="N11C56">Si trium magnitudinum centra grauitatis in re<lb></lb>cta linea fuerint poſita, & magnitudines æqualem <lb></lb>habuerint grauitatem, acrectæ lineæ inter centra <lb></lb>fuerint æquales, magnitudinis ex omnibus magni<lb></lb>tudinibus compoſitæ centrum grauitatis erit <expan abbr="pũ">pum</expan> <lb></lb>ctum, quod & ipſarum mediæ centrum grauitatis <lb></lb>exiſtit. </s> </p> <figure id="id.077.01.054.1.jpg" xlink:href="077/01/054/1.jpg"></figure> <p id="N11C6B" type="main"> <s id="N11C6D"><emph type="italics"></emph>Sint tres magnitudines ACB. ipſarum autem centra grauitatis ſint <lb></lb>puncta ACB in resta linea<emph.end type="italics"></emph.end> ACB <emph type="italics"></emph>poſita. </s> <s id="N11C79">ſint verò magnitudines ACB <lb></lb>æquales; rectæquè lineæ AC CB<emph.end type="italics"></emph.end> inter centra ipſarum <emph type="italics"></emph>aquales. </s> <s id="N11C83">Di<lb></lb>co magnitudinis ex omnibus<emph.end type="italics"></emph.end> ACB <emph type="italics"></emph>magnitudinibus compoſitæ <expan abbr="centrũgra">centrungra</expan> <lb></lb>uitatis eſſe punctum C.<emph.end type="italics"></emph.end> quod eſt centrum grauitatis mediæ ma<lb></lb>gnitudinis. <emph type="italics"></emph>Quoniam enim magnitudines AB æqualem habent graui<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg37"></arrow.to.target> <emph type="italics"></emph>tatem<emph.end type="italics"></emph.end>; magnitudinis ex vtriſ〈que〉 AB compoſitæ <emph type="italics"></emph>centrum graui<lb></lb>tatis erit punctum C: cùm ſint AC CB æquales.<emph.end type="italics"></emph.end> ſitquè propterea <lb></lb>punctum C medium rectæ lineę AB. <emph type="italics"></emph>Sed & magnitudinis C <expan abbr="cē">cem</expan> <lb></lb>trum grauitatis est<emph.end type="italics"></emph.end> idem <emph type="italics"></emph>punctum C.<emph.end type="italics"></emph.end> punctum ergo C <expan abbr="triũ">trium</expan> ma<lb></lb>gnitudinum ABC centrum quo〈que〉 grauitatis erit. <emph type="italics"></emph>Quare pa<lb></lb>tet magnitudinis ex omnibus magnitudinibus<emph.end type="italics"></emph.end> ACB <emph type="italics"></emph>compoſitæ centrum <lb></lb>grauitatis eſſe punctum, quod &<emph.end type="italics"></emph.end> magnitudinis <emph type="italics"></emph>mediæ centrum graui<lb></lb>tatis existit.<emph.end type="italics"></emph.end> quod demonſtrare oportebat. </s> </p> <pb xlink:href="077/01/055.jpg" pagenum="51"></pb> <p id="N11CE6" type="margin"> <s id="N11CE8"><margin.target id="marg37"></margin.target>4 <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N11CF1" type="head"> <s id="N11CF3">COROLLARIVM. I.</s> </p> <p id="N11CF5" type="main"> <s id="N11CF7">Ex hoc autem manifeſtum eſt, ſi quotcunquè <lb></lb>magnitudinum, & numero imparium, centra <arrow.to.target n="marg38"></arrow.to.target> gra<lb></lb>uitatis in recta linea conſtituta fuerint; & magni<lb></lb>tudines æqualem habuerint grauitatem; rectæquè <lb></lb>lineæ inter ipſarum centra fuerint æquales, ma<lb></lb>gnitudinis ex omnibus magnitudinibus compoſi<lb></lb>tæ centrum grauitatis eſſe punctum, quod & ipſa<lb></lb>rum mediæ centrum grauitatis exiſtit. </s> </p> <p id="N11D0B" type="margin"> <s id="N11D0D"><margin.target id="marg38"></margin.target>*</s> </p> <p id="N11D11" type="head"> <s id="N11D13">SCHOLIVM.</s> </p> <figure id="id.077.01.055.1.jpg" xlink:href="077/01/055/1.jpg"></figure> <p id="N11D18" type="main"> <s id="N11D1A">Ex demonſtratione colligit Archimedes ſi plures fuerint <lb></lb>magnitudines, <expan abbr="quã">quam</expan> tres; dummodo ſint numero impares, vt <lb></lb>ABCDE; quarum centra grauitatis ABCDE reperiantur in li<lb></lb>nea recta AE. fuerint autem hę magnitudines æquales in gra<lb></lb>uitate. </s> <s id="N11D28">inſuper rectę lineę AB BC CD DE, quę ſunt inter <expan abbr="cẽ-tra">cen<lb></lb>tra</expan> grauitatis, fuerint æquales: magnitudinis ex omnibus ma<lb></lb>gnitudinibus ABCDE compoſitæ centrum grauitatis eſſe <lb></lb>punctum C. quod eſt centrum grauitatis magnitudinis <lb></lb>mediæ. </s> </p> <p id="N11D36" type="main"> <s id="N11D38">Eodem enim modo, ac primùm quidem ex demonſtratio <lb></lb>ne patet <expan abbr="punctũ">punctum</expan> C centrum eſſe grauitatis <expan abbr="triũ">trium</expan> <expan abbr="magnitudinũ">magnitudinum</expan> <lb></lb>BCD, & quoniam AB BC ſunt æquales ipſis CD DE, <pb xlink:href="077/01/056.jpg" pagenum="52"></pb>erit AC ipſi CE ęqualis. </s> <s id="N11D4E">cùm què ſit grauitas magnitudinis <lb></lb> <arrow.to.target n="marg39"></arrow.to.target> A ęqualis grauitati ipſius E, erit itidem punctum C magni<lb></lb>tudinum AE centrum grauitatis. </s> <s id="N11D58">ergo punctum C magni<lb></lb>tudinis ex omnibus magnitudinibus ABCDE compoſitæ <lb></lb>centrum grauitatis exiſtit. </s> </p> <p id="N11D5E" type="margin"> <s id="N11D60"><margin.target id="marg39"></margin.target>4 <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N11D69" type="main"> <s id="N11D6B">Quòd ſi fuerint ad huc plures magnitudines, impares verò <lb></lb>extiterint; quæ ita ſe habeant, vt expoſitum eſt; ſimiliter <expan abbr="oſtẽ">oſtem</expan> <lb></lb>detur, centrum grauitatis mediæ magnitudinis centrum eſſe <lb></lb>grauitatis magnitudinis ex omnibus magnitudinibus com<lb></lb>poſitæ. </s> </p> <p id="N11D79" type="main"> <s id="N11D7B"> <arrow.to.target n="marg40"></arrow.to.target> In hoc corollario, verba illa, <emph type="italics"></emph>& magnitudines æqualem habue<lb></lb>rint grauitatem<emph.end type="italics"></emph.end> in greco codice ita habentur. <foreign lang="grc">εἵκα τατε ἴσον ἀπέχον<lb></lb>τα ἀπὸ τοῦ μέσου μεγέθεος ἴσον βάρος ἔχωντι</foreign> quorum multa ſuperuaca<lb></lb>nea nobis viſa ſunt; loco quorum (vt arbitror) rectè <expan abbr="congruẽt">congruent</expan> <lb></lb><foreign lang="grc">καὶ τὰ μεγέθεα ἴσον βάρος ἔχωντι</foreign>, vt vertimus. </s> <s id="N11D9B">Nam ſi ordinis at〈que〉 <lb></lb><expan abbr="cõditionum">conditionum</expan> propoſitę propoſitionis ratio habenda eſt, opor<lb></lb>tet vt magnitudines ęqualem habeant grauitatem; Nam & <lb></lb>Archimedes in ſe〈que〉ntibus demonſtrationibus ijs vtitur, ut <lb></lb>ſunt æ〈que〉graues. </s> <s id="N11DA8">Adhuc tamen veritatem habebit ſi cæteris <lb></lb>conditionibus illud quo〈que〉 addere voluerimus, nempe ſi <emph type="italics"></emph>ma<lb></lb>gnitudines à media magnitudine æqualiter diſtantes æqualem habuerint <lb></lb>grauitatem<emph.end type="italics"></emph.end> eodem modo punctum C centrum erit grauitatis <lb></lb> <arrow.to.target n="fig23"></arrow.to.target><lb></lb>magnitudinis ex omnibus ABCDE compoſitę, Nam ſi ma<lb></lb>gnitudines à media magnitudine ſunt ę〈que〉graues; ęqualem <lb></lb>quo〈que〉 habebunt grauitatem magnitudines AE; veluti ma<lb></lb>gnitudines BD, quæ æqualiter à media magnitudine C di<lb></lb>ſtant. </s> <s id="N11DC5">& quam uis non ſint omnes æ〈que〉graues, ſufficit, vt AE <lb></lb>quæ ęqualiter à media magnitudine diſtant, ſint ę〈que〉graues. <lb></lb>ſimiliter BD ę〈que〉graues. </s> <s id="N11DCB">Eadem enim ratione, quoniam <lb></lb>BD ſunt æ〈que〉graues, & diſtantiæ BC CD ęquales; erit C ipſa- <pb xlink:href="077/01/057.jpg" pagenum="53"></pb>rum BD centrum grauitatis. </s> <s id="N11DD3">pari què ratione C erit centrum <lb></lb>grauitatis magnitudinum AE ę〈que〉grauium. </s> <s id="N11DD7">cum ſint AC <lb></lb>CE ęquales, & idem C eſt grauitatis centrum magnitudinis <lb></lb>C. ergo punctum C magnitudinis ex omnibus magnitudini<lb></lb>bus ABCDE compoſitę centrum grauitatis exiſtit. </s> </p> <p id="N11DDF" type="margin"> <s id="N11DE1"><margin.target id="marg40"></margin.target>*</s> </p> <figure id="id.077.01.057.1.jpg" xlink:href="077/01/057/1.jpg"></figure> <p id="N11DE9" type="head"> <s id="N11DEB">COROLLARIVM. II.</s> </p> <p id="N11DED" type="main"> <s id="N11DEF">Si verò magnitudines fuerint numero pares; <lb></lb>& ipſarum centra grauitatis in recta linea extite<lb></lb>rint, magnitudineſquè æqualem habuerint graui <arrow.to.target n="marg41"></arrow.to.target><lb></lb>tatem, rectæ què lineæ inter centra fuerint æqua<lb></lb>les: magnitudinis ex omnibus magnitudinibus <expan abbr="cõ">com</expan> <lb></lb>poſitæ centrum grauitatis erit medium rectæ li<lb></lb>neæ, quæ magnitudinum centra grauitatis <expan abbr="coniũ-git">coniun<lb></lb>git</expan>. vt in ſubiecta figura. </s> </p> <p id="N11E0A" type="margin"> <s id="N11E0C"><margin.target id="marg41"></margin.target>*</s> </p> <figure id="id.077.01.057.2.jpg" xlink:href="077/01/057/2.jpg"></figure> <p id="N11E13" type="head"> <s id="N11E15">SCHOLIVM.</s> </p> <p id="N11E17" type="main"> <s id="N11E19">Colligit præterea Archimedes ſi magnitudines ABCDEF <lb></lb>fuerint numero pares, quarum centra grauitatis ABCDEF in <lb></lb>recta linea AF ſint conſtituta; magnitudineſquè ſint æquales <lb></lb>in grauitate; ſintquè inter centra lineę AB BC CD DE EF <lb></lb>æ quales. </s> <s id="N11E23">diuidatur autem AF bifariam in G. erit punctum <lb></lb>G centrum grauitatis magnitudinis ex omnibus compoſi<lb></lb>tæ quod quidem, figura tantùm inſpecta, perſpicuum eſt. <lb></lb>Cùm enim magnitudines AF ſint æ〈que〉graues, & AG GF <pb xlink:href="077/01/058.jpg" pagenum="54"></pb> <arrow.to.target n="marg42"></arrow.to.target> ſint æquales, erit G centrum grauitatis magnitudinis ex AF <lb></lb>compoſitæ. </s> <s id="N11E35">quia verò AB eſt ipſi EF æqualis, reliqua BG <lb></lb>ipſi GE æqualis exiſtet. </s> <s id="N11E39">& ſunt magnitudines BE ę〈que〉gra<lb></lb>ues, erit idem G centrum grauitatis <expan abbr="magnitudinũ">magnitudinum</expan> BE. ſimili<lb></lb>ter cùm ſit BC æqualis DE, relin〈que〉tur CG ipſi GD ęqua<lb></lb>lis; magnitudinesquè CD ſunt ę〈que〉graues. </s> <s id="N11E45">ergo <expan abbr="pũctum">punctum</expan> G <expan abbr="cẽ">cem</expan> <lb></lb>trum eſt quo〈que〉 magnitudinum CD. Vnde ſequitur, <expan abbr="punctũ">punctum</expan> <lb></lb>G magnitudinis ex omnibus magnitudinibus ABCDEF <expan abbr="cõ-poſitæ">con<lb></lb>poſitæ</expan> centrum grauitatis exiſtere. </s> </p> <p id="N11E5D" type="margin"> <s id="N11E5F"><margin.target id="marg42"></margin.target>4 <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N11E68" type="main"> <s id="N11E6A"> <arrow.to.target n="marg43"></arrow.to.target> Hoc quo〈que〉 loco verba illa <emph type="italics"></emph>magnitudineſquè æqualem habuerint <lb></lb>grauitatem.<emph.end type="italics"></emph.end> Græcus codex ita mendosè legit. <foreign lang="grc">καὶ τὰ μέσα αὔτης ἴσον <lb></lb>βάρος ἔχωντι</foreign>, quæ quidem verba hoc modo reſtitui poſſunt. <lb></lb><foreign lang="grc">καὶ τὰ μεγέθεα ἴσον βάρος ἔχωντι. </foreign></s> </p> <p id="N11E82" type="margin"> <s id="N11E84"><margin.target id="marg43"></margin.target>*</s> </p> <p id="N11E88" type="main"> <s id="N11E8A">In præcedenti propoſitione oſtendit Archimedes, quomo<lb></lb>do ſe habet centrum grauitatis magnitudinis ex duabus ma<lb></lb>gnitudinibus ęqualibus compoſitæ. </s> <s id="N11E90">In hac autem <expan abbr="demõſtrat">demonſtrat</expan>, <lb></lb>vbi ſimiliter grauitatis centrum reperitur inter plures magni<lb></lb>tudines æ〈que〉graues, & inter ſe ęqualiter diſtantes. </s> <s id="N11E9A">ex quibus <lb></lb>tandem colliget fundamentum ſæpiùs dictum. </s> <s id="N11E9E">nempè ſi ma<lb></lb>gnitudines ę〈que〉ponderare debent; ita ſe habebit magnitudi<lb></lb>num grauitas ad grauitatem, ut ſe habent diſtantiæ permuta<lb></lb>tim, ex quibus ſuſpenduntur. </s> <s id="N11EA6">& hoc demonſtrat Archimedes <lb></lb>in duabus ſe〈que〉ntibus propoſitionibus. </s> <s id="N11EAA">nam magnitudines, <lb></lb>vel ſunt commenſurabiles interſeſe, vel incommenſurabiles. <lb></lb>de commenſurabilibus aget in ſe〈que〉nti: de incommenſurabi<lb></lb>libus verò in ſeptima propoſitione. </s> <s id="N11EB2">& Archimedes duas <expan abbr="ſe〈quẽ〉-tes">ſe〈que〉n<lb></lb>tes</expan> propoſitiones ueluti coniunctas proponit. </s> <s id="N11EBA">Nam in ſexta <lb></lb>inquit <emph type="italics"></emph>Magnitudines commenſurabiles,<emph.end type="italics"></emph.end> &c. </s> <s id="N11EC4">in ſeptima uerò in<lb></lb>quit, <emph type="italics"></emph>Si autem magnitudines ſuerint incommenſurabiles,<emph.end type="italics"></emph.end> quaſi vna <expan abbr="tã">tam</expan> <lb></lb>tùm ſit propoſitio in duas partes diuiſa. </s> <s id="N11ED6">ita ut ne〈que〉 numeris <lb></lb>eſſent diſtinguende, ſed pro vna tantùm propoſitione <expan abbr="ſummẽ-dæ">ſummen<lb></lb>dæ</expan>, obſe〈que〉ntis autem demonſtrationis faciliorem <expan abbr="intelligẽ-tiam">intelligen<lb></lb>tiam</expan> hęc priùs præmittimus. </s> </p> <p id="N11EE7" type="head"> <s id="N11EE9">LEMMA.</s> </p> <p id="N11EEB" type="main"> <s id="N11EED">Si duę fuerint magnitudines in æquales, quarum maior ſit <lb></lb>alterius dupla, tertia verò quędam magnitudo minorem me- <pb xlink:href="077/01/059.jpg" pagenum="55"></pb>tiatur. </s> <s id="N11EF5">maiorem quo〈que〉 in partes numero pares metietur. </s> </p> <p id="N11EF7" type="main"> <s id="N11EF9">Sint duę in ęquales magni<lb></lb><arrow.to.target n="fig24"></arrow.to.target><lb></lb>tudines AB, ſitquè A ipſius <lb></lb>B duplex. </s> <s id="N11F04">magnitudo <expan abbr="autẽ">autem</expan> <lb></lb>C <expan abbr="magnitudinẽ">magnitudinem</expan> B metia<lb></lb>tur. </s> <s id="N11F12">Dico C <expan abbr="magnitudinẽ">magnitudinem</expan> <lb></lb>A metiri, menſurationesquè numero pares eſſe. </s> <s id="N11F1A">Quoniam <lb></lb>enim C metitur B, eodem numero C metietur medietates <lb></lb>ipſius A, quæ ſuntipſi B æquales. </s> <s id="N11F20">ergo duplo plures erunt nu<lb></lb>mero menſurationes ipſius A, quàm ipſius B. quare menſu<lb></lb>rationes ipſius A ſunt numero pares. </s> <s id="N11F26">duplum enim ſemper <lb></lb>paritatem ſecum affert. </s> <s id="N11F2A">quod demonſtrare oportebat. </s> </p> <figure id="id.077.01.059.1.jpg" xlink:href="077/01/059/1.jpg"></figure> <p id="N11F30" type="main"> <s id="N11F32">Porrò maxima in his duabus ſe〈que〉ntibus propoſitionibus <lb></lb>adhibenda eſt diligentia; quibus tota rerum Mechanicarum <lb></lb>ratio in nititur. </s> <s id="N11F38">Quocirca vt harum propoſitionum demon<lb></lb>ſtrationes perfectè intelligere poſſimus; præter eos argumen<lb></lb>tandi modos, quorum ante quintam huius propoſitionem <lb></lb>meminimus; alterum quo〈que〉 modum, quo Archimedes in <lb></lb> <arrow.to.target n="fig25"></arrow.to.target><lb></lb>hac ſexta propoſitione vtitur, nouiſſe oportet. </s> <s id="N11F47">vt ſcilicet, ſi ma<lb></lb>gnitudo A æ〈que〉ponderatipſis BC facta ſuſpenſione ex <expan abbr="pũ-cto">pun<lb></lb>cto</expan> D; ita ſcilicet, vt D ſit centrum grauitatis magnitudinis <lb></lb>ex omnibus ABC magnitudinibus compoſitæ; ipſarum verò <pb xlink:href="077/01/060.jpg" pagenum="56"></pb>magnitudinum BC, hoc eſt magnitudinis ex BC compoſi<lb></lb>tæ centrum grauitatis ſit punctum E; auferantur verò BC <lb></lb>à linea EA, & ipſarum loco ponatur in E magnitudo; <lb></lb>quæ ſit vtriſ〈que〉 ſimul BC ęqualis, vt in ſecunda figura. </s> <s id="N11F5D">Dico <lb></lb>eodem modo pondera ABC ę〈que〉ponderare in prima figu<lb></lb>ra, veluti grauia AE in ſecunda. </s> </p> <figure id="id.077.01.060.1.jpg" xlink:href="077/01/060/1.jpg"></figure> <p id="N11F67" type="main"> <s id="N11F69">Primum autem, vthoc recte per <lb></lb> <arrow.to.target n="fig26"></arrow.to.target><lb></lb>pendamus, intelligantur pondera <lb></lb>BC (vt in tertia figura) ſeorſum <lb></lb>à linea CA, & penes diſtantias EC <lb></lb>EB conſtituta. </s> <s id="N11F78">quorum quidem <expan abbr="põ-derum">pon<lb></lb>derum</expan> ſit centrum grauitatis E. ſi igitur intelligatur poten <lb></lb> <arrow.to.target n="marg44"></arrow.to.target> tia in E ſuſtinere pondera BC, hoc eſt pondus exipſis BC <lb></lb>compoſitum: pondera uti〈que〉 manebunt. </s> <s id="N11F88">quòd ſi ambo pe<lb></lb>penderint, vt quinquaginta, potentia in E tantùm quinqua <lb></lb>ginta ſuſtinebit. </s> <s id="N11F8E">quoniam totum ſuſtinebit pondus ex ipſis <lb></lb>compoſitum, auferantur verò pondera BC à ſitu BC, intelli <lb></lb>ganturquè pondera eſſe in E conſtituta; hoc eſt vnum ſit <lb></lb>pondus ex ipſis ſimul iunctis compoſitum, cuius <expan abbr="cẽtrum">centrum</expan> gra<lb></lb>uitatis ſit in E conſtitutum; tunc eadem potentia in E eo<lb></lb>dem modo hoc pondus ſuſtinebit; propterea quod <expan abbr="eodẽ">eodem</expan> mo<lb></lb>do quinquaginta tantùm ſuſtinebit. </s> <s id="N11FA4">Quare pondera BC <expan abbr="tã">tam</expan> <lb></lb>ex diſtantijs EC EB grauitant, quàm ſi vtra〈que〉 in E con <lb></lb>ſtituta fuerint; vel quod idem eſt, quàm pondus ipſis BC ſi<lb></lb>mul æquale in E poſitum. </s> <s id="N11FB0">Ex quo patetid, quod initio prę<lb></lb>fati ſum us, nempe, vnumquodquè graue in eius centro gra<lb></lb>uitatis propriè grauitare. </s> <s id="N11FB6">Quocum 〈que〉 enim modo <expan abbr="eadẽ">eadem</expan> gra<lb></lb>uia ſeſe habent, eodem ſemper modo in eius grauitatis <expan abbr="cẽtro">centro</expan> <lb></lb>grauitant. </s> </p> <p id="N11FC4" type="margin"> <s id="N11FC6"><margin.target id="marg44"></margin.target><emph type="italics"></emph>per def. <lb></lb>cent. </s> <s id="N11FCE">grau.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.060.2.jpg" xlink:href="077/01/060/2.jpg"></figure> <p id="N11FD6" type="main"> <s id="N11FD8">Quibus cognitis, intelligantur nunc grauia BC in linea <lb></lb>CA poſita eſſe; ut in ſuperiori figura: & ut quod propoſitum <lb></lb>fuit, oſtendatur; hoc modo argumentari licebit. </s> <s id="N11FDE">Quoniam <lb></lb>enim magnitudines BC ſuam habent grauitatem in E, ſiqui <lb></lb>dem pro vna tantùm intelliguntur magnitudine ex BC com<lb></lb>poſita, cuius punctum E centrum grauitatis exiſtit. </s> <s id="N11FE6">in <expan abbr="ſecũ">ſecum</expan> <lb></lb>da verò figura magnitudo E ſimiliter ſuam habet <expan abbr="grauitatẽ">grauitatem</expan> <lb></lb>in puncto E; quod eſt eius <expan abbr="centrũ">centrum</expan> grauitatis. </s> <s id="N11FF8">at〈que〉 magnitu <pb xlink:href="077/01/061.jpg" pagenum="57"></pb>do E eſtipſis BC ſimul ſumptis ęqualis. </s> <s id="N11FFE">diſtantię verò AD <lb></lb>DE ſunt æquales, cum ſint ęedem; erit vti〈que〉 punctum D in <lb></lb>ſecunda figura centrum grauitatis magnitudinis ex AE com<lb></lb>poſitæ, veluti D in prima figura ipſarum ABC centrum gra<lb></lb>uitatis exiſtit. </s> <s id="N12008">ac propterea in vtra〈que〉 figura pondera æ〈que〉<lb></lb>ponderabunt: </s> </p> <p id="N1200C" type="main"> <s id="N1200E">Cæterum hoc quo〈que〉 oſtendemus hoc pacto. </s> </p> <figure id="id.077.01.061.1.jpg" xlink:href="077/01/061/1.jpg"></figure> <p id="N12013" type="main"> <s id="N12015">Iiſdem nam〈que〉 poſitis; æ〈que〉ponderarent ſcilicet grauia <lb></lb>ABC facta ex D ſuſpenſione. </s> <s id="N12019">ſitquè punctum E <lb></lb>centrum grauitatis ponderum CB. quæ quidem pondera <lb></lb>CB grauitatis centrum habeant in linea CB. Dico pondus <lb></lb>A ponderi ipſis CB ſimul ſumptis æquali in E conſti<lb></lb>tuto æ〈que〉ponderare. </s> <s id="N12023">Mente concipiamus diſtantias EC <lb></lb>EB, manente centro E, circa ipſum circumuerti poſſe; <lb></lb>vt modò ſint in FEG, modò in HEK. ſimiliter in<lb></lb>telligantur pondera CB, modò in FG, modò in HK <lb></lb>exiſtere. </s> <s id="N1202D">Quoniam igitur punctum E. centrum eſt <lb></lb>grauitatis ponderum CB; erit idem E (cùm ſitum <lb></lb>nonmutet) centrum grauitatis ponderum in ſitu FG, ac <lb></lb>ponderum in HK exiſtentium. </s> <s id="N12035">Quiaverò vnumquod<lb></lb>〈que〉 pondus (ex dictis) propiè in eius centro grauitatis graui<lb></lb>tat; pondera ſimul CB ſiue ſint in FG, ſiue in HK, proprie <lb></lb>in puncto E grauitabunt. </s> <s id="N1203D">At verò quoniam idem <pb xlink:href="077/01/062.jpg" pagenum="58"></pb>pondus vnam & eandem ſemper habet grauitatem; erit <expan abbr="põdus">pondus</expan> <lb></lb>ex CB compoſitum æ〈que〉graue, tam in ſitu CB, quàm in <lb></lb>FG, & in ſitu HK. conſiderando nempe pondera CB (ut <lb></lb>revera ſunt) nilaliud eſſe niſi vnum tantùm pondus ex CB <lb></lb>compoſitum. </s> <s id="N1204F">Ex quibus perſpicuum eſt, punctum E eodem <lb></lb>ſemper modo grauitare. </s> <s id="N12053">Quare quoniam pondera CB in ſi<lb></lb>tu CB ipſi A ę〈que〉ponderant, ſuamquè habent grauitatem <lb></lb>in puncto E; eadem pondera CB ſiue ſint in FG, ſiue in <lb></lb>HK, eidem ponderi A æ〈que〉ponderabunt. </s> <s id="N1205B">ſiquidem propriè <lb></lb>ſemper grauitant in E, & eandem ſemper habent <expan abbr="grauita-tẽ">grauita<lb></lb>tem</expan> Intelligatur deni〈que〉 HEK in centrum mundi tendere; e<lb></lb>runtvti〈que〉 vtra〈que〉 pondera HK, tanquam in puncto E <expan abbr="cõ">com</expan> <lb></lb>ſtituta, vt ex prima propoſitione noſtrorum Mechanicorum <lb></lb>elici poteſt, quamuis perſe notum ſit. </s> <s id="N1206F">ſiquidem ſeorſum pon<lb></lb>dus H ſecundùm eius centrum grauitatis propriè grauitat ſu<lb></lb>per puncto E; pondus verò K eſt, tanquam ex E appenſum; <lb></lb>vndè & in eodem puncto E quo〈que〉 grauitat. </s> <s id="N12077">Ita〈que〉 <expan abbr="quoniã">quoniam</expan> <lb></lb>ambo propriè grauitant in E, erunt pondera HK perinde, <lb></lb>acſi vnum eſſet pondusipſis HK, hoc eſtipſis CB æquale, cu<lb></lb>ius centrum grauitatis ſit in E conſtitutum. </s> <s id="N12083">atverò pondus <lb></lb>A ipſis CB in ſitu HK exiſtentibus æ〈que〉ponderat. </s> <s id="N12087">ergo <expan abbr="idẽ">idem</expan> <lb></lb>pondus A ipſis CB in E conſtitutis, hoc eſt ponderi ipſis CB <lb></lb>ſimul ſumptis ęquali in E poſito æ〈que〉ponderabit. </s> <s id="N12091">quod de<lb></lb>monſtrare oportebat. </s> </p> <p id="N12095" type="main"> <s id="N12097">Quod idem quo〈que〉, ſi plura eſſent pondera, ſimiliter o<lb></lb>ſtendetur. </s> </p> <p id="N1209B" type="main"> <s id="N1209D">Valetita〈que〉 conſe〈que〉ntia, punctum D centrum eſtgra<lb></lb>uitatis magnitudinis ex ponderibus ABC compoſitę; ergoi<lb></lb>dem punctum D centrum eſt grauitatis ponderis in A, & <expan abbr="põ">pom</expan> <lb></lb>derisipſis BC ſimul ęqualis in E conſtituti. </s> <s id="N120A9">ex quo conſequi<lb></lb>tur, quòd ſi magnitudines ABC ex D æ〈que〉ponderant, ergo <lb></lb>ex eodem D magnitudo ipſis BC ſimul æqualis in E poſita, <lb></lb>& magnitudo A æ〈que〉ponderabunt. </s> <s id="N120B1">quòd ſi rectè perpenda<lb></lb>mus, nil aliud ſunt pondera in BC, niſi magnitudo in E con<lb></lb>ſtituta. </s> <s id="N120B7">ſiquidem punctum E ipſius centrum grauitatis <lb></lb>exiſtit </s> </p> <p id="N120BB" type="main"> <s id="N120BD">In noſtro autem Mechanicorum libro in quinta propoſi- <pb xlink:href="077/01/063.jpg" pagenum="59"></pb>tione tractatus de libra duas attulimus demon ſtrationes <expan abbr="oſtẽ-tes">oſten<lb></lb>tes</expan> duo pondera vt CB tam in punctis CB ponderare, quàm ſi <lb></lb>vtra〈que〉 ex puncto E ſuſpendantur. </s> <s id="N120CB">At verò quo niam demon <lb></lb>ſtrationes ibi allatæ ijs indigent, quę Archimedes in ſe〈que〉n<lb></lb>ti ſexta propoſitione demonſtrauit, idcirco demonſtrationes <lb></lb>illæ huic loco non ſunt oportunæ; vt ex ipſisſumi poſſit tan<lb></lb>quam demonſtratum pondera CB, tam in punctis CB pon<lb></lb>derare, quàm ſi vtra〈que〉 ex E ſuſpendantur. </s> <s id="N120D7">Quare hoc loco hę <lb></lb>tantùm ſufficiant rationes, quæ dictæ ſunt. </s> <s id="N120DB">Ex quibus poteſt <lb></lb>Archime des diſtam conſe〈que〉ntiam colligere; nempè magni<lb></lb>tudines ABC ex D æ〈que〉ponderant, auferantur autem BC, <lb></lb>& loco ipſarum vtriſ〈que〉 ſimul ę〈que〉grauis ponatur magnitu<lb></lb>do in E; ſimiliter hęc magnitudo ipſi A æ〈que〉ponderabit. </s> <s id="N120E5">Po<lb></lb>ſtea verò ex ijs, quæ Archimedes demonſtrauit, fieri poteſt re <lb></lb>greſſus; v<gap></gap>apertiùs, manifeſtiùſ què cognoſcere valeamus, pon<lb></lb>dera BC ita ponderare, ac ſi vtra〈que〉 ex puncto E ſuſpen<lb></lb>dantur. </s> </p> <figure id="id.077.01.063.1.jpg" xlink:href="077/01/063/1.jpg"></figure> <p id="N120F4" type="main"> <s id="N120F6">Cęterum hoc loco Archimedes non ſolùm de duobus, <expan abbr="verũ">verum</expan> <lb></lb>etiam de pluribus ponderibus idipſum <expan abbr="intelligendũ">intelligendum</expan> admittit. <lb></lb>vt ſi magnitudines STVXZM æ〈que〉ponderent facta <expan abbr="ſuſpẽſio">ſuſpenſio</expan> <lb></lb>ne ex puncto C. ſitquè magnitudinum MZ <expan abbr="centrũ">centrum</expan> grauitatis <lb></lb>D; ipſarum verò STVX ſit centrum grauitatis E. ſi ita〈que〉 ma <lb></lb>gnitudines STVX, & ZM ex C æ〈que〉ponderant; auferantur <lb></lb>STVX, quarum loco ponatur in E magnitudo ipſis STVX ſi <lb></lb>mul ſumptis ęqualis: auferanturquè ZM, at〈que〉 <expan abbr="ipſarũ">ipſarum</expan> loco po <lb></lb>natur in D magnitudo ipſis ZM ſimul ęqualis; tunclicetinfer <lb></lb>re, ergo hæ magnitudines in ED poſitæ ę〈que〉pondera<lb></lb>bunt. </s> <s id="N12120">Quod quidem ijsdem prorſus modis oſtendentur. <lb></lb>præſertim ſi mente concipiamus diſtantias ES EX, <pb xlink:href="077/01/064.jpg" pagenum="60"></pb>nec non magnitudines STVX in ſuis diſtantijs circa <expan abbr="centrũ">centrum</expan> <lb></lb>grauitatis E circumuerti poſſe; veluti diſtantias DZ DM, ma <lb></lb>gnitudineſquè ZM circacentrum D. moueantur autem <lb></lb>SEX, & ZDM, donec in centrum mundi vergant. </s> <s id="N12132">ſimiliter <lb></lb>oſtendetur magnitudines STVX eſſe, ac ſi in E eſſent appen <lb></lb>ſę, ſiue conſtitutę; magnitudines verò ZM ac ſi in D poſi<lb></lb>tæ fuerint. </s> <s id="N1213A">&c. </s> <s id="N1213C">Ex quibus ſequitur, ſi punctum C centrum <lb></lb>eſt grauitatis magnitudinum STVXZM. ponatur magnitu<lb></lb>do ipſis STVX ſimul ſumptis ęqualis in E; magnitudo au<lb></lb>tem ipſis ZM ſimul æqualis in D; punctum C ſimiliter <lb></lb>ipſarum quo〈que〉 centrum grauitatis exiſtet. </s> <s id="N12146">vnde vtro〈que〉 mo <lb></lb>do æ〈que〉ponderabunt. </s> <s id="N1214A">& ita in alijs, ſi plures fuerint magni<lb></lb>tudines. </s> </p> <p id="N1214E" type="head"> <s id="N12150">PROPOSITIO. VI.</s> </p> <p id="N12152" type="main"> <s id="N12154">Magnitudines commenſurabiles ex diſtantijs <lb></lb>eandem permutatim proportionem habentibus, <lb></lb>vt grauitates, æ〈que〉ponderant. </s> </p> <p id="N1215A" type="main"> <s id="N1215C"><emph type="italics"></emph>Commenſurabiles ſint magnitudines AB quarum centra<emph.end type="italics"></emph.end> grauita<lb></lb>tis <emph type="italics"></emph>AB, & quædam ſit diſtantia E D, & vt<emph.end type="italics"></emph.end> ſe habet grauitas ma<lb></lb>gnitudinis <emph type="italics"></emph>A ad<emph.end type="italics"></emph.end> grauitatem magnitudinis <emph type="italics"></emph>B, ua ſit <expan abbr="diſtãtia">diſtantia</expan> <lb></lb>DC ad distantiam CE. <expan abbr="ostendẽdũ">ostendendum</expan> eſi<emph.end type="italics"></emph.end>, ſi centra grauitatis AB fue <lb></lb>rint in punctis ED conſtituta, hoc eſt A in E, & B in D; <lb></lb><emph type="italics"></emph>magnitudinis ex vtriſquè<emph.end type="italics"></emph.end> magnitudinibus <emph type="italics"></emph>AB compoſitæ centrum <lb></lb>grauitatis eſſe punctum C. Quoniam enim ita est<emph.end type="italics"></emph.end> magnitudo <emph type="italics"></emph>A ad<emph.end type="italics"></emph.end><lb></lb>magnitudinem <emph type="italics"></emph>B, vt DC ad CE. eſt autem<emph.end type="italics"></emph.end> magnitudo <emph type="italics"></emph>A ipſi <lb></lb> <arrow.to.target n="marg45"></arrow.to.target> B commenſurabilis; erit & CD ipſi CE commenſurabilis; hoc eſt <lb></lb>recta linea rectæ lineæ<emph.end type="italics"></emph.end> commenſurabilis exiſtet. <emph type="italics"></emph>Quare ipſarum EC <lb></lb>CD communis reperitur menſura. </s> <s id="N121B4">quæ quidem ſit N. deinde ponatur <lb></lb>ipſi EC æqualis vtra〈que〉 DG DK; ipſi verò DC æqualis EL. & <lb></lb>quoniam æqualis est DG ipſi CE<emph.end type="italics"></emph.end>, communi addita CG, <emph type="italics"></emph>erit DC <lb></lb>ipſi EG æqualis<emph.end type="italics"></emph.end>; ſed DC eſt ipſi EL ęqualis: <emph type="italics"></emph>erit igitur LE æqua<lb></lb>lis ipſi EG.<emph.end type="italics"></emph.end> quare vtra〈que〉 LE EG ęqualis eſt ipſi DC. <emph type="italics"></emph>ac propte<emph.end type="italics"></emph.end> <pb xlink:href="077/01/065.jpg" pagenum="61"></pb><emph type="italics"></emph>rea dupla est LG ipſius DC.<emph.end type="italics"></emph.end> quia verò vtra〈que〉 DG DK æqualis <lb></lb>facta eſt ipſi CE, erit <emph type="italics"></emph>& ipſa quo〈que〉 GK ipſius CE<emph.end type="italics"></emph.end> dupla. <emph type="italics"></emph>Quare <lb></lb>N <expan abbr="vtrã〈que〉">vtran〈que〉</expan> LG Gk metitur, cùm & ipſarum medietates<emph.end type="italics"></emph.end> DC CE <lb></lb> <arrow.to.target n="fig27"></arrow.to.target><lb></lb>metiatur. <emph type="italics"></emph>Et quoniam<emph.end type="italics"></emph.end> magnitudo <emph type="italics"></emph>A ita eſt ad<emph.end type="italics"></emph.end> magnitudinem <lb></lb><emph type="italics"></emph>B, vt DC ad CE, ut autem DC ad CE, ita eſt LG ad G<emph.end type="italics"></emph.end>K, <emph type="italics"></emph>utra〈que〉 <lb></lb>enim vtriuſ〈que〉 duplex exiſtit<emph.end type="italics"></emph.end> (ſiquidem LG dupla eſt ipſius DC, <lb></lb>& GK itidem ipſius CE duplex) <emph type="italics"></emph>erit<emph.end type="italics"></emph.end> magnitudo <emph type="italics"></emph>A ad<emph.end type="italics"></emph.end> <arrow.to.target n="marg46"></arrow.to.target> magni<lb></lb>tudinem <emph type="italics"></emph>B, ut LG ad G<emph.end type="italics"></emph.end>k; & conuertendo magnitudo B ad <lb></lb>magnitudinem A, vt KG ad GL. <emph type="italics"></emph>Quotuplex autem est LG ipſius <lb></lb>N, totuplex ſit<emph.end type="italics"></emph.end> magnitudo <emph type="italics"></emph>A ipſius F, erit vti〈que〉 LG ad N, vt<emph.end type="italics"></emph.end><lb></lb>magnitudo <emph type="italics"></emph>A ad F, atqui est KG ad LG, vt<emph.end type="italics"></emph.end> magnitudo <emph type="italics"></emph>B ad<emph.end type="italics"></emph.end><lb></lb>magnitudinem <emph type="italics"></emph>A:<emph.end type="italics"></emph.end> LG verò ad N eſt, vt magnitudo A ad <arrow.to.target n="marg47"></arrow.to.target> <expan abbr="i-psã">i<lb></lb>psam</expan> F, <emph type="italics"></emph>ex æquali igitur erit KG ad N, vt<emph.end type="italics"></emph.end> magnitudo <emph type="italics"></emph>B ad F quare æ<lb></lb>〈que〉multiplex eſt<emph.end type="italics"></emph.end> kG <emph type="italics"></emph>ipſius N, veluti<emph.end type="italics"></emph.end> magnitudo <emph type="italics"></emph>B ipſius F. demon <lb></lb><expan abbr="ſtratũ">ſtratum</expan> <expan abbr="aũt">aunt</expan> eſt<emph.end type="italics"></emph.end> <expan abbr="magnitudinẽ">magnitudinem</expan> <emph type="italics"></emph>A ipſius F multiplicem eſſe<emph.end type="italics"></emph.end>, ſiquidem eſt <lb></lb>magnitudo A ad ipſam F, vt LG ad N, quæ quidem LG mul <lb></lb>tiplex eſt ipſius N. <emph type="italics"></emph>qua propter F ipſarum AB communis existit men <lb></lb>ſura. </s> <s id="N12290">Jta〈que〉 diuiſa LG in partes<emph.end type="italics"></emph.end> LH, HE, EC, CG, <emph type="italics"></emph>ipſi N aquales<emph.end type="italics"></emph.end>, <lb></lb>cadent vti〈que〉 diuiſiones in punctis EC, quoniam <expan abbr="Nipsã">Nipsam</expan> EC <arrow.to.target n="marg48"></arrow.to.target><lb></lb>metitur, nec non ipſam quo〈que〉 LE metitur; cùm ſit LE ipſi <lb></lb>CD æqualis. </s> <s id="N122A8">eruntquè diuiſiones LH, HE, EC, CG, numero <lb></lb>pares; cùm N dimidiam ipſius LG, hoc eſt CD metiatur. <pb xlink:href="077/01/066.jpg" pagenum="62"></pb><emph type="italics"></emph>Averò<emph.end type="italics"></emph.end> ſimiliter diuiſa <emph type="italics"></emph>in partes<emph.end type="italics"></emph.end> OP QR <emph type="italics"></emph>ipſi F æquales; ſectio<lb></lb>nes<emph.end type="italics"></emph.end> LH, HE, EC, CG <emph type="italics"></emph>in LG existentes magnitudini N æqua<lb></lb>les, erunt numero æquales ſectionibus<emph.end type="italics"></emph.end> OPQR <emph type="italics"></emph>in<emph.end type="italics"></emph.end> magnitudine <emph type="italics"></emph>A <lb></lb>existentibus ipſi F æqualibus.<emph.end type="italics"></emph.end> Diuidantur ſectiones LH, HE, EC, <lb></lb> <arrow.to.target n="fig28"></arrow.to.target><lb></lb>CG bifariam in punctis STVX. <emph type="italics"></emph>ſi it a〈que〉 in vnaqua〈que〉 ſestio <lb></lb>ne ipſius LG apponatur magnitudo æqualis ipſi F, quæ centrum gra<lb></lb>uitatis babeat in medio ſectionis<emph.end type="italics"></emph.end>; vt ſi in LH ponatur magnitudo <lb></lb>S, in HE magnitudo T, in EC magnitudo V, & in <lb></lb>CG magnitudo X; ipſarum què vna quæ〈que〉 STVX ſit ipſi <lb></lb>F æqualis: habeat verò magnitudo S ſuum grauitatis <expan abbr="centrũ">centrum</expan>, <lb></lb>quod ſit punctum S, in medio ſectionis LH, nempè in <expan abbr="pũ-cto">pun<lb></lb>cto</expan> S; ſimiliter cæteræ magnitudines TVX habeant <expan abbr="cẽrra">cerrra</expan> <lb></lb>grauitatis; quæ ſint puncta TVX, in medio ſectionum HE, <lb></lb>EC, CG, in punctis nempè TVX, erunt centra grauitatisma <lb></lb>gnitudinum STVX in recta linea conſtituta, & quoma<gap></gap>o <lb></lb>SH dimidia eſt ipſius LH, veluti HT ipſius HE, erit ST, <lb></lb>ipſius LE dimidia, vnaquæ〈que〉 verò LH HE dimidia <lb></lb>quo〈que〉 eſt ipſius LE, ſiquidem LH, HE inter ſe ſunt ęqua <lb></lb>les; erit igitur ST vnicui〈que〉 LH, & HE æqualis. </s> <s id="N12310">eodem què <lb></lb>prorſus modo oſtendeturi TV ęqualem eſſe vnicui〈que〉 HE <lb></lb>EC. & VX æqualem EC. & CG. & quoniam omnes <pb xlink:href="077/01/067.jpg" pagenum="63"></pb>LH, HE, EC, CG, inter ſe ſunt æquales; erunt ST TV VX in<lb></lb>terſe æquales. </s> <s id="N1231C">quare lineæ inter centra grauitatis magnitudi<lb></lb>num STVX exiſtentes ſunt inter ſe ęquales. <emph type="italics"></emph>omnes verò magni<lb></lb>tudines<emph.end type="italics"></emph.end> STVX ſimul <emph type="italics"></emph>ſunt æquales ipſi A<emph.end type="italics"></emph.end>, quandoquidem ipſis <lb></lb>OPQR, & numero, & magnitudine ſunt ęquales; ergo <emph type="italics"></emph>magni<lb></lb>tudinis ex omnibus<emph.end type="italics"></emph.end> magnitudinibus STVX <emph type="italics"></emph>compoſitæ centrumgra <lb></lb>uitatis erit punstum E. cùm omnes<emph.end type="italics"></emph.end> magnitudines STVX <emph type="italics"></emph>ſint nu<lb></lb>mero pares.<emph.end type="italics"></emph.end> quippe cùm ſint in ſectionibus LH HE EC CG nu<lb></lb>mero paribus. </s> <s id="N1234A">& <emph type="italics"></emph>LE ipſi EG æqualis exiſtat.<emph.end type="italics"></emph.end> quòd ſi LE eſtipſi <lb></lb>EG æqualis, demptis æqualibus LS GX æqualibus, ſiquidem <lb></lb>ſunt dimidiæ ſectionum LH CG æqualium: erunt SE EX <arrow.to.target n="marg49"></arrow.to.target> in<lb></lb>terſe æquales, vnde ex præcedenti colligitur, punctum E cen<lb></lb>trum eſſe grauitatis magnitudinum STVX. <emph type="italics"></emph>ſimiliter autem <expan abbr="oſtẽ">oſtem</expan> <lb></lb>detur, quòd ſi<emph.end type="italics"></emph.end> diuidatur GK in partes GD DK ipſi N æquales; <lb></lb>cadetvti〈que〉 diuiſionum aliqua in <expan abbr="pũcto">puncto</expan> D; ſiquidem Nipſas <lb></lb>GD DK metitur; cùm vtra〈que〉 ſit æqualisipſi EC. diuiſioneſ<lb></lb>què GD DK numero pares erunt; cùm N dimidiam ipſius <arrow.to.target n="marg50"></arrow.to.target><lb></lb>GK, ipſam ſcilicet EC metiatur. </s> <s id="N12379">ſi ita〈que〉 diuidatur GD DK <lb></lb>bifariam in punctis ZM. deinde diuidatur magnitudo B <lb></lb>in partes ipſi F æquales; ſectiones GD DH in GK exiſtentes <lb></lb>ipſi N æquales, erunt numero æquales ſectionibus in ma <lb></lb>gnitudine B exiſtentibus ipſi F æqualibus. </s> <s id="N12383">quare <emph type="italics"></emph>vnicui〈que〉 <lb></lb>partium ipſius GK apponatur magnitudo æqualis ipſi F; centrum gra<lb></lb>uitatis habens in medio ſectionis<emph.end type="italics"></emph.end>; vt <expan abbr="ponãtur">ponantur</expan> magnitudines ZM in <lb></lb>ſectionibus GD DK, ita vt magnitudinum centra grauita<lb></lb>tis, quæ ſint ZM, in medio ſectionum GD DK, in punctis <lb></lb>nempè ZM ſint conſtituta, <emph type="italics"></emph>omnes autem magnitudines<emph.end type="italics"></emph.end> ZM ſi <lb></lb>mul <emph type="italics"></emph>ſunt æquales ipſi B. magnitudinis ex omnibus<emph.end type="italics"></emph.end> magnitudinibus <lb></lb>ZM <emph type="italics"></emph>compoſitæ centrum grauitatis erit punctum D.<emph.end type="italics"></emph.end> cùm ſit ZD <lb></lb>ęqualis DM. <emph type="italics"></emph>ſed<emph.end type="italics"></emph.end> magnitudines STVX ſunt magnitudini A <lb></lb>æquales, & ZM ipſi B ergo <emph type="italics"></emph>magnitudo A eſt<emph.end type="italics"></emph.end> tanquam <emph type="italics"></emph>impoſita <lb></lb>ad E, ipſa verò B ad D.<emph.end type="italics"></emph.end> eodem ſcilicet modo ſe habebit ma<lb></lb>gnitudo A impoſita ad E, vt ſe habent magnitudines STVX; <lb></lb>ipſa verò B ſe habebit ad D, vt magnitudines ZM. <emph type="italics"></emph>ſunt au<lb></lb>tem magnitudines<emph.end type="italics"></emph.end> STVXZM <emph type="italics"></emph>inter ſe æquales<emph.end type="italics"></emph.end>, cùm vnaquæ 〈que〉 ſit <lb></lb>ipſi F ęqualis: ſuntquè omnes, (hoc eſt ipſarum centra graui<lb></lb>tatis) <emph type="italics"></emph>inrecta linea poſitæ; quarum centragrauitatis poſita ſunt inter ſe<emph.end type="italics"></emph.end> <pb xlink:href="077/01/068.jpg" pagenum="64"></pb><emph type="italics"></emph>æqualiter diſtantia;<emph.end type="italics"></emph.end> ſiquidem oſtenſum eſt ST TV VX inter<lb></lb>ſe æquales eſſe. </s> <s id="N123EE">Eodemquè modo oſtendetur XZ ZM cæteris <lb></lb>æquales eſſe. <emph type="italics"></emph>& ſunt<emph.end type="italics"></emph.end> magnitudines STVXZM <emph type="italics"></emph>numero pares,<emph.end type="italics"></emph.end><lb></lb>cùm ſectiones totius LK, ( in quibus inſunt) ipſi N æquales <lb></lb>ſint inter ſe ęquales, & numero pares. </s> <s id="N12401">cùm oſtenſum ſit ſectio <lb></lb> <arrow.to.target n="marg51"></arrow.to.target> nes in LG, & in Gk exiſtentes numero pares eſſe. <emph type="italics"></emph>conſtat magni<lb></lb>tudinis ex omnibus<emph.end type="italics"></emph.end> STVXZM magnitudinibus <emph type="italics"></emph>compoſitæ centrum<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg52"></arrow.to.target> <emph type="italics"></emph>grauitatis eſſe medietatem restæ lineæ, in qua centra grauitatis magnitu<lb></lb>dinum habentur. </s> <s id="N12421">Ita〈que〉 cùm LE ſit æqualis C D, EC verò ipſi D<emph.end type="italics"></emph.end>k, <lb></lb><emph type="italics"></emph>tota LC æqualis erit CK.<emph.end type="italics"></emph.end> cùm autem ſint LHDK æquales; ſi<lb></lb>quidem ſunt eidem N æquales, & harum medietates, hoc eſt <lb></lb>LS ipſi MK ęqualis erit. </s> <s id="N12431">& ob id SC ipſi CM eſt æqualis. <lb></lb>at verò linea SM magnitudinum centra grauitatis <expan abbr="coniũgit">coniungit</expan>, <lb></lb><emph type="italics"></emph>ergo magnitudinis ex omnibus<emph.end type="italics"></emph.end> STVXZM magnitudinibus <emph type="italics"></emph>compoſi <lb></lb>tæcentrum grauitatis est punctum C. Quare<emph.end type="italics"></emph.end> loco magnitudinum <lb></lb>STVX, <emph type="italics"></emph>poſito<emph.end type="italics"></emph.end> centro grauitatis <emph type="italics"></emph>A ad E, B verò<emph.end type="italics"></emph.end> loco ipſarum <lb></lb>ZM poſito <emph type="italics"></emph>ad D,<emph.end type="italics"></emph.end> erit punctum C grauitatis centrum ma<lb></lb>gnitudinis ex vtriſ〈que〉 magnitudinibus AB compoſitæ. </s> <s id="N12460">ac <lb></lb>prop terea <emph type="italics"></emph>ex puncto C æ〈que〉ponderabunt.<emph.end type="italics"></emph.end> ergo magnitudines AB <lb></lb>ex diſtantijs DC CE, quę permutatim eandem habent pro. <lb></lb>portionem, vt grauitates, ę〈que〉ponderant. </s> <s id="N1246E">quod demonſtrare <lb></lb>oportebat. </s> </p> <p id="N12472" type="margin"> <s id="N12474"><margin.target id="marg45"></margin.target><emph type="italics"></emph>ex<emph.end type="italics"></emph.end> 3 <emph type="italics"></emph>de<lb></lb>cimi.<emph.end type="italics"></emph.end></s> </p> <p id="N12484" type="margin"> <s id="N12486"><margin.target id="marg46"></margin.target>11 <emph type="italics"></emph>quinti. <lb></lb>cor.<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>quin<lb></lb>ti.<emph.end type="italics"></emph.end></s> </p> <p id="N12499" type="margin"> <s id="N1249B"><margin.target id="marg47"></margin.target>22. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N124A4" type="margin"> <s id="N124A6"><margin.target id="marg48"></margin.target><emph type="italics"></emph>iemme.<emph.end type="italics"></emph.end></s> </p> <p id="N124AE" type="margin"> <s id="N124B0"><margin.target id="marg49"></margin.target><emph type="italics"></emph>ex<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>cor.<emph.end type="italics"></emph.end></s> </p> <p id="N124BE" type="margin"> <s id="N124C0"><margin.target id="marg50"></margin.target><emph type="italics"></emph>lemma.<emph.end type="italics"></emph.end></s> </p> <p id="N124C8" type="margin"> <s id="N124CA"><margin.target id="marg51"></margin.target>2.<emph type="italics"></emph>cor. </s> <s id="N124D1">quin<lb></lb>tæ huius.<emph.end type="italics"></emph.end></s> </p> <p id="N124D7" type="margin"> <s id="N124D9"><margin.target id="marg52"></margin.target>*</s> </p> <figure id="id.077.01.068.1.jpg" xlink:href="077/01/068/1.jpg"></figure> <figure id="id.077.01.068.2.jpg" xlink:href="077/01/068/2.jpg"></figure> <p id="N124E5" type="head"> <s id="N124E7">SCHOLIVM.</s> </p> <p id="N124E9" type="main"> <s id="N124EB"> <arrow.to.target n="marg53"></arrow.to.target> Circa finem Gręcus codex habet, <foreign lang="grc">τα κέντ<10>α τῶν μέσων μεγεθῶν</foreign>, <lb></lb>quaſi dicat, centrum grauitatis magnitudinis ex omnibus <lb></lb>magnitudinibus STVXZM compoſitę medietatem eſſe rectę <lb></lb>lineę VX, quę centra mediarum magnitudinum VX coniun<lb></lb>git; quòd cùm ſint omnes magnitudines numero pares; <expan abbr="itidẽ">itidem</expan> <lb></lb>eſſet punctum C, & quamuis hoc ſit verum, non tamen ad hoc <lb></lb>reſpexit Archimedes duabus de cauſis. <expan abbr="Nãin">Nanin</expan> ſecudo corollario <lb></lb>pręcedentis oſtendit centrum grauitatis omnium magnitu<lb></lb>dinum eſſe medietatem rectę lineę, quę grauitatis centra om<lb></lb>nia coniungit. </s> <s id="N1250F">Deinde concludere volens punctum C <expan abbr="centrũ">centrum</expan> <lb></lb>eſſe grauitatis omnium magnitudinum, ſtatim inquit hoc ſe <lb></lb>qui, quia LC eſt ipſi CK ęqualis, quę ſunt medietates totius <pb xlink:href="077/01/069.jpg" pagenum="65"></pb>rectælineę LK. Et non dixit, quia VC ſitipſi CX ęqualis. <lb></lb>Quare codicem græcum ita reſtituendum cenſeo. <foreign lang="grc">τὰκέντ<10>κ τῶν <lb></lb>τοῦ βὰ<10>εος μεγεθῶν</foreign>, vt vertimus. </s> </p> <p id="N12525" type="margin"> <s id="N12527"><margin.target id="marg53"></margin.target>*</s> </p> <p id="N1252B" type="main"> <s id="N1252D">Ob ſe〈que〉ntis verò demonſtrationis cognitionem, hoc pro <lb></lb>blema priùs oſtendemus. </s> </p> <p id="N12531" type="head"> <s id="N12533">PROBLEMA.</s> </p> <p id="N12535" type="main"> <s id="N12537">Duarum expoſitarum magnitudinum incommenſurabi<lb></lb>lium altera vtcum〈que〉 ſecetur; magnitudinem tota ſecta ma<lb></lb>gnitudine minorem, & altero ſegmentomaiorem, alteri ve<lb></lb>rò expoſitæ magnitudini commenſurabilem inuenire. </s> </p> <p id="N1253F" type="main"> <s id="N12541">Sint duæ magnitudi<lb></lb>nes incommenſurabiles <lb></lb> <arrow.to.target n="fig29"></arrow.to.target><lb></lb>AE BC. ſeceturquè ipſa<lb></lb>rum altera, putà BC, vt<lb></lb>cum〈que〉 in D. oportet <lb></lb>magnitudinem inuenire <lb></lb>minorem quidem BC, <lb></lb>maiorem verò BD, quæ ſitipſi AE commenſurabilis. </s> <s id="N12556">Au<lb></lb>feratur ab AE pars dimidia, rurſus dimidiæ partis ipſius AE <lb></lb>dimidia auferatur; & eius, quæ remanet, adhuc dimidia; idquè <lb></lb>ſemper fiat, donec relinquatur magnitudo minor, quàm DE. <lb></lb>quod quidem perſpicuum eſt poſſe fieri ex prima decimi Eu<lb></lb>clidis propoſitione. </s> <s id="N12562">ſitita〈que〉 AF, quæ minor exiſtat, quàm <lb></lb>DC. quippe quę AF, cùm ſit abla ta ex AE ſemper per dimi <lb></lb>diam partem, metietur vti〈que〉 AF ipſam AE. Deinde mul<lb></lb>tiplicetur AF ſuper BD, tum demum multiplicatio vltima, <lb></lb>vel in puncto D cadet, vel minus. </s> <s id="N1256C">ſi cadet; ſeceturex DE <lb></lb>magnitudo DG ęqualis AF. quod quidem fiet, <expan abbr="quoniã">quoniam</expan> AF <lb></lb>minor eſt DC. Quoniam igitur AF metitur BD, & DG; <lb></lb>metietur AF totam BG. Sed & ipſam AE metitur; etgo <lb></lb>AF ipſarum BG AE communis exiſtit menſura, ac propte<lb></lb>rea BG ipſi AE commenſurabilis exiſtir; quæ quidem BG <lb></lb>minor eſt BC, maior verò BD. Si verò vltima <arrow.to.target n="marg54"></arrow.to.target> multi<lb></lb>plicatio ipſius AF ſuper BD non cadet in D. ſed in H, <lb></lb>erit vti〈que〉 HD minor AF. nam ſi HD ipſi AF eſſet ęqualis, <pb xlink:href="077/01/070.jpg" pagenum="66"></pb>vltima multiplicatio caderet in D. ſi verò maior eſſet HD, <lb></lb>quàm AF tunc non eſſet vltima multiplicatio. </s> <s id="N1258C">quare cùm ſit <lb></lb>DC maior AF; erit & HC ipſa FA maior. </s> <s id="N12590">ſi ita〈que〉 fiat HK <lb></lb>æqualis AF; erit punctum K inter puncta DC. BK igitur <lb></lb>minor erit, quàm BC, & maior BD; eodemquè modo o<lb></lb>ſtendetur AF ipſarum Bk AE communem eſſe menſu<lb></lb>ram. </s> <s id="N1259A">& obid BK ipſi AF commenſurabilem exiſtere. </s> <s id="N1259C">quod <lb></lb>facere oportebat. </s> </p> <p id="N125A0" type="margin"> <s id="N125A2"><margin.target id="marg54"></margin.target>1.<emph type="italics"></emph>def.deci<lb></lb>mi.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.070.1.jpg" xlink:href="077/01/070/1.jpg"></figure> <p id="N125B1" type="main"> <s id="N125B3">Cùm autem verba ſe〈que〉ntis demonſtrationis aliquantu<lb></lb>lum ſint obſcura, vt vim demonſtrationis rectè petcipiamus, <lb></lb>hoc quo〈que〉 theorema ex ijs, quæ ab Archimede hactenus de<lb></lb>monſtrata ſunt, oſtendemus. </s> <s id="N125BB">ad quod demonſtrandum com <lb></lb>muni notione indigemus, quam nos in noſtro Mechanico<lb></lb>rum libro poſuimus. </s> <s id="N125C1">Nempè. </s> </p> <p id="N125C3" type="main"> <s id="N125C5">Quæ eidem æ〈que〉pondeiant, inter ſe æquè ſunt grauia. </s> </p> <p id="N125C7" type="head"> <s id="N125C9">PROPOSITIO.</s> </p> <p id="N125CB" type="main"> <s id="N125CD">Si commenſurabiles magnitudines minorem habuerint <lb></lb>proportionem, quàm diſtantię permutatim habent; vt ę〈que〉<lb></lb>ponderent, maiori opus erit magnitudine, quàm ſit ea, quę <lb></lb>ad alteram magnitudinem minorem proportionem habet. </s> </p> <figure id="id.077.01.070.2.jpg" xlink:href="077/01/070/2.jpg"></figure> <p id="N125D8" type="main"> <s id="N125DA">Sint magnitudines AC commenſurabiles, diſtantię ve<lb></lb>rò ſint ED EF. minorem autem habeat pro- <pb xlink:href="077/01/071.jpg" pagenum="67"></pb>portionem A ad C, quàm ED ad EF. Dico, vt magnitu<lb></lb>dines ex diſtantijs ED EF æ〈que〉ponderent, maiori o<lb></lb>pus eſſe magnitudine in F, quàm ſit magnitudo A; <lb></lb>ita vt ipſi C in D æ〈que〉ponderare poſſit. </s> <s id="N125E8">fiat ED <lb></lb>ad EG, vt magnitudo A ad magnitudinem C. <lb></lb>Deindefiat EK æqualis EG. exponaturquè altera ma<lb></lb>gnitudo L ipſi A ęqualis. </s> <s id="N125F0">Quoniam igitur minorem <lb></lb>habet proportionem A ad C, quàm ED ad EF, & <lb></lb>vt A ad C, ita ED ad EG; habebit ED ad <lb></lb>EG minorem proportionem, quàm ad EF. ac propterea <arrow.to.target n="marg55"></arrow.to.target><lb></lb>EF minor eſt, quàm EG. quoniam ausem A ad C <lb></lb>eſt, vt ED ad EG, commenſurabiles magnitudines <lb></lb>AC ex diſtantijs ED EG æ〈que〉ponderabunt. </s> <s id="N12601">Cùm <arrow.to.target n="marg56"></arrow.to.target><lb></lb>verò EK ſit æqualis EG, magnitudines AL æ<lb></lb>quales ex diſtantis æqualibus EK EG ſimiliter æ〈que〉<lb></lb>ponderabunt. </s> <s id="N1260C">At verò quoniam C in D æ〈que〉<lb></lb>ponderat ipſi A in G, ſimiliter L in K eidem A in <lb></lb>G ę〈que〉ponderat; ęqualem habebit grauitatem C in D, vt <arrow.to.target n="marg57"></arrow.to.target><lb></lb>L in K. Ita〈que〉 quoniam diſtantia EG æqualis eſt diſtan<lb></lb>tiæ Ek, longitudo EK maior erit longitudine EF. ergo <lb></lb>magnitudines AL ęquales ex inæqualibus diſtantijs EK <arrow.to.target n="marg58"></arrow.to.target><lb></lb>EF non ę〈que〉ponderabunt. </s> <s id="N12620">ſed magnitudo L deorſum ver<lb></lb>get. </s> <s id="N12624">ſi igitur in F collocanda ſit magnitudo, quæ æ〈que〉pon<lb></lb>deret ipſi L in K, proculdubiò hęc magnitudine A ma<lb></lb>ior exiſtet. </s> <s id="N1262A">Inæqualia enim grauia, nempè L, & magnitu <arrow.to.target n="marg59"></arrow.to.target><lb></lb>do maior, quàm A, exinæqualibus diſtantijs EK EF æ<lb></lb>〈que〉ponderant, dummodo maius, hoc eſt magnitudo maior, <lb></lb>quàm A, ſit in diſtantia minori EF. minusverò, hoc eſt ma<lb></lb>gnitudo L, ſit in minori EK. Quoniam ita〈que〉 magnitudo <lb></lb>C in D eſt ę〈que〉grauis, vt L in K, magnitudo, quæ in F <lb></lb>ipſi L in K æ〈que〉ponderat, eadem quo〈que〉 in F ipſi C in D <lb></lb>æ〈que〉ponderabit maior verò magnitudo, quàm ſit A, in F ipſi <lb></lb>L in K æ〈que〉ponderat, ergo maior magnitudo, quàm A in <lb></lb>F, ipſi C in D æ〈que〉ponderabit. </s> <s id="N12641">quod demonſtrare opor<lb></lb>tebat. </s> </p> <p id="N12645" type="margin"> <s id="N12647"><margin.target id="marg55"></margin.target>10. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N12650" type="margin"> <s id="N12652"><margin.target id="marg56"></margin.target>6. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N1265B" type="margin"> <s id="N1265D"><margin.target id="marg57"></margin.target><emph type="italics"></emph><expan abbr="cõm">comm</expan>. not.<emph.end type="italics"></emph.end></s> </p> <p id="N12668" type="margin"> <s id="N1266A"><margin.target id="marg58"></margin.target>2. <emph type="italics"></emph>poſt bu<lb></lb>ius.<emph.end type="italics"></emph.end></s> </p> <p id="N12675" type="margin"> <s id="N12677"><margin.target id="marg59"></margin.target>3. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N12680" type="main"> <s id="N12682">His cognitis poſſumus ad Archimedis demonſtrationem <lb></lb>accedere. </s> </p> <pb xlink:href="077/01/072.jpg" pagenum="68"></pb> <p id="N12689" type="head"> <s id="N1268B">PROPOSITIO. VII.</s> </p> <p id="N1268D" type="main"> <s id="N1268F">Si autem magnitudines fuerint incommenſura <lb></lb>biles, ſimiliter æ〈que〉ponderabunt ex diſtantijs per <lb></lb>mutatim eandem, at〈que〉 magnitudines, propor<lb></lb>tionem habentibus. </s> </p> <figure id="id.077.01.072.1.jpg" xlink:href="077/01/072/1.jpg"></figure> <p id="N1269A" type="main"> <s id="N1269C"><emph type="italics"></emph>Sint incommenſurabiles magnitudines AB C. Distantiæ verò <lb></lb>DE EF. Habeat autem AB ad C proportionem eandem, quam di <lb></lb>stantia ED ad ipſam EF. Dico,<emph.end type="italics"></emph.end> ſi ponatur AB ad F, C ve<lb></lb>rò ad D, <emph type="italics"></emph>magnitudinis ex vtriſ〈que〉 AB C compoſitæ centrum gra<lb></lb>uitatis eſſe punctum E. ſi enim non æ〈que〉ponderabit<emph.end type="italics"></emph.end> (ſi fieri poteſt) <lb></lb><emph type="italics"></emph>AB poſita ad F ipſi C poſitæ ad D; velmaior est AB, quàm C, ita <lb></lb>vt<emph.end type="italics"></emph.end> AB ad F <emph type="italics"></emph>æ〈que〉ponderet ipſi C<emph.end type="italics"></emph.end> ad D; <emph type="italics"></emph>vel non. </s> <s id="N126C3">Sit maior<emph.end type="italics"></emph.end>; ſitquè <lb></lb>exceſſus HL; ita vt KH ad F, & C ad D ę〈que〉ponderent. <lb></lb> <arrow.to.target n="marg60"></arrow.to.target> <emph type="italics"></emph>auferaturquè ab ipſa AB<emph.end type="italics"></emph.end> magnitudo NL, quæ ſit <emph type="italics"></emph>minor exceſſu<emph.end type="italics"></emph.end><lb></lb>HL, <emph type="italics"></emph>quo maior est<emph.end type="italics"></emph.end> tota <emph type="italics"></emph>AB, quàm C, ita vt æ〈que〉ponderent<emph.end type="italics"></emph.end>; vt <expan abbr="dictũ">dictum</expan> <lb></lb>eſt. <emph type="italics"></emph>& ſit quidem reſiduum A,<emph.end type="italics"></emph.end> hoc eſt KN, <emph type="italics"></emph>commenſurabile ipſi C.<emph.end type="italics"></emph.end><lb></lb>Et quoniam minor eſt kN quàm KM, minorem quo〈que〉 <pb xlink:href="077/01/073.jpg" pagenum="69"></pb>habebit proportionem kN ad C, quàm kM ad eandem <lb></lb>C. tota verò KM ad C eſt, vt DE ad EF; ergo KN ad <lb></lb>C minorem habet proportionem; quàm DE ad EF. <emph type="italics"></emph>Quo <lb></lb>niam igitur magnitudines AC,<emph.end type="italics"></emph.end> hoc eſt KN C, <emph type="italics"></emph>ſunt commenſurabi<lb></lb>les, & minorem habet proportionem A,<emph.end type="italics"></emph.end> hoc eſt kN <emph type="italics"></emph>ad C, quàm DE <lb></lb>ad EF; non æ〈que〉ponderabunt A C,<emph.end type="italics"></emph.end> hoc eſt KN C, <emph type="italics"></emph>ex distantiis<emph.end type="italics"></emph.end> <arrow.to.target n="marg61"></arrow.to.target><lb></lb><emph type="italics"></emph>DE EF, poſito quidem A,<emph.end type="italics"></emph.end> hoc eſt KN <emph type="italics"></emph>ad F, C verò ad D.<emph.end type="italics"></emph.end> & <lb></lb>vt æ〈que〉ponderent, oporter, vt in F maior ſit magnitudo, <lb></lb>quàm KN; ita vt ipſi C in D æ〈que〉ponderate poſſit. </s> <s id="N12736">Ac <lb></lb>propterea cùm ſit kH adhuc minor, quàm KN, ſi igitur <lb></lb>KH ponatur ad F, & C ad D, nullo modo æ〈que〉ponde<lb></lb>rabunt. </s> <s id="N1273E">quod tamen fieri non poteſt. </s> <s id="N12740">ſupponebatur enim eas <lb></lb>æ〈que〉ponderare. </s> <s id="N12744">Non igitur magnitudo minor, quàm tota <lb></lb>KM in F magnitudini C in D æ〈que〉ponderat. <emph type="italics"></emph>Eadem au<lb></lb>tem ratione, ne〈que〉 ſi C maior fuerit, quàm vt æ〈que〉ponderet ipſi A<emph.end type="italics"></emph.end>B, <lb></lb>hoc eſt ipſi KM. etenim grauiore <expan abbr="exiſtẽte">exiſtente</expan> C ad D, quàm KM <lb></lb>ad F. primùm auferatur ex C exceſſus, quo C grauior eſt, <lb></lb>quàm KM, ita vt æ〈que〉ponderet ipſi KM. Deinde rurſus <lb></lb>auferatur quædam magnitudo minor exceſſu, quo grauior <lb></lb>eſt C, quàm kM, ita vt æ〈que〉ponderent; reſiduum verò ſit <lb></lb>ipſi KM commenſurabile, & c. </s> <s id="N12760">ſimiliter oſtendetur <expan abbr="nullã">nullam</expan> <lb></lb>magnitudinem ipſa C minorem poſitam ad D vllo modo <lb></lb>æ〈que〉ponderare ipſi KM ad F poſitæ. </s> <s id="N1276A">Quare magnitudo <lb></lb>C ad D, kM verò ad F ę〈que〉ponderant. </s> <s id="N1276E">Vnde ſequitur ma <lb></lb>gnitudinis ex vtriſ〈que〉 magnitudinibus compoſitæ centrum <lb></lb>grauitatis eſſe punctum E. ac propterea incommenſurabiles <lb></lb>magnitudines AB C ex diſtantiijs ED EF, quæ permutatim <lb></lb>eandem habent proportionem, vt magnitudines, æ〈que〉pon<lb></lb>derare. </s> <s id="N1277A">quod demonſtrare oportebat. </s> </p> <p id="N1277C" type="margin"> <s id="N1277E"><margin.target id="marg60"></margin.target><emph type="italics"></emph>ex proxi<lb></lb>mo proble<lb></lb>mate.<emph.end type="italics"></emph.end><lb></lb>8. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N12791" type="margin"> <s id="N12793"><margin.target id="marg61"></margin.target><emph type="italics"></emph>ex præce<lb></lb>denti. <lb></lb>ex prima <lb></lb>propoſitio<lb></lb>ne.<emph.end type="italics"></emph.end></s> </p> <p id="N127A3" type="head"> <s id="N127A5">SCHOLIVM.</s> </p> <p id="N127A7" type="main"> <s id="N127A9">In demonſtratione occurrit obſeruandum, quòd ſi exceſ<lb></lb>ſus HL ita diuideret magnitudinem KM, vt reſiduum KH <lb></lb>fuerit commenſurabile ipſi C; tunc abſ〈que〉 alia conſtructio<lb></lb>ne, magnitudines commenſurabiles KH C ex diſtantijs DE <lb></lb>EF æ〈que〉ponderarent; quod fieri non poteſt. </s> <s id="N127B3">cùm minorem <pb xlink:href="077/01/074.jpg" pagenum="70"></pb>habeat proportionem KH ad C, quàm ED ad EF. <expan abbr="ſiquidẽ">ſiquidem</expan> <lb></lb>ſupponitur KM ad C ita eſſe, vt ED ad EF. Archimed es ve <lb></lb>iò, vt demonſtratio abſ〈que〉 diſtinctione ſit vniuerſalis, prę<lb></lb>cipit (exiſtente KH ipſi C commenſurabili, ſiue incommen <lb></lb>ſurabili) vt auferatur pars aliqua minor exceſſu HL, ut AL, <lb></lb>ita tamen, vt reliqua KN ſit commenſurabilis ipſi C. quod qui <lb></lb>dem fieri poſſe oſtenſum eſt in proximo problemate. </s> <s id="N127C9">ex tota <lb></lb>enim magnitudine KM partem abſcindere poſſumus, vt KN <lb></lb>minorem quidem tota KM, maiorem verò KH, quæ ipſi <lb></lb>C commenſurabilis exiſtat. </s> </p> <p id="N127D1" type="main"> <s id="N127D3">Cognita Archimedis demonſtratione de incommenſura<lb></lb>bilibus magnitudinibus, idem alio quo〈que〉 modo oſtendere <lb></lb>poſſumus, applicando nempè diuiſibilitatem, & commenſura <lb></lb>bilitatem non magnitudinibus, verùm diſtantijs. </s> <s id="N127DB">hac autem <lb></lb>priùs demonſtrata propoſitione. </s> </p> <p id="N127DF" type="head"> <s id="N127E1">PROPOSITIO.</s> </p> <p id="N127E3" type="main"> <s id="N127E5">Si commenſurabiles diſtantię maiorem habuerint pro<lb></lb>portionem, quàm magnitudines permutatim habent; vt <lb></lb>ę〈que〉ponderent, maiori opus erit longitudine, quàm ſit <lb></lb>ea, ad quam altera longitudo maiorem habet proportio<lb></lb>nem. </s> </p> <figure id="id.077.01.074.1.jpg" xlink:href="077/01/074/1.jpg"></figure> <p id="N127F2" type="main"> <s id="N127F4">Sint diſtantiæ DE EH commenſurabiles, magnitudines <lb></lb>verò ſint A C. habeatquè ED ad EH maiorem proportio<lb></lb>nem, quàm A ad C. Dico vt AC ę〈que〉ponderent, maiori opus <pb xlink:href="077/01/075.jpg" pagenum="71"></pb>eſſe longitudine, quàm ſit EH. exponatur altera magnitu<lb></lb>do G, quæ ad C eandem habeat proportionem, quàm habet <lb></lb>DE ad EH. erunt vti〈que〉 magnitudines GC inter ſe <expan abbr="commẽ">commen</expan> <lb></lb>ſurabiles. </s> <s id="N12808">Deinde fiat EK æqualis EH, exponaturquè ma<lb></lb>gnitudo L ipſi G æqualis. </s> <s id="N1280C">Quoniam igitur G ad C eſt, <lb></lb>vt DE ad EH, ob commenſurabilitatem æ〈que〉pondera bunt <arrow.to.target n="marg62"></arrow.to.target><lb></lb>G in H, & C in D. ſimiliter æ〈que〉pondera bunt magnitudi<lb></lb>nes æquales GL ex æqualibus diſtantijs EK EH. Cùm igitur <lb></lb>C in D ipſi G in H æ〈que〉ponderet; L verò in K ipſi quo<lb></lb>〈que〉 G in H æ〈que〉ponderet; eandem habebit grauitatem C <arrow.to.target n="marg63"></arrow.to.target><lb></lb>in D, ut L in K. Quoniam autem maiorem habet propor<lb></lb>tionem DE ad EH, quàm A ad C, & vt DE ad EH, ita eſt <lb></lb>G ad C; maiorem habebit proportionem G ad C, quàm A <lb></lb>ad C. ergo maior eſt G, quàm A. ac propterea magnitudo A <arrow.to.target n="marg64"></arrow.to.target><lb></lb>minor eſt magnitudine L. poſita igitur magnitudine L in K, <lb></lb>& A in H, non æ〈que〉pondera bunt; & vt ę〈que〉ponderent, o<lb></lb>portet, vt A in longiori ſit diſtantia, quàm ſit EH: Inęqualia <lb></lb>enim grauia LA ex inęqualibus diſtantijs ę〈que〉ponderant, <arrow.to.target n="marg65"></arrow.to.target><lb></lb>maius quidem L in minori diſtantia EK, minus verò graue <lb></lb>A in maiori, quàm ſit EK, hoc eſt in maiori, quàm ſit EH. <lb></lb>Ita〈que〉 cùm ſit C in D æ〈que〉grauis, vt L in k; longitudo, <lb></lb>quæ efficit, vt A æ〈que〉ponderetipſi L in K; eadem prorſus <lb></lb>efficiet, vt A ipſi C in D ę〈que〉ponderare poſſit. </s> <s id="N1283E">A verò in <lb></lb>maiori diſtantia, quàm EH, ipſi L in K ę〈que〉ponderat; ergo <lb></lb>in maiori diſtantia, quàm EH, magnitudo A ipſi C in D <lb></lb>ę〈que〉ponderabit. </s> <s id="N12846">quod demonſtrare oportebat. </s> </p> <p id="N12848" type="margin"> <s id="N1284A"><margin.target id="marg62"></margin.target>6. <emph type="italics"></emph>buius.<emph.end type="italics"></emph.end></s> </p> <p id="N12853" type="margin"> <s id="N12855"><margin.target id="marg63"></margin.target><emph type="italics"></emph><expan abbr="cõmunis">communis</expan> no <lb></lb>tio ſupradi <lb></lb>cta.<emph.end type="italics"></emph.end></s> </p> <p id="N12864" type="margin"> <s id="N12866"><margin.target id="marg64"></margin.target>10. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N1286F" type="margin"> <s id="N12871"><margin.target id="marg65"></margin.target>3. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N1287A" type="main"> <s id="N1287C">Hoc demonſtrato Archimedis propoſitionem de incom<lb></lb>menſurabilibus magnitudinibus aliter oſtendemus hoc <lb></lb>pacto. </s> </p> <p id="N12882" type="head"> <s id="N12884">ALITER.</s> </p> <p id="N12886" type="main"> <s id="N12888">Incommenſurabiles magnitudines ex diſtantijs permuta<lb></lb>tim eandem, at〈que〉 magnitudines, proportionem habenti<lb></lb>bus; ę〈que〉ponderant. </s> </p> <pb xlink:href="077/01/076.jpg" pagenum="72"></pb> <p id="N12891" type="main"> <s id="N12893">Sint incom<lb></lb> <arrow.to.target n="fig30"></arrow.to.target><lb></lb><expan abbr="mẽſurabiles">menſurabiles</expan> ma <lb></lb>gnitudines AC, <lb></lb>diſtantiæ verò <lb></lb>DE EF. ſitquè vt <lb></lb>A ad C, ita DE <lb></lb>ad EF. Dico A <lb></lb>in F, C verò in <lb></lb>D æ〈que〉ponde<lb></lb>rare. </s> <s id="N128AF">Si autem (ſi fieri poteſt) non æ〈que〉pondera bunt; <expan abbr="diſtã">diſtam</expan> <lb></lb>tiæ DE EF aliter ſeſe habere debebunt, vt magnitudines AC <lb></lb>ę〈que〉ponderent. </s> <s id="N128B9">Quocirca vel longior eſt EF, quàm opus <lb></lb>ſit, vel longior eſt ED. ſit EF longior. </s> <s id="N128BD">ſitquè exceſſus GF, ita <lb></lb>vt poſita magnitudine A in G ipſi C in D æ〈que〉ponde<lb></lb> <arrow.to.target n="marg66"></arrow.to.target> ret. </s> <s id="N128C7">Fiat EH maior EG, minor verò EF. ſit autem EH <lb></lb>ipſi ED commenſurabilis. </s> <s id="N128CB">Quoniam igitur DE ad EH <lb></lb>maiorem habet proportionem, quàm ad EF; & vt DE ad <lb></lb>EF, ita eſt A ad C; maiorem habebit proportionem DE <lb></lb>ad EH, quàm A ad C. ſuntquè longitudines ED EH in<lb></lb>terſe commenſurabiles; ergo magnitudo A in H ipſi C in <lb></lb> <arrow.to.target n="marg67"></arrow.to.target> D non æ〈que〉ponderabit, ſed vt ę〈que〉ponderet, maiori opus <lb></lb>eſt longitudine, quàm ſit EH; ita vt A ipſi C in D æ〈que〉 <lb></lb>ponderare poſſit. </s> <s id="N128DF">at〈que〉 adeò cùm adhuc minor ſit EG, quàm <lb></lb>EH; magnitudo A in G magnitudini C in D nullo modo <lb></lb>æ〈que〉ponderabit. </s> <s id="N128E5">quod fieri non poteſt. </s> <s id="N128E7">ſupponebatur enim <lb></lb>A in G, & C in D ę〈que〉ponderare. </s> <s id="N128EB">eademquè prorſus ra<lb></lb>tione, ſi ED longior fuerit, quàm opus ſit, ita vt magnitu<lb></lb>dines æ〈que〉ponderent, oſtendetur <expan abbr="magnitudinẽ">magnitudinem</expan> C nullo pa<lb></lb>cto æ〈que〉ponderare poſſe ipſi A in F in minori diſtantia, <lb></lb>quàm DE. Quare magnitudines in commenſurabiles AC ex <lb></lb>diſtantijs ED EF, quæ eandem permutatim habent propor<lb></lb>tionem, vt magnitudines, æ〈que〉ponderant. </s> <s id="N128FD">quod demonſtra<lb></lb>re oportebat. </s> </p> <p id="N12901" type="margin"> <s id="N12903"><margin.target id="marg66"></margin.target><emph type="italics"></emph>problema <lb></lb>ante<emph.end type="italics"></emph.end> 7. <emph type="italics"></emph>bu<lb></lb>ius<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>quinti<emph.end type="italics"></emph.end></s> </p> <p id="N1291B" type="margin"> <s id="N1291D"><margin.target id="marg67"></margin.target><emph type="italics"></emph>ex pxima <lb></lb>ppoſitione<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.076.1.jpg" xlink:href="077/01/076/1.jpg"></figure> <p id="N1292B" type="main"> <s id="N1292D">In prioribus ſermonibus ante quintam propoſitionem ha<lb></lb>bitis, diximus propoſitionum præcedentium demonſtratio<lb></lb>nes planiores euadere, ſi intelligamus magnitudines eiuſdem <lb></lb>eſſe ſpeciei, & homogeneas. </s> <s id="N12935">Quòd quidem ſi Archimedem <pb xlink:href="077/01/077.jpg" pagenum="73"></pb>his, vel de rectilineis tantùm demonſtrationes attuliſſe (vt <expan abbr="nõ-nulli">non<lb></lb>nulli</expan> fortaſſe falsò exiſtimarunt) intelligeremus; ita vt ex Ar<lb></lb>chimedis demonſtrationibus non ſit adhuc vniuerſaliter de<lb></lb>monſtratum hoc pręcipuum fundamentum; nempè magni<lb></lb>tudines ex diſtantijs permutatim <expan abbr="proportionẽ">proportionem</expan> habentibus, vt <lb></lb>ipſarum grauitates, ę〈que〉ponderare; in hoc certè rationes ab <lb></lb>Archimede allatas, ipſarum què demonſtrationum vim mini<lb></lb>mè percipiemus. </s> <s id="N12951">Quapropter ea, quæ demonſtrauit, omni<lb></lb>bus magnitudinibus vniuerſaliter competere ipſum voluiſſe <lb></lb>nullatenus eſt dubitandum. </s> <s id="N12957">Ne〈que〉 enim, vt perfectè, & vni<lb></lb>uerſaliterſciamus, magnitudines ç〈que〉ponderare ex diſtantijs <lb></lb>permutatim proportionem habentibus, vt ipſarum grauita<lb></lb>tes, alijs, quàm pręcedentibus propoſitionibus indigemus. <lb></lb>In hoc enim fundamento demonſtrando minimè diminu<lb></lb>tus extitit Archimede. </s> <s id="N12963">Nam ſi ad propoſitiones ab ipſo alla<lb></lb>tas, pręcipuèquè ad vim demonſtrationum reſpiciamus, ſiuè <lb></lb>magnitudines intelligantur eiuldem ſpeciei, ſiue diuerſę, ſi<lb></lb>ue homogeneę, ſiue heterogeneę, ſiue planę, ſiue ſolidę, & <lb></lb>hę quidem, ſiue rectilineę, ſiue quom odocun〈que〉 mixtę; ni<lb></lb>hilominus demonſtrationes idem prorſus concludent, ita vt <lb></lb>Archimedes non de aliquibus magnitudimbus tantùm de<lb></lb>monſtrationes attulerit; ſed de omnibus prorſus demonſtra<lb></lb>uerit. </s> <s id="N12975">In his enim Archimedes non ad magnitudines tantùm, <lb></lb>verùm ad magnitudinum grauitates potiſſimùm reſpexit. <lb></lb>quandoquidem loco grauium magnitudines nominat; vt <lb></lb>poſt quartam huius propoſitionem adnotauimus. </s> <s id="N1297D">quod qui<lb></lb>dem facilè ex verbis ipſius rectè intellectis apparere poteſt. <expan abbr="Nã">Nam</expan> <lb></lb>in quærta propoſitione cùm inquit, <emph type="italics"></emph>ſi duæ fuerint magnitudines <lb></lb>æquales<emph.end type="italics"></emph.end>, vt antea diximus, intelligendum eſt eas ęquales <lb></lb>eſſe grauitate. </s> <s id="N12991">quod non ſolùm ex eius demonſtrationeli<lb></lb>〈que〉t, verùm etiam ex modo lo〈que〉ndi, quo vſus eſt Archime<lb></lb>des in alijs propoſitionibus. </s> <s id="N12997">In quinta enim propoſitione, <lb></lb>quę eiuſdem eſt cum quarta ordinis, & naturę, in quit; <lb></lb><emph type="italics"></emph>Sitrium magnitudinum centra grauitatis in recta linea fuerint poſi<lb></lb>ta, & magnitudines æqualem habuerint grauitatem.<emph.end type="italics"></emph.end> ſimlli<lb></lb>ter poſt quintam demonſtrationem bis quoquè eodem v<lb></lb>titur lo〈que〉ndi modo, nempè cùm adhuc proponit <pb xlink:href="077/01/078.jpg" pagenum="74"></pb>plures magnitudines, inquit, <emph type="italics"></emph>& magnitudines æqualem habuerint <lb></lb>grauitatem.<emph.end type="italics"></emph.end> ex quibus conſtat Archimedem ad magnitudinum <lb></lb>grauitates omnino reſpexiſſe. </s> <s id="N129B6">ita vt quando Archimedes in<lb></lb>quit, <emph type="italics"></emph>& magnitudines æquales<emph.end type="italics"></emph.end>, idem eſt, ac ſi dixiſſet, <emph type="italics"></emph>& magnitu<lb></lb>dines æqualem habuerint grauitatem.<emph.end type="italics"></emph.end> Præterea in ſexta propoſitio <lb></lb>ne inquit magnitudines ę〈que〉ponderare ex diſtantijs permu<lb></lb>tàtim proportionem habentibus, vt grauitates. </s> <s id="N129CC">ita ut cauſa <lb></lb>huius æ〈que〉ponderationis ſit (vt reuera eſt) magnitudinum <lb></lb>grauitas. </s> <s id="N129D2">& <expan abbr="quãquam">quanquam</expan> in hac ſeptima propoſitione dicat, ma <lb></lb>gnitudines æ〈que〉ponderare ex diſtantijs permutatim propor<lb></lb>tionem habentibus, vt magnitudines, & non dixit, vt grauita <lb></lb>tes; intelligendum tamen eſt, ac ſi dixiſſet, eas ę〈que〉pondera<lb></lb>re, vt magnitudinum grauitates. </s> <s id="N129E0">hęc enim ſeptima propoſi<lb></lb>tio eſt pars ſextæ propoſitionis, vt iam pręfati fum^{9}; vnde ſi in <lb></lb>ſexta magnitudines ę〈que〉ponderant ob earum grauitatem, ob <lb></lb>eandem quo〈que〉 cauſam & in hac ſeptima æ〈que〉ponderare de <lb></lb>bent. </s> <s id="N129EA">Pręterea in ſe〈que〉nti etiam propoſitione dum proponit <lb></lb>oſtendere quam proportionem habere debent ſectiones lineę <lb></lb>intercentra grauitatum diuiſę magnitudinis <expan abbr="exiſtẽtes">exiſtentes</expan>, inquit, <lb></lb><emph type="italics"></emph>quam habet grauitas magnitudinis ablatæ ad grauitatem reſiduæ<emph.end type="italics"></emph.end> hoc <lb></lb>autem deinceps exponens, <expan abbr="nõ">non</expan> inquit oportere ſectiones lineæ <lb></lb>eam habere proportionem, quàm grauitas ad grauitatem ha<lb></lb>bet; ſed horum loco inquit, quàm magnitudo ad magnitudi <lb></lb>nem. </s> <s id="N12A07">ex quibus omnibus clarè perſpicitur, quòd quando Ar<lb></lb>chimedes magnitudines nominat, omnino magnitudinum <lb></lb>grauitates vult intelligere. </s> </p> <p id="N12A0D" type="main"> <s id="N12A0F">Ad eorum autem <expan abbr="intelligentiã">intelligentiam</expan>, quę dicta ſunt in ſexta, ſepti <lb></lb>maquè propoſitione, <expan abbr="earũquè">earunquè</expan> <expan abbr="demõſtrationibus">demonſtrationibus</expan>, <expan abbr="obſeruandũ">obſeruandum</expan> <lb></lb>eſt, quòd in ſexta propoſitione pro magnitudinibus commen <lb></lb>ſurabilibus intelligere oportet magnitudines grauitate com<lb></lb>menſurabiles; ita nempe, vt numeris exprimi poſſint; quam<lb></lb>quam non ſint mole, & magnitudine commenſurabiles, vt <lb></lb>in figura ſextę propoſitionis magnitudo A ponderet exempli <lb></lb>gratia vt XVI. B verò vt VIII. <expan abbr="intelligaturq́">intelligatur〈que〉</expan>; F <expan abbr="magnitudinũ">magnitudinum</expan> <pb xlink:href="077/01/079.jpg" pagenum="75"></pb>AB <expan abbr="cõmunis">communis</expan> menſura in grauitate, ita vt ſit æ〈que〉grauis vni<lb></lb>cui〈que〉 parti OPQR, quæ quidem, & ſi non ſint magnitu<lb></lb>dine inter ſe ęquales, ſufficit, vt ſint æ〈que〉graues: veluti magni<lb></lb> <arrow.to.target n="fig31"></arrow.to.target><lb></lb>tudines quo〈que〉 STVX inter ſe, <expan abbr="ipſisq́">ipſis〈que〉</expan>; OPQR tantùm ę〈que〉 <lb></lb>graues; ita ut vnaquæ〈que〉 ponderet, vt IIII. veluti etiam par <lb></lb>tes ipſius B, & vnaquæ〈que〉 ZM. hiſquè ita poſitis <expan abbr="demõſtra">demonſtra</expan> <lb></lb>tio rectè concludet. </s> </p> <figure id="id.077.01.079.1.jpg" xlink:href="077/01/079/1.jpg"></figure> <p id="N12A5C" type="main"> <s id="N12A5E">In hacverò ſeptima Archimedis propoſitione ſimiliter <arrow.to.target n="marg68"></arrow.to.target> in<lb></lb>telligantur magnitudines kMC incommenſurabiles graui<lb></lb>tate, vt in eius figura grauitas ipſius C ponderet, vt XII. gra<lb></lb>uitas verò ipſius KM maior ſit, quàm XX. ita vthę graui<lb></lb>tates ſint in commenſurabiles. </s> <s id="N12A6C">auferaturquè grauitas exceſſus <lb></lb>HL, quæ ſit vt IIII. ita vt quæ relinquiturgrauitas, ipſius <expan abbr="nẽ-pè">nen<lb></lb>pè</expan> KH, quę quidem maior eſt, quàm XVI, in F poſita, gra<lb></lb>uitati ipſius C, quæ eſt XII, in D poſitæ æ〈que〉ponderet, <lb></lb>Auferatur deinde NL minor exceſſu HL; cuius quidem gra<lb></lb>uitas ſit maior, quàm II. ita vt grauitas reſidui KN, quæ <lb></lb>nimirum ſit XVIII, ſit commenſurabilis grauitati <lb></lb>XII. ipſius C. & <expan abbr="quãuis">quamuis</expan> magnitudines KM C, & KN C ſint, <lb></lb>vel <expan abbr="nõ">non</expan> ſint inter ſe magnitudine <expan abbr="cõmenſurabiles">commenſurabiles</expan>, vel incom <pb xlink:href="077/01/080.jpg" pagenum="76"></pb>menſurabiles; eadem prorſus demonſtratio idem concludet. <lb></lb>quæ quidem omnia in ſe〈que〉nti quo〈que〉 propoſitione <expan abbr="conſi-derãda">conſi<lb></lb>deranda</expan> occurrunt. </s> <s id="N12A9A">Vnde perſpicuum eſt has Archime dis pro <lb></lb>poſitiones, ac demonſtrationes vniuerſaliſſimas eſſe, ar〈que〉 o<lb></lb>mnibus, & quibuſcun〈que〉 magnitudinibus conuenientes. </s> </p> <p id="N12AA0" type="margin"> <s id="N12AA2"><margin.target id="marg68"></margin.target><emph type="italics"></emph>reſpice <expan abbr="fi-gurã">fi<lb></lb>guram</expan> ſepti<lb></lb>mæ propoſi <lb></lb>tionis Ar<lb></lb>chimedis.<emph.end type="italics"></emph.end></s> </p> <p id="N12AB6" type="main"> <s id="N12AB8">Iacto hoc pręcipuo, ac pręſtantiſſimo mechanico funda<lb></lb>mento; in ſe〈que〉nti propoſitione colligit ex hoc Archimedes, <lb></lb>quomodo ſe habent centra grauitatis magnitudinis diuiſæ. </s> </p> <p id="N12ABE" type="head"> <s id="N12AC0">PROPOSITIO. VIII.</s> </p> <p id="N12AC2" type="main"> <s id="N12AC4">Si ab aliqua magnitudine magnitudo aufera<lb></lb>tur; quæ non habeat idem centrum cum tota; re<lb></lb>liquæ magnitudinis centrum grauitatis eſt in re<lb></lb>cta linea, quæ coniungit centra grauitatum to tius <lb></lb>magnitudinis, & ablatæ, ad eam partem produ<lb></lb>cta, vbi eſt centrum to tius magnitudinis, ita vt aſ<lb></lb>ſumpta aliqua ex producta, quæ coniungit <expan abbr="cẽtra">centra</expan> <lb></lb>prædicta eandem habeat proportionem ad eam, <lb></lb>quæ eſt inter centra, quam habet grauitas magni<lb></lb>tudinis ablatæ ad grauitatem reſiduæ, centrum e<lb></lb>rit terminus aſſumptæ. </s> </p> <p id="N12ADE" type="main"> <s id="N12AE0"><emph type="italics"></emph>Sit alicuius magnitudinis AB centrum grauitatis C. auferatur<lb></lb>què ex AB magnitudo AD; cuius centrum grauitatis ſit E. coniuncta <lb></lb>verò EC, &<emph.end type="italics"></emph.end> ex parte C <emph type="italics"></emph>producta, aſſumatur CF, quæ ad CE <expan abbr="eã">eam</expan> <lb></lb>dem habeat proportionem, quam habet magnitudo AD ad DG. osten<lb></lb>dendum est, magnitudinis DG centrumgrauitatis eſſe punctum F. <expan abbr="Nõ">non</expan> <lb></lb>ſit autem; ſed, ſi fieri potest, ſit punctum H. Quoniam igitur magnitudi<lb></lb>nis AD centrum grauitatis est punctum E; magnitudinis verò DG <lb></lb>eſt punctum H; magnitudinis ex vtriſ〈que〉 magnitudinibus AD DG,<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg69"></arrow.to.target> <emph type="italics"></emph>compoſitæ centrum grauitatis erit in linea EH, ita diuiſa, ut pirtes ipſius <lb></lb>permutatim eandem <expan abbr="habeãt">habeant</expan> proportionem, vt magnitudines. </s> <s id="N12B11">Quare non<emph.end type="italics"></emph.end> <pb xlink:href="077/01/081.jpg" pagenum="77"></pb><emph type="italics"></emph>erit punctum C ſecundùm diuiſionem proportione reſpondentem prædi<lb></lb>etæ.<emph.end type="italics"></emph.end> vt ſcilicet ſit HC ad CE, vt AD ad DG. etenim ut AD <lb></lb>ad DG; ita <expan abbr="factũ">factum</expan> fuit FC ad CE. ſi igitur ſecetur linea EH ſe <lb></lb>cundùm proportionem ipſius AD ad DG; non terminabit <lb></lb> <arrow.to.target n="fig32"></arrow.to.target><lb></lb>diuiſio ad punctum C. cùm ſit impoſſibile eandem habere <lb></lb>proportionem FC ad CE, quam. </s> <s id="N12B32">HC ad eandem CE. di<lb></lb>uiſio igitur ad aliud terminabitur punctum, vt K; ita vt HK <arrow.to.target n="marg70"></arrow.to.target><lb></lb>ad KE ſit, vt AD ad DG. vnde ſequitur punctum K cen<lb></lb>trum eſſe grauitatis magnitudinis ex AD DG compoſitæ. <lb></lb><emph type="italics"></emph>Non eſt igitur punctum C centrum magnitudinis ex AD DG compo <lb></lb>ſitæ; hoc est ipſius AB. eſt autem; ſuppoſitum eſt enim<emph.end type="italics"></emph.end> ipſum eſſe. <emph type="italics"></emph>er<lb></lb>go ne〈que〉 punctum H centrum est grauitatis magnitudinis DG.<emph.end type="italics"></emph.end> eſt <lb></lb>igitur punctum F; quod quidem eſt terminus productę lineę <lb></lb>CF; quæ eandam habet proportionem ad lineam CE inter <lb></lb>centra exiſtentem; quam habet grauitas magnitudinis AD <lb></lb>ad grauitatem ipſius DG. quod demonſtrare oportebat. </s> </p> <p id="N12B56" type="margin"> <s id="N12B58"><margin.target id="marg69"></margin.target><emph type="italics"></emph>ex præce<lb></lb>dentibus.<emph.end type="italics"></emph.end></s> </p> <p id="N12B62" type="margin"> <s id="N12B64"><margin.target id="marg70"></margin.target><emph type="italics"></emph>ex præce<lb></lb>dentibus.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.081.1.jpg" xlink:href="077/01/081/1.jpg"></figure> <p id="N12B72" type="head"> <s id="N12B74">SCHOLIVM.</s> </p> <p id="N12B76" type="main"> <s id="N12B78">In hac demonſtratione intelligendum eſt etiam punctum <lb></lb>H eſſe poſſe extra lineam EF, ita vt EFH non ſitirecta linea. <lb></lb>quòd ſi H non eſſet in linea EF, idem ſequi abſurdum adeò <lb></lb>perſpicuum eſt; vt nec demonſtratione egeat. </s> <s id="N12B80">Quoniam ſi in<lb></lb>telligatur H extra lineam EF; iuncta EH, & ita diuiſa intel<lb></lb>ligatur, vt ipſius partes permutatim grauitatibus magnitudi<lb></lb>num AD DG reſpondeant; eſſet vti〈que〉 hoc punctum <expan abbr="inuẽ-tum">inuen<lb></lb>tum</expan>, quod extra lineam EF reperiretur, centrum grauitatis to <pb xlink:href="077/01/082.jpg" pagenum="78"></pb>tius AB quod fieri non poteſt. </s> <s id="N12B92">ſiquidem eſt punctum C, vt <lb></lb>ſuppoſitum fuit. </s> <s id="N12B96">Vnde ne〈que〉 illud punctum H ipſius DG <expan abbr="cẽ">cem</expan> <lb></lb>trum grauitatis exiſteret. </s> </p> <p id="N12B9E" type="main"> <s id="N12BA0">Hic eſt terminus primę partis principalis, in qua Archime <lb></lb>des (vt initio dixim^{9}) de magnitudinib^{9}, & degrauibus in <lb></lb>communi pertractauit; quandoquidem propoſitiones, ac de<lb></lb>monſtrationes tam planis, quàm ſolidis quibuſcun〈que〉 ſunt <lb></lb>accomodatæ; vt manifeſtum fecimus. </s> </p> <p id="N12BAA" type="main"> <s id="N12BAC">Nunc ita 〈que〉 ſe conuertit Archimedes ad <expan abbr="inueſtigandũ">inueſtigandum</expan> cen<lb></lb>tra grauitatis planorum. </s> <s id="N12BB4">primùm què perquirit centrum gra<lb></lb>uitatis parallelogrammorum; oſtendetquè centrum grauitatis <lb></lb>cuiuſlibet parallelogrammi eſſe in recta linea, quæ coniungit <lb></lb>oppoſita latera bifariam diuiſa. </s> <s id="N12BBC">ob cuius intelligentiam hæc <lb></lb>priùs lemmata in vnum collecta nouiſſe erit valdè vtile. </s> </p> <p id="N12BC0" type="head"> <s id="N12BC2">LEMMA.</s> </p> <p id="N12BC4" type="main"> <s id="N12BC6">Sit parallelogrammum ABCD, cuius oppoſita latera AB <lb></lb>CD ſint bifariam diuiſa in EF. connectaturquè EF, quæ ni <lb></lb>mirum æquidiſtans erit ipſis AC BD. Deinde diuidatur v<lb></lb> <arrow.to.target n="fig33"></arrow.to.target><lb></lb>naquæ〈que〉 AE EB in partes numero pares, & inuicem ęqua <lb></lb>les; vt in AG GE; & EH HB. <expan abbr="ducãturquè">ducanturquè</expan> GK HL ipſi <lb></lb>EF ęquidiſtantes. </s> <s id="N12BDB">ſit verò centrum grauitatis ipſius AK pun<lb></lb>ctum M. ipfius verò GF punctum N, & ipſius EL pun<lb></lb>ctum O deniquè ipſius HD punctum P. Dico primùm <expan abbr="pũ">pum</expan> <lb></lb>cta MNOP eſſe in linea recta. </s> <s id="N12BE7">deinde lineas MN NO OP <lb></lb>inter centra exiſtentes inter ſe æquales eſſe. </s> <s id="N12BEB">Deni〈que〉 centrum <lb></lb>grauitatis parallelogrammi AD eſſe in linea NO, quę con <lb></lb>iungit centra grauitatis ſpatiorum mediorum; parallelogram <lb></lb>morum ſcilicet GF EL. <pb xlink:href="077/01/083.jpg" pagenum="79"></pb>Ducantur à punctis MN ipſi AGE ęquidiſtantes QMR <lb></lb>SNT. erunt vti〈que〉 AQRG, & GSTE parallelogramma. <lb></lb>Quoniam igitur parallelogramma AK GF in æqualibus <lb></lb>ſuntbaſibus AG GE, & in ijſdem parallelis; erunt AK GF <arrow.to.target n="marg71"></arrow.to.target><lb></lb>inter ſe ęqualia. </s> <s id="N12C02">& quoniam AC GK EF ſunt <expan abbr="ęquidiſtãtes">ęquidiſtantes</expan>; <lb></lb>erit angulus CAG ipſi KGE ęqualis, & KGA ipſi FEG <arrow.to.target n="marg72"></arrow.to.target><lb></lb>æqualis; & horum oppoſiti inter ſe ſunt ęquales; ergo <arrow.to.target n="marg73"></arrow.to.target> paralle<lb></lb>logrammum GF ipſi AK ęquale, & ſimile exiſtit. </s> <s id="N12C15">Ita〈que〉 <lb></lb>ſi GF colloceturſuper AK, rectè congruet: eruntquè paral<lb></lb>lelogramma inuicen coaptata. </s> <s id="N12C1B">lineęquè GE AG, GK AC, & <lb></lb>reliquæ coaptatæ erunt. </s> <s id="N12C1F">quare eorum centra grauitatis <arrow.to.target n="marg74"></arrow.to.target> inui<lb></lb>cem coaptata erunt. </s> <s id="N12C27">hoc eſt N erit in puncto M. Quoniam <lb></lb>autem à punctis MN (quod nunc intelligitur vnum tantum <lb></lb>eſſe punctum) ductæ fuerunt ST QR ipſi AGE æquidi<lb></lb>ſtantes, linea ST coaptabitur cum QR, quippe cùm ambæ <lb></lb>hæ lineæ ab vno puncto prodeuntes ipſi AG ęquidiſtantes <lb></lb>eſſe debeant. </s> <s id="N12C33">punctum igitur S in Q, & T in R coaptabi<lb></lb>tur. </s> <s id="N12C37">eritquè QM ipſi SN ęqualis, & MR ipſi NT. ac pro <lb></lb>pterea linea GS parallelogrammi GT erit coaptata in <expan abbr="Aq;">A〈que〉</expan> <lb></lb>& ET coaptata erit in GR parallelogrammi AR. Vnde e<lb></lb>rit AQ ęqualis GS, cùm ſint coaptatæ; & GR ipſi ET ę<lb></lb>qualis; cùm ſint quo〈que〉 coaptatę. </s> <s id="N12C45">Quocirca quoniam <arrow.to.target n="marg75"></arrow.to.target> pa<lb></lb>rallelogramma AR GT ſunt inuicem coaptata, paral<lb></lb>lelogrammorumquè oppoſita latera ſunt inter ſe ęqualia, <expan abbr="erũt">erunt</expan> <lb></lb>AQ GS GR ET inter ſe ęqualia. </s> <s id="N12C55">Nunc autem <expan abbr="intelligãtur">intelligantur</expan> <lb></lb>parallelogramma AK GF non ampliùs coaptata. </s> <s id="N12C5D">& <expan abbr="quoniã">quoniam</expan> <lb></lb>lineę QMR, & SNT ſuntipſi AGE parallelę; & AQ GR, <lb></lb>GS ET, inter ſe ſuntæquales, & ęquidiſtantes; puncta RS in <lb></lb>vnum coincident punctum. </s> <s id="N12C69">eritquè QST linea recta. </s> <s id="N12C6B">ex qui <lb></lb>bus patet, rectam <expan abbr="lineã">lineam</expan>, quæ coniungit centra grauitatis MN <lb></lb>ipſi AGE æquidiſtantem exiſtere. </s> <s id="N12C75">eodemquè modo oſtende<lb></lb>tur rectas lineas, quæ coniungunt grauitatis centra NO, cen<lb></lb>traquè OP, ipſi AB <expan abbr="æquidiſtãtes">æquidiſtantes</expan> eſſe. </s> <s id="N12C7F">Vnde ſequitur lineam <lb></lb>MNOP rectam eſſe. </s> <s id="N12C83">Quare primùm conſtat grauitatis <expan abbr="cẽtra">centra</expan> <lb></lb>in recta linea exiſtere. </s> </p> <p id="N12C8B" type="margin"> <s id="N12C8D"><margin.target id="marg71"></margin.target>36. <emph type="italics"></emph>primi.<emph.end type="italics"></emph.end></s> </p> <p id="N12C96" type="margin"> <s id="N12C98"><margin.target id="marg72"></margin.target>29. <emph type="italics"></emph>primi.<emph.end type="italics"></emph.end></s> </p> <p id="N12CA1" type="margin"> <s id="N12CA3"><margin.target id="marg73"></margin.target>34. <emph type="italics"></emph>primi.<emph.end type="italics"></emph.end></s> </p> <p id="N12CAC" type="margin"> <s id="N12CAE"><margin.target id="marg74"></margin.target>5. <emph type="italics"></emph>post, hu<lb></lb>ius.<emph.end type="italics"></emph.end></s> </p> <p id="N12CB9" type="margin"> <s id="N12CBB"><margin.target id="marg75"></margin.target>34. <emph type="italics"></emph>primi.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.083.1.jpg" xlink:href="077/01/083/1.jpg"></figure> <p id="N12CC8" type="main"> <s id="N12CCA">Quoniam autem oſtenſum eſt QM æqualem eſſe ipſi SN, <lb></lb>& MR ipſi NT, eodem quo〈que〉 modo oſtendetur OT ęqua- <pb xlink:href="077/01/084.jpg" pagenum="80"></pb>lem eſſe ipſi SN. Quoniam igitur OT NS ſunt ęquales, iti<lb></lb>demquè TN SM æquales, erit ON ipſi NM æqualis. </s> <s id="N12CD4">ea<lb></lb>demquè ratione oſtendetur OP ęqualem eſſe ipſi ON. vn<lb></lb>de colligitur lineas MN NO OP inter centra exiſtentes in<lb></lb>rerſe ęquales eſſe. </s> </p> <p id="N12CDC" type="main"> <s id="N12CDE">Poſtremò quoniam parallelogramma AK GF EL HD <lb></lb>ſunt inuicem æqualia, & numero paria, centraquè grauitatis <lb></lb>ſunt in recta linea poſita. </s> <s id="N12CE4">lineęquè MN NO OP inter cen<lb></lb>tra ſunt ęquales, magnitudinis ex omnibus AK GF EL HD <lb></lb> <arrow.to.target n="marg76"></arrow.to.target> magnitudinibus compoſitæ centrum grauitatis eſt in linea <lb></lb>MP bifariam diuiſa. </s> <s id="N12CF0">Et quoniam MN eſt æqualis ipſi OP, <lb></lb>punctum, quod bifariam diuidit MP cadet in linea NO. <lb></lb>centrum ergo grauitatis omnium magnitudinum AK GF <lb></lb>EL HD, hoc eſt parallelogrammi AD eſt in linea NO, quę <lb></lb>coniungit centra ſpatiorum mediorum GF EL. quę <expan abbr="quidẽ">quidem</expan> <lb></lb>omnia oſtendere oportebat. </s> </p> <p id="N12D00" type="margin"> <s id="N12D02"><margin.target id="marg76"></margin.target>2.<emph type="italics"></emph>cor. </s> <s id="N12D09">quin<lb></lb>tæ huius.<emph.end type="italics"></emph.end></s> </p> <p id="N12D0F" type="main"> <s id="N12D11">Quoniam autem centrum grauitatis <expan abbr="parallelogrãmi">parallelogrammi</expan> AD <lb></lb>eſt in linea NO, & in linea MP bifariam diuiſa; non repu<lb></lb>gnare videtur, quin inferri poſſit, hoc centrum eſſe in puncto <lb></lb>T, in linea EF exiſtente. </s> <s id="N12D1D">Quòd tamen falſum eſt. </s> <s id="N12D1F">nam poſ <lb></lb>ſet quidem concludi centru eſſe in medio lineę NO (<expan abbr="ſiquidẽ">ſiquidem</expan> <lb></lb>eſt in medio lineę MP, vt <expan abbr="dictũ">dictum</expan> eſt) ſed <expan abbr="nõ">non</expan> in <expan abbr="pũcto">puncto</expan> T; ex <expan abbr="demõ">demom</expan> <lb></lb>ſtratione enim oſtenditur NS æqualem eſſe ipſi TO. at verò <lb></lb>NT ęqualem eſſe ipſi TO, nullo modo demonſtrari poteſt; <lb></lb>niſi ſupponeremus centra grauitatis MNOP in parallelogra <lb></lb>mis ita ſe habere, vt MQ MR, & MR RN, & RN NT & <lb></lb>NT TO, &c. </s> <s id="N12D43">inter ſe ęquales eſſent. </s> <s id="N12D45">quod nullo modo ſup<lb></lb>poni poteſt nam hoc modo centra grauitatis parallelogram<lb></lb>morum AK GF &c. </s> <s id="N12D4B">eſſent in lineis, quę bifariam ſecant op <lb></lb>poſita latera. </s> <s id="N12D4F">eſſent quippè in lineis à punctis MN OP du<lb></lb>ctisipſis AC GK EF &c. </s> <s id="N12D53">æquidiftantibus, quæ oppoſita la <lb></lb>tera AG CK, GE KF, EH FL, &c. </s> <s id="N12D57">bifariam ſecarent. </s> <s id="N12D59">quod <lb></lb>eſt id, quod Archimedes demonſtrare in <expan abbr="ſe〈quẽ〉ti">ſe〈que〉nti</expan> nititur. </s> <s id="N12D61">quod <lb></lb>quidem in cauſa eſt, vt demonſtratione ad impoſſibile id de<lb></lb>ducat. </s> <s id="N12D67">ſuppoſuimus autem (vt pareſt) parallelogramma cen- <pb xlink:href="077/01/085.jpg" pagenum="81"></pb>tra grauitatis habere; ac centra grauitatis MNOP intra pa<lb></lb>rallelogramma exiſtere, quoniam parallelogramma ſunt <arrow.to.target n="marg77"></arrow.to.target> fi<lb></lb>guræ ad eaſdem partes concauæ. </s> <s id="N12D75">quod quidem eodem modo <lb></lb>ab Archimede in ſe〈que〉nti ſupponitur. </s> </p> <p id="N12D79" type="margin"> <s id="N12D7B"><margin.target id="marg77"></margin.target>9. <emph type="italics"></emph>poſt hu<lb></lb>ius.<emph.end type="italics"></emph.end></s> </p> <p id="N12D86" type="head"> <s id="N12D88">PROPOSITIO. IX.</s> </p> <p id="N12D8A" type="main"> <s id="N12D8C">Omnis parallelogrammi centrum grauitatis <lb></lb>eſt in recta linea, quæ oppoſita latera parallelo<lb></lb>grammi bifariam diuiſa coniungit. </s> </p> <figure id="id.077.01.085.1.jpg" xlink:href="077/01/085/1.jpg"></figure> <p id="N12D95" type="main"> <s id="N12D97"><emph type="italics"></emph>Sit parallelogrammum ABCD, linea verò EF bifariam diuidat la <lb></lb>tera AB CD. Dico parallelogrammi ABCD centrum grauitatis eſſe<emph.end type="italics"></emph.end> <arrow.to.target n="marg78"></arrow.to.target><lb></lb><emph type="italics"></emph>in linea EF. Non ſit quidem, ſed, ſi fieri poteſt, ſit H. &<emph.end type="italics"></emph.end> ab ipſo <expan abbr="vſq;">vſ〈que〉</expan> <lb></lb>ad lineam EF <emph type="italics"></emph>ducatur H<gap></gap> æquidistansipſi AB. Diuiſa verò EB <lb></lb>ſemper bifariam<emph.end type="italics"></emph.end> in G. rurſuſquè EG brfariam in K; idèquè <lb></lb>ſemper fiat, tandem <emph type="italics"></emph>quædam relin〈que〉tur linea,<emph.end type="italics"></emph.end> putà EK, <emph type="italics"></emph>minor <lb></lb>ipſa HI. Diuidaturquè vtra〈que〉 AE EB in partes<emph.end type="italics"></emph.end> AN NM ML <arrow.to.target n="marg79"></arrow.to.target><lb></lb>LE GO OB <emph type="italics"></emph>ipſi EK æquales.<emph.end type="italics"></emph.end> quod quidem fieri poteſt, quia <lb></lb>diuiſa eſt EB in partes ſemper ęquales. <emph type="italics"></emph>& ex<emph.end type="italics"></emph.end> his <emph type="italics"></emph>diuiſionum pun<lb></lb>ctis ducantur<emph.end type="italics"></emph.end> NP MQ LR kS GT OV <emph type="italics"></emph>ipſi EF æquidistantes. <lb></lb>diuiſum enim erit totum parallelogrammum in parallelogramma æqualia <lb></lb>& ſimiliaipſi<emph.end type="italics"></emph.end> k<emph type="italics"></emph>F.<emph.end type="italics"></emph.end> cùm enim ſint parallelogrammorum baſes <lb></lb>EL LM MN NA KG GO OB ipſi KE æquales, <arrow.to.target n="marg80"></arrow.to.target> parallelo<lb></lb>grammaquè in ijſdem ſint parallelis AB CD conſtituta; <lb></lb>erunt parallelogramma æqualia. </s> <s id="N12DFF">ſimilia verò, quoniam <lb></lb>ſunt ęquiangula. <emph type="italics"></emph>Parallelogrammis igitur æqualibus, at〈que〉<emph.end type="italics"></emph.end> <pb xlink:href="077/01/086.jpg" pagenum="82"></pb><emph type="italics"></emph>ſimilibus ipſi KF inuicem coaptatis, & centra grauitatis inter ſe conue<lb></lb>nient.<emph.end type="italics"></emph.end> quia verò in EB facta eſt diuiſio ſemper in duas partes <lb></lb>ęquales erunt parallelogramma in ED numero paria. </s> <s id="N12E16">ac per <lb></lb>conſe〈que〉ns & quę ſunt in EC numero paria. </s> <s id="N12E1A">vnde & quę <expan abbr="sũ">sunt</expan> <lb></lb>in toto AD numero paria <expan abbr="erũt">erunt</expan>. <emph type="italics"></emph>Jta〈que〉 quædam erunt magnitudi<lb></lb>nes æquidiſtantium laterum æquales ipſi KF numero pares,<emph.end type="italics"></emph.end> hoc eſt o<lb></lb> <arrow.to.target n="marg81"></arrow.to.target> mnes, quæ ſunt in AD, <emph type="italics"></emph>centraquè grauitatis ipſarum in recta linea<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg82"></arrow.to.target> <emph type="italics"></emph>ſunt conſtituta, & lineæ inter centra ſunt a quales magnitudinis ex ipſis <lb></lb>omnibus compoſitæ centrum grauitatis erit in recta linea, quæ coniungit <lb></lb>centra grauitatis mediorum ſpatiorum,<emph.end type="italics"></emph.end> parallelogrammorum ſcili<lb></lb>cet LF KF. <emph type="italics"></emph>Non est autem; punctum enim H,<emph.end type="italics"></emph.end> quod ſupponitur <lb></lb>eſſe centrum grauitatis omnium magnitudinum, hoc eſt pa <lb></lb>rallelogrammi AD, <emph type="italics"></emph>extra media parallelogramma<emph.end type="italics"></emph.end> LF KF <emph type="italics"></emph>exiſtit.<emph.end type="italics"></emph.end><lb></lb>etenim cùm ſit EK minor HI, linea KS ipſi EF <expan abbr="ęquidiſtãs">ęquidiſtans</expan> <lb></lb>lineam HI ipſi EK æquidiſtantem ſecabit, quippè quæ re<lb></lb>lin〈que〉t punctum H extra figuram KF, ac per conſe〈que〉ns ex<lb></lb>tra media parallelogramma LF KF. quare punctum H non <lb></lb>eſt centrum grauitatis parallelogrammi AD, vt ſupponeba<lb></lb>tur. <emph type="italics"></emph>ergo conſtat, centrum grauitatis parallelogrammi ABCD eſſe in re <lb></lb>cta linea EF.<emph.end type="italics"></emph.end> quod demonſtrare oportebat. </s> </p> <p id="N12E74" type="margin"> <s id="N12E76"><margin.target id="marg78"></margin.target>*</s> </p> <p id="N12E7A" type="margin"> <s id="N12E7C"><margin.target id="marg79"></margin.target><emph type="italics"></emph>ex prima <lb></lb>pręcedenti<emph.end type="italics"></emph.end></s> </p> <p id="N12E86" type="margin"> <s id="N12E88"><margin.target id="marg80"></margin.target>36. <emph type="italics"></emph>primi.<emph.end type="italics"></emph.end></s> </p> <p id="N12E91" type="margin"> <s id="N12E93"><margin.target id="marg81"></margin.target>*</s> </p> <p id="N12E97" type="margin"> <s id="N12E99"><margin.target id="marg82"></margin.target><emph type="italics"></emph>lemma.<emph.end type="italics"></emph.end></s> </p> <p id="N12EA1" type="head"> <s id="N12EA3">SCHOLIVM.</s> </p> <p id="N12EA5" type="main"> <s id="N12EA7"> <arrow.to.target n="marg83"></arrow.to.target> Græcus codex poſt verba, <emph type="italics"></emph>centraquè grauitatis ipſarum in recta <lb></lb>linea ſunt constituta,<emph.end type="italics"></emph.end> habet, <foreign lang="grc">καὶ τὰ μὲσα ἴσα, καὶ ω̄ὰντα τὰ εφ̓ εκάτεζα <lb></lb>τῶν μἐσων αυτά τε ἴσα ἐντί</foreign>, quæ quidem omnino ſuperflua nobis <lb></lb>ui<gap></gap>a ſunt, & <expan abbr="tanquã">tanquam</expan> ab aliquo addita. </s> <s id="N12EC3">Nam ſi Archimedes di<lb></lb>xit omnia parallelogramma eſſe inter ſe, & ęqualia, & ſimilia; <lb></lb>non opus eſt addere, media LF ES eſſe inter ſe ęqualia, & <lb></lb>quę ab his ſunrad vtram〈que〉 partem, vt MR KT, NQ GV, <lb></lb>AP OD, eſſe inter ſe æqualia; cum omnia (vt dictum eſt) ſint <lb></lb>ęqualia. </s> <s id="N12ECF">quare verba hęc (meo quidem iudicio) delenda ſunt. <lb></lb>demonſtrationes enim mathematicę nullum admittunt ſu<lb></lb>perfluum. </s> <s id="N12ED5">& Archim edes non tantùm ſuperfluus, quin potiùs <lb></lb>ob cius breuitatem diminutus ferè videatur. </s> </p> <pb xlink:href="077/01/087.jpg" pagenum="83"></pb> <p id="N12EDC" type="margin"> <s id="N12EDE"><margin.target id="marg83"></margin.target>*</s> </p> <p id="N12EE2" type="main"> <s id="N12EE4">Ex hac nona propoſitione duo corolloria elicere poſſum^{9}; <lb></lb>quæ quidem tanquam valde nota fortafſe videtur omiſiſſe Ar <lb></lb>chimedes. </s> <s id="N12EEA">quamuis <expan abbr="primũ">primum</expan> in ſe〈que〉nti <expan abbr="demõſtratione">demonſtratione</expan> inſeruit. </s> </p> <p id="N12EF4" type="head"> <s id="N12EF6">COROLLARIVM. I.</s> </p> <p id="N12EF8" type="main"> <s id="N12EFA">Ex hoc perſpicuum eſt cuiuſlibet parallelogrammi <expan abbr="cẽtrum">centrum</expan> <lb></lb>grauitatis eſſe punctum, in quo coincidunt rectæ lineæ, quæ <lb></lb>oppoſita latera bifariam ſecant. </s> </p> <p id="N12F04" type="main"> <s id="N12F06">Nam (vt Archimedes etiam ſe <lb></lb> <arrow.to.target n="fig34"></arrow.to.target><lb></lb>〈que〉nti demonſtratione inquit) <lb></lb>ſi parallelogrammi ABCD lineę <lb></lb>EF GH bifariam diuident late<lb></lb>ra oppoſita AB DC, & AD BC. <lb></lb>patet in EF centrum eſſe graui<lb></lb>tatis parallelogrammi AC. ſimi <lb></lb>liter conſtat idem centrum eſſe <lb></lb>in linea GH, quæ oppoſita latera AD BC bifariam ſecat. </s> <s id="N12F1D">e<lb></lb>ritigitur in K, vbi EF GH ſeinuicem ſecant. </s> </p> <figure id="id.077.01.087.1.jpg" xlink:href="077/01/087/1.jpg"></figure> <p id="N12F25" type="head"> <s id="N12F27">COROLLARIVM. II.</s> </p> <p id="N12F29" type="main"> <s id="N12F2B">Ex hoc patet etiam, cuiuſlibet parallelogrammi <expan abbr="centrũ">centrum</expan> gra<lb></lb>uitatis eſſe in medio rectæ lineę, quæ bifariam oppoſita latera <lb></lb>diſpeſcit. </s> </p> <p id="N12F35" type="main"> <s id="N12F37">Cùm enim oſtenſum ſit centrum grauitatis parallelogram <lb></lb>mi AC eſſe punctum K. & ob parallelogrammum EH eſt <lb></lb>EK æqualis BH. propter parallelogrammum verò KC <arrow.to.target n="marg84"></arrow.to.target><lb></lb>linea KF eſt æqualis HC. ſuntquè BH HC æqua<lb></lb>les. </s> <s id="N12F44">erit EK ipſi KF æqualis. </s> <s id="N12F46">punctum ergo K eſt in medio <lb></lb>rectæ lineę EF, quæ oppoſita latera AB DC bifariam diui<lb></lb>dit. <expan abbr="Eodẽq́">Eoden〈que〉</expan>; prorſus modo <expan abbr="oſtẽdetur">oſtendetur</expan>, K <expan abbr="mediũ">medium</expan> eſſe rectę lineę <lb></lb>GH, quæ bifariam ſecat oppoſita latera AD BC. </s> </p> <p id="N12F5A" type="margin"> <s id="N12F5C"><margin.target id="marg84"></margin.target>34. <emph type="italics"></emph>primi.<emph.end type="italics"></emph.end></s> </p> <p id="N12F65" type="main"> <s id="N12F67">In ſe〈que〉nti Archimedes adhuc perſiſtit in inuentione cen<lb></lb>tri grauitatis parallelogrammorum, alia tamen methodo. <lb></lb>nam hoc peripſorum parallelogrammorum diametros duo<lb></lb>bus modis aſſequitur. </s> </p> <pb xlink:href="077/01/088.jpg" pagenum="84"></pb> <p id="N12F72" type="head"> <s id="N12F74">PROPOSITIO. X.</s> </p> <p id="N12F76" type="main"> <s id="N12F78">Omnis parallelogrammi centrum grauitatis <lb></lb>eſt punctum, in quo diametri coincidunt. </s> </p> <p id="N12F7C" type="main"> <s id="N12F7E"><emph type="italics"></emph>Sit parallelogrammum <lb></lb>ABCD. & in ipſo ſit li<lb></lb>nea EF<emph.end type="italics"></emph.end> bifariam <emph type="italics"></emph><expan abbr="ſecãs">ſecans</expan><emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="fig35"></arrow.to.target><lb></lb><emph type="italics"></emph>latera AB CD. itidem<lb></lb>què ſit KL <expan abbr="ſecãs">ſecans</expan> AC BD<emph.end type="italics"></emph.end><lb></lb>bifariam. </s> <s id="N12FA3">conueniant<lb></lb>què EF kL in H. <emph type="italics"></emph>est <lb></lb>vti〈que〉 parallelogrammi<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg85"></arrow.to.target> <emph type="italics"></emph>ABCD centrum grauita <lb></lb>tis in linea EF. hoc enim <lb></lb>oſtenſum eſt. </s> <s id="N12FBB">eadem verò de cauſa<emph.end type="italics"></emph.end> centrum grauitatis ipſius AD <emph type="italics"></emph>est <lb></lb>etiam in linea<emph.end type="italics"></emph.end> K<emph type="italics"></emph>L. quare punctum H<emph.end type="italics"></emph.end> parallelogrammi AD <emph type="italics"></emph>cen<lb></lb>trum grauitatis existit. </s> <s id="N12FD3">Verùm in puncio H diametri parallelogram<lb></lb>mi concurrunt.<emph.end type="italics"></emph.end> ductis enim lineis AH HB CH HD; quoniam <lb></lb>lineæ AE EB EF FD inter ſe ſunt ęquales. </s> <s id="N12FDC">ſimiliter quo〈que〉 <lb></lb>AK KC BL LD inter ſe ęquales; erit EH ipſi HF ęqua <lb></lb>lis, cùm ſint ipſis BL LD ęquales. </s> <s id="N12FE2">duæ igitur AE EH dua <lb></lb> <arrow.to.target n="marg86"></arrow.to.target> bus DF FH ſunt æquales, & angulus AEH angulo DFH <lb></lb> <arrow.to.target n="marg87"></arrow.to.target> ęqualis; erit triangulum AEH triangulo DFH ęquale. </s> <s id="N12FF0">ac <lb></lb>propterea angulus EHA angulo FHD æqualis. </s> <s id="N12FF4">cùm igitur <lb></lb>ſit EHF recta linea, eruntangnli EHA FHD adverticem, <lb></lb>& obid AHD recta exiſtit linea. </s> <s id="N12FFA">ac per conſe〈que〉ns diame<lb></lb>ter parallelogrammi AD. pariquè ratione oſtendetur BHC <lb></lb>rectam eſſe lineam. </s> <s id="N13000">ex quibus patet in puncto H <expan abbr="vtrã〈que〉">vtran〈que〉</expan> dia <lb></lb>metrum conuenire. </s> <s id="N13008">centrum igitur grauitatis parallelogram<lb></lb>mi AD eſt <expan abbr="pũctum">punctum</expan>, in quo diametri concurrunt. <emph type="italics"></emph>Quare demon <lb></lb>stratumeſt, quod propoſitum fuit.<emph.end type="italics"></emph.end></s> </p> <pb xlink:href="077/01/089.jpg" pagenum="85"></pb> <p id="N1301A" type="margin"> <s id="N1301C"><margin.target id="marg85"></margin.target>9 <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N13025" type="margin"> <s id="N13027"><margin.target id="marg86"></margin.target>29, <emph type="italics"></emph>primi.<emph.end type="italics"></emph.end></s> </p> <p id="N13030" type="margin"> <s id="N13032"><margin.target id="marg87"></margin.target>4. <emph type="italics"></emph>primi.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.089.1.jpg" xlink:href="077/01/089/1.jpg"></figure> <p id="N1303F" type="main"> <s id="N13041">ALITER. </s> </p> <p id="N13043" type="main"> <s id="N13045"><emph type="italics"></emph>Hoc autem aliter quo<lb></lb>〈que〉 oſtendetur. </s> <s id="N1304B">ſit paralle<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="fig36"></arrow.to.target><lb></lb><emph type="italics"></emph>logrammum ABCD. <lb></lb>ipſius verò diameter ſit<emph.end type="italics"></emph.end> <arrow.to.target n="marg88"></arrow.to.target><lb></lb><emph type="italics"></emph>B D. triangula<emph.end type="italics"></emph.end> vti〈que〉 <lb></lb>ABD BDC <emph type="italics"></emph>erunt in<lb></lb>terſe æqualia, & ſimilia. <lb></lb>quare triangulis inuicem <lb></lb>coaptatis; centra quo〈que〉 <lb></lb>grauitatis ipſorum inuicem coaptabuntur. </s> <s id="N13074">Sit autem trianguli ABD cen<emph.end type="italics"></emph.end> <arrow.to.target n="marg89"></arrow.to.target><lb></lb><emph type="italics"></emph>trum grauitatis punctum E; lineaquè BD bifariam ſecetur in H. con <lb></lb>nectaturquè EH, & producatur. </s> <s id="N13082">ſumaturquè FH æqualisipſi HE. <lb></lb>Ita〈que〉 coaptato triangulo ABD cumtriangulo B DC, poſitoquè latere <lb></lb>AB in DC,<emph.end type="italics"></emph.end> hoc eſt A in C, & B in D. <emph type="italics"></emph>AD autem<emph.end type="italics"></emph.end> poſito <emph type="italics"></emph>in <lb></lb>BC;<emph.end type="italics"></emph.end> A ſcilicet in C, & D in B. vnde & BD cum ipſamet <lb></lb>DB coaptatur, B ſcilicet in D, & D in B. quia verò pun<lb></lb>ctum H ſibi ipſi coaptatur, cùm fitmedium lineę BD. & an <lb></lb>guli EHD FHB ad verticem ſunt æquales; lineaquè EH eſt <lb></lb>ipſi HF ęqualis; <emph type="italics"></emph>congruet etiam recta HE cum recta FH, & <expan abbr="pũ-ctum">pun<lb></lb>ctum</expan> E cum F conueniet, ſed<emph.end type="italics"></emph.end> quoniam punctum E centrum <lb></lb>eſt grauitatis trianguli ABD idem punctum E <emph type="italics"></emph>cum centro e<lb></lb>tiam grauitatis trianguli B DC<emph.end type="italics"></emph.end> conueniet. </s> <s id="N130B7">ergo punctum F <expan abbr="cẽ-trum">cen<lb></lb>trum</expan> eſt grauitatis trianguli BDC. Nunc verò intelligantur <lb></lb>triangula non ampliùs coaptata. <emph type="italics"></emph>Quoniam igitur centrum graui<lb></lb>tatis trianguli ABD eſt punctum E, ipſius verò DBC est punctum F,<emph.end type="italics"></emph.end><lb></lb>triangulaquè ABD DBC ſunt ęqualia, <emph type="italics"></emph>patet magnitudinis ex v<lb></lb>triſ〈que〉 triangulis compoſit<gap></gap> centrum grauitatis eſſe medium rectæ lineæ<emph.end type="italics"></emph.end> <arrow.to.target n="marg90"></arrow.to.target><lb></lb><emph type="italics"></emph>EF; quod eſt punctum H,<emph.end type="italics"></emph.end> vt factum furt. </s> <s id="N130DE">Quoniam autem dia<lb></lb>metri cuiuſlibet parallelogrammi ſeſe bifariam diſpeſcunt, e<lb></lb>rit punctum H, vbi diametri parallelogrammi ABCD con<lb></lb>currunt. </s> <s id="N130E6">ergo punctum H, in quo diametri coincidunt; ipſius <lb></lb>ABCD centrum grauitatis exiſtit. </s> <s id="N130EA">quod demonſtrare opor<lb></lb>rebat. </s> </p> <pb xlink:href="077/01/090.jpg" pagenum="86"></pb> <p id="N130F1" type="margin"> <s id="N130F3"><margin.target id="marg88"></margin.target><emph type="italics"></emph>ex<emph.end type="italics"></emph.end> 34.<emph type="italics"></emph>pri <lb></lb>mi.<emph.end type="italics"></emph.end></s> </p> <p id="N13103" type="margin"> <s id="N13105"><margin.target id="marg89"></margin.target>5. <emph type="italics"></emph>post hu<lb></lb>ius.<emph.end type="italics"></emph.end></s> </p> <p id="N13110" type="margin"> <s id="N13112"><margin.target id="marg90"></margin.target>4. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.090.1.jpg" xlink:href="077/01/090/1.jpg"></figure> <p id="N1311F" type="head"> <s id="N13121">SCHOLIVM.</s> </p> <p id="N13123" type="main"> <s id="N13125">Cognito centro grauitatis cuiuſlibet parallelogrammi, <lb></lb>vult Archimedes oſtendere centrum grauitatis triangulorum. <lb></lb>& quoniam in hac poſtrema demonſtratione aſſumpſit cen<lb></lb>trum grauitatis trianguli ABD eſſe punctum E, videtur or <lb></lb>dinem peruertiſſe, & per ignotiora doctrinam tradidiſſe; cùm <lb></lb>non ſit adhuc oſtenſum, in quo ſitu dictum centrum in <expan abbr="triã-gulis">trian<lb></lb>gulis</expan> reperiatur. </s> <s id="N13137">quod tamen ſi rectè perpendamus, non ita ſe <lb></lb>habet. </s> <s id="N1313B">Nam vis demonſtrationis eſt in hoc conſtituta, vt <lb></lb>ſupponatur triangulum habere centrum grauitatis, idquè tan <lb></lb> <arrow.to.target n="marg91"></arrow.to.target> <gap></gap>ùm eſſe intra ipsum triangulum, quod quidem ſupponi po<lb></lb>teſt. </s> <s id="N13149">cùm triangulum ſit figura ad eaſdem partes concaua. </s> <s id="N1314B">ne<lb></lb>〈que〉 enim refert, ſiuè centrum ſit in E, ſiuè in alio ſitu, dum<lb></lb>modo intra triangulum exiſtat. </s> <s id="N13151">demonſtratio enim <expan abbr="eodẽ">eodem</expan> mo<lb></lb>do ſemper concludet punctum H centrum eſſe grauitatis pa <lb></lb>rallelogrammi AC, quod idem obſeruandum eſt in <expan abbr="nõnullis">nonnullis</expan> <lb></lb>alijs demonſtrationibus. </s> <s id="N13161">vt in ſecunda demonſtratione deci<lb></lb>mæ tertiæ, hui^{9} & in prima ſecundilibri. </s> <s id="N13165">Antequam <expan abbr="autẽ">autem</expan> Ar<lb></lb>chimedes centrum grauitatis triangulorum oſtendat, nonnul<lb></lb>las pręmittit propoſitiones. </s> </p> <p id="N1316F" type="margin"> <s id="N13171"><margin.target id="marg91"></margin.target>9. <emph type="italics"></emph>post hu<lb></lb>ius.<emph.end type="italics"></emph.end></s> </p> <p id="N1317C" type="head"> <s id="N1317E">PROPOSITIO. XI.</s> </p> <p id="N13180" type="main"> <s id="N13182">Si duo triangula inter ſe ſimilia fuerint, & in i<lb></lb>pſis ſint puncta ad triangula ſimiliter poſita & alre <lb></lb>rum punctum trianguli, in quo eſt, centrum fue<lb></lb>rit grauitatis, & alterum punctum trianguli, in <lb></lb>quo eſt, centrum grauitatis exiſtet. </s> </p> <pb xlink:href="077/01/091.jpg" pagenum="87"></pb> <p id="N1318F" type="main"> <s id="N13191">Dicimus quidem puncta in ſimilibus figuris eſſe <lb></lb>ſimiliter poſita, è quibus ad æquales angulos du<lb></lb>ctæ rectæ lineæ, æquales efficiunt angulos ad ho<lb></lb>mologalatera. </s> <s id="N13199">Vt dictum fuit in ſeptimo poſtulato. </s> </p> <figure id="id.077.01.091.1.jpg" xlink:href="077/01/091/1.jpg"></figure> <p id="N1319E" type="main"> <s id="N131A0"><emph type="italics"></emph>Sint duo triangula ABC DEF<emph.end type="italics"></emph.end> ſimilia. <emph type="italics"></emph>ſit què AC ad DE, vt <lb></lb>AB ad DE, & BC ad EF. & in præfatis triangulis ABC DEF <lb></lb>ſint puncta HN ſimiliter poſita ſitquè punctum H centrum grauitatis <lb></lb>trianguli ABC. Dico & punctum N centrum eſſe grauitatis trianguli <lb></lb>DEF. non ſit quidem, ſed, ſi fieripoteſt, ſit punctum G centrum grauita <lb></lb>tis trianguli DEF. <expan abbr="connectãturquè">connectanturquè</expan> HA HB HC, DN EN FN, <lb></lb>DG EG FG. Quoniamigitur ſimile eſt triangulum ABC triangulo <lb></lb>DEF, &<emph.end type="italics"></emph.end> ipſorum <emph type="italics"></emph>centra grauitatum ſunt puncta HG. ſimi<lb></lb>lium autem figurarum centra grauitatum ſunt ſimiliter poſita; ita vt<emph.end type="italics"></emph.end> <arrow.to.target n="marg92"></arrow.to.target><lb></lb>ab ipſis ad ęquales angulos ductæ rectæ lineę <emph type="italics"></emph>æquales faciant <lb></lb>angulos ad homologa latera, vnum〈que〉mquè vnicuiquè; erit angulus <lb></lb>GDE ipſi HAB aqualis. </s> <s id="N131D3">at verò anguius HAB aqualis est angulo <lb></lb>EDN, cùm ſint puncta HN ſimiliter poſita: angulus igitur EDG <lb></lb>angulo EDN æqualis existit. </s> <s id="N131D9">maior minori quòd fierinon potest. </s> <s id="N131DB">Non <lb></lb>igitur punctum G centrum eſt grauitatis trianguli DEF. Quare eſt <lb></lb>punctum N. quod demonstrare oportebat.<emph.end type="italics"></emph.end></s> </p> <pb xlink:href="077/01/092.jpg" pagenum="88"></pb> <p id="N131E6" type="margin"> <s id="N131E8"><margin.target id="marg92"></margin.target>6.& 7 <emph type="italics"></emph>poſt <lb></lb>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N131F3" type="head"> <s id="N131F5">SCHOLIVM.</s> </p> <p id="N131F7" type="main"> <s id="N131F9">In hac propoſitione ſupponit Archimedes dari poſſe pun<lb></lb>cta in triangulis ſimilib^{9} ſimiliter poſita, qd <expan abbr="quidẽ">quidem</expan> ſieri poſſe <lb></lb>oſtendimus in ſcholijs ſeptimi poſtulati. </s> <s id="N13203">Præterea idem vide<lb></lb>tur Archimedes in triangulis demonſtrare, quod in ſexto po<lb></lb>ſtulato vniuerſaliter in figuris ſuppoſuit. </s> <s id="N13209">Nam ſi centra gra<lb></lb>uitatis ſupponuntur in ſimilibus figuris eſſe ſimiliter poſita; <lb></lb>& in ſimilibus triangulis quo〈que〉 erunt ſimiliter poſita. </s> <s id="N1320F">In<lb></lb>ter hęc tamen maxima eſt differentia, nam in poſtulato inquit, <lb></lb>centra grauitatum in ſimilibus figuris eſſe ſimiliter poſita; cu<lb></lb>ius quidem conuerſum, nempè puncta in ſimilibus figuris ſi<lb></lb>militer poſita eſſe ipſarum centra grauitatis, eſt falium. </s> <s id="N13219">quod <lb></lb>eſt quidem manifeſtum abſ〈que〉 alio exemplo. </s> <s id="N1321D">ac propterea <lb></lb>Archimedes hoc in loco inquit, ſi duo erunt punſta in ſimi<lb></lb>libus triangulis ſimiliter poſita, & alterum ipſorum fuerit <expan abbr="cẽ-trum">cen<lb></lb>trum</expan> grauitatis. </s> <s id="N13229">& alterum quo〈que〉 <expan abbr="cẽtrum">centrum</expan> grauitatis exiſtet. <lb></lb>Vnde propoſitio hęc potiùs eſt conuerſa poſtulati, quàm <lb></lb>eadem. </s> </p> <p id="N13233" type="main"> <s id="N13235">Ob demonſtrationem autem nouiſſe oportet, quòd ſi pun<lb></lb>ctum G fuerit in linea DN, tuncanguli EDG EDN eſſent in<lb></lb>terſe ęquales, ac propterea demonſtratio nihil abſurdi conclu<lb></lb>deret. </s> <s id="N1323D">In hoc autem caſu oſtendendum eſſet, angulum EFG <lb></lb>ipſi EFN ęqualem eſſe, vel FEG ipſi FEN. quæ quidem eo<lb></lb>dem prorſus modo oſtendentur. </s> <s id="N13243">comparando nempè angu<lb></lb>los EFG EFN angulo BCH; angulos verò FEG FEN ipſi <lb></lb>CBH. Quòd ſi G fuerit in alio ſitu, vt in triangulo EDN, <lb></lb>tuncanguli FDG FDN oſtendentur ęquales. </s> <s id="N1324B">& ita in alijs <lb></lb>caſibus, vbicun〈que〉 ſcilicet fuerit punctum G, ſemper ali<lb></lb>quod inuenietur huiuſmodi abſurdum. </s> <s id="N13251">quæ quidem omni<lb></lb>nò fieri non poſſunt. </s> </p> <pb xlink:href="077/01/093.jpg" pagenum="89"></pb> <p id="N13258" type="head"> <s id="N1325A">PROPOSITIO. XII.</s> </p> <p id="N1325C" type="main"> <s id="N1325E">Si duo triangula ſimilia fuerint, alterius verò <lb></lb>trianguli centrum grauitatis in rectalinea fuerit, <lb></lb>quæ ſit ab aliquo angulo ad dimidiam baſim du<lb></lb>cta; & alrerius trianguli centrum grauitatis erit in <lb></lb>linea ſimiliter ducta. </s> </p> <figure id="id.077.01.093.1.jpg" xlink:href="077/01/093/1.jpg"></figure> <p id="N1326B" type="main"> <s id="N1326D"><emph type="italics"></emph>Sint duo triangula ABC DEF<emph.end type="italics"></emph.end> ſimilia <emph type="italics"></emph>ſitquè AC ad DF, vt <lb></lb>AB ad DE, & BC ad FE. Diuiſaquè AC bifariam in G, iunga <lb></lb>tur BG. centrum verò grauitatis trianguli ABC ſit punctum H in li <lb></lb>nea BG. Dico centrum grauitatis trianguli EDF eſſe in recta linea ſi <lb></lb>militer ducta. </s> <s id="N1327F">ſecetur DF bifariam in puncto M. & iungatur EM. <lb></lb>& vt BG ad BH, ita fiat ME ad EN. connectanturquè AH <lb></lb>HC, DN NF. Quoniam enim<emph.end type="italics"></emph.end> eſt BA ad ED, vt AC ad DF, & <lb></lb><emph type="italics"></emph>AG dimidia eſt ipſius AC; ipſius verò DF dimidiaest DM; erit BA <lb></lb>ad ED, vt AG ad DM.<emph.end type="italics"></emph.end> Quoniam autem ob <expan abbr="triãgulorum">triangulorum</expan> <arrow.to.target n="marg93"></arrow.to.target><lb></lb>ABC DEF ſimilitudinem angulus BAC angulo EDF eſt ę<lb></lb>qualis. </s> <s id="N1329C">& vt AB ad DE, ita AG ad DM; <expan abbr="permutandoq́">permutando〈que〉</expan>; AB ad <arrow.to.target n="marg94"></arrow.to.target><lb></lb>AG, vt DE ad DM; erit <expan abbr="triangulũ">triangulum</expan> ABG <expan abbr="triãgulo">triangulo</expan> DEM ſimile. <lb></lb><expan abbr="ſimiliũ">ſimilium</expan> <expan abbr="ãt">ant</expan> <expan abbr="triãgulorũ">triangulorum</expan> <expan abbr="ãguli">anguli</expan> <expan abbr="sũt">sunt</expan> ęquales, <emph type="italics"></emph>et circa æquales <expan abbr="ãgulos">angulos</expan> late<emph.end type="italics"></emph.end> <pb xlink:href="077/01/094.jpg" pagenum="90"></pb><emph type="italics"></emph>ra sut proportionalia. </s> <s id="N132D4">erit <lb></lb>igitur angul^{9} AGB angulo<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="fig37"></arrow.to.target><lb></lb><emph type="italics"></emph>DME aqualis, et<emph.end type="italics"></emph.end> ABG ip <lb></lb>ſi DEM æqualis quare <lb></lb><emph type="italics"></emph>vt AG ad DM, ita eſt BG<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg95"></arrow.to.target> <emph type="italics"></emph>ad EM,<emph.end type="italics"></emph.end> & vt AB ad DE, <lb></lb>ita BG ad EM; & pmu<lb></lb>tado AB ad BG, vt DE <lb></lb>ad EM. <emph type="italics"></emph>eſt autem BG ad<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg96"></arrow.to.target> <emph type="italics"></emph>BH, vt ME ad EN, erit igitur ex æquali<emph.end type="italics"></emph.end> AB ad BH, vt DE ad EN. <lb></lb> <arrow.to.target n="marg97"></arrow.to.target> rurſuſquè permutando <emph type="italics"></emph>AB ad DE, vt BH ad EN.<emph.end type="italics"></emph.end> <expan abbr="quoniã">quoniam</expan> <lb></lb> <arrow.to.target n="marg98"></arrow.to.target> autem anguli ABH DEN, quos ipſæ lineę continent, ſunt <lb></lb>æquales, erit triangulun. </s> <s id="N13329">ABH triangulo DEN ſimile. </s> <s id="N1332B">qua <lb></lb>re anguli ſunt inter ſe æquales, <emph type="italics"></emph>& circa a quales angulos latera ſunt <lb></lb>proportionalia ſi autem hoc, angulus BAH angulo EDN est æqualis. <lb></lb>Vnde & reliquus angulus HAC angulo NDF æquolis exiſtit.<emph.end type="italics"></emph.end> <gap></gap>qui<lb></lb>dem totius BAC ipſi EDF eſt æqualis. <emph type="italics"></emph>Eademquè ratione an-<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg99"></arrow.to.target> <emph type="italics"></emph>gulus BCH ipſi EFN est æqualis. </s> <s id="N1334B">& angulas HCG angulo NFM <lb></lb>æqualis, oſtenſum est autem angulum ABH ipſi DEM aqualem eſſe.<emph.end type="italics"></emph.end><lb></lb>ob ſimilitudinem autem riangulorum ABC DEF totus an <lb></lb> <arrow.to.target n="marg100"></arrow.to.target> gulus ABC eſtipſi DEF ę ualis: <emph type="italics"></emph>ergo & reliquus angulus HBC <lb></lb>ipſi NEF æqualis exiſtit. </s> <s id="N1335E">Porrò ex his omnibus patet puncta HN ad <lb></lb>homologa latera eſſe ſimiliter poſita, &<emph.end type="italics"></emph.end> cum ipſis <emph type="italics"></emph>angulas æquales effi<lb></lb>cere. </s> <s id="N1336A">Cùm igitur puncta HN ſint ſimiliter poſita; & punctum H cen<lb></lb>trum eſt grauitatis trianguli ABC, & puncium N trianguli DEF <expan abbr="cẽ-trum">cen<lb></lb>trum</expan> grauitatis existet.<emph.end type="italics"></emph.end> exiſtente igitur centro grauitatis H in li <lb></lb>nea BG ab angulo ad dimidiam baſim ducta. </s> <s id="N13379">& alterum gra<lb></lb>uitatis centrum N in linea EM ſimiliter ducta reperitur. <lb></lb>quod demonſtrare oportebat. </s> </p> <p id="N1337F" type="margin"> <s id="N13381"><margin.target id="marg93"></margin.target>16. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N1338A" type="margin"> <s id="N1338C"><margin.target id="marg94"></margin.target>6.<emph type="italics"></emph>ſeati.<emph.end type="italics"></emph.end></s> </p> <p id="N13395" type="margin"> <s id="N13397"><margin.target id="marg95"></margin.target>16. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N133A0" type="margin"> <s id="N133A2"><margin.target id="marg96"></margin.target>22. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N133AB" type="margin"> <s id="N133AD"><margin.target id="marg97"></margin.target>16. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N133B6" type="margin"> <s id="N133B8"><margin.target id="marg98"></margin.target>6. <emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N133C1" type="margin"> <s id="N133C3"><margin.target id="marg99"></margin.target>7. <emph type="italics"></emph>post hu<lb></lb>ius.<emph.end type="italics"></emph.end></s> </p> <p id="N133CE" type="margin"> <s id="N133D0"><margin.target id="marg100"></margin.target>11.<emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.094.1.jpg" xlink:href="077/01/094/1.jpg"></figure> <p id="N133DD" type="head"> <s id="N133DF">SCHOLIVM.</s> </p> <p id="N133E1" type="main"> <s id="N133E3">In ſe〈que〉nti Archimedes oſtendet, in qua linea reperitur <expan abbr="cẽ">cem</expan> <lb></lb>trum grauitatis cuiuſlibet trianguli. </s> <s id="N133EB">quod quidem duobus aſ<lb></lb>ſequitur medijs. </s> <s id="N133EF">Diligenter autem omnia ſunt conſideranda; <lb></lb>quoniam in hoc conſiſtit tota perſcrutatio centri grauitatis <lb></lb>triangulorum. </s> <s id="N133F5">Quapropter vt prior demonſtratio appareat <lb></lb>perſpicua, hęc antea demonſtrabimus. </s> </p> <pb xlink:href="077/01/095.jpg" pagenum="91"></pb> <p id="N133FC" type="main"> <s id="N133FE">LEMMA. I. </s> </p> <p id="N13400" type="main"> <s id="N13402">Æquidiſtantes lineæ lineas in eadem proportione diſpe<lb></lb>ſcunt. </s> </p> <p id="N13406" type="main"> <s id="N13408">Sintlineę AB CD, quas ſecent æqui<lb></lb> <arrow.to.target n="fig38"></arrow.to.target><lb></lb>diſtantes lineæ AC EF BD. Dico ita eſ<lb></lb>ſe BE ad EA, vt DF ad FC. primùm <lb></lb>quidem AB CD vel ſunt ęquidiſtantes, <arrow.to.target n="marg101"></arrow.to.target><lb></lb>vel minùs. </s> <s id="N1341A">ſi ſunt æquidiſtantes, iam habe <lb></lb>tur intentum. </s> <s id="N1341E">Nam BE erit æqualis DF, <lb></lb>& EA ipſi FC. vnde ſequitur ita eſſe BE <lb></lb> <arrow.to.target n="fig39"></arrow.to.target><lb></lb>ad EA, vt DF ad FC. </s> </p> <p id="N13429" type="margin"> <s id="N1342B"><margin.target id="marg101"></margin.target>34. <emph type="italics"></emph>primi.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.095.1.jpg" xlink:href="077/01/095/1.jpg"></figure> <figure id="id.077.01.095.2.jpg" xlink:href="077/01/095/2.jpg"></figure> <p id="N1343C" type="main"> <s id="N1343E">Si verò AB CD non fuerint æquidi<lb></lb>ſtantes, concurrant in G, vt in ſecunda fi<lb></lb> <arrow.to.target n="fig40"></arrow.to.target><lb></lb>gura, & quoniam BD EF ſunt <arrow.to.target n="marg102"></arrow.to.target> æquidi<lb></lb>ſtantes, erit GB ad BE, vt GD ad DF. <arrow.to.target n="marg103"></arrow.to.target><lb></lb>& <expan abbr="cõponendo">componendo</expan> GE ad EB, vt GF ad FD. <arrow.to.target n="marg104"></arrow.to.target><lb></lb>conuertendoquè BE ad EG, vt DF ad <lb></lb>FG, rurſus quoniam EF AC ſunt æquidi <lb></lb>ſtantes; erit GE ad EA, vt GF ad FC, e<lb></lb>ritigitur ex æquali BE ad EA, vt DF ad FC. </s> </p> <p id="N13463" type="margin"> <s id="N13465"><margin.target id="marg102"></margin.target>2.<emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N1346E" type="margin"> <s id="N13470"><margin.target id="marg103"></margin.target>18.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N13479" type="margin"> <s id="N1347B"><margin.target id="marg104"></margin.target><emph type="italics"></emph>cor.<emph.end type="italics"></emph.end>4.<emph type="italics"></emph><expan abbr="quĩti">quinti</expan><emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.095.3.jpg" xlink:href="077/01/095/3.jpg"></figure> <p id="N1348F" type="main"> <s id="N13491">Secent verò ſeſe lineæ AB CD, vt in tertia figura, ob <arrow.to.target n="marg105"></arrow.to.target> ſimi<lb></lb>litudinem triangulorum BGD EGF, it a erit BG ad GE, vt <arrow.to.target n="marg106"></arrow.to.target><lb></lb>DG ad GF. & componendo BE ad EG, vt DF ad FG. eſt <arrow.to.target n="marg107"></arrow.to.target><lb></lb>verò GE ad EA, vt GF ad FC. ergo ex æquali BE ad EA <lb></lb>erit, vt DF ad FC. quod demonſtrare oportebat. </s> </p> <pb xlink:href="077/01/096.jpg" pagenum="92"></pb> <p id="N134A8" type="margin"> <s id="N134AA"><margin.target id="marg105"></margin.target><emph type="italics"></emph>ex<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N134B8" type="margin"> <s id="N134BA"><margin.target id="marg106"></margin.target>18. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N134C3" type="margin"> <s id="N134C5"><margin.target id="marg107"></margin.target>2. <emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N134CE" type="main"> <s id="N134D0">LEMMA. II. </s> </p> <p id="N134D2" type="main"> <s id="N134D4">Sit A ad B, vt C ad D; rurſus A ad E ſit, vt C ad F. <lb></lb>Dico primùm A ad BE ſimul ita eſſe, vt C ad DF. </s> </p> <figure id="id.077.01.096.1.jpg" xlink:href="077/01/096/1.jpg"></figure> <p id="N134DB" type="main"> <s id="N134DD"> <arrow.to.target n="marg108"></arrow.to.target> Quoniam enim A eſt ad B, vt C ad D, erit conuertendo <lb></lb> <arrow.to.target n="marg109"></arrow.to.target> B ad A, vt D ad C. eſt autem A ad E, vt C ad F; ergo ex ę<lb></lb> <arrow.to.target n="marg110"></arrow.to.target> quali B erit ad E, vt D ad F. quare componendo BE ad <lb></lb> <arrow.to.target n="marg111"></arrow.to.target> E, vt DF ad F. quoniam autem A eſt ad E, vt C ad F; e <lb></lb> <arrow.to.target n="marg112"></arrow.to.target> rit conuertendo E ad A, vt F ad C. rurſus igitur ex ęquali <lb></lb>erit BE ad A, vt DF ad C. ac deni〈que〉 conuertendo A e<lb></lb>rit ad BE, vt C ad DF. </s> </p> <p id="N134FF" type="margin"> <s id="N13501"><margin.target id="marg108"></margin.target><emph type="italics"></emph>cor.<emph.end type="italics"></emph.end>4.<emph type="italics"></emph><expan abbr="quĩti">quinti</expan><emph.end type="italics"></emph.end></s> </p> <p id="N13511" type="margin"> <s id="N13513"><margin.target id="marg109"></margin.target>22. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N1351C" type="margin"> <s id="N1351E"><margin.target id="marg110"></margin.target>18. <emph type="italics"></emph>qninti.<emph.end type="italics"></emph.end></s> </p> <p id="N13527" type="margin"> <s id="N13529"><margin.target id="marg111"></margin.target><emph type="italics"></emph>cor.<emph.end type="italics"></emph.end>4.<emph type="italics"></emph><expan abbr="quĩti">quinti</expan><emph.end type="italics"></emph.end></s> </p> <p id="N13539" type="margin"> <s id="N1353B"><margin.target id="marg112"></margin.target>22. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N13544" type="main"> <s id="N13546">Si verò fuerint quattuor magnitudines; vt adhue A (in ea<lb></lb>dem figura) ad G ſit, vt C ad H. ſimili<lb></lb> <arrow.to.target n="fig41"></arrow.to.target><lb></lb>ter oſtendetur A ad omnes BEG ſimul <lb></lb>ſumptas ita eſſe, vt C ad omnes ſimul <lb></lb>DFH. ſumendo vt in ſecunda figura BE <lb></lb>pro vna tan ùm magnitudine, & DF pro <lb></lb>alia; erunt〈que〉 ex vtra〈que〉 parte tres <expan abbr="tãtùm">tantùm</expan> <lb></lb>magnitudines; eritquè A ad BE ſimul, <lb></lb>vt C ad DF ſimul, vt oſtenſum eſt, dein<lb></lb>de A ad G eſt, vt C ad H, erit igitur <lb></lb>A ad BEG ſimul, vt C ad DFH. </s> </p> <pb xlink:href="077/01/097.jpg" pagenum="93"></pb> <figure id="id.077.01.097.1.jpg" xlink:href="077/01/097/1.jpg"></figure> <p id="N1356C" type="main"> <s id="N1356E">Pariquè ratione ſi quin〈que〉 fuerint magnitudines, eodem <lb></lb>modo tres mediæ <expan abbr="iũgatur">iungatur</expan> ſimul, ita vttres ſint <expan abbr="dũtaxat">duntaxat</expan> magni<lb></lb>tudines. </s> <s id="N1357C">& ſic in infinitum. </s> <s id="N1357E">quod demonſtrare oportebat. </s> </p> <p id="N13580" type="head"> <s id="N13582">COROLLARIVM.</s> </p> <p id="N13584" type="main"> <s id="N13586">Ex hoc elici poteſt. </s> <s id="N13588">quòd ſi fuerint quotcun〈que〉 magnitudi <lb></lb>nes proportionales; & alię ipſis numero æquales, & in eadem <lb></lb>proportione, vt ſcilicet ſit (vt in prima figura) A ad B, vt C <lb></lb>ad D, B verò ad E, vt D ad F. deinde vt E ad G, ſic F <lb></lb>ad H, & ita deinceps, ſi plures fuerint magnitudines, ſi<lb></lb>militer erit A ad omnes BEG ſimul ſumptas, vt C ad om<lb></lb>nes ſimul DFH. </s> </p> <p id="N13596" type="main"> <s id="N13598">Primùm quidem A eſt ad B, vt C ad D. & quoniam ma <lb></lb>gnitudines ſunt proportionales, ex ęquali erit A ad E, vt C <arrow.to.target n="marg113"></arrow.to.target><lb></lb>ad F. ſimiliter A ad G, vt C ad H. Ex quibus ſequitur <lb></lb>A ad BE ſimul ita eſſe, vt C ad DF. A verò ad omnes <lb></lb>BEG ſimul, vt C ad omnes ſimul DFH. & ita ſi plures fue<lb></lb>rint magnitudines. </s> </p> <p id="N135A7" type="margin"> <s id="N135A9"><margin.target id="marg113"></margin.target>22. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N135B2" type="main"> <s id="N135B4">LEMMA. III. </s> </p> <p id="N135B6" type="main"> <s id="N135B8">Sit triangulum ABC, cuiuslatus BC in quotcun〈que〉 di<lb></lb>uidatur partes æquales BE ED DF FC. & a punctis EDF <lb></lb>ipſi AB equidiſtanres ducantur EG DH FK. rurſus à pun<lb></lb>ctis GHK ipſi BC ęquidiſtantes ducantur GL HM KN. <lb></lb>Dico triangulum ABC ad omnia triangula ALG GMH <lb></lb>HNK KFC ſimulſumpta eandem habere proportionem, <lb></lb>quam habet CA ad AG. </s> </p> <pb xlink:href="077/01/098.jpg" pagenum="94"></pb> <p id="N135C9" type="main"> <s id="N135CB"> <arrow.to.target n="marg114"></arrow.to.target> <expan abbr="Quoniã">Quoniam</expan> enim FK ęquidiſtans eſtipſi DH; erit CF ad FD, <lb></lb>vt CK ad KH. <expan abbr="ſuntq́">ſunt〈que〉</expan> CF FD æquales; ergo & CK KH in<lb></lb>terſe ſunt æquales. </s> <s id="N135DD">ſimiliter propter lineas æquidiſtantes FK <lb></lb> <arrow.to.target n="marg115"></arrow.to.target> DH EG, ita eſt KH ad HG, vt FD ad DE; eſt autem FD <lb></lb>æqualis DE; erit igitur KH ipſi HG æqualis. </s> <s id="N135E7">Pariquè ra<lb></lb> <arrow.to.target n="fig42"></arrow.to.target><lb></lb>tione oſtendetur ob ęquidiſtantes lineas DH EG BA, <expan abbr="lineã">lineam</expan> <lb></lb>HG ipſi GA æqualem eſſe. </s> <s id="N135F6">Ex quibus patet CK KH HG <lb></lb>GA inter ſe æquales eſſe. </s> <s id="N135FA">Quoniam autem trianguloru ABC <lb></lb>kFC angulus ad C eſt vtri〈que〉 communis; & ABC ipſi kFC, <lb></lb> <arrow.to.target n="marg116"></arrow.to.target> & BAC ipſi FKC æqualis, cum ſit Fk ipſi AB æquidiſtans; <lb></lb>erit triangulum ABC ipſi KFC ſimile. </s> <s id="N13606">& quonian NK FC, <lb></lb>& HN KF ſunt ęquidiſtantes, erunt anguli KCFCkF angu<lb></lb>lis HkN KHN ęquales; ac propterea reliquus CFK reliquo <lb></lb>KNH ęqualis: latus verò CK lateri KH eſt ęquale; erit igi<lb></lb> <arrow.to.target n="marg117"></arrow.to.target> tur triangulum KFC triangulo HNK ſimile, & ęquale. </s> <s id="N13614">ſimi <lb></lb>literquè <expan abbr="oſtẽdetur">oſtendetur</expan> omnia triangula ALG GMH HNK KFC <lb></lb>interſeſe ſimilia, & æqualia eſſe. </s> <s id="N1361E">& obid ipſi ABC ſimilia eſſe. <lb></lb>Fiat igit vt AC ad AG, ita AG ad alia O. ſimiliterv AC ad GH, <lb></lb>ita GH ad P. rurſusvt AC ad Hk, ita HK ad <expan abbr="q.">〈que〉</expan> deniquè <lb></lb>vt AC ad Ck, ita CK ad R. & quoniam AG GH HK KC <lb></lb> <arrow.to.target n="marg118"></arrow.to.target> ſunt æquales, eadem AC ad vnamquam〈que〉 ipſarum ean<lb></lb>dem habebit proportionem, ergo eandem quo〈que〉 habebit <lb></lb>propoſitionem AG ad O, vt GH ad P, & HK ad Q, & <pb xlink:href="077/01/099.jpg" pagenum="95"></pb>KC ad R. ac propterea lineæ OPQR inter ſe ſunt æquales. <lb></lb>Atverò quoniam ita eſt AC ad AG, vt AG ad O, & vt <lb></lb>AC ad GH, ita GH, hoc eſt AG ipſi ęqualis, ad P. rurſus <lb></lb>vt AC ad HK, ita HK, hoc eſt AG ad <expan abbr="q.">〈que〉</expan> ac tandem vt <lb></lb>AC ad KC, ita KC, hoc eſt AG ipſi ęqualis, ad R. erit AC <arrow.to.target n="marg119"></arrow.to.target><lb></lb>ad omnes conſe〈que〉ntes ſimul ſumptas AG GH HK KC, <lb></lb>hoc eſt erit AC ad eandem AC, vt AG ad omnes ſimul <lb></lb>OPQR. vnde ſequitur omnes ſimul OPQR ipſi AG ęqua <lb></lb>les eſſe. </s> <s id="N1364F">Ita〈que〉 quoniam ſimilia triangula in dupla ſunt <arrow.to.target n="marg120"></arrow.to.target> pro<lb></lb>portione laterum homologorum, erit triangulum ABC ad <lb></lb>ALG, vt AC ad O. eodemquè modo erit triangulum ABC <lb></lb>ad GMH, vt AC ad P. rurſus ABC ad HNK, vt AC ad <lb></lb>Q, & vt idem ABC ad KFC, ita AC ad R. triangulum <lb></lb>igitur ABC ad omnes conſe〈que〉ntes, videlicet ad omnia <expan abbr="triã">triam</expan> <arrow.to.target n="marg121"></arrow.to.target><lb></lb>gula ſimul ſumpta ALG GMH HNK KFC, eritvt AC ad <lb></lb>omnes ſimul OPQR. hoc eſt ad AG. oſtenſum eſt igitur, <lb></lb>quod propoſitum fuit. </s> </p> <p id="N1366C" type="margin"> <s id="N1366E"><margin.target id="marg114"></margin.target>2. <emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N13677" type="margin"> <s id="N13679"><margin.target id="marg115"></margin.target>1. <emph type="italics"></emph>lemma.<emph.end type="italics"></emph.end></s> </p> <p id="N13682" type="margin"> <s id="N13684"><margin.target id="marg116"></margin.target>29. <emph type="italics"></emph>primi.<emph.end type="italics"></emph.end></s> </p> <p id="N1368D" type="margin"> <s id="N1368F"><margin.target id="marg117"></margin.target>76. <emph type="italics"></emph>primi.<emph.end type="italics"></emph.end></s> </p> <p id="N13698" type="margin"> <s id="N1369A"><margin.target id="marg118"></margin.target><emph type="italics"></emph>ex<emph.end type="italics"></emph.end> 17 <emph type="italics"></emph><expan abbr="quĩi">quini</expan>.<emph.end type="italics"></emph.end></s> </p> <p id="N136AB" type="margin"> <s id="N136AD"><margin.target id="marg119"></margin.target><emph type="italics"></emph>ex <expan abbr="præcedẽ">præcedem</expan> <lb></lb>ti lemmate<emph.end type="italics"></emph.end></s> </p> <p id="N136BB" type="margin"> <s id="N136BD"><margin.target id="marg120"></margin.target>19.<emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N136C6" type="margin"> <s id="N136C8"><margin.target id="marg121"></margin.target><emph type="italics"></emph>ex <expan abbr="præcedẽ">præcedem</expan> <lb></lb>ti lemmate<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.099.1.jpg" xlink:href="077/01/099/1.jpg"></figure> <p id="N136DA" type="head"> <s id="N136DC">PROPOSITIO. XIII.</s> </p> <p id="N136DE" type="main"> <s id="N136E0">Omnis trianguli centrum grauitatis eſt in recta <lb></lb>linea ab angulo ad dimidiam baſim ducta. </s> </p> <p id="N136E4" type="main"> <s id="N136E6"><emph type="italics"></emph>Sit triangulum ABC. & in ipſo ſit AD<emph.end type="italics"></emph.end> ab angulo A <emph type="italics"></emph>ad dimi<lb></lb>diambaſim BC ducta. </s> <s id="N136F2">oſtendendum est, centrum grauitatis trianguli <lb></lb>ABC eſſe in linea AD. Non ſit quidem, ſed ſi fieri potest ſit punctum <lb></lb>H. & ab ipſo ducatur HI æquidiſtansipſi BC,<emph.end type="italics"></emph.end> quæ ipſam AD ſecet <arrow.to.target n="marg122"></arrow.to.target><lb></lb>in I. <emph type="italics"></emph>Deinde diuiſa DC bifariam, idquè ſemper fiat, dones relinqua<lb></lb>tur linea<emph.end type="italics"></emph.end> D<foreign lang="grc">ω</foreign> <emph type="italics"></emph>minor ipſa HI. Diuidaturquè ipſarum vtra〈que〉 BD DC <lb></lb>in partes æquales<emph.end type="italics"></emph.end> D<foreign lang="grc">ω</foreign>; parteſquè in DC exrſtentes ſint D<foreign lang="grc">ω ωβ <lb></lb>β</foreign>Z ZC; quibus reſpondeant æquales partes D<foreign lang="grc">ααζζ</foreign>O OB. <emph type="italics"></emph>& <lb></lb>a ſectionum punctis ducantur<emph.end type="italics"></emph.end> OE <foreign lang="grc">ζ</foreign>G <foreign lang="grc">α</foreign>L <foreign lang="grc">ω</foreign>M <foreign lang="grc">β</foreign>K ZF <emph type="italics"></emph>æquidictan<lb></lb>tes ipſi AD. & connectantur EF G<emph.end type="italics"></emph.end>k <emph type="italics"></emph>LM quæ nimirum ipſi BC <lb></lb>æquidistantes erunt.<emph.end type="italics"></emph.end> cùm enim ſint BD DC interſe equales, iti<lb></lb>dem OB ZC æquales; erit DO ipſi DZ ęqualis. </s> <s id="N1374C">quare DO <lb></lb>ad OB eſt, vt DZ ad ZC. Quoniam autem EO FZ ſunt <pb xlink:href="077/01/100.jpg" pagenum="96"></pb>ipsi AD æquidiſtantes, erit AE ad EB, vt DO ad OB; & vt <lb></lb> <arrow.to.target n="marg123"></arrow.to.target> DZ ad ZC, ſic AF ad FC. at〈que〉 DO ad OB eſt, vt DZ ad <lb></lb>ZC. erit igitur AE ad EB, vt AF ad FC. quare EF ipſi BC <lb></lb> <arrow.to.target n="marg124"></arrow.to.target> eſt æquidiſtans, eodemquè modo oſtendetur, ita eſſe AG ad <lb></lb> <arrow.to.target n="fig43"></arrow.to.target><lb></lb>GB, vt AK ad KC, & AL ad LB, vt AM ad MC. ex quib^{9} <lb></lb>ſequitur LM GK EF non ſolùm ipſi BC, verùm etiam inter<lb></lb>ſeſe parallelas eſſe. </s> <s id="N1376D">ſecct EF lineas G<foreign lang="grc">ζ</foreign> K<foreign lang="grc">β</foreign> in X<foreign lang="grc">ε</foreign>. ipſam verò <lb></lb>AD in T. lineaquè GK ſecet L<foreign lang="grc">α</foreign> M<foreign lang="grc">ω</foreign> in N<foreign lang="grc">δ</foreign>, & AD in Y. <lb></lb>linea deniquè LM ipſam AD in S diſpeſcat. </s> <s id="N1378B">Quoniam au<lb></lb>tem D<foreign lang="grc">ω</foreign> eſt ipſi HI æquidiſtans, eſtquè D<foreign lang="grc">ω</foreign> minor <expan abbr="quã">quam</expan> HI, li <lb></lb>nea <foreign lang="grc">ω</foreign>M ipſi AL ęquidiſtans ipſam HI ſecabir. </s> <s id="N137A1">ac propterea <lb></lb>punctum H centrum grauitatis trianguli ABC extra paral<lb></lb> <arrow.to.target n="marg125"></arrow.to.target> lelogrammum DM reperitur. </s> <s id="N137AB">At verò quoniam LD DM <lb></lb>ſunt para lelogramma, erunt LS <foreign lang="grc">α</foreign>D inter ſe æquales, ſimili<lb></lb>ter SM D<foreign lang="grc">ω</foreign> ęquales. </s> <s id="N137B9">ſuntverò <foreign lang="grc">α</foreign>D D<foreign lang="grc">ω</foreign> ęquales: ergo & LS <lb></lb>SM inter ſe ſunt ęquales. </s> <s id="N137C5">eademquè rarione NY Y<foreign lang="grc">δ</foreign> inter ſe<lb></lb>ſe, & ipſis LS SM ęquales exiſtent. </s> <s id="N137CD">quarelinea SY bifariam <lb></lb>diuiditlatera oppoſita parallelogrammi MN. pariquè ratio<lb></lb>ne oſtendetur lineam YT bifariam diuidere oppoſita latera <lb></lb>parallelogrammi KX; lineamquè TD latera oppoſita paral- <pb xlink:href="077/01/101.jpg" pagenum="97"></pb>lelogrammi FO bifariam quo〈que〉 diuidere. <emph type="italics"></emph>Ita〈que〉 parallelogrà <lb></lb>mi MN centrum grauitatis est in linea <foreign lang="grc">Υ</foreign>S. parallilogrammi ver<gap></gap><lb></lb>KX grouitatis centrum est in linea T<foreign lang="grc">Υ</foreign>. parallelogrammi autem FO in <lb></lb>linea TD; magnitudinis igitur ex<emph.end type="italics"></emph.end> his <emph type="italics"></emph>omnibus<emph.end type="italics"></emph.end> parallelogrammi <lb></lb>MN KX FO <emph type="italics"></emph>compoſitæ centrum grauitatis eſt in recta linea S D. ſiv <lb></lb>ita〈que〉 punctum R.<emph.end type="italics"></emph.end> quod quidem erit centrum grauitatis figura <lb></lb>LNGXEOZF <foreign lang="grc">ε</foreign>K<foreign lang="grc">δ</foreign>M. <emph type="italics"></emph><expan abbr="lũgaturq́">lungatur〈que〉</expan>; RH, & producatur,<emph.end type="italics"></emph.end> quæ ipsa <foreign lang="grc">ω</foreign>M <lb></lb>ſecet in P. <emph type="italics"></emph>ipſiquè AD<emph.end type="italics"></emph.end> a puncto C <emph type="italics"></emph>æqui diſtans ducatur CV,<emph.end type="italics"></emph.end> qu<gap></gap><lb></lb>ipſi RH occurrat in V. <emph type="italics"></emph><expan abbr="triangulũ">triangulum</expan> ita〈que〉 ADC ad omnia triangu<lb></lb>la ex AM MK<emph.end type="italics"></emph.end> k<emph type="italics"></emph>F FC deſcripta ſimiliaipſi ADC,<emph.end type="italics"></emph.end> hoc eſt ad tria <lb></lb>gula ASM M <foreign lang="grc">δ</foreign>K K<foreign lang="grc">ε</foreign>F FZC ſimul ſumpta <emph type="italics"></emph>eandem habet propor <lb></lb>tionem, quam habet CA ad AM. ſiquidem ſunt AM MK<emph.end type="italics"></emph.end> k<emph type="italics"></emph>F FC<emph.end type="italics"></emph.end> <arrow.to.target n="marg126"></arrow.to.target><lb></lb><emph type="italics"></emph>æquales quia verò & triangulum ADB ad omnia ex AL LG GE <lb></lb>EB deſcripta triangula ſimilia<emph.end type="italics"></emph.end> ALS LGN GEX EFO <emph type="italics"></emph>eandem ha <lb></lb>bet proportionem, quam ‘BA ad AL<emph.end type="italics"></emph.end>: & antecedentes ſimul ad <arrow.to.target n="marg127"></arrow.to.target><lb></lb>omnes conſe〈que〉ntes, hoc eſt totum triangulum ABC ad on <lb></lb>nia triangula ſimul ſumpta, quæ ſunt in AB, & in AC conſti<lb></lb>tuta, eandem habebit proportionem, quam habet AC AB ſi<lb></lb>mul ad AM AL ſimul, quia verò ob <expan abbr="ſimilitudinẽ">ſimilitudinem</expan> <expan abbr="triangulorũ">triangulorum</expan> <lb></lb>ABC ALM CA ad AM eſt, vt BA ad AL; erit CA ad AM, vt <lb></lb>CA BA ſimul ad AM AL ſimul. <emph type="italics"></emph>triangulum igitur ABC ad omnia<emph.end type="italics"></emph.end> <arrow.to.target n="marg128"></arrow.to.target><lb></lb><emph type="italics"></emph>prædicta triangula eandem habet proportionem quam habet CA ad AM. <lb></lb>At〈que〉 CA ad AM maiorem habet proportionem quàm VR ad RH; e<lb></lb>tenim proportio ipſius CA ad AM eſt eadem, quæ est totius VR <expan abbr="adipsã">adipsam</expan> <lb></lb>R. p. <expan abbr="quãdoquidẽ">quandoquidem</expan> triangula<emph.end type="italics"></emph.end> ACD MC<foreign lang="grc">ω</foreign> <emph type="italics"></emph>ſunt ſimilia.<emph.end type="italics"></emph.end> <expan abbr="ſintq́">ſint〈que〉</expan>; AD & <arrow.to.target n="marg129"></arrow.to.target><lb></lb>M<foreign lang="grc">ω</foreign> ęquidiſtantes, ſitquè propterea CA ad AM, vt CD ad <lb></lb>D<foreign lang="grc">ω</foreign>. & quoniam VR DC àlineis DR <foreign lang="grc">ω</foreign>p CV æquidiſtantib^{9} <arrow.to.target n="marg130"></arrow.to.target><lb></lb>diuiduntur; erit C<foreign lang="grc">ω</foreign> ad <foreign lang="grc">ω</foreign>D, vt VP ad PR. & <expan abbr="cõponendo">componendo</expan> CD <arrow.to.target n="marg131"></arrow.to.target><lb></lb>ad D<foreign lang="grc">ω</foreign>, vt VR ad RP. quare vt CA ad AM, ita VR ad RP. <arrow.to.target n="marg132"></arrow.to.target><lb></lb>quia verò VR ad RP maiorem habet proportionem, quàm <arrow.to.target n="marg133"></arrow.to.target><lb></lb>ad RH. maiorem quo〈que〉 habebit proportionem CA ad <lb></lb>AM, quàm VR ad RH. eſt autem CA ad AM, vt <expan abbr="triangulũ">triangulum</expan> <lb></lb>ABC ad omnia triangula in lineis AC AB. (vt dictum eſt) <lb></lb>conſtituta; ergo <emph type="italics"></emph>& triangulum ABC adprædicta<emph.end type="italics"></emph.end> triangula <emph type="italics"></emph>maio <lb></lb>rem habet proportionem, quàm VR ad RH. Quare & diuidendo pa-<emph.end type="italics"></emph.end> <arrow.to.target n="marg134"></arrow.to.target><lb></lb><emph type="italics"></emph><expan abbr="rallelogrāma">rallelogramma</expan> MN<emph.end type="italics"></emph.end> k<emph type="italics"></emph>X FO<emph.end type="italics"></emph.end> hoc eſt figura LNGXEOZF <foreign lang="grc">ε</foreign>K <foreign lang="grc">δ</foreign>M) <emph type="italics"></emph>ad <lb></lb>circumrelicta triangula<emph.end type="italics"></emph.end> in lineis AC AB conſtituta <emph type="italics"></emph>maiorem ha-<emph.end type="italics"></emph.end> <pb xlink:href="077/01/102.jpg" pagenum="98"></pb><emph type="italics"></emph>bent proportionem, quam NH ad HR.<emph.end type="italics"></emph.end> linea igitur, quæ eandem <lb></lb>habeat proportionem ad HR, quam parallelogramma MN <lb></lb>kX FO ad circumrelicta triangula, maior erit, quàm VH <lb></lb><emph type="italics"></emph>Fiat itaquè in eademproportione QH ad HR, ut parallelogramma ad <lb></lb>triangula;<emph.end type="italics"></emph.end> erit vti〈que〉 QH maior, quam VH. <emph type="italics"></emph>Quoniam igitur eſt <lb></lb>magnitudo ABC, cuius centrum grauitatis est H, & ab ea magnitudo<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="fig44"></arrow.to.target><lb></lb><emph type="italics"></emph>auferatur compoſita ex MN<emph.end type="italics"></emph.end> k<emph type="italics"></emph>X FO parallelogrammis; & magnitudi <lb></lb>nis ablatæ centrum grauitatis eſt punctum R; magnitudinis reliquæ ex <lb></lb>circumrelictis triangulis compoſitæ centrum grauitatis erit in recta li-<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg135"></arrow.to.target> <emph type="italics"></emph>nea RH<emph.end type="italics"></emph.end> ex parte H <emph type="italics"></emph>producta, aſſumptaquè aliqua<emph.end type="italics"></emph.end> vt, QH, <emph type="italics"></emph>quæ ad <lb></lb>HR eam habeat proportionem, quam habet magnnudo<emph.end type="italics"></emph.end> ex parallelo<lb></lb>grammis MN KX FO conſtans <emph type="italics"></emph>ad reliquum,<emph.end type="italics"></emph.end> hoc eſt ad reli<lb></lb>qua triangula, <emph type="italics"></emph>ergo punctum Q centrum est grauitatis magnitudinis <lb></lb>ex ipſis circumrelictis<emph.end type="italics"></emph.end> triangulis <emph type="italics"></emph>compoſitæ. </s> <s id="N1397F">quoa fieri non poteſi aucta <lb></lb>enim recta linea <foreign lang="grc">θκ</foreign> per Q ipſi AD æquidistante in<emph.end type="italics"></emph.end> eodem <emph type="italics"></emph>plano<emph.end type="italics"></emph.end> <expan abbr="triã">triam</expan> <lb></lb>guli ABC, <emph type="italics"></emph>in ipſa eſſent omnia centra<emph.end type="italics"></emph.end> grauitatis trian<lb></lb>gulorum, <emph type="italics"></emph>hoc est in vtram〈que〉 partem<emph.end type="italics"></emph.end> Q<foreign lang="grc">θ</foreign> Q<foreign lang="grc">κ</foreign>, centraquè <lb></lb>grauitatis trianguli ALM, ac centrum magnitudinis ex vtriſ<lb></lb>què triangulis LGN MK <foreign lang="grc">δ</foreign> <expan abbr="cōpoſitę">compoſitę</expan> in parte Q<foreign lang="grc">θ</foreign> eſſe <expan abbr="deberẽt">deberent</expan>. <pb xlink:href="077/01/103.jpg" pagenum="99"></pb>centra verò grauitatis magnitudinis ex GEX K<foreign lang="grc">ε</foreign>F compo<lb></lb>ſitę, ac magnitudinis ex. </s> <s id="N139CA">EBO FZC compoſſtæ, eſſent in par <lb></lb>te Q<foreign lang="grc">κ</foreign>, ita vt punctum Q magnitudinis ex omnibus trian<lb></lb>gulis compoſitæ centrum eſſet grauitatis. </s> <s id="N139D4">quæ <expan abbr="quidẽſunt">quidenſunt</expan> om<lb></lb>nino abſurda. </s> <s id="N139DC">Quòd ſi ducta linea per Q, non fuerit etiam <lb></lb>ipſi AD ęquidiſtans, eadem ſe〈que〉ntur in conuenientia. <emph type="italics"></emph>Ma <lb></lb>niſestum eſt igitur; quod propoſitum fuerat.<emph.end type="italics"></emph.end></s> </p> <p id="N139E7" type="margin"> <s id="N139E9"><margin.target id="marg122"></margin.target><emph type="italics"></emph>ex<emph.end type="italics"></emph.end> t. <emph type="italics"></emph>deci<lb></lb>mi.<emph.end type="italics"></emph.end></s> </p> <p id="N139F9" type="margin"> <s id="N139FB"><margin.target id="marg123"></margin.target>2. <emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N13A04" type="margin"> <s id="N13A06"><margin.target id="marg124"></margin.target>2. <emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N13A0F" type="margin"> <s id="N13A11"><margin.target id="marg125"></margin.target>34. <emph type="italics"></emph>primi.<emph.end type="italics"></emph.end></s> </p> <p id="N13A1A" type="margin"> <s id="N13A1C"><margin.target id="marg126"></margin.target>3. <emph type="italics"></emph>lemma.<emph.end type="italics"></emph.end></s> </p> <p id="N13A25" type="margin"> <s id="N13A27"><margin.target id="marg127"></margin.target><emph type="italics"></emph>ex<emph.end type="italics"></emph.end>12.<emph type="italics"></emph><expan abbr="quĩti">quinti</expan><emph.end type="italics"></emph.end></s> </p> <p id="N13A37" type="margin"> <s id="N13A39"><margin.target id="marg128"></margin.target><emph type="italics"></emph>ex<emph.end type="italics"></emph.end>12.<emph type="italics"></emph><expan abbr="quĩti">quinti</expan><emph.end type="italics"></emph.end></s> </p> <p id="N13A49" type="margin"> <s id="N13A4B"><margin.target id="marg129"></margin.target><emph type="italics"></emph>ex<emph.end type="italics"></emph.end> 4.<emph type="italics"></emph>ſexti<emph.end type="italics"></emph.end></s> </p> <p id="N13A59" type="margin"> <s id="N13A5B"><margin.target id="marg130"></margin.target>1. <emph type="italics"></emph>lemma.<emph.end type="italics"></emph.end></s> </p> <p id="N13A64" type="margin"> <s id="N13A66"><margin.target id="marg131"></margin.target>8. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N13A6F" type="margin"> <s id="N13A71"><margin.target id="marg132"></margin.target>11. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N13A7A" type="margin"> <s id="N13A7C"><margin.target id="marg133"></margin.target>8. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N13A85" type="margin"> <s id="N13A87"><margin.target id="marg134"></margin.target>20. <emph type="italics"></emph>quinti <lb></lb>add.<emph.end type="italics"></emph.end></s> </p> <p id="N13A92" type="margin"> <s id="N13A94"><margin.target id="marg135"></margin.target>8.<emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.103.1.jpg" xlink:href="077/01/103/1.jpg"></figure> <figure id="id.077.01.103.2.jpg" xlink:href="077/01/103/2.jpg"></figure> <p id="N13AA5" type="head"> <s id="N13AA7">SCHOLIVM.</s> </p> <p id="N13AA9" type="main"> <s id="N13AAB">Id ipſum vult ad huc Archimedes aliter oſtendere. </s> <s id="N13AAD">ob <expan abbr="ſe〈quẽ〉">ſe〈que〉m</expan> <lb></lb>tem verò demonſtrationem hoc priùs cognoſcere oportet. </s> </p> <p id="N13AB5" type="head"> <s id="N13AB7">LEMMA.</s> </p> <p id="N13AB9" type="main"> <s id="N13ABB">Si intra triangulum vni lateri ęquidiſtans ducatur, ab op<lb></lb>poſito autem angulo intra triangulum quoquè recta ducatur <lb></lb>linea, æquidiſtantes lineas in eadem proportione diſpeſcet. </s> </p> <p id="N13AC1" type="main"> <s id="N13AC3">Hoc in ſecundo noſtrorum planiſphęriorum libro in ea <lb></lb>parte oſtendimus, vbi quomodo conficienda ſit ellipſis, inſtru<lb></lb>mento à nobis inuento demonſtrauimus. </s> <s id="N13AC9">hoc nempè modo, <lb></lb> <arrow.to.target n="fig45"></arrow.to.target><lb></lb>Sit triangulum ABC, ipſiquè BC in<lb></lb>tra triangulum ducatur vtcumquè æ<lb></lb>quidiſtans DE. à punctoquè A intra <lb></lb>triangulum ſimiliter quocum〈que〉 du<lb></lb>catur AF; quæ lineam BC ſecet in F; <lb></lb>lineam verò DE in G. Dico ita oſſe <lb></lb>CF ad FB, vt EG ad GD. <expan abbr="Quoniã">Quoniam</expan> <lb></lb>enim GE FC ſunt æquidiſtantes, erit <lb></lb>triangulum AFC triangulo AGE æquiangulum, vt igitur <arrow.to.target n="marg136"></arrow.to.target><lb></lb>AF ad AG, ita CF ad EG. ob eandemquè cauíam ita eſt FA <lb></lb>ad AG, vt FB ad GD. quare vt CF ad EG, ita eſt FB ad GD. <arrow.to.target n="marg137"></arrow.to.target><lb></lb>ac permutando, vt CF ad FB, ita EG ad GD. quod demon <arrow.to.target n="marg138"></arrow.to.target><lb></lb>ſtrare oportebat. </s> </p> <pb xlink:href="077/01/104.jpg" pagenum="100"></pb> <p id="N13AFA" type="margin"> <s id="N13AFC"><margin.target id="marg136"></margin.target><emph type="italics"></emph>ex<emph.end type="italics"></emph.end> 4.<emph type="italics"></emph>ſexti<emph.end type="italics"></emph.end></s> </p> <p id="N13B0A" type="margin"> <s id="N13B0C"><margin.target id="marg137"></margin.target>11.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N13B15" type="margin"> <s id="N13B17"><margin.target id="marg138"></margin.target>16.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.104.1.jpg" xlink:href="077/01/104/1.jpg"></figure> <p id="N13B24" type="head"> <s id="N13B26"><emph type="italics"></emph>IDEM ALITER.<emph.end type="italics"></emph.end></s> </p> <p id="N13B2C" type="main"> <s id="N13B2E"><emph type="italics"></emph>Sit triangulum ABC, ducaturquè AD<emph.end type="italics"></emph.end> ab angulo A <emph type="italics"></emph>ad <expan abbr="dimidiã">dimidiam</expan><emph.end type="italics"></emph.end><lb></lb>baſim <emph type="italics"></emph>BC. Dico in linea AD centrum eſſe grauitatis trianguli ABC. <lb></lb>N on ſit autem, ſed ſi fieri poteſt; ſit H. iunganturquè AH HB HC, & <lb></lb>ED<emph.end type="italics"></emph.end> DF <emph type="italics"></emph>FE ad dimidias BA<emph.end type="italics"></emph.end> BC <emph type="italics"></emph>AC<emph.end type="italics"></emph.end> ducantur, ſecetquè EF ip<lb></lb>ſam AD in M. & <emph type="italics"></emph>ipſi AH æquidistantes ducantur EK FL. &<emph.end type="italics"></emph.end> <lb></lb><arrow.to.target n="fig46"></arrow.to.target><lb></lb><emph type="italics"></emph>iungantur KL LD Dk DH<emph.end type="italics"></emph.end>; ſecetquè DH ipſam KL in N. <lb></lb>iungaturquè <emph type="italics"></emph>MN. Quoniam igitur triangulum ABC ſimile est <expan abbr="triã">triam</expan> <lb></lb>gulo DFC, cùm ſit BA ipſi FD æquidistans<emph.end type="italics"></emph.end>; ſiquidem ſunt late<lb></lb> <arrow.to.target n="marg139"></arrow.to.target> ra CA CB bifariam diuiſa, ideoquè ſit CF ad FA, vt CD <lb></lb>ad DB. <emph type="italics"></emph>trianguliquè ABC centrum grauitatis est punctum H; &<emph.end type="italics"></emph.end> <lb></lb> <arrow.to.target n="marg140"></arrow.to.target> <emph type="italics"></emph>trianguli FDC centrum grauitatis erit punctum L. puncta enim HB <lb></lb>intra vtrumquè triangulum ſunt ſimiliter poſita. </s> <s id="N13B8E">etenim ad homologa <lb></lb>latera angulos efficiunt æquales. </s> <s id="N13B92">hoc enim perſpicuum. </s> <s id="N13B94">est<emph.end type="italics"></emph.end> cùm enim <lb></lb>ſint triangulorum ABC DFC homologa latera AC FC, <lb></lb> <arrow.to.target n="marg141"></arrow.to.target> AB FD, BC DC, ſintquè AH FL æquidiſtantes; erit an<lb></lb>gulus LFC angulo HAC ęqualis. </s> <s id="N13BA3">ſed angulus CFD eſt ipſi <pb xlink:href="077/01/105.jpg" pagenum="101"></pb>CAB æqualis; reliquus igitur angulus LFD reliquo HAB <lb></lb>æqualis exiſtit. </s> <s id="N13BAB">& quoniam ita eſt CF ad FA, vt CL ad LH, <arrow.to.target n="marg142"></arrow.to.target><lb></lb>cùm ſint FL AH ęquidiſtantes. </s> <s id="N13BB2">CF verò dimidia eſt ipſius <lb></lb>CA, erit & CL ipſius quo〈que〉 CH dimidia. </s> <s id="N13BB6">at CD ipſius <lb></lb>CB dimidia exiſtit; erit igitur DL ipſi BH ęquidiſtans. </s> <s id="N13BBA">ac <arrow.to.target n="marg143"></arrow.to.target><lb></lb>propterea angulus LDC eſt ipſi HBC ęqualis, & LDF ipſi <arrow.to.target n="marg144"></arrow.to.target><lb></lb>HBA ęqualis. </s> <s id="N13BC6">cùm ſittotus CDF toti CBA ęqualis; anguli <lb></lb>verò ACH & HCB tam ſunt trianguli ABC, quàm FDC. <lb></lb><emph type="italics"></emph>Obeandem autem rationem trianguli EBD centrum grauitatis est <expan abbr="pũ-">pun-</expan><emph.end type="italics"></emph.end> <arrow.to.target n="marg145"></arrow.to.target><lb></lb><emph type="italics"></emph>ctum K.<emph.end type="italics"></emph.end> ſimiliter enim oſtendetur punctum K in triangu<lb></lb>lo EBD eſſe ſimiliter poſitum, vt H in triangulo ABC. <lb></lb><emph type="italics"></emph>Quare magnitudinis ex vtriſquè triangulis EBD FDC compoſitæ <lb></lb>centrum grauitatis eſt in medietate lineæ<emph.end type="italics"></emph.end> k<emph type="italics"></emph>L. cum triangula EBD<emph.end type="italics"></emph.end> <arrow.to.target n="marg146"></arrow.to.target><lb></lb><emph type="italics"></emph>FDC ſint æqualia.<emph.end type="italics"></emph.end> ſunt enim in ęqualibus baſibus BD DC, <arrow.to.target n="marg147"></arrow.to.target><lb></lb>& in ijſdem parallelis EF BC, ſiquidem eſt AE ad EB, vt <arrow.to.target n="marg148"></arrow.to.target><lb></lb>AF ad FC. quippè cùm latera AB AC ſint bifariam diui<lb></lb>ſa. <emph type="italics"></emph>medium veròipſius<emph.end type="italics"></emph.end> k<emph type="italics"></emph>L eſt punctum N; cùm ſit<emph.end type="italics"></emph.end> KE ipſi AH <lb></lb>ęquidiſtans, & ob id ſit <emph type="italics"></emph>BE ad EA, vt B<emph.end type="italics"></emph.end>k <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> k<emph type="italics"></emph>H.<emph.end type="italics"></emph.end> & vt BE <arrow.to.target n="marg149"></arrow.to.target><lb></lb>ad EA, ita CF ad FA; <emph type="italics"></emph>vt autem CF ad FA, ſic CL ad LH.<emph.end type="italics"></emph.end><lb></lb>quare vt BK ad KH, ita CL ad LH. <emph type="italics"></emph>Si autem hoc. </s> <s id="N13C34">æquidi-<emph.end type="italics"></emph.end> <arrow.to.target n="marg150"></arrow.to.target><lb></lb><emph type="italics"></emph>ſtans est BC ipſi<emph.end type="italics"></emph.end> k<emph type="italics"></emph>L, & iuncta est DH, erit igitur BD ad DC, vt<emph.end type="italics"></emph.end> <arrow.to.target n="marg151"></arrow.to.target><lb></lb><emph type="italics"></emph>KN ad NL.<emph.end type="italics"></emph.end> D verò medium eſt ipſius BC. ergo & N <arrow.to.target n="marg152"></arrow.to.target> me<lb></lb>dium eſt ipſius KL. <emph type="italics"></emph>Quare magnitudinis ex vtriſquè <expan abbr="dictorũ">dictorum</expan> trian <lb></lb>gulorum<emph.end type="italics"></emph.end> EBD & FDC <emph type="italics"></emph>compoſitæ centrum<emph.end type="italics"></emph.end> grauitatis <emph type="italics"></emph>est punctum<emph.end type="italics"></emph.end> <arrow.to.target n="marg153"></arrow.to.target><lb></lb><emph type="italics"></emph>N. parallelogrammi verò AEDF centrum grauitatis eſt punctum M,<emph.end type="italics"></emph.end><lb></lb>vbi ſimiliter diametri concurrunt, <emph type="italics"></emph>ac propterea magnitudinis ex<emph.end type="italics"></emph.end> <arrow.to.target n="marg154"></arrow.to.target><lb></lb><emph type="italics"></emph>omnibus<emph.end type="italics"></emph.end> triangulis EBD FDC vna <expan abbr="cũ">cum</expan> parallelogramo AEDF <lb></lb><emph type="italics"></emph>compoſitæ centrum grauitatis eſt in linea MN. Verùm<emph.end type="italics"></emph.end> <expan abbr="triangulorũ">triangulorum</expan> <lb></lb>EBD FDC, ſimulquè parallelogrammi AEDF, hoc eſt totius <lb></lb><emph type="italics"></emph>trianguli ABC grauitatis centrum est punctum H; linea igitur MN pro<emph.end type="italics"></emph.end> <arrow.to.target n="marg155"></arrow.to.target><lb></lb><emph type="italics"></emph>ducta tranſibit per punctum H. quod eſſe non poteſt.<emph.end type="italics"></emph.end> etenim cùm ſit <lb></lb>KN ipſi BD æquidiſtans; erit BK ad KH, vt DN ad <lb></lb>NH: vt autem BK ad KH, ita eſt BE ad EA, & vt BE ad <lb></lb>EA, ita eſt DM ad MA, cùm ſit EM ipſi BD æquidiſtans. <lb></lb>erit igitur DM ad MA, vt DN ad NH. quare MN ipſi AH <lb></lb>eſt ęquidiſtans; ideoquè MN numquam cùm AH conueni<lb></lb>re poteſt. <emph type="italics"></emph>Non est igitur<emph.end type="italics"></emph.end> punctum <emph type="italics"></emph>H centrum grauitatis trianguli<emph.end type="italics"></emph.end> <pb xlink:href="077/01/106.jpg" pagenum="102"></pb><emph type="italics"></emph>ABC. quare non eſt extra lineam AD. in ipſi igitur exiſtit.<emph.end type="italics"></emph.end> Quod <lb></lb>demonitrare oportebat. </s> </p> <p id="N13CD1" type="margin"> <s id="N13CD3"><margin.target id="marg139"></margin.target>2.<emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N13CDC" type="margin"> <s id="N13CDE"><margin.target id="marg140"></margin.target>11.<emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N13CE7" type="margin"> <s id="N13CE9"><margin.target id="marg141"></margin.target>29. <emph type="italics"></emph>primi.<emph.end type="italics"></emph.end></s> </p> <p id="N13CF2" type="margin"> <s id="N13CF4"><margin.target id="marg142"></margin.target>2. <emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N13CFD" type="margin"> <s id="N13CFF"><margin.target id="marg143"></margin.target>2.<emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N13D08" type="margin"> <s id="N13D0A"><margin.target id="marg144"></margin.target>29. <emph type="italics"></emph>primi.<emph.end type="italics"></emph.end></s> </p> <p id="N13D13" type="margin"> <s id="N13D15"><margin.target id="marg145"></margin.target>11. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N13D1E" type="margin"> <s id="N13D20"><margin.target id="marg146"></margin.target>4.<emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N13D29" type="margin"> <s id="N13D2B"><margin.target id="marg147"></margin.target>38. <emph type="italics"></emph>primi.<emph.end type="italics"></emph.end></s> </p> <p id="N13D34" type="margin"> <s id="N13D36"><margin.target id="marg148"></margin.target>2.<emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N13D3F" type="margin"> <s id="N13D41"><margin.target id="marg149"></margin.target>2.<emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N13D4A" type="margin"> <s id="N13D4C"><margin.target id="marg150"></margin.target>11.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N13D55" type="margin"> <s id="N13D57"><margin.target id="marg151"></margin.target>2.<emph type="italics"></emph>ſexti. <lb></lb>lemma.<emph.end type="italics"></emph.end></s> </p> <p id="N13D62" type="margin"> <s id="N13D64"><margin.target id="marg152"></margin.target>*</s> </p> <p id="N13D68" type="margin"> <s id="N13D6A"><margin.target id="marg153"></margin.target>11.<emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N13D73" type="margin"> <s id="N13D75"><margin.target id="marg154"></margin.target>*</s> </p> <p id="N13D79" type="margin"> <s id="N13D7B"><margin.target id="marg155"></margin.target>A</s> </p> <figure id="id.077.01.106.1.jpg" xlink:href="077/01/106/1.jpg"></figure> <p id="N13D83" type="head"> <s id="N13D85">SCHOLIVM.</s> </p> <p id="N13D87" type="main"> <s id="N13D89"> <arrow.to.target n="marg156"></arrow.to.target> Inquit Archimedes <emph type="italics"></emph>linea igitur MN producta tranſibit per pun<lb></lb>ctum H. quod eſſe non poteſt,<emph.end type="italics"></emph.end> nempè, vt non ipſamet linea MN, <lb></lb>ſed eius pars, ſiuead M, ſiue ad N producta cum H conue <lb></lb>nireoporteat. </s> <s id="N13D9B">cùm tamen ipſamet linea MN per punctum <lb></lb>H tranſire debeat. </s> <s id="N13D9F">ita vt punctum H ſit inter puncta MN; <lb></lb>hoc eſt in linea MN, & non in eius parte producta. </s> <s id="N13DA3">Nam ſi <lb></lb>punctum H centrum eſt grauitatis totius trianguli ABC. <lb></lb>punctum verò N centrum grauitatis magnitudinis ex <expan abbr="triãgu">triangu</expan> <lb></lb>lis EBD FDC compoſitę; at〈que〉 punctum M centrum gra<lb></lb>uitatis parallelogrammi AEDF; oportet vt punctum H ita li<lb></lb>neam diuidat MN; vt eius partes magnitudinibus permuta<lb></lb>tim reſpondeant. </s> <s id="N13DB5">vt nimirum pars ad M ad partem ad N ſit, <lb></lb>vt magnitudo ex triangulis EBD FDC conſtans ad parallelo <lb></lb>grammum AEDF. vt ex ſexta, & octaua huius propoſitione <lb></lb>perſpicuum eſt. </s> <s id="N13DBD">Quare punctum H in linea MN eſſe debe<lb></lb>ret; vt ipſemet Atchimedes paulò ſuperiùs affirmauit; cùm in<lb></lb> <arrow.to.target n="marg157"></arrow.to.target> quit. <emph type="italics"></emph>ac propterea magnitudinis ex omnibus compoſitæ contrum grauita<lb></lb>tis eſt in linea MN.<emph.end type="italics"></emph.end> & non dixit in eius parte producta. </s> <s id="N13DCF">Quodiv <lb></lb>ca vel del<gap></gap>dum eſt verbum illud <emph type="italics"></emph>producta,<emph.end type="italics"></emph.end> tanquam ab aliquo <lb></lb>additum, vel ideo tamen hoc dixiſſe voluit Archimedes, vt o<lb></lb>ſtenderet lineam MN nullo modo (etiam ſi produceretur) <expan abbr="cũ">cum</expan> <lb></lb>H conuenire poſſe. </s> </p> <p id="N13DE5" type="margin"> <s id="N13DE7"><margin.target id="marg156"></margin.target>A</s> </p> <p id="N13DEB" type="margin"> <s id="N13DED"><margin.target id="marg157"></margin.target>*</s> </p> <p id="N13DF1" type="head"> <s id="N13DF3">PROPOSITIO. XIIII.</s> </p> <p id="N13DF5" type="main"> <s id="N13DF7">Omnis trianguli centrum grauitatis eſt <expan abbr="punctũ">punctum</expan> <lb></lb>in quo rectæ lineæ ab angulis trianguli ad dimidia <lb></lb>later a ductæ concurrunt. </s> </p> <pb xlink:href="077/01/107.jpg" pagenum="103"></pb> <p id="N13E04" type="main"> <s id="N13E06"><emph type="italics"></emph>Sit triangulum ABC, &<emph.end type="italics"></emph.end> ab angulo A <emph type="italics"></emph>ducatur AD ad dimi<lb></lb>diam BC. BE verò<emph.end type="italics"></emph.end> ab angulo B <emph type="italics"></emph>ad dimidiam AC.<emph.end type="italics"></emph.end> quę quidem <lb></lb>lineę AD BE ſeinuicem ſecent in <expan abbr="pū">pum</expan> <lb></lb> <arrow.to.target n="fig47"></arrow.to.target><lb></lb>cto H. <emph type="italics"></emph>Quoniam igitur centrum grauita<lb></lb>tis trianguli ABC est in vtra〈que〉 linea <lb></lb>AD BE; hoc enim demonstratum eſt<emph.end type="italics"></emph.end> in <lb></lb>pręcedenti. </s> <s id="N13E34">erit vti〈que〉 centrum graui<lb></lb>tatis, vbilineç AD BE ſe <expan abbr="inuicẽ">inuicem</expan> <expan abbr="ſecãt">ſecant</expan>. <lb></lb>ſecant verò ſeſe in H. <emph type="italics"></emph>ergo punctum <lb></lb>H centrum eſt grauitatis<emph.end type="italics"></emph.end> trianguli ABC. <lb></lb>quod demonſtrare oportebat. </s> </p> <figure id="id.077.01.107.1.jpg" xlink:href="077/01/107/1.jpg"></figure> <p id="N13E50" type="head"> <s id="N13E52">SCHOLIVM.</s> </p> <p id="N13E54" type="main"> <s id="N13E56">Similiter ſi ducta fuerit CH, & producta, bifariam ſecaret <lb></lb>AB. In hac enim linea eſſet centrum grauitatis trianguli; <expan abbr="cẽ">cem</expan> <lb></lb>trum verò eſt in linea ab angulo ad dimidiam baſim ducta: <lb></lb>ergo hæc linea ab angulo C ad dimidiam AB ducta eſſet. <lb></lb>Præterea ſi linea à puncto C ad dimidiam AB ducta <expan abbr="nõ">non</expan> tran <lb></lb>ſiret per H; eſſet vti〈que〉 in hac linea centrum grauitatis; ſed <arrow.to.target n="marg158"></arrow.to.target> <expan abbr="cẽ-trum">cen<lb></lb>trum</expan> quo〈que〉 grauitatis eſt in linea AD, & in linea BE, ut in <lb></lb>H; vnius igitur figurę plura darentur centra grauitatis. </s> <s id="N13E76">quod <lb></lb>fieri non poteſt. </s> <s id="N13E7A">quod quidem, cùm ſit in con ueniens, nos in <lb></lb>noſtro Mechanicorum libro dari non poſſe ſuppoſuimus. <lb></lb>Quare linea CH indirectum ducta, bifariam ſecaret AB. <lb></lb>quod quidem paulò infra aliter quo〈que〉 oſtendemus, <expan abbr="nõnul">nonnul</expan> <lb></lb>lis prius demonſtratis; quæ Archimedes ob ſe〈que〉ntem <expan abbr="demõ-ſtrationem">demon<lb></lb>ſtrationem</expan>, tanquam demonſtrata ſupponit. </s> <s id="N13E8E">Vult enim Ar<lb></lb>chimedes, poſtquam inuenit centrum grauitatis cuiuſlibet <lb></lb>trianguli, centrum quo〈que〉 grauitatis quærere trapetij duo la<lb></lb>tera ęquidiſtantia habentis. </s> <s id="N13E96">quod eſt quidem pars trianguli, <lb></lb>& tanquam fruſtum a triangulo abſciſſum. </s> <s id="N13E9A">ſupponitquè den<lb></lb>trum grauitatis cuiuſlibet trianguli eſſe in recta linea baſi du<lb></lb>cta ęquidiſtante, quæ latera ita diuidat, vt partes ad uerticem <lb></lb>ſint reliquarum partium duplæ. </s> <s id="N13EA2">quod quidem ortum ducit <lb></lb>ex cognitione alterius theorematis oſtendentis centrum gra- <pb xlink:href="077/01/108.jpg" pagenum="104"></pb>uitatis cuiuſlibet trianguli eſſe in recta linea ab angulo ad di<lb></lb>midiam baſim ducta (vt Archimedes demonſtrauit) & inſu<lb></lb>per in eo puncto, quod dictam lineam diuidatita, vt pars ad <lb></lb>angulum reliquę ad baſim ſit dupla. </s> <s id="N13EB0">Quare hoc prius ita <expan abbr="oſtẽ">oſtem</expan> <lb></lb>demus. </s> </p> <p id="N13EB8" type="margin"> <s id="N13EBA"><margin.target id="marg158"></margin.target>13.<emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N13EC3" type="head"> <s id="N13EC5">PROPOSITIO.</s> </p> <p id="N13EC7" type="main"> <s id="N13EC9">Omnis trianguli centrum grauitatis eſt punctum in recta <lb></lb>linea ab angulo ad dimidiam baſim ducta exiſtens, quod li<lb></lb>neam diuidat, ita vt poitio ad angulum reliquæ ad baſim, ſit <lb></lb>dupla. </s> </p> <p id="N13ED1" type="main"> <s id="N13ED3">Sit triangulum ABC, in quo ab an <lb></lb> <arrow.to.target n="fig48"></arrow.to.target><lb></lb>gulo A ad dimidiam baſim BC re<lb></lb>cta ducatur linea AD. Ducaturquè <lb></lb>ab angulo B ad dimidiom baſim <lb></lb>AC linea BE, quæſecet AD in F. Et <lb></lb>quoniam centrum grauitatis <expan abbr="triãgu-">triangu<lb></lb></expan> <arrow.to.target n="marg159"></arrow.to.target> li ABC eſt punctum F; <expan abbr="oſtendendũ">oſtendendum</expan> <lb></lb>eſt lineam FA ipſius FD duplam eſ<lb></lb>ſe. </s> <s id="N13EF5">iungatur FC. quoniam enim AE <lb></lb>eſt equalis ipſi EC, erit triangulum <lb></lb> <arrow.to.target n="marg160"></arrow.to.target> ABE triangulo EBC æquale, cùm <lb></lb>ſint ſub eadem altitudine. </s> <s id="N13F01">Ob eandemquè cauſam <expan abbr="triangulũ">triangulum</expan> <lb></lb>AFE triangulo EFC exiſtit æquale. </s> <s id="N13F09">ſi igitur à triangulo ABE <lb></lb>auferatur triangulum AFE, & à triangulo EBC triangulum <lb></lb>auferatur EFC; relin〈que〉tur triangulum ABF triangulo BFC <lb></lb>æquale. </s> <s id="N13F11">Rurſus quoniam BD eſt æqualis ipſi DC; erit trian<lb></lb> <arrow.to.target n="marg161"></arrow.to.target> gulum BFD triangulo DFC æquale, ſiquidem candem ha<lb></lb>bentaltitudinem. </s> <s id="N13F1B">duplum igitur eſt triangulum BFC <expan abbr="triãgu-li">triangu<lb></lb>li</expan> BFD. Quare & triangulum ABF trianguli BFD duplum <lb></lb> <arrow.to.target n="marg162"></arrow.to.target> exiſtit. </s> <s id="N13F29">quia verò triangula ABF FBD in eadem ſunt altitudi <lb></lb>ne, idcirco ſeſe habebunt, vt baſes AF FD. at〈que〉 triangulum <lb></lb>ABF. duplum eſt ipſius FBD; ergo portio AF ipſius FD dupla <lb></lb>exiſtit. </s> <s id="N13F31">quod demonſtrare oportebat. </s> </p> <pb xlink:href="077/01/109.jpg" pagenum="105"></pb> <p id="N13F36" type="margin"> <s id="N13F38"><margin.target id="marg159"></margin.target>14.<emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N13F41" type="margin"> <s id="N13F43"><margin.target id="marg160"></margin.target>1.<emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N13F4C" type="margin"> <s id="N13F4E"><margin.target id="marg161"></margin.target>1.<emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N13F57" type="margin"> <s id="N13F59"><margin.target id="marg162"></margin.target>1.<emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.109.1.jpg" xlink:href="077/01/109/1.jpg"></figure> <p id="N13F66" type="main"> <s id="N13F68">ALITER. </s> </p> <p id="N13F6A" type="main"> <s id="N13F6C">Sit rurſus triangulum ABC, & AD BE ab angulis ad di <lb></lb>midias baſes ductæ ſint erit vti〈que〉 punctum, F (vbi ſe in ui <arrow.to.target n="marg163"></arrow.to.target><lb></lb>cen fecant) centrum grauitatis triangulb ABC. Drco AF a<lb></lb>pſius FD duplam eſſe. </s> <s id="N13F77">Iungatur DE. Quoniam enim BC <lb></lb> <arrow.to.target n="fig49"></arrow.to.target><lb></lb>AC in punctis DE bifariam ſecantur; erit <lb></lb>CD ad DB, vt CE ad EA. linea igitur <lb></lb>DE ipſi AB eſt æquidiſtans. </s> <s id="N13F84">quare <arrow.to.target n="marg164"></arrow.to.target> trian<lb></lb>gulum ABC ſimile eſt triangulo EDC. <arrow.to.target n="marg165"></arrow.to.target><lb></lb>ac propterea ita eſt BC ad CD, vt AB <lb></lb>ad DE. eſt autem. </s> <s id="N13F93">BC dupla ipſius CD <lb></lb>(ſiquidem punctum D bifariam diuidit <lb></lb>BC) erit igitur AB dupla ipſius DE. At <lb></lb>vero quoniam AB DE ſunt parallelæ, erit triangulum AFB <lb></lb>triangulo EFD ſimile. </s> <s id="N13F9D">& vt AB ad ED, ita AF ad FD, eſt <arrow.to.target n="marg166"></arrow.to.target><lb></lb>autem AB ipſius ED dupla, ergo AF ipſius FD dupla <lb></lb>exiſtit. </s> <s id="N13FA6">quod demonſtrare oportebat. </s> </p> <p id="N13FA8" type="margin"> <s id="N13FAA"><margin.target id="marg163"></margin.target>14. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N13FB3" type="margin"> <s id="N13FB5"><margin.target id="marg164"></margin.target>2. <emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N13FBE" type="margin"> <s id="N13FC0"><margin.target id="marg165"></margin.target>4. <emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N13FC9" type="margin"> <s id="N13FCB"><margin.target id="marg166"></margin.target>4.<emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.109.2.jpg" xlink:href="077/01/109/2.jpg"></figure> <p id="N13FD8" type="main"> <s id="N13FDA">Exijs, quæ demonſtrata ſunt, oſtendemus, quod paulò an<lb></lb>te propoiuimus, nempè cùm lineæ AD BE bifariam ſecent <lb></lb>BC CA. Dico lineam CF productam bifariam quo〈que〉 ſe<lb></lb>care ipſam AB. </s> </p> <p id="N13FE2" type="main"> <s id="N13FE4">Producatur enim (ijsdem poſitis) CFGH; quæ lineam <lb></lb> <arrow.to.target n="fig50"></arrow.to.target><lb></lb>AB ſecet in G. & à puncto B <lb></lb>ipſi AD æquidiſtans ducatur <lb></lb>BH. quæ ipſi CG occuriat in <lb></lb>H. Quoniam igitur FD, eſt i<lb></lb>pſi BH ęquidiſtans, erit CD <lb></lb>ad DB, vt CF ad FH. CD <arrow.to.target n="marg167"></arrow.to.target> ve<lb></lb>rò eſt æqualis BD; ergo CF ipſi <lb></lb>FH æqualis exiſtit. </s> <s id="N13FFF">ac propterea <lb></lb>CH dupla eſt ipſius (F. At ve<lb></lb>rò quoniam ob ſimilitudinem <lb></lb><expan abbr="triangulorũ">triangulorum</expan> CBH CDF, ita eſt <lb></lb>HC ad CF, vt BH ad DF; erit & BH ipſius FD duplex. <pb xlink:href="077/01/110.jpg" pagenum="106"></pb>verùm & AF (ex proximè demonſtratis) ipſius FD duplex <lb></lb>exiſtit. </s> <s id="N14012">erunt igitur BH FA inter ſe ęquales. </s> <s id="N14014">Quoniam autem <lb></lb>BH eſt ęquidiſtans ipſi AF, æquiangula erunt triagula GBH <lb></lb> <arrow.to.target n="marg168"></arrow.to.target> GAF. quare vt BH ad AF, ita BG ad GA, quia verò BH eſt <lb></lb>ipſi AF æqualis; erit & BG ipſi GA æqualis. </s> <s id="N14020">ergo recta li<lb></lb>nea EFG bifariam diuidit AB. quod demonſtrare oporte<lb></lb>bat. </s> </p> <p id="N14026" type="margin"> <s id="N14028"><margin.target id="marg167"></margin.target>2. <emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N14031" type="margin"> <s id="N14033"><margin.target id="marg168"></margin.target><emph type="italics"></emph>ex<emph.end type="italics"></emph.end> 4.<emph type="italics"></emph>ſexti<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.110.1.jpg" xlink:href="077/01/110/1.jpg"></figure> <p id="N14045" type="main"> <s id="N14047">Reliquum eſt, vt ob ſe〈que〉ntem demonſtrationem alteram <lb></lb>propoſitionem oſtendamus. </s> </p> <p id="N1404B" type="head"> <s id="N1404D">PROPOSITIO.</s> </p> <p id="N1404F" type="main"> <s id="N14051">Centrum grauitatis cuiuſlibet trianguli eſt in recta linea <lb></lb>baſi ducta æquidiſtante, quæ latus ita diuidat, vt pars ad an<lb></lb>gulum reliquæ ad baſim ſit dupla. </s> </p> <p id="N14057" type="main"> <s id="N14059">In trianagulo enim ABC ducta <lb></lb>ſit DE baſi BC æquidiſtans, quæ <lb></lb> <arrow.to.target n="fig51"></arrow.to.target><lb></lb>latus AB diuidat in D, ita vt DA <lb></lb>ipſius DB ſit duplex. </s> <s id="N14066">Dico in linea <lb></lb>DE centrum eſſe grauitatis triangu<lb></lb>li ABC. Ducatur ab angulo A ad <lb></lb>dimidiam BC linea AF, quæ di<lb></lb> <arrow.to.target n="marg169"></arrow.to.target> uidat DE in G. erit AD ad DB, <lb></lb>vt AG ad GF, ac propterea erit <lb></lb>AG ipſius GF dupla. </s> <s id="N14078">punctum er<lb></lb>go G centrum eſt grauitatis trian<lb></lb>guli ABC. Quare conſtat <expan abbr="centrũ">centrum</expan> <lb></lb>eſſe in linea DE. quod demonſtra<lb></lb>re oportebat </s> </p> <pb xlink:href="077/01/111.jpg" pagenum="107"></pb> <p id="N14089" type="margin"> <s id="N1408B"><margin.target id="marg169"></margin.target>2. <emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.111.1.jpg" xlink:href="077/01/111/1.jpg"></figure> <p id="N14098" type="head"> <s id="N1409A">COROLLARIVM.</s> </p> <p id="N1409C" type="main"> <s id="N1409E">Ex hoc elici poteſt centrum grauitatis cuiuſlibet trianguli <lb></lb>eſſe in medio ductæ lineæ baſi æquidiſtantis, quę latus diui<lb></lb>datita, vt portio ad verticem ſit reliquę ad baſim dupla. </s> </p> <p id="N140A4" type="main"> <s id="N140A6">Eſt enim DG ad GE, vt BF ad FC. ſunt verò BF FC <arrow.to.target n="marg170"></arrow.to.target> æ<lb></lb>quales; ergo & DG GE inter ſe ſunt æquales. </s> <s id="N140AE">quare grauita<lb></lb>tis centrum G eſt medium lineę DE. </s> </p> <p id="N140B2" type="margin"> <s id="N140B4"><margin.target id="marg170"></margin.target><emph type="italics"></emph>lemm.<emph.end type="italics"></emph.end><lb></lb>2. <emph type="italics"></emph>der <lb></lb>ſtratic<emph.end type="italics"></emph.end><lb></lb>13.<emph type="italics"></emph>hi<emph.end type="italics"></emph.end></s> </p> <p id="N140CC" type="head"> <s id="N140CE">PROPOSITIO. XV.</s> </p> <p id="N140D0" type="main"> <s id="N140D2">Omnis trapezij duo latera inuicem habentis æ<lb></lb>quidiſtantia centrum grauitatis eſt in recta linea, <lb></lb>quæ latera æquidiſtantia bifariam ſecta <expan abbr="cõiungit">coniungit</expan>; <lb></lb>ita diuiſa, vt ipſius portio terminum habens mino <lb></lb>rem parallelam bifariam diuiſam ad <expan abbr="reliquã">reliquam</expan> por<lb></lb>tionem eandem habeat proportionem, quam ha <lb></lb>bet vtra〈que〉 ſimul, quæ ſit æqualis duplæ maioris <lb></lb>parallelarum cum minore ad <expan abbr="duplã">duplam</expan> minoris cum <lb></lb>maiore. </s> </p> <p id="N140F0" type="main"> <s id="N140F2"><emph type="italics"></emph>Sit trapezium ABCD habens latera AD BC parallela. </s> <s id="N140F6">linea <lb></lb>verò EF bifariam diuidat AD BC. Quòd igitur in linea EF ſit cen<lb></lb>trum grauitatis trapezii, perſpicuum est. </s> <s id="N140FC">productis enim CDG FEG <lb></lb>BAG, li〈que〉t in idem punctum,<emph.end type="italics"></emph.end> putà G <emph type="italics"></emph>concurrere.<emph.end type="italics"></emph.end> propterea quòd <lb></lb>cùm ſit AD æquidiſtans ipſi BC, neceſſe eſt proportionem <arrow.to.target n="marg171"></arrow.to.target><lb></lb>ipſius BA ad AG, ipſiusquè FE ad EG, & CD ad DG, quæ <expan abbr="ni-mirũ">ni<lb></lb>mirum</expan> in omnibus <expan abbr="eadẽ">eadem</expan> eſt, in <expan abbr="vnũ">vnum</expan> & <expan abbr="idẽ">idem</expan> <expan abbr="pũctũ">punctum</expan> terminare. <emph type="italics"></emph><expan abbr="eritq́">erit〈que〉</expan>; <lb></lb>trianguli GBC centrum grauitatis in linea GF. ſimiliter〈que〉 trianguli<emph.end type="italics"></emph.end> <arrow.to.target n="marg172"></arrow.to.target> <pb xlink:href="077/01/112.jpg" pagenum="108"></pb> <arrow.to.target n="marg173"></arrow.to.target> <emph type="italics"></emph>AG D centrum grauitatis in linea EG. ergo reliqui trapezii ABC <lb></lb>centrum grauitatis erit in linea EF. iungatur ita〈que〉 BD, quæ int <lb></lb>æqua in punctis<emph.end type="italics"></emph.end> K<emph type="italics"></emph>H diuidatur. </s> <s id="N1414A">ac per ea <expan abbr="ducãtur">ducantur</expan> LHM N<emph.end type="italics"></emph.end>k<emph type="italics"></emph>T<gap></gap><lb></lb>BC æquidiſtantes<emph.end type="italics"></emph.end>; quæ lineam EF in punctis RS diſpeſcant <lb></lb><emph type="italics"></emph>lungantur〈que〉 DF BE,<emph.end type="italics"></emph.end> ſecetquè DF lineam LM in X. ip <lb></lb>verò EB ſecet NT in O. Iungaturquè <emph type="italics"></emph>OX<emph.end type="italics"></emph.end>, quæ lineam EF <lb></lb> <arrow.to.target n="fig52"></arrow.to.target><lb></lb> <arrow.to.target n="marg174"></arrow.to.target> P ſecet. <emph type="italics"></emph>erit ita〈que〉 trianguli DBC centrum grauitatis in linea H <lb></lb>cùm ſit HB tertia pars ipſius B D<emph.end type="italics"></emph.end>; ſitquè propterea DH ipſi <lb></lb>HB dupla. <emph type="italics"></emph>& per punctum H ducta ſit baſi<emph.end type="italics"></emph.end> BC <emph type="italics"></emph>æquidiſtans M<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg175"></arrow.to.target> <emph type="italics"></emph>eſt autem centrum quo〈que〉 grauitatis trianguli DBC in linea DF<emph.end type="italics"></emph.end>; q <lb></lb>eſt ab angulo D ad dimidiam BC ducta. <emph type="italics"></emph>Quare dicti triang <lb></lb>centrum grauitatis est punctum X. Eademquè ratione<emph.end type="italics"></emph.end> cùm ſit D <lb></lb>tertia pars ipſius DB, ac proptcrea ſit BK ipſius KD dup <lb></lb>ſitquè KN æquidiſtans ipſi AD; erit centrum grauitatis tri <lb></lb>guli ABD in linea KN; idem verò centrum reperitur quo <lb></lb>in linea BE, cùm ſit ab angulo B ad dimidiam AD duc <lb></lb>ergo <emph type="italics"></emph>punctum O<emph.end type="italics"></emph.end>, vbi ſe inuicem ſecant, <emph type="italics"></emph>centrum eſt grauitatist <lb></lb>guli ABD. magnitudinis igitur ex vtriſ〈que〉 triangulis ABD BI <lb></lb>compoſitæ, quæ eſt trapezium<emph.end type="italics"></emph.end> ABCD, <emph type="italics"></emph>centrum grauitatis est in rect<emph.end type="italics"></emph.end> <pb xlink:href="077/01/113.jpg" pagenum="109"></pb><emph type="italics"></emph>nea OX. dicti autem trapezii centrum gauitatis est etiam in li<lb></lb>nea EF, quare trapezii ABCD centrum grauitatis est punctum <lb></lb>P. At verò triangulum BCD ad ABD proportionem habet eam, quam <arrow.to.target n="marg176"></arrow.to.target><lb></lb>OP ad P<emph.end type="italics"></emph.end>X. cùm ſint puncta OX triangulorum centla graui<lb></lb>tatis, ac punctum P vtrorum〈que〉 commune centrum. <emph type="italics"></emph>Sed vt <lb></lb>triangulum BDC adtriangulum ABD, ita eſt<emph.end type="italics"></emph.end> quo〈que〉 baſis <emph type="italics"></emph>BC<emph.end type="italics"></emph.end> <arrow.to.target n="marg177"></arrow.to.target><lb></lb><emph type="italics"></emph>ad<emph.end type="italics"></emph.end> baſim <emph type="italics"></emph>AD.<emph.end type="italics"></emph.end> cùm triangula eandem habeant altitudinem, <lb></lb>ſiquidem ſunt in ijsdem parallelis AD BC. quare vt BC ad <lb></lb>AD, ita OP ad PX. <emph type="italics"></emph>Sed<emph.end type="italics"></emph.end> quoniam anguli RPO SPX ad <arrow.to.target n="marg178"></arrow.to.target> ver<lb></lb>ticem ſunt ęquales, & angulus PRO ipſi PSX, veluti angulus <arrow.to.target n="marg179"></arrow.to.target><lb></lb>ROP angulo PXS eſt ęqualis, erit triangulum OPR triangu<lb></lb>lo XPS ſimile; quare <emph type="italics"></emph>vt OP ad PX, ſic PR ad PS.<emph.end type="italics"></emph.end> eſt autem <arrow.to.target n="marg180"></arrow.to.target><lb></lb>BC ad AD, vt OP ad PX<emph type="italics"></emph>; vt igitur BC ad AD, ita RP ad PS.<emph.end type="italics"></emph.end> <arrow.to.target n="marg181"></arrow.to.target><lb></lb>& antecedentium dupla, duæ ſcilicet BC ad AD, vt duæ PR <lb></lb>ad PS. & componendo duæ BC cum AD ad AD; vt duæ <arrow.to.target n="marg182"></arrow.to.target><lb></lb>PR cum PS ad PS. & ad conſe〈que〉ntium dupla, vt ſcilicet <lb></lb>duæ BC cum AD ad duas AD, ita duæ PR cum PS ad duas <lb></lb>PS. dictum eſt autem BC ad AD ita eſſe, vt PR ad PS. quare <lb></lb>conuerrendo AD ad BC erit, vt PS ad PR. & antecedentium <arrow.to.target n="marg183"></arrow.to.target><lb></lb>dupla. </s> <s id="N14232">hoc eſt duæ AD ad BC, vt duæ PS ad PR. Ita〈que〉 in <lb></lb>eadem ſunt proportione duç BC cum AD ad duas AD, vt <lb></lb>duę PR <expan abbr="cũ">cum</expan> PS ad duas PS. ſicut verò duę AD ad BC, ita duę <lb></lb>PS ad PR. antecedentes igitur ad ſuas ſimul conſe〈que〉ntes in <arrow.to.target n="marg184"></arrow.to.target><lb></lb>eadem erunt proportione. <emph type="italics"></emph>Quare ſicut duæ BC cum AD ad duas <lb></lb>AD cum BC, ita duæ RP cum PS ad duas P S cum PR, <lb></lb>verùm duæ quidem RP cum PS eſt vtra〈que〉 ſimul SR RP.<emph.end type="italics"></emph.end> bis <lb></lb>enim aſſumitur PR, ſemel verò PS. Cum autem lineæ DH ES <lb></lb>à lineis diuidantur ęquidiſtantibus ED OT HM, erit DK ad <arrow.to.target n="marg185"></arrow.to.target><lb></lb>KH, vt ER ad CS; kD verò eſt æqualis KH, erit ER ipſi <lb></lb>RS ęqualis. </s> <s id="N14258">erit igitur ER cum RP, <emph type="italics"></emph>hoc est PE<emph.end type="italics"></emph.end> ipſis SR RP <lb></lb>ęqualis. <emph type="italics"></emph>duæ verò PS cum PR eſt vtra〈que〉 PS SR.<emph.end type="italics"></emph.end> bis enim aſ<lb></lb>ſumitur PS, ſemel què PR. & quoniam FS eſt ęqualis ipſi SR. <lb></lb>quod quidem eodem modo oſtendetur, cùm ſit FS ad SR, vt <lb></lb>BH ad Hk. </s> <s id="N1426E">erit FS cum SP, <emph type="italics"></emph>hoc est PF<emph.end type="italics"></emph.end> ipſis PS SR æqualis. <lb></lb>Quare ita ſehabet PE ad PF, vt duæ BC cum AD ad duas <lb></lb>AD cum BC. Centrum igitur grauitatis P trapezij ABCD <lb></lb>in linea eſt EF, quæ <expan abbr="cõiungit">coniungit</expan> parallelas AD BC bifariam di <pb xlink:href="077/01/114.jpg" pagenum="110"></pb>uiſas; ita vt pars PE, quæ eſt ad minorem parallelam AD <lb></lb>reliquampartem PF eam habet proportionem, quam du <lb></lb>ipſius BC, quæ eſt maior æquidiſtantium, vna cum min <lb></lb>AD, ad duplam minoris AD cum maiore BC, <emph type="italics"></emph>ergo demons<gap></gap><lb></lb>ta ſunt, quæ propoſita fuerant.<emph.end type="italics"></emph.end></s> </p> <p id="N14292" type="margin"> <s id="N14294"><margin.target id="marg171"></margin.target><emph type="italics"></emph>ex<emph.end type="italics"></emph.end> 2.<emph type="italics"></emph>ſ<gap></gap><emph.end type="italics"></emph.end></s> </p> <p id="N142A3" type="margin"> <s id="N142A5"><margin.target id="marg172"></margin.target>13.<emph type="italics"></emph>hu<gap></gap><emph.end type="italics"></emph.end></s> </p> <p id="N142AF" type="margin"> <s id="N142B1"><margin.target id="marg173"></margin.target>8. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N142BA" type="margin"> <s id="N142BC"><margin.target id="marg174"></margin.target><emph type="italics"></emph>ex proxi<lb></lb>me demon <lb></lb>ſtratis.<emph.end type="italics"></emph.end></s> </p> <p id="N142C8" type="margin"> <s id="N142CA"><margin.target id="marg175"></margin.target>* <lb></lb>13. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N142D5" type="margin"> <s id="N142D7"><margin.target id="marg176"></margin.target>6. <emph type="italics"></emph>hu<emph.end type="italics"></emph.end></s> </p> <p id="N142E0" type="margin"> <s id="N142E2"><margin.target id="marg177"></margin.target>1. <emph type="italics"></emph>ſe.<emph.end type="italics"></emph.end></s> </p> <p id="N142EB" type="margin"> <s id="N142ED"><margin.target id="marg178"></margin.target>15. <emph type="italics"></emph>p<emph.end type="italics"></emph.end></s> </p> <p id="N142F6" type="margin"> <s id="N142F8"><margin.target id="marg179"></margin.target>29. <emph type="italics"></emph>p<emph.end type="italics"></emph.end></s> </p> <p id="N14301" type="margin"> <s id="N14303"><margin.target id="marg180"></margin.target><emph type="italics"></emph>ex<emph.end type="italics"></emph.end> 4.</s> </p> <p id="N1430C" type="margin"> <s id="N1430E"><margin.target id="marg181"></margin.target>11. <emph type="italics"></emph><expan abbr="q.">〈que〉</expan><emph.end type="italics"></emph.end></s> </p> <p id="N14319" type="margin"> <s id="N1431B"><margin.target id="marg182"></margin.target>18. <gap></gap></s> </p> <p id="N14320" type="margin"> <s id="N14322"><margin.target id="marg183"></margin.target><emph type="italics"></emph>corol <lb></lb>quint<emph.end type="italics"></emph.end></s> </p> <p id="N1432C" type="margin"> <s id="N1432E"><margin.target id="marg184"></margin.target><emph type="italics"></emph>cor.<emph.end type="italics"></emph.end>2 <lb></lb><emph type="italics"></emph>ma a<gap></gap><lb></lb>huius<emph.end type="italics"></emph.end></s> </p> <p id="N14340" type="margin"> <s id="N14342"><margin.target id="marg185"></margin.target>1. <emph type="italics"></emph>l. <lb></lb>in<emph.end type="italics"></emph.end> 13</s> </p> <figure id="id.077.01.114.1.jpg" xlink:href="077/01/114/1.jpg"></figure> <p id="N14352" type="head"> <s id="N14354">SCHOLIVM.</s> </p> <p id="N14356" type="main"> <s id="N14358"> <arrow.to.target n="marg186"></arrow.to.target> Græcus codex poſt ea verba, <emph type="italics"></emph>cùm ſit HB tertia pars ipſius<emph.end type="italics"></emph.end> Z <lb></lb>habet <foreign lang="grc">και δια τοῡ θ σαμ<gap></gap>ιου ω̄αζἀλλ<gap></gap>λος τἀ βὰσει ὀυχ τᾱς ἁ μθ</foreign>, qu<gap></gap><lb></lb>quidem verba illa <foreign lang="grc">οὐκ τὰς</foreign> perperam leguntur; quorum l<gap></gap><lb></lb>ponerem <foreign lang="grc">α<gap></gap>ομὶνα ἐσὶ</foreign>, ita vt ſint hoc modo reſtituenda, <foreign lang="grc">κα<gap></gap> δια <lb></lb><gap></gap> σαμε̄ιου ω̄αζάλλ<gap></gap>λως τᾱ βὰσει α<gap></gap>ομὲνα ισὶ ἁ μθ. </foreign></s> </p> <p id="N1438A" type="margin"> <s id="N1438C"><margin.target id="marg186"></margin.target>*</s> </p> <p id="N14390" type="main"> <s id="N14392">Hæc ſunt, quæ de centro grauitatis figurarum rectiline <lb></lb>Archimedes ſcripta reliquit. </s> <s id="N14396">Ex quibus maxima certè vtil <lb></lb>habetur; ne〈que〉 ampliùs de rectilineis figuris Archimedes p <lb></lb>tractare voluit. </s> <s id="N1439C">ex dictis enim alia omnia dependent. </s> <s id="N1439E">Nan <lb></lb>tra grauitatis rectilinearum figurarum, quæ æquales angu<lb></lb>latera〈que〉 æqualia habent, ex his in uenire poterimus. </s> <s id="N143A4">quæ <lb></lb>dem figurę in circulo inſcribi poſſunt. </s> <s id="N143A8">Quod ſanè Federi <lb></lb>Comandinus in eius libro de centro grauitatis ſolidorum <lb></lb>prioribus propoſitionibus præſtitit. </s> <s id="N143AE">aliaquè nonnulla, vt<gap></gap><lb></lb>tragrauitatis rectilinearum figurarum in ellipſi, deindè ip<gap></gap><lb></lb>circuli, & ellipſis centra grauitatis in uenit. </s> <s id="N143B6">omneſquè dem<lb></lb>ſtrationes in ijs, quæ in hoc libro iam demonſtrata ſunt, <lb></lb>dauit. </s> <s id="N143BC">præterea ex his etiam idem Commandinus in com <lb></lb>tarijs libri Archimedis de quadratura paraboles, (quo ad p <lb></lb>xim) grauitatis centrum cuiuſlibet figurę rectilineæ ad in<lb></lb>nit. </s> <s id="N143C4">Quod quidem nos quo〈que〉, vt initio polliciti fuimus, <lb></lb>nullis mutatis idem oſtendemus. </s> <s id="N143C8">hoc prius ſuppoſito. </s> </p> <p id="N143CA" type="main"> <s id="N143CC">Triangula in eadem baſi conſtituta eam inter ſe propo<gap></gap><lb></lb>nem habent, quam eorum altitudines. </s> </p> <p id="N143D1" type="main"> <s id="N143D3">Hoc autem demonſtratum eſt ab excellentiſsimis viris, <lb></lb>riſquè Euclidis interpretibus, Federico <expan abbr="Cõmandino">Commandino</expan>, & Cl <lb></lb>ſtophoro Clauio; qui hanc propoſitionem poſt primam <lb></lb>ti libri Euclidis demonſtrarunt. </s> </p> <pb xlink:href="077/01/115.jpg" pagenum="111"></pb> <p id="N143E2" type="head"> <s id="N143E4">PROBLEMA.</s> </p> <p id="N143E6" type="main"> <s id="N143E8">Cuiuſlibet rectilineę figurę centrum grauitatis inuenire. </s> </p> <p id="N143EA" type="main"> <s id="N143EC">Triangulorum centrum grauitatis iam ab Archimede de<lb></lb>monſtratum eſt. </s> </p> <p id="N143F0" type="main"> <s id="N143F2">Sit ita〈que〉 primùm quadri <lb></lb> <arrow.to.target n="fig53"></arrow.to.target><lb></lb>laterum ABCD, cuius opor<lb></lb>teat centrum grauitatis inue <lb></lb>nire. </s> <s id="N143FF">Ducatur AC, quæ qua <lb></lb>drilaterum in duo triangula <lb></lb>ABC ACD diuidet. </s> <s id="N14405">à <expan abbr="pũctiſ-què">punctiſ<lb></lb>què</expan> BD ad AC perpendicu<lb></lb>lares ducantur BE DF. In<lb></lb>ueniantur deinde ex dictis <expan abbr="cẽ">cem</expan> <lb></lb>tra grauitatis triangulorum <lb></lb>ABC ACD. ſintquè puncta <lb></lb>GH. iungaturquè GH, quæ diuidatur in K, ita vt GK <lb></lb>ad KH ſit, vt DF ad BE. Dico punctum K centrum <lb></lb>eſſe grauitatis quadrilateri ABCD. Quoniam enim triangu<lb></lb>la ABC ACD in eadem ſunt baſi AC, erunt inter ſeſe, vt al<lb></lb>titudines. </s> <s id="N14423">quare triangulum ACD ita ſe habet ad <expan abbr="triangulũ">triangulum</expan> <lb></lb>ABC, vt DF ad BE. hoc eſt GK ad KH. <expan abbr="punctũ">punctum</expan> ergo K <expan abbr="cẽ">cem</expan> <lb></lb>trum eſt grauitatis magnitudinisex vtril què triangulis ABC <arrow.to.target n="marg187"></arrow.to.target><lb></lb>ACD compoſitæ; hoc eſt quadrilateri ABCD. </s> </p> <p id="N1443A" type="margin"> <s id="N1443C"><margin.target id="marg187"></margin.target><emph type="italics"></emph>ex 6.h<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.115.1.jpg" xlink:href="077/01/115/1.jpg"></figure> <p id="N14448" type="main"> <s id="N1444A">Sit autem pentagonum <lb></lb> <arrow.to.target n="fig54"></arrow.to.target><lb></lb>ABCDE. <expan abbr="iungãturquè">iunganturquè</expan> AC <lb></lb>AD. inueniaturquè <expan abbr="triãgu">triangu</expan> <lb></lb>li ABC centrum grauitatis <lb></lb>H. quadrilateri verò ACDE <lb></lb>ex proximè <expan abbr="demõ">demom</expan> ſtratis <expan abbr="cẽ-trum">cen<lb></lb>trum</expan> grauitatis inueniatur <lb></lb>Iam vti〈que〉 conſtat (du<lb></lb>cta HK) centrum grauita <lb></lb>tis totius ABCDE in linea <pb xlink:href="077/01/116.jpg" pagenum="112"></pb>HK exiſtere. </s> <s id="N14477">Rurilus trianguli ADE centrum inueniatur F <lb></lb>quadrilateri verò ADCB punctum G. iungaturquè GF. e<gap></gap><lb></lb>eodem modo centrum grauitatis totius ABCDE in linea F<gap></gap><lb></lb>ſed eſt quo〈que〉 in linea HK, ergo vbrſe inuicem ſecant, vt <lb></lb>L, centrum erit grauitatis pentagoni ABCDE. </s> </p> <figure id="id.077.01.116.1.jpg" xlink:href="077/01/116/1.jpg"></figure> <p id="N14487" type="main"> <s id="N14489">In hexagonis ſimiliter. <lb></lb> <arrow.to.target n="fig55"></arrow.to.target><lb></lb>vt ABCDEF iungantur <lb></lb>AC AE, deinceps inuenia <lb></lb>tur trianguli ABC <expan abbr="cẽtrum">centrum</expan> <lb></lb>grauitatis G, pentagoni <lb></lb>verò ACDEF ex dictis cen<lb></lb>trum ſit H. ductaquè GH <lb></lb>centrum grauitatis totius <lb></lb>ABCDEF erit in linea GH <lb></lb>ſimiliter centrum grauita<lb></lb>tis trianguli AFE ſit K, <expan abbr="pẽ">pem</expan> <lb></lb>tagoni verò AEDCB ſit L, iunctaquè KL, erit centrum gr <lb></lb>uitatis totius hexagoni in linea KL. verùm eſt etiam in lin <lb></lb>GH. ergo errt in M. in quo GH <emph type="italics"></emph>K<emph.end type="italics"></emph.end>L ſe inuicem ſecant. </s> </p> <figure id="id.077.01.116.2.jpg" xlink:href="077/01/116/2.jpg"></figure> <p id="N144BC" type="main"> <s id="N144BE">Nequè aliter in heptago <lb></lb> <arrow.to.target n="fig56"></arrow.to.target><lb></lb>no ABCDEFG, in quo du<lb></lb>cantur BG CE. trianguli <lb></lb>verò ABG centrum graui<lb></lb>tatis ſit H. hexagoni <expan abbr="autẽ">autem</expan> <lb></lb>GBCDEF, ſit K. deinde <lb></lb>trianguli CDE <expan abbr="centrũ">centrum</expan> gra<lb></lb>uitatis ſit L, hexagoni ve<lb></lb>rò CEFGAB ſit M. iun<lb></lb>ctiſquè HK ML, eadem ra <lb></lb>tione centrum grauitatis <lb></lb> <arrow.to.target n="marg188"></arrow.to.target> totius heptagoni erit in vtraquè linea Hk LM. ergo erit in </s> </p> <p id="N144E7" type="margin"> <s id="N144E9"><margin.target id="marg188"></margin.target>*</s> </p> <figure id="id.077.01.116.3.jpg" xlink:href="077/01/116/3.jpg"></figure> <p id="N144F1" type="main"> <s id="N144F3">Eodemquè prorſus modo in octagono, & in alijs demc<gap></gap><lb></lb>figuris centrum graui tatis inuenietur. </s> <s id="N144F8">quæ quidem facere <lb></lb>portebat. </s> </p> <pb xlink:href="077/01/117.jpg" pagenum="113"></pb> <p id="N144FF" type="main"> <s id="N14501">Curautem hoc modo centra grauitatum in præfatis figu<lb></lb>ris poſitione tantùm, & non determinatè ea indeterminata, <lb></lb>linea, & in tali ſitu exiſtere inuenerimus, vt in parallelogram <lb></lb>mis & in triangulis factum fuitab Archimede; explicabitur in <lb></lb>ſecundo libro poſt tertiam proportionem; vbi oſtendemus, <lb></lb>in quibus figuris determinatè inueniri poteſt centrum graui<lb></lb>tatis. </s> </p> <p id="N1450F" type="main"> <s id="N14511">Antequam autem finem primolibro imponamus, <expan abbr="reliquũ">reliquum</expan> <lb></lb>eſt; vt ea quæ in præfatione ſuppoſuimus, oſtendamus. </s> <s id="N14519">pri<lb></lb>mùm què quando ſecundùm rectam lineam aliqua diuiditur <lb></lb>figura per centrum grauitatis, aliquando diuidi in partes ſem<lb></lb>per ęquales, & aliquando in partes inæquales. </s> </p> <p id="N14521" type="head"> <s id="N14523">PROPOSITIO.</s> </p> <p id="N14525" type="main"> <s id="N14527">Figura dari poteſt, quę per centrum grauitatis recta li<lb></lb>nea diuiſa, ſemper in partes diuidatur æquales. </s> </p> <p id="N1452B" type="main"> <s id="N1452D">Sit <expan abbr="parallelogrammũ">parallelogrammum</expan> <lb></lb> <arrow.to.target n="fig57"></arrow.to.target><lb></lb>ABCD, cuius <expan abbr="centrũ">centrum</expan> gra<lb></lb>uitatis E. Ducaturquè per <lb></lb>E <expan abbr="vtcunq́">vtcun〈que〉</expan>; linea GEF, quę <lb></lb>vel diameter eſt, vel min^{9}. <lb></lb>ſi eſt diameter, iam <expan abbr="cõſtat">conſtat</expan> <lb></lb><expan abbr="parallelogrãmum">parallelogrammum</expan> in duo <lb></lb>ęqua eſſe diuiſum. </s> <s id="N14555">Si verò non eſt diameter, <expan abbr="ducãtur">ducantur</expan> diametri <arrow.to.target n="marg189"></arrow.to.target><lb></lb>AC BD, quæ per E tranſibunt. </s> <s id="N14560">Quoniam igitur AF eſt æqui<lb></lb>diftans ipſi CG, eritangulus EAF ipſi ECG, & EFA ipſi EGC <arrow.to.target n="marg190"></arrow.to.target><lb></lb>æqualis, eſt autem AEF ipſi GEC ad verticem æqualis, <expan abbr="latusq́">latus〈que〉</expan>; <arrow.to.target n="marg191"></arrow.to.target><lb></lb>AE ipſi EC æquale; erit triangulum AEF triangulo CEG ęqua <lb></lb>le. </s> <s id="N14574">eodemquè modo oſtendetur triangulum FEB triangulo <lb></lb>EGD. & triangulum AED ipſi BEC æquale. </s> <s id="N14578">Ex quibus patet. <lb></lb>figuram ex tribus triangulis compoſitam, hoc eſt figuram <lb></lb>FGDA ipſi FGCB æqualem eſſe. </s> <s id="N1457E">diuiditurergo <expan abbr="parallelogrã-mum">parallelogran<lb></lb>mum</expan> à linea per centrum grauitatis ducta in partes ſem perç<lb></lb>quales. </s> <s id="N14588">quod demonſtrare oportebat. </s> </p> <pb xlink:href="077/01/118.jpg" pagenum="114"></pb> <p id="N1458D" type="margin"> <s id="N1458F"><margin.target id="marg189"></margin.target>34.<emph type="italics"></emph>primi<emph.end type="italics"></emph.end></s> </p> <p id="N14598" type="margin"> <s id="N1459A"><margin.target id="marg190"></margin.target>29. <emph type="italics"></emph>primi<emph.end type="italics"></emph.end></s> </p> <p id="N145A3" type="margin"> <s id="N145A5"><margin.target id="marg191"></margin.target>15. <emph type="italics"></emph>primi<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.118.1.jpg" xlink:href="077/01/118/1.jpg"></figure> <p id="N145B2" type="main"> <s id="N145B4">Hoc idem multis alijs figuris accidet, vt pentagonis, he <lb></lb>gonisæquiangulis, & æquilateris, & alijs. </s> </p> <p id="N145B8" type="head"> <s id="N145BA">PROPOSITIO.</s> </p> <p id="N145BC" type="main"> <s id="N145BE">Figura dari poteſt, quæ per centrum grauitatis recta li <lb></lb>diuiſa, non ſemper in partes diuidatur ęquales. </s> </p> <p id="N145C2" type="main"> <s id="N145C4">Habeat triangulum ABC <lb></lb> <arrow.to.target n="fig58"></arrow.to.target><lb></lb>latera AB AC æqualia. </s> <s id="N145CD">trian <lb></lb>guliverò centrum grauitatis ſit <lb></lb>D. à quo ipſi BC ęquidiſtans <lb></lb>Ducatur FDG. Dico partem <lb></lb>AFG <expan abbr="minorẽ">minorem</expan> eſſe parte BFGC. <lb></lb>ducatur ADE, quæ bifariam <lb></lb> <arrow.to.target n="marg192"></arrow.to.target> BC diuidet. </s> <s id="N145E3">& à puncto G <lb></lb>ipſi AE ęquidiſtans ducatur <lb></lb>HGK. compleantur〈que〉 figurę <lb></lb>EH KF. Quoniam enim FG <lb></lb> <arrow.to.target n="marg193"></arrow.to.target> ęquidiſtans eſt ipſi BC, erit FD ad DG, vt BE ad E<gap></gap><lb></lb>& eſt BE ipſi EC æqualis. </s> <s id="N145F4">erit igitur FD ipſi DG ęqua <lb></lb>vt etiam paulò ante 15. huius oſtendimus. </s> <s id="N145F8">quare FG ip <lb></lb>DG dupla. </s> <s id="N145FC">eſt. </s> <s id="N145FE">ac propterea <expan abbr="parallelogrãmum">parallelogrammum</expan> FK dupi <lb></lb>eſt parallelogrammi DK. quia verò AD ipſius DE du <lb></lb>exiſtit, erit quoquè parallelogrammum DH ipſius DK <lb></lb>plum. </s> <s id="N1460A">Quare DH ipſi FK eſt æquale. </s> <s id="N1460C">At verò quoni <lb></lb> <arrow.to.target n="marg194"></arrow.to.target> FG dupla eſt ipſius DG. erit triangulum AFG parallelog <lb></lb>mo DH æquale. </s> <s id="N14616">triangulum igitur AFG parallelog<gap></gap><lb></lb>FK eſt æquale. </s> <s id="N1461B">Quare pars AFG parte BFGC minor <gap></gap><lb></lb>ſtit. </s> <s id="N14620">quod demonſtrare oportebat. </s> </p> <p id="N14622" type="margin"> <s id="N14624"><margin.target id="marg192"></margin.target><emph type="italics"></emph>ex<emph.end type="italics"></emph.end> 13. <emph type="italics"></emph>hui'<emph.end type="italics"></emph.end></s> </p> <p id="N14632" type="margin"> <s id="N14634"><margin.target id="marg193"></margin.target><emph type="italics"></emph>lemma an<lb></lb>te <expan abbr="ſecundã">ſecundam</expan> <lb></lb><expan abbr="demonſtra-tionẽ">demonſtra<lb></lb>tionem</expan><emph.end type="italics"></emph.end> 13 <emph type="italics"></emph>bu <lb></lb>ius.<emph.end type="italics"></emph.end></s> </p> <p id="N14650" type="margin"> <s id="N14652"><margin.target id="marg194"></margin.target><emph type="italics"></emph>ex<emph.end type="italics"></emph.end> 41.<emph type="italics"></emph>pri. <lb></lb>mi.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.118.2.jpg" xlink:href="077/01/118/2.jpg"></figure> <p id="N14666" type="main"> <s id="N14668">Hinc perſpicuum eſt, eandem figuram per centrum gra<lb></lb>tatis diuiſam, aliquando in partes in æquales, aliquando in <lb></lb>tes æquales diuidi poſſe. </s> <s id="N1466E">in partes inęquales iam oſtenſum <lb></lb>hocaccidere <expan abbr="perlineã">perlineam</expan> FG. in partes verò æquales patet pe <lb></lb>neam ADE, quæ triangulum ABC in duo ęqua diuidi<gap></gap>. t<gap></gap><lb></lb> <arrow.to.target n="marg195"></arrow.to.target> gulum enim ABE triangulo: AEC eſt ęquale, cùm ſint<gap></gap><lb></lb>eadem altitudine, baſeſquè BE EC inter ſe ſint æquales. </s> </p> <pb xlink:href="077/01/119.jpg" pagenum="115"></pb> <p id="N14687" type="margin"> <s id="N14689"><margin.target id="marg195"></margin.target>1. <emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N14692" type="main"> <s id="N14694">Adhuc (veluti initio quo〈que〉 diximus) ſi fuerit prisma, vt <lb></lb>AB, cuius altera baſis ſit AC. tale verò ſit prisma, vt pl mum <lb></lb>AC planis CH CK &c. </s> <s id="N1469A">ſit erectum. </s> <s id="N1469C">ſit autem ipſius baſis <lb></lb>AC centrum grauitatis E. Dico ſi prima ſuſpendatur ex pu<lb></lb> <arrow.to.target n="fig59"></arrow.to.target><lb></lb>cto E, baſim AC horizonti æquidiſtantem permanere. </s> <s id="N146A7">vt co <lb></lb>gnoſcamusea, quæ his libris pertractantur, ad praxim poſſe <lb></lb>reduci. </s> <s id="N146AD">& ne aliquid abſ〈que〉 demonſtratione confirmatum re <lb></lb>linquamus. </s> <s id="N146B1">hoc quo〈que〉 oſtendemus. </s> <s id="N146B3">hoc pacto. </s> </p> <figure id="id.077.01.119.1.jpg" xlink:href="077/01/119/1.jpg"></figure> <p id="N146B9" type="main"> <s id="N146BB">Primùm quidem exijs, quæ demonſtrata ſunt, rectilineæ <lb></lb>figuræ AC centrum granitatis inueniatur E. eodemquè mo <lb></lb>do figuræ BD centrum grauitatis ſit F. Iungaturquè EF, <lb></lb>quæ bifariam diuidatur in G. Iam patet punctum G cen<lb></lb>trum eſſe grauitatis priſmatis AB, ex octaua propoſitione Fe<lb></lb>derici <expan abbr="Cõmandini">Commandini</expan> de centro grauitatis ſolidorum, & ex corol<lb></lb>lario quintæ propoſitionis eiuſdem libri, lineam EF late<lb></lb>ribus AD CB ęquidiſtantem eſſe. </s> <s id="N146CF">quoniam <expan abbr="autẽ">autem</expan> plana CH <lb></lb>CK ad rectos ſuntangulos plano AC, erit CB eorum commu <arrow.to.target n="marg196"></arrow.to.target><lb></lb>nisſectio eidem plano AC perpendicularis. </s> <s id="N146DC">acpropterea EF <lb></lb>ipſi CB æquidiſtans plano AC perpendicularis exiſtit. <pb xlink:href="077/01/120.jpg" pagenum="116"></pb>Ita〈que〉 intelligatur ſolidum AB ex E ſuſpenſum; tunc ex <lb></lb>ma propoſitione de libra noſtrorum mechanicorum pon <lb></lb>AB ex E ſuſpenſum <expan abbr="numquã">numquam</expan> manebit, niſi recta EG fu <lb></lb>horizonti perpendicularis. </s> <s id="N146EE">Quando autem EF erit horizc <lb></lb>ti perpendicularis, erit planum AC horizonti æquidiſtan <lb></lb> <arrow.to.target n="marg197"></arrow.to.target> tunc. <expan abbr="n.">enim</expan> EF tum horizonti, tum plano AC perpendicul<gap></gap><lb></lb>exiſtet. </s> <s id="N146FF">Inuento igitur centro grauitatis E ipſius baſis A <lb></lb>ſi AB ſuſpendatur ex E, linea EGF in centrum mundi to <lb></lb>det; planumquè AC horizonti erit æquidiſtans. </s> <s id="N14705">quod de<gap></gap><lb></lb>ſtrare oportebat. </s> </p> <p id="N1470A" type="margin"> <s id="N1470C"><margin.target id="marg196"></margin.target>19. <emph type="italics"></emph>v <lb></lb>mi.<emph.end type="italics"></emph.end></s> </p> <p id="N14717" type="margin"> <s id="N14719"><margin.target id="marg197"></margin.target>14.<emph type="italics"></emph>vndeci <lb></lb>mi.<emph.end type="italics"></emph.end></s> </p> <p id="N14724" type="main"> <s id="N14726">PRIMI LIBRI FINIS. </s> </p> <pb xlink:href="077/01/121.jpg" pagenum="117"></pb> <p id="N1472B" type="head"> <s id="N1472D">GVIDIVBALDI <lb></lb>E MARCHIONIBVS <lb></lb>MONTIS.</s> </p> <p id="N14733" type="head"> <s id="N14735">In Secundum Archimedis æ〈que〉ponderan<lb></lb>tium Librum.</s> </p> <p id="N14739" type="head"> <s id="N1473B">PRÆFATIO.</s> </p> <p id="N1473D" type="main"> <s id="N1473F">Secundus Archimedisliber, vtinitio primi <lb></lb>libri præfati ſumus, ſubtiliſſima theo<lb></lb>remata ſpeculatur. </s> <s id="N14745">Vultenim Archimedes <lb></lb>inueſtigare centrum grauitatis plani coni<lb></lb>cæſectionis, quæ parabole paſſim vocatur. <lb></lb>quamuis Archimedes alio nomine, ac po<lb></lb>tiùs deſcriptione quadam <expan abbr="ſectionẽ">ſectionem</expan> <expan abbr="hãc">hanc</expan> <expan abbr="nũ-cuparit">nun<lb></lb>cuparit</expan>: veluti portio recta linea <expan abbr="rectãguliq́">rectanguli〈que〉</expan>; coniſectione <expan abbr="cõ">com</expan> <expan abbr="tẽ">tem</expan> <lb></lb>ta. </s> <s id="N1476B">Refert enim Eutocius Aſcalonita in principio ſui <expan abbr="commẽ-tarij">commen<lb></lb>tarij</expan> in libros conicorum Apollonij Pergęi, ex ſententia Ge<lb></lb>mini (cui Pappus etiam ex Ariſtęi ſententia aſſentire videtur) <lb></lb>quòd qui ante Apollonium fuerunt, perfectam, & abſolutam <lb></lb>conorum <expan abbr="cognitionẽ">cognitionem</expan> <lb></lb> <arrow.to.target n="fig60"></arrow.to.target><lb></lb>non habuerunt; inter <lb></lb>quos reſpoſuit Archime <lb></lb>de. <expan abbr="Nã">Nam</expan> inquit <expan abbr="conũ">conum</expan> deſi <lb></lb>nientes, ipſum per <expan abbr="rectã">rectam</expan> <lb></lb>guli <expan abbr="triãguli">trianguli</expan> circumuo<lb></lb>lutionem manente vno <lb></lb>eorum, quæ circa <expan abbr="rectũ">rectum</expan> <lb></lb><expan abbr="angulũ">angulum</expan> ſunt, latere <expan abbr="cõſi-derarunt">conſi<lb></lb>derarunt</expan>. vt habetur in <lb></lb>definitionibus Euclidis <lb></lb>vndecimi libri elem <expan abbr="en-torũ">en<lb></lb>torum</expan>. vt Conus ABC fit <lb></lb>ex <expan abbr="circũuoluto">circumuoluto</expan> triangulo rectangulo ADC. conus verò EBC <lb></lb>ex triangulo EDC, & conus FBC ex rectangulo triangulo <pb xlink:href="077/01/122.jpg" pagenum="118"></pb>FDC. & ſi AD fuerit i<lb></lb> <arrow.to.target n="fig61"></arrow.to.target><lb></lb>pſi DC æqualis, conus <lb></lb>ABC vocabit rectan<lb></lb>gulus. </s> <s id="N147D0">nam vtcumquè <lb></lb>ducto plano per axem, <lb></lb> <arrow.to.target n="marg198"></arrow.to.target> quod triangulum faciat <lb></lb>ABC; erit angulus BAC <lb></lb>ad coniverticem rectus: <lb></lb>ſiquidem DAC recti di <lb></lb>midius exiſtit, veluti <lb></lb>DAB. pari ratione ſi ED <lb></lb>fuerit ipſa DC minor; <lb></lb>erit conus EBC obtuſi <lb></lb>angulus:nam ducto per axem plano, habebit triangulum <lb></lb>EBC angulum ad verticem coni BEC obtuſum; cùm ſit <lb></lb> <arrow.to.target n="marg199"></arrow.to.target> BEC maior BAC. exiſtenteautem FD ipſa DC maiori, co <lb></lb>nus FBC acutiangulus nuncupabitur; quoniam <expan abbr="triangulũ">triangulum</expan> <lb></lb>per axem FBC angulum ad verticem coni F acutum poſſide <lb></lb>bit; ſiquidem minor eſt BFC, quam BAC. Refert deinde, <lb></lb>quòd vnum〈que〉mquè <lb></lb>horum conorum <expan abbr="eo-dẽ">eo<lb></lb>dem</expan> modo piſci ſecue<lb></lb> <arrow.to.target n="fig62"></arrow.to.target><lb></lb>runt; vt ſit rectangu<lb></lb>lus conus ABC; trian <lb></lb>gulum verò per axem <lb></lb>ſit ABC. in latere au<lb></lb>tem AC quoduis ſu<lb></lb>matur punctum D; <lb></lb>ducaturquè DE ad <lb></lb>AC perpendicularis; <lb></lb>& per DE ducatur pla <lb></lb>num plano ABC ere <lb></lb>ctum, quod quidem conum ſecet, ſectio autem ſit FDG. quę <lb></lb>ſanè eſt ſe ctio, quæ abipſis vocatur rectanguli coni ſectio, <lb></lb>quippè quæ ſi intelligatur terminata recta linea FG, nuncupa <lb></lb>tur portio recta linea, rectanguli〈que〉 coni ſectione contenta. </s> </p> <pb xlink:href="077/01/123.jpg" pagenum="119"></pb> <p id="N1482A" type="margin"> <s id="N1482C"><margin.target id="marg198"></margin.target>3. <emph type="italics"></emph>primi co <lb></lb>mcorum A <lb></lb>pol.<emph.end type="italics"></emph.end></s> </p> <p id="N14839" type="margin"> <s id="N1483B"><margin.target id="marg199"></margin.target>21. <emph type="italics"></emph>primi.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.123.1.jpg" xlink:href="077/01/123/1.jpg"></figure> <figure id="id.077.01.123.2.jpg" xlink:href="077/01/123/2.jpg"></figure> <figure id="id.077.01.123.3.jpg" xlink:href="077/01/123/3.jpg"></figure> <p id="N14850" type="main"> <s id="N14852">Si verò conus <lb></lb> <arrow.to.target n="fig63"></arrow.to.target><lb></lb>ABC fuerit obtu <lb></lb>ſiangulus, ſitquè <lb></lb>triangulum per <lb></lb>axem ABC, <expan abbr="eo-dẽ">eo<lb></lb>dem</expan> modoà quo<lb></lb>uis puncto D, du<lb></lb>cta DE ad re<lb></lb>ctos angulos ipſi <lb></lb>AC, acper DE <lb></lb>ducto plano ad <lb></lb>planum ABC erecto, quod conum ſecet, vt FDG; erit FDG <lb></lb>obtuſianguli coni ſectio, quæ vnà cum recta FG vocatur por<lb></lb>tio recta linea, obtuſianguliquè coni ſectione contenta. </s> </p> <figure id="id.077.01.123.4.jpg" xlink:href="077/01/123/4.jpg"></figure> <p id="N1487B" type="main"> <s id="N1487D">Similiter <expan abbr="exiſtẽte">exiſtente</expan> co<lb></lb> <arrow.to.target n="fig64"></arrow.to.target><lb></lb>no acutiangulo ABC, <lb></lb>cuius triangulum per a<lb></lb>xem ſit ABC. & à <expan abbr="pũcto">puncto</expan> <lb></lb>D ducta ſit DE perpen<lb></lb>dicularis ipſi AC, du<lb></lb>ctoquè plano per DE ad <lb></lb>planum ABC erecto, e<lb></lb>rit DFEG acutianguli <lb></lb>coni ſectio. </s> </p> <figure id="id.077.01.123.5.jpg" xlink:href="077/01/123/5.jpg"></figure> <p id="N148A2" type="main"> <s id="N148A4">Apollonius au-<lb></lb>tem Pergęus, qui ab<lb></lb>ſolutiſſima commenta<lb></lb>ria de conicis ſcripſit, <lb></lb>huiuſmodi conos omnesvocauit rectos; ad differentiam coni <lb></lb>ſcaleni. </s> <s id="N148B0">coni enim rectiaxes habent baſibus erectos. </s> <s id="N148B2">ſcaleni ve <lb></lb>rò nequaquam. </s> <s id="N148B6">& in ſcalenis latera triangulorum per axem <lb></lb>non ſunt ſemper æqualia. </s> <s id="N148BA">quod ſemper conis rectis contingit. </s> </p> <p id="N148BC" type="main"> <s id="N148BE">Preterea ſectionem rectanguli coni parabolen nominauit; <lb></lb>obtuſianguli verò coni ſectionem hyperbolen; ſectionem au<lb></lb>tem acutianguli coni ellipſim nuncupauit. </s> <s id="N148C4">& in vnoquo〈que〉 <lb></lb>cono tàm recto, quàm ſcaleno has tres ineſſe ſectiones <expan abbr="demõ">demom</expan> <pb xlink:href="077/01/124.jpg" pagenum="120"></pb>ſtrauit. </s> <s id="N148D0">Ex quibus colligit Geminus (〈que〉m Eutocius, alijquè <lb></lb>complures ſecuti ſunt) eos, qui ante Apollonium extitere, <lb></lb>conostantùm rectos cognouiſſe. </s> <s id="N148D6">& in vnoquo〈que〉 cono <expan abbr="vnã">vnam</expan> <lb></lb>tantùm ſectionem animaduertiſſe. </s> <s id="N148DE">quod quidem ſi de ijs, qui <lb></lb>ante Archimedem fuere intelligatur; ad mitti fortaſſe poterit; <lb></lb>ac præſertim de Euclide. </s> <s id="N148E4">vt patet ex definitione coni abeo <lb></lb>tradita. </s> <s id="N148E8">At verò de Archimede, qui poſt Euclidem, ante verò <lb></lb>Apollonium fuit, non ita facilè concedendum videtur. <expan abbr="Nã">Nam</expan> ex <lb></lb>ijs, quæ ſcripta reliquit. </s> <s id="N148F2">eum non ſolùm notitiam ha-<lb></lb>buiſſe de conis rectis; verùm <expan abbr="etiã">etiam</expan> de ſcalenis facilè ex i-<lb></lb>pſius ſcriptis conijci poteſt. </s> <s id="N148FC">In primo enim librode ſphæ<lb></lb>ra, & cylindro multis in locis, vt in ſeptima, octaua, no <lb></lb>na, decimaquarta, decimaquinta propoſitione; alijsquè in <lb></lb>locis conos nominat ęquicrures, quod quidem ſecundum i<lb></lb>pſum ſunt, qui in eius ſuperficie æquales habent rectas lineas <lb></lb>à vertice coni ad baſim ductas. </s> <s id="N14908">item in epiſtola quo〈que〉 libri <lb></lb>de conoidibus & ſphęroidibus, quam Archimedes Deſitheo <lb></lb>ſcribit. </s> <s id="N1490E">cùm de obtuſiangulo conoideverba facit, conum vo<lb></lb>catæquicrurem. </s> <s id="N14912">Quòd ſi Archimedes hos conos vocauit æ<lb></lb>quicrures, cui dubium, ipſum eosad differentiam eorum, qui <lb></lb>non ſunt æquicrures ita nuncupaſſe? </s> <s id="N14918">qui verò non ſunt æ<lb></lb>quicrures ex ipſomet Apollonio ſunt ſcaleni; nam æquicrures <lb></lb>hoc modo coni axes habent baſibus erectos. </s> <s id="N1491E">qui igitur non <lb></lb>erunt æquicrures, eorum axes ſuis baſibus nunquàm erunt e<lb></lb>recti. </s> <s id="N14924">Præterea idem quo〈que〉 confirmari poteſt ex demon<lb></lb>ſtratione vigeſimæquintæ propoſitionis eiu<gap></gap>dem libri, in qua <lb></lb>cùm nominet Archimehes conum rectum proculdubiò ad <lb></lb>differentiam eorum, qui non ſuntrecti ita eum nuncupauit. <lb></lb>nam ſi Aichimedes (ex illorum ſententia) conos tan ùm re<lb></lb>ctos cognouiſſet; quorſum his in locis conum rectum, vel æ<lb></lb>quicrurem nominaſſet? </s> <s id="N14934">ſat ſibi fuiſſet conum tantum dixiſſe. <lb></lb>Ne〈que〉 verò dicendum eſt Archimedem per cono recto intel<lb></lb>lexiſſe conum rectangulum eo modo, 〈que〉m ſupra expoſui<lb></lb>mus. </s> <s id="N1493C">nam in ea propoſitione, dum conſtituit hunc conum, <lb></lb>non conſurgit conus rectangulus, ſed obtuſiangulus quapro <lb></lb>pter conum rectum nominatad differentiam coni ſcaleni. </s> <s id="N14942">Cę <lb></lb>terùm ut manifeſtè oſtendamus Archimedem conos cogno- <pb xlink:href="077/01/125.jpg" pagenum="121"></pb>uiſſe ſcalenos, conſideranda eſt octaua propoſitio libri de co<lb></lb>noidibus, & ſph æroidibus, in qua proponit Archimedes co<lb></lb>num conſtituere, & inuenire, in quo ſitſectio ellipſis data, ver <lb></lb>tex autem coni in linea exiſtat a centro ellipſis ad<gap></gap>ectos angu<lb></lb>los ellipſis plano erecta. </s> <s id="N14954">Exqua conſtructione planè apparet, <lb></lb>Archimedem (vt ex eius demonſtratione conſtat) hoc in lo<lb></lb>co 〈que〉rere, & inuenire conum proculdubio ſcalenum. </s> <s id="N1495A">vt <expan abbr="etiã">etiam</expan> <lb></lb>ex nona eiuſdem libri propoſitione perſpicuum eſſe poteſt; in <lb></lb>qua vt plurimùm conus inuenitur ſcalenus. </s> <s id="N14964">Ex quibus mani<lb></lb>feſtiſſimè patet Archimedem non ſolùm de conis rectis, <expan abbr="verũ">verum</expan> <lb></lb>etiam de conis ſcalenis notitiam habuiſſe. </s> <s id="N1496E">Porrò ea verba, quę <lb></lb>refert Eutocius ex ſententia Heraclij, qui Archimedis vitam <lb></lb>literis mandauit; idipſum ſatis manifeſtant. </s> <s id="N14974">Heraclius enim <lb></lb>inquit Archimedem quidem <expan abbr="primũ">primum</expan> conica theoremata fuiſſe <lb></lb>aggreſſum; Apollonium verò, cùm ea inueniſſetab Archime <lb></lb>de nondum edita; tanquam eius propria edidiſſe. </s> <s id="N14980">quod qui<lb></lb>dem etiam exipſiusmet Archimedis ſcriptis <expan abbr="cõfirmari">confirmari</expan> poteſt. <lb></lb>in libro nam〈que〉 de conoidibus, & ſphæroidibus ante <expan abbr="quartã">quartam</expan> <lb></lb>propoſitionem vbi Archimedes theorema proponit alibi de<lb></lb>monſtratum, inquit, <emph type="italics"></emph>Hoc autem oſten ſum eſt in conicis elementis.<emph.end type="italics"></emph.end> in <lb></lb>principio etiam libri de quadratura paraboles, cùm nonnulla <lb></lb>propoſuiſſet; poſt tertiam propoſitionem ſcilicet, inquit <emph type="italics"></emph>De<lb></lb>monſtrata autem ſunt hæc in elementis conicis.<emph.end type="italics"></emph.end> nonneigitur conſtat <lb></lb>Archimedem <expan abbr="elemẽta">elementa</expan> conica ſcripſiſſe? </s> <s id="N149AA">Obijciet verò aliquis, <lb></lb>non propterea conſtare, hęc elementa eonica, quorum me<lb></lb>minit Archimedes, ipſiusmet eſſe Archimedis; cùm non affir <lb></lb>met, hæcfuiſſe ab ipſo demonſtrata. </s> <s id="N149B2">verùm illud in primis ma <lb></lb>nifeſtum eſt, tempore Archimedis conica elementa extitiſſe. <lb></lb>vt nonnulli Euclidem quatuor conicorum libros edidiſſe <expan abbr="af-firmãt">af<lb></lb>firmant</expan>; ſicut Pappus in ſeptimo <expan abbr="Mathematicarũ">Mathematicarum</expan> <expan abbr="collectionuũ">collectionuum</expan> <lb></lb>libro aſſerit. </s> <s id="N149C8">Sed ex modo lo〈que〉ndi Archimedis planè <expan abbr="cõſtat">conſtat</expan> <lb></lb>hæc fuiſſe ab ipſo conſcripta. </s> <s id="N149D0">Nam quando Archimedes ali<lb></lb>qua ſupponitab alijs demonſtrata, <expan abbr="tũc">tunc</expan> addere conſueuit, illa <lb></lb>ab alijs demonſtrata eſſe; vt in vndecima propoſitionedeco<lb></lb>noidibus, & ſphæroidibus; cùm inquit. <emph type="italics"></emph>omnis coni ad conum pro<lb></lb>portionem compoſitam eſſe ex proportione baſium, & proportione altitu<lb></lb>dinum,<emph.end type="italics"></emph.end> quod quidem, quia ab alijs demonſtratum fuerat, ſta <pb xlink:href="077/01/126.jpg" pagenum="122"></pb>tim inquit, <emph type="italics"></emph>demonſtratum eſt ab iis, qui ante nos fuerunt.<emph.end type="italics"></emph.end> ſimiliter <lb></lb>in libro de ſphęra, & cylindro ante propoſitionem decimam <lb></lb>ſeptimam, cùm nonnulla ſuppoſuerit ab alijs demon ſtrata in <lb></lb>quit. <emph type="italics"></emph>Hæc autem omnia à ſuperioribus ſunt demonſtrata.<emph.end type="italics"></emph.end> In ſecunda <lb></lb>verò parte <expan abbr="quĩtę">quintę</expan> propoſitionis hui^{9} ſecudi libri cu inquit, <emph type="italics"></emph>De <lb></lb>monſtratum eſt enim aliis in locis portiones ſeſquitertias eſſe <expan abbr="triangulorũ">triangulorum</expan>.<emph.end type="italics"></emph.end><lb></lb>quod quia ipſemet aſſecutus eſt in libro de quadratura para<lb></lb>boles, idcircò non addit ab ipſomethoc oſtenſum fuiſſe. </s> <s id="N14A11">A<lb></lb>liaquè huiuſmodi loca breuitatis ſtudio omitto oſtendentia <lb></lb>ea, quæ Archimedes ſupponit tanquam demonſtrata, <expan abbr="quãdo">quando</expan> <lb></lb>non additab alijs oſtenſa eſſe, à ſe ipſo demonſtrata fuiſſe, vt <lb></lb>in demonſtratione decimæ quartę propoſitionis primi libri, <lb></lb>nec non ex octaua huius ſecundi libri demonſtratione; alijſ<lb></lb>què locis perſpicuum eſſe poteſt. </s> <s id="N14A23">Quare tùm ex præfntis Archi <lb></lb>medis locis, tùm Heraclij teſtim onio manifeſtè elicipoteſt, <lb></lb>Archimedem elementa conica ſcrip ſiſſe. </s> <s id="N14A29">Ne〈que〉 verò quicqua <lb></lb>nos turbare debet, quòd Apollo nius coni ſectionibus nomina <lb></lb>impoſuerit; ſi tamen ipſe prim us fuit; cùm eas proprijs nomi<lb></lb>nibus, vt potè parabolen, hyperbolen, & ellipſim nuncupet; <lb></lb>& in quolibet cono omnes agnouerit ſectiones. </s> <s id="N14A33">Nam quam<lb></lb>uis vſ〈que〉 ad Archimedis tempus hi termini nondum extite<lb></lb>rint; & in ſingulis conis priſci illi vnicam <expan abbr="tãtùm">tantùm</expan> cognouerint <lb></lb>ſectionem; Archimedes tamen vlteriùs progreſſus eſt. </s> <s id="N14A3F">etenim <lb></lb>hæc quo〈que〉 <expan abbr="ſectionũ">ſectionum</expan> nomina ipſi fortaſse minùs ignota fue<lb></lb>runt: quandoquidem in demonſtratione nonæ propoſitio<lb></lb>nis de conoidibus, & ſphęroidibus ellipſim nominat. </s> <s id="N14A4B">Pręte<lb></lb>rea non ſolùm cognouit Archimedes conos ſecari poſſe pla<lb></lb>nis lateribus coni erectis, verùm etiam alijs modis: quod qui<lb></lb>dem exemplo ellipſis manifeſtari optimè poteſt. </s> <s id="N14A53">Nam in o<lb></lb>ctaua propoſitione eiuſdem libri ellipſes latus coni ad angu<lb></lb>los rectos minimè ſecant. </s> <s id="N14A59">veluti quo〈que〉 in nona propoſitione <lb></lb><expan abbr="idẽ">idem</expan> ſępè <expan abbr="cõtingit">contingit</expan>. At verò in <expan abbr="eodẽ">eodem</expan> adhuc libro ante <expan abbr="primã">primam</expan> pro <lb></lb>poſitionem inquit Archimedes. <emph type="italics"></emph>Si conus plano ſecetur cum omnibus <lb></lb>eius lateribus coeunti, ſectio vel erit circulus, vel acutianguli coni ſe<lb></lb>ctio.<emph.end type="italics"></emph.end> Vnde perſpicuum eſt non in vno duntaxat cono acutian <lb></lb>gulo, verùm in omnibus conisſectionem ellipſis cognouiſſe. <lb></lb>Præterea ex hoclo〈que〉ndi modo li〈que〉t ipſum ſectionem quo <pb xlink:href="077/01/127.jpg" pagenum="123"></pb>〈que〉 nouiſſe ſubcontrariam; quæ cùm ſit baſi ſubcontraiſè po <lb></lb>ſita, <expan abbr="oĩa">oina</expan> latera coni ſecat; & <expan abbr="tñ">tnm</expan> <expan abbr="nō">non</expan> eſt ellipſis, ſed circulus. <arrow.to.target n="marg200"></arrow.to.target> qua<lb></lb>propter ſi in omnibus conis ellipſis nouit ſectionem; cur in i<lb></lb>pſis, & parabolas, & hyperbolas minùs animaduertit? </s> <s id="N14A96">cùm <lb></lb>ſit manifeſtum ex dictis in cono obtuſiangulo & <expan abbr="hyperbolẽ">hyperbolem</expan> <lb></lb>& ellipſim; in rectangulo autem parabolem, ellipſimquè co<lb></lb>gnouiſſe? </s> <s id="N14AA2">hòc certè non eſt aſſerendum. </s> <s id="N14AA4">Ex hoc enim perſpi<lb></lb>cuum eſt Archimedem cognouiſſe conos ſecari poſſe planis, <lb></lb>quæ non ſint ſemper ad coni latus erecta. </s> <s id="N14AAA">dormitaſſequè Eu<lb></lb>tocium Geminum, & alios ſecus hac in parte de Archimede <lb></lb>ſentientes. </s> <s id="N14AB0">Ampliùs <expan abbr="nõ">non</expan> ne cognouit etiam Archimedes ſeca<lb></lb>ri poſſe rectangulos conoides, itidemquè & <expan abbr="obtuſiãgulos">obtuſiangulos</expan> pla <lb></lb>nis, quæ ne〈que〉 ſint per axem ducta, ne〈que〉 axi æquidiſtantia; <lb></lb>ne〈que〉 ſuper axem erecta. </s> <s id="N14AC0">vt in duodecima, decimatertia, & <lb></lb>decima quarta propoſitione eiuſdem libri patet. </s> <s id="N14AC4">quomodo i<lb></lb>ta〈que〉 his quo〈que〉 modis 〈que〉mlibet conum ſecari poſſe igno<lb></lb>rauit? </s> <s id="N14ACA">Non eſt igitur ambigendum Archimedem cognouiſ<lb></lb>ſe conos ſecari poſſe planis ad latus coni differentem inclina<lb></lb>tionem habentibus. </s> <s id="N14AD0">Ex quibus perſpicuum eſt, ipſum in om<lb></lb>nibus conis omnes ineſſe ſectiones omnino animaduertiſſe. <lb></lb>At ſi concedamus etiam ſua tempeſtate nondum ſectioni<lb></lb>bus ipſis propria fuiſſe impoſita nomina; tam eam parabo<lb></lb>lem, quæ erat rectanguli coni ſectio; quàm quæ erat ſectio <lb></lb>alterius coni, cùm ſit eadem ſectio, eodem nomine nuncu<lb></lb>pabat; nempè rectanguli coni ſectionem. </s> <s id="N14ADE">Et hoc, quia <lb></lb>priùs hæc ſectio cognita ſuit in cono rectangulo (vnde ſi<lb></lb>bi nomen vindicauit) quam in alio. </s> <s id="N14AE4">quod idem dicen<lb></lb>dum eſt de alijs ſectionibus. </s> <s id="N14AE8">Vt manifeſtum eſſe poteſt <lb></lb>exemplo ſectionis acutianguli coni. </s> <s id="N14AEC">Archimedes enim eo<lb></lb>dem loco, anteprimam ſcilicet propoſitionem de conoidi <lb></lb>bus, & ſphęroidibus inquit, <emph type="italics"></emph>Si cylindrus duobus planis æquidi<lb></lb>stantibus ſecetur; quæ cum omnibus ipſius lateribus coeant, ſectio<lb></lb>nes, uelerunt circuli; uel conorum acutiangulorum ſectiones.<emph.end type="italics"></emph.end> vo<lb></lb>catigitur Archimedes acutianguli coni ſectionem, tam coni <lb></lb><expan abbr="ſectionẽ">ſectionem</expan>, quàm <expan abbr="ſectionẽ">ſectionem</expan> cylindri. </s> <s id="N14B07">veluti <expan abbr="etiã">etiam</expan> in decimatertia, <lb></lb>& decimaquarta propoſitione <expan abbr="eiuſdē">eiuſdem</expan> libri <expan abbr="acutiãguli">acutianguli</expan> coni ſe<lb></lb>ctio ab ipſo ea <expan abbr="nūcupatur">nuncupatur</expan> ſectio, quæ <expan abbr="oīa">oina</expan> latera tam conoidis <pb xlink:href="077/01/128.jpg" pagenum="124"></pb>rectanguli, quàm obtuſianguli abſcindit. </s> <s id="N14B25">dum modo non ſit <lb></lb>ad axem erecta. </s> <s id="N14B29">nullaquè alia de cauſa hæ ſectiones omnes i<lb></lb>dem acutianguli coni ſectionis nomen obtiuerunt; niſi quia <lb></lb>priùs hæc ſectio à cono acutiangulo nomen accepit, quando<lb></lb>quidem in ipſo fortaſse primùm cognita fuit, quaàm in alijs. <lb></lb>Ex dictis ita〈que〉 manifeſtum eſt, ſententiam Heraclij veram <lb></lb>eſſe poſſe, & rationi valdè conſentaneam; Archimedem ſcili <lb></lb>cet elementa conica ſcripſiſſe; Apollonium què, cùm ea ab Ar <lb></lb>chimede nondum edita inueniſſet, ſicut propria ſua edidiſſe. <lb></lb>Omitto interim multa ab Archimede in eius libris ſupponi, <lb></lb>quæ non niſi in conicis eſſe dcbebant, quæ quidem <expan abbr="habẽtur">habentur</expan> <lb></lb>ſolùm in conicis Apolloni. </s> <s id="N14B43">Negandum tamen non eſt, vt <lb></lb>Eutocius quo〈que〉 affirmat, ipſum Apollonium multa auxiſſe, <lb></lb>multaquè ad conica ſpectantia adinueniſſe. </s> <s id="N14B49">vt ipſemet Apol<lb></lb>lonius in epiſtola ad Eudemum fatetur. </s> <s id="N14B4D">cùm tamen non ſit <lb></lb>ſemperfacilè inuentis addere. </s> <s id="N14B51">Sed de his hactenus. </s> <s id="N14B53">ſat ſit au<lb></lb>tem nouiſſe, Archimedem, <expan abbr="quãdo">quando</expan> in hoclibro nominat por <lb></lb>tionem recta linea, rectanguliquè coni ſectione contentam, <lb></lb>eam ſignificare fectionem, quæ parabole nuncupatur. </s> </p> <pb xlink:href="077/01/129.jpg" pagenum="125"></pb> <p id="N14B62" type="margin"> <s id="N14B64"><margin.target id="marg200"></margin.target>5. <emph type="italics"></emph>primi co <lb></lb><expan abbr="nicorũ">nicorum</expan> A<lb></lb>poll.<emph.end type="italics"></emph.end></s> </p> <p id="N14B74" type="head"> <s id="N14B76">GVIDIVBALDI <lb></lb>E MARCHIONIBVS <lb></lb>MONTIS.</s> </p> <p id="N14B7C" type="head"> <s id="N14B7E">IN SECVNDVM ARCHIMEDIS <lb></lb>ÆQVEPONDERANTIVM <lb></lb>LIBRVM.</s> </p> <p id="N14B84" type="head"> <s id="N14B86">PARAPHRASIS <lb></lb>SCHOLIIS ILLVSTRATA.</s> </p> <figure id="id.077.01.129.1.jpg" xlink:href="077/01/129/1.jpg"></figure> <p id="N14B8D" type="head"> <s id="N14B8F">PROPOSITIO. I.</s> </p> <p id="N14B91" type="main"> <s id="N14B93">Si duo ſpacia recta linea, & re <lb></lb>ctanguli coni ſectione conten<lb></lb>ta, quæ ad datam rectam <expan abbr="lineã">lineam</expan> <lb></lb>applicare poſſumus, non ha <lb></lb>beantidem grauitatis <expan abbr="centrũ">centrum</expan>; <lb></lb>magnitudinis ex vtriſ〈que〉 i<lb></lb>pſorum compoſitæ centrum <lb></lb>grauitatis erit in recta linea, quæ ipſorum centra <lb></lb>grauitatis coniungit; ita diuidens dictam rectam li <lb></lb>neam, vt ipſius portiones permutatim eandem ad <lb></lb>inuicem proportionem habeant, vt ſpacia. </s> </p> <pb xlink:href="077/01/130.jpg" pagenum="126"></pb> <figure id="id.077.01.130.1.jpg" xlink:href="077/01/130/1.jpg"></figure> <p id="N14BB7" type="main"> <s id="N14BB9"><emph type="italics"></emph>Sint duo ſpacia AB CD, qualia dicta ſunt. </s> <s id="N14BBD">ipſorum autem centra <lb></lb>grauitatis ſint puncta EF.<emph.end type="italics"></emph.end> iungaturquè EF, quæ diuidatur in <lb></lb>H; <emph type="italics"></emph>& quam proportionem habet AB ad CD, <expan abbr="eãdem">eandem</expan> habeat FH <lb></lb>ad HE. oſtendendum eſt magnitudmis ex utriſquè AB CD ſpa<lb></lb>ciis compoſitæ centrum grauitaias eſſe punctum H. ſit quidemipſi EH <lb></lb>utra〈que〉 ipſarum FG FK æqualis; ipſi autem FH, hocest GE<emph.end type="italics"></emph.end><lb></lb>(ſuntenim EH GF æquales, à quibus dempta communi <lb></lb>GH remanent EG HF ęquales) <emph type="italics"></emph>ſit æqualis EL.<emph.end type="italics"></emph.end> & <expan abbr="quoniã">quoniam</expan> <lb></lb>FH eſt æqualis LE, & FK ipſi EH, <emph type="italics"></emph>erit & LH ipſi KH <lb></lb>æqualis.<emph.end type="italics"></emph.end> Cùm autem ſit FH ad HE, vt AB ad CD; ipſi <lb></lb>verò FH vtra〈que〉 ſit æqualis LE EG. ipſi autem HE vtra<lb></lb>〈que〉 æqualis GF FK, <emph type="italics"></emph>erit <expan abbr="etiã">etiam</expan> ut LG ad G<emph.end type="italics"></emph.end>k, <emph type="italics"></emph>ita AB ad CD.<emph.end type="italics"></emph.end><lb></lb>cùm ſit LG ad GK, vt FH ad HE; <emph type="italics"></emph>aupla enim est utra〈que〉<emph.end type="italics"></emph.end><lb></lb>EG GK <emph type="italics"></emph>utriuſ〈que〉<emph.end type="italics"></emph.end> FH HE. <emph type="italics"></emph>At uerò circa punctum<emph.end type="italics"></emph.end> E <emph type="italics"></emph>ipſius <lb></lb>AB,<emph.end type="italics"></emph.end> quod eſt eius centrum grauitatis, <emph type="italics"></emph>ex utra〈que〉 parte lineæ LG, <lb></lb>ipſi LG æquidistantes ducantur<emph.end type="italics"></emph.end> MO QN, quæ æqualiter ab <lb></lb>LG diſtent, ductis ſcilicet MQ ON æquidiſtantibus, ſint <lb></lb>LM LQ GO GN inter ſe æquales; <emph type="italics"></emph>ita ut ſpacium MN ſit <lb></lb>ſpacio AB æquale<emph.end type="italics"></emph.end>: quod quidem applicatum eſt ad <expan abbr="lineã">lineam</expan> LG. <lb></lb> <arrow.to.target n="marg201"></arrow.to.target> <emph type="italics"></emph>erit uti〈que〉 ipſius MN centrum grauitatis punctum E.<emph.end type="italics"></emph.end> cùm ſit <expan abbr="pũ-ctum">pun<lb></lb>ctum</expan> E in medio lineæ LG, quæ bifariam diuidit latera <lb></lb>oppoſita MQ ON parallelogrammi MN. <emph type="italics"></emph>compleatur ita<lb></lb>〈que〉 ſpacium NX. habebit quidem MN. ad NX proportionem,<emph.end type="italics"></emph.end> <pb xlink:href="077/01/131.jpg" pagenum="127"></pb><emph type="italics"></emph>quam<emph.end type="italics"></emph.end> habet QN ad NP, hoceſt <emph type="italics"></emph>LG ad GK. habet autem & <lb></lb>AB ad CD proportionem ipſius LG ad G<emph.end type="italics"></emph.end>K. <emph type="italics"></emph>ut igitur AB ad<emph.end type="italics"></emph.end> <arrow.to.target n="marg202"></arrow.to.target><lb></lb><emph type="italics"></emph>CD, ſic est MN ad NX. & permutando<emph.end type="italics"></emph.end> vt AB ad MN, ita <lb></lb>CD ad NX. <emph type="italics"></emph>æquale autem est AB ipſi MN, erit igitur & CD <lb></lb>ipſi NX æquale. </s> <s id="N14C79">Centrum autem grauitatisipſius<emph.end type="italics"></emph.end> NX <emph type="italics"></emph>est <expan abbr="punotũ">punotum</expan> <lb></lb>F.<emph.end type="italics"></emph.end> propterea quod eſt in medio lineæ GK, quæ <arrow.to.target n="marg203"></arrow.to.target> parallelo<lb></lb>grammi NX oppoſita latera ON XP bifariam ſecat. <emph type="italics"></emph>& <lb></lb>quoniam æqualis eſt LH ipſi HK, totaquè LK appaſita latera<emph.end type="italics"></emph.end> MQ <lb></lb>XP <emph type="italics"></emph>bifariam diuidit, totius PM <expan abbr="centrũ">centrum</expan> grauitatis erit punctum Hr <lb></lb>Verùm ipſum MP æquale est utriſ〈que〉 MN NX,<emph.end type="italics"></emph.end> quorum, cùm <lb></lb>ſint centra grauitatis EF, æ〈que〉pondera bunt ſpacia MN <lb></lb>NX ex diſtantijs FH HE. ſi igitur loco parallelo gram mo<lb></lb>rum MN NX ponatur AB in E, & CD in F, cùm ſit <lb></lb>AB ipſi MN, & CD ipſi NX æquale; ſpacia AB CD ex <arrow.to.target n="marg204"></arrow.to.target><lb></lb>diſtantijs FH HE æ〈que〉ponderabunt. <emph type="italics"></emph>ac propterea magnitudi <lb></lb>nis ex utriſ〈que〉 AB CD<emph.end type="italics"></emph.end> compoſitæ <emph type="italics"></emph>centrum grauitatis <expan abbr="eſtpunctũ">eſtpunctum</expan> <lb></lb>H.<emph.end type="italics"></emph.end> quod quidem propoſitum fuit. </s> </p> <p id="N14CC7" type="margin"> <s id="N14CC9"><margin.target id="marg201"></margin.target>2. <emph type="italics"></emph>cor.<emph.end type="italics"></emph.end> 9. <lb></lb><emph type="italics"></emph>primihui<emph.end type="italics"></emph.end>^{9}.</s> </p> <p id="N14CDA" type="margin"> <s id="N14CDC"><margin.target id="marg202"></margin.target>16.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N14CE5" type="margin"> <s id="N14CE7"><margin.target id="marg203"></margin.target>2.<emph type="italics"></emph>cor.<emph.end type="italics"></emph.end> 9. <lb></lb><emph type="italics"></emph>primihui<emph.end type="italics"></emph.end>^{9}.</s> </p> <p id="N14CF8" type="margin"> <s id="N14CFA"><margin.target id="marg204"></margin.target>8.<emph type="italics"></emph>poſthui<emph.end type="italics"></emph.end>^{9}</s> </p> <p id="N14D04" type="head"> <s id="N14D06">SCHOLIVM.</s> </p> <p id="N14D08" type="main"> <s id="N14D0A">Cùm ſit intentio Archimedis non nulla pertractare ad pa<lb></lb>rabolen ſpectantia; primùm iacit fundamentum, parabolas <lb></lb>nempe ita ſe habere, vt permutatim diſtantiæ, ex quibus <lb></lb>ſuntcollocatæ, ſe habent. </s> <s id="N14D12">& <expan abbr="quãuis">quamuis</expan> vniuerſim, atquè in om<lb></lb>nibus mutuam hanc conuenientiam ex dictis ex primo libro <lb></lb>depræhendere liceat, hoc tamen loco peculiariter voluitad <lb></lb>huberiorem do ctrinam id ipſum in parabolis demonſtrare. <arrow.to.target n="marg205"></arrow.to.target><lb></lb>& quamuis in primo libro dixerit Archimedes magnitudi<lb></lb>nes æ〈que〉ponderare, quando ita ſe habent inter ſe, ut diſtan<lb></lb>tiæ permutatim ſe habent; hocautem loco quærit <expan abbr="centrũ">centrum</expan> gra<lb></lb>uitatis magnitudinis ex parabolis compoſitæ; non ſunt <expan abbr="tamẽ">tamen</expan> <lb></lb>propoſitiones diuerſæ. </s> <s id="N14D33">nam & in primo libro dum in demon<lb></lb>ſtratio ne quærit proportionem diſtantiarum, oſtendit, vbi <lb></lb>nam ſit centrum grauitatis magnitudinum. </s> <s id="N14D39">quare <expan abbr="quãnis">quannis</expan> pro <lb></lb>poſitiones videantur diuerſæ, non ſunt tamen diuerſæ, ete<lb></lb>nim vt poſt tertiam primi libri propoſitionem adnotauimus, <pb xlink:href="077/01/132.jpg" pagenum="128"></pb>hæc planèſe conſequuntur, vt exempli gratia in figura pun<lb></lb>ctum H centrum eſt grauitatis magnitudinis ex vtriſ〈que〉 <lb></lb>AB CD compoſitæ. </s> <s id="N14D4B">ergo AB, & CD ex diſtantijs HEHF <lb></lb>æ〈que〉ponderant. </s> <s id="N14D4F">& è contra. </s> <s id="N14D51">hoc eſt AB CD æ〈que〉ponde<lb></lb>rant ex diſtantijs EH HF. ergo punctum H centrum eſt <lb></lb>grauitatis magnitudinis ex vtriſ〈que〉 AB CD compoſrtæ; <expan abbr="cũ">cum</expan> <lb></lb>ſit EHF recta linea. </s> <s id="N14D5D">Solent autem mathematici aliquando <lb></lb>eandem propoſitionem pluribusmedijs demonſtrare; idcirco <lb></lb>conſiderandum eſt, Archimedem in hac propoſitione alio v<lb></lb>ti medio ad oſtendendum punctum H centrum eſie graui<lb></lb>tatis, quo uſus eſt in ſexta propoſitione primi libri. </s> <s id="N14D67">cùm in pri <lb></lb>mo libro per diuiſionem magnitudinum, diuiſio nem què di <lb></lb>ſtantiarum vniuerſaliter domonſtret centrum grauitatis ma<lb></lb>gnitudinum. </s> <s id="N14D6F">hoc autem loco per parallelogramma MN <lb></lb>NX parabolis æqualia, & circa centra grauitatis EF conſti<lb></lb>tuta, in uenit centrum grauitatis magnitudinis ex vtriſ〈que〉 pa <lb></lb> <arrow.to.target n="marg206"></arrow.to.target> rallelogrammis MN NX compoſitæ. </s> <s id="N14D7B">quod eſt <expan abbr="quidẽ">quidem</expan> pun<lb></lb>ctum H. medium nempè totius parallelogrammi MP. <lb></lb>quod idem punctum H centrum eſt grauitatis vtriuſ〈que〉 pa <lb></lb>raboles AB CD in EF collocatæ. </s> </p> <p id="N14D87" type="margin"> <s id="N14D89"><margin.target id="marg205"></margin.target>6.7.<emph type="italics"></emph>primi <lb></lb>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N14D94" type="margin"> <s id="N14D96"><margin.target id="marg206"></margin.target><emph type="italics"></emph>ex<emph.end type="italics"></emph.end> 9.<emph type="italics"></emph>&<emph.end type="italics"></emph.end> 10 <lb></lb><emph type="italics"></emph>primihui<emph.end type="italics"></emph.end>^{9}.</s> </p> <p id="N14DAC" type="main"> <s id="N14DAE">Ex his obſeruandum occurrit, hanc eſſe peculiarem metho <lb></lb>dum, qua poſſumus quorumlibet planorum æ〈que〉pondera<lb></lb>tionem oſtendere; hoc eſt plana ex diſtantijs eandem permu<lb></lb>tatim proportionem habentibus, vt eadem met plana, æ〈que〉<lb></lb>ponderare; dum modo ipſis æqualia parallelogramma conſti <lb></lb>tuere poſſimus. </s> <s id="N14DBA">ac propterea ſupponit Archimedes, nos poſſe <lb></lb>applicare ad rectam lineam ſpacium æquale ſpacio recta li<lb></lb>nea, rcctanguliquè coni ſectione contento. </s> <s id="N14DC0">quod <expan abbr="quidẽ">quidem</expan> ſpa<lb></lb>cium ſupponit parallelogram mum exiſtere, cùm pun<lb></lb>ctum E centrum ſit grauitatis ſpacij MN, eſt F <lb></lb>ſpacij NX. punctum verò H totius PM. quòd ſi MN <lb></lb>NX & MP non eſſent parallelogramma, ne〈que〉 puncta EFH <lb></lb>eorum centra grauitatis exiſterent. </s> <s id="N14DD0">vt ex demonſtranone pa<lb></lb>tet. </s> <s id="N14DD4">ſuppoſuit tamen Archimedes nos poſſe applicare ad re<lb></lb>ctam lineam parallelogrammum æquale ſpacio recta linea, <lb></lb>rectanguliquè coniſectione contento; quia duplici medio in <pb xlink:href="077/01/133.jpg" pagenum="129"></pb>libro de quadratura paraboles, propoſitione ſcilicet decimaſe <lb></lb>ptima, & vigeſimaquarta, docuit quamlibet portionem recta <lb></lb>linea, rectanguliquè coni ſectione contentam ſeſquitertiam <lb></lb>eſſe trianguli eandem ipſi baſim habentis, & <expan abbr="altitudinẽ">altitudinem</expan> ęqua <lb></lb>lem. </s> <s id="N14DEA">Ex qua propoſitione facilè conſtat nos parabolę <expan abbr="ſpaciū">ſpacium</expan> <lb></lb>ad rectam lineam applicare poſſe, vt propoſitum fuit hoc <lb></lb>modo. </s> </p> <p id="N14DF4" type="head"> <s id="N14DF6">PROBLEMA.</s> </p> <p id="N14DF8" type="main"> <s id="N14DFA">Ad datam rectam lineam datę parabolę ęquale parallelo<lb></lb>grammum applicare, ita vt data linea oppoſita <expan abbr="parallelogrã-mi">parallelogran<lb></lb>mi</expan> latera biſariam diuidat. </s> </p> <figure id="id.077.01.133.1.jpg" xlink:href="077/01/133/1.jpg"></figure> <p id="N14E07" type="main"> <s id="N14E09">Data ſit parabole <lb></lb>ABC, ſitquè data recta <lb></lb>linea GK. oportet ad <lb></lb>GK <expan abbr="parallelogrãmum">parallelogrammum</expan> <lb></lb>applicare æquale por<lb></lb>tioni ABC, ita vt GK <lb></lb>bifariam diuidat oppo <lb></lb>ſita parallelogram mi <lb></lb>latera. </s> <s id="N14E1F">Conſtituatur ſu<lb></lb>per AC <expan abbr="triãgulũ">triangulum</expan> ABC, <lb></lb>qd baſim habeat AC, <lb></lb>eandem〈que〉 portionis <lb></lb><expan abbr="altitudinẽ">altitudinem</expan>; quod <expan abbr="quidẽ">quidem</expan> <lb></lb>fiet, <expan abbr="inuẽta">inuenta</expan> diametro DB, quæ parabolen in B ſecet, <expan abbr="iunctiſq́">iunctiſ〈que〉</expan>; <arrow.to.target n="marg207"></arrow.to.target><lb></lb>AB BC. eritvti〈que〉 parabole ABC trianguli ABC ſeſquitertia. <lb></lb>Ita〈que〉 diuidatur AC in tria ęqualia, quarum vna pars ſit CH. <arrow.to.target n="marg208"></arrow.to.target><lb></lb>producaturquè AC. fiatquè CL ipſi CH ęqualis<gap></gap> erit ſanè AL <lb></lb>ipſius AC ſeſq uitertia. </s> <s id="N14E4E">Et obid (iuncta BL) erit triangulum <lb></lb>ABL trianguli ABC ſeſquitertium. </s> <s id="N14E52">ſunt quippè triangula ABL <arrow.to.target n="marg209"></arrow.to.target><lb></lb>ABC inter ſe, vt baſes AL AC. ac per conſe〈que〉ns triangulum <lb></lb>ABL patabolę ABC exiſtit ęquale. </s> <s id="N14E5B">Applicetur ita〈que〉 ad linea <arrow.to.target n="marg210"></arrow.to.target><lb></lb>GK <expan abbr="parallelogrãmũ">parallelogrammum</expan> GS ęquale <expan abbr="triãgulo">triangulo</expan> ABL. erit GS parabo <pb xlink:href="077/01/134.jpg" pagenum="132"></pb> <arrow.to.target n="fig65"></arrow.to.target><lb></lb>læ ABC ęquale. </s> <s id="N14E73">deinceps ducatur NP ipſi GK <lb></lb>ęquidiſtans, quę bifariam diuidat oppoſita latera GR <lb></lb>KS. producanturquè RG SK. fiantquè GO KX ę<lb></lb>quales ipſis GN KP. iungaturquè OX; erit nimi-<lb></lb>rum parallelogram mum OP ipſi GS ęquale. </s> <s id="N14E7D">qua<lb></lb>re parallelogram mum OP parabolę ABC exiſtit ę<lb></lb>quale. </s> <s id="N14E83">Applicatum eſt igitur ad GK parallelogram<lb></lb>mum expoſitę parabolę ęquale. </s> <s id="N14E87">lineaquè GK paralle<lb></lb>logrammi OP bifariam diuidit oppoſita latera ON <lb></lb>XP. quod fieri oportebat. </s> </p> <p id="N14E8D" type="margin"> <s id="N14E8F"><margin.target id="marg207"></margin.target>44. <emph type="italics"></emph><expan abbr="ſecũdi">ſecundi</expan> <lb></lb>conicorum <lb></lb>Apoll.<emph.end type="italics"></emph.end></s> </p> <p id="N14E9F" type="margin"> <s id="N14EA1"><margin.target id="marg208"></margin.target>17. 24. <emph type="italics"></emph>Ar <lb></lb>ch. </s> <s id="N14EAA">dquad. <lb></lb>patab.<emph.end type="italics"></emph.end></s> </p> <p id="N14EB0" type="margin"> <s id="N14EB2"><margin.target id="marg209"></margin.target>1.<emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N14EBB" type="margin"> <s id="N14EBD"><margin.target id="marg210"></margin.target><emph type="italics"></emph>ex<emph.end type="italics"></emph.end> 44.<emph type="italics"></emph>pri<lb></lb>mi.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.134.1.jpg" xlink:href="077/01/134/1.jpg"></figure> <p id="N14ED1" type="main"> <s id="N14ED3">Si in portione recta linea rectanguliquè coni <lb></lb>ſectione contenta triangulum inſcribatur, <expan abbr="eandẽ">eandem</expan> <lb></lb>baſim cum portione habens, & altitudinem æqua <lb></lb>lem: & rurſus in reliquis portionibus triangula in<lb></lb>ſcribantur, quæ eaſdem baſes cum portionibus <lb></lb>habeant, & altitudinem æqualem; ſemper què in <lb></lb>reſiduis portionibus triangula eodem modo <lb></lb>inſcribantur: figura, quæ in portione oritur, <lb></lb>planè inſcribi dicatur. </s> <s id="N14EE9">Patet quidem lineas <pb xlink:href="077/01/135.jpg" pagenum="131"></pb>huius figuræ inſcriptæ angulos, qui ſunt vertici <lb></lb>portionis proximi, eoſquè deinceps coniungen<lb></lb>tes, baſi portionis æquidiſtantes eſſe; bifariamquè <lb></lb>à diametro portionis diuidi; diametrum verò in <lb></lb>proportione diuidere numeris deinceps impari<lb></lb>bus. </s> <s id="N14EF9">vno deno minato ad verticem portionis. </s> <s id="N14EFB">Hoc <lb></lb>autem ordinate oſtenſum eſt. </s> </p> <p id="N14EFF" type="head"> <s id="N14F01">SCHOLIVM.</s> </p> <p id="N14F03" type="main"> <s id="N14F05">Scopus Archimedis in hoc ſecundo libio, vt initio primi <lb></lb>diximus, eſt inuenire centrum grauitatis paraboles. </s> <s id="N14F09">& vt de<lb></lb>ducatnos in hanc cognitionem, quadam vtitur figura rectili<lb></lb>nea in parabole inſcripta, quę plurimùm conducit, & eſt <expan abbr="tã">tam</expan> <lb></lb>quam medium ad inueniendum hoc grauitatis centrum. </s> <s id="N14F15">his <lb></lb>igitur verbis docet, quo modo in parabole in ſcribenda ſit hęc <lb></lb>figura; in quibus multa quo 〈que〉 proponit tanquam ſit pro<lb></lb>poſitio quædam; in qua multa ſint oſtendenda. </s> <s id="N14F1D">quorum ta<lb></lb>męn demonſtrationem omiſit, ac tanquam ab eo alibi de<lb></lb>monſtratam. </s> <s id="N14F23">Horum autem ex Apollonij Pergęi conicis <lb></lb>demonſtrationem elicere quidem potuiſſemus. </s> <s id="N14F27">at quoniam <lb></lb>Archimedes ipſe non nulla ad hæ cſpectantia alijs in locis de<lb></lb>monſtrauit ideo Archimedem per Archimedem declarare o<lb></lb>portunum magis nobis viſum eſt. </s> </p> <p id="N14F2F" type="main"> <s id="N14F31">Sit portio contenta recta linea, rectanguliquè coni ſectio<lb></lb>ne ABC, cuius diameter BD. Iunganturquè AB BC, diuida<lb></lb>tur deinde AB bifariam in E, a quo ipſi BD æquidiſtans <pb xlink:href="077/01/136.jpg" pagenum="132"></pb>ducatur EF, eritvti 〈que〉 punctum F vertex portionis AFB. <lb></lb>vt Archimedes demonſtrauit in libro de quadratura parabo<lb></lb>les propoſitione decimaoctaua. </s> <s id="N14F3F">iungantur〈que〉 AF FB. rur <lb></lb>fus bifariam diuidantur AF FB in punctis GH, à quibus <lb></lb>ipſi BD ducantur æquidiſtantes GI HK <gap></gap>b eandem cau<lb></lb>ſam erit punctum I vertex portionis AIF. K verò portio<lb></lb>nis FKB. connectanturquè AI IF FK KB. eademquè pror <lb></lb>fus ratione ad alteram partem inſcribantur triangula CLB <lb></lb> <arrow.to.target n="fig66"></arrow.to.target><lb></lb>CML, & LNB. Primùm <expan abbr="quidẽ">quidem</expan> triangulum ABC dicitur <lb></lb>planè inſcriptum, vt Archimedes ipſe infra in demonſtratio<lb></lb>nibus quintæ, ſextæ, & octauæ propoſitionis nominat. </s> <s id="N14F5C">Dein<lb></lb>de figura AFBLC, figuraquè AIFKBNLMC dicuntur in <lb></lb>portione planè inſcriptæ. </s> <s id="N14F62">figuraquè AFBLC vna cum AC <lb></lb><expan abbr="pentagonũ">pentagonum</expan> in portione planè <expan abbr="inſcriptũ">inſcriptum</expan> dici <expan abbr="põt">pont</expan>. vt Archime <lb></lb>des in ſecunda parte demonſtrationis quintæ propoſitionis <lb></lb>huius libri nuncupat. </s> <s id="N14F75">ideòquè erit AIFKBNLMC nonago<lb></lb>num in portione planè inſcriptum. </s> <s id="N14F79">& ita in alijs. <expan abbr="Connectã">Connectam</expan> <pb xlink:href="077/01/137.jpg" pagenum="133"></pb>tur KN FL IM, quæ diametrum BD ſecent in punctis <lb></lb>STV. oſtendendum eſt, lineas KN FL IM baſi AC ęqui <lb></lb>diſtantes eſſe. </s> <s id="N14F87">deinde diametrum BD lineas KN FL IM <lb></lb>bifariam in punctis STV diuidere poſtremo lineas KN F<gap></gap><lb></lb>IM ita diametrum BD diſpeſcere, vt poſito vno BS, linea ST <lb></lb>ſit tria, TV quin〈que〉; & VD ſeptem. </s> <s id="N14F90">Producantur FE KH <lb></lb>ad RX. quoniam enim FR eſt æquid<gap></gap>tans BD, erit AE ad <arrow.to.target n="marg211"></arrow.to.target><lb></lb>EB, vt AR ad RD; eſt〈que〉 AE ipſi EB æqualis ergo AR i<lb></lb>pſi RD æqualis exiſtit. </s> <s id="N14F9D">eodem què modo oſtendetur FX æ<lb></lb>qualem eſſe XT. quandoquidem eſt FX ad XT, vt FH ad <lb></lb>HB. ſimiliterquè ad alteram partem, exiſtentibus LO NP i<lb></lb>pſi BD æquidiſtantibus, erit DO ipſi OC æqualis, & TP <lb></lb>ipſi PL. quod quidem eodem prorſus modo demonſtrabi<lb></lb>tur. </s> <s id="N14FA9">Quoniam autem AC bifariam à diametro diuiditur in <lb></lb>puncto D, erit DR ipſi DO æqualis, cùm vnaquæ〈que〉 ſit <lb></lb>dimidia ipſarum AD DC æqualium. </s> <s id="N14FAF">eſt igitur RD dimidia <lb></lb>ipſius AD, quæ dimidia eſt baſis AC. quod idem euenit ipſi <lb></lb>DO. quare BD ſeſquitertia eſt ipſius FR, & ipſius LO, ex de<lb></lb>cimanona Archimedis de quadratura paraboles. </s> <s id="N14FB7">ac propterea <lb></lb>eandem habet proportionem BD ad FR, quam ad LO. vnde <arrow.to.target n="marg212"></arrow.to.target><lb></lb>ſequitur FR æqualem eſſe ipſi LO. & obid FL ipſi AC <expan abbr="æ-quidiſtantẽ">æ<lb></lb>quidiſtantem</expan> eſſe. </s> <s id="N14FC6">& FT ipſi RD, & TL ipſi DO ęqualem. <lb></lb>vnde FT ipſi TL ęqualis exiſtit. </s> <s id="N14FCA">eadem quèratione prorſus in <lb></lb>portione FBL oſtendetur KN ipſi FL, ac per conſe〈que〉ns i<lb></lb>pſi AC ęquidiſtantem eſſe. </s> <s id="N14FD0">& KS ipſi SN æqualem exiſte<lb></lb>re. </s> <s id="N14FD4">Producatur IG ad Z, quæ ipſam AB ſecet in 9. linea ve<lb></lb>rò LO ſecet BC in <expan abbr="q;">〈que〉</expan> ductaquè MY ipſi BD æquidiſtans <lb></lb>ipſam ſecet BC in <foreign lang="grc">α</foreign>. & quoniam IZ eſt æquidiſtans FR, e<lb></lb>rit AG ad GF, ut A9 ad 9E, & AZ ad ZR. & eſt AG ipſi <arrow.to.target n="marg213"></arrow.to.target><lb></lb>GF æqualis, erit igitur A9 ipſi 9E, & AZ ipſi ZR æquaiis. <lb></lb>Eodemquè modo oſtendetur C<foreign lang="grc">α</foreign> ipſi <foreign lang="grc">α</foreign>Q, & CY ipſi YO ę<lb></lb>qualem eſſe. </s> <s id="N14FF5">quo niam autem in portione AFB a dimidia baſi <lb></lb>ducta eſt LF, à puncto autem 9, hoc eſt à dimidia dimidię ba <lb></lb>ſis AB (eſt enim E9 dimidia ipſius AE, quæ dimidia eſt baſis <lb></lb>AB) ducta eſt 9I diametro æquidiſtans, erit EF ſeſquitertiai<lb></lb>pſius I9 pari〈que〉 ratione oſtendetur QL ſeſquitereiam eſſe i<lb></lb>pſius M<foreign lang="grc">α</foreign> quare vt FE ad I9, ita LQ ad M<foreign lang="grc">α</foreign>. obſimilitudinem <pb xlink:href="077/01/138.jpg" pagenum="134"></pb> <arrow.to.target n="marg214"></arrow.to.target> autem triangulorum ABD AER ita eſt BD ad ER, vt DA <lb></lb>ad AR. eadem〈que〉iatione ita ſehabet BD ad QO, vt DC <lb></lb>ad CO. Sed vt DA ad AR, ita eſt DC ad CO, eſt quip <lb></lb>pe DA ipſius AR dupla, veluti DC ipſius CO. quare i<lb></lb> <arrow.to.target n="marg215"></arrow.to.target> ta erit BD ad ER, vt BD ad QO. ac propterea ER ipſi <lb></lb> <arrow.to.target n="marg216"></arrow.to.target> QO ęqualis exiſtit. </s> <s id="N15023">oſtenſa verò eſt RF ęqualis OL, reli<lb></lb>quaigitur EF reliquæ QL eſt æqualis, quia verò ita eſt FE <lb></lb> <arrow.to.target n="marg217"></arrow.to.target> ad I9, vt QL ad M<foreign lang="grc">α</foreign>, erit permutando FE ad QL, vt I9 <lb></lb> <arrow.to.target n="fig67"></arrow.to.target><lb></lb>ad M<foreign lang="grc">α</foreign>. ſuntquè FE QL ęquales, ergo I9 ipſi M<foreign lang="grc">α</foreign> ęqua<lb></lb>lis exiſtit. </s> <s id="N15042">quoniam autem ob trianguſoium ſimilitudinem <lb></lb>AER A9Z, ita eſt AR ad AZ, vt ER ad 9Z. ob ſimili<lb></lb>tudinem vero triangulorum QOC <foreign lang="grc">α</foreign>YC ita eſt CO ad CY, <lb></lb>vt QO ad <foreign lang="grc">α</foreign>Y: & eſt RA ad AZ, vt OC ad CY, cùm <lb></lb> <arrow.to.target n="marg218"></arrow.to.target> vtrę〈que〉 in dupla exiſtant proportione; e<gap></gap>t ER ad 9Z, vt <lb></lb>QO ad <foreign lang="grc">α</foreign>Y. & permutando ER ad QO vt 9Z ad <foreign lang="grc">α</foreign>Y. eſt <lb></lb>vero ER ipſi QO, æqualis, ergo 9Z ipſi <foreign lang="grc">α</foreign>Y ęqualis exiſtit. </s> <s id="N1506A">at <lb></lb>vero oſtenſa eſt I9 ęqualis M<foreign lang="grc">α</foreign>; to ta igitur IZ ipſi MY eſt ę- <pb xlink:href="077/01/139.jpg" pagenum="135"></pb>æqualis, quæ cùm ſintipſi BD æquidiſtantes, erunt & inter ſe<lb></lb>ſe parallelæ. </s> <s id="N15078">quare IM ipſi AC eſt æquidiſtans. </s> <s id="N1507A">Quoniam <arrow.to.target n="marg219"></arrow.to.target> ita<lb></lb>〈que〉 AR eſt æqualis CO, & horum dimidia, hoc eſt RZ ipſi <lb></lb>OY æqualis erit. </s> <s id="N15084">atqui DR eſt ipſi DO æqualis; ergo DZ ipſi <lb></lb>DY exiſtit æqualis. </s> <s id="N15088">ipſi verò DZ eſt æqualis IV, & ipſi DY æ<lb></lb>qualis VM. eruntigitur IV VM inter ſe equales. </s> <s id="N1508C">Iam ita〈que〉 <arrow.to.target n="marg220"></arrow.to.target><lb></lb>oſtenſum eſt, lineas KN FL IM, quę coniunguntangulos fi <lb></lb>guræ in parabole planè inſcriptæ, ipſi AC æquidiſtantes eſſe. <lb></lb>Diametrum què BD ipſas in punctis STV bifariam diſpeſcere. </s> </p> <p id="N15097" type="margin"> <s id="N15099"><margin.target id="marg211"></margin.target>2. <emph type="italics"></emph>lemma.<emph.end type="italics"></emph.end></s> </p> <p id="N150A2" type="margin"> <s id="N150A4"><margin.target id="marg212"></margin.target>9. <emph type="italics"></emph>quinti. <lb></lb>ex<emph.end type="italics"></emph.end> 33.34 <lb></lb><emph type="italics"></emph>primi.<emph.end type="italics"></emph.end></s> </p> <p id="N150B6" type="margin"> <s id="N150B8"><margin.target id="marg213"></margin.target>2.<emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N150C1" type="margin"> <s id="N150C3"><margin.target id="marg214"></margin.target><emph type="italics"></emph>ex<emph.end type="italics"></emph.end> 4.<emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N150D1" type="margin"> <s id="N150D3"><margin.target id="marg215"></margin.target>11. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N150DC" type="margin"> <s id="N150DE"><margin.target id="marg216"></margin.target>9. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N150E7" type="margin"> <s id="N150E9"><margin.target id="marg217"></margin.target>16. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N150F2" type="margin"> <s id="N150F4"><margin.target id="marg218"></margin.target><emph type="italics"></emph>ex<emph.end type="italics"></emph.end> 11.<emph type="italics"></emph>quin<lb></lb>ti<emph.end type="italics"></emph.end> 16.<emph type="italics"></emph>qu<gap></gap>u<emph.end type="italics"></emph.end></s> </p> <p id="N1510C" type="margin"> <s id="N1510E"><margin.target id="marg219"></margin.target>33.<emph type="italics"></emph>primi.<emph.end type="italics"></emph.end></s> </p> <p id="N15117" type="margin"> <s id="N15119"><margin.target id="marg220"></margin.target>34.<emph type="italics"></emph>primi.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.139.1.jpg" xlink:href="077/01/139/1.jpg"></figure> <figure id="id.077.01.139.2.jpg" xlink:href="077/01/139/2.jpg"></figure> <p id="N1512A" type="main"> <s id="N1512C">Quoniam ita〈que〉 in portione FBL à dimidia baſi ducta eſt <lb></lb>TB, a dimidia verò dimidiæ baſis ducta eſt XK, erit BT <arrow.to.target n="marg221"></arrow.to.target> ſeſ<lb></lb>quitertia ipſius KX, hoc eſt ipſius ST. eſt enim KT parallelo<lb></lb>grammum, & ST ipſi KX æqualis. </s> <s id="N15138">Si igitur ponatur BT <lb></lb>quattuor, erit ST tria, & BS vnum. </s> <s id="N1513C">ſimiliter quoniam BD <arrow.to.target n="marg222"></arrow.to.target><lb></lb>ſeſquitertia eſt ipſius FR, hoc eſt ipſius TD, cùm ſit TD ipſi <lb></lb>FR ęqualis. </s> <s id="N15145">ſi ita 〈que〉 ponatur BD ſexdecim, erit vnaquæ〈que〉 <lb></lb>FR TD duodecim. </s> <s id="N15149">& TB quattuor, vt poſitum fuit. <expan abbr="Quoniã">Quoniam</expan> <lb></lb>autem (vt diximus) eſt BD ad ER, vt DA ad AR, erit BD du<lb></lb>pla ipſius RE. quare ſi BD eſt ſexdecim, erit RE octo. </s> <s id="N15153">& quo<lb></lb>niam eſt FR duodecim, erit EF quatuor. </s> <s id="N15157">eſt autem FE ipſius <lb></lb>I9 ſeſquitertia, erit igitur I9 tria. </s> <s id="N1515B">& quoniam eſt ER ad 9Z, vt <lb></lb>RA ad AZ, erit ER dupla ipſius 9Z. ac propterea erit 9Z quat <lb></lb>tuor, cum ſit ER octo, & eſt 9I tria, tota ergo IZ, hoc eſt DV, <lb></lb>ſeptem exiſtet. </s> <s id="N15163">ſed quoniam eſt DT duodecim, cuius pars <lb></lb>DV eſt ſeptem, eritreliqua VT quin〈que〉. </s> <s id="N15167">Poſito igitur BS v<lb></lb>no, erit ST tria, TV quin〈que〉, & VD ſeptem. </s> <s id="N1516B">quod erat quo<lb></lb>〈que〉 demonſtrandum. </s> <s id="N1516F">Et hæc ſunt quę ab Archimede pro<lb></lb>poſita fucrant. </s> </p> <p id="N15173" type="margin"> <s id="N15175"><margin.target id="marg221"></margin.target>19.<emph type="italics"></emph>Archi<lb></lb>medis de <lb></lb>quad. </s> <s id="N15180">pa<lb></lb>rab.<emph.end type="italics"></emph.end></s> </p> <p id="N15186" type="margin"> <s id="N15188"><margin.target id="marg222"></margin.target>34. <emph type="italics"></emph>primi.<emph.end type="italics"></emph.end></s> </p> <p id="N15191" type="main"> <s id="N15193">Ex his tamen nonnulla quo〈que〉 colligemus ad ea, quæ ſe<lb></lb>quuntur neceſſaria. </s> <s id="N15197">ac primùm quidem conſtat BD quadru<lb></lb>plam eſſe ipſius BT, & ipſius FE. </s> </p> <pb xlink:href="077/01/140.jpg" pagenum="136"></pb> <p id="N1519E" type="main"> <s id="N151A0">Oſtenſum eſt enim BD ſexdecim eſſe, & BT quatuor, & FE <lb></lb>itidem quatuor exiſtere. </s> <s id="N151A4">Ex demonſtratione autem Archime <lb></lb>dis decimæ nonæ ptopoſitionis de quadratura paraboles cla<lb></lb>rè elicitur BD quadruplam eſſe ipſius BT. </s> </p> <p id="N151AA" type="main"> <s id="N151AC">Ex quibus etiam ſequitur FE QL inter ſe æquales eſſe. </s> <s id="N151AE">am<lb></lb>bo enim ſunt, vt quatuor. </s> </p> <figure id="id.077.01.140.1.jpg" xlink:href="077/01/140/1.jpg"></figure> <p id="N151B5" type="main"> <s id="N151B7">Præterea oſtendendum eſt triangulum AFB <expan abbr="triãgulo">triangulo</expan> BLC <lb></lb>ęquale eſſe, portionem què paraboles AFB portiom BLC ęqua <lb></lb>lem. </s> <s id="N151C1">Ampliùs triangulum AIF triangulo CML, & portio<lb></lb>nem AIF portioni CML æqualem eſſe, & reliqua triangula <lb></lb>reliquis triangulis, acportiones portionibus ęquales eſſe. </s> </p> <p id="N151C7" type="main"> <s id="N151C9">Ex vigeſima prima propoſitione Archimedis de quadratu<lb></lb>ra paraboles triangulum ABC vniuſcuiuſ〈que〉 trianguli AFB <lb></lb> <arrow.to.target n="marg223"></arrow.to.target> BLC eſt <expan abbr="octuplũ">octuplum</expan>. ergo ad ambo <expan abbr="eandẽ">eandem</expan> <expan abbr="hẽt">hent</expan> <expan abbr="proportionẽ">proportionem</expan>. qua <lb></lb>re triangula AFB BLC inter ſe ſunt ęqualia. </s> <s id="N151E5">At vero <expan abbr="quoniã">quoniam</expan> <pb xlink:href="077/01/141.jpg" pagenum="137"></pb>portio AFB trianguli AFB eſt ſeſquitertia, 〈que〉madmodum <arrow.to.target n="marg224"></arrow.to.target><lb></lb>portio BLC trianguli BLC, eritportio AFB ad triangulum <lb></lb>AFB, vt portio CLB ad triangulum CLB, & permutando <lb></lb>portio AFB ad portionem CLB, vt triangulum AFB ad <arrow.to.target n="marg225"></arrow.to.target><lb></lb>ipſum CLB <expan abbr="triãgula">triangula</expan> verò ſunt æqualia; ergo portiones AFB <lb></lb>CLB inter ſe ſunt æquales. </s> <s id="N15203">Eademquè ratione <expan abbr="triangulũ">triangulum</expan> AFB <lb></lb>octuplum eſt trianguli AIF, & triangulum CLB octuplum <lb></lb>ipſius CML. vnde triangula AIF CML ſunt æqualia. </s> <s id="N1520D">et ea<lb></lb>rum quo〈que〉 portiones AIF CML ſunt æquales, ſiquidem <lb></lb>ſunt triangulorum ſeſquitertiæ. </s> <s id="N15213">Et hoc modo reliqua trian<lb></lb>gula FKB LNB, & portiones FKB LNB <expan abbr="oſtendẽtur">oſtendentur</expan> æqua<lb></lb>les. </s> <s id="N1521D">cùm ſit triangulum FBL dictorum triangulorum octu<lb></lb>plum. </s> <s id="N15221">quod oportebat quo〈que〉 demonſtrate. </s> </p> <p id="N15223" type="margin"> <s id="N15225"><margin.target id="marg223"></margin.target>9. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N1522E" type="margin"> <s id="N15230"><margin.target id="marg224"></margin.target>17.24. A<emph type="italics"></emph>r <lb></lb>chimedis <lb></lb>de quad. <lb></lb>parab.<emph.end type="italics"></emph.end></s> </p> <p id="N1523F" type="margin"> <s id="N15241"><margin.target id="marg225"></margin.target>16. <emph type="italics"></emph>quimi<emph.end type="italics"></emph.end><lb></lb>21.<emph type="italics"></emph>Archi<lb></lb>medis de <lb></lb>quad. </s> <s id="N15253">pa<lb></lb>rab.<emph.end type="italics"></emph.end></s> </p> <p id="N15259" type="main"> <s id="N1525B">His demonſtratis ſequitur Archimedes quaſi connectens ſe <lb></lb>〈que〉ntem propoſitionem cumijs, quæ ſuppoſita ſunt, inqui<lb></lb>ens, <emph type="italics"></emph>ſi autem & in portione<emph.end type="italics"></emph.end> &c. </s> </p> <p id="N15267" type="head"> <s id="N15269">PROPOSITIO. II.</s> </p> <p id="N1526B" type="main"> <s id="N1526D">Si autem & in portione rectalinea, rectangu<lb></lb>li〈que〉 coni ſectione contenta, figura rectilinea pla <lb></lb>ne inſcribatur, inſcriptæ figuræ centrum grauita<lb></lb>tis erit in diametro portionis. </s> </p> <pb xlink:href="077/01/142.jpg" pagenum="138"></pb> <p id="N15278" type="main"> <s id="N1527A"><emph type="italics"></emph>Sit portio ABC, qualis dicta eſt, & in ipſa planè inſcribatur recti<lb></lb>linea figura AEFGBHIKC. portionis verò diameter ſit BD. <expan abbr="oſtẽ-">oſten-</expan><emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg226"></arrow.to.target> <emph type="italics"></emph>dendum eſt, rectilineæ figuræ centrum grauitatiseſſe in linea BD.<emph.end type="italics"></emph.end> <expan abbr="iũ">ium</expan> <lb></lb>gantur GH FI EK. quę ipſi AC, & inter ſe ęquidiſtantes <lb></lb>erunt. </s> <s id="N15299">hę verò lineæ diametrum BD ſecent in punctis NML <lb></lb> <arrow.to.target n="fig68"></arrow.to.target><lb></lb><emph type="italics"></emph>Quoniam enim<emph.end type="italics"></emph.end> lineæ GH FI EK bifariam ſunt à diame<lb></lb>tro BD diuiſæ in punctis NML, trapezium AEKC duas <lb></lb> <arrow.to.target n="marg227"></arrow.to.target> habebit line as æquidiſtantes AC EK, quas bifariam diuidit <lb></lb>DL, quare <emph type="italics"></emph>trapezii AEKC centrum grauitatis est in LD. at<emph.end type="italics"></emph.end> ob <lb></lb>eandem cauſam <emph type="italics"></emph>trapezii EFIK centrum est in ML; trapezii verò <lb></lb>FGHI centrum est in MN.<emph.end type="italics"></emph.end> lineæ enim LM MN bifariam <lb></lb> <arrow.to.target n="marg228"></arrow.to.target> diuidunt parallela latera EK FI GH, <emph type="italics"></emph>ſed & trianguli etiam <lb></lb>GBH centrum grauitatis eſt in BN.<emph.end type="italics"></emph.end> quippè cùm BN ipſam <lb></lb>GH bifariam diuidat. <emph type="italics"></emph>perſpicuum eſt totius rectilineæ figuræ<emph.end type="italics"></emph.end><lb></lb>AEFGBHIKC <emph type="italics"></emph>centrum grauitatis eſſe in linea BD.<emph.end type="italics"></emph.end> quod de<lb></lb>monſtrare oportebat. </s> </p> <pb xlink:href="077/01/143.jpg" pagenum="139"></pb> <p id="N152E3" type="margin"> <s id="N152E5"><margin.target id="marg226"></margin.target><emph type="italics"></emph><expan abbr="exdemõ">exdemom</expan> <lb></lb>stratis.<emph.end type="italics"></emph.end></s> </p> <p id="N152F2" type="margin"> <s id="N152F4"><margin.target id="marg227"></margin.target>15. <emph type="italics"></emph>primi <lb></lb>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N152FF" type="margin"> <s id="N15301"><margin.target id="marg228"></margin.target>13. <emph type="italics"></emph>primi <lb></lb>huius.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.143.1.jpg" xlink:href="077/01/143/1.jpg"></figure> <p id="N15310" type="head"> <s id="N15312">SCHOLIVM.</s> </p> <p id="N15314" type="main"> <s id="N15316">Ecce qúo Archimedes incipit inueſtigare centrum graui<lb></lb>tatis paraboles. </s> <s id="N1531A">nam ex hoc, quod oſtendit centrum grauita<lb></lb>tis figuræ in portione planè inſcriptæ eſſe in diametro por<lb></lb>tionis, ſtatim colliget in quarta propoſitione centrum graui<lb></lb>tatis paraboles in diametro quo〈que〉 ipſius portionis exiſtere. <lb></lb>interponit autem Archimedes ſe〈que〉ntem propoſitionem. <expan abbr="nã">nam</expan> <lb></lb>antequam inueniat centrum grauitatis paraboles, opus habet <lb></lb>prius oſtendere centra grauitatis duarum, & vt ita dicam om <lb></lb>nium parabol<gap></gap>rum diametros in eadem proportione ſecare. <lb></lb>ad quod demonſtrandum, hanc <expan abbr="paſſionẽ">paſſionem</expan> figuris planè inſcri<lb></lb>ptis priùs accidere <expan abbr="oſtẽdit">oſtendit</expan>. potuiſſetquè Archimedes priùs quar <lb></lb>tam propoſitionem oſtendere, quam tertiam; ſe〈que〉ntem ve<lb></lb>rò propoſitionem immediatè poſuit poſt ſecundam, ordo e<lb></lb>nim ſic poſtulat. </s> <s id="N15342">etenim ambæ deijs pertractant, quæ rectili<lb></lb>neis figuris plane inſcriptis accidunt. </s> <s id="N15346">Pręterea earum demon <lb></lb>ſtrationes ferè circa eadem verſantur, cùm ijsdem rectis lineis <lb></lb>in portionibus eodem modo ductis vtantur; ob ſe〈que〉ntis ve<lb></lb>rò propoſitionis intelligentiam hęc priùs oſtendemus. </s> </p> <figure id="id.077.01.143.2.jpg" xlink:href="077/01/143/2.jpg"></figure> <p id="N15351" type="head"> <s id="N15353">LEMMA I.</s> </p> <p id="N15355" type="main"> <s id="N15357">Eandem habeat proportionem AB ad CD, quam habet <lb></lb>GH ad KL. CD verò ad EF <expan abbr="eã">eam</expan>, <expan abbr="quã">quam</expan> habet kL ad MN. ſintquè <pb xlink:href="077/01/144.jpg" pagenum="136"></pb>AB CD EF inter ſe ęquidſtantes. </s> <s id="N15367">ſimiliter GH KL MN <lb></lb>æquidiſtantes, ſintantem ductæ BDF HLN rectæ lineæ; ſit<lb></lb>què BD ad DF, vt HL ad LN. ſitquè maior AB quàm <lb></lb>CD, & CD, quàm EF. vnde erit quoquè GH maior KL, <lb></lb>& KL, quam MN. iunctiſquè AC CE, & GK KM. <lb></lb>Dico ſpacium ACDB ad ſpacium CEFD eandem habere <lb></lb>proportionem, quam ſpacium GKLH ad ſpacium KMNL. </s> </p> <figure id="id.077.01.144.1.jpg" xlink:href="077/01/144/1.jpg"></figure> <p id="N15378" type="main"> <s id="N1537A">Producantur AC CE, quæ cum BF conueniant in OP. <lb></lb>productæquè GK KM cum HN conueniant in QR. <lb></lb>concurrentenim, quoniam CD KL ſunt minores ipſis AB <lb></lb> <arrow.to.target n="marg229"></arrow.to.target> GH, & EF MN minores ipſis CD KL. Fiatquè vt AB <lb></lb>ad CD, ita CD ad V. & vt GH ad kL, ita KL ad X. <lb></lb>deinceps CD ad EF, ita EF ad Y. & vt KL ad MN, <lb></lb>ita MN ad Z. Quoniam igitur triangulum ABO ſimile <lb></lb>eſt triangulo CDO, cùm ſit CD æquidiſtansipſi AB. ha <lb></lb> <arrow.to.target n="marg230"></arrow.to.target> bebit triangulum ABO ad CDO, proportionem, quam ha <lb></lb>bet AB ad CD duplicatam. </s> <s id="N15396">hoc eſt quam hab et AB ad <lb></lb>V. Eodemquè modo oſtendetur <expan abbr="triangulũ">triangulum</expan> GHQ ad KLQ <lb></lb>ita eſſe, vt GH ad X<gap></gap> quia verò AB CD V ita ſe <expan abbr="habẽt">habent</expan>, <lb></lb> <arrow.to.target n="marg231"></arrow.to.target> vt GH kL X, erit ex æquali AB ad V, & GH ad X. <lb></lb>triangulum igitur ABO eandem habet proportionem ad <pb xlink:href="077/01/145.jpg" pagenum="129"></pb>CDO, quam triangulum GHQ ad <expan abbr="KLq.">KL〈que〉</expan> quare diuiden<lb></lb>do ſpacium ACDB ad triangulum CDO eſt, vt ſpacium <arrow.to.target n="marg232"></arrow.to.target><lb></lb>GKLH ad triangulum <expan abbr="kLq.">kL〈que〉</expan> Rurſus quoniam ob triangu<lb></lb>lorum ſimilitudinem ABO CDO, ita eſt AB ad CD, vt <arrow.to.target n="marg233"></arrow.to.target><lb></lb>BO ad OD. ſimiliter ob ſimilitudinem <expan abbr="triangulorũ">triangulorum</expan> GHQ <lb></lb>KLQ ita eſt GH ad kL, vt HQ ad QL. & eſt AB ad CD, <lb></lb>vt GH ad KL, erit BO ad OD, vt HQ ad QL. & <arrow.to.target n="marg234"></arrow.to.target> diui<lb></lb>dendo BD ad DO, vt HL ad <expan abbr="Lq.">L〈que〉</expan> deinde <expan abbr="conuertẽdo">conuertendo</expan> DO <lb></lb>ad DB, vt LQ ad LH. & eſt BD ad DF, vt HL ad LN, erit <arrow.to.target n="marg235"></arrow.to.target><lb></lb>ex ęquali DO ad DF, vt LQ ad LN. Quoniam autem ſimi <lb></lb>lium triangulorum CDP EFP latus CD ad latus EF ita ſe <lb></lb>habet, vt DP ad PF. ſimiliter exiſtentibus ſimilibus triangu<lb></lb>lis KLR MNR ita eſt KL ad MN, vt LR ad RN, & vt CD <lb></lb>ad EF, ita eſt KL ad MN, erit DP ad PF, vt LR ad RN. <arrow.to.target n="marg236"></arrow.to.target><lb></lb>& per conuerſionem rationis PD ad DF, vt RL ad LN. & <lb></lb>conuertendo DF ad DP, vt LN ad LR. diximus <expan abbr="autẽ">autem</expan> OD <lb></lb>ad DF ita eſſe, vt QL ad LN, & eſt DF ad DP, vt LN ad <lb></lb>LR. ergo ex ęquali erit OD ad DP, vt QL ad LR. At verò <arrow.to.target n="marg237"></arrow.to.target><lb></lb>quoniam ita eſt OD ad DP, vt triangulum OCD ad PCD, <lb></lb>& vt QL ad LR, ita eſt triangulum QKL ad <expan abbr="triangulũ">triangulum</expan> RKL, <lb></lb>erit OCD ad PCD, vt QKL ad RKL. Quoniam <expan abbr="autẽ">autem</expan> <expan abbr="triã">triam</expan> <lb></lb>gula CDP EFP ſunt ſimilia, triangulum CDP ad triangulum <arrow.to.target n="marg238"></arrow.to.target><lb></lb>EFP proportionem habebit, quam CD ad EF duplicatam, <lb></lb>hoc eſt quam habet CD ad Y, cùm ſint CD EF Y propor<lb></lb>tionales. </s> <s id="N1541C">ſimiliter ob triangulorum KLR MNR ſimilitudi<lb></lb>nem triangulum KLR ad MNR, ita erit vt KL ad Z, eſt au<lb></lb>tem CD ad Y, vt KL ad Z, erit igitur <expan abbr="triãgulum">triangulum</expan> CDP ad <lb></lb>EFP, vt KLR ad MNR, & diuidendo <expan abbr="ſpaciũ">ſpacium</expan> CEFD ad trian <arrow.to.target n="marg239"></arrow.to.target><lb></lb>gulum EFP, vt ſpacium KMNL ad triangulum MNR. & <expan abbr="cõ">com</expan> <arrow.to.target n="marg240"></arrow.to.target><lb></lb>uertendo triangulum EFP ad ſpacium CEFD, vt <expan abbr="triangulũ">triangulum</expan> <lb></lb>MNR ad ſpacium KMNL. Ita〈que〉 quoniam oſtenſum eſt i<lb></lb>ta eſſe ſpacium ACDB ad triangulum CDO, vt ſpacium <lb></lb>GKLH ad triangulum <expan abbr="KLq.">KL〈que〉</expan> & vt <expan abbr="triangulũ">triangulum</expan> CDO ad trian <lb></lb>gulum CDP, ita triangulum KLQ ad <expan abbr="triangulũ">triangulum</expan> KLR, dein<lb></lb>de, vt triangulum CDP ad triangulum EFP, ita <expan abbr="triãgulum">triangulum</expan> <lb></lb>KLR ad triangulum MNR; deniquè vt triangulum EFP ad <lb></lb>ſpacium CEFD, ita triangulum MNR ad ſpacium kMNL, <pb xlink:href="077/01/146.jpg" pagenum="142"></pb> <arrow.to.target n="marg241"></arrow.to.target> erit ex æquali à primo ad vltimum ſpacium ACDB ad <expan abbr="ſpaciũ">ſpacium</expan> <lb></lb>CEFD, vt ſpacium GKLH ad ſpacium KMNL. quod <expan abbr="demõ">demom</expan> <lb></lb>ſtrare oportebat. </s> </p> <p id="N15470" type="margin"> <s id="N15472"><margin.target id="marg229"></margin.target>11. <emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N1547B" type="margin"> <s id="N1547D"><margin.target id="marg230"></margin.target>9. <emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N15486" type="margin"> <s id="N15488"><margin.target id="marg231"></margin.target>22 <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N15491" type="margin"> <s id="N15493"><margin.target id="marg232"></margin.target>17. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N1549C" type="margin"> <s id="N1549E"><margin.target id="marg233"></margin.target><emph type="italics"></emph>eſt<emph.end type="italics"></emph.end> 4.<emph type="italics"></emph>ſexti<emph.end type="italics"></emph.end></s> </p> <p id="N154AC" type="margin"> <s id="N154AE"><margin.target id="marg234"></margin.target>17.<emph type="italics"></emph>quinti. <lb></lb>cor.<emph.end type="italics"></emph.end>4. <emph type="italics"></emph><expan abbr="quī">quim</expan> <lb></lb>ti.<emph.end type="italics"></emph.end></s> </p> <p id="N154C4" type="margin"> <s id="N154C6"><margin.target id="marg235"></margin.target>22. <emph type="italics"></emph>quinti<emph.end type="italics"></emph.end></s> </p> <p id="N154CF" type="margin"> <s id="N154D1"><margin.target id="marg236"></margin.target><emph type="italics"></emph>ex<emph.end type="italics"></emph.end> 11. <emph type="italics"></emph><expan abbr="quĩ">quim</expan> <lb></lb>ti. <lb></lb>cor.<emph.end type="italics"></emph.end> 19. <lb></lb><emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N154ED" type="margin"> <s id="N154EF"><margin.target id="marg237"></margin.target>22. <emph type="italics"></emph>quinti <lb></lb>ex<emph.end type="italics"></emph.end> 1.<emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N15500" type="margin"> <s id="N15502"><margin.target id="marg238"></margin.target>19. <emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N1550B" type="margin"> <s id="N1550D"><margin.target id="marg239"></margin.target><emph type="italics"></emph>ex quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N15515" type="margin"> <s id="N15517"><margin.target id="marg240"></margin.target><emph type="italics"></emph>cor.<emph.end type="italics"></emph.end>4. <emph type="italics"></emph><expan abbr="quī">quim</expan> <lb></lb>ti.<emph.end type="italics"></emph.end></s> </p> <p id="N1552A" type="margin"> <s id="N1552C"><margin.target id="marg241"></margin.target>22. <emph type="italics"></emph>quinti<emph.end type="italics"></emph.end></s> </p> <p id="N15535" type="head"> <s id="N15537">LEMMA II.</s> </p> <p id="N15539" type="main"> <s id="N1553B"><expan abbr="Æquidiſtãtes">Æquidiſtantes</expan> verò lineę AB CD ita ſe habeant, vt æquidi<lb></lb>ſtantes EF GH, ſitquè maior AB, quàm CD, & EF, quam <lb></lb>GH. & ſuper CD GH ſint triangula CDP GHR, <expan abbr="ſintq́">ſint〈que〉</expan>; BDP <lb></lb>FHR rectæ lineæ, & vt BD ad DP, ita ſit FH ad HR. <expan abbr="iunctisq́">iunctis〈que〉</expan>; <lb></lb>AC EG. Dico ſpacium ACDB ad <expan abbr="triangulũ">triangulum</expan> CDP ita eſſe, vt <lb></lb>ſpacium EG HF ad triangulum GHR. </s> </p> <figure id="id.077.01.146.1.jpg" xlink:href="077/01/146/1.jpg"></figure> <p id="N15559" type="main"> <s id="N1555B">Eadem enim prorſus ratione productis AC EG, quæ cum <lb></lb>BP FR conueniant in OQ, oſtendetur ſpacium AD ad trian <lb></lb>gulum CDO ita eſſe, vt ſpacium EH ad triangulum <expan abbr="GHq.">GH〈que〉</expan> & <lb></lb>eſſe OD ad DB, ut QH ad HF. & quoniam eſt BD ad DP, vt <lb></lb> <arrow.to.target n="marg242"></arrow.to.target> FH ad HR, erit ex ęquali OD ad DP, vt QH ad HR. & vt OD <lb></lb>ad DP, ita eſt triangulum CDO ad triangulum CDP, & vt <lb></lb>QH ad HR, ita triangulum GHQ ad GHR. cùm ita〈que〉 ſit <lb></lb>AD ad CDO, vt EH ad GHQ, & vt CDO ad CDP, ita <lb></lb> <arrow.to.target n="marg243"></arrow.to.target> GHQ ad GHR. ex æquali erit ſpacium AD ad triangulum <lb></lb>CDP, vt ſpacium EH ad triangulum GHR. quod demonſtra <lb></lb>re oportebat. </s> </p> <pb xlink:href="077/01/147.jpg" pagenum="143"></pb> <p id="N15580" type="margin"> <s id="N15582"><margin.target id="marg242"></margin.target>22 <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end><lb></lb><gap></gap>. <emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N15593" type="margin"> <s id="N15595"><margin.target id="marg243"></margin.target>22. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N1559E" type="head"> <s id="N155A0">LEMMA. III.</s> </p> <figure id="id.077.01.147.1.jpg" xlink:href="077/01/147/1.jpg"></figure> <p id="N155A5" type="main"> <s id="N155A7">Sit A ad CD, vt E ad FG, diuidan <lb></lb><expan abbr="turq́">tur〈que〉</expan>; CD FG in <expan abbr="eadẽ">eadem</expan> proportione in HK, <lb></lb>ita vt ſit CH ad HD, vt FK ad KG. <lb></lb>Dico A ad DH ita eſſe, vt E ad KG. <lb></lb>A verò ad CH, vt E ad Fk. </s> </p> <p id="N155B8" type="main"> <s id="N155BA">Quoniam enim ita eſt CH ad HD, vt FK ad kG; e<lb></lb>rit componendo CD ad DH, vt FG ad GK. eſt autem A <arrow.to.target n="marg244"></arrow.to.target><lb></lb>ad CD, vt E ad FG; CD verò eſt ad DH, vt FG ad G<emph type="italics"></emph>K<emph.end type="italics"></emph.end>; er <lb></lb>go ex æquali A erit ad DH, vt E ad GK. Deinde <arrow.to.target n="marg245"></arrow.to.target> quo<lb></lb>niam eſt GH ad HD, vt FK ad kG; erit conuertendo <arrow.to.target n="marg246"></arrow.to.target><lb></lb>DH ad HC, vt GK ad KF. rurſus igitur ex æquali A e<lb></lb>rit ad CH, vt E ad FK. quod oſtendere oportebat. </s> </p> <p id="N155D8" type="margin"> <s id="N155DA"><margin.target id="marg244"></margin.target>18.<emph type="italics"></emph>qumti.<emph.end type="italics"></emph.end></s> </p> <p id="N155E3" type="margin"> <s id="N155E5"><margin.target id="marg245"></margin.target>22 <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N155EE" type="margin"> <s id="N155F0"><margin.target id="marg246"></margin.target><emph type="italics"></emph>cor.<emph.end type="italics"></emph.end>4.<emph type="italics"></emph><expan abbr="quĩ-ti">quin<lb></lb>ti</expan>.<emph.end type="italics"></emph.end></s> </p> <p id="N15603" type="head"> <s id="N15605">PROPOSITIO. III.</s> </p> <p id="N15607" type="main"> <s id="N15609">Si in <expan abbr="vtraq;">vtra〈que〉</expan> <expan abbr="duarũ">duarum</expan> <expan abbr="ſimiliũ">ſimilium</expan> <expan abbr="portionũ">portionum</expan> recta linea re <lb></lb>ctanguliquè coni ſectione contentarum rectili<lb></lb>neæ figuræ planè inſcribantur; figuræ verò inſcri<lb></lb>ptæ latera inter ſe multitudine æqualia habeant; <lb></lb>rectilinearum centra grauitatum portionum dia<lb></lb>metros ſimiliter ſecabunt. </s> </p> <pb xlink:href="077/01/148.jpg" pagenum="144"></pb> <figure id="id.077.01.148.1.jpg" xlink:href="077/01/148/1.jpg"></figure> <p id="N1562B" type="main"> <s id="N1562D"><emph type="italics"></emph>Sint duæ portiones ABC XOP, in ipſiſquè planè in ſcribantur recti<lb></lb>lineæ figuræ<emph.end type="italics"></emph.end> AEFGBHIKC XSYQOZVTP; <emph type="italics"></emph>quæ omnia latera <lb></lb>inter ſe numero æqualia habeanta, Diametri verò portionum ſint BD<emph.end type="italics"></emph.end> <pb xlink:href="077/01/149.jpg" pagenum="145"></pb><emph type="italics"></emph>OR. <expan abbr="iungãturq́">iungantur〈que〉</expan>; E<emph.end type="italics"></emph.end>k <emph type="italics"></emph>FI GH.<emph.end type="italics"></emph.end> quæ inter ſe, & ipſi AC <expan abbr="çquidiſtãtes">çquidiſtantes</expan> <arrow.to.target n="marg247"></arrow.to.target><lb></lb>erunt; bifariam què à diametro BD in punctis LMN diuiſæ e<lb></lb>runt. </s> <s id="N1565C">Iungantur ſimiliter <emph type="italics"></emph>& ST YV QZ<emph.end type="italics"></emph.end>, quas bifariam dia<lb></lb>meter OR in punctis 9<foreign lang="grc">αβ</foreign> diuidet. </s> <s id="N1566A">eruntquè ductæ lineæ ipſi <lb></lb>XP, & inter ſe æquidiſtantes. <emph type="italics"></emph>Quoniam igitur BD diuiditur à lineis <lb></lb>æquidiſtantibus<emph.end type="italics"></emph.end> GH FI EK <emph type="italics"></emph>in proportionibus numeris deinceps impa<lb></lb>ribus;<emph.end type="italics"></emph.end> poſito enim vno BN, eſt quidem NM tria, ML quin〈que〉, <lb></lb>& LD ſeptem. </s> <s id="N15680">ſed <emph type="italics"></emph>& RO ſimiliter<emph.end type="italics"></emph.end> à lineis QZ YV ST in pro<lb></lb>portionibus diuiditur numeris deinceps imparibus, <expan abbr="eadẽ">eadem</expan>. <expan abbr="n.">enim</expan> <lb></lb>ratione ſi ponatur O<foreign lang="grc">β</foreign> vnum, erit <foreign lang="grc">βα</foreign> tria, <foreign lang="grc">α</foreign>9 <expan abbr="quinq́">quin〈que〉</expan>;, & 9R <lb></lb>ſeptem. <emph type="italics"></emph>& portiones ipſorum<emph.end type="italics"></emph.end> diametrorum BD OR <emph type="italics"></emph>ſunt numero æ<lb></lb>quales.<emph.end type="italics"></emph.end> quot.n ſunt BN NM ML LD, tot ſunt O<foreign lang="grc">β βα α</foreign> 9 9R. <emph type="italics"></emph>pa <lb></lb>tet diametrorum portiones in eadem eſſe proportione<emph.end type="italics"></emph.end>, vt 〈que〉m <expan abbr="admodũ">admodum</expan> <lb></lb>eſt BN ad NM, & NM ad ML, & ML ad LD, ita eſſe O<foreign lang="grc">β</foreign> ad <lb></lb><foreign lang="grc">βα</foreign>, & <foreign lang="grc">βα</foreign> ad <foreign lang="grc">α</foreign>9, & <foreign lang="grc">α</foreign>9 ad 9R. Atverò quoniam ita eſt DB ad BL, <lb></lb>vt RO ad O9; (ſunt.n.ut ſexdecim ad nouem) & ut DB ad BL, <arrow.to.target n="marg248"></arrow.to.target><lb></lb>ita eſt quadratum ex AD ad <expan abbr="quadratũ">quadratum</expan> ex EL; & vt RO ad O9, <lb></lb>ita eſt <expan abbr="quadratũ">quadratum</expan> ex XR ad quadratum ex S<emph type="italics"></emph>9<emph.end type="italics"></emph.end>; erit <expan abbr="quadratũ">quadratum</expan> ex <lb></lb>AD ad <expan abbr="quadratũ">quadratum</expan> ex EL, vt <expan abbr="quadratũ">quadratum</expan> ex XR ad ex S9 <expan abbr="quadratũ">quadratum</expan>. <lb></lb>ergo ut AD ad EL, ita XR ad S9. & horum dupla <expan abbr="nẽpè">nempè</expan> AC ad <lb></lb>EK, vt XP ad ST: <expan abbr="eademq́">eadem〈que〉</expan>; prorſus <expan abbr="rõne">ronne</expan>, quoniam ita eſt LB <arrow.to.target n="marg249"></arrow.to.target><lb></lb>ad BM, vt 9O ad O<foreign lang="grc">α</foreign> (ſunt.n.ut nouem ad quatuor) oſtendetur <lb></lb>EL ad FM ita eſſeut S9 ad Y<foreign lang="grc">α</foreign>, & horum dupla, ſcilicet EK ad FI <lb></lb>ita eſſe, ut ST ad YV. <expan abbr="Cùmq́">Cùm〈que〉</expan>; ſit MB ad BN, vt <foreign lang="grc">α</foreign>O ad O<foreign lang="grc">β</foreign>, ut ſci <lb></lb>licet quatuor ad vnum; ſimiliter oſtendetur FM ad GN ita eſſe <lb></lb>vt Y<foreign lang="grc">α</foreign> ad Q<foreign lang="grc">β</foreign>; FI uerò ad GH, vt YV ad QZ. vnde colligitur <expan abbr="nõ">non</expan> <lb></lb>ſolùm portiones diametrorum (ut dixim us) in eadem eſſe pro<lb></lb>portione, ſed <emph type="italics"></emph>& parallelas<emph.end type="italics"></emph.end> AC EK FI GH, & XP ST YV QZ <emph type="italics"></emph>in <lb></lb><expan abbr="eadē">eadem</expan> eſſe proportione. </s> <s id="N15753">& T rapeziorum ipſius quidem AE<emph.end type="italics"></emph.end>k<emph type="italics"></emph>C, & ipſius<emph.end type="italics"></emph.end> <arrow.to.target n="marg250"></arrow.to.target><lb></lb><emph type="italics"></emph>XSTP centra grauitatum eſſe in lineis LD 9R ſimiliter poſita, cùm <lb></lb>eandem habeant proportionem AC EK, quam XP ST.<emph.end type="italics"></emph.end> lineæquè <lb></lb>LD 9R bifariam diuidant ſuas æquidiſtantes AC EK. <lb></lb>& XP ST. etenim ſi ponatur trapezij AK centrum graui<lb></lb>tatis <foreign lang="grc">γ</foreign>, ipſius vcrò XT centrum grauitatis <foreign lang="grc">δ</foreign>, erit L<foreign lang="grc">γ</foreign> ad <foreign lang="grc">γ</foreign>D, <lb></lb>vt dupla ipſius AC cum EK ad duplam ipſius EK <arrow.to.target n="marg251"></arrow.to.target><lb></lb>cum AC. & 9<foreign lang="grc">δ</foreign> ad <foreign lang="grc">δ</foreign>R erit, vt dupla ipſius XP cum <lb></lb>ST ad duplam ST cum XP. quoniam autem ita eſt AC ad EK, <pb xlink:href="077/01/150.jpg" pagenum="146"></pb> <arrow.to.target n="fig69"></arrow.to.target> <pb xlink:href="077/01/151.jpg" pagenum="147"></pb>vt XP ad ST, & antecedentium dupla, hoc eſt dupla i<lb></lb>pſius AC ad EK erit, vt dupla ipſius XP ad ST. <lb></lb>& componendo dupla ipſius AC cum EK, vt dupla <arrow.to.target n="marg252"></arrow.to.target> i<lb></lb>pſius XP cum ST ad ST. At verò EK ad duplam <lb></lb>ipſius EK, ita eſt, vt ST ad duplam ipſius ST, ſed EK <arrow.to.target n="marg253"></arrow.to.target><lb></lb>ad AC eſt, vt ST ad XP, erit EK ad vtraſ〈que〉 conſe<lb></lb>〈que〉ntes ſim ul ſumptas, hoc eſt ad duplam ipſius EK cum <lb></lb>AC, vt ST ad ſuas conſe〈que〉ntes, nempe ad duplam ipſius <lb></lb>ST cum XP. Ita〈que〉 quoniam ita eſt dupla ipſius AC <lb></lb><expan abbr="cũ">cum</expan> EK ad Ek, vt dupla ipſius XP cum ST ad ST, & eſt EK <lb></lb>ad duplam ipſius EK cum AC, vt ST ad duplam ipſius <lb></lb>ST cum XP. erit ex ęquali dupla ipſius AC cum EK ad du <arrow.to.target n="marg254"></arrow.to.target><lb></lb>plam ipſius EK cum AC, vt dupla ipſius XP cum ST ad <lb></lb>duplam ipſius ST cum XP. ac propterea ita eſt L<foreign lang="grc">γ</foreign> ad <foreign lang="grc">γ</foreign>D, <lb></lb>vt 9<foreign lang="grc">δ</foreign> ad <foreign lang="grc">δ</foreign>R, & ob id centra <foreign lang="grc">γδ</foreign> erunt in lineis LD 9R ſi<lb></lb>militer poſita. <emph type="italics"></emph>Rurſus<emph.end type="italics"></emph.end> eodem modo (ne eadem ſæpiùs repetan<lb></lb>tur) <emph type="italics"></emph>Trapeziorum EFI<emph.end type="italics"></emph.end>k <emph type="italics"></emph>S<emph.end type="italics"></emph.end><foreign lang="grc">Γ</foreign><emph type="italics"></emph>VT centragrauitatum<emph.end type="italics"></emph.end>, quæ ſint <foreign lang="grc">εζ</foreign>, <emph type="italics"></emph>ſi <lb></lb>militer<emph.end type="italics"></emph.end> hoc eſt in eadem proportione <emph type="italics"></emph>diuident lineas LM<emph.end type="italics"></emph.end> 9<foreign lang="grc">α</foreign>, i<lb></lb>ta vt ſit L<foreign lang="grc">ε</foreign> ad <foreign lang="grc">ε</foreign>M, vt 9<foreign lang="grc">ζ</foreign> ad <foreign lang="grc">ζα</foreign>. <emph type="italics"></emph>& in trapezits FH<emph.end type="italics"></emph.end> <foreign lang="grc">Γ</foreign><emph type="italics"></emph>Z centra <lb></lb>grauitatum<emph.end type="italics"></emph.end> <foreign lang="grc">Ηκ</foreign> <emph type="italics"></emph>ſimiliter diuident MN<emph.end type="italics"></emph.end> <foreign lang="grc">αβ</foreign>, ita ut M<foreign lang="grc">Η</foreign> ad <foreign lang="grc">Η</foreign>N ſit, vt <lb></lb><foreign lang="grc">ακ</foreign> ad <foreign lang="grc">κβ</foreign> <emph type="italics"></emph>ſed & triangulorum GBH QOZ centra grauitatum<emph.end type="italics"></emph.end> <foreign lang="grc">λμ</foreign><lb></lb><emph type="italics"></emph>in lineis B N<emph.end type="italics"></emph.end> O<foreign lang="grc">β</foreign> <emph type="italics"></emph>erunt ſimiliter poſita<emph.end type="italics"></emph.end>, ſiquidem B<foreign lang="grc">λ</foreign> ad <foreign lang="grc">λ</foreign>N eſt, vt <arrow.to.target n="marg255"></arrow.to.target><lb></lb>O<foreign lang="grc">μ</foreign> ad <foreign lang="grc">μβ</foreign>; quippè cùm in dupla ſint proportione. <emph type="italics"></emph>eandem au<lb></lb>tem habent proportionem Trapezia, & triangula:<emph.end type="italics"></emph.end> Nam cùm <lb></lb>ſit AD ad EL, vt XR ad S9, & ut EL ad FM, ita S9 ad Y; <lb></lb>eſtquè DL ad LM, ut R9 ad 9<foreign lang="grc">α</foreign>, cùm ſint, vt ſeptem ad quin <lb></lb>〈que〉; erit ſpacium AL ad ſpacium EM, vt ſpacium X9 ad <arrow.to.target n="marg256"></arrow.to.target> ſpa<lb></lb>cium S. ſimiliterquè oſtendetur DK ad LI ita eſſe, vt RT <lb></lb>ad 9V. quare totum trapezium AK ad EI eſt, vt XT ad SV. <lb></lb>pariquè ratione oſtendeturita eſſe trapezium EI ad FH, vt <lb></lb>SV ad YZ. quia verò ita eſt FM ad GN, vt Y<foreign lang="grc">α</foreign> ad Q<foreign lang="grc">δ</foreign>, <lb></lb>eſt autem MN ad NB, vt <foreign lang="grc">αβ</foreign> ad <foreign lang="grc">β</foreign>O, ſunt quippè ut tria ad <lb></lb>vnum, erit ſpacium FN ad triangulum GBN, vt ſpacium <arrow.to.target n="marg257"></arrow.to.target><lb></lb>Y<foreign lang="grc">β</foreign> ad triangulum Q<foreign lang="grc">β</foreign>O. codemquè modo oſtendetur ita <lb></lb>eſſe ſpacium IN ad triangulum BNH, vt ſpacium V<foreign lang="grc">β</foreign> ad <lb></lb>triangulum O<foreign lang="grc">β</foreign>Z. Ex quibus ſequitur ita eſſe <expan abbr="trapeziũ">trapezium</expan> FH <lb></lb>ad triangulum BGH, vt trapezium YZ ad <expan abbr="triangulũ">triangulum</expan> OQZ. <pb xlink:href="077/01/152.jpg" pagenum="148"></pb> <arrow.to.target n="fig70"></arrow.to.target> <pb xlink:href="077/01/153.jpg" pagenum="149"></pb>ſi ita〈que〉 diuidatur <foreign lang="grc">γε</foreign> in <foreign lang="grc">ν</foreign>, ita ut ſit <foreign lang="grc">εν</foreign> ad <foreign lang="grc">νγ</foreign>, vt <expan abbr="trapeziũ">trapezium</expan> AK <lb></lb>ad EI. erit punctum <foreign lang="grc">ν</foreign> centrum grauitatis figurę AEFIKC. <arrow.to.target n="marg258"></arrow.to.target><lb></lb>ſimiliquè modo diuidatur <foreign lang="grc">δζ</foreign> in <foreign lang="grc"><10></foreign>, ita vt ſit <foreign lang="grc">ζ<10></foreign> ad <foreign lang="grc"><10>δ</foreign>, vt trape <lb></lb>zium XT ad SV; erit punctum <foreign lang="grc"><10></foreign> grauitatis centrum figuræ <lb></lb>XSYVTP. quia verò ita eſt AK ad EI, vt XT ad SV, erit <foreign lang="grc">εν</foreign><lb></lb>ad <foreign lang="grc">νγ</foreign>, vt <foreign lang="grc">ζ<10></foreign> ad <foreign lang="grc"><10>δ</foreign>. Diuidatur <expan abbr="aũt">aunt</expan> deinceps <foreign lang="grc">λΗ</foreign> in <foreign lang="grc">σ</foreign>, <expan abbr="ſitq́">ſit〈que〉</expan>; <foreign lang="grc">λσ</foreign> ad <foreign lang="grc">σΗ</foreign>, vt <lb></lb>FH ad triangulum BGH, erit punctum <foreign lang="grc">σ</foreign> centrum grauitatis <lb></lb>figuræ FGBHI. eademquè ratione diuidatur <foreign lang="grc">μκ</foreign> in <foreign lang="grc">τ</foreign>, ſitquè <lb></lb><foreign lang="grc">μτ</foreign> ad <foreign lang="grc">τκ</foreign>, vt YZ ad triangulum OQZ; erit punctum <foreign lang="grc">τ</foreign> cen<lb></lb>trum grauitatis figuræ YQOZV. ſed eſt FH ad BGD, vt YZ <lb></lb>ad OQZ, erit igitur <foreign lang="grc">λσ</foreign> ad <foreign lang="grc">ση</foreign>, vt <foreign lang="grc">μτ</foreign> ad <foreign lang="grc">τκ</foreign>. Quoniam autem <lb></lb>ita eſt Ak ad EI, vt XT ad SV, erit componendo AEFIKC <arrow.to.target n="marg259"></arrow.to.target><lb></lb>ad EI, vt figura XSYVTP ad SV; & eſt EI ad FH, vt SV ad <arrow.to.target n="marg260"></arrow.to.target><lb></lb>YZ. ergo ex æquali figura AEFIKC erit ad FH, vt figura <lb></lb>XSYVTP ad YZ. eſt autem FH ad BGH, vt YZ ad OQZ. e<lb></lb>ritigitur figura AEFIKC ad ſuas conſe〈que〉ntes, ad figuram <arrow.to.target n="marg261"></arrow.to.target><lb></lb>ſcilicet FGBHI, vt figura XSYVTP ad ſuas conſe〈que〉ntes, hoc <lb></lb>eſt ad figuram YQOZV. Diuidatur ita〈que〉 <foreign lang="grc">σν</foreign> in <foreign lang="grc">χ</foreign>, ita ut <foreign lang="grc">σχ</foreign><lb></lb>ad <foreign lang="grc">χ</foreign> ſit, vt figura AEFIKC ad figuram FGBHI, erit punctum <arrow.to.target n="marg262"></arrow.to.target><lb></lb><foreign lang="grc">χ</foreign> <expan abbr="centrũ">centrum</expan> grauitatis totius figurę AEFGBHIKC. ſimiliter di<lb></lb>uidatur <foreign lang="grc">τ<10></foreign> in <foreign lang="grc">ξ</foreign>, ſit〈que〉 <foreign lang="grc">τξ</foreign> ad <foreign lang="grc">ξ<10></foreign>, ut figura XSYVTP ad figu<lb></lb>ram YQOZV, erit punctum <foreign lang="grc">ξ</foreign> centrum grauitatis totius fi<lb></lb>guræ XSYQOZVTP. quia verò ita eſt figura AEFIKC ad fi <lb></lb>guram FGBHI, vt figura XSYVTP ad figuram YQOZV. e<lb></lb>rit <foreign lang="grc">σχ</foreign> ad <foreign lang="grc">χν</foreign>, vt <foreign lang="grc">τξ</foreign> ad <foreign lang="grc">ξ<10></foreign>. Ita〈que〉 quoniam BD ad DL eſt, vt <foreign lang="grc">σν</foreign><lb></lb>ad R9, cùm ſin^{4} utſexdecim ad ſeptem. </s> <s id="N159D8">& eſt L<foreign lang="grc">γ</foreign> ad <foreign lang="grc">γ</foreign>D, vt 9<foreign lang="grc">δ</foreign><lb></lb>ad <foreign lang="grc">δ</foreign>R, erit BD ad L<foreign lang="grc">γ</foreign>, vt <foreign lang="grc">σν</foreign> ad 9<foreign lang="grc">δ</foreign>. & vt BD ad <foreign lang="grc">γ</foreign>D, ita OR ad <arrow.to.target n="marg263"></arrow.to.target><lb></lb><foreign lang="grc">δ</foreign>R. rurſus quoniam BD ad LM eſt, vt OR ad 9<foreign lang="grc">α</foreign>, nempe vt ſex <lb></lb>decim ad quin〈que〉; & eſt L<foreign lang="grc">ε</foreign> ad <foreign lang="grc">ε</foreign>M, ut 9<foreign lang="grc">ζ</foreign> ad <foreign lang="grc">ζα</foreign>, erit BD ad <foreign lang="grc">ε</foreign>L, <lb></lb>vt OR ad 9<foreign lang="grc">ζ</foreign>. eſt verò BD ad L<foreign lang="grc">γ</foreign>, vt OR ad 9<foreign lang="grc">δ</foreign>; erit igitur BD ad <lb></lb>vtram 〈que〉 ſimul <foreign lang="grc">ε</foreign>L L<foreign lang="grc">γ</foreign>, hoc eſt ad <foreign lang="grc">εγ</foreign>, vt OR ad <foreign lang="grc">ζδ</foreign>. ſed <expan abbr="quoniã">quoniam</expan> <arrow.to.target n="marg264"></arrow.to.target><lb></lb>eſt <foreign lang="grc">γν</foreign> ad <foreign lang="grc">νε</foreign>, vt <foreign lang="grc">δ<10></foreign> ad <foreign lang="grc"><10>ζ</foreign>, erit BD ad <foreign lang="grc">γν</foreign>, vt OR ad <foreign lang="grc">δ<10></foreign>. eſt <expan abbr="autẽ">autem</expan> BD <lb></lb>ad D<foreign lang="grc">γ</foreign>, vt OR ad R<foreign lang="grc">δ</foreign>, vt dictum eſt, ergo BD ad D<foreign lang="grc">ν</foreign> eſt, vt OR <lb></lb>ad R<foreign lang="grc"><10></foreign>. ſimiliterquè <expan abbr="oſtẽdetur">oſtendetur</expan> BD ad BA ita eſſe, vt OR ad O<foreign lang="grc">τ</foreign>. <lb></lb>Cùm ita〈que〉 ſit BD ad DR, & ad B<foreign lang="grc">σ</foreign>, ut OR ad R<foreign lang="grc"><10></foreign>, & ad O<foreign lang="grc">τ</foreign>; e<lb></lb>rit BD ad DR B<foreign lang="grc">σ</foreign> ſimul, vt OR ad R<foreign lang="grc"><10></foreign> O<foreign lang="grc">τ</foreign> ſimul, & permutan<lb></lb>do tota BD ad totam OR, vt ablata D<foreign lang="grc">ν</foreign>B<foreign lang="grc">σ</foreign> ad ablatam R<foreign lang="grc"><10>οτ</foreign>. <pb xlink:href="077/01/154.jpg" pagenum="150"></pb> <arrow.to.target n="marg265"></arrow.to.target> ergo & reliqua <foreign lang="grc">σν</foreign> ad reliquam <foreign lang="grc">τ<10></foreign> eſt, ut tota BD ad <expan abbr="totã">totam</expan> OR. <lb></lb>rurſuſquè permutando <foreign lang="grc">σν</foreign> ad BD ut <foreign lang="grc">τ<10></foreign> ad OR, <expan abbr="conuertendoq́">conuertendo〈que〉</expan>; <lb></lb>BD ad <foreign lang="grc">σν</foreign> eſt, ut OR ad <foreign lang="grc">τ<10></foreign>, Quia verò ita eſt <foreign lang="grc">σχ</foreign> ad <foreign lang="grc">χν</foreign>, ut <foreign lang="grc">τξ</foreign> ad <foreign lang="grc">ξ<10></foreign>; <lb></lb> <arrow.to.target n="marg266"></arrow.to.target> erit BD ad <foreign lang="grc">σχ</foreign>, vt OR ad <foreign lang="grc">τξ</foreign> atverò BD ad b<foreign lang="grc">σ</foreign> eſt, vt OR ad O<foreign lang="grc">τ</foreign>. <lb></lb>erit igitur BD ad B<foreign lang="grc">χ</foreign>, ut O<foreign lang="grc">γ</foreign> ad O<foreign lang="grc">ξ</foreign>. ac propterea diuidendo D<foreign lang="grc">χ</foreign><lb></lb>ita ſe habet ad <foreign lang="grc">χ</foreign>B, vt R<foreign lang="grc">ξ</foreign> ad <foreign lang="grc">ξ</foreign>O. <emph type="italics"></emph>Quare manifestum est totius recti<lb></lb>lineæ figuræ in portione ABC inſcriptæ centrum grauitatis<emph.end type="italics"></emph.end> <foreign lang="grc">χ</foreign> <emph type="italics"></emph>in eadem <lb></lb>proportione diuidere BD, veluti centrum grauitatis<emph.end type="italics"></emph.end> <foreign lang="grc">ξ</foreign> <emph type="italics"></emph>figuræ rectilineæ <lb></lb>in portione XOP<emph.end type="italics"></emph.end> inſcriptæ <emph type="italics"></emph>ipſam OR<emph.end type="italics"></emph.end> diametrum. <emph type="italics"></emph>quod demonstra<lb></lb>re oportebat.<emph.end type="italics"></emph.end></s> </p> <p id="N15B46" type="margin"> <s id="N15B48"><margin.target id="marg247"></margin.target><emph type="italics"></emph>ex iis quę <lb></lb>poſt <gap></gap> pri<lb></lb>mi huius <lb></lb>demonſtra <lb></lb>ta ſunt.<emph.end type="italics"></emph.end></s> </p> <p id="N15B5A" type="margin"> <s id="N15B5C"><margin.target id="marg248"></margin.target>3. A<emph type="italics"></emph>rchi. <lb></lb>de quad. <lb></lb>parab. </s> <s id="N15B67">&<emph.end type="italics"></emph.end><lb></lb>20, <emph type="italics"></emph>primi <lb></lb>conicorum <lb></lb>Apoll.<emph.end type="italics"></emph.end></s> </p> <p id="N15B76" type="margin"> <s id="N15B78"><margin.target id="marg249"></margin.target>22. <emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N15B81" type="margin"> <s id="N15B83"><margin.target id="marg250"></margin.target>15. <emph type="italics"></emph>primi <lb></lb>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N15B8E" type="margin"> <s id="N15B90"><margin.target id="marg251"></margin.target>15. <emph type="italics"></emph>primi <lb></lb>buius.<emph.end type="italics"></emph.end></s> </p> <p id="N15B9B" type="margin"> <s id="N15B9D"><margin.target id="marg252"></margin.target>18. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N15BA6" type="margin"> <s id="N15BA8"><margin.target id="marg253"></margin.target>2. <emph type="italics"></emph><expan abbr="lẽma">lemma</expan> an<lb></lb>te<emph.end type="italics"></emph.end> 13. <emph type="italics"></emph>pri <lb></lb>mi huius.<emph.end type="italics"></emph.end></s> </p> <p id="N15BBE" type="margin"> <s id="N15BC0"><margin.target id="marg254"></margin.target>22. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N15BC9" type="margin"> <s id="N15BCB"><margin.target id="marg255"></margin.target><emph type="italics"></emph><expan abbr="ãte">ante</expan><emph.end type="italics"></emph.end> 13.<emph type="italics"></emph>pri <lb></lb>mi huius.<emph.end type="italics"></emph.end></s> </p> <p id="N15BDD" type="margin"> <s id="N15BDF"><margin.target id="marg256"></margin.target>1.<emph type="italics"></emph>lemma.<emph.end type="italics"></emph.end></s> </p> <p id="N15BE8" type="margin"> <s id="N15BEA"><margin.target id="marg257"></margin.target>2.<emph type="italics"></emph>lemma.<emph.end type="italics"></emph.end></s> </p> <p id="N15BF3" type="margin"> <s id="N15BF5"><margin.target id="marg258"></margin.target><emph type="italics"></emph>ex<emph.end type="italics"></emph.end> 6. <emph type="italics"></emph>pri<lb></lb>mi huius.<emph.end type="italics"></emph.end></s> </p> <p id="N15C05" type="margin"> <s id="N15C07"><margin.target id="marg259"></margin.target>18. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N15C10" type="margin"> <s id="N15C12"><margin.target id="marg260"></margin.target>22.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N15C1B" type="margin"> <s id="N15C1D"><margin.target id="marg261"></margin.target><emph type="italics"></emph>cor.<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>lem <lb></lb>ma m<emph.end type="italics"></emph.end> 13. <lb></lb><emph type="italics"></emph>primi hui<emph.end type="italics"></emph.end>^{9}</s> </p> <p id="N15C35" type="margin"> <s id="N15C37"><margin.target id="marg262"></margin.target><emph type="italics"></emph>ex<emph.end type="italics"></emph.end> 6. <emph type="italics"></emph>pri <lb></lb>mi huius.<emph.end type="italics"></emph.end></s> </p> <p id="N15C47" type="margin"> <s id="N15C49"><margin.target id="marg263"></margin.target>3. <emph type="italics"></emph>lemma.<emph.end type="italics"></emph.end></s> </p> <p id="N15C52" type="margin"> <s id="N15C54"><margin.target id="marg264"></margin.target>2. <emph type="italics"></emph><expan abbr="lẽma">lemma</expan> an<lb></lb>te<emph.end type="italics"></emph.end> 13. <emph type="italics"></emph>pri <lb></lb>mi huius.<emph.end type="italics"></emph.end><lb></lb>3. <emph type="italics"></emph>lcmma.<emph.end type="italics"></emph.end><lb></lb>2. <emph type="italics"></emph><expan abbr="lẽma">lemma</expan> an<lb></lb>te<emph.end type="italics"></emph.end> 13. <emph type="italics"></emph>pri<lb></lb>mi huius<emph.end type="italics"></emph.end><lb></lb>16.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N15C8C" type="margin"> <s id="N15C8E"><margin.target id="marg265"></margin.target>19.<emph type="italics"></emph>quinti. <lb></lb>co.<emph.end type="italics"></emph.end>4.<emph type="italics"></emph><expan abbr="quīti">quinti</expan>.<emph.end type="italics"></emph.end><lb></lb>3.<emph type="italics"></emph>lemma.<emph.end type="italics"></emph.end></s> </p> <p id="N15CA9" type="margin"> <s id="N15CAB"><margin.target id="marg266"></margin.target>2. <emph type="italics"></emph>lemma <lb></lb>ante<emph.end type="italics"></emph.end> 13. <lb></lb><emph type="italics"></emph>primi hui<emph.end type="italics"></emph.end>^{9} <lb></lb>18. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.154.1.jpg" xlink:href="077/01/154/1.jpg"></figure> <figure id="id.077.01.154.2.jpg" xlink:href="077/01/154/2.jpg"></figure> <p id="N15CCD" type="head"> <s id="N15CCF">SCHOLIVM.</s> </p> <p id="N15CD1" type="main"> <s id="N15CD3">Hinc colligere licet parabolas omnes inter ſe ſimiles eſſe. </s> <s id="N15CD5">Re <lb></lb>fert enim Eutocius hoc in loco, Apollonium pergęum in ſex <lb></lb>to Conicorum libro. (qui nondum in lucem prodijt) ſimiles <lb></lb>coni ſectiones dixiſſe eas eſſe, quando in vnaqua〈que〉 ſectione <lb></lb>lineę <expan abbr="ducũtur">ducuntur</expan> baſi <expan abbr="æquidiſtãtes">æquidiſtantes</expan> numero pares; hoc eſt tot in v<lb></lb>na, quot in alia; vt in ſuperioribus figuris ductæ fuerunt, in v<lb></lb>na quidem EK FI GH ipſi AC æquidiſtantes; & in altera ST <lb></lb>YV QZ ipſi PX æquidiſtantes; quę quidem efficiant, vt dia<lb></lb>metri in eadem proportione diuiſæ proueniant; vt ſunt BN <lb></lb>NM ML LD; & O<foreign lang="grc">β βα α</foreign>9 9R. Deinde <expan abbr="æquidiſtãtes">æquidiſtantes</expan> AC EK <lb></lb>FI GH in eadem ſint proportione ipſarum XP ST YV QZ. <lb></lb>& quoniam hæ conditiones in omnibus poſſunt accidere pa <lb></lb>rabolis; vt ex ijs, quæ demonſtrata ſunt, manifeſtum eſt; id<lb></lb>circo parabolæ omnes ſunt ſimiles. </s> <s id="N15D01">Ne〈que〉 verò <expan abbr="exiſtimandũ">exiſtimandum</expan> <lb></lb>eſt, quoniam parabolæ ſunt ſimiles, figur as quo〈que〉 planè <lb></lb>inſcriptas, vt AEFGBHIKC & XSYQOZVTP ſimiles eſſe in<lb></lb>ter ſe, ea præſertim ſimilitudine, qua ſunt figuræ rectilineæ; <lb></lb>vt ſcilicet anguli ſint æquales, & circum ęquales angulos late<lb></lb>ra proportionalia. </s> <s id="N15D11">in parabolis <expan abbr="nõ">non</expan> attenditur hęc ſimilitudo. <lb></lb>ſatenim eſt, vt præfatæ adſint conditiones; ex quibus ſequi<lb></lb>tur (vt oſtendimus) trapezia AK EI FH, triangulum què <lb></lb>BGH in eadem eſſe proportione trapeziorum XT SV YZ, ac <pb xlink:href="077/01/155.jpg" pagenum="151"></pb>trianguli OQZ. ac propterea quando Archimedes in propo<lb></lb>ſitione inquit <emph type="italics"></emph>ſi in vtra〈que〉 ſimilium portionum rectalmea, rectangu<lb></lb>liquè coni ſectione contentarum,<emph.end type="italics"></emph.end> non propterda exiſtimandum eſt <lb></lb>reperiri poſſe aliquas parabolas recta linea terminatas no eſſe <lb></lb>ſimiles inter ſe; ea nimirumiam explicata ſimilitudine. </s> <s id="N15D2F">ſunte<lb></lb>nim Archimedis verba hoc modo intelligenda, nempè, ſi in <lb></lb>vtra〈que〉 portionum recta linea rectanguliquè coni ſectione <lb></lb>contentarum, quæ omnes ſunt ſimiles, & c. </s> <s id="N15D37">veluti ſi dicere<lb></lb>mus. </s> <s id="N15D3B">In ſimilibus ſemicirculis anguli omnes ſuntrecti. </s> <s id="N15D3D">non <lb></lb>eſt intelligendum nonnullos ſemicirculos inter ſe diſſimiles <lb></lb>exiſtere poſſe. </s> <s id="N15D43">ſed hoc modo; in ſemicirculis, qui omnes ſunt <lb></lb>ſimiles, anguliſunt recti. </s> <s id="N15D47">Et hoc modo ſemperintelligere o<lb></lb>portet, quando in ſe〈que〉ntibus Archimedes parabolas ſimiles <lb></lb>nominat. </s> <s id="N15D4D">Nam & Archimedes cognouit omnes parabolas <lb></lb>inter ſe ſimiles eſſe; vt ipſe in demonſtratione octauæ propoſi <lb></lb>tionis huius ſupponere videtur. </s> <s id="N15D53">Oportebatenim aliquam in <lb></lb>parabolis demonſtrare ſimilitudinem, vt demonſtrari poſſet <lb></lb>centrum grauitatis in omnibus parabolis eſſe in certo, ac de<lb></lb>terminato ſitu ipſius figuræ. </s> <s id="N15D5B">in figuris enim, quæ aliquam in<lb></lb>terſe non habent ſimilitudinem, in ipſis centrum grauitatis <lb></lb>determinari minimè poſſe videtur. </s> <s id="N15D61">Dicet autem fortaſſe ali<lb></lb>quis, determinatur tamen centrum grauitatis in omnibus <expan abbr="triã">triam</expan> <lb></lb>gulis, quæ quidem interſe non ſuntſimilia. </s> <s id="N15D6B">Cui reſponden<lb></lb>dum; triangula omnia inter ſe ſimilia non eſſe ſimilitudine <lb></lb>rectilinearum figurarum, nempè vt anguli ſintæquales, & cir<lb></lb>cum æqualesangulos latera proportionalia. </s> <s id="N15D73">quòd tamen nul<lb></lb>lam inter ſeſe habeant conuenientiam, omnino negatur. <expan abbr="nã">nam</expan> <lb></lb>triangula omnia ſimul quodam modo illam habent conue<lb></lb>nientiam, & ſimilitudinem; quæ parabolis accidit. </s> </p> <p id="N15D7F" type="main"> <s id="N15D81">In triangulis enim ABC DEF ductę ſint AG DH ab angu<lb></lb>lis ad dimidias baſes. </s> <s id="N15D85">ſintquè diuiſa triangulorum latera in ea <lb></lb>dem proportione, in punctis kL, OP. & vt AK KL LB, ita ſit <lb></lb>AM MN NC, & DQ QR RF. ductiſquè KM LN OQ PR, <arrow.to.target n="marg267"></arrow.to.target><lb></lb>quæ lineas AG DH ſecent in punctis ST VX; primùm <expan abbr="quidẽ">quidem</expan> <lb></lb>erunt KM LN OQ PR baſibus BC EF æquidiſtantes; quas <lb></lb>lineæ AG DH in punctis ST VX bifariam diuident, cùm ſit <pb xlink:href="077/01/156.jpg" pagenum="152"></pb>BG ad GC, vt LT ad TN, & KS ad SM. & ut EH ad HF ita <lb></lb>PX ad XR, & OV ad <expan abbr="Vq.">V〈que〉</expan> Deinde erunt AG DH à lineis KM <lb></lb>LN OQ PR in eadem proportione diuiſæ; ſiquidem ita eſt <lb></lb>AS ST TG, ut DV VX XH. cùm ſint, ut expoſitæ propor<lb></lb>tiones AK KL LB, & DO OP PE. Præterea erit ſpacium, <lb></lb>BN ad LM, vt ER ad PQ, & LM ad triangulum AK M, <lb></lb> <arrow.to.target n="fig71"></arrow.to.target><lb></lb>vt PQ ad triangulum <expan abbr="DOq.">DO〈que〉</expan> Nam quoniam triangulu AEC <lb></lb>ſimile eſt triangulo ALN, oblatus LN ipſi BC æquidiſtans; <lb></lb>erit ABC ad ALN, ut AB ad AL duplicata. </s> <s id="N15DB9">eodemquè modo <lb></lb>erit DEF ad DPR, vt DE ad DP duplicata; eandem aut<gap></gap>m, <lb></lb>habet proportionem AB ad AL, quam DE ad DP: quadoqui <lb></lb>dem latera AB DE in eadem ſunt proportione diuiſa; erit igi<lb></lb>tur triangulum ABC ad ALN, vt triangulum DEF ad DPR. <lb></lb>ſimiliterquè oſtendetur ALN ad AkM ita eſſe, ut DPR ad <lb></lb><expan abbr="DOq.">DO〈que〉</expan> Quoniam autem ABC eſt ad ALN, ut DEF ad DPR, <lb></lb> <arrow.to.target n="marg268"></arrow.to.target> diuidendo erit BN ad ALN, ut ER ad DPR. Atverò <expan abbr="quoniã">quoniam</expan> <lb></lb>ALN ad AKM eſt, vt DPR ad <expan abbr="DOq;">DO〈que〉</expan> erit per conuerſio<lb></lb>nem rationis ALN ad LM, vt DPR ad <expan abbr="Pq.">P〈que〉</expan> qua<lb></lb> <arrow.to.target n="marg269"></arrow.to.target> re ex ęquali BN eſt ad LM, ut ER ad <expan abbr="Pq.">P〈que〉</expan> Cùm au<gap></gap>em ſit <lb></lb>ALN ad AKM, ut DPR ad <expan abbr="DOq;">DO〈que〉</expan> erit diuidendo LM ad <lb></lb>AKM, vt PQ ad <expan abbr="DOq.">DO〈que〉</expan> Quocirca erit ſpacium BN ad <lb></lb>LM, vt ER ad PQ, & LM ad triangulum AKM, <lb></lb>vt PQ ad triangulum <expan abbr="DOq.">DO〈que〉</expan> Ex quibus perſpicuum <lb></lb>eſt omnia triangula aliquam inter ſe habere ſimilitudinem, <lb></lb>ex qua poſſibile fuit determinare in omnibus ſitum, vb<gap></gap>epe- <pb xlink:href="077/01/157.jpg" pagenum="153"></pb>ritur centrum graurtatis. </s> <s id="N15E0C">Quòd ſi figurę nullam conuenien<lb></lb>tiam, nullamquè ſimilitudinem inter ſe habuerint; ut in qua <lb></lb>drilateris, pentagonis, & reliquis figuris, quæ inter ſe ne〈que〉 <lb></lb>latera ne〈que〉 angulos ęquales <expan abbr="habeãt">habeant</expan>; & propterea nullam in<lb></lb>terſe conuenientiam, & ſimilitudinem habere poſſunt; im<lb></lb>poſſibile quidem eſſet in ipſis determinare ſitum <expan abbr="cẽtri">centri</expan> grauita <lb></lb>tis; ita vt omnibus quadrilateris, ac omnibus pentagonis quo <lb></lb>modo cun〈que〉 factis, & ita cęteris figuris deſeruire poſſit. </s> <s id="N15E24">Cum <lb></lb>exempli gratia in pentagonis modò in vno, modò in alio ſi<lb></lb>tu centrum reperiatur; prout ſunt diuerſę figuræ. </s> <s id="N15E2A">Poſſumus <lb></lb>quidem in vnaqua〈que〉 figura reperire punctum poſitione, <lb></lb>quod ſit quidem centrum grauitatis illius determinatæ figu<lb></lb>ręt. </s> <s id="N15E32">vt in fine primilibri oſtendimus. </s> <s id="N15E34">eſſet tamen impoſſibile <lb></lb>in omnibus proprium certum, ac determinatum ſitum repe<lb></lb>rire; vt ſcilicet ſit in tali linea, taliquè modo diuiſa, vtomnib^{9} <lb></lb>pentagonis, & hexagonis, cæteriſquè huiuſmodi deſeruire <lb></lb>poſſit. </s> <s id="N15E3E">vt determinatur in triangulis, & vt determinari poteſt <lb></lb>in quadrilateris; quæ vel ſint parallelogramma, vel duo <expan abbr="ſaltẽ">ſaltem</expan> <lb></lb>latera ſint æquidiſtantia. </s> <s id="N15E48">cùm in his conuenientia, quàm <lb></lb>triangulis accidere oſtendimus, reperiatur; quandoquidem <lb></lb>ſunt <expan abbr="triãgulorum">triangulorum</expan> portiones. </s> <s id="N15E52">ſimiliter in parallelogrammis fa <lb></lb>cilè erit oſtendere aliquam inter ſe ſimilitudinem exiſtere. <expan abbr="pẽ-tagona">pen<lb></lb>tagona</expan> verò hexagona, & cæteræ figuræ, quæ angulos æqua<lb></lb>les, & æqualia latera habent; iam conſtat ſimilia eſſe inter ſe. <lb></lb>præterea circuliomnes ſunt ſimiles. </s> <s id="N15E60">Ellipſes quo〈que〉 inter ſe <lb></lb>aliquam habent ſimilitudinem, in quibus deſcribitur figura, <lb></lb>planè inſcripta. </s> <s id="N15E66">vt perſpicuum eſt in libro Federici Comman<lb></lb>dini de centro grauitatis ſolidorum. </s> <s id="N15E6A">ac propterea in his, & in <lb></lb>alijs, quibus inter ſe aliqua ſimililudo reperiri poteſt, centrum <lb></lb>quo〈que〉 grauitatis determinari poterit. </s> </p> <p id="N15E70" type="margin"> <s id="N15E72"><margin.target id="marg267"></margin.target><emph type="italics"></emph>ex<emph.end type="italics"></emph.end> 2.<emph type="italics"></emph>ſexti <lb></lb>ex lèmate <lb></lb><expan abbr="ĩ">im</expan> <expan abbr="ſecũdã">ſecundam</expan> d <lb></lb><expan abbr="mõſtratio-ne">monſtratio<lb></lb>ne</expan><gap></gap>. pri<lb></lb>mi huius.<emph.end type="italics"></emph.end></s> </p> <p id="N15E95" type="margin"> <s id="N15E97"><margin.target id="marg268"></margin.target>17. <emph type="italics"></emph>quinti. <lb></lb>coro.<emph.end type="italics"></emph.end> 19. <lb></lb><emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N15EA9" type="margin"> <s id="N15EAB"><margin.target id="marg269"></margin.target>22. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.157.1.jpg" xlink:href="077/01/157/1.jpg"></figure> <p id="N15EB8" type="head"> <s id="N15EBA">LEMMA.</s> </p> <p id="N15EBC" type="main"> <s id="N15EBE">Sint quatuor magnitudines ABCD. ſitquè A maior B; <lb></lb>&C maior D. Dico A ad D maiorem habere proportio<lb></lb>nem, quàm habet B ad C. </s> </p> <pb xlink:href="077/01/158.jpg" pagenum="154"></pb> <p id="N15EC7" type="main"> <s id="N15EC9">Hoc à nobis oſtenſum fuitinitio tractatus devecte in no<lb></lb>ſtris mechanicishoc pacto. </s> </p> <figure id="id.077.01.158.1.jpg" xlink:href="077/01/158/1.jpg"></figure> <p id="N15ED0" type="main"> <s id="N15ED2"> <arrow.to.target n="marg270"></arrow.to.target> Quoniam enim A ad C maiorem habet pro<gap></gap><lb></lb>portionem, quam B ad C; & A ad D maiorem <lb></lb>quo〈que〉 habet proportionem, quàm habetad C; <lb></lb>A igitur ad D maiorem habebit, quàm B ad C. <lb></lb>quod demonſtrare oportebat. </s> </p> <p id="N15EE1" type="margin"> <s id="N15EE3"><margin.target id="marg270"></margin.target>8. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N15EEC" type="head"> <s id="N15EEE">PROPOSITIO. IIII.</s> </p> <p id="N15EF0" type="main"> <s id="N15EF2">Omnis portionis recta linea, rectanguliquè co <lb></lb>ni ſectione contentæ, centrum grauitatis eſt in dia<lb></lb>metro portionis. </s> </p> <figure id="id.077.01.158.2.jpg" xlink:href="077/01/158/2.jpg"></figure> <p id="N15EFB" type="main"> <s id="N15EFD"><emph type="italics"></emph>Sit portio, vt dicta eſt, ABC; cuius diameter ſit BD. demon<lb></lb>ſtrandum est dictæ portionis centrum grauitatis eſſe in linea BD. ſi.n. <lb></lb>non, ſit punctum E. & ab ipſo ducatur ipſi BD aquidistans EF; at<lb></lb>〈que〉 in portione inſcribatur triangulum ABC eandem baſim<emph.end type="italics"></emph.end> AC <lb></lb><emph type="italics"></emph>habens, & altitudinem<emph.end type="italics"></emph.end> portioni <emph type="italics"></emph>æqualem. </s> <s id="N15F14">& quam proportionem <lb></lb>habet CF ad FD, eandem habeat triangulum ABC ad ſpacium<emph.end type="italics"></emph.end> <pb xlink:href="077/01/159.jpg" pagenum="155"></pb>k. <emph type="italics"></emph>in portione autem planè inſcribatur figura rectilinea<emph.end type="italics"></emph.end> AGBNC, <emph type="italics"></emph>ita <lb></lb>vt relictæ portiones<emph.end type="italics"></emph.end> AOG GPB BQN NRC ſimul <emph type="italics"></emph>ſint minores<emph.end type="italics"></emph.end> <arrow.to.target n="marg271"></arrow.to.target><lb></lb><emph type="italics"></emph>ipſo K. inſcriptæ quidem rectilineæ figuræ centrum grauitatis est in linea <lb></lb>B D. ſit punctum H. connectaturquè HE, & producatur; &<emph.end type="italics"></emph.end> à pun<lb></lb>cto C <emph type="italics"></emph>ipſi B D ducatur æquidistans CL.<emph.end type="italics"></emph.end> Quoniam autem por <lb></lb>tiones AOG GPB BQN NRC ſimul ſunt ipſo K mino<lb></lb>res; maiorem habebit proportionem triangulum ABC ad <arrow.to.target n="marg272"></arrow.to.target> di<lb></lb>ctas portiones, quàm ad K; inſcripta verò figura AGBNC ma <lb></lb>ior eſt triangulo ABC, K verò maius eſt reliquis portionibus. <lb></lb><emph type="italics"></emph>Maniſeſtum est<emph.end type="italics"></emph.end> igitur <emph type="italics"></emph>figuram rectilineam<emph.end type="italics"></emph.end> ACBNC <emph type="italics"></emph>in portione in-<emph.end type="italics"></emph.end> <arrow.to.target n="marg273"></arrow.to.target><lb></lb><emph type="italics"></emph>ſcriptam <expan abbr="maiorẽ">maiorem</expan> habere proportionem adreliquas portiones<emph.end type="italics"></emph.end> AOG GPB <lb></lb>BQN, NRC, <emph type="italics"></emph>quàm triangulum ABC ad K. ſed vt triangulum <lb></lb>ABC ad K, ita est CF ad FD; figura igitur inſcripta ad reliquas por<lb></lb>tiones maiorem habebit proportionem, quam CF ad FD; hoc eſt LE ad <lb></lb>EH.<emph.end type="italics"></emph.end> Cùm ſint LH CD à lineis æquidiſtantibus LC EF <arrow.to.target n="marg274"></arrow.to.target><lb></lb>HD druiſæ. </s> <s id="N15F87">quare cùm figura inſcripta ad reliquas portio<lb></lb>nes maiotem habeat proportionem, quàm LE ad EH; linea, <lb></lb>quæ ad EH eandem habeat <expan abbr="proportionẽ">proportionem</expan>, quàm figura inſcri<lb></lb>pta ad reliquas portiones, maior erit, <expan abbr="quã">quam</expan> LE. <emph type="italics"></emph>Habeat igitur ME<emph.end type="italics"></emph.end> <arrow.to.target n="marg275"></arrow.to.target><lb></lb><emph type="italics"></emph>ad EH <expan abbr="proportionẽ">proportionem</expan> eam, <expan abbr="quã">quam</expan> figura inſcripta ad portiones. </s> <s id="N15FAC">Quoniam igi<lb></lb>tur punctum E centrum eſt grauitatis totius portionis, figuræ <expan abbr="autẽ">autem</expan> in ipſa <lb></lb>inſcriptæ<emph.end type="italics"></emph.end> centrum grauitatis <emph type="italics"></emph>est punctum H: constat reliquæ magni<lb></lb>tudinis ex circumrelictis portionibus compoſitæ centrum grauitatis eſſe in <lb></lb>linea HE producta; ita vt aſſumpta aliqua recta linea<emph.end type="italics"></emph.end> ME <emph type="italics"></emph>eam proportio <lb></lb>nem habeat ad EH, quam figura inſcripta ad circumrelictas portiones. <lb></lb>Quare magnitudinis ex circumrelictis portionibus compoſitæ centrum gra<lb></lb>uitatis eſt punctum M. quod est abſurdum. </s> <s id="N15FCC">Ducta enim linea<emph.end type="italics"></emph.end> ST <emph type="italics"></emph>per <lb></lb>punctum M ipſi BD æquidiſtante, in ea omnes circumrelictæ portiones <lb></lb>centra grauitatis habebunt.<emph.end type="italics"></emph.end> hoc eſt magnitudinis ex portioni<lb></lb>bus BPG-BQN compoſitæ centrum grauitatis eſſet in parte <lb></lb>MS. centrum verò grauitatis portionum AOG CRN eſſet in <lb></lb>parte MX; ita ut M omnium dictarum portionum eſſet gra<lb></lb>uitatis centrum. </s> <s id="N15FE3">quæ ſuntquidem inconuenientia. </s> <s id="N15FE5">quippè <lb></lb>quæ etiam eodem modo ſe〈que〉ntur, ſi ST ipſi BD <expan abbr="æquidiſtãs">æquidiſtans</expan> <lb></lb>non eſſet. <emph type="italics"></emph>Patet igitur centrum grauitatis<emph.end type="italics"></emph.end> portionis ABC <emph type="italics"></emph>eſſe in <lb></lb>linea BD.<emph.end type="italics"></emph.end> quod demonſtrare oportebat. </s> </p> <pb xlink:href="077/01/160.jpg" pagenum="156"></pb> <p id="N16000" type="margin"> <s id="N16002"><margin.target id="marg271"></margin.target>2. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N1600B" type="margin"> <s id="N1600D"><margin.target id="marg272"></margin.target>8. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N16016" type="margin"> <s id="N16018"><margin.target id="marg273"></margin.target><emph type="italics"></emph>lemma.<emph.end type="italics"></emph.end></s> </p> <p id="N16020" type="margin"> <s id="N16022"><margin.target id="marg274"></margin.target><emph type="italics"></emph>1: <expan abbr="tem-ĩ">tem-im</expan><emph.end type="italics"></emph.end> 13. <lb></lb><emph type="italics"></emph>primi hui<emph.end type="italics"></emph.end></s> </p> <p id="N16034" type="margin"> <s id="N16036"><margin.target id="marg275"></margin.target>8. <emph type="italics"></emph>primi <lb></lb>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N16041" type="head"> <s id="N16043">SCHOLIVM.</s> </p> <p id="N16045" type="main"> <s id="N16047">In hac demonſtratione obſeruandum eſt; quòd <expan abbr="quãdo">quando</expan> Ar<lb></lb>chimedes inquit, <emph type="italics"></emph>in portione autem planè inſcribatur figura<emph.end type="italics"></emph.end> &c. </s> <s id="N16055">in<lb></lb>telligendum eſt, inſcribatur primò pentagonum AGBNC <lb></lb>in portione planè inſcriptum; quod quidem relin〈que〉t por<lb></lb>tiones AOG GPB BQN NRC, quæ ſimul uel erunt minores <lb></lb>ſpacio K, vel minùs. </s> <s id="N1605F">ſi non, rurſus planè adhuc inſcribatur <lb></lb>in portione ABC nonagonum; deinde alia figura; idquè ſem<lb></lb>per fiat, donec circumrelictæ portiones ſimul ſint ſpacio K <lb></lb>minores. </s> <s id="N16067">quod quidem fieri poſſe ex prima decimi Euclidis <lb></lb> <arrow.to.target n="marg276"></arrow.to.target> patet. </s> <s id="N1606F">Aufertur enim ſemper maius, <expan abbr="quã">quam</expan> dimidium. </s> <s id="N16075">Cùm quæ <lb></lb>libet portio paraboles trianguli plane in ipſa inſeripti ſit ſeſ<lb></lb>quitertia. </s> <s id="N1607B">Vnde triangulum ABC maius eſt, quàm <expan abbr="dimidiũ">dimidium</expan> <lb></lb>portionis ABC. triangulum què AGB maius, quàm <expan abbr="dimidiũ">dimidium</expan> <lb></lb>portionis AGB. ſimiliter triangulum BNC portionis BNC & <lb></lb>ita in alijs. </s> <s id="N1608B">Quæ quidem omnia ſuntquo〈que〉 manifeſta ex vi <lb></lb>geſima propoſitione, eiuſquè demonſtratione de quadratura <lb></lb>paraboles Archimedis. </s> </p> <p id="N16091" type="margin"> <s id="N16093"><margin.target id="marg276"></margin.target>17. <emph type="italics"></emph>Archi. <lb></lb>de quad. <lb></lb>parab.<emph.end type="italics"></emph.end></s> </p> <p id="N160A0" type="main"> <s id="N160A2">Demonſtrato centro grauitatis cuiuſlibet paraboles in eius <lb></lb>diametro exiſtere; oſtendet Archimedes, (vt diximus) in pa<lb></lb>rabolis grauitatum centra in eadem proportione diametros <lb></lb>diſpeſcere. </s> <s id="N160AA">antequam autem hoc demonſtret, duas pręmittit <lb></lb>ſe〈que〉ntes propoſitiones ad demonſtrationem neceſſarias. </s> </p> <p id="N160AE" type="head"> <s id="N160B0">PROPOSITIO. V.</s> </p> <p id="N160B2" type="main"> <s id="N160B4">Si in portione recta linea, rectanguliquè coni <lb></lb>ſectione contenta rectilinea figura planè inſcriba <lb></lb>tur, totius portionis <expan abbr="centrũ">centrum</expan> grauitatis <expan abbr="propĩquius">propinquius</expan> <lb></lb>eſt vertici portionis, <expan abbr="quã">quam</expan> <expan abbr="centrũ">centrum</expan> figuræ inſcriptæ. </s> </p> <pb xlink:href="077/01/161.jpg" pagenum="157"></pb> <p id="N160CF" type="main"> <s id="N160D1"><emph type="italics"></emph>Sit portio ABC, qualis dictaest, ipſius verò diameter ſit BD. <lb></lb>primùmquè in ipſa planè inſeribatur triangulum ABC. & diuidatur<emph.end type="italics"></emph.end> <arrow.to.target n="marg277"></arrow.to.target><lb></lb><emph type="italics"></emph>BD in E, ita vt dupla ſit BE ipſius ED. erit vtiquè trtanguli ABC <lb></lb>centrum grauitatis punctum E. Diuidatur ità〈que〉 biſariam vtra〈que〉 <lb></lb>AB BC in punctis FG. & <gap></gap>punctis FG ipſi BD ducantur æquidi<lb></lb>ſtantes FK GL. erit ſanè portionis A<emph.end type="italics"></emph.end>k<emph type="italics"></emph>B centrum grauitatis in linea<emph.end type="italics"></emph.end> <arrow.to.target n="marg278"></arrow.to.target><lb></lb><emph type="italics"></emph>F<emph.end type="italics"></emph.end>k. <emph type="italics"></emph>portionis verò BLC centrum grauit atis erit in linea GL. ſint ita<lb></lb>〈que〉 puncta HI. connectanturquè HI FG.<emph.end type="italics"></emph.end> quæ BD ſecent in QN. <lb></lb> <arrow.to.target n="fig72"></arrow.to.target><lb></lb>erit vti〈que〉 punctum Q vertici B propinquius, quàm N. quia <arrow.to.target n="marg279"></arrow.to.target><lb></lb>verò eſt BF ad FA, vt BG ad GC, erit FG <expan abbr="æquidiſtãsipſi">æquidiſtansipſi</expan> AC, <lb></lb>eritquè FN ad NG, vt AD ad DC. eſt verò AD ipſi DC æqua<lb></lb>lis, ergo FN NG inter ſe ſunt æquales. </s> <s id="N16118">quoniam autem FN <lb></lb>eſt ipſi AD æquidiſtans, erit AF ad FB, vt DN ad NB. eſt au <arrow.to.target n="marg280"></arrow.to.target><lb></lb>tem AF dimidia ipſius AB; cùm ſint AF FB ęquales ergo & <lb></lb>DN dimidia eſt ipſius DB. at verò quoniam DE terria eſt <lb></lb>pars ipſius DB, ſiquidem eſt BE ipſius ED dupla, erit pun<lb></lb>ctum N propinquius vertici B portionis, quàm pun<lb></lb>ctum E. <emph type="italics"></emph>Et quoniam parallelogrammum est HFGI. & æqualis est <lb></lb>FN ipſi NG, erit QH ipſi QI æqualis. </s> <s id="N1612E">ac propterea magnitudinis ex <lb></lb>vtriſ〈que〉 A<emph.end type="italics"></emph.end>k<emph type="italics"></emph>B BLC portionibus compoſitæ centrum grauitatis eſt in<emph.end type="italics"></emph.end> <arrow.to.target n="marg281"></arrow.to.target><lb></lb><emph type="italics"></emph>medio lineæ HI, cùm portiones<emph.end type="italics"></emph.end> AKB BLC <emph type="italics"></emph>ſint æquales. </s> <s id="N16148">erit ſcilicet <lb></lb>punctum <expan abbr="q.">〈que〉</expan> Quoniam autem trianguli ABC centrum grauitatis eſt <lb></lb>punctum E, magnitudinis verò ex vtriſquè A<emph.end type="italics"></emph.end>k<emph type="italics"></emph>B BLC compoſisæ<emph.end type="italics"></emph.end> <pb xlink:href="077/01/162.jpg" pagenum="158"></pb><emph type="italics"></emph>eſt punctum <expan abbr="q.">〈que〉</expan> conſtat totius portionis ABC centrum grauitatis eſſe<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg282"></arrow.to.target> <emph type="italics"></emph>in linea QE. hoc est inter puncta QE. Quare totius portionis <expan abbr="cētrum">centrum</expan> <lb></lb>grauitatis propinquius eſt vertici portionis, quam<emph.end type="italics"></emph.end> centrum grauitatis <lb></lb><emph type="italics"></emph>trianguli planè inſcripti.<emph.end type="italics"></emph.end></s> </p> <p id="N1617F" type="margin"> <s id="N16181"><margin.target id="marg277"></margin.target><emph type="italics"></emph>ante pri<lb></lb>mi huius.<emph.end type="italics"></emph.end></s> </p> <p id="N1618B" type="margin"> <s id="N1618D"><margin.target id="marg278"></margin.target>4. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N16196" type="margin"> <s id="N16198"><margin.target id="marg279"></margin.target>2. <emph type="italics"></emph>ſexti<lb></lb>lemma ta <lb></lb>aliter<emph.end type="italics"></emph.end> 13. <lb></lb><emph type="italics"></emph>primi hui^{9}<emph.end type="italics"></emph.end></s> </p> <p id="N161AC" type="margin"> <s id="N161AE"><margin.target id="marg280"></margin.target>2. <emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N161B7" type="margin"> <s id="N161B9"><margin.target id="marg281"></margin.target>4. <emph type="italics"></emph>primi <lb></lb>buius. <lb></lb>ex its quæ <lb></lb>ante<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>hu<lb></lb>ius demon <lb></lb>ſtrata ſunt. <lb></lb>ex<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>pri<lb></lb>mi huius.<emph.end type="italics"></emph.end></s> </p> <p id="N161DC" type="margin"> <s id="N161DE"><margin.target id="marg282"></margin.target>*</s> </p> <figure id="id.077.01.162.1.jpg" xlink:href="077/01/162/1.jpg"></figure> <figure id="id.077.01.162.2.jpg" xlink:href="077/01/162/2.jpg"></figure> <p id="N161E9" type="main"> <s id="N161EB"><emph type="italics"></emph>Rurſus in portione pent agonum rectilineum AKBLC planè inſcri<lb></lb>batur. </s> <s id="N161F1">ſitquè totius portionis diameter BD, vtriuſ〈que〉 autem portionis<emph.end type="italics"></emph.end><lb></lb>AKB. BLC <emph type="italics"></emph>diameter ſit vtra〈que〉 KF LG. & quoniam in portione <lb></lb>AKB planè inſcripta est figura rectilinea<emph.end type="italics"></emph.end> trilatera AKB, <emph type="italics"></emph>totius por <lb></lb>tionis<emph.end type="italics"></emph.end> AKB <emph type="italics"></emph>centrum grauitatis est propinquius vertici<emph.end type="italics"></emph.end> K, <emph type="italics"></emph>quam <lb></lb>centrum rectilineæ figuræ<emph.end type="italics"></emph.end> AKB. <emph type="italics"></emph>ſit ita〈que〉 portionis A<emph.end type="italics"></emph.end>k<emph type="italics"></emph>B centrum <lb></lb>grauitatis punctum H; trianguli verò punctum 1. Rurſus autem ſit por <lb></lb>tionis BLC centrum grauitatis punctum M. trianguli verò<emph.end type="italics"></emph.end> BLC <emph type="italics"></emph>pun<lb></lb>ctum N. iunganturquè HM JN<emph.end type="italics"></emph.end>; quæ BD ſecent in punctis <lb></lb>QT. erit vti〈que〉 punctum Q vertici B propinquius, <expan abbr="quã">quam</expan> <lb></lb>T. & quoniam (ſi ducta eſſet FG) lineæ HM IN FG ab æ<lb></lb> <arrow.to.target n="marg283"></arrow.to.target> quidiſtantibus lineis KF BD LG in eadem <expan abbr="diuidũtur">diuiduntur</expan> pro<lb></lb>portione. </s> <s id="N16241">FG verò, vt oſtenſum eſt, bifariam à linea BD di<lb></lb>uideretur; ergo & lineæ HM IN bifariam diuiſę <expan abbr="proucniẽt">proucnient</expan>. <lb></lb><emph type="italics"></emph>æqualis est igitur HQ ipſi QM; & IT ipſi TN. ſed triangulo <lb></lb>AKB æquale est triangulum BLC; portio vero A<emph.end type="italics"></emph.end>k<emph type="italics"></emph>B portioni <lb></lb>BLC eſt æqualis. </s> <s id="N16257">Demonstratum eſt enim alis in loçis portiones<emph.end type="italics"></emph.end> <pb xlink:href="077/01/163.jpg" pagenum="159"></pb><emph type="italics"></emph>ſeſquitertias eſſe triangulorum, erit igitur magnitudinis ex vtriſ〈que〉 por-<emph.end type="italics"></emph.end> <arrow.to.target n="marg284"></arrow.to.target><lb></lb><emph type="italics"></emph>tionibus A<emph.end type="italics"></emph.end>k<emph type="italics"></emph>B BLC compoſitæ centrum grauitatis punctum <expan abbr="q.">〈que〉</expan> magni<lb></lb>tudinis verò ex vtriſ〈que〉 triangulis AKB BLC compoſitæ punctum <lb></lb>T. Rurſus ita〈que〉 quoniam trianguli ABC centrum grauitatis eſt <expan abbr="punctū">punctum</expan> <lb></lb>E, magnitudinis verò ex vtriſ〈que〉 A<emph.end type="italics"></emph.end>k<emph type="italics"></emph>B BLC portionibus punctum <lb></lb><expan abbr="q.">〈que〉</expan> manifestum eſt totius portionis A<emph.end type="italics"></emph.end>B<emph type="italics"></emph>C centrum grauitatis eſſe in linea <lb></lb>QE ita diuiſa<emph.end type="italics"></emph.end> in O puncto, <emph type="italics"></emph>vt quam proportionem habet trian<lb></lb>gulum ABC ad vtraſ〈que〉 portiones A<emph.end type="italics"></emph.end>k<emph type="italics"></emph>B BLC, eandem habeat por<emph.end type="italics"></emph.end> <arrow.to.target n="marg285"></arrow.to.target><lb></lb><emph type="italics"></emph>tio ipſius terminum habens punctum Q,<emph.end type="italics"></emph.end> hoc eſt OQ <emph type="italics"></emph>ad portionem <lb></lb>minorem<emph.end type="italics"></emph.end> OE. <emph type="italics"></emph>pentagoni autem AKBLC,<emph.end type="italics"></emph.end> hoc eſt magnitudinis <lb></lb>ex triangulo ABC, trianguliſquè AKB BLC compoſitæ <lb></lb><emph type="italics"></emph>centrum grauitatis eſt in linea ET ſic diuiſa<emph.end type="italics"></emph.end> in S, <emph type="italics"></emph>vt quam habet <lb></lb>proportionem triangulum ABC ad triangula AKB BLC, eande ha<lb></lb>beat portio ipſius ad T terminata,<emph.end type="italics"></emph.end> hoc eſt ST <emph type="italics"></emph>ad reliquam<emph.end type="italics"></emph.end> SE. <lb></lb><emph type="italics"></emph>Quoniam igitur maiorem habet proportionem triangulum ABC ad <expan abbr="triã">triam</expan><emph.end type="italics"></emph.end> <arrow.to.target n="marg286"></arrow.to.target><lb></lb><emph type="italics"></emph>gula KAB LBC, quam ad portiones<emph.end type="italics"></emph.end> AKB BLC; minora enim <lb></lb>ſunt triangula portionibus. </s> <s id="N162EB">habebit TS ad SE <expan abbr="miorẽ">miorem</expan> pro<lb></lb>portio nem, quam QO ad OE ac propterea erit <expan abbr="punctũ">punctum</expan> S <lb></lb>propinquiusipſi E, quàm O. Nam ſi punctum S primùm <lb></lb>eſſet in eodem puncto O, tunc TO ad OE, non quidem <lb></lb>maiorem, ſed minorem haberet proportionem, quàm QO <arrow.to.target n="marg287"></arrow.to.target><lb></lb>ad OE, cùm ſit TO minor QO. ſimiliter ob eadem cau<lb></lb>ſam ſi punctum S eſſet inter OT, minorem haberet <arrow.to.target n="marg288"></arrow.to.target> pro<lb></lb>portionem TS ad SE, quàm QS ad SE, quare & ad huc <lb></lb>maiorem haberet proportionem QO ad OE, quàm TS <lb></lb>ad SE. neceſſe eſt igitur punctum S eſſe inter puncta OE. <lb></lb>Itaquè cùm punctum O ſit <expan abbr="centrũ">centrum</expan> grauitatis portionis ABC, <lb></lb>punctum verò S centrum ſit grauitatis rectilineæ figuræ <lb></lb>AK BLC; <emph type="italics"></emph>constat portionis ABC centrum grauitatis propinquius <lb></lb>eſſe vertici B, quàm centrum rectilineæ figuræ inſcriptæ. </s> <s id="N1631D">Et in om<lb></lb>nibus rectilineis figuris in portionibus planè inſcriptis eadem eſt ratio.<emph.end type="italics"></emph.end><lb></lb>quod demonſtrare oportebat. </s> </p> <pb xlink:href="077/01/164.jpg" pagenum="160"></pb> <p id="N16328" type="margin"> <s id="N1632A"><margin.target id="marg283"></margin.target><emph type="italics"></emph>prima lem <lb></lb>ma in<emph.end type="italics"></emph.end> 13. <lb></lb><emph type="italics"></emph>primi bui^{9}.<emph.end type="italics"></emph.end></s> </p> <p id="N1633B" type="margin"> <s id="N1633D"><margin.target id="marg284"></margin.target>4. <emph type="italics"></emph>primi <lb></lb>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N16348" type="margin"> <s id="N1634A"><margin.target id="marg285"></margin.target><emph type="italics"></emph>ex<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>pri<lb></lb>mi huius.<emph.end type="italics"></emph.end></s> </p> <p id="N1635A" type="margin"> <s id="N1635C"><margin.target id="marg286"></margin.target>8. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N16365" type="margin"> <s id="N16367"><margin.target id="marg287"></margin.target>8. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N16370" type="margin"> <s id="N16372"><margin.target id="marg288"></margin.target>8.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N1637B" type="head"> <s id="N1637D">SCHOLIVM.</s> </p> <p id="N1637F" type="main"> <s id="N16381"> <arrow.to.target n="marg289"></arrow.to.target> In fine primæ demonſtrationis in vltima concluſione <expan abbr="quã-do">quan<lb></lb>do</expan> inquit Archimedes. <emph type="italics"></emph>Quare totius portionis centrum propinquius <lb></lb>eſt vertici portionis, <expan abbr="quã">quam</expan> trianguli planè in ſcripti<emph.end type="italics"></emph.end> Gra cus codexita ſe <lb></lb>habet <foreign lang="grc">ὢς τ̓ ἒιηκα ἐγγυτε<10>ον τᾶς τοῦ τμάματος κο<10>υφᾶς τὸ κέντ<10>ον τοῦ ὂλου <lb></lb>τμάματος, ἤ τοῦ ἐγγ<10>αφομένου τ<10>ιγώνου γνω<10>ίμως</foreign>. verbaquè <foreign lang="grc">ἔιη κα</foreign> malè in<lb></lb>terpoſita ſunt, nullumquè cum alijs rectum ſenſum habent, <lb></lb>quare horum loco ponerem <foreign lang="grc">ἐσί</foreign>, vt ſenſus ſit, <foreign lang="grc">ὤς τἔγγύτε<10>ον ἐσι τᾶς <lb></lb>τοῦ τμάματος</foreign>, &c. </s> </p> <p id="N163B3" type="margin"> <s id="N163B5"><margin.target id="marg289"></margin.target>*</s> </p> <figure id="id.077.01.164.1.jpg" xlink:href="077/01/164/1.jpg"></figure> <p id="N163BC" type="main"> <s id="N163BE">Obſeruandum autem occurrit in demonſtrationibus, ab <lb></lb>Archimede allatis; quòd in prima demonſtratione ſupponit <lb></lb>Archimedes, HFGI eſſe parallelogrammum. </s> <s id="N163C4">quòd vt ſit pa<lb></lb>rallelogrammum, neceſſe eſt ſupponere centra grauitatis HI <lb></lb>ſecare lineas KF LG in partes inuicem proportionales. </s> <s id="N163CA">quod <lb></lb>tamen ſupponi poſſe minimè videtur. </s> <s id="N163CE">Et ſi quis ex quinto <lb></lb>poſtulato obijceret, centragrauitatis in æqualibus, ſimilibuſ<lb></lb>què figuris eſſe æqualiter poſita; admitti quidem poteſt; quo- <pb xlink:href="077/01/165.jpg" pagenum="161"></pb>niam figuræ, ipforum què centra inter ſe coaptari poſſunt. </s> <s id="N163D8">vt <lb></lb>omnibus figuris rectilineis ęqualibus, & ſimilib^{9} accidere po<lb></lb>teſt. </s> <s id="N163DE">Hoc tamé contingere poſſe in parabolis, vt AKB BLC, vi <lb></lb>detur in <expan abbr="cõueniés">conueniés</expan>. <expan abbr="Nã">Nam</expan> quamuis AKB BLC ſint æquales, & ſint <lb></lb><expan abbr="etiã">etiam</expan> ſimiles; non ſunt tamen ſimiles ea ſi militudine, vt ſuntre <lb></lb>ctilineæ figuræ; vtantea diximus. </s> <s id="N163F1">Quod etiam <expan abbr="perſpicuũ">perſpicuum</expan> fit ex <lb></lb>hoc, quia non ſemper coaptari poreiſt portio AKB <expan abbr="cũ">cum</expan> portio<lb></lb>ne BLC. <expan abbr="nõ">non</expan>. <expan abbr="n.">enim</expan>ſemper recta linea BC erit æqualisipſi BA; <expan abbr="neq́">ne〈que〉</expan>; <lb></lb>ſectionis linea BLC ſectionis lineę BKA ęqualis exiſtet. <expan abbr="Cũ">Cum</expan> <expan abbr="nõ">non</expan> <lb></lb>ſemper AC, & quæ ſuntipſi AC æquidiſtates ad rectos ſint an <lb></lb>gulos diametro BD. ſi.n. </s> <s id="N16419">ęquidiſtantes lineę diametro fuerint <lb></lb>perpendiculares, tunc AB BC inter ſe ęquales eſſent; <expan abbr="portioq́">portio〈que〉</expan>; <lb></lb>AKB <expan abbr="cũ">cum</expan> portione BLC coaptari poſſet: ſecùs autem minimè. <lb></lb>Quare centra grauiratis HI lineas KFLG in eadem proportio <lb></lb>ne ſecare minimèſupponi poſſe videtur; tùm exijs, quæ dicta <lb></lb>ſunt; tú quia hoc oſtendet Archimedes in ſeptima propoſitio <lb></lb>ne. </s> <s id="N1642F">quòd ſi adhuc non eſt <expan abbr="demõſtratú">demonſtratú</expan>, <expan abbr="nõ">non</expan> poteſt <expan abbr="quoq́">quo〈que〉</expan>; ſuppo <lb></lb>ni; præſertim cùm ſit demonſtrabile. </s> <s id="N1643F">ac propterea <expan abbr="demõſtra-tio">demonſtra<lb></lb>tio</expan> nullam videturvim haberead <expan abbr="oſtendendũ">oſtendendum</expan>, quod propoſi<lb></lb>tú fuit. </s> <s id="N1644D">Huic <expan abbr="tamẽ">tamen</expan> occurri poſſevidetur <expan abbr="cũ">cum</expan> Eutocio in exphca <lb></lb>tione huiusloci dicendo, hoc ſupponere Archimedé, quia por <lb></lb>tiones AKBBLC ſuntęquales, quarú diametri KFLG ſunt ę<lb></lb>quales, & <expan abbr="ęquidiſtãtes">ęquidiſtantes</expan>, quæ ſimiliter diuiduntur à punctis HI; <lb></lb>vnde erit kG ad HF, vt LI ad IG. ex quibus colligit HF ipſi IG <lb></lb><expan abbr="æqualẽ">æqualem</expan> eſſe; ac propterea HG <expan abbr="parallelogrãmũ">parallelogrammum</expan> exiltere. </s> <s id="N1646C">Quæ <expan abbr="tñ">tnm</expan> <lb></lb>reſponſio <expan abbr="nõ">non</expan> eſt Eutocio digna. </s> <s id="N16478">cùm ex dictis <expan abbr="nõ">non</expan> ſit omninò <lb></lb>demonſtratiua, vtres mathematicę <expan abbr="requirũt">requirunt</expan>; quapropter omit <lb></lb>tenda eſt.hac.n.rationeſupponitur centra HI lineas KFLG in <lb></lb>eadem proportione ſecare.quod nullo modo ſupponi poteſt. <lb></lb>Quare dici poterit, & fortaſle rectiùs, quòd vis demonſtratio<lb></lb>nis videtur in hoc eſſe conſtituta, vt ſupponatur puncta HI <expan abbr="v-bicunq́">v<lb></lb>bicun〈que〉</expan>; eſſe poſſe in lineis KFLG; ita vt ſiue ducta HI fuerit, <lb></lb>ſiue etiam non fuerit ipſi FG æquidiſtans, demonſtratio <expan abbr="tamẽ">tamen</expan> <lb></lb>ſuam ſemper habebit vim, <expan abbr="idẽq́">iden〈que〉</expan>; concludet. </s> <s id="N1649E">Nam ex <expan abbr="præcedẽ">præcedem</expan>. <lb></lb>ti patet centra grauitatis portionum AKB BLC eſſe in lineis <lb></lb>KF LG; hoceſt inter puncta KF, & LG. <expan abbr="ſupponãturitaq́">ſupponanturita〈que〉</expan>; <expan abbr="cẽ-tra">cen<lb></lb>tra</expan> grauitatis <expan abbr="portionũ">portionum</expan> AKB BLC eſſe puncta HI <expan abbr="vbicũq́">vbicun〈que〉</expan>; po <pb xlink:href="077/01/166.jpg" pagenum="162"></pb>ſita, <expan abbr="dũmodo">dummodo</expan> ſint in lineis KF LG, veluti Archimedes ipſe in <lb></lb>demonſtratione ſupponit. <expan abbr="Ducaturq́">Ducatur〈que〉</expan>; HI; quæ vel ipſi FG æ<lb></lb>quidiſtans erit, vel minùs: ſi eſt æquidiſtans, <expan abbr="parallelogrãmũ">parallelogrammum</expan> <lb></lb>eſt HFGI, & vera eſt demonſtratio Archimedis. </s> <s id="N164D0">ſi verò <expan abbr="nõ">non</expan> eſt <lb></lb><expan abbr="æquidiſtãs">æquidiſtans</expan>, nihilominus veriſſima eſt eadem <expan abbr="demõſtratio">demonſtratio</expan>. <expan abbr="Nã">Nam</expan> <lb></lb>ſi HI ipſi FG <expan abbr="nõ">non</expan> eſt <expan abbr="ęquidiſtãs">ęquidiſtans</expan>, patet in primis <expan abbr="pũctũ">punctum</expan> Q propin<lb></lb>quius eſſe vertici B portionis ABC, <expan abbr="quã">quam</expan> <expan abbr="punctũ">punctum</expan> N, ac per con<lb></lb>ſe〈que〉ns, <expan abbr="quã">quam</expan> punctum E centrum grauitatis trianguli ABC. <lb></lb>Etquoniam lineæ HI FG à lineis diuiduntur KF BN LG ę <lb></lb> <arrow.to.target n="fig73"></arrow.to.target><lb></lb> <arrow.to.target n="marg290"></arrow.to.target> quidiſtantibus, erit HQ ad QI, vt FN ad NG. eſt autem FN i<lb></lb>pGNG ęqualis, ergo HQ ipſi QI ęqualis quo〈que〉 erit. </s> <s id="N16510">ita〈que〉 <lb></lb>quoniam portiones AKBBLC ſunt æquales, erit magnitudi<lb></lb>nis ex vtriſ〈que〉 AKB BLC portionibus compoſitę <expan abbr="centrũ">centrum</expan> gra<lb></lb>uitatis in medio lineę HI. ergo eritpunctum <expan abbr="q.">〈que〉</expan> quo cognito <lb></lb>eadem demonſtratio Archimedis oſtendet centrum grauita<lb></lb>tis portionis ABC eſſe inter puncta <expan abbr="Eq.">E〈que〉</expan> Nam ex verbis ipſius, <lb></lb>cùm ait, <emph type="italics"></emph>Quoniam autem trianguli ABC centrum grauitatis est <lb></lb>punctum E magnitudinis verò ex vtriſ〈que〉 AkB BLC compoſicæ <lb></lb>est punctum <expan abbr="q;">〈que〉</expan> constat totius portionis ABC centrum grauitatis <lb></lb>eſſe in in linea QE. hoc est inter puncta QE. Quare totius portionis <lb></lb>centrum grauitatis propinquius eſt vertici portionis, quàm trian<lb></lb>guli planè inſcripti.<emph.end type="italics"></emph.end> <expan abbr="manifeſtũ">manifeſtum</expan> eſt igitur centrum grauitatis por <lb></lb>tionis ABC, ſiuè ſit HI ipſi FG æquidiſtans, ſiue non æ. <lb></lb>quidiſtans, propinquius eſſe vertici B portionis, quàm <expan abbr="cẽtrum">centrum</expan> <pb xlink:href="077/01/167.jpg" pagenum="163"></pb>grauitatis trianguli ABC<gap></gap> Quare cuca <gap></gap>erba demonſtratio<lb></lb>nis, cùm inquit Archimedes, <emph type="italics"></emph>& quoniam parallelogrammum est <lb></lb>HFGJ, & æqualisest FN ipſi NG.<emph.end type="italics"></emph.end> &c. </s> <s id="N1655C">immitando ſecun<lb></lb>dam Archimedis demonſtrationem huius propoſitionis, vel <lb></lb>delenda ſuntverba, <emph type="italics"></emph>parallelogrammum eſt HFGI, &<emph.end type="italics"></emph.end> tamquam <lb></lb>ab aliquo ad dita; ita vt verba ſint hoc modo vniuerſalia, <emph type="italics"></emph>& <lb></lb>quoniam æqualis eſt FN ipſi NG,<emph.end type="italics"></emph.end> & quæ ſequuntur. </s> <s id="N16572">vel ſat for<lb></lb>taſſe Archimedi viſum eſt. </s> <s id="N16576">ſe oſtendiſſe hoc contingere exi<lb></lb>ſtente HI ipſi FG æquidiſtante. </s> <s id="N1657A">quòd ſi etiam non fuerit HI <lb></lb>æquidiſtans FG, idem ſequi tanquam notum omiſit. </s> <s id="N1657E">cùm per <lb></lb>facilis ſit demonſtratio, vt dictum eſt. </s> <s id="N16582">Archimedeſquè res val <lb></lb>dè notas ſępè prætermittereſolet. </s> </p> <p id="N16586" type="margin"> <s id="N16588"><margin.target id="marg290"></margin.target>1.<emph type="italics"></emph><expan abbr="lẽwaĩ">lenwaim</expan><emph.end type="italics"></emph.end> 15 <lb></lb><emph type="italics"></emph>primu hu<lb></lb>ius.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.167.1.jpg" xlink:href="077/01/167/1.jpg"></figure> <p id="N165A0" type="main"> <s id="N165A2">Hocidem etiam conſiderari poteſt in ſecunda demonſtra <lb></lb>tione quamuis verba hanc difficultatem non habeant. <expan abbr="nã">nam</expan> ea<lb></lb>dem ſequltur demonſtratio, ſiuèſit HM lineæ IN ęquidiſtás, <lb></lb>vel non æquidiſtans, vt ex verbis Archimedis perſpicuum eſt. <arrow.to.target n="marg291"></arrow.to.target><lb></lb>etenim manifeſtum eſt centra grauitatis portionum AKB <lb></lb>BLC eſſeinlineis KF LG. ſimiliter centra grauitatis <arrow.to.target n="marg292"></arrow.to.target> trian<lb></lb>gulorum AKB BLC in ijsdem eſſe lineis KF LG. vt in <expan abbr="pũ-ctis">pun<lb></lb>ctis</expan> IN; quæ neceſſariò diuidunt KF LG in partes propor<lb></lb>tionales, vnde FI GN euadunt æquales. </s> <s id="N165C3">& quoniam por<lb></lb>tionum centra HM ſunt propinquiora verticibus KL, quam <lb></lb>triangulorum centra IN; ideo neceſſe eſt <expan abbr="pũcta">puncta</expan> HM in lineis <lb></lb>KI LN exiſtere. </s> <s id="N165CF">quare ſint puncta HM vbicú〈que〉 in lineis KI <lb></lb>LN conſtituta; <expan abbr="ductaq́">ducta〈que〉</expan>; HM, quæ ſiuè ſit ipſi IN ęquidiſtans, <lb></lb>ſiuenon æquidiſtans, ſem per erit <expan abbr="pũctum">punctum</expan> Qpropinquius ver <lb></lb>tici B, quam T. eodem què modo erit punctum Q <expan abbr="mediũ">medium</expan> li<lb></lb>neæ HM <expan abbr="centrũ">centrum</expan> grauitatis magnitudinis ex portionib^{9} AKB <lb></lb>BLC compoſitæ. </s> <s id="N165EB">ſiquidem portiones ſunt ęquales. </s> <s id="N165ED">quę <expan abbr="quidẽ">quidem</expan> <lb></lb>omnia ex ipſamet demonſtratione ſunt manifeſta. </s> <s id="N165F5">ſuntquè <lb></lb>hæc <expan abbr="eadẽ">eadem</expan> <expan abbr="obſeruãda">obſeruanda</expan> in duabus <expan abbr="ſe〈quẽ〉tibus">ſe〈que〉ntibus</expan> <expan abbr="demõſtrationib^{9}">demonſtrationib^{9}</expan>. </s> </p> <p id="N16609" type="margin"> <s id="N1660B"><margin.target id="marg291"></margin.target>4. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N16614" type="margin"> <s id="N16616"><margin.target id="marg292"></margin.target><emph type="italics"></emph>ante<emph.end type="italics"></emph.end> 15. <lb></lb><emph type="italics"></emph>primi hu<lb></lb>ius.<emph.end type="italics"></emph.end></s> </p> <p id="N16627" type="head"> <s id="N16629">PROPOSITIO. VI.</s> </p> <p id="N1662B" type="main"> <s id="N1662D">Data portione rectalinea, rectanguliquè coni <lb></lb>ſectione <expan abbr="cõtenta">contenta</expan>, in portione figurarectilinea pla <lb></lb>ne inſcribi poteſt; ita vt linea inter centrum graui <pb xlink:href="077/01/168.jpg" pagenum="164"></pb>tatis portionis, & figuræ rectilineæ inſcriptæ, mi<lb></lb>nor ſit propoſita recta linea. </s> </p> <figure id="id.077.01.168.1.jpg" xlink:href="077/01/168/1.jpg"></figure> <p id="N16640" type="main"> <s id="N16642"><emph type="italics"></emph>Data ſit portio ABC, qualis dicta est. </s> <s id="N16646">cuius centrum grauitatis ſit <lb></lb>punctum H. & in ipſa planè inſcribatur triangulum ABC. ſitquè pro <lb></lb>poſita recta linea F. & quam proportionem habet BH ad F, eandem <lb></lb>habeat triangulum ABC ad ſpacium<emph.end type="italics"></emph.end> k. <emph type="italics"></emph>inportione autem ABC pla<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg293"></arrow.to.target> <emph type="italics"></emph>nè inſcribatur figura rectilinea AGB LC, ita vt circumrelictæ portio <lb></lb>nes<emph.end type="italics"></emph.end> ANG GOB BPL LQC ſimul ſumptę <emph type="italics"></emph>ſint minoresipſo<emph.end type="italics"></emph.end> k<emph type="italics"></emph>: <lb></lb>ipſiuſquè figuræ inſcriptæ centrum grauitatis ſit punctum E. Dico li<lb></lb>neam HE minorem eſſe ipſa F. N amſi non, vel æqualis est, vel <lb></lb>maior. </s> <s id="N16673">Quoniam autem<emph.end type="italics"></emph.end> maior eſt figura rectilinea AGBLC, <lb></lb>quàm triangulum ABC, maius verò eſt ſpacium K portio<lb></lb>nibus ANG GOB BPL LQC ſimul ſumptis, ideo <emph type="italics"></emph>rectili-<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg294"></arrow.to.target> <emph type="italics"></emph>nea figura AGBLC ad circumrelictas portiones maiorem habet pro<lb></lb>portionem, quàm triangulum ABC ad K. hoc est HB ad F. at ue <lb></lb>rò BH nonhabet minorem proportionem ad F, quàm habet ad HE. <lb></lb>cùmnon ſit minor HE ipſa F.<emph.end type="italics"></emph.end> ſi enim ponatur HE ipſi F <pb xlink:href="077/01/169.jpg" pagenum="165"></pb>æqualis, eandem habebit proportionem BH ad HE, <expan abbr="quã">quam</expan> <arrow.to.target n="marg295"></arrow.to.target><lb></lb>ad F. quæ eſt proportio trianguli ABC ad. </s> <s id="N166A0">K. vnde figu<lb></lb>ra rectilinea AGBLC ad circumrelictas portiones maiorem, <lb></lb>habebit proportionem, quàm BH ad HE. ſi verò ponatur <lb></lb>HE maior, quàm F, habebit BH ad F, hoc eſt <expan abbr="triangulũ">triangulum</expan> <arrow.to.target n="marg296"></arrow.to.target><lb></lb>ABC ad K maiorem proportionem, quàm BH ad HE. <lb></lb><emph type="italics"></emph>multo igitur maiorem habet proportionem figura rectilinea AGBLC ad <lb></lb>circumrelictas portiones, quàm BH ad HE. Quare ſi fiat ut rectili<lb></lb>linea figura AGBLC ad circumrelictas portiones, ſic alia quædam li<lb></lb>nea ad HE. erit maior, quàm BH. ſitquè HM. Cùm enim portio<lb></lb>nis ABC centrum grauitatis ſit H. figuræ verò rectilineæ AGBLC <lb></lb>punctum E. producta EH, aſſumptaquè aliqua recta linea proportione <lb></lb>babente ad EH, quam rectilineum AGBLC ad circumtelictas por<lb></lb>tiones; maior erit quàm HB. habeat igitur<emph.end type="italics"></emph.end> (vt dictum eſt) <emph type="italics"></emph>MH ad <lb></lb>HE<emph.end type="italics"></emph.end> proportionem eam, quam habet figura AGBLC ad reli <arrow.to.target n="marg297"></arrow.to.target><lb></lb>quas portiones, <emph type="italics"></emph>ergopunctum M centrum est grauit atis magnitudi<lb></lb>nis ex circumrelictis portionibus compoſitæ. </s> <s id="N166D8">quod eſſe non poteſt. </s> <s id="N166DA">Ducta <lb></lb>enimrecta linea<emph.end type="italics"></emph.end> RS <emph type="italics"></emph>per M ipſi AC æquidistante, inipſa ſunt centra <lb></lb>grauitatis vnicuiquè portioni reſpondentia<emph.end type="italics"></emph.end>; ita ſcilicet vt centrum <lb></lb>magnitudinis ex portionibus ANG GOB compoſitæ ſit in <lb></lb>linea RS. ſed in parte MR. in parteverò MS ſit grauitatis <lb></lb>centrum magnitudinis ex reliquis portionibus BPL LQC <lb></lb>compoſitæ; ita vt punctum M magnitudinis ex omnibus <lb></lb>portionibus compoſitæ centrum grauitatisexiſtat. </s> <s id="N166F3">quæ <expan abbr="tamẽ">tamen</expan> <lb></lb>eſſe non poſſunt. </s> <s id="N166FB">quod idem accideret, ſi etiam RS ipſi AC <lb></lb>æquidiſtans non eſſet. <emph type="italics"></emph>Patetigitur HE minorem eſſe, quam F.<emph.end type="italics"></emph.end><lb></lb>cùm ne〈que〉 maior, ne〈que〉 ęqualis eſſe poſſit. <emph type="italics"></emph>quod quidem de<lb></lb>monſtrare oportebat.<emph.end type="italics"></emph.end></s> </p> <p id="N1670D" type="margin"> <s id="N1670F"><margin.target id="marg293"></margin.target>A</s> </p> <p id="N16713" type="margin"> <s id="N16715"><margin.target id="marg294"></margin.target><emph type="italics"></emph><expan abbr="lẽma">lemma</expan> in<emph.end type="italics"></emph.end> 4. <lb></lb><emph type="italics"></emph><expan abbr="ſecũdi">ſecundi</expan> hui<emph.end type="italics"></emph.end>^{9}</s> </p> <p id="N1672B" type="margin"> <s id="N1672D"><margin.target id="marg295"></margin.target>7. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N16736" type="margin"> <s id="N16738"><margin.target id="marg296"></margin.target>8.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N16741" type="margin"> <s id="N16743"><margin.target id="marg297"></margin.target>8.<emph type="italics"></emph>primi hu<lb></lb>ius.<emph.end type="italics"></emph.end></s> </p> <p id="N1674E" type="head"> <s id="N16750">SCHOLIVM.</s> </p> <p id="N16752" type="main"> <s id="N16754">In hac quo〈que〉 demonſtratione obſeruandum eſt, quod <arrow.to.target n="marg298"></arrow.to.target><lb></lb>poſt quartam huius adnotauimus; nimirum ſi pentagonum <lb></lb>AGBLC in portione planèinſcriptum relin〈que〉ret portiones <lb></lb>ANG GOB BPL LQC, quæ ſimul maiores, vel etiam æ- <pb xlink:href="077/01/170.jpg" pagenum="166"></pb>quales eſſent ſpacio K. Rurſus planè adhuc inſ cribatur in <lb></lb>portione ABC nonagonum, deinde altera figura, idquè ſem<lb></lb>per fiat, donec circumrelictę portiones ſimul ſint ſpacio K <lb></lb>minores. </s> <s id="N16769">quod quidem fieri poſſe ibidem oſtendimus: </s> </p> <p id="N1676B" type="margin"> <s id="N1676D"><margin.target id="marg298"></margin.target>A</s> </p> <p id="N16771" type="head"> <s id="N16773">PROPOSITIO. VII.</s> </p> <p id="N16775" type="main"> <s id="N16777">Duabus portionibus ſimilibus recta linea, re<lb></lb>ctanguliquè coni ſectione contentis, centra gra<lb></lb>uitatum diametros in eadem proportione diſpe<lb></lb>ſcunt. </s> </p> <figure id="id.077.01.170.1.jpg" xlink:href="077/01/170/1.jpg"></figure> <p id="N16782" type="main"> <s id="N16784"><emph type="italics"></emph>Sint duæ portiones, quales dictæ ſunt ABC EFG. quarum diame<lb></lb>tri BD FH. ſitquè portionis ABC centrum grauitatis punctum K. <lb></lb>ipſius verò EFG punctum L. Demonstrandum est, puncta<emph.end type="italics"></emph.end> k<emph type="italics"></emph>L in <lb></lb>eadem proportione diametros diuidere,<emph.end type="italics"></emph.end> ita vt BK ad KD ſit, vt FL <pb xlink:href="077/01/171.jpg" pagenum="167"></pb>ad LH. <emph type="italics"></emph>ſi autemnon.<emph.end type="italics"></emph.end> ſi fieri poteſt, <emph type="italics"></emph>ſit BK ad<emph.end type="italics"></emph.end> k<emph type="italics"></emph>D, vt FM ad <lb></lb>MH. & in portione EFG rectilineum planè inſcribatur, ita vt linea <lb></lb>inter centrum<emph.end type="italics"></emph.end> grauitatis <emph type="italics"></emph>portionis, &<emph.end type="italics"></emph.end> centrum grauitatis <emph type="italics"></emph>figuræ<emph.end type="italics"></emph.end> <arrow.to.target n="marg299"></arrow.to.target><lb></lb><emph type="italics"></emph>inſcriptæ minor ſit, quàm LM. ſitquè figuræ inſcriptæ centrum graui<lb></lb>tatis punctum X.<emph.end type="italics"></emph.end> eritvtiquè punctum L propinquius vertici <arrow.to.target n="marg300"></arrow.to.target><lb></lb>F, quàm punctum X. & quoniam LX minor eſt, quàm <lb></lb>LM, erit quo〈que〉 punctum X vertici F propinquius, quàm <lb></lb>M. <emph type="italics"></emph>Jn portione autem ABC inſcribatur figura rectilinea ſimilis figu<lb></lb>ræ in portione EFG inſcriptæ. </s> <s id="N167D7">hoc est ſimiliter planè,<emph.end type="italics"></emph.end> (ita nempè vt <lb></lb>figurę latera multitudine ęqualia habeant) <emph type="italics"></emph>cuius centrum graui<lb></lb>tatis<emph.end type="italics"></emph.end> ſit punctum N. & quoniam figuræ in porrionibus pla<lb></lb>nèinſcriptę habentlatera multitudine æqualia, ipſarum cen<lb></lb>tra grauitatis diametros BD FH in eadem proportione diſpe<lb></lb>ſcent. </s> <s id="N167EC">quare erit BN ad ND, vt FX ad XH. poſitum <expan abbr="autẽ">autem</expan> <arrow.to.target n="marg301"></arrow.to.target><lb></lb>fuitita eſſe FM ad MH, vt BK ad KD. ſi ita〈que〉 <expan abbr="punctũ">punctum</expan> <lb></lb>X propinquius eſt ipſi F, quàm M; erit & punctum N i<lb></lb>pſi B propinquius, quàm K. eſtverò punctum K <expan abbr="centrũ">centrum</expan> <lb></lb>grauitatis portionis ABC, punctum verò N centrum figuræ <lb></lb>inſcripte; ergo centrum grauitatis figurę inſcriptæ <emph type="italics"></emph>propinquius <lb></lb>erit vertici portionis,<emph.end type="italics"></emph.end> quam centrum ipſius portionis. <emph type="italics"></emph>quod fieri <expan abbr="nõ">non</expan> <lb></lb>potest. </s> <s id="N16818">Manifeſtum est igitur eandem habere proportionem BK ad KD. <lb></lb>quam FL ad LH.<emph.end type="italics"></emph.end> quod demonſtrare oportebat. </s> </p> <p id="N1681F" type="margin"> <s id="N16821"><margin.target id="marg299"></margin.target>6. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N1682A" type="margin"> <s id="N1682C"><margin.target id="marg300"></margin.target>5. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N16835" type="margin"> <s id="N16837"><margin.target id="marg301"></margin.target>3. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N16840" type="head"> <s id="N16842">SCHOLIVM.</s> </p> <p id="N16844" type="main"> <s id="N16846">Pręſens demonſtratio ea tantùm ratione eſſicax eſſe vide<lb></lb>tur, quatenus ſupponitur punctum L vertici F propinqui^{9} <lb></lb>eſſe, quàm M. ex hoc enim ſequitur punctum X eſſe ipſi F <lb></lb>propinquius, quàm M. vnde euenitabſurdum, nempè, <expan abbr="pũ">pum</expan> <lb></lb>ctum N eſſevertici B propinquius, quàm K. Quòd ſi ſup <lb></lb>poſitum fuerit Bk ad KD ita eſſe, vt FP ad PH; fuerit <lb></lb>autem P inter LF; tunc centrum grauitatis figurę in EFG <pb xlink:href="077/01/172.jpg" pagenum="168"></pb>planè inſcriptæ eſſetinter puncta PH; vnde centrum ctiam <lb></lb>figurę in ABC ſimiliter planè inſcriptę inter KD eueniret; <lb></lb>eſſetquè centrum grauitatis portionis ABC vertici B propin<lb></lb>quius, quam centrum figuræ planè inſcriptæ. </s> <s id="N16862">ideoquè <expan abbr="nullũ">nullum</expan> <lb></lb>accideret abſurdum. </s> <s id="N1686A">Quare ſi ſuppoſitum fuerit FP ad PH <lb></lb>eſſe, vt BK ad KD, tunc (vt eadem demonſtratio rei propo <lb></lb>ſitæ inſeruire poſſet) diuidenda eſſet diameter BD in <expan abbr="q;">〈que〉</expan> i<lb></lb>ta vt BQ ad QD ſit, vt FL ad LH. & quoniam maio<lb></lb> <arrow.to.target n="marg302"></arrow.to.target> rem habet proportionem FL ad LH, quàm FP ad PH; ſiqui<lb></lb>dem maior eſt FL, quàm FP, & PH maior, quàm LH. Vtverò <lb></lb>FL ad LH, ita eſt BQ ad QD; & vt FP ad PH. ita BK ad KD; <lb></lb>maiorem quo〈que〉 habebit proportionem BQ ad QD, quàm <lb></lb> <arrow.to.target n="marg303"></arrow.to.target> BK ad KD. & componendo BD ad DQ maiorem, quàm ea<lb></lb> <arrow.to.target n="marg304"></arrow.to.target>dem BD ad Dk. </s> <s id="N1688E">Quare maior eſt DK, quàm <expan abbr="Dq.">D〈que〉</expan> & ob id <lb></lb>punctum K propinquius erit vertici B, quàm <expan abbr="q.">〈que〉</expan> Deinde <lb></lb>planè inſcribenda eſſet figura in portione ABC, ita vt linea <lb></lb>inter centrum figuræ inſcriptæ, & centrum portionis minor <lb></lb>eſſet, quàm <expan abbr="Kq;">K〈que〉</expan> & reliqua quæ ſequuntur, ita tamen, vt quę <lb></lb>facta ſunt in EFG, fiant in ABC; & quæ in ABC, <expan abbr="fiãt">fiant</expan> in EFG. <lb></lb>oſtendeturquè centrum figurę inſcriptę in portione EFG pro <lb></lb>pinquius eſſe vertici F, quàm centrum grauitatis ipſius portio <lb></lb>nis EFG. quod quidem fieri non poteſt. </s> <s id="N168B0">Ex quibus perlpi<lb></lb>cuum fit demonſtrationem eſſe vniuerſalem. </s> <s id="N168B4">& hanc <expan abbr="demõ">demom</expan> <lb></lb>ſtrationis partem Archimedem omiſiſſe, vt notam. </s> <s id="N168BC">Etvt an<lb></lb>tea admonuimus, quòd centra grauitatis diametros in eadem <lb></lb>proportione diuidunt, omnibus parabolis competere intelli<lb></lb>gendum eſt. </s> <s id="N168C4">ſiquidem omnes ſuntſimiles. </s> <s id="N168C6">quo demonſtrato, <lb></lb>in ſe〈que〉nti, quo in loco, & in qua diametri parte reperitur <expan abbr="cẽ">cem</expan> <lb></lb>trum grauitatis paraboles demonſtrat, quòd vt res perſpicua <lb></lb>reddatur; hæc priùs demonſtrabimus. </s> </p> <p id="N168D2" type="margin"> <s id="N168D4"><margin.target id="marg302"></margin.target><emph type="italics"></emph><expan abbr="lẽma">lemma</expan> in<emph.end type="italics"></emph.end> 4. <lb></lb><emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N168E6" type="margin"> <s id="N168E8"><margin.target id="marg303"></margin.target>28.<emph type="italics"></emph>quinti. <lb></lb>addi.<emph.end type="italics"></emph.end></s> </p> <p id="N168F3" type="margin"> <s id="N168F5"><margin.target id="marg304"></margin.target>10.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N168FE" type="head"> <s id="N16900">LEMMA. I.</s> </p> <p id="N16902" type="main"> <s id="N16904">Si magnitudo magnitudinis fuerit quadrupla, minorverò <lb></lb>magnitudo alterius magnitudinis ſit tripla, maior magnitu<lb></lb>do vtrarum què ſimul magnitudinum tripla erit. </s> </p> <pb xlink:href="077/01/173.jpg" pagenum="169"></pb> <p id="N1690D" type="main"> <s id="N1690F">Quadrupla ſit magnitudo A magnitudinis BC. <lb></lb> <arrow.to.target n="fig74"></arrow.to.target><lb></lb>ſit verò BC alterius magnitudinis CD tripla. </s> <s id="N16918">Di <lb></lb>co magnitudinem A vtrarumquè ſimul BC CD, <lb></lb>hoc eſt BD triplam eſse. </s> <s id="N1691E">Quoniam enim BC tri<lb></lb>pla eſt ipſius CD, erit componendo BC cum CD, <lb></lb>hoc eſt BD ipſius CD quadrupla. </s> <s id="N16924">ſed magnitudo <lb></lb>quo〈que〉 A quadrupla eſt ipſius BC, eandem igitur <lb></lb>habetproportionem A ad BC, vt BD ad CD. & <lb></lb>permutando A ad BD, vt BC ad CD. & eſt <arrow.to.target n="marg305"></arrow.to.target> qui<lb></lb>dem BC tripla ipſius CD, ergo A ipſius BD tri<lb></lb>pla exiſtit. </s> <s id="N16934">quod demonſtrare oportebat. </s> </p> <p id="N16936" type="margin"> <s id="N16938"><margin.target id="marg305"></margin.target>16.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.173.1.jpg" xlink:href="077/01/173/1.jpg"></figure> <p id="N16945" type="head"> <s id="N16947">LEMMA. II.</s> </p> <p id="N16949" type="main"> <s id="N1694B">Si magnitudo magnitudinis fuerit ſeſquitertia, erit magni<lb></lb>tudo minor ipſius exceſſus tripla. </s> </p> <p id="N1694F" type="main"> <s id="N16951">Sit magnitudo AB magnitudinis C ſeſquiter <lb></lb> <arrow.to.target n="fig75"></arrow.to.target><lb></lb>tia; exceſſus verò, quo AB ſuperat C, ſit BD. Dico <lb></lb><expan abbr="magnitudinẽ">magnitudinem</expan> C ipſius BD triplam eſſe. </s> <s id="N1695F">quod qui <lb></lb>dem ex ſe patet. </s> <s id="N16963">Nam quoniam BD eſt exceſ<lb></lb>ſus, quo AB ſuperat C. magnitudo autem AB i<lb></lb>pſam C ſuperat tertia ipſius C parte, cum ſit AB <lb></lb>ipſius C ſeſquitertia. </s> <s id="N1696B">erit igitur BD tertia pars i<lb></lb>ſius C. quare magnitudo C ipſius BD tripla <lb></lb>exiſtit. </s> <s id="N16971">quod oſtendere oportebat. </s> </p> <figure id="id.077.01.173.2.jpg" xlink:href="077/01/173/2.jpg"></figure> <p id="N16977" type="head"> <s id="N16979">LEMMA III.</s> </p> <p id="N1697B" type="main"> <s id="N1697D">Sit magnitudo AB ipſius BC ſextupla. </s> <s id="N1697F">ſit verò AD tripla <lb></lb>ipſius AC. Dico BD ipſius BA ſeſquialteram eſse. </s> </p> <pb xlink:href="077/01/174.jpg" pagenum="170"></pb> <p id="N16986" type="main"> <s id="N16988"><expan abbr="Quoniã">Quoniam</expan>. <expan abbr="n.">enim</expan> AD multiplex eſt ipſius AG, erit AC pars ipſi^{9} <lb></lb>AD. ac propterea ipſam AD metictur. </s> <s id="N16993">rurſus quoniam AB, <lb></lb>hoc eſt AC vnà cum CB ſextupla eſt ipſius BC, erit <expan abbr="diuidẽdo">diuidendo</expan> <lb></lb>AC ipſius CB quintupla. </s> <s id="N1699D">vndè CB ipſam AC, ac propterea <expan abbr="etiã">etiam</expan> <lb></lb>ipſam AB metietur. </s> <s id="N169A5">Vta utem AC ad AD, ita fiat <lb></lb> <arrow.to.target n="fig76"></arrow.to.target><lb></lb>CB ad aliam <expan abbr="magnitudinẽ">magnitudinem</expan> G; eritvti <expan abbr="q́">〈que〉</expan>; CB ipſius <lb></lb>G pars tertia, cùm ſit AC ipſius AD pars quo〈que〉 <lb></lb>tertia. </s> <s id="N169BA">Ita〈que〉 quoniam CB ad G eſt, vt AC ad AD, <lb></lb> <arrow.to.target n="marg306"></arrow.to.target> erit perm utando CB ad CA, vt G ad AD. BC verò <lb></lb>ipſam CA metitur, eiuſquè eſt pars quinta; ergo <lb></lb>Gipſam quo〈que〉 AD metietur, eritquè ipſius pars <lb></lb>quinta. </s> <s id="N169C8">Quoniam autem BC ipſam BA metitur, <lb></lb>eademquè BC ipſam quo〈que〉 G metitur, erit BC <lb></lb>ipſarum AB G communis menſura. </s> <s id="N169CE">quia verò AB <lb></lb>ſextupla eſt ipſius CB, G verò eſt eiuſdem CB tri<lb></lb>pla, erit AB ad G, ut ſextupla ad triplam. </s> <s id="N169D4">hoc eſt <lb></lb>ſe habebunt in dupla proportione. </s> <s id="N169D8">quapropter <lb></lb>AB dupla eſt ipſius G; ac per conſe〈que〉ns Gipſam <lb></lb>AB metitur. </s> <s id="N169DE">Quoniam igitur G totam AD metitur, & <lb></lb>ablatam AB quo〈que〉 metitur; metietur G reliquam BD. G <lb></lb>igitur ipſarum AB BD communis exiſtit menſura. </s> <s id="N169E4">& <expan abbr="quoniã">quoniam</expan> <lb></lb>AB dupla eſt ipſius G, tota verò AD eiuſdem G quintupla <lb></lb>exiſtit, erit reliqua BD tripla ipſius G. Ex quibusſequitur <lb></lb>DB ad BA ita ſe habere, vt tripla ad duplam. </s> <s id="N169F0">Quare DB <lb></lb>ipſius BA ſeſquialtera exiſtit. </s> <s id="N169F4">quod oſtendere oportebat. </s> </p> <p id="N169F6" type="margin"> <s id="N169F8"><margin.target id="marg306"></margin.target>16,<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.174.1.jpg" xlink:href="077/01/174/1.jpg"></figure> <p id="N16A05" type="head"> <s id="N16A07">PROPOSITIO. VIII.</s> </p> <p id="N16A09" type="main"> <s id="N16A0B">Omnis portionis recta linea, rectanguliquè co <lb></lb>ni ſectione contentæ centrum grauitatis diame<lb></lb>trum portionis ita diuidit, vt pars ipſius ad verti<lb></lb>cem portionis reliquæ ad baſim ſit ſeſquialtera. </s> </p> <pb xlink:href="077/01/175.jpg" pagenum="171"></pb> <p id="N16A16" type="main"> <s id="N16A18"><emph type="italics"></emph>Sit portio ABC, qualis dicta est. </s> <s id="N16A1C">ipſius verò diameter ſit BD. cen<lb></lb>trum autem grauitatis ſit punctum H. oſtendendum eſt BH ipſius HD <lb></lb>ſeſquialteram eſſe. </s> <s id="N16A22">Planè inſcribatur in portione ABC triangulum ABC. <lb></lb>cuius centrum grauitatis ſit punctum E. biſariamquè diuidatur vtra<lb></lb>què AB BC in punctis FG. & ipſi BD æquidiſtantes ducantur F<emph.end type="italics"></emph.end>k <lb></lb><emph type="italics"></emph>GL. erunt vti〈que〉<emph.end type="italics"></emph.end> FK GL <emph type="italics"></emph>diametri portionum A<emph.end type="italics"></emph.end>k<emph type="italics"></emph>B BLC. ſit ita<lb></lb>〈que〉 portionis A<emph.end type="italics"></emph.end>k<emph type="italics"></emph>B centrum grauitatis M; portionis verò BLC pun<lb></lb>ctum N. connectantur〈que〉 FG MN<emph.end type="italics"></emph.end> k<emph type="italics"></emph>L<emph.end type="italics"></emph.end>, quæ diametrum BD ſe<lb></lb> <arrow.to.target n="fig77"></arrow.to.target><lb></lb>cent in punctis OQS. Quoniam igitur puncta MN in <expan abbr="eadẽ">eadem</expan> <lb></lb>proportione diuidunt KF LG, erit KM ad MF, vt LN ad <arrow.to.target n="marg307"></arrow.to.target><lb></lb>NG. & componendo KF ad FM, vt LG ad GN. & <arrow.to.target n="marg308"></arrow.to.target> per<lb></lb>mutando KF ad LG, vt FM ad GN. ſuntquè KF LG <lb></lb>æquales; erit FM ipſi GN ęqualis; & reliqua Mk reliquæ <arrow.to.target n="marg309"></arrow.to.target><lb></lb>LN æqualis. </s> <s id="N16A6D">& quoniam FM GN, & Mk NL ſunt <arrow.to.target n="marg310"></arrow.to.target> ęqui<lb></lb>diſtantes, erunt FG MN KL inter ſe ęquales, & <arrow.to.target n="marg311"></arrow.to.target> <expan abbr="æquidiſtã-tes">æquidiſtan<lb></lb>tes</expan>. & eſt BD æquidiſtans KF, erit igitur SQ ipſi KM æ<lb></lb>qualis. </s> <s id="N16A81">quia verò KF BD LG ſunt æquidiſtantes, erit MQ ad <arrow.to.target n="marg312"></arrow.to.target><lb></lb>QN, vt FO ad OG. Cùm autem ſit BF ad FA, vt BG ad GC, <pb xlink:href="077/01/176.jpg" pagenum="172"></pb> <arrow.to.target n="marg313"></arrow.to.target> crit FG ipſi AC ęquidiſtans. </s> <s id="N16A90">& vt AD ad DC, ita FO ad <lb></lb>OG. ſunt autem AD DC æquales, ergo FO OG, ac per con<lb></lb>ſe〈que〉ns MQ QN inter ſe ſunt æquales. </s> <s id="N16A96">ita〈que〉 quoniam por <lb></lb> <arrow.to.target n="marg314"></arrow.to.target> tiones AKB BLC ſunt æquales, <emph type="italics"></emph>magnitudinis ex vtriſ〈que〉 portio<lb></lb>nibus<emph.end type="italics"></emph.end> AKB BLC <emph type="italics"></emph>compoſitæ centrum grauitatis erit<emph.end type="italics"></emph.end> in medio li<lb></lb> <arrow.to.target n="marg315"></arrow.to.target> neç MN; hoc eſt erit <emph type="italics"></emph>punctum <expan abbr="q.">〈que〉</expan> & quoniam BH ad HD est,<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg316"></arrow.to.target> <emph type="italics"></emph>vt KM ad MF<emph.end type="italics"></emph.end> (centra enim grauitatum portionum in ea<lb></lb> <arrow.to.target n="fig78"></arrow.to.target><lb></lb>dem proportione diametros ſecare neceſſe eſt) <emph type="italics"></emph>& componendo<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg317"></arrow.to.target> BD ad DH, vt KF ad FM. <emph type="italics"></emph>permutandoquè vt BD ad KF,<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg318"></arrow.to.target> <emph type="italics"></emph>ita HD ad MF. at verò BD quadrupla est ipſius KF. Hoc enim<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg319"></arrow.to.target> <emph type="italics"></emph>in fine demonſtratum est, vbi est ſignum hoc, H. quadrupla igitur est<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg320"></arrow.to.target> <emph type="italics"></emph>& DH ipſius MF. Quare & reliqua BH reliquæ<emph.end type="italics"></emph.end> k<emph type="italics"></emph>M, hoc est i<lb></lb>pſius SQ, quadrupla existit.<emph.end type="italics"></emph.end> exiſtente autem tota BH, quæ <expan abbr="cõ">com</expan>. <lb></lb>poſita eſt ex BS QH, & SQ, quadrupla ipſius <expan abbr="Sq;">S〈que〉</expan> dempta <lb></lb>SQ ab ipſis BS QH SQ, <emph type="italics"></emph>reliqua igitur ex vtriſ〈que〉 BS QH<emph.end type="italics"></emph.end><lb></lb>conſtans <emph type="italics"></emph>tripla est ipſius <expan abbr="Sq.">S〈que〉</expan> ſit BS tripla ipſius SX.<emph.end type="italics"></emph.end> & <expan abbr="quoniã">quoniam</expan> <lb></lb>tota HQ cum SB ad totam QS eſt, vt ablata BS ad ab<lb></lb> <arrow.to.target n="marg321"></arrow.to.target> latam SX; ſunt quidem triplę; erit reliqua HQ ad <expan abbr="reliquã">reliquam</expan> <lb></lb> <arrow.to.target n="marg322"></arrow.to.target> QX in eadem proportione. <emph type="italics"></emph>ergo & QH ipſius XQ eſt tripla. <lb></lb>Et quoniam quadrupla est BD ipſius BS. hoc enim demonſtratum<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg323"></arrow.to.target> <emph type="italics"></emph>eſt. </s> <s id="N16B4B">ipſa verò BS ipſius SX eſt tripla<emph.end type="italics"></emph.end>; erit BD ipſius BX tripla. <pb xlink:href="077/01/177.jpg" pagenum="173"></pb>ac propterea <emph type="italics"></emph>erit XB ipſius BD pars tertia. </s> <s id="N16B57">Verùm ED ipſius <lb></lb>DB parstertia existit. </s> <s id="N16B5B">Cùm centrum grauitatis trianguli ABC ſit <lb></lb>p<gap></gap>nctum E.<emph.end type="italics"></emph.end> quod ita diuidit BD, vt BE ipſius ED ſitdupla. <arrow.to.target n="marg324"></arrow.to.target><lb></lb>At verò quoniam totius lineæ BD (quæ compoſita eſt ex DE <lb></lb>EX XB) tertia pars eſt ipſa DE. & tertia quo〈que〉 ipſa BX; <lb></lb><emph type="italics"></emph>reliqua igitur XE tertia est pars ipſius BD. & quoniam totius por<lb></lb>tionis centrum grauitatis est punctum H; magnitudinis verò ex v<lb></lb>tr<gap></gap>〈que〉 portionibus A<emph.end type="italics"></emph.end>k<emph type="italics"></emph>B BLC compoſitæ centrum grauitatis est pun<lb></lb>ctum <expan abbr="q;">〈que〉</expan> trianguli verò ABC est punctum E; erit triangulum ABC <lb></lb>ad circumrelictas portiones<emph.end type="italics"></emph.end> AKB BLC, <emph type="italics"></emph>vt QH ad HE, <expan abbr="triplũ">triplum</expan><emph.end type="italics"></emph.end> <arrow.to.target n="marg325"></arrow.to.target><lb></lb><emph type="italics"></emph>autem eſt triangulum ABC portionum. </s> <s id="N16B96">Cùm totaportio<emph.end type="italics"></emph.end> ABC <emph type="italics"></emph>ſeſqui<lb></lb>tertia ſit trianguli ABC<emph.end type="italics"></emph.end>, exceſſus verò, quo portio ABC <arrow.to.target n="marg326"></arrow.to.target> ſupe<lb></lb>rat triangulum ABC, ſint portiones AKB BLC ſimul ſum<lb></lb>ptæ. <emph type="italics"></emph>tripla igitur est QH ipſius HE. ostenſa verò eſt etiam QH <lb></lb>tripla ipſius QX.<emph.end type="italics"></emph.end> quare erit QX ipſi HE æqualis. </s> <s id="N16BB3">& <arrow.to.target n="marg327"></arrow.to.target> quo<lb></lb>niam HQ eſt tripla ipſius QX, erit HQ cum QX, hoc <lb></lb>eſt tota BX quadrupla ipſius QX, hoc eſt ipſius HE. ſi<lb></lb>militer quoniam XH quadrupla eſt ipſius HE; <emph type="italics"></emph>quintupla i<lb></lb>gitur eſt<emph.end type="italics"></emph.end> XH cum HE, tota ſcilicet <emph type="italics"></emph>XE ipſius EH; hoc est <lb></lb>DE ipſius EH. inuicem enim ſunt æquales<emph.end type="italics"></emph.end> EX ED, vt oſten<lb></lb>ſum eſt. </s> <s id="N16BD1">Cùm ita〈que〉 ſit DE ipſius EH quintupla; erit DE <lb></lb>cum EH ſextupla ipſius EH. <emph type="italics"></emph>Quare ſextupla est<emph.end type="italics"></emph.end> tota <emph type="italics"></emph>DH <lb></lb>ipſius HE. & eſt BD ipſius DE tripla; ſequialtera igitur eſt BH<emph.end type="italics"></emph.end> <arrow.to.target n="marg328"></arrow.to.target><lb></lb><emph type="italics"></emph>ipſius HD.<emph.end type="italics"></emph.end> Quare centrum grauitatis H ita diuidit diame<lb></lb>trum BD, vtpars BH ad HD ſeſquialtera exiſtit. </s> <s id="N16BEF">quod de <lb></lb>monſtrare oportebat. </s> </p> <p id="N16BF3" type="margin"> <s id="N16BF5"><margin.target id="marg307"></margin.target>7. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N16BFE" type="margin"> <s id="N16C00"><margin.target id="marg308"></margin.target>18.16 <emph type="italics"></emph><expan abbr="quĩti">quinti</expan><emph.end type="italics"></emph.end></s> </p> <p id="N16C0B" type="margin"> <s id="N16C0D"><margin.target id="marg309"></margin.target><emph type="italics"></emph>poſt <expan abbr="primã">primam</expan> <lb></lb>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N16C1B" type="margin"> <s id="N16C1D"><margin.target id="marg310"></margin.target>33. <emph type="italics"></emph>primi<emph.end type="italics"></emph.end></s> </p> <p id="N16C26" type="margin"> <s id="N16C28"><margin.target id="marg311"></margin.target>34, <emph type="italics"></emph>primi<emph.end type="italics"></emph.end></s> </p> <p id="N16C31" type="margin"> <s id="N16C33"><margin.target id="marg312"></margin.target>1. <emph type="italics"></emph>lemma <lb></lb>in<emph.end type="italics"></emph.end> 13. <emph type="italics"></emph>pri <lb></lb>mi huius<emph.end type="italics"></emph.end></s> </p> <p id="N16C46" type="margin"> <s id="N16C48"><margin.target id="marg313"></margin.target><emph type="italics"></emph><expan abbr="lẽma">lemma</expan> in ali <lb></lb>ter<emph.end type="italics"></emph.end> 13 <emph type="italics"></emph>pri <lb></lb>mi huius<emph.end type="italics"></emph.end></s> </p> <p id="N16C5D" type="margin"> <s id="N16C5F"><margin.target id="marg314"></margin.target><emph type="italics"></emph>poſt <expan abbr="primã">primam</expan> <lb></lb>huius<emph.end type="italics"></emph.end></s> </p> <p id="N16C6D" type="margin"> <s id="N16C6F"><margin.target id="marg315"></margin.target>4. <emph type="italics"></emph>primi hu<lb></lb>ius.<emph.end type="italics"></emph.end></s> </p> <p id="N16C7A" type="margin"> <s id="N16C7C"><margin.target id="marg316"></margin.target>7. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N16C85" type="margin"> <s id="N16C87"><margin.target id="marg317"></margin.target>18.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N16C90" type="margin"> <s id="N16C92"><margin.target id="marg318"></margin.target>16.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N16C9B" type="margin"> <s id="N16C9D"><margin.target id="marg319"></margin.target>A</s> </p> <p id="N16CA1" type="margin"> <s id="N16CA3"><margin.target id="marg320"></margin.target>19 <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N16CAC" type="margin"> <s id="N16CAE"><margin.target id="marg321"></margin.target>19.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N16CB7" type="margin"> <s id="N16CB9"><margin.target id="marg322"></margin.target>B</s> </p> <p id="N16CBD" type="margin"> <s id="N16CBF"><margin.target id="marg323"></margin.target>1.<emph type="italics"></emph><expan abbr="lẽma">lemma</expan> hui^{9}<emph.end type="italics"></emph.end></s> </p> <p id="N16CCB" type="margin"> <s id="N16CCD"><margin.target id="marg324"></margin.target><emph type="italics"></emph>ante<emph.end type="italics"></emph.end> 1;.<emph type="italics"></emph>pri <lb></lb>mi huius.<emph.end type="italics"></emph.end></s> </p> <p id="N16CDD" type="margin"> <s id="N16CDF"><margin.target id="marg325"></margin.target>8.<emph type="italics"></emph>primi hu<lb></lb>ius.<emph.end type="italics"></emph.end></s> </p> <p id="N16CEA" type="margin"> <s id="N16CEC"><margin.target id="marg326"></margin.target>2.<emph type="italics"></emph>lemma <lb></lb>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N16CF7" type="margin"> <s id="N16CF9"><margin.target id="marg327"></margin.target>9.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N16D02" type="margin"> <s id="N16D04"><margin.target id="marg328"></margin.target>3.<emph type="italics"></emph>lemma <lb></lb>huius.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.177.1.jpg" xlink:href="077/01/177/1.jpg"></figure> <figure id="id.077.01.177.2.jpg" xlink:href="077/01/177/2.jpg"></figure> <p id="N16D17" type="head"> <s id="N16D19">SCHOLIVM.</s> </p> <p id="N16D1B" type="main"> <s id="N16D1D">Ea verba in demonſtratione poſita nempè <emph type="italics"></emph>Hoc enim in fine<emph.end type="italics"></emph.end> <arrow.to.target n="marg329"></arrow.to.target><lb></lb><emph type="italics"></emph>demonſtratum eſt, vbi est ſignum hoc, H.<emph.end type="italics"></emph.end> ita credo eſſe intell igen<lb></lb>da, quòd ſcilicet Archimedes alicubi, & in fine, ſiue huius, ſi<lb></lb>ue alicuius alterius demonſtrationis, demonſtrauerit linea in <pb xlink:href="077/01/178.jpg" pagenum="174"></pb>BD quadruplam eſſe ipſius KF. & vbi hoc demonſtratum <lb></lb>erat, ibi quo〈que〉 pro ſigno poſita fuerit littera H. quod qui<lb></lb>dem oſtenſum eſt à nobis paulò ante ſecundam huius propoſi <lb></lb>tionem; vbi etiam appoſuim us pro ſigno hanc literam H. </s> </p> <p id="N16D3D" type="margin"> <s id="N16D3F"><margin.target id="marg329"></margin.target>A</s> </p> <p id="N16D43" type="main"> <s id="N16D45"> <arrow.to.target n="marg330"></arrow.to.target> Rurſum in demonſtratione paulò infra Archimedes dixit, <lb></lb><emph type="italics"></emph>Hoc enim demonstratum eſt<emph.end type="italics"></emph.end>, ſcilicet BD ipſius BS quadruplam <lb></lb>eſſe. </s> <s id="N16D54">ſupponit autem hoc tanquam demonſtratum poſt pri<lb></lb>mam <expan abbr="propoſitionẽ">propoſitionem</expan> huius, vbi tota BD eſt ſexdccim, & BS qua <lb></lb>tuor, vt eodem in loco oſtenſum fuità nobis. </s> <s id="N16D5E">Vel ad ea re<lb></lb>ſpexit Archimedes, quæ ab ipſo in decimanona propoſitione <lb></lb>de quadratura paraboles demonſtra ta fuerunt. </s> <s id="N16D64">vbi circa <expan abbr="finẽ">finem</expan> <lb></lb>demonſtrationis oſtendit BD quadruplam eſſe ipſius BS. </s> </p> <p id="N16D6C" type="margin"> <s id="N16D6E"><margin.target id="marg330"></margin.target>B</s> </p> <p id="N16D72" type="main"> <s id="N16D74">Inuento ita〈que〉 centro grauitatis paraboles, vult Archime<lb></lb>des in ueſtigare centrum grauitatis fruſti à parabole abſciſſi. <lb></lb>〈que〉madmodum in primo libro poſt inuentionem centri gra<lb></lb>uitatis trianguli, adinuenit etiam centrum grauitatis trapezij. <lb></lb>quod eſt tan quam fruſtum à triangulo abſciſsum. </s> <s id="N16D7E">quare duo <lb></lb>adhuc theoremata proponit, in quorum poſtremo, vbi ſit <expan abbr="cẽ">cem</expan> <lb></lb>trum grauitatis fruſti demonſtrat. </s> <s id="N16D88">in ſe〈que〉nri verò quædam <lb></lb>demonſtrat neceſſaria, vt huiuſmodi centrum determinare <lb></lb>poſſit. </s> <s id="N16D8E">Quoniam autem ſe〈que〉ns theorema arduum, difficile<lb></lb>què ſeſe offert; non nulla priùs quibuſdam lemmatibus oſten<lb></lb>demus, ne ſi in demonſtratione ea inſererentur, longa nimis <lb></lb>euaderet, ac tædioſa demonſtratio. </s> <s id="N16D96">quæ quidem ſumma indi<lb></lb>get attentione. </s> <s id="N16D9A">quamquàm in hoc theoremate explicando ad <lb></lb>vitandam obſcuritatem copioſum ſermonem adhibendum <lb></lb>curauimus; ne breuitati ſtudentes obſcuriores eſſemus. </s> </p> <p id="N16DA0" type="head"> <s id="N16DA2">LEMMA. I.</s> </p> <p id="N16DA4" type="main"> <s id="N16DA6">Si qua tuor magnitudines in continua fuerint proportione, <lb></lb>& earum exceſſus in eadem erunt proportione <expan abbr="magnitudinũ">magnitudinum</expan>. </s> </p> <pb xlink:href="077/01/179.jpg" pagenum="175"></pb> <p id="N16DB1" type="main"> <s id="N16DB3">Sint quatuor magnitudines AF BH CL D in continua <lb></lb>proportione; vt ſcilicet ſit AF ad BH, vt BH ad CL; & CL <lb></lb>ad D. exceſſus verò, quo AF ſuperat BH, ſit EF. & exceſſus, quo <lb></lb>BH ſuperat CL, ſit GH. exceſſus deni〈que〉, quo CL ſuperat <lb></lb>D, ſit KL. eruntuti〈que〉 AE BH inter ſe ęquales, itidemquè <lb></lb> <arrow.to.target n="fig79"></arrow.to.target><lb></lb>BG CL æquales. </s> <s id="N16DC4">Dico EF GH KL in eadem eſſe proportio <lb></lb>ne, vt ſunt magnitudines AF BH CL, & vt BH CL D. Quo<lb></lb>niam enim tota AF ad totam BH eſt, vt BH ad CL; hoc eſt <arrow.to.target n="marg331"></arrow.to.target><lb></lb>vt ablata EA ad ablatam GB. erit reliqua EF ad reliquam GH; <lb></lb>vt AF ad BH. Pariquè ratione oſtendetur GH ad kL ita eſ<lb></lb>ſe, vt BH ad CL. ergo exceſſus EF GH KL in eadem ſunt <lb></lb>proportione, vt magnitudines AF BH CL. quæ cùm ſint, vt <lb></lb>magnitudines BH CL D, ſiquidem omnes in continua ſunt <lb></lb>proportione; exceſſus igitur EF GH KL in eadem quo〈que〉 <lb></lb>ſunt proportione, vt magnitudines BH CL D. quæ quidem <lb></lb>demonſtrare oportebat. </s> </p> <pb xlink:href="077/01/180.jpg" pagenum="176"></pb> <p id="N16DE0" type="margin"> <s id="N16DE2"><margin.target id="marg331"></margin.target>19.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.180.1.jpg" xlink:href="077/01/180/1.jpg"></figure> <p id="N16DEF" type="head"> <s id="N16DF1">LEMMA. II.</s> </p> <p id="N16DF3" type="main"> <s id="N16DF5">Si tres fuerint magnitudines, & aliæ ipſis numero æquales, <lb></lb>& in eadem proportione, in primis magnitudinibus prima; <lb></lb>& ſecunda ad tertiam erunt, vt in ſecundis magnitudinibus <lb></lb>prima & ſecunda ad tertiam. </s> </p> <figure id="id.077.01.180.2.jpg" xlink:href="077/01/180/2.jpg"></figure> <p id="N16E00" type="main"> <s id="N16E02">Sint tres magnitudines ABC, & aliæ tres DEF in <expan abbr="eadẽ">eadem</expan> pro<lb></lb>portione. </s> <s id="N16E0A">Dico AB ſimul ad C ita eſſe, vt DE ſimul ad F. <lb></lb> <arrow.to.target n="marg332"></arrow.to.target> Quoniam enim A ad B eſt, ut D ad E, erit <expan abbr="componẽdo">componendo</expan> AB <lb></lb> <arrow.to.target n="marg333"></arrow.to.target> ad B, ut DE ad E. ſed vt B ad C, ita eſt E ad F. ergo ex ęquali <lb></lb>AB ſimul ad C eſt, vt DE ſimul ad F. quod demonſtrare opor <lb></lb>tebat. </s> </p> <p id="N16E20" type="margin"> <s id="N16E22"><margin.target id="marg332"></margin.target>18,<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N16E2B" type="margin"> <s id="N16E2D"><margin.target id="marg333"></margin.target>22.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N16E36" type="head"> <s id="N16E38">LEMMA. III.</s> </p> <p id="N16E3A" type="main"> <s id="N16E3C">Si fuerit AB ad AC, vt DE ad DF. Dico exceſſum BC ad <lb></lb> <arrow.to.target n="marg334"></arrow.to.target> CA ita eſſe, vt exceſſus EF ad FD. </s> </p> <p id="N16E44" type="margin"> <s id="N16E46"><margin.target id="marg334"></margin.target><emph type="italics"></emph>cor.<emph.end type="italics"></emph.end>4.<emph type="italics"></emph><expan abbr="quĩti">quinti</expan><emph.end type="italics"></emph.end></s> </p> <p id="N16E56" type="main"> <s id="N16E58">Quoniam enim eſt AB ad AC, vt DE ad DF, erit con- <pb xlink:href="077/01/181.jpg" pagenum="177"></pb> <arrow.to.target n="fig80"></arrow.to.target><lb></lb>uertendo CA ad AB, vt FD ad DE. & per conuer <lb></lb>ſionem rationis AC ad CB, vt DF ad FE. & rurſus <arrow.to.target n="marg335"></arrow.to.target><lb></lb>conuertendo CB ad CA, vt FE ad FD. quod <expan abbr="demõ-ſtrare">demon<lb></lb>ſtrare</expan> oportebat. </s> </p> <p id="N16E70" type="margin"> <s id="N16E72"><margin.target id="marg335"></margin.target><emph type="italics"></emph>co.<emph.end type="italics"></emph.end>4.<emph type="italics"></emph><expan abbr="quĩti">quinti</expan>.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.181.1.jpg" xlink:href="077/01/181/1.jpg"></figure> <p id="N16E87" type="head"> <s id="N16E89">ALITER.</s> </p> <p id="N16E8B" type="main"> <s id="N16E8D">Quoniam enim AB eſt ad AC, vt DE ad DF, erit conuer<lb></lb>tendo AC ad AB, vt DF ad DE. diuidendoquè CB ad BA, vt <lb></lb>FE ad ED. eſt autem AB ad AC, vt DE ad DF, erit igitur <arrow.to.target n="marg336"></arrow.to.target><lb></lb>ex æquali BC ad CA, vt EF ad FD. quod demonſtrare opor <arrow.to.target n="marg337"></arrow.to.target><lb></lb>tebat. </s> </p> <p id="N16E9D" type="margin"> <s id="N16E9F"><margin.target id="marg336"></margin.target>17.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N16EA8" type="margin"> <s id="N16EAA"><margin.target id="marg337"></margin.target>22,<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N16EB3" type="head"> <s id="N16EB5">LEMMA IIII.</s> </p> <figure id="id.077.01.181.2.jpg" xlink:href="077/01/181/2.jpg"></figure> <p id="N16EBA" type="main"> <s id="N16EBC">Si fuerint quotcun〈que〉 magnitudines ABC, & nlię ipſis nu<lb></lb>mero æquales DEF, & in <expan abbr="eadẽ">eadem</expan> proportione. </s> <s id="N16EC4">Dico vtram〈que〉 <lb></lb>ſimul AD ad vtram〈que〉 ſimul BE, & vtram〈que〉 ſimul BE ad v<lb></lb>tram〈que〉 ſimul CF eandem habere proportionem, quam ha<lb></lb>bet A ad B, & B ad C. </s> </p> <pb xlink:href="077/01/182.jpg" pagenum="178"></pb> <p id="N16ECF" type="main"> <s id="N16ED1"> <arrow.to.target n="marg338"></arrow.to.target> Quoniam enim eſt A ad B, ut D ad E; erit AD ſimul ad <lb></lb>BE ſimul, vt A ad B. ſimiliter quoniam B ad C eſt, vt E ad <lb></lb>F, erit BE ſimul ad CF ſimul, vt B ad C. in eadem igitur ſunt <lb></lb>proportione AD ſimul, & BE ſimul, & CF ſimul, vt ABC. <lb></lb>quod demonſtrare oportebat. </s> </p> <p id="N16EDF" type="margin"> <s id="N16EE1"><margin.target id="marg338"></margin.target>12.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N16EEA" type="head"> <s id="N16EEC">LEMMA. V.</s> </p> <p id="N16EEE" type="main"> <s id="N16EF0">Si magnitudo magnitudinis fuerit ſeſquialtera ad tres quin<lb></lb>tas eiuſdem erit duplex ſeſquialtera. </s> </p> <figure id="id.077.01.182.1.jpg" xlink:href="077/01/182/1.jpg"></figure> <p id="N16EF7" type="main"> <s id="N16EF9">Sit AB ipſius CD ſeſquialtera. </s> <s id="N16EFB">ſit uerò CE tres quintæ <lb></lb>ipſius CD. Dico AB ad CE ita eſſe, vt quin〈que〉 ad duo. </s> <s id="N16EFF">Fiat EF <lb></lb>ęqualis EC, erit CF ſex quintæ ipſius CD. & quoniam AB i<lb></lb>pſius CD eſt ſeſquialtera, ſuperabit AB ipſam CD dimidia <lb></lb>ipſius CD. erit igitur AB ſeptem quintæ cum dimidia i<lb></lb>pſius CD. quare CF minor eſt AB. fiat igitur AG æqua<lb></lb>lis CF. erit vti〈que〉 AG ſex quintę ipſius CD. & ob id GB <lb></lb>ipſius CD quinta eſt pars cum dimidia. </s> <s id="N16F0D">& quoniam CE eſt <lb></lb>eiuſdem CD tres quintæ, erit BG dimidia ipſius CE. qua<lb></lb>re GB ipſam CE bis metietur. </s> <s id="N16F13">Et quoniam EF eſt æqua<lb></lb>lis ipſi EC, ipſa BG bis quo〈que〉 metietur ipſam EF. quare <pb xlink:href="077/01/183.jpg" pagenum="179"></pb>totam CF, hoc eſt ipſam AG quater metietur. </s> <s id="N16F1B">at verò GB ſei<lb></lb>pſam ſemel metitur ipſa igitur GB totam AB quinquies metie<lb></lb>tur. </s> <s id="N16F21">Ex quibus li〈que〉t GB ipſarum ABCE communem eſſe <lb></lb>menſuram. </s> <s id="N16F25">Et eſt quidem AB quintupla ipſius BG; ipſa verò <lb></lb>CE eiuſdem BG dupla. </s> <s id="N16F29">erit AB ad CE, vt quintupla ad <expan abbr="duplã">duplam</expan>. <lb></lb>hoc eſt duplex ſeſquialtera. </s> <s id="N16F31">quod demonſtrare oportebat. </s> </p> <p id="N16F33" type="head"> <s id="N16F35">PROPOSITIO. VIIII.</s> </p> <p id="N16F37" type="main"> <s id="N16F39">Si quatuor lineæ in continua fuerint proportio<lb></lb>ne, & quam proportionem habet minima ad exceſ <lb></lb>ſum, quo maxima minimam ſuperat; eandem ha<lb></lb>beat quædam aſſumpta linea ad tres quintas exceſ<lb></lb>ſus, quo maxima proportionalium tertiam exce<lb></lb>dit: quam verò proportionem habet linea æqualis <lb></lb>duplæ maximæ proportionalium, & quadruplæ ſe <lb></lb>cundæ, & ſextuplæ tertiæ, & triplæ quartæ ad <expan abbr="lineã">lineam</expan> <lb></lb>æqualem quintuplæ maximæ, & decuplæ ſecundæ, <lb></lb>& decuplæ tertiæ, & quintuplæ quartæ, ean-<lb></lb>dem habeat quædam aſſumpta linea ad ex ceſſum, <lb></lb>quo maxima proportionalium tertiam ſuperat; <lb></lb>vtræ〈que〉 ſimul aſſumptæ lineæ erunt duæ quin<lb></lb>tæ maximæ. <lb></lb></s> </p> <pb xlink:href="077/01/184.jpg" pagenum="180"></pb> <p id="N16F5D" type="main"> <s id="N16F5F"><emph type="italics"></emph>Sint quatuor lineæ proportionales AB BC BD BE,<emph.end type="italics"></emph.end> ita vt AB <lb></lb>ad BC ſit, vt BC ad BD. & vt BC ad BD, ita ſit BD ad BE. <emph type="italics"></emph>& <lb></lb>quam proportionem habet BE ad E A, eandem habeat FG adtres quin<lb></lb>tas ipſius AD. quam autem proportionem habet linea æqualis duplæ i<lb></lb>pſius AB, & quidruplæ ipſius BC, & ſextuplæ ipſi^{9} BD, & triplæ ipſi^{9} <lb></lb>BE, ad <expan abbr="lineã">lineam</expan> <expan abbr="æqualẽ">æqualem</expan> <expan abbr="quĩtuplæ">quintuplæ</expan> ipſi^{9} AB, ot decuplæ ipſi^{9} CB, & decuplæ <lb></lb>ipſi^{9} B D, & quintuplæ ipſius BE, eandem habeat GH ad AD. Oſteden<lb></lb>dum est FH duasquintas eſſe ipſius AB. Quoniam enim proportiona<lb></lb>les ſunt AB BC BD BE, &<emph.end type="italics"></emph.end> ipſarum exceſſus <emph type="italics"></emph>AC CD DE in<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="fig81"></arrow.to.target><lb></lb> <arrow.to.target n="marg339"></arrow.to.target> <emph type="italics"></emph>eadem erunt proportione. </s> <s id="N16F9B">&<emph.end type="italics"></emph.end> quoniam magnitudines AB BC BD <lb></lb>in continua ſunt proportione, & earum exceſſus AC CD DE <lb></lb>in eadem erunt proportione. </s> <s id="N16FA4">quia verò tres ſunt magnitudi<lb></lb>nes proportionales AB BC BD; & alię ipſis numero çquales, & <lb></lb> <arrow.to.target n="marg340"></arrow.to.target> in eadem proportione AC CD DE, erit in primis magnitu<lb></lb>dinibus prima, & ſecunda ad tertiam, vt in ſecundis magni<lb></lb>tudinibus prima, & ſecunda ad tertiam; hoc eſt <emph type="italics"></emph>vtra〈que〉 ſimul <lb></lb>AB BC ad BD eandem habebit proportionem, quam<emph.end type="italics"></emph.end> vtra〈que〉 ſimul <lb></lb> <arrow.to.target n="marg341"></arrow.to.target> AC CD, hoc eſt <emph type="italics"></emph>AD ad DE; &<emph.end type="italics"></emph.end> ob eandem rationem cum <lb></lb> <arrow.to.target n="marg342"></arrow.to.target> tres ſint proportionales magnitudines AC CD DE, aliçquè <lb></lb>eodem modo proportionales BC BD BE; crit vtra〈que〉 ſimul <pb xlink:href="077/01/185.jpg" pagenum="181"></pb>AC CD, hoc eſt AD ad DE, vt <emph type="italics"></emph>vtra〈que〉 ſimul BC BD ad EB. <lb></lb>& omnes adomnes,<emph.end type="italics"></emph.end> quoniam ſcilicet eſt vtra〈que〉 ſimul AB BC <lb></lb>ad BD, vt horum dupla; erit vtra〈que〉 ſimul AB BC ad BD, vt <lb></lb>dupla vtriuſ〈que〉 ſimul AB BC ad duplam ipſius BD. eſt <expan abbr="autẽ">autem</expan> <lb></lb>vtra〈que〉 ſimul AB BC ad BD, vt AD ad DE. erit igitur AD ad <lb></lb>DE, vt dupla vtriuſ〈que〉 ſimul AB BC ad duplam ipſius BD. <lb></lb>quia veròita etiam eſt AD ad DE, vtvtra〈que〉 ſimul CB BD ad <lb></lb>BE; erit dupla vtriuſ〈que〉 ſimul AB BC ad duplam ipſius BD, vt <arrow.to.target n="marg343"></arrow.to.target><lb></lb>vtra〈que〉 ſimul CB BD ad BE. & vtra〈que〉 antecedentia ad <arrow.to.target n="marg344"></arrow.to.target> vtra<lb></lb>〈que〉 conſe〈que〉ntia in eadem erunt proportione: eruntquè in <lb></lb>antecedenti duę AB, tres BC, & ſola BD. in conſe〈que〉nti verò <lb></lb>erunt duæ BD cum ſola BE. erit igitur dupla ipſius AB, & tri <lb></lb>pla ipſius CB cum ſola BD ad duplam ipſius BD cum ſola BE, <lb></lb>vt vtra〈que〉 ſimul CB BD ad BE. vtra〈que〉 verò ſimul CB BD <lb></lb>ad BE eſt, vt AD ad DE. <emph type="italics"></emph>eandem ergo proportionem habet AD ad<emph.end type="italics"></emph.end> <arrow.to.target n="marg345"></arrow.to.target><lb></lb><emph type="italics"></emph>DE, quam linea æqualis duplæ ipſius AB, & triplæipſius CB, &<emph.end type="italics"></emph.end> ſoli <lb></lb><emph type="italics"></emph>DB adlineam æqualem duplæ ipſius BD &<emph.end type="italics"></emph.end> ſoli <emph type="italics"></emph>BE.<emph.end type="italics"></emph.end> Quoniam au<lb></lb>tem linea compoſita ex dupla ipſius AB, & quadrupla ipſius <lb></lb>CB, & quadrupla ipſius BD, & dupla ipſius BE, maior eſt ea, <lb></lb>quæ compoſita eſt ex dupla ipſius AB, & tripla ipſius CB, & <lb></lb>ſola BD; maiorem habebit proportionem compoſita ex <arrow.to.target n="marg346"></arrow.to.target> du<lb></lb>pla ipſius AB, & quadrupla ipſius CB, & quadrupla ipſius BD, <lb></lb>& dupla ipſius BE ad compoſitam ex dupla ipſius BD cum <lb></lb>ſola BE, quam compoſita ex dupla ipſius AB, & tripla ipſius <lb></lb>CB cum ſola BD ad eandem compoſitam ex dupla ipſius BD <lb></lb>cum ſola EB. compoſita verò ex dupla ipſius AB, & tripla <lb></lb>ipſius BC cum ſola BD ad duplam ipſius BD cum ſola BE ita <lb></lb>oſtenſa eſt ſe habere AD ad DE. compoſita igitur ex dupla i<lb></lb>pſius AB, & quadrupla ipſius BC, & quadrupla ipſius BD, & <lb></lb>dupla ipſius BE ad compoſitam ex dupla ipſius BD cum ſola <lb></lb>BE maiorem habebit proportionem, quam AD ad DE. <emph type="italics"></emph>Quam <lb></lb>ita〈que〉 proportionem habet linea æqualis duplæ ipſius AB, & quadruplæ <lb></lb>ipſius BC, & quadruplæ ipſius BD, & duplæ ipſius BE ad <expan abbr="lineã">lineam</expan> <expan abbr="æqualẽ">æqualem</expan> <lb></lb>duplæ ipſius DB, & ad EB, eandem habebit AD adminorem ipſa DE.<emph.end type="italics"></emph.end> <arrow.to.target n="marg347"></arrow.to.target><lb></lb><emph type="italics"></emph>habeat igitur ad DO.<emph.end type="italics"></emph.end> & <expan abbr="quoniãita">quonianita</expan> ſe habet AD ad DO, vt <expan abbr="cõpo">compo</expan> <lb></lb>ſita ex dupla ipſius AB, & quadrupla ipſius BC, & quadrupla <lb></lb>ipſius BD, & dupla ipſius BE, hoc eſt <expan abbr="cõpoſita">compoſita</expan> ex dupla vtriuſ- <pb xlink:href="077/01/186.jpg" pagenum="182"></pb>〈que〉 ſimul AB BE, & quadrupla vtriuſ〈que〉 ſimul BC BD. (bis <lb></lb>enim aſſumitur AB, & bis BE, quater verò BC, & quater BD) <lb></lb> <arrow.to.target n="marg348"></arrow.to.target> ad compoſitam ex dupla ipſius BD cum ſola BE; erit conuer<lb></lb>rendo, ut OD ad DA, ita compoſita ex dupla ipſius BD <expan abbr="cũ">cum</expan> ſo<lb></lb>la BE ad <expan abbr="cõpoſitam">compoſitam</expan> ex dupla utriuſ〈que〉 ſimul AB BE, & qua<lb></lb> <arrow.to.target n="marg349"></arrow.to.target> drupla vtriuſ〈que〉 ſimul BCBD. <emph type="italics"></emph>et vtræ〈que〉 ad primas eandem habe <lb></lb>bunt proportionem.<emph.end type="italics"></emph.end> hoc eſt componendo erit OA ad AD, vt <expan abbr="cõ-poſita">con<lb></lb>poſita</expan> ex dupla ipſius BD cum ſola BE, & dupla vtriuſ〈que〉 ſi<lb></lb>mul AB BE, & quadrupla vtriuſ〈que〉 ſimul BC BD ad compo<lb></lb> <arrow.to.target n="fig82"></arrow.to.target><lb></lb>ſitam ex dupla vtriuſ〈que〉 ſimul AB BE, & quadrupla <expan abbr="vtriusq́">vtrius〈que〉</expan>; <lb></lb>ſimul BC BD. In hoc autem antecedente bisſumitur AB, qua <lb></lb>ter BC, ſexies verò BD, & ter BE. <emph type="italics"></emph>habebit igitur OA ad AD ean<lb></lb>demproportionem, quam linea æqualis duplæipſius AB, et quadruplæi<lb></lb>pſius CB, et ſextuplæ ipſius BD, ettriplæ ipſius BE ad lineam compoſi<lb></lb>tam ex dupla vtriuſ〈que〉 ſimul AB EB, et quadrupla vtriuſ〈que〉 ſimul <lb></lb>CB BD. babet autem<emph.end type="italics"></emph.end> (vt ſuppoſitum eſt) GH ad AD eandem <lb></lb>proportionem, quam linea æqualis duplæ ipſius AB, & qua<lb></lb>druplæ ipſius BC, & ſextuplæ ipſius BD, & triplæ ipſius BE <lb></lb>ad lineam æqualem quintuplæ ipſius AB, & decuplæ ipſius <lb></lb>CB, & decuplæ ipſius BD, & quintuplæ ipſius BE, hoc eſt ad <pb xlink:href="077/01/187.jpg" pagenum="183"></pb><expan abbr="quintuplã">quintuplam</expan> vtriuſ〈que〉ſimul AB BE <expan abbr="cũ">cum</expan> decupla vtriuſquè ſimul <lb></lb>CB BD. In <expan abbr="cõſe〈quẽ〉ti">conſe〈que〉nti</expan>.n.quinquies <expan abbr="aſsũpta">aſsumpta</expan> eſt AB, & quinquies <lb></lb>BE, decies CB, & decies BD. & conuettendo habebit <emph type="italics"></emph>AD ad<emph.end type="italics"></emph.end> <arrow.to.target n="marg350"></arrow.to.target><lb></lb><emph type="italics"></emph>GH eandem proportionem, quam quintupla vtriuſ〈que〉 ſimul AB BE <lb></lb><expan abbr="cũ">cum</expan> decupla vtriuſ〈que〉 ſimul CB BD ad lineam compoſitam ex dupla i<lb></lb>pſius AB, & quadrupla ipſius CB, & ſextuplaipſius BD, & triplai<lb></lb>pſius EB. Diſsimiliter autem quàm in proportionibus ordinatis, hoc est <lb></lb>in perturbata proportione<emph.end type="italics"></emph.end> quoniam in primis magnitudinibus ita <lb></lb>ſe habet antecedens OA ad conſe〈que〉ns AD, vt in ſecundis ma <lb></lb>gnitudinibus antecedens compoſita nempè ex dupla ipſius <lb></lb>AB, & quadrupla ipſius BC, & ſextupla ipſius BD, & tripla <lb></lb>ipſius BE, ad conſe〈que〉ns lineam ſcilic et compoſitam ex du<lb></lb>pla vtriuſ〈que〉 ſimul AB BE, & quadrupla vtriuſ〈que〉 ſimul CB <lb></lb>BD: ut autem in primis magnitudinibus conſe〈que〉ns AD ad <lb></lb>aliud quippiam GH, ita in ſecundis magnitudinibus aliud <lb></lb>quippiam, nempèlinea compoſita ex quintupla vtriuſ〈que〉 ſi<lb></lb>mul AB BE cum decupla vtriuſ〈que〉 ſimul CB BD ad antece<lb></lb>dens, hoc eſt ad compoſitam ex dupla ipſius AB, & quadru<lb></lb>pla ipſius CB, & ſextupla ipſius BD, & tripla ipſius BE. quare <lb></lb><emph type="italics"></emph>ex æquali eandemhabet proportionem OA ad GH, quam quintupla v-<emph.end type="italics"></emph.end> <arrow.to.target n="marg351"></arrow.to.target><lb></lb><emph type="italics"></emph>triuſ〈que〉 ſimul AB BE cum decupla<emph.end type="italics"></emph.end> vtriuſ〈que〉 ſimul <emph type="italics"></emph>CB BD ad <lb></lb><expan abbr="cõpoſitã">compoſitam</expan> ex dupla <expan abbr="vtriusq́">vtrius〈que〉</expan>; ſimul AB BE, et quadrupla <expan abbr="vtriusq́">vtrius〈que〉</expan>; ſimul <lb></lb>CB BD. At verò<emph.end type="italics"></emph.end> quoniam quintupla ipſius AB ad duplam <lb></lb>eiuſdem AB eſt, vt quin〈que〉 ad duo; ſimiliter quintupla ipſi^{9} <lb></lb>BE ad duplam eiuſdem BE eſt, vt quin〈que〉 ad duo, erit quin<lb></lb>tupla vtriuſ〈que〉 ſimul AB BE ad duplam vtriuſ〈que〉 ſimul AB <lb></lb>BE, vt quin〈que〉 ad duo. </s> <s id="N17133">pariquè ratione decupla vtriuſ〈que〉 ſi<lb></lb>mul CB BD ad quadruplam vtriuſ〈que〉 ſimul CB BD eſt, vt <lb></lb>decem ad quatuor, hoc eſt vt quin〈que〉 ad duo. </s> <s id="N17139">& <expan abbr="antecedẽtia">antecedentia</expan> <arrow.to.target n="marg352"></arrow.to.target><lb></lb>ad conſe〈que〉ntia in eadem erunt proportione, hoceſt <emph type="italics"></emph>compoſi<lb></lb>ta ex quintupla vtriuſ〈que〉 ſimul AB BE cum decupla vtriuſ〈que〉 ſimul <lb></lb>CB BD ad compoſitam ex dupla vtriuſ〈que〉 ſimul AB BE, & quadru<lb></lb>pla vtriuſ〈que〉 ſimul CB BD proportionem habet, quam quin〈que〉 ad duo <lb></lb>Quare OA ad GH proportionem habet, quam quin〈que〉 ad duo. </s> <s id="N1714F">Rurſus<emph.end type="italics"></emph.end><lb></lb>factum fuit AD ad DO, vt compoſita ex dupla vtriuſ〈que〉 ſi<lb></lb>mul AB BE cum quadrupla vtriuſ〈que〉 ſimul CB BD ad <expan abbr="lineã">lineam</expan> <lb></lb>BE vnà cum dupla ipſius BD. conuertendo etiam <emph type="italics"></emph>quoniam<emph.end type="italics"></emph.end> <arrow.to.target n="marg353"></arrow.to.target> <pb xlink:href="077/01/188.jpg" pagenum="184"></pb>in primis magnitudinibus antecedens <emph type="italics"></emph>OD ad<emph.end type="italics"></emph.end> conſe〈que〉ns <emph type="italics"></emph>DA <lb></lb>eandem habet proportionem, quam<emph.end type="italics"></emph.end> in ſecundis magnitudinibus an<lb></lb>tecedens <emph type="italics"></emph>EB cum dupla ipſius BD ad<emph.end type="italics"></emph.end> conſe〈que〉ns, <emph type="italics"></emph>lineam<emph.end type="italics"></emph.end> ſcilicet <emph type="italics"></emph>æ<lb></lb>qualem lineæ compoſitæ ex dupla vtriuſ〈que〉 ſimul AB BE cum quadru<lb></lb>pla vtriuſ〈que〉 ſimul CB BD; est autem<emph.end type="italics"></emph.end> (vt antea oſtenſum eſt) & <lb></lb>in primis magnitudinibus conſe〈que〉ns <emph type="italics"></emph>AD ad<emph.end type="italics"></emph.end> aliud <expan abbr="quippiã">quippiam</expan> <lb></lb><emph type="italics"></emph>DE, vt<emph.end type="italics"></emph.end> in ſecundis magnitudinibus aliud quippiam, linea <lb></lb>ſcilicet <emph type="italics"></emph>compoſita ex dupla ipſius AB, & tripla ipſius CB, &<emph.end type="italics"></emph.end> ſola <emph type="italics"></emph>BD <lb></lb>ad<emph.end type="italics"></emph.end> antecedens, nempè <emph type="italics"></emph>lineam <expan abbr="æqualẽ">æqualem</expan> ipſi EB, & duplæ ipſius BD.<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="fig83"></arrow.to.target><lb></lb><emph type="italics"></emph>Non igitur perinde, vt in proportione ordinata; hoc est, perturbata <expan abbr="exiſtẽ">exiſtem</expan><emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg354"></arrow.to.target> <emph type="italics"></emph>te proportione, ex æqualiest OD ad DE, vt duplaipſius AB cum tripla <lb></lb>ipſius BC &<emph.end type="italics"></emph.end> ſola <emph type="italics"></emph>BD ad <expan abbr="cõpoſitam">compoſitam</expan> ex dupla vtriuſ〈que〉 ſimul AB BE, <lb></lb>& quadrupla vtriuſ〈que〉 ſimul CB BD.<emph.end type="italics"></emph.end> ſuperat verò DE ipſam <lb></lb>DO exceſſu OE; linea verò <expan abbr="cõpoſita">compoſita</expan> ex dupla vtriuſ〈que〉 ſimul <lb></lb>AB BE, & quadrupla vtriuſ〈que〉 ſimul CB BD lineam excedit <lb></lb>compoſitam ex dupla ipſius AB cum tripla ipſius BC, ac ſola <lb></lb>BD, exceſſu lineæ, quæ ſit æqualis ſoli CB cum tripla ipſius <lb></lb> <arrow.to.target n="marg355"></arrow.to.target> BD, & dupla ipſius BE. <emph type="italics"></emph>Quare est EO ad ED, vt CB cum tripla <lb></lb>ipſius BD, & dupla ipſius EB ad duplam vtriuſ〈que〉 ſimul AB BE, <lb></lb>& quadruplam vtriuſ〈que〉 ſimul CB BD. est autem<emph.end type="italics"></emph.end> in lineis pro- <pb xlink:href="077/01/189.jpg" pagenum="185"></pb>portionalibus initio expoſitis; cùm in continua ſint propor<lb></lb>tione, tertia in ordine BD ad quartam BE, vt prima AB ad <lb></lb>ſecundam BC, quare diuidendo vt DE ad EB, ita AC ad <arrow.to.target n="marg356"></arrow.to.target><lb></lb>CB. Rurſus quoniam in lineis proportionalibus ob eandem <lb></lb>cauſam CB ad BD ita eſt, vt DB ad BE; erit diuidendo, vt <lb></lb>CD ad DB, ita DE ad EB. ego <emph type="italics"></emph>vt DE ad EB, ita AC ad<emph.end type="italics"></emph.end> <arrow.to.target n="marg357"></arrow.to.target><lb></lb><emph type="italics"></emph>CB, & CD ad DB. ac propterea ſecundum<emph.end type="italics"></emph.end> multiplicem <emph type="italics"></emph>compoſitio <lb></lb>nemtripla ipſius CD, adtriplam ipſius DB<emph.end type="italics"></emph.end> eſt, vt ſola CD ad ſo<lb></lb>lam DB. <emph type="italics"></emph>& dupla ipſius DE ad duplam ipſius EB<emph.end type="italics"></emph.end> eſt, <lb></lb>vt DE ad EB. eſt verò CD ad DB, vt DE ad <lb></lb>EB, & AC ad CB; erit igitur AC ad CB, vt tripla ipſius <lb></lb>CD ad triplam ipſius DB; & vt dupla ipſius DE ad <lb></lb>duplam ipſius EB. <emph type="italics"></emph>Quare &<emph.end type="italics"></emph.end> tria antecedentia ſimul ad <arrow.to.target n="marg358"></arrow.to.target><lb></lb>tria ſimul conſe〈que〉ntia, hoc eſt, <emph type="italics"></emph>compoſita ex AC, & <lb></lb>tripla ipſius CD, & dupla ipſius DE ad compoſitam ex CB, <lb></lb>& tripla ipſius DB, & dupla ipſius EB<emph.end type="italics"></emph.end> ita erit, vt AC <lb></lb>ad CB, hoc eſt, DE ad EB. <emph type="italics"></emph>Rurſus ita〈que〉 diſsimili modo, <lb></lb>quàm in proportionibus ordinatis, hoc est in perturbata proportione,<emph.end type="italics"></emph.end><lb></lb>quoniam eſt in primis magnitudinibus antecedens OE ad <lb></lb>conſe〈que〉ns ED, ita in ſecundis magnitudinibus an <expan abbr="tecedẽs">tecedens</expan> <lb></lb>compoſita ſcilicet ex CB, cum tripla ipſius BD, & dupla ip<lb></lb>ſius EB, ad conſe〈que〉ns nem pè compoſitam ex dupla vtriuſ<lb></lb>〈que〉 ſimul AB BE, cum quadrupla vtriuſ〈que〉 ſimul CB BD: <lb></lb>in primis verò magnitudinibus conſe〈que〉ns DE ad aliud quip <lb></lb>piam EB eſt, vt in ſecundis magnitudinibus aliud quippia, <lb></lb>hoc eſt compoſita ex AC cum tripla ipſius CD, & dupla ip<lb></lb>ſius DE ad antecedens, lineam ſcilicet compoſitam ex CB cum <lb></lb>tripla ipſius BD, & dupla ipſius EB. <emph type="italics"></emph>ex æquali eandem<emph.end type="italics"></emph.end> <arrow.to.target n="marg359"></arrow.to.target><lb></lb><emph type="italics"></emph>habebit proportionem EO ad EB, quam AC cum tri <lb></lb>pla ipſius CD, & dupla ipſius DE ad duplam vtriuſ <lb></lb>〈que〉 ſimul AB BE cum qnadrupla vtriuſ〈que〉 ſimul CB <lb></lb>BD.<emph.end type="italics"></emph.end> & componendo erit OB ad BE, vtlinea AC <arrow.to.target n="marg360"></arrow.to.target><lb></lb>cum tripla ipſius CD, & dupla ipſius DE, & dupla <lb></lb>vtriuſ〈que〉 ſimul AB BE, & quadrupla vtriuſ〈que〉 ſi<lb></lb>mul CB BD, ad duplam vtriuſ〈que〉 ſimul AB BE <lb></lb>cum quadrupla vtriuſ〈que〉 ſrmul CB BD. In hoc autem <pb xlink:href="077/01/190.jpg" pagenum="186"></pb>antecedente aſſumitur ſola AC, ter CD, bis DE, bis AB, <lb></lb>bis BE, quater CB, & quater BD. Duæ verò AB vnà <lb></lb>cum ſola AC, & ſola. </s> <s id="N17299">CB, ex quatuor vicibus, quibus ip<lb></lb>ſa CB ſumitur, ſunt æquales tribus AB. tres autem CB, <lb></lb>quæ relictæ ſunt, vnà cum tribus CD, & tribus BD <lb></lb>ex quatuor vicibus, quibus ipſa BD ſumitur, ſunt æ<lb></lb>quales ſex CB. ſola verò BD, quæ relicta fuit, vnà <lb></lb>cum duabus DE, & duabus BE, eſt æqualis tribus <lb></lb>BD. linea nimirum AC cum tripla ipſius CD, & <lb></lb>dupla ipſius DE, & dupla vtriuſ〈que〉 ſimul AB BE, <lb></lb>& quadrupla vtriuſ〈que〉 ſimul CB BD, æqualis erit tri<lb></lb>plæ ipſius AB, cum ſextupla ipſius CB, & tripla ip<lb></lb>ſius BD. <emph type="italics"></emph>Tota igitur OB ad EB eandem habet proportio<lb></lb>nem, quam linea æqualis triplæ ipſius AB cum ſextupla ip<lb></lb>ſius CB & tripla ipſius BD ad duplam vtriuſ〈que〉 ſimul <lb></lb>AB BE cum quadrupla vtriuſ〈que〉 ſimul CB BD. & <lb></lb>quoniam<emph.end type="italics"></emph.end> initio oſtenſum fuit lineas AC CD DE in eadem <lb></lb>eſſe proportione, vt ſunt quatuor lineæ continuè pro<lb></lb>portionales AB BC BD BE; erunt tres AC CD <lb></lb>DE, & tres AB BC BD, & tres BC BD BE <lb></lb> <arrow.to.target n="marg361"></arrow.to.target> in eadem proportione. </s> <s id="N172C9">conuertendo igitur in eadem quo<lb></lb>〈que〉 erunt proportione. </s> <s id="N172CD">quare tres <emph type="italics"></emph>ED DC CA,<emph.end type="italics"></emph.end> & <lb></lb>tres BE BD BC, & tres BD BC BA <emph type="italics"></emph>in eadem ſunt proportione.<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg362"></arrow.to.target> Quoniam autem BE BD BC ita ſe habent, vt BD BC BA; <lb></lb>vtra〈que〉 ſimul BE BD advtram〈que〉 ſimul BD BC, & <lb></lb>vtra〈que〉 ſimul BD BC ad vtram〈que〉 ſimul BC BA <lb></lb>ita ſe habebunt, vt BE BD BC. hæ verò <emph type="italics"></emph>B<emph.end type="italics"></emph.end>E <emph type="italics"></emph>B<emph.end type="italics"></emph.end>D <lb></lb>BC ſunt, vt ED DC CA. ergo <emph type="italics"></emph>& vtra〈que〉 ſimul <lb></lb>vna〈que〉〈que〉 ipſarum EB BD, DB BC, CB BA<emph.end type="italics"></emph.end>, ita ſe <lb></lb> <arrow.to.target n="marg363"></arrow.to.target> habebunt, vt ED DC CA. quare <emph type="italics"></emph>erit &<emph.end type="italics"></emph.end> antecedens <lb></lb><emph type="italics"></emph>ED<emph.end type="italics"></emph.end> ad ſuas conſe〈que〉ntes DC CA ſimul ſumptas, <lb></lb>hoc eſt <emph type="italics"></emph>ad DA, vt<emph.end type="italics"></emph.end> antecedens <emph type="italics"></emph>vtra〈que〉 ſimul EB BD<emph.end type="italics"></emph.end><lb></lb>ad ſuas conſe〈que〉ntes, nempè <emph type="italics"></emph>ad <expan abbr="vtrā〈que〉">vtran〈que〉</expan> ſimul DB BC<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg364"></arrow.to.target> <emph type="italics"></emph>cum vtra〈que〉 ſimul CB BA. & componendo EA ad AD, <lb></lb>vt vtra〈que〉 ſimul EB BD cum vtra〈que〉 ſimul AB BC, <lb></lb>& vtra〈que〉 ſimul CB BD<emph.end type="italics"></emph.end> ad vtram〈que〉 ſimul BD BC <pb xlink:href="077/01/191.jpg" pagenum="187"></pb>cum vtra<gap></gap>ue ſimul CB BA. In hoc autem antecedenti ſemel <lb></lb>ſumitur EB, & ſemel AB, bis BD, & bis BC. in conſe〈que〉ntive <lb></lb>rò ſumitur <gap></gap>ola BD, ſolaquè BA, & bis BC. Proportio igitur <lb></lb>ipſarum EA AD eſt eadem, <emph type="italics"></emph>quæ est vtra〈que〉 ſimul EB BA cum du<lb></lb>pla vtriuſ〈que〉 ſimul DB BC ad vtram〈que〉 ſimul BD BA cum dupla <lb></lb>ipſius BC. Quare & dupla ad duplam eandem habebit <expan abbr="proportionẽ">proportionem</expan> hoc <lb></lb>est, vt EA ad AD, ita dupla vtriuſ〈que〉 ſimul EB BA cum quadru<lb></lb>pla vtriuſ〈que〉 ſimul CB BD ad duplam vtriuſ〈que〉 ſimul AB BD cum<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="fig84"></arrow.to.target><lb></lb><emph type="italics"></emph>quadrupla ipſius CB. Quapropter EA adtres quintas ipſius AD eſt, vt <lb></lb>compoſita ex dupla vtriuſ〈que〉 ſimul AB BE, & qua-<emph.end type="italics"></emph.end> <arrow.to.target n="marg365"></arrow.to.target><lb></lb><emph type="italics"></emph>drupla utrivs〈que〉 ſimul CB BD ad tres quintas lineæ com<lb></lb>poſitæ ex dupla vtriuſ〈que〉 ſimul AB BD, & quadruplaipſius CB. Ve<lb></lb>rùm<emph.end type="italics"></emph.end> quia initio aſſumptum fuitita eſſe BE ad EA, vt FG ad <lb></lb>tres quintas ipſius AD, erit conuertendo EA ad EB, vt <arrow.to.target n="marg366"></arrow.to.target><lb></lb>tres quintæ ipſius AD ad FG; permutandoquè <emph type="italics"></emph>vt EA ad <lb></lb>tres quintasipſius AD, ſic eſt EB ad FG, vtigitur EB ad FG, <lb></lb>ſic dupla vtriuſ〈que〉 ſimul AB BE cum quadrupla vtriuſ〈que〉<emph.end type="italics"></emph.end> <pb xlink:href="077/01/192.jpg" pagenum="188"></pb><emph type="italics"></emph>ſimul DB BC ad tres quintas lineæ compoſitæ ex dupla vtriuſ〈que〉 ſi<lb></lb>mul AB BD cum quadrupla ipſius CB. osten ſum eſt aut<gap></gap> OB ad EB <lb></lb>ita eſſe, vt<emph.end type="italics"></emph.end> tripla ipſius AB cum ſextupla ipſius CB, & tripla i<lb></lb>pſius BD ad duplam vtriuſ〈que〉 ſimul AB BE cum quadrupla <lb></lb>vtriuſ〈que〉 ſimul CB BD. At in hoc antecedente ter aſſumpta <lb></lb>eſt AB, terquè BD, & ſexies CB. erit ita〈que〉 in primis magni<lb></lb>tudinibus antecedens OB ad conſe〈que〉ns EB, vt in ſecundis <lb></lb>magnitudinibus an recedens <emph type="italics"></emph>tripla<emph.end type="italics"></emph.end> ſcilicet <emph type="italics"></emph>vtriuſ〈que〉 ſimul AB <lb></lb>BD cum ſextupla ipſius CB ad<emph.end type="italics"></emph.end> conſe〈que〉ns nempè <emph type="italics"></emph>duplam v<lb></lb>triuſ〈que〉 ſimul AB BE, & quadruplam vtriuſ〈que〉 ſimul CB BD.<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="fig85"></arrow.to.target><lb></lb>in primis verò magnitudinibus eſt conſe〈que〉ns EB ad aliud <lb></lb>quippiam FG, ut in ſecundis magnitudinibus conſe〈que〉ns, <lb></lb>hoc eſt dupla vtriuſ〈que〉 ſimul AB BE cum quadrupla vtriuſ<lb></lb>〈que〉 ſimul DB BC ad aliud quippiam, nempè ad tres quintas <lb></lb>lineæ <expan abbr="cõpoſitę">compoſitę</expan> ex dupla vtri^{9} <expan abbr="q́">〈que〉</expan>; ſimul AB BD <expan abbr="cũ">cum</expan> quadrupla i<lb></lb> <arrow.to.target n="marg367"></arrow.to.target> pſi^{9} CB. <emph type="italics"></emph>Ex æquali igitur eſt, ut OB ad FG, ita linea compoſita ex tripla<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg368"></arrow.to.target> <emph type="italics"></emph><expan abbr="utriusq́">utrius〈que〉</expan> ſimul AB BD, et ſextuplaipſi^{9} CB ad tres <expan abbr="quĩtas">quintas</expan> lineæ <expan abbr="cõpoſi">compoſi</expan> <lb></lb>tæ ex dupla utri^{9} <expan abbr="q́">〈que〉</expan>; ſimul AB BD, & quadrupla ipſius CB. at uerò<emph.end type="italics"></emph.end> tri <lb></lb>pla ipſius AB ad <expan abbr="duplã">duplam</expan> <expan abbr="eiuſdẽ">eiuſdem</expan> AB eſt, vt tria ad duo. </s> <s id="N17400">ſimiliter <lb></lb>tripla ipſius BD ad duplam eiuſdem BD eſt, vt tria ad duo. <pb xlink:href="077/01/193.jpg" pagenum="189"></pb>pariquè ratione ſextupla ipſius CB ad quadruplam ciuſdem, <lb></lb>CB ita ſe habet, vt ſex ad quatuor, hoceſt tria ad duo, & om<lb></lb>nesad omnes, hoc eſt <emph type="italics"></emph>compoſita ex tripla vtriuſ〈que〉 ſimul AB BD, <lb></lb>et ſextupla ipſius CB ad compoſitam ex dupla vtriuſ〈que〉 ſimul AB BD, <lb></lb>& quadrupla ipſius CB proportionem habet, quam tria ad duo.<emph.end type="italics"></emph.end> vt exem <arrow.to.target n="marg369"></arrow.to.target><lb></lb>pli gratia quindecim ad decem, <emph type="italics"></emph>ſed<emph.end type="italics"></emph.end> eadem compoſita ex tri<lb></lb>pla vtriuſ〈que〉 ſimul AB BD, & ſextupla ipſius CB <emph type="italics"></emph>ad tres quin<lb></lb>tas eiuſdem<emph.end type="italics"></emph.end> compofitæ ex dupla vtriuſ〈que〉 ſimul AB BD, & qua <arrow.to.target n="marg370"></arrow.to.target><lb></lb>drupla ipſius, CB, quæ poſita eſt decem, <emph type="italics"></emph>proportionem habet, quam <lb></lb>quin〈que〉 ad duo.<emph.end type="italics"></emph.end> hoc eſt ut quindecim ad ſex, tres enim quintæ <lb></lb>ipſius decem ſunt ſex. </s> <s id="N1743A">at verò proportio, quam habet linea <expan abbr="cõ">com</expan> <lb></lb>poſita ex tripla vtriuſ〈que〉 ſimul AB BD, & ſextupla ipſius CB <lb></lb>ad tres quintas lineæ compoſitę ex dupla vtriuſ〈que〉 ſimul AB <lb></lb>BD cum quadrupla ipſius CB, eſt æqualis ei, quam habet OB <lb></lb>ad FG. ergo erit OB ad FG, vtquin〈que〉 ad duo. <emph type="italics"></emph><expan abbr="Demonstratū">Demonstratum</expan> <lb></lb>autem eſt, & AO ad GH proportionem habere, quam quin〈que〉 ad duo; <lb></lb>totaigitur BA ad totam FH proportionem habet, quam quin〈que〉 ad duo.<emph.end type="italics"></emph.end> <arrow.to.target n="marg371"></arrow.to.target><lb></lb><emph type="italics"></emph>ſiautem hoc, eſt quidem FH duæ quintæ ipſius AB. Quod oportebat <lb></lb>demonſtrare.<emph.end type="italics"></emph.end></s> </p> <p id="N17460" type="margin"> <s id="N17462"><margin.target id="marg339"></margin.target>1.<emph type="italics"></emph><expan abbr="lẽma">lemma</expan> hu<lb></lb>ius.<emph.end type="italics"></emph.end></s> </p> <p id="N17470" type="margin"> <s id="N17472"><margin.target id="marg340"></margin.target>2. <emph type="italics"></emph>lemma <lb></lb>buius.<emph.end type="italics"></emph.end></s> </p> <p id="N1747D" type="margin"> <s id="N1747F"><margin.target id="marg341"></margin.target>1.<emph type="italics"></emph><expan abbr="lẽma">lemma</expan> hu<lb></lb>ius.<emph.end type="italics"></emph.end></s> </p> <p id="N1748D" type="margin"> <s id="N1748F"><margin.target id="marg342"></margin.target>2. <emph type="italics"></emph>lemma <lb></lb>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N1749A" type="margin"> <s id="N1749C"><margin.target id="marg343"></margin.target>11. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N174A5" type="margin"> <s id="N174A7"><margin.target id="marg344"></margin.target>12. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N174B0" type="margin"> <s id="N174B2"><margin.target id="marg345"></margin.target>11, <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N174BB" type="margin"> <s id="N174BD"><margin.target id="marg346"></margin.target>8.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N174C6" type="margin"> <s id="N174C8"><margin.target id="marg347"></margin.target><emph type="italics"></emph>ex<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph><expan abbr="quĩti">quinti</expan><emph.end type="italics"></emph.end></s> </p> <p id="N174D8" type="margin"> <s id="N174DA"><margin.target id="marg348"></margin.target><emph type="italics"></emph>co.<emph.end type="italics"></emph.end>4.<emph type="italics"></emph><expan abbr="quĩti">quinti</expan><emph.end type="italics"></emph.end></s> </p> <p id="N174EA" type="margin"> <s id="N174EC"><margin.target id="marg349"></margin.target>18, <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N174F5" type="margin"> <s id="N174F7"><margin.target id="marg350"></margin.target><emph type="italics"></emph>co.<emph.end type="italics"></emph.end>4.<emph type="italics"></emph><expan abbr="quĩti">quinti</expan>.<emph.end type="italics"></emph.end></s> </p> <p id="N17508" type="margin"> <s id="N1750A"><margin.target id="marg351"></margin.target>23. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N17513" type="margin"> <s id="N17515"><margin.target id="marg352"></margin.target>12, <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N1751E" type="margin"> <s id="N17520"><margin.target id="marg353"></margin.target><emph type="italics"></emph>co.<emph.end type="italics"></emph.end>4.<emph type="italics"></emph><expan abbr="quĩti">quinti</expan><emph.end type="italics"></emph.end></s> </p> <p id="N17530" type="margin"> <s id="N17532"><margin.target id="marg354"></margin.target>23.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N1753B" type="margin"> <s id="N1753D"><margin.target id="marg355"></margin.target>3.<emph type="italics"></emph><expan abbr="lẽma">lemma</expan> hu<lb></lb>ius.<emph.end type="italics"></emph.end></s> </p> <p id="N1754B" type="margin"> <s id="N1754D"><margin.target id="marg356"></margin.target>17. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N17556" type="margin"> <s id="N17558"><margin.target id="marg357"></margin.target>A</s> </p> <p id="N1755C" type="margin"> <s id="N1755E"><margin.target id="marg358"></margin.target>12.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N17567" type="margin"> <s id="N17569"><margin.target id="marg359"></margin.target>23.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N17572" type="margin"> <s id="N17574"><margin.target id="marg360"></margin.target>18.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N1757D" type="margin"> <s id="N1757F"><margin.target id="marg361"></margin.target><emph type="italics"></emph>cor.4.quĩ <lb></lb>ti.<emph.end type="italics"></emph.end></s> </p> <p id="N17589" type="margin"> <s id="N1758B"><margin.target id="marg362"></margin.target>4.<emph type="italics"></emph>lema hu<lb></lb>ius.<emph.end type="italics"></emph.end></s> </p> <p id="N17596" type="margin"> <s id="N17598"><margin.target id="marg363"></margin.target><emph type="italics"></emph>cor.2.lem. <lb></lb>in<emph.end type="italics"></emph.end> 13. <emph type="italics"></emph>pri<lb></lb>mi huius.<emph.end type="italics"></emph.end></s> </p> <p id="N175AA" type="margin"> <s id="N175AC"><margin.target id="marg364"></margin.target>18.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N175B5" type="margin"> <s id="N175B7"><margin.target id="marg365"></margin.target>B</s> </p> <p id="N175BB" type="margin"> <s id="N175BD"><margin.target id="marg366"></margin.target><emph type="italics"></emph>co.<emph.end type="italics"></emph.end>4.<emph type="italics"></emph><expan abbr="quĩti">quinti</expan>.<emph.end type="italics"></emph.end><lb></lb>16,<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end><lb></lb>11. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N175DC" type="margin"> <s id="N175DE"><margin.target id="marg367"></margin.target>22.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N175E7" type="margin"> <s id="N175E9"><margin.target id="marg368"></margin.target>C</s> </p> <p id="N175ED" type="margin"> <s id="N175EF"><margin.target id="marg369"></margin.target>D</s> </p> <p id="N175F3" type="margin"> <s id="N175F5"><margin.target id="marg370"></margin.target>5.<emph type="italics"></emph>lemma <lb></lb>huius.<emph.end type="italics"></emph.end></s> </p> <p id="N17600" type="margin"> <s id="N17602"><margin.target id="marg371"></margin.target>12.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.193.1.jpg" xlink:href="077/01/193/1.jpg"></figure> <figure id="id.077.01.193.2.jpg" xlink:href="077/01/193/2.jpg"></figure> <figure id="id.077.01.193.3.jpg" xlink:href="077/01/193/3.jpg"></figure> <figure id="id.077.01.193.4.jpg" xlink:href="077/01/193/4.jpg"></figure> <figure id="id.077.01.193.5.jpg" xlink:href="077/01/193/5.jpg"></figure> <p id="N1761F" type="head"> <s id="N17621">SCHOLIVM.</s> </p> <p id="N17623" type="main"> <s id="N17625">Græcus codex poſt ea verba, <emph type="italics"></emph>vt DE ad EB, ita AC ad CB,<emph.end type="italics"></emph.end> <arrow.to.target n="marg372"></arrow.to.target><lb></lb>non habet, <emph type="italics"></emph>& CD ad DB,<emph.end type="italics"></emph.end> quæ ob ea, quæ ſequuntur, omninò <lb></lb>neceſſaria videntur. </s> <s id="N1763A">ideo poſt gręca verba, <foreign lang="grc">ἔσιδὲκα<gap></gap> ὡς δε ω̄<10>ὸς εβ, <lb></lb>οὔτως ἄτε αγ ω̄<10>ὸς, γβ</foreign> deſiderarividentur. <foreign lang="grc">κα<gap></gap> ἇ γδ ω̄<10>ός δβ. </foreign></s> </p> <p id="N17649" type="margin"> <s id="N1764B"><margin.target id="marg372"></margin.target><emph type="italics"></emph>A<emph.end type="italics"></emph.end></s> </p> <p id="N17653" type="main"> <s id="N17655">Vbiautem ſuntverba, <emph type="italics"></emph>vt <expan abbr="cõpoſita">compoſita</expan> ex dupla vtriuſ〈que〉 ſimul,<emph.end type="italics"></emph.end> Græ <arrow.to.target n="marg373"></arrow.to.target><lb></lb>cus codex tantùm habet, <foreign lang="grc">οὒτως ἀ συγκειμένα ἒχτε τᾶς συυαμφοτε<10>ου</foreign>. <lb></lb>In quibus deſideratur illa particula, <emph type="italics"></emph>dupla,<emph.end type="italics"></emph.end> ideo corrigendus eſt <lb></lb>hoc modo, <foreign lang="grc">οὔτως ὰ συγκειμένα ἒκτε τᾶς β συυαμφοτέ<10>ου</foreign>, &c. </s> </p> <pb xlink:href="077/01/194.jpg" pagenum="190"></pb> <p id="N1767B" type="margin"> <s id="N1767D"><margin.target id="marg373"></margin.target>B</s> </p> <p id="N17681" type="main"> <s id="N17683">Præterea cùm inquit, <emph type="italics"></emph>ex æqualiigitur eſt vt OB ad FG,<emph.end type="italics"></emph.end> Græ<lb></lb> <arrow.to.target n="marg374"></arrow.to.target> cus non habet, <emph type="italics"></emph>ad FG,<emph.end type="italics"></emph.end> idcirco poſt ea verba <foreign lang="grc">καὶ δὶ<gap></gap>σου ἄ<10>α ἐσιν ξὁς <lb></lb>α<gap></gap> οβ</foreign> addenda ſunt <foreign lang="grc">ω̄<10>ὸς ζκ. </foreign></s> </p> <p id="N176A4" type="margin"> <s id="N176A6"><margin.target id="marg374"></margin.target>C</s> </p> <p id="N176AA" type="main"> <s id="N176AC">Similiter quando in quit <emph type="italics"></emph>ad compoſitam ex dupla vtriuſ〈que〉 ſimul<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg375"></arrow.to.target> <emph type="italics"></emph>AB BD, & quadrupla ipſius CB,<emph.end type="italics"></emph.end> græca verba ſunt <foreign lang="grc">ω̄<10>ο̂ς μὲν τὰν συγ<lb></lb>κειμ<gap></gap>ναν ἔκτε τᾶς β συναμφοτὲ<10>ου τᾶς αβ βδ τᾶς Γβ</foreign>, in quib^{9} ſimiliter deli<lb></lb>deratur, <emph type="italics"></emph>& quadrupla.<emph.end type="italics"></emph.end> quare ita corrigendus videtur. <foreign lang="grc">ω̄<10>ὸς μὲν τάν <lb></lb>συγκειμὲναν ἔ κ τε τας β συναμφοτέ<10>ου τᾶς αβ βδ, καὶ δ τἄς Γβ</foreign>, </s> </p> <p id="N176D5" type="margin"> <s id="N176D7"><margin.target id="marg375"></margin.target>D</s> </p> <p id="N176DB" type="main"> <s id="N176DD">Poſtremum theorema, & ſi non habeat <expan abbr="tãtam">tantam</expan> <expan abbr="obſcuritatẽ">obſcuritatem</expan>, <lb></lb>veluti pręcedens, non eſt tamen ſine aliqua obſcuritate, ob cu<lb></lb>ius intelligentiam hanc priùs propo ſitionem oſtendemus. </s> </p> <p id="N176EB" type="head"> <s id="N176ED">PROPOSITIO.</s> </p> <p id="N176EF" type="main"> <s id="N176F1">Si duæ fuerint rectæ lineę in para bolc ad diametrum ordi <lb></lb>natim applicatæ, erit maior parabole ad <expan abbr="minorẽ">minorem</expan>, vt cubus ex <lb></lb>dimidia lineę maioris ad cubum ex dimidia minoris. </s> </p> <figure id="id.077.01.194.1.jpg" xlink:href="077/01/194/1.jpg"></figure> <p id="N176FE" type="main"> <s id="N17700">In parabole ABC, cuius diameter BF, duæ ſint rectæ lineæ <lb></lb>ad diametrum applicatæ AC DE. Dico parabolen ABC ad <lb></lb>parabolen DBE eandem habere proportionem, quam cub^{9} <lb></lb>ex AF ad cubum ex DG. lungantur AB BC DB BE; ſecet- <pb xlink:href="077/01/195.jpg" pagenum="191"></pb>què AB ipſam DG in H. Quoniam enim parabole ABC <arrow.to.target n="marg376"></arrow.to.target><lb></lb>ſeſquitertia eſt trianguli ABC, itidemquè parabole DBE <lb></lb>trianguli DBE ſeſquitertia exiſtit, erit parabole ABC ad trian <lb></lb>gulum ABC, vt parabole DBE ad triangulum DBE. & <arrow.to.target n="marg377"></arrow.to.target> per<lb></lb>mutando parabole ABC ad parabolen DBE, vt triangulum <lb></lb>ABC ad triangulum DBE. Quoniam autem AC ordina<lb></lb>tim eſt applicata, vnde AF ipſi FC eſt æqualis, ac per conſe<lb></lb>〈que〉ns AF eſt ipſius AC dimidia. </s> <s id="N17721">erit triangulum ABF dimi<lb></lb>dium trianguli ABC. cùm vtraquè ſub eadem ſint altitudine. <arrow.to.target n="marg378"></arrow.to.target><lb></lb>eademquè ratione triangulum DBG trianguli DBE dimi<lb></lb>dium exiſtit. </s> <s id="N1772C">quare vt triangulum ABF ad triangulum <lb></lb>DBG, ita eſt triangulum ABC ad DBE triangulum, ac pro<lb></lb>pterea triangulum ABF ad DBG triangulum eſt, vt parabo<lb></lb>le ABC ad parabolen DBE. Cùm autem ſit HG æquidiſtans <lb></lb>ipſi AF, ſiquidem ſunt ordinatim applicatæ, ob <expan abbr="triangulorũ">triangulorum</expan> <arrow.to.target n="marg379"></arrow.to.target><lb></lb>ſimilitudinem ABF HBG, ita erit FB ad BG, vt AF ad HG <lb></lb>vt autem FB ad BG, ita quadratum ex AF ad quadratum ex <arrow.to.target n="marg380"></arrow.to.target><lb></lb>DG, erit igitur quadratum ex AF ad quadratum ex DG, vt AF <lb></lb>ad HG. quare lineę AF DG HG ſunt proportionales. </s> <s id="N17748">Pro<lb></lb>ducatur FB, ducaturquè à puncto D ipſi AB æquidiſtans <lb></lb>DK, erit vtiquè triangulorum ABF DKG anguli ABF <lb></lb>DHG æquales, & angulus AFB angulo DGK eſt æqualis, erit <lb></lb>igitur, & reliquus BAF reliquo KDG æqualis, ac propterea <lb></lb>triangulum ABF eſt triangulo DKG ſimile. </s> <s id="N17754">quare triangu<lb></lb>lum ABF ad triangulum DKG eam habet proportionem, <lb></lb>quàm AF ad DG duplicatam, hoc eſt quàm AF ad HG, quę <lb></lb>eſt ea, quàm habet FB ad BG. atqui triangulum ABF ad <lb></lb>DKG eam quo〈que〉 habet proportionem, quam FB ad GK <lb></lb>duplicatam. </s> <s id="N17760">tres igitur lineę FB GK GB ſunt proportiona<lb></lb>les. </s> <s id="N17764">ex quibus ſequiturita eſſe FB ad GK, vt AF ad DG; & <lb></lb>GK ad GB, vt DG ad GH. ſed quoniam triangulum GDK <lb></lb>ad GDB (cùm ſint ſub eadem altitudine) ita eſt, vt KG ad <arrow.to.target n="marg381"></arrow.to.target><lb></lb>BG, ſi igitur fiat HG ad L, vt KG ad BG, erit triangulum <lb></lb>GDK ad triangulum GDB, vt HG ad L. Cùm autem ſit <expan abbr="triã">triam</expan> <lb></lb>gulum ABF ad DKG, vt AF ad HG, eſtquè <expan abbr="triangulũ">triangulum</expan> DKG <lb></lb>ad DBG, vt HG ad L, erit ex ęquali triangulum ABF ad <lb></lb>triangulum DBG, vt AF ad L. ac propterea parabole ABC <pb xlink:href="077/01/196.jpg" pagenum="192"></pb>ad parabolen DBE eam habet proportionem, quam linea <lb></lb>AF ad lineam L. Quoniam autem ita eſt KG ad GB, vt <lb></lb>HG ad L, & vt eadem KG ad GB, ita eſt DG ad GH. vt <lb></lb>verò DG ad GH, ita eſt AF ad DG; crunt quatuor lineæ AF <lb></lb>DG HG L in continua proportione. </s> <s id="N1778B">& quoniam cubi in tri<lb></lb>pla ſunt proportione laterum, erit cubus ex AF ad cubum ex <lb></lb>DG, vt AF ad L. cubus ergo ex AF ad cubum ex DG eam <lb></lb>habet proportionem, quam parabole ABC ad parabolen <lb></lb>DBE. quod demonſtrare oportebat. </s> </p> <p id="N17795" type="margin"> <s id="N17797"><margin.target id="marg376"></margin.target>17.34. A<emph type="italics"></emph>r <lb></lb>ch.de qua. <lb></lb>par.<emph.end type="italics"></emph.end></s> </p> <p id="N177A4" type="margin"> <s id="N177A6"><margin.target id="marg377"></margin.target>16. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N177AF" type="margin"> <s id="N177B1"><margin.target id="marg378"></margin.target><emph type="italics"></emph>ex prima <lb></lb>ſextt.<emph.end type="italics"></emph.end></s> </p> <p id="N177BB" type="margin"> <s id="N177BD"><margin.target id="marg379"></margin.target><emph type="italics"></emph>ex<emph.end type="italics"></emph.end> 4.<emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N177CB" type="margin"> <s id="N177CD"><margin.target id="marg380"></margin.target>20. <emph type="italics"></emph>primi <lb></lb>conicorum <lb></lb>Apoll. </s> <s id="N177D8">& <lb></lb>ex<emph.end type="italics"></emph.end>3. A<emph type="italics"></emph>rch. <lb></lb>de quad. <lb></lb>parab. <lb></lb>ex cor.<emph.end type="italics"></emph.end> 20. <lb></lb><emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N177F1" type="margin"> <s id="N177F3"><margin.target id="marg381"></margin.target>1.<emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end><lb></lb>11.<emph type="italics"></emph>quintl.<emph.end type="italics"></emph.end></s> </p> <p id="N17803" type="main"> <s id="N17805">Oportet autem banc quoquè <expan abbr="propoſitionẽ">propoſitionem</expan> nobis eſſe cogni <lb></lb>tam, nem pè quòd ſolida parallelepipeda in eadem baſi conſti <lb></lb>tuta eam inter ſe proportionem habent, quam ipſarum alti<lb></lb>tudines. </s> </p> <p id="N17811" type="main"> <s id="N17813">Hoc quidem à Federico Commandino in eius libro de cen<lb></lb>tro grauitatis ſolidorum propoſitione decimanona demon<lb></lb>ſtratum fuit. </s> </p> <p id="N17819" type="head"> <s id="N1781B">PROPOSITIO. X.</s> </p> <p id="N1781D" type="main"> <s id="N1781F">Omnis fruſti à rectanguli coni portione abſciſſi <lb></lb>centrum grauitatis eſt in recta linea, quæ fruſti dia<lb></lb>meter exiſtit, ita poſitum, vt diuiſa linea in quin<lb></lb>〈que〉 partes æquales, ſit in quinta parte media; ita <lb></lb>vt ipſius portio propinquior minoribaſi fruſti ad <lb></lb>reliquam portionem eandem habeat proportio<lb></lb>nem, quam habet ſolidum baſim habens quadra<lb></lb>tumex dimidia maioris baſis fruſti, altitudinem au<lb></lb>tem lineam æqualem vtri〈que〉 ſimul duplæ mino<lb></lb>ris baſis, & maiori ad ſolidum baſim habens qua<lb></lb>dratum ex dimidia minoris baſis fruſti, <expan abbr="altitudinẽ">altitudinem</expan> <lb></lb>autem lineam æqualem vtri〈que〉 duplæ maioris, & <lb></lb>minori. </s> </p> <pb xlink:href="077/01/197.jpg" pagenum="193"></pb> <figure id="id.077.01.197.1.jpg" xlink:href="077/01/197/1.jpg"></figure> <p id="N17843" type="main"> <s id="N17845"><emph type="italics"></emph>Sit in rectanguli coni portione<emph.end type="italics"></emph.end> ABC <emph type="italics"></emph>duæ rectæ lineæ AC DE<emph.end type="italics"></emph.end><lb></lb>æquidiſtantes. <emph type="italics"></emph>diameter verò portionis ABC ſit BF.<emph.end type="italics"></emph.end> Intelli<lb></lb>gaturquè fruſtum ADEC à portione ABC abſciſſum. </s> <s id="N1785B">om<lb></lb>nes vti〈que〉 lineæ ipſis AC DE æquidiſtantes in fruſto AD <lb></lb>EC ductæ, erunt à linea GF bifartam diuiſæ, ex quo <emph type="italics"></emph>pa<lb></lb>tet quidem & ipſius ADEC diametrum eſſe GF, lineasquè AC <lb></lb>DE lineæ portionem in B contingenti æquidistantes eſſe. </s> <s id="N17868">Recta<emph.end type="italics"></emph.end> <arrow.to.target n="marg382"></arrow.to.target><lb></lb><emph type="italics"></emph>verò linea FG in quin〈que〉 partes æquales diuiſa, quinta pars me<lb></lb>dia ſit HK. at〈que〉<emph.end type="italics"></emph.end> diuidatur HK in I, ita vt <emph type="italics"></emph>HI ad <lb></lb>IK eandem habeat proportionem, quam habet ſolidum baſim habens <lb></lb>quadratum ex AF, altitudinem verò lineam æqualem vtriſ〈que〉 <lb></lb>ſimul duplæ ipſius DG, & ipſi AF, ad ſolidum, quod <lb></lb>baſim habeat quadratum ex DG, altitudinem autem lineam æqua-<emph.end type="italics"></emph.end> <pb xlink:href="077/01/198.jpg" pagenum="194"></pb> <arrow.to.target n="fig86"></arrow.to.target><lb></lb><emph type="italics"></emph>lem vtriſ〈que〉 duplæ ipſius AF, & ipſi DG. ostenden<lb></lb>dum est frusti ADEC centrum grauitatis eſſe punctum 1.<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg383"></arrow.to.target> <emph type="italics"></emph>ſit quidem ipſi FB æqualis MN, ipſi verò GB æqualis NO. <lb></lb>ſumaturquè ipſarum MN NO media proportionalis NX. <lb></lb>quarta verò proportionalis TN.<emph.end type="italics"></emph.end> lineæ nimirum MN NX <lb></lb>NO NT in continua erunt proportione. <emph type="italics"></emph>& vt TM <lb></lb>ad TN, ita<emph.end type="italics"></emph.end> fiat <emph type="italics"></emph>FH ad quandam lineam à puncto I, vt <gap></gap>R, vbi<lb></lb>cun〈que〉 perueniat alterum punctum<emph.end type="italics"></emph.end> R. <emph type="italics"></emph>nihil enim refert, ſiue inter <lb></lb>FG, ſiue inter GB cadat. </s> <s id="N178BF">& quoniam in portione rectanguli coni<emph.end type="italics"></emph.end><lb></lb>ABC <emph type="italics"></emph>diameter portionis est FB; at verò BF, vel prin<lb></lb>cipalis est diameter portionis, vel ducta diametro æquidistans. <lb></lb>lineæ verò AF DG ad ipſam ordinatim ſunt ap<lb></lb>plicatæ, cùm ſint æquidistantes contingenti portionem<emph.end type="italics"></emph.end> <pb xlink:href="077/01/199.jpg" pagenum="195"></pb><emph type="italics"></emph>in puncto B. ſi autem hoc, est vt AF ad DG potentia,<emph.end type="italics"></emph.end> <arrow.to.target n="marg384"></arrow.to.target><lb></lb><emph type="italics"></emph>ſic FB ad BG longitudine, hoc est MN ad NO. <lb></lb>vt autem MN ad NO longitudine, itaest MN ad Nx potentia.<emph.end type="italics"></emph.end><lb></lb>quandoquidem treslineæ MN NX NO ſunt proportio<lb></lb>nales. <emph type="italics"></emph>vt igitur AF ad DG potentia, ita est MN ad N X<emph.end type="italics"></emph.end> <arrow.to.target n="marg385"></arrow.to.target><lb></lb><emph type="italics"></emph>potentia. </s> <s id="N178F6">quare, & longitudine in eadem ſunt proportione<emph.end type="italics"></emph.end>; vt ſcili <lb></lb>cet AF ad DG, ita MN ad NX. <emph type="italics"></emph>ſieist ita〈que〉 cubus ex AF<emph.end type="italics"></emph.end> <arrow.to.target n="marg386"></arrow.to.target><lb></lb><emph type="italics"></emph>ad cubum ex DG, ita cubus ex MN ad cubum ex NX. Verùm<emph.end type="italics"></emph.end> <arrow.to.target n="marg387"></arrow.to.target><lb></lb><emph type="italics"></emph>vt cubus ex AF adcubum ex DG, ſic portio ABC ad portio<lb></lb>nem DBE.<emph.end type="italics"></emph.end> vtigitur cubus ex MN ad cubum ex NX, ita <lb></lb>portio ABC ad portionem DBE. <emph type="italics"></emph>ſicut autem cubus ex MN <lb></lb>ad culum ex Nx, ita MN ad NT.<emph.end type="italics"></emph.end> cùm ſint quatuor lineæ <lb></lb>MN NX NO NT in continua proportione. </s> <s id="N17925">ac propterea <lb></lb>eritportio ABC ad portionem DBE, vt MN ad NT. <lb></lb><emph type="italics"></emph>Quare & diuidendo frustum ADEC ad portionem DBE eſt, vt<emph.end type="italics"></emph.end> <arrow.to.target n="marg388"></arrow.to.target><lb></lb><emph type="italics"></emph>MT ad NT.<emph.end type="italics"></emph.end> Quia vero, vt factum fuit, ità eſt MT ad TN, <lb></lb>vt FH ad IR, eſt verò FH ipſius FG tresquintæ, erit fru<lb></lb>ſtum ADEC ad portionem DBE, vt FH ad IR <emph type="italics"></emph>hoc est <lb></lb>tres quintæ ipſius GF ad IR. & quoniam ſolidum baſim habens qua<lb></lb>dratum ex AF, altitudinem verò lineam compoſitam ex dupla ipſius <lb></lb>DG, & ipſa AF, ad cubum ex AF proportionem habet,<emph.end type="italics"></emph.end> quam ſo <lb></lb>lidi altitudo ad altitudinem cubi, ſiquidem ſunt in eadem ba <lb></lb>ſi. </s> <s id="N1794E">tàm emm ſolidum, quàm cubus baſim habet quadratum <lb></lb>ex AF. idcirco ſolidum baſim habens quadratum ex AF, <lb></lb>altitudinem verò lineam compoſitam ex dupla ipſius DG, & <lb></lb>ipſa AF ad cubum ex AF eam proportio nem habebit, <emph type="italics"></emph>quam<emph.end type="italics"></emph.end><lb></lb>ſolidi altitudo, <emph type="italics"></emph>dupla,<emph.end type="italics"></emph.end> ſcilicet <emph type="italics"></emph>ipſius DG cumlinea AF<emph.end type="italics"></emph.end> ad alci<lb></lb>tudinem cubi, hoc eſt <emph type="italics"></emph>ad FA.<emph.end type="italics"></emph.end> Atverò quoniam oſtenſum eſt <lb></lb>ita eſſe AF ad DG, vt MN ad NX, eritconuertendo DG <lb></lb>ad AF, vt NX ad MN, & antecedentium dupla, hoc eſt du<lb></lb>pla ipſius DG ad AF, vt dupla ipſius NX ad MN. & com<lb></lb>ponendo dupla ipſius DG cum AF ad AF, vt dupla ipſius <arrow.to.target n="marg389"></arrow.to.target><lb></lb>NX cum MN ad MN. <emph type="italics"></emph>Quare & vt<emph.end type="italics"></emph.end> ſolidum baſim habens <lb></lb>quadratum ex AF, altitudinem verò lineam compoſitam ex <lb></lb>dupla ipſius DG cum AF ad cubum ex AF, ita <emph type="italics"></emph>dupla ipſius NX <lb></lb>cum linea NM ad NM. est autem<emph.end type="italics"></emph.end> cubus ex AF adcubum <lb></lb>ex DG, vt cubus ex MN ad cubum ex NX, vt oſtenſum eſt, <pb xlink:href="077/01/200.jpg" pagenum="196"></pb> <arrow.to.target n="fig87"></arrow.to.target><lb></lb><emph type="italics"></emph>cubusverò ex MN ad cubum ex NX eſt, vt MN ad N<emph.end type="italics"></emph.end>T; <lb></lb>erit <emph type="italics"></emph>& vt cubus ex AF ad cubum ex DG, ita MN ad NT. <lb></lb>ſicut autem cubus ex DG ad ſolidum baſim habens quadratum ex DG, <lb></lb>altitudinem verò lineam compoſitam ex dupla ipſius AF, cum linea <lb></lb>DG,<emph.end type="italics"></emph.end> ita altitudo cubi ad altitudinem ſolidi, cum ſint in ea<lb></lb>dem baſi, quadrato nempè ex DG. erit igitur vt cubus ex <lb></lb>DG ad ſolidum baſim habens quadratum ex DG, altitudi<lb></lb>nem verò lineam compoſitam ex dupla ipſius AF cum linea <lb></lb>DG, <emph type="italics"></emph>ita<emph.end type="italics"></emph.end> cubi altitudo <emph type="italics"></emph>DG ad<emph.end type="italics"></emph.end> altitudinem ſolidi, ad <lb></lb>lineam ſcilicet <emph type="italics"></emph>compoſitam ex dupla ipſius AF, & linea <lb></lb>DG.<emph.end type="italics"></emph.end> Quoniam autem ita eſt AF ad DG, vt <lb></lb>MN ad NX, vt verò MN ad NX, ita NO <lb></lb>ad NT. cùm ſint MN NX NO NT in continua proportio <lb></lb> <arrow.to.target n="marg390"></arrow.to.target> ne, crit AF ad DG, vt NO ad NT. & antecedentium dupla, <pb xlink:href="077/01/201.jpg" pagenum="197"></pb>hoc eſt, dupla ipſius AF ad DG, vt dupla ipſius NO ad <lb></lb>NT, & componendo, dupla ipſius AF cum DG ad <arrow.to.target n="marg391"></arrow.to.target><lb></lb>DG, vt dupla ipſius NO cum NT ad NT. & conuer<lb></lb>tendo DG ad duplam ipſius AF cum DG, vt NT ad <arrow.to.target n="marg392"></arrow.to.target> du<lb></lb>plam ipſius NO cum NT. <emph type="italics"></emph>Quare & vt<emph.end type="italics"></emph.end> ſe habet cubus ex <lb></lb>DG ad ſolidum baſim habens quadratum ex DG, altitu<lb></lb>dinem verò compoſitam ex dupla ipſius AF cum DG, ita <lb></lb>eſt <emph type="italics"></emph>TN ad compoſitam ex dupla ipſius ON, & linea TN.<emph.end type="italics"></emph.end> Ita<lb></lb>〈que〉 ex ijs, quæ dicta ſunt, ita ſe habet ſolidum baſim ha<lb></lb>bens quadratum ex AF, altitudinem verò lineam com<lb></lb>poſitam ex dupla ipſius DG, & linea AF ad cubum <lb></lb>ex AF, vt dupla ipſius NX cum NM ad MN, <lb></lb>cubus verò ex AF ad cubum ex DG eſt, vt MN ad <lb></lb>NT; ita deinde ſe habetcubus ex DG ad ſolidum ba<lb></lb>ſim habens quadratum ex DG, altitudinem verò lineam <lb></lb>compoſitam ex dupla ipſius AF, & ipſa DG, vt <lb></lb>NT ad compoſitam ex dupla ipſius NO, & ipſa NT. <lb></lb><emph type="italics"></emph>Sunt igitur quatuor magnitudines ſolidum baſim habens quadratum <lb></lb>ex AF, altitudinem verò lineam compoſitam ex dupla ipſius <lb></lb>DG, & linea AF, & cubus ex AF, & cubus ex <lb></lb>DG, & ſolidum baſim habens quadratum ex DG, altitu<lb></lb>dinem verò lineam compoſitam: ex dupla ipſius AF, & ipſa <lb></lb>DG, quatuor magnitudinibus proportionales, duabus ſimul ſumptis <lb></lb>tineæ compoſitæ ex dupla ipſius NX<gap></gap> & ipſa NM; & alte<lb></lb>ri magnitudini MN; aliiquè deinceps NT, ac tandem lineæ <lb></lb>compoſitæ ex duplaipſius NO, & ipſa NT. ex æquali igitur <lb></lb>erit, vt ſolidum baſim habens quadratum ex AF, altitudinem<emph.end type="italics"></emph.end> <arrow.to.target n="marg393"></arrow.to.target><lb></lb><emph type="italics"></emph>autem lineam compoſitam ex dupla ipſius DG, & ipſa AE, ad <lb></lb>ſolidum baſim habens quadratum ex DG, altitudinem verò lt<lb></lb>neam compoſitam ex dupla ipſius AF, & ipſa DG, ita <lb></lb>compoſita ex dupla ipſius NX, & ipſa MN ad compoſitam <lb></lb>ex dupla ipſius NO, & ipſa NT ſed vt præfatum ſoii<lb></lb>dum<emph.end type="italics"></emph.end> baſim habens quadratum ex AF, altitudinem verò <lb></lb>lineam compoſitam ex dupla ipſius DG, & ipſa AF <emph type="italics"></emph>ad <lb></lb>dictum ſolidum<emph.end type="italics"></emph.end> baſim habens quadratum ex DG, altitudi<lb></lb>nem verò compoſitam ex dupla ipſius AF & ipſa DG, <arrow.to.target n="marg394"></arrow.to.target><lb></lb><emph type="italics"></emph>ita<emph.end type="italics"></emph.end> factum fuit <emph type="italics"></emph>HI ad IK. vt igitur HI ad IK, ſu<emph.end type="italics"></emph.end> <pb xlink:href="077/01/202.jpg" pagenum="198"></pb> <arrow.to.target n="fig88"></arrow.to.target><lb></lb><emph type="italics"></emph>compoſita<emph.end type="italics"></emph.end> ex dupla ipſius NX cum MN <emph type="italics"></emph>ad compoſitam<emph.end type="italics"></emph.end> ex dupla <lb></lb> <arrow.to.target n="marg395"></arrow.to.target> ipſius NO cum NT. <emph type="italics"></emph>quare & componendo<emph.end type="italics"></emph.end> HK ad KI, vt <lb></lb>dupla ipſius NX cum MN, & dupla ipſius NO cum NT ad <lb></lb>compoſitam ex dupla ipſius NO cum NT, quia verò in hoc <lb></lb>antecedenti ſemel ſumitur MN, & ſemel NT, bis verò NX, <lb></lb>& bis NO, erit HK ad KI, vt vtra〈que〉 ſimul MN NT, & du<lb></lb>pla vtriuſ〈que〉 ſimul NX NO ad duplam ipſius NO, & ipſam <lb></lb>NT. <emph type="italics"></emph>& antecedentium quintupla.<emph.end type="italics"></emph.end> quintupla verò antecedentis <lb></lb>HK eſt FG, quintupla verò alterius antecedentis MN NT, <lb></lb>& duplæ vtriuſ〈que〉 ſimul NX NO eſt quintupla vtriuſ〈que〉 ſi<lb></lb>mul MN NT, & decupla vtriuſ〈que〉 ſimul NX NO. decu<lb></lb>pla enim eſt quintupla duplæ. <emph type="italics"></emph>eſt igitur FG ad IK, vt quintupla <lb></lb>vtriuſ〈que〉 ſimul MN NT, & decupla vtriuſ〈que〉 ſimul NX NO ad du<lb></lb>plam ipſius ON, & ipſam NT. & vt FG ad FK, quæeſt duæ quin<lb></lb>tæ ipſius<emph.end type="italics"></emph.end> FG, <emph type="italics"></emph>ita quintupla vtriuſ〈que〉 ſimul MN NT, & decupla <lb></lb>vtriuſ〈que〉 ſimul NX NO ad duplam vtriuſ〈que〉 ſimul MN NT,<emph.end type="italics"></emph.end> <pb xlink:href="077/01/203.jpg" pagenum="199"></pb><emph type="italics"></emph>& quadruplam vtriuſ〈que〉 ſimul NX NO.<emph.end type="italics"></emph.end> cùm hoc quidem con <lb></lb>ſe〈que〉ns ſitduæ quintæ ipſius antecedentis. </s> <s id="N17AB1">etenim dupla v<lb></lb>triuſ〈que〉 ſimul MN NT quintuplæ earumdem ſimul MN <lb></lb>NT duæ quintæ exiſtit. </s> <s id="N17AB7">& quadrupla vtriuſ〈que〉 ſimul NX <lb></lb>NO eſt duæ quintæ decuplæ earumdem NX NO. quadru<lb></lb>pla enim decuplæ eſt duæ quintæ. </s> <s id="N17ABD">Quoniam ita〈que〉 ita eſt FG <lb></lb>ad FK, vt quintupla vtriuſ〈que〉 ſimul MN NT, & decupla <lb></lb>vtriuſ〈que〉 ſimul NX NO ad duplam vtriuſ〈que〉 ſimul MN <lb></lb>NT, & quadruplam vtriuſ〈que〉 ſimul NX NO, & vt FG ad <lb></lb>KI, ita quintupla vtriuſ〈que〉 ſimul MN NT, & decupla vtriuſ <lb></lb>〈que〉 ſimul NX NO ad duplam ipſius ON, & ipſam NT: <lb></lb>erit FG ad ſuas conſe〈que〉ntes ſimul ſumptas FK KI, hoc <arrow.to.target n="marg396"></arrow.to.target><lb></lb>eſt FI, vt quintupla vtriuſ〈que〉 ſimul MN NT, & decupla <lb></lb>vtriuſ〈que〉 ſimul NX NO ad duplam vtriuſ〈que〉 ſimul MN <lb></lb>NT, & quadruplam vtriuſ〈que〉 ſimul NX NO, & duplam <lb></lb>ipſius ON, & ipſam NT. ſed in hoc conſe〈que〉nti bis ſumi<lb></lb>tur MN, quater NX, ſexies NO, & ter NT. <emph type="italics"></emph>erit igitur vt <lb></lb>FG æd FI, ita quintupla vtriuſ〈que〉 ſimul MN NT, & decupla v<lb></lb>triuſ〈que〉 ſimul NX NO ad compoſitam ex dupla ipſius MN, & qua<lb></lb>drupla ipſius NX, & ſextupla ipſius NO, & tripla ipſius NT.<emph.end type="italics"></emph.end> & <lb></lb>conuertendo FI ad FG, vt compoſita ex dupla ipſius MN, <arrow.to.target n="marg397"></arrow.to.target><lb></lb>& quadrupla ipſius NX, & ſextupla ipſrus NO, & tripla ip<lb></lb>ſiús NT ad quintuplam vtriuſ〈que〉 ſimul MN NT, & decu<lb></lb>plam vtriuſ〈que〉 ſimul NX NO. <emph type="italics"></emph>Quoniam ita〈que〉 quatuor rectæ li <lb></lb>neæ MN NX NO NT ſunt continuè proportionales.<emph.end type="italics"></emph.end> factaquè <lb></lb>fuit MN æqualis ipſi FB, & NO ipſi GB; crit reliqua OM <lb></lb>ipſi FG æqualis. </s> <s id="N17AFB">& vt TM ad TN ita factum fuit FH, <lb></lb>hoc eſt tres quintæ ipſius FG, tres ſcilicet quintæ ipſius MO <lb></lb>ad IR. quare & conuertendo <emph type="italics"></emph>vt NT ad TM, ita quædam aſſum<lb></lb>pta linea NI ad tres quintas ipſius FG, hoc eſt ipſius MO. vt autem <lb></lb>compoſita ex dupla ipſius NM, & quadrupla ipſius NX, & ſextupla ip<lb></lb>ſius NO & tripla ipſius NT ad lineam compoſitam ex quintupla vtrius<lb></lb>〈que〉 ſimul MN NT, & decupla vtriuſ〈que〉 ſimul XN NO, ſic altera quæ <lb></lb>dam aſſumpta linea IF ad FG, hoc est ad MO, erit ex ſuperioribus RF<emph.end type="italics"></emph.end> <arrow.to.target n="marg398"></arrow.to.target><lb></lb><emph type="italics"></emph>duæ quintæ ipſius MN, hoc est ipſius FB.<emph.end type="italics"></emph.end> ac propterea reliqua RB <lb></lb>erit tres quintæ ipſius FB. & obid BR ad. </s> <s id="N17B1D">RF eſt, vt tria ad <arrow.to.target n="marg399"></arrow.to.target><lb></lb>duo. <emph type="italics"></emph>Quare punctum R centrum est grauitatis portionis ABC. ſit<emph.end type="italics"></emph.end> <pb xlink:href="077/01/204.jpg" pagenum="200"></pb> <arrow.to.target n="fig89"></arrow.to.target><lb></lb> <arrow.to.target n="marg400"></arrow.to.target> <emph type="italics"></emph>quidem portionis DBE centrum grauitatis punctum Q frusti AD <lb></lb>EC centrum grauitatis erit in linea QR<emph.end type="italics"></emph.end> producta, <emph type="italics"></emph>quæ<emph.end type="italics"></emph.end> quiden QR <lb></lb><emph type="italics"></emph>adipſain<emph.end type="italics"></emph.end> productam <emph type="italics"></emph>eandem habeat proportionem quam habet fruſium<emph.end type="italics"></emph.end><lb></lb>ADEC <emph type="italics"></emph>ad reliquam portionem<emph.end type="italics"></emph.end> DBE. <emph type="italics"></emph>est autem punctum I. nam.<emph.end type="italics"></emph.end><lb></lb>cùm ſit tota FB ad totam BR, vt ablata BG ad ablatam <lb></lb> <arrow.to.target n="marg401"></arrow.to.target> BQ, ſunt enim vt quin〈que〉 ad tria, erit & reliqua FG ad reli<lb></lb>quam QR, vt FB ad BR. ita〈que〉 <emph type="italics"></emph>quoniam tres quintæ ipſius FB <lb></lb>linea eſi BR; ipſius verò GB tres quintæ linea est <expan abbr="Bq.">B〈que〉</expan> & reliquæ <lb></lb>igitur GF est tres quintæ QR. quoniamigitur est, vt fruſtum AD <lb></lb>EC adportionem DBE, ita MT ad NT,<emph.end type="italics"></emph.end> vt oſtenſum fuit; <emph type="italics"></emph>ſed vt <lb></lb>MN ad NT, ſic<emph.end type="italics"></emph.end> factum fuit FH ad IR, hoc eſt <emph type="italics"></emph>tres quintæ ipſius <lb></lb>GF; quæ est QR ad RI. erit igitur vt fruſtum ADEC adportionem <lb></lb>DBE, ita QR ad RI. & est quidem totius portionis<emph.end type="italics"></emph.end> ABC <emph type="italics"></emph>centrum<emph.end type="italics"></emph.end><lb></lb> <arrow.to.target n="marg402"></arrow.to.target> <emph type="italics"></emph>grauitatis punctum R; ipſius verò DBE centrum grauitatis punctum <lb></lb>Q: manifeſtum est igitur fruſti ADEC centrum grauitatis eſſe <expan abbr="pun-ctũ">pun<lb></lb>ctum</expan> l.<emph.end type="italics"></emph.end> quod <expan abbr="quidẽ">quidem</expan> eſt in quinta parte media HK ipſius FG ab <pb xlink:href="077/01/205.jpg" pagenum="201"></pb>eo ita diuiſa, vt HI ad IK ſit, vt ſolidum baſim habens qua<lb></lb>dratum ex AF, altitudinem autem duplam ipſius DG cum <lb></lb>AF ad ſolidum baſim habens quadratum ex DG, altitudinem <lb></lb>verò duplam ipſius AF <expan abbr="cũ">cum</expan> DG. quod demonſtrare oportebat. </s> </p> <p id="N17BB5" type="margin"> <s id="N17BB7"><margin.target id="marg382"></margin.target>1 <emph type="italics"></emph>Arch de <lb></lb>quad. </s> <s id="N17BC0">pa<lb></lb>rab. </s> <s id="N17BC4">& <lb></lb><expan abbr="ſecũdi">ſecundi</expan> coni <lb></lb>corum A<lb></lb>poll.<emph.end type="italics"></emph.end></s> </p> <p id="N17BD1" type="margin"> <s id="N17BD3"><margin.target id="marg383"></margin.target>13.<emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N17BDC" type="margin"> <s id="N17BDE"><margin.target id="marg384"></margin.target>3.<emph type="italics"></emph>Arch.de <lb></lb>quad. </s> <s id="N17BE7">pa<lb></lb>rab. </s> <s id="N17BEB">&<emph.end type="italics"></emph.end> 20. <lb></lb><emph type="italics"></emph>pilmi coni <lb></lb>corum A<lb></lb>poil.<emph.end type="italics"></emph.end></s> </p> <p id="N17BFA" type="margin"> <s id="N17BFC"><margin.target id="marg385"></margin.target>2.<emph type="italics"></emph>cor.<emph.end type="italics"></emph.end> 20. <lb></lb><emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N17C0C" type="margin"> <s id="N17C0E"><margin.target id="marg386"></margin.target>22.<emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></s> </p> <p id="N17C17" type="margin"> <s id="N17C19"><margin.target id="marg387"></margin.target>37. <emph type="italics"></emph>vndeci <lb></lb>mi.<emph.end type="italics"></emph.end></s> </p> <p id="N17C24" type="margin"> <s id="N17C26"><margin.target id="marg388"></margin.target>17.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N17C2F" type="margin"> <s id="N17C31"><margin.target id="marg389"></margin.target>18.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N17C3A" type="margin"> <s id="N17C3C"><margin.target id="marg390"></margin.target>11.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N17C45" type="margin"> <s id="N17C47"><margin.target id="marg391"></margin.target>18.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N17C50" type="margin"> <s id="N17C52"><margin.target id="marg392"></margin.target><emph type="italics"></emph>cor<emph.end type="italics"></emph.end> 4.<emph type="italics"></emph>quin<lb></lb>ti.<emph.end type="italics"></emph.end></s> </p> <p id="N17C62" type="margin"> <s id="N17C64"><margin.target id="marg393"></margin.target>22.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N17C6D" type="margin"> <s id="N17C6F"><margin.target id="marg394"></margin.target>11.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N17C78" type="margin"> <s id="N17C7A"><margin.target id="marg395"></margin.target>18.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N17C83" type="margin"> <s id="N17C85"><margin.target id="marg396"></margin.target><emph type="italics"></emph>cor.<emph.end type="italics"></emph.end>2.<emph type="italics"></emph>lem<lb></lb>in<emph.end type="italics"></emph.end> 13. <emph type="italics"></emph>pri<lb></lb>mi huius.<emph.end type="italics"></emph.end></s> </p> <p id="N17C9D" type="margin"> <s id="N17C9F"><margin.target id="marg397"></margin.target><emph type="italics"></emph>cor.<emph.end type="italics"></emph.end>4.<emph type="italics"></emph>quin<lb></lb>ti.<emph.end type="italics"></emph.end></s> </p> <p id="N17CAF" type="margin"> <s id="N17CB1"><margin.target id="marg398"></margin.target><emph type="italics"></emph>ex præce<lb></lb>denti.<emph.end type="italics"></emph.end></s> </p> <p id="N17CBB" type="margin"> <s id="N17CBD"><margin.target id="marg399"></margin.target>8.<emph type="italics"></emph>buius.<emph.end type="italics"></emph.end></s> </p> <p id="N17CC6" type="margin"> <s id="N17CC8"><margin.target id="marg400"></margin.target>8.<emph type="italics"></emph>prim hu<lb></lb>ius.<emph.end type="italics"></emph.end></s> </p> <p id="N17CD3" type="margin"> <s id="N17CD5"><margin.target id="marg401"></margin.target>19.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N17CDE" type="margin"> <s id="N17CE0"><margin.target id="marg402"></margin.target>8 <emph type="italics"></emph>prim.hu<lb></lb>ius.<emph.end type="italics"></emph.end></s> </p> <figure id="id.077.01.205.1.jpg" xlink:href="077/01/205/1.jpg"></figure> <figure id="id.077.01.205.2.jpg" xlink:href="077/01/205/2.jpg"></figure> <figure id="id.077.01.205.3.jpg" xlink:href="077/01/205/3.jpg"></figure> <figure id="id.077.01.205.4.jpg" xlink:href="077/01/205/4.jpg"></figure> <p id="N17CFB" type="head"> <s id="N17CFD">SCHOLIVM.</s> </p> <p id="N17CFF" type="main"> <s id="N17D01">In hoc Theoremate primùm obſeruanda occurrunt verba <lb></lb>propoſitionis, quibus Archimedes pręcipit pottionem HK <lb></lb>in I ita diuiſam eſſe oportere, vt HI ad IK eam habeat pro<lb></lb>portionem, quam habet ſolidum baſim habens quadratum <lb></lb>ex dimidia maioris baſis fruſti, altitudinem autem lineam æ<lb></lb>qualem vtri〈que〉 ſimul duplæ minoris baſis, & maiori ad ſoli<lb></lb>dum baſim habens quadratum ex dimidia minoris baſis fru<lb></lb>ſti, altitudinem autem lineam æqualem vtriſ〈que〉, duplæ ſcili<lb></lb>cet baſis maioris, & minori. </s> <s id="N17D13">hoc eſt ſit HI ad IK, vt ſolidum <lb></lb>baſim habens quadratum ex AF, altitudinem verò lineam æ<lb></lb>qualem duplæ ipſius DE cum AC ad ſolidum baſim habens <lb></lb>quadratum ex DG, altitudinem verò lineam æqualem <expan abbr="vtriq;">vtri〈que〉</expan> <lb></lb>ſimul duplæ ipſius AC, & ipſi DE. In conſtructione autem <lb></lb>hunc propoſitionis locum explicans, & in pergreſſu totius <expan abbr="de-mõſtrationis">de<lb></lb>monſtrationis</expan>, inquit HI ad IK <expan abbr="eã">eam</expan> debere proportionem habe<lb></lb>re, quam habet ſolidum baſim habens quadratum ex AF, alti <lb></lb>tudinem verò lineam æqualem <expan abbr="vtriq;">vtri〈que〉</expan> ſimul duplæ ipſius DG, <lb></lb>& ipſi AF ad ſolidum baſim habens quadratum ex DG, al<lb></lb>titudinem verò lineam æqualem vtri〈que〉 ſimul duplæ ipſius <lb></lb>AF, & DG. Quoniam autem ſolida parallelepipeda (vt præ<lb></lb>fata ſolida ſunt) in eadem baſi exiſtentia ita ſe habent interſe, <lb></lb>vt corum altitudine; ſolidum, quod baſim habet quadratum <lb></lb>ex AF, altitudinem autem duplam ipſius DE cum AC, du<lb></lb>plum erit ſolidi baſim habentis quadratum ex AF, altitudi<lb></lb>nem verò duplam ipſius DG cum AF. Nam hæc ſolida ean<lb></lb>dem habent baſim, quadratum nempè ex AF; ipſorumquè <lb></lb>alterum habet altitudinem duplam. </s> <s id="N17D49">quia cùm ſit DE dupla <lb></lb>ipſius DG, erit dupla ipſius DE dupla ipſius duplæ DG; <pb xlink:href="077/01/206.jpg" pagenum="202"></pb>& AC dupla eſt ipſius AF. altitudines igitur horum <expan abbr="ſolidorũ">ſolidorum</expan> <lb></lb>in dupla ſunt proportione. </s> <s id="N17D57">hoc eſt altitudo, linea ſcilicet du<lb></lb>pla ipſius DE cum AC altitudinis nempè lineæ duplæ ipſius <lb></lb>DG cum AF dupla exiſtit. </s> <s id="N17D5D">Quare ſolidum baſim habens qua<lb></lb>dratum ex AF, altitudinem verò duplam ipſius DE cum AC <lb></lb>duplum eſt ſolidi, quod baſim habeatidem quadratum ex AF, <lb></lb>altitudinem verò duplam ipſius DG cum AF. cademquè ratio <lb></lb>neoſtendetur <expan abbr="ſolidũ">ſolidum</expan> baſim habens quadratum ex DG, altitu<lb></lb>dinem verò duplam ipſius AC cum DE duplum eſſe ſolidi ba <lb></lb>ſim habentis quadratum ex eadem DG, altitudinem autem du<lb></lb>plam ipſius AF cum DG. ſolidum igitur baſim habens qua<lb></lb>dratum ex AF, altitudinem autem duplam ipſius DE cum AC <lb></lb>ad ſolidum quadtatum habens baſim ex AF, altitudinent verò <lb></lb>duplam ipſius DG cum AF eam habet proportionem, quam <lb></lb>habet ſolidum baſim habens quadratum ex DG, altitudinem <lb></lb>verò duplam ipſius AC cum AE ad ſolidum baſim <expan abbr="habẽs">habens</expan> qua <lb></lb>dratum ex DG, altitudinem verò duplam ipſius AF cum DG. <lb></lb> <arrow.to.target n="marg403"></arrow.to.target> quare permutando <expan abbr="primũ">primum</expan> ſolidum baſim habens quadratum <lb></lb>ex AF, altitudinem verò duplam ipſius DE cum AC ad ſecun<lb></lb>dum ſolidum baſim habens quadratum ex DG, altitudinem <lb></lb>autem duplam ipſius AC cum DE eandem habet proportio<lb></lb>nem, quam habet tertium ſolidum baſim habens quadratum <lb></lb>ex AF, altitudinem autem duplam ipſius DG cum AF ad quar <lb></lb>tum ſolidum baſim habens quadratum ex DG, altitudinem ve <lb></lb>rò duplam ipſius AF cum DG. Quapropter Archimedes loco <lb></lb>primi, & ſecundi ſolidi in propoſitione propoſiti rectè potuit <lb></lb>in demonſtratione accipere tertium, & quartum ſolidum. </s> <s id="N17D9D">co <lb></lb>dem enim modo, & in eadem proportione linea HK in pun<lb></lb>cto I diuiſa prouenit: quod quidem punctum fruſti ACED <lb></lb>centrum grauitatis exiſtit. </s> </p> <p id="N17DA5" type="margin"> <s id="N17DA7"><margin.target id="marg403"></margin.target>16.<emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> </p> <p id="N17DB0" type="head"> <s id="N17DB2">Secundi libri Finis.</s> </p> <pb xlink:href="077/01/207.jpg"></pb> <p id="N17DB6" type="head"> <s id="N17DB8">Erratorum quorundam reſtitutio.</s> </p> <p id="N17DBA" type="main"> <s id="N17DBC">Pagina 8, verſu 18, Archimedes. </s> <s id="N17DBE"><33> 10, 7, ſione. </s> <s id="N17DC0"><33> 18, 20, conducenti. </s> <s id="N17DC2"><33> 21, 14, per <lb></lb>diſcere ipſum. </s> <s id="N17DC6"><33> 39, 25, hoc eſt AB. <33> 43, 19, lineam. </s> <s id="N17DC8"><33> 47, 20, cúm inquit, <33> 63, <lb></lb>20, GD DK in. </s> <s id="N17DCC"><33> 65, 21, DC. Ibidem, 27, ex DC. <33> 67, 29, in maiori. </s> <s id="N17DCE"><33> 69, in <lb></lb>poſtil: ex proxima propoſitione. </s> <s id="N17DD2"><33> 70, 5, vt NL <33> 73, 1, de his, vel. </s> <s id="N17DD4"><33> 84, 8, AEEB <lb></lb>CF FD. <33> 90, 17, totus. </s> <s id="N17DD8"><33> 98, 1, quam VH. Ibidem, 7, aufertur. </s> <s id="N17DDA"><33> 11<gap></gap>, 21, repo<lb></lb>ſuit. </s> <s id="N17DE0"><33> 124, 19, <expan abbr="ſectionẽ">ſectionem</expan>, <33> 140, 1, <expan abbr="æquidiſtãtes">æquidiſtantes</expan> <33> 143, 11, eſt CH <33> 147, 3, <expan abbr="cũ">cum</expan> EK ad EK, vt. <lb></lb>Ibide, 25, ſta S 9, ad Y<foreign lang="grc">α</foreign> <33> 149, 19, ad <foreign lang="grc">χν</foreign>. Ibidem, 25, eſt, vt OR. Ibidem, 27, L<foreign lang="grc">Γ</foreign>, vt <lb></lb>OR ad. </s> <s id="N17DFE">Ibidem, 31, vt OR ad <foreign lang="grc">ζδ</foreign> Ibidem, 32, vt <foreign lang="grc">δ<10></foreign> ad <foreign lang="grc"><10>ζ</foreign> Ibidem, 34, BD ad B<foreign lang="grc">σ</foreign>, <lb></lb>ita. </s> <s id="N17E12">Ibidem, 35, ſit BD ad D<foreign lang="grc">ν</foreign> Ibidem, 36, BD ad D<foreign lang="grc">ν</foreign> B<foreign lang="grc">σ</foreign>. <33> 150, 5, vt OR ad O<foreign lang="grc">ξ</foreign> <33> 153, <lb></lb>13, ræ, vt. </s> <s id="N17E26"><33> 157, in poſtill ante 15, primi Ibidem, 17, maiorem. </s> <s id="N17E28"><33> 161, 24, erit KH. <lb></lb><33> 167, 34, efficax. </s> <s id="N17E2C"><33> 170, 1, ipſius AC erit. </s> <s id="N17E2E"><33> 181, 36, ex dupla ipſius AB, <33> 191, <lb></lb>21, erunt. </s> <s id="N17E32">Ibidem, 22, DKG æquales. </s> </p> <p id="N17E34" type="head"> <s id="N17E36">REGISTRVM.</s> </p> <p id="N17E38" type="main"> <s id="N17E3A"><12> ABCDEFGHIKLMNOPQRSTVXYZ, <lb></lb>AA BB. </s> </p> <p id="N17E3E" type="head"> <s id="N17E40">Omnes duerniones, præter, BB, ternionem.</s> </p> <p id="N17E42" type="head"> <s id="N17E44">PISAVRI. <lb></lb>Apud Hieronymum Concordiam, <lb></lb>M. D. LXXXVII.</s> </p> </chap> </body> <back></back> </text> </archimedes>